**Lecture Notes in Mechanical Engineering**

Saman K. Halgamuge Hao Zhang Dingxuan Zhao Yongming Bian Editors

The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering ICACMVE 2023

# **Lecture Notes in Mechanical Engineering**

### **Series Editors**

Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia

Francesco Gherardini , Dipartimento di Ingegneria "Enzo Ferrari", Università di Modena e Reggio Emilia, Modena, Italy

Vitalii Ivanov, Department of Manufacturing Engineering, Machines and Tools, Sumy State University, Sumy, Ukraine

Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia

### **Editorial Board**

Francisco Cavas-Martínez , Departamento de Estructuras, Construcción y Expresión Gráfica Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain

Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany

Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA

Tullio A. M. Tolio, Department of Mechanical Engineering, Politecnico di Milano, Milano, Italy

Justyna Trojanowska, Poznan University of Technology, Poznan, Poland

Robert Schmitt, RWTH Aachen University, Aachen, Germany

Jinyang Xu, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China

**Lecture Notes in Mechanical Engineering (LNME)** publishes the latest developments in Mechanical Engineering—quickly, informally and with high quality. Original research or contributions reported in proceedings and post-proceedings represents the core of LNME. Volumes published in LNME embrace all aspects, subfields and new challenges of mechanical engineering.

To submit a proposal or request further information, please contact the Springer Editor of your location:

**Europe, USA, Africa:** Leontina Di Cecco at Leontina.dicecco@springer.com **China:** Ella Zhang at ella.zhang@springer.com

**India:** Priya Vyas at priya.vyas@springer.com

**Rest of Asia, Australia, New Zealand:** Swati Meherishi at swati.meherishi@ springer.com

Topics in the series include:


#### **Indexed by SCOPUS, EI Compendex, and INSPEC.**

All books published in the series are evaluated by Web of Science for the Conference Proceedings Citation Index (CPCI).

To submit a proposal for a monograph, please check our Springer Tracts in Mechanical Engineering at https://link.springer.com/bookseries/11693.

Saman K. Halgamuge · Hao Zhang · Dingxuan Zhao · Yongming Bian Editors

# The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering

ICACMVE 2023

*Editors*  Saman K. Halgamuge Department of Mechanical Engineering School of Electrical, Mechanical and Infrastructure Engineering University of Melbourne Melbourne, VIC, Australia

Dingxuan Zhao Yanshan University Qinhuangdao, Hebei, China Hao Zhang College of Electronic and Information Engineering Tongji University Shanghai, China

Yongming Bian College of Mechanical Engineering Tongji University Shanghai, China

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-97-1875-7 ISBN 978-981-97-1876-4 (eBook) https://doi.org/10.1007/978-981-97-1876-4

© The Editor(s) (if applicable) and The Author(s) 2024. This book is an open access publication.

**Open Access** This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Paper in this product is recyclable.

# **Committee Member**

# **Honorary Chairman**

Laide Shi, Tongji University, China

### **General Chairman**

Yongming Bian, Tongji University, China

### **Co-Chairman**

Dingxuan Zhao, Yanshan University, China Marcus Geimer, Karlsruhe Institute of Technology, Germany

### **Publication Chair**

Yongming Bian, Tongji University, China Dingxuan Zhao, Yanshan University, China Hao Zhang, Tongji University, China Saman K. Halgamuge, University of Melbourne, Australia

### **Academic Committee Chairman**

Zhihua Zhong, Chinese Academy of Engineering, China Huayong Yang, Zhejiang University, China

### **Academic Committee Co-Chairman**

Xuedong Chen, Hefei General Machinery Research Institute, China Hehua Zhu, Tongji University, China Jianlong Zheng, Changsha University of Science and Technology, China Xinping Yan, Wuhan University of Technology, China Shirong Ge, China University of Mining and Technology-Beijing, China Zongquan Deng, Harbin Institute of Technology, China Jue Zhong, Central South University, China Bangchun Wen, Northeastern University, China Baojun Lin, University of China Academy Sciences, China Guobiao Wang, Tianjin University, China Pei'en Feng, Zhejiang University, China Huiqiang Zheng, Tongji University, China Jürgen Fleischer, German Academy of Engineering, Germany Saman K. Halgamuge, The University of Melbourne, Australia Michael Abramovici, Ruhr University Bochum, Germany

### **Program Committee Chairman**

Yuhou Wu, Shenyang Jianzhu University, China Xianbiao Zhou, Chinese Construction Machinery Society, China

### **Program Committee Co-Chairman**

Weijian Mi, Shanghai Maritime University, China Hanbin Xiao, Wuhan University of Technology, China Gening Xu, Taiyuan University of Science and Technology, China Ming Ye, Chang'an University, China Jianxin Zhu, Central South University, China Zhencai Zhu, China University of Mining and Technology, China Sanip Kunar, Aditya Engineering College, India Veeresh Fuskele (JAIN), Government Engineering College, India

Hassan Elahi, Sapienza University of Rome, Italy Hongfu Yu, Sany Heavy Industry Co., Ltd., China Puzhou Zhuo, China Railway Hi-Tech Industry Co., Ltd., China Suoyun Li, Xuzhou Construction Machinery Group Co., Ltd., China Ling Fu, Zoomlion Heavy Industry Science and Technology Co., Ltd., China Xinghua Huang, Chinese Construction Machinery Society, China

# **Program Committee Member**

Weihua Bao, Shanghai Automation Instrumentation Co., Ltd., China Hui Cai, China Railway Engineering Equipment (Tianjin) Co., Ltd., China Ming Cao, University of Shanghai for Science and Technology, China Donghui Cao, SANY Heavy Industry Co., Ltd., China Gengxiang Cao, CCCC Third Harbor Engineering Bureau Co., Ltd, China Yuanwen Cao, Chongqing Jiaotong University, China Li Chen, Tongji University, China Haiying Cheng, Chang'an University, China Wenming Cheng, Southwest Jiaotong University, China Xiaoming Cheng, Ningbo University of Technology, China Feng Chi, Shandong Lingong Construction Machinery, China Jinling Cong, Shihezi University, China Xichen Dong, Wuhan University of Technology, China Wenjun Duan, China Railway Engineering Services Limited, China Maohua Fu, Shanghai Zhenhua Heavy Industries Co., Ltd., China Rongzhi Gao, Zoomlion Concrete Pumping Machinery Branch, China Shunde Gao, Dalian University of Technology, China Yanbei Gao, Chang'an University, China Mingde Gong, Yanshan University, China Rui Guo, Yanshan University, China Chuanxin Guo, Beijing Institute of Construction Mechanization Co., Ltd., China Wenwu Guo, Shijiazhuang Tiedao University, China Xiaohong Guo, Chongqing Jiaotong University, China Rujiang Hao, Shijiazhuang Tiedao University, China Qin He, Shandong Jianzhu University, China Lin Hu, Changsha University of Science and Technology, China Yongbiao Hu, Chang'an University, China Hui Huang, Huaqiao University, China Ming Huang, Guangxi LiuGong Machinery Co., Ltd., China Yi Huang, China Communications Construction Company, China Haibo Huang, Ningbo University, China Heting Huang, Xiamen XGMA International Trading Co., Ltd., China Qiufang Huang, Fujian Longma Sanitation Equipment Co., Ltd., China

Yunpeng Huang, Shenji (Shanghai) Intelligent System R&D and Design Co., Ltd., China Ziming Kou, Taiyuan University of Technology, China Peng Lan, Xi'an University of Architecture and Technology, China Hong Li, Yunnan Construction Machinery Industry Association, China Zheng Li, Inner Mongolia University of Science and Technology, China Zong Li, Xuzhou Construction Machinery Group Co., Ltd., China Anhu Li, Tongji University, China Suoyun Li, Xuzhou Construction Machinery Group Co., Ltd., China Taijie Li, Da Gang Holding Group Co., Ltd., China Yunhua Li, Beihang University, China Zhonggang Li, Lanzhou Jiaotong University, China Deng Lin, Zhejiang Yongan Construction Machinery Co., Ltd., China Mingzhi Lin, Guangxi LiuGong Machinery Co., Ltd., China Guangjun Liu, Tongji University, China Xiangdong Liu, Inner Mongolia University of Technology, China Xinhui Liu, Jilin University, China Yuxin Liu, Zoomlion Construction Crane Branch, China Yuanhong Liu, Beijing Institute of Construction Mechanization Co., Ltd., China Zhengfu Liu, Kunshi Control Systems (Shanghai) Co., Ltd., China Liang Long, Changsha Zoomlion Environmental Industry Co., Ltd., China Liang Lu, Tongji University, China Nianli Lu, Harbin Institute of Technology, China Biao Ma, Shanghai Municipal Engineering Design Institute (Group) Co., Ltd., China Xianjun Meng, China Railway Tunnel Bureau Group Company Limited Equipment Branch, China Junying Min, Tongji University, China Tao Ni, Yanshan University, China Lisheng Qin, Shanghai Tongxin Electromechanical Control Technology Co., Ltd., China Jiang Qiu, Shanghai Environmental Engineering Design and Research Institute Co., Ltd., China Long Quan, Taiyuan University of Technology, China Haibo Shang, Tianjin Construction Machinery Research Institute Co., Ltd., China Jijun Shen, China Construction Machinery Co., Ltd., China Jinliang Sheng, Tongji University, China Xianxin Shi, XCMG Xuzhou Heavy Machinery Co., Ltd., China Hong Sun, Shenyang Jianzhu University, China Ruming Teng, Dalian University of Technology, China Zhicheng Tian, CIMIC (Beijing) Vehicle Inspection Engineering Research Institute Co., Ltd, China Anlin Wang, Tongji University, China Caiying Wang, Huicong Construction Machinery Network, China Dajiang Wang, Qinhuangdao Tianye Tolian Heavy Industry and Technology Co., Ltd, China

Jiru Wang, Hangzhou Huaxin Mechanical and Electrical Engineering Co., Ltd, China Jiangbo Wang, Jiangsu Donghua Testing Technology Co., Ltd., China Zhiyong Wang, Caofeidian Port Group Co., Ltd., China Zhongbin Wang, China University of Mining and Technology, China Chong Wu, Tongji University, China Binxing Wu, Zoomlion Heavy Industry Co., Ltd., China Renzhi Wu, Tongji University, China Haibo Xie, Zhejiang University, China Ligang Xin, Yantai Hyde Specialized Vehicle Co., Ltd., China Bing Xu, Zhejiang University, China Huaiyu Xu, Xuzhou Xugong Schwing Machinery Co., Ltd., China Zhaohong Xun, Anhui Taiyuan Construction Machinery Co., Ltd., China Bin Yang, Phoenix (China) Investment Co., Ltd., China Wenbiao Yang, Anhui Taiyuan Construction Machinery Co., Ltd., China Hong Yi, Lanzhou University of Technology, China Chenbo Yin, Nanjing Tech University, China Haoshan Yu, Chinese Academy of Geological Sciences, China Ye Yuan, Zoomlion Heavy Industry Co. Earthmoving Machinery Company, China Wenhui Yue, University of Science and Technology, China Qingliang Zeng, Shandong Normal University, China Hao Zhang, Tongji University, China Li Zhang, Anhui Taiyuan Construction Machinery Co., Ltd., China Qing Zhang, Tongji University, China Yuan Zhang, LIBO Heavy Industries Science and Technology Co., Ltd., China Daqing Zhang, Shanhe Intelligent Equipment Co., Ltd., China Guosheng Zhang, Tianjin Construction Machinery Research Institute, China Junhui Zhang, Zhejiang University, China Shejun Zhang, China Railway Construction Heavy Industry Corporation Limited, China Shengjun Zhang, Beijing Institute of Construction Mechanization Co., Ltd., China Weiguo Zhang, Shanghai Maritime University, China Xuhui Zhang, Xi'an University of Science and Technology, China Yanwei Zhang, Wuhan University of Technology, China Zhonghai Zhang, Xugong Foundation Construction Machinery Co., Ltd., China Zili Zhang, Jiangsu Tianzhou Testing and Inspection Ltd., China Bin Zhao, Jiangsu Xugong Research Institute, China Jingyi Zhao, Yanshan University, China Keli Zhao, Jilin University, China Rongyong Zhao, Tongji University, China Shuen Zhao, Chongqing Jiaotong University, China Youjun Zhao, Xi'an Coal Mining Machinery Co., Ltd., China Jianping Zhao, Xinjiang University, China Qiping Zhu, Shijiazhuang Tiedao University, China Yutian Zhu, Tongji University, China Hongquan Zhuang, Tsinghua University Press, China

# **Organizing Committee Chairman**

Xianbiao Zhou, China Construction Machinery Society, China

# **Organizing Committee Co-Chairman**

Li Chen, China Construction Machinery Society, China

### **Organizing Committee Secretary**

Jianli Chen, China Construction Machinery Society, China

# **Organizing Committee Member**

Wanghui Bu, Tongji University, China Wei Cai, Yanshan University, China Jianli Chen, China Construction Machinery Association, China Maolin Chen, Tongji University, China Qianpeng Chen, Lanzhou University of Technology, China Qihuai Chen, Huaqiao University, China Zhe Chen, Tongji University, China Min Cheng, Chongqing University, China Christopher Ehrmann, Tongji University, China Ruqi Ding, East China Jiaotong University Changhong Fu, Tongji University, China Xiangyu He, Changsha University of Science and Technology, China Zeran Hou, Tongji University, China Jiahai Huang, Taiyuan University of Technology, China Tianliang Lin, Huaqiao University, China Changhui Liu, Tongji University, China Liang Lu, Tongji University, China Junying Min, Tongji University, China Morteza Mohammadzaheri, Birmingham City University, UK Muhammad Saleem, Imam Abdulrahman Bin Falsal University, Saudi Arabia Youcheng Shi, Lanzhou University of Technology, China Bo Sun, Tongji University, China Yuantao Sun, Tongji University, China Tang Tang, Tongji University, China

Yichao Tang, Tongji University, China Valerian Croitorescu, University Politehnica of Bucharest, Romani Jianwei Wang, Yanshan University, China Kun Wang, Tongji University, China Liang Wang, Tongji University, China Yajing Wu, Tongji University, China Yao Xiao, Tongji University, China Dongdong Xu, Tongji University, China Meng Yang, Tongji University, China Chunlong Yu, Tongji University, China Zhicheng Yuan, Tongji University, China Cuixiang Zhu, Tongji University, China

# **Preface**

This proceedings volume *Design, Manufacture and Intelligent Development of Construction Machinery and Vehicle Engineering* presents the select peer-reviewed proceedings of the 2023 8th International Conference on Advances in Construction Machinery and Vehicle Engineering (ICACMVE 2023) held on October 13–16, 2023, in Shanghai, China.

ICACMVE 2023 examines a wide spectrum covering the latest trends in the fields of construction machinery and vehicle engineering and conducts in-depth and extensive exchanges on relevant theories and methods in view of the development trend, research hotspots, and difficulties in the fields. The topics covered include modern design methods, advanced manufacturing technology, transmission and drive control, civil engineering method integration, new energy, intelligence, cloud application, etc.

The conference agenda was composed of opening speech, keynote speech, oral presentation, poster presentation, award ceremony, as well as academic tour. We have invited many well-known professors in specific research areas to shared their research experience and special insights with us, providing new ideas and directions for the young researchers and others interested in the researches of these fields. Among them, Professor Hehua Zhu is one of our keynote speakers sharing his research and views with us. He has been teaching and researching in the field of tunneling and underground space for over 30 years and was selected as the chief scientist of 973 National Project.

All the contributions presented at ICACMVE 2023 and published in the proceedings have gone through a rigorous review process including scientific integrity check, initial check, peer review, review check, final decision, etc., by the Program Committee and top experts in related fields. Their originality, validity, quality, and readability were evaluated by at least three independent reviewers.

We would like to give sincere gratitude to all authors and speakers who have made their contributions to ICACMVE 2023, editors and reviewers who have guaranteed the quality of papers with their expertise, and the committee members who have devoted themselves to the success of the conference.

The proceedings volume puts forward the hot spots and difficulties in construction machinery and vehicle engineering, gives new design ideas and manufacturing technologies, and extends to the direction of intelligent application. It is believed to be a valuable reference for researchers and professionals interested in construction machinery and vehicle engineering, and allied fields.

The Committee of ICACMVE 2023

# **Contents**




Contents xvii




Contents xix



#### Contents xxi







# **Mechanical Design and Power System Modeling**

# **Rigid-Flexible Coupling Dynamics Analysis of Boom-Hoisting System of Wind Power Crane**

**Xuyang Cao, Guoyang Xu, Yongchang Hu, Jihong Zhou, and Jian Kang** 

**Abstract** Based on the slenderness of wind power crane boom and the complexity of working environment, a mathematical model of rigid-flexible coupling dynamics including wind load and vibration characteristic analysis is carried out for the boomhoisting system. The equivalent spring-damping system, spatial pendulum system and elastic double-force rod are used to simulate the elastic vibration of boom, the swing of lifting weight and the elastic vibration of wire rope, respectively. The kinematic characteristics of each component are described using a hybrid coordinate system. Simulation of wind loads through an instantaneous wind model consisting of the average and pulsating wind. Derivation of dynamic model for wind power crane under wind load based on the Lagrange equation. Taking a certain type of wind power crane as the research object, based on the derived mathematical model and ADAMS model, the response curves of boom and lifting weight are solved in MATLAB and ADAMS respectively for the rotary working condition, to verify the rationality and accuracy of the mathematical model and to obtain the influence of wind load on the swing angle of lifting weight. The analysis results provide a certain theoretical basis for crane structure design, control system design and wind turbine hoisting.

**Keywords** Wind power crane · Rigid-flexible coupling model · Lagrange equation · Wind load · Lifting weight deflecting and oscillating

X. Cao · G. Xu (B) · Y. Hu

J. Zhou · J. Kang Taiyuan Heavy Industry CO., LTD., Shanxi Taiyuan 030024, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_1

School of Mechanical Engineering, Dalian University of Technology, Dalian Liaoning 116024, China e-mail: xuguoyang158@163.com

### **1 Introduction**

Wind power crane is a type of lattice boom crawler crane, which has irreplaceable importance in the field of engineering machinery as the cornerstone of wind turbine installation. The working environment of wind power crane has the characteristics of large lifting scale and long-term operation in strong wind environments [1], which will inevitably cause changes in the deflection and oscillation angle of the lifting weight during hoisting operations. In the early stages, cranes use rigid body dynamics and kinematic elasticity analysis methods to estimate the swing trend of the lifting weight, and it tends to be a large slenderness ratio, which makes the boom vibration gradually obvious during hoisting, so that the coupling effect between the swing of the lifting weight and the boom vibration cannot be ignored.

Multi-body dynamics has made great progress in the field of cranes; however, the development of rigid-flexible coupling dynamics in boom cranes still has numerous problems. Fan et al. [2] established a four-degree-of-freedom dynamics model for the lifting weight of a truck crane, and analyzed the effects of rope length and rotational angular acceleration on the swing of the lifting weight. Ouyang et al. [3] established a nonlinear dynamic model of rotary crane considering the double pendulum effect, and deduced that the swing angle has stronger coupling under the double pendulum effect. All of the above models characterize the dynamics of the lifting weight in terms of a rigid-body dynamics model, ignoring the coupling effect of the boom vibration on the swing of the lifting weight. In order to more realistically reflect the dynamic characteristics of the boom-lifting system, Zheng et al. [4] used the multibody system transfer matrix method to establish a rigid flexible coupling model of a truck crane, which better reflects the coupling relationship between boom vibration and lifting weight deflection and oscillation, but the equation is too complicated and difficult to solve. Yan et al. [5], Kwang-Phil et al. [6] and Jerman et al. [7] used modal analysis and the finite segment method to describe the elastic vibration of the boom, but it is difficult to apply them to the model with medium and large deformation. Anja et al. [8] utilized a Euler–Bernoulli beam equivalent armature and derive a Euler–Lagrange flexible multibody system model, and the simulation results have a high degree of matching with the experimental measurements.

In summary, scholars at home and abroad have conducted numerous studies on the dynamic characteristics of boom cranes, but the existing dynamic models still have problems in the description of system structure and component motion. Based on this, this paper establishes a mathematical model of rigid-flexible coupling dynamics (hereinafter referred to as the mathematical model) including wind load for the boomhoisting system. The equivalent spring-damping system, spatial pendulum system and elastic double-force rod are used to simulate the elastic vibration of the boom, the swing of the lifting weight and the elastic vibration of the wire rope, respectively. The kinematic characteristics of each component are described using a hybrid coordinate system. The wind load borne by the model can be simulated using an instantaneous wind model. Establishing a rigid-flexible coupling dynamic model for the rotation, variable amplitude and lifting motion of wind power crane based on the Lagrange equation. Numerical analysis software is applied to solve the response curves of the model under corresponding inputs, and compare them with the ADAMS model results to verify the rationality and accuracy of the mathematical model.

### **2 Mathematical Modeling of Dynamical Equations**

Figure 1 shows the boom-lifting system of wind power crane, and Fig. 2 shows the abstract rigid-flexible coupling dynamic model of wind turbine crane. The global coordinate system *o–xyz* is established at the center of the crane turntable.

In order to simplify the establishment process of Lagrange equations, the following assumptions are made for the crane: (1) Neglect the mass of the wire rope; (2) The lifting weight is simplified as a mass; (3) Neglect the friction between the various institutions in the system; (4) Consider only the stable static wind load on the horizontal plane. Based on the research content of this paper, the Lagrange equation is as follows:

**Fig. 1** Boom-hoisting system

**Fig. 2** Rigid-flexible coupling dynamic model

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}\_i}\right) - \frac{\partial L}{\partial q\_i} + \frac{\partial F\_\mathbf{c}}{\partial \dot{q}\_i} = \mathcal{Q}\_{\text{out}} \tag{1}$$

### *2.1 Dynamic Analysis of Boom*

As in Fig. 2, the equivalent spring-damper system concentrates the boom mass at the equivalent mass point (*m*r) at the head of the boom; taking the variable amplitude rotation center as the origin, the inertial coordinate system *o*1–*x*1*y*1*z*1 at the tail of the boom is established; based on the projection of the boom on the rotary plane at *o*1, the following-rotation coordinate system *o*1–*x*2*y*2*z*2 is established; and with the spatial position of the boom as the reference, the following-rotation coordinate system *o*2 *x*3*y*3*z*3 is established at the equivalent mass point (*m*r) to describe the spatial position of the head of the boom. Using the follower coordinate system *o*j–*x*i*y*i*z*i, it is known that the vector diameter of the equivalent mass point (*m*r) in the global coordinate system *o*–*xyz* is given by

$$r\_{\mathbf{r}} = R\_a \left[ R\_\beta (r\_\mathbf{b} + \mathbf{u}) + r\_\mathbf{a} \right] \tag{2}$$

where, *r***b** =[ *l*b 00 ] T is the position vector of the mass point when the boom is not deformed in the coordinate system *o*2–*x*3*y*3*z*3, *l*b is the length of the boom; *r***a** =[ *a* 0 0 ] T is the position vector of the amplitude center *o*1 in the coordinate system *o–xyz*, *a* is the linear distance between the amplitude center *o*1 and the center of the turntable *o*; *u* = *u*(*t*) =[ *u*x(*t*) *u*y(*t*) *u*z (*t*)] T is the vibration displacement of the mass point in the coordinate system *o*2–*x*3*y*3*z*3 at time *t*; *R***<sup>α</sup>** is the coordinate transformation matrix between the coordinate system *o*1–*x*2*y*2*z*2 and *o*1–*x*1*y*1*z*1; *R***<sup>β</sup>** is the coordinate transformation matrix between the coordinate system *o*1–*x*2*y*2*z*<sup>2</sup> and *o*2–*x*3*y*3*z*3.

The vectorial diameter *r*r of the mass point (*m*r) is a first-order derivative with respect to time *t*, and its velocity in the inertial system can be obtained as

$$\dot{r}\_r = R\_a A\_a \left[ R\_\flat (r\_\flat + \mu) + r\_\natural \right] + R\_a R\_\flat A\_\flat (r\_\flat + \mu) + R\_a R\_\flat \dot{u} \tag{3}$$

where, *Λ***<sup>α</sup>** = α(˙ *t*)*Γ* **<sup>α</sup>** is the angular velocity correlation matrix of the boom rotation, α(*t*) is the rotation angle of the boom at time *t*; *Λ***<sup>β</sup>** <sup>=</sup>β(˙ *<sup>t</sup>*)*<sup>Γ</sup>* **<sup>β</sup>** is the angular velocity correlation matrix of the boom luffing, β(*t*) is the luffing angle of the boom at time *t*.

Let *R*β**r** = *R***β**(*r***b** + *u*), *r***bu** = *r***b** + *u*. From Eq. (3), the kinetic energy of the equivalent spring-damped system is given by

$$T\_{\mathbf{r}} = \frac{1}{2} \dot{\mathbf{r}}\_{\mathbf{r}}^{\mathrm{T}} m\_{\mathbf{r}} \dot{\mathbf{r}}\_{\mathbf{r}} = \frac{1}{2} \begin{bmatrix} \dot{\alpha} \ \dot{\beta} \ \dot{\mathbf{u}}^{\mathrm{T}} \end{bmatrix} \begin{bmatrix} \mathbf{M}\_{\alpha\alpha}^{\mathrm{r}} \ M\_{\alpha\beta}^{\mathrm{r}} \ M\_{\alpha\mathbf{u}}^{\mathrm{r}} \\\ \mathbf{M}\_{\beta\alpha}^{\mathrm{r}} \ M\_{\beta\beta}^{\mathrm{r}} \ M\_{\beta\mathbf{u}}^{\mathrm{r}} \\\ \mathbf{M}\_{\mathbf{u}\alpha}^{\mathrm{r}} \ M\_{\mathbf{u}\beta}^{\mathrm{r}} \ M\_{\mathbf{u}\mathbf{u}}^{\mathrm{r}} \end{bmatrix} \begin{bmatrix} \dot{\alpha} \\\ \dot{\beta} \\\ \dot{\mathbf{u}} \end{bmatrix} \tag{4}$$

where, *M***<sup>r</sup>** αα=*m*r *R*<sup>T</sup> <sup>β</sup>**<sup>r</sup>***Γ* <sup>T</sup> **<sup>α</sup>***Γ* **<sup>α</sup>***R*β**<sup>r</sup>**; *M***<sup>r</sup>** αβ =  *M***<sup>r</sup>** βα <sup>T</sup>= *m*r *R*<sup>T</sup> <sup>β</sup>**<sup>r</sup>***Γ* <sup>T</sup> **<sup>α</sup>***R***β***Γ* **<sup>β</sup>** *r***bu**; *M***<sup>r</sup>** <sup>α</sup>**u** = *M***<sup>r</sup> u**α <sup>T</sup>= *m*r *R*<sup>T</sup> <sup>β</sup>**<sup>r</sup>***Γ* <sup>T</sup> **α***R***β**; *M***<sup>r</sup>** ββ = *m*r*r*<sup>T</sup> **bu***Γ* <sup>T</sup> **<sup>β</sup>***Γ* **<sup>β</sup>** *r***bu**; *M***<sup>r</sup>** <sup>β</sup>**u** =  *M***<sup>r</sup> u**β <sup>T</sup>= *m*r*r*<sup>T</sup> **bu***Γ* <sup>T</sup> **β** ; *M***<sup>r</sup> uu** = *m*r *I* **<sup>3</sup>**×**<sup>3</sup>**, *I* **<sup>3</sup>**×**<sup>3</sup>**is the unit matrix. Let the boom-head mass correlation array be *M***r**.

The damping of the equivalent spring-damped system is given by

$$F\_{\mathbf{cr}} = \frac{1}{2} \dot{\boldsymbol{\mu}}^{\mathrm{T}} \mathbf{C}\_{\mathbf{r}} \dot{\boldsymbol{\mu}} \tag{5}$$

where, *C***r** = *diag*(*c*x, *c*y, *c*z) is the damping coefficient matrix of the equivalent system, *c*i = 2*m*rξrωnr.

The equivalent spring-damped system considering its elastic potential energy, it is given by

$$\mathbf{V\_r = \frac{1}{2}u^\mathrm{T}K\_\mathrm{r}u} \tag{6}$$

where, *K***r**=*diag*(*k*x, *k*y, *k*z) is the matrix of stiffness coefficients of the equivalent system, *k*i = *F*i/*u*ri.

### *2.2 Dynamic Analysis of Lifting Weight*

As in Fig. 2, the spatial pendulum system concentrates the mass of the lifting weight at the equivalent mass point (*m*q); at the equivalent mass point (*m*q), the coordinate system *o*1–*x*2*y*2*z*2 is used as the reference, and the follower coordinate system *o*1– *x*4*y*4*z*4 is established so that it is always parallel to the coordinate system *o*1–*x*2*y*2*z*2, which is used to describe the spatial position of the lifting weight. Using the rotation coordinate system *o*j–*x*i*y*i*z*i and the body coordinate system *o*1–*x*4*y*4*z*4, the vector diameter of the equivalent mass point (*m*q) in the global coordinate system *o–xyz* is given by

$$r\_{\mathbf{q}} = R\_{\mathbf{a}} \Big| R\_{\mathbf{\bar{\beta}}}(r\_{\mathbf{b}} + \mathbf{u}) + l\_{\mathbf{q}} + r\_{\mathbf{a}} \Big| \tag{7}$$

where, *l***q** = (*l* + δ)sin θ*x* cos θ*y* (*l* + δ)sin θ*y* −(*l* + δ) cos θ*x* cos θ*<sup>y</sup>* T is the spatial position vector of the lifting weight in the follower coordinate system *o*1– *x*4*y*4*z*4, *l* = *l*(*t*) is the length of the wire rope at time *t*, δ = δ(*t*) is the elastic deformation of the wire rope at time *t*, let *φ*(*t*) = *l*(*t*) δ(*t*) T ; θx = θx(*t*), θy = θy(*t*) is the oscillation angle in the amplitude plane and the deflection angle in the rotary plane, respectively, let *ϑ*(*t*) = θx(*t*) θy(*t*) T .

The vectorial diameter *r*q of the mass point (*m*q) is a first-order derivative with respect to time *t*, and its velocity in the inertial system can be obtained as

$$\begin{split} \dot{r}\_{\mathsf{q}} &= \mathsf{R}\_{\mathsf{a}} \mathsf{A}\_{\mathsf{a}} \Big[ \mathsf{R}\_{\mathsf{\bar{\theta}}} (r\_{\mathsf{b}} + \mathsf{u}) + l\_{\mathsf{q}} + r\_{\mathsf{a}} \Big] + \mathsf{R}\_{\mathsf{a}} \mathsf{R}\_{\mathsf{\bar{\theta}}} \mathsf{A}\_{\mathsf{\bar{\theta}}} (r\_{\mathsf{b}} + \mathsf{u}) \\ &+ \mathsf{R}\_{\mathsf{a}} \mathsf{R}\_{\mathsf{\bar{\theta}}} \dot{\mathsf{u}} + \mathsf{R}\_{\mathsf{a}} \mathsf{R}\_{\mathsf{\bar{\theta}}} \dot{\mathsf{\bar{\theta}}} + \mathsf{R}\_{\mathsf{a}} \mathsf{R}\_{\mathsf{\bar{\theta}}} \dot{\mathsf{y}} \end{split} \tag{8}$$

where, *R***f** = sin θ*x* cos θ*y* sin θ*y* − cos θ*x* cos θ*<sup>y</sup>* sin θ*x* cos θ*y* sin θ*y* − cos θ*x* cos θ*<sup>y</sup>* T is the wire rope velocity matrix; *<sup>R</sup>***J** = (*l* + δ) cos θ*x* cos θ*<sup>y</sup>* 0 sin θ*x* cos θ*<sup>y</sup>* − sin θ*x* sin θ*y* cos θ*y* cos θ*x* sin θ*<sup>y</sup>* T is the swing velocity matrix of lifting weight.

Let *R*β**<sup>q</sup>**= *R***β**(*r***b** + *u*) + *l***q** + *r***a**. From Eq. (8), the kinetic energy of the spatial pendulum system is given by

$$\begin{aligned} T\_{\mathbf{q}} &= \frac{1}{2} \dot{\mathbf{r}}\_{\mathbf{q}}^{\mathrm{T}} m\_{\mathbf{q}} \dot{\mathbf{r}}\_{\mathbf{q}} \\ &= \frac{1}{2} \begin{bmatrix} \dot{\boldsymbol{\alpha}} & \dot{\boldsymbol{\beta}} \end{bmatrix} \dot{\boldsymbol{\alpha}}^{\mathrm{T}} \dot{\boldsymbol{\phi}}^{\mathrm{T}} \dot{\boldsymbol{\phi}}^{\mathrm{T}} \dot{\boldsymbol{\phi}}^{\mathrm{T}} \begin{bmatrix} \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\alpha}\boldsymbol{\alpha}} \ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\alpha}\boldsymbol{\beta}} \ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\alpha}\boldsymbol{\beta}} \ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\alpha}\boldsymbol{\beta}} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{\alpha}} \\ \dot{\boldsymbol{\beta}} \\ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\alpha}\boldsymbol{\alpha}} \ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\beta}\boldsymbol{\beta}} \ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\alpha}\boldsymbol{\beta}} \ \boldsymbol{M}^{\mathrm{q}}\_{\boldsymbol{\beta}\boldsymbol{\beta}} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{\alpha}} \\ \dot{\boldsymbol{\beta}} \\ \dot{\boldsymbol{\beta}} \\ \dot{\boldsymbol{\alpha}} \\ \dot{\boldsymbol{\beta}} \\ \dot{\boldsymbol{\beta}} \end{bmatrix} \end{aligned} (9)$$

where, *M***<sup>q</sup>** αα=*m*q *R*<sup>T</sup> <sup>β</sup>**<sup>q</sup>***Γ* <sup>T</sup> **<sup>α</sup>***<sup>Γ</sup>* **<sup>α</sup>***R*β**<sup>q</sup>**;*M***<sup>q</sup>** αβ = *M***<sup>q</sup>** βα<sup>T</sup> <sup>=</sup>*m*q *R*<sup>T</sup> <sup>β</sup>**<sup>q</sup>***Γ* <sup>T</sup> **<sup>α</sup>***R***β***Γ* **<sup>β</sup>** *r***bu**;*M***<sup>q</sup>** <sup>α</sup>**<sup>u</sup>**= *M***<sup>q</sup> u**α <sup>T</sup>= *m*q *R*<sup>T</sup> <sup>β</sup>**<sup>q</sup>***Γ* <sup>T</sup> **α***R***β**;*M***<sup>q</sup> <sup>α</sup>***<sup>φ</sup>* = *M***<sup>q</sup>** *φ***α** T <sup>=</sup>*m*q *R*<sup>T</sup> <sup>β</sup>**<sup>q</sup>***Γ* <sup>T</sup> **α***R***f**; *M***<sup>q</sup> <sup>α</sup>***<sup>ϑ</sup>* =  *M***<sup>q</sup>** *ϑ***α** <sup>T</sup><sup>=</sup> *m*q *R*<sup>T</sup> <sup>β</sup>**<sup>q</sup>***Γ* <sup>T</sup> **α***R***J**; *M***<sup>q</sup>** ββ = *m*<sup>q</sup> *r*<sup>T</sup> **bu***Γ* <sup>T</sup> **<sup>β</sup>***<sup>Γ</sup>* **<sup>β</sup>** *<sup>r</sup>***bu**;*M***<sup>q</sup>** <sup>β</sup>**<sup>u</sup>**= *M***<sup>q</sup> u**β T = *m*<sup>q</sup> *r*<sup>T</sup> **bu***Γ* <sup>T</sup> **<sup>β</sup>** ; *M***<sup>q</sup> <sup>β</sup>***<sup>φ</sup>* = *M***<sup>q</sup>** *φ***β** T = *m*<sup>q</sup> *r*<sup>T</sup> **bu***Γ* <sup>T</sup> **<sup>β</sup>** *R***<sup>T</sup> <sup>β</sup>** *<sup>R</sup>***f**; *M***<sup>q</sup> <sup>β</sup>***<sup>ϑ</sup>* = *M***<sup>q</sup>** *ϑ***β** T = *m*<sup>q</sup> *r*<sup>T</sup> **bu***Γ* <sup>T</sup> **<sup>β</sup>** *R***<sup>T</sup> <sup>β</sup>** *R***J**;*M***<sup>q</sup> uu** = *m*q *I* **<sup>3</sup>**×**<sup>3</sup>**; *M***<sup>q</sup> <sup>u</sup>***<sup>φ</sup>* = *M***<sup>q</sup>** *φ***u** T <sup>=</sup>*m*q *R***<sup>T</sup> <sup>β</sup>** *<sup>R</sup>***f**; *M***<sup>q</sup> <sup>u</sup>***<sup>ϑ</sup>* =  *M***<sup>q</sup>** *ϑ***u** <sup>T</sup>= *m*q *R***<sup>T</sup> <sup>β</sup>** *<sup>R</sup>***J**; *M***<sup>q</sup>** *φφ* = *m*q *R***<sup>T</sup> f** *R***f**;*M***<sup>q</sup>** *φϑ* = *M***<sup>q</sup>** *ϑφ*<sup>T</sup> <sup>=</sup>*m*q *R***<sup>T</sup> f** *R***J**; *M***<sup>q</sup>** *ϑϑ* <sup>=</sup>*m*q *R***<sup>T</sup> <sup>J</sup>***R***J**. Let the boom-head mass correlation array be *M***q**.

The damping of the spatial pendulum system mainly considers its internal and external damping of the system; the internal damping is replaced by the equivalent damping, and the external damping is mainly the air friction damping, then

$$\begin{cases} \text{Internal damping: } \boldsymbol{F}\_{\text{cq}}^{1} = \frac{1}{2} \boldsymbol{\dot{\mathfrak{b}}}^{T} \boldsymbol{C}\_{\boldsymbol{\vartheta}} \boldsymbol{\dot{\vartheta}} \\ \text{External damping: } \boldsymbol{F}\_{\text{cq}}^{2} = \boldsymbol{\mu}\_{\text{g}} \boldsymbol{\dot{\mathfrak{b}}}^{T} \boldsymbol{M}\_{\boldsymbol{\vartheta}\boldsymbol{\vartheta}}^{q} \boldsymbol{\dot{\mathfrak{b}}} \end{cases} \tag{10}$$

where, *Cϑ*=*diag*(*c*<sup>ϑ</sup> , *c*<sup>ϑ</sup> ) is the damping coefficient matrix of the equivalent system, *c*<sup>ϑ</sup> = 2*mq* ξ*q* ω*nq* , μg is the air damping coefficient.

The potential energy of the system mainly considers the gravitational potential energy of the lifting weight and the elastic potential energy of the wire rope, then

$$\begin{cases} \text{Lifting weight } \colon V\_{\neq}^{1} = m\_{\text{q}} \text{g} \{ (l\_{\text{b}} + u\_{\text{x}}) \sin \beta + u\_{\text{z}} \cos \beta - (l + \delta) \cos \theta\_{\text{x}} \cos \theta\_{\text{y}} \} \\ \text{Wire rope: } V\_{\neq}^{2} = \frac{1}{2} k\_{\text{g}} \delta^{2} \end{cases} \tag{11}$$

where, *k*g is the wire rope stiffness coefficient, *k*g = *n*ε*E*g *A*g/*l*.

### *2.3 Model of Wind Load*

"Crane Design Code" GB/T3811-2008 in the provisions of the crane working state of the wind load calculation formula [9] as follows

$$P\_{\text{w}} = CpA = \frac{1}{2}\rho\_{\text{u}}v^2CA\tag{12}$$

where, *C* is the wind coefficient; *p* is the working state of the calculated wind pressure; *A* is the actual windward area of the crane;ρa is air density; *v* is the instantaneous wind speed.

The instantaneous wind speed consists of the mean wind speed (v) and the turbulent wind speed (vturb) obeying the Gaussian distribution, i.e. v = v + vturb. The turbulent velocity is modeled as follows

$$\begin{cases} \frac{d\upsilon\_{\text{turb}}}{dt} = \frac{1}{T\_{\text{v}}} \upsilon\_{\text{turb}} + m\_{\text{wind}}\\\ T\_{\text{v}} = \frac{10.5z}{\overline{v}} \end{cases} \tag{13}$$

where, *m*wind is the standard distribution white noise generator; *T*v is the time constant; *z* is the height of the mass center of the entity.

### *2.4 Dynamic Model of Boom-Hoisting System*

Taking the generalized coordinates as *q*i = <sup>α</sup>˙ <sup>β</sup>˙ *<sup>u</sup>*˙ <sup>T</sup>*φ*˙ <sup>T</sup>*ϑ*˙ <sup>T</sup> <sup>T</sup> . Substituting Eqs. (4)– (6) and Eqs. (9)–(12) into Eq. (1) and simplifying it, the system dynamics equation is given by

$$\mathbf{M}\ddot{q}\_{\text{i}} + \mathbf{C}\dot{q}\_{\text{i}} + \mathbf{K}q\_{\text{i}} = \mathbf{Q}\_{\text{out}} + \mathbf{Q}\_{\text{v}} \tag{14}$$

where, *M* = *M***r** *O O O* + *M***q** is the matrix of system stiffness coefficients; *C* = *diag*(0, 0, *c*x, *c*y, *c*z, 0, 0, *c*<sup>ϑ</sup> , *c*<sup>ϑ</sup> ) is the matrix of system damping coefficients; *K* = *diag*(0, 0, *k*x, *k*y, *k*z, *k*g, 0, *k*<sup>ϑ</sup> , *k*<sup>ϑ</sup> )is the matrix of system stiffness coefficients, *k*<sup>ϑ</sup> = *mq g*(*l* + δ); *Q***out**=[ *TM*α(*t*) *TM*β(*t*) 0 *Tuy* (*t*) *Tuz*(*t*) *Fl* (*t*) 0 *T*<sup>θ</sup> *<sup>x</sup>*(*t*) *T*<sup>θ</sup> *<sup>y</sup>* (*t*)] T is the system input signal, *Q***v** is the system inertial force.

In summary, a seven-input nine-degree-of-freedom underdriven rigid-flexible coupling dynamic mathematical model is established. Among them, 3 degrees of freedom describe the rotation, variable amplitude and lifting motions of the system; 3 degrees of freedom describe the boom head vibration of the boom; 1 degree of freedom describes the elastic vibration of the wire rope; and 2 degrees of freedom describe the spatial swing of the lifting weight. The numerical solution of the multibody dynamics Eq. (14) is realized using the fourth-order Lunga-Kuta method and MATLAB.

### **3 Establish ADAMS Simulation Model**

Based on a certain type of wind power crane, the flexible boom model is established by ANSYS, as shown in Fig. 3. In ADAMS, the CABLE module is used to establish the lifting wire rope; the boom tension plate is equivalently replaced by two wire ropes with larger stiffness; the rigid body turntable, chassis and support plate of boom are modeled by SOLIDWORKS; the contact force is set between the self-built ground and the lifting weight, and the flexible boom in ANSYS is used to replace the rigid boom in the original ADAMS model. The simulation model is shown in Fig. 4.

**Fig. 3** Boom in ANSYS

**Fig. 4** ADAMS rigid-flexible coupling model

### **4 Example Analysis**

In the actual hoisting operation, there is only one of the three working conditions of rotation, variable amplitude and lifting motion of the wind power crane. This paper takes the rotary condition as an example, and analyzes the dynamic response curves of the above mathematical model and simulation model in MATLAB and ADAMS.

In this paper, when verifying the rationality and accuracy of the mathematical model, both models do not apply wind load. The rotary signal of the boom is shown in Fig. 5. When verifying the effect of wind load on the swing of the lifting weight and vibration of the boom head, the wind load is applied in the mathematical model, and the rotary signal of the boom is shown in Fig. 6. The acceleration and deceleration processes of the rotary signal are relatively smooth during the actual rotary operation, and the sinusoidal signal is used to approximate the simulated acceleration model during the solution process [9].

### *4.1 Model Parameter*

For the above type of crane, the boom consists of 14 section booms, with a total length of 130 m when fully assembled. In the case analysis, the wire rope multiplication is 8 and the boom amplitude angle is 75°. The structural parameters of the equivalent spring-damping system of the boom are obtained by using ANSYS and ADAMS

**Fig. 5** Rotary signal 1

**Fig. 6** Rotary signal 2

**Table 1** Structural parameters of the equivalent spring damping system


analysis results, as shown in Table 1. The equivalent stiffness of the elastic two-force bar is *k*g = 9.07 × 106 N m−1.

For wind load, the standard of crawler crane stipulates that the working wind speed of lattice boom crane should satisfy: when the boom length is more than 50 m, the wind speed should not be more than 9.8 m/s [10]. The wind speed in grade 5 is 8.0–10.7 m/s. In this paper, the influence of wind load on swing angle is studied using five-stage average wind speed of 9.35 m/s.

### *4.2 Dynamic Response Analysis of the System*

Figure 7 shows the Z-direction (the direction perpendicular to the boom in the variable amplitude plane) response characteristic curve of the boom head at different moments when the lifting weight is 50 t. In 0–50 s, the lifting weight is placed on the ground in the ADMAS model, the boom head vibrates freely, and the Z-direction displacement of the boom head is 1000 mm when it is stable. In 50–60 s, the lifting weight of the ADAMS model is lifted off the ground, and at the same time, the boom head of the mathematical model starts to vibrate freely under the action of the crane weight

**Fig. 7** Z-direction vibration of boom head

of 50 t. In 60–250 s, the two models do the free vibration without excitation and with damping. When the rotary signal of Fig. 7 is applied at 250 s, the mathematical model shows a more obvious stimulated vibration with increased amplitude in the acceleration and deceleration stages, and the ADAMS model also has a corresponding trend but is not obvious. At the end of the rotation at 268 s, the vibration of the two models gradually decays to 0, and the Z-direction deformation of the models at the time of stabilization is 1767.3 mm and 1764.6 mm, respectively. In ANSYS, applying the boundary conditions and the 50 t lifting weight to the boom, the Zdirection deformation of the boom head is *u*z = 1798.94 mm, as shown in Fig. 8; the comparison of the deformation results verifies the accuracy of the model construction.

Figures 9 and 10 describe the oscillation angle in the variable amplitude plane and the deflection angle in the rotary plane of the two models under the action of the signals in Fig. 5, respectively. The oscillation angle in the variable amplitude plane is mainly caused by the centrifugal force during rotation of the lifting weight and the Coriolis force of the lifting weight deflecting on the rotary plane. From the graph, it can be seen that the response gradually increases to the peak value in the uniform motion stage, at this time, the oscillation angle of the ADAMS model is 2.10° and that of the mathematical model is 2.31°; the oscillation angle of the mathematical model increases by 15.2% compared with that of the ADAMS model at the maximum value of the difference. The mathematical model has a superior following to the ADAMS model, but the mathematical model decays faster than the ADAMS model due to the consideration of the effect of air resistance. The deflection angle of the rotary plane is mainly caused by the inertia force during rotation of the

#### **Fig. 8** Ansys results

lifting weight, and the amplitude is obtained in the acceleration and deceleration phases. The maximum deflection angles of the two models are 2.54° and 2.60°, respectively, and the deflection angle of the mathematical model is 14.1% higher than that of the ADAMS model at the maximum difference.

### *4.3 Dynamic Response of the System Under Wind Load*

Based on the description in Sect. 4.1, the instantaneous wind direction is specified to be 30° and blowing from *y*− to *x*+ in the global coordinate system *o–xyz*. Due to the specificity of the model in this paper, the change of the windward area of the model during rotation is not considered. Using the wind load model in Sect. 2.3 to solve the wind load signals of the boom head and the lifting weight, the lifting weight is 10 t during the wind load impact analysis. The rotary signal is shown in Fig. 6.

Figure 11 shows the Z-direction vibration of the boom in the mathematical model without and with wind load. The model applies wind load at 100 s, and then the amplitude of the boom vibration increases and the equilibrium position is shifted downward by 21 mm, which is also consistent with the ANSYS model results.

Figures 12 and 13 describe the swing angles of the lifting weight in the absence and presence of wind loads. In the absence of wind load, the maximum deflection angles of the crane in the variable amplitude plane and the rotary plane are 5.19° and 4.29°, respectively. In comparison with the results of the analysis in Sect. 3.2, it shows that the swing angle of the lifting weight increases greatly under the conditions of increased rotary speed and reduced lifting weight. In the case of wind load, the

**Fig. 9** Oscillation angle of the lifting weight in variable amplitude plane

**Fig. 10** Deflection angle of the lifting weight in rotary plane

**Fig. 11** Comparison of Z-direction deformation of boom head in mathematical model without and with wind load applied

balance position of the lifting weight is shifted to the zero position, and the oscillation angle in the variable amplitude plane and the deflection angle in the slewing plane increase by 0.19° and 0.29°, respectively. The change in oscillation and deflection angles under wind load is small, but it has caused a large positional deviation for the ultra-long flexible boom crane that needs precise positioning. Based on the simulation data in this paper, the maximum offset of the lifting weight is 385 mm, which makes it difficult to ensure the accuracy of the hoisting if no intervention is made in the actual operation.

**Fig. 12** Oscillation angle of the lifting weight in variable amplitude plane under wind load

**Fig. 13** Deflection angle of the lifting weight in rotary plane under wind load

# **5 Conclusion**

In this paper, based on the wind power crane with a large slenderness ratio, a rigidflexible coupling dynamic mathematical model is established by using the equivalence method, and constructing the ADAMS rigid-flexible coupling model. The accuracy and rationality of the mathematical model are verified by the numerical simulation analysis of the two models. Based on this, the transient wind model is used to simulate the effect of wind load on the swing of the lifting weight. The results obtained are as follows:


**Acknowledgements** This work is supported by National Natural Science Foundation of China No. 52275088 and the Fundamental Research Funds for the Central Universities No. DUT22LAB507.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Parametric Design Method of Hydraulic Buffer System for Low-Speed Heavy-Load Trailer**

**Guanyu Min, Qing Zhang, and Liwei Xu** 

**Abstract** To address the problem of high-pressure low-frequency hydraulic impact on the hydraulic buffer system mounted on self-propelled hydraulic trailer, a parametric simulation and design framework is provided for the widespread use of integrated accumulator systems in the low-speed heavy-load vehicle. A mathematical model of the accumulator was established for theoretical analysis, and numerical simulation were conducted on corresponding parameters using AMESim. Finally, a routine method was proposed to achieve feasible arrangement of accumulators. In this study, for the problem of high-pressure impact conditions caused by severe fluctuations, the system performance could be increased by arranging different combinations of accumulators with different parameter configurations. Taking a commercial model of heavy trailer as an actual case, a reasonable parameterized design was carried out for its comprehensive accumulator buffer system, and verification was conducted.

**Keywords** Accumulator · Hydraulic buffering · Low-speed heavy-load · Parametric design · AMESim simulation

# **1 Introduction**

The low-speed heavy-load vehicle such as self-propelled hydraulic trailer could experience periodic and severe fluctuations in external loads during working operation, causing rapid reversal or stagnation of the fluid flow in the hydraulic buffer system, which results in low-frequency and high-strength hydraulic impacts [1].

As an important component of the hydraulic buffer system, the working principle is to convert excess energy into potential energy for storage based on energy balance. When the hydraulic buffering system needs it, the stored energy can be released [2].

Tongji University, Shanghai 200092, China e-mail: zhqing\_tj@126.com

G. Min · Q. Zhang (B) · L. Xu

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_2

The airbag type accumulator has relatively small inertia, fast response, and the characteristics of oil gas separation, less leakage, and easy maintenance, which make it widely used in the design of hydraulic buffer system [3, 4]. Since WABCO Company proposed the hydropneumatic suspension technology in the 1950s, hydraulic buffer systems based on accumulators have been widely used in low-speed and heavy-load vehicles and construction machinery. For example, various types of mining dump trucks, all terrain cranes, wheeled excavators, and self-propelled hydraulic trailers.

The use of computer simulation technology to study the actual dynamic characteristics of hydraulic system and achieve parameterized design is an important tool for developing modern hydraulic system research technology. For a certain type of system family, including single component design selection and system combination optimization configuration, has become a new idea in the current hydraulic buffer system design [5]. This study takes a type of heavy-load transport trailer as an actual case and provides a parameter simulation path and a combination optimization framework for the design of hydraulic buffer system family.

### **2 Mathematical Analysis**

Taking the airbag type accumulator as the research object, the simplified model [6] of the accumulator is shown in Fig. 1.

The inflation gas in the airbag accumulator is nitrogen, which is regarded as an ideal gas and satisfies the equation:

$$p\_{a0}V\_{a0}^{n} = p\_{a2}V\_{a2}^{n} = \text{C} \tag{1}$$

**Fig. 1** Simplified model of energy accumulator

where *pa*0 (MPa) is initial pressure and *pa*2 (MPa) is the final pressure, *Va*0 (m3) is the initial inflation volume and *Va*2 (m3) is the final inflation volume in the accumulator. n is polytropic exponent, which is between 1 and 1.4 in practical gas compression process.

Simplifying airbag type accumulator into a system with only axial motion indicates that the system is a spring damping system. The force equation of the charging chamber of the accumulator is deduced as

$$(p\_b - p\_a)A\_a = k\_e \frac{V\_a}{A\_a} + c\_e \frac{1}{A\_a} \frac{dV\_a}{dt} \tag{2}$$

where *pb* (MPa) is oil pressure and *pa*(MPa) is gas pressure in the accumulator. *Aa*  (m2) is section area of the accumulator. *ke* is stiffness coefficient, *Va* (MPa) is volume and *ce* is damping coefficient of the gas in the accumulator. *ke* and *ce* are defined [7] by:

$$k\_e = \frac{\Delta F}{\Delta x} = \frac{\Delta p \times A}{\Delta V / A} = A\_a^{\;\;2} \frac{dp}{dV} = A\_a^{\;\;2} \frac{np\_{a0}}{V\_{a0}} \tag{3}$$

$$c\_{\epsilon} = 8\pi \,\mu\_1 l = 8\pi \,\mu \frac{V\_a}{A\_a} \tag{4}$$

where μ1 is the viscosity coefficient of the inflated gas in the accumulator.

Due to the fact that the liquid stiffness of the hydraulic oil in the accumulator is much greater than the gas stiffness, the elastic modulus of the oil is not considered in the force analysis of the hydraulic oil in the airbag accumulator. The force equation of the oil inlet chamber of the accumulator is deduced as:

$$p\_1 A\_n - p\_a A\_a = m \frac{d^2 V\_a}{dt^2} \frac{1}{A\_a} + B\_e \frac{d V\_a}{dt} \frac{1}{A\_a} \tag{5}$$

where *p*1 (MPa) is inlet pressure of the accumulator. *m* (kg) is quality and *Be* is viscous damping coefficient of the hydraulic oil. *Be* has the form:

$$B\_{\epsilon} = 8\pi \,\mu\_2 (l\_a + l\_n) \tag{6}$$

where μ2 is dynamic viscosity coefficient of hydraulic oil, *la* (m) is length of oil inlet and *ln* (m) is pipe length between hydraulic system and accumulator.

Now assuming *An* = *k Aa* and substituting into Eq. 2 and Eq. 5, the force equation of the accumulator is obtained as:

$$kp\_1 - p\_a = \frac{1}{A\_a^2} \left( m \frac{d^2 V\_a}{dt^2} + B\_\epsilon \frac{dV\_a}{dt} + c\_\epsilon \frac{dV\_a}{dt} + k\_\epsilon V\_a \right) \tag{7}$$

With reference to Eq. 1, there is rewritten as:

$$p\_{a0}V\_{a0}^{n} = p\_aV\_a^{n} \tag{8}$$

$$\frac{dV\_a}{dt} = -\frac{V\_a}{np\_a}\frac{dp\_a}{dt} = -\frac{V\_{a0}}{np\_{a0}}\frac{dp\_a}{dt} \tag{9}$$

Since the elastic modulus of the oil is not taken into account, the change in the gas chamber is equal to the change of the hydraulic oil entering the liquid chamber in the accumulator. The oil flow rate of system *Q* has the form:

$$\mathcal{Q} = -\frac{dV\_a}{dt} = \frac{V\_{a0}}{np\_{a0}} \frac{dp\_a}{dt} \tag{10}$$

By Laplace transform, Eq. 11 is rewritten as:

$$\mathcal{Q}(\mathbf{s}) = \frac{V\_{a0}}{np\_{a0}} \mathbf{(s)}\tag{11}$$

By combining Eqs. 7 and 11, the whole system transfer function *G*(*s*) can be derived as:

$$G(s) = \frac{V\_a(s)}{p\_1(s)} = \frac{kV\_a^2}{ms^2 + (B\_e + c\_e) - (\frac{np\_{a0}A\_a^2}{V\_a^2} - k\_e)}\tag{12}$$

which reduces to:

$$G(s) = \frac{kV\_a^2}{np\_{a0}A\_a^2 - k\_eV\_{a0}} \cdot \frac{\alpha\_n^2}{ns^2 + 2\xi\alpha\_n + \alpha\_n^2} \tag{13}$$

where ω*n* is the natural frequency and ξ is equivalent damping ratio of airbag accumulators, having the form:

$$
\omega\_n = \sqrt{\frac{n p\_{a0} A\_a^2}{V\_{a0} m} - \frac{k\_e}{m}} \tag{14}
$$

$$\xi = \frac{B\_{\epsilon} + c\_{\epsilon}}{2\sqrt{\frac{np\_{o0}A\_{g}^{2}m}{V\_{o0}} - k\_{\epsilon}m}} \tag{15}$$

### **3 System Simulation and Analysis**

In this paper, the hydraulic buffering system is mainly used for vibration buffering of low-speed heavy-load trailers. In the vibration analysis, vibration generally could be simplified and analyzed. Vehicle vibration can be simplified into models with multiple degrees of freedom. This article adopts a two-degree-of-freedom model. As shown in Fig. 2, the impact of tire elasticity on vibration is more considered than the single-degree-of-freedom model.

Take a certain commercial model of heavy-load trailer as an example shown in Fig. 3. Based on the physical model of the accumulator shown in Fig. 1 and the suspension buffering model shown in Fig. 2, a simulation system [7] is established as shown in Fig. 4. The parameters of each component in the system are shown in Table 1.

The impact buffering efficiency of accumulator group is now evaluated. Due to the long-term impact conditions of low-speed heavy-load trailers, the buffering effect is more significantly reflected in the maximum displacement of the trailer body after each impact compared to the decay time of a single step response. Similarly, external impact load from road will also be reflected in the form of impact displacement of the trailer body.

Based on the maximum displacement of the trailer body, under a certain load pressure, the impact displacement rate *kr* and the buffering efficiency η are defined by:

**Fig. 3** A certain commercial heavy-load trailer as example



$$k\_r = \frac{|\boldsymbol{x}\_{\max} - \boldsymbol{x}\_{\min}|}{|\boldsymbol{x}\_{end}|} \tag{16}$$

$$
\eta = \frac{k\_{r0} - k\_{r1}}{k\_{r0}} \times 100\% \tag{17}
$$

where *xmax* and *xmin* are related to the highest and lowest height of the trailer body in response to impact load. *xend* is the final height of the trailer body. *kr*0 is the impact displacement rate without accumulator group. *kr*1 is the impact displacement rate with accumulator group. The higher η, the better the buffering effect of the system compared to the working condition without accumulator.

In engineering, the pre-inflation volume of an accumulator is determined by its geometric parameters, and the only parameter that can be easily adjusted for existing

AMESim

accumulators is the pre-inflation pressure. For the single accumulator system determined in the previous part, a series of pre-inflation pressures were set for buffering performance estimate.

As shown in Fig. 5, when the external impact displacement is 0.25 m and 0.5 m, the optimal pre-inflation pressures are 3.7 MPa and 2.1 MPa, respectively. At this time, the buffering performance of the whole system is the best, reaching 39.7% and 49.8% respectively.

**Fig. 5** Single accumulator buffering test

**Fig. 6** The different combination of three accumulators

For accumulator group combination system, the first to consider is that as the number of accumulators increases, there could be different series and parallel combinations, which may affect the overall performance of the accumulator group. As shown in Fig. 6, for situations where the arrangement is on the same main pipelines, these three series and parallel combinations are essentially equivalent without significant influence on the overall performance of the accumulator group.

When designing the number of accumulators in an accumulator group under the same pre-inflation pressure, it is important to consider that their buffering efficiency does not differ significantly at each load pressure, in order to achieve a globally balanced and optimal buffering performance of the accumulator group. A combination of 1 to 6 accumulators was set up for this purpose, selecting a pre-inflation pressure of 9 MPa, with an external impact displacement of 0 to 0.5 m, and each interval of 0.02 m was used to load the system with one impact. The experimental results are shown in Fig. 7.

From Fig. 7, it can be seen that when the external impact displacement is below about 0.22–0.26 m, the fewer the number of accumulators, the better the overall system buffering performance. Moreover, at around 0.1 m, the global performance of a single accumulator reaches its optimal level, which is about 13.2%. When the external impact displacement is higher than 0.25 m, as the number of accumulators increases, the overall system buffering performance increases and tends to a stable value. At the maximum impact displacement, the final buffering performance of the six accumulators is about 9%, much higher than the 4.4% of single accumulator.

**Fig. 7** Accumulator group buffering test

### **4 Discussion and Conclusion**

The simulation results in Figs. 5 and 7 can be summarized as follows: As the final design parameter for the accumulator group, each pre-inflation pressure corresponds to an external impact displacement range where the accumulator has the optimal buffering performance. In addition, under the same pre-inflation pressure, the number of accumulators and buffering performance are not directly proportional. When the external impact is small, an increase in the number of accumulators will actually reduce the overall buffering performance of the system.

Based on this issue, a concept of establishing a combination design program is proposed, which combines the combination optimization algorithm to design an optimization program for pre-inflation pressure and the number of accumulators, so as to achieve a global equilibrium and optimal buffering performance of the accumulator group. In actual algorithm selection, genetic algorithm, ant colony algorithm, or simulated annealing algorithm [8] can be used. The program diagram is shown in Fig. 8. Record the number of accumulators as *N* , and the pre-inflation pressure of each accumulator as *Pi* .

Based on the actual parameters of a type of heavy-load transport trailer, the design of accumulator combinations was studied. A mathematical model of a single accumulator system was established, and based on this, an AMESim simulation model of a combined accumulator system was established.

With a series of tests conducted on the simulation model, the comprehensive effects of the number of accumulators in the group and the pre-inflation pressure

**Fig. 8** Combination design program chart

on the performance of the accumulator group were found. In the discussion section, the comprehensive impact was evaluated and the process of combination design was obtained. The system performance could be increased by configuring suitable number of accumulators with each different parameter configurations.

**Acknowledgements** This research was supported by a project from the National Natural Science Foundation of China (No. 52075389).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Algorithm Design of a Variable Height Wheel-Legged Robot with Fuzzy Theory and PID Fusion Control**

**Xuyang Cao and Changlin Pu** 

**Abstract** A balancing control algorithm for a wheel-legged robot is designed for current logistics and distribution mobile robots. Since the wheel-legged robot is a nonlinear underactuated system, it is crucial to realize the balanced control and robustness of the wheel-legged robot in the absence of an accurate dynamics model. In this paper, an optimal control scheme for the wheel-legged robot based on the fusion control of variable-height adaptive fuzzy PID and conventional PID is designed. The balance of the wheel-legged robot is controlled by adaptive fuzzy PID control, the lifting and attitude changes of the two sides of the wheel-legs are controlled with high precision by traditional PID control, and the speed and steering control of the wheel-legged robot is not affected by the structure of the model, and is controlled by traditional PID control with linearization. The attitude of the robot is detected in real time using the BMI088 attitude sensor to realize the positional control. Experimental and simulation results show that the control algorithm of the wheel-legged robot designed in this paper is reliable, and the robot runs smoothly and robustly.

**Keywords** Wheel-legged robot · Adaptive fuzzy PID · Optimization and optimal control · Robustness

### **1 Introduction**

A two-wheeled legged mobile robot is a naturally unstable system with high order, nonlinearity, strong coupling, underactuated, etc., and the design of the balance controller is a difficult task, and most of the existing works use the model-based Linear Quadratic Regulator (LQR) technique to solve the balance problem, whose performance is highly dependent on the accuracy of the technique regarding the dynamic model. Scholars at home and abroad have made relevant research in the

X. Cao · C. Pu (B)

School of Mechanical Engineering, Dalian University of Technology, Dalian Liaoning 116024, China

e-mail: 2423637061@qq.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_3

balance control of wheel-legged robots, and Sanyo Electric Company [1] proposed a way to indirectly change the overall center of gravity of the robot through the forward and backward probing of the head, so as to achieve the purpose of controlling the robot's forward and backward movement. Tencent Robotics Robotics X [2] used the output regulation of a linear system to estimate the error between the actual forward tilt angle of the robot and the forward tilt angle in the inverted pendulum model so that the robot can maintain balance. Wang xiaoyu [3] et al. established a complete dynamics model, and an adaptive control algorithm was applied to the controller in order to improve the anti-interference ability in the balanced state. Harbin Institute of Technology [4] has designed a gain adjustment controller based on self balancing, consisting of a real-time centroid calculator and a feedback controller based on centroid adjustment.

Due to the complex mechanical structure of wheel-legged robots, it is difficult to establish an accurate dynamic model for wheel-legged robots. In addition, when the physical parameters of the robot change, the model-based controller parameters need to be manually adjusted again to ensure system stability, which affects the automation of the robot. In contrast, with the adaptive fuzzy PID control system, the robot height and tilt angle will be adjusted in time when they change to keep the car body stable, which can solve the problem of balance and stability of the wheel-legged robot [5]. At the same time, due to the changes in the attitude of the two sides of the wheel-legs and the speed and steering of the body need to be accurately controlled by using the traditional PID. Therefore, a control system based on the fusion control of adaptive fuzzy PID and traditional PID is designed to improve the balance stability of the wheel-legged robot.

### **2 Structural Design**

In order to meet the needs of intelligent logistics, the logistics and distribution mobile robot is designed into the structure shown in Fig. 1. It consists of a seven-link mechanism to form a unilateral wheel leg, two sets of three-link symmetrically distributed, and the angle is adjusted by the leg drive rods *lAB* and *lHG* to keep the body smooth, while the legs are added at the bottom end to improve the stability and robustness of the operation of the mobile robot.

The robot needs to constantly adjust its own attitude to maintain the smooth operation of the vehicle body, which requires constant calculation of the E-point coordinates to obtain the deviation of the center of gravity of the body and the height of the body, and change the position of the center of gravity by constantly adjusting the angle of the wheel legs so that it falls on the axes of the driving wheels on both sides. It can be seen from the structure sketch in Fig. 1. To get the height of the robot leg, it is necessary to solve the kinematics of the planar seven-link mechanism of the robot leg.

**Fig. 1** Wheel-legged robot model and structural diagram

$$\begin{cases} \mathbf{x}\_G = l\_1 + l\_9 \cos \phi\_5 \\ \mathbf{x}\_D = \mathbf{x}\_G - (l\_4 + l\_8) \cos \phi\_7 \\ \mathbf{x}\_E = \mathbf{x}\_D + l\_5 \cos \phi\_3 \\ \mathbf{y}\_B = l\_2 \sin \phi\_1 \\ \mathbf{y}\_F = \mathbf{y}\_B + (l\_3 + l\_7) \sin \phi\_2 \\ \mathbf{y}\_E = \mathbf{y}\_F + l\_6 \sin \phi\_4 \end{cases} \tag{1}$$

The coordinates of point E can be solved by calculation:

$$\begin{cases} \mathbf{x}\_E = l\_1 + l\_9 \cos \phi\_5 - (l\_4 + l\_8) \cos \phi\_7 + l\_5 \cos \phi\_3 \\ \mathbf{y}\_E = l\_2 \sin \phi\_1 + (l\_3 + l\_7) \sin \phi\_2 + l\_6 \sin \phi\_4 \end{cases} \tag{2}$$

### **3 System Design**

The system adopts the balance ring, speed ring and steering ring in series, and the speed of the wheel legs is controlled by the current ring of the control motor, as shown in Fig. 2, which shows the program structure of the wheel-legged robot system.

### *3.1 Conventional PID Control*

The left and right side joints of the robot use digital motors with position control, and the wheeled motion uses the PID control method to design the robot movement and steering motion controllers respectively. The wheeled motion motor control adopts the classic three-loop PID control method, the innermost loop is the balance loop, which adopts fuzzy control, and controls the balance of the body by reading the BMI088 attitude sensor data for PD operation; The second ring is the speed ring,

**Fig. 2** Program structure of wheel-legged robot system

which realizes the control of the motor speed by performing PI operation on the motor speed feedback deviation; The outermost ring is the position ring, which realizes the control of the motor position by performing PD operation on the motor position feedback deviation. The height change of the car body is controlled by fuzzy PID, the system monitors the height of the car body in real time and changes the PID parameters to adapt to the balance of the car body of different heights.

The input–output relation equation of the PID controller is:

$$\mathbf{u}(t) = K\_p e(t) + K\_i \int\_0^t e(t)dt + K\_d \frac{de(t)}{dt} \tag{3}$$

where *Kp* is the scaling factor, *Ki* is the integration factor, *Kd* is the differentiation factor, and *e*(*t*) and *u*(*t*) are the input and output quantities respectively.

The driving wheel is a DC brushless motor, the robot speed loop control is mainly through the measurement of the rotational speed values of the wheels on both sides, the calculation of the difference between the actual speed of the wheel-legged robot and the theoretical speed, low-pass filtering of the speed deviation, filtering out the high-frequency interference, preventing the speed of the sudden change, so as to make the waveform smoother and the operation more stable, and finally according to the output of the difference to change the real-time speed of the robot, the calculation formula is shown in Eq. (4):

$$Speed\\_Out = Kp \times Speed\\_Error\\_Lowout + Ki \times Speed\\_Integral \tag{4}$$

The wheel-legged robot steering loop control calculates the difference between the forward direction of the robot and the theoretical steering angle through the yaw angle and yaw angular velocity, and changes the steering of the wheel-legged robot according to the output difference, which is calculated as shown in Eq. (5):

$$Turn\\_Out = Kp \times (Target - Angle) + Kd \times gryo \tag{5}$$

The wheel-legged robot leg control calculates the difference between the robot body angle and the horizontal angle through the pitch angle and the pitch angle velocity, and changes the body tilt angle of the wheel-legged robot according to the output difference to keep the wheel-legged robot body in a horizontal state, and the calculation formula is shown in Eq. (6):

$$Height\\_Out = Kp \times (Pitch - Target) + Kd \times \text{g}ryo \tag{6}$$

### *3.2 Adaptive Fuzzy PID Control*

According to the control requirement of variable height adaptive, the fuzzy adaptive PID control of wheel-legged robot is realized by changing the PID coefficient factor. The main design idea is that the system obtains the body inclination error according to the difference between the data of BMI088 attitude sensor and the set equilibrium angle, and takes the error e and the rate of change of the error ec as the inputs, Kp, Ki, Kd as the outputs, and uses the fuzzy control rules to regulate the PID parameters so as to realize the purpose of adaptive control. The parameters of the traditional pid are fixed values, and not the adjusted parameters are the best parameters, while the fuzzy adaptive PID has the advantage of flexibility and stability, especially for the variable height wheel-legged robot, which is a large time-varying and nonlinear controlled object, its advantages are more prominent.

According to the advantages and disadvantages of fuzzy adaptive control and traditional PID, this paper combines the advantages of the two to achieve optimal control, such as Fig. 3 shows the composite structure of fuzzy PID and traditional PID.

According to the characteristics of the control system of the wheel-legged robot, the angle value measured by the angle sensor and the theoretical value at the time of

**Fig. 3** Fuzzy PID and conventional PID composite structure

balancing are used to find out the deviation e, and the deviation is derived to obtain the deviation rate of change ec, and the deviation e and the deviation rate of change ec are used as the input values of the system, and since the balancing loop adopts the fuzzy PD control, the Kp and the Kd are used as the output parameters of the fuzzy control. In order to ensure the stability of the wheel-legged robot and the safety of the body in the face of unexpected situations, the maximum controllable angle is specified as [-24°,24°], the input linguistic variable parameters of the system are set as E and EC, and the output linguistic variable parameter is set as U. The thesis domain of the deviation E is {NB, NM, NS, Z, PS, PM, PB}, and then the quantization factor is K1=0.25. According to the body simulation graph and the actual body swing, the value range of ec is determined as [-3,3]rad/s, and the same set the thesis domain of EC as {NB, NM, NS, Z, PS, PM, PB}, then the quantization factor is K2=2.

Fuzzy inference rules are developed using Simulink simulation toolbox. Based on the engineering experience, the fuzzy rule table as shown in Tables 1 and 2 is developed.

The Simulink model is built in MATLAB software, the FUZZY submodule is established to realize parameter fuzzification, and the PID submodule completes the PID control operation. As Fig. 4 shows the adaptive fuzzy control output surface


**Table 1** Fuzzy rules for Kp

**Table 2** Fuzzy rules for Kd


**Fig. 4** Kp and Kd output surface

diagram. By establishing the fuzzy theory PID control system simulation model helps to improve the stability and robustness of the control system of the wheel-legged robot.

### **4 Experimental Analysis**

In this paper, ADAMS and MATLAB joint simulation is used to analyze the balance and robustness of the wheel-legged robot. The physical model of the two-wheeled legged robot is built by writing C language program for testing to check the research theory.

### *4.1 Simulation Analysis*

The 3D model of the wheel-legged robot is established by SolidWorks software, imported into ADAMS software to impose constraints and add function relationships, and finally ADAMS software and MATLAB software are used for joint simulation. As Fig. 5 shows the simulation structure of the wheel-legged robot, the left side shows the state of the wheel-legged robot when it descends to the lowest position and the right side shows the state when it rises to the highest position.

The curve shown in Fig. 6 is obtained through simulation, and the robustness of the wheel-legged robot is examined by adding a disturbance test at the 5th and 10th seconds in the equilibrium state, applying a 0.8 N disturbance force, and the angular velocity in the equilibrium direction produces a sharp oscillation under the disturbance of the external force as shown in Fig. 7, which is quickly restored to the equilibrium state under the regulation of fuzzy control. Figure 8 shows the Roll-Pitch angle change, the body occurred a small angle change, do not affect the balance state of the body. From the 11th to the 20th second during the rise of the body, the yaw

**Fig. 5** Wheel-legged robot simulation location map

angle do not change, keeping the forward direction, the Roll-Pitch angle is adjusted at a small angle, according to the structure of the body and the distribution of the mass of each part, the balance of the body was maintained by adjusting the angle during the balance. From Fig. 9 the body rise height is 550 mm. Figure 10 shows the yaw angular velocity profile which fluctuates near 0 to maintain the stability of the body direction. In Fig. 11, the wheel speeds on both sides keep near 0. The speeds of the wheels on both sides are the same at equilibrium, so the figure shows a unilateral wheel speed curve diagram, and the simulation results show that the wheel-legged robot has good stability and robustness [6–8].

**Fig. 8** Roll angle and pitch angle

### *4.2 Motion Analysis*

On the basis of theoretical research, the construction of the physical model was completed. First of all, the wheel-legged robot motor speed response speed test, Fig. 12 shows the DC brushless motor speed response curve, it can be seen that the wheels can respond very quickly under different speed switching. The velocity response curve of the wheel-legged robot running under no load is shown in Fig. 13. It can be seen from the curve graph that different interference forces are constantly applied to the wheel-legged robot during its operation at different rotational speeds, and the running speed can be recovered in a very short time after being interfered

**Fig. 10** Pitch angular velocity

with, which indicates a better velocity response characteristic of the robot as well as a good robustness.

In the equilibrium state, Fig. 14 shows the curves of roll angle and yaw angle of the wheel-legged robot, and Fig. 15 shows the robustness test graph of the wheellegged robot, which shows that the robot has been subjected to external forces for many times, and the roll angle has been deflected by a large angle relative to the equilibrium state, and it is still able to return to the stable state in a relatively short period of time by continuously adjusting the robot's position, while keeping the roll angle fluctuating up and down in the equilibrium position, and the yaw angle

#### **Fig. 9** Body height

**Fig. 11** Wheel speed on both sides

**Fig. 12** Drive wheel motor speed response

fluctuating up and down in the fixed angle all the time, indicating that the robot's forward direction has not been changed because of the action of the external forces, and it has a higher degree of smoothness and robustness.

The wheel-legged robot can adapt to different terrain, mainly by changing the height of the two sides of the wheel legs to keep the body smooth. Figure 16 shows the physical picture of the wheel-legged robot passing through the uneven ground, Fig. 17 shows the output curve of the PWM value of the four drive motors on both sides of the wheel-legged, and Fig. 18 shows the change of the pitch angle of the wheel-legged robot during operation. According to the requirements of the smoothness of the Vehicle body, always keep the center of gravity of the Vehicle body in the lowest state, with the constant change of the body's tilt angle, the PWM output

**Fig. 13** Wheel speed under interference force

**Fig. 14** Curve plot of roll angle and yaw angle

**Fig. 15** Applying interference force to test robustness

value of the drive motors on both sides of the wheel-legged robot is also changing, adjusting the posture to ensure the balance of the body. The body can realize the lift function through the movement of the two sides of the wheel legs, Fig. 19 shows the process of the wheel-legged robot body rising, this process is mainly through the fuzzy control to realize the smoothness of the body, the two sides of the wheel legs to assist in the control, so that the center of gravity of the projection as far as possible to maintain in the wheel axis.

**Fig. 16** Wheel-legged robot passes through uneven ground

**Fig. 17** Drive motor PWM output value

**Fig. 19** Wheel-legged robot body lifting

# **5 Conclude**

This paper focuses on the design of a wheel-legged robot system algorithm based on the fusion control of variable height adaptive fuzzy PID and traditional PID, for the liftable double wheel-leg structure, it is not possible to accurately establish the system model, and it is not suitable for a single control method with a linear model, so the control method of fusion of a linear model and a nonlinear model is used to verify the feasibility of the method, and the experimental results are as follows:


The controller based on the fusion control of adaptive fuzzy PID and traditional PID has good control effect, which can meet the wheel-leg balancing system with variable center of gravity height, and greatly improves the stability and robustness of the wheel-leg robot. And the system principle is clear, easy to implement, does not need to establish an accurate system model, and can meet the design requirements of modern wheel-legged robot intelligent control system.

**Acknowledgements** This work is supported by National Natural Science Foundation of China under Grant No. 52275088 and the Fundamental Research Funds for the Central Universities under Grant No. DUT22LAB507.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Multidisciplinary Design Optimization of Excavator Positive Flow Control System**

**Yuanliu Chen, Anlin Wang, and Xiaotian Li** 

**Abstract** Due to the obstacle to the global optimal solution through the conventional sequential design, the multidisciplinary design optimization (MDO) of an excavator positive flow control system was carried out with hydraulic components and controller programs as design objects simultaneously. The positive flow control system model was established on the basis of bench and field tests to describe the coupling relationship among different subsystems. The comprehensive performance function was proposed and the MDO was carried out with the multiway valve's throttling characteristics, positive flow control parameters and spool geometries as design variables. Compared with the sequential design that only the control parameters were adjusted, the MDO searched a wider design space. The comprehensive working performance has been improved with the MDO of an excavator's swing system, which is an example for forward designs of other complicated electro-mechanical-hydraulic systems.

**Keywords** Excavator · Multiway valve · Positive flow control · MDO

# **1 Introduction**

As one of the most functional construction machineries, excavators are widely applied in fields like architectural engineering, transportation construction and so on. Excavators can achieve different complex actions by regulating the speed of actuators such as boom and arm cylinders. The positive flow control system is one of the commonest excavator systems, whose main function is to supply the demanded flow rate by controlling the displacement of the main pump. The speed of actuators is determined by the inlet flow rate. On the one hand, it is influenced by the flow rate of the main pump, namely the positive flow control. On the other hand, it is related to the fluid distribution of the multiway valve, namely the throttling characteristics. A

© The Author(s) 2024

Y. Chen · A. Wang · X. Li (B)

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: lixiaotian@tongji.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_4

conventional sequential design is usually structure designs of components followed by adjustments of controller programs. However, the global optimal solution, even the feasible solution can hardly be reached if the relationship among subsystems is cut off by decomposing the coupling hardware and controller program artificially. It is necessary to carry out an integrated design of excavator positive flow control systems with the increasing demand for the comprehensive performance.

The multidisciplinary design optimization (MDO) is a methodology to design complicated engineering systems and subsystems by fully exploring and utilizing the cooperative interaction mechanism in systems, which is widely applied in fields like aerospace and so on [1–3]. Qian [4] carried out the MDO of the composite cooling structure for a nickel-based alloy turbine blade. Nscimento [5] proposed the hybrid optimization algorithm for preliminary design of multistage launch vehicles through the application of MDO.

In the field of construction machineries, Sun [6] optimized the bucket shape line of an excavator. The bucket weight and excavation resistance were decreased. Ranjan et al. [7, 8] utilized hydro-pneumatic accumulators to store the substantial potential energy during downward movement of the boom and arm. The linear position control of them was accomplished by proportional-integral-derivative control. The mentioned researches only considered the individual design of excavator's hardware or controller. The coupling relationship between them was neglected, which made it difficult to reach the global optimal solution. Yuan [9] optimized the structural and control parameters of a bucket wheel reclaimer at the same time, reducing the energy consumption and achieving smaller vibration amplitude than the conventional sequential solution.

Hydraulic components are one of the most significant components that determine the operation performance of excavators. There have been few researches on their integrated design with the controller. This paper takes an excavator with the positive flow control system as an example, carrying out the MDO of its hydraulic component and controller.

### **2 Materials and Methods**

### *2.1 Model of the Positive Flow Control System*

The system schematic of a positive flow excavator is shown in Fig. 1. The driver controls the stroke of the multiway valve spool through the joystick. The controller adjusts the displacement of the main pump according to the pilot pressure of the joystick, making the main pump supply required flow rate. The multiway valve distributes the fluid from the main pump into the actuator or back to the tank. The inlet flow rate of the multiway valve can be changed through adjusting the displacement of the main pump, thus influencing the speed of the actuator. Meanwhile, the fluid

distribution varies with the stroke of the multiway valve spool, so that the speed of actuators is changed.

Normal operations of excavators require accurate speed regulating. As mentioned above, the speed of actuators is closely related to both the multiway valve and the positive flow control. A model of the positive flow control system was developed through a commercial software AMESim to describe the relationship among the actuator, the multiway valve and the positive flow control. The model is showed in Fig. 2.

This model described the swing circuit of the excavator as an example. Field and bench tests had been carried out to calibrate the model, as described in Chen et al. [10, 11]. The stable flow rate of the swing motor *q*m was obtained under different valve spool stroke *x* by setting different pilot pressure in the joystick of the swing circuit system model. The results are demonstrated in Fig. 3.

According to Fig. 3, when the valve spool stroke stays within the dead zone, the swing motor remains still. If the dead zone is too large, the micro movement performance will be influenced, which goes against the delicate operations. Within the speed regulating zone, the flow rate increases nonlinearly with the spool stroke. There will be differences between the required and actual speed due to the nonlinear characteristics. Drivers have to adjust the joystick repeatedly to achieve the required speed, which will lead to tiredness.

**Fig. 2** System model of the positive flow control system (1—swing motor; 2—multiway valve; 3—positive flow controller; 4—main pump; 5—joystick)

### *2.2 Flow-Number Model*

The fluid through the multiway valve adheres to the throttling equation, as shown in Eq. (1).

$$q = \mathcal{C}\_{\mathrm{d}} \mathcal{A} \sqrt{2 \Delta p / \rho} \tag{1}$$

where *q* is the flow rate, *C*d is the flow coefficient, *A* is the orifice area, *p* is the pressure drop and ρ is the density of the fluid.

The product of *C*d and *A* was defined as flow-number. The throttling characteristics of the multiway valve were described with the flow-number in the model of the positive flow control system. The flow-number model had been established in Checn et al. [12] to predict the flow-number of a single groove. The flow-number of the multiway valve could be obtained by a simple linear superposition of each groove's flow-number. Therefore, the mapping relationship between the spool geometries and the flow-number was established.

### **3 Discussion**

### *3.1 Objective Function*

The comprehensive performance was evaluated from three aspects of micro movement, speed regulating and energy consumption.

The dead zone as well as the flow rate at the beginning should be as little as possible to improve the micro movement performance. The micro movement performance was evaluated by the distance from the origin when the swing motor just started. The micro movement performance function is shown in Eq. (2).

$$f\_1 = \left[ (q\_{\rm m1}/q\_{\rm m\,max})^2 + (\mathbf{x}\_1/\mathbf{x}\_{\rm max})^2 \right]^{0.5} \tag{2}$$

where *f* 1 is the micro movement performance function. *q*m1 and *x*1 are the swing motor's flow rate and the spool stroke at the beginning. *q*mmax and *x*max are the two at the maximum spool displacement. The difference of order of magnitudes and dimensions can be cleared through normalization.

The speed regulating performance was evaluated by the linearity of the flow rate, as shown in Eq. (3). The less the value of *f* 2, the more linear the swing motor's flow rate.

$$f\_2 = \max(|q\_{\rm mi} - \mathbf{y}\_i|) / q\_{\rm m\,max} \tag{3}$$

where *f* 2 is the speed regulating performance function. *q*m*<sup>i</sup>*is the swing motor's flow rate at the *i*th spool stroke. *yi* is the linear fitting value of *q*m*<sup>i</sup>*. *f* 2 represents the maximum difference between the flow rate curve and its fitting line within the speed regulating zone.

The energy consumption performance was evaluated by the power of the main pump, as demonstrated in Eq. (4).

$$f\_3 = \sum\_{i=1}^{n} p\_{\text{pump}}(\mathbf{x}\_i) \cdot q\_{\text{pump}}(\mathbf{x}\_i) \tag{4}$$

where *f* 3 is the energy consumption function. *p*pump and *q*pump are the pressure and flow rate of the main pump respectively. *xi* is the *i* th spool stroke.

The comprehensive performance of excavators was evaluated by the weighted mean of *f* 1, *f* 2 and *f* 3, as shown in Eq. (5).

$$F = \alpha\_1 f\_1 + \alpha\_2 f\_2 + \alpha\_3 f\_3 \tag{5}$$

where *F* is the comprehensive performance function. α1, α2 and α3 are weight coefficients.

### *3.2 Sequential Design*

In conventional sequential design, every subsystem is designed independently. The comprehensive performance was optimized with the positive flow control parameters, leaving the throttling characteristics unchanged. The sequential design procedure is illustrated in Fig. 4.

The optimizer only changed the parameters of the positive flow controller in the system model. Then the comprehensive performance function *F* was computed as the objective function. The optimal control parameters were obtained through the sequential design.

**Fig. 5** MDO procedure

### *3.3 Multidisciplinary Design Optimization*

The MDO of the positive flow control system was carried out with the control parameters, multiway valve's throttling characteristics and the spool geometries as design variables simultaneously. The MDO procedure is demonstrated in Fig. 5.

The optimizer 1 optimized the comprehensive performance with both the control parameters and multiway valve's flow-number changeable. The optimizer 2 designed the spool structures to meet the required throttling characteristics. The collaborative optimizer was responsible for minimizing the interdisciplinary discrepancies while satisfying specific local constraints.

### **4 Results**

The optimized and original flow rate is shown in Fig. 6.

Both the sequential design and MDO can improve the linearity within the speed regulating zone. However, the dead zone can only be minimized through the MDO strategy. The detailed performance functions are listed in Table 1.

**Fig. 6** Optimized and original flow rate


**Table 1** Detailed performance functions

The speed regulating performance and energy consumption performance were improved by both the sequential design and MDO. The micro movement performance was mainly modified through the MDO. Decreasing the dead zone would cause more throttling loss by the sequential design because only the displacement of the main pump was adjusted. In contrast, the MDO took control parameters, multiway valve's throttling characteristics and spool geometries as design variables at the same time. The micro movement performance could be optimized with the coordination of different subsystems, with little influence on the speed regulating or energy consumption performance.

### **5 Conclusions**

Main conclusions are as follows.


**Acknowledgements** This work was supported by the Key Technology Research and Development Program of Shandong (Grant No. 2020CXGC011005).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Development of CDC Shock Absorber Design Software**

**Zhitao Liang, Zhiwei Zhao, Luyou Yue, Xinlong Yang, Xinxing Zhang, and Bing Zhang** 

**Abstract** In response to the low design efficiency caused by the numerous design parameters of existing shock absorbers, this research presents the design of an efficient software tool for obtaining the external characteristics of Continuous Damping Control (CDC) shock absorbers. Firstly, a mathematical model for the damping force of the shock absorber was established based on principles from fluid dynamics, elasticity mechanics, and throttle orifice models in the rebound and compression valve systems. Secondly, a Graphical User Interface (GUI) was developed to visualize the external characteristics and conduct simulation studies of the CDC shock absorber. Finally, experiments were performed on a shock absorber test bench to validate the external characteristics of the CDC shock absorber using initial structural parameters. The GUI software interface enables direct adjustment of the shock absorber's structural parameters and signal excitation, thereby enhancing the practical efficiency of the shock absorber. A comparison between simulation and experimental results reveals that the relative error rate is generally high when the velocity amplitude is 0.052m/s, with a maximum relative error rate of 24.44%. For other excitation velocity amplitudes, the relative error rate remains within 10%. This demonstrates the high accuracy and reliability of the established mathematical model for the CDC shock absorber, providing a solid theoretical foundation for studying the shock absorber's external characteristics.

**Keywords** CDC shock absorber · External characteristics · Graphical user interface · Damping force · Simulation model

Z. Liang

© The Author(s) 2024

Suzhou City University, Suzhou 215104, Jiangsu, China

Z. Zhao · L. Yue · X. Yang · X. Zhang · B. Zhang (B) Jiangsu University, Zhenjiang 212013, Jiangsu, China e-mail: zbnicai@126.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_5

### **1 Introduction**

In recent years, with the progress of science and technology and the improvement in living standards, the demand for vehicles has gradually improved, and the automotive industry has also witnessed rapid development. Due to these factors, the smoothness of driving as an important investigation index for automotive performance has received extensive attention from people [1]. Research results show that strong vibration will not only lead to fatigue damage of important parts of the vehicle, reduce the service life, reliability, and driving safety of the vehicle, but also seriously damage human organs [2–4]. Therefore, vibration shock absorbers are widely used in the design of various types of vehicles to provide good smoothness in various complex dynamic environments. Passive shock absorbers are still widely used because of their low cost and simple structure [5]. However, this type of shock absorber has the obvious disadvantage that its external characteristics (damping characteristics) are fixed and cannot be adapted to different road conditions [6]. Active shock absorbers need to be equipped with high-precision servo devices, sensors, complex control systems, and additional power sources, which are costly and consume a lot of energy, therefore reducing the performance of the whole vehicle [7]. The viscosity of the "current fluid" and "magnetorheological fluid" of the electrodynamic and magnetorheological shock absorbers, which are adjustable, can be changed according to the strength of the physical field, and therefore the damping force can be continuously adjusted [8–10]. Continuous Damping Control (CDC), as a kind of semi-active shock absorber, can theoretically realize any damping force within the working interval formed by the maximum and minimum damping force and can achieve continuous adjustment, which is a more advanced kind of shock absorber in semi-active shock absorber. In recent years, some domestic enterprises and scientific research institutes have conducted research on CDC shock absorber, but due to the late start and weak technology, many of them are still in the initial stage of research, and the research literature is relatively small. A lot of key technologies have not yet been solved, nor have reliable products been launched for the market [11]. The external characteristic curve, also known as the damping characteristics curve, is the most important indicator of the shock absorber performance, which can judge the shock absorber performance comprehensively, and is also an important part of the characteristic parameters of the vehicle suspension system [12]. Therefore, in this paper, a graphical user interface (GUI) with MATLAB/Simulink simulation tool and visualization function is used to visualize the external characteristics of the CDC shock absorber, which can help shock absorber developers to design and analyze the CDC shock absorber. It is of great significance to improve the design efficiency of the shock absorber and to reduce the research and development time and cost.

# **2 Establishment of Mathematical Model of CDC Shock Absorber Damping Force**

### *2.1 CDC Shock Absorber*

The CDC shock absorber studied in this paper is shown in Fig. 1. The normal working process of the shock absorber is to switch back and forth between the recovery stroke and the compression stroke, while the oil is circulated inside the shock absorber. During the circulating work, the fluid is throttled by the valve holes to form the damping force of the shock absorber, which cuts down the excitation from the road surface to the frame or body, and improves the smoothness, stability, and comfort of the vehicle, and then through the control valve to change the throttling effect of the valve holes according to the size of the different control currents to control the shock absorber's work performance to achieve the desired effect.

1-plastic cover; 2-guide oil seal assembly; 3-inner cylinder barrel normally open throttle hole; 4-intermediate chamber; 5-reservoir chamber; 6-outer cylinder barrel; 7-Limit ring; 8-Piston; 9-Extension valve; 10-CDC control valve assembly; 11- Bottom valve assembly; 12-Piston rod; 13-Working cylinder rod chamber; 14- Intermediate cylinder; 15-Inner cylinder (working cylinder); 16-Flow valve support; 17-Flow valve plate and flow valve adjusting spring; 18-Extension valve adjusting spring; 19-Extension valve adjusting nut; 20-Rodless chamber of working cylinder; 21-Fixed lugs.

1-plastic cover; 2-guide oil seal assembly; 3-inner cylinder barrel normally open throttle hole; 4-intermediate chamber; 5-reservoir chamber; 6-outer cylinder barrel; 7-Limit ring; 8-Piston; 9-Extension valve; 10-CDC control valve assembly; 11-Bottom valve assembly; 12-Piston rod; 13-Working cylinder rod chamber; 14-Intermediate cylinder; 15-Inner cylinder (working cylinder); 16-Flow valve support; 17-Flow valve plate and flow valve adjusting spring; 18-Extension valve adjusting spring; 19-Extension valve adjusting nut; 20-Rodless chamber of working cylinder; 21-Fixed lugs

**Fig. 1** CDC shock absorber structure diagram

### *2.2 CDC Shock Absorber Working Stroke Oil Flow Modelli*

The CDC shock absorber in the recovery stroke of the working process of the oil flow through the piston assembly, the bottom valve assembly and the CDC control valve assembly and other structures as a series–parallel connection of different valves and throttle orifices, to facilitate the analysis and understanding of the problem, and then build a mathematical model of the CDC shock absorber damping force, such as Fig. 2 for the recovery stroke of the oil circulation diagram.

S1-working cylinder rodless chamber; S2-working cylinder rodded chamber; S3-intermediate cylinder inner chamber; S4-oil reservoir chamber; a-compensation valve; b-extension valve; c-Rod chamber of the working cylinder; d-Stacked valve disc; d1-Stacked valve disc constant throttling orifice; d2-Stacked valve disc variable flow throttling orifice; e-Valve disc constant throttling orifice (through-hole I); f-Valve spool adjustable flow orifice; g-pilot valve constant flow throttling orifice (through hole II); h-equivalent relief valve; k-CDC shock absorber control valve; y-equivalent adjustable relief valve.

Similarly, through the compression stroke working process of the oil through the piston assembly, bottom valve assembly and CDC control valve assembly, and other structures such as different valves and throttle orifice series–parallel connection, you can get the compression stroke oil circulation path.

S1-working cylinder rodless chamber; S2-working cylinder rodded chamber; S3-intermediate cylinder inner chamber; S4-oil reservoir chamber; a-compensation valve; b-extension valve; c-Rod chamber of the working cylinder; d-Stacked valve disc; d1-Stacked valve disc constant throttling orifice; d2-Stacked valve disc variable flow throttling orifice; e-Valve disc constant throttling orifice (through-hole Ⅰ); f-Valve spool adjustable flow orifice; g-pilot valve constant flow throttling orifice (through hole Ⅱ); h-equivalent relief valve; k-CDC shock absorber control valve; y-equivalent adjustable relief valve

**Fig. 2** CDC shock absorber recovery stroke oil circulation path diagram

### *2.3 CDC Shock Absorber Damping Force Modelling*

Since the recovery stroke of the CDC shock absorber is modeled along roughly the same lines as the compression stroke, only the recovery stroke will be analyzed later.

Since the shock absorber will have multiple valves and throttle ports during a single operating cycle under normal operating conditions, forming a series–parallel oil circuit, assuming that the total pressure loss in series and parallel is ∆*pc*,*<sup>z</sup>*, ∆*pb*,*<sup>z</sup>*, and the total flow rate is *Qc*,*z*, *Qb*,*z*, respectively, and if there are n series-connected throttle ports, with the same flow rate through each of them, then the total pressure loss is the sum of the pressure loss of each throttle port, i.e.

$$\begin{cases} \mathcal{Q}\_{c,\boldsymbol{z}} = \mathcal{Q}\_{c,1} = \mathcal{Q}\_{c,2} = \dots = \mathcal{Q}\_{c,n} \\ \Delta p\_{c,\boldsymbol{z}} = \Delta p\_{c,1} + \Delta p\_{c,2} + \dots + \Delta p\_{c,n} \end{cases} \tag{1}$$

With n throttle ports in parallel, the pressure difference between the two sides of each throttle is equal and the total flow rate is equal to the sum of the flow rates of the individual throttles, i.e.:

$$\begin{cases} \mathcal{Q}\_{b,z} = \mathcal{Q}\_{b,1} + \mathcal{Q}\_{b,2} + \dots + \mathcal{Q}\_{b,n} \\ \Delta p\_{b,z} = \Delta p\_{b,1} = \Delta p\_{b,2} = \dots = \Delta p\_{b,n} \end{cases} \tag{2}$$

The modeling of the throttle orifices in each valve system for the shock absorber recovery and compression stroke is too lengthy and can be accomplished by combining fluid and elastic mechanics, so it will not be repeated here.

When the piston rod is given a speed of movement, it will drive the piston and the internal oil flow of the shock absorber, the oil through the various valve groups and holes for throttling the formation of damping force, according to the law of conservation of flow, the total amount of oil inflow and outflow from the various chambers remains unchanged, which is the basic idea for the establishment of the mathematical model of damping force of CDC shock absorber.

# 1. When ∆*ph* ≤ ∆*ph* max in the equivalent relief valve

At this time, the equivalent relief valve is not open, assuming that the damping fluid is incompressible, the fluid density is uniformly distributed, ignoring the effect of temperature on the viscosity of the fluid, etc., so the shock absorber in the work of the tandem oil circuit in the oil through each throttle hole, in turn, is equal to the flow rate, and the total amount of fluid in the shock absorber is unchanged, and the flow rate of *Qc* is due to the piston and rod to do the restoration of the movement when squeezing the rod cavity fluid due to the result of the so that it can be obtained:

$$\mathcal{Q}\_c = (A\_{piston} - A\_{rod})v\_f = \frac{\pi (d\_{piston}^2 - d\_{rod}^2)}{4} v\_f \tag{3}$$

where: *Qc*- the flow rate through the normally open throttle orifice of the rod chamber of the working cylinder; *Apiton*, *Arod* -Area of piston and piston rod; v *<sup>f</sup>* -Velocity of the piston when it is in recovery motion; *dpitondrod* -Diameter of piston and piston rod.

Since ∆*ph* ≤ ∆*ph* max when the flow *Qh* = 0, at this time in the shock absorber internal oil circulation path in the overall general direction can be regarded as a series oil circuit.

so *Qc* <sup>=</sup> *Qd* <sup>=</sup> *Qd*1 <sup>+</sup> *Qd*2 <sup>=</sup> *Qy* <sup>=</sup> *Qe* <sup>=</sup> *Q f* <sup>=</sup> *Qg* <sup>=</sup> <sup>π</sup>(*d*<sup>2</sup> *piston*−*d*<sup>2</sup> *rod* ) <sup>4</sup> v *<sup>f</sup>*, the following relationship can be obtained by associating ∆*pc*, ∆*pd* , ∆*pe*, ∆*p f* , ∆*pg*, and stacking the differential pressure ∆*pc*, ∆*pd* , ∆*pe*, ∆*p f* , ∆*pg* to get the differential pressure between the rod chamber and the oil storage chamber ∆*p*24. In the recovery stroke of the shock absorber, due to the existence of the rod, the oil in the rodless chamber increases more than the oil in the rod chamber decreases more than π*d*<sup>2</sup> *rod*  <sup>4</sup>*x f* , while the total oil inside the shock absorber remains unchanged, to prevent the occurrence of air travel distortion phenomenon and excessive differential pressure during the working process, the internal part of the shock absorber has to be filled with a certain amount of nitrogen, and the volume of nitrogen increases. Therefore, the volume of nitrogen increases during the recovery stroke <sup>π</sup>*d*<sup>2</sup> *rod*  <sup>4</sup>*x f* , which can be obtained from the following equation:

$$P\_{N\_2}V\_{N\_2}^n = P\_4 \left(V\_{N\_2} + \frac{\pi d\_{rod}^2}{4} x\_f\right)^n \tag{4}$$

Due to the smaller throttling effect of the compensating valve, it is known that the rodless cavity pressure *P*1 = *P*4, viz.

$$P\_1 = \left(\frac{V\_{N\_2}}{V\_{N\_2} + \frac{\pi d\_{vol}^2}{4} x\_f}\right)^n P\_{N\_2} \tag{5}$$

where: *P*1-Pressure in rodless cavity; *VN*<sup>2</sup> -Initial volume of nitrogen inside the shock absorber; *PN*<sup>2</sup> -Initial pressure of nitrogen inside the shock absorber; *n*-Multivariate index; *x f* -Displacement of the piston rod movement during the recovery stroke.

In summary, an expression for the damping force of the shock absorber during the recovery stroke can be established as:

$$\begin{split} F\_{fuyuan} &= \Delta p\_{21} A\_{piston} - P\_2 A\_{rod} \\ &= \Delta p\_{21} A\_{piston} - (P\_2 - P\_1 + P\_1) A\_{rod} \\ &= \Delta p\_{21} A\_{piston} - (\Delta p\_{21} + P\_1) A\_{rod} \end{split} \tag{6}$$

Since it is known that ∆*p*24, and ∆*p*24 ≈ ∆*p*21. .(2.58) Substituting (2.59) the mathematical model of the damping force at the recovery stroke when ∆*ph* ≤ ∆*ph* max in the equivalent relief valve can be obtained.

2. When ∆*ph*>∆*ph* max in the equivalent relief valve.

When the pressure reaches the equivalent relief valve opening threshold ∆*ph* max, the valve disc opens a certain amount of *fh*, and the oil begins to overflow.

so *Qc* <sup>=</sup> *Qd* <sup>=</sup> *Qd*1 <sup>+</sup> *Qd*2 <sup>=</sup> *Qy* <sup>=</sup> *Qe* <sup>+</sup> *Qh* <sup>=</sup> *Q f* <sup>+</sup> *Qh* <sup>=</sup> *Qg* <sup>+</sup> *Qh* <sup>=</sup> <sup>π</sup>(*d*<sup>2</sup> *piston*−*d*<sup>2</sup> *rod* ) <sup>4</sup> v *<sup>f</sup>*, solving the joint equation for ∆*pc*, ∆*pd* , ∆*pe*, ∆*p f* , ∆*pg*, and the same will be the pressure difference ∆*pc*, ∆*pd* , ∆*pe*, ∆*p f* , ∆*pg* superposition of the differential pressure between the rod chamber and the oil storage chamber can be obtained ∆*p*24, and then use the same method of calculating the recovery stroke damping force model in the equivalent relief valve at ∆*ph* ≤ ∆*ph* max, to derive a mathematical model of the recovery stroke damping force when the equivalent relief valve at ∆*ph*>∆*ph* max.

# **3 Visualisation of the External Characteristics of the Shock Absorber**

# *3.1 Implementation of Simulation Flow and Operation Interface*

Based on the mathematical model of shock absorber recovery and compression stroke established above, its damping force simulation model can be constructed using MATLAB/Simulink software.

As shown in Fig. 3, the flow chart of the simulation of the external characteristics of the CDC shock absorber is shown. Firstly, the geometrical (structural) parameters of the shock absorber structure and the fluid parameters are given and the excitation motion parameters are also given, the speed of the piston rod (v) is used to judge whether the shock absorber is in the recovery stroke or the compression stroke; secondly, according to the magnitude of the relationship between ∆*ph* and ∆*ph* max(whether the equivalent relief valve is open or not), the damping force solution is carried out for the four cases, and the data are obtained to draw the external characteristic diagrams.

Due to the complex structure of the CDC shock absorber model and a large number of design parameters, the actual tuning process is cumbersome and prone to errors, and the simulation output results are not intuitive, so it is necessary to process and plot the obtained simulation data to obtain the external characteristic curve diagram. For this reason, the next section will deal with this problem.

As shown in Fig. 4, the flowchart of the visualisation of the external characteristics is implemented, and the design of the manipulation interface and the writing of the callback functions are carried out with the Simulink simulation model of the shock absorber built out.

As shown in Fig. 5, a layout editor view of the visual control interface for the external characteristics of the shock absorber was created using GUIDE. The following five types of controls are used in the design of the control interface:

**Fig. 3** Simulation flow chart of external characteristics of CDC shock absorber

Static text controls (Text): such as "Continuous Adjustment of Current", "Piston Diameter", "Piston Rod Diameter", etc. in the interface; Edit box controls (Edit): here, the callback function is used to write the initial geometric structure parameters, the excitation signal and the oil parameters of such shock absorber as the default values. The Edit Box control (Edit): In the Edit Box, the callback function is used to write the initial geometrical structure of the shock absorber, the excitation signal under normal conditions, and the oil parameter values as the default values, so that when running the program, all the editable textboxes will have the values, which is convenient for adjusting the parameters and prompting for the initial structural parameters of the shock absorber; Push Button Controls (Push Button): For example, the Push Button Controls (Push Button) can be used to adjust the parameters of the shock absorber. Push Button: Push buttons such as "Start Simulation", "Save Diagram", "Exit", etc.; Panel: A panel such as "Menu Bar", which is used to illustrate that different functions in the menu can be selected for the application. Axes: The Axes control is used to visualise the external characteristics of the vibration shock absorber by plotting the power diagrams and the velocity characteristic curves respectively.

Finally, after the control interface is designed, it needs to be supported by corresponding callback functions, i.e., a callback function is associated with a process in advance, so that when the process is running, it can be called on the preset callback function and used to activate other processes, and the callback functions are different for different controls.

**Fig. 4** Flow chart for visualisation of external characteristics


**Fig. 5** GUI control interface design

### *3.2 Visual Simulation of External Characteristics*

Click Run in the designed GUI interface to enter the execution page of the program, at this time, the blank editable text box is automatically embedded in the initial default value of the callback function compilation, and then click the "Choosefile" button to select the Simulink simulation model that needs to be simulated, and then enter the size of the signal value that needs to be simulated, and then click the "Start Simulation" to simulate. After that, click the "Choosefile" button to select the Simulink simulation model to be simulated, and then input the size of the signal value to be simulated as needed, and then click the "Start Simulation" button to start the simulation. The external characteristic curve will be drawn automatically and displayed in the graphic plotting area, thus realising the visualisation of the external characteristics of the CDC shock absorber. If the oscilloscope corresponding to the damping force of the Simulink shock absorber is opened while the programme is running, the effect of some parameter adjustments on the damping force can be observed in real time. As shown in Fig. 6, the maximum speed is 0.524m/s, the working stroke is −0.05*m* ∼ +0.05m, and the input current values are 0A, 0.3A, 0.6A, 0.9A, 1.2A, and 1.5A, respectively, for the visualisation of the external characteristics of the simulation.

From the velocity characteristic curve in Fig. 6, it can be seen that the recovery and compression stroke is mainly divided into two stages, the first stage is: when ∆*ph* ≤ ∆*ph* max, that is, the equivalent relief valve did not carry out the overflow effect, the oil pressure rises more steeply; when ∆*ph*>∆*ph* max, the equivalent relief valve opens, and begins to overflow part of the oil resulting in the slowing down of the rate of rise of the damping force, this is the second stage.

**Fig. 6** GUI simulation diagram for visualisation of external characteristics

# **4 CDC Shock Absorber External Characteristics Experiment**

In this paper, the experimental rig for the external characteristics of the CDC shock absorber mainly includes mechanical structure, hydraulic system and electrical control system. The mechanical structure is mainly used to fix the shock absorber, actuator and sensors; the hydraulic system is mainly used to make the actuator produce excitation with different frequencies and amplitudes; and the electrical control system is mainly responsible for the processing of the signals.

According to the cartridge shock absorber bench test requirements, before the test, the CDC shock absorber to be tested needs to stand for at least 6 h at a temperature of 20 ± 3 °C, and five exhaust processes are completed under the test conditions of a stroke of ± 50 mm and an excitation frequency of 1.67 Hz (0.524 m/s). Afterwards, in the recovery of the maximum stroke of 0.05 m, i.e., the excitation amplitude is limited to ± 50 mm, the frequency is 0.166 Hz, 0.417 Hz, 0.815 Hz, 1.67 Hz, respectively, and the corresponding excitation of the speed amplitude of 0.052 m/ s, 0.131 m/s, 0.256 m/s, 0.524 m/s respectively under the conditions of the control valve, the measured drive current of each drive current ( 0A, 0.3A–1.2A, 1.5A) under the corresponding damping force and piston rod speed experimental data, as shown in Fig. 7.

From the figure can be obtained, at the same speed, the damping force with the increase in the value of the current and reduce;damping force and piston rod speed of the relationship have a common feature, that is, the control valve under the drive current of the damping force and the corresponding piston rod speed have a clear turning point, turning point after the damping force increase rate decreases, here the turning point for the equivalent relief valve relief pressure due to unloading. However, in general, the damping force increases with the increase of piston rod speed for the same drive current. Based on the experimental data measured in Fig. 7, the relative error rates are calculated as shown in Tables 1, 2, 3, 4, compared with the simulation data of the relationship between piston rod speed and damping force under different control current conditions in Fig. 6.

From the above table, it can be seen that the relative error rates between the experimental data and the simulation data are generally large when the excitation velocity amplitude is 0.052 m/s, and the largest relative error rate reaches 24.44%. At other excitation speed amplitudes, the relative error rate is below 10%, and the relative error rates of the simulation and experimental results are within the acceptable range, so it can be shown that the mathematical model of the vibration shock absorber has a high accuracy and reliability. From the above table, it can be seen that, due to the neglect of some influencing factors during the mathematical modeling of the damper damping force, the peak values of the recovery and compression damping forces obtained from the simulation are smaller than those obtained from the experiments, e.g., the friction force between the piston and the working cylinder inside the damper. In addition, the gap between the piston rod guiding fixture and the piston rod on the experimental bench is small, which is easy to cause the eccentricity of the piston rod

**Fig. 7** The relationship between current-piston rod speed-damping force


**Table 1** Comparison of experimental and simulated damping forces when the velocity amplitude is 0.052 m/s

when the actuator vibrates up and down, resulting in the friction force, especially when the velocity amplitude is 0.052 m/s, the denominator is very small, the numerator is unstable, and the friction force is stronger, which is very easy to cause a large relative error, and the friction force increases as the velocity increases and the control current decreases, the friction force increases, and the friction force increases as the


**Table 2** Comparison of experimental and simulated damping forces when the velocity amplitude is 0.131 m/s

**Table 3** Comparison of experimental and simulated damping forces when the velocity amplitude is 0.256 m/s


**Table 4** Comparison of experimental and simulated damping forces when the velocity amplitude is 0.524 m/s


control current decreases. As the speed increases and the control current decreases, the damping force increases and the relative role of friction is weakened, so most of the relative error rate values in the table tend to decrease as the speed increases and the current decreases.

### **5 Summary**


### **References**

1. Ma R (2021) Analysis and optimal control of vehicle Semi-Active air suspension system based on damping Multi-Mode Switching. Jiangsu Univ, Zhenjiang


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Dynamic Simulation of Crane Movable Pulley System with Defects**

**Shuo Li, Hongsheng Zhang, and Xiangxiang Wang** 

**Abstract** The movable pulley block failure can lead to catastrophic crane accidents, so the dynamic performance of a defective movable pulley block system is essential. Based on ANSYS contact analysis through APDL command, a pulley block simulation platform is developed, which can be used for dynamic analysis of defective pulley systems. Firstly, a parameterized 2D contact model is established and validated through static analysis. Then, dynamic simulation analysis for the intact pulley block under two working conditions is performed: lifting from the ground and sudden unloading. Finally, COMBIN 37 is used as a defective element, and dynamic simulations are analyzed for the faulty pulley system with hook or wire rope rupture. The analysis results show that the longer the unloading time and the shorter the wire rope length will lead to a minor impact under hook rupture and sudden unloading conditions. Meanwhile, if the unloading time is consistent, the dynamic analysis of the intact pulley block under sudden unloading and defective pulley block with hook rupture are equivalent. It is worth noting that when the steel wire ropes on both sides of the moving pulley break simultaneously in a defective system, it will cause a more significant horizontal impact force. This study demonstrates that ANSYS contact analysis with COMBIN 37 as the defective element can accurately and efficiently apply the dynamic simulation of a crane movable pulley system with defects.

**Keywords** Pulley block · Dynamic simulation · Defective system · Contact analysis · Crane

S. Li · H. Zhang (B) · X. Wang School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China e-mail: zhanghs@hit.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_6

### **1 Introduction**

The movable pulley block is a vital component of the crane lifting mechanism. The study of a defective movable pulley block system's dynamic performance has significant theoretical and practical importance. Many researchers have conducted dynamic analysis on different types of cranes, among which Adamiec-Wójcik [1] has conducted dynamic theoretical derivation and simulation research on offshore multiboom cranes by establishing static and dynamic models. Through the use of ADAMS, Yin Xiao-lei [2] developed a rigid-flexible coupling simulation model for cranes and optimized the vibration of the crane. Cibicik and Egeland [3] took the steering arm of a floating crane as the research object, extended the Kane motion equation to modeling, and studied the pitch motion dynamics of the flexible arm driven by a hydraulic cylinder. With the FZQ40 tower crane, Lu Yan [4] investigated the impact coefficient of a jib tower crane's sudden unloading and concluded that several parameter changes, including amplitude, balance weight, and a ratio of unloading mass to lifting mass, affect the impact coefficient. Based on the stiffness equivalence of truss structures, mass equivalence, and rotational inertia equivalence of mechanism dynamic analysis, Zong Hao [5] constructed an equivalent model for the rotary operation of a 2800 t. m tower crane, then dynamic analysis was then applied to the simplified tower crane structure. Using ADAMS software, Wang Xin et al. [6] dynamically analyzed the sudden unloading of the crane boom, whose research showed that under the same lifting capacity conditions, the larger the elevation angle of the crane's boom, the greater the inclination angle of the boom after sudden unloading, meanwhile, the shorter the unloading time, the greater the caster angle of the jib.

There are also many studies on the working state of steel wire ropes. Li Bo-lun [7] and Jiao Qian-qian [8] odelled and simulated the fracture dynamics of a pulley block system with two wire ropes, respectively. Additionally, when a wire rope suddenly fails, the stress on the remaining wire rope doubles. Qing Huang et al. [9] studied the multi-body dynamics of lifting steel wire ropes. In their respective simulation analyses of the winding and swaying of crane steel wire rope, Yan Jing-Feng [10] and Cao Xu-yang [11] found that the lifting weight has no effect on deviation but that the fixed pulley's position does, with the deviation decreasing as it gets closer to the fixed pulley's distance from the drum. Wang Yong-long [12] built the vertical vibration dynamics model of crane hoisting weight when the hoisting weight is displaced horizontally and the wire rope amplitude is changed through the rope compensation method and proposed a theoretical method to reduce the vibration by the speed regulation of the reel. Although there are many simulations and analyses of crane wire ropes, but dynamic simulation of crane movable pulley systems with defects needs more in-depth and comprehensive research.

This study aims to establish a pulley block simulation platform based on ANSYS contact analysis using the APDL command. Through simulation and comparison, the relationship between the impact of the pulley block system and the input load is obtained, and factors affecting the recoil force will be analyzed, which provides theoretical guidance for optimizing the dynamic performance of the defective movable pulley block system. This paper is organized as follows: Sect. 2 builds and verifies parametric models of the BEAM and PLANE element rope pulley block through static simulation. In Sect. 3, dynamic simulations are performed on an intact pulley block model under two dangerous working conditions: lifting off the ground and sudden unloading. The relationship between the impact force of the pulley block and the input load is investigated, and the factors influencing the impact force are analyzed. Section 4 establishes models of defective wire rope and defective hook, and dynamic analyses are implemented on the hook-defective model for hook rupture and the wire rope-defective model for double-sided wire rope rupture., then the factors influencing the recoil force are analyzed. Finally, the research conclusion is summarized in Sect. 5.

# **2 Establishment and Verification of Parameterized Model for Pulley Block**

### *2.1 Establishment of Parameterized Model for Pulley Block*

To reduce the number of elements, simulation time, and modeling workload, the pulley block model needs to be simplified by building a 2D plane pulley block model. The pulley is simulated using the plane element PLANE182, while the wire rope is simulated using both the beam element BEAM3 and the plane element PLANE182. The pulley block model is illustrated in Fig. 1.

The established pulley block model is parameterized using a number of adjustable parameters, including pulley diameter, wire rope diameter, pulley thickness, material elastic modulus, density, and Poisson's ratio. These parameters are defined using the APDL command and are associated with the relevant dimensions as variables. The

(a) using plane elements for ropes (b) using beam elements for ropes

key points in the model are defined by these parameters, allowing for the size of the pulley block to be changed as a whole by altering the values of the parameters. This parameterization enables the model to accommodate different sizes and materials for the pulley block.

### *2.2 Verification of Parameterized Model for Pulley Block*

First, a contact model based on contact elements is established for the pulley block. The CONTA171 element is selected as the contact element, and the TARGE169 element as the target element. Contact pairs and models are formed based on the previously established beam element rope pulley block model. The pulley block's static structure is modeled by constraining all degrees of freedom at the rope ends. Then, a concentrated force of 10000N is applied downward at the center of the movable pulley, and static simulation is performed. As illustrated in Fig. 2, the contour displacement plot in the X and Y directions, the equivalent stress diagram of the wire rope, and the support reaction force of the pulley block are all recorded. The theoretical reaction force is 5000N, and the combined force with the maximum deviation from the theoretical value is 4960.5N. The maximum relative error of the beam element rope model based on contact elements is 0.79%.

### **3 Dynamic Simulation of Intact Pulley Block System**

### *3.1 Dynamic Analysis Under Lifting from Ground Conditions*

A condition with large fluctuations in the lifting load is selected from the observed lifting load data because it has a more noticeable effect on the crane and its operation. The selected relationship curve between lifting load and time is illustrated in Fig. 3.

After applying the lifting load, a transient dynamic analysis is carried out, and the displacement–time curve of the center of the movable pulley, the stress-time curve of the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 4.

Data analysis shows that the maximum load applied in lifting off the ground condition is 27.77 KN, and the maximum vertical recoil force is 27.998 KN, with an impact amplification factor of 1.0082. The force of the horizontal impact is insignificant and can be disregarded. The maximum load during lifting off the ground occurs at 51.498 s, and the maximum vertical recoil force occurs at 51.560 s. Due to the damping in the pulley block system, the response of the pulley block lags behind the lifting load.

(a) Figure of pulley block support reaction force

(b) Figure of equivalent stress of steel wire rope

(d) Contour displacement plot in the Y direction

direction

**Fig. 2** Static simulation result

**Fig. 3** The relationship curve between lifting load and time

**Fig. 4** Dynamic simulation results under lifting load

### *3.2 Dynamic Analysis Under Sudden Unloading Conditions*

The sudden unloading condition is simulated by setting multiple load steps. The initial load P0 is set to 10000N and at 0.1s the initial load was unloaded within 1e-6s. After setting the load steps, transient dynamic analysis is implemented, and the displacement–time curve of the center of the movable pulley, the stress-time curve of the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 5.

It is clear from examining the force–time curve of the support reaction at the fixed pulley's center that abruptly unloading the pulley block causes a quick drop in the force of the vertical reaction, which ultimately causes unfavorable vibrations that have a significant negative impact on the crane's overall functionality. The graph shows a maximum vertical recoil force of 7438.55N and an impact coefficient of 0.7438. The horizontal recoil force is negligible and can be ignored.

(b) Stress curve of the steel wire rope

direction of the fixed pulley

direction of the fixed pulley

**Fig. 5** Dynamic simulation results under sudden unloading conditions

### *3.3 Analysis of Influencing Factors*


**Fig. 6** The relationship curve between the maximum recoil force and the unloading time

that there is generally a positive correlation between rope length and maximum recoil force. The longer the rope length, the greater its maximum recoil force generated in the pulley block.

**Fig. 7** The relationship curve between the maximum recoil force and the pulley distance (wire rope length)

### **4 Dynamic Simulation of Defective Pulley Block System**

### *4.1 Dynamic Simulation of Hook System with Defects*

In order to simulate the defect, the COMBIN37 element's switch function is employed to turn off the spring element at a predetermined time point, simulating the rupture or disengagement of the crane hook. The hook rupture or disengagement is achieved by setting multiple load steps, with different steps such as applying the initial load P0 = 10000N, breaking the hook at 0.1 s, and applying subsequent analysis in different load steps. A transient dynamic analysis is carried out, and the displacement–time curve of the center of the movable pulley, the stress-time curve of the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 8.

(a) Displacement curve of the center of the moving pulley

(c) The support reaction curve in the Y direction of the fixed pulley

(b) Stress curve of the steel wire rope

(d) Support reaction curve in the X direction of the fixed pulley

**Fig. 8** Dynamic simulation results of hook fracture

**Fig. 9** Comparison diagram of vertical support reaction force between hook fracture and sudden unloading of the fixed pulley

By comparing the dynamic simulation results of the pulley block hook rupture impact and the sudden unloading condition, it is found that the pulley block responses are consistent as long as the hook rupture time and sudden unloading time are the same. Figure 9 shows the comparison diagrams of the displacement–time curve of the center of the movable pulley and the vertical support reaction force of the fixed pulley under the two conditions.

# *4.2 Dynamic Simulation of Steel Wire Rope System with Defects*

The spring element COMBIN37 in ANSYS simulates the defect and establishes a model with breakpoints on the wire ropes on both sides of the moving pulley. The model contains two COMBIN37 elements, the first COMBIN37 element has one node connected to the right end of the rope, and the other node uses a fixed constraint as support, while the other COMBIN37 unit is connected to the left wire rope and is symmetrical with the right one. The defective model of the wire rope is shown in Fig. 10.

Considering self-weight, an external load of P0 = 10000N is applied as a concentrated force at the node below the movable pulley of the COMBIN37 element and makes the wire rope break at 0.1s. A transient dynamic analysis is conducted under the condition of wire rope rupture unloading, and the displacement–time curve of the center of the movable pulley, the stress-time curve on the wire rope, and the support reaction force–time curve of the center of the fixed pulley are recorded, as shown in Fig. 11.

The vertical support reaction force rapidly decreases and produces vibrations based on the pulley's self-weight when the wire rope breaks at 0.1 s, which can

**Fig. 11** Simulation analysis curve of broken rope in systems with defects

have a significant impact on the crane behavior, according to the force–time curve of the vertical support reaction at the fixed pulley's center. The maximum impact force recorded is 7421.3N, with an impact coefficient of 0.74213. In contrast to sudden unloading and hook rupture conditions, where the horizontal impact force is negligible and can be ignored, when the steel wire ropes on both sides of the moving pulley break simultaneously in a defective system will cause a more significant horizontal impact force, resulting in vibrations and oscillations in the horizontal support reaction force.

### **5 Conclusion**

This paper establishes a parametric model of the crane pulley block using APDL commands of ANSYS. Based on contact analysis, static simulation verification is carried out on the contact model of the beam element rope pulley block. Dynamic simulations are conducted on the intact pulley block model under two dangerous working conditions: lifting off the ground and sudden unloading. Dynamic simulations are also performed on the hook-defective and the wire rope-defective pulley block models for hook and double-sided wire rope rupture, respectively. Through simulation comparison, it is concluded that the longer the unloading time, the smaller the impact generated in the pulley block under sudden unloading conditions. The longer the wire rope length, the greater the impact force generated in the pulley block under sudden unloading conditions. When the unloading time is consistent, the dynamic analysis of the intact pulley block under sudden unloading and the defective pulley block with hook rupture are equivalent. Unlike the sudden unloading and hook rupture conditions, where the horizontal impact force can be ignored, the steel wire ropes on both sides of the moving pulley break simultaneously in a defective system will cause a more significant horizontal impact force. This study demonstrates that ANSYS contact analysis with COMBIN37 as the defective element can accurately and efficiently apply dynamic simulation to a crane movable pulley system with defects.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design and Dynamic-Static Characteristics Analysis of Electric Fuel Pump Boost Valve**

**Guolei Si, Binjie Li, Mengru Li, Caibo Zhou, Liang Lu, and Zize Zhang** 

**Abstract** In response to the challenges of significant power loss and the difficulty in quickly blocking fuel reverse backflow in one-way valves used in electric fuel systems for aerospace engines, this paper proposes a pump boost valve with a main valve spool designed as a baffle structure and equipped with a backflow damping mechanism. The aim of this boost valve is to achieve flow distribution between the main and auxiliary fuel circuits during engine startup, thereby reducing ineffective power loss. The baffle spool design is implemented to block reverse fuel backflow, while the backflow damping mechanism is incorporated to regulate the dynamic and static characteristics of the boost valve. Based on the proposed design, the working principle is analysed, and a dynamic model is established using AMESim for the modelling and simulation study of the boost valve's dynamic and static characteristics. Comparative experiments are conducted between the newly designed boost valve and traditional boost valves, and the accuracy of the simulation model is validated through experimental verification. The experimental results confirm that the newly designed boost valve outperforms traditional valves in terms of reducing power loss and effectively blocking fuel reverse backflow. Additionally, through the analysis of parameters such as spring stiffness, spool diameter, and backflow damping hole, this study optimises the static and dynamic characteristics of the electric fuel pump. By optimizing these structural parameters, the performance of the electric fuel pump is improved, leading to enhanced stability and responsiveness.

**Keywords** Optimised design · Dynamic and static characteristics · Boost valve · Electric fuel pump systems

G. Si (B) · B. Li

M. Li · C. Zhou · L. Lu · Z. Zhang School of Mechanical Engineering, Tongji University, Shanghai 200092, China

© The Author(s) 2024

Aerospace Fenghuo Servo Control Technology Corporation, Chengdu 611130, China e-mail: siguolei@foxmail.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_7

### **1 Introduction**

With the continuous advancement of high-power electronic technology and highpower density electromechanical systems, More Electric Engine (MEE) technology has been propelled to the forefront of development [1]. Due to its capability to comprehensively optimize the efficiency and performance of engines, the MEE technology has garnered significant attention from developed countries worldwide [2–4]. The electric fuel pump which plays a crucial role as a component of the More Electric Engine, due to their advantages such as high power density, rapid response, and precise flow rate control over a wide range, have been widely used in the fuel systems of turbofan engines in the field of unmanned aerial vehicles, cruise missiles, and other weapon systems [5]. Therefore, researching low-cost, high-precision, and highly reliable electric fuel pumps is a focal point for achieving domestic production in China's turbofan engine fuel systems. Domestic research on fuel pumps has primarily focused on conventional aviation engines, with limited attention to their application in small-scale turbofan engines.

In the field of aerospace engines, the efficient and reliable operation of electric fuel systems is of paramount importance. These systems are responsible for delivering fuel to meet the demanding requirements of aircraft propulsion. However, traditional one-way valves used in electric fuel systems have been associated with significant power losses and difficulties in quickly blocking fuel reverse backflow, leading to inefficiencies and potential hazards during engine operation.

To address these challenges, this study proposes a novel design of a pump boost valve specifically tailored for electric fuel systems in aerospace engines [6]. The main valve spool is designed as a baffle structure, and a backflow damping mechanism is incorporated to enhance the valve's performance. The primary objective of this design is to achieve efficient flow distribution between the main and auxiliary fuel circuits during the engine startup phase, thereby reducing ineffective power loss and improving overall system efficiency.

The proposed pump boost valve design aims to overcome the limitations of traditional one-way valves by introducing the baffle spool to effectively block reverse fuel backflow and implementing the backflow damping mechanism to regulate the valve's dynamic and static characteristics. Furthermore, the study explores the influence of key parameters such as spring stiffness, spool diameter, and backflow damping hole on the static and dynamic characteristics of the electric fuel pump.

The findings of this research contribute to the advancement of electric fuel systems in aerospace engines by addressing the challenges associated with power losses and fuel reverse backflow. The proposed pump boost valve design offers a promising solution to improve the efficiency and reliability of electric fuel systems. By optimizing the valve's static and dynamic characteristics, this study provides valuable insights for the design and optimization of electric fuel systems in aerospace applications.

### **2 Design Principles of Boost Valve**

### *2.1 Working Principle of Traditional Boost Valve*

As shown in Fig. 1, the electric fuel pump consists of a gear pump, a main fuel pressure valve, an overflow valve, and a motor. The motor is directly connected to the gear pump [7], driving it to rotate and supply fuel to the fuel system. When the motor speed is low, the pressure in front of the pressure valve is insufficient to push open the valve core, and at this time, the auxiliary fuel circuit supplies fuel separately. As the motor speed increases, the pressure in front of the pressure valve exceeds the opening pressure of the valve, allowing both the main fuel circuit and the auxiliary fuel circuit to supply fuel simultaneously. When the speed reaches a certain value, the auxiliary fuel circuit is manually closed, and at this point, the main fuel circuit supplies fuel alone.

The traditional fuel pump employs a one-way valve as the boost valve [8]. In order to enhance the power-to-weight ratio and reliability of aviation engines, modern electric fuel pump systems incorporate two fuel circuits: the auxiliary circuit and the main circuit. The main circuit exhibits relatively lower flow resistance compared to the auxiliary circuit. During the engine startup phase, there exists a certain flow distribution relationship between the main and auxiliary circuits. Typically, a boost valve is installed in the main circuit to satisfy the aforementioned flow distribution relationship, as depicted in Fig. 1. Once the engine startup is complete, the auxiliary circuit is closed, and the engine is supplied with fuel solely through the main circuit.

However, the presence of a high-flow-resistance boost valve in the main circuit results in significant power losses for the electric fuel pump. This not only doubles the required input power for the motor but also severely impacts the endurance of the aircraft. Furthermore, when the motor ceases rotation, there is a risk of reverse backflow of fuel. The conventional one-way valve structure of the boost valve encounters

**Fig. 1** Schematic diagram of a conventional electric fuel pump

difficulties in promptly blocking the fuel flow, thereby affecting the normal operation of the engine fuel system.

# *2.2 New Structure and Working Principle of the Novel Boost Valve*

In order to address the issues associated with conventional solutions, a new boost valve needs to be investigated [9]. This valve core should be capable of maintaining a certain flow distribution relationship between the main fuel circuit and the auxiliary fuel circuit, without causing significant pressure loss when the main fuel circuit supplies a high flow rate. The schematic diagram of the new structure of the boost valve for the electric fuel pump is shown in Fig. 2. This boost valve introduces a completely redesigned valve core.

The boost valve consists of an inlet, an outlet, a return port, a housing, a valve seat, a valve core, and a spring assembly. The valve seat is designed in a nozzle structure, while the valve core adopts a baffle structure. The valve core is installed inside the boost valve, with one end connected to one end of the spring assembly and the other end forming a throttle orifice with the valve seat, which acts as a nozzle baffle. The other end of the spring assembly is connected to the base of the housing.

During the engine starting phase, the throttle port formed by valve seat 2 and valve core 3 remains closed, and the starting fuel circuit is supplied independently as Fig. 3. As the fuel pressure increases, it overcomes the spring force and hydraulic pressure acting on the end face of the boost valve outlet core, causing a certain opening in the throttle port formed by valve seat 2 and valve core 3. The fuel flows through the throttle port and exits through the boost valve outlet 6, achieving flow matching between the starting fuel circuit and the main fuel circuit.

1-motor; 2- gear pump; 3-boost valve; 4- overflow valve

**Fig. 2** Schematic diagram of the electric fuel pump

1-Valve Body 2-Valve Seat 3-Valve Core 4-Return Oil Damping Port 5-Fuel Pump Inlet 6-Return Oil Damping Port 7-Boost Valve Return Port

The fuel medium in the chamber where the boost valve spring assembly 4 is located flows back to the fuel pump inlet 5 through the boost valve return port 7, passing through the return oil damping port 4. The role of the return oil damping port 4 is to provide damping for the movement of the valve core 3, improving the dynamic characteristics and robustness of fuel flow control in the electric fuel pump system.

After the engine has completed the starting phase, the starting fuel circuit is closed, and all fuel from the boost valve outlet 6 must pass through the boost valve inlet 5. At this point, only the main fuel circuit supplies fuel. As the motor speed increases, the area of the throttle port increases, and the damping at the throttle port decreases.

When the throttle port reaches its maximum opening, even if the motor speed continues to increase, the opening no longer changes. At this point, the throttle port is in its lowest resistance state. Compared to traditional boost valves, the pressure in the spring chamber of this boost valve is lower, facilitating the movement of the valve core to its extreme position.

When the electric fuel pump receives a parking command, the throttle port formed by the nozzle plate and valve closes rapidly under the action of the spring force and reverse fuel pressure, achieving a quick shutdown function. After closing, the throttle port maintains a high level of sealing.

### **3 Mathematical Modelling of Boost Valve**

### *3.1 Mathematical Model of Traditional Boost Valve*

Modelling using a cone valve as the spool of a check valve, denoted as valve A, as shown in Fig. 4 [10].

**Fig. 4** Check valve port and pressure on both sides

$$d\_a = d\_s - 2\mathfrak{x}\_v \sin \alpha \cos \alpha \tag{l}$$

$$A\_v = \pi \ge\_v d\_a \text{ sin } \alpha \tag{2}$$

As in Eqs. (1) and (2), *da* is the diameter of the valve core at the flow section, *ds* is the diameter of the upstream channel in the valve port, *x*<sup>v</sup> is the valve port opening, α is the cone angle of the valve core, and *A*<sup>v</sup> is the cross-sectional area of the valve port. Equation (3) represents the flow equation of the valve port.

$$\mathcal{Q} = C\_d A\_v \sqrt{\frac{2(p\_1 - p\_2)}{\rho}} \tag{3}$$

In the Eq. (3), *Cd* is the flow coefficient, ρ is the density of the fuel, *p*1 is the pressure in the upstream chamber of the valve port, and *p*2 is the pressure in the downstream chamber of the valve port.

$$F\_s = 2C\_v C\_d A\_v (p\_1 - p\_2) \cos \theta \tag{4}$$

In the Eq. (4), *Fs* represents the steady-state hydrodynamic force acting on the valve core, *C*<sup>v</sup> is the flow velocity coefficient ranging from 0.98 to 0.99, and θ is the jet direction angle. The ideal jet angle is 69°.

$$F\_1 = \frac{\pi}{4} p\_1 (d\_s - 2x\_v \sin \alpha \cos \alpha)^2 \tag{5}$$

$$F\_2 = \frac{\pi}{4} p\_2 (d\_p^2 - d\_a^2) \tag{6}$$

In the Eqs. (5) and (6), *F*1 represents the pressure exerted on the valve core due to the action of the upstream fluid, while *F*2 represents the pressure exerted on the valve core due to the action of the downstream fluid. The parameter *dp* refers to the diameter of the cylindrical section of the valve core.

The driving force on the valve core is determined by the pressure in the front chamber of the pressure relief valve. The driving force on the valve core will overcome the inertial force, spring force, and steady-state fluid dynamic force acting on the valve core. Neglecting transient fluid dynamic forces, the differential equation that governs the force balance on the valve core during dynamic processes is given by:

$$F\_1 = m\frac{d^2\mathbf{x}}{dt^2} + K\_1(\mathbf{x} + \mathbf{x}\_0) + F\_s + F\_2 \tag{7}$$

In the Eq. (7), *x*0 represents the compression of the spring when the valve opening is zero, and *K*1 is the spring stiffness.

### *3.2 Mathematical Model of the New Boost Valve*

The main difference between this boost valve and the one-way valve is that the valve core is changed from a cone valve to a flat plate structure, and there is damping in the spring chamber to create a certain back pressure. We refer to this valve as Valve B, as shown in Fig. 3.

To analyse the dynamic and static characteristics of the new boost valve, a mathematical model is established based on its structural parameters and operating principles. The mathematical model takes into account the forces acting on the valve spool, including the inertial force, spring force, steady-state fluid dynamic force, and transient fluid dynamic force.

$$\mathcal{Q} = C\_d \pi D \chi\_v \sqrt{\frac{2(p\_1 - p\_2)}{\rho}} \tag{8}$$

In the Eq. (8), *D* represents the diameter of the pressure acting surface in the front chamber of the pressure relief valve. The steady-state hydrodynamic force acting on the valve spool is.

$$F\_{s1} = 2C\_v C\_d A (p\_1 - p\_2) \cos \theta \tag{9}$$

The driving force of the valve spool is determined by the pressure in the front chamber of the boost valve. It overcomes the inertial force, spring force, and steadystate fluid dynamic force acting on the spool. Neglecting the transient fluid dynamic force, the differential equation that describes the force equilibrium of the spool during the dynamic process is given by [11].

$$p\_1 A = m \frac{d^2 \mathbf{x}}{dt^2} + K\_1(\mathbf{x} + \mathbf{x}\_0) + F\_{x1} + p\_2 A\_0 \tag{10}$$

In the Eq. (10), *A*0 represents the effective area of the pressure acting on the spring chamber of the boost valve. *m* denotes the equivalent mass of the moving components of the valve, taking into account the mass of the valve spool and the spring.

The mathematical model provides insights into the dynamic behaviour of the new boost valve, allowing for the analysis of its response characteristics and performance optimisation. By solving the differential equation, the dynamic response of the valve spool can be obtained under different operating conditions and structural parameters, facilitating the design and evaluation of the new boost valve.

Next, this paper will discuss the simulation and analysis of the dynamic and static characteristics of the new boost valve based on the established mathematical model.

# **4 Simulation and Comparative Analysis of Static and Dynamic Characteristics of Traditional and New Boost Valves in AMESim**

In this section, we will build the mathematical models of the traditional boost valve, referred to as Valve A in Fig. 4, and the new boost valve, referred to as Valve B in Fig. 3, in the AMESim simulation environment. By utilizing the simulation results, we will compare the static and dynamic characteristics of these two different boost valves. The key simulation parameters are shown in Table 1.

### *4.1 Comparison of Static Characteristics of Boost Valves*

First, we compared the static characteristics of these two valves. The static characteristics primarily include the flow-pressure curve and the flow-valve position curve. By simulating under different operating conditions and plotting the corresponding curves, we can observe the performance differences between the two valves.

Figure 5 shows the comparison of the pressure- flow curves between Valve A and Valve B. By comparing the curves, we can observe that Valve B has a higher flow rate at the same pressure, indicating that Valve B has better flow control capability. The curve of Valve A is smoother, indicating that it can maintain a more stable flow


output across different pressure ranges. By comparing the two sets of curves, we find that the new type of booster valve has lower pressure losses at the same flow rate. This means that using the new type of booster valve in the fuel system can reduce energy consumption and heat loss in the system.

Figure 6 illustrates the comparison of the flow-valve position curves between Valve A and Valve B. From the graph, it can be observed that at the same valve position, Valve B exhibits a significantly higher flow rate compared to Valve A, indicating that Valve B is capable of providing a greater flow output under the same control signal. Furthermore, the curve of Valve B is steeper, indicating that it is more sensitive to small adjustments in the valve position and possesses higher responsiveness.

**Fig. 5** Pressure-flow comparison chart

**Fig. 6** Flow-valve spool position comparison chart between new type boost valve and traditional boost valve

**Fig. 7** Comparison chart of dynamic characteristics between new type boost valve and traditional boost valve

# *4.2 Comparison of Dynamic Characteristics of the Boost Valve*

By inputting a unit step velocity into the electric fuel pump system, the unit step response of the flow rate through the boost valve is obtained, as shown in Fig. 7. It can be observed that the traditionally designed boost valve, Valve A, exhibits significant overshoot with a steady-state accuracy of 1.3% and a settling time of 41 ms. In contrast, the new-type boost valve, Valve B, has a much smaller overshoot of only 2.7%, a steady-state accuracy of 1.4%, and a settling time of 40 ms. From the comparison, it is evident that Valve A has a significant overshoot, while both valves demonstrate similar steady-state accuracy and settling time. This indicates that the new-type boost valve has better dynamic performance compared to the traditionally designed boost valve, providing a smoother operation during startup.

### *4.3 Analysis of Simulation Results for Valve Closing Speed*

In the simulation experiment, we recorded the variations of the valve position over time for the conventional boost valve and the new boost valve. The closing speed of Valve A was measured to be 2.35 mm/s, while Valve B exhibited a closing speed of 2.55 m/s. It is evident that the new boost valve demonstrates a significant advantage in terms of closing speed. It can close the valve more rapidly, reducing the possibility of backflow and liquid reflux. This contributes to the improved stability and response speed of the fuel supply system. Through quantitative analysis of the simulation data, we calculated the valve closing time for both the conventional boost valve and the new boost valve. The results clearly indicate that the new boost valve exhibits significantly shorter closing time, highlighting its superiority in terms of rapid response and backflow prevention.

Based on the analysis of the simulation results, we can draw the following conclusions: Firstly, the new boost valve demonstrates a clear advantage in terms of pressure loss. Compared to the conventional boost valve, it can minimize the energy loss in the system, thereby improving the efficiency and energy utilization of the fuel system. Secondly, the new boost valve exhibits a remarkable improvement in valve closing speed. It can close the valve more rapidly, effectively preventing backflow and liquid backflow, thus enhancing the stability and response speed of the system. Considering comprehensive analysis, the new boost valve demonstrates excellent performance in terms of pressure loss and valve closing speed. This makes it as an ideal choice for the electric fuel pump system, capable of improving the system's operational efficiency and stability.

However, it is important to note that the simulation experiments were conducted under ideal conditions, and there may be other factors and constraints in practical systems. Therefore, further experimental validation and engineering optimization are required before applying the new boost valve to real-world systems. In conclusion, the simulation results and analysis presented in this chapter validate the improvements achieved by the new boost valve in terms of pressure loss and valve closing speed. This provides strong theoretical and experimental evidence for the application and promotion of the new boost valve.

# *4.4 Comparison of Boost Valve Simulation and Experimental Results*

The new boost valve was applied to an electric fuel pump, and a prototype of the electric fuel pump was constructed for bench testing. The pressure-flow characteristic curve of the electric fuel pump under no-load conditions was measured, and the experimental results were compared with the simulation results, as shown in Fig. 8. The maximum error was 8.6%, and the average error was 1.9% according to the Mean Absolute Error (MAE) as in Eq. (11). Therefore, the experimental results largely validated the correctness of the simulation model.

$$MAE = \frac{1}{m} \sum\_{i=1}^{m} |\mathbf{y}\_i - f(\mathbf{x}\_i)| \tag{11}$$

**Fig. 8** Comparison of experimental and simulation results for boost valve pressure-flow curves

# **5 Analysis and Optimization of Factors Influencing Boost Valve Static and Dynamic Characteristics**

# *5.1 Influence of Spring Stiffness on Boost Valve Static and Dynamic Performance*

Without changing the preloading force of the regulating spring, only the stiffness coefficient is varied to 6N/mm, 7N/mm, and 8N/mm, respectively. From Fig. 9, it can be observed that as the stiffness coefficient increases, the inlet pressure increases at the same flow rate [12].

**Fig. 9** Comparison of pressure-flow curves for boost valves with different spring stiffness

**Fig. 10** Comparison chart of dynamic characteristics of boost valves with different spring stiffness

By applying a step input of full flow rate, the dynamic response curves of the boost valve under three different spring stiffness values are obtained, as shown in Fig. 10. It can be observed that the curves almost overlap, indicating that the influence of spring stiffness on the dynamic performance is minimal, except for a slight variation in the steady-state value of the inlet pressure.

# *5.2 Influence of Main Spool Diameter on Boost Valve Static and Dynamic Performance*

The diameter of the main spool has a significant impact on the static characteristics of the boost valve, as shown in Fig. 11. Three different main spool diameters of 5 mm, 10 mm, and 15 mm are considered. As the main spool diameter increases, the inlet pressure gradually decreases at the same flow rate.

In terms of dynamic characteristics, an increase in the main spool diameter results in a decrease in the output pressure during dynamic response, as shown in Fig. 12. When the main spool diameter is 5 mm, there is a significant overshoot. As the spool diameter increases, the overshoot, settling time, and steady-state accuracy all decrease. Considering the design requirements of the electric fuel pump and the static characteristics, it is important to select an appropriate main spool diameter.

**Fig. 11** Pressure-flow curve at different main valve spool diameters

**Fig. 12** Comparison chart of dynamic characteristics of boost valves with different spool diameter

# *5.3 Influence of Return Damping Orifice Diameter on Static and Dynamic Performance of the Boost Valve*

By varying the diameter of the return damping orifice, it is possible to alter the back pressure in the spring chamber, thereby affecting the static and dynamic performance of the electric fuel pump. As shown in Fig. 13, increasing the diameter of the return damping orifice reduces the back pressure in the spring chamber, resulting in a lower opening pressure of the boost valve. Having a certain back pressure ensures that the valve core closes faster when the electric fuel pump stops working, preventing backflow of the fuel.

**Fig. 13** Comparison of pressure-flow curves of boost valves at different orifice diameters

**Fig. 14** Comparison of flow Step response of boost valves at different orifice diameters

The step response of the electric fuel pump flow under different damping orifice diameters is shown in Fig. 14. It can be observed that changing the diameter of the damping orifice has a minimal impact on the system's dynamic response.

### **6 Conclusion**

In conclusion, through the study and comparison of the static and dynamic characteristics of the traditional and new-type boost valves in the electric fuel pump system, the following conclusions can be drawn. The new-type boost valve exhibits better flow control capability and lower pressure loss. In terms of dynamic performance, the newtype boost valve demonstrates faster response time, smaller overshoot, and higher steady-state accuracy. Factors such as spring stiffness, main valve core diameter, and damping hole diameter have certain influences on the performance of the boost valve. However, the damping hole diameter has minimal effect on the system's dynamic response. Overall, the new-type boost valve shows excellent performance in reducing pressure loss, improving response speed, and enhancing stability. It provides valuable insights for the design and optimization of electric fuel pump systems. Further research is needed to validate and optimise the application of the new-type boost valve in practical systems.

**Acknowledgements** This work was supported by the Key Technology Research Program (Grant No. JSYF-JT-2201) on Low Energy Consumption, High Precision, and Wide Range Flow Control Electric Fuel Pump, the Fifteenth Excellent Experimental Project of Tongji University, the National Natural Science Foundation of China (Grant No. 52075387), the National Key Research and Development Program of China (Grant No. 2019YFB2005102), the Shanghai Natural Science Foundation (Grant No. 22ZR1464400).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Study on the Parameters Matching of Dynamics System Electric Bulldozer**

**Congfeng Tian, Piqiang Tan, Wenpu Wang, Bin Hu, Zhaoliang Wang, Jinbao Song, and Hanxiao Xi** 

**Abstract** In this paper, the parameter matching of dynamics system electric bulldozer was proposed. Firstly, Structure and parameters of the electric bulldozer were introduced. Secondly, the speed and power of the motor were calculated. Finally, power battery parameter was calculated. The results show the peak torque of the drive motor is 800 N·m, the peak power is 100 kW, the maximum rotational speed is 3000 rpm, the rated rotational speed is 1200 rpm and discharge power of the power battery are 235kw.

**Keywords** Bulldozer · Electric · Dynamics system · Parameter matching

# **1 Introduction**

The current energy crisis and increasingly stringent emission regulations have made electric technologies the best way to improve the economy of construction machinery [1–5]. Some of the electric products of construction machinery were developed by domestic and foreign construction machinery groups [6–10], but the research and application in bulldozers are still relatively small. In this paper, the parameter matching of drive motors are carried out, and the effectiveness of parameter matching is especially analyzed.

C. Tian (B) · P. Tan School of Automaotive Studies, Tongji University, Shanghai 201804, China e-mail: tiancf@shantui.com

C. Tian · B. Hu · Z. Wang · J. Song Shantui Construction Machinery Co. Ltd., Jining 272073, China

W. Wang · H. Xi Chang'an University, Xi'an 710064, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_8

105

### **2 Structure and Parameters of Electric Bulldozers**

Figure 1 is a schematic diagram of the structure of an electric bulldozer. The dual motor independent drive bulldozer eliminates the horizontal mechanical connection between the drive wheels on both sides. By controlling the motor drive systems on both sides separately, the battery energy is converted into mechanical energy to drive the track. In addition, a separate motor drives the transfer box to distribute power to various pumps. Tables 1 and 2 show the parameters and of electric bulldozers respectively.

**Fig. 1** Dual motor independent drive bulldozer configuration scheme


Blade capacity 8.8m3

### **3 Motor Matching Calculation**

### *3.1 Motor Speed Matching*

Considering the safe working speed of the drive motor, the maximum speed of the bulldozer drive motor is taken to be 3000 rpm. The side drive can be calculated from Eqs. (1) and (2).

$$n\_{\text{max}} = \frac{u\_{\text{max}}i}{0.377r} \tag{1}$$

$$i = i\_0 \* i\_1 \tag{2}$$

In the above equation, *r* is the radius of the driving wheel 0.4553 m; *i* is the side transmission ratio; μmax is the maximum driving speed of the bulldozer 10km/h; *n*max is the maximum working speed of the motor 3000rpm.This gives a calculated sidetrack ratio of 88. Since the speed of bulldozer in horizontal ground operation is generally 2~6km/h, the average driving speed of 4km/h is selected as the basis for matching the rated speed of the drive motor, and the rated speed of the motor is obtained by the calculation of Eq. (3).

$$m\_e = \frac{\mu\_e i}{0.377r} \tag{3}$$

### *3.2 Matching of Torque*

The calculated working condition of bulldozer operating resistance is the maximum working condition of operating resistance during normal bulldozing, i.e., the bulldozer is traveling at a constant speed on the horizontal road surface, the shovel blade operates at the maximum depth of cutting, the largest pile of soil is formed in front of the shovel blade, and the end of the cutting is about to lift the shovel blade at the instant of the end of the shovel blade. At this time, the unilateral track resistance *R*  consists of two parts: the rolling resistance and the operating resistance. The unilateral track rolling resistance *Ff* is half of the rolling resistance of the whole vehicle, which is

$$F\_f = \frac{fG}{2} \tag{4}$$

where *f* is the rolling resistance coefficient, taken as 0.1, and *G* is the weight. Bulldozer operating resistance *FT* mainly includes: tangential cutting resistance *F*1, push resistance of accumulated soil in front of the shovel blade *F*2, friction resistance between the blade and the soil *F*3 and the horizontal component of the friction resistance of the soil debris as it rises along the shovel blade *F*4. One-side track operating resistance is half that of the whole vehicle.

$$F\_T = \frac{(F\_1 + F\_2 + F\_3 + F\_4)}{2} \tag{5}$$

(1) Tangential cutting resistance *F*<sup>1</sup>

$$F\_1 = 10^6 B\_1 h\_p k\_b \tag{6}$$

where *B*1 is the width of the shovel 3.4 m; *h p* is the depth of cut of the bulldozer shovel 0.2 m; and *kb* is the cutting specific resistance 0.15 MPa.

(2) Pushover resistance of soil accumulated in front of the shovel blade *F*<sup>2</sup>

$$F\_2 = G\_t \mu\_1 \cos \alpha = \frac{V \gamma \mu\_1 \cos \alpha}{k\_s} \tag{7}$$

$$V = \frac{B\_1 (H - h\_p)^2 k\_m}{2 \tan \alpha\_0} \tag{8}$$

where, *Gt* is the gravity of the soil pile in front of the bulldozer plate (N); *V* is the volume of the soil pile in front of the bulldozer plate (m3); *ks* is the loosening coefficient of the soil, which is generally taken as 1.06; *km* is the filling coefficient of the soil, which is generally taken as 1.0; *H* is the height of the shovel blade of 1.2 m; μ1 is the friction coefficient between the soil and the soil, which is taken as 1.0; γ is the gravitational degree of the soil (N/m3), which is taken as 17,700; α is the slope (°), calculated according to 0°; α0 is the natural slope angle of soil (°),which is taken as 28°.

(3) Blade-soil friction resistance *F*<sup>3</sup>

$$F\_3 = 10^3 B\_1 X \mu\_2 k\_y \tag{9}$$

where, *ky* is the specific resistance (MPa) of the cutting edge pressed into the soil after the cutting edge is worn, taken as 0.6; *X* is the grounding length (m) of the cutting edge after the cutting edge is worn, taken as 0.01; μ2 is the coefficient of friction between soil and steel, taken as 0.5.

(4) Horizontal component of the frictional resistance of the soil chip as it rises along the shovel blade *F*<sup>4</sup>

$$F\_4 = G\_t \mu\_2 (\cos \delta)^2 \cos \alpha \tag{10}$$

where, δ is the cutting angle (°) of the push shovel, taken as 25°. Then there is unilateral track resistance.

$$R = F\_f + F\_T \tag{11}$$

Unilateral track torque:

$$T = Rr\tag{12}$$

Unilateral motor torque:

$$T\_{motor} = \frac{Rr}{i\eta} \tag{13}$$

where η is the motor-to-track transmission efficiency. By calculating the corresponding unilateral track resistance under different cutting depths.

Figure 2 is curve of resistance moment of unilateral motor changing with cutting depth. Therefore, the corresponding resistance moment of 800N·m when the cutting depth is 200mm is taken as the basis for matching the torque of the motor.

### **4 Motor Power Matching**

Since torque, speed and power have the following relationship,

$$T = 9549 \frac{p}{n} \tag{14}$$

Where is *n* = 1200*r*/ min,*T* = 800*N* · *m*. The matched motor power is *p* = 100*kW* .

### **5 Power Battery Parameter Matching**

### *5.1 Power Battery Parameter Matching Calculation*

The selection of power battery should ensure the requirements of bulldozer power and its range. This paper will take the grader 10km/h driving range of 80km as the design goal to match the parameters of its power battery. Considering the capacity consumed by the pure electric drive bulldozer operating device, set the drive motor power accounted for 85% of the power of the vehicle, then the discharge power of the power battery is

$$P\_b = \frac{nP\_e}{0.85} \frac{2 \times 100}{0.85} = 235 kW \tag{15}$$

Where *Pb* is the power battery discharge power, kW; n is the number of motors, which is taken as 2; and *Pe* is the individual motor power, kW, which is taken as 100kW.

### **6 Conclusion**

In this paper, based on the working conditions of the electric drive bulldozer, the drive motor and side transmission parameters are matched, the peak torque of the drive motor is 800*N* · *m*, the peak power is 100 kW, the maximum rotational speed is 3000 rpm, the rated rotational speed is 1200 rpm, and the side transmission ratio is 88, discharge power of the power battery is 235kW.

**Acknowledgements** This work was supported by Shandong Province Major Science and Technology Innovation Project (grant No. 2023CXGC010210)

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design and Application of Hydraulic Inverted Pendulum**

**Guyue Ding, Yongming Bian, and Meng Yang** 

**Abstract** This paper briefly describes the designing process of a hydraulic inverted pendulum including hardware and software design. First, the mechanical structure, including the components of the platform will be introduced. Second, the electrical system including controllers for receiving signals from sensors which measure the variables important for controlling inverted pendulum is about to be shown. Afterwards, the paper will present a mathematical model of the whole platform, then shows up an open loop simulation established by AMESim and Simulink in order to analysis its dynamic characteristic. By comparing the simulation result and reality, the rationality of mathematical model is finally verified.

**Keywords** Hydraulic inverted pendulum · System design · Simulink · AMESim

# **1 Introduction**

Closed-circuit hydraulic system has already been implemented not only in application of steering system for wheel loader but also in other construction vehicles [1, 2]. It shows greater efficiency and higher power density compared to opened-loop hydraulic system. In particular, Direct-Drive Volume Control (DDVC) system has gained sufficient attentions in studying its dynamic characteristics and control strategies [3, 4]. Compared with the traditional valve control system, the DDVC system has the technical advantages of a high power–gain ratio, high integration, environmental friendliness and high efficiency and energy saving [5, 6]. Since pump-controlled motor servo system is similar to conventional DDVC system since its actuator is hydraulic motor instead of hydraulic cylinder, its study of dynamic characteristic is

G. Ding · Y. Bian · M. Yang (B)

© The Author(s) 2024

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: yangmeng@tongji.edu.cn

Y. Bian · M. Yang Shanghai Engineering Research Center for Safety Intelligent Control of Building Machinery, Shanghai 200032, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_9

still about to be explored. Meanwhile, the inverted pendulum has become a hot topic as a typical nonlinear and unstable system. It is a simple pendulum whose mass is located in the air. The system presents an unstable equilibrium in a vertical position. This position is maintained by the control of a movable cart [7]. Its control strategy varies from reinforcement learning to fuzzy control [8, 9], which mainly used to test the effectiveness of various control methods, such as their ability to handle nonlinear and unstable issues effectively etc.

In addition, the inverted pendulum has been applied in many different research fields, particularly in robotics. Since the balance control of robots has the similar objective compared to the inverted pendulum, the mathematical model of certain robots can be considered as a variation of inverted pendulum. For instance, the inverted pendulum robot [10] uses wheels in order to maintain the balance. By limiting the robot's posture, the wheeled inverted pendulum can replace the tradition model which requires more complicated control methods to maintain the posture's stability [11, 12]. This mathematical model has been applied not only in electric-driven robot [13] but also in hydraulic powered robot [14].

However, research combining hydraulic system and inverted pendulum is not as extensive as expected. Many researchers focus mainly on either the control strategies for the former one or the improvement of the control method towards the latter one powered by electric motor, even the control strategy for triple inverted pendulum has already used neural networks to control the complete system within 3.0 s [15]. Let alone building the actual platform in order to verify the control method. In this paper, hardware design for hydraulic inverted pendulum, the method of establishing mathematical model of the system will be provided afterwards. Then, the paper presents an open-loop control using co-simulation between AMESim and Simulink in order to study its dynamic characteristic under step disturbance. Finally, by comparing the result between simulation and field test of step response of the system the paper verifies the rationality of mathematical model.

# **2 Hardware Composition of Hydraulic Inverted Pendulum Platform**

# *2.1 Mechanical Structure of Hydraulic Inverted Pendulum Platform*

The mechanical structure of the proposed hydraulic inverted pendulum platform using pump-controlled motor servo system includes two parts: the pump station and the motor-inverted pendulum platform. The 3D models for both of the parts are showed in Figs. 1 and 2. The length of the whole platform is about 1000 mm at maximum, while its full width is about 215 mm and the height is approximately 128 mm.

The pump station includes servo motor, oil tank, and valve blocks. The motorinverted pendulum platform contains not only inverted pendulum itself, but also hydraulic motor and the valve blocks which are used to connect between hydraulic motor. Apart from that, there are connectors which are applied for installing encoders for measuring angle displacement and linear displacement for pendulum rod and slider.

One of the advantages of applying hydraulic closed circuit in driving inverted pendulum is that closed circuit provides higher power and efficiency compared to open hydraulic circuit. Furthermore, since the shaft speed of hydraulic motor is controlled eventually by the servo motor, transitions of the speed and torque produced by hydraulic motor can be accomplished directly through reducer, and pulleys in the guide rail. So, it's obvious that the volume of the whole hydraulic system hardware is much smaller compared to the way of using double piston rod hydraulic cylinder and servo valve.

# *2.2 Hydraulic Circuit of the Hydraulic Inverted Pendulum Platform*

The hydraulic circuit is the most critical part of the hydraulic inverted pendulum platform, which schematic diagram is shown in Fig. 3.

The shaft speed of hydraulic motor is controlled directly by the pump, while the flow rate of the pump is dominated by the speed changing from the servo motor. The check valves are acquired to replenish oil from the oil tank since the pump and motor both have external leakage. The cracking pressure of the check valves is small enough for the hydraulic system in order to accomplish replenishing oil without using accumulator or slippage pump. Finally, the relief valves are applied to control the maximum pressure of the whole system.

# *2.3 Electrical System of Hydraulic Inverted Pendulum Platform*

The electrical system includes sensors, motion controller, servo controller and PC, the components of which are shown in Fig. 4. The limit switches ensure the extreme displacement of the slider, giving protection to the guide rail. The motion controller is the key part of the electrical system, it receives signals not only from the encoders but also from the limit switches. The connection between PC and motion controller is accomplished by Ethernet, while the latter using Ether CAT to establish the communication to the servo controller. Last but not least, the power supply and the control signal towards the servo motor, were all provided by the servo controller.

**Fig. 4** Components of electrical system used in hydraulic inverted pendulum

### **3 System Modelling and Dynamic Characteristics**

### *3.1 Mathematical Model of Hydraulic Inverted Pendulum*

Assuming that both the car and the pendulum are considered as rigid bodies, the conveyor belt is not retractable, and ignoring air resistance and friction resistance moment of driving components, the differential equations of the inverted pendulum are:

$$\begin{cases} \ddot{\boldsymbol{x}} = -\frac{m^2 l^2 g}{(M+m)J + Mml^2} \boldsymbol{\theta} - \frac{(J+ml^2)b}{(M+m)J + Mml^2} \dot{\boldsymbol{x}} + \frac{J+ml^2}{(M+m)J + Mml^2} \boldsymbol{F} \\ \boldsymbol{\theta} = \frac{mlb}{(M+m)J + Mml^2} \dot{\boldsymbol{x}} - \frac{(m+M)mgl}{(M+m)J + Mml^2} \boldsymbol{\theta} - \frac{ml}{(M+m)J + Mml^2} \boldsymbol{F} \end{cases} \tag{1}$$

where *x* is the displacement of the slider, θ is the angle displacement of the rod, *M*  and *m* are the masses of the rod and the slider, *F* is the external force given to the slider, *J* is the rotational inertia of the rod, and *l* is the distance from the centre of mass of the rod to the axis of rotation.

As for the hydraulic system, ignoring the flow pulsation of hydraulic pump and hydraulic motor and pressure loss in pipeline and frictional resistance loss in the hydraulic system, The flow continuity of the hydraulic pump (high pressure side) can be described as:

118 G. Ding et al.

$$\begin{cases} q\_{ph} = d\_p n\_p - C\_{tp} p\_h \\ q\_{ph} - C\_{tm} p\_h = D\_m \frac{d\theta\_m}{dt} + \frac{V\_h}{\beta\_e} \frac{dp\_h}{dt} \end{cases} \tag{2}$$

Similarly, the flow continuity of the hydraulic pump (low pressure side) can be considered as follows:

$$\begin{cases} q\_{pl} = -d\_p n\_p - C\_{lp} p\_l \\ q\_{pl} - C\_{tm} p\_l = -D\_m \frac{d\theta\_n}{d\_t} + \frac{V\_l}{\beta\_e} \frac{dp\_l}{dt} \end{cases} \tag{3}$$

The balance equation of the shaft torque of hydraulic motor can be described as this:

$$D\_m p\_h = J\_t \frac{d^2 \theta\_m}{dt} + B\_{tm} \frac{d\theta\_m}{dt} + T\_L \tag{4}$$

Due to disregarding the frictional resistance moment, the load torque of the reducer is directly applied to the slider of the inverted pendulum through the external force converted by the pulley, so the equations from the load torque to the torque generated by hydraulic motor are:

$$\begin{cases} T\_d = F \cdot r \\ T\_m - T\_d = I \frac{d\omega\_m}{dt} \\ T\_m = n \cdot T\_L \\ \dot{x} = \omega\_m r \end{cases} \tag{5}$$

Finally, the differential equation of the whole system can be described as follows:

$$\begin{cases} \ddot{\mathbf{x}} = Z\theta + V\dot{\mathbf{x}} + O\dot{P}\_h\\ \ddot{\theta} = N\theta + M\dot{\mathbf{x}} + S\dot{P}\_h \end{cases} \tag{6}$$

The value of Z, V, O, W, N, M, S, T are shown as follows:

$$\begin{cases} Z = \frac{r^2 m^2 g l^2}{\Theta} \\ O = \frac{nr D\_n (J + ml^2)}{\Theta} \\ V = \frac{(J + ml^2)(r^2 b + n^2 B\_{tm})}{\Theta} \\ S = \frac{1}{r\Omega} \left[ \frac{(n^2 J\_t + I)(J + ml^2)}{\Theta} - 1 \right] \\ N = \frac{1}{\Omega} \left[ (m + M) mgl - \frac{(n^2 J\_t + I) m^3 g l^3}{\Theta} \right] \\ M = \frac{1}{\Omega} \left[ mlb - \frac{ml(n^2 J\_t + I)(J + ml^2)(r^2 b + n^2 B\_{tm})}{r^2 \Theta} + \frac{n^2 mlB\_{tm}}{r^2} \right] \end{cases} (7)$$

While:


**Table 1** Meanings of variables from Eqs. (2) to (7)

$$\begin{cases} \Omega = (M+m)J + Mml^2\\ \Theta = r^2 \Omega + (n^2 J\_t + I)(J + ml^2) \end{cases} \tag{8}$$

The meanings of the variables of the equations above are shown in Table 1.

### *3.2 Dynamic Characteristics of Hydraulic Inverted Pendulum*

The form of co-simulation using AMESim and Simulink is shown in Fig. 5.

The parameters of the system are showed in Table 2. Given a step input to the system (500 rpm), the corresponding relationship between the angle of the inverted pendulum rod and the displacement of the slider can be obtained in Figs. 6 and 7.

**Fig. 5** Scheme of co-simulation for hydraulic inverted pendulum

And the simulation result of rotary velocity of servo motor and hydraulic motor are shown in Fig. 8.

From the simulation results, by giving a step input, it can be concluded that the inverted pendulum itself is an unstable system, and the hydraulic system has a certain degree of hysteresis compared to the reference step input. By combining the characteristic of inverted pendulum and hydraulic system, the traditional control strategy such as PID or LQR are no longer valid for hold the rod to maintain balance.


**Table 2** Parameters used for co-simulation

**Fig. 6** Slider displacement under step input

**Fig. 7** Angle displacement under step input

**Fig. 8** Simulation result of rotary velocity of servo motor and hydraulic motor

**Fig. 9** Hydraulic inverted pendulum platform in kind

### **4 Field Test on Hydraulic Inverted Pendulum Platform**

### *4.1 Platform Establishment*

The actual platform in kind as mentioned before is displaced in Fig. 9, including pump station, inverted pendulum, and electrical system including motion controller and servo controller.

### *4.2 Program Design*

In order to achieve hardware-in-loop control (HLP) of hydraulic inverted pendulum, the article implements level 2 MATLAB s-function to achieve programming for motion controller. The flow chart of the codes for controlling motion controller is shown in Fig. 10.

### *4.3 Results and Discussions*

By giving a step input (500 rpm), the comparison between simulation result, referenced input and actual rotary velocity of the hydraulic motor is shown in Fig. 11. According to the result, t because of the errors generated by the sensor, the actual value of the rotary velocity of hydraulic motor can't remain constant. Also, some of the variables are actually ignored such as the moment of inertia of the joints connecting between hydraulic pump and servo motor or the joints connecting between the reducer and pulley etc. The rotary of hydraulic motor is able to stabilize at approximately

**Fig. 10** Flow chart of the codes for controlling motion controller

500 rpm, which can approximately fit into the simulation result presented in Fig. 11. It can be seen from this that the established mathematical model can basically conform to the actual system.

**Fig. 11** Comparison between actual value and simulation result

### **5 Conclusion**

In this paper, an introduction of the hardware design of hydraulic inverted pendulum is presented, including mechanical structure, hydraulic system and electrical system. The mathematical model of the hydraulic inverted pendulum is showed afterwards, from inverted pendulum to hydraulic system. The simulation result given by step input shows that the hydraulic system contains hysteresis when driving the inverted pendulum. Finally, the paper presents the field test in order to verify the characteristic of the model, from the result, it can be concluded that the mathematical model can generally fit into the actual system to show its characteristic. In the future, close loop control strategy will be established in order to achieve the swing up and balance control.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Mixed Linear Quadratic Regulator Controller Design for Path Tracking Control of Autonomous Tracked Vehicles**

**Baiyu Tian, Tianliang Lin, Chunhui Zhang, Zhongshen Li, Shengjie Fu, and Qihuai Chen** 

**Abstract** With the advancements in autonomous driving technology, artificial intelligence and computer science have facilitated the autonomous operation of machinery. Industrial production efficiency is enhanced by this autonomy in construction machinery. Furthermore, personal safety for machine operators is also improved. Among these advancements, path tracking control emerges as a critical component for the automation of construction machinery. It primarily focuses on the lateral and longitudinal control of the machinery, enabling stable tracking of a predefined path even under challenging operational conditions. To enhance the accuracy and robustness of path tracking, this paper proposes a mixed linear quadratic regulator (LQR) controller specifically designed for autonomous tracked vehicles. This approach uses the LQR control algorithm to suppress the noise generated by complex construction environments and accurately track the desired path. The effectiveness and practicality of this method are validated through simulation experiments.

**Keywords** Path tracking · Autonomous tracked vehicle · Linear quadratic regulator control

# **1 Introduction**

Tracked vehicles are regarded as commonly ground unmanned mobile platforms in the domain of off-road strategies owing to their excellent traversability and maneuverability, to enhance navigation proficiency across diverse terrains. These scenarios encompass snow-covered terrains, unconsolidated sandy substrates, adhesive mud matrices, steep gradients, terrain laden with rubble, as well as intricate amalgamation thereof [1]. Nowadays, tracked vehicles are used in various industries, such as

College of Mechanical Engineering and Automation, Huaqiao University, Xiamen 361021, Fujian, China e-mail: tt070707@126.com

B. Tian (B) · T. Lin · C. Zhang · Z. Li · S. Fu · Q. Chen

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_10

agriculture and military. Therefore, the study of automatic navigation and operation of tracked vehicles becomes imperative. As a pivotal technology in the realm of autonomous driving, trajectory tracking control also serves as a prerequisite for unmanned tracked vehicles to carry out missions effectively [2].

In previous studies, different strategies for vehicle trajectory tracking control have been proposed in the literature to achieve high-precision tracking, such as geometric and kinematic controllers, model based controllers, adaptive and intelligent controllers, and so on [3].

Initially, pure pursuit represents one of the foremost geometric controllers extensively employed by researchers [4]. It's important to note that while geometric controllers offer these advantages, they may also have limitations, such as challenges in handling complex maneuvers or uncertainties in certain scenarios [4].

Model predictive control (MPC) serves as a technique capable of addressing system constraints and future projections. MPC aims to minimize tracking errors by utilizing a vehicle dynamics model to forecast the forthcoming vehicle nalysed within a defined prediction horizon [3]. Tang et al. [6] amalgamated model-based and data-driven control techniques, introducing a novel approach for trajectory tracking control in bidirectional independent electric-driven unmanned tracked vehicles. Model-based controllers also have limitations, including the need for accurate models, potential sensitivity to nalysed errors, and increased complexity in model development and maintenance.

Adaptive and intelligent controllers are typically employed in research or applications that demand a high degree of robustness against disturbances and variations. The fuzzy control rules of the agricultural machinery path tracking method utilizing the fuzzy adaptive pure pursuit model proposed by Li et al. [6] are established based on expert experiential knowledge, resulting in notable tracking errors that pose challenges for expeditious rectification. However, these controllers' successful implementation requires proper design, training, validation, and consideration of potential limitations and challenges [8, 9].

In this paper, a trajectory tracking control method for unmanned tracked vehicles is proposed. The remainder of the paper is organized as follows. The kinematics analysis and nalysed of the tracked vehicle are carried out, and the expression of the controller is presented in Sect. 2. To demonstrate the performance of the proposed controller in path tracking, nalysed, and simulation experiments are carried out in Sect. 3, and the experimental results are nalysed. Finally, conclusions are given in Sect. 4.

### **2 Problem Description**

In this paper, the trajectory tracking control of unmanned tracked vehicles is studied. For the subsequent construction of a path following controller and experimental verification, kinematic modeling is used in this paper, not only because it provides a simplified and accurate representation of vehicle motion, especially in cases where the primary concern is a path following and position accuracy rather than complex mechanical interactions, but also because it improves computational efficiency and allows for real-time implementation of control algorithms.

### *2.1 Modeling Premises and Assumptions*

This section undertakes a kinematic analysis of tracked vehicles. To facilitate the analysis of tracked vehicle kinematics, specific assumptions are necessary regarding the research subject. Throughout the motion process, this study makes the following assumptions [10].


### *2.2 Kinematic Analysis of Tracked Vehicles*

It is imperative to establish a suitable coordinate system that accurately describes the motion state, pose information, and relative position details of the tracked vehicle to successfully carry out tasks such as path planning, obstacle avoidance, and tracking. This paper introduces a tracked vehicle kinematic model based on the instantaneous center of rotation, as depicted in Fig. 1.

Tracked vehicles commonly involve two types of coordinate systems: the absolute coordinate system *F*(*XOY* ) and the body coordinate system *f* (*xoy*). In Fig. 1, ICC represents the instantaneous rotation center of the tracked vehicle, *R* represents the instantaneous rotational motion radius, *l* is the center distance of both tracks, ω represents the lateral angular velocity of the vehicle, −→v is the instantaneous velocity vector, θ is the heading angle and α represents the angle between the velocity vector and the X-axis, the side-slip angle. (*X*, *Y*, θ) is the pose information of the tracked vehicle in the absolute coordinate system; (*X* ˙ , *Y* ˙, θ )˙ is the linear velocity and rotational angular velocity of the tracked vehicle in the X and Y axis directions in the absolute coordinate system.

Under conditions of low-speed motion, the slip motion and slip angle of tracked vehicles are disregarded. Based on Fig. 1, formula (1) can be derived [3].

$$
\begin{bmatrix}
\dot{X} \\
\dot{Y} \\
\dot{\theta}
\end{bmatrix} = \begin{bmatrix}
\cos\theta & 0 \\
\sin\theta & 0 \\
0 & 1
\end{bmatrix} \begin{bmatrix}
v \\
\omega
\end{bmatrix} \tag{1}
$$

In tracked vehicles, the forward velocity of the vehicle's center point is equal to the average speeds of the left and right tracks:

$$v = \frac{v\_{\rm r} + v\_{\rm l}}{2} \tag{2}$$

When the time increment is very small and the change in the vehicle's rotation angle is minimal, the approximate formula is as follows:

$$
\Delta\theta \approx \sin\theta = \frac{(v\_\text{r} + v\_\text{l})\Delta t}{l} \tag{3}
$$

where, vl and vr represent the speed of the left and right track, respectively.

From Eq. (3), the angular velocity of the tracked vehicle's motion around the center of rotation ω can be obtained, which is also the rate of change in the heading angle.

$$
\omega = \frac{\Delta \theta}{\Delta t} = \frac{v\_\mathbf{r} + v\_\mathbf{l}}{l} \tag{4}
$$

The curved motion radius of the tracked vehicle can be derived from Eqs. (2) and (4).

Mixed Linear Quadratic Regulator Controller Design for Path Tracking … 131

$$R = \frac{v}{\omega} = \frac{l(\upsilon\_{\rm r} + \upsilon\_{\rm l})}{2(\upsilon\_{\rm r} - \upsilon\_{\rm l})} \tag{5}$$

The relationship between the left and right wheel speeds of the tracked vehicle, the center point linear velocity, and the angular velocity can be obtained from Eqs. (2), (4), and (5) as follows.

$$
\begin{bmatrix} \upsilon \\ \upsilon \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ -\frac{1}{I} & \frac{1}{I} \end{bmatrix} \begin{bmatrix} \upsilon\_{\text{l}} \\ \upsilon\_{\text{r}} \end{bmatrix} \tag{6}
$$

### *2.3 Pose Error Model for Tracked Vehicles*

Tracking error is one of the most commonly used quantitative measures for assessing the effectiveness of a controller. It is typically characterized as the difference between the desired path and the actual vehicle position [3, 11]. The parameters for tracking error, as depicted in Fig. 2, are illustrated.

Equation (7) is one of the commonly used formulas for determining tracking error, encompassing longitudinal, lateral, and heading errors.

$$
\begin{bmatrix} e\_\mathbf{x} \\ e\_\mathbf{y} \\ e\_\theta \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin\theta \ 0 \\ -\sin\theta \cos\theta \ 0 \\ 0 & 0 \ 1 \end{bmatrix} \begin{bmatrix} X\_{\text{ref}} - X \\ Y\_{\text{ref}} - Y \\ \theta\_{\text{ref}} - \theta \end{bmatrix} \tag{7}
$$

The differential equations for the pose error can be obtained from Eqs. (6) and (7).

$$\begin{cases}
\dot{e\_{\chi}} = e\_{\chi} - v + v\_{\text{ref}} \cos e\_{\theta} \\
\dot{e\_{\chi}} = e\_{\chi} \omega + v\_{\text{ref}} \sin e\_{\theta} \\
\dot{e\_{\theta}} = \frac{v\_{\text{ref}}}{R} - \omega
\end{cases} \tag{8}$$

### *2.4 Controller Design*

In this section, the proposed LQR controller is introduced in detail. By employing Eq. (9) as the differential equation for pose error, with lateral error and heading error as state variables, and rotational angular velocity as input, a linearized equation can be derived:

$$\begin{cases} \begin{bmatrix} \dot{e\_\mathbf{y}} \\ \dot{e\_\theta} \end{bmatrix} = \begin{bmatrix} 0 \ v\_{\text{ref}} \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} e\_\mathbf{y} \\ e\_\theta \end{bmatrix} + \begin{bmatrix} -L \\ -1 \end{bmatrix} \omega + \begin{bmatrix} 0 \\ v\_{\text{ref}} \end{bmatrix} \frac{1}{\mathbb{R}} \\ \mathbf{y}(t) = \begin{bmatrix} 1 \ 0 \\ 0 \ 1 \end{bmatrix} \begin{bmatrix} e\_\mathbf{y} \\ e\_\theta \end{bmatrix} \end{cases} \tag{9}$$

$$\text{where}, \mathbf{x}(t) = \begin{bmatrix} e\_{\mathbf{y}} \\ e\_{\theta} \end{bmatrix}, \mathbf{x}(t) = \begin{bmatrix} e\_{\mathbf{y}} \\ e\_{\theta} \end{bmatrix}, \boldsymbol{\mu} = \boldsymbol{\omega}, \boldsymbol{A} = \begin{bmatrix} 0 \\ 0 & 0 \end{bmatrix}, \boldsymbol{B} = \begin{bmatrix} -I \\ -1 \end{bmatrix}$$

### **3 Illustrative Examples**

In this section, the proposed LQR controller is applied to a tracked vehicle simulation model to demonstrate its effectiveness and practicality as shown in Fig. 3. The simulation environment is MATLAB R2022b. The simulation results are shown in Figs. 4–7.

Figure 4 presents a comparison of steering trajectories, illustrating the actual path followed by the tracked vehicle when different controllers are employed under the reference path. It can be observed that when the traditional PID controller is used, there is a certain amount of overshoot after the straight-line travel phase ends. Moreover, during sharp turns, the traditional PID controller struggles to accurately track the curvature, resulting in noticeable errors. In contrast, the LQR controller demonstrates improved steering performance, particularly when entering turns. Figure 5 presents the overall trajectory map used in the simulation experiments, illustrating the performance of different control algorithms throughout the complex route.

In Fig. 6, the actual vehicle motion under control signals is depicted, where "krz" represents the discrete input for the speed ratio. Compared to the output signal under

**Fig. 3** Track vehicle driving simulation

**Fig. 5** Trajectory tracking

**Fig. 6** Comparison of motion states

PID control, the LQR control signal achieves superior control performance with a smaller overshoot and shorter settling time. Notably, the settling time is reduced by 5 s, and the overshoot is diminished to one-third. Furthermore, from the tracked vehicle's Y-coordinate data, it is evident that LQR initially exhibits a slight deviation but subsequently achieves stability more rapidly through angle adjustments.

Figure 7 displays a comparison of tracking performance, demonstrating the control signals generated by the controllers under different error inputs. While the PID control system exhibits faster error reaching its peak value, it experiences a longer overall settling time and shorter oscillation periods. In contrast, the LQR control allows the error to reach zero within a shorter time frame of 15 s. Additionally, regarding the control signal for angular velocity output, LQR results in a smaller overshoot. This enables the tracked vehicle system to achieve the desired pose more rapidly when the tracking path undergoes sudden changes.

**Fig. 7** Tracking performance comparison

### **4 Conclusion**

This paper establishes a tracked vehicle kinematic model based on the instantaneous center of rotation. A path tracking controller using a Linear Quadratic Regulator is designed, which simultaneously takes into account vehicle tracking accuracy and reduces the demand for model precision.

Through an analysis of the results obtained from MATLAB simulation experiments, it is observed that the approach presented in this paper achieves higher path tracking accuracy compared to traditional control methods. Under the same driving trajectory, notably improved steering performance is attained during corner entry. Overshoot is significantly reduced, the settling time is shortened, and steady-state error is eliminated.

**Acknowledgements** Key projects of natural science foundation of Fujian Province (2021J02013), Xiamen Major Science and Technology Plan Projects: 3502Z20231013

Fujian University industry university research joint innovation project plan (2022H6007)

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design Method of Rail Grinding Profile Based on Wheel Rail Contact Relationship**

**Rui Song, Dilai Chen, and Gang Shen** 

**Abstract** When the railway track is damaged, it is essential to grind it in in a timely manner to enhance the wheel-rail contact performance. This paper relies primarily on the rolling radius difference function as the main design basis, taking the expected distribution between the wheel and the rail as the boundary condition. In cases where the abscissa of the rail is known, the Euler method is used to solve the differential equation to obtain the optimized ordinate of the rail. The profile of the rail is optimized, and the feasibility of the design method is verified through programmed algorithm. Based on the application results of the rail grinding profile design on the Wuhan-Guangzhou line, the rail light strip position and contact point distribution are more reasonable after grinding, the dynamic performance of the vehicle passing is improved, and the wheel-rail wear rate is reduced.

**Keywords** Wheel-rail relationship · Profile design · Rolling radius difference function · Dynamic performance

# **1 Introduction**

In the process of vehicle operation, the rail will gradually develop concave wear, fatigue, wave wear and other adverse conditions that require maintenance. As an important method of railway maintenance, rail grinding technology can eliminate and mitigate surface damage on rails, thereby extending the rail's service life [1, 2].

R. Song · G. Shen

R. Song

D. Chen (B)

Railway and Urban Rail Transit Research Institute, Tongji University, Shanghai 201804, China

China Railway Test and Certification Center Limited, Beijing 100081, China

School of Railway Transportation, Shanghai Institute of Technology, Shanghai 201418, China e-mail: chendilai@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_11

In railway operation, the dynamic interaction between wheels and rails, rail wear and train operation safety are closely linked to the geometric shape matching between wheels and rails. By reasonably improving the wheel rail contact state can ensure that train operation safety, dynamic performance and wheel and rail wear are in a more reasonable state, so as to effectively extend the service life of rails [3, 4]. Chen [5] studied the rail profile in the turnout area of passenger lines and heavy haul lines, and proposed design methods under different application scenarios. Shevtsov et al. [6] proposed the optimal design of wheel tread based on the function of rolling circle radius difference (RRD, the same below). The numerical analysis method was used to optimize the design of wheel tread, improving vehicle running performance and reducing wheel rail wear. Shen et al. [7] took the RRD function as the design goal to push back the wheel tread of railway vehicles, reducing the wheel rail contact stress and making the wear more uniform, and developed the corresponding calculation program. Jiang [8] used NURBS curve fitting method to parametrically reduce the grinding of the rail profile before grinding, and proposed a grinding mode optimization decision-making method for the quality of rail grinding profile. This method can effectively select the optimal grinding scheme from the rail grinding mode library. Ha [9] aimed at the phenomenon of serious rail wear in small radius curve sections, genetic optimization algorithm was used to optimize the asymmetric profile design, and the verification showed that the optimized profile shows good dynamic performance and wear reduction effect.

Most of the existing references carry out wheel rail optimization design for a single optimization objective, with relatively simplistic considerations. The traditional design of rail profile based on wheel diameter difference function deduces the position of rail contact point based on the position of contact point on the wheel as known condition. In this way, one-to-one correspondence between design objectives and known conditions is established, which will lead to large errors in design results. In order to make up for this defect, this paper proposes directly deducing the rail profile based on the rail contact point position (target contact light band) to obtain a better rail grinding profile, and comparisons are made between the performance of vehicle system dynamics and wheel-rail contact mechanical characteristics before and after optimization.

### **2 Mathematical Modeling**

First, given the wheel profile, the rail profile to be optimized, and the basic wheel rail contact parameters, calculate the RRD curve. Then, optimize the RRD curve based on the requirements of vehicle operation. Take the optimized RRD curve as the main objective function of the design, and deduce the design rail profile based on the expected distribution of wheel rail contact. Subsequently, obtain the whole optimized rail shape by splicing the original rail profile. After that, the designed rail profile is tested for wheel rail geometric contact, contact stress, dynamics performance and other indicators. If it does not meet the requirements, return to modify the parameters and redesign the rail profile until it meets the requirements. The design flow chart is shown in Fig. 1.

The optimized RRD function must satisfy three conditions: First when the wheelset has a slight lateral displacement, the RRD should be minimal, to ensure that the vehicle can travel steadily on a straight track. Second, when the wheelset has a large lateral displacement, the RRD should be increased in order to enhance the curve passing performance and reduce the wheel-rail wear; third, the RRD function curve should be as smooth as possible.

The definitions of wheel-rail coordinate systems are established, according to the right-hand rule, as shown in Fig. 2. Black solid lines represent the wheelset, the blue solid line represents the rail.

The wheel-rail contact points must coincide at the same point in space [10]. Therefore,

$$
\begin{pmatrix} \mathbf{y}\_{rl} \\ \mathbf{z}\_{rl} \end{pmatrix} = \begin{bmatrix} \cos \phi\_w - \sin \phi\_w \\ \sin \phi\_w \ \cos \phi\_w \end{bmatrix} \begin{pmatrix} \mathbf{y}\_{wl} \\ \mathbf{z}\_{wl} \end{pmatrix} + \begin{pmatrix} \mathbf{y}\_{w0} \\ \mathbf{z}\_{w0} \end{pmatrix} \tag{1}
$$

$$
\begin{pmatrix} \mathbf{y}\_{rr} \\ \mathbf{z}\_{rr} \end{pmatrix} = \begin{bmatrix} \cos \phi\_w - \sin \phi\_w \\ \sin \phi\_w \ \cos \phi\_w \end{bmatrix} \begin{pmatrix} \mathbf{y}\_{wr} \\ \mathbf{z}\_{wr} \end{pmatrix} + \begin{pmatrix} \mathbf{y}\_{w0} \\ \mathbf{z}\_{w0} \end{pmatrix} \tag{2}
$$

**Fig. 1** Flow chart of grinding rail profile design process

**Fig. 2** Wheel-rail contact coordinates

There must be a common section and a common normal at the contact point due to the overlap of contact points in space, such that

$$\frac{d\mathbb{Z}\_{rl}}{d\mathbf{y}\_{rl}} = \tan\left(a\tan\frac{d\mathbb{Z}\_{wl}}{d\mathbf{y}\_{wl}} + \phi\_w\right) \tag{3}$$

$$\frac{dz\_{rr}}{dy\_{rr}} = \tan\left(a \tan\frac{dz\_{wr}}{dy\_{wr}} + \phi\_w\right) \tag{4}$$

The geometric relationship between the roll angle and other contact parameters can be written as follows:

$$\tan\phi\_w = \frac{z\_{wl} - z\_{wr} - \frac{z\_{rl} - z\_{rr}}{\cos\phi\_w}}{|\chi\_{wr}| + |\chi\_{wl}|}\tag{5}$$

The RRD curve is defined as:

$$
\Delta \mathcal{R} = z\_{wl} - z\_{wr} \tag{6}
$$

These equations hold true in situations where there is a solitary contact point between the wheel and the rail in space:

$$z\_{wl}(\mathbf{y}\_w) - z\_{wr}(\mathbf{y}\_w) = \min \left\{ z\_{wl}(\mathbf{y}\_w) - z\_{wr}(\mathbf{y}\_w)|\_{\mathbf{y} = \mathbf{y}w} \right\} \tag{7}$$

$$\operatorname{sign}\left(\frac{dz\_{wl}}{d\mathbf{y}\_{wl}}\right) = \operatorname{sign}\left(\frac{dz\_{rl}}{d\mathbf{y}\_{rl}}\right) \equiv 1;\\\operatorname{sign}\left(\frac{dz\_{wr}}{d\mathbf{y}\_{wr}}\right) = \operatorname{sign}\left(\frac{dz\_{rr}}{d\mathbf{y}\_{rr}}\right) \equiv -1 \qquad (8)$$

Based on assumption (2), the rail is a convex curve such that:

$$\operatorname{sign}\left(\frac{d^2 z\_{rl}}{d\mathbf{y}\_{rl}^2}\right) = \operatorname{sign}\left(\frac{d^2 z\_{rr}}{d\mathbf{y}\_{rr}^2}\right) \equiv 1\tag{9}$$

To address the problem using numerical integration, it is necessary to transform the differential–algebraic equations (DAE) given in Eqs. (1–9) into ordinary differential equations (ODE). Simultaneously, derivatives with respect to *yw* must be taken at both ends of Eqs. (1–9). There are:

$$\begin{cases} \frac{d\mathcal{L}\_f(\mathbf{y}\_w)}{d\mathbf{y}\_w} = \frac{d\mathbf{y}\_{wf}(\mathbf{w}\_w)}{d\mathbf{y}\_w} \sin\phi\_{\mathbf{w}}(\mathbf{y}\_w) + \mathbf{y}\_{wf}(\mathbf{y}\_w)\cos\phi\_{\mathbf{w}}(\mathbf{y}\_w)\frac{d\boldsymbol{\theta}\_{\mathbf{w}}(\mathbf{y}\_w)}{d\mathbf{y}\_w} + \frac{d\boldsymbol{\mathcal{L}}\_{wf}(\mathbf{y}\_w)}{d\mathbf{y}\_w}\cos\phi\_{\mathbf{w}}(\mathbf{y}\_w) \\\\ \frac{d\boldsymbol{\varepsilon}\_{fw}(\mathbf{y}\_w)}{d\mathbf{y}\_w} = \frac{d\boldsymbol{\varepsilon}\_{fw}(\mathbf{y}\_w)}{d\mathbf{y}\_w}\sin\phi\_{\mathbf{w}}(\mathbf{y}\_w) + \mathbf{y}\_{wf}(\mathbf{y}\_w)\cos\phi\_{\mathbf{w}}(\mathbf{y}\_w)\frac{d\boldsymbol{\theta}\_{\mathbf{w}}(\mathbf{y}\_w)}{d\mathbf{y}\_w} + \frac{d\boldsymbol{\varepsilon}\_{wf}(\mathbf{y}\_w)}{d\mathbf{y}\_w}\cos\phi\_{\mathbf{w}}(\mathbf{y}\_w) \\\\ -\boldsymbol{\varepsilon}\_{wf}(\mathbf{y}\_w)\sin\phi\_{\mathbf{w}}(\mathbf{y}\_w)\frac{d\boldsymbol{\theta}\_{\mathbf{w}}(\mathbf{y}\_w)}{d\mathbf{y}\_w} \\\\ \frac{d\boldsymbol{\varepsilon}\_{f(\mathbf{w})}}{d\mathbf{y}\_w} = \tan\left(a\operatorname{\operatorname{\operatorname{\bf$$

The column vectors of all unknowns are separated into matrices in the form of:

$$A \times \mu = b$$

To solve the system of obtained six independent differential algebraic equations with six unknowns, the above equations are converted into ordinary differential equations, which are then solved by applying the Euler method:

$$\begin{cases} \mathbf{y}\_{k+1} = \mathbf{y}\_k + h \times f(\mathbf{x}\_k, \mathbf{y}\_k) \\ \mathbf{y}\_0 = \mathbf{y}(\mathbf{x}\_0) \end{cases} \tag{12}$$

Indeed, the design challenge related to rail profiles in turnout areas is a multiobjective optimization problem. To ensure the dynamic performance of the vehicle, as discussed in this paper, the objective function can be formulated as follows:

$$Obj: f\_1 = \min \left\{ \frac{\int\_{-\text{y}\_{w\text{max}}}^{\text{y}\_{w\text{max}}} \left( \left| \Delta R\_{real}(\text{y}\_w) - \Delta R\_{opt}(\text{y}\_w) \right| \right) d\mathbf{y}\_w}{\int\_{-\text{y}\_{w\text{max}}}^{\text{y}\_{w\text{max}}} \Delta R\_{opt}(\text{y}\_w) d\mathbf{y}\_w} 100\% \right\} \tag{13}$$

where Rreal is the computed RRD after the optimization, while Ropt is the targeted optimization of RRD.

### **3 Application Examples**

In order to verify the reliability of this paper, a section of rail on the Wuhan-Guangzhou line railway was used as an example to optimize the grinding profile. The original rail profile is paved with 60 kg/m standard profile (Type: CN60, gauge 1435 mm, cant: 1/40), and the LMA type tread with the most passes was selected as the wheel reference profile.

The measured rail surface state before grinding is shown in Fig. 3. From the figure, there are double light bands on the rail, and the rail contact light band is too wide. According to the feedback from the railway section, the vehicle shaking alarm often occurs when running on this section.

Ignoring the influence of the wheelset yaw angle on the geometric contact characteristics, the assembly set is selected as the given wheel profile and average worn rail profile. The wheel rail geometric contact characteristics are shown in Fig. 4. Figure 4 shows that there is an obvious jumping phenomenon at the contact points during the lateral movement of the wheelset, which will cause multi-point contact during the movement of the vehicle, resulting in double light bands, uneven wear and large impact, thus shortening the service life of the rail.

The RRD curve and equivalent conicity of wheel rail original contact are shown in Fig. 5. Figure 5 shows that the RRD and equivalent conicity, which are the main indicators of the vehicle system dynamics, exhibit obvious linearity in the section where the wheelset lateral displacement is small ([−3, 3] mm), and the equivalent conicity slope is low (the equivalent conicity is about 0.01). This is not favourable for the stability of the wheelset when running on a straight line, and it may cause the vehicle to shake due to excessive wheelset lateral displacement. Figure 4 shows that the wheel rail contact points are obviously concentrated within this range, which is not conducive to uniform wear. When the lateral displacement of the wheelset is large ([±8, ±10] mm), there is an obvious step phenomenon in the wheel diameter difference curve, which may cause a large impact when the vehicle passes through the curved section, posing potential safety risks. Combined with Figs. 3 and 4, when

**Fig. 3** Rail surface state

**Fig. 4** Original contact distribution of LMA and worn CN60 rail

**Fig. 5** RRD function (**a**) and equivalent conicity (**b**) of LMA and worn CN60 rail

the wheel rail contact point transitions from the rail head area to the gauge angle side contact area, the wheel rail contact point has a large jump phenomenon, which leads to multi-point contact when the wheelset displacement is large, resulting in uneven wear, thus reducing the service life of the wheel rail.

According to the above analysis, a reasonable RRD curve should have the following characteristics:


The optimized RRD curve is shown in Fig. 6.

According to the optimized RRD curve, the rail grinding profile is obtained, as shown in Fig. 7.

The wheel rail contact calculation and analysis of the optimized profile are performed again, as shown in Fig. 8. From the figure, the distribution of wheel rail contact points after the shape of the rail is matched with the LMA tread is relatively uniform and continuous, which overcomes the defect that the wheel rail contact points jump with the lateral displacement, and avoids the occurrence of excessive local stress and local wear.

Figure 9a shows the verification results of RRD curve. The overall difference between the target RRD and the calculated RRD is less than 4%, and the area with the largest difference has a small probability of wheel rail contact when the wheel set traverse is ±12 mm. In general, the design effect meets the design requirements. From the perspective of wear, the optimized profiles tend to have longer service life due to the increase of contact pieces and better control of local stress concentration.

**Fig. 7** Comparison of left and right rail profile before and after griding

**Fig. 8** Optimized contact distribution of LMA and designed CN60 rail

**Fig. 9** Error statistics of the calculated RRD (**a**) and equivalent conicity (**d**) of LMA and designed CN60 rail

The optimized equivalent conicity is shown in Fig. 9b. When the transverse displacement of the wheel set is within ±3 mm, the equivalent conicity is about 0.05–0.06, indicating that the stability on the straight line is good and can suppress the shaking phenomenon. In a large range of lateral displacement, different equivalent conicity calculation methods maintain a high value, which is conducive to the passage of the curve.

