Focusing on innovation, these proceedings present recent advances in the field of mechanical design in China and offer r

*1,349*
*145*
*160MB*

*English*
*Pages XVI, 1154
[1170]*
*Year 2020*

- Author / Uploaded
- Jianrong Tan

*Table of contents : Front Matter ....Pages i-xvi Multi-hierarchy Carbon Footprint Analysis and Low-Carbon Design Improvement Method (Hong Bao, Sheng Guo, Qing Di Ke)....Pages 1-11 The Design and Dynamics Analysis of Cylindrical Roller Surface Unfolding Mechanism (Yudong Bao, Xiaojian Chen, Chengyi Pan, Yanling Zhao, Liqun He)....Pages 12-27 A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning of the Large Storage Tank (Wanghui Bu, Weiqi Zhou, Laixin Fang, Jing Chen, Xianghua An, Jingkai Huang)....Pages 28-40 A Novel Biomimetic Design Method Based on Biology Texts Under Network (Bowen Chen, Liang Chen, Xiaomin Liu, Hao Dou)....Pages 41-51 Numerical Simulation on Effect of Graphene Doped Morphology on Heat Transfer Efficiency of Anti-/deicing Component (Long Chen, Zhanqiang Liu)....Pages 52-62 Numerical Analysis on Load Sharing Characteristics of Multistage Face Gears in Planetary Transmission (Xingbin Chen, Qingchun Hu, Chune Zhu)....Pages 63-83 Reachable Matrix and Directed Graph – Based Identification Algorithm of Module Change Propagation Path for Product Family (Xianfu Cheng, Liyun Wan, Jian Zhou)....Pages 84-92 6-DOF Industrial Manipulator Motion Planning Based on RRT-Connect Algorithm (Chengren Yuan, Guifeng Liu, Wenqun Zhang)....Pages 93-101 A Precise Identification and Matching Method for Customer Needs Based on Sales Data (Xingpeng Chu, Jian Zhang, Uday Shanker Dixit, Peihua Gu)....Pages 102-112 Optimization Design of the Non-magnetic Drill Rod for Directional Drilling in the Coal Mine (Dayong Tang, Lu Liu)....Pages 113-124 Research on Product Redesign Process Based on Functional Analysis (Yafan Dong, Runhua Tan, Peng Zhang, Wei Liu, Ruiqin Wang)....Pages 125-138 Analysis of Characteristics and Structure Optimization of Anti-rolling Torsion Bar (Yahong Dong, Yuejin Shang)....Pages 139-150 Coupling Mechanism of Errors in the Planetary Roller Screw Mechanism (Chuanming Du, Geng Liu, Junqi Liu, Shangjun Ma)....Pages 151-161 A Novel Tooth Contact Analysis Method Based on Value Iteration (Jinfu Du, Zhengrong Wang, Kai Liu, Yiteng Gao)....Pages 162-169 Rigidity Synthesis and Machining Error Analysis of Machine Tool Chain (Pengyu Hao, Gongxue Zhang, Zijian Pei, Xianming Gao)....Pages 170-188 Thermodynamic Lubrication Performance and Stability for a Deep/Shallow Pocket Hybrid Bearing Considering Bubbly Oil (Hong Guo, Shuai Yang, Ningning Wu, Ruizhen Li)....Pages 189-201 Time Delay Chen System Analysis and Its Application (Hongjun He, Yan Cui, Chenhui Lu, Guan Sun)....Pages 202-213 The Driver-in-the-Loop Simulation on Regenerative Braking Control of Four-Wheel Drive HEVs (Hexu Yang, Xiaopeng Li, Pengxiang Li, Yu Gao)....Pages 214-222 Strength and Modal Analysis of High Speed EMU Gearbox Housing (Haijun Huang, Bichao Yin, Chaowen Wang, Haiyan Zhu)....Pages 223-228 Indirect Adaptive Fuzzy Sliding-Mode Control for Hydraulic Manipulators (Xin Huang, Yi Wan, Yao Sun, Jiarui Hou)....Pages 229-242 Improvement and Research of Tennis Training Machine (Huiqiang Guo, Jianye Pan, Wen Liu, Yiling Yue)....Pages 243-252 Design of Flagstone Transport Device Based on TRIZ Theory (Fan Jiang, Jian Shen, Tao Zhu, Jinfeng Wen)....Pages 253-266 Fault Analysis and Structure Optimization of End Face Seal of Hydraulic Oscillator (Haixiang Jiang)....Pages 267-273 Innovative Design of Leeward Surface of Pin Fin in Flame Arresters Fitted in Explosion Relief Valve (Lanfang Jiang, Shuyou Zhang, Xin Shu, Weina Hao, Yun Ren, Zhiya Chen)....Pages 274-287 Analysis and Optimization of Performance Under Operating Condition of Thrust Aerostatic Bearing with Vacuum Pre-load (Mengyang Li, Qiu Hu, PinKuan Liu, Ming Huang)....Pages 288-301 A Study on Innovative Design of Rotary Pile Foundation Drilling Machine Based on TRIZ Theory (Fuxing Li)....Pages 302-309 Design and Optimization of Focusing X-Ray Telescope Based on Intelligent Algorithm (Liansheng Li, Zhiwu Mei, Jihong Liu, Fuchang Zuo, Jianwu Chen, Hanxiao Zhang et al.)....Pages 310-324 Research of Dormitory Furniture Design Based on Group Interaction (Lin Li, Xupeng Wang, Chunqiang Zhang)....Pages 325-341 Analysis and Extraction of Consumer Information for the Evaluation of Design Requirement Depending on Consumer Involvement (Shipei Li, Dunbing Tang, Qi Wang, Haihua Zhu)....Pages 342-353 Reduction in Aerodynamic Resistance of High-Speed Train Nose Based on Kriging Model and Multi-objective Optimization (Tian Li, Deng Qin, Le Zhang, Jiye Zhang, Weihua Zhang)....Pages 354-370 Research and Simulation on Pilot Configuration in Multi-antenna System Based on Kalman Filter (Ying Li, Lei Cui, Zhe Zhang)....Pages 371-379 Development and Experiment of an XθYθZ Micro-motion Stage Based on a Straight-Beam Three-Quarter Round Type Flexure Hinge (Junlang Liang, Lanyu Zhang, Jian Gao, Gengjun Zhong, Guangtong Zhao, Jindi Zhang)....Pages 380-393 An Improved PC-Kriging Method for Efficient Robust Design Optimization (Qizhang Lin, Chao Chen, Fenfen Xiong, Shishi Chen, Fenggang Wang)....Pages 394-411 Electro-Mechanical Response of a Cracked Piezoelectric Cantilever Beam (Chao Liu, Wenguang Liu, Yaobin Wang)....Pages 412-423 A Time-Variant Reliability Analysis Method Considering Maintenance (Jingfei Liu, Chao Jiang, Xiangyun Long)....Pages 424-446 Research on Tooth Profile Error of Non-circular Gears Based on Complex Surface Theory (Yongping Liu, Fulin Liao, Changbin Dong)....Pages 447-455 Topology Optimization Design of the Monocoque Bus Body Structure (Zhuli Liu, Xiangyu Huo, Hao Zhou, Xiaoguang Wu)....Pages 456-465 Analysis of Wind Vibration Response of Transmission Tower (Zhuli Liu, Beibei Liu, Zhuan You, Mengli Li)....Pages 466-476 Design of a Stabilize Device for Heavy Oil Transportation in Water Ring (Haoran Lu, Fan Jiang, Yuliang Chen, Xiaolong Qi, Zhongmin Xiao)....Pages 477-492 Design of Spatial Lissajous Trajectory Vibrating Screen (Zhipeng Lyu, Sizhu Zhou)....Pages 493-498 Part Retrieval Technology Based on Geometric Shape and Topological Correlation (Ning Ma, Bo Yang, Junzhi Shang)....Pages 499-507 Analysis of Tool Wear and Wear Mechanism in Dry Forming Milling for Rail Milling Train (Chao Pan, Xiaoshan Gu, Mulan Wang, Baosheng Wang)....Pages 508-517 Wear Numerical Simulation and Life Prediction of Microstructure Surface Unfolding Wheel for Steel Ball Inspection (Chengyi Pan, Xia Li, Hepeng Wang, Yanling Zhao, Jingzhong Xiang, Sihai Cui et al.)....Pages 518-530 Design of Passive Gravity Balance Mechanism for Wearable Exoskeleton Suit (Chenxi Qu, Peng Yin, Xiaohua Zhao, Liang Yang)....Pages 531-545 Design and Analysis of Underwater Drag Reduction Property of Biomimetic Surface with Micro-nano Composite Structure (Xuezhuang Ren, Lijun Yang, Chen Li, Guanghua Cheng, Nan Liu)....Pages 546-559 A Method of Parametric Stability Region Determination for Non-linear Gear Transmission System (Dongping Sheng, Xiaozhen Li, Rupeng Zhu)....Pages 560-569 Performance and Parameter Sensitivity Analysis of Finger Seal with Radial Clearance (Hua Su)....Pages 570-591 Calculation and Experimental Study on Comprehensive Stiffness of Angular Contact Ball Bearings (Peng Sun, Weifang Chen, Chuan Su, Yusu Shen)....Pages 592-601 Design of a New Hydraulic Manipulator with Kinematic and Dynamic Analysis (Yao Sun, Yi Wan, Xichang Liang, Xin Huang, Ziruo Liu)....Pages 602-614 Research on Electro-Hydraulic Servo System Based on BP-RBF Neural Network (Tao Chen, Wenqun Zhang, Jianggui Han)....Pages 615-622 Collision Simulation of GQ70 Light Oil Tank Car at the Level Crossing (Maopeng Tian, Ronghua Li, Xiujuan Zhang, Miao Jin)....Pages 623-633 Reason Study of Collision Between Valves and Piston of Diesel Engine Valve Train (Dameng Wang, Xiujuan Zhang, Deyu Yue, Weipeng Fan)....Pages 634-646 Macroscopic Topology Optimization of Fusion Cages Used in TLIF Surgery (Hongwei Wang, Yi Wan, Xinyu Liu, Bing Ren, Zhanqiang Liu, Xiao Zhang et al.)....Pages 647-660 Fuzzy Optimization Design of Disc Brakes Based on Genetic Algorithm (Jianbin Wang, Jishu Yin)....Pages 661-668 Modeling Design and Evaluation of Rotary Tiller Based on Multidisciplinary Optimization (Wang Jianwei, Jianmin Zhang, Qin Yang)....Pages 669-685 Experimental Investigation of Thermal Elasto-Hydrodynamic Lubrication Based on Temperature Control (Shuaihong Yu, Yazhen Wang, Jiacheng Shen, Huihui Yue)....Pages 686-696 Study on Nonlinear Characteristics of Spatial Spreading Mechanism with Multiple Clearance Joints (Wenzhou Lin, Xupeng Wang, Xiaomin Ji, Chunqiang Zhang)....Pages 697-707 A Novel Design of Tapered Sub-array Structure for SSPS-OMEGA (Tong Wu, Yingzhong Tian, Meng Li, Long Li)....Pages 708-718 Research on Multiresponse Robustness Optimization for Unmanned Aerial Vehicle Electrostatic Spray System (Yangdong Wu, Jiajie Lu, Yiquan Wang)....Pages 719-728 Research on Trajectory Optimization of Six-Axis Manipulator Based on Watchcase Polishing (Xiang Wang, Ying Xi, Chao Gu, Mengru Li)....Pages 729-739 Reverse Modeling of the Helix Roller in the Omni-Directional Wheel Based on NURBS (Xie Xia)....Pages 740-748 Research on Automatic Matching Model of Power Battery (Chuanfu Xin, Fengxia Zhao, Yujin Wu, Jianshe Gao)....Pages 749-759 Random Vibration Analysis of Optical Adjustable Frame Based on ANSYS Workbench (Nan Xu, Yuan Wang, Kelin Xu)....Pages 760-768 Analysis of Influence of Eccentricity Error on Transmission Performance of Micro-segment Gears (Rui Xu, Lielong Wang, Peidao Pan, Kang Huang)....Pages 769-784 Research on the Design of Wearable Equipment for Intelligent Emotion Detection for Empty Nester (Yan-min Xue, Chang Ge, Shi-yi Xu, Xu-yang Zhang, Bo-xin Xiao)....Pages 785-794 Narrative Design of Old Brand Image: A Case Study of Demaogong (Yanmin Xue, You Wu, Yihui Zhou, Yang Liu)....Pages 795-803 Study on the Theory and Practice of Mechanical Design (Yunna Xue, Xuehui Shen, Baolin Wang)....Pages 804-810 Simulation Analysis of Pipe Bending Under Multiple Conditions (Guodong Yi, Zhenan Jin, Shaoju Zhang)....Pages 811-822 Simulation Analysis of Sealing Performance of Double-Offset Butterfly Valve (Guodong Yi, Shaoju Zhang, Zhenan Jin)....Pages 823-831 Motion Performance Analysis of the Sawyer Ankle Rehabilitation Robot (Yongfeng Wang, Xiangzhan Kong, Jing Yang, Guoru Zhao)....Pages 832-846 Automated Sustainable Low-Carbon Design of Offshore Platform for Product Life Cycle (Qianyi Yu, Bin He)....Pages 847-863 Comfort of Minors’ Sitting Posture in Learning Based on Motion Capture (Chunqiang Zhang, Xiaomin Ji, Yanmin Xue, Chunmei Zhang)....Pages 864-876 Lightweight Design of Dump Truck Frame Based on Finite Element Method (Gongxue Zhang, Sen Yang, Shanshan Guo, Qichen Niu)....Pages 877-887 Topological Optimization of the Front Beam in Metal Extruders (Xugang Zhang, Peilin Yang, Xiaole Cheng, Yi Hou)....Pages 888-897 Car Radiator Upper Beam Stamping Process Design and Numerical Simulation (Zhiyuan Wan, Yinping Chen)....Pages 898-903 A Fast Tacho-Less Order Tracking Method for Gear Fault Diagnosis Under Large Rotational Speed Variation Conditions Based on Multi-stage Generalized Demodulation (Qi Zhou, Wenjin Liu, Chaoqun Wu)....Pages 904-917 Gear Fault Diagnosis Under the Run-Up Condition Using Fractional Fourier Transform and Hilbert Transform (Qi Zhou, Chaoqun Wu, Qingrong Fan)....Pages 918-943 An Empirical Study of the Effect of the Completeness of Hands-on Training Design Projects on the Preservation of Tacit Knowledge (Yihui Zhou, Wenzhou Lin, Zhi Qiao, Xiaozhou Li)....Pages 944-951 Research on Structural Size Optimization of 3-TPT Parallel Mechanism Based on Stiffness Characteristics (Chunxia Zhu, Chengzhu Hu)....Pages 952-970 Vibration Response Analysis of Gearbox Housing of High Speed Train Under Wheel and Rail Excitation (Haiyan Zhu, Xiao Su, Lei Tao, Qian Xiao)....Pages 971-976 An Intelligent Fatigue Life Prediction Method for Aluminum Welded Joints Based on Fatigue Characteristics Domain (Li Zou, Hongxin Li, Wei Jiang)....Pages 977-989 Optimum Design of Radiation Well Horizontal Drilling Rig Based on TRIZ and Bionics (Chenghao Liu, Changqing Gao, Bo Yang, Zhenghe Xu)....Pages 990-1001 Innovative Design of Reversing Device and Rod Loading Device of Horizontal Drilling Rig Based on TRIZ (Shifeng Sun, Changqing Gao, Bo Yang, Zhenghe Xu)....Pages 1002-1009 AGV Trolley Tray Based on Honeycomb Paper-Based New Material (Jingjing Yang, Xiaoyi Jin, Anran Wang)....Pages 1010-1017 The Study of Electric Field Distribution on Small Holes in Pulse Electrochemical Machining (Zhaolong Li)....Pages 1018-1029 Reliability Optimization Design of New Automatic Tensioning Belt-Gear Transmission (Chengyi Pan, Guanqun Cao, Yuanqi Tong)....Pages 1030-1043 Design and Simulation Analyses of a Five-Wheeled Stair-Climbing Mechanism Based on TRIZ Theory (Chengyi Pan, Yuanqi Tong)....Pages 1044-1053 Extensible Innovation Design of Globoidal Cam Deceleration Mechanism Based on Knowledge (Shengyang Tian, Qingshan Gong, Guangguo Zhang, Mingmao Hu, Yuemin Wu)....Pages 1054-1069 Tension Analysis of Small Motor Stator Winding Tensioning Process (Yanling Zhao, Zhao Zhang, Jingzhong Xiang, Yudong Bao)....Pages 1070-1081 Study on Contact Dynamics of Cylindrical Roller Feeding Mechanism (Yanling Zhao, Xinyue Wang, Yudong Bao)....Pages 1082-1092 Trajectory Planning for Winding Process of Small-Sized Motor Stator Winding Robot (Yanling Zhao, Linqiang Wang, Jingzhong Xiang, Yudong Bao)....Pages 1093-1108 Experimental Study of a Solid-Liquid Mixing and Conveying Pump with Variable Flow and Proportion (Wei Liu, Qian Tang, Wenzhe Cai, Pinghua Liang, Zongmin Liu, Xiaojie Fan)....Pages 1109-1123 Effect of Pore Parameters on Lubrication Performance of Oil-Containing Cage (Tingting Yin, Yuanyuan Li, Ke Yan, Pan Zhang, Yongsheng Zhu, Jun Hong)....Pages 1124-1135 Defects Detection System for Fluorescent Coating of Metal Plate Based on Machine Vision (Yujin Wu, Fengxia Zhao, Chuanfu Xin, Jianshe Gao, Zhigao Chen)....Pages 1136-1149Back Matter ....Pages 1151-1154*

Mechanisms and Machine Science 77

Jianrong Tan Editor

Advances in Mechanical Design Proceedings of the 2019 International Conference on Mechanical Design (2019 ICMD)

Mechanisms and Machine Science Volume 77

Series Editor Marco Ceccarelli, Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy Editorial Board Members Alfonso Hernandez, Mechanical Engineering, University of the Basque Country, Bilbao, Vizcaya, Spain Tian Huang, Department of Mechatronical Engineering, Tianjin University, Tianjin, China Yukio Takeda, Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan Burkhard Corves, Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany Sunil Agrawal, Department of Mechanical Engineering, Columbia University, New York, NY, USA

This book series establishes a well-deﬁned forum for monographs, edited Books, and proceedings on mechanical engineering with particular emphasis on MMS (Mechanism and Machine Science). The ﬁnal goal is the publication of research that shows the development of mechanical engineering and particularly MMS in all technical aspects, even in very recent assessments. Published works share an approach by which technical details and formulation are discussed, and discuss modern formalisms with the aim to circulate research and technical achievements for use in professional, research, academic, and teaching activities. This technical approach is an essential characteristic of the series. By discussing technical details and formulations in terms of modern formalisms, the possibility is created not only to show technical developments but also to explain achievements for technical teaching and research activity today and for the future. The book series is intended to collect technical views on developments of the broad ﬁeld of MMS in a unique frame that can be seen in its totality as an Encyclopaedia of MMS but with the additional purpose of archiving and teaching MMS achievements. Therefore, the book series will be of use not only for researchers and teachers in Mechanical Engineering but also for professionals and students for their formation and future work. The series is promoted under the auspices of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). Prospective authors and editors can contact Mr. Pierpaolo Riva (publishing editor, Springer) at: [email protected] Indexed by SCOPUS and Google Scholar.

More information about this series at http://www.springer.com/series/8779

Jianrong Tan Editor

Advances in Mechanical Design Proceedings of the 2019 International Conference on Mechanical Design (2019 ICMD)

123

Editor Jianrong Tan Zhejiang University Hangzhou, China

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-981-32-9940-5 ISBN 978-981-32-9941-2 (eBook) https://doi.org/10.1007/978-981-32-9941-2 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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

Preface

Since prehistory times, the ancient inhabitants of the earth have made the design as a way to meet needs and increasingly complex. In the days of the great scientist James Watt, who worked on the improvisation of the steam engine, mechanical engineering started developing rapidly and systematically. And nowadays, the design factor has become one of the most important aspects of mechanical engineering. With a detailed design and engineering process, it can always save lots of costs and improve efﬁciency. China is moving away from a country with a large manufacturing scale to one with strong manufacturing capabilities. In this process, the ability of design is a key to the role translation from a follower to an initiator in the ﬁeld of manufacturing for China. Every step to realize the goal of manufacturing enhancing national strength strategy needs the improvement of design capabilities. With the development of artiﬁcial intelligence, intelligent design becomes possible and provides an excellent opportunity for leaping development for China. Further development requires us to insist on sustainable development as the basic principle. Therefore, green manufacturing will be an inevitable choice for the industry’s development, and it is becoming one of the factors to measure the manufacturing comprehensive strength of a country. In the tide of manufacturing development, it needs us to seize the opportunities and take actions, so that green manufacturing can be developed with the help of intelligent design and the national manufacturing strength will be improved relying on green manufacturing. In order to promote the development and integration of mechanical design-related disciplines, information exchange on the latest developments in the ﬁeld of mechanical design, and mutual understanding of demands with industry companies. The 2019 International Conference on Mechanical Design (2019 ICMD) has one theme as Intelligent Design Green Manufacturing. The conference is going to be held in Huzhou, China, during October, and it is a leading conference in the ﬁeld of mechanical design in China and aims to provide an international platform for researchers, scholars, and scientists to present their research advances and exchange their ideas. Based on the annual meeting of Mechanical Design Branch of China Mechanical Engineering Society, the v

vi

Preface

conference is proposed once every two years since 2017 and we hope to establish a brand name of its kind for the development of the mechanical design. With over 148 submissions, the rigorous review process was held with detailed and in-depth comments on each paper. This resulted in the acceptance of 94 papers with an additional period for authors to revise their papers following the reviewers’ comments. We are very pleased with the overwhelming support from authors and from the communities and with the excellent work from the authors and their serious effort to improve and ﬁnalize the papers. This book presents a vivid development of mechanical engineering and is a collection of the work in design and development across the disciplines. With the contributions from the authors, this book delivers a lasting impact on the study and development of the mechanical design. In compiling this book, we are pleased to see the variety of the topics, the depth of the study, and the wide range of the applications of innovative design theory. The development of innovative mechanisms is seen in the book contributing to the development of the mechanical design and presents a strong part for advancing the knowledge in the ﬁeld and for the economical development. We thank all the authors for their contributions and meticulous manner in preparing their manuscripts and thank all the reviewers for their rigorous review and detailed comments to help authors to improve their papers. Further, we thank Jingjun Yu for his dedication and persistence in checking every manuscript and contacting the authors for their revision of the manuscripts and for ﬁnalizing this book. We have received immense support from the China Association for Science and Technology (CAST), The National Natural Science Foundation of China (NSFC), the Chinese Mechanical Engineering Society (CMES), and Chinese Academy of engineering. In organizing the 2019 international conference of mechanical design, we are grateful to members of the scientiﬁc committee, the program committee, and the chairs/co-chairs for rigorous peer review of papers.

Committees

General Chair Jianrong Tan

Zhejiang University, China

Program Committee Zongquan Deng Yanmin Zhang Feng Gao Peihua Gu Xu Han Zhifeng Liu Stephen Lu Xianmin Zhang Nam P. Suh Moshe Shpitalni Rupeng Zhu Caichao Zhu Renbin Xiao Sami Kara Yimin Zhang Guanghong Duan Yoram Koren Jack Hu Zhenqiang Yao Wei Chen Runhua Tan

Harbin Institute of Technology, China Chinese Mechanical Engineering Society, China Shanghai Jiao Tong University, China Shantou University, China Hebei University of Technology, China Hefei University of Technology, China University of Southern California, USA South China University of Technology, China Massachusetts Institute of Technology, USA Technion - Israel Institute of Technology, Israel Nanjing University of Aeronautics and Astronautics, China Chongqing University, China Huazhong University of Science and Technology, China University of New South Wales, Australia Shenyang University of Chemical Technology, China Tsinghua University, China University of Michigan, USA University of Michigan, USA Shanghai Jiao Tong University, China Northwestern University, USA Hebei University of Technology, China

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Tetsuo Tomiyama Liyang Xie Xilun Ding Hongsheng Ding Guofu Yin Paul Maropoulos Chao Jiang Shuxin Wang Jian S. Dai Yinan Lai Xiangfeng Liu Weizhong Guo Shijing Wu Rainer Stark Hui Liu Jun Hong Xiongping Du Ying Xi Peiqi Ge Zhong You Xiangyang Xin Jingjun Yu Wei Sun Zhongrong Zhou Geng Liu

Committees

Cranﬁeld University, UK Northeastern University, China Beihang University, China Beijing Institute of Technology, China Sichuan University, China Aston University, UK Hunan University, China Tianjin University, China King’s College London, UK National Natural Science Foundation of China, China Tsinghua University, China Shanghai Jiao Tong University, China Wuhan University, China TU Berlin, Germany Beijing Institute of Technology, China Xi’an Jiaotong University, China Missouri University of Science and Technology, USA Tongji University, China Shandong University, China Oxford University, UK Jiangnan University, China Beihang University, China Dalian University of Technology, China Southwest Jiaotong University, China Northwestern Polytechnical University, China

Contents

Multi-hierarchy Carbon Footprint Analysis and Low-Carbon Design Improvement Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hong Bao, Sheng Guo, and Qing Di Ke

1

The Design and Dynamics Analysis of Cylindrical Roller Surface Unfolding Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yudong Bao, Xiaojian Chen, Chengyi Pan, Yanling Zhao, and Liqun He

12

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning of the Large Storage Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wanghui Bu, Weiqi Zhou, Laixin Fang, Jing Chen, Xianghua An, and Jingkai Huang

28

A Novel Biomimetic Design Method Based on Biology Texts Under Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bowen Chen, Liang Chen, Xiaomin Liu, and Hao Dou

41

Numerical Simulation on Effect of Graphene Doped Morphology on Heat Transfer Efﬁciency of Anti-/deicing Component . . . . . . . . . . . . Long Chen and Zhanqiang Liu

52

Numerical Analysis on Load Sharing Characteristics of Multistage Face Gears in Planetary Transmission . . . . . . . . . . . . . . . Xingbin Chen, Qingchun Hu, and Chune Zhu

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Reachable Matrix and Directed Graph – Based Identiﬁcation Algorithm of Module Change Propagation Path for Product Family . . . Xianfu Cheng, Liyun Wan, and Jian Zhou

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6-DOF Industrial Manipulator Motion Planning Based on RRT-Connect Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chengren Yuan, Guifeng Liu, and Wenqun Zhang

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Contents

A Precise Identiﬁcation and Matching Method for Customer Needs Based on Sales Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Xingpeng Chu, Jian Zhang, Uday Shanker Dixit, and Peihua Gu Optimization Design of the Non-magnetic Drill Rod for Directional Drilling in the Coal Mine . . . . . . . . . . . . . . . . . . . . . . . . 113 Dayong Tang and Lu Liu Research on Product Redesign Process Based on Functional Analysis . . . 125 Yafan Dong, Runhua Tan, Peng Zhang, Wei Liu, and Ruiqin Wang Analysis of Characteristics and Structure Optimization of Anti-rolling Torsion Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Yahong Dong and Yuejin Shang Coupling Mechanism of Errors in the Planetary Roller Screw Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Chuanming Du, Geng Liu, Junqi Liu, and Shangjun Ma A Novel Tooth Contact Analysis Method Based on Value Iteration . . . . 162 Jinfu Du, Zhengrong Wang, Kai Liu, and Yiteng Gao Rigidity Synthesis and Machining Error Analysis of Machine Tool Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Pengyu Hao, Gongxue Zhang, Zijian Pei, and Xianming Gao Thermodynamic Lubrication Performance and Stability for a Deep/Shallow Pocket Hybrid Bearing Considering Bubbly Oil . . . . . . 189 Hong Guo, Shuai Yang, Ningning Wu, and Ruizhen Li Time Delay Chen System Analysis and Its Application . . . . . . . . . . . . . 202 Hongjun He, Yan Cui, Chenhui Lu, and Guan Sun The Driver-in-the-Loop Simulation on Regenerative Braking Control of Four-Wheel Drive HEVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Hexu Yang, Xiaopeng Li, Pengxiang Li, and Yu Gao Strength and Modal Analysis of High Speed EMU Gearbox Housing . . . 223 Haijun Huang, Bichao Yin, Chaowen Wang, and Haiyan Zhu Indirect Adaptive Fuzzy Sliding-Mode Control for Hydraulic Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Xin Huang, Yi Wan, Yao Sun, and Jiarui Hou Improvement and Research of Tennis Training Machine . . . . . . . . . . . . 243 Huiqiang Guo, Jianye Pan, Wen Liu, and Yiling Yue Design of Flagstone Transport Device Based on TRIZ Theory . . . . . . . 253 Fan Jiang, Jian Shen, Tao Zhu, and Jinfeng Wen

Contents

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Fault Analysis and Structure Optimization of End Face Seal of Hydraulic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Haixiang Jiang Innovative Design of Leeward Surface of Pin Fin in Flame Arresters Fitted in Explosion Relief Valve . . . . . . . . . . . . . . . . . . . . . . . 274 Lanfang Jiang, Shuyou Zhang, Xin Shu, Weina Hao, Yun Ren, and Zhiya Chen Analysis and Optimization of Performance Under Operating Condition of Thrust Aerostatic Bearing with Vacuum Pre-load . . . . . . . 288 Mengyang Li, Qiu Hu, PinKuan Liu, and Ming Huang A Study on Innovative Design of Rotary Pile Foundation Drilling Machine Based on TRIZ Theory . . . . . . . . . . . . . . . . . . . . . . . . 302 Fuxing Li Design and Optimization of Focusing X-Ray Telescope Based on Intelligent Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Liansheng Li, Zhiwu Mei, Jihong Liu, Fuchang Zuo, Jianwu Chen, Hanxiao Zhang, Hao Zhou, and Yingbo He Research of Dormitory Furniture Design Based on Group Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Lin Li, Xupeng Wang, and Chunqiang Zhang Analysis and Extraction of Consumer Information for the Evaluation of Design Requirement Depending on Consumer Involvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Shipei Li, Dunbing Tang, Qi Wang, and Haihua Zhu Reduction in Aerodynamic Resistance of High-Speed Train Nose Based on Kriging Model and Multi-objective Optimization . . . . . . . . . . 354 Tian Li, Deng Qin, Le Zhang, Jiye Zhang, and Weihua Zhang Research and Simulation on Pilot Conﬁguration in Multi-antenna System Based on Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Ying Li, Lei Cui, and Zhe Zhang Development and Experiment of an XhYhZ Micro-motion Stage Based on a Straight-Beam Three-Quarter Round Type Flexure Hinge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Junlang Liang, Lanyu Zhang, Jian Gao, Gengjun Zhong, Guangtong Zhao, and Jindi Zhang An Improved PC-Kriging Method for Efﬁcient Robust Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Qizhang Lin, Chao Chen, Fenfen Xiong, Shishi Chen, and Fenggang Wang

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Electro-Mechanical Response of a Cracked Piezoelectric Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 Chao Liu, Wenguang Liu, and Yaobin Wang A Time-Variant Reliability Analysis Method Considering Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Jingfei Liu, Chao Jiang, and Xiangyun Long Research on Tooth Proﬁle Error of Non-circular Gears Based on Complex Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Yongping Liu, Fulin Liao, and Changbin Dong Topology Optimization Design of the Monocoque Bus Body Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Zhuli Liu, Xiangyu Huo, Hao Zhou, and Xiaoguang Wu Analysis of Wind Vibration Response of Transmission Tower . . . . . . . . 466 Zhuli Liu, Beibei Liu, Zhuan You, and Mengli Li Design of a Stabilize Device for Heavy Oil Transportation in Water Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Haoran Lu, Fan Jiang, Yuliang Chen, Xiaolong Qi, and Zhongmin Xiao Design of Spatial Lissajous Trajectory Vibrating Screen . . . . . . . . . . . . 493 Zhipeng Lyu and Sizhu Zhou Part Retrieval Technology Based on Geometric Shape and Topological Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Ning Ma, Bo Yang, and Junzhi Shang Analysis of Tool Wear and Wear Mechanism in Dry Forming Milling for Rail Milling Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 Chao Pan, Xiaoshan Gu, Mulan Wang, and Baosheng Wang Wear Numerical Simulation and Life Prediction of Microstructure Surface Unfolding Wheel for Steel Ball Inspection . . . . . . . . . . . . . . . . . 518 Chengyi Pan, Xia Li, Hepeng Wang, Yanling Zhao, Jingzhong Xiang, Sihai Cui, and Yudong Bao Design of Passive Gravity Balance Mechanism for Wearable Exoskeleton Suit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Chenxi Qu, Peng Yin, Xiaohua Zhao, and Liang Yang Design and Analysis of Underwater Drag Reduction Property of Biomimetic Surface with Micro-nano Composite Structure . . . . . . . . 546 Xuezhuang Ren, Lijun Yang, Chen Li, Guanghua Cheng, and Nan Liu A Method of Parametric Stability Region Determination for Non-linear Gear Transmission System . . . . . . . . . . . . . . . . . . . . . . . 560 Dongping Sheng, Xiaozhen Li, and Rupeng Zhu

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Performance and Parameter Sensitivity Analysis of Finger Seal with Radial Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 Hua Su Calculation and Experimental Study on Comprehensive Stiffness of Angular Contact Ball Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Peng Sun, Weifang Chen, Chuan Su, and Yusu Shen Design of a New Hydraulic Manipulator with Kinematic and Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Yao Sun, Yi Wan, Xichang Liang, Xin Huang, and Ziruo Liu Research on Electro-Hydraulic Servo System Based on BP-RBF Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 Tao Chen, Wenqun Zhang, and Jianggui Han Collision Simulation of GQ70 Light Oil Tank Car at the Level Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Maopeng Tian, Ronghua Li, Xiujuan Zhang, and Miao Jin Reason Study of Collision Between Valves and Piston of Diesel Engine Valve Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 Dameng Wang, Xiujuan Zhang, Deyu Yue, and Weipeng Fan Macroscopic Topology Optimization of Fusion Cages Used in TLIF Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 Hongwei Wang, Yi Wan, Xinyu Liu, Bing Ren, Zhanqiang Liu, Xiao Zhang, and Mingzhi Yu Fuzzy Optimization Design of Disc Brakes Based on Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Jianbin Wang and Jishu Yin Modeling Design and Evaluation of Rotary Tiller Based on Multidisciplinary Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 Wang Jianwei, Jianmin Zhang, and Qin Yang Experimental Investigation of Thermal Elasto-Hydrodynamic Lubrication Based on Temperature Control . . . . . . . . . . . . . . . . . . . . . . 686 Shuaihong Yu, Yazhen Wang, Jiacheng Shen, and Huihui Yue Study on Nonlinear Characteristics of Spatial Spreading Mechanism with Multiple Clearance Joints . . . . . . . . . . . . . . . . . . . . . . 697 Wenzhou Lin, Xupeng Wang, Xiaomin Ji, and Chunqiang Zhang A Novel Design of Tapered Sub-array Structure for SSPS-OMEGA . . . 708 Tong Wu, Yingzhong Tian, Meng Li, and Long Li

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Research on Multiresponse Robustness Optimization for Unmanned Aerial Vehicle Electrostatic Spray System . . . . . . . . . . . . . . . . . . . . . . . 719 Yangdong Wu, Jiajie Lu, and Yiquan Wang Research on Trajectory Optimization of Six-Axis Manipulator Based on Watchcase Polishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Xiang Wang, Ying Xi, Chao Gu, and Mengru Li Reverse Modeling of the Helix Roller in the Omni-Directional Wheel Based on NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 Xie Xia Research on Automatic Matching Model of Power Battery . . . . . . . . . . 749 Chuanfu Xin, Fengxia Zhao, Yujin Wu, and Jianshe Gao Random Vibration Analysis of Optical Adjustable Frame Based on ANSYS Workbench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 Nan Xu, Yuan Wang, and Kelin Xu Analysis of Inﬂuence of Eccentricity Error on Transmission Performance of Micro-segment Gears . . . . . . . . . . . . . . . . . . . . . . . . . . 769 Rui Xu, Lielong Wang, Peidao Pan, and Kang Huang Research on the Design of Wearable Equipment for Intelligent Emotion Detection for Empty Nester . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 Yan-min Xue, Chang Ge, Shi-yi Xu, Xu-yang Zhang, and Bo-xin Xiao Narrative Design of Old Brand Image: A Case Study of Demaogong . . . 795 Yanmin Xue, You Wu, Yihui Zhou, and Yang Liu Study on the Theory and Practice of Mechanical Design . . . . . . . . . . . . 804 Yunna Xue, Xuehui Shen, and Baolin Wang Simulation Analysis of Pipe Bending Under Multiple Conditions . . . . . . 811 Guodong Yi, Zhenan Jin, and Shaoju Zhang Simulation Analysis of Sealing Performance of Double-Offset Butterﬂy Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Guodong Yi, Shaoju Zhang, and Zhenan Jin Motion Performance Analysis of the Sawyer Ankle Rehabilitation Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 Yongfeng Wang, Xiangzhan Kong, Jing Yang, and Guoru Zhao Automated Sustainable Low-Carbon Design of Offshore Platform for Product Life Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 Qianyi Yu and Bin He Comfort of Minors’ Sitting Posture in Learning Based on Motion Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864 Chunqiang Zhang, Xiaomin Ji, Yanmin Xue, and Chunmei Zhang

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Lightweight Design of Dump Truck Frame Based on Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 Gongxue Zhang, Sen Yang, Shanshan Guo, and Qichen Niu Topological Optimization of the Front Beam in Metal Extruders . . . . . 888 Xugang Zhang, Peilin Yang, Xiaole Cheng, and Yi Hou Car Radiator Upper Beam Stamping Process Design and Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 Zhiyuan Wan and Yinping Chen A Fast Tacho-Less Order Tracking Method for Gear Fault Diagnosis Under Large Rotational Speed Variation Conditions Based on Multi-stage Generalized Demodulation . . . . . . . . . . . . . . . . . . 904 Qi Zhou, Wenjin Liu, and Chaoqun Wu Gear Fault Diagnosis Under the Run-Up Condition Using Fractional Fourier Transform and Hilbert Transform . . . . . . . . . . . . . . 918 Qi Zhou, Chaoqun Wu, and Qingrong Fan An Empirical Study of the Effect of the Completeness of Hands-on Training Design Projects on the Preservation of Tacit Knowledge . . . . . 944 Yihui Zhou, Wenzhou Lin, Zhi Qiao, and Xiaozhou Li Research on Structural Size Optimization of 3-TPT Parallel Mechanism Based on Stiffness Characteristics . . . . . . . . . . . . . . . . . . . . 952 Chunxia Zhu and Chengzhu Hu Vibration Response Analysis of Gearbox Housing of High Speed Train Under Wheel and Rail Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 971 Haiyan Zhu, Xiao Su, Lei Tao, and Qian Xiao An Intelligent Fatigue Life Prediction Method for Aluminum Welded Joints Based on Fatigue Characteristics Domain . . . . . . . . . . . . 977 Li Zou, Hongxin Li, and Wei Jiang Optimum Design of Radiation Well Horizontal Drilling Rig Based on TRIZ and Bionics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 Chenghao Liu, Changqing Gao, Bo Yang, and Zhenghe Xu Innovative Design of Reversing Device and Rod Loading Device of Horizontal Drilling Rig Based on TRIZ . . . . . . . . . . . . . . . . . . . . . . . 1002 Shifeng Sun, Changqing Gao, Bo Yang, and Zhenghe Xu AGV Trolley Tray Based on Honeycomb Paper-Based New Material . . . 1010 Jingjing Yang, Xiaoyi Jin, and Anran Wang The Study of Electric Field Distribution on Small Holes in Pulse Electrochemical Machining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018 Zhaolong Li

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Reliability Optimization Design of New Automatic Tensioning Belt-Gear Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030 Chengyi Pan, Guanqun Cao, and Yuanqi Tong Design and Simulation Analyses of a Five-Wheeled Stair-Climbing Mechanism Based on TRIZ Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 Chengyi Pan and Yuanqi Tong Extensible Innovation Design of Globoidal Cam Deceleration Mechanism Based on Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054 Shengyang Tian, Qingshan Gong, Guangguo Zhang, Mingmao Hu, and Yuemin Wu Tension Analysis of Small Motor Stator Winding Tensioning Process . . . 1070 Yanling Zhao, Zhao Zhang, Jingzhong Xiang, and Yudong Bao Study on Contact Dynamics of Cylindrical Roller Feeding Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 Yanling Zhao, Xinyue Wang, and Yudong Bao Trajectory Planning for Winding Process of Small-Sized Motor Stator Winding Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 Yanling Zhao, Linqiang Wang, Jingzhong Xiang, and Yudong Bao Experimental Study of a Solid-Liquid Mixing and Conveying Pump with Variable Flow and Proportion . . . . . . . . . . . . . . . . . . . . . . . 1109 Wei Liu, Qian Tang, Wenzhe Cai, Pinghua Liang, Zongmin Liu, and Xiaojie Fan Effect of Pore Parameters on Lubrication Performance of Oil-Containing Cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124 Tingting Yin, Yuanyuan Li, Ke Yan, Pan Zhang, Yongsheng Zhu, and Jun Hong Defects Detection System for Fluorescent Coating of Metal Plate Based on Machine Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 Yujin Wu, Fengxia Zhao, Chuanfu Xin, Jianshe Gao, and Zhigao Chen Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151

Multi-hierarchy Carbon Footprint Analysis and Low-Carbon Design Improvement Method Hong Bao1,2(&), Sheng Guo1, and Qing Di Ke1 1

HeFei University of Technology, 193 Tunxi Road, Hefei 230009, China [email protected] 2 The State Key Laboratory of Fluid Power and Mechatronic Systems of Zhejiang University, No. 38 Zheda Road, Xihu District, Hangzhou 310027, China

Abstract. Multi-hierarchy carbon footprint analysis for products is proposed to support product low-carbon design. A multi-hierarchy carbon footprint analysis model is established. The analysis model of product carbon footprint is established by analyzing the factors which influence product carbon footprint. Carbon footprint of module unit is calculated and allocated through function-structure mapping. By integrating sensitivity analysis into multi-hierarchy carbon footprint analysis for products, the application of this method in design improvement is discussed. Finally a case study of refrigerator is done to illustrate the feasibility and applicability of this method. Keywords: Multi-hierarchy carbon footprint analysis Function-structure mapping Low-carbon design Integrating sensitivity analysis Design improvement

1 Introduction With the growing tension of the global environment, energy and resource use, technical barriers in the international trade has become increasingly harsh, the green and lowcarbon attributes of products becomes as important as function, performance and quality control. Low carbon design based on product life cycle, which aims to improve the performance of its carbon emissions, has become one of the important trends in the ﬁeld of product design. On the premise of not reducing the effectiveness of products, low carbon design based on product life cycle considers energy efﬁciency improvements and resource conservation at the design stage, and minimizes the carbon footprint of products by the key technologies of energy and material saving design, which is of great signiﬁcance in supporting and guiding the design process of products. The performance analysis of product carbon emission is the basis of low-carbon design based on life cycle. Many scholars have carried on the research. Haapala et al. [1, 2] built an energy consumption prediction model of the product manufacturing process by using basic product information and processing information. By analyzing environmental impact of products, energy and raw material consumption in the life cycle as well as the emissions of waste are reduced. Kainuma et al. [3] predicted the demand for energy services with an indicator of ecological and economic factors by © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 1–11, 2020. https://doi.org/10.1007/978-981-32-9941-2_1

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using Asian-paciﬁc integrated model, and made comparisons and decisions on the existing technical data and energy data. Venkatesh et al. [4] built an uncertainty assessment model of the emissions of greenhouse gases in the life cycle of fossil fuel by using the method of process framework and statistical modeling. On the basis of product life cycle energy analysis, Qi [5] built a life cycle energy model, described the dynamic characteristics of energy consumption in product life cycle, and gave the formula of energy consumption. Zhang [6] conducted a life cycle assessment of greenhouse gas emissions in the use process of mechanical products, established the parameters and evaluation boundary for life cycle inventory analysis, and then carried out the inventory calculation and the results comparison. Xu [7] established a cold chain carbon footprint model based on the product life cycle time dimension. By analyzing and calculating the model, the accurate method of quantifying the carbon footprint of cold chain workstation in unit time was given. Zhang [8] proposed a lowcarbon product design information model based on carbon footprint, and a dual progressive carbon footprint feature location method based on product structure tree and detailed design parameters after analyzing the content of low-carbon design decision information. Inakollu [9] proposed a practical life cycle assessment method to realize the carbon footprint calculation of ﬁber production. Restrepo [10] proposed a life cycle assessment (LCA) method for carbon footprint analysis to analyze carbon emissions from bamboo production. However existing researches are mainly focused on the calculation of carbon emissions of product and the quantitative analysis on the perspective of energy consumption while the research on the mapping between product carbon emissions and design parameters is still not enough, which can not provide a better guide for the designer to improve the low-carbon design. This article builds a product model of multi-hierarchy carbon footprint analysis and proposes a multi-hierarchy carbon footprint analysis method on the basis of the carbon footprint analysis of product layer and module unit layer. It combines the carbon footprint analysis method with sensitivity analysis and discusses its application into low-carbon design improvements, which provides practical analytical tools and application methods for low-carbon design.

2 Multi-hierarchy Carbon Footprint Analysis Model Product carbon footprint is the sum of greenhouse gas emissions generated in all stages of the product life cycle which is used to measure their contribution on global warming in particular system boundaries. Product carbon footprint mainly includes four steps: selection of functional unit; determination of system boundary; data collection and calculation of carbon footprint [11]. On the basis of the acquisition of customer’s demand for products, product layer can be decomposed to the module unit layer according to product function decomposition and function structure mapping. Through the level mapping of carbon emissions of the acquisition of raw materials, manufacturing, using and recycling etc. in each stage of the life cycle emissions, all levels of carbon footprint can be distributed and quantiﬁed. On the basis of the acquisition of the product layer and the result of module unit layer carbon footprint analysis, it is necessary to identify the key life cycle

Multi-hierarchy Carbon Footprint Analysis

3

stage and module, research the mapping relationship between key design parameters and carbon emissions based on the sensitivity analysis to support the following low carbon design improvements. Multi-hierarchy carbon footprint analysis model is shown in Fig. 1.