Figure 10 shows the comparison of the normal contact forces. The optimized non-Hertz contact stress is greatly reduced in the rail top contact area, and the combined contact points are more evenly distributed in this area, indicating that the wheel rail wear will be more uniform, and the service life of the rail will be extended.

**Fig. 10** Comparisons of normal stress of LMA and designed CN60 rail

### **4 Dynamic Verification**

A vehicle turnout coupling dynamic model was established based on MATLAB and Simulink platform to verify the dynamic response of the rail profile in the turnout area before and after optimization. Using the LMA wheel tread, a rail vehicle passage of a CN60 single straight turnout was simulated, as shown in Fig. 11. From the dynamic simulation results, the optimized vehicle has better performance when going through the turnout, and the wear index between the wheel and rail is lower.

### **5 Conclusions**

This paper introduced a rail grinding profile design method for the heavy-haul train. It is based on the wheel-rail contact relationship and aims at improving vehicle dynamic performance and wheel rail wear rate by controlling wheel rail contact distribution. It utilizes the idea of inverse design method and multigoal optimization. The results of computer software program simulation and the application results of rail grinding profile design on the Wu-Guang line show that the grinding profile design using this algorithm can optimize the dynamic performance of the wheel and rail while ensuring a good match with the target wheel diameter difference curve, and provide a feasible and efficient profile customization method for rail grinding. With the help of the proposed method, it can be expected that the rail could have longer service life under proper grinding operation.

**Fig. 11** The comparison of dynamic interaction. **a** Lateral displacement of wheelset, **b** lateral acceleration, **c** lateral force, **d** lateral force Wear index

**Acknowledgements** 1. The project of Shanghai Science and Technology Commission Local Capacity Building (No. 22YF1447600: Study on variable cross section profile optimization of high-speed turnout oriented to wear control).

2. Key project of Science and Technology Research and Development Plan of China National Railway Group Co., LTD. (N2022G043, Research on Detection and Evaluation of rail wear profile and testing technology under high-speed conditions).

3. Key project of Science and Technology Research and Development Plan of China National Railway Group Co., LTD. (L2022J003, Evaluation index and evaluation method of high-speed wheel tread concave grinding).

4. China Academy of Railway Science Group Co., LTD. Research Project. (2022YJ264, Railway vehicle inertial suspension system and vibration reduction application research).

5. The project of Shanghai Science and Technology Commission Local Capacity Building (No. 21210750300: International Joint Laboratory of China Laos Railway Engineering under the "the Belt and Road" Initiative)

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Review on Safety Management Strategies: Theory and Practical Application of Lithium-Ion Power Batteries**

### **Xiaojian Yi, Lin Hu, Shuang Liu, and Changfu Zou**

**Abstract** Battery safety in electric vehicles is a comprehensive engineering endeavor that requires meticulous consideration at every stage, including battery materials, battery pack design, and battery management systems (BMS). This review focuses on safety management strategies and practical applications of lithium-ion power batteries. The management of battery safety primarily encompasses charge and discharge safety, high-voltage safety, and thermal safety. Among these, charge and discharge safety management aims to prevent battery damage or safety incidents caused by overcharge or over discharge. High-voltage safety management involves detecting insulation faults, overcurrent, and other potential risks to prevent electrical hazards. Thermal safety management ensures individual battery cells, modules, and the battery pack maintain an optimal operating temperature range and uniform temperature distribution, thereby preventing thermal runaway.

**Keywords** Electric vehicle · Battery safety management · Lithium-ion battery · Safety regulations

X. Yi (B)

L. Hu · S. Liu

X. Yi · L. Hu · S. Liu

C. Zou

© The Author(s) 2024

School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha 410114, China e-mail: csust\_yxj@163.com

School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China

Hunan Key Laboratory of Safety Design and Reliability Technology for Engineering Vehicle, Changsha University of Science and Technology, Changsha 410114, China

Department of Electrical Engineering, Chalmers University of Technology, 41296 Gothenburg, Sweden

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_12

### **1 Power Battery Safety Regulations**

### *1.1 International Regulations*

In the field of power battery international standardizations, Organizations as ISO and IEC have launched standards for lithium-ion batteries, lead-acid batteries, alkaline batteries, and fuel cells in succession. America, Japan and South Korea have combined their own technological development paths and national conditions, established their own power battery standard system based on the requirements of ISO and IEC standards. The Table 1 below presents the safety testing standards for international lithium-ion power batteries.

### *1.2 Domestic Regulations*

The in-progress standard in China is GB 38,031–2020- Electric vehicles traction battery safety requirements. Chinese power battery safety standards, starting from the automotive industry standard QC/T 743–2006-Lithium-ion batteries for electric vehicles [4], to the recommended standards GB/T 31,485–2015-Safety requirements and test methods for traction battery of electric vehicle [5] and GB/T 31,467–2015 [6], and finally to the national mandatory standard GB 38,031–2020 [7].

GB 38,031–2020 benchmarked international standards IEC 62,660–2 Part 2- Reliability and abuse testing, IEC 62,660–3 Secondary lithium-ion cells for the propulsion of electric road vehicles Part 3- Safety requirements, and has been upgraded to a mandatory standard based on industry needs. Table 2 presents the main power battery safety standards of GB 38,031–2020.

### **2 The Inequality of Lithium-Ion Batteries**

The inequality of batteries can be attributed to two factors. Firstly, during the manufacturing process, variations in material composition and process issues lead to differences in the activation level, thickness, porosity, tab connections, and separators of battery electrode materials. Additionally, the solid electrolyte interface film near the electrodes during battery formation is random and can contribute to battery inequality. Secondly, during installation and use, factors such as temperature, ventilation conditions, self-discharge level, and electrolyte density in the battery pack can affect the voltage, internal resistance, and capacity, thereby increasing the inequality of these parameters among individual batteries.

At present, common positive electrode active materials for lithium-ion batteries include LiCoO2, LiNiO2, LiMn2O4, LiFePO4, et al. The safety of positive electrode materials mainly includes thermal stability and overcharging safety. During


**Table 1** International safety testing requirements for lithium-ion power batteries

overcharge process of lithium-ion batteries, the reaction equation is as follows.

$$Li\_{0.5}COO\_2 \rightarrow 1/6CO\_3O\_4 + 1/6O\_2 + 1/2Li\_{0.5}COO\_2 \tag{1}$$

The consistency of capacity in practical applications is that the remaining amount of electricity in the battery during discharge is not equal, and the remaining amount of electricity in the battery can be expressed as follows:

$$C = C\_0 - \int I\_b(t)d(t) \tag{2}$$


**Table 2** Power battery safety standards of GB 38,031–2020

In the formula, *C*0 is the initial capacity of the battery, and *Ib* is the discharge current related to time.

As shown in Fig. 1, the "barrel effect" shows that during charging, the middle battery has reached the cut-off voltage, while the left and right batteries are not fully charged. If there is no balanced control and safety management mechanism, continuing to charge the battery will cause the middle battery to overcharge and cause unnecessary safety accidents.

Currently, there are two approaches to address battery inequality: active equalization and passive equalization. Active equalization involves transferring energy from cells with higher energy to cells with lower energy or supplementing the energy of the entire pack to the lowest cell. Its advantages include high efficiency and minimal losses. Passive equalization, on the other hand, typically involves discharge the higher voltage cells through resistors, releasing energy in the form of heat, and allowing other cells to have more charge time. In this way, the overall system's energy is limited by the cell with the lowest capacity [8, 9].

In addition to the impact of battery inequality on battery charge safety, charge stations also affect battery charge safety. Due to space limitations, a brief overview is provided here. The reliability and security of the communication system in charge stations have a significant impact on the safety of electric vehicle charge. If the communication protocols between the vehicle and the charge station are mismatched or incompatible, the electric vehicle will not be able to initiate the charge process [10]. During the charge process, transmission or reception errors can lead to charge interruptions, overcharge, and the risk of fire in the electric vehicle or charge equipment. The aging and failure of components in charge equipment also affect charge safety [11–14].

### **3 High Voltage Safety Management**

High voltage inter-lock (HVIL) is a safety feature in electric vehicles that ensures the electrical integrity of high-voltage components and their connections to the highvoltage power lines by using low-voltage signals. When the Battery management system (BMS) detects an abnormal circuit open, it is necessary to promptly disconnect the high-voltage power to ensure the personnel safety and vehicle equipment operation [15, 16]. To ensure the safe use of batteries, monitoring the current inside the battery pack can provide short-circuit and over current protection. Current monitoring is commonly achieved using shunt resistors and Hall sensors. Shunt resistors offer high precision but introduce voltage drop and energy loss. Additionally, their resistance can vary with temperature, leading to measurement errors [17]. Hall sensors, on the other hand, offer advantages such as high accuracy and linearity, independent of electrical isolation devices, fast response time, and no voltage loss. Those are widely used for current monitoring inside power battery packs. Hall sensors should be installed before the main negative relay so that the current can be monitored regardless of whether the power battery is in a charge or discharge state. This arrangement is also based on the consideration of monitoring the current during the battery's self-heating process, as it is essential for ensuring the heating process safety [18].

Figure 2 shows lithium-ion power battery protection circuit model. Overcurrent protection is achieved by detecting the voltage across a sampling resistor caused by the discharge current and comparing it with the overcurrent threshold voltage. The overcurrent protection adopts a multi-level protection mode, firstly, the current

**Fig. 2** A type of lithium-ion power battery protection circuit

I flowing through both ends of resistor R, converted to voltage through a current voltage conversion circuit.

$$U\_1 = \frac{IR\_S}{R\_2}(R\_2 + R\_3) \tag{3}$$

In the formula, the voltage *U*1 is sent to the I/O port ANL5, and after A/D conversion, it is compared with the overcurrent threshold voltage. When the value exceeds the overcurrent threshold voltage and the duration exceeds the corresponding overcurrent delay time, turned off the MOSFET at the overvoltage control end, and cut off the discharge circuit. Suitable resistor values are selected based on sampling accuracy and circuit power consumption considerations. To balance sampling accuracy and short-circuit protection response time, the values of capacitors should not be too large. Multiple levels of overcurrent protection with different threshold voltages and corresponding delay times are used, where larger threshold voltages have shorter delay times [19].

### **4 Thermal Runaway Warning**

Regarding the warning of battery thermal runaway, it mainly focuses on monitoring and warning critical conditions when the battery experiences thermal runaway [20]. By monitoring temperature, smoke, and combining parameters such as voltage, current, and internal resistance, the occurrence of thermal runaway can be determined. By combining multiple parameters for warning, the reliability of the warning system Can effectively reduce false positives and improve system reliability.

Figure 3 shows the conventional practical application of battery thermal management development process.

**Fig. 3** Battery thermal management development and design process

When the battery operates at normal temperatures, its internal resistance decreases as the temperature rises. However, when the temperature exceeds the normal operating range and thermal runaway occurs, the internal resistance of the battery increases significantly. Srinivansan [21] proposed a thermal runaway warning method for lithium-ion batteries based on impedance phase rapid monitoring. The internal impedance of the battery is divided into amplitude Z and phase shift θ. the monitoring of internal battery temperature and prediction of thermal runaway can be achieved. Since a sudden change in internal resistance does not necessarily indicate thermal runaway, as the battery can experience changes in resistance due to external disturbances or poor contacts, it is necessary to combine multiple characteristic parameters for warning [22].

The characteristics of gases generated during battery thermal runaway are more suitable as the basis for early warning judgments. Research has shown that the detection sensors have advantages such as high reliability and low cost compared to other combustible gas sensors. Therefore, it has been determined that gas and temperature can serve as early signals for battery thermal runaway warning. In order to explore more effective warning methods, many scholars have combined multiple parameters for analysis, which can further enhance the safety and reliability of lithium-ion battery systems. For example, Ma Wei [23] built a warning system for lithium-ion batteries, it serves as a warning for thermal runaway, considering the temperature and voltage parameters in abnormal operating conditions. Wang Fang [24] concluded a warning method for thermal runaway of individual lithium-ion batteries based on temperature, smoke, and combustible gas data, using the Dempster-Shafer evidence theory, and implemented the warning of thermal runaway.

To conduct scientific research and engineering applications, simulation and experimental methods are commonly used, including ADVISOR based simulation platforms, hardware in the loop, and model in the loop experiments. Shown as in Fig. 4.

**Fig. 4** Lithium-ion battery simulation model in ADVISOR

# **5 Conclusion**

This article starts with the factors influencing battery safety and provides a detailed introduction to the safety management of electric vehicle batteries. Power batteries are complex systems combined electrochemistry, mechanics, heat, and control management. Those are closely interconnected, and failure in any aspect can have a significant impact on the safety of power batteries. To facilitate the understanding of lithium-ion power battery safety design methods, this includes battery selection, module assembly design, battery pack safety protection, and design strategies of lithium-ion battery safety management.

**Acknowledgements** This work was supported by the National Natural Science Foundation of China (Grant No. 52211530054); the Scientific Research Fund of Hunan Provincial Education Department (Project No. 21A0193); Changsha Natural Science Foundation Project (Project No. KQ2208235) and Graduate science and Hunan Graduate Students Research Innovation Project (Project No. QL20220193).

### **References**

1. Yang Q, Li Q (2016) The comparison of Li-ion battery national standard GB 31241 and UN 38.3. Batter Bimon, 46(01):46–48. CNKI:SUN:DCGY.0.2017–03–013


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Parameter Optimization Methods of the Hydraulic Regenerative Braking System Based on the Actual Working Condition**

**Tingting Luo, Nianning Luo, Jing Yang, and Bo Huang** 

**Abstract** Hydraulic regenerative braking system is a kind of complex MIMO (Multiple Input Multiple Output) system which is strongly nonlinear and multivariable, especially affected observably by the working condition. Moreover, it possesses the typical hybrid system characteristics since there are a multitude of nonlinear components internal which should result to continuous dynamic variation and discrete decision process. Hence, the traditional calculating method is difficult to analyze and evaluate the system accurately while operating on the actual working conditions. In order to solve the parameters optimization problem of the system which has significantly hybrid system characteristics constrained by the actual working conditions, this paper presents a new type of genetic algorithm by solving fitness function based on operation simulation, which provides a promising full condition optimization method for parameter optimization according to this kind of system. Simulation results show that this algorithm could optimize system configuration and enhance 3% for the braking energy recovery efficiency.

**Keywords** Hydraulic · Regenerative braking · Hybrid system · Parameter optimization

# **1 Introduction**

Regenerative braking is one of the main methods for improving vehicle fuel economy. The hydraulic regenerative braking technology makes widely attention due to the hydraulic energy storage possesses multiply characteristics including inflation speed, large power density, high efficiency, and long service life [1–4]. Wu et al. investigated

© The Author(s) 2024

T. Luo (B) · N. Luo · B. Huang

Harbin Institute of Technology, Harbin 150001, China e-mail: luo991122@163.com

J. Yang Taiyuan University of Technology, Taiyuan 030024, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_13

the dynamic response of vehicles in hydraulic regenerative braking mode [5]. Zhou Shilei et al. investigated the regenerative braking control strategy for medium duty trucks and optimized the parameter design of their hydraulic drive system [6]. Oza et al. investigated the performance impact of hydraulic regenerative braking system on school buses and public transportation [7]. Li Ning et al. studied the application of hydraulic regenerative braking in electric hydraulic hybrid vehicles, and Shilei Zhou et al. studied the optimal design of medium-sized truck based on typical urban working conditions. However, how to optimize the parameters is becoming one hot spot problem since the complicated system structure of Hydraulic regenerative braking system.

After investigating working condition of the vehicle, it can be got that traffic environment is uncertainty. Hence, hydraulic regenerative braking system needs to adapt to the changing state of vehicle and working condition requirements. Both of the requirements for cooperation with rapid response of the friction brake system and slow response engine system should be met to recycle braking energy and adjust working area of the engine to reduce the fuel consumption. Moreover, the two modes work alternately. From the aspect of the control theory, the system possesses the continuous state variables dynamic process who obeys Newton mechanics law, at the same time it also follows decision-making information logic principle of discrete event driven process, is a typical hybrid system characteristics of multivariable strongly coupled multiple input multiple output nonlinear dynamic systems [8, 9]. Furthermore, since coupling with the hydraulic system characteristic of strong nonlinear and parameter perturbation, the traditional optimization method is arduous to practice for describing and evaluating the objective function of hydraulic regenerative braking system.

In order to solve the problem, this paper presents a genetic algorithm which is based on the conditions simulation for fitness function, and it provides a full process optimization method for the parameter optimization of this kind of system.

# **2 Working Principle of Hydraulic Regenerative Braking System**

Hydraulic hybrid system can be divided into the string, parallel and mixing structure according to the relationship topology of the main and auxiliary power system which are engine and hydraulic accumulator. However, no matter how complex the system is, regenerative braking system is the same radically, and can be simplified as shown in Fig. 1. It includes controller, variable hydraulic pump/motor 1 and its operating mechanism 2, andhydraulic accumulator 3, typical system of pipe and accessories.

There are two modes for the pump/motor, which are pump and motor mode. The pump/motor is working as a pump controlling by the variable hydraulic pump/motor operating mechanism during braking, and the controllable brake torque is provided. At the same time, the braking energy can be transformed into hydraulic energy

and stored in the hydraulic accumulator. Similarly, the pump/motor is working as a motor mode which obtaining the energy from the accumulator to assist the engine or individual drive vehicle during driving condition [1–4].

In order to maximize the recovery of braking energy, hydraulic regenerative braking system (pump mode) is preferred to use generally. And the insufficient braking force is assisted by the original vehicle or the improved friction braking system [10–12]. The control strategy is described in diagram 2.

Where *r*p, *q,* are the displacement and flow rate of hydraulic pump/motor, *p*  is hydraulic accumulator pressure, *r*f is the mechanical friction control signal, *u*  is speed, *T*f, *T*p, respectively friction braking torque and the regenerative braking torque, finally, β is the brake pedal stroke.

Mathematical description of control algorithm are as follows.

$$T\_{k,req} = k\beta \tag{l}$$

$$T\_{\mathfrak{p}} = \min(T\_{k,req}, T\_{p,max})\tag{2}$$

$$T\_{\rm f} = \begin{cases} T\_{\rm k,req} - T\_{\rm p}, \ T\_{\rm k,req} > T\_{\rm p} \\ 0, & T\_{\rm k,req} \le T\_{\rm p} \end{cases} \tag{3}$$

where *T*k,req means the desired braking force of the driver, and *k* is the gain of the braking system, the maxim braking torque of the pump/motor is presented by *T*p, max.


#### **Table 1** Factors-levels tables

# **3 Experimental Design of Hydraulic Regenerative Braking System**

# *3.1 Hydraulic Regeneration System Test Program and Parameter Selection*

For the system shown in Fig. 1, when the displacement of the hydraulic pump/motor and the rated volume of the accumulator are smaller, and the working pressure is lower, the system's reliability is higher, the modification cost is lower, and the required installation space is smaller. To achieve this, a multi-factor orthogonal experimental method was adopted to simulate the braking of the vehicle under the initial speed of 60 km/h, as shown in Table 1, with a design of 4 factors and 3 levels in a full-factorial experiment.

### *3.2 Analysis of Experimental Results*

The summary of the hydraulic regenerative system test results is presented in Table 2.

In order to determine the extent of the influence of different factors on the energy recovery rate of the hydraulic regenerative braking system, based on the experimental results in Table 2, a range analysis of the braking energy recovery rate was conducted, as shown in Table 3.

In Table 3, *Ji* represents the mean of the braking energy recovery rate at the same level for each factor, with the optimal level being *Ji* max. The range analysis results indicate that the influence of hydraulic regenerative braking system parameters on the utilization of braking energy, from greatest to least, is as follows: accumulator pre-charge pressure > system's minimum operating pressure > system's maximum operating pressure > accumulator rated volume > hydraulic pump/motor rated displacement.


**Table 2** Test results for hydraulic regenerative system braking energy recovery rate

**Table 3** Range analysis results for the hydraulic regenerative braking system


### **4 Optimization Problem Resolution**

Obviously, the maximum braking energy recovery is the most important optimization goal of regenerative braking process. It is significant for vehicle safety, easy recycling braking that how to optimize the parameters of the hydraulic matching [13–15]. From the mathematics point of view, the optimization of hydraulic auxiliary power system, can be described as, choosing appropriate accumulator rated capacity and inflation pressure, working pressure and displacement of hydraulic pump/motor, under the typical vehicle braking mode. And it also needs to consider reliability, cost, and installation space constraints for recycling braking energy as much as possible, making the braking energy recovery to be the largest. Hence, the equation can be obtained as below,

$$\begin{cases} \min \, \, f(X) & X \in D \\ \text{s.t. } g\_i(X) \le 0, \quad i = 1, \, 2, \ldots, m \\ \quad h\_j(X) = 0, \quad j = 1, \, 2, \ldots, p(p < n) \\ \quad X = \begin{bmatrix} \chi\_1, \chi\_2, \chi\_3, \ldots, \chi\_n \end{bmatrix}^T \end{cases} \tag{4}$$

where, *f* (*x*), *gi*(*x*), *hj*(*x*) are the objective function, constraint inequalities and constraint equation, respectively. And *X*, *D* are design variables of the feasible region.

### *4.1 Design Variable*

In this paper, the design variables of choosing hydraulic inflation pressure *p*0, the lowest pressure of the system *p*1 and the largest pressure *p*2, volume of the accumulator*V*0, displacement of the pump/motor *Vg*, and the relationship of theses parameters can be listed as,

$$X = \begin{pmatrix} p\_0, p\_1, \ p\_2, \ V\_0, \ V\_g \end{pmatrix}^T \tag{5}$$

### *4.2 Objective Function*

According to the analysis above, braking energy recovery is the only optimization goal. And the objective function can be got from Ref. [11].

$$f(X) = -\frac{\int pq\mathbf{dt}}{\frac{1}{2}mu\_1^2 - \frac{1}{2}mu\_2^2 - \int (F\_f - F\_w - F\_i)\mu\mathbf{dt}} \times 100\% \tag{6}$$

where *m*, *u*1, *u*2 are the mass, initial and final speed of the vehicle during braking, and *Ff* , *Fw*, *Fi* are rolling resistance, wind resistance and gradient resistance of the vehicle.

In fact, (6) is difficult to obtain by theory calculation, because the hydraulic hybrid vehicle has a variety of working modes, this kind of working mode is made up of the vehicle control strategy based on the driver demand and is determined by the state of the vehicle. Hence, the working mode of the system is also different among different working conditions. Furthermore, the brake of the depth of the initial conditions and the friction brake intervention is also different, even under the same test conditions, the vehicle is also a process of alternating between driving and braking. Hence it should be more difficult to examine different configuration of the vehicle under different control strategies in different working environment by using a mathematical expression.

In this paper, according to DD6141S02 bus, then the simulation model is constructed according to Ref. [15] by using the control strategy shown in Fig. 2. And the working condition is chosen from Ref. [15], which is CCBUS (China City Bus). Then, the braking energy recovery rate got from the simulation is transferred to MATLAB environment as optimization objective function. The objective function evaluation model is shown in Fig. 3.

**Fig. 3** The evaluation model of object function

### *4.3 Constraint Condition*

**Displacement of hydraulic pump/motor**. The greater the displacement of hydraulic pump/motor, the greater the braking torque should be provided under low back pressure of the system; Moreover, the faster that the back pressure rises and the wider that regenerative braking system of braking force adjustment allowance under the same accumulator volume. However, it will lead to the increasing in the cost of the system, and the demand of the installation space is bigger and *V*g ≤ 500 mL/r is chosen in this paper.

**Rating volume of the accumulator**. About accumulator volume, the greater the regeneration system of liquid volume is larger, more conducive to the storage of hydraulic energy. However, with increasing accumulator volume, weight and cost of the system is also enhanced, the more important is that it should also be constrained by limited installation space of vehicles. Therefore, it is guaranteed that there is enough volume to recover typical braking mode braking energy, its volume should be as small as possible. And the *V*0 ≤ 500L according to the above.

**Maximum working pressure**. The power density should be higher with the increasing maximum working pressure. However, the maximum system pressure is determined as *P*2 ≤ 30 MPa by considering with the working condition and safety of the system.

**Minimum working pressure**. After analyzing the regenerative braking process, it can be obtained the higher the system pressure, the regenerative braking system can provide larger regenerative braking, conducive to the recovery of braking energy. However, the established accumulator should lead to the effective work of accumulator volume decreases if the pressure is higher, and in turn affects the recovery efficiency of braking energy. On the contrary, the lower the working pressure of system, can lead to a large compression ratio, and the large changing amount of volume for the skins of accumulator, combining with the disadvantage of reducing service life after Severe temperature change. There are,

$$t\_2 = \left(\frac{p\_2}{p\_1}\right)^{\frac{n-1}{n}} (273 + t\_1) - 273\tag{7}$$

where, *t*1, *t*2 are accumulator temperatures under the minimum and maximum pressure respectively.

**Prefill pressure of the accumulator.** Accumulator prefilled pressure is mainly associated with the minimum working pressure of system pressure, when pressure to meet minimum working pressure, a small amount of oil should be reserved between the skin and check valve, to prevent contact with the shell wall completely in the working process of the accumulator skins, avoid in the process of expansion to the outlet valve. And *V*1 ≤ 0.9 *V*0.

In addition, this part of the reserve oil also has a purpose, to ensure enough oil to provide servo oil in the hydraulic system and it should meet the requirement that,

Parameter Optimization Methods of the Hydraulic Regenerative … 167

$$\begin{cases} -p\_1^{\frac{1}{\tilde{\boldsymbol{\eta}}}}(V\_0 - \boldsymbol{v}\_{\text{min}}) + p\_0^{\frac{1}{\tilde{\boldsymbol{\eta}}}}V\_0 \le \mathbf{0} \\\ p\_1^{\frac{1}{\tilde{\boldsymbol{\eta}}}}(V\_0 - \boldsymbol{v}\_{\text{max}}) - p\_0^{\frac{1}{\tilde{\boldsymbol{\eta}}}}V\_0 \le \mathbf{0} \end{cases} \tag{8}$$

where *v*min, *v*max are minimum and maximum reserved oil volumes respectively.

**Limit allowable working volume.** According to the HYDAC company suggestion, bag type accumulator capacity utilization rate of 75% of the actual gas capacity allowed. According to the character of an ideal gas.

$$(p\_0 p\_1)^{\frac{1}{s}} - (p\_0 p\_2)^{\frac{1}{s}} - 0.75(p\_1 p\_2)^{\frac{1}{s}} \le 0 \tag{9}$$

**Working condition of constraint**. Effect of regenerative braking energy recovery factors in addition to the associated with the configuration of the hydraulic system, also with the initial braking speed and braking intensity and the duration of the brake, the same configuration of regenerative braking system under different traffic conditions, the braking energy recovery rate is different. Optimization should, therefore, under the environment of drive with typical characteristics.

### **5 Optimization Algorithms and Optimization Results**

In this paper, the genetic algorithm is calculated with GADS toolbox in MATLAB, MATLAB environment GADS algorithm with Simulink data exchange process as shown in Fig. 4.

In GADS interface Optimization Tool, the choice of based Algorithm solver, the number of design variables is 5, and the variable scope is listed below,

$$\begin{aligned} Lb &= \begin{bmatrix} 12 \ 13 \ 25 \ 200 \ 200 \end{bmatrix} \\ Ub &= \begin{bmatrix} 16 \ 17 \ 30 \ 500 \ 500 \end{bmatrix} \end{aligned} \tag{10}$$

Linear inequality coefficient matrix:

**Fig. 4** The data exchange process between matlab and Simulink in the optimization

process

$$A = \begin{bmatrix} 1 & -0.9 & 0 & 0 \ 0 \\ -1 & 0.85 & 0 & 0 \ 0 \\ 0 & -1.9869 & 1 & 0 \ 0 \\ 0 & 1 & -1 & 0 \ 0 \end{bmatrix} \tag{11}$$

Nonlinear constraints are determined by Eqs. (8) and (9).

Running genetic algorithm, the objective function of optimization search process is shown in Fig. 5. It can be found form the diagram, in the early evolution of the fitness function value and the optimal value decay faster, average rapidly toward the optimal value, show that the convergence rate of the population is very obvious; To 40 generations later, the optimal value of fitness function is basic remained unchanged, at this time and achieve the optimal population size can be thought of to get the optimal solution of the problem. The parameter for the hydraulic system, *p*0 = 15.159, *p*<sup>1</sup> = 16.861, *p*2 = 29.992, *V*0 = 269.404, *V*g = 499.935. The objective function is minimum as *J* = −85.441%.

Considering the practical market products supply, *Vg* = 500, *V*0 = 250, *p*2 = 30 are confirmed. In order to get the optimal matching parameters, the introduced genetic algorithm above is used, and after thirty iterations, we can get *p*0 = 14.380, *p*1 = 15.977, and the minimum objective function value *J* = -85.131%. The final parameters are listed in Table 4. An additional quality hydraulic auxiliary system of about 500 kg is added after optimization configuration of the vehicle modification.

**Fig. 5** Optimization process of objective function


**Table 4** The optimized parameters of hydraulic system

### **6 Simulation**

Simulation is made under 25, 50, 75, 50% and 75% rated load for the hydraulic hybrid vehicle in CCBUS condition. When the accumulator prefilled pressure is 15 MPa, average recyclable energy, average braking energy recovery and the accumulator average braking pressure with the change trend of accumulator configuration is shown in Fig. 6. According to the figure, the 250L accumulator hydraulic system under CCBUS all conditions, can obtain higher braking energy recovery, which conforms to the above optimization results.

Furthermore, the optimized parameters are input into the Simulink to run a simulation, Figs. 7, 8, 9 show the comparison before and after the optimization of the accumulator pressure *p*, hydraulic regenerative braking system of *T*p and friction dynamic imitation of *T*f. It can be seen that the optimized system average work pressure increased significantly, the hydraulic regenerative braking system can provide more regenerative braking torque, and the optimized friction braking torque are

**Fig. 6** The relationship between the recovery rate of braking energy and the volume of accumulator under CCBUS working condition

significantly reduced, both on braking times and values. Based on the actual recovery of hydraulic energy statistics, the working condition of the whole driving cycle braking energy recovery rate enhances from 82.07% to 84.84%. Rated volume of the hydraulic accumulator is decreased to 250L compared with the estimate value of 400L, it should also reduce system cost and installation requirements of space. At the same time it also reduces the additional load because of the hydraulic hybrid modification.

### **7 Conclusion**

Through orthogonal experimental analysis, it can be determined that in the hydraulic regenerative braking system, the influence weights of relevant parameters on the braking energy recovery rate are ranked from high to low as follows: accumulator pre-charge pressure, system's minimum operating pressure, system's maximum operating pressure, accumulator rated volume, and hydraulic pump/motor rated displacement. Through the simulation analysis, for the system possesses the characteristics of both continuous and discrete features, at the same time constrained by actual operation condition, the genetic algorithm which based on the working condition of simulation for fitness function is easy to get the result of the global optimization. From the optimization results of hydraulic pump/motor rated volume and system working pressure values are close to the limit, this suggests that their value is bigger, the better, because of the large hydraulic pump/motor displacement and higher prefilled pressure can provide larger regenerative braking, is conducive to recycle more regenerative braking energy. Moreover, the higher braking energy recovery could be obtained only after choosing suitable accumulator volume in the actual working condition. This is because the larger accumulator capacity should lead to the braking pressure rise slowly during braking, and then the friction braking should be intervened early which will result in kinetic energy loss.

**Acknowledgements** This research was supported by National Key R&D Program of China [2021YFB2011900].

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design and Development of Dynamic Test Bench for Electric Wheel Load Sensing Suspension**

**Yan Li, Pengbo Sun, Jiawei Chen, and Xinbo Chen** 

**Abstract** Based on the modular design idea, a new type of dynamic test bench for electric wheel load sensing suspension was designed and developed, including load-bearing platform module, vertical force loading module, electric wheel module, spring mass simulation module, etc., and the test bench has the functions of wheel jump simulation, steering simulation and wheel vertical force and lateral force detection. The finite element simulation results show that the strength of the test bench meets the requirements. The results of the virtual prototype simulation show that the test bench can accurately measure the vertical and lateral forces experienced during the dynamic operation of the wheels.

**Keywords** Electric wheels · Test bench · Vertical force · Lateral force

# **1 Research Background and Significance**

Road test is the most important means for car companies to carry out vehicle performance testing and suspension system testing, which requires a larger site and high cost, and many small and medium-sized car companies do not have the conditions for road testing. In order to meet the needs of small and medium-sized car companies for suspension system testing, many researchers have begun to develop and research suspension test benches for different test objectives. Literature [1, 2] develops an air suspension experimental bench to solve the problem of nonlinear dynamics of air suspension; XUE et al. developed and designed a single-wheel test bench of the anti-lock braking system, which verified the effectiveness of the all-electric ABS algorithm [3]; Literature [4, 5] developed a test bench for suspension performance testing and vehicle performance matching, and studied the dynamic characteristics of

Y. Li · P. Sun

School of Mechanical Engineering, Tongji University, Shanghai 201804, China

J. Chen · X. Chen (B)

Automotive College, Tongji University, Shanghai 201804, China e-mail: chenxinbo@tongji.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_14

suspension components and vehicle characteristic matching; Literature [6, 7] designs an oil and gas suspension test bench to study the electro-hydraulic ratio control of oil and gas suspension; Literature [8, 9] The fatigue life of the suspension system is studied by test bench.

Considering that distributed drive electric vehicles are the most ideal models for autonomous vehicles, and the perception of wheel load can provide the necessary wheel load data for the automatic driving system, there is a lack of research on the test bench for dynamic load measurement of electric wheels, so the dynamic test bench of electric wheel load sensing suspension is developed and designed.

# **2 The Main Structure and Working Principle of the Test Bench**

### *2.1 The Main Structure of the Test Bench*

The overall design of the test bench adopts the modular design idea, as shown in Fig. 1, the main structure of the test bench includes the bearing platform module, the vertical force loading module, the electric wheel module, the spring load mass simulation module, the profile bracket and other structures. The load-bearing platform module is connected to the profile bracket by the vertical guide rails on the front and rear sides, and connected with the vertical force loading module through the horizontal guide rails at the bottom; The electric wheel module is connected to the body simulation board in the spring mass simulation module through the connector, and the wheel is directly pressed on the roller in the load-bearing platform module; The sprung mass simulation module, the vertical force loading module and the profile holder are fixed to the T-slot base.

The structure of the load-bearing platform module is shown in Fig. 2, the four rollers are opened at both ends of the bearing housing bore, the deep groove ball bearings are installed, the bearing inner ring is preloaded through the shaft shoulder of the lock nut to support the roller in the middle of the frame.

Lateral force detection principle: the frame front and rear are connected with the outer connector through two transverse guide rails, to ensure that the frame and the outer connector have a lateral degree of freedom of movement, the inner connector is fixed on the frame, and the tension pressure sensor is connected between the inner and outer connectors, because the outer connector is connected to the profile bracket through a vertical guide rail before and after, ensuring that the bearing platform module only has the freedom of vertical movement, the tension pressure sensor only bears the tensile pressure generated by the lateral movement of the frame, there is no additional torque, and the wheel side declination angle does not exceed 5 [10], the lateral force on the wheel can be measured directly. The seat plate is bolted to the frame.

The vertical force loading module structure is shown in Fig. 3, using servo motor to directly drive the ball screw form, different from the test bench in the literature [11], considering to simulate the process of wheel jumping during automobile driving, the lead screw nut and its connecting parts will produce a displacement of 250mm along the direction of the lead screw axis. The ball screw is supported by a fixed end and free at the other, and the fixed end is supported by a pair of angular contact ball bearings arranged back-to-back.

The connecting plate in the vertical force loading module is connected with the seat plate in the load-bearing platform module through the transverse guide rail, and its direction is consistent with the direction of the transverse guide rail in the loadbearing platform module, so the connecting plate only has the freedom of movement in the vertical direction, because it does not limit the rotational freedom of the lead screw nut, when the lead screw rotates, under the action of friction, the lead screw nut will have a tendency to rotate, and additional torque will be generated on the two pressure sensors, affecting the measurement accuracy and life of the sensor, so

**Fig. 3** Vertical force loading module 1. Bracket, 2. Servo motors, 3. Coupling, 4. Bearing end covers, 5. Ball screw pair, 6. Connect the cylinder, 7. Pressure sensors, 8. Connect plate, 9. Guide parts

two guides are arranged, In order to eliminate the effect of additional torque on the pressure sensor.

The structure of the electric wheel module is shown in Fig. 4, which adopts the form of hub motor integrated MacPherson suspension system, and the wheel is directly driven by the hub motor; The steering of the wheels is realized by pushing the steering knuckle by the electric actuator, the stroke of the electric actuator is 150mm, the running speed is 7mm/s, and the output thrust is 2500N. In addition, different types of suspension can be replaced on the test bench by designing different connectors, and then the performance of different suspension systems can be tested.

### *2.2 The Working Principle of the Test Bench*

The working principle of the test bench is shown in Fig. 5, with black lines representing motion transfer and red lines representing drives. The working principle is outlined as follows: the hub motor directly drives the wheel to rotate; During the wheel operation, the road surface excitation is input to the servo motor, the servo motor drives the ball screw pair, the movement is transmitted to the bearing platform module, the bearing platform drives the vertical movement of the wheel, simulates the wheel jumping process, the wheel runout is transmitted to the spring-loaded

mass simulation module through the suspension system, so that the vehicle weight simulates the vertical movement of the board, simulates the vertical movement of the body, and at the same time, the pressure sensor measures the vertical force of the wheel; After the servo motor is reset, the electric actuator pushes the wheel to deflect a certain angle, simulating the steering movement of the wheel, at this time, there is a certain angle between the wheel speed direction and the roller rotation direction, the wheel is affected by the lateral force, the lateral force is transmitted to the tension pressure sensor through the roller and frame, to achieve the measurement of the lateral force of the wheel, the principle of lateral force generation is shown in Fig. 6, the roller is subjected to the positive pressure FN applied by the wheel, when the wheel turns, the lateral reaction force FY is generated on the roller, The lateral reaction force FY can be decomposed into mutually orthogonal components FYH and FYV, and when the wheel deflection angle is small, FY can be considered approximately equal to FYH, and FYH can be directly measured by the tensile pressure sensor of the carrier platform module.

### **3 Strength Check of Key Components**

### *3.1 Acquisition of Road Load Spectra*

CARSIM as one of the commonly used vehicle simulation software, can more accurately simulate the vehicle in different road conditions during the wheel load, its parametric modeling characteristics can allow users to quickly establish a more

**Fig. 6** Schematic of lateral force generation. 1. Roller, 2. Wheel

accurate vehicle model, a model of full load mass of 1410 kg, the maximum speed of 80 km/h, in order to simulate the bumpy working conditions during the car driving, set in the straight road 6 speed bumps, each speed bump interval of 10 m, through the speed bump speed bump set to 20 km/h; The radius of the curve is set to 12 m, and the speed of the cornering is set to 30 km/h. The CARSIM simulation results are shown in Table 1.


### *3.2 Strength Check of Key Components*

The bracket in the vertical force loading module as a member to bear all vertical forces, its bottom surface is fixed, the upper four bolt holes apply a total of 7500N force downward, the finite element analysis results are shown in Fig. 7, the maximum stress is 68.6 MPa, much less than the yield stress of Q235, and the strength meets the requirements.

The bearing platform module is directly in contact with the wheel and rotates with the wheel, which is the key component to bear the vertical force of the wheel, and the bearing mounting surface at both ends of the single roller is fixed constraint, and the outer surface in contact with the wheel in the middle is applied along the radial force of 3750N, the finite element analysis results are shown in Fig. 8, the maximum stress is 38.531 MPa, which is much less than the yield stress of Q235, and the strength meets the requirements.

**Fig. 7** Stress cloud plot of the bracket

**Fig. 8** Stress cloud of a roll

**Fig. 9** Stress cloud of internal connectors

The middle and outer connections of the bearing platform module mainly bear the lateral force, and the 8 bolt mounting holes on the front and rear side plates are fixed and constrained, and the lateral force sensor in the middle is connected to the bolt hole to apply a transverse force of 3000N, and the finite element analysis results are shown in Fig. 9, the maximum stress is 178.36 MPa, which is less than the yield stress of Q235, and the strength meets the requirements.

The finite element analysis results show that the strength of the main load-bearing structure meets the requirements, and the strength of the overall structure of the test bench meets the requirements.

### **4 ADAMS Dynamics Simulation Analysis**

### *4.1 Test Bench Virtual Mockup Model*

ADAMS software is a commonly used mechanical system dynamics simulation software, through the construction of the virtual prototype of the mechanical system, the mechanical system structural rationality, drive control and structural force can be analyzed, in the early stage of design optimization and upgrade, the use of ADAMS software to build a virtual prototype of the test bench, as shown in Fig. 10. The drive of the virtual prototype is divided into three parts, including the drive of the hub motor, the drive of the servo motor and the drive of the electric actuator, and the drive control is set in the form of load steps, as follows.

$$step(time,\ time1,\ data1,\ time2,\ state2) \tag{1}$$

where time is the independent variable, time1 is the start time, data1 is the increment driven at the start time, time2 is the end time, data2 is the increment driven at the end time, and the drive changes linearly in the time interval.

**Fig. 10** Test bench virtual mockup model

### *4.2 Analysis of Simulation Results*

The simulation of the vertical load on the wheels in the virtual prototype is shown in Fig. 11a, it shows that the value of the pressure sensor is about 1200N greater than the vertical load at the center of the wheel, because the pressure sensor not only bears the vertical load at the center of the wheel, but also bears the gravity of the wheel, hub motor and the bearing platform, after correcting the value of the pressure sensor, the resulting vertical load simulation is shown in Fig. 11b.

**Fig. 11** Vertical load simulation **a** Vertical load simulation, **b** Corrected vertical load simulation

**Fig. 12** lateral load simulation

The simulation of the wheel lateral load in the virtual prototype is shown in Fig. 12, and the simulation results show that the tension pressure sensor in the load-bearing platform module can accurately measure the side load at the wheel center.

### **5 Conclusion**

In this paper, aiming at the lack of research on 1/4 suspension dynamic test bench and the rapid development of distributed drive technology, this paper designs and develops an electric wheel load sensing suspension dynamic test bench, which can not only test the performance of different types of suspension systems, but also accurately measure the vertical load and lateral load on the wheel. The simulation results show that the test bench has achieved the expected effect, which can effectively solve the problem of high cost of road test for collecting road load spectrum.

**Acknowledgements** Sincere thanks to Prof. Yan Li and Prof. Xinbo Chen for their guidance on the paper. Thanks for the support of The National Natural Science Foundation of China (NO.52275123).

### **References**

1. Young JK, et al. Testing device for active-type air suspension system of commercial vehicle, has solenoid valve module that is provided to control air pressure generated in air spring, Jeonbuk Automot Technol Inst (JEON-Non-standard)


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Dynamical Simulation Analysis of Faulty Gearbox in Quay Crane Under Dynamic Load**

**Wenzong Feng, Qing Zhang, Zhuoxiang Chen, Jianqun Zhang, and Haoyu Wang** 

**Abstract** Dynamics simulations with faults can elucidate fault vibration characteristics, yet the vibrational properties of the quay crane lifting gearboxes under dynamic load excitation remain unclear. Based on multi-body dynamics theory, a multi-body dynamical model of the quay crane gearbox is established, simulating dynamic load excitation caused by cables and containers during the operation of a quay crane. The vibration responses under various working conditions and load types of different gear states are analyzed, and the corresponding fault frequency features are extracted by envelope spectrum. Simulations indicate that local gear faults enlarge the amplitude of gearbox vibrations, inducing the phenomenon of gear mesh frequency and its harmonics modulated by gear fault frequency. Based on these studies, a testbed for the quay crane gearbox is constructed. The experiment verifies the accuracy of the dynamic model and reveals that the simulation signal of load-asdynamic-load is more consistent with reality than static load. The results provide a basis for fault diagnosis of quay crane lifting gearboxes under dynamic load, and can offer simulation data support for intelligent diagnosis models lacking fault samples.

**Keywords** Mechanical transmission system · Dynamic load · Multibody dynamics · Fault diagnosis · Construction machinery

# **1 Introduction**

Gearboxes are common transmission components in mechanical systems and are widely used in quay cranes. However, since quay cranes are often in normal operation, many ports now adopt regular inspection strategies, making it difficult to obtain various fault signals. Therefore, it is necessary to model and simulate the quay crane gearbox to obtain simulated signals.

W. Feng · Q. Zhang (B) · Z. Chen · J. Zhang · H. Wang

School of Mechanical Engineering, Tongji University, Shanghai 200092, China e-mail: zhqing\_tj@126.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_15

Currently, common methods for obtaining simulation signals can be mainly divided into several categories: using finite element simulation to generate data, constructing vibration models based on phenomenological theory, setting up laboratory test rigs to obtain experimental data, and solving data based on dynamic models, etc. Jiawei Xiang [1] took mechanical transmission systems such as bearings, gear transmissions, and rotor systems as examples, constructed a complete structural finite element model, carried out model corrections, and obtained a simulation model with a certain degree of accuracy. Ming Zhang [2] and others proposed a digital twin framework, which provides a fault signal generation mechanism for rolling bearings based on phenomenological theory. Liang Guo [3] set up a laboratory test rig and used experimental data to implement unsupervised transfer learning. Yiming Xiao [4] and others proposed a bearing dynamic model and successfully applied the simulation signals to local transfer learning. Jing Liu [5] and others considered the lateral, longitudinal, and torsional vibrations of the shaft, and established a rigid shaft gear transmission dynamic system model for multi-stage transmission systems. Zhaoyang Tian [6] and others considered time-varying meshing stiffness, backlash, gyroscopic effects, and transmission error excitation, and established a finite element nodal dynamic model for the gear-bearing-Squeeze Film Damper system. Yuhang Hu [7] and others established a six-degree-of-freedom interval dynamic model for the gear system, considering the internal excitation changes caused by shaft misalignment, which can better predict the actual dynamic characteristics of the gear system compared to traditional models. Josef Koutsoupakis [8] and others used multibody dynamics software to build a dynamic model of a two-stage gearbox and used simulation signals to validate the proposed algorithm. Jianbo Yu [9] and others built a dynamic model of a planetary gear system through multibody dynamics software, and proposed a signal fusion model based on digital twins, which can accurately identify gearbox faults at the signal level.

When the aforementioned scholars model transmission systems, they often set the load as a constant torque. However, in the operation of quay cranes handling containers, the output shaft of the gearbox is connected to the wire rope, wheel slide group, and the container, resulting in a coupling phenomenon between the gearbox, wire rope, and container. If there are accelerations, decelerations, or if the container sways in the air, or even changes in stiffness during the meshing process of the gears, it will cause changes in the internal excitation of the gearbox, leading to dynamic loads. Furthermore, changes in the weight of the container itself can also cause variations in dynamic loads.

In response to this situation, this paper proposes a multibody dynamics model of the gearbox based on dynamic load simulation, used to generate simulation signals. First, the gearbox is modeled, then meshing is performed on key components, and finally, the multibody dynamics simulation software is used to build the dynamic model. Two dynamic model control schemes, dynamic load and static load, were designed, and an actual scaled-down quay crane test rig was set up. Through the dynamic modeling of normal and faulty gears and the analysis of simulation signals, it was proven that the dynamic model considering dynamic loads can better predict the actual dynamic characteristics of the gearbox system than the dynamic model with constant loads.

### **2 Common Fault Characteristics of Gears**

During the operation of a gearbox, it transfers a very large load. When gears mesh and rotate, the number of teeth involved in the meshing alternates, resulting in significant alternating changes in the meshing stiffness of the gears. When a pair of gears (with the number of teeth for Gear 1 and Gear 2 being *z*1 and *z*2, and their rotation frequencies being *f* 1 and *f* 1, respectively) engage in transmission, several frequencies will appear in the vibration signal spectrum collected from Shaft 1. One is the rotation frequency *f* 1 of Gear 1 and Shaft 1. The second is the meshing frequency of each pair of teeth, *f* m = *f* 1 × *z*1 = *f* 2 × *z*2. When Gear 1 has a localized fault (such as pitting, broken teeth, etc.), it will produce periodic impacts when meshing with Gear 2. The spectral line of the rotation frequency *f* 1 in the collected vibration signal spectrum will increase. At the same time, the gear fault frequency will also induce a gear meshing modulation effect in the vibration signal [10]. As shown in Fig. 1b, the meshing frequency *f* m and the sidebands around its harmonics are the fault frequency *f* 1. If one uses the spectrum to observe whether the bearing has a fault, it is time-consuming and labor-intensive. Envelope spectrum analysis of the vibration signal can demodulate the vibration signal. As shown in Fig. 1c, compared to the spectrum, the envelope spectrum of the faulty bearing can more accurately locate the fault frequency characteristic *f* 1 and its harmonics.

### **3 Proposed Dynamic Modeling Method**

### *3.1 Construction of Dynamic Equations*

Without considering the elastic deformation of the shaft, bearings, casing, etc., in the gearbox transmission process, the cylindrical gear system can be simplified as a pure torsional vibration system. Its vibration can be simplified as a single degree of freedom vibration equation. Using the lumped mass method to establish the dynamic model of the auxiliary gear system's torsional vibration, as shown in Fig. 2, the dynamic model of Gear 1 (driving gear) in the meshing process with Gear 2 (driven gear) is as follows:

$$[m\ddot{x} + c\dot{x} + k(t)][x - e(t)] = (M\_2 - M\_1)/r\_1 \tag{1}$$

**Fig. 2** Schematic diagram of gear meshing. In the diagram, the blue part corresponds to the constant torque in Eq. 1, while the red part corresponds to the wire rope + container section considered in Eq. 2

where *x* is the relative displacement of the gears on the meshing line; *m* is the equivalent mass, *m* = *<sup>m</sup>*1*m*<sup>2</sup> *m*1+*m*<sup>2</sup> ; *c* is the meshing damping; *k*(*t*) is the time-varying meshing stiffness; *e*(*t*) is the transmission error; *M*1 and *M*2 are the torques acting on the gears; *r*1 is the base circle radius of Gear 1.

During the operation of quay cranes, containers are transported using a wire rope + pulley system. The weight of the transported container is unknown and might be much larger than the total weight of the transmission system. The wire rope also has associated stiffness and mass properties. In the actual operation of the quay crane, there is a coupling phenomenon between the wire rope + container and the gearbox transmission system. In Eq. 1, the load on Gear 2 is a constant torque *M*2, which does not consider important influences such as the inertia of the load in the transmission system and cannot directly reflect the real response state of the gearbox.

If the influence of the rope and the weight of the container is considered, the dynamic model is as follows.

$$(m\ddot{x} + c\dot{x} + k(t)[x - e(t)] = ((m\_c g + k\_c x\_c) \times r\_2 - M\_1)/r\_1 \tag{2}$$

where *mc* represents the total mass of the wire rope and the container; *g* is the gravitational acceleration; *rc* is the drum radius where the wire rope and container are connected to the gear; *xc* is the elongation of the wire rope.

### *3.2 Construction of the Multibody Dynamics Model*

Considering the limitations of the gear dynamic equation Formula 1 in Sect. 3.1 and the difficulty in solving Eq. 2, this paper proposes a digital modeling method for the multibody dynamics of the quay crane gearbox transmission system. The modeling process is illustrated in Fig. 3.

Step 1: Based on the actual quay crane, use 3D modeling software to model the gearbox components. Perform interference checks on the model to ensure that there is no interference between components during system operation.

Step 2: Use finite element software to mesh the key components of the gearbox and produce flexible body files that can be used by multibody dynamics simulation software.

**Fig. 3** Flowchart of the modeling method


**Table 1** Gear parameters of the quay crane hoisting mechanism gearbox

Step 3: Import the models and flexible body files created in Steps 1 and 2 into the multibody dynamics simulation software. Create constraints for the imported models, solve the calculations, and obtain the required vibration signals.

#### **3.2.1 Component Modeling**

Based on the actual object, this paper has drawn a 1:4 scaled model of the quay crane gearbox hoisting mechanism. The hoisting mechanism adopts a dual-input and dualoutput structure, with a total of four shafts and three pairs of meshing helical gears. The gearbox input comes from two motors, and the output leads to two double drums, with a wire rope winding ratio of 2. Detailed parameters of the gearbox gears are shown in Table 1.

#### **3.2.2 Meshing**

To obtain vibration simulation signals from the dynamic model that are similar to the actual ones, it's necessary to perform meshing on the gearbox established in Sect. 3.2.1. After meshing the key components of the quay crane gearbox and generating the flexible body file, it's essential to set key points that contact other objects. In this paper, 23 key points were set for the flexible body file of the gearbox, including 6 fixed points with the ground, 8 fixed points of the casing with the transmission shaft, and 9 sensor fixed points. The mesh model of the gearbox and its key points are shown in Fig. 4.

**Fig. 4 a** Meshing diagram of the gearbox **b** Key points of the gearbox flexible body file; yellow points represent fixed points with the ground, red points represent connection points of the casing with the transmission shaft, and green points represent sensor fixed points

#### **3.2.3 Construction of the Dynamic Model**

When modeling the quay crane gearbox transmission system using multibody dynamics software, the gearbox components established in Sect. 3.2.1 and the flexible body files created in Sect. 3.2.2 are imported. The specific construction process is as follows.

Step 1: Modify the material properties of the imported model. Step 2: Create constraints. Establish fixed pairs for the 6 key points on the gearbox with the ground. Create two motor dummies at both ends of the input shaft and set rotational pairs to simulate the two input motors. Constrain the shaft and the gear on the shaft with fixed pairs. Also, constrain the output shaft and the drum with fixed pairs. Step 3: Define motion. Apply rotational motion to the two rotational pairs of the motor dummies created in Step 2. Step 4: Define forces. Add contact forces between each pair of driving and driven gears. Add a rotational torque to the two drums on the output shaft. Apply a sleeve force between the input shaft and the motor dummy that only restricts the rotation direction of the motor. Step 5: Define the bearing module. Use the virtual bearing module provided by the multibody dynamics software to create virtual bearings connecting the gearbox shaft and the casing. Step 6: Define the rope module. To address the issue of dynamic load simulation difficulty raised in Sect. 3.1, use the cable module provided by the multibody dynamics software to simulate the rope. Create a dummy at each end of the rope as an anchor point, with one dummy fixed and the other having a translational pair and force applied. Define the mutual conversion relationship between the rope module and the output shaft and drum, as shown in Fig. 5.

By following the above steps, the dynamic modeling of the quay crane hoisting mechanism gearbox is completed. The dynamic model is shown in Fig. 6.

**Fig. 6** Schematic diagram of the quay crane's dynamic model

# **4 Simulation Experiment**

# *4.1 Introduction to the Simulation Conditions of the Multibody Dynamics Model*

To verify the effectiveness of the modeling method proposed in this paper, the constructed dynamic model was simulated under different state categories, different loading methods, and different operating conditions. Vibration acceleration signals from key sensor points on the gearbox casing were obtained across various dimensions.

**Fig. 7** Model of the driven gear 1 on intermediate shaft 1. **a** Normal state **b** Missing tooth state

**Fig. 8 a** Static load gearbox dynamic model **b** Dynamic load dynamic model

#### **4.1.1 Types of Gear Faults**

This paper depicts different states of the driven gear 1 on the intermediate shaft 1, as shown in Fig. 7, mainly divided into normal state and missing tooth state.

#### **4.1.2 Different Load Weights**

To verify the impact between different loads as proposed in Sect. 3.1, in addition to creating a dynamic load dynamic model based on the rope, this paper also produced a dynamic model with an equivalent constant torque as a static load, as shown in Fig. 8.

#### **4.1.3 Different Operating Conditions**

During the actual operation of the quay crane, the transported container is first lifted to a certain height and then lowered onto the truck transporting the container. In the quay crane hoisting mechanism gearbox, there are two different operating conditions: forward rotation and reverse rotation. To simulate the relevant operating process of the gearbox, this paper set the weight of the transported container to 1000 kg. The motor and input shaft speeds are as shown in the Fig. 9. The gearbox first increases the input shaft speed from 0 to 900Rpm between 0.2 s and 0.3 s (with a maximum

**Fig. 9** Motor and input shaft speeds

angular velocity of -5400 degrees/sec in the figure). Between 2.6 s and 2.9 s, the speed drops from 900Rpm to -900Rpm (with a maximum angular velocity of 5400 degrees/sec in the figure). The total simulation time is 5 s, with 50,000 simulation steps.

### *4.2 Verification with the Built Test Rig*

To verify the effectiveness of the method proposed in this paper, a scaled test rig of an actual quay crane hoisting mechanism was constructed. Sensors were installed at corresponding positions, and faulty components were fabricated, as shown in Figs. 10 and 11. The sensor sampling frequency *f* s = 10 kHz.

**Fig. 10** Scaled test rig of the quay crane gearbox and sensor installation positions (numbers in the picture indicate sensor labels)

**Fig. 11** Scaled test rig of the quay crane gearbox and the faulty gear component

### *4.3 Analysis of Simulation Results*

Since the driven gear 1 of the Intermediate shaft 1 was selected as the faulty gear, the vibration acceleration signal collected from sensor channel 3 was chosen as the research subject. Under the conditions proposed in this paper, when this gear is faulty, the gear rotation frequency *f r* = 5.455 Hz and the meshing frequency *f m* = 300Hz. The simulation signal from 0.5 s to 2.5 s is taken as the container lifting phase signal, and the simulation signal from 3 to 5 s is taken as the container lowering phase signal.

By observing and comparing the simulation signals of the dynamic load dynamic model during the rising and falling phases, as shown in Figs. 12 and 13, it can be found that compared to the container lowering phase, the container lifting phase can more accurately and clearly capture the modulation phenomenon of the rotation frequency and meshing frequency in the envelope spectrum. The rotation frequency and its harmonics can be more precisely captured in the envelope spectrum.

**Fig. 12** Simulation signals of the dynamic load dynamic model **a** Container lifting phase **b** Container lowering phase

**Fig. 13 a** Simulation signal of the static load dynamic model during the rising phase **b** Signal from the scaled test rig

### **5 Conclusion**

This paper proposes a gearbox multibody dynamic model based on dynamic load simulation and has experimentally proven the effectiveness of the proposed method. It also reveals that the simulation signals with dynamic loads are more in line with actual conditions than those with static loads. Additionally, during the experimental process, it was found that the fault characteristics of the gearbox during the rising phase are more pronounced than those during the lowering phase. The above results can provide a basis for fault diagnosis of quay crane hoisting gearboxes under dynamic loads and can offer simulation data support for intelligent diagnostic models that lack fault samples.

**Acknowledgements** This research was supported by a project from the National Natural Science Foundation of China (No. 52075389).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Optimal Design of Frame Structure of Center Axle Trailer Under Heavy Load Conditions**

**Dongyu Gao, Jinyu Ma, Huisai Zheng, Menzhe Zhang, and Jingyi Zhao** 

**Abstract** In recent years, the mid axle train has gradually entered the transportation market because of its large loading capacity and flexible steering. The static simulation of full load bending, full load torsion and full load braking were carried out for the upper and lower platforms of the trailer frame, and the stress and deformation were obtained. Through the topology optimization design, the pseudo density cloud image with material removal as the optimization objective was obtained. The response surface optimization method was used to adjust the main optimization parameters of the frame platform, and the sensitivity value of its influence on the strength, stiffness and mass of the lightweight frame was analyzed. Finally, the actual value of the optimization parameters was determined. The results showed that the strength, stiffness, and stability of the lightweight frame met the requirements, and the total weight loss of the lightweight frame is 437.07 kg, and the weight loss ratio reached 14.51%. Finally, the static strength test of the lightweight frame was carried out, and the results showed that the lightweight frame met the use requirements.

**Keywords** Center axle train · Topological optimization · Lightweight · Static analysis

D. Gao

China Academy of Launch Vehicle Technology, Beijing 100076, China

J. Ma (B) · H. Zheng Beijing Institute of Precision Mechatronics and Controls, Beijing 100076, China e-mail: 493536925@qq.com

M. Zhang · J. Zhao Hebei Key Laboratory of Special Delivery Equipment, Yunshan University, Qinhuangdao 066004, China

Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, Yanshan University, Qinhuangdao 066004, China

J. Zhao

Key Laboratory of Advanced Forging and Stamping Technology and Science, Yanshan University, Qinhuangdao 066004, China

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_16

### **1 Introduction**

With the rapid development of China's automobile industry, the problem of commodity vehicle transportation has gradually become prominent [1, 2]. The midaxle car carrier is connected by the tractor and trailer through couplers, which reduces the turning radius while improving the steering flexibility of the vehicle, and can give full play to the advantages of large loading capacity and high transportation efficiency [3, 4]. As the key part of the mid-axle car carrier, the working state of the car frame directly affects the safety of transportation, so the optimization of the frame is particularly important. Zhang Kaicheng et al. [5] conducted modal analysis and typical working condition static analysis on a commercial frame, and optimized both the size and material of the frame. The results showed that while ensuring the strength and modal characteristics, the weight of the new frame was reduced by 5.6%. Hu Zhaohui et al. [6] proposed a lightweight design method for truck frame based on interval separation. Based on this method, a truck frame was optimized, and the results showed that the optimized frame weight decreased by more than 10%. Xin Yong et al. [7] used orthogonal experimental method and compromise programming method to improve the stiffness, stability and lightweight of an SUV frame, resulting in a weight reduction of 6.7 kg. Wang Xinyan et al. [8] used stress test method and finite element analysis method to determine the dangerous parts of a certain type of frame, and used MATLAB to obtain the reliability index, and realized the weight reduction of the frame. Jiang Jinxing et al. [9] adopted SIMP variable density method to carry out multi-objective optimization of 300t mining dump truck, and the results showed that the performance of the new vehicle frame was greatly improved. Wang Shuting et al. [10] established a comprehensive weight sensitivity analysis model for frame stiffness, modal frequency, and structural response to analyze the life of the frame, forming an analysis system for frame lightweight. Fan Wenjie et al. [11] adopted the compromise programming method and the average frequency method to obtain a vehicle frame multi-objective topology optimization method suitable for continuum structure. Ding Xiaolin et al. [12] obtained the random stress spectrum of vehicle frame through finite element numerical simulation and the fourth strength theory, and accurately predicted the fatigue life of the mid-axle sedan trailer. Chen Wuwei et al. [13] used kriging surrogate model and multi-objective genetic algorithm to optimize the dynamic characteristics of a pickup truck frame, and the results showed that the frame not only avoided low-order modal resonance, but also realized lightweight and stiffness optimization of the frame. Based on genetic algorithm, Wei Zhong et al. [14] proposed a multi-objective optimization method for vehicle frame through static and modal analysis, and the results showed that the frame weight was reduced by 3.43%. Guo Z Q et al. [15] carried out static and modal analysis of vehicle frame through computer-aided design and finite element platform, and obtained the optimization method of the static and dynamic characteristics of the frame.

In this paper, a mid-axle trailer frame is taken as the research object, the finite element software is used to nalyse the stress state of the frame under three typical working conditions, and the topological optimization is carried out under the goal of "removing materials", the lightweight design of the frame is carried out, and a new frame with good performance is obtained, which provides certain engineering practical significance for the lightweight design of the mid-axle train.

# **2 Frame Performance Analysis**

A mid-axle trailer frame designed and produced by Jiangsu Tianming Company is taken as the research object, as shown in Fig. 1a. Figure 1b is the specific crossbeam position of the upper platform frame, and the specific measurement position of the lower platform frame is also arranged and named from left to right, so we will not go into details here.

(a) Photos of a central axle trailer

(b) The specific beam position of the upper platform frame

The main material of the frame is Q235 steel, and the main failure form is plastic deformation. The fourth strength theory is used as the evaluation theory of the strength of the frame, and the frame is analyzed under three working conditions: full load bending, torsion, and braking.

### *2.1 Full Load Bending Condition Analysis*

Full load bending condition means that when the train is full load, the torsion bending deformation is small, and the frame is mainly vertical bending deformation. Through static analysis, the large stress on the upper platform is mainly concentrated at the junction of the front end of the vehicle frame connecting the base and the column oil cylinder, and the maximum value is as high as 50.58 MPa, as shown in Fig. 2a; the lower platform stress is mainly concentrated at the connection between the rear part of the frame and the support part of the axle, with a maximum value of 83.48 MPa, as shown in Fig. 2b. These parts of the stress are more concentrated, easy to cause damage.

The analysis suggests that the upper platform vehicle frame is finally borne by the four points at the junction of the front end of the frame connecting the base and the column oil cylinder, which causes the stress concentration here. The back half of the lower platform is in a cantilever beam state without support, so the connection between the back part of the vehicle frame and the support part of the axle has the greatest stress. The rest of the vehicle frame is well stressed and has a large redundancy.

### *2.2 Full Load Torsional Condition Analysis*

In this analysis, the full load torsional working condition of the upper platform simulated the simultaneous failure of the right front pillar oil cylinder and safety pin, and the loss of support in the right front part of the upper platform caused torsional deformation of the frame. The full load torsional working condition of the lower platform simulated the right-side collapse deformation caused by the suspension of the right wheel. Through static analysis, the stress in the side beam area connected to the left front connecting base and the right rear connecting base of the upper platform is relatively concentrated, and the maximum value is 155.31 MPa, as shown in Fig. 3a. The maximum stress at the connection between the left longitudinal beam of the lower platform and the plate spring of the central axle wheel group reached 157.3 MPa, as shown in Fig. 3b.

The analysis suggests that the right connecting base of the upper platform loses support and sinks, and the load-bearing task of the left front connecting base and the right rear connecting base increases, resulting in the stress concentration in this part. Similarly, the connection between the middle and back section of the left longitudinal

(a) Stress distribution diagram of upper platform frame

(b) Stress distribution diagram of lower platform frame

**Fig. 2** Stress distribution of frame under full load bending condition

beam responsible for support at the lower platform and the side beam bears a large load, resulting in increased stress there.

### *2.3 Full Load Braking Condition Analysis*

The full load braking condition simulates the strength and stiffness of the mid-axle car under the emergency braking condition, and this braking acceleration is calculated according to a = −0.7 g. Through static analysis, the maximum stress on the upper platform is located at the junction between the front end of the frame connecting base and the column oil cylinder, and the maximum stress is 50.98 MPa, as shown in Fig. 4a. The maximum stress of the lower platform is in the connecting part of the rear longitudinal beam of the axle and the plate spring of the wheel group, and the maximum value is 78.88 MPa, as shown in Fig. 4b.

(a) Stress distribution diagram of upper platform frame

(b) Stress distribution diagram of lower platform frame

**Fig. 3** Stress distribution diagram of frame under full load torsion condition

The analysis shows that the stress under full load braking conditions of the upper and lower platforms is like that under full load bending conditions, which is far lower than the yield limit of 235MPa, and the performance of the frame is far beyond the use requirements.

### *2.4 Frame Structural Strength Evaluation*

The safety factor n specified in material mechanics is.

$$n = \frac{\sigma\_s}{\sigma\_{\text{max}}} \tag{1}$$

In the above formula, σ*s* is the material yield limit of 235MPa and σmax is the maximum composite stress of the frame.

The calculated safety factor of the frame is shown in Table 1.

(a) Stress distribution diagram of upper platform frame

(b) Stress distribution diagram of lower platform frame

**Fig. 4** Frame stress distribution diagram under full load braking condition


**Table 1** Frame safety factor table

The maximum allowable safety factor of special vehicle materials should be above 1.5, leaving a certain redundancy, that is, when n <1.5, fatigue damage may occur to the frame. The analysis shows that there are hidden dangers in the maximum stress structure of the platform under full load torsion condition. But at the same time, the safety factor of other structural parts of the frame under the three working conditions is above 5.0, which has a large redundancy and a large lightweight design space.

### **3 Frame Optimization Design**

### *3.1 Frame Topology Optimization*

Based on the SIMP topology optimization model, "removal of 40, 50, 60% of frame materials" was adopted as the optimization objective under normal working conditions (full-load bending conditions), and the results were shown in Figs. 5 and 6.

Under the fully loaded bending condition on the upper platform, the fourth, eighth and ninth crossbeams of the frame have the most significant changes on the platform, and their redundancy is relatively large. Therefore, it is considered to remove or merge with other structures. During the process of increasing the material removal from 40 to 60%, the red area in the middle of the edge beam gradually increases, and it is considered to open a weight reduction hole in this area. In addition, the changes in the third, fourth, fifth, and eighth groups of longitudinal bars have also begun to be significant. Considering that this part mainly plays a connecting role in the actual work process and does not directly bear the load, it is considered to reduce the cross-sectional area during the optimization process to reduce weight.

Similarly, consider reducing the weight of the eleventh crossbeam in the lower platform. Consider cutting the seventh crossbeam and adjusting the eighth crossbeam to the center to consider the weight reduction effect and mechanical properties. For the four column parts, it can be considered to increase the weight reduction holes or reduce the cross-sectional area. Appropriately remove one of the first and second connecting beams, and reduce the cross-section area of the third connecting beam.

### *3.2 Frame Structure Improvement Design*

Based on the topological optimization results, the outer section size of the vertical bar of the upper platform was adjusted from 80 × 60 mm to 60 × 48 mm, a weight reduction hole was drilled in the middle of the side beam, and the fourth, seventh and tenth crossbeams were removed, and the positions of the fifth, eighth and ninth beams were adjusted to make the eighth and ninth crossbeams serve as the bearing beams. The optimized upper platform was shown in Fig. 7.

Remove the seventh and eleventh crossbeams of the lower platform and adjust the eighth crossbeam to the middle position. Based on the analysis results and practical needs, the low-density entities such as the first connecting beam at the front end of the longitudinal beam and the third connecting beam at the middle front are removed. At the same time, adjust the cross-sectional area of the column and drill weight reduction holes every 600 mm. After adjustment, the cross-sectional area of the column is shown in Fig. 8.

The above optimization greatly adjusted the structure of the upper platform. In order to obtain accurate topological optimization structure parameters, the response

(a) Removal of 40% of the material

(b) Removal of 50% of the material

(c) Removal of 60% of the material

(a) Removal of 40% of the material

(b) Removal of 50% of the material

(c) Removal of 60% of the material

**Fig. 7** The overall structure of the upper platform frame after optimization

surface optimization method was selected to adjust the 7 main optimization parameters of the upper platform to obtain the best combination of strength, stiffness, and mass. Beam spacing, thickness and the hole diameter of the side beam were selected as optimization variables, and the optimization variables of beam spacing were shown in Fig. 9.

**Fig. 9** Beam spacing optimization variables

In addition, the thickness of the entire section of the beam is selected as *x*6, and the diameter of the weight reduction hole of the side beam is selected as *x*7. The optimization objective is the best combination of strength, stiffness and mass, and the optimization variable is expressed as: *X* = *x*1, *x*2, *x*3, *x*4, *x*5, *x*6, *x*7.

The constraint conditions are: *x* <sup>0</sup> *<sup>i</sup>*≤ *xi* ≤ *x* <sup>1</sup> *i* .

The objective function is: ⎧ ⎨ ⎩ max *K* = *f* (*X*) max *S* = *f* (*X* ) min *M* = *f* (*X*)

In the above formula: *xi*—Optimize the parameters of the variable; *x* <sup>0</sup> *<sup>i</sup>x* <sup>1</sup> *<sup>i</sup>x* <sup>0</sup> *<sup>i</sup>x* <sup>1</sup> *<sup>i</sup>*— Value range of *xi xi* , *K*—Stiffness; *S*—Intensity; *M*—Quality.

The Custom method was used to screen 80 sample points for normalization processing, and the response surface model of the neural network was established. The fitting degree between the predicted value and the actual value was shown in Fig. 10 below.

After the response surface is generated, the correlation sensitivity between input and output parameters is constructed. The closer the sensitivity value is to 1, the stronger the positive correlation between input parameters and corresponding output parameters, and the closer the sensitivity is to −1, the greater the negative correlation between the two, and the closer the sensitivity is to 0, the weaker the correlation between the two. The sensitivity histogram is shown in Fig. 11. It can be seen from the analysis in Fig. 11 that x2 has the most significant influence on the strength. x3

**Fig. 10** Fit degree between predicted value and actual value of response surface model

and x5 have the most significant influence on stiffness. The x6 has the most significant impact on quality.

The multi-objective genetic algorithm was selected, and the initial population number was set to 100, the sample number of each genetic iteration was set to 100, the maximum genetic algebra was set to 20, the maximum Pareto percentage was 70%, and the maximum number of iterations was set to 20. The initial optimization values and results on the platform were shown in Table 2.

After optimization, the unloading frame is shown in Fig. 12.

**Fig. 11** Parameter sensitivity histogram


**Table 2** Parameter optimization initial values and results

Set the same boundary conditions as the original frame, and analyze full-load bending conditions, full-load torsion conditions and full-load braking conditions. The results are shown in Tables 3 and 4 below.

As can be seen from Tables 3 and 4, the overall weight loss of the upper platform frame after optimization is 197.87 kg, which is about 16.6% of the original frame. After the optimization of the lower platform frame, the weight is reduced by 239.20 kg, about 13.1% of the original frame. At the same time, in addition to the maximum braking stress of the lower frame, the optimized stress is far below the yield limit, and the error of the maximum stress calculated by the simulation of the new frame under the remaining three working conditions is less than 3%, and the


**Table 3** Comparison of the corresponding data of the upper platform frame before and after optimization

**Table 4** Comparison of the corresponding data of the lower platform frame before and after optimization


strength and stiffness performance of the optimized frame are not decreased, while the weight is greatly reduced.

### **4 Frame Static Test Verification**

Due to the large size of the frame, it is impossible to test all positions, so the stress concentration point and the load point are selected as the test points. The test points are shown in Figs. 13 and 14 below.

The strain gauge is arranged in a right-angle strain flower scheme, as shown in Fig. 15.

The stress calculation results of strain gauge with three angles of strain flower are σ0, σ45 and σ90, respectively. The maximum, minimum and direction Angle of the principal stress are calculated according to the following formula:

$$
\sigma\_{\text{max}} = \frac{1}{2} \left[ \frac{\sigma\_0 + \sigma\_{90}}{1 - \mu} + \frac{\sqrt{2}}{1 + \mu} \sqrt{\left(\sigma\_0 - \sigma\_{45}\right)^2 + \left(\sigma\_{45} - \sigma\_{90}\right)^2} \right] \tag{2}
$$

$$\sigma\_{\rm min} = \frac{1}{2} \left[ \frac{\sigma\_0 + \sigma\_{\rm 0}}{1 - \mu} - \frac{\sqrt{2}}{1 + \mu} \sqrt{(\sigma\_0 - \sigma\_{45})^2 + (\sigma\_{45} - \sigma\_{90})^2} \right] \tag{3}$$

$$\alpha = \frac{1}{2} t g^{-1} \frac{2\_{\sigma\_{\theta\\$}} - (\sigma\_0 + \sigma\_{\theta 0})}{\sigma\_0 - \sigma\_{\theta 0}} \tag{4}$$

In the above formula.

σmax——Maximum principal stress, MPa;

σmin——Minimum principal stress, MPa;

μ——Poisson's ratio, 0.3;

α——Principal stress direction Angle (relative to 0 °).

**Fig. 13** Schematic diagram of the strain measurement position of the upper platform frame

**Fig. 14** Schematic diagram of the strain measurement part of the lower platform frame

**Fig. 15** Right angle rosettes

Strain flowers are evaluated by equivalent stress, as shown in formula 5.

$$\sigma\_{\sigma} = \sqrt{\frac{\sigma\_{0} + \sigma\_{00}}{2(1 - \mu)} \Big)^{2} + 3 \left(\frac{1}{2(1 + \mu)}\right)^{2} \left[ (\sigma\_{0} - \sigma\_{\emptyset 0})^{2} + (2\sigma\_{4\S} - (\sigma\_{0} + \sigma\_{\emptyset 0}))^{2} \right]} \tag{5}$$

In the above formula:

σ*e*——Equivalent stress, MPa;

After the arrangement of strain gauges is completed, part of the position is shown in Fig. 16.

After the preparation of the test, the static strength test under bending and torsion conditions was carried out. The test was tested three times under each working condition, and the average value was calculated as the result. The test stress values obtained at the 12 test points tested by the upper platform frame under the two working conditions were shown in Tables 5 and 6. The test stress values obtained at 20 test points tested by the lower platform frame under two working conditions are shown in Tables 7 and 8.

**Fig. 16** Strain gauge after fixation






**Table 7** Stress value of lower platform under full load bending condition (unit: MPa)

**Table 8** Stress value of lower platform under full load torque condition (unit: MPa)


The elastic modulus of Q235A steel is E = 2.06×105 MPa, Poisson's ratio is 0.3, and the yield strength is 235 MPa. When the safety factor n is 1.5, [δ] can be obtained by calculation as 157 MPa.Therefore, the stress of the above measuring points is less than the maximum allowable safety stress, so the structure of the frame meets the requirements. However, the test result value is smaller than that of the simulation, because the stress peak under the simulation condition is inside the solid structure, while the test is on the surface of the component, so certain errors will be generated.

### **5 Conclusion**

The results show that the stress value of the frame under torsion condition is much higher than the other two conditions, but the value is much lower than the material properties, and it has a large space for lightweight design. Through topology optimization, the redundant structure and unreasonable layout of the frame were evaluated and optimized. Simulation and test results showed that the lightweight trailer frame lost 437 kg, accounting for 15.7% of the original weight, while maintaining the original strength and stiffness performance. The weight reduction effect was remarkable, and the lightweight goal was achieved.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation and Analysis of the Internal Flow Field of Mining Solenoid Pilot Valve Based on Fluent**

**Zhaoqiang Wang, Lijing Zhu, Lei Guo, Yangjun Lu, Peixing Li, and Chenhun Lu** 

**Abstract** During the commutation process, the electromagnetic pilot valve experiences turbulent kinetic energy dissipation, leading to unstable internal flow and generating noise in the valve. Based on computational fluid dynamics (CFD), the phenomena of impact and energy consumption in a new type of high-pressure and high-speed electromagnetic pilot valve were investigated. Then flow field simulations were conducted under various differential pressure and valve opening conditions, in order to obtain distribution maps of turbulent kinetic energy, pressure, and velocity. Finally, the simulation results were compared, and the results showed that increasing the outlet pressure and enlarging the valve port opening have a positive impact on mitigating the structural issues in the pilot valve.

**Keywords** Solenoid pilot valve · Ball valve · Fluent · Flow field simulation

# **1 Introduction**

Electromagnetic pilot valve is one of the key components of mining electro-hydraulic control system. It is commonly used in hydraulic and pneumatic systems to control the internal fluid flow through electromagnetic force, thereby achieving pressure and flow control of the main valve. The performance of electromagnetic pilot valves directly affects the safety and reliability of hydraulic mining machinery. However, the internal structure of pilot valves is complicated, and complex flow paths such as sudden expansion, contraction and bending can occur, which in turn generate phenomena such as vortex, reflux, de-wallization and re-attachment to the wall [1]. Improper design can lead to excessive energy loss, cavitation, and thermal deformation, thereby reducing the service life and performance of pilot valve components and affecting the operation of the entire system. Therefore, it is necessary to study the working process of electromagnetic pilot valves for improvement and optimization.

Z. Wang · L. Zhu (B) · L. Guo · Y. Lu · P. Li · C. Lu

Shanghai University of Engineering Science, Shanghai 201620, China e-mail: zhulijing126@126.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_17

Choi et al. [2] were numerically investigated the flow characteristics of butterfly valves with different sizes under different valve opening percentages. They determined that increasing turbulence effect could cause many discrepancies between turbulence models, especially in areas with large pressure drop and velocity increase. In addition, sensitivity analysis of flow properties was conducted to determine the effect of constants used in each turbulence model. Huovinen et al. [3] investigated a flow past a choke valve by experimental and numerical means. The flow profile after a choke valve with high Reynolds number of approximately 1,000,000 was measured using a LDV and computed using RANS simulations. Gabel et al. [4] studied constitutes a coupled computational and experimental methodology to analyse the complex flow within a choke valve under laminar inflow conditions. The studies reveal that, at low valve openings, when only the small ports are engaged, there forms a four-lobed vortical structure which is established upon collision of the incoming jets. The vortical structure of the flow is much more complex at higher valve openings when both the small and large ports are engaged. Xavier and Ortiz [5] investigated cavitation characteristics in Hollow-Jet valves and possible solutions. Three-dimensional numerical simulations—CFD (Computational Fluid Dynamics) were carried out in an unsteady state, considering homogenous multiphase flow, to identify the phenomenon in these components. The results confirmed the occurrence of cavities with a mixture of vapor and liquid at the valve tip. The cavities are followed by a vortex generation near the same region.

In summary, domestic and foreign scholars attach great importance to the study of the internal fluid characteristics of hydraulic valves. Energy consumption, cavitation and other problems need to be solved in-depth understanding of its internal flow characteristics. Therefore, the research in this paper provides an important support for the optimisation of the structure of this new foreign electromagnetic pilot valve in order to obtain better flow control performance and to achieve the prediction of the liquid flow characteristics required by the system. This is to improve the existing products and develop more new electromagnetic pilot valves in line with China's needs has an important role in promoting.

### **2 Principle of Pilot Valve Structure**

The solenoid pilot valve is to effectively convert the pulse electrical signal from the electronic control system into a hydraulic signal to drive the corresponding main control valve spool in the main control unit, so as to realize the precise control of the actuator [6].

The structure schematic diagram of the electromagnetic pilot valve studied in this paper is shown in Fig. 1. Under the action of the return spring, the left ball valve remains normally closed and the right ball valve remains normally open, with an opening of 0.1 mm. After the solenoid valve is powered on, electromagnetic force will be generated. The electromagnetic force pushes lever 8, driving the top rod 7 to move to the left, thereby pushing the ceramic ball valve core 6 to move 0.1 mm

to the left. At the same time, the right valve port is blocked and the left valve port opens. Port P and Port A are connected, and high-pressure hydraulic oil enters Port A through a flow channel. When the electromagnetic signal disappears, under the action of the reset spring, the left ceramic ball valve core is pushed to move to the right. The left valve port is blocked, and the opening of the right valve port is 0.1 mm. At this time, hydraulic oil flows into port T from port A, that is, hydraulic oil flows to the oil tank to achieve unloading.

### **3 Mathematical Model for Flow Field Simulation**

This paper quantitatively analyses the characteristics of the flow field and the distribution of flow parameters in the solenoid pilot valve through numerical simulation to provide theoretical support for design optimization.

The flow field simulation analysis in this paper is based on the Realizable K–ε model in the Fluent model library. This model mainly aims to modify the normal stress of the standard model under conditions of extremely high strain rates, in order to avoid the generation of negative stresses and make the computed results reasonable. The expression is as follows: where *C*1 = max- 0.43, <sup>η</sup> η+5 ; <sup>η</sup> <sup>=</sup>*Sk*  <sup>ε</sup> ; the involved constant parameters are: C1<sup>ε</sup> = 1.44; C2 = 1.9; σk = 1.0; σε = 1.2 [7, 11].

$$\rho \frac{DK}{Dt} = \rho \left(\frac{\partial K}{\partial t} + \overline{\mu}\_j \frac{\partial K}{\partial \mathbf{x}\_j}\right) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left(\mu + \frac{\mu\_t}{\sigma\_\varepsilon}\right) \frac{\partial K}{\partial \mathbf{x}\_j} \right] + G\_k + G\_b - \rho \varepsilon - Y\_M \tag{1}$$

$$D\varepsilon \qquad (\partial \varepsilon \qquad \partial \varepsilon \qquad \text{i.e.}) \qquad \text{j.e.} \quad \Box \quad (\varepsilon \qquad \mu\_\varepsilon) \text{ (see } \mathbb{T})$$

$$\rho \frac{D\varepsilon}{Dt} = \rho \left(\frac{\partial \varepsilon}{\partial t} + \overline{\mu}\_j \frac{\partial \varepsilon}{\partial x\_j}\right) = \frac{\partial}{\partial x\_j} \left[ \left(\mu + \frac{\mu\_t}{\sigma\_\varepsilon}\right) \frac{\partial \varepsilon}{\partial x\_j} \right]$$

$$+\rho C\_1 S \varepsilon - \rho C\_2 \frac{\varepsilon^2}{k + (v\varepsilon)^{1/2}} + C\_{1\varepsilon} \frac{\varepsilon}{k} G\_{3\varepsilon} G\_b \tag{2}$$

Compared to the standard K–ε model, this model no longer utilizes the original turbulence viscosity formula. Instead, it derives a new viscosity formula by considering the rotational flow and jet curvature generated during fluid motion, achieving a closer approximation to the actual flow state of the fluid and more accurate computation results [12].

### **4 Model Preprocessing**

**Fig. 2** Flow field domain model

### *4.1 Flow Domain and Grid Partitioning*

As the solenoid pilot valve possesses a symmetrical structure, the fluid flow within the valve cavity also exhibits symmetry. Leveraging the principle of symmetry, it is possible to simulate only half of the structure, thereby enhancing the computational speed of numerical calculations and conserving computational time and memory [8]. Fig. 2 illustrates the utilization of SpaceClaim, a pre-processing software within the Workbench, to generate the Fluent flow field domain model. Subsequently, the mesh generation is performed using the confined geometry workflow provided by the software, followed by transferring it to Fluent for the solution setup. During the meshing process in Fig. 3, the local mesh encryption is executed due to the significant variation in pressure and velocity gradient at the inlet and outlet of the valve port, as well as the presence of a complex vortex field. Additionally, a relatively larger mesh is employed in regions where the gradient is minimal, thereby reducing the computation time [9, 10].

**Fig. 3** Mesh division

### *4.2 Simulation Boundary Condition Setting*

The pressure at port A of the pilot valve needs to be calculated first based on the force required to move the main valve. The main valve incorporates a spring with a stiffness of k = 6.48 N/mm, which is compressed from 46.50 mm to 31.91 mm. The effective area on the main valve is A = 314.16 mm2, while the area at the P port is A0 = 144.60 mm2. Frictional forces are neglected in the analysis. Using Eq. (3), the pressure for actuating the main valve is derived as P = 14.8 MPa.

$$P = \frac{F}{A} = \frac{k\Delta x + P\_0 \times A\_0}{A} \tag{3}$$

The boundary conditions for the fully open valve are defined as follows: the fluid model is selected as Realizable K–ε turbulence model, while the wall adjacent to the fluid is considered stationary. The fluid medium, in this case, is an emulsion with a density of 997.7 kg/m3 and a viscosity of 0.00149 kg/(m·s). The inlet pressure is set to 31.5 MPa. The outlet pressure is varied as 0 MPa, 5 MPa, 10 MPa, and 15 MPa, with a valve opening of 0.1 mm. The right valve port is blocked.

For the partially open valve, the following boundary conditions are set: the fluid model is again selected as Realizable K–ε turbulence model, while the wall adjacent to the fluid is considered stationary. The fluid medium remains the same as the fully open valve scenario. The inlet pressure at the P port is maintained at 31.5 MPa, while the outlet pressures at the A and T ports are set to 14.8 MPa and 0 MPa, respectively. The left valve port opening sizes are set to 0.08 mm, 0.06 mm, and 0.04 mm.

### **5 Results and Discussion**

Turbulent flow is a physical quantity that reflects the amplitude of fluid velocity fluctuations. The greater the kinetic energy of turbulence, the more intense the velocity fluctuations generated, and the greater the energy loss generated. Therefore, the reason for energy dissipation inside the valve can be evaluated based on the magnitude and distribution of turbulent kinetic energy.

# *5.1 Effect of Different Pressure Differences on Turbulent Kinetic Energy Dissipation*

From the turbulent kinetic energy distribution plots in Fig. 4(a)–(d), it can be observed that significant energy dissipation occurs at the valve orifice. This is due to the rapid increase in flow velocity and turbulence intensity at the contracted area of the orifice, leading to an increase in turbulent kinetic energy. Significant energy dissipation also occurs at the corners of the flow passage inside the valve chamber. This is caused by the change in flow streamline and the generation of vortices, which increase the turbulent kinetic energy. However, under a constant inlet pressure and varying outlet pressures of 0 MPa, 5 MPa, 10 MPa, and 15 MPa, the maximum turbulent kinetic energy reaches 3349 m2/s2 at 0 MPa, and it decreases as the pressure difference decreases, indicating a reduction in turbulence intensity and energy dissipation.

# *5.2 Effect of Different Ball Valve Openings on Turbulent Energy Dissipation*

From the distribution of turbulent kinetic energy in Fig. 5(a)–(c), it can be concluded that significant energy dissipation occurs due to the contraction of the valve ports on both sides, leading to an increase in turbulent kinetic energy. Energy dissipation also occurs at the flow passage corners in the middle section of the valve chamber. Under constant pressure differential conditions, the valve aperture was set at 0.04 mm, 0.06 mm, and 0.08 mm. The turbulence kinetic energy reached a minimum of 2470 m2/s2 at 0.06 mm aperture. As the aperture gradually increased, the turbulence intensity of the pilot valve weakened and then strengthened, resulting in a decrease in turbulence kinetic energy, which in turn led to reduced energy dissipation.

**Fig. 4** Turbulent kinetic energy distribution for different differential pressure conditions. (**a**) 0 MPa. (**b**) 5 MPa. (**c**)10 MPa. (**d**) 15 MPa

**Fig. 5** Distribution of turbulent kinetic energy for different openings. (**a**) 0.08 mm. (**b**) 0.06 mm. (**c**) 0.04 mm

# **6 Conclusion**

In this study, the internal flow field of the electromagnetic pilot valve during operation was simulated and analyzed using Fluent. The distribution map of turbulence kinetic energy and the internal flow state of the pilot valve were obtained. By analyzing and comparing different pressure differential conditions and valve aperture sizes, the variation patterns of turbulence kinetic energy and vortex intensity were determined. The results are as follows:


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on EMC Simulation of Electric Drive System of Electric Engineering Machinery**

**Qiaohong Ming, Yangyang Wang, and Meiyu Zong** 

**Abstract** In order to suppress the electromagnetic interference in electric construction machinery, improve the stability and safety of the vehicle on the influence factors of electric construction machinery EMC system analysis, electric drive system due to the internal power electronic equipment for a long time in the high voltage, high current conditions and become the main influence factors of electric construction machinery EMC. The main methods to reduce electromagnetic interference of electric drive system grounding, shielding and filtering are expounded, which leads to the emulated research of EMC simulation of electric drive system of electric engineering machinery. The equivalent circuit model of battery, electronic control, motor and test system is established, and the low-pass filter composed of inductor and capacitor is designed. And the combined three electric system, test system, filter circuit composed of the model of simulation analysis, with or without filter input current, different frequencies of input and output voltage signals compared, the results show that adding filter can effectively improve the conducted interference, the use of RLC filter composed of four RLC components, can effectively improve the signal low frequency bias and high frequency distortion.

**Keywords** Electric construction machinery · Electric drive system · EMC · Filter · Simulation

# **1 Overview**

Electric construction machinery has achieved intelligent control through high electrification, but how to achieve electromagnetic compatibility (EMC) with so many electronic components becomes a hot topic. EMC of electric construction machinery

Q. Ming · M. Zong Sany Heavy Machinery Co., Ltd, Kunshan 215300, China

© The Author(s) 2024

Q. Ming (B) · Y. Wang

School of Automotive Engineering, Tongji University, Shanghai 201800, China e-mail: 1811124@tongji.edu.cn; mingqh.2007@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_18

is a hot research topic at present. The EMC performance of electric vehicles in the world has a strict standard system, such as GB/T 18,387-2017, GB/T 14,023-2011, ECE R10.05 and so on [1].

EMC refers to the ability of electrical equipment and systems to work acceptably in their electromagnetic environment [2]. EMC mainly includes the following two aspects: 1. Electromagnetic compatibility; The electromagnetic interference (EMI) generated by the equipment during normal operation cannot exceed a certain limit value. 2. The equipment has a certain immunity to electromagnetic interference in the environment, that is, electromagnetic Susceptibility (EMS) [3]. Electromagnetic interference that causes performance degradation or failure of equipment must have three elements at the same time: 1. Interference; 2, sensitive equipment; 3, coupling path [4].

For the electric engineering machinery three electric system, because the battery pack only outputs direct current, so it can be excluded from the interference source [5], the electric drive system (electronic control and motor) contains a large number of high-frequency switches and other electronic components, is the main source of EMI, in the switching process of these semiconductor devices, The rate of change of current and voltage is very large (can be as high as 10 kV/μs or 10 kA/μs speed switch, the frequency can be as high as several hundred kHz), through the circuit or component of the parasitic inductance or parasitic capacitance caused by conduction and radiation electromagnetic interference [6]. According to the three elements of electromagnetic interference, methods to reduce electromagnetic interference generally include grounding, shielding and filtering [7].

### *1.1 Shielding*

For high-voltage equipment, the method of shielding and grounding is generally used. Aluminum foil can be covered in places where there may be electromagnetic leaks, such as cables, to shield interference [8]. Figure 1 shows a dedicated cable with a shielded connector [9].

Communication cables generally use shielded twisted pair, the twisted pair mode can make the induced magnetic field cancel each other, and there is a metal shield

**Fig. 1** Cable with shielded connector

between the twisted pair and the outer insulating sheath, which can reduce radiation. Figure 2 shows Communication cables use shielded twisted pair.

### *1.2 Filtering*

In 2021, Tan et al. [10, 11] proposed a simulation model of multi-in-one electric drive system that integrates conducted emission risk prediction and interference suppression, as shown in Fig. 3. Based on this model, they studied the effect of filter circuit Y capacitive parasitic inductance on the system.

In this study, as shown in Fig. 4 π-type filters are used, where Cy1 is the front-end capacitor near the power supply and Cy2 is the back-end capacitor near the IGBT.

In order to determine the effect of the parasitic inductance of the filter Y-type capacitor in the model on the conducted interference voltage, the parasitic inductance

**Fig. 3** Simulation model of electromagnetic interference in electric drive system

**Fig. 4** Equivalent circuit of π-type filter

**Fig. 5** Simulation results of different parasitic inductors

is set to 5 nH, 50 nH and 500 nH, respectively. The simulation results are as follows Fig. 5.

It can be concluded from the simulation results that the larger the parasitic inductance, the smaller the conducted noise voltage generated in the high frequency band (5–30 MHz).

### **2 Three-Power System Modeling**

The three-power system simulation model based on LTspice was established, and the effect of the filter was simulated.

**Fig. 6** Simulation model of battery system

### *2.1 Battery System Simulation Modeling*

As shown in Fig. 6, the battery system is simplified into a high voltage source (HV voltage source) V2 and its internal resistance Rbat. The HV voltage source can continuously output 355V DC, and the corresponding R is bat0.15 Ω.

### *2.2 MCU Simulation Modeling*

The role of MCU is to convert the DC current of the battery into the AC power supply of the motor. Therefore, the MCU can be thought of as an inverter consisting of six IGBTs, as shown in Fig. 7. The MCU controls the input voltage and current by controlling the on–off of the switching tube, thus controlling the speed and direction of the motor; IGBT parameters are set as follows: internal inductance = 4 nH, internal capacitance = 500 pF, internal resistance = 1 mΩ; Since the motor controller gets the RMS or peak pulse current from the battery pack, it will generate a high pulse voltage on the DC bracket, which the motor controller can not bear, so it is necessary to select a capacitor (C7) to connect the battery and MCU, and the capacitor of C7 is set to 650 uF.

### *2.3 Motor Simulation Modeling*

Ac after MCU inverter, before supplying power to the motor, it is necessary to reduce the terminal voltage of the motor through the inductor (L1, L2, L3), so as to reduce the current flowing through the motor; Capacitors (C1, C2, C3) are the starting capacitors of the motor, which can make the current of the secondary winding 90 degrees ahead

**Fig. 7** MCU simulation model

of the current phase Angle of the main winding, thus generating magnetic field torque and starting the rotor; The inductor in the motor (L4, L5, L6) is affected by the rotating magnetic field, causing the current flow lag, generating torque and generating power; The internal resistance of the motor is R1, R2, R3; The parameters of the motor are as follows: inductance of L1, L2, L3 = 200 uH, inductance of L4, L5, L6 = 10 nH, capacitance of C1, C2, C3 = 1 nF, resistance of R1, R2, R3 = 1 mΩ.

The simulation model of Motor is shown in Fig. 8.

### *2.4 LISN Simulation Modeling*

Line Impedance stabilization network (LISN) is an important auxiliary equipment for EMC test of power system. When the equipment is in normal operation, it is directly powered by the battery. Therefore, it is possible to test the EMC performance of the device by connecting the test instrument, however, if EMI is detected during this test, it is impossible to distinguish whether it is the cause of the device under test (DUT) itself, or the influence of the environment or the battery. As shown in Fig. 9 EMC measurement method.

As shown in Fig. 10 LISN test topology. LISN can isolate conducted emissions outside the DUT and only measure conducted emissions from the DUT. At the same time, it can provide a stable impedance to the product's power cord over the entire frequency range of the conducted emission measurement.

Figure 11 shows the internal structure of LISN. The battery and DUT are connected in series on either side of the LISN. In the LISN, the functions and parameters of the electronic components are set as follows:

Capacitor (C1) 1 uF, inductor (L1) 5 uH, used to filter grid-side interference to provide consumers with maximum undisturbed power;

Resistors (Rmess, 50 Ω) are located inside the receiver and their role is to convert the noise current into noise voltage;

**Fig. 10** LISN test topology

**Fig. 11** Internal structure of LISN

The resistor R31 kΩ is used for 0.1 uF capacitor discharge; Capacitor (C2) 0.1 uF series capacitor, Rmess and R3 parallel branches, used to block DC and prevent DC components from damaging the receiver. Figure 12 shows LISN symbol in LTspice.

**Fig. 12** LISN symbol in LTspice


### *2.5 Three-Power System Simulation Model*

To sum up, the three-power system simulation model is shown in Fig. 13.

### **3 Design of Filter**

In EMC design, filter is an important method. The filter usually refers to a lowpass filter composed of inductors and capacitors, and the effectiveness of the filter is not only related to the structure, but also to the impedance [12] of the connected network. Capacitors are commonly used for decoupling, filtering, bypass, and voltage regulation of power busbars. Inductors can increase the impedance of the loop to reduce the interference current in the loop, so as to achieve the purpose of suppressing interference.

The structural design of the filter complies with the "maximum mismatch principle", that is, in any filter, there is a high impedance at both ends of the capacitor and a low impedance at both ends of the inductor. In order to suppress differential mode interference, a filter capacitor is connected between the two power lines, which has a low impedance to high-frequency interference, so the high-frequency interference between the two power lines can pass through it, and has a high impedance to the power frequency signal, so there is no effect [13] on the transmission of the power frequency signal.

**Fig. 13** Three-power system simulation model

### *3.1 The Self-Resonant Frequency of the Decoupled Capacitor*

Actual capacitors all have parasitic inductors. The size of the lead basically depends on the length of the lead, for the round wire type of lead, the typical value is 10 nH/ cm [14] the typical ceramic capacitor's lead is about 6 mm, will introduce about 15 nH of inductance.

The inductance of the lead is estimated as follows [15]:

$$L\_s = \frac{\mu\_0}{2\pi} l \left\{ \left[ \frac{l}{r} + \sqrt{\left(\frac{l}{r}\right)^2 + 1} + \frac{r}{l} - \sqrt{\left(\frac{r}{l}\right)^2 + 1} \right] \right\} \tag{1}$$

In this equation,

*l* Represents the length of the cable,

*r* Represents the radius of the cable.

The parasitic inductors and capacitors will create a series resonance, or selfresonance [16]. Below the self-resonant frequency, the capacitor remains capacitive; Above the self-resonant frequency, the capacitor is inductive, and the impedance becomes larger as the frequency increases, so that the decoupling or bypass function of the capacitor is greatly reduced, therefore, at the self-resonant frequency, the impedance presented by the decoupling capacitor is the smallest, and the decoupling effect is the best. The self-resonant frequency of the decoupling capacitor *f*0 should be selected according to the highest frequency of noise *f*max, and the best value

$$f\_{\text{max}} = f\_0 \tag{2}$$

Commonly used power filter, when the electrical sensing resistance rL, the selfresonant frequency are respectively:

$$f\_0 = \frac{1}{2\pi\sqrt{LC}}\tag{3}$$

### *3.2 Choice of Capacitance Capacity*

The capacity of the decoupling capacitor is usually estimated as follows:

$$C = \frac{\Delta I}{\Delta V / \Delta t} \tag{4}$$

In this equation,

∆*I* Is a transient current,

∆*V* Is the allowable change in the power supply voltage of the logic device,

∆*t* Is the switching time.

The choice of decoupling capacitor is based on:


$$
\Delta V\_0 = \frac{L\Delta I}{\Delta t} \le V\_{NI} \tag{5}
$$

(3) From the perspective of decoupling capacitors to provide the required current for the chip, its capacity should meet:

$$C \ge \Delta I \Delta t / \Delta V \tag{6}$$

(4) The discharge speed of the chip switching current *ic* must be less than the maximum discharge speed of the decoupling capacitor current:

$$\frac{di\_c}{dt} \le \frac{\Delta V}{L} \tag{7}$$

In addition, when the power lead is relatively long, the transient current will cause a large voltage drop, at this time, it is necessary to add a holding capacitor to maintain the voltage required by the device.

Figure 14 shows the RLC filter used in this paper. In order to verify the effect of the filter, two sets of parameters are set for the filter in this paper. The first set of parameters is inductance C1 of 390 nF, capacitance C2 of 1.5 uF, inductance L1 of 4 uH and resistance of 0.17 Ω in Fig. 14. Inductance L1 in parameter group 2 is changed to 8 uH, and other parameters remain unchanged with parameter group 1.

Figure 15 shows a tri-electric system with an RLC filter.

### **4 Simulation and Conclusion**

First of all, it is necessary to measure the input current of the motor to confirm whether the input of the entire three-power system simulation model is correct, and then measure the output voltage and input signal of the LISN in the loop, and compare the waveform to analyze the influence of electromagnetic interference on the signal in the three-power system.

The simulation results of no filter are as follows.

As shown in Fig. 16, the DC output of the battery is converted into AC through the motor controller, which conforms to the working principle of the system.

Figure 17 shows the input and output signals of LISN at different frequencies. It can be seen that when the frequency is low, the offset between the input voltage and the output signal is small, and the same waveform is basically maintained; When

**Fig. 15** Three-electric system with RLC filter2

the frequency is greater than 8 kHz, the waveform of the input signal begins to have a large deviation; When the frequency is greater than 100 kHz, the input signal is completely distorted.

Add filter parameter: C(390 nF)-L(4 uH)-R(0.17 Ω), simulation results are as follows:

**Fig. 16** Input current of the motor without filter3

**Fig. 17** Input and output voltage signals of LISN at different frequencies4

As shown in Fig. 18, after filtering, the distortion degree of the input signal is improved. When the frequency is greater than 500 kHz, the input signal is offset by about 15 dB, but the waveform is basically the same as the output signal.

Add filter parameter: C(390 nF)-L(8 uH)-R(0.17 Ω), simulation results are as follows:

It can be seen from Fig. 19 that the waveforms of the input signal and the output signal are basically the same except for a few peak signals.

To sum up, adding filters can effectively improve conducted interference. This paper uses an RLC filter composed of four RLC components, which can effectively improve the low frequency bias and high frequency distortion of the signal.

Electromagnetic interference EMI research methods for motor drive systems and vehicle chargers can be classified as follows from the perspective of suppression technology:

**Fig. 18** LISN input and output voltage signals at different frequencies5

**Fig. 19** LISN input and output voltage signals at different frequencies6

➀ Add EMI filter device.

By adding an EMI filter containing capacitors, inductors and other energy storage components between the interference source and the output end, the conducted interference current is suppressed or attenuated, and the propagation path of electromagnetic interference is blocked.

➁ Improve the PWM control strategy or change the internal winding connection mode.

Common-mode voltage suppression from the perspective of optimal control strategy, to reduce common-mode interference from the source, has become the focus of optimization control research. The main method is to reduce the amplitude or pulse number of the common mode voltage by changing the switching sequence of the DC/AC inverter or the operation law of the switching device.

➂ Active cancellation suppression method.

Active cancellation means to reduce or eliminate interference by superimposing or injecting reverse common mode current or voltage to the part that needs to be compensated in the power converter. The cost of this suppression method is small, the space volume is relatively small, and the suppression effect can be obtained.

The EMC compatibility design for the integrated controller of the electric excavator can refer to the EMI suppression technology of the electric vehicle drive system and charging system mentioned above, that is, the internal suppression method is adopted: the corresponding filter or common mode choke is designed according to the resonance point inside the controller to adjust the resonance frequency; Or external suppression method: Design effective active or passive EMI filters.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design of Expandable Electric Vehicle Drive Demonstration Teaching Platform**

**Zifeng Zhao, Zele Chen, Shaojia Huang, Jianguo Fan, and Lijie Gao** 

**Abstract** In recent years, the new energy vehicle industry has been vigorously developed, and the demand for new energy vehicle practitioners and their knowledge of electric vehicles has been continuously increasing. In the undergraduate education of vehicle engineering, there is an urgent need for an electric drive demonstration teaching platform. This design aims to solve the teaching demonstration function of multiple configurations of electric vehicle drive systems on the same platform. The electric vehicle drive demonstration teaching platform is designed as an expandable way, which can meet the requirements of various combinations of different motor forms, battery types and capacities, control methods, etc. for platform loading and demonstration, achieving diversification of the teaching platform. It can also conduct experiments on various types of new energy vehicle configurations through connecting racks The expansion of the demonstration teaching platform meets the needs of teaching demonstrations and experiments related to electric vehicle drive systems.

**Keywords** Electric vehicle · Electric drive system · Expandable demonstration platform

# **1 Introduction**

The development of new energy vehicles has now become a national strategy, which is of great significance in alleviating environmental and energy pressures, promoting the transformation and upgrading of the automotive industry, It is particularly important

© The Author(s) 2024

Z. Zhao · Z. Chen · S. Huang · L. Gao (B)

Department of Mechanical Engineering, Zhuhai College of Science and Technology, Zhuhai 519040, China e-mail: 56019874@qq.com

J. Fan

Department of Defense Industry, Inner Mongolia North Heavy Industries Group Corp. LTD, Baotou 014030, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_19

to carry out research and development of new energy vehicles [1–4]. In recent years, China's new energy vehicle industry has achieved significant development, with the production and sales of new energy vehicles continuously increasing and the market share increasing year by year. In 2022, the production and sales of new energy vehicles for the entire year were 7.058 million and 6.887 million, respectively, with year-on-year growth of 96.9% and 93.4% [5, 6], maintaining the world's first place for 8 consecutive years. The sales of new energy vehicles accounted for 25.6% of the total sales of new vehicles.

With the transformation and development of automotive power from traditional fuel to new energy, the original curriculum system of Vehicle Engineering can no longer fully meet the training needs of talents. Vehicle Engineering majors in various universities are actively exploring and reforming, introducing courses related to the research and development, manufacturing, and other aspects of new energy vehicles, so that students can initially master the theoretical knowledge and practical abilities of new energy vehicles in their studies. The number of new energy vehicle products continues to grow, the learning experience gained from using Electric Vehicle Drive Demonstration Teaching Platform will prove to be very valuable for further development in this field, and it is of great significance for the progress of electric vehicle technology education [7, 8]. For application-oriented undergraduate education, while learning relevant theoretical knowledge, it is particularly urgent to focus on practical abilities, combine theory with practice, and design a teaching demonstration platform that corresponds to the theory of new energy vehicles to meet students' learning needs [9, 10].

The motor drive system is an important part of the three electric system of new energy vehicles. The learning, understanding and application of the motor drive system is the key point in the teaching process of new energy vehicles for applicationoriented undergraduates. The development and design of the demonstration teaching platform of the motor drive system can enable students to deepen their learning and understanding in building the motor drive system, completing the function demonstration, understanding the working mode of the motor, and better integrate with theoretical knowledge.

Electrification in the automotive industry appears in the form of Electric Vehicles (EVs) and Hybrid Electric Vehicles (HEVs). Different configurations of HEVs, namely series, parallel, and series–parallel [11, 12], for the above reason, as well as the various configurations of electric vehicles such as two wheel drive and four wheel drive, as well as various types of driving motors such as DC and AC motors, as well as various forms of batteries such as lead-acid batteries and lithium batteries, the battery capacity, volume, and interface size of different systems also vary, and the controller form, quantity, and connection method of different systems also vary. If a driving demonstration teaching platform is designed for each different situation, A large amount of resources need to be invested, resulting in the waste of teaching and experimental sites and an increase in equipment management and other aspects of work. Designing the electric vehicle drive demonstration teaching platform as an expandable approach can meet the compatibility of the teaching platform with different motor forms, battery types and capacities, and control methods, achieve diversification of the teaching platform, better meet the needs of electric vehicle drive system related demonstrations and experiments, reduce resource and venue waste, and reduce management and maintenance costs.

# **2 Composition and Principle of Electric Vehicle Drive System**

The motor drive system of electric vehicles is one of the core components of electric vehicles. Its core components are generally composed of motors, controllers, power converters, mechanical transmission devices, etc. in order to ensure that the motor drive system can work according to the driver's intention, it is necessary to connect the corresponding operating mechanism and sensors, and at the same time, there is a battery pack for its energy supply [13–15]. Its basic composition structure is shown in Fig. 1.

As shown in Fig. 1, the battery pack provides energy for the motor drive system. The controller receives signals such as throttle, brake and speed, judges the driver's intention, and sends out control signals. The control power converter provides the corresponding voltage or frequency to drive the motor, and finally drives the mechanical transmission device to meet the driving requirements of the vehicle.

The display and understanding of the principle of electric vehicle drive system, the connection between various physical components and the realization of control strategies are the key contents of undergraduate education. Especially for applicationoriented undergraduate education, students should be able to correspond the corresponding components with the physical object, connect them according to the principle, complete the drive control, and deeply understand the realization and control of motor drive system. The expandable demonstration teaching platform for electric vehicle drive can carry and demonstrate different batteries, controllers, motors and drive devices to meet the teaching needs.

**Fig. 1** Composition of electric vehicle drive system

# **3 Design Requirements of Electric Vehicle Drive Demonstration Teaching Platform**

For teaching experiments of courses related to new energy vehicle technology, the driving demonstration teaching platform shall be able to display various functional modules, and students shall be able to conduct physical connection of mechanical and electrical aspects according to the operating principle, so as to realize the driving function, monitor relevant parameters and analyze the driving state.

The new-energy vehicle includes a variety of design forms such as pure electric vehicle, hybrid vehicle and extended-process vehicle. The structure of pure electric vehicle is typical, and the composition is relatively simple, which is more conducive to students' learning and understanding. Later, the driving device of driving pure electric vehicle serves as a demonstration teaching platform, and the structure is simpler, the composition is clearer, and it is more consistent with the requirements of application-oriented undergraduate experimental teaching. From the analysis of the composition of pure electric vehicles, pure electric vehicles have different compositions in terms of driving mode, control method, battery capacity, etc. Therefore, the design of driving demonstration teaching platform shall comprehensively consider the realization of various functions, and shall consider its extensibility to meet the needs of demonstration and teaching, and make corresponding preparation for the future platform expansion.

# **4 Scheme Design of Electric Vehicle Drive Demonstration Teaching Platform**

Based on the demonstration and teaching requirements of the electric vehicle drive demonstration teaching platform, the specific design of the program shall first meet the hardware installation and functional connection, and then shall be able to control the drive system through the operating device after the completion of the connection, and display the operation status on the instrument. The schematic composition of the system is shown in Fig. 2.

For the convenience of control, the demonstration teaching platform uses DC motor as the driving device. The motor is connected with the driving wheel of the axle through the reducer and differential, so as to complete the driving of the tire to simulate the running of the electric vehicle. The control of the motor is completed through the controller. The control signal can be input by the hand throttle or the foot throttle. The deceleration is completed by the mechanical brake of the hand brake or the foot brake. The energy is provided by a group of batteries, the voltage and capacity required by the matched motor, motor speed, speed, current, voltage and other information are displayed on the combination instrument to analyze the corresponding working conditions. The 3D design drawing is shown in Fig. 3.

**Fig. 2.** Principle composition of electric vehicle drive demonstration teaching platform system

1 control display console, 2 support platform, 3 vehicle controller, 4 power battery, 5 motor drive system, 6 reduction differential

**Fig. 3** Three dimensional design of electric vehicle rear drive debugging teaching platform

# **5 The Design Scheme of Expanded Capacity of Electric Vehicle Drive Demonstration Teaching Platform**

In the design of an electric vehicle drive demonstration teaching platform, full consideration should be given to its expansion ability. The implementation of expansion ability can generally be achieved by increasing the installation points and holes on the support platform, lengthening the design platform, and improving the loading capacity of the platform. A dedicated mechanism should be designed to enable the support platform to have lifting function, meeting the installation needs of more components of different specifications and dimensions. At the same time, there is sufficient expansion space and installation location reserved at the control display desk to meet the installation and connection requirements of control and display equipment.

In order to meet the expandable design requirements of the teaching platform for electric vehicle rear drive debugging, different specifications of motor drive axles and batteries are installed in teaching to form different driving modes. The speed and torque of the motor on the motor drive axle are controlled by the drive motor controller. In the overall design, the mounting base and support platform are designed respectively. The support platform is composed of fixed connecting frame and mounting frame, A number of mounting holes are equally spaced on the surface of the mounting frame and the connecting frame to facilitate the installation of motor controllers, display instruments, control devices and other components of different sizes on the mounting frame. The motor drive axle is fixed by bolts at the lower end of the mounting frame, and batteries or other energy storage and energy supply devices are installed on the fixed connecting frame. The components are connected by wire harness to achieve relevant experimental debugging and demonstration functions, Its basic structure is shown in Fig. 4. At the same time, in order to realize the adjustable height and position of the bench, an adjustable supporting platform height adjusting mechanism is designed on the inner wall of the mounting base to meet the installation requirements of different motors and drive axles.

In order to realize the adjustability of the electric vehicle rear drive debugging teaching platform, a special structural design was carried out. The handle can be turned to drive the screw column to rotate, which can drive the screw transmission between the nut seat and the screw column, so as to drive the slider to move, so that the first support rod and the second support rod can push the support platform up and down, so as to adjust the height of the mounting bracket, which is convenient for the installation of the motor drive axle and other components, It can also meet the installation of motor drive axles and other components of different specifications and models. The specific mechanism is shown in Fig. 5.

The design configurations of new energy vehicles are diverse. Pure electric vehicles, hybrid vehicles and some new types of new energy vehicles are constantly

1 mounting base, 2 support platform, 3 connection frame, 4 mounting frame, 5 motor controller, 6 motor drive axle, 7 threaded column

Front view Section view 1 mounting base, 2 supporting platform, 3 connecting frame, 4 threaded column, 5 nut seat, 6 connecting rod, 7 sliding rod, 8 sliding block, 9 first supporting rod, 10 second supporting rod

**Fig. 5** Basic structure of support frame

developing new technologies. This platform can also further expand the structure at the connecting frame and connect more external devices, laying the foundation for the subsequent expansion of various types of new energy vehicle driving teaching demonstration platform.

### **6 Conclusion**

With the rapid development of China's new energy vehicle industry, the demand for employees of new energy vehicles continues to rise. Due to the diverse configurations of new energy vehicles, it is particularly necessary to design and develop relevant teaching and experimental equipment with good expansion performance, which has become the hardware basis for cultivating students' practical ability. On the basis of meeting the basic principles of electric vehicles, debugging and demonstration, the extensible electric vehicle drive debugging teaching platform designed and developed in this design has deeply considered the extended transformation performance to meet the requirements of subsequent experimental demonstration of different configurations of electric vehicles, and has played a corresponding role in promoting the teaching of electric vehicles.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design and CFD Simulation of Heat Transfer in Circular Pipes**

**Furman Ali, Munawwar Ali Abbas, Bo Sun, Li Chen, and Shahid Hussain** 

**Abstract** A major issue for human growth is the energy crisis. In the current study, circular pipes are considered an energy-efficient technology viable in all seasons. It uses non-constant soil temperature and water flow in coldest regions. This model reports the thermal performance of pipes with different results in hot and cold climates. To degree thermal performance, pipe version changed into advanced and simulated with ANSYS 18.1 Fluid Flow (Fluent). Data was collected from two modelled summer and winter seasons. Continuity, momentum, and energy equation have been used for the simulation. Comparison analysis is also being carried out about the obtained results with previously published articles. The effects of summer and winter temperature and speed on performance were also evaluated. Further, the graphical interpretation is also carried for velocity and pressure distribution. The study highlights a drastic drop in temperature and velocity near the inlet and outlet region while Pressure is inversely proportional to velocity in these regions. It is also obtained from the simulation that the temperature pressure and velocity remain uniform in the fully develop region.

**Keywords** Temperature · Velocity assessment · Cylindrical coordinates · Heat transfer · CFD

F. Ali · M. A. Abbas · S. Hussain

Department of Mathematics, University of Baltistan, Skardu 16200, Pakistan

L. Chen (B)

#### B. Sun

School of Mechanical Engineering, Tongji University, Shanghai 201804, China

© The Author(s) 2024

School of Automotive Studies, Tongji University, Shanghai 201804, China e-mail: lilychen@tongji.edu.cn

Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Shanghai 201804, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_20

### **1 Introduction**

The use of energy is an integral part of the progress of human society, playing a role in controlling and adapting to the environment. There is a gradual increase in the consumption of various types of energy around the world. Several researchers studied about the Potential for supply temperature reduction of existing district heating substations [1–3]. This rise in energy demand is mostly caused by population and income growth [4]. The heat transfers and pressure drop of the perpendicular GHE is dissembled with different inflow rates, pipe compasses, and well depths. Li et al. [5] investigated the thermic response tests evaluation of borehole heat exchanger thermal short and effective. The liquid absorbs or rejects heat with the surface through the downhole leg (DLP) and top of the tube (ULP) inside a perpendicular drag heat exchanger (BHE). Since the periphery of the hole is only 0.11−0.2 m, the temperature difference between DLP and ULP will inescapably beget a thermal short circuit probing the effect of different geometric features on the short circuit. The heat transfer between the two legs was developed into a 2-D model, after which the most suitable expression for the thermal resistance of the short circuit was presented without dimension. Sotgia et al. [6] experimental analyzed the inflow contours and pressure drop reduction of oil painting-water fusions. They performed an experimental study of nonstop oil painting-water inflow in vertical pipes with mineral oil painting and valve water with a density rate of roughly 900 and a viscosity rate of 0.9. Seven different pyrex and plexiglass tubes with a periphery of 21-0 mm were used. Kim et al. [7] suggested that numerical analysis of the evolution of laminar flows extending from a spiral pipe to a straight pipe. Li et al. [8] discussed heat transfer capacity in deeply buried coaxial pipes. They concluded that heat transfer can be improved by modifying the size and structure of the inner pipes.

In this study it is Numerically investigated the fluorescent laminate inlet domain of a straight pipe that evolves from a fully grown laminar flow in a spiral tube are presented. The evolution of the flow is studied in the modified dean number range from 1 to 400. The thermal performance of these pipes in different climates, specifically hot and cold climates. To assess thermal performance, a thermal pipe version is developed and simulated using ANSYS 18.1 Fluid Flow (Fluent), which is software for fluid dynamics simulations. The Continuity, momentum, and energy equations were used for the simulation, which are fundamental equations in fluid dynamics.

### **2 Mathematical Model**

The geometry was created in Ansys 18.1 as shown in Fig. 1. A pipe with a radius of 10 cm and a length of 50 cm. Material used in the pipes of Thalay powerhouse.

**Fig. 1** Geometry of the pipe

# *2.1 Meshing of the Geometry*

A typical network for an above-ground piping system is shown in Figs. 2, 3, 4. Mesh generated with Design Modeler in ANSYS 18.1. The mesh creates a large number of cells in the far-field Cartesian layout and provides accurate fluid flow results. An optimal number of 3093 nodes and 11,548 elements were generated in the mesh to obtain perfect results for this case.

**Fig. 3** Mesh at the pipe's

### *2.2 Mathematical Equations*

The mathematical equation of continuity Navier Stoke's equation and energy equation are given as:

$$\frac{\partial \rho\_1}{\partial t} + \frac{1}{r\_1} \frac{\partial}{\partial r\_1} (\rho\_1 r\_1 u\_{r\_1}) + \frac{1}{r\_1} \frac{\partial}{\partial \varphi\_1} (\rho\_1 u\_{\varphi\_1}) + \frac{\partial}{\partial z} (\rho\_1 u\_{z\_1}) = 0 \tag{1}$$

Navier–Stoke equation is used. Cylindrical coordinates.

$$\begin{aligned} \frac{\partial u\_{\mathcal{L}\_{1}}}{\partial t} + u\_{r} \frac{\partial u\_{\mathcal{L}\_{1}}}{\partial r\_{2}} + \frac{u\_{\phi 3}}{r} \frac{\partial u\_{\mathcal{L}\_{1}}}{\partial \phi\_{3}} + u\_{\mathcal{L}\_{1}} \frac{\partial u\_{\mathcal{L}\_{1}}}{\partial z\_{1}} &= -\frac{1}{\rho} \frac{\partial p}{\partial z\_{1}} + g\_{1} \\ &+ v \left[ \frac{1}{r\_{2}} \frac{\partial}{\partial r\_{2}} \left( r\_{2} \frac{\partial u\_{\mathcal{L}\_{1}}}{\partial r\_{2}} \right) + \frac{1}{r\_{2}^{2}} \frac{\partial^{2} u\_{\mathcal{L}\_{1}}}{\partial \phi\_{3}^{2}} + \frac{\partial^{2} u\_{\mathcal{L}\_{1}}}{\partial z\_{1}^{2}} \right] \end{aligned} \tag{2}$$

The temperature difference in the (low speed) flow is small enough for K to be assumed constant.

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial x} = k \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} \right) \tag{3}$$

where *k* = *k*/ρ*c*ρ is the thermometric conductivity. It is Use the energy equation to change boundary conditions temperature such as water as well as surrounding temperature. We obtained these results from Ansys 18.1.

### **3 Results and Discussion**

Inlet temperature profiles with constant velocity v = 0.425 m/s, the constant inlet temperature of the water is 278 k, and different walls of the temperature such as 272 k.

The effect of low liquid cargo on colorful inflow characteristics and problems related to inflow resistance, like pipe erosion, suggest that further analysis of inflow is demanded. In this study, CFD simulations were performed on a vertical pipe where liquid and gas are fed independently at constant haste at the bay. The analysis substantially focuses on the shape of the interface, the haste fields in the liquid and gas phases, the liquid retention, and the shear stress profile. Tests are performed as a liquid phase with water or oil painting, the liquid volume bit in the bay sluice is 0.0005 to 0.00020. Overall, the results help to understand the miracle of low liquid charge inflow. It can be seen from Figs. 5 and 6 the inlet of the pipe temperature/ velocity contours are given below. Inlet temperature profiles as seen in Fig. 7, with constant velocity v = 0.425 m/s the constant inlet temperature of the water is 278 k, and different walls of the temperature such as 272 k, 267 k, and 262 k.

**Fig. 7** Inlet temperature profiles with v = 0.425 m/s Inlet T w (K) = 278 k, Wall of the T p (K) = 272 k

Simulations are performed with Ansys Fluent 18.1 using Volume of Fluid. Inlet velocity profiles as seen in Fig. 8, with constant velocity v = 0.425 m/s, a constant inlet temperature of the water is 278 k, and different walls of the temperature such as 272 k, 267 k*,* and 262 k.

Figure 7 indicates that the temperature of the water increases gradually and maximum in the grown region of the pipe. The wall temperature of the pipe is 272 k or (−1 ◦C). When water flows in the pipelines initially temperature gradually increases. When the fluid reaches a fully developed region then the temperature is maximum.

Figure 8 displays that the velocity of the water increases gradually and maximum in the fully grown region of the pipe. Wall temperature of the pipe is 272 k or (−<sup>1</sup> ◦C). When water flows in the pipelines initially velocity gradually increases. When the fluid reaches a fully developed region then velocity is maximum.

The outlet of the pipe temperature/velocity contours are given as Figs. 9 and 10. Express that the surrounding or wall of the pipe temperature recorded 272 k or (−<sup>1</sup> ◦C). When the temperature of flowing water is 278 k*.* The temperature of the water flow is maximum to the fully developed region of the pipe. Now the temperature of the water gradually decreases at exit point, as shown in Fig. 11. As a result, the water temperature dropped. The lowest temperature is recorded. Display that the

surrounding or wall of the pipe temperature recorded 278 k or (−1 ◦C). When the temperature of flowing water is 272 k*.* The velocity of the water flow is maximum to the fully developed region of the pipe. Now the velocity of the water gradually decreases at exit point, as shown in Fig. 12. As a result, the water velocity dropped. The lowest velocity is recorded at exit point.

**Fig. 10** Outlet velocity profile, v = 0.425 m/s

### **4 Conclusion**

The study developed the model of the simulation and design of water circular pipe. The following points are the concluded from the study as:


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Thermal Robustness Redesign of Electromagnet Under Multi-Physical Field Coupling**

**Chenyu Liu, Anlin Wang, and Xiaotian Li** 

**Abstract** Aiming at the durability problem of the proportional electromagnet used in the proportional valve of engineering machinery, in order to improve its thermal failure resistance under random load conditions, a parametric redesign model of the proportional electromagnet was proposed based on the multi-physics coupling theory and robust optimization theory. This article takes the proportional electromagnet with a basin-type suction structure as the research object. The parameter model was verified through steady-state proportional electromagnet tests and temperature distribution tests. On the premise of ensuring the accuracy of electromagnetic calculation force, the conductivity and heat transfer parameters with fuzzy magnitude in the system were calibrated. Taking the key structural parameters of the proportional electromagnet and coil as the control factors, and the enameled wire diameter of the coil caused by the uncertainty of the production process conditions as the noise factor, an orthogonal experiment was designed based on the Taguchi method, and the thermal robustness redesign evaluation function of the proportional electromagnet was defined. Multi-factor weighted form. The thermal load of the proportional electromagnet obtained from the excavator field test was used as the response to calculate the heat source. Under the constraint of allowable temperature rise that can not cause coil insulation failure, a redesign method for key structural parameters that minimizes changes in system response under noise interference is given. Studies have shown that coil length and number of turns are the main factors affecting the thermal robustness of proportional electromagnets. The window shape of the coil is determined by the winding process and determines the magnetic properties and heat transfer capabilities

A. Wang e-mail: wanganlin@tongji.edu.cn

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_21

265

C. Liu (B) · A. Wang · X. Li School of Mechanical and Energy Engineering, Tongji University, Shanghai 201804, China e-mail: 1910420@tongji.edu.cn

X. Li e-mail: lixiaotian@tongji.edu.cn

of the system. The thermal robustness redesign method of proportional electromagnets proposed in this article has engineering reference value for the customized design of electromechanical products under magnetothermal coupling conditions.

**Keywords** Proportional electromagnet · Coil · Multi-physics coupling · Thermal robustness redesign · Taguchi method

### **1 Introduction**

With the rapid popularization and application of electro-hydraulic proportional pilot technology, the layout of hydraulic excavators is more flexible, the dynamic response is improved, the noise source is reduced, and the combination of control accuracy and position closed loop provides conditions for the automation of construction machinery, thereby enabling dangerous situations It is possible to realize remote control and wireless remote control for construction machinery operating in regional areas. The electro-hydraulic proportional multi-way valve mainly uses a cartridge proportional pressure valve as the pilot stage, and controls the system flow by controlling the displacement of the main spool through the pilot valve.

Pproportional solenoid electromagnet (PSE) is the main component of proportional solenoid valve (PSV). In decades, researchers have conducted lots of research on proportional electromagnets. The main research directions are optimization strategies, fault detection and control technology. Technical means are computer simulations and engineering tests. Liu [1] and others modeled finite element model of PSV based on FE theory. Thermal deformation and temperature distribution of PSV under different conditions were simulated, and compared with the experimental results to verify the reliability of model and provide a theoretical basis for the reliability design of the PSV. Wang [2] and others verified the numerical model of the PSV based on the finite element theory, then used genetic algorithms and multi-objective optimization methods to optimize the shape design parameters of the electromagnet. The influence of the optimized parameters on the dynamic and static performance of the electromagnet was verified through experiments. Fan [3] modeled a multi-physics performance model considering the influence of temperature field. The purpose is solving the problem of non-uniformity between multiphysics fields during model establishing process. The correctness of the model was verified through relevant experimental design, and an evaluation method of electromagnetic force was proposed, which provides theory basis for rapid and customized design and engineering processing of PSVs.

Because of large design parameters of the proportional electromagnet structure, it is hard to design the PSE by changing single variable, and most of the design parameters are nonlinear with the steady-state PSE force [4–7]. In actural practice, traditional design methods can no longer fully meet the requirements of reliability and robustness. From perspective of design quality, design process not only meet the requirements of functional reliability, but also satisfy the requirements of robustness and be able to fit complex conditions [8, 9]. This article takes the PSE as the research object. Parametric redesign model under magnetothermal coupled condition is proposed. Finite element theory, multi-physics coupling simulation and Taguchi test methods are applied to the robust redesign of the complex non-homogeneous proportional electromagnet magnetothermal coupling system, and thermal robustness redesign of PSE is defined in the form of multi-factor weighting. Designed evaluation function. The combination of redesign method, experiment and approximate model technology broadens the idea of research on the durability of proportional electromagnets.

### **2 Parametric Modeling of Proportional Electromagnet**

### *2.1 Structural Characteristics of Proportional Electromagnet*

PSE has direct-acting electromagnetic actuator and special magnetic isolation structure. As shown in Fig. 1, 1/2 cross-sectional structural diagram of an proportional electromagne, which consists of an PSE end cover, magnet steel, shell, bobbin, seal, armature, gasket, coil, and sleeve.

The coil is the key component to provide power input for the electromagnet. The coil properties are directly related to the electromagnetic performance, thermal reliability and life of the electromagnet and even the proportional solenoid valve. The parameter expression of proportional coil is necessary in optimization of proportional electromagne. Turns of the coil is *N*, excitation current in the coil is *I*, inner radius is *rxqinner*, outer radius is *rxqouter*, and winding length is *2L*. The initial design range of each characteristic variable has been shown in Table 1.


# *2.2 Magnetic Field Response Based on Coil Optimal Shape Optimization*

The coil model parameters have been normalized: *X* = *rxqouter* /*rxqinner* , *Y* = *L* / *rxqinner* .

According to the theory of electromagnetics, there are solenoid coil center field strength as shown in Eq. (1):

$$dH\_0 = \frac{2\pi}{10} j\lambda \frac{r^2}{(r^2 + z^2)^{3/2}} dr dz \tag{1}$$

In Eq. (1), λ is filling factor, *j* is current density.

Furthermore, power function of the PSE coil, as shown in Eq. (2), the field intensity at every point on central axis of PSE coil could be simplified by Eq. (3). Determinant G representing shape type of PSE is shown as Eq. (4).

$$P = \int dp = \int j^2 \rho dv = \frac{\pi \rho N^2 I^2}{2 \lambda r\_{xqinner}} \frac{X+1}{Y(X-1)}\tag{2}$$

$$H\_z = C\_1 \sqrt{\frac{P}{R\_1}} G(X, Y, \overline{z}) \tag{3}$$

$$\begin{aligned} G(X,Y,\overline{z}) &= \frac{1}{2\sqrt{2\pi Y(X^2 - 1)}} \cdot \\ \left[ (\overline{z} + Y) \ln \frac{X + \sqrt{X^2 + (\overline{z} + Y)^2}}{1 + \sqrt{1 + (\overline{z} + Y)^2}} - \right. \\ (\overline{z} - Y) \ln \frac{X + \sqrt{X^2 + (\overline{z} - Y)^2}}{1 + \sqrt{1 + (\overline{z} - Y)^2}} \, \Big| \, \end{aligned} \tag{4}$$

The isosurface and contour of the morphological parameter factor G are shown in Fig. 2. The results show that, the maximum value of G is 0.1426 when X = 3.051, Y = 1.864, which is consistent with previous research [10].

Without considering changing the structure of the electromagnet on the magnetic circuit and the structure of the coil skeleton, keeping inner diameter *rxqinner*  unchanged, combined result of the shape factor G, the shape constraint space is obtained as shown in Eq. (5):

$$28.0\,\mathrm{mm} < r\_{x\,\mathrm{outer}} < 54.9\,\mathrm{mm}, \quad 28.1\,\mathrm{mm} < L < 33.9\,\mathrm{mm} \tag{5}$$

The approximation process and mathematical model of the relationship between wire and turns of windings are set by Fig. 4 and Eq. (6):

$$N = \frac{2kL(r\_{xqouter} - r\_{xqinner})}{s} \tag{6}$$

In Eq. (6), *k* is stacking coefficient, *s* is cross-sectional area, Fig. 3 gives coil windings stacking shape, where *d* is the diameter of wire. Number of coil turns *N* is 930, and limit value of *k* is 0.907 [11].

**Fig. 3** The stacking pattern of coil windings

**Fig. 4** Electromagnetic finite element calculation result

# **3 Analytical Verification of Multi-Physical Field Coupling of Parametric Model**

### *3.1 Multi-Field Coupling Theory*

Field distribution only exists in circuit of electromagnet. Considering the symmetry of the structure of PSE, when establishing the finite element model of electromagnet. In order to simplify the calculation, spiral oil tank and shell on the armature were removed. For electrical interface on the body, a 2-dimensional numerical model is established in RZ coordinate. The numerical calculation is divided into two parts: steady state and transient state. Based on the sequential coupling theory, steady-state and transient electromagnetic finite element calculations were first performed on the electromagnet. The obtained transient electromagnetic calculation results are applied as loads to steady state temperature field simulation, that is, the electromagnetic calculation results are transmitted to the temperature field calculation module with average heating power, and then distribution of temperature on each component of PSE is obtained.

### *3.2 Coupling Field Analysis and Model Verification*

The material selection corresponding to every component of PSE is shown in Table 2.

Figure 4 shows the parameterized scanning calculation results of armature displacement and current magnitude in steady-state electromagnetic calculations.


**Table 2** Material selection of electromagnet parts

The current interval is set 0–700 ma, displacement interval is set 0 mm to 2.5 mm, and calculation step is set to 0.5 mm.

Figure 5 shows that temperature distribution of the electromagnet calculated by the finite element method of the magneto and thermal couped field. The initial calculated temperature and ambient temperature are both set to 30 °C.

### **4 Validation Testing and Results Analysis**

### *4.1 Electromagnetic Characteristic Test*

To test displacement and current with force characteristics of the electromagnet, peripheral components that affect PSE force characteristics test were removed. The test device is shown in Fig. 6.

# *4.2 Temperature Distribution Test*

The above calculation model is based on analysis of the magnetothermal coupled field. However, it is hard to conduct further detailed tests on temperature inside distribution PSE. The test was conducted in high and low temperature alternating damp thermal test chamber and being shown in Fig. 7. The relationship between internal temperature simulated by magnetothermal coupled simulation and the surface temperature of PSE obtained by experiment was established, and then the distribution of the internal temperature of PSE was inferred.

In actual operation of the electromagnet, coil is loaded with PWM modulated current for excitation, and the armature opens and closes repeatedly to achieve working heating. Short-period action process cannot quickly meet the thermodynamic steady state inside electromagnet under uncertain external thermal dissipation conditions. Experimental process in this article was under constant conditions. Test steady-state temperature of PSE is surface ambient temperatures (−20 °C, −10 °C, 0 °C, 60 °C, 80 °C, 120 °C).

### *4.3 Excavator Field Test*

In order to truly reflect the performance of proportional valve in the installed state and obtain real, reliable, and effective data samples, the excavator field test method was used to obtain original data samples. Parameters characterizing the dynamic traction performance of the excavator are directly measured and recorded while the excavator is operating on site. The field test took a 20t medium-sized hydraulic excavator as the subject. As shown in Fig. 8.

The temperature of electromagnetic coil rising fast due to the thermal effect during the operation, which causes the coil resistance increasing. The measured working pressure should be linearly proportional to the input current in the control loop, where the feedback loop in the control system performs feedback regulation on the input. Therefore, the linear mapping relationship between input and output can be constructed, and the measured pressure data can be converted into coil current data which is not easy to be measured directly in the installed state for thermal load calculation, as shown in Eq. (7). The heating power of the proportional electromagnet is 7.156 W, which is used as the heat source for thermal robustness analysis and calculation in the following.

**Fig. 8** Field test environment and object

$$P\_A = 0.0591 \cdot I\_F - 10.751\tag{7}$$

### *4.4 Result Analysis*

In Fig. 9 and Table 3, comparison of the test and simulationof electromagnetic force and surface temperature distribution shows that the error is at the level of ±10%, which is within the design acceptance range. From this, the validity of the above proportional electromagnet parameterized model can be determined. In the later thermal robustness redesign process, the finite element simulation calculation results based on the numerical calculation model can be used as the system response of the orthogonal test, thus replacing the complex test process.

**Fig. 9** Comparison of current-electromagnetic force experiment and simulation results

**Table 3** Comparison of temperature distribution test and simulation results


# **5 Characteristic Parameter Analysis and Thermal Robustness Redesign of Electromagnet**

Taguchi method is put forward by Dr. Taguchi. Taguchi design is an experimental design that selects a product or process that will perform more stably in an operating environment [12].

### *5.1 Comprehensive Performance Evaluation Method*

The rated suction, linearity and steady-state temperature rise of the proportional solenoid for valves are closely related to its structural form and parameters. It is an key index to evaluate the performance of the solenoid, whether it meets the industry design specifications, and whether it meets the product-level customization design requirements of the user enterprise. In the specific design process, the above three performance indicators conflict with the determination of design parameters, which is manifested in the performance of multiple sets of design parameters, that is, the rated suction, linearity, and steady-state temperature rise cannot reach the optimal level at the same time. In order to achieve the goal of robust and customized redesign, the weighted average method is used to determine the comprehensive evaluation value of multiple performance indexes of the proportional electromagnet, as shown in Eq. (8):

$$Z = \alpha F + \beta V + \gamma T, \; \alpha + \beta + \gamma = 1 \tag{8}$$

Among them, *Z* is the comprehensive performance index of the proportional electromagnet, *F*, *V* and *T* are the parameters of the performance index of the rated suction, linearity and steady-state temperature rise of the proportional electromagnet respectively, α, β and γ are the weight factors, and the determination of the weight value contains certain subjective factors of the designer. Standardization of performance evaluation parameters, as shown in Eq. (9).

$$F = \frac{\overline{F} - \overline{F}\_{\text{min}}}{\overline{F}\_{\text{max}} - \overline{F}\_{\text{min}}}, \ V = \frac{CV - CV\_{\text{max}}}{CV\_{\text{min}} - CV\_{\text{max}}}, \ T = \frac{\overline{T} - \overline{T}\_{\text{min}}}{\overline{T}\_{\text{max}} - \overline{T}\_{\text{min}}} \tag{9}$$

# *5.2 Approximate Calculation Model of Electromagnetic Characteristics*

According to mechanical analysis, displacement of armature is the result of force. Total electromagnetic attraction consists two components. Electromagnetic attraction generated by main air gap permeability on the armature end face and its main magneto field energy changing. The additional electromagnetic force generated by the armature side leakage magnetic field energy change [13–15]. The solution process of electromagnetic force is very complex and contains many parameters and nonlinear problems. Therefore, steady-state temperature rise of the PSE surface is also a nonlinear process, which is not easy to solve using simple symbolic calculations. Therefore, by identifying and analyzing the characteristic parameters of electromagnets, the response surface method is an effective method for calculating electromagnetic forces based on structural parameters.

The structural parameters are selected as the armature cone angle (a), the gap of the magnetic isolation ring (b), the height of the magnetic isolation ring (c), the cone angle of the pole shoe (d), the diameter of the armature (e), the thickness of the annular yoke (f), and the non-working air gap (g). The sensitivity analysis of the parameters is carried out by the range analysis method. Through the sensitivity analysis of structural parameters, six parameters were determined to construct the response surface model of electromagnetic force. The response surface method fits the response surface equation through limited experimental data. Considering the number of design factors and the characteristics of design points, the Box-Behnken design is selected. The items with poor fitting effect can be eliminated by R-square analysis, and the final response surface function can be obtained by re-fitting. The final response surface function of the electromagnetic force is as shown in Eq. (10):

$$\begin{aligned} F &= 21.09 - 18.6a - 49.33b - 6.207c + 3.874e + 2.1678f \\ &- 18.39g + 18.79a^2 + 64.86b^2 + 0.6592c^2 \\ &- 0.3445e^2 - 0.1550f^2 + 7.83g^2 + 40.37ab + 1.353ac \\ &- 0.865ae - 0.513af + 7.03ag - 1.523bc - 0.845be \\ &- 0.298bf + 7.7bg + 0.25ce + 0.065cf + 1.2cg + 0.43ef \end{aligned} \tag{10}$$

### *5.3 Thermal Robustness Redesign*

The key structural component affecting the thermal performance of PSE is coil. The shape of coil winding rules of copper wire directly affect the steady-state temperature rise of PSE. Therefore, coil inner radius *rxqin*, outer radius *rxqout*, length *hxq*, coil turns *N*, wire duty cycle *k* caused by different winding rules, and the total working air gap δ are set as the control factors of the test. The diameter of the enameled copper wire is set as the noise factor of the test, and the input current *I* is set as the signal factor. The maximum steady-state temperature rising on the surface of the solenoid valve is system response. The control factor and its level setting are shown in Table 4. The L8 (2 ^ 6) Taguchi table is designed, including 6 control factors and 1 signal factor, which are response sequences of two levels and two noise levels.


**Table 4** Control factors at different level settings

The results of Taguchi test are shown in Fig. 10, which are given from three aspects: signal-to-noise ratio, slope and standard deviation. In these results, we want to make the standard deviation as small as possible, and the signal-to-noise ratio and slope as large as possible.

Signal-to-noise ratio is a measure of robustness and is used to determine control factor settings that minimize the impact of noise on the response. In these results, the main effect plot of the signa-noise ratio shows that the coil length has the greatest impact on the signal-to-noise ratio, followed by the number of coil turns, and the smallest is the working air gap. In all cases we want to maximize the signal-to-noise ratio. For dynamic designs, the signal-to-noise ratio is approximated by the phase "lookout" signal–noise ratio. The main rendering shows how each factor affects the response characteristics (signal-to-noise ratio, slope, standard deviation). Main effects exist when factors at different levels have different effects on a trait. Through experimental design, it can be concluded that the combination of factor levels shown in Table 5 can make the steady-state temperature rising of PSE remain stable under the influence of noise, that is, it is robust.

According to the Eq. (8), the thermal robust performance evaluation function under the influence of the rated suction, linearity and steady-state temperature rise of the proportional electromagnet is shown in Eq. (11), where α1, α2 and α3 are weighting coefficients, which are 0.3,0.3 and 0.4 respectively.

$$Z = \alpha\_1 F + \alpha\_2 V + \alpha\_3 T \tag{11}$$

Based on the results of sensitivity analysis and thermal robustness analysis of electromagnet pull-in structure parameters, the optimal factor combination is obtained. The value of Z is 0.897, which is the closest to 1 compared with other factor combinations, indicating that it is more robust. The calculation result of the Z value depends on the numerical distribution of the weighting coefficient, and the selection rules of different electromagnets are different.

**Fig. 10** Diagram of main effect by Taguchi test


**Table 5** The highest signal-to-noise ratio factor combination

### **6 Conclusion**

In this article, Taguchi and FE theory are combined to the processing of redesign thermal robustness of the electromagnet and the coil. From the analysis of design results, the robust parameter factor combination of steady state temperature rise of PSE is obtained on premise of ensuring the robustness of the electromagnetic characteristics. The design method improves the efficiency of redesign and analysis calculation effectively, and the operation is more flexible.

Inner and outer diameter, length, turns, filling rate and working air gap of PSE are key parameters that affect thermal characteristics of PSE. Appropriate parameter combination selection can improve the aging capability of the system and quickly achieve thermal balance, thereby reducing the excessive temperature rising of PSE caused by local heat accumulation caused by temperature inertia. The coil structure parameters get reck with the wire diameter of the enameled wire. Random error in wire diameter caused by the winding process is transmitted in the system, affecting the transfer vector of the inter-turn microstructural force in the system, resulting in excessive local stress and insulation failure.

The thermal robust redesign method is used to release sensitivity of PSE coil to factors which are uncontrollable. It could be used to solve insulation failure and stagnation caused by PSE thermal effects. The redesign theory play a key role in improving durability of PSE.

**Acknowledgements** The research of this article was supported by the Key Research and Development Plan of Shandong Province (Major Scientific and Technological Innovation Project) (2020CXGC011005).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Reliability Optimization Design Method for Firearms Automaton Mechanism**

**Yichuan Fang, Yongjuan Wang, Pengchao Li, Tongguang Gu, and Xin'an Gao** 

**Abstract** In order to deal with the reliability optimization design problem of mechanisms containing multiple motion phases and multiple collisions, this paper takes the firearms automaton as an example and gives a general method to deal with such problems. Firstly, the motion stages are divided according to the automatic cycle diagram of the firearms automaton, and the discrete dynamics model of the firearms is established. Then the reliability model of the automaton mechanism is established based on the performance margin theory. Finally, taking the pressure coefficient of air chamber, the friction coefficient of the guide, the stiffness of the counter-recoil spring and the stiffness of the hammer spring as design variables, the optimized design variables are obtained through the calculation of the sequential quadratic programming (SQP) algorithm. The results of the optimized design are close to the idealized results of the existing products, which proves the rationality and effectiveness of the mechanism reliability optimization design method proposed in this paper.

**Keywords** Firearms automaton · Reliability optimization design · Discrete dynamics · Performance margin theory · Sequential quadratic programming

# **1 Introduction**

In recent years, many modern design methods and corresponding sciences have appeared in the field of mechanical design [1–4]. At present, reliability design and optimization design have reached a certain level in theory and methodology, but

P. Li

NO.208 Research Institute of China Ordnance Industries, Beijing 102202, China

T. Gu

Chongqing Changan Wangjiang Industrial Group Co., LTD, Chongqing 401120, China

© The Author(s) 2024

Y. Fang · Y. Wang (B) · X. Gao

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: 13951643935@139.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_22

it is impossible to exert the great potential of reliability design and optimization design no matter whether reliability design or optimization design is carried out unilaterally. On the one hand, reliability design is sometimes not equal to optimization design. Thus, reliability design of mechanical products does not guarantee their best performance or parameters. On the other hand, optimization design does not necessarily include reliability design. Likewise, mechanical products in the absence of considering reliability optimization design cannot guarantee that it will complete the specified functions within a specified condition and time. Besides, even failures and malfunctions may occur. [5–8].

The firearms automaton is mainly to transform the chemical energy of gunpowder into the mechanical kinetic energy of the automaton, to complete the action cycle of firing, uncocking, recoiling, cartridge case ejecting, counter-recoiling, feeding and locking, etc. Due to the existence of numerous design parameters in the mechanism of the firearms automaton, the simple reliability design method becomes impotent to determine multiple design parameters at the same time. Therefore, it is very important to carry out the research on optimal design of firearms automatic mechanism reliability. In order to make the firearms automaton not only to ensure the reliability requirements, but also to ensure the best performance and parameters, it is necessary to combine the reliability design with the optimal design. In this way can we give full play to the great potential of the reliability design and optimal design. while giving full play to the strengths of the two design methods. Therefore, the best reliability performance of firearms automaton can be achieved.

In this paper, a reliability optimization design method based on SQP for firearms automaton mechanism is proposed as shown in Fig. 1. The main steps are:


# **2 Dynamics and Reliability Modeling of Firearms Automaton**

The firearms automaton has the dynamic characteristics of variable mass and multiple motion phases and collisions, which needs to be described by discrete dynamics. Taking a gas operating automatic rifle as the research object, based on its automatic cycle diagram, the motion of the automaton can be represented as a point and line diagram shown in Fig. 2, in which the line segments denote the motion process, and the nodes denote the collision combination or collision separation.

Due to the differences in the mass of the moving parts corresponding to the different stages of motion of the automaton and the magnitude of the combined

**Fig. 1** A reliability optimization design method based on SQP for firearms automaton mechanism

**Fig. 2** The point-line diagram of the motion of the automatic mechanism

forces applied, the dynamical equations are divided into 13 stages based on the nodes, where each parameter of the next stage of motion is updated at the nodes. The equations for each stage can be written as follows:

$$\stackrel{\cdots}{M}\stackrel{\cdots}{x} = F \tag{1}$$

where *x* denotes the displacement of the moving parts, *M* denotes the vector of the mass of the corresponding moving parts in each stage, and *F* denotes the vector of the magnitude of the combined force applied in each stage.

*M* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *m*1 *<sup>m</sup>*1 <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *tr*1 <sup>η</sup> *m*- <sup>3</sup><sup>+</sup> *<sup>k</sup>*<sup>2</sup> *tr*2 <sup>η</sup> *m*<sup>3</sup> *m*1 + *m*2 + *m*3 + *m*<sup>5</sup> *m*1 + *m*2 + *m*3 + *m*<sup>5</sup> *m*1 + *m*2 + *m*<sup>3</sup> *m*1 + *m*2 + *m*<sup>3</sup> *m*1 + *m*2 + *m*<sup>3</sup> *m*1 + *m*2 + *m*3 + *m*<sup>4</sup> *m*1 + *m*3 + *m*<sup>4</sup> *m*1 + *m*3 + *m*<sup>4</sup> *<sup>m</sup>*1 <sup>+</sup> *<sup>k</sup>*<sup>2</sup> *tr*1 <sup>η</sup> *m*- <sup>3</sup><sup>+</sup> *<sup>k</sup>*<sup>2</sup> *tr*2 <sup>η</sup> *m*<sup>3</sup> *m*1 *m*2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , *F* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *F*1 − *k*1*x*1 + μ*m*1*g <sup>F</sup>*1 <sup>−</sup> *<sup>k</sup>*1*x*1 <sup>+</sup> μ(*m*1 <sup>+</sup> *<sup>m</sup>*3)*g* <sup>+</sup> *ktr*<sup>1</sup> <sup>η</sup> *Ftr*1 <sup>+</sup> *ktr*<sup>2</sup> <sup>η</sup> *Ftr*<sup>2</sup> *F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) + μ *m*1 + *m*2 + *m*3 + *m*<sup>5</sup> *g F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) + μ *F*0 − *m*1 + *m*2 + *m*3 + *m*<sup>5</sup> *g F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) + μ *F*0 − (*m*1 + *m*2 + *m*3)*g F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) + μ(*m*1 + *m*2 + *m*3)*g F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) − μ(*m*1 + *m*2 + *m*3)*g F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) − μ *F*0 − (*m*1 + *m*2 + *m*3 + *m*4)*g F*1 − *k*1*x*1 + *F*2 + *k*2(*d*0 − *x*1) − μ *F*0 − *m*1 + *m*3 + *m*<sup>5</sup> *g F*1 − *k*1*x*1 − μ(*m*1 + *m*3 + *m*4)*g <sup>F</sup>*1 <sup>−</sup> *<sup>k</sup>*1*x*1 <sup>−</sup> μ(*m*1 <sup>+</sup> *<sup>m</sup>*3)*g* <sup>−</sup> *ktr*<sup>1</sup> <sup>η</sup> *Ftr*1 <sup>−</sup> *ktr*<sup>2</sup> <sup>η</sup> *Ftr*<sup>2</sup> *F*1 − *k*1*x*1 + μ*m*1*g F*2 − *k*2*x*2 + μ*m*2*g*  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2)

Firearms automaton mechanism reliability modeling can be achieved by the performance margin equation [9]:

$$M = G(\mathbf{P}, \mathbf{P}\_{th}) > 0\tag{3}$$

where *M* is the performance margin vector, *P* is the performance parameter vector, and *Pth* is the critical value vector of the performance parameter. When the performance parameter vector does not exceed its critical value, i.e. *M* > 0, the system can work reliably.

The reliability model of the mechanism based on performance margin is:

$$R\_J = P\left(\tilde{M} > 0\right) = P\left(\frac{\varepsilon - \left|P\_j^r - P\_j\right|}{\varepsilon}\right) > 0\tag{4}$$

where *Pj* is the performance parameter of the mechanism in state *J*, *P<sup>r</sup> <sup>j</sup>*is the actual performance parameter of the mechanism, δ is the performance parameter error, and ε is the maximum value of the performance parameter error.

# **3 SQP-Based Optimal Design Method for Mechanism Reliability**

The optimal design of firearms automaton mechanism reliability studied in this paper is a nonlinear planning problem, so a sequential quadratic programming algorithm is used to solve it. The algorithm is very suitable for solving small and medium-sized smooth nonlinear problems containing constraints because its fast convergence, timesaving, and high accuracy [10, 11]. A firearms automaton is a system that consists of subsystems such as the case ejecting mechanism, feeding mechanism and locking mechanism. Therefore, the objective function of reliability optimization is generally the maximum system reliability, and the reliability of the subsystems is used as a constraint. The mathematical description of the optimization model is assumed as follows:

Objective function: max *R*(*X* )

Constraints: *Gi* = 0, *Gi* ≤ 0

where *X* = [*X*1, *X*2,..., *Xn*] *<sup>T</sup>*are the design variables. *G*(*X* ) <sup>=</sup> [*g*1(*X* ), *g*2(*X* ),..., *gm*(*X* )] *<sup>T</sup>*is the constraint function. By introducing Lagrange multipliers, the SQP optimization algorithm firstly transforms the above constrained optimization problem into an unconstrained optimization problem.

$$L(X, \lambda) = R(X) + \sum^{m} \lambda\_i g\_i(X) \tag{5}$$

where λ is the Lagrange multiplier.

The constraints are then linearized to obtain the subproblem of quadratic programming:

Objective function:

$$\min \frac{1}{2} d^T H\_k d + \nabla R(X\_k)^T d \tag{6}$$

Constraints:

$$\nabla \mathbf{g}\_i(X)^T d + \mathbf{g}\_i(X) = 0, i = 1, 2, \dots, m\_e \tag{7}$$

$$\nabla g\_i(X)^T d + g\_i(X) \le 0,\\ i = m\_{\epsilon+1}, m\_{\epsilon+2}, \dots, m \tag{8}$$

where *d* is the search direction of the global variable; ∇ denotes the gradient; and *Hk* is the positive definite proposed Newtonian approximation of the Hession matrix of the Lagrange function.


**Table 1** Main design variables and their values

The SQP optimization algorithm implements nonlinear constrained planning precisely by solving quadratic programming subproblems with the quadratically approximated Lagrange function. The transformed objective function can be solved using any quadratic programming algorithm.

### **4 Results and Discussion**

### *4.1 Design Variables of Firearms Automata*

The three most common failures of firearms automaton are case jam, cartridge jam, and insufficient recoiling. And the corresponding mechanisms are the case-ejecting mechanism, feeding mechanism, and locking mechanism, respectively. In this study, the reliability of the firearms automaton is the product of the reliability of the above three mechanisms, and the design variables are the pressure coefficient of air chamber *x*1, friction coefficient of the guide *x*2, stiffness of the counter-recoiling spring *x*3, and stiffness of the hammer spring *x*4, and the main design variables and their values are shown in Table 1.

# *4.2 Objective Function and Constraints of Reliability Optimization*

The goal of the reliability optimization of the firearms automaton is to maximize the reliability of the system. Hence, the objective function can be expressed as:

$$\max \, R(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = R\_1 \cdot R\_2 \cdot R\_3 \tag{9}$$

Where *R*1, *R*2 and *R*3 represent the reliability of case-ejecting mechanism, feeding mechanism and locking mechanism respectively. For firearms in the design stage, different mechanisms need to be allocated a corresponding reliability level requirement, this paper takes the reliability level of each mechanism is not less than 0.998, thus obtaining the reliability optimization constraints are as follows:

$$R\_1(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = P(\upsilon\_1 \ge \mathbf{3.5}) \ge 0.998 \tag{10}$$

$$R\_2(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = P(\upsilon\_2 \ge 2.8) \ge 0.998 \tag{11}$$

$$R\_3(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = P(\upsilon\_3 \ge \mathcal{D}) \ge 0.998 \tag{12}$$

Where *v*1, *v*2 and *v*3 represent the case ejecting velocity, cartridge feeding velocity and recoil in place velocity, respectively. Through the calculation of the dynamics model of the firearms automaton, the velocity of each characteristic point can be represented by the following surrogate model, respectively:

$$\begin{aligned} v\_1 &= -13.4 + 27.2 \cdot \mathbf{x\_1} - 2.8 \cdot \mathbf{x\_2} + 0.02 \cdot \mathbf{x\_3} + 0.06 \cdot \mathbf{x\_4} \\ &+ 1.3 \cdot \mathbf{x\_1} \cdot \mathbf{x\_2} + 0.01 \cdot \mathbf{x\_1} \cdot \mathbf{x\_3} - 0.04 \cdot \mathbf{x\_1} \cdot \mathbf{x\_4} - 0.01 \cdot \mathbf{x\_2} \cdot \mathbf{x\_3} \\ &+ 0.03 \cdot \mathbf{x\_2} \cdot \mathbf{x\_4} - 9.4 \cdot \mathbf{x\_1} \cdot \mathbf{x\_1} - 0.23 \cdot \mathbf{x\_2} \cdot \mathbf{x\_2} - 0.0001 \cdot \mathbf{x\_3} \cdot \mathbf{x\_3} - 0.0002 \cdot \mathbf{x\_4} \cdot \mathbf{x\_4} \end{aligned} \tag{13}$$

$$\begin{aligned} v\_2 &= -3\mathbf{3} + 6.6 \cdot \mathbf{x}\_1 + 0.6 \cdot \mathbf{x}\_2 + 0.01 \cdot \mathbf{x}\_3 + 0.02 \cdot \mathbf{x}\_4 \\ &+ 0.25 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_2 + 0.002 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_3 - 0.005 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_4 - 0.004 \cdot \mathbf{x}\_2 \cdot \mathbf{x}\_3 \\ &- 0.016 \cdot \mathbf{x}\_2 \cdot \mathbf{x}\_4 - 2 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_1 + 0.79 \cdot \mathbf{x}\_2 \cdot \mathbf{x}\_2 \end{aligned} \tag{14}$$

$$\begin{aligned} v\_3 &= -16.4 + 30 \cdot \mathbf{x}\_1 - 0.5 \cdot \mathbf{x}\_2 + 0.037 \cdot \mathbf{x}\_3 + 0.05 \cdot \mathbf{x}\_4 \\ &+ 1.19 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_2 + 0.012 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_3 - 0.017 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_4 - 0.013 \cdot \mathbf{x}\_2 \cdot \mathbf{x}\_3 \\ &- 0.003 \cdot \mathbf{x}\_2 \cdot \mathbf{x}\_4 - 11.5 \cdot \mathbf{x}\_1 \cdot \mathbf{x}\_1 + 0.19 \cdot \mathbf{x}\_2 \cdot \mathbf{x}\_2 \end{aligned} \tag{15}$$

### *4.3 Reliability Optimization Results*

Keeping other conditions unchanged, when the objective function is set as the maximum reliability of case ejecting, the maximum reliability of cartridge feeding, the maximum reliability of sufficient recoil and the maximum reliability of system, the optimization results are shown in Table 2 respectively.

As can be seen from Table 2, it is advantageous to increase the pressure coefficient of air chamber appropriately to improve the reliability of each mechanism and system. A smaller friction coefficient (0.149) is advantageous to improve the reliability of


**Table 2** Reliability optimization results obtained with different objective functions

case ejecting mechanism, feeding mechanism, and locking mechanism, but the ideal value of friction coefficient corresponding to the maximum system reliability is 0.259.

Increasing the stiffness of counter-recoil spring and the hammer spring is advantageous to feeding and the recoiling, but it would increase the resistance in the recoil process, so it will reduce the reliability of case ejecting mechanism. From the point of view of the maximum reliability of the system, the ideal value of the stiffness of the counter-recoil spring and the hammer spring are 230 N/m and 80 N/m, respectively.

The optimized design variables are close to the values of the actual existing products, and the maximum relative error is 1.3%, which proves that the optimization method given in this paper is effective.

### **5 Conclusion**

In this paper, a mechanism reliability optimization design method of firearms automaton is proposed. The dynamics and reliability model of a gas operating firearm automaton are established. And the reliability optimization design of firearms automaton is completed based on the SQP optimization algorithm, and the main conclusions obtained are as follows:


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Study on Kinetic Energy Conversion of Perforating Shaped Charge Jet in Perforating Completion**

**Zhenxiang Li, Fayong Yuan, Ruifeng Guo, Zhihang Chen, and Zhengjin Zhang** 

**Abstract** The energy of a perforating shaped charge is the cause of transient pressure fluctuations in a wellbore. Based on the law of energy conservation, the energy can be divided into the kinetic energy of a jet, the residual energy of a wellbore, and energy dissipation. The jet kinetic energy is used to penetrate a perforating gun, casing, and formation. The residual energy of a wellbore is used to cause wellbore pressure fluctuations. Based on the fluid–structure coupling principle, a jet penetration model was developed to improve the conversion rate of the jet kinetic energy, reduce the residual energy of a wellbore, protect wellbore safety, and reduce the downhole perforating completion accident. This model took into account a penetrating charge shell, explosive, liner, perforating gun, and casing. Meanwhile, the penetration process and kinetic energy conversion of a perforating shaped charge jet were studied. The obtained results indicated that the kinetic energy conversion of a perforating shaped charge jet is significantly affected by the cone angle of a liner, the thickness of a liner, and explosive mass. The 70°cone angle of a liner, its 1 mm thickness, and the 25 g explosive mass have the maximum kinetic energy conversion in the research range.

**Keywords** Perforation · Jet penetration · Jet kinetic energy · Fluid–structure coupling

Z. Li (B)

F. Yuan · R. Guo · Z. Chen · Z. Zhang Downhole Testing Company, Sinopec Oilfield Service Jianghan Corporation, Wuhan 430000, China

© The Author(s) 2024

Sinopec Exploration Branch, Chengdu 610041, China e-mail: lqs0313@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_23

### **1 Introduction**

Perforation completion is a widely used completion method [1–3]. The energy of a perforating shaped charge can be divided into the kinetic energy of a jet, the residual energy of a wellbore, and energy dissipation after the perforating shaped charge explosion. During the perforating explosion, the liner gradually liquefies to form a perforating shaped charge jet. A perforating gun, casing, and formation are penetrated by the perforating shaped charge jet to make a channel that connects a reservoir to a wellbore [4, 5]. To increase the production of oil and gas wells and lower operating costs, various construction parameters should be set fairly according to different operating environments while constructing the perforation scheme on site. However, with various perforating conditions, the kinetic energy conversion of the perforating shaped charge jet change. Therefore, studying the conversion relationship and influence of the perforating shaped charge structure on the kinetic energy of the perforating shaped charge jet is critical to improve the conversion rate of the jet kinetic energy and increase the production.

The perforating shaped charge jet has attracted much attention among researchers, and some studies have been performed. In their studies, Brown et al. [6] demonstrated the derivative mechanism of a perforating shaped charge jet and offered suggestions for further research. Lee [7] used a simplified two-dimensional numerical calculation model combined with the Euler algorithm to simulate a perforating shaped charge jet process. Jin et al. [8] established a two-dimensional perforating shaped charge jet simulation model by using the ALE algorithm to study the formation mechanism, velocity evolution, and penetration characteristics of a perforating shaped charge jet. The obtained results indicated that the cone angle of the liner has a great influence on the velocity and shape of a perforating shaped charge jet, the mass of the jet head and the slug, and the penetration depth of a perforating shaped charge jet. Cao et al. [9] established a two-dimensional jet simulation model of two perforating shaped charges to analyze the influence of interference between perforating shaped charges on a perforating shaped charge jet and the factors causing interference between perforating shaped charges. Kang and Sheng [10] established a three-dimensional simulation model to study how a perforating shaped charge explosion crushes the liner and penetrates the casing. The change laws of the velocity and energy of a perforating shaped charge jet were carried. Suneson [11] used the LS-DYNA software to conduct a finite element simulation on the conversion of a solid metal liner covert into a high-speed metal fluid and the penetration of the high-speed metal fluid into a concrete medium. Liu et al. [12] established a two-dimensional simulation model of a perforating shaped charge jet to analyze the formation process of a perforating shaped charge jet and the proportion of the jet kinetic energy for different explosive types and densities. He [13] conducted an experiment on a new perforating shaped charge liner, and the results of the explosive test showed that the perforating shaped charge liner could achieve comparable or even a greater penetration performance. Elshenawy [14] and Du [15] studied the effect of the target strength on the shaped charge jet penetration. The findings demonstrate that when the target yield strength increases, The penetration depth of the shaped charge jet in the target significantly decreases.

Numerous studies on the generation of a perforating shaped charge jet, its penetrating properties, and the velocity have been conducted by academics. However, little research has been performed on the conversion relationship and influence of the perforating shaped charge structure on the jet kinetic energy and the improvement of the conversion rate of the jet kinetic energy. In this research, a jet penetration model is developed based on the ALE algorithm. The factors including the cone angle and thickness of the liner and the explosive mass affecting the energy conversion law of the jet kinetic energy are systematically studied.

### **2 Jet Kinetic Energy of a Perforating Shaped Charge**

The liner is liquefied into a thinner jet in the front and a thicker slug in the back due to the explosive of a perforating shaped charge. A perforating gun, casing, and formation are penetrated by a perforating shaped charge jet created by the jet and the slug to create a channel that connects a reservoir to a wellbore. Meanwhile, the kinetic energy of the jet and slug gradually decreases during the penetration process. Figure 1 depicts a wellbore and a typical perforating shaped charge structure. The figure shows that a typical perforating shaped charge structure consists of a shell, explosive, and liner.

Based on the kinetic energy theorem, the jet kinetic energy could be expressed as:

$$E\_J = \frac{1}{2} (m\_j v\_j^2 + m\_c v\_c^2) \tag{1}$$

**Fig. 1** Wellbore structure and typical perforating shaped charge structure

where *mj* is the jet mass; *vj* is the jet velocity; *mc* is the slug mass; *vc* is the slug velocity.

In this research, a jet penetration model was established to study the proportion of the total energy of a perforating shaped charge converted into the jet kinetic energy and the influence of various factors on the kinetic energy of a perforating shaped charge jet.

### **3 Jet Penetration Model**

### *3.1 ALE Algorithm and Geometric Model*

The penetration simulation of a perforating shaped charge jet is a large deformation and fluid–structure coupling problem. The mesh of the jet penetration model will be deformed and the calculation will be simple to stop if the Lagrange algorithm is adopted. The Euler algorithm requires a high level of mesh precision, resulting in a high calculation cost. To effectively prevent grid distortion, The ALE algorithm can track the motion of a material's boundary and allow material to move freely within a space grid [16]. Therefore, the ALE algorithm was adopted to simulate the penetration process of a perforating shaped charge jet in this paper.

For the structure of a perforating shaped charge, the geometric model of the perforating shaped charge was established, as shown in Fig. 2. The cone angle of the liner was 60°. The thickness of the liner was 1.5 mm. The explosive mass was 25 g. The outer diameter of a casing was 127 mm. The wall thickness of a casing was10.36 mm. The outer diameter of a perforating gun was 86 mm. The wall thickness of a perforating gun was 6.45 mm. A blind hole diameter was 40 mm. The depth of a blind hole was 4 mm. In the meshing process of the jet penetration model, the solid mesh and the ALE mesh overlap. Therefore, a penalty function was used to realize the fluid–structure coupling of the jet penetration simulation. When the cell size of the mesh is 1 mm, the calculation time is 5 h, and the energy ratio is only 0.1% less than that of 0.8 mm. To improve computing efficiency, 1 mm was selected as the basic mesh size.

### *3.2 Material Constitutive and State Equation*

The HNS explosive was used in the perforating shaped charge of this model. The specific parameters are displayed in Table 1. The High-Explosive-Burn material model was selected to define the HNS explosive properties. The JWL equation was conducted to calculate shockwave pressure due to the HNS explosive explosion, whose expression was as follows:

#### **Fig. 2** Geometric model


$$P\_1 = A \left(1 - \frac{\alpha}{R\_1 V}\right) e^{-R\_1 V} + B \left(1 - \frac{\alpha}{R\_2 V}\right) e^{-R\_2 V} + \frac{\alpha E}{V} \tag{2}$$

where *P1* is an isentropic pressure; *V* = ρ*0*/ρ = *v*/*v0* is a relative specific volume; *A*, *B*, *R1*, *R2*, ω are undefined constants, respectively; *E* is the internal explosion energy per unit initial volume of the explosive.

Air parameters are shown in Table 2. The Mat-Null material model and The Eos-Linear-Polynomial state equation were conducted to defined air properties. The pressure linear polynomial of the state equation was expressed as:

$$P\_2 = C\_0 + C\_1\mu + C\_2\mu^2 + C\_3\mu^3 + \left(C\_4 + C\_5\mu + C\_6\mu^2\right)E\tag{3}$$

where, *P2* is air pressure; μ is air density ratio; If μ < 0, *C2*μ*<sup>2</sup>*= *C6*μ*<sup>2</sup>*= 0, *C0* ~ *C6* are the state equation coefficient.

Perforating fluid parameters are shown in Table 3. The Mat-Null material model and The Gruneisen state equation were conducted to defined perforating fluid properties. The Gruneisen state equation of a perforating fluid was expressed as:

$$p = \frac{\rho\_0 C^2 \mu\_k \left[1 + (1 - \gamma\_0/2)\mu\_k - \frac{\alpha}{2}\mu\_k^2\right]}{\left[1 - (S\_1 - 1)\mu\_k - S\_2 \frac{\mu\_k}{\mu\_k + 1} - S\_3 \frac{\mu\_k^3}{(\mu\_k + 1)^2}\right]^2} + (\wp\_0 + \alpha \mu\_k) E\_s \tag{4}$$



**Table 3** Perforating fluid parameters


where, ρ*0* is the initial density of perforating fluid; *C* is the intercept of the wave velocity curve; μ*k* is the compressibility factor of perforating fluid; γ *<sup>0</sup>*is the Grueisen constant; *S1, S2, S3* are the slope coefficients of the wave velocity curve; α is the first order volume correction; *Es* is the internal energy.

The liner was made of a high-conductivity oxygen-free copper. The parameters are shown in Table 4. The Mat-Steinberg material model and the Gruneisen state equation were conducted to defined liner properties. The Mat-Steinberg material model was expressed as:

$$G = G\_0 \left[ 1 + b\_i p V^{\frac{1}{7}} - h \left( \frac{E\_i - E\_c}{3R'} - 300 \right) \right] \frac{-f E\_i}{E\_m - E\_i} \tag{5}$$

$$
\sigma\_s = \sigma\_0' \left[ 1 + b\_i p \, V^{\frac{1}{\tau}} - h \left( \frac{E\_i - E\_c}{3R'} - 300 \right) \right] \frac{-fE\_i}{E\_m - E\_i} \tag{6}
$$

where, *p* is pressure; *V* is the relative volume; *Ec* is the cold compression energy; *Em*  is the melting energy; *R*' = *R*ρ*/A0*, *R* is a gas constant; *A0* is an atomic weight; As the material melts, σ*s* and *G* are reduced to half of their initial values as the material melts.

Casing was made of 25CrMnMo. The parameters are shown in Table 5. The Johnson–Cook material model and the Gruneisen state equation were conducted to defined casing properties.

$$
\sigma\_\circ = \left( A + B \overline{\varepsilon}^{p^m} \right) \left( 1 + C \ln \dot{\varepsilon}^\* \right) \left( 1 - T^{\*m} \right) \tag{7}
$$

$$\varepsilon^{f} = \left(D\_1 + D\_2 \exp D\_3 \sigma^\*\right) \left(1 + D\_4 \ln \dot{\varepsilon}^\*\right) \left(1 + D\_5 T^\*\right) \tag{8}$$

**Table 4** Liner parameters


**Table 5** Casing parameters



**Table 6** Perforating gun and shell parameters

where, σ*y* is a yield stress; *A, B, C, n, m* are the input constant, ε*<sup>P</sup><sup>n</sup>*is an equivalent plastic strain; ε˙<sup>∗</sup> = ε˙ *p e* /ε˙0 is a relative equivalent plastic strain rate; *T\** is a homologous temperature. σ*\** is a ratio of pressure to the Von-Mises equivalent stress; *D1*–*D5*  are damage coefficients.

The material 32CrMO4 was used to make a perforating gun and perforating bullet shell. The parameters are shown in Table 6. The Plastic-Kinematic material model was conducted to defined perforating gun and perforating bullet shell properties.

# **4 Process of Perforating Shaped Charge Jet and Jet Kinetic Energy Conversion Analysis**

### *4.1 Penetration Process of Perforating Shaped Charge Jet*

Figure 3 depicts the forming and penetrating process of a perforating shaped charge jet a perforating. When t = 0 µs, the explosive started to explode from the top, producing a large amount of high temperature and high pressure gas that liquefied the liner and propagated the shockwave formed by the gas along the axis, as shown in Fig. 3a. When t = 10 µs, the shockwave swept along the liner and the liner was liquefied into a thinner jet in the front and a thicker slug in the back, as shown in Fig. 3b. When t = 40 µs, a perforating shaped charge jet still moved along the axis direction at a great speed and a violent collision occurred at a perforating gun and casing, which penetrated the perforating gun and casing and formed a channel connecting a reservoir and wellbore, as shown in Fig. 3c. In this process, due to the resistance of the above factors, the jet kinetic energy gradually decreased in this process until the energy was completely consumed and the jet penetration process also ended.

**Fig. 3** Penetration process of perforating shaped charge jet

# *4.2 Kinetic Energy Conversion Analysis of Perforating Shaped Charge Jet*

Figures 4 and 5 show the relationship between the total energy of the explosive and the kinetic energy of a perforation shaped charge jet over time. With the penetrating process of a perforating shaped charge jet, the total energy of the explosive gradually decreased from 110.92 kJ. The kinetic energy of a perforation shaped charge jet reached its maximum of 17.185 kJ at 12 µs and then decreased gradually due to air resistance and penetration influence. The kinetic energy conversion rate of a perforation shaped charge jet was 15.49%.

### **5 Analysis of Influencing Factors of Jet Kinetic Energy**

### *5.1 Cone Angle of Liner*

By changing the size of the cone angle of the liner and keeping all other conditions constant. The influence of the cone angle of the liner on the kinetic energy of a perforation shaped charge jet was studied. The cone angles were 45°, 50°, 55°, 60°, 65°, and 70°. Its energy conversion rate is shown in Fig. 6.

As shown in Fig. 6, with an increase in the cone angle of the liner, the kinetic energy ratio of a perforation shaped charge jet increase. The obtained results indicated that a perforation shaped charge jet is divided into the front jet and the rear slug. With an increase in the cone angle of the liner, the front jet becomes thicker. The mass proportion of the front jet gradually increases and the front jet velocity is increased. Therefore, according to the kinetic energy theorem, the kinetic energy conversion rate of a perforation shaped charge jet is increased with an increase in the cone angle of the liner.

### *5.2 Thickness Angle of Liner*

By changing the thickness of the liner and keeping all other conditions constant. The influence of the thickness of the liner on the kinetic energy of a perforation shaped charge jet was studied. the thicknesses of the liner were 1 mm, 1.25 mm, 1.5 mm, 1.75 mm, 2 mm, and 2.25 mm. Its energy conversion rate is shown in Fig. 7.

As shown in Fig. 7, with an increase in the thickness of the liner, the kinetic energy ratio of a perforation shaped charge jet decreases. The obtained results indicated that the energy derived from the explosive is used partly to melt the liner and partly

to maintain the kinetic energy of a perforation shaped charge jet. With an increase in the thickness of the liner, the energy required to melt the liner is increased, the kinetic energy of a perforation shaped charge jet is decreased, and the kinetic energy conversion rate of a perforation shaped charge jet is decreased.

### *5.3 Explosive Mass*

By changing the explosive mass and keeping all other conditions constant. The influence of the explosive mass on the kinetic energy of a perforation shaped charge jet was studied. Explosive masses were 25 g, 30 g, 35 g, 40 g, 45 g, and 50 g. Its energy conversion rate is shown in Fig. 8.

As shown in Fig. 8, with an increase in the explosive mass, the kinetic energy ratio of a perforation shaped charge jet decreases. The obtained results indicated that the explosive total energy and the kinetic energy of a perforation shaped charge jet increase with an increase in explosive mass. But the increase rate in the explosive total energy is significantly greater than the kinetic energy of a perforation shaped

charge jet. Therefore, the ratio of the two decreases. the kinetic energy conversion rate of a perforation shaped charge jet is decreased.

### **6 Conclusions**

A jet penetration model was established based on the kinetic energy ratio of a perforation shaped charge jet. The typical perforating charge—perforating gun—casing structure, the penetration process of a perforation shaped charge jet, the kinetic energy conversion relationship of a perforation shaped charge jet, and its influencing factors were analyzed. The following are the conclusions:

In the angle range of 45° to 70°, the kinetic energy conversion of the 70° cone angle of the liner reached a maximum of 16.26%. In the thickness range of 1 mm to 2.25 mm, the kinetic energy conversion of the 1 mm thickness of the liner reached a maximum of 16.24%. In the mass range of 25 g to 50 g, the kinetic energy conversion of the 25 g explosive mass reached a maximum of 15.49%.

The cone angles and thicknesses of a liner and explosive mass all affected the kinetic energy of a perforation shaped charge jet. In the actual production operation, the effect of a perforating charge structure on the jet kinetic energy should be fully considered. It is helpful to improve the kinetic energy conversion rate of a perforation shaped charge jet and the use efficiency of explosives. Meanwhile, the ratio of the kinetic energy of a perforation shaped charge jet to its total energy is determined, providing pre-data support for future predictions of the transient pressure of perforating detonation.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Characterization of Solder Mechanical Properties Based on Nanoindentation**

**Wenjun Sheng, Zhihai Wang, and Huiming Cheng** 

**Abstract** This article systematically elaborates on the in-situ measurement of solder using nanoindentation technology, and combines numerical inversion methods to analyze its elastic–plastic and creep mechanical properties. The elastic–plastic and creep constitutive equations of solder are constructed, and the influence of temperature on the elastic–plastic and creep characteristic of solder is studied. This provides an effective in-situ measurement and analysis method for characterizing the mechanical properties of high-density micro solder interconnection structures in radar electronic equipment.

**Keywords** Nanoindentation · Solder · Mechanical properties · Constitutive model · Elastic–plastic · Creep

# **1 Introduction**

With the rapid development of radar electronic technology, the functionality and integration of radar systems are becoming stronger and higher, and the reliability requirements for radar electronic equipment are also increasing. The more complex and integrated the radar electronic system is, the more small interconnected structures such as BGA (Ball Grid Array Package) solder jointsin related electronic modules, especially in the transceiver components. Typical tantalum capacitor solder joints with SMT (Surface Mount Technology) process are shown in Fig. 1. Related studies have shown that 70% of the reasons for electronic device failure are caused by solder interconnect failure, and mechanical connection is one of the most important functions of solder. The mechanical properties of solder greatly affect the failure rate of electronic products, especially flexible electronic systems. The development of

W. Sheng · Z. Wang

No. 38 Research Institute of China Electronics Technology, Hefei, China

H. Cheng (B)

303

No. 34 Research Institute of China Electronics Technology, Guilin, China e-mail: coraldevil@126.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_24

relevant research requires an accurate grasp of the mechanical properties of solder joints and the provision of their mechanical constitutive models. With the miniaturization and versatility of electronic modules such as transceiver components, the size of solder interconnection structures for electronic components is becoming smaller and smaller. Under the submillimeter size conditions of actual connection status, the forming conditions are significantly different from that of bulk solder. Whether there is a change in the mechanical properties of solder requires relevant in-situ testing technology research to accurately characterize the mechanical properties of solder. Thus providing a reliable basis for the interconnection design of electronic modules such as high-density and highly integrated transceiver components is important.

The traditional uniaxial tensile/compressive testing methods have specific requirements for the size of the sample, and are often macroscopic standard sizes. This type of method is insufficient for the testing and characterization of the mechanical properties of small-scale solder joints. The micro nano indentation testing method developed based on contact mechanics theory has been widely used in the field of surface engineering, microelectromechanical systems, and bioengineering to test and characterize the mechanical properties of functional materials. This method uses hard indenters of different shapes (usually diamonds) to in-situ press the surface of the tested material, and the instrument automatically records the real-time pressing load and depth (P–h) during the pressing process, and the mechanical performance parameters of the material are obtained through relevant mechanical models. This technology has the advantages of small required testing area, high testing accuracy, and the ability to achieve in-situ testing. Compared to traditional tensile/compressive

**Fig. 1** Tantalum capacitor solder joints

testing, the indentation method does not require too much material shape and size, and only needs to provide a smooth and flat surface, making it very suitable for characterizing the mechanical properties of small size materials/structures. The study of the mechanical properties of electronic packaging solder through micro nano indentation testing technology has attracted widespread attention in the industry. Marques [1] studied the high-temperature mechanical properties of lead-free solder joints through micro nano indentation testing of high-temperature modules. The results showed that the hardness and elastic modulus of SnAgCu lead-free solder exhibited temperature sensitivity, while the mechanical properties of its intermetallic compounds exhibited anisotropy. Tabor [2] found through extensive experimental research that there is a specific level of plastic strain for the indentation test of conical indenters ε*r* . Just ensure that the uniaxial stress–strain curve of the material passes through - ε*y* + ε*r* , σ*<sup>r</sup>* , the hardness values obtained are all equal, and thus the hardness values and characteristic stresses are established σ*r* . The one-to-one correspondence between r greatly simplifies the inverse analysis problem of indentation. Yu Blanchard et al. [3] combined elastic theory and tangent slip theory to establish the relationship between hardness, yield stress, elastic modulus, and Poisson's ratio. On this basis, Dao [4] established a series of dimensionless functions based on dimensional analysis to describe the relationship between metal indentation response and its elastic–plastic parameters. This method uses the least squares method to

perform dimensional analysis on the indentation response of multiple sets of materials with known mechanical properties and obtain characteristic plastic strains. Viola et al. [5–8] established an improved method for determining the plastic performance parameters of metal materials based on the nanoindentation method and numerical simulation. By successively determining the characteristic stress, characteristic plastic strain, strain strengthening index, and initial yield limit, the power constitutive relationship of the metal is obtained. Some scholars have conducted research on extracting plastic parameters of various metal materials through micro nano indentation testing [9–14]. Due to the low melting point of solder, significant creep usually occurs at room temperature. The above research indicates that the combination of micro nano indentation method and numerical simulation back analysis provides an effective way to obtain the elastic–plastic constitutive relationship of electronic packaging interconnection metals through in-situ measurement.

### **2 Experimentation**

Due to the local action mode of micro nano indentation testing, the stress–strain field of the material below the indenter (usually conical) is relatively complex. The difficulty in this field is to effectively extract the elastic–plastic parameters and related constitutive relationships of metal materials through this technology. Establishing a suitable mechanical model is the key to effectively extract the elastic– plastic parameters of materials through indentation testing. The current analysis model for indentation testing is based on the theoretical framework of continuum mechanics. Considering the complexity of indentation contact problems, it is difficult to strictly derive its analytical solution through forward analysis. Using indenters with different cone angles for indentation testing and conducting systematic research through numerical inversion analysis is an effective way to extract the elastic–plastic parameters of solder alloys. Related studies introduce characteristic stresses (σ*r* ) and characteristic plastic strain (ε*r* ) for back analysis. This method indicates that by using the same (ε*r* , σ*r* ) the elastic–plastic constitutive relationship curves of points with different power hardening exponents exhibit consistent indentation load depth curves under specific cone angle indenters, indicating the existence of uniqueness issues in inferring the power hardening relationship curve through a single cone angle indenter. In view of this, this article uses a cylindrical indenter combined with two conical indenters with different cone angles to conduct micro nano indentation testing on the solder alloy. The load control method is "fast loading + holding + fast unloading". This loading method can avoid the viscous deformation of the solder alloy in the loading and unloading sections, improve the testing accuracy of the solder alloy's elastic–plastic mechanical properties, and characterize the microscale solder mechanical properties through this method. The nanoindentation experiment in this article was conducted on Sn63Pb37 solder thin films in actual interconnected states, and elastic–plastic and creep tests were conducted under different temperature conditions.

### *2.1 Sample Preparation*

Using a laser scribing machine, cut 0.2 mm thick MoCu thin sheets and Sn63Pb37 soldering lug into 3 mm \* 3 mm sizes. Clean the MoCu thin sheets and Sn63Pb37 soldering lug with alcohol solvent, and apply soldering flux on the surface of the MoCu thin sheets. As shown in Fig. 2a, stack them in the order of MoCu thin sheets, Sn63Pb37 soldering lug, and MoCu thin sheets. As shown in Fig. 2b, place the stacked module in the groove of the aluminum reflow welding fixture, Finally, perform reflow soldering in the reflow soldering system (SMT Quattro Peak M), with a maximum reflow soldering temperature of 220 ºC. After the welding is completed, the welding sample is fixed with a metal frame and inserted with a hot inlay resin. After the welding is completed, the molybdenum copper welding sample is fixed with a metal frame, and inserted with a hot inlay resin. The cross section of the sample is further cross polished with sandpaper, and polished with flannelette and diamond spray polishing agent as shown in Fig. 2c. Before the experiment, use ethanol ultrasonic cleaning to avoid the influence of surface pollutants.

**Fig. 2 a** Welding sample profile, **b** Welding fixture, **c** Sample profile after polishing

### *2.2 Nano Indentation Testing*

The mechanical properties of solder alloy samples were tested and characterized using the Nano Indenter G200 type (Agilent Technologies Inc.) micro nano indentation testing system (as shown in Fig. 3a). The system is equipped with a resistance heating high-temperature testing module, with load and displacement resolutions of 50 nN and <0.01 nm, respectively. The loading frame stiffness is ≈ 1.0 × 107 N/ m. The conical indenter used for nanoindentation testing is a diamond indenter with equivalent half cone angles of 61.43º and 70.32º, respectively. The tip radius of the conical indenter is ≤20 nm, and the diameter of the cylindrical diamond indenter is 5.5 μm. The nanoindentation test adopts the Continuous Stiffness Measurement Method (CSM) and adopts the "fast loading +holding+fast unloading" load loading method. During the test, the indenter slowly moves down to the surface of the contact sample, and the maximum indentation load is determined through multiple prediction tests. The loading time, holding time, and unloading time of the cylindrical indenter are 5 s, 10 s, and 2 s, respectively. During testing, the compressive load linearly increases from 0 to the maximum value within 5 s, then maintains the maximum compressive load value for 10 s, and finally decreases from the maximum value to 0 within 2 s. The loading time, holding time, and unloading time of the conical indenter are 2 s, 400 s, and 2 s respectively. During testing, the indentation load linearly increases from 0 to the maximum value within 5 s, and then maintains the maximum indentation load value for 400 s to obtain the creep behavior of the solder alloy. Finally, the indentation load decreases from the maximum value to 0 within 2 s. The allowable thermal drift for room temperature indentation testing (25 ºC) is set to ≤0.1 nm/s, and the allowable thermal drift for high temperature indentation testing (50 ºC, 75 ºC, 100 ºC, and 125 ºC) is set to ≤5 nm/s. This results in indentation testing results at different temperature points, as shown in Fig. 3b for the intermediate welding fracture surface. As shown in Fig. 4, the P–h curve obtained from nanoindentation testing using three types of indenters at five temperatures is shown.

The P–h curves obtained from the nanoindentation tests of three types of indenters are shown in the following figure.

**Fig. 3** Nanoindentation indentation testing equipment and indentation

**Fig. 4** P–h curve of nanoindentation test **a** Cylindrical indenter **b** 61.43° conical indenter **c** 70.32° conical indenter

### **3 Calculation of Constitutive Relationship**

In the inverse calculation of the mechanical properties of solder, the unloading section data of cylindrical indenters and the loading section data of 61.43° and 70.32° triangular pyramid indenters are used to analyze the elastic–plastic constitutive relationship of solder, and the holding section data of 61.43° and 70.32° triangular pyramid indenters are used to analyze the creep constitutive relationship of solder.

### *3.1 Calculation of Elastic–plastic Constitutive Relationship*

The elastic–plastic constitutive relationship of common metal/alloy materials can be expressed in the form of a power function, and the specific expression is shown as Eq. (1).

**Fig. 5** Elastic modulus fitting curve

$$\sigma = \begin{cases} E\varepsilon, \varepsilon \le \sigma\_\text{y}/E \\ R\varepsilon^n, \varepsilon \ge \sigma\_\text{y}/E \end{cases} \tag{1}$$

In the formula, *R* = σ*<sup>y</sup> E*  σ*y n*  is the strength coefficient. The parameters to be determined are: elastic modulus E, Poisson's ratio υ, yield stress σ*<sup>y</sup>* , and strain hardening index n, with Poisson's ratio assumed to be 0.3.

Due to the constant contact area of the cylindrical indenter, its unloading load and depth are theoretically linear. Therefore, based on the slope value of the unloading section, the reduced modulus Eris obtained based on Eq. (2), and the elastic modulus E of the solder alloy is further obtained. In the equation, a is the radius of the cylindrical indenter.

$$\frac{dP}{dh} = 2E\_r a \tag{2}$$

As shown in Fig. 5, it is the fitting result of the elastic modulus, and the slope of the straight line is the elastic modulus at the corresponding temperature.

Based on the loading section data obtained under the action of two different conical indenters with different cone angles, a quadratic fitting is performed to obtain the loading curvature constant based on Eq. (3).

$$C = \frac{dP\_0}{dh^2} \tag{3}$$

On this basis, the power hardening elastic–plastic constitutive relationship is degenerated into an ideal elastic–plastic case (i.e. hardening index n = 0), and combined with the obtained elastic modulus value, ansys finite element software is used to model the indentation process, and numerical simulations are conducted to continuously change the input value of yield stress until the simulated loading curvature value of the loading section matches the experimental loading curvature value. At this point, the input value of yield stress is the characteristic stress of the solder alloy under a specific cone angle pressure head. Further based on the empirical

**Fig. 6 a** Fitting results, **b** Finite element simulation model of indenter pressing

relationship between characteristic plastic strain and the equivalent half cone angle of the indenter ε*r* = 0.0319*cot*α, the characteristic plastic strain under two different cone angle indenters is obtained. The fitting results and finite element simulation model are shown in Fig. 6.

Characteristic plastic strain and characteristic stress of solder alloy under specific cone angle indenter (ε*r* , σ*r* ) are got into the power strengthened elastic–plastic constitutive relationship yields as shown in Eq. (4).

$$
\sigma\_r = \mathbb{R} \left( 2 \frac{\sigma\_r}{E} + 2 \varepsilon\_r \right)^n \tag{4}
$$

Further based on the characteristic plastic strain and characteristic stress of the solder alloy obtained under two different cone angle indenters (ε*r* , σ*r* ) (with known elastic modulus E), the strength coefficient R, strain hardening index n, and yield stress σ*y* with Eqs. (4) and (5). Furthermore, the power hardening elastic–plastic constitutive relationship of the solder alloy was determined.

$$
\sigma\_\circ = \left(\frac{E^n}{R}\right)^{\frac{1}{n-1}}\tag{5}
$$

### *3.2 Calculation of Creep Constitutive Relationship*

The common tensile creep process is mainly divided into three stages, with the first stage where the creep rate continuously decreases and the material hardens, known as the unstable creep stage; Entering the second stage, when the creep rate reaches its minimum and lasts for a long time, it is a creep deformation stage of great concern in engineering, known as the steady-state creep stage; In the third stage, the creep rate rapidly increases, and the material begins to experience necking and failure. The steady-state creep stage accounts for the majority of the material life, especially for commonly used solder materials, which usually undergo creep at room temperature. During the creep process, the strain rate sensitivity of materials can be represented by a simple power-law relationship, which can be described based on the Wertman Dorn equation over a large range of strain rates as shown in Eq. (6).

$$
\dot{\epsilon} = A \sigma^n exp(\frac{-Q}{RT}) \tag{6}
$$

In the equation ε˙ is the strain rate, σ Is the applied creep stress, n is the creep stress index, Q is the creep activation energy, R is the ideal gas constant, T is the thermodynamic temperature, and A is a constant related to the alloy composition and structure.

When the temperature is constant, Eq. (6) is simplified as:

$$
\dot{\varepsilon} = C\_1 \sigma'' \tag{7}
$$

Given the geometric self similarity of the pyramid indenter, the indentation strain rate is defined as the ratio of the instantaneous indentation rate to the indentation depth, namely:

$$
\dot{s}\_i = \frac{dh}{dt} \frac{1}{h} \tag{8}
$$

Simultaneously pressing hardness and flow stress approximately satisfy a linear relationship *H* ∝ σ. Eq. (7) can be expressed as:

$$
\dot{\varepsilon}\_i = C\_2 H^n \tag{9}
$$

The compressive creep stress index is:

$$m = \frac{d\ln\dot{x}\_i}{d\ln H} \tag{10}$$

Based on the results of conical indenter indentation tests with different solder alloys at different temperatures, the corresponding creep stress index is obtained by processing and analyzing the data of the holding section, as shown in Fig. 7, which is the fitting result of the creep stress index.

Take the logarithm of both sides of Eq. (10) to obtain:

$$
\ln \dot{\varepsilon} = -\frac{\mathcal{Q}}{RT} + \ln A + n \ln \sigma \tag{11}
$$

Same flow stress σ at (i.e. hardness H), the activation energy of creep can be expressed as:

$$\mathcal{Q} = -R \cdot \frac{d \ln \dot{\varepsilon}}{d\left(\frac{1}{T}\right)}\tag{12}$$

Based on the data of Sn63Pb37 solder alloys with conical indenter pressure tests at different temperatures, the corresponding creep activation energy can be obtained through processing and analysis.

On the basis of obtaining the creep stress index and creep activation energy, the finite element software Ansys is used to perform numerical inversion analysis on the creep constant A by inputting the steady-state creep constitutive relationship. The simulated compressive creep curve is compared with the experimental curve until they match to determine the creep constant A.

### **4 Results and Discussion**

The elastic-plasticand creep constitutive parameters of Sn63Pb37 solder are obtained through inverse analysis at different temperature. The characteristic curves are shown in the following Fig. 8.

From the analysis results, it can be seen that in the elastic–plastic constitutive parameters, the higher the temperature, the smaller the elastic modulus, the larger the strain hardening index, and the smaller the yield strength. As the temperature increases, the strength coefficient first decreases and then increases, and reaches its minimum at 75 °C; In creep constitutive parameters, the higher the temperature, the smaller the creep stress index. As the temperature increases, the elastic–plastic and creep properties of the material are deteriorating.

This article used the same method to test the elastic–plastic properties of bulk Sn63Pb37 solder at room temperature. The constitutive parameters are shown in the Table 1 below. It can be seen that compared with the real welded sample, its elastic modulus increases and the yield strength decreases, indicating significant differences.

**Fig. 8 a** Elastoplastic characteristic curves, **b** Creep characteristic curves

**Table 1** Elastoplastic Constitutive Parameters of BulkSn63Pb37Solder


Therefore, it is necessary to conduct in-situ measurements of the mechanical properties of the solder in the real welded sample for more accurate stiffness and strength analysis in the future.

### **5 Conclusion**

This article systematically elaborates on the in-situ measurement of solder using nanoindentation technology, and combines numerical inversion methods to analyze its elastic–plastic and creep mechanical properties. The mechanical constitutive relationship of solder is constructed, and the influence of temperature on the elastic– plastic and creep properties of solder is studied. The mechanical properties of bulk materials and micro interconnected structures are compared. This provides an effective in-situ measurement and analysis method for the analysis of micro solder interconnection structures in high-density electronic components.

**Acknowledgements** This work was supported by the National Defense Basic Research Project (No. JCKY2019210B005).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **The Research on Structural Design of Vehicle Integrated Display and Control Console**

#### **Xiancheng Luo**

**Abstract** The display console and vehicle platform designed in this paper adopt an integrated structure design, the structure is simple and beautiful, easy to operate and easy to maintain, and the standard function module is selected as far as possible to meet the standardization requirements. Based on the design idea of generalization, serialization and combination, the display console is divided into unit design, which can be successively divided into: display unit, control unit, display and control processing unit and cabinet unit. Among them, the display unit and control unit learn from the mature design technology to ensure the design quality; The display and control processing unit is designed separately, with rigid connections to both the display unit and control unit externally, and the internal air duct is set up to meet the heat dissipation requirements. The cabinet unit is arranged under the display and control processing unit and the two are rigidly connected to ensure the overall structural rigidity and provide required installation space for user devices. The display console adopts split designs to support quick installation, disassembly and handling of four units. 3D design software is used to analyse the installation space, visual field, hand operation space, knee space and maintenance space of the display console to ensure that the display console has a good human–computer interaction function. The internal space of the display console is designed with three-dimensional wiring, the routing path of each cable in each unit is planned, and the cable can be customized in advance, which greatly improves the efficiency of on-site electrical assembly. Mechanical analysis software is used to simulate the main structure of the display console to ensure that the whole display console meets the mechanical performance requirements. The above design and analysis have certain guiding significance for similar design work of vehicle display console.

**Keywords** Display console · Integrated structure design · Split design · Three dimensional wiring

X. Luo (B)

© The Author(s) 2024

315

Department of Electromechanics, Jiangsu Automation Research Institute, Lian Yungang, Jiangsu 222006, China e-mail: luoxianchengcssc@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_25

### **1 Introduction**

In recent years, countries all over the world attach great importance to the application of advanced technical groups with digital information technology as the core in the military field in the construction of military quality, and compete to develop high-tech weapons and equipment, and the key technology to realize this strategy is battlefield digitalization. The application of this technology in the field of armored vehicle platforms is reflected in the field of vehicle electronics. The application purpose of vehicle electronics technology is to transform the information among subsystems of combat vehicles from self-contained and disconnected to computer-based interconnection among systems in the vehicle (see [1]). In the field of armored fighting vehicles, the prerequisite for digitalization is vehicle electronics technology with digital transmission capabilities, including systems and components such as power control, communications, information/data management, computers, sensors, signal processing and fire control. With the rapid development of weapons and equipment in the world, the electronic system of combat and logistics vehicles is more and more valued by the military forces of various countries. The highly integrated and high-tech equipment and equipment forces are an inevitable trend and the electronic system of vehicles will also play an increasingly important role in the battlefield. The on-board display and control terminal, as the control centre of the vehicle, will monitor all the equipment in the vehicle in real time.

However, in different carriers, electronic equipment to withstand a variety of different degrees of vibration environment for a long time, the structure of electronic equipment will appear fatigue damage, causing its performance to decline, and even cause the damage of electronic equipment. In 1956, Lunney and Crede performed vibration analysis and fatigue analysis of electronic devices based on test data from real-world environments (see [2]). In 1973, Steinberg conducted a systematic analysis of the mechanical characteristics of electronic components from component level to system level electronic chassis in the book of vibration analysis for electronic equipment (see [3]). Finite element analysis technology was used to analyze and study the dynamic effects of surface-mount components (see [4]). Finite element analysis technique was used to study the dynamic response, dynamic strain and dynamic stress of printed boards (see [5]). Finite element analysis technology and experimental technology were used to analyze and calculate the dynamic response of printed boards under falling environment (see [6]). The research work carried out abroad mainly focuses on the object of components or printed boards, and the results obtained are also in these aspects, while the research on the system or product object is relatively rare, and the reference is limited. Domestic electronic equipment design and manufacturing managers have achieved a great development. The author systematically expounds the structural design and dynamic theory of the structure of electronic equipment, and systematically introduces the theory of vibration, shock and effective measures to reduce vibration in [7]. The author studies how to improve the anti-vibration and shock capability of electronic equipment from the perspective of the overall layout of electronic equipment and the structural design of the whole machine, and carried out optimization analysis on it in [8]. It can be seen that the domestic work on the structural dynamic response of electronic equipment is mainly focused on the system or the whole of the product, and the structural details of the components are relatively little considered, and the information available is limited.

### **2 Display Console Structure Scheme**

Aiming at vehicle-mounted combat system and combat environment, this paper carries out research on vehicle-cabin integration, natural human–machine interaction, modeling design, human factor design, cable layout and environmental adaptability. The structure is simple and beautiful, the operation is convenient, and the maintenance is convenient. According to the installation space in the vehicle platform, the display console is adapted to change the leg part of the display console into an electronic cabinet unit, so as to provide installation space for user equipment and adopt the bottom cabinet design to improve the strength and stiffness of the overall structure of the display console. The 3D design software is used to simulate the human factor engineering of the display console. The finite element analysis software is used to simulate the mechanical test conditions of the display console. In addition, noise control and thermal design of the equipment are also carried out. In the design, we strictly follow the principles of generalization, serialization and combination design to ensure universality and extensibility.

The display console implements the design idea of generalization, serialization, modularization and combination. The display unit is composed of display unit, control unit and display control processing unit. The display unit mainly carries touch display and provides graphic image display and multi-touch function. The control unit mainly carries voice terminals, general control modules, etc., which can provide good human–computer interaction functions. The display and control processing unit carries the display and control computer and the power supply control module, and the cabinet unit mainly carries the task computer, network switch and user computer modules. In addition, the design of the display console fully considers the human factor engineering elements, adopts rounded corner design, and has no screws on the front, which can effectively improve the safety and comfort during use [9]. The layout of each unit of the display console is shown in Fig. 1.

Due to the limited space in the vehicle, in order to reduce the impact of equipment distribution on the space, the integrated design idea is adopted. The structural modeling of the display console is designed according to the space in the vehicle, and the wall-mounted installation method is adopted. The display console is connected and fixed by the bottom vibration isolator and the bottom plate of the vehicle, and the back vibration isolator and the frame beam are connected and fixed. Through the decorative parts, the display console can be integrated with the vehicle shell, the appearance is beautiful and simple, and the interior space can be maximized. The interior installation is shown in Fig. 2.

(a) Main view (b) Left view

(c) Top view (d) Rear view

**Fig. 1** The layout view of the display console

**Fig. 2** The view of the display console installed in the vehicle

**Fig. 3** Top view of chair spacing in human-factor integrated model

The display console carries out structural modeling design for the vehicle platform, adopts three-dimensional modeling technology to analyze the model of the vehicle display console, and focuses on the analysis of human factors including cockpit space, operator's body data measurement, field of vision, hand operation range, knee space and maintenance space. It can be seen that, this type of display console basically meets the requirements of integrated human factor engineering design. As shown in Fig. 3, the top view of the human-factor integration model shows that the spacing between the two cross-row seats is about 409 mm, which does not affect the traffic of personnel. In addition, the seat also has a folding or moving function to facilitate personnel access.

As can be seen from Fig. 4, the overall layout of the display console is reasonable, and the operation angle and hands of the personnel are accessible, which meets the requirements of human factors engineering design [10]. In addition, the operation table of the display console fully considers human factors engineering elements, adopts streamlined and rounded corners, and has high safety and comfort without screws on the front.

Figure 5 is a schematic diagram of the knee space operated by the human integrated model personnel. It can be seen that the knee height is about 725 mm, the width is about 762 mm, and the depth is about 515 mm, which meets the requirements of "the height of the knee area is not less than 640 mm, the width is not less than 510 mm, and the depth is not less than 460 mm" in the specification. In addition, the display console operating table is sprayed with skin-friendly coating, which has the effect of anti-freezing and greatly improves the comfort of operators.

(a) Stereo view

(b) Main view

**Fig. 4** The view of operation in the human factor integration model

(c) Top view

**Fig. 4** (continued)

# **3 Electrical Cable Design**

# *3.1 Routing Design of Display and Control Processing Unit*

The following mechanism is arranged at the rear of the display and control computer, and the cable is bundled and fixed through a series of holes on the part. The two ends of the wire following mechanism are connected and fixed with the backplane of the display and control computer and the wall plate of the platform through hinges. The internal cable routing method of the display and control processing unit is attached to the cable binding belt at the rear of the chassis. Figure 6 shows the cable routing method.

# *3.2 Routing Design of Control Unit*

The module cables embedded on the control console are routed along the cable channel on the lower surface of the control mesa panel. Standard cable fixation clips are set to facilitate cable fixation. The cables extend to the rear of the display console and connect to the rear connector of the di splay and control computer. The internal wiring mode of the control unit is shown in Fig. 7.

(a) Height space

(b) Width space

**Fig. 5** The view of the knee area in the human factor integration model

# *3.3 Routing Design of Display Unit*

The display cable extends downward along the cable trough and fixing device on the side wall of the display unit support, and enters the platform body through the connecting board at the rear of the display unit, and is connected with the corresponding navigation plug of the display and control computer. The back of the display unit is routed as shown in Fig. 8.

(c) Depth space

#### **Fig. 5** (continued)

### *3.4 Routing Design of Cabinet Unit*

Figure 9 shows the internal cabling of the cabinet unit.

### **4 Mechanical Design**

### *4.1 Shock Analysis*

According to the technical requirements document of the display console, the shock test conditions refer to the test grade 1 (10 g/3000 times) in GJB4.8-83 Environmental Test of Ship Electronic Equipment [11]. According to the regulations, the repetition frequency is 60–80 rpm and the shock pulse duration is ≥16 ms. In the calculation, the period duration of a single shock T is taken as 1 s, and the half-sine shock duration T are taken as 16 ms. The acceleration loading load within a single shock period is shown in Fig. 10 below, and the solution target is the stress curve at the position where the maximum stress occurs within a period.

Figure 11 shows the overall equivalent stress nephogram of the display console under the action of shock load. It can be seen that after the shock impact of the display console, the maximum stress of the overall structure is 57.62 MPa, which is far less than the allowable strength of the material, the maximum stress position is located in the lower left part of the display unit box and the control table bolt connection position, the overall structure is not in danger of damage, and the vast majority of the position stress is less than 5 MPa.

(a) View 1

(b) View 2

**Fig. 6** The view of internal wiring mode of display and control processing unit

**Fig. 7** The view of internal wiring mode of control unit

**Fig. 8** The view of the wiring mode on the back of display unit

**Fig. 9** The view of the wiring mode of cabinet unit (using the right side as an example)

In a shock period, the stress generated by the impact attenuates well after the impact. The whole structure can meet the strength requirements under the action of shock load.

**Fig. 10** Input curve of the shock load

**Fig. 11** Equivalent stress nephogram of shell during shock analysis

### *4.2 Impact Analysis*

According to the technical requirements document of the display console, the impact load is applied by referring to the test impact response spectrum used in GJB150.18A-2009 military equipment laboratory environmental test method impact test [12] when no measurement data is available—ground equipment Functional test (40 g). As shown in Fig. 12, the impact time is 15–23 ms, and here it is 21 ms, and the half-sine impact load is applied.

Vertical and 30-degree oblique impact response analysis was adopted for the display console. Figure 13 is the overall equivalent stress nephogram after the display

**Fig. 12** Input curve of half sinusoidal impact load

console is subjected to vertical impact, and Fig. 14 is the overall equivalent displacement nephogram after the display console is subjected to vertical impact. After the display console is impacted in the vertical direction, the maximum stress of the overall structure is 162.99 MPa, which is less than the allowable strength of the material. The maximum stress is located at the bolt connection position of the stage body and the stage body, and the overall structure does not fail to damage.

Compared with the initial position, the maximum displacement is 22.71 mm, located on the upper panel of the lower box, within the safe displacement range.

Figure 15 is the overall equivalent stress nephogram after the display console is subjected to 30° oblique impact, and Fig. 16 is the overall equivalent displacement

**Fig. 13** Equivalent stress nephogram of shell during vertical impact analysis

**Fig. 14** Equivalent displacement nephogram of shell during vertical impact analysis

nephogram after the display console is subjected to 30° oblique impact. After the display console is impacted by the tilt of 30°, the maximum stress of the overall structure is 187.97 MPa, which is less than the allowable strength of the material. The maximum stress is located at the bolt connection position of the stage body and the stage body, and the overall structure does not fail to damage.

Compared with the initial position, the maximum displacement is 24.92 mm, located on the upper panel of the lower box, within the safe displacement range.

Under the impact load, the stress produced by the impact decays well after the impact. The overall structure of the display console can meet the strength requirements under the impact load.

### **5 Conclusions**

Based on the mature design technology of display console, make full use of 3D design software and simulation analysis software, optimize and analyze the main design parameters of display console, and the technology is reliable. The scheme meets the technical requirements of the display console, strictly follows the design principles of universality, serialization and combination, and strictly follows the design requirements of reliability, maintainability, environmental adaptability, electromagnetic compatibility and structural manufacturability. The scheme is reasonable and feasible.

**Fig. 15** Equivalent stress nephogram of shell during 30° oblique impact analysis

**Fig. 16** Equivalent displacement nephogram of shell during 30° oblique impact analysis

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation Study on Driving Dynamic Characteristics of a Light Rocket Launcher**

**Qingtian Ma, Cungui Yu, Guanglei Li, Cheng Zhou, Shuang Liu, and Junyi Ma** 

**Abstract** Taking a light rocket launcher as the research object, based on random road excitation and applying the theory of multi-rigid-body system dynamics, the multi-body dynamics model of a light rocket launcher is established, and its driving dynamic characteristics are studied. Considering the load-bearing characteristics of leaf spring and the use of tires, the driving dynamics simulation of light rocket launcher on C-class virtual road is carried out. Considering the vibration impact of the road surface on the rocket launcher, the vibration characteristic results of the rocket launcher are analyzed, and the ride comfort index is used to evaluate the driving reliability of the rocket launcher.

**Keywords** Light rocket launcher · Leaf spring · Random road spectrum · Ride comfort

# **1 Introduction**

In the aspect of rocket launcher dynamic modeling, Ren [1] and others built a rigidflexible coupling driving dynamic model of the vehicle and gun after flexible treatment of the frame, and only considered the role of tire and ground excitation in the vibration. Li [2] studied the influence of structural damping and installation position on the ride comfort of the whole gun. Yao [3] established the driving dynamic characteristics of rocket launcher with hydro-pneumatic suspension, and compared with the simulation results with linear spring. Wang [4] proposed a method of experiment combined with numerical simulation to complete the bench test of the rocket launcher.

Ride comfort is one of the important performances of vehicles. When the ride comfort is poor, the system vibrates violently. For the vehicle-mounted rocket

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, China

Q. Ma · C. Yu (B) · G. Li · C. Zhou · S. Liu · J. Ma

e-mail: yu\_cungui@sina.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_26

launcher, on the one hand, the vibration of the vehicle itself reduces the service life, passability and operational stability; on the other hand, excessive vibration will reduce the life and reliability of the rocket launcher, especially for some electrical equipment, the harm caused by vibration is greater. Based on the above reasons, it is necessary to ensure the ride comfort of the rocket launcher in the design process.

The suspension system of the light rocket launcher studied in this paper adopts plate spring non-independent suspension with nonlinear characteristics and good damping capacity. In this paper, the driving dynamics and ride comfort of the rocket launcher are studied on the basis of considering the nonlinear characteristics of the plate spring suspension.

### **2 Establishment of Driving Model**

### *2.1 Description of Rocket Launcher Model*

The structure of the light rocket launcher consists of a rocket launcher loading system and a chassis system. The rocket launcher loading system is composed of directional tube, rocket, landing gear, electric cylinder, rotary body and base, and the suspension system is composed of leaf spring, shock absorber, U-shaped bolt and buffer block. Figure 1 shows the topology of the rocket launcher model.

In this paper, the multi-rigid-body dynamic model of self-propelled rocket launcher is simplified as follows [5]:


### *2.2 Rocket Launcher Loading System*

According to Fig. 1, there is a fixed pair (F1) between the rocket and the directional tube, between the directional tube and the landing gear (f2), between the landing gear and the rotary body (f3), between the electric cylinder push rod and the landing gear (f4), between the electric cylinder push rod and the electric cylinder seat is the moving pair and the linear spring force (f7), and between the electric cylinder seat and the rotary body is the rotating pair (f5). Between the rotary body and the base

**Fig. 1** Topology of rocket launcher

is the rotating pair and the torsional spring force (f6), and between the base and the frame is the fixed pair (f8).

### *2.3 Chassis System*

The light rocket launcher is a military off-road chassis. For the driving dynamics model, the chassis can be simplified into frame, spring suspension, tires and so on. The cab and the frame are fixed by a fixed pair (f9) simulation bolt, the frame and the front and rear axle are leaf spring non-independent suspension, the frame and the leaf spring ear are rotating pairs and torsion springs (f10, f11), while the lower end of the leaf spring and the axle are simulated by bushing force (f12, f13), and the axle and the wheel are represented by rotating pair (f13-f16) and angular speed drive. The contact force (f17-f20) is between the wheel and the ground.

**Fig. 2** Leaf spring suspension structure and simplified model of three-link

#### **Plate Spring Suspension**

The front and rear suspension is composed of leaf spring, lifting lug, U-shaped bolt, buffer block and shock absorber. The front leaf spring is a constant stiffness spring, and the rear leaf spring is a gradual leaf spring, which is composed of a main spring and a secondary spring.

Firstly, the three-link leaf spring model is established [6]. The three-link method is a leaf spring kinematics calculation model recommended by the American Society of Automotive Engineers (SAE), as shown in Fig. 2.

In the figure: *L* is the total arc length of the leaf spring; *a* and *b* are the arc length of the front and rear segments of the leaf spring; *c* is the length of the lifting lug; *m*  and *n* are the distance between the front and rear segments of U-shaped bolts and the center of the leaf spring; *Ra*, *Rb* and *Rc* are the lengths of the front, middle and rear rods of the three-link model, respectively; *d* is the distance between the front section of the middle rod and the axle axis. The calculation formula of three-link is as follows:

$$\begin{cases} \begin{aligned} \mathcal{R}\_a &= 0.75(a - m) \\ \mathcal{R}\_b &= 0.75(b - n) \\ \mathcal{R}\_c &= L - (\mathcal{R}\_a + \mathcal{R}\_b) \\ d &= a - \mathcal{R}\_a \end{aligned} \end{cases} \tag{1}$$

In order to simulate the load-bearing characteristics of the leaf spring model, a rotating pair and a torsion spring are established between the connecting rods to ensure the stiffness characteristics of the three-link model. The stiffness of torsion spring is expressed by cubic polynomial about displacement:

$$\begin{cases} T\_1 = \left[ \begin{array}{cc} a\_1 \ a\_2 \ a\_3 \end{array} \right] \left[ \begin{array}{cc} \theta\_1 \ \theta\_2^2 \ \theta\_3^3 \end{array} \right]^T \\ T\_2 = \left[ \begin{array}{cc} b\_1 \ b\_2 \ b\_3 \end{array} \right] \left[ \begin{array}{cc} \theta\_1 \ \theta\_2^2 \ \theta\_3^3 \end{array} \right]^T \end{cases} \tag{2}$$

The reverse simulation method is used to identify the parameters of the torsion spring stiffness in the three-link equivalent model, and the equivalent stiffness of the leaf spring model is obtained. Figure 3 shows the stiffness test data of leaf spring and the results of reverse simulation method. The load-bearing characteristics of the three-link model are very close to the test data, and the simulation results meet the requirements.

**Fig. 3** Comparison of the results of reverse simulation of front and rear leaf springs

The damping force–deformation velocity curve of the front and rear shock absorbers is shown in Fig. 4 and is added to the equivalent spring in the form of Spline function.

#### **Tire Model**

Tire is an important part of a vehicle, on the one hand, it supports the weight of the whole vehicle, ensures that the vehicle has good adhesion to the ground, and transmits driving torque and braking torque. on the other hand, it works with the suspension system to alleviate the impact caused by the uneven road surface and attenuate the resulting vibration.

The selection of tire types and parameters has an important influence on the analysis of vehicle driving performance. The force between tire and ground is based on UA tire model. The main characteristic parameters of UA model are lateral stiffness, tilting stiffness, vertical stiffness, longitudinal stiffness, rolling resistance coefficient and vertical damping coefficient. Tire model parameters are provided according to the data provided by the factory, as shown in Table 1.

**Fig. 4** Damping force–velocity curves of front and rear shock absorbers


**Table 1** Tire parameters

### *2.4 Road Model*

The general description method of road roughness is power function power spectral density. According to the content of Chinese national standard GB/T7031-2005 "Mechanical Vibration Road pavement Spectrum Measurement data report" [7], the displacement power spectral density of pavement roughness can be fitted by the following formula.

$$G\_d(n) = G\_d(n\_0) (\frac{n}{n\_0})^{-W} \tag{3}$$

In the formula, *n* is the spatial frequency, *n*0 is the reference spatial frequency, *n*0 = 0.1*m*−1, *Gd* (*n*)is the displacement power spectral density of road roughness, *Gd* (*n*0) is the displacement power spectral density under the reference spatial frequency *n*0, *W* is the frequency index and the slope in double logarithmic coordinates.

The harmonic superposition method is used to simulate the random pavement [8], and the function of the vertical displacement of the random pavement with respect to the longitudinal displacement can be obtained by adding the sine wave function of each small frequency interval.

$$Z(\mathbf{x}) = \sum\_{i=1}^{N} \sqrt{2Gq(f\_{mid,i}) \cdot \Delta f\_i} \cdot \sin(2\pi f\_{mid,i}t + \theta\_i) \tag{4}$$

In the formula, *Z* (*x*) is the road roughness function, ∆*fi* is the spatial frequency interval, θ*i* is the sinusoidal function phase, which is randomly distributed in [0, 2π], and *x* is the road length.

In this paper, the triangulation method is used to establish the rdf file that can be recognized by recurdyn, and the C-level simulation road is generated for subsequent virtual driving simulation. The road is shown in Fig. 5.

**Fig. 5** C level simulation road

### **3 Analysis of Vibration Characteristics of Rocket Launcher**

### *3.1 Driving Simulation Analysis*

In the dynamic model of the rocket launcher, x is the positive direction of the vehicle, y is the lateral direction, and z is the vertical direction. The uniform driving time of the rocket launcher is set as 25 s, and the driving simulation is carried out on the C-class road at the speed of 40 km/h, 50 km/h and 60 km/h respectively. The vibration acceleration of the rocket launcher base in three directions is obtained, as shown in Fig. 6.

Based on the spectrum analysis of the z-direction acceleration curve of the rocket launcher base, the vertical acceleration power spectrum density curve is obtained. As shown in Fig. 7.

From the acceleration power spectral density curve, it can be seen that the acceleration power spectral density in the vertical direction of the rocket launcher base is mainly concentrated between 2.11 Hz and 4.38 Hz, while the peak appears in several different positions, the more obvious are 2.31 Hz, 2.96 Hz and 3.50 Hz. For the rocket launching process, the minimum vibration frequency [9] is about 7 Hz, when the lowest order frequency of the launcher is relatively high, which well avoids the vibration frequency of the whole vehicle at different speeds.

The 1st to 6th order modes of the rocket launcher itself can be known by finite element calculation, as shown in Table 2.

The natural frequencies of the rocket launcher in the second and third modes are 2.00 Hz and 4.18 Hz, as shown in Fig. 8. In the driving process of the rocket launcher, the frequency of the peak value is different from the natural frequency of the rocket launcher, which avoids the danger of resonance.

### *3.2 Ride Comfort Analysis*

According to the ISO2631-1 1997 (E) standard, when the peak coefficient of vibration waveform is less than 9, the basic evaluation method-weighted root mean square of

(a) X direction acceleration curve (b) Y direction acceleration curve

(c) Z direction acceleration curve

**Fig. 6** Vibration acceleration curve of rocket launcher base

**Fig. 7** Acceleration power spectral density curve of rocket launcher base in z

direction


**Table 2** Modal calculation results of rocket launcher

**Fig. 8** Second and third order vibration modes of rocket launcher

acceleration [10] is used to evaluate the effect of vibration on human comfort and health. When using this method, the axial acceleration curve, the power spectral density of acceleration and the root mean square formula of acceleration should be calculated.

$$a\_w = \left[\frac{1}{T} \int\_0^T a\_w^2(t)dt\right]^{\frac{1}{2}}\tag{5}$$

When considering three axial vibrations at the same time, the root mean square of the total weighted acceleration of the three axes can be calculated according to the following formula:

$$a\_v = \left[ (1.4a\_{xw})^2 + \left(1.4a\_{yw}\right)^2 + a\_{zw}^2 \right]^{\frac{1}{2}} \tag{6}$$

In the formula: *ax*<sup>w</sup> is the root mean square of x axis weighted acceleration, *ax*<sup>w</sup> is the root mean square of y axis weighted acceleration, and *az*<sup>w</sup> is the root mean square of z axis weighted acceleration.

This paper analyzes the ride comfort of the rocket launcher and studies the bad vibration of the rocket launcher at different speeds on the C-class road surface. Table 3 shows the total weighted acceleration root mean square of the rocket launcher base.


**Table 3** Total weighted acceleration root mean square of rocket launcher base

As can be seen from Table 3, with the increase of vehicle speed, the ride comfort of the rocket launcher base becomes worse and worse, but the weighted root mean square of acceleration an is less than the specified value [10] 2.5 m/s2. For the rocket launcher structure, the bad degree of vibration acceleration will not affect the rocket launcher parts and electronic components, and can well meet the ride comfort requirements.

### **4 Conclusion**

In this paper, the dynamic model of rocket launcher with plate spring suspension is established, the generation program of C class road spectrum is compiled, and the driving dynamics simulation of the rocket launcher is carried out at different speeds. the vibration acceleration curve of the rocket launcher base in 3 direction and the acceleration power spectrum density curve in z direction are obtained. Through the simulation analysis, we can get the following conclusions:


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Piecewise Linear Recursion Iterative Method for Form Finding and Force Analysis of Suspension Cable**

**Renjie Shang, Linghao Ren, and Pianpian Huang** 

**Abstract** The suspension cable structure has the advantages of saving steel, light weight, beautiful in shape, and has been widely used in long-span bridges. In recent years, the suspension cable structure has been used in the flexible photovoltaic supports of small and medium-sized spans. The existing piecewise catenary method and finite element method have the problems of initial value sensitivity and easy divergence due to the highly geometric nonlinearity of suspension cable structure, and a practical calculation method of piecewise linearization of the suspension cable is proposed in this paper. The suspension cable is divided into n (n ≥ 40) segments. First, the initial values of inclination angle and tension of the left end of the suspension cable are given roughly, and the position of the right end of the suspension cable can be obtained by piecewise recursive method according to the external forces. Based on the deviation between the calculated position and the target position of the right end of the cable, the values of the inclination angle and tension of the left endpoint of the suspension cable are modified, and the second iteration is carried out. In this way about 3–6 iterations, the calculated position of the right end of the cable can converge to the target position, and the exact geometry and internal force of the suspension cable can be obtained. The method can be used to calculate the internal force and deformation of suspension bridges under static loads, and the results accuracy fully meets the engineering design requirements.

**Keywords** Suspension cable · Piecewise linearization · Recursive method · Iterative method

e-mail: shangrenjie@aliyun.com

R. Shang (B) · L. Ren · P. Huang

Central Research Institute of Building and Construction Co., Ltd, MCC Group, Beijing 100088, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_27

### **1 Introduction**

Suspension cable structure is a geometric nonlinear flexible structure system, which is not only used in large-span suspension bridges [1], but also widely used in flexible photovoltaic supports with spans about 20 m in the recent years [2, 3]. The key point to the design and construction of suspended cable structure is to find the balance form of the cable under certain loads and calculate the deformation and internal forces caused by the load after the form finding balance. The calculation of the suspension cable should consider not only the self-weight of the cable body evenly distributed along the cable, but also other concentrated loads on the cable. Catenary equation can be solved according to differential equation under uniform load, but when the load is complex, the suspension cable leaves the theoretical catenary position, it would be calculated by finite element method or numerical method.

The segmented catenary method is a form finding method of suspension bridges [4] proposed in 2003, and has been widely applied and developed in the following years [5–9]. The method takes the catenary shape within the segment into consideration and calculate the cable shape under various load conditions by recursion and iteration. Segmented catenary method can be applied to spatial cable form finding [5], and the friction between cable saddle and main cable can be considered [6]. The calculation efficiency was improved by considering the form changing stiffness of the main cable [7], the suspender stiffness is considered in the form-finding of the suspension bridge adaptively [8]. Steffens-Newton method was used for iterative solution to improve iteration efficiency [9].

Finite element method is a general method for structural calculation, beam elements and truss elements were used to conduct finite element form-finding calculation for suspension cables [10, 11], and segment catenary was combined with finite element method to calculate cable form-finding [12–15]. The suspender forces were equivalent to concentrated loads, and the finite element model of the main cable was established by ANSYS software to calculate the shape of the cable [16, 17]. Because of the strong geometric nonlinearity of the suspension cable, the segmented catenary method has strong dependence on the initial value, and the finite element method is often difficult to converge. In addition to segmental catenary method and finite element method for suspension cables calculation, there are also numerical methods for dynamic problem considering the suspension cable as a continuum according to the differential equation [18, 19].

In the paper, a practical calculation method of piecewise linearization of the suspension cable is proposed. The suspension cable is divided into n (n ≥ 40) segments. The form of the suspension cable depends only on the inclination angle and tension of the left end. If the values of the angle and tension of the left end of the suspension cable are given roughly, the position of the right end of the cable can be obtained by recursion considering the loads. Based on the deviation between the calculated position and the target position of the right end of the cable, the values of the inclination angle and tension of the left end of the suspension cable are modified, and the new iteration is carried out. After about 3–6 iterations, the exact solution can be obtained. The method has the advantages of simple calculation, fast convergence and accurate results.

### **2 Piecewise Linearity and Recurrence of Suspension Cables**

### *2.1 Piecewise Linearity and Node Balance*

The suspension cable is divided into n segments and numbered from left to right, with the left end numbered as 1 and the right end numbered as n + 1. When the length of the cable segment is small enough relative to the total length, such as the cable is divided into 100 segments, the geometric nonlinear influence caused by the sag caused by the weight of the cable of a segment can be ignored.

For node i, the stress-free length of the left cable of the node i is *L*i, and the inclination angle between the cable segment and the horizontal plane is θi. θi takes a negative value when the segment tilts downward from left to right, takes a positive value if not. The tension horizontal component in the cable is *H*i, and the vertical component is *V*i= − *H*i tan θi. The stress-free length of the right segment of the node i is *L*i+1, the inclination angle is θi+1(Fig. 1), and the tension horizontal component in the cable is *H*i+1, and the vertical component is *V*i+1= − *H*i+1 tan θi+1. The total vertical force *G*i of node i is obtained including the weight of the left half and right half cable segments and the load on node i. The horizontal force *F*i in the vertical plane of the suspension cable is also applied to node i. The equilibrium equation of node i in the x and y directions is obtained where x is horizontal to the right and y is the vertical upward:

$$H\_{\rm i+1} = H\_{\rm i} - F\_{\rm i} \tag{1}$$

$$H\_{\rm i+1} \tan \theta\_{\rm i+1} = H\_{\rm i} \tan \theta\_{\rm i} + G\_{\rm i} \tag{2}$$

We can calculate the values of *H*i and θi of each segment of the cable when we know the initial values of *H*0 and θ 0 using recursion Eqs. (1) and (2). The approximation initial values of tension horizontal component *H*0 and inclination angle θ 0 of the left endpoint are given. Half of the total vertical loads can be taken as, and −45°– −30°can be taken as θ 0. The approximation initial values will affect the convergence speed but will not affect the final convergence.

The height of the suspension bridge is about 1/10 of the span, the inclination angle of the endpoint of the cable end is about 20°–45°, so we choose the initial valued θ <sup>0</sup>= 20°–45°. Because the tension vertical component *V*0 of the endpoint of the cable is about half the total vertical loads, and the tension horizontal component is

$$H\_0 = V\_0 / \tan \theta\_0\text{, so we choose the initial value of } H\_0 \text{ is } H\_0 = 0.5 \sum\_{i=1}^n G\_i.\text{}$$

**Fig. 1** Equilibrium of node i

### *2.2 Suspension Coordinate Recursion*

Divide both sides of formula (2) by *H*i+1:

$$\tan \theta\_{\mathbf{i}+1} = \frac{H\_{\mathbf{i}}}{H\_{\mathbf{i}+1}} \tan \theta\_{\mathbf{i}} + \frac{G\_{\mathbf{i}}}{H\_{\mathbf{i}+1}} \tag{3}$$

When the angle θ1 and the tension horizontal component *H*1 of the first segment of the left end is known, the tension horizontal component *H*i of the tension and angle θ i of each segment can be obtained by recursion of formula (1) and (3).

For the left segment to node i, the tension horizontal component is *H*i, and the inclination angle is θ i, then the tension of the segment is *F*i = *H*i/ cos θi.

The section area of the cable is *A*, the elastic modulus is *E*, and the temperature expansion coefficient is α. The elongation of the cable segment under tension *F*i and temperature change ∆*T* is as follows:

$$
\Delta L\_i = \left(\frac{H\_{\text{i}}}{EA\cos\theta\_{\text{i}}} + \alpha\Delta T\right)L\_i \tag{4}
$$

The coordinate increments in the x and y directions of segment i are:

$$
\Delta \mathbf{X}\_{\mathbf{i}} = \left( 1 + \frac{H\_{\mathbf{i}}}{EA \cos \theta\_{\mathbf{i}}} + \alpha \Delta T \right) L\_i \cos \theta\_{\mathbf{i}} \tag{5}
$$

$$
\Delta \mathbf{Y}\_i = \left( 1 + \frac{H\_{\mathbf{i}}}{EA \cos \theta\_{\mathbf{i}}} + \alpha \Delta T \right) L\_i \sin \theta\_{\mathbf{i}} \tag{6}
$$

Assuming the coordinate of left endpoint A of the cable is (XA, YA), the calculated coordinates of right endpoint B of the cable can be obtained:

Piecewise Linear Recursion Iterative Method for Form Finding … 347

$$\mathbf{X\_{BC}} = \sum\_{i=1}^{n} \left( 1 + \frac{H\_{\mathbf{i}}}{EA \cos \theta\_{\mathbf{i}}} + \alpha \Delta T \right) L\_{i} \cos \theta\_{i} + \mathbf{X\_{A}} \tag{7}$$

$$\mathbf{Y\_{BC}} = \sum\_{i=1}^{n} \left( 1 + \frac{H\_{\mathbf{i}}}{EA \cos \theta\_{\mathbf{i}}} + \alpha \Delta T \right) L\_{i} \sin \theta\_{i} + \mathbf{Y\_{A}} \tag{8}$$

# **3 Iteration Method for Form Finding and Force Calculation**

### *3.1 Iteration Method for Form Finding*

According to Eq. (7) and Eq. (8), the calculated coordinate (XBC,YBC) of right endpoint B of the cable is obtained. The target coordinate of point B is (XBT,YBT) and has been given by design. The difference between the calculated coordinates and the target coordinates are:

$$
\Delta \mathbf{X} = \mathbf{X\_{BT}} - \mathbf{X\_{BC}} \tag{9}
$$

$$
\Delta \mathbf{Y} = \mathbf{Y\_{BT}} - \mathbf{Y\_{BC}} \tag{10}
$$

The steps to solve the horizontal component of the tension and inclination angle of the left end of the cable by iterative method are described below.

**Preparation**: Given the total length *L* of the zero-stress cable, and the cable is divided into n segments. Sufficient accuracy can be achieved if n is not less than 40. When an external force is not on the segment node of the cable, a new node can be added at the point external force applied. The elastic elongation of the cable and the temperature strain are considered in Eqs. (5)–(8).

**Step 1**: Take the tension horizontal component *H*1 = *H*0 and inclination angle θ <sup>1</sup> = θ 0 as the first segment cable, and obtain the angle θi of all cable segments by recursion Eqs. (1) and (3); The calculated coordinates XBC and YBC of point B can be obtained by Eqs. (7) and (8), denoted as XB0 = XBC and YB0 = YBC respectively.

**Step 2**: Calculate the difference (∆X, ∆Y) between the calculated coordinates (XBC,YBC) and the target coordinates (XBT,YBT) of point B by formula (9) and (10). When ∆X and ∆Y meet the accuracy requirements, the form of the cable and the tension of each segment are the target values; otherwise, the following calculations are performed.

**Step 3**: Keeping the inclination angle of the left endpoint A remains the same as in step 1 which is θ <sup>1</sup>= θ 0. Giving the tension horizontal component *H*1 = *H*0 a small

increment δ*H* such as δ*H* = 0.0001*H*0, we can get *H*1 = *H*0 + δ*H*. The angles of all cable segments are obtained by recursion Eqs. (1) and (3). The calculated coordinates XBC and YBC of point B can be obtained by formula (7) and (8), denoted by XBH = XBC and YBH = YBC respectively. The influence coefficients of *H* on the coordinates of point B can be obtained:

$$\frac{\partial \mathbf{X}}{\partial H} = \frac{\mathbf{X\_{BH}} - \mathbf{X\_{B0}}}{\delta H} \tag{11}$$

$$\frac{\partial \mathbf{Y}}{\partial H} = \frac{\mathbf{Y\_{BH}} - \mathbf{Y\_{B0}}}{\delta H} \tag{12}$$

**Step 4**: Keeping the tension horizontal component the same as step (1) which is *H*1=*H*0.Giving the inclination angle θ1 = θ0 a small increment δθ such as δθ = 0.0001θ0, we can get θ1 = θ0 + δθ. The angles of all cable segments are obtained by recursion Eqs. (1) and (3). The calculated coordinates XBC and YBC of point B can be obtained by formula (7) and (8), respectively denoted by XB<sup>θ</sup> = XBC and YB<sup>θ</sup> = YBC. The influence coefficients of θ on the coordinates of point B can be obtained:

$$\frac{\partial \mathbf{X}}{\partial \theta} = \frac{\mathbf{X}\_{\text{B}\theta} - \mathbf{X}\_{\text{B0}}}{\delta \theta} \tag{13}$$

$$\frac{\partial \mathbf{Y}}{\partial \theta} = \frac{\mathbf{Y}\_{\text{B}\theta} - \mathbf{Y}\_{\text{B0}}}{\delta \theta} \tag{14}$$

**Step 5**: Using the coordinate differences ∆X and ∆Y obtained by Eq. (9)–(10), influence coefficients <sup>∂</sup><sup>X</sup> <sup>∂</sup> *<sup>H</sup>*, <sup>∂</sup><sup>Y</sup> <sup>∂</sup> *<sup>H</sup>*, ∂X ∂θ and <sup>∂</sup><sup>Y</sup> ∂θ obtained by Eq. (11)–(14), the corrections of *H*0 and θ0 can be obtained by solving the linear equations:

$$
\Delta H = \left(\Delta \mathbf{X} \frac{\partial \mathbf{Y}}{\partial \theta} - \Delta \mathbf{Y} \frac{\partial \mathbf{X}}{\partial \theta}\right) \Bigg/ \left(\frac{\partial \mathbf{X}}{\partial H} \frac{\partial \mathbf{Y}}{\partial \theta} - \frac{\partial \mathbf{Y}}{\partial H} \frac{\partial \mathbf{X}}{\partial \theta}\right) \tag{15}
$$

$$
\Delta\theta = \left(\Delta\mathbf{X}\frac{\partial\mathbf{Y}}{\partial H} - \Delta\mathbf{Y}\frac{\partial\mathbf{X}}{\partial H}\right) \Bigg/ \left(\frac{\partial\mathbf{X}}{\partial\theta}\frac{\partial\mathbf{Y}}{\partial H} - \frac{\partial\mathbf{Y}}{\partial\theta}\frac{\partial\mathbf{X}}{\partial H}\right) \tag{16}
$$

**Step 6**: Taking *H* = *H*0 + ∆*H* as *H*0 which is the tension horizontal component at the left endpoint of the new iteration, and θ = θ0 + ∆θ as θ0 which is the inclination angle at the left endpoint of the new iteration, the calculation of a new iteration from step 1 is carried out.

When the calculated coordinates of the right endpoint B of the cable in step 2 differ within a certain range from the target coordinates, such as not greater than 0.01 mm, the iteration is completed. The tensions and the form shape of the cable can be obtained using *H*0 and θ 0. The piecewise linear recursion iteration method makes the highly nonlinear problem of the suspension cable structure become a linear problem with only two parameters per step iteration, and the calculation difficulty is significantly reduced. The iterative convergence process can be seen through Eqs. (9) and (10).

### *3.2 Excel Implementation of Iterative Process*

The iterative process can be realized by two Excel tables.

Table 1 the suspension cable is divided into n segments and fill the length *L*<sup>i</sup> of each segment into column 2. Data in the non-shaded part is filled according to known conditions: vertical force *G*i and horizontal force *F*i of each node are filled into column 3 and column 4. The target coordinate (XBT,YBT) of the right endpoint B, the tensile stiffness EA, and the temperature effect α∆*T* .

The initial tension horizontal component *H*0 and inclination angle θ 0 are given according to "preparation", Data in the shaded part can be calculated automatically by excel using Eqs. (1)–(6). *H*i in column 5 can be calculated by Eq. (1), and θi in column 6 can be obtained by finding the inverse function on both sides of Eq. (3):

$$\theta\_{\rm i+1} = \operatorname{a} \tan \left( \frac{H\_{\rm i}}{H\_{\rm i+1}} \tan \theta\_{\rm i} + \frac{G\_{\rm i}}{H\_{\rm i+1}} \right) \tag{17}$$

∆*x*i in column 7 and ∆*y*i in column 8 can be obtained by Eqs. (5) and (6), the calculated coordinates XBC and YBC of point B can be obtained by Eqs. (7) and (8), and the differences ∆X and ∆Y between the calculated coordinates and the target coordinates of point B can be obtained by Eqs. (9) and (10).

Table 2 the values in the "initial value" row are obtained from Table 1 when the initial values *H*0 and θ 0 were given. Giving *H*0 a small increment δ*H* = 0.0001*H*0, we can get 1.0001*H*0 as the new value of *H*0. When the new value *H*0 is filled in Table 1 in the position *H*0, we can get the row of "influence coefficients of δ*H*" of Table 2 from Table 1. Giving angle θ0 a small increment δθ such as δθ = 0.0001θ0, we can get 1.0001θ0 as the new value of θ0. When the new value θ0 is filled in Table 1


**Table 1** Piecewise linear recursion


**Table 2** Excel iteration calculate

in the position θ 0, we can get the row of "influence coefficients of δθ" of Table 2 from Table 1. According to the data of the shaded part in Table 2, the "iterative correction value" can be obtained by formula (15) and (16) and the "new value" can be obtained for the next iteration calculation

The data in Table 2 can be obtained automatically by calling Table 1. In Table 2, if the initial values *H*0 and θ 0 are given, the new values *H*0 + ∆*H* and θ <sup>0</sup>+ ∆θ after iteration can be automatically obtained; Enter *H*0 + ∆*H* and θ <sup>0</sup>+ ∆θ as new initial values *H*0 and θ 0 into Table 2 to obtain *H*0 + ∆*H* and θ <sup>0</sup>+ ∆θ after a new iteration. Examples show that convergence to the stable value can be achieved after about 3–6 iterations.

### **4 Example**

### *4.1 Example 1*

To verify the feasibility and accuracy of the method, the calculation example is a standard catenary. The cable length is *L* = 120 m and neglecting the elastic extension. The left end point of the cable is A(−50,0), the right end point of the cable is B(50,0), and the cable weight is 11 N/m, as shown in Fig. 2.

The standard catenary equation of is:

$$y = \frac{\cosh(\alpha x)}{\alpha} - \frac{\cosh(50\alpha)}{\alpha} \tag{18}$$

Parameter α is the solution of the length constraint equation:

$$1\sinh(\mathfrak{F}0\alpha) = 60\alpha\tag{19}$$

**Fig. 2** Standard catenary

We can get α = 0.02129737 and the theory value of the sag is *f* t = *y*(0) = 29.23433 m = 29,234.3 mm.

The piecewise linear recursive iterative method is used to solve the problem. The total weight of the cable is G0 = 1320 N, the approximation initial values *H*0 = 0.5G0 = 660 N and θ <sup>0</sup>= −30°are selected. Taking n = 40, 60, 80 and 100 for calculation, all the four cases converge to within 0.01 mm after the 5th iteration. The iteration and convergence processes of *H*0 and θ 0 are shown in Table 3, Figs. 3 and 4.

The sag value of the cable calculated by the piecewise linear recursive iterative method is *f* p, and calculate the error of the method is ∆*f* = *f*p − *f*t. we get ∆*f* =


**Table 3** Iterative and convergence process

1.5 mm when n = 100, ∆*f* = 2.4 mm when n = 80, ∆*f* = 4.2 mm when n = 60, and ∆*f* = 9.4 mm when n = 40. This example verifies the convergence speed and accuracy of the method.

### *4.2 Example 2*

The method can be used to calculate the internal force and deformation of suspension bridges under concentrated loads [20]. The suspension cable span is 1000 m, the vector height is 100 m, the section area is A = 0.2 m2, the elastic modulus is E = 2 × 105 N/mm2, and the evenly distributed load along the horizontal direction is 5 KN/m. After the suspension cable reaches self-balance, the horizontal force of the suspension cable and the deformation of the loading point when the concentrated force is applied at the mid-span and 1/4 span positions are calculated respectively.

The total vertical load of the suspension cable before applying concentrated force is G0 = 5 × 1000 = 5000 KN. The cable is divided equally along the horizontal direction and n = 100. After form finding using the method provided in the paper, the tension horizontal component *H*0 = 6249.534 KN, and the inclination angle θ <sup>0</sup> = −21.8014°. The height of the midspan is 100 m, the vertical height of the 1/4 span position is 75 m.

When additional load apply on the mid-span point or 1/4 span, we can carry out the recursion iterative method and calculate the tension horizontal component *H*<sup>0</sup> and the inclination angle θ 0 of the left endpoint of the cable. The increment of *H*<sup>0</sup> and the vertical displacement of the loading point can be calculated as ∆*H* and ∆*y*.

∆*H*/*H*0 increases with the increase of PK/G0 can be seen in Fig. 5. The incremental ratio of tension horizontal component ∆H/H0 is basically linear with the incremental ratio of load PK/G0, and the slope of secant line to initial tangent line decreases by about 3.3% when PK/G0 = 0.20. Under the same load value, the tension horizontal component increment caused by mid-span load is 32% larger than that caused by 1/ 4 span load.

∆*y*/*f* 0 increases with the increase of PK/G0 can be seen in Fig. 6. There are obvious nonlinearities between the incremental ratio ∆*y*/*f* 0 and the load ratio PK/ G0. The slope ratio of secant line to initial tangent line increases by about 33% when

PK/G0 = 0.20. Under the same load value, the vertical displacement of the loading point in 1/4 span is 38% higher than that in mid-span.

This example proposed to compare to the equations of different scholars to calculate the internal force and vertical displacement of suspension bridge [20]. The formulas are complicated, and the difference between different formulas is large, some results difference between them can reach 15–20% [20]. The method in this paper fully satisfies the balance equation of each node of the cable after divided, and the error is only due to the influence of the curve in a single segment. When n = 100, the calculation error of midspan sag is only 0.005% relative to the theoretical mid-span sag, which fully meets the engineering design requirements.

# **5 Conclusion**


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Design of Fiber Winding for Thick-Walled Cylinder with Uniform Residual Tension**

**Zilong Yang and Fujiang Yang** 

**Abstract** Fiber winding with metal cylinders as liners is the primary method used for producing high-speed rotors, flywheels, and pressure vessels. A design model was established for thick-walled cylindrical fiber winding by using the layer-bylayer stacking method of thin composite material rings in this article. The effect of liner structure and fiber type on prestress was analyzed. And the phenomenon of tension loss in no-twist fibers was studied. The winding tension of each layer was calculated accurately under the condition of uniform residual tension, which was the optimal process in engineering. Compared with the experimental results, the computational error of the model does not exceed 5%. The winding model developed in this article, which takes into account tension loss, can more accurately guide the design of winding tensions for thick-walled cylinders.

**Keywords** Thick-walled cylinder · Fiber winding · Residual tension · Prestress

# **1 Introduction**

The main technique in the formation of high-speed rotors, energy storage flywheels, pressure vessels, and other mechanical equipment is fiber tension winding with metal cylinders as liners. The longevity and dependability of the spinning or bearing elements are directly impacted by whether the tension distribution of each layer and the winding prestress fulfill the design criteria. In recent years, with the development of new pressure vessels that are resistant to high pressure, corrosion and irradiation, the wall thickness of the lining has been designed to be thicker and thicker, and fibre-wound reinforcement technology based on thick-walled cylinders has become an industry requirement.

Z. Yang (B) · F. Yang

National Key Labortory of Particle Transport and Separation Technology, 300180 Tianjin, China e-mail: yangzilong1214@163.com

Research Institute of Physical and Chemical Engineering of Nuclear Industry, 300180 Tianjin, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_28

Since the 1990s, several academics have studied fiber winding in great detail. For the solution of residual tension, Springer and Calius [1, 2] suggested a multilayer composite thin-walled cylinder overlaid model. A more precise fiber winding model that takes the resin flow process into account was created by Gutowski [3]. Cohen [4, 5] examined how factors including resin, fiber type, winding tension distribution, and winding angles affected the structure's stress. Zhao [6] performed the FEA modeling of fiber wet winding and examined several factors throughout the winding and curing process, including temperature, degree of curing, resin viscosity, and others. Based on the thin-shell model, Wang [7, 8] derived the tension formula for fiber winding with a thin-walled cylinder. The impact of matrix penetration, threshold tension, and other variables on the precision of prestress calculations for thin-walled cylinders was determined by Lu [9]. As a result of his investigation into the connection between interference amount and prestress, Qin [10] arrived at the composite flywheel winding's simplified prestress model. Ren [11, 12] suggested an intricate and very precise iterative approach for calculating the fiber tension of each layer under the assumption of uniform residual stress. Based on the idea of elastic deformation of an anisotropic winding layer and thick-walled cylinder with isotropic liner, Kang [13] suggested an analytical technique of residual tension. In a research on the dynamic properties of composite fiber tension wound prestress fields, Su [14] provided an explanation of the stress saturation phenomena.

In general, a lot of studies has been done on fiber winding in the theoretical, modeling, and experimental realms, but there are still two issues that have not received thorough investigation. First, thin-walled cylinders are the subject of the majority of investigations, and the thin-shell model is used to determine their winding tension. There are few studies on thick-walled cylinders. For thick-walled cylinders, the designing technique for winding tension often relies on intricate, layer-by-layer iterative calculations or repetitive process testing. It lacks a precise formula that can be applied to direct calculations. Second, while designing winding tension, the tension loss issue is typically disregarded. It is necessary to increase the precision of prestress design, particularly for non-twist fibers like carbon fiber whose tension loss is more pronounced than that of twist fibers like glass fiber. A technique was provided in the study to determine the starting tension of each layer for a thick-walled cylinder under the assumptions of uniform residual tension in order to address the aforementioned issues. A series of process tests led to the establishment of the winding tension loss model and a modification of the tension designing formula, which served as guidelines for designing the winding tension of thick-walled cylinders.

### **2 Fiber Winding Theory**

### *2.1 Winding Model*

After each layer of winding in the fiber winding process, the surface pressure of the liner and the stress distribution of the wound fibers alter. Figure 1 depicts the fiber winding geometric model.

Elastic modulus *Em*, Poisson's ratio ν*m*, inner radius *rmi*, and outer radius *rmo* are the parameters of a metal liner. Inner radius *rfi*, outer radius *rfo*, radial and circumferential modulus *E*<sup>θ</sup> and *Er*, Poisson's ratio μθ*r* and μ*<sup>r</sup>*<sup>θ</sup> are the parameters of the winding layer.

The thickness ∆*h* of the fiber layer is wound under tension *T*(*h*), resulting in new displacement and stress to fulfill the equation.

$$\frac{d\Delta\sigma\_r}{dr} + \frac{\Delta\sigma\_r - \Delta\sigma\_\theta}{r} = 0 \tag{1}$$

Orthogonal anisotropic composite ring's geometric and physical equations are,

$$\begin{aligned} \Delta \varepsilon\_r(r) &= d \Delta u / dr \\ \Delta \varepsilon\_\theta(r) &= \Delta u / r \\ \Delta \varepsilon\_r &= \Delta \sigma\_r / E\_r - \mu\_{\theta r} \Delta \sigma\_\theta / E\_\theta \\ \Delta \varepsilon\_\theta &= \Delta \sigma\_\theta / E\_\theta - \mu\_{r\theta} \Delta \sigma\_r / E\_r \end{aligned} \tag{2}$$

The border conditions are,

$$\begin{aligned} \Delta \sigma\_r(h) &= -T(h)\Delta h/h\\ \Delta \sigma\_r(r\_i) &= \Delta \sigma\_{rm}(r\_{mo}) \end{aligned} \tag{3}$$

**Fig. 1** Fiber winding geometric model

The variation of radial displacement, stress, and surface pressure of each wound layer may be solved using Eq. (1). By integrating, one may determine the total interface pressure (*P*), radial displacement (*u*), and stress (σ).

$$P(h) = 2\lambda r\_{mo}^{\lambda - 1} f(r\_{mo})$$

$$u(r) = \left[ (\lambda - \mu\_{\theta r} - K)(\lambda + \mu\_{\theta r}) \left( \frac{r\_{mo}}{r} \right)^{\lambda} - (\lambda + \mu\_{\theta r} + K)(\lambda - \mu\_{\theta r}) \left( \frac{r}{r\_{mo}} \right)^{\lambda} \right] \frac{r\_{mo}^{\lambda}}{E\_{\theta}} f(r)$$

$$\sigma\_{r}(r) = - \left[ (\lambda - \mu\_{\theta r} - K) \left( \frac{r\_{mo}}{r} \right)^{\lambda + 1} + (\lambda + \mu\_{\theta r} + K) \left( \frac{r}{r\_{mo}} \right)^{\lambda - 1} \right] r\_{mo}^{\lambda - 1} f(r) \tag{4}$$

$$\sigma\_{\theta}(r) = \left[ (\lambda - \mu\_{\theta r} - K) \left( \frac{r\_{mo}}{r} \right)^{\lambda + 1} - (\lambda + \mu\_{\theta r} + K) \left( \frac{r}{r\_{mo}} \right)^{\lambda - 1} \right] \lambda r\_{mo}^{\lambda - 1} f(r) + T(r)$$

In the equation above, the integral function's expression is,

$$f(r) = \int h^{\lambda} T(h) / [(\lambda + \mu\_{\vartheta r} + K)h^{2\lambda} + (\lambda - \mu\_{\vartheta r} - K)r\_{mo}^{2\lambda}] dh \tag{5}$$

Each coefficient's expression is,

$$K = E\_{\theta} \left[ (1 - \upsilon\_m) + \xi^2 (1 + \upsilon\_m) \right] / E\_m (1 - \xi^2)$$

$$\begin{array}{l} \xi = r\_{mi} / r\_{mo} \\ \lambda = \sqrt{E\_{\theta} / E\_r} \end{array} \tag{6}$$

From the surface pressure *P*(*h*), it is simple to compute the displacement and stress of an isotropic mental liner [15]. Wet winding is currently a widely utilized production process. The equivalent stiffness of the uncured wound layer may be determined using the empirical formula provided by Cai and Gutowski.

$$E\_{\theta} = \pi E\_f V\_f / 4V\_a$$

$$E\_r \frac{A\_s \sqrt{V\_a / V\_0} \left[ 5 - \sqrt{V\_f / V\_a} - 4\sqrt{V\_0 / V\_f} \right]}{2\left(\sqrt{V\_a / V\_f} - 1\right)^5} \tag{7}$$

Taking into account the variation in fiber volume content with radius, λ in formula (6) should be recast as λ(*h*), which is more precise as an integral function, but the computation is also trickier.

### *2.2 Design with Uniform Residual Tension*

The inner fiber relaxes once the outer fiber has been wrapped. Making the residual tension of each layer after winding equal is the technological goal sought in practical engineering. *N* represents the overall number of winding layers. The illness is defined as,

$$
\sigma(r\_i) = \sigma(r\_{i-1}) \quad (i = 2, 3, \dots, N) \tag{8}
$$

By combining (4) and (8), equations relating to the winding tension may be rewritten as,

$$MT = H\tag{9}$$

*T* represents each layer's winding tension.

$$T = \begin{bmatrix} T\_1 \ T\_2 \ \dots \ T\_i \ \dots \ T\_N \end{bmatrix}^T \tag{10}$$

*M* is the coefficient matrix.

⎡⎣*B*1 *B*2 *B*3 ... *Bi* ... *BN*  − *A*1 *B*1 + 1 (*A*2 − *A*1)*B*2*r*λ2 − 1 (*A*2 − *A*1)*B*3 ... (*A*2 − *A*1)*Bi* ... (*A*2 − *A*1)*BN*  ... ... ... ... ... ... −*A j B j* + 1 (*A j* − *A j*−1 )*B j r*λ*j* − 1 ... (*A j* − *A j*−1 ) *Bi* ... (*A j* − *A j*−1 ) *BN*  ... ... ... ... ... − *A N*−2 *B N*−2 + 1 ( *A N*−1 − *AN* −2 ) *B N*−1*r*λ *N*−1 − 1 ( *A N*−1 − *A N*−2 ) *BN*  0 − *A N*−1 *B N*−1 + 1 ( *AN*  − *A N*−1 ) *BN r*λ*N*  − 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (11)

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

The expression of matrix *H* is,

$$H = \begin{bmatrix} \frac{1 - r\_{ml}^2 / r\_{mo}^2}{1 + r\_{ml}^2 / r\_{mo}^2} \frac{\sigma\_0}{2\lambda r\_{mo}^{\lambda - 1}}\\0\\\cdots\\0\end{bmatrix} \tag{12}$$

The definition of Coefficient *A* and *B* is,

$$A\_i = \left[ (\lambda - \mu\_{\theta r} - K) \left( \frac{r\_{mo}}{r\_l} \right)^{\lambda + 1} - (\lambda + \mu\_{\theta r} + K) \left( \frac{r\_l}{r\_{mo}} \right)^{\lambda - 1} \right] \lambda r\_{mo}^{\lambda - 1} \tag{13}$$

$$B\_i = \frac{r\_l - r\_{i-1}}{(\lambda + \mu\_{\theta r} + K) r\_l^{\lambda} + (\lambda - \mu\_{\theta r} - K) r\_{mo}^{\lambda}} r\_i^{\lambda}$$

A fiber tension algorithm that satisfies the requirement of uniform residual tension was developed. The real tension of the fiber is lower than the technical tension during the winding process of no-twist fiber, such as carbon fiber. The utilization coefficient of fiber tension is defined as,

$$\delta\_i = \frac{T\_{acll}}{T\_{tech}} = \sqrt{\frac{1 + (i - 1)\chi}{1 + n\chi}}\tag{14}$$

The equivalent thickness of the resin matrix χ is an empirical value that is established by tests. A more precise algorithm for designing tension has been developed by transforming the technological tension (*T*tech) into the actual tension (*T*actl).

### **3 Numerical Examples**

### *3.1 Constant Technological Tension Winding*

The distribution law of the prestress in a thick-walled cylinder was examined using constant technical tension winding as an example. Table 1 displays the winding process's parameters.


**Table 1** Winding process's parameters

Figure 2 depicts the stress distribution under various winding tensions. It is demonstrated that whereas the circumferential stress jumps at the interface between the liner and the wound layer, the radial stress fluctuates constantly throughout the radial direction. With a rise in winding tension, the liner's prestress grows quickly.

Figure 3 depicts the relationship between the prestress, the thickness of the liner, the number of wrapped layers, and the fiber modulus. It is seen in Fig. 3:

(1) The bigger the prestress, the thinner the liner and the higher the winding tension. In this instance, the wound prestress is decreased by nearly half while the liner thickness is raised from 1 to 2 mm. The thick-walled cylinder's prestressing is quite challenging.

**Fig. 2** Stress distribution of liner and wound layers

**Fig. 3** Relationship between prestress and winding parameters

(2) Stress saturation is the situation where the impact of prestressing the liner by winding steadily decreases with the number of wrapped layers. It is more important for the method when the wound layer's circumferential modulus is higher and the liner modulus is lower. It is best to wind fiber with a low modulus to increase prestress.

Combining the process circumstances, liner material, liner thickness, working load, structural strength, and failure mode will help you choose the right fiber and tension.

### *3.2 Variable Volume Content of Fiber*

Take the linear distribution of fiber volume content along the radial direction as an example. Figure 4 depicts the impact of the change in volume content on the stress distribution.

We may infer the following from Fig. 4:


**Fig. 4** Influence of fiber volume content on stress


**Table 2** Comparison of winding tension results

### *3.3 Uniform Residual Tension*

The design scheme for the thick-walled cylinder example from the literature [11] is illustrated in Table 2.

The results in this article are in good agreement with the literature, and the highest variance is less than 1%, demonstrating the high precision of the algorithm used here, which is also easier to use than the iterative approach used in the literature.

### **4 Experimental Verification of the Tension Loss Model**

Three groups of winding experiments with various liner thicknesses and fiber types were created in order to validate the accuracy of the tension loss model put out in the research. Table 3 compares the experimentation circumstances in each group.

The dynamic strain measurement apparatus was utilized to assess how prestress changed layer by layer while the fiber was wound (Fig. 5). The inner wall of the liner's liner was evenly covered with four strain gauges, A, B, C, and D. The mean value of the strain was taken as the liner's circumferential strain.

The circumferential strain of the liner layer by layer during fiber winding was calculated using the tension loss model described in the study, and the comparison with the test results is shown in Fig. 6.

Figure 6 demonstrates that, with practically no tension loss, the real tension of glass fiber winding is nearly equivalent to the technical tension. However, the carbon fiber winding tension loss is rather evident. There is a significant discrepancy between


**Table 3** Comparison of experiment parameters

**Fig. 5** Dynamic strain monitoring device

**Fig. 6** Test results of strain

the real prestress and the design if the winding tension is estimated while disregarding tension loss. The largest strain divergence between the modified designs and the testing in #2 and #3, according to the tension loss model, was 4.1% and 3.3%, respectively. The experiments and the theoretical prediction correspond rather well.

# **5 Conclusions**

Winding tension was designed for thick-walled cylinders with uniform residual tension in this thesis. The following are the primary conclusions:


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on the Design and Verification Process of Mechanical Penetrations in Reactor Compartment**

**Qian Zhang, Zuoqin Qian, Qiang Wang, and Xinyu Wang** 

**Abstract** Mechanical penetrations, as important pressure pipelines penetrating the reactor compartment, withstand high temperatures and pressures. The current complete design and verification process for mechanical penetrations. This article focuses on the problem of stress concentration and easy damage of the penetration components in the reactor compartment under high temperature and high pressure environment. Combining with the existing regulations of nuclear power plants and ships, finite element analysis method is used to analyze the stress of the penetration components under specific high temperature and high pressure and ship ultimate load coupling. At the same time, based on the simulation analysis results, the structural dimensions of the penetration components are optimized, and a mechanical penetration verification process is designed. The coupled thermal stress results of the penetration indicate that the stress of the penetration is too large at the tail of the sleeve, with the values of primary film stress Pm and primary bending stress Pb being 228.2 and 275.91 MPa, respectively. From this, it can be seemed that there is obvious stress concentration at the junction of the support ring and sleeve, as well as at the transition point of the insulation layer, which is the weakest area of the penetration.

**Keywords** Mechanical penetration parts · Nozzle load · Finite element method · Thermal mechanical coupling analysis

# **1 Introduction**

Floating nuclear power plant (FNPP), also known as floating nuclear power plant, is a mobile small nuclear power plant on the ocean. It combines small nuclear reactors with maritime transportation engineering, which can meet the needs of cooling and heating in urban areas, reducing haze weather, and energy supply in remote areas [1].

Q. Zhang · Z. Qian · Q. Wang · X. Wang (B)

Wuhan University of Technology, Wuhan 430063, China e-mail: 292814@whut.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_29

Mechanical penetrations are part of the reactor containment vessel and are components that connect the internal and external pipelines of the containment vessel [2]. According to the relevant requirements of nuclear safety, mechanical penetrations still need to maintain the ability to perform their functions under severe accident conditions [3].

Mechanical penetrations are installed on the cabin walls to transport fluid inside and outside the cabin. As an important component of the cabin structure, they, together with electrical penetrations, shielding doors, and the main structure of the cabin, form the third barrier for radioactive shielding in the cabin [4]. The penetration is connected to the cabin bulkhead and is located in a structurally discontinuous position, where stress is easily concentrated. Therefore, the structural strength design of the penetration should not only ensure the airtightness and integrity of the safety shell, but also meet the mechanical performance requirements under various working conditions. At present, many experts and scholars at home and abroad have conducted research on it. Li [5] used finite element software ABAQUS to simulate the steadystate and transient thermal coupling of pressure vessels, analyzed the law of coupled thermal stress during radial and circumferential changes in the vessel structure, and explored the influence of temperature on heat transfer; Hazizi [6] combined with ASME specifications and designed a cylindrical pressure vessel that will not fail by analyzing a pressurized liquefied petroleum gas vertical pressure vessel with a capacity of 10 cubic meters. Jasion [7] studied a pressure vessel with a special cylindrical shape, established an analytical model of this container composed of a rotating shell, and based on membrane theory, analyzed, and determined the stress state in the container. Chen [8] designed a set of loading test equipment to meet the testing requirements of penetrations under extreme load. It was found that when the ultimate load was applied, the stress was mainly concentrated at the root of the penetration; Tong [9] analyzed the stress and deformation of the containment vessel, as well as its ultimate bearing capacity, using a prestressed finite element model. Song [10] conducted vulnerability analysis and probabilistic safety performance evaluation of the nuclear containment structure under the condition of prestressed tendon fracture based on the finite element model of the nuclear containment. They discussed the changes in concrete compressive stress state and the ultimate bearing capacity of the nuclear containment structure, and analyzed the vulnerability curve and probabilistic safety margin of the nuclear containment structure under the condition of prestressed tendon fracture.

Currently, there are complete regulations in domestic and foreign regulations regarding the evaluation standards for nuclear grade pressure vessels. However, there is no clear regulation on the design verification process and evaluation standards for nuclear grade pressure vessels on ships, such as mechanical penetrations in storage tanks. Therefore, this study combines ships with nuclear power plants. Based on the relevant specifications of nuclear grade pressure vessels and ship structures, combined with theoretical research and specific engineering examples, finite element software is used to conduct thermal coupling simulation analysis on the penetration components of the reactor compartment. Based on the results, the structural dimensions of the penetration components are optimized, and a set of process verification methods is designed. In this paper, a reactor tank penetration with a size of ϕ 273.0 × 25.4 [11] was designed and checked through thermal simulation and stress deformation simulation analysis and calculation.

### **2 Thermodynamic Coupling Theory**

The coupling field can be divided into different physical fields: magnetic field and electromagnetic field, magnetic field and stress field, temperature field and stress field, temperature field and flow field coupling, etc. The coupling methods are divided into direct coupling and sequential coupling. Direct coupling involves introducing coupling effects into the control equation through element matrices or load vectors, and then solving the control equation. The calculation process is convenient but the calculation accuracy is not high, and it is mostly suitable for theoretical analysis of coupled fields; Sequential coupling is the process of solving a single physical field within each incremental step, and then applying the solution of the previous physical field as an external load to the latter physical field to complete the coupling of the two physical fields. Although the calculation process is relatively complex, this method has high computational accuracy and is suitable for engineering calculations of coupled fields [12].

This article adopts the sequential coupling method, and the finite element equation of sequential coupling is:

$$
\begin{bmatrix} K\_{11}X\_1 + K\_{11}X\_2 \ 0 \\ 0 \end{bmatrix} \begin{bmatrix} X\_1 \\ X\_2 \end{bmatrix} = \begin{Bmatrix} F\_1X\_1 + F\_1X\_2 \\ F\_2X\_1 + F\_2X\_2 \end{Bmatrix} \tag{1}
$$

In the formula: *X*1 and *X*2 represent the displacement matrices of two physical fields; *F*1 and *F*2 represent the equivalent load matrices of two physical fields; *K*<sup>11</sup> and *K*22 represent the equivalent stiffness matrices when two physical fields exist separately.

Bring the single stress field control equation and the single temperature field control equation into the sequential coupling control equation to obtain the thermal coupling control equation:

$$
\begin{bmatrix} M \ 0 \\ 0 \ 0 \end{bmatrix} \begin{Bmatrix} \ddot{u} \\ \ddot{T} \end{Bmatrix} + \begin{bmatrix} C \ 0 \\ 0 \ C' \end{bmatrix} \begin{Bmatrix} \dot{u} \\ \dot{T} \end{Bmatrix} + \begin{bmatrix} K & 0 \\ 0 \ K' \end{bmatrix} \begin{Bmatrix} u \\ T \end{Bmatrix} = \begin{Bmatrix} F \\ Q + Q^p \end{Bmatrix} \tag{2}
$$

In the formula: *u* and *T* are the element displacement matrix and element temperature matrix, respectively; *C*,*C<sup>t</sup>*is the unit damping matrix and the unit specific heat matrix, respectively; *K* , *K <sup>t</sup>*is the element stiffness matrix and the element heat conduction matrix, respectively; *M* is the unit quality matrix; *F* is the unit load matrix; *Q* is the unit heat generation and unit heat flow rate; *Q <sup>p</sup>*is the unit plastic heat generation rate.

**Fig. 1** Thermal analysis method process

When the temperature of the penetration changes, deformation will occur, resulting in thermal stress in the structure. The thermal stress of high-energy penetration belongs to secondary stress, which needs to be considered under various working conditions. Therefore, the strength of the penetration must meet the requirements under ultimate load. At the same time, in order to ensure that the temperature of the shielding material is within its working temperature range, thermal coupling analysis of the penetration is necessary. This article uses Ansys software and sequential coupled thermal analysis method to calculate the stress components of each component of the penetration, and verifies the strength of each component of the penetration according to the stress assessment criteria of ASME code [13]. The flow chart of the sequential coupled thermal analysis method in this article is shown in Fig. 1.

### **3 Model and Mesh**

A certain reactor compartment is 1 m × 1m × 1 m enclosed space, pipeline penetrations are key components for fluid transportation inside and outside the reactor compartment, mainly including structural parts and other ancillary structures. The geometric model of the penetration is 1 m × 1m × 1 m 3D model including the penetration. The structure of the penetration includes the middle pressure pipe, the head between the pipe and the sleeve, the sleeve and the support ring; The remaining components are auxiliary structures, including the insulation layer between the sleeve and the penetrating pipe fittings, fixed steel bars, angle steels, and resin plates, radiation resistant materials such as lead plates and lead boron polyethylene, and fire-resistant materials such as polycrystalline wires; The rest is filled with air. The three-dimensional model of the penetration is shown in Fig. 2.

The geometric grid size of the bulkhead and T-shaped support plate is set to 15 mm, the web plate is set to 10 mm, the bar steel, angle steel, and resin plate are set to 5 mm, and the polycrystalline wire is set to 7.5 mm. For penetrations, sleeves, and support rings, the grid size is set to verify the thickness direction of three units. For welding points, the grid size is set to 1 mm, and for other areas, the grid size is

**Fig. 2.** 3D geometric modeling of penetration

set to default. At the same time, select slow transition areas for the grid transition, set the resolution for small areas to 3, and the number of grid cells is 6,865,833. The 3D mesh model is shown in Fig. 3.

**Fig. 3** Mesh division of penetrations

**Fig. 4** Setting of temperature boundary conditions for penetrations

### **4 Boundary Condition**

### *4.1 Temperature Field Boundary Conditions*

The interior of the penetration pipe is a high-temperature and high-pressure fluid. During the steady-state simulation of the temperature field of the penetration and its affiliated structures, the inner wall of the penetration pipe is assumed to be convective heat transfer. When the internal fluid is liquid, the convective heat transfer coefficient is taken as 2000 w/m2 K, and when the internal fluid is gas, the convective heat transfer coefficient is taken as 300 w/m2 K, with an ambient temperature of 25 °C and a convective heat transfer coefficient of 15 w/m2 K; The bulkhead and T-shaped support platform are connected to the storage tank as an insulated wall surface; The contact between other external surfaces and the environment is assumed to be natural convection, and the internal fluid is set as gas. The specific boundary conditions are set as shown in Fig. 4.

### *4.2 Stress Field Boundary Conditions*

The loads borne by the penetration are mainly the design internal pressure, temperature load, and nozzle load. The design internal pressure is set as surface pressure, which acts on the inner wall of the connecting piece along the internal normal direction, at a pressure of 7.0 MPa; There are two commonly used methods for determining the nozzle load of nuclear vessels: ➀ conducting mechanical analysis of the pipeline system to obtain the nozzle load of the vessel; ➁ The allowable load of the nozzle (which is the upper limit value of the nozzle load obtained from the mechanical analysis of the pipeline system) is determined based on the geometric parameters, material characteristics, force torque relationship, and stress assessment criteria under various load conditions of the container nozzle. Method ➁ is not affected by the progress of piping analysis. In the preliminary stage of nuclear vessel design, in order to accelerate the engineering progress, a container-based nozzle load analysis method can be used.

According to the provisions of Section III of the ASME Boiler and Pressure Vessel Code and the Construction Rules for Nuclear Facility Components, the allowable nozzle load of the vessel is calculated to ensure that the maximum stress caused by the design pressure and external load does not exceed the limit of primary stress under design, critical, and accident conditions. Therefore, based on the geometric parameters and material characteristics of the nozzle structure, the stress intensity values at each section of the nozzle can be obtained. Then, according to the overall primary membrane stress intensity Pm, local primary membrane stress intensity PL, Pm (or PL) + primary bending stress intensity Pb stress limit value specified in the ASME code under various working conditions, the allowable nozzle load can be determined by calculating the assumed nozzle load of each type [14]. According to elasticity, the stress calculation of the evaluated section is as follows:

Axial stress:

$$
\sigma\_z = \frac{P}{A} + \frac{p\_i r\_i}{2t} + \frac{M\_{\rm Br} r\_o}{I} \tag{3}
$$

In the formula: *p*i is the internal pressure, Pa; *A* is the cross-sectional area, m2; *r*i is the inner radius, m; *r*o is the outer radius, m; *t* is the wall thickness, m; *I* is the moment of inertia, m4; *M*B is the total bending moment, N m.

Circumferential stress:

$$
\sigma\_0 = \frac{p\_i r\_i}{t} \tag{4}
$$

Radial stress:

$$
\sigma\_r = -\frac{p\_i}{2} \tag{5}
$$

Shear stress:

$$\begin{array}{c} \mathsf{r} = \mathsf{r}\_1 + \mathsf{r}\_2\\ \mathsf{r}\_1 = \frac{\mathsf{V}\mathcal{Q}}{2It}; \mathsf{r}\_2 = \frac{M\_{\mathsf{I}\mathcal{I}^0}}{2I} \end{array} \tag{6}$$

In the formula: Q is the shape coefficient, m3. Section 3 principal stresses σ1, σ2, σ3 is:

$$\begin{aligned} \sigma\_1 &= \frac{\sigma\_\circ + \sigma\_\theta}{2} + \left[ \left( \frac{\sigma\_\circ - \sigma\_\theta}{2} \right)^2 + \pi^2 \right]^{0.5} \\ \sigma\_2 &= \frac{\sigma\_\circ + \sigma\_\theta}{2} - \left[ \left( \frac{\sigma\_\circ - \sigma\_\theta}{2} \right)^2 + \pi^2 \right]^{0.5} \\ \sigma\_3 &= \sigma\_r \end{aligned} \tag{7}$$

The stress difference on the section is:

$$\begin{aligned} S\_{12} &= |\sigma\_1 - \sigma\_2| \\ S\_{23} &= |\sigma\_2 - \sigma\_3| \# \\ S\_{13} &= |\sigma\_3 - \sigma\_1| \end{aligned} \tag{8}$$

The total stress intensity is:

$$S\_{\rm I} = \max(S\_{\rm I2}, S\_{\rm 23}, S\_{\rm 13}) \tag{9}$$

In solid mechanics simulation, the outer wall and connecting plate are set as fixed constraint points, and the nozzle load is calculated according to the formula, which is divided into four types: Tensile force FT = 253.652 kN; Shear force FV = 253.652 kN; torque MV = 126.826 kN m, and Bending moment MB = 126.826 kN m. The torque is clockwise. Due to the connection between both ends of the penetration and the connecting pipeline, four types of nozzles load under normal operating conditions are applied at both ends of the penetration. The loading method is shown in Fig. 5.

### **5 Calculation and Result Analysis**

After setting the boundary conditions, conduct a steady-state analysis and solution of the temperature field of the penetration; Import the temperature field into the mechanical analysis interface for thermal stress analysis, and then load the mechanical boundary conditions again to enter the stress evaluation analysis.

There are two main verification standards for temperature field simulation: 1. Whether the maximum temperature of the lead plate shielding layer exceeds 300 °C; 2. Does the maximum temperature of the lead boron polyethylene shielding layer exceed 120 °C.

For the stress field, set the stress path and extract the membrane stress and bending stress on the stress path. The stress path is shown in Fig. 6.

Calculate the allowable stress of different components according to the specifications, verify each stress component according to the allowable stress, and complete stress assessment. The stress assessment criteria for penetrations are shown in Table 1.

**Fig. 5** Setting of mechanical boundary conditions for penetrations

**Fig. 6** Penetration stress path


**Table 1** ASME evaluation standards

### *5.1 Temperature Field Evaluation*

The temperature field distribution of the penetration after steady-state temperature simulation is shown in Figs. 7, 8 and 9. The overall maximum temperature of the

**Fig. 7** Overall temperature distribution of penetration

penetration is 299.4 °C, and the minimum temperature is 25.151 °C; The maximum temperature of the lead plate shielding layer is 29.103 °C, and the minimum temperature is 26.044 °C; The maximum temperature of the lead boron polyethylene shielding layer is 58.226 °C, and the minimum temperature is 26.157 °C. The calculation results indicate that the temperature evaluation of the penetration meets the design requirements.

### *5.2 Stress Field Evaluation*

The stress field distribution of the penetration after steady-state stress simulation is shown in Fig. 10. The maximum overall stress of the penetration is 651.44 MPa, and the stress is mainly concentrated at the root.

According to ASME Section II specifications, compare the calculated stress components with the corresponding allowable stresses. Under normal working conditions, take a 90° profile stress, and the evaluation results are shown in Table 2. The calculation results show that the stress of the penetration is at paths 11 and 12, with

**Fig. 8** Temperature distribution of lead plate shielding layer

**Fig. 9** Temperature distribution of lead boron polyethylene shielding layer

**Fig. 10** Overall stress distribution of penetrations

the values of primary membrane stress Pm and primary bending stress Pb being 228.2 and 275.91 MPa respectively, which do not meet the evaluation requirement of 207 MPa. The local stress component of the penetration is relatively large, mainly concentrated at the transition between the insulation layer and the support ring of the penetration. The thickness of the sleeve can be increased to strengthen it.

### *5.3 Verification Process for Overall Design of Penetrations*

For the overall design verification of mechanical penetrations, based on the theory of thermal coupling, the ultimate load calculation is combined with the ASME "Boiler and Pressure Vessel Code" Section III and "Nuclear Facility Component Construction Rules". The evaluation criteria are in accordance with ASME Section II specifications, and a complete set of processes is designed. The specific design verification process is shown in Fig. 11.


**Table 2** Profile stress assessment results

**Fig. 11** Verification process for overall design of penetrations

# **6 Conclusion**

This article focuses on the problem of stress concentration and easy damage of the penetration components in the reactor compartment under high temperature and high pressure environment. Combining with the existing regulations of nuclear power plants and ships, finite element analysis method is used to analyze the stress of the penetration components under specific high temperature and high pressure and ship ultimate load coupling. At the same time, based on the simulation analysis results, the structural dimensions of the penetration components are optimized, and a mechanical penetration verification process is designed. Based on the above results, the following conclusions can be drawn:


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Mechanical Control and Fault Monitoring Analysis**

# **A Machine-Vision-Based Hub Location Detection Technique for Installing Wind Turbines**

**Xuyang Cao, Yongchang Hu, Guoyang Xu, Shuai Song, and Xiaochun Tie** 

**Abstract** A position detection approach based on machine vision is developed to address the issue that the traditional wind turbine hub lifting procedure relies on the location of engineers and cannot provide quick feedback on the position information. A camera and a range sensor are mounted on the suspended object to enable real-time detection of the fan hub's posture state in relation to the engine room. A monocular camera positioning technique based on circular features is created for this detection method. The relative pose coordinates of the hub are computed using the position data given by the range sensor. A filter for the range sensor is created based on the hoisting features of the actual working circumstances. The filter applies a filter to the data from the ranging sensor, considerably reducing measurement error and enhancing detection technique stability.

**Keywords** Wind turbine hoisting · Machine vision · Pose detection · Data processing

# **1 Introduction**

The advancement of wind turbine-related technology has raised the bar for crane efficiency, lifting precision, real-time detection, and precise positioning control in wind power construction projects as well as the installation and maintenance of wind turbines. At present, the commonly used domestic wind turbine installation method is to install the nacelle on the tower first, and then the blade and hub are combined on the ground, and the blade and hub assemblies are lifted by large cranes in the high altitude for the installation method. Hub blade assembly in the lifting process, the

X. Cao · Y. Hu (B) · G. Xu

e-mail: huyongchangd@163.com

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_30

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

S. Song · X. Tie Taiyuan Heavy Industry Co., Ltd., Taiyuan 030024, Shanxi, China

need for multiple cranes to work together, through the rope pulling and other ways to adjust the position of the hub blade assembly.

With the development of wind turbine related technologies, the capacity of wind turbines has been continuously improved, and the installation height of wind turbines has also reached more than 100 m, as shown in Fig. 1, for the installation of wind turbine hubs. Under this working condition, it is difficult to work at high altitude, there are many people involved in installation, and the cost is high. When the hub is installed and aligned, due to factors such as wind power or technical personnel cooperation, the installation period is long and the installation time is greater than 70 h. With the rapid development of China's wind power industry, it is necessary to develop a pose detection method with less direct participation, high safety, high detection efficiency and high detection accuracy.

Compared with other pose detection methods, visual positioning can detect the position of the measured object under non-contact conditions, and can adapt to various complex environments under sufficient illumination. Scholars at home and abroad have accumulated relevant research foundations in machine vision. Liu et al. [1] designed a robot intelligent unstacking system based on visual positioning. The corresponding world coordinates are obtained by using the coordinate transformation of the target pixel center. The position error is 1.1 mm and the angle error is 1.2°. Chen et al. [2] proposed a screw automatic fastening assembly system based on visual positioning. The visual positioning technology based on sub-pixel edge is introduced to accurately compensate the positioning deviation of screw assembly, and the visual positioning accuracy is better than 0.02 mm.

Currently, the detection of camera fixation and the movement of the measured item are the basic foundations of visual positioning research. This study suggests a pose

**Fig. 1** Wind turbine hub lifting

detection technique in which the measured object and the camera are both in a state of erratic motion. This study offers a fan hub based on a circle fitting algorithm that integrates a laser range sensor with machine vision to address the complex working environment.

# **2 The Wheel Hub Pose Detection Scheme Under High Altitude Condition**

Since the characteristics of the cabin itself are difficult to be directly used as positioning features, it is necessary to add feature color blocks as positioning markers, as shown in Fig. 3. The selection of positioning features is the key to the realization of visual detection. For the needs of practical engineering, two square color blocks of a certain size are installed in different directions of the transmission shaft as the positioning feature of the secondary treatment, and the bolt hole of the engine room shaft is used as the feature of the primary treatment.

The camera is used to collect and process the image to obtain the position information of the target point. Combined with the distance information of the ranging sensor, the pixel information on the image is converted into the actual distance information, and the pose state of the fan hub is obtained.

The position of the camera and the ranging sensor is shown in Fig. 2, and the installation position of the ranging sensor and the industrial camera is that the hub is facing the cabin surface. After the camera is installed, the relative position matrix of the camera and the hub center should be obtained to initialize the pose matrix.

a) Camera mounting position b) Industrial camera

**Fig. 2** Sensor mounting

**Fig. 3** Characteristic color block

### **3 Visual Orientation Program**

### *3.1 Image Segmentation Scheme of Secondary Processing*

Since the cabin itself has no obvious features that can be used as features for visual recognition and positioning, it is necessary to add feature color blocks for positioning [3]. The square color block features are prominent, and when performing contour fitting, it can effectively eliminate the influence of the external environment on visual recognition. Therefore, two square color blocks of a certain size are installed in different directions of the drive shaft as the positioning feature of the secondary processing, as shown in Fig. 3. The bolt hole of the nacelle shaft is used as a feature of one-time processing.

The quality of the image the camera captures cannot be guaranteed due to the intricacy of the real operating conditions. As a result, during image processing, secondary processing is used to segment the images [4]. The captured image is first simply processed to separate the main recognition portion from the remainder portion, which is then binarized. The picture is then re-segmented following one procedure: binarization processing is used to produce the desired contour after screening by area, contour, and other parameters [5]. It is a flow chart for secondary processing, as seen in Fig. 4.

This image processing method increases the computational pressure of the host computer to a certain extent and increases the time of image processing, but it can effectively overcome the influence of the environment on image processing and improve the accuracy of visual positioning [6]. In addition, during the installation of

**Fig. 4** Secondary image processing flow

the fan hub, the movement speed is slow, and the influence of the algorithm delay on the installation process can be ignored.

### *3.2 Circle Fitting Algorithm Based on Least Square Method*

When the wheel hub and the engine room are far apart, the area of the positioning color block in the image collected by the camera is small, and it is difficult to extract the features. Aiming at the situation that the color block cannot be identified and located at a long distance, a circle fitting algorithm based on the least square method is introduced [7].

After the image is binarized, the edge of the image will be highlighted. The highlighted pixels can be regarded as points in the two-dimensional coordinate system (x*<sup>i</sup>* , y*i* ).

First, the mathematical algebraic expression of the circle is:

$$(\mathbf{x} - \mathbf{A})^2 + (\mathbf{y} - \mathbf{B})^2 = R^2 \tag{1}$$

According to (1), another form of the equation of the circle can be obtained:

$$ax^2 + y^2 + ax + by + c = 0\tag{2}$$

That is to say, the parameters of the circle can be obtained by calculating a, b, c. The distance from each point to the center of the circle *di* :

$$\left|\mathbf{d}\right|^2 = \left(\mathbf{x}\_i - \mathbf{A}\right)^2 + \left(\mathbf{y}\_i - \mathbf{B}\right)^2\tag{3}$$

The square difference between the square of the distance from each point to the edge of the circle and the radius is:

$$
\delta\_i = \mathbf{d}\_i^2 - \mathbf{R}\_i^2 \tag{4}
$$

Let Q (a, b, c) be the sum of squares of δ*<sup>i</sup>* :

<sup>Q</sup><sup>=</sup> *n i*=1 δ*i*  2 (5)

The parameters (a, b, c) are obtained to minimize the value of Q.

Figure 5 shows the binary image of the interface shape of the wind turbine. When the camera has a large angle deflection or the camera cannot collect a complete engine room image, as shown in Fig. 6, this circle fitting algorithm can realize the detection and feedback of the deflection.

**Fig. 5** Cabin docking surface

**Fig. 6** Simulation of large angle deflection

When the camera only captures part of the cabin image, the simple color feature or circular feature algorithm cannot reflect the correct position relationship [8, 9]. However, the circle fitting algorithm based on the least square method can solve this kind of problem to a certain extent.

Firstly, the center coordinates of each bolt hole are obtained by image processing, as shown in Fig. 7.

Then, the circle fitting of the center coordinates of each bolt hole is carried out, and the pixel coordinates of the center of the engine room are obtained, as shown in Fig. 8.

According to the returned information, the position and posture of the hub are adjusted in time to prevent collision accidents and ensure the safety of installation work.

#### **Fig. 8** Fitting the center of the circle

# *3.3 Ranging Sensor Filtering Algorithm Based on Mean Filtering*

The measurement range of the laser ranging sensor used in the experiment is 0.01– 8 m, the measurement blind area is 10 mm, and the measurement error is 2%. When the laser ranging sensor performs distance detection, the detected distance information is unstable and there is a certain error. The software is used to simulate the random error distribution of the ranging sensor within the range, as shown in Fig. 9.

The ranging sensor data is collected at different distances, and several data are randomly collected at the same distance, as shown in Fig. 10, which is the measurement data at a distance of 1138 mm.

Based on the data collected several times, it is concluded that the size of the error of the ranging sensor increases with the increase of the distance when the data measurement is performed. According to the working principle of the range sensor and statistics related theories, the filtering algorithms such as minimum filter, median

**Fig. 9** Random error simulation of ranging sensor

**Fig. 10** Error floating of ranging sensor

filter, mean filter and Kalman filter [10] are used to perform the filtering experiments on the range sensor data in static and dynamic situations respectively, as shown in Fig. 11.

By measuring a number of data at a fixed distance and taking the mean value of a number of data as a reference value. According to the data obtained from many experiments, it can be seen that these filtering methods can well reduce the floating trend of the measurement data. Moreover, the effect of mean filtering is the best.

According to the experimental data under different conditions, when the motion speed is greater than 0.5 m/s, the filtering effect of median filter and Kalman filter is better, and the value is closer to the reference value. When the motion speed is less than 0.2 m/s, the mean filter has the best filtering effect, and the data is stable and close to the reference value.

In addition, the size of the filter also affects the filtering performance. Taking mean filtering as an example, when the motion speed is slow, the larger the filter is, the better the filtering effect is. When the motion speed is fast, the smaller the filter, the better the filtering effect.

**Fig. 11** Comparison of minimum and mean filtering

### *3.4 Coordinate Conversion Based on Image DPI*

In the traditional coordinate transformation method, the internal and external parameters of the camera are calculated by camera calibration, and then the image coordinate system is transformed into the world coordinate system through the camera internal and external parameter matrix and pixel coordinates [11]. When the camera is stationary, this coordinate conversion method can achieve high accuracy and good stability. However, during the installation of the fan hub, the hub and the nacelle are in an unstable state of motion. Therefore, the traditional coordinate transformation method does not fully meet the positioning requirements under this working condition.

Aiming at the inapplicability of traditional coordinate transformation methods, a camera coordinate transformation method based on distance and camera DPI relationship is adopted. At a fixed distance, the number of pixels per inch of the actual distance is fixed. During the movement, the DPI of the camera changes with the distance. By randomly changing the distance between the camera and the measured object within a certain distance range, the relevant data of the camera DPI and distance are obtained.

Then, the curve fitting of camera DPI and distance is carried out by polynomial approximation. Finally, the relationship between camera DPI and distance is obtained, as shown in Fig. 12, Distance in mm.

Through multiple sets of data fitting, the quadratic polynomial equation of camera DPI with distance transformation is obtained.

$$\mathbf{D} = 6.13 \ast 10^{-8} \ast \mathbf{x}^2 + 1.38 \ast 10^{-3} \ast \mathbf{x} - \mathbf{5} \ast 10^{-2} \tag{6}$$

x is the vertical distance between the center of the camera imaging plane and the plane of the cabin, in mm. The measurement for D is mm/pixel, which stands for the

**Fig. 12** Camera DPI and actual distance fitting

reciprocal of the camera's DPI and represents the actual distance that each pixel at a given distance represents.

The derived equation's error squares are equal to 0.02, and the R-square (coefficient of determination) is equal to 0.998.

On the basis of this, it is possible to determine the real distance L of the item that was measured using image processing:

$$\mathbf{L} = \mathbf{D} \ast \mathbf{l} \tag{7}$$

l: The number of pixels in the radius of the image, in pixels.

Taking the engine room center as the origin of the world coordinate system, the actual distance between the engine room center and the camera center is obtained by calculation. Taking the direction perpendicular to the nacelle section as the y-axis, the relative position of the camera relative to the nacelle can be obtained, that is, the position matrix T of the hub, as shown in Fig. 13.

Then the rotation matrix R of the hub is obtained by the gyroscope, and the pose matrix P of the hub can be obtained:

**Fig. 13** Pose coordinate system

$$\mathbf{P} = \mathbf{R} \ast \mathbf{T} \tag{8}$$

R is the initialized rotation matrix. T is the initialized position matrix.

### **4 Experimental Analysis**

Firstly, the error of the visual algorithm itself is evaluated, and the evaluation results can provide reference for the subsequent accuracy improvement. The positioning mark used in the experiment is a blue circular mark with an actual radius of 85 mm, as shown in Fig. 14. The radius of the positioning mark is measured at a fixed distance. The pixel size of the radius of the circle is 47.7 based on the principle of averaging multiple measurements. The visual algorithm is used to process the positioning mark multiple times, and the pixel size of the radius of the circle is measured, and the error of the visual algorithm can be obtained, as shown in Fig. 15.

**Fig. 15** Algorithm error

a) Visual inspection device b) Model of wind turbine nacelle

According to the data obtained from several experiments, the error of the algorithm in detecting the radius is between 0.15 pixels when the conditions are fixed. According to the working conditions of wheel installation, the introduction of mean value filter to optimize the algorithm can keep the error of the algorithm within 0.1 pixels.

On this basis, a ranging sensor is introduced for practical measurement experiments. The positioning markers are subjected to reciprocating motion and arbitrary angle deflection at random speeds within a range of 1 m in front of the camera to simulate the instability of the actual lifting process, as shown in Fig. 16.

The image processing algorithm, ranging sensor filtering algorithm and position detection algorithm shown in the paper are used to carry out the measurement of the radius of the positioning markers for many times, as shown in Fig. 17. The experimental results are compared with the actual radius of the positioning markers to find out the error, and the measured error fluctuation curve is obtained, as shown in Fig. 18.

According to the comprehensive results of multiple sets of experimental data, the position detection method in this paper can realize real-time radius detection of positioning markers in dynamic situations, and the detection error is within 5 mm. The results of multiple experiments show that the position detection method described in this paper has good repeatability. In the process of fan hub installation, the distance between the hub and the nacelle is within 1 m, so the fan hub position detection method described in this paper has reference value for the actual fan hub installation process.

### **5 Conclusion**

In this paper, a method for detecting the position and posture of the fan hub based on machine vision and ranging sensor is proposed. The camera is used to collect the image, and the image is processed twice. The circle fitting algorithm based on the least square method is used to obtain the pixel coordinates of the positioning mark. The relative position relationship between the hub and the cabin is obtained by the relationship between the ranging sensor and the fitted camera DPI with the change of distance. The obtained data is processed and optimized by various filtering algorithms, which improves the accuracy of visual positioning and can meet the positioning requirements under actual working conditions. The feasibility and stability of the wheel pose detection method are verified by experiments, which meets the accuracy requirements of engineering hoisting. The feasibility of applying machine vision to construction machinery is verified.

This visual positioning method can reduce the direct participation of personnel in the process of fan hub hoisting, and return the real-time position information to the crane console and commander. When there is a large angle deflection or abnormal position, the technical personnel can adjust the hoisting process in time to eliminate the risk. It provides a technical reference for the research and development of intelligent hoisting or intelligent crane.

**Acknowledgements** This work is supported by National Natural Science Foundation of China under Grant No. 52275088 and the Fundamental Research Funds for the Central Universities under Grant No. DUT22LAB507.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Fault Simulation and Experimental Validation of Accessory Transmission System**

**Yu Sun, Xuyang Cao, Yunbo Yuan, Guang Zhao, Song Ma, and Haofan Li** 

**Abstract** As an important part of the aero-engine, the accessory transmission system affects the operation status of the engine, and it is of great significance to carry out fault mechanism analysis and fault identification. This paper takes a certain type of aero-engine accessory transmission system as the research object, flexes the box using ANSYS, establishes a rigid-flexible coupling model under normal working conditions and fault conditions through ADAMS, and studies the vibration characteristics in single fault and multi-fault combination modes, such as unbalance of the shaft system, gear misalignment and gear broken tooth. Considering different load loading conditions, the dynamic simulation of the transmission system is carried out, the shafting displacement and box acceleration response are extracted, the time– frequency domain feature information of the vibration signal is analyzed, and the fault characteristics and fault types are corresponded one-to-one. The test bench for the principle prototype of the accessory transmission system is designed and built, and the experimental research is carried out. The results show that the simulation results have good consistency with the principle prototype test results, which verifies the rationality of the simulation model.

**Keywords** Accessory transmission system · Rigid-flexible coupling · Vibration characteristics · Fault diagnosis

Y. Sun · X. Cao

Y. Yuan (B) · S. Ma School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China e-mail: sy18103645250@163.com

G. Zhao

School of Energy and Power, Dalian University of Technology, Dalian 116024, Liaoning, China

H. Li AVIC Shenyang Aircraft Design and Research Institute, Shenyang 110850, Liaoning, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_31

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

### **1 Introduction**

The accessory transmission system is a key component of the aero-engine, which works in an environment of high-speed, high-temperature and high-load for a long time, and will be subjected to strong impact. Gears, bearings and other components are prone to fatigue damage. Therefore, the study of vibration characteristics under fault conditions is of great practical application significance.

Domestic and foreign scholars have widely used dynamic modeling methods to study the vibration signal characteristics of gear systems. Hong et al. comprehensively considered the time-varying meshing stiffness, meshing damping, and the integrated error of wheel tooth meshing, and simulated the dynamic characteristics of a multi-stage gear transmission system based on ADAMS [1]. Lu et al. established a rigid-flexible coupling model of the gearbox and compared and analyzed the vibration signal characteristics of the gearbox under normal and broken tooth conditions [2]. Loutas et al. established a two-stage gear system model and investigated the effects of the meshing stiffness and tooth flank clearance on the vibration response of the box [3]. Helsen et al. studied the vibration characteristics of the pure torsion model, the rigid six-degree-of-freedom multi-body dynamics model and the rigidflexible coupling model of the planetary gearbox, respectively, and concluded that the rigid-flexible coupling model affects the vibration modes and transfer function of the gearbox [4]. Dąbrowski et al. used ADAMS to establish a rigid-flexible coupling model of planetary gearboxes, and studied the vibration response generated by planetary gear transmission errors and manufacturing errors through the meshing force between gear pairs [5]. Chen et al. quantitatively analyzed the effects by time domain indexes for the rigid-flexible coupling model of planetary gearboxes with different tooth root crack degrees [6]. Wu et al. constructed a fixed-axis rigid-flexible coupling model based on LMS Virtual Lab, and studied the vibration response characteristics of the gearbox under normal, broken tooth and eccentric faults [7].

At present, most of the research focuses on the single-stage gear system and the single gear failure condition of the transmission system vibration response. For the multi-stage transmission system, the coupled fault condition of the box vibration response in research is less, and the lack of test. This paper considers the single-fault and multi-fault combination conditions of the accessory transmission system, such as shafting imblance, gear misalignment and gear broken teeth. The rigid-flexible coupling model of the accessory transmission system is established by ANSYS and ADAMS. The signal characteristics are extracted and compared with the vibration response of the box under normal conditions. Finally, the simulation data is compared with test bench data to verify the rationality of the model.

# **2 Rigid-Flexible Coupling Model of Accessory Transmission System**

Taking a certain type of aero-engine accessory transmission system as the research object, its transmission principle is shown in Fig. 1. Using SOLIDWORKS to establish a three-dimensional model including the gear shaft, spline shaft and box, as shown in Fig. 2, and adding the appropriate coordinate system to determine the relative position of various components of the accessory transmission system.

In order to balance simulation accuracy and model complexity, the model of the accessory transmission system is simplified. Retaining the support hole of the box, ignoring the oil hole, through hole and bolt hole; ignoring the smaller steps and chamfers on the gear shaft; ignoring the key slot on the input shaft and output shaft; and fixing the gearbox body and the gearbox cover as a single unit. According to the actual operation characteristics of the accessory transmission system, respectively, set the revolute joint of the gear shaft and spline shaft relative to the box, the fixed joint of the box and the earth, and the solid–solid contact of the gear pair and the spline pair. The contact collision stiffness coefficient K is related to the shape of the

**Fig. 1** Transmission principle

**Fig. 2** Three-dimensional model

contact surface and the material properties, which are calculated by the formula:

$$\begin{cases} K = \frac{4}{3} \rho^{\frac{1}{2}} E^\*\\ E^\* = \frac{1 - \upsilon\_1^2}{E\_1} + \frac{1 - \upsilon\_2^2}{E\_2} \\ \frac{1}{\rho} = \frac{1}{\rho\_1} + \frac{1}{\rho\_2} \end{cases} \tag{1}$$

where ρ is the comprehensive radius of curvature, *E*<sup>∗</sup> is the integrated elasticmodulus, v1 and v2 is the Poisson's ratio of the two meshing gears, *E*1 and *E*<sup>2</sup> is the modulus of elasticity of the two meshing gears, and ρ1 and ρ2 is the equivalent radius of curvature of the two meshing gears at the meshing point.

For a pair of spur gears meshing with each other, the radii of curvature of the two tooth profiles at the node are calculated as:

$$\begin{cases} \rho\_1 = \frac{d\_1^{\prime} \sin \alpha^{\prime}}{2} \\ \rho\_2 = \frac{d\_2^{\prime} \sin \alpha^{\prime}}{2} \end{cases} \tag{2}$$

$$d\_1' = d\_1 \frac{\cos \alpha}{\cos \alpha'} \tag{3}$$

where *d* is the diameter of the gear pitch circle, *d* is the diameter of the gear indexing circle, α is the pressure angle of the pitch circle, and α is the pressure angle of the indexing circle.

Considering that the bearings can have a great impact on the system performance and the large number of bearings, the contact model would lead to excessive computation, so it is decided to use the bushing force to simulate the bearings [8]. When using the bushing force to simulate the bearing, the stiffness coefficient *K* and damping coefficient *C* need to be determined. According to the actual situation, the stiffness and the damping in the direction of the rotational degree of freedom are set to be 0, and the stiffness value of the rest of the five degrees of freedom is 1e5 N/mm, and the damping value is 50 Ns/mm.

Compared with the rigid body, the flexible box is more likely to show the vibration characteristics of gear meshing excitation in the simulation process. Therefore, ANSYS is used to flexible the box, which is imported into ADAMS to replace the rigid body structure in the model, and construct the rigid-flexible coupling model of the accessory transmission system, as shown in Figs. 3 and 4.

**Fig. 4** Rigid flexible coupling mode

### **3 Simulation Results and Analysis**

### *3.1 Normal Condition*

The drive is loaded on the revolute joint at the input end of the internal spline shaft, and the load is added on the end of the output shaft in the form of torque. The simulation end time is 0.3 s, and the simulation steps are 25,000. Simulate the normal condition of the accessory transmission system with a rotational speed of 3000 r/min and a load of 20 Nm, extract the displacement of the output shaft and the input shaft, as well as the acceleration of the three measuring points on the box for the time–frequency domain analysis, and position of the measuring points as shown in Fig. 4.

Through simulation, the output shaft speed is 1843.01 r/min, and the error between the theoretical speed and the simulated speed is 0.023% by calculation, theoretical speed as shown in Table 1. The speed of the accessory transmission system under normal conditions is shown in Fig. 5. Analyze the frequency domain of the output shaft speed, as shown in Fig. 6, the larger spectral line of spectrogram amplitude corresponds to the gear meshing frequency ( *f* m32 = 644.82 Hz) and its multiple meshing frequencies, which can verify the feasibility of the model.

According to Fig. 7, it can be seen that the gear meshing vibration response is relatively smooth, and the spectral lines with larger amplitude in the frequency


**Table 1** Operating parameters of the accessory transmission system

**Fig. 7** The time–frequency domain diagram of vibration acceleration

domain correspond to the meshing frequency and its multiple meshing frequencies of the multi-stage gear transmission system. For example, the spectrum components of measuring point 3 are mainly the meshing frequency of 05 gear shaft and 04 gear shaft ( *f* m54 = 996.58 Hz), the double meshing frequencies of 06 gear shaft and 05 gear shaft (2*f* m65 = 2300 Hz), the double meshing frequencies of 03 gear shaft and 02 gear shaft (2*f* m32 = 1289.64 Hz), the triple meshing frequencies (3*f* m32 = 1934.46 Hz), quadruple meshing frequencies (4*f* m32 = 2579.28 Hz).

### *3.2 Unbalance of the Shaft System*

The instability of the transmission system caused by rotor mass imbalance is one of the typical vibration faults of the aero-engine. The test is realized by installing a balancing disk with threaded holes near the gear end of the output shaft to add the amount of unbalance, and the simulation is set up according to the position of the bolts to simulate the unbalance fault of the shaft system with a solid ball with mass. The mass of the counterweight bolt is 0.79 g, the amount of unbalance is 31.6 g mm, and the drive and load are the same as in the normal condition.

**Fig. 8** Time–frequency domain diagram of vibration acceleration at the measuring point 1 and 2

Comparing the output shaft displacement obtained from the normal condition and the unbalanced fault condition as shown in Fig. 10a, the amplitude of the displacement under the fault condition increases in the x and y directions, and the shaft trajectory is elliptical. Extracting the vibration acceleration of the accessory transmission system measuring points 1 and 2 as shown in Fig. 8, compared with the normal condition, the spectral components of the two measuring points are the same as in the normal condition, which are the meshing frequency of the gear drive system and its multiple meshing frequencies.

Due to the small amount of unbalance, the unbalance fault added by the simulation has no obvious effect on the time domain course and spectrogram of the vibration acceleration response of the box. In order to further analyze the fault characteristics, the root mean square (RMS), kurtosis (Ku), waveform factor (WF), crest factor (CF) and impulse factor (IF) of five time-domain indicators are selected to calculate the characteristic differences between the fault condition and the normal condition. The RMS reflects the vibration amplitude, the Ku represents the degree of smoothness of the waveform, and the WF, CF, and IF are used to check whether there is a shock in the signal.

According to the comparison of indicator characteristics at measuring point 1 near the unbalance fault in Fig. 9, when the unbalance fault occurs, other than the RMS, the Ku, WF, CF, and IF of measuring point 1 increase slightly compared with the normal condition.

**Fig. 9** Comparison of time-domain indicators of acceleration signals under unbalanced shafting conditions

**Fig. 10** The displacement comparison results of normal and fault conditions

### *3.3 Gear Misalignment*

In the gear misalignment fault simulation, the fault is simulated by modifying the position and direction of the Marker point where the revolute joint is located. The 05 gear shaft is deflected in the plane formed by its axis and the axis of the 06 gear shaft. The 05 gear shaft near the gear end bearing is deflected by 1 mm in the direction away from the 06 shaft, and the inclination misalignment between the two shafts is 0.57°.

Comparing the input shaft displacement obtained from the normal condition and the misalignment fault condition as shown in Fig. 10b, the shaft trajectory is still elliptical. According to Fig. 11, the time-domain amplitude of measuring point 3 near the 05 fault shaft decreases compared with the normal condition. The spectrum of measuring point 2 generates the meshing frequency ( *f* m65 = 1150 Hz) and triple meshing frequencies (3*f* m65 =3450 Hz) of 06 gear shaft and 05 faulty gear shaft on the basis of the spectrum component of the normal condition. The spectrum of measuring point 3 generates the 06 gear shaft and 05 faulty gear shaft triple meshing frequencies (3*f* m65 = 3450 Hz) and the triple meshing frequencies (3*f* m54 = 2989.74 Hz) of the 05 faulty gear shaft with the 04 gear shaft on the basis of the spectral components of the normal condition.

According to the comparison of indicator characteristics at measuring point 3 near the misalignment fault in Fig. 12, when the misalignment fault occurs, the RMS, Ku, CF, and IF decrease slightly compared with the normal condition.

**Fig. 11** Time–frequency domain diagram of vibration acceleration at the measuring point 2 and 3

**Fig. 13** Time–frequency domain diagram of acceleration at measuring points 1 and 2

### *3.4 Gear Broken Tooth*

Considering the characteristics of processing and assembly, a broken tooth fault is manually set up in the 03 gear shaft, the normal gear in the model is replaced by a broken tooth at the root of one of its gear teeth, and the rigid-flexible coupling model of the accessory transmission system with a broken tooth fault is constructed.

Comparing the output shaft displacement obtained from the normal condition and the gear broken tooth fault condition as shown in Fig. 10c, the amplitude of the displacement under the fault condition increases in the x and y directions, and the shaft trajectory is elliptical. According to Fig. 13, compared with the normal condition of the box vibration time-domain, the broken teeth fault condition of the vibration acceleration has a significant impact, and the impact signal interval time is 0.048 s, which corresponds to the reciprocal of the rotation frequency ( *f* z3 = 20.8 Hz) of the 03 fault gear shaft. The spectral components of the fault condition are similar to those of the normal condition, and the amplitude corresponding to the meshing frequency and its multiple meshing frequency spectral lines increases. From the refined spectrum near the meshing frequency (3*f* m32 = 1934.46 Hz), it can be seen that there exists a faulty shaft rotation frequency ( *f* z3 = 20.8 Hz) and its frequency doubling modulation at the meshing frequency. The modulation sidebands are asymmetrically distributed.

As can be seen from Fig. 14, the values of the indicators at measuring points 1 and 2 close to the fault increase.

### *3.5 Unbalance of the Shaft System and Gear Broken Tooth*

When implementing the coupled fault, refer to the single fault implementation mode. A coupled fault rigid-flexible model is constructed by adding a broken 03 gear shaft tooth fault to the accessory transmission system, while a sphere with mass is added to the 02 gear shaft end.

**Fig. 14** Comparison of time-domain indicators of acceleration signals at measuring points 1 and 2

Comparison of normal conditions and coupled fault conditions of the output and input shaft displacement, as shown in Fig. 15, shows that coupled fault conditions of the input and output shaft displacement increase the amplitude in the x and y directions, and the shaft trajectory is elliptical. As can be seen from Fig. 16, due to the small amount of unbalance, the shaft system imbalance and gear broken teeth coupling fault time–frequency domain analysis more intuitively show the characteristics of a broken teeth fault. In order to further explore its fault characteristics, calculate the time-domain indicators. According to Fig. 17, the five indicator values of measuring points 1 and 2 have increased. Compared with the single gear broken fault, the coupling fault adds the unbalance mass, and the increase at measuring point 1 near the shaft system imbalance fault is slightly reduced.

**Fig. 15** Output shaft, input shaft displacement

**Fig. 16** Time–frequency domain diagram of vibration acceleration at measuring points 1 and 2

**Fig. 17** Comparison of time-domain indicators of acceleration signals under coupled fault conditions

### **4 Experimental Verification**

In order to verify the simulation results, the test bench for the accessory transmission system is constructed as shown in Fig. 18. The arrangement of the acceleration sensor on the box is consistent with the position of the measuring point of the simulation

**Fig. 18** Test bench sensor installation position

model, and the eddy current displacement sensor is set at the end of the output and input shafts, respectively. The loading conditions are the same as in the simulation.

From Fig. 19, it can be seen that the axial trajectory is elliptical, and the amplitude in the x and y directions increases slightly under broken tooth and coupling faults. According to Fig. 20, under normal conditions, the time-domain simulation results of the vibration acceleration of the three measuring points of the box are consistent with the main test results, the main components in the spectrum are the gear shaft meshing frequency and multiple meshing frequencies. Under the fault condition, extracting the acceleration of the measuring points close to the fault shaft for comparison. The gear broken fault and coupling fault in the simulation results of the impact of measuring point 2 is not obvious. The sidebands appear on both sides of the spectrum meshing frequency of the test results, and the comparison results of the remaining measuring points meet the consistency of simulation and test.

**Fig. 19** Displacement test results

**Fig. 20** Vibration acceleration simulation and test results at measuring points

# **5 Conclusion**

In this paper, by constructing the rigid-flexible coupling model of the accessory transmission system, the time–frequency domain characteristics under fault excitation are studied. The conclusions are as follows:


**Acknowledgements** This work is supported by the National Natural Science Foundation of China (No. 52275088) and the Fundamental Research Funds for the Central Universities (No. DUT22LAB507).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Adaptive Control of Drilling Rig Power Head Speed**

**Xinxin Xu and Wanzhi Cai** 

**Abstract** In the drilling process, the adjustment of drill pipe rotational speed is essential to accommodate the variations encountered in different geological strata. However, the existing low automation level in drilling rigs and heavy operator reliance often result in suboptimal speed regulation, leading to low drilling efficiency. An adaptive control approach for power head speed based on a fuzzy clustering algorithm was proposed, which utilizes the fuzzy clustering algorithm (FCM) to cluster drilling parameters, including drilling pressure, rotational speed, and drilling speed, to identify of distinct strata. Furthermore, the rotational speeds of the clustering centers of different strata were used as the input rotational speeds of its control system, the simulation results show that the speed control can be realized by controlling the opening of the valve port, and the fuzzy PID control is used to realize the adaptive control of the speed and the tracking of the speed curve.

**Keywords** Stratigraphic recognition · Adaptive control · Clustering algorithm · Fuzzy PID · Power head

# **1 Introduction**

In the drilling process, due to the different strata encountered, it is necessary to adjust the rotational speed of the drill pipe to adapt to different formations. The automation degree of the drilling rig is not high, and the drilling effect is highly dependent on the operator. When the encountered formation changes are complicated, owing to the

X. Xu (B) · W. Cai

#### X. Xu

Key Laboratory of Intelligent Manufacturing of Construction Machinery, Anhui 230009, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_32

National Engineering Research Center for Highway Maintenance Equipment, Chang'an University, Xi'an 710064, China e-mail: 442977939@qq.com

Henan Key Laboratory of High. Grade Highway Detection and Maintenance Technology, Xinxiang 453003, China

limitations of the operator's experience and the influence of the change of the formation environment; it is often difficult for the operator's reaction speed to adapt to the change of the working conditions. The speed regulation process may have hysteresis, and the speed regulation effect is not ideal, resulting in poor drilling efficiency [1]. The proximity of the operating table to the orifice is a safety hazard. At present, the control of drilling parameters is still in the state of theoretical research, mainly due to the complexity of the drilling process mechanism, the correlation among the variables, is the nonlinear and uncertain process. The interference factors are many random and difficult to accurately measure and there is no clear functional relationship between the monitoring parameters and the drilling parameters. However, it is impossible to describe the drilling process with a precise mathematical model. In order to improve the adaptive ability of drilling rigs in complex rock formations, Xie et al. [1], used the ratio gong method to identify the formation, which provided conditions for realizing the adaptive control of rotational speed. Xu et al. [2], analyzed the drilling parameters collected by a real-time monitoring system and showed that it is effective to judge the formation based on the follow-drilling parameters. Wu et al. [3], established a propulsion adaptive control system, which improved the adaptive ability of the drilling rig in complex formations and controlled the degree of drilling deflection. Cheng et al. [4], proposed a synergistic adaptive control method between the propulsion system and the rotary system, and the optimal rotational speed was determined by using the collected parameters of the drilling with the drill. Lu et al. [5], designed a fuzzy PID control system to realize the stable control of drilling pressure of drilling rig. Chang et al. [6], used BP neural network technology to analyze historical drilling data and realize adaptive control of drilling pressure. H. Graham investigated a system identification method using ROP and RPM as inputs and WOB and Torque as outputs for identification. The method can be applied in real time during the drilling process to recognize typical drilling dynamics for adaptive control [7].

Machine learning algorithm is the research focus of lithology identification, which has the advantages of high precision, intelligence, high efficiency, accuracy and realtime acquisition of stratigraphic information compared with the traditional method of lithology identification by monitoring the follow-drill parameters. Yue et al. [8], summarized the current status of the application of various types of commonly used machine learning algorithms in the field of lithology identification, and compared the advantages and disadvantages of various types of algorithms in the application of lithology identification. Gao et al. [9], established a K-means based lithology classification model to realize the classification of lithology. Zhang et al. [10], proposed a clustering algorithm to classify rock types on core data to realize the classification and identification of lithology. Zhang et al. [11]. established a group clustering algorithm to recognize volcanic lithology and compared it with K-means and other algorithms, and the results showed that this method is a more effective lithology recognition algorithm with high accuracy.

L. A. Castañeda et al. used adaptive control for solving problems with uncertain dynamic models. The results show that the adaptive controller also outperforms the PID controller [12]. In this paper, the fuzzy clustering algorithm (FCM) technology is introduced into the control system of power head speed to realize the discrimination of strata, and then realized the adaptive control of speed, Laying the theoretical foundation for automation and intelligence of drilling rigs. AMESim and Simulink are used for joint simulation to realize closed-loop speed regulation and speed curve tracking by fuzzy PID algorithm.

# **2 Modeling and Analysis of the Powerhead Hydraulic System**

The power head rotary circuit consists of Rexroth's A8VO variable double pumps, multi-way valves, and variable motors. The main pump A8VO double pump supplies oil to the LUDV valve. The LUDV system can be divided into three parts, including the throttling part, the pressure compensation part and the reversing part, in which the throttling part regulates the flow size of the system by controlling the size of the throttling port area; the pressure compensation part maintains the stability of the pressure difference between the front and rear of the valve; and the reversing part controls the flow direction of the oil to control the motor's forward and reverse rotation. Due to the complexity of the power head system, in order to simplify the system model to speed up the simulation, the main pump is represented by a variable pump, and the main component parameters are shown in Table 1.

The modeling of the hydraulic system of the power head is shown in Fig. 1, because the different loads faced by the power head in different strata, the rotational speed of the motor of the power head in different strata (sand and gravel layer, coal layer, and rock layer) is analyzed through the simulation of AMESim software, and the results of simulation are shown in Fig. 2. From 0 to10 s, the higher motor speed is 1414.4 r/min for load of 168.8 Nm. From 10 to 30 s, the motor speed is 675.25 r/ min for load of 976.7 Nm. From 30 to 40 s, the motor speed is 466.2 r/min for load of 1012.8 Nm. The results show that the rotational speeds of the motor of the motor in different strata are different, with the lowest rotational speed of the sand and gravel layer, the rotational speed of the coal layer is higher than the one of the sand and gravel layer, and the rotational speed of the rock layer is the highest.


**Fig. 1** Hydraulic system model

### **3 Stratigraphic Identification Methods**

In the drilling process, the different mechanical properties possessed by different strata, and the drilling parameters can obviously reflect their differences [13, 14]. The drilling parameters such as drilling pressure, rotational speed and drilling speed are selected as the characteristic parameters. In this paper, the relevant drilling parameters are obtained through indoor drilling experiment mainly based on the variability of rotational speed in different formations. The drilling test rig consists of drilling rig, hydraulic pumping station, water circulation system and other parts, and the hydraulic system of the drilling rig consists of relief valves, hydraulic cylinders, oil tanks and other components. The relief valve controls the pressure of the whole hydraulic system. The drilling rig test bench is shown in Fig. 3, and the drilling test bench is utilized for the acquisition of parameters with drilling, including rotational speed, drilling speed, and drilling pressure.

In order to adapt the power head of the drilling rig to different strata during the drilling process, it is necessary to manually adjust the rotational speed in the past, disadvantages of speed lag and low efficiency. Clustering is an unsupervised machine learning method, the goal is to collect and categorize data, and the clustering algorithm can collect the drilling parameters to analyze and judge the stratum. In order to select the appropriate rotational speed, it overcomes the disadvantage of manually adjusting the speed. In the clustering process, data objects similar to each other will be divided into the same subset, and the whole observation data will be divided into several subsets. Therefore, FCM algorithm is used as the algorithm for strata identification [15].

A number of problems in different areas have been effectively solved through the use of FCM and its different variants.

Fuzzy C-mean clustering algorithm (FCM) is to classify *n* vectors into *c* classes, the clustering center of is *V*{*v*1,*v*2…*v*n}, this is accomplished by finding the minima of the objective function of Eq. (1).

$$J(U,V) = \sum\_{i=1}^{c} J\_i u\_{ij}^m = \sum\_{i=1}^{c} \sum\_{j=1}^{n} u\_{ij}^m d\_{ij}^2 \tag{1}$$

Affiliations satisfy the condition,

**Fig. 3** Drilling test rig

PDC drill bit Drill pipe Slewer Racks

$$\sum\_{i=1}^{c} u\_{ij} = 1, 0 \le u\_{ij} \le 1, i = 1, 2, \dots, n \tag{2}$$

The algorithm obtains the clustering centers by minimizing the objective function, it is solved by the Lagrange multiplier method so that its minimum value satisfies the condition:

$$V\_i = \frac{\sum\_{j=1}^{n} u\_{ij}^m \mathbf{x}\_i}{\sum\_{j}^{n} u\_{ij}^m} \tag{3}$$

$$U\_{ij} = \frac{1}{\sum\_{k=1}^{c} (\frac{d\_{ij}}{d\_{kj}})^{\frac{1}{m-1}}} \tag{4}$$

FCM algorithm steps as follows,


The clustering results of the drilling parameters based on the FCM algorithm are shown in the Fig. 4, the results show that the typical drilling parameters of rotational speed, drilling speed and drilling pressure are different for different formations, with obvious partitions, it can be concluded that the application of the clustering algorithm to the identification of formations in the drilling process is reliable and has obvious effects. The cluster center obtained by the clustering algorithm is the optimal drilling parameter for a certain formation.

In the drilling process, through the clustering algorithm of the drilling parameters collected by the sensor (rotational speed, drilling pressure, drilling speed, etc.) to analyze and process the information of the current stratum, and then select the optimal rotational speed for the realization of adaptive control of rotational speed to provide the conditions.

# **4 Fuzzy Adaptive PID Control Strategy for Power Head Speed**

When facing the nonlinear system and complex working conditions with variable parameters, PID control is less effective. Fuzzy control can control the nonlinear system without obtaining an accurate mathematical model, but the control accuracy is not satisfactory. The use of fuzzy adaptive PID control integrates the advantages of

**Fig. 4** Clustering results

the two, and achieves the purpose of online adaptive adjustment of PID parameters, enhanced the control performance. The principle of fuzzy adaptive PID control is to take the error and the rate of change of the error as the input, through the rules obtained from expert experience, to establish the proportional, differential and integral three coefficients of the fuzzy control table for the control of the system [16], and its principle is shown in Fig. 5.

In this paper, a typical two-dimensional fuzzy controller is used, the inputs are motor speed deviation e and the deviation change rate ec, and the three output parameters *Kp*, *Ki* and *Kd* are obtained through four processes: quantization, fuzzification, fuzzy inference and defuzzification. Set the fuzzy domain of input deviation *e* = [−3,3], deviation change rate *ec* = [−3,3], the fuzzy domain of output *Kp* = [−0.3,0.3], the fuzzy domain of *Ki* = [−0.06,0.06], and the fuzzy domain of *Kd*  = [−3,3]. The input and output quantities are divided into seven fuzzy subsets {*NB*, *NM*, *NS*, *ZO*, *PS*, *PM*, *PB*}.

**Fig. 5** Fuzzy adaptive controller principle

# **5 AMESim and Simulink Co-simulation**

### *5.1 Control Modeling*

Simulink is a modular graphical environment, a visual simulation tool used for model building and simulation analysis of results in many fields, with the advantages of clear structure and high efficiency. Moreover, MATLAB algorithms can be integrated into the model in Simulink, and the simulation results can be exported to MATLAB for further research and analysis [17].

In this paper, visual, MATLAB and AMESim are used. A joint simulation model is created in AMESim for simulation to generate S-functions [18]. In Simulink environment, the S-function is called and the simulation model is created. The control system built in Simulink is shown in Fig. 6.

**Fig. 6** Simulink model

**Fig. 7** Three stratigraphic rotational speed responses

### *5.2 Response Under Three Kinds of Rotational Speeds*

Taking the rotational speeds of three different strata as input signals, by adjusting the three parameters of the fuzzy PID, the system can achieve a better control effect when *KP* = 0.1, *Ki* = 1, and *Kd* = 0. When the simulation time is 40 s, the response time is 2 s, the simulation results shown in Fig. 7.

### *5.3 Speed Curve Tracking Characteristics*

The drilling characteristics of different strata are different, in order to improve the drilling efficiency, it is necessary to make the dynamic response characteristics of the power head system is good, when the stratum change is the power head can realtime speed adjustment to adapt to different formations. The power head system uses sinusoidal signal and square wave signal as input signals to simulate the rotational speed change of different formations to observe the following effect of the power head, and it can be seen that the dynamic tracking effect of the speed is better through the adjustment of the three parameters of fuzzy PID. Figure 8 shows the opening of the valve port under the sinusoidal signal, and Fig. 9 shows the pressure change. Figure10 shows the following curves of power head speed under sinusoidal and square wave signals. Figure 11 shows the opening of the valve port under the square wave signal, and Fig. 12 shows the pressure change.

**Fig. 9** Motor inlet pressure with sinusoidal signal

# **6 Conclusions**


**Fig. 10** Power head speed tracking response

**Fig. 12** Motor inlet pressure under square wave signal

dynamic response of the system was simulated and analyzed with sinusoidal and square wave signals as input signals, and the opening of the valve port and the change of pressure under sinusoidal signals were also analyzed.

(3) Under the same PID parameters, the powerhead speed tracks the input signal better, both in the sine signal and in the square wave model. A smaller amount of overshooting, about 3.08%, was present only in the case of signaling mutations. The power head speed achieves a good speed curve tracking effect.

**Acknowledgements** The authors would like to thank Shaanxi Provincial Science and Technology Department (No. 2022PT-30), Key Technological Special Project of Xinxiang city (No. 22ZD013) and Key Laboratory of Intelligent Manufacturing of Construction Machinery (No. IMCM2021KF02).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Method of Assembling Clearance Measuring of Oil Film Bearing in Tandem Cold Mill**

**Bo Sun, Chuanjian Liu, Weimin Zhou, Zhan Xu, and Heng Cao** 

**Abstract** In order to reduce the certain number of oil film bearing burning accident, this paper proposes a novel method used to test static pressure pipes and measure the assembling clearance between sleeve and bush inside the bearing. On account of the theory of lubrication and the working mechanism for oil film bearing, through the development of a new set of hydrostatic oil supply system and test platform, provide the static supporting pressure and the disassembling and locking pressure. And based on this method, the assembling clearance measuring was carried out. The results show that the proposed measuring method has the ability to support the disassembling and assembling process of the oil film bearing. In addition, this method provides an important basis for the engineers of cold rolling mill on site to adjust the assembling clearance of oil film bearing. The application of this method effectively reduces the online burning accidents of the oil film bearing.

**Keywords** Cold rolling · Oil film bearing · Sleeve and bush · Assembling clearance

# **1 Introduction**

Cold rolling is a form of cold working process performed below the recrystallization temperature. Undergoing multi-stage processing including pickling, cold rolling, heat treatment and coating, hot-rolled strip eventually transforms into cold-rolled products [1]. The majority of modern large-scale tandem cold mills characterized by

© The Author(s) 2024

B. Sun (B) · C. Liu

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: bo.sun@tongji.edu.cn

W. Zhou · Z. Xu Baoshan Iron and Steel Co., Ltd., Shanghai 201900, China

H. Cao

School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_33

large size, heavy load, continuous operation and automation, use oil film bearing as the heart of whole machine [2]. With the function to provide stable loading capacity for rolling process, oil film bearing has the feature of high working load, high operating speed, and harsh working environments [3]. If the running stability of the oil film bearing is compromised, their periodic errors would be beyond the compensation capability of various automatic control systems and directly impact the quality of the finished sheet. Furthermore, in case of an online burning incident in the oil film bearing, the entire mill would be forced to shut down, resulting in significant economic losses [4].

Consider a hybrid bearing with a design of double static pressure chambers for oil film bearing. The failure of one of the chambers will cause the clearance between sleeve and bush to tilt, accelerating the wear of Babbitt alloy [5], and indirectly reducing the service life of the bearing [6]. Therefore, ensuring the state of the static pressure chambers is particularly important. To ensure no leakage in the static pressure pipe system during each assembling process, a roller grinding workshop used to adopt a kind of conical tool recommended by Morgan Company to mutually support the upper and lower hydrostatic oil inlet, as a sealing measure. This method was effective when initially used, but after multiple uses, it caused certain damage to the inlet and resulted in the bump of nearby Babbitt alloy. The bump sealed off the inlet, causing severe negative impact. Morgan also recommends an online test method of the static pressure chamber using an online hydrostatic pump. However, this method occupies production time and is not accurate due to the mutual influence of the upper and lower rollers.

Besides, the clearance is a principal assembling parameter of the bearing. It plays a crucial role in load carrying, temperature rise and other factors during the operation of the bearing [7, 8]. So far, only the outer diameter of the sleeve and the inner diameter of the bush are measured in the workshop before assembling, while the actual assembling clearance after assembling is unable to measure. When the clearance is too small, the heat generated from the friction between the sleeve and bush accumulates continuously, causing sudden locking of the high-speed rotating roller immediately, thus causing burning accidents [9]. As the internal sleeves and bushes gradually age, most of the bushes in the workshop have undergone multiple repairs, leading to an increasing trend in the clearances. According to recent records for several years, the average number of sleeve damages per year is 19, and this number even skyrocketed to 28 in the last recorded year. What is particularly serious is that 8 of them were online damages, which bring out the shut-down of the machines to replace the support rollers. The total downtime reached 22 h during which the direct economic loss from sleeve repairs amounted to 2.05 million yuan. Therefore, it is necessary to propose a novel method for the static pressure pipe system test and the assembling clearance measuring, to avoid leakage in the system and master the corresponding relationship between actual assembling clearance and on-site use status.

### **2 Theory of Lubrication for Oil Film Bearing**

Bearings are essential components in mechanical systems. Based on the hydrodynamic principles, once the shaft starts rotating, it continuously squeezes lubricating oil with a certain viscosity into the converging gap. As the shaft sucks the lubricating oil into the wedge-shaped gap, the compression of the oil generates distributed resistance, which is termed the hydrodynamic pressure [10]. When the sum of the hydrodynamic pressures in the load direction is able to balance the external load, a complete layer of pressure oil film is formed between the shaft and the bearing. The oil film causes the surfaces of the shaft and the bearing to separate from each other, therefore resulting in pure liquid friction. The whole process above is known as the hydrodynamic lubrication mechanism [11].

The fluid lubrication film that exists between Surface 1 and Surface 2, and the velocity of two surfaces are shown in Fig. 1. The Reynolds equation can be derived as follows from the velocity components *U*1, *V*1 and *W*1 on surface 1, and the velocity components *U*2, *V*2 and *W*2 on surface 2 [12]:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{y}} \right) = 12 \frac{\partial (\overline{U}h)}{\partial \mathbf{x}} + 12 \frac{\partial h}{\partial t} \tag{1}$$

Here, *U* = <sup>1</sup> <sup>2</sup>(*U*1 + *U*2), *h* = *h*2 − *h*1, η represents the viscosity of the lubricating oil.

For oil film bearing, it is usually a fixed bearing and a rotating journal, implying that *U*1 = 0 and *U*2 =*U*. Considering the steady state situation, the Reynolds equation has the form which is commonly used in engineering:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( \frac{h^3}{\eta} \frac{\partial p}{\partial \mathbf{y}} \right) = 6U \frac{\partial h}{\partial \mathbf{x}} \tag{2}$$

Meanwhile, introduce the Reynolds boundary condition, assuming that the breakthrough of the oil film occurs at a divergent gap downstream from the minimum oil film thickness. At this point, the pressure and pressure gradient are both zero. So, the condition for complete termination boundary of the oil film is defined by:

$$p = p\_a, \quad \text{and} \quad \frac{\partial p}{\partial \alpha} = 0 \tag{3}$$

When adopting hybrid journal bearing as oil film bearing, the hybrid journal bearing is supplied with oil through hydrostatic system during the startup, braking or low-speed operation of the bearing. The hydrostatic oil film separates the Babbitt alloy layer on the surface of sleeve and bush, achieving the effect of hydrostatic levitation to improve performance of the bearing under low-speed operation [13]. As the rotational speed of the bearing gradually increases, the bearing reaches a hydrostatic-hydrodynamic hybrid lubrication state due to the oil film formation. During the operation of the bearing, static pressure is mainly utilized, while dynamic pressure act as a supplement to compensate for static pressure loss with hydrodynamic effect [14].

### **3 Test Method**

The test method mainly consists of four parts, (1) static clearance test stand for oil film bearing, (2) electric-hydraulic control loading trolley, (3) loading trolley and (4) assembling clearance measuring module. Considering the complexity of the work environment in industrial sites, it is necessary to pre-layout the equipment involved in the testing method. In accordance with the dimensions obtained from surveying and mapping, a three-dimensional solid model of the work area is established as Fig. 2.

Figure 3 shows the process of assembling clearance measuring. The oil film bearing is placed on a bracket after its complete assembly. Then, communicate the hydrostatic outlet of the electric-hydraulic control loading trolley with the hydrostatic inlet on the bearing seat through a hydraulic hose. The lubricating oil is injected by means of hydrostatic oil supply, to prevent the sleeve and bush from moving up and down, which would cause collisions and wear on roller. Simultaneously, push the loading trolley below the bearing seat, and output control commands through test stand to make the piston rod on the loading trolley move up and down, thereby driving the bearing seat to move similarly. When the roller is stationary, the displacement of the bearing seat from its natural resting position to the highest position shall be the target assembly clearance. The clearance value is calculated using a digital dial gauge installed on the bearing seat, which has remote transmission capability to measure the clearance.

# *3.1 Static Clearance Test Stand for Oil Film Bearing*

As an operation terminal, the test stand is the primary human–machine interface device for performing test tasks and can collect test data in real-time. It is equipped with a computer analysis module inside for processing data, as well as a liquid crystal display with digital voltmeter display module on the front panel. The buttons on the control panel are responsible for process control throughout the entire test process. In addition, the test stand has a PLC control module and possesses data printing functionality.

### *3.2 Electric-Hydraulic Control Loading Trolley*

The electric-hydraulic control loading trolley is a movable hydrostatic oil supply system that provides test pressure for various test conditions. According to the test requirements, the trolley needs to fulfill the following functions: be able to provide a pressure of 8 MPa for locking the oil film bearing; provide a pressure of 10 MPa for disassembling the bearing; and provide a pressure for static pressure test of the bearing, with a theoretically designed value of up to 80 MPa.

The principle diagram of the hydraulic system for the trolley is shown in Fig. 4. The maximum working pressure of the system reaches 80 MPa. The system uses a highpressure ball valve to switch pressure between ultra-high and high, with the secondary pressure being controlled by a proportional relief valve to avoid the impact of primary pressure. The primary pressure is directly provided by a piston pump for the static pressure test. In terms of unloading, use the median function of electromagnetic directional valves for the process, with overflow valves serving as safety valves.

The whole specific test process can be subdivided into 9 conditions with corresponding action sequence of each electromagnet under different states, as shown in Table 1. When pump 1–1 starts, electromagnet 7YA is energized and the system starts running in low pressure mode. The system remains in an unloading condition until any operation is performed. At this time, all electromagnets except for 7YA and 8YA are in a power-off state and all electromagnetic valves are in the median function state. Currently with the system being in condition 7, the system pressure can be adjusted by proportional relief valve 1–4. The states of all electromagnets in condition 2, 4 and 6 are same as those in condition 7.

When the system is loading for locking or disassembling, with three electromagnets adjusting the pressure, test oil passes through electromagnetic directional valves

**Fig. 4** The hydraulic principle diagram of the hydrostatic oil supply system


**Table 1** Sequence table for the actions of electromagnets

1–2 and 1–5. Then, a power-on electromagnet 6YA would introduce the oil into the rodless chamber of locking cylinder 1–9, leading to condition 1; or a power-on electromagnet 5YA would introduce the oil into the rod side, leading to condition 3. In condition 5, three energized electromagnets adjust system pressure for static pressure test, ensuring that the test oil enters static pressure chamber 1–10. In condition 8, the system pressure is adjusted by a high-low pressure system consisting of overflow valves. The operations in condition 9 is similar to those in condition 5. When the oil pressure lifts the bearing seat, the head of the digital dial gauge moves accordingly, thus measuring the clearance.

### *3.3 Loading Trolley*

If the loading pressure of the oil pressure control module is inadequate to lift the bearing seat, it is necessary to assist with the loading pressure output by the auxiliary oil pressure control module on the loading trolley. Owing to the load-bearing surface of the oil film bearing being on the underside, the force applied to the bearing seat by the loading trolley needs to be upward. The difference between the applied force and the self-weight of the bearing seat is balanced by the oil film pressure, which is supplied by hydrostatic system to avoid rigid contact between sleeve and bush.

### **4 Analysis of Test Result**

This testing method has been conducted on-site at a cold rolling plant, as the photos in Fig. 5 show. Table 2 presents the relationship between the assembling clearance and burning accident. The following ten assembling clearance test values are listed, and the tracking records of whether there were burning accidents during the operation of the oil film bearing under these assembling clearances are provided.

For oil film bearings, the initial design value of assembling clearance is generally taken between 0.3 and 2.0‰ of the nominal diameter. The permissible range for the assembling clearance is between 0.2355 and 1.57 mm, calculated based on a 785 mm outer diameter of the sleeve. However, it needs to be noted that this range is too broad and vague so that it can only serve as a reference.

Figure 6 shows the actual data on assembling clearance collected using an offline test device for oil film bearing. There are a total of 33 assembling data points, with each point representing a clearance value. The triangles in Fig. 6 represent the clearance values where burning accidents occurred, while the dots represent the values without accidents. Taking 1.1 mm as the critical threshold, it is evident that all assembling clearance values resulting in burning accidents are below 1.1 mm. The 7 triangles in 21 data points indicate a staggering probability of 33.33% for burning accidents. In contrast, there have been no burning accident above 1.1 mm. Therefore, after analyzing with on-site engineers, it can be concluded that the danger zone for assembling clearance is below 1.1 mm, while the safe zone is above 1.1 mm. It is suggested to set the assembling clearance for oil film bearing starting from 1.1 mm,

**Fig. 5** The industrial site test photos


**Table 2** Table for the measurement results of the clearance between sleeve and bush

**Fig. 6** The relationship between the assembling clearance and burning accident

with an upper limit not exceeding the maximum value 1.4 mm in the diagram. If the safety factor is set to a larger value, it can be between 1.2 and 1.4 mm, but must not exceed 1.57 mm.

### **5 Conclusion**

Here are a few suggestions for selecting assembling clearance between sleeve and bush:


In conclusion, it is a feasible solution to increase the minimum value of the assembling clearance from below 1.1 mm to above the safety threshold of 1.1 mm without exceeding the maximum clearance value, which can prevent the clearance from being in the danger zone. The clearance can be effectively controlled by the measuring device for the clearance between sleeve and bush.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Fluctuation of Near-Wall Pressure During the Cavitation Bubble Collapse**

**Zhicheng Yuan, Haowen Li, Meng Yang, Yongming Bian, and Li Chen** 

**Abstract** Cavitation bubble collapse, which generates strong shock waves and highvelocity liquid jets, is responsible for the erosive damage to hydraulic components. In order to assess the fluctuation of near-wall pressure, in this work, an open-source package OpenFOAM is utilized for solving the Navier–Stokes equation. To track the liquid–air interface, the volume of fluid (VoF) method-based *compressibleInterFoam*  solver is selected, and its shipped dynamic contact angle model is modified to obtain better accuracy when considering the wettability of substrates. Numerical methods are first validated by comparing with experiment, and then it is extended to study the effect of bubble diameter, pressure difference, and surface wettability on the fluctuation of near-wall pressure. Simulation results show that the initial sphere bubble goes through three stages of growth, shrinkage, and collapse near the wall. A larger bubble size leads to higher impact pressure due to the higher speed of the liquid jet. The difference in initial pressure in and out of the bubble has a great effect on the collapse behaviour. In addition, a hydrophobic surface, meaning hard liquid pining, can speed up the damping of near wall pressure. The findings in this work will be a guide to designing hydraulic components for limiting the erosive damages of cavitation bubble collapse.

**Keywords** Cavitation bubble collapse · Near-wall pressure · Simulation · Hydraulic components

M. Yang · Y. Bian

L. Chen

H. Li

© The Author(s) 2024

Z. Yuan · M. Yang (B) · Y. Bian School of Mechanical Engineering, Tongji University, Shanghai 201804, P. R. China e-mail: yangmeng@tongji.edu.cn

National Engineering Technology Research Center for Prefabrication Construction in Civil Engineering, Tongji University, Shanghai 200092, P. R. China

School of Automotive Engineering, Tongji University, Shanghai 201804, P. R. China

School of Mechanical and Automotive Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou, Zhejiang 310018, P. R. China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_34

### **1 Introduction**

Cavitation bubble forms as the instantaneous pressure in the hydraulic valve falling below the vapor pressure. The collapse of those bubbles often generates shock wave and the high-speed jet, which are supposed as the major reasons for mechanical damage [1]. So, it is of great importance to understand the physics of bubble collapse and explore the method for limiting the cavitation erosion.

Cavitation involves complex two-phase flow process, including formation, growth, shrinkage and collapse of bubbles in liquid field. Cavitation bubble collapses when it closes to a rigid surface, causing considerable interface and pressure fluctuation. Philipp and Lauterborn [2] found that the velocity of high-speed jet in the centre of toroidal bubbles is about 100 m/s for millimetre bubbles, which can be significantly reduced by increasing the bubble-wall distance. By an ultra-fast high-speed camera, Lauterborn and Vogel [3] obtained breakdown shock waves with velocities of up to 5000 m/s missioned by a laser-induced bubble, and the pressure around the bubble is up to 10 GPa. Kadivar [4] experimentally demonstrated that a shark skin inspired micro structured riblet surface can passively control the cavitation-induced erosion created by collapse of single bubbles. However, the physics behind has not been fully studied due to the complexity and instantaneity of collapse. In generally, it is extremely hard to explore the physics on the complex flow field by only experiments, especially for the final sharp change stage of collapse. Hence, numerical simulation should be conducted to investigate the complex process of cavitation bubble collapse.

By using Lattice Boltzmann Method (LBM), Liu and Peng [1] simulated the collapse process of cavitation bubbles in three typical boundaries, namely, an infinite domain, a straight wall and a convex wall. They found that no second collapse happened if the distance between the wall and the bubble is relatively long. Also, the distance between bubble and wall has influence on the temperature inside the bubble. Other LBM simulation results can be fund in papers [5–9]. Since Computational Fluid Dynamics (CFD) with the volume of fluid (VoF) model generally provides higher spatial and temporal resolution compared to LBM, Koch et al. [10] conducted numerical study using OpenFOAM for the dynamics of a laser-induced bubble on the top of a solid cylinder. The result illustrated that the bubble collapses in the shape of a mushroom. While numerous studies have investigated the deformation behavior of collapsing bubbles, there has been limited research on the pressure fluctuation on the wall caused by the collapsing process.

In this work, therefore, our purpose is to investigate the dynamics of near-wall pressure under the influence of initial bubble diameter, pressure difference, and surface wettability. This work is organized as follows: The methodology utilized is presented in Sect. 2, in which are governing equations, interface tracking method, and dynamic contact angle model. In Sect. 3, we show the problem statement to make the simulation system and cases clear. Finally, the simulation results are given including the process of bubble deformation, bubble diameter effect, pressure difference effect and surface wettability effect.

### **2 Methodology**

### *2.1 Governing Equations*

Considering the compressible, immiscible, non-isothermal phases in this bubble collapse system, the *compressibleInterFoam* multiphase solver shipped with OpenFOAM-v2212 is utilized to simulation the compressible two-phase flows. The solve includes the governing equations of continuity, momentum, and energy as,

$$\frac{\partial \rho}{\partial t} + \nabla(\rho \mathfrak{u}) = 0 \tag{1}$$

$$\frac{\partial \rho \mathbf{u}}{\partial t} + \nabla \cdot (\rho \mathbf{u} \mathbf{u}) = -\nabla P + \nabla \cdot \left[ \mu \left( \nabla \mathbf{u} + \nabla^T \mathbf{u} \right) \right] + \mathbf{g} + F\_{\sigma} \tag{2}$$

$$\frac{\partial \left(\rho \mathbf{C}\_p T\right)}{\partial t} + \nabla \cdot \left(\rho \mathbf{u} \mathbf{C}\_p T\right) = -\nabla (k \nabla T) + \mathbf{S}\_T \tag{3}$$

where ρ is the density, *u* the velocity, *t* the time, *P* the pressure, μ the dynamic viscosity, *g* the gravity, *F*<sup>σ</sup> the surface tension force that calculated by continuum surface force (CSF) method proposed by Brackbill et al. [11], *F*<sup>σ</sup> = σ κ*n*α, σ the surface tension force coefficient, κ the surface curvature, κ = −(∇ · *n*α), *n*<sup>α</sup> the normal vector, *n*<sup>α</sup> = ∇α/|∇α|, α the volume fraction of liquid in each cell, *Cp* the specific heat, *T* the temperature, *k* the thermal conductivity, *ST* the additional sources in the energy equation. The present work does not consider the phase change, and the transport properties such as density, dynamic viscosity, and thermal conductivity are regarded as the weighted average of α that can be calculated as below,

$$
\rho = \rho\_l a + \rho\_v (1 - a) \tag{4}
$$

$$
\mu = \mu \alpha + \mu\_v (1 - \alpha) \tag{5}
$$

$$k = k\_l \alpha + k\_v (1 - \alpha) \tag{6}$$

where the subscript *l* and v indicate the liquid and vapor properties, respectively.

### *2.2 Interface Tracking Method*

The *compressibleInterFoam* solver uses the volume of fluid (VoF) method to track the liquid–vapor interface by 0 < α < 1, where α = 1 indicates the liquid property, α = 0 means vapor phase. The transport equation for volume fraction α for a two-phase compressible system is given by,

$$\frac{\partial \alpha}{\partial t} + \nabla \cdot (\alpha \mathbf{u}) + \nabla \cdot [\mathbf{u}\_r \alpha (\mathbf{l} - \alpha)] = \alpha (\mathbf{l} - \alpha) \left[ \frac{\psi\_v}{\rho\_v} - \frac{\psi\_l}{\rho\_l} \right] \frac{Dp}{Dt} \tag{7}$$

where ∇·[*ur* α(1 − α)] is the 'compression term' to compress the free surface towards a sharper one ranging from 0 to 1, *ur* is the velocity at cell face that can be estimated by

$$\mathfrak{u}\_r = \mathfrak{n}\_f \min \left[ C\_a \frac{|\phi|}{\mathcal{S}\_f}, \max \left( \frac{|\phi|}{\mathcal{S}\_f} \right) \right] \tag{8}$$

where φ is face volume flux, *C*<sup>α</sup> is a user defined value, *n f* is the face unit normal flux, *S f* is the face area vector. ψ is the compressibility of the medium and correlated with the sound speed *C*,

$$
\psi = 1/C^2 \tag{9}
$$

Considering the compressibility, the liquid and air density used in Eq. (4) can be a function of ψ,

$$
\rho\_l = \psi\_l \cdot P + \rho\_{l0} \tag{10}
$$

$$
\rho\_v = \psi\_v \cdot P + \rho\_{v0} \tag{11}
$$

where ρ*l*0 and ρv0 are the initial value of liquid and vapor density.

### *2.3 Dynamic Contact Angle Model*

For free liquid–vapor surface, the VoF method is capable to track the interface position. However, the three-phase contact line (TPCL) forms when liquid–vapor interface meets the solid wall. Hence, the contact angle, characterizing the surface wettability, should be imposed into the boundary condition at the TPCL by a notable method proposed by Brackbill et al. [11],

$$
\mathfrak{n}\_{\alpha} = \mathfrak{n}\_{\perp}\cos\theta + \mathfrak{n}\_{\parallel}\sin\theta \tag{12}
$$

where, *n*<sup>⊥</sup> is the normal vector of the wall surface, and *n* is the parallel vector to the wall surface, and θ is the contact angle.

In general, contact angle for a dynamic interface is not a constant, whereas it varies and depends on the combined effect of equilibrium contact angle θ*e*, contact line velocity *ucl* , and surface roughness. OpenFOAM uses the dynamic contact angle model of Eq. (13),

$$
\theta\_d = \theta\_\epsilon + (\theta\_a - \theta\_r) \times \tanh\left(\frac{\mu\_{cl}}{\mu\_\theta}\right) \tag{13}
$$

where θ*a* and θ*r* are the dynamic advancing and receding contact angles, respectively. *ucl* is the contact line speed near the wall, *u*<sup>θ</sup> the dynamic contact angle velocity scale, its value is chosen as 0.1 in this work. In addition, our previous study shows that the original dynamic contact angle model of Eq. (13) cannot limit to θ*a* and θ*r* as expected at large and small velocities. Hence, a modified dynamic contact angle of Eq. (14) is implemented to this work,

$$\theta\_d = \begin{cases} (1 - t\_u)\theta\_e + t\_u\theta\_u, & if t\_u \ge 0 \\ (1 + t\_u)\theta\_e - t\_u\theta\_r, & if t\_u < 0 \end{cases} \tag{14}$$

where *tu* <sup>=</sup> tanh *ucl u*θ . Based on our previous validation study, the modified dynamic contact angle model of Eq. (14) with *u*<sup>θ</sup> = 0.1 is utilized in the present work [8].

### **3 Problem Statement**

To explore the physics behind the bubble collapse near a solid wall, a simplified computational system (1/4 of a complete domain) is constructed as shown in Fig. 1, which takes advantage of the symmetric calculation, thus limiting the computational cost. Also, only a quarter of bubble with diameter *D*0 locates above the bottom wall with stand-off distance *h*. As for the boundary conditions, left and front patches are symmetry boundary conditions (BCs). Right, back, and top surfaces are set as zero gradient BCs, as well as free slip BCs for the bottom surface, respectively. The initial pressure and temperature in and out of the bubble are *P*v, *T*<sup>v</sup> and *Pl* , *Tl* , respectively. A uniform mesh with = 6.25 µ m is used in this system. Thus, there are more than 32 grids along the diameter of bubble, which is fine enough for the two-phase flow simulation. Cases studied in this work are shown in Table 1.

### **4 Results and Discussions**

### *4.1 Numerical Validation*

To validate the ability of *compressibleInterFoam* solver for bubble cavitation, one case is conducted and the comparison between experiment [12] and simulation is shown in Fig. 2. It is seen that simulation well reproduces the bubble shrinkage

**Table 1** Cases simulated in this study and their respective conditions


behaviour. Moreover, the high-speed liquid jet in the centre of bubble is also obtained in simulation. Therefore, simulation results are in agreement with experiment, and the *compressibleInterFoam* solver can be extended to the following cases.

### *4.2 Bubble Deformation*

Figure 3 shows the snapshots for bubble deformation against time. Arrows in the 1st row indicate the velocity at the interface, and 2nd row gives the pressure contour on the wall. It is clear that the initial small bubble experiences growth, shrinkage, and

**Fig. 2** Snapshot comparison between experiment (1st row) and simulation (2nd row)

**Fig. 3** Variations of the velocity at the interface (1st row) and the pressure on the wall (2nd row) with bubble deformation

collapse stages. In the bubble growth stage, due to the larger pressure of vapour phase, the initial bubble grows big rapidly shown from 0.01 to 0.06 ms, when the pressure on the wall is very small. At the bubble shrinkage stage, then, a high-speed liquid jet at 0.08 ms forms at the upper interface and goes through the bubble with velocity larger than 10 m/s. Because of the high-speed impact of liquid-jet, the pressure on the wall increases with the impingement. After that, the bubble starts to touch the solid wall, and spreads around radically. With the collapse of the bubble, the pressure on the wall decreases. The high-pressure point is obtained at the centre of bubble impact area.

Former papers demonstrate that the high-speed liquid jet and high-pressure during the bubble collapse are the main reasons leading to mechanical damage [1]. Therefore, in Fig. 4, we plot the average pressure on the wall. It is seen that the pressure goes up to its peak more than 250 kPa at around 0.1 ms, which corresponds to the high-speed jet impact stage around 0.09 ms in Fig. 3. Then, the value of pressure shows the damping behaviour. It happens due to the diffusion of impact energy.

### *4.3 Bubble Diameter Effect*

To evaluate the effect of bubble diameter on the wall-pressure, we conducted Case 2 with a smaller diameter of *D*0 = 0.16 mm. Compared with the Case 1 with *D*0 = 0.2 mm, the dynamics of averaged wall-pressure is plotted in Fig. 5, in which

the black and red curves are for bubbles with D0 = 0.2 mm and D0 = 0.16 mm in Case 1 and Case 2, respectively. As shown in Fig. 5, larger bubble imposes the wall a larger pressure than that for smaller bubbles. In addition, wall-pressure for the larger bubble exhibits a smaller damping ratio than that for smaller bubbles after 0.2 ms. Interestingly, the frequency of oscillation for smaller bubble is larger than that for larger bubble. This suggests that reducing the bubble diameter may be a more effective method for limiting the wall-pressure, and thus decrease the mechanical damage of high-speed jet.

### *4.4 Pressure Effect*

Figure 6 shows the damping curves of wall-pressure for bubbles with *P*<sup>v</sup> = 1e6 Pa and *P*<sup>v</sup> = 2e5 Pa in Case 1 and Case 3. The purpose of this figure is to explore the pressure difference (*P*<sup>v</sup> − *Pl*) effect on the wall-pressure. It can be seen that smaller pressure difference generates smaller peak of wall-pressure. Also, smaller pressure difference leads to a larger frequency and damping of oscillation of pressure. This finding suggests that limiting the pressure difference in and out of the bubble can also reduce the mechanical damage of high-speed jet.

### *4.5 Surface Wettability Effect*

Surface wettability refers to the water repellent of a solid wall. For hydrophobic surface, meaning the contact angle larger than 90◦, water cannot pin on but slid away from the substrate. For hydrophilic surface with contact angle smaller than 90◦, however, water sticks on the surface and moves hardly. Figure 7 gives the schematics of bubble interface at 0.2 and 0.4 ms for Case 1 (θ*e* = 10◦ in black) and Case 4 (θ*e* = 170◦ in red), respectively. At 0.2 ms, bubble starts touching the bottom wall, when the shape of bobbles for Case 1 and Case 4 differs due to the wettability difference. Especially at 0.4 ms, the bobble in black is away from the substrate caused by the thin liquid film. However, the bubble pins on the wall on hydrophobic surface.

Figure 8 illustrates the impact of surface wettability on the fluctuation of near-wall pressure. Prior to 0.4 ms, both Case 1 and Case 4 exhibit similar pressure variations due to the influence of the high-speed jet impact. However, at 4 ms, the pressure for the hydrophobic surface reaches a higher peak and subsequently attenuates rapidly to a relatively constant value of approximately 100 kPa after 4 ms. These findings

indicate that a hydrophobic surface restricts the fluctuation of near-wall pressure once the bubble comes into contact with the substrate.

### **5 Conclusion**

In this study, an improved compressible two-phase flow solver was utilized to simulate bubble cavitation near a rigid surface. The primary focus was to investigate the dynamics of near-wall pressure under varying conditions, including initial bubble diameter, pressure difference, and surface wettability. Simulation results revealed that the near-wall pressure quickly reaches its maximum value upon the impact of the first jet, followed by a subsequent dampening effect. Notably, the highestpressure point was observed at the centre of the bubble impact area. Furthermore, it was observed that smaller bubble diameters, lower pressure differences, and larger contact angles contribute to a more rapid dampening of the near-wall pressure. These findings hold significant implications for the design of valves aimed at mitigating cavitation collapse erosion. By considering factors such as bubble diameter, pressure difference, and surface wettability, designers can optimize valve designs to effectively control and limit the detrimental effects of cavitation.

**Acknowledgements** This research was supported by the Fundamental Research Funds for the Central Universities (10002150058), National Key Research and Development Program of China (2023YFF0613200), China National Postdoctoral Program for Innovative Talents (BX20190242), and the Open Foundation of the National Engineering Technology Research Center for Prefabrication Construction in Civil Engineering (2021CPCCE-K02).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Equivalent Wind Load Spectrum Acquisition and Remaining Fatigue Strength Method of Double Active Arm Type Holding Pole**

#### **Wenwu Liu, Hongsheng Zhang, and Nianli Lu**

**Abstract** In order to study the wind load fatigue residual strength in in-service holding poles for preventing disasters, and improving load-bearing capacity reliability. Firstly, using the Davenport wind speed spectrum to simulate the wind speed, it will be the input load of the finite element dynamics model of the 2 × 40 kN double active arm type holding pole (DAATHP). The stress time history curve at the welding section of the main chord at the bottom of the tower shaft is obtained at the working condition with the wind angle of 0°, 45°, 90°, and 135°. The rain flow counting method derives the statistical characteristics of the mean, amplitude, and frequency of stress at four wind angles. Secondly, the frequency of wind load under four wind angles is extended, and the synthesized fatigue load spectrum is used as the equivalent wind load spectrum. Finally, the Paris-Eadogan equation is used to derive the calculation formula of fatigue residual strength of the main chord at the bottom of the tower shaft, and the functional function of the main chord is established and then the failure probability formula is derived by combining the equivalent wind load spectrum. It is found that, compared with the ANSYS calculation results, the error of the predicted stress spectrum of the proposed method is minor, only 7.95%. It is verified that the proposed method can efficiently obtain the equivalent load spectrum of the DAATHP. The results indicate that it is feasible and practical to apply the proposed method to calculate the remaining fatigue life of the DAATHP.

**Keywords** Double active arm type holding pole · Wind load · Equivalent load spectrum · Remaining fatigue life · Rain flow counting method

W. Liu

H. Zhang (B) · N. Lu School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China e-mail: zhanghs@hit.edu.cn

© The Author(s) 2024

College of Engineering, Northeast Agricultural University, Harbin 150030, Heilongjiang, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_35

### **1 Introduction**

A holding pole is an everyday lifting and hoisting of mechanical equipment when lifting large components for ultra-high voltage (UHV) system transmission towers. Using a holding pole can shorten the construction period and reduce costs, so its reliability is directly related to the safety and quality of tower construction and is an essential guarantee for the smooth progress of the transmission project. Chi et al. [1] used finite element software to analyzed the overall stress state of the inner suspension holding pole used in the 1000 kV project. Lin et al. [2] established a finite element model of the 40 m all-steel internal suspended holding pole, and obtained the stress distribution of each part of the holding pole. Huang [3] developed a detection system to prevent the internal suspension holding pole from tilting due to inclination angle and tensile force exceeding the limit. Ding et al. [4] considered the influence of self-weight, lifting load, wind load, and lifting deflection, and find the weak parts in the structure. Zhang et al. [5] conducted finite element analysis on the inner suspension and external tension holding poles at different inclination angles, and the results showed that the tensile force, displacement, and stress increased with the increase of the inclination angle. Tan et al. [6] used the finite element method to analyze the mechanical properties of the DAATHP. The results showed that to ensure construction safety, the height of the free section should be reduced as much as possible. Wang et al. [7] used ANSYS software to analyze the static force of the holding pole under various typical working conditions to study its bearing law. The results showed that the axial force of the main chord of the tower structure was large, and the axial force of the oblique bar was relatively small.

Most existing studies focus on the holding pole's stability, nonlinearity, and strength. In contrast, research on wind-induced fatigue damage and fatigue residual strength of the holding pole is rarely reported, which increases the unsafety of the holding pole structure. DAATHP is a common tool used in the lifting and assembly process of UHV transmission towers, so studying its working reliability under wind load has important practical and theoretical significance to reducing construction risks. From the above analysis, the main chord of the pole holding tower shaft, under the action of alternating stress, the initial crack of the welding section, easily expands and makes the pole fatigue failure. Hence, this paper takes the welding section of the main chord at the bottom of the tower shaft of 2 × 40 kN DAATHP as the research object, and the residual strength of its fatigue is studied.

Load spectrum is a key factor in calculating the remaining fatigue strength of structures. Due to the high randomness of wind load during the operation, the uncertainty of stress at the weld section of the main chord at the bottom of the tower is increased. This makes it impossible to directly use the measured wind load for theoretical analysis and engineering practice. Furthermore, due to the limitations of field test conditions, it is difficult to conduct a large number of actual load spectrum tests. There needs to be more research on the load spectrum of tower structures domestically and internationally, with most existing research focused on other structural fields, such as cranes. Joo et al. [8] conducted specimen tests and statistical analysis on carbon fiber epoxy resin composite materials, and established standards for the optimal load spectrum's reliable truncation level and load coefficient. Wang et al. [9, 10] found a load spectrum based on damage consistency, aiming to reproduce the damage of the frame in working conditions by considering the difference between measured fatigue damage and calculated damage as the objective function. Li et al. [11, 12] obtained the load's average frequency and amplitude-frequency matrix through the rain flow counting method, established load spectra for the main shaft and other structures, and applied them to fatigue life prediction. Through experiments, Pokorný et al. [13] determined the load spectrum under cyclic amplitude load for railway bearings. Zhu et al. [14] proposed a Bayesian method for correcting fatigue load spectrum for ship life estimation. Xu et al. [15] established a nonlinear mapping relationship between crane working cycles and different rated weights and lifting loads using a support vector machine model, which was the equivalent load spectrum of the crane.

The methods mentioned above to obtain load spectra are targeted at single or simple structures. At the same time, the holding pole belongs to a complex truss structure composed of many beams and rod components. The formation mechanism of the dangerous section load spectrum under wind load is still unknown. Therefore, it is urgently necessary to seek a quick and efficient method to obtain the load spectrum of the tower brace to evaluate its fatigue remaining strength.

In this study, the Davenport wind speed spectrum is used to simulate the wind speed, and the wind load during a working period is collected as the external load of the finite element dynamic simulation model. The axial stress time history at the weld section of the main chord at the bottom of the tower shaft is obtained considering the geometric nonlinearity effect at 0°, 45°, 90°, and 135° wind directions. The statistical characteristics of stress mean, amplitude, and frequency in the four wind directions are obtained using the rain flow counting method. The mean frequency of stress amplitude in the four wind directions is expanded, and the synthesized fatigue load spectrum is used as the wind load equivalent load spectrum. Based on linear elastic fracture mechanics theory and Miner fatigue damage accumulation theory, the theoretical formula for fatigue remaining strength of main chord components is derived using the Paris-Eadogan equation. The effectiveness of this method is verified by comparing the failure probability of the main chord obtained through the ANSYS stochastic finite element module.

# **2 DAATHP Tower Shaft Wind Load Equivalent Load Spectrum Acquisition**

### *2.1 Equivalent Load Statistical Analysis*

The 2 × 40 kN DAATHP is used as the research object, as shown in Fig. 1. The height of the holding pole *H* is 24.35 m, the self-weight is 16.883 kN, the component material is Q345B, the active arms on both sides of the holding pole are lifted at

7 left rope

the same time, the lifting angle is maintained at 10°, the lifting load on both sides is 40 kN, the main specifications of the holding pole are as follows: (1) tower shaft: height 12 m, section 656 mm × 656 mm; (2) Tower cap: height 11.571 mm, bottom section size 970 mm × 970 mm, top section 650 mm × 650 mm; (3) Active arms: the length of the single arm is 11.37 m, the section of the active section is 420 mm × 520 mm, and the section of the middle arm section is 465 mm × 565 mm.

Based on the Davenport wind speed spectrum Eq. (1), the harmonic synthesis method simulates the pulsating wind speed. The average wind speed at the height of 10 m is 26.8 m/s on the *B*-level ground and a working cycle is 300 s.

$$S\_v(n) = 4K\overline{v}\_{10}^2 \frac{\mathbf{x}^2}{n(1+\mathbf{x}^2)^{(4/3)}}\tag{1}$$

where, v10 is the average wind speed vector representation of pulsating wind force at an altitude of 10 m above the ground; *x* is the coefficient, *x* <sup>=</sup> <sup>1200</sup>*<sup>n</sup>* <sup>v</sup>10 ; *n* is the pulsating wind frequency, HZ; *K* is the roughness factor of the ground.

In Fig. 1, *F* is the lifting load of the holding pole, θ is the angle between the active arm and the horizontal direction, *v* is the wind speed, and α is the angle between the wind speed and the DAATHP.

Wind load *P*(*t*) to which the pole is exposed:

$$P(t) = \frac{\mathcal{V}}{2g} \mu\_s(z) A(z) [\overline{v}(z) + \upsilon\_d(z, t)]^2 \tag{2}$$

where, γ is the air bulk density, kg/m3; *g* is the acceleration due to gravity, kg/m3; μ*s*(*z*) is the body shape coefficient at structural height *z*; v(*z*) is the average wind speed at height, m/s; v*d* (*z*, *t*) is the pulsating wind speed at height *z*, m/s; *A*(*z*) is the effective windward area at structural height *z*, m2.

**Fig. 2** Axial stress time history curve of the main chord at the bottom of the tower shaft

The wind load is applied to the finite element model of the holding pole according to different heights. The axial stress time history curve at the welding section of the main chord at the bottom of the tower shaft is shown in Fig. 2 when α is 0°, 45°, 90° and 135°.

The axial stress time history curve corresponding to the wind direction angle of 0°, 45°, 90°, and 135° is calculated by using the rain flow counting algorithm, and the amplitude, average value, and frequency of axial stress at the welding section of the main chord at the bottom of the tower shaft are obtained as shown in Fig. 3.

The lognormal, normal, and Weibull distributions test the axial stress relationship between amplitude-frequency and the mean-frequency at different wind angles. The results show that the relationship between the mean-frequency of axial stress under the four wind angles follows the normal distribution, and the relationship between the amplitude-frequency of axial stress obeys the Weibull distribution.

When the amplitude and mean distributions are independent, the amplitudemean combined probability density function can determine the frequency of the load spectrum at all levels. According to Fischer's theorem, two random variables approximately obey the chi-square χ2 distribution with (*r* − 1) · (*s* − 1) degrees of freedom:

$$\chi^2 = \frac{1}{\lambda} \sum\_{i=1}^{r} \sum\_{j=1}^{s} \frac{(\lambda \lambda\_{ij} - \lambda\_i \lambda\_j)^2}{\lambda\_i \lambda\_j} \tag{3}$$

**Fig. 3** Rain flow counting results of axial stress

where, λ is the sample size; *r*, *s* are the mean, magnitude grading; λ*i* is the frequency at which the mean is at level *i*; λ*j* is the frequency of the amplitude at the *j*-th level; λ*ij* is the frequency with the mean at level *i* and the amplitude at class *j*.

According to the rain flow calculation results and the chi-square test results of the mean amplitude at the four wind angles, it can be seen that the amplitude and mean of the axial stress at the four wind angles are independent of each other when the significance level is 0.05, and the mean amplitude is combined with the probability density function *f* (*X*, *Y* )

$$f(X,Y) = \frac{1}{\sigma\sqrt{2\pi}} \frac{\alpha}{\beta} (\frac{Y-\varepsilon}{\beta})^{\alpha-1} e^{-(\frac{Y-\varepsilon}{\beta})^a - (\frac{X-\mu}{\sigma\sqrt{2}})} \tag{4}$$

where, *X*, *Y* are the stress mean and amplitude; μ, σ are the normal distribution mean and standard deviation; α, β, ε are the shape, scale parameters, and threshold parameters of the Weibull distribution.

**Fig. 4** The mean of stress-frequency normal distribution fitting

According to the normal distribution and Weibull distribution, the mean-frequency and amplitude-frequency histograms of the axial stress time history of the main chord at the bottom of the 2 × 40 kN DAATHP under four wind directions were curved fitted, as shown in Figs. 4 and 5, respectively.

The average and amplitude distribution fitting parameters of the axial stress of the main chord at the bottom of the tower shaft at four wind angles of 2 × 40 kN double rocker holding poles are shown in Table 1.

### *2.2 DAATHP Wind Load Synthesis and Extrapolation*

To include the wind load condition with a small probability of occurrence, it is necessary to extend the frequency of the wind load and extrapolate the extreme value. The cumulative frequency is extended to 1.0 × 106 cycles, and the wind load expansion frequency under the four wind direction angles is calculated as follows:

**Fig. 5** The mean of stress-frequency Weibull distribution fitting


**Table 1** Fitting parameters for mean axial stress and amplitude distribution

$$N\_k = N'\left(\frac{m\_k \lambda\_k}{s\_k}\right) / \sum\_{k=1}^4 \left(\frac{m\_k \lambda\_k}{s\_k}\right) \tag{5}$$

where, *Nk* is the frequency of the spread of the wind angle k; *N* is the cumulative number of cycles in which extreme loads occur once; *mk* is the number of cycles obtained by counting the wind direction angle k rain flow; λ*k* is the proportion of time occupied by the wind direction angle *k*; *sk* is the actual time occupied by the wind angle *k*.

After the wind load is synthesized, the axial stress extremum and amplitude extremum of the main chord at the bottom of the tower shaft is the maximum values in each wind direction angle. The Ref. [16] of this paper divides the load spectrum into eight levels, the mean interval is divided by equidistant intervals based on the mean extremes, and the amplitude interval is divided by the ratio coefficient method based on the extreme amplitude of the non-equidistant interval. The ratio coefficients of each level are 1, 0.95, 0.85, 0.725, 0.575, 0.425, 0.275, and 0.125.

The axial stress cycle number *Nkab* corresponding to the amplitude interval of class a and class b at four wind angles is obtained by the mean amplitude combined probability density function *f* (*X*, *Y* ) by the following equation:

$$N\_{kab} = N\_k \int\_{-\mathfrak{r}\_a}^{\mathfrak{r}\_{a+1}} \int\_{-\varphi\_b}^{\varphi\_{b+1}} f(X,Y)dXdY \tag{6}$$

where, τ *<sup>a</sup>*, τ *<sup>a</sup>*+1 are the lower and upper limits of the mean stress of class *a*; ϕ*b*, ϕ*<sup>b</sup>*+1 are the lower and upper limits of the amplitude stress of class *b.* 

The cyclic number *Nab* of the mean of class *a* and the amplitude of class *b* in the eight-stage two-dimensional wind load equivalent load spectrum synthesized from multiple wind angles is obtained by the following equation:

$$N\_{ab} = \sum\_{k=1}^{4} N\_{kab} \tag{7}$$

# **3 Theoretical Derivation of Fatigue Residual Strength of the Main Chord of the Tower Shaft**

Aiming at the fatigue fracture, which is the primary failure form of the metal structure of the DAATHP, this paper takes the welding section of the main chord at the bottom of the tower shaft as the research object and uses the previous section method to obtain the two-dimensional wind load equivalent load spectrum, and uses the Paris-Eadogan equation to derive the fatigue residual strength calculation formula based on the linear elastic fracture mechanics theory and the Miner fatigue damage accumulation theory.

# *3.1 Miner Linear Damage Accumulation Theory and Equivalent Stress Amplitude*

The DAATHP lifts the work piece while being subjected to random wind load, and the welding section of the main chord at the bottom of the tower shaft is subjected to cyclic stress during this process. Miner's linear damage accumulation theory holds that under a given stress or strain level, the damage is linearly accumulated in the number of stress or strain cycles, and failure will occur when the damage accumulation reaches a particular critical value [17].

This paper uses the Miner linear damage accumulation theory to evaluate fatigue damage of the welding section of the main chord at the bottom of the tower shaft. Under the action of stress σ*i*, the damage of the main chord of the holding pole after each stress cycle is 1/*Ni*, and the damage value after the *ni* cycle is:

$$D\_i = \frac{n\_i}{N\_i} \tag{8}$$

The total damage accumulation value of the welding section of the main chord at the bottom of the tower shaft during fatigue failure is:

$$D = \sum\_{i=1}^{M} \frac{n\_i}{N\_i} \tag{9}$$

where, *ni* is the actual number of Class *i* stress cycles; *Ni* is the number of failure cycles under Class *i* stress cycles*.* 

When fatigue failure occurs in the main chord at the bottom of the tower shaft, *D* = 1. In classical fatigue theory, the S–N curve at a specific stress ratio *R* can be described by the Basquin equation:

$$
\sigma\_1^m N\_1 = \sigma\_2^m N\_2 = \mathcal{C} \tag{10}
$$

where, σ*i* is the stress amplitude of Class *i*, *i* = 1, 2; *Ni* is the number of failure cycles under Class *i* stress cycles; *m*, *C* are the constants related to material, stress ratio, and loading mode.

When the welding section of the main chord of the holding tower shaft fails, the simultaneous (9) and (10) can be obtained:

$$\sum \sigma\_i^m n\_i = \sigma\_j^m N\_j \tag{11}$$

To simplify the complex calculation, this paper adopts the Miner stress amplitude equivalence method, that is, replacing the stress amplitude of each stage in the variable amplitude stress with equivalent force amplitude, and the equivalent force amplitude formula is:

Research on Equivalent Wind Load Spectrum Acquisition … 471

$$
\sigma\_{eq} = \sqrt[m]{\sum \sigma\_i^m \frac{n\_i}{N\_f}} \tag{12}
$$

where, σ*eq* is the equivalent force amplitude; σ*i* is the stress amplitude of Class *i*; *N f* is the total number of cycles at failure; *ni* is the actual number of class *i* stress cycles; *m* is the material constant, due to the lack of holding pole experimental data, this paper draws on the crane data in the literature [18], that is, Q235, *m* = 3.214, Q345, *m* = 7.806.

# *3.2 Fatigue Residual Strength Calculation Formula Based on Linear Elastic Fracture Mechanics Theory*

The DAATHP is a welded structure, and the full fatigue strength of most welded structural parts is determined by the crack expansion stage [19, 20]. Therefore, according to the literature [21], the fatigue crack growth strength at the welding section of the main chord of the tower shaft is calculated by the fracture mechanics method, which is used as the fatigue residual strength of the DAATHP. The correlation between crack propagation velocity and stress intensity factor amplitude is obtained by the linear elastic fracture mechanics Paris formula:

$$\frac{dl}{dN} = C(\Delta K)^n\tag{13}$$

where, *l* is the crack length; *N* is the number of stress cycles; *dl dN* is the crack propagation velocity; *n*, *C* is the material constant; *K* is the Stress intensity factor amplitude.

$$
\Delta K = K\_{\text{max}} - K\_{\text{min}} = Y \sigma \sqrt{\pi l} \tag{14}
$$

where, *Y* is the stress intensity factor correction coefficient; σ is the amplitude of stress at the crack.

By substituting Eq. (14) into Eq. (13) and integrating to obtain the number of stress cycles experienced when the welding section of the main chord at the bottom of the pole holding tower shaft is extended from the initial crack *l*0 to the failure crack *l f* , that is, the fatigue residual strength of the holding pole, *N f* the formula is as follows:

$$N\_f = \frac{2(l\_f^{1-0.5n} - l\_0^{1-0.5n})}{\left(2 - n\right)C\pi^{n/2}\left(Y\sigma\right)^n} \tag{15}$$

Under the action of the wind load equivalent load spectrum, the functional function of the main material at the bottom of the tower shaft can be obtained from Eq. (16):

$$Z = N\_f - N\_{ab} \tag{16}$$

According to the stress intensity interference theory, the failure probability of the main material at the bottom of the tower shaft is *Pf* :

$$P\_f = P(Z < 0) = \int\_{-\infty}^{0} \frac{1}{\sigma\_z \sqrt{2\pi}} \exp\left[ -\frac{1}{2} \left( \frac{z - \mu\_z}{\sigma\_z} \right)^2 \right] dz \tag{17}$$

where, μ*z* is the mean of the functional function; σ*z* is the standard deviation of the functional function.

### **4 Example**

A finite element model of the main chord at the bottom of the 2 × 40 kN DAATHP is built, as shown in Fig. 6. The length is 1000 mm, one end is fixed, the other is free, and the free end is subjected to the axial force *P* = 40 kN, the transverse wind load is applied, and the cross-sectional size is 63 mm × 63 mm × 5 mm × 5 mm, considering the uncertainty of cross-section size, elastic modulus, and density, the random conditions are as follows: (a) the cross-sectional edge length and thickness are uniformly distributed by ±0.1; (b) The mean modulus of elasticity is 210 GPa, following the Gaussian distribution and the standard deviation is 0.05 times the mean; (c) The density follows a uniform distribution, with an average value of 7800 kg/m3; (d) Transverse wind velocity *v* adopts the Davenport wind speed spectrum, which obeys the Weibull distribution.

The working time of the DAATHP is taken as 60 min, of which the time ratio of 0°, 45°, 90°, and 135° wind direction angle is 2:3:3:2. The two-dimensional wind load equivalent load spectrum in Eq. (7) is sorted out into a one-dimensional wind equivalent load spectrum by variable mean method and low–high–low loading sequence, as shown in Table 2.


**Table 2** One-dimensional wind load equivalent load spectrum

According to the literature [20], for the main chord material is Q345B, take *n* = 3.5, *C* = 2.52 × 10–13, for the general weld size, the stress concentration correction coefficient *Y* = 1.12 is used the initial crack length *l*0 of the weld is 0.68 mm by measured and statistical analysis, according to the fracture test of the main chord of the holding pole, the standard lifting load of the DAATHP is taken as 40 kN. The literature shows that the weld's failure crack length *l* s is 95 mm [21]. The above parameters are combined with the stress amplitude-frequency and cumulative frequency substitution in Table 2 into Eqs. (11) and (12) to obtain the wind load equivalent stress spectrum and the equivalent force value at the welding section of the main chord at the bottom of the rod-holding tower. The residual fatigue strength at the welding section of the main chord at the bottom of the rod-holding tower is obtained by substituting the equivalent force value into Eq. (15). The comparison results of the failure probability of the primary metal and ANSYS stochastic finite element are obtained from Eqs. (16) and (17), shown in Table 3.

It can be seen from Table 3 that by comparing the probability of failure of the main chord at the bottom of the tower shaft calculated by the ANSYS stochastic finite element method and the proposed method, it is found that the failure probability obtained by the proposed method is larger, but the error between the two methods is only 7.95%, indicating that the proposed method is feasible and effective.



# **5 Conclusions**

The wind speed spectrum of Davenport is used to simulate the wind speed, the axial stress time history of the welding section of the main chord at the bottom of the tower shaft of the 2 × 40 kN DAATHP at the wind angle of 0°, 45°, 90°, and 135° is obtained by finite element method. The stress amplitude, average value, and frequency are statistically analyzed by the rain-flow counting method, the wind load frequency of the four wind directions is extended, the fatigue strength formula at the welding section of the main chord at the bottom of the tower shaft is derived, and the functional function is established. Based on this, the failure probability formula of the main chord at the bottom of the tower shaft is further derived, and specific conclusions are as follows:


**Acknowledgements** This research is supported by the Guangxi Science and Technology project of open competition mechanism to select the best candidates (Grant: Research on fundamental techniques of wind-resisting tower cranes under strong typhoons).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Lateral Control of Autonomous Concrete Mixer Truck Based on Multiple Look Ahead Distances and Fuzzy Controller**

**Ling Fu, Qiangrong Yang, and Yanbin Liu** 

**Abstract** Automated driving and working have been attracted considerable attention in the construction machinery industry due to their safety and efficiency. In this paper, multiple look ahead distances are calculated for lateral error and heading-angle error and multiple fuzzy inference engines are proposed with vehicle speed and the abovementioned errors. Moreover, a kinematic model of mixer truck was proposed in this paper to address the length and width of the concrete mixer truck. Satisfactory simulation and experimental results have been obtained for different reference paths.

**Keywords** Lateral control · Multiple look ahead distances · Fuzzy inference engines

# **1 Introduction**

For a autonomous vehicle, the control system utilized the real-time position and the kinematic model to track the target path which is given by a planning system precisely. The control system is composed of the lateral control and longitudinal control system, and the lateral control attracts more attention due to its difficulty in the previous researches and most of the researches for lateral control are proposed based on classical control theory [1–4]. In [1], Fenton proposed a lateral control system based on the root-locus. In [5], Kosecka proposed a vehicle sensing system model and estimate the dynamic property of the lateral error and heading-angle error based on the look ahead distance, which is widely used in later researches [5–8]. In [9], a time-variant look ahead distance is proposed to adapt to different speed. In [10], a fuzzy control strategy is proposed to handle the nonlinear characteristics and uncertainty of the system.

© The Author(s) 2024

L. Fu · Q. Yang (B) · Y. Liu

Zoomlion Heavy Industry Science and Technology Company, Changsha 410006, China e-mail: bit20070042@163.com

State Key Laboratory of Crane Technology, Changsha 410006, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_36

In most of the previous researches, a certain moving direction (forward or backward) was considered or constrained and only one look ahead distances was proposed to estimate the control errors. In this paper, multiple look ahead distances are calculated for lateral error and heading-angle error and multiple fuzzy inference engines are proposed with vehicle speed and the above-mentioned errors. Moreover, a kinematic model of mixer truck was proposed in this paper to address the length and width of the concrete mixer truck.

### **2 Kinematic Modeling**

In previous researches, a linear model was proposed as a simplified model of the real kinematic model of the vehicle. But in most cases, the simplified linear model can not capture the real dynamic properties of the vehicle. Especially when the length of the vehicle is very large (e.g. a concrete truck), the errors of the simplified linear model is unacceptable.

In this paper, a Unicycle model is proposed to describe the vehicle which is shown in Fig. 1. X axis refers to the east and the Y axis refers to the North; P0(X0, Y0) is the center point of the rear axis of the vehicle; P0(X0, Y0) is the center point of the front axis of the vehicle; L is the length between different axis; ϕ is the heading angle of the vehicle and θ is the steering angle of the vehicle.

As such, the dynamic property of the vehicle can be described as follows:

Lateral Control of Autonomous Concrete Mixer Truck Based … 479

$$\begin{array}{|c|c|c|}
\hline
\dot{X}\_0 & & \nu \sin \varphi \\
\dot{Y}\_0 & & \\
\dot{X}\_1 & = \\
\dot{Y}\_1 & = \\
\dot{\varphi} & & \nu \cos(\varphi + \theta) \\
& \nu \cos(\varphi + \theta) \\
& & \frac{\nu \tan \theta}{\mathcal{L}} \\
\end{array} \tag{1}$$

### **3 Control**

### *3.1 Match Point*

The match point Pnow(Xnow, Ynow) is the point on the reference path which meets the following requirements: (1) the heading of Pnow on the path should match the heading angle of vehicle; (2) the distance between the match point and the real position should be the smallest.

### *3.2 Look Ahead Points*

Multiple look ahead points are calculated in this section to estimate the corresponding look-ahead distances.

The lateral error look ahead point Ppos(Xpos, Ypos) and the heading-angle look ahead point Pang(Xang, Yang) are defined as follows:

$$N\_{pos} = \operatorname\*{argmin} \left| \sum\_{i=N\_{now}}^{i^\*} \sqrt{(\mathbf{x}\_{i+1} - \mathbf{x}\_i)^2 + (\mathbf{y}\_{i+1} - \mathbf{y}\_i)^2} - d\_p \right|$$

$$N\_{ang} = \operatorname\*{argmin} \left| \sum\_{i=N\_{now}}^{i^\*} (\arctan \frac{\mathbf{x}\_{i+1} - \mathbf{x}\_{now}}{\mathbf{y}\_{i+1} - \mathbf{y}\_{now}}) - d\_a \right| \tag{2}$$

dp and da are lateral error look ahead and heading-angle look ahead distance respectively, and Npos, Nang are the sequence number of Ppos and Pang in the reference path respectively.

### *3.3 Lateral Error and Heading-Angle Error*

The lateral error and heading-angle error are calculated as follows:

$$E\_{pos} = \frac{1}{N\_{pos} - N\_{now}} \sum\_{i=N\_{now}}^{N\_{pos}} x\_i \tag{3}$$

$$E\_{ang} = \sum\_{i=N\_{now}}^{N\_{ang}} \arcsc \tan(\frac{\mathbf{x}\_i - \mathbf{x}\_{now}}{\mathbf{y}\_i - \mathbf{y}\_{now}}) \tag{4}$$

### *3.4 Fuzzy Inference Engines*

The fuzzy inference engines are utilized to calculate the target steering with three input (speed v, heading angle error Eang, lateral error Epos) and two outputs (θ*ang* and θ*pos*), and every parameter is divided into 5 fuzzy sets according to their own value set as shown in Fig. 2.

And the target wheel angle of the vehicle is given below:

$$
\theta = \wp \ast \theta\_{\text{ang}} + (1 - \wp)\theta\_{\text{pos}} \tag{5}
$$

### **4 Experiments**

### *4.1 Simulations*

In this section, two simulations are conducted to show the effectiveness of our work. The original one look ahead distance method is compared with our two look ahead distances method in the first simulation; And the PID control method is compared with our fuzzy method to show the effectiveness of the proposed work.

In the first simulation, the reference path is shown as below, which includes a Sshape curve and a high-frequency sine noise to estimate the precision of the proposed control method. And the simulation is shown in Fig. 3.

**Fig. 3** Simulation result of one look ahead distance method (left) and the proposed two look ahead distance method (right)

In the simulation, two special condition (S-shape curve and a high-frequency sine noise) are zoomed in to show the precision of the proposed work. The simulation results shows that the original one look ahead distance method can't handle the noise quite precisely and a cut-corner is observed when the look ahead distance is quite big. However, our work can handle the noise quite well and the precision is also acceptable.

In the second simulation, the comparison results of PID control method and our method is given. The above two look ahead distance method is used for both control method. The parameter of both control method are fine-tuned with our best effort. The reference path of the second simulation include an S-shape curve, a sharp turn and a straight-way. The simulation result is shown in Fig. 4.

**Fig. 4** Simulation result of PID control method (left) and the proposed fuzzy control method (right)

### *4.2 On-Board Experiments*

The proposed lateral control method has been experimented on our unmanned concrete mixer truck (Fig. 5) for about 500 km and the maximum lateral error for a straight-way is less than 30 cm and for a sharp curve is less than 80 cm, which is acceptable for most conditions of the concrete mixer truck, and the results are shown in Figs. 6 and 7.

**Fig. 5** Our unmanned concrete mixer truck

**Fig. 6** Tracking errors of our model for a U-shape curve (blue line is the reference path, red line is the real trajectory)

**Fig. 7** Tracking errors of our model for a backward reference path (blue line is the reference path, red line is the real trajectory)

### **5 Conclusions and Future Work**

In this paper, multiple look ahead distances are calculated for lateral error and heading-angle error and multiple fuzzy inference engines are proposed with vehicle speed and the above-mentioned errors. Moreover, a kinematic model of mixer truck was proposed in this paper to address the length and width of the concrete mixer truck. Satisfactory simulation and on-board experiments results are obtained for our model.

Future work may include longitudinal control model and an analysis of the effect of speed for our model with more on-board experiments.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Bilateral Servo Control Strategy for the 3T Redundant Freedom Parallel Robot**

**Mingde Gong, Kongming Hu, and Yue Zhang** 

**Abstract** In this paper, a force feedback position bilateral servo control strategy with position difference compensation for a 3T redundant degree of freedom parallel robot is proposed. We have designed master and slave controllers based on fuzzy control, and conducted single motor bilateral servo control experiments and single branch bilateral servo control experiments on the experimental platform. The experimental results show that this method can effectively reduce the error accumulation caused by using non redundant inverse kinematics methods to solve redundant parallel robots, and maintain good position following performance of the master slave robot, Improved the transparency of force displacement control in the master–slave remote control robot system.

**Keywords** Bilateral servo control · Force feedback · Inverse kinematics · Remote control

# **1 Introduction**

The bilateral servo control technology was first proposed by Masato Kishiri of Hitachi Corporation in 1963, to achieve true remote operation between master and slave robotic arms [1]. Subsequently, INOUE, Lee and Andriot et al. pushed bilateral servo control technology to new heights [2–10]. Although various improved bilateral servo control strategies emerged later, they can be generally classified into four categories: powerless feedback type, position symmetry type, force position synthesis type, and force feedback position type [11]. The force feedback position type is an improvement

K. Hu

M. Gong (B) · Y. Zhang

School of Mechanical Engineering, Yanshan University, Qinhuangdao 066000, Hebei, China e-mail: gmd@ysu.edu.cn

School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, Jilin, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_37

of the force position comprehensive type, which is easier to achieve [12], has high stability, and is suitable for force telepresence robot systems.

The inverse kinematics solution is the foundation of robot control, but the 3T redundant degree of freedom parallel robot cannot obtain accurate analytical solutions. In order to achieve control, the inverse kinematics results of a non redundant 3D translational parallel robot are used for control, and then the control algorithm is used for real-time correction to eliminate accumulated errors, achieving relatively accurate position control of a 3T redundant degree of freedom parallel robot.

A force feedback position type bilateral servo control strategy with position difference compensation was designed for the working condition of using a 3T redundant degree of freedom parallel robot as a remote control handle. The master slave hand force feedback position bilateral servo control experiment and force feedback position bilateral servo control experiment with position difference compensation were conducted on the experimental platform. The experimental results showed that, This method can effectively reduce the error accumulation caused by using non redundant inverse kinematics methods to solve redundant parallel robots, maintain good position following performance of the master–slave robot, and improve the transparency of force displacement control in the master–slave remote control robot system.

### **2 Mechanism Design**

The overall structure of the 3T redundant parallel robot is shown in Fig. 1, and the range of motion of the driving pair and the length of the connecting rod are shown in Table 1.

In the Fig. 1, 1—handle; 2—Three dimensional force sensor; 3—Mobile platform; 4—Connection head; 5—Short pole; 6—Long pole; 7—Drive rod; 8—Rotating drive components; 9—Linear drive components; 10—Guide rail plate; 11—Fixed platform.

### **3 Bilateral Servo Control Strategy**

The inverse kinematics of a 3T redundant degree of freedom parallel robot can only be solved offline, which increases the difficulty of controlling the slave hand. So a force feedback with position difference compensation—position bilateral servo control strategy—was proposed, using the inverse kinematics solution method of non redundant parallel robots to calculate the driving amount. Then, through position difference compensation, the consistency of motion between the master and slave hands was achieved. The principle of the force feedback position bilateral servo control strategy with position difference compensation is shown in Fig. 2.

In the Fig. 2, *Fh* is the operating force, *Ff* is the feedback force, *Fs* is the environmental contact force, *F*0 is the set threshold, *qm* is the actual angular displacement of

**Fig. 1** Design schematic diagram of a 3T redundant degree of freedom parallel robot


**Table 1** Range of motion and connecting rod parameters of the motion pair

**Fig. 2** Schematic diagram of control strategy

the main hand rotating motor, *qms* is the actual angular displacement of the main hand linear motor, *qmd* is the driving angular displacement of the main hand motor, *qs* is the actual angular displacement of the slave hand rotating motor, *qss* is the actual angular displacement of the slave hand linear motor, *qsd* is the driving angular displacement of the slave hand rotating motor *qbs* and the driving angular displacement of the linear motor *qsz*, *Xm* is the spatial pose of the main manual platform center, and *Xs*  is the spatial pose of the manual platform center, *EX* is the difference information between the master and slave hands, *Fs* is the difference between the environmental contact force *Fs* and the threshold *F*0.

When there is no contact between the hand and the environment, *qmd* is the driving angular displacement *qmz* of the main hand linear motor; When the hand comes into contact with the environment, *qmd* includes both the driving angular displacement *qbm* of the rotating motor and the driving angular displacement *qmz* of the main hand linear motor; *EX* includes not only the position difference *eX* between the master and slave hands, but also the position difference *eXm* between the master hand and the previous moment, as well as the position difference *eXs* between the slave hand and the previous moment, as well as the difference *eXms* between *eXm* and *eXs*.

When the hand comes into contact with the environment, the environmental force *Fs* received from the hand suddenly increases, instantly exceeding the threshold *F*0. The slave controller controls the slave motor and linear motor to immediately stop rotating, and then determines the displacement, speed, and direction of the retraction based on the size and direction of *Fs*, completes the retraction command, and transmits the retraction command to the master controller; Then continue to determine the size of *Fs* and *F*0. If *Fs* is greater than *F*0 at this time, continue to retreat and transmit retreat commands to the main controller; If *Fs* is less than *F*0 at this time, terminate the rollback and transmit the rollback commands, and return to the master hand to control the operation of the slave hand.

In the retraction state, the main controller first controls the relay to turn on, causing the main hand to turn on the motor, and then executes the retraction task according to the retraction command. At this point, the operator will feel the direction of the feedback force and feedback force, and immediately stop the forward operation. After waiting for the retraction task to be completed, according to the direction of the feedback force, operate the main hand handle to keep the secondary hand handle away from the elastic contact position, avoid obstacles, and complete the operation task.

The spatial postures of the center point of the master–slave manual platform are *Xm* and *Xs*, respectively. The spatial postures of the previous moment are *Xm* − <sup>1</sup> and *Xs* − 1, respectively. Therefore, the position difference *eXm* and *eXs* between the master–slave manual platform and the previous moment are

$$e\_{Xm} = X\_m - X\_{m-1} \tag{1}$$

$$e\_{Xs} = X\_s - X\_{s-1} \tag{2}$$

The difference *eX* between the spatial pose of the center point of the master–slave manual platform and the difference *eXms* between *eXm* and *eXs* are

$$e\_X = X\_m - X\_s \tag{3}$$

$$e\_{Xms} = e\_{Xm} - e\_{Xs} \tag{4}$$

Difference between hand force and threshold

$$
\Delta F\_s = F\_s - F\_0 \tag{5}
$$

When there is no contact between the hands and the environment, *Fs* is less than 0; When in contact with the environment from the hand, *Fs* is greater than 0.

### **4 Testing and Verification**

The experimental platform of the assembled bilateral servo control system is shown in Fig. 3.

The initial position is set as the initial position of the linear motor by placing the nut sleeve of the main slave hand on the outermost end of the lead screw (θ*i* = 157 mm), The top of the drive lever is located at the lowest point (θ*<sup>i</sup>*2 = 65°) serves as the initial angular displacement of the rotating motor. Respectively apply the force feedback position bidirectional servo control strategy and the force feedback position bidirectional servo control strategy with position difference compensation to control the slave hand to follow the master hand's movement. From the hand handle, first reach near the rubber ball, and then make elastic contact with the rubber ball. After the operator feels the feedback force, detach the rubber ball from the hand handle and walk around the rubber ball for half a circle. Finally, the master slave operating handle returns to its initial position. Used to detect the master–slave following performance

**Fig. 3** Bilateral servo control test platform

(a) Force Feedback – Position (b) Force feedback – position with compensation

**Fig. 4** Spatial displacement following curve

and obstacle avoidance performance of redundant isomorphic master–slave control systems.

Collect angular displacement data of 12 motors and substitute the angular displacement data into the forward kinematics solution formula to obtain the displacement curve of the center of the master slave robot's moving platform. The results are shown in Fig. 4.

Comparing Fig. 4a, b, it can be seen that both control strategies can achieve simple obstacle avoidance capabilities. Regardless of whether elastic contact occurs or not, the tracking error of the control strategy without position difference compensation is greater than that of the control strategy with position difference compensation, so the tracking performance of the control strategy with position difference compensation is better.

### **5 Conclusion**

Based on the above research, the following conclusions can be drawn.


**Acknowledgements** The authors gratefully acknowledge the financial support of Natural Science Foundation of Hebei Province (E2021203145).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Study on Weighting Factors in Cost Function of Model Predictive Control Algorithms**

**Zhou You, Yongming Bian, Zhangjie Ding, and Li Chen** 

**Abstract** With the growing emphasis on environmental protection, the engineering construction industry is facing new demands. The widespread electrification of construction machinery has emerged as an inevitable trend. To align with this trend, this paper conducts research on the model predictive control algorithm (MPC) applied to permanent magnet synchronous motors (PMSM). The MPC algorithm primarily consists of the motor's mathematical model and the cost function. This study focuses on analyzing the weight factor within the cost function. The spectrum of weight factor combinations concerning the dq-axis current is formulated across a broad range of intervals. The algorithm's performance is assessed under various weight factors using experimental data from three-phase current operation, motor torque, rotational speed, and other parameters. Ultimately, a novel evaluation function is proposed in this study to quantitatively assess the algorithm's performance across various combinations. The process involves identifying the optimal combination of weight factors for the cost function. This effort contributes significantly to establishing a robust groundwork for the general electrification of construction machinery.

**Keywords** Permanent magnet synchronous motor (PMSM) · Model predictive control (MPC) · Cost function · Weight factor

Z. You · Y. Bian · Z. Ding

School of Mechanical Engineering, Tongji University, Shanghai 201804, China

L. Chen (B)

Y. Bian

© The Author(s) 2024

School of Automotive Studies, Tongji University, Shanghai 201804, China e-mail: lilychen@tongji.edu.cn

Shanghai Engineering Research Center for Safety Intelligent Control of Building Machinery, Shanghai 200032, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_38

### **1 Introduction**

In the era of new infrastructure, the cornerstone industry of engineering construction confronts novel demands and challenges [1, 2]. As a vital component of engineering construction, engineering machinery must also align with the imperative for green and low-carbon solutions by transitioning toward electrification [3]. Within the journey toward the comprehensive electrification of construction machinery, the permanent magnet synchronous motor (PMSM) emerges as a pivotal subject of research, with its related investigations progressively delving deeper. As part of the electrification of construction machinery, the reliability of PMSM puts forward new requirements, and the researchers [4] use the fast digital twin system constructed in Simulink to construct a PMSM monitoring system, and use it as a basis for determining the remaining structural life of the permanent magnet. PMSM can suffer from undervoltage during sudden changes in torque, which reduces the dynamic performance of the motor. Researchers [5] proposed a predictive trajectory control strategy with improved dynamic performance, combining trajectory optimisation with a dynamic overmodulation algorithm, verifying that the proposed control scheme has good performance and low computational cost.