The existing product model

Product layer

Product carbon footprint

Function structure mapping

Carbon footprint map

Module layer

Module unit carbon footprint

BOM

Target definition and scoping Inventory analysis

Impact assessment

Impact assessment Multi-hierarchy carbon footprint analysis

based on the sensitivity analysis of key design parameters and the mapping relationship and improvement direction of the various levels of carbon emissions

Fig. 1. Multi-hierarchy carbon footprint analysis model

3 Multi-hierarchy Carbon Footprint Analysis Method 3.1

Product Carbon Footprint Analysis

Product carbon footprint can be regarded as the superposition of carbon emissions generated in each stage of life cycle, carbon emissions in each stage can be analyzed through decomposing the life cycle. Carbons emissions in life cycle can be divided into direct carbon emissions generated by material flow and indirect carbon emissions caused by energy flow. Here ﬁve stages are considered including acquisition of raw materials and energy, manufacturing, distribution, use, end of life. We need to ignore the life cycle stages according to the speciﬁc circumstance in the actual calculation process. The product carbon footprint based on the energy flow and material flow is shown in Fig. 2. The quantitative analysis model of product carbon footprint can be expressed as: G Gm þ Ge þ Gp þ Gd þ Gu þ Gr

ð1Þ

Gi ¼ Gienergy þ Gimaterial

ð2Þ

Where Gm , Ge , Gp , Gd , Gu , Gr respectively represents the carbon emissions in stage of acquisition of raw materials, stage of acquisition of energy, stage of manufacturing, stage of distribution, stage of use, stage of recycling. Gi represents a stage of life cycle,

4

H. Bao et al. Natural resources

Raw materials, energy

Exploitat … … -ion

Manufacturing

Process Smelting

Component assembly

……

Product assembly

*

Transportion

Transport Total carbon emissions

Remanufacturing reuse Materials recovery

Dismantling Energy recovery Scrap use

recycling Carbon emissions generated by energy consumption energy flow

*

Carbon emissions generated by material consumption material flow

Fig. 2. The products processes of carbon footprint based on the energy flow and material flow

Gienergy and Gimaterial respectively represents the direct carbon emissions generated by material flow and indirect carbon emissions caused by energy flow. Carbon emission in each stage of product life cycle is based on the indirect calculation of carbon emission factors. Its quantitative methods of carbon emission and factors which are needed to be considered are introduced as follows. • Acquisition of raw materials and energy: the carbon emission in the stage of acquisition of raw materials mainly depends on the energy consumption and the direct emission of greenhouse gases in the process of material production. The carbon emission generated by the energy consumption mainly depends on the type and physical quantity of energy consumption and the emission factors corresponding to the energy in production activities. The carbon emission caused by the direct emission of greenhouse gases depends on the type and physical quantity of material consumption and the emission factors corresponding to the material in production activities. The carbon emission in the stage of acquisition of energy mainly depends on the type and physical quantity of energy consumption and the emission factors corresponding to the energy in production activities. Here it is conﬁned to production of the primary energy. • Manufacturing: The carbon emissions in this stage mainly depends on the energy consumption and the direct emission of greenhouse gases, which is reflected in the energy consumption of manufacturing of basic component, the direct emissions of greenhouse gases in the process of manufacturing, energy consumption of product assembly and the direct emission of greenhouse gases in the process of assembling. The energy consumption here is conﬁned to the combustion of primary energy and the use of secondary energy. The carbon emission can be calculated by the physical quantity of each kind of energy in production activities multiplied by the sum of emission factors corresponding to the energy production. The carbon emission caused by the emissions of greenhouse gases can be calculated by the physical

Multi-hierarchy Carbon Footprint Analysis

5

quantity of each kind of greenhouse gas in production activities multiplied by the sum of equivalent corresponding to the greenhouse gas. • Distribution: The carbon emission in this stage is generated when basic components are sent to manufacturers; products are distributed to vendors, products are distributed to customers. It mainly depends on the types of distribution, distribution distance, the amount of energy consumption per km and the corresponding energy emission factors. • Use: The carbon emission in this stage mainly depends on the electricity consumption and direct emissions of greenhouse gases. The carbon emissions caused by the electricity consumption on average running time can be calculated by the power consumption measured per day multiplied by average running time and average power emission factor of national grid. The carbon emission caused by emission of greenhouse gases can be calculated by the physical quantity of each kind of greenhouse gases in the stage of using multiplied by the sum of equivalent corresponding to the greenhouse gas. The formula of speciﬁc products also varies. Take the refrigerator as an example: The carbon emissions caused by emission of greenhouse gases can be calculated by the weight of refrigerant multiplied by leakage factors in the stage of using and the equivalent of carbon dioxide of the refrigerant. • End of life: The carbon emission in this stage mainly depends on the energy consumption and the direct emissions of greenhouse gases. The carbon emissions caused by the energy consumption can be calculated by the physical quantity of each kind of energy in recycling activities multiplied by the sum of the emission factors corresponding to energy production. The carbon emissions caused by the emissions of greenhouse gases can be calculated out by the leakage of each kind of greenhouse gases in recycling activities multiplied by the sum of equivalent corresponding to the greenhouse gas. 3.2

The Quantitative and Distribution Method of Carbon Footprint of Module Unit Layer

Through product function structure mapping, product family is divided into a number of module units of different functions. According to the conventional and green demand of customers, each module can be derived from a number of instances of module units which have the same function and different carbon footprint through the selection and combination of components inside module unit. The quantitative analysis of carbon footprint of module unit mainly includes the selection of functional unit, the determination of system boundary, data collection, the calculation of carbon footprint and the distribution of product carbon footprint. Functional unit is selected as a modular unit. Component elements are the components inside module unit. System boundary is deﬁned as the acquisition of raw materials, the acquisition of energy, manufacturing, distribution, use and end of life. The quantitative model of carbon footprint of module unit can be expressed as:

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Gkj ¼

m X h¼1

GkjMh

þ GkjEh

þ

n X

! kj EPhi

b þ GkjRh

þ GkjPmaterial þ GkjU þ GkjD

ð3Þ

i¼1

Where: Gkj represents the carbon footprint of j-th instance derived from k-th module unit. GkjMh , GkjEh , GkjRh represents respectively the carbon emissions of the h-th component of j-th instance derived from k-th module unit in the stage of the acquisition of raw materials, the stage of the acquisition of energy and the stage of recycling. kj EPhi , b, GkjPmaterial represents respectively the energy consumption, emission factors and emission of greenhouse gas of the h-th component of j-th instance derived from k-th module unit in the i-th manufacturing process. GkjU , GkjD respectively represents the carbon emissions of the jth instance derived from kth module unit in the stage of use and the stage of distribution. The carbon emission of the module unit instance produced in acquisition of raw materials and energy, manufacturing and recycling stages can be indirectly calculated by using the carbon emission factors according to the product carbon footprint quantitative methods afore mentioned, the carbon emission produced in the distribution stage needs to be identiﬁed according to carbon emissions contribution ratio in the entire product system in and the carbon contribution ratio h of this stage can be expressed by the ratio of the weight of the module unit instance to total product weight. Since the function of each module unit are independent, and the influence of carbon emissions in use phase on the entire product system is not the same, the carbon emissions of the module unit instance in use phase mainly from two ways, one is the allocation of indirect carbon emissions resulting from the energy consumption in the operation process of the products, the other is its internal greenhouse gas leak caused by the direct carbon emissions. The allocation ratio of carbon emissions η produced by energy consumption during the use phase of each module unit instance mainly depends on the proportion of its energy consumption during operation of the product in total energy consumption. The energy consumption of each module unit instance can be determined through the energy storage of the system and the coupling and superposition of energy function of all component consumption based on the behavioral modeling of energy [12]. The internal greenhouse gas leak caused by the direct carbon emissions of the module unit instance in use phase can be quantiﬁed through calculating the internal containing greenhouse gas leakage. The formula to quantify the carbon emissions of the module unit instance system allocated by the product system can be expressed as: GkjA ¼ gkj EUenergy b þ

m X

kj WUi li pi þ hkj GD

ð4Þ

i¼1

Where: GkjA —The allocation of carbon emissions in the j-th instance derived by the k-th module unit of product in the product system;

Multi-hierarchy Carbon Footprint Analysis

7

EUenergy , GD —Represent the energy consumption the use phase and the carbon emissions generated by the distribution phase of the product; gkj , hkj —Represent the indirect carbon emissions produced by power consumption in the use phase of product system in the j-th instance derived by the k-th module unit of product, and the distribution ratio of carbon emissions in the distribution phase; kj , li , pi —Represent the weight, the leakage rate and the carbon dioxide WUi equivalent of i-th class greenhouse gases within the j-th instance derived by the k-th module unit of product.

4 Sensitivity Analysis Based Carbon Design Improvement Multi-hierarchy carbon footprint analysis result can be used to guide low-carbon design improvements of products. Firstly carbon footprint analysis on the product level should been done, if the result is higher than the set target value, the carbon emission performance requirement of product meets the design goal, there is no need for low-carbon design improvements; otherwise, there is necessary for module unit-level carbon footprint analysis. By comparing the carbon footprint quantitative analysis results of each modular unit, we can determine the module unit with higher impact on the carbon footprint of product and its distribution of the carbon emissions in various stages of the life cycle and ﬁnd a signiﬁcant life cycle stage. By analyzing the relationship between major changes in structural parameters and direct or indirect carbon emissions change in the signiﬁcant life-cycle stage of the key module unit, we will ﬁnd that the structural parameters are sensitive or not to the increase of carbon emissions of module unit, which can provide a direction for life cycle carbon design improvement of products from the aspects of energy efﬁciency improvement and resource conservation. To calculate the degree of sensitivity, the functions between life-cycle carbon emissions of the module unit and relevant structural parameters must be established, which involve the principle of product and related theories. For example, the function relation between the indirect carbon emissions due to energy consumption in the use phase and the relevant structural parameters of the refrigerator door foam layer is: GUenergy ¼

kDT t1 be x

ð5Þ

Where, k, DT, x, t1 and be —Respectively, the conductivity of the foam layer, the door body temperature difference between both sides, the foam layer thickness, the door body working hours and the electricity emission factors. To obtain the sensitivity of the impact of GUenergy during the changes of the structural parameters, a further mathematical transformation is necessary. For example, in order to ﬁnd x’s impact on the sensitivity of GUenergy , the mathematical derivation is as follows:

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H. Bao et al.

@GUenergy kDT ¼ 2 t1 be x @x

ð6Þ

When the premise of other variables remain constant,, the sensitive degree between indirect carbon emissions and structure parameters x can be gotten by calculating the speciﬁc change value of GUenergy after each additional unit of x (0.01). The sensitivity between the other parameters can be gotten by the same method. To guide the lowcarbon design improvements according to the result of sensitivity analysis, the improvement direction of key design parameters can be further deﬁned until the product level carbon footprint is higher than the set target.

5 Examples Demonstration Take a refrigerator which is a low-carbon design product of a company for the example, we verify the methodology of multi-hierarchy carbon footprint analysis. A refrigerator can be divided into six module units through function-structure mapping, which includes door system, cabinet system, refrigeration system, attachment and package. 5.1

Product and Carbon Footprint Analysis of Module Unit

System boundary includes use of raw material, manufacturing, distribution, use, end of life. The major system boundary for the carbon footprint analysis of the refrigerator is shown in Fig. 3. In the process of gaining the raw materials, direct greenhouse gases emissions and indirect emissions which is caused by energy consumption in the production process of steel, iron, copper, plastic, foam, freezing medium and etc. are mainly considered. Cyclopentane is used as foam beater and R134a is used as freezing medium. Obtaining carbon emission factor data of the main materials production, energy consumption and greenhouse gas emissions of the processes in the production stage should be considered, which include the manufacturing of key parts in every system, door pre-installed, cabinet pre-installed, the foaming of door, the foaming of box, the assembly of door, the assembly of cabinet, the packing of refrigerant, phosphate coating and the splice between the processes. Energy source is set to power grid in eastern China and the power consumption is about 12kwh in this stage. The foaming process of carbon leakage is negligible because the GWP value of the used cyclopentane is approximately zero. The GWP value of R134a is 1300, its amount is about 50 g, the leak rate is 0.8% in the packing process, and the leakage of the substance can be calculated as 0.4 g. In the distribution stage, the distribution mode is deﬁned as rail and road, and the average distribution distance is 1000 km. The power consumption of the product in use stage is 0.45 kwh and its life is 10 years. Boundary conditions of the recycling stage include refrigerant recovery, crushing, materials recovery and etc. Refrigerant recovery and the crushing process not only require calculating the indirect carbon emissions caused by energy consumption, but also considering the direct carbon emissions caused by the refrigerant leak. Synthesizing the above analysis, carbon emission in life cycle of this refrigerate can be calculated as

Multi-hierarchy Carbon Footprint Analysis

9

1690.09 kgCO2e by using carbon footprint analysis and allocation method of product and module unit. Carbon emissions in recovery stage is negative, which indicates that the stage counteract the carbon emissions of the entire life-cycle. The result of carbon footprint analysis of product and module unit is shown in Fig. 4.

raw material acquisition

manufacturing

distrbution Foam leakage

steel

side plate door tank

iron

side plate cabinet tank

copper foam

energy

PUR recycle

refrigeration attachment

greenhouse gas emissions Lubricant recycle outsourcing Wind classifi cation

door foaming

cabinet preinstalled

Cabinet foaming

coal oil

energy

bracket bracket

assembly

Oil and gas classificati on Magenetism classification

railway

assembly Coolant leakage

energy

assembly

foam leakage

compressor

Coolant plastic

door preinstalled

Coolant leakage Coolant recycling House shredding

road

energy

Manual disassembly

The use of electricity

energy energy

foam leakage recycling

use

Fig. 3. The major system boundaries of carbon footprint analysis for the refrigerator

Fig. 4. The result of carbon footprint analysis of product and its module unit (kgCO2e)

10

5.2

H. Bao et al.

Low-Carbon Design Improvement Based on Sensitivity Analysis

According to the proportion of carbon emissions of life cycle stages in product carbon footprint, the proportion of carbon emissions in the use phase and the use of raw materials are higher than the other life cycle stages, 86% and 15% respectively, which need our attention. By way of carbon footprint analysis of module units, refrigeration system and door are the module units which have the strongest influence on carbon emissions in the use phase of the refrigerator. The proportion in the total carbon emissions in the use phase are 38% and 34% respectively, which are the key module units to be improved. Taking door system as an example, low-carbon design improvement is applied by sensitivity analysis. According to the steps described in Sect. 3, the sensitivity of the relevant structure parameters with the indirect carbon emissions that are caused by energy consumption in the use phase of the foaming layer, which is shown as Table 1.

Table 1. The sensitivity of the structure parameters with the indirect carbon emissions of the foaming layer Number Change

GUenergy

1 2 3 4

−0.188 −0.036 −0.011 −0.042

x increases 1% DT reduces 1% t1 reduces 1% k reduces 1%

From Table 1, the thickness of the faming layer x is the most sensitive design parameters that influence the indirect carbon emissions caused by energy consumption in the use phase of the foaming layer. If the thickness of the faming layer x increases by 1%, carbon emissions will reduce 0.188 kg. Considering the design constraints, the indirect carbon emissions of the foaming layer caused by energy consumption during the use phase can be reduced signiﬁcantly by increasing the thickness.

6 Conclusion A multi-hierarchy carbon footprint analysis model for products is established. On the basis of carbon footprint analysis of product and module unit, a multi-hierarchy carbon footprint analysis method is proposed, and the application of sensitivity analysis in low-carbon design improvement is discussed. The case study illustrated this method can identify the key parameters of low-carbon design and support product low-carbon design effectively. The result of carbon footprint analysis is influenced by some factors which include usage modes by consumer, energy types, service life, there is some uncertainty. The process of life cycle is simpliﬁed, and some data input items are assumed. The application of this method requires the support of abundant basis data, the accumulation of basis data at the present stage is not enough. Therefore the data accumulation and assumption of input parameters is the important condition before the

Multi-hierarchy Carbon Footprint Analysis

11

application of this method. The range extension of product applied by this method and the development of computer aided design tools will be the research activities in the future. Acknowledgment. This work was ﬁnancially supported by the National Science Foundation in China (51505119) and the Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems.

References 1. Haapala, K.R., Khadke, K.N., Sutherland, J.W.: Predicting manufacturing waste and energy for sustainable product development via WE-FabSoftware. In: Proceedings Global Conference on Sustainable Product Development and life Cycle Engineering, pp. 243–250 (2004) 2. Haapala, K.R., Sutherland, J.W., Rivera, J.L.: Reducing environmental impacts of steel product manufacturing. Trans. NAMRI/SME 37, 419–426 (2009) 3. Kainuma, M., Matsuoka, Y., Morita, T.: The AIM/end-use model and its application to forecast Japanese carbon dioxide emissions. Eur. J. Oper. Res. 122, 416–425 (2000) 4. Venkatesh, A., Grifﬁn, W.M., Jaramillo, P.: Uncertainty analysis of life cycle greenhouse gas emissions from petroleum-based fuels and impacts on low carbon fuel policies. Environ. Sci. Technol. 45(1), 125–131 (2010) 5. Qi, Y., Huang, H., Liu, G.: Energy analysis method based on dynamic life cycle. Chin. J. Mech. Eng. 43(8), 129–134 (2007). (in Chinese) 6. Zhang, C., Pu, G., Wang, C.: Comparison of life cycle assessment between e-bike and motorbike. Mach. Des. Res. 19(4), 69–71 (2003). (in Chinese) 7. Xu, X., Li, R.W., Wu, X.L., Zhao, Y., Wang, X.Z., Bao, S., Nyberg, T.: Carbon footprint model and calculation of cold chain workstation based on product life cycle time dimension. Comput. Integr. Manuf. Syst. 24(2) (2018) 8. Zhang, Y., Li, W.Q., Li, Y., Ma, J.L.: Product low carbon innovation design based on carbon footprint information model. Chin. J. Eng. Des. 24(2), 141–148 (2017) 9. Inakollu, S., Morin, R., Keefe, R.: Carbon footprint estimation in ﬁber optics industry: a case study of OFS Fitel, LLC. Sustainability 9(5), 865 (2017) 10. Restrepo, Á., Becerra, R., Tibaquirág, J.E.: Energetic and carbon footprint analysis in manufacturing process of bamboo boards in Colombia. J. Clean. Prod. 126, 563–571 (2016) 11. British Standards Institute (BSI), PAS 2050-Speciﬁcation for the Assessment of the Life Cycle Greenhouse Gas Emissions of Goods and Services (2008) 12. Zhang, J., Wei, X., Zhang, D.: Principle scheme design of mechanical systems based on energy interaction model. China Mech. Eng. 15(9), 820–823 (2004). (in Chinese)

The Design and Dynamics Analysis of Cylindrical Roller Surface Unfolding Mechanism Yudong Bao(&), Xiaojian Chen, Chengyi Pan, Yanling Zhao, and Liqun He School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, China [email protected]

Abstract. In order to design an efﬁcient and stable cylindrical rollers surface unfolding mechanism, and know about the motion of the cylindrical rollers surface during the unfolding process. The surface unfolding model of the cylindrical roller is established. The time taken for roller to achieve the uniform rotation and the rollers vibration on the direction of x-axis are used to evaluate the roller unfolding effect. The factors affecting the roller unfolding efﬁciency and the unfolding stability are determined. The roller accelerating timeconsuming equation and the deformation equations of the unfolding wheel are established. The dynamics of the surface unfolding model for the cylindrical roller is studied. The simulation of the roller unfolding motion is carried out by the Adams software to the roller which is U32 52 mm. The simulation results are consistent with the theoretical analysis results. In the roller surface unfolding process, the roller accelerates ﬁrstly, the vibration of roller is large, and then the uniform rotation is achieved, and the vibration of roller is small. According to the simulation data, the influence of the material of the unfolding wheel, the bottom radius of the unfolding wheel and the minimum distance between the outer edges of the two unrolling wheels on the stability and unfolding efﬁciency of the roller unfolding motion are determined. It is concluded that the bottom radius of the unfolding wheel is 16 mm, the material is silicone rubber and the minimum distance between the outer edges of the two unfolding wheels is 8 mm, the cylindrical roller has the best unfolding effect. According to the simulation results, the cylindrical roller surface unfolding mechanism is designed to meet the requirements of use, and the rationality of the cylindrical roller surface unfolding model is veriﬁed. It provides a theoretical basis for the research of the surface unfolding equipment for the column workpieces. Keywords: Cylindrical roller Vibration

Surface unfolding model Dynamics

This project is supported by Research Special Fund Project of Harbin Science and Technology Innovation Talent (Grant Nos. 2017RAQXJ060, 2016RAXXJ001). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 12–27, 2020. https://doi.org/10.1007/978-981-32-9941-2_2

The Design and Dynamics Analysis of Cylindrical Roller Surface

13

1 Introduction With the development of Chinese rail transportation, the reliability of cylindrical roller bearings has been paid more attention. Cylindrical roller as an important part of the bearing, its surface quality is an important factor affecting the normal operation of the cylindrical roller bearing [1]. In order to get higher detection efﬁciency, machine vision technology has been applied to defect detection of cylindrical rollers surface widely [2]. The effectively unfolding of the cylindrical rollers surface is the premise to detect cylindrical rollers surface defects by machine vision. Therefore, it is necessary to have a reasonable design and dynamic analysis for the cylindrical rollers surface unfolding mechanism. The cylindrical rollers surface unfolding system has been studied by some scholars. A cylindrical rollers surface unfolding system that relies on rollers gravity to accelerate off the slope and then roll on the horizontal raceway at a constant speed is designed by Wang [3]. A cylindrical rollers surface unfolding system that uses electromagnetic force to ﬁx cylindrical rollers on a turntable is designed by Zhang [4]. A cylindrical work piece surface unfolding system is designed by rotating the wheel to drive the roller [5]. Wu presents a twin-screw transmission scheme for surface unfolding of cylindrical roller [6]. Li from Harbin University of Science and Technology designs a cylindrical rollers dimension measuring system [7], which uses a pusher to push the roller rolling on the horizontal raceway for the surface unfolding of cylindrical roller. Tan et al. designs a battery cylinder surface unfolding system, which working principle is that the battery is rotating on its revolution [8], which provided a reference for the design of cylindrical roller unfolding system. Bo et al. proposes a new method is proposed for developable surface reconstruction [9]. Koukash et al. proposes a new cylindrical roller unfolding method, which the laser beam is focused and cleaned by the spatial ﬁlter, the expanded beam is passed through the square wave amplitude grating to form a fringe pattern, which is projected onto the surface of the cylindrical roller to complete the surface unfolding of the cylindrical roller [10]. In this paper, a new cylindrical roller surface unfolding mechanism is designed, and a two-dimensional cylindrical roller surface unfolding model is constructed. The model is used to determine the factors that affecting the roller vibration and acceleration time in the process of the acceleration rotation and uniform rotation. The dynamic characteristic of the cylindrical rollers surface unfolding model is analyzed. The physical prototype is established and the surface unfolding efﬁciency is veriﬁed. The research has provide a reference for the relevant research of cylinder surface defect detection.

2 The Cylindrical Rollers Surface Unfolding Mechanism The cylindrical rollers surface unfolding mechanism consists of stepping motor, gear I, gear II, gear III, bearing assembly, unfolding wheel I, unfolding wheel II and support frame, which is shown in Fig. 1. The unfolding wheel I and the unfolding wheel II are installed on the support frame through the bearing assembly. The gear II is connected to the unfolding wheel I, and the gear III is connected to the unfolding wheel II. The gear I is connected to the output shaft of the stepping motor, and engaging with the gear II and the gear III. When it is working, the cylindrical roller is placed on the middle of

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two unfolding wheels, and the stepping motor is energized. The rotating unfolding wheels drive the cylindrical roller to rotate around its own axis, and unfold the full surface of the cylindrical roller. Unfolding wheel

Gear

Cylindrical roller Gear

Stepping motor

Gear

Unfolding wheel

Bearing

Support frame

Fig. 1. Cylindrical roller unfolding mechanism

3 Establishment and Dynamic Analysis of Surface Unfolding Model of Cylindrical Roller 3.1

Establishment of the Two-Dimensional Model of Cylindrical Rollers Surface Unfolding

In order to understand the dynamic law of the cylindrical rollers surface unfolding process, a two-dimensional model of the cylindrical rollers surface unfolding is established, as shown in Fig. 2.

ylindrical roller nfolding wheel

nfolding wheel

Fig. 2. Two-dimensional model of cylindrical rollers surface unfolding

The Design and Dynamics Analysis of Cylindrical Roller Surface

15

The cylindrical roller, which rotates around its own axis, is driven by the frictional force between the roller and the two unfolding wheels. In order to ensure the stationarity and simpleness of the roller unfolding mechanism, the roller unfolding mechanism should be easy to install and maintain. So the two unfolding wheels have the same material, size and surface roughness, and their axis are placed in the same horizontal plane. Therefore, the cylindrical rollers surface unfolding mechanism could be simpliﬁed into a two-dimensional model in the xoy plane. Through analysis of the two-dimensional model, it can be seen that the axis C of the cylindrical roller with the axis A and the axis B of the two unfolding wheels constitute an isosceles triangle in the xoy plane. The point E which is the contact point of the circle C and the circle A is on the straight line CA, as well the point H which is the contact point of the circle C and the circle B is on the straight line CB. h1 is the angle between the line BC and the vertical direction, h2 is the angle between the line AC and the vertical direction, both h1 and h2 is equal to h which could be calculated by the following equation. h ¼ arcsin

R þ d=2 Rþr

ð1Þ

where R is the bottom radius of the unfolding wheel, and r is the bottom radius of the cylindrical roller, and d is the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II. 3.2

Dynamic Analysis of the Two-Dimensional Model of Cylindrical Rollers Surface Unfolding

3.2.1 The Acceleration Rotation of Cylindrical Roller At the beginning of the cylindrical roller unfolding motion, the cylindrical roller is placed on the two unfolding wheels which rotates at the same angular velocity, and the roller starts to accelerate. The unfolding model is shown in Fig. 3.

θ1 r Δx1 ω2

FAC E FCA

G

R

Δx2 FBC

Ft2 ω2

H

Ft1

A

y o

θ2 MC α C

B

R

d

x

Fig. 3. Unfolding model of the accelerated rotation for cylindrical roller

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At the beginning of the unfolding movement, the cylindrical roller is greatly vibrated when contacted with the two unfolding wheels at an instant. After the roller is fully contacted with the two unfolding wheels, the roller is accelerated. The force of the cylindrical roller satisﬁes the following equation. G ¼ FAC cos h þ FBC cos h þ Ft2 sin h Ft1 sin h

ð2Þ

0 ¼ Ft1 cos h þ Ft2 cos h þ FAC sin h FBC sin h

ð3Þ

MC ¼ Ft1 r þ Ft2 r

ð4Þ

Ft1 ¼ FAC l

ð5Þ

Ft2 ¼ FBC l

ð6Þ

where Ft1 is the friction that the unfolding wheel I provides for the cylindrical roller, and Ft2 is the friction that the unfolding wheel II provides for the cylindrical roller, and FAC is the support force to the cylindrical roller which is accelerated rotating, and FBC is the support force to the cylindrical roller, and MC is the cylindrical rollers moment of force, and G is the weight of the cylindrical roller, and l is the friction coefﬁcient between the cylindrical roller and the unfolding wheels. The following relationships could be obtained by the formulas (2) to (6). FAC ¼

Gðtan h 1Þ 2ðl2 þ 1Þ sin h

ð7Þ

FBC ¼

Gðl þ tan hÞ 2ðl2 þ 1Þ sin h

ð8Þ

Glrð2 tan h þ l 1Þ 2ðl2 þ 1Þ sin h

ð9Þ

MC ¼

It can be seen from Eqs. (7) and (8) that the cylindrical roller is subjected to different supporting forces during the accelerated rotation phase, which causes the cylindrical rollers to vibrate greatly on the direction of x-axis and the direction of yaxis. The angular acceleration during the rotation of the roller can be obtained by the following formula. MC ¼ Ja

ð10Þ

mr2 2

ð11Þ

J¼

Where a is the angular acceleration of the cylindrical roller, and J is the moment of inertia to the cylindrical roller, and m is the mass of the cylindrical roller.

The Design and Dynamics Analysis of Cylindrical Roller Surface

17

The following relationship can be obtained by the formulas (9) to (11). a¼

glð2 tan h þ l 1Þ rðl2 þ 1Þ sin h

ð12Þ

Where g is the gravitational acceleration. Since the roller does not always accelerate during the unfolding process, when the linear velocity at the contact point of the unfolding wheel and the roller is equal, the angular acceleration of the roller disappears, and the acceleration phase of the roller is completed. The acceleration time of the roller can be calculated by the following formula. Rx2 ¼ rx1 ¼ rat

ð13Þ

Where x1 is the angular velocity of the roller which is rotating at uniform speed, x2 is the angular velocity of the two unfolding wheels, t is the time taken for the roller to accelerate the rotation phase. It can be seen from the Eqs. (12) and (13) that the time t taken for the roller to accelerate the rotation phase can be calculated by the following formula. t¼

x1 rðl2 þ 1Þ sin h glð2 tan h þ l 1Þ

ð14Þ

It can be seen from Eqs. (1) and (14) that if the speciﬁcation of the roller and the detection speed of the roller are known, the radius of the unfolding wheel, the material of the unfolding wheel, and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II are main factors that affecting the accelerate time to the roller. Through the two-dimensional unfolding model of the cylindrical roller during acceleration, it can be seen that the unfolding wheel I and the unfolding wheelII support the cylindrical roller, and also cause deformation by the reaction force from the roller, the deformation amount can be calculated by the following formula. FAC ¼ FCA ¼ K1 Dx1

ð15Þ

FBC ¼ FCB ¼ K2 Dx2

ð16Þ

Where FCA is the force of the cylindrical roller on the unfolding wheel I, and FCB is the force of the cylindrical roller on the unfolding wheel II, and K1 is the stiffness coefﬁcient of the unfolding wheel I, and K2 is the stiffness coefﬁcient of the unfolding wheel II.

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Since the material, size and surface roughness of the unfolding wheel I and the unfolding wheel II are the same, both K1 and K2 are equal to K. The amount of deformation of the unfolding wheel I and the unfolding whee l II can be obtained by the Eqs. (7), (8), (15), and (16). Dx1 ¼

Gðtan h 1Þ 2Kðl2 þ 1Þ sin h

ð17Þ

Dx2 ¼

Gðl þ tan hÞ 2Kðl2 þ 1Þ sin h

ð18Þ

Where Dx1 and Dx2 are the deformation amount of the unfolding wheel I and II respectively during the roller acceleration rotation phase. It can be seen from the Eqs. (1), (17) and (18) that during the accelerated rotation phase of the cylindrical roller, the deformation amount of the unfolding wheel I and II at the contact of the roller is different, which makes the roller have a serious vibration on the direction of x-axis and the direction of y-axis. If the roller speciﬁcation is known, the bottom radius of the unfolding wheel, the material of the unfolding wheel and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II are main factors affecting the deformation of the unfolding wheel. 3.2.2 The Uniform Rotation of Cylindrical Roller When the linear velocity at the point of contact between the cylindrical roller and the unfolding wheel is equal, the roller is rotating at uniform speed, and the roller unfolding model during the uniform rotation is as shown in Fig. 2. The forces relationship of the roller unfolding model following these equation. 0 0 G ¼ FAC cos h þ FBC cos h

ð19Þ

0 0 0 ¼ FAC sin h FBC sin h

ð20Þ

0 is the support force to the cylindrical roller which is uniformly rotation, Where FAC 0 and FBC is the support force to the cylindrical roller which is uniformly rotation. It is known from the formulas (19) and (20) that the support forces to the roller which is rotating at uniform speed could be calculated by the following formula. 0 0 FAC ¼ FBC ¼

G 2 cos h

ð21Þ

The Design and Dynamics Analysis of Cylindrical Roller Surface

19

It can be seen from the formula (21) that if the roller rotates at a constant speed, the support forces on the roller are equal. It can be known from the Newton’s third law that the reaction force to the unfolding wheel I and II is also equal, so the deformation of the unfolding wheel I and II can be calculated by the following formula. Dx01 ¼ Dx02 ¼

G 2K cos h

ð22Þ

Where Dx01 and Dx02 are the deformation amount of the unfolding wheel I and II respectively, when the roller rotates at a constant speed. It can be seen from the Eqs. (1) and (22) that when the roller rotates at a constant speed, the deformation amount of the unfolding wheel I is equal to deformation amount of the unfolding wheel II, and the roller’s vibration is smaller. If the roller’s speciﬁcation is known, the bottom radius of unfolding wheel, the material of the unfolding wheel, and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II are main factors that affecting the deformation of the unfolding wheel. According to the above analysis, when the roller rotated acceleratory, the roller rotates unsteadily and vibrates sharply, which is not suitable for image acquisition. When the roller rotates at a constant speed, the roller rotates steadily and the vibrates less, which is suitable for image acquisition. The time taken for the roller to accelerate and the vibration of the roller which is rotating are affected by the material of unfolding wheel, the bottom radius of the unfolding wheel, and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II.

4 The Surface Unfolding Model Simulation and Data Analysis for Cylindrical Roller 4.1

The Design of Orthogonal Experiment for the Surface Unfolding Model of Cylindrical Roller

In order to verify the analysis of the roller surface unfolding model is correctly and know about the influence of various factors on the time taken for roller to accelerate and roller’s vibration, the bottom radius of unfolding wheel, the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II and the material of unfolding wheel were selected as factors A, B and C used to design a three-factor and three-level orthogonal experiment, as shown in Table 1. The cylindrical roller used for orthogonal experiment has a speciﬁcation of U32 52 mm and it’s material is bearing steel. The stable rotation speed of the roller is required to be 1.2 r/s, and the accuracy of roller surface detection is 0.01 mm. Under the premise of ensuring that the roller reaches the predetermined surface unfolding speed, it is often

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used for prevent the contact deformation between the unfolding wheel and the roller from being too large that the bottom radius of the unfolding wheel is as much as the bottom radius of the roller, so 10 mm, 16 mm and 22 mm are selected as Levels A1, A2 and A3 of the bottom radius for the unfolding wheel. In order to facilitate the installation and transportation of the equipment, the cylindrical rollers surface unfolding mechanism should be as small and light as possible, so the distance of the outer edges between the two deployment wheels should be less than 10 mm, as well as a reliable distance can prevent the two unfolding wheels from crashing caused by installation errors, so the distance of the outer edges between the two unfolding wheels should be greater than 2 mm, therefore, 8 mm, 6 mm and 4 mm are selected as levels B1, B2 and B3 for the outer edges minimum distance between the two unfolding wheels. In order to ensure the unfolding wheel have a good abrasive resistance and low cost, steel, nylon and silicone rubber were selected as the three levels C1, C2 and C3 for the material of unfolding wheel. At the same time, it is necessary to ensure that the two unfolding wheels have the same surface roughness. Table 1. The orthogonal test scheme Experiment number A 1 A1 2 A1 3 A1 4 A2 5 A2 6 A2 7 A3 8 A3 9 A3

B B1 B2 B3 B1 B2 B3 B1 B2 B3

C C1 C2 C3 C3 C1 C2 C2 C3 C1

When the bottom radius of the unfolding wheel is 10 mm, 16 mm and 22 mm respectively, in order to achieve the rotation speed of 1.2 r/s to the roller, the rotation speed of the unfolding wheel should be set at 1.92 r/s, 1.2 r/s and 0.87 r/s respectively. 4.2

The Simulation of Surface Unfolding Model for Cylindrical Roller Based on the Orthogonal Experiment

The orthogonal experiment for the surface unfolding model of cylindrical roller has been designed and based on the analysis above in this paper. It can be known that the friction coefﬁcients of steel, nylon and silicone rubber with the bearing steel are 0.15, 0.5 and 0.85, respectively. The dynamics simulation is carried out in Adams software base on the orthogonal experiment of the surface unfolding model for the roller, and the simulation time is set to 0.15 s and the number of steps is set to 300 steps. The simulation results are shown in Fig. 4.

The Design and Dynamics Analysis of Cylindrical Roller Surface

Fig. 4. The simulation results of the surface unfolding model for the roller

21

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Fig. 4. (continued)

Figure 4 is the line chart showing the displacement of the roller’s center of mass on the direction of x-axis and the direction of y-axis in the process of roller’s rotation. It can be seen from the simulation results in Fig. 4 that two phases are contained in the roller unfolding process. The ﬁrst phase is the roller rotates acceleratingly phase, the roller and the unfolding wheel just start to contact, resulting in the rotation speed of the roller instability, and the rotation speed of the roller reaches 1.2 r/s gradually. At the same time, the vibration of the roller’s center of mass on the direction of x-axis and

The Design and Dynamics Analysis of Cylindrical Roller Surface

23

y-axis becomes stable gradually. The second phase is the uniform rotation of the roller, the roller’s rotation speed is stable at 1.2 r/s, and the vibration of the roller’s center of mass on the direction of x-axis and y-axis is much smaller than the ﬁrst phase. The simulation results shown in Fig. 4 are consistent with the analysis results of the roller surface unfolding model. 4.3

The Analysis and Veriﬁcation of the Simulated Data Based on Roller Surface Unfolding Model

The prerequisite for using machine vision to detect cylindrical roller’s surface defects is having a clear image of the cylindrical roller. When the roller rotates at a constant speed, the time taken for the roller to reach a uniform speed and the vibration on the direction of x-axis are the two main factors affecting the efﬁciency and sharpness for image capturing. According to the simulation results of the orthogonal experiment, the motion of the cylindrical roller during the unfolding process is known, as well as, it is necessary to assess the effect of the bottom radius of the roller, the material of the roller, and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II on the vibration of the roller’s center of mass and the time taken for roller to rotate acceleratingly by the range analysis and the variance analysis based on the simulated data. According to the orthogonal experiment scheme, the test indicator of time taken for roller to reach a uniform speed is T, and the test indicator of the vibration of the roller’s center of mass on the direction of x-axis when the roller rotates uniformly is D. The time taken for the roller to reach the uniform rotation and the vibration value in the direction of x-axis when the roller is rotated at a constant speed are recorded in Table 2. Table 2. Orthogonal experiment design array L9 for 3 key factors Experiment number A 1 A1 2 A1 3 A1 4 A2 5 A2 6 A2 7 A3 8 A3 9 A3

B B1 B2 B3 B1 B2 B3 B1 B2 B3

C C1 C2 C3 C3 C1 C2 C2 C3 C1

T(s) 0.115 0.0385 0.0355 0.022 0.028 0.0365 0.039 0.0265 0.126

C(mm) 1.29E−05 3.95E−04 2.66E−03 2.68E−04 1.29E−04 2.84E−05 7.54E−06 4.19E−05 2.69E−06

For understanding the effect of factors A, B and C on the test indicators T and D, a range analysis was performed according to the data in Table 2, as shown in Table 3. T1, T2 and T3 in Table 3 indicate the effect of level 1 level 2 and level 3 on the orthogonal experimental results, respectively, and R indicates the effect of the level change on the experimental results. The less time it takes for the roller to reach a uniform speed, the

24

Y. Bao et al. Table 3. The range analysis based on the simulation data of the orthogonal experiment Influence T1(T) T2(T) T3(T) R(T) T1(D) T2(D) T3(D) R(D)

A 0.189 0.0865 0.1915 0.105 0.0030679 0.0004254 0.00005213 0.00301577

B 0.176 0.093 0.198 0.105 0.00028844 0.0005659 0.00269109 0.00240265

C T 0.269 T(T) = 0.467 T(D) = 0.003545 0.114 0.084 0.185 0.00014459 0.00043094 0.0029699 0.00282531

higher efﬁciency of roller unfolding. Therefore, the optimal combination of factors is A2B2C3. The order of influence for the factors on the time taken for the roller to reach a uniform speed is the unfolding wheel’s material, the radius of unfolding wheel, and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II. The smaller vibration on the direction of x-axis when the roller rotates at a constant speed, the more stable for the motion roller unfolding. So the optimal combination of factors is A3B1C1, and the order of influence of various factors on the vibration of the roller on the direction of x-axis when the roller rotates at a constant speed is the bottom radius of unfolding wheel, the material of unfolding wheel and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II. As the bottom radius of the unfolding wheel increasing, the time taken for the roller to reach a uniform speed is reduced ﬁrstly and then increased. When the bottom radius of the unfolding wheel is 16 mm, the time taken for the roller to reach a uniform speed is the least. With increasing of the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II, the time taken for the rollers to reach a uniform rotation is reduced ﬁrstly and then increased. When the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II is 6 mm, the time taken for the roller to rotate at a constant speed is the least. The greater coefﬁcient of friction between the unfolding wheel and the roller, it takes less time for the roller to reach a uniform speed. With increasing of the bottom radius of the unfolding wheel, the vibration on the direction of x-axis is smaller when the roller rotates at a constant speed. With the reducing of the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II, the vibration on the direction of x-axis is greater when the roller rotates at a uniform speed. With increasing of the rigidity of the unfolding wheel’s material, the vibration on the direction of x-axis is smaller when the roller rotates at a constant speed. According to the range analysis, the order of the influence was determined that the radius of unfolding wheel, the material of unfolding wheel and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II to the time taken for roller to reach a constant speed and the vibration of the roller on the direction of x-axis when the roller rotates at a constant

The Design and Dynamics Analysis of Cylindrical Roller Surface

25

speed. In order to know about the signiﬁcance that the factors A, B and C affect the vibration of the roller on the direction of x-axis and the time taken for the roller to reach a constant speed, it is necessary to make variance analysis of simulation data based on the orthogonal experiment. As shown in Table 4. Table 4. The analysis of variance based on the simulation data of the orthogonal experiment Factor A

Index T D B T D C T D Error range T D Total T D

Quadratic sum 2.393 10−3 1.802 10−6 2.044 10−3 1.152 10−6 6.572 10−3 1.612 10−6 1.425 10−3 1.360 10−6 3.667 10−2 7.323 10−6

NDOF 2 2 2 2 2 2 2 2 9 9

Mean square 1.197 10−3 9.009 10−7 1.022 10−3 5.759 10−7 3.286 10−3 8.061 10−7 7.13 10−4 6.801 10−7

F 1.679 1.325 1.434 0.847 4.611 1.185

P 0.373 0.430 0.411 0.541 0.178 0.458

The influence of the three factors A, B and C can be determined from Table 4. When the level of signiﬁcance a = 0.05, the bottom radius of the unfolding wheel has a signiﬁcant effect on the time taken for the roller to reach a uniform speed (P < 0.05), and it has an effect on the x-axis vibration when the roller rotates at a constant speed (P is close to 0.05). The smallest distance from the outermost point on the unfolding wheelI to the outermost point on the unfolding wheel II has an effect on the time taken for the roller to reach a uniform speed (P is close to 0.05), and have no effect on the vibration of the x-axis when the roller rotates at a constant speed (P > 0.05). The material of unfolding wheel has a signiﬁcant effect on the time taken for the roller to reach a uniform speed (P < 0.05), and has an effect on the x-axis vibration when the roller rotates at a constant speed (P is close to 0.05). This is consistent with the results of the range analysis based on the simulation data of the orthogonal experiment. The results of simulation and data analysis show that the vibration of the roller on the x-axis is within the required range of detection accuracy, so the scheme with the shortest time for the roller to achieve a uniform speed is selected as the best design scheme. The bottom radius of the unfolding wheel is 16 mm, the outer minimum distance of two unfolding wheels is 8 mm, and the material of the unfolding wheels is silicone rubber, the unfolding efﬁciency is the highest, and the vibration range of the roller is less than 0.01 mm. So the fourth experiment is chosen as the best design, a set of surface unfolding mechanism of cylindrical roller is designed, as shown in Fig. 5, the mechanism can unfold cylindrical rollers effectively and help linear camera to obtain clear image of the rollers surface.

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Fig. 5. Physical prototype of cylindrical roller surface unfolding mechanism and roller unfolding effect

5 Conclusions (1) The surface unfolding model of two-dimensional cylindrical roller is established and the position relation between the two unfolding wheels and the cylindrical roller is determined. According to the analysis of the model, the influence can be known of the unfolding wheel material, the bottom radius of the unfolding wheel and the minimum distance between the outer edges of the two unfolding wheels on the time taken for the roller to reach a uniform speed and the vibration of the roller. It can be found that in the process of the roller unfolding, the roller accelerates ﬁrstly, the vibration of roller is large, and then the uniform rotation is achieved, and the vibration of roller is small. (2) The cylindrical roller with the speciﬁcation of U32 52 mm and density of 7.9 g/cm3 and bearing steel material is selected as the experiment object. The orthogonal experiment of three factors and three levels is designed according to the unfolding model of cylindrical roller. The experimental data analyses and simulation results show that the optimal bottom radius of the unfolding wheel in the cylindrical roller unfolding mechanism is 16 mm, the material of the unfolding wheel is silicone rubber, and the smallest distance from the outermost point on the unfolding wheel I to the outermost point on the unfolding wheel II is 8 mm.