For PMSM, their control schemes such as field oriented control (FOC), sliding mode control (SMC) and model predictive control (MPC) are hotspots in current research [6–8]. Among them, the MPC algorithm has attracted much attention because of its good dynamic performance, high adaptability and multi-objective optimization. In the recent past, researchers [9] proposed a novel control strategy based on MPC algorithm and used it for distributed drive electric vehicles, which will be possibly applied to construction machinery in the next step.

Within the context of the MPC algorithm for PMSM, a pivotal element is the cost function, serving to designate the algorithm's controlled entity and its respective weightage. To advance the exploration of the MPC algorithm, this study centers its attention on investigating the cost function linked to dq-axis current and its associated weight factor. Experimental results encompassing dq-axis current, motor output torque and speed, and three-phase current are also provided. Facilitating a quantitative analysis, this study constructs a novel evaluation function employing current ripple and current harmonics. This evaluation function assesses the performance of the MPC algorithm across diverse weight factor combinations, culminating in the synthesis of multiple outcomes to identify an optimal weight factor composition. The efforts in this study establish a foundational framework for the prospective widespread adoption of electric drive construction machinery.

### **2 Simulation Model**

For PMSM, the following assumptions are made when establishing its mathematical model when it is located in a rotating coordinate system:


Under the above assumptions, the current differential equation of PMSM can be obtained after finishing as:

$$\begin{cases} i\_d(k+1) = i\_d(k) + \frac{T\_s}{L\_d} \left[ \mu\_d(k) + w\_r(k)L\_d i\_q(k) - R\_S i\_d(k) \right] \\\\ i\_q(k+1) = i\_q(k) + \frac{T\_s}{L\_q} \left[ \mu\_q(k) - w\_r(k)L\_q i\_d(k) - R\_S i\_q(k) - w\_r(k)\Psi\_f \right] \end{cases} (1)$$

In the Eq. (1), Ld and Lq are the corresponding inductances of the d and q axes; RS is the stator resistance of the motor; Ts is the sampling period; uq(k) is the torque voltage at the moment of k, which is the q-axis component of the stator terminal voltage; ud(k) is the magnetic chain voltage at the moment of k, which is the d-axis component of the stator terminal voltage; id(k) and iq(k) are the dq-axis currents at the moment of k; f is the permanent magnet chain. The Eq. (1) is used to build the mathematical model of PMSM. For MPC algorithm, the cost function also needs to be constructed to complete the control closed loop. In the algorithm, the cost function is utilized to constrain the output variables to obtain the switching state of the inverter at the next moment. The traditional cost function is generally the sum of squares of the dq-axis current error, which can be written as:

$$\begin{cases} e\_{i\_d}(k+1) = i\_d(k+1) - i\_d^{ref} \\ e\_{i\_q}(k+1) = i\_q(k+1) - i\_q^{ref} \\ f = e\_{i\_d}^2(k+1) + e\_{i\_q}^2(k+1) \end{cases} \tag{2}$$

In Eq. (2), eid (k + 1) and eiq (k + 1) are the errors between the predicted current and the set value of the dq-axis at the k + 1 moment in the discrete domain. In order to further improve the performance of MPC algorithm, this paper constructs the cost function containing the dq-axis weight factor, which is expressed as:

$$f = \lambda\_d e\_{i\_d}^2 (k+1) + \lambda\_q e\_{i\_q}^2 (k+1) \tag{3}$$

Differing from the conventional cost function, this paper introduces specific weight factors prior to squaring the dq-axis error. In particular, λd signifies the weight factor pertaining to the squared summation of chain current error, while λq denotes the weight factor linked to the squared summation of torque current error. Varied combinations of weight factors are employed to conduct simulation experiments, ultimately identifying the optimal combination of weight factors for PMSM's MPC. In the course of the study, the paper explores seven diverse weight factor combinations within an expansive scope, facilitating experimental investigations. To streamline presentation, the designations of corresponding working conditions in Table 1 are employed to signify distinct weight factor combinations.

Equations (1) and (3) are employed to form a MPC system for PMSM, and the simulation system shown in Fig. 1 is built within Simulink.

To provide a more comprehensive demonstration of the various combinations of weight factors, a current ripple measurement module is incorporated into the simulation system. This module can be represented as follows:

$$\Delta \, i\_{RMS} = \left| \left\{ \frac{1}{N} \sum\_{k=0}^{N-1} \left\{ \left[ i\_d(k) - \overline{i\_d} \right]^2 + \left[ i\_q(k) - \overline{i\_q} \right]^2 \right\} \right\} \right| \tag{4}$$


**Table 1** Working conditions designations of weight factor combinations

**Fig. 1** Simulation system

### **3 Results**

The MPC algorithm model for PMSM is built in Simulink, and the specific simulation parameters are shown in Table 2.

Simulation experiments are carried out for the working conditions corresponding to different weighting factors, and the results obtained are shown below. In the simulation system, PMSM is accelerated to 1200 rpm and decelerated to −1200 rpm at 0.2 s. In order to further verify the performance of the algorithms, the system applies an external load of 10 n\*m at 0.35 s. Figure 2 shows the operation of the dq-axis current under different working conditions. It can be seen that the q-axis current fluctuates relatively less under the working conditions represented by C, D and E, while the fluctuation of the d-axis current is more obvious, which is also related to the fact that the weighting factor correspondingly enlarges the proportion of the torque current in the cost function; and the opposite is true for the working conditions corresponding to B, F and G. Taken together, only the A case can suppress all the dq-axis current fluctuations well, which proves the excellent performance of the algorithm to the A case.

For a deeper investigation into the MPC algorithm's performance across varying weight factors, Fig. 3 displays both motor speed and torque. Analysis of the Fig. 3 results reveals that when the weighting factor for torque current is increased (cases C, D, and E), motor torque exhibits enhanced stability with reduced fluctuations arising from torque oscillations. This consistency corresponds to the performance of torque currents shown in Fig. 2. Nevertheless, the augmentation of the torque current weighting factor concurrently impacts the chain currents, resulting in minor motor speed fluctuations. Conversely, the A case exerts superior control over PMSM speed and torque fluctuations, facilitating a more balanced and consistent torque and speed output.

Building upon the preceding analysis outcomes, Fig. 4 illustrates the behavior of motor three-phase currents across distinct weight factor scenarios. Observing the figure, it becomes apparent that the A case yields a three-phase current output with reduced irregularities and a narrower fluctuation span compared to others. Particularly, at 0.35 s when an external load is imposed, all conditions exhibit a substantial upsurge in A-phase current. However, the A case generates a comparatively


**Fig. 2** Simulation results: **a** q-axis current **b** d-axis current

**Fig. 3** Simulation results: **a** motor speed **b** motor torque

minor increase, implying enhanced system robustness when confronted with external perturbations. This attribute proves invaluable when constructing machinery under intricate operational settings or challenging environments.

Many of the aforementioned findings are derived from qualitative analyses and may lack robust persuasiveness. To provide deeper insight into the algorithm's performance, we compute and present in Table 3 the current ripple, current harmonics, and average switching frequency across the entire simulation timeframe, encompassing a range of operational scenarios.

**Table 3** Comparison of algorithm performance under different working conditions



Table 3 demonstrates a clear trend where the A case exhibits the smallest current ripple and harmonics. This observation substantiates the algorithm's efficacy under the A case and aligns with findings regarding three-phase currents. Regarding the switching frequency, the A case remains viable; however, a notable surge in current ripple and harmonics is evident when contrasted with the lower switching frequency linked to the A case. This escalation compromises the MPC algorithm's efficacy and contributes to heightened losses during PMSM system operation. To provide further insight into the Table 3 outcomes, the article introduces an evaluation function M for validation, computed as follows:

$$M = \,\_{\Delta}i\_{RMS} + I\_{THD} \tag{5}$$

It's important to emphasize that the numerical value embodied by the evaluation function M entails a level of intricacy. For now, we consider it solely as a dimensionless parameter, initially utilized for comparing the performance of the MPC algorithm across diverse working conditions. We refrain from delving into its current physical interpretation. Refer to Table 4 for its specific values.

The data presented in Table 4 reveals that the A case yields the lowest evaluation function value. While the C case results in a slightly higher evaluation function value compared to the A case, an analysis considering preceding parameters like dq-axis current fluctuation, three-phase current operation, and average switching frequency further supports the conclusion that the A case stands as the optimal weight factor combination for application in the MPC algorithm for PMSM.

### **4 Conclusion**

This paper delves into the cost function of the MPC algorithm for PMSM. The model is built within a simulation system, exploring and contrasting various combinations of weight factors for the dq-axis current within the cost function. Beyond examining parameters such as dq-axis current, motor speed, and three-phase current, this study introduces current ripple and current harmonics to formulate a novel evaluation function. Leveraging this approach alongside the average switching frequency during algorithm execution, the study quantitatively assesses the working conditions depicted by distinct weight factors, ultimately deriving optimal performance from the amalgamation of weight factor combinations. In a comprehensive sense, this paper's research content holds specific implications for the advancement of electric power-driven construction machinery.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Study on Multiple Hydraulic Cylinders Synchronous Control Under Impact Load**

**Jianjun Wang, Sulan Hao, and Wenlei Li** 

**Abstract** In engineering, it is often necessary to use multiple hydraulic cylinders to drive the load at the same time. Under the impact load, because of the compressibility and leakage of hydraulic oil is inevitable, it is difficult to accurately control the speed of the actuator. Firstly, the compression and leakage of the hydraulic oil under the impact load are analyzed theoretically, and the main factors affecting the speed stability of the hydraulic cylinder are obtained. A multiple hydraulic cylinders synchronous control experimental circuit with electro-hydraulic proportional PID control is designed, which is composed of pump station, oil cylinders, displacement detection sensors and computer control system. The master–slave control strategy, electro-hydraulic proportional closed-loop control and fuzzy PID weighted control algorithm are adopted, and the simulation results show that this method can realize the stable control of hydraulic cylinder speed, and improve the synchronous control accuracy of multiple hydraulic cylinders, and the synchronous accuracy is within 0.5 mm. Finally, the experimental platform is built for practical verification, and through data acquisition and displacement monitoring of the system, it can be proved this method can realize the accurate synchronization of displacement driven by four hydraulic cylinders. The results show that the system is easy to control and has high synchronization accuracy, which can provide a reference for the design of multiple actuators synchronization control.

**Keywords** Synchronous control · PID control · Master–slave control strategy

J. Wang (B) · S. Hao

© The Author(s) 2024

College of Mechanical Engineering, Anhui Technical College of Mechanical and Electrical Engineering, Wuhu 241000, China e-mail: ahjdwjj@ahcme.edu.cn

W. Li

Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, Yanshan University, Qinhuangdao 066004, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_39

### **1 Introduction**

With the development of industry and engineering, multiple hydraulic cylinders are widely used to drive loads in various fields, such as mechanical manufacturing, marine engineering, national defense, lifting equipment of large stage, synchronous lifting system of large bridge and so on [1, 2]. And the requirement of synchronization accuracy is relatively high.

In the hydraulic system, the large impact load will cause large speed fluctuation of the actuator, especially the change of the comprehensive oil elastic modulus after the gas escaping from the oil caused by the load change, which will cause greater harm to the system. The speed stability of hydraulic actuator is affected by the elastic modulus of oil, which mainly depends on the air content and working pressure of oil [3].

In general, the stiffness of the hydraulic system is often ignored when the load changes suddenly. In some cases, the requirement of the speed stability of the actuator is relatively high, so the influence of the oil elastic modulus should be considered. Through the analysis of the elastic modulus of the hydraulic oil under the pressure change, it is found that in order to make the output speed of the hydraulic cylinder tend to be stable, it is necessary to rely on the control algorithm to reduce the interference of the impact load. Through the mathematical model and simulation verification, it is concluded that the fuzzy PID control weighted algorithm can significantly reduce the impact of load and keep the speed of hydraulic cylinder with high control accuracy [4].

The closed-loop control of electro-hydraulic servo valve or electro-hydraulic proportional valve is often used in multiple cylinders synchronous control system [5]. Hydraulic servo and proportional synchronous control system is composed of various electro-hydraulic proportional valves, electro-hydraulic servo valves or digital valves [6]. The advantage of the synchronous system is that it usually adopts closed-loop control mode, which can continuously detect and feedback the output variables, so that the errors caused by the input signal, system type and system interference can be suppressed in time, and high control accuracy can be obtained while meeting the requirements of rapidity and system stability. At present, the commonly used control algorithms include PID control and related improved control algorithm, adaptive control, sliding mode variable structure control and intelligent control [7, 8].

### **2 Speed Stability Analysis**

### *2.1 Hydraulic Oil Bulk Modulus of Elasticity*

In the work process of hydraulic system, the volume of oil will shrink under a large pressure, which is called the compressibility of oil. The compressibility of hydraulic oil can be expressed by the following equation [9].

Study on Multiple Hydraulic Cylinders Synchronous Control Under … 507

$$E = -\frac{\Delta p}{\Delta V} \times V \tag{1}$$

where P is the change amount of the pressure, *V* is the volume change amount of the hydraulic oil, and *V* is initial volume of the hydraulic oil.

The change of oil bulk elastic modulus will directly affect the speed rigidity of hydraulic actuator [10, 11]. In general or under low pressure, the relative compression volume of hydraulic oil is very small, which can be ignored and considered as incompressible. But for heavy load system or system with large impact load, the compressibility of oil seriously affects the stability of system speed. The oil bulk elastic modulus is an important parameter for dynamic analysis, modeling and Simulation of hydraulic system. The size of hydraulic oil bulk elastic modulus has a great relationship with the type, air content, working pressure and temperature of hydraulic oil [12].

When the hydraulic oil model is selected, the volume elastic modulus of the hydraulic oil will increase with the increase of pressure according to Eq. (1), and the volume elastic modulus of the hydraulic oil will decrease with the increase of temperature. Assuming that the bulk elastic modulus of the pure hydraulic oil is *E*0, when the pressure is *P*, the bulk elastic modulus of oil can be expressed by the following equation [13].

$$E = \begin{cases} \frac{E\_0}{aP\_0 E\_0 (\frac{1}{P^2} - \frac{1}{P\_c^2})}, & P\_0 < P < P\_c\\ & E\_0, & P \ge P\_c \end{cases} \tag{2}$$

where *a* is the air content in the fluid, *P*0 is the standard atmospheric pressure and *P*c is the critical pressure.

It can be seen that the bulk elastic modulus of hydraulic oil is greatly affected by pressure. With the change of pressure, the state of gas in the oil will also change. There are two states of gas in the oil, which are compression and dissolution. The volume change of gas in two states of the hydraulic oil can be expressed by the following equation [14].

$$\frac{P\_1V\_1}{T\_1} = \frac{(P\_1 + \Delta P)(V\_1 + \Delta V\_p + \Delta V\_d)^\gamma}{T\_2} \tag{3}$$

where *P1* is the initial state pressure, *V1* is the volume of gas in initial state, Δ*Vp* is the volume change of gas under pressure, Δ*Vd* is the volume change of gas dissolution, *T1* is the temperature of the hydraulic oil in initial state, *T2* is the temperature of the hydraulic oil in working condition, γ is the adiabatic index of gas.

### *2.2 Hydraulic Cylinder Speed Analysis*

The hydraulic cylinder speed mainly depends on the volume of hydraulic oil entering the hydraulic cylinder. Assuming that the pipeline and valve body through which the hydraulic oil flows are rigid bodies without deformation, the continuity equation of the input hydraulic cylinder flow can be expressed as follows [15].

$$q = Av + \lambda (P + \Delta P) + \frac{V\_l}{E} \frac{d(P + \Delta P)}{dt} \tag{4}$$

where *A* is the piston area of the hydraulic cylinder, *v* is the speed of hydraulic cylinde, λ is the Hydraulic cylinder leakage coefficient, *P* is the hydraulic cylinder input pressure, *Vt* is the input liquid volume of hydraulic cylinder.

Based on the above theoretical analysis, if an impact load is encountered, it will cause the impact pressure, and the volume elastic modulus of hydraulic oil will change, which will cause the speed of hydraulic cylinder to be unstable and nonlinear. In order to improve the speed stability of the hydraulic cylinder and reduce the influence of the impact pressure and the change of the hydraulic oil bulk elastic modulus, the hydraulic oil bulk elastic modulus and the control algorithm can be used to improve the stability of the speed control. To improve the bulk modulus of elasticity of hydraulic oil, it is necessary to reduce the air content in the oil, which can be achieved by vacuuming [16].

### **3 Control Algorithm Selection**

The displacement control scheme of one single hydraulic cylinder is shown in Fig. 1. The electro-hydraulic proportional directional valve is used to control the displacement of the hydraulic cylinder. The displacement sensor of the hydraulic cylinder collects the displacement signal. The difference between the displacement signal and the set displacement value is used as the control signal of the controller.

**Fig. 1** The displacement control scheme of hydraulic cylinder

**Fig. 2** Schematic block diagram of fuzzy PID control

**Fig. 3** Membership function

### *3.1 Controller Design*

Fuzzy-PID control can effectively improve the control accuracy. Fuzzy-PID control is developed on the basis of ordinary PID control, and it is the product of the combination of fuzzy theory and PID control. Fuzzy-PID control is also a control method often used in industry. Fuzzy-PID control can change the three parameters (proportional coefficient, integral coefficient and differential coefficient) according to the change of error signal, thus improving the control accuracy.

The input objects of Fuzzy-PID controller are *e*(error) and *ec*(error change rate). If they change, it will trigger the fuzzy controller to adjust the three coefficients (Δ*Kp*, Δ*Ki*, Δ*Kd* ). The control block diagram is shown in Fig. 2.

For the hydraulic cylinder speed control system, the fuzzy rule control table can be formulated by summing up the previous experience, and the membership function relationship of parameters can be formulated according to the actual deviation requirements of the project. According to the control accuracy requirements, the 7 segment fuzzy universe is selected, and the triangular membership function shown in Fig. 3. It is selected as the input and output membership function.

According to the characteristics of hydraulic cylinder system, the Fuzzy-PID control rules are determined as shown in Table 1.

### *3.2 Simulation Analysis*

The displacement control system of single hydraulic cylinder is established by using AMESim software, as shown in Fig. 4.


**Table 1** Fuzzy control rules

**Fig. 4** AMESim model of hydraulic cylinder displacement control system

The fuzzy controller is built in Simulink, and the final combined simulation model is shown in Fig. 5.

The hydraulic cylinder will encounter impact load during its movement, which will cause hydraulic impact on the system and affect the stability of the hydraulic cylinder speed. In the simulation model, the load pressure is set to 5 MPa, and the air content of the hydraulic oil is set to 10 and 20%. When the system runs to the

**Fig. 5** The combined simulation model

10th second, the load will suddenly change to 2 times of the original. The simulation results are shown in Fig. 6.

**Time/s** 

From the simulation results, it can be seen that the greater the air content, the worse the speed stability of the hydraulic cylinder. Due to the sudden change of the load, the system began to oscillate at the 10th second, but then the system returned to the equilibrium state in a short time, indicating that the Fuzzy-PID control has a good ability to resist external interference.

### **4 Synchronization Control Strategy**

Theoretically, synchronization can be achieved with the same input flow of the same hydraulic cylinder, but in practice, different manufacturing errors of the hydraulic cylinder, different loads, and different characteristics of the control elements will lead to non-synchronization of multiple hydraulic cylinders.

The general hydraulic synchronization circuit limits the synchronization accuracy of multiple hydraulic cylinders. It is necessary to use advanced controllers to realize the synchronization of hydraulic cylinders with uneven load. The control of each hydraulic cylinder needs to be controlled separately [17].

**Fig. 7** "Leader–follower" control scheme

There are many control strategies for hydraulic synchronization. Open loop control adopts synchronous loop, and closed loop control adopts "parallel synchronization" and "leader–follower" control strategies. Closed loop control often can achieve high control accuracy. In order to achieve high-precision synchronization, many researchers have done a lot of work and achieved relevant research results [18–20]. In order to obtain better synchronization accuracy and dynamic response, referring to a large number of relevant literature and similar research experience, the combination of "leader–follower" control strategy and Fuzzy-PID is planned to be adopted.

The "leader–follower" control strategy is to select one of the leader cylinders and the other cylinders as the follower cylinders, and take the output displacement of the leader cylinder as the input of other hydraulic cylinders, so as to track and achieve synchronization. The control scheme is shown in Fig. 7.

### *4.1 Design of Fuzzy-PID Controller*

Fuzzy-PID control has excellent robustness and good dynamic and static control performance in principle, and can meet the requirements of online adjustment of various parameters in the PID controller when the structural characteristics of the system change [21, 22], which can achieve the best control effect of synchronization. A two-dimensional input and a three-dimensional output fuzzy controller are selected, with the deviation *e* and the deviation change rate *e*c as the input values, and the increment of the three parameters of the PID controller as the output values of the fuzzy controller. The flow quantity is controlled by controlling the spool displacement of the electro-hydraulic proportional valve, so the displacement of the hydraulic cylinder can be controlled.

According to the synchronization characteristics of multiple hydraulic cylinders, a Fuzzy-PID control rule is determined. In the actual synchronization process, according to the control quantity *u*, the common weighted average method is used to solve ambiguity [23]. For example *K p*, the following equation is established.

$$
\Delta K\_p = \frac{\sum\_{i}^{n} u\_i \mu(i)}{\sum\_{i=1}^{n} \mu(i)} \tag{5}
$$

where *ui* is the number of control quantity, *Kp* is the output values and μ(*i* ) is the membership of control quantity.

Therefore, the modified PID parameters can be obtain by adding the calculated output values and the initial parameters. The calculation equation is as follows.

$$K\_p = K\_{p0} + \{e, ec\}\\ K\_p = K\_{p0} + \Delta K\_p \tag{6}$$

$$K\_i = K\_{i0} + \{e, ec\}\\ K\_i = K\_{i0} + \Delta K\_i \tag{7}$$

$$K\_d = K\_{d0} + \{e, ec\}\\ K\_d = K\_{d0} + \Delta K\_d \tag{8}$$

### *4.2 Simulation and Analysis of Synchronous Control*

The Simulink simulation model of fuzzy PID control is established for the synchronous loop of four hydraulic cylinders, as shown in Fig. 8.

The displacement of the hydraulic cylinder is set to 280 mm. After simulation, the results are shown in Fig. 9. The maximum synchronization error of the piston during movement is about 0.4 mm.

### **5 Synchronization Control Experiment**

The designed four cylinders synchronous circuit is shown in Fig. 10. Each of the four hydraulic cylinders is equipped with position sensors. The data acquisition card collects the displacement signals of the hydraulic cylinder, and the operator console sends instructions to control the speed and displacement of the first hydraulic cylinder. The output displacement signals are used as input instructions for other hydraulic cylinders. The displacement sensor continuously detects the displacement signal, and the controller continuously corrects the displacement error, until the synchronization

**Fig. 8** Four hydraulic cylinders synchronization simulation model

of each hydraulic cylinder is achieved. An experimental platform was built based on the hydraulic circuit, as shown in Fig. 11.

In the circuit, a closed-loop control system consisting of an electro-hydraulic proportional directional valve and a hydraulic cylinder is used. The displacement signal of one hydraulic cylinder is used as input to the other three hydraulic cylinders, and the proportional directional valve current is controlled through displacement

**Fig. 10** Four cylinder synchronous hydraulic circuit

**Fig. 11** The experimental platform

feedback to keep the four cylinders synchronized. The instantaneous pressure change of the electromagnetic overflow valve is used as the impact load. The displacement curves of the four hydraulic cylinders obtained through the data acquisition card are shown in Fig. 12. It can be seen that the displacement curves are basically coincident, with a synchronization error of about 0.5 mm.

### **6 Conclusions**


**Acknowledgements** This research was funded by the National Natural Science Foundation of China (51675461), Academic support project for top professional talents in Colleges and Universities in Anhui Province (gxbjZD2021100), the Scientific Research Project Foundation of Anhui Education Department (KJ2020A1108).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Autonomous Operation Motion Control of Excavator**

**Mingde Gong, Zhong Jin, Yue Zhang, Wenbin Liu, and Yue Ning** 

**Abstract** Accurate motion control is the premise for the unmanned excavator to complete the operation. In this paper, the motion control of a 22 t hydraulic excavator is studied. By establishing the dynamic model of the 4-degree-of-freedom excavator working device, the external interference caused by gravity, centripetal force, Coriolis force and inertia to the hydraulic cylinder is compensated. The bucket load and related parameters in the excavation process are identified online, and the motion control accuracy of the excavator is improved. Based on the dynamic model, the motion control method of the unmanned excavator is given. The simulation and experimental results show that this motion control strategy can effectively improve the autonomous operation accuracy of the excavator.

**Keywords** Autonomous excavator · Motion control · Online identification · Dynamic model

# **1 Introduction**

Hydraulic excavator is one of the commonly used engineering machinery, which has the characteristics of typical function and strong adaptability. Since China's reform and opening up, in order to meet the national infrastructure, mining, engineering water conservancy and other fields, the country's research and exploration of hydraulic excavators and other engineering sports machinery and industrial development support is very strong. At present, China has become the international largest

Y. Ning

School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China

M. Gong (B) · Z. Jin · Y. Zhang · W. Liu

School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China e-mail: gmd@ysu.edu.cn

Z. Jin Guangxi Liugong Machinery Co., Ltd., Liuzhou 545027, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_40

producer and consumer of hydraulic excavators, and the market is increasingly demanding the working performance of excavators. After the development of traditional hydraulic excavators has matured, the academic research direction of excavators has begun to move towards unmanned and autonomous. In the research process of excavator autonomous operation, the two main problems are motion planning and control accuracy. In the above two problems, the research on the control accuracy is the premise of the motion planning problem. The problem that the control accuracy cannot reach the standard increases the huge safety hazard when the excavator carries out the autonomous operation. At present, domestic and foreign scholars and insiders have initially formed a scale for the research on the motion control accuracy of hydraulic excavators during unmanned operation. He et al. [1] used the adaptive control method to control the flow of the boom hydraulic cylinder. The simulation results are compared with the conventional PID control, and the simulation speed and accuracy are improved. Wang et al. [2] used the cross-coupled pre-compensation (CCP) method to combine its compensation with the nonlinear proportional integral controller of each actuator to achieve coordinated control of unmanned excavators. The model predictive control method was utilized by Zheng et al. [3] to track the predefined trajectory under diverse working conditions. The results show that the model predictive control can provide more accurate feedforward control force under complicated working situations, achieve accurate trajectory tracking, effectively minimize control delay caused by high inertia conditions, and has better robustness. Hanh et al. [4] proposed a fuzzy self-tuning controller based on neural network to control a small electro-hydraulic excavator. The neural network method is used to tune the output signal online, which has certain anti-interference ability and adaptability to improve the tracking performance of the system. Kadu et al. [5] proposed a time-delay sliding mode control strategy for the motion control of unmanned excavators so that the working device performs well under the influence of external disturbances and uncertainties. Precision control approach for industrial hydraulic excavator robot movements developed by Lee et al. [6] based on data-driven model inversion. To boost the system's learning speed, a modular-driven model inversion control technique is proposed. When compared to PI control, simulation and experiment demonstrate the control system's accuracy.

In this paper, the motion control accuracy of the self-operated excavator is studied from the perspective of the motion control of the valve-controlled asymmetric hydraulic cylinder. The online compensation control of the excavator's valvecontrolled asymmetric hydraulic cylinder is carried out by using the method of online identification and compensation, and the modified unmanned excavator is used for excavation experiment verification.

# **2 Establishment of Mathematical Model for Working Device**

# *2.1 The Establishment of Kinematic Model for Working Equipment*

The kinematic model is shown in Fig. 1.

Where θi and θ˙ i is joint angle and joint angular velocity. The joint torque of the excavator manipulator is positive in the counterclockwise direction.

The positive kinematics relationship is shown in Eq. (1), and the inverse kinematics relationship is shown in Eq. (2). The unspecified parameters in the formula can be derived from the geometric relationship. In [7] Koivo gave a detailed description of kinematics.

$$\begin{cases} x = c\_0(a\_0 + a\_1c\_1 + a\_2c\_{12} + a\_3c\_{123}) \\ y = s\_0(a\_0 + a\_1c\_1 + a\_2c\_{12} + a\_3c\_{123}) \\ z = a\_1s\_1 + a\_2s\_{12} + a\_3s\_{123} + d\_0 \\ \xi = \theta\_1 + \theta\_2 + \theta\_3 \end{cases} \tag{1}$$

$$\begin{cases} \theta\_0 = \arctan\frac{y}{x} \\ \theta\_1 = \alpha + \beta + \gamma \\ \theta\_2 = \pi - \cos^{-1}\frac{a\_1^2 + a\_2^2 - BJ^2}{2a\_1a\_2} \\ \theta\_3 = \xi - \theta\_1 - \theta\_2 \end{cases} \tag{2}$$

### *2.2 The Establishment of Dynamic Model of Working Device*

The working device of the autonomous excavator has 4 degrees of freedom. The dynamic model is established as shown in Fig. 2. The model mainly includes the boom, the arm, the bucket and its corresponding action cylinder. The influence of system friction and external load on the dynamic characteristics of the system is ignored and the mass of each hydraulic cylinder is equivalent to the working device.

The joint torque of the manipulator can be expressed as follows:

$$
\pi = M(\theta)\ddot{\theta} + c(\theta, \dot{\theta})\dot{\theta} + G(\theta) \tag{3}
$$

**Fig. 2** Dynamic model of the working device of autonomous excavator

where *τ* is the joint moment (N m), *M* R3×3 is the inertia matrix of rigid body (kg m2), *c* R3×3 is centripetal force matrix (N m), *G* R3×3 is the gravity matrix (N m).

# **3 Online Parameter Identification and Compensation Terms**

### *3.1 Gravity Identification Compensation Method*

The manipulator of the excavator is heavy. The dynamic features of the boom raising, arm, and bucket are strongly influenced during the operation. If the excavator boom, arm, and bucket online gravity compensation computation is performed, the parameters, τ1, τ2, τ3, *m*1, *m*2, *m*3, *l*1, *l*2, *l*3, φ1, φ2, φ3 must be obtained. This work completes the calculation of kinetic parameters using the techniques outlined below.

The pressure sensor is used in the experiment to measure the hydraulic cylinder's driving force *f* 1, *f* 2, *f* 3 under different attitudes and no-load conditions. The measured value of the pressure sensor at no load is almost equal to the external interference caused by the gravity of the working device, that is *f* g1, *f* g2, *f* g3. In this instantaneous state, the excavator manipulator's driving force is as follows:

$$\begin{cases} f\_1 = f\_{\mathfrak{g}^1} \\ f\_2 = f\_{\mathfrak{g}^2} \\ f\_3 = f\_{\mathfrak{g}^3} \end{cases} \tag{4}$$

At this time, the dynamic equation of the three joint torques τ - = τ - <sup>1</sup>τ - <sup>2</sup>τ - 3 *<sup>T</sup>*of the manipulator generated by gravity is shown as follows:

$$
\begin{bmatrix} \mathbf{r}\_1' \\ \mathbf{r}\_2' \\ \mathbf{r}\_3' \end{bmatrix} = \begin{bmatrix} m\_1 l\_1 c\_1 + m\_2 a\_1 + m\_3 a\_1 \ m\_2 l\_2 c\_2 + m\_3 a\_2 \ m\_3 l\_3 c\_3 \\ 0 & m\_2 l\_2 c\_2 + m\_3 a\_2 \ m\_3 l\_3 c\_3 \\ 0 & 0 & m\_3 l\_3 c\_3 \end{bmatrix} \begin{bmatrix} c\theta\_1 \\ c\theta\_{12} \\ c\theta\_{123} \end{bmatrix} \mathbf{g} \\
$$

$$
\tag{5}
$$

In the formula, c1 = cos φ1, c2 = cos φ2, s1 = sin φ1, s2 = sin φ2, cθ1 = cosθ1, cθ12 = cos(θ1 + θ2), sθ1 = sinθ1, sθ12 = sin(θ1 + θ2). τ - <sup>1</sup>τ - <sup>2</sup>τ - 3 *<sup>T</sup>*is the driving torque acting on the boom, arm and bucket only under the action of gravity, *m*1, *m*2, *m*3 are the mass of the boom, arm and bucket respectively.

The driving torque is generated by the hydraulic cylinder's driving force, and the force arm serves as an intermediary medium for the conversion of the driving force **Fig. 3** Structure analysis diagram of working device

and torque. The geometric method is used in Fig. 3 to derive the relationship between the driving force arm or the equivalent length of each cylinder and the length of the cylinder.

The driving force arms of *E*1, *E*2, and *E*3, each consisting of three joints, can be derived from geometric relationships

According to θ 1, θ 2, θ 3 derived from inverse kinematics, the external forces *f* g1, *f* g2, *f* g3 caused by self-weight are calculated as follows:

$$\begin{cases} f\_{\mathfrak{g}1} = \mathfrak{r}\_1'/E\_1 \\ f\_{\mathfrak{g}2} = \mathfrak{r}\_2'/E\_2 \\ f\_{\mathfrak{g}3} = \mathfrak{r}\_3'/E\_3 \end{cases} \tag{6}$$

And then the displacement *y*1, *y*2, *y*3 and driving force *f* 1, *f* 2, *f* 3 of the boom, arm, and bucket hydraulic cylinders produced in the preceding phases are used to compute the joint angles θ 1, θ 2, θ 3 and driving torque τ - 1, τ - 2, τ - 3. The least square approach is used to calculate and identify the parameters in Eq. (5). The parameter identification process is as Fig. 4.

All the parameters in the R<sup>3</sup>×3 matrix of Eq. (5). can be derived. The obtained matrix is processed to obtain the required parameters *m*1, *m*2, *m*3, *l*1, *l*2, *l*3 etc. The online gravity compensation torque can be obtained by substituting the parameter identification results into formula Eq. (5).

**Fig. 4** Derivation process diagram of relevant parameters

# *3.2 Online Centripetal Force Term and Inertial Force Term Compensation*

The inertia force and centripetal force generated by the large gravity of the manipulator affect the motion control accuracy of the hydraulic cylinder, so it is necessary to compensate the centripetal force and inertia force online.

The inertial force term as:

$$M(\theta) = \sum\_{i=1}^{N} \left( \mathbf{m}\_i \boldsymbol{J}\_v^{i^T} \boldsymbol{J}\_v^i + \boldsymbol{I}\_i \boldsymbol{J}\_w^{i^T} \boldsymbol{J}\_w^i \right) = \begin{bmatrix} \boldsymbol{M}\_{11} \ \boldsymbol{M}\_{12} \ \boldsymbol{M}\_{13} \\ \boldsymbol{M}\_{21} \ \boldsymbol{M}\_{22} \ \boldsymbol{M}\_{23} \\ \boldsymbol{M}\_{31} \ \boldsymbol{M}\_{32} \ \boldsymbol{M}\_{33} \end{bmatrix} \tag{7}$$

The centripetal force term as:

$$c(\theta, \theta) = \mathcal{C}\_{ijk} = \frac{1}{2} \sum\_{jk} \left( \frac{\partial M\_{ij}}{\partial \theta\_k} + \frac{\partial M\_{ik}}{\partial \theta\_j} - \frac{\partial M\_{jk}}{\partial \theta\_i} \right) \theta\_j \theta\_k \tag{8}$$

The relevant parameters identified in the previous section are substituted into Eqs. (7) and (8) to obtain the specific inertial force term and centrifugal force term.

Combined with the previous section, the online compensation model of gravity, centripetal force and inertial force can be obtained. The driving force of the boom, arm and bucket hydraulic cylinder measured by the force sensor during operation is *f* 1, *f* 2, *f* 3. The driving force after the external interference force *f* e1, *f* e2, *f* e3 in the direction of the compensated hydraulic cylinder can be used as the driving force *f* R1, *f* R2, *f* R3 of the hydraulic cylinder under actual conditions, which as:

$$\begin{cases} f\_{R1} = f\_1 - f\_{\epsilon1} \\ f\_{R2} = f\_2 - f\_{\epsilon2} \\ f\_{R3} = f\_3 - f\_{\epsilon3} \end{cases} \tag{9}$$

### **4 Experiment**

### *4.1 Control Objective*

According to the task requirements, trajectory planning was carried out for the autonomous excavator. This section presents the Cartesian trajectory path of the working device bucket tip through the excavation trajectory of the excavator driver. The Cartesian path of the excavator can be transformed through inverse kinematics to obtain the joint angle planning for excavation, as shown in Fig. 5.

**Fig. 5** Trajectory planning of bucket tooth tip

The experimental equipment of the self-operated excavator is shown in Fig. 6. The joint angle measurement units at the boom and stick are installed on their respective outer surfaces. The bucket uses a built-in displacement sensor, and the pressure sensor is installed at the inlet and outlet of the hydraulic cylinder pipeline.

### *4.2 Experimental Results*

In order to verify the effectiveness of the control strategy, the prototype test was carried out using the above-mentioned self-operated excavator. The tracking effect of the experimental results under no-load are shown in Fig. 7. The spatial distance error of the excavator bucket tooth tip is shown in Fig. 8.

In the case of compensation, the average error of the boom joint angle tracking is reduced by 69.72%, the maximum error of the arm joint angle and the bucket joint angle tracking is reduced by 58.44 and 51.05% respectively, and the maximum error of the tooth tip space position tracking is reduced by 57.42%. It can be seen that the on-line compensation control strategy can effectively improve the control accuracy of the working device system of the autonomous operation excavator, and increase the operation accuracy and tracking performance of the manipulator of the autonomous operation excavator.

**Fig. 6** Autonomous excavator test platform

# **5 Conclusions**

In this paper, a motion control method of excavator autonomous operation based on online identification method is proposed to improve the motion control accuracy of excavator. The motion control accuracy of the autonomous excavator manipulator after compensating the gravity term, centripetal force term and inertial force term is studied. The effectiveness of the motion control strategy is verified by tracking the fitted excavator trajectory. The experiments prove that the accuracy of the excavator's autonomous operation can be improved by compensating the gravity of the excavator's working device and the inertial force and centripetal force generated during the movement.

(a) Time domain planning of boom joint angle

(c) Time domain planning of bucket joint angle

**Fig. 7** Tracking under no-load

**Fig. 8** Cartesian space position error

(b) Time domain planning of arm joint angle

(d) Time domain planning of Rotary joint angle

**Acknowledgments** The authors gratefully acknowledge the financial support of Natural Science Foundation of Hebei Province (E2021203145).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Review of Intelligent Vehicle Trajectory Tracking Control Methods Based on Overtaking Lane Change Scenarios**

**Dongzhao Yang, Lin Hu, Xiaojian Yi, and Xin Zhang** 

**Abstract** In order to achieve safe, comfortable, and stable control of lane-changing trajectories for intelligent vehicles, this paper reviews various trajectory tracking control methods for intelligent vehicles. Focusing on overtaking lane-changing trajectories, the paper provides a comprehensive review of research progress in PID, LQR, MPC, and SMC control methods both domestically and internationally. Striking a balance between addressing the challenges posed by non-integrity and multiple constraints of the vehicle, coping with the time-variability of the vehicle dynamics model, and ensuring the accuracy and computational efficiency of the vehicle model is essential for fostering the continuous development and enhancement of intelligent vehicle technology.

**Keywords** Intelligent vehicle · Trajectory tracking control · Lane change scenarios · Review

# **1 Introduction**

In recent years, with the growing popularity of intelligent driving technology, expectations for intelligent cars have risen significantly. One of the core aspects of intelligent cars is the precise and secure tracking of planned trajectories, a pivotal element

D. Yang · L. Hu · X. Zhang

X. Yi (B)

D. Yang · L. Hu · X. Yi · X. Zhang

School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China

School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha 410114, China e-mail: ydz6768520@163.com

Hunan Key Laboratory of Safety Design and Reliability Technology for Engineering Vehicle, Changsha University of Science and Technology, Changsha 410114, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_41

in achieving autonomous driving, enhancing traffic efficiency, and reducing accidents. Rear-end collisions, inadvertent contact with vehicles in the opposite lane, and even direct frontal collisions with oncoming vehicles are significant contributors to accidents, the field of overtaking and lane-change control for intelligent vehicles faces numerous challenges, including complex and evolving road conditions, traffic regulations, vehicle dynamics, and environmental perception. To address these challenges, researchers have diligently worked on developing innovative methods and technologies to enhance the trajectory tracking performance of intelligent vehicles. Trajectory tracking, as the core technology in intelligent vehicle decision-making, has consistently been a central and intricate issue, representing a critical aspect of safe intelligent vehicle operation. Figure 1 shows the schematic diagram of lane change control based on overtaking.

This paper aims to comprehensively review and analyze the current research status and future development trends in intelligent vehicle trajectory tracking control methods. It will delve into classical trajectory tracking control methods, examining their respective strengths and weaknesses. Additionally, it will explore the challenges encountered in intelligent vehicle trajectory tracking and propose potential avenues for future research. This endeavor will contribute to the continuous advancement and refinement of intelligent vehicle technology.

### **2 Review of Research Methods**

### *2.1 Research on PID Control*

In lane change trajectory control, PID control primarily focuses on utilizing displacement, velocity, and acceleration as control parameters. However, PID control is often combined with other methods to enhance tracking performance. For instance, the fusion of fuzzy control with PID control is employed to enhance the path tracking control strategy within collision avoidance systems [1]. Depending on the speed level of the intelligent vehicle, PID control is employed under low-speed conditions, while MPC is utilized under high-speed conditions [2]. This approach significantly enhances tracking precision and driving stability. Li et al. [3] have developed a PID control method based on the traditional preview error model. To ensure both followability and comfort, Huang et al. [4, 34] have proposed a controller that relies on personalized spacing strategies from an adaptive cruise system's modeswitching strategy. This controller integrates feedforward and PID control algorithms at the lower level. Recognizing the insufficient responsiveness of the vehicle steering system, Xu et al. [5] have introduced real-time steering system status to establish a dual closed-loop trajectory tracking control structure. The outer loop employs MPC, while the inner loop incorporates a feedback control law using the PID method to compensate for the steering system's inadequate response.

### *2.2 Research on LQR Control*

Linear Quadratic Regulator (LQR) control is a closed-loop optimal control system, as depicted in Fig. 2. It is particularly effective when highly accurate trajectory control is necessary, such as in lane change maneuvers. Utilizing the optimal preview control approach for longitudinal vehicle tracking, the nonlinear optimal control problem is transformed into an augmented LQR problem, leading to an analytical solution that reduces computational complexity [6]. Compared to PI control, optimal preview control offers superior tracking accuracy and smoother control action. Furthermore, Zhang et al. [7] introduced a cooperative control method combining LQR and dual PID controllers, which incorporates feedforward control and angle compensation. This approach enhances the stability and robustness of drive or braking force output during trajectory tracking. In a similar vein, Hu et al. [8] developed a fuzzy LQR controller with preview PID angle compensation for path tracking control. Their aim was to eliminate steady-state errors and enhance tracking accuracy, addressing the control precision and stability requirements of intelligent vehicles at various speeds.

### *2.3 Research on MPC Control*

Model predictive control (MPC) has the ability to systematically consider predictive information and deal with multi-constraint optimization problems, as shown in Fig. 3. The model predictive control of unmanned vehicle published by Beijing Institute of Technology systematically introduces the basic method of vehicle motion control using MPC [9]. In [10] and [11], a linear time-varying model predictive controller was designed to tackle issues related to low control accuracy and limited real-time performance, demonstrating strong adaptability in complex and challenging operational environments. Chen et al. [12] proposed an integrated longitudinal and lateral trajectory tracking control method based on nonlinear model predictive control to enhance tracking accuracy, especially during high-curvature turning conditions. Lin et al. [13] introduced a novel model predictive path planning controller, combined with potential field methods, to address intricate traffic scenarios. Zhang et al. [14] developed an adaptive model predictive control approach that takes into account the calculation errors associated with tire lateral forces. This approach aims to address the challenge of maintaining adaptability in vehicle trajectory tracking due to constant changes in road adhesion conditions. Dong et al. [15] introduced a multi-objective Model Predictive Control strategy based on a track tracking model for predicting

longitudinal motion control. This strategy effectively mitigates pronounced overshooting and oscillations, thereby improving track tracking performance, energy efficiency, and ride comfort.

### *2.4 Research on SMC Control*

Sliding Mode Control (SMC) directs the system state to a specific sliding mode surface in lane change trajectory control and maintains the state trajectory on that surface, as illustrated in Fig. 4. Zhang et al. [16] developed a novel autonomous tracking control system for intelligent electric vehicles based on lane detection and SMC. Utilizing a 2-degree-of-freedom (2-DOF) vehicle model, Hu et al. [17] employed lateral position deviation to steer the vehicle and keep it on the desired trajectory. Wu et al. [18] introduced a robust path-following control strategy for unmanned vehicles, incorporating non-singular terminal sliding mode and active disturbance rejection control. This strategy enables quick and precise path following while exhibiting strong robustness. To address trajectory tracking control robustness in the presence of various interference factors, Li et al. [19] proposed a robust controller featuring enhanced sliding mode control. To mitigate the jitter problem often associated with sliding mode control, Norouzi et al. [20] designed an adaptive

sliding mode control approach based on a fuzzy boundary layer. This approach effectively circumvents the upper limits of system uncertainty, significantly improving the existing jitter issue and enhancing tracking control capabilities.

### **3 Research Method Analysis**

The control accuracy can be greatly improved by integrating kinematics and dynamics into the control method. The kinematics model includes position, velocity, acceleration and yaw angular velocity, etc. Considering that the dynamic characteristics of the vehicle are ignored in the kinematics model, the kinematics controller is not suitable for the control of the vehicle under high-speed conditions [29]. The controller based on dynamic model has better tracking performance than the controller based on kinematic model at high speed, because the dynamic characteristics of the vehicle are considered, but the system complexity is increased.

Overtaking lane change is a relatively complex part of intelligent vehicle behavior decision-making. The tracking control of its trajectory involves both horizontal

**Fig. 5** Horizontal and vertical control system

control and longitudinal control, so that the longitudinal and lateral displacement errors of the vehicle tend to converge at the same time [28]. As shown in Fig. 5, the structure of its control system is basically the same, mainly due to the differences in controller design. The analysis will be carried out from PID, LQR, MPC and SMC four control methods.

PID controller has been widely used in autonomous vehicles, compared with other controllers, it has a great advantage, that is, it does not need to know the mathematical model or object, has a strong practicality, and sometimes may cause the integration operation to diverge due to the saturation of the actuator capacity, but it is not suitable for nonlinear systems.

SMC has strong robustness to the uncertainty of parameters and interference caused by complex driving conditions of autonomous vehicles [21]. SMC control system equation is shown in Eq. (1). SMC can be integrated with other control algorithms to make up for each other's weaknesses [22]. A key problem of SMC is buffeting caused by switching motion of sliding surfaces [21], which was also improved by researchers in the following years. See references [20] and [23].

$$X^{(\mathfrak{n})} = f(\mathfrak{x}, t) + b(\mathfrak{x}, t)u \tag{1}$$

where X, u and x are output, input and state quantities of the control system, and functions f and b are nonlinear unknown dynamic functions of the system.

Compared with PID control, LQR can obtain the optimal control law of linear feedback of the state, which is easy to form a closed-loop optimal control, The cost function is shown in Eq. (2). LQR optimal control can make the original system achieve better performance indicators by using low cost, and the method is simple and easy to realize in essence, it is an unconstrained optimization method for linear system, and the control quantity is assumed to be unconstrained when solving. However, depending on a more accurate mathematical model, the robustness of the system cannot be guaranteed in the presence of time-varying parameters and external disturbances, so the tracking control accuracy will be reduced in the limit working conditions with significant nonlinear characteristics.

$$J(\mathbf{x}, u) = \int\_0^\infty (X^T Q X + \boldsymbol{\mu}^T R u) dt \tag{2}$$

where x is the system state, u is the control input, and Q and R are positive definite weight matrices, which are used to adjust the weight of the state and control input.

MPC is a powerful state feedback controller, which uses the mathematical model of the vehicle to predict the future condition of the vehicle in the predicted range. MPC can deal with the constraint control problems of uncertain and nonlinear systems, and can also deal with the possible interaction between the input and output of the multiinput multi-output system, which is also difficult to achieve with PID controller. In addition, MPC is considered as a development of the optimal control strategy, which solves the optimization problem by using the cost function. However, MPC needs to solve the online optimization problem of each sampling time, which requires the computer to have powerful processing power, and the relationship between its computing power and system performance is contradictory, The general form of the objective function is shown in Formula (3).

$$J(k) = \sum\_{i=1}^{N\_p} \left\| X(k+i) - X\_{ref}(k+i) \right\|\_{\mathcal{Q}}^2 + \sum\_{i=1}^{N\_c-1} \left\| \Delta u(k+i) \right\|\_{\mathcal{R}}^2 \tag{3}$$

where X(k + i) represents state prediction, u(k + i) represents control input prediction, Xref is the reference trajectory, Nc is the control time domain; Np is the prediction time domain; Q and R are weight matrices.

The advantages and disadvantages of each trajectory tracking control are summarized in Table 1.


**Table 1** Comparison of various control schemes

### **4 Challenges**

There are still many challenges to achieve accurate trajectory tracking [24–27]. Ensuring good safety, comfort and stability in the control process is the goal of realizing intelligent driving technology [31].

### *4.1 Face the Challenges of a Complex Driving Environment*

The complex external driving environment is full of multi-layered challenges, including the perception of the complex environment, high-precision position positioning, dynamic obstacle handling, and strict traffic law compliance. Perceive road traffic environment information in advance to detect possible traffic conflicts [32, 33], so as to avoid potential accidents or keep driving on the predetermined path. Therefore, smart cars must be equipped with advanced sensing systems to detect and predict these external disturbances in real time and take appropriate actions. Therefore, the accuracy and robustness of environmental perception are critical to ensuring safe driving [30].

# *4.2 Overcoming the Problem of Vehicle Incompleteness and Multiple Constraints and Coping with the Time-Varying of Vehicle Dynamics Model*

Vehicles often exhibit non-integrity characteristics, which means that they are limited by degrees of freedom in their movement, and in addition, vehicles are constrained by multiple physical constraints, including maximum rotation angles, maximum acceleration, and so on. The existence of these non-integrity characteristics and constraints makes the vehicle control problem more complex. At the same time, road and vehicle dynamics models often exhibit nonlinear characteristics, which are difficult to be accurately described by simple mathematical equations. Therefore, to achieve efficient trajectory tracking control, it is necessary to consider non-holonomic characteristics, physical constraints, and uncertainties, which is also an important challenge in the field of autonomous driving.

# *4.3 Balance Between Vehicle Model Accuracy and Computational Efficiency*

To achieve high precision trajectory tracking, the control algorithm needs to adopt a more complex vehicle dynamics model. In the design of control algorithms, the use of more accurate vehicle dynamics models to achieve higher accuracy of trajectory tracking and better capture the actual behavior of vehicles often leads to the sacrifice of computational efficiency, which means that a reasonable trade-off point needs to be found between model complexity, computational resources and performance to ensure that the control system can operate reliably under real-time requirements. This is a critical issue that needs to be continuously optimized and adjusted to meet the needs of different application scenarios.

### **5 Conclusion**

In view of real driving conditions, from the perspective of control, various necessary constraints should be considered. For example, increasing the control freedom conditions such as direct yaw moment control can improve the lateral response speed of the vehicle under extreme working conditions and expand the control margin. Meanwhile, most trajectory control studies focus on normal driving scenarios and should be expanded from conventional working conditions to extreme working conditions. Due to the high-speed, highly nonlinear and time-varying characteristics of emergency driving scenarios, there are few researches on motion control under extreme conditions, and most of them stay in the simulation verification and typical working conditions test verification stage. The applicability and robustness of the algorithm under complex working conditions are still not sufficiently verified. Meanwhile, the research focuses on multi-objective, dynamic and multi-space scenarios.

**Acknowledgements** This work was supported by the National Natural Science Foundation of China (Grant No. 52172399); the Scientific Research Fund of Hunan Provincial Education Department (Project No. 21A0193); Changsha Natural Science Foundation Project (Project No. KQ2208235).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Accelerated Degradation Test and Reliability Evaluation Method for Hydraulic Pumps Based on Parallel Energy Saving**

#### **Jianjun Qi, Shaojia Chen, Dongfeng Wang, Liqin Wang, and Rui Guo**

**Abstract** Hydraulic pump is a product with high reliability and long-life, there exist many urgent problems like long test period, high cost and a larger power consumption in the reliability evaluation method of pump. A new kind of reliability test bench is built combined with parallel power-saving design concept, which based on principles of energy saving and power recycle. Multi-pumps reliability accelerated degradation test is achieved by applying step stress on the pump under test. Volumetric efficiency is selected as performance degradation evaluating index, the degradation of whose is much closed to real physics model. The accretion model for hydraulic pump reliability test is built based on inverse moment estimation method, by which hydraulic pump reliability level under different stress conditions is given. In the meantime, the method of concurrent accelerated degradation testing can serve as references of other core hydraulic components.

**Keywords** Hydraulic pump · Reliability evaluation method · Reliability accelerated degradation test · Concurrent · Volumetric efficiency

### **1 Introduction**

As the key basic parts of the equipment manufacturing industry, the hydraulic pump is known for its high reliability and long lifespan, among them, especially its reliability plays a decisive role in the reliability of host products. It is necessary to conduct the reliability test to evaluate, improve, and enhance the reliability of the hydraulic pump.

Beijing Special Engineering Design Institution, Beijing 100036, China

J. Qi · D. Wang · L. Wang

S. Chen · R. Guo (B) Yanshan University, Qinhuangdao 066004, China e-mail: guorui@ysu.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_42

At present, hydraulic reliability life tests in developed countries (such as Sundstron in the United States [1], Nippon Steel [2], UK National Laboratory, Korea Institute of Metallurgy and Mechanical Engineering, French Center for Mechanical Industry Research (ETIT), etc.) usually adopt longer cycle full life tests.

There is a lager gap between the whole life test method of foreign countries and the condition of our country, and the accelerated experimental method is favored. Among them, accelerated degradation test has absolute advantages in solving the evaluation problem of high reliability and long life [3]. Hydraulic pump is a typical system with degraded performance, and performance degradation data actually contain a large amount of life information. The performance reliability analysis of plunger pump was carried out with plunger clearance and outlet valve opening as performance criteria, and it was pointed out that the flow characteristics of plunger pump were affected by the degradation of component design parameters [4]. The oil return flow rate was used as a sensitive signal to reflect the wear condition of the aircraft hydraulic pump, and its wear condition was predicted [5]. Beijing Aeronautical Engineering and Technology Research Center has well predicted the remaining life of a hydraulic pump by analyzing and studying the failure mode and degradation parameters of a certain type of hydraulic pump [6]. The Second Artillery Institute of Engineering used the trained SRELM model to extrapolate the vibration signal of the hydraulic pump and predict the health status of the hydraulic pump [7]. On the basis of comprehensive consideration of numerous parameters (vibration, pressure, flow, temperature, oil information, etc.) affecting the life of hydraulic pump and their correlation, vibration signals are used to predict the life of hydraulic pump [8].

Based on the current hydraulic pump reliability accelerated degradation test method, on the premise of shortening the test cycle and reducing the test energy consumption, a new type of parallel energy-saving reliability test method of multiple hydraulic pumps is proposed in this paper, the parallel energy-saving accelerated test bench is built, and the in-depth theoretical and practical research work is carried out.

### **2 Hydraulic Pump Reliability Test**

### *2.1 Selection of Test Conditions*

Load pressure, medium temperature, higher speed and medium pollution will affect the service life of hydraulic pump, according to the principle of the choice of test conditions, and combining with the use of the hydraulic pump and field test conditions, choose hydraulic pump load pressure and rotation speed as the accelerated stress, control the medium temperature and cleanness of the oil in the process of test in the test within the prescribed scope of the standard. The reliability test object of this paper is A4VS series axial piston pump produced by a domestic enterprise. The rated pressure of this series of products is 35 MPa, and the peak pressure is 40 MPa. The four selected samples cover the open-loop hydraulic pump with a displacement of


**Table 1** Standard for failure detection and judgment of axial piston pump

**Table 2** 30 MPa pressure condition hydraulic pump life parameters


**Table 3** 20 MPa pressure condition hydraulic pump life parameters


250 ml/r, the closed-loop hydraulic pump with a displacement of 250 ml/r, the openloop hydraulic pump with a displacement of 125 ml/r, and the closed-loop hydraulic pump with a displacement of 125 ml/r. Failure detection and determination.

In this paper, volumetric efficiency is selected as the main performance monitoring parameter. In addition, outlet pressure pulsation and sealing performance of dynamic seals are also tested during the test. The failure criteria are shown in Table 1.

### *2.2 Parallel Reliability Testing System and Scheme Design*

According to the reliability life requirements of hydraulic pumps, a multi machine parallel energy-saving reliability testing device for hydraulic pumps has been developed. The hydraulic system schematic diagram is shown in Fig. 1. Compared with the conventional hydraulic pump reliability testing device, this testing device and testing plan adopt a parallel design concept. Four A4V series domestic hydraulic piston pumps are tested simultaneously using a dual axis extension motor, with a sample size of four in a single test.

In the test sample, the first hydraulic pump is an open circuit hydraulic pump with a displacement of 250 ml/r, the second hydraulic pump is a closed circuit hydraulic pump with a displacement of 125 ml/r, the third hydraulic pump is an open circuit hydraulic pump with a displacement of 125 ml/r, and the fourth hydraulic pump is

**Fig. 1** Working principle of parallel reliability test bench for axial piston pump

a closed circuit hydraulic pump with a displacement of 250 ml/r. The experimental subjects are both large displacement and medium displacement hydraulic pumps, which are mainstream products of hydraulic axial piston pumps and include two forms of hydraulic pumps applied to open and closed circuits.

The experiment adopts the step-by-step stress acceleration method, with pressure as the acceleration factor. The test is conducted for 1030 h at a stress of 35 MPa in the first stage, and 846 h at a stress of 41 MPa in the second stage. Using the volumetric efficiency of the tested plunger pump as the performance degradation parameter and failure criterion, the final life data in terms of time was obtained.

Because axial piston motors and axial piston pumps are structurally reversible, the 2nd and 4th closed hydraulic piston pumps can be used as hydraulic motors. One end of the dual axis extension motor is connected to the No. 1 hydraulic pump through a torque tachometer, and it is dragged to continue rotating. The other end of the dual axis extension motor is connected to the No. 4 hydraulic pump; The pressure oil output from the 1st hydraulic pump drives the 2nd hydraulic pump. The 2nd hydraulic pump is connected to the torque tachometer as a hydraulic motor, and the torque tachometer is then connected to the 3rd hydraulic pump; The second hydraulic pump is used to drive the third hydraulic pump to rotate, the pressure oil output by the third hydraulic pump is used to drive the fourth hydraulic pump, and the fourth hydraulic pump serves as a hydraulic motor to drive the motor.

In the experimental system, while the No. 4 hydraulic pump is the subject, it completes the power recovery of the hydraulic system and uses the recovered energy

to drive the motor to rotate. On the one hand, it can reduce the input power of the motor and reduce electrical energy consumption; On the other hand, due to the extremely small flow rate passing through the loading valve during high-pressure operation, the heat generation of the hydraulic system is greatly reduced, which reduces the requirements for the heat dissipation system and also reduces the energy consumption of the heat dissipation system.

The total input power of the system is calculated from the collected input torque and speed of pump 1. The recovered power is calculated from the collected output flow and pressure of pump 4. The lost power can be obtained by subtracting the recovered power from the total input power of the system. After matching the flow of the whole test with the speed of the tested pump, adjust the displacement of No. 1 pump to V = 165 ml/r, the displacement of No. 4 pump to V = 111.6 ml/r, and the motor speed to n = 1500 r/min, where the total efficiency of No. 4 pump is n = 0.907. When the test pressure is 35MPa, the measured input torque T = 849.3Nm, and the power distribution diagram of the test is shown in Fig. 2.

### **3 Hydraulic Pump Reliability Test Data Processing**

Combined with the time conversion equations, and substituting the failure data into *tj* (*b*), the estimated value of shape parameter *m* ˆ = 2.0138, the estimated value of characteristic life ηˆ = 4657.9, the average life of 4127.5 h, and the median life of 3882.8 h can be calculated.

By putting the estimated values of the parameters into Equation, the probability density function of the hydraulic pump can be obtained as:

$$f(t) = \frac{2.0138}{4657.9} \left(\frac{t}{4657.9}\right)^{1.0138} e^{-\left(\frac{t}{4657.9}\right)^{2.0138}}\tag{1}$$

The probability density function curve is shown in Fig. 3b.

**Fig. 3** 35 MPa reliability index function curve

Putting the estimated values of the parameters into Eq. (2) and setting γ = 0, the distribution function of the hydraulic pump life can be written as:

$$F(t) = 1 - e^{-(\frac{t}{46S7.9})^{2.0138}} \tag{2}$$

The life distribution function curve is described in Fig. 3a.

Putting the estimated values of the parameters into Eq. (3) and setting γ = 0, the reliability function of the hydraulic pump can be obtained as:

$$R(t) = e^{-(\frac{t}{46S7.9})^{2.0138}}\tag{3}$$

The reliability function curve is shown in Fig. 3a.

Putting the estimated values of the parameters into Eq. (4) and setting γ = 0, the failure rate function of the hydraulic pump can be obtained as:

$$
\lambda(t) = \frac{2.0138}{4657.9^{2.0138}} t^{1.0138} \tag{4}
$$

The failure rate function curve is shown in Fig. 3b. Calculate the average life of hydraulic pump:

$$E(t) = \int\_0^\infty t f(t) \mathrm{d}t = \hat{\eta} \Gamma(1 + \frac{1}{\hat{m}}) = 4127.5 \mathrm{h} \tag{5}$$

In addition, according to the acceleration model equation, the reliability index of the hydraulic pump under the pressure condition of 30 and 20 MPa can be obtained, as shown in Tables 2 and 3. The reliability function, distribution function, probability density function and failure rate function curves are shown in Figs. 4 and 5.

functions

rate function

**Fig. 4** 30 MPa reliability index function curve

functions

rate function

**Fig. 5** 20 MPa reliability index function curve

### **4 Conclusion**


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Analysis and Experiment Study About Vibration Compaction Based on Drucker-Prager Model**

**Liying Ma, Macheng Gui, Shaoxiong Gui, and Yuanwen Cao** 

**Abstract** The relationship and the applicable conditions between Mohr–Coulomb model and linear Drucker-Prager model were discussed, and the results showed that when the friction angle was less than 22° Drucker-Prager model was more suitable for modelling the soil unit; When the friction angle was more than 22°, the Mohr–Coulomb model should be used. In order to further study the compaction characteristics of vibratory roller and the dynamic relationship between vibratory roller and soil, the finite element model of "vibrating roller-soil" was established. The simulation results showed that vertical stress of the soil distributed symmetrically along the axial direction of the vibrating drum, shift forward along the direction of the vibration drum, and decreased sharply with the increasement of the depth. Then experiments carried out showed that acceleration of the vibrating drum was positively correlated with the compaction times, which verified the model was basically correct. Moreover, the regression equation between the compaction degree and the effective value of acceleration was obtained, which provided the idea for the new compaction degree monitoring system.

**Keywords** Compaction degree · Drucker-prager model · Stress distribution · Acceleration · Regression equation

L. Ma (B) · Y. Cao

School of Mechantronics and Vehicle Engineering, Chongqing Jiaotong University, Chongqing 400074, P. R. China e-mail: 623663698@qq.com

M. Gui

S. Gui

© The Author(s) 2024

Hanhong College, Southwest University, Chongqing 400715, P. R. China

College of Traffic and Transportation, Chongqing Jiaotong University, Chongqing 400074, P. R. China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_43

### **1 Introduction**

Road traffic plays an important role in regional economic growth. In the inspection items of roadbed, subbase and asphalt layer, the weight rating of compaction degree was the highest [1]. It can be seen that the quality of the highway largely depended on the compaction effect. Traditional compaction methods detected a few sections of road after compaction, which is poor representation, time-consuming and destructive structure. With the improvement of construction efficiency, the traditional testing method can not meet the requirements of real-time construction guidance [2]. The development of real-time, accurate and low-cost new compaction monitoring technology has become a research hotspot. Therefore, further study on compaction characteristics of vibratory roller and dynamic response relationship between vibratory drum and soil can also provide a theoretical basis for new compaction detection technology.

Over the years, the interaction between vibrating drum and soil has been studied extensively. Sheng Qiu simulated and analyzed the process of vibration compaction, which showed that the nonlinear response of vibration compaction process can be well simulated by the finite element model of vibration drum-soil [3]. Fu Huang established the dynamics model of vibration pressure system, and obtained the relation between the vibration drum acceleration and the stiffness coefficient and damping coefficient of pavement material [4]. The effect of the sliding rate of steel wheel, the compacting times, the friction coefficient of the steel wheel and the soil and the wheel weight on the deformation of the sticky soil were studied by Maciejewski and Jarzebowski [5]. The results showed that the quality of the steel wheel was the most important factor affecting the degree of soil compaction.

### **2 Constitutive Model of Soil**

### *2.1 Yielding Criterion*

The common yield criterion of geotechnical materials was Mohr–Coulomb yield criterion, whose governing equation was illustrated in Eq. (1) as follows:

$$\frac{\sigma\_1 - \sigma\_3}{2} + \frac{(\sigma\_1 + \sigma\_3)\sin\phi}{2} - c\cos\phi = 0\tag{1}$$

where σ1, σ2 and σ3 the first, second and third principal stresses, respectively; *c* = cohesion; and ϕ = the internal friction angle.

The Mohr–Coulomb criterion, however, didn't reflect the effect of intermediate principal stress on yield and failure. Moreover, the yield surface of Mohr–Coulomb criterion was hexagonal conical, and the direction of plastic strain increment was not unique at corner, which made it difficult to converge in plastic analysis and was not good for numerical calculation.

Drucker and Prager constructed the classical Drucker-Prager model to solve the problems above. It was the conical yield surface of the hexagonal cone which was tangent or internally tangent to the Mohr–Coulomb criterion, and the D-P yield criterion was proposed [6, 7]. The governing equation of the yield criterion was illustrated in Eq. (2) as follows:

$$D\_1 I\_1 + \sqrt{J\_2} - D\_2 = 0 \tag{2}$$

$$I\_1 = \sigma\_1 + \sigma\_2 + \sigma\_3 \tag{3}$$

$$J\_2 = \frac{1}{6} \left[ \left( \sigma\_1 - \sigma\_2 \right)^2 + \left( \sigma\_2 - \sigma\_3 \right)^2 + \left( \sigma\_1 - \sigma\_3 \right)^2 \right] \tag{4}$$

$$D\_1 = \frac{2\sin\phi}{\sqrt{3}(3-\sin\phi)}\tag{5}$$

$$D\_2 = \frac{6c\cos\phi}{\sqrt{3}(3-\sin\phi)}\tag{6}$$

where I1 = the first invariant of the stress tensor; J2 = the second invariants of the stress bias; and the definition of the other parameters are the same as in Eq. (1).

### *2.2 The Selection of Constitutive Model*

Figure 1 shows yielded curve of different yielded criterion on the π plane. A large number of constitutive models for geotechnical materials were provided in ABAQUS. The yield line of the Mohr–Coulomb plastic model, the classical Drucker-Prager plasticity model, and the extended Drucker-Prager plasticity model on the partial plane.

Through calculation and simulation, authors analyzed the relationship and the applicable conditions between the Mohr–Coulomb model and linear Drucker-Prager model.

As Fig. 1 shows, with the same definition of tension and compression failure for the two models, the linear Drucker-Prager model can be re-expressed as another form [8, 9], which was shown under the condition of triaxial compression and triaxial tensile illustrated as follows: Under the condition of triaxial compression,

$$(\sigma\_1 - \sigma\_3 + \frac{\tan \beta}{2 + \frac{1}{3} \tan \beta} (\sigma\_1 + \sigma\_3) - \frac{1 - \frac{1}{3} \tan \beta}{1 + \frac{1}{6} \tan \beta} \sigma\_c^0 = 0 \tag{7}$$

where β = the friction angle in Drucker-Prager model, and σ<sup>0</sup> <sup>c</sup>= yield stress. Under the condition of triaxial tensile,

$$(\sigma\_1 - \sigma\_3 + \frac{\tan \beta}{\frac{2}{K} - \frac{1}{3} \tan \beta} (\sigma\_1 + \sigma\_3) - \frac{1 - \frac{1}{3} \tan \beta}{\frac{1}{K} - \frac{1}{6} \tan \beta} \sigma\_c^0 = 0 \tag{8}$$

For all (σ1, σ3), in order to make Eqs. (7) and (8) be consistent with the control equation of the Mohr–Coulomb model (1), Coefficient K was described in Eqs. (9), (10), (11), and (12) as follows:

$$K = \frac{1}{1 + \frac{1}{3}\tan\beta} \tag{9}$$

$$\tan \beta = \frac{6 \sin \phi}{3 - \sin \phi} \tag{10}$$

$$
\sigma\_c^0 = 2c \frac{\cos \phi}{1 - \sin \phi} \tag{11}
$$

According to the analysis and calculation above, the results can be obtained.

$$K = \frac{3 - \sin\phi}{3 + \sin\phi} \tag{12}$$

Where K is the ratio of triaxial tensile strength to triaxial compression strength.

When K = 1, the yield surface is Von Mises circle in the deflection stress plane, and the yield stress of triaxial tension and compression is the same. In order to guarantee the yield surface protruding, it is required that K ≥ 0.778. According to Eq. (12), it is required ϕ ≤ 22°. That is to say, when ϕ ≤ 22°, the two models can be well fitted. At this time, the Drucker-Prager model is applied to analyze and calculate. When ϕ ≥ 22°, the fitting effect of the two models is not good, so it is not suitable for parameter conversion. Mohr–Coulomb plastic model should be adopted directly at this time.

The soil parameters of the Mohr–Coulomb model can be measured by different conventional experiments, which is convenient and practical. The Drucker-Prager model overcomes the disadvantages of Mohr–Coulomb model, but its model parameters need to be converted from Mohr–Coulomb model. In a comprehensive view, the Drucker-Prager model is more suitable for the soil unit in this paper, but it should satisfy the condition that the friction angle is less than or equal to 22°. When the friction angle is greater than 22°, the Mohr–Coulomb model should be more useful.

### **3 The Establishment of a Finite Element Model**

### *3.1 The Determination of Size Parameters*

The principle of vibration roller is that the centrifugal force is driven by the hydraulic device to drive the eccentric block in the steel wheel to rotate at high speed, and the centrifugal force compels the steel wheel to vibrate to achieve the purpose of compaction [10]. In finite-element analysis, complex structure of vibratory drum did not only lead to modeling difficulties, but also increased the amount of calculation and was not easy to converge, so the model should be simplified.

In the simplified model, the quality of the front wheel distribution of the vibratory roller was directly reflected by the vibration drum, and the excitation force acted directly on the center of mass. Considering the analysis time of the model, the running speed of the vibratory roller and the geometric parameters of the actual compacted soil, the finite-element model size of the soil in this paper was taken as follows: 20 m\*8 m\*0.8 m. This paper chose the small vibratory roller of XD82E of Xugong.

### *3.2 Determination of Material Parameters*

Based on the project of Jiangxi Provincial Communications Department, the parameters of soil under different compaction times in a compaction operation were determined by combining experimental research with experience. The main parameters of the soil were shown in Table 1.

It is found that the soil has elastoplastic deformation at the beginning of compaction. With the soil compacted, the plastic deformation of the soil was decreased gradually. When the soil is completely compacted, it is only elastic deformation. Therefore, the soil is elastic–plastic model, the elastic part of the soil is defined by the linear elastic model, and the plastic part linear Drucker-Prager plastic model.


**Table 1** The main parameters of soil

### *3.3 Division of Units*

The unit partition directly affects the speed, the convergence, and the accuracy of analysis and calculation. The soil was composed of 120,000 C3D8R units with reduced integral. Vibration drum adopts C3D4 element, with a total of 2000 units. The soil belongs to semi space infinite body, but the finite-element model can only calculate the finite problem. In modeling, the soil is truncated so that the pressure wave can not propagate to infinity, but the truncation is reflected back, which affects the accuracy of the calculation. In order to reduce the influence of truncating infinite space, a higher precision viscous artificial boundary is applied at the truncation point. The "vibratory drum-soil" model was shown in Fig. 2.

### **4 Simulation Analysis of Finite-Element Model**

# *4.1 Specificity Analysis of Soil Stress Distribution Under Vibration Drum*

According to the finite-element calculation of the established model parameters and material parameters, the cloud chart of soil stress distribution was obtained shown in Fig. 3.

As Fig. 3 shown, the region with larger vertical stress was similar to the ellipsoid extending from the vibration drum axis to the downward direction, and along the direction of the ellipsoid, the vertical stress of the soil becomes smaller and smaller

**Fig. 3** Cloud chart of vertical stress distribution characteristics along the width and moving direction of the vibration drum

from the contact surface of the vibration wheel to the soil. The maximum vertical stress of the soil appears on the vertical line of the vibrating drum shaft. The vertical stress is symmetrically distributed along the width of the vibration drum, and the vertical stress is asymmetrical along the direction of the vibration drum, and it is larger in the direction of the drum moving forward.

# *4.2 Influence of Soil Parameters on Vertical Acceleration of Vibrating Drum*

The parameters of the vibration drum were consistent. Finite-element analysis of soil I, soil II, and soil III was carried out, respectively, which obtained the acceleration curve of the result shown in Fig. 4.

As Fig. 4 shown, the parameters of the vibration drum were consistent. Finiteelement analysis of soil I, soil II, and soil III was carried out, respectively, which obtained the acceleration curve of the result showed that the vibration wheel produced a large positive and negative acceleration at the beginning of vibration, and then the vibration acceleration increased gradually, and then tended to be relatively stable. With the change of soil parameters, the amplitude of vertical acceleration of vibration drum changed, and the amplitude of vertical acceleration was positively related to compaction times. There was obvious irregularity in the acceleration curve corresponding to the soil III, which inferred that the soil compaction degree has reached the optimum value of this condition due to the increasement of compaction times, and the vibration wheel became irregularly.

### **5 Test Verification**

In order to verify the correctness of the finite element model, the experiments were carried out. The main items of test were: working speed of roller, vertical acceleration of vibration drum, and compaction degree of roadbed.

**Fig. 4** Vertical acceleration curve of drum

Because the period of vibration acceleration, usually 0.02 to 0.05 s, was short, and the speed of the roller was slow, therefore, 20 periodic signals were taken as an testing object period in this paper. After filtering the vibration acceleration signal and eliminating the trend item, the effective acceleration value of 20 period signals was calculated, and then the average value of the acceleration effective value in one compaction operation was taken as the acceleration value of this compaction. After each compaction, the compaction degree of roadbed was measured by sand filling method. The relationship between acceleration, compactness and compaction times was shown in Fig. 5, and the relation between acceleration and compaction in Fig. 6.

As Fig. 5 shown, the vertical acceleration of the vibration drum was positively correlated with compacted times, which was basically consistent with the conclusion of the model. The compaction degree also increased by the increase of compaction

times. With the further analysis of acceleration and compaction degree, Eq. (13) can be obtained by the linear fitting of points in Fig. 6.

$$\mathbf{y} = 0.8665\mathbf{x} + 61.339, \quad \mathbf{R}^2 = 0.9684\tag{13}$$

where y = compaction degree, x = acceleration, and R2 = correlation coefficient.

The correlation coefficient of Eq. (13) was 0.9684, which indicates that the correlation between the vertical vibration acceleration of the vibration drum and the compaction degree were well correlated. Therefore, the acceleration were monitored, and compactness value was obtained by using correlation formula between the acceleration of vibration wheel and compaction degree.

### **6 Conclusion**


**Acknowledgements** The authors are grateful to Dr. Naixing Liang for his help with the preparation of projects in this paper. This paper is supported by the Project of Henan Provincial Department of Transportation: Hnxrc2018001 and the project of Department of Transport of Yunnan Province: 2017 (A) 15. The authors would like to thank the associate editor and the reviewers for their useful feedback that improved this paper.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Visual Detection Method for Train Couplers Based on YOLOv8 Model**

**Wenning Zhao, Xin Yao, Bixin Wang, Jiayi Ding, Jialu Li, Xiong Zhang, Shuting Wan, Jingyi Zhao, Rui Guo, and Wei Cai** 

**Abstract** Accurately identifying the coupler operating handle during the operation of the hook-picking robot has a significant impact on production activities. This article is based on the YOLOv8 model. Due to the limited variety of on-site coupler operating handles and working environment, it is difficult to ensure the richness of image categories in the dataset. Before the experiment, a series of expansion operations were performed on the dataset, such as rotation, translation, and brightness adjustment. Use the expanded images to simulate the images detected by the hookpicking robot in harsh environments. This model performs feature extraction and target recognition on the expanded coupler handle dataset, thereby achieving recognition of the coupler handle. The experimental results show that the accuracy of the model for the coupler handle in complex environments is 98.8%, which effectively reduces the time required for training and testing. Compared with the commonly used SSD300 model and YOLOv4Tiny model, it not only has higher accuracy, but also shows obvious advantages in parameter quantity, weight file size, and other aspects, which can be well deployed in actual production.

**Keywords** Deep learning · Target identification · YOLOv8n · Coupler handle

W. Zhao · B. Wang

CHN Energy Hebei Dingzhou Power Generation Co., Ltd, Dingzhou 073000, Hebei, China

X. Yao Beijing Huily Intelligent Technology Co., Ltd, Beijing 100070, China

J. Ding · J. Li · X. Zhang (B) · S. Wan

Department of Mechanical Engineering, North China Electric Power University, Baoding 071003, China e-mail: hdjxzx@ncepu.edu.cn

J. Zhao · R. Guo · W. Cai School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China

© The Author(s) 2024

561

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_44

### **1 Introduction**

Dumper is a frequently used equipment in China's coal transportation industry. It fixes and rotates a carriage of a train to a certain angle, dumping all the coal inside. Because each carriage of the train is connected by a coupler, it is particularly important to promptly and effectively remove the coupler before the tippler works. When the train coupler is removed, there are two types of operating handles: up and down, and the removal methods for different handles are also different. The extraction of train couplers was initially carried out manually, but due to issues such as low manual operation efficiency and high operational risks. The ability to accurately and quickly locate the coupler operating rod and accurately identify the type of operating rod directly affects the quality of decoupling and operational efficiency. Compared to manual operations, this robot has strong adaptability and high work efficiency. Therefore, it is of great significance to develop a new type of fully automatic decoupling robot for target detection of the coupler control lever.

With the advancement of technology, computer vision technology has been widely applied in moving object detection and tracking and has gradually affected various aspects of our lives [1–3]. Its importance is becoming increasingly prominent, attracting more and more scholars and research institutions from home and abroad to participate in research in this field. Moving object detection and tracking based on computer vision have wide applications in multiple fields. It achieves public safety monitoring and management, accident prevention, detection and processing, and other important functions by analyzing moving objects in videos or images. This technology has also been applied in fields such as virtual reality, human–computer interaction, planetary exploration, and behavioral understanding. In addition, it also plays an important role in monitoring the elderly, young, sick, and disabled, as well as autonomous navigation. The method of object detection is mainly divided into two stages: the traditional object detection stage (HOG [4] and SIFT [5]), and the deep learning-based object detection stage (Faster R-CNN [6], SSD [7] and YOLO [8]). The traditional target detection method uses an exhaustive method to determine the target position, which not only takes a long time but also has a high error detection rate in practical applications. Therefore, it is gradually replaced by deep learning methods. There are two main types of object detection methods based on deep learning. One type is to first form a series of candidate boxes for feature extraction and target position judgment, known as the Two-stage algorithm, The other type does not generate candidate boxes separately but completes the selection of target positions and feature extraction together to achieve end-to-end target detection tasks, known as the One stage algorithm. YOLO is one of the most commonly used technologies in recent years, which facilitates user usage through continuous updates (YOLOv4 [9], YOLOv5 [10], and YOLOv7 [11]). Jun [9] proposed a small object detection algorithm based on YOLOv4, multi-scale contextual information, and Soft-CIOU loss function, called MCS-YOLOv4. This method adds scale detection to the existing three scales to obtain rich location information. An attention module has been introduced into the neck of YOLOv4 to reduce the impact of irrelevant information on small objects in the image. In addition, a Soft-CIOU loss function has been proposed to further improve the detection accuracy of small objects. This method achieved better detection performance on publicly available small object datasets. Arifando [10] proposed the Improved YOLOv5 method, which integrates GhostConv and C3Ghost modules into the YOLOv5 network to reduce model parameters while maintaining detection accuracy. At the same time, we use the SimSPPF module to replace the SPPF in the YOLOv5 backbone to improve computational efficiency and accurate object detection capabilities. Through comparative analysis, we found that the Improved YOLOv5 method has smaller FLOPS and parameter count, uses less memory, and has faster inference time, thus achieving higher efficiency. Yanyun [11] proposed an improved detector called YOLOv7 sea, which is an improvement on the original model. This improvement adds a prediction head to enable the model to better detect small people or objects. In addition, YOLOv7 sea also integrates a simple parameterless attention module (SimAM) for searching for regions of interest in the scene. To further improve the performance of the model, researchers have adopted methods such as data augmentation, test time augmentation (TTA), and bundling box fusion (WBF) for improvement. The calculation results show that the improved YOLOv7 sea model has improved performance and performs better than the original model. Susu [12] proposed an improved SSD object detection algorithm called DF-SSD, which is based on DenseNet and feature fusion technology. We will replace the original backbone network VGG-16 with DenseNet-S-32-1 to enhance the feature extraction capability of the model. In the multi-scale detection section, we introduce a multi-scale feature layer fusion mechanism to organically combine low-level visual features and high-level semantic features. In addition, we also introduced residual blocks before target prediction to further improve model performance. The experimental results indicate that our improved model has higher accuracy in detecting small objects and objects with specific relationships. Zhang [13] proposed an object detection model called MA ResNet, which is based on the design of a multi-attention residual network. Specifically, they introduced spatial attention, channel attention, and self-attention mechanisms into the residual network, enabling the network to more accurately focus on the target region and extract key features. It is worth noting that they replaced the original VGG-16 feature extractor in Faster RCNN with MA ResNet. In the experiment, it was found that MA ResNet showed significant advantages in convergence speed, accuracy, and small target classification accuracy. This proves the excellent performance of MA ResNet in feature extraction, which helps to improve the performance and accuracy of object detection. Therefore, MA ResNet based on multi-attention residual networks can be regarded as an effective feature extraction model and exhibits superior performance in object detection tasks [14].

The object detection method has been widely applied in various fields, but due to the difficulty in capturing the dataset of the coupler operating handle, it has been less applied in the field of train uncoupling. In response to this issue, this article constructs a dataset of train coupler operating handles and expands the data. Simulate and analyze the complex visual situation during the hook-picking process using an expanded dataset. The target detection method using the YOLOV8n model was used for recognition and achieved a detection effect of 98.8%, demonstrating its practical engineering value [15].

### **2 YOLOv8 Model**

YOLOv8, as the latest YOLO model, can be used for object detection, image classification, and instance segmentation tasks. This article is intended for object detection tasks. Based on the consideration of model size, this article selects the YOLOv8n network with small volume and high accuracy. The YOLOv8n model detection network mainly consists of three parts (as shown in Fig. 1): Backbone, Neck, and Head.

The backbone is mainly used for feature extraction, which includes modules such as Conv, C2f, SPPF, etc. The main function of the Conv module is to perform convolution, BN, and SiLU activation function operations on the input image. YOLOv8n has designed a brand new C2f structure, which is the main module for learning residual features. Its structure allows YOLOv8n to have rich gradient flow information while ensuring lightweight, The SPPF module, also known as spatial pyramid pooling, can convert feature maps of any size into fixed-size feature vectors.

The main function of the Neck network is to fuse multi-scale features and generate feature pyramids. The Neck part adopts a PANet structure, and its core structure consists of two parts: feature pyramid network FPN and path aggregation network PAN. FPN first extracts feature maps from convolutional neural networks to construct feature pyramids, and then uses upsampling and coarser-grained feature maps fusion

**Fig. 1** YOLOv8 Model structure diagram

from top to bottom to achieve the fusion of features at different levels. However, if only FPN is used, it will lack the position information of the target. PAN is a supplement to FPN, using a bottom-up structure to fuse feature maps from different levels through the use of a convolutional layer, accurately preserving spatial information. The combination of FPN and PAN fully achieves the fusion of upstream and downstream information flows in the network, improving the detection performance of the network.

As the final prediction part, the Head output terminal obtains the category and position information of target objects of different sizes based on feature maps of different sizes [16].

### **3 Basic Theory**

### *3.1 Loss Function*

YOLOv8 is an improved version of the YOLO series of object detection algorithms. In YOLOv8, the loss function consists of multiple parts, including classification loss (VFL Loss) and regression loss (CIOU Loss and DFL) [17]. The classification loss (VFL Loss) is used to measure the accuracy of object categories in the prediction box. VFL stands for Variable Focal Loss and is an improvement on Focal Loss. It solves the problem of sample imbalance by adjusting the weight of positive and negative samples, making the model more focused on difficult-to-classify samples. VFL Loss can be used to measure the error of classification prediction results.

Regression loss consists of two parts: CIOU Loss and DFL. CIOU Loss stands for Complete Intersection over Union Loss, which is a regression loss used to measure the accuracy of prediction box bounding boxes. CIOU Loss not only considers the position offset between the predicted box and the real box but also considers the shape similarity between them, which can better measure the matching degree of the bounding box. This can help improve the accuracy of object detection.

DFL stands for Distribution Focal Loss, which is mainly used to enhance the detection ability of small targets. DFL utilizes the scale distribution information of the target to adjust the weight of positive and negative samples, making the model more focused on detecting small targets. DFL Loss can help improve the detection performance of the model on small targets.

Overall, the loss function of YOLOv8 consists of classification loss (VFL Loss) and regression loss (CIOU Loss and DFL), which are used to measure the accuracy of classification prediction and bounding box regression, respectively. The introduction of these loss functions can help improve the performance of the YOLOv8 model in object detection tasks.

### *3.2 Evaluation Indicators*

In the process of evaluating deep learning models, accuracy (A), accuracy (P), recall (R), and F1 values can be used to evaluate the model [18].

(1) Accuracy is the proportion of correctly classified samples to the total number of samples, calculated as follows Eq. (1).

$$A = \frac{TP + TN}{TP + TN + FP + FN} \tag{1}$$

(2) The accuracy rate represents the proportion of true positive cases in the samples predicted as positive cases, as shown in Eq. (2).

$$P = \frac{TP}{TP + FP} \tag{2}$$

(3) Recall rate represents the proportion of samples predicted to be positive in a true positive example, as shown in Eq. (3).

$$R = \frac{TP}{TP + FN} \tag{3}$$

In the formula: TP represents that the predicted value is 1, the actual value is 1, and the predicted value is correct, which is the true example; FP indicates that the prediction is 1, the actual value is 0, and the prediction is incorrect, which is a false negative example; FN indicates that the prediction is 0, the actual value is 1, and the prediction is incorrect, which is a true negative example [19].

In summary, accuracy represents the percentage of correctly predicted results in the total sample; Accuracy represents the degree to which the average measured value matches the true value, and is used to represent the magnitude of system error; The recall rate is used to evaluate the quality of results, representing the proportion of correctly classified results.

The analysis of experimental results in this article will comprehensively consider the above indicators to evaluate the predicted results.

**Fig. 2** Original coupler unhook operating handle, **a** Top acting handle. **b** Down acting handle

### **4 Experiment Validation**

### *4.1 Experimental Datasets*

There are very few datasets on the uncoupling operation handle of the coupler in object detection, making it difficult to verify with public datasets. Therefore, this experiment used a dataset of self-built coupler operating handles taken from the onsite photos of the automatic uncoupling robot. The handle types are mainly divided into two types: upper-acting and lower-acting, as shown in Fig. 2. However, due to the limited variety of operating handles and working environment for on-site coupler uncoupling, it is difficult to ensure the richness of image categories in the dataset. Before experimenting, this article first expanded the dataset, including a series of operations such as rotation, translation, and brightness adjustment, as shown in Fig. 3. Through data expansion, the diversity of experimental data has been effectively increased, and the recognition and localization ability of experimental algorithms for target objects in complex visual situations can be verified. Annotate the expanded dataset and divide the annotated images and label files into training, validation, and testing sets, with a ratio of 8:1:1. The results obtained during the training validation process are shown in Fig. 4.

### *4.2 Practice and Result Analysis*

The training results are shown in Fig. 5. The abscissa in the figure represents the number of training rounds (epochs). From the figure, it can be seen that the loss rate curve of model training and validation gradually decreases with the increase of training times, especially in the first 50 rounds where the descent rate is the fastest, and the gradient decreases as the model gradually stabilizes, The four indicators of accuracy, recall rate, mAP50, and mAP50-95 are opposite, increasing with the

$$\mathbf{(a)}$$

(a) (b)

(c) (d)

(e)

**Fig. 3** Expansion of handle image data, **a** Rotation processing. **b** Translation processing. **c** Brightness adjustment processing. **d** Obfuscation. **e** Signal-to-noise ratio processing

**Fig. 4** Training validation process image, **a** Training process. **b** Verification process

increase of training times and eventually stabilizing, indicating that the training data is stable and normal [20].

After the model training is completed, the entire indicator results are shown in the table below. From Table 1, it can be seen that the average accuracy of the model is 0.996, the average recall rate is 0.986, the average value of mAP50 is 0.988, the average value of mAP50-95 is 0.943. The accuracy, recall rate, mAP50, and mAP50-95 values of the Down action handle and Top action handle categories can all achieve good accuracy results, which can be applied to actual coupler handle detection systems. To see its effect, make Table 1 as Fig. 6. In the following figure, 1 represents the overall model, and 2 represents the down-acting handle, 3 represents the top-acting handle.

### *4.3 Compare with Other Mainstream Algorithms*

Using the expanded dataset as samples, the algorithm proposed in this paper was compared with SSD300 and YOLOv4Tiny in experiments. The evaluation indicators were volume (MB), parameter quantity Params (M), computational quantity (GFLOPs), and mAP%. The results are shown in the table below.

According to Table 2, the YOLOv8n algorithm has the smallest number of parameters, volume, and computational complexity. Secondly, data augmentation techniques are used to simulate the images detected by the hook-picking robot in complex environments. The YOLOv8n method was used for object detection of the coupler handle in complex situations, and excellent results were achieved, fully meeting the accuracy requirements of object detection. In addition, its algorithm saves the smallest optimal weight volume, making it easier to deploy to resource-constrained platforms such as embedded or mobile devices. At the same time, a smaller volume can also achieve faster inference speed, thus completing prediction tasks faster. Therefore,

**Fig. 5** Training validation results, **a** Training process loss function. **b** Verification process loss function. **c** Model training metrics. **d** Model training metrics


**Table 1** Model evaluation indicators

compared to other models, the YOLOv8n model is more suitable for complex task scenarios with high timeliness requirements such as coupler handle detection.

**Fig. 6** Model evaluation indicators

**Table 2** Comparative experiments with other algorithms


### **5 Conclusions**

To accurately identify and locate the operating handle of the train automatic uncoupling robot, a method based on the YOLOv8n model for object detection of the coupler handle is proposed. This method uses the YOLOv8n model as the benchmark model for detection, ensuring the accuracy of the model for detecting simple targets while effectively reducing the number of model parameters. Due to the limited variety of on-site coupler operating handles and working environment, it is difficult to ensure the richness of the image categories in the dataset. Before experimenting, the dataset was expanded and processed, including a series of operations such as rotation, translation, and brightness adjustment to simulate complex visual environments. The YOLOv8n model was used to detect and analyze different types of coupler handles. The experimental results show that this method has excellent detection accuracy, reaching 98.8%. In addition, its algorithm saves the smallest optimal weight volume, making it easier to deploy to resource-constrained platforms such as embedded devices or mobile devices. At the same time, a smaller volume can also achieve faster inference speed, thus completing prediction tasks faster. Therefore, it has high engineering application value.

**Acknowledgements** This work is supported by National Natural Science Foundation of China (52105098), Natural Science Foundation of Hebei Province (E2021502038) and the Fundamental Research Funds for the Central Universities (2023MS130).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Dynamic Synovial Control Method of Four-Cable-Driven Parallel Robot Based on Interference Observer**

**Ke Zhang, Jiabao Xu, Shenghao Tong, Huaitao Shi, Zhiqiang Duo, and Cai He** 

**Abstract** Aiming at the problem of high-precision trajectory tracking of four-cabledriven parallel robot, a dynamic non-singular fast terminal sliding mode surface based on the estimation output of interference observer is proposed. Firstly, based on the combination of traditional fast terminal sliding mode control and non-singular terminal synovial control, the estimation value of interference observer is introduced, and a new nonlinear sliding mode surface is established to suppress the influence of unknown disturbance, and at the same time track the desired trajectory from unstable state to steady state. Based on the Lyapunov stability theory of the controller, the effectiveness of the control scheme is proved. Finally, in the Matlab/Simulink environment, a comparative simulation with the existing linear synovial film and nonsingular fast terminal sliding mode is carried out, and the designed control method realizes high-precision fast tracking of the desired trajectory of the end grab under the uncertainty of modeling error, external interference and joint friction, which verifies the effectiveness and feasibility of the proposed control method.

**Keywords** Four-cable drive parallel robot · Interfering observer · Synovial membrane control · Tracking control

# **1 Introduction**

With the development of construction machinery manufacturing industry, the characteristics of building types becoming more and more complex have become increasingly prominent. For the capture, identification and handling of target building materials in the construction environment, most of them are still carried out manually. As a typical parallel robot that realizes the movement of the end effector through

K. Zhang

575

K. Zhang · J. Xu · S. Tong (B) · H. Shi · Z. Duo · C. He

School of Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China e-mail: tongshenghao@sjzu.edu.cn

School of Mechanical Engineering, Shenyang University of Technology, Shenyang 110870, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_45

the retracting and releasing control of the flexible cable, the flexible rope parallel drive robot has many advantages such as large working space, small dynamic inertia and fast movement speed [1], and is widely used in many fields such as astronomical observation [2], field video [3], 3D printing [4] and so on. However, the soft rope is also exposed to the external disturbance caused by the flexibility, elasticity, nonlinearity and unidirectional constraint characteristics of the soft cable. Therefore, ensuring the high-precision trajectory tracking of the end effector of the robot driven in parallel has become an urgent problem to be solved.

Many scholars at home and abroad have conducted research on motion control of soft rope parallel robot. Yan-Lin Wang et al. [5] proposed a new controller consisting of an inner loop fuzzy sliding mode controller and an outer loop variable admittance controller to control the cable-driven lower limb rehabilitation robot. Ji, Y.F., et al. [6] established an eight-rope traction robot in a low-speed wind tunnel, and verified the phenomenon of aerodynamic retardation through experiment. The above literature adopts various control strategies for high-precision. Peng Liu et al. [7] proposed a robust adaptive fuzzy control strategy, which proved the reliability of trajectory tracking of the soft-cable-driven picking robot through Simula trajectory tracking of soft cable-driven robots. An, B., et al. [8] proposed an adaptive terminal sliding mode based on interference observer in the aircraft model, which combines the interference signal with the terminal synovial membrane to construct and eliminate the singularity in the system. Zhang, Q. W., et al. [9] proposed a non-singular terminal sliding mode control method for redundant drive parallel mechanism system, which realized highprecision trajectory tracking control. In this paper, based on the combination of traditional fast terminal sliding mode control and non-singular terminal synovial control, the estimation value of interference observer is introduced, a new nonlinear sliding mode surface is established to suppress the influence of unknown disturbance, and enable the system to track the expected trajectory from an unstable state to a stable state.

### **2 Kinematics and Dynamics Models of Rope-Driven Robots**

### *2.1 Rope-Driven Robot Kinematics*

In this paper, the principle of vector closure is used to establish the kinematic model of cable length space and end material grasping space of four-cable robot, and the end grab is regarded as an abstract point P(x, y, z). Establish a Cartesian coordinate system O-XYZ, OX horizontal direction, OY direction determined according to the right-hand rule, OZ perpendicular to the corresponding platform. a, b and h represent the spatial structure dimensions of the four-cable robot, respectively. The center of the pulley set in the coordinate system is represented as *Ai* (*xi* , *yi* ,*zi* ),i = 1,2,3,4. Then each length of the cord can be expressed as:

**Fig. 1** Schematic diagram of kinematics of a four-cable parallel robot

$$L\_i = \sqrt{(\mathbf{x} - \mathbf{x}\_i)^2 + (\mathbf{x} - \mathbf{y}\_i)^2 + (\mathbf{x} - \mathbf{z}\_i)^2} \\ (i = 1, 2, 3, 4) \tag{1}$$

The kinematics diagram of the four-cable driven robot was analyzed (see Fig. 1). Equation (1) describes the mapping relationship between the end grab space and the cable length space of the robot. In robot kinematics, the Jacobi matrix represents the linear mapping of the robot's operational space velocity and joint space velocity. In a cable-driven parallel robot, it can also be expressed between the running speed of the end grab and the speed at which the cord changes, as shown in Eq. (2)

$$
\dot{L}\_i = JV \tag{2}
$$

Formula *L*˙*i* <sup>=</sup> [*L*˙ <sup>1</sup> *<sup>L</sup>*˙ <sup>2</sup> *<sup>L</sup>*˙ <sup>3</sup> *<sup>L</sup>*˙ <sup>4</sup>] *<sup>T</sup>*, V represents the velocity vector of the end grab, V = [*x*˙ *y* ˙ *z*˙] *<sup>T</sup>*. J represents the Jacobi matrix. J <sup>=</sup> [ <sup>∂</sup>*Li*  ∂*x*  ∂*Li*  ∂*y*  ∂*Li*  <sup>∂</sup>*<sup>z</sup>*] (i = 1,2,3,4).

### *2.2 Rope-Driven Robot Dynamics*

In this paper, the end grab of the four-cable robot is the research object, and the driving force of each soft cable applied by the facility to the end grab is *T*i, and the force is shown in Fig. 1. Thus, its dynamic model can be obtained by the Euler–Lagrange method:

$$M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + \mathfrak{r}\_f + \mathfrak{r}\_d = J^T T \tag{3}$$

Among them, *q*, *q*˙, *q*¨, ∈ *R<sup>n</sup>*is the motion position, speed and acceleration of the end gripper. *M*(q) ∈ *R<sup>n</sup>*×*<sup>n</sup>*is the positively definite symmetric mass inertia matrix, *C*(q, ˙q) ∈ *R<sup>n</sup>*×*<sup>n</sup>*is the Coriolis force coupling matrix, *G*(q) ∈ *R<sup>n</sup>*×*<sup>n</sup>* are the gsravitational vectors, τ*d* , τ *<sup>f</sup>*∈ *R<sup>n</sup>*is the external disturbance and the friction force vector, T is the driving force of each rope, and J is the Jacobi structure Considering the modeling error of the Flexo robot, there is *M*(*q*) = *M*0(*q*) + *M*(*q*);*C*(*q*, *q*˙) = *C*0(*q*, *q*˙) + *C*(*q*, *q*˙);*G*(*q*) = *G*0(*q*) + *G*(*q*). Corresponding estimate is *M*0(*q*),*C*0(*q*, *q*˙), *G*0(*q*). System modeling error is *M*(*q*), *C*(*q*, *q*˙), *G*(*q*). Definition *d* = *M*(*q*)*q* ¨ + *C*(*q*, *q*˙)*q* ˙ + *G*(*q*) + τ *<sup>f</sup>*+ τ*<sup>d</sup>* .

Therefore, The kinetic model can be simplified to:

$$M\_0(q)\ddot{q} + C\_0(q,\dot{q})\dot{q} + G\_0(q) + d = J^T T \tag{4}$$

### **3 Controller Design**

### *3.1 Interference Observer Design*

Aiming at the problem of composite interference in the control system of the flexible drive robot, a nonlinear interference observer is used to numerically estimate the interference signal, and then use it as a feedback signal to adjust the control torque of the system to reduce the influence of the interference signal on the system.

Definition *d* ˆ is the estimate of d, then the estimated error F of interference is defined as *F* <sup>=</sup> *<sup>d</sup>*<sup>−</sup> *<sup>d</sup>*ˆ. The interference observer can be temporarily set to

$$
\dot{\hat{d}} = L(q, \dot{q})(d - \hat{d})\tag{5}
$$

Formula, *L*(*q*, *q*˙) is the gain matrix to be designed in the interference observer.

Assuming that the disturbance d has a very small or negligible dynamic change rate compared to a nonlinear observer ( *d* ˙ <sup>=</sup> 0).it can be obtained: *F* ˙ = −*L*(*q*, *<sup>q</sup>*˙)*F*. As the estimation error can be exponentially converged by designing a properly designed gain matrix. Since the acceleration signal of the end grab is not easy to obtain in practice, it is also necessary to define new auxiliary variables to build a new observer

$$z = \hat{d} - p(q, \dot{q})\tag{6}$$

Formula, *z* ∈ *R*<sup>2</sup>**;** *p*(*q*, *q*˙) is the nonlinear function to be designed. Moreover, The gain matrix *L*(*q*, *q*˙) and nonlinear functions *p*(*q*, *q*˙) need to be satisfied:

Dynamic Synovial Control Method of Four-Cable-Driven Parallel … 579

$$L(q, \dot{q})M(q)\ddot{q} = \frac{dp(q, \dot{q})}{dt} \tag{7}$$

The combined formula (3)~(7) can obtain a nonlinear interference observer:

$$\begin{cases} \dot{z} = -L(q, \dot{q})z + L(q, \dot{q})[C(q, \dot{q})\dot{q} + G(q) - T - p(q, \dot{q})] \\ \dot{d} = z + p(q, \dot{q}) \end{cases} \tag{8}$$

# *3.2 Dynamic Non-Singular Fast Terminal Sliding Mode Surface Constructed Based on the Estimated Output of Interference Observer*

When constructing the traditional terminal sliding mode controller, it is faced with the problem of singularity, and in order to overcome the problem of singularity. Presented by References [10]. it is proposed that a dynamic non-singular sliding mode surface be constructed on this basis

$$\mathbf{s} = \mathbf{e} + \frac{1}{\alpha} |e|^{\gamma + 1} \text{sgn}e + \frac{1}{\beta} (\dot{e} + \int M^{-1} \hat{d})^{\frac{p}{q}} \tag{9}$$

Formula, α、β、γ > 0 is a design constant; p、 q is positive odd and satisfied 1 < *p*/*q* < 2; e is the difference between the desired trajectory and the actual trajectory *e* <sup>=</sup> *<sup>q</sup>*<sup>−</sup> *qd* ; *d* <sup>ˆ</sup> is an estimate of the interference. The synovial control law is proposed:

$$\begin{split} T &= (J^T)^{-1}(-\frac{\beta q}{p}M(\dot{e} + \int M^{-1}\dot{d})^{(2-\frac{\beta}{q})} - ML\text{sgn}(s) + d - \hat{d} + \text{g} \\ &+ M\ddot{q}\_d - \frac{\beta q}{p}M(\dot{e} + \int M^{-1}\dot{d})^{(1-\frac{p}{q})}(\frac{\mathcal{V}+1}{\alpha}|e|^\mathcal{V}\dot{e})) \end{split} \tag{10}$$

Formula, *L* > 0 is the constant to be designed. This is obtained by Eq. (10).

$$\begin{split} \dot{s} &= \dot{\mathbf{e}} + \frac{\mathcal{V} + 1}{\alpha} |e|^{\mathcal{V}} \dot{e} + \frac{p}{\beta q} (\dot{e} + \int M^{-1} \hat{d})^{(\frac{p}{q} - 1)} (\ddot{e} + M^{-1} \hat{d}) \\ &= \dot{\mathbf{e}} + \frac{\mathcal{V} + 1}{\alpha} |e|^{\mathcal{V}} \dot{e} + \frac{p}{\beta q} (\dot{e} + \int M^{-1} \hat{d})^{(\frac{p}{q} - 1)} (M\_0^{-1} (q)) J^T T \\ &- C\_0(q, \dot{q}) \dot{q} - G\_0(q) - d) - \ddot{q}\_d + M^{-1} \hat{d} ) = -\frac{p}{\beta q} (\dot{e} + \int M^{-1} \hat{d})^{(\frac{p}{q} - 1)} L \text{sgn}(s) \end{split} \tag{11}$$

### *3.3 Lyapunov Proved*

Consider the Lyapunov function as follows

$$V(t) = \frac{1}{2}\mathbf{s}^2\tag{12}$$

Derive the above equation:

$$\begin{split} \dot{V}(t) &= \mathbf{s}^{T}\dot{\mathbf{s}} = \mathbf{s}^{T} [-\frac{p}{\beta q}(\dot{\mathbf{e}} + \int M^{-1}\hat{d})^{(\frac{p}{q}-1)} L \text{sgn}(\mathbf{s})] \frac{\partial^{2} \Omega}{\partial u^{2}} \\ &\leq -|\mathbf{s}| \frac{p}{\beta q}(\dot{\mathbf{e}} + \int M^{-1}\hat{d})^{(\frac{p}{q}-1)} L \end{split} \tag{13}$$

Formula, β, *q*, *p*, *L* > 0. Hypothesis ˙ *<sup>e</sup>*˜ <sup>=</sup> *<sup>e</sup>*˙ <sup>+</sup> *<sup>M</sup>*−1 *d*ˆ. when ˙ *<sup>e</sup>*˜<sup>=</sup> 0, From 0 <sup>&</sup>lt; *<sup>p</sup> <sup>q</sup>*<sup>−</sup> <sup>1</sup><sup>&</sup>lt; 1 you get (*e* ˙ <sup>+</sup> *<sup>M</sup>*−1 *d*ˆ) ( *p <sup>q</sup>*−1) > 0. , according to Reference [11], When a sufficiently large *L* > 0 is satisfied, When ˙ *e* ˜ ≤ −*L* is then *s* > 0, The opposite is the same, Therefore, ˙ *e* ˜ = 0 has little effect on the stability of the sliding surface, and it can also be said that ˙ *e*˜ < δ exists near the ˙ *e* ˜ = 0 range. So when *s* > 0 is then *e* ˜ ≤ −*L*, The opposite is the same, Therefore, the intersection of dynamic synovial trajectories when *s* > 0 is *e* ˜ = δ from *e* ˜ = −δ. So it can be launched

$$\dot{V}(t) = \mathbf{s}^{\top}\dot{\mathbf{s}} \le \mathbf{0} \tag{14}$$

### **4 Simulation Experiments**

The simulation of all the examples in this simulation was completed under MATLAB2023aSimulink. The simulation object selects the four-cable parallel drive robot model. The matrix parameters in the kinetic model (3) are: *M*(*q*) = *diag*(*m*, *m*, *m*); *C*(*q*, *q*˙) = *diag*(0, 0, 0); *G*(*q*) = [*mg*; 0; 0] *<sup>T</sup>*. Four-cable drive parallel robot structure parameters a = 10 m,b = 5 m,h = 5 m,m = 1 kg,g = 9.8N/ kg. Take the perturbation item *d* = [0.25 sin(*t*); 0.25 sin(*t*); 0.25 sin(*t*)] *<sup>T</sup>*. In the interference observer *p*(*q*, *q*˙) = ρ*q*˙, ρ = 0.5. The initial position of the end gripper of the four-cable-driven parallel robot is set q = [0;0;0];The simulated motion trajectory is as follows: *x* = 2 sin(0.5π*t*)+1;*y* = sin(0.5π*t*)+1;*z* = 0.5 sin(0.5π*t*)+1. When the expected trajectory of the end grab of the four-cable parallel drive robot is the above trajectory, the synovial control rate is used to track and control the motion trajectory of the end grab, and it can be seen from the figure that the threedimensional diagram of the expected spiral trajectory and the actual trajectory is obtained for simulation. Taking values on the sliding surface: p = 7, q = 5, α = 3, β = 1000, γ = 0.5.

**Fig. 2** Trajectory tracking

From the observation of the above figure (see Fig. 2), it can be found that the dynamic non-singular fast terminal sliding mode control strategy constructed based on the estimated output of the interference observer has a good trajectory tracking effect on the motion trajectory of the four-cable parallel drive robot end grab.

As shown in the figure (see Fig. 3) above, the speed tracking trajectory change diagram of the end gripper in the three directions of x, y, and z can clearly see that the overall fluctuation is not large and is in a stable change. Among them, the speed error in the x direction is absolutely within 3.3%, the velocity error in the y direction is within 2.3%, and the speed error in the z direction is within 2.1%.

As shown in the figure (see Fig. 4) above, the trajectory error of the proposed controller and the fast terminal synovial membrane controller is compared, indicating that the trajectory error of this observer is smaller, stable and faster.

**Fig. 3** Velocity error

**Fig. 4** Track tracking comparison

# **5 Conclusion**

Aiming at the interference such as external disturbance and modeling error of the fourcable parallel drive robot, a dynamic model is established based on the Newton–Euler method, and a dynamic non-singular fast terminal sliding mode surface based on the estimated output construction of the interference observer is designed. Lyapunov was used to prove the stability of the design control system, and the trajectory tracking effect was good and the error was within the acceptable range through simulation, and it was compared with the fast terminal synovial controller to prove that it had a better tracking trajectory and tracking error.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Multi-mode Sliding Mode Control of Four-Cable Parallel Robot Based on Wind Disturbance Observation**

**Shenghao Tong, Long Zhao, Huaitao Shi, Zhiqiang Duo, and Cai He** 

**Abstract** Aiming at the problem of accurate trajectory tracking of four-cable parallel robot under wind disturbance in the process of material handling in buildings, a multimode sliding mode control (MSMC) method based on wind disturbance observer is proposed. Firstly, the wind field model is established, and the comprehensive wind speed model is introduced into the accurate dynamic model as a wind disturbance factor. Secondly, the wind disturbance observer is introduced to estimate the total disturbance of wind disturbance error in real time, which effectively reduces the switching gain, thus effectively reducing the vibration and improving the control accuracy of the system. Combined with the dynamic performance of multimode sliding mode controller, it can be switched at will to reduce chattering. Compared with the traditional sliding mode control method SMC, the anti-interference ability of MSMC is verified. The results show that the designed multimode sliding mode controller can effectively suppress the influence of wind disturbance on the vibration of the end effector.

**Keywords** Cable parallel robot · Wind · Disturbance observer · Multimodal · Sliding mode

# **1 Introduce**

The parallel robot with flexible cables drives the hanging object to move through four or more flexible cables, and can move freely in the direction in the building space. The flexible cable driving device is arranged above or around the construction site, and the transported hanging objects are placed in the three-dimensional space through a plurality of driving flexible cables. This kind of parallel robot with flexible cables moves fast, and only needs remote control by people, which can ensure the safety of people in the process of lifting objects. This robot can take multi-mode shots in many

S. Tong · L. Zhao (B) · H. Shi · Z. Duo · C. He

Institute of Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China e-mail: zhaolong\_sjzu@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_46

occasions [1, 2]. In Ref. [3], a robust switching integral sliding mode control method is proposed for mismatched uncertain systems, which makes each subsystem robust and stable to uncertain disturbances. References [4–6] all use sliding mode control based on disturbance observer to compensate the mismatch uncertainty. However, the above two methods all rely on the assumption that the mismatch uncertainty satisfies the time-invariant or slow-time-varying assumption. For the more general uncertainty of the function model, the output of the control system cannot converge to zero strictly, but only to the neighborhood near zero. Different from the traditional second-order sliding mode and high-order sliding mode control methods, In Ref. [7], the upper bound function type of mismatched uncertainty in mismatched uncertain systems is extended from constant type to more general positive function type, and the corresponding Second ordersliding mode (SOSM) control law is designed based on this type of disturbance boundary, but the chattering phenomenon cannot be well suppressed. Literature [8] Nonlinear Extended State Observer (NLESO) provides good tracking performance for uncertain systems, but the selection of nonlinear functions has no clear theoretical basis, and in most case it depends on the empirical judgment of researchers. Therefore, a four-cable parallel machine based on multimode sliding mode control is designed to meet the demand of efficient load and suppress buffeting caused by wind disturbance [9, 10].

### **2 Mathematical Model of Four-Cable Parallel Robot**

### *2.1 Wind Speed Model*

The basic wind selects the average wind speed value, which is 5 m/s.

$$V\_{\mathfrak{b}} = K \tag{1}$$

Gust is used to describe the sudden change of wind speed:

$$V\_{\mathbf{g}} = \begin{cases} 0, (t < t\_1) \\ \frac{V\_{\mathbf{g}\max}}{2} \left\{ 1 - \cos \left[ 2\pi \left( \frac{t - t\_1}{T\_{\mathbf{g}}} \right) \right] \right\}, (t\_1 \le t \le t\_1 + T\_{\mathbf{g}}) \\ 0, (t > t\_1 + T\_{\mathbf{g}}) \end{cases} \tag{2}$$

Gradual wind is used to describe the characteristics of wind speed gradient:

$$V\_r = \begin{cases} 0, (t < t\_{r1} \text{ or } t > t\_{r2} + t\_{r3}) \\ V\_{r \max} \frac{t - t\_{r1}}{t\_{r2} - t\_{r1}}, (t\_{r1} \le t \le t\_{r2}) \\ V\_{r \max}, (t\_{r2} < t \le t\_{r2} + t\_{r3}) \end{cases} \tag{3}$$

**Fig. 1** Graph of measured total wind speed

Random wind speed is used to describe the randomness of wind speed in wind field;

$$V\_n = V\_{n\max} R\_{am}(-1, 1)\cos(\omega\_n t + \varphi\_n) \tag{4}$$

Based on the wind speed descriptions in the above four parts, the simulated wind speed acting on the impeller can be obtained from Eqs. (1)–(4) as follows: integrating the wind shape to (see Fig. 1).

### *2.2 Dynamic Model of Four Cable Parallel Robot*

The working range of the four cables parallel robot motion platform is a threedimensional space, which has the function of moving up, down, left, right, and can also achieve three motion directions simultaneously. The overall geometric model of the four cable parallel robot system has been established, as shown in Fig. 2.

In the Cartesian coordinate system, the length L of each flexible cable can be solved by the position vector of the end effector and the position vector of the pulley. The position vector of the end effector, the position vector of the pulley, k = 1, 2, 3, 4, and the length of any cable can be calculated according to the following equation:

$$L\_k = P - B\_k \tag{5}$$

Under the Lagrange function of the reference frame, a four cables parallel robot can be constructed:

**Fig. 2** Geometric model of four-cable parallel robot

$$M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = F\_d - F \tag{6}$$

$$\begin{cases} F\_d = \omega pA \\ p = \frac{1}{2} r V\_w^2 \\ V\_w = V\_b + V\_g + V\_r + V\_n \end{cases} \tag{7}$$

The wind coefficient is 1.2, P is the calculated wind pressure, r is the air density of 1.25 kg/m, V is the calculated wind speed, and A is the windward area of the load. The dynamic model of a four cables parallel robot can be obtained:

$$M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = J^T T + F\_d \tag{8}$$

# **3 Four Cable Parallel Multi-mode Sliding Mode Control and Stability Analysis**

### *3.1 Design of Disturbance Observer*

Based on the previously obtained dynamic model of a four cables parallel human:

$$M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau + F\_d \tag{9}$$

Among them is the Jacobian matrix of the robot system, where the cable force of the T-drive parallel robot is the control signal vector. According to the mathematical model observer of the four cables parallel robot:

Multi-mode Sliding Mode Control of Four-Cable Parallel Robot Based … 589

$$\dot{Z} = -L(q, \dot{q})z + L(q, \dot{q})(C(q, \dot{q})\dot{q} + G(q) - \tau - p(q, \dot{q})) \tag{10}$$

$$
\hat{F}\_d = Z + p(q, \dot{q}) \tag{11}
$$

After adding a disturbance observer, the disturbance torque of the system decreases from Fd to observation error, indicating that adding a disturbance observer can effectively reduce disturbance momentum.

# *3.2 Design and Stability Analysis of Multimodal Sliding Mode Control*

The so-called multimodality refers to the design of a sliding mode motion path connected by multiple sliding mode regions with s(0) = 0 and s(0) = 0, allowing the system state points to move from one mode to another along a determined path. The final approaching target point SMC consists of two parts: ISMC and SMC. Switching control may exhibit chattering due to the discontinuous sign of the term. Sign is replaced by a smooth function, and stability is demonstrated using Lyapunov [11].

Set *qd* as a reference signal, defined as:

$$e = q\_d - q \tag{12}$$

Construct a sliding surface, called a sliding function, and select the sliding surface:

$$\begin{cases} s\_1 = \dot{e} + \mu e \\ \dot{s}\_1 = \ddot{e} + \mu \dot{e} \end{cases} \tag{13}$$

*uk* Satisfy the arrival condition and implement sliding motion control when manifold s = 0. Set a high gain saturation function to approximate the sign function to reduce chattering. Set the control law:

$$
\mu\_1 = m\ddot{q}\_d + c\_1\dot{q} + k\_1q + \mu m\dot{e} + d\_1\text{sign}(\mathbf{s}\_1) + d\_2\mathbf{s}\_1 \tag{14}
$$

If the initial condition is not on the sliding surface and due to wind disturbance, a controller is proposed to drive the output to the sliding surface to be 0 in a finite time. Under this condition, the output will reach the sliding surface. The Lyapunov function represents:

$$\begin{aligned} V\_1(t) &= \frac{1}{2} s\_1^2, \quad V\_1(t) > 0, \quad \text{and } \mathbf{s}\_1 \neq 0\\ S\_1(t) &= s\_1 \left[ \ddot{q}\_d - \frac{1}{m} (u\_1(t) - c\_1 \dot{q} - d\_1 q + \mu \dot{e}) \right] = \frac{1}{m} s\_1 \left[ -d\_1 \text{sign}(s\_1) - d\_2 s\_1 \right] \end{aligned} \tag{15}$$

When the error gradually decreases, the integration method is introduced into the SMC to eliminate the error, thereby improving the tracking performance of the system. The integration part is added to the SMC, making the sliding surface an integral augmented SMC. In the figure, when the error is less than or equal to the threshold e, the control system will adopt ISMC.

$$\begin{cases} s\_2 = \dot{e} + \mu\_1 e + \mu\_2 \int\_0^t e(\tau) d\tau\\ s\_2 = \ddot{e} + \mu\_1 \dot{e} + \mu\_2 e \end{cases} \tag{16}$$

The control law is:

$$\begin{aligned} u\_2 &= m\ddot{q}\_d + c\_1\dot{q} + k\_1q + \mu m\dot{e} + \mu\_2 me\\ &+ d\_3 \frac{s\_2}{|s\_2| + \delta} + d\_4 s\_2 \quad (d\_3 > 0, d\_4 > 0) \end{aligned} \tag{17}$$

When the error threshold is greater than, u1 is used, and when the error threshold is less than or equal to, u2 is used as the control input u.

### **4 Simulation Analysis**

In the construction of MATLAB/SIMULINK, this chapter constructs a system simulation model and writes the S function in MATLAB language to better describe the system, control algorithms, and improve system performance. Use the above method to control the trajectory of the four cables parallel robot. For the four cables parallel control platform, the length a = 20 m, width b = 12 m, and the mass of the end effector is m = 5 kg, g = 9.8 m/s. a1 = a2 = 0.1, d1 = d2 = d3 = d4 = 0.01. In order to verify the performance advantages of the multimodal sliding mode control based on wind disturbance observer proposed in this article, simulation analysis will be conducted from two aspects: speed and displacement tracking observation results, and compared with the classical sliding mode control method SMC. The classic sliding mode controller used is:

$$\begin{aligned} e &= qd - q \quad T\\ de &= dqd - dq\\ s &= ce + de\\ u &= m\*(n\*sign(s) + p\*sS + c\*(dqd - dq)\\ &+ ddqd + (k/m)\*q) \end{aligned} \tag{18}$$

The simulation diagrams of velocity and displacement for comparison between MSMC and SMC controllers, including x, y, and z, as well as the local comparison diagrams of velocity tracking and displacement tracking, as shown in Figs. 3 and 4.

From the simulation results, it can be seen that under the influence of wind disturbance observer compensation, the speed and displacement are approximately close, and there is a slight vibration in the speed. After being compensated by disturbance observer, the tracking effect is very good through the action of multimodal sliding mode controller. Compared with traditional sliding mode control methods, the displacement and velocity are closer to the expected trajectory, and the vibration is reduced, resulting in better control effect (see Figs. 5 and 6).

**Fig. 3** Speed tracking x, y, z y and z comparison chart

**Fig. 4** x, y, z y and z comparison diagram of displacement tracking

**Fig. 5** Speed tracking diagram

**Fig. 6** Displacement tracking diagram

### **5 Conclusion**

This article establishes a refined wind field synthesis model for a four cables parallel robot system with wind disturbance. A wind disturbance observer is introduced to estimate and compensate for the changes in the total disturbance in real time, and a (MSMC) controller is designed in combination with a multimodal sliding mode control with small chattering and fast response speed. The simulation results show that the control algorithm proposed in this paper can achieve precise trajectory control after compensating for wind disturbance by the wind disturbance observer. The numerical simulation results show that the position optimization rate of the four cables parallel robot system is closer to the numerical value and better than that of the traditional SMC controller under wind disturbance, indicating that the system has strong anti-interference ability and robustness. It has high reference value for the design of the actual intelligent storage flexible cable handling mechanism controller.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Finite Element Simulation of Loss Stroke Phenomenon of Shock Absorber Based on Fluid Structure Interaction**

**Zhen-huan Yu and Na Zhang** 

**Abstract** To provide a design reference with guaranteed performance for vehicle shock absorbers, the loss stroke mechanism is studied in this paper based on fluid– structure interaction (FSI) numerical method. According to the parametric model of cavitation mechanism and by applying fluid–structure interaction (FSI) numerical methods, the structural and fluid finite element mesh shock absorber models with high precision are both established. Furthermore, simulations for the proposed models are carried out based on ADINA software. By analyzing the simulation results, the specific position and the distribution of loss stoke, and the difference of cavitation phenomenon under diverse shock absorber loading speeds are revealed. The results indicate that cavitation of the shock absorber primarily centers around the shock absorber valve system, and this phenomenon becomes increasingly pronounced as the piston speed rises. Moreover, the cavitation effect is higher apparent when the shock absorber oil possesses higher viscosity. These findings offer valuable insights for the design and implementation of measures aimed at preventing performance distortion in shock absorbers.

**Keywords** Shock absorber · FSI · Finite element method

### **1 Introduction**

Hydraulic shock absorber are an important component of automotive suspension systems [1–3], which can convert mechanical energy into thermal energy, thereby attenuating the vibration and shock caused by road excitation and improving the

Z. Yu

N. Zhang (B) Department of Mechanical Engineering, Changchun Technical University of Automobile, Jilin Changchun 130011, China e-mail: yuzhenhuan@caii.edu.cn

Department of Automotive Engineering, Changchun Technical University of Automobile, Jilin Changchun 130011, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_47

handling stability of the vehicle. However, the hydraulic shock absorber is subjected to frequent, high-speed vibration impacts during which the shock absorber fluid dissolves into air bubbles and cavitation occurs [4–7]. This phenomenon reacts to the shock absorber operation called shock absorber cavitation, which can lead to the failure of the shock absorber in a certain section of the stroke, causing an increase in vehicle vibration, or a bird-like noise, and the cavitation phenomenon can reduce the life cycle of the shock absorber [8]. Therefore, it is significant to study the air travel phenomenon of hydraulic shock absorbers.

At present, there are few studies on the cavitation phenomenon of the shock absorber, especially the detailed theory and the location where it occurs. In this paper, to solve this problem of the cavitation phenomenon of the shock absorber, high-precision hydraulic shock absorber solid and fluid finite element models are established respectively, and the finite element models are simulated in ADINA software to determine the specific location of the cavitation phenomenon inside the shock absorber. At the same time, the influence of different fluid parameters on the hydraulic shock absorber cavitation phenomenon was analyzed, and effective measures and methods to weaken the cavitation phenomenon were proposed and verified through experiments. The cavitation phenomenon of the hydraulic shock absorber is weakened.

# **2 Analysis of the Causes of Hydraulic Shock Absorber Loss Stroke**

### *2.1 Shock Absorber Oil Cavitation Principle*

Since the shock absorber is equipped with shock absorber oil inside, there is a certain amount of gas mixed in the oil, and as the pressure and temperature field inside the shock absorber changes, the air dissolved inside the oil will increase or decrease [9, 10].

As shown in Fig. 1, the shock absorber produces a cavitation phenomenon: the shock absorber recovery valve will be divided into the upper and lower chambers of the shock absorber, when the piston rod upward or downward movement, the upper and lower cavity fluid is compressed, the oil pressure becomes larger, this cavity fluid dissolved into more gas, when the throttle piece open valve pressure, the pressure of the compressed chamber instantly decreases, when the pressure drops to the saturated vapor pressure of the fluid, the fluid has been pressed into the gas will be released, so that the shock absorber fluid produces a large number of bubbles, this phenomenon for the hydraulic shock absorber cavitation phenomenon [11].

### *2.2 Parametric Model of Shock Absorber Cavitation [12]*

At constant temperature, the cavitation coefficient σ is usually used to characterize the extent to which cavitation occurs. It is expressed by Eq. (1).

$$
\sigma = \frac{P\_2 - P\_\mathrm{v}}{\rho v^2 / 2} \tag{l}
$$

P2—shock absorber lower cavity pressure; Pv—shock absorber fluid cavitation critical pressure; ρ—shock absorber fluid density; v—fluid flow average speed.

When the hydraulic shock absorber moves up and down, the shock absorber fluid passes through the recovery valve system and reciprocates between the upper and lower chambers of the shock absorber, generating a differential pressure P. P can be defined as:

$$
\Delta P = P\_1 - P\_2 = \rho v^2 / 2 \tag{2}
$$

P1-shock absorber upper chamber pressure. Pv-the critical pressure of shock absorber fluid cavitation,can be neglected compared to P2 and P1, i.e., Pv = 0, which can be simplified by Eqs. (1) and (2) as:

$$
\sigma = \frac{P\_2}{P\_1 - P\_2} \tag{3}
$$

When σ is less than 0.4, cavitation phenomenon occurs [13]. σ = 0.4 is the critical point at which the cavitation phenomenon occurs in the oil-hydraulic shock absorber. As a result, Eq. (3) is simplified as.

$$
\xi = \frac{P\_1}{P\_2} = 3.5\tag{4}
$$

ξis the oil-hydraulic shock absorber cavitation critical pressure ratio.

### **3 FSI Finite Element Analysis Method Based on ADINA**

In order to more intuitively observe the cavitation phenomenon of the oil-hydraulic shock absorber, this paper establishes a finite element model of the cavitation phenomenon generated by the oil-hydraulic shock absorber. When the host plant tests the shock absorber, the common test temperature requirement is 22 degrees Celsius, so this paper selects this temperature value for the saturation vapor pressure of the shock absorber fluid.

### *3.1 FSI Finite Element Mathematical Model [14]*

FSI finite element analysis is used to solve the problem of the state of motion between a fluid and a solid, and its basic principle is to satisfy both the fluid-coupled boundary dynamics and kinematic equations, and the solid-coupled boundary dynamics and kinematic equations.

$$d\_f = d\_s \tag{5}$$

The kinetic conditions:

$$n \cdot \mathfrak{r}\_f = n \cdot \mathfrak{r}\_s \tag{6}$$

where df and ds are the fluid and solid boundary displacements, respectively, τf and τs are the fluid and solid stresses, respectively, and n is the unit external normal vector. From the kinematic conditions, the fluid velocity conditions can be derived as follows.

$$n \cdot v\_f = n \cdot \mathbf{v}\_s \tag{7}$$

### *3.2 ADINA-based Flow-solid Coupling FEM Analysis*

When using the finite element FSI theory to analyze the flow-solid coupling problem of the oil-hydraulic shock absorber, two key points need to be addressed.

(1) Accurate transfer of interaction forces on the fluid–solid coupling surface.The solid model and fluid model of ADINA software are established separately, and the boundary conditions are defined on the coupling surfaces of the two models to solve the flow-solid coupling problem. The meshes of the fluid and solid do not need to be identical, as long as a certain tolerance is achieved [15]. Therefore, the fluid nodal displacements on the fluid–solid coupling surface are obtained by interpolation of the nearby solid nodal displacements, while the solid nodal forces are obtained by integration of the surrounding fluid stresses, i.e.

$$F(t) = \int h^d \mathbf{r}\_f dS\tag{8}$$

Where hd is the solid imaginary displacement

(2) Realizing efficient solution of coupled systems ALE (Arbitrary Lagrangian Euleria) is currently used to solve the problem of fluid mesh movement due to solid deformation.

ADINA software has both iterative and direct solution methods, both of which need to ensure the consistency of the integration points of the fluid and solid models during dynamic analysis. Both methods require iterative solutions in order to solve the fluid and solid equations separately. The direct solution method puts the fluid model and the solid model in the same matrix, and the finite element equations are as follows:

$$
\begin{bmatrix} A\_{ff} & A\_{fs} \\\\ A\_{sf} & A\_{ss} \end{bmatrix} \begin{bmatrix} \Delta X\_f^k \\\\ \Delta X\_s^k \end{bmatrix} = \begin{bmatrix} B\_f \\\\ B\_s \end{bmatrix} \tag{9}
$$

$$X^{k+1} = X^k + \Delta X^k \tag{10}$$

$$A\_{ff} = \frac{\vartheta F\_f^k}{\vartheta X\_f} A\_{fs} = \lambda\_d \frac{\vartheta F\_f^k}{\vartheta X\_s} \tag{11}$$

$$A\_{sf} = \lambda\_{\pi} \frac{\vartheta F\_f^k}{\vartheta X\_f} A\_{ss} = \frac{\vartheta F\_s^k}{\vartheta X\_s} \tag{12}$$

where the displacement relaxation factor is λd(0<λ d ≤ 1)and the stress relaxation factor is λτ(0<λτ ≤ 1) the direct solution method has high accuracy when solving non-contact transient models where the computational volume is not very large.

# *3.3 Establishment of the Finite Element Model of the Oil-hydraulic Shock Absorber Superimposed Throttling Recovery Valve System*

In order to build a superimposed throttling recovery valve system for the oil shock absorber, the overall valve system needs to be simplified in consideration of the accuracy of the finite element model and the computing capability of the computer.


Figure 2 shows the fluid finite element mesh model, and Fig. 3 shows the solid finite element mesh model. The fluid model is an 8-node hexahedral cell with all variables defined at each corner node. This cell satisfies the inf–sup stability and optimality conditions, and has good second-order accuracy when calculating variable interpolation, which is more accurate than the first-order accuracy of the tetrahedral cell. Considering the computational power of the computer and the solution time, the mesh is subdivided only in the multi-piece throttle piece and the fluid coupling part, which can ensure that the relative displacement is within the tolerance range when the fluid–solid coupling surface is calculated. To ensure the density compatibility of the fluid–solid coupling surface, the solid model is subdivided by 8-node hexahedral unitization. The solid model is divided into 8-node hexahedron cells, with 11,277 solid meshes and 29,733 fluid meshes, to ensure density compatibility of the fluid– solid coupling surface.

**Fig. 2** Fluid finite element model

**Fig. 3** Solid finite element model

The velocity is applied at the fluid inlet of the recovery valve, the loaded velocity is equal in magnitude to the piston moving velocity but in the opposite direction, and the loaded amplitude is 1/4 sine wave of 1 m/s for 1.56s, as shown in Fig. 4. No boundary conditions are set at the outlet of the fluid model, and the default pressure value is zero. Set the fluid–solid coupling surface boundary conditions and set to symmetric properties, the rest of the fluid boundary are set to wall, so set because the sliding interface of the FCBI unit, can only be solved with a sparse solver, and multiple throttle pieces in the fluid motion will produce deformation, is a transient nonlinear motion, in the ADINA software using the leader–follower command, this command can make the fluid flow The leader–follower command is used in the ADINA software to keep the fluid flow in the same direction, and the integration can be calculated for the entire fluid level in post-processing.

In the solids model, there are five valves forming a group of stacked multi-piece throttling valves, which are defined as contact between the valves, and the entire boundary of the stacked multi-piece throttling valves is defined as contact with the piston. The implicit dynamic method is set as the solids analysis type.

### *3.4 Analysis of Simulation Results*

The simulation results are shown in (Figs. 4 and 5).

Figure 5 for t = 0.66 s, v = 0.42 m/s, v = 0.77 m/s, and 1 m/s. The values are all sinusoidal, with an amplitude of 30 mm, corresponding to t = 0.66 s, t = 1.2 s, and t = 1.56 s. The simulation results are shown in Fig. 5 for t = 0.66 s, v = 0.42 m/s, and the graph shows that at this time the valve has just been opened, and the pressure drop is not obvious. So the cavitation phenomenon is not obvious, at this time the oil shock absorber work in the normal state. As shown in Fig. 6,t = 1.2 s, v = 0.77 m/s, as can be seen in the figure, at this time the valve has opened, the pressure drop is larger, at this time the cavitation phenomenon around the valve system is clearly presented, at this time the oil shock absorber work destabilization is not obvious. As shown in Fig. 7,t = 1.56 s, v = 1 m/s, at this time the superimposed

**Fig. 4** Fluid velocity loaded sine wave

throttle valve system completely open, the pressure drop is the largest, the oil shock absorber cavitation phenomenon is the most obvious, cavitation phenomenon with the valve system mouth down the cavity to move, at this time the oil shock absorber work part of the stroke failure state. From Figs. 5, 6 and 7 can be seen, the cavitation phenomenon initially.

distributed around the superimposed throttle piece, the formation of the lower cavity spread, with the oil-hydraulic shock absorber piston speed increases, the more obvious the shock absorber cavitation phenomenon.

# **4 Fluid Viscosity on the Shock Absorber Air Range Phenomenon of the Impact**

The lower the kinematic viscosity of the shock absorber fluid, the smaller the pressure loss, the less likely it is to be below the saturated vapor pressure of the fluid, and the less obvious the oil shock absorber air travel phenomenon.

In this paper, three different kinematic viscosities of the shock absorber fluid are used for simulation comparison, the piston hole diameter is 1.5mm, and the fluid viscosities are 12.3 mm2/s, 13.3 mm2/s and 14.3 mm2/s respectively. The simulation results are compared under the same conditions.

The simulation results are shown in Fig. 8. When the kinematic viscosity is 14.3 mm2/s, the cavitation phenomenon is more pronounced than the other two kinematic viscosities in the three time regions of t = 0.66 s, t = 1.2 s, and t = 1.56 s, proving that the high kinematic viscosity of the shock absorber fluid is prone to produce the shock absorber cavitation phenomenon. When the oil kinematic viscosity is 12.3 mm2/s, in the three time regions of t = 0.66 s, t = 1.2 s and t = 1.56 s, respectively, compared with the other two oil kinematic viscosities, the shock absorber cavitation phenomenon is the least obvious, indicating that the lower the oil viscosity, the less

**Fig. 8** Pressure cavitation cloud of shock absorber oil with different kinematic viscosities. **a** Oil kinematic viscosity of 14.3 mm2/s. **b** Oil kinematic viscosity of 13.3 mm2/s. **c** Oil kinematic viscosity is 12.3 mm2/s

obvious the shock absorber cavitation phenomenon, indicating that low viscosity oil has a certain effect on reducing the shock absorber cavitation phenomenon.

### **5 Conclusion**


**Funding** This work was supported by the Changchun Technical University of Automobile Science Foundation of P.R. C. (No. 2024002) and the Jilin provincial Natural Science Foundation of P.R. C. Grant No.JJKH20221345KJ).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on TBM Disc Cutter Wear Prediction Based on CSM Model**

**He Wang, Junkun Wei, Guangyu Yan, Haigen Zhao, and Jianing Xu** 

**Abstract** The wear of disc cutters significantly impacts the efficiency and cost of TBM operations. In this study, we established a calculation model for radial wear based on the CSM formula and abrasive wear mechanisms. Experimental validation of the wear mechanism confirmed that abrasive wear is the primary mechanism responsible for disc cutter wear. The predictive model was subjected to engineering validation, where the calculated values from the model were compared and analyzed against actual on-site measurements. This validation process confirmed the accuracy and applicability of the model.

**Keywords** CSM · Abrasive wear · Radial wear

# **1 Introduction**

The disc-shaped cutter is a primary component of Tunnel Boring Machines (TBM), and its cutting ring is a consumable part during construction. The cutting ring is prone to wear due to its interaction with the rock surface to be excavated and the crushed rock particles, leading to a decrease in the excavation performance of the disc-shaped cutter. Studying the forces acting on the cutter during rock rolling and assessing the degree of cutter wear are of significant importance for improving the rock-breaking efficiency of the excavation machine and reducing construction costs. Maidl et al. [1] derived an empirical formula relating the average lifespan of 17-in. cutters to the

H. Wang · G. Yan

© The Author(s) 2024

H. Wang · J. Wei · G. Yan (B) · H. Zhao · J. Xu

School of Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China e-mail: 775083630@qq.com

Joint International Research Laboratory of Modern Construction Engineering Equipment and Technology, Shenyang Jianzhu University, Shenyang 110168, China

H. Wang

State Local Joint Engineering Laboratory of High-Grade Stone CNC Processing Equipment and Technology, Shenyang 110168, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_48

uniaxial compressive strength and the CAI index of the rock by studying various rock properties. Yu et al. [2] established a mapping model between the new cutter health index and specific field parameters based on one-dimensional convolutional neural networks through the analysis and selection of specific parameters from actual construction sites. Estimating the cutter health index provides information about the cutter's wear state. Koppl and Thuro, as well as Koppl et al. [3], developed empirical predictive models for initial and secondary wear of cutting tools caused by abrasive soil erosion. Li et al. [4] validated the stress of cutters in weathered mudstone formations based on the CSM prediction model and concluded that increasing the ratio of penetration rate to effective torque and thrust could reduce cutter wear. Ko et al. [5] studied various TBM cutter prediction models, including the Gehring model, CSM model, and NTNU model, and analyzed their characteristics. The above-mentioned predictive method in the study has a significant margin of error and cannot accurately forecast disc cutter wear. This paper primarily focuses on the positive cutter, and based on abrasive wear mechanisms, calculates the sliding distance through the CSM cutter stress model and cutter motion analysis to establish a calculation model for radial cutter wear. This model facilitates the assessment of wear severity and is validated through wear volume calculations. Therefore, this research is meaningful for guiding the timely replacement of disc-shaped cutters in TBM excavation processes.

### **2 Force Analysis During Rock Cutting by the Cutter**

The stress state of the cutter during rock breaking is depicted in Fig. 1. In Fig. 1, *Fn*  represents the thrust force generated by the TBM propulsion system, while *Ft* denotes the rolling force induced by the cutterhead torque system. Additionally, when the cutter rotates around the cutterhead center, it also generates a lateral force. However, due to the typically low rotational speed of the cutterhead, the lateral force is usually negligible [6]. When rolling against the rock, in order to balance the effects of *Fn*  and *Ft*, the rock exerts a reaction force *Fr* on the cutter, and *Fr* is one of the primary causes of cutter wear.

The calculation model for Fr is typically based on the (Colorado School of Mines, CSM) CSM model [7], and its expression is as follows:

$$\begin{cases} F\_r = \frac{P\_0 \phi R T}{1 + \psi} \\\\ \phi = \arccos\left(\frac{R - h}{R}\right) \\\\ P\_0 = C \sqrt[3]{\frac{S}{\phi \sqrt{R T}} \sigma\_c^2 \sigma\_t} \end{cases} \tag{1}$$

In the equation, Fr represents the normal reaction force acting on the cutter, ψ is the distribution coefficient of the cutter's outer edge, φ is the contact angle of the cutter, *R* is the cutter's radius, *h* represents penetration depth, *T* is the cutter's blade width, *S* represents the spacing between cutters, σc is the uniaxial compressive strength of the rock, σt is the uniaxial tensile strength of the rock, *C* is a dimensionless constant, typically around 2.12, and *P*0 is the crushing zone pressure, which is related to rock strength and the size and shape of the cutter.

### **3 Radial Wear Analysis**

Among the various failure modes of cutters, wear consumption is the primary reason, and its predominant form is abrasive wear. The degree of cutter wear primarily refers to the change in the cutter ring's radius, and the wear depth of the cutter ring radius is referred to as "radial wear," which is a crucial criterion for determining whether a cutter needs replacement. For abrasive wear, a simplified model often used is the one proposed by Rabinowicz [8], as shown in Fig. 2.

The calculation formula for Rabinowicz wear is as follows:

$$V = kL\_g \frac{P \cot \beta\_g}{\pi H} = k\_s \frac{PL\_g}{H} = k\_d P L\_g \tag{2}$$

In the equation, *P* represents the applied load, *H* is the hardness of the metal material, β*g* is the half-angle of the abrasive grain, *Lg* is the sliding distance, and *ks*  is the abrasive wear coefficient, which is the product of the abrasive grain's geometric *cot*β*g* and the probability constant *k*. Typically, its value ranges from 10−1 to 10–6 [9].

**Fig. 2** Rabinowicz simplified model

From Eq. (2), it can be observed that the wear amount is primarily related to the magnitude of the load and the distance traveled. Therefore, based on the stress analysis, calculating the sliding distance of the cutter can yield a more accurate estimation of cutter wear.

### *3.1 Calculation of Sliding Distance*

To achieve the rock-breaking operation of the Tunnel Boring Machine (TBM), the cutterhead is subjected to torque. In addition to revolving around the axis of the cutterhead, the disc cutters mounted on the cutterhead also undergo rotational motion. This, in conjunction with the advancement of the system, allows for penetration into the rock to complete the rock-breaking process. Let ω represent the angular velocity of the cutterhead, ω0 denote the angular velocity of the disc cutter, and *ri* be the installation radius of the disc cutter. When the disc cutter rotates by an angle φ − θ, the relative rotation angle of the cutterhead is ϕ When the cutter edge is located at position *N*, the linear velocity generated by the rotation of the disc cutter is:

$$v\_a = a\_0 R \tag{3}$$

The linear velocity generated under the effect of the cutterhead's revolution is:

$$
v\_b = a\mathcal{R}\tag{4}$$

The penetration velocity under the influence of the propulsion system is:

$$
v\_c = 
ha\_0/2\pi = 
ha r\_i/2\pi R\tag{5}$$

In accordance with the concept of "one-time penetration displacement" as described in Ref. [10], and considering the dynamic variation of the disc cutter position during rock breaking, we establish the coordinate system shown in Fig. 3.

**Fig. 3** Sliding speed diagram of disc cutter

Assuming the sliding distance of the disc cutter within the rock-breaking time t is denoted as *L*, its components along the *x*, *y*, and *z* axes are as follows:

$$\begin{cases} L\_x = \int\_{\theta\_1}^{\theta\_2} (-a\rho\_0 R \cos\theta \cos\varphi + \omega r\_i \cos\varphi) dt \\\ L\_\gamma = \int\_{\theta\_1}^{\theta\_2} (a\rho\_0 R \cos\theta \sin\varphi - \omega r\_i \sin\varphi) dt \\\ L\_z = \int\_{\theta\_1}^{\theta\_2} (a\rho\_0 R \sin\theta + h\omega\_0 / 2\pi) dt \end{cases} (6)$$

From the relationship between the rotation of the cutter and the rotation of the cutter, we can get ω0 = ω*ri* /*R*, the relative rotation angle *R*(φ − θ ) = *ri* ϕ. The relationship between θ and t is θ = φ − ω*ri t*/*R*, so the *d*θ = −(ω*ri* /*R*)*dt d*θ = −(ω*ri* /*R*)*dt* transformed to

$$dt = -\frac{Rd\theta}{\alpha r\_i} \tag{7}$$

Hence, by combining the sliding distances in three directions, the total sliding distance of the disc cutter for one rock-breaking cycle is obtained as follows:

$$L = \sqrt{L\_x^2 + L\_y^2 + L\_z^2} \tag{8}$$

### *3.2 Calculation of Radial Wear Amount*

As shown in Fig. 3, when the disc cutter makes one full rotation from initial contact with the rock to the deepest point of rock breaking at *M*, the relative rotation angle is ϕ. Therefore, the integration limits θ2 and θ1 are 0 and ϕ, respectively. Substituting Eqs. (1) and (8) into Eq. (2), we obtain the volume of wear at the point M from initial contact with the rock to the deepest point of rock breaking as V:

$$V = k\_d F \times L = k\_d \times F\_r \times \sqrt{L\_x^2 + L\_y^2 + L\_z^2} \tag{9}$$

Considering the basic uniformity of the equivalent stress peak values at different points on the cutter ring contact surface, and the relatively stable rotational angular velocity of the disc cutter, it is assumed that the radial wear is consistent at all points on the blade. The expression for the contact area A between the cutter ring and the rock, when the disc cutter completes one rock-breaking cycle, is given by:

$$A = 2\pi R$$

Hence, the magnitude of the radial wear amount δ is:

$$\delta = \frac{V}{A\_{01}} = \frac{k\_d F\_r}{2\pi R T} \times \sqrt{L\_x^2 + L\_y^2 + L\_z^2} \tag{11}$$

Assuming that the disc cutter completes a certain distance of excavation with Z rotations, the post-wear disc cutter radius is denoted as *Ri*+1 = *Ri* −δ*i* , and the blade width as *Ti*+1 <sup>=</sup> *Ti* +2δ*i* tan <sup>α</sup> <sup>2</sup>, where i represents the number of rotations of the disc cutter (i = 1, 2, 3, … Z), and α is the blade angle. Therefore, the cumulative radial wear amount H after Z rotations is given by:

$$H = \sum\_{i=1}^{Z} \delta\_i(i = 1, 2, 3...Z) \tag{12}$$

### **4 Analysis of Experimental Results**

### *4.1 Wear Mechanism Experiment*

In this study, we utilized a disc cutter wear experimental setup to conduct tool wear experiments under different environmental conditions. Through observations using a scanning electron microscope (SEM), we obtained the wear morphology of the tool materials. The wear results indicate that in dry/wet conditions, the primary cause of wear failure in excavation machine tools is abrasive wear. Based on the wear experimental results, this paper derives a formula for calculating tool wear, enabling the prediction and calculation of tool wear. Additionally, Fig. 4 presents the design of the disc cutter rock-breaking simulation experimental setup.

During the rock-breaking process of the tunnel boring machine, in order to reduce the heat generated during rock fragmentation, water spray devices are installed on the disc cutters to achieve cooling and suppress the generation of dust. Utilizing the established disc cutter wear experimental platform, wear tests of the disc cutter materials were conducted with the water spray apparatus. The maximum penetration depth of the disc cutter was set to 2 mm, the rotational speed of the disc cutter specimen was 30 rpm, and granite was chosen as the rock material. The post-wear morphology of the disc cutter is shown in Fig. 5.

**Fig. 4** Disc cutter rock-breaking simulation experimental apparatus

**Fig. 5** Tool wear profile under dry/wet conditions

**Fig. 6** Three-dimensional wear morphology

From Fig. 5, the wear morphology of the disc cutter material reveals that under dry/wet conditions, abrasive wear predominates on the surface of the disc cutter material. Under dry friction conditions, the wear of the disc cutter material is more severe, and there is a noticeable increase in the width of abrasive particle cutting depth.

Based on the three-dimensional morphology analysis in Fig. 6, it can be observed that the addition of water and similar mediums reduces the wear depth of the disc cutter material, indicating a lubricating effect of water. During the cutting of rock, the disc cutter generates numerous cutting furrows on the contact surface, along with a few occurrences of pits and surface oxidation fatigue wear. Under dry friction conditions, the number of furrows on the disc cutter surface is most pronounced, leading to more severe wear. With the introduction of water, the wear level of the disc cutter material is somewhat alleviated. However, abrasive wear remains the primary wear mechanism, albeit with a reduction in the extent of micro-cutting by rock abrasives.

In summary, abrasive wear is the primary cause of disc cutter wear, and it primarily occurs on the disc cutter surface in the form of micro-cutting.

### *4.2 Engineering Case Validation*

In a construction section of a tunnel within a mountain range in Zhejiang [11], the predominant geological composition of the strata consists mainly of granite and quartzite. The average uniaxial compressive strength of the rock is approximately 110.9 MPa, and the value of *kd* is taken as 2 × 10–6 mm3/(N mm). There are a total of 9 positive disc cutters installed on the cutterhead. The average penetration depth achieved by one rotation of the disc cutter for rock breaking is 4.12 mm. The arrangement of the disc cutters on the cutterhead is shown in Fig. 7, and the basic parameters of the disc cutters are listed in Table 1.

Figure 8 depicts the relationship between the cutter position number and the wear amount of the positive disc cutters, which was measured after the tunnel boring machine had operated for 471.16 m. As shown in Fig. 8, the predicted values of radial wear amount exhibit a trend that is generally consistent with the actual wear

**Fig. 7** Installation position diagram of disc disc cutter on cutterhead


**Table 1** Basic parameters of disc cutter

amount. Due to the complex underground construction conditions and factors such as rock anisotropy and joint characteristics, the predicted values may exhibit some errors when compared to actual wear amounts. However, the average error in the calculation results does not exceed 6%, indicating that the established model can reasonably predict the radial wear amount of the disc cutters during their operation.

# **5 Conclusion**

Wear mechanism experiments were conducted using a disc cutter wear test rig, confirming that abrasive wear is the primary mechanism responsible for disc cutter wear. By synthesizing the three movements of disc cutters, namely penetration motion, revolution of the disc cutter, and self-rotation, the sliding distance of the disc cutter after one rotation was determined. This formed the basis for establishing a calculation model for radial disc cutter wear. By inputting actual TBM construction parameters into the predictive model and comparing the results with actual tool wear values, the average error between theoretical values and actual values was within 6%.

**Acknowledgements** Support from the Liaoning Provincial Natural Fund Program [grant number 2019-KF-04-01] is gratefully acknowledged.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Path Tracking Control of Driverless Trucks**

**Wei Song, Junying Min, Tao Zhang, Yong Zhang, and Fengkui Zhao** 

**Abstract** Path tracking is a crucial function for achieving unmanned driving. This paper addresses the challenge of low tracking accuracy and poor stability in driverless trucks caused by uncertain model parameters and steady-state errors during path tracking. A linear quadratic regulator (LQR) controller optimization by an improved genetic algorithm has been designed. Firstly, the paper formulates the dynamic model of a two-degree-of-freedom vehicle as well as the model for tracking error. Subsequently, path tracking control is achieved through the utilization of feedforward control and LQR feedback control algorithms. Secondly, the weight coefficient of the LQR controller is enhanced through the utilization of an improved GA in order to boost the precision of path tracking. Ultimately, the devised controller undergoes simulation and validation in the TruckSim-Matlab/Simulink platform across diverse operational circumstances. The findings from the simulation demonstrate that the controller, optimized through improvements in the genetic algorithm, exhibits excellent tracking accuracy and stability.

**Keywords** Driverless trucks · Path tracking control · LQR · Genetic algorithm

# **1 Introduction**

In the past few years, the persistent expansion of global trade volume has resulted in a shortage of truck drivers in both domestic and foreign logistics and transportation industries. This shortage is evident in nations such as the United States, the United

W. Song · J. Min

Tongji University, Shanghai 200092, China

W. Song NAVECO, Nanjing 210037, Jiangsu, China

T. Zhang · Y. Zhang · F. Zhao (B) Nanjing Forestry University, Nanjing 210037, Jiangsu, China e-mail: zfk@njfu.edu.cn

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_49

619

Kingdom, Germany, China, Japan, India, and Brazil [1]. The deficit of truck drivers in the United States has intensified during the recent years, despite the truck transportation industry's efforts to meet the growing demand. Consequently, the utilization of driverless truck technology has surfaced as a substantial remedy to tackle the shortage of truck drivers in numerous countries. As a fundamental and critical component of autonomous driving, the main aim of path tracking control is to make the actual route of the vehicle as close as possible to the pre-planned route, minimizing lateral and heading errors. As a result, improving the precision and firmness of the trajectory tracking regulator becomes imperative and urgent.

The development of unmanned driving path tracking control began early, and currently the main methods for tracking control include Proportional–Integral– Differential (PID) control [2], Sliding Mode control [3], Fuzzy logic control [4], Model Predictive control (MPC) [5], and LQR control [6]. Among them, LQR control is a feedback control method designed based on optimization principles [7, 8]. The primary principle is to establish a linearized model for the path tracking error of the vehicle system and solve the optimal feedback control parameters through quadratic optimization within the cyclic control cycle [9]. The LQR regulator is widely utilized in unmanned vehicles to accurately follow paths due to its exact performance, easyto-follow algorithm, suitability for non-linear systems, and ability to take into account dynamic limitations. Currently, numerous scholars have conducted research on this topic and proposed various solutions. Hu et al. [10] introduced a fuzzy LQR with preview PID angle compensation that exhibits high tracking accuracy while maintaining stability across varying vehicle speeds. Xu et al. [11] developed a multi-point preview lateral controller based on the monorail dynamics model, implementing a LQR approach. They utilized the system's disturbance term as a state vector to construct an augmented LQR problem, resulting in enhanced path tracking accuracy. Wang et al. [12] presented an enhanced lateral path control strategy for vehicle navigation. The foundation of this strategy is constructed using an LQR algorithm while integrating a fuzzy control approach to actively adapt the weighting coefficients of LQR in real-time. This, in turn, enhances the precision of tracking, stability of steering, and effectiveness of computation. Although the aforementioned enhancements to the LQR have yielded favorable outcomes, there exists potential for additional advances in both control accuracy and stability.

This paper proposes the utilization of an improved genetic algorithm to optimize a LQR controller, aiming to enhance the tracking precision and stability of the controller. Firstly, a classic LQR controller was designed using the Two-Degreeof-Freedom vehicle dynamics lateral tracking error model. In order to eradicate the persistent deviation of the model's equilibrium state, feedforward control compensation was implemented based on this model. Secondly, this paper analyzes the effect of weight coefficients of the LQR controller on the vehicle's tracking capability. In order to achieve a balance between control stability and tracking accuracy, an enhanced genetic algorithm is applied for optimize the LQR weighting factors. Finally, the study concludes by conducting simulation experiments on both the preoptimized and post-optimized LQR controllers. The results consistently demonstrate the superior performance of the GA-LQR controller devised in this research.

# **2 Vehicle Path Tracking Control Model**

### *2.1 Vehicle Dynamic Model*

The vehicle's lateral dynamics are influenced by a multitude of factors. The complexity of the vehicle model in controller design needs to be minimized for theoretical analysis, requiring certain assumptions to be made for driverless trucks:


Considering the aforementioned assumptions, driverless trucks exhibit only two degrees of freedom: motion in the horizontal direction along the y-axis and movement in the direction of orientation around the z-axis. Here is the corresponding linear model depicting the two degrees of freedom, as depicted in Fig. 1, where O-x-y-z is the coordinate system of the vehicle.

Assuming a uniform speed for the vehicle and considering a small front wheel angle δ (less than 5°), expressed as cos δ ≈ 1, the vehicle dynamics equation can be

**Fig. 1** Vehicle dynamics model

represented as:

$$\begin{cases} ma\_{\rm y} = F\_{\rm yf} + F\_{\rm yr} = C\_f \alpha\_f + C\_r \alpha\_r = C\_f \left(\frac{\dot{\varphi} a + v\_y}{v\_x} - \delta\_f\right) + C\_r \left(\frac{v\_y - \dot{\varphi} b}{v\_x}\right) \\\\ I\_z \ddot{\varphi} = aF\_{\rm yf} - bF\_{\rm yr} = aC\_f \alpha\_f - bC\_r \alpha\_r \\\\ \quad = aC\_f \left(\frac{\dot{\varphi} a + v\_y}{v\_x} - \delta\_f\right) - bC\_r \left(\frac{v\_y - \dot{\varphi} b}{v\_x}\right) \end{cases} \tag{1}$$

where the moment of inertia about the z-axis is denoted as *Iz*; the vehicle mass is represented by *m*; the vehicle's heading rate is given by ϕ˙; the lateral forces of the front and rear wheels are indicated as *Fyf* and *Fyr* ; the values representing the distances from the vehicle's center of mass to the front and rear axles are identified as *a* and *b*; the lateral stiffness of the front and rear wheels is represented as *C f* and *Cr* ; the longitudinal speed of the vehicle is denoted by v*<sup>x</sup>* ; and finally, the lateral velocity of the vehicle is given by v*<sup>y</sup>* .

Simplify the above equation and define v*y* = *y*˙, thus obtaining the matrix form of the vehicle dynamics equation:

$$
\begin{pmatrix} \ddot{\mathbf{y}} \\\\ \ddot{\boldsymbol{\varphi}} \end{pmatrix} = \begin{pmatrix} \frac{\mathcal{C}\_f + \mathcal{C}\_r}{mv\_x} & \frac{a\mathcal{C}\_f - b\mathcal{C}\_r}{mv\_x} - \boldsymbol{v}\_x \\\\ \frac{aC\_f - b\mathcal{C}\_r}{I\_z v\_x} & \frac{a^2\mathcal{C}\_f + b^2\mathcal{C}\_r}{I\_z v\_x} \end{pmatrix} \begin{pmatrix} \dot{\mathbf{y}} \\\\ \dot{\boldsymbol{\varphi}} \end{pmatrix} + \begin{pmatrix} -\frac{\mathcal{C}\_f}{m} \\\\ -\frac{a\mathcal{C}\_f}{I\_z} \end{pmatrix} \delta\_f \tag{2}
$$

### *2.2 Lateral Error Model for Vehicle Path Tracking*

By applying coordinate transformation to the existing model depicting the vehicle's dynamics with two degrees of freedom, we can obtain a lateral error model specifically designed for path tracking, illustrated in Fig. 2.

Among them, the lateral error *ed* is defined as the shortest distance between the vehicle centroid and the reference path projection point. The heading angle error *e*<sup>ϕ</sup> = ϕ−θ*r* may be described as the difference between the true heading angle θ of the vehicle and the specified reference heading angle θ*r* . In real-life control applications, it becomes essential to develop a controller capable of eradicating these two errors instantaneously. This guarantees real-time accurate tracking of the intended path by the vehicle. The first derivatives of the lateral error *e*˙*d* and heading angle error *e*˙<sup>ϕ</sup> are computed by differentiating the lateral error *ed* and heading angle error *e*<sup>ϕ</sup> with respect to time.

$$
\dot{e}\_d = v\_\mathbf{x} e\_\varphi + v\_\mathbf{y} \tag{3}
$$

$$
\dot{e}\_{\varphi} = \dot{\varphi} - \dot{\varphi}\_r \tag{4}
$$

By combining Eq. (1), we can obtain the second derivatives of both the lateral error *e*¨*d* and the heading angle error *e*¨ϕ.

$$
\begin{split}
\ddot{e}\_d &= \left(\frac{C\_f + C\_r}{m\upsilon\_x}\right)\dot{e}\_d + \left(-\frac{C\_f + C\_r}{m}\right)e\_\varphi + \left(\frac{aC\_f - bC\_r}{m\upsilon\_x}\right)\dot{e}\_\varphi \\
&+ \left(\frac{aC\_f - bC\_r}{m\upsilon\_x} - \upsilon\_x\right)\dot{\theta}\_r + \left(-\frac{C\_f}{m}\right)\delta\_f
\end{split}
\tag{5}
$$

$$
\begin{split}
\ddot{e}\_{\varphi} &= \left(\frac{aC\_f - bC\_r}{I\_\varepsilon v\_x}\right)\dot{e}\_d + \left(-\frac{aC\_f - bC\_r}{I\_\varepsilon}\right)e\_\varphi + \left(\frac{a^2C\_f + b^2C\_r}{I\_\varepsilon v\_x}\right)\dot{e}\_\varphi \\
&+ \left(\frac{a^2C\_f + b^2C\_r}{I\_\varepsilon v\_x}\right)\theta\_r + \left(-\frac{aC\_f}{I\_\varepsilon}\right)\delta\_f
\end{split}
\tag{6}
$$

By further modifying the above mentioned formula, the derivation of state space equations capturing lateral and heading errors in unmanned vehicle steering is achievable. The obtained equations are as follows:

$$
\begin{pmatrix}
\dot{e}\_d \\
\ddot{e}\_d \\
\dot{e}\_\varphi \\
\ddot{e}\_\varphi
\end{pmatrix} = \begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & \frac{C\_f + C\_r}{mv\_x} & -\frac{C\_f + C\_r}{m} & \frac{aC\_f - bC\_r}{mv\_x} \\
0 & 0 & 0 & 1 \\
0 & \frac{aC\_f - bC\_r}{I\_\varepsilon v\_x} & -\frac{aC\_f - bC\_r}{I\_\varepsilon} & \frac{a^2C\_f + b^2C\_r}{I\_\varepsilon v\_x}
\end{pmatrix} \begin{pmatrix}
e\_d \\
\dot{e}\_d \\
e\_p \\
\dot{e}\_\varphi
\end{pmatrix}
$$

$$+\begin{pmatrix} 0\\ -\frac{C\_f}{m} \\ 0 \\ -\frac{aC\_f}{I\_\natural} \end{pmatrix} \delta\_f + \begin{pmatrix} 0\\ \frac{aC\_f - bC\_r}{mv\_x} - v\_x \\ 0 \\ \frac{a^2C\_f + b^2C\_r}{I\_\natural v\_x} \end{pmatrix} \dot{\theta}\_r \tag{7}$$

The state space equation for the tracking error of vehicle dynamics can be derived as follows:

$$
\dot{\mathbf{X}} = \mathbf{A}\mathbf{X} + \mathbf{B}\mathbf{U} + \mathbf{C}\dot{\theta}\_r \tag{8}
$$

### **3 Design of Path Tracking Controller**

The algorithm flowchart of the path tracking controller is depicted in Fig. 3. By combining the given path planning information with real-time vehicle status and pose information, the path tracking error can be calculated. To address the hysteresis in algorithm control, a prediction module is introduced following the real-time vehicle status information. This enables the algorithm to predict road conditions, thereby enhancing control accuracy.

### *3.1 LQR Controller Design*

The essence of the LQR controller is to seek the optimal control law for control variables and state variables while operating within linear constraints. Even when the following objective functions reach their minimum values.

**Fig. 3** Path tracking controller

Research on Path Tracking Control of Driverless Trucks 625

$$\min \mathbf{J} = \sum\_{k=0}^{\infty} \left( \mathbf{X}\_{\mathbf{k}}^{\mathrm{T}} \mathbf{Q} \mathbf{X}\_{\mathbf{k}} + \mathbf{U}\_{\mathbf{k}}^{\mathrm{T}} \mathbf{R} \mathbf{U}\_{\mathbf{k}} \right) \tag{9}$$

$$\mathbf{Q} = \text{diag}[q\_1, q\_2, q\_3, q\_4] \tag{10}$$

$$\mathbf{R} = [q\_{\mathcal{S}}] \tag{11}$$

where the control variable of the system is denoted as **U**; the system's state parameter is denoted as **X**; the weighted coefficient **Q** representing the error represents the extent of significance attached to the corresponding control target; the control weighting matrix is denoted as **R**.

The feedback control law of LQR controller is:

$$\mathbf{U(k)} = -\mathbf{K}\mathbf{X(k)}\tag{12}$$

In the formula: **K** = (*R* + **BTPB**)−<sup>1</sup>**BTPA**, where the control gain coefficient is denoted as **K**; **P** is a constant matrix, consisting of a positive definite solution to the Riccati equation as follows.

$$\mathbf{P} = \mathbf{Q} + \mathbf{A}^T \mathbf{P} \mathbf{A} - \mathbf{A}^T \mathbf{P} \mathbf{B} (\mathbf{R} + \mathbf{B}^T \mathbf{P} \mathbf{B})^{-1} \mathbf{B}^T \mathbf{P} \mathbf{A} \tag{13}$$

Substituting Eq. (12) into Eq. (9) yields:

$$\min J = \sum\_{k=0}^{\infty} \mathbf{X}^T (\mathbf{Q} + \mathbf{K}^T \mathbf{R} \mathbf{K}) \mathbf{X} \tag{14}$$

By introducing a comprehensive deviation of lateral distance deviation and heading angle deviation *err* as the system state variables, and controlling the front wheel angle δ *<sup>f</sup>*, the LQR controller can acquire the ideal control law for the front wheel angle:

$$\delta\_f(k) = -\mathbf{K} \mathbf{e}\_{\mathbf{r}\mathbf{r}}(\mathbf{k})\tag{15}$$

where **K** = [*k*1, *k*2, *k*3, *k*4] is the feedback gain matrix of the LQR controller; **err** = [*ed* , *e*˙*d* , *e*ϕ, *e*˙ϕ] *<sup>T</sup>*is the comprehensive deviation amount.

### *3.2 Feedforward Control*

Substituting Eq. (15) into Eq. (8) yields the following result:

$$
\dot{\mathbf{X}} = (\mathbf{A} - \mathbf{B}\mathbf{k})\mathbf{X} + \mathbf{C}\dot{\theta}\_r \tag{16}
$$

Currently, irrespective of the magnitude of **k**, it is impermissible for **X** ˙ to be zero. Solely employing LQR feedback control will induce an enduring stagnant error. As a consequence, to eradicate the effect of the **C** θ˙ *<sup>r</sup>*parameter, the feedforward control variable δ *ff* is implemented. The system's state variable, with the inclusion of feedforward control, can be expressed as:

$$\mathbf{U} = -\mathbf{k}\mathbf{X} + \delta\_{ff} \tag{17}$$

In the event that there is zero steady-state error, When **X** ˙ <sup>=</sup> 0, the equation for the state variable of the system is as follows:

$$\mathbf{X} = -(\mathbf{A} - \mathbf{B}\mathbf{k})^{-1}\mathbf{A}(\mathbf{B}\delta\_{ff} + \mathbf{C}\dot{\theta}\_r) \tag{18}$$

According to Eq. (18), it is inferred that attaining the optimal control effect requires determining a suitable value δ *ff* that ensures the system exhibits zero steady-state error. The system's equation for steady-state error is determined by utilizing Eq. (7) and is given by the following expression:

$$
\begin{bmatrix} e\_d \\ \dot{e}\_d \\ e\_\varphi \\ \dot{e}\_\varphi \end{bmatrix} = \begin{bmatrix}
\frac{1}{k\_l} \left[ \delta\_{ff} - \frac{\dot{\vartheta}\_r}{v\_x} \left[ a + b - bk\_3 - \frac{mv\_x^2}{a+b} \left( \frac{b}{C\_f} + \frac{a}{C\_r} k\_3 - \frac{a}{C\_r} \right) \right] \\\ 0 \\\ -\frac{\dot{\vartheta}\_r}{v\_x} \left( b + \frac{a}{a+b} \frac{mv\_x^2}{C\_r} \right) \\\ 0 \\\ \end{bmatrix} \tag{19}
$$

From Eq. (19), it is evident that:

$$e\_{\varphi} = -\frac{\dot{\theta}\_r}{v\_x} \left( b + \frac{a}{a+b} \frac{mv\_x^2}{C\_r} \right) = -\beta \tag{20}$$

In order to attain sideways deviation *ed* = 0, the feedforward control amount δ *ff*  would be:

$$\delta\_{ff} = \rho \left[ a + b - bk\_3 - \frac{mv\_x^2}{a+b} \left( \frac{b}{C\_f} + \frac{a}{C\_r} k\_3 - \frac{a}{C\_r} \right) \right] \tag{21}$$

where <sup>ρ</sup> <sup>=</sup> <sup>θ</sup>˙ *r*  <sup>v</sup>*x* is the path curvature.

### *3.3 LQR Controller Optimized Based on GA*

Genetic algorithm [13, 14] is a series of search algorithms inspired by natural evolution theory. The algorithm's exceptional convergence, efficient computational time, and robustness make it a popular choice for parameter optimization problems. And there are not many mathematical requirements for the optimization problem being solved. Owing to its characteristics, the search process does not rely on the intrinsic properties of the problem and can accommodate any type of objective function and constraint, regardless of whether they are linear or nonlinear, discrete or continuous in nature. In the design of LQR controllers, choosing appropriate Q and R values is crucial for achieving effective tracking with LQR controllers. The specific process of optimizing matrices Q and R using genetic algorithm is shown in Fig. 4.

Firstly, encode the parameters, including the fundamental parameters of the vehicle along with the coefficients of the LQR state space equation. Secondly, a population is generated, and each individual population value is assigned sequentially to the randomly generated weight coefficients of LQR, within a feasible range. Reuse the generated weight coefficients Q and R to calculate the optimal feedback

**Fig. 4** Genetic algorithm


**Table 1** The optimal solution of LQR weight matrix

gain matrix K of LQR. Finally, the Simulink model is then executed to compute the pertinent parameters, and based on these parameters, the fitness function is evaluated to determine the fitness value of each particle. Once the termination condition is satisfied, the optimal value will be displayed.

Based on the optimization design method described above, the Table 1 displays the optimal values of the final LQR weight matrix.

### **4 Simulation Verification**

In order to confirm the efficiency of the LQR controller, optimized using the proposed improved genetic algorithm, a joint simulation experiment was conducted using TruckSim and Matlab/Simulink. The primary parameters of the vehicle are shown in Table 2.

In this article, the double lane change scenario is employed as the benchmark path, depicting the standard driving conduct of cars navigating through barriers. The automobile is put to the test on a road with a friction coefficient of 0.85, experiencing two different velocities: 30 and 60 km/h. Simulations are conducted for the LQR and GA-optimized LQR controllers at both speeds. Figure 5 shows the tracking effect of vehicle lateral displacement. The diagram demonstrates that the car exhibits satisfactory performance when following the designated route at both velocities. Nevertheless, the LQR controller designed in this study, using GA-optimized weight coefficients, exhibits superior tracking performance and enhanced stability.


**Fig. 5** Lateral displacement tracking effect

In Fig. 6, the tracking accuracy of the LQR controller and the GA-LQR controller in managing lateral errors is showcased in the context of double line shifting situations, precisely at velocities of 30 and 60 km/h. According to the results findings, the LQR controller has the ability to consistently restrict the maximum tracking discrepancy to 0.2848 m at different velocities. The lateral error of the GA-LQR controller designed in this article is considerably lower compared to that of the LQR controller. At a velocity of 30 km/h, the GA-LQR controller can effectively limit the maximum lateral error to 0.0704. This value, however, is approximately 49.7% greater than that of traditional LQR controllers. At a velocity of 60 km per hour, the GA-LQR controller attains a peak lateral deviation of 0.1429. This represents an enhancement of approximately 49.8% compared to the effectiveness of the traditional LQR controller.

This implies that the LQR controller designed in this article can more efficiently minimize errors and enhance the precision of vehicle tracking during path tracking.

Figure 7 illustrates the impact of dual line shifting conditions on the yaw error tracking for both the LQR and the GA-LQR controller at speeds of 30 and 60 km/ h. The simulation results demonstrate that the yaw error remains within 0.1145 rad across various vehicle speeds, and the observed variation is insignificant. These findings indicate the vehicle's tracking process maintains a relatively stable performance. The GA-LQR controller successfully limits the maximum yaw error to 0.0984 at a speed of 30 km/h, and 0.1114 at a speed of 60 km/h. These values demonstrate an improvement over the performance of traditional LQR controllers.

Figure 8 depicts the performance of the front wheel angle tracking for both the LQR controller and the GA-optimized LQR controller under dual lane shifting conditions, at speeds of 30 and 60 km/h. The simulation results reveal a gradual increase in the front wheel angle as the vehicle speed rises, without any step changes. And the front wheel angle is effectively controlled within a range of −10° to 10° at various vehicle speeds, which strongly suggests a stable tracking process. Overall,

**Fig. 6** Lateral error tracking effect

**Fig. 7** Yaw error tracking effect

the GA-LQR controller demonstrates satisfactory levels of tracking accuracy and stability.

### **5 Conclusions**

This study addresses the issue of low tracking accuracy and poor stability in unmanned trucks during path tracking under traditional LQR controllers. To combat these problems, the inclusion of feedforward control methods is proposed to compensate for the front wheel angle and mitigate steady-state errors resulting from the

**Fig. 8** Front wheel angle tracking effect

simplification of the vehicle dynamics model. The weight coefficients are optimized and adjusted using a genetic algorithm to achieve higher tracking accuracy while simultaneously addressing the vehicle's driving stability. The outcomes of the simulation demonstrate that the GA-LQR controller, developed in this article, exhibits commendable tracking accuracy and stability when subjected to the double lane reference path. This research achievement offers innovative insights for designing and optimizing path tracking controllers for driverless trucks. In future research, it is necessary to conduct further validation of the proposed control strategy using an actual vehicle.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Prediction of Mechanical Properties and Analysis of Damage Evolution of Fiber Bundles in Carbon Fiber Reinforced Composite Materials**

**Rongjiao Guo and Renjun Yan** 

**Abstract** Fiber bundles are an important component of woven composite materials, and predicting the mechanical properties of fiber bundles can provide a basis for the study of the mechanical properties of woven composite materials. This paper establishes the micro representative volume element (RVE) model of composite materials, and obtains the equivalent elastic constant of yarn through the model homogenization theory and periodic boundary conditions. Strength prediction is performed through the VUMAT user subroutine of ABAQUS. This paper uses the maximum stress standards and Von Mises standards to predict the damage initiation of TC33 carbon fiber and epoxy resin matrix, respectively. Combined with the constant degradation method, the simulation of the damage behaviors of the micro model is achieved, and the equivalent strength of the fiber bundle is obtained. The effectiveness and correctness of this method are verified by comparing the numerical model results with the Chamis theoretical model results. The accurate prediction of mechanical properties and damage process of fiber bundles provides theoretical support for the analysis of mechanical properties of composites, and has guiding significance for the performance design of composite materials.

**Keywords** Composites · Microstructure · VUMAT subroutine · Equivalent modulus · Progressive damage · Ultimate strength

R. Guo

R. Yan (B)

© The Author(s) 2024

School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China

Educational Ministry Key Lab of High Performance Ship Technology, Wuhan University of Technology, Wuhan 430063, China e-mail: yan\_renjun@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_50

### **1 Introduction**

Composite material is a kind of multiphase material composed of two or more materials. Compared with raw material, composite material has the advantages of high specific stiffness, high specific strength and designability [1]. The fibers are mixed with the matrix to form fiber bundles, which is woven and mixed with the matrix to form a woven composite material, as shown in Fig. 1.

In order to make better use of composite materials, it is necessary to understand the stiffness and strength properties of composite materials, and the relationship between these properties and the components of composite materials. In order to meet the requirements of engineering, the mechanical properties of fiber bundles can be predicted to predict the properties of composites, which can provide reference for the design and optimization of composites.

For predicting the mechanical properties of fiber bundles, there are mainly theoretical formulas and finite element methods. In the classical RVE, fiber distribution is considered to be uniform and uniform. Jha [2] established a cell model with uniform fiber distribution, studied the effects of fiber geometry and fiber volume fraction on the mechanical properties of composites. It was found that the fiber geometry had an effect on the transverse Young's modulus, the influence on the axial Young's modulus is negligible. Wang [3] used the finite element method to establish the unit cell model of micro-fiber bundle with uniform and random distribution of fibers in carbon fiber bundles, and carried on the analysis to it, the equivalent elastic modulus and strength of the fiber bundle are predicted, and the difference of the damage caused by the arrangement is compared. Vignoli [4] established the prediction model of tensile strength and shear strength of unidirectional fiber composites, and compared with other prediction models, it was found that the prediction accuracy of this method was higher. Han [5] established a microscopic RVE model of random distribution of fibers and studied the damage initiation and evolution of unidirectional fiber composites under various loads, the main failure modes of axial compression are fiber buckling and matrix collapse. Therefore, It is important to predict the modulus and strength of fiber bundles and analyze the damage evolution of fiber bundles by using the fiber random distribution cells.

In this paper, the mechanical properties of fiber bundles are obtained by establishing a representative volume element finite element model and applying periodic boundary conditions, compared with the classical Chamis model, the method is proved to be effective. The damage evolution of the micro-scale cell is analyzed, and the generation and development of the damage of the fiber bundle are explained.

### **2 Periodic Boundary Conditions**

For the periodic arrangement of micro fiber bundle RVE models, two conditions should be met simultaneously at the cell boundary: (1) deformation coordination, and (2) stress continuity, as shown in Fig. 2.

For a cell model with parallel and paired boundaries, Xia [6] proposed a periodic boundary conditions suitable for finite element analysis, which requires that the meshes on opposite surfaces are the same. A periodic displacement field is applied to the surface of the RVE representative volume element, which can obtain the relevant effective elastic coefficiency based on the microscopic mechanical response. The expression form of periodic displacement field is:

$$
\mu\_i = \varepsilon\_{ik}\mathbf{x}\_k + \mu\_i^\* \tag{1}
$$

where ε*ik* is the average strain of the single cell structure, *xk* is the position function at any position within the single cell, and *u*<sup>∗</sup> *<sup>i</sup>*is the strain correction. The boundary conditions on one side of the rectangular cuboid cell structure can be expressed as:

$$
\mu\_i^{j+} = \varepsilon\_{ik} \mathbf{x}\_k^{j+} + \mu\_i^\* \tag{2}
$$

The boundary conditions of its opposite surface are:

$$
u\_i^{j-} = \varepsilon\_{ik} \mathbf{x}\_k^{j-} + \mathbf{u}\_i^\* \tag{3}$$

Subtracting the boundary conditions on one side from the boundary conditions on the other side yields:

**Fig. 2** Periodic boundary conditions of RVE

$$
\mu\_i^{j+} - \mu\_i^{j-} = \varepsilon\_{ik} \left( \mathbf{x}\_k^{j+} - \mathbf{x}\_k^{j-} \right) \tag{4}
$$

# **3 The Solution of Equivalent Elastic Constants of RVE Model**

### *3.1 Establishment of RVE Model*

The research object of this article is carbon fiber reinforced resin based fiber bundles, usually assuming that carbon fibers are unidirectionally distributed in the matrix and have a circular cross-section. Fiber bundles can be regarded as many single cell structures arranged periodically in three spatial directions. Therefore, a representative volume element (RVE) can be selected to establish a single cell model with random fiber distribution to calculate the performance of fiber bundles.

Using the Random Sequence Adsorption (RSA) method [7], a cubic RVE model was established for the random distribution of fibers in the matrix, as shown in Fig. 3, with a unit cell size of 35 μm × 35 μm × 35 μm. This article studies the case where the volume content of fibers in fiber bundles is 65.12%. It is specified that the fiber arrangement direction is in the x direction and the vertical fiber direction is in the y and z directions. The model is meshed by voxel grid, and the element type is C3D8R (eight node linear Hexahedron element, reduced integral, hourglass control), a total of 27,000 elements. The RVE model modeling parameters are shown in Table 1.

The RVE includes the matrix phase and the fiber phase, as shown in Fig. 3, the matrix is an isotropic material, and the carbon fiber is a transverse anisotropic

**Fig. 3** Micro-scale RVE


**Table 1** Modeling parameters of the RVE




material. All material parameters are taken from reference [8, 9], as shown in Tables 2 and 3.

### *3.2 Solution of Equivalent Elastic Constant*

Apply the periodic boundary conditions mentioned in Sect. 2 to the RVE model to obtain the stress nephogram in all directions as shown in Fig. 4.

From Fig. 4a, it can be seen that when subjected to tension along the fiber direction, the RVE has the maximum modulus, and the fiber bears most of the load. Compared with the other stress nephogram in Fig. 4, the tensile modulus in Y and Z directions and the shear modulus in XY, XZ and YZ directions are relatively small, and the fiber is no longer subject to most forces. From the Stress–strain curve of the results, the prediction data of equivalent elastic constants of carbon fiber bundles can be obtained as shown in Table 4.

The Chamis theoretical model [10] is a classic model for the equivalent mechanical constants of composite materials, in which the prediction formula for the equivalent elastic constants of unidirectional composite materials is as follows:

**Fig. 4** RVE stress cloud diagram: **a** X-direction stretching, **b** Y-direction stretching, **c** Z-direction stretching, **d** XY direction shearing, **e** XZ direction shearing, **f** YZ direction shearing


**Table 4** Prediction of elastic constants of fiber bundles in all directions

$$\begin{cases} E\_1 = V\_f E\_{f1} + (1 - V\_f) E\_m \\ E\_2 = E\_3 = \frac{E\_n}{1 - \sqrt{V\_f} \left(1 - E\_m / E\_{f2} \right)} \\ G\_{12} = G\_{13} = \frac{G\_n}{1 - \sqrt{V\_f} \left(1 - G\_n / G\_{f12} \right)} \\ G\_{23} = \frac{G\_n}{1 - \sqrt{V\_f} \left(1 - G\_n / G\_{f23} \right)} \\ \mu\_{12} = \mu\_{13} = V\_f \mu\_{f12} + \left(1 - V\_f \right) \mu\_m \\ \mu\_{23} = \frac{E\_2}{2G\_{23}} - 1 \end{cases} (5)$$

where *E*f1, *E*f2 are axial and transverse elastic modulus of fiber, respectively. *G*f12, *G*f23 are axial and transverse shear modulus of fiber, respectively. *E*m is matrix elastic modulus. *G*m is matrix shear modulus. μm is matrix Poisson ratio. *V*f is fiber volume fraction.

The predicted elastic constants in all directions of the fiber bundle obtained from the theoretical model are shown in Table 4. Because the established finite element model is a random distribution of fibers, the predicted fiber bundles are anisotropic materials. However, in Chamis theory, it is assumed that the fibers are uniformly arranged, and the predicted fiber bundles are transversely isotropic materials. So for finite element models, there are differences in modulus between the transverse and axial directions, while in theoretical predictions, the transverse modulus is equal to the axial modulus. Compare the average values of the axial and transverse modulus results of the finite element method with the Chamis theory results, and the numerical values and relative errors are shown in Table 4.

From the relative errors in Table 4, it can be seen that the results of finite element simulation have a certain degree of accuracy. The error along the fiber direction is the smallest, while the error in the vertical fiber direction is slightly larger. The reason is that in the unit cell model built in this article, the fibers are randomly distributed, while in Chamis theory, the fibers are assumed to be uniformly and neatly arranged, and the random arrangement of fibers is closer to the actual situation. Moreover, the Shear modulus does not consider the influence of the interface between the fiber and the matrix, which will lead to a certain gap in the results.

### **4 RVE Model Strength Analysis**

# *4.1 Material Performance Degradation Plan and Analysis Process*

Fiber bundles are composed of fibers and matrices, and VUMAT subroutines are written to first determine the type of unit and run different strength standards for different units. Among them, epoxy resin matrix is considered an isotropic linear elastic material, and various strength standards applicable to isotropic materials can determine matrix damage. If it is a matrix unit, the Von Mises standards [11] is used to determine the failure of the unit. The formula is:

$$\sqrt{\frac{1}{2}\left[\left(\sigma\_1-\sigma\_2\right)^2+\left(\sigma\_2-\sigma\_3\right)^2+\left(\sigma\_3-\sigma\_1\right)^2\right]} \le \left[\sigma\right] \tag{6}$$

If it is judged as a fiber unit, use the maximum stress standards to determine the damage of the fiber. When any of the three directions in space reaches the strength value of the fiber in that direction, the fiber begins to fail. The specific formula is as follows:

$$\max\left\{\frac{\sigma\_1}{X}, \frac{\sigma\_2}{Y}, \frac{\sigma\_3}{Z}\right\} = 1\tag{7}$$

where *Xi f t* , *Xi f c*, *Sij f* are the tensile strength, compressive strength and shear strength of the fiber in three directions.

Based on the above damage evolution theory, the fiber and matrix constitutive equation, failure criteria and stiffness reduction programs were written in Fortran language, and the damage evolution process of cells was simulated using the VUMAT subroutine in ABAQUS. The damage evolution of materials is a process of gradual accumulation. After loading the model, calculate the strain, stiffness matrix and stress, and judge whether the material fails according to the failure criteria. If there is no failure, update the stress and strain, and if there is failure, update the stiffness matrix.

### *4.2 Analysis of Numerical Simulation Results*

Based on the above research, the microscopic unit cell model is established, periodic boundary conditions are applied, and the connection between the loading point (reference point RP) and the corresponding node is established, as shown in the figure. Apply X, Y, and Z directions to a single cell, with a displacement of 20% of the cell's side length, which is 7 μm setting unit deletability and setting unit deletion variables, using explicit analysis, can effectively simulate the degradation and failure of such materials. The obtained stress–strain results are shown in the Figs. 5, 6, 7, and 8.

The stress–strain curve in the axial direction is almost linear, because the fibers are the main force under axial force, and the strength is also determined by the fibers. When subjected to lateral forces, the matrix is mainly subjected to force, so the stress–strain curve exhibits a non-linear characteristic. The highest point of the Stress–strain curve is taken as the strength value, and the predicted results are shown in Table 5.

From Fig. 9, it can be seen that due to the much greater tensile strength of TC33 carbon fiber in the axial direction than the resin matrix, during the progressive damage process, the resin matrix will first undergo damage, followed by subsequent fiber damage and failure. Strain in Fig. 9a ε = 0.54%, the fibers and matrix have not yet

**Fig. 5** Axial tensile stress–strain curve

**Fig. 6** Axial compression stress–strain curve

**Fig. 7** Transverse tensile stress–strain curve

**Fig. 8** Transverse compressive stress–strain curve


**Table 5** Fiber bundle strength in all directions

been damaged, and the fibers bear most of the tensile load. The performance of the matrix gradually deteriorates, and the elastic modulus slowly decreases. When strain ε = 1.31%, as shown in Fig. 9b, the matrix has been damaged, but the material still has load-bearing capacity as the fibers have not been damaged. When loaded into ε = 1.49%, the carbon fiber loses its load-bearing capacity, and the fiber bundle immediately fails after the fiber breaks, resulting in a decrease in stress to 0. As shown in Fig. 9c, when ε = 1.51%, the single cell model has completely failed, and both the fiber and matrix have lost their load-bearing capacity.

The Chamis theoretical model also proposes a strength prediction formula for composite materials:

**Fig. 9** Damage evolution of the RVE under axial tension: **a** ε = 0.54%, **b** ε = 1.31%, **c** ε = 1.51%

$$\begin{cases} X\_{\rm T} = V\_{\rm f} X\_{\rm f\Gamma} \\ X\_{\rm C} = V\_{\rm f} X\_{\rm f\Gamma} \\ Y\_{\rm T} = S\_{\rm m\Gamma} \left( 1 - \left( \sqrt{V\_{\rm f}} - V\_{\rm f} \right) (1 - E\_m / E\_{\rm f2}) \right) \\ Y\_{\rm C} = S\_{\rm m\Gamma} \left( 1 - \left( \sqrt{V\_{\rm f}} - V\_{\rm f} \right) (1 - E\_m / E\_{\rm f2}) \right) \\ S\_{12} = S\_{13} = S\_{\rm m\S} \left( 1 - \left( \sqrt{V\_{\rm f}} - V\_{\rm f} \right) (1 - G\_m / G\_{\rm f12}) \right) \\ S\_{23} = S\_{\rm m\S} \left( 1 - \sqrt{V\_{\rm f}} / (1 - G\_m / G\_{\rm f23}) \right) / (1 - V\_{\rm f} (1 - G\_m / G\_{\rm f23})) \end{cases} (8)$$

where *X*f*<sup>T</sup>*, *X*fC are fiber tensile strength and compressive strength, *S*mT, *S*mC, *S*mS are tensile strength, compressive strength and Shear strength of matrix.

According to the Chamis strength prediction model, the strength of the fiber bundle in all directions was calculated, and the results are shown in Table 5. X, Y, Z are directions, subscript T is the tensile strength, C is compressive strength, S is the shear strength, 1, 2, 3 are shear in x, y, z directions.

For the prediction of axial strength, the finite element simulation results are in good agreement with the Chamis theory results, as the tensile and compressive loads along the fiber direction are mainly borne by the fibers and are not significantly related to the cross-sectional shape and distribution of the fibers. For the prediction of lateral strength, the Chamis model assumes that the fibers are uniformly and neatly arranged, while the fiber random arrangement model in this article is closer to the actual situation. However, the influence of the interface between fibers and matrix on strength was not considered in the model in this article, which will result in certain errors. Overall, the relative error between the strength prediction results in each direction and the theoretical model is within an acceptable range, verifying the feasibility of the prediction method in this paper.

And from Table 5, it can be seen that the axial modulus and axial tensile strength of fiber bundle materials are the highest, with 149.86 GPa and 2239.74 MPa, respectively, while the transverse modulus and tensile strength are only 9.97 GPa and 41.14 MPa. It indicates that fiber bundles have good mechanical properties in the fiber direction, while other directions have poor performance. In engineering applications, it is necessary to consider the differences in mechanical properties in different directions and make more effective use of materials.

### **5 Conclusions**

This article uses the representative volume unit method to obtain the equivalent modulus of fiber bundle cells, and combines the constant degradation method to predict the progressive damage process. The damage evolution of the microscopic model and the equivalent strength of the fiber bundle are obtained, and the following conclusions are drawn:


### **References**


11. Giraldo-Londoño O, Paulino GH (2020) A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler–Pister and Willam-Warnke. Proc R Soc A Math Phys Eng Sci 476(2238):20190861. https://doi.org/10.1098/rspa.2019.0861

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Kalman Filter-based Vibration State Estimation and Optimal Control of a Special Vehicle**

**Bowen Zhang, Ming Zhang, and Yunbo Zhou** 

**Abstract** Take a certain type of special vehicle as the research object, establish the four-degree-of-freedom suspension vibration time-domain simulation model of the semi-vehicle, and use the Kalman filter algorithm for the vibration state of the vehicle in the driving process. By designing the Kalman observer of the linear suspension system, the state estimation of the vertical displacement and vertical acceleration of the front and rear suspensions and the body of the vehicle travelling on the uneven road surface is carried out, and the simulation analysis of the estimation effect is carried out, and the simulation results show that: the algorithm is able to estimate the relevant parameters of the vehicle vibration more accurately under the random road surface of E-level. Based on the state estimation value, a linear-quadratic optimal control method is adopted for the active control of suspension, which provides a basis for the design and optimal control of active suspension for subsequent special vehicles.

**Keywords** Automobile suspension · Four degrees of freedom model for half car · Kalman filtering · State estimation

# **1 Introduction**

The driving conditions of special vehicles are relatively complex, usually driving on emergency military roads, rural dirt roads, and roadless areas. In such off-road environments, the excitation generated by road roughness can seriously affect the smoothness, safety, and handling stability of special vehicles [1]. Active suspension can dynamically adaptively adjust the stiffness and damping characteristics of the suspension system based on the driving conditions of the vehicle, effectively reducing the impact and vibration transmitted from the road surface to the vehicle

B. Zhang · M. Zhang · Y. Zhou (B)

School of Mechanical Engineering, Nanjing University of Science and Technique, Nanjing 210094, Jiangsu, China e-mail: yunbo31983@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_51

body. Therefore, research on suspension control strategies is of great significance for improving the overall performance of special vehicles.

In the research of active suspension control, vehicle state estimation is crucial and is also the foundation of suspension controller design. In most studies, acceleration sensors or displacement sensors are used to measure the acceleration of spring loaded/ non spring loaded masses or the relative displacement of the suspension, and then these signals are processed in the back-end to obtain the required state variables. The Kalman filtering algorithm is a relatively classic algorithm in vehicle system state estimation. Reference [2] uses the Kalman filtering algorithm to estimate the vehicle's center of mass sideslip angle in real-time and eliminate yaw rate errors; Reference [3] uses extended Kalman filtering to estimate vehicle state parameters such as yaw rate, center of mass sideslip angle, and longitudinal velocity, but there is currently little research applied to suspension state estimation.

Aiming at the problem of state variables required in the optimal control of active suspension of special vehicles, this paper establishes a vehicle vibration state estimation model based on Kalman filter algorithm, which estimates the state of the vertical displacement and vertical acceleration of the front and rear suspension and the body of the vehicle traveling on uneven roads. Based on the state estimation value, the linear quadratic form optimal control method is used to actively control the suspension, Provide a foundation for optimal control of active suspension of special vehicles.

### **2 Road Input Model and Vehicle Model**

### *2.1 Pavement Input Model*

The variation of the height q of the pavement relative to the datum plane along the length I of the road alignment is called the pavement longitudinal profile or pavement unevenness function q(I).

When measuring the unevenness of the pavement, the professional pavement meter or level can be used to measure on the actual pavement to get the unevenness value on the longitudinal section of the pavement, and the large amount of data obtained from the measurement can be imported into the computer for processing to get the statistical characteristic parameters such as the power spectral density function of the unevenness of the pavement, Gq(n), or the variance, σ<sup>2</sup> q, and so on.

As the vehicle vibration input of the road surface unevenness, the road surface power spectral density function is mainly used to describe its statistical characteristics, the road surface power spectral density function fitting expression is as follows:

$$G\_q(n) = G\_q(n\_0) \left(\frac{n}{n\_0}\right)^{-W} \tag{1}$$

where: Gq(n)-spatial power spectral density of pavement; Gq(n0)-pavement unevenness coefficient at reference spatial frequency; n-spatial frequency; n0-reference spatial frequency, taking the value of 0.1 m−1; W-frequency index, taking the value of 2.

The more common method to simulate the pavement unevenness excitation is to generate the pavement unevenness excitation by filtering white noise, which is suitable to meet the international and national standards of pavement unevenness excitation generation [4], and its time domain expression is as follows:

$$\dot{q}(t) = -2\pi n\_q \mu q(t) + 2\pi n\_0 \sqrt{G\_q(n\_0)} \mu w(t) \tag{2}$$

where: u-vehicle speed; w(t)-standard Gaussian white noise with mean 0 and variance 1; q(t)-excitation of the road surface unevenness generated; nq-spatial lower cut-off frequency, which takes the value of 0.01/m. Considering that the driving environment of special vehicles is relatively harsh, the simulation is carried out by using the working condition of 30 km/h through the E-class road surface, in which Gq(n0) is 4096 × 10–6 m3.

### *2.2 4-Degree-of-Freedom Semi-Vehicle Model*

A 4-degree-of-freedom 1/2 vehicle dynamics model is established, whose four degrees of freedom are the vertical motion of the body, the pitching motion, and the vertical motion of the front and rear wheels, respectively [5]. In the Fig. 1, ms is the mass of the half-vehicle model body, J<sup>ϕ</sup> is the moment of inertia, zs is the displacement of the center of mass of the half-vehicle model body, ϕ is the pitch angle, l1 and l2 are the distances from the front and rear axles to the center of mass, zsf and zsr are the displacements of the concentrated mass on the front and rear axles, ksf and ksr are the stiffness coefficients of the front and rear suspension, csf and csr are the damping coefficients of the dampers of the front and rear suspensions, zuf and zur are the displacement of unsprung mass of the front and rear axle, ktf, ktr are the stiffness coefficients of the front and rear tires, and ztf, ztr are the road surface unevenness excitation of the front and rear wheels.

According to D'Alembert's principle, the differential equations of motion for the passive suspension dynamics of the semi-vehicle model can be listed, where Eqs. (3) and (4) are the differential equations of motion for the unsprung masses of the front and rear suspensions, Eq. (5) is the differential equation of motion for the center of mass of the vehicle body, and Eq. (6) is the differential equation of motion for the pitch angle.

$$m\_{uf}\ddot{z}\_{uf} = k\_{tf}(z\_{tf} - z\_{uf}) - c\_{sf}(\dot{z}\_{uf} - \dot{z}\_{sf}) - k\_{sf}(z\_{uf} - z\_{sf}) \tag{3}$$

$$m\_{ur}\ddot{z}\_{ur} = k\_{tr}(z\_{tr} - z\_{ur}) - c\_{sr}(\dot{z}\_{ur} - \dot{z}\_{sr}) - k\_{sr}(z\_{ur} - z\_{sr}) \tag{4}$$

**Fig. 1** 4-degree-of-freedom 1/2 vehicle suspension model

$$m\_s \ddot{z}\_s = c\_{sf}(\dot{z}\_{uf} - \dot{z}\_{sf}) + k\_{sf}(z\_{uf} - z\_{sf}) + c\_{sr}(\dot{z}\_{ur} - \dot{z}\_{sr}) + k\_{sr}(z\_{ur} - z\_{sr}) \tag{5}$$

$$\begin{split} J\_{\psi}\ddot{\boldsymbol{\varphi}} &= -l\_{f} \Big[ c\_{sf} \left( \dot{\boldsymbol{z}}\_{uf} - \dot{\boldsymbol{z}}\_{sf} \right) + k\_{sf} \left( \boldsymbol{z}\_{uf} - \boldsymbol{z}\_{sf} \right) \Big] \\ &+ l\_{r} \left[ c\_{sr} \left( \dot{\boldsymbol{z}}\_{ur} - \dot{\boldsymbol{z}}\_{sr} \right) + k\_{sr} \left( \boldsymbol{z}\_{ur} - \boldsymbol{z}\_{sr} \right) \right] \end{split} \tag{6}$$

Let the state variable of the system *X* = *zuf zur zs* ϕ *z*˙*uf z*˙*ur z*˙*s* ϕ˙ *T*  , The interference input is *U* = *ztf ztr <sup>T</sup>*, The output is *Y* = *<sup>z</sup>*¨*uf z*¨*ur z*¨*s* ϕ¨ *zuf* <sup>−</sup> *ztf zur* <sup>−</sup> *ztr zsf* <sup>−</sup> *zuf zsr* <sup>−</sup> *zur <sup>T</sup>*, Transform the above differential equations of motion into state space form:

$$\begin{aligned} \dot{X} &= AX + BU\\ Y &= CX + DU \end{aligned} \tag{7}$$

where A and B are the state and input matrices, respectively, and C and D are the output and direct transfer matrices, respectively. By building a four degree of freedom


**Table 1** Parameters required for vehicle simulation

model of the semi vehicle in matlab/simulink, the vehicle parameters are shown in the table below (Table 1).

### **3 Kalman Filtering Algorithm**

Kalman filtering consists of two main processes: the time update process (prediction) and the measurement update process (correction). The prediction process mainly uses the time updating equation to establish a priori estimates of the current state, project the values of the current state variables and the error covariance estimates forward in time, and construct a priori estimates for the next time state; the calibration process is responsible for the feedback, and uses the measurement updating equation to establish improved a posteriori estimates of the current state based on the a priori estimates from the prediction process as well as the current measurement variables [6].

Firstly, the state-space model of the above equation is discretized discrete by means of the Laplace transform.

$$\begin{aligned} \mathbf{x}(k) &= A\mathbf{x}(k-1) + Bw(k) \\ \mathbf{y}(k) &= C\mathbf{x}(k) + Dw(k) \end{aligned} \tag{8}$$

where x(k) and x(k−1) are the state vectors at moments k and (k−1), respectively; and y(k) is the observation vector at moment k.

The specific algorithm flow is as follows [7]:

a. Time updating process

State prediction equations:

$$
\hat{X}(k|k-1) = A\hat{x}(k-1|k-1) + Bw(k) \tag{9}
$$

Error covariance prediction:

$$AP(k|k-1) = AP(k-1|k-1)AP^T + \mathcal{Q}(k) \tag{10}$$

where Q(k) is the covariance of the system noise; P(k|k−1) is the propagation form of the covariance under the a priori state estimation, i.e. The time-updated expression for the covariance under the a priori state estimation.

b. Measurement update process

Gain equations:

$$K\_{\mathcal{S}}(k) = P(k|k-1)C^{\top} \Big/ \left( C P(k|k-1)C^{\top} + R \right) \tag{11}$$

State update equations:

$$\mathbf{x}(k|k) = \mathbf{x}(k-1|k-1) + K\_{\mathcal{S}}(k)[\mathbf{y}(k) - \mathbf{C}\mathbf{x}(k|k-1)] \tag{12}$$

Error covariance update expression:

$$P(k|k) = \left[I - K\_{\mathcal{S}}(k)C\right]P(k|k-1) \tag{13}$$

### **4 LQR Control Based on State Estimation**

Linear quadratic regulator (LQR) is one of the most fundamental optimal control problems for control problems and is commonly used to solve linear system control problems with Gaussian white noise inputs, i.e., to solve the optimal feedback control law problem that minimises a quadratic objective function [8]. It is mainly applicable to linear time-invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that is easy to compute and implement. The state space model of the passive suspension is transformed into a multivariate linear system state equation for the active suspension [9].

In the process of automobile suspension design, the performance index of the automobile usually refers to some parameters related to its safety and driving smoothness, which are mainly reflected in the tire dynamic displacement, body acceleration and suspension dynamic deflection. The linear-quadratic optimal control is used for the active suspension, in which the state variables of the suspension are all estimated using the Kalman algorithm, and the above parameters are selected as the performance evaluation indexes, and the integral values of their weighted squares can be obtained [10].

$$J = \lim\_{T} \frac{1}{T} \int\_{0}^{T} \left[ q\_1(\ddot{z}\_s)^2 + q\_2(z\_{sf} - z\_{uf})^2 + q\_3(z\_{sr} - z\_{ur})^2 \right. $$

$$+ q\_4(z\_{uf} - z\_{tf})^2 + q\_5(z\_{ur} - z\_{tr})^2 \Big] dt \tag{14}$$

where q1, q2, q3, q4 and q5 are the weighting coefficients for body acceleration, front suspension dynamic deflection, rear suspension dynamic deflection, front tire dynamic displacement and rear tire dynamic displacement respectively.

Call the LQR toolbox in matlab, find the control rate [K] = lqr(A, B, Q, R, N), K is the optimal feedback control rate sought, change the value of q1, q2, q3, q4, q5, through repeated debugging until a good control effect occurs, to get q1 = 8000, q2 = 100, q3 = 100, q4 = 10, q5 = 10. The active suspension predictive control system was modeled and simulated, and the simulation results for each performance metric are shown in Fig. 2.

According to the simulation verification results, compared with the previous passive suspension, the active suspension based on Kalman filter LQG control reduces the vertical acceleration of the vehicle by 35.36%, increases the dynamic deflection of the front and rear suspensions by 27.92% and 22.38% respectively, and increases the dynamic travel of the front and rear tires by 28.43 and 16.12%.

**Fig. 2** Comparison of simulation results for each performance index; **a** is the vertical acceleration of the car body; **b** is the dynamic deflection of the front suspension; **c** is the dynamic deflection of the rear suspension; **d** is dynamic displacement of front tires; **e** is dynamic displacement of rear tires

Although LQR control significantly reduces the vertical acceleration of the vehicle body, it reduces the bumpiness of the vehicle during driving and improves ride comfort. At the same time, the increase in suspension dynamic deflection also makes the suspension system more flexible during driving and better responsive to changes in the road surface. However, the optimization also sacrifices a portion of the tire travel, making the vehicle more bumpy and the riding experience worse when passing through off-road roads.

### **5 Conclusion**

The Kalman filter algorithm is able to estimate the vibration state variables of special vehicles better, and also has good reliability under poor road conditions, which can be used for the active suspension control of special vehicles. In addition, based on the optimal control of the active suspension, the state variables estimated by Kalman filtering are used to replace the observed values of the sensors, which can satisfy the expected control effect within a certain permissible error range and reduce the installation of sensors while lowering the cost.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Analysis on the Precooler Outlet Compensation Pipe Rupture of Civil Aircraft**

#### **Nannan Du**

**Abstract** This paper focused on the rupture problem of precooler outlet compensation pipe of civil aircraft, the design concerning was discussed. Based on the design proposal, aging mechanical from two aspects of temperature and high velocity sand dust flow were calculated and tested. Root cause of pipe rupture is identified and the result shows silicon rubber could not suffer high temperature up to 250 °C and velocity up to 150 m/s.

**Keywords** Compensation pipe · Aging · High temperature · Sand dust

# **1 Introduction**

The civil aircraft Pneumatic system usually use the intermediate-pressure or highpressure ports to bleed air, the maximum bleed temperature could be higher than 500 °C [1]. To cooling down the temperature, the fan air is also bled as the cold source [2]. In order to maximize the heat exchange capability in limited space, usually a cross-flow precooler is used. The fan air is heated and then discharged by the exhaust louver; the system diagram is shown in Fig. 1.

A compensation pipe should be used to connect the precooler and louver in order to compensate the manufacture and installation tolerance. Based on the engineering experience, The pipe design requirement should consider following issues:


N. Du (B)

Shanghai Aircraft Design and Research Institution, Shanghai, China e-mail: dunannan@comac.cc

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_52

**Fig. 1** Diagram of heat exchanger system of pneumatic system


To meet all the requirement listed above, usually the material of silicone rubber reinforced with glass fiber is used. This material combines the advantage of silicon rubber and glass fiber [3, 4], however the rubber aging problem would also appear.

### **2 Design Requirement**

Tables 1 and 2 shown a type of compensation pipe design requirement and related material information.


**Table 2** Material information


### **3 Analysis**

### *3.1 Thermal Aging Analysis*

**Thermal Aging mechanism**. Silicon rubber will occur high temperature aging failure in the long-term high temperature environment. According to the domestic and foreign scholars, the high temperature resistance of silicon rubber is mainly affected by two factors: one mainly occurs in an oxygen-free high temperature environment, the main chain will break and rearrange the silicon-oxygen bond, so that the silicon rubber will soften and aging. Another mainly occurs in an aerobic high temperature environment, and the organic side group will have thermal oxygen aging, so that the silicon rubber will harden and aging.

The compensation pipe is normally exposed to the air, and combined with the phenomenon of failure component losing elasticity and hardening, it can be determined that the product has hardened and aged, the reaction equation is shown in Fig. 2 [5, 6].

**Fig. 2** Equation of hot oxygen reaction


**Table 3** Parameter of thermal analysis

**Fig. 3** CFD calculation of compensation duct

**Thermal Analysis**. System performance analysis shows the average temperature could meet the material requirement, but the cross-flow precooler will lead to the temperature stratification, so the CFD should also performed to analysis the temperature field. The temperature stratification was supposed as linear distribution and the boundary conditions and turbulence model are shown in Table 3.

As Fig. 3 shows, the maximum temperature could reach to 279 °C,which is beyond the limit temperature 232 °C.

### *3.2 High-Speed Air with Sand Dust*

**Abrasion mechanism**. According to the research, both sand dust and high-speed air flow can cause abrasion on the surface of silicon rubber, but the abrasion mechanism of the two effects is not the same. The abrasion under the sand dust is caused by the impact of particles and friction and wear, and its abrasion level is affected by the hardness and concentration of particles. The abrasion under the impact of highspeed air flow is caused by the surface damage and stripping wear caused by the gas molecules strike [7, 8].

**Flow field analysis**. In order to identify the exact speed in the compensation pipe, flow field analysis is performed as Table 4. Three critical cases show the average speed in the pipe is about 200 m/s, the maximum could reach to 300 m/s as Fig. 4. The comparison in Fig. 5 shows the simulated high-speed area is in good agreement with the actual white area of the product.

**Abrasion test**. In order to verify the impact of the high-speed air with sand dust, an abrasion test is performed, the test requirement is accordance with ISO12103 [9]. Test instruments and results show in Fig. 6 and Table 5.

The test result shown the blowing surface became white and the internal glass fiber exposed, this phenomenon is the same as the airline products.

The test evidences that the high velocity and dust flow indeed wear the silicon rubber. When the velocity reach to 200 m/s, never less the room temperature or high


**Table 4** Boundary conditions of flow field calculation

**Fig. 4** Flow field calculation of compensation duct

**Fig. 5** CFD result compared with the component

**Fig. 6** Diagram of **s**and dust test

temperature, the air flow with sand dust has a serious impact on the abrasion of silicon rubber, and at the velocity of 150 m/s, the abrasion is significantly weakened.

# **4 Conclusion**

Aiming at the problem of pneumatic compensation pipe rupture problem, this paper focuses on the impact of thermal aging and sand dust. Based on the CFD calculation and abrasion test, the root cause of rupture is identified. Because of the technology limitation, the area which to suffer high temperature up to 250 °C and velocity up to 150 m/s is not suitable to use silicon rubber.


**Table 5** Test results

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **An Efficient Strategy for Creep-Fatigue Assessment of Pipelines with Complex Local Structures and High Computational Demands**

#### **Mingda Han, Hetong Liu, Yujian Tang, Yibo Zhang, and Weixu Zhang**

**Abstract** Thermal pipelines are extensively utilized in power plants, and they operate in high-temperature and high-pressure environments. Consequently, creep and fatigue are two crucial factors that need to be considered in the design and operation of main pipelines. Assessing the safety of main pipelines considering creep-fatigue is of paramount importance. However, the current assessment of main pipelines faces challenges such as large computational requirements and susceptibility to complex local structures, making efficient evaluation a longstanding problem. Furthermore, there is a lack of commercially available software capable of automating the creep-fatigue assessment for structurally discontinuous pipelines. To address these issues, this paper proposes an efficient strategy for evaluating creepfatigue in pipelines with complex local structures and significant computational requirements. By approximating non-critical irregular pipelines with equivalent stiffness of regular pipelines, computational accuracy and efficiency are improved. The proposed approach also enables the identification of critical locations prone to creepfatigue. Additionally, a post-processing software based on ASME standards and utilizing existing commercial finite element software has been developed to automate the assessment of main pipelines. Compared to other computational strategies, this approach significantly enhances computational efficiency.

**Keywords** Creep-fatigue assessment · Pipelines · Efficient · High computational demands

M. Han · Y. Zhang · W. Zhang (B)

State Key Laboratory for Strength and Vibration of Mechanical Structures, Department of Engineering Mechanics, School of Aerospace Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China e-mail: zhangwx@mail.xjtu.edu.cn

H. Liu · Y. Tang China Nuclear Power Engineering Co., Ltd, Beijing 100840, PR China

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_53

665

### **1 Introduction**

The main pipelines in power plants play a critical role in transporting coolant from boilers to power generation equipment [1]. These pipelines, which consist of multiple layers, are responsible for simultaneously conveying both cold and hot working fluids. Due to temperature differences between the inner and outer layers, the main pipelines experience significant temperature gradients and high thermal stress. Figure 1 illustrates a schematic diagram of the main pipeline's structure and its operational concept. Operating in high-temperature and high-pressure environments, the main pipelines undergo creep and increasing the risk of leaks or ruptures. Therefore, it is crucial to consider the influence of creep on the main pipelines. Similarly, during the operation of main pipelines, they are subjected to cyclic loading, such as pressure fluctuations caused by temperature variations or changes in operating modes, which result in fatigue failure. Additionally, main pipelines typically are often very long and contain complex local structures which significantly increase the computational requirements for their analyses. Consequently, the assessment of main pipelines becomes challenging and lengthens the development cycle.

There has been a great deal of research on the mechanisms of creep-fatigue. In 1992, Kitamura et al. [2] proposed a simplified stochastic model to predict the distribution of crack initiation and early-stage propagation under creep and creep-fatigue loading conditions. This model was constructed based on the concept of critical damage necessary for damage accumulation and intergranular failure. Wei et al. [3] developed a creep-fatigue damage accumulation model for creep-brittle materials to predict the propagation of creep and fatigue cracks under periodic and continuous creep and fatigue loading. Farrahi et al. [4] performed deterministic calculations on pressurized pipes with external longitudinal semi-elliptical surface cracks to determine the propagation of creep-fatigue cracks. However, these studies have focused primarily on microscale phenomena, limiting their practical guidance for creepfatigue assessment in engineering applications. In engineering practice, creep-fatigue

**Fig. 1** Schematic diagram of main pipeline

assessment is typically conducted according to the corresponding formulas specified in the ASME Boiler and Pressure Vessel Code [5], which is a set of standards established by the American Society of Mechanical Engineers (ASME).

Although, many studies have been conducted, however, currently, there lack automated creep-fatigue assessment capabilities for structures with discontinuous pipes in commercial software [6] for engineers. This paper proposes an efficient strategy for evaluating creep-fatigue in main pipelines with complex local structures and large computational requirements. To simplify the assessment process, non-critical irregular sections in the pipelines are approximated using regular pipelines with equivalent stiffness. This simplification improves both the computational accuracy and efficiency of the evaluation, while also identifying critical locations prone to creepfatigue. Additionally, a post-processing software based on ASME standards has been developed, utilizing existing commercial finite element software to automate the assessment of main pipelines. This approach significantly enhances computational efficiency compared to alternative strategies.

### **2 Method**

To achieve automated and efficient assessment, it is crucial to consider reducing the overall computational requirements while ensuring accuracy from the modelling stage. During the geometric modelling phase, it is essential to identify the components of irregular structures and determine whether they are critical components of the main pipeline. If these irregular structures are non-critical components, an approach for finding equivalent replacements should be employed to simplify the assessment process. This not only reduces the computational requirements for assessing the main pipeline but also eliminates interference caused by irregular components, thus improving computational efficiency.

In the assessment phase following the computational stage, for continuous straight pipelines, evaluation can be limited to key locations such as structural discontinuities or material transitions. This approach eliminates the need to assess all cross-sections exhaustively, greatly reducing the computational workload. Taking the main pipeline with the structure shown in Fig. 1 as an example, the assessment steps and strategies will be introduced.

### *2.1 Simplified Components of the Main Pipeline*

Taking the example of the corrugated tube as an irregular structure within the main pipeline, the corrugated tube is considered a non-critical component. The structure of the corrugated tube is irregular. Existing commercial finite element software does not provide sufficiently fine meshing for this region [7], resulting in reduced computational accuracy and even loss of physical significance in the results. Therefore, it is

**Fig. 2** Meshing diagrams of insulation layer before and after simplification

necessary to simplify the bellows tube by considering it as an equivalent cylindrical pipe. As shown in Fig. 2, Fig. 2a represents the original three-dimensional view of the bellows tube, while Fig. 2b shows the completed equivalent three-dimensional view. Figure 2c and d illustrate the meshing of the insulation layer of the main pipeline before and after simplifying the bellows tube, revealing a noticeable improvement in mesh quality.

In order to maintain similar properties to the original bellows expansion joint, an equivalent method is required to determine the elastic modulus E of the simplified bellows expansion joint. The equivalence is achieved by ensuring that the simplified cylindrical pipe and the original bellows expansion joint exhibit the same macroscopic displacement under identical loads, thereby achieving stiffness equivalence. Based on the axial stiffness formula [8], the following equation can be derived.

The axial force of the corrugated tube before simplification is represented by Eq. (1).

$$F\_P = K\_Z \times \Delta L \tag{1}$$

The equivalent axial force of the simplified tube is represented by Eq. (2).

$$F\_J = \sigma A = E\varepsilon A \tag{2}$$

The axial force before and after simplification is represented by Eq. (3).

$$F\_P = F\_J \tag{3}$$

Substituting Eqs. (1) and (2) into Eq. (3), we obtain:

$$E = \frac{K\_Z l\_0}{A} \tag{4}$$

*KZ* is axial stiffness of the bellows expansion joint. *FP* is axial force applied to the bellows expansion joint. *FJ* is axial force of the simplified bellows expansion joint. -*L* is axial elongation of the expansion joint. E is elastic modulus of the simplified expansion joint. *l*0 is initial length of the expansion joint. *A* is area of the simplified expansion joint.

In the calculation of the temperature field distribution of the Main Pipeline, the actual operating conditions of the conduit are taken into account. The inner wall of the hot air duct carries high-temperature fluid, while the outer wall of the hot air duct and the inner surface of the duct shell carry low-temperature fluid. Therefore, the boundary conditions for the temperature field of the Main Pipeline are set as convective heat transfer with high-temperature fluid on the inner surface of the inner wall of the hot air duct, and convective heat transfer with low-temperature fluid on the inner surface of the outer wall of the hot air duct and the inner surface of the duct shell. The boundary conditions for the temperature field analysis are illustrated in Fig. 3.

For the structural analysis of the Main Pipeline, it is necessary to apply the results obtained from the temperature field analysis of the pipeline to the nodes of the structural field. In the analysis of thermal stress, in order to avoid insufficient degrees of freedom constraints leading to rigid body displacement in the finite element method, the three directions of freedom at both ends of the Main Pipeline are restricted. In the analysis considering the structural field under self-weight, a standard gravity acceleration along the negative Y-axis direction is applied, while also restricting the three directions of freedom at both ends of the Main Pipeline. The boundaries for the structural field analysis are illustrated in Fig. 3.

**Fig. 3** Boundary condition settings for the temperature field and structural field

**Fig. 4** Temperature contour plot of the main pipeline

**Fig. 5** Stress contour plot of the main pipeline

After the simplification steps described above, the temperature distribution of the main pipeline is obtained, as shown in Fig. 4. It can be observed that the inner layers of the pipeline have higher temperatures, while the outer layers have lower temperatures. The conical barrel in the middle experiences significant temperature gradients. The thermal stress distribution of the main pipeline is illustrated in Fig. 5. Due to the slip joint structure, the stress in the inner layers is released. In order to obtain the resultant moment *MA* caused by self-weight, it is necessary to calculate the stress field distribution of the main conduit under self-weight.

According to Eq. (4), the equivalent elastic modulus E of the structurally irregular corrugated tube (i.e., expansion joint) after simplification can be determined, ensuring consistency between the results before and after simplification. By employing the approach of utilizing non-critical and irregular components in the pipeline, the quality of the mesh is enhanced, and there is a significant reduction in computational requirements, resulting in a 50% reduction in computation time. Therefore, it is evident that the adoption of the aforementioned strategy of equivalent simplification is highly necessary.

### *2.2 Creep-Fatigue Assessment of the Main Pipeline*

In the assessment of creep-fatigue for the main pipeline, creep refers to the deformation phenomenon that occurs in a material under prolonged high-temperature conditions. When the main pipeline operates at high temperatures, the material may experience creep, resulting in changes in the shape and dimensions of the pipeline. This deformation can lead to stress concentration and material fatigue, increasing the risk of leakage or rupture in the pipeline. Therefore, creep behavior must be considered in the design and material selection of the main pipeline to ensure structural stability throughout its operational lifespan [9].

Fatigue refers to the progressive damage and failure that occurs in a material under repeated loading or stress cycles. In nuclear power plants, the main pipeline is often subjected to cyclic loading, such as pressure fluctuations caused by temperature changes or variations in operating modes. These cyclic loads can cause stress concentration and the formation of cracks in the main pipeline. Over time, these cracks may propagate and lead to pipeline failure. Therefore, fatigue analysis and assessment are essential for the design and operation of the main pipeline to ensure reliability and safety under repeated loading [10].

After the aforementioned simplification steps, the structure becomes more regular, and the mesh quality significantly improves. To quantify the assessment of creepfatigue on the safety of the main pipeline, the following sections describe the ASME standard provisions used in the mechanical analysis and assessment of the main pipeline.

In the assessment of thermal expansion stress considering creep-fatigue, Formula (5) from ASME Specification HCB-3634(b) is required. This formula is used for the design of nuclear secondary pipelines. The formula is presented as follows [5].

$$\begin{split} \mathbf{S\_{TE}} &= \frac{\mathbf{PD}\_0}{4\mathbf{t\_h}} + \mathbf{0.75i} \left(\frac{\mathbf{M\_A}}{Z}\right) + \mathbf{i} \left(\frac{\mathbf{M\_C}}{Z}\right) \\ &\leq \begin{bmatrix} \text{lesser of} (\mathbf{S\_A} + \mathbf{S\_h}) \text{ or } \mathbf{0.75S\_{yc}} + \mathbf{0.25S\_h} \end{bmatrix} \end{split} \tag{5}$$

In Eq. (5), *MA* represents the resultant moment of the pipeline cross-section caused by self-weight. *MC* is the resultant moment of the pipeline caused by thermal stress.

It can be observed that the aforementioned equation requires the *MC* and *MA* data on the cross-sections of the pipeline. However, employing an exhaustive method to obtain the resultant moment data for each cross-section would undoubtedly lead to excessive computational requirements and increased computation time. Therefore, this paper proposes an assessment strategy that utilizes the binary search method to identify critical locations (such as structural discontinuities or material transitions) for evaluation. Compared to the original assessment method, this approach significantly improves computational efficiency. This is of great significance for pipeline evaluations in the engineering development process, as it shortens the development cycle.

### **3 Conclusion**

This paper proposes an efficient strategy for creep-fatigue assessment of pipelines with complex local structures and high computational requirements. The strategy involves simplifying non-critical irregular sections of the pipeline by replacing them with equivalent regular sections in terms of stiffness. This approach enhances both the computational accuracy and efficiency. Additionally, a post-processing software based on the ASME standards has been developed using existing commercial finite element software. This software automatically assesses the creep-fatigue conditions at critical locations, enabling the automation of the main pipeline assessment tasks. Compared to other computational strategies, this approach significantly improves computational efficiency. In summary, the developed program in this study is of great significance for pipeline development assessments in engineering projects, as it shortens the pipeline development cycle.

**Acknowledgements** This work is supported by National Natural Science Foundation of China or by NSFC (Grant Nos. U21B2058, 11972025, 11772246) National Science and Technology Major Project(2019-VII-0007-0147), by Innovative Scientific Program of CNNC (J202107006-025).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Fatigue Life Assessment of Ship Hatch Corner Based on Hot Spot Stress Method**

**Langhao Qiao, Renjun Yan, and Yu Qiu** 

**Abstract** The hatch corner of ship is the most prone area for stress concentration, which makes the fatigue problem of hatch corner particularly serious, and becomes a position that is easily damaged. Firstly, the hot spot stress method based on surface extrapolation is introduced, and the hatch corner of a bulk carrier is selected as the research object, the scale model test is designed and carried out, the finite element model is established, and the parameters of the corner form and different transition radii are analyzed. The CCS calculation specification evaluates the fatigue strength of hatch corner, and compares the test results with the finite element calculation results. The results show that the hot spot stress method has good adaptability to the fatigue assessment of hatch corner, and the fatigue strength evaluation of hatch corner needs to select the corresponding *S–N* curve according to the position of the maximum hot spot stress. The form of corner and the increase of transition radius are of great significance for alleviating stress concentration effect and improving fatigue life of corner, which provides reference for fatigue resistance and optimal design of hatch corner structure.

**Keywords** Hatch corner · Fatigue test · Hot spot stress · Fatigue life

# **1 Introduction**

Because the cabin opening of a large-opening hull is usually large, its structural continuity is destroyed, which is not conducive to stress transmission. When deformed by the bending moment, stress concentration is easily formed at the hatch corners,

L. Qiao · R. Yan (B) · Y. Qiu

© The Author(s) 2024

School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China e-mail: q1301580236@163.com

R. Yan

Educational Ministry Key Lab of High Performance Ship Technology, Wuhan University of Technology, Wuhan 430063, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_54

which makes the fatigue problem of hatch corner particularly serious [1]. In view of the irreplaceable role of large ship in maritime transportation, the fatigue assessment of hatch corner of large-opening ship is a very meaningful research problem. Aiming at this problem, domestic and foreign scholars have carried out a series of related research. Yu et al. [2] used the semi-probability design wave method to calculate the fatigue cumulative damage of the hatch corner near the cargo hold of an 8530TEU large container ship, and at the same time predict the fatigue life. Yu et al. [3] used spectral analysis method combined with finite element method and crack propagation path to predict and evaluate the fatigue life of a large container ship hatch corner. Li et al. [4] used the global model method to evaluate the fatigue life of the circular hatch corner of bulk carriers, and analyzed the factors affecting the fatigue life. Cheng et al. [5] used finite element analysis and inertia release technology to evaluate the fatigue life and strength of hatch corner on container decks, and propose optimization measures.

The above studies on the fatigue performance of hatch corner are mostly designed to combine finite elements with different methods. The nominal stress method has some obvious shortcomings, and the fatigue performance evaluated at the same time has strong divergence. Therefore, Wang et al. [6] applied sub-model technology to evaluate the fatigue life of welded frames. Wang et al. [7] used hot spot stress to study the fatigue of FPSO welded structures.

In this paper, a typical joint of a large bulk carrier-hatch corner is selected as the research object, and scaled model tests of different forms and sizes of transition corner are carried out. The finite element model of the hatch corner structure is established, and its parameters are analyzed. The fatigue strength evaluation of hatch corner is carried out, and the test and finite element results are compared to provide reference for fatigue resistance of similar structures in hatch corner.

### **2 Hot Spot Stress Method**

### *2.1 Hot Spot Stress Method Based on Surface Extrapolation*

In a welded structure, the weld toe is the most prone to fatigue failure of the structure, and the hot spot stress usually refers to the structural stress at the weld toe. The nominal stress at the weld toe of the member and the corresponding structural stress concentration factor are used to obtain the hot spot stress of the welded joints [8]. For structures with complex forces in the field of ocean engineering, the nominal stress method is difficult to solve hatch corner. Therefore, this paper adopts the hot spot stress method based on surface extrapolation.

In welded structures, the surface linear extrapolation method usually uses the structural stress at a certain distance from the weld toe to perform linear or quadratic interpolation calculations to determine the hot spot stress of the weld toe. Its schematic diagram is shown in Fig. 1. The interpolation calculation formula of surface

**Fig. 1** Schematic diagram of hot spot stress method based on surface linear extrapolation

linear extrapolation method is shown in (1).

$$\delta\_{\rm hr} = \frac{[\mathbf{x}\_2 \mathbf{x}\_3 (\mathbf{x}\_3 - \mathbf{x}\_2) \delta\_1 + \mathbf{x}\_1 \mathbf{x}\_3 (\mathbf{x}\_1 - \mathbf{x}\_3) \delta\_2 + \mathbf{x}\_1 \mathbf{x}\_2 (\mathbf{x}\_2 - \mathbf{x}\_1) \delta\_3]}{[\mathbf{x}\_2 \mathbf{x}\_3 (\mathbf{x}\_3 - \mathbf{x}\_2) + \mathbf{x}\_1 \mathbf{x}\_3 (\mathbf{x}\_1 - \mathbf{x}\_3) + \mathbf{x}\_1 \mathbf{x}\_2 (\mathbf{x}\_2 - \mathbf{x}\_1)]} \tag{1}$$

Fricke et al. [9] studied the hot spot stress of FPSO structural details, by comparing the hot spot stress solved by different extrapolation points, the results showed that the hot spot stress solved by applying the extrapolation points of 0.5t and 1.5t had strong convergence. (2) is the formula for solving hot spot stress with the extrapolation points being 0.5t and 1.5t respectively.

$$\delta\_{\rm hs} = 1.5\delta\_{0.5t} - 0.5\delta\_{1.5t} \tag{2}$$

### *2.2 Hot Spot Stress S–N Curve*

When performing fatigue life assessments on complex welded structures, using the nominal stress method to check the component fatigue is a relatively traditional method, but this method needs to select different values according to the different structural forms and complexity of the joints That is, multiple *S–N* curves. When calculating the fatigue life of components, (3) is a formula related to the *S–N* curve.

$$\lg N = \lg K - m \lg(\Delta \sigma) \tag{3}$$

### **3 Static Load and Fatigue Test of Hatch Corner**

### *3.1 Overview of the Test Piece*

The research object of this test is a typical joint hatch corner of a large bulk carrier, and the test model range is selected from three rib lengths in the ship's length direction and three longitudinal frame spacing lengths in the ship's width direction. The geometric size of the test piece is scaled by 1:4. The thickness of the test piece except the arc is 4 mm, and the thickness of other plates is 6 mm. Figure 2 and Table 1 show the specific dimensions and specifications of the test pieces.

The material of the model is AH36 steel, and the welding construction is carried out according to the requirements for welds in CCS "The rules and regulations for the construction and classification of sea-going steel ships". Basic electrodes are used for the welding rods, and the welds are continuous welding. After the welding is completed, the corner of the hatch and other welds are to be polished to ensure that there are no visible defects. Material yield strength δ*s* = 355 MPa, tensile strength δ*b* = 490–630 MPa.

**Table 1** Hatch corner model specification


### *3.2 Static Load and Fatigue Test*

Considering that the overall stress and local stress are mainly evaluated in the fatigue assessment process of the hull structure, the overall stress mainly presents a tendency to spread to both sides at the corner, and the simplified method of fixing one end and loading the other end is adopted to realize the hatch corner For loading, the specimen is fixed on the test bench by bolts, and the upper end of the specimen is connected to the MTS actuator chuck through bilateral double-hole chucks to realize the vertical loading of the entire model. The test loading device is shown in Fig. 3. The measuring points of the test are mainly distributed near A and B, that is, the connection between the arc and the free edge of the corner, to monitor the stress distribution near it. The measuring points are all unidirectional strain gauges, and the layout positions are shown in Fig. 4.

Before formal loading, it is necessary to preload the specimen several times to eliminate the influence of residual stress. According to the test results, the static load is determined to be 50 kN. After the static load test, the tension-tension fatigue test was carried out on the three groups of specimens in turn. The fatigue test was carried out on the QBS-350A fatigue testing machine. Table 2 is the fatigue test parameters.

**Fig. 3** Test loading device

**Table 2** Fatigue test parameters


# **4 Numerical Simulation Verification of Hatch Corner Structure**

The commercial finite element software ABAQUS is used to simulate and analyze the hatch corner model, and the steps are as follows:


# *4.1 Comparative Analysis of Numerical Simulation and Test Results of Hatch Corner Structure*

Static load test adopts step-by-step loading, and the loading load increases step by step according to 10, 30 and 50 kN. After the loading load stabilizes for a period of time, data will be collected. See Fig. 6 for the comparison between the axial stress test and the finite element method at R200 hatch corner.

From the analysis in Fig. 6, it can be obtained that: under all levels of load, the finite element numerical calculation and the experimental measurement results are basically consistent in trend and value, and the error is within 8%. Under this relatively small error, the accuracy and reliability of the finite element model are verified.

**Fig. 6** Experimental and finite element axial stress comparison diagram for R200 hatch corner model

### *4.2 Hot Spot Stress Calculation*

Wang et al. [10] compares the effects of different hot spot stress selection methods on fatigue damage, and should choose the maximum principal stress interpolation to calculate hot spot stress. Figure 7 is the cloud diagram of the first principal stress of each model of the hatch corner when the load is 50 kN.

It can be seen from Fig. 7 that the hot spot area of the R200 and (200 × 120) hatch corner model is located near A, while the hot spot area of the R120 hatch corner model is located near C. The stress values σ<sup>0</sup>.5*<sup>t</sup>*and σ<sup>1</sup>.5*<sup>t</sup>*, substituting the hot spot stress results obtained in (2), and counting them in Table 3.

R120 hatch corner

**Fig. 7** The first principal stress cloud diagram of hatch corner models


**Table 3** Hot spot stress results

### **5 Parametric Analysis**

The size and shape of the transition corner may have a great influence on the stress state in the corner area. Based on the test results, this paper explores the influence of corner transition radius and main shape parameters on stress distribution in the hot spot area and the fatigue life of corner. The applied load is 50 kN.

### *5.1 Stress Analysis in Hot Spots*

In order to accurately capture the stress distribution state at the corner of the model, the regional grid at the corner is refined, and that relationship between the principal stress of each specimen and the weld toe distance is drawn in the Fig. 8.

Compared with the stress value at hot spot A of the R120 hatch corner model, the stress value of the R200 hatch corner model at hot spot A is reduced by 10.4%; (200 mm × 120 mm) hatch corner model is reduced to 19.5%, indicating that the transition corner Radius size and shape play an important role in alleviating stress concentrations caused by structural discontinuities between cabins. Therefore, (200 mm × 120 mm) hatch corner is the best for alleviating the stress concentration effect of the corner. At the same time, for the arc hatch corner, the stress concentration effect at the corner is alleviated with the increase of the transition corner radius.

**Fig. 8** Relationship diagram between principal stress and toe distance of each specimen

To sum up, R120 hatch corner has no maximum stress at the welding toe. Due to the influence of weld defects, fatigue damage will first occur at the weld toe. Therefore, the fatigue strength analysis and evaluation of hatch corner will be carried out next.

### *5.2 Fatigue Strength Analysis*

Reference [11] stipulates that for welded joints, the fatigue strength evaluation adopts the D curve; for the free edge of base metal, the fatigue strength evaluation adopts the C curve. See Table 4 for the relevant parameters of the *S–N* curve referring to CCS. Use (3) to calculate the fatigue evaluation results of C and D curve, σ is the hot spot stress value in Table 3, *m* and *K* are the values in Table 4 respectively. The fatigue evaluation results of hatch corner specimens are shown in Table 5, and the test results are plotted in Fig. 9.

From the results in Table 5 and Fig. 9, The fatigue test data points of each specimen are above the D curve. Among them, data point of the R120 hatch corner is above the C curve, and the R200 hatch angle Compared with (200 mm × 120 mm) hatch corner, the fatigue life of R200 hatch corner and R120 hatch corner are reduced by 20.4% and 33.2% respectively, and (200 mm × 120 mm) hatch corner has the best fatigue performance.


**Table 4** *S–N* curve related parameters


**Table 5** Fatigue life evaluation results

**Fig. 9** Hot spot stress data points in hatch corner

### **6 Conclusions**

This paper first introduces the hot spot stress method, designs and carries out hatch corner scale model tests, establishes its finite element model, conducts parameter analysis on the hatch corner form and transition radius, the fatigue strength of corner structure is evaluated by using CCS calculation specification, and the test results are compared with the finite element calculation results, and the following conclusions are obtained:


# **Appendices**


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Intelligent Manufacturing and Mechanical Performance Research**

# **Research on the Swing Characteristics of Lifting Load Based on the Contact Relationship Between Pulley and Wire Rope**

**Xuyang Cao, Zhihe Xu, and Siqi Wang** 

**Abstract** In the process of lifting operations, the complex contact relationship of the rope-pulley system is inevitably related to the swing of the lifting load. However, in previous research, the quality of the pulley and wire rope is often overlooked and the wire rope is simplified as a spring. The object of this paper is to use virtual prototyping technology to establish the rope-pulley system model that the mass and contact relationship of the pulley and wire rope are considered, and to investigate the influence of the contact relationship of the rope-pulley system on the swing characteristics of the lifting load. According to the characteristics of the rope-pulley system, the matching parameters and winding ratio of the rope-pulley system are selected as the research variables. The research method of this paper is to establish three discrete system simulation models of the rope-pulley system with different winding ratio by ADAMS/Cable module. Simulation results show that as the winding ratio of the rope-pulley system increases, the swinging amplitude of the lifting load increases; and the change in wire rope diameter has a more obvious effect on the sway of the lifting load than the change in the angle of pulley groove. The results of the study can provide a certain reference for the study on the swing of lifting loads with the rope-pulley system.

**Keywords** Rope-pulley system · Swing characteristics of lifting load · ADAMS/ Cable · Winding ratio

# **1 Introduction**

Rope-pulley system, which is widely used in the crane industry and plays a significant role, has the advantages of a simple structure, a large bearing capacity, and smooth transmission as a component of the crane hoisting mechanism. The inevitable

X. Cao · Z. Xu (B) · S. Wang

School of Mechanical Engineering, Dalian University of Technology, Dalian 116023, Liaoning, China

e-mail: 15265402003@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_55

swaying phenomena of the lifting load has a direct impact on the precision and safety of the operation when the crane performs lifting and lowering operations. With the increasing demand for lifting positioning accuracy, the study of the swing characteristics and influencing factors of lifting weights during the lifting process has become more and more essential.

In the majority of the current studies on lifting load sway and anti-sway control, the weights and dimensions of the ropes and pulleys are typically ignored when the model is established, and the influence of the contact relationship between the rope and pulley is not considered, or the ropes are rigidized. However, the contact relationship between the wire rope and the pulley block in different situations is complex and varied [1], and there must exist a certain relationship with the swing of the lifting load. Therefore, it is necessary to study the influence of the rope-pulley system's contact relationship in different states on the swing of lifting load.

The contact relationship between wire rope and pulley has been thoroughly studied by a large number of academics both domestically and internationally. Based on Archard's wear rule, Argatove et al. investigated the wear pattern of wire ropes under pressure effect and wear coefficients [2]. Kong et al. proposed a modified simplex method using contact solutions to simulate micromotion wear [3]. Qi et al. established the link between material velocity and spatial velocity for the rope-pulley system, and based on this, constructed a dynamic node rope element [4]. Guo et al. established a discrete wire rope model and a contact force model between pulleys through the macro-command method, which realistically responded to the dynamic characteristics of the mechanism in the operation process [5]. Zhang et al. used ADAMS/Cable module to establish the simulation model of the pulley wire rope system, and derived the dynamic response of the wire rope in the transition stage and stable operation [6].

In this article, the research object is the rope-pulley system considering the mass, winding ratio and matching size of the wire rope and pulley. The dynamics simulation model of the rope-pulley system is established by using the ADAMS/Cable module. By changing the winding ratio and matching size of the rope-pulley system, the influence of the contact relationship on the swing characteristics of the lifting load under different conditions is explored. The research content can provide a certain reference for the study including rope-pulley system.

### **2 Rope-Pulley System Modeling**

To study the influence of different winding ratios on the swing characteristics of lifting load, models of rope-pulley system with four-fold winding ratio, six-fold winding ratio and eight-fold winding ratio are established. By changing the size of the eight-fold winding ratio rope-pulley system model, the influence of the change of the matching size on the swing of the lifting load is studied.

### *2.1 Modeling Theory*

The ADAMS/Cable module creates the rope-pulley system model in two ways, one is a simplified model and the other is a discrete model as shown in Figs. 1 and 2. The simplified model uses a dummy object with pure motion constraints to track the tangent between the pulleys [7]. The simulation speed is fast but the wire rope has no solid model, and the influence of wire rope quality and inertia is not considered. The discrete model is to discretize the wire rope into a finite number of rigid spheres. The contact between the wire rope and the pulley is continuous. Considering the mass and inertia effects of the wire rope, the vibration of the wire rope and the friction between the pulleys can be truly simulated, which is more accurate than the simplified model. Therefore, this paper uses the discrete modeling method of ADAMS/Cable module to establish the pulley wire rope system.

The discrete model uses the finite element idea, and the finite rigid spheres are flexibly connected by the sliding pair, the revolute pair and the one-way force. Compared with the cylindrical discrete model, the spherical discrete model has higher accuracy. The calculation formula of the connection force is shown in Eq. (1).

$$\mathbf{F} = -\mathbf{K}X - \mathbf{C}\dot{X} + \mathbf{F}\_0 \tag{1}$$

where F is the force acting on the rope system, K is the stiffness, C is the damping force of the system, X is displacement, X ˙ is the speed and F0 is the initial force of the rope system.

The wire rope composed of rigid spheres is wound by contact force with the pulley. The contact force consists of two parts, the elastic force generated by the mutual incision between the rigid sphere and the pulley, and the damping force

#### **Fig. 1** Simplified model

#### **Fig. 2** Discrete model

generated by the relative velocity between the two. The contact force between the wire rope and the pulley is calculated by the impact function method. The generalized form of the impact function is shown in Eq. (2).

$$\mathbf{F}\_{\rm ni} = \mathbf{K} \mathbf{\delta}\_{\rm i}^{\rm e} + \mathbf{C} \mathbf{V}\_{\rm i} \tag{2}$$

where Fni is the normal contact force between two components, K is the stiffness of the contact surface and δ<sup>e</sup> i is the normal penetration depth of the contact point.

When the wire rope and the pulley contact, the damping value will quickly reach the maximum and remain constant, so the penetration depth is very small. However, considering the convergence of the values in ADAMS, the recommended value of 0.1 mm is used.

### *2.2 Selection of Main Parameters*

According to the specification standard of pulley and wire rope, the pulley of model WJ4202 is selected as an example for modeling, and its size parameters are shown in Table 1.

Where D is the inner diameter of the pulley, D1 is the outer diameter of the pulley, R is the radius of the pulley groove, b is the width of the pulley and A is the angle of the pulley groove.

The force index E is the coefficient of material stiffness in the instantaneous normal force when the wire rope and the pulley contact. For the metal contact between the


**Table 1** Specification standard of the pulley and wire rope

#### **Table 2** Contact parameters of the pulley


**Table 3** Parameters of the wire rope


pulley and the wire rope, the force index is 1.5. The contact parameters of the pulley surface are shown in Table 2.

Where K is the stiffness coefficient, E is the force index, Cm is the damping coefficient, μ is the friction coefficient, Vt is the friction velocity.

The parameters of the wire rope are shown in Table 3.

Where D is the diameter of the wire rope, ρ is the density of the wire rope, Cm is the damping coefficient.

The three models of the rope-pulley system are shown in Fig. 3. Where (a) is the four-fold winding ratio model, (b) is the six-fold winding ratio model and (c) is the eight-fold winding ratio model. The system is composed of movable pulley, fixed pulley, wire rope, drum, hook and lifting load. The size of the drum, the mass of the lifting load, the winding way of wire rope and other parameters in the system are all the same except for different winding ratio.

### **3 Simulation Analysis of Different Winding Ratio**

### *3.1 Driving Function*

STEP function is used as the driving function, and its format is STEP (x, x0, h0, x1, h1). In the function expression, x is the independent variable, x0 is the initial value of the independent variable, h0 is the function value at the initial point, x1 is the end value of the independent variable, and h1 is the function value at the end point.

The driving functions of the three models are shown in Table 4. In a certain period of time, the drum drive wire rope stops after the lifting weight is lifted to the same height at a constant speed.

**Fig. 3** Simulation model of the rope-pulley system


**Table 4** The driving functions of different models

### *3.2 Simulation Result*

In the established model of the rope-pulley system, the lifting weight load in the xoz plane and rises in the y direction. The simulation time-domain result curves of the lifting load in x direction and z direction are shown in Figs. 4, 5 and 6.

**Fig. 4** Swing curves of the lifting load centroid of the four-fold winding ratio system

**Fig. 5** Swing curves of the lifting load centroid of the six-fold winding ratio system

**Fig. 6** Swing curves of the lifting load centroid of the eight-fold winding ratio system

Through the analysis of the curve, it can be seen that in the three cases, the lifting load starts to rise at the seventh second and then swings around the equilibrium position with a regular positive spinning tendency. The amplitude of the oscillation reaches its maximum during the lifting process and begins to decrease at the end of the lifting motion. The vibration decay rate of the eight-fold winding ratio system is faster compared to the other two cases.

The equilibrium position coordinates of the lifting load are calculated by extracting the data in the curve, and then the maximum offset of the center of mass of the lifting load in the x direction and the z direction is calculated according to the equilibrium coordinates. The calculation results are shown in Table 5.



According to the data in the table, the maximum offset of the four-fold winding ratio system is the smallest, and the maximum offset of the eight-fold winding ratio system is the largest. Therefore, it can be concluded that with the increase of the winding ratio, the maximum offset of the center of the lifting load increases continuously, but the swing amplitude of the eight-fold winding ratio system attenuates faster and the ability to resist swing is better.

### **4 Simulation Analysis of Different Matching Parameters**

### *4.1 Matching Parameter Selection*

In the actual operation process, it is necessary to choose different types of the ropepulley mechanism for lifting operation according to the working conditions. Through the matching parameters in Sect. 2.2, it can be seen that the same pulley specification can be matched with different diameters of wire ropes, and different matching parameters will make the contact relationship between the pulley and the wire ropes change, and will also affect the swinging of the lifting weight in the process of operation.

According to the pulley specification standard and the matching size relationship between the pulley and the wire rope, this paper selects nine cases of three groups of matching relationships, as shown in Table 6.

### *4.2 Simulation Result*

In the simulation model of the eight-fold winding ratio rope-pulley system, the parameters are changed according to Table 6. Simulation analysis is carried out under the same other variables such as lift speed and lift height. The simulation time-domain result curves are shown in Table 7.

**Table 7** The swing curves of the lifting load centroid with different matching parameters

(continued)

**Table 7** (continued)

By analysing the curve, it can be concluded that when the angle of the pulley groove is unchanged, the model with a wire rope diameter of 25 mm has the fastest attenuation of the swing amplitude in the x direction, and the time to approach a stable state is about 90 s. The models with wire rope diameters of 24 mm and 26 mm reached a nearly stable state in about 350 s and 500 s, respectively. This shows that when the diameter of the wire rope is 25 mm, the wire rope is more suitable for the pulley, and its contact relationship makes the lifting load's stability better. When the diameter of the wire rope is constant and the angle of the pulley groove is changed, the swing curve of the load is relatively close, and there is no obvious difference.

Similarly, by extracting the data in the curves, the maximum offset of the lifting load centroid in the x direction and z direction is calculated and the results are shown in Table 8.


**Table 8** The maximum offset of the lifting load centroid with different matching parameters

It can be obtained from the data in the table that when the angle of the pulley groove is unchanged, with the increase of the diameter of the wire rope, the maximum offset of the lifting load centroid increases gradually in the x-direction and the z-direction. When the diameter of the wire rope is constant and the angle of the pulley groove is changed, the maximum offset of the lifting load centroid is similar and almost unchanged.

In summary, when other conditions remain unchanged, the diameter of the wire rope has a significant effect on the swing characteristics of the lifting load.

### **5 Conclusion**

In this paper, the discrete simulation model of the rope-pulley system is established by using ADAMS/Cable module. The rope-pulley system model with three kinds of winding ratio and three sets of matching parameters is selected as the research object. Through simulation analysis, the maximum offset and swing curves of the lifting load centroid are discovered. The effect of the contact relationship between the pulley and the wire rope in different states on the swing characteristics of the lifting load is investigated. The conclusions obtained are as follows:


**Acknowledgements** This work is supported by National Natural Science Foundation of China under Grant No. 52275088 and the Fundamental Research Funds for the Central Universities under Grant No. DUT22LAB507.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **The Influence of Parameters and Load on the Spin Angle of Multilayer Strand Wire Rope**

**Siqi Wang, Xuyang Cao, Dianlong Wang, and Zhihe Xu** 

**Abstract** The spin characteristics of multi-layer strand wire ropes can reduce the control accuracy of the crane lifting system, resulting in the crane being unable to meet the high-precision operation requirements. An object of this paper is to know about the influence of different parameters on the spin characteristics of multi-layer strand wire ropes. This article first theoretically analyzes the effects of wire rope twist angle, wire diameter, material, and load on the rotational characteristics of multi-layer strand wire ropes. In addition, we use CREO to establish a parameterized model for multi-layer strand wire ropes, and then set the material properties, contact parameters, and boundary conditions of the model in ABAQUS. Finally, the dynamic simulation of the multi-layer strand wire ropes model was completed. The results indicate that when the multi-layer strand wire rope is in a steady state, the twist angle has the greatest impact on the rotation angle, and under the same conditions, the rotation angle increases with the increase of the twist angle. The rotation angle at steady state is basically consistent with the theoretical calculation results, indicating that the theoretical influence of wire rope twist angle, wire diameter, material, and load on the rotation characteristics of multi-layer strand wire ropes is accurate.

**Keywords** Multi-layer strand · Wire ropes · Spin characteristics · CREO · ABAQUS

# **1 Introduction**

Multi-layer strand wire ropes can withstand large axial loads, with relatively low torsional and bending stiffness [1], and are an important component of cranes. With the improvement of the operational indicators of engineering projects, the lifting

S. Wang (B) · X. Cao · D. Wang · Z. Xu

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

e-mail: wsq691301@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_56

system of cranes needs to be precisely controlled to achieve higher operational accuracy. However, most existing control technologies overlook the flexible characteristics of steel wire ropes [2, 3], and do not consider the changes in the torsion angle of steel wire ropes under the combined effects of their own parameters and loads. As a result, existing control methods are difficult to meet higher accuracy requirements.

In previous studies, Nawrocki et al. [4] analyzed the mechanical theoretical model of steel wire ropes earlier, studying the mechanical characteristics of single steel wire rope and entire steel wire rope from both static and dynamic responses. Erdönme et al. [5] used nested spiral geometric parameter equations to define a nearly realistic analysis model for the core of steel wire rope. Zhang et al. [6] studied the dynamic torsional characteristics and internal rotational state of steel wire ropes by considering the changes in geometric parameters and mechanical properties of spiral elements in the axial tensile layer of the steel wire rope. Zhang et al. [7] conducted research on shallow well low load mine hoists and used dynamic theory and friction transmission theory to obtain the dynamic tension of steel wire ropes at different positions from the container. Fu et al. [8] applied the theory of differential geometry to derive the spatial curve parameter equation for compiling anti torsion steel wire ropes. Meng et al. [9, 10] proposed a parameterized model for the structure of circular steel wire ropes with arbitrary center lines, especially regarding the direction and type of strand wires, and derived a series of recursive formulas for the spatial winding equations of strand wires.

In response to the shortcomings of the above research content, this article starts from the basic parameters of steel wire rope, such as material, number of strands, wire diameter, twisting method, and twisting angle, and summarizes the variation law of the rotation angle of the steel wire rope in steady state based on the load borne by the steel wire rope.

# **2 Mathematical Modeling of Multi-layer Strand Wire Ropes**

Multi-layer strand steel wire ropes have complex spatial spiral structures, and existing modeling methods use the Frenet standard framework to represent the parameter equations of the centerline of each layer of strand steel wire and complete threedimensional modeling. This method mainly completes the mathematical modeling of multi-layer steel wire ropes through mathematical calculations, ignoring the role of steel wire rope structure in the modeling. Based on the parameters and structural characteristics of steel wire ropes, this article adopts a combination of numbers and shapes to complete the mathematical modeling of multi-layer strand steel wire ropes. Figure 1 shows the distribution of the basic elements that make up ropes and strands on the cross-section of the steel wire rope.

The centerline of the central strand core line is a straight line, and its parameter equation can be written as

**Fig. 1** 18 × 7 + IWS multi-layer strand wire rope cross section strand

$$\begin{cases} x = 0\\ y = 0\\ z = c \cdot t \end{cases} \tag{1}$$

where *c* is the coefficient of rope length, *t* is changeable parameter.

The centerline of the strand side steel wire and the centerline of the strand side core wire are first-order spirals, and their parameter equations can be written as

$$\begin{cases} x = r\_1 \cdot \cos\left(\frac{z \cdot \tan\beta\_1}{r\_1} \cdot \frac{360}{2\pi} + \theta\_{1i}\right) \\ y = r\_1 \cdot \sin\left(\frac{z \cdot \tan\beta\_1}{r\_1} \cdot \frac{360}{2\pi} + \theta\_{1i}\right) \\ z = c \cdot t \end{cases} \tag{2}$$

where *r*1 is the distance between the centerline of the central strand side wire and the centerline of the central strand core wire, β1 is the twist angle of the central strand, θ1*<sup>i</sup>*is the starting position angle of the centerline of the central strand side wire or the centerline of the side strand core wire,θ1*i* <sup>=</sup> <sup>360</sup><sup>×</sup> *<sup>i</sup> <sup>m</sup>*, *i* = 1 ∼ *m*, *m* is the number of side strands or cores of side strands.

The centerline of the side strands is a quadratic helix, and its parameter equation can be written as

$$\begin{cases} x = r\_1 \cdot \cos\left(\frac{z \cdot \tan\beta\_1}{r\_1} \cdot \frac{360}{2\pi} + \theta\_{1i}\right) + r\_2 \cdot \sin\left(\frac{\left(\left(z \cdot \tan\left(90^\circ - \beta\_1\right)\right)^2 + z^2\right)^{\frac{1}{2}} \cdot \tan\left(90^\circ - \beta\_2\right)}{r\_2} \times \frac{360}{2\pi} + \theta\_{2i}\right) \\\\ y = r\_1 \cdot \sin\left(\frac{z \cdot \tan\beta\_1}{r\_1} \cdot \frac{360}{2\pi} + \theta\_{1i}\right) + r\_2 \cdot \sin\left(\frac{\left(\left(z \cdot \tan\left(90^\circ - \beta\_1\right)\right)^2 + z^2\right)^{\frac{1}{2}} \cdot \tan\left(90^\circ - \beta\_2\right)}{2\pi \cdot r\_2} \times \frac{360}{2\pi} + \theta\_{2i}\right) \\\\ z = c \cdot t \end{cases} \tag{3}$$

where *r*1 is the distance between the centerline of the strand side core wire and the centerline of the central strand core wire, β1 is the twist angle of the central line of

the strand side core wire, *r*2 is the distance between the centerline of the side strand side wire and the centerline of the side strand core wire, β2 is the twist angle of the centerline of the side strands, θ2*<sup>i</sup>*is the starting position angle of the centerline of the lateral strands, θ1*i* <sup>=</sup> <sup>360</sup><sup>×</sup> *<sup>i</sup> <sup>n</sup>*, *i* = 1 ∼ *n*, n is the number of side strands and side wires.

The research object selected in this article is 18 × 7 + IWS multi-layer strand steel wire rope. Its central strand core wire has a radius of 1 mm, the central strand side wire has a radius of 0.8 mm, a twist angle of 13.46°, right-handed. The inner side strand core wire has a radius of 1 mm, a twist angle of 16.15°, left-handed, and the inner side strand side wire has a radius of 0.8 mm, a twist angle of 13.46°, left-handed. The outer core wire has a radius of 1 mm, a twist angle of 16.15°, right-handed, and the outer side wire has a radius of 0.8 mm, a twist angle of 13.46°, right-handed. We defined the parameters of the wire rope based on CREO 3D software and established a multi-layer strand wire rope model with a rope length of 54.25 mm. The established 3D model is shown in Fig. 2.

# **3 Relationship Between Load and Deformation of Multi-layer Strand Steel Wire Ropes**

### *3.1 Load Analysis of a Single Steel Wire*

The spatial force situation of any object can be decomposed into rotation and motion along the coordinate axis. The spatial stress state of steel wire includes the load on the cross-section and the load per unit length. Therefore, the load on a single steel wire can be decomposed into 12 components. The load acting on a single steel wire is shown in Fig. 3. Where *S* is the arc length along the wire.

In Fig. 3, we established a Frenet framework on a single wire cross-section. Where *N* and *N* respectively represent the shear force components in the *x*-axis and *y*-axis directions of the steel wire cross-section, *T* represents the axial tensile force on the steel wire, *G* and *G* respectively represent the bending moment components in the x-axis and y-axis directions of the cross-section, *H* represents the torsional moment on the steel wire, *X*, *Y* and *Z* respectively represent the line load components per unit

length of the steel wire centerline in the *x*, *y*, and *z* directions, *K*, *K* and θ respectively represent the external torque per unit length of the steel wire centerline in the *x, y*, and *z* directions.

The steel wire is in equilibrium under the combined action of these forces and moments. We sum the forces and moments along the coordinate axis of the Frenet framework to obtain the mechanical equilibrium equation for a single metal wire. The equation can be written as

$$\begin{cases} \frac{dN}{ds} - N'\mathbf{\tau} + T\kappa' + X = 0 \\\\ \frac{dN'}{ds} - T\kappa + N\tau + Y = 0 \\\\ \frac{dT}{ds} - N'\kappa' + N'\kappa + Z = 0 \\\\ \frac{dG}{ds} - G'\mathbf{\tau} + H\kappa' - N' + K = 0 \\\\ \frac{dG'}{ds} - H\kappa + G\tau + N + K' = 0 \\\\ \frac{dH}{ds} - G\kappa' + G'\kappa + \theta = 0 \end{cases} (4)$$

where κ and κ respectively represent the curvature components in the *x*-axis and *y*-axis directions on the Frenet frame; τ is the torsion angle per unit length of the steel wire.

# **4 The Relationship Between Load and Deformation of Single Strand Steel Wire Rope**

The initial structure of a single outer wire before and after loading is shown in Fig. 4, and the force relationship between the wire before and after loading is shown in Fig. 5. Then, based on the geometric relationship, we can obtain the deformation parameters of the single strand steel wire rope after being loaded.

The axial strain of steel wire rope and its core wire can be written as

$$\varepsilon = \varepsilon\_{11} = \frac{\overline{h\_{11}} - h\_{11}}{h\_{11}} \tag{5}$$

The axial strain of the force triangle of the wire rope core can be written as

$$
\varepsilon\_{11}^{\prime} = \frac{\overline{h\_{11}^{\prime}} - h\_{11}^{\prime}}{h\_{11}^{\prime}} = \frac{h\_{11}}{h\_{11}^{\prime}} \varepsilon\_{11} \tag{6}
$$

The triangular axial strain of the wire rope side force can be written as

$$
\varepsilon\_{12}^{\prime} = \varepsilon\_{11}^{\prime} + \frac{\alpha\_{12} - a \sin\left(\frac{\varepsilon\_{11}^{\prime} + 1}{\varepsilon\_{11}^{\prime} + 1/\sin \alpha\_{12}}\right)}{\tan \alpha\_{12}} \tag{7}
$$

The torsion angle per unit length of the side wire of a steel wire rope can be written as

**Fig. 4** Structure of single strand steel wire rope

$$\begin{split} \pi\_{\text{s12}} &= \frac{\overline{(\theta\_{12} + 1) - (\theta\_{12} + 1)}}{h} = \frac{\varepsilon\_{12}'}{r\_{12}\tan\alpha\_{12}} - \frac{1}{r\_{12}}a\sin\left(\frac{\varepsilon\_{11}' + 1}{\varepsilon\_{12}' + 1/\sin\alpha\_{12}}\right) + \frac{\alpha\_{12}}{r\_{12}} \\ &+ \upsilon \frac{R\_{11}\varepsilon\_{11}' + R\_{12}\varepsilon\_{12}'}{r\_{12}^2\tan\alpha\_{12}} \end{split} \tag{8}$$

The torsion angle per unit length of steel wire rope and its core wire can be written as

$$
\pi\_s = \pi\_{s11} = \pi\_{s12} \cdot \frac{6W\_{t12}}{W\_{t11} + 6W\_{t12}} \tag{9}
$$

In work, one end of a steel wire rope can be considered fixed while the other end is in a free state. As shown in Fig. 6, the free end generates a component *px*  perpendicular to the axial direction of the wire rope under the influence of the weight and the structure of the wire rope, causing the wire rope to rotate. The turning angle of the steel wire rope increases with the increase of the twisting angle of the steel wire rope. The expression for *px* can be written as

$$p\_x = p\_y \cdot \tan \mathfrak{h} \tag{10}$$

Figure 7 shows the working state of the steel wire rope. Figures 8 and 9 show that the load of the wire rope can be decomposed into axial force F11 and axial torque M11 on the central wire, as well as axial force F12 and axial torque M12 on the external wire. According to Eq. (4), the torque balance equation of the steel wire rope can be written as

**Fig. 6** Schematic diagram of wire rope rotation

**Fig. 7** Steel wire rope for hanging heavy objects

The force balance equation of steel wire rope can be written as

$$\begin{cases} \begin{aligned} \mathcal{Q} &= F\_{11} + 6 \times F\_{12} \\ F\_{11} &= \mathbb{E}\pi \, R\_{11}^{2} \varepsilon\_{11} \\ F\_{12} &= A\_{F12} \varepsilon\_{11}^{\prime} + B\_{F12} \varepsilon\_{12}^{\prime} \\ A\_{F12} &= \frac{(1 + 2v\sin^{2}\alpha\_{12})\pi r\_{12}\cos^{3}\alpha\_{12}\tan\alpha\_{12} - \pi v^{2}\sin\alpha\_{12}\cos^{4}\mathcal{R}\_{11}}{4(1 + v\frac{r\_{12}}{r\_{12}})R\_{12}^{2}} \, E\mathcal{R}\_{12}^{2} \\ B\_{F12} &= \left[\pi\sin\alpha\_{12} - \frac{(1 + 2v\sin^{2}\alpha\_{12})\pi r\_{12}\cos^{3}\alpha\_{12}\tan\alpha\_{12} - \pi v^{2}\sin\alpha\_{12}\cos^{4}\mathcal{R}\_{12}}{4(1 + v)r\_{12}^{3}/R\_{12}^{2}}\right] \end{aligned} \end{cases} (12)$$

# *4.1 Relationship Between Load and Deformation of Multilayer Strand Steel Wire Rope*

Based on the torque balance equation of a single strand steel wire rope, we have obtained the torque balance equation of a multi-layer strand steel wire rope, which can be written as

$$\begin{cases} M\_{in} = -(M\_{11} + 6 \times M\_{12} + 6 \times M\_{21} + 36 \times M\_{22} \cos \beta\_{22} + 12 \times M\_{31} \\ \qquad + 72 \times M\_{32} \cos \beta\_{22}) \\\ M\_{11} = \frac{E}{2(1+v)} M\_{11} \cdot \text{tr}\_{511} \cdot W\_{i11} \\\ M\_{ij} = \left( A\_{Mij} + B\_{Mij} R\_{xin\_{ij}} \right) \varepsilon\_{xim\_{ij}}^{'} + \left( -A\_{Mij} + B\_{Mij} R\_{ij} \right) \varepsilon\_{12}^{'} + \pi \frac{r\_{ij}}{R\_{ij}} \cos a\_{ij} \varepsilon\_{ij}^{'} \\\\ A\_{Mij} = \frac{\left[ \left( 1 + \cos^2 a\_{ij} \right) \cos 2a\_{ij} - 2(1+v) \left( 1 + \sin^2 a\_{ij} \right) \cos^2 a\_{ij} \right] \pi r\_{ij} \sin a\_{ij} \tan a\_{ij}}{4(1+v)r\_{ij}^2 / R\_{ij}} \, ER\_{ij}^3 \\\\ B\_{Mij} = \frac{\left[ \left( 1 + \cos^2 a\_{ij} \right) \sin^2 a\_{ij} - (1+v) \left( 1 + \sin^2 a\_{ij} \right) \cos^2 a\_{ij} \right] \pi v \cos a\_{ij}}{4(1+v)r\_{ij}^2 / R\_{ij}} \, ER\_{ij}^3} \end{cases} \tag{13}$$

The force balance equation of multi-layer wire rope can be written as

$$\begin{cases} Q = F\_{11} + 6 \times F\_{12} + 6 \times F\_{21} + 36 \times F\_{22} \cos \beta\_{22} + 12 \times F\_{31} + 72 \times F\_{32} \cos \beta\_{32} \\ F\_{11} = E \pi R\_{11}^2 \epsilon\_{11} \\ F\_{ij} = A\_{Fij} \epsilon\_{xi\_{i \text{in}\_{ij}}}^{'} + B\_{Fij} \epsilon\_{ij}^{'} \end{cases}$$

$$\begin{cases} A\_{Fij} = \frac{\left(1 + 2v \sin^2 \alpha\_{ij}\right) \pi r\_{ij} \cos^3 \alpha\_{ij} \tan \alpha\_{ij} - \pi v^2 \sin \alpha\_{ij} \cos^4 R\_{ij}}{4(1+v)r\_{ij}^3 / R\_{ij}^2} E R\_{ij}^2 \\\\ B\_{Fij} = \left[\pi \sin \alpha\_{ij} - \frac{\left(1 + 2v \sin^2 \alpha\_{ij}\right) \pi r\_{ij} \cos^3 \alpha\_{ij} \tan \alpha\_{ij} - \pi v^2 \sin \alpha\_{ij} \cos^4 R\_{ij}}{4(1+v)r\_{ij}^3 / R\_{ij}^2}\right] E R\_{ij}^2 \end{cases} (14)$$

### **5 Theoretical Calculation and ABAQUS Simulation**

To verify the correctness of the relationship between load and deformation of multilayer wire ropes, We calculated the deformation of 18 × 7 + IWS multi-layer strand steel wire rope under load. The parameters of the multi-layer wire rope are the same as those of the multi-layer wire rope in Fig. 2. The lifting weight is 70 kg, the elastic modulus is 202,000 Mpa, and the Poisson's ratio is 0.3. The deformation results of the free end of the multi-layer strand steel wire rope under a load of 70 kg are shown in Table 1 through calculation.


**Table 1** Deformation results of free end of multilayer strand steel wire rope

We established a three-dimensional model of multi-layer wire rope in CREO, and then imported the model into ABAQUS to add heavy loads and set material parameters. Finally, we conducted finite element analysis on the model. Figure 10 show the solid finite element model of the steel wire rope. Figure 11 show the finite element analysis of the solid finite element model of the steel wire rope with a heavy object. Figure 12 show the finite element displacement analysis results of the overall model of multi-layer stranded wires. Figure 13 show the deformation results of the free end of the central stranded wire in the *z* direction.

**Fig. 12** Displacement results of the overall model

**Fig. 13** *z*-direction deformation results of the free end of the central strand core wire

In the drawing function of ABAQUS, we have plotted the displacement curve of the top left corner vertex in the *x* and *y* directions in Fig. 12. The plotted curve is shown in Fig. 14, and the stable displacement value is shown in Fig. 15.

The displacement of the top left corner vertex of the heavy object in the *x* direction is −0.310654 mm, and the displacement in the *y* direction is −0.311836 mm. After calculation, the torsion angle of the steel wire rope during balance is displayed as

$$\tau\_s = 0.178^\circ \text{ (right rotation)}$$

**Fig. 14** Displacement curve of the top left corner vertex of the heavy object


The error between the theoretical calculation results and the simulation calculation results of the torsion angle during the balance of multi-layer wire ropes is written as

$$\delta = \frac{0.185^{\circ} - 0.178^{\circ}}{0.178^{\circ}} \times 100\% = 3.93\%$$

### **6 Conclusion**

This article establishes a parameterized model of steel wire rope based on the relationship between its structure and parameters, providing an effective mathematical analysis model for studying new steel wire rope structures. Then we explored the relationship between load and deformation of single-layer and multi-layer steel wire ropes, and summarized the relationship between torsion angle and deformation of steel wire ropes under gravity. After verification, we found that the relative error between the deformation calculation results of multi-layer wire ropes under load and the finite element calculation results is 3.93%. The research results provide theoretical guidance for exploring the accuracy impact of steel wire ropes during crane operation.

**Acknowledgements** This work is supported by National Natural Science Foundation of China under Grant No. 52275088 and the Fundamental Research Funds for the Central Universities under Grant No. DUT22LAB507.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Moment-Frequency Characteristics of Limited-Angle Torque Motors for Direct-Drive Servo Rotary Valve**

**Junfeng Shan, Youcheng Shi, Hong Ji, Shengbing Cao, Zheng Li, and Hucheng Zhang** 

**Abstract** Direct-drive rotary valve (DDRV) is directly driven by a limited-angle torque motor (LATM), which is characterized by anti-pollution, high efficiency and high frequency response. A LATM is designed for the DDRV. The main dimensions and electromagnetic parameters of the motor are calculated by analytical expression of electromagnetic torque. The torque–angle characteristics and torque characteristics of LATM were analyzed by using finite element method for ensure that the motor can have stable torque output in a wide range of turning angle, the results of theoretical and simulation agree well. On the basis, the motor was improved. The results show that the motor has a working angle of ≥±20°, the torque fluctuation within the working angle is less than 5%, and is suitable for driving the DDRV.

**Keywords** Direct drive servo valves · Limited-angle torque motors · Rotary valve · Torque fluctuation · Moment angle characteristic

# **1 Introduction**

With the characteristics of fast response and high accuracy, electro-hydraulic servo valve is widely used in the aerospace field and engineering machinery. It is basically composed of a torque motor and a hydraulic amplifier, and can drive the highpower hydraulic energy through electronic signals. At present, the most widely used domestic product are nozzle flapper valve and jet pipe valve. The nozzle flapper valve has small flow channel, sensitive to oil pollution. The pre-stage hydraulic amplifier structure of jet pipe valve is complex, difficult to manufacture, and has large internal leakage. In recent years, the demand for miniaturization and high-frequency response

© The Author(s) 2024

J. Shan · Y. Shi (B) · H. Ji · Z. Li · H. Zhang

College of Energy and Power Engineering, Lanzhou University of Technology, Lanzhou 730050, China

e-mail: shiyc@lut.edu.cn

S. Cao AVIC Qingan Group Co., Ltd., Xian 710000, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_57

of servo valves has led to rapid development of direct drive valves (DDV) [1]. DDV includes slide valve and rotary valve. The rotary valve uses the rotary movement of the valve spool relative to the valve body or valve sleeve to realize the opening and closing, reversing and flow regulation of the oil circuit. Compared with the slide valve, the rotary valve has better high-frequency response characteristics, can realize the control of small flow, and has higher resolution; It can be directly connected with the driving device, which greatly simplifies the mechanism and improves the anti-pollution performance; And the valve spool movement has no acceleration zero drift, and the control accuracy is high [2].

With the development of new materials, the driving structure of DDV is more diversified, such as piezoelectric ceramics, giant magnetostrictive material driven servo valve has been widely used, but its limitations are also very obvious, the disadvantage of small strain makes it can only be used in small flow field. Yu et al. developed a piezoelectric ceramic DDV with a hydraulic micro-displacement amplification. It can be applied to servo systems with large flow and high-frequency [3]. In view of the poor tensile capacity of the pressure ceramic actuator, which affects the performance of the DDV, Guan and Jiao proposed a piezoelectric ceramic DDV with a new multi-body contacting valve spool-driving mechanism [4]. Giant magnetostrictive materials have excellent properties such as large magnetostrictive strain, high output force, high electromechanical coupling coefficient, and fast response speed. They are widely used in electro-hydraulic servo valves. When applied to DDV, the valves have a high-frequency response. However, due to the limited magnetostrictive strain, the output flow rate is relatively small. If they are to be used in large flow applications, an amplification mechanism is required [5]. Yuan et al. has developed a DDV by LATM. The rotational motion of the motor is transformed into the linear motion of the valve spool through the eccentric mechanism at the end of the motor shaft, and the corresponding flow is output. The control of the output flow is transformed into the control of the angular displacement of the motor, but the design of the conversion mechanism is relatively complex [6]. Some DDVs also use voice coil motor as driving mechanism. Wu et al. developed a DDV using high-frequency voice coil motor and advanced digital controller, which further improved the performance of the DDV in steady and transient states [7]. Shih et al. developed a fuzzy self-tuning controller for the DDV driven by voice coil linear force motor. After verification, the controller is superior to the PID controller, making the response of the DDV faster than the general servo valve [8].

The development of rotary valves has a long history in countries such as the United Kingdom, the United States, and Germany. The related technological and products are already mature. Rotary valve is commonly found in patents and company product introductions. In 2015, MOOG Inc developed a DDRV, valve spool can be driven by a torque motor, and the control of the oil circuit is achieved by changing the position of the valve spool relative to the valve sleeve. This valve has a simple principle, is easy to manufacture, and effectively balances the hydraulic forces on the valve spool [9]. In 2018, Zhu et al. proposed a novel servo valve which is driven by two servo motors. The valve sleeve and valve plate can rotate. It can increase the operating frequency and effectively balance the hydrodynamic forces [10]. In 2022, DOMIN Inc developed a mini rotary valve. The valve spool can rotate and the valve sleeve is omitted [11]. Research on rotary valves in China started relatively late. Cui developed rotary valve is driven by an electromagnet. The geometric shapes of the valve spool and the valve sleeve are symmetric, which offsets the radial unbalanced force on the valve spool and greatly reduces the driving torque [2]. Wang et al. proposed a theoretical model to calculate the flow torque in rotary valve which is driven by a servo motor. The experimental results showed that the model is effective [12].

In practical applications, DDVs driven by LATM mainly utilize a conversion mechanism to transform the rotational motion of the LATM into linear motion of the valve spool. However, this design increases the volume of the valve and complexity of the structure, and difficult to manufacturing. LATM is a kind of motor that can rotate within a certain angle range around their axis without additional mechanical devices. They had advantages such as convenient control, high torque density, simplicity, and reliability. However, because of cogging torque is existence, they had torque fluctuations. These fluctuations can affect the high accuracy and smoothness of DDVs. This article developed a LATM for DDRV and conducted simulation analysis using the finite element method. This approach significantly reduces torque fluctuations and enhances the accuracy and stability of the rotary valve.

### **2 Method**

In this article, LATM is used to drive the rotary valve, the motor shaft is directly connected to the valve spool, and the valve spool and the motor shaft are rotate synchronously within a certain range. Linear Hall will rotor position feedback to the controller to realize the valve spool position closed-loop control. Structure shown in Fig. 1. Eliminated the motion conversion device, the valve spool and valve sleeve are relative rotate, make the valve had ability of low friction, anti-pollution.

LATM on a similar principle to permanent magnet brushless DC motors, both based on ampere forces. Structurally, the windings are embedded at the stator and the permanent magnets are placed at the rotor, the difference is that the latter has multi-phase winding, while the former only has single-phase winding [13]. Cogging torque is one of the unique problems of permanent magnet motor, which must be considered and solved in the design and manufacture of permanent magnet motor.

Cogging torque is a pulsating torque generated by the interaction between the permanent magnets and the tooth slots in the core when the winding is not energized. The cogging torque varies with the position of the rotor. It arises from the tangential force between the permanent magnets and the tooth of the stator, which always tries to align the magnetic field axis of the permanent magnets with the axis of the stator tooth, thereby causing the rotor to have a tendency to be positioned at a certain location [14].

From an energy perspective, the cogging torque is caused by the changes in magnetic field energy generated by the permanent magnets. The magnetic field energy is related to the angular position of the rotor. In fact, the cogging torque

*Tcog* can be approximated as the rate of change of the static magnetic field energy *W*  in the motor air gap with respect to the rotor angle. Therefore, the cogging torque can be defined as the negative reciprocal of the magnetic field energy *W* with respect to the relative position angle α between the stator and rotor when the motor is not energized. This relationship is given by Eq. (1).

$$T\_{\rm cog} = -\frac{\partial W}{\partial \alpha} \tag{1}$$

In surface-mounted permanent magnet brushless DC motors, assuming the relative permeability μ*r* of the armature iron core is infinite, and the permeability of the permanent magnet material is the same as air, which is μ0, the magnetic field energy can be approximately considered as the sum of the energy of the permanent magnet and the energy of the air gap.

$$W = \frac{1}{2\mu\_0} \int\_v B\_r^2(\theta) \left( \frac{h\_m}{h\_m + g(\theta, \alpha)} \right)^2 dV \tag{2}$$

In the above equation: *Br* (θ) is the distribution of permanent magnet residual magnetism along the circumference; g(θ, α) is the distribution of the effective air gap length along the circumference when the angle between the pole centerline and the tooth centerline is α; *hm* is the length of permanent magnet magnetization direction.

Fourier decomposition of *B*<sup>2</sup> *<sup>r</sup>*(θ):

$$B\_r^2 = B\_{r0} + \sum\_{n=1}^{\infty} B\_{rn} \cos(2np\theta) \tag{3}$$

In the above equation: *p* is the number of pole pairs; *n* is the positive number that makes *nz*  <sup>2</sup>*<sup>p</sup>*an integer.

Fourier decomposition of ( *hm hm*+*g*(θ ,α)) 2 :

$$\left(\frac{h\_m}{h\_m + g(\theta, \alpha)}\right)^2 = G\_0 + \sum\_{n=1}^{\infty} G\_n \cos(nz(\theta + \alpha))\tag{4}$$

The final expression for the cogging torque is given by

$$T\_{\rm cog}(\alpha) = \frac{\pi z L\_a}{4\mu\_0} \left(R\_2^2 - R\_1^2\right) \sum\_{n=1}^{\infty} n G\_n B\_{r \frac{n\omega}{2\rho}} \sin(n z \alpha) \tag{5}$$

In the above equation: *R*1 is the outer diameter of armature; *R*2 is the inner diameter of stator; *La* is the length of armature core.

The expression of the cogging torque can be seen that the cogging torque is mainly related to *Brn* and *Gn*. Therefore, from reducing the amplitude and number of Fourier decomposition coefficients, weakening the cogging torque is mainly achieved by reasonable pole-slot fits, changing the pole parameters and changing the armature parameters [15].

A reasonable pole-slot fit can change the number and amplitude of *Brn* and *Gn*  to weaken the cogging torque. The concentrated winding LATM can significantly reduce the effect of cogging torque on motor performance [16]. The concentrated winding LATM has an equal number of poles and slots, and there is only one cogging torque cycle within one electromagnetic torque cycle, and the amplitude of the cogging torque is very small within the constant torque interval of the LATM, so the cogging torque has little influence on the constant torque range. The use of a concentrated winding makes the winding end length smaller than that of the conventional cogging type, which effectively reduces the use of copper. Meanwhile, due to the relatively small number of slots in the concentrated winding LATM, the slot area utilization is high and the winding coefficient is higher than that of the distributed winding, resulting in a higher power density.

In addition, changing the pole parameters can change the amplitude of Brn. Therefore, the design of unequal thickness of permanent magnets can also reduce the impact of cogging torque on the motor performance. The shape of the permanent magnets affects the distribution of the air-gap magnetic density, which affects the amplitude of the cogging torque, in the design of this article, the rotor is of conventional surfacemounted type, but the shape of the permanent magnets is in the shape of a bread-like,



which reduces amplitude of cogging torque and thus improves the performance of the motor [17].

Based on the above analysis, the general structure of the motor was determined to be a new structure combining a concentrated winding and a bread-like shape of permanent magnets, as shown in Fig. 2. Subsequently, the motor was further designed based on the design parameters. The specific design parameters are listed in Table 1.

The specific design process is as follows:

First, based on Eq. (6) [16], select the materials for each component and estimate the size of the motor;

$$T\_c = \frac{2P\mu\_0\mu\_r H\_c h\_m IN\_c l\_{ef}}{\ln(r\_2/r\_1) + \mu\_r \ln(r\_3/r\_2)}\tag{6}$$

In the above equation: *P* is the number of pole pairs, μ0 is the permeability of air, μ*r* is the permeability of permanent magnet, *Hc* is the coercivity of permanent magnet, *hm* is the thickness of permanent magnet, *I* is the current, *Nc* is the number of windings, *lef* is the effective length of the motor, *r*1 is the inner diameter of the permanent magnet, *r*2 is the outer diameter of the permanent magnet, *r*3 is the inner diameter of the stator.

Secondly, according to the design requirements, the outer diameter of stator and the number of windings are determined, and the slot full factor of the motor is calibrated; the rated torque and constant torque range of the motor under the initial scheme are obtained by using the finite element method.


Then, under the premise of meeting the design requirements, the simulation results were analyzed and compared using the finite element method. The analysis primarily focused on the fluctuation and mean values of the cogging torque and electromagnetic torque under different tooth parameters and magnetic pole parameters (such as tooth width, slot width, tooth tip thickness, and permanent magnet width). By comparing the results, the parameter combinations that minimized the fluctuation and maximized the mean values can be determined.

Finally, the overall assembly scheme of the motor is designed, and a sealing structure is specifically designed for the working environment of the LATM to ensure that the motor armature is not affected by hydraulic oil.

By calculating and calibrating the slot full factor, the motor parameters obtained are shown in Table 2.

### **3 Simulation Results and Comparisons**

Torque fluctuation not only affects the positioning accuracy of the motor, but also triggers vibration and noise, so in the process of motor design, it is necessary to use finite element method to simulate and analyze the motor, and through the comparative of various parameters, minimize the torque fluctuation of the motor as much as possible.

Improvement target: Motor with less torque fluctuation in the ±20° torque range of the motor.

The parameters of the motor tooth and magnetic poles are shown in Table 3. During the improvement process, it is important to ensure both the motor performance and outer diameter dimensions while also ensuring sufficient slot area. Considering these factors together, the following ranges for the tooth parameters can be determined: tooth width range of 5–6 mm, slot width range of 1–2 mm, tooth tip thickness range of 1–2 mm, and permanent magnet width range of 9–11 mm.

# *3.1 Tooth Width*

First of all, to determine the appropriate tooth width parameters, when the current with an RMS value of 1.5 A is passed, the electromagnetic torque varies with the *T*<sup>w</sup> as shown in Fig. 3, and the cogging torque varies with the tooth width as shown in Fig. 4.

From the simulation curves of electromagnetic torque, it can be observed that the three curves obtained are essentially identical when the tooth width varies between 5 and 6 mm. Regarding the simulation curves of cogging torque, it is evident that the tooth width affects the crest value of the cogging torque. As the tooth width increases, the crest value of the cogging torque decreases. In order to achieve a desired slot fill factor while ensuring that the motor has strong saturation resistance capability, the tooth width is selected as 5.5 mm.

### *3.2 Slot Width*

Then, the slot width of the LATM is analyzed. The variation of electromagnetic torque and cogging torque with the slot width is shown in Figs. 5 and 6.

As shown in the Fig. 5, the slot width affects the asymmetry of electromagnetic torque on the left and right sides of the rotor position, and the symmetry of electromagnetic torque is optimal at mm. As the slot width varies from 1 to 1.5 mm, the amplitude of the cogging torque decreases. However, as the slot width further increases from 1.5 to 2 mm, the amplitude of the cogging torque increases in the opposite direction. After comprehensive consideration, the slot width is selected as 1.5 mm.

### *3.3 Tooth Tip Thickness*

The tooth tip thickness is further analyzed. The variation of electromagnetic torque and cogging torque with tooth tip thickness is shown in Figs. 7 and 8.

When the tooth tip thickness changes from 1 to 1.5 mm, it affects the asymmetry of the electromagnetic torque on the left and right sides of the rotor position θ = 0◦ and the average value in the working range; when the tooth tip thickness changes from 1.5 to 2 mm, there is no significant change in the curve. And with the change of tooth tip thickness, the change of cogging torque is very small and basically overlapped. The final value of tooth tip thickness is 1.5 mm.

### *3.4 Permanent Magnet Width*

Finally, based on the determined tooth width, slot width, and tooth tip thickness, a comparison is made of the electromagnetic torque curves obtained from different permanent magnet widths to determine the optimal permanent magnet width parameter. The variation of electromagnetic torque and cogging torque with the width of permanent magnet is shown in Figs. 9 and 10. As can be seen from the figures, the width of permanent magnet has a greater influence on the electromagnetic torque and cogging torque. When the width of permanent magnet changes from 9 to 10 mm, the electromagnetic torque curve is smoother and the average value is larger. When the width of permanent magnet changes from 10 to 11 mm, the electromagnetic torque curve changes little. The permanent magnet width also affects the crest value of the cogging torque. It can be seen that when the permanent magnet width is 9.5 mm, the fluctuation is smaller. The permanent magnet width is selected as 9.5 mm.

After the above comparative analysis, the final parameter results are *T*<sup>w</sup> = 5.5 mm, *t*1 = 1.5 mm, *t*2 = 1.5 mm, *bm* = 9.5 mm.

The general assembly of the LATM developed in this article is shown in Fig. 11, which mainly consists of stator, rotor, coil winding, end-cap assembly, angular displacement sensor, steel sleeve, and shell. The rotor consists of a rotor shaft, a magnetic yoke and permanent magnet, and the end-cap assembly includes an inner end cap, an outer end cap and an angular displacement sensor. The rotor shaft transmits the angular displacement to the valve spool. The steel sleeve and O-ring ensure that the hydraulic oil can only enter the inside of the steel sleeve, providing cooling for the rotor portion and lubrication for the bearings without affecting the coil [18]. The comparison of the results before and after the improvement is shown in Fig. 12. The average value of the improved torque within the working angle ±20° is 93.5 mNm, and the torque fluctuation rate is 4.27%, which meets the requirements for use as a rotary valve driving mechanism.

**Fig. 10** Variation of cogging torque with permanent magnet width

**Fig. 11** LATM overall assembly diagram

### **4 Conclusion**

A LATM is developed for rotary valve with simple overall structure, strong antipollution ability and high frequency response. The finite element simulation results show that the motor can output torque (>90 mNm) stably in the operating range of ±20°, and the torque fluctuation is <5% (4.27%), which meets the design requirements.

By combining the concentrated winding and the permanent magnet of breadlike design, the amplitude of the cogging torque is reduced to near zero within the working range, thereby weakening the effect of the cogging torque on the motor's performance.

The sealing structure composed of steel sleeve and O-ring avoids the influence of hydraulic oil on the coil, so that the working process of the motor is not disturbed. At the same time, hydraulic oil can lubricate the bearings, rotor part of the cooling, compact structure.

**Acknowledgements** Project supported by the National Key Research and Development Program of China (Grant No. 2022YFC2805703) and the Natural Science Foundation of Gansu Province, China (Grant No. 21JR7RA270).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation Research on Cutting of Shield Machine Cutter Tool Based on Anisotropic Composite Materials**

**Qiuping Wang, Wanli Li, and Daozhi Wang** 

**Abstract** In order to study the cutting force and variation of anisotropic composite materials such as plastic vertical drainage board left in soft soil foundation by the cutter on the cutterhead of shield machine. Finite element model of orthogonal cutting theory based on macroscopic anisotropic composite materials was established by numerical simulation. Based on the above model, the finite element analysis software LS-DYNA was used to numerically simulate the cutting of plastic vertical drainage board by shield machine cutter in a project in Singapore. Meanwhile, the single cutter test device and triaxial force sensor were used to build a test platform to focus on the cutting ability test of the cutting tool on the plastic vertical drainage board under the cutting conditions of 0° and 90°. The validity and rationality of the simulation model are verified by comparing the simulated cutting force value with the experimental value. The results show that the cutting process of the plastic vertical drainage board is from the notch generation to the notch expansion and then to the final breaking process. So it can be concluded that the sharp-edge cutter is more suitable for the cutting of the plastic vertical drainage board. In the actual construction, ensuring the wear resistance and impact resistance of the sharp-edge cutter is the key to improve the service life. Sponsored by Shanghai Rising-Star Program (22QB1401900).

**Keywords** Shield machine · Cutterhead · Anisotropy · Cutting simulation · Sharp-edge cutter

W. Li e-mail: cnlwl@tongji.edu.cn

D. Wang e-mail: wang\_daozhi@tongji.edu.cn

#### Q. Wang

© The Author(s) 2024

Q. Wang (B) · W. Li · D. Wang School of Mechanical Engineering, Tongji University, Shanghai 201800, China e-mail: wangqiuping@stecmc.com

Equipment Design and Research Institute, Shanghai Urban Construction Tunneling Equipment Co., Ltd., Shanghai 200137, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_58

### **1 Introduction**

Shield machine is a large high-tech construction equipment dedicated to underground tunnel excavation. Modern shield machine integrates optical, mechanical, electrical, hydraulic, sensor and information technology, involving geology, civil, mechanical, mechanics, hydraulic, electrical, control and measurement and other disciplines [1].

In the large water content, large pore ratio, high compressibility, deep soft land foundation highway, railway, airport and other facilities, the shield machine often encounters the plastic vertical drainage(PVD) board left by the soft land foundation treatment [2–4]. The middle of the PVD board is wavy or harmonica shape extruded plastic core board shown in Fig. 1, which is mixed with polypropylene (PP) and polyethylene (PE). Plastic core board surrounded by non-woven filter layer which is the skeleton and channel of the PVD board. PVD board is a commonly used geotechnical material for soft foundation treatment. It can increase the drainage way of soil layer, shorten the drainage distance. Under the action of the upper load, additional stress is generated to make the pore water through the drainage board, reduce the pore ratio and water content, increase the soil compaction. It can accelerate the consolidation and settlement of the foundation for achieving a higher consolidation degree in a short time and improving the bearing capacity of the foundation soil [5–7].

PVD board has good toughness, corrosion resistance, so it is difficult to degrade. PVD board can remain underground for decades. When the shield machine cross through this section, it would block the cutterhead, screw conveyor and reduce the tunneling efficiency [1]. According to the literature, the current research on PVD board cutting is mostly in the qualitative simulation test study in the laboratory, but the theoretical research on the damage mechanism of the composite matrix is not deep enough [8].

**Fig. 1** Plastic core board of PVD board

The finite element model based on the macroscopic anisotropic composite, predict the plastic drainage plate and the method, and reveal the matrix damage, the evolution of matrix damage and the surface damage mechanism [9, 10].

### **2 3D Model Establishment**

### *2.1 Material Setting*

LS-DYNA is both explicit and implicit algorithm, the main advantage is to deal with material failure and large deformation problems. LS-DYNA calculation process based on ANSYS Workbench environment.

As shown in the Fig. 2, Singapore T316 project may have PVD board from Changi Bound CH99+950 to CH100+170 and Woodlands Bound CH99+970 to CH100+120 about 220 m zone with a spacing of 1.2 m. The stratum in this area is soft soil which is mainly fine sand and clay strata through PVD board during tunneling.

The cutterhead is a composite cutterhead, and the front disc cutter spacing is 115 mm. Disc cutter can be replaced with PVD special single-edged tear cutter according to the construction needs. In this project, the 6.7 m shield machine is analyzed by numerical simulation. During the single cutter cutting process, the blade material mold steel is T13. According to the characteristics of single-cutter cutting, the cutter is assumed to be rigid. Mechanical property parameters of the PVD board materials are shown in Table 1. The PVD board is assumed to be an anisotropic elastic oplastic ribbon material with a sectional size of 100 mm × 2 mm in order to simplify the calculation model. It's worth noting that segmented linear elastoplastic material is very commonly used because the data we measure experimentally is often of this type.

**Fig. 2** PVD zone


**Table 1** Mechanical property parameters of the PVD board materials

**Fig. 3** Basic strength of single-layer composite materials

Unidirectional reinforced composites are orthogonal anisotropic materials. As shown in the Fig. 3, the load along the main direction of the material is called in the main direction load, and the corresponding stress is called in the main direction stress. The strength of the following characteristics:


X—axial or longitudinal strength (along the main material direction 1);


In this strength theory, the stress in the main direction of the material must be less than the strength of the respective direction, otherwise the damage occurs. For tensile stress is:

$$\begin{cases} \sigma\_1 < X\_1 \\ \sigma\_2 < Y\_1 \\ |\mathbf{r}\_{12}| < S \end{cases} \tag{1}$$

The model was imported into ANSYS-WorkBench, and the surface grid was refined when dividing the grid. If the cutterhead speed is 0.85 r/min, the simulated cutter speed is 150 mm/s. And the cutting depth is 5 mm, 10 mm, 15 mm, 20 mm, 25 mm and 30 mm respectively [11].

**Fig. 4** PVD cutter

### **3 Cutting Force Experiment**

### *3.1 Cutter Selection*

The selection of cutting force experimental cutter based on Singapore T316 project engineering demand. The main design purpose is to improve the cutting effect of the PVD board, because the ground is soft soil layer. So the cutter edge with sharp Angle design improve the cutting ability of the tool. The wider cutting area on cutter could improve the impact resistance and durability of the cutter. At the same time, the cutter which is easy to meet the positive and reverse running requirements of the cutterhead is designed as a double direction blade type. The slice blade design is shown in Fig. 4, referred to as PVD cutter.

### *3.2 Cutter Wear Calculation*

The relationship between the wear height t (mm) and operating distance λ (km) can be based by the accumulated experience of Komatsu in the following [12]:

$$\mathbf{t} = \mathbf{K}\lambda \times 10^{-3} \tag{2}$$

In the formula:

K—Wear coefficient of PVD cutter. According to the similar strata and similar cutterl use experience in China, the wear coefficient of the steel wear resistant cutter of the mold K is about 25–45.

The tunnel distance that can be excavated when reaching PVD cutter wear limit

$$L = 10,000 \times \text{Pe} \times \text{t} (2\pi \text{R} \times K) \tag{3}$$

In the formula:

L—The tunnel distance that can be excavated when reaching PVD cutter wear limit, m.

K—Wear coefficient of PVD cutter. Combined with the similar project— Zhoushan Lujiazhi project and the formation factors, the wear coefficient is precalculated according to the highest value K = 45.

t—Allow wear height of PVD cutter, mm.

R—Installation radius of the outermost peripheral cutter, m.

Pe—The cutting depth of PVD cutter, cm r–1.

$$P\mathbf{e} = V/N\tag{4}$$

V—The excavation speed of the shield machine, cm min–1.

N—Cutterhead rotation speed, r m–1.

According to the analysis, when the wear height of the cutter head is calculated at 25 mm, the tear cutter is expected to reach 526 mm.

### *3.3 Cutting Test Equipment*

The instrument performs single cutting test using the linear cutting device shown in Fig. 2. The data acquisition system uses FC3D three-axial force sensor fixed between the cutter pedestal and the planer roof-plate to test the force values of X, Y and Z. The X axis side force direction is perpendicular to the side of the cutterl; the Y-axis normal force direction is perpendicular to the surface of the cutting material; the Z axis cutting force direction is parallel to the feed direction. The sensor range is 10 kN shown in Fig. 4. The auxiliary load measuring device uses the S-type tension sensor installed on the fixed edge of the drainage plate. The sensor range is 7 kN shown in Fig. 5 [13].

According to the experience of cutting the PVD board, the main failure forms are expected to present three main states shown in Fig. 6: (1) complete cutting off; (2) complete pulling failure on the fixed side; (3) cutting and pulling failure. For the cutting test of the PVD board, the placement state of the test device has the following two types. According to the cutterhead rotation speed of 0.85 r/min, the simulated cutting speed is 150 mm/s. The different placement PVD boards are tested mainly by changing the cutting depth, and the cutting depth is kept unchanged during the single cutting process [14].

A comparison of the different placement PVD boards cutting peak forces are made in Table 2. By observing the cutting state under horizontal placement shown in Figs. 7 and 8, the pulling failure of the PVD board accounts for a large part of the fracture of the drainage plate. The common damage process is: (1) cutting produces a cut; (2) the proportion of the pulling fracture increases; (3) pulling failure.

**Fig. 5** Cutting test equipment

**Fig. 6** Two types placement state

By observing the cutting state under vertical placement shown in Figs. 9 and 10, the cutting failure of the PVD board accounts for a large part of the fracture of the drainage plate. The common damage process is: (1) cutting produces a cut; (2) the proportion of the cutting fracture increases; (3) pulling failure.

It can be seen that the cutting failure process of PVD board is from the incision to the expansion of the incision and then to the final pull damage.


**Table 2** Cutting working condition data

**Fig. 7** Before cutting state under H placement

### **4 Conclusions**

The validity and rationality of the simulation model are verified by comparing the simulated cutting force value with the experimental value. The results show that the cutting process of the plastic vertical drainage board is from the notch generation to the notch expansion and then to the final breaking process. So it can be concluded that the sharp-edge cutter is more suitable for the cutting of the plastic vertical drainage board. In the actual construction, ensuring the wear resistance and impact resistance of the sharp-edge cutter is the key to improve the service life.

**Fig. 8** After cutting state under H placement

**Fig. 9** Before cutting state under V placement

#### **Fig. 10** After cutting state under V placement

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation of the Influence of Rotational Speed on the Crushing of Cement Agglomerates**

**Binqiang Wang, Jianjun Shen, Feng Jia, and Zehang Sun** 

**Abstract** The uniform dispersion of cement agglomerates during the concrete mixing process affects the overall homogeneity of concrete and adversely affects its strength. To investigate the effect of mixing speed on the crushing of cement agglomerates, this article uses the discrete element software EDEM to simulate the process of crushing of cement agglomerates by collision with aggregates during the mixing process. Within the traditional mixing speed, three different mixing speeds are set to study the crushing ratio of cement agglomerates. The results show that, when other parameters are unchanged and the mixing speed is certain, the breaking of cement bonds shows a rapid increase in time and then a slow increase in time and finally tends to stabilize. To make the cement agglomerates uniformly dispersed, the mixing time should be maintained for more than 15 s at a speed of more than 60 rpm. When the speed is below 50 rpm, the mixing time should be extended and the mixing intensity should be increased.

**Keywords** Cement aggregates · Rotating speed · Crushing · EDEM

# **1 Introduction**

Cement concrete is an artificial stone made of cement, water, sand, stone and additives according to a certain proportion, which is widely used in China's infrastructure construction industry [1]. Mixing is a process to make the overall dispersion of concrete mixture more uniform, and it is a key procedure in concrete production [2]. After mixing ordinary concrete, it is found that the cement slurry in the homogeneous concrete is observed macroscopically. Under the microscope, it is found that 10–30% of the cement agglomerates will agglomerate together [3], forming tiny flocculent aggregates and dry powder aggregates, which are not evenly dispersed in the water,

B. Wang · J. Shen (B) · F. Jia · Z. Sun

Key Laboratory of Road Construction Technology and Equipment of Ministry of Education, Chang'an University, Middle-Section of Nan'er Huan Road, Xi'an 710064, Shaanxi, China e-mail: sjjun7406@sina.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_59

affecting the overall uniformity of the final concrete, and having a very negative impact on the strength, durability and workability of the concrete [4].

The mixing process plays a role in breaking up agglomerates and dispersing particles, which is essential for improving the fluidity of the slurry [5, 6]. The fluidity of the slurry depends on the microstructure of the slurry, and cement agglomeration and dispersion directly affect the flow of the slurry [7–10]. Mixing speed is the main operating parameter imposed during the mixing process and has an important influence in the agglomeration and dispersion of cement agglomerates. Therefore, it is important to study the effect of speed on cement agglomerates to improve the strength of concrete.

Numerical simulation is an effective means to study the collision, crushing and dispersion of materials in the mixing process. On the basis of molecular dynamics, Cundall proposed the discrete element method (DEM) as an analysis method of discrete materials in 1971 [11]. DEM is a common method to study the flow behaviour of particles. The macroscopic flow behaviour of mud and cement agglomerates is reproduced by reconstructing the internal meso-structure [12, 13]. Compared with computational fluid dynamics (CFD), which is difficult to reproduce the movement process of particles, DEM has certain advantages in dealing with the interaction between leaves and particles under mixed conditions and characterizing the agglomeration and dispersion process of particles [14, 15]. Therefore, the aim of this article is to model cement agglomerates using the discrete element software EDEM and the discrete cell method to study the flow behaviour of particles during the mixing process more clearly. By simulating different rotational speeds, the important effects of rotational speed on cement agglomerate crushing and on concrete strength enhancement can be deeply investigated.

### **2 Establishment of Discrete Element Simulation Model**

In order to simulate the crushing of cement aggregates in the mixer, it is necessary to model the mixer and cement aggregates respectively. When the mixer is modelled, the solid mixer is simplified, and the mixing device part is retained and modelled in the three-dimensional modelling software SolidWorks. Using the API (Application Programming Interface) secondary development interface provided by EDEM, the particle replacement and modelling of cement aggregates are carried out. Mixer model and cement aggregates model constitute the basis of the mixing model in this article. Set different mixer speeds to study the crushing of cement aggregates at different speeds.

### *2.1 EDEM and Discrete Element Method*

EDEM is a general discrete element solution software, which consists of three parts: pre-processing module (Creator), solving module (Simulator) and post-processing analysis module (Analyst). The pre-processing mainly completes the model import and setting, the definition of particle properties, the establishment and setting of particle model and particle factory. The solver mainly completes the simulation of the motion process of the geometric model. The post-processing analysis module mainly completes the analysis and processing of the calculation results, and completes the graphical and visual expression of the results.

In the study of particle flow problems, the discrete element method usually divides the studied area into spherical particles, and then based on Newton's second law, calculates the displacement, velocity, acceleration, force and other variables of the particles in each time step, and then enters the next operation process through contact judgment. As shown in Fig. 1. There are usually two different types of processing methods for soft sphere and hard sphere models. The hard sphere model completely ignores the size of the particle contact force and the deformation details of the particle surface, and the contact process is simplified to an instantaneous collision process. The cement group studied in this article has the surface deformation of the particles, so the soft ball model is selected. The soft sphere model simplifies the contact process between particles into the damping motion of the spring oscillator, and its motion equation is calculated according to Eq. (1).

$$m\ddot{x} + \eta \ddot{x} + kx = 0\tag{1}$$

In the Eq. (1), *x* is the displacement deviating from the equilibrium position; *m* is the mass of vibrator; η and *k* are spring damping coefficient and elastic coefficient respectively.

**Fig. 1** Soft sphere model and hard sphere model

### *2.2 Cond Model*

In the EDEM software, the cement particles in a cement agglomerate are connected to each other by means of bonds. The bonds are subjected to tangential and normal forces. When the critical value is reached, the bonds breaks, and the particles are treated as rigid bodies in subsequent calculations. When the particles have not been bonded, the calculation of the particles is calculated by the standard contact model. When the bonding occurs, the ratio of the normal force and moment of the particles is 0, and the following equations are used to calculate the moment and normal force of the particles in each time step.

$$
\delta F\_n = -\upsilon\_n S\_n A \delta t \tag{2}
$$

$$
\delta F\_t = -\upsilon\_t S\_t A \delta t \tag{3}
$$

$$
\delta M\_n = -\omega\_n S\_t J \delta t \tag{4}
$$

$$
\delta M\_t = -\omega\_t S\_n \frac{J}{2} \delta t \tag{5}
$$

$$A = \pi R\_B^2\tag{6}$$

$$J = \frac{1}{2}\pi R\_B^4\tag{7}$$

In the above equation, *Sn*, *St* are the normal stiffness and tangential stiffness of the particles, respectively; *Fn*, *Ft* are the normal force and tangential force of the particles, respectively; δ*t* is the time step; υ*n*, υ*t* are the normal and tangential velocity of particles, respectively; ω*n*, ω*t* are the normal and tangential angular velocity of particles, respectively; *Mn*, *Mt* are the normal and tangential torque of the particles, respectively; *RB* is the radius of the bonds.

The critical values of normal and tangential shear stresses can be calculated by the following equation:

$$
\sigma\_{\text{max}} < \frac{-F\_t}{A} + \frac{2M\_t}{J} R\_B \tag{8}
$$

$$
\tau\_{\text{max}} < \frac{-F\_t}{A} + \frac{2M\_n}{J} R\_B \tag{9}
$$

The bonds parameters between particles and particles are shown in Table 1. The replacement time in API compilation is 0.4 s, and the particles are bonded immediately after replacement. Therefore, the bond time is also set to 0.4 s.


**Table 1** Bond parameter



**Table 3** Material construction contact attribute property


In the simulation process, the constructed cement aggregates should be as close as possible to the actual situation to ensure the accuracy of the simulation. Therefore, it is one of the factors to ensure the simulation results to reasonably determine the intrinsic parameters, basic contact parameters and contact model parameters of aggregates, cement, cement agglomerates and mixing equipment geometric model. In order to simplify the calculation and solution process, two kinds of spherical particles are used to represent single cement particle and cement agglomerates respectively. The particle radius is 3 mm and 20 mm respectively, and the bonds radius between cement particles in the cement agglomerate is 4 mm. Aggregates particles are continuous random distribution with particle size distribution of 15–40 mm. The intrinsic parameters of the material are shown in Table 2, and the attribute parameters between the materials are shown in Table 3.

### *2.3 Cement Aggregates Modeling*

The cement slurry in the mixing process involves two kinds of particle phases that need to be mixed, one is a single cement, and the other is a cement aggregates. EDEM software is often used to build a geometric model of cement agglomerates by

**Fig. 2** Cement aggregates modeling

covering and dispersing a single cement particle in the geometric model of cement agglomerates, as shown in Fig. 2. The average particle size of cement aggregates was selected as 20 mm. The steps of EDEM to construct cement aggregates model are as follows:


When EDEM software is used for simulation calculation, the software cannot directly produce cement aggregates and needs to run API plug-ins to replace them. As shown in Fig. 3a, in the simulation of the process of crushing cement agglomerates by mixing equipment, the particles before replacement are first generated from the particle factory, that is, a complete cement agglomerate. Under the action of the API plug-in, EDEM removes the previously generated replaced particles and then replaces them with cement aggregates composed of many small cement particles, as shown in Fig. 3b. At the same time, a bond (blue cylinder) is generated between small

cement particles, as shown in Fig. 3c. The cement ball collides with the aggregates and the machine during the mixing process, and the bonds breaks, as shown in Fig. 3d.

### *2.4 The Establishment of Mixer Model*

As shown in Fig. 4, the double horizontal shaft forced mixer is the main model in the mixing equipment at present. In this article, this model is used as a vessels machine for mixing cement aggregates and aggregates. As shown in Fig. 5, this article simplifies the mixer, only considering the mixing device, leaving the mixing cylinder, mixing shaft and mixing blade. The number of blades installed on the single stirring shaft is 5, and the blade installation angle α is 37°. The mixing blades are arranged in staggered arrangement to avoid interference, and the arrangement is positive and negative. The two shaft speeds are the same, and the steering is opposite. Save it as IGS format and import it into EDEM.

### **3 The Effect of Rotational Speed on Cement Aggregates**

This article mainly studies the influence of stirring speed on the crushing process of cement agglomerates. The rotation speed of traditional cement mixing equipment is usually below 100 rpm. The rotation speed of mixing equipment studied in this article is set to 50 rpm, 60 rpm and 70 rpm respectively. The crushing situation of cement agglomerates under three rotation speeds is analyzed, and the retention of bonds between cement agglomerates under three rotation speeds are comprehensively analyzed to explore the influence of stirring speed on the crushing process of cement agglomerates. In order to control the variables and ensure the accuracy of

**Fig. 4** Double horizontal shaft mixer

the test comparison scheme, the number of cement agglomerates generated in each comparison test is set to be the same. Considering that the cement agglomerates are replaced by small particles, the number of cement agglomerates is fixed, and the number of bonds between small particles are also the same.

Figure 6 shows the simulation of the crushing process of cement aggregates at different rotational speeds. The dark yellow arrow indicates the movement direction and collision of each aggregate during the mixing process. The colored balls represent the cement aggregates produced during the mixing process, and the balls with different colors represent the magnitude of the force. It can be seen that there is a large amount of collision between cement particles, aggregates, and machines under the action of mixing. The bonds of cement aggregates in the middle and bottom areas of the mixer break, and the cement aggregates are broken and stirred more evenly. At the same time, the number of broken cement aggregates increases with the increase of the rotational speed of the mixer.

During the mixing process, there is a large amount of collision between cement particles and aggregates, as well as between the mixer and the mixing shaft, which

**Fig. 6** Simulation of crushing process diagram

is affected by various forces in different directions. In the process of cement particle crushing, there are both normal and tangential forces. In the process of cement particle crushing, there are both internal stress caused by normal force exceeding the strength, which leads to fragmentation, and shear stress caused by tangential force exceeding the cutting strength, which leads to fragmentation. As shown in Figs. 7 and 8, the average normal force and average tangential force acting on the cement paste at different rotational speeds are respectively shown.

With the operation of the mixer, the average normal force and average tangential force on the cement agglomerate at each rotational speed increased slowly with time. At 0–25 s, the difference between the normal force and tangential force at each rotational speed is not big, and both of them are growing slowly and steadily. There is a clear difference between the normal force and tangential force applied at 50 rpm and the other two rotational speeds after 25 s. At 25–50 s, the normal force and tangential force at 50 rpm rotational speed are no longer growing upward, but keeping a slight fluctuating stability. The normal force and tangential force at 60 and 70 rpm are keeping a slow and steady growth. The normal force at 60 rpm is smaller than that at 70 rpm and the tangential force is larger than that at 70 rpm.

As shown in Fig. 9 is the breakage of cement bonds with time for cement agglomerates at 50 rpm, 60 and 70 rpm mixing speeds, respectively. The number of cement bond breakage over time shows a rapid rise and then a slow rise and finally stabilizes. The number of bond breakage at all three speeds rises rapidly in the early stage, reaches the upper limit of the number of breakage and remains unchanged until the mixing stops. The number of breakages at 50 rpm gradually rises from 0 to 20 s, and reaches the upper limit at 20 s and remains unchanged until the end of the mixing process. The number of breakages at 60 and 70 rpm rises rapidly from 0 to 15 s, and

**Fig. 7** Average normal force on cement particles at different rotational speeds

**Fig. 8** Average tangential force on cement particles at different rotational speeds

reaches the upper limit at 15–50 s until the end of the mixing process. At 60 rpm and 70 rpm, the fracture number rises rapidly in 0–15 s, and reaches the upper limit in 15–50 s and remains stable. The faster the mixing speed, the faster the pre-breakage number grows, and the first to reach the upper limit of breakage.

**Fig. 9** Bonds changes with time at different speeds

In this article, the number of bonds fracture in cement aggregates before and after mixing are used as a method to evaluate the crushing of cement aggregates, and the parameter of fracture percentage is introduced. The more the number of bonds fracture after mixing, the better the crushing effect of cement aggregates, the more conducive to the improvement of the microscopic uniformity of cement slurry, and the better the mixing effect. Therefore, by comparing the changes in the number of bonds before and after the completion of mixing, the crushing effect of cement aggregates and the mixing effect can be roughly judged.

Since the difference in the number of breaks at the later stages of the three rotational speeds was not significant, it was not possible to compare the advantages and disadvantages. Therefore, the point where the fracture number did not reach the upper limit was chosen as the end point to compare the percentage of fracture of the stirring. This article selects 15 s as the final point for comparison, as shown in Table 4. With the increase of stirring speed, the crushing effect of cement aggregates is correspondingly enhanced. The fracture rate of cement aggregates bonds in the mixer at 60 and 70 rpm is more than 90%, and the fracture rate of cement aggregates in the mixer at 50 rpm is less than 80%. Therefore, when the cement mixing operation is carried out, the stirring time should be maintained at least 15 s when the rotation speed is above 60 rpm. When the rotation speed is below 60 rpm, the stirring time should be prolonged accordingly, and the stirring strength should be improved.


# **4 Conclusion**

This article uses discrete element software to simulate the process of cement agglomerates crushing at different speeds in a mixer, and the following conclusions are obtained:


**Acknowledgements** This research was supported by the Key Research and Development Program of Shaanxi (No. 2023-YBGY-327) and the Fundamental Research Funds for the Central Universities of CHD (No. 300102253203).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Study on Wind-Induced Response and Wind Vibration Coefficient of Tower Crane Under Different Wind Speed Spectra Excitation**

#### **Xiangxiang Wang, Hongsheng Zhang, and Wenwu Liu**

**Abstract** The wind-induced response and wind vibration coefficient of tower cranes under different wind speed spectra and wind direction angles are investigated to reveal the influence of wind speed distribution on the wind vibration characteristics of tower cranes. Firstly, the theoretical formula for the wind vibration coefficient based on random vibration theory is derived. Then, the APDL modeling method establishes a parametric finite element analysis model with the QTZ25 tower crane as a research object, and wind speed time history at various heights is simulated using the harmonic synthesis method. Finally, the wind speed is converted into wind loads, and the finite element model of the QTZ25 crane is subjected to random vibration analysis and spectra analysis under wind loads. The displacement time history response and displacement power spectral density at different heights are obtained, and the wind vibration coefficients are derived accordingly. The analysis results show that the wind vibration coefficient in descending order is Simiu, Davenport, and Harris wind speed spectra excitation (WSSE), respectively. The maximum deviation of the wind vibration coefficient is 12.01% at 26.09 m height with a 45° wind direction angle. In contrast, the maximum deviation of the wind vibration coefficient is 18.57% at the same height with a 0° wind direction angle. This study demonstrates that the windinduced response of the QTZ25 tower crane remained the same under different wind speed spectra. Meanwhile, the wind vibration coefficient varied for different heights and wind direction angles, and the wind vibration coefficient was higher under Simiu WSSE and lower under Harris WSSE.

**Keywords** Tower crane · Wind speed spectra · Wind-induced response · Wind vibration coefficient · Parametric modelling

X. Wang · H. Zhang

W. Liu (B)

© The Author(s) 2024

759

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China

College of Engineering, Northeast Agricultural University, Harbin 150030, Heilongjiang, China e-mail: liuwenwu963@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_60

### **1 Introduction**

Tower cranes are important lifting and transportation tools in the construction industry. With the deepening of urbanization, tower cranes are showing a trend of being taller, larger, and heavier. The slight vibrations arising from the truss structure of tower cranes under wind load can cause fatigue in local components and decrease their reliability. Therefore, the study on the wind-induced response and wind vibration coefficients of tower cranes under wind load excitation has significant theoretical and engineering value.

Due to the randomness of wind loads, accurately simulating wind loads is crucial for obtaining accurate structural wind-induced responses. Xu et al. [1] simulated pulsating wind loads using the MATLAB linear filtering method, performed wind vibration response analysis on an inner-suspended outer-tensioned holding pole, and revealed the wind vibration characteristics of the holding pole structure. Ke et al. [2, 3] used dynamic time-domain dynamic calculation methods to analyze the distribution of wind vibration coefficients and their values in chimney structures under different objectives. Based on the modal decoupling principle and detailed time history integration method, Pan et al. [4, 5] proposed a multi-mode detailed time history integration method to derive wind-induced response and wind vibration coefficient calculation formulas for steel pipe truss structures. Deng et al. [6] considered the contribution of the first four vibration modes of the mast structure to the wind-induced response, and a simplified formula for the wind vibration coefficient of the pole was proposed. Wang et al. [7] conducted wind tunnel tests on aerial elastic models to study the aerodynamic response for high-rise single-pole transmission towers under different wind speeds. Chu et al. [8, 9] studied the wind-induced response of wind turbine towers under different WSSE based on random vibration theory. Huang et al. [10] obtained the form factor coefficients of holding poles under different wind direction angles and arm postures through high-frequency balance force tests in the wind tunnel. Ma and Rezaiee-Pajand et al. [11, 12], combining the harmonic superposition method with the autoregressive method, developed a new wind time history hybrid simulation method and studied the wind-induced dynamic response of typical tower crane structures using the Newmark-β method. Luisa [13] based on the idea of gust factor, considering the cross-contribution of different turbulence conditions and wake excitation, a numerical method suitable for cranes is proposed to evaluate the wind-induced response of slender structures. Hamada and Chang et al. [14, 15] studied the aerodynamic characteristics and dynamic response of transmission tower line systems, indicating that wind-induced dynamic response plays a major role in the vibration characteristics of multi-span guyed transmission tower line systems.

In summary, current research mainly focuses on mast-like structures such as transmission towers and holding poles. Meanwhile, there is limited study on the windinduced response of tower cranes under different wind speed spectra. This paper will use different wind speed spectra to simulate wind speeds, and then investigate the wind-induced response and wind vibration coefficient of tower cranes. It will reveal the impact of wind speed distribution on the wind vibration characteristics of tower cranes.

# **2 Theoretical Analysis of Wind Response of Tower Crane Structure**

Based on the theory of structural random vibration, the structural motion equation of tower cranes can be expressed as:

$$\mathbf{M}\ddot{\boldsymbol{x}}(t) + \mathbf{K}\boldsymbol{x}(t) + \mathbf{C}\dot{\boldsymbol{x}}(t) = P(t) \tag{1}$$

where

M, C, and K are the mass, damping, and stiffness matrix of the structure, respectively,

*x*¨(*t*), *x*˙(*t*), *x*(*t*) are the node's acceleration, velocity, and displacement, respectively,

*P*(*t*) is the Vector representation of pulsating wind force.

$$\alpha\_i(t) = \sum\_{j=1}^{n} \phi\_{ij} q\_j(t) \tag{2}$$

where

*xi* (*t*) is the displacement of node *i*,

φ*ij* is the *jth* mode shape coefficient of node *i*,

*q j* (*t*) is the *j-*th mode shape.

The solution of substituting Eq. (2) into Eq. (1) is deduced as:

$$M\_j^\* \{ \ddot{q}\_j(t) + 2\xi\_j \omega\_j \dot{q}\_j(t) + \omega\_j^2 q\_j(t) \} = P\_j^\* f(t) \tag{3}$$

where

ω*j* , ξ *<sup>j</sup>*are the natural frequency and damping ratio of *j-*th mode shape,

*f* (*t*) is the pulsating wind load,

*M*<sup>∗</sup> *j* , *P*<sup>∗</sup> *<sup>j</sup>*are the generalized mass and generalized force in mode shape coordinate system:

$$M\_j^\* = \sum\_i m\_i \phi\_{ji}^2,\; P\_j^\* = \sum\_i \kappa\_i A\_i \phi\_{ji} \; .$$

where

κ*i* is the pulsating surface force,

*Ai* is the windward area corresponding to node *i*,

*mi* is the mass of the section of the structure.

According to the random vibration theory, the generalized integral open square of the output power spectral density (PSD) of the tower crane in the frequency spectra is the root variance σ*x* of the pulsation response with a mean of zero:

$$\begin{split} \sigma\_{xi} &= \sqrt{\int\_{-\infty}^{\infty} S\_{\mathbf{x}}(z\_{i}, \omega) d\omega} = \sqrt{\sum\_{j=1}^{n} \sigma\_{xij}^{2}} \\ &= \sqrt{\sum\_{j=1}^{n} \frac{\phi\_{ji}^{2}}{(M\_{j}^{\*})^{2}} \int\_{-\infty}^{\infty} \sum\_{i} \sum\_{k} \kappa\_{i} A\_{i} \kappa\_{k} A\_{k} \rho\_{ik}(\omega) \phi\_{ji} \phi\_{jk} \sqrt{S\_{f}(z\_{i}, \omega) S\_{f}(z\_{k}, \omega)} \left| H\_{j}(i\omega) \right|^{2} d\omega} \end{split} \tag{4}$$

where

*x*, *j*, ω are the response displacement, mode shape, and circular frequency of pulsating wind, respectively,

*i*, *k* are the node number,

*zi* is the node height,

σ*xi* , σ*xji* are the root variance of the nodal displacement response, corresponding to the jth mode of vibration;

*Sx* (*zi* , ω) is the displacement response output PSD,

*S f* (*zi* , ω) is the input excitation PSD,

ρ*ik* (ω) is the spatial correlation coefficient between nodes *i* and *k*.

The total displacement of the crane structure is expressed as:

$$
\alpha\_i = \beta\_z \bar{\mathbf{x}}\_i \tag{5}
$$

$$
\beta\_{\varepsilon} = 1 + \frac{\mu \sigma\_{yi}}{\overline{x}\_i}
$$

where

*xi* is the total displacement response of the structure,

*x*¯*i* is the static displacement average,

μ is the peak factor under probabilistic conditions,

β*z* is the wind vibration coefficient.

# **3 Establishment of a Parametric Finite Element Model for a Typical Tower Crane**

The QTZ25 tower crane is selected as the typical research object. The parameterized finite element analysis model of the QTZ25 tower crane was established using ANSYS Parametric Design Language (APDL), as shown in Fig. 1. The tower shaft

**Fig. 1** QTZ25 parametric finite element model

has ten standard segments, and the lifting arm has seven segments. The tower top height of the QTZ25 tower crane is 30,470 mm, and the tower shaft cross-section parameters are 1300 mm × 1300 mm. The lifting arm cross-section has a bottom edge length of 1000 mm and a height of 1000 mm. The balance arm has a cross-section of 300 mm × 100 mm, and the balance arm height above the ground is 25,000 mm. The balance arm length is 10,000 mm. The tower crane is divided into 12 parts along the height direction, and the middle position of each part is taken as the height value for the wind speed in that part, as shown in Fig. 1a.

The wind direction angle θ is treated as a variable, and the finite element numerical simulations are performed for two calculation conditions with θ values of 0° and 45°, as shown in Fig. 1b. In these two calculation conditions, the shape factor μ*s* is 1.628 and 2.04, respectively. In Fig. 1a, *V* represents the wind speed.

# **4 QTZ25 Tower Crane Wind Response Analysis and Wind Vibration Coefficient Calculation**

### *4.1 Simulation of the Time History of Pulsating Wind Loads*

Considering the excitation of different wind speed spectra, the formulas of various wind speed spectra are shown in Eqs. (6)–(8).

(1) Davenport WSSE

$$\frac{nS\_v(n)}{v\_\*^2} = \frac{4x^2}{\left(1+x^2\right)^{4/3}}, x = 1200\frac{n}{\overline{v}\_{10}}\tag{6}$$

(2) Harris WSSE

$$\frac{nS\_v(n)}{\upsilon\_\*^2} = \frac{4x}{\left(2+x^2\right)^{5/6}}, x = 1800\frac{n}{\overline{v}\_{10}}\tag{7}$$

(3) Simiu WSSE

$$\frac{nS\_v(n)}{v\_\*^2} = \frac{200f}{(1+50f)^{5/3}},\\ f = \frac{nz}{\bar{v}\_{10}}\tag{8}$$

where

v<sup>∗</sup> is the shear flow rate: v<sup>2</sup> <sup>∗</sup> = *K* (v¯ <sup>10</sup>) 2 ,

*S*v(*n*) is the pulsating wind speed PSD,

*n* is the frequency,

*K* is the wind spectrum coefficient,

v¯ 10 is the average wind speed at 10 m above the ground.

From the pulsating wind speed PSD, the pulsating wind pressure PSD is:

$$S\_w(n) = 4\left(\frac{\varpi}{\bar{v}\_{10}}\right)^2 S\_v(n) \tag{9}$$

where

is the average wind pressure,

*S*w(*n*) is the pulsating wind pressure PSD.

The wind pressure *p* at the windward face node of the tower crane can be obtained by multiplying the average wind pressure by the body size factor μ*s* and height factor μ*z*:

$$p = \mu\_s \mu\_z \varpi \tag{10}$$

By selecting Class B landforms, the average wind speed v¯ <sup>10</sup>= 10.0 m/s at a height of 10 m, and the average wind pressure in the working state of the tower crane is 150 N/m2. When the height above the ground is 10 m, the three pulsating wind speed PSD curves and pulsating wind pressure PSD curves are shown in Fig. 2.

The wind speed time history (WSTH) at different heights of the Tower crane is simulated by the harmonic synthesis method and considers the effect of different wind speed spectra.

The wind speed simulation at the height of 10.8 m and 23.85 m of the QTZ25 Tower crane is shown in Fig. 3, Fig. 4, and Fig. 5, respectively.

The pulsating wind load of each node of the tower crane is calculated as follows:

$$P(z,t) = 0.5\mu\_s \rho A \text{ [}V(z,t)\text{]}^2\tag{11}$$

where

μ*s* is the body size factor, <sup>ρ</sup> is the air density, <sup>ρ</sup> <sup>=</sup> <sup>1</sup>.25 kg/m3 , *A* is the member projection area,

**Fig. 2** Wind speed PSD and wind pressure PSD curves with different wind speed distribution laws

**Fig. 3** The WSTH curve of Davenport WSSE

*V* (*z*, *t*) is the wind speed time history at height z.

### *4.2 Wind-Induced Response Calculation and Analysis*

For the wind-induced response of the Tower crane under different WSSE, Eq. (11) is applied to the finite element model to obtain the displacement time history response of the QTZ25 Tower crane at different heights under different wind directions as shown in Figs. 6 and 7.

**Fig. 5** The WSTH curve of Simiu WSSE

It can be seen from the analysis of displacement time history response in Fig. 6 and Fig. 7 that under the excitation of different wind speed spectra when the wind direction angle is 0° and 45°, respectively, the displacement response amplitude increases with the increase of height. The displacement time history of the QTZ25 Tower crane structure at typical heights of 10.8 and 23.85 m is consistent, indicating that the wind-induced displacement response of the Tower crane under the excitation of different wind speed spectra is almost the same.

**Fig. 6** Displacement time history at different heights at 0° wind angle

**Fig. 7** Displacement time history at different heights at the 45° wind angle

### *4.3 Calculation and Analysis of Wind Vibration Coefficient*

The displacement PSD of the QTZ25 Tower crane at different wind directions and heights under different wind speed spectral excitation is shown in Fig. 8 and Fig. 9, respectively.

It can be seen from the analysis in Figs. 8 and 9 that with the increase of excitation frequency, the displacement PSD at different heights changes significantly, and the displacement PSD values at the same height at different wind directions are also quite different. At the same height, the displacement PSD value under the excitation of the Simiu wind speed spectrum is the largest, followed by the Davenport wind speed spectrum, and the displacement PSD value under the action of the Harris wind speed spectrum is the smallest. It can also be seen from Figs. 8 and 9 that under

**Fig. 8** Displacement PSD at different heights at 0° wind angle

**Fig. 9** Displacement PSD at different heights at a 45° wind angle

the excitation of different wind speed spectra, the QTZ25 Tower crane is prone to resonance at the first three natural frequency ranges, producing more significant displacement.

According to the national reliability index regulations, consider the value of the peak factor μ under a certain probability of crossing is 3.241. According to the displacement response and displacement PSD of the QTZ25 Tower crane at different wind directions and heights under the Davenport, Harris, and Simiu WSSE, the comparison of structural wind vibration coefficients is shown in Fig. 10.

It can be seen from Fig. 10 that with the increase of height, the wind vibration coefficient of the QTZ25 Tower crane structure under different WSSE gradually increases. At the same height, the wind vibration coefficient in descending order is Simiu, Davenport, and Harris WSSE, respectively. Compared with the Simiu wind speed spectrum, The wind vibration coefficient values under Davenport WSSE are closer to those under Harris WSSE because the Simiu wind speed spectrum considers

**Fig. 10** Wind vibration coefficient of QTZ25 structure at different heights

the impact of height changes on wind speed. The higher the height, the greater the wind speed. It can also be seen from Fig. 10 that different wind directions have a specific impact on the wind vibration coefficient of the QTZ25 Tower crane structure. Under the excitation of the three wind speed spectra, when the wind direction angle is 0°, the maximum deviation of the wind vibration coefficient at 26.09 m height is 18.57%. When the wind direction angle is 45°, the maximum deviation of the wind vibration coefficient at 26.09 m height is 12.01%.

### **5 Conclusion**

Based on the random vibration theory, the theoretical formula of the wind vibration coefficient of the Tower crane structure is derived. Taking the QTZ25 Tower crane as the research object, the parametric finite element analysis model of its beam structure is established. The wind speed distribution law is simulated by the Davenport, Harris, and Simiu wind speed spectra. The wind-induced response and spectra analysis of the QTZ25 Tower crane are carried out accordingly. The displacement response and displacement PSD at different wind directions and heights are obtained. Then the wind vibration coefficient formula is used to obtain the wind vibration coefficient at different heights and wind directions under three wind speed spectral excitations. The specific conclusions are as follows:

(1) Under the excitation of different wind speed spectra, the displacement response of Tower cranes at different heights is consistent, which means nearly the same displacement time response can be obtained with different wind speed distribution laws.


**Acknowledgements** This research is supported by the Guangxi Science and Technology project of an open competition mechanism to select the best candidates (Grant no: 2022JBGS039).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Structural Stability Analysis of a High Power PEMFC Stack**

**Ting Chen, Jie Pan, Xiao Zhang, Dong Guan, and Zhen Chen** 

**Abstract** An automobile fuel cell stack mounted horizontally will be subjected to a large impact in the direction parallel with the cells. This may cause interfacial slippage between cells, showing a downward bowing phenomenon, which may cause decay in the performance, and even stack leakage. Therefore, ensuring structural and mechanical integrity is very important in stack assembly design, especially for high power stacks with hundreds of unicells. In this paper, the authors presented numeric analysis for a 300 kW stack with 550 unicells, and analyzed the local vibration from the stack global modes based on finite element analysis (FEA), further evaluated the vibration responses, such as deformed shape and directions, then modified the stack structurally by applying the reinforcement bars, and finally confirmed the robustness of the reinforced stack. It is demonstrated that with proper design, a single stack with hundreds of unicells can be produced for vehicle applications.

**Keywords** PEMFC · Fuel cell stack · FEA · Shock and vibration · Robustness

# **1 Introduction**

Due to lack of space in vehicle compartments, an automotive PEM fuel cell stack is normally mounted on its side or mounted horizontally. It is known that this mounting style could cause the slippage among the stack cells during vehicle operations; under certain conditions, cell slippage could be observed as a downward shift in the middle of the stack [1, 2], even hydrogen leak [3], and degradation of catalyst layer in

Jiangsu Sunshine New Energy Co., Ltd., Yangzhou 225000, China e-mail: ting.chen@nowogen.com

D. Guan College of Mechanical Engineering, Yangzhou University, Yangzhou 225127, China

© The Author(s) 2024

T. Chen (B) · J. Pan · X. Zhang · Z. Chen

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_61

MEA [4]. The assembly design has great impact on the reliability of fuel cell stacks, especially for large power stack with high output voltage. Therefore, it is utmost important to ensure the safe operation of an automotive PEM fuel cell stack by providing a structurally sound assembly design.

There is an increasing demand on high power fuel cell stacks for applications [5], such as mining trucks, ship, high-speed trains, and hydrogen-based power generators. In comparison with the common integration approach involving multiple small stacks, a single PEM fuel cell stack has the advantage of system integration due to simplified piping and matching voltage with inverters. However, for high power stacks, this means an assembly of over hundreds or thousands of unicells which posts challenges in manufacturing and operations. In addition, vibration during operation or shipment [6–8] could expose structural vulnerability of these larger stacks, in comparison to smaller stacks [9].

In fact, the stack integrity highly depends on the friction force between the cells which was generated by the stack compression force and surface texture of the connecting cells. When being induced by vehicle road vibration, gravity acceleration could overcome the friction force between the cells and cause the relative shift between the cells. In fact, a study by Hou et al. showed that the hydrogen leakage rate of the stack increased by a factor of 1.5 during the test, and the hydrogen utilization rate decreased by 30.7%, with a maximum decrease of 21.0% in actual efficiency [10, 11]. A sound design principle of fuel cell stacks should be limiting the impact of gravity acceleration with three approaches: by eliminating the space between the cells and side plates, limiting the shape-change of side plates with high strength structural design, and designing stack assembly with lower intrinsic frequencies [12].

Finite element analysis (FEA) is a numerical analysis technique for obtaining approximate solutions to complex engineering problems. A model can be built to replace an object with an assemblage of discrete elements, and then analytically to obtain solutions in approximation to the governing equations. FEA is used extensively for structural evaluation and improvement, and in this paper, the structure of a highpower PEM fuel cell stack is analyzed and optimized.

Despite an increasing number of FEA analysis, there is not yet studies for structural integrity of large stack in vibration environment. Academic publications are often limited to small stack application, such as stacks of less than 100 cells. In this paper, we present a first-of-its kind overview of vibration simulation of a 300 kW single stack with 550 cells. We derive a FEA methodology and reviewed technology parameters and application requirements. The resulting method and data are then used to calculate the deformation potential under highway vibration conditions. By incorporating FEA, this study can help to reduce uncertainty around the future application of large fuel cell stacks.

**Fig. 1** The initial stack assembly

### **2 Method and Materials**

### *2.1 Fuel Cell Stack*

A fuel cell stack with 550 unicells is assembled and selected as the object of FEA analysis. The unicells are sandwiched between anode and cathode end plates (Fig. 1), insulation plates and current collectors, and then were compressed to 100 psi over the active area. The cells and plates are then assembled by adding four pieces of upper and lower side mounting plates. The edges of the side mounting plates are bent 90° to increase the strength of the overall side plates. The side mounting plates are bolted to the end plate to complete the initial stack assembly.

To ensure the robustness of initial stack structure, a reinforced stack assembly was designed (Fig. 2a, b). In the updated version, a total of 14 side bars are added to join the top and bottom side plates. In addition, electrical insulation blocks are inserted strategically into the space between the cells and side plates to limit the movement of cells in the middle of the stack. The addition of side bars is expected to increase the strength of the overall stack structure.

### *2.2 FEA Analysis*

**Parameters:** The following parameters and are used for FEA analysis with Ansys Mechanical APDL: Bolt preload: 20,000; End plate bearing force: 11,150 N \* 2; Stack compression force: 22,300 N; Properties of materials are listed in Table 1; Vibration conditions are adapted from GJB150A—2009 Lab Environmental Test Methods for Military Equipment, and then used in this analysis (Table 2).

**Boundary setup:** To simplify the model, the parts of the end hardware are set to binding (reject normal and tangential movement), and the contact surfaces between the bipolar plates, between the bipolar plate and end plate are set to friction contact, with the friction coefficient set at 0.76 [13]. However, the Ansys software will default

**Fig. 2 a** The reinforced stack assembly (with 550 unicells), **b** the reinforced stack assembly (without 550 unicells)


**Table 1** Materials used in the fuel cell stack

contact as binding in actual calculation, and only mass participation is considered in the displacement caused by the vibration excitation, so the contact type has little influence on the result.

The model adopts a hybrid grid with hexahedral dominant scheme, using a multiarea structured meshing scheme for bolts, bolt gaskets and end hardware, and locally enhanced meshing scheme (enhanced meshing resolution at 2) for the screw holes and their imprint surfaces. The overall element size is 5 mm, the node order is set to 2 to obtain more accurate integration interpolation results, the total number of cells is 1,244,154, the number of nodes is 2,076,272, the total number of meshes in modal computing is 699,432, and the number of nodes is 1,352,436.


**Table 2** Highway truck vibration testing method

Adapted from GJB150A—2009 Lab Environmental Test Methods for Military Equipment

The bolt thread surface and its contact surface are equivalent to binding constraints, the smooth axial surface and its contact surface are set to friction contact with the friction coefficient at 0.7 under standard gravity [13], and the remaining contact surfaces are set to no separation under actual working conditions. In order to prevent the penetration of the contact surface during the calculation, the permeation tolerance is set to 0.9. To further prevent the gap in the 3D model causing the contacting pair to diverge in calculations, the pinball search is enabled and the radius is controlled by the software.

**Ansys solver settings:** The frictional contact is a nonlinear contact, and the nonseparation contact is linear in nature, but a nonlinear algorithm is used in the calculation. To ensure nonlinear result convergence and computational efficiency, augmented Lagrangian algorithm and full Newton–Raphson iterative methods are enabled.

In nonlinear calculation, even if the boundary conditions are set to ensure that force is balanced on both sides of the end plates, a slight deviation of the vector of force during iteration could cause the accumulation of errors in the form of diverging results. Therefore, a weak spring as correction force is introduced to counter the iterative cumulative error.

### **3 Results and Discussions**

### *3.1 Modal and Vibration Simulation Results*

Modal analysis was conducted with the stack mode. In the analysis, the compression force of the fuel cell stack is set at 22,300 N, which is the same as the stack compression force in its manufacturing process. The first 10 orders of the modal analysis results are listed in Table 3.

The results showed that at all orders, the frequencies are above the vehicle resonance frequency at 30 Hz, which suggests the stack structure is safe against the risk of resonance. In addition, by reviewing the mode shape at each order, one could conclude that most of the potential shape change occurs in the middle of the stack, with much lower shape change at both ends of the stack. This conclusion is consistent with the fact that the cells in very large stack are connected through surface friction between the cells, and the middle of the stack is vulnerable under road vibration.

In the second step, the harmonic response deformation cloud was reviewed. The setting is at vertical acceleration of 9 g [14, 15] (Fig. 3). It is concluded that the maximum harmonic response is at 34 Hz which is associated with the largest phase change. Again, the harmonic response deformation result showed the largest shape change potential at this condition is at the middle of the stack (data not shown).

In the third step, the acceleration density spectrum PSD in the vertical direction was conducted according to Highway truck vibration testing method (Table 2). At scale factor of 1 sigma, the deformed cloud diagram (Fig. 4) showed that the largest potential shape change is in the middle of the stack.

To summarize the above modal analysis results, the path to improve the structure of a large fuel cell stack with hundreds of connected cells is to reduce the shape change in the middle of the stack. This could be achieved by limiting the free space around the cells in the middle of the stack, and by strengthen the mechanical structure of stack assembly.

### *3.2 Structural Statics Analysis of Large Fuel Cell Stacks*

In this section, statics analysis is used to improve the structure of the stack assembly, with the emphasis on comparison between the initial stack design (Fig. 1) and the reinforced stack design (Fig. 2a). It is expected that potential design weakness could be illustrated through the FEA analysis, and it could be addressed in the followingup design. Once the weakness is addressed, an experimental vibration test could be carried out for validation of the reinforced design.

Lateral load analysis. A load is equally added to both sides of the stack end plates (Fig. 1), and structural deformation is then observed. Under lateral load, the maximum deformation and stress distribution of the assembly is concentrated in the center of the upper and lower side plates in the original stack design (Fig. 5a). It is

**Table 3** The first 10 order of modal analysis

(continued)

**Table 3** (continued)

**Fig. 3** Shape-change in relation to frequency

**Fig. 4** Deformed cloud diagram of the fuel cell stack under vibration simulation

obvious that in the original stack design, a wider area of the two upper side plates is under higher stress than the two lower side plates.

To apply the same lateral load to the reinforced design, the high stress areas on the upper and lower side plates are greatly reduced (Fig. 5b). However, the upper side plates are still subject to more stress than the lower side plates. In addition, the maximum deformation on the side plates is also reduced from 0.32 mm in the original design to 0.14 mm in the reinforced design. Therefore, the reinforced design with the side bars strengthened the structure of the stack assembly.

Normal load analysis. In the second step, a normal load of 1000 N is applied to the side plates of the stack, and the degree of deformation was observed and compared between the original and reinforced designs. The results show that under normal load, the middle of the side plates is experienced larger deformation at 3.62 mm, while in the reinforced version, it is reduced to 1.02 mm (Fig. 6a, b). In addition, the area of the maximum deformation in the middle of the side plates are also reduced in the reinforced version of stack design. However, the small deformation suggests that the exist of localized Von Mises stress (results not shown), and this will be an area of further design improvement.

### **4 Conclusions**

Modal analysis and load analysis have been conducted to illustrate the structural weakness of a large fuel cell stack. It was found that the weakest area of the stack is in the middle of the stack, and the reinforced stack design with side bars greatly improved its stability.

Results indicated that the first 10 orders of modal analysis of the stack have modal frequency over 30 Hz, so the design of the large fuel cell stack is safe to be used for highway automotive applications. Results from both harmonic response deformation cloud and acceleration density spectrum in the vertical direction point to the potential deformation in the middle of the stack. Therefore, strengthening the

**Fig. 5 a** Lateral load analysis on the original stack design, **b** lateral load analysis for the reinforced stack design

stack assembly structure through design improvement and limiting the middle section of cell movement by removing free space is of ultimate important for robustness of a fuel cell stack.

Results through structural statics analysis confirmed the weak area of the stack assembly is in the middle of upper and lower side plates. By adding 14 side bars (Fig. 2b), the structure of the stack assembly is greatly stabilized with reduced deformation and Von Mises stress. The concept of side bars points out directions for further design improvement and optimization. In the next phase, the authors will conduct shock and vibration tests to validate the FEA model done in this paper, and further structural improvement is expected.

**Fig. 6 a** Normal load analysis results of the original stack assembly, **b** normal load analysis results of the reinforced stack assembly

**Acknowledgements** This work is funded by the National Key R&D Program of China (2022YFB4301400) and Carbon Peak and Neutrality Science and Technology Innovation Fund of Jiangsu (SBE2023090082).

### **References**

1. Zhao J, Li X (2019) A review of polymer electrolyte membrane fuel cell durability for vehicular applications: degradation modes and experimental techniques. Energy Convers Manag 119:112022. https://doi.org/10.1016/j.enconman.2019.112022


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Characteristics of Three-Chamber Hydraulic Cylinder Driving Loader Boom**

**Huidian Zhu, Jiangjiang Feng, and Jing Yang** 

**Abstract** When the loader boom is lifted and lowered, the hydraulic pump operates at high peak power levels. In this process, the hydraulic valve port dissipates the gravitational potential energy of the boom. Consequently, the dissipated energy is converted into thermal energy, resulting in elevated hydraulic oil temperature and reduced energy efficiency. This paper proposes a gravitational potential energy recovery system based on three-chamber cylinder and a hydraulic accumulator. The system utilizes the hydraulic accumulator to balance the weight of the loading boom and achieve energy recovery for the boom. The co-simulation model of wheel loader based on the three-chamber cylinder was built in SimulationX, then the energy consumption was analyzed under two kinds of typical operating modes. The simulation results illustrate that when the initial pressures of accumulators are 6 and 8 MPa with heavy-load and without load, the system has the highest energy utilization rate. By establishing a loader prototype and investigating the operating characteristics and energy efficiency of the boom driven by a two-chamber cylinder and a three-chamber cylinder, the experimental results illustrate that the new system operates smoothly, reducing energy consumption by 39.24% and peak power by 27.41%.

**Keywords** Three-chamber hydraulic cylinder · Loader arm · Hydraulic-gas energy storage · Potential energy recovery

# **1 Introduction**

The loader is one of the most widely used construction machinery. In this working process, the boom lifts and falls frequently. When the boom is lifted, the hydraulic pump drives the high-pressure oil flow to the rodless cavity of the hydraulic cylinder to raise the work-equipment. When the boom is fallen, the gravitational potential

H. Zhu · J. Feng · J. Yang (B)

Key Lab of Advanced Transducers and Intelligent Control System of Ministry of Education and Shanxi Province, Taiyuan University of Technology, Taiyuan 030024, China e-mail: yangjing001@tyut.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_62

energy of the working device is converted into thermal energy in the form of throttling loss, resulting in the increase of the oil temperature and the reduction of the working efficiency of the system.

To reclaim and utilize the stored potential energy during lifting and falling of the working equipment of construction machinery, many researches have been carried out on excavators, forklifts [1–5]. The existing potential energy recovery methods can be categorized into electrical recovery and hydraulic recovery based on the energy storage forms. In the electrical recovery method, the electrical energy storage components are mainly ultra-capacitors and batteries. Anderaen set up a test prototype of forklift truck, using the secondary component hydraulic motor—generator recovery when the lifting mechanism is lowered. The potential energy is transformed into electric energy and subsequently reserved in the battery, and the energy recovery efficiency is obtained by about 40% [6]. Wang and Lin used a similar method to recover the potential energy of the excavator boom. By adding an accumulator in the potential energy recovery loop, the potential energy recovery rate was increased up to 41% and the peak power of the generator was decreased.

If batteries or super-capacitor are used to recover potential energy, the overall recovery efficiency will be reduced due to many more energy conversion links and the complex transfer path of energy recovery. In addition, although the battery has a high energy density, its power density is very low, only about 30–100 W/kg, which cannot meet the demand of instantaneous high power of construction machinery. As an alternative method, the power density of hydraulic accumulator can reach up to 50–1000 W/kg. As a result, the use of hydraulic energy recovery mode can better fulfill the high-power demands of construction machinery across various operating conditions. Ge utilized an asymmetric pump having three ports and an accumulator to regenerate the potential energy directly. The potential energy recovery rate was 82.7% [7]. Ranjan transformed the potential energy of the boom into the stored pressure energy within the accumulator. The pressure oil in the accumulator and the main pump were driven together as the boom was lifted. The proportional flow valve, the accumulator and the model prediction controller were used to realize the position control of the boom cylinder. The efficiency of the system was increased by 10% [8]. Hippalgaonkar introduced the high-pressure oil into the accumulator to convert the gravitational potential energy into energy in the form of hydraulic power when the boom falls. The recovered energy can be used to drive the cooling system or other auxiliary devices. The fuel consumption of the system is decreased by 27% [9]. Zhao adopted the method of energy regeneration, introduced the recovered high-pressure oil from the accumulator into the inlet of the hydraulic pump, and then used the boom potential energy, which reduced the energy consumption of the system by 14.8%.

The advantage of hydraulic energy recovery can be integrated easily into the hydraulic system. However, nearly all conventional potential energy recovery methods possess a multitude of energy conversion links, resulting in a lengthy chain for energy transfer, and the recovered hydraulic energy is not easy to be reused. Xia established a prototype of hydraulic excavator by using the method based on self-weight liquid–gas balance of the three-chamber hydraulic cylinder, and realized about 48.5% of energy consumption reduction.

To solve the problem of large energy loss during the loader boom lifting and falling process, this paper proposes an energy recovery system based on the threechamber hydraulic cylinder driving the loader boom. A good energy-saving effect had been achieved on excavators, however excavators have different external load characteristics from loaders, and the excavator's ratio of the external load to the weight is much smaller than the loader's. As a result, the balance position of the hydraulic and gas balance system of the excavator is basically unchanged when the bucket is lifted with full load and is fallen without load, while the balance position of the loader changes greatly. Therefore, it is necessary to study the dynamic and energy consumption characteristics of the liquid–gas balance energy recovery and reuse system of the three-chamber cylinder of the loader under different load conditions. When the initial working pressure of the accumulator is different, the energy recovery efficiency of the system is also different. The energy recovery efficiency of the liquid– gas balance system can be optimized through experiments. The supreme merit of this method is that it does not require to change the original boom structure and the original hydraulic system. The three-chamber cylinder is used to substitute the traditional single lever hydraulic cylinder, which can directly transform the hydraulic energy and potential energy, and it has high recycling efficiency.

In this paper, the mechanical-hydraulic joint simulation model of the threechamber hydraulic cylinder driving the loader boom is firstly established. The initial pressure of the accumulator is set as 6 MPa and 8 MPa respectively when the loader is under heavy and no-load conditions. A test prototype of a loader with the three-chamber hydraulic cylinder driving the boom is entrenched. The operating characteristics and energy efficiency characteristics of the system are analyzed and compared by the experimental analysis when the two-chamber hydraulic cylinder and the three-chamber hydraulic cylinder are used to drive the boom.

The key contribution of this paper is to propose a method of potential energy of loader boom recovery and reuse based on the three-chamber hydraulic cylinder. In the proposed method, the potential energy recovery chamber in the three-chamber hydraulic cylinder is connected with the accumulator, and the gravitational potential energy is directly recovered and utilized. The loader is the construction machinery for shovel excavation and the piston stroke of the boom cylinder is basically the same in each working cycle. Therefore, the initial pressure of the accumulator is optimized to obtain higher energy efficiency.

The following is the compositional structure of the remainder of the paper. In Sect. 2, the working principle of the three-chamber cylinder driving loader boom hydraulic system with gravitational potential energy recovery is analyzed. In Sect. 3, the co-simulation model is established and the energy consumption is analyzed. In Sect. 4, A prototype of loader is established, Experimental studies were conducted to investigate the operational characteristics and energy efficiency of the boom driven by a two-chamber cylinder and a three-chamber cylinder. Finally, Sect. 5 offers some Summary recommendations about this study.

### **2 System Working Principle**

The hydraulic circuit principle of the three-chamber hydraulic cylinder driving the boom is shown in Fig. 1. The signal test system is also shown in Fig. 1. Chamber C of the three-chamber hydraulic cylinder is composed of hollow piston rod and fixed plunger on the cylinder body, the original chamber without piston-rod and chamber with piston-rod are chambers A and B separately. Chambers A and B are joined with the original hydraulic circuit, which realize the oil inlet and outflow of chambers A and B of the three-cavity hydraulic cylinder. Chamber C is directly connected with the hydraulic accumulator, which realized the direct the recuperation and utilization of the potential energy stored in the boom.

In order to further explain the change of dynamic characteristics in the working process of the new system and the operation of the boom by replacing the twochamber hydraulic cylinder with the three-chamber hydraulic cylinder, the transfer function of the valve-control three-chamber hydraulic cylinder is established. The influence of initial pressure, hydraulic frequency and damping ratio of accumulator on operating characteristics of the boom are analyzed by the transfer function.

Flow continuity equation of a chamber of the three-chamber cylinder is:

$$\frac{V\_A}{\beta\_e} \frac{dp\_A}{dt} = q\_A - A\_A \frac{dx\_P}{dt} - c\_{ep} p\_A \tag{1}$$

where *VA* is the volume of chamber A of three-chamber hydraulic cylinder (including valve and connecting pipe), β*e* is the effective bulk elastic modulus of hydraulic oil (including hydraulic oil, mechanical flexibility of connecting pipes and cylinder blocks), *pA* is the pressure in chamber A, *qA* is the flow of chamber A, *AA* is the area

**Fig. 1** Boom driving system based on three-chamber cylinder

of chamber A, *xp* is the piston displacement of the three-chamber hydraulic cylinder, *cep* is the external leakage coefficient of hydraulic cylinder.

Flow continuity equation of chamber B of the three-chamber cylinder is:

$$\frac{V\_B}{\beta\_c} \frac{dp\_B}{dt} = A\_B \frac{dx\_P}{dt} - c\_{ep} p\_B - q\_B \tag{2}$$

where *VB* is the volume of chamber B (including valve and connecting pipe), *pB* is the pressure in chamber B, *AB* is the area of chamber B, *qB* is the flow of chamber B.

Flow continuity equation of chamber C of the three-chamber cylinder is:

$$\frac{V\_{\mathcal{C}}}{\beta\_{\mathcal{C}}} \frac{dp\_{\mathcal{C}}}{dt} = A\_{\mathcal{C}} \frac{dx\_P}{dt} - q\_{\mathcal{C}} - c\_{ep} p\_{\mathcal{C}} \tag{3}$$

where *VC* is the volume of chamber C (including valve and connecting pipe), *pC*  is the pressure in chamber C, *AC* is the area of chamber C, and *qc* is the flow from accumulator to chamber C.

Chamber C of the three-chamber hydraulic cylinder is directly connected to the accumulator through pipelines, ignoring the pressure loss, thus the pressure in chamber C is equivalent to the oil pressure in the accumulator. Due to the use of bladder accumulator, the relationship between pressure and gas volume in the accumulator satisfies the gas state equation, as shown in the formula (4). Since the accumulator acts as an auxiliary oil source to release energy rapidly, This is commonly recognized as an adiabatic process., so for *n* = 1.4.

$$p\_0 V\_0^{\prime\prime} = p\_c V\_c^{\prime\prime} = \text{Const} \tag{4}$$

where *p*0 is the accumulator initial state pressure, *V*0 is the volume of gas at the initial state of the accumulator, *V*' *<sup>c</sup>*is the volume of gas in the accumulator when the pressure being *pC*, where*V* ' *<sup>c</sup>*= *V*0 − Δ*V* , Δ*V* is the oil volume released by chamber C during operation.

The equation of force balance for this hydraulic cylinder is,

$$A\_A p\_A + A\_C p\_C = m\_t \frac{d^2 x\_p}{dt^2} + B\_p \frac{d x\_p}{dt} + F\_L \tag{5}$$

where *mt* is the total effect of the three-chamber hydraulic cylinder piston mass and the mass converted to the piston, *Bp* is the viscous damping coefficient, *FL* is the external load acting on the piston.

By applying Laplace transform to the Eqs. (1)–(5), we can get:

$$
\mathcal{Q}\_L = K\_q X\_v - K\_C P\_L \tag{6}
$$

$$Q\_L = A\_P s X\_P + C\_{tp} P\_L + \frac{V\_t}{4\beta\_c} s P\_L \tag{7}$$

$$A\_A P\_A + A\_C P\_C = m\_t \mathbf{s}^2 X\_P + B\_P \mathbf{s} X\_P + F\_L \tag{8}$$

where *Vt* is the total compression volume.

The velocity transfer function of three-chamber cylinder is obtained by:

$$X\_P = \frac{P\_0 A\_C + \frac{A\_A K\_q X\_v}{C\_{tp} + \frac{V\_t}{4\rho\_v}s + K\_c} - F\_L}{(m\_t + m)s^2 + 2B\_P s + \frac{A\_A^2 s}{K\_c + C\_{tp} + \frac{V\_t}{4\rho\_v}s}} \tag{9}$$

Natural frequency of hydraulic system and damping ratio are:

$$w\_n = \sqrt{\frac{1}{(m\_t + m)\frac{v\_t}{4\beta\_e}}}\tag{10}$$

and

$$\zeta\_n = \frac{K\_{cc}(m\_t + m) + \frac{B\_F V\_t}{2\beta\_e}}{2\sqrt{(m\_t + m)\frac{V\_t}{4\beta\_e}}} \tag{11}$$

where *m* is the mass of the oil in the accumulator, and *Kce* is the total flow-pressure coefficient.

Increasing the initial pressure *p0* of the accumulator can improve the response speed of the boom. Reducing the oil mass *m* (i.e., reducing the accumulator volume *V0*) in the accumulator can improve the natural frequency of the system and make the system respond quickly, where it reduces the damping ratio and weakens the stability of the system. Therefore, on the basis of comprehensive consideration of the dynamic response speed, stability and energy consumption of the system, the volume of the accumulator should be reduced as much as possible. Therefore, the accumulator volume is preliminarily determined to be 4 L. The initial pressure of the accumulator is determined according to the following simulation study.

### **3 Simulation Study**

Firstly, the 3D model of the loader's working device was established in Pro/E software. Meanwhile, The hydraulic system model of the whole machine was established by using simulation and analysis software SimulationX. SimulationX is a universal CAE tool for modeling, simulation and analysis in a multidisciplinary field. It has a powerful library of standard components and provides a high-effective method of co-simulation between mechanical devices, hydraulic system and control system.

The 3D model was imported into the simulation software SimulationX, in which the 3D model and the simulation model were connected through the three-chamber hydraulic cylinder model to establish the mechanic-hydraulic co-simulation model of the three-chamber hydraulic cylinder driving the boom. The co-simulation model is shown in Fig. 2.

According to the load characteristics and reserved installation space of the boom hydraulic cylinder, the geometrical parameters of the three-chamber hydraulic cylinder are selected, as shown in Table 1. In Table 1. Da is the diameter of chamber A, Db is the diameter of chamber B, Dc is the plunger diameter of chamber C, and *L*  represents the stroke of three-chamber hydraulic cylinder. In the simulation model, three plunger hydraulic cylinders are used to replace the three-chamber hydraulic cylinder, and other hydraulic components are introduced directly from the component library of software SimulationX.

The lifting and falling process of the loaded boom is simulated under no-load and heavy-load conditions separately. The power and energy simulation curves of the accumulator within a single period of lifting and falling process of the boom under different initial pressures of the accumulator are shown in Fig. 3. The simulation result of the hydraulic pump output power is shown in Fig. 4. The power *Ps* and energy *Es*

**Fig. 2** Co-simulation model


**Table 1** The main parameters of the three-chamber cylinder

**Fig. 3** Simulation result of power and energy of the accumulator

**Fig. 4** Simulation result of power and energy of the pump outlet

stored in the accumulator are calculated by Eqs. (12) and (13) respectively, and the output power *Pp* and energy *Ep* of the hydraulic pump are calculated by Eqs. (14) and (15) separately.

$$P\_s = p\_c A\_c v$$

$$E\_s = \int p\_C A\_C vdt\tag{13}$$

$$P\_p = p\_p q\_p \tag{14}$$

$$E\_p = \int p\_p q\_p dt\tag{15}$$

where *pp* represents the output pressure of the hydraulic pump. *qp* represents the output flow of hydraulic pump.

The energy consumption characteristics of the system are shown in Table 2 for the initial working pressures of the accumulator at 4, 6, and 8 MPa. Based on the data in Table 2, it is observed that the hydraulic pump has the minimum peak power and energy consumption under a no-load condition when the initial working pressure of the accumulator is 6 MPa. Additionally, the hydraulic pump exhibits the minimum


**Table 2** Simulation results of energy consumption characteristics under different pressures in the accumulator

peak power and energy consumption under a heavy-load condition when the initial pressure of the accumulator is 8 MPa. Therefore, based on these results, the following experiments were conducted: no-load and heavy-load tests were carried out at the initial working pressure of the accumulator of 6 MPa and 8 MPa, respectively.

### **4 Experimental Study**

XG916 II loader test prototype photos is shown in Fig. 5. The test principle is shown in Fig. 1. Atos pressure sensor is used to test the pressure in each chamber of the hydraulic cylinder. The wire-drawing displacement sensor tests the displacement of the hydraulic cylinder. In the meanwhile, the output power of hydraulic pump also needs to be tested. Considering the throttling loss caused by the flow sensor, the realtime rotation speed of diesel crankshaft is measured by photoelectric sensor. The diesel engine drives the hydraulic pump through the splitter box (the transmission ratio is 1:1), and the real-time rotation speed of the hydraulic pump shaft can be obtained. The hydraulic pump is a gear fixed displacement pump, whose nominal displacement is 50 ml/r, rated pressure is 16 MPa, and rated speed is 2000 r/min. Volumetric efficiency of the hydraulic pump is approximately 92%. When the loader boom is lifted, if the outlet pressure of the hydraulic pump does not reach 16 MPa (i.e., the set value of the overflow valve), all the output oil of the hydraulic pump is supplied to the mobile arm hydraulic cylinder.

The output power of hydraulic pump in energy consumption analysis can be calculated by the following equation:

$$P\_0 = \frac{pVn\tau\_c}{60} \tag{16}$$

where *P*0 is the output power of the hydraulic pump in the test, *p* is the output pressure of the hydraulic pump in the test, τ *<sup>c</sup>*is the volumetric efficiency, *V* is the displacement of the pump, and *n* is the rotating speed of the pump.

**Fig. 5** The test bench

The original hydraulic cylinder of the loader boom is a single-rod piston hydraulic cylinder, of which the area of rod-less chamber is 6362 mm2 and the area of rod chamber is 4398 mm2. The size of the three-chamber hydraulic cylinder is determined by the installation space and simulation results. The area of chamber A is 4938 mm2, the area of chamber B is 1944 mm2, and the area of chamber C is 1964 mm2*.* Thus, the area of chamber B and that of chamber C is approximately equal. The volume of the hydraulic accumulator is 4 L, and the initial pressure of the hydraulic accumulator is determined by the optimization of simulation results.

Two-chamber hydraulic cylinders and three-chamber hydraulic cylinders are used to drive the loader boom under no-load and heavy-load conditions.

### *4.1 No-Load Operation Characteristics*

#### **4.1.1 Two-Chamber Hydraulic Cylinder Driving the Boom**

The displacement and velocity of the hydraulic cylinder are shown in Fig. 6 during the three working cycles when the two-chamber hydraulic cylinder is used to drive the loader boom.

The dynamic working process and energy consumption characteristics of the boom are analyzed in detail for one working cycle. The falling-lifting cycle of the

boom is selected from 10 to 28 s in Fig. 6, the pressure curve of each chamber when the two-chamber hydraulic cylinder drives the boom in this cycle are shown in Fig. 7. When the boom starts to fall, the pressure in chambers A and B fluctuates obviously because of the impact of high-pressure oil on the rod-less chamber when the hydraulic valve core moves. The piston rod of the hydraulic cylinder is retracted by the gravity of the boom itself in the steady descending stage, so the pressure of the rod chamber (chamber B) keeps low, while the pressure of the rodless chamber (chamber A) decreases from 2.1 MPa to about 1.5 MPa to balance the gravity of the arm. At the end of the boom falling, the contact between the bucket and ground results in the impact of external load on the system, and the pressure of chamber A fluctuates sharply. When the boom lifts, the pressure of chamber A fluctuates obviously at the moment of starting, which causes the velocity fluctuation of the hydraulic cylinder. After that, the pressure of chamber A rises to about 2.5 MPa to drive the boom lifting.

The output power and energy curve of the hydraulic pump are shown in Fig. 8 when the two-chamber hydraulic cylinder drives the boom under no-load condition. The power and energy of the hydraulic pump are calculated by Eq. (16) according to the pressure data collected by the pressure sensor and the crankshaft speed of the diesel engine collected by the speed sensor. At the initial stage of the boom falling, the output power of the hydraulic pump was instantly increased to drive the boom to accelerate falling, and then remained at about 0.51 kW. The peak power of the

hydraulic pump in this stage was 4.10 kW, and the total output energy of the hydraulic pump was 4.47 kJ.

#### **4.1.2 Three-Chamber Hydraulic Cylinder Driving the Boom**

The displacement and velocity curves of the hydraulic cylinder is shown in Fig. 9 when the three-chamber hydraulic cylinder is used to drive the boom without load. In the new system, the accumulator is joined with chamber C, so the system damping increases. When comparing three-chamber hydraulic cylinders to twochamber hydraulic cylinders, it can be observed that the former experiences less peak velocity oscillation, and thus the operation is relatively stable.

The pressure of each chamber when the three-chamber hydraulic cylinder drives the boom in a working cycle without load is shown in Fig. 10. Chamber C is joined with the accumulator, and the working pressure of the accumulator is 6Mpa when the boom is at the initial position. When the piston rod of the hydraulic cylinder extends to the maximum displacement, the oil volume of the accumulator is the smallest, and the minimum pressure is about 2.8 MPa. During the descent of the boom, the oil in chamber C enters the accumulator under the action of the gravity

of the boom. The accumulator pressure rises with the increase of the oil volume and rises to 6 MPa at the end of the descent. During this process, The potential energy of the boom is transformed into hydraulic energy and reserved in the accumulator. Since the pressure offered by the accumulator in the descending stage of the boom can basically balance the weight of the boom, The pressure in chamber A is relatively low. The pressure in chamber B fluctuates obviously when the boom begins to drop, and increases with the increase of the pressure in the chamber C. During the boom lifting process, the accumulator releases high-pressure oil into chamber C, and the pressure in chamber A is very low. With the extension of piston rod, the volume of chamber C increases, the oil volume in the accumulator decreases, and the pressure decreases gradually. Therefore, the pressure in chamber A rises gradually to drive the boom lifting, while the pressure in chamber B remains low.

The output power and energy curve of the hydraulic pump when the three-chamber hydraulic cylinder drives the boom without load are shown in Fig. 11. The power and energy curves of accumulator charging and discharging are shown in Fig. 12. In the boom falling stage, as the accumulator recovers potential energy, chambers A and B maintain low pressure, so the hydraulic pump needs to output energy to drive the boom falling, and the peak output power of this stage is 2.25 kW, and the output energy is 6.82 kJ. In the boom lifting stage, the self-weight of the boom is balanced by the accumulator oil hydraulic pressure at the initial position, so the pressure of chamber A is low. As its area is small, so the flow rate is small, and the output power of hydraulic pump is not large. With the decrease of hydraulic oil pressure in the accumulator, the pressure of chamber A of hydraulic cylinder increases, and the output power of hydraulic pump increases gradually. In this stage, the peak output power of hydraulic pump is 1.43 kW and the output energy is 4.41 kJ. The total output energy of the hydraulic pump is 12.46 kJ when the boom is lowered and lifted in one cycle under no-load condition.

# *4.2 Comparison of No-Load Energy Consumption Characteristics*

The energy consumption of the new system and the original system in a lifting cycle is shown in Table 3. In the boom falling stage, the output energy of the hydraulic pump in the original system is 4.47 kJ. Due to an accumulator is used in the new system, more output energy is needed to drive the boom falling, and the hydraulic pump outputs 6.82 kJ. When lifting the boom, the energy output of the hydraulic pump in the new system is 4.41 kJ, while that of the original system is 8.55 kJ, and the energy output of the accumulator is 0.68 kJ. In a complete work cycle of the boom, the peak power output of the hydraulic pump decreased from 3.63 to 1.43 kW, reducing by 60.61%. The energy output of the hydraulic pump decreased from 16.55 to 12.46 kJ, reducing the energy consumption by 24.71%.


**Table 3** Comparison of energy consumption under no-load condition

### *4.3 Heavy-Load Performance*

#### **4.3.1 Conventional Hydraulic Cylinder Driving Boom**

The velocity, the displacement and the pressure of the two-chamber hydraulic cylinder under heavy load are shown in Figs. 13 and 14. As can be seen from Fig. 13, when the throttle angle of the diesel engine remains unchanged, compared with the no-load condition, in the heavy-load condition the speed of the boom decreases slightly when it rises, but becomes faster when it falls, and the operation process is relatively stable. It can be seen from Fig. 14 that under different working conditions, the pressure of chamber B does not change much, while the pressure of chamber A increases obviously under heavy load.

The output power and energy of the hydraulic pump under heavy load condition are shown in Fig. 15. When the boom is lifted and fallen, the output power of the hydraulic pump fluctuates instantly. In the boom falling stage, the peak value of hydraulic pump is 2.77 kW and the output energy is 1.87 kJ. In the boom lifting stage, the peak value of the hydraulic pump is 3.72 kW and the output energy is 13.31 kJ. The total output energy of the hydraulic pump is 21.56 kJ under heavy load condition within a cycle of boom fall-lift.

**Fig. 15** Power and energy of pump outlet under heavy load

#### **4.3.2 Three-Chamber Hydraulic Cylinder Driving the Boom**

The displacement, the velocity and the pressure of the three-chamber hydraulic cylinder under heavy load are shown respectively in Fig. 16 and Fig. 17. During the boom lifting stage, the pressure of chamber A increases as the pressure of chamber C decreases gradually, and the output power of the hydraulic pump also increases. At the initial stage of boom falling, the pressure of chamber B raises briefly and then maintains at a low pressure. During the movement of the boom, the pressure of each chamber of the hydraulic cylinder fluctuates slightly.

Figure 18 shows the output power and energy of the hydraulic pump under heavy load condition are shown in Fig. 18. The power and energy curves of accumulator charging and discharging liquid are shown in Fig. 19. When the boom is falling, the peak power of the hydraulic pump is 0.51 kW and the output energy is 1.89 kJ. When the boom is lifted, the peak power of the hydraulic pump is 3.72 kW and the output energy is 12.31 kJ. The total output energy in one cycle of the hydraulic pump is 13.10 kJ under heavy load condition within one cycle of the boom fall-lift.

Pressure, [MPa]

Time, [s]


**Table 4** Energy consumption under heavy-load condition

# *4.4 Comparison of Energy Consumption Characteristics Under Heavy Load Conditions*

Table 4 shows the energy consumption of the new system and the original system within a cycle of boom fall-lift. The peak output power of the hydraulic pump decreased from 3.72 to 2.70 kW, i.e., a decrease of 27.41% is obtained. The output energy of hydraulic pump decreased from 21.56 to 13.10 kJ, and the energy consumption is decreased by 39.23%.

### **5 Conclusion**

This paper proposes a potential energy recovery system based on three-chamber hydraulic cylinder driving loader boom to recover and reuse the potential energy of gravity. Chamber C is directly connected to accumulator. As the boom is fallen the gravitational potential energy is transformed into hydraulic energy and reserved in the accumulator; as the boom is lifted, the stored energy is released into chamber C and lift the boom auxiliary. Through theoretical analysis and experimental research, the following conclusions are obtained.

The three-chamber hydraulic cylinder is adopted to replace the two-chamber hydraulic cylinder in the original system to drive the loader boom lifting. The three-chamber hydraulic cylinder and the accumulator constitute a liquid–gas energy storage system that can balance the weight of the boom. Through this, the potential energy of the boom can be recovered and reused directly, and the energy consumption and the installed power can be significantly reduced. The output energy of the hydraulic pump is 39.24% less than the former system in one working cycle, the peak power output of the hydraulic pump was reduced by 27.41% compared with the original system in the boom lifting stage. With the benefits of the proposed method, this study made the following contributions. First, the running speed of the system slightly fluctuates to decrease the peak value because the access of the accumulator provides damping for the new system. The required flow rate is reduced at the same speed condition, which further reduces the throttling loss because the area of the working oil chamber is reduced. Second, the Hydraulic-Gas balancing energy recovery system has the characteristics of the reduced number of energy conversion links and shorter energy transfer chains, the system exhibits high efficiency in energy utilization.

In the proposed system, Due to the mechanical connection between the inlet and outlet orifices, a valve control system with a single degree of freedom is used to control chamber A and chamber B in the three-chamber hydraulic cylinder with poor controllability, so that the pressure and the flow of chambers A and B cannot be controlled in the mean time, The throttle is applied simultaneously to both the inlet and outlet of the valve, so the pressure loss is large.

In view of the above limitations, in order to minimize the throttling losses in the system, The next step of the research work involves implementing and testing the use of pumps, valves, synergistic control, and the flow matching method in the hydraulic excavator system to achieve precise control of throttle points for fluid intake and outflow on the basis of the three-chamber hydraulic cylinder, and to realize the independent adjustment of the pressure and flow of chamber A and chamber B in the three-chamber hydraulic cylinder.

**Acknowledgements** The authors gratefully acknowledge the financial support of National Natural Science Foundation of China (U1510206), and the Natural Science Foundation of Shanxi Province of China(201801D121179).

**Funding** National Key Research and Development Program (2021YFB2011900).

**Declaration of Competing Interest** The submission of this manuscript does not create a conflict of interest with any organization or individual and has been unanimously approved for publication by all authors.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research Progress of Remanufacturing Technology in the Field of Construction Machinery**

**Kaiming Wang, Wei Liu, Xiaotong Pang, Yongle Hu, and Yonggang Tong** 

**Abstract** Excavator bucket teeth are often in direct contact with abrasive particles in the soil during work, resulting in the tip being severely worn. This study used laser cladding to synthesize (Ti, Nb)C reinforced coatings on the surface of Q550 steel of bucket teeth of commonly used excavators to improve the wear resistance, and researches the effects of different Ti, Nb, Cr3C2 powder ratios on the hardness, wear resistance, and wear loss of laser cladding (Ti, Nb)C reinforced coatings. The results show that the optimal powder ratio is 80% Ni-based + 1.26% Ti + 7.54% Nb + 11.2% Cr3C2. Under the optimal powder ratio, the hardness of the coating is 213.4 HV, the wear amount is 32.7 mg, and the wear failure form is abrasive wear and slight adhesive wear.

**Keywords** Remanufacturing technology · Construction machinery · Laser cladding · Research progress

# **1 Introduction**

Bucket tooth is a very important part of excavator, and wear failure is the main failure form of bucket tooth [1]. The common methods to improve the surface wear resistance mainly include laser cladding, surfacing welding, thermal spraying, etc. Compared with the other surface modification measures, the coatings prepared by laser cladding technology have the advantages of dense structure and small dilution rate [2].

As an advanced surface strengthening technology, laser cladding technology has been widely used in the automobile industry [3]. It significantly improves the wear resistance and hardness of the substrate by depositing the cladding material onto

K. Wang · W. Liu · X. Pang · Y. Hu · Y. Tong (B)

College of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China e-mail: tongyonggang\_csust@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_63

the substrate [4]. This technology can strengthen the part and extend its service life, thereby reducing costs, improving economy and efficiency.

With the development of industry, single metal materials can no longer meet the production requirements, and their properties can be significantly improved by introducing reinforcing phases. There are two methods to introduce the enhanced phase: direct addition and in situ synthesis, in which the enhanced phase produced by the in-situ synthesis method is better matched to the substrate at the interface [5]. For example, Li [6] et al. synthesized TiB/TiC reinforced titanium-based composite coatings in situ on Ti–6Al–4V, and the coatings were free of porosity. Wang [7] et al. synthesized TiB-reinforced titanium-based composite coating in situ on Ti– 8Al–1Mo–1V, and the results show that the TiB phase generated by in situ synthesis can effectively increase the hardness of the coatings.

In summary, the effects of different Ti, Nb and Cr3C2 powder ratios on the hardness and wear resistance of laser cladding (Ti, Nb)C reinforced nickel-based coating were studied to improve the wear resistance of bucket teeth, and the best powder ratio parameters were obtained.

### **2 Experimental Process**

### *2.1 Experimental Materials and Parameters*

The substrate is Q550 steel, and the specimen size is 100 mm × 100 mm × 12 mm. Nickel-based nickel alloy is used as the self-melting alloy powder, and different contents of Ti, Nb and Cr3C2 powders are added as the reinforcing phase powders. The specific configuration ratio of the powder is shown in Table 1. The laser cladding process parameters are shown in Table 2.


**Table 1** Powder configuration ratio


**Table 2** The laser cladding process parameters

### *2.2 Hardness Test*

Hardness testing was used by HVST-1000Z hardness tester. The measuring load was 300 g and the measuring time was 10 s. Five points were randomly selected in the coatings for testing, and finally the average value was taken to obtain the microhardness.

# *2.3 Wear Test*

MDW-05 high frequency reciprocating wear tester was used to carry out the wear test. The reciprocating stroke was 5 mm, the load was 50 N, and the test time was 30 min.

### **3 Results and Analysis**

### *3.1 Microhardness*

The microhardness of the cladding layer and substrate is shown in Fig. 1, the hardness of the substrate is 150 HV, and the hardness of the coating A to E is 206.2 HV, 205.7 HV, 218.5 HV, 210.2 HV, 213.4 HV, respectively. Overall increase in hardness of the coatings compared to the substrate. This is due to the fact that Cr can act as a solution strengthening effect.

### *3.2 Wear Resistance*

Wear experiments were carried out on the coatings and the substrate, and the wear data are shown in Fig. 2. The wear amount of the substrate is 59.2 mg, and the wear amount of the coating A to E is 48.9 mg, 41.5 mg, 36.3 mg, 37.1 mg, 32.7 mg, respectively. The wear amount of the coatings E is much smaller than the coating A to D. The reason is that the content of Nb in powder E is much higher than that of other powders, and more NbC being generated during the laser cladding process, which improves the wear resistance of the coatings.

**Fig. 1** Microhardness of the coatings and substrate

**Fig. 2** The wear amount of coatings and substrate

The average friction coefficients of the substrate and coatings are shown in Fig. 3. The average friction coefficient of the substrate is 0.536, and the average friction coefficient of coatings A to E are 0.428, 0.499, 0.405, 0.481, 0.441, respectively.

### *3.3 Wear Mechanism*

The worn morphology under different Ti:Nb ratios is shown in Fig. 4. There are obvious furrows on the worn surface, indicating that the wear mechanism is mainly abrasive wear. Furthermore, there is a small amount of adhesive on the worn surface, indicating that there is slight adhesive wear. As can be seen from Fig. 5, the coatings did not undergo oxidative wear during the wear process.

Figure 6 is the surface morphology of the grinding ball after wear, there is a large block of adhesive on the surface of the grinding ball, mainly adhesion wear occurs. EDS analysis of the adhesive and grinding ball shows that the oxidation degree of the grinding ball surface is very low, and the distribution of elements on the surface of the grinding ball is similar to the worn surface of the coatings.

The wear debris morphology is shown in Fig. 7. The wear debris are mainly composed of large blocks, indicates that severe wear of the coatings occurred during the wear process. EDS analysis of the wear debris found that the main chemical composition of the wear debris is basically similar to the worn surface of the coatings and the worn surface of the grinding ball.

**Fig. 4** Worn morphology of C/Ni-based coatings under different Ti:Nb ratios: **a** and **b** Ti:Nb = 6:1; **c** and **d** Ti:Nb = 3:1; **e** and **f** Ti:Nb = 1:1; **g** and **h** Ti:Nb = 1:3; **i** and **g** Ti:Nb = 1:6

**Fig. 5** EDS spot scanning results of Ti:Nb = 1:6 worn surface **a** region 1; **b** region 2

**Fig. 6** Surface morphology and EDS results of grinding ball **a** Surface morphology of grinding ball under low magnification; **b** surface morphology of grinding ball under high magnification; **c** EDS point scan results for point 1; **d** EDS point scan results for point 2

**Fig. 7** Wear debris and EDS results of coatings **a** morphology of wear debris; **b** EDS spot scan results of wear debris

### **4 Conclusion**

In this paper, the effects of different Ti:Nb ratios on the hardness, and wear resistance of Ni-based coatings are studied, and the results are as follows:


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Fast Vision-Based Algorithm for Automated Container Pose Measurement System**

**Yujie Zhang and Chao Mi** 

**Abstract** Addressing the current issues of low accuracy in container positioning and posture recognition, as well as long response times during the port automation loading and unloading process, this paper designs a rapid container target recognition and measurement device and method for automated loading and unloading, thereby optimizing the acquisition of key parameters in automated loading and unloading operations. This method combines advanced convolutional neural networks and traditional image processing algorithms to achieve precise detection and tracking of container corner fittings. Furthermore, this paper proposes a high-speed response method for small target measurement, which integrates minimized deep learning network technology and fuzzy image morphology matching algorithms to enhance the accuracy and stability of corner fitting detection. Through experimental verification, this method effectively improves the speed of single detection and reduces the localization error of small targets.

**Keywords** Vision-based measurement · SSD · Container lifting operation

# **1 Introduction**

In modern port automation systems, gantry rail-mounted container cranes play a critical role, responsible for efficiently and swiftly transferring containers between transportation equipment and container storage areas. Container hoisting operations primarily involve two key alignment steps, as shown in Figure (see Fig. 1).

The increasing demand for automation technology in container docks stems from the inefficiencies of manual alignment operations, characterized by low precision and heightened error probability due to the significant distance between the driver's

Y. Zhang · C. Mi (B)

Shanghai Maritime University, Shanghai 201306, China e-mail: chaomi@shmtu.edu.cn

Y. Zhang e-mail: 202130210133@stu.shmtu.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_64

**Fig. 1** Container lifting operation

cabin and hoisting equipment. Current technologies predominantly utilize laser radar for container positioning, identifying and measuring geometric shapes. Despite its resistance to adverse weather and lighting, the technology is marred by high costs and limited accuracy.

Machine vision and its associated algorithms have significantly advanced dock container automation, facilitating tasks such as container identification and safety inspections [1]. However, the existing research, including the stereo vision-based positioning method proposed by Yoon et al., encounters limitations in accuracy, with errors sometimes reaching up to 60 mm, primarily due to the baseline length between cameras and confined space on lifting equipment [2]. In real dock settings, image processing algorithms grapple with environmental factors affecting image clarity, such as container color variations and surface contamination. Current methods, although capable of locating containers within cabins, necessitate substantial computation time, extending single image processing to 0.6 s [3]. Moreover, existing vision tracking methods, despite their speed, fail to meet the stringent accuracy requirements of container hoisting operations due to simplistic algorithm structures [4].

Addressing these challenges, machine learning emerges as a promising avenue. Recent advancements in deep learning have revolutionized industrial detection and recognition. Studies by Qi and others [5], Kazmi and others [6], and Liu and Wang [7] underscore the potential of deep learning in enhancing product quality and manufacturing efficiency. New methodologies, such as those proposed by Zhang and others [8], He and Liu [9], and Yu and others [10], are paving the way for more precise and efficient container positioning technologies, heralding a new era in port automation systems.

The aforementioned methods primarily target object recognition, detection, and measurement in the industrial sector, mainly applied within factory indoor environments utilizing specialized cameras and computing equipment. However, container automation loading and unloading necessitate adapting to all-weather outdoor scenarios, and due to limited installation space within port equipment, high-computational power devices cannot be employed. Therefore, implementing an efficient all-weather lightweight recognition network under limited computational power conditions presents a current challenge. Given that recognition and measurement methods based on deep learning still exhibit slow response speeds and poor accuracy in container target detection, this paper proposes a fast container target identification and measurement device and method for automated loading and unloading. By compressing and optimizing traditional deep learning networks and integrating container appearance features, the device and method described in this paper achieve higher detection efficiency compared to conventional methods.

### **2 Vision-Based Measuring System**

In this study, a vision-based measurement system structure is proposed, as shown in Figure (see Fig. 2). The primary objective is to accurately determine the pose information of the container. The first step of the system involves detecting the container corners in the image, followed by utilizing this data to calculate the exact position of the container. Upon completion of all steps, the positional data will be transmitted to the ACCS (Automated Crane Control System) to facilitate precise control of the hoisting actions.

### *2.1 Image Capture Device*

The system proposed in this paper relies on multiple cameras to capture images of the top of the container, as illustrated in the figure (see Fig. 3) depicting the device installation method.

### *2.2 Improved SSD Image Processing Section*

An image processing section based on the improved SSD, a model grounded on Convolutional Neural Networks (CNN), is utilized. Figure (see Fig. 4) provides a

**Fig. 2** A vision-based measurement system structure

**Fig. 3** Installation of cameras and the image captured by cameras

detailed display of the basic model based on SSD-300 (with an input image size of 300× 300) and its structural components. To further optimize the SSD model to adapt to the target detection of container corners, the following two main improvements were made to the model based on SSD-300:

**Backbone Layer Update**: The DSSD (Deconvolutional Single Shot MultiBox Detector) is adopted as the improved SSD detector, which enhances the representation ability of shallow features and increases the recognition rate of small targets by replacing VGG-16 with the updated ResNet-50 as the backbone layer. The deeper ResNet can retain more feature information, thereby enhancing the robustness to small targets.

**Feature Map Layer Adjustment**: The feature map layer of the basic SSD model was optimized. Since the original high-level feature map layer is relatively insensitive to the recognition of small targets, the Conv10\_2 and Conv11\_2 layers were removed to accelerate detection speed, while adding a feature map layer with higher resolution to enhance the recognition ability for small targets at container corners.

# *2.3 High-Precision Calculation Method for Container Pose Parameters*

To precisely adjust the control strategies of lifting equipment, the pose parameters of the container are subdivided into two core parts: firstly, the offset vector of the container's geometric center on the horizontal plane, and secondly, the deviation angle of the container on the horizontal plane relative to its geometric center. Based on this, further consideration was given to the issue of measurement data fusion in the dual-camera system. Specifically, when calculating the pose parameters of the container, the average of the measurement results obtained from two independent cameras will be used to derive more accurate overall pose information.

The displacement vector of the container position is denoted as *l* = (Δ*x*, Δ*y*), which characterizes the offset difference on the plane between the current detected position of the container and its theoretical position under standard working conditions. As shown in figure (see Fig. 5), to more comprehensively utilize the information obtained from the dual cameras, a method was designed to calculate the displacement vector based on the Euclidean distance between the spatial coordinates of the four corners of the detected container and their calibrated positions under standard working conditions, as detailed in Eq. (1)and (2).

$$
\Delta \mathbf{x} = \frac{1}{4} (\mathbf{x}\_{a1} + \mathbf{x}\_{b1} + \mathbf{x}\_{c1} + \mathbf{x}\_{d1} - \mathbf{x}\_{a0} - \mathbf{x}\_{b0} - \mathbf{x}\_{c0} - \mathbf{x}\_{d0}) \tag{1}
$$

$$\Delta \mathbf{y} = \frac{1}{4} (\mathbf{y}\_{a1} + \mathbf{y}\_{b1} + \mathbf{y}\_{c1} + \mathbf{y}\_{d1} - \mathbf{y}\_{a0} - \mathbf{y}\_{b0} - \mathbf{y}\_{c0} - \mathbf{y}\_{d0}) \tag{2}$$

On the other hand, the deviation angle of the container is defined as the counterclockwise central angle of deviation relative to its standard working state (i.e., parallel to the observation plane). As revealed in Figure (see Fig. 6).

The calculation of this parameter also integrates the lock hole coordinate information from the images of the front and rear cameras. The angle is determined with reference to the inclination angle of the short side of the container in the detection state relative to the standard state, with specific calculation details shown in equation.

$$\begin{array}{c|c} \text{(\huge{1cm})}\\ \text{(\huge{1cm})}\\ \text{(\huge{1cm})}\\ \text{(\huge{1cm})}\\ \text{(\huge{1cm})} \end{array} \xrightarrow[\text{D(\harrow\text{1cm})}]{\text{A}\_{\text{A}}(\text{x}\_{\text{O(x}}\text{Y}\_{\text{O}}))} \xrightarrow[\text{D(\harrow\text{1cm})}]{\text{B}\_{\text{B}}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O}}))} \xrightarrow[\text{C(\harrow\text{1cm})}]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O}}))} \text{B} \xrightarrow[\text{C(\harrow\text{1cm})}]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O}}))} \text{B} \xrightarrow[\text{C}(\harrow\text{1cm})]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O}}))} \text{B} \xrightarrow[\text{C}(\harrow\text{1cm})]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O(x}})))} \text{B} \xrightarrow[\text{C}(\harrow\text{1cm})]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O(x}})))} \text{B} \xrightarrow[\text{C}(\harrow\text{1cm})]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O(x}})))} \text{B} \xrightarrow[\text{C}(\harrow\text{1cm})]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O(x}})))} \text{B} \xrightarrow[\text{C}(\harrow\text{1cm})]{\text{B}(\text{X}\_{\text{O(x}}\text{Y}\_{\text{O(x}})))} \text{B} \xrightarrow$$

$$\theta = \arccos\left[\frac{1}{4}\left(\frac{a\_1 \cdot a\_3}{|a\_1| \cdot |a\_3|} + \frac{a\_2 \cdot a\_4}{|a\_2| \cdot |a\_4|}\right)\right] \tag{3}$$

$$a\_1 = (\mathbf{x}\_{a2} - \mathbf{x}\_{d2}, \mathbf{y}\_{a2} - \mathbf{y}\_{d2}), a\_2 = (\mathbf{x}\_{b2} - \mathbf{x}\_{c2}, \mathbf{y}\_{b2} - \mathbf{y}\_{c2})\tag{4}$$

$$a\_3 = (\mathbf{x}\_{a0} - \mathbf{x}\_{d0}, \mathbf{y}\_{a0} - \mathbf{y}\_{d0}), a\_4 = (\mathbf{x}\_{b0} - \mathbf{x}\_{c0}, \mathbf{y}\_{0b} - \mathbf{y}\_{c0})\tag{5}$$

Through this approach, it is anticipated that the real-time pose parameters of the container can be calculated more accurately, thereby providing more accurate and stable control instructions for the hoisting equipment.

### **3 Experimental Result**

The device described in this paper has been deployed in a certain railway automated container yard. Experiments are conducted based on this platform, where the image data used in the experiments is captured by high-resolution dome cameras installed at designated positions, with hardware parameters as shown in the Table 1. Detailed configurations of the operating platform relied upon by the measurement algorithm can be seen in Table 2, with its hardware performance matching the capabilities of industrial computers commonly used in the current field of industrial control.


**Table 1** Hardware parameters


**Table 2** Computing platform

# *3.1 Verification of the Improved SSD Image Processing Section*

The optimized SSD algorithm was trained based on 8700 images of the upper surfaces of containers, which were captured during the actual operation process of container lifting equipment. These images encompass the upper surface images of containers under different environmental conditions of day and night lighting, with each image encompassing visual data of about 2 to 6 container corner areas.

The performance evaluation of the improved SSD detector is divided into two stages. The first stage involves evaluation by comparing the detection performance of the SSD model before and after modification. The experimental group for this stage includes our optimized version of the improved SSD-300 model proposed in this paper and the original SSD-300 model. 500 sample pictures taken at different operating times are used to evaluate the detection performance of these two networks, with multiple container corner features annotated in each image. As shown in Table 3, detection performance parameters are calculated through Average Precision (AP). The experiments proved that our optimized SSD network proposed in this study surpasses the original version in terms of detection accuracy and speed, reducing computation time by 5.35 ms and increasing AP by 3.45%. The corner area detection results of the optimized SSD algorithm are shown in the figure (Fig. 7).

The second stage involves localization error analysis, where the deviation between the detection results and image calibration results is statistically analyzed, and its distribution fitting curve is calculated using normal distribution statistical analysis. The maximum error values at 95% and 90% confidence intervals are taken as the maximum error of the calibration results, and the actual error values are calculated, with results shown in Table 4.


**Table 3** The performance of improved SSD detection

**Fig. 7** Lightweight SSD detector corner area detection results

**Table 4** Positioning error of


The standard dimensions of the container corner lock holes are 124 mm × 63.5 mm. The measurement method implemented in this study has an error distribution of 21.3 mm × 15.9 mm at a 95% confidence level. The final measurement accuracy of this method can satisfy the practical application requirements for container pose measurement tasks.

# **4 Conclusion**

This paper proposes a fast container pose measurement device and method for automated loading and unloading. By compressing and optimizing traditional deep learning networks and integrating container appearance features, precise detection and tracking of container corners are achieved. This system reduces the single detection time by 5.35 ms, with a high detection rate of up to 90%, capable of achieving a positioning error between 14.3 and 19.6 mm at a frame rate of 10 fps. This research paves the way for further advancements in port automation, potentially fostering more efficient, safer, and cost-effective operations through the integration of sophisticated detection and tracking technologies in container handling processes.

**Acknowledgements** This research was supported by the Science and Technology Commission of Shanghai Municipality (No. 122ZR1427700 and No. 223692106900)

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **The Relationship Between "Nominal Pressure" and Pressure Terms Related to Lightweight Design and Manufacture of Hydraulic Cylinders**

#### **Yao Liu, Jingyi Zhao, Rui Guo, and Yingda Tang**

**Abstract** Lightweight design and manufacture of hydraulic cylinders is receiving increasing attention in construction machinery and vehicle engineering. Hydraulic cylinders, designed based on some current standards such as JB/T 10205-2010 *hydraulic cylinder*, have a common issue that they need to be built with a large size. This "thick and heavy" is not conducive to improving equipment performance and achieving energy conservation and emission reduction. This paper identified the root causes leading to that issue and managed to make clear the interrelation between terms used in the "nominal pressure" system and the lightweight design and manufacturing method of hydraulic cylinders. A set of novel specifications to design and test hydraulic cylinders have also been proposed which adopts the "rated pressure" as the base parameter, after referring to some related standards. Furthermore, this approach would solve the lightweight problem of hydraulic cylinder manufacturing and lay a reliable foundation for the promotion of hydraulic cylinder technology in China.

**Keywords** Hydraulic cylinder · Nominal pressure · Rated pressure · Lightweight design and manufacture

Y. Liu (B) · R. Guo Yanshan University, Qinhuangdao, Hebei 066004, P. R. China e-mail: 314907883@qq.com

J. Zhao

# Y. Tang

MFR Advanced Material Technology Co., Ltd., Changshu, Jiangsu, P. R. China

© The Author(s) 2024

827

Qinhuangdao Yanda Yihua Mechanical and Electrical Engineering Technology Research Institute Co., Ltd., Qinhuangdao, Hebei, P. R. China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_65

### **1 Introduction**

The lightweight of hydraulic components and systems is one of the important approaches to achieve weight reduction in construction machinery and vehicle engineering. This not only helps to reduce the weight of the equipment itself, but also improves the overall mobility and load-bearing capacity of the machinery. Furthermore, it aligns with the requirements for energy conservation and emission reduction. In all kinds of hydraulically driven mobile equipment: 1% weight reduction of the pump truck can achieve 0.6–1% [1] reduction in fuel consumption; each 1% reduction in aircraft weight improves performance by 3–5%, and can reduce fuel consumption, improve load, and reduce operating costs[2].

At present, in the construction machinery and vehicle engineering equipment, hydraulic cylinders for one of the most lightweight potential hydraulic components, some developed countries have been through Carbon fiber, compact design [3, 4] and other new materials and technology means to achieve lightweight hydraulic cylinders (including integrated in the hydraulic cylinder on the oil block, etc.). However, according to some current hydraulic cylinder related standards, such as JB/T10205- 2010 *Hydraulic cylinder* [5] designed and manufactured hydraulic cylinder "thick and bulky" problem still cannot be ignored. However, the issue of "thick and bulky" hydraulic cylinder design and manufacturing still cannot be disregarded in compliance with some of the most recent hydraulic cylinder standards, such as JB/T 10205- 2010 *Hydraulic Cylinder*. One of the reasons for this problem is the "nominal pressure" as the basic parameters of the hydraulic cylinder design and manufacture of hydraulic cylinders.

Hydraulic cylinder is one of the hydraulic components with the most potential for lightweight, and it is a project involved in the national key research and development plan. Avoid using "nominal pressure" as the basis for hydraulic cylinder design, testing, and using. Instead, use "rated pressure" as the fundamental hydraulic cylinder design and manufacturing criteria in order to achieve the lightweight hydraulic cylinders that is a necessary condition for advancing Chinese hydraulic cylinder technology.

### **2 Why Hydraulic Cylinders Are "Big and Bulky"?**

An example of hydraulic cylinders design is given to illustrate the point. For instance, there is a valve-controlled hydraulic cylinder system, the choice of the rated pressure of the hydraulic valve (or rated supply pressure) of 21 MPa, the user also needs to give the basic parameters of the hydraulic cylinder in order to carry out the purchase of hydraulic cylinders. Because the 21 MPa pressure value is not the GB/T 2346- 2003 [6] "Fluid Transmission System and Components Nominal Pressure Series" in the specified pressure value, can only be selected in the GB/T 2346-2003 specified 25 MPa pressure value as the nominal pressure of this hydraulic cylinder.

Hydraulic cylinder design and manufacture based on the basic hydraulic cylinder parameter of 25 MPa nominal pressure. According to JB/T 10205-2010 "6.2.7 Pressure resistance: Hydraulic cylinder block should be able to withstand the nominal pressure of 1.5 times the pressure, there shall be no external leakage and damage to parts and other phenomena." and JB/T 11718-2013 *Hydraulic Cylinders-Technical conditions for cylinder tubes* [7] given in the calculation formula, cylinder material strength requirements of the minimum wall thickness δ01 is:

$$\delta\_{01} \ge \frac{p\_{\text{max}}D}{2\lbrack\sigma\rbrack} = \frac{1.5 \, p\_{\text{n}}D}{2\lbrack\sigma\rbrack} = 18.75 \frac{D}{\lbrack\sigma\rbrack} \text{ (mm)}\tag{1}$$

where

*p*max—cylinder pressure, MPa;

*p*n—nominal pressure, MPa, *p*n = 25 MPa;

*D*—cylinder diameter, mm;

[σ]—permissible stress of cylinder material, MPa.

As stipulated in JB/T3818-2014 *Hydraulic Press-Technical requirement* [8] "The self-made hydraulic cylinder should be subjected to pressure test, the pressure holding time is not less than 10 min, and there should be no leakage, permanent deformation and damage. The compressive test pressure shall comply with the following requirements: (a) When the rated pressure is less than 20 MPa, the test pressure shall be 1.5 times the rated pressure. (b) When the rated pressure is greater than or equal to 20 MPa, the pressure test pressure should be 1.25 times." According to the design and calculation of the above requirements, cylinder material strength requirements of the minimum wall thickness δ02 is:

$$\delta\_{02} \ge \frac{p\_{\text{max}}D}{2\lbrack\sigma\rbrack} = \frac{1.25 p\_{\text{e}}D}{2\lbrack\sigma\rbrack} = 13.125 \frac{D}{\lbrack\sigma\rbrack} \tag{2}$$

where

*p*e—rated pressure, MPa, *p*e = 21 MPa;

Other parameters have the same meaning as above.

If the same material is chosen for the above two design calculations and the cylinder diameter is the same, comparing Eq. (1) with Eq. (2):

$$\frac{\delta\_{01}}{\delta\_{02}} = \frac{18.75 \frac{D}{\lceil \sigma \rceil}}{13.125 \frac{D}{\lceil \sigma \rceil}} = 1.429 \tag{3}$$

From Eq. (3), it can be seen that δ01 calculated in accordance with the design of JB/T 10205-2010 is 1.429 times larger than δ02 calculated in accordance with the design of JB/T 3818-2014. It could be concluded that the hydraulic cylinder designed according to JB/T10205-2010 has the problem of "thick and heavy".