References 1. Su, J., Zhang, S., Yuan, J., et al.: Research on precision polishing technology of cylindrical roller of bearing. Mech. Electr. Eng. 35(10), 1073–1076+1093 (2018) 2. Li, X., Zhang, Z., Bai, R.: Visual inspection method for surface defects of bearing cylindrical rollers. Autom. Instrum. 35(12), 58–62 (2014)

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3. Wang, H.: Machine vision based cylindrical roller surface defect detection system. Henan University of Science and Technology (2012) 4. Zhang, B.: Research on surface defect detection of bearing roller based on machine vision. Nanchang Hangkong University (2018) 5. Zhang, J., Ye, Y., Xie, Y., et al.: Photoelectric detection of defects in metal cylindrical workpieces. Opt. Precis. Eng. 22(07), 1871–1876 (2014) 6. Wu, J.: Machine vision-based bearing roller surface defect detection system. Zhejiang University (2018) 7. Li, H.: Research on comprehensive measurement method and measurement system of bearing cylindrical roller size. Harbin University of Science and Technology (2018) 8. Tan, W., Wen, Q., Duan, F., et al.: Visual inspection of curved surface defects of cylindrical batteries based on HV&VHS. Control Eng. 26(01), 17–22 (2019) 9. Bo, P., Yuan, Y., Zhang, C.: Automatic reconstruction of developable surfaces. J. Comput. Aided Des. Comput. Graph. 28(09), 1428–1435 (2016) 10. Koukash, M., Hobson, C.A., Lalor, M.J., Atkinson, J.T.: Detection and measurement of surface defects by automatic fringe analysis. Opt. Lasers Eng. 87(07), 125–135 (1986)

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning of the Large Storage Tank Wanghui Bu1(&), Weiqi Zhou1, Laixin Fang1, Jing Chen2, Xianghua An3, and Jingkai Huang1 1

2

School of Mechanical Engineering, Tongji University, Shanghai 201804, China [email protected] School of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou 310018, China 3 School of Mechanical and Power Engineering, Dalian Ocean University, Dalian 116023, China

Abstract. Large vertical tanks are widely used in the storage of slag and powder in metallurgical industry. For inner wall cleaning of these large storage tanks, traditional working platforms such as hanging baskets and scaffolds have disadvantages of high labor intensity, high danger and low efﬁciency. Hence, this paper presents a novel cleaning robot working in the large tank based on the cable-driven parallel mechanism. A novel kinematic modeling method based on lifting point coordinates for the cable-driven parallel mechanism is proposed, which need not directly calculate the position and orientation of the center of the moving platform, but just indirectly analyzes the position and orientation of the moving platform through the positions of lifting points. In this way, the kinematic analysis becomes concise, and the workspace of the moving platform is convenient to obtain. For the cleaning robot working in a large storage tank with 50 m high and 18 m diameter, the forward and inverse kinematic solutions of the parallel mechanism with three cables are studied under the kinematic modeling method based on lifting point coordinates. Finally, the speciﬁc structure of the cable-driven parallel robot is designed. Keywords: Cable driven Kinematic analysis

Parallel mechanism Cleaning robot

1 Introduction Solid wastes such as dust and mud are inevitably produced in iron and steel production process. These by-products can be used for building materials. Iron and steel mills store all the dust in the large storage tank, which will harden and stick to the tank wall during the long storage process, so it needs to be cleaned regularly. The tank studied in this This work was supported by National Natural Science Foundation of China (Grant Nos. 51475331, 61703127, 51605067), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F020026), and Fundamental Research Funds for the Central Universities. © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 28–40, 2020. https://doi.org/10.1007/978-981-32-9941-2_3

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning

29

paper is 50 m high and 18 m in diameter. The clearance work in this suspended environment is dangerous. At present, the cleaning of the tank wall is mainly carried out by artiﬁcial hanging baskets. The labor intensity, high risk and low efﬁciency of the workers are urgently needed. Therefore, it is urgent to research a new type of clearance device to meet the large load and sufﬁcient space for activities. The cable-driven parallel mechanism has the advantages of high rigidity and high precision of the rigid parallel mechanism, and also has the advantages of small mass, inertia and large length range of the cable. By controlling the length change of the cable, the cable-driven parallel mechanism has a great working space advantage; Through the synchronous control of multiple ropes, the dynamic platform orientation of the cable parallel mechanism can be flexibly adjusted. And this kind of mechanism mainly uses light weight, high strength and low inertia polymer rope [1], in order to meet the need for high sensitivity occasions. These advantages make the cable-driven parallel mechanism an ideal substitute for rigid link manipulators in many industries or combined with connecting rods to produce light components [2]. Therefore, it is a good application potential and prospect to realize the cleaning operation of large vertical storage tanks by using cable-driven parallel mechanism. The force and motion of the driving unit of a cable-driven parallel robot are transmitted to the moving platform through the cable [3–5], Compared with the rigid link parallel robot, the position and direction of the moving platform are determined by the cable length. However, due to the special characteristics of the cable, it can only bear tension but not pressure, that is to say, the tensile force of each driving cable must be greater than zero [6–8]. Such unilaterality means that the existing analysis and design methods of rigid linkage mechanisms cannot be directly applied to cable-driven parallel robot. This is also a difﬁcult point in the application of cable-driven parallel robots. At present, Diao et al. [9] discussed the singularity analysis of a fully constrained planar cable robot with four or more cables. And based on the rank analysis of Jacobian matrix, a set of Jacobian singularities were proved mathematically. Yang et al. [10] carried out kinematics and singularity analysis of 4-cable 3-DOF cable-driven parallel robot in the plane, and calculated the singularity of robot motion. Carricato and Merlet [11] analyzed the dynamic and static characteristics of under-constrained 3cable parallel robot under crane conditions. Soon afterwards Abbasne and Carricato [12] extended the research to underconstrained n-cable-driven robots and looked for cable tensile force distribution under equilibrium conditions. While Xu et al. [13] analyzed the dynamics and control of a cable-driven hyper-redundant manipulators Barrette et al. [14] introduced a new concept of dynamic workspace to analyze workspace. Merlet [15] establishes a new model and gives a general solution to the inverse solution of 6-cable-driven robot. To perform the static and kinematic analysis, the screw theory with wrenches and twists can be adopted [16–18]. In above papers, the rigid parallel mechanism kinematical modeling method is usually used to establish the kinematics model of the flexible parallel mechanism. The process of modeling and solving is complex and complicated, and the structural characteristics of the flexible mechanism are not utilized. Therefore this paper presents a lifting point coordinate modeling method for kinematics analysis of cable-driven parallel mechanism. This method does not directly calculate the position and orientation of the center of the moving platform, but indirectly analyzes the position and

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orientation of the moving platform through the position of each lifting point of the moving platform. And it has a simple and intuitive kinematics modeling process, which is convenient for analyzing the advantages of the moving platform workspace.

2 Statics of the Cable-Driven Mechanism According to the working conditions in practical application, the stereogram and structure of the 3-cable-driven parallel robot are given at ﬁrst in Fig. 1. B2

z

B3

B1 O

L2

y

x

L3

L1 P2 P P3 P1

Fig. 1. Structure of the 3-cable-driven parallel robot

Assuming that the mass of the cable is neglected, that is, the weight of the cable is not taken into account and the cable is considered to be straight at rest. As shown in the ﬁgure, the global ﬁxed coordinate system (XO, YO, ZO) is established on the stationary platform and the local coordinate system (XP, YP, ZP) is established on the moving platform. B1, B2, and B3 denote the anchor points on the stationary platform; P1, P2, and P3 denote the lifting points on the moving platform. According to the static equilibrium relationship of the moving platform, the following equations can be obtained. 8 3 P > > < ui Ti þ f = 0 i¼1

3 P > > : (R ri Þ ui Ti þ M = 0

ð1Þ

i¼1

where ui denotes the unit vector of the ith cable; Ti denotes the tensile force of the ith cable; ri denotes the vector from the center of moving platform to the anchor point described in the coordinate system P; f and M denote the external force and moment on the moving platform; and R denotes the rotation matrix of the coordinate system P relative to coordinate system O, and

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning

2

cos b cos c R ¼ 4 cos c sin a sin b þ cos a sin c sin a sin c cos a cos c sin b

31

3 sin c cos b sin a 5 cos a cos b

cos b sin c cos a cos c sin c sin a sin b cos a sin c þ cos a sin b sin c

where a, b, and c are three Euler angles. Equation (1) can be rewritten in the matrix form: JT ¼ W where J¼

u1 ðR r1 Þ u1

u2 ðR r2 Þ u2

T ¼ ð T1

T2

u3 ðR r3 Þ u3

T3 ÞT

W¼ (f MÞT In the global coordinate system, the unit vector of the cable is as follows. ui ¼

OBi OPi jjOBi OPi jj

3 Kinematical Analysis 3.1

Inverse Kinematic Solution

Inverse kinematic solution refers to the solution of each cable length with the position of the moving platform known. In this paper, a lifting point coordinate modeling method is proposed. This method does not directly calculate the position and orientation of the center of the moving platform, but indirectly analyses the position and orientation of the moving platform through the position of each hanging point of the moving platform. In this way, instead of calculating the transformation between the dynamic and static coordinate systems, only a global coordinate system is needed on the static platform, which reduces the amount of calculation and improves the accuracy of calculation. We know the coordinates of the anchor points Bi of the cable on the stationary platform, and the distribution of the lifting points Pi on the moving platform. In order to calculate the cable length through the center of moving platform P (Px, Py, Pz), the formula of cable length is as follows. Li ¼ OBi OPi where Li denotes the vector of cable i.

ð2Þ

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Equation (2) shows that li ¼ kOBi OPi k. Then we requires coordinates of P1, P2 and P3. Because the three-cable parallel mechanism studied in this paper is only affected by gravity of the moving platform, if the gravity is G, then the following equations hold. f ¼ð 0

0

G ÞT

M ¼ f OP The static equilibrium equations can be obtained by Eq. (1). In order to avoid the influence of direction vectors on the calculation, we express the tensile force of each cable as follows. Ti ¼ ti kOBi OPi k

ð3Þ

Si ¼ OPi Li

ð4Þ

Where Ti represents the tensile force, li represents the length. Substituting Li and Si into Eq. (1), the following equation is obtained.

LT ¼f ST ¼M

ð5Þ

where L ¼ ½ L1 S ¼ ½ S1

L2 L3 S2 S3

i represent the vector matrix of cable. From Eq. (3), we know ti ¼ kOBiTOP , and ik

T ¼ ½ T1 T2 T3 T . In this way, we avoid the calculation of unit vectors and treat as an unknown number. Since the length of the cable is inversely decomposed by the center coordinate of the platform, and the position of three anchor points of the moving platform is uniformly ﬁxed on the moving platform and the size of the moving platform is known. Hence, the position relation of three points can be mathematically described as follows. 8 > < ðP1x þ P2x þ P3x Þ=3 ¼ Px ðP1y þ P2y þ P3y Þ=3 ¼ Py > : ðP1z þ P2z þ P3z Þ=3 ¼ Pz

ð6Þ

8 2 2 > < kOP1 OP2 k ¼ p12 kOP1 OP3 k 2 ¼ p213 > : kOP2 OP3 k 2 ¼ p223

ð7Þ

and

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning

33

where Pix ; Piy ; Piz denote the x, y, z coordinates of the Pi point, and pij denotes the distance between Pi and Pj. From the previous illustration, it can be seen that there are 9 unknowns from three lifting points, and 3 unknowns ti . Hence, there are totally 12 unknows. By combining Eqs. (5), (6), and (7), a total of 12 scalar equations can be conveniently solved through mathematics software. 3.2

Forward Kinematic Solution

Forward kinematic solution refers to solving the position and orientation parameters of the moving platform with the given length of each cable. For the analysis of the positive kinematics solution of the parallel mechanism, it is often necessary to solve a set of nonlinear equations coupled with position and orientation, which is rather difﬁcult. The Newton-Raphson method may be adopted, where the length of the cable obtained by the inverse solution approximates the target length of the cable and ﬁnally obtains the forward kinematic solution. Although this method is effective, the error will always accumulate to a large extent. In this paper, the lifting point coordinate modeling method is established to avoid these problems and simplify the calculation steps. Solving the position and orientation parameters of the moving platform by the cable length, the coordinates of the center point of the platform and the corresponding X-YZ Euler angles need to be obtained. The relationship of cable length has been expressed in Eq. (2). Thus, substituting l1 ; l2 ; l3 ; p12 ; p13 ; p23 into Eqs. (2), (5), and (7), results in the forward position solution. After solving the coordinates of P1 ; P2 ; P3 , the center point can be mathematically expressed from Eq. (6). And the orientation of the moving platform can be obtained through three lifting points. Then the corresponding X-YZ Euler angles of the moving platform can be calculated.

4 Orientation Analysis of the Moving Platform The motion of moving platform is limited by cables. Only when the path of moving platform is planned in the workspace can the required motion be realized. The workspace is to measure the translational and rotational performance of the cable-driven parallel robot, which is deﬁned as the set of all position and posture points of the moving platform satisfying the cable tensile force constraints in space. In traditional research, workspace is usually computed by traversing all required points in a range. But these methods are computationally expensive and inaccurate. In this paper, based on the singular condition that the cable tensile force is zero, assuming one of the three cables has a cable tensile force of 0, that is T1 = 0, the following equations hold.

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82 3 B2x P2x B3x P3x > > > > 4 B2y P2y B3y P3y 5 t2 ¼ f > > t3 > > B2z P2z B3z P3z > > > < t ½ S2 S3 2 ¼ M t 3 > > > > > kOP1 OP2 k2 ¼ p212 > > > > kOP1 OP3 k2 ¼ p213 > > : kOP2 OP3 k2 ¼ p223

ð8Þ

We take the coordinates of x, y and z of Pi as output, take t1 and t2 as input, and the functional relationship between them is expressed by f. The polynomial expression can be obtained.

Pix ; Piy ; Piz ¼ fi ðt2 ; t3 Þ

So that

1X f i ðt 2 ; t 3 Þ Px ; Py ; Pz ¼ 3

According to the working conditions of the cable, the tensile force of each cable is positive when the moving platform is subjected to external force ðf M ÞT at any point in the working space. And a minimum positive force is needed to ensure that the cable is not relaxed during the movement. In addition, the maximum tensile force of the cable cannot be inﬁnite, it should be more practical, and the maximum value should be less than the ultimate tensile force of the cable. By combining Eqs. (3) and (4), the range of ti can be obtained. Tmin \tkOBi OPi k\Tmax According to the range of a singular surface with tensile force of 0 for each cable, the three-sided closed part is the workspace of the three-cable-driven mechanism. On the basis of workspace, there are various paths that can be planned. For the purpose of research, we choose the path as shown in the Fig. 2. This path can be described as rðaÞ ¼ ð 9 sin a

4:5 sin a

a Þ; a 2 ½0; 2p

Set the starting point of motion on point P0(9, 9, 25), the position and orientation of the moving platform can be obtained as shown in Fig. 3.

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning

35

Fig. 2. The path of moving platform

Fig. 3. The position and orientation of the moving platform

5 Structural Design The main function of the cable-driven parallel robot designed in this paper is to clear the tank. Considering the influence of the workspace of the mechanism, the top view of the cleaning robot is shown in the Fig. 4. The moving platform is driven by three cables. One end of the cable is connected to a guide rail driven by a motor at the top of the tank, and the other end is connected to three ﬁxed joints connected by hooks and rings on the moving platform as shown in the Fig. 5. Four saw blades for cleaning work are installed under the moving platform. The four saw blades are interlaced to ensure that no matter which side of the moving platform moves to the tank wall, they can work normally. In the working state, the dust adhering to the tank wall is cut by high-speed rotating gear. The electric saw is controlled to rotate in the way that the direction of the cutting force acting on the tank wall is upward. Therefore, the direction of the reaction force acting on the moving platform is downward, which is consistent with the gravity direction. Hence, the tensile

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W. Bu et al.

Guide rail and mobile table

Moving platform with saw blade

Fig. 4. The top view of the cleaning robot

forces of three cables, the gravity and the reaction force produced by cutting can ensure the force closure of the moving platform when the electric saw rotates. At this time, fasteners will be subject to greater vertical tensile force, so the bolts between the motor and the installation plate, and the bolts between the installation plate and the moving platform should be fastened.

Fig. 5. The structure of the moving platform

In view of the actual working conditions, this paper proposes a method of installing guide rail along the wall of the tank, which can move the ﬁxed hanging point at the upper end, and make the mechanism clean the wall completely in a limited number of times of movement. Motors are installed on the mobile table and gears are installed under the mobile table. In order to prevent undercutting, the modiﬁed gears should be used here. Rollers are installed on the worktable to control the collection and release of

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning

37

cables. The motor drives the gear to rotate. With the movement of the gear, the position of the lifting points on the wall can be changed (Fig. 6).

Fig. 6. The structure of the guide rail and the mobile table

Considering the actual situation, three lifting points of the tank are evenly distributed in a quarter of the circle area, and the center of the moving platform is located in the center of the gravity center vertical line of the isosceles triangle formed by the three lifting points. Overhead view of the overall structure is shown in the Fig. 7.

B2

P2 B3

P3

P1

B1

Fig. 7. The top view sketch

The minimum distance between moving platform and tank wall is as follows. DP ¼ D=2 jjOP3 jj

ð9Þ

Where D denotes the diameter of the tank, The center of the moving platform is the center of gravity of DB1B2B3. By substituting coordinates of three lifting points B1, B2 and B3 of static coordinate into Eq. (6), coordinates P1, P2 and P3 can be obtained, then substituting coordinates B1, B2 and B3 into Eq. (8), we can obtain that DP = 1.21 m. Figure 8 shows the change of the triangle orientation of lifting points on the moving platform. The solid line graph shows the initial distribution position of three lifting

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W. Bu et al.

points on the moving platform, and the dotted line graph shows the distribution of lifting points under the working condition. From calculation we can see that the z-Euler angle of the moving platform is 15°, that is to say, the moving platform rotates 15° around the z-axis. At this time, the center of the moving platform is 1.21 m away from the tank wall and the radius of the moving platform is 1 m. Therefore, in order to ensure that the moving platform keeps close to the horizontal posture, the radius of the saw blade can be designed to be about 0.25 m.

Fig. 8. The position of lifting points on the moving platform

In order to ensure that the orientation angle of the working process is not too large, and the dust can be cleaned smoothly. In practice, we change the cleaning position by moving the mobile station on the guide rail. Starting from the upper boundary of the initial position, the moving platform moves vertically along the tank wall through the extension and shortening of the cable. When the moving platform moves to the lower boundary, a cleaning step is completed. Then the motor drives the moving platform to move a certain distance along the circumferential direction of the tank wall, and the moving platform continues to move vertically. In this way, the whole tank is cleaned up after the moving platform circles the tank.

6 Conclusions In this paper, a cable-driven parallel robot for cleaning large vertical storage tank is proposed. A kinematic model based on lifting point coordinates is established, and the forward and inverse kinematic solutions of the three-cable parallel mechanism are analyzed. Finally, the mathmatica software can be used to quickly calculate the platform posture and cable length corresponding to the position of each moving platform. On the basis of calculation, the workspace constraints of the cable structure are obtained, and the traveling path of the moving platform is planned in the space under the condition of equal lifting points, and shows the changing trend of the position and posture of the moving platform. For storage tanks with a diameter of 18 m and a height of 50 m, a control device using track control motion is designed. Saw blades can be

A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning

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used to achieve the action of clearing dust up and down. Saw blade diameter is designed to keep the moving platform close to the horizontal posture, so that the effective working space can cover the whole tank wall that needs to be cleaned by sliding smoothly through the lifting point position. Acknowledgement. This work was supported by National Natural Science Foundation of China (Grant Nos. 51475331, 61703127, 51605067), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F020026), and Fundamental Research Funds for the Central Universities.

References 1. Schmidt, V., Pott, A.: Increase of position accuracy for cable-driven parallel robots using a model for elongation of plastic ﬁber ropes. In: New Trends in Mechanism and Machine Science. Springer, Cham (2017) 2. Li, C., Rahn, C.D.: Design of continuous backbone, cable driven robots. J. Mech. Des. 124 (2), 265–271 (2002) 3. Tang, X.: An overview of the development for cable-driven parallel manipulator. Adv. Mech. Eng. 6, 823028 (2015) 4. Gouttefarde, M., Collard, J.F., Riehl, N., et al.: Geometry selection of a redundantly actuated cable-suspended parallel robot. IEEE Trans. Robot. 31(2), 501–510 (2017) 5. Lin, Y., Ban, W., Hai, H., et al.: An adaptive mechanism for space tetherreel. J. Astronaut. 35(12), 1379–1387 (2014). (in Chinese) 6. Yu, L., Qiu, Y., Yu, S.: Dynamic workspace of a high-speed cable-driven camera robot. Eng. Mech. 30(11), 245–250 (2013). (in Chinese) 7. Bo, O., Shang, W.: Efﬁcient computation method of force-closure workspace for 6-DOF cable-driven parallel manipulators. J. Mech. Eng. 49(15), 34–41 (2013). (in Chinese) 8. Wei, Y., Deng, Z., Li, Q., et al.: Analysis of dynamic response of tethered space solar power station. J. Astronaut. 37(9), 1041–1048 (2016). (in Chinese) 9. Diao, X., Ma, O., Lu, Q.: Singularity analysis of planar cable-driven parallel robots. In: IEEE Conference on Robotics. IEEE (2008) 10. Yang, G., Yeo, S.H., Pham, C.B.: Kinematics and singularity analysis of a planar cabledriven parallel manipulator. In: Proceedings of 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems, (IROS 2004). IEEE (2004) 11. Carricato, M., Merlet, J.P.: Direct geometrico-static problem of underconstrained cabledriven parallel robots with three cables. In: IEEE International Conference on Robotics & Automation. IEEE (2011) 12. Abbasnejad, G., Carricato, M.: Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots with Five Cables. Computational Kinematics. Springer, Dordrecht (2014) 13. Xu, W., Liu, T., Li, Y.: Kinematics, dynamics and control of a cable-driven hyper-redundant manipulator. IEEE/ASME Trans. Mechatron. 23, 1693–1704 (2018) 14. Barrette, G., Gosselin, C.M.: Determination of the dynamic workspace of cable-driven planar parallel mechanisms. J. Mech. Des. 127(2), 242 (2005) 15. Merlet, J.P.: A new generic approach for the inverse kinematics of cable-driven parallel robot with 6 deformable cables. In: Springer Proceedings in Advanced Robotics: Advances in Robot Kinematics (2018)

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A Novel Biomimetic Design Method Based on Biology Texts Under Network Bowen Chen, Liang Chen(&), Xiaomin Liu, and Hao Dou Institute of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China [email protected], [email protected], [email protected], [email protected]

Abstract. Most of works on biomimetic design methods are devoted to provide an artiﬁcially built biological information database and make engineering design depending on inspiring. Though using these approaches can improve work efﬁciency on biomimetic design, it sometimes limits the scope of thinking. Here we provide an approach that using natural language text of biological from network to expand design ideas. In addition, a new biomimetic design mode is proposed by building tree model of engineering object. This paper descript how to retrieve biological texts from network and make text processing, then introduce a text matching algorithm combining with TRIZ to map engineering design problems to biological texts. After obtaining biological texts by using match algorithm, a novel method to guide product design with these texts is proposed. At the end of this paper, an example of dust collector design is used to explain the whole process. Keywords: Biomimicry Biologically inspired design Natural language processing Text processing Design theory Innovation Computer aided design

TRIZ

1 Introduction Biologically inspired design was ﬁrst proposed on the late of last century (French 1997). It refers to a method of combining exploration and practice with reference to organisms in nature. A series of innovators were created that inspired from nature creatures. Dating back to 2002, foreign scholar proposed the idea of introducing TRIZ [1] to realize the technical transformation from biological to engineering system and proposed a PRIZM matrix for comparing the similarity between engineering problems and biological examples [2]. A mapping between the functions of biology and engineering was established by making an engineering-biological dictionary method [3, 4]. Combining the traditional TRIZ method with the functional basis is a research hotspot in recent years. A bio-inspired design software was developed, which combine semantic relationship dictionary among various functional keywords, then stored in the ontology database [5]. By abstracting engineering and biology information into functional ontology, a bionic design website was schemed, through searching the ontology database by inputting key statements in a speciﬁc format then got relative biological © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 41–51, 2020. https://doi.org/10.1007/978-981-32-9941-2_4

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prototypes [6]. Receiving revelation by information engineering, material flow, information flow, and energy flow was combined as the functional relationship, the 39 engineering parameters in TRIZ are re-divided, and then FB-TRIZ was proposed [7]. The contradiction matrix with the 40 invention principles of TRIZ was combined and the genetic algorithms in the ParaGen framework to implement a mapping based on the design process [8]. The relationship between TRIZ innovation principle and biological case was combined, and then a system optimization strategy for the design of product service system been proposed [9]. In the last few parts, we will introduce an innovative approach, which using biology texts scrawling from Internet and other sources like biological authority books, using information extract from these texts to guide us on engineering design. At ﬁrst, several concepts and theories are proposed to introduce the working process of this novel biomimetic design method. Second, we explain the core theories behind the whole processing. Then, based on the workflow illustrated, corresponding code and Graphical User Interface (GUI) is written and debugged. After that, we give an example of optimization of cyclone dust collector to show the practicality of this biomimetic design method. Finally, a conclusion is proposed to summarize our current research on biology inspired design and our future plan of work on it.

2 Overall Approaches The whole processing of the biomimetic design could be separated as three parts, been sorted as engineering ﬁeld, biological ﬁeld and intermediate, as shown in Fig. 1. Designer get design object based on a speciﬁc mission plan or customer request at the beginning. To complete the ﬁrst step on design, as a concept design, successfully, a tree-like model based on TRIZ has to be constructed, which serves as a part of our design method, will be illustrated in detail on Sect. 3. After construct the initial model, we get one or several pairs of contradiction and map them into principle keywords of expanded TRIZ based on intermediate. Searching on the local biological database with principle keywords and customize keywords, which are decided by designer’s own experiential knowledge, several biological texts in database are matched. Based on corresponding algorithm, which introduce in Sect. 3, designer could get operators to modiﬁed current engineering model. Repeat this whole process continuously until model satisﬁed designer’s intention, and then get a feasible solution on this stage.

Fig. 1. Whole processing of biomimetic design

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To make this process work successfully, some preparations in the early stage should be done, including crawling raw texts data and restore them normally in the local database; expanding traditional TRIZ 40 principle keywords to make retrieving database in a large range. Meanwhile, what algorithm based on search database and how to build the engineering model and translate biological texts into operators to modiﬁed model are also parts of question. All of them would immediately explained in Sect. 3.

3 Core Theories 3.1

Biological Data

The method presented in this paper using normalized biologically texts from network. First, we need to scrawl raw texts from websites about biological in a certain scale; and then, after ﬁlter the stop words and stem each word, we embedding words in documents into a vector space [10], then, a bi-LSTM layer and a CRF layer is used to construct part of speech tagging neural network model [11], trained with Conll2000 corpus. The documents are represented via a bag of words (BOW) encoding technique, which is regarded as x, and the output is the part of speech tags, which is regarded as y. As Fig. 2 shows, totally 2033198 parameters are need to be trained. After hours of training and parameter tuning, the model could be used to extract nouns and verbs in sentences of each text and deposit into the database. The speciﬁc steps is described in Fig. 3.

Fig. 2. Neural network model

Fig. 3. Normalizing raw texts based on neural network

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Engineering Model

To map engineer problems to biological texts and then translate the information from biological text to product design again, a speciﬁc corresponded model based on TRIZ is introduced. The engineering model is separated into four parts as function, principle, behavior and structure. When we get a certain design object at the beginning, we use certain words as nodes and add them onto this tree-like model, inclusion relationships could be expressed as layers of model. The number of nodes in each branch determined by actual situation and designer’s own experience. The more nodes, the lower the design freedom, on the contrary, the fewer nodes, the higher the design freedom. After construct our design object properly, designer should consider the contradiction of every leaf nodes and determine the principle keywords based on TRIZ contradict matrix. The initialized engineering model then been constructed. The kind of tree model structure is shown in Fig. 4.

Fig. 4. A tree model structure

3.3

TRIZ Principle Words Expand

On the other side, we expand the traditional 40 principle words into 40 principle sets, based on pre-trained 300 dimensions Word2Vec [10], which is a popular words embedding model, calculating the words distance in embedding vector space and extract nearest 10 words of each principle word into one set. The aim of expanding words is explained in Sect. 3.4. 3.4

Searching Method and Algorithm

After determine the certain principle words, we search the database with using a list of principle words and a list of custom words deﬁned by designer for each contradiction. Figure 5 shows the keywords to be searched on a speciﬁc contradiction. When input principle words, we actually input sets of words expanded before potentially. After designer set weight for each pair of word list, Term Frequency-Inverse Document Frequency (TF-IDF) algorithm [12] is used to calculate scores for texts in database and count score for each texts and return texts sorted by score.

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Fig. 5. The keywords to be searched for contradiction i

3.5

Modifying Engineering Model by Texts

Before we get close to strategy in detail, we ﬁrstly deﬁne “operators”: (1) Add: add a child node from current node; (2) Delete: delete current node and its child node; (3) Change: change the word in current node; (4) Merge: delete all child nodes from parent of current node; (5) Split: add a child node from parent of current node; (6) Connect: connect current node with other node; (7) Disconnect: disconnect current node with other node. We extract the most high score text from return result to modify current engineering model. The strategy used here is to map noun into node in engineering model, and map verb into operator to deﬁne how to modify the model. The steps in detail is to get sentences in text, then get the verbs and nouns in each sentence. Based on Word2Vec model, the distance of nouns in sentence and node in engineering model are calculated, and return the most nearest word pair, which are regarded as operating node. Meanwhile, the distance of verbs in sentence and operators are calculated, return the nearest operator and operate on matched node to modify the engineering model. The process is shown in Fig. 6. Because the matching mechanism not always return the correct way to modify the model, designer plays an important part in selecting right node and operator, this depends on the experience of designer. If designer concerned the text cannot meet the design needs, the second high score of text can be selected and analyze.

Fig. 6. Use biological texts to modify engineering model

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After each time of modifying model, we may get new contradictions, search the database with new principle and custom keywords, and modify model again, loop this process until the model satisﬁed speciﬁc needs, then the feasible solution is got.

4 Software Develop Based on core theories introduced in Sect. 3, relative software is developed, the main user interface is shown in Fig. 7. The interface is obviously separated into two parts, the left side is for constructing and modifying engineering model, the principle keyword sets of model is automatically given while designer input related contradict parameters for leaf nodes.

Fig. 7. Interface of software

At the beginning of design, the graphic on left side is set with “function”, “principle”, “behavior” and “structure” initially. Based on speciﬁc application situation, the graphic is updatable when user set an operator and input the correct format of string, after we generate pairs of contradictions, we can input a list of custom keywords, and set the weight of custom keywords for each pair of contradiction. Waiting for few seconds, the corresponding biological texts with verbs and nouns would generate automatically for each contradiction. As introduced in Sect. 3, repeating this process until a feasible solution is got.

5 Case Study In order to verify the validity of this method and explain the processing more speciﬁcally, the case of the cyclone dust collector design is used to illustrate. 5.1

Physical Model of Cyclone Dust Collector

Cyclone dust collector is used to enhance the quality of air released by collecting dust and other impurities from air or gas. We depart the cyclone dust collector model into nodes of

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four dimensions. For the dimension of “function”, we only consider the direct function to avoid limiting freedom of design, thus the only one node “dust collect” is added as a child node of “function”. Cyclone dust collector separates air and dust by centrifugal force, and it generates by inertia, a node names “centrifugal force” under “principle”, and derivative a node names “inertia” as a child node of “centrifugal force”. The act of cyclone dust collector during operating is to keep inner rotate to divide air and dust, dust been separated is sink and ﬁll into the collector, and air is risen and pollute outside of dust collector, so the node “rotate” is added under “behavior”, “sink” and “rise” is generated as child nodes under “rotate”. The main parts of cyclone dust collector are entrance aisle, main body, exit aisle. And reference the traditional structure [13], we add “entrance”, “body”, “exit” under node “structure”, add “area” under “entrance”, add “wall” and “tilt” under “body”, add “area” under “exit”, add “diameter” and “height” under “wall”. After the model is initialized, analyzing the current model, “loss of substance” and “use of energy” are as a pair of contradiction generated by “dust collect”, “force” and “area of moving object” are as a pair of contradiction under “inertia”, “speed” and “area of stationary object” are as a pair of contradiction under “diameter”. After analyzing the function, principle, behavior and structure of a traditional cyclone dust collector, the initial engineering model and three contradictions appear, three lists of custom keywords are all set with weights 0.5. The user interface with initial model we build shown as Figs. 8 and 9 shows the keywords and potential word sets we used to search.

Fig. 8. Initial model and keywords to search

Fig. 9. Workflow of keywords used to search

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Modify Engineering Model

After a few second, several texts of biological species are obtained. Considering that the nouns and verbs extracted may not always relate to texts, spending time on ﬁlter invalid words should be done. The operator and operating nodes to be selected depends on designer. At that, two operators are used to modify engineering model, which is shown in red boxes in Fig. 10. The operation “add” structure “diameter” and “split” behavior “rise”. We conceive design plan at the text of species spike trisetum and heartleaved foamflower, the picture of their images are shown in Fig. 11. For the current engineering model, we could make different plans of the design of the dust collector.

Fig. 10. Select nodes and operators

Fig. 11. Spike trisetum and heartleaved foamflower

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After modifying model, we concern the mean contradiction in model is eliminated, the current engineering model satisﬁed preliminary requirements, then we get a feasible solution. Because the map from biology text to engineering product based on intrinsic association of word similarity, the relationship between them can not be expressed intuitively. 5.3

Physical Product Modeling

Based on parameters of a practical dust collector [13] and the engineering model modify of product, we could come up with many options to adjust our plan to practical dust collector. Inspired by the modifying of engineering model, we consider two types of dust collector as example. (1) By operator “add”, we add nodes “upper diameter” and “lower diameter” as child nodes of “diameter”; and by operator “split”, we split the node “rise” into nodes “elevate” and “uplift”. The inlet of cyclone dust collector are designed with a tilt angle to increase the uplift of gas to be ﬁltered. Consider of more centrifugal force we need to separate the gas and impurities and innovate by added nodes “upper diameter” and “lower diameter”, the diameter closed to exit is expanded. The product model of biological inspired cyclone dust collector at the conceptual design stage is built. (2) By operator “add”, we add nodes “changeable diameter” as child nodes of “diameter”; and by operator “split”, we split the node “rise” into nodes “elevate” and “uplift”. On the inner wall of cyclone dust collector, the tilt shape also assist to make a separation between gas and dust, combining with the idea of adding diameter of inner wall, we could also we sacriﬁce some performance of centrifugal force to increase the tilt of wall. As the part of nodes “elevate” and “uplift”, we also use a tilt angle of inlet to increase the uplift of gas to be ﬁltered.

Fig. 12. Biological inspired cyclone dust collector

The kinds and amounts of product design results is different among designers because of various ideas and experiences. The design of the dust collector on the step of concept design is shown in Fig. 12. Now we have completed the preliminary

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physical design step, designer could regard it as a basic model for further optimization such as performance and service life. 5.4

Design Comparison

Comparing with traditional design structure, the two models above using biomimetic design methods are difference on aspect of air inlet and the shape of inner wall. Although deep analysis like simulation should be done after numerical optimization, we could theoretically analysis the feasibility of the new designs. Air with dust enter the ﬁlter body of cyclone dust collector from inlet. Traditional cyclone dust collector only provides horizontal tangential speed for inner air and dust, as the innovative cyclone dust collector, the tilt angle is used for providing an axial speed for air and dust, and reduces the influence of gravity, to thus increase the rotate time of air and dust in the body, and to more completely distinct dust. Two different kinds of inner wall are used in the innovative design. One of the plans, diameter of the lower part of the straight cylinder is reduced, to make the downer part of cylinder providing more centrifugal force while the upper part of body could not depart the dust from air. Another plan is to replace the cylinder to a whole cone, although it would increase the friction between air and wall, it provides an axial force above from inner side, and meanwhile increase the centrifugal force at lower part.

6 Conclusions Traditional biologically inspired design methods focus on product designers to drive the processing and these is no doubt that using these methods may accomplish tasks of biomimetic design well, but it usually wasting lots of times to get the appropriate way on design. As is shown in Sect. 5, with a computer aided method could improve the efﬁciency of engineering design at a certain extent, on other hand, the texts information could be from network or arthritic books, which means that it provides a broader range of biomimetic design and prevent directional thinking of designer. The source of data could continuously added into database and the process of raw texts processing is fast because of the trained neural network model, making the software easy to maintenance and adjustment. On the case of dust collector optimize design, the method works well on mapping engineering problems to biological texts, but works a little worse while modifying engineering model using operators, which is translated from texts. The software at the current stage requires the designer using own experience of product designing on selecting reasonable verb-operator and noun-node pairs, huge difference of model modiﬁcation among designers, the advance of the method is that respecting subjective of different designers, but sometimes the uniform standard is hard to be given. Future work will focus on providing a second way to guide designers to make biological inspired design with more automated operation pipeline, and different training model is considered to be integrated in the current neural network model for extracting nouns and verbs more accurately.

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References 1. Altshuller, G.S.: Creativity as an Exact Science: The Theory of the Solution of Inventive Problems. Gordon and Breach, Amsterdam (1984) 2. Vincent, J.F.V., Mann, D.L.: Systematic technology transfer from biology to engineering. Philos. Trans. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 360(1791), 159–173 (2002) 3. Nagel, J.K.S., Stone, R.B., McAdams, D.A.: An engineering-to-biology thesaurus for engineering design. In: ASME 2010, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 117–128. American Society of Mechanical Engineers (2010) 4. Nagel, J.K.S., Nagel, R.L., Stone, R.B.: Abstracting biology for engineering design. Int. J. Des. Eng. 4(1), 23–40 (2011) 5. Trotta, M.G.: Bio-inspired design methodology. Int. J. Inf. Sci. 1(1), 1–11 (2011) 6. Kozaki, K., Mizoguchi, R.: A keyword exploration for retrieval from biomimetics databases. In: Joint International Semantic Technology Conference, pp. 361–377. Springer, Cham (2014) 7. Nix, A.A., Sherrett, B., Stone, R.B.: A function based approach to TRIZ. In: ASME 2011, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 285–295. American Society of Mechanical Engineers (2011) 8. Khodadadi, A.: Synergy of a genetic algorithm and TRIZ in conceptual design. In: Proceedings of IASS Annual Symposia, no. 2, pp. 1–8. International Association for Shell and Spatial Structures (IASS) (2018) 9. Chen, J.L., Hung, S.C.: Eco-innovation by TRIZ and biomimetics design. In: 2017 International Conference on Applied System Innovation (ICASI), pp. 40–43. IEEE (2017) 10. Mikolov, T., Sutskever, I., Chen, K., et al.: Distributed representations of words and phrases and their compositionality. Im: Advances in Neural Information Processing Systems, pp. 3111–3119 (2013) 11. Xu, K., Zhou, Z., Hao, T., et al.: A bidirectional LSTM and conditional random ﬁelds approach to medical named entity recognition. In: International Conference on Advanced Intelligent Systems and Informatics, pp. 355–365. Springer, Cham (2017) 12. Salton, G., Buckley, C.: Term-weighting approaches in automatic text retrieval. Inf. Process. Manag. 24(5), 513–523 (1988) 13. Chi, Y., Huo, H., Dai, Y.: Cyclone dust collector optimization design and CFD numerical veriﬁcation. Mech. Des. Manuf. 2018(8), 33–35, 40 (2018)

Numerical Simulation on Effect of Graphene Doped Morphology on Heat Transfer Efﬁciency of Anti-/deicing Component Long Chen1,2(&) and Zhanqiang Liu1,2 1

2

Key Laboratory of High Efﬁciency and Clean Manufacturing, Shandong University, Jinan 250061, China {chenlong,melius}@sdu.edu.cn School of Mechanical Engineering, Shandong University, Jinan 250061, China

Abstract. Graphene doping can effectively improve the heat transfer capacity and anti-/deicing efﬁciency of anti-/deicing component. The main factors that affecting the improvement of heat transfer efﬁciency of graphene-doped anti-/ deicing component are the graphene-doped morphology, which including the distribution and location in the composites and the size and shape of the graphene sheet. In this paper, the heat transfer characteristics of graphene-doped anti-/deicing component were studied by numerical simulation. By establishing the heat transfer mathematical model of graphene-doped anti-/deicing component, the cross-scale heat transfer numerical simulation with different graphenedoped forms was proposed to solve the temperature ﬁeld distribution of graphene-doped anti-/deicing component. Finally, based on the results of temperature ﬁeld calculation of graphene-doped anti-/deicing component, the effect of graphene doping morphology on the heat transfer efﬁciency of anti-/deicing component was studied, which were beneﬁcial to heat transfer of anti-/deicing component. By employing the proposed cross-scale heat transfer model for graphene doping of composite materials, the heat transfer characteristics of graphene doping was qualitatively described, and it is shown that the doping of graphene can optimize the heat transfer of the composite anti-/deicing component. Keywords: Graphene doping Numerical simulation Anti-/deicing component Composites

1 Introduction With the continuous prominence of engineering safety problems, more and more engineering safety technologies have been developed and studied extensively. Wind turbine blades and aircraft will ice under extreme conditions, which causing serious safety accidents. Icing will affect the power supply efﬁciency of wind turbine blades, This project is supported by Fundamental Research Funds of Shandong University (Grant No. 2018GN034), State Key Laboratory of Mechanical System and Vibration (Grant No. MSV2019-13), and 13th Five-Year Plan Equipment Pre-Research Fund (Grant No. 61402060404). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 52–62, 2020. https://doi.org/10.1007/978-981-32-9941-2_5

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especially the flight control and flight safety of aircraft. At present, the research on improving heat transfer of composite materials is more extensive: Wang et al. [1] proposed a composite porous wick with spherical-dendritic powders. Two evaporative heat transfer modes are observed through political and legal experiments, which can improve the heat transfer characteristics. Buffone [2] proposed a paradigm shift in improving the heat transfer rate between the surface of the radiator and the surrounding fluid by a new design method of composite radiator. This approach consists in using high thermally conductive coatings on top of the ﬁnned substrate in order to increase the local temperature along the ﬁn washed surface. Lei et al. [3] proposed a decomposition algorithm which using a composite cylindrical surface source solution to deal with the composite heat conduction along the radius and a coil line source solution to deal with the heat conduction between tube loops in depth direction. Two computationally efﬁcient and accurate solution methods for heat transfer analysis in composite metal cylindrical vessel subjected to internal thermal load are presented by Masumi [4]. The methods were veriﬁed by numerical simulation, which are more accurate. In Krysko’s work, the topological optimization of composite structures is widely used while tailoring materials to achieve the required engineering physical properties [5]. Considering the competitive mechanics and thermal properties of the composites, the materials with the maximum effective bulk modulus and thermal conductivity were constructed. Wang et al. [6] developed a grille-sphere composite structured packed bed (GSCSPB). The new structure aims to overcome the shortcomings of random packed bed and traditional packed bed. The pressure drop and heat transfer in GSCSPB were measured by naphthalene sublimation experiment, and the comprehensive heat transfer performance of GSCSPB with random packed bed and structured packed bed was evaluated. Aadmi et al. [7] studied the thermo-physical properties and melting process of phase change material composites numerically and experimentally based on epoxy resin parafﬁn. The above research optimizes and improves the heat transfer of composite materials through numerical simulation, experimental veriﬁcation and theoretical derivation, which providing research methodological guidance and research reference for improving anti-/deicing efﬁciency and solving icing problems. In recent years, graphene, as a new multifunctional material, has shown excellent performance in heat conduction and conduction [8–11]. In terms of thermal conductivity, graphene-related studies show that graphene can improve the thermal conductivity of composites and metal-based materials, which also has toughening effect on mechanical properties. Hussien et al. [12] studied the forced convective heat transfer of water-borne Al2O3 nanofluids mixed with graphene nanofluids. They selected volumetric concentrations of Al2O3/water nanofluids, and graphene nanofluids were compared. Liao et al. [13] theoretically investigated near-ﬁeld heat transfer between graphene–SiC–graphene–metamaterial (GSGM) multilayer structures. The results show that heat transfer between GSGM structures is signiﬁcantly larger than that between SiC-coated metamaterials when the chemical potential of graphene is not very high. Anhua et al. [14] established a coupled thermo-mechanical model of graphene ﬁlm and obtained an accurate analytical solution of the thermoacoustic radiation of graphene ﬁlm. By comparing with the experimental results, the correctness of the

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theoretical model is veriﬁed. Zhou et al. [15] studied the heat transfer performance of graphene nanoplate (GNP) fluid oscillating heat pipe (OHPS). The results showed that GNP nanofluid as a working fluid could improve the heat transfer performance of OHP compared with deionized water (DW). Bharadwaj et al. [16] studied the application of mixed nanofluids in radiators to improve their heat transfer performance. Because graphene has high thermal conductivity, carboxyl graphene and graphene oxide nanoparticles were selected as doping particles of nanofluids. The results show that heat transfer efﬁciency and cooling rate are improved. In summary, the above researches show that graphene has a signiﬁcant effect on improving the heat transfer efﬁciency. In this paper, graphene is doped in composite materials to improve the heat transfer efﬁciency of composite anti-icing module, and the heat transfer effect of graphene doping is studied by numerical simulation. In this research, the graphene doping technology is used to modify the composite anti-/deicing component in order to improve the heat transfer efﬁciency of the composite anti-/deicing component and reduce the heat loss in the composite during the heat transfer process. Through cross-scale modeling of graphene-doped composites, multi-ﬁeld coupling simulation was used to solve the problem, and different morphologies of graphene-doped composites were analyzed and studied. The temperature ﬁeld distribution of graphene-doped composites was predicted by numerical simulation. Using the advantage of numerical simulation for reference, the number of experiments can be reduced effectively, and the research and result prediction of graphene-doped composites can be improved. The relationship between the temperature ﬁeld distribution and the graphene doping morphology was established. The effect of graphene doping morphology on the heat transfer characteristics of composites was revealed, which provided a theoretical basis for the optimization and control of graphene doping process.

2 Numerical Simulation Process The micro-morphology of composite anti-/deicing component doped with graphene is shown in Fig. 1. Graphene is distributed in the gap between epoxy resin and glass ﬁber in lamellar form and is evenly dispersed in epoxy resin. However, within the graphenedoped composites, the graphene lamellae within the unit structure were found to be randomly distributed by scanning electron microscopy. The mathematical model of micro graphene doped composites was established by cross-scale modeling method, and the temperature ﬁeld of the model was solved by numerical simulation. The temperature ﬁeld distribution of graphene-doped composites under the coupling of flow ﬁeld, temperature ﬁeld and thermal stress ﬁeld was analyzed by solving the multi-ﬁeld coupled ﬁnite element model. Intelligent meshing technology is used to mesh graphene-doped composites. The partitioned grid satisﬁes the condition of multi-ﬁeld coupling and improves the accuracy of numerical simulation.

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Fig. 1. Schematic of cross-scale modeling process for graphene doped composites

In the graphene-doped composites model, the size of graphene sheet length is 300– 600 lm, and the dispersion form is random distribution. Therefore, the three-dimensional model of graphene doping is established according to the microstructure of the composites. The bottom of the simulation unit is a glass ﬁber paving structure. The glass ﬁber is preimpregnated in epoxy resin, and the space between epoxy resin and glass ﬁber is doped with graphene sheets. After the three-dimensional model of graphene doping is established, the geometric model of graphene-doped composites was meshed (as shown in Fig. 2) and imported in the commercial ﬁnite element software ANSYS Workbench.

Fig. 2. Schematic of cross-scale modeling meshing

In the process of numerical simulation, micro-scale cross-scale modeling of composite anti-/deicing component with different graphene doping morphology was carried out. The heat transfer effect of composites with different graphene doping morphology was analyzed under the equivalent thermal boundary condition, and the anti-icing efﬁciency of composite anti-/deicing component was analyzed. Then the influence of graphene doping on the heat transfer effect of composite anti-/deicing component was expounded. Graphene doping morphology includes graphene doping content and graphene distribution. The heat transfer characteristics of composite anti-icing module under different doping conditions were studied by simulating different graphene doping morphologies. The steps of numerical simulation are as follows: • Step 1: Firstly, the geometric model of graphene-doped composites is established. • Step 2: Then the multi-ﬁeld coupling boundary conditions of graphene-doped composites are determined.