# **3 Discuss the Term "Nominal Pressure"**

(1) Differences and similarities in the definition of the term "nominal pressure" in various standards

In the current standards, the term "nominal pressure" and its definition can be found mainly in:


Although there are differences between the above two definitions, such as the definition of GB/T 17446-2012 does not specify the "pressure series" or "nominal pressure series", but there are some common points:


It should be evident from the definitions of the terms that the "nominal pressure" is solely for labeling and/or indication purposes, and hydraulic cylinders cannot be designed or operated based on this value.

(2) Evolution of the "nominal pressure" in the hydraulic cylinder standard

JB/T 10205-2010 *Hydraulic Cylinder* was released on 02-11-2010, implemented since 07-01-2010, and replaced JB/T 10205-2000 *Hydraulic fluid power cylinders-Technical specifications*.

In JB/T 10205-2010, it states, "3 Terms and definitions: The terms and definitions established in GB/T 17446 and the following terms and definitions apply to this standard." "5.3 Basic parameters: The basic parameters of hydraulic cylinders shall include cylinder bore, piston rod diameter, nominal pressure, stroke, and mounting dimensions." "6.2.7 Pressure resistance: The cylinder of the hydraulic cylinder shall be able to withstand a pressure of 1.5 times the nominal pressure without external leakage and damage to parts, etc."

After checking, in JB/T 10205-2000 *Hydraulic fluid power cylinders-Technical specifications* (has been replaced) there are no provisions on hydraulic cylinder parameters or basic parameters of hydraulic cylinders. Therefore, it can be determined that the "nominal pressure" as the basic parameter of the hydraulic cylinder should be from JB/T 10205-2010 *Hydraulic fluid power cylinders-Technical specifications*.

In JB/T 10205-2000 defined "3.1 nominal pressure: The nominal value of the working pressure of the hydraulic cylinder. That is, under the specified conditions of continuous operation, the design life of the working pressure can be guaranteed." And stipulates: "4.1.1 Nominal pressure series shall comply with the provisions of GB/T 2346." "4.2.6 Pressure resistance: The block of a hydraulic cylinder shall be able to withstand a pressure of 1.5 times its maximum working pressure without external leakage or damage to parts, etc."

After the above comparison, the definition of "nominal pressure" in JB/T 10205- 2000 is obviously different from that in GB/T 2346-2003 and GB/T 17446-2012. However, it is basically the same as the definition of "nominal pressure" in GB/ T17446-1998 *Fluid power systems and components–Vocabulary* (which has been replaced).

Therefore, JB/T 10205-2010 changed the definition of terms "nominal pressure" as the basic parameters of the hydraulic cylinder. This is the cause of the design and manufacture of hydraulic cylinders in accordance with JB/T 10205-2010 prevailing "thick and bulky" the root cause of the problem.

It should be noted that the definition of the term "nominal pressure" in JB/T 10205-2000 is similar to the definition of the term "rated pressure" in GB/T 17446- 2012. However, it is unclear why JB/T 10205-2000 selected "nominal pressure" and "maximum working pressure" instead of "rated pressure" at that time. While, in GB/ T 17446-2012, the "maximum working pressure" is categorized under "Pressure terminology related to fluid drive systems".

In GB/T17446-2012, the term "3.2.464 nominal pressure" is defined as "see ISO2944 and Fig. 1. "However, Fig. 1 does not include "nominal pressure," and the relation between "nominal pressure" and other stress terms cannot be established.

In ISO/DIS 5598:2020(E) [10], the definition of "3.2.478 nominal pressure" includes a reference to "see also ISO 2944 and Figure 20". However, "nominal pressure" is not depicted in "Figure 20", making it difficult to establish the connection between "nominal pressure" and other pressure terms.

It remains unclear whether this is due to an omission or other reasons. Moreover, according to the definition of the term "nominal pressure" in GB/T 17446-2012, it should also be the term for pressure associated with components and piping, but it is not indicated under the definition of the term "3.2.464 Nominal pressure" or in Fig. 2.

**Fig. 1** S2 hydraulic cylinder shape diagram

**Fig. 2** Hydraulic barrel weight comparison

Taking the pressure terms displayed in Figs. 1 and 2 of GB/T 17446-2012 as an example, Table 1 specifies the pressure terms and definitions necessary for the design, manufacturing (testing), sale, and utilization of hydraulic cylinders.

Refer to the terms and definitions related to pressure stipulated in GB/T 150.1- 2011 *Pressure vessels—Part 1: General requirements* [11], as shown in Table 2.

An attempt can be made to give some of the pressure terms closely related to the design and manufacture of hydraulic cylinders have the following relationships:


Based on the above findings, the S2 hydraulic cylinders with sliding bearings on the rear ears in standard JB/T6134-2006 *Hydraulic cylinders for metallurgical equipment* [12] are designed, and the shape diagram is given in Fig. 1. According to the nominal pressure of 25 MPa and the area ratio ϕ1.4 given by the standard, the weight of the cylinder barrel at the limit stroke section is calculated, and the data can be obtained as shown in Table 3.


**Table 1** Pressure-related terms and definitions in GB/T 17446-2012

a There is no "burst pressure" and description in Fig. 2 b The definitions of terms are broader than those set forth in Figs. 1 or 2 c There is no "rated pressure" and description in Fig. 2 d Also in Fig. 2 and its description "8—minimum rated pressure (pneumatic)"


**Table 2** Pressure-related terms and definitions in GB/T 151.1-2011

**Table 3** Cylinder barrel weight at nominal pressure of 25 MPa


Without changing other conditions, the rated pressure of 21 MPa as a design parameter, based on the above formula to calculate the limit stroke of the hydraulic cylinder barrel weight as shown in Table 4.

Comparison of the data in Tables 3 and 4 leads to Fig. 2, in which the barrel weight and cylinder wall thickness are drastically lowered while the other conditions remain the same. It shows that the use of rated pressure design and manufacture may


**Table 4** Cylinder barrel weight at rated pressure of 21 MPa

significantly reduce the weight of hydraulic cylinders, to help solve the problem of hydraulic cylinders "thick and heavy".

With the "rated pressure" as the basic parameter of the hydraulic cylinder, the relationship between the "rated pressure" and the "proof pressure "is determined according to the standards observed. Meanwhile, according to the relationship given in item (b) above to calculate the specified "nominal pressure", which may become a new hydraulic cylinder design, test specifications.

### **4 Conclusion**

Through the above analysis and demonstration, the following conclusions could be drawn:


The current "thick and bulky" status quo of hydraulic cylinders, although it is advantageous for manufacturers to sell hydraulic cylinders by weighing and pricing. From a long-term perspective, it will be unfavorable for hydraulic industry and even the national level and the effective use of limited resources.

At the standard level to establish the "rated pressure" as the basic parameter of the hydraulic cylinder, although there are still different opinions, but in order to realize the lightweight design and manufacture of hydraulic cylinders is also imperative.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Comparative Analysis of Mathematical Models of Hydro-pneumatic Suspension Damping**

**Zeguang Li, Laiping Li, and Wei Huang** 

**Abstract** At present, for the study of the dynamic characteristics of the hydropneumatic suspension of vehicles, the elastic force is mainly modeled by the variable gas equation of state, and the damping force is modeled by thin-walled orifice theory, which only considers the turbulent flow. Here, based on expressing the whole flow field including laminar flow, transition flow, and turbulence with piecewise function, the turbulence region is modeled by the Brasius formula and thin-walled orifice theory respectively. By applying vibration signals collected from real roads, the responses of two piecewise function damping force models and traditional thinwalled orifice model of 1/4 suspension system in the time domain and frequency domain respectively are calculated. The average absolute error MAE and root mean square error RSME are used to compare them with the real upper fulcrum data of the suspension cylinder. The results show that different models can simulate suspension vibration well in the low-frequency range, but there are obvious deficiencies in the middle and high-frequency range, while the short-hole flow theoretical model in the form of a piecewise function is closer to the real value in the frequency domain.

**Keywords** Hydro-pneumatic suspension cylinder · Damping force · Short-hole flow · Transition zone · Piecewise function

# **1 Introduction**

Suspension design is crucial for ensuring the safety of personnel and extending the lifespan of vehicles, making it a key aspect of vehicle design. Accurately analyzing the dynamic characteristics of the suspension using precise mathematical models is highly significant. Currently, hydro-pneumatic suspension cylinders with exceptional elasticity and damping are utilized for mine vehicle suspensions, specifically designed to tackle complex road conditions. A lot of basic and systematic

Z. Li (B) · L. Li · W. Huang

Inner Mongolia North Hauler Joint Stock Co., Ltd., Inner Mongolia 014000, China e-mail: lizg@chinanhl.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_66

researches have been done on hydro-pneumatic suspension, and the formation mechanism and modeling method of nonlinear stiffness and damping characteristics have been analyzed in detail [1, 2]. The elastic force is modeled by the variable state equation of gas [3, 4], while the damping force is modeled by thin-walled orifice theory, whether it is 1/4 vehicle suspension simulation [5, 6], complete vehicle suspension vibration reduction optimization [7, 8], multi-axle vehicle suspension modeling [9], or active suspension adjustment research [10, 11]. However, it has been observed that the performance of these models is less than ideal in both the frequency domain and time domain [7, 12]. Recently, the use of CFD technology has emerged to obtain nonparametric results and fit them to modify damping characteristics [13, 14]. Although this method provides results, it is not conducive to the subsequent optimization of suspension cylinder design. Therefore, it is necessary to further investigate the dynamic model of the suspension cylinder. In the past, only the flow in the orifice was usually considered to be in a turbulent state. Still, the suspension cylinder was always in a frequent up-and-down bumping state during the driving process of the vehicle. The flow field in the orifice constantly developed from laminar flow to transition and then entered the turbulent zone. The changing direction is reciprocated, so the actual situation should include laminar flow and transition zone. Especially according to Nikolaz's graph or Moody diagram [15], the resistance along the transition zone has a violent change process, including chaos in this area, which was narrow and ignored in the past. However, considering that the sharp change of flow field in this area will affect the dynamic characteristics in the frequency domain, it is necessary to consider the whole change process of flow in the damping orifice and one-way valve from static to complete development of turbulence. In addition, because the ratio of the length and diameter of the damping hole of the suspension cylinder is not thin-walled, and it is actually in the range of short-hole flow theory, a more realistic model should be constructed according to the short-hole flow theory. This paper aims to reconstruct the damping force using the piecewise function expression method of the short-hole flow theory. Additionally, two theoretical models are adopted to represent the turbulent region: the Blasius formula, recommended by the short-hole flow theory, which offers high accuracy and broad applicability, and the traditional thin-walled orifice theoretical model in the time and frequency domains are compared, and the accuracy of several theories, especially their performance in the frequency domain, is analyzed and discussed.

### **2 Damping Force Mathematical Model**

The structural diagram of the conventional suspension cylinder is shown in Fig. 1.

The fluid resistance of the damping hole and one-way valve is generated by the pressure difference between chamber I and chamber II, and the nonlinear damping Fc is:

$$F\_C = \mathbf{A} \Delta P \tag{1}$$

**Fig. 1** Structural diagram of oil pneumatic spring

In the formula: Δ*P* is the pressure difference between cavity I and cavity II; A is the radial area of the buffer chamber II. The main differences between the two mathematical models of nonlinear damping forces are as follows.

### *2.1 Thin-Walled Orifice Theory*

The relationship between the pressure difference between chamber I and chamber II and the flow rate Q through the damping hole and one-way valve is:

$$\mathbf{Q} = \mathbf{C}\_{\mathbf{d}} A\_{1} \sqrt{\frac{2\Delta P}{\rho}} \tag{2}$$

In the formula, Cd is the flow coefficient; ρ is the density of the oil; A1 is the total flow area through the damping hole and one-way valve.

Bringing Formula (2) into (1), while considering that the flow rate Q is the flow rate from cavity I to cavity II, it can be obtained by multiplying the velocity difference between the upper and lower support points of the suspension cylinder by A. Assuming that the flow rate in the damping hole and the one-way valve is the same, we obtain the relationship between the final suspension cylinder damping force and the velocity difference between the upper and lower support points:

$$\text{F}\_C = \frac{\rho A^3 \upsilon^2 \text{sign}\upsilon}{2 \{ \text{nC}\_d \left[ A\_d + A\_\upsilon (0.5 + 0.5 \text{sign}\upsilon) \right] \}^2} \tag{3}$$

In the formula: n is the number of damping holes and one-way valves; Ad is the cross-sectional area of the damping hole; Av is the effective flow area of the one-way valve; υ is the speed of the cylinder barrel relative to the piston; Sign υ is the sign function of velocity, which is compressed as positive and restored as negative based on the direction of velocity.

### *2.2 Short-Hole Flow Theory*

According to formulas 8.32 in [15], the head losses of cavity I and cavity II are composed of three parts: inlet head loss, short-hole overflow head loss, and outlet head loss. The relationship between pressure difference and flow velocity through damping holes and one-way valves is as follows:

$$
\Delta P = (\xi\_{inlet} + \xi\_{outlet} + \xi\_{channel}) \frac{\rho \upsilon\_H^2}{2} \tag{4}
$$

In the formula, ζinlet, ζoutlet and ζchannel are the local head loss coefficients along the inlet, outlet, and short hole respectively; υ*H* is the flow rate inside the hole. The inlet resistance coefficient ζ*inlet* can be based on the data obtained from the Weissbach experiment, and considering the shrinkage situation, the value is 0.5 when a container with a large cross-section flows into the pipeline. The outlet resistance coefficient ζ*outlet* is based on the Borda formula and is considered when the fluid flows into a container with a large cross-section under submerged conditions. Therefore, the local head loss is taken as 1. The local head loss coefficient along the short hole ζchannel is calculated in sections based on the changes in the flow field inside the small hole.

### *2.3 Damping Force in Laminar Flow Zone*

The laminar flow zone is the initial stage of the flow field, with Re ranging from 0 to 2300. The local head loss coefficient in this range is inversely proportional to Re:

$$
\zeta\_{\text{channel}} = \frac{64}{Re} \frac{L}{d} \tag{5}
$$

Here *L* is the wall thickness of the small hole, and d is the diameter of the small hole. *R*e is the Reynolds number, indicating the turbulent state of flow. When this value is less than 2300, it is laminar flow, which is related to velocity, flow field size, and viscosity:

$$\text{Re} = \frac{\upsilon\_H d}{\nu} \tag{6}$$

In the formula: ν is the kinematic viscosity of the fluid oil in the suspension cylinder; *dH* is the diameter of the small hole.

Formula (6) is brought into (4) to get the relationship between the damping force and the flow velocity in the small hole:

$$F\_C = \left(1.5 + \frac{64}{Re} \frac{\text{L}}{\text{d}}\right) \frac{\rho \,\upsilon\_{\text{H}}^2}{2} A \tag{7}$$

Similar to the form of the Formula (3), a formula for constructing a compression state that can be applied to both the open and closed recovery states of a one-way valve is constructed. The relationship between the flow in the small hole υH and the velocity difference υ between the upper and lower support points of the suspension cylinder during the recovery and compression processes is:

$$\upsilon\_{\rm H}(\upsilon) = \frac{A\upsilon}{\rm nC\_d \left[A\_d + A\_\upsilon (0.5 + 0.5 \text{sign}\,\upsilon) \right]} \tag{8}$$

The relationship between the damping force and the speed difference between the upper and lower support points of the suspension cylinder is obtained by bringing formula (8) into (6) and then into (7) as follows:

$$F\_C = \left(1.5 + \frac{64v\mathcal{L}}{\upsilon\_H(\upsilon)\mathcal{d}^2}\right)\frac{\upsilon\_H^2(\upsilon)}{2}\rho A\tag{9}$$

### *2.4 Damping Force in the Transition Zone*

It is generally believed that the transition zone, due to its narrow range and limited practical significance, is not discussed. However, the transition zone produces a significant change in frictional coefficient along the way under a small Reynolds number change, which inevitably leads to an increase in fluid noise. Therefore, it is necessary to analyze its impact. The range of Re in the transition zone is between 2300 and 3000, and it can be seen from the Nicholas curve that there is a clear chaotic state. Therefore, there has been little research in the past. According to the empirical formula provided by [15], the relationship between the local head loss coefficient ζchannel and Re is as follows:

$$
\zeta\_{\text{channel}} = 0.0025 R\_{\text{c}}^{1/3} \tag{10}
$$

The relationship between damping force and flow velocity in the small hole:

$$F\_C = \left(1.5 + 0.0025 R\_\epsilon^{1/3} \frac{\mathbf{L}}{\mathbf{d}}\right) \frac{\rho \upsilon\_\mathcal{H}^2}{2} A \tag{11}$$

Similarly, Formulas (8) and (6) are introduced into (11) to obtain the relationship between the damping force and the velocity difference υ between the upper and lower support points of the suspension cylinder. A formula is constructed for the compression state with both the open and closed one-way valves, and the relationship between the damping force and the velocity υ between the upper and lower support points of the suspension cylinder becomes:

$$F\_C = \left(1.5 + 0.0025 \left(\frac{\upsilon\_H(\upsilon)\mathbf{d}}{\upsilon}\right)^{\frac{1}{3}} \frac{\mathbf{L}}{\mathbf{d}}\right) \frac{\upsilon\_H^2(\upsilon)\rho}{2} A \tag{12}$$

### *2.5 Damping Force in the Turbulent Zone*

The damping force in the turbulent zone adopts two models, namely the damping force formula for thin-walled orifice (3) and the Blasius model in the short-hole flow theory. Brasius is a widely applied and mature empirical formula provided by the short-hole flow theory for turbulent regions. The calculation of the resistance loss coefficient ζchannel along the path in the range of 3000 < Re < 10e5 using this formula has high accuracy. The relationship between the local head loss coefficient ζchannel and Re is as follows:

$$
\zeta\_{channel} = \frac{0.3164}{\mathrm{Re}^{0.25}} \frac{L}{d} \tag{13}
$$

Similarly, Formulas (8) and (6) are taken into (13) and then taken into (4) to obtain the relationship between the damping force with both compression and recovery states and the velocity difference υ between the upper and lower support points of the suspension cylinder:

$$F\_C = \left(1.5 + 0.3164 \left(\frac{\upsilon}{\upsilon\_H(\upsilon)\mathrm{d}}\right)^{\frac{1}{4}} \frac{\mathrm{L}}{\mathrm{d}}\right) \frac{\upsilon\_H^{\frac{2}{2}}(\upsilon)\rho}{2} \mathrm{A} \tag{14}$$

### *2.6 Summary of Piecewise Function Damping Force Models*

The damping force of the suspended cylinder is expressed using a piecewise function based on the short-hole flow theory, which is laminar flow in the early development stage of Re ≤ 2300, the transition of 2300 < Re < 3000, and turbulence of 3000 ≤ Re. According to Sect. 2.3 and 2.4, the laminar damping force and transition zone damping force are calculated using formula (3) and Brasius' formula for the turbulent zone:

```
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
  − − − − − − − − − − − − − − −−
  flow progress expressed by piecewise: 
  Re < 2300 laminar flow zone: 
  formula (9)
  2300 < Re < 3000 transient zone: 
  formula (12)
   Re > 3000 turbulence zone with 2 model: 
  formula (14)
  formula (3)
  − − − − − − − − − − − − − − −−
  the entire flow progress: 
  thin-walled model (3)
                                                                  (15)
```
So, two piecewise function models are established here, with different formulas used in the turbulent zone. Finally, the calculation results are compared with those of the traditional thin-walled orifice theory model (3) used throughout the entire flow field, and the influence of different damping force models on the results is observed.

#### **2.6.1 Dynamics Model of Suspension Cylinder**

The lower fulcrum of the suspension cylinder is excited by external input shock or road displacement, and the vibration reduction of the upper fulcrum is achieved through the elasticity of the suspension cylinder's air chamber and fluid damping. The dynamic model of the upper fulcrum based on Newton's third law of motion is:

$$\mathbf{m}\dot{\boldsymbol{\upsilon}} = F\_{\mathcal{C}} + F\_{\text{gas}} \tag{16}$$

In the formula, m represents the mass on the spring, υ˙ represents the acceleration of the support point on the suspension cylinder, *F*c represents the damping force in Formula (15), and *Fgas* represents the elastic force derived from the gas variable state equation. The weight on the spring is reflected by the gas compression amount in the elastic equation in the static equilibrium state.

# **3 Simulation Comparison of Three Kinds of Damping Force Mathematical Models**

The common suspension simulation generally uses pulse input to verify the impact characteristics or a random road surface model constructed by the power spectral density function for suspension dynamics simulation in the absence of road test data [16]. A more accurate method is to collect the suspension vibration and driver's body acceleration when the vehicle is driving on real mining roads, using the standards for evaluating vehicle smoothness specified in ISO2631 [14], grading to evaluate the vehicle's suspension damping capability. This method has been widely promoted and applied in the automotive industry.

Here, the lower fulcrum of the suspension cylinder under real road conditions is used as the model input, and the sampling results of the upper fulcrum are compared with the simulation calculation results to verify the accuracy and difference between the two damping force models. The data collection of real road conditions adopts the method of installing an acceleration sensor at the top of the front axle, which is rigidly connected to the front suspension cylinder barrel, to obtain the acceleration of the lower fulcrum of the oil and gas suspension cylinder. An acceleration sensor is installed at the top of the piston rod to obtain the acceleration of the upper fulcrum of the oil and gas suspension cylinder, with a sampling period of 5e-4 s. As the input of Formula (15), the motion of the lower fulcrum can obtain the acceleration data of the upper fulcrum, which can be compared with the actual sampling results. Figure 2 shows the comparison between the calculation results of three models in the time domain and the real fulcrum values. The horizontal axis represents the time, and a typical time-domain comparison chart from 1.55 to 1.95 s is taken to display the simulation results.

Firstly, from the overall trend, it can be seen that all three models can follow the excitation signal to make a response that is close to the real one. However, there is a lag delay phenomenon in the wave peaks of all three models, and the thin-walled orifice theory model is more evident at individual moments, such as at 1.7 s. In addition, it can be seen that the two piecewise functions models have better tracking performance, and most of the time remain within the fluctuation range of the true value. At 1.75–1.8 s, it can also better simulate small fluctuations in real vibration. The two piecewise functions do not exhibit particularly significant differences in the time domain. However, all three models exist: they cannot be simulated on finer vibrations, i.e. losing the high-frequency part; Calculate the situation where the vibration amplitude is lower than the true value, as shown in the peaks and valleys in Fig. 2.

Figure 3 shows the comparison between the three theoretical models in the frequency domain and the actual sampling results. The frequency domain is obtained by adding a recovery coefficient of 1.633 and a power recovery coefficient of 2 through Fourier transform after the hamming window. It can be seen from the figure that both models can reflect the low-frequency vibration, but the attenuation is fast. After about 15 Hz, the calculated results show a significant attenuation of amplitude,

**Fig. 2** Comparison of calculation results and real values between two models in the time domain

which cannot simulate the real situation. After 20 Hz, the real frequency cannot be simulated.

Figure 4 is an enlargement of the low-frequency region of Fig. 3, showing that the calculated values of the three models at the maximum vibration value of the suspended cylinder are all greater than the real upper fulcrum values. The results of the thin-walled orifice theory model are more detached from the actual situation, and the two piecewise function models are relatively close to the actual values. The attenuation degree of the two piecewise function models above 6 Hz is also better than that of the thin-walled orifice theory, and they can still calculate a certain acceleration amplitude. The performance of the two piecewise functions in the frequency domain is almost identical visually, with no significant differences.

To quantitatively evaluate the accuracy of the three models, the root mean square error (RMSE) and mean absolute error (MAE) of the corresponding frequency values in the frequency domain are introduced to measure the degree to which the two models deviate from the true value. MAE (Mean Absolute Error) is the average absolute value of the error between the calculated value and the true value. The formula is:

$$MAE = \frac{1}{n} \sum\_{i=1}^{n} |\mathbf{\hat{y}\_i} - \mathbf{y}\_i|$$

The Root Mean Square Error (RMSE) is concentrated on the order of magnitude for comparison, such as RMSE = 10. It can be considered that the regression effect has an average difference of 10 compared to the true value.

**Fig. 3** Frequency domain comparison between the calculated results of three models and the true values

**Fig. 4** Comparison of three models and real fulcrum values in the low-frequency range

Comparative Analysis of Mathematical Models of Hydro-pneumatic … 849

$$RMSE = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left(\mathbf{\hat{y}}\_i - \mathbf{y}\_i\right)^2}$$

When the calculated value completely matches the actual value, it is equal to 0, which is a perfect model; the larger the error, the greater the value. Two indicators are used to describe the error between calculated values and true values. The difference is that if the dispersion of the error is high and the maximum deviation value is large, RMSE will amplify compared to MAE, which can better reflect the degree of deviation. In Table 1, the input parameters and output performance of the three models are listed as "×" indicating that the parameter is not required.

In Table 1, the performance of three models in the time domain MAE and RMSE under a duration of 20 s was compared, while the performance in the frequency domain was compared within the range of 0–100 Hz. From Table 1, it can be seen that the damping force constructed using piecewise functions has better accuracy than the thin-walled orifice theory model. The difference between the two piecewise functions is not significant. From the perspective of formula complexity, Formula


**Table 1** Comparison of the advantages and disadvantages of three damping force models

(3) can be selected as the piecewise function form of the turbulent zone model to construct the damping force.

### **4 Conclusion**

This article modifies the dynamic characteristics of the hydraulic and pneumatic suspension of mining vehicles, especially the damping force model, according to the actual structural characteristics of the suspension cylinder. The short-hole flow theory model is used to consider the entire process of the flow field development in the small hole, which includes the initial laminar flow, the transient transition in the middle, and the turbulence in the later stage to reconstruct the piecewise damping force model. By comparing the system response under real road excitation with the traditional thin-walled orifice theoretical model, the average absolute error MEA root mean square error RMSE was compared in both time and frequency domains. Compared with thin-walled orifices, the piecewise short-hole flow model improved the accuracy in the time domain by 24–25%, frequency domain accuracy MAE by about 15.8%, and RMSE by 37.4%. It is verified that the damping force of the piecewise short-hole flow model is closer to the real situation in the frequency and time domains. From the perspective of simplicity and ease of use, using the piecewise short-hole flow model using Formula (3) in the turbulent zone is recommended. Its time-domain and frequency-domain MEA and RMSE are close to the Formula (14) but with low complexity.

Through the results in the time domain and frequency domain, it can be found that the performance of the three damping force models in the middle and high-frequency parts, except the low frequency has an obvious suppression phenomenon, which may be related to three chaotic phenomena: (1) Chaotic phenomenon in the transition zone shown by Nikolaz graph or Moody graph; (2) Chaotic motion of steel ball in flow field in one-way valve [14]; (3) The oil sloshing in the oil–gas suspension cylinder causes the internal flow field to be in a chaotic state. The above three situations may be the main reasons that lead to the poor simulation of the dynamic characteristics of suspension cylinders by the existing theoretical models, and further research is needed to find a more accurate mathematical model.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Experimental Study on the Cohesive Model of Steel-Carbon Fiber Reinforced Plastic Interface by Laser Treatment**

**Hao Teng, Hailang Wan, and Junying Min** 

**Abstract** The interfacial bonding performance between steel and CFRP significantly influences the mechanical properties of steel-CFRP hybrid structures. Surface treatment is commonly employed to enhance the bonding interface of steel-CFRP. Laser surface treatment is particularly advantageous due to its high efficiency, automatic production capabilities, and widespread use in enhancing interfacial bonding performance. However, little attention has been given to the impact of laser surface treatment on the property parameters that describe the cohesive mode's mechanical behavior at the steel-CFRP interface. This study examined the cohesive zone modes of both original and laser-treated steel-CFRP joints through a double lap shear test following ASTM D3528-96 (2016) standards. Non-contact strain measurement was conducted using 3D digital image correlation techniques. The analysis indicates that the bilinear cohesive model effectively describes the mechanical behavior of the steel-CFRP interface. Laser surface treatment resulted in a respective increase of 83.8% in maximum shear strength, 111.6% in the relative slip corresponding to maximum shear strength and 116.8% in maximum relative slip. Consequently, this study showcases the efficacy of laser surface treatment in improving the mechanical performance at the steel-CFRP interface while quantitatively assessing these improvements through performance parameters within the cohesive zone model.

**Keywords** Laser surface treatment · Steel-CFRP interface · Bond behavior · Cohesive model

# **1 Introduction**

In pursuit of energy-saving and environmentally friendly development strategies, the adoption of lightweight materials has become crucial for automotive manufacturers to achieve these goals. Carbon fiber reinforced polymers (CFRP) exhibit superior

H. Teng · H. Wan · J. Min (B)

© The Author(s) 2024

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: junying.min@tongji.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_67

mechanical and physical properties compared to steel, such as high specific strength, high specific stiffness, fatigue resistance, and low density, indicating their vast potential as lightweight materials. However, they also present certain drawbacks, including poor toughness and high costs [1]. The localized application of CFRP to enhance metal components, forming steel-CFRP hybrid structures, successfully retains the original component's ductility while simultaneously improving rigidity and ultimate load capacity. Steel-CFRP hybrid structures have been widely utilized in the automotive and aerospace industries [2–4], advancing remarkable performance benefits and prospects in applications such as aircraft wings and fuselage skin, automotive drive shafts, and vehicle bodies. To circumvent additional mechanical damage, thermal damage, and weight associated with conventional joining techniques like riveting and welding, steel and CFRP are bonded using adhesives, resulting in a more even stress distribution and flexible design. The mechanical performance of steel-CFRP interfaces has been identified as a critical factor in determining the overall mechanical properties of steel-CFRP composite components [5]. Understanding the underlying bonding mechanisms at the steel-CFRP interface is instrumental for facilitating the successful application of steel-CFRP composites in vehicle body structures.

To elucidate the impact of various factors on the mechanical performance of adhesive interfaces, it is imperative to accurately characterize the influence of stress and fracture energy during the interfacial crack propagation process. The cohesive zone model (CZM) is widely adopted by researchers due to its strong physical significance, numerical stability, and simulation compatibility [6–9]. Wang et al. [7] investigated the influence of adhesive properties and thickness on the shear performance of steel-CFRP interfaces, revealing that ductile adhesive interfaces are well-suited for trapezoidal CZM characterizations with thickness impacting shear strength, whereas brittle adhesive interfaces are more amenable to bilinear CZM representations, but adhesive layer thickness has less pronounced effects on shear strength. Pang et al. [8] found that the shear strength of steel-CFRP interfaces bonded with Sikadur-30 (CN) adhesive increases with an increasing loading rate. Zhu [9] demonstrated that extreme temperatures can alter the mechanical properties of adhesive layers, subsequently reducing the mechanical performance of steel-CFRP interfaces and changing the interfacial failure modes.

To further enhance the mechanical performance of bonded interfaces, surface modification of the adherend is one of the most common approaches, including techniques such as grinding [10, 11], sandblasting [11, 12], plasma treatment [13–15], mechanical scoring [16], and laser processing. Russian et al. [11] demonstrated that increasing the surface roughness of steel through sandblasting and grinding treatments led to enhanced interfacial strength, with the strength post-sandblasting being 1.62 times. Manuel et al. [12] revealed that shot peening, compared to sandblasting, can further improve the shear strength. Studies by Williams et al. [14] and Lin et al. [15] showed that plasma-treated surfaces not only exhibit altered surface morphology but also host reactive chemical functional groups, resulting in an improved interfacial bonding performance. Guo et al. [16] investigated the impact of cutting mechanical grooves on the steel surface on the shear behavior of steel-CFRP interfaces, finding that resin flowed into the grooves and induced mechanical interlocking during shearing, consequently enhancing the shear strength of the interface.

Laser processing, characterized by its excellent controllability and capability to generate specific surface morphologies, has been widely investigated for creating dimples, linear grooves, and crossed grooves on metal surfaces. In studies focusing on dimpled microstructures, Zhang Yulong [17], Zou et al. [18], and Li et al. [19] conducted laser treatments with varying power and spacing on metal surfaces, revealing that surface roughness, surface energy after treatment were closely related to the bonding performance. Feng et al. [20] examined the influence of three structural morphologies on steel adhesive bonding performance, and found that although dimple morphology did not significantly enhance interfacial adhesive strength, both groove and grid morphologies notably increased the strength, with failure modes transitioning to cohesive failure.

Laser processing is a rapid and controllable surface treatment method. Current research indicates that laser processing can significantly enhance the interfacial strength of steel-CFRP interface. However, existing mechanical characterizations are limited to shear strength measurements, which cannot adequately describe the failure process. This paper investigates the influence of groove microstructures, created through laser treatment, on the interfacial shear strength, stress distribution, and failure modes of steel-CFRP double-lap shear specimens. Furthermore, we elucidate the mechanisms underlying the improved shear strength at the steel-CFRP interface using a cohesive zone model.

# **2 Double Lap Shear Experiment of Steel-CFRP Based on DIC**

### *2.1 Material Properties*

The steel plates utilized in this research were 5.0 mm thick Q235, with a plate width of 5.0 mm and an elastic modulus of 205GPa. The cured CFRP plate used in this study, with a width of 25 mm and a thickness of 0.36 mm, was supplied by Composite Materials Easy Purchase (Beijing) Technology Co., Ltd. According to ASTM D3039, the tensile strength and elastic modulus were determined to be 2934 MPa and 147 GPa.

### *2.2 Laser Treatment of Steel Surface*

To investigate the impact of laser treatment on bond behavior of the steel-CFRP interface, an IR laser (Wuhan Raycus Fiber Laser Technologies Co., Ltd.) with a wavelength of 1064 nm and spot diameter of 50 μm was used on steel plate. Linear

**Fig. 1 a** Schematic diagram of linear groove microstructure of the steel surface; **b** schematic illustration showing the laser spot size and overlap

groove microstructures were prepared on the steel surface through a filling method, as illustrated in Fig. 1a. The steel surface can be divided into two distinct areas: the lasertreated area and the untreated area. The laser-treated area has a width of W, while the untreated area has a width of S. The process parameters were controlled by adjusting the galvanometer, which was based on the computer numerical control (CNC) system. In the laser processing area, the laser scanning strategy was configured with a power of 24 W, a scanning frequency of 40 kHz, a scanning speed of 500 mm/s, and a scanning interval of 0.05 mm within the laser filling area, resulting in the achieved effect shown in Fig. 1b. The direction of laser scanning was perpendicular to the length direction of the sample. In order to quantify the enhancement of interfacial performance on the surface of steel after laser treatment, a control group was established in this study, which group solely employed the use of alcohol spray to remove surface impurities from the steel.

# *2.3 The Preparation of Steel-CFRP Double Lap Shear Specimens*

In this study, steel-CFRP double lap shear specimens were used to investigate the interfacial shear performance, as illustrated in Fig. 2a. The specimens were loaded at a rate of 1 mm/min using an MTS universal testing machine, and surface axial strain measurements were conducted using a three-dimensional digital image correlation (3D-DIC) system [21]. The testing area is located on the right side of the specimen, with a CFRP width of 25 mm and a length of 200 mm. The left end of the scatter region was taken as the origin and positive to the right. The first strain extraction region is positioned at 10 mm along the central axis, and the remaining strain extraction

**Fig. 2 a** Construction of the steel-CFRP double lap shear specimen; **b** vacuum bag pressure process for the steel-CFRP double lap shear specimens

region are spaced at 20 mm intervals along the central axis. The steel-CFRP doubleshear specimens were fabricated using the vacuum bagging process, as shown in Fig. 2b. The first step in the manufacturing process is to remove impurities from the surface of the steel plate using alcohol, which is then polished with sandpaper and treated with a laser. Subsequently, the prepared steel plate is coated with CFRP prepreg. The steel-CFRP double lap shear specimen is then placed into a vacuum bag and subjected to vacuum extraction. Finally, the vacuum bag is positioned within a heating furnace at a temperature of 120 °C for a curing duration of 2 h. Due to the resin in the prepreg, the CFRP is bonded to the steel plate during the curing process.

### *2.4 Observation of Steel Surface Morphology*

Surface topography observation of the laser-treated steel was conducted under a digital microscope (VHX-1000) to obtain dimensional parameters of linear groove microstructures and to observe the characteristics of the interface after failure.

### *2.5 Cohesion Model Calculation*

The interfacial shear stress and relative slip for the steel-CFRP double lap shear joint are determined by the integral and difference of strain values from the strain extraction regions, whose Computational methods have been used in various literature [7, 10]. Assuming that the CFRP section's stiffness is significantly lower than that of the steel section, and that the strain varies linearly between two adjacent strain extraction region. The interfacial shear stress and relative slip of steel-CFRP double lap shear joint at the middle point between two adjacent strain extraction region can be expressed as

$$
\pi\_{i+1/2} = E\_f t\_f \frac{\varepsilon\_{i+1} - \varepsilon\_i}{\chi\_{i+1} - \chi\_i} \tag{1}
$$

$$\delta\_{i+1/2} = \frac{\delta\_{i+1} + \delta\_i}{2} = \frac{\varepsilon\_i + \varepsilon\_{i+1}}{4}(\mathbf{x}\_{i+1} - \mathbf{x}\_i) + \sum\_{i=1}^{n} \frac{\varepsilon\_i + \varepsilon\_{i+1}}{2}(\mathbf{x}\_{i+1} - \mathbf{x}\_i) \tag{2}$$

where τ*<sup>i</sup>*−1/2 and δ*<sup>i</sup>*+1/2 are the interfacial shear stress and relative slip at the middle point between the *i*th strain extraction region and the (*i* +1)th strain extraction region; *E f* and *t f* are the elastic modulus and the thickness of CFRP; ε*i* and ε*<sup>i</sup>*+1 are the *i*th and (*i* + 1)th strain extraction region, respectively; *xi* and *xi*+1 are the distances of the *i*th and (*i* + 1)th strain extraction region from free end, respectively; and n is the total number of strain extraction regions.

### **3 Experimental Results and Analysis**

# *3.1 Surface Morphology of Steel Before and After Laser Treatment*

Figure 3 presents the surface morphology of the steel prior to and following laser treatment. As show as Fig. 3a, the steel surface exhibits primarily smooth characteristics, albeit with size variations attributed to manufacturing process complications. After laser treatment, the current microstructure of the steel surface is generated with a depth of D 25.2 μm, a width of W 110.6 μm, and a spacing of S 489.4 μm. Meanwhile, the metal undergoes a phase of melting due to the high pulse, followed by propulsion and reshaping caused by the impact of the laser pulses. Consequently, an accumulation with an average height of 6.6 μm forms on one side of the groove, while a stack of 18.6 μm is generated on the other side. Simultaneously, the metal spatter generated by the laser pulses envelops the entirety of the untreated area, as illustrated in Fig. 3b, c. The preexisting pits in the untreated region have been filled by the spatter, resulting in a color transformation from the metal's original silver hue to a yellow oxide.

### *3.2 Load–displacement Curve and Failure Mode*

In Fig. 4, the load–displacement curves of the original and laser-treated steel-CFRP double-lap shear specimens are presented, whose behavior can be divided into three stages: an initial stage with linear load increase, an elastic–plastic stage with nonlinear

**Fig. 3 a** The original surface of steel; **b** laser treated steel surface; **c** 3D morphology of steel surface after laser

load increase indicating the beginning of interface failure, and a platform stage suggesting equilibrium sustained by interlocking, friction, or interfacial bonding. During the initial stage, there was no significant difference in the initial stiffness of both specimens. However, after laser treatment, the load and displacement at the onset of the steel-CFRP interface failure increased by 133.1 and 229.4%, respectively. In the plateau stage, the load and displacement of the laser-treated specimens increased by 81.6 and 81.7% respectively. This analysis indicates that laser treatment does not alter the stiffness of the steel-CFRP interface, but it can enhance the maximum stress and relative slip at the interface. As shown in Fig. 4b, c, the failure mode of the original specimens is interface failure, while the laser-treated specimens exhibit debonding crack parallel to the longitudinal direction of the CFRP throughout the entire region, indicating CFRP delamination failure. The incorporation of Fig. 4b demonstrates that the presence of linear grooves and metal splatters causes a shift in the failure mode from interface failure to cohesive failure, owing to mechanical contraction and frictional influences.

**Fig. 4 a** Load–displacement curve of steel-CFRP double lap shear specimen; typical failure modes of steel-CFRP double lap shear specimens: **b** original specimens; **c** laser-treated specimens

**Fig. 5** CFRP axial strain distribution of **a** original specimen; **b** laser treated specimen; **c** Strain history at x = 140 mm

### *3.3 The Axial Strain Distribution of CFRP*

Figure 5a, b show the axial strain distribution of CFRP for the original and lasertreated specimens during the entire test. Throughout the strain evolution process, similarities can be observed between the two: during the initial loading stages, as the applied load increases, the strain initially increases continuously on the side farthest from the free end. Once it reaches its maximum value and remains unchanged, it indicates debonding of CFRP and steel at this specific interface. At the same time, the region where the strain reaches its Upper limit continues to expand towards the free end until the entire CFRP surface, that is, the steel and CFRP are completely unbonded. Figure 5c displays the strain history of the strain extraction region located 140 mm from the free end for both the original and laser-treated samples. The strain at the interface debonding time of the laser-treated sample exhibits a significant increase from 4058 με to 7829 με, representing a 92.9% increase, while the bearing time also shows a 60.2% increase. Furthermore, the results demonstrate that the stress transferred between the CFRP and steel interfaces is significantly higher after laser treatment, leading to an increase in relative slip.

### *3.4 The Steel-CFRP Bond-Slip Relationship*

The figure presented in Fig. 6 illustrates the bond-slip relationship at the steel-CFRP interface for untreated and laser-treated specimens, as obtained using Eqs. (1) and (2). Both the original and laser-treated steel-CFRP interfaces exhibit similar distribution trends in terms of bond-slip relationship. These trends can be divided into two stages: (1) the elastic stage, where the interface shear stress gradually increases with increasing slip until reaching the maximum interface shear stress, and (2) the plastic softening stage, where the interface shear stress gradually decreases and approaches zero with increasing slip. Based on the features of the interfacial bond-slip curves in Fig. 6, the bond-slip response for the steel-CFRP interface can be simplified as follow:

Experimental Study on the Cohesive Model of Steel-Carbon Fiber … 861

$$\tau(s) = \begin{cases} \frac{\mathfrak{r}\_{\text{max}}}{s\_{\text{l}}} s & if \quad 0 \le s \le s\_{\text{l}}\\ \frac{\mathfrak{r}\_{\text{max}}}{s\_{\text{max}} - s\_{\text{l}}} (s\_{\text{max}} - s) & if \ s\_{\text{l}} < s \le s\_{\text{max}}\\ 0 & if \quad s > s\_{\text{max}} \end{cases} \tag{3}$$

where τmax is interfacial maximum shear stress, *S*1 is the relative slip corresponding to τmax, *S*max is maximum relative slip.

The specific values for τmax, *S*1, *S*max and fracture energy of the interface *G*  (the area enclosed by the two linear segments) are provided in Table 1. The research demonstrates that the enhancement of shear performance in the steel-CFRP interface due to laser treatment is attributed to several factors, including a 83.8% increase in maximum shear strength, a 111.6% increase in relative sliding at the moment of maximum stress, a 116.8% increase in relative sliding, and a 298.4% increase in fracture energy.

**Table 1** Key parameters of the bilinear bond-slip relationship


# **4 Conclusion**

This study employs the 3D-DIC-assisted strain measurement method to investigate the influence of laser treatment on the failure mode, ultimate load, and shear performance of the steel-CFRP interface. Additionally, a cohesive force model is established. The following conclusions can be drawn:


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Study on Blast Resistance of Armored Steel Welded Joint Structures**

**Song Wu, Ming Zhang, and Yunbo Zhou** 

**Abstract** In order to study the anti-detonation performance of welded joint structure of armored steel, the finite element analysis of welded joint structure under explosion impact was carried out by using material assignment method and structured ALE algorithm (S-ALE), and the accuracy of the model was verified by explosion impact test. The central deformation and failure form of each weld structure under different equivalent loads of a single explosion are studied. Repeated explosive loading of weld structure based on complete restart technology; The cumulative analysis of plastic deformation of weld structure under multiple explosions is realized, and the relationship between central deformation and explosion times and explosion equivalent is obtained.

**Keywords** Weld connection · Blast impact · Complete restart · Center deformation

# **1 Introduction**

Welding as a common joining method for mechanical structures, residual stresses and certain thermal deformations exist in welded structural members in practical engineering, and the weld and heat affected zone (HAZ) will be destroyed and fail when large incidental loads are encountered. Gaudreault is one of the few who have presented a study dedicated to welded armor steel joints subjected to explosions [1]. His welded joints were modeled by shell units with HAZ material properties and did not take into account the weld geometry, so it was not possible to truly predict stress concentrations and plate fracture. Cerik investigated the dynamic plastic response and failure mechanisms of fusion-welded aluminum quadrilateral plates and orthogonally-reinforced plates subjected to blast loading, and performed a rigorous

S. Wu · M. Zhang · Y. Zhou (B)

School of Mechanical Engineering, Nanjing University of Science and Technique, Nanjing 210094, Jiangsu, China e-mail: yunbo31983@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_68

numerical parametric study in order to propose new empirical equations for welded aluminum plates subjected to blast loading, with a detailed investigation of effects of various geometrical aspects, heat affected zone location and impact strength on the overall deformation of the plate [2]. Xu et al. presented an experimental and numerical study of a near-field air blast loaded aluminum plate in order to evaluate its deformation pattern and failure mode [3]. Few papers dealt with the material behavior of armor steel with welded seams used to protect vehicles under high strain rates or the mechanical strength of welded substructures under impact loading. Based on the studies of many researchers mentioned above, this paper verifies the accuracy of the established model by establishing a finite element model of welded seamconnected armor steel under blast loading and using blast tests. Based on this, the structural strength of welded joints under single and repeated blast loading is further investigated, so the results of this paper can be used to evaluate the strength and safety of welded joints of armored steel under blast loading.

### **2 Explosive Impact Test of Armor Steel with Welded Joints**

### *2.1 Test Setup and Results*

In order to verify the structural strength of the welded joint armor steel under the explosion load, an explosion test rig was designed as shown in Fig. 1. The weld target plate is fixed in the front end of the steel bracket by M24 bolts all around, and the top and back side of the steel bracket have cement counterweights to make it fixed in the corresponding position. Three kinds of weld structures were adopted in this test: Sample 1 was butt-welded; Sample 2 was butt-welded with an additional 2 mm thick reinforcing beam; Sample 3 was overlap welded. The deformation of the target plate can be visualized by drawing an 80 × 80 mm grid on the backside for mesh visualization. The size of each target plate is 400 mm × 400 mm × 4 mm, of which the target plate material is bulletproof steel plate 6252, and the reinforcement beam is SAPH440. In Fig. 2, The distance between the explosive center and the center of the target plate is 560 mm, and the explosive equivalent is 2 kg, adopting the cylindrical charging mode, with the explosive diameter of 110 mm, height of 135 mm, and igniting from the 1/3 of the height.

The test results are shown in Fig. 3, the three target plates formed a bulge in the back convex, none of which showed cracks. The maximum deformation of the center of the backplane of the three samples was 29.7 mm, 25.9 mm and 27.8 mm, respectively, in which no fracture failure occurred in the weld portion of sample 1 and sample 3, while the deformation of the reinforcing beam of sample 2 was more intense, and some spalling occurred in the nearby spot weld edges.

**Fig. 2** Schematic of explosives in the test

**Fig. 3** Explosive test results

### *2.2 Finite Element Models*

The corresponding finite element model was developed in LS-dyna simulation software based on the blast impact test. The structured ALE algorithm (S-ALE) is adopted, and the air flow field is simulated with structured grid, with a cell size of 20 mm, which is generated by the keywords ALE\_STRUCTURED\_MESH and ALE\_STRUCTURED\_MESH\_CONTROL\_POINTS [4]; and the keywords \*CONSTRAINED\_LAGRANGE\_IN\_SOLID are utilized to deal with the fluid– structure coupling problem [5]. The target plate is taken as a hexahedral body cell, with different cell sizes and mesh transitions for different regions, and is connected by a common node. The results are shown in Figs. 4, 5, and 6.

The simplified model of the three groups of welded joints is shown in Fig. 7, according to the actual measurements of the test, the width of the butt weld is 4 mm, and the width of the heat-affected zone is taken as 6 mm according to the literature,

**Fig. 4** Model schematic

**Fig. 5** Grid transition

so the total width of the welded area is 16 mm [6]. The elliptical cross-sectional region of the fillet weld is simplified to a regular polygon, and the simplified weld model must be close to the volume of the weld in the experimental study. Generally the strength of the heat affected region is about 60% to 80% of the base material, and in this paper the material parameter of the heat affected region is taken as 70% of the base material [7]. The Johnson–Cook material model is used to simulate the mechanical ontological relationship between the weld region and its base material, and the J–C dynamic parameters of some materials are shown in Table 1 [8].

**Fig. 6** Target plate meshing

### *2.3 Simulation Model Accuracy Verification*

The displacement cloud and deformation results of each sample part obtained by simulation are shown in Fig. 8, the maximum deformation of sample part 1 is 29.74 mm, the maximum deformation of sample part 2 is 25.67 mm, and the maximum deformation of sample part 3 is 27.61 mm, and the weld seams of the three samples do not have rupture failure. The error of the test and simulation of each sample is within the allowable range, which verifies the accuracy of the finite element model.

### **3 Structural Strength Analysis of Welded Joints**

# *3.1 Effect of a Single Blast Loading on the Strength of a Structure*

This numerical analysis simulated 1, 1.5, 2, 2.5, 3, 3.5 kg a total of six explosion cases, as a comparison in the original model based on the addition of pure steel plate without welds. As shown in Figs. 9 and 10, sample 1 is the worst resistance to a single explosion, sample 2 is second, sample 3 is the best. The pure steel plate is more resistant to a single blast than any of the weld-connected plates, but the amount of deformation may be larger than that of the weld-connected plates. Figure 11 gives the failure form of each sample, in the structural optimization can be considered as far as possible to make the weld region away from the blast wave (sample 1), add a higher strength protective components (sample 2), there is an overlap in the region of the weld form can be changed to bolt connection (sample 3).

**Fig. 7** Simplification of weld and heat affected zone


**Table 1** Main J–C model material parameters

**Fig. 8** Displacement maps and deformation results for each specimen

**Fig. 10** Center deformation vs. explosive equivalent

**Fig. 11** Forms of failure for each sample

# *3.2 Effect of Repeated Explosive Loading on the Strength of Structures*

In order to simulate repeated explosive loading, this paper adopts the complete restart technique of LS-dyna [9], and sets four explosive equivalents of 0.65 kg, 0.85 kg, 1 kg and 1.15 kg, and the number of explosive loading times for each sample is 3 times. As shown in Fig. 12, the amount of center deformation brought about by using steel plates with different forms of weld connections varies considerably. The use of welded joints greatly reduces the structural strength, but has less effect on the amount of structural deformation within the range of non-failure. It can also be found that when the sum of the explosive equivalents of the repeated explosions is not much different from the explosive equivalents of the single explosion, the center deformation of all samples under the effect of repeated explosions is lower than under the effect of a single explosion.

As shown in Figs. 13 and 14, the center deformation of each part at the end of each blast load action increases linearly with increasing blast equivalent. With the increase in the number of explosions, in the range of non-failure (i.e., small equivalent stage) in the center of each piece of the deformation of the fitted line of the gradient

**Fig. 12** Deformation in the center of each sample of repeated explosive loads

is smaller than that of the pure steel plate, indicating that the use of welded seams can reduce the center of the repeated explosive load deformation, similar to the role of reinforcing bars. Define the difference between the center deformation between two consecutive repetitive explosive loads as the center asymptotic deformation, and power function curve fitting [10], it can be found that with the increase in the number of explosions is an exponential decrease, taking into account the residual stresses of the plate and the work-hardening phenomenon, so at the end of each explosion load will be further reduced, and the magnitude of the decrease will be smaller and smaller until the failure.

**Fig. 13** Plot of center deformation versus number of blasts at different explosive equivalents

**Fig. 14** Plot of center asymptotic deformation versus number of blasts

# **4 Conclusion**

This paper establishes a solid element finite element model of steel plate including weld region and heat-affected region, and adopts the material assignment method combined with J–C material intrinsic and structured ALE algorithm, and obtains the following conclusions in the experimental and numerical studies:


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Plastic Constitutive Relation for Improving the Calculation Accuracy of Mechanical Performance of Die-Cast Al-Si Aluminium Alloy Products**

**Xueqiang Wang, Liming Peng, and Siping Li** 

**Abstract** With the interaction in automobile manufacturing technology, products made of integrated die-cast aluminum alloy are becoming more and more widespread. However, engineers frequently ignore the impact of structural features on mechanical properties when utilizing simulation software to determine a product's strength, and current constitutive models do not account for structural flaw studies. To examine the correlation between structural flaws and mechanical properties of the die-cast aluminum alloy, quasi-static tensile tests were performed on JDA1b alloy specimens. The defect rates were varied by adding circular holes with varying diameters at the center of the specimens. The results showed that the JDA1b alloy's tensile strength and elongation significantly decreased as the fault rate increased. A constitutive model with defect rates is proposed, which has higher accuracy than the J–C model. Simulations and experimental findings effectively validated the accuracy of the proposed constitutive model. The proposed model provides support for high-precision computing for analyzing the mechanical performance of materials.

**Keywords** Die-cast · Light-alloy · Constitutive modeling · Defect rate · Structural defect

X. Wang · S. Li (B)

L. Peng

© The Author(s) 2024

Department of Engineering Mechanics, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China e-mail: lisp@sjtu.edu.cn

National Engineering Research Center of Light Alloy Net Forming and Key State Laboratory of Metal Matrix Composites, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_69

### **1 Introduction**

The advancement of integrated die-cast conception and lightweight design in new energy vehicles has propelled the use of die-cast aluminum alloy [1], with an increasing number of die-cast parts found in automobiles. The transition from tiny and basic die-cast parts to massive structural die-cast elements has already been applied [2]. Hence, die-cast parts become intricate and varied, with unique structural attributes. Throughout the car manufacturing process, numerous structural holes are needed for machining, coating, and final assembly during the car production process. These structural holes, referred to as structural defects, significantly affect the strength and stiffness of die-cast parts used in automobiles. The implementation of accurate material constitutive relation is essential for part performance analysis, which is a critical stage in simulation. Given the intricacy of parts, higher simulation accuracy is demanded. The effects of structural defects on parts were disregarded in the investigation of mechanical performance; therefore, to overcome this limitation, calculation accuracy improvements are needed.

J–C constitutive model is a succinct empirical model that explains the link between temperature, strain rate, and stress and strain [3–5]. The J–C model is commonly used to predict the mechanical performances of materials [6, 7]. However, the accuracy of the empirical models, which are derived from curve fitting, is limited. Structural features and constitutive models with defect rates have not been investigated thoroughly yet but are urgently needed to understand the mechanical behavior of parts with structural features. Therefore, an agreement on the constitutive model that takes structural features into account needs to be reached.

JDA1b an acronym for Shanghai Jiao Da Aluminum 1#, is a high-strength, exceptionally ductile, non-heat-treated die-cast aluminum alloy [8]. It has been utilized in numerous automotive components. Compared to traditional die-cast aluminum alloys, the alloy does not experience rectification or dimensional instability. This study aims to obtain the defect rate effects and create a constitutive model with defect rates of die-cast aluminum alloys.

### **2 Materials and Experiments**

### *2.1 Materials and Specimens*

The cast and specimen are presented in Fig. 1. Figure 1a shows the JDA1b alloy ingots. A TOYO BD-350-V5 die-casting machine was employed for manufacturing the castings of JDA1b alloy (Fig. 1b). The casting pressure was 75 MPa. The square thin plates with a thickness of 3.0 mm were obtained after removing the runner on the castings as shown in Fig. 1c and d. Specimens for this study were cut from the square plates using the computerized numerical control wire-cut electric discharge technique. The specimens were polished to achieve a smooth surface. Figure 1e

**Fig. 1** Cast and specimen of the JDA1b alloy


**Table 1** Chemical compositions of the JDA1b aluminum alloy (wt.%)

displays several specimens that were created in compliance with Chinese standard GBT 2281–2021. There are two types of specimens: those without a hole and those with a circular hole in the center. The dimensions are displayed in Fig. 1f. Here, the structural features are supposed to be circular holes with varying sizes in the center and the defect rate (denoted with λ) is the ratio of the maximum projected area of the circular hole along the stretching direction to the contour area of the cross-section in the specimen's gauge. Since the specimen's thickness is homogeneous, the defect rate λ may be written as ρ/*b*, where ρ is the radius of holes and *b* = 7.5 mm is the specimen's half-width (Table 1).

### *2.2 Experimental Apparatus and Method*

A total of eight groups of specimens (five specimens per group, 40 specimens in total) were tested at room temperature (25 °C) at a tensile rate of 1 mm/min using a Zwick Roell 8506 universal electrical testing machine (100 kN). The quasi-static tensile experiments were performed in accordance with the international standard ASTM E8/E8M-16a. The gauge length strain was measured using the extensometer. The Zwick Roell information processing systems were utilized to obtain the true stress– strain curves. When calculating stresses, the reference area is the cross-sectional area of the specimen's gauge that is devoid of holes. Five specimens were repeatedly tested for each defect rate and took the average value as the experimental outcome.

### **3 Experimental Result**

The true stress–strain curves for the JDA1b alloy per defect rate are shown in Fig. 2, suggesting these curves are stable within each group and have comparable hardening characteristics. The results of each experimental group are thereafter represented by the average curves, which are indicated by solid curves. Strong nonlinearity exists in the work hardening. The stresses rose swiftly in the early stages of deformation due to the generation and multiplication of dislocations. After a quick rise, the stresses turn to a slow increase. In the later stage of deformation, the stress steadily stabilizes. Figure 2i superimposed the average true stress–strain curves at various defect rates, ranging from 0 to 46.67%. Defect rates have a noticeable impact on stress. The stresses under the same strain decrease as defect rates rise. The yield strength, ultimate tensile strength, and elongation all decrease as the defect rate rises. Several fractured specimens are displayed in Fig. 2i, and all specimens were fractured at the smallest cross-section.

### **4 Constitutive Model**

The J–C model is expressed as:

$$
\sigma = (A + B\varepsilon^{\prime})(1 + C \ln \dot{\varepsilon}^{\*})(1 - T^{\*m}) \tag{l}
$$

where ε is the plastic strain; *A*, *B*, *C*, *n*, and *m* are undetermined parameters; ε˙ is the plastic strain rat, ε˙0 is the reference plastic strain rate; ε˙<sup>∗</sup> = ε/˙ ε˙0; *T* \* = (*T*-*T*r)/ (*T*m-*T*), where *T*r is the room temperature; and *T*m is the melting temperature. The current work focuses on strain-strengthening effects, hence the last two terms in Eq. (1) are ignored.

The true stress–plastic strain data of intact specimens acquired from the quasistatic uniaxial tensile test are shown in Fig. 3. The yield stress is 163.25 MPa, or A = 163.25 MPa. Strain strengthening coefficients, *B* = 392.93 MPa and *n* = 0.41345, are calculated by the method from references [9, 10]. The stress of JDA1b stabilized eventually. However, the stress did not converge in the J–C model. This indicates that precise flow stress predictions are not possible using the original J–C model. Therefore, to better predict stress, convergence must be considered in constitutive models.

**Fig. 2** True stress–strain curves of JDA1 alloy at defect rates of **a** 0, **b** 6.67%, **c** 13.33%, **d** 20.00%, **e** 26.67%, **f** 30.00%, **g** 40.00%, **h** 46.67%, and **i** average curves

The proposed constitutive model is expressed as

$$\sigma = \left[ \mathcal{Q} - \frac{\mathcal{Q} - A}{1 + \left( \varepsilon / \varepsilon\_0 \right)^p} \right] (1 + C \ln \dot{\varepsilon}^\*) (1 - T^{\*m}) (1 - \lambda^k) \tag{2}$$

where *Q* is the tensile stress limit value; ε0 is the critical strain; λ presents the defect rate of specimens with a circular hole in the center; λ = ρ/*b*, *p* and *k* are unknown coefficients, and the remaining parameters the same as in the J–C model. The parameters *Q*, *A*, and ε0 show clear physical significance, and their values have been previously discussed in our work [11]. Here, *Q* = 366.70 MPa, ε0 = 0.03425, *p* = 0.94521, and *k* = 1.52873.

The proposed new constitutive model introduces the (1-λk) term to represent the influence of defects and uses stress limit value and critical strain to describe the stress–strain curve. The convergence of the stress is represented by the inverse proportional function *Q* <sup>−</sup> *<sup>Q</sup>*−*<sup>A</sup>* <sup>1</sup>+(ε/ <sup>ε</sup>0)*<sup>p</sup>*term. The predicted curve of the proposed model is displayed in Fig. 3. The J–C model's form is changed by the proposed model. Based on the comparisons, the proposed model fits the tendency of the stress better and shows better predictability than the J–C model.

### **5 Results and Discussion**

### *5.1 Effect of Defect Rates on Strength and Elongation*

The Effect of defect rates on strength and elongation are summarized in Fig. 4. Here, YS stands for yield strength, UTS stands for ultimate tensile strength, and EL stands for elongation. As defect rates rise, the YS, UTS, and EL all nonlinearly decline. When defect rates are less than 13.33%, the EL fluctuates widely, from 7.9 to 14.7%, but when defect rates are more than 13.33%, the EL fluctuates narrowly. Furthermore, YS fluctuates quite little.

### *5.2 Analysis of Constitutive Model Accuracy*

The J–C model can be used to express the defect rate effect by multiplying the defect term (1-λk) behind Eq. (1). The proposed model and the J–C model's projected stresses were compared with the results of experiments. As can be seen in Fig. 5, the J–C model's prediction accuracy is not optimal for all strains. Only a weak correlation between the expected and experimental stresses. When the defect rates are higher than 13.33%, the predicted values exhibit a significant deviation from the experimental results. The J–C model does not adequately capture the trend of the stresses, thus it appears unreasonable. From Fig. 6, for all defect rates, the true stress–strain curves predicted by the proposed model show a consistent trend with the experimental curves. The results demonstrate that, over the whole range of strain and defect rates, the proposed model outperforms the J–C model in describing the deformation behavior of die-cast aluminum alloys.

**Fig. 4** Effect of defect rates on strength and elongation

The accuracy of the two constitutive models is assessed using the average absolute relative error (*AARE*) between the experimental data and the predicted results. The *AARE* values for the J–C model and the proposed model are 2.66~12.54% and 0.21~0.36%, respectively. Compared to the J–C model, the proposed model's *AARE*  is significantly lower. When the defect rates rose, the J–C model's error increased gradually. The primary reason for this inaccuracy was the J–C model's failure to converge. The suggested model's predicted values were more closely correlated with the experimental values, and its accuracy was noticeably higher than the J–C model's accuracy. Thereby resolving the J–C model's poor prediction accuracy problem.

# *5.3 Validation with a Die-Cast Part*

To verify the proposed constitutive model, we created a JDA1b die-cast part with a complicated cross-section and conducted tensile tests on it. The die-cast part is numerically simulated using the proposed constitutive model with and without defect rates. The experimental setups and three-dimensional model are shown in Fig. 7. Figure 7a presents the tensile experimental setup. A unique fixture is designed to clamp the parts. The die-cast part's cross-sectional shape and dimensions are displayed in Fig. 7b. The part's hole measures 7 mm in diameter, while the side and bottom walls have widths of 32 mm and 67 mm, respectively. As a result, the defect rates of the entire side wall and bottom wall are 21.87% and 10.77%, respectively. Figure 7c displays a three-dimensional finite element model.

The deformation in tests and simulations is displayed in Fig. 8. Figure 8a compares the parts before and after tests. The largest displacement, measuring 9 mm in tests, is found at the top of the part. The displacement simulated by the proposed model

**Fig. 7** Tensile experiments and simulations of the JDA1b die-cast part

**Fig. 8** Experiment and simulation of the JDA1b die-cast part

with defect rates is 9.06 mm, which is near to the displacements obtained experimentally. As shown in Fig. 8b, the applied maximum tensile load is roughly 33 kN. The simulation's fracture form and position are consistent with the test's (see Fig. 8c). However, the fracture does not occur, and the constitutive model without defect rates only simulates a displacement of 5.37 mm (see Fig. 8d). Therefore, applying a constitutive model that considers the defect rate will produce more realistic simulation results. The weakening phenomenon of holes to the mechanical performance is reflected in the proposed constitutive model.

### **6 Conclusions**

The mechanical performances of die-cast aluminum alloys with structural defects were investigated experimentally and numerically. A constitutive model of die-cast aluminum alloy with defect rates was proposed and verified. One can infer the following conclusions.

(1) The structural features of the JDA1b alloys are connected to their tensile mechanical performances. Yield strength, ultimate tensile strength, and elongation all decrease with increasing defect rates. Performance deterioration can be severe even with a small structural defect.


This work offers suggestions for investigating the mechanical behavior of alloys with structural features. It facilitates accurate numerical simulations and studies relevant to engineering products besides die-cast aluminum parts.

**Acknowledgements** This work was supported by the National Key Research and Development Program (grant number 2021YFB3701000); the Basic Strengthening Program (grant number 2022- JCJQ-ZD-173-048); the National Natural Sciences Foundation of China (grant number U2102212, 51878407).

### **References**


11. Wang XQ, Yuan LY, Xiao G, Peng LM, Li SP (2023) A high-accuracy dynamic constitutive relation of die-cast Al–Si aluminium alloy. Int J Mech Sci 251:108304

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Modelling and Finite Element Simulation of Ball-End Milling for Nickel-Based Superalloy Inconel 718**

**Yaoman Zhang and Jin Zheng** 

**Abstract** Comprehensive exploration of ball-end milling processes is presented in this paper, with a primary focus on the modelling of milling forces and the execution of finite element analysis during the machining of Inconel 718, a material known for its challenging machinability. A detailed milling force model, considering various parameters such as cutting speeds, feed rates, and depths of cut, has been developed, providing valuable insights into the optimization of machining parameters. Temperature and stress distributions within the tool during milling, particularly in the context of difficult-to-machine materials like Inconel 718, were investigated through finite element analysis. Critical temperature profiles at the tool tip, rake face, and flank face, which have an impact on tool wear and lifespan, were identified through the temperature field analysis. Notably, a maximum tool tip temperature of 682 °C was observed during the machining of Inconel 718. Challenges posed by difficult materials were unveiled through the stress field analysis, aiding in stress mitigation and enhancing the understanding of machining processes. In conclusion, a significant contribution is made by this paper to the understanding of ball-end milling processes.

**Keywords** Nickel-based superalloy · Finite element analysis · Modelling of cutting force

# **1 Introduction**

Inconel 718, a nickel-based superalloy, boasts remarkable attributes such as exceptional heat resistance [1], corrosion resistance [2], fatigue resistance [3], and fracture strength [4]. It has proven its mettle by enduring extreme environments with temperatures soaring up to 600 °C. Consequently, it has emerged as the material of choice

e-mail: zhangyaoman@mail.neu.edu.cn

© The Author(s) 2024

889

Y. Zhang (B) · J. Zheng

School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, Liaoning, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_70

across a spectrum of industries, most notably in aerospace [5, 6] and shipbuilding [7], where its applications hold immense promise.

In the realm of aerospace engineering, Inconel 718 has firmly established its reputation as an indispensable component. It plays a pivotal role in the manufacturing of aviation jet engines [8] and finds extensive use in various industrial gas turbines [9]. Impressively, this alloy often accounts for up to half of the total weight of these high-performance engines.

Milling is the primary method employed for the manufacture of Inconel 718; however, it poses several challenges, including severe tool wear, high thermal stress, substantial cutting forces, and chattering [10, 11]. These issues must be effectively addressed to fully harness the potential of Inconel 718 in various applications. To tackle these challenges, a comprehensive investigation into the milling mechanism of Inconel 718 is imperative. The utilization of modelling and finite element simulation techniques for analyzing the cutting forces in Inconel 718 offers a powerful solution. Through these methods, the cutting process of Inconel 718 can be accurately simulated under different cutting parameters. This approach enables the swift assessment of the influence of varying cutting conditions on critical performance metrics such as cutting forces and tool wear. Importantly, it obviates the need for costly physical experiments. By leveraging these techniques, manufacturers can determine the optimal combination of machining parameters. This optimization is instrumental in mitigating tool wear, extending the service life of cutting tools, suppressing chattering, and ultimately enhancing manufacturing efficiency. Therefore, investigating the milling mechanism of Inconel 718 through modelling and finite element simulation serves as a key strategy to overcome the challenges associated with machining this high-performance alloy.

Lee et al. [12] used a series of orthogonal milling test data at different milling speeds and feeds to derive a calculation model for the milling force of a ball-end milling cutter. And for the first time, the milling force generated during the milling performed by the ball-end milling cutter was divided into edge force and shearing force. When calculating the milling force, the blade of the ball-end milling cutter was divided into a series of oblique milling elements, and considerable experiments were carried out to verify the correctness of the proposed model. Sonawane et al. [13] established a theoretical calculation model of milling force for Inconel 718 with ball-end milling cutters. This model can be used to predict the milling force, chip geometry and instantaneous shear angle during the milling process. The relationship between the shear strength of the workpiece material and the strain, strain rate and temperature were taken into account when the theoretical model was established, and the Johnson–Cook material model was used to describe the above relationship. Kao et al. [14] decomposed the milling force generated during the milling process of the ball-end milling cutter into axial, radial and tangential forces. In addition, the authors drew the stability lobe diagram of the milling process of the ball-end milling cutter through the CUTPRO software, determined the stability conditions of the milling process of the ball-end milling cutter, and verified the reliability of the milling force model of the ball-end milling cutter through experiments. Sethupathy et al. [15] established a statistical model through the response surface method, and predicted the milling force based on tool geometric parameters and processing parameters. A second-order mathematical model for calculating the milling force of AL7075-T6 aluminum alloy with a ball-end milling cutter was developed, and the correctness of the model was verified by experiments. Zhao et al. [16] used orthogonal tests to obtain the milling force coefficients and established an empirical model of milling force. Li et al. [17] established an empirical model for the prediction of milling force and cutting temperature based on the multiple linear regression method. Li et al. [18] used ABAQUS software to analyze the effects produced by polycrystalline diamond tools for machining thin walls of SiCp/Al composite material. Jin et al. [19] established an optimized finite element model based on pendulum motion, considered the performance parameters of the workpiece, and used ABAQUS software to study the rule of change of milling force during the milling process of aerospace thin-wall component. Davoudinejad et al. [20] studied the chip formation and milling force when milling Ti–6Al–4V titanium alloy under dry and cryogenic cooling conditions by using the AdvantEdge finite element cutting software. When establishing the finite element model, the Coulomb's friction law is used to calculate the friction stress in the milling process, and the accuracy of the model is verified by comparison with the experimental results. Mebrahitom et al. [21] and Bhopale et al. [22] simulated the milling process of Aluminum 6010 and AISI 1018 cold-rolled steel using Abaqus software, and analyzed the milling force and processing parameters. Yameogo et al. [23] used Abaqus software to establish a finite element model of milling of Ti6Al4V alloy, and proposed a new Multi-Branch (MB) theory to consider the dynamic recrystallization during milling. Rahul et al. [24] used the finite element method to analyze the residual stress of the subsurface during micro-milling Ti–6Al–4V alloy.

In summary, based on abovementioned research background, in this study, novel contributions have been introduced in this paper, encompassing milling force modelling, finite element analysis, the influence of machining parameters, and practical applications. Collectively, these contributions serve to advance the understanding of ball-end milling processes and their optimization, especially for challenging materials like Inconel 718.

### **2 Modelling of Milling Force of Ball-End Milling Cutter**

A well-developed model of milling force of ball-end milling cutter allows for a systematic analysis of how milling forces change under different conditions, such as varying cutting speeds, feed rates, and depths of cut. This information is crucial for optimizing machining parameters, leading to increased efficiency and reduced tool wear.

# *2.1 Establishment of Milling Force Model of Ball-End Milling Cutter*

According to the theory proposed by Lee et al. [12], if the tip radius of the ball-end milling cutter could be neglected, the microelement of cutting force generated by the ball-end milling cutter during machining can be presented as:

$$\begin{cases} dF\_{t,j}(\phi\_j(z)) = K\_{tc}dS(\phi\_j(z)) + K\_{tc}h\_j(\phi\_j(z))db \\ dF\_{r,j}(\phi\_j(z)) = K\_{r\epsilon}dS(\phi\_j(z)) + K\_{rc}h\_j(\phi\_j(z))db \\ dF\_{a,j}(\phi\_j(z)) = K\_{a\epsilon}dS(\phi\_j(z)) + K\_{ac}h\_j(\phi\_j(z))db \end{cases} \tag{1}$$

In Eq. (1), the parameters *Ktc*, *Krc*, *Kac* are named as shearing force coefficients, and parameters *Kte*, *Kre*, *Kae* are called edge force coefficients. According to method outlined by Kao et al. [14], the edge and shearing force coefficients can be calculated by using equation given below:

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *Kac* = *Fzc* (*C*3*C*7*C*15−*C*1*C*9*C*15)−*C*7*C*13*C*15 *F f c*+*C*1*C*13*C*15 *Fnc C*3*C*7*C*<sup>2</sup> 15−*C*5*C*7*C*13*C*15−*C*1*C*9*C*<sup>2</sup> <sup>15</sup>+*C*1*C*11*C*13*C*<sup>15</sup> *Kae* = *Fze* (*C*4*C*8*C*16−*C*2*C*10*C*16)−*C*14*C*8*C*16 *F f e*+*C*2*C*14*C*16 *Fne C*4*C*8*C*<sup>2</sup> 16−*C*6*C*8*C*14*C*16−*C*2*C*10*C*<sup>2</sup> <sup>16</sup>+*C*2*C*12*C*14*C*<sup>16</sup> *Krc* = *C*7*C*15 *F f c*−*C*1*C*15 *Fnc*+(*C*1*C*11−*C*5*C*7)*Fzc C*3*C*7*C*15−*C*5*C*7*C*13−*C*1*C*9*C*15+*C*1*C*11*C*<sup>13</sup> *Kre* = *C*8*C*16 *F f e*−*C*2*C*16 *Fne*+(*C*2*C*12−*C*8*C*6)*Fze C*4*C*8*C*16−*C*6*C*8*C*14−*C*2*C*10*C*16+*C*2*C*12*C*<sup>14</sup> *Ktc* = *F f c*−*C*3 *Krc*−*C*5 *Kzc C*1 *Kte* = *F f e*−*C*4 *Kre*−*C*6 *Kze C*2 (2)

Finally, the average milling force per rotation of the ball-end milling cutter can be calculated using the Eq. (3). The parameters *C*1–*C*16 in Eq. (3) are explained in Ref. [14].

$$\begin{cases} \overline{F}\_{f} = (C\_{1}K\_{tc} + C\_{3}K\_{rc} + C\_{5}K\_{ac})f\_{t} + C\_{2}K\_{tc} + C\_{4}K\_{r\epsilon} + C\_{6}K\_{a\epsilon} \\\\ \overline{F}\_{n} = (C\_{7}K\_{tc} + C\_{9}K\_{rc} + C\_{11}K\_{ac})f\_{t} + C\_{8}K\_{t\epsilon} + C\_{10}K\_{r\epsilon} + C\_{12}K\_{a\epsilon} \\\\ \overline{F}\_{z} = (C\_{13}K\_{rc} + C\_{15}K\_{ac})f\_{t} + C\_{14}K\_{r\epsilon} + C\_{16}K\_{a\epsilon} \end{cases} (3)$$

### *2.2 Influence of Machining Parameters on Milling Force*

The average milling force can be computed by employing Eq. (3), and simultaneously adjusting the machining parameters will result in changes to the milling force, thereby revealing the influence and effect of machining parameters on milling force is crucial.

The feed per tooth plays a pivotal role in determining both the magnitude of the milling force and the quality of the machined surface. Consequently, to investigate

**Fig. 1** Influence of processing parameters on milling force of X-Axis: **a** Influence of feed per tooth; **b** Influence of tool helix angle

the influence of the feed per tooth on the milling forces, the different feed rate per tooth is set separately as *fz* = 0.1 mm/z, 0.2 mm/z, 0.3 mm/z, 0.4 mm/z, while keeping the radial depth of cut and axial depth of cut unchanged, namely *ae* = 3 nm, *ap* = 0.5 mm. In addition to feed per tooth, the tool helix angle is another important parameter that affects the milling process. Change in helix angle in ball-end milling cutter affects the cutting angle and feed angle, which in turn affects the contact between tool and workpiece. The tool helix angle is set separately as β = 15°, β = 25°, β =35°, β = 45° in the case of ensuring other parameters remain unchanged. The results of above studies are shown in Figs. 1, 2 and 3. When the feed per tooth of the ball-end milling cutter increases, there is a corresponding upward trend observed in the milling force for each axis. Similarly, an increase in the helix angle of the ball-end milling cutter results in a proportional rise in the cutting force. This can be attributed to the expansion of the contact area between the tool and the workpiece, leading to an overall mounting trend in cutting force along all axes. Due to the significant impact of the ball-end milling cutter's helix angle on the generated milling force in each axis, careful consideration and selection of the appropriate tool helix angle are imperative to meet specific machining requirements.

### **3 Finite Element Analysis of Milling of Inconel 718**

In this study, a finite element model has been established, encompassing both the workpiece and the tool. Subsequently, the temperature and stress in the milling process are calculated.

**Fig. 2** Influence of processing parameters on milling force of Y-Axis: **a** Influence of feed per tooth; **b** Influence of tool helix angle

**Fig. 3** Influence of processing parameters on milling force of Z-Axis: **a** Influence of feed per tooth; **b** Influence of tool helix angle

### *3.1 Temperature Field Analysis of Milling of Inconel 718*

During the milling process, the temperature at the tool tip, rake face, and flank face of the milling tool assumes paramount importance in the determination of tool wear and lifespan. Generally, it can be asserted that the higher the cutting temperature of the tool, the faster the tool will experience wear and eventual failure. Consequently, by means of simulating and analyzing the temperature profiles at the tool tip, rake face, and flank face, the tool's performance and lifespan can be effectively predicted and assessed. To investigate the temperature variations at the tool tip, rake face, and flank face throughout the milling process, as depicted in Fig. 4, the finite elements situated within the tool tip (L1), rake face (L2), and flank face (L3) have been chosen, and a curve illustrating the time-dependent variations in the average temperature of these selected elements in L1, L2, and L3 has been generated.

**Fig. 4** Temperature curves of each part of the tool

### *3.2 Stress Field Analysis of Milling of Inconel 718*

Difficult-to-machine materials, such as nickel-based superalloys, give rise to significant tool wear and thermal damage. Tool stress analysis enables the determination of both the maximum stress value and the distribution of stress within the tool, mitigating the risk of tool breakage or harm to the workpiece. Through an examination of the stress distribution within the tool, the stress-bearing regions and stress patterns can be ascertained. Furthermore, the analysis of tool stress unveils physical phenomena occurring during milling, including friction and thermal deformation, contributing to a more profound understanding of the machining process. To investigate the stress variations at the tool tip, rake face, and flank face throughout the milling process, as depicted in Fig. 5, the finite elements situated within the tool tip (L1), rake face (L2), and flank face (L3) have been chosen, and a curve illustrating the time-dependent variations in the average stress of these selected elements in L1, L2, and L3 has been generated.

### *3.3 Results of Analysis of Temperature Field and Stress Field*

From the curve presented in Fig. 4, it is evident that during the progression of the milling process, the temperatures of the tool tip, rake face, and flank face exhibit an initial increase followed by a subsequent decrease. This phenomenon is associated with plastic deformation, accompanied by the generation of a substantial amount of cutting heat. The chips come into direct contact with the tool, transferring cutting heat into the tool, consequently resulting in a significant rise in tool temperature. As

**Fig. 5** Stress curves of each part of the tool

the chips eventually fracture and disengage from contact with the tool, a substantial portion of the cutting heat is carried away by the chips, leading to a reduction in tool temperature. In addition, it is observed that the temperature of the tool tip directly involved in the cutting process exceeds that of the rake face. This phenomenon arises because the tool tip is actively engaged in material removal, leading to significant plastic deformation and the generation of the highest amount of heat, resulting in the highest temperature, reaching a maximum of 682 °C. Furthermore, the temperature of the cutting edge's rake face involved in the cutting process is higher than that of the flank face. This discrepancy is attributed to the pronounced extrusion and friction occurring between the rake face and the chips, which themselves possess elevated temperatures, leading to a sharp temperature increase. In contrast, the extrusion and friction experienced by the flank face against the machined surface are less severe, resulting in a lower temperature than that of the rake face. The maximum temperature recorded for the rake face is 614 °C, while the maximum temperature for the flank surface is 460 °C.

As shown in Fig. 5, the maximum stress region on the flank is larger than that on the rake face, with the stress in the section of the tool tip exceeding that on the flank face. Because the flank face of the tool is in contact with the workpiece's surface, which means it experiences a significant amount of friction and resistance as the tool moves across the material. This results in higher stresses on the flank face compared to the rake face. The tool tip is where the actual material removal happens, and it experiences intense localized stress as it cuts through the superalloy. This stress can exceed that experienced by the flank face, especially if the cutting conditions are not optimized or if the tool is not designed for efficient chip formation. Additionally, the average stress on the flank is slightly higher than that on the rake face, which could be due to the cumulative effect of cutting forces, friction, and heat generated during machining. The rake face, while it does experience some stress, is primarily responsible for providing clearance and reducing the cutting forces.

### **4 Conclusions**

In this study, an in-depth exploration of ball-end milling processes was undertaken, offering insights into the modelling of milling forces, finite element analysis of machining of Inconel 718, and the subsequent evaluation of temperature and stress within the tool. Significant findings underscored the pivotal role played by feed per tooth and tool helix angle in the determination of milling forces, with their impact on efficiency and tool performance being highlighted. The application of finite element analysis unveiled dynamic temperature and stress patterns during milling. Tool tip temperatures and stress exhibited peaks attributed to tool tip's active engagement in material removal. This newfound comprehension aids in the mitigation of tool wear and damage.

In summary, this research advances our understanding of ball-end milling processes, allowing for more efficient machining and enhanced tool longevity, particularly in the context of challenging materials such as Inconel 718.

**Acknowledgements** The authors wish to thank the National Natural Science Foundation of China for granting the funding (grant number: 62273081), and this research is also funded by the Liaoning Provincial Department of Education 2021 Scientific Research Funding Project, grant number: LJKZ0001.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Ultimate Strength Test and Numerical Simulation Analysis of Typical Cabin Made of a Novel Steel**

**Chen Luan, Pengyu Lou, Hongfu Wang, and Qi Wan** 

**Abstract** A novel steel was proposed to apply to a ship, thereby enabling ships to safely complete their missions. But it is different in the ultimate bearing capacity between the novel material and traditional low-alloy high-carbon steels. There is little research on it. In this paper, the ultimate strength and the destruction mode of the typical cabin made of the novel steel under the hazardous condition are successfully obtained through a four-point bending test. Numerical simulation of the typical cabin at its limit state is verified by the test. Based on the numerical simulation method, theoretical initial deformation and ideal constitutive relation are introduced, which contributes to more conservative prediction for the ultimate strength of the ship made of the novel steel. This paper has guiding significance for subsequent tests and provides technical support for the structure design of the ship made of the novel steel.

**Keywords** Novel steel · Ultimate strength · Destruction mode · Numerical simulation

# **1 Introduction**

Most ships needed to be composed of different materials for different missions to ensure tasks successful completion. A novel steel composed of Ni-Cr-Mn-Mo-N alloys had excellent magnetic stability and exhibited low magnetic conductivity of 1.001–1.003, which helped ships successfully complete their tasks. However, there were many differences in the ultimate bearing capacity between the novel steel and traditional steels. Therefore, the ultimate strength assessment of the ship made of the novel steel was urgently needed for guaranteeing life safety at sea [1].

Ultimate strength of hull structure was used to measure the ability to resist destruction, which reflected the reliability of ships [2]. Research about ultimate strength of

C. Luan (B) · P. Lou · H. Wang · Q. Wan

Marine Design and Research Institute of China, Shanghai, China e-mail: luanchen1997@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_71

hull structure was widely attracted by attention for a long time [3–5]. Most of experimental research was conducted on box beams, multi-span continuous beams and scaled modes instead of full ships [6, 7]. Theoretical calculation based on system theory and numerical analysis mainly contained progressive collapse methods and nonlinear finite element methods [8]. Wang compared the test results and the theoretical calculation, which suggested that the nonlinear finite element analysis was more accurate than the progressive collapse method [9].

Based on the certain ship, we carried out a four-point bending test to explore the ultimate bearing capacity of the typical cabin in the sagging state by a scaled model, and went on numerical simulation analysis by the nonlinear finite element method. We also investigated the effect of initial deformation and constitutive relation on ultimate strength. Our results helped the follow-up tests on the large-scale models and predicted the ultimate strength of the ship made of the novel steel.

### **2 Experimental Design**

### *2.1 Experimental Model*

Hulls had two states: sagging and hogging. According to different standards, the margin of ultimate strength of the certain ship in the sagging state was lower than that in the hogging state. Thus the sagging state was chosen as the hazardous state. For typical cabins, the number of spans had a great impact on the ultimate strength under the sagging condition. Based on our previous experiments, when the tests were carried out by the three-span continuous beams, the effect of the boundary constraints and the web frames could be considered, which facilitated the results close to the actuality. The web frames were designed as rigid support in order to stabilize structure. Considering the factors such as manufacture, transportation and loading capacity, we conducted the test by a scaled model of the typical cabin. The guidelines the scaled model (0.78 × 1.318 × 0.9 m) followed were as follows: (a) The structural geometry was designed with a certain scale ratio; (b) The scaled model was made of the novel steel; (c) The stress and strain of the scaled model was consistent with those of the full ship. Therefore, the destruction mode and order of the scaled model were consistent with those of the full ship. In addition, the longitudinals of the scaled model were used by flat steel for convenience.

### *2.2 Experimental Layout*

In order to adapt the loading conditions in the laboratory, the scaled model was stretched as shown in Fig. 1. The two ends of the scaled model were placed on two supports to copy simple support. The extended sections were strong enough to ensure

**Fig. 1** Experimental layout

that the tested section was destroyed first. The high-strength transverse bulkheads were arranged at the loading and support locations to avoid structural instability. The connections between the tested section and the extended section were flanges by sixty-eight M24 screw-bolts. The brackets were added at the junction of the extended section and the flanges in order to ensure structural strength and avoid localized stress. We also processed the shear check of the screw-bolts and they were all satisfied test requirements. We made sure the loading capacity by designing the suitable force arms.

### *2.3 Experimental Points*

**Displacement test points**. To monitor the overall deformation, five displacement test points were placed at the junction of the bottom shell and the central longitudinal section as shown in Fig. 2a. To monitor the out-of-plane deformation of the decks, four displacement test points were placed on the E deck at the B cross section as shown in Fig. 2b.

**Fig. 2** Experimental points. **a** Displacement test points at the bottom shell; **b** Displacement test points on the E deck; **c** Strain test points on the E deck

**Strain test points**. Destruction locations were usually concentrated on decks or the side shell close to decks under sagging condition. There were eighty unidirectional strain gauges and eight triaxial strain gauges arranged the locations where damage might occur. The triaxial strain gauges were marked with circles as indicated in Fig. 2c.

### *2.4 Experimental Tests*

The steel plates with the actual thickness of 3 and 5.3 mm were conducted uniaxial tensile tests for guiding numerical simulation. Due to various machining accuracy, various deformation was occurred, which affected ultimate bearing capacity of constructions. The initial deformation of localized deformation of plates, light side deformation of stringers and columnar deformation of plates could be calculated by the Smith method [10]. In order to obtain more precise estimate about initial deformation, the scaled model was tested by the handheld 3D laser scanner. The load was loaded through the multi-channel static loading system. The static strain instrument automatically recorded strain and displacement. The speed of loading was 50 kN/min. The frequency of the recording about loading was 10 Hz and the frequency of the recording about strain and displacement was 2 Hz. The initial load was 2 kN. When large nonlinear structural deformation occurred, the load was uninstalled until zero.

### **3 Test Results and Analysis**

### *3.1 Constitutive Relation*

As seen in Fig. 3, the constitutive relation of the two steel plates was not ideal elastic– plastic relationship. The specification required their yield strength to be no less than 400 MPa (ideal constitutive relation). The actual yield strength of processed steel was usually higher and they were 534 MPa (3 mm) and 491 MPa (5 mm) (actual constitutive relation). The following stress tested were all obtained through the ideal constitutive relation in terms of the actual strain.

### *3.2 Initial Deformation*

The average initial deformation by the Smith method was ±2.2 mm (theoretical initial deformation). But the actual initial deformation of the decks was −4.2 ~ 4.2 mm and

**Fig. 3** Constitutive relation of the two steel plates with different thickness. **a** 3 mm; **b** 5 mm

the actual initial deformation of the side shell was −3.1 ~ 5.3 mm by the handheld 3D laser scanner.

### *3.3 Stress and Displacement Results*

**Unidirectional stress on the E deck**. The relationship between the stress on the three cross section and the test points was observed in Fig. 4. The A-33, the B-60 and the C-86 on the E deck were placed on the central longitudinal section and they were all the centres of the plates. When the load increased to 800 N, the stress on the A-33 and the C-86 transformed from tensile to compressive, however, the stress on the B-60 increased continuously, which manifested that the localized plates buckled and the buckling direction of the neighbouring localized plates was reverse.

**Triaxial stress on the E deck**. The B4-0°, the B4-45°and the B4-90° represented triaxial stress on the E deck as indicated in Fig. 5a. During sagging, the E deck was compressed. The B4-90° and B5-90° that stood for longitudinal stress were all compressed. But the B5-90° was higher than the B4-90° due to the reverse buckling direction among the neighbouring localized plates. When buckled, the localized

**Fig. 4** Curves of stress on the three cross section

**Fig. 5** Stress and displacement results. **a** Curves of stress on the E deck; **b** Curves of absolute displacement on the E deck; **c** Curves of stress on the side shell

plates were subjected to transverse stress. The B4-0° was tensile and the B5-0° was compressed because of the reverse buckling direction. Furthermore, the transverse stress started from beginning.

**Displacement on the E deck**. The B cross section moved down along with increasing loading. For analysis of out-of-plane deformation, the absolute displacement based on the junction of the B cross section and the central longitudinal section (No.7) increased during loading as shown in Fig. 5b. The results indicated that the out-of-plane deformation occurred from beginning. When the load increased to 800 kN, the absolute displacement increased or reduced quickly, suggesting that the buckling direction was reverse among the adjacent localized plates.

**Unidirectional stress on the side shell**. As indicated in Fig. 5c, the unidirectional stress on the side shell varied with the distance between the test points and the baseline. In the sagging state, the decks were compressed and the bottom shell was tensile. The stress was zero at the neutral axis. The height of the neutral axis was 435 mm in the designed model. The distribution of the stress at the port and the starboard was symmetric because of structural symmetry. The height of the neutral axis stabilized 413 mm during loading. When the load was high up to 850 kN, the neutral axis moved to the baseline rapidly and the relationship between the stress and the load transformed from linear to nonlinear.

**Unidirectional stress on the bottom shell**. Because the bottom shell was thick enough to resist stretching, it was still in an elastic state though the model was under the limit.

### **4 Numerical Simulation Analysis**

### *4.1 Nonlinear Finite Element Method*

When the hull structure generated large deformation or geometric nonlinearity due to buckling, they were in the plastic state and nonlinear behaviour occurred. This paper simulated the ultimate strength through the nonlinear finite element method by ABAQUS. For computational convergence, the theoretical initial deformation was introduced.

The finite element model was composed of quadrilateral shell elements and a small number of triangular shell elements as shown in Fig. 6. The constrains were imposed at the connection of the model and the support. On the one side: Ux = 0, Uy = 0, Uz = 0, θy = 0 and θz = 0; On the other side: Ux = 0, Uy = 0, θy = 0 and θz = 0. The concentrated force was loaded at the position consistent with the test. Bolt joints were simulated by MPC constrains. Meshing is important for calculation accuracy and workload. Although increasing the numbers of the element is beneficial for improving computational accuracy, the computational workload expanded. Therefore, we designed different element sizes in terms of different sections. The element size of the tested section was 9 mm and the element size of the extended section was 20 mm.

In order to provide reference for future simulation, we designed a simplified finite element model, which facilitated to reduce modelling and computing burden. The simplified model was the tested section extracted from the complete model. The simplified model gradually reached its limit by imposing an angle around x-axis.

**Fig. 6** Finite element model

### *4.2 Comparison of Test and Numerical Simulation*

#### **Ultimate Strength**

The comparison of the test result and the simulation result was observed in Fig. 7. The two curves had similar trend. In terms of the test, there was an inflection point and the displacement was 12.5 mm when the load was up to 850 kN. The model was at the limit state and the displacement was 23 mm when the load was high up to 944 kN. The displacement increased by 46% and the load only increased by 10% from the inflection point to the limit point, which manifested that the hull structure buckled and yielded. The model was subjected to the bending moment calculated as follows:

$$M = P \times 2.17m \tag{l}$$

The P represented the load. The ultimate bending moment was 2.05E + 09 N·mm. In terms of the simulation result, there was an inflection point and the displacement was 13 mm when the load was up to 850 kN. The model was at the limit state and the displacement was 24 mm when the load was high up to 893 kN, which was lower by 5.4% than the test result. The deviation was caused by the welding. The tested model

**Fig. 7** Curves of load and displacement


had higher stiffness than the finite element model due to high strength of welding. The results also declared that the numerical simulation was more reliable.

As indicated in Table 1, the ultimate bending strength calculated by the simplified model was lower by 7.2% than that calculated by the complete model, which manifested that the simplified model simulated accurately enough to apply for subsequent numerical calculation.

**Destruction Mode**. At the limit state, the E deck farthest from the neutral axis exhibited significant buckling and generated large deformation at the middle span as seen in Fig. 8a, which agreed well with the test result in Fig. 8b. The buckling direction was inverse among the adjacent localized plates. The F deck was in the stage of plastic deformation and underwent yield failure under limit. The side shell between the E deck and the F deck was compressed during sagging. It exhibited significant buckling and generated large deformation at the middle span as seen in Fig. 8c, which agreed well with the test result in Fig. 8d. In general, the destruction mode calculated by the numerical simulation was all accordant to that observed by the test.

### *4.3 Interfering Factors on Ultimate Strength*

**Initial deformation**. Based on the aforementioned simplified model, the theoretical initial deformation was introduced and the comparison was exhibited in Table 2. The ultimate bending moment by the theoretical initial deformation was higher by 2.8% than that by the actual initial deformation. Although the actual initial deformation was almost double, the effect on the ultimate bearing capacity was little. So, the numerical simulation by the theoretical initial deformation was adopted to avoid obtaining the initial deformation through experiments.

**Constitutive relation**. Based on the aforementioned simplified model, the actual constitutive relation was introduced and the comparison was exhibited in Table 3 and Fig. 9. The ultimate bending moment by the actual constitutive relation was higher by 16.7% than that by the ideal constitutive relation, which implied that the constitutive relation had great effect on the ultimate strength. However, the more conservative results were obtained by the ideal constitutive relation.

**Fig. 8** Comparison of test and numerical simulation. **a** Distribution of stress of the simplified model; **b** Deformation of the E deck; **c** Distribution of displacement of the simplified model; **d** Deformation of the side shell


**Table 3** Comparison of ideal constitutive relation and actual constitutive relation


### **5 Conclusions**

The typical cabin under the limit was taken as the research object. By comparing the test results and the numerical simulation analysis, the conclusions were as below:

**Fig. 9** Numerical comparison of ideal constitutive relation and actual constitutive relation


This paper can provide technical support for the structural design of the ship made of the novel steel and offer important reference significance for the ships made of other steels.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Aerodynamic Characterization of a Ballistic-Correction Bullet**

**Hongyue Zhang and Zhilin Wu** 

**Abstract** Obtaining the aerodynamic parameters of a ballistic-correction bullet is crucial for improving ballistic correction efficiency and enhancing shooting accuracy. In this study, a numerical simulation method is utilized to analyze and calculate the aerodynamic characteristics of the bullet. The focus is on investigating the changes in aerodynamic parameters when the rudder wings are in both unexpanded and expanded states, as well as analyzing the impact of deflection angles on the bullet's aerodynamic characteristics. The simulation results demonstrate that the bullet exhibits favorable aerodynamic performance and static stability when the rudder wings are unexpanded, leading to improved ballistic correction efficiency during flight. Moreover, the results indicate that in the expanded state of the rudder wings, increasing the deflection angle enhances the impact on the bullet's aerodynamic characteristics, resulting in more pronounced changes and stronger correction capabilities. This research provides valuable insights into the aerodynamic characteristics of ballistic-correction bullets.

**Keywords** Aerodynamic characteristics · Numerical simulation · Ballistic-correction · Deflection angle

# **1 Introduction**

After leaving the muzzle, traditional bullets are in an uncontrolled state and are susceptible to various external factors [1], making it difficult to guarantee accuracy [2]. With the development of advanced technologies such as inertial microsystems, new materials, and optoelectronics, ballistic correction techniques widely used in rocket projectiles can also be applied to small-caliber bullets. This makes it possible

H. Zhang · Z. Wu (B)

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China e-mail: wuruinan-1994@njust.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_72

to achieve guided capabilities for bullets. As a type of smart ammunition, ballisticcorrection bullets achieve significantly higher accuracy than conventional bullets through the combined efforts of a guidance section for detection and positioning and an actuation mechanism for controlling the bullet's orientation [3, 4]. For this type of bullet, there have been numerous related studies both domestically and internationally. The University of Gothenburg in Germany has studied a tube-launched self-adaptive ammunition that utilizes piezoelectric ceramic materials to control the deviation of the bullet. Sandia National Laboratories in the United States has designed a tail-fin deflection smart bullet and conducted live-fire tests. Jonathan et al. proposed a new concept for the actuation mechanism of bullets used in light weapons based on the technique of transforming the center of mass. Chen Ying et al. studied the effect of tail wings on micro-guided projectiles using numerical simulations of the external flow field. Liu Wei et al. proposed a folding duck tail mechanism based on small-caliber smart ammunition and demonstrated the reliability of the folding rudder mechanism. Zhou et al. [5] studied the influence of changes in the external shape dimensions of the projectile tail on its aerodynamic characteristics based on computational fluid dynamics. Liu et al. [6] proposed a folding duck tail mechanism based on small-caliber smart ammunition and demonstrated the reliability of the folding rudder mechanism.

Studying the aerodynamic characteristics of ballistic correction bullets allows for more accurate acquisition of the external ballistics performance parameters of the bullets. This, in turn, enables further optimization and calculation of the external ballistics to achieve higher shooting accuracy [7, 8]. With the rapid development of computational fluid dynamics, numerical simulation has become an important tool for studying the aerodynamic characteristics of projectiles and arrows. Liu et al. [9] conducted an analysis of the aerodynamic characteristics of a two-dimensional correction projectile using fluid dynamics methods and examined the influence of rudder deflection angle. They concluded that the lift coefficient is positively correlated with the equilibrium angle of attack and rudder deflection angle. Zhong et al. [10] based on numerical simulation methods, analyzed the lateral aerodynamic effects of differential control surfaces on rotating two-dimensional ballistic correction projectiles. The results indicated a linear relationship between the cross-stream force, yawing moment, and angle of attack. Shao et al. [11], using combined fluid dynamics and dynamics simulation, analyzed the impact of different rudder heights and deflection angles on the correction ability of canard correction mechanisms. They found that the range and elevation of the projectile were positively correlated with the rudder deflection angle. Kou et al. [12], based on fluid dynamics methods, analyzed the influence of canard fins on the aerodynamic characteristics of spinning missiles. They found that canard fins can provide a greater rolling moment to some extent, and the rolling moment is positively correlated with the height of the canard fins. Zhang et al. [13], based on numerical analysis, studied the aerodynamic characteristics of a movable canard on mortar projectiles. The results showed that the control surface force increases with the deflection angle and Mach number. The aforementioned studies mainly focus on large-caliber rocket projectiles and highlight the significance of analyzing aerodynamic characteristic parameters for enhancing projectile performance. Similarly, investigating the external flow field aerodynamic characteristics of small-caliber ballistic correction bullets is also crucial for improving bullet performance.

In this paper, numerical simulation method is adopted to analyze and calculate the aerodynamic characteristics of a ballistic-correction bullet, and the aerodynamic characteristics of the rudder wing in the unexpanded state and the rudder wing in the expanded state are obtained respectively, and the effect of the change of the rudder wing's deflection angle on the aerodynamic characteristics of the projectile is analyzed.

### **2 Model Building**

### *2.1 Shape of the Bullet*

This paper investigates a bullet that can perform ballistic correction, and the overall structure includes a guide head, a circular arc section, a cylindrical section, rudder wings, a contraction and expansion section, tail wings and other parts, and the overall structure is shown in Fig. 1. The rudder wing, as the actuator of the projectile, is located in the cylindrical part of the projectile. Through the expansion of the rudder wing to change the aerodynamic shape of the bullet, so as to realize the ballistic correction. When the rudder wing is not unfolded, the rudder wing can be closed inside the bullet, and the rudder wing will be opened when two-dimensional correction of the ballistic trajectory is required.

**Fig. 1** Schematic diagram of bullet structure

(1)

### *2.2 Grid Partitioning and Parameter Setting*

Due to the intricate geometry of the bullet model at its tail and the presence of four symmetrically arranged tail fins, the computational domain for this study is partitioned using the Poly-Hexcore mesh. This particular mesh type offers an optimal trade-off between mesh quality and memory usage, yielding comparable accuracy to polyhedral meshes while considerably improving solution speed and demonstrating excellent adaptability to complex geometries. Furthermore, the volume mesh undergoes appropriate refinement in regions characterized by significant variations in flow field or fluid variable gradients, including the bullet head, wing sections, and nearfield shock wave areas. To better emulate real-world conditions, a rectangular outer flow field calculation domain is adopted, encompassing 1000 mm ahead of the bullet, 5000 mm behind the bullet, and 1500 mm surrounding the bullet. In order to capture the fluid characteristics throughout the flight process more effectively, a boundary layer with a y+ value of 1 is implemented on the surface of the bullet. The first layer of the boundary layer has a height of 0.0013 mm, and a total of 20 layers are employed, exhibiting a grid growth rate of 1.2, resulting in an approximate total of 6 million grids. The computational domain mesh around the bullet body is shown in Fig. 2.

Calculation using three-dimensional compressible flow Reynolds-averaged Navier–Stokes (RANS) equations, with its non-dimensionalized form as follows.

∂ *Q*  ∂*t*  + ∂ *E*  <sup>∂</sup>*x* <sup>+</sup> ∂ *F*  ∂*y*  + ∂*G*  ∂*z* = <sup>∂</sup> *<sup>E</sup>*<sup>v</sup> <sup>∂</sup>*x* <sup>+</sup> ∂ *F*<sup>v</sup> ∂*y*  + ∂*G*<sup>v</sup> ∂*z* 

**Fig. 2** Computational grid for ballistic correction bullet

In the formula: *Q* represents the conservation vector; *E*, *F*, *G* represent the inviscid fluxes in the *x*, *y*, and *z* coordinate directions respectively; *Ev*, *Fv*, *Gv* represent the viscous fluxes in the *x*, *y*, and *z* coordinate directions respectively.

The selection of a turbulence model is crucial for accurately simulating the flow field. In this study, the SST *k*-ω turbulence model is employed for calculations. This model takes into account the exchange of shear stress within the turbulent flow, resulting in more realistic simulation effects for both near-wall regions and regions far from the wall. The specific equations are as follows:

$$\begin{cases} \frac{\partial u\_i}{\partial x\_i} = 0\\ \frac{\partial \boldsymbol{x}\_i}{\partial t} + \boldsymbol{u}\_j \frac{\partial \boldsymbol{u}\_j}{\partial \boldsymbol{x}\_j} = -\frac{1}{\rho} \frac{\partial \rho}{\partial \boldsymbol{x}} \left( \upsilon \frac{\partial \boldsymbol{u}\_i}{\partial \boldsymbol{x}\_j} \right) \\ \frac{\partial}{\partial t} (\rho \boldsymbol{u}\_i) + \frac{\partial}{\partial \boldsymbol{x}\_i} (\rho \boldsymbol{u}\_i \boldsymbol{u}\_j) = -\frac{\partial p}{\partial \boldsymbol{x}\_i} + \frac{\partial}{\partial \boldsymbol{x}\_j} \left( \Gamma \frac{\partial \boldsymbol{u}\_i}{\partial \boldsymbol{x}\_j} \right) + S\_i \\ \frac{\partial}{\partial t} (\rho \boldsymbol{k}) + \frac{\partial}{\partial \boldsymbol{x}\_j} (\rho \boldsymbol{k} \boldsymbol{u}\_j) = \frac{\partial}{\partial \boldsymbol{x}\_j} \left( \Gamma\_k \frac{\partial \boldsymbol{k}}{\partial \boldsymbol{x}\_j} \right) + \overline{G}\_k - Y\_k + S\_k \\ \frac{\partial}{\partial t} (\rho \boldsymbol{\omega}) + \frac{\partial}{\partial \boldsymbol{x}\_j} (\rho \boldsymbol{\omega} \boldsymbol{u}\_j) = \frac{\partial}{\partial \boldsymbol{x}\_j} \left( \Gamma\_w \frac{\partial \boldsymbol{\omega}}{\partial \boldsymbol{x}\_j} \right) + G\_w - Y\_w + D\_w + S\_w \end{cases} (2)$$

In the equation: *xi* and *xj* represent spatial coordinates (*i*, *j* = 1, 2, 3); *t* denotes time; ρ represents air density; ν is the kinematic viscosity; *p* stands for pressure; *k* and ω respectively represent turbulent kinetic energy and turbulent dissipation rate; Г, Г*<sup>k</sup>* , and Г<sup>ω</sup> are effective diffusion coefficients for velocity u (*v* or *w*), turbulent kinetic energy *k*, and turbulent dissipation rate ω; *G*¯ *<sup>k</sup>*and *G*<sup>ω</sup> are production terms for *k* and ω; *Yk* and *Y*<sup>ω</sup> are dissipation terms for *k* and ω; *D*<sup>ω</sup> represents the cross-diffusion term; *Si*, *Sk* , and *S*<sup>ω</sup> are user-defined terms for various transport equations.

In the process of numerical computation, both the governing equations and the turbulence model are discretely solved using a second-order upwind scheme. The fluxes are discretized using the Roe-Flux Difference Splitting (Roe-FDS) method, which offers higher numerical simulation accuracy.

The pressure far-field boundary condition is selected for the boundary conditions, and the surface of the body is set as a no-slip wall boundary. The initial condition is set as a far-field inflow at the outer boundary, allowing the flow to start from the outer boundary and move towards the inner boundary.

Other settings are as follows: the solver is selected as a density-based solver using an explicit scheme. The fluid physical properties are defined as an ideal gas, and viscosity is calculated using the three-coefficient Sutherland's law. The relaxation factors are set to their default values.

# **3 Aerodynamic Characterization of Rudder Wing in Undeployed Condition**

To achieve a more accurate simulation of the bullet's aerodynamic characteristics, the flight Mach number of the bullet is deliberately chosen to range from 0.5 to 2.25 Ma, encompassing the entire spectrum of subsonic, transonic, and supersonic regimes. Moreover, the bullet predominantly operates within a near-zero or small angle of attack regime, leading to the selection of 0°, 1°, 2°, 3°, and 4° as the designated angle of attack values. By meticulously examining and comparing the drag coefficient, lift coefficient, and pitching moment coefficient, it becomes possible to discern and comprehend the underlying trends in the evolution of the bullet's aerodynamic properties. The specific configurations and corresponding parameters employed for this investigation are presented in Table 1.

### *3.1 Analysis of Aerodynamic Parameter Characteristics*

Figure 3 shows the variation curve of drag coefficient with Mach number and angle of attack under the rudder wing unexpanded state, from which it can be seen that the drag coefficient of the bullet shows a trend of increasing and then decreasing with Mach number, and the drag coefficient grows the most rapidly when the flight speed of the bullet grows from 0.75 to 1.03 Ma, and the drag coefficient reaches the maximum at 1.25 Ma. With the increase of the angle of attack, the drag coefficient also shows an increasing trend, but the change is small.

Figure 4 shows the variation curve of the lift coefficient with Mach number and angle of attack under the rudder wing not extended state, from which it can be seen that when the angle of attack is 0°, the lift of the bullet is basically 0, and the lift coefficient increases gradually with the increase of the angle of attack, and the approximate linear trend is presented. When the angle of attack is certain, with the increase of Mach number, the lift coefficient shows the trend of increasing and then decreasing, and the lift coefficient reaches the maximum value at 1.03 Ma.


**Table 1** Model calculation parameters

**Fig. 3** Drag coefficient variation curve

**Fig. 4** Lift coefficient variation curve

Figure 5 shows the variation curve of the pitching moment coefficient with Mach number and angle of attack under the rudder wing not extended state, from which it can be seen that when the angle of attack is 0°, the pitching moment of the projectile is basically zero, and with the increase of the angle of attack, the pitching moment coefficient also increases gradually, presenting the same trend as that of the coefficient of lift; when the angle of attack is certain, the pitching moment coefficient shows the tendency to increase and then decrease with the increase of the Mach number, and the maximum value of pitching moment coefficient is reached at 1.03 Ma when the pitching moment coefficient reaches the maximum value. The pitching moment characteristic of the bullet mainly reflects the static stability of the bullet, and the larger the pitching moment coefficient is, the better the static stability of the bullet is.

In summary, when the rudder wing of the bullet is not extended, the aerodynamic characteristics of the bullet are basically consistent with those of ordinary bullets, but due to the design of the blunter head and tail wings, the bullet has a larger drag coefficient and lift coefficient, and good longitudinal static stability.

**Fig. 5** Pitching moment coefficient variation curve

# **4 Aerodynamic Characterization of Rudder Wing in Deployed Condition**

Due to the insignificant impact of the variable factors on the respective Mach numbers, we opt for flight velocities of 0.5 Ma, 1.03 Ma, and 1.5 Ma to represent the subsonic, transonic, and supersonic flight regimes, respectively. Subsequently, we conduct simulation calculations on bullets with deflection angles of 1°, 3°, 5°, and 7°, while analyzing the influence of the deflection angle on the aerodynamic characteristics of the bullet.

Figure 6 shows the variation of the cross-stream force coefficient and yawing moment coefficient of the bullet with Mach number and angle of attack for different rudder deflection angles when the rudder wing is subjected to lateral correction. From Fig. 6a, it can be seen that the l cross-stream force coefficient of the bullet decreases and then increases with the increase of Mach number at different rudder deflection angles, and gradually increases with the increase of angle of attack. The special thing is that at the rudder deflection angle of 1°, the correction direction of the bullet is opposite to other directions. From Fig. 6b, it can be seen that the yawing moment coefficient of the bullet first increases and then decreases with the increase of Mach number, and increases with the increase of the angle of attack, and when the rudder deflection angle is 7°, the yawing moment coefficient of the bullet shows a faster decreasing trend with the gradual increase of the angle of attack. The larger the cross-stream force coefficient and yawing moment coefficient, the stronger the lateral correction ability of the bullet.

Figure 7a shows the curves of lift coefficient variation with rudder deflection angle for different Mach numbers when the rudder elevator is expanded. Where the rudder deflection angle 0° corresponds to the case when the rudder is not expanded. From Fig. 7a, different rudder deflection angles have more different effects on the lift of the bullet. When the rudder wing is not expanded, the lift of the bullet is smaller than the rudder deflection angles of +1° and −1° at all three Mach numbers. When the bullet flight speed is 0.5 Ma, and the rudder deflection angle is negative, with the increase of

**Fig. 6** Cross-stream force coefficient and yawing moment coefficient variation curves

the rudder wing deflection angle, the lift coefficient of the bullet gradually decreases, and the negative lift provided by the rudder wing for the bullet gradually increases, in which the rudder deflection angles of −5° and −7°, the lift coefficient of the bullet is smaller than that of the case of the rudder wing is not expanded, and the bullet has a tendency to fall faster; when the rudder deflection angle is positive, with the increase of the rudder deflection angle, the lift coefficient of the bullet gradually increases, with the increase of the rudder wing deflection angle, the lift coefficient of the bullet gradually increases, with the increase of the rudder wing deflection angle. When the rudder deflection angle is positive, with the increase of rudder deflection angle, the lift coefficient of the bullet gradually increases, and the positive lift provided by the rudder wing gradually increases, and the bullet has an upward trend. When the flight speed of the projectile is 1.03 and 1.5 Ma, the change of the lift coefficient of the bullet has the same trend, when the rudder deflection angle is negative, the rudder deflection angle increases, the lift coefficient decreases, and the rudder provides additional negative lift for the bullet when the rudder deflection angle is −7°. When the rudder wing deflection angle is positive, the rudder deflection angle increases and the lift coefficient decreases, and when the rudder deflection angle is +1°, the rudder wing still provides positive lift for the bullet.

Figure 7b shows the variation curves of the lift coefficient with the rudder deflection angle at different angles of attack. From the figure, when the angle of attack is 0°, the total lift of the bullet is 0 when the rudder wing is not expanded; when the rudder deflection angle is negative, the rudder wing will provide positive lift at −1° rudder deflection angle, and then the bulled lift decreases as the rudder wing deflection angle increases; when the rudder deflection angle is positive, the rudder wing will provide negative lift at +1° rudder deflection angle, and then the bullet lift increases as the rudder wing deflection angle increases; the bullet lift increases as the rudder wing deflection angle increases. Angle of attack of 2° shows the same tendency of angle of attack of 0°. The bullet at angle of attack of 2° shows the same trend as that at angle of attack of 0°. At angle of attack of 4°, with the increase of rudder deflection

**Fig. 7** Lift coefficient variation curve

angle, the bullet lift shows a decreasing trend, and at rudder deflection angles of − 7°, +5° and +7°, the bullet has additional negative lift.

The variation curves of the pitching moment coefficient with the deflection angle of the rudder wing at different Mach numbers when the rudder wing elevator and rudder are deployed are shown in Fig. 8a. As can be seen from the figure, the pitching moment coefficient of the bullet maintains a similar trend at different Mach numbers. The pitching moment coefficient of the bullet gradually decreases when the rudder deflection angle is negative to the maximum angle and deflected to the positive to the maximum angle. The pitching moment coefficient of the bullet increases when the bullet is flying at the speeds of 0.5 and 1.03 Ma without rudder wings expanded, and the pitching moment coefficient of the bullet decreases at the speed of 1.5 Ma. When the deflection angle of the rudder wing is +7°, the pitching moment coefficient is greater than 0. Since the lift is positive at this time, the pressure centre of the bullet is located in front of the centre of mass, and the static stability of the bullet can not be guaranteed at this time, and it is prone to out-of-control phenomenon. In the longitudinal comparison, when the speed of the bullet is 1.03 Ma, the pitching moment coefficient of the bullet is the largest, which corresponds to the better flight stability of the bullet at this time.

Figure 8b shows the variation curves of the pitching moment coefficient with the rudder wing deflection angle at different angles of attack. The larger the rudder wing deflection angle, the larger the additional pitching moment provided to the bullet, and when the rudder deflection angle is negative, the pitch-down moment of the bullet becomes larger, and when the rudder deflection angle is formal, the pitch-up moment becomes larger. When the angle of attack is 0°, the change of moment provided by the negative rudder deflection is more obvious, and the bullet is in static instability when the rudder wing deflection is positive; when the angle of attack is 4°, the change of moment provided by the positive rudder deflection is more obvious.

In summary, the different deflection angle of the rudder wing will have a greater effect on the force and moment of the bullet, and the larger the deflection angle of the rudder wing, the greater the ability of the rudder wing to correct the ballistic of

**Fig. 8** Pitching moment coefficient variation curve

the bullet, and at the same time, the different deflection angle of the rudder wing will also have a greater effect on the static stability of the bullet.

### **5 Conclusion**

In this paper, for a certain ballistic correction bullet, the aerodynamic characteristics of the bullet are investigated by numerical simulation, the changes of the aerodynamic characteristics of the bullet in the unexpanded and expanded states of the rudder wing are investigated respectively, and the influence of the rudder wing deflection angle on the aerodynamic characteristics of the bullet is analysed, and the conclusions are as follows:


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **High Temperature Creep Property of Nanoparticle Reinforced Composite by Discrete Dislocation Dynamic Method**

**Jian Wang, Zhengwei Zhang, Kai Zhang, and Hua Zhang** 

**Abstract** Metal-matrix composites (MMCs) reinforced by nanoparticles exert much fine mechanical performance such as high strength, high modulus and good conductivities. The creep property of nanoparticle reinforced MMCs is investigated by discrete dislocation dynamic (DDD) method in this study. Both motions of dislocation glide and climb are involved in the present scheme and a dual time step strategy is adopted to deal with the velocity gap between dislocation glide and climb. The results reveal a transition of creep mechanism as the creep stress increases, and a negative correlation between the threshold stress and the environmental temperature.

**Keywords** Metal-matrix composites · Creep · Discrete dislocation dynamics · Dislocation climb

# **1 Introduction**

Metal-matrix reinforced with elastic particles composite materials have drawn much interests due to their high strengthss, high modulus and good conductivities, which can be served effectively for the extreme environments, especially at elevated temperature [1–4]. It has been found that the mechanical performance of metal-matrix composites (MMCs) tends to be improved with the size of particles decreasing, especially when the particle size reduces to nanometer scale [5]. It was found that the tensile strength of aluminum-matrix composites with 1 vol% nanoparticles is equivalent to that with 15 vol% microparticles [6]. The creep property of particulatereinforced MMCs at high temperature has also been investigated by experimental tests [1–5, 7], and the results reveal high values of stress exponent and activation energy held in these composite materials. By introducing the concept of threshold

927

J. Wang (B) · Z. Zhang · K. Zhang · H. Zhang

China Ship Science Research Center, Wuxi 214082, China e-mail: wangjianhj@cssrc.com.cn

Z. Zhang · K. Zhang · H. Zhang Taihu Laboratory of Deep Sea Technological Science, Wuxi 214082, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_73

stress σ0, the creep data can be well rationalized to the values for the common alloys, and thus the power law relation for creep of MMCs can be regulated as the form [8]:

$$
\dot{\varepsilon} \propto (\sigma - \sigma\_0)^n \tag{1}
$$

where ε˙ is the creep strain rate, σ the applied stress and *n* the stress exponent. It has been pointed out that different values of the stress exponent *n* corresponds to different creep mechanism [5]: *n* = 3, 5, 8 corresponds to the mechanism of viscous dislocation glide controlled creep, dislocation climb controlled creep and lattice diffusion controlled creep, respectively. And the threshold stress σ0 in Eq. (1) is proved to be strongly dependent on temperature by in-situ creep experiments [5].

To figure out the intrinsic mechanical creep mechanisms of nanoparticle reinforced MMCs at elevated temperature, numerical researches are urgently required. Discrete dislocation dynamics (DDD), which tracks the motions of dislocations, can be taken as an effective numerical method to study the mechanical mechanism of micro/nano materials. Since the size of dislocations in the matrix and the size of particles are both fallen into the range of mesoscale, DDD method is suitable to model the creep deformation. In practical, the plastic deformation of MMCs almost originates from the plastic deformation in matrix while the particles are usually assumed to be rigid, so that the interactions between dislocations and particles are critical in the simulation. In recent years, DDD method has been widely applied to investigate the mechanical behaviors of MMCs, such as particle reinforced single crystals [9] and nickel-based single crystal superalloy [10]. Traditional DDD methods only take dislocation glide motion into account, as done by [11–15]. Yet at high temperature the motion of dislocation climb coupled with vacancy diffusion also exerts dominated effects on the plastic deformation of metal materials [16–19]. The time separation between dislocation glide and climb is the major challenge held in DDD simulation [16, 20–22], which dues to that the glide velocity is about 4~5 magnitudes larger than the climb velocity. For creep simulations, an adaptive, staggered time stepping approach is adopted by several researcheres [5, 10, 20, 21] to simulate the creep behavior of single crystal materials and nickel-based single crystal superalloy. This approach assumes the motions of dislocation climb and glide are carried out separately and a frozen state for dislocations is attained when dislocation climb is under modeling, which is appropriate because the applied loading is fixed at a desired value.

However, litter work has been conducted to investigate the creep behaviors and the mechanical mechanism of nanoparticle reinforced MMCs. Thus, this paper focus on the creep behavior of nanoparticle reinforced MMCs at elevated temperature by DDD method, and the results of the creep simulations are carefully analyzed.

### **2 Theoretical Method**

A two-dimensional (2D) single crystal model containing numerous randomly distributed nanoparticles is built, with an aspect ratio *H*/*D* = 2, as showed in Fig. 1. Pure edge dislocations are assumed to glide on two slip systems oriented at ±60◦, and the motion of dislocation climb is also involved to assist the dislocations cross the nanoparticles. The climb velocity for dislocations can be given by [20, 21]:

$$V\_c = -\frac{2\pi D\_v}{b\ln\left(\left(\sqrt{D\_v t^i} + b\right)/b\right)} \left[\mathbf{C\_0} \exp\left(-\frac{f\_c \Omega}{bkT}\right) - \mathbf{C}(\mathbf{R}\_i, t)\right] \tag{2}$$

where *D*<sup>v</sup> = *D*0 exp( <sup>−</sup> *Em kT* ) is the diffusion constant with *Em* the migration energy of vacancy, and *b* is the magnitude of burgers vector, Ω the atomic volume, and *fc* = **s** · σ · **b** the Peach–Koehler force on the climb direction, with **s** the unit vector along the slip direction and **b** the burgers vector. The parameter **C0** <sup>=</sup> exp( <sup>−</sup> *E f kT* ) is the equilibrium vacancy concentration at temperature *T* , with *E f* the formation energy of vacancy, *k* the Boltzmann constant. As showed in Eq. (2), *C*(*Ri* , *t*)in the last term of the bracket represents the remote vacancy concentration for *i* th dislocation, which can be obtained by interpolating the vacancy concentration fields at time *t* to the position of *i* th dislocation. The diffusion equation of vacancy concentration can be given by [15, 20]:

$$\begin{cases} \dot{C} = D\_v \nabla^2 C - \frac{D \Omega\_v}{kT} \nabla \cdot (C \nabla p) \ln V \\\\ C = C\_0 \exp\left(\frac{\sigma\_0 \Omega}{kT}\right) \text{on } \partial V \end{cases} \tag{3}$$

where Ωv is the vacancy relaxation volume, *p* the hydrostatic stress, and σ0 is the applied stress, which can be taken as the creep stress in the present creep problem. To account for the contribution of dislocation climb on the vacancy concentration fields,

**Fig. 1** 2D single crystal model containing nanoparticles

an additional term Δ*Cs* = −*b*<sup>2</sup> ∑ *i*∈*Ve si <sup>c</sup>*/*Ve* should be introduced into the numerical module of vacancy diffusion equation. This term Δ*Cs* represents the contributed quantity of vacancy concentration for every dislocation *i* in element *Ve* during the climb process, which should be distributed into the mesh nodes of element *Ve* at the end of staggered climb step as means of initial conditions for next climb process.

Besides climb, dislocations can also cross the particles by the thermally activated mechanism: dislocations are allowed to jump over the particles and continue to glide if the thermally activated probability *P* satisfies [20]:

$$P = \exp\left[-\frac{\Delta F}{kT}\left(1 - \frac{\left|f\_{\S}\right|}{\tau\_p b}\right)\right] \le RN \tag{4}$$

where *RN* ∈ [0, 1] [0, 1] is a random number generated at every glide increment, Δ*F* = 2μ*b*3 the activation energy for dislocations to jump over the nanoparticles, and τ*p* <sup>=</sup> <sup>μ</sup>*<sup>b</sup> <sup>l</sup>*= μ*b* √ρ represents the athermal strength for particles, with *l* the particle spacing and ρ the particle density, respectively. As described above, the time separation exists between the motions of dislocation glide and climb. In the present scheme, a dual time step procedure is adopted to overcome this challenge. Two time steps are introduced into the model: dislocation glide is simulated with time step Δ*tg* while dislocation climb as well as the evolutions of vacancy concentration fields is simulated with a lager time step Δ*tc*, and the detailed strategy is stated below: the nanoparticle reinforced composite structure containing numerous initial dislocations and sources is constructed and performed under a stress free condition to relax the microstructure of dislocations until the system reaches an equilibrium state. In this process, only the motion of dislocation glide is considered with the glide velocity *Vg* <sup>=</sup> *fg <sup>B</sup>*, where *fg* = **n** · σ · **b** is the Peach–Koehler force along the slip direction and *B* = 3.2 μ Pa · s is the drag coefficient, and the time step is set to be Δ*tg*; a uniform strain-controlled load is applied on the vertical direction until reaching the desired creep stress. The motion of dislocation glide as well as the short range effect between dislocations like dislocation nucleation, annihilate is simulated. Dislocations can cross the obstacles only by the thermally activated mechanism in this glide-restricted motion. The data transition is holds in this process: the plastic strain induced by dislocation glide is obtained and distributed into the Guassian points of the FEM module, and the stress fields are solved and transferred back to the DDD module to drive the evolutions of dislocations; when the equilibrium state of the system is attained (i.e. the frozen state for dislocations is reached), the dislocations involving the model are assuming to be immobile and the time step is set to be Δ*tc*. This state means the dislocations are obstructed by the nanoparticles, and the motion of climb starts to exert its effects by assisting dislocations cross the nanoparticles. The vacancy diffusion Eq. (3) is solved to get the vacancy concentration fields which would be used to determine the climb velocity of dislocations, and the disturbed term due to dislocation climb is introduced into the diffusion equation after the climb process is ended. When the climb distance of any dislocations reaches the spacing of slip plane, the frozen state for dislocations is relieved. The time step is switched back to Δ*tg* and the iterative step for dislocation glide is repeated.

### **3 Results and Discussion**

In the present simulation, the equilibrium state is considered to be attained if the average strain rate is less than a threshold value over 100 glide increments. The composite is assumed to be elastically isotropic for both matrix and particles, with the elastic constant μ = ( 3 × 10<sup>4</sup>− 16 × *T* ) MPa [23] and ν = 0.35. As noted, the shear modulus for the composite is related to the environmental temperature. The density of Frank-Read sources is set to be 120 μ*m*2. A Gaussian distribution is assigned to the nucleation strength of the sources with the mean strength τ *nuc* = 50 MPa and the standard deviation δτ = 1 MPa; the density of initial dislocations is 80 μ*m*2. The composite contains a distribution of nanoparticles with a density 113 μ*m*2 and diameter 7.5 nm to match the 2% volume fraction for nanoparticles. The time step for glide is Δ*tg* = 0.5 ns. Compared with the velocity of dislocation glide, the dislocation climb velocity is much slower so that the time step for dislocation climb Δ*tc* is at least 102 larger than the glide time step. The material parameters for vacancy diffusion are given by: *D*0 = 1.51 × 10−5 m2/s, *E f* = 0.67 ev, *Em* = 0.61 ev.

The tensile responses for the nanoparticle reinforced MMCs are first studied, and the results are obtained by executing the simulations without dislocation climb being considered under the applied strain rate ε = 100 s−1. The stress–strain curves of the model with *D* = 2 μ*m* are plotted in Fig. 2a, with and without nanoparticles containing in the model both being considered for comparison. As seen, the values of yields stress for the model with nanoparticles and without them are 68 MPa and 50 MPa, respectively. The determination of yields stress provides the range for choosing creep stress for nanoparticle reinforced MMCs, that is, the desired creep stress should be lower than the yields stress. The strain hardening is captured for both cases, which is common for small-size materials. In addition, it can be clearly found in Fig. 2a that the both the initial yields stress and the flow stress are enhanced when the nanoparticles are involved in the model, and this is because the existence of nanoparticles improves the resistance to dislocation glide, leading to the strengthening of materials. The evolutions of dislocation density plotted in Fig. 2b indicated that the evolution rates of dislocation density are larger in the model with nanoparticles contained than that without nanoparticles, and this is due to the block effects of nanoparticles on dislocations keeping more dislocations inside the model.

Figure 3a-c shows the creep response of MMCs with same size for different values of the creep stress in the range 20~30 Mpa at temperature *T* = 673 K, *T* = 623 K and *T* = 573 K. The values of creep stress chosen for nanoparticle reinforced MMCs are all lower than the yields stress, and the computation strategy described above is carried out to obtain these results. Primary creep (stage 1) and steady state creep (stage 2) are both captured in Fig. 3. With the temperature being fixed, the creep

**Fig. 2 a** Stress–strain curves and **b** Evolutions of dislocation density for model with and without nanoparticles contained

strain is higher for higher creep stress at a given time. As the temperature rises, the creep strain at a given creep stress is higher at higher temperature. In addition, at any one of these three chosen temperatures the steady creep rate (creep rate in stage 2) is small when the creep stress is small while in the high stress region the steady creep rate turns to be extremely large.

To obtain the stress exponent *n*, the logarithmic variations of steady creep rate versus stress for nanoparticle reinforced MMCs at different values of temperature are plotted in Fig. 3d and the scatter for the obtained creep rates is under calculations at least three realizations of the initial configurations of dislocations, sources and obstacles. A two stage creep is captured for all temperatures chose: in the low stress region, the steady stress exponent *n* is approaching to 2; in the high stress region, the steady stress exponent *n* exceeds 10. For those of pure aluminum and solid solution aluminum alloys, the stress exponent *n* for creep is fallen in the range *n* = 3 ∼ 5 [22], which is higher than the value in the low stress region and smaller than the value in the high stress region. This suggests the existence of threshold stress, which is resulted from the effective barriers of nanoparticles for dislocation motion. In the low stress region, dislocations cannot break away the obstruction of nanoparticles and the climb rate of dislocations is also slow, while in the high stress region, the thermally activated probability as well as the climb rate of dislocations increases significantly. The threshold stress at every temperature can be obtained by linearly fitting the ε˙<sup>1</sup>/*<sup>n</sup>*∼ σ in high stress region, where *n* = 3, 5, 8 corresponding to viscous dislocation glide controlled creep, dislocation climb controlled creep, and lattice diffusion controlled creep, respectively. In the present simulation, *n* is chosen to be 5 because neither viscous dislocation glide deformation mechanism nor lattice diffusion deformation mechanism is considered. With *n* being fixed, the linearly fitting lines ε˙<sup>1</sup>/*<sup>n</sup>*∼ σ can be get and the threshold stress yields the value of stress at zero strain rate (i.e. the stress of the intersection point between the fitting line and the line ε˙ = 0). According to this method, the values of threshold stress at temperature

**Fig. 3** Creep curves for nanoparticle reinforced MMCs at **a** *T* = 673K, **b** *T* = 623K and **c** *T* = 573K, and **d** logarithmic plots of ε˙ ∼ σ

*T* = 673 K, *T* = 623 K and *T* = 573 K are 38.25 MPa, 41.86 MPa and 42.48 MPa, respectively. It is clear that that the threshold stress tends to decrease with increasing temperature.

### **4 Conclusion**

In summary, the creep property of nanoparticle reinforced MMCs at high temperature is investigated by DDD method in this paper. Vacancy diffusion coupled dislocation climb is involved in the scheme and a dual time step strategy is adopted to overcome the velocity gap between dislocation climb and glide. The results show a transition of creep mechanism as the creep stress increases and it is revealed that the threshold stress tends to decrease with increasing temperature.

**Acknowledgements** Authors wishing to acknowledge assistance from colleagues and financial support from Jiangsu Provincial Double-Innovation Doctor Program Grant No. JSSCBS20221782.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Experimental Studies on the Load Characteristics of Low-Speed Droplets Impinging onto Surface**

**Guandong Li, Qiulin Qu, and Peiqing Liu** 

**Abstract** Droplet impingement on a wall is a fundamental scientific problem with wide engineering applications. When a droplet impacts the surface of an aircraft, it generates shock waves, airflow disturbances, and splashing phenomena. This not only has a negative impact on the aerodynamic performance and stability of the aircraft but also obstructs the field of view of optical sensors or causes distortion in optical devices. It can also damage the aircraft's structure, thus it's vital to assess the droplet impact force for flight safety. However, droplets are often treated as rigid spheres for simplicity, but this does not reflect the real physical situation. In this paper, we utilized high-precision force sensors and high-speed imaging technology to experimentally investigate the impact dynamic of droplet impingement on a dry wall. The temporal evolution of force, the associated morphology changes and their relationship during collisions were analyzed systematically, we also elucidated the physical mechanisms underlying flow phenomenon. An unified and accurate mechanical model were established for droplet impingement, providing guidance for related engineering designs.

**Keywords** Drops · Impact dynamics · Force model

G. Li (B)

Q. Qu · P. Liu Key Laboratory of Fluid Mechanics Ministry of Education, Beihang University, Beijing 100191, China

Key Laboratory of Aero-Acoustics Ministry of Industry and Information Technology (Lu Shijia Laboratory), Beihang University, Beijing 100191, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_74

Beijing Electro-Mechanical Engineering Institute, Beijing 100074, China e-mail: zy2005309@buaa.edu.cn

### **1 Introduction**

The collision between a liquid drop and a dry substrate is a dynamic, complicated physical process and widespread in nature, industrial and agricultural production, therefore it has received extensive attention from scientists. As a common basic scientific problem in many engineering applications, various outcome of drop impact [1–3] and complex phenomenon [4–6] during collision have been investigated widely. Nowadays we have already acquired a systematic understanding of these phenomenon and underling mechanisms.

The impact force exerted by droplets plays an important role in aerospace, energy, chemical and other technical fields: the take-off and landing of aircraft on waterpolluted runways, aircraft/missiles flying in rainy circumstance, the erosion of soil and steam turbine blades. In these fields, the impact force is an essential factor for designing materials and structures and has become one of the most concerned issues for engineers. However, there is a relative scarcity of experimental studies that specifically examine the evolution of impact force, there are still numerous cognitive gaps.

Impact force is can be written as *F* = *f* (ρ , *V*0, *D*0, μ, σ, *t*), where ρ , *V*0, *D*0, σ, μ is droplet density, impact velocity, droplet diameter, surface tension and dynamic viscosity respectively. A majority of the existing studies have primarily concentrated on analyzing the maximum impact force exerted by liquid drops [7–9]. Such measurements revealed that impact forces increase with the increase of droplet diameter, density and velocity, which is in accordance with intuitions. Dimensionless analysis indicates that *F* ˜ <sup>=</sup> *<sup>f</sup>*(*Re*, *W e*, τ ), where *Re* <sup>=</sup> <sup>ρ</sup>*V*0 *D*/μ, *We* <sup>=</sup> <sup>ρ</sup>*V* <sup>2</sup> <sup>0</sup>*D*/σ, *<sup>F</sup>*˜ <sup>=</sup> *<sup>F</sup>*/ρ*V* <sup>2</sup> 0 *D*<sup>2</sup> <sup>0</sup> , τ = *V*0*t*/*D*0.

Zhang et al. [9] investigated the influence of *Re* and *We* on the peak forces systematically for the first time. It has been shown that the dimensionless impact force is hardly influenced by Weber number, whereas the Reynolds number is the key factor that affects impact behaviors, the conclusion were reinforced in subsequent studies [10, 11]. Therefore, impact force characteristics are only related to Reynolds number, namely:

$$
\tilde{F} = f(Re, \mathfrak{r}) \tag{1}
$$

Impact regimes can be categorized into two zones by Reynolds numbers: an inertial-dominated and a viscous regime. The value of critical *Re* for classification is between 200–280 [9–11]. When considering finite Reynolds numbers, the viscous effects cannot be disregarded, the force evolution varies with the Reynolds number, however, the normalized peak force collapse into a universe curve at high *Re*. Experimental results show that the impact force first increases to the peak value in a short time, and then decays to zero slowly during the impact duration. Philippi et al. [12] analyzed the semi-similar flow structure and deduced the normal force of liquid droplet impinging onto the wall at early times, the dimensionless form is given by:

Experimental Studies on the Load Characteristics of Low-Speed … 939

$$\text{cerists of Low-Speed }\dots\tag{939}$$

$$\widetilde{F} = \frac{3\sqrt{6}}{2}\sqrt{\pi}\tag{200}$$

However, the analytical solution is only valid for a very short time (τ < 0.03), it deviates rapidly from measurements as time increased. In the refined research of Mitchell et al. [10], they developed an empirical model that represents the entire force evolution of droplet impacts for high *Re*: √

$$
\tilde{F} = \sqrt{\frac{1000\pi\tau}{243}} e^{-10\tau/3} \tag{3}
$$

However, the early time dynamics and exponential decay law phase in impact force require a theoretical understanding, as the maximum impact force occurs during the transition and the underlying physical mechanisms are still uncertain. With regard to impact force dynamics in viscous regime, Gordillo et al. [11] analyzed the early impact force characteristics at finite *Re* via a perturbation method based on Philippi et al. [12], they found that the propagation speed of the self-similar filed increases due to viscous boundary layer in early stages, nevertheless, our understanding of the entire evolution of drop impact forces in viscous regime is still limited and requires further exploration because of the complex nature of the problem. The existence of viscous boundary layer makes it harder to predict impact force in viscous regime theoretically and there is still a lack of model equation to represent the force profiles at finite *Re*.

In this paper, through analyzing the impact forces and the associated morphology change in impact process across different Reynolds numbers, we verify the interaction of inertial force, viscous force and surface tension and their roles played in force– time history, providing a theoretical support for exponential decay law at long times, and an uniform expression is constructed for the scaling of force profiles of different Reynolds numbers due to the similar physical mechanisms and therefore connect the two impact regimes successfully.

### **2 Experiments**

### *2.1 Experimental Setup*

Schematic diagram of the experimental apparatus is shown in Fig. 1. The experiment setup consists of four main systems: three-axis mobile platform, droplet generator and control device, visualization and force measurement system.

Droplets were produced using a high-precision syringe pump, in which stainless steel needles of varying sizes were attached to control the diameter of the droplets. Droplets were released under gravity and impacted a 2.25 g, 25 mm diameter quartz glass plate, the surface was designed to be larger than the droplets' maximum

**Fig. 1** Experimental apparatus; **a** Image, and **b** Schematic

spreading diameter to ensure that the droplets did not flow out of the substrate. The time evolution of the droplet morphology during impact was recorded by a highspeed camera (FASTCAM AX200) at 20,000 frames per second. We used a PCB model 209C01 force transducer to measure force profiles. The sensitivity and force revolution of the transducer were 489.2 mV**/**N and 0.1 mN respectively. The signal acquisition equipment (NIPXLE-4499) was employed and the sample rate was set to 200 kHz.

In the paper, silicone oil with different viscosity (10–1000 cst) and distilled water were used to vary the physical properties of the impacting droplets. The inner and outer diameters of needle were 0.41 mm and 0.72 mm respectively. The fall height was adjusted to change the impact velocity, the silicon oil droplets were kept at 220 mm while water droplets were released from 530 mm approximately, the precise control of x–y plane location made sure of that the droplets fall on the center of the substrate. We synchronized the force evolution and high-speed imaging by triggering the highspeed camera using a photo-interrupter (KEYENCE LV-N10) activated by the falling drops, which enabled us to investigate the kinematics and dynamics of drop impacts simultaneously. Droplet diameter and impact velocity were determined from image analysis using 50 frames up to the last frame before impact. It's noted that droplets deform as they fall and may not be a perfect spherical shape, droplet equivalent diameter is determined by *D*0 = <sup>3</sup> √ *D*2 *<sup>h</sup>DV* [3]*D*<sup>2</sup> *<sup>h</sup>DV* , where *Dh*, *D*<sup>v</sup> denotes droplet horizontal diameter, vertical diameter respectively. The vibration of system resulted from the droplet impact leads to low-amplitude, high-frequency oscillations in force signals, we applied s filter to eliminate small oscillations to enhance its reliability.

The test conditions are listed in Table 1. In the experiment, the temperature of room was controlled 25 °C, we mainly pay attention to the Reynolds number, the advantage of silicon oil lies in its nearly constant surface tension and thus the Weber numbers are kept constant (approximately 36). The accuracy of the experiments was validated by conducting a comparison between the measured impulse and the initial momentum of droplets. Theoretically, we have <sup>∫</sup> <sup>∞</sup> <sup>0</sup>*F*(*t*) *dt* <sup>=</sup> *mV*0 <sup>=</sup> <sup>ρ</sup> · <sup>4</sup> <sup>3</sup>π *R*3*V*0, in dimensionless form: ∫

$$\int\_0^\infty \tilde{F}(\tau) \, d\tau = \frac{\pi}{6} \tag{4}$$


**Table 1** Test conditions of the present experiments

### **3 Results and Discussion**

### *3.1 Impact Dynamics*

In this section the shape deformation of droplets and the corresponding force profiles are combined to investigate the impact dynamics and kinetics during collision. We mainly analyze the interplay of inertial force, viscous force and surface tension, as well as their varying significance in the impact process.

The normalized profiles are displayed in Fig. 2, the results reveal the existence of a viscous and inertial self-similar regime. Dimensionless peak force decreases rapidly at small *Re* number, when *Re* number increases above 200, all force profiles almost collapse into one curve, which has been confirmed by previous studies as well.

Figure 3 exhibits the corresponding temporal evolution of spreading factor and dimensionless apex height for test conditions, the two parameters are defined as <sup>β</sup> <sup>=</sup> *<sup>d</sup>*/*D*0, *h*˜*max* <sup>=</sup> *hmax*/ *<sup>D</sup>*0 respectively, where *d* denotes wetting length. *h*˜*max* is good choice to reflect impact force dynamics since it characterizes the momentum transfer rate between droplets and substrate. The results show that viscosity plays a significant role in preventing droplet deformation at finite *Re*, the formation of liquid sheet is limited due to high radical shear stress near the initial point of contact, the droplet can be approximated as a rigid ball in the scenario where the liquid viscosity approaches infinite, therefore leading to a shorter impact duration and a greater peak force.

The impact behavior of droplets is affected by inertial force, viscous force and surface tension, which is a complex dynamic phenomenon. The entire impingement process can be divided into early kinetic phase and later spreading phase. In kinetic phase, inertial force dominates the initial stage of collision, droplet can be regarded as a truncated sphere in early stages, the apex height follows the same trend regardless of the Reynolds number; viscosity is a key factor in the later spreading phase, the central height slows down gradually and approaches boundary thickness, central height spends less time reaching steady state and remains unchanged,

**Fig. 2** Non-dimensional force profiles; **a** Viscous regime, and **b** Self-similar inertial regime

**Fig. 3** Non-dimensional morphology change; **a** Normalized apex height, and **b** Spreading factor

although the surface tension is responsible for retraction and spreading behaviors such as maximum spreading factor, it exerts no evident effects on impact forces.

### *3.2 Impact Force Model*

Regardless of Reynolds number, the drop keeps falling at the initial-impact velocity in early phases as if it had not experienced any impact at all and early pre-peak force is proportional to the square root of time, this unusual phenomenon can be attributed to the finite propagation speed of the self-similar physical fields at high Reynolds numbers [12], the existence of the viscous boundary layer makes similar pressure field travels faster than that in the inviscid flow, thus the apex velocity deviates in advance, which induces a faster rise of impact force and a shorter peak time [11].

In the later stages of deformation, the inertial force weakens, impact force can be well approximated with an exponential decay, which is verified by several researchers [10, 13]. Here we provide a theoretical support for the conclusion briefly by analogy. Considerating that the surface tension and viscous force dominate the behaviors of impacting liquid drops at long times, which can be equivalent to a spring and a damper respectively in post-peak stages (see Fig. 4). Furthermore, we have acknowledged that force profiles are independent of We numbers and it has little effect on impact forces. Therefore the influence of surface tension during decline stage can be ignored.

For the spring-damper model, the Newton's second law can be expressed as: *m dV dt* + *kV* = 0, where *m* and *k* are the mass of the droplet and damping factor respectively. Simple dimensional analyses indicts that *m* ∼ ρ*D*<sup>3</sup> <sup>0</sup> , *k* ∼ μ*D*0.The impact force exerts on the wall is considered equal to the damping force, thus we have: *d F* ˜ *<sup>d</sup>*<sup>τ</sup> ∼ − *<sup>D</sup>*0*<sup>k</sup> <sup>U</sup>*0*<sup>m</sup>F* ˜ ∼ − <sup>μ</sup> <sup>ρ</sup>*U*0 *D*0 *F*˜. In other words, *d F* ˜ *<sup>d</sup>*<sup>τ</sup> = −*c F*˜, where *c* is a function of Reynolds numbers. At given *Re*, we have: ∫∫

$$\int\_{\tilde{F}\_{\text{decay}}}^{\tilde{F}} \frac{d\tilde{F}}{F} = \int\_{\mathbf{r}\_{\text{decay}}}^{\tau} -c(Re)d\tau \tag{5}$$

τ*decay* , *F*˜ *decay* are recession moment and the corresponding dimensionless impact force respectively, integrating both sides of the equation by separation of variables:

$$
\tilde{F}(\mathfrak{r}) = \tilde{F}\_{decay} \ast e^{-c(\mathfrak{r} - \mathfrak{r}\_{decay})} \tag{6}
$$

Equation (6) indicts the exponential decay law of impact force at long times successfully, furthermore, it predicts that impact force declines faster at relatively low *Re* result from the increase of viscosity and therefore the force curves exhibit a shorter non-dimensional time duration. The speculation is in line with measurements, which proves the effectiveness of the analog damper model. In self-similar regime, droplet impact dynamics is almost independent of *Re* and *c* gradually approaches constant.

In spite of the fact that a self-similar regime and a viscous regime which can be classified by dimensionless peak force, from the perspective of the impact process, the evolution of impact force exhibits similar trends and there are same physical mechanisms across different *Re* numbers during collision: early impact force can be scaled as the square root of time and long time exponential decay. Therefore the empirical model equation form proposed in reference [10] used for inertial impact is still applicable to finite Reynolds numbers:

$$
\tilde{F}(\mathbf{r}) = c\_1 \sqrt{\mathbf{r}} e^{-c\_2 \mathbf{r}} \tag{7}
$$

where the physical meanings represented by *c*1, *c*2 can be regarded as the propagation speed of the similar field in the early stage and the decay rate at long times, and the two parameters vary with *Re* because of the existence of boundary layer.

In early stages, *F* ˜ <sup>∼</sup> *<sup>c</sup>*<sup>1</sup> <sup>√</sup><sup>τ</sup> , the initial-impact theory developed by Gordillo et al. [11] reveals that early impact force can be scaled as 1/ <sup>√</sup>*Re*, and gives the coefficient theoretically using a perturbation method, namely:

**Fig. 5** Comparison of peak force and corresponding peak time over a wide range of Reynolds numbers

$$c\_1 = \frac{3\sqrt{6}}{2} (1 + \frac{8\sqrt{6}}{3\pi^{3/2}} \frac{1}{Re^{1/2}}) \tag{8}$$

The parameter *c*2 can be determined by the fact that the impulse is equal to the initial momentum of the droplet numerically (see Eq. (4)), we then fit the parameter quantitatively as follows:

$$c\_2 = \frac{10}{3}(1 + \frac{0.825}{Re^{1/2}})\tag{9}$$

We can see that the two parameters decrease with the increase of Reynolds number and then gradually tend to constant values, which is in line with our cognition. In order to demonstrate the validity of the force formula to represent the evolution of impact force, the comparison of peak force and corresponding peak time over a wide range of Reynolds numbers are exhibited in Fig. 5, the results reveal that the model equation are in good well with experiments very well when *Re* > 10.

### **4 Conclusion**

We investigate the influence of the Reynolds numbers on the impact dynamics of liquid drops impingement and divide our discussion into two parts: the inertiadominated kinetic phase and long decay regime. An analogous damper model characterizing viscosity-dominant effect at sufficient long times is put forward to explain the exponential decay law successfully.

Based on the fact that impact force is proportional to the square root of time in early stages and obeys exponential decay law at long times regardless of Reynolds number, the differences lies in that it rises more rapid and decays faster at finite *Re*. The similar physical mechanisms underlying flow phenomenon across different Re are explained and thus allow us to develop an unified semi-empirical force model to represent the entire force evolution for various Reynolds numbers,providing guidance for related engineering designs.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Load and Deformation Distribution Along the Bolt During Assembly Process**

**Zhixiang Li, Zhen Zhao, and Jiaying Zhang** 

**Abstract** Bolt connection structure is widely utilized in modern large-scale equipment. Understanding the stress distribution along the bolt during assembly process is essential for improving its reliability. In this paper, assembly torque model is established to formulate the relationship between tightening torque and preload force, and a linear proportionality between them is obtained. The stress distribution during assembly process is modeled. Rotation angle and displacement distribution can be obtained from the proposed method. To validate our model, numerical simulation analysis is carried out to obtain the load and deformation distribution under different conditions. Comparison with the previous literature confirms the accuracy of the proposed model.

**Keywords** Bolt · Deformation distribution · Assembly process · Numerical simulation

# **1 Introduction**

Bolt connection structure is widely utilized in modern large-scale equipment, such as aviation spacecraft, wind turbine spindles, petrochemical pipeline docking and so on. In practical applications, fasteners are inevitably affected by external excitation, and the behavior of the contact surface is extremely intricate. Understanding the stress distribution in the bolt during the assembly process is essential for improving its reliability. However, due to the complexity of inter-thread contact and friction, accurately modeling the bolt assembly process is challenging.

Extensive studies have been done along the bolt contact problem. For the bolt head contact problem, Motosh developed a friction model that accurately establishes the relationship between tightening torque and axial force for threaded fasteners [1]. According to this model, the tightening torque is the sum of the thread contact torque

© The Author(s) 2024

Z. Li · Z. Zhao (B) · J. Zhang

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China e-mail: bhzhaozhen@buaa.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_75

and the bolt head friction torque. Nassar et al. modeled the stress distribution on a simplified bolt head surface and derived four distinct friction distribution, considering the effect of the tightening speed [2].

For the screw thread contact problem, Sopwith developed a widely accepted load distribution model [3]. In this model, the load between the threads is treated as a concentrated force acting in the middle of the screw thread cross-section. The strain in the bolt and nut threads is divided into three components: bending deflection in the threads of bolt and nut, axial recession due to radial compression in the threads of bolt and nut, axial recession due to radial contraction of bolt and expansion of nut caused by radial pressure between bolt and nut. Yamamoto further expanded the analysis of the deformation of the screw thread cross-section, categorizing the elastic deformation into five types: bending deformation δ1, shearing deformation δ2, inclination deformation of the thread root δ3, shearing deformation of the thread root δ4, and deformation due to radial expansion (nut) and radial shrinkage (bolt) δ5 [4]. Besides, Nassar proposed five pressure distribution models for internal and external threads [5]. Stoeckley and Macke investigated the effect of tapered threads on load distribution [6].

In terms of experimental investigations into bolt stress distribution, Goodier employed an extensometer to measure both radial and axial deformation of nuts [7]. Goodier's experiments revealed that the load at the free end or in the middle of bolt is small, but becomes concentrated near the loaded end. It is important to note that both Goodier's experiments and earlier photo-elasticity studies have certain limitations, as their experimental models fail to accurately represent the 3D geometry of nuts and bolts. Hetenyi conducted 3D photo-elastic experiments using a series of nuts and identified stress concentration phenomenon [8]. In Hetenyi's experiments, the stress near the loaded face tends to decrease, which contrasts with Sopwith's model and Goodier's experiments showing a parabolic increase tread, as shown in Fig. 1. Brown and Hickson replicated Hetenyi's experiments using a new photoelastic material called Fosterite [9]. Their results were consistent with Hetenyi's findings, showing a reduction in load distribution at the loaded face of the nut, as also observed in Kenny and Patterson's experiments [10]. This reduction may be attributed to the fact that the threads on the nut and bolt are not fully formed at the loaded face, resulting in lower thread stiffness and load capacity.

In addition, several other researchers have used FE software to investigate bolt stress [12]. Maruyama employed FE analysis and copper plating method to study the stress in nut-bolt joints [13]. Honglin Xu et al. studied the stress distribution of casing threaded joints according to the Yamamoto method, and established a 2D finite element model [14]. The theoretical results were found to be generally consistent with the FE simulation. Shikun Lu compared the stress distribution obtained from Yamamoto model with that from a 3D finite element model [15]. The findings revealed that the load and stress distribution results from both methods were essentially identical.

In previous theoretical models, the bolt axial stress distribution model was established, but the effect of the addition of tightening torque on the stress distribution

during the actual assembly process was not considered. Similarly, in previous experiments and simulations, the axial force was added to simulate the preload force, without considering the intricacies of the assembly process. To address these limitations, this article aims to investigate the stress distribution of bolt fasteners during and after the assembly process, combined with Yamamoto thread deformation model.

The rest of this paper is organized as follows. In Sect. 2, the relationship between the tightening torque *T* and the preload force *F* is deduced. In Sect. 3, stress distribution along the bolt during the tightening process is studied. In Sect. 4, numerical simulations and experiments are carried out and relevant results are compared with the proposed model. Finally, the conclusions are summarized.

### **2 Assembly Torque Model**

The purpose of this section is to establish the relationship between tightening torque *T* and preload force *F*. A brief analysis shows that during the bolt assembly process, as depicted in Fig. 2, the tightening torque *T* is equal to the sum of the bolt head contact torque *Tt* and the thread contact torque *Tb*. In the subsequent discussion, *Tt*  and *Tb* are analyzed separately.

# *2.1 Bolt Head Contact Torque Tt*

Before analyzing the bolt head contact torque, it is necessary to determine the stress distribution on the contact surface. The following assumptions are made:

i. The bolt head and the support surface are considered as rigid bodies.

**Fig. 2** Assembly torque analysis

ii. The contact stress between the bolt head and the support surface is uniformly distributed.

On the contact surface, the moment generated by friction can be expressed as

$$T\_t = \iint \mu\_t \sigma r \mathbf{d} A\_t$$

where normal stress σ = *F*/*Ah*, *Ah* is the contact area of the bolt head, μ*t* is the friction coefficient of the bolt head contact surface. Then the above equation can be formulated as

$$T\_t = \mu\_t F r\_t,\tag{1}$$

where *rt* = ˜ *r*d*A*/*Ah*.

# *2.2 Thread Contact Torque Tb*

According to Yamamoto method, the bolt thread can be simplified as a circular helix with radius *a* (*a* is effective radius of the thread), on which the thread contact force is assumed to act. The helix equation can be expressed as

$$x = a\cos\theta, \ y = b\cos\theta, \ z = b\theta, \ \theta$$

where θ denotes the angular coordinate of a point on the helix, *b* = *p*/(2π), *p* is the pitch of screw.

The global and local coordinate systems are established on the helix. As shown in Fig. 3, *e* (*x*, *y*, *z*) is the global coordinate system with the origin at the bottom center of the bolt, *z*-axis along the axial direction of the bolt, and *x*-axis pointing to the start of the helix. For any point *Ol* on the helix, there are three local coordinates. The direction of *e***1** (*x*1, *y*1, *z*1) can be obtained by rotating *e* counterclockwise about *z*-axis by θ*l* (θ*l* is the coordinate of the helix angle corresponding to point *Ol*). *e***2** (*x*2, *y*2, *z*2) is obtained by rotating *e***1** counterclockwise about *x*1-axis by α (α is thread

lead angle). *e***3** (*x*3, *y*3, *z*3) is obtained by rotating *e***2** counterclockwise about *y*2-axis by β (β is half of thread included angle).

The contact forces acting on the thread surface are assumed to consist of a normal force *f* and a friction force μ*f* , where μ is the friction coefficient between the threads. In *e***3** coordinate system, the thread contact force can be expressed as (0, μ*f* , −*f* ). According to the coordinate transformation relationship, in *e***1** coordinate system, the projections of the contact forces on the coordinate axes are

$$f\_{\varepsilon1} = (\mu \sin \alpha - \cos \alpha \cos \beta)f = -\left(\sqrt{\mu^2 + \cos^2 \beta} \cos(\alpha + \xi)\right)f,$$

$$f\_{\text{y1}} = (\mu \cos \alpha + \sin \alpha \cos \beta)f = \left(\sqrt{\mu^2 + \cos^2 \beta} \sin(\alpha + \xi)\right)f,$$

where ξ = μ/ cos β.

The thread contact torque and the preload force satisfy

$$T\_b = \int\_0^s f\_{\text{y1}} a \text{ds}, \; F = \int\_0^s f\_{\text{z1}} \text{ds},$$

where *s* is the arc length coordinate of a point on the helix. Thus, we can obtain the relationship between thread contact torque *Tb* and preload force *F*.

$$T\_b = Fa \tan(\alpha + \xi) \tag{2}$$

Then the tightening torque can be expressed as

$$T = T\_b + T\_t = k\_{FT} F,\tag{3}$$

where *kFT* = μ*trt* + *a*tan(α + ξ ).

### **3 Stress Distribution Along the Bolt During Assembly**

In Lu's study, it is evident that the nut stress is very small compared to the bolt stress [15]. Consequently, when solely considering stress distribution along the bolt, the overall compressive deformation of the nut can be disregarded.

### *3.1 Thread Contact Stiffness*

As illustrated in Fig. 4, according to Yamamoto model [4], the thread contact deformation is categorized into five types [5]: bending deformation δ1, shearing deformation δ2, inclination deformation of the thread root δ3, shearing deformation of the thread root δ4, and deformation due to radial expansion (nut) and radial shrinkage (bolt) δ5.

The formulas for these five kinds of deformations are as follows.

**Fig. 4** Thread contact deformation

Load and Deformation Distribution Along the Bolt During Assembly … 953

$$
\delta\_1 = \frac{3f \sin \beta (1 - \upsilon^2)}{4E} \left[ \left[ 1 - \left( 2 - \frac{d\_m}{d} \right)^2 + 2 \ln \left( \frac{d}{d\_m} \right) \right] \cot^3 \beta - 4 \left( \frac{h}{d} \right)^2 \tan \beta \right],
$$

$$
\delta\_2 = \frac{6f \sin \beta \cot \beta (1 + \upsilon)}{5E} \ln \left( \frac{d}{d\_m} \right),
$$

$$
\delta\_3 = \frac{12f \cos \beta (1 - \upsilon^2) h}{\pi E d^2} \left( h - \frac{d\_m}{2} \tan \beta \right),
$$

$$
\delta\_4 = \frac{2f \cos \beta (1 - \upsilon^2)}{\pi E} \left[ \frac{p}{d} \ln \left( \frac{p + d/2}{p - d/2} \right) + \frac{1}{2} \ln \left( 4 \frac{p^2}{d^2} - 1 \right) \right],
$$

$$
\delta\_{5s} = \frac{f d\_p (1 - \upsilon\_s) \cos \beta \tan^2 \beta}{2 p E\_3},
\delta\_{5m} = \frac{f d\_p \cos \beta \tan^2 \beta}{2 p E\_m} \left( \frac{d\_0^2 + d\_p^2}{d\_0^2 - d\_p^2} + \upsilon\_m \right).
$$

where *E* is Young's modulus, υ is Poisson's ratio, *h* is the thread pitch line height, *d* is the thread root width, *dm* is the thread width at the pitch line, *dp* is the effective diameter of the thread, *d*0 is the outer diameter of the nut, and *p* is the pitch of thread. The subscripts *m* and *s* denote the nut and bolt, respectively.

Consequently, the thread contact stiffness of the bolt and nut can be expressed as follows.

$$k\_i = f \cos \beta / \delta\_i,$$

where δ*i* = δ1*i* + δ2*i* + δ3*i* + δ4*i* + δ5*<sup>i</sup>*, *i* represents *s* and *m*, respectively.

### *3.2 Tensile and Torsional Equilibrium Equations*

After calculating the thread contact stiffness, we can proceed to establish the tensile and torsional equilibrium equations for the bolt.

Referring to Fig. 5, the length of the connected part is *lm*1, the length of the nut is *lm*2, the length of the free section of the bolt is *lm*3, and the origin of the global coordinate system is located at the bottom center of the nut. The axial coordinates for the material points on the bolt and nut are denoted as *zs* and *zm*, respectively.

The tensile and torsional equilibrium equations of the bolt can be obtained from the element force analysis.

$$\mathbf{d}F + f\_{z1}\mathbf{ds} = 0,\\ \mathbf{d}T - f\_{\mathbf{y}1}\mathbf{ads} = \mathbf{0}.$$

Combining the coordinate transformation equations and the relationship between stress and strain,

**Fig. 5** Element force analysis

$$F = EA \frac{\mathrm{d}u}{\mathrm{d}z\_s}, \quad T = GI\_p \frac{\mathrm{d}\varphi}{\mathrm{d}z\_s}, \quad \mathrm{d}s = \frac{\mathrm{d}z}{\mathrm{sin}\alpha} = 0,$$

the above equilibrium equations can be expressed as follows.

$$\begin{aligned} EA \frac{\mathbf{d}^2 u}{\mathbf{d}z\_s^2} + \left(\frac{\mu}{\cos \beta} - \cot \alpha \right) k\_s \delta\_s &= 0, \\ GI\_p \frac{\mathbf{d}^2 \varphi}{\mathbf{d}z\_s^2} - \left(\mu \frac{\cot \alpha}{\cos \beta} + 1\right) a k\_s \delta\_s &= 0, \end{aligned}$$

where *A* denotes the bolt cross sectional area, *Ip* denotes the second polar moment of area of the bolt cross-section, *G* stands for the shear modulus. *u* and ϕ represent the displacement and rotation angle of the bolt, respectively.

### *3.3 Supplementary Deformation Conditions*

During the process of bolt tightening, the material points along the bolt and nut satisfy the following deformation criteria:

i. Matching material points occupy the same position along the *z*-axis.

$$z\_s + \mu - \delta\_s \cos \alpha = z\_m + \delta\_m \cos \alpha \tag{4}$$

ii. Assuming no deformation of the reference helix, the corresponding angles of the matching material points are the same.

Load and Deformation Distribution Along the Bolt During Assembly … 955

$$
\theta\_m(z\_m) = \theta\_s = \theta\_{s0}(z\_s) - \varphi(z\_s), \tag{5}
$$

where

$$
\theta\_m = 2\pi z\_m \Big/ p, \quad \theta\_{s0} = 2\pi z\_s \Big/ p.
$$

### *3.4 Boundary Conditions*

At the boundary *zm* = 0, the tension and torque are both zero.

$$EA\frac{d\mathbf{u}}{d\mathbf{z}\_s}|\_{\mathbf{z}\_m=0} = 0, \quad GI\_p \frac{d\boldsymbol{\varphi}}{d\mathbf{z}\_s}|\_{\mathbf{z}\_m=0} = 0 \tag{6}$$

At the boundary *zm* = *lm*2, the displacement and torque satisfy

$$\mu|\_{z\_m = l\_{m2}} = -\frac{Fl\_{m1}}{EA}, \quad GI\_p \frac{\mathbf{d}\varphi}{\mathbf{d}z\_s}|\_{z\_m = l\_{m2}} = T\_b \tag{7}$$

Combining the complementary conditions with the boundary conditions, the displacement and angle of the bolt in the contact section can be obtained. Afterwards, the deformation information of the bolt in the non-contact section can be solved according to the thread contact torque, and then the deformation of the whole bolt can be deduced.

### **4 Numerical Simulation Analysis**

### *4.1 Load Percentage Analysis*

After establishing the theoretical model, the following numerical simulation analysis is carried out. The loading on each thread turn is analyzed and compared with the existing literature. In order to validate the accuracy of this model and also to compare the results with the existing literature, M10 and M20 bolts are used in the simulation. The different bolt parameters are listed in Table 1. The cross-section of M10 bolt head is a circle with a diameter of 16 mm, and the cross-section of M20 bolt head is a square hexagon with a side length of 17 mm.

M10 bolts with only five turns of thread are used to study the load distribution on each turn of thread. Comparison of the results of this model with those of other studies [12] is shown in Fig. 6. A satisfactory agreement between different calculation methods can be found. The first three turns of threads near the loaded face account for almost 80% of the total preload. The last turn of the thread carries little force.


**Table 1** Formatting sections, subsections and subsubsections

In order to investigate the load percentage of the bolt with more turns of threads, M20 bolts with 20 turns of effective thread are use. The load proportion analysis of the first five turns of threads is shown in Fig. 7a. It can be seen that the axial force on the first five turns of threads constitutes approximately 70% of the total preload, and this proportion gradually increases as the coefficient of friction μ increases. This phenomenon suggests that as the thread friction coefficient increases, the load concentration becomes more pronounced. Then the loads on the first five turns of threads are represented by the histogram in Fig. 7b, which are consistent with Lu's results [15].

### *4.2 Rotation Angle and Displacement Distribution*

With this model, if the tightening torque applied to the bolt is known, the deformation information of the whole bolt can be obtained. In order to perform simple calculation, assuming an applied torque of 1.4 N m for the M10 bolt, the coefficient of friction at the contact interface is 0.1. The preload force can be calculated as 1 kN. The values of *lm*1, *lm*2, *lm*3 are 0.2 mm, 0.3 mm, 0.2 mm, respectively. Then the rotation angle and displacement distribution are depicted in Fig. 8.

#### (b) Load histogram

**Fig. 7** The load on the first five turns of threads

**Fig. 8** The rotation angle and displacement distribution

Therefore, based on the tightening torque of the bolt, we can obtain information about the angle of the bolt head and the overall displacement of the free section of the bolt. This provides theoretical support for the use of corner control to add bolt preload in engineering design.

### **5 Conclusions**

In this paper, the theoretical modeling and numerical simulation of the axial force and torque distribution during bolt assembly are performed. Assembly torque model is established. The linear relationship between tightening torque *T* and preload force *F*  is formulated. Stress distribution along the bolt during assembly process is modeled. The tensile and torsional differential equations are developed by element force analysis of the bolt. Combined with the supplementary deformation conditions and boundary conditions, the force and displacement distribution can be deduced. Numerical simulation analysis is carried out. The loading on each thread turn is analyzed and compared with the existing literature. A satisfactory agreement between different calculation methods can be found. The study of load and deformation distribution of the bolt is of guiding significance for the assembly process in practical engineering, and at the same time provides initial conditions for the study of vibration dynamics.

**Acknowledgements** The research is supported by Fast Track Enabling Program (80910020104).

### **References**

1. Motosh N (1976) Development of design charts for bolts preloaded up to the plastic range. J Eng Ind 98(3):849–851. https://doi.org/10.1115/1.3439041


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research of the Flow Field Characteristics of Flake Graphite in the Fan Shaped Nozzle of Premixed Water Jet**

#### **Xing Dong, Chenhao Guo, Jifeng Chen, and Haorong Song**

**Abstract** Graphite is an important strategic resource, in order to explore the flow performance of flake graphite in the flow field of the premixed water jet fan nozzle, numerical simulation of nozzle flow field was carried out by FLUENT software. According to the jet characteristics, the Euler model is selected to simulate the twophase flow law of water and flake graphite, and standard k-ε turbulence model is selected as the turbulence model, the influence of different inlet pressure on the flow field distribution of flake graphite was analyzed. The results show that the axial velocity distribution of flake graphite flow field under different inlet pressures is similar, and the distribution cloud chart has significant plane symmetry, the jet plane diffuses outward at a certain angle. With the increase of target distance, the jet section changes from a flat structure to a symmetrical "groove" structure with high ends and low middle, the axial velocity of flake graphite flow field has secondary acceleration in the internal flow field and primary acceleration in the outward flow field, the maximum axial velocity appears near the nozzle outlet.

**Keywords** Pre mixed water jet · Fan shaped nozzle · Flake graphite · Flow field characteristics

X. Dong (B) · J. Chen · H. Song School of Mechanical Engineering, Heilongjiang University of Science and Technology, Harbin, Heilongjiang 150022, China e-mail: dongxingwrh@163.com

C. Guo

© The Author(s) 2024

961

School of Safety Engineering, Heilongjiang University of Science and Technology, Harbin, Heilongjiang 150022, China e-mail: 18145974766@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_76

### **1 Introduction**

Graphite has good thermal shock resistance, high self-lubricating strength, good thermal conductivity and electrical conductivity. It finds extensive application across various sectors including metallurgy, machinery, national defense, military industry and so on. It is an important strategic resource. Flake graphite is an important branch of graphite, which has few resources, wide uses and high value, the research on its high purification technology is very important.

Ma F Y et al. [1] proposed a new grinding flotation process for the beneficiation of flake graphite, which can improve the coarse graphite ore to a higher grade graphite concentrate. Thair A et al. [2] used ball mill and mill to grind raw flake graphite ore, and obtained high grade flake graphite through alkaline roasting and acid leaching process. Mustafa A [3] applied FLUENT software to analyze the jet characteristics of three kinds of nozzles with different shapes, including rectangular, circular and two-dimensional profiles, and obtained the central velocity distribution laws of the three kinds of nozzle jets within the range of 0.2 ~ 13 mm from the target distance, the correctness of the numerical simulation results is proved by experiments. Juan W et al. [4] used CFD method to explore the atomizing performance of fan shaped nozzle, and obtained the relationship between nozzle spray drift and wind speed. Chunzhao Z et al. [5] used FLUENT software to conduct numerical simulation on the fan-shaped nozzle with round head, truncated cone and innovative outlet, it was found that the comprehensive performance of the innovative nozzle was better than that of the other two nozzles. Mireia A et al. [6] applied CFD method to analyze the internal and outward flow fields of three industrial fan-shaped atomizing nozzles, and obtained the characteristic values such as nozzle flow coefficient.

To sum up, some scholars have studied the grinding and floating of flake graphite and the jet characteristics of fan shaped nozzles and have achieved some research results [7, 8]. However, there are few reports on the numerical simulation research of the whole flow field of the fan-shaped liquid–solid two-phase jet formed by the ejection of flake graphite slurry through the fan-shaped nozzle. In this article, the self-designed fan-shaped nozzle is used as the energy conversion element, and the whole flow field of flake graphite slurry flowing through the fan nozzle is numerical simulated by FLUENT software. The flow properties of water flow field and flake graphite flow field are obtained, which lays a further technical foundation for the separation of flake graphite by jet.

### **2 Basic Model and Calculation Method**

### *2.1 Geometric Model*

The geometric model for numerical simulation of the fan shaped nozzle with premixed water jet is established by using SolidWorks software, which is consisted of four parts: the inlet cone convergence section, the middle cylinder section, the outlet spherical V-groove and the outward flow field cylinder. In this case, the inlet diameter of the conical convergence section of the nozzle is 7 mm, the convergence angle is 30º, and the length of the convergence section is 10.3 mm; The diameter of the cylindrical section is 2 mm and the length is 5 mm; The diameter of the outlet spherical end is 2 mm, and the V-groove chamfer is 45º. The outward flow field is set as a cylinder with a diameter and length of 20 mm.

### *2.2 Finite Element Model*

The geometry model is meshed by using the preprocessing module Gambit in FLUENT software. In order to enhance the stability and accuracy of the calculation process, the tetrahedral unstructured mesh is selected, and the local mesh refinement is carried out at the connection position of the internal flow field and the outward flow field of the nozzle. Through grid independence test, the number of grid cells after division is 1331257, and the number of nodes is 237238. Figure 1 shows the finite element model of the fan shaped nozzle.

**Fig. 1** Finite element model

### *2.3 Mathematical Model*

#### **2.3.1 Governing Equation**

The premixed water jet of flake graphite is a liquid–solid two-phase high speed jet, and the liquid–solid two-phase medium in the jet flow field is similar to a continuous medium, and the volume fraction of solid particle flake graphite in the conveying process of flake graphite is more than 10%, which is suitable for Euler multiphase flow model. Therefore, Euler model is selected to analyze the liquid–solid two-phase flow of water and flake graphite, and its control equation is [9].

The mass conservation equations of water and flake graphite are

$$\frac{\partial}{\partial t}(\alpha\_w \rho\_w) + \nabla \cdot (\alpha\_w \rho\_w \upsilon\_w) = \sum\_{s=1}^n (\vec{m}\_{sw} - \vec{m}\_{ws}) + S\_w \tag{1}$$

$$\frac{\partial}{\partial t}(\alpha\_s \rho\_s) + \nabla \cdot (\alpha\_s \rho\_s v\_s) = \sum\_{w=1}^n (\vec{m}\_{ws} - \vec{m}\_{sw}) + S\_s \tag{2}$$

The momentum equations of water and flake graphite are

$$\begin{aligned} \frac{\partial}{\partial t}(\alpha\_w \rho\_w v\_w) + \nabla(\alpha\_w \rho\_w v\_w v\_w) &= \alpha\_w \nabla p - \nabla p\_w + \nabla \cdot \tau w + \alpha\_w \rho\_w g \\ + \sum\_{s=1}^n \left[ K\_{sw}(v\_s - v\_w) + \stackrel{\rightarrow}{m} v\_{sw} - \stackrel{\rightarrow}{m} v\_{ws} \right] + \left( F\_w + F\_{lift,w} + F\_{wl,w} + F\_{vm,w} + F\_{td,w} \right) \end{aligned} \tag{3}$$

$$\begin{aligned} \frac{\partial}{\partial t}(\mathbf{a}\_s \rho\_s v\_s) + \nabla \cdot (\mathbf{a}\_s \rho\_s v\_s v\_s) &= -\alpha\_s \nabla p - \nabla p\_s + \nabla \cdot \mathbf{r}\_s + \mathbf{a}\_s \rho\_s \mathbf{g} \\ + \sum\_{w=1}^n \left[ K\_{ws} (v\_w - v\_s) + \vec{m}\_{ws} v\_{ws} - \vec{m}\_{sw} v\_{sw} \right] &+ \left( F\_s + F\_{lif,s} + F\_{wl,s} + F\_{vm,s} + F\_{td,s} \right) \end{aligned} \tag{4}$$

The interphase velocity of water to flake graphite and the interphase velocity of flake graphite to water *vws* and *vsw* are respectively

$$v\_{ws} = \begin{cases} v\_{ws} = v\_w, \ \vec{m}\_{ws} > 0\\ v\_{ws} = v\_s, \ \vec{m}\_{ws} < 0 \end{cases} \tag{5}$$

$$v\_{sw} = \begin{cases} v\_{sw} = v\_s, \ \vec{m}\_{sw} > 0 \\ v\_{sw} = v\_w, \ \vec{m}\_{sw} < 0 \end{cases} \tag{6}$$

The energy equations of water and flake graphite are

$$\begin{aligned} &\frac{\partial}{\partial t}(\alpha\_w \rho\_w h\_w) + \nabla \cdot (\alpha\_w \rho\_w v\_w h\_w) \\ &= \alpha\_w \frac{\mathrm{d}\rho\_w}{\mathrm{d}t} + \mathfrak{r}\_w : \nabla v\_w - \nabla \cdot \vec{q}\_w + S\_w \\ &+ \sum\_n^{s=1} (Q\_{sw} + \vec{m}\_{sw} h\_{sw} - \vec{m}\_{sw} h\_{ws}) - \nabla \cdot \sum\_j h\_{j,w} \vec{J}\_{j,w} \\ &\frac{\partial}{\partial t}(\alpha\_s \rho\_s h\_s) + \nabla \cdot (\alpha\_s \rho\_s v\_s h\_s) \\ &= \alpha\_s \frac{\mathrm{d}\rho\_s}{\mathrm{d}t} + \mathfrak{r}\_s : \nabla v\_s - \nabla \cdot \vec{q}\_s + S\_s \\ &+ \sum\_n^{w=1} (Q\_{ws} + \vec{m}\_{sw} h\_{ws} - \vec{m}\_{sw} h\_{sw}) - \nabla \cdot \sum\_j h\_{j,s} \vec{J}\_{j,s} \\ \end{aligned} \tag{8}$$

#### **2.3.2 Turbulence Model**

The liquid–solid two-phase jet of flake graphite slurry with fan-shaped nozzle is a high turbulent flow, and the internal and outward flow fields have high Reynolds numbers, so the turbulence model selection standard *k*- ε Model. The turbulence model contains two basic equations, which are turbulent kinetic energy *k* and dissipation rate ε of mixed fluid transport equation [10].

Turbulent kinetic energy *k* of mixed fluid is

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial x\_i}(\rho k u\_i) = \frac{\partial}{\partial x\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_k} \right) \frac{\partial k}{\partial x\_j} \right] + G\_k + G\_b - \rho \varepsilon - Y\_M + S\_k \tag{9}$$

Dissipation rate ε of mixed fluid is

$$\frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial x\_l}(\rho \varepsilon u\_l) = \frac{\partial}{\partial x\_l} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial x\_j} \right] + C\_{1\varepsilon} \frac{\varepsilon}{k} (G\_k + C\_{3\varepsilon} G\_b) - C\_{2\varepsilon} \rho \frac{\varepsilon^2}{k} + S\_\varepsilon \tag{10}$$

### *2.4 Calculation Method*

The numerical simulation of fan-shaped nozzle jet flow field is solved by a separation solver. The first-order accuracy upwind difference scheme is selected, and the turbulence control equation is based on the volume fraction of liquid–solid multiphase flow. The convergence criterion of numerical simulation is residual *R* ≤ 10–4, the relaxation factor is the default value.

### *2.5 Boundary Condition*

The pressure inlet and outlet for the fan-shaped nozzle are positioned at the nozzle's inlet and the right end face of the outward flow field's outlet, respectively. The pressure of the fan-shaped nozzle outward flow field is designated as the standard atmospheric pressure in its respective domain. Additionally, the standard wall function without velocity slip is applied to the inner wall surface of the nozzle. The density of the main phase, water, is established at 1000 kg/m3, whereas its viscosity is 1 × 10–3 Pa·s. For the secondary phase, flake graphite, its density is set at 2100 kg/m3. The viscosity of this phase is measured at 1 × 10–5 Pa·s, and it has a diameter of 0.1 mm. In order to analyze how variations in the inlet pressure affect the properties of the jet flow field.

### **3 Results and Discussion**

# *3.1 Effect of Inlet Pressure on the Flow Field of Flake Graphite*

Figure 2 shows the axial velocity distribution cloud diagram of the flake graphite flow field in the fan-shaped nozzle when the inlet pressure *p* is 2 MPa, 4 MPa, 6 MPa, 8 MPa, and 10 MPa.

From Fig. 2, it can be seen that the axial velocity distribution of the flake graphite flow field has significant surface symmetry at different inlet pressures, and the jet plane diffuses outward at a certain angle. Due to the acceleration of the conical contraction structure of the fan nozzle, the axial velocity of the flake graphite flow field increases continuously. In the cylindrical section of the middle of the nozzle, the axial velocity of the flake graphite flow field remains unchanged due to the constant cross-sectional area. At the nozzle outlet, due to the joint action of the spherical structure at the fan-shaped nozzle outlet and the V-groove, the cross-sectional area of the jet decreases rapidly, and the flow field of flake graphite is accelerated for the second time, and its axial velocity reaches the maximum at the V-groove at the nozzle outlet. Similarly, the axial dynamic pressure distribution rule of flake graphite flow field obtained by analysis is similar to the axial velocity.

# *3.2 Effect of Target Distance on the Cross-Sectional Shape of the Flow Field of Scaled Graphite*

Figure 3 shows the shape distribution of axial velocity section of flake graphite flow field of fan-shaped nozzle when the inlet pressure *p* is 10 MPa and the target distance *l* is 0, 5, 10, 15 and 20 mm respectively.

From Fig. 3, it can be seen that when the target distance is 0~10 mm, the axial velocity of flake graphite is relatively large, the regular flat structure of the jet can be basically maintained and gradually developed. When the target distance is 0 mm, 5 mm and 10 mm respectively, the width *b* and height *h* of the axial velocity flat section are 3.84 mm and 0.61 mm, 9.70 mm and 1.62 mm, 14.75 mm and 2.02 mm respectively. Because the flat structure of the jet is in contact with the air, with the increase of the target distance, the disturbance of the outflow field air to the jet becomes more intense, and the uniformity of the flow field decreases. When the target distance is 15 mm, the jet begins to fluctuate obviously, the shape of the flat section of the jet changes, and the height direction gradually becomes a symmetrical "groove" structure with high ends and low middle. At this time, the width *b* and height *h* on both sides of the axial velocity section are 18.99 mm and 3.05 mm respectively. When the target distance is 20 mm, the air moves with jet entrainment to the outlet of the outward flow field. At this time, the axial velocity section width of flake graphite reaches the maximum, and its width *b* and height *h* are 19.60 mm and 4.02 mm, respectively.

# *3.3 Flake Graphite Flow Field Axial Velocity Change Rule of Law*

Figure 4 shows that the axial velocity curve of the flake graphite flow field on the nozzle axis changes with the abscissa *x* when the inlet pressure is 10 MPa.

**Fig. 3** Axial velocity section shape of flake graphite outward flow field

From Fig. 4, it can be seen that in the jet flow field, due to the small volume fraction of flake graphite, the water flow field has a good acceleration effect on the flake graphite flow field, and the kinetic energy of the water flow field is fully utilized by the flake graphite flow field. In the inner flow field, the abscissa is 0 mm (at the entrance of the fan-shaped nozzle), and the axial velocity of the flake graphite flow field is 4.45 m/s. Due to the increase of abscissa, the axial velocity of flake graphite

flow field increases rapidly and gets the first acceleration. At the abscissa position of 9.3 mm (the outlet of nozzle cone convergence section), the axial velocity of flake graphite flow field is 40.53 m/s. When the abscissa is in the range of 9.3~14.3 mm (the cylindrical section of the nozzle), the axial velocity of the flake graphite flow field remains basically unchanged due to the constant cross-sectional area of the cylindrical section. When the abscissa is in the range of 14.3~15.3 mm (the

**Fig. 4** Axial velocity curve of flake graphite flow field

spherical V-groove at the outlet of the nozzle), due to the joint action of the spherical structure at the outlet of the fan-shaped nozzle and the V-groove, the jet changes from cylindrical to flat fan-shaped jet, the cross-sectional area of the jet decreases rapidly, the axial velocity of the flake graphite flow field increases sharply, and the second acceleration is obtained. At the abscissa of 15.3 mm (the V-groove outlet), the axial velocity of the flake graphite flow field is 101.24 m/s. In the outward flow field, in the range of 15.3~15.8 mm on the abscissa, because the flow field of flake graphite at the nozzle outlet lags behind the water flow field, the flake graphite gets the third acceleration under the action of water. At the position of 15.82 mm on the abscissa, the axial velocity of the flow field of flake graphite reaches the maximum, which is 104.38 m/s. Then, with the increase of target distance, the axial velocity of flake graphite flow field decreases rapidly.

### **4 Conclusion**


**Acknowledgements** This article is one of the phased achievements of the National Natural Science Foundation project 'Mechanism Research on Dissociation of Flake Graphite by Multistage Cavitation Water Jet Basing on the Friction and Shearing' (52075151).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Analysis of Factors Affecting Critical Speed of Multi-span Rotor System Coupled with Diaphragm Coupling**

**Zhihua Wu, Zhanwei Li, Mengfan Shi, and Qing Zhang** 

**Abstract** Taking the multi-span rotor system composed of two double diaphragm couplings coupled with three rotors as the object, the finite element model of the highspeed multi-span rotor system is established, and the critical speed of the rotor system is solved using ANSYS Workbench software. The effects of the radial stiffness of the diaphragm coupling and the bearing support stiffness on the critical speed of the rotor system in the first three orders are analyzed. The results show that, for the model established in this paper, within a certain range, the radial stiffness of the diaphragm coupling has a greater influence on the third-order critical speed of the multi-span rotor system, and the increase of the radial stiffness of the diaphragm coupling and the increase of the bearing support stiffness will increase the critical speed of the multi-span rotor system.

**Keywords** Critical speed · Coupling radial stiffness · Bearing stiffness

# **1 Introduction**

When a rotor operates in a high-speed rotating system, there are certain specific rotational speeds at which, when the rotor operates at or near these speeds, the rotor shaft deflects and generates large unbalanced forces and moments that cause the rotor to vibrate violently. These speeds are known as critical speeds. Operating at these speeds, the rotor system may vibrate with serious consequences, such as failure of the rotating machinery to operate properly or damage to components. Therefore, it is important to accurately calculate the critical rotational speed of the rotor to control the vibration of the rotor system and to ensure the safe operation of the rotor.

Z. Wu · M. Shi · Q. Zhang (B)

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China e-mail: dazhang@njust.edu.cn

#### Z. Li

Nanjing Chenguang Group Co. Ltd., 215, Nanjing 210006, Jiangsu, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_77

Most of the current critical speed calculation methods are for structurally regular stepped shafts, and the transfer matrix method and the finite element method are used to calculate the critical speed of a rotor [1–3], and the analysis objects are mostly individual single rotors with continuous structure [4–6]. However, for most practical scenarios, the rotor system usually does not contain only one rotor, but at least the motor rotor as the power supply, and the output rotor to be driven, and the coupling as the connection between the input and output components, the stiffness characteristics of which have a non-negligible impact on the entire rotor system. Pirogova et al. [7] analyzed the intrinsic and critical frequencies of the turbocharger rotor and the starter-generator rotor when they are connected into a rotor system by means of an elastic coupling. Zeng et al. [8] took the slender tandem shaft system of submersible oil pump as the research object and analyzed the effect of torsional stiffness of coupling on the torsional vibration characteristics of the slender tandem shaft system. Kim et al. [9] developed a novel coupled modeling approach to analyze the effects of non-beam behavior on the dynamics of a rotor system brought about by local joint flexure in a preloaded coupling. Ren et al. [10] analyzed the flywheel rotor supported by electromagnetic bearings based on ANSYS Workbench software, and investigated the effect of damping and support stiffness on the critical speed of the flywheel rotor system, and calculated the optimum support stiffness range adjustment.

This paper takes the rotor system composed of two double diaphragm couplings coupling three rotors as the object, establishes the finite element model of the multispan high-speed rotor system, carries out the solution of the critical speed, and analyzes the influence of the radial stiffness of the coupling and the bearing support stiffness on the critical speed of the rotor system.

### **2 Modeling of Multi-span Rotor Systems**

The multi-span rotor system comes from the high-speed transmission part of a spaceuse gearbox load test stand, and consists of a motor rotor shaft, a speed and torque sensor shaft, a gearbox input shaft, two double-diaphragm couplings, and three shafts each with their own support bearings in a total of ten parts. The rotor shaft and couplings are modeled in Creo as shown in Fig. 1, from left to right, the motor shaft, diaphragm coupling 1, speed and torque sensor shaft, diaphragm coupling 2, and reducer input shaft. The length of the motor shaft is 400 mm; the length of the speed and torque sensor shaft is 164 mm; the input shaft of the reducer is 92 mm; the length of the coupling 1 is 76 mm and the diameter is 68 mm; the length of the coupling 2 is 50 mm and the diameter is 44 mm.

Considering the structural complexity of the double diaphragm coupling, when performing the dynamic analysis of the rotor system, ANSYS Workbench is used directly to perform simulation calculations, ignoring the structures in the coupling that are not easy to calculate, such as chamfers, small holes, asymmetric cross sections, and screws used for locking in the bushings on both sides, and ANSYS

**Fig. 1** Multi-span rotor system assembly

Workbench can perform the dynamic analysis of the rotor for the axisymmetric rotor directly. The material of the three rotor shafts is 40Cr, while the materials of the diaphragm coupling are aluminum alloy (bushings), stainless steel (diaphragms), and alloy steel (pins). The material parameters are shown in Table 1.

For the model in this paper, the free mesh division mode is used, and appropriate mesh refinement is carried out at the coupling diaphragm as well as at the place where the rotor shaft and coupling are in contact, and the mesh division is carried out after the setup is completed, and the number of nodes is finally obtained to be 308,696, and the number of cells is 166,485, and the rotor system after the division of the mesh is shown in Fig. 2.


**Table 1** Rotor system material parameters

**Fig. 2** Grid division diagram of rotor

**Fig. 3** Bearing support position diagram

The constraints are set after the meshing is completed. The three rotors are supported by their respective bearings, and the bearing support positions are shown in Fig. 3. The Stiffness K11 and K22 of the motor shaft bearings are defined as 4000 N/mm, and the Stiffness K11 and K22 of the speed and torque sensor shaft and the gearbox input shaft bearings are defined as 3000 N/mm. 2 N s/mm is defined for Damping C11 and C22. After lifting the five bearings to the remote points, remote displacement constraints are used to constrain the axial displacements of the five remote points and the axial displacement of the five remote points, use the remote displacement constraints to constrain the axial displacement as well as the axial rotation of these 5 remote points.

In the solution setup, set Damped in Solve Controls to Yes, Solver Type to Program Controlled; Coriolis Effect in Rotordynamics Controls to on, Campbell Diagram to on, Number of Points to 5; Stiffness Coefficient in Damping Controls to 8e-4.

When Workbench solves the critical speed of a rotor, the rotational speed must be defined. Insert five rotational speed points: 100, 1000, 5000, 10,000, 60,000 rpm. after defining the direction of rotation, modal analysis is carried out to solve the critical rotational speed, and the critical rotational speed of this multi-span rotor system is calculated to obtain the Campbell diagram as shown in Fig. 4, the firstorder critical rotational speed is 7180 rpm, the second-order critical rotational speed is 12,919 rpm and the third order critical speed is 33,083 rpm.

### **3 Analysis of Critical Speed Influencing Factors**

# *3.1 Influence of the Radial Stiffness of the Coupling on the Critical Speed*

In order to quantitatively analyze the effect of the radial stiffness of the coupling on the critical speed of the rotor system, the coupling is simulated by using a bushing coupling, which can be used to simulate a flexible coupling to connect two shafts in ANSYS Workbench. The data in the parameter table in Fig. 5 can be changed to

**Fig. 4** Campbell diagram

simulate different couplings with different properties. The first two elements of the diagonal can be changed to simulate couplings with different radial stiffnesses. To connect the end faces of two neighboring shafts by means of a Bushing coupling, the reference and moving surfaces must be carefully defined, the reference surface being the right end face of the first shaft and the moving surface being the left end face of the second shaft.

The design value of radial stiffness of coupling 1 and 2 is 400, 300 N/mm respectively. The radial stiffness of coupling 1 and 2 are studied as variables, and 20, 40, 60, 80, 100, 150, 200, 250, 300% are set as adjustment factors. In addition, the radial stiffnesses of couplings 1 and 2 are adjusted proportionally at the same time. Calculate the critical rotational speed under each adjustment factor according to the above method, and plot the change curve as shown in Figs. 6 and 7.

Analyzing the data, it is learned from Fig. 6a that in the process of increasing the radial stiffness of coupling 1, the first three orders of critical speed of the rotor system are slowly increased with a low growth rate, and the overall critical speed remains stable; it is learned from Fig. 6b that in the process of increasing the radial stiffness of coupling 2, the first two orders of critical speed of the rotor system are almost unchanged, whereas the third order of the critical speed is significantly increased with a very high growth rate; From Fig. 7, when the radial stiffness of couplings 1 and 2 are adjusted proportionally at the same time, the first two critical speeds of the rotor system increase slowly with the increase of stiffness, and the overall tendency is stable, while the third critical speed is significantly increased with a high growth rate, and the overall value is higher than that under the influence of coupling 1 alone.

**Fig. 5** Table of stiffness coefficients for bushing connecting pairs

**Fig. 6 a** Critical rotor speed with radial stiffness variation curve of coupling 1; **b** Critical rotor speed with radial stiffness variation curve of coupling 2

Therefore, it can be learned that for the coupled rotor system analyzed in this paper, the radial stiffness of coupling 1 mainly slightly affects the first two orders of critical speed of the rotor system, while the radial stiffness of coupling 2 significantly affects the third order of critical speed of the rotor system.

**Fig. 7** Critical speed variation curves for simultaneous changes in radial stiffness of couplings 1 and 2

### *3.2 Effect of Bearing Support Stiffness on Critical Speed*

Bearing support stiffness refers to the stiffness provided by the bearing in the radial and axial directions, which determines the stiffness characteristics of the bearing system. The larger the bearing support stiffness is, the stiffer the bearing system is, and the stronger the support ability for the rotor [11]. In high speed operation, a larger bearing support stiffness can reduce the deflection and deformation of the rotor, thus reducing the risk of resonance and improving the stability and reliability of the system.

In this paper, only the effect of bearing radial stiffness on the critical speed of the rotor system is considered, the initial stiffness of the two bearings of the motor shaft is 4000 N/mm, the stiffness of the bearings of the sensor rotor shaft is 3000 N/mm, and the stiffness of the bearings of the reducer input shaft is 3000 N/mm, as in the previous section, 20, 40, 60, 80, 100, 150, 150, 150, and 300% of the bearing radial stiffness are studied as adjustment factors, and the radial stiffness of the five bearings is adjusted synchronously and proportionally. 200, 250, and 300% as the adjustment coefficients are investigated to adjust the stiffness of the five bearings synchronously and proportionally. The variation curves of the first three orders of critical speed of the rotor system with bearing stiffness are calculated as shown in Fig. 8.

From the data in the Figure, it can be seen that the first three critical speeds of the rotor system increase with the increase of bearing support stiffness, and the first two critical speeds of the rotor system in the range of the adjustment factor of 20–150%, the rate of change of the rotor system is relatively high; the adjustment factor of 20–80% of the range of the critical speed of the rotor system of the third order of the bearing stiffness of the influence of the rotor system is more significant; in the subsequent change of the bearing stiffness, the rotor system of the critical speeds of

**Fig. 8** Variation curve of the first three orders of critical speed of rotor system with bearing stiffness

the first three orders has been increasing, but the overall growth rate slows down and gradually tends to be stabilized.

### **4 Conclusion**

In this paper, a multi-span rotor system consisting of two double diaphragm couplings coupling three rotors is taken as the research object, and a finite element model of the multi-span rotor system is established, and the critical speed is solved using ANSYS Workbench to obtain the first three orders of the critical speed of the rotor system and Campbell's diagrams, and meanwhile, the effects of the radial stiffness of the diaphragm couplings and the stiffness of the bearing supports on the critical speed of the multi-span rotor system are analyzed. The results show that:


(3) This paper analyzes the influence factors of the critical speed of the coupled rotor, which can provide a reference for the lateral vibration of rotating machinery, and provide a reference basis for the selection of flexible couplings and bearings in engineering applications.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Stiffness and Strength Analysis of 2D Woven Composite Materials Based on Multi-scale Finite Element Method**

**Weiye Wang, Renjun Yan, Kang Liu, and Yu Qiu** 

**Abstract** This study is based on the multi-scale finite element method and establishes RVE model of 2D woven fiber yarn and woven structure. The Stiffness matrix of plain, satin and twill 2D braided composites is calculated by applying Periodic boundary conditions and tensile shear loads in six directions. Write a UMAT subroutine to perform finite element simulation of the strength of woven RVE based on the three-dimensional Hashin failure criterion, and compare and verify it with existing experimental data. By comparing the load displacement curves of three types of braided composite materials under the same volume fraction, the stiffness and strength characteristics of the three types of braided composite materials, as well as the mechanical performance similarity issues caused by structural similarity, are studied.

**Keywords** Woven composites · Multi-scale · Representative volume element · Stiffness and strength

W. Wang · R. Yan · Y. Qiu

K. Liu (B)

School of Navigation, Wuhan University of Technology, Wuhan 430079, China

W. Wang · R. Yan · Y. Qiu Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China

K. Liu

© The Author(s) 2024

School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430079, China

Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education, Chongqing Jiaotong University, Chongqing 430074, China e-mail: liukang\_88@whut.edu.cn

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_78

### **1 Introduction**

At present, the application of sandwich composite materials is becoming increasingly common in the field of ship building. Sandwich panel panels mainly use unidirectional fiber composite materials and woven composite materials. Compared with unidirectional fiber composite materials, the performance of woven composite materials is affected by factors such as fiber bundle bending and staggered weaving. Its structure is more complex than unidirectional fiber composite materials, and the mechanical properties such as stiffness and strength research methods are also different from traditional unidirectional fibers [1]. However, in the production and application of woven composite materials, it is necessary to conduct in-depth research on the stiffness and strength of woven composite materials starting from the material constitutive model.

Woven composite materials and unidirectional fiber reinforced composite materials have similarities in structure, with the main difference being that the fiber bundles inside the woven composite materials are crisscrossed and rub against each other [2]. This structure makes the mechanical properties of the woven composite materials more consistent in the layer plane. At present, researchers mainly obtain the stiffness and strength of woven composite materials through experimental methods and finite element methods. The experimental method measures their tensile and shear properties through tensile experiments, compression tests, in-plane shear tests, and out of plane shear tests, and calculates their engineering constants through experimental data. The multi-scale finite element simulation method, on the other hand, predicts the material parameters of unidirectional fiber reinforced composite materials by numerically simulating RVE. Both methods lack accurate understanding of the constitutive properties of woven composite materials [3, 4].