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• Step 3: Solving flow ﬁeld distribution of graphene-doped composites and applying the results on geometric model as initial boundary conditions of thermal module. • Step 4: Thus the temperature distribution of graphene-doped composites is solved. • Step 5: Using the temperature ﬁeld results as initial conditions to solute of thermal stress ﬁeld distribution. • Step 6: Solving thermal stress ﬁeld distribution of graphene-doped composites. • Step 7: Finally, the temperature and stress ﬁelds of graphene-doped composites are solved by coupling method.

3 Calculation of Effective Thermal Conductivity According to the law of thermal resistance and the equivalent thermal conductivity, a series-parallel heat transfer model of graphene doped composites was established. The integral composite anti-/deicing component was considered to consist of a large number of square elements. An arbitrary element is selected from the composite graphene doped anti-/deicing component (see Fig. 3). The equivalent thermal resistance of the whole structure is established by calculating the equivalent thermal conductivity of the layered structure in the unit.

Fig. 3. Schematic of unit and equivalent thermal resistance

Through the analysis of heat transfer unit, the composite graphene doped anti-/ deicing component is composed of glass ﬁber and epoxy resin matrix. kb and kf are set as the thermal conductivity of epoxy resin matrix and glass ﬁber respectively. According to Fourier’s law, the thermal resistance is as follows: R¼

d kA

ð1Þ

Where d is the unit thickness, and A is the cross-sectional area of the unit. The equivalent thermal resistance model of Fig. 3 is analyzed. The model divides the unit into model 1, model 2 and model 3. Due to the randomness of the shape and content of graphene doping between glass ﬁber and epoxy resin, it is approximately considered that the graphene sheet doped in the space formed by glass ﬁber and epoxy resin are uniformly distributed. Therefore, the equivalent thermal resistance of

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graphene in epoxy resin is set as Rg , and the equivalent thermal resistances of the corresponding models are R1 , R2 , and R3 respectively. Then the equivalent thermal resistance model of the element body is a parallel model, and the equivalent thermal resistance Re of the element body can be expressed as: 1 1 1 1 ¼ þ þ Re R1 þ Rg R2 þ Rg R3 þ Rg

ð2Þ

Therefore, based on the equivalent thermal resistance model in Ref. [17], considering the influence of graphene equivalent thermal resistance on each model, the equivalent thermal resistance model of model 1, model 2 and model 3 are deduced as follows: 2 kb D " #9 8 > > 2kf 3kb > > > a tan 1 > > > < 2 2 = 2k k 3k ð f b bÞ 1 2 R2 ¼ þ 1 2kb D D2 > > > 2kf kb 3k2b 2 > > > > > ; : R1 ¼ ¼

" #9 8 > > 2k 3k f b > > > a tan 1 > > > < 2 2 = ð2kf kb 3kb Þ 8 R3 ¼ 2 1 D > > > 2kf kb 3k2b 2 > > > > > ; :

ð3Þ

ð4Þ

ð5Þ

Equation (2) is sorted out and the equivalent thermal resistance model of the unit body Re is deduced as follows: Re ¼

ðR1 þ Rg ÞðR2 þ Rg ÞðR3 þ Rg Þ R1 R2 þ R1 R3 þ R3 R2 þ Rg ðR1 þ R2 þ R3 þ 3Rg Þ

ð6Þ

R1 R2 R3 R1 R2 þ R1 R3 þ R2 R3

ð7Þ

R0e ¼

Equation (6) is compared with Eq. (7) by division, and Eq. (8) is obtained: Rg ðR1 þ R3 þ R2 þ 3Rg Þ Re R1 R2 R3 1 þ ¼ R1 R2 þ R1 R3 þ R2 R3 R0e ðR1 þ Rg ÞðR2 þ Rg ÞðR3 þ Rg Þ

ð8Þ

Based on the above deduced equivalent thermal resistance model, the doping heat transfer of graphene in composite materials was analyzed theoretically. The expression can be used to analyze that, because the thermal resistance of graphene Rg \ R1 ; R2 ; and R3 , thus, the ratio of Eq. (8) is less than 1. The above formula shows

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that the equivalent thermal resistance of the whole unit doped with graphene composite anti-/deicing component is less than that of the composite anti-/deicing component unit without graphene.

4 Results and Discussion The heat transfer of composite graphene doped anti-/deicing component was solved by establishing a mathematical model of cross-scale graphene doping heat transfer. The heat transfer solutions of graphene-doped anti-/deicing component were compared with that temperature distribution of undoped graphene. Table 1 shows the relevant thermophysical parameters of composite anti-/deicing component. According to the relevant parameters in the table, the mathematical model of heat transfer of composite graphene-doped anti-/deicing component is solved. The influence of graphene doping on heat transfer characteristics of composite anti-/deicing component is analyzed, and the effect of graphene sheet on heat transfer characteristics is expounded.

Table 1. Material thermal physical parameters

2700

Thermal conductivity (W/m K) 1.46

Coefﬁcient of thermal expansion (1/K) 2.7e−6

2300

0.21

5.6e−7

1760

5300

−3e−5

Layer

Density (Kg/m3)

Glass ﬁber Epoxy resin Graphene

The heat transfer of composite graphene doped anti-/deicing component was solved by establishing a mathematical model of cross-scale graphene doping heat transfer. The heat transfer solutions of graphene-doped anti-/deicing component were compared with that undoped graphene. Graphene doping played a signiﬁcant role in promoting the temperature ﬁeld of composite anti-/deicing component. The numerical results show that graphene doping has a signiﬁcant effect on the homogenization of temperature ﬁeld of composite anti-/deicing component. As shown in Fig. 4, by comparing the temperature distribution of graphene-doped anti-/deicing component under the same thermal boundary conditions, the temperature ﬁeld of nondoped graphene composite anti-/deicing component forms a large area of low temperature region, while that of doped graphene anti-/deicing component is smaller, which shows that the heat transfer effect of doped graphene composite anti-/deicing component is optimized.

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Fig. 4. Temperature ﬁeld distribution of composite anti-/deicing component: (a) graphene doped; (b) graphene undoped.

As shown in Fig. 5, by analyzing the temperature ﬁeld of graphene-doped composite anti-/deicing component, the area where the surface temperature of composite anti-/deicing component meets the anti-icing temperature requirement (the surface temperature is above 2 °C) is shown in the following ﬁgure. From the distribution of temperature ﬁeld, it can be found that the areas are the zone where graphene sheets are densely distributed. The results indicate that graphene doping plays a positive role in heat transfer of composites.

Fig. 5. Anti-icing area distribution of graphene doped anti-/deicing component

The heat flux of graphene-doped composites anti-/deicing component was analyzed (as shown in Fig. 6). The numerical simulation results of graphene-doped composites show that the graphene layer doped in the composites has a signiﬁcant effect on the heat flux in the local area. It shows that the heat transfer characteristics of graphene layer have promoted the heat transfer performance of the entire composites. The numerical simulation results show that the heat flux at the agglomeration of graphene sheet is signiﬁcantly higher than that in the space region composed of epoxy resin and glass ﬁber.

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Fig. 6. Heat flux distribution of graphene doping model

The heating efﬁciency curve as shown in Fig. 7 is drawn by collecting the surface heating area of the composite graphene doping anti-/deicing component and without doping. It can be found from the curve that the proportion of the area where the surface temperature of graphene-doped anti-/deicing component reaches the anti-icing temperature increases with the heating time. The increasing speed of graphene-doped anti-/ deicing component is higher than that of the non-doped graphene, which indirectly veriﬁes that the anti-icing ability of graphene-doped composite anti-/deicing component has been improved.

Fig. 7. The heating efﬁciency curve of composite anti-/deicing component

5 Conclusions 1. By establishing a cross-scale heat transfer model of graphene doping in composite materials, the numerical simulation of graphene doping was carried out to reveal the effect of graphene doping on heat transfer of composite materials. The proposed

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cross-scale heat transfer model for graphene doping of composite materials can qualitatively describe the heat transfer characteristics of graphene doping. 2. Based on the assumption of uniform distribution of graphene sheets, the equivalent thermal resistance of graphene doped glass ﬁber and epoxy resin matrix was deduced. The expression of equivalent thermal resistance of graphene doped composites was obtained. 3. The numerical model of graphene doping is solved by multi-ﬁeld coupling numerical solution method, and the temperature ﬁeld distribution of graphene doping numerical simulation is obtained. By comparing the temperature distribution of the components of the composite anti-/deicing component with that of the undoped graphene, it is shown that the doping of graphene can optimize the heat transfer of the composite anti-/deicing component. 4. The numerical simulation results show that graphene doping can improve the local surface temperature and heat flux of composite anti-/deicing component, which effectively increase the surface temperature of composite anti-/deicing component. The anti-icing surface area ratio of composite anti-/deicing component was also increased by graphene doping. Through graphene doping, the surface heating speed of composite anti-/deicing component was increased, and the anti-icing efﬁciency can be improved, which promoting the anti-icing reaction rate of anti-/deicing component.

References 1. Wang, D., Wang, J., Bao, X., et al.: Evaporation heat transfer characteristics of composite porous wick with spherical-dendritic powders. Appl. Therm. Eng. 152, 825–834 (2019) 2. Buffone, C.: A novel approach for heat transfer enhancement in composite ﬁns. Int. J. Heat Mass Transf. 130, 650–659 (2019) 3. Lei, F., Hu, P., Huang, X.P.: Hybrid analytical model for composite heat transfer in a spiral pile ground heat exchanger. Appl. Therm. Eng. 137, 555–566 (2018) 4. Masumi, A.A., Rahimi, G.H., Liaghat, G.H.: The use of the layerwise theory in heat transfer analysis of metal composite vessel by DQM. Int. J. Therm. Sci. 132, 14–25 (2018) 5. Krysko, A.V., Awrejcewicz, J., Pavlov, S.P., et al.: Topological optimization of thermoelastic composites with maximized stiffness and heat transfer. Compos. B Eng. 158, 319–327 (2019) 6. Wang, J., Yang, J., Cheng, Z., et al.: Experimental and numerical study on pressure drop and heat transfer performance of grille-sphere composite structured packed bed. Appl. Energy 227, 719–730 (2017). S0306261917310152 7. Aadmi, M., Karkri, M., El, H.M.: Heat transfer characteristics of thermal energy storage of a composite phase change materials: Numerical and experimental investigations. Energy 72, 381–392 (2014) 8. Tan, Y.W., Zhang, Y., Bolotin, K., et al.: Measurement of scattering rate and minimum conductivity in graphene. Phys. Rev. Lett. 99, 246803 (2007) 9. Kim, H., Miura, Y., Macosko, C.W.: Graphene/polyurethane nanocomposites for improved gas barrier and electrical conductivity. Chem. Mater. 22(11), 3441–3450 (2010) 10. Esfahani, M.R., Languri, E.M.: Exergy analysis of a shell-and-tube heat exchanger using graphene oxide nanofluids. Exp. Therm. Fluid Sci. 83(Complete), 100–106 (2017)

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11. Tian, L., Wang, Y., Li, Z., et al.: The thermal conductivity-dependant drag reduction mechanism of water droplets controlled by graphene/silicone rubber composites. Exp. Therm. Fluid Sci. 85, 363–369 (2017) 12. Hussien, A.A., Yusop, N.M., Moh’d, A.A.N., et al.: Numerical study of heat transfer enhancement using Al2O3–graphene/water hybrid nanofluid flow in mini tubes. Iran. J. Sci. Technol. Trans. A: Sci. 1–12 (2019) 13. Liao, Y.F., Wang, G.Y.: Active control of near-ﬁeld radiative heat transfer via multiple coupling of surface waves with graphene plasmon. Eur. Phys. J. B: Condens. Matter Complex Syst. 92, 65 (2019) 14. Anhua, B., Shuang, L.I., Qianhe, X., et al.: Thermoacoustic theory of graphene ﬁlms considering heat transfer of substrate materials. Chin. J. Acoust. 42(2), 755–761 (2018) 15. Zhou, Y., Cui, X., Weng, J., et al.: Experimental investigation of the heat transfer performance of an oscillating heat pipe with graphene nanofluids. Powder Technol. 332, 371–380 (2018). S003259101830175X 16. Bharadwaj, B.R., Sanketh, M.K., Manjunath, D.M., et al.: CFD analysis of heat transfer performance of graphene based hybrid nanofluid in radiators. In: IOP Conference Series: Materials Science and Engineering, vol. 346, p. 012084 (2018) 17. Long, C., Yidu, Z., Qiong, W.: Heat transfer optimization and experimental validation of anti-icing component for helicopter rotor. Appl. Therm. Eng. 127, 662–670 (2017)

Numerical Analysis on Load Sharing Characteristics of Multistage Face Gears in Planetary Transmission Xingbin Chen1(&), Qingchun Hu1, and Chune Zhu2 1 School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, China [email protected] 2 Institute for Biomedical and Pharmaceutical Sciences, Guangdong University of Technology, Guangzhou 510006, China

Abstract. Planetary transmission devices that use multistage face gears comprise structural components such as bevel gears, cylindrical gears, and face gears. Because of the novel structure and innovative operating principle of variable speed transmissions, inevitably, there are some problems such as manufacturing, installation errors and need for geometric parameter optimization. Therefore, it is necessary to study the static and dynamic load sharing characteristics of the planetary system to improve the transmission reliability under variable speed conditions. In this study, the calculating methods of the static and dynamic load sharing coefﬁcients are studied by constructing the dynamic structure model and moment equilibrium equation. The influences of the manufacturing and installation errors on the load sharing coefﬁcient among the various stages of the transmission system are analyzed. The influences of various component errors on the dynamic load sharing characteristics under the conditions of different stages with various speeds are studied by the dynamic response analysis. The influence of working condition, physical, and geometric parameters on the load sharing characteristics are analyzed. The results are beneﬁcial in reducing the vibration and noise of the planetary transmission caused by uneven load distribution. They also provide a theoretical basis for improving the stability and loading capacity. Keywords: Multistage face gears Load sharing Dynamic characteristics Planetary gears

Variable speed

1 Introduction The load sharing characteristics of planetary gears reflect the uniformity of the transmission power of each branch, which is directly related to the design rationality of the system and the reliability of the transmission. In order to study the dynamic performance of planetary gears more deeply, many domestic and foreign scholars have This work was supported by the National Natural Science Foundation of China under Grant No. 51575191. © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 63–83, 2020. https://doi.org/10.1007/978-981-32-9941-2_6

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analyzed the load sharing characteristics of planetary gears from the static and dynamic perspectives respectively. Shuai et al. studied the three split static load sharing characteristics of the two-stage external meshing planetary transmission system of an aeroengine, and obtained the load sharing coefﬁcient under manufacturing and installation error conditions [1]. Kahraman et al. analyzed the load sharing performance of planetary gears, and proposed to characterize the load sharing effect with static load sharing coefﬁcient, dynamic load sharing coefﬁcient and dynamic load coefﬁcient [2]. Hu et al. established a static load-sharing calculation model for the aeronautical twostage ﬁve-branch planetary gear train, and introduced loaded tooth contact analysis method. Combined with the coordination condition of torsion angle deformation, the torque of each gear pair is obtained. The load-sharing coefﬁcient of the system is further calculated [3, 4]. Viadero et al., studied the quasi-static characteristics of threeplanet gear transmission system by numerical deﬁnition and ﬁnite element modeling, and analyzed the influence of planetary radial mounting errors on the load sharing characteristics under different pressure angles [5, 6]. Yi et al. analyzed the initial position’s influence on the load sharing of transmission system to provide a theoretical basis of load sharing control [7]. Zhu R.P. et al. established a coupled dynamic analysis model considering the time-varying meshing stiffness, mesh errors and mesh damping to obtain the distribution law of the sensitivities of load sharing coefﬁcient for the split torque transmission system of the traditional epicyclic gear transmission of the helicopter [8–10]. Li et al. established a model to predict the reliability of helicopter planetary gear train under the condition of partial load. The bad influence of unequal load sharing on planetary gear train is shown by prediction result and the reliability model is veriﬁed through a statistical analysis method of random censored data [11]. Du et al. proposed a load sharing analysis methodology for a new type of power split spiral bevel gearing system based on the closed-loop characteristic of the power flow. A general formula for calculating the simpliﬁed engineering load distribution is derived. An experiment is designed to demonstrate the feasibility of this method [12]. In addition, Zhang [13], and Huang et al. established the equilibrium equations of the corresponding planetary gear system from factors such as the planetary gear eccentricity error, number of planets and gear clearance, and calculated the non-load-sharing coefﬁcient of planetary transmission system [14]. Singh [15], Shu [16], Gu [17], Chaari et al. studied the load sharing characteristics of planetary gear systems from different aspects, such as positional deviation of the gear, tooth frequency error, tooth proﬁle error, meshing stiffness of gear pair, support stiffness and floating quantity of each component [18, 19]. Presently available results pertain mainly to single-stage planetary gear systems, and there are signiﬁcant differences among the various models, so it is difﬁcult to use each other. The paper studies a planetary multistage face gears train, which has provision for multiple stages to allow for variable speeds. Static and dynamic load sharing models are established to analyze the influence of working conditions, geometry and corresponding physical parameters on the load sharing characteristics, which can provide a theoretical basis for improving the stability and load carrying capacity of the system. It also has considerable practical signiﬁcance for the optimization of planetary multistage face gears devices.

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2 Model of the Multispeed Transmission Device In the previous research, a novel multistage transmission was proposed that with face gears introduced into the planetary multistage transmission for the ﬁrst time and the novel multistage transmission was constructed by the structural advantages of the face gear transmission, as shown in Fig. 1A. Compared to an ordinary multistage system, the novel transmission mode can reduce the complexity and irregularity of the mechanism which has the features of stable transmission, high transmission efﬁciency, flexible space layout and simple structure, as shown in Fig. 1B. The motion is input from driving bevel gear (splined joint), and then split into four branches by driven bevel gears which are connected with the cylindrical gears by spline shafts, the cylindrical gear meshes with the multistage face gears at the same time. Among that, the number of the driving bevel gear is 36, the driven bevel gear is 19, the cylindrical gear is 17, the face gears are temporarily set to three stages, and the numbers of teeth are 56, 68 and 80 respectively. The teeth in the axial direction of the face gears are reserved for meshing with the shifting brake mechanism. If one stage of face gears is braked, the cylindrical gears revolve around the face gear corresponding to this stage, causing the driven bevel gears to rotate and revolve with the cylindrical gears through the spline shaft. If no brake is applied, face gears all freely idling, power transmission does not take place. Under the action of the spline shaft, the planets drive the output shaft to rotate. In summary, it is the revolution confluence of four planetary axes. The input stage of the system is the splitting transmission of face gear. Considering the meshing characteristics of the face gear transmission, the synchronous meshing request of the input gear can be satisﬁed as long as the teeth numbers of all stages face gears are the multiple of the number of meshing gears in the split-twist conﬁguration of the face gear transmission system.

Fig. 1. A: Structure schematic diagram of multistage face gears planetary transmission B: Assembly diagram of key core structure of planetary multistage face gears (1. Input shaft; 2. Driving bevel gear; 3. Driven bevel gear; 4. Planetary gear; 5. Planetary shaft; 6-8. Face gears; 9. Output shaft)

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3 Analysis of the Load Sharing Mechanism According to the structural principle of the planetary multistage face gears transmission device (MFGTD), the swing arm is an output component, and it must be assumed that the driving bevel gear and face gear are combined as input component. The speed of the driving bevel gear is constant, while the speed of the face gear is adjustable (0 njmax ), the direction of the rotation speed is unchanged, and the distribution coefﬁcient is 0 K 1. Then the speed of the face gear can be expressed as K njmax at any time. P1 T1 n1 T1 n1 ¼ ¼ Pj Tj nj Tj K njmax n1 1 R ¼ ¼ ¼ i3j i27 K njmax K w K

ð1Þ

i i Kn

In this equation, w ¼ 3j 27 n1 jmax is speed regulation magnitude and R ¼ w1 is the differential ratio. According to the power equilibrium equation: (

P1 ¼ R þR K PH Pj ¼ P3 ¼ R þK K PH

ð2Þ

The magnitude of the distribution coefﬁcient varies with the speed of the face gear, and is an important factor affecting the power ratio. If the working speed is different, the power ratio of two input components is also different. But the truly decisive factors are the speed regulation magnitude w and the differential ratio R. w is larger, R is smaller, the power transmitted by the face gear consumes a greater proportion of the total input power, and vice versa. Similarly, if the face gear is assumed to have a constant speed and the driving bevel gear as shifting, the results so obtained are consistent. 3.1

Transmission Ratio Analysis

As the planetary gear is a moving shaft transmission, the transmission ratio cannot be calculated simply by the equation of ﬁxed axis driving, but usually by the ﬁxed planetary carrier method, graphic method, vector method and moment method, etc. For studying simpliﬁcation, the planetary MFGTD can be considered as a differential planetary gear mechanism conforming to a 2 K-H (NW) type. Among that, face gears are composed of multiple circular rings, which can be unlimited in number theoretically, however, considering the loading capacity and general gear box dimension, this paper sets the face gears combination j to level 3 with a, b and c, as shown in Figs. 2 and 3.

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67

The transmission ratio of the transfer mechanism is obtained according to the general equation of a ﬁxed planetary carrier: (

j ¼ 1 iH i1H 1j ZZ

j 7 iH 1j ¼ ðn1 nH Þ=ðnj nH Þ ¼ Z3 Z2

Fig. 2. Kinematic relationship schematic about planetary transmission system

Fig. 3. Structure schematic of transmission ratio

ð3Þ

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j In this equation, i1H is the transmission ratio of the input shaft 1 and the tumbler H rotating relative to the face gears j. iH 1j is the transmission ratio of the input shaft 1 and the face gears j rotating relative to the tumbler H. n1 is the speed of the input shaft 1. nH is the speed of the tumbler H. nj is the speed of the face gears j. Zj represents the teeth number of the face gear. Z7 represents the teeth number of the driven bevel gear. Z3 represents the teeth number of the cylindrical gear. Z2 represents the teeth number of the driving bevel gear. So: j i1H ¼ 1þ

Zj Z7 Z3 Z2

ð4Þ

Deﬁning the speed ratio relations of the gear pairs as i27 ¼ Z7 =Z2 , i3j ¼ Zj =Z3 , the corresponding kinematic relation of each component are: 8 ZZ ZZ > n1 ¼ Z3j Z72 nj þ ð1 Z3j Z72 ÞnH ¼ i3j i27 nj þ ð1 i3j i27 ÞnH > > < ZZ i i Z2 nH ¼ Z3 ZZ23Z n1 þ Z3 Z2jZ7 j Z7 nj ¼ 1i13j i27 n1 þ 1i3j 3j27i27 nj j Z7 > > > : n ¼ Z3 Z2 Zj Z7 n Z3 Z2 n ¼ ði i 1Þn 1 n j H 3j 27 H i3j i27 1 Zj Z7 Zj Z7 1

ð5Þ

When selecting the relevant tooth numbers to determine the transmission ratio, the corresponding concentric condition, assembly condition, adjacency condition and other additional conditions should be considered. For example, the teeth number on every stage of the face gears should be an integral multiple of the number of planetary gears. uan qan ¼ 2p=Z ¼n 2

j i1H Z2 np

¼ Znj Z2 Z3 np Z2 Z7 ¼ N; ðN ¼ 1; 2; Þ

ð6Þ

In this equation, qan is the ratio between the installation angle and the corresponding center angle a of two teeth with the adjacent center gear. uan ðn ¼ 1; 2; Þ is the angle turned to match the installation angle of corresponding planetary gear. np is the number of planetary gears. n cannot be a multiple of np . N is an integral multiple of the number. If the numbers of driving and driven bevel gears are Z2 ¼ 36 and Z7 ¼ 19, based on the above conditions, the relationship between the teeth numbers of cylindrical gears and face gears and the planetary system transmission ratio is shown in Table 1. Table 1. Relationship about tooth numbers of cylindrical gears, face gears and transmission ratios of planetary system Z3 multipleofthenumberofplanetary 44 56 68 80 17 2.3660 2.7386 3.1111 3.4837 19 2.2222 2.5556 2.8889 3.2222 21 2.1058 2.4074 2.7090 3.0106

92 3.8562 3.5556 3.3122

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3.2

69

Force Analysis

In order to study the relationship between the distribution of power flow and the structural parameters in the system, it is necessary to analyze the forces in each component to determine the magnitude of the load torque. Dynamic meshing force equation: Fn ðtÞ ¼ km ðtÞfb ðxÞ þ cm ðtÞ_eðtÞ

ð7Þ

Normalize the meshing force of differential gears train and planetary gears train: (

Fbi ðtÞ ¼ kmb ðtÞDbR Fcfi ðtÞ ¼ kmcf ðtÞDcfR

ð8Þ

In this equation, kmb ðtÞ, kmcf ðtÞ are the time-varying mesh stiffness. DbR , DcfR are the relative displacements of ideal meshing line containing the integrated errors. Among that: 8 bi i cosðwb t þ ubi Þ k ðtÞ ¼ kbs þ kbs > > > m > > < kcfi ðtÞ ¼ kcfs þ ki cosðwc t þ ucfi Þ m cfs > Dbi > R ¼ f ðdb Þ þ Db > > > : cfi DR ¼ f ðdcf Þ þ Dcf

ð9Þ

In this equation, kbs , kcfs are the average meshing stiffness of the bevel gear pair and i i the cylinder-face gear pair respectively. kbs , kcfs are the transient meshing stiffness fluctuation amplitude of the bevel gear pair and the cylinder-face gear pair respectively. db , dcf are the clearance functions of the bevel gear pair and the cylinder-face gear pair respectively. f ðdb Þ, f ðdcf Þ are the integrated error displacement of the bevel gear pair and the cylinder-face gear pair respectively. Db , Dcf are the integrated error of the bevel gear pair and the cylinder-face gear pair respectively. xb ¼ 2pnb Zb =60, xc ¼ 2pnc Zc =60 are the meshing Angle frequency of the bevel gear pair and the cylinderface gear pair respectively. ubi ¼ ði 1Þ 2pi , ucfi ¼ ði 1Þ 2pi are the initial installation angles of each bevel gear in the bevel gear pair and each cylindrical gear in the cylinder-face gear pair respectively. 3.3

Torque and Power Analysis

Before analyzing the power flow of each component, it is necessary to clarify the rules for determining the input and output power: (1) When the torque direction is consistent with the motion direction, the power of the component can be deﬁned as the input power. (2) When the torque direction is opposite to motion direction, the power of the component can be deﬁned as the output power. (3) When the torque or speed is zero, the component does not transmit power.

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Based on the structural of the planetary MFGTD, it is assumed that the friction loss is neglected in the torque equilibrium equation: T 1 þ Tj þ T H ¼ 0

ð10Þ

In this equation, T1 is the torque of the driving bevel gear (sun gear). Tj is the torque of the face gear (gear ring). TH is the torque of the swing arm (planetary carrier). The power equilibrium equation is: T1 n1 þ Tj nj þ TH nH ¼ 0

ð11Þ

The torque relationships of the components obtained from the Eqs. 10 and 11 are: T1 ¼ ð1 þ i3j i27 ÞTH ¼

1 Tj i3j i27

ð12Þ

Then the corresponding power ratio relationship is given by: 8P i 1 ¼ TT1j nn1j ¼ i3j1ji27 > > < Pj P1 T1 n 1 PH ¼ TH nH ¼ ð1 þ i3j i27 Þi1H > > : Pj Tj n j PH ¼ TH nH ¼ ð1 þ i3j i27 Þi3j i27 ijH

ð13Þ

In this equation, P1 is the power delivered by the driving bevel gear. Pj is the power delivered by the face gear. PH is the power delivered by the swing arm. When the driving bevel gear 1 and the face gear j are all input components, the power transmitted by both can be categorized as input power, if P1 =Pj [ 0, then, i1j \0, indicating that the driving speed of the driving bevel gear is opposite to that of the face gear. Assuming that the face gear rotates in the positive direction, according to Eq. 5, both the swing arm and the driving bevel gear are in the negative direction. Therefore, it should be ensured that the motion direction of the driving bevel gear is opposite to that of the face gear, so as to ensure the output power converging of the planetary gear. When the driving bevel gear 1 and the swing arm H are all input components, the power transmitted by both are can be categorized as input power, if P1 =PH [ 0, then, i1H [ 0, indicating that the driving speed of the driving bevel gear is the same as that of the swing arm. Assuming that the driving bevel gear rotates in the positive direction, according to Eq. 3, the swing arm moves in the positive direction, but the motion of the face gear is in the negative direction. Therefore, it should be ensured that the motion direction of the driving bevel gear is the same as that of the swing arm, so as to ensure the output power converging of the planetary gear. When the face gear j and the swing arm H are all input components and the power transmitted by both can be categorized as input power, if Pj =PH [ 0, then, ijH \ 0, indicating that the driving speed of the face gear is opposite to that of the swing arm. Assuming that the swing arm moves in the positive direction, according to Eq. 3, the driving bevel gear rotates in the positive direction, but the face gear rotates in the

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negative direction. Therefore, it should be ensured that the motion direction of the face gear is opposite to that of the swing arm, so as to ensure the output power converging of the planetary gear. The relationship between the input powers of the system and each branch when ignoring the meshing power loss is given by: P1 ¼ P21 þ P22 þ P23 þ P24

ð14Þ

The relationship between the powers of the each branch and output power of system when ignoring the meshing power loss: P31 þ P32 þ P33 þ P34 ¼ PH

ð15Þ

On the planet carrier, the driven bevel gear and cylindrical gear are ﬁxed by a spline shaft, which can be considered: P21 ¼ P31 ; P22 ¼ P32 ; P23 ¼ P33 ; P24 ¼ P34

ð16Þ

Deﬁne the meshing efﬁciencies of the four bevel gear pairs as a1 , a2 , a3 and a4 , the meshing efﬁciencies of four pairs with cylindrical gears and different stage face gears as b1j , b2j , b3j and b4j (j corresponds to the ﬁrst, second, and third stage), when considering the meshing power loss, the relationship between the input power of the system and the power of each branch is given by: P1 ¼ a1 P21 þ a2 P22 þ a3 P23 þ a4 P24

ð17Þ

When considering the meshing power loss, the relationship between the power of each branch and the output power of the system is given by: b1j P31 þ b2j P32 þ b3j P33 þ b4j P34 ¼ PH

ð18Þ

4 Calculation of Static Load Sharing Characteristics Since there are inevitably different errors in design, manufacture and installation of each individual gear, the uniformity of load distribution among the planet gears directly affects the stability, reliability, load capacity and service life of the transmission system. Therefore, it is great signiﬁcance to study on the load sharing characteristics of the planetary MFGTD. The load sharing characteristics of each train of the planetary system can be characterized by the corresponding load sharing coefﬁcient. The socalled load sharing coefﬁcient can be deﬁned as the ratio between the maximum load experienced by a differential gear/planetary gear and the average load experienced by

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all the gears in each component when the load distribution of the differential gear/planetary gear component is not uniform. The load sharing coefﬁcients can be obtained by solving the static equilibrium equation: 8 < Xbi ¼ Fbi ðNrbbi Þ 1 þ 1 T1 ð19Þ : Xcfi ¼ Fcfi ðNrbcfi Þ 1 þ 1 T3 In this equation, Xbi is the load sharing coefﬁcient of the i driven bevel gear. Fbi is the meshing force of the i driven bevel gear. N is the number of planetary gears or differential gears. rbbi is the base radius of the i driven bevel gear. Xcfi is the load sharing coefﬁcient of the i cylindrical gear. Fcfi is meshing force of the i cylindrical gear. rbcfi is the number of the i cylindrical gear. The static load sharing coefﬁcient is the maximum value of the load sharing coefﬁcient among all the tooth frequencies in each gear component, which is given by:

Xb ¼ maxðXb1 ; Xb2 ; ; XbN Þ Xcf ¼ maxðXcf 1 ; Xcf 2 ; ; XcfN Þ

ð20Þ

According to the relevant dynamic model, several main structural parameters of the planetary system are extracted to study the influence on the static load characteristics, such as the number of differential gears, number of planetary gears, teeth number, interstage load sharing characteristics and input torque. 4.1

Influence of the Number of Differential/Planetary Gears on Static Load Sharing Characteristics

Based on the structural principle and design parameters of the optimized planetary system, it is assumed that the input torque of the driving bevel gear is 100 Nm, the modulus of the driving and driven bevel gears are 3, the teeth number of the driving bevel gear is 36, and the teeth number of the driven bevel gear is 19. The modulus of the cylindrical-face gears pairs are 2, the teeth number of the cylindrical gear is 17, the second stage of the face gear is selected for the study, and the teeth number is 68. The corresponding discrete value curves of the static load sharing characteristics are shown in Figs. 4 and 5. From Figs. 4 and 5, no matter in the differential gear train or the planetary gear train, as the numbers of related gears increase, the static load sharing characteristics are ﬁrst decreased and then increased, when the number of bevel gears is selected as 3 or that of cylindrical gears as 2, the load sharing characteristics are the smallest. However, considering the loading capacity, transmission stability, and variable speed smoothness, the numbers of the bevel gears and cylindrical gears are still arranged in four.

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Fig. 4. Influence of static load sharing characteristics about the driven bevel gears numbers

Fig. 5. Influence of static load sharing characteristics about the cylindrical gears numbers

4.2

Influence of the Teeth Numbers on the Static Load Characteristics

It is assumed that the input torque of the driving bevel gear is still 100 Nm, the modulus of bevel gears pair are 3, the teeth numbers of the driving bevel gears are all 36, the teeth numbers of driven bevel gears are 17–25. The modulus of the cylindricalface gears pairs are 2, the teeth numbers of the cylindrical gears are 17–25, the face gears would be selected 3 stages for study and the teeth numbers are 56, 68, and 80 respectively. The corresponding discrete values curves of the static load sharing characteristics are shown in Figs. 6 and 7. From Figs. 6 and 7, in the differential gear train, as the teeth numbers of the bevel gears increase, the load sharing characteristic also increases. As the teeth numbers of cylindrical gears increase, the load sharing characteristic decreases. When switching through stages, the larger the teeth number of the face gear, the larger the load sharing characteristic.

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Fig. 6. Influence of static load sharing characteristics about the driven gears tooth numbers

Fig. 7. Influence of static load sharing characteristics about the cylindrical gears tooth numbers

4.3

Influence of Input Torque on Static Load Sharing Characteristics

The other parameters remain unchanged, and the input torque is varied in the range 100 Nm–3000 Nm. The corresponding discrete values curves of the static load sharing characteristics are shown in Figs. 8 and 9. From Figs. 8 and 9, regardless of the differential or planetary gear train, the load sharing coefﬁcient does not change as the input torque changes. Otherwise, since static analysis does not consider the influence of acceleration or damping, the main static load excitation parameters of support stiffness and friction damping have only insigniﬁcant influence on the static load sharing characteristics, therefore, this study does not make further analysis.

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Fig. 8. Static load sharing characteristics of driven bevel gears with different input torque

Fig. 9. Static load sharing characteristics of cylindrical gears with different input torque

5 Analysis of Dynamic Load Sharing Characteristics In the process of dynamic meshing transmission, the planetary system is affected not only by design, manufacturing, and installation errors, but also by contact deformation, meshing abrasion, and inter-stage coupling. In the analysis of static load sharing characteristics, the influence of acceleration and deceleration, friction damping and time-varying meshing stiffness in transient meshing that arise from these errors are not considered, which cannot completely reflect the actual load sharing state of the transmission system. Consequently, this section tries to solve the dynamic meshing force linearly, and establish the corresponding dynamic load sharing coefﬁcient equation in order to analysis the influences of system parameters or excitation factors such as the error, stiffness, friction, and working conditions on the dynamic load sharing characteristics.

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Calculation of Dynamic Load Coefﬁcient Analysis of Transmission System

The load sharing coefﬁcient of the single tooth frequency of each gear can be deﬁned as the ratio between the maximum load of this gear and the mean load of all the planet gears in the planetary component: 8 N P > D > < Xbi ¼ N ðFbi ðtÞÞmax = ðFbi ðtÞÞmax i¼1

N P > > : XD ðFcfi ðtÞÞmax cfi ¼ N ðFcfi ðtÞÞmax =

ð21Þ

i¼1

The dynamic load sharing coefﬁcient is the maximum value of the load sharing coefﬁcient in each tooth frequency cycle given by: (

D XD b ¼ Xbi 1 max þ 1 D D Xcf ¼ Xcfi 1 þ 1

ð22Þ

max

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Influence of Arrangement Number or Installation Angle of Differential Gears/Planetary Gears

According to the structural device scheme of the universal planetary gear, in order to guarantee transmission stability, high load capacity and variable speed smooth, when the structure is arranged with three or four differential/planetary gears respectively, it is necessary to arrange them symmetrically around the periphery at intervals of 90° or 120°, the corresponding dynamic load sharing characteristics are shown in Figs. 10, 11, 12 and 13.

Fig. 10. Dynamics load sharing characteristics of three driven bevel gears with different layout angles

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Fig. 11. Dynamics load sharing characteristics of three cylindrical gears with different layout angles

Fig. 12. Dynamics load sharing characteristics of four driven bevel gears with different layout angles

Fig. 13. Dynamics load sharing characteristics of four cylindrical gears with different layout angles

By comparing Figs. 10, 11, 12 and 13, it can be seen that in the differential gear train or the planetary gear train, as the number of gears increase and the arrangement angle decreases, the load sharing coefﬁcient increases, but the fluctuation periods remain unchanged. Similarly, the behavior of the gear train corresponding to different arranged numbers of the cylindrical gears also has a similar trend. In addition, comparing the fluctuation characteristics of the dynamic load sharing coefﬁcient of each bevel gear in the differential gear train, the load sharing coefﬁcient of each cylindrical gear in the planetary gear train is more even and stable as the face gears meshing with the cylindrical gears are used for braking and have static motion characteristics.

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Influence of Integrated Error

The design error present in the system mainly arises from the difference between the meshing curve of the double crown face gear and the ideal conjugate curve. The bevel gear is a standard gear with straight tooth structure, the design error is assumed in the ideal state without deviation. Manufacturing errors include crown machining deviation, radix-node error, tooth pitch error, radial run-out error of gear ring, and tooth flank spacing. Installation errors mainly include offset error, and shaft angle error, etc. In order to simplify the analysis, three kinds of errors can be combined to introduce a comprehensive error sensitivity function represented by:

Db ¼ db ð1 þ 2 sin ab Þ=mb Dcf ¼ dcf ð1 þ 2 sin acf Þ=m

ð23Þ

In this equation, db and dcf are the linear distance between the theoretical contact point and the actual contact point of the bevel gear pair or the cylindrical-face gear pair (which also can be further simpliﬁed to the maximum deformation) respectively. ab and acf are the angles between the axis and the dedendum line of the bevel gear pair or the cylindrical-face gear pair respectively. mb and m are the modulus at the midpoints of tooth width of the bevel gear pair and the cylindrical-face gear pair respectively. Assuming that the input torque is 300 Nm and rotational speed is 300 r/min, based on the simulation results of the dynamic model by the ﬁnite element method, which are computed that the average meshing stiffness of the bevel gear pair is 2.7747e+8 N/m, the fluctuation amplitude of the meshing stiffness is 1.414e+9 N/m; and the average meshing stiffness of the cylindrical-face gear pair is 5.3825e+8 N/m, the fluctuation amplitude of the meshing stiffness is 0.6145e+9 N/m. Considering a bevel gear and a cylindrical gear connected by a planetary spline shaft, it is assumed that the meshing deviation values in the integrated error are 0.01 mm, 0.06 mm and 0.11 mm respectively, and the offset angles are respectively 1°, 6° and 11°, so the dynamic load sharing characteristics are shown in Figs. 14 and 15.

Fig. 14. Dynamics load sharing characteristics of driven bevel gears with different integrated errors

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Fig. 15. Dynamics load sharing characteristics of cylindrical gears with different integrated errors

The integrated error fluctuates with the rotation of the gear shaft, which has signiﬁcant periodic characteristics. For the same differential or planetary gear, when considering different errors values, the difference in dynamic load characteristics are not obvious, but as the integrated error values increase, the load sharing characteristics also increase. 5.4

Influence of Meshing Stiffness

Considering a bevel gear and a cylindrical gear connected by a planetary spline shaft, it is computed that the average meshing stiffness of the driven bevel gears are 0.9497e+8, 2.7747e+8, 4.1471e+8 and 5.9883e+8 N/m respectively, the fluctuation amplitudes of transient stiffness are 0.4223e+9, 1.4140e+9, 2.4803e+9 and 5.3165e+9 N/m respectively. Besides, it is also computed that the average meshing stiffness of the cylindrical gears are 3.1887e+8, 5.3825e+8, 5.6359e+8 and 5.2064e+8 N/m respectively, the fluctuation amplitudes of transient stiffness are 0.1835e+9, 0.6145e+9, 1.0778e+9 and 2.3104e+9 N/m respectively, so the dynamic load sharing characteristics are shown in Figs. 16 and 17.

Fig. 16. Dynamics load sharing characteristics of driven bevel gears with different meshing stiffness

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Fig. 17. Dynamics load sharing characteristics of cylindrical gears with different meshing stiffness

For the same stage of a differential gear and planetary gear, the dynamic load sharing coefﬁcient increases with the increase in meshing stiffness, which indicates that the load sharing capacity optimizing the system load transmission through the meshing axis of the differential gear/planetary gear is reduced. Since there is effect of decelerating and torque increasing on the system, the loading capacity of driven gear in the differential gear train is smaller than that of cylindrical gear in the planetary gear train. Therefore, in comparison with the differential gear train, the load distribution in the planetary gear train is smoother and the dynamic load sharing coefﬁcient is much smaller. 5.5

Influence of Input Speed

Other parameters remaining the same, it is assumed that the input speeds are 100, 300, 500 and 1000 r/min respectively. Considering a bevel gear and cylindrical gear connected by a planetary spline shaft, the dynamic load sharing characteristics are shown in Figs. 18 and 19.

Fig. 18. Dynamics load sharing characteristics of driven bevel gears with different input speed

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Fig. 19. Dynamics load sharing characteristics of cylindrical gears with different input speed

For the same stage of a differential and planetary gear, the dynamic load sharing coefﬁcient increases with the increase in the input speed, and there is a notable periodicity to the characteristic plots. When the rotation speed is low, the variation amplitude of the load sharing coefﬁcient is gentler and less affected by the errors. For the load sharing characteristics of the cylindrical gear, when the speed is 300 r/min, the change trend of the load sharing coefﬁcient is close to the standard fluctuation curve, which indicates that the transmission stability is better at this speed.

6 Conclusion This study has examined the calculation methods of static load sharing coefﬁcient and dynamic load sharing coefﬁcient of the system, analyzing the influence of key structural parameters on the load sharing coefﬁcient between stages of the transmission system, in order to quantify the influence of the key system parameters on the load sharing characteristic, which provides a theoretical basis for improving the stability and loading capacity of the transmission. (1) In order to maximize the transmission efﬁciency and optimize the transmission performance of the planetary transmission system under different loads or working conditions, the transmission mechanism and transmission characteristics of a planetary MFGTD are studied from the aspects of speed ratio selection and load sharing or splitting. (2) The mechanism, size and property of the dynamic excitation of the transmission system are studied through the dynamic meshing force of the gear teeth, to determine the dynamic load and dynamic load coefﬁcients of the gear teeth and thereby improve the loading capacity of the transmission system. (3) The effects of working condition parameters (input speed, input torque), as well as physical and geometric parameters (central components support stiffness, numbers of planet gears, tooth surface friction) on the load sharing characteristics of the system stages are studied. The vibration and noise caused by the uneven load distribution in planetary gear transmission are reduced to improve the stability and reliability of the device. This work analyzes the feasibility of a planetary MFGTD. The results are beneﬁcial to mitigate unfavorable phenomenon, such as gear wear and damage, transmission

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discontinuity and jerky motion caused by the uneven load distribution. It is also helpful to understand the influence of the dynamic excitation mechanism, size, and properties on transmission efﬁciency to provide a reference for the dynamic design and structural optimization of the transmission system. Acknowledgements. The authors would like to thank Miss Tiancui Xu and Dr. Ding Yao for free help in professional writing. The authors would also like to thank editor and reviewers for their helpful comments and suggestions to improve the manuscript. Conflicts of Interest. The authors declare that there are no conflicts of interest regarding the publication of this paper.