This study focuses on this current situation and adopts multi-scale finite element analysis method. Firstly, the accuracy of two theoretical calculation formulas for the engineering constants of unidirectional fiber composite materials was evaluated at the micro scale (referring to the scale that can reflect the structural characteristics of the fiber matrix, where the fiber diameter is 10–100 μm, and the fiber matrix RVE size is 50–200 μm). Subsequently, based on the engineering constant results of unidirectional fiber composite materials, plain, twill, and Three kinds of RVE (Representative Volume Element) models of 2D braided composites commonly used in satin Naval architecture are used to analyze woven composites from the mesoscopic scale (referring to the scale that can reflect the specific braided structure of braided composites, and the RVE model size of braided structures is 1–5 mm), and then obtain the stiffness and strength of woven composites [5, 6].

# **2 Method for Analyzing the Mechanical Properties of Composite Material Cells**

# *2.1 Representative Volume Element and Homogenization Theory*

The characterizing volume unit RVE is a typical unit selected from composite materials, which is a unit cell volume unit that can represent the microstructure characteristics of composite materials. Through the assumption of homogenization theory, its stress and strain can be considered uniform to reflect the average properties of the material. Therefore, studying the equivalent mechanical properties of macroscopic homogenization models for composite materials is equivalent to studying the mechanical properties of homogenized equivalent media corresponding to representative volume units in the material [7].

### *2.2 Periodic Boundary Conditions*

When the RVE is loaded, the stress and strain on its symmetrical interface should be periodic and continuous rather than a simple free boundary [8]. This constraint condition can be achieved by applying Periodic boundary conditions. The Periodic boundary conditions requires that the symmetric nodes of the cell meet the following formula:

$$u\_i - u\_i' = \overline{\varepsilon}\_{ik} \cdot \Delta x\_k \tag{l}$$

Among them, *ui* , *u*- *<sup>i</sup>*is the displacement of two symmetrical nodes, ε*ik* is the average strain component of a single cell, and *xk* is a fixed value, which is equal to the relative position vector of the corresponding nodes of the single cell. This boundary condition not only satisfies the continuity of displacement on the boundary, but also ensures the continuity of stress at the boundary of the single cell [9, 10].

# *2.3 Finite Element Analysis Method for Woven Composite Materials*

Woven fiber composite materials are composed of woven fiber bundles and matrices, and properties need to be defined separately during finite element analysis. The fiber bundles in the woven structure are infiltrated by fiber bundles and matrices, which can be regarded as transversely isotropic materials (*E*2 = *E*3, *G*12 = *G*13, ν12 = ν13), the matrix is defined as an isotropic material. For woven composite materials, their woven structure makes the elastic modulus of the warp and weft lines in the fiber cloth yarn equal in both directions. In finite element calculations, defining their material properties often simplifies them as *E*1 = *E*2, *G*13 = *G*23, ν13 = ν23. First: Treat the fibers and matrix as isotropic materials and measure material parameters through experiments; Second: through the analysis of the micro RVE element (fiber bundle RVE), solve the Stiffness matrix and calculate the material properties; Third: solve the Stiffness matrix of the woven structure RVE in the second step, and calculate the material parameters. The fiber bundle material properties are calculated in the second step, and the matrix material properties remain unchanged [7].

### *2.4 Finite Element Analysis of Mesoscale Stiffness*

According to the composite Strength of materials, the composite Stiffness matrix is solved, and the composite stress–strain relationship is as follows:

$$
\sigma = C\varepsilon, \varepsilon = S\sigma \tag{2}
$$

$$
\boldsymbol{\varepsilon} = S\boldsymbol{\sigma} \Rightarrow \begin{Bmatrix} \boldsymbol{\sigma}\_{\boldsymbol{x}} \\ \boldsymbol{\sigma}\_{\boldsymbol{y}} \\ \boldsymbol{\sigma}\_{\boldsymbol{z}} \\ \boldsymbol{\tau}\_{\boldsymbol{xy}} \\ \boldsymbol{\tau}\_{\boldsymbol{xz}} \\ \boldsymbol{\tau}\_{\boldsymbol{yz}} \end{Bmatrix} = \begin{bmatrix} \frac{1}{E\_{1}} & -\frac{\boldsymbol{\nu}\_{12}}{E\_{2}} - \frac{\boldsymbol{\nu}\_{13}}{E\_{3}} & 0 & 0 & 0 \\ -\frac{\boldsymbol{\nu}\_{21}}{E\_{1}} & \frac{1}{E\_{2}} & -\frac{\boldsymbol{\nu}\_{23}}{E\_{3}} & 0 & 0 & 0 \\ -\frac{\boldsymbol{\nu}\_{31}}{E\_{1}} - \frac{\boldsymbol{\nu}\_{21}}{E\_{2}} & \frac{1}{E\_{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G\_{31}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G\_{11}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G\_{12}} \end{bmatrix} \times \begin{Bmatrix} \boldsymbol{\varepsilon}\_{x} \\ \boldsymbol{\varepsilon}\_{y} \\ \boldsymbol{\varepsilon}\_{z} \\ \boldsymbol{\chi}\_{xy} \\ \boldsymbol{\chi}\_{xz} \\ \boldsymbol{\chi}\_{yz} \end{Bmatrix} \tag{3}$$

Abaqus is used to establish the RVE model of the meso fiber matrix, apply Periodic boundary conditions, and apply tensile loads in *x*, *y*, *z* directions and shear loads in *xy*, *yz*, *xz* directions to the cell through the displacement of the coupling point of the control surface. In the post-processing, the reaction force and displacement at the coupling point of the control surface are extracted, and the six groups of stress and strain states of the cell under six working conditions are calculated to solve the Stiffness matrix of the fiber matrix RVE. The material properties in other literature are selected for this study, and the specific material properties are shown in Table 1.

The generated unit cell is a cube with a length of 16.3 μm, a width of 6.145 μm, and a height of 3 μm, in which the fiber volume fraction is 70%. The Hexahedron unit is selected in Abaqus for grid division. No interface layer attribute is defined


**Table 2** Parameters of cellular materials with 70% fiber volume fraction calculated by finite element method


between the fiber and the node, and the fiber bundle and the matrix are connected in a common node manner (Fig. 1).

Calculate the unit cell flexibility matrix when the fiber volume fraction is 70%. Calculate the strain in each direction of RVE based on the displacement of the coupling point and the unit cell size. Calculate the stress in the corresponding direction under each working condition based on the support reaction force and RVE size. Calculate the nine material parameters of RVE, and obtain the RVE material parameters with a fiber volume fraction of 70%, as shown in Table 2.

### **3 Finite Element Analysis of Mesoscopic Scale Stiffness**

# *3.1 Establishment and Calculation of RVE Finite Element Model for Woven Structures*

Analyzing the RVE units of 2D woven composite materials at a mesoscale, three weaving methods are shown in Fig. 2, with the specified warp direction of the yarn being in the 1 direction, the weft direction being in the 2 direction, and the plane normal direction of the woven fiber cloth being in the 3 direction.

**Fig. 2** Schematic diagram of three 2D woven structures RVE

In the finite element calculation, material properties are assigned to the fiber bundles and matrix of the RVE of the woven structure. According to the scanning results of the electron microscope, the internal yarns of the plain weave woven composite material are long ellipses, so an ellipse is selected as the cross-section of the finite element model yarn. The fiber bundles are transversely isotropic materials, and the matrix is isotropic materials [12]. The woven structure of the yarn causes the yarn itself to have a bending arc, Therefore, it is necessary to define the direction for yarns at different arc positions to maintain *E*1 always along the fiber axis in the yarn.

# *3.2 Calculation Results of RVE Finite Element Model for Woven Structures*

Generate a unit cell model with a fiber volume fraction of 45% in Digimat software, solve the stiffness and flexibility matrix of the RVE, and calculate the nine engineering constants of the RVE. The calculation results of the woven RVE with a fiber volume fraction of 45% are as follows (Fig. 3).

# **4 Finite Element Analysis of Tensile Strength of 2D Woven Composite Materials**

# *4.1 Establishment of Finite Element Model for 2D Woven Composite Materials*

Due to the complex periodic boundary application of woven RVE, three types of voxel mesh models were established for woven RVE, with voxel mesh sizes of (0.129 mm × 0.0069 mm × 0.0069 mm), with a grid quantity of 14,400, 57,600 and 90,000, respectively. After the grid components are generated in Digit, they are imported into ABAQUS for post-processing. Since the catastrophic failure of woven RVE mainly occurs when the fiber bundle breaks, the damage of the matrix near the fiber bundle

**Fig. 3** Finite element results of engineering constants for woven RVE

is not considered, and the Periodic boundary conditions conflicts with the ABAQUS display dynamics algorithm, Statics is comprehensively considered for calculation, and the UMAT subroutine based on 3D Hashin failure criterion is compiled for failure simulation, where the 3D Hashin failure criterion has four failure modes, They are: fiber axial tensile failure, fiber axial compression failure, fiber normal tensile failure, and fiber normal compression failure.

This study selected the material parameters and their sizes from the literature for the simulation of the woven RVE model. The specific dimensions and material parameters are shown in Tables 3 and 4 (with a fiber volume fraction of 0.722 in the fiber bundle and 0.453 in the woven composite material). Given the volume fraction and component material parameters, the material parameters of the fiber bundle can be obtained. The stiffness of the fiber bundle is directly calculated based on Chamis's equations [13], and the strength is directly calculated based on the statistical strength theory of Zhou et al. [14]. The theory suggests that the strength of the fiber bundle is distributed according to the Weibull rule, taking into account existing formulas, and no further micro mechanical performance analysis of the fiber bundle will be conducted separately.


**Table 3** Material properties of the constituents [15]

# **Table 4** Strength parameters


The load displacement curves of the finite element calculation results and experimental results are shown in Fig. 4, indicating a good agreement between the experimental curve and the simulation calculation curve.

For different weaving methods, failure simulation calculations were conducted on three types of woven composite materials under the same volume fraction. The load displacement curves of plain weave, twill weave, and satin weave were obtained, as shown in Fig. 5. Catastrophic failures occurred on the fiber bundle in the tensile direction, and the main failure mode was tensile failure along the fiber axis direction. Among them, the stress distribution and damage situation of three types of woven fiber composite materials during failure are shown in Fig. 6 (with hidden resin matrix element). From the figure, it can be seen that tensile failure in the axial direction mainly occurs at the intersection of fiber bundles, where there is obvious stress concentration, and failure also occurs first in these areas.


**Fig. 6** Stress distribution and damage situation of three types of woven composite materials during failure

# **5 Conclusions**

This study simulates the stiffness and strength of three 2D woven composite materials by establishing a composite material RVE finite element model. The stiffness and strength of three types of woven composite materials were compared under the same volume fraction, and the following conclusions were drawn:


**Acknowledgements** This work was supported by a grant from the Open Fund Project of Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education, Chongqing Jiaotong University (No. SLK2023A01), the Open Fund Project of State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (No. LP2314), and Natural Science Foundation of Hubei Province (No. 2023AFB120).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation Study of Shear Stress Distribution in Bolted Connection Structures of Sandwich Composite Plate**

**Ruizhang You, Renjun Yan, Haowen Zhu, and Ziwei Zhang** 

**Abstract** For the shear problem of the bolt connection structure of Sandwich Composite Plate, shear test is carried out on two kinds of Bolt Connection Structure of Sandwich Composite Plate, namely, Pre-embedded countersunk bolt and Assembled countersunk bolt, and the shear strength and stress distribution of the two kinds of bolt connection structure are investigated by using ABAQUS, and the finite element simulation results coincided with the test phenomenon well. Using this model, the stress distribution under shear damage of the Bolt Connection Structure of Sandwich Composite Plate is analysed, and the influence of bolt preload is also analysed, and the results showed that: the stress of the Sandwich Composite Plate is mainly concentrated in the Skin on the bottom side, and gradually decreased to the top surface; the influence of bolt preload could be disregarded in the analysis of the shear strength and stress peak of the Sandwich Composite Plate.

**Keywords** Sandwich composite plate · Bolt connection structure · Finite element simulation · Shear stress distribution · Bolt preload

# **1 Introduction**

Sandwich Composite Plate is composed of high-strength Composite Plate Skins and lightweight Core Materials, which are widely used in the manufacture of ships and marine structures by virtue of their light weight, high stiffness, high strength, and

R. You · R. Yan (B) · H. Zhu

School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan 430063, China e-mail: yan\_renjun@163.com

R. Yan

© The Author(s) 2024

Educational Ministry Key Lab of High Performance Ship Technology, Wuhan University of Technology, Wuhan 430063, China

Z. Zhang Changsha BYD Auto Co, Ltd., Changsha 410116, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_79

superior acoustic stealth properties [1]. In the ship structure, the steel structure as the main structure, Sandwich Composite Plate is often used as a component fixed to the main structure by bolts [2], Sandwich Composite Plate, due to the characteristics of its constituent structure, the strength of the various components of the larger differences, the damage mode is more complex and serious compared to the Metal Materials and Composite Materials. At the same time for the demand of bolt connection structure, the Sandwich Composite Plate for the open hole or cross-section mutation processing, inevitably cause the phenomenon of stress concentration, so the mechanical properties of the Sandwich Composite Plate and steel skeleton bolt joints research is very important.

In recent years, a large number of scholars have carried out meticulous experimental and numerical studies on the bolted connection structure of composite laminates, Chishti, Chen, and Liu [3–5] studied the countersunk bolt connection structure of composite laminates, using experimental, numerical simulation, and theoretical analysis to study different bolt countersunk sizes, analysis the force state of the countersunk bolts, and reveal the damage mechanism of composite plywood connection structure, Sen and Ascione [6, 7] conducted experimental studies on the effects of bolt hole size and the distance from the hole to the edge of composite plywood on the bolted structure of sandwich composite plywood, Xu [8] carried out experimental studies on the bolted structure of composite plywood of L-type and ∏-type, An [9] carried out fatigue tests on a number of 3-nailed carbon fibre reinforced Plastic and titanium alloy countersunk bolt connection structures with bushings, and established a fatigue numerical analysis model to compare with the connection model without bushings, which showed that the bolt connection structure with bushings had higher fatigue life.

And less research on Sandwich Composite Plate [10, 11], this paper for the actual situation of Sandwich Composite Deck Structure, Pre-embedded countersunk bolts and Assembled countersunk bolts two kinds of connection structure for finite element simulation, using commercial finite element software ABAQUS and the different bolt preload under the shear strength and stress distribution of the analysis and comparison, to provide a research foundation for exploring the design and application in bolted connection structures of Sandwich Composite Plate.

# **2 Structural Shear Test in Bolted Connection Structures of Sandwich Composite Plate**

### *2.1 Test Specimen*

Reference to the typical parts of the hull deck and rigid skeleton connection, the structural shear specimen of sandwich composite and steel skeleton connection assembled with pre-buried countersunk head bolts and assembled countersunk bolts is designed, which consists of two parts: the Sandwich Composite Plate and the steel skeleton plate, and is connected by a special bushing and high strength bolt. Pre-embedded countersunk head bolts are buried to the inner side of the skin, and Assembled countersunk head bolts run through the Sandwich Composite Plate, and the bolts are tightened with a torque of 20 N m during installation. The specific dimensions of the connection structure are as follows: the Sandwich Composite Plate is a 300 mm × 150 mm × 66 mm rectangle; the sandwich Plate Skin layup material is glass fibre, with a thickness of about 3 mm; the Core Material is 60 mm of polyethylene foam; the special bushing is T-shaped, with an outer diameter transitioning from 45 to 30 mm and an inner diameter transitioning from 25 to 22 mm. The center of the bolt hole is located 45 mm from the short edge of the Sandwich Composite Plate, and bolt diameter is 20 mm.

### *2.2 Test Equipment and Process*

The hull deck boundary is generally supported by strong components (e.g. steel skeleton, hull wall, etc.), when the hull deck is loaded, the bolts at the connecting structure are in the shear condition, and through the design of reasonable tooling, the loading conditions in the laboratory are as much as possible in line with the actual working conditions. Changchun Qianbang test equipment QBS-350A electrohydraulic servo fatigue testing machine is used to carry out the shear strength test, the test steps are as follows: (1) numbering of the specimen before the test; (2) determining the position of the tooling, bolting the tooling to the test platform; (3) fixing the specimen on the tooling, loosening all the fixing bolts, and using the test machine to preload the specimen, so that the test machine loading collet is vertically corresponds to the loading position, packing the test specimen, and then the test specimen is preloaded. (4) fix all the constraints, debug the test system, and complete all the preparatory work; (5) The shear strength test adopts displacement control, and the test piece is loaded at a displacement rate of 0.1 mm/s until the end of damage.

### *2.3 Experimental Phenomena*

Through the shear test of the Sandwich Composite Plate and steel skeleton connection structure, it is found that the bolts located in the lower Skin appeared to be bent and deformed, as well as the lower side of the skin bolt holes appeared to be extruded damage, while the upper side of the Skin and the Core Material did not appear to be damaged, as shown in Fig. 1.

(a) Pre-embedded countersunk head bolt connection structure

(b) Assembled countersunk head bolt connection structures

**Fig. 1** Shear test phenomena in the bolt connection structures of Sandwich Composite Plate

### **3 Finite Element Model**

### *3.1 Geometric and Material Model*

The finite element software ABAQUS is used to carry out finite element simulation analysis on the bolt connection structure of Sandwich Composite Plate. According to the actual size of the test specimen, the bolt, bushing, composite Skin, Core Material and rigid connecting plate are modeled according to the actual size of the test specimen, with the intersection point of the bolt centerline and the lower surface of the sandwich composite material plate as the origin, the bolt centerline as the yaxis, and the axis of symmetry of the short side of the sandwich composite material as the x-axis. Detailed dimensions of the finite element simulation model are given in Sect. 2.1, and the structural characteristics of the connection are shown in Fig. 2.

The Skin of Sandwich Composite Plate is made of glass fibre reinforced material, and the Core Material is made of polyethylene foam. The lamination sequence of Skin is [0/90/0/90] s. Based on the structural characteristics of Sandwich Composite Plate, the lamination direction is not established when modelling the composite material Skin. The Skin is modeled as an anisotropic laminate. The relevant material parameters of composite material Skin are shown in Table 1, and the relevant material parameters of polyethylene Core Material are shown in Table 2, See Table 3 for material parameters of connecting bolts.

Modulus of elasticity (MPa) Poisson's ratio Modulus of rigidity (MPa) *E*<sup>11</sup> *E*<sup>22</sup> *E*<sup>33</sup> μ<sup>12</sup> μ<sup>13</sup> μ<sup>23</sup> *G*<sup>12</sup> *G*<sup>13</sup> *G*<sup>23</sup> 20 20 8 0.12 0.12 0.12 6 6 4

**Table 1** Material parameters of composite material skin

**Table 2** Material parameters of polyethylene core material


**Table 3** Material parameters of connecting bolts


### *3.2 Interaction and Loading*

The model is set up with two kinds of interactions of "Tie" constraints and contact pairs. According to the principle of "main surface principle", the main surfaces are selected according to the order of bolts, connecting Plate, bushing, Skins and Core Materials. In order to study the effect of bolt connection structure on the shear strength of the Sandwich Composite Plate, contact pairs are set on the contact surfaces of the bolt and the bushing, the bushing and the Core Material, the bushing and the Skin, the bolt and the Skin, and the Skin and the connecting plate; and "Tie" constraints are set on the contact surfaces of the Skin and the Core Material, and the bolt and the connecting surface. The contact constraints are tangential behaviour with a friction coefficient of 0.1 and normal behaviour for hard contact. The gap between the bolt and the bushing is realised by means of the Interference Fit function in the contact settings.

The fixed constraint is applied to the right end of the sandwich composite material, and a displacement of 5 mm is applied to the left side of the connection plate. The bolt tightening torque is applied by the Bolt load command in ABAQUS [12], and its preload magnitude is calculated by the formula equation as (1).

$$F = \frac{T}{1.2\mu d} \tag{l}$$

When *T* = 20 N m, *d* = 26.08 mm and μ = 0.2, then *F* = 3.195 kN.

The Static, Risk analysis step is used to perform the shear strength analysis, and the Static, general analysis step is added to add the bolt preload to the model prior to the shear strength analysis.

### *3.3 Mesh Division Convergence Analysis*

Eight-node hexahedral linear reduced integral solid elements (C3D8R) are used to divide the model, and according to the research focus, the model is divided into three parts, the first part is the bolt and the bushing; the second part is the focus area of the Sandwich Composite Plate, which is a square area with a side length of 90 mm centered on the bolt axis; and the third part is the non-focus area of the Sandwich Composite Plate, which is all the other areas except the focus area. In order to reduce the computation time, smaller element sizes are set in the first and second parts, while larger element sizes are set in the third part. Comparison of the number of elements and calculated shear strength of the models with different element sizes is shown in Table 4, from which it can be seen that the shear strength of the connecting structure is greatly affected by the element sizes of the bolts in the first part, and the stress distribution of the Sandwich Composite Plate is greatly affected by the element sizes of the first part and the second part, so in order to save the computation time and to


**Table 4** Comparison table of different modelsa

a The CPU of computer is Intel Core i7-12700 (2.10 GHz)

ensure the accuracy of the calculation, model 3 (the first part of the element sizes of 2 mm, the second part of the element sizes of 3 mm, the third part of the element sizes of 5 mm) is selected as the model of the bolted connection structure of the Sandwich Composite Plate.

# **4 Finite Element Analysis in Bolted Connection Structures of Sandwich Composite Plate**

### *4.1 Shear Stress Distribution Analysis*

Shear analysis is conducted on the bolted connection structures of two types of Sandwich Composite Plate. The Mises stress cloud maps of each component at the maximum shear force are taken as shown in Figs. 3, 4 and 5. The deformation of the bolts in Fig. 3 is basically consistent with the deformation of the bolts in the experimental phenomenon in Fig. 1. The Mises stress above 1000 MPa appeared in the Skin on bottom side of Fig. 5, which is consistent with the phenomenon of compression failure in the Skin on bottom side of the experimental phenomenon. It can be verified that the numerical simulation technology used is true and effective.

Figure 6 shows the stress variation curve along the length of Sandwich Composite Plate, where the horizontal coordinate is the x-axis coordinate of the finite element model, and the vertical coordinate is the Mises stress. It can be seen from Fig. 6 that the stress distribution trend of Sandwich Composite Plate of the pre-embedded countersunk bolt connection structure and the assembled countersunk bolt connection structure is basically the same, which manifests itself in that the stress on the bottom surface is much larger than that on the top surface, and the Skin on bottom side and the lower surface of Core Material reach peak value on the side of the bolt hole wall

(a) Pre-embedded countersunk head bolt connection structure

(b) Assembled countersunk head bolt connection structures

**Fig. 3** Stress distribution and deformation of bolts in the bolt connection structures of Sandwich Composite Plate

**Fig. 4** Stress distribution of core material in the bolt connection structures of Sandwich Composite Plate

close to the loading end. And the Skin on top side and upper surface of Core material reach the peak at the side of the bolt hole wall near the fixed end. From Fig. 6b, it can be found that the peak stress of Core Material of assembled countersunk head bolt connection structure is larger than that of pre-embedded countersunk head bolt connection structure.

Figure 7 shows the stress variation curve along the width of Sandwich Composite Plate, where the horizontal coordinate is the z-axis coordinate of the finite element model, and the vertical coordinate is the Mises stress. It can be found from Fig. 7 that the stress distribution trend of Sandwich Composite Plate of the pre-embedded countersunk head bolted connection structure and the assembled countersunk head bolted connection structure is basically the same, which manifests itself in the fact that the stress on bottom surface is much greater than that on top surface, and the stress reaches the peak in the wall of the bolted holes, and the pre-embedded countersunk head bolted connection structure exhibits a greater peak of the stress on the Skin on the bottom side and the lower surface of the Core Material, whereas the assembled countersunk head bolted connection structure exhibits a greater peak of the stress on the Skin on the upper side and the upper surface of the Core Material.

Figure 8 shows the stress variation curve along the thickness of Sandwich Composite Plate, where the horizontal coordinate is the y-axis coordinate of the finite element model and the vertical coordinate is the Mises stress of the bolt hole wall. From Fig. 8, it can be observed that the stress on the bolt hole wall inside the Core Material gradually decreases from bottom to top, while the stress of the Skin on bottom side is much greater than that of the Core Material, and the stress of the Skin on top side is slightly greater than that of the Core Material.

### *4.2 Analysis of the Influence of Bolt Preload*

Figure 9 shows the variation curve of shear strength of the bolt connection structures of Sandwich Composite Plate with bolt preload, where the horizontal coordinate is the bolt preload, and the vertical coordinate is the shear strength of the connection structure, from the figure it can be seen that the shear strength of the two kinds of bolt connection structure is similar, and is less affected by the bolt preload, and under the action of different bolt preloads, the shear strength is all between 133–135 kN. Figure 10 shows the variation curve of maximum stress of sandwich composite plate connection structure with bolt preload, where the horizontal coordinate is the bolt preload, and the vertical coordinate is the maximum stress of core and skin when the connection structure reaches the shear strength. Figure 10 shows the variation curve of maximum stress of connection structure with bolt preload, where the horizontal coordinate is the bolt preload, and the vertical coordinate is the maximum stress of Core Material and Skin when the connection structure reaches the shear strength, and it can be found in the Fig. 10 that the maximum stress of the bolt connection structure of the Sandwich Composite Plate is less affected by the bolt preload, and the maximum stress of the pre-buried countersunk head bolted connection structure

(a) Pre-embedded countersunk head bolt connection structure

**Fig. 5** Stress distribution of skin in the bolt connection structures of Sandwich Composite Plate

is larger than that of the assembled countersunk head bolt connection structure, in which the maximum stress of skin is about 13.88% greater than that of the assembled countersunk head bolt connection structure, and the maximum stress of the Core Material is about 9.20% larger.

# **5 Conclusions**

The shear failure phenomenon of the bolt connection structure of Sandwich Composite Plate is studied through test, and then the finite element simulation model is established to study the pre-embedded countersunk bolt connection structure and assembled countersunk bolt connection structure. The bolt deformation of finite element simulation and the stress concentration phenomenon of the Skin of Sandwich Composite Plate coincide well, based on which the stress distribution of the bolt connection structure of Sandwich Composite Plate and the influence of the bolt preload on the shear strength and maximum stress of the connection structure are analyzed, and the following conclusions are obtained:

(1) The structural shear failure modes of the bolt connection structure of Sandwich Composite Plate is bolt shear damage and extrusion damage at the Skin openings on the bottom side.


**Fig. 7** Stress variation curve along the width of Sandwich Composite Plate

**Fig. 10** Variation curve of maximum stress of the bolt connection structure of Sandwich Composite Plate with bolt preload

### **Appendices**


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Intelligent Technology Application and Safety Management**

# **Research on Path Planning of Concrete Pouring Construction Robot Based on Online Obstacle Avoidance Algorithm**

**Siwen Fan, Wanli Li, and Ruihao Yin** 

**Abstract** Aiming at the problem of harsh concrete construction environment and low level automation of concrete pouring operation, this paper proposes an autonomous concrete pouring planning method based on real-time obstacle avoidance algorithm. A new 3-DoF articulated concrete pouring robot is designed and kinematically modeled. Rectangular fitting is adopted to simplify irregular obstacles. Then generate obstacle maps and apply intelligent obstacle avoidance algorithms based on the maps, which adopt the reward mechanism in reinforcement learning. The effectiveness of the proposed method in autonomous obstacle avoidance path planning for pouring robots is demonstrated through simulation and prototype experiments.

**Keywords** Concrete pouring construction robot · Pouring path planning · Online obstacle avoidance · Reinforcement learning

# **1 Introduction**

Concrete pouring refers to the process of pouring concrete material into a designed mold until it is plasticized [1], and now concrete pump trucks and pumping pouring technology are mostly used for pouring operations [2]. In the real concrete pump truck pouring process, the cooperative control of the pump truck boom often depends on the rich experience of the construction site operators [3], through manual control of the end joint hose to ensure that the end of the boom pouring mouth in accordance with the expected trajectory to move, so as to better complete the concrete pouring [4]. However, under this operation method, there are certain safety troubles, which not only directly affect the quality of pouring, but also threaten personal safety. And as the scale of urban construction is expanding, in order to alleviate the situation of tense construction land, the development of high building is accelerating, and at the

© The Author(s) 2024

S. Fan (B) · W. Li · R. Yin

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: 1910416@tongji.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_80

same time, the complexity of the building structure is also increasing, especially in the ultra-long, large-volume concrete pouring surface, ultra-high-rise buildings, the concrete pouring of underground steel sinkholes, the application of the traditional boom-type concrete pump trucks has a certain test [5, 6]. Therefore the robotization of the concrete pouring process is particularly urgent and important for the development of construction engineering and intelligent construction.

In the early stage, the research on concrete pump trucks mainly focused on the structural analysis and vibration suppression control of the boom. Guo et al. carried out a finite element analysis of the horizontal working condition of the material handling mechanism and obtained the dynamic response of the material handling system structure, as well as the distribution of the stress and deformation of the system at various moments of the time [7]. Khulief et al. designed a vibration control method of the boom by establishing a finite element model of a flexible arm and analyzing the dynamics characteristics [8]. Qiu et al. studied the synchronized positioning control and vibration suppression of a new type of flexible robotic arm [9]. Henikl et al. took the concrete pump truck boom as the research object, applied the modal control theory, and put forward the concept of active damping and decentralized control of the vibration of the flexible arm of the pump truck. Applications are ineffective and considered an active control method to suppress boom vibration [10]. Cazzulani et al. concluded that the method of vibration suppression through external passive compensating devices is not effective in concrete placing applications and considered an active control method to suppress boom vibration [11].

Concrete pouring construction robot is the latest improvement of intelligent construction machinery based on the truck mounted concrete pump which is now heavily used in the field of concrete pouring [12]. Recently, more and more researchers pay attention to the new construction machinery because it has both the practicality of the traditional concrete pouring machinery and the intelligence of the robot. Due to the complex and changing environment of concrete construction, the existence of pouring exclusion zones is often faced during the concrete pouring process, including the area where the pouring has been completed and the multiple construction equipments that exist in the construction site. Therefore, it is essential to plan the pouring path of this construction robot in real time, particularly with regard to construction safety. Huang et al. proposed a pseudo-distance based obstacle avoidance method for the end obstacle avoidance and boom joint limitation problems of concrete pouring robots, which is currently a more advanced research in obstacle avoidance of pouring robots [13]. In addition, in terms of path planning based on obstacle avoidance, reinforcement learning is currently a more advanced intelligent algorithm, and current applications have been able to achieve good autonomous planning performance, especially in the field of mobile robots [14].

Considering the impact of the construction environment on concrete pouring, an intelligent trajectory planning method based on grid map is proposed for a 3-DoF articulated concrete pouring construction robot, which applies an advanced real-time obstacle avoidance algorithm with improvements for the first time.

In this paper, we will combine the algorithmic ideas in reinforcement learning and the characteristics of concrete pouring construction to establish an autonomous path planning system for concrete pouring robots. The new designed robot is firstly analyzed for kinematics and a complete kinematic model with D-H parameter method is also developed. Then, the simplified pouring map with obstacles is building based on the construction environment. Aiming at the obstacle avoidance in the map, the Bellman equation in reinforcement learning is used as the obstacle avoidance policy.

Moreover, experiments were conducted on a prototype robot based on the obstacle avoidance method. Finally, the results of simulation and experiment show that the presented method in this research is well performed to avoid obstacles autonomously, and this research has a prospective impact on the field of autonomous concrete construction machinery.

### **2 Robot Modeling and Analysis**

In order to better achieve the intelligence of concrete pouring construction, in this study, a 3-DoF articulated concrete pouring construction robot is designed according to the existing structure of the concrete pouring boom, and it is used as the research object of pouring trajectory planning.

### *2.1 Design of Robot Structure*

As shown in Fig. 1, the robot structure refers to the design of simple rotary joints for concrete pouring boom to simplify motion control, reduce the complexity of the combined motion of arm stretching and rotating in traditional concrete pouring machinery. It is suitable for later integration with intelligent platforms and the application of algorithms to facilitate the control of multiple robots. The robot structure consists of 4 parts, including: 1-column, 2-pouring pipeline, 3-horizontal support part, 4-slewing ring bearing. The column assembly adopts lightweight truss structure, which plays a reinforcing role, and it is easy to disassemble and adapt to pouring planes of different heights, especially for ultra-high level planes which cannot be reached by traditional concrete pump trucks. Concrete and fluid are transported to the pipeline section through the pump and output pouring at the end of the pipeline. The pipe section is supported by part 3 and part 4 and connected by part 4, which is equivalent to the robot rotary joints and driven by servo motors.

Figure 2 shows the working range of the concrete pouring construction robot, with 2 degrees of freedom in the pouring plane x–y and 3 degrees of freedom for the 3-joint pouring robot. Therefore, an additional redundant degree of articulation increases the number of possible pouring path solutions and simplifies the structural complexity of the robot. It is suitable for the more complex construction environment and working conditions, which can further improve the construction efficiency and reduce the construction cost by enhancing the operational flexibility of the pouring

**Fig. 1** The structure of concrete pouring construction robot

robot and strengthening the stability of the mechanism under the adverse working conditions.

### *2.2 Kinematic Modeling and Analysis*

Considering that the robot concrete pouring work is done in Cartesian space and the motion control of the robot is based on Joint space, kinematic modeling of the robot is needed to complete the conversion from Cartesian space to Joint space. Based on the above new designed robot structure and D-H parameter representation, the model under the Cartesian space is established in the Robotics Toolbox in MATLAB, as shown in Fig. 3.

The robot has four coordinate systems, which are including a base coordinate system and three joint coordinate systems. Then the positive kinematic model of the robot is established based on the coordinate systems, and the relationship between neighboring coordinates is deduced as formula (1).

**Fig. 3** The corresponding structure in MATLAB

$$\begin{aligned} \,^i\_i \,^{i-1}T = \begin{bmatrix} \cos\theta\_i - \sin\theta\_i \ 0 \ 0 \\ \sin\theta\_i & \cos\theta\_i \ 0 \ 0 \\ 0 & 0 & 1 \ 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \\ 0 \ 0 \ 1 \ 0 \\ 0 \ 0 \ 0 \ 1 \end{bmatrix} \end{aligned} \tag{1}$$

i is the number of joints of the robot, here i = 3. To simplify the equation, ci and si denote a and b respectively. Substitution can be made to obtain the relationship between the pouring actuator of the robot and the fixed base as formula (2).

$$\begin{aligned} \,^0\_3 T &= \,^0\_1 T \times \,^1\_2 T \times \,^2\_3 T \\ &= \begin{bmatrix} \mathbf{n}\_x \ \mathbf{o}\_x \ a\_x \ \mathbf{p}\_x \\ \mathbf{n}\_y \ \mathbf{o}\_y \ a\_y \ \mathbf{p}\_y \\ \mathbf{n}\_x \ \mathbf{o}\_z \ a\_z \ \mathbf{p}\_z \\ 0 \ 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} \mathbf{c}\_{123} & -\mathbf{s}\_{123} & 0 \ I\_1 \mathbf{c}\_1 + I\_2 \mathbf{c}\_{12} + I\_3 \mathbf{c}\_{123} \\ \mathbf{s}\_{123} & \mathbf{c}\_{123} & 0 \ I\_1 \mathbf{s}\_1 + I\_2 \mathbf{s}\_{12} + I\_3 \mathbf{s}\_{123} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{2}$$

As shown in the simplified schematic of the robot in Fig. 4, the inverse kinematics model of the robot is solved based on the above positional relationship, so that the rotation joint angles of the robot θ1, θ2, θ3 can be obtained from the known coordinates of the pouring point P*x* , P*<sup>y</sup>* , and the inverse kinematics model is calculated by the geometric method as formula (3).

$$\begin{cases} P\_x = l\_1 \times \cos \alpha\_1 + l\_2 \times \cos \alpha\_2 + l\_3 \times \cos \alpha\_3 \\ P\_y = l\_1 \times \sin \alpha\_1 + l\_2 \times \sin \alpha\_2 + l\_3 \times \sin \alpha\_3 \\ \alpha\_1 = \theta\_1 \\ \alpha\_2 = \alpha\_1 + \theta\_2 = \theta\_1 + \theta\_2 \\ \alpha\_3 = \alpha\_2 + \theta\_3 = \theta\_1 + \theta\_2 + \theta\_3 \end{cases} \tag{3}$$

After solve the equations simultaneously, the following joint angles are obtained for each pouring point.

# **3 Concrete Pouring Path Planning**

In the construction site, the pouring exclusion zone is usually irregular, considering that the precision requirement of concrete pouring operation is not high, in order to facilitate the faster application of obstacle avoidance algorithms, irregular obstacles are fitted into rectangles, as shown in Fig. 5a, and the orange part is the irregular obstacle. The input obstacle is subjected to image processing, using MATLAB Image Processing Toolbox to obtain the edge contour of the irregular obstacle, dispersing it into a series of coordinate points (*xbi* , *ybi* ) and seeking its extreme value, and then, as in formula (4), the difference of the extreme value obtained in its x and y coordinates respectively is used as the length a and width b to automatically fit a rectangular obstacle enclosing all coordinate points.

$$\begin{cases} \mathbf{a}\_{j} = |\max(\mathbf{x}\_{bi}) - \min(\mathbf{x}\_{bi})| \\ \mathbf{b}\_{j} = \left| \max(\mathbf{y}\_{bi}) - \min(\mathbf{y}\_{bi}) \right| \end{cases} \tag{4}$$

**Fig. 5** Simplified fitting obstacle

map

For obstacles with a certain height, the mid-arm avoidance also needs to be considered. In order to simplify the pouring path computational solution, the concept of safety margin is utilized to form a safe avoidance area outside the simplified obstacle, as shown in Fig. 5b. The width s of the margin area consists of the reaction distance *sre* and braking distance of the robot *s*br, and since each arm may not have the same motion speed, the maximum value of the all arms width is taken as s as shown in formula (5).

$$\begin{cases} \mathbf{s} = \max(\mathbf{s}\_{re} + \mathbf{s}\_{br}) \\ \mathbf{s}\_{re} = |\mathbf{v}\_{cui} \cdot \mathbf{t}| \\ \mathbf{s}\_{br} = \begin{vmatrix} \frac{\mathbf{v}\_{cui}}{2 \cdot \mu \cdot \mathbf{a}\_{rti}} \end{vmatrix} \end{cases} \tag{5}$$

In the above formula, v*cui* is the current end running speed of robot arm i, t is the reaction time, a*bri* is the braking acceleration, and μ is the braking damping coefficient. Based on the above method, obstacle fitting is performed to obtain a simplified construction environment map with obstacles as shown in Fig. 6.

Obstacle avoidance adopts an intelligent algorithm based on a reinforcement learning reward mechanism, where the core idea in reinforcement learning is the

solution of the Bellman equation as shown in formula (6) [15].

$$V\_{\pi}(\mathbf{s}) = \sum\_{a} \pi(a|\mathbf{s}) \cdot \left[ \sum\_{r} p(r|\mathbf{s}, a) \cdot r + \chi \cdot \sum\_{\mathbf{s'}} V\_{\pi}(\mathbf{s'}) \cdot p(\mathbf{s'}|\mathbf{s}, a) \right] \tag{6}$$

a, s, s' , r, π represent the important components of reinforcement learning action, state, next moment state, reward and policy respectively. The state value *V*π(*s*) is obtained by solving the Bellman equation after substituting all the values of given quantity. The state space S of the pouring robot consists of the angle values of each joint of the robot and can be expressed as in formula (7).

$$\mathbf{s}\_{i} = [\theta\_{1i}, \theta\_{2i}, \theta\_{3i}], \mathbf{s}\_{i} \in \mathbb{S}, \mathbb{S} \in \left[ -\Im 60^{\circ}, \Im 60^{\circ} \right] \tag{7}$$

The action a of the pouring robot is composed of 3 actions, forward motion, stationary and backward motion, which has the value of unit joint rotation or 0. Since the robot has 3 joints, the action matrix has the specification of 3×3×3, which represents 27 states under all feasible actions of the robot. To prevent falling into a long term static state dead zone, state [0, 0, 0] are removed, thus the action matrix a includes 26 action states that are feasible for the concrete pouring robot. In the reward r setting, r consists of *r*1, *r*2, *r*3 as shown in formula (8).

$$\mathbf{r} = r\_1 + r\_2 + r\_3 \tag{8}$$

In the setting of *r*1, when the robot arm passes through the obstacle area, *r*1 = −∞, and vice versa *r*2 = 0, which ensures the obstacle avoidance function of the robot arm. In the setting of *r*2, *r*2 is the sum of all unit joint rotation, which ensures the optimal energy target of the robot arm motion. In the setting of *r*3, when the end of the robot arm reaches the target pouring position, a larger reward value scalar Rtar is given to *r*3, which ensures that the robot can reach the correct pouring area.

Then solve formula (6) after predefine discount factor γ, and iteratively update it to find the maximum state value and output the policy as the optimal solution. As shown in Fig. 7, after applying the obstacle avoidance algorithm based on the simplified obstacle map, the frobot can achieve good obstacle avoidance performance.

### **4 Experiment and Verification**

According to the intelligent pouring path planning method proposed in this research, the planning system of the robot was established, as shown in Fig. 8, and experiments were carried out on the prototype of the concrete pouring robot.

The concrete pouring construction robot planning system is divided into 3 parts, including path planning, trajectory generation and prototype experiment of the robot.

**(a)** Obstacle avoidance algorithm before use **(b)** Obstacle avoidance algorithm after use

**Fig. 8** Algorithm-based concrete pouring construction robot planning system

In the path planning part, the obstacles are first fitted into rectangles using the planning method proposed in this research, and it is used to build a simplified construction map. In the prototype experiments, colorful stickers are used to represent the fitting obstacles. The robot then generates a series of actions in the trajectory generation to conform to the optimally planned path and forms a joint angle curve θ*i* (t) from the values of this action. The joint motion signal is transmitted to the servo controller, performs the servo motor control of the robot three rotary joints. Finally, the experimental results showed the feasibility of the planning method proposed in this research.

### **5 Conclusion**

In this paper, a new 3-DoF articulated concrete pouring construction robot is designed, and a robot autonomous pouring path planning method is proposed based on this robot structure. For the complex and changeable construction environment, a simplified obstacle map is established, and an obstacle avoidance algorithm based on reinforcement learning idea is applied. In the simulation and prototype experiments, the method shows obstacle avoidance feasibility, which is a good reference value for the intelligence of concrete construction robots.

### **References**


15. Sutton RS, Barto AG (1998) Reinforcement learning: an introduction. IEEE Trans Neural Netw 9(5):1054

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research and Practice on the Validation Testing Method of Safety of the Intended Functionality for High Way Assist Function**

**Kuiyuan Guo, Jiarui Zhang, Juan Shi, Zhiqiang Zhang, and Guotian Ji** 

**Abstract** With the continuous development of the level of driving automation, the need for safety of the intended functionality (SOTIF) has gradually become prominent. As an important part of ensuring that vehicles meet the requirements of SOTIF, unknown scenario verification methods urgently need research and breakthroughs. A customized SOTIF verification test method for highway Assist (HWA) function based on Design Operation Domain (ODD) and a test route formulation method based on multi-dimensional key elements are studied and proposed. This method considers the data dimensions and environmental characteristics involved when deriving the acceptance criteria, and effectively meets the effective and practicable requirements of unknown scenario tests.

**Keywords** HWA · Safety of the intended functionality · Confirmation test · Test methodology · Unknown scenario

# **1 Introduction**

With the continuous development of the level of driving automation, the SOTIF need for driving automation system has become increasingly prominent [1]. As an important link to ensure that vehicles meet the requirements of SOTIF, unknown scenario verification method urgently needs research and breakthrough. As an important pillar of intelligent connected vehicle safety, expected functional safety is mentioned in many domestic and foreign intelligent connected vehicle policies and regulations

CATARC Automotive Test Center (Tianjin) Co., Ltd., Tianjin 300300, China

G. Ji (B)

© The Author(s) 2024

1025

K. Guo · J. Zhang · J. Shi · Z. Zhang

China Automotive Technology & Research Center Co., Ltd., Tianjin 300300, China e-mail: jiguotian@catarc.ac.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_81

such as Opinions on Strengthening intelligent Connected Vehicle Production Enterprises and Product Access Management [2], and UN R157. At present, there are mainly the following problems in the confirmation test of expected functional safety:


At present, there is a lack of custom formulation methods based on target market for the formulation of acceptable criteria for expected functional safety. This paper mainly studies the second-level acceptance criteria and related validation tests. This paper studies and proposes a customized SOTIF validation test method for highway assistance (HWA) functions based on Design Operation Domain (ODD). A validation test method based on multi-dimensional key elements of the target market is designed. Based on the HWP function, experiments are carried out to verify the effectiveness and practicability of the method.

# **2 Brief Description of Expected Functional Safety Validation Test Requirements**

For autonomous driving functions, it is crucial to demonstrate that the system does not create unreasonable risks at SOTIF level. In order to meet the requirements of SOTIF, autonomous driving systems generally carry out safety demonstration from two layers of acceptance criteria. Acceptance criteria consist of first-level acceptance criteria and second-level acceptance criteria [3]. The first level of acceptance criteria is the hazard behavior acceptance criteria, which is used to determine whether the vehicle's behavior constitutes a hazardous behavior [4]. The criteria here are "controllable" (C = 0) or severity rated "no harm" (S = 0). The second level of acceptance criteria is the residual risk acceptance criteria., which is used to determine whether the residual risk is at a reasonable level during the operation of the vehicle. The first level of acceptance criteria is the micro-level acceptance criteria. The second level of acceptance criteria is an acceptance criterion that evaluates the risks at the vehicle level from a macro perspective. The second layer of acceptance criteria mainly considers whether unknown scenarios can accept risks at the vehicle level. Validation testing for SOTIF is primarily a validation of the second level of acceptance criteria, i.e. testing for residual risks in unknown scenarios.

After the first layer of acceptance criteria defines hazardous behavior at the micro level, the second layer accepts that the criteria constrain the residual risk at the vehicle level at the macro level. The confirmation verification strategy is the strategy of actual verification and confirmation testing of the two-layer acceptance criteria. This article focuses on a detailed study of the confirmation test methods for the second acceptance criteria, taking into account the characteristics of the target market.

### **3 Expected Functional Safety Confirmation Test Method**

The SOTIF validation test method mainly includes the formulation of the confirmation goal and the confirmation test execution method.

### *3.1 Identification Targets Based on Target Markets*

Based on the acceptance criteria, the export process for confirming the target is:


The second level of acceptance criteria mainly deals with the residual risk at the vehicle level from the macro level, mainly to determine whether the hazard risk accumulated during operation is at a reasonable acceptable level. The second level of acceptance criteria can be demonstrated in terms of the number of vehicle accidents and/or the number of casualties. Starting with the official statistics of the target market is a recommended path, and this article will provide a target market data derivation idea to demonstrate the implementation of the second level acceptance standard at the macro level.

Considering the information of the target market in the process of formulating the second-level acceptance criteria can ensure that the acceptance criteria are formulated in line with the conditions of the target market. The derivation process of the second level acceptance criterion mainly has the following steps: 1. Hazard scenarios with unreasonable risk need to be considered; 2. Data sources derived from the definition of acceptance of guideline statistical information, in the general case accident statistics B; 3. Consider the safety margin Y; 4. Provide an acceptance criterion AH with a safety margin, and finally derive an acceptance criterion with a safety margin AH = 1*/(*b ∗ Y*)*.

The second level of acceptance criteria is a risk assessment of the autonomous driving system at a holistic level, so the derivation data should be based on the information that the target is placed on the market. At present, there is a lack of systematic combing of the method of generating second-tier acceptance criteria based on target city customization, and this study provides a second-layer acceptance criterion setting method based on target market.

In the data sources for the second tier of acceptance criteria, the data need to consider the following factors:


The resources that can be used include: data from the National Bureau of Statistics, cooperation and exchange with local traffic management departments, road test development, literature, network data, etc. Here, the HWA function target ODD is used as an example in Chongqing. First, it is determined that the ODD of the HWA function is a high-speed structured road.

Taking Chongqing as an example, the following data were obtained from the national statistical yearbook and literature (see Tables 1 and 2):

It can be seen that on the expressway in Chongqing, the average mileage of driver accidents is 10159635.08 km. From the GAMAB principle, autonomous driving


**Table 1** Statistics on targeted acceptance criteria in target cities

**Table 2** Examples of second-tier acceptance criteria in target cities


functions need to be at least better than the driver's ability, i.e. less average accident mileage.

From the above table, it can be obtained that with a safety margin of 2 and a confidence level of 0.99, and an ESC probability of 0.1, Chongqing's confirmed target is 93,573,500 km.

### *3.2 Requirements for Confirming the Test Method*

Confirming the test method requires, on the one hand, whether the system meets the residual risk acceptance criteria, that is, whether there is sufficient anticipation of the risk of unknown scenarios. On the other hand, it is necessary to fully discover the expected functional safety risks of the system and make timely functional modifications to ensure that the system fully meets the expected functional safety requirements. Therefore, for confirmatory testing, it is critical to meet the test adequacy of unknown scenarios and the data traceability of functional modifications for expected functional safety issues.

### *3.3 Select Development Mode for the Test Method*

The confirmation strategy of unknown scenarios mainly assigns mileage and duration under different test methods according to the key points of simulation and real vehicle testing (see Fig. 1).

Simulation test: According to the ODD function under test, set the distribution ratio of environmental requirements and traffic vehicle types, and automatically generate simulation test scenarios. The simulation test of unknown scenarios focuses on the random combination and coverage of different scenarios and scene parameters.

Real vehicle test: consider the project cycle and mileage, covering different road types, time types, weather types, and driver types. For unknown scenario testing, open road testing is necessary, and the route of open road needs to fully cover the key elements of ODD (duration, road type, etc.).

In order to explore the unknown hazard scenario of the HWP function and confirm that the probability of encountering the unknown hazard scenario is low enough, according to the limitation of its ODD, the large mileage test of the open road is the most close to the user use scenario and the most realistic test method, so this method is selected.

In view of the test adequacy of unknown scenarios, it is necessary to pay attention to the route selection of the confirmation test and the selection of the time of the confirmation test in the way of confirming the test method.

To confirm the selection of road sections for the test, it is necessary to traverse the expressway according to the distribution of high-speed routes in the target city

**Fig. 1** Confirm method for unknown scenarios

and the needs of scene traversal, and evenly distribute the time on each highway according to the distribution of traffic flow.

To confirm the selection of road sections for the test, it is necessary to traverse the expressway according to the distribution of high-speed routes in the target city and the needs of scene traversal, and evenly distribute the time on each highway according to the distribution of traffic flow. The time period for the confirmation test was selected to be distributed according to the percentage of day, night, and dusk available from local highway traffic statistics.

### *3.4 Test Equipment for Unknown Scenarios*

In terms of the data traceability required for functional modification of expected functional safety issues, it is especially critical to confirm the equipment mode, data recording method, data statistics and other methods of testing in unknown scenarios.

The types of data that need to be recorded to confirm the test are: external environment information, vehicle motion information, driver operation in the car, internal interaction information of the vehicle, and vehicle controller signals. In order to ensure the traceability and accuracy of the data, the test is carried out using a combination sensor and a data synchronization device.

Confirm the equipment method of the test (see Fig. 2): use external cameras to record the surrounding environment information of the vehicle, internal cameras to record the driver's hand and foot operations, and use the in-car camera to record the human–computer interaction display, use the inertial navigation system to record the movement information of the car, use the intelligent camera to record the relative position of the wheels and lane lines, use lidar to record the relative distance from the vehicle in front, and finally use the terminal equipment to synchronize all nodes and record the key nodes and information of the test for labeling.

During the test, the data is recorded through the truth value of the vehicle and the subjective point of the safety officer. The different risky behaviors are recorded as follows (see Table. 3).

The subjective data is the safety officer's continuous labeling of the scene in the passenger seat during the test execution, and the objective data is the camera, lidar and related controller signals.

### **4 Experimental Validation**

According to the above process, this paper carries out a confirmation test based on a prototype of HWP, selects the local highway to carry out forward and reverse bidirectional traversal according to the method proposed in this paper, and the test duration distribution is defined according to the statistical definition of the highway traffic time period, and finally carries out the verification test according to this test

#### **Fig. 2** Test equipment

#### **Table 3** Data recording


strategy. The high-speed test period has been opened for 1989.7 km and has been tested for a total of 1940 min. There are 182 times hazard behavior detected, and the average mileage of hazard behavior was 10.9 km (see Table. 4).

The distribution of risky behaviors and the main causes are shown in the Table 5. Next, more vehicles with HWA systems on the market will be tested and verified to further confirm the feasibility of the test method.



**Table 5** Distribution of hazard behaviors and scene elements

# **5 Conclusion**

In this paper, a customized SO verification test method for highway assistance (HWA) function based on design operation domain (ODD) is studied and proposed, and a test route formulation method based on multi-dimensional key elements is designed. This method fully considers the data dimensions and environmental characteristics involved when deriving acceptance criteria and confirming targets, and effectively meets the adequacy requirements of unknown scenario experiments. This paper designs a vehicle test method for HWA for expected functional safety, explains the selection principle of the function to be tested, the basis of the test method—safety analysis and the design of the test method, and conducts verification tests. The experimental results show that the test method designed in this paper is reasonable and implementable.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Intelligent Construction Machinery SLAM with Stereo Vision and Inertia Fusion**

**Tianliang Lin, Zhongyuan He, Jiangdong Wu, Qihuai Chen, and Shengjie Fu** 

**Abstract** Positioning technology is the foundation of intelligent construction machinery, the current mainstream positioning solution is simultaneous localization and mapping (SLAM) technology, which is mainly divided into lidar SLAM and visual SLAM. Due to the high cost of lidar, it is easy to degrade or even fail in scenes with a single environmental texture; while the cost of vision sensors is low and has a wealth of environmental texture information acquisition capabilities, which can effectively avoid degradation problems. In order to reduce the localization cost of intelligent construction machinery and improve the positioning accuracy, based on the VINS-Fusion stereo visual-inertial tightly coupled system framework, an improved Random Sampling Consensus (RANSAC) algorithm is used to reduce feature mismatch, and the Huber kernel function is used to IMU residuals and visual residuals are constrained to improve the effect of the SLAM system. Compared with the mainstream VINS-Fusion algorithm, the positioning root mean square error of this method on the EuRoC dataset is reduced by an average of 12.41%, which improves the positioning accuracy; simultaneously, the experimental results in the actual scene show that the motion trajectory of the algorithm, it is closer to the real trajectory than VINS-Fusion, which verifies the effectiveness of the method.

**Keywords** Construction machinery · SLAM · Visual and inertial fusion · Tight coupled

T. Lin · Z. He (B) · J. Wu · Q. Chen · S. Fu

College of Mechanical Engineering and Automation, Huaqiao University, Xiamen 361021, China e-mail: 2551847813@qq.com

Fujian Key Laboratory of Green Intelligent Drive and Transmission for Mobile Machinery, Xiamen 361021, China

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_82

### **1 Introduction**

Simultaneous localization and mapping (SLAM) [1–3] technology is widely used in autonomous driving, drones, AR/VR and other fields. SLAM technology is mainly divided into lidar SLAM based on lidar and visual SLAM based on camera. Lidar is not affected by changes in illumination and can accurately measure the depth information of the environment. It has been widely used in the field of autonomous driving in recent years. However, high-precision lidar has a high cost and is prone to distortion during high-speed movement. Scenes with single environmental texture information such as indoor corridors are easy to degrade. In contrast, the cost of the camera is much lower than that of the lidar, and it can make full use of the environmental texture information. The effect in the low-texture environment is better than that of the lidar. The difficulty of visual SLAM lies in how to minimize the mismatch of features and the error in pose estimation process.

The camera and the inertial measurement unit (Inertial Measurement Unit, IMU) both have good complementarity. IMU measurement is not affected by environmental characteristics, and its accurate estimation of fast motion in a short period of time can effectively compensate for the camera's state estimation of fast motion, while camera measurement can correct the cumulative error of IMU. At the same time, the IMU can also be used to correct the point cloud distortion caused by the high-speed movement of the lidar. Therefore, in recent years, SLAM with fusion of vision or lidar and inertia has become a research hotspot in the field of SLAM. There are usually two schemes for fusion with inertial sensors, one is a filtering scheme based on EKF (extended Kalman filter) or ESKF (error state Kalman filter) [4, 5]. Such as MSCKF [6, 7] and FAST-LIO [8], et al. Another optimization-based scheme [9, 10] represented by VINS-Mono [11] and LIO-SAM [12].

However, there are always some problems in various current SLAM schemes.


In response to the above problems, this paper proposes a SLAM scheme based on the tight coupling of stereo vision and IMU based on the VINS-Fusion [13, 14] framework. The main innovations of this paper are as follows:


The method proposed in this paper was tested on the public data set, which verified that the system has good accuracy and robustness in various scenarios. At the same time, a real vehicle platform was built to test it, proving that The effectiveness of the proposed method.

### **2 Overview**

The structure of the stereo vision and inertial fusion SLAM system proposed in this paper is shown in Fig. 1. The system starts with front-end data preprocessing, including feature extraction and tracking, feature point triangulation to obtain depth information, and IMU measurement between two consecutive frames based on the camera frame do pre-integration. In the back-end nonlinear optimization part, various observation information extracted by the front-end are used to construct error constraints, including visual re-projection error, IMU pre-integration error, and prior error of sliding window marginalization. Loop detection uses the DBoW2 [15] bag-of-words model as the basis.

### *2.1 Visual Processing*

For each new image, FAST corner points are extracted, and the existing features are tracked by the KLT [16] sparse optical flow method, and the minimum pixel threshold between two adjacent features is set to achieve a uniform distribution of feature points. The improved RANSAC [17] method is used to eliminate feature mismatches. First, the cross-validation method is used for rough elimination, and then the two-point RANSAC method is used for fine elimination to obtain correctly matched feature points.

Another task in this step is the selection of key frames in the sliding window, and there are two selection criteria. The first one is to judge whether the average disparity between the current frame and the previous key frame exceeds the set threshold, and if it exceeds the threshold, it is judged as a new key frame. Another criterion is to judge

**Fig. 1** Frame diagram of stereo vision and inertial fusion system

whether the number of features of the current frame is lower than a certain threshold, and this frame is also regarded as a key frame if it is lower than the threshold. This criterion is to avoid the complete loss of feature tracks.

### *2.2 IMU Preintegration*

The Acceleration *a* - *<sup>t</sup>*and angular velocity ω - *<sup>t</sup>*with noise measured by the IMU are modeled as follows.

$$\begin{cases} \hat{a}\_t = a\_t + b\_{a\_t} + n\_a + R\_w^t g^w \\ \hat{a}\_t = a\_t + b\_{a\_t} + n\_w \end{cases} \tag{1}$$

where *at* and ω*t* are the true valuesof acceleration and angular velocity. *na* and *n*<sup>ω</sup> are Gaussian white noise from accelerometer and gyroscope measurements, *na* ∼ *N* (0, σ<sup>2</sup> *<sup>a</sup>*), *n*<sup>ω</sup> ∼ *N* (0, σ<sup>2</sup> ω). *bat* and *b*<sup>ω</sup>*t* are acceleration and angular velocity deviations modeled by random walk, respectively, and their derivatives are Gaussian white noise. *R<sup>t</sup>* <sup>w</sup> is the rotation matrix from the world coordinate system to the current IMU coordinate system. g<sup>w</sup> is the gravity acceleration vector in the world coordinate system.

Taking the image frame *bk* as a reference, the position, velocity and rotation measured by the IMU at time *bk*+1 are modeled as follows [18].

$$\begin{cases} p\_{b\_{k+1}}^{b\_k} = \mathcal{R}\_w^{b\_k} (p\_{b\_k}^w + v\_{b\_k}^w \Delta t\_k - \frac{1}{2} \mathcal{g}^w \Delta t\_k^2) + \alpha\_{b\_{k+1}}^{b\_k} \\ v\_{b\_{k+1}}^{b\_k} = \mathcal{R}\_w^{b\_k} (v\_{b\_k}^w - \mathcal{g}^w \Delta t\_k) + \beta\_{b\_{k+1}}^{b\_k} \\ q\_{b\_{k+1}}^{b\_k} = \mathcal{\nu}\_{b\_{k+1}}^{b\_k} \end{cases} \tag{2}$$

where

$$\begin{cases} \boldsymbol{a}\_{b\_{k+1}}^{b\_{k}} = \iint\_{\boldsymbol{t} \in \left[\boldsymbol{t}\_{k}, \boldsymbol{t}\_{k+1}\right]} \boldsymbol{R}\_{\boldsymbol{t}}^{b\_{k}} (\hat{\boldsymbol{a}}\_{\boldsymbol{t}} - \boldsymbol{b}\_{a\_{\boldsymbol{t}}}) d\_{\boldsymbol{t}^{2}}\\ \boldsymbol{\beta}\_{b\_{k+1}}^{b\_{k}} = \int\_{\boldsymbol{t} \in \left[\boldsymbol{t}\_{k}, \boldsymbol{t}\_{k+1}\right]} \boldsymbol{R}\_{\boldsymbol{t}}^{b\_{k}} (\hat{\boldsymbol{a}}\_{\boldsymbol{t}} - \boldsymbol{b}\_{a\_{\boldsymbol{t}}}) d\_{\boldsymbol{t}}\\ \boldsymbol{\gamma}\_{b\_{k+1}}^{b\_{k}} = \int\_{\boldsymbol{t} \in \left[\boldsymbol{t}\_{k}, \boldsymbol{t}\_{k+1}\right]} \frac{1}{2} \boldsymbol{q}\_{\boldsymbol{t}}^{b\_{k}} \otimes (\hat{\boldsymbol{a}}\_{\boldsymbol{t}} - \boldsymbol{b}\_{a\_{\boldsymbol{t}}}) d\_{\boldsymbol{t}} \end{cases} (3)$$

Since the IMU measurement frequency is higher than that of the camera, there are multiple inertial measurement values in the time interval *tk* ∈ *tk* , *tk*+<sup>1</sup> , so the position *p bk bk*+<sup>1</sup> , velocity v *bk bk*+<sup>1</sup>and rotation change *q bk bk*+<sup>1</sup>of the IMU during this period are preintegrated. Where: α *bk bk*+<sup>1</sup>is the position preintegration amount. β*bk bk*+<sup>1</sup>is the speed preintegration amount. γ *bk bk*+<sup>1</sup>is the rotation preintegration amount. Through the above formula, the constantly changing state change can be obtained by using frame *bk* as a reference.

When the estimation of the IMU bias has a small change, the preintegration measurement value is corrected by the following first-order approximation, so as to obtain a more accurate pre-integration measurement value. This method saves a lot of computing resources for the optimization-based algorithm, because Measurements that do not require repeated propagation of the IMU.

$$\begin{cases} \alpha\_{b\_{k+1}}^{b\_k} \approx \hat{\alpha}\_{b\_{k+1}}^{b\_k} + J\_{b\_a}^{\alpha} \delta b\_{a\_k} + J\_{b\_{a}}^{\alpha} \delta b\_{a\_k} \\ \beta\_{b\_{k+1}}^{b\_k} \approx \hat{\beta}\_{b\_{k+1}}^{b\_k} + J\_{b\_a}^{\beta} \delta b\_{a\_k} + J\_{b\_u}^{\beta} \delta b\_{a\_k} \\ \gamma\_{b\_{k+1}}^{b\_k} \approx \hat{\gamma}\_{b\_{k+1}}^{b\_k} \otimes \begin{bmatrix} 1 \\ \frac{1}{2} J\_{b\_u}^{\gamma} \delta b\_{a\_k} \end{bmatrix} \end{cases} \tag{4}$$

where *J* is the partial derivative Jacobian matrix of the error term with respect to the variable to be optimized.

### **3 Tightly Coupled Stereo VIO**

When the visual measurement and IMU measurement are preprocessed and the system is initialized successfully, the nonlinear optimization link based on sliding window tight coupling can be performed, as shown in Fig. 2.

### *3.1 Optimization Variables*

All variables χ to be optimized in the sliding window are defined as follows.

$$\begin{cases} \chi = [\chi\_0, \chi\_1, \dots, \chi\_n, \lambda\_0, \lambda\_1, \dots, \lambda\_m] \\ \chi\_k = [p\_{b\_k}^w, v\_{b\_k}^w, q\_{b\_k}^w, b\_a, b\_a], k \in [0, n] \end{cases} \tag{5}$$

where *xk* is the state of the IMU at the *k*th frame of the camera, including the position, velocity and rotation of the IMU in the world coordinate system at this time, as well as the acceleration deviation and gyroscope deviation of the IMU; *n* is the total key

**Fig. 2** Diagram of a tightly coupled sliding window

in the sliding window The number of frames; *m* is the number of feature points in the sliding window; λ*i* is the reciprocal of the first observed depth value of the *i*th feature point.

Optimizing the state variable by minimizing the sum of all residuals by modeling an overall optimization model within a sliding window.

$$\min\_{\boldsymbol{\chi}} \left\{ \left\| \boldsymbol{r}\_{\mathcal{P}} - \boldsymbol{H}\_{\mathcal{P}} \boldsymbol{\chi} \right\|^{2} + \sum\_{k \in \mathcal{B}} \rho \left\| \boldsymbol{r}\_{\mathcal{B}} \left( \boldsymbol{\hat{z}}\_{b\_{k+1}}^{b\_{k}}, \boldsymbol{\chi} \right) \right\|\_{\boldsymbol{P}\_{b\_{k+1}}^{b\_{k}}}^{2} + \sum\_{(l,j) \in \mathcal{C}} \rho \left\| \boldsymbol{r}\_{\mathcal{C}} \left( \boldsymbol{\hat{z}}\_{b\_{l}}^{\mathcal{C}\_{j}}, \boldsymbol{\chi} \right) \right\|\_{\boldsymbol{P}\_{l}^{c\_{j}}}^{2} \right\} \tag{6}$$

where ρ is the Huber norm.

$$\rho(e) = \begin{cases} e, & |e| \le 1 \\ 2\sqrt{e} - 1, \ |e| > 1 \end{cases} \tag{7}$$

*rB*(*z* - *bk bk*+<sup>1</sup> , χ) and *rC* (*z* -*C j bl* , χ) are the residual terms of the IMU and visual measurements, respectively. *B* is the set of all IMU measurements, and *C* is the set of feature points within the current sliding window. *r <sup>p</sup>*, *Hp* is the prior information related to marginalization, which is used to construct the prior residual term. Use the Ceres solver [19] to solve the entire nonlinear optimization problem.

### *3.2 IMU Residual Block*

Construct the residual term of the IMU measurement between two consecutive frames *bk* and *bk*+1 within the sliding window according to the IMU preintegration.

$$r\_{B}\left(\boldsymbol{z}\_{b\_{k+1}}^{b\_{k}},\boldsymbol{\chi}\right) = \begin{bmatrix} R\_{w}^{b\_{k}}\left(\boldsymbol{p}\_{b\_{k+1}}^{w} - \boldsymbol{p}\_{b\_{k}}^{w} - \boldsymbol{v}\_{b\_{k}}^{w}\Delta t\_{k} + \frac{1}{2}\boldsymbol{g}^{w}\Delta t\_{k}^{2}\right) - \hat{\boldsymbol{\alpha}}\_{b\_{k+1}}^{b\_{k}}\\\ R\_{w}^{b\_{k}}\left(\boldsymbol{v}\_{b\_{k+1}}^{w} - \boldsymbol{v}\_{b\_{k}}^{w} + \boldsymbol{g}^{w}\Delta t\_{k}\right) - \hat{\boldsymbol{\beta}}\_{b\_{k+1}}^{b\_{k}}\\\ 2\left[\boldsymbol{q}\_{b\_{k+1}}^{w}\otimes\left(\boldsymbol{q}\_{b\_{k}}^{w}\right)^{-1}\otimes\left(\hat{\boldsymbol{\gamma}}\_{b\_{k+1}}^{b\_{k}}\right)^{-1}\right]\_{\mathrm{xyz}}\\\ \boldsymbol{b}\_{a\_{k+1}} - \boldsymbol{b}\_{a\_{k}}\\\ \boldsymbol{b}\_{a\_{k+1}} - \boldsymbol{b}\_{a\_{k}}\end{bmatrix} \tag{8}$$

where [·]*xyz* means to extract the vector part of the quaternion.

### *3.3 Visual Residual Block*

Assuming that the observed points of the feature point *l* on the normalized plane of frame *i* and frame *j* are respectively *p Ci <sup>l</sup>*(*uCi* , v*Ci* ) and *p C j <sup>l</sup>*(*uC j* , v*C j* ), the visual reprojection error is constructed by back-projecting *p Ci <sup>l</sup>*to frame *j.* 

$$\begin{cases} r\_C \left( \hat{z}\_l^{C\_l}, \chi \right) = \overline{p}\_l^{C\_l} - p\_l^{C\_l} \\ \overline{p}\_l^{C\_l} = T\_b^C T\_w^{b\_l} T\_{b\_l}^w T\_C^b \frac{1}{\lambda\_l} p\_l^{C\_l} \end{cases} \tag{9}$$

where *p C j <sup>l</sup>*is the back-projected point of *p Ci <sup>l</sup>*onto the *j*th frame. *T* is the transformation matrix between different coordinate systems. λ*l* is the depth value observed for the first time by feature point *l.* 

### *3.4 Marginalization*

In order to reduce the computational complexity of optimizing VIO based on sliding windows, a marginalization strategy is adopted, and the marginalized measurements are converted into prior information. The number of key frames in the sliding window is guaranteed to be the set value.

As shown in Fig. 3, when a new frame arrives, if the latest frame in the sliding window is a key frame at this time, the oldest frame in the sliding window and the corresponding measurement. Otherwise, the latest frame in the current sliding window is marginalized, but the IMU measurement of this frame is retained, and the current frame enters the sliding window to become the new latest frame.

Marginalization was achieved using Schur complement [20]. Construct prior residuals based on all marginalized measurements to achieve further optimization of state variables.

**Fig. 3** Illustration of marginalization strategy

### **4 Experimental Results**

In order to evaluate the accuracy and robustness of the system proposed in this paper, the algorithm proposed in this paper was tested and verified on the public data set and the real vehicle platform built by ourselves. Select the EuRoC public dataset [21] to compare the stereo vision and inertial fusion algorithm VINS-Fusion with the algorithm in this paper. At the same time, use the self-built real vehicle platform to verify the effectiveness of the algorithm in this paper.

### *4.1 Dataset Experiment*

In order to verify the localization accuracy of the algorithm proposed in this paper, four motion sequence data of MH\_04\_difficult, MH\_05\_difficult, V1\_03\_difficult and V2\_03\_difficult of the EuRoC visual-inertial dataset were used for testing. The data in this dataset is collected using UAV-mounted sensors, including binocular images, IMU measurements, and ground-truth data. In this experiment, the algorithm in this paper is compared with the VINS-Fusion algorithm, and the comparison results are shown in Table 1.

It can be seen from Table 1 that the relative pose error (RPE) of the algorithm proposed in this paper is smaller than that of VINS-Fusion on the four selected sequences, and the root mean square error (RMSE) is reduced by 12.41% on average, which proves the effectiveness and robustness of the algorithm in this paper.

Figure 4 is a comparison of the trajectories between the algorithm of this paper and VINS-Fusion in the above four sequences. In the figure, ground truth is the real motion trajectory provided by the EuRoC dataset, and Ours represents the method of this paper. It can be seen from Fig. 4 that the running trajectory of the algorithm in this paper is closer to the real trajectory, and the error is smaller.


**Table 1** Algorithm error comparison in dataset experiment

**Fig. 4 a** Trajectory in V1\_03\_difficult. **b** Trajectory in V2\_03\_difficult. **c** Trajectory in MH\_04\_ difficult. **d** Trajectory in MH\_05\_difficult

### *4.2 Real Scene Experiment*

In order to verify the effectiveness of the algorithm in this paper, experiments are carried out in actual scene. The experimental platform is a crawler chassis equipped with ZED2i stereo camera and GNSS, as shown in Fig. 5. The camera is used to collect images and IMU data. It has a built-in 6-axis IMU. While collecting images and IMU data, it records GNSS data for making ground truth in real scenes. In order to facilitate the comparison and ensure that the two algorithms run on the same trajectory, this paper adopts the method of recording data sets to conduct comparative experiments in real scene.

The experimental environment Fig. 6 is a scene with good lighting conditions during the day. The crawler chassis test platform is used to drive on a random path in the open space inside the electromechanical test building, and the raw camera image,

**Fig. 5** Real vehicle test platform

IMU raw data and GNSS positioning data are collected at the same time. The total length of the path is about 113 m.

Figure 7(a) is the comparison between the trajectory of the method in this paper and VINS-Fusion after running the recorded real data set, and the real trajectory of GNSS. It can be seen that the overall positioning effect of the algorithm in this paper is slightly better than that of VINS-Fusion, it can achieve basic and real trajectories. Figure 7b, c shows the relative pose error between our improved method and the original algorithm in the actual environment. Table 2 shows the RPE comparison between the proposed method and the original method in real scene. It is obvious that the error of the proposed algorithm is significantly lower than that of the original algorithm. This coincides with the validity of the method in this paper.

**Fig. 6** Test environment

$$\bf{(a)}$$

**Fig. 7 a** Trajectory comparison chart in real scene. **b** The relative pose error of our method in real scenes. **c** The relative pose error of VINS-Fusion algorithm in real scene


**Table 2** Algorithm error comparison in real scene experiment

### **5 Conclusion**

Based on the VINS-Fusion system framework, this paper proposes to use the improved RANSAC algorithm to reduce mismatching in the feature tracking step, and iterates only in the matching points whose Hamming distance is less than the set threshold. At the same time, in the nonlinear optimization link based on the sliding window, the Huber kernel function is used to preprocess the larger residual items for both the IMU residual and the visual residual. The algorithm comparison test was carried out on the public dataset EuRoC. The experimental results show that the positioning root mean square error of the improved method is reduced by an average of 12.41% compared with the original algorithm, which effectively improves the accuracy of the SLAM system; the experiment in the actual scene is also verified the effectiveness of the method.

**Acknowledgements** This work was supported by the Key projects of natural science foundation of Fujian Province (2021J02013), Fujian University industry university research joint innovation project plan (2022H6007) and Industry Cooperation of Major Science and Technology Project of Fujian Province (2022H6028).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Improved Bidirectional Dynamic JPS Algorithm for Global Path Planning of Mobile Robot**

**Ronghua Liu, Xin Wang, Di Wu, and Chunyuan Xie** 

**Abstract** This paper proposed to improve the bidirectional dynamic JPS algorithm in order to address the issues with the jump point search algorithm in the process of pathfinding, such as numerous path inflection points, numerous intermediate search hop points, numerous extended nodes, and lengthy pathfinding time in the process of finding jump points. The algorithm was based on the two-way dynamic hopping point search algorithm, defined the target points in the forward and reverse expansion directions dynamically, defined the heuristic function dynamically. In the process of searching each other from the starting point and the target point, the dynamic constraint ellipse was constructed to limit the expansion area of the jump point. The simulation results show that the optimized bidirectional dynamic JPS algorithm has a certain performance in general maps, especially in indoor environment maps with fewer obstacles and the target point is close to the starting point.

**Keywords** Path planning · Jump point search · Two-way dynamic search · Mobile robot

# **1 Introduction**

Path planning, which acts as a bridge between robot perception and control, is a critical technology for achieving autonomous navigation of mobile robots [1]. Using the route planning algorithm to create an optimal or nearly optimal path from the beginning point to the goal location based on previously known environmental factors is known as global path planning for mobile robots. The intended route should

R. Liu · X. Wang (B) · C. Xie

e-mail: wangx@dlut.edu.cn

D. Wu

© The Author(s) 2024

1049

School of Mechanical Engineering, Dalian University of Technology, Liaoning, Dalian 116000, China

School of Computer Science and Technology, Dalian University of Technology, Liaoning, Dalian 116000, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_83

make it possible to avoid obstacles with ease and meet the robot's numerous performance parameters [2–4]. The Dijkstra algorithm [5], A\* algorithm [6], artificial potential field method [7], RRT algorithm [8], and jump point search algorithm are now the more popular global path planning algorithms. The A\* algorithm extends many unessential intermediary nodes during the path-finding process and regularly accesses the Openlist to gather node information, requiring a huge calculation and a lengthy path-finding time [9]. Harabor [10] et al. created a jump point search algorithm on the basis of keeping the heuristic function of the A\* algorithm in order to address the inadequacies of the A\* algorithm.

Numerous scholar and international experts have considerably improved the jump point search algorithm and improved the method's performance. Based on the improvement of the A\* algorithm, Duchon [11] conducted comparative experiments on JPS and RSR algorithms using different environment maps. He proved that the JPS algorithm exhibits obvious advantages in terms of wayfinding efficiency. A planning algorithm combining the improved JPS and A\* algorithm was proposed by Jin [12] to address the issue of multiple and chaotic jump-points in the processing of large-scale maps with complex environmental information. This algorithm is intended to reduce redundant intermediate jump-points by enhancing the jump-point screening rules. Ma [13] proposed a bidirectional jump point search algorithm by improving the jump point screening rules and conducting path search from two directions to address problems such as insecurity, long path search time, and large memory consumption in the pre-processing rules of the jump point search algorithm in the global path planning of mobile robots. Miao [14] proposed a parallel—alternate jump point search algorithm by adding the central hot spot region between the starting point and the target point because paths in two directions cannot meet under certain conditions in the bidirectional jump point search algorithm, resulting in an increase in redundant paths. The parent node of the current jump point was introduced as the target point of the jump point expansion in the relative direction, and a two-way dynamic jump point search algorithm was proposed by Qin [15] to address the issue of two-way jump search. Ma [16] et al. integrated the target point's direction vector and the node's initial search direction, constructed the priority of the jump point search direction, and proposed a directional jump search algorithm. Yakovlev [17] formally presents the angle-constrained path planning problem and introduces a new algorithm, LIAN, specifically designed to smooth the path obtained through the JPS algorithm. To somewhat boost the path finding speed of the jump point search algorithm, the majority of their solutions, however, focus on bettering the jump point filtering rules or searching the path in two separate directions. The pathfinding process is not optimized for the algorithm's inherent problems, such as the algorithm's excessive expansion nodes, intermediate search hops, unnecessary inflection points, and overly long pathfinding periods.

Therefore, this paper proposes an improved bidirectional dynamic JPS algorithm to address the aforementioned issues. This algorithm judges the obstacle information between the minimum cost point in the two directions while expanding jump points, and it limits the expansion range of jump points in the forward and reverse directions to a certain extent through dynamic constraint ellipses.

### **2 Improved Bidirectional Dynamic JPS Algorithm**

# *2.1 Dynamically Define Target Points and Heuristic Functions*

The bidirectional dynamic JPS method extends the jump point from the starting and ending points, and the goal point and heuristic function in the two extension directions change dynamically as the two extension sides approach each other. Figure 1 depicts the definition of certain dynamic target sites. In the figure, black grids are obstacles, gray grids are jump points extended in both positive and negative directions, S1 is the starting point, and G1 is the target point. The first step is to add S1 and G1 to the forward open list and the reverse open list respectively, take the minimum cost point S1 in the forward open list, take the minimum cost point G1 in the reverse open list as the target point for forward expansion, carry out forward expansion and add the expanded jump point to the forward open list, and take the minimum cost point S2. In the second step, taking G1 as the starting point of the reverse expansion, the reverse expansion is carried out to the target point S2, which is the minimum forward cost obtained in the first step. The expanded jump point is added to the reverse open list, and the minimum reverse cost point G2 is taken out. In the third step, the minimum point of the reverse cost obtained in the second step is taken as the new forward target point, and the forward expansion is carried out to obtain the minimum point of the forward cost S3. In the fourth step, the minimum forward cost S3 obtained in the third step is taken as the new reverse target point, and the reverse cost minimum G3 is obtained by the reverse extension. In the fifth step, the minimum point of the positive cost (G3) obtained in the fourth step is taken as the new forward target point, and the forward expansion is carried out to obtain the minimum point of the positive cost (S4). In the sixth step, the forward cost minimum point S4 obtained in the fifth step is used as the new reverse target point for reverse expansion, and the reverse cost minimum point S4 is obtained. At this time, the forward and reverse cost minimum points coincide, and the wayfinding ends.

The heuristic function used to describe the estimated distance between the current least cost point and the target point will change along with the dynamic change of the forward and backward expansion target point. Equations (1) and (2) are examples of positive and reverse dynamic heuristic functions that were obtained in this study using Euclidean distance.

$$h\_f = \sqrt{\left(\mathbf{x}\_{f\text{ min}} - \mathbf{x}\_{r\text{ min}}\right)^2 + \left(\mathbf{y}\_{f\text{ min}} - \mathbf{y}\_{r\text{ min}}\right)^2} \tag{1}$$

$$h\_r = \sqrt{\left(\chi\_{r\text{ min}} - \chi\_{f\text{ min}}\right)^2 + \left(\chi\_{r\text{ min}} - \chi\_{f\text{ min}}\right)^2} \tag{2}$$

The horizontal and vertical coordinates of the forward cost minimum point in the current iteration are represented in the equation by the letters *xfmin* and *yfmin*, respectively, whereas the horizontal and vertical coordinates of the reverse cost minimum point are represented by the letters *xrmin* and *yrmin*.

### *2.2 Environment Complexity and Dynamic Constraint Ellipse*

The calculation region for environmental complexity is defined as the rectangle range with the diagonal line of the point with the lowest cost of forward and backward expansion under the present iteration. The amount of obstacle grids in the entire rectangle is calculated to estimate the environmental complexity k for the current iteration.

The gray grids in Fig. 2 serve as an illustration of the current minimum cost points for each iteration in the two expansion directions, and the black grids represent obstacles. The line between the start and goal points is used as the diagonal line of the rectangle in the algorithm's first iteration to establish the calculation area for environmental complexity, which is represented by a dashed line box. This figure shows that there are 28, that is 19.58% of all the grids in the wire frame, are black. This means that the environmental complexity k1 in the first iteration is 0.196, and that the determination of environmental complexity in subsequent iterations is similar to that in the first step, so further explanation is not necessary.

Dynamic constraint ellipse is introduced via an improved bidirectional dynamic JPS method. The elliptic focus F1 and the elliptic focus F2 are the points of minimal current cost in the expansion process, respectively, in one expansion direction and the other. The dynamic constraint ellipse for the current iteration can be generated by

**Fig. 2** Determination of environment complexity

using the heuristic function to calculate the focal length of the ellipse and then taking the focal length of ζ times as the major axis length of the ellipse. The following is the standard equation for the dynamic constraint ellipse.

$$\frac{(\mathbf{x} - m)^2}{a^2} + \frac{(\mathbf{y} - n)^2}{b^2} = 1,\\
(a > b > 0) \tag{3}$$

where *m* <sup>=</sup> *<sup>x</sup>*min(1)+*x*min(2) <sup>2</sup> stands for the ellipse's center point's horizontal coordinate, while *xmin* (1) and *xmin* (2) represent the points' horizontal coordinates with the lowest current costs in the two expansion directions, respectively. *n* <sup>=</sup> *<sup>y</sup>*min(1)+*y*min(2) 2 stands for the ellipse's center point's ordinate, while *ymin* (1) and *ymin* (2) represent the points' ordinates with the lowest current costs in the two expansion directions, respectively. *a* = ζ *h*, represents the length of the major axis of the ellipse, where ζ is the coefficient of elliptic expansion, determined by the relationship between the environmental complexity and the coefficient of elliptic expansion in Table 1, and *h* represents the focal length of the ellipse, which can be obtained by the heuristic function (1) or (2) under the current iteration. *b* <sup>=</sup> *<sup>h</sup>* <sup>√</sup><sup>4</sup><sup>ζ</sup> <sup>2</sup>−<sup>1</sup> <sup>2</sup> represents the minor axis length of the ellipse.

Figure 3 illustrates the precise creation process of the dynamic constraint ellipse. The minimum cost points produced by each iteration in the two expansion directions


**Fig. 3** Dynamic constraint ellipse

are gray nodes. Beginning with Start and Goal as the ellipse's two focal points and ζ *1* times the focal length as the length of the major axis, the constrained ellipse 1 is created. The minimum cost points in the subsequent iteration, S1 and G1 are utilized as the new ellipse focus points and the focal length ζ *2* times as the ellipse's major axis length to create a new restricted extension ellipse known as ellipse 2. Up till the path finding is finished, the appeal process is repeated.

The expansion range of the node under the current iteration is constrained by the dynamic constraint ellipse. Figures 4 and 5 show the effects of the constraint ellipse on the expansion range of nodes during the first round of jump point expansion, with gray grids representing the expansion nodes in two directions, and Δ and ◯ indicating the beginning point and target point respectively. The grid intersecting the ellipse's boundary is extracted in Fig. 4, and the grid outside and on the boundaries of the ellipse is given an extended weight that is smaller than the one without obstacles but larger than the one with obstacles, dividing the environment map into the high-priority extension area inside the ellipse and the lower-priority extension area outside the ellipse. Compared with the unconstrained extension in Fig. 4, the algorithm gives priority to the expansion inside the ellipse when looking for jump points. Because the elliptic boundary is not an impediment, more jump points are not introduced, and the intermediate expansion nodes and intermediate jump points in the path finding process are reduced due to the elliptic limitation, thus speeding up the speed of bidirectional search to each other and improving the algorithm's convergence efficiency.

# *2.3 Judgment of Obstacles Between the Minimum Cost Points of Forward and Reverse Direction*

JPS algorithm can only plan courses along straight lines and grid diagonals in raster maps, as demonstrated in Fig. 6a black path, which introduces meaningless inflection points in the process. The introduction of the idea of forced neighbor points near obstacles also results in the production of a significant number of meaningless intermediate inflection points in some situations, as demonstrated in Fig. 6b black path. The bidirectional dynamic JPS method may evaluate the environmental data between the two directions of the minimum cost points in each iteration because it grows from two directions to each other. If there are no barriers in the road connecting the two

**Fig. 6** Correction of meaningless inflection points. **a** Constraints on the direction of expansion; **b** Effects of obstacles

minimum cost points, the path connecting two points (shown by the gray path) forms a complete pathfinding path with the planned path.

# *2.4 Improved Bidirectional Dynamic JPS Algorithm Path Planning*

By using a dynamic constraint ellipse, the improved bidirectional dynamic JPS algorithm reduces the number of intermediate nodes that must be extended in order to find the jump points in each iteration and avoids adding jump points that are highly improbable to be far from the line connecting the positive and negative cost points to the open list. The pathfinding path of the algorithm is also made less complicated by eliminating pointless inflections by determining, after each iteration, if the line of least cost points in two directions goes through the obstruction.

The improved bidirectional dynamic jump point search algorithm's pathfinding procedure is as follows:

**Step 1:** Obtain the starting point, target point, and obstacle coordinate information from the raster environment, initialize the raster map, add the starting point to the forward the Openlist, and add the target point to the reverse the Openlist.

**Step 2:** Verify that the reverse the Openlist and forward the Openlist are both empty. Pathfinding is completed if any list is empty. If neither list is empty, remove the least expensive point from the two open lists and make it the extension's new target point.

**Step 3:** Verify that the goal points for the two expansion directions are parallel. The path finding process is finished if the target points match. If the goal points are not congruent, see if there is an obstruction in the path that connects them.

**Step 4:** Create a constrained ellipse with the focus at the point with the lowest cost in both directions, along with the focal length ζ times as the ellipse's major axis length, and look for jump points in the high-priority extension region. The jump points are added to the open list if there are any jump points in the ellipse that satisfy the requirements. The jump points are extended outside the ellipse and the jump points that satisfy the conditions are also added to the open list if there are no jump points that fit the criteria.

**Step 5:** Go to Step 2.

### **3 Simulation and Experimental Analysis**

Two different environment maps, 20 × 20 and 40 × 40, are utilized to simulate and compare the classic JPS algorithm, the bidirectional dynamic JPS algorithm, and the proposed method in order to assess the efficacy of the improved bidirectional dynamic JPS algorithm. Windows 10 64-bit operating system, Core i7-10700F processor, main frequency of 2.9 GHz, installation memory of 16 GB, and MATLAB R2020b simulation running platform make up the device configuration employed in the experiment.

Figures 7 and 8 display the simulation environment. In the grid map, Δ and ◯ stand in for the beginning point and target point, respectively. Obstacles are represented by black grids, intermediate expansion points by light gray grids, and jump points by deep gray grids. The comparison of the three methods' path lengths, algorithm running times, number of intermediate expansion nodes, number of jump points, and number of inflection points under two distinct raster maps is shown in Tables 2 and 3, respectively.

Bidirectional dynamic JPS algorithm shows no improvement in pathfinding time when compared to standard JPS algorithm, as shown by Figs. 7, 8, Tables 2, and

**Fig. 7** Path comparison under 20 × 20 raster map. **a** JPS algorithm; **b** Bidirectional dynamic JPS algorithm; **c** Improved algorithm

**Fig. 8** Path comparison under 40 × 40 raster map. **a** JPS algorithm; **b** Bidirectional dynamic JPS algorithm; **c** Improved algorithm


**Table 2** Comparison of 20 × 20 raster map results


**Table 3** Comparison of 40 × 40 raster map results

3, among other data. This is due to the fact that, despite the bidirectional dynamic JPS algorithm's ability to expand in two directions, the expansion in each direction does not occur simultaneously but rather alternately. The bidirectional dynamic JPS algorithm has the property of approaching each other with the expansion in two directions, but it does not address the issue of the large number of middle invalid expansion nodes and invalid jump points in the path finding process of the traditional JPS algorithm. This feature is combined with the introduction of the dynamic constraint ellipse to divide the priority of the expanding range of each algorithmic step. The bidirectional dynamic JPS method is proposed to be enhanced. As can be seen, when the path is optimal or tends to be optimal, the optimized algorithm can significantly reduce the invalid expansion nodes and jump points produced in the expansion process of the algorithm, so as to improve the convergence speed and increase the pathfinding efficiency of the algorithm.

# **4 Conclusion**

In this paper, an improved bidirectional dynamic jump point search algorithm is proposed. The algorithm extends the jump point to each other in both forward and reverse directions. In the process of jump point expansion, the target point and the heuristic function are defined dynamically. At the same time, the obstacle information between the least cost points in two directions is judged, so as to solve the inherent problems of traditional jump search algorithm such as an excess of jump points, intermediate expansion nodes and invalid inflection points. Experimental results show that the improved method speeds up the convergence speed and the efficiency of the algorithm.

**Acknowledgements** This work described in this paper is fully supported by a grant from the Basic Research Fund of Central Universities (No.DUT22LAB507).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Customer Needs Assessment and Screening for Transmission Solution Selection**

**Hegen Wang** 

**Abstract** When selecting technology for transmission in the process of product development and design, various methods available for choose ideal solution from many different technical alternatives. However, this does not guarantee that a very good developed product will achieve ideal market result. This is often because there is a difference between the functionalities provided by the product and the actual customer demands. This article explores the use of the KANO model to analyze customer requirements in the early stages of product development. By categorizing customer needs and selecting the ideal solution based on different categories of customer needs, it ensures that the product development aligns with customer demands.

**Keywords** Transmission · Transmission selection · Function and customer needs · KANO model · AHP

# **1 Introduction**

The function of transmission is to transfer power. A transmission system refers to a complete set of devices or mechanisms that transfer power from a power source to an actuator. In the era of animal power and steam engines, there was almost no difference in speed between the power source output and the actuator output, so there was no need for a complex transmission system with deceleration. With the development of power sources from steam engines to IC engines, the output speed gap of IC engine and actuator increased, making it necessary to add deceleration between the power source and the actuator. As a result, transmission system with deceleration and other developed functions are widely used.

© The Author(s) 2024

1061

H. Wang (B)

School of Mechanical Engineering, Tongji University, Shanghai 200092, China e-mail: genhwang@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_84

At early stage, people focused on the practical significance of transmission systems and the impact of the deceleration on the working performance of actuator. There were limitations on the layout space, structural shape, and weight of the deceleration mechanism between the power source and the actuator, and the feasibility of different deceleration mechanisms was also very important. At the same time, cost was also concerned. Gear ratio, performance, feasibility and cost became important indicators for the evaluation of transmission systems. Later on, the requirements for the handling experience of transmission systems and the auxiliary functions that could be achieved by transmission systems increased. The requirements for smooth speed switching, impact resistance, precise control, load holding capability, and power density were also introduced as evaluation indicators. In recent years, the development of safety, environmental protection, and intelligence has become deeply rooted. Young users in particular are looking for personalised experiences. Efficiency, energy saving, environmental friendliness, and the completeness of support facilities are becoming increasingly important. Environmental requirements and the uneven distribution of energy resources have elevated transmission efficiency to a key evaluation criterion in the process of selecting transmission technologies.

As technology evolves, there may be more complex requirements and functions that will be included in the evaluation indicators when considering different application scenarios, different market segments, and different customer groups.

In the current product development process, target functionalities and requirements are passed directly to design engineers from the relevant departments. Engineers do not analyse and prioritise the list of features and requirements. As a result, engineers may select the optimal technical solution based on the target functionalities and requirements, but this does not guarantee that the product will meet market and customer expectations. This can result in a very well-designed product that fails to gain customer acceptance in the marketplace [1], Fig. 1 try to state the relationship.

This article attempts to introduce the KANO requirements analysis model to the evaluation of functionality and requirements before product development. It also

**Fig. 1** Connection of determining actors and customer satisfaction

presents the application of three evaluation methods. By evaluating and filtering functionalities and requirements, the aim is to help engineers determine the reference indicators in engineering design and increase the success rate of projects.

### **2 Literature Review**

The transmission system is a mechanical device that enables the transmission of power and motion. It acts as a connection between the power source and the executing mechanism. In the actual design process, there are many transmission solutions available to meet different customer needs.

Chen Yijun and Wu Kejian [2] considered how to select a transmission solution among numerous options for a machine. They found that applying fuzzy comprehensive evaluation theory to study systems composed of a limited number of components can effectively avoid the complexity and computational burden of traditional evaluation methods. The Analytic Hierarchy Process (AHP) [3], proposed by American operations researcher T L Saaty, is a method based on human judgement that can effectively quantify qualitative problems. It is concise, flexible, and logically strong, making it a feasible solution for multi-level, multi-objective optimization problems. Xiong Rui and Cao Kunsheng [4] introduced a decision-making method using "indicator matrices" to construct "judgment matrices" for solving quantitative or mixed-type multi-objective problems. To avoid the subjectivity and bias in the AHP method, Ma Yundong and Hu Mingdong [3] developed an improved Analytic Hierarchy Process called I-AHP, using the optimal transfer matrix. Song Baowei, Zeng Wenhua, Mao Zhaoyong, and Liang Qingwei [5] introduced a combined evaluation method that integrates the Analytic Hierarchy Process with fuzzy mathematics and incorporates fuzzy Analytic Hierarchy Process based on entropy weight, forming a subjective–objective combined evaluation method. Yao Huibo [6] studied four transmission scheme selection and evaluation models: fuzzy comprehensive judgment method, support vector machine method, uncertain measure model, and entropy weight-based fuzzy AHP method. He believed that the fuzzy comprehensive evaluation method can provide a relatively comprehensive quantitative analysis of all major influencing factors in the evaluation of mechanical transmission schemes, enabling objective and effective selection of more reasonable solutions.

As machinery and equipment are tools used to accomplish specific tasks, the transmission system is a part of the machinery or equipment. Functional requirements are defined by the tasks to be performed by the machinery and equipment and may have special characteristics due to different individual customer needs. The degree of success of a product which is reflected in customer satisfaction depends on the product's ability to meet actual functional requirements, and can only be achieved by meeting customer needs. If engineers rely on the target functional requirements provided by the relevant departments to complete the development and design, without verifying whether the target functional requirements match reality. Customer satisfaction will unable to be guaranteed, refer to Fig. 1.

According to Kelsey Miller's definition [7], customer needs are the problems that individuals are trying to solve, which drive them to seek products or services to address these problems. Customer needs include functional needs, social needs, and emotional needs. Jeffrey Bussgang [8] defines the consistency between a product and the target customer needs as product-market fit (PMF).

Without a method to analyse and classify target function before product development, it is inevitable that there may be discrepancies or deviations between the function provided by the product and the actual customer needs.

### **3 Solution**

In the process of new product development, enterprises emphasize division of labour and collaboration, ask engineers designing products according to the requirements of the demand department. However, for some new products, it is not until the marketing stage that the market response is discovered and no product is sold, causing the new product programme to lose its intended value. Engineers may feel frustrated when they have found a solution based on reasonable evaluation methods but receive a negative market response. This situation may also lead to conflict with the collaborating departments. The product demand department expects engineers to design products that are closer to the market and more grounded.

The Fuzzy Analytic Hierarchy Process [5] based on entropy weighting can effectively solve the problem of selecting technical solutions at the evaluation level. However, it cannot weigh whether the function achieved by the right technical solution is the function required by the client. This indicates that product designers need to evaluates the product functional requirements before selecting technical solutions. Ranking product attributes based on how they are perceived by customers and their impact on customer satisfaction [9–12] by those who understand the technical options and functional requirements is an essential approach to success.

The Kano model [13, 17] established by Noritaki Kano which associates product design and production with customer demands. As a tool for enhancing products and services based on customer situations, the Kano model correlates customer satisfaction with product attributes. Based on the principle that different customer reactions are obtained depending on the degree to which the product meets their needs, the influencing factor attributes of the product are classified into five categories of customer satisfaction: attractive attributes (referred to as *A* attributes), onedimensional attributes (referred to as *O* attributes), must-be attributes (referred to as *M* attributes), indifferent attributes (referred to as *I* attributes), and reverse attributes (referred to as *R* attributes).


The principles to be followed in product development and design are as follows: When designing a product, it is important to avoid indifferent attributes and reversal attributes as much as possible. After addressing the must-be attributes and onedimensional attributes, efforts should be made to achieve attractive attributes for the product. This approach can effectively help engineers evaluate product requirements, obtain a combination of product requirements with higher overall customer satisfaction, and improve market fit.

There are two ways to collect customer feedback: by questionnaire or by customer interview, according to Sauerwein Elmar et al. [14], individual interviews seem to be more favourable, but interviews need to include questions for questionnaire and record properly. All the questions must be written clearly [16]. After collecting, the feedback of each customer must be sorted out with the following table.

Table 1 is one customer's feedback, according to Griffin/Hauser, 20 to 30 customer interviews in homogenous segments is suffice to determine approximately 90–95% of all possible product requirements [14, 20]

For multi-customer sample data, there are three methods for analysing and filtering customer feedback.

Discrete Analysis: In discrete analysis, the survey results are categorized based on the total count of attributes for each function or requirement. If there is a category


**Table 1** Satisfaction rating Table—The feedback of customer LI

aQ = Questionable. bF-III = Function III, N-I = Needs-I

with the highest frequency, the function or requirement can be classified into that category. If the results are close for different categories, they can be prioritized as follows [14]: *M*ust-be > *O*ne-dimensional > *A*ttractive > *I*ndifferent. The importance of the function or requirement can also be determined by considering the feedback from the survey.

Quantitative Analysis (Better-Worse Coefficient): In this method, each function or requirement is classified based on its attributes, and the Better-Worse coefficient is calculated to represent its impact. The satisfaction coefficient when provided is calculated as Better/SI = (A + O)/(A + O + M + I) [14]. The Better/SI value is usually positive, indicating that satisfying a particular function or requirement will increase satisfaction. The closer the value is to 1, the stronger the impact of that function or requirement on satisfaction, making the product more favourable. The dissatisfaction coefficient when not provided is calculated as Worse/DSI = -1\*(O + M)/(A + O + M + I) [14]. The Worse/DSI value is usually negative, indicating that not meeting a certain function or requirement will decrease satisfaction. The closer the value is to -1, the greater the reverse impact of that function or requirement on satisfaction, suggesting it should be included in the standard configuration.

Continuous Analysis: In continuous analysis, the results are converted into scores ranging from -2 to 4 based on the level of functional [15, 16, 18, 19]:—2 (I dislike it that way), -1 (I can live with it that way), 0 (I am neutral), 2 (It must be that way), 4 (I like it that way) when functional. -2 (I like it that way), -1 (It must be that way), 0 (I am neutral), 2 (I can live with it that way), 4 (I dislike it that way) when dysfunctional. A higher score reflects a greater preference for that function by customers.

### *3.1 An Example*

Collected 30 feedback results from customers, for the Needs XI, by attribute evaluation: 15 evaluation bit *A*, 6 evaluation bit *O*, 6 evaluation bit *I*, 3 evaluation bit *M*, the details in the following label.

Then, based on the data of Table 2, the results of the three analytical methods will be.

The result of Discrete Analysis is presented in Table 3.

The results of Quantitative Analysis.

Satisfaction coefficient while functional: better/SI = (A + O)/(A + O + M + I) = (15 + 6)/ (15 + 6 + 6 + 3) = 0.7, which is close to 1 and has a stronger impact.

Dissatisfaction coefficient while Dysfunctional: Worse/DSI = -1\*(O + M)/(A + O + M + I) = -1\*(6 + 6)/ (15 + 6 + 6 + 3) = -0.4, which is far away from -1 and has a weak influence.

The Better-worse coefficients are more consistent with the characteristics of the desired attribute (O).

The results of Continuous Analysis.

Functional: 15\*4 + 6\*4 + 6\*2 + 3\*0 = 96, which is 80% of the total score of 120(=4\*30).


**Table 2** Satisfaction Rating Table—The feedback of Customer 1 ~ Customer 30

aC-(8,12,20) represents the feedback results from Customer 8, Customer 20 and Customer 12

**Table 3** Results of discrete analysis


Dysfunctional: 6\*(-1) + 3\*0 + 3\*0 + 6\*2 + 3\*2 + 6\*4 + 3\*4 = 48, accounting for 40% of the total score of 120 (=4\*30).

If 50% is identified as the cut-off point for strong or weak influence, then influence is stronger when functional and weaker when dysfunctional. It is more consistent with the desired attribute (*O*) characteristic.

### *3.2 Discussion on the Results*

Among the three analysis methods, the result of the Discrete Analysis is *A*, while the results of the Quantitative Analysis and the Continuous Analysis are both *O*. The results obtained from these three methods show discrepancies. According to the KANO model, category *A* represents "the satisfaction level of users will not decrease if dysfunctional, but will greatly increase if functional," while category *O* represents "the satisfaction level of users will increase if functional, but will decrease if dysfunctional." These two classifications have different impacts on customer satisfaction. Therefore, it is necessary to use the KANO model to analyze all the functions or needs of the product again.

Among all feedback *A* in Table 2, there are various scenarios ranging from " It must be that way" to "I dislike it that way" for the feedback if dysfunctional. It is evident that the categorization of all these results as category *A* lacks sufficient discrimination.

The results of the Quantitative Analysis and the Continuous Analysis are almost similar, but the Quantitative Analysis also suffers from the same problem as the Discrete Analysis. The Continuous Analysis scores the different feedback results in category *A* from two dimensions, making the results more reliable.

### **4 Conclusion**

Based on customer's different personalized understanding of functions or needs, an example of function and need analysis using the KANO model shows that it is crucial to analyze and select the set of functions or needs for a product before its development. This analysis is expected to be beneficial in effectively utilizing resources to prioritize key functions or needs. By using the KANO model, the functions or needs of a product can be analyzed based on customer feedback. Among the three analysis methods, the Continuous Analysis has a higher ability to differentiate customer feedback results, making it more advantageous in obtaining reliable results.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Study on Intelligent Driving School Vehicle System Based on Digital Twin Technology**

**Xinran Zhang, Yongming Bian, Meng Yang, and Jie Shao** 

**Abstract** The development of intelligent vehicle systems has progressed through multiple stages, evolving from initial basic functionalities to the integration of intelligent and automated technologies, achieving significant advancements. With technological advancements, intelligent vehicle systems have gradually introduced driving assistance features such as autonomous driving assistance and adaptive cruise control. In response to the developmental needs of intelligent driving school vehicle systems, a comprehensive research proposal is presented based on digital twin technology. By incorporating a multi-source data acquisition module, real-time collection and monitoring of various sensor data within the driving school vehicles are achieved, ensuring the acquisition of crucial information during the driving process. To facilitate efficient communication between the data monitoring center and the driving school vehicles, a driving school local area network is established, providing a solid foundation for data transmission and real-time monitoring. A user-friendly and intuitive front-end interface is designed to visualize vehicle status information and student driving data for instructors, enhancing the efficiency and effectiveness of driving school education. Finally, based on practical application tests, data processing of coordinate information is conducted and showcased in the front-end interface, validating the feasibility and practicality of the designed system in driving school vehicle management and educational training.

**Keywords** Digital twin · Intelligent vehicles · Frontend interface design

Y. Bian · M. Yang

© The Author(s) 2024

X. Zhang · Y. Bian · M. Yang (B) · J. Shao

Tongji University School of Mechanical Engineering, Shanghai 201804, China e-mail: yangmeng@tongji.edu.cn

Tongji University National Engineering Technology Research Center for Prefabrication Construction in Civil Engineering, Shanghai 200092, China

Shanghai Engineering Research Center for Safety Intelligent Control of Building Machinery, Shanghai 200032, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_85

### **1 Introduction**

With the rapid development of technology, digital and intelligent technologies have demonstrated strong potential and broad prospects across various domains. In the field of transportation, especially in driving training and vehicle management, the application of digital technology is gradually transforming traditional driving school vehicle systems. As one of the core technologies of digital transformation, Digital Twin technology boasts prominent advantages in simulating the digital image, operational status, and performance characteristics of physical objects [1]. Through Digital Twin technology, the actual driving school vehicles can be combined with their virtual counterparts in the digital realm, enabling real-time monitoring, analysis, and optimization [2]. This technology not only provides an accurate understanding of the driving school vehicle's status but also simulates various driving scenarios, aiding students in practicing driving within a safe environment and effectively enhancing their driving skills.

In China, satellite navigation systems such as GPS and Bei Dou have been widely applied in the field of vehicle positioning for tasks such as vehicle navigation, realtime positioning, and path planning. With the expansion of vehicle-to-everything (V2X) technology, the integration of intelligent vehicle positioning systems and the Internet has become increasingly seamless in China. Vehicles can transmit real-time location information through the network, providing more real-time data for drivers and systems. In foreign countries, high-precision positioning technologies like Real-Time Kinematic GPS (RTK-GPS) are extensively utilized, offering sub-meter or even higher accuracy positioning services, particularly applicable to areas like autonomous driving [3]. The application of intelligent vehicle positioning systems in foreign countries has grown more prevalent in fields such as autonomous driving and vehicleto-vehicle communication, enabling vehicle-to-vehicle communication and location information to facilitate fleet coordination and traffic flow optimization [4].

In reference [5], a collaborative underwater network for an ocean observation system was deployed using Digital Twin methodology. Reference [6] explores the necessity of Digital Twins as decision-making tools from a demand perspective. In conclusion, Digital Twin technology holds substantial practical significance for advancing the development and evaluation of intelligent vehicle functions. However, there are relatively few research reports on the specific implementation, validation, and applications of Digital Twin technology in intelligent vehicles. This paper aims to investigate an intelligent driving school vehicle system based on Digital Twin technology. By integrating physical vehicles with Digital Twin models, real-time vehicle monitoring, vehicle positioning, and driving training optimization can be achieved. This study will delve into three aspects: multi-source data acquisition, local network establishment, and data visualization interface design. Application testing will be conducted on actual driving school vehicles, aiming to provide innovative solutions for driving school vehicle management and training. This research is expected to contribute new ideas and approaches to the development and application of intelligent driving school vehicle systems.

# **2 Multi-Source Data Acquisition for Intelligent Driving School Vehicles**

The vehicle sensor data acquisition module utilizes both serial ports and Controller Area Network (CAN) to collect and parse the serial output information from the P3DU receiver unit in the driving school vehicle's electronic control system. This includes location data and velocity data. Additionally, it collects and interprets CAN messages from the On-Board Diagnostics (OBD) interface, gear position sensor (not equipped in automatic transmission training vehicles), and brake/clutch/accelerator pedal angle sensors (not equipped in automatic transmission training vehicles), thereby obtaining signals from various sensors.

The CAN messages from the driving school vehicle can be categorized into three types: OBD interface information, CAN messages from the gear position sensor, and CAN messages from the brake/clutch/accelerator pedal angle sensors. The CAN messages from the OBD interface encompass status information such as handbrake, turn signals, and driver's door. For the RTK-GPS system, data collection involves obtaining the \$GPHCD position information from the P3 receiver unit's serial port 1, as well as the \$GNVTG data from serial port 2.

### **3 IoT-Based Data Transmission Architecture**

This section aims to propose an IoT-based data transmission architecture to meet the functional requirements of the smart driving school coach robot hardware system. By establishing a local area network (LAN) communication between the Data Monitoring Center and the smart driving school coach, real-time monitoring of vehicle status information, training data storage, and audio–video communication functionalities are achieved. Specifically, the network architecture employs the AP + AC mode, with the Data Monitoring Center serving as the core to establish a comprehensive LAN for centralized management and control of the smart driving school coach.

To realize the data transmission architecture, this paper adopts the AP + AC networking mode to build the driving school's local area network. As shown in Fig. 1, the router is directly connected to the Internet, and the switch is used to assign IP addresses to various connected devices in the lower layer. The AC functions as a centralized management device, overseeing the configuration of parameters such as IP addresses, channels, and power for the APs. Terminal devices access the Internet through the wireless network generated by the APs.

To establish an internal local area network (LAN) for the intelligent driving school, this study employs router devices that support Access Controller (AC) functionality, connecting multiple Access Points (APs) via switches to create a wireless network within the training area. The instructor vehicles connect to the wireless network

within the training area, enabling local area network communication between the vehicles and the data monitoring centre.

# **4 Designing a Vehicle Data Visualization Interface**

To achieve real-time monitoring and positioning of vehicle status within the driving school training area, this section presents an interface design solution based on the ThingJS platform. The interface functionality encompasses displaying vehicle operational status, visually presenting vehicle travel data, and providing real-time vehicle location tracking. Through this interface design, driving school administrators can promptly grasp vehicle operational status and location information, offering timely and effective data support to instructors.

### *4.1 Vehicle State Presentation*

The in-vehicle pad student training interface establishes a WebSocket connection with the onboard computer to receive vehicle operating status information, including driving speed, steering wheel angle, left and right turn signal status, clutch pedal angle, brake pedal angle, throttle pedal angle, driver's door status, seatbelt status, handbrake status, and ignition status.

### *4.2 Vehicle Location Display*

The objective of the GPS coordinate positioning module is to take the positioning information received by the vehicle sensor data acquisition module, calculate it in the coordinate system of the training area, and determine the position coordinates of vehicle key points (such as headlights and wheels) based on the vehicle's heading angle. This is done to determine the current training subject for the student, generate corresponding instructions for the start or completion of the training subject, and transmit them to the onboard pad. Figure 2 shows an example of the in-vehicle Pad trainee training interface.

**Fig. 2** Vehicle and training area Three-Dimensional models

# **5 Application Examples of Intelligent Driving Instructor Car Systems**

Taking the Intelligent Driving Instructor Robot Project at a Driving School as a Case Study, Demonstrating the Application Achievements of Intelligent Vehicle Systems in the Context of the Intelligent Driving Instructor Robot Project.

# *5.1 System Deployment*

By setting up a local area network, communication between the instructor's vehicle and the data monitoring center is established, allowing the transmission of instructor vehicle status information to the centre. Sensors. The OBD-CAN interface, gear position sensor, and brake/clutch/throttle pedal angle sensors are installed at corresponding positions within the driver's cabin to achieve the collection of vehicle status information. Auxiliary brake control system. The auxiliary brake control system is placed on the vehicle roof. It comprehensively analyzes ultrasonic radar signals, millimeter-wave radar signals, speed information, reverse light signals, ignition signals, handbrake signals, shielded brake signals, etc. It then executes corresponding control strategies to ensure the safety of the driver. Figure 3 illustrates the main hardware devices added to the vehicle hardware modification.

The main design section showcases the vehicle's positional status, depicting its location within the driving school training ground model. Additionally, auxiliary practice buttons and information display boxes are integrated into the surroundings. Through the frontend interface's vehicle positioning display, vehicle status presentation, and auxiliary function design, learners can observe the vehicle's operational location and status during the driving skill learning process. This provides a more intuitive understanding of the impact of driving manoeuvres on vehicle control.

**Fig. 3** Driving school vehicle hardware modification

**Fig. 4** In-Car and exterior pad training interfaces

Figure 4 demonstrates a test screen after the hardware and software systems have been deployed.

### *5.2 Data Analysis and Processing*

Through the intelligent driving school car system, vehicle positioning data is collected and subjected to coordinate system transformation processing. As a result, coordinates in the training ground's coordinate system are obtained. As shown in the figure, the initial data obtained is relatively coarse, requiring further smoothing and approximation for display on the front-end interface. Due to the high real-time requirements of vehicle positioning, three data processing methods are proposed.

$$\mathbf{x} = \mathbf{0}.25 \times \mathbf{x}\_{t-2} + \mathbf{0}.25 \times \mathbf{x}\_{t-1} + \mathbf{0}.5 \times \mathbf{x}\_t \tag{1}$$

$$\mathbf{x} = round(\mathbf{x}\_t \times \mathbf{50})/50\tag{2}$$

$$\mathbf{x} = round(\mathbf{x}\_t \times 12.5) / 12.5 \tag{3}$$

Here, x represents the coordinate value processed at the current time step, *xt* denotes the raw data of the coordinate value at the current time step, *xt*−1 corresponds to the raw data of the coordinate value at the previous time step, *xt*−2 represents the raw data of the coordinate value received two time steps ago, and 'round' indicates rounding to the nearest integer. The subsequent method 1, method 2 and method 3 correspond to data processing approaches in Formulas (1), (2) and (3), respectively. Dividing vehicle status into two categories, motion and stationary, based on velocity information.

For the motion scenario, taking the example of original data during a vehicle's movement on a straight road segment, the x-axis data gradually increases while the y-axis data is subjected to three different processing methods successively to create

**Fig. 5** Coordinate data line graph during vehicle motion

**Fig. 6** Coordinate data scatter plot during vehicle standstill

plots. The resulting graph is shown in Fig. 5 below. From the graph, it is evident that the curve obtained from processing method 3 fits the original data less accurately compared to processing methods 1 and 2. The smoothness of the curve obtained from processing method 2 is not as good as that of processing method 1. Therefore, processing method 1 is more suitable for data processing under motion status.

In the case of vehicle at rest, taking the original data of the vehicle stationary at a certain point in the ignition state as an example, data processing was conducted using three different methods for both x and y-axis data. The resulting scatter plots are shown in the figure below. It can be observed from Fig. 6 that method 1 still yields a set of points with significant data fluctuations, method 2 ultimately yields 4 coordinate points, and method 3 yields 2 coordinate points. Therefore, for the stationary state, data processing method 2 is more suitable.

### **6 Conclusions**

• This paper introduced the hardware and software components of an intelligent vehicle system, realizing the collection of multi-source data from sensors, its transmission through the network to the computer, and subsequent interface display;


**Acknowledgements** This research was supported by the National Natural Science Foundation of China (52205279), the Open Foundation of the National Engineering Technology Research Center for Prefabrication Construction in Civil Engineering (2021CPCCE-K02), and the Top Discipline Plan of Shanghai Universities-Class I.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Architecture of Intelligent Simulation System for Automatic Quay Crane Training Based on Embedded Digital Twin Technology**

#### **Mengjie He, Yujie Zhang, Yang Shen, and Chao Mi**

**Abstract** The training of automated container gantry crane drivers faces with challenges, such as high safety risks, difficult training, and high learning cost. Therefore, we have designed an intelligent simulation system architecture for automated container quay cranes training based on embedded digital twin technology. This system can be directly embedded into the real quay crane remote control console without changing the original quay crane hardware equipment. It enables real-time switching between virtual quay crane remote control operation simulations and real gantry crane remote control operation, providing training for quay bridge drivers while also accounting for the real quay crane operation in time. To achieve seamless switching between real control and simulation training, we have designed two types of controllers: an automatic quay bridge remote control operation simulation system based on OPC UA communication protocol, and a real quay crane automation controller based on PLC control bus. Both controllers can communicate bidirectionally with the remote control room operating station PLC. Practical application at a wharf has demonstrated that our designed system offers high safety, real operation experience, and significant training effectiveness, effectively improving automated quay crane training outcomes, efficiency, and effectiveness.

**Keywords** Digital twin technology · Intelligent simulation system · Virtual simulation

© The Author(s) 2024

1081

M. He · Y. Zhang · Y. Shen · C. Mi (B) Shanghai Maritime University, Shanghai 201306, China e-mail: chaomi@shmtu.edu.cn

Y. Shen e-mail: yangshen@shmtu.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_86

### **1 Introduction**

With the development of technologies such as 5G communication, big data, and artificial intelligence, the automation level of ports has been rapidly improved [1]. Automated remote operation port loading and unloading equipment has been widely applied to port enterprises at home and abroad. Given this new type of automated remote control mode, the existing operating driver training is faced with high safety risks, difficult training, and high learning cost, so this paper proposes an intelligent simulation system architecture for automated quay crane training based on embedded digital twin technology.

The concept of a digital twin includes both physical structures in physical space and digital models in cyberspace, aiming to realize real-time interaction between physical entities and virtual models [2]. At present, digital twin technology has been widely used in automated container terminals. For example, there are applications in port structure and auxiliary monitoring systems [3], port crane operation status monitoring [4], intelligent scheduling of port logistics [5], and the construction of ship digital twins [6].

In recent years, scholars at home and abroad have also done a lot of research on virtual crane simulators. For example, the immersive virtual reality simulator proposed by Patrao and Menezes [7]. Juang et al. [8] developed a crane simulator called SimCrane 3D. Noda et al. [9] studied a training system that enables operators to safely and effectively master the operation of electric bridge cranes while suppressing load sway. In addition, there are bridge crane simulator systems [10, 11], both of which are effective in training crane drivers.

However, the virtual crane simulator and the real crane are independent of each other, so the operation of the real crane cannot be taken into account in time when the virtual crane simulator is used for training. However, the intelligent simulation system architecture of automatic quay crane training based on embedded digital twin technology proposed in this paper to use digital twin technology to establish a digital twin model for the structure and function simulation of the real automatic quay crane remote control system. And a digital virtual simulation System with the functions of the Chassis Parking System and Ship Profile Scanning System (SPSS) is designed. Moreover, the system can be directly embedded into the real quay crane remote control operation platform without changing the original automatic quay crane control system hardware, and realize the real-time switch between the virtual quay crane remote control operation simulation and the real quay crane remote control operation. It can not only realize the training function of the opposite quay crane driver but also take into account the real gantry crane operation in time, to achieve the effect of idle training and busy operation.

# **2 Implementation**

### *2.1 Automated Quay Crane Remote Control System*

The overall architecture of the system is shown in Fig. 1. When the driver remotely controls the actual quay crane, the on-board PLC of the shore container needs to transmit the sensor signals from the field operations, various mechanical actions, field control signals, and other signals to the PLC of the remote control room operator console through the control bus. Then, the received signal data is analyzed and processed by the PLC and presented to the CMS system. Simultaneously, the CMS system reads and analyzes the task instructions, and returns them to the PLC for the next specific operation of the quay crane.

The remote console is the use terminal of the remote control system and the visualization device that interacts with the driver. As shown in Fig. 2, it is composed of

**Fig. 1** The overall architecture diagram

6 Closed-Circuit Television (CCTV) screens, the remote port machinery monitoring system, the driver's console and the GP touch screen. According to the CCTV screen and the remote port machinery monitoring system, the driver in front of the remote operation platform sets the work task information of the quay crane with the button on the operation platform. GP touch screen is used to display PLC operation status, such as unlocking, locking, spreader size, car speed, position, etc. In addition, the driver can also switch the video screen through the GP touch screen, turn on the projector light, adjust the camera shooting parameters, etc.

### *2.2 Intelligent Simulation Sensing System*

The automated quay crane remote control system has a ship profile scanning system, chassis parking system, optical character recognition, and other sub-systems instead of on-site manual operation of the link. To achieve the same effect as the real quay crane remote control control system, our automated quay crane remote control operation of the intelligent simulation system is also equipped with a large number of virtual sensor modules. For example, the realization of the SPSS function is through the system's virtual laser range finder constantly launching rays on the ship shape scanning, when the rays hit containers or obstacles, real-time detection of distance information, and then the ship shape data obtained will be transmitted to the PLC for interaction, the spreader to limit the speed of deceleration of the action, the realization of the quay crane hoisting, and the direction of the trolley spreader collision protection and intelligent deceleration function. Figure 3 shows the schematic diagram of the 2D LiDAR installation position in the virtual quay crane simulation system.

The realization of the container truck self-positioning function in this system mainly relies on the built-in collision detection technology of Unity3D to set up the collision body on each lane and the corresponding collector truck, as shown in Fig. 4, and the collision official body in the actual system is transparent, which adjusts the position of container truck by detecting the collision between the container truck and

**Fig. 4** Schematic diagram of virtual set card perception

the front and rear limits and the lane limits and sends the position of container truck to the PLC when the container truck reaches the specified position. Send the position of the container truck to the PLC, in addition, according to the collision detection information of arriving at the front and rear limits, call the function of the container truck to move forward or backward, so that the container truck arrives at the specified position, to realize the virtual simulation of the container truck positioning system.

### *2.3 Control System*

In order to embed this system into the real quay crane remote control console without changing the hardware of the existing automated quay crane control system, so as to realize the one-key switching function between the real quay crane remote control operating system and the virtual quay crane remote control operating system, the data transmission between the automated quay crane remote control operation intelligent simulation system and the remote control room operating console PLC needs to be considered when carrying out the hardware design. In the virtual quay crane remote control operating system, the system integrates a large number of virtual sensors, which not only highly reproduces the real quay crane in terms of visual and physical effects, but also reproduces the actual operating conditions. The specific process is to realize bi-directional communication with the PLC of the remote control room operator console through OPC UA communication protocol, i.e., the system can not only receive the mechanism drive signal from the PLC of the remote control room operator console, but also transmit the signal from the virtual sensors to the PLC of the remote control room operator console.

Through the above operation, the remote control room console PLC can not only obtain the relevant data information of the real shoreline remote control operation but also read the data information of the virtual shoreline remote control operation and carry out the same processing as the real shoreline remote control operation, so that it can realize the function of switching between the real shoreline remote control operation and the virtual shoreline remote control operation with one key on the same console.

### **3 Experiment**

Based on the overall architecture diagram in Fig. 1, it is evident that the driver can familiarize themselves with the operational procedures of the console through the intelligent simulation system. This allows them to safely and efficiently acquire the necessary skills for controlling the remote quay crane. These skills encompass basic operations such as loading and unloading ships and trucks, utilizing the spreader guide effectively, as well as managing car acceleration and deceleration. Alongside these foundational skills, the system also trains drivers in specialized tasks like lifting heavy parts and handling dangerous goods or special containers under challenging conditions such as rain or wind. Furthermore, the system records historical training data which facilitates the assessment of the driver's performance. This enables us to evaluate the standardization of their operation, optimize their working methods, enhance their efficiency, and assess their readiness to operate real remote control quay cranes.

As shown in Fig. 5, this paper also conducted functional tests of the system in six aspects, namely basic function, communication function, operational task module, operation perception, performance and stability, and one-click switching function, at a marina, as shown in Table 1.

The above test results show that the system can not only realize the training function of the quay crane driver but also realize the real-time switching function between the virtual quay crane remote control operation simulation and the real quay crane remote control operation after being directly embedded in the real quay crane remote control console. Thus, the effect of training in idle time and operation in busy time can be achieved.

**Fig. 5** System functional test

**Table 1** System function test status


# **4 Conclusion**

This paper proposes an intelligent simulation system architecture based on embedded digital twin technology for automated quay crane training, which has been practically applied in a terminal, and the results show that the system architecture can realize the real automated quay crane operation and operation simulation with one-click switching function by directly embedding it into the real remote control console without modifying the hardware of the existing automated quay crane control system.

It can realize the training of novice drivers and the daily training of other drivers, and it can not delay the normal operation of the automated remote control quay crane, and the transformation cost is low and the safety risk is low. However, due to time constraints, certain aspects of the system have not been fully investigated. To further enhance the virtual quay crane simulation system, future plans involve incorporating an automatic evaluation module to provide feedback on driver training. Additionally, a study on the training effectiveness of novice drivers in relation to spreader swing amplitude and time in both virtual and real quay cranes is also planned.

**Acknowledgements** This research was supported by the Science and Technology Commission of Shanghai Municipality (Nos. 122ZR1427700 and 223692106900)

### **References**


Research on Architecture of Intelligent Simulation System … 1089

11. Guoqian W, Zhenyan Z, Haoming D (2009) Research and implement of the training simulator for overhead cranes. In: 2009 IEEE 10th international conference on computer-aided industrial design & conceptual design, Wenzhou, China, pp 1049–1054. https://doi.org/10.1109/CAI DCD.2009.5375328

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Research of Different Energy Management Strategies of Lithium-ion Battery-Ultracapacitor Hybrid Energy Storage System**

#### **Dongjie Zhang, Lin Hu, Qingtao Tian, and Changfu Zou**

**Abstract** Given the exacerbating effect of fossil fuel use in conventional vehicles on the greenhouse effect, the imperative development of electric vehicle technology becomes evident. To address the high energy and power density demands of electric vehicles, a lithium-ion battery-ultracapacitor hybrid energy storage system proves effective. This study, utilizing ADVISOR and Matlab/Simulink, employs an electric vehicle prototype for modeling and simulating both logic threshold and fuzzy logic control strategies. It aims to analyze the average output power and state of charge (SOC) of the lithium-ion battery, as well as the SOC of the ultracapacitor, within hybrid energy storage systems governed by these differing strategies. The findings indicate that the fuzzy logic control strategy results in a reduction of 2.73 kW in the average output power of the lithium-ion battery and a 20% increase in the SOC drop rate of the ultracapacitor compared to the logic threshold control strategy. Under the logic threshold control strategy, lithium-ion batteries demonstrate superior output stability, albeit within a broader amplitude range. Conversely, the fuzzy logic control strategy maximizes the utilization of ultracapacitors but leads to frequent fluctuations in the output power of lithium-ion batteries, thereby exhibiting reduced stability. These results underscore the inherent trade-offs between stability and utilization efficiency in hybrid energy storage systems for electric vehicles under different control strategies. The selection of a control strategy should be contingent upon specific

D. Zhang · L. Hu · Q. Tian (B)

#### Q. Tian

School of Traffic and Transportation Engineering, Changsha University of Science and Technology, Changsha 410114, China

C. Zou

© The Author(s) 2024

D. Zhang · L. Hu

School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China

Hunan Key Laboratory of Safety Design and Reliability Technology for Engineering Vehicle, Changsha University of Science and Technology, Changsha 410114, China e-mail: tianqt\_csust@163.com

Department of Electrical Engineering, Chalmers University of Technology, 41296 Gothenburg, Sweden

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_87

performance priorities and objectives within the context of electric vehicle design and operation.

**Keywords** Hybrid energy storage system · Energy management · Logical threshold control strategy · Fuzzy logic control strategy

### **1 Introduction**

In recent years, the gas emitted by traditional fuel vehicles has aggravated air pollution and fossil energy is non-renewable. Therefore, it's essential to develop and enter a period of accelerated development of electric vehicles (EV).

In [1], it indicates that high discharge rates and depth of discharge will cause high rate of capacity fade and a shorter useful life of lithium-ion battery (LiB). Studies have shown that when the LiB faces a high-power impact, it may lead to over-discharge of the LiB, leading to the reduction of the number of charge and discharge cycles of the LiB. Ultracapacitors (UC) can meet the high-power requirements of EV, but their low energy density makes it difficult for vehicles to have a long life. The hybrid energy storage system (HESS) composed of LiB and UC plays a role of "peak cutting and valley filling" for LiB. In [2], the results show that HESS with appropriate size and enabled energy management can significantly reduce the battery degradation rate by about 40% compared to battery energy storage systems (ESS), and at only 1/8 of the additional cost of the system.

HESS primarily encompass three types of topologies: passive topology, semiactive topology, and active topology [3]. Pratim Bhattacharyya et al. [4] proposed an improved LiB and UC hybrid semi-active structure for EVs where the size and space of the energy storage system (ESS) are critical. Zhu et al. [5] proposed that bidirectional DC/DC converters can be divided into isolated (IBDC) and non-isolated bidirectional DC/DC converters (NBDC) according to whether electrical isolation is realized between ports. Li et al. [6] conducted an optimization design for a UC-based semi-active HESS and interleaved parallel bidirectional Buck/Boost converters, and analyzed the reasons behind the optimization results.

A good energy management strategy (EMS) of HESSs can improve performance in different and complex driving conditions and reduce driving costs. HESS energy management methods [7–9] in EV applications can be summarized into the following two categories, namely model-based methods and rule-based methods. Model-based energy management methods usually cooperate with other control methods for power distribution, such as the global optimization algorithm dynamic programming (DP) [10], which can obtain the optimal control input by minimizing the cost function, and it's suitable for offline calculation to provide parameter setting values for determining rule control. After analyzing different data sets according to the optimal power distribution strategy, Shen et al. [11] proposed a neural network-based (NN-based) HESS control method for medium-sized EVs. Rule-based methods consist of a set of predefined, empirical control rules [12]. Wang et al. [13] proposed a rule-based control method to realize mode selection and current distribution of multi-mode HESS in EV applications. Yin et al. [14] proposed an adaptive fuzzy logic control (FLC) scheme for EV energy management, where the output membership functions are periodically refreshed to adapt to changing driving patterns. The experimental results in An et al. [15] conducted under the HWFET condition demonstrate that the fully active dual-energy source HESS, along with the EMS based on FLC, effectively safeguards the LiB against the detrimental effects of substantial current fluctuations, consequently prolonging the battery's lifespan. Based on the advantages of flexibility and robustness of the fuzzy logic controller, Gao et al. [16] proposed an optimization method of fuzzy controller hybridization (DOH) and membership function based on the golden section rule. In [17], The optimal logical threshold control (LTC) can fully leverage the high-power UC's characteristics and conduct hardware-in-theloop (HIL) experiments to further validate the real-time and dependable nature of the near-optimal LTC. The gray wolf optimization (GWO) is used in Hu et al. [18] to optimize the battery output power upper limit and UC charging upper limit of mature multi-mode control (MMC). In [19], frequency-based energy management distributes high- and low-frequency power requirements to batteries and ultracapacitors respectively. Min [20] proposed a multi-objective EMS for EV HESS based on separating load power. The impact of different sorting methods on the results by using elite strategy non-dominating sorting genetic algorithms (NSGA-II).

With the widespread adoption of EVs, experts and scholars have initiated research into artificial intelligence power distribution strategies. On the basis of existing algorithms, it's necessary to improve the robustness of EV under driving conditions and optimize the control strategy algorithm. The sections of this article are as follows. In the second section, we introduce vehicle parameters, HESS components, and the simulation model. Section 3 provides an overview of the LTC and FLC strategies. Sections 4 and 5 present the simulation results and draw conclusions, followed by future prospects.

### **2 Parameters and Hybrid Energy Storage System Model**

### *2.1 Vehicle Parameters*

The parameters of the electric vehicle are designed according to a benchmark model. The main vehicle parameters are shown in Table 1.

### *2.2 Lithium-ion Battery*

The equivalent circuit models of LiB include Rint model, RC model, PNGV model, etc. Due to its concise structure and convenient calculation method, the Rint model


was selected in this research. There is a certain relationship between the open circuit voltage UOC and the load voltage Ub, as shown in equation: *Ub* = *UOC* − *IL* · *R*0. The parameter settings of the LiBs are shown in Table 2.

### *2.3 Ultracapacitor*

Under high power requirements, ultracapacitor can provide excellent output power, effectively reduce the load of LiB, and significantly improve the efficiency of the system. The parameter settings of the UCs are shown in Table 3.


# *2.4 Topology Structure*

Figure 1 shows the specific classification of HESS topology. We used a UC semiactive topology in this study for the following reasons: passive topology makes it difficult to achieve energy conversion between LiB and UC, while active topology is more expensive and harder to control. In the semi-active topology of LiB, the linear charge and discharge characteristics of UC can lead to sharp fluctuations in DC circuit voltage. The UC semi-active topology effectively improves capacity efficiency. The formed topology model is presented in Fig. 2.

**Fig. 2** UC semi-active topology simulation model

**Fig. 3** Logic threshold control strategy flowchart

# **3 Control Strategy**

### *3.1 Logical Threshold Control Strategy*

In this study, writing an integrated program in Matlab to calculate the energy demand according to the required power of the UDDS model of an electric vehicle with a single power supply, to get the Pave\_p logic threshold, the related parameters are listed in Table 4. The logic flow of the LTC strategy is presented in Fig. 3.

### *3.2 Fuzzy Logic Control Strategy*

This research employs the Madamni structure fuzzy logic controller, where the input signals consist of requested power, SOCuc, and SOCbat. The output is denoted as the power allocation factor Kbat. The domains and subsets of the membership functions are presented in Table 5.


(a) Preq>0, under driving conditions (b) Preq<0, under braking conditions

**Fig. 4** FLC rule surface under driving and braking conditions

**Fig. 5** FLC simulation model

By dividing the intervals as outlined above, we derive a total of 108 control rules. The surface of FLC rules under both driving and braking conditions is illustrated in Fig. 4, and its model is further visualized in Fig. 5.

### **4 Simulation Results**

In this study, we conducted simulation verification within the Matlab and ADVISOR environments to align vehicle parameters and EV control models. Both the LiB and UC were initialized with a State of Charge (SOC) of 1. As depicted in Fig. 6, the speed request curves generated by the two distinct control strategies closely match the actual speed curves. This substantiates that the HESS control strategy proposed in this paper adequately fulfills the vehicle's speed requirements.

**Fig. 6** Vehicle speed tracking results of two control strategies in UDDS

In Fig. 7, within the single-power controlled EMS, the peak output power of the LiB reaches 50.72 kW. This is notably higher, by 40.52 and 39.33 kW, than the values obtained using LTC and FLC strategies in the HESS, as detailed in Table 6. In Fig. 8, the SOC drop of the LiB within the HESS is more than 30% lower compared to the EMS relying on a single power supply. This observation underscores the crucial role of well-distributed power in the HESS, effectively extending the lifespan of the LiB and potentially increasing the mileage range of EVs.

In LTC, the output power of the LiB is precisely controlled around the threshold value of 10.20 kW, signifying superior output stability of the LiB with a wider

**Table 6** Output peak and average power of LiB and UC under driving conditions in UDDS

energy management

strategies


amplitude range. In contrast, FLC employs 108 flexible rules to control the LiB's output power. Since this paper focuses on UC's role in recovering braking energy in LTC, we have compared the average LiB output power under driving conditions in Table 6. In the FLC, the average LiB output power is 2.73 kW lower than that in LTC, highlighting a narrower amplitude range. Figure 7 further illustrates that the LiB's output power curve exhibits more frequent fluctuations in FLC, indicating inferior stability compared to LTC.

The output power of the UC in the FLC strategy consistently surpasses that of LTC during most driving instances. Additionally, the SOC of the UC in FLC and LTC decreases by approximately 37 and 17%, respectively, signifying a deeper utilization and better alignment of capacity parameters in FLC. However, it's important to note that this approach may ultimately render the HESS ineffective in the long term, as depicted in Fig. 9.

# **5 Conclusions and Prospect**

In this paper, we investigate the EMS for both single power supply and hybrid power supply configurations. The LTC and FLC power distribution strategies effectively regulate the output power of the LiB within an appropriate range. The high-power discharge characteristics of the UC efficiently mitigate excessive LiB discharge, thus ensuring its prolonged service life, exemplifying the significance of the LiB-UC HESS.

In the future, we will consider utilizing the LiB to supply power to the UC in order to meet the power requirements of the UC. Concurrently, further research is required to enhance the alignment of UC capacity parameters and control strategies, incorporating diverse driver behavior patterns and various system topologies into our algorithm design. This will enable us to optimize system costs and control effectiveness while facilitating the flexible selection and application of control strategies based on real-world driving scenarios.

**Acknowledgements** This work was supported by the National Natural Science Foundation of China (Grant No. 52211530054); the National Natural Science Foundation of China (Grant No. 52172399); the Scientific Research Fund of Hunan Provincial Education Department (Project No. 21A0193); Changsha Natural Science Foundation Project (Project No. KQ2208235).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Recognition Method for Train Coupler Handle Based on YOLOv5 Model**

**Zhiyuan Liu, Yan Li, Zhanmou Xu, Jialu Li, Jiayi Ding, Xiong Zhang, Shuting Wan, Jingyi Zhao, Rui Guo, and Wei Cai** 

**Abstract** To solve the problem of identifying different types of car couplers during the operation of the automatic uncoupling robot of a tippler, a method for recognizing the handle of a car coupler based on the YOLOv5 model has been proposed. This method selects YOLOv5n, which is relatively simple in the YOLOv5 series, as the benchmark model for the detection network. The overall structure is more concise, effectively reducing the number of model parameters while ensuring detection accuracy. The YOLOv5n model used for feature extraction and target recognition on two types of coupler datasets: upper action and lower action, greatly reducing the time required for training and testing, and achieving extremely high recognition accuracy. Compared with the commonly used SSD300 model and Faster R-CNN model, it shows significant advantages in terms of parameter quantity, computational complexity, predictive inference speed and weight file size.

**Keywords** Target identification · Coupler handle · YOLOv5 · Unhooking

CHN Energy Hebei Dingzhou Power Generation Co.Ltd., Dingzhou 073000, Hebei, China

Y. Li

Beijing Huily Intelligent Technology Co., Ltd., Beijing 100070, China

J. Li · J. Ding · X. Zhang (B) · S. Wan Department of Mechanical Engineering, North China Electric Power University, Baoding 071003, China e-mail: hdjxzx@ncepu.edu.cn

J. Zhao · R. Guo · W. Cai School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China

© The Author(s) 2024 S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_88

1103

Z. Liu · Z. Xu

### **1 Introduction**

As an important tool for unloading goods, tipplers are widely used in the coal transportation industry. Its working principle is to fix and rotate a carriage of the train at a certain angle, thereby dumping the coal out of it. However, there are couplers between the train carriages for connection, so it is crucial to timely and effectively remove the couplers before the tippler works. The operating handles for uncoupling train couplers are divided into two types: upper acting and lower acting, and different handles require different methods of uncoupling operation. The removal of train couplers was initially completed by manpower, but due to the low efficiency of manual removal of couplers and the danger of workers during the process, the research on automatic decoupling robots has emerged. Compared to manual labor, automatic hook picking robots can adapt to more harsh working environments and have higher work efficiency. The ability to accurately and quickly determine the position of the coupler and identify the type of coupler has a significant impact on the completion and work efficiency of the uncoupling robot. Therefore, the research on target detection of the coupler operating handle is of great significance.

The development of target detection methods is mainly divided into two stages: the stage of using traditional methods for target detection and the stage of using deep learning methods for target detection. The traditional target detection method uses exhaustive methods to select the target location, which not only takes a long time, but also has a high false detection rate in practical applications, so it is gradually replaced by the deep learning method. In deep learning, commonly used target detection methods are mainly divided into two types. One type is to first form a series of candidate boxes for feature extraction and target position judgment, known as the two stage algorithm, such as R-CNN algorithm and Faster R-CNN algorithm. Another type of algorithm, which does not generate candidate boxes separately, completes the selection of target positions and feature extraction together to achieve end-to-end target detection tasks, is called one stage algorithm, such as YOLO algorithm and SSD algorithm [1–3]. Both algorithms can effectively complete various complex object detection tasks and are widely used by a large number of scholars. He et al. [4] introduced an attention module, balance module, and context module to construct an intelligent detection model in Mask R-CNN, which is used for welding quality detection of subway car bodies. According to experimental verification, the detection accuracy of this method is 4.5 percentage points higher than the traditional Mask R-CNN algorithm. Zhang et al. [5] applied the Faster R-CNN algorithm to equipment recognition and status detection tasks in power rooms, and performed well in both image and video tests, proving the effectiveness of the algorithm. Li et al. [6] combines deep learning with unmanned aerial vehicle (UAV) bridge crack detection, uses UAV to obtain high-quality images of long distance bridges, and then uses Faster R-CNN algorithm based on transfer learning to identify cracks, and obtains high detection accuracy and efficiency. Dai et al. [7] proposed a multi task detector based on the Faster R-CNN algorithm, which uses an improved ResNet-50 architecture to estimate distance and detect pedestrians during autonomous driving. It combines infrared cameras and LiDAR with target detection algorithms, achieving good detection results in real nighttime road scenes. Luo et al. [8] optimized the Faster R-CNN algorithm and combined it with feature enhancement methods to detect different vehicles, effectively solving the problem of vehicle detection in complex traffic environments. Li et al. [9] combines YOLOv4 and YOLO-GGCNN for object recognition in positional environments, facilitating the robotic arm to selectively grasp objects in unknown environments. Jiang et al. [10] used the YOLO model to extract features from videos and images captured by infrared cameras, and achieved good results in complex environments, verifying the effectiveness of the YOLO model in drone target detection tasks. Mushtaq et al. [11] applied the deep learning method to the identification of important components of aerospace vehicles, used YOLOv5 algorithm to identify and classify different components on the assembly line, and achieved high detection accuracy. Ji et al. [12] improved the traditional YOLOv4 algorithm by introducing the extended perception module and the attention mechanism module, and improved the CIOU Loss function. It was verified that it has better detection effect in small target detection than other models through public data sets.

The above methods have achieved good detection results, but they have not taken into account the complexity and timeliness of the model. Most existing target detection models have complex structures in order to achieve high detection accuracy, but this also leads to an increase in the computational time of the model, an increase in the demand for hardware computing power, and a longer inference time when making predictions in practical applications. The detection objects studied in this article are only two types of coupler operating handles, which belong to relatively simple target detection tasks and do not require extremely complex model structures and calculation steps. Using common models for detection and analysis can waste a lot of time and burden hardware. In response to this issue, this article uses the simplest YOLOv5n model in the YOLO series as the detection model, which ensures accuracy while obtaining fewer model parameters and faster operation speed. It has high engineering application value for simple target detection tasks.

This article is summarized as follows. Section 2 introduces the diagnostic methods used in this article. Section 3 introduces the experimental process and result analysis of target recognition for the coupler handle. Section 4 summarizes the diagnostic methods used in this article.

### **2 YOLOv5 Model**

YOLO series algorithms are representative methods in one stage target detection algorithms. Due to their fast detection speed, strong generalization ability, and high robustness, they have been widely used in various object detection scenarios. YOLOv5 is one of the most widely used detection methods. Compared to the previous YOLOv4 method, this method has fewer parameters, faster calculation speed, and higher accuracy [13]. Its model structure is shown in Fig. 1. According to the

**Fig. 1** YOLOv5 model structure diagram

complexity of the models, the YOLOv5 series proposes a total of five official models, namely YOLOv5x, YOLOv5l, YOLOv5m, YOLOv5s, and YOLOv5n [14–16]. The YOLOv5n model has the fastest computational speed and the smallest number of parameters, making it more suitable for simpler target recognition applications.

The Input section is the input end of the model, used to input the original image dataset. This section mainly focuses on mosaic image enhancement, adaptive anchor box calculation, and adaptive image scaling processing [17, 18]. The mosaic method can randomly select several different images for random stitching and combination to form a large image, thereby improving the multiplicity of training images and the robustness of the model network. Adaptive anchor box can automatically calculate the most suitable anchor box parameters for the input image, improving the accuracy of detection results. Adaptive image scaling refers to standardizing the size of the original image to reduce information redundancy in the input image and improve the inference speed of the algorithm.

The Backbone section consists of CBS module, C3\_1 module and SPPF module, mainly used for feature extraction of images [19]. Among them, CBS module and C3\_1 module can repeatedly analyze the input images and gradually extract useful information. The SPPF module can achieve local feature fusion and to some extent solve the multi-scale problem of the target. Compared to the original SPP module, it has faster computational speed and higher efficiency. In the overall framework of Backbone, the first convolutional layer usually has two structures: Focus and CBS. The Focus module has stronger feature extraction capabilities, but the Focus structure is relatively complex in implementation, which can lead to increased computational complexity. The CBS module has a simple structure and is more suitable for simple target recognition tasks. Therefore, the model in this article chooses the CBS module as the first convolutional layer, which can simplify the model structure and improve computational efficiency.

**Fig. 2** FPN + PAN structure

The Neck section continues to use the traditional FPN + PAN structure, as shown in Fig. 2. FPN can transmit deep semantic features to shallow layers, while PAN can conversely transmit shallow localization information to deep layers. The effective combination of the two can enhance the ability of network feature extraction and fusion. Additionally, the Neck section introduces C3\_2 modules to further enhance the network's feature fusion capability [20].

The Head section is the output end of the model, which is divided into three scales for prediction at different scales [21].

### **3 Experiments and Result Analysis**

### *3.1 Experimental Introduction*

#### **Experimental Dataset**

The automatic uncoupling robot needs to recognize and locate the operating handle of the coupler during the uncoupling process, which requires it to learn and memorize different types of coupler operating handles. This experimental dataset was collected from different connecting parts of train carriages, and images of coupler operating handles of different models and backgrounds were collected. The operating handle is mainly divided into two types: top acting and down acting, and its schematic diagram is shown in Figs. 3 and 4. Use the "Labelimg" tool to label all images and set the labels for the two types of couplers to "Top acting handle" and "Down acting handle", respectively. Finally, the annotated images are randomly divided into validation, testing and training sets in a ratio of 1:1:8, with all three datasets having different images [22, 23].

**Fig. 3** Top acting handle

**Fig. 4** Down acting handle

#### **Experimental Evaluation Indicators**

In terms of model effectiveness detection, this article selects indicators such as *P*  (Precision), *R* (Recall) and *mAP* (Mean average precision) to evaluate the effectiveness of the model. Accuracy represents the proportion of correctly predicted positive samples in the predicted data to the actual positive samples, as shown in Eq. (1), where *TP* represents the number of predicted positive samples in the positive samples and *FP* represents the number of predicted positive samples in the negative samples [24]. The recall rate represents the probability that the correct category in the sample is predicted correctly, as shown in Eq. (2), where *FN* represents the number of predicted negative categories in the positive sample. Drawing with *P* as the vertical axis and *R* as the horizontal axis can obtain the *PR* curve. *AP* (Average Precision) represents the area enclosed under the *PR* curve, as shown in Eq. (3). The larger the *AP* value, the better the model's data processing performance. At present, *mAP* is often used to measure the model detection effect, which represents the average accuracy of all categories [25, 26], as shown in Eq. (4), where *n*class represents the number of categories, and the larger the value of *mAP*, the better the model detection effect.

$$P = \frac{TP}{TP + FP} \tag{1}$$

$$R = \frac{TP}{TP + FN} \tag{2}$$

$$AP = \int\_0^1 P(R)dR\tag{3}$$

$$mAP = \frac{1}{n\_{\text{class}}} \int\_0^1 P(R)dR. \tag{4}$$

### *3.2 Network Training*

Input the dataset into the network model for training. During the training process, the learning rate is set to 0.01, the batch size is set to 8, and the epoch is set to 200. After training, the curves of various loss values gradually changing with continuous iteration are shown in Fig. 5. In the picture, box\_loss represents the loss of the bounding box, obj\_loss represents whether there is a corresponding object loss in the bounding box, cls\_loss represents classification loss, train represents training set, and val represents validation set. As shown in the figure, the loss curves of both the training and validation sets can effectively converge, ultimately approaching 0, indicating that this method can achieve the best effect after 200 iterations.

Calculate the accuracy, recall rate and *mAP* values and plot the change curve, as shown in Fig. 6. As shown in the figure, after 200 iterations, the curves of accuracy, recall rate, and mean average precision all converge completely and are infinitely

**Fig. 5** Various loss curves

**Fig. 6** Change curve of various evaluation indicators. **a** Accuracy curve. **b** Recall rate curve. **c** mAP curve

close to 1, indicating that the method of this article can classify, locate, and recognize different types of train coupler operating handles, and has achieved excellent results.

Figure 7 shows the confusion matrix of the validation results. As shown in the figure, the recognition accuracy for both top acting handle and down acting handle categories is 1, and there is no recognition at the background category. This indicates that the method of this article can effectively recognize both types of coupler handles, and the positioning effect is good, without confusing the handle with other background parts.

### *3.3 Visualization of Test Results*

Use the trained model to detect and recognize the images of the coupler operating handle in the test set, and the results are shown in Fig. 8. From the figure, it can be seen that this method can accurately select different styles of coupler operating

**Fig. 8** Visualization of detection results. **a** Top acting handle detection results. **b** Down acting handle detection results

handles and has a high recognition probability, which once again demonstrates the good engineering application value of this method.

### *3.4 Comparison of Different Models*

To test the advantages of the YOLOv5n method used for detecting the uncoupling handle of car couplers in this article, it was compared with SSD300 and Faster R-CNN, and analyzed from the aspects of detection accuracy, parameter quantity, computational complexity, prediction speed, and saved weight file size. In terms of detection accuracy, draw the *mAP* change curves of the three methods, as shown in Fig. 9. From the figure, it can be seen that under a threshold of 0.5, the *mAP* values of YOLOv5n, SSD300, and Faster R-CNN methods can all rapidly increase with iteration and eventually converge. The best *mAP* values can reach 99.50, 99.95, and 96.88%, respectively. The *mAP* value for target detection of the coupler handle using the YOLOv5n method, although not the highest among all models, fully meets the accuracy requirements of target detection. Calculate the parameter quantity, computational complexity and prediction speed of the three methods, as shown in Table 1. From the table, it can be seen that compared to the SSD300 and Faster R-CNN algorithms, the YOLOv5n algorithm has the least number of parameters and the smallest computational complexity, thus requiring less hardware computing power during the operation process. In terms of prediction speed, the FPS value of YOLOv5n algorithm is higher than the other two algorithms, indicating that it has the fastest inference speed in the prediction process and can more quickly identify task targets in practical applications. In terms of model weight preservation, draw a bar chart comparing the optimal weight file sizes of the three methods, as shown in Fig. 10. As shown in the figure, the YOLOv5n algorithm saves the smallest optimal weight volume, making it easier to deploy to resource constrained platforms such as embedded or mobile devices. At the same time, a smaller volume can also achieve faster inference

**Table 1** Comparison of the

**Fig. 9** Different methods mAP@0.5 curve


speed, thus completing prediction tasks faster. Based on the above analysis, it can be concluded that compared to other models, the YOLOv5n model is more suitable for task scenarios with simple targets and high timeliness requirements such as coupler handle detection.

# **4 Conclusions**

To solve the problem of identifying and positioning the operating handle of the train automatic uncoupling robot during the uncoupling process, a method based on the YOLOv5 model for object detection of the coupler handle is proposed. This method uses the relatively simple YOLOv5n model as the reference model for detection, ensuring the detection accuracy of the model for simple objects while effectively reducing the parameter quantity of the model. The YOLOv5n model is used to detect and analyze different types of coupler handles. The experimental results show that this method has high detection accuracy, fast speed, small weight file volume, and smaller hardware requirements, thus having high engineering application value.

**Acknowledgements** This work is supported by National Natural Science Foundation of China (52105098), Natural Science Foundation of Hebei Province (E2021502038) and the Fundamental Research Funds for the Central Universities (2023MS130).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Numerical Study of Heat Transfer Effects on Non-Newtonian Nanofluid Flow Between Two Parallel Plates in the Presence of Darcy Brinkman Forchheimer**

**Muhammad Bashir, Munawwar Ali Abbas, Bo Sun, Li Chen, and Saima Muhammad** 

**Abstract** Due to its vast industrial applications and biological context, the investigation of the inconsistent heat and mass transfer that drives the flow of squeezing viscous nanofluids between two plates is a fascinating topic. In this study, we investigated the heat transfer analysis of unsteady viscous nanofluid between two parallel plates. The partial differential equations illustrating the flow model are converted to nonlinear ordinary differential equations by suggesting similarity transformations. The resulting dimensionless and nonlinear ODEs of temperature functions and velocity are solved using the well know numerical technique shooting method by transforming the problem into initial value problem from boundary value problem. The results found are consistent with this numerical solution. Graphically explore the impacts of different parameters on temperature profiles and velocity. The results are compared with the results solved by HPM. The results concurred with this numerical solution. These findings are considered much be useful in the application of polymer processing, power transmission, compression, temporary loading of mechanical parts, food processing, cooling water, gravity machinery, modeling of plastic transport in vivo, chemical processing instruments, and demolition due to freezing.

**Keywords** Heat transfer · Squeezing flow · Shooting method · RK4 · Nanofluid

Department of Mathematics, University of Baltistan, Skardu 16200, Pakistan

B. Sun

#### L. Chen

School of Automotive Studies, Tongji University, Shanghai 201804, China

Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Shanghai 201804, China

© The Author(s) 2024

M. Bashir · M. A. Abbas · S. Muhammad

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: lilychen@tongji.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_89

### **1 Introduction**

Nanofluids are mainly found in industrial and technological applications, particularly in melts of polymer, paints, asphalts and glues, biological solutions, etc. It has been observed that Nanofluids have the potential to significantly enhance the rate of transfer of heat in various areas. The term Nanofluids is a liquid containing an array of nanometer-sized (less than 100 nm in diameter) particles mixed in a base liquid. Nanoparticles used in Nanofluids are typically metals (Al, Cu) while oxides are includes (CUO, Al2O3, SiO2, TiO2), carbides (SiC), nitrites (SiN, AlN), and nonmetals (graphite, carbon nanotubes). The base liquid is typically a conductive liquid for example ethylene glycol or water. Toluene, oils, biological fluids, other lubricants, and polymer solutions are also base fluids. Nanoparticles occur in nanofluids with volume fractions up to 5%. Conventional heat exchangers are considered to have poor thermal conductivity. In this regard, Nanofluids are better since it has the capability of a high heat transfer rate. In energy supply, these fluids play an important role to establish energy heat transfer equipment. These nano-sized conducting metals are used to increase the thermal conductivity of these fluids. Thus, a proper understanding of these particles is very important to make their efficient utilization. The greatest uses of nanofluids occur in electrochemical cells, microelectronics, and pharmaceutical procedures. Choi and Eastman [1] pioneered the use of the word nanofluid for nanoscale particles with low thermal conductivity.

Industrial applications of nanofluids include polymer processing, power transmission, compression, temporary loading of mechanical parts, food processing, cooling water, gravity machinery, modeling of plastic transport in vivo, chemical processing instruments, and demolition due to freezing. Due to its vast industrial applications and biological context, the investigation of the inconsistent heat and mass transfer that drives the flow of squeezing viscous nanofluids between two plates is a fascinating topic. Stefan [2] was the first to study squish flow under the lubrication approximation. Domairry and Hatami [3] studied the analytical flow of nanofluids between two parallel plates of copper–water-compressed nanofluids. Khan et al. [4] studied the flow of Cu-water (or kerosene) nanofluids between two parallel plates with the influence of viscous fluid and velocity sliding. On the other hand, Sheikholeslami and Ganji [5] performed an analytical study on heat transfer in compressed nanofluid flows between parallel plates with the help of the Homotopy Perturbation Method. They found a direct correlation between the Nusselt number and nanoparticles in percentage. It is reported in their studies that the Nusselt number is directly proportional to the volume fraction of nanoparticles, whereas when the two parallel plates are separated, the Nusselt number is inversely related to the number squeezed.

Most engineering-related problems are solved using numerical methods but are also resolved by using different well known analytical techniques such as Homotopy Perturbation Method, Variational Iteration Method, and Homotopy Analysis Method and to exclude small parameters, some new methods have recently been developed. HPM is a semi-precise method that does not require small parameters [6]. In most cases, this method produces very fast convergence of the solution series. HPM has demonstrated its ability to efficiently solve several nonlinear problems with accuracy, ease, and approximate convergence. By iteration, this leads to a highly accurate solution. It is mainly used in engineering research. HPM was developed to solve nonlinear problems such as the MHD Jeffery-Hamel problem by Moghimi et al. [7]. Further, to analyze the influence of nanofluid flow between parallel plates by Mustafa et al. [8]. A Similar method is used by Siddique et al. [9] to solve nonlinear problems including Newtonian and non-Newtonian fluids. For a detailed understanding of the useful use of the Homotopy Perturbation method in the solution of problems regarding fluid dynamics can be viewed in the available Refs. [10–12].

Keeping in view the above discussion, this current work aims to study nanofluid flow and heat transfer between two parallel plates the rough numerical technique. The impact of Darcy Brinkman Forchheimer number on the flow of Nanofluid between two parallel plates of infinite extent. Although the numerical solution of the Darcy Brinkman Forchheimer equation is valid for forced convective fluid through a porous medium, we will find the impact of Darcy Brinkman Forchheimer numbers on the Nanofluid flowing between parallel palates. The classical Darcy's law is a model used to describe fluid flow in porous media, but it becomes inaccurate for high flow rates and larger porosity. Forchheimer and Muskat expanded Darcy's law to account for the effects of inertia and boundary by adding a square velocity term called the Forchheimer term. To the expression of Darcy's velocity, the Darcy-Forchheimer theory has been applied in a number of different areas including mixed convective flow, hydro-magnetic flow, and heat transfer in porous media. Recent studies have also investigated the Darcy-Forchheimer flow in materials with temperature-dependent thermal conductivity.

### **2 Mathematical Model**

In the current study, the configuration of the flow model is explained as the plates are separated by a separation *y* = ±*h*(*t*) = ±*l* √1 <sup>−</sup> <sup>α</sup>*t* where *l* is the initial position between the two plates (when time *t* = 0). Also, if α > 0, it means plates are pushed together until t = 1/α, and if α < 0, the plates become far away. To illustrate this problem, consider a Cartesian coordinate system with the x-axis along the axial flow. The y-axis and direction are taken perpendicular to the axial direction. Clearly, Fig. 1 shows the geometrical structure and coordinate system of the problem in question.

This type of fluid flow can be explained with the help of momentum, continuity, vorticity, and energy equation as below:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{1}$$

$$
\rho\_{n\_f} \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \mu\_{n\_f} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) \tag{2}
$$

**Fig. 1** Geometry of present work

$$
\rho\_{n\_f} \left( \frac{\partial v}{\partial t} + \mu \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial \mathbf{y}} + \mu\_{n\_f} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 v}{\partial \mathbf{y}^2} \right) \tag{3}
$$

$$\begin{split} \frac{\partial T}{\partial t} + \mu \frac{\partial T}{\partial \mathbf{x}} + v \frac{\partial T}{\partial \mathbf{y}} &= \frac{K\_{nf}}{\left(\rho\_{cp}\right)\_{nf}} \left( \frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial \mathbf{y}^2} \right) \\ &+ \frac{\mu\_{nf}}{\left(\rho\_{cp}\right)} \left[ 4 \left(\frac{\partial u}{\partial \mathbf{x}}\right)^2 + \left(\frac{\partial u}{\partial \mathbf{y}} + \frac{\partial v}{\partial \mathbf{x}}\right)^2 \right] \end{split} \tag{4}$$

$$
\mu \frac{\partial w}{\partial \mathbf{x}} + v \frac{\partial w}{\partial \mathbf{y}} = \frac{\mu\_{nf}}{\rho\_{nf}} \left( \frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial \mathbf{y}^2} \right) \tag{5}
$$

Equation (5) is known as the vorticity equation (equation of curl of v) is obtained by doing; ∂(3) <sup>∂</sup>*x* <sup>−</sup> ∂(2) ∂*y*  with <sup>∂</sup>*<sup>u</sup>* <sup>∂</sup>*x* <sup>−</sup> ∂v ∂*y*  where *u* and v are the components of velocities respectively along x and y, ρ*n f* is the "effective density" of Nanofluid, "dynamic effective viscosity" of nanofluid is represented by μ*nf* , ρ*cp nf* denotes the "heat capacity" of the nanofluid and *Knf* represents the "nanofluid thermal conductivity" which are written as:

$$
\rho\_{nf} = (1 - \phi)\rho\_f + \phi\rho\_p \tag{6}
$$

$$(\rho Cp)\_{\mathfrak{n}f} = (1 - \phi)(\rho Cp)\_f + \phi(\rho Cp)\_p \tag{7}$$

$$
\mu\_{nf} = \frac{\mu\_f}{(1-\phi)^{2.5}}\tag{8}
$$

$$
\mu\_{nf} = \frac{K\_s + 2K\_f - 2\phi(K\_f - K\_s)}{K\_s + 2K\_f + 2\phi(K\_f - K\_s)} K\_f \tag{9}
$$

For the time-dependent squeezing flow boundary conditions are given by:

$$u = 0, \ v = 0, \ T = T\_1 \text{ and } \mathbf{y} = \mathbf{0} \tag{10}$$

Numerical Study of Heat Transfer Effects on Non-Newtonian Nanofluid … 1119

$$u = 0, \ v = \frac{dh}{dt}, \ T = T\_2 \text{ and } \mathbf{y} = h(t) \tag{11}$$

Equation (11) gives that <sup>v</sup> <sup>=</sup> *dh dt* indicates the velocity with which the relative motion of the upper plate with respect to the lower plate is constant at a distance *<sup>y</sup>*<sup>=</sup> 0 from the upper plate at *y* <sup>=</sup> *<sup>h</sup>*(*t*). The parameter <sup>v</sup> is defined as −α*l*  2(1−α*t*) 1 2 . Whereas Eqs. (4)–(5) combined using the conditions (10) and (11) governing nanofluids squeeze flow are highly nonlinear in nature, this cannot be solved analytically. Therefore, the system of PDE, corresponding from (4) to (5) is reduced to ordinary ODE with help of the similarity transformation.

$$\eta = \frac{y}{l(1 - \alpha t)^{\frac{1}{2}}} = \frac{y}{h(t)}\tag{12}$$

$$
\mu = \frac{\alpha x}{2(1 - \alpha t)} f'(\eta) \tag{13}
$$

$$v = \frac{-\alpha l}{2(1 - \alpha t)^{\frac{l}{2}}} f(\eta) \tag{14}$$

$$
\Theta = \frac{T - T\_2}{T\_1 - T\_2} \tag{15}
$$

Substituting Eq. (12) into Eqs. (5) and (6) yields the reduced governing equations. Fourth-order nonlinear ODE for the momentum equation.

$$f'''' - S\frac{A\_1}{A\_2} \left( 3f'' + \eta f'' + f'f'' - ff'' \right) - Daf' - Ff'^2 = 0 \qquad (16)$$

$$\left(\theta'' + SPr\frac{A\_4}{A\_3}\left(f\theta' - \eta\theta'\right) + \left(P\_r E\_c \frac{A\_2}{A\_4}\right)\left(f'^2 + 4\delta^2 f'^2\right) = 0\tag{17}$$

The second-order differential equation for the energy equation. Where *A*1, *A*2 and *A*3 are dimensionless constants defined as follows:

$$A\_1 = \frac{\rho\_{nf}}{\rho\_f} \tag{18}$$

$$A\_2 = \frac{\mu\_{nf}}{\mu\_f} \tag{19}$$

$$A\_3 = \frac{K\_{nf}}{K\_f} \tag{20}$$

$$A\_4 = \frac{\left(\rho c\_p\right)\_{nf}}{\left(\rho c\_p\right)\_f} \tag{21}$$

Using boundary condition (10) in the terms of similarity transformation (11) becomes:

$$f'(0) = 0, \ f(0) = 0, \ \theta = 1 \ at \ \eta = 0 \tag{22}$$

$$f'(1) = 0, \ f(1) = 1, \ \theta = 0 \ at \ \eta = 1 \tag{23}$$

where *S* <sup>=</sup> <sup>α</sup>*<sup>l</sup>* 2 <sup>2</sup><sup>υ</sup> *<sup>f</sup>*is the squeeze number, *Pr* <sup>=</sup> (ρ*C p*)*<sup>f</sup> K f* υ *<sup>f</sup>*is the Prandtl number, *Ec* = α<sup>2</sup> *x*<sup>2</sup> (*T*1−*T*2)*c p f*  , Darcy-Brinkman number is *Da* <sup>=</sup> <sup>μ</sup>*<sup>d</sup> <sup>k</sup>*and Forchheimer parameter is *<sup>F</sup>*<sup>=</sup> *C f* ρ*ur k*  1 2 *d*2.

The other required physical quantities under consideration are the Nusselt number *N*<sup>υ</sup> , and skin-friction coefficient *C f* , are given by:

$$c\_f = \frac{\tau\_w}{\rho\_{nf}\upsilon\_w} \tag{24}$$

$$N\_{\upsilon} = \frac{lq\_w}{k\_f(T\_1 - T\_2)}\tag{25}$$

where

$$
\pi\_{\rm w} = \mu\_{nf} \left( \frac{\partial \mu}{\partial \mathbf{y}} \right) \tag{26}
$$

$$q\_w = \left(k\_{nf}\frac{\partial T}{\partial \mathbf{y}}\right)\_{\mathbf{y}=\mathbf{0}}^{\mathbf{y}=\mathbf{0}}\tag{27}$$

Using (7) and (11) in (10), we get:

$$C\_f^\* = \frac{\chi^2}{l^2} (1 - \alpha t) R\_{\text{ex}} C\_f = \frac{f''0}{\left(1 - \phi\right)^{2.5} A\_1} \tag{28}$$

$$N\_{ux}^{\*} = \sqrt{1 - \alpha t} N\_{ux} = -A\_3 \Theta'(0) \tag{29}$$

where *Rex* <sup>=</sup> <sup>α</sup>*<sup>l</sup>* 5 2*x*3(1−α*t*) 1 <sup>2</sup>υ *<sup>f</sup>* is the local Reynolds number.

### **3 Solution of the Problem**

To convert the boundary value problem into initial value problems the best technique is the shooting method in numerical analysis. The goal is to solve the initial value problem by utilizing the given various initial conditions until an accurate solution is found which must be fulfil the boundary conditions of the problem. In layman's terms, we "shoot" out trajectories from one boundary in different directions until it finds the trajectories that "shoot" into other boundary conditions.

In numerical analysis, Runge–Kutta (R K) method is a series of implicit and explicit iterative methods which together Euler method is used in time to make a discrete form of approximate solutions to simultaneous nonlinear equations.

Applying the Runge–Kutta method. Suppose the mathematical problem:

$$\frac{dl}{dt} = f(t, l), \quad l(t\_0) = l\_0$$

Here *y* is a function of time *t* which is to be determined, it may be scalar or vector. To find the approximate solution of *l* using *dl dt* . The initial time is represented by *t*<sup>0</sup> who's corresponds to the v*alue is l*0. At the initial condition (*t*0,*l*0) the function *f*  is given.

Now we select a step-size of *h* > 0 and define it as:

$$l\_{n+1} = l\_n + \frac{1}{6}(k\_1 + 2k\_2 + 2k\_3 + k\_4)h$$

$$t\_{n+1} = t\_n + h$$

For *n* = 0, 1, 2, 3, 4,... .

$$\begin{array}{c} k\_1 = f(t\_n, l\_n) \\ k\_2 = f\left(t\_n + \frac{h}{2}, l\_n + h\frac{k\_1}{2}\right) \\ k\_3 = f\left(t\_n + \frac{h}{2}, l\_n + h\frac{k\_2}{2}\right) \\ k\_4 = f(t\_n + h, l\_n + hk\_3) \end{array}$$

The approximate value of *y*(*tn*+<sup>1</sup>) is *yn*+1 in RK4 and the second term (*ln*+<sup>1</sup>) is calculated using the first value of (*ln*) plus the weighted mean of four small increments where each small increment represents the product of interval size "h". Finally, an estimated slope given by function *f* on the right-hand side of the differential equation.

From Eqs. (16) and (17), it can be obtained the following systems of equations.

$$f = f(\mathbf{l})\tag{30}$$

$$f' = f(\mathbf{2})\tag{31}$$

$$f'' = f(\mathbf{3})\tag{32}$$

$$f''' = f(4) \tag{33}$$

$$f^{\prime\prime\prime} = f(\mathfrak{F}) = \frac{SA\_1}{A\_2} (3f(\mathfrak{I}) + \eta f(4) - f(\mathfrak{I})f(\mathfrak{I}) - f(\mathfrak{I})f(4)) \tag{34}$$

$$
\theta' = f(6) \tag{35}
$$

$$\boldsymbol{\theta}'' = \boldsymbol{f}(7) = -\left(\boldsymbol{S}\boldsymbol{P}\boldsymbol{r}\frac{A\_4}{A\_3}(\boldsymbol{f}(1)\boldsymbol{f}(6) - \eta\boldsymbol{f}(6)) + \left(\boldsymbol{P}\boldsymbol{r}\boldsymbol{E}\boldsymbol{c}\frac{A\_2}{A\_4}\right)(\boldsymbol{f}(3)^2 + 4\delta^2\boldsymbol{f}(2)^2)\right) \tag{36}$$

Now Eqs. (30)–(36) are solved by MATLAB software. The obtained graphical results of velocity and temperature have been discussed in the next section.

### **4 Results and Discussion**

The unsteady squeezed nanofluid flow between the two parallel plates is solved by utilizing the numerical technique. To support the present numerical solution, we compared our results with results given by using HPM. They are in an excellent agreement as they have been demonstrated in Table 1. Further from the current study, the temperature distribution graphs are same response as obtained by [8] for the various parameters.

The results are in outstanding agreement as they have been shown in Table 1. The graphical results also illustrated for various pertinent parameters. Figure 1 demonstrates that the axial velocity profile is inversely related to the Darcy number, i.e., by increasing the Darcy number velocity profile decreases. Figure 2 depicts that


**Table 1** Comparison the present results with previously published results using HP

by increasing Darcy's number the redial velocity behaves different way. Initially, it reduces however from the mid of the plates it is going larger for various values of *Da*. Figure 3 displayed the behavior of the Forchheimer number and axial velocity profile. It can be seen from the figure that for larger values of F the radial velocity decreases slightly. While Fig. 4, shows that by increasing the Forchheimer number the radial velocity is increasing. Figures 5 and 6 produces graphs of the temperature profile for different values of the Prandtl and Eckert number. It can see that the temperature description improves when the Prandtl number increases in the flow regime. The change in the temperature description is mostly dependent on a large decrease in thermal diffusivity with larger values of *Pr* and a decrease in the thickness of the thermal boundary layer. This increases the thermal profile of the flow regime. Further, Fig. 7 shows the impact of the Eckert number on the temperature profile. From Fig. 7 it can be viewed that the temperature description improves as the value of the Eckert number increases. Whereas this increase in temperature description is expected as it directly affects the heat loss process and increases the temperature field in the flow domain.

*f* **-**

From Table 1 we concluded that for promoting values of S, the coefficient of skin friction is upgrading while local Nusselt number is decreased at static values of Eckert and Prandtl numbers, i.e. (Ec = 0.01, Pr = 6.2).

### **5 Conclusions**

This paper investigates the effects of heat transfer in an unsteady flow of viscous and incompressible nanofluid in two dimensions between infinitely expanding parallel plates with collisions caused by viscous dissipation. The associated nonlinear PDEs are converted into a set of ODEs and are solved analytically by using the RK4. From the discussion above, we got the following notable results are:


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Accurate Finite Element Modeling of Bolted Joints and Modified IWAN Model**

**Jiuyi Li, Yunhou Sun, Qiang Yan, Huaiqiang Kang, Haoxiang Wang, and Shangwei Dong** 

**Abstract** A three-dimensional finite element model of helical bolted joints is established in this paper, and the validity of the model establishment scheme is verified by a numerical examples of "the stress distribution on the bolted joint". Then, Hysteresis curves of the resultant torque versus the applied torsion angle are obtained from the finite element analysis. The results indicate that: (1) The contact threads have experienced three states of adhesion, partial slip and macro slip during the tightening and loosening process; (2) In the state of macro sliding, the resultant torque exhibits an upward trend due to the increase of the clamping pressure in tightening process. On the contrary, the resultant torque exhibits a downward trend due to the decrease of the clamping pressure in the loosening process. A modified IWAN model is established for the three-dimensional helical bolted joint under torsional load by adding the residual stiffness and correction of torque based on the results of finite element analysis. The results of the modified IWAN model for the M12 bolted joint present that the modified IWAN model can reproduce the hysteresis curves obtained by finite element analysis accurately, and the modified IWAN model composed of three Jenkins elements can cover the dissipated energy precisely.

**Keywords** Helical bolted joints · Torsional load · Finite element model · Hysteresis curve · Modified IWAN model

# **1 Introduction**

The loosening of bolted joints is a complex nonlinear process and will be affected by many factors such as the friction coefficient between the clamped parts, the friction coefficient between the contact threads of bolted joints, the preload of bolted joints, the amplitude of external load, temperature, and the material of the structures [1].

National Defense Engineering Research Institute of Academy of Military Science of PLA, Beijing 100850, China e-mail: houyunsun@126.com

© The Author(s) 2024

J. Li · Y. Sun (B) · Q. Yan · H. Kang · H. Wang · S. Dong

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_90

Some research on the mechanism of loosening behavior of bolted joints has been carried out for a long time, and many valuable research results have been summarized. Nassar et al. [2] reported that when the amplitude of axial load exceeds a critical value, the clamped parts will be separated, and the bolts joints will bear all the axial loads, resulting in severe plastic deformation and eventually loosening of the bolt joints. Junker [3] found that dynamic shear load has a greater impact on the looseness of bolted joints than axial load. When bolted joints are excited by shear load, the contact threads of bolted joints will produce a relative torsional angle. As the torsional angle increases, the clamping pressure decreased when the slippage between the contact surfaces occurred. Jiang et al. [4] reproduced the loosening process of bolted joints under cyclic shear load. They found the loosening process can be divided into two stages. In the first stage, there is no obvious slip between the contact threads, and the clamping pressure decreased slowly due to the plastic deformation of the bolted joints; In the second stage, as the clamping pressure decreased to a critical value, clamping pressure dropped rapidly due to the relative rotation between the contact threads. Therefore, it is of great significance to study the mechanical properties of bolted joints under torsional load. In recent years, the finite element method has been used to investigate the mechanical properties of bolted joints. However, it is difficult to generate high-quality finite element meshes because of the complex geometric shape of the helical bolted joints. To improve the finite element mesh quality, the effect of the helix angle of threads of a bolted joint is neglected in most studies. Verwaerde et al. [5] investigate the stress distribution on the bolted joints by using a three-dimensional bolted joint model without threads. Zhao et al. [6] analysed stress concentration at thread roots using an axisymmetric bolted joint model. The finite element model established by the above two methods of bolted joint modeling can not accurately simulate the mechanical properties of the bolted joints with the helix angle of threads. It is an effective method to accurately predict the stress distribution at the thread of bolted joints by using the model with helix angle to carry out finite element analysis. However, For different sizes of bolted joints, the bolted joints need to be remodeled and the efficiency is low [7, 8]. Therefore, the quest for a finite element modeling method to quickly construct the bolted joints with the helix angle is of great significance. It takes a considerable amount of time to conduct a finite element simulation for obtaining the hysteresis curve of the resultant torque versus the applied rotation angle for bolted joints. Hence, it is of great significance to develop a corresponding mathematical model with clear physical significance for the bolted joints [9]. With the development of contact mechanics and tribology, many theoretical models have been developed to predict the nonlinear mechanical behavior of bolted joints. Gaul et al. [10] examined the dynamic response of bolted joints under torsional load. Subsequently, they established a model for bolted joints by utilizing the Valanis model with several Jenkins elements arranged in parallel to one another. Liu [11] developed the modified Valanis model based on experimental and finite element analysis, and proved the effectiveness of the modified Valanis model. At present, the most representative models include Valanis model [12], Bouc-Wen model [13, 14] and IWAN model [15, 16], etc. Among them, due to the clear and intuitive physical meaning of the IWAN model, it has been widely used in nonlinear model establishment and dynamic behavior analysis of rivet connection and thread connection [17], etc. The remainder of this paper is organized as follows: In Sect. 2, the three-dimensional finite element modeling method of the bolted joints with the helix angle of threads is presented, and the parametric modeling platform based on this method is briefly introduced. In Sect. 3 the axial load on the M12 bolted joints are simulated in ABAQUS. Meanwhile, the stress distribution and initial location of plastic deformation for the M12 bolted joint are given. The dynamic responses of bolted joints under torsional load are studied in 9 different working conditions, and their hysteretic curves are obtained. In Sect. 4, the classical IWAN model is modified to obtain hysteresis curves of the resultant torque versus the applied rotation angle according to the simulation results. The correctness of the modified IWAN model is verified. Concluding remarks are provided in Sect. 5.

# **2 Accurate Finite Element Model of Bolted Joints with the Helix Angle of Threads**

### *2.1 Mathematical Expressions of Thread Profile*

Here, we will briefly introduce the mathematical model of the thread profile, which has been widely described in the literature [18, 19]. Figure 1a illustrates the profile of the external thread of a bolt along the central axis of the bolt. The specification of the thread profile is given in ISO 68, 261, 262, and 724. To prevent excessive stress concentration at the thread root, the thread root radius ρ is required to be greater than 0.125. The symbol *r* is the external radial coordinate; is the circumferential coordinate; *d*1 is the minor diameter of thread; *d* is the nominal diameter of thread, *H* is the thread overlap, *P* is the thread pitch, *ABCD* and *A*' *B*' *C*' *D*' are symmetrical points about θ = 0 on the thread profile. The profile of the external thread of the bolt starts at point *A* and can be divided into three parts from θ = 0 to θ = π, *A*-*B*  (thread flank), *B*-*C* (thread tooth side) and *C*-*D* (thread crest).

The thread profile of these three parts is projected onto a plane perpendicular to the bolt axis. Thus, the cross-section of the external thread profile can be obtained, as shown in Fig. 1b. The mathematical expression of the external thread profile can be obtained:

$$r = \begin{cases} \frac{d}{2} - \frac{7}{8}H + 2\rho - \sqrt{\rho^2 - \frac{p^2}{4\pi^2}\theta^2} & (0 \le \theta \le \theta\_1) \\\frac{H}{\frac{d}{2}}\theta + \frac{d}{2} - \frac{7}{8}H & (\theta\_1 \le \theta \le \theta\_2) \\\frac{d}{2} & (\theta\_2 \le \theta \le \pi) \end{cases} \tag{1}$$

where θ1 <sup>=</sup> <sup>√</sup>3<sup>π</sup> *<sup>P</sup>*ρ, θ2 <sup>=</sup> <sup>7</sup> <sup>8</sup>π, <sup>ρ</sup> <sup>≤</sup> <sup>√</sup><sup>3</sup> <sup>12</sup>*P*, *H* <sup>=</sup> <sup>√</sup><sup>3</sup> <sup>2</sup>*P*. In this way, the mathematical expression of the internal thread profile of the nut can be obtained:

(a) Cross-section of the thread along the central axis

(b) Cross-section of the thread perpendicular to the central axis

$$r = \begin{cases} \frac{d\_1}{2} & (0 \le \theta \le \theta\_1) \\\frac{H}{\pi}\theta + \frac{d}{2} - \frac{\gamma}{8}H & (\theta\_1 \le \theta \le \theta\_2) \\\frac{d}{2} + \frac{1}{8}H - 2\rho\_n + \sqrt{\rho\_n^2 - \frac{P^2}{4\pi^2}(\pi - \theta)^2} & (\theta\_2 \le \theta \le \pi) \end{cases} \tag{2}$$

where θ1 <sup>=</sup> <sup>π</sup> <sup>4</sup>, θ2 = π ( 1 − <sup>√</sup>3ρ<sup>n</sup> *P* ) , ρn <sup>≤</sup> <sup>√</sup><sup>3</sup> <sup>24</sup>*P*, ρ and ρ*n* are the upper limits of the root radius of external thread and internal thread respectively.

### *2.2 Finite Element Modeling Platform for Bolts and Nuts*

To construct the finite element model of bolted joints quickly with different sizes and types in practical engineering, we developed the bolts and nuts finite element modeling platform with C++ language in the Qt development framework. The main interface of the "modeling platform" is shown in Fig. 2.

The flowchart of constructing the finite element model of bolts and nuts is described in Fig. 3. At the beginning of the program, we need to input the basic parameters of bolts and nuts, i.e., thread pitch *P*, number of pitches *N*, nominal diameter *d*, and the number of elements in single turn thread *ne*, etc. Then, the node coordinates of elements are calculated by Eqs. (1) and (2). The obtained node coordinates data and the index of nodes are stored in the predefined VCORG matrix space. Finally, the hexahedral element is constructed by the index of each element's nodes, and the index of the elements and nodes are all stored in the predefined NELEM matrix space.


**Fig. 2** The main interface of the "modeling platform"

**Fig. 3** Flowchart of constructing the model

### **3 The Finite Element Model**

In this section, the finite element model for the ISO M12 bolted joint with a performance grade of 5.8 is selected for numerical analysis in commercial finite element software ABAQUS. The results are compared with the experimental results to verify the effectiveness of the finite element modeling scheme for bolted joints. The finite element model for the M12 bolted joint is shown in Fig. 4, the model consists of four components: upper plate, lower plate, bolt, and nut. These components are all meshed by hexahedral reduced integral elements, with a total of 87,245 nodes and 77,712 elements. Before finite element analysis, we define the following four contact pairs: (Cont1) between the bottom surface of the bolt head and the top surface of the upper plate, (Cont2) between the bottom surface of the upper plate and the top surface of the lower plate, (Cont3) between the bottom surface of the lower plate and the top surface of the nut, (Cont4) between the contact threads. The above four contact pairs are all defined as surface-surface constraints (Standard) in ABAQUS. The contact behavior is set as tangential and normal, where the normal behavior of the bolt and nut surface is set as an exponential soft contact constraint, and the normal behavior of the other contact surfaces is set as a hard contact constraint. The tangential behavior is constrained by the penalty formula. A torsional load is applied to the nut to rotate it upwards to clamp the two plates.

In the model, the bolt material is 35CrMn with an elastic modulus of about 213,000 MPa and Poisson ratio of about 0.286; The nut is made of 45 Steel with an elastic modulus of about 209,000 MPa and Poisson ratio of about 0.269. The Plane material is Carbon Steel with an elastic modulus of about 210,000 MPa and Poisson ratio of about 0.280; The density of the above materials is 7.87e−9 t/mm3. The plastic strain behavior of the bolt and nut are given in Table 1. The general static analysis step is established, and the loading time is set as 1 s. The number of maximum time steps is set to 100, the initial time step is 0.01, and the influence of geometric nonlinearity is considered. The field output selects stress, strain, etc.

**Fig. 4** Overall finite element model of bolted joints


Regarding the application of preload to bolt joints, ABAQUS provides a method to simulate bolt preload by imposing BOLT LOAD over the cross-section of the bolted joints. However, this method is unable to simulate the friction torque applied to the bolt joints during the pre-tightening process. Therefore, this paper proposes the imposition of a rotation angle on the outside of the nut to simulate the pre-tightening process of the bolted joints. We get the equivalent stress distribution and maximum plastic strain distribution on the bolt, as shown in Fig. 5 and Fig. 6 respectively. It can be found from Fig. 5 that the first three turns of threads bear the largest load, and the larger value of stress is mainly concentrated at the root of the thread, which is consistent with the previous experimental results [20]. It can be seen from Fig. 6 that the plastic deformation first occurs at the root of the first turn of thread of the bolt, and the fatigue fracture of the bolt often occurs at the root of the first turn of thread in practical engineering [21].

**Fig. 5** The contour of the equivalent stress

**Fig. 6** The contour of the plastic strain

# *3.1 Dynamic Behavior of a Bolted Joint Subjected to Torsional Excitation*

The influence of different friction coefficients of Cont2 and Cont4 on the dynamic response of bolted joints is being studied using three levels of friction coefficients, 0.05, 0.1, and 0.15 respectively. It is ensured that other variables remain constant. In order to investigate the influence of different preloads, denoted as *FP* , on the dynamic response of a bolted joint, the preloads of 5 kN, 10 kN, and 20 kN were applied to the bolted joints, respectively. The numerical calculations were carried out for each preload value of *FP* . To study the influence of different torsional load amplitudes on the dynamic response of bolted joint. A reference point, denoted as *pre f erence*, is set directly below the nut along the bolt axis. Nodes on the outer surface of the nut are bound to Preference using distributive coupling. Torsional amplitudes of 0.002, 0.0035, and 0.005 rad are then externally imposed on the reference point *pre f erence*, to evaluate the numerical response of the bolted joints to torsional excitation from the nut. The different cases of finite element simulation are shown in Table 2.

### *3.2 The Hysteretic Curves*

The hysteretic curves of torque *T* versus torsion angle θ at the reference point *pre f erence* under the control of various factors are shown in Fig. 7. As shown in the hysteretic curve, there are three stages that could occur throughout the tightening and loosening process adhesion, partial slip, and macroscopic slip. The gradual alteration of the sliding contact area is indicated in the tightening process by the smooth transition from the adhesive stage to the partial slip stage. Sliding begins in the region of low contact pressure. The torque steadily grows as the torsional angle increases


**Table 2** Summary of finite element simulation cases

and moves to the region of high contact pressure at the mating hole. On the other hand, the change from sliding to adhesion is not as seamless when the speed direction is opposite. This can be explained as follows: Sliding gradually starts on the contact surface as the torsional Angle rises and comes to an end when the velocity direction is reversed. The friction coefficient between the clamped parts μ*c* and the friction coefficient between the contact threads μ*<sup>t</sup>* , as shown in Fig. 7c, d, have a significant impact on the friction dissipation energy of the bolted joints. Cont2 has a more pronounced effect on the friction dissipation of the bolted joints than Cont4, which causes the friction dissipation energy of the bolted joints to change more significantly and is the primary cause of the loosening of the bolted joints collectively.

# **4 Modified IWAN Model**

According to the finite element analysis, the hysteresis curves provided by the classical IWAN model are insufficient to describe the hysteresis characteristics of bolted joints the helix angle of threads under torsional load. The classical IWAN model needs to be modified to account for these differences.

### *4.1 Residual Stiffness in Macro Slip Stage*

The classical IWAN model is shown in Fig. 8a. It is composed of *N* Jenkins elements in parallel. Each Jenkins element is composed of a linear spring with stiffness *ki* and a coulomb element with yield force *Ri* in series. In its adhesive stage, the Coulomb element provides resistance that is both equal to and in opposition to the external load. The Jenkins element reaches the sliding stage when its friction capacity is achieved, which occurs when the external load is larger than the friction coefficient

**Fig. 7** The influence of different variables on hysteresis curve

times the normal force. The load that imposed on the sliding element is equal to the friction coefficient multiplied by the sliding element's natural resistance. When only some Jenkins elements slip during the periodic harmonic motion of the model, it is considered partial slip. If all Jenkins elements slip, the model has reached the stage of macro slip. The classical IWAN model incorporates a spring with an elastic coefficient *ks*, that is coupled in parallel to take into account the residual stiffness to accurately replicate the macro slip process. In Fig. 8b presents the IWAN model with residual stiffness.

Figure 9a illustrates the hysteresis curve of *a-b-c-d-e-f-a*, which is bounded by a small area and drawn using the classical IWAN model. In the loosening part of *a-b-c-d* and the tightening part of *d-e-f-a*, the model is in a state of adhesion. Figure 9b illustrates the hysteretic curve enclosed by *a*' *-b*' *-c*' *-d*' *-e*' *-f* ' *-a*' drawn using the IWAN model that incorporates residual stiffness. In this instance, as the external load increases, all Jenkins elements in the model will eventually slide, resulting in macro slip phenomena. The model is in a state of adhesion at the loosening part

(a) The classical IWAN model

(b) The IWAN model is supplemented with residual stiffness

**Fig. 8** The IWAN model

*a*' *-b*' and tightening part *d*' *-e*' . At this stage, all the Jenkins elements in the IWAN model do not yield, making the model equivalent to a linear spring. The middle part, *b*' *-d*' of loosening part and *e*' *-f* ' of tightening part, are the segments where the model experiences partial slip. At this stage, some Jenkins elements in the IWAN model yield. On the other hand, in the loosening part *c*' *-d*' and tightening part *f* ' *-a*' , the model is in a state of macro slip. The slope that corresponds to *e*' *-d*' and *f* ' *-a*' parts specifies the residual stiffness. The IWAN model's Jenkins elements yield at this point, and only the residual stress acts, resulting in macro slip of the model. The relationship between torque and torsion angle of the bolted joints can be determined after accounting for the residual stiffness *ks*:

$$T\_j = k\_s \theta + \sum\_{l}^{m} \begin{cases} r\_i(t) & \text{abs}(r\_i(t)) < R\_i \\ R\_i \text{sgn}(\dot{\theta}) \text{ else} \end{cases} \tag{3}$$

and

$$r\_i(t) = k\_i(\theta - \theta\_{\text{rev}}) + R\_i \text{sgn}(\dot{\theta}\_{\text{rev}}) \tag{4}$$

where θ and θ*re*<sup>v</sup> denote the torsion angle of the bolted joints during forward rotation and reverse rotation, respectively. θ˙ *re*<sup>v</sup> denotes the angular velocity of the bolted joints before the speed reversal. *m* is the number of Jenkins elements in the modified IWAN model. sgn(θ˙ rev) can be written as:

(a) Hysteresis curve of classical IWAN model

(b) Hysteresis curve of the IWAN model withresidual stiffness

$$\text{sgn}(\dot{\theta}\_{\text{rev}}) = \begin{cases} 1 & \text{if} \quad \dot{\theta}\_{\text{rev}} > 0 \\ 0 & \text{if} \quad \dot{\theta}\_{\text{rev}} = 0 \\ -1 & \text{if} \quad \dot{\theta}\_{\text{rev}} < 0 \end{cases} \tag{5}$$

### *4.2 The Overall Offset of Torque*

The hysteretic curve in Fig. 9 is symmetrical about the origin because the classical IWAN model is only applied to the case of the same reciprocating friction conditions. In the case of reciprocating friction between thread pairs having helix angles, the friction force would increase upon the ascent of the nut. As a result, the hysteretic curve in Fig. 7 obtained via finite element simulation is non-symmetrical about the origin. This feature causes an overall upward torque offset in the resulting curve. In an IWAN model having residual stiffness, the stress relationship meets Masin's assumption [16]. As a result, the relationship between the loosening torque of the model and the tightening torque can be expressed as follows:

$$T\_r(\theta) = -T\_u(-\theta) \tag{6}$$

where the footmark *r* represents the tightening process and *u* represents the loosening process.

Obviously, Eq. (6) is inconsistent with the relationship between torque and torsion angle of the bolted joint described in real practice. The value of the torque offset can be estimated by calculating the difference in torque between the tightened nut and the loosened nut. The relationship between forces acting on rectangular threads is studied. Unfold the external thread as shown in Fig. 10a along the pitch diameter *d*<sup>2</sup> of the thread to obtain an inclined plane with an angle of inclination ϕ as shown in

(a) Torque balance ofthe bolted joint

**Fig. 10** Force diagram of the bolted joint

Fig. 10b. When the sliding block is subjected to axial force is pushed at a horizontal and uniform speed with force *Ft* if the sliding block rises along the inclined plane, its tightening torque *Tr* needs to overcome the thread friction moment *T*1 of the thread pair and the end face friction moment *T*2 between the nut and the supporting surface of the bolted joint at the same time. The tightening torque can be obtained:

$$T\_r = T\_1 + T\_2 = F \tan(\varphi + \rho\_v) \frac{d\_2}{2} + \frac{F\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \tag{7}$$

where *D*<sup>w</sup> and *d*0 denote the outer diameter and inner diameter (hole diameter) of the indirect contact surface between the nut and the connected piece respectively. The equivalent friction angle of the thread is:

$$\rho\_v = \arctan \frac{f}{\cos \chi} \tag{8}$$

where *f* denotes the friction coefficient between threads and γ denotes the half angle of the thread profile. Considering that the rising angle of thread ϕ is quite small, the tightening torque Eq. (8) is simplified as follows:

$$T\_r = T\_1 + T\_2 = F \left[ \frac{d\_2 \mu\_t}{2 \cos \nu} + \frac{P}{2\pi} + \frac{\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \right] \tag{9}$$

Similarly, the loosening torque can be deduced:

(a) The modified IWAN model

(b) The hysteretic curve

**Fig. 11** The modified IWAN model after incorporating the overall torque offset *U* 

$$T\_u = T\_1 + T\_2 = F \left[ \frac{d\_2 \mu\_l}{2 \cos \chi} - \frac{P}{2 \pi} + \frac{\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \right] \tag{10}$$

When the torsion angle is small, the change of axial force *F* is very small, so it is regarded as a fixed value. Then the average difference between tightening torque and loosening torque is expressed as the upward correction, and the following can be obtained:

$$U = \frac{T\_r - T\_u}{2} = F \frac{P}{2\pi} \tag{11}$$

Add the overall torque offset into Eq. (3), and the equation of IWAN model incorporated into the overall torque offset can be obtained:

$$T\_j = U + k\_s \theta + \sum\_{l}^{m} \begin{cases} r\_i(t) & \text{abs}(r\_i(t)) < R\_i \\ R\_i \text{sgn}(\dot{\theta}) \text{ else} \end{cases} \tag{12}$$

The modified IWAN model after incorporating the overall torque offset *U* and its hysteretic curve are shown in Fig. 11b.

### *4.3 Modified Residual Stiffness Slope*

As the average stiffness increases with the increase of the torsion angle in the macro slip stage, the gradient of torque to the torsional angle in the macro slip tightening stage is higher than that in the loosening stage. The residual stiffness values *Kr* and *Ku* can be obtained by estimating the torque when tightening and loosening the nut and then deriving the torque from the torsion angle. Rotating the nut will stretch the bolt while compressing the clamped part, which is equivalent to two springs in a series. The system stiffness of the series spring is *Kc* = *Kb K j* /(*Kb*+*K j* ). According to Hooke's law, the axial force on the bolt is:

$$F = K\_c P \theta / 2\pi = K\_b K\_j P \theta / [2\pi (K\_b + K\_j)] \tag{13}$$

Substitute Eq. (13) into Eq. (9) and Eq. (10) respectively to obtain:

$$T\_r = K\_c P \theta / 2\pi \left[ \frac{d\_2 \mu\_l}{2 \cos \chi} + \frac{P}{2\pi} + \frac{\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \right] \tag{14}$$

$$T\_u = K\_c P \theta / 2\pi \left[ \frac{d\_2 \mu\_1}{2 \cos \chi} - \frac{P}{2\pi} + \frac{\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \right] \tag{15}$$

take the derivative of Eqs. (14) and (15) with respect to torsional angle θ:

$$K\_r = \frac{dT\_r}{d\theta} = \frac{K\_c P}{2\pi} \left[ \frac{d\_2 \mu\_t}{2 \cos \chi} + \frac{P}{2\pi} + \frac{\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \right] \tag{16}$$

$$K\_{u} = \frac{dT\_{u}}{d\theta} = \frac{K\_{c}P}{2\pi} \left[ \frac{d\_{2}\mu\_{t}}{2\cos\chi} - \frac{P}{2\pi} + \frac{\mu\_{c}}{3} \times \frac{D\_{w}^{3} - d\_{0}^{3}}{D\_{w}^{2} - d\_{0}^{2}} \right] \tag{17}$$

according to the above equation, the residual stiffness value of the classical IWAN model is:

$$k\_s = \frac{K\_c P}{2\pi} \left[ \frac{d\_2 \mu\_t}{2 \cos \chi} + \frac{\mu\_c}{3} \times \frac{D\_w^3 - d\_0^3}{D\_w^2 - d\_0^2} \right] \tag{18}$$

there is a constant stiffness difference between the tightening process and the loosening process, i.e.,

$$K\_{\rm T} - K\_{\rm L} = \frac{K\_c P^2}{2\pi^2} \tag{19}$$

In the process of tightening and loosening the nut, because the angular displacement is opposite, the stiffness symbol reflected on the hysteretic curve in the tightening and loosening process is opposite. That is, based on the residual stiffness, add the stiffness value of *<sup>k</sup> <sup>p</sup>*= − *Ks P*<sup>2</sup> <sup>4</sup>π2 in the unloading stage and the stiffness value of *<sup>k</sup> <sup>p</sup>*<sup>=</sup> *Ks P*<sup>2</sup> <sup>4</sup>π2 in the loading section. Add the corrected residual stiffness slope into the residual stiffness term in Eq. (12), and the model expression can be rewritten as:

$$T\_j = (k\_s + \text{sgn}(\dot{\theta})k\_p)\theta + U + \sum\_{l}^{m} \begin{cases} r\_l(t) & \text{abs}(r\_l(t)) < R\_i \\ R\_i \text{sgn}(\dot{\theta}) \text{ else} \end{cases} \tag{20}$$

**Fig. 12** The modified IWAN model after corrected residual stiffness slope

where *k <sup>p</sup>*is the corrected value of residual stiffness. The modified IWAN model after corrected residual stiffness slope and its hysteretic curve are shown in Fig. 12.

### *4.4 Solution Process of IWAN Modified Model*

By modifying the classical IWAN model, the new model (hereinafter referred to as the modified IWAN model) is able to describe the force of the bolted joints with helix angle under torsional load. The expression of the modified IWAN model is Eq. (20). By solving the following parameters: *ks*, *ki* , *k <sup>p</sup>*, *U* and *Ri* . Finally, the relationship between the reaction torque and the torsion angle of the modified IWAN model can be obtained. Parameter extraction results of the modified IWAN model with different number Jenkins and their dissipated energy are shown in Table 3.


**Table 3** Parameter extraction results of IWAN model with different number Jenkins elements

According to the hysteresis curve fitted by the modified IWAN model in Fig. 13, it is not difficult to find that the approximation of fitting increases gradually with the increase of Jenkins units in the IWAN model. When the classical IWAN model is used, the calculation results of dissipated energy match well with the finite element simulation results. However, the expression of mechanical relation can not reflect the mechanical behavior of the bolted joints under torsional load. By using the modified IWAN model, the force relation of the bolted joints with helix Angle can be more truly described, the equivalent dissipation behavior of the bolted joints can be simulated without distortion, and the hysteresis curve of the bolted structure can be obtained. Compared with the finite element model, the modified IWAN model can greatly reduce the computational cost without reducing the accuracy. The modified IWAN model consisting of only three Jenkins unit models can reproduce the hysteretic curve obtained in the finite element analysis accurately enough, and further save the processing time.

(a) Hysteresis curves of the classical IWAN model

with two Jenkins elements

(b) Hysteresis curves of the modified models with one Jenkins element

(d) Hysteresis curves of the modified models with three Jenkins elements

**Fig. 13** Hysteresis curves of the modified models with different Jenkins elements

# **5 Conclusion**

Based on the mathematical expressions of thread profile, the effective finite element model for bolted joints with helix angle is obtained. Using the finite element model thus obtained, the hysteresis curves on the thread interface of the resultant torque versus the applied rotation angle are reproduced under different variables. The modified IWAN model is obtained by analyzing the hysteresis curve and the mechanical properties of the bolted joints. The hysteresis curves of the bolted joint can be well reproduced by using a third order modified IWAN model, and good agreement was achieved for the energy dissipation from the finite element model and the third order modified model.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Standard Components Query System Based on Logical Filtering and Semantic Retrieval**

**Ziyan Huang, Yongming Bian, and Meng Yang** 

**Abstract** The establishment of a robust standard components database is essential in various industries to streamline product development and ensure quality. This paper presents a system for querying standard components data, leveraging the power of logical filtering and semantic retrieval. The structured approach of this system includes a well-defined database structure, logical filtering capabilities at different data levels, and advanced semantic retrieval techniques. The outputs of the system demonstrate its effectiveness in handling user queries, analysing unstructured data, and providing meaningful feedback based on logical filtering outcomes. This research contributes to the efficient utilization of standard components data through an innovative and powerful digital query system.

**Keywords** Standard components · Logical filtering · Semantic search · Query system

# **1 Introduction**

In modern industries, effectively managing and utilizing standard components is crucial for achieving high-quality products, ensuring cost-effectiveness, and meeting project deadlines. The fundamental point is to establish a comprehensive standard components database. While collecting comprehensive components data is too difficult to be achieved, this paper focuses on developing a query system for the database, which is expandable for effortless data integration. Logical filtering and semantic

Y. Bian

© The Author(s) 2024

Z. Huang · Y. Bian · M. Yang (B)

School of Mechanical Engineering, Tongji University, Shanghai 201804, China e-mail: yangmeng@tongji.edu.cn

National Engineering Technology Research Center for Prefabrication Construction in Civil Engineering, Tongji University, Shanghai 200092, China

Shanghai Engineering Research Center for Safety Intelligent Control of Building Machinery, Shanghai 200032, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_91

**Fig. 1** The query system functional architecture

searching are integrated to enhance system's functionality. Data of SKF [1] hydraulic seals is used as an illustrative example in the system developing process.

A robust standard components database acts as a centralized repository, promoting consistency across projects, reducing duplication, and facilitating team collaboration. Similar to MEGARes 2.0 [2], which aids identifying antimicrobial resistance genes in metagenomic data for epidemiological investigations. Such a database speeds up the design process by eliminating the need for manual handbook searches, making it indispensable for the application of AI in industrial settings.

Creating a database for standard components, akin to PubChem's [3] inter-linked Substance, Compound, and BioAssay databases, needs a well-organized data structure, robust search functionalities (including logical and semantic filtering), and accessible through APIs for programmatic use.

The query system for the database should contain the following parts as shown in Fig. 1. The user interface is the entry point for users to interact with the query system. It comprises two main components:


The query processor is responsible for handling user inputs and transforming them into database processable requests. It comprises two key modules:

• Logical Filtering Module: Allows users to filter components based on attributes such as size, material, or other technical specifications. Supports cross-logical filtering, combining criteria from different data tables to identify components that meet complex requirements.

• Semantic Retrieval Module: Leveraging advanced natural language processing as showed in the vitrivr [4] and SOSRepair [5]. Instead of relying on precise keyword matches, it interprets query descriptions based on content, delivering results that harmonize better with the user's intended context. It's used for exploration of standard components descriptions and related textual materials.

The standard components database is the core of the architecture, housing a collection of standardized parts, specifications, and related data. It is structured to accommodate different data types:


# **2 Standard Components Database**

### *2.1 Structured Data*

The standard components database contains three types of structured data tables: basic data tables, associated data tables, and multilevel data tables. Figure 2 provides a visual representation of how these three types of data tables are interconnected, covering all the structured data related to the components in this comprehensive framework.

Basic Data Tables: The fundamental building blocks of the database, containing essential structured information about individual standard components.


**Table 1** Seal types and basic features

Associated Data Tables: Providing additional context information for components in different basic data tables, enhancing the abilities of components cross-reference and components filtering based on associated data.

Multilevel Data Tables: Capturing the hierarchical relationships within standard components data, which enables representing assemblies, sub-components, and other intricate structures, and retrieving information at various levels of detail.

#### **2.1.1 Basic Data Tables**

A basic data table is designed as a simple bivariate chart, in which columns stand for distinct features or attributes, while rows correspond to individual standard components. For instance, Table 1 displays a basic data table illustrating the types and basic features of seal parts.

#### **2.1.2 Associated Data Tables**

An associated data table is a set of basic data tables, with each of them focusing on specific sets of attributes that are related in components filtering.

For example, Table 1 records seal types and materials, while Table 2 records hydraulic fluids and seal material compatibility. The synergy between these tables is crucial in choosing an appropriate seal for a specific application, as the compatibility of hydraulic fluids with seal materials (from Table 2) and the corresponding seal types (from Table 1) collectively determine the optimal choice. When a particular hydraulic fluid is specified, the system utilizes associated data from both Tables 1 and 2 to deliver logical filtering outcomes. This ensures that the chosen seal not only aligns with the hydraulic fluid but also correlates with other attributes detailed in the associated data tables.

#### **2.1.3 Multilevel Data Tables**

A multilevel data table is a basic data table and its sub-tables, capturing how different attributes of the component interact with one another.

For instance, consider a scenario where Table 3 represents basic seal installation dimensions, while Table 4 serves as a sub-table, capturing installation dimensions that are specifically related to pressure considerations. While Table 3 provides



**Table 3** Basic seal installation dimensions

**Table 4** Maximum extrusion gape


fundamental installation dimensions, it is Table 4 that refines these dimensions in the context of pressure requirements. Tables 3 and 4 collaboratively define the complete installation dimensions of a seal under varying pressure conditions.

# *2.2 Unstructured Data*

The standard components database contains two main types of unstructured data: component descriptions and multimedia assets. Component descriptions consist of Textual narratives that offer detailed information about the characteristics, features, and possible uses of standard components. Multimedia assets encompass visual and multimedia resources like pictures, videos, and CAD drawings, which help users visualize physical attributes of standard components.

### **2.2.1 Component Descriptions**

Employs JSON to store textual descriptions that can not fit into SQL data table. Python can be used to manipulate JSON files, supporting programmatic interactions and implementing advanced semantic search. Here is an example describing the temperature condition of the hydraulic seals.

> { "component": "seal", "object": "temperature condition of the hydraulic seals", "description": "Recommended …"}

#### **2.2.2 Multimedia Assets (Images, CAD Drawings, Etc.)**

Document-oriented NoSQL databases are well-suited for handling multimedia assets. They excel in managing and storing unstructured data in flexible, JSONlike documents. Storing multimedia files alongside associated metadata and unique identifiers ensures easy retrieval, categorization, and access control.

### **3 Query Processor**

### *3.1 Logical Filtering Module*

In the query system design, all structured information finds its place in basic data tables. These tables comprise 'Attributes' as column names, 'Entries' as unique row values, and 'Values' as the contents within. In essence, structured data can be represented as triples denoted as (A, E, V), where A stands for Attributes, E for Entries, and V for Values. The logic filtering is the process of completing triple from these tables based on user-query conditions, such as (A, E, ?), (A, ?, V) and (?, E, V).

For logical filtering of basic data tables, the data of a (A, E, V) triple can be get from a single table using SQL query commands.

For an associated data table, logical filtering requires a series of (A, E, V) triples to be completed, crossing different basic data tables in the set. Take selecting types of seals that are compatible with hydraulic fluids material according to Tables 1 and 2, as described in Sect. 2.1.2, for example. As shown in Fig. 3, one basic table corresponds to one triple, where the user-query condition is (A2, ?, V2), the user-query target is (A1, E1, V1), and the associated table's matching condition is E2 ⇐⇒ (A1, V1). So, the mathematical description of the logical filtering is shown in Eq. (1).

**Fig. 3** Associated data table logical filtering

**Fig. 4** Multilevel data table logical filtering

$$\text{(A2, ?, V2)} \xrightarrow{\text{Table 1}} \text{(A2, E2, V2)} \xrightarrow{\text{assoculated data table}} \text{(A1, ?, V1)} \xrightarrow{\text{Table 2}} \text{(A1, E1, V1)}\text{(1)}$$

For a multilevel data table, logical filtering also involves completing a series of (A, E, V) triples, while the matching condition between basic tables is different. As exemplified in the description of parent Tables 3 and 4 in Sect. 2.1.3, Fig. 4 shows that the matching condition is (A3' , V3' ) ⇒ E4 and V4 ⇒ (A3, V3). The goal is to obtain the correct value for Table 3 from Table 4. Therefore, the user-query condition is (A3, E3, ?) and (A4, ?, ?), the user-query target is (A3, E3, V3), and the mathematical description of the logical filtering is shown in Eq. (2).

$$\text{(A3, E3, ?)} \xrightarrow{\text{Table 3}} \text{(A3', E3, V3')} \xrightarrow{\text{multilevel}} \text{(A4, E4, ?)} \xrightarrow{\text{Table 4}} \text{(A4, E4, V4)} \xrightarrow{\text{multilevel}} \text{(A3, E3, V3)}\tag{2}$$

In the query system, all the basic data tables should be connected using the described relationships, forming a complete ontology. This means that starting from any point within the ontology, the user should be able to obtain a clear query result with sufficient conditions. Figure 5 illustrates an example ontology from Tables 1 to 4.

### *3.2 Semantic Retrieval Module*

In the query system design, the semantic retrieval module is used for user contentbased query analyse and Textual description analyse.

Content-based query analysis aims to convert unstructured textual queries into a structured format for the query system's comprehension. This involves initial text

parsing to identify relevant elements like keywords, phrases, and entities, using techniques like tokenization, part-of-speech tagging, and named entity recognition. The next steps include structuring the query by extracting subject, predicate, and object information, which often results in the creation of (A, E, V) triples. Entity resolution is then performed to link entities to specific database tables or records, ensuring the system knows where to retrieve data. Finally, the structured query conditions and targets are used to generate a formal query, typically in SQL or a similar query language, for execution against the database. Logical filtering can be applied to further refine the results based on user-defined criteria or constraints.

Textual description analysis is used to analyse component descriptions described in Sect. 2.2.1. It can extract valuable information from text, and generate meaningful natural language responses. It involves identifying key information, such as facts, entities, and relationships through techniques like sentiment analysis and topic modelling. It enables the query system to handle unstructured data and generate contextually appropriate responses by assembling information and providing human-like answers.

### **4 User Interface**

An example of the user interface for the query system is shown in Fig. 6, featuring logical filtering and semantic search as inputs, along with a preview of structured data and unstructured data (models and figures) from the standard components database.

**Fig. 6** User interface

**Acknowledgements** This research was supported by the National Natural Science Foundation of China (52205279), the Open Foundation of the National Engineering Technology Research Center for Prefabrication Construction in Civil Engineering (2021CPCCE-K02), and the Top Discipline Plan of Shanghai Universities-Class I.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research and Practice on Verification Test of Intended Functional Safety Awareness System Based on HWA System**

**Zhiqiang Zhang, Shaohua Liu, Zhaoyuan Shi, and Juan Shi** 

**Abstract** With the development of driving automation technology, more and more vehicles are equipped with intelligent driving functions. The Ministry of Industry and Information Technology (MIIT) issued in 2021 "Intelligent connected Vehicle production Enterprises and Product Access Management Guide (Trial)" (draft for comment) proposed that "intelligent connected vehicle products should meet the process assurance requirements of functional safety, intended functional safety and cybersecurity." As the "eyes" of the HWA functional system, the perception system, due to its complexity, may still have safety problems caused by functional limitations such as sensors or algorithms in the absence of faults. Therefore, the intended functional safety analysis and verification methods for the perception system are particularly important to ensure that the system does not have unreasonable safety risks. This paper describes the analysis method of the intended functional safety function deficiency, and puts forward the test verification strategy and test verification method of known hazard scenarios. Taking the HWA system as an example, this paper analyzes the hazard scenario of the HWA perception system based on camera and millimeter wave radar, conducts testing and verification evaluation for typical hazard scenarios, identifies the insufficient function of the HWA perception system, and puts forward functional modification suggestions to ensure that the perception system does not have unreasonable risks.

**Keywords** HWA sensing system · Intended functional safety · Verification test method

Z. Zhang

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_92

1159

Jilin University, Changchun, Jilin 130015, China

Z. Zhang · S. Liu (B) · J. Shi CATARC Automotive Test Center (Tianjin) Co., Ltd, Tianjin, China e-mail: liushaohua@catarc.ac.cn

Z. Shi Duke University, Durham, NC 27708, USA

### **1 Introduction**

With the continuous improvement of vehicle intelligence level, intelligent driving system adopts a large number of perception sensors, perception fusion technology, planning algorithms, etc. In the complex vehicle operating environment, even if the intelligent driving system does not have functional safety-related failure problems, there may still be safety problems caused by functional limitations such as sensors or algorithms. The concept of anticipatory functional safety comes into being, which is mainly used to solve the safety problems caused by insufficient functions and human misuse of intelligent driving vehicles. In June 2022, the International Organization for Standardization (ISO) published the intended Functional Safety standard ISO 21448 [1].In order to accelerate the application and implementation of intended functional safety technologies and standards in the country, The National Automotive Standards Technical Committee has formulated the "China Functional Safety (Functional Safety) and intended Functional Safety (SOTIF) technology and standards Research Medium and Long Term Plan (2020–2025)" and "China Functional Safety (Functional Safety)Safety) and Intended Functional Safety (SOTIF) technology and standard system, the planning and standard system to the national standards GB/T34590 "Road vehicle functional safety", GB/T "Road vehicle intended functional safety" as the guidance and research main line, Based on China's national conditions, research on technologies and standards applicable to the functional safety and intended functional safety of new energy vehicles, autonomous vehicles, traditional vehicle vehicles and key electronic control systems in China is carried out. At the same time, the concept of intended functional safety has been introduced by the United Nations, the European Union and other relevant organizations into the safety related regulations of intelligent connected vehicles, and the "Intelligent Connected Vehicle Production Enterprises and Product Access Management Guide (Trial)" (draft for comment) issued in 2021 in China puts forward the requirements of intended functional safety for enterprises and products. This makes the design, development and testing of intended functional safety become an unavoidable technical problem for enterprises.

In 2016, the autopilot Tesla rear-ended the road sweeper, resulting in the death of the driver [2]; In 2018, an Uber self-driving car hit a woman crossing the road during a road test[3].In 2020, an ideal car collided with a right-front lane changing truck when the driver assistance system was turned on [4];The analysis of a number of autonomous vehicle accidents found that the performance limitation of the autonomous driving perception system could not identify the target in front of it, which was the main cause of many accidents[5]. By applying the thought of intended functional safety analysis, this paper puts forward the method of analysis and test verification of the intended functional safety hazard scenario of HWA perception system, and realizes the intended functional safety goal through systematic and comprehensive analysis and test guarantee system to ensure that there is no unreasonable risk.

# **2 Intended Functional Safety Test Validation Scenario Analysis Method**

Intended functional safety mainly protects the hazards caused by insufficient performance of electronic and electrical systems of intelligent driving vehicles or personnel misuse. The operation scenarios of intelligent driving vehicles are divided into four areas, namely known safety, known insecurity, unknown insecurity and unknown safety. The overall research idea is to find the scenarios that may cause harm by the system through systematic analysis. Then the function improvement and test verification of the system are carried out to make the unsafe scenarios as little as possible, and finally the risk acceptance criteria are met through the test verification. The methods used to analyze the intended functional safety performance deficiency and trigger conditions of the system include HAZOP, FTA, STPA, etc. Leveson proposed the STPA analysis method based on system theory in 2011[6], which is used to evaluate the safety of complex systems and identify safety constraints and requirements. The interaction between system, scenario and human can be regarded as the source of harm, and this method is more suitable for the analysis of intended functional safety trigger conditions of intelligent driving system. The method regards the whole system as a whole, systematically analyzes the possible unsafe control behavior, and combines the unsafe control behavior with trigger conditions to form a test scenario for the intended functional safety test verification.

The analysis of the intended functional safety hazard scenario is completed through two steps: hazard analysis, performance deficiency and trigger condition analysis. The specific analysis contents and steps are shown in Fig. 1. Firstly, the hazard analysis of the vehicle level is carried out according to the functional specifications and the corresponding vehicle level safety objectives are determined. The safety objectives are the basic guidance of the subsequent analysis and the target of the intended functional safety testing and verification. After that, it is necessary to establish a system control architecture by using STPA method according to functional specifications and system architecture design to analyze unsafe control behaviors that may lead to hazards. At the same time, it is necessary to analyze the trigger conditions that may lead to hazards in various parts of the system at the technical level, including perception layer, control layer, execution layer and human–computer interaction, and finally form a hazard test scenario.

**Fig. 1** Analysis steps of the intended functional safety test scenario

# *2.1 Vehicle-Level Hazard Analysis and Definition of System Safety Objectives*

Highway Assist (HWA) can be opened on structured roads such as expressways and urban expressways. It has the functions of lateral and longitudinal control such as lane keeping, car following, lane changing, overtaking, etc. It is one of the mainstream intelligent driving systems currently studied. This paper takes 5R1V architecture to realize HWA function perception system as an example to analyze and study.

Generally, the safety objective of the intended functional safety of intelligent driving system is put forward according to the functional specification according to the requirements of horizontal control, vertical control and human–machine interaction system respectively. For HWA sstems with horizontal and longitudinal control capabilities, the requirements for horizontal control include no unintended steering, insufficient steering force, excessive or insufficient acceleration during steering, etc. The requirements of longitudinal control capability include no unanticipated acceleration, unanticipated braking and insufficient braking force. In addition to the control requirements related to vehicle motion, the intended functional safety also focuses on the hazards of human misuse, and the safety objectives of the human–computer interaction system include the absence of inappropriate switching between the system and the driver's control control and inappropriate switching between the system and other systems. Table 1 below shows the unacceptable whole-vehicle hazard analysis of the HWA function and the corresponding safety objectives.

### *2.2 Analysis of Trigger Conditions of Sensing System*

Trigger conditions are the direct causes of harmful events in the system. One or more trigger conditions may lead to harmful events in a specific operation scenario. In this paper, the analysis of trigger conditions is divided into two parts. The first step is to analyze the situation that may cause vehicle harm due to the sensor components carried by the intelligent driving car; the second step is to identify unsafe control behaviors using STPA, and form a hazard test scenario after combination. In this 5R1V sensing system, five millimeter wave radars and one camera sensing system are used.

Based on the sensors and system components carried by autonomous vehicles, the analysis of system and sensor interference elements, accumulation of expert experience, accumulation in existing databases, component-level safety analysis and other aspects are analyzed in Fig. 2. Among them, it is mainly based on the working principle of the sensor to analyze and summarize the interference elements of the sensor, for example, according to the camera mainly based on the working principle of image recognition, easy to be affected by the environment and light, the trigger conditions of the camera consider heavy rain, night, fog and other special environments; Millimeter wave radar echo signal is easy to be affected by metal material


**Table 1** Hazard analysis and safety objectives of H vehicle class

objects and misidentification occurs, and its triggering conditions consider static targets, speed limit signs, manhole covers, etc. According to the analysis of expert experience, when the target object is similar to the background color of the camera, the camera may produce missing recognition, such as a white truck in front of a sunny day; One of the limitations of millimeter wave radar is that the signal from the target is very different: when the motorcycle and truck are driving in front of the vehicle, because the motorcycle is relatively weak in reflection compared to the truck, the reflected signal of the motorcycle may be buried by the strong reflected signal of the truck, resulting in missing identification of the motorcycle. Analysis from available accident databases or test results, such as from an autonomous vehicle accident, suggests that the limited vertical separation capability of millimeter-wave radars may lead to a lack of differentiation between road infrastructure and stationary targets. Component level safety analysis is also performed, such as the automatic Emergency braking system (AEB) based on the camera, the processing error during image extraction and filtering, and the unintended opening of the AEB. The trigger conditions of human–computer interaction are analyzed according to the personnel misuse analysis method.

**Fig. 2** Trigger condition identification method

### *2.3 Hazard Scenario Analysis Based on STPA*

STPA analysis method mainly consists of four steps, namely defining analysis purpose, establishing control structure, identifying unsafe control behavior, and identifying causative scenarios. Defining the purpose of the analysis This part of the content analysis was performed in 1.1. The STPA method analyzes the unsafe control behavior of the system based on the control structure, in which the control flow chart of the system needs to specify the control components of the system, the executive components, the control or display instructions sent by the control components to the executive components, and the feedback signals sent by the executive components to the control components. This paper focuses on scenario generation and uses the HWP system control model built by Feng Hao [7].

According to the system architecture and the Unsafe Control behavior guide words, the corresponding Unsafe Control Action (UCA) is obtained. UCA= scenario + design operation scenario + guide word + control behavior + hazard event, for example, on the highway, the car is following the car, and the perception module does not detect the deceleration of the vehicle in front, resulting in a collision with the vehicle in front. Finally, the cause scenario is identified. The occurrence of unsafe control behavior is in a certain scenario, and there are performance limitations that will lead to harm. Therefore, according to the system architecture, it is necessary to filter the related items that cause the occurrence of hazards from the trigger conditions such as components and systems, and form the occurrence scenario of hazards, that is, the hazard scenario to be tested. Table 2 is part of harm scenarios of HWA sensing system.


**Table 2** Harm scenario of HWA sensing system

# **3 Intended Functional Safety Test Validation Method**

### *3.1 Testing and Verification Policies*

For intended functional safety testing methods, a "three-pillar" approach is typically used for testing, consisting of simulation testing, closed site testing, and actual road testing, and the three pillar testing methods work together to ensure that the product does not present unreasonable risks.

Simulation test refers to the digital restoration of intelligent connected vehicles and their application scenarios by means of actual collection, computational reasoning and other modeling methods, and the establishment of vehicle and environment simulation models as close as possible to the real world. Through the analysis of vehicle operation in the simulation environment, the safety of vehicles in ODC is evaluated. In order to achieve the purpose of effective testing of intelligent connected vehicles [8]. The simulation test is mainly based on the product ODC to test the safety of the automatic driving system in nominal scenarios, dangerous scenarios and edge scenarios. On the basis of considering the reliability assessment of simulation model tools, tool chains such as SIL, HIL and VIL are applied for testing.

Closed site test refers to the real vehicle test for intelligent connected vehicles carried out in a closed site, which is used to verify the functions and performance of vehicles in typical scenarios [9]. The closed site test should focus on the ability of vehicles to cope with typical traffic environment [10].

The actual road test refers to the real car test for intelligent connected vehicles carried out on the open road. The test road should be selected according to the design and operation range of the automatic driving system, and the test mileage or time should be determined to ensure the full coverage of ODC scenario elements. Meanwhile, the test vehicle should be monitored to verify the ability of the declared automatic driving function to cope with the real traffic environment.

Based on the three-pillar test method, this paper proposes the intended functional safety test verification strategy combined with the three-pillar method for known scenarios, as shown in Fig. 3 below. The known hazard scenarios analyzed in the intended functional safety development stage are distributed rationally and verified through simulation test and real vehicle test. Simulation test can test all known scenarios, including test scenarios that can not be realized by real vehicles; The closed site test mainly tests the extreme danger scenario, and verifies the reliability of the simulation test. The actual road test can cover a large number of known scenarios, ensure the test coverage of known scenarios, and find unknown unsafe scenarios.

### *3.2 Test and Evaluation of Known Hazard Scenarios*

According to the intended functional safety hazard analysis and trigger condition analysis of the HWA perception system mentioned above, fog is one of the influencing factors of the perception system. Therefore, this paper focuses on the HWA highway cruise assistance perception system, and designs the vehicle recognition test scenario of the front vehicle stationary accident under different visibility conditions in foggy environment. Since the foggy environment cannot be simulated by the real environment, therefore, simulation tests were carried out to verify the recognition ability of the perception system to the vehicle hazard scenario in a stationary accident in front of the foggy day, to verify the insufficient function of the HWA perception system, and to propose modification measures.

The test scenario was built through the simulation software Carmaker, and the sensor system under test was integrated by the dSPACE hardware in the loop simulation test system, where the front camera was installed in the video black box and the millimeter-wave radar was installed in the millimeter-wave radar echo simulator. The test equipment is shown in Fig. 4, and the test scenario is shown in Fig. 5. In a fog-filled environment, the vehicle under test is driving at a speed of 80 km/h on the express road when the HWA function is enabled, and there is a stationary accident vehicle in front of the vehicle under test in the lane, visibility of 50 m, 100 m and 200 m are set respectively in the fog-filled environment.

The test results under different visibility conditions in fog were obtained through simulation test, as shown in Fig 6. Read test data from CANoe, including lane line information identified by the sensing system, type of vehicle ahead, speed, and position information, Table 3 is the test results under different visibility conditions in foggy days.

It can be seen from the test results that fog has a great impact on the recognition ability of the perception system, and under different visibility, the recognition ability of the perception system does not meet the passing requirements. From the analysis

**Fig. 4** dSPACE hardware-in-the-loop simulation test system

**Fig. 5** Test scenario of the front static vehicle under different visibility in foggy days

**Fig. 6** Test result reading


**Table 3** Test results under different visibility conditions in foggy days

of the expected functional safety processes, functional modifications should be made for the sensing system. The purpose of functional modifications is to identify and implement measures to address the risks associated with SOTIF and to update the input information for the Specification Definitions and Design. Therefore, propose functional modifications for this test results: add additional sensing device to improve the recognition of fog days; or improved sensor recognition algorithm; or limit the system operation in the functional specification.

### **4 Summary**

Based on the thought of intended functional safety analysis and test verification, this paper proposes analysis and verification strategies and methods for the intended functional safety hazard scenarios of intelligent driving vehicles. In addition, based on the perception system of HWA functions, the hazard scenarios are obtained through systematic analysis methods, and reasonable verification strategies are selected for testing and verification. Through testing the recognition ability of the sensing system to the vehicle hazard scenario in a stationary accident in front of the foggy day, the function deficiency of the HWA sensing system is verified, and the modification measures are proposed. This method can reduce the risk of hazards in intelligent driving system by ensuring the sufficiency of hazard scenario analysis, which is of great significance for ensuring the intended functional safety of intelligent driving vehicles.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Energy Management Control Strategy of Series Hydraulic Hybrid Vehicle**

**Jie Gong and Jinyi Zuo** 

**Abstract** Aiming at a series hydraulic hybrid vehicle, the mathematical model and Matlab/Simulink simulation model of the vehicle are established, and the energy management strategy based on rules is put forward. The simulation results show that the control strategy not only realizes four working modes of hydraulic hybrid vehicle, but also improves its fuel economy by 11.9%. It is found that the volume of accumulator and the working pressure range of accumulator are two important parameters that affect the fuel economy of hydraulic hybrid vehicle. Under the condition of a certain accumulator volume, increasing the working pressure range of the accumulator can not only increase the energy storage of the accumulator, but also reduce the idling times of the engine and further improve the fuel economy of the vehicle.

**Keywords** Series hydraulic hybrid vehicle · Energy management control strategy · Fuel economy · Accumulator pressure range

# **1 Introduction**

Energy problems and environmental pollution have become the focus of attention in today's world, and developing a low-carbon economy with low energy consumption and low emissions is becoming the common choice of all countries in the world. With the rapid increase of automobile ownership in China, it is of great practical significance to solve the problem of energy saving and environmental protection of vehicles.

Series hydraulic hybrid power is one of the main energy-saving and emissionreducing technologies in the automotive industry in the world today, and energy

J. Gong (B)

J. Zuo Shanghai Huawei Technology Co., Ltd., Shanghai, China

© The Author(s) 2024

1171

School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China e-mail: gongjie@jlu.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_93

management strategy is the core technology to realize energy-saving and emissionreducing of series hydraulic hybrid power vehicles. At present, the control strategies for energy management of hybrid electric vehicles include rule-based energy management control strategy, real-time optimization-based energy management control strategy, global optimization-based energy management control strategy, and intelligent control strategies such as fuzzy logic, neural network or genetic algorithm, among which the rule-based energy management control strategy is the simplest algorithm [1–9].

In this paper, the mathematical model of a series hydraulic hybrid vehicle and the whole vehicle simulation model of Matlab/Simulink are established, and the energy management strategy based on rules is put forward, which gives full play to the advantages of charging and discharging energy of the accumulator in the control rules and verifies its practicability.

# **2 Composition and Working Principle of Series Hydraulic Hybrid Vehicle**

The series hydraulic hybrid vehicle consists of an engine, a closed volume speed regulating circuit with an accumulator and the transmission system of a traditional vehicle, as shown in Fig. 1. The power output by the engine is transmitted to the variable pump through the clutch, and the variable pump converts mechanical energy into hydraulic energy. The hydraulic oil in the hydraulic pipeline drives the variable pump/motor to convert hydraulic energy into mechanical energy, which is transmitted to the gearbox and the drive axle, and finally the wheels are driven. The accumulator arranged between the variable pump and the variable pump/motor is an energy storage device, which can drive the vehicle independently and recover the braking energy when parking.

**Fig. 1** Schematic diagram of power system of series hydraulic hybrid vehicle

### **3 Vehicle Dynamics Model**

### *3.1 Engine Torque Output Model*

The output torque of the engine output shaft is

$$T\_{\text{out}} = T\_{\text{max}} \cdot \alpha\_{\text{throulle\\_position}} \tag{1}$$

where *T*out is the output torque of the engine output shaft (N • m), and *T*max is the maximum torque of the engine, αthrottle\_position is the throttle opening of the engine.

The torque balance equation of the engine output shaft is

$$T\_{\rm out} = T\_{\rm load} + J\_{\rm engine} \alpha \tag{2}$$

where *T*Load is the load torque, *J*engine is the inertia of the engine integrated accessories, and α is the angular acceleration of the engine output shaft.

### *3.2 Pump/Motor Dynamic Model*

The torque balance equation of the pump/motor output shaft is

$$T\_{\rm p/m} = \Delta P D \mathbf{x} + J\_{\rm p/m} \alpha + T\_{\eta} \tag{3}$$

where, *T*p/m is the input torque of the pump/motor, *P* is the pressure difference between the inlet and outlet of the pump/motor, *D* is the maximum displacement of the pump/motor, *x* is the variable coefficient, *J* p/m is the moment of inertia of the output shaft of the pump/motor, α is the angular acceleration of the output shaft of the pump/motor and *T*<sup>η</sup> is the moment of efficiency loss.

### *3.3 Accumulator Model*

According to Boyle-Mariotte's law, the relationship between the pressure and volume of the gas of the airbag accumulator can be expressed as follows

$$p\_0 V\_0'' = p\_1 V\_1'' = p\_2 V\_2'' = \mathbf{C} \tag{4}$$

where *p*0 is the inflation pressure, *p*1 is the lowest working pressure of the system, *p*<sup>2</sup> is the rated working pressure of the system, *V*0, *V*1 and *V*2 are the volume of the gas under the corresponding pressure, *n* is the thermal coefficient, and during adiabatic process *n* = 1.4, C is the gas constant.

The amount of liquid that the accumulator can store is called effective working volume *V*, and its value is

$$
\Delta V = V\_1 - V\_2 \tag{5}
$$

Substituting the values of *V*1 and *V*2 obtained from Eq. (4) into Eq. (5), it can be obtained that the accumulator volume *V*0 is in adiabatic state, that is, under the condition of fast charge and fast discharge.

$$V\_0 = \Delta V / \left[ \left( p\_0 / p\_1 \right)^{1/n} - \left( p\_0 / p\_2 \right)^{1/n} \right] \tag{6}$$

The range of inflation pressure is 0.25*p*2 ≤ *p*0 ≤ 0.9*p*1.

The simulation model of accumulator ignores the influence of temperature and high pressure (when *p*2 ≥ 20 MPa) on volume *V*0.

### *3.4 Vehicle Driving Model*

The total driving resistance of the vehicle is

$$\sum F = F\_{\text{f}} + F\_{\text{w}} + F\_{\text{i}} + F\_{\text{j}} \tag{7}$$

Among

$$F\_{\rm f} = m \mathbf{g} \, f \tag{8}$$

$$F\_{\mathrm{w}} = \frac{1}{2} C\_{\mathrm{D}} A \rho u\_{\mathrm{r}}^2 \tag{9}$$

$$F\_{\mathbf{i}} = m \mathbf{g} \sin \alpha \tag{10}$$

$$F\_{\mathbf{j}} = \delta m (\mathbf{d}u/\mathbf{d}t) \tag{11}$$

where, *F*f is the rolling resistance, *F*w is the air resistance, *F*i is the ramp resistance and *F*j is the acceleration resistance; *m* is the mass of the car; g is acceleration of gravity; *f* is the rolling friction coefficient. *C*D is the air resistance coefficient, A is the windward area, ρ is the air density, and *u*r is the relative speed between the vehicle and the air, and the actual driving speed of the vehicle when there is no wind. α is *i* = *h*/*s* = tanα, where *i* represents the road slope, which is equal to the ratio of the slope height h to the bottom length s. δ is the conversion coefficient of the rotating mass of the automobile, δ > 1, and d*u*/d*t* is the driving acceleration.

The dynamic equation of vehicle driving is

Energy Management Control Strategy of Series Hydraulic Hybrid Vehicle 1175

$$\frac{M}{\mathbf{r}} = m\mathbf{g}f + \frac{1}{2}\mathbf{C}\_{\text{D}}\mathbf{A}\rho v^2 + m\mathbf{g}\sin\alpha + \delta m \frac{\mathbf{d}u}{\mathbf{d}t} \tag{12}$$

where *M* is the driving torque that the pump/motor needs to provide for the vehicle, and r is the wheel radius.

# **4 Energy Management Control Strategy and Working Mode**

# *4.1 Control Strategies*

Strategy 1.

Make the engine work in the best efficiency area and emission area.

Strategy 2.

Under the condition of a certain accumulator volume, the working pressure range of the accumulator is increased, that is, the energy storage capacity of the accumulator is increased, so as to give full play to the energy storage and discharge efficiency of the accumulator in the hydraulic hybrid electric vehicle, thus reducing the idling times of the engine and obtaining better fuel economy of the whole vehicle.

# *4.2 Energy Management Control Rules and Working Mode of Vehicles*

The ultimate goal of energy management strategy is to realize the energy saving of vehicles. Therefore, during the whole journey of vehicles, the optimal energy saving effect of vehicles can be achieved by controlling the engine to work cyclically in the two working conditions of optimal fuel consumption and idle speed, and at the same time, making it cooperate with the charging and discharging of accumulators. During the whole journey, the vehicle works in the following four modes:

1. Vehicle starting mode

When the vehicle speed is 0 and the working pressure of the accumulator is greater than its minimum working pressure, the accumulator discharges energy and drives the vehicle to start.

2. Accumulator working alone mode

When the working pressure of the accumulator is greater than its maximum working pressure, the engine works at idle speed and the output torque is zero; The accumulator discharges energy and drives the vehicle independently.

**Fig. 2** Vehicle system simulation model

3. The engine is driven and the accumulator is charged

When the working pressure of the accumulator is less than its minimum working pressure, the engine is controlled to work at the optimal fuel consumption point. At this time, the engine charges the accumulator while driving the vehicle.

4. Braking energy recovery

When the vehicle decelerates or brakes, the pump/motor works in the pump condition, the accumulator is charged, the engine is in the idle condition or stopped, and the output torque is zero.

### **5 Simulation of Series Hybrid Vehicle System**

### *5.1 Simulation Model*

The simulation model of hybrid vehicle is established in Matlab/Simulink, which includes engine model, clutch model, variable pump model, variable pump/motor model, accumulator model, safety valve model, reducer model, vehicle running model, energy management control model, engine control model, variable pump control model, variable pump/motor control model and driver model. The simulation model is shown in Fig. 2.

# *5.2 Simulation Conditions*

The main parameters of series hydraulic hybrid vehicle are shown in the Table 1.


**Table 1** Main parameters of series hydraulic hybrid vehicle

### *5.3 Analysis of Simulation Results*

In this paper, CYC\_HWFET is selected as the simulation road condition to test the fuel economy of the rule-based energy management strategy. The simulation results are shown in Figs. 3, 4, 5, 6 and 7.

Figures 3, 4, 5, 6 and 7 respectively show the changes of vehicle speed, pressure of accumulator, engine speed, actual output torque of pump/motor and fuel usage when the vehicle runs according to CYC\_HWFET.

As can be seen from Fig. 4, the vehicle is started by discharging energy from the accumulator. After discharging the accumulator, it can be seen from Fig. 5 that at this time, the engine starts to work at the optimal fuel consumption point, charging the accumulator on the one hand and driving the vehicle on the other. When the accumulator is charged, the engine works at idle speed, and the accumulator is discharged to drive the vehicle alone. This process is circulated throughout the whole journey of the vehicle, until the end of the journey, the pump/motor works in the pump working

**Fig. 3** HWFET high-speed cycle condition

**Fig. 4** Working pressure change of accumulator

**Fig. 5** Engine speed

**Fig. 6** Actual output torque of pump/motor

**Fig. 7** Fuel consumption

condition and starts to recover the braking energy, and the accumulator completes the energy recovery synchronously.

Compared with Figs. 4 and 5, it can be seen from Fig. 7 that when the accumulator works independently, the engine is at idle speed and consumes only a small amount of oil. When the engine works at the best fuel consumption point, it consumes a lot of oil because it charges the accumulator and drives the vehicle at the same time.

Referring to Fig. 3, it can be seen from Fig. 6 that whenever the vehicle is in the deceleration state, the pump/motor will work in the pump condition and recover the excess energy to the accumulator.

### *5.4 Fuel Economy Analysis*

As can be seen from Fig. 7, the hydraulic hybrid vehicle runs according to CYC\_ HWFET road conditions, and the total fuel consumption is 723.5 g. However, the distance travelled by the vehicle according to CYC\_HWFET road condition is 16.51 km [4], and the fuel consumption of the vehicle is 1.00 L according to the gasoline density of 0.725 kg/L. From this, it can be further obtained that the average fuel consumption per 100 kms of hydraulic hybrid vehicles is 6.06 L. Compared with the average fuel consumption of small cars of 6.88 Ls per 100 kms [10], the fuel economy of hydraulic hybrid vehicles has improved by 11.9%.

It can be predicted that the fuel economy will be further improved due to the frequent start and stop of vehicles if the urban circular road simulation is adopted.

### **6 Conclusion**


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Leveling Strategy of Suspension System of Roadway Heavy-Load Transport Vehicle**

**Jianjun Dai, Hao Zhang, Wenlei Li, Rui Guo, and Jingyi Zhao** 

**Abstract** As a transport device for underground equipment, the roadway heavyload transport vehicle needs to ensure its safety and stability during loading and transportation. In order to reduce the overturning risk and improve the stability during transportation, the improved position error leveling method is proposed, and the effectiveness of the scheme is verified by simulation analysis and actual results.

**Keywords** Roadway heavy-load transport vehicle · Position error · Four-point supporting · Leveling strategy

# **1 Introduction**

In the underground construction operation of coal mine, the mining efficiency is limited by the transportation speed of mining hydraulic supports, continuous mining machines, shearers and other equipment in the roadway. Although some equipment, such as continuous mining machines, can perform short-distance transportation by their own hydraulic drive system, considering the moving speed and damage to the road surface, the continuous mining machine itself cannot perform long-distance transportation [1, 2] . Therefore, it is necessary to carry out research on Heavy-load transport vehicles in roadways.

In order to suppress the body overturning problem caused by different suspension forces when the heavy-load transport vehicle is transporting, the leveling strategy,

J. Dai

H. Zhang (B) · W. Li · R. Guo

© The Author(s) 2024

Jiangsu Haipeng Special Vehicles Co., Ltd, Jingjiang 214521, Jiangsu, China

College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, Hebei, China e-mail: zh\_ysu\_stu@163.com

J. Zhao

Limited Company of Yanda-Yihua Electromechanical Engineering Technology Research Institute, Qinhuangdao 066004, Hebei, China

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_94

after the heavy-load transport vehicle is fully loaded, is studied to reduce the risk of accidents.

Based on the angle error leveling method, Wang et al. adopted the leveling strategy of 'only lifting without lowering' and 'low-leg raising' to solve the weak leg in the leveling process [3]. Guo Yakui et al. used the angle error leveling strategy to control the opening of the proportional valve to perform leveling by the measurement of the two-axis horizontal sensor [4]. Wang Tianhui et al. adopted the 'height-by-height method' control strategy to improve the problem of weak leg in leveling through pressure feedback [5]. Liu Zhi et al. used pressure sensors to make the four legs of the aerial work platform support separately and ensure no weak leg, and then used the chasing leveling strategy to control the platform leveling [6]. Li Xiangyu proposed a fuzzy PID control scheme based on particle swarm optimization, which improved the response speed and adaptive ability of the leveling control system [7]. Based on the integral separation PID control method and the chasing leveling strategy, Xia Xin proposed the automatic leveling control system of radar vehicle [8]. Jonnavittula used a two-axis horizontal sensor to obtain the tilt angle, and realized the leveling control of the omnidirectional vehicle platform through the Arduino Mega 2850 micro-controller [9]. Foley invented an intelligent platform with all-terrain automatic leveling. By analyzing the current working conditions, the platform predicts the future working conditions to prevent or reduce the influence of working conditions on the stability of the platform [10].

It can be seen from the above researches that the weak leg, which may occur in the leveling process, is still the main concern. At the same time, considering the disadvantage of long overall leveling time in the angle error leveling strategy, the improvement research on the position error leveling strategy is carried out.

### **2 Research on Leveling Strategy**

### *2.1 Common Leveling Strategies*

At present, the leveling strategies used for engineering transport equipment mainly include position error leveling strategy and angle error leveling strategy. The two leveling control strategies have their own advantages and disadvantages [11, 12].

The position error leveling strategy contains three different forms, namely, the highest point fixation, the center point fixation and the lowest point fixation. In the process of leveling with different methods, the position of the corresponding point is kept unchanged, so that it becomes the target point. By reducing the position error between the other legs and the target point in the vertical direction, it is gradually close to the target point. Usually, in the process of adjusting other legs close to the reference point, the speed of each leg is proportional to its position error relative to the target point in the vertical direction, so as to eliminate the position error and achieve the purpose of leveling. The advantage of this method is that the inertia force acting on the plunger cylinder is small, and the speed mutation is relatively small, which is conducive to the stability and accuracy of leveling. However, it cannot maintain the uniform force of each fulcrum during the adjustment process, so it is prone to occur the failure of the weak leg of the plunger cylinder.

The principle of the angle error leveling strategy is that the platform plane, parallel to the ground, of the large engineering transport vehicle is used as the datum plane, and the platform is divided into X and Y directions. Then, α is used to represent the offset angle between the platform plane and the X axis direction, and β is used to represent the offset angle between the platform plane and the Y axis direction. Before leveling, α and β are generally not equal to 0, so it is necessary to adjust α and β to achieve the purpose of rebalancing the vehicle platform. Compared with the position error leveling strategy, the principle and implementation algorithm of the angle error leveling strategy are simpler, and the coupling relationship between each support point is ignored. However, in the leveling process, when one of the directions of X and Y is adjusted, it will always interfere with one or two support points in the other direction, which will affect the offset angle in the other direction. Therefore, this method needs to repeat the leveling in both directions many times, makes the overall leveling time longer. And this method is only applicable to the three-point or four-point leveling system, which cannot complete the multi-point participation leveling task.

### *2.2 Improved Position Error Leveling Strategy*

The coordinate system before and after the leveling of the four-point support platform is shown in Fig. 1. OXYZ is the coordinate system when the platform is horizontal, which is called the reference coordinate system. The center point of the coordinate system is the geometric center point of the body platform, and the coordinate system remains horizontal. The platform coordinate system is OX1Y1Z1, which rotates with the rotation of the body platform. The coordinate center point always coincides with the OXYZ center point, and the coordinate system is called the platform coordinate system. The inclination angles of the platform coordinate system relative to the reference coordinate system in two directions are α and β respectively, that is, the angle of rotation around the X axis is α, and the angle of rotation around the Y axis is β. The angle can be measured by a biaxial sensor installed at the geometric center of the body platform.

If the body platform rotates angle α around the X-axis and then rotates angle β around the Y-axis relative to the reference coordinate system, the conversion matrix between the platform coordinate system and the reference coordinate system is as follows:

$$A\_1 = \operatorname{Rot}(Y,\alpha)\operatorname{Rot}(X,\alpha) = \begin{bmatrix} \cos\beta & \sin\alpha\sin\beta & \sin\beta\cos\alpha \\ 0 & \cos\alpha & -\sin\alpha \\ -\sin\beta & \sin\alpha\cos\beta & \cos\alpha\cos\beta \end{bmatrix}.$$

**Fig. 1** Four-point support platform coordinate system

If the body platform rotates angle β around the Y-axis and then rotates angle α around the X-axis relative to the reference coordinate system, the conversion matrix between the platform coordinate system and the reference coordinate system is as follows:

$$A\_2 = \operatorname{Rot}(X, \alpha) \operatorname{Rot}(Y, \alpha) = \begin{bmatrix} \cos \beta & 0 & \sin \beta \\ \sin \alpha \sin \beta \cos \alpha & -\sin \alpha \cos \beta \\ -\sin \beta & \sin \alpha & \cos \alpha \cos \beta \end{bmatrix}.$$

Considering that the inclination angle of the body platform is very small in the actual working process, the trigonometric functions can be transformed according to the following relations: *sin*α = α, *sin*β = β, *cos*α = *cos*β = 1, *sin*α*sin*β = 0, *cos*α*cos*β = 1 by means of Limit Thought and equivalent infinitesimal substitution. Therefore, the transformation matrix *A*1 and the transformation matrix *A*2 are simplified to the following forms:

$$A = A\_1 = A\_2 = \begin{bmatrix} 1 & 0 & \beta \\ 0 & 1 & -\alpha \\ -\beta & \alpha & 1 \end{bmatrix}$$

The scope of application of the small angle approximation is briefly discussed, and the following conclusions are drawn through simple calculations: (1) When the arc degree is <0.244, the error between sinα and α is <1%. (2) When the arc degree is <0.141, the error between cosα and 1 is <1%. Therefore, it is considered that when the arc degree is <0.141, the small angle approximation can be performed.

It can be seen that in the leveling process of the body platform, the platform firstly rotates around the X axis or firstly rotates around the Y axis has no effect on the conversion matrix, so the two processes can be seemed as one case. For any point B on the platform, assuming that the body platform is horizontal, its coordinates in the platform coordinate system are (*x*1, *y*1,*z*1). When the platform rotates relative to the X-axis rotation angle α and relative to the Y-axis rotation angle β, the coordinates of point B in the reference coordinate system are (*x*, *y*,*z*), and the transformation relationship between the two is as follows:

$$
\begin{bmatrix} x \\ y \\ z \end{bmatrix} = A \cdot \begin{bmatrix} x\_1 \\ y\_1 \\ z\_1 \end{bmatrix}
$$

B is the point on the platform, that is *z*1 = 0, so the coordinates of any point on the platform in the vertical direction (i.e., Z axis) can be obtained as follows:

$$z = -\beta x\_1 + ay\_1 + z\_1 = -\beta x\_1 + ay\_1$$

In order to overcome the weak leg of the plunger cylinder that may occur in the position error leveling strategy, the position error leveling strategy is analyzed from the geometric direction. It is known that in space, three points which are not on the same line, can only determine one plane; in the actual working process, the inclination arc degree of the body platform is very small. Therefore, it can be considered that in the three-point support, the position error leveling strategy will not appear the weak leg of the plunger cylinder. Therefore, the improvement of the position error leveling strategy is to convert the four-point support into three-point support. The specific scheme is as follows.

The four supporting points of the body platform are recorded as the highest point A, the second highest point B, the lowest point C and the second lowest point D. During the leveling process, the second highest point B and the second lowest point D move at a speed proportional to the error value of their respective vertical direction with the highest point A. A plane can be determined by the three points A, B and D, and the lowest point C pursuing surface ABD can ensure that the four points are located on the same plane. In addition, the pressure of each plunger cylinder is monitored by the pressure sensor, which can effectively prevent the weak leg of plunger cylinder in the leveling process.

### **3 Simulation Verification**

The leveling process of the suspension system using the improved position error leveling strategy is established by MatLab. The model is shown in Fig. 2. The four points of A, B, C and D correspond to the piston cylinder 1–3 respectively. Through the input step signal simulation system, the platform lifting signal is issued. The signal is processed by the PID controller and output to the control valve. The mathematical model of the valve converts the displacement signal into a flow signal and outputs it to the plunger cylinder 1. After the sensor 1 monitors the displacement and pressure signals, the signals are fed back to the controller to complete the closed-loop control, and it is also used as the input signal of the piston cylinder 2 and 3 to control the action of the two cylinders. The displacement and pressure signals collected by sensors 2 and 3 are input into the controller together with the signals of sensor 1. The controller converts the position coordinate of the piston cylinder 4 obtained by the operation

**Fig. 2** Suspension system leveling control simulation model

into signal to control the action of the piston cylinder 4. In addition, the displacement and pressure signals collected by sensors 1–4 are plotted into corresponding curves.

The lifting and falling processes of the suspension system are calculated by the model, and the results are shown in Figs. 3 and 4.

It can be seen from the simulation results that the displacement error between the suspension groups is kept within 2.0 mm, and the pressure error is <0.1 MPa, whether in the lifting process or the falling process. It can be judged that there is no weak leg.

### **4 Experimental Verification**

The rated load of the roadway transporter is up to 150 t, the maximum speed is 10 km/ h, the length of the whole vehicle is 6200 mm, and the width of the whole vehicle is 3500 mm. The test load is 100 t.

**Fig. 3** Leveling error of lifting process

In the real machine test, after loading, the whole-body platform lifted by 170 mm and stayed for 20 s, and then the platform dropped to the initial position. The displacement and pressure curves during the synchronous lifting process were recorded by the sensor, as shown in Fig. 5.

It can be seen from the curves that the displacement error is <5.0 mm in the actual operation process, which is close to the simulation results. At the same time, according to the vehicle length of 6200 mm and the vehicle width of 3500 mm, the inclination arc degree of the body platform in the length direction is calculated to be 8.06 × 10–4, and the inclination arc degree in the length direction is 1.43 × 10–3, both of which are <0.141. Therefore, it can be seen that in actual work, the small angle approximation is still valid. The pressure error is <2.5 MPa, and there is a certain error between the simulation results. After analysis, it was found that the pressure error was mainly caused by the different tire pressure between the tires. The leveling control accuracy meets the requirements.

**Fig. 4** Leveling error of falling process

**Fig. 5** Synchronous lifting leveling error

# **5 Conclusion**


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation of Nanoindentation Technology Based on Rough Surface**

**Zihao Ye, Yonghang Wang, Jiasheng Yao, and Jiankang Jiang** 

**Abstract** In order to study the influence of roughness on the nanoindentation test, finite element method was used to simulate the nanoindentation test process, and secondary indentation method was further proposed to suppress the influence of roughness on the indentation. After optimizing the indentation parameters, the relative error between the results of nanoindentation simulation for rough surface samples and smooth surface samples is stable within 5%. The secondary indentation method can effectively reduce the influence of roughness on nanoindentation test results.

**Keywords** Nanoindentation · Surface roughness · Finite element analysis · Mechanical property

# **1 Introduction**

With the development of modern industry technology, the research of precision and ultra-precision machining technology is extensive, and the requirements for surface properties of materials are also widely, especially the mechanical properties of materials at micro and nano scale have attracted widespread attention [1–3]. The application and research of micro-nano structured materials require properties characterization and fracture analysis. Meanwhile, with the development of various surface treatment technologies, the scale of material properties is getting smaller [4–6],

School of Mechanical and Electrical Engineering, Soochow University, Suzhou, China e-mail: suda\_yzh@163.com

Y. Wang e-mail: 906734837@qq.com

J. Yao e-mail: byjs011621@163.com

1191

Z. Ye (B) · Y. Wang · J. Yao · J. Jiang

J. Jiang e-mail: 1445158122@qq.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_95

due to the size effect, there is a great difference between the mechanical properties of materials at nano scale and bulk materials, thus traditional mechanical properties characterization methods have been unable to accurately obtain the mechanical properties of materials at nano scale. Nano-indentation method is very suitable for characterizing the mechanical properties at nano scale that does not need to process materials into standard size [7–9]. By recording the indenter load and displacement data during the indentation process, based on the Oliver-Pharr theory, the mechanical properties of the material can be calculated, such as hardness, elastic modulus, strain hardening effect, creep, and so on [10–12]. However, the roughness of the sample will affect the evaluation of the geometric parameters of the residual indentation and directly affect the analysis of load–displacement data, resulting in poor reliability and repeatability of the test results [13, 14]. Therefore, the application of nano-indentation technology theory based on smooth surface will be limited. In order to reduce the cost and improve the test efficiency, this paper studied the influence of roughness on the nanoindentation test based on finite element simulation, and further proposed the secondary indentation method nanoindentation test technology to suppress the influence.

# **2 Nanoindentation Simulation by Secondary Indentation Method**

### *2.1 Modeling of Rough Surface Samples*

There are two main criteria for evaluating surface roughness: the arithmetic mean deviation method *R*a and the maximum height method *Rz*. The arithmetic mean deviation *R*a refers to the arithmetic mean value of the absolute value between the point on the contour line of the measurement direction and the reference line in the sampling length, calculated by the equations:

$$Ra = \frac{1}{l} \int\_0^l |\mathbf{y}| dx \tag{1}$$

$$Ra = \frac{1}{n} \sum\_{i=1}^{n} |\mathbf{y}\_i| \tag{2}$$

*Rz* refers to the distance between the peak line and the bottom line within the sampling length *l*, as shown in Fig. 1.

The arithmetic mean difference method uses the average of the height difference between the sampling point and the reference contour to define the roughness, which does not reflect the characteristics of the highest and lowest points of the surface. The definition of maximum height method reflects the characteristics of the maximum **Fig. 1** Maximum height method to evaluate roughness

height difference of the surface, which is more beneficial to the determination of the indentation parameters.

In the simulation, the maximum height method *Rz* is used as the standard for evaluating roughness, and *Z*<sup>v</sup> = *Z p* simplifies the rough sample surface into a surface composed of continuous identical peaks and bottoms. According to the definition of roughness, different roughness sample models were built and assembled into ABAQUS software. Pure copper is selected as the simulation material. Considering the actual machining accuracy and wear, the tip of the indenter was set a radius of 400 nm. In order to reduce the total number of meshes and shorten the calculation time, the mesh density near the contact area was encrypted by means of edge planting.

### *2.2 Secondary Indentation Nanoindentation Technology*

The nanoindentation test process is divided into two steps, as shown in Fig. 2. Step 1: flat indenter is carried out on the rough surface to reduce roughness. Step 2: Conventional nanoindentation testing is performed in the residual indentation area.

### *2.3 Simulation of Secondary Indentation Method*

In order to understand the plastic deformation and stress distribution of the first indentation, the flat indenter nanoindentation simulation was carried out, and the smooth surface nanoindentation simulation was used to analyze the influence of surface roughness and maximum indentation depth. The diameter of the flat indenter *D f* = 2 mm, and the samples with different surface roughness (*Rz*) were simulated with different maximum indentation depth (*h*1 max).The simulation results show that under the same indentation depth, with the roughness increase, the relative error of load–displacement curve is greatly affected. When *Rz* = 120 µm, *h*1 max = 400 µm, the relative error of maximum load is 6.32%, as show in Fig. 3.

**Fig. 2** Nanoindentation test process using the secondary indentation method

On the other hand, the plastic deformation of the material is independent of the surface roughness. Figure 4 shows the residual stress distribution and the comparison of plastic deformation regions in indentation of flat indenters with different roughness. The residual stress is mainly distributed just below the indentation surface, and the stress and plastic deformation area gradually increase with the increasing of the indentation depth. The stress distribution is mainly concentrated in the area near the surface, while the lower region is basically the same as that in the plastic region. This means that at the same indentation depth, the stress–strain properties of the central region away from the surface of the sample are not affected by the roughness, which means that the stress–strain relationship at a representative point under the surface of the residual indentation is not affected by the surface roughness.

**Fig. 4** Residual stress distribution and plastic deformation region by indentation of flat indenter

### *2.4 Optimization of Simulation Parameters*

According to the definition of roughness by maximum height method, when the maximum indentation depth of step 1 is *h*1 max ≥ 2*Rz*, the residual indentation satisfies the conditions of Step 2. For the sample with roughness *Rz* = 120 µm, the secondary indentation simulation was carried out. In Step 1, the diameter of the flat indenter *D f* = 2 mm, the maximum indentation depth *h*1 max = 240 µm, *h*2 max = 400 µm. As show in Fig. 5, the results of the secondary indentation method are different from the smooth surface with Berkovich indenter, and the relative error of the maximum load is 11.65%.

The Step 2 is carried out in the central area of residual indentation. However, when the depth of indentation increases, the contact radius between the edge of the Berkovich indenter and the surface of the residual indentation gradually rises. If the area of the residual indentation is not large enough, the stress distribution and plastic deformation in the surrounding area will affect the results. Therefore, the diameter *D f* of Step 1 is an important parameter of second indentation method, and further analysis is necessary.

Simulation using the secondary indentation method was carried out for the diameters of *D f* = 4, 8,10 mm respectively. Table 1 lists the relative error values between the maximum load of the second indentation method and smooth surface indentation test under different *D f* . As show in Fig. 6, with the increase of *D f* , the relative errors between the second indentation method and smooth surface decrease gradually, and tend to be stable after reaching a certain degree. This indicates that the area of the first

stage residual indentation is large enough, and the influence of the residual stress and plastic deformation on the surface of the material residual indentation is significantly reduced.

Define geometric parameter γ = *h*2 max/*D f* . when *h*2 max = 400 µm, the results show that the relative error with the maximum indentation load on the smooth surface tends to be stable at γ ≤ 5%, and a similar trend is also present in the indentation

**Table 1** Maximum load and relative error values of the second indentation method and smooth surface indentation under different *D f* 


**Fig. 6** Comparison between the indentation results of the second indentation method with different flat indenter diameters *D f* and those of the smooth surface (*h*2 max = 400 µm): **a** *D f* = 4 mm; **b** *D f* = 8 mm; **c** *D f* = 10 mm

**Fig. 7** Comparison of simulation results of the second indentation method and smooth surface indentation for different geometric parameters (γ = 20, 10, 5, 2%): **a** = 300 µm; (b)*h*2 max = 500 µm


**Table 2** Maximum load and relative error for different geometric parameters

simulation with the second indentation method at *h*2 max = 300 µm and *h*2 max = 500 µm, as show in Fig. 7. Which means the smaller geometric parameter γ is beneficial to the secondary indentation method. The data in Table 2 further show that when γ ≤ 5%, the secondary indentation method can effectively reduce the influence of material surface roughness on the nanoindentation test results.

### **3 Conclusion**

This study aims at the problem of poor repeatability of traditional nanoindentation testing technology, based on the finite element simulation technology, the influence of roughness on the results of nanoindentation testing is studied, and the secondary indentation method is further proposed to suppress the influence of roughness on the results of indentation testing. The indentation parameter γ = *h*2 max/*D f* was defined by the secondary indentation method, and multiple indentation simulations were carried out for different values of γ . According the simulation, when γ ≤ 5%, the relative error of the maximum load decreases from 15% to <5%, indicating that the secondary indentation method can effectively reduce the influence of material roughness on nanoindentation testing.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Research on Wading Mobility of a Certain Type of Special Vehicle Based on MPS Method**

**Yiyang Hao, Yunbo Zhou, and Ming Zhang** 

**Abstract** In order to study the wading mobility of a certain type of special vehicle, a high-precision dynamic model of the special vehicle is built. The vehicle dynamic model is verified by in-situ acceleration test and traction test. Through the dam-break experiment and wet road traction test, the feasibility and reliability of calculating the water impact pressure, buoyancy and solving the fluid–solid coupling problem based on the moving particle semi-implicit algorithm (MPS) are verified. Then, based on the MPS algorithm, a multi-body dynamics-fluid analysis simulation model is built to simulate and analyze the vehicle mobility under the static water wading condition and the water flow lateral impact condition. The relationship between the maximum speed of the vehicle at different water depths and gears under the static water wading condition the and the relationship between the lateral offset of the vehicle and the flow velocity and the deflection angle of the vehicle entering the water in the lateral impact condition are obtained.

**Keywords** MPS · Multi-body dynamics · Vehicle wading · Mobility · Co-simulation

# **1 Introduction**

Military vehicles will encounter various extremely complex road conditions on the battlefield, such as rapid rivers or bridge decks submerged by rivers. Due to the buoyancy of the water and the lateral impact of the water flow, whether the vehicle can pass safely and quickly is of great significance to the safety and combat efficiency of the occupants. Vehicle mobility can measure the ability of the vehicle to maintain rapid passage under various road and terrain conditions (such as water-covered roads) that the vehicle may encounter [1, 2]. Studying the vehicle's wading mobility can

Y. Hao · Y. Zhou (B) · M. Zhang

© The Author(s) 2024

1201

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: yunbo31983@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_96

effectively evaluate and optimize the actual mobility performance of military vehicles under wading conditions.

The research methods of vehicle wading maneuverability mainly include test analysis and simulation. The vehicle wading test involves the safety of vehicle occupants and the regulation of water flow velocity, which is difficult to implement and costly. The simulation method can reduce the cost while ensuring the calculation accuracy. Tison et al. [3] established a six-degree-of-freedom model of an amphibious vehicle. Based on the finite element method, the water-exit process of the vehicle was simulated and analyzed. At this time, the finite element method needs to perform mesh reconstruction to ensure the mesh quality [4], and it is difficult to simulate the fluctuation and splash of the water area, so the particle algorithm is proposed and gradually widely used.

Particle method, also known as meshless method, is mainly divided into smooth particle algorithm (SPH) and moving particle semi-implicit algorithm (MPS). Yreux [5] simulated the dam-break experiment and vehicle wading conditions through the SPH algorithm in LS-DYNA. The MPS algorithm was first proposed by Koshizuka et al. [6]. The implicit method was used to solve the pressure equation to ensure the incompressibility of the fluid. At the same time, the semi-implicit solution method was introduced to shorten the calculation time. It has a wide range of applications in simulating water flow. Chixin et al. [7] used the MPS algorithm to visually analyze the splash lubrication characteristics of the gearbox. Jing and Jinbiao [8] improved the MPS algorithm and verified the feasibility of the MPS algorithm in fluid–solid coupling simulation.

Based on the MPS algorithm, this paper tests, numerically simulates and analyzes the mobility of a special vehicle under wading conditions through the joint simulation of multi-body dynamics software and fluid analysis software. The maximum speed under different water depths and the body offset under the lateral impact of water flow are obtained, and the influence of water depth and water flow velocity on the mobility of the vehicle is analyzed.

### **2 Moving Particle Semi-Implicit Algorithm**

The moving particle semi-implicit algorithm (MPS) characterizes the interaction between particles by kernel function, and uses gradient model and Laplace model to simulate the control equation.

### *2.1 Control Equation*

The MPS algorithm mainly calculates and simulates incompressible fluid, and its control equations are continuous equations and momentum conservation equations: Research on Wading Mobility of a Certain Type of Special Vehicle … 1203

$$\frac{d\rho}{dt} = 0\tag{1}$$

$$
\rho \frac{d\mathbf{u}}{dt} = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g} \tag{2}
$$

where *u* is the velocity of the fluid, ρ is the density of the fluid, μ is the dynamic viscosity of the fluid, *p* is the pressure, *g* is the acceleration of gravity, and *t* is the time.

### *2.2 Kernel Function*

The kernel functions commonly used in the MPS algorithm are:

$$w(r\_p) = \begin{cases} \frac{r\_e}{r\_p} - 1, 0 < r\_p < r\_e\\ 0, r\_p > r\_e \end{cases} \tag{3}$$

$$r\_e = RD\_i \tag{4}$$

where *r <sup>p</sup>*is the distance between particles, *re* is the effective radius of particles, *R* is the effective radius coefficient, and *Di* is the radius of particles.

### *2.3 Gradient Model and Laplace Model*

The gradient model and the Laplace model in the MPS algorithm can discretize the control equations. The gradient model and the Laplace model are as follows:

$$[\nabla \varphi]\_i = \frac{d}{n\_0} \sum\_{j \neq i} \left[ \frac{\varphi\_j - \varphi\_i}{\left| r\_j - r\_i \right|^2} (r\_j - r\_i) \omega(\left| r\_j - r\_i \right|) \right] \tag{5}$$

$$\left[\nabla^2 \varphi\right]\_i = \frac{2d}{n^0 \lambda} \sum\_{j \neq i} (\varphi\_j - \varphi\_i) \rho(|r\_j - r\_i|) \tag{6}$$

In the formula, *d* is the dimension of the solution space, *n*0 is the particle number density, *ri* is the coordinate vector of the i th particle, ϕ is the scalar value of the physical parameter of the particle, λ is the correction factor, and its expression is:

$$\lambda = \frac{\sum\_{j \neq i} \alpha(\left| r\_j - r\_i \right|) \left| r\_j - r\_i \right|^2}{\sum\_{j \neq i} \alpha(\left| r\_j - r\_i \right|)}. \tag{7}$$

a. Real vehicle tire b. Tire model c. Vehicle dynamics model

**Fig. 1** Tire and vehicle model

# **3 Vehicle Multi-Body Dynamics Modeling and Verification**

### *3.1 The Construction of Dynamic Model*

In the vehicle dynamics model, the parameters that have the greatest influence on the calculation accuracy of wading conditions include engine performance parameters, body shape characteristics and tire-ground adhesion coefficient. In order to ensure the simulation accuracy, the torque-speed curve on each driving wheel of the simulation model is obtained from the engine characteristic curve of the original vehicle and the overall speed ratio of the transmission system. The body shape is consistent with the original vehicle, and the three-dimensional tire model is exactly the same as the original vehicle (Fig. 1). The tire model file adopts UA tire, and the tire parameters are adjusted by the test data. At the same time, the suspension system is built, and a high-precision multi-degree-of-freedom dynamic vehicle model is established. The vehicle dynamics model is shown in Fig. 1.

### *3.2 Validation of Vehicle Dynamic Model*

The reason for the decrease of power and the deflection of the vehicle body during wading and lateral impact is the slip of the tire, so the correct and reliable contact parameters between the tire and the ground in the dynamic model are very important to the accuracy of the simulation results. It is verified by hard road traction test and in-situ starting acceleration test.

According to the test standard, in the hard road traction test (Fig. 2.a), the test vehicle is connected to the load trailer through the traction rod, and the load trailer steps on the brake until the test vehicle is completely skidded. The maximum traction force of the test vehicle on the hard road is obtained by the force sensor on the traction rod. The load trailer brake is equivalent to a fixed end. The test vehicle continues to move forward to tighten the traction rod until the maximum traction force is obtained. Therefore, the load trailer can be equivalent to a fixed end in the simulation model

a. Hard road traction test b. Simulation model of hard road traction test

**Fig. 2** Hard road traction test and simulation model

of the traction force test (Fig. 2. b). The vehicle is connected to the fixed end by a spring, and the spring is tightened until the wheel slip rate reaches 100%, and the maximum traction force of the simulation is obtained.

The traction curve of hard road is shown in Fig. 3.a. Comparing the maximum traction force of the test and simulation, the test result is 80.78 kN, the simulation result is 82.32 kN, and the calculation error is 1.9%. It shows that the simulation model can well simulate the characteristics of vehicle tire and ground mechanics on hard road.

In order to ensure the dynamic performance of the vehicle when it is wading, it is usually driven in first gear. Therefore, the model is compared with the speed change curve of the original vehicle in the first gear (Fig. 3.b). It can be seen that the established dynamic model reaches the maximum speed of 24.5km/h in the first gear at 2.4s, and the acceleration time and the maximum speed can be well fitted with the test curve.

**Fig. 3** Hard road traction curve and in-situ starting acceleration curve

### **4 Verification of MPS Algorithm**

Based on the MPS algorithm, the simulation analysis of the vehicle wading condition needs to ensure the correctness of the MPS algorithm in simulating the water pressure, buoyancy and fluid–solid coupling calculation. The dam impact test and the vehicle wet slippery road traction test are used for verification.

### *4.1 Dam Break Experiment Based on MPS Algorithm*

Under the conditions of wading and lateral impact of water flow, the water force that has the greatest impact on mobility is the impact pressure and buoyancy of water flow. The essence of buoyancy is the pressure difference caused by different water layer heights, and then the longitudinal force acting on the surface of the object is generated. Therefore, It is necessary to verify whether the MPS algorithm is reliable for the simulation of water pressure. This is verified by the dam break experiment done by Kleefsman et al. [9] and Sug et al. [10].

Kleefsman et al. built an experimental platform for dam break, and measured the impact pressure of water flow at different positions of the baffle by sensors. Firstly, a 1:1 dam break simulation model with the original experiment is built, as shown in Fig. 4. The particle spacing of the model is set to 10 mm. In order to accurately capture the pressure peak point, the output data spacing is set to 0.01 s.

The curves of simulation data and experimental data are shown in Fig. 4b. Comparing the data results, the simulation model reaches the peak pressure point at 0.43 s, and the relative error of the peak pressure is 9.8%. The pressure change trend is basically the same as the experimental data, but due to the size of the model particles, the calculation results fluctuate significantly. In addition, the peak of about 5s has a time delay, which is caused by the echo after the impact of water flow, and has little effect on the wading and water flow impact conditions of the vehicle in the open water.

**Fig. 4** Dam break simulation model

**Fig. 5** Wet road traction test and data curve

### *4.2 Traction Test of Wet Road Based on MPS Algorithm*

The vehicle wet road traction test (Fig. 5) can not only verify whether the tire-ground mechanics of the vehicle dynamics model is reliable under the influence of the water layer, but also verify whether the calculation of the fluid–solid coupling problem based on the MPS algorithm is accurate. The simulation model of wet road traction also equivalents the load trailer to a fixed end, and establishes a joint simulation of dynamic software and fluid analysis software.

The vehicle wet road traction curve is shown in Fig. 5. The maximum traction force obtained by simulation is 39.41 kN, while the maximum traction force measured by the test fluctuates around 38.37 kN, and the calculation error is 2.7%. The existence of the water layer makes the adhesion coefficient of the tire-ground decrease, resulting in the maximum traction force and the time to reach the complete slip of the tire are reduced compared with the hard road conditions.

# **5 Simulation Analysis of Vehicle Wading Condition**

### *5.1 Simulation Analysis of Vehicle Wading in Static Waters*

The simulation of vehicle wading conditions in static waters can calculate and analyze the mobility of this type of special vehicle in different water depths. The mobility index involved is the maximum speed that the vehicle can reach. The water depth is set to 0.5m and 1m. The vehicle first accelerates to the maximum speed in the first or second gear on the hard road, and then drives into the water area, and finally reaches the stable maximum speed of wading.

The components in the vehicle dynamics model that are in contact with the water are exported as interface files and coupled with the fluid analysis software to obtain a simulation model of static water wading as shown in Fig. 6. The horizontal area (not including uphill and downhill) of the water area is 40 m long, 10 m wide. Due to the

a. Vehicle wading simulation model

b. Vehicle into the water c. Vehicle fully launching into the water

**Fig. 6** Simulation model and simulation process of vehicle wading condition

large volume of the water area, in order to reduce the calculation cost, the particles are appropriately enlarged, and the particle spacing is set to 50 mm.

When the water depth is 1 m, the curves of vehicle speed, resistance and buoyancy are shown in Fig. 7. The simulation results of the first gear are shown in Fig. 7a. Due to the influence of water resistance, the actual maximum speed is 16 km/h. The simulation results of the second gear are shown in Fig. 7b. It can be seen that when the vehicle accelerates on the hard road to the maximum speed (42 km/h) and enters the water, the speed drops rapidly to 10 km/h, reaching the actual maximum speed of the second gear wading.

When the vehicle wades in the second gear, the longitudinal resistance and buoyancy do not increase gradually as the first gear enters the water, but will produce a large peak at the moment when the vehicle enters the water. This is because the water entry speed of the second gear is 1.7 times that of the first gear. The increase in speed makes the body have a greater impact on the water body. It can be seen that the vehicle speed has a great influence on the longitudinal resistance and buoyancy of the vehicle.

**Fig. 7** Simulation results of wading conditions (water depth 1 m)

**Fig. 8** Simulation results of wading conditions (water depth 0.5 m)


When the water depth is 0.5 m, the speed, longitudinal resistance and buoyancy of the vehicle are shown in Fig. 8. Compared with the simulation results of the water depth of 1 m, it can be seen that the decrease of the water depth makes the longitudinal resistance and buoyancy of the vehicle significantly reduced. In the stable stage of vehicle speed, the buoyancy value is reduced from 35 to 10 kN, while the longitudinal resistance value is reduced from 15 to 8 kN. The maximum speed has increased, when the vehicle to a gear into the waters, the stable maximum speed of 19 km/h; when the vehicle enters the waters with a second gear, the maximum speed that can be achieved is 24 km/h. The maximum speed of the vehicle under different water depths and gears is shown in Table 1.

### *5.2 Simulation Analysis of Water Flow Lateral Impact*

When military vehicles pass through the bridge deck submerged by the river, the external force on the body is not only the buoyancy of the water body, but also the lateral impact force caused by the river. In order to ensure the trafficability of the vehicle, it is necessary to ensure that the lateral offset (Δy) of the body under the lateral impact of the water flow is controlled within an appropriate range. The lateral offset refers to the displacement of the body center of mass in the direction perpendicular to the center line of the bridge deck. Its range is determined by the width of the vehicle and the bridge deck. The purpose is to ensure that the vehicle will not be impacted by the water flow and fall off the bridge deck. The wheel track of the comprehensive test model (1.97 m) and the assumed bridge width (8 m), the body offset should not be >6m. Considering the general river width, the length of the water area is set to 30 m.

The dynamic model does not consider the driver's control of the steering wheel, so only the lateral impact of the water flow when the vehicle is fixed on the steering wheel is simulated. Considering the influence of water flow, the water depth of 0.5 and 1 m are compared and analyzed. The water flow impact model is shown in Fig. 9a, and the particle spacing of the model is maintained at 50 mm. When the water depth is 0.5 m, the lower water flow rate will not have a significant impact on the body, so the flow rate begins to increase from 1.5 m/s. When the water depth is 1 m, the flow velocity begins to increase from 0.5 m/s until the body offset is greater than the set target value, and the limit flow impact velocity is obtained. The simulation process is shown in Fig. 9b.

In the actual driving process, when the vehicle encounters the lateral impact of the river, it often changes the direction of the driving in advance, deflects the direction of the vehicle, and maintains a certain angle with the center line of the bridge deck to enter the water area. Here, the lateral impact is simulated with the deflection angles of 0°, 5° and 10° respectively. The relationship between the lateral offset of the vehicle and the flow velocity and angle under different water depths is shown in Fig. 10.

According to the simulation results, whether the water depth is 0.5 or 1 m, when the vehicle enters the water in the direction parallel to the center line of the bridge deck (0°), the lateral offset of the vehicle will be improved to varying degrees compared to the deflection of a certain angle into the water at the same flow rate. When the flow

a. Water flow lateral impact model b. Simulation process

**Fig. 9** Simulation model and simulation process of water flow lateral impact condition

**Fig. 10** Simulation results of lateral impact of water flow

rate increases, the difference is more obvious. Taking the impact condition of 1 m water depth as an example, when the flow velocity is 0.5 m/s, the lateral offsets of the three water entry angles are basically the same. When the flow velocity increases to 1 m/s, the lateral offset difference between 0°and 10°reaches 3.55 m, which is much larger than the wheelbase of the vehicle itself. It can be seen that changing the angle of the vehicle into the water has an important influence on improving the trafficability and safety of the vehicle under the impact of higher flow velocity.

In terms of the limit flow velocity to ensure the safe passage of vehicles, when the water depth is 0.5 m, the limit flow velocity of 0° water entry is 1.9 m/s, and when the deflection angle is 10°, the limit flow velocity increases to 2 m/s. When the water depth is 1 m, due to the increase of water flow, the lateral impact force of water flow on the body increases. At this time, when the vehicle enters the water at 0°, the allowable limit flow rate is only 0.7 m/s. After increasing the deflection angle, the trafficability of the vehicle is improved, and the allowable maximum flow rate at 10° is 0.9 m/s.

In terms of the relationship between the lateral offset of the vehicle body and the flow velocity, the simulation results of the lateral offset of the water entering in the 0° direction at a water depth of 1 m are taken as an example. When the flow velocity increases to 1 m/s, the lateral offset suddenly increases from 7.86 to 10.89 m, and the increase is significantly increased. This is because when the flow velocity reaches 1 m/s, the lateral impact force of the water flow is greater than the lateral adhesion of the wheel, and the tire becomes sliding in the lateral direction, resulting in a significant increase in the lateral offset. At this time, in order to improve the mobility of the vehicle driving through the river, it is necessary to improve the adhesion coefficient of the tire, that is, to change the material, pattern and width of the tire until the requirements of wading are met, so as to ensure the trafficability and safety of the vehicle.

### **6 Conclusion**

A multi-body dynamics model of a special vehicle is built, and the vehicle model and MPS algorithm are verified according to the vehicle tests and dam break experiment in this paper. Then, the maximum speed of the vehicle under different water depths and different gears is obtained by joint simulation. In addition, the lateral impact conditions of water flow at different water depths are simulated and analyzed with the deflection angles of 0°, 5° and 10° respectively. The variation law of lateral offset under different flow velocities and deflection angles is obtained.

**Acknowledgements** Postgraduate Research & Practice Innovation Program of Jiangsu Province.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **A Study of the Effect of Frontal Crash Seat Belts on Driver Injury in a Certain Type of Wheeled Tactical Vehicle**

**Tao Wang, Yunbo Zhou, and Ming Zhang** 

**Abstract** A finite element model of frontal collision of a certain type of wheeled tactical vehicle is established, and the collision test method is designed concerning the test requirements of Chinese frontal collision standard GB11551 and Chinese side collision standard GB2007 to get the acceleration inside the vehicle as well as the damage of the occupants, and the model accuracy is verified according to the results of the actual collision test and the simulation test, as well as the change of the energy in the process of the simulation. The Hybrid III 50% male dummy is placed in the driver's seat, and under the action of different conditions and ways of restraint systems, the damage of each part of the dummy is obtained in the simulation test. The study shows that multi-point seat belts provide better occupant protection compared to traditional three-point seat belts in response to high-impact crash environments. It can improve the direction of the force on the occupant and effectively reduce the occupant injury. The final occupant WIC decreased by 25.9% relative to the initial four-point seat belt position, improving the survival rate of the occupants.

**Keywords** Crash · Specialty vehicles · Restraint systems · Occupant injury

### **1 Introduction**

Wheeled tactical vehicles, as fast maneuvering platforms and multi-purpose combat platforms, play an important role in the modern ground battlefield, and their battlefield safety has always been a focus of attention due to the huge amount of equipment and the complex environment of use [1]. In current research on active safety in vehicle collisions, seat belts can reduce occupant injuries [2]. According to the National Highway Traffic Safety Administration (NHTSA), seat belt use can reduce fatal

T. Wang · Y. Zhou (B) · M. Zhang

1213

School of Mechanical Engineering, Nanjing University of Science and Technique, NanjingJiangsu 210094, China e-mail: yunbo31983@163.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_97

injuries by nearly 45%, and severe injuries by 67% [3]. Data analysis of Crashworthiness Data System (CDS) shows that the head and chest are the most common areas causing serious injuries during vehicle crashes, and compared to common two-point and three-point seat belts, four-point and five-point seat belts can provide restraints in different orientations to better restrain the occupant in the seat, thus improving occupant safety. According to the literature [4] under 400N preload force makes the seat belt 93% effective for occupant safety. At this stage, there are experimental studies on collision are mostly concentrated in the field of civil vehicles, literature [5] on the use of seat belt pre-tensioners to analyze, expanding the scope of research on seat belt pre-tensioners and load limiters; literature [6] Liu Xin et al. will be the position of the seat belt hanging point, elongation and initial strain as a design variable for the optimization of the seat belt restraining system, fast and effective to obtain the optimal matching parameters of the seat belt restraining system to ensure the safety of automobile occupants; literature [7] on the impact of special vehicle seat belt form on space safety, mainly to study the application of multi-point seat belts in the explosion impact environment. In this paper, the seat belt restraint system under the collision conditions of wheeled tactical vehicles is studied, and the damage to the driver's seat occupant is verified by comparing the test and simulation results to verify the accuracy of the simulation model. According to the simulation restraint system select the appropriate form of seat belt, to select the best restraint program.

### **2 Establishment and Analysis of the Whole Vehicle Model**

### *2.1 Finite Element Modeling*

The 3D model of the wheeled tactical vehicle is simplified, after which the mesh is divided using HyperMesh software for pre-processing. The body is mainly sheet metal parts, using two-dimensional shell cells for simulation, the cell size of the model division according to the size of the impact of the components on the calculation of the working conditions, the model cell size of this paper is 10–20 mm, the shape of the quadrilateral and triangular-based, the minimum side of the mesh is >5 mm, the cell warping degree is <15°, the triangle cell internal angle to be in the range of 20°–120°, the quadrilateral cell maximum/minimum angle to be in the range of 35°–145°, the cell Jacobian is >0.65, etc. The collision process mainly relies on the plastic deformation of the material to absorb energy, so the body parts are mostly simulated by MAT24, the bolts are simulated by RBE2 and BEAM, and the welds are simulated by rigid and weld cells. The vehicle model consists of 1,826,899 nodes and 1,855,398 grid cells. The finite element model of the whole vehicle is shown in Fig. 1.

The Hybrid III 50% dummy model was used for the test. Before the simulation, the dummy was placed in the appropriate position of the seat, and the seat belt was built to adjust the dummy's sitting posture and foot placement. The seat belt in contact

with the dummy is divided by a two-dimensional grid, and the rest of the seat belt is divided by a one-dimensional cell, as shown in Fig. 2.

### *2.2 Simulation of the Two-Vehicle Collision*

Since there is no current collision standard for military vehicles, refer to the civilian vehicle collision standard while considering the later experimental verification of the situation, set the initial speed of the vehicle collision \*INITIA\_VELOCITY\_ GENERATION for 50 km/h to hit another stationary vehicle lateral B-pillar. At the same time, the two vehicles will be given the vertical direction of the gravitational acceleration g = 9.8 m/s2, and the vehicle and the ground kinetic friction factor is set to 0.3, the basic situation is shown in Fig. 3.

**Fig. 3** Simulation model illustration

# *2.3 Simulation Results and Test Verification*

To verify the accuracy of the whole vehicle collision test model, two protective assault vehicles were selected by the finite element model test for a real vehicle collision. One of the vehicles, code F2135, collided with another static parked vehicle, code S819, at a speed of 50 km/h, as shown in in Fig. 4. The Hybrid III 50% male test dummy was placed at the driver's position of the collision vehicle. A Hybrid III 50% male test dummy was placed at the driver's position of the collision vehicle, and accelerometers were placed at the B-pillar and the center of mass of the vehicle, which were consistent with the simulation model as shown in Fig. 5. The accuracy of the simulation model was verified by comparing the vehicle's B-pillar and centerof-mass X-direction accelerations, as well as the injury responses of the dummy's head, chest, and left and right thighs.

According to the results of the actual collision as shown in Fig. 6, the injury response of the occupants under the whole vehicle collision is obtained to be more consistent with the occupant response in the simulation, and according to the peak response of the acceleration at the center of mass and the B-pillar of the vehicle and the time of emergence comparison, there is a high degree of similarity, which indicates that the vehicle collision simulation model has a high degree of accuracy.

**Fig. 4** Crash test chart of two vehicles

**Fig. 5** Accelerometer and driver's seat dummy

Under the rigid impact, the X-direction maximum acceleration at the center of mass position in the vehicle is 29.15 g during the actual collision, and the X-direction maximum acceleration at the center of mass position in the simulation is 26.36 g, which is easy to cause impact injuries to the occupants.

The change of system energy during the collision process is shown in Fig. 7, the system energy is gradually converted from the initial kinetic energy of the vehicle to the internal energy generated during the collision process. When the hourglass energy is <5% of the total energy can be considered that the simulation model is credible [8]. Fig. 8 shows the change in the ratio of the hourglass energy to the total energy during the collision process, the maximum is 1.7%, which is <5%.

### *2.4 Criteria for Evaluating Occupant Injury*

In order to comprehensively evaluate the overall performance of the restraint system, the weighted injury criterion WIC [9] (Weighted Injury Criterion) is used to evaluate the occupant injury,

$$WIC = 0.6\frac{HIC}{1000} + 0.35\left(\frac{C\_{3ms}}{60} + \frac{D}{75}\right) / 2 + 0.05\left(\frac{F\_{FL} + F\_{FR}}{20.0}\right)$$

The lower the WIC value, the better the protection performance. Among them, HIC is the comprehensive performance index of the head, and the limit value of this index is 1000; *C*3*ms* is the 3 ms acceleration value of the chest (unit G), and the limit value of the acceleration is 60G; D is the compression volume of the chest (unit mm), and the limit value of the compression volume is 75 mm; *FFL* and *FFR* are the axial force of the left and right thighs (unit kN), and the limit value of the axial force is 10 kN.

**Fig. 6** Comparison plot of test simulation response

# **3 Study on the Effect of Different Seat Belt Styles on Driver Protection**

To improve the spatial safety of the occupants, the pre-tensioned seat belts, which are primarily used in current civilian vehicles, as well as the four-point and five-point seat belts will be investigated for their effects on the occupant's limb motion response and injury. Occupant injury results as shown in Table 1 are obtained after simulation model validation.

Based on the simulation results, it was obtained that an increase in the preload of the seat belt decreases the head injury of the dummy with a decrease in HIC by 3.9%, and an increase in the preload using the retractor increases the tension of the seat belt, resulting in an increase in the amount of compression in the chest of the dummy as well as an increase in the axial force in the right leg in the vicinity of the


**Table 1** Restraint effects of different forms of seat belts

belt buckle, which increases by 0.5 and 3.9%, respectively. The four-point harness was more effective in reducing the head HIC value, the cumulative 3 ms synthetic acceleration of the chest, and the chest compression by 4.6 and 60.3%, respectively, compared to the three-point harness. Five-point seat belts have better restraint effects, mainly in the lower body, the actual wearing process is more cumbersome, which is not conducive to the vehicle occupants to deal with unexpected situations. For multipoint seat belts to increase the pre-tensioning, two pre-tensioning devices need to be added, the cost is higher, and at present most single pre-tensioning force-limiting seat belts are mainly used [10]. To comprehensively evaluate the effect of different forms of seat belts, the occupant injury numerical analysis, to establish a comprehensive performance evaluation of different seat belts on the occupant injury, the value of the safety coefficient varies from 0 to 1, the larger the value of the better the safety and protection performance [11], of which

safety coefficient = Standard Injury Threshold − Maximum Simulation Test Value Standard Injury Threshold <sup>×</sup>100%

According to Fig. 9 multipoint seat belts have a better restraining effect, after that, the occupant protection in different installation situations is further investigated.

**Fig. 9** Occupant restraint safety factor curve

### **4 Conclusion**

In this paper, a 50 km/h frontal collision driver seat occupant damage verification was completed for a certain type of wheeled tactical vehicle, and based on the simulation results, the occupant damage study was carried out by increasing the preload force and multi-point seat belt. Based on the simulation results, the following conclusions were obtained: Multi-point seat belts have better crash safety constraints for the occupants of this type of particular vehicle, especially for the upper body constraints of the occupants, which can be selected according to the actual four-point or fivepoint full belts inside the special vehicle. In the case of the same form of seat belt, the use of the roll-up pre-tensioning force has less impact on the dummy injury, and only increases the upper body part of the occupant's restraint, but will correspondingly increase the probability of injury to the occupant's chest.

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **The Perfect Fluid Characteristic of the Quark Gluon Plasma**

**Ke Li, Cheng Ma, Jiahua Qu, and Jiayi Zhang** 

**Abstract** This article explores the unique characteristics of the Quark Gluon Plasma (QGP) by analyzing open data obtained from the ALICE experiment for Pb–Pb collisions and from the CMS experiment for Xe–Xe collisions at the Large Hadron Collider (LHC). The total integrated luminosity of the analyzed data is 3.42 μ*b*−1. The findings indicate that there are similar patterns in the correlation between the transverse momentum (*Pt*) and the flow coefficients (v2 and v3) in both Xe–Xe and Pb–Pb collisions. Additionally, the paper estimates the shear viscosity to entropy density ratios of the QGPs by comparing the *Pt* dependence of v2 and v3, as obtained from the experiments, with the calculations derived from relativistic hydrodynamics equations using various shear viscosity to entropy density ratios.

**Keywords** QGP · Perfect liquid · Shape fluctuations · Anisotropic flow · Shear viscosity

#### K. Li

C. Ma (B)

J. Qu

J. Zhang

© The Author(s) 2024

1223

Experimental Middle School, Affiliated to Beijing Normal University, Beijing 100032, China e-mail: Jim\_Leek@hotmail.com

College of Mechanical Engineering, Tianjin University, Tianjin 300254, China e-mail: 3020001568@tju.edu.cn

Beijing 80Th Middle School Wangjing Campus, Beijing 100102, China e-mail: michaelquu070308@icloud.com

Winchester Thurston School, Pittsburgh, PA 15213, USA e-mail: ZhangJ@winchesterthurston.org

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_98

### **1 Introduction**

### *1.1 QGP*

About 15 billion years ago, the Big Bang occurred. The Big Bang created an environment of extremely high temperature and density, which produced a large number of quarks, antiquarks and gluons that interacted strongly to form the quark-gluon plasma, after which the Universe expanded and cooled down to form the material world we now live in.

Free quarks have never been observed, and the theory suggests that quarks and gluons are confined within the hadron by strong interactions. At the same time, the theory predicts that hadrons will be produced out of confinement at extremely high temperatures and high baryon densities, and the QCD phase diagram (Fig. 1) graphically demonstrates that phase transitions of nuclear matter can occur at the extremes of high temperatures and high baryon densities [1].

Experimental methods involving relativistic heavy-ion collisions are employed for the purpose of generating quark-gluon plasmas (QGPs). This involves accelerating two nuclei close to the speed of light and subsequently causing them to collide. The subsequent collision results in high temperatures and pressures, leading to the deconfinement of the nuclei and the formation of a quark-gluon plasma. A visual representation of this process is depicted in Fig. 2, showcasing a cross-section of a relativistic heavy ion pre- and post-collision [2].

Figure 3 illustrates the schematic representation of the evolution of relativistic heavy ion collisions [3]. During the collision of high-speed "discs", the energy

**Fig. 1** The Phases of QCD [1]

**Fig. 2** Schematic cross section before and after relativistic heavy ion collisions [2]

**Fig. 3** A schematic of the evolution of relativistic heavy-ion collisions [3]

density in the collision region rapidly increases, leading to the decoupling of nuclei and the generation of a significant number of quarks, antiquarks, and gluons. The intense interactions between these elementary particles drive the system towards thermal equilibrium, resulting in the formation of a quark-gluon plasma. As the collision system expands and cools down, some of the hadrons undergo hadronization, followed by their subsequent scattering. Eventually, the hadrons escape the reaction system and are detected.

For the past experiments, RHIC (relativistic heavy ion collider) and LHC (large hadron collider) separately imposed <sup>√</sup>*SNN* <sup>=</sup> 200 GeV to *Au* <sup>+</sup> *Au* collision and <sup>√</sup>*SNN* <sup>=</sup> <sup>5</sup>.02 TeV to *Pb*<sup>+</sup> *Pb* collision. Those collisions have revealed horrendous characteristics of the QGP. For example, the most ideal fluid, high vorticity and strong magnetic field.

### *1.2 Anisotropic Flow*

Anisotropic flow is a result of the initial geometry of the hot dense matter produced by heavy ion collisions and is therefore sensitive to the early properties of the evolution of the hot dense matter. The quantification of this property can be achieved through **Fig. 4** Schematic diagram of elliptical flow produced by relativistic heavy ion non-central collisions [4]

the Fourier expansion of the azimuthal distribution of particles with respect to the reaction plane.

where φ is the azimuth of the particle and Ψ*n* is the azimuth of the reaction plane in the laboratory system. The first three coefficients of the Fourier expansion are the direct flows, elliptic flows, and triangle flows, i.e., *v*1 denotes the strength of the direct flows, *v*2 denotes the strength of the elliptic flows, and *v*3 denotes the strength of the triangle flows. *v*3 denotes the strength of triangle flows. In fact, the anisotropic flow originates from the transformation of the anisotropy of the initial space to the anisotropy of each of the azimuthal angles in the momentum space, as shown in Fig. 4 [4].

# **2 Calculation of** *v***2 and** *v***<sup>3</sup>**

In the measurement of elliptic flows using azimuthal correlations, some of the correlations come from correlations other than the initial geometry, such as decays of resonance states, jets, fragmentation of hard scattering partons, and so on (Fig. 5). The strength of the non-flows is considered to be related to the multiplicity of charged particles *N*ch with 1/*N*ch in the two-particle measurement process [5], while in the four-particle measurement process, the contribution from the non-flow is related to the charged particle multiplicity by 1/*N* <sup>3</sup> ch [5].

Therefore, in large-system collisions (e.g., Au–Au, Pb–Pb), the noncurrent contribution is greatly compressed due to the larger particle multiplicity *Nch* produced by the collision. In contrast, during small-system collisions (e.g., proton-proton, protonlead), a relatively large contribution to nonflow is produced due to the smaller number of particles produced by the collisions and thus the relatively large contribution to nonflow. In two-particle correlation measurements of the collective flow, the noncurrent component has an impact on the collective flow size, since the collective flow size is given by the Fourier series expansion of the azimuthal equation. Therefore, twoparticle correlations with a large rapidity interval (Δη) are usually only considered for the purpose of compressing the non-flow (Fig. 6).

The final emitted particles can be expressed within two coordinates: the configuration coordinate (*x*, *y*,*z*), and the momentum coordinate (*Pt* , η, Φ) (Fig. 7).

In order to find the momentum along each direction within the configuration coordinate, we can adopt some simple calculations with measure data. With the transverse momentum *Pt* , and the angle Φ, we can calculate the *x* and *y* components of the total momentum as (3) and (4) shows:

$$P\_x = P\_t \times \cos\ \Phi \tag{1}$$

$$P\_y = P\_t \times \sin\ \Phi \tag{2}$$

With the measured data, η, we can find θ:

$$\eta = -lntan\left(\frac{\theta}{2}\right)\tag{3}$$

$$P\_z = P \times \cos\,\theta\tag{4}$$

$$P = \sqrt{P\_x^2 + P\_y^2 + P\_z^2} \tag{5}$$

Knowing *Px*, *Py* and θ, we can solve this equation set to find the *z* component of the total momentum.

The Fourier expansion of the azimuthal correlation function is Bleibel et al. [6]:

$$\frac{dN}{d\Delta\phi} = N[1 + 2v\_2^2 \cos(2\Delta\phi) + 2v\_3^2 \cos(3\Delta\phi)]\tag{6}$$

The *dN* /*d*Δφ from the open data of ALICE Pb-Pb collision experiment at LHC and the fitted correlation between *dN* /*d*Δφ and Δφ using Eq. (9) are shown in Fig. 8. While taking the first two terms of the Fourier decomposition of the azimuthal distribution function for fitting to obtain Fig. 8a, where v2 = 1.48211340*e*−01; while taking the first three terms for fitting to obtain Fig. 8b, where v3 = 1.48275351*e*−01, v3 = −5.31178879*e* − 02. It could be observed that the curve generated using v<sup>2</sup> and v3 fits the experimental data better than the curve generated using the term v<sup>2</sup> solely. The gaps in the figure are probably caused by the detector's inability to detect particles moving in certain azimuthal angles.

The Fourier coefficients v2 and v3:

$$v\_2 = \langle \cos(2(\phi - \psi\_2)) \rangle \tag{7}$$

$$v\_3 = \langle \cos(\Im(\phi - \psi\_3)) \rangle \tag{8}$$

In (9) and (10), ψ2 and ψ3 represents the minor axis of the elliptic flow and the minor axis of the triangular flow that are shown in Fig. 3.

$$\psi\_2 = \frac{\arctan2\left(\langle r^2 \sin\{2\phi^{part}\} \rangle, \langle r^2 \cos\{2\phi^{part}\} \rangle\right) + \pi}{2} \tag{9}$$

$$\psi\_3 = \frac{\arctan2\left(\langle r^2 \sin\left(3\phi^{part}\right)\rangle, \langle r^2 \cos\left(3\phi^{part}\right)\rangle\right) + \pi}{3} \tag{10}$$

By plugging (9) and (10) into (7) and (8), we can easily approach the values of the elliptic flow and triangular flow, which act as the final conditions of our calculation.

### **3 The Perfect Fluid and Viscosity**

Since the density of QGP is around ten to the twenty times larger than water which makes it hard to compare the viscosity of QGP with other daily fluid, the shear viscosity to entropy density ratio (η/*s*) is introduced. Placing the entropy density at the denominator effectively wipes out the overwhelming density of QGP when comparing the viscous effect. Considering the modern physics perspective, the Kubo formulae can obtain the shear viscosity by analyzing the linear response and fluidgravity coupling [7]:

$$\eta = \lim\_{\omega \to 0} \left( \frac{1}{2\omega} \int dt d\overline{x} e^{i\alpha t} \langle \left[ T\_{\text{xy}}(t, \overline{x}), T\_{\text{xy}}(0, \overline{0}) \right] \rangle \right) \tag{11}$$

This equation can be used to calculate *h*/*s* under the circumstance of the AdS/ CFT correspondence. In order to do that one starts from a CFT with gravity dual. The dual theory is a CFT (conformal field theory) at temperature *T* equal to the Hawking temperature of the black-brane and entropy *S* = *A*/4*GN*. As the graviton is polarized within x–y plane, the imaginary part in Green's function is coupled with the metric. Then we can get [8]:

$$\eta = \frac{\sigma(0)}{16\pi G\_N} \tag{12}$$

where *s*(0) is the graviton absorption cross-section at zero energy.

As the result η/*s* ratio with gravity dual is yielded, the Kovtun, Son and Starinets (KSS) proposed that for a broad range of systems, including those that can only be explained by the quantum field theory, the lower bound of the η/*s* is 1/4π[9] . The experiments that RHIC conducted may have created a significant amount of the collective flow before the plasma area rebound themselves into hadrons. The lowest η/*s* ratio of the collective flow in the QGP area is around 0.08 which is the least value ever presented (Fig. 9).

With the value of the ε2 and ε3, v2 and v3, we can put the η/*s* value and determine if the ratio fits in the situation. The shear viscosity causes the anisotropic deviation from the equilibrium. Its main effect is that it reduces the gap of the expansion along each direction when the QGP is formed by using the shear viscous tensor *πij* to reduce the longitudinal pressure and promotes the transverse pressure.

**Fig. 9** η/s value under different temperature, including water, helium and QGP

Using different η*<sup>s</sup> <sup>S</sup>*, we will obtain different correlation between *pt* , v2, v3. We will determine η*<sup>s</sup> S* by finding out which value of η*<sup>s</sup> <sup>S</sup>*gives the *pt* , v2, v3 dependence that best fits with our experiments. Figure 10a shows the correlation between *Pt* and v<sup>2</sup> and v3 and Fig. 10b shows the correlation between *Pt* and v2 for different <sup>η</sup>*<sup>s</sup> <sup>s</sup>*obtained by solving relativistic hydrodynamic equations (adapted from Schenke et al. [10]).

The *Pt* dependence of v2 and v3 for Pb-Pb collision and Xe-Xe collision with centrality between 40 and 50% is compared in Fig. 10a. It could be observed that the *Pt* dependence of v2 and v3 follows similar patterns.

The range of low transverse momentum (*Pt* < 3*GeV* ), the elliptic flow parameter v2 and the delta flow parameter v3 for Xe–Xe and Pb–Pb collisions both show a monotonically increasing trend with *Pt* , whereas, after the value of *Pt* exceeds 3*GeV* , v2 and v3 for both Pb–Pb and Xe–Xe start to decrease with the increase of the value of Pt and the value of v2 is larger than v3 for both Pb–Pb and Xe–Xe are greater than v3. And for different collisional centripetal and transverse momentum conditions, the values of the elliptic flow parameter v2 decrease as the value of the approximated viscosity coefficient η/*s* increases.

It is also observed that the η/*s* ratio of Pb–Pb and Xe–Xe collisions is approximately 0.08.

### **4 Conclusion**

This paper presents a discussion on the characteristics of anisotropic flow and low viscosity observed in the quark gluon plasma (QGP), which behaves as a perfect fluid. The anisotropic flow of the QGP originates from the conversion of the initial anisotropy in shape-space within the interaction region to the anisotropies in the momentum space of the final-state particles after a nucleus-nucleus collision. These anisotropic flows can be categorized as direct, elliptic, and triangular flows, with their magnitudes determined by the coefficients of the first three orders obtained through Fourier decomposition of the azimuthal distribution function of the final-state particles.

The state information of the end-state particles is described using the transverse momentum (*Pt*), azimuthal angle (Φ), and pseudo-rapidity (η) within a laboratory coordinate system. By using these variables, we can compute the elliptic flow parameter (v2) and the triangular flow parameter (v3).

The results by analyzing public data from the ALICE experiment for Pb–Pb collisions and the CMS experiment for Xe–Xe collisions at the Large Hadron LHC reveal consistent trends in the relationship between transverse momentum (*Pt*) and flow coefficients (v2 and v3) for both Xe–Xe and Pb–Pb collisions. It is estimated that the QGP's shear viscosity to entropy density ratios is approximately 0.08 by comparing the experimentally observed Pt dependence of v2 and v3 with the computed values derived from relativistic hydrodynamics equations utilizing η/*s*.

### **References**

1. Akiba Y et al (2015) The hot QCD white paper: exploring the phases of QCD at RHIC and the LHC


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Thermal Error Modeling Method of Machining Center Linear Axis for Heat Conduction Mechanism**

**Qiangqiang Ding, Shijie Guo, Geng Chen, and Shufeng Tang** 

**Abstract** To address the thermal deformation of machine tool components, a thermal error prediction model based on the ROA-LSSVM network was proposed. First of all, the heat transfer mechanism of the linear feed system was analyzed. By analyzing temperature distribution characteristics during the heat transfer process, the best temperature measurement point position was determined to ensure that the thermal error could be accurately predicted. Secondly, in order to build a prediction model with high accuracy and strong robustness, Raccoon optimization algorithm (ROA) was proposed to optimize the hyperparameters of the least square support vector machine (LSSVM) network model, which was difficult to determine the kernel function and penalty function. Finally, the experiment was measured on a VDL-600A machining center, and the accuracy and practicability of the proposed thermal error prediction model were verified by the thermal deformation in the measurement process. The experimental results show that The ROA-LSSVM model reduces the RMSE by 42% compared with the LSSVM network and 45% compared with the SVM network.

**Keywords** Machining centers · Heat conduction mechanism · Thermal error of linear shaft · ROA-LSSVM model · Prediction of thermal error

Q. Ding · S. Guo (B) · G. Chen · S. Tang

School of Mechanical Engineering, Inner Mongolia University of Technology, Huhhot 010051, China e-mail: sjguo@imut.edu.cn

Q. Ding e-mail: 1466743737@qq.com

G. Chen e-mail: 1442921605@qq.com

S. Tang e-mail: tangshufeng@imut.edu.cn

© The Author(s) 2024

1235

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_99

### **1 Introduction**

CNC machine tools are known as "industrial mother-machines", which significantly impact manufacturing advancement [1]. Many studies have shown that the influence of thermal error on the machining accuracy of machine tools is as high as 40–70% [2]. Therefore, it is important to reduce the influence of thermal error on the machining accuracy of the machine tool.

Researchers [3–6] at home and abroad have conducted in-depth research on thermal error compensation technology. To effectively implement thermal error compensation, it is crucial to establish a precise thermal error model. Scholars [7–10] generally use neural networks, support vector machines, multiple regression and grey theory to establish thermal error models. however, a substantial number of temperature sensors must be strategically positioned during testing, and the subsequent sncreeing of heat sensitive points should be carried out.

However, there are many problems in the selection of conventional temperature measurement points, more sensor wiring and collinearity between measurement points, therefore, to address these issues, this study examines the heat source of the linear feed system using the heat transfer mechanism and identifies the optimal thermal sensitive point to enhance the precision of the thermal error model. Moreover, A new algorithm (ROA) combined with the thermal error prediction model of least square support vector Machine (LSSVM) is proposed to optimize the kernel function, penalty function and relaxation variables of the network. Finally, thermal error measurement tests are carried out on a VDL-600A vertical machining center to verify the reliability of the proposed method.

# **2 Temperature Field Analysis of Feed System Based on Heat Conduction Mechanism**

### *2.1 Temperature Field Analysis of the Feed System*

As depicted in Fig. 1, the ball screw is dispersed into m lead screw units S*i* (*i* = 1 2… m), each segment length is S, the length of the non-moving end of the lead screw motor end is S0, and the length of the non-moving end of the other side is SM. the heat transfer mode is shown in Fig. 2.

Take Δ*s* within the travel range of the lead screw, and the energy conservation relationship within Δ*t* time is:

$$
\Delta U = \mathcal{Q}\_f(\mathbf{s}, t) + \mathcal{Q}\_{cd}(\mathbf{s}, t) + \mathcal{Q}\_{cv}(\mathbf{s}, t) + \mathcal{Q}\_r(\mathbf{s}, t) \tag{1}
$$

where, Δ*U* represents the change of internal energy in the range of Δ*s*; *Qf* (*s,t*) represents the friction heat generated by the nut; *Qcd* (*s,t*) represents the heat conduction of the micro end of the lead screw Δ*s* to the adjacent sides; *Qcv*(*s,t*) represents the

**Fig. 1** Structureof the feed system

**Fig. 2** Lead screw heat transfer diagram

heat pair flow between the lead screw and the surrounding air in the state of motion; *Qr*(*s,t*) represents the amount of thermal radiation between Δ*s* and the surrounding air.

Take a *Si* segment of the lead screw for analysis, and the change in internal energy of the lead screw between time (*t*−Δ*t*, *t*) is::

$$
\Delta U(\mathbf{s}, t) = c \rho \pi R^2 \frac{\partial T(\mathbf{s}, t)}{4 \partial t} \Delta \mathbf{s} \Delta t \tag{2}
$$

where, *c* is the specific heat capacity and the value is 460 J/kg · °C; ρ indicates the density of the lead screw, the value is 7850 (kg/m3); *R* is the screw diameter value of 40 mm, *Qf* (*s, t*) calculation formula is as follows:

$$Q\_{\rm f}(\mathbf{s}, t) = \frac{\pi \, R^2}{4} q\_f(\mathbf{s}, t) \Delta \mathbf{s} \Delta t \tag{3}$$

where, *qf* (*s,t*) represents the average heat generated within the travel of the lead screw, and its value is 42 W.

The calculation for heat conductivity *Qcd* (*s, t*) is as follows:

$$\mathcal{Q}\_{cd}(\mathbf{s},t) = \pi \,\mathrm{R}^2 k \frac{\partial \frac{\partial T(\mathbf{s},t)}{\partial \mathbf{s}}}{4\partial \mathbf{s}} \Delta \mathbf{s} \,\Delta t \,\tag{4}$$

where, *k* value is 71.568 (W/m · °C), The formula for *Qr* is:

$$\mathcal{Q}\_r(\mathbf{s}, t) = \varepsilon \pi \,\mathrm{Re}\_r(T^4(\mathbf{s}, t) - T\_a^4(t)) \Delta \mathbf{s} \,\Delta t \tag{5}$$

where, *Ta*(*t*) represents the ambient temperature around the lead screw at time *t*. the value of *cr* is 5.667 × 10–8 W/(m2 · K4). The calculation formula of *Qcv*(*s,t*) is as follows:

$$\mathcal{Q}\_{cv}(\mathbf{s}, t) = \pi \, R h\_{cv}(T(\mathbf{s}, t) - T\_a(t)) \Delta \mathbf{s} \Delta t \tag{6}$$

where, *hcv* value is 60.47 (W/m · °C). after calculation and analysis, considering the influence of heat source on bearing seat with fixed end, *T*(0, *t*) = *TS*<sup>0</sup>−<sup>τ</sup> *<sup>j</sup>*is required; considering the influence of free end bearing seat, *T*(*L, t*) = *TSM*−<sup>τ</sup> *<sup>j</sup>*, and *Ta*(*t*) = *Ta*−<sup>τ</sup> *<sup>j</sup>*should be guaranteed.

### **3 Thermal Error Measurement Experiment of Feed Shaft**

### *3.1 Measurement of Thermal Error of Linear Feed Shaft*

This paper takes the X-axis of machining center of VDL-600A vertical milling machine as the research object to measure the thermal positioning error of linear feed shaft of machine tool. field measurement is shown in Fig. 3, the location of the temperature sensor is shown in Fig. 4.

In the process of experimental measurement, the NI-5910 temperature acquisition system was used for temperature collection, and the XL-80 laser interferometer was used for displacement collection.

The specific test process is as follows:


**Fig. 3** Experimental site diagram

**Fig. 4** Location of the temperature sensor

In this experiment, three kinds of experiments at different constant speed are designed to investigate the impact of temperature on the thermal error of the feed system at various speeds.

Table 1 displayed the test conditions in detail.


#### **Table 1** Experimental design

### *3.2 Results of Thermal Error Measurement*

Taking a measurement result as an example, The collected data includes temperature values and corresponding thermal error values as follows.

As can be seen from Fig. 5, With the increase of time, the temperature change of the bearing seat at the fixed end increases the most, reaching a maximum of 33.6 °C, while the temperature rise of the free end changes little. and finally becomes stable after 220 min. It can be seen from Fig. 6 that the displacement curve and temperature change rule of the machine tool are basically the same during operation. With the passage of time, the thermal positioning error increases gradually. The displacement caused by heat tends to be stable after 200 min and reaches a maximum of 70 µm.

### **4 Heat Error Modeling with Raccoon-Optimized LSSVM**

LSSVM is more sensitive to noise during model training, and the processing of noise is relatively poor. This may cause the training results to be affected by noise and the generalization performance is poor. Therefore, this paper uses raccoon optimization algorithm (ROA) to address the aforementioned challenges, Through the

**Fig. 6** Thermal error curve

iterative process, the algorithm can search the optimal parameter combination of the least squares support vector machine model, thus improving the accuracy and generalization ability of the model.

### **5 Model Prediction and Evaluation**

With the temperature variable and position of the best thermal sensitive points were determined as the input, and the established ROA-LSSVM model, LSSVM model and SVM neural network model were used to predict the thermal positioning error of the X feed axis. Thermal errors of X axis in 60 and 240 min predicted results and residual errors of each network model are shown in Fig. 7:

As Fig. 7 illustrates, it is evident that the predicted residual value of SVM network model is 10.9 µm at most. The maximum residual predicted by LSSVM model is 8.6 µm. After adjusting the LSSVM model using ROA algorithm, the fluctuation becomes significantly smaller, and the maximum residual is 1.9 µm.

The model's goodness of fit was assessed according to the root-mean-square error RMSE, fitting coefficient R2, mean absolute error MAE and mean deviation MBE. The goodness of fit of each model is shown in Table 2.

As indicated in Table 2, Compared with SVM network, the RMSE of ROA-LSSVM model is reduced by 45%. Compared with the LSSVM network, the RMSE of the ROA-LSSVM model is reduced by 42%. It is proved that the thermal error prediction model of CNC machine tool linear axis based on ROA-LSSVM network has higher prediction accuracy.

**Fig. 7** Comparison of thermal error prediction results

(a)ROA-LSSVM network prediction results.

(c) SVM network prediction results.


**Table 2** Goodness of Fit evaluation of each network

### **6 Conclusions**


**Acknowledgements** Foundation Project: National Natural Science Foundation of China (52065053); Basic Research Funds for Universities directly under the Inner Mongolia Autonomous Region (ZTY2023005).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Tolerance Analysis Method of Aircraft Door Sealing Structure Based on Linear Simplified Model**

**Jinlin Ke, Xiongfei Lv, and Cuncun Jiang** 

**Abstract** As a moving component of the aircraft, the sealing state of the door will directly affect the aircraft's pressurization level and aerodynamic performance. The existing analysis methods of sealing structures mainly consider the compression deformation and contact stress of the seals. However, in practical engineering applications, it has been found that the sealing performance of cabin door depends on the interaction between the sealing structure and the seal. On the one hand, the tolerance fluctuation and elastic deformation of the sealing structure will cause the deviation between the actual compression ratio of the seal from the theoretical value; On the other hand, the deformation caused by the resilience of the seal will change the tolerance distribution of the sealing structure. In this paper, based on the local linear simplification of loading curve of the seal, the calculation efficiency of the large-size sealing structure's deformation is improved, so that the compression ratio fluctuation of the seal is converted into the deformation tolerance which is included in the tolerance simulation of the sealing structures to obtain the distribution characteristics of step difference value on the aerodynamic surface as well as the different compression states of the seal. The validity of this analysis method is confirmed by comparing with the commercial software calculation and the practical measured data of the aircraft door.

**Keywords** Sealing performance · Linear simplification · Deformation analysis · Tolerance analysis

X. Lv e-mail: lvxiongfei@comac.cc

C. Jiang e-mail: jiangcuncun@comac.cc

© The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_100

1245

J. Ke (B) · X. Lv · C. Jiang Aircraft Structure and Stress Division, Shanghai Aircraft Design and Research Institute, Shanghai 201210, China e-mail: kejinlin@comac.cc

### **1 Introduction**

Aircraft doors are the accesses for personnel to enter and exit the cabin as well as for equipment maintenance. While the inadequate compression ratio of the seal or its separation from the sealing structure can lead to a decline in cabin sealing performance, potentially resulting in aircraft pressure loss and even posing serious flight hazards [1]. On the contrary, an excessively high compression ratio will lead to some problems such as increase in the operation force of the door, decrease in the service life of the seal and deterioration of aerodynamic surface flatness which is usually seen as a step difference between the door skin and the fuselage skin. Therefore, the seal properties and the relative deformation of the sealing structure have always been important factors in the design of the airframe structure. The existing structural sealing performance analysis is more concerned with the mechanical properties and compression state of the seal [2, 3], but in engineering applications, the state of the seal often has a coupling effect with the sealing structure. On the one hand, due to the elastic deformation of the sealing structure as well as the manufacturing and installation tolerance, the compression state of the seal will deviate from the theoretical design state in varying degrees. On the other hand, the seal under different compression states will cause the fluctuations in the deformation of the sealing structure. Therefore, the sealing performance of the door is determined by both.

The seal usually adopts a composite structure of fabric and rubber that is why traditional commercial software requires more calculation time for the deformation and contact analysis of the seal and sealing structure. In engineering applications, typical section analysis [2] or integral modelling analysis methods [4] are often used, while the former has a faster analysis speed, but its applicability is limited and not suitable for complex structures; The latter divides the structure and seal into 3D elements for calculation directly which has strong universality, but low computational efficiency and high hardware requirements. In order to balance calculation accuracy and analysis efficiency, a simplified analysis method applicable to large-size sealing structure of the door was proposed in reference [5]. While for the sealing structure, the tolerance not only leads to the compression ratio dispersion of the seal, but also affected by the seal will causes varying degrees of step difference problems between the door and the fuselage. Since Monte–Carlo method has been widely used in the analysis of tolerance characters of structural and mechanism [6, 7], this method can be used to analyse the relative relationship between parts tolerance and structural deformation at the door area.

In this paper, local linear simplification of the seal is firstly carried out based on the fitting of the loading curve, then the tolerance related to the compression of the seal is converted into the deformation tolerance of the structure. Furthermore, this deformation tolerance is incorporated into the step difference dimension chain formula between the door and the fuselage by Monte–Carlo method. The difference of the compression state of the seal is also analysed through the simulation of the example. Finally, the Bootstrap method is adopted to expand the measured data of the step difference values at different points of the door which can be used to analyse the distribution characteristics of practical case and verify the effectiveness of this method by comparing with the theoretical calculation results.

# **2 Linear Simplification of Seal and Deformation Analysis of Structure**

# *2.1 The Relationship Between Mesh Size and Calculation Accuracy of the Seal*

The computational efficiency and convergence will encounter many problems because of the non-linearity of materials and the contact algorithm when calculating the deformation of the sealing structure at large-sized level. Although using methods such as segmented modelling [4] can improve computational efficiency, the analysis strategy of 3D elements still restricts the selection of the minimum mesh size for the seal while the size affects the accuracy of the seal deformation and contact analysis. Taking an Ω-shape seal as an example, its section geometric dimension is shown in Fig. 1.a. which with a length of 140 mm. The uniaxial compression test data of its rubber material is shown in reference [1]. The Ogden model is used to describe the constitutive relationship of this rubber material, and the plane-strain model is used for analysis. The loading curves of the seal under different mesh sizes are calculated that are shown in Fig. 1.b.

As can be seen from Fig. 1b, with the increase of the compression value of the seal, the difference of the resilience calculated by different mesh sizes gradually increases, and the maximum deviation ratio increases to 32.63% at 9.5 mm. The compression ratio of aircraft door seals generally set in range 25–30% which means the corresponding resilience deviation ratio is 17.89–19.12% if adopting this Ωshape seal. Therefore, when analysing the deformation of the sealing structure, no matter plan elements or 3d elements are used, the non-fine mesh will produce a

**Fig. 1** Calculating results of the seal (**a** model of compression; **b** loading curves)

large resilience deviation, while the excessively fine mesh has an extremely high requirement of the calculation step difference size during the contact process.

### *2.2 Linear Simplified Modeling Method for Sealing Structure*

To solve the above problems, the simplified seal unit can be obtained by fitting the nonlinear loading curve shown in Fig. 1 [5]. Moreover, by locally fitting the estimated compression range of the seal, sufficient accuracy can be obtained even if it is just linear fitting.

The local linearization function of the loading curve of the seal with unit length is:

$$F = kx + c \tag{1}$$

where *x* is the compression value of the seal, *F* is the corresponding resilience, *k*  and *c* are the coefficients of fitting function. The energy *Us* which is released by the seal before and after deform-ation in the door can be expressed as:

$$U\_s = \int\_{\Omega} \int\_{x\_0 - \Delta u}^{x\_0} (kx + c) dx d\Omega \tag{2}$$

*x*0 and Δ*u* indicate the initial and the change of the compression value respectively, Ω is the area where the seal is deployed. Assuming that the seal and sealing structure remain contacting before and after deformation, which means there is no separation or penetration between them. After discretizing the sealing structure with finite element, the compression value change Δ*u* can be expressed by node displacement vector **u***<sup>e</sup>* and the shape function **N** of the sealing structure that is generally consist of holder component and compressor for the seal, using **u***<sup>e</sup> <sup>A</sup>*indicate the node displace vector for the holder and **u***<sup>e</sup> <sup>C</sup>*for the compressor, then the relationship between these vectors can be get:

$$\begin{cases} \mathbf{u}^{\epsilon} = \left[ \mathbf{u}\_{C}^{\epsilon} \, \mathbf{u}\_{A}^{\epsilon} \right]^{T} \\ \Delta u = \mathbf{N} \mathbf{u}\_{C}^{\epsilon} - \mathbf{N} \mathbf{u}\_{A}^{\epsilon} = \mathbf{N} \Big[ \mathbf{I} - \mathbf{I} \Big] \mathbf{u}^{\epsilon} \end{cases} \tag{3}$$

where **I** is the unit matrix. Combinate Eq. (2) with (3) and using the variational calculus of **u***<sup>e</sup>*, it can be seen:

$$\delta U\_s = \delta \mathbf{u}^{\mathbf{e}\mathbf{T}} \sum\_{el\epsilon} \int\_{\Omega\_{\epsilon}} (k\mathbf{x}\_0 + c) \mathbf{N}^{\mathbf{T}} \left[\mathbf{I} - \mathbf{I}\right]^T - k \mathbf{N}^{\mathbf{T}} \mathbf{N} \begin{bmatrix} \mathbf{I} & -\mathbf{I} \\ -\mathbf{I} & \mathbf{I} \end{bmatrix} \mathbf{u}^{\epsilon} d\Omega\_{\epsilon} \tag{4}$$

Through equation above, the effect from the seal to the sealing structure is equivalent to adding a pair of generalized forces which have the same magnitude but opposite direction. The magnitude expression is:

$$\mathbf{F}\_{\mathbf{s}} = \sum\_{ele} \left( \int\_{\Omega^{\varepsilon}} \mathbf{N}^{\mathbf{T}}(k\mathbf{x}\_{0} + c) d\Omega^{\varepsilon} - \int\_{\Omega^{\varepsilon}} k \mathbf{N}^{\mathbf{T}} \mathbf{N} d\Omega^{\varepsilon} \left(\mathbf{u}\_{C}^{\varepsilon} - \mathbf{u}\_{A}^{\varepsilon}\right) \right) \tag{5}$$

where Ω*e* is the elements that the seal is deployed and *ele* is the number of elements.

# **3 Tolerance Analysis Method of Sealing Structure Considering Deformation**

Taking the typical aircraft door structure which is shown in Fig. 2.a. as an example: Two hinge arms are hinged to the fuselage and when the door is closed in position, the lug is also hinged to the fuselage. The boundary size of the door is 180 × 180 mm, and all the parts are made of aluminium alloy except for the seal which is deployed around the quadrilateral of the door panel. The section and material are the same as the seal shown in Fig. 1.a. In this model, the preset seal's compression state which the value is 4.3 mm is characterized by setting the interference between the door and the seal. The outer edge of the door panel that is also the compressor of the seal is 1.27 mm thick. The thickness of the holder is 2.03 mm, and the thickness of the fuselage skin is 1.52 mm. And the position relationship between the sealing structure and the seal is shown in Fig. 2b. The left side is the fuselage, and the right side is the door structure. The surface contour of the skin edges on both sides will directly affect the external step difference. At the same time, the positioning dimension tolerance of the seal's holder component at the left side will affect the compression ratio of the seal which will change the deformation state of the cabin door.

Due to the profile tolerance of a surface is the upper and lower limits of a series of spherical diameters that are tolerance values *t* [8], combined with the linear simplified analysis method of sealing structure, the Monte-Carlo method can be used to simulate the deformation of sealing structure and the influence of skin contour deviations on the step difference as well as the relative distribution characteristic.

**Fig. 2** The schematic diagram of structure (**a** model of door; **b** section of sealing structure)

In the Fig. 2b. the profile of the fuselage skin edge and the door panel is *F*1 = 0.00 ± 0.80 and *F*2 = 0.00 ± 0.70 respectively, while the dimension requirement between holder and compressor is *F*3 = 12.60 ± 0.60. Then the dimension chain calculation formula of the step difference between the two skin is:

$$F\_0 = G(F\_3) + F\_2 - F\_1 \tag{6}$$

The above equation *G*(*F*3) represents the deformation tolerance of the door skin caused by the resilience of the seal which is a function of the installation size *F*3. In order to obtain the distribution characteristics of the deformation tolerance, FEM analysis of the seal structure shown in Fig. 2 was carried out in which the seal was adopted the linear simplified model, and the analysis software was ABAQUS.

Except the hinge arms are modelled with C3D4 element, all other parts are modelled with 4-node/5-DOF shell element. Due to the independent interpolation of the translational and rotational DOF of the Mindlin shell element [9], the shape functions of shell element which is related to the normal displacement is relatively simple. Assuming the seal loads apply on the seal structure is line contact pressure, and incorporating the shape function into Eq. (5), it can be concluded that:

$$\mathbf{F}\_s = \sum \left( \begin{bmatrix} \frac{1}{2} l^e q \\ \frac{1}{2} l^e q \end{bmatrix} - \frac{kl^e}{6} \begin{bmatrix} 2 \ 1 \\ 1 \ 2 \end{bmatrix} \begin{pmatrix} \Delta \mathbf{u}\_1 \\ \Delta \mathbf{u}\_2 \end{pmatrix} \right) \tag{7}$$

where Δu1 = *uC*1 − *u A*1 and Δu2 = *uC*2 − *u A*2, the *l <sup>e</sup>*is the length of element, (*u*∗<sup>1</sup>, *u*∗<sup>2</sup>) *<sup>T</sup>*indicate node displacement of the elements at both ends of the holder and the compressor. From Eq. (5), it can be seen that the loading effect of the seal can be regarded as two parts. First is the uniformly normal load corresponding to the initial compression value, and the other is the supports stiffness on the nodes. Furthermore, the second support stiffness on the right in Eq. (7) can be decomposed into:

$$\frac{kl^{\epsilon}}{6} \begin{bmatrix} 2 \ 1 \\ 1 \ 2 \end{bmatrix} \begin{pmatrix} \Delta u\_1 \\ \Delta u\_2 \end{pmatrix} = \frac{kl^{\epsilon}}{6} \begin{bmatrix} 3 \ 0 \\ 0 \ 3 \end{bmatrix} \begin{pmatrix} \Delta u\_1 \\ \Delta u\_2 \end{pmatrix} + \frac{kl^{\epsilon}}{6} \begin{pmatrix} \Delta u\_2 - \Delta u\_1 \\ \Delta u\_1 - \Delta u\_2 \end{pmatrix} \tag{8}$$

The first item on the right of Eq. (8) can be realized by applying a spring element at the corresponding node between the holder and compressor elements. While the second item is the coupling of Δ*u*1 and Δ*u*2 which can be realized through MPC element. However, for the purpose of simplifying the analysis model, this item can be ignored when the mesh size is small enough due to the magnitude of the difference between Δ*u*1 and Δ*u*2 is smaller compared with themselves.

The deformation analysis of this structure adopts a progressive local linearization method, which is first using linear fitting of the loading curve that the mesh size is 0.5 mm in Fig. 1 within a large interval. After preliminary analysis of the finite element model, the deformation interval of the seal is obtained, and then updated the linearization simplified element by fitting the deformation interval to obtain the final structural deformation result. In theoretical state (compression value is 4.3 mm)

After the upper limit of 4.9 mm and lower limit of 3.7 mm compression of the seal are brought in, the minimum step difference is calculated as *G*(*F*3) min <sup>=</sup> <sup>0</sup>.12<sup>+</sup>0.<sup>03</sup> <sup>−</sup>0.02, and the maximum is *G*(*F*3) max <sup>=</sup> <sup>0</sup>.98<sup>+</sup>0.<sup>15</sup> <sup>−</sup>0.14. Usually, the positioning tolerance *F*<sup>3</sup> follows a normal distribution, while the absolute values of upper and lower limits of deformation tolerance *G*(*F*3) are close to each other so the normal distribution can also be used for simulation. Relevant parameters are brought into Eq. (6) and Monte-Carlo method is used to simulate step difference for 3000 times. Then the maximum/minimum value of step difference as well as the probability distribution of step difference without seals are obtained, as shown in Fig. 4.

Due to the deformation of the sealing structure, the mean value of step difference distribution of the door is uneven around the edge, and the position tolerance of the holder *F*3 is compensated by the deformation of the seal at the cost of 1.2 mm compression fluctuation of the seal. Combined with the deformation of the sealing structure, the actual compression ratio range of the seal is 2.76–4.53 mm which means the fluctuation of the seal compression ratio archives 9.83%. The corresponding

**Fig. 5** Contact pressure of the seal

contact stress of the seal at the minimum and maximum compression state is shown in Fig. 5. It shown that due to the deformation and positioning tolerance of the seal structure, the maximum contact stress of the seal is reduced from 4.59 MPa to 2.73 MPa, with a reduction ratio of 40.52%. By improving the stiffness of the seal structure and the installation and positioning accuracy, the contact stress loss can be reduced.

### **4 Comparison with Measured Data**

The structural design and analysis of an airliner APU door [10] had been discussed in reference [11], but it did not discuss the rebound effect of seals on the edge of the composite door which was further analysed in the reference [5]. However, due to the manufacturing tolerance of the door components, there was a certain deviation between the analysis results and the physical data. At the same time, restricted by the number of samples, the validity of analysis results needs to be further discussed and confirmed. The configuration of the door is shown in Fig. 6.

The composite material parameters are consistent with reference [11] which is carbon fibre and benzoxazine resin composites solidified at 185 °C. And the core material is the same as the in reference [5] which is GILLCORE HD 332O type aryl-paper honeycomb. Take the lower door as the analysis object in consideration of symmetry configuration. The analysis software is NASTRAN with CQUAD4 element used for the inner and outer skins as well as laminated plate. And CHEXA/ CPENTA solid element used for the honeycomb core. The door is hinged to the

fuselage through the titanium hinges and in the middle of these two doors, four shear pins fix the door to the fuselage when the door is closed.

As the seal holders are made of alloy material and directly connected to the frame of the fuselage, the stiffness is stronger than the composite panel, so only the deformation of the door is considered in the analysis. The local linear fitting equation of the seal and the calculation results of the deformation of the door are given in reference [5] in which the loading curve of the seal is obtained by experiment. Fitted within 0.6–5.0 mm, it can obtain that *F* = 0.184*x* + 0.022 while the initial compression *x*0 = 4.60 mm. The results of the door deformation are shown in Fig. 7 in which large deformation occurs in the two-corner area, with the maximum deformation value of 3.75 mm in the right corner and 3.03 mm in the left corner.

The sealing structure for this door is similarly with Fig. 2 [11], the profile of the fuselage skin edge and the door panel is *F*1 = 0.00 ± 1.00 and *F*2 = 0.00 ± 0.75 respectively, while the dimension requirement between holder and compressor is

**Fig. 7** The deformation contour plot of the APU door


*F*3 = 12.30 ± 0.60 through which the lower edge deformation of the door left corner is calculated as *GL* (*F*3) <sup>=</sup> <sup>3</sup>.07<sup>+</sup>0.<sup>55</sup> <sup>−</sup>0.34 and the right corner deformation is *G R*(*F*3) <sup>=</sup> <sup>3</sup>.75+0.<sup>54</sup> <sup>−</sup>0.48。

As *GL* (*F*3) and *G R*(*F*3) is presenting right skew distribution while the square root of the data is approximately normal distribution. Therefore, convert the Eq. (6) into:

$$F\_0 = \left(\sqrt{G(F\_3)}\right)^2 + F\_2 - F\_1 \tag{9}$$

Monte-Carlo method is used to simulate Eq. (9) by 5000 times, and the statistical information of the step difference distribution of two corner points are listed in Table 1. The probability density distribution of step difference at these two locations is shown in Fig. 8.

As a comparison, the actual measurement data of step difference of the left and right corner are given in Table 2.

**Fig. 8** Distribution of the step difference at two corners


**Table 2** Measured data of two corners' step difference (unit: mm)


**Table 3** Static information of step difference distribution

Since the sample size of the measured data is relatively small, assuming that the step difference of the left and right corner follows the normal distribution then the parametric Bootstrap method [12–14] is used to resampling the step difference sample, and the confidence estimation of the distribution parameters is performed. The number of Bootstrap resample size at each corner is 2000 and the distribution parameters of the Bootstrap group of the step difference is shown in Table 3 which the confidence probability is 99%.

By comparing the data in Tables 1 and 3, the average value deviation of the average value and the theoretical analysis is 0.29 mm at left corner and 0.46 mm at right corner respectively, while the average value deviation of the standard deviation is 0.38 and 0.01 relatively. The static character data obtain by simulation are all in the confidence interval and the deviation of the average value is sufficiently accurate for estimating in engineering cases which shows there is a good agreement between theoretical analysis and actual measure data.

### **5 Conclusion**

In this paper, the linear simplified method of the seal and Monte-Carlo method are used to analyse the factors affecting the deformation and step difference of the sealing structure for aircraft cabin door during which the finite element modelling strategy and calculation method are expounded. And the validity of this analysis method is verified by the actual data which static information is obtained by Bootstrap estimate method. The conclusions are as follows:

(1) The loading curve of the seal is variated in the different analysis mesh sizes. And the linear simplified method of seal can used to analysis the deformation of large-size level sealing structure. Duo to the loading curve of the seal can be derived from planar structure simulation or experimental measurement, it won't increase the calculation cost of the door seal structure compared to the traditional 3D finite element model.


# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Experimental Study on the Impact of Thermal Stress on Aircraft Structural Performance**

**Xiaoli Bo** 

**Abstract** The local heat source of a civil aircraft affects the temperature of the lower panel structure of the center wing. In this paper, a local heat source is arranged on the lower panel structure of the center wing. The strain of the center wing lower panel skin under different temperature gradients under local thermal loads and the corresponding temperature gradients around the heat source were investigated. The natural mesh model and the fine finite element model were used to analyze the thermal load of the center wing lower panel. The temperature on the node was applied according to the real temperature in the test, and the analysis results were compared with the test values. After comparative analysis, the strain value of the natural mesh model has a high degree of fit with the test value, and the analysis results of the natural mesh model can be used to analyze the thermal stress intensity of the center wing lower panel.

**Keywords** Thermal stress · Aircraft · Experimental study

# **1 General**

The ambient temperature envelope of civil aircraft is large, and the influence of thermal stress on its static strength and fatigue strength cannot be ignored [1, 2]. Usually the calculation of the skin surface temperature field often only considers the external pneumatic environment, which is not consistent with the actual skin overall thermal environment. Therefore, it is necessary to establish a reasonable thermal analysis model and calculation method to analyze the skin temperature field and influencing factors [3–5].

© The Author(s) 2024

1259

X. Bo (B)

Shanghai Aircraft Design and Research Institute of COMAC, Shanghai 201210, China e-mail: boxiaoli@comac.cc

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_101

# **2 Test Data**

The lower panel of the center wing of a civil aircraft is formed by connecting the skin of 2024HDT-T351 and the stringer of 2026-T3511 through fasteners, and 15 stringers are arranged in total (the number from the front spar to the rear spar is stringer 1 to stringer 15 respectively). The heading length is 2890 mm, the span width is 3664 mm.

This test mainly investigates the strain of the center wing lower panel skin under different temperature gradients under local thermal load. The investigation area is 1000 mm × 560 mm, as shown in Fig. 1. The circle in the inspection area is a schematic diagram of the location of the heat source (heat lamp used in the test).

Distributed optical fiber sensors are arranged in the main inspection area of the center wing lower panel for temperature field and strain field reconstruction, and fiber grating sensors (FBG) are arranged on the central skin of the main inspection area for point-type high-frequency real-time measurement, as shown in Fig. 2. The monitored strains are all wing spanwise.

In the test, a heat lamp was used to irradiate and load, as shown in Fig. 3. During the test, a total of 6 heat lamps were used and three heating-cooling cycles were performed. The position of the heat lamp is the same in the first two cycles, and the position of the heat lamp is slightly adjusted in the third cycle, so that the temperature of the fiber grating sensor (FBG) at the center skin of the monitoring area increases.

The temperature loading and unloading process curve of the three cycles is shown in Fig. 4, and the thermal strain change curve of the inner and outer surfaces is shown in Fig. 5.

As can be seen from Figs. 4 and 5, the outer surface temperature of the skin in the first two cycles is up to about 72 °C. In the third cycle, the position of the heat lamp is adjusted so that the heat lamp directly hits the central area of the test, that is,

**Fig. 2** Monitoring area and sensor arrangement

**Fig. 3** Hot lamp loading photo

**Fig. 4** Heating cycle temperature change curve

**Fig. 5** Heating cyclic strain change curve

the position of the FBG sensor, so The maximum temperature reaches about 80 °C, while the maximum temperature of the inner skin is about 60 °C. Correspondingly, the maximum thermal strain of the outer skin in the first two cycles is about 544 με, the maximum thermal strain of the inner skin is about 444 με, the maximum thermal strain of the outer skin in the third cycle is about 552 με, and the maximum thermal strain of the inner skin is about 452 με.

According to the temperature-strain change data of the inner and outer surfaces of the three cycles, the corresponding graphs are drawn as shown in Figs. 6, 7, 8, 9, and 10.

It can be seen from Figs. 6, 7, and 8 that in all three cycles, the inner surface temperature-strain curve has a good linearity, while the outer surface is a typical

**Fig. 6** Temperature-strain change curve of inner and outer surfaces in the first cycle

**Fig. 7** Temperature-strain curves of inner and outer surfaces in the second cycle

**Fig. 8** Temperature-strain change curve of inner and outer surfaces in the third cycle

hysteresis loop. The reason for the analysis may be that the outer surface is directly irradiated by a heat lamp, which is more sensitive to temperature, causing the temperature to rise and fall faster than the structural strain change rate.

As can be seen from Figs. 9 and 10, the three cycle cooling processes on the outer surface have good linearity and repeatability, and the heating process is different due to the different position of the heat lamp; except for the first cycle heating process, the rest of the heating and cooling processes on the inner surface have good linearity and repeatability.

The temperature distribution results of the three cycles of the distributed optical fiber sensor are shown in Figs. 11, 12, and 13.

The thermal strain distribution results of the three cycle temperatures of the distributed optical fiber sensor are shown in Figs. 14, 15, and 16.

**Fig. 9** Temperature-strain variation curves of three cycles on the outer surface

**Fig. 10** Temperature-strain variation curves of three cycles on the inner surface

**Fig. 11** Outboard temperature results of center wing lower panel (first cycle)

**Fig. 12** Outboard temperature results of center wing lower panel (second cycle)

**Fig. 13** Outboard temperature results of center wing lower panel (third cycle)

**Fig. 14** Thermal strain results of center wing lower panel (first cycle)

**Fig. 15** Thermal strain results of center wing lower panel (second cycle)

**Fig. 16** Thermal strain results of center wing lower panel (third cycle)

### **3 Finite element analysis**

A natural mesh model and a fine finite element model are used to analyze the thermal load of the center wing lower panel. The temperature on the node is applied according to the real temperature in the test. Among them, the temperature applied at the nodes in the natural mesh model (GFEM0, GFEM1) is taken from the average value of the temperature of the fiber measuring point at the adjacent position of the node in the test, as shown in Fig. 17; the fine model (DFEM0) The temperature applied at the node is basically consistent with the temperature of the fiber measuring point at the corresponding position of the node in the test, as shown in Fig. 18.

As shown in Figs. 19 and 20, the strain value of the natural mesh model (GFEM0) has a high degree of fit with the test value.

**Fig. 17** Schematic diagram of temperature applied at nodes in natural mesh model of center wing lower panel

**Fig. 18** Temperature applied at nodes in fine model of center wing lower panel

**Fig. 19** Strain comparison of center wing lower panel at monitoring area 4

**Fig. 20** Strain comparison of center wing lower panel at monitoring area 6

# **4 Conclusions**

After comparative analysis, the strain value of the natural mesh model (GFEM0) has a high degree of fit with the test value, and the analysis results of the natural mesh model can be used to analyze the thermal stress intensity of the center wing lower panel.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Analysis of Deformation in Aircraft Skin Vacuum Adsorption Clamping**

#### **Wenhao Xing, Aimin Wang, Long Wu, Baode Xu, Jiayu Zhang, and Yuan Yu**

**Abstract** Large lightweight alloy skin panels are extensively utilized in the aerospace industry and serve as crucial components constituting the outer shells of aircraft, launch vehicles, manned spacecraft, and other equipment. However, due to their thin-walled nature and limited stiffness, they are susceptible to clamping deformation and machining-induced deformations. Vacuum adsorption technology is widely employed in aircraft manufacturing to mitigate part deformation during machining through the utilization of profiling molds. However, to achieve the milling and drilling tasks in a single clamping process, it is necessary to reserve drilling positions in the vacuum adsorption mold. Unfortunately, this leads to significant deformation of the aircraft skin at the reserved drilling positions during processing. This article utilizes finite element simulation technology to analyze the deformation of the aircraft skin during the vacuum adsorption clamping process. The simulation results indicate that the maximum deformation reaches 5.602 mm, which primarily occurs at the middle hole of the vacuum adsorption mold. To address this issue, the article proposes a solution of adding sealing strips around the reserved holes. This solution effectively reduces the deformation of the skin during the processing stage.

**Keywords** Thin walled skin · Vacuum adsorption · Clamping deformation · FEM · Clamping optimization

# **1 Introduction**

The primary function of skin panels is to maintain the shape of the aircraft's outer shell and ensure optimal aerodynamic performance. To enhance structural strength, reinforcing ribs are strategically distributed on the inner surface of the skin panels. It

W. Xing · A. Wang (B) · L. Wu

B. Xu · J. Zhang · Y. Yu Beijing Xinghang Mechanical-Electric Equipment Co., Ltd., Beijing 100074, China

© The Author(s) 2024

School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China e-mail: wangam@bit.edu.cn

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_102

is desirable for these panels to be as thin as possible to reduce weight and minimize flight costs [1, 2]. Considering the above, large lightweight alloy skin panels are usually high-precision, large-sized, and weakly rigid large thin-walled parts. For instance, large passenger aircraft typically require over 200 medium to large skin panels, with most panels measuring 3 m in length and 2 m in width. These panels have a wall thickness requirement of 1.2 ± 0.1 mm, and the aspect ratio reaches 2500. In the aerospace industry, there is a growing trend towards integration and thinness of structural components to facilitate maintenance and reduce weight. Consequently, the demand for large skin panels with high precision is continuously increasing, making it of significant importance to explore efficient and precise machining methods for such panels [3].

The traditional methods for processing large skin panels mainly include chemical milling and CNC milling [4]. Among these methods, chemical milling is the primary approach for processing large skin panel parts. Chemical milling is a specialized machining process that utilizes chemical corrosion to achieve the desired shape and size of parts [5]. It eliminates the need for cutting tools and does not produce chips or stress. However, this method has certain drawbacks. The process steps are cumbersome, and the machining accuracy and efficiency are relatively low. Moreover, it generates high levels of pollution, which can be detrimental to the health of workers. On the other hand, CNC milling requires the use of auxiliary support to ensure sufficient rigidity of the process system. Common auxiliary support methods include profiling support and flexible multi-point support [6]. Flexible multi-point support involves using a support unit with adjustable telescopic rods and vacuum suction cup angles [7, 8]. This arrangement allows for adaptation to different workpiece shapes through array arrangement. However, when processing the suspended areas between the support points, the skin wall panel may deform and vibrate, resulting in poor precision and surface quality of the processed parts. The profile support method utilizes a profile that perfectly matches the shape of the workpiece, providing surface contact support for thin-walled components [9, 10]. This approach effectively prevents part processing deformation. However, it is worth noting that manufacturing molds for profile support is a time-consuming process that can potentially decrease overall production efficiency. To mitigate this issue, when processing large batches of parts, cost reduction can be achieved by combining profile support with vacuum adsorption technology. This combination enables rapid assembly and disassembly, resulting in a more uniform fit between the workpiece and the mold. By leveraging vacuum adsorption, the fitting process becomes more efficient and consistent.

When utilizing vacuum adsorption profiling fixtures for skin clamping, it's essential to consider the impact of reserved processing slots on the deformation of thinwalled skin clamping. Prior to processing, a thorough analysis is necessary to prevent excessive clamping deformation during the clamping process. This can lead to uneven residual wall thickness and potentially compromise the performance and safety of the aircraft structure. This study aims to nalyse the aircraft skin's deformation during the vacuum adsorption clamping process using finite element simulation technology and propose corresponding solutions.

# **2 Mechanical Analysis of Vacuum Adsorption**

The aircraft skin parts that require processing are indicated in Fig. 1, with a size parameter of 4000×1600×2 mm. It is important to note that besides being processed into a uniformly distributed rectangular lower wire slot, this skin part also needs to be punched at four positions.

To ensure that both milling and drilling tasks can be completed in a single clamping state, it is necessary to reserve a machining suspension area on the vacuum adsorption mold for drilling, as shown in Fig. 2. The presence of these suspended areas may cause deformation of the skin during vacuum adsorption. It is necessary to analyze whether this clamping deformation will affect the processing accuracy of the skin before proceeding.

Figure 3 illustrates the schematic diagram of the adsorption clamp between the skin and the vacuum profiling tool before processing. In this figure, q represents the vacuum negative pressure, which is the pressure difference between the inside and outside of the vacuum during adsorption. As depicted in Fig. 3, it can be observed that when the skin is clamped with the vacuum profiling suction tool, the inner surface of the profiling suction tool comes into surface contact with the outer surface of the skin. By activating the vacuum negative pressure q, adsorption clamping is achieved. The surface of the suction device provides surface support for the skin and generates a supporting force. Additionally, the inner surface of the skin experiences uniform air

**Fig. 1** The aircraft skin and its hole structure that needs

**Fig. 3** Schematic diagram of vacuum adsorption

**Fig. 4** Analysis of skin stress state

pressure, resulting in the generation of an equivalent force F, leading to the clamping deformation of the spherical shell. Considering the symmetry, the stress state of the skin is shown in Fig. 4.

Within the sealed volume formed by the suction tool and the workpiece, a certain amount of gas is extracted through a vacuum source to generate a vacuum negative pressure q. The equivalent adsorption force generated by the suction tool is

$$F = kCq \frac{s}{N} \tag{1}$$

where *k* is the coefficient of effective vacuum adsorption force, generally taken as 0.9; *C* is the conversion coefficient, and when the units of each parameter are MPa, mm2, and N, the value is 1; *N* is the safety factor during adsorption, with *N* ≥ 4 for horizontal clamping and *N* ≥ 8 for vertical clamping; *S* is the effective adsorption area of the adsorbent.

### **3 Analysis of Clamping Deformation Based on FEM**

To enhance the efficiency of simulation analysis, we opted for a simplified structure for the skin parts. As shown in Fig. 5 and Fig. 6, it is a simplified model of the parts and the vacuum profiling tool.

After simplifying the model and completing the necessary modeling operations, we imported the model into ABAQUS software for analysis of vacuum adsorption clamping deformation. The material properties of the aluminum alloy skin and

vacuum adsorption mold are presented in Table 1, and the corresponding material definitions and parameter settings were implemented in ABAQUS based on the provided data.

Once the model import and parameter settings were done, it was essential to establish appropriate fit constraints. Setting up the contact constraints involved selecting the contact and target surfaces, determining the contact stiffness, and choosing the appropriate contact types and algorithms. Contact analysis falls under the category of nonlinear problems. In ABAQUS, contact is typically categorized as rigid to flexible or flexible to flexible. The skin contact belongs to the flexible to rigid contact category. For the contact analysis, the surfaces in contact with the skin and the mold were chosen as the contact and target surfaces, respectively.

Figure 7 shows the establishment of parts and the application of constraint conditions. As we can see that full constraints were applied to the bottom surface of the profiling suction cup model, while lateral displacement and rotation constraints were applied to the sides of the skin to prevent slippage. Additionally, during the simulated vacuum adsorption process, a surface adsorption force perpendicular to the skin contact surface was applied outward.

In order to make the analysis results more reliable, the model is first finely divided into global grids during grid division, and then local grids are refined using contact surface size control. The aim is to generate a consistent grid in the contact area, thereby making the calculation more accurate. Figure 8 shows the situation of the parts after grid partitioning.

After simulation analysis using ABAQUS software, the simulation results are shown in Fig. 9. From Fig. 9, it is evident that the deformation of the aircraft skin at non-perforated areas is negligible after using vacuum adsorption for clamping, as it


**Table 1** Material parameters

**Fig. 7** Establishing part and mold constraints

**Fig. 8** Mesh

fits well with the vacuum adsorption fixture. However, significant deformation can be observed at the four punching points, with the square hole reserved for punching in the middle experiencing the maximum deformation of 5.602 mm.

Cross-sectional views of the aircraft skin at the maximum deformation position are shown in Figs. 10 and 11. Figure 10 illustrates the cross-sectional view of the maximum deformation position, indicating significant deformation of the skin under the pressure difference. The overall deformation pattern is characterized by a bowlshaped depression inward, with the central area of the square hole experiencing

**Fig. 9** Simulation analysis results of overall deformation of skin parts

the maximum deformation of 5.602 mm. In the figure, the cyan color represents a deformation amount of 1.867–3.208 mm. It can be observed that more than 80% of the square hole area of the aircraft skin falls within the cyan or deeper color range, indicating that the deformation amount in this area is generally greater than 1.867 mm. As previously mentioned, the machining requirement for the wall thickness of the aircraft skin is 1.2 ± 0.1 mm. This indicates that during the machining process, the aircraft skin will not be milled in the square hole area, which does not align with the machining requirements.

In addition, Fig. 11 shows the sectional view of the slice at the maximum deformation position. The cross-sectional slice diagram visually demonstrates the impact of the vacuum adsorption clamping process on the deformation of the skin at the reserved hole position. From the simulation cross-section, it can be observed that due to the vacuum adsorption force, the parts sag downwards in the suspended area, resulting in insufficient cutting amount. At the edge of the hole cavity, the concave deformation in the middle of the hole can cause local warping, leading to over-cutting during the machining process. This uneven residual wall thickness of the skin can have adverse effects on the performance and safety of the aircraft structure.

The deformation at the reserved hole position caused by the vacuum adsorption clamping method primarily arises from the skin being in a suspended state at that location, resulting in inconsistent pressure on the inner and outer surfaces of the skin. This leads to a downward adsorption force perpendicular to the surface of the part at the suspended position, causing deformation.

To address these issues, two feasible solutions are proposed:

• Adding sealing strips around the reserved hole to maintain consistent pressure in that area, preventing stress deformation caused by pressure differences;

**Fig. 10** Cross section view of maximum deformation position

**Fig. 11** Sectional view of maximum deformation position slice

• Connecting the chamber at the reserved hole position to the outside, ensuring that the pressure at the reserved hole position matches atmospheric pressure and avoiding skin deformation during the adsorption process due to pressure differences.

Considering that opening local areas will reduce the adsorption effect, we opted for the first solution, which involves adding sealing strips at the reserved hole positions. According to experimental results, the addition of sealing strips effectively mitigates local overfitting caused by clamping deformation, thereby improving the machining accuracy and reliability of the skin parts.

### **4 Conclusion**

This study aims to analyze the deformation problem that occurs in aircraft skin during the vacuum adsorption clamping process using finite element simulation analysis. The findings from the analysis reveal that the presence of large reserved holes in the adsorption mold during clamping can lead to deformation issues like localized depression and distortion of the skin. These deformations can have negative implications on the performance and safety of the skin structure. To mitigate this problem, we propose the addition of sealing rings around the reserved holes during the actual aircraft manufacturing process. This approach helps to reduce the pressure difference between the inner and outer surfaces of the skin, thereby minimizing the impact of clamping deformation caused by vacuum adsorption.

**Acknowledgements** The authors gratefully acknowledge the financial support of the National key research and development program (Grant No. 2020YFB1713100, No. JCKY2020204B017).

### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Analysis on Random Vibration and Impact Response of Vehicle-Borne Electronic Chassis**

**Zenghui Zhu and Qingqin Meng** 

**Abstract** In order to meet lightweight design requirement of vehicle-borne electronic chassis, vibration and impact response of chassis was analyzed based on the theory of random vibration and transient impact dynamics. Considering the influence of heat dissipation, corrosion resistance, material strength and economy of processing, weight reduction design was carried out according to vibration and impact response stress nephogram of chassis. The computation results show that the weight of chassis frame decreased by 16.4% and the first-order modal frequency of chassis decreased by 14.2% after weight reduction. Impact response had a lag relative to input impact spectrum. The increase of vibration stress was obviously higher than that of impact stress in different directions, which was still less than the corresponding material yield strength. That is, the structural strength of chassis after weight reduction meets the requirements of vibration and impact resistant design. The research results can provide theoretical basis of electronic equipment lightweight design under vehicle vibration and impact environment.

**Keywords** Vehicle-borne electronic chassis · Random vibration · Transient impact · Lightweight design

### **1 Introduction**

Due to the limitation of installation space and vehicle carrying capacity, the design of vehicle-borne electronic chassis should consider miniaturization, integration and lightweight. Meanwhile, when vehicle is driving on the road, the tire collision with the ground obstacles generates non-periodic excitation load, which is transferred to chassis through car body. That is, chassis is subjected to vibration and impact load generated by vehicle transportation. Lightweight design of chassis reduces structural rigidity and strength, which is not conducive to improving its vibration and impact

© The Author(s) 2024

Z. Zhu (B) · Q. Meng

The 724 Research Institute of CSSC, Nanjing, China e-mail: zhuzenghui1120@163.com

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_103

resistance. At present, experimental and simulation method is applied to lightweight research of electronic equipment. In recent years, based on a large number of vibration and impact test data, dynamics algorithm of finite element analysis (FEA) software is constantly improving and computation accuracy is greatly improved. Compared with experimental method, finite element method (FEM) can shorten the development cycle and reduce the cost, and is widely used in lightweight design of electronic equipment [1].

Liu [2] performed numerical calculation of random vibration response and fatigue life estimation for BGA package. Simulation results of vibration displacement response were in good agreement with experimental results, which effectively verified the reliability of FEA model. The results show that response stress increased with the increase of vibration load, and screw loosening made BGA package more likely to be damaged. Xie [3] used Adams software for kinematic simulation of RV reducer to obtain acceleration time history of spur gear. PSD function was obtained by solving autocorrelation function and Fourier transform of acceleration time history. The first six-orders modal frequencies of reducer were computed in ANSYS Workbench software, and random vibration analysis was carried out by input PSD function. According to the stress nephogram of gear, reducer had high reliability. Huang [4] established dynamic analysis model of battery pack in electric vehicle (EV) and computed its vibration response based on random vibration theory. To verify the accuracy of computation model, free modal computation data and modal test results were compared. The errors of first 4-orders modal frequencies were less than 3.3%, and modal shapes were the same, indicating that computation modal had certain reliability. According to vibration and impact experiment conditions specified in national standard of EV battery pack, steady-state random excitation load and half sine wave impact load were applied respectively. Vibration and impact stress of battery pack was less than the material strength, that is, structural strength of battery pack met the requirements of vibration and impact environment. At present, there are few researches on lightweight design of vehicle-borne electronic equipment under random vibration and transient impact.

In this paper, random vibration and transient impact of vehicle-borne electronic chassis are nalysed theoretically, and FEA model of chassis is established. Based on modal computation, random vibration spectrum and half sine wave impact spectrum are input into FEA software to nalyse vibration and impact response. Vibration and impact stress nephogramis nalysed to carry out weight reduction design. Vibration and impact response of chassis after weight reduction is computed, and the changes of stiffness and strength before and after weight reduction are compared to determine the weight reduction scheme of chassis.

### **2 Description of Analysis Model**

Figure 1 shows the geometric model of vehicle-borne electronic chassis. The chassis consists of frame, modules, axial flow fan and switching board. The frame carries the weight of modules, axial flow fan and switching board. To meet the requirement of lightweight design, the material is aluminum alloy whose yield strength is 165 MPa. The weight of chassis is 45 kg. The weight of frame is 14.6 kg.

### **3 Excitation Spectrum**

### *3.1 Vibration Excitation Spectrum*

The vibration excitation of chassis mainly comes from the interaction between tire and ground. Figure 2 shows the random vibration spectrum of chassis in different directions. Vibration frequencies ranged from 10 to 500 Hz. The vertical, transversal and longitudinal root mean square (RMS) of acceleration was 1.04 g, 0.204 g and 0.74 g respectively.

### *3.2 Impact Excitation Spectrum*

In addition to continuous random vibration load, chassis is subjected to impact load with short pulse width. The impact environment of chassis was half sine wave, peak acceleration was 15 g, pulse width was 11 ms, and impact spectrum in different directions was the same.

**Fig. 2** Random vibration spectrum of chassis

### **4 Analytical Method**

### *4.1 Computation Method of Random Vibration*

When chassis is excited by acceleration whose PSD is *P A*¨(ω), the dynamic equation of random vibration is defined as [5]

$$M\ddot{Z} + C\dot{Z} + KZ = MI\ddot{A}(t) \tag{1}$$

where *M* is the mass matrix, *C* the damping matrix, *K* the stiffness matrix, *Z* the displacement vector, *I* the acceleration vector, *A*¨(*t*) the acceleration excitation.

Random vibration analysis, namely PSD analysis, is different from sinusoidal vibration which expresses dynamic equation with specific function. Probability statistics method is used to describe its stochastic process related characteristic quantities. Based on modal analysis, vibration stress, strain and deformation are computed by combining modal response statistics of each order. Input acceleration load and response data of random vibration are random variables [6, 7]. In Eqs. (1), the value of *C* and *A*¨(*t*) is 0, and the i order modal participation factor (*Qi*) is obtained by modal calculation of chassis. *Qi* = β*T MI* , β is the mass-normalized mode shape. According to Wiener-Khintchine theorem, response PSD (*PZZ* (ω)) is defined as

$$P\_{ZZ}(\omega) = \sum\_{i=1}^{n} \left| \mathcal{Q}\_i^2 \beta\_i \beta\_i^T \right| F\_i(\omega) \left| \right.\\ \left. \left. \begin{aligned} ^\*P\_{\bar{A}}(\omega) \end{^\* \right.} \right. \tag{2}$$

where ω is the circular frequency, F*i*(ω) the frequency transfer function.

RMS of random vibration response stress (σ *<sup>Z</sup>* ) is defined as

$$\sigma\_Z = \left[\int\_{-\infty}^{+\infty} P\_{ZZ}(\omega) d\omega\right]^{1/2} = \left[\sum\_{i=1}^n Q\_i^2 \beta\_i \beta\_i^T \int\_{-\infty}^{+\infty} |F\_i(\omega)|^2 P\_{\bar{A}}(\omega) d\omega\right]^{1/2} \tag{3}$$

### *4.2 Computation Method of Transient Impact*

Impact response of chassis is computed by time domain analysis, whose input impact spectrum is half sine wave. Newmark iteration method in the FEA software transient structural analysis module is applied for impact computation. Newmark iteration method is defined as [8, 9].

$$z\_{k+1} = z\_k + \Delta t \underset{k}{\dot{\bar{z}}} + (0.5 - \eta)\Delta t^2 \ddot{\bar{z}} + \eta \Delta t^2 \underset{k+1}{\ddot{\bar{z}}} \tag{4}$$

$$\underset{k+1}{\dot{\boldsymbol{z}}}\_{k+1} = \dot{\boldsymbol{z}} + (1 - \boldsymbol{\gamma})\Delta t \,\ddot{\boldsymbol{z}} + \boldsymbol{\gamma}\Delta t \,\overset{\ddot{\boldsymbol{z}}}{\underset{k+1}{\ddot{\boldsymbol{z}}}} \tag{5}$$

where *z* is the displacement, *t* the time, *k* the iterative steps, η and γ the Newmark constant.

The principle of Newmark iteration method is to divide the impact pulse width into several uniformly distributed time substeps. Firstly, strain rate is gained by solving element node velocity of chassis model, and element strain increment is derived by integrating strain rate. Then, total element strain is acquired by summing strain of previous time step with strain increment. Finally, on the basis of the relationship between stress and strain, total element strain is transformed into stress. From the foregoing, impact stress computed by time domain analysis is close to measured stress. Impact computation has high accuracy and reliability, but is limited by the number of iterative steps, resulting in a large amount of computation.

### **5 Vibration and Impact Computation for Chassis**

### *5.1 FEA Model*

On the premise of not affecting the overall stiffness of chassis, the hole, fillet, boss, chamfering and other features of model were simplified. The front door, back door, roof, switching board, modules and axial flow fans were replaced by equivalent mass points [10]. Figure 3 shows the FEA model of chassis. Tetrahedral and hexahedral element was applied for meshing, whose element type was Tet10 and Hex20 respectively. The number of FEA elements was 95,511.The number of FEA nodes was 195,001 [11]. The boundary condition was defined as the inner surface of the 12 bolt holes at the bottom of chassis adopted fixed constraints.

**Fig. 3** FEA model of chassis

### *5.2 Modal Analysis*

Random vibration analysis was based on modal analysis, and modal calculation was carried out first [12]. Figure 4 shows the first six-orders mode shapes. The first 6 orders modal frequencies (Hz) were 37.518, 85.189, 145.16, 183.55, 207.06 and 247.18. It can be seen from mode shapes that the upper and lower beams had large modal deformation, whose rigidity was weak.

# *5.3 Random Vibration Response Analysis*

On the basis of modal computation of chassis, vertical, transversal and longitudinal random vibration spectrum was input respectively in the random vibration module of FEA software to compute vibration response. Random vibration spectrum was applied to the entire chassis. Figure 5a, b and c show the vertical, transversal and longitudinal random vibration 3σ stress nephogram respectively. The stress was normally distributed with a probability of 99.73% between -3σ and 3σ. Because the beams were used to support modules with large weight, vertical, transversal and longitudinal maximum stress was located in the upper beam, and maximum stress was 65.926 MPa, 25.592 MPa and 50.326 MPa respectively. Vertical stress was the largest, which was less than yield strength of beam.

**Fig. 4** The first six-orders mode shapes

# *5.4 Transient Impact Response Analysis*

Half sine wave impact spectrum in different directions was added to FEA software to compute stress. Impact spectrum was applied to the entire chassis. The number of substeps was 50. Figure 6a, b and c show the time history of vertical, transversal

**Fig. 5** Random vibration 3σ stress nephogram

and longitudinal maximum stress respectively. Because impact spectrum in different directions was the same, time history of stress had the same trend, with only one peak point. Stress reached its peak at 9.86 ms, 11.11 ms and 9.33 ms, and maximum stress was 50.124 MPa, 120.59 MPa and 37.852 MPa respectively. Transversal stress was the highest. The maximum stress occurred after the peak time of input half sine wave impact, and the stress gradually decreased with the increase of time after the impact. It shows that impact response had a lag relative to input impact spectrum. Figure 7a, b and c show the vertical, transversal and longitudinal transient impact stress nephogram respectively. Similar to the results of vibration computation, maximum impact stress in three directions was located in the upper beam of chassis, which was smaller than yield strength of beam.

**Fig. 6** Time history of maximum stress

**Fig. 7** Transient impact stress nephogram

### **6 Lightweight Design of Chassis**

It can be seen from Sect. 5 that the beam carried the weight of modules, resulting in high stress, which indicates that beam was not suitable for weight reduction design. The stress of sidewall, roof and floor of chassis was low, which had design space to reduce the weight. If the chassis wall was too thin in thickness, machining distortion **Fig. 8** FEA model of chassis after weight reduction

will be greatly affected and process performance will decrease. If the weight reduction holes were designed on the side wall of chassis, the heat dissipation performance of axial flow fan will be reduced. Because the size and number of modules remained the same, weight cannot be reduced by reducing the size of chassis. Considering the influence of corrosion resistance, material strength and economy of processing, material whose density was less than aluminum alloy cannot meet the above requirements. It can be seen from the above that reducing the thickness of chassis wall was a feasible weight reduction scheme. The thickness of sidewall, roof and floor of chassis was changed from 3 to 2 mm. The weight of frame after weight reduction was 12.2 kg, which decreased by 16.4%. The weight of chassis was 42.6 kg, which decreased by 5.3% [13].

# **7 Vibration and Impact Computation for Chassis After Weight Reduction**

# *7.1 FEA Model*

Figure 8 shows the FEA model of chassis after weight reduction. The grid division method remained unchanged. The number of FEA elements was 82,481. The number of FEA nodes was 172,876.

# *7.2 Modal Analysis*

Figure 9 shows the first six-orders mode shapes after weight reduction. The first 6-orders modal frequencies (Hz) were 32.208, 66.554, 106.45, 137.02, 176.11 and 189.46.

**Fig. 9** The first six-orders mode shapes after weight reduction


Table 1 shows the first six-orders mode frequencies before and after weight reduction, *f* 1 is the mode frequency before weight reduction, *f* 2 is the mode frequency after weight reduction, *p*1 is the percentage of modal frequency reduction after weight reduction. The first-order modal frequency decreased by 14.2% after weight reduction, and all modal frequencies decreased, indicating that reducing the thickness of chassis wall reduced the stiffness.

### *7.3 Random Vibration Response Analysis*

Figure 10 shows the random vibration 3σ stress nephogram after weight reduction in different directions. Vertical, transversal and longitudinal maximum stress was 140.33 MPa, 57.792 MPa and 88.37 MPa respectively, which was smaller than yield strength.

Table 2 shows the maximum vibration stress of chassis before and after weight reduction, σ1 is the maximum vibration stress before weight reduction, σ2 is the maximum vibration stress after weight reduction, *p*2 is the percentage of stress increase after weight reduction. Vertical, transversal and longitudinal stress increased significantly after weight reduction. Transversal stress increased by 125.8%, which increased the most.

### *7.4 Transient Impact Response Analysis*

Figure 11 shows the time history of maximum stress after weight reduction in different directions. Stress reached its peak at 9.53 ms, 11.66 ms and 9.28 ms, and maximum stress was 53.075 MPa, 133.23 MPa and 39.066 MPa respectively, which was smaller than yield strength. Figure 12 shows the impact stress nephogram after weight reduction in different directions. Similar to the results of vibration computation, maximum impact stress in different directions was located in the upper beam of chassis.

**Fig. 10** Random vibration 3σ stress nephogram after weight reduction

**Fig. 11** Time history of maximum stress after weight reduction

**Fig. 12** Transient impact stress nephogram after weight reduction

Table 3 shows the maximum impact stress of chassis before and after weight reduction, σ3 is the maximum impact stress before weight reduction, *t*1 is the time when impact stress reached its peak before weight reduction, σ4 is the maximum impact stress after weight reduction, *t*2 is the time when impact stress reached its peak after weight reduction, *p*3 is the percentage of stress increase after weight reduction. Vertical, transversal and longitudinal stress increased. Transversal stress increased by 10.5%, which increased the most. The increase of vibration stress was obviously higher than that of impact stress, the results show that the strength of chassis decreased after weight reduction, the change of transversal vibration and impact strength was obvious, and weight reduction had little influence on the trend of time history of maximum stress.


**Table 3** Maximum impact stress of chassis before and after weight reduction

# **8 Conclusions**

Random vibration and transient impact response of vehicle-borne electronic chassis before and after weight reduction was compared and analyzed. The weight of chassis frame decreased by 16.4%. All modal frequencies decreased, and the firstorder modal frequency of chassis decreased by 14.2% after weight reduction. The maximum stress of vibration and impact was located in the upper beam. Vertical, transversal and longitudinal maximum vibration stress increased by 112.9%, 125.8% and 75.6% respectively. Impact stress reached its peak at 9.53 ms, 11.66 ms and 9.28 ms, and maximum impact stress increased by 5.9%, 10.5% and 3.2% respectively. Impact response had a lag relative to input impact spectrum. The increase of vibration stress was obviously higher than that of impact stress, which was still less than the corresponding material yield strength. The change of transversal vibration and impact strength was obvious, and weight reduction had little influence on the trend of time history of maximum stress. It indicates that the stiffness and strength of chassis after weight reduction were reduced, but it still met the requirements of vibration and impact resistant design. The design scheme for weight reduction was feasible. Lightweight design of vehicle-borne electronic equipment based on FEM can shorten the development cycle and reduce the test cost.

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **The Influence of Longitudinal Oil Tank Bodies on the Vehicle's Rollover Mechanics**

**Yonggang Zuo, Fuze Chen, Jiansheng Huang, Yuliang Zhang, and Meichun Wu** 

**Abstract** In order to determine the influence of longitudinal oil tank bodies with different cross-sectional shapes on the vehicle's center of gravity position and the lateral overturning moment generated during cornering and braking, this paper conducts oscillation simulation calculations and analysis on longitudinal oil tanks with three different cross-sectional shapes. The optimal cross-section of the longitudinal tank is ultimately determined, providing a basis for the stability calculation of oil tankers.

**Keywords** Cross-sectional shape · Longitudinal oil tank · Stability

# **1 Introduction**

When oil tankers are in motion, the overall center of gravity of the vehicle changes in response to the sloshing of the liquid within the tank. Different tank shapes, as well as variations in the size and shape of the free liquid surface within the tank, result in varying degrees of liquid motion. Consequently, different cross-sectional tank shapes can impact the safety and stability of the vehicle differently during cornering and emergency braking processes [1].

# **2 Vehicle Rollover State Conditions**

During the vehicle's turning and braking process, the critical indicators that influence the risk of vehicle rollover include lateral acceleration *a*, roll angle θ, and two-wheel load transfer rate *LTR*. Among these indicators, the LTR exhibits the highest level of reliability in determining whether a vehicle will roll over or not [2].

Y. Zuo · F. Chen (B) · J. Huang · Y. Zhang · M. Wu

Army Logistics University, Chongqing 401331, China e-mail: 2501273575@qq.com

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_104

Denoting the bearing force of the inner wheels as *FL* and the bearing force of the outer wheels as *FR*, we then derive the following equation [3]:

$$LTR = \frac{F\_R - F\_L}{F\_R + F\_L} \tag{1}$$


As mentioned above, the occurrence of vehicle rollover involves the vehicle initially being in a state between semi-unstable and unstable, the load transfer rate *LTR* greater than zero (*LTR* > 0), as well as the lateral torque generated by the lateral forces produced by the liquid's movement at that moment [6].

### **3 Vehicle Rollover Mechanics Calculation**

Based on the previously discussed conditions for rollover, it is evident that when *LTR* = 1, the vehicle is in an unstable state, with the inner wheel's load-bearing capacity as *FL* = 0. The occurrence of rollover is then determined by the lateral impact forces generated by the movement of the liquid. The critical point at which a refueling truck experiences a rollover is when the clockwise moment *M*z equals the counter-clockwise moment *M*−z. To prevent a rollover, the condition of *M*z < *M*−<sup>z</sup> should be satisfied. According to the equilibrium conditions, it can be concluded as: The Influence of Longitudinal Oil Tank Bodies on the Vehicle's Rollover … 1299

$$\begin{cases} (G\_{0\text{y}} - F\_{0\text{y}}) \cdot \frac{L}{2} + (G\_{1\text{y}} - F\_{1\text{y}}) \cdot L\_1 = M\_{\text{-}x} \\ (G\_{0\text{x}} - F\_{0\text{x}}) \cdot H + (G\_{1\text{x}} - F\_{1\text{x}}) \cdot H\_1 = M\_z \\ M\_z < M\_{-z} \end{cases} \tag{2}$$

The rollover moment can be calculated as:

$$M\_{F\_1} = F\_{1\text{y}}L\_1 - F\_{1\text{x}}H\_1 \\ < (G\_{0\text{y}} - F\_{0\text{y}}) \cdot \frac{L}{2} - (G\_{0\text{x}} - F\_{0\text{x}}) \cdot H - G\_{1\text{x}}H\_1 + G\_{1\text{y}}L\_1 \tag{3}$$

where:

$$\begin{cases} G\_{0\mathbf{y}} = G\_0 \cos(\theta) \\ G\_{0\mathbf{x}} = G\_0 \sin(\theta) \end{cases} \begin{cases} G\_{1\mathbf{y}} = G\_1 \cos(\theta) \\ G\_{1\mathbf{x}} = G\_1 \sin(\theta) \end{cases} \begin{cases} F\_{0\mathbf{y}} = F\_0 \cos(\theta) \\ F\_{0\mathbf{x}} = F\_0 \sin(\theta) \end{cases} \begin{cases} F\_{1\mathbf{y}} = F\_1 \cos(\theta) \\ F\_{1\mathbf{x}} = F\_1 \sin(\theta) \end{cases}$$

### **4 FLUENT Numerical Computation**

The fluid computational model is imported into the ANSYS FLUENT computational module after being processed. Within the FLUENT computational environment, a User-Defined Function (UDF) program is invoked to input the initial conditions for simulating the emergency turning and braking process. This includes the initialization of the free liquid surface. The simulation calculations are then performed for two sets of modular refueling truck storage units with three different cross-sectional shapes, operating at load rates of 40, 55, 70, 85, and 95%.

Under the same filling rate condition, the liquid surface heights for three tank shapes are concluded in the Table 1:


**Table 1** The height of the liquid level of the three tanks at the same filling rate

# **5 Simulation Results for Longitudinal Tank at Different Filling Rates**

### *5.1 Simulation of Liquid Sloshing in a 40% Filling Rate*

Under a 40% filling rate condition, the variation in impact forces exerted by the liquid on the tank walls for the three tank shapes, identified as Type a', Type b' , and Type c' , is illustrated in Fig. 1. The trends in the alteration of impact forces in the lateral, longitudinal, and vertical directions exhibit a general similarity. The vertical direction experiences the most significant impact force, particularly at the tank bottom. The presence of transverse and longitudinal baffles inside the tank influences the impact forces differently, with the lateral impact force affecting the tank sidewall and the longitudinal impact force influencing the tank's front head to varying degrees. The maximum values of impact forces experienced in each direction are summarized in Table 2. When the maximum lateral impact force is generated, Fig. 2 depicts the fluid state, wherein the intensity of liquid sloshing is notably strong, and significant fluctuations in impact force are observed. Despite the limited lateral force, its effect on the stability of the tank truck is relatively minor [7].

During the process of emergency turning and braking, the liquid exerts significant stress impact in the outward direction on the outer bottom of the tank as it turns and creates a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is illustrated in Fig. 2. When examining the distribution

**Fig. 1** Tank body force change curve under 40% filling rate within 0–3 s


**Table 2** Maximum impact value and tank bottom pressure value of 40% filling rate

**Fig. 2** Flow pattern and pressure distribution at the maximum pressure of 40% filling rate

of the maximum pressure area, the tank with a Type a' tank has the largest maximum pressure area, followed by the Type b' tank, with the Type C tank showing the lowest maximum pressure on the tank bottom. Looking at the maximum pressure values on the tank bottom, the tank with a Type A tank has the largest, followed by the Type B tank, and the Type C tank has the smallest maximum pressure on the tank bottom. Specific values for the maximum pressure on the tank bottom can be found in Table 2.

### *5.2 Sloshing Conditions at a 55% Filling Rate*

Under a 55% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for Type a' , Type b' , and Type c' , are shown in Fig. 3. As depicted, the trends in the variation of impact forces in the lateral, longitudinal, and vertical directions are generally consistent. The impact force has the most significant effect in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head varied in their degree of influence. The maximum values of impact forces experienced in each direction are summarized in Table 2. When the maximum lateral impact force is generated,

a significant accumulation of liquid at the front section of the tank can be observed (Fig. 4). Throughout this process, the intensity of liquid sloshing is notably strong, with significant fluctuations in impact force, while the lateral force is of moderate magnitude with insignificant effect on the stability of the tank [8].

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a zone of negative pressure at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 4. Based on the distribution of the maximum pressure area, the tank with a Type b' tank has the largest maximum

**Fig. 3** Tank body force change curve under 55% filling rate within 0–3 s

**Fig. 4** Flow pattern and pressure distribution diagram at a maximum pressure of 55% filling rate


**Table 3** Maximum impact value and tank bottom pressure value of 55% filling rate

pressure area, followed by the Type a' tank, with the Type c' tank demonstrating the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type c' tank experiences the highest maximum pressure, followed by the Type b' and the Type a' tank, which has the lowest maximum pressure on the tank bottom. The maximum pressures on the tank bottom are summarized in Table 3.

### *5.3 Sloshing Conditions at a 70% Filling Rate*

Under a 70% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for the three tank shapes, designated as Type a' , Type b' , and Type c' , are depicted in Fig. 5. The graphs illustrate the general consistency in the trends of variation in impact forces in the lateral, longitudinal, and vertical directions. The most significant impact force occurs in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head are varied. The maximum values of impact forces experienced in each direction are summarized in Table 4. When the maximum lateral impact force is generated, a significant accumulation of liquid can be observed at the front section of the tank. Throughout this process, the intensity of liquid sloshing is notably strong, with significant fluctuations in impact force. The lateral force is relatively large, which has a significant impact on the stability of the tank truck [9].

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 6. Based on the distribution of the maximum pressure area, the tank with a Type b' cross-section had the largest maximum pressure area, followed by the Type a' tank, with the Type c' tank having the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type c' tank experienced the highest maximum pressure, followed by the Type a' , and the lowest maximum pressure on the tank bottom is observed in the Type b' tank. The maximum pressures on the tank bottom are summarized in Table 4.

**Fig. 5** Tank body force change curve under 70% filling rate within 0–3 s


**Table 4** Maximum impact value and tank bottom pressure value of 70% filling rate

**Fig. 6** Flow pattern and pressure distribution diagram at maximum pressure of 70% filling rate

**Fig. 7** Tank body force change curve under 85% filling rate within 0–3 s

### *5.4 Sloshing Conditions at an 85% Filling Rate*

Under an 85% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for Type a' , Type b' , and Type c' tanks are depicted in Fig. 5. The graphs illustrate the general consistency in the trends of variation in impact forces in the lateral, longitudinal, and vertical directions. The most significant impact force occurs in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head are varied. The maximum values of impact forces experienced in each direction are summarized in Table 5. When the maximum lateral impact force is generated, a significant accumulation of liquid can be observed at the front section of the tank. Throughout this process, the intensity of liquid sloshing is notably strong, with significant fluctuations in impact force. The lateral force is relatively large, which significantly impacts the tank truck's stability [10].

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 8. Based on the maximum pressure area distribution, the tank with a Type b' cross-section has the largest maximum pressure area, followed by the Type c' tank, with the Type a' tank having the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type a' tank experiences the highest maximum pressure, followed by the Type c' , and the


**Table 5** Maximum impact value and tank bottom pressure value of 85% filling rate

**Fig. 8** Flow pattern and pressure distribution diagram maximum pressure of 85% filling rate

lowest maximum pressure on the tank bottom is observed in the Type b' tank. The maximum pressures on the tank bottom are summarized in Table 5.

### *5.5 Sloshing Conditions at a 95% Filling Rate*

Under a 95% filling rate condition, the changes in impact forces exerted by the liquid on the tank walls over time for the Type A, Type B, and Type C tanks are presented in Fig. 9. The graphs illustrate that the trends in the variation of impact forces in the lateral, longitudinal, and vertical directions are entirely consistent. The most significant impact force occurs in the vertical direction on the tank's bottom. Due to the presence of transverse and longitudinal baffles inside the tank, the impact of lateral force on the tank sidewall and the impact of longitudinal force on the tank's front head are varied. The maximum values of impact forces experienced in each direction are summarized in Table 6. The fluid state at the time that the maximum lateral impact force is generated is illustrated in Fig. 10, in which a substantial accumulation of liquid at the front section of the tank can be observed. Throughout this process, the intensity of liquid sloshing is notably strong, and there are significant fluctuations in impact force. The lateral force is exceptionally significant, and it substantially impacts the tank truck's stability.

**Fig. 9** Tank body force change curve under 95% filling rate within 0–3 s


**Table 6** Maximum impact value and tank bottom pressure value of 95% filling rate

**Fig. 10** Flow pattern and pressure distribution diagram at maximum pressure of 95% filling rate

During emergency turning and braking, the liquid normally exerts substantial stress impact in the outward direction on the outer bottom of the tank as it turns, creating a negative pressure zone at the rear inside of the tank. The distribution of pressure on the tank's bottom is depicted in Fig. 10. Based on the distribution of the maximum pressure area, the tank with a Type c' cross-section had the largest maximum pressure area, followed by the Type a' tank, with the Type b' tank having the smallest pressure area. In terms of the maximum pressure values on the tank bottom, the Type a' tank experienced the highest maximum pressure, followed by the Type c' , and the lowest maximum pressure on the tank bottom is observed in the Type b' tank. The maximum pressures on the tank bottom are summarized in Table 6.

# **6 Comparative Analysis of Longitudinal Tank Sloshing Conditions at Different Filling Rates**

# *6.1 Comparative Analysis of Lateral Impacts for Three Tank Shapes at Different Filling Rates*

As illustrated in Fig. 11, the magnitude of lateral impact forces on the tank truck varies with changes in tank cross-section and filling rates. At a 40% filling rate, the lateral impact forces generated by the tanks with Type a' , Type b' , and Type c' crosssections are as follows: 22,115 N, 23,842 N, and 25,798 N, respectively. At a 55% filling rate, the lateral impact forces are as follows: 41,342 N, 41,918 N, and 44,163 N, respectively. At a 70% filling rate, the lateral impact forces are as follows: 66,500 N, 79,740 N, and 94,477 N, respectively. At an 85% filling rate, the lateral impact forces are as follows: 97,298 N, 108,000 N, and 108,294 N, respectively. Finally, at a 95% filling rate, the lateral impact forces are as follows: 152,831 N, 161,803 N, and 184,284 N, respectively. The maximum lateral impact force experienced by the tanks is positively correlated with the filling rate; a higher filling rate results in greater lateral impact forces, while a lower filling rate leads to smaller lateral impact forces. Specifically, when the filling rate is less than 85%, the Type c' tank exhibits the highest lateral impact force, while when the filling rate exceeds 85%, the Type b' tank experiences the highest lateral impact force.

The flow pattern distribution diagrams reveal that the lateral force fluctuations resulting from tank sloshing exhibit similar patterns under different filling rate conditions. In the case of the modular tank truck, starting from the initiation of emergency braking during a turn and extending to approximately 2.00 s, the lateral impact force initially increases, then gradually decreases until it disappears. When the maximum lateral impact force occurs, the motion of the liquid inside the tank becomes highly turbulent. The liquid surges from the lower part to the upper part and from the rear to the front in the direction of the turn's outer side. A substantial amount of liquid accumulates on the outer side of the front half in the direction of the turn. Subsequently, the liquid undergoes a rolling motion, briefly lifting off and descending, releasing

**Fig. 11** Comparison of the lateral impact force changes of three tanks under different filling rates


**Table 7** Maximum lateral force appearance time at different filling rates

potential energy. With multiple iterations of rolling motion, the lateral impact force gradually subsides. As the tanker truck decelerates due to braking, the sloshing of the oil gradually ceases. Therefore, the maximum lateral impact force is closely related to the intensity of sloshing and the filling rate. The timing of the occurrence of the maximum lateral impact force is roughly consistent, as indicated in Table 7.

# *6.2 Comparison of Lateral Impact Forces Over Time for Different Cross-Sectional Tank Shapes at Equal Filling Rates*

A comparative analysis of three distinct tank shapes, denoted as Type a' , Type b' , and Type c' , reveals variations in lateral impact forces at different filling rate conditions (40, 55, 70, 85, and 95%) over 0–3 s. As depicted in Fig. 12, the Type a' tank units generally exhibit slightly lower impact forces than the Type b' tank units. In comparison, the Type c' tank units experience slightly higher forces than Type b' . From a lateral force perspective, the modular refueling truck comprised of Type a' tanks exhibits the least detrimental influence on the tank structure during the turning

**Fig. 12** Comparison of the lateral impact force of three tanks with time at the same filling rate

and braking, followed by Type b' tank, and Type c' tank has the most pronounced impact. Examining the variational tendency, it can be inferred that the timing of the maximum lateral impact force is correlated with the filling rate. When operating under lower filling rate conditions, the maximum lateral impact force tends to manifest at a later point in time, resulting in an extended average duration of lateral impact forces. Conversely, in situations characterized by higher filling rates, the maximum lateral impact force can be reached earlier, leading to a diminished average duration of lateral impact forces.

# *6.3 The Variations in the Maximum Lateral Rollover Moments for Different Tank Cross-Sectional*

The variations in the maximum lateral rollover moments for different tank crosssectional shapes at various filling rates can be calculated using the torque formulas. When the fluid motion continues until around 0.65 s under 40% filling rate conditions, the lateral rollover moments for the three tank shapes reach approximately 1.86×10<sup>4</sup> N/m, 2.17 × 104 N/m, and 2.35 × 104 N/m, respectively. In the case of a 55% filling rate, the lateral rollover moments at approximately 0.65 s are 3.60 × 104 N/m, 4.02×104 N/m, and 4.24×104 N/m for the respective tank shapes. At a 70% filling rate, the lateral rollover moments that occur at around 0.42 s are 6.06 × 104 N/m, 7.97 × 104 N/m, and 9.46 × 104 N/m for the different tank shapes. For an 85% filling rate, the lateral rollover moments at approximately 0.37 s are 9.26 × 104 N/ m, 11.2 × 104 N/m, and 11.2 × 104 N/m, corresponding to the various tank shapes. Lastly, at a 95% filling rate, the lateral rollover moments occur around 0.21 s and are 14.93 × 104 N/m, 21.35 × 104 N/m, and 19.54 × 104 N/m for the different tank shapes.

From the curves in Fig. 13 that depict the variation in maximum lateral rollover moments of tanks with Type a' , Type b' and Type c' cross-sections at different filling rates, it is evident that as the filling rate increases, the maximum lateral forces for all three tank shapes continuously rise. When the filling rate is below approximately 55%, the maximum lateral rollover moments of the three tank shapes are nearly equal. In the filling rate range of 55% to 85%, tanks with the Type b' cross-section exhibit larger maximum lateral rollover moments than those with the Type c' crosssection. For filling rates exceeding 85%, tanks with the Type c' cross-section surpass those with the Type b' cross-section in terms of maximum lateral rollover moments. Notably, for filling rates above 55%, tanks with the Type a' cross-section consistently demonstrate smaller maximum lateral rollover moments compared to the Type b' and Type c' tanks, with the disparity becoming more pronounced as the filling rate increases.

**Fig. 13** Variation of the maximum lateral moment of the three tanks with the filling rate

# **7 Conclusion**

This study introduces the simulations of oil tank trucks composed of tanks with three different cross-sectional shapes, investigating the sloshing behaviors of the internal oil in the tanks under different filling rates (40%, 55%, 70%, 85%, and 95%) during emergency braking while making sharp turns. The research focuses on factors that directly impact the tanks, including the impact forces, tank structural stress, and the lateral rollover moments during the turning and braking process. The following conclusions are drawn:


### **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Simulation and Comparative Analysis of Welding of the Safety End of the Pipe Based on Different Body Heat Source Models**

**Wenjie Chen and Dongmei Ji** 

**Abstract** Based on ANSYS APDL, this paper simulates the welding process of the safety end of the pipe using different heat source models. The results show that the shape of the moving heat source under the body heat rate heat source model and the double ellipsoid heat source model is ellipsoidal shape. The body heat rate heat source has a larger high temperature area than the double ellipsoidal heat source. The variation law of temperature field and stress field is basically the same, but the overall stress field under double ellipsoid heat source load is about 20 MPa smaller than the total stress field under body heating rate. The comparison of different heat source models can provide some help for the numerical simulation of welding.

**Keywords** Component · Welding · Numerical simulation · Temperature field: stress field

# **1 Introduction**

Welding process is a key process in the manufacture and production of nuclear power pressure equipment, in the nuclear power plant primary circuit system, there are many nozzle safety ends, mainly including the hot section and cold section on the reactor pressure vessel to take over the safety end, the fluctuating tube on the regulator to take over the safety end, etc., the corresponding geometry, materials, welding processes are generally similar [1–3]. It has been observed that the coupling of residual stress formed after welding and service load will accelerate the stress corrosion phenomenon of components, promote the germination and propagation of cracks, and thus threaten the safe operation of nuclear power plants. Therefore, it is necessary to pay close attention to the residual stress changes of the welded joint [4, 5].

1315

W. Chen · D. Ji (B)

College of Energy and Mechanical Engineering, Shanghai University of Electric Power, 1851, Hucheng Huan Road, Pudong New District, Shanghai 201306, China e-mail: jdm@shiep.edu.cn

<sup>©</sup> The Author(s) 2024

S. K. Halgamuge et al. (eds.), *The 8th International Conference on Advances in Construction Machinery and Vehicle Engineering*, Lecture Notes in Mechanical Engineering, https://doi.org/10.1007/978-981-97-1876-4\_105

In the current research, there are few three-dimensional overall simulations of welded joints at the safety end of the pipe. In the process of welding temperature field calculation, the correct selection of heat source model is an important guarantee for the accuracy of the calculation results [6, 7]. Based on ANSYS APDL, this paper simulates the welding safety end of the joint by using ellipsoid heat source and body heat rate heat source model, and analyzes the changes of welding temperature field and stress field.

### **2 Geometry and Meshing**

The object studied in this paper is the model of welded joints of the safety end of the pressure vessel of the primary circuit reactor of nuclear power plants, with a total length of 1628.7 mm, an inner and outer radius of 417.3 mm and 500.3 mm, respectively, and a single-sided V-shaped groove with an angle of 15°. In addition, considering the calculation of the three-dimensional model, it is necessary to simplify the weld part reasonably, and relevant studies have shown that for the simulation of multi-pass welding problems, the intermediate process has little effect on the residual stress, therefore, under comprehensive consideration, the total weld area is divided into 11 layers, and the thickness of the weld in the cover welding stage at the end of the welding is 11.5 mm, and the rest are 8 mm, as shown in Fig. 1. In this paper, the thermal–mechanical indirect coupling is used for simulation, the solid70 unit is selected for thermal analysis, and the structural analysis is converted to the solid185 unit by etchg, tts command. The meshing adopts a combination of mapping surface mesh and sweep mesh, which is easier to obtain a good quality hexahedral mesh, the weld and heat affected zone are the places with the highest degree of mesh fineness, and the base metal area can be properly coarseened for computational efficiency, as shown in Fig. 2.

**Fig. 2** Finite element model of welded joint at the safety end of a reactor takeover

### **3 Welding Process and Heat Source Model**

According to the welding process of nuclear power equipment, for the welding joint of the safety end of the pipe with isolation layer, the preheating treatment is generally not done before welding, and the temperature between the welding layers is controlled at 100–200 °C, and the excessive interlayer temperature will adversely affect the nickelbased alloy. In this simulation, the welding speed is set to 20 mm/s, the current is set to 630 A, and the voltage is set to 35 V. The main driving factor for the welding process to proceed is determined by the welding heat source, which causes phase changes, thermal strains, thermal stresses. For welding engineering applications, Gaussian surface heat source, double ellipsoidal heat source, bulk heat rate model, conical heat source, and strip heat source are commonly used [8, 9]. For large cylindrical models, the double ellipsoidal heat source model and the bulk heat rate model are usually used in the numerical simulation of welding, and the expression is as follows. The shape parameters of the double ellipsoid heat source have a great influence on the temperature field, and it is necessary to obtain the ideal parameters through repeated experiments [10], and *a f* = 8 mm, *ar* = 12 mm, b = 10 mm, c = 11.5 mm are used in this paper.

$$q(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{6\sqrt{3}f\_f \eta UI}{aba\_f \pi^{3/2}} \exp\left(\frac{-3x^2}{a\_f^2}\right) \exp\left(\frac{-3y^2}{b^2}\right) \cdot \exp\left(\frac{-3z^2}{c^2}\right) \tag{1}$$

$$q(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{6\sqrt{3}f\_r\eta UI}{aba\_r\pi^{3/2}} \exp\left(\frac{-3x^2}{a\_r^2}\right) \exp\left(\frac{-3y^2}{b^2}\right) \cdot \exp\left(\frac{-3z^2}{c^2}\right) \tag{2}$$

$$HGEN = \frac{K \* U \* I}{A \* V \* DT} \tag{3}$$

In Eqs. (1) and (2), q is the heat generation rate, w m-3; *af* , *ar*, *b*, *c* are the shape parameters of the double ellipsoidal heat source, m; *f f* and *f r* are the energy distribution coefficients before and after the heat source model, where HGEN represents the bulk heat generation rate (w/m3), K represents the welding efficiency, A represents the weld cross-sectional area (m2), V represents the welding speed (m/s), and DT represents the time step size of each substep (s).

### **4 Results and Discussions**

# *4.1 Comparison of Welding Temperature Field Analysis Results*

Two different heat source models are used as heat load inputs and analyzed using Ansys apdl. Because of the large number of weld layers, Only the temperature field distribution of the first and last welds are captured in this paper, the results are shown in the Figs. 3 and 4. From the temperature field analysis results, it can be seen that the welding heat source center moves continuously along the circumferential direction of the weld. The temperature is the highest from the welding heat source center and the heat gradient is distributed in an ellipsoidal shape. The welding time of each weld is set to 157 s, and the cooling time of 10,000 s is set after welding is completed to cool it to room temperature.

Comparing the instantaneous temperature field of the two bulk heat source models at the same time, the temperature field distribution trend is basically the same. Due to the combined effect of heat conduction and heat source movement, the temperature field of the high temperature zone of the bulk heat rate model is also ellipsoidal, so the shape of the two molten pools is basically similar. From the temperature field of the first weld, the highest temperature under the action of the body heat rate model is 1595 °C, the highest temperature under the action of the double ellipsoidal heat source is 1513 °C. The maximum temperature in the high temperature area is about 60 °C higher than that of the double ellipsoid heat source because the energy of the

body heat rate heat source is more concentrated. This is shown in Figs. 5 and 6, from the temperature field of the last weld, the maximum temperature is reduced due to the reduction of the interlayer temperature during the welding process, and the effect of the double ellipsoid heat source is more obvious from the figure.

### *4.2 Analysis and Comparison of Welding Stress Field Results*

When performing stress field analysis, it is necessary to fix constraints on both ends of the welded joint, otherwise there will be a situation where the calculation does not converge. The calculation results of the temperature field of the two are taken as the thermal load, that is, read the.rth file, and the \*do cycle statement is also used to apply it one by one, and finally obtain the welding residual stress after the cooling

of the two. As shown in the Figs. 7 and 8, the overall welding residual stress size

when using double ellipsoid heat source is 395 MPa, and the welding residual stress size when using bulk heat rate heat source is 417 MPa, the difference between the two is about 20 MPa, indicating that the use of different heat sources has a certain impact on the welding residual stress field, but the residual stress distribution of the two is basically the same. The stress at the weld and the heat-affected zone is large, and the stress influence at the base metal away from the weld is small. Select the 0° direction as the path to observe the distribution of welding residual stress after the cooling of the entire welded joint. As can be seen from Fig. 9, the peak residual stress after cooling is located at the junction of the weld and the base metal, and the stress gradually decreases far from the weld.

**Fig. 7** Distribution of residual stress field after welding under the action of bulk heat generation rate heat source

**Fig. 6** Temperature field distribution of final weld under the action of double ellipsoid heat source

**Fig. 8** Distribution of residual stress field after welding under the action of double ellipsoid heat source

**Fig. 9** Distribution of residual stress under the action of two heat sources

# **5 Conclusion**

Based on the results and discussions presented above, the conclusions are obtained as below:


**Acknowledgements** This work was financially supported by the Applied Research Project of National Natural Science Foundation of China under Grant (No. 52175343).

# **References**


**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.