References 1. Shuai, M., Yidu, Z., Qiong, W.: Research on multiple-split load sharing of two-stage star gearing system in consideration of displacement compatibility. Mech. Mach. Theory 88, 1– 15 (2015) 2. Hong, J., Talbot, D., Kahraman, A.: A generalized semi-analytical load distribution model for clearance-ﬁt, major-ﬁt, minor-ﬁt, and mismatched splines. Proc. Inst. Mech. Eng. C-J. Mech. 230(7–8), 1126–1138 (2016) 3. Dong, H., Liu, Z.Y., Duan, L.L., Hu, Y.H.: Research on the sliding friction associated spurface gear meshing efﬁciency based on the loaded tooth contact analysis. PloS ONE 13(6), e0198677 (2018) 4. Hao, D., Fang, Z.D., Hu, Y.H.: Study on the load-sharing characteristics of an aeronautical II-stage ﬁve-branching planets gear train based on the loaded tooth contact analysis. Math. Probl. Eng., 1–18 (2018). https://doi.org/10.1155/2018/5368294 5. Iglesias, M., del Rincon, A.F., de-Juan, A., Garcia, P., Diez-Ibarbia, A., Viadero, F.: Planetary transmission load sharing: manufacturing errors and system conﬁguration study. Mech. Mach. Theory 111, 21–38 (2017) 6. Iglesias, M., Del Rincon, A.F., De-Juan, A.M., Garcia, P., Diez, A., Viadero, F.: Planetary gear proﬁle modiﬁcation design based on load sharing modelling. Chin. J. Mech. Eng. 28(4), 810–820 (2015) 7. Yi, P.X., Zhang, C., Guo, L.J., et al.: Dynamic modeling and analysis of load sharing characteristics of wind turbine gearbox. Adv. Mech. Eng. 7(3), 1–16 (2015). https://doi.org/ 10.1177/1687814015575960 8. Jin, G.H., Xiong, Y.P., Gui, Y.F., Zhu, R.P.: Sensitive parameter and its influence law on load sharing performance of double input split torque transmission system. J. Vibr. Eng. Technol. 5(6), 583–595 (2017) 9. Sheng, D.P., Zhu, R.P., Jin, G.H., Lu, F.X., Bao, H.Y.: Dynamic load sharing behavior of transverse-torsional coupled planetary gear train with multiple clearances. J. Cent. South Univ. 22(7), 2521–2532 (2015) 10. Sheng, D.P., Zhu, R.P., Jin, G.H., Lu, F.X., Bao, H.Y.: Dynamic load sharing characteristics and sun gear radial orbits of double-row planetary gear train. J. Cent. South Univ. 22(10), 3806–3816 (2015) 11. Li, M., Xie, L.Y., Ding, L.J.: Load sharing analysis and reliability prediction for planetary gear train of helicopter. Mech. Mach. Theory 115, 97–113 (2017) 12. Du, J.F., Mao, J., Cui, Y.H., Liu, K., Zhao, G.R.: Theoretical and experimental study on load sharing of a novel power split spiral bevel gear transmission. Adv. Mech. Eng. 10(6) (2018)

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13. Jing, L.B., Zhang, T., Lin, Y., Cheng, J.: Design, analysis, and realization of a magnetic force transmission PM brushless motor. IEEJ Trans. Electr. Electron. Eng. 13(5), 791–798 (2018) 14. Huang, C.H., Tsai, S.J.: A study on loaded tooth contact analysis of a cycloid planetary gear reducer considering friction and bearing roller stiffness. J. Adv. Mech. Des. Syst. 11(6) (2017) 15. Singh, A.: Epicyclic load sharing map - development and validation. Mech. Mach. Theory 46(5), 632–646 (2011) 16. Shu, R.Z., Liu, Z.J., Liu, C.Z., Lin, X.Y., Qin, D.T.: Load sharing characteristic analysis of short driving system in the long-wall shearer. J. Vibroeng. 17(7), 3572–3585 (2015) 17. Gu, X., Velex, P.: A dynamic model to study the influence of planet position errors in planetary gears. J. Sound Vibr. 331(20), 4554–4574 (2012) 18. Hammami, A., Del Rincon, A.F., Chaari, F., Santamaria, M.I., Rueda, F.V., Haddar, M.: Effects of variable loading conditions on the dynamic behaviour of planetary gear with power recirculation. Measurement 94, 306–315 (2016) 19. Hammami, A., Del Rincon, A.F., Chaari, F., Rueda, F.V., Haddar, M.: Dynamic behaviour of back to back planetary gear in run up and run down transient regimes. J. Mech. 31(4), 481–491 (2015)

Reachable Matrix and Directed Graph – Based Identiﬁcation Algorithm of Module Change Propagation Path for Product Family Xianfu Cheng(&), Liyun Wan, and Jian Zhou School of Mechatronics and Vehicle Engineering, East China Jiaotong University, Nanchang 33001, China [email protected]

Abstract. Product family design is the core of mass customization and has been identiﬁed as an effective tool to quickly respond to the customers’ demands, and achieve the proliferation of product variants by diverse market niches. There are often coupling dependence relationships between modules, which will increase the difﬁculty of product design. For this reason, an identiﬁcation algorithm of change propagation path for modular product family. Instigating component and affected component between association modules are determined, and the latter is regarded as feeding component inside the module. Applying reachable matrix, the target components influenced by affected component can be conﬁrmed. Then the directed graph combined, the search process of change propagation path is presented. Keywords: Product family Identiﬁcation algorithm

Module change Change propagation path

1 Introduction Product family design is a complex task. There is not only a contradiction between commonality and diversity, but also often exist association relationship between modules, which bring about the coupling of product family design and increases the difﬁculty of product design. In view of engineering change, design change in modular product considers mainly the increase, deletion or replacement of module, as well as the change of characteristic parameter or assembly constraints of the component in modules, which is called module change. The increase, deletion or replacement of module belong to module overall change, which causes the change of components in other modules through the change module level parameters, thus a new modular product is acquired. Therefore, the problem on module change can be transformed into that of change propagation between components. Change propagation paths are analyzed and change influence degree is measured, which can provide methodological guidance for reducing iteration of modular product family design. Clarkson et al. [1] used likelihood and impact of change to predict the risk of change propagation, traced potential propagation paths among components base on DSM, and outlined the methods to compute the risk of direct and indirect change © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 84–92, 2020. https://doi.org/10.1007/978-981-32-9941-2_7

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propagation. Tang et al. [2] proposed the concrete method for analyzing direct and indirect engineering change impacts based on design structure matrix, and presented mathematical model to predict the comprehensive impact of engineering change. Yang and Tang [3] analyzed the propagation of change from characteristic linkage perspective and categorized the characteristic relations were into two types. They discussed propagation features and proposed the process model of change propagation. Qiao et al. [4] applied graph theory to develop an adjacency matrix to describe the relevance between different parameters, organized changing paths and related information into structured storage ﬁles to improve the extendibility of the changing paths and their related information, and then integrated the structured information and fully constrained module to develop the self-adaption module. Koh et al. [5] presented a method matrix-based to assess the effects of engineering change propagation, which set up the HoQ and the CPM to model the effects of potential change propagation caused by different change options. Yang and Duan [6] constructed a product model from the parameter linkage perspective, and discussed propagation mechanism of change prediction. Then they proposed a searching method of change propagation paths. Ullah et al. [7] proposed a change propagation mathematical model in a product family design, and progressive change propagation algorithm, which was viewed as a modiﬁed breadth ﬁrst search method, to ﬁnd the shortest possible route between the reachable nodes. Cheng et al. [8] discussed the coupling in product family design, identiﬁed coupling incidence path between modules, and calculated correlation impact degree. For simple modular product, if the size of modular product is small, change propagation path between modules can be intuitively analyzed. However, with the increase of the size of module and the coupling between modules, identifying the indirectly related information will become more and more difﬁcult. For this reason, it is necessary to develop an identiﬁcation algorithm of change propagation path, in order to compute effectively the change influence degree between modules.

2 Determining Target Components of Change Propagation Before analyzing the change propagation path, ﬁrstly, the instigating component and affected component of the two interaction modules should be determined. Assume there exists an interaction between module Mp and Mq, and Ci in Mp has impact on Cj in Mq, then Ci is called instigating component of Mp affecting Mq (denoted as Mp! Mq), and Cj is called affected component. For the module in which affected component is located, if there is a relationship between other components and affected component, then the affected component will have a change propagation effect on other components. In order to search for all coupling correlation paths within the module, the affected component should be used as the starting point, and identify all components affected by change propagation. According to the characteristics of this task, searching can be done by means of reachable matrix, because the reachable matrix is a matrix to describe the reachability between nodes in a directed graph, and other components within module are determined that are reachable by above-mentioned affected component through calculating reachable matrix corresponding to a module of the product

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family, which are the target components affected by change propagation. Since this method is universal, therefore, a module with less design components and simpler coupling correlation is illustrated as an example, and its form is shown in Fig. 1. C

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In order to calculate the reachable matrix corresponding to the module, the corresponding adjacency matrix should be derived according to its change propagation directed graph. Suppose C4 is the affected component of the module for other modules, when analyzing the impact of other modules on the propagation of changes to this module, C4 can be regarded as feeding component inside the module. As can be seen intuitively from Fig. 1, change in C4 can directly affect C1, C2 and C3, and does not directly affect C5, C6 and C7. Whereas change of C3 can directly affect C7, therefore, the change of C4 can affect C7 indirectly. According to this method, the directed graph of change propagation path between components can be drawn, as shown in Fig. 2.

Fig. 2. The directed graph of change propagation path

From the knowledge of graph theory, directed graph D can be represented by a binary array , i.e., D = . In this formula, V = {v1, v2, ……, vn}, it is a set of non-empty numbers, called the vertex set of directed graph D, the elements in V are all vertices of directed graph D (or known as node), and n is the number of vertices.

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E = {}, it is a multiple subset of Cartesian product, the elements in E are called directed edges of directed graph D (can be abbreviated as edge). denotes directed edges with vertex vi and vertex vj as starting and ending points respectively. The adjacency matrix A corresponding to directed graph D can be expressed as A = {aij}. It is an n-order Boolean matrix. When there is a directed edge between vertex vi and vertex vj, then aij = 1; When there is no directed edge between vertex vi and vertex vj or i = j, then aij = 0. Therefore, the adjacency matrix A corresponding to the directed graph of the change propagation path shown in Fig. 2 can be obtained, and its form is shown in Fig. 3. No.

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Comparing the form of Figs. 1 and 3, it can be seen that the adjacency matrix is actually derived from the variant of the original design incidence matrix, that is, the diagonal values in original matrix are all changed to 0, the non-zero values in the nondiagonal elements are all changed to 1, and the zero values in the non-diagonal element remains unchanged. Therefore, when there are many components in the module and the coupling relationship between components is complex, the adjacency matrix can be obtained without drawing the directed graph of the change propagation path. The reachable matrix P corresponding to the module shown in Fig. 1 can be represented as P = {pij}. It is also an n-order Boolean matrix. When vertex vi is reachable to vertex vj, pij = 1. When vertex vi is not reachable to vertex vj, pij = 0. Because the reachable matrix P is a Boolean matrix, it can not be obtained directly by calculation. The precursor matrix M of adjacent matrix A should be calculated ﬁrstly, and then the reachable matrix P is obtained by the variant of the precursor matrix M. The precursor matrix M can be expressed by the following formula: M ¼ I þ A þ A2 þ . . .. . . þ An ¼ ðI þ AÞn

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The elements in the precursor matrix M represent the degree of accessibility between components in the module. Non-zero elements represent reachability, while 0 elements represent inaccessibility. According to the value rule of the row and column elements pij in the reachable matrix, all non-zero elements in M are modiﬁed to 1, while 0 elements remain unchanged. Thus the obtained matrix is the reachable matrix P, and its form is shown in Fig. 4.

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As can be seen from the reachable matrix shown in Fig. 4, changes in C4 propagated to C1, C2, C3 and C7, so these components are the target components affected by change propagation. However, the reachable matrix can only represent the reachability between components, and can not be used to judge the nodes and the length of the corresponding change propagation path. That is to say, it is impossible to identify the direct or indirect coupling relationship between components.

3 Identiﬁcation Algorithm of Change Propagation Path In the reachable matrix, if there is a zero element in the column in which the affected component is located, it means that the component corresponding to the row with 0 element will not appear in the change propagation path. These components can be removed from the single module to which they belong, that is to say, it is not considered when searching for changing propagation path, so as to simplify the reachable

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matrix. Here, the matrix obtained by this way is called reduced dimension reachable matrix. For instance, in the reachable matrix shown in Fig. 4, there are two zero elements in the column where C4 is located, their rows correspond to C5 and C6 respectively, that is to say, the change of C4 will not affect C5 and C6, at this time, the reachable matrix can be reduced in dimension. When analyzing all possible paths of change propagation in C4, C5 and C6 can be ignored. The corresponding form of reduced dimension reachable matrix is shown in Fig. 5. The larger the size of the module and the higher the position of the components in the module, the wider the scope of the impact of change propagation and the more complex the propagation path. There are two ways of change propagation for components, one is that one component propagates directly to another component, the other is that one component propagates to another component through at least one intermediate component, that is, indirect change propagation. Different correlations among components will make change propagation diffuse in different directions and form multiple change propagation routes. Different propagation routes are connected according to the relationship between components in the module and the propagation behavior characteristics, which will form a change propagation tree, as shown in Fig. 6.

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Assume a module is composed of m components, and has n components through dimension reduction, where Cj is a feeding component. The DSM of change impact is denoted with B. How to determine change propagation paths of Cj? This section addresses a search algorithm of change propagation path based on design incidence matrix and directed graph. In the presented algorithm, a vertex denotes a product component. The feeding component is deemed as start vertex V0 of directed graph, the adjacent vertex directly related to V0 are visited. They will be taken as new vertices, and the vertices directly related to them are continuously visited. Repeat the procedure until access traverses all vertices. Suppose the feeding component Cj is regarded as start vertex V0 of directed graph, the target component of change propagation Ck is regarded as target vertex Vtarget, the number of the level of change propagation tree is n-1, initial propagation level t = 0, initial number of propagation paths S = 0, change impact degree of each propagation path P = 0, as well as change impact degree of each level of change propagation tree L = 0. The procedure of searching algorithm is as follows: Step 1: input matrix B, set queue Q = 0. Step 2: search adjacent vertex directly related to V0, record its direct change impact degree ri;j ; L ri;j ; If there exists Vtarget (i.e. i = k) in all adjacent vertices, S S þ 1, PðSÞ rk;j , t t þ 1, then these adjacent vertices will be placed in the queue Q(t). Step 3: Take out a vertex that is not Vtarget from queue Q(t) as the current vertex in turn, search its adjacent vertex that is not visited in propagation path, and record its direct change impact degree rp;I , L L rp;I ; If adjacent vertex is Vtarget, S S þ 1, PðSÞ L Step 4: t t þ 1, above adjacent vertices will be continuously placed in the queue Q(t). If Q(t) is empty, the search ends. Step 5: If t n 1, the algorithm terminates. Otherwise, go to step 3. The search process of change propagation path from one feeding component to one target component is shown as Fig. 7 and the procedure of the searching method is illustrated in Fig. 8. rk,i ri,j Cj

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4 Conclusion Product family design is a complex task. There is not only a contradiction between commonality and diversity, but also often exist association relationship between modules, which bring about the coupling of product family design and increases the

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difﬁculty of product design. For simple modular product, if the size of modular product is small, change propagation path between modules can be analyzed intuitively. However, with the increase of the size of module and the coupling between modules, identifying the indirectly related information will become more and more difﬁcult. This paper proposed reachable matrix and directed graph – based identiﬁcation algorithm of module change propagation path for product family. We utilized dimension reduction reachable matrix to determine the target components of change propagation. The search process of change propagation path was built on change propagation tree. It can provide methodological guidance for reducing iteration of modular product family design. Acknowledgements. This project was supported by the National Natural Science Foundation of China (Grant No. 51765019 and No. 71462007).

References 1. Clarkson, P.J., Simons, C., Eckert, C.: Predicting change propagation in complex design. J. Mech. Des. 126(5), 788–797 (2004) 2. Tang, D.B., Xu, R.H., Tang, J.C., et al.: Analysis of engineering change impacts based on design structure matrix. J. Mech. Eng. 46(1), 154–161 (2010). (in Chinese) 3. Yang, F., Tang, X.Q.: Propagation of engineering change based on characteristic linkage perspective. J. Beijing Univ. Aeronaut. Astronaut. 38(8), 1032–1039 (2012). (in Chinese) 4. Qiao, H., Mo, R., Xiang, Y.: Adaptive module modeling by predicting change path. J. Comput.-Aided Des. Comput. Graph. 12, 2358–2366 (2015). (in Chinese) 5. Koh, E.C.Y., Caldwell, N.H.M., Clarkson, P.J.: A method to assess the effects of engineering change propagation. Res. Eng. Des. 23(4), 329–351 (2012) 6. Yang, F., Duan, G.J.: Developing a parameter linkage-based method for searching change propagation paths. Res. Eng. Des. 23(4), 353–372 (2012) 7. Ullah, I., Tang, D., Wang, Q., et al.: Exploring effective change propagation in a product family design. J. Mech. Des. 139(12), 1–13 (2017) 8. Cheng, X., Xiao, R., Wang, H.: A method for coupling analysis of association modules in product family design. J. Eng. Des. 29(6), 327–352 (2018)

6-DOF Industrial Manipulator Motion Planning Based on RRT-Connect Algorithm Chengren Yuan1, Guifeng Liu2, and Wenqun Zhang1(&) 1

Power Engineering Department, Naval University of Engineering, Wuhan 430033, China [email protected], [email protected] 2 Ofﬁce of Educational Administration, Naval University of Engineering, Wuhan 430033, China [email protected]

Abstract. The motion planning and obstacle avoidance of industrial manipulator are researched in the V-REP platform. Virtual motion control and motion simulation of PUMA-560 manipulator are realized in the simulation environment. The manipulator STL ﬁle drawn by Solidworks is imported into the VREP platform. The embedded thread script of the LUA language uses OMPL as a plug-in to provide motion planning functions, enabling more flexible implementation of robot motion planning. The RRT-Connect algorithm called by the motion planning library in the embedded script form API is implemented in the simulation environment. The algorithm gets fast convergence in motion planning and maintains good accuracy and rapid response. But the simulation trajectory is random and blind. For the defects of simulation, this paper gives the direction of future work to realize trajectory optimization. Keywords: Industrial manipulators RRT-Connect algorithm Fast convergence Motion planning

1 Introduction Motion planning is an important topic in robotics, especially in the study of the manipulator. The purpose in motion planning is that the manipulator can search a collision-free path to operate in a three-dimensional space under certain constraints [1] from the start-state to goal-state. For the manipulator motion planning problem, it can be divided into the method of graph search, the method of artiﬁcial potential ﬁeld, the method of random sampling and the method of intelligent optimization. The motion planning of the random sampling method has obvious advantages over other methods. In Cartesian space where the manipulator is located, the collision-free path is obtained by randomly sampling the unknown space and then forming the connection graph from the sampling points. However, Probabilistic Roadmap Methods (PRM) and Rapidly-exploring Random Tree (RRT), which ﬁrst proposed by LaValle [2], are two most successful planning methods based on random sampling method.

© Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 93–101, 2020. https://doi.org/10.1007/978-981-32-9941-2_8

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In order to improve the search speed, many improved RRT algorithms have proposed. The RRT algorithm was improved the adaptiveness of random tree expansion by Melchior et al. introduced Particle Filter [3]. AlDahak et al. proposed the KD tree concept and improved the search efﬁciency [4]. Yershova et al. added extended feedback information to suppress extended point range [5]. Burns et al. proposed a dynamic RRT algorithm based on predictive models, which reduced planning time [6]. Kuffner et al. improve search efﬁciency and response speed by increase the connection tree [7]. The performance of the improved algorithm in path planning has been signiﬁcantly improved. Especially, the performance of the RRT-Connect algorithm is outstanding. The RRT-Connect algorithm can effectively avoid falling into local optimum. It can also improve the efﬁciency of trajectory planning. In this paper, the PUMA-560 robot arm is selected as the simulation object. The simulation environment is built in V-REP, and the RRT-Connect algorithm is used to realize motion planning. In particular, V-REP is a flexible robot simulator. And not widely use in domestic, only the ofﬁcial manual can be referred. At the same time, the reliability of the simulation process was veriﬁed.

2 Manipulator and Simulation Platform 2.1

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PUMA-560 industrial manipulator is used as a case, which structures is shown in the Fig. 1. PUMA-560 is an articulated manipulator that has 6 rotating joints. The ﬁrst three joints determine the position of the wrist reference point, and the last three joints determine the orientation of the wrist. The axis of the joint 1 is in the vertical direction, and the axes of the joint 2 and the joint 3 are horizontal and parallel by a2 . The axes of joint 1 and joint 2 intersect perpendicularly, and the axes of joint 3 and joint 4 are vertically staggered with a distance of a3 . The distance between the joints 1 and 2 in the x-axis direction is d2 , and the distance between the joints 3 and 4 in the x-axis direction is d4 [8].

Fig. 1. Schematic diagram of the PUMA-560 industrial manipulator.

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Virtual Robot Experiment Platform (V-REP) is an experimental platform for robot motion simulation. It supports six programming methods such as embedded scripts, and six programming languages such as C/C++, Python and Lua. In addition, simulations support path planning, shortest distance calculation, and collision detection etc. V-REP uses the Open Motion Planning Library (OMPL) as a plug-in to replace the previous method for motion planning by calling the API. This approach can solve motion planning problems more flexibly and conveniently. When developed offline programming software that integrates programming and simulation, V-REP can be selected to embed into as a separate simulation environment [9]. To realize the simulation of the motion planning, setting up the simulation environment in V-REP requires the following steps: (1) Set up the simulation environment; (2) Import the standard CAD format of the manipulator into the simulation environment; (3) Add kinematic and dynamic properties to the manipulator; (4) Add dummy, including the start and goal position of the motion planning point (dummy_start and dummy_goal) and the handle position of the end of manipulator (dummy_handel); (5) Write control script code and communication interface to model. There are usually some constraints (obstacles) in the motion planning process of the manipulator. After setting up the simulation environment, the STL format of the manipulator is imported into the created environment, as shown in Fig. 2. It should be noted that the introduced manipulator has no kinematics and dynamics. So, in the next process, the properties of kinematics and dynamics need to be set. Finally, determine the starting point and end point of the motion planning, and add dummy to the end of the arm, as shown in Fig. 3. In this way, the manipulator can achieve the accuracy position of motion planning.

Fig. 2. V-REP simulation environment layout.

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Lua and Simulation Experiment

V-REP supports 6 programming languages. This article uses embedded thread scripts in Lua. The advantage is that it can be executed cyclically and can continue to accept external commands during the run. V-REP uses OMPL as a plug-in to provide motion planning functions, enabling more flexible implementation of robot motion planning. In this paper, the RRT-Connect algorithm is called by the motion planning library in the embedded script form API, which is implemented in the simulation environment. Writing programs mainly includes: create tasks, select appropriate algorithms, create state space and create a state space for each joint, set state space, set collision pairs, set start and target states, calculate appropriate paths, and so on. In the programming process, the ofﬁcial manual can be referred [10]. In addition, this paper involves seven rotating joints, six of which are used to drive the simulated motion of the manipulator. The last one is set at the end of the manipulator. The end handle is used to accurate the position between the start-point and the goal-point, as shown in Fig. 3.

Fig. 3. Simulation result.

3 RRT-Connect Algorithm Traditional path planning algorithms include artiﬁcial potential ﬁeld method [11], fuzzy rule method [12], genetic algorithm [13], neural network [14], simulated annealing algorithm [15], ant colony optimization algorithm [16]. However, these methods need to model the obstacles in a certain space. The computational complexity is exponential with the degree of freedom of manipulator. It is not suitable for solving the multidegree-of-freedom manipulator in complex environments [17]. The planning principle of the RRT algorithm is to perform collision detection on sampling points in the state space. Furthermore, the modeling of space is avoided, and the path planning problem of high-dimensional space with complex constraints can be

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effectively solved. RRT algorithm is suitable for motion planning of multi-degree-offreedom robots in complex environments. Compared with the RRT algorithm, the branch of RRT-Connect expansion method is improved. The basic idea is to make the distance of the single step expansion as far as possible. The RRT-Connect algorithm improves the basic one-step extension to a greedy multi-step extension that ends an extended iteration until it encounters an obstacle or reaches the target point. This method can effectively avoid the possibility of falling into a local optimum, and can quickly expand to the unexplored area in a relatively open zone. The result can greatly improve the efﬁciency of the spanning tree. There are Figs. 4 and 5 of the RRT and RRT-Connect algorithms.

Fig. 4. The way of RRT path planner.

Fig. 5. The way of RRT-Connect path planner

The RRT-Connect planner is designed speciﬁcally or path planning problems that involve no differential constraints. The method is based on two ideas: the connect heuristic that attempts to move over a longer distance, and the growth of RRTs from both qstart and q goal [16]. qnew is a random point, and qrand is a random invalid point. The RRT-Connect algorithm initializes two trees Ta and Tb respectively. Ta and Tb alternately expand toward each other in the C space, iterating over and over until the two trees meet, and successfully ﬁnd a path from the initial state to the ﬁnal state. If the number of iterations is still not met, the pathﬁnding fails. For each iteration, ﬁrst select a random point for T1 and select a point on T that is closest to the random point as the growth point. Then select one of all the inputs to make the new pose point closest to the

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random point as the best input, and use this input to multi-step expansion until the new generation point is no closer to the random point, then the latest generated point. As a random point of Ta , Tb grows in the same manner. Then judge whether the latest node distance of the two trees meets the error requirement, that is, whether they meet. When encountering, record the shortest path length; if they do not meet, exchange Ta and Tb sequentially into the next iteration. Once again, judge whether you meet. When the encounter meets, the path length is calculated and compared with the previous path to retain the shortest path. After the number of iterations, the output path is reached: if the number of encounters is still not met, the exit is failed [7].

4 Results and Discussion Since the RRT-Connect algorithm has a certain degree of blindness and randomness, the motion planning trajectory generated for each time is different. Figure 3 shows a simulation process from the start-point to the goal-point. The corresponding graphic in the simulation process were drawn in Figs. 6, 7, 8 and 9. As shown in Fig. 6, the end of manipulator was represented the movement of three curves in X-axis, Y-axis, and Z-axis. It can be seen that the manipulator is moving slowly without violent shaking.

Fig. 6. Manipulator tip track in three axes.

Angular velocity and angular acceleration of manipulator joints were described in Figs. 7 and 8. As shown as Fig. 7, the joints of the manipulator are slow in motion which cost about 20 s. In this period, there is no large speed change. In Fig. 7, the overall movement of the manipulator is relatively gentle. But at the start and near the goal-point has a large speed change, which is caused by the blindness and randomness of the RRT-Connect algorithm.

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Fig. 7. Joint angular velocity of each arm.

Fig. 8. Joint angular acceleration of each arm.

As shown in Fig. 9, the joint rotation speed of manipulator is generally stable, also the rotation amplitude is small. It is indicating that RRT-Connect algorithm gets fast convergence in motion planning and maintains good accuracy and rapid response.

Fig. 9. Joint rotation speed of each arm.

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5 Conclusions In this paper, the virtual simulation of the PUMA-560 manipulate is carried out in VREP software. RRT-Connect algorithm is used to realize the motion planning problem of the manipulate. The properties of the manipulator in motion planning under such conditions are analyzed. The existed problems and future work prospects are raised. As we can see, the motion planning is well solved. The key of motion planning, convergence speed and accuracy, can be obtained from the simulation. The RRTConnect algorithm converges quickly and manipulator ﬁnd target in the speciﬁc motion planning. But the algorithm has certain blindness and randomness, each search path is different from the ideal path. It can also be seen from the ﬁgure that the generated motion trajectory has inflection points. There are also large variations in angular velocity and acceleration ﬁgure. Reflected in the actual problem, it is easy to produce a certain impact and energy loss on the manipulator itself. For future work, Artiﬁcial Neural Networks can be integrated with the RRT-Connect algorithm to increase the manipulator motion planning to achieve the ideal state and trajectory optimization.

References 1. Liu, G., Xie, H., Li, C.: Method of MOBILE robot path planing in dynamic environment based on genetic algorithm (2003) 2. LaValle, S.M., Kuffner Jr., J.J.: Rapidly-exploring random trees: progress and prospects (2000) 3. Melchior, N.A., Simmons, R.: Particle RRT for path planning with uncertainty. In: Proceedings 2007 IEEE International Conference on Robotics and Automation, pp. 1617– 1624. IEEE (2007) 4. AlDahak, A., Elnagar, A.: A practical pursuit-evasion algorithm: detection and tracking. In: Proceedings 2007 IEEE International Conference on Robotics and Automation, pp. 343– 348. IEEE (2007) 5. Yershova, A., Jaillet, L., Siméon, T., et al.: Dynamic-domain RRTs: efﬁcient exploration by controlling the sampling domain. In: 2005 IEEE International Conference on Robotics and Automation, pp. 3856–3861. IEEE (2005) 6. Burns, B., Brock, O.: Single-query motion planning with utility-guided random trees. In: Proceedings 2007 IEEE International Conference on Robotics and Automation, pp. 3307– 3312. IEEE (2007) 7. Kuffner Jr., J.J., LaValle, S.M.: RRT-connect: an efﬁcient approach to single-query path planning, p. 2. ICRA (2000) 8. Cai, Z.: Robotics (2000) 9. Zhao, H., Qian, W., Sun, F.: Research of articulated robot motion simulation based on VREP. Electron. Sci. Technol. 30(4), 53–55 (2017) 10. http://www.coppeliarobotics.com/helpFiles/index.html 11. Chen, Y., Luo, G., Mei, Y., et al.: UAV path planning using artiﬁcial potential ﬁeld method updated by optimal control theory. Int. J. Syst. Sci. 47(6), 1407–1420 (2016) 12. Mohanty, P.K., Parhi, D.R.: A new intelligent motion planning for mobile robot navigation using multiple adaptive neuro-fuzzy inference system. Appl. Math. Inf. Sci. 8(5), 2527 (2014)

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13. Asadi, H., Mohamed, S., Rahim Zadeh, D., et al.: Optimisation of nonlinear motion cueing algorithm based on genetic algorithm. Veh. Syst. Dyn. 53(4), 526–545 (2015) 14. Singh, B., Marks, T.K., Jones, M., et al.: A multi-stream bi-directional recurrent neural network for ﬁne-grained action detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1961–1970 (2016) 15. Dowsland, K.A., Thompson, J.M.: Simulated annealing. In: Handbook of Natural Computing, pp. 1623–1655 (2012) 16. Lazarowska, A.: Ship’s trajectory planning for collision avoidance at sea based on ant colony optimisation. J. Navig. 68(2), 291–307 (2015) 17. Feng, L., Jia, J.: Improved algorithm of RRT path planning based on comparison optimization. Comput. Eng. Appl. 47(3), 210–213 (2011)

A Precise Identiﬁcation and Matching Method for Customer Needs Based on Sales Data Xingpeng Chu1, Jian Zhang1(&), Uday Shanker Dixit2, and Peihua Gu3 1

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Department of Mechatronics Engineering, Shantou University, Shantou 515063, China {17xpchu,jianzhang}@stu.edu.cn 2 Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India [email protected] College of Mechanical Engineering, Tianjin University, Tianjin 300072, China [email protected]

Abstract. In the stage of product design, it is very helpful to understand customer needs accurately to improve product performance. The current customer needs identiﬁcation scheme cannot meet the dynamic and precise characteristics of customer needs. A precise identiﬁcation architecture of customer demand based on online sales data and probability theory is proposed. Online sales data provide potential customer demand information, the data is used to build the relationship between customer satisfaction and product function. Vector similarity is used to match user needs and product functions. At the same time, a ranking method of product satisfaction recommended to users is proposed by using the concepts of probability and statistics. The characteristic of this method is that it can feedback user’s needs in real time. The customer eventually gets a product that meets the demand and the customer is most satisﬁed with the product. Finally, this paper demonstrates the implementation process of this method by taking the multi-functional desk as an example. Keywords: Customer requirements Product design Customer satisfaction K nearest neighbor

1 Introduction In modern society, product design is facing challenges like shortening production cycle and rapid iteration. This requires designers to identify customers’ demands as accurately as possible during the early stage of design. On the other hand, with the rapid development of e-commerce, a rich supply of product information and sales records can be quickly and easily obtained from the Internet (especially e-commercial websites)

This project is supported by National Natural Science Foundation of China (No: 51505269), the National Key R&D Program of China (No: 2018YFB1701701). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 102–112, 2020. https://doi.org/10.1007/978-981-32-9941-2_9

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using data mining techniques. These lay a foundation for the research of customer needs using online purchasing data. The motivation of this paper is the following problems present in customer needs identiﬁcation. On the one hand, customers and designers have different understandings of products. Customers’ descriptions of needs and Designers’ descriptions of needs are different. Using traditional tools such as Quality Function Deployment (QFD) for modeling and analysis requires a high skill from designers. On the other hand, in the process of requirement deﬁnition, problems such as inaccurate content, omission, ambiguity and inconsistency of description occur very often. On these basis, the objective of this paper is to implement a method of customer needs identiﬁcation. This method builds up a real-time system utilizing online sales data and probability theory to accurately identify customers’ requirements and match the corresponding product feature combinations. The structure of this paper is arranged as follows. The following part in this section introduces the related work on customer needs identiﬁcation. The second section introduces the basic process of this method, including building up the relationship between customer satisfaction and product characteristics by using the existing product data, analyzing the user’s needs and forecast using vector similarity search, and improving the prediction by receiving customers’ feedback. The third section is the case study of this proposed method, taking students’ multi-functional desk as an example, to verify the practicability of the scheme. The fourth section gives a conclusion and some future works. In the process of acquisition, analysis, and identiﬁcation of customer needs (CNs), relevant research has given a lot of inspiration in the study of data sources for user needs. Wang et al. [1] developed a high-end equipment customer requirement analysis method based on online reviews. Ireland [2] used online product reviews to identify a series of customer needs. Haug [3] developed a conceptual framework based on 10 industrial designers’ interviews and studies on reference projects. Many attempts have been made on how to process the collected user data. Ni et al. [4] used extended quality function deployment (QFD) and data-mining (DM) techniques to develop a supplier selection methodology. Wang and Chin [5] proposed a linear goal programming (LGP) approach to evaluate the relative weight of customer requirements (CRs) in QFD. Juang et al. [6] use fuzzy reasoning and expert systems to propose and develop a customer requirement information system (CRIS) in the machine tool industry. Guo et al. [7], based on the fuzzy set theory and Euclidean space distance, puts forward a method for customer requirement weight calculation called Euclidean space distances weighting ranking method. Chan et al. [8] used the genetic programming to generate accurate nonlinear models in QFD systems to relate the CR and the engineering characteristics. For the constraints of product characteristics on customer needs, Xie et al. [9] proposed a constraint model as an extension to constraint satisfaction problems (CSPs) with the independent and dependent variables, Sheng et al. [10] studied the customer requirement modeling and its mapping to a numerical control machine design. Shiau [11] considered the ﬁnite resource constraint to the rapid changing CRs.

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Ma et al. [12] gives out a new method for obtaining and describing the common understanding of CRs and more importantly transferring them into a detailed and accurate product design speciﬁcations (PDS) to interact with different team members effectively. At present, the related research mainly focuses on the classiﬁcation of user reviews or questionnaires QFD method and expert system, and the establishment of the relationship between customer needs and product feature functions. Little consideration is given to the impact of product feature combination on product performance. At the same time, due to the lack of user participation in the demonstration process, relevant research cannot guarantee to eliminate the impact of user demand changes on product performance. The method introduced in this paper solves the uncertainty caused by the change of user’s demand or repeated results prediction by real-time user participation. This method can accurately identify user needs. Users can get the product feature combination of their needs in real time.

2 The Proposed Method 2.1

Framework

The implementation of this method can be divided into offline and online stage. The offline stage is set up because collecting and processing products’ data takes a long time. It can save time by collecting in advance. By setting the online stage, customers can feedback the product feature combination in real-time, and determine the convergence condition of the user demand by each input after the user accepts the feedback. Offline stage: common needs of customer could be presented by the statistical results of network sales data. By collecting historical sales data of a certain product through e-commercial websites, including the collection of product information and sales information, the relationship between product characteristics and customer satisfaction can be constructed to facilitate product retrieval in the offline part. Online stage: The user’s understanding of the product is usually vague and incomplete. Therefore, the customer needs usually do not completely and accurately represent the product of the customers’ real needs. A set of programs is set up, which can compare the similarity between the customer demand information proposed by the user and the actual product characteristics in real time and obtain some results most similar to the features currently proposed by the user. At the same time, by establishing a feedback mechanism, the process of obtaining a satisfactory product for the user is as short as possible (Fig. 1).

A Precise Identiﬁcation and Matching Method for Customer Needs Based on Sales Data Offline stage Dataset preparation

Product Feature Library

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Online Stage Customer initial needs

Match with Product Feature Library

Customer browse recommendations

Relationship Modeling

The system recommends similar specification combinations to customers in CSI order

Customer Specification Combination-Cust omer Satisfaction Index Relationship

Given the convergence condition, judge whether the user needs converge

and give feedback

Non-convergence

Judge Convergence

End

Fig. 1. Framework of the proposed method

2.2

Dataset Preparation for Online Shopping Data

2.2.1 Online Shopping Data Collection The e-commerce website has rich product information including sales-related attributes and product-related attributes, such as product’s price, sales volume, cumulative evaluation, style, weight, etc. Using web crawling techniques, product information can be obtained from the e-commerce website in large-scale batches. The raw data of the obtained information is generally in the presence of duplicates, vacancies, over deﬁnitions, and the like. By using data cleaning techniques, an XML document containing all the information about a product is obtained. The constructed product feature can describe a product in the form of a column vector Pi, as is shown in Eq. 1. Each item of the vector Si represents a feature of the product, and the feature is described by the value Vi of the feature. After the feature is extracted and transformed, the product is converted into a vector in the n-dimensional vector space; here n represents number of the product’s speciﬁcation. Thus, the customer’s needs can be represented by a vector containing the feature values of the following form:

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Pi ¼ ½ S1 V1

S2 V1

S3 V1 . . . Sn V1

ð1Þ

2.2.2 Building Up Feature Library The behavior of predicting the products that the user may need can be converted into a problem of comparing the similarity between user requirements and product features, i.e. product speciﬁcation combinations. Following Sect. 2.2.1, any product Pi can be represented by a point in the vector space D, as is shown in Eq. 2. The product information library collected offline in this paper can be regarded as a M-Dimensional vector set in the vector space D: D ¼ ½ P1

P2

...

Pm

ð2Þ

Limited by the collection conditions, the collected product data does not fully cover all the forms of the product’s feature performance. At this time, the performance of each feature of the product is arranged and combined. By setting feature-related constraints and excluding feature combinations that are not physically feasible, a vector space of all product speciﬁcation combinations can be obtained. 2.3

Building Up Relationship Between Product Speciﬁcations and Customer Satisfaction Index

In this step, Customers satisfaction index (CSI) is used for the quantiﬁcation of customer’ satisfaction on a speciﬁc product [13]. Product sales number is used to indicate Customer satisfaction index. Since CSI and product feature library has been obtained, the relationship between speciﬁcation combinations and CSI is established using machine learning (ML) models. 2.4

Matching Customer Needs and Product Features

2.4.1 Distance Measurement Customers initial need could be described as a vector lower than the dimension of the product’s full speciﬁcation combination. It is possible that the customer’s need P0x does not include all the speciﬁcations. The user’s request P0x could be expressed as P0x ¼ ½ Si Vi

Sj V j

Sk V k . . .

Sl Vl

ð3Þ

Before the comparison, the user’s initial requirement P’x is expand to the same dimension as the vector in the product feature library, those speciﬁcations that the customer does not mention is added to 0. Then it is compared with the product feature library. By comparing vectors of the same dimension, the distance between two vectors can be used to describe the degree of similarity.

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2.4.2 Type of Query For different application ﬁelds, the query mode of multidimensional database is mainly divided into two types: range query and k-nearest neighbor query. Given the query point and query distance threshold, according to the distance measurement method, the range query will query all the points with less distance from the point [14]. In K nearest neighbor query, we select K points closest to the query point from the database as the result set [15]. For range queries, if the size of the feature library cannot be determined in advance, a user cannot determine how many query results can be obtained after deﬁning the query distance. In this case, it is possible to get no results, or to get most points in the database. In this case, the k-nearest neighbor query method greatly reduces the amount of computation because it deﬁnes the number of query results in advance. It is selected as the method used to query similar requirements in this method. The selection of K depends on the number of samples in the feature library. 2.5

Result Ranking and Customer Feedback

After getting K nearest product speciﬁcation combinations, the CSI of each combination could be obtained from Sect. 2.3. Thus, these speciﬁcation combinations are sort out by referring to their CSI. Customers see each of the speciﬁcation combination with its CSI and similarity with their own needs and give their feedback in forms of a new set of needs. By calculating the distances between optimal results for each iteration, a threshold is set to detect convergence. The threshold is set on the basis of the minimum distance between speciﬁcation combinations

3 Case Study In this paper, the multi-functional desks for school students are selected because the needs of customers for the desks can be intuitively transformed into the form of physical components. Therefore, this product is selected as an example to verify the implementation process of this method. 3.1

Data Acquisition and Preprocessing

The data used in this article is derived from data publicly available on the China Ecommerce website (Taobao). Using the keyword “desk, students, multi-function” to get a list of products that meet the search criteria and using the crawler software to customize the rules. Sales information and product information on the target product page can be obtained in batches. A sample data is shown in Table 1.

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Value 2149 6500 42728 4900 154179 …

Product speciﬁcation Lifting mode Facing material Target users Furniture structure Assembled or not …

Value Hand-held Fireproof board Children scaffold structure Assembled …

The obtained product information is organized in the form of matrix, and the original data is 80 groups. The original data obtained by the crawler may have some problems, such as missing data, vague description, misleading data and so on. It is necessary to use data cleaning techniques in data science ﬁeld to process the original data. The evaluation criteria for a set of qualiﬁed data is as follows. A set of data corresponds to a product independently, and each feature information of the product is fully represented by the data. After cleaning up the original data, a set of product feature information database is obtained. Each group of data has 15 features, including price, monthly sales, whether to support customization and so on. Due to the limitation of acquisition conditions, 80 sets of data collected cannot fully represent the whole feature combination of this product. All the eigenvalues of each feature should be reassembled to obtain a complete feature information base of 1474560 combinations. However, the constraints between product features make not all feature combinations satisfy the requirements of product feasibility. It is assumed here that after feature constraints, only 80 sets of product feature combinations can meet the requirements of product feasibility. After users put forward their requirements, the recommended similar product portfolio will be given from the collected 80 sets of data. 3.2

Establishing the Relationship Between Product Customer Satisfaction and Product Feature Portfolio

After cleaning, the data contains 80 groups of data, each group of data has 15 characteristics. Firstly, the customer satisfaction index (CSI) of each group of data is constructed. Then, the regression model is established by using the neural network technique. Finally, all features are rearranged. The relationship between feature combination and customer satisfaction is obtained. In this case, the ANN Fit Neural Network Model of Mathematical Modeling Software MATLAB is used to construct the model. The model uses 10 hidden layers, and the activation function uses Bayesian model. The results are validated by using all product feature combinations in mentioned in Sect. 3.1, as shown in Fig. 2. It can be seen that some feature combinations have higher customer satisfaction than other feature combinations.

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Fig. 2. Product speciﬁcation combinations and their CSI

3.3

Costumer Real-Time Feedback

A random customer who is willing to buy a multi-functional student desk is chosen to utilize this method. Accurate identiﬁcation of user requirements is carried out according to the process described in Sect. 2. Customer present initial needs, as is shown in Table 2. Customer feedback is shown in Table 3. Table 2. Customer initial needs Assembled or not (1: yes, 2: no) 1 Is it customizable (1: yes, 2: no) 2 Is it taxi transportable (1: yes, 2: no) 1 Lift mode (1: pneumatic, 2: no lift, 3: hand pull, 4: hand held) 4 Style orientation (1: economic type, 2: luxury quality type, 3: artistic style type) 2 Furniture structure (1: scaffold structure, 2: frame structure) 1 Expansion - computer table (1: support, 2: not support) 1

Table 3. Customer feedback Assembled or not (1: yes, 2: no) Is it customizable (1: yes, 2: no) Is it taxi transportable (1: yes, 2: no) Lift mode (1: pneumatic, 2: no lift, 3: hand pull, 4: hand held) Style orientation (1: economic type, 2: luxury quality type, 3: artistic style type) Furniture structure (1: scaffold structure, 2: frame structure) Decoration materials (1: PVC, 2: wood-based panels, 3: other, 4: ﬁre-proof panels, 5: none) Expansion - Computer Table (1: Yes, 2: No)

1 2 1 1 2 1 2 1

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The user requirements are organized according to the form in the product feature library, and the vector representing the initial customer needs is obtained as Px1 ¼ ½ 0 0 1 2 1 4 0 0 2 1 0 0 1 0

ð4Þ

According to the size of the data set (80), we set K = 3, closest product feature vectors. The K-nearest neighbor search algorithm is programmed and three closest product feature combinations are obtained as is shown in Table 4. Table 4. Matching results Product speciﬁcations Price Gross weight Assembled or not (1: yes, 2: no) Is it customizable (1: yes, 2: no) Is it taxi transportable (1: yes, 2: no) Lift mode (1: pneumatic, 2: no lift, 3: hand pull, 4: hand held) Style orientation (1: economic type, 2: luxury quality type, 3: artistic style type) Decoration materials (1: PVC, 2: wood-based panels, 3: other, 4: ﬁre-proof panels, 5: none) Style orientation (1: economic type, 2: luxury quality type, 3: artistic style type) Furniture structure (1: scaffold structure, 2: frame structure) Main material (1: solid wood, 2: wood-based panels, 3: metal, 4: other) Expansion - bookshelf (1: yes, 2: no) Expansion - computer table (1: yes, 2: no) Expansion - division plate (1: yes, 2: no) CSI Similarity

First round matching results >5000 2000– 1000– 3000 2000 20–30 20–30 >40 1 1 1 1 2 1 1 1 1 1 1 1

Second round matching results >5000 2000– < 440 x 760; y ¼ 370 fA ðx; y; zÞ ¼ ; 12 z 80 ð27Þ 50 y 370; x ¼ 440 > > : 50 y 370; x ¼ 760

Fig. 10. Process error analysis of milling square façade

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The A side is the milling process, and the corresponding cutting force calculation model is:

F ðx; y; zÞ

8 > > > > > > > < > > > > > > > :

Fx ¼

x

y

u

CF apF fz F ae F z kFc q d0 F nwF

d a

x

n

0d0 p CFx apFx f yFx vc Fx KFx sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x y u xF nF C a Ff Fa Fz 0:5d0 ap Fy ¼ F2dp qFznwFe kFc 1 0:5d CFy ap y f yFy vc y KFy ð28Þ 0 0 Mc ¼

Fz ¼ 0 x y u CF apF fz F ae F zkFc d0 q 3d F nwF 0

210

Where d0 –Milling cutter outer diameter ðmmÞ; n–Milling cutter revolution ðr=minÞ; fz –Milling cutter movement distance per tooth table, that is, feed per tooth ðmm=zÞ; z–Number of milling cutter teeth; ae –Milling width ðmmÞ; ap –Milling depth ðmmÞ; kFc –Cutting force correction factor when cutting conditions are changed; The other characteristics and the cutting loads corresponding to the features are sequentially modeled according to the above formula. Due to space limitations, this article will not repeat them. 4.3

Standard Parts Error Analysis

For the same machining feature, the machining error is mainly due to the difference in machine tool rigidity at different positions on the machined surface, which in turn causes machining errors on the machined surface, and the machining error is inversely proportional to the rigidity. The machining error caused by the static rigidity of the machine tool mainly includes roundness error, straightness error, flatness error, contour error, and other shape errors. The above shape error produces a position error with a plurality of processing surfaces, and mainly includes positioning errors such as parallelism error, verticality error, and inclination error. The cylindricity error is primarily due to the weak rigidity of the machine tool’s corner direction. Under the action of the cutting force, the angle of the tool changes, and the cylindricity error is caused by the oblique feed. The errors such as straightness and flatness are mainly due to the rigidity of the machined surface. The ﬁeld is uneven, and the rigidity ﬁeld mentioned here is the axial rigidity, and the direction is the normal direction of the machined surface (Fig. 11).

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Fig. 11. Main error analysis results of standard test pieces

Error modeling all the working surfaces of the whole standard test piece, introducing the processing parameters, and completing the error analysis of the surface of the part. The results of the XYZ analysis are shown below. From the analysis results, when the part is processed, when the tool is from Y direction When it is thought to increase, the machining error increases and the accuracy decreases. At the same time, the amount of error of the part on the X-axis during the machining of the section is small, which is mainly due to the symmetrical structure design of the machine tool to enhance the rigidity of the X-direction. According to the analysis of machining error between parts, it can be known that this type of machine tool should ensure the lateral feed machining of the tool when arranging the machining process. For the too high longitudinal parts, it can be processed horizontally, thus avoiding the error-sensitive direction and improving the machining precision (Fig. 12).

Fig. 12. X-direction error results of the square façade

Taking the square façade of the standard test piece as an example, the influence of the static rigidity of the machine tool on the machining error is analyzed. According to the requirements of the national standard accuracy inspection manual, the primary detection accuracy of the square façade is the straightness of the side. Comparing the

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analysis results of the whole standard test piece, the deformation of the square façade in the Y direction is quite different. Therefore, the straightness of the square YZ façade is analyzed, and the X-direction machining error analysis result is extracted. As shown in the ﬁgure, the plane is in the Y direction. There is a signiﬁcant straightness error, the straightness error is 0.004, and the tolerance is 0.010, so the straightness error meets the design requirements. Since ANSYS cannot achieve the above-mentioned continuous analysis of part feature error, to verify the accuracy of the error analysis of this method, 6 points are selected on the surface of the part, and the error analysis is performed point by point using ANSYS to extract the Z-direction deformation in the analysis result. Analysis results and calculation efﬁciency and accuracy of the error analysis model (Fig. 13 and Table 2).

Fig. 13. Comparative analysis points

Table 2. Z-direction error analysis comparison Operation time P1 (650, 50, 50) P2 (650, 114, 50) P3 (650, 178, 50) P4 (650, 242, 50) P5 (650, 306, 50) P6 (650, 370, 50)

Error synthesis model ANSYS 0.83 (s) 137(s) 8.29 lm 7.83 lm 9.10 lm 8.75 lm 9.85 lm 9.55 lm 10.56 lm 10.30 lm 11.22 lm 11.03 lm 11.83 lm 11.73 lm

Comparing the calculation results of the two, the accuracy of the error synthesis model is consistent with the results of the ﬁnite element analysis. Comparing the operation time of this model with ANSYS, the time taken to calculate 6 points simultaneously in this model is 0.83 s, and only the ﬁnite analysis time of single point after ANSYS meshing is 137 s. At the same time, in the ANSYS analysis, the analysis

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of another point requires adjustment of the component position in the model phase, remeshing, and then ﬁnite element error analysis. Therefore, the computational efﬁciency of the method far exceeds that of ANSYS, and the technique is not limited to the processing characteristics and processing technology of the parts and has broad applicability (Fig. 14).

Fig. 14. Z-direction comparison analysis results

5 Conclusions Through the establishment of the chain-spinning synthesis model of the machine tool, the reasonable characterization of the space rigidity ﬁeld of the machine tool was realized. The spatial rigidity ﬁeld characterization model of the machine tool with integrated rigidity chain could directly describe the 6-direction rigidity variation in the sufﬁcient machining space of the machine tool. At the same time, the model had realized the 6-direction machining error prediction of the parts through the processing technology of the associated components. The rigidity synthesis model could solve the machining error under any proportional load process, thus an effective prediction of the machining error of the machine tool integrating multiple machining processes was realized. By comparing the results of ANSYS analysis, the method greatly had improved the analysis efﬁciency on the basis of ensuring accuracy. Finally, the space rigidity ﬁeld was established with the precision coordinate boring machine as the object. Furthermore, the machining error of the standard proﬁle test piece was analyzed. The analysis showed that the machining accuracy of the machine tool had met the design requirements. The method greatly improved the analysis efﬁciency on the basis of ensuring the accuracy, and had wide applicability. It provided a good performance reference for the process design of the machine tool and the design of the machine tool structure was based on the machining process.

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References 1. Lu, B.: Technical Basis of Machinery Manufacturing. Machinery Industry Press, Beijing (2007) 2. Schellekens, P., Rosielle, N., Vermeulen, H., et al.: Design for precision: current status and trends. CIRP Ann. 1998-Manuf. Technol. 47(2), 557–586 (1998). Annals of the international institution for production engineering research 3. Huang, T., Lee, J.: On obtaining machine tool rigidity by CAE techniques. Int. J. Mach. Tools Manuf. 41, 1149–1163 (2001) 4. Wang, J., Wang, L., Li, T., et al.: Study on the rigidity of a 5-DOF hybrid machine tool with actuation redundancy. Mech. Mach. Theory 44(2), 289–305 (2009) 5. Gosselin, C.: Rigidity mapping for parallel manipulators. IEEE Trans. Robot. Autom. 6(3), 377–382 (1990) 6. Yan, R., Chen, W., Peng, F., Lin, S., Li, B.: Modeling and stiffness performance analysis of closed-chain stiffness ﬁeld for multi-axis machining system. J. Mech. Eng. 48(1), 177–184 (2012) 7. Salgado, M.A., Lo pez de Lacalle, L.N., et al.: Evaluation of the rigidity chain on the deflection of end-mills under cutting forces. Int. J. Mach. Tools Manuf. 45, 727–739 (2005) 8. Lo pez de Lacalle, L.N., Lamikiz, A.: Machine Tools for High Performance Machining. Springer, London (2009) 9. Chanal, H., Duc, E., Ray, P.: A study of the impact of machine tool structure on machining processes. Int. J. Mach. Tools Manuf 46(2), 98–106 (2006) 10. Zhao, Y., Lin, Z., Wang, H.: Operational performance analysis of heavy forging operator. J. Mach. Tool Mech. Eng. (6), 69–75 (2010) 11. Liu, Q., Zhang, Y., Lin, J., Li, H., Zhao, F.: Series stiffness ﬁeld of CNC machine tools and its application. Manuf. Technol. Mach. Tools (2), 29–32 (2011) 12. Liu, H., Zhao, W.: Dynamic characteristics analysis of machine tools based on the concept of generalized processing space. J. Mech. Eng. 46(21), 54–60 (2010)

Thermodynamic Lubrication Performance and Stability for a Deep/Shallow Pocket Hybrid Bearing Considering Bubbly Oil Hong Guo, Shuai Yang(&), Ningning Wu, and Ruizhen Li School of Mechanical Engineering, Zhengzhou University, Zhengzhou 450001, China {gghhletter,lrzh}@zzu.edu.cn, [email protected], [email protected]

Abstract. In order to research on the influence of thermal effect and bubbly lubricant oil on the bearing performance and rotor system stability. A hybrid bearing with deep and shallow pockets was selected as a study object, and its thermodynamic models including Reynolds equation, energy equation, viscosity-bubble volume fraction equation and temperature-viscosity equation were established. The pressure distribution, temperature and viscosity distribution were calculated considering the influence of the bubble using the classical numerical method, and the lubricant characteristics were analyzed. The motion model was derived for a single mass flexible rotor supported in two hybrid bearings, the system thermal instability was simulated and discussed further when the bubble volume fraction is lower than 5.0%. The results show that there exists an obvious non-uniformed temperature distribution in the lubricant ﬁlm, and the bearing lubrication parameters are climbing with the rising of the bubble volume fraction, and the thermal effect weakens the bearing load capacity and friction loss, whereas it increases the flow volume rate. The stability parameters drop when the thermal effect is considered, and they would be improved with the consideration of bubble within the limited bubble proportion range. Keywords: Hybrid bearing Thermodynamic lubrication Thermal instability Bubbly lubricant oil

1 Introduction The sliding bearings are widely used in high-speed and high-accurate rotational machineries due to their high stability and lower friction [1–3]. The hybrid bearings as a type of bearings which combine the advantages of the hydrodynamic bearing and hydrostatic bearing are paid attention to increasingly [4, 5]. With the development of research, the influence of the thermodynamic (THD) behaviors on the bearing characteristics have become a hot issue, and there have been a large number of related work. Yuansheng et al. [6], used the Newton-Raphson method to solve the bearing This project is supported by National Natural Science Foundation of China (Grant No. 51575498). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 189–201, 2020. https://doi.org/10.1007/978-981-32-9941-2_16

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pressure and temperature, and analyzed the thermal dynamic performance. Lu, et al. [7], aimed to ceramic water-lubricated bearing to simulate the thermal expansion and verify the thermal boundary conditions in their theoretical studies through experiments. Jintai and Yonggang [8] gave a comprehensive THD analysis about a dynamically rolling piston loaded bearing, and they further compared the bearing eccentricity and axis orbits between the isothermal model and THD model. Donghyun et al. [9], associated the increasing temperature with the lubrication failure mechanism including the minimum ﬁlm thickness and load capacity for a tilting pad journal bearing. Shi et al. [10], calculated the temperature rise of the floating ring bearing and the impact of operating conditions on the lubrication performance of the turbochargers. Besides these, Lin et al. [11], took account the surface texture into the slide bearing THD analysis. During the accelerating of shaft, lubricant ﬁlm would appear some gas bubbles due to the surround air vibration and oil vaporization. It would change the properties of the lubricant oil to affect some important performance parameters of bearing. The scholars made many researches about this. Nikolajsen [12, 13] studied the impact of bubble on the lubricant density and viscosity considering bubble surface tension, and analyzed how the bubbly oil affect the load capacity of bearing. Rust [14] explained the mechanism of bubble deformation, and deigned the experiments to measure the speciﬁc impact of bubble on the viscosity of corn syrup. Zebin et al. [15], researched the relationship between the bubble proportion and dynamic viscosity of the lubricant oil through the experiment. The models of two-phase fluid including gas and oil were applied to the lubricant simulation of the bearing in these existing studies. Hekmat and Biukpour [16] predicted the effect of the shaft speed and external load on the instability mechanism of the bearing lubricant ﬁlm under the influence of the cavitation. Lili and Qingliang et al. [17], focused on the speciﬁc boundary conditions affecting the cavitation behaviors for a three oil wedges bearing based on the computational fluid dynamics software. Above all, there are rare studies about coupling the bubbly oil and thermal effect on bearing characteristics and stabilities. In this research, a THD model and bearing-rotor system motion model are established considering the bubble in the lubricant ﬁlm for a hybrid bearing with deep/shallow pockets. The ﬁnite element method and ﬁnite difference method are used to solve the THD models, and to simulate the temperature distribution in the lubricant ﬁlm. It is discussed that the influence of the thermal effect and bubble content on the lubrication performance involving load capacity, friction loss and volume and stability for this bearing. This research provides a reference for lubrication theory modelling and sliding bearing design.

2 The Basic Simulating Models The hybrid bearing structure is presented in Fig. 1. The lubricant oil from the external pump to the bearing interior through the deep pocket which shows the static effect. The shallow pocket and bearing land show the strong hydrodynamic effect when journal begin rotating under the external load.

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External load Shallow pocket Deep pocket

Φ

Bearing shell Lubricant film

θ Ω

Journal

Fig. 1. Conﬁguration of the hybrid bearing with the deep/shallow pockets

2.1

Reynolds Equation

The lubricant ﬁlm is assumed to be the laminar and Newtonian flow, and the journal misalignment and bearing shell elastic deformation are ignored. The U represents the clockwise circular direction and the k represents the axial direction. The dimensionless Reynolds equation which governing the lubricant ﬁlm pressure can be expressed as follow: ! ! 3 3 @ h @p d 2 @ h @p 3 @h þA þ ¼ BM @U l @U l @k l @k 2 @U

ð1Þ

p ¼ pps ; h ¼ hc ; e0 ¼ ec0 ; u ¼ U h; BM ¼ lp0 ds c2X ; 2

A ¼ 3BM cos u þ e0 h_ sin u where d is the bearing diameter, l is the bearing axial length, c is the bearing clearance, h is lubricant ﬁlm thickness, X is the journal angular speed, p is the pressure distribution of the lubricant ﬁlm, and ps is the external supply pressure. 2.2

Thermodynamic Governing Equations

2.2.1 Energy Equation The quantities of heat convection is dominant comparing with heat conduct in the lubricant ﬁlm, the heat conduct is neglected. So, the simpliﬁed dimensionless energy equation which describes the temperature distribution along the bearing circular and axial directions is presented [18]: 3 d 2 2 h3 @p @T 2l 2 h @p @T h 3BM l @U @U l 3BM l @k @k ¼ h þ 2 3 d 2 @p2 @p 8 h þ l @U @k 3BM 2 l T¼

T l ;l¼ T0 l0

ð2Þ

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where T and l is the temperature distribution and viscosity distribution of the lubricant ﬁlm, respectively. The T0 is the bearing inlet temperature and l0 is the dynamic viscosity at this temperature. 2.2.2 Isothermal Model In order to simplify the complex thermal analysis process, the lubricant ﬁlm temperature distribution is assumed to be uniformed. Therefore, the bearing outlet temperature is regarded to be equal to the effective temperature of the whole lubricant ﬁlm. This effective temperature model is expressed as follow: DT ¼

Hf cv qQ

ð3Þ

Teff ¼ T0 þ DT

ð4Þ

where the Hf is the ﬁction power loss, Q is the lubricant oil leakage from the bearing ends, cv is the lubricant speciﬁc heat, q is the lubricant density. The D T and Teff are the average temperature rise and effective temperature of the whole lubricant ﬁlm, respectively. 2.2.3 Viscosity Equation The lubricant viscosity is affected by the bubble existing in the lubricant ﬁlm, and the impact of bubble on the density is ignored, so the dependent relationship equation between the viscosity and bubble volume fraction equation is [15]: lb ¼ 1 þ j/2

ð5Þ

where the / is the bubble volume fraction, and the j is the viscosity-bubble volume fraction coefﬁcient. The Reynolds temperature-viscosity equation is used to express the influence of the temperature on viscosity: l ¼ eaðTT 0 Þ

ð6Þ

where the a is the temperature-viscosity dependence coefﬁcient. The isothermal model cannot describe the real non-uniformed temperature distribution, so the THD models analyze the bearing lubrication parameters relating the thermal effect more accurately. 2.3

Lubricant Film Thickness 8 < 1 þ e cosðU hÞ h ¼ 1 þ e cosðU hÞ þ hd : 1 þ e cosðU hÞ þ hs

Bearing land Deep pocket Shallow pocket

ð7Þ

where hd ¼ hcd , hs ¼ hcs , hd is the deep pocket depth, and hs is the shallow pocket depth.

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Boundary Conditions

As the Fig. 2 is shown, the deep pocket has the internal feedback restrictor which shows the static effect, so the hybrid bearing pressure distribution have to obey the Reynolds boundary condition and restrictor flow equations simultaneously.

Fig. 2. Boundary conditions

The pressure conditions expression is 8 j 2 C1 > < pj ¼ 0; pj ¼ prm ; j 2 C2 > : p ¼ 0; @pj ¼ 0; j 2 C3 j @Uj

ð8Þ

where 1 prm ¼ ðqout qin þ qcout þ qcout Þm Rrm 3 R k 2 3 h @p qin ¼ k23 dl BMh 2 l @U U¼U dk 2 3 R k3 l 2 3 @p qout ¼ k2 d 2 BMh hl @U dk U¼U3 R U3 h3 @p qcout ¼ U2 l @k dU k¼k2

The temperature in the deep pocket is regarded to be the inlet temperature, so the temperature condition is (

T j ¼ T 0;

j 2 C2

@T j @Uj

j 2 C3

¼ 0;

ð9Þ

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Lubrication Characteristic Parameters

The lubricant characteristic parameters can be obtained through the lubricant ﬁlm pressure integration: Dimensionless load capacity: 8 Z Z l 2p 1 > > > F ¼ p sin UdkdU < x d 0 1 Z Z > l 2p 1 > > p cos UdkdU : Fy ¼ d 0 1

ð10Þ

Dimensionless friction power loss: Z Hf ¼

2p

Z

0

1

l h @p BM þ dkdU 2 @U 4h

1

ð11Þ

Dimensionless flow volume rate: Z

2p

Q¼ 0

3

h @p l @k

! dU

ð12Þ

k¼1

3 Stability Modelling The Fig. 3 presents a flexible symmetric rotor supported in two hybrid bearings, the rotor mass is 2mr, and the shaft flexural stiffness is ks.

Fig. 3. Flexible symmetric bearing-rotor system

The stiffness coefﬁcients and damping coefﬁcients are kmn and bmn (m, n = x, y), respectively. The lubricant ﬁlm can be seen as the springs and dampers. The motion equation of the bearing-rotor system is established as follow:

Thermodynamic Lubrication Performance and Stability

8 ks n ¼ DFx > > < ks g ¼ DFy DFx ¼ kxx x þ kxy y þ bxx x_ þ bxy y_ > > : DFy ¼ kyx x þ kyy y þ byx x_ þ byy y_

195

ð13Þ

where DFx and DFy are the lubricant ﬁlm force increment, x and y are the displacement of the shaft away from the mass center, n and g are the displacement of the rotor away from the shaft center. The general solution form of the Eq. (13) is: x ¼ x0 em t , y ¼ y0 em t , n ¼ n0 em t , g ¼ g0 em t , so the Eq. (13) is transferred as follow:

keq ixbxx kxx x0 ixbxy þ kxy y0 ¼ 0 ixbyx þ kyx x0 þ keq ixbyy kyy y0 ¼ 0

ð14Þ

The system remains stable when both of the real and image parts of the Eq. (14) solution are 0, so the rotor dimensionless critical mass Mst can be obtained as follows: Mst ¼

keq c2st

ð15Þ

where the system effective stiffness coefﬁcient k eq is keq ¼

kxx byy þ kyy bxx k xy byx kyx bxy bxx þ byy

ð16Þ

The system instability whirl frequency cst is

c2st

keq kxx k eq kyy k xy k yx ¼ bxx byy bxy byx

ð17Þ

The rotor system produce oil whirl/oscillation when the dimensionless rotor mass exceeds the Mst. The viscosity is variable at the different spots in the lubricant ﬁlm, so the threshold speed has to be calculated though iteration using Routh-Hurwitz method rather than above stability analysis method [19, 20].

4 Results and Discussion The ﬁnite element is used to solve the Reynolds equation and the ﬁnite difference method is used to solve the energy equation. Table 1 gives the main parameters during the simulation process.

196

H. Guo et al. Table 1. The main parameters in the calculation example Parameters Number of pockets Diameters/(mm) Bearing lengths/(mm) Axial length of bearing land/(mm) Deep pocket angle/(degree) Shallow pocket angle/(degree) Depth of deep pocket/(mm) Depth of shallow pocket/(mm) Bearing clearance/(mm) Oil supply pressure/(MPa) Inlet temperature/(°C) Viscosity at 40 °C/(mPa s) Speciﬁc heat capacity/(J kg−1 °C−1) Lubricant oil density/(kg/m3) Temperature-viscosity coefﬁcient Viscosity-bubble volume fraction coefﬁcient

4.1

Value 4 70 80 10/10 15 55 0.25 0.022 0.035 2.5 40 4.475 1800 889 0.03568 269.1

Temperature Distribution

Figure 4 presents the 3-D temperature distributions of the lubricant ﬁlm under different bubble volume when the rotational speed is 12000 rpm and the eccentric ratio is 0.3. The lubricant temperature is climbing from the deep pocket end to the bearing land along the clockwise circular direction. Because the cold oil in the deep pocket cool the lubricant ﬁlm, and the temperature is dropped dramatically after its peaks along this direction. In the axial direction, the heat is transferred from the pocket center to the bearing ends with lubricant leakage. The difference between temperature peak in the pocket center and temperature at the bearing end is about 7–12 °C. The bubble increases the inner shear fore of the lubricant ﬁlm, so it is seen clearly that temperature is increasing due to the increment of bubble volume fraction.

(a) φ = 0.0%

(b) φ = 5.0%

Fig. 4. The temperature distribution of the lubricant ﬁlm

Thermodynamic Lubrication Performance and Stability

4.2

197

Lubrication Characteristics

The Figs. 5, 6 and 7 describe the influence of the thermal effect and bubbly oil on the bearing lubrication performance parameters when the shaft rotational speed is 12000 rpm. The load capacity is increasing while the eccentric ratio gets climbing. The thermal effect is considered, the load capacity is diminished due to the reduction of lubricant viscosity. For example, when / = 0 and e = 0.2, the dimensionless load capacity is 1.318 with isothermal model, and it goes to 1.165 with THD model which drops by 11.5%. While the bubble volume fraction gets increasing, and the capacity gets larger. So the bubbly oil is beneﬁcial to improve the load capacity and alleviate the bad influence of viscous dissipation.

Dimensionless load capacity Fr

9

THD model, Isothermal model, THD model, Isothermal model, THD model, Isothermal model,

8 7 6 5 4 3 2 1 0

0.1

0.2

0.3 0.4 Eccentric ratio ε

0.5

0.6

Fig. 5. The dimensionless load capacity F r

The dimensionless friction power loss goes up with the increasing of the shaft rotational speed rapidly, however, there is almost stable trend to the eccentric ratio except the high eccentric ratio situations. The bubble increase the friction loss, with the isothermal model, when e = 0.1 and / = 0, the dimensionless friction loss is 21.89, and it goes up to 36.74 dramatically which climbs by 67.8% while / = 5.0 at the same operation. Therefore, the influence of the bubble on the bearing friction should not be ignored. While thermal effect of the lubricant ﬁlm is considered, the friction loss gets lower.

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Fig. 6. The dimensionless power loss H f

Considering the influence of thermal effect, it is easier to flow out from the bearing for the lubricant ﬁlm. Especially at the high bubble fraction relatively, the difference between the THD model and Isothermal model reaches about 30%. Moreover, the lubricant volume flow rate rises with the bubble fraction is climbing, and the impact of the bubble in the oil has exceeded the influence of the eccentricity on the fluid volume flow rate while the rotational speed of the shaft remains constant.

Fig. 7. The dimensionless volume flow rate Q

Thermodynamic Lubrication Performance and Stability

4.3

199

Stability Analysis

For the general slide bearings, the system would be more stable if the keq is larger and the c2st is smaller. From the Figs. 8 and 9, it can be found that keq growing with the increment of the bearing eccentricity. When the thermal effect is considered, k eq is fallen, and this tendency is obvious increasingly while the growth of the eccentric ratio and bubble fraction. The c2st with THD model shows an growth than isothermal model. Therefore, the temperature rise of the lubricant ﬁlm would suffer the system stability. Additionally, it can be seen obviously that the keq increase rapidly when the bubble fraction is rising from 0.0% to 5.0% using THD and isothermal models. But the c2st dose not change signiﬁcantly with the variation of bubble fraction.

Fig. 8. The dimensionless effective stiffness keq

Fig. 9. Instability whirl frequent square c2st

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The Fig. 10 illustrates the variation of rotor dimensionless critical mass Mst under different operating conditions when the rotational speed of the shaft is 12000 rpm. It is showed that the hybrid bearing-rotor system has an excellent stability at high eccentricities. Below the threshold speed, when the thermal effect is considered, the Mst drops comparing with the isothermal assumption, especially at the high eccentricities and bubble fraction. As bubble increase the lubricant viscosity which could improve the stiffness effect and the lubricant ﬁlm resistance on external vibration and shock, the increment of the bubble fraction would increase the Mst within the given range.

Fig. 10. The dimensionless critical mass Mst

5 Conclusions (1) There are consecutive temperature peaks in the lubricant ﬁlm along the fluid main flow direction for the hybrid bearing, the temperature at bearing ends is higher than pocket center. The non-uniformed temperature distribution should not be neglected at high rotational speeds and eccentricities, and the maximum temperature is increasing while the bubble volume fraction is going up. (2) With the considering of the thermal effect on the lubricant ﬁlm, the load capacity and friction loss are fallen, and the flow volume rate gets larger. And they become increasing with the increment of bubble volume fraction particularly at high eccentricities. (3) The parameters relating to the stability including the effective stiffness, whirl frequency and critical mass are dropped due to the reduction of the lubricant viscosity caused by temperature rise. However, the bubble existing in the lubricant could improve the system stability when the bubble volume fraction is lower than 5.0%.

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References 1. Abali, E., Uckan, E.: Parametric analysis of liquid storage tanks base isolated by curved surface sliding bearings. Soil Dyn. Earthq. Eng. 30(1-2), 21–31 (2010) 2. Amamou, A., Chouchane, M.: Nonlinear stability analysis of long hydrodynamic journal bearings using numerical continuation. Mech. Mach. Theory 72, 17–24 (2014) 3. Zhang, K., Zhao, X., Feng, K., et al.: Thermohydrodynamic analysis and thermal management of hybrid bump-metal mesh foil bearings: experimental tests and theoretical predictions. Int. J. Therm. Sci. 127, 91–104 (2018) 4. Linjamaa, A., Lehtovaara, A., Larsson, R., et al.: Modelling and analysis of elastic and thermal deformations of a hybrid journal bearing. Tribol. Int. 118, 451–457 (2018) 5. Jianbin, Z., Changqing, C.: Analysis of thermal effects in shallow pocket-step hydrostatic and hydrodynamic oil bearings. J. Beijing Univ. Aeronaut. Astronaut. 23(3), 390–394 (1997) 6. Yuansheng, L., Ming, Y., Liangbo, A., et al.: Thermohydrodynamic lubrication analysis on equilibrium position and dynamic coefﬁcient of journal bearing. Trans. Nanjing Univ. Aeronaut. Astronaut. 29(3), 227–236 (2012) 7. Lu, D., Kejia, K., Wanhua, Z., et al.: Thermal characteristics of water-lubricated ceramic hydrostatic hydrodynamic hybrid bearings[J]. Tribol. Lett. 63(2), 23 (2016) 8. Jintai, M., Yonggang, M.: THD analysis of rolling piston and journal bearings in rotary compressors. Tribol. Trans. 59(2), 195–207 (2016) 9. Donghyun, L., Kyung, H., Byungock, K.: Thermal behavior of a worn tilting pad journal bearing: thermohydrodynamic analysis and pad temperature measurement. Tribol. Trans. 61 (6), 1074–1083 (2018) 10. Shi, Z., Kang, Y., Wang, J., et al.: An investigation on static lubricating characteristics of floating ring bearing considering thermal effects. Lubr. Eng. 44(1), 36–41 (2019). (in Chinese) 11. Lin, W., Zhanhai, H., Guoding, C., et al.: Thermo-hydrodynamic analysis of largeeccentricity hydrodynamic bearings with texture on journal surface. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. 232(19), 3564–3569 (2018) 12. Nikolajsen, J.L.: The effect of aerated oil on the load capacity of a plain journal bearing. Tribol. Trans. 42(1), 58–62 (1999) 13. Nikolajsen, J.L.: Viscosity and density models for aerated oil in fluid-ﬁlm bearings. Tribol. Trans. 42(1), 186–191 (1999) 14. Rust, A.C., Manga, M.: Effects of bubble deformation on the viscosity of dilute suspensions. J. Non-Newton. Fluid Mech. 104(1), 53–63 (2002) 15. Zebin, Z., Yong, L., Rognshang, C., et al.: Effect of void fraction on viscosity of lubricating oil. J. Henan Univ. Sci. Technol.: Nat. Sci. 2, 23–27 (2019). (in Chinese) 16. Hekmat, M.H., Biukpour, G.A.: Numerical study of the oil whirl phenomenon in a hydrodynamic journal bearing. J. Braz. Soc. Mech. Sci. Eng. 41(5), 218 (2019) 17. Lili, W., Qingliang, Z., Changhou, L., et al.: A numerical analysis and experimental investigation of three oil grooves sleeve bearing performance. Ind. Lubr. Tribol. 71(2), 181– 187 (2019) 18. Laraqi, N., Rashidi, M.M., Garcia, J.M., et al.: Analytical model for the thermohydrodynamic behaviour of a thin lubricant ﬁlm. Lubr. Int. 44(9), 1083–1086 (2011) 19. Hong, G., Zhiming, Z., Shaolin, Z., et al.: Multi stable regions of hydrodynamic floating ring journal bearing-rotor system. J. Vib. Shock 35(2), 168–172 (2016). (in Chinese) 20. Zhiming, Z.: Fluid Hydrodynamic Lubrication Theory. Higher Education Press, Beijing, pp. 90–120 (1986). (in Chinese)

Time Delay Chen System Analysis and Its Application Hongjun He, Yan Cui(&), Chenhui Lu, and Guan Sun Shanghai University of Engineering Science, Shanghai 201620, China [email protected]

Abstract. This paper analyzes the stability of Chen system equilibrium point and Hopf bifurcation parameter with a time delay term. According to routhhurwitz criterion, the asymptotic stability of the system near the zero equilibrium point is determined. The distribution of characteristic roots of the system is calculated, and the time delay parameters of Hopf bifurcation are calculated by using the distribution results. Based on the calculation results, the path of the wireless robot with the Chen system with added parameter terms as the navigation equation is solved, and the path map is drawn. The actual operation results show that the time delay parameters determined by this method can increase the coverage rate of robot travel path from 60% to more than 80%. Keywords: Time delay

Chen system Hopf bifurcation Wireless robot

1 Introduction Chaos theory points out the initial value sensitivity, ergodicity and long-term unpredictability of nonlinear systems, which makes the nonlinear systems have been concerned by the academic community. Since lorenz modeled and proposed the ﬁrst chaotic system [1] in 1963, chaotic systems have been applied in various ﬁelds [2–4], such as memristor circuit [5–7], chaotic synchronization [8–10], secure communication [11], industrial robots [12–15] and so on. In recent years, applying chaos theory to robot research has become a new research direction. In 2012, Volos et al. [16] designed a random signal generator using a chaotic system with double vortex-attractors to control the humanoid robot. Experimental results show that the bit sequence generated by chaos generator can be converted into four or eight motion steps, and the robot has good stability. In the same year, Ni et al. [17] proposed a new chaos algorithm (FCGA) for target search in the unknown environment of robots. When there is no target information or the information density around the robot is the same, the algorithm can avoid the disordered motion of the robot and greatly reduce the searching time. Experiments show that the chaotic genetic algorithm enables the robot to ﬁnd the target effectively. There are also many robots that use chaotic systems as navigation equations, such as sweeping robot [18] and ﬁre robot [19, 20]. In 2016, Sambas et al. [21] took the sweeping robot as the prototype, This project is supported by National Natural Science Foundation of China (Grant No.11604205). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 202–213, 2020. https://doi.org/10.1007/978-981-32-9941-2_17

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established the kinematics model of robot motion, and successfully realized the line control. Combined with a new network system of three-dimensional impulse circuit, the driving strategy of mobile robot is studied to produce the most unpredictable trajectory, but the coverage rate is low. In 2017, they changed the wire control to wireless control [22] on the basis of the previous one, and increased the track coverage to 67% at the same time. The research shows that the Hopf bifurcation characteristic of time-delay Chen system as a navigation equation can improve the coverage of mobile robot. In this paper, according to the existing research literature [23–25], Hopf bifurcation condition of Chen system with time delay was analyzed, time delay parameters were calculated and its dynamic characteristics were veriﬁed by Matlab simulation. Based on the analysis results, the time-delay parameters were determined and the navigation equation was established. Based on the experimental model used in literature [21], the mobile robot was controlled and its real coverage was tested.

2 Time Delay Chen System Analysis The state equation of the single time delay class Chen system studied in this report is as follows: 8 < x_ ðtÞ ¼ aðyðtÞ xðtÞÞ y_ ðtÞ ¼ ðc aÞxðt sÞ xðtÞzðtÞ þ cyðtÞ ð1Þ : z_ ðtÞ ¼ xðtÞyðtÞ bzðtÞ Where [x, y, z]T is the parameter of system (1), a; b and c are the parameters of system (1). By Ref. [20] know when a [ 2c, system (1) has a unique equilibrium for Að0; 0; 0Þ and when a\2c there are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ bð2c aÞ; bð2c aÞ; bð2c aÞÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cð bð2c aÞ; bð2c aÞ; 2c aÞ Bð

and the equilibrium B and C are symmetric about the Z axis. 2.1

Stability Analysis

Firstly, the equilibrium point Að0; 0; 0Þ is analyzed. According to the method proposed by the canonical lemma and Ref. [15], the system (1) can be expressed as after linearization at that point. 8 < x_ ðtÞ ¼ aðyðtÞ xðtÞÞ y_ ðtÞ ¼ ðc aÞxðt sÞ þ cyðtÞ : z_ ðtÞ ¼ bzðtÞ

ð2Þ

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Corresponding jacobian matrix of the system a JðAÞ ¼ ðc aÞekt 0

(2) is: a c 0

0 0 b

The corresponding characteristic equation of the jacobian matrix is: k3 þ p1 k2 þ p2 k p3 ðp4 k þ p5 Þeks ¼ 0

ð3Þ

Where the parameter is: p1 ¼ a þ b c; p2 ¼ ab a b; p3 ¼ ab p4 ¼ aðc aÞ; p5 ¼ abðc aÞ Lemma 1. The system (1) is asymptotically stable at the equilibrium point Að0; 0; 0Þ. Prove: When the time delay is equal to 0, we can get from Eq. (3) that k3 þ p1 k2 þ ðp2 p4 Þk p3 p5 ¼ 0 According to the routh-hurwitz criterion, Eq. (3) has a negative real part of all characteristic roots, which should satisfy the following conditions: p1 [ 0; p3 þ p5 \0; p1 ðp2 p4 Þ þ p3 þ p5 [ 0 The corresponding parameters are taken into the equation. The system is stable at the equilibrium point a when a > 2c. So at that time, when the delay was equal to 0, the system (1) was stable at equilibrium A. 2.2

Hopf Bifurcation Analysis

Let the s [ 0, k ¼ ixðx [ 0Þ is a pair of pure virtual roots of the characteristic equation. Bring it into Eq. (3): ix3 p1 x2 þ p2 xi p3 ðp4 xi þ p5 Þðcos xs i sin xsÞ ¼ 0

ð4Þ

After simplifying Eq. (4), the following equations can be obtained:

p1 x2 þ p3 þ p5 cos xs þ p4 x sin xs ¼ 0 x3 þ p2 x p4 x cos xs þ p5 sin xs ¼ 0

ð5Þ

Time Delay Chen System Analysis and Its Application

205

First square the two sides of Eq. (5), and then add the following six-member parametric equation for x: x6 þ p21 2p2 x4 þ p22 þ 2p1 p3 p24 x2 p25 þ p23 ¼ 0

ð6Þ

Lemma 2. Equation 6 contains at least one positive real root Prove: Let x ¼ x2 , then Eq. (6) can be converted into the following form x3 þ p21 2p2 x2 þ p22 þ 2p1 p3 p24 x p25 þ p23 ¼ 0 Of the above equation is changed as follows: gð xÞ ¼ x3 þ p21 2p2 x2 þ p22 þ 2p1 p3 p24 x p25 þ p23

ð7Þ

The problem of the root of the equation is then transformed into a positive zero in Eq. (7). Perform the following transformation on Eq. (7): gð x Þ ¼

1 þ p21 2p2 1x þ p22 þ 2p1 p3 p24 x12 p25 x13 þ p23 x13 1 x3

Obviously there is: gð0Þ ¼ p25 þ p23 ; lim gð xÞ ¼ þ 1 [ 0 x! þ 1

For gð0Þ ¼ p25 þ p23 ¼ ðp3 p5 Þðp3 þ p5 Þ, according to condition 1 p3 þ p5 \0, then gð0Þ\0. Therefore, gð xÞ must have a little x0 on ð0; þ 1Þ so that gðx0 Þ ¼ 0 is established and it is veriﬁed. Based on the above veriﬁcation, we set x0 to be a positive real root of the equation, then we get according to Eq. (5) cos xs ¼

p2 p4 x2 p1 p5 x2 p3 p5 p4 x4 p25 þ p24 x2

We can available the time-delay parameter s is: s¼

1 p2 p4 x2 p1 p5 x2 p3 p5 p4 x4 2np cos1 þ ðn ¼ 0; 1; 2. . .Þ x0 x0 p25 þ p24 x2

ð8Þ

According to Eq. (8), the solution of Eq. (3) is ðx0 ; sn Þ, k ¼ ix0 (x0 > 0) is a pair of pure imaginary roots of the characteristic equation. Take the time-delay parameter s ¼ s0 , which is the minimum time-delay parameter. The following gives a bifurcation condition for the hopf bifurcation of the system at this point. Let the characteristic root of the equation be ks ¼ aðsÞ þ ixðsÞ.

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ð sÞ Lemma 3. If g0 x20 ¼ 3x4 þ 2 p21 2p2 x2 þ p22 þ 2p1 p3 p24 [ 0, then dRek [ 0. ds Prove: Finding the derivative of the delay s on both sides of (3) is available: 2 dk ¼ kðp4 k þ p5 Þeks 3k þ 2p1 k þ p2 p4 eks sðp4 k þ p5 Þeks ds

ð9Þ

The following equation is obtained from Eq. (3): ðp4 k þ p5 Þeks ¼ k3 þ p1 k2 þ p2 k p3 Bring it into Eq. (9): 1 dk 3k2 þ 2p1 k þ p2 p4 s þ ¼ 3 ds kð p4 k þ p5 Þ k k k þ p1 k2 þ p2 k p3 Bringing its characteristic root k ¼ ix0 into the equation above: "

# dk 1 Re js ¼ s n ¼ ds

"

# 3k2 þ 2p1 k þ p2 p4 þ Re 3 js ¼ sn kð p4 k þ p5 Þ k k þ p1 k2 þ p2 k p3 p2 3x2 þ 2p1 xi p4 ¼ Re 4 þ Re x p2 x2 ðp1 x3 þ p3 xÞi p4 x2 þ p5 xi 2 2 3 ðx p2 Þðp2 3x Þ þ 2p1 ðp1 x þ p3 xÞ p24 ¼ p24 x2 þ p25 ðx2 p2 Þ2 x2 þ ðp1 x3 þ p3 xÞ2 ð10Þ

Bringing its eigenvalue k ¼ ix0 into the characteristic equation yields the following equation: x3 i p1 x2 þ p2 xi p3 ðp4 xi þ p5 Þeixs ¼ 0 Because eixs ¼ cos xs i sin xs and jeixs j ¼ 1, so we take the absolute value of both sides of the equation: 3 x i p1 x2 þ p2 xi p3 ¼ jp4 xi þ p5 j According to Eqs. (10) and (11), the following equation can be obtained: "

# dk 1 gð x Þ Re [0 j s ¼ sn ¼ 2 2 ds p4 x þ p25

ð11Þ

Time Delay Chen System Analysis and Its Application

Since Re

h dk 1 ds

207

i js ¼ sn and Re dk ds js ¼ sn have the same sign, Lemma 3 holds.

Combining Hopf’s bifurcation lemma, the following conclusion can be drawn: if g0 x20 [ 0, then: (a) When s 2 ½0; s0 , system (1) is asymptotically stable at point Að0; 0; 0Þ. (b) When s [ s0 system (1) is unstable at Að0; 0; 0Þ. (c) When s ¼ sn , system (1) generates Hopf bifurcation at Að0; 0; 0Þ and generates a limit cycle.

3 Numerical Simulation In this section, we use matlab to simulate the system (1) with time-delay term to verify the correctness of the theoretical analysis results in the previous section. According to the previous analysis, the system parameters a ¼ 5; b ¼ 1; c ¼ 1, the system (1) is as follows: 8 < x_ ðtÞ ¼ 5ðyðtÞ xðtÞÞ y_ ðtÞ ¼ 4xðt sÞ xðtÞzðtÞ þ yðtÞ : z_ ðtÞ ¼ xðtÞyðtÞ zðtÞ

ð12Þ

According to Eq. (6) we can get the equation about x: x6 þ 27x4 349x2 375 ¼ 0 this equation yields the only positive root x ¼ 3:213 and Solving 2 g x0 ¼ 527:78 [ 0. According to the conclusion (8), the time-delay coefﬁcient s ¼ 0:2176 can be calculated. The above conclusion is translated into: When the system parameter a ¼ 5; b ¼ 1; c ¼ 1 (b) When s 0:2176 þ 0:6225npðn ¼ 0; 1; 2. . .Þ, system (1) generates Hopf bifurcation at Að0; 0; 0Þ and generates a limit cycle. Combining the above conclusions, using matlab simulation software, the numerical value is brought into the system for simulation, and the state phase diagram of the system under different time-delay terms is given to verify the correctness of the conclusion. Figure 1 below shows the time-series diagram of the system when the lag coefﬁcient s ¼ 0:21. It can be observed from the ﬁgure that only about 70 integration times have elapsed from the integration starting point ð1; 1; 1Þ to stabilization, and ﬁnally it is stable. Balance point ð0; 0; 0Þ, this result shows that the conclusion (a) is correct. 0

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x

1 0 -1

0

10

20

30

40

50 t

60

70

80

90

100

0

10

20

30

40

50 t

60

70

80

90

100

0

10

20

30

40

50 t

60

70

80

90

100

y

2 0 -2

z

1 0.5 0

Fig. 1. System time series with lag coefﬁcient s = 0.21

Figure 2 below shows the system time-series diagram when the lag coefﬁcient s ¼ 0:22. Since the time-delay coefﬁcient has passed the bifurcation point s ¼ 0:2176, according to the conclusion (b), the limit cycle has already been generated. It is also not difﬁcult to observe from the graph that starting from the starting point of the integration ð1; 1; 1Þ, about 5 integration times have entered a stable oscillation, that is, a limit cycle has been generated and continues to stabilize in this state. Conclusion (b) Get a preliminary veriﬁcation.

x

2 0 -2

0

10

20

30

40

50 t

60

70

80

90

100

0

10

20

30

40

50 t

60

70

80

90

100

0

10

20

30

40

50 t

60

70

80

90

100

y

2 0 -2

z

2 1 0

Fig. 2. Time series with lag s = 0.22

In order to more intuitively observe the actual state of the system under various time-delay coefﬁcients, the projections of the system phase diagrams in the xy plane with three different time-delay coefﬁcients are also given in Fig. 3, and are distinguished by different colors. From Fig. 3, it can be observed that when the time lag coefﬁcient s ¼ 0:21, the system phase diagram is projected to be red, and the system gradually approaches the equilibrium point ð0; 0; 0Þ from the iterative starting point ð1; 1; 1Þ and eventually stabilizes. Balance point. When the lag coefﬁcient s ¼ 0:22, the phase diagram of the system is projected to be blue, and the system does not move closer to the equilibrium point from the iterative starting point ð1; 1; 1Þ. Instead, the system generates a limit cycle, enters the oscillating state, and ﬁnally stabilizes. On the limit ring. When the lag coefﬁcient s ¼ 0:25, the phase diagram of the system is projected to be green. At this time, the process experienced by the system is similar to

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209

s ¼ 0:22. The only difference is that the resulting limit cycles have different amplitudes. After repeated experiments, it was veriﬁed that when s 0:2176, the system generates limit cycles and the stability is good, which further veriﬁes the correctness of the conclusion (b). 5 4 3 2 1 0 -1 -2

τ=0.21

-3

τ=0.22 τ=0.25

-4 -5 -3

-2

-1

0

1

2

3

Fig. 3. xy plane projection when s = 0.21, 0.22, and 0.25

4 Application in Wireless Mobile Robot 4.1

The Mathematical Model of a Wheeled Robot

In recent years, many scholars have successfully applied chaotic systems to mobile robots. In 2015, M.J.M et al. [19] successfully developed a chaotic ﬁre ﬁghting robot. There are many other applications, such as cleaning robots [18], mobile robot patrols [20] and so on. The robot’s wire control was successfully implemented in [21]. Soon after, their team implemented wireless control of the robot in reference [22], making the design more practical. However, the simulated trajectories of robots in the existing References all show that the coverage is generally low. Our team is working hard to improve the report on robots in this report. In addition, we also added an infrared obstacle avoidance module to the robot so that the robot can avoid obstacles in actual motion. The world recognized three-wheeled mobile robot navigation equation is [21]: 8 < X_ ¼ vðtÞ cos hðtÞ ð13Þ Y_ ¼ vðtÞ sin hðtÞ : h_ ¼ wðtÞ Where vðtÞ ¼ 12 ðvr ðtÞ þ vl ðtÞÞ in the above Eq. (13), vr ðtÞ is the speed of the left

l ðt Þ wheel and vl ðtÞ is the speed of the right wheel, wðtÞ ¼ vr ðtÞv is angular velocity, L is L the direct distance between two active wheels. The principle of navigation using chaotic system is to replace the left and right wheel linear velocity in the navigation equation with two state quantities in the chaotic equation. The navigation equation and chaotic system are combined into a 6dimensional system, and the motion trajectory of the robot is solved by the fourth-order runge-kutta method. In this report, vr ðtÞ is replaced by xðtÞ, vl ðtÞ by yðtÞ, and the navigation equation is changed into the following equation:

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(

vðtÞ ¼ 12 ðxðtÞ þ yðtÞÞ ðtÞ wðtÞ ¼ xðtÞy L

ð14Þ

In order to better demonstrate the characteristics of the path, we simply transform the Chen system into the following form: 8 < x_ ðtÞ ¼ ayðtÞ bxðtÞÞ y_ ðtÞ ¼ dxðtÞ xðtÞzðtÞ þ cyðtÞ ð15Þ : z_ ðtÞ ¼ xðtÞyðtÞ ezðtÞ By combining Chen’s Eqs. (15) and (14), we can get the following equation: 8 x_ ðtÞ ¼ ayðtÞ bxðtÞÞ > > > > _ ðtÞ ¼ dxðtÞ xðtÞzðtÞ þ cyðtÞ y > > < z_ ðtÞ ¼ xðtÞyðtÞ ezðtÞ X_ ¼ vðtÞ cos hðtÞ > > > > Y_ ¼ vðtÞ sin hðtÞ > > :_ h ¼ wðtÞ

ð16Þ

The above equation describes the navigation of the robot about the Chen system. The motion trajectory image of the robot can be obtained by solving the above equation with matlab. In the previous References, the selected chaotic equation is in the state of chaos, and the coverage rate of robot is very low. In this paper, it is found that the chaotic system in Hopf bifurcation state can also cause the chaotic trajectory of the robot. While the chaotic trajectory is ensured, the motion coverage of the robot is also improved, which is about 80%. In previous experiments, the trajectory of the robot was scattered, while when the system was in the Hopf bifurcation state, the trajectory of the robot was in the center attachment. The trajectory of the robot is similar to that of a chaotic attractor, moving at a certain point. Figure 4 shows the motion trajectory of the simulated robot. 8

7

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7

Fig. 4. Simulation results of navigation route of mobile robot

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In the above equation a ¼ 35; b ¼ 25; c ¼ 20; d ¼ 8; e ¼ 3. The starting point of the iteration is ðx; y; zÞ ¼ ð0:2; 0:2; 0:1Þ and ðX0 ; Y0 ; h0 Þ ¼ ð0; 0; 0Þ. 4.2

Control Circuit Design

The path simulation results in the previous section are used to control the robot. First, the computer establishes a Bluetooth connection with the robot, and then sends the path data to the Bluetooth module of the robot through the Bluetooth serial port assistant. The microprocessor controls the left and right wheels after receiving the data. The path coverage shown in Fig. 4 is around 82%. This coverage is higher than in previous Ref. Through multiple experiments, we found that when other systems are in Hopf bifurcation state, they can still maintain a high coverage rate. The communication methods used in this report are those in Ref. [25]. We use an external bluetooth to sent chaos signal to wireless mobile robot. After receiving the path signal from the bluetooth module, arduino sends the path data to the left and right wheels in the form of PWM. The control loop selects the existing loop in [22]. First establish a connection between the PC and the HC05 Bluetooth module, use the Bluetooth management software to establish a serial port with the HC05, and send the prepared path data to the HC05 through the serial debugging assistant. The preprogrammed Arduino program on the mainboard will be in sequence. The data is read, then the left and right wheels are turned with the PWM pulse of 0–255, and the above operation is repeated until the drawing of all paths is completed. Due to the measurement error of the vehicle’s track distance L, the trajectory of the vehicle will be different from the numerical simulation during the experiment. The following Fig. 5 shows the experimental results.

Fig. 5. Navigation experiment results of mobile robot

5 Conclusion This paper analyzes the stability of Chen system equilibrium point and Hopf bifurcation parameter with a time delay term. From the theoretical analysis, Hopf bifurcation still occurs in the case of parameters with time-delay. The speciﬁc bifurcation parameters are listed in the text, and the dynamics simulation of the bifurcation shows that the theoretical analysis is correct. The combination of chaos theory and mobile robot navigation is currently less studied, mainly because its motion trajectory is not

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predictable. The previous research has successfully realized the wireless control of mobile robots. This paper successfully improves the robot’s motion coverage by changing the navigation equation. From the experimental results, the coverage of the motion trajectory is signiﬁcantly improved compared with the previous study. The theoretical coverage rate in Matlab simulation is 80%. Due to experimental errors and other factors, the actual coverage may decrease, and it needs to be improved in future research. The nonlinear system with time-delay term has very rich dynamic characteristics, mainly in the high concentration of its attraction domain, which is conducive to navigation research. At present, nonlinear systems have been successfully applied to military UAV navigation, path planning for sweeping robots, etc. I believe that this technology will have better development in the future due to its particularity.

References 1. Lorenz, E.N.: The mechanic of vacillation. J. Atmos. Sci. 20(5), 448–465 (1963) 2. Vaidyanathan, S., Volos, C.: Advances and Applications in Chaotic Systems. Springer, Heidelberg (2016) 3. Azar, A.T., Vaidyanathan, S.: Advances in Chaos Theory and Intelligent Control. Springer, Heidelberg (2016) 4. Vaidyanathan, S., Volos, C.: Advances and Applications in Nonlinear Control Systems. Springer, Heidelberg (2016) 5. Azar, A.T., Vaidyanathan, S., Ouannas, A.: Fractional Order Control and Synchronization of Chaotic Systems. Studies in Computational Intelligence, vol. 688 (2017) 6. Li, J.: Dynamic analysis and sliding mode control of fractional-order memristor chaotic circuits. Anhui University (2017) 7. Min, F., Wang, Z., Cao, G.: Analysis of multi-steady state characteristics of chaotic circuit of bimemristor based on hyperbolic function. Acta Electronica Sinica 46(2), 486–494 (2018) 8. Liang, L.I., Qingbin, L.I., Beixing, M.: Chaos synchronization for fractional-order synchronous motor systems. Henan Sci. (2017) 9. Jingbo, C., Ping, F., Hong, T., et al.: Chaos secure communication research based on feedback regulating function projective synchronization. Comput. Digit. Eng. (2017) 10. Lu, L., Li, Y., Wei, L.: Chaos synchronization of regular network based on sliding mode control. Acta Physica Sinica 61(12), 120504 (2012) 11. Lin, S., Jiang, L., Wang, C.: A three-dimensional encryption orthogonal frequency division multiplexing passive optical network based on dynamic chaos-iteration. Acta Physica Sinica 67(2), 282–290 (2018) 12. Dongsheng, X., Yongtao, L.: Study on application of chaotic dynamic behavior in multi degree of freedom robot. Comput. Measur. Control 25(1), 70–73 (2017) 13. Li, J.X., Meng, R.: Global positioning method of intelligent warehousing robot. Ind. Control Comput. 29(9), 85–86 (2016) 14. Ren, J., Yu, Q., Yao, J.: Study on nonlinear dynamic characteristics of multi-stage planetary gear train of shield tunneling machine. Mech. Drive (9), 45–50 (2017) 15. Zhang, C.: Path planning for robot based on chaotic artiﬁcial potential ﬁeld method. Sci. Technol. Eng. 317(1), 012056 (2011) 16. Volos, C., Kyprianidis, I.M., Stouboulos, I.N.: Motion control of robots using a chaotic truly random bits generator. J. Eng. Sci. Technol. Rev. 5(2), 6–11 (2012)

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17. Ni, J., Yang, S.X.: A fuzzy-logic based chaos ga for cooperative foraging of multi-robots in unknown environments. Int. J. Robot. Autom. 27(1), 15–30 (2012) 18. Palacin, J., Salse, J.A., Valganon, I., Clua, X.: Building a mobile robot for a floor cleaning operation in domestic environments. IEEE Trans. Instrum. Meas. 53, 1418–1424 (2004) 19. Tavera, M.J.M., Dutra, M.S., Diaz, E.Y.V., Lengerke, O.: Implementation of chaotic behaviour on a ﬁre ﬁghting robot. In: Proceedings of the 20th International Congress of Mechanical Engineering, Gramado, Brazil, November 2015 20. Tavera, M.J.M., Lengerke, O., Dutra, M.S.: Implementation of chaotic behavior on a ﬁre ﬁghting robot. In: Mechatronics Series. Intelligent Transportation Vehicles (2012) 21. Sambas, A., Vaidyanathan, S., Mamat, M.: A 3D novel jerk chaotic system and its application in secure communication system and mobile robot navigation. Studies in Computational Intelligence, vol. 636, pp. 283–310 (2016) 22. Sundarapandian, V., Aceng, S., Mustafa, M.: A new three-dimensional chaotic system with a hidden attractor, circuit design and application in wireless mobile robot. Arch. Control Sci. 27(4), 541–554 (2017) 23. Li, D., Lian, Y.: Hopf bifurcation analysis of the delayed Lorenz-like system. J. Math. 30(3), 7–11 (2015) 24. Wenjuan, L., Xiaomeng, N., Xvchao, L., et al.: Hopf bifurcation analysis of the disturbed Lorenz-like System with the delayed. Pure Appl. Math. 33(5), 475–485 (2017) 25. Chen, G., Lv, J.: Dynamic Analysis, Control and Synchronization of Lorenz System Family. Science Press, Beijing (2003)

The Driver-in-the-Loop Simulation on Regenerative Braking Control of Four-Wheel Drive HEVs Hexu Yang1,2(&), Xiaopeng Li1, Pengxiang Li2, and Yu Gao2 1

School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China [email protected], [email protected] 2 School of Mechanical Engineering, Ningxia Institute of Science and Technology, Shizuishan 753000, China

Abstract. Nowadays, the environmental problem is becoming more and more serious, attracting the attention of most people. Automobiles emission pollution has a great influence on environment, so the development of vehicles requires more efﬁcient and cleaner. Regenerative braking is an effective method for hybrid electric vehicle (HEV) to improve fuel efﬁcient. In this paper, ﬁrstly, the dynamics of the target vehicle was veriﬁed according to the selected parameters, including acceleration and climbing conditions. Then, a control strategy based on the parallel hybrid electric vehicle was proposed. Finally, in order to verify the control strategy’s effectiveness and real-time performance, a driver-in-theloop real-time simulation platform for HEVs was built up based on the Development to Production (D2P) product-level controller, which can reduce development costs and is easy to implement. The results show that the proposed regenerative brake control strategy has good real-time performance. Keywords: Hybrid electric vehicle Driver in the loop simulation

Control strategy D2P

1 Introduction The problems of environmental pollution and energy crisis are becoming more and more serious. Environmental issues require the development of the vehicle must be toward to the direction of low pollution and high efﬁciency [1]. Including hybrid electric vehicle, pure electric vehicle, and fuel cell electric vehicle, when the vehicle is braking, compared to conventional cars, these types of electriﬁed vehicles can be recharged by generators and stored in the batteries. Thereby, these types of electriﬁed are an effective way for the vehicle to extend the mileage compared with traditional automobile [2].

This project is supported by National Natural Science Foundation of China (Grant No. 51875092), The Fundamental Research Funds for the Central Universities (N170302001) © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 214–222, 2020. https://doi.org/10.1007/978-981-32-9941-2_18

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Nowadays, regenerative braking technologies is widely used in electriﬁed vehicles, in order to improve the efﬁciency of this technology, more and more scholar study about regenerative braking. HAN etc. proposed an adaptation regenerative brake torque optimization method using under-steer index to recover optimal braking energy for front wheel drive HEV, and by CARSIM software to verify the effects [3]. Maia etc. proposed a fuzzy logic model of regenerative braking to distribute the ratio of regenerative braking force [4]. Xu etc. proposed a braking system using only electric motors/generators as the actuators with a hierarchical control structure [5]. Kim etc. aimed at a four-wheel-drive hybrid electric vehicle using rear motor driving as the research object and proposed a stability enhancement control algorithm, using ADAMS and MATLAB Simulink simulations [6]. All of the above papers are off-line simulation. Using off-line simulation can short the development cycle and reduce costs. But it can’t reflected the real-time of the control strategy and ignore the driver’s impact on the vehicle. Based on the issue discussed in the article, the proposed effect of regenerative braking is accurately studied. This paper takes a four-wheel hybrid vehicle as the research object. In the study, author carried out the-driver-in-the-loop simulation for HEV using rear motor control. This paper mainly studies the of the control usage strategy of the parallel hybrid electric vehicle. A driver-in-the-loop real-time simulation platform for HEVs was built up based on the development to Production (D2P) of product-level controller. And the regenerative brake control strategy is proposed and simulated under acceleration and climbing conditions. Compared with offline simulation, the driver-in-the-loop real-time simulation can verify the real-time effect. Therefore, it is of great importance to study this kind of model in the loop simulation. The remainder of this paper is organized as follows: the analysis of vehicle dynamics is described in Sect. 2. The braking force distribution strategy and driver in the loop real time simulation system are proposed in Sect. 3. Simulation results and analysis are carried out in Sect. 4, and conclusions are given in Sect. 5.

2 Analysis of Vehicle Dynamics In this paper, the target model is different from the conventional parallel hybrid electric vehicle. The electric motor and the engine drive the front and rear wheels respectively, and the torque is coupled by the ground. This structure simpliﬁes the torque coupling structure and that can achieve four-wheel drive under certain conditions. Figure 1 shows the structure of a hybrid electric vehicle with rear axle electric drive vehicle that is proposed in this study. In this section, ﬁrstly, according to the driving equation of vehicle, the selected parameters of the vehicle were shown on the Table 1. Then, in order to verify the selected parameters meeting the performance requirements of vehicle, we passed through simulation on the different road conditions in this paper. The results were shown in the Figs. 2 and 3.

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Charge

Transimission

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Fig. 1. The structure of rear axle electric drive vehicle Table 1. Key parameters of target vehicle Name Maximum engine power Electric motor power Battery capacity Vehicle loaded mass Gearbox Air resistance coefﬁcient Windward area Number of batteries Rolling resistance coefﬁcient Wheel radius

Value 40 KW 49 KW 45 Ah 1400 kg 3.52, 2.04, 1.40, 1 0.301 2.05 m2 25 0.012 0.315 m

250

adhering coefficient is 0.6 adhering coefficient is 0.3

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Road slope is 16° Road slope is 9°

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Fig. 3. Climbing map of rear axle electric drive vehicle

Figure 2 is an acceleration diagram of a rear axle hybrid electric vehicle on different road surfaces with varying coefﬁcients of adhesion. From the experimental data, it can be seen that the acceleration value of rear-wheel drive electric vehicle increases with time in 0–60 s when the vehicle is on the road with good adhesion. This changing trend is related to the size of the adhesion coefﬁcient. The larger the adhesion coefﬁcient, the more obvious the acceleration increase value. The slope of change trend decreases obviously after 60–80 s, and gradually reaches the critical value. After the 80 s, the critical value of acceleration remains constant, and the higher the adhesion coefﬁcient, the higher the critical value of acceleration. The blue line shown in Fig. 2 shows that the vehicle has high acceleration, and the maximum speed can reach 200 km/h. Even if the road surface does not have good adhesion conditions, such as the adhesion coefﬁcient shown by the red line is only 0.3, the maximum speed can reach 150 km/h. Therefore, we can conclude that the selected parameters can meet the acceleration requirements of the vehicle. Figure 3 is a climbing map of the target vehicle on different slopes of the road. The climbing ability of rear wheel drive electric vehicle is related to the gradient of the road surface. The smaller the gradient of the road surface, the stronger the climbing ability, the larger the gradient of the road surface, the weaker the climbing ability of the vehicle. As can be seen from Fig. 3, when the gradient of the road surface is 9°, the climbing ability of the vehicle is strong, and the maximum speed can reach 100 km/h. When the gradient of the road surface increases to 16°, the climbing ability of the vehicle is slightly weaker, but the maximum speed can also reach 60 km/h. Therefore, the matching parameters can meet the climbing requirements of the vehicle. From what has been discussed above, we can get the conclusion that the target model can get better power when the road conditions are poor or the torque requirements are large. Therefore, the study of this type of vehicle is very necessary.

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3 Regenerative Braking Control Strategy In this paper, development of control strategy must ensure the braking safety as much as possible on the recovery of braking energy. The speciﬁc process is shown in Fig. 4. When the vehicle is braking, the Electronic Control Unit (ECU) calculates the required braking force through the change of the brake pedal travel. Then, brake force distribution between front and rear axles is determined by the braking strength z. The braking force of the front wheels is supplied by mechanical brake system braking, and the braking force of the rear wheels is supplied by regenerative braking force and mechanical braking force. The size of the regenerative brake force is determined by many factors such as motor maximum braking power and maximum charge power of the battery. Start Total braking force F b Distribution strategy of braking force

> ϕ

≤

G ϕ (a − ϕ × hg) L Fbf = Fb − Fbr

G (a − Z × hg) L Fbf = Fb − Fbr

Fbr = GZ

Fbr =

Fbr = Z

Fbf = Fb − Fbr

Rear wheels braking force

Front wheels braking force

Limiting factor Motor maximum braking power Pmotor

Maximum charge power of the battery Pbat

SoC fs Pv > fs Ps > Pv Ps > Ps > Pv. Relation diagram of effective curves is shown in Fig. 6. It can be seen that if interaction effect is not considered, stiffness volatility is optimal when fs values level 3, Pv values level 4 and Ps values level 2. Namely, stiffness volatility has a minimum value of 4.272 N/lm when area ratio values 30%, vacuity values −50 kPa, and positive pressure values 0.4 MPa. This value is better than any of the cases calculated above. It means that we do obtain a relatively optimization.

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12345 12345 12345 12345 12345 12345 factor level

Fig. 6. Effective curve on Sv

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Optimization of Average Stiffness Under Operating Conditon

Change of average stiffness is analyzed below. Average (Ki) and extreme difference (R) of each level are shown in Table 5. Table 5. Visual analysis of results of K

K1 K2 K3 K4 K5 R

Factor ns 304.1 275.9 328.4 333.3 251.5 81.8

Pv 218.4 279.7 281.1 343.1 371 152.6

ns Pv 340.7 301.1 255.9 292.4 303.2 84.7

Ps 204.7 287.4 323.1 342.1 336 137.5

ns Ps 311.1 294.3 320.8 290.5 276.6 44.2

Pv Ps 320.2 301.8 319.5 268.4 283.3 51.8

Extreme difference of vacuity Pv and positive pressure Ps are signiﬁcantly larger than other factors. That means that these two factors have more influence on average stiffness. Degree of effect ranks like this: Pv > Ps > fs Pv > fs > Pv Ps > fs Ps. Relation diagram of effective curves is shown in Fig. 7. It can be seen that if interaction effect is not considered, average stiffness is optimal when fs values level 4, Pv values level 5 and Ps values levels 4. Maximum of average stiffness is 499 N/lm when area ratio values 35%, vacuity values −60 kPa, and positive pressure values 0.6 MPa. This value is also better than any of the cases calculated above. 380 360 340

K (N/ μm)

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Fig. 7. Effective curve on K

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Synthesis Optimization

Optimization of the two performance objects are been obtained respectively in previous analysis (summarized in Table 6). It shows that there are different choices of parameter combination when pursuing different requirements. Too large stiffness is not necessary in some applications. Conversely, a small performance difference of stiffness under different operating condition is required to make equipment predictable and stable. In this situation, we choose the ﬁrst optimization in Table 6 which has little stiffness volatility. If we need sufﬁcient stiffness, the second optimization is better. Table 6. Optimum scheme of the VPL pad N Optimization object Parameter combination Results (N/lm) Pv (KPa) Ps (MPa) Sv fs K 30% −50 0.4 4.272 334.07 1 Sv 2 K 35% −60 0.6 23.64 499.17

5 Conclusions In this paper, a new index is deﬁned to express performance of VPL pad under operating condition. A better static performance is obtained by orthogonal method. The ﬁndings are summarized as below. (1) Stiffness volatility is considerable while parameters of VPL pad are incongruous. (2) Vacuity, area ratio and positive pressure distinctly influence performance of VPL pad. Stiffness volatility grows when value of parameters is relatively small or large. (3) Orthogonal method is effective to optimize performance of VPL pads with multiply parameters. Computing quantity is reduced obviously. (4) Extreme value of average value and stiffness volatility infrequently emerge together. Selection of optimization of performance object is based on actual application.

References 1. Raparelli, T., Viktorov, V., Colombo, F., et al.: Aerostatic thrust bearings active compensation: critical review. Precis. Eng. 44, 1–12 (2016) 2. Chen, C.H., Tsai, T.H., Yang, D.W., et al.: The comparison in stability of rotor-aerostatic bearing system compensated by oriﬁces and inherences. Tribol. Int. 43(8), 1360–1373 (2010) 3. Miyatake, M., Yoshimoto, S.: Numerical investigation of static and dynamic characteristics of aerostatic thrust bearings with small feed holes. Tribol. Int. 43(8), 1353–1359 (2010) 4. Zhao, G., Belyaev, A., Wolters, C.H., et al.: Stabilizing a substrate using a vacuum preload air bearing chuck: US, US7607647, 27 October 2009

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5. Yandayan, T., Akgoz, S.-A., Asar, M.: Calibration of high-resolution electronic autocollimators with demanded low uncertainties using single reading head angle encoders. Measur. Sci. Technol. 25(1), 303–309 (2014) 6. White, J.: Air bearing slider-disk interface for single-sided high speed recording on a metal foil disk. J. Tribol. 129(3), 562 (2007) 7. Schenk, C., Buschmann, S., Risse, S., et al.: Comparison between flat aerostatic gas-bearing pads with oriﬁce and porous feedings at high-vacuum conditions. Precis. Eng. 32(4), 319– 328 (2008) 8. Zhu, H., Li, Y., Lin, Y., et al.: CFD investigation on the performance of aerostatic thrust bearing with exhaust slots used in low-vacuum condition. Mems-12 (2012) 9. Liu, S.-S., Chen, H., Chen, X.-D.: Experimental investigation of static characteristics of a vacuum preloaded aerostatic bearing. Adv. Mater. Res. 311–313, 1012–1016 (2011) 10. Huang, M., Xu, Q., Li, M., et al.: A calculation method on the performance analysis of the thrust aerostatic bearing with vacuum pre-load. Tribol. Int. 110(Suppl. C), 125–130 (2017) 11. Wang, N., Kong, P.-H.: A simulated air bearing analysis by design of experiments and its applications in optimization. Tribol. Trans. 44(4), 597–602 (2001) 12. Shinde, A.B., Pawar, P.M.: Multi-objective optimization of surface textured journal bearing by Taguchi based grey relational analysis. Tribol. Int. 114, 349–357 (2017) 13. Jia, X., Guo, F., Huang, L., et al.: Parameter analysis of the radial lip seal by orthogonal array method. Tribol. Int. 64(3), 96–102 (2013) 14. Wang, N., Tsai, C.M., Cha, K.C.: Optimum design of externally pressurized air bearing using cluster Openmp. Tribol. Int. 42(8), 1180–1186 (2009) 15. Wang, N., Cha, K.C.: Multi-objective optimization of air bearings using hypercube-dividing method. Tribol. Int. 43(9), 1631–1638 (2010)

A Study on Innovative Design of Rotary Pile Foundation Drilling Machine Based on TRIZ Theory Fuxing Li(&) Qilu University of Technology (Shandong Academy of Sciences), Jinan, China [email protected]

Abstract. Whereas such problems as small borehole diameter and low efﬁciency occur in the operation process of rotary pile foundation drilling machine, this paper puts forward an innovative design method based on the TRIZ theory and comes up with the detailed solution to the conflicts, through the use of the contradiction conflict solution matrix comprised by 39 engineering parameters and 40 pieces of inventive principles pertaining to TRIZ, in the aspects of borehole diameter enlargement and operation efﬁciency improvement. Furthermore, this paper has manifested the feasibility to conduct TRIZ-theory-based innovative design in the ﬁeld of engineering machinery, thus providing important theoretical tools for design ideas in the ﬁeld of traditional engineering innovation. Keywords: TRIZ theory Rotary pile foundation drilling machine Technical conflict Innovative theory Conflict solution matrix

1 Introduction As is well-known to all, prior to the constructions of house, high tower or bridge, etc., pile foundation should be constructed using drilling machine. Rotary-digging pile foundation drilling machine is one of the most frequently-used machines. However, due to the limitation of drill’s diameter, it’s hard to carry out project with largediameter drill hole. Besides, during drilling work, drill at the deﬁned position ﬁrst, and then elevate the drill to the outside of hole to unload the clump, and ﬁnally move the drill back to the hole to continue the drilling process. Repeat the preceding working process until the drill reaches the scheduled depth. In the whole process, to align the “original hole” consumes too much time and seriously lowers the drilling efﬁciency. To this end, this paper aims to carry out innovative design of rotary drilling rig based on TRIZ theory, so as to enhance its working capacity.

2 TRIZ Theory Summary TRIZ, the Russian abbreviation for “Teoriya Resheniya Izobreatatelskikh Zadatch” (which means Theory of Inventive Problem Solving in English), is the theory system formed through the analysis of nearly 2.5 million high-level patents for invention © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 302–309, 2020. https://doi.org/10.1007/978-981-32-9941-2_26

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worldwide and integration of the theories and rules of multi-disciplinary ﬁelds, which was made by the research institution with a former Soviet Union expert named Altshuller as the leader. 2.1

Constitution of TRIZ Theory System

With dialectics, system theory and epistemology as philosophical guidance, the analytic research results of natural science, system science and noetic science as the foundation, and technical system evolution rule as the theoretical basic and core concept, TRIZ theory system includes all kinds of analytic methods, solution tools and algorithm flow that are needed for solving engineering conflict problems and complicated inventive problems [1], as seen in Fig. 1. The application of TRIZ theory can expedite the process of creation and invention to get high-quality innovative products. As regards the innovation design of rotary-digging pile foundation drilling machine, the contents of TRIZ theory, including 39 general engineering parameters, 40 pieces of inventive principles, 39 39 conflict solution matrix, etc., will be used.

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Fig. 1. TRIZ theory system

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Determination of Technical Conflict and Its Solution Procedure

StyFunction is a kind of understanding of product from the angle of technical realization and a kind of abstract description of the changes to parameter or condition when product inputs or outputs under given conditions [2]. Product innovation design starts

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from product function ﬁrst, and through the analysis of function, points out the ideal function and disadvantageous function, so as to determine the technical conflict. The so-called function: it means an action can result in both advantageous and disadvantageous results at the same time [3], and the results can be solved by utilizing the contradiction solution matrix of TRIZ theory. TRIZ believes the product evolution process is to ﬁnd out and solve the conflicts constantly so as to boost the product to develop ideally [4]. Therefore, the TRIZtheory-based design process will center upon how to analyze the problem and solve the conflict. TRIZ theory serves to solve the complicated inventive problems which are characterized by technical conflict and physical contradiction, and, via the systemic problem-solving process, shows the designers the most effective solution [5]. Figure 2 shows the product design flow by applying the conflict solution principle of TRIZ theory: ﬁrstly, analyze the product function, ﬁnd out the ideal function and disadvantageous function, and determine its technical conflict; then, use the 39 engineering parameters of TRIZ theory to describe the pre-determined technical conflict; ﬁnally, obtain the related inventive principle by querying the conflict solution matrix [6], and conduct the innovation design of product scheme with the inventive principle as the guiding ideology.

Fig. 2. Design process model for TRIZ theory

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3 TRIZ Theory Based Innovation Design Example 3.1

Analysis of Problems in Rotary-Digging Pile Foundation Drilling Machine

Rotary-digging pile foundation drilling machine crushes the rock-soil by rotating the bucket drill which has a valve at the bottom. Put it directly into the drill bucket, and then use the hoisting unit and bumper sub to lift the drill bucket up to unload the clump. Repeat the process to load clump ﬁrst and then unload it. When the hole has reached the scheduled depth, cast concrete and then a pile is built. In drilling work, the process goes as follows: drill a hole ! lift the drill to unload ! return to the original hole ! drill again. However, it really consumes too much time and energy to pinpoint the original hole, thus severely reducing the work efﬁciency. In addition, the rotary-digging drilling machine relies on lifting the drill bucket for dislodging. So when the drilling depth is too deep, the drill pipe has to be enlarged, and then time spent on drilling in the hold will be much less than that on lifting the drill and unloading the clump, which will reduce work efﬁciency. Then, to improve the work efﬁciency, the drill pipe has to be shortened, while innovative design should be made to the structure of rotary-digging drilling machine to guarantee the drill effect. As a result, the whole set of drilling equipment will be much more complicated. Besides, to enlarge the diameter of drill hole, the diameters of drill pipe and drill bucket should be enlarged as well, i.e., the cross sectional area will be enlarged, which will undoubtedly add to its weight and thus require larger momentum. 3.2

Determination of Technical Conflict

In the working process of rotary drilling rig, the cross section area of drill pipe and its weight constitute technical conflict, while the improvement of drilling efﬁciency and the complexity of the whole set of drilling equipment also constitute technical conflict. Then, turn the technical conflicts into the 39 engineering parameters, namely: Area of No. 5 moving object: it refers to the geometric measure inside or outside of the moving object, which may be the area in plane proﬁle and also that in threedimensional surface. In this example, it refers to the cross section area of drill and belongs to the improved parameter. Weight of No. 1 moving object: it refers to the force acted by the moving object in gravitational ﬁeld on the support that prevents the former from free falling. In this example, it refers to the deteriorated parameter [7]. Production rate of No. 39 unit: it refers to the functions performed or quantity operated by the system in unit time; or it refers to the time to complete a function or an operation, which should be the improved parameter. Complexity of No. 36 unit: it refers to the number and diversity of system element and its mutual relationship, which belongs to the deteriorated parameter [8].

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Determination of Inventive Principle

As regards the technical conflict produced during the working process of rotary-digging pile foundation drilling machine, the corresponding inventive principle can be obtained via querying the contradiction solution matrix provided in TRIZ theory, as seen in Table 1.

Table 1. Conflict solution matrix Improved general Deteriorated general engineering parameter engineering 1 Weight of 2 Weight 3 Size of … 36 parameter moving of static moving Complexity object object object of unit 1 Weight of – – 15, 8, 38, … 26, 30, 36, moving object 34 34 2 Weight of static – – – … 1, 10, 26, 39 object 3 Size of moving 8, 15, 29, – – … 1, 19, 26, 24 object 34 4 Size of static – 35, 28, 40, – … 1, 26 object 29 5 Area of moving 2, 17, 29, 4 – 14, 15, 18, … 14, 1, 13 object 4 … … … … … … 39 Productivity of 35, 26, 24, 28, 27, 15, 18, 4, 28, … 12, 17, 28, unit 37 3 38 24

… 39 Productivity of unit … 35, 3, 24, 37 … 1, 28, 15, 35 … 14, 4, 28, 29 … 30, 14, 7, 26 … 10, 26, 34, 2 … … … –

As regards the conflict constituted by “area of moving object” and “weight of moving object”, the 2th, 17th, 29th and 4th inventive principles can be selected. The speciﬁc explanations for each inventive principle can be seen in Table 2.

Table 2. Inventive principle and explanation (one) Serial number 2 4 17

29

Principle name

Explanation

Separation principle Asymmetric Dimensional change

Select or separate the key parts of the object

Pneumatic and hydrodynamic

Change the object shape from symmetric to asymmetric Turn the moving or static object in one-dimensional space into that in two-dimensional space, and change the object in two-dimensional space to that in three-dimensional space The solid parts of object can be replaced with pneumatic or hydrodynamic parts, with gas or liquid used for expansion or damping

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As regards the conflict constituted by “productivity” and “complexity of unit”, the 12th, 17th, 28th and 24th inventive principles can be selected. The speciﬁc explanations for each inventive principle can be seen in Table 3. Table 3. Inventive principle and explanation (two) Serial number 12

Principle name

Explanation

Equipment principle

24

Mediator principle

28

Replacement of mechanical system

Change the work condition so that the object doesn’t need to be raised or lowered ① Use mediator to transmit certain object or certain medium process; ② Integrate one easily-removable object with another one temporarily Use the electric ﬁeld, magnetic ﬁeld or electromagnetic ﬁelds that have interaction with the object

With regards the to-be-solved problems in rotary digging rig, screen out the inventive principles and then the 2th, 17th, 24th, 29th inventive principles are selected, which will be the guiding ideology for innovation design of rotary digging rig. No. 29 pneumatic and hydrodynamic structure principle: use the pressure difference produced from power plant, suck in the drilled rock-soil to unload to the outside of drill hole so that the drill hole and soil unloading can work at the same time to improve the efﬁciency of drill work; No. 17 dimensional change principle: the drill can rotate in the vertical direction and revolve in the horizontal direction, so as to enlarge the diameter of drill hole; No. 2 separation principle: place the dislodging passage outside of the pile foundation drilling machine to separate it from engine body par, so as to facilitate the elimination of muck and control the muck; No. 24 mediator principle: use the support frame to put up the drilling machine and wind-power unit, so as to leave work space for drill bit and suction headpiece and other units and enable the whole set of equipment to go down along with the increase of drilling depth.

4 Design Scheme for Ultra-large Diameter Pile Foundation Drilling Machine To use the ultra-large diameter pile foundation drilling machine, ﬁx the position of cement caisson circle (18) ﬁrst on the ground, place the track frame (9) and rotary support frame (4) sequentially on cement caisson circle (18), adjust the earthling positions of drill bit (17) and sucker (10); then, unlock the adjustable-speed motor (3) and wind motor (8), via the driving gear (12), the adjustable-speed motor (3) can drive the input gear (13) and drill spindle (2) to rotate; at the same time, via the carrier gear (1), the input gear (13) can drive the running gear (15) to rotate, so as to rotate and revolve around the central axis at the same time; wind motor (8) can drive the wind wheel (6) to rotate and form the high negative pressure, which sucks the drilled rocksoil into the air conveying pipe (5) and thus exhausts the soil outside of the pipe; when

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the drill hole reaches a certain depth, new cement caisson circle (18) can be added, however, the second cement caisson circle (18) should leave the support part of track frame (9) empty so that the drill rig can dig deep repeatedly. When the scheduled depth is reached, use the hoist to dismantle the drill rig step by step, as seen in Fig. 3. The ultra-large diameter pile foundation drilling machine principle is relatively novel, which changes the traditional drilling principle and can ﬁnish projects with ultra-large diameter. However, the whole set of pile foundation drilling machine is bulky, while its structure is not compact. Therefore, more improvements will be made.

Fig. 3. Ultra-large diameter pile foundation drilling machine structure principle

5 Conclusions By applying the conflict solution principle of TRIZ theory, conduct innovation design on traditional rotary-digging pile foundation drilling machine to enable the drill to rotate in the vertical direction and meanwhile revolve in the horizontal direction of the center axis of rotary table, so as to enlarge the diameter of drill hole; use wind power installation to discharge the drilled rock-soil directly, thus reducing the time spent on repetitive soil-casting. Besides, adjust the activity range of drill and suction headpiece on the pile foundation drilling machine so as to meet with the requirement for diameter in actual working condition and thus make it the orientation for improvement. The application of TRIZ theory to conduct innovation design on rotary-digging pile

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foundation drilling machine veriﬁes the feasibility to use this theory to conduct innovation design in the ﬁeld of engineering machinery, so as to provide a new theoretical tool for the development and design of related product.

References 1. Ren, G.: Research the effect of industrial design to product maturity based on TRIZ. Mach. Des. Manuf. 3, 264–265 (2009) 2. Lu, X.: Product innovation design based on the theory of TRIZ and functional analysis. Mach. Des. Manuf. 12, 255–257 (2010) 3. Zhang, J.: Innovation design of products based on TRIZ theory. J. Mach. Des. 2, 35–37 (2009) 4. Fu, M.: Reviews on integrated innovation methods for TRIZ-based product design. Chin. J. Constr. Mach. 2, 170–174 (2013) 5. Hou, L.: Optimal design for dry ﬁlter clip based on TRIZ theory. Mach. Des. Manuf. 7, 240– 242 (2011) 6. Chen, S.: Innovative design of new wire-saw winding equipment based on TRIZ theory. Mach. Des. Manuf. 1, 4–6 (2013) 7. Tan, R.: Study on the conceptual design process based on QFD and TRIZ. J. Mach. Des. 9, 1– 4 (2002) 8. Li, F.: Innovation design of road roller based on the conflict solving principle of the TRIZ. J. Mach. Des. 98–111 (2014)

Design and Optimization of Focusing X-Ray Telescope Based on Intelligent Algorithm Liansheng Li1(&), Zhiwu Mei1, Jihong Liu2, Fuchang Zuo1, Jianwu Chen1, Hanxiao Zhang1, Hao Zhou1, and Yingbo He1 1

2

Beijing Institute of Control Engineering, Beijing 100191, China [email protected] School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China

Abstract. Focusing X-ray telescope (FoXT) is widely used in space science detection, pulsar navigation and timing, space X-ray communication, etc. In order to improve the performance and reduce the weight of FoXT, this paper proposes a FoXT design and optimization method based on intelligent algorithm. Firstly, based on the X-ray total reflection theory, a multi-objective optimization model with maximum effective area and light weight is formulated. The Pareto non-inferior solution set of the product is obtained using the nondominated sorting genetic algorithm (NSGA-II). With data mining methods such as cluster analysis and association analysis, the implicit correlation characteristics and variation rules between design parameters and optimized solution sets are studied. The influence of parameters such as X-ray optic length, optical aperture and nested layer on effective area is revealed. The distribution of focal length and optical aperture ratio, effective collection area and weight ratio are implemented with statistical analysis. Finally, ﬁve sets of design optimization solutions are selected according to practical engineering development capability and cost, which provides theoretical basis for design and optimization of FoXT. Keywords: Focusing X-ray telescope Intelligent algorithm

Multi-objective optimization

1 Introduction With the advantages of high angle resolution, low background noise, and large collecting areas, the focusing grazing incidence X-ray telescope (FoXT) gained more and more attentions. Especially, due to recent advancements in fabrication technology, multilayer mirrors open a wide range of possibilities in the ﬁeld of X-ray astronomical telescope, X-ray pulsar navigation and timing [1], space scientiﬁc exploring, and deep space exploration, etc. [2]. As an X-ray astronomical instrument, how to improve the inherent characteristics and performances is a very import thing. With the actual demand for aerospace This project is supported by National Natural Science Foundation of China (Grant no. 51175019), The National Key Research and Development Program of China (Grant No. 2017YFB0503300). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 310–324, 2020. https://doi.org/10.1007/978-981-32-9941-2_27

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engineering, this requires an instrument with the large effective area, high sensitivity and the lowest possible weight, volume, and power. Actually, the focusing grazing incidence X-ray telescope is a typical complex space product which involves coupled structure, optics, thermal, electrics etc. In addition, the enhancement of effective collecting area that means the telescope has longer focal, bigger optical aperture and more multi-layer nested mirrors. As a result, the weight of FoXT will increase dramatically with the improvement of efﬁcient collecting area. Obviously, the optimization of FoXT is a multi-objective optimization problem (MOOP) due to conflicting nature of effective collecting area and weight. Currently, researches of focusing X-ray telescope are mainly focus on the optimization of optical performance or other single objective. A ﬁgure-of merit that accounts for the coating response over a speciﬁed range of energies and off-axis angles is developed by Mao et al. [3], which presents a deep understand of optical performance and coating thickness. Take the reflectivity of X-ray mirrors as an optimization objective, an extensive numerical method is used by Biskach et al. [4] to obtain the optical solutions based on analytical method with oversimpliﬁed analytical and semiempirical formulation. A ray-tracing simulation method is proposed by Mitsuishi [5] to investigate the optical performance of ultra-lightweight X-ray optics prepared for the future Jupiter exploration mission. Conconi et al. [6] employs a weighted merit function method to carry out the multi-objective optimization of wide ﬁeld X-ray telescope, considering the angular resolution and effective area simultaneously. To improve focusing performance of large X-ray observatory, the size optimization of mirror segments for X-ray optics has been implemented by Zhao et al. [7]. With the method of ray tracing and ﬁnite element analysis considering the thickness, axial length, azimuthal span and mass density. Obviously, conventional single-objective optimization methods are always employed to optimization the grazing incidence X-ray telescope, which may obtain a local or a subjective optimal result that far from the engineering tradeoff solutions. In engineering design, for a designer, it is important to know a number of optimal solutions that gives lot of insight into the design thereby providing lot of feasible and useful design solutions. Over the past decades, a number of multi-objective evolutionary algorithms (MOEAs) have been suggested [8]. The NSGA-II is adopted in this paper to implement multi-objective optimization of the focused X-ray telescope. Meanwhile, the influence of multivariate on the effective collection area is analyzed based on data mining method. Using mathematical statistics to analyze the distribution characteristics of focal length optical aperture ratio and effective collecting weight ratio, this is expected to summarize the theoretical basis for similar design optimization. Finally, ﬁve sets of optimization solutions were selected considering the development cost and capability, and then the correctness of the optimization results was veriﬁed by simulation.

2 Focusing X-Ray Telescope FoXT is a complex product involving multiple disciplines such as opto-mechanical, electronics and thermal, which includes Wolter-I optical system, detector, electronics and structure, as shown in Fig. 1. Based on the total reflection theory, the Wolter-I

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optics reflect and focus the X-ray photons onto a small detector. The high efﬁciency of FoXT and the small size of the detector’s size help to minimize background radiation, allowing for a relatively high signal-to-noise readout.

Fig. 1. Focusing grazing incidence X-ray telescope

(1) Focusing X-ray optics. The focusing X-ray optics is made up of concentric foil mirrors. The mirrors concentrate X-rays onto a small detector area using a single bounce, in contrast to typical imaging X-ray optics that require two bounces and thus incur a signiﬁcant efﬁciency penalty to achieve quality imaging. Unlike the typical Wolter-I, the FoXT is not imaging optics, X-rays undergo a single reflection. The absence of secondary mirrors increases efﬁciency and decreases weight and assembly complexity. (2) Detector. The silicon drift detector is employed in this paper, which has very high quantum efﬁciency over the photon energy range of interest. Due to the fact that the detector is read out by dual-channel electronics chains, therefore, both high time resolution (100 ns) and excellent spectral resolution (150eV) with very low dead time. In addition, the SDD include integrated thermoelectric coolers (TECs) and thermal/optical-blocking ﬁlters. SDDs offer energy resolutions typical of silicon-based detectors, approaching the Fano limit. (3) Electronics subsystem. The electronics subsystem includes the detector readout, slow and fast shaper, peak detect, event logic and power electro circuit, which aims at obtaining high signal-to-noise ratio in dealing with the faint X-rays radiated from pulsars. (4) Mechanical structures. The mechanical structures of FoXT are composed of mirror structures, optical bench and flange, which holds the optical and detector systems in a compact, co-aligned assembly, providing mechanical rigidity, thermal stability.

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3 Multiobjective Optimization Model 3.1

Design Variables and Parameters

Large effective collecting area can be achieved by nesting multiple co-aligned, co-axial grazing-incidence mirror pairs in order to optimize the available aperture. Consequently, large-effective-area X-ray telescopes must use thin, lightweight mirrors to achieve a high degree of nesting and acceptably low weight. In addition, the focal length and weight should be considered during the design and development simultaneously due to the some exhibitions of the rocket. As a result, both minimum of reciprocal of area (f1) and weight (f2) are chosen as the optimization objectives. The design variables and parameters are shown in Tables 1 and 2. Table 1. Design variables of FoXT w t h n L* Rd [120, 150] [3, 5] [0.3, 0.5] [2, 4] [1.0, 1.5] [5, 30] *L: Length of mirrors (mm) Rd: Radius of detector (mm) w: Thickness of mirrors (mm) t: Thickness of tube (mm) h: Grazing incidence angle (°) n: number of multilayer mirrors

Table 2. Design parameters of FoXT qtube FOV* h qm 0.125° 0.6 [0.3, 0.5] [2, 4] *FOV: Field of view h: Reflectivity (mm) qm: Density of Ni qs: Density of Al

3.2

Optimization Model

In order to formulate the multi-objective function, we give the schematic of multi-layer nested Wolter-I mirrors that arranged from inner to outer in Fig. 2. The ri1 denotes the inner radius of the ith bottom layer, whereas the ri2 represents the inner radius of the ith upper layer.

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r12 r22

L

w

r21=r12-w r11

w

Fig. 2. The schematic of multi-layer nested X-ray mirrors

3.2.1 Effective Aera All the income X-ray photons from the FOV of FoXT should be focused on the small detector after reflected by the Wolter-I mirrors. Therefore, the focal length of FoXT can be calculated by the Eq. (1) when both the radius of detector and FOV are known. f ¼

Rd tanðxÞ

ð1Þ

The focusing X-ray mirrors is a single reflection focusing mirror, because it is used to collect the number of focused X-ray photons only, but not imaging. As a result, the X-ray mirrors includes a parabola mirror, the surface formula of which can be depicted by Eq. (2). y ¼

1 rci x2 2rci 2

ð2Þ

where, rci is the curvature radius of each nested mirror, i denotes the number of nested layer. Generally, the surface formula of parabola can be simpliﬁed as Eq. (3) in engineering design, because the curvature radius are always tiny. y ¼

x2 2rci

ð3Þ

When x equals to ri1, the function value of Eq. (3) is just the focal length. ri1 ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2rci f

ð4Þ

And also, the ri2 can be derived when the length of mirror (L) is known, which can be expressed as following.

Design and Optimization of FoXT Based on Intelligent Algorithm

ri2 ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2rci ðf LÞ

315

ð5Þ

Obviously, there is a recursive relationship between ri1 and ri2, shown in Eq. (6). ri2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ L ¼ ri1 1 f

ð6Þ

Meanwhile, the FOV of FoXT should not be obscured by the nested mirrors. We can derive the following relationship between any two layered nested mirrors. rði þ 1Þ;2 ¼ rðiÞ;1 w

ð7Þ

As a result, the total effective collecting area of FoXT which has n layered nested mirrors can be represented by Eq. (8). 8 n < Aeff ¼ h P pr 2 r 2 i1 i2 i¼1 :f ¼ 1 1

ð8Þ

Aeff

3.2.2 Weight In this paper, only the weight of mirrors and structures are considered. f2 ¼ Min Woptics þ Wtube

ð9Þ

where, f2 is the weight of FoXT, Woptics is the weight of multi-layer nested mirrors, Wtube denotes the weight of optical tube. According to [9], the ratio of effective area and weight of nickel (twice reflected Xray optics) is about 10. Since the single reflection of X ray optics is adopted in this paper, the ratio of effective area and weight of which can be viewed equal to 33.3, which the reflectivity of X-ray optics is 0.6. So, the total weight of X-ray optics can be calculated by the following Eq. (10). Woptics ¼

n 2 3 X 2 g p ri1 ri2 100 i¼1

ð10Þ

The mirror structure is a cylinder minus a circular truncated cone, the weight of which can be derived by Eq. (11). h i Wtube ¼ pf qtube ðRto þ tÞ2 R2to

ð11Þ

Where, Rto is the inner radius of mirror structure, t is the thickness of tube. As a result, the multi-objective optimization model of FoXT can be formulated as Eq. (12).

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Min : ðf1 ; f2 Þ 8 1 f1 ¼ P > n > > h p < ðr2 r2 Þ i1

> > > : f2 ¼

i¼1

3 100 h

i2

h

n 2 P 2 p ri1 ri2 þ pf qtube ðRto þ tÞ2 R2to

i

ð12Þ

i¼1

4 Nondominated Sorting Genetic Algorithm II 4.1

Main Idea

With the aim of minimizing the weight and reciprocal effective area of FoXT, the Nondominated sorting genetic algorithm-II (NSGA-II) is adopted to obtain the Pareto frontier. With the elitist strategy, the NSGA-II sorts the set of solutions according to their ﬁtness (or objective values) and then selects the best solutions as parents for the next generation in every generation. Non-dominated Pareto frontier at each generation are found. The main characteristic of NSGA-II includes: (1) the computational complexity is reduce dramatically with the adoption of ranging strategy-based fast non-dominated sorting method. (2) the elitist preserve strategy is adopted to make sure the paternal progeny population have competition capability, and thus avoid of losing the non-dominated solution. (3) using a module of multi-dimensional vector to avoid local minima. 4.2

Procedures

The procedure of NSGA-II is shown in Fig. 3.

Initialization of DV Setting Pareto MOOP parameters

Create/Update current Pareto frontier

Stopping criteria is met? Yes

No

Create new population by genetic manipulation

Output the final Pareto frontier End

Fig. 3. The procedures of NSGA-II

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The steps of procedures are as followings: Step1: Initialization. Setting the design parameters of FoXT. Step2: Pareto optimization. Setting the parameters of population type, number of population, iterations, cross function, Pareto front coefﬁcient, and then implementing the global optimization. Step2.1: Select the reasonable ﬁtness of individual distribution, facilitating the continuation of optimization and near to the Pareto front gradually. Assign the ﬁtness for each individual and thus they have individual replication probability. The calculation of ﬁtness can be expressed as Eq. (13). Fitn V ð pÞ ¼ 2 ps þ 2ðps 1Þ

p1 Mc 1

ð13Þ

where ps is a selecting coefﬁcient, which has the range of values [1, 2], p is the position of individual in the ranking population, FitnV(p) is the ﬁtness of the individual at the p position, Mc represents the number of individual in the population. Step2.2: Elitist preserve strategy. Select Nc individuals that ranked ahead of the elitist storage library according to the ranking result of Step2.1. After the genetic manipulation of the residual populations, the new produced elitist will be combined with the selected Nc individuals, which are viewed as the new populations that to be implemented with the genetic algorithm. Step2.3: To improve the efﬁciency of genetic algorithm, it must try to generate as diverse a population as possible in order to avoid local minima. The modiﬁed NSGA II uses a module of multi-dimensional vector, the individual ﬁtness value of which is viewed as the wash out rule. The individual ﬁtness value can be expressed as Eq. (14). qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Fit V ðYi Þ ¼ kfi k ¼ ðfi1 ð xÞÞ2 þ ðfi2 ð xÞÞ2 þ þ ðfin ð xÞÞ2

ð14Þ

where i and j are two different individuals, when there are lots of objectives have the same ranking number, the minimum distance among them should be calculate accordingly through Eq. (13). The individual will be washed out if the individual ﬁtness value of which less than the given value. This method resolves the problem that when lots of individuals have the same ﬁtness value and ranking order, in which it is difﬁcult to evaluate how to wash out individual according to the ﬁtness value only. Namely, when any two individuals in the same ranking and also the distance between them is smaller enough than the given value, the ﬁtness of each module of multi-dimensional vector can be viewed as the evaluation criterion. Step2.4: Setting the NSGA-II optimization parameters, in which includes the population, selection, reproduction, mutation, crossover, migration and generations. The Pareto solutions can be obtained after implementing the NSGA-II. Step3: End.

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5 Optimization and Data Mining 5.1

Multiobjective Optimization

The design variables and parameters are shown in Tables 1 and 2. The optimization parameters of NSGA-II are as follows: Population (200), iterations number (300), the scattered is employed in crossover function, the Pareto front population fraction is 0.7, and selecting coefﬁcient is 1.5. The Pareto solution of FoXT is shown in Fig. 4. Totally, there are about 350 arrays solutions. 12

Weight W/kg

10 8 6 4 2 0

0.005

0.01 0.015 0.02 1/Effective Aera (1/cm2)

0.025

0.03

Fig. 4. Pareto solutions of FoXT

It is can be seen from Fig. 4, both the weight and the reciprocal of effective area are conflicted optimization objectives. 210 Pareto solutions have been obtained from this optimization, some of which are shown in Table 3. Obviously, both of the two optimization objectives have order of magnitude difference, which provide more chances for designers and engineers to select personal preference solution. Table 3. Pareto frontier L/mm 149.9998 120.0015 147.8684 149.9906 149.9753 149.9755 149.9918 147.8679 149.9995 149.9991 149.9997 144.1223

Rd/mm 3.0000 3.0119 3.0059 3.0083 4.6033 3.3675 3.7808 3.1861 4.1387 3.0009 3.4827 3.0000

w/mm 0.3001 0.3001 0.3157 0.3028 0.3000 0.3001 0.3001 0.3000 0.3000 0.3000 0.3000 0.3001

t/mm 2.0002 2.0000 2.0000 2.0065 2.0000 2.0001 2.000 2.0000 2.0000 2.0237 2.0000 2.0000

h/° 1.4059 1.2000 1.2036 1.2970 1.4941 1.4903 1.4910 1.4915 1.4999 1.4687 1.4999 1.2000

n 29 18 18 24 30 30 29 29 30 18 29 5

A/cm2 125.4304 77.9245 84.2729 104.4371 315.9749 176.8945 220.0687 158.6052 263.1243 126.3477 191.0850 43.453

W/kg 4.1976 3.4570 3.5433 3.8764 9.9467 5.5165 6.8612 4.9591 8.1779 4.2532 5.9132 3.1758

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Data Mining and Analysis

Cluster analysis, correlation analysis are adopted in this paper to mining the optimization results. The statistical characteristics of focal, diameter of X-ray optics, ratio of effective area and weight, and the law of influence some design variables exerted on the multi-objective have been researched. 5.2.1 The Influence Design Variables on Effective Area According to Eq. (8), the effective area of FoXT is closely related to the aperture and length of X-ray optics, number of nested layers. (1) Length of X-ray optics The Pareto solution that have the approximately same values are selected to implement the data analysis, and with the hope of ﬁnding some law of influences. The function shown in Fig. 5 can be ﬁtted according to the optimization solutions. The maximum error, minimum error and mean square error of ﬁtting errors are 0.0203 mm, 0.01887 mm and 0.009151 respectively. yAeff ¼ 0:0024x2L þ 0:9352xL þ 87:2141

ð15Þ

Effective Aera of X-ray Optics (A/cm2)

174 172 170 168 166 164 120

125

140 145 130 135 Length of X-ray Optics (L/mm)

150

Fig. 5. Length of X-ray optics and effective area

(2) Diameter of X-ray optics Through statistical analysis of optimized data, we can see that, the aperture of the X-ray optics is proportional to the product of the detector radius and the tangent value of grazing incidence angle. The law of change is shown in Fig. 6. The larger the effective area of each layer corresponding to the large diameter, and the larger the aperture of the optical system, the more the number of nesting layers. If the lengths of the mirrors are the same, the effective collecting area is larger.

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Diameter of X-ray optcis(mm)

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250 225 200 175 150 125 100 1.5 1.45

1.4 1.35

1.3 1.25

1.2 1.15

Grazing incidence angle (°)

1.1

3

3.25

3.5

3.75 4

4.25 4.5

4.75 5

Radius of detector (mm)

Fig. 6. Influence of detector radius and grazing incidence angle on optical aperture

Number of nesting layers: set the other parameters to be the same, the optical aperture selects the maximum and minimum values of the optimized solution are 240.2038 mm, and 115.2672 mm. The variation of effective collecting area from nesting layer from 5 to 30 layers is shown in Fig. 7. There is a common law between the optical effective area and the number of nested layers, which is the effective area gradually increases with the number of nested layers, and the increasingly rate is becoming from faster to slower, and gradually becomes gentle. This is mainly because the outermost optical aperture is large, and the effective collecting area of each layer contributes greatly, and as the optical aperture from the outside is gradually reduced, the effective area of the single layer is increasingly reduced. In the case of the same number of nesting layers, as the optical aperture increases, the effective area increases sequentially, with increases of 50.34%, 50.92%, 53.27%, 57.22%, 60.46%, 63.21%, and 65.19%, respectively.

Effective Aera of X-ray Optics

400 350 300 250

D=240.2mm D=219.3mm D=195.8mm D=173.0mm D=152.2mm D=130.2mm D=115.3mm

200 150 100 50 0 0

4

8 12 16 20 Cases of Pareto Frontiers

24

28

Fig. 7. The influence of the number of nesting layers on the effective area

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5.2.2 Ratio of Focal Length and Effective Area According to the statistical analysis of 210 sets of disaggregation, the ratio of focal length to optical aperture is statistically signiﬁcant and the distribution is signiﬁcant, mainly distributed in the following three intervals (as shown in Fig. 8). (1) 9 Df 10 : 174 (2) 10 Df 11 : 10 (3) Df 11 : 26

12

Focal/Diameter of X-ray optics

11 10 9 8 7 6 5 4 3 2 1 0 0

30

60

90 120 Pareto frontier

150

180

210

Fig. 8. Distribution of focal length to optical aperture ratio

It can be seen that before designing the focused X-ray telescope, the focal length of FoXT can be determined according to the launch rocket envelope or the spacecraft system, and the focal length and optical aperture of the telescope can be initially determined, and the selection can be basically between 9 and 11. In addition, by deﬁning other parameters, statistical analysis shows that the ratio has a reciprocal relationship with the grazing incidence angle. Therefore, in the design of the scheme, the reflectance angle of the different X-ray energy segments can be determined according to the requirements, and the focal length and optical are determined. 5.2.3 Ratio of Effective Area and Weight According to the Pareto optimization results, there are differences in the magnitude of the effective area and weight optimization values. In the ﬁeld of optical telescope design, the performance of the product is often measured by the aspect ratio. To this end, the ratio of the effective detection area to the weight is analyzed as a statistic, and the statistical result is shown in Fig. 9.

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Effective Aera/ Weight

30 25 20 15 10 5 0 0

30

60

90 120 150 Pareto Frontiers

180

210

Fig. 9. Effective area and weight ratio distribution

As can be seen, the corresponding effective area and weight ratio is the largest when f/D belong to [9 10], the minimum value is 31.0266, the maximum value is 32.3156, and the average value is 32.0322. When f/D becomes to greater or equal to 11, the minimum value is 18.2538, the maximum value is 23.7834, and the average value is 20.2247. Through in-depth analysis, it can be seen that the X-ray mirror length and the grazing incidence angle corresponding to the design scheme with relatively large surface weight basically reach the maximum value of the design interval, and the number of Pareto optimization solutions conforming to this type of distribution is 174 groups. 5.3

Decision-Making

How to select the design program that meets the requirements of engineering development from the above three types of Pareto frontier solutions is the key. It is well known that X-ray optical reflection efﬁciency is directly related to the grazing incidence angle. Generally, the grazing incidence angle when the product of the surface reflectance and the grazing incidence angle is maximum is the optimum average grazing incidence angle. Therefore, the optimal average grazing incidence angle of the X-ray optical system of the 0.2 keV–12 keV energy section is about 1.25° [9]. According to the analysis of 210 sets of optimization solutions, there are 26 sets of solutions satisfying the grazing incidence angle constraint, that is, the Pareto optimal solution corresponding to f/D 11, the focal length is basically between 1370 mm– 1380 mm, and the optical aperture range is 115 mm–125 mm. From the perspective of engineering development, the design scheme with relatively large surface weight has certain advantages. Therefore, this paper selects 5 competitive design schemes in 26 sets of optimization solutions, as shown in Table 4.

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Table 4. Selected ﬁve Pareto results Pareto frontiers h(°) Num. Aeff(cm2) 1 1.20 5 37.46 2 1.23 7 59.20 3 1.26 24 98.19 4 1.26 18 94.01 5 1.29 24 104.43

Weight(kg) 3.14 3.37 3.74 3.69 3.83

L(mm) 120.00 149.96 149.99 149.99 149.99

D(mm) 115.27 118.41 121.23 121.66 124.94

f(mm) 1375.10 1376.83 1375.18 1377.00 1378.89

Considering the difﬁculty and cost of engineering development (ultra-precision manufacturing and assembly), the design is carried out with the fourth group of optimization solutions. Table 5 shows the large and small port diameters of the nested 18-layer system and their parabolic curvature radius. Figure 10 is a simulation of optimized FoXT. Table 5. Optical Aperture Pareto frontier (i = 1, 2, 3…18) ni 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

ri1/mm 60.8255 57.1168 53.6161 50.3115 47.1921 44.2475 41.4679 38.8441 36.3673 34.0293 31.8224 29.7391 27.7726 25.9162 24.1639 22.5098 20.9484 19.4745

ri2/mm 57.4167 53.9161 50.6115 47.4921 44.5476 41.7679 39.1441 36.6674 34.3294 32.1224 30.0391 28.0726 26.2162 24.4639 22.8098 21.2483 19.7744 18.3831

Fig. 10. Simulation of optimized FoXT

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6 Conclusions (1) A multi-objective optimization method based on intelligent algorithm for FoXT is proposed. Based on X-ray total reflection theory, a multi-objective optimization model of FoXT is constructed. The non-dominated sorting genetic algorithm (NSGA-II) is used to obtain Pareto solutions. (2) Data mining methods such as cluster analysis and association analysis are used to explore the implicit correlation characteristics and variation laws between design parameters and optimization solutions. The effects of optical lens length, optical aperture, nesting layer and other parameters on the effective area are revealed. The distribution characteristics of focal length and optical aperture ratio and surface weight ratio are statistically analyzed. It is concluded that the optical aperture is proportional to the product of the detector radius and the grazing incidence angle. (3) According to the data mining and analysis results, ﬁve sets of optimization schemes with design competitiveness were selected, and one of them was designed and implemented. The number of nesting layers was 18, the aperture was 121.65 mm, the focal length was 1377 mm, and the effective area and weight were 94.01 cm2 and 3.69 kg respectively.

References 1. Sheikh, S.I., Pines, D.: Spacecraft navigation using X-ray pulsar. J. Guid. Control Dyn. 29(1), 49–63 (2006) 2. Li, L.S., Mei, Z.W., Lv, Z.X., et al.: Grazing incidence focusing X-ray pulsar telescope and analysis of in-orbit observation data. J. Ordnance Equip. Eng. 38(12), 175–179 (2017). (in Chinese) 3. Mao, P.H., Bellan, L.M., Harrison, F.A., et al.: Evaluation and optimization of multilayer design for astronomical X-ray telescope using a ﬁeld-of-view and energy-dependent ﬁgure of merit. In: Proceedings of the International symposium on Optical Science and Technology, Xray Optics, Instruments and Missions IV, pp. 126–133 (2000) 4. Biskach, M.P., Mcclelland, R.S., Saha, T., et al.: Size optimization for mirror segments for Xray optics. In: International Society for Optics and Photonics, pp. 814711–814711-9 (2011) 5. Mitsuishi, I., Ogawa, T., Sato, M., et al.: Ray-tracing simulations for the ultra-lightweight Xray optics towards a future Jupiter exploration mission. Adv. Space Res. 57(1), 320–328 (2016) 6. Zhao, D.C., Chen, B., Liu, P., et al.: Image quality evaluation of Wolter X-ray nested telescope. Acta Optica Sinca 36(3), 84–90 (2016). (in Chinese) 7. Conconi, P., Pareschi, G., Campana, S., et al.: Design optimization and trade-off study of WFXT optics. In: Proceedings of the SPIE-the International Society for Optical Engineering, Optics for EUV, X-Ray, and Gamma-Ray Astronomy IV, San Diego, pp. 1–10 (2009) 8. Kalyanmoy, D., Amerit, P., Sameer, A., et al.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002) 9. Li, L.S., Deng, L.L., Mei, Z.W., et al.: Pareto-based multi-objective optimization of focusing X-Ray pulsar telescope and multi-physics coupling analysis. J. Mech. Eng. 54(23), 174–184 (2018). (in Chinese)

Research of Dormitory Furniture Design Based on Group Interaction Lin Li(&), Xupeng Wang, and Chunqiang Zhang Department of Industrial Design, Xi’an University of Technology, Xi’an 710054, China {lilin15158,wangxupeng}@xaut.edu.cn, [email protected]

Abstract. In order to improve Chinese university students’ life and learning quality through the design and layout of dormitory furniture in a limited space, a dormitory furniture design concept for university students based on group interaction is researched in this paper. Firstly, the current relevant data in China of student dormitory furniture arrangement collected based on ﬁeld visits, online surveys, and conducts scientiﬁc analysis in order to summarize both the main problems of living environment and dormitory furniture, as well as the main factors that affect the needs of group communication in the dormitory. Secondly, a design concept for college dormitory furniture is introduced to meet the needs of group communication. Finally, two dormitory furniture design schemes are used as example to validate the design concept proposed in this paper. Keywords: Dormitory furniture design Spatial function region Group interaction Communication space Design concept

1 Introduction As an important place for college students to study and live, the dormitory has important application signiﬁcance and theoretical value. The choice of accommodation for foreign college students is diversiﬁed because except student residences, many students choose to rent out of school. And the number of student living in the dormitory is small, the living environment is relatively relaxed and comfortable, so students have higher satisfaction with dormitory furniture. Foreign scholars’ research on these area mainly refer to: (1) Research on classroom desks and chairs based on anthropometric analysis and ergonomics [1–3]. (2) Dormitory space design based on indoor physical environment of dormitory (such as indoor temperature, humidity, illumination, etc.) and interpersonal relationship [4]. (3) Renovation of old dormitory space that takes advantages and controls disadvantages [5]. (4) Evaluation of dormitory facilities from a technical and functional perspective to promote the improvement of dormitory facilities [6].

This project is supported by MOE (Ministry of Education in China). Project of Humanities Social Sciences (Project NO. 14YJCZH199). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 325–341, 2020. https://doi.org/10.1007/978-981-32-9941-2_28

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In China, student dormitory is almost the only accommodation choice for college students. The ratio of student number and dormitory area is large, and the per capita accommodation area is small. Therefore, how to improve college students’ life and learning quality through the design and layout of dormitory furniture in a limited space is particularly important and urgent. At present, the domestic scholars’ research on dormitory furniture is mainly to solve the problems of living and storage. The structure, function, man-machine size and storage space of the furniture were discussed [7–9], and some design methods of dormitory furniture were put forward. From the research of Yu, it can be concluded that according to the students’ behavioral habits in the dormitory and the anthropometric analysis, the function of furniture is reﬁned, design details are concerned, and the humanized design idea is embodied [7]. Yang studied the relationship between furniture and space, and concluded that the variability and applicability of the dormitory function is improved by the combination of different furniture, or the space separation by the soft materials for home decoration [8]. Through the analysis of the human status data when using furniture, the ideal humanmachine size is obtained, which is applied to design practice [9]. The satisfaction of individual psychological needs in dormitory has an important influence on the students’ personal emotions, interpersonal communication with roommates, as well as construction of the dormitory culture. At present, personal privacy requirements have been paid more attention to the study of dormitory furniture design [10], and the author has explored a design concept of dormitory furniture based on privacy requirements [11]. The satisfaction of the group interaction needs is based on the communication space in the dormitory, and the research on the communication space is mainly from the perspective of dormitory space planning and design [12–14]. Although dormitory furniture is an important way to realize the dormitory space planning, there is almost no relevant literature published refers to the relationship between dormitory furniture and communication space. The paper intends to explore a design concept of college dormitory furniture, which meets the group interaction and communication needs. The research framework is as follows. In Sect. 2, the current situation of dormitory environment and dormitory furniture are comprehensive understood by ﬁeld survey and network investigation. The main problems in the living environment, dormitory furniture and the main factors affecting the group interaction needs in the dormitory are determined by questionnaire research. In Sect. 3, the design concept of dormitory furniture based on the needs of group interaction and communication is summarized. Two dormitory furniture design cases are put forward to verify the effectiveness of the research process in Sect. 4. Finally, the conclusion of this study is drawn in Sect. 5.

2 Investigation of University Dormitory Environment and Dormitory Furniture in China The investigation consists of two parts: (1) The current situation of dormitory environment and dormitory furniture are comprehensive understood by ﬁeld survey and network investigation. According to the regional distribution, 32 colleges and universities in China were selected for survey research. The research team found that many

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students are dissatisﬁed with the dormitory living environment, and the insufﬁcient function of dormitory furniture is very prominent. (2) According to the previous survey research, the online questionnaire survey was used to further clarify the main problems of dormitory furniture, as well as the main factors affecting group communication in the dormitory. 2.1

Questionnaire Design

Dormitory space, dormitory furniture, and furniture users are closely connected and interact with each other. Therefore, the dormitory furniture design questionnaire is carried out in three aspects: living environment, status of dormitory furniture and the effect of dormitory furniture on group interaction in the dormitory. The questionnaire has ﬁve parts, and a total of 18 questions as depicts in Table 1. Table 1. Questionnaire content Number of questions Part 1 3 Part 2 5

Questionnaire content

Basic information of questionnaire respondents Dormitory living environment, dormitory life satisfaction, dormitory function effectiveness Part 3 3 Dormitory furniture type, dormitory furniture satisfaction, concrete problems of dormitory furniture Part 4 5 Demand degree on group interaction and communication, factors affecting the interaction between roommates, forming elements of communication space Part 5 2 (open questions) In-depth information on the insufﬁciency of dormitory furniture, ideal dormitory life

2.2

Data Collection

A questionnaire was sent out on the questionnaire website, and a total of 265 questionnaires were retrieved, of which 257 were valid ones. The basic information of the sample is shown in Table 2.

Table 2. Sample basic information (N = 257) Variable Option Frequency Proportion Gender Male 85 33.2% Female 172 66.8% Grade Freshman 57 22.27% Sophomore 64 24.61% Junior 57 22.27% Senior 62 24.22% Master student 17 6.64%

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Samples from all over the country, as shown in Fig. 1.

Fig. 1. Sample source basic information

2.3

Data Analysis

2.3.1 Reliability Analysis The Cronbach’s coefﬁcient method [15] was adopted for the reliability analysis, which is calculated as follows: P 2 k s a¼ ð1 2 i Þ k1 sx

ð1Þ

where, k is the number of test questions, s2i indicates the variance of the test results on a certain scale, and s2x indicates the total variance of all test results. Quantifying 257 questionnaires and using them as sample data into SPSS software for reliability analysis, the results can be obtained as following: Table 3. Reliability statistics Cronbach’s Alpha Cronbach’s Alpha based on Standardized item 0.675 0.675

It is clear as shown in Table 3 that the Cronbach’s Alpha is 0.675, which is close to 0.7, it means that the questionnaire has high internal consistency, high reliability, and the results are credible and effective. 2.3.2 Validity Analysis The validity analysis was carried out by the KMO sample measurement method and the Bartlett spherical test method [15]. The results are as follows: Table 4. Inspection of KMO and Bartlett Kaiser-Meyer-Olkin metric Sampling enough Bartlett’s sphericity test Approximate chi square Approximate chi square Approximate chi square

0.724 2375.7177 780 780

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As depicts in Table 4, the KMO value is 0.724, and the signiﬁcant level value of the chi-square statistical value of the Bartlett spherical test is 0.000 < 0.01. Both of them have reached the test standard of factor analysis, indicating that the sample data selected in this study is more suitable for factor analysis. 2.4

Analysis of the Relationship Between Dormitory Furniture and Student Dormitory Life Satisfaction

In this paper, H0 represents no signiﬁcant impact on dormitory furniture and student dormitory satisfaction, and H1 represents a signiﬁcant impact on dormitory furniture and student dormitory satisfaction. The statistic formula is: F ¼

SA =s 1 Fðs 1; n sÞ SE =n s

ð2Þ

where, SA is the sum of the squares of the regression, SE is the sum of squared errors. The speciﬁc analysis is as follows: Table 5. Analysis of furniture type and dormitory life satisfaction Furniture type

Number of cases

Average value

Standard Standard deviation error

95% conﬁdence Minimum interval for the value mean Lower Upper limit limit

Maximum value

Bunk beds, tables and cabinets are independent Upper bed, lower table, cabinet, integral Beds, tables, cabinets are independent Other Total

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3.80

.886

.077

3.65

3.95

1

5

84

3.02

.776

.085

2.86

3.19

1

5

16

3.56

.727

.182

3.17

3.95

2

4

24 256

4.04 3.55

.690 .905

.141 .056

3.75 3.44

4.33 3.66

3 1

5 5

As depicts in Table 5, 1 indicates that it is very satisﬁed, 2 expressed satisfaction, 3 indicates basic satisfaction, 4 indicates dissatisfaction, and 5 indicates that it is very dissatisﬁed. It can be seen from Table 6 that the signiﬁcance level is 0.279, which is greater than 0.05. Therefore, the assumption of the homogeneity test of variance is established, and the results of one-way ANOVA (Analysis of Variance) are valid and credible.

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From Table 7, it can be seen that the signiﬁcance level is 0.000, less than 0.05, rejecting H0, accepting H1, which indicates that the improvement of dormitory furniture design plays an important role in improving student dormitory life satisfaction. Table 7. One-way ANOVA Satisfaction Sum of square Degree of freedom Mean square F Signiﬁcance Between groups 37.174 3 12.391 18.188 0 Within groups 172.367 253 0.681 Total 209.541 256

2.5

Investigation and Analysis of College Dormitory Environment and Dormitory Furniture

From Figs. 2 and 3, it can be concluded that the problems existing in the current dormitory of Chinese universities are mainly concentrated in the following four aspects: (1) The dormitory space is small, the living environment is crowded, and the storage space is insufﬁcient, as well as the items are piled up and messy. (2) Dormitory space planning focuses on the realization of living functions, while social functions are neglected, which is not conducive to promoting the development of dormitory community activities. (3) The lack of dormitory furniture, insufﬁcient variability and combination of furniture are difﬁcult to meet the multi-level needs of dormitory members. (4) Dormitory furniture is old-fashioned and monotonous in color, which cannot meet the aesthetic needs of young people, nor can it adapt to the pace of modern campus construction. Crowded 200 150 Other shortcomings

100

Unreasonable spatial planning

50 0

Disordered item storage

Roommates don't get along well

Simple dormitory facilities

Fig. 2. Problems in the dormitory environment

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Few types of furniture 200 150

Other shortcomings

Lack of functional diversity

100 50 0 Insufficient storage space

Inappropriate size of furniture

Monotonous color

Outdated shape

Fig. 3. Problems with dormitory furniture

The above problems have seriously affected the achievement and satisfaction of the function for dormitory furniture. Especially the spatial planning and furniture conﬁguration are mainly focused on the realization of daily living functions, and neglect the needs of group communication in the dormitory. 2.6

Main Factors Affecting the Needs of Group Communication in the Dormitory

As illustrated in Figs. 4 and 5, the main factors which affect the development of group communication activities in the dormitory are the following four aspects: (1) Dormitory space planning is mainly based on sleeping space, learning space and storage space, but social space is not included in the planning system. (2) Dormitory furniture is conﬁgured to achieve daily living functions, whereas, casual furniture that meets the needs of group communication rarely appears. (3) Dormitory furniture is used in a single way, and the correlation between furniture is weak, which limits the expansion of dormitory functions. (4) Insufﬁcient storage space, messy living environment and dull dormitory atmosphere are not conducive to promote good interpersonal interaction for dormitory members.

Lack of communication space

150

other problems

100

Lack of leisure furniture

50 0 Different hobbies

Unharmonious roommate relationship

Single use of furniture

Dormitory atmosphere is dull

Fig. 4. Factors affecting the interaction between roommates

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100 50

Multifunctional furniture

0 Furniture accessories and decorations

Modular furniture

Dividing function space with color

Fig. 5. Construction method of the communication space

From Figs. 6 and 7, it can be concluded that the warm, harmonious, clean and orderly dormitory life is the yearning for students, and the dormitory furniture based on group interaction helps to promote good interpersonal interaction in the dormitory, as well as create a harmonious and orderly dormitory environment.

Fig. 6. Word cloud diagram of the deepening of dormitory furniture design

Fig. 7. Word cloud diagram of ideal dormitory life

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3 Dormitory Furniture Design Concept Based on Group Interaction and Communication Needs According to the research conclusions above, the activity of group interaction among roommates is influenced not only by psychological factors such as personal personality, hobbies et al., but also the dormitory environment. And effect of the dormitory environment on the group interaction activities in the dormitory is mainly reflected in the two aspects as follow, the one is the facility allocation, and the other is space atmosphere. It need to be highlighted that facility allocation includes communication space, leisure furniture et al., which provide a basic guarantee for the dormitory members’ group interaction and communication activities. The space atmosphere includes the cleanliness, aesthetics, comfort and other elements of the dormitory environment, which enhance the enthusiasm and the harmony of communication between the roommates. It is clear that dormitory furniture plays an important role on the layout of dormitory space, the realization of dormitory functions, the beautiﬁcation of indoor environment and the comfortability of residence. It can be concluded that in order to improve the enthusiasm and initiative of communication, and interaction among roommates, the aspects of dormitory facility allocation and space atmosphere should be considered seriously in design concept of dormitory furniture. 3.1

Design Concept of Dormitory Furniture Based on Living Environment and Usage Requirements

The dormitory leisure furniture design should suitable for the living environment and usage requirements. According to the previous investigation and research, it can be concluded as follow. (1) Living environment and usage requirements should be considered in the design of dormitory leisure furniture. And the dormitory leisure furniture should be small and simple, the structure should not be complicated, and the size should not be too large, as well as the number should be limited. (2) In order to provide the facilities for the group interaction activities, dormitory furniture design can be combined with other functions to bring more convenience to dormitory life. (3) The flexible usage expands the function of dormitory furniture, and creates a multifunctional space for both living and leisure in a limited space. It need to be noted that there is a small population and relatively large space in postgraduate dormitory, so the group interaction needs can be considered by setting up the social space and equipping the leisure furniture. Contrary to postgraduate dormitory, the undergraduate dormitory has small living space but large population. Therefore, it is useful to fully expand the usage of dormitory furniture of undergraduate through design methods, such as multi-functional design and modular combination, to meet the diverse living needs.

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Design Concept of Dormitory Furniture to Create Clean Environment and Harmonious Atmosphere

(1) Expanding the storage space and optimizing the storage method to create a clean and bright living environment. The dormitory storage space should be planned based on the user’s storage needs, and the type of items to be stored. While providing storage furniture with sufﬁcient storage space, it becomes possible to expand the storage space with high-altitude space and multi-functional storage furniture. (2) Furniture accessories, decorations and other items can emphasize the casual atmosphere of the interaction and communication space. Appropriate furniture accessories can expand the functional scope of the main furniture. Wall or space decoration can beautify the environment and enhance the harmonious atmosphere of the dormitory which is conducive to the development of personal mental health. (3) Appropriate form design, reasonable structure, and good color design can create a warm and comfortable dormitory atmosphere, which can promote the positive interaction between roommates It need to be highlighted that Chinese college dormitory furniture is mainly made of wood, with a simple shape and natural color. However, this style of furniture has been used for many years, lacking visual attraction and becoming difﬁcult to be recognized by young people. The indoor environment affects people’s emotional changes, and dormitory furniture plays an important role in shaping the dormitory environment. The fashionable style and pleasant color matching can meet the aesthetic needs of young people, and enhance the desire for communication between roommates.

4 Dormitory Furniture Design Examples Based on Group Communication Needs According to the design concept of dormitory furniture based on the group communication needs, two design examples are presented as follow. 4.1

Example 1: The Fusion of Movement and Static

According to the questionnaire survey, most students hope that the number of people living in a dormitory is less than four, and the dormitory space is spacious and bright. This example sets the dormitory area to 25 m2 and the number of residents is three. The design results not only provide reasonable solutions for daily dormitory life, but also meet the needs of group interaction among roommates. The style and color of the scheme should match the aesthetic needs of young people. 4.1.1 Dormitory Furniture Design In Figs. 8, 9, 10, 11, 12 and 13, the design contents of college dormitory furniture are shown based on the needs of group communication.

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Fig. 8. Dormitory furniture

Fig. 9. Top view of dormitory furniture

Fig. 10. Bed, desk and bookcase

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Fig. 11. Leisure table, lounge chair, shared bookshelf, wardrobe and shoe cabinet

Fig. 12. Leisure exchange area

Fig. 13. Rotating table and rack

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Design Features

(1) The spatial planning of “moving and quiet” is the basis for realizing the multilevel functional requirements of the dormitory. The dormitory space is divided into a static space area and a dynamic space area (as shown in Figs. 8 and 9). The static space satisﬁes the needs of individual activities, such as learning and rest, in which mainly places furniture such as beds, desks, and personal bookcases are placed. The dynamic space satisﬁes the collective communication and storage needs, mainly including leisure stools, leisure tables, shared bookshelves, wardrobes, shoe cabinets and other furniture. (2) The leisure furniture structure is simple and exquisite, economical and practical. The structures of lounge chair, small table and bookshelves are simple and delicate which not only provides convenience for interaction between students,but also can save costs and space (as shown in Fig. 11). (3) Small furniture accessories enhance the character of the space and create a relaxed and welcoming dormitory atmosphere. The design details reflect human care and enhance the quality of life in the dormitory. Two lounge chairs with storage functions, a small table, a TV set, and a row of shared bookshelves form a small and comfortable space for casual interaction (Fig. 12). The rotatable tabletop mounted on the side of the bed provides convenience for students to read or write in the bed (as illustrates in Fig. 13). (4) Reasonable storage forms a clean and orderly dormitory environment. The storage space of the dormitory furniture adopts a box structure, and the articles are stored inside the cabinet without being directly exposed to the external environment, thereby solving the problem of disordered personal items and making the dormitory environment orderly. The desk design of the foldable tabletop improves the space utilization and further improves the cleanliness of the dormitory environment (as depicts in Fig. 10). (5) The multi-functional storage furniture effectively utilizes the structural space of the furniture, expands the functional scope of the furniture, and expands the storage space. The four storage compartments arranged vertically on the side of the desk, which can store personal small items and can also be used as bed ladders (as depicts in Fig. 10). The combination of the leisure chair and the shoe cabinet in the leisure exchange area has both practicality and aesthetics while saving space (as shown in Fig. 12). 4.2

Example 2: Beating Melody

According to the survey and analysis, two styles of dormitory environment are yearn for by students, which are the fashionable and novel dormitory space, the warm and harmonious dormitory atmosphere. This example sets the dormitory area to 30 m2 and the number of students is four. The design scheme is novel and lively, and it can create a warm feeling of “home”. The usage of furniture should be flexible and varied to meet different needs.

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4.2.1 Dormitory Furniture Design In Figs. 14, 15, 16, 17, 18 and 19, the design contents of college student dormitory furniture are shown based on the needs of group communication.

Fig. 14. Combo desk

Fig. 15. Vertical cabinet

Fig. 16. Desk usage scenarios for group activities

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Fig. 17. Desk usage scenarios for personal events

Fig. 18. Overall environment of the dormitory

Fig. 19. Foldable bunk

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4.2.2

Design Features

(1) The new and lively style creates a comfortable and inviting living environment. The whole set of dormitory furniture uses circular shape as a modeling element in order to create a fashionable and modern space style. The combinable curved desk, the semi-transparent cabinet, and the simple and exquisite wall cabinet are all blended with the circular elements, which make people feel refreshed (as illustrates in Fig. 18), creating a relaxed and pleasant “home” atmosphere. (2) Functional variability provides a flexible and versatile way to use furniture. The combinable personal desk features an arc shape as shown in Fig. 14, which can be placed for different usage needs. When the desk is placed in a line, the users are staggered in position, which reduces mutual interference and facilitates the development of personal activities (as illustrates in Fig. 17). The four desks can also be combined into a large round table to provide a discussion and chat area for roommates (as depicts in Fig. 16). (3) The foldable bunk bed can be transferred into a sofa which creates a small leisure space in the dormitory with limited space. The upper bunk bed is foldable and the lower mattress is a pillow top mattress. When the upper bunk is folded up, the double bottom mattress of the lower bunk is unfolded, and the lower bunk becomes a simple and comfortable leisure sofa (as see in Fig. 19). (4) The semi-transparent cabinet separates the bed, but maintains the visibility between roommates which is convenient for communication. A hollow design method is used for the vertical cabinet between the two sets of beds (as illustrates in Fig. 15). The cabinet separates the beds on both sides, creating a relatively independent personal sleeping space. As one of the favorite ways of communication, dormitory symposiums are an indispensable part of college life in China. The hollow structure maintains the permeability of sight on both sides of the bed, which is conducive to bedtime communication between roommates.

5 Conclusions Based on the social function of the dormitory and the needs of group communication, a design concept and speciﬁc methods of dormitory furniture have been presented and discussed in this paper, and two design examples are used to validate the design concept. The main conclusions can be obtained as follows: (1) Through extensive investigation and research, the main factors that influence the living environment and dormitory furniture, as well as the community interaction in the dormitory have be summarized as follow: rare consideration of social space in dormitory space planning, less leisure furniture, weak correlation between furniture and insufﬁcient storage space for items. (2) The design concept of college dormitory furniture that meets the needs of group communication is proposed, which mainly includes ﬁve aspects as follow: rationally planning the functional division of dormitory space, strengthening the

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association and variability of furniture, increasing furniture accessories, and expanding storage space, as well as optimizing modeling and color design. (3) Based on the design concept proposed in this paper, two dormitory furniture design schemes “the fusion of movement and static” and “the melody of beating” are given, which show that the reasonable dormitory furniture design can not only promote the communication between roommates, but also improve other functions of the dormitory. It is need to be highlighted that, this paper puts forward the design concept of dormitory furniture from the perspective of psychological needs, but does not study the use comfort, durability, production process and cost of dormitory furniture. The sample data of the questionnaire is mainly based on Shaanxi Province, and the data of other provinces is relatively small. So the extensiveness of samples is relatively insufﬁcient. So these will be researched seriously in our future researches.

References 1. Asif, S., Qutubuddin, S.M., Hebbal, S.S.: Anthropometric analysis of classroom furniture used in colleges. Int. J. Eng. Res. Dev. 3, 1–7 (2012) 2. Qutubuddin, S.M., Hebbal, S.S., Kumar, C.S.: Anthropometric consideration for designing students desks in engineering colleges. Int. J. Curr. Eng. Technol. 3, 1179–1185 (2013) 3. Odunaiya, N.A., Owonuwa, D.D., Oguntibeju, O.O.: Ergonomic suitability of educational furniture and possible health implications in a university setting. Adv. Med. Educ. Pract. 5, 1–14 (2014) 4. Kılıcaslan, H.: Design of living spaces in dormitories. Procedia-Soc. Behav. Sci. 92, 445– 451 (2013) 5. Khajehzadeh, I., Vale, B.: Shared student residential space: a post occupancy evaluation. J. Facil. Manag. 14, 102–124 (2016) 6. Hassanain, M.A.: On the performance evaluation of sustainable student housing facilities. J. Facil. Manag. 6, 212–225 (2008) 7. Yu, L.: Humanized design of college dormitory furniture-A case study of Tsinghua university. Decoration 5, 96–99 (2018) 8. Yang, C.: Overall furniture design of college dormitory with outstanding space reconstruction. Art J. 6, 116–120 (2014) 9. Ma, G., Wang, M.: Research on man-machine scale of modular multi-functional furniture in college apartment. J. Shenyang Constr. Univ. (Nat. Sci.) 4, 734–740 (2014) 10. Li, M., Chen, S.: Research on humanistic design creativity of college student dormitory furniture. Packag. Eng. 22, 115–122 (2013) 11. Li, L.: Research on dormitory furniture design based on personal privacy requirements. Decoration 2, 136–137 (2018) 12. Wei, W., Liu, Y.: Interaction space design of college student dormitories. J. Anhui Inst. Archit. Technol. (Nat. Sci. Ed.) 4, 94–96 (2008) 13. Sun, Z., Wang, H.: Talking about the design of college student dormitories from the perspective of space communication. Shanxi Archit. 24, 37–38 (2006) 14. Cronbach, L.J.: Coefﬁcient alpha and the internal structure of tests. Psychometrika 16(3), 297–334 (1951) 15. Li, H., Jiang, H.: SPSS Data Analysis Course, pp. 251–252. The Posts and Telecommunications Press, Beijing (2012). (in Chinese)

Analysis and Extraction of Consumer Information for the Evaluation of Design Requirement Depending on Consumer Involvement Shipei Li, Dunbing Tang(&), Qi Wang, and Haihua Zhu College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China [email protected], [email protected], [email protected]

Abstract. In the product design, identiﬁcation of critical design requirements (DRs) is essential to the success of the product. Usually, DRs are rated according to CRs for allocating resources reasonably as well as improving consumer satisfaction. With the popularization of the Internet, consumers can involve in open design (OD) and describe their requirements freely. It is a challenge to rate DRs due to a great quantity of CRs that are expressed as free text in OD. In existing approaches, CRs are collected by the method of survey or questionnaire in the market, and it cannot amply reflect the CRs in OD. To this end, this paper focuses on analyzing CRs for rating DRs in OD. The CRs are captured and rated in OD by a group-organization approach and a fuzzy Delphi method. Based on the importance weight of CR, a machine learning approach and a fuzzy QFD method are proposed to rate DRs according to the expression of CRs in OD. A case study is presented to demonstrate the effectiveness of the method. Keywords: Open design Consumer requirement Group-organization Fuzzy QFD

Design requirement

1 Introduction As market competition increases and product life-cycles decrease signiﬁcantly, companies have been forced to develop new products for meeting the highly varied and rapidly changing consumer requirements (CRs). It can be argued that the products can be developed successfully by the full understanding of CRs [1]. Based on the sale network and market survey of the company, Li et al. [2] integrated analytical hierarchy process (AHP), Kano’s model, rough set theory, and scale method for evaluating the importance weight of CR. Considering both consumer preference and consumer satisfaction, Nahm et al. [3] proposed two sets of evaluating methods to analyze the This work was supported by National Natural Science Foundation of China (Grant Nos. 51575264, 51805253); Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 2018K017C); and Qinglan Project of Jiangsu Province of China. © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 342–353, 2020. https://doi.org/10.1007/978-981-32-9941-2_29

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importance and competitive advantage of CRs. Zheng et al. [4] put forward a weighted interval rough number method for rating CRs while the importance weights of consumers are considered. Dou et al. [5] obtained CRs using a fuzzy Kano model and benchmarking theory to enhance the consumer satisfaction of each product attribute. In these methods, interview and questionnaire were used to collect CRs. With the development of the Internet, consumers can participate in the product design process and express their requirements freely in open design (OD) [6]. The number of CRs is increasing as consumers continuously to involve in OD. The ability of identifying crucial CRs is considered as a signiﬁcant element in the product design process [7]. Traditionally, the methods of collecting and analyzing CRs are survey or questionnaire in the market, and they are inappropriate for dealing with a great quantity of CRs that are expressed freely [8]. In the product design process, it is essential to estimate the ﬁnal priority ratings of design requirements (DRs). The importance weights of DRs can be rated based on the weights of CRs and the relation intensity between CRs and DRs. Quality Function Deployment (QFD) is a widely used planning methodology for interpreting CRs into DRs. Normally, human subjective judgments are used as the inputs of QFD [9]. Owing to the vagueness and uncertainty of human judgment, the fuzzy evaluation method was widely used in rating DRs in QFD [10]. Chen, Fung, and Tang [11] applied a fuzzy weighted average method with fuzzy expected value operator to rate DRs in fuzzy QFD. Chen and Weng [12] used the fuzzy goal programming models to deduce the different fulﬁll levels of DRs considering the fuzzy coefﬁcients in QFD. Wang and Chin [13] applied a pair of nonlinear programming models and two equivalent pairs of linear programming models to evaluate the importance weights of DRs in fuzzy QFD. Wang [14] utilized a fuzzy-normalisation-based group decision-making approach to evaluate DRs, and their technical importance was aggregated in fuzzy QFD. Miao et al. [15] applied two uncertain chance-constrained programming models to determine the importance of DRs in QFD with the condition of minimizing the design cost and maximizing consumer satisfaction. In these methods, CRs were collected by survey or questionnaire, and it cannot understand CRs sufﬁciently from the consumer’s perspective. Additionally, the relationship between CR and DR was assumed or evaluated by designers manually, and it is hard to rate the DRs exactly using these methods due to a great quantity of CRs that are expressed freely in OD. Although there have been many methods for collecting and analyzing CRs, they are not suitable for dealing with a great quantity of CRs. In addition, most fuzzy evaluation methods are difﬁcult to rate the DRs exactly according to the CRs that are expressed freely in OD. In this paper, to evaluate DRs in QFD, CRs are collected and analyzed in OD by a group-organization approach and a fuzzy Delphi method. Then a machine learning approach and a fuzzy QFD method are proposed to rate DRs on the basis of the expression of CRs in OD.

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2 Process Model With the development of Internet technology, product design patterns are being challenged as consumers can involve in the process of OD. In OD, consumers can participate in product design process and describe their requirements freely. As more and more consumers involved in OD, an increasing number of CRs can be captured. Precise information about CRs is helpful to rate DRs accurately. Designers can distribute resources properly according to the importance weights of DRs in the product design process. To rate DRs, CRs are collected depending on consumers involvement in OD. The framework of the analyzing CRs for the evaluation of DRs is shown in Fig. 1. Initially, an OD platform is established by the Internet. Consumers can involve in product design to describe their requirements and evaluate CRs through the OD platform. Then, a group-organization approach is put forward to eliminate the evaluation values of CRs that are very different from others. Based on the processed evaluation data, a fuzzy Delphi method is used to rate CRs. Meanwhile, the relation intensity between CR and DR is identiﬁed by a machine learning approach. Finally, the importance weights of DRs are evaluated based on the weights of CRs and the relation intensity between DRs and CRs in fuzzy QFD.

Fig. 1. The framework of analyzing CRs for the evaluation of DRs in OD

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3 Processing Consumer Requirements in OD The consumer who participants in OD can describe the requirements freely and interact with others. As participants can exchange their opinions about the CRs with each other in OD, their initial requirements may change. Consumers who participant in OD expect to get their ideal product, and it is similar to the economic issues in negotiations. Nash equilibrium is widely used for dealing with the collective beneﬁcial relationship among participants [16]. The dynamic CRs in OD can also be analyzed using the Nash Equilibrium. Because consumers involve in OD to get their desired product, the similar CRs will tend to reach the common expectation with the increasing of their communication. 3.1

Collecting Consumer Requirement in OD

Though Internet, OD platform can be constructed for collecting CRs. Consumers with different age or background can describe requirements and communicate with each other in the OD platform. Because consumers are given the freedom to involve in OD, there will be a large number of CRs. The qualitative method is more suitable for extracting the most important information of CRs. Usually, the importance of CRs can be rated by consumers using numerical values. In OD, consumers can evaluate the CRs that they are interested in. As anyone who participates in OD can evaluate CRs, there may be someone who evaluates CRs maliciously. Therefore, a ﬁltering method is proposed for eliminating the abnormal values of the evaluation data. 3.2

Analyzing Consumer Requirement

Through OD platform, consumer can evaluate the CRs by using numerical values. The evaluation value of a requirement that is evaluated by the ith consumer is denoted as xi . To eliminate the abnormal values of the evaluation data, a group-organization approach (GOA) is proposed in this section. The evaluation values of CRs in OD can be organized into different groups on the basis of time series. The outliers of each group are ﬁltered by the GOA. In GOA, based on the similarity analysis approach given by Yang and Wu [17], the similarity relation S xi ; xj can be expressed as follows:

S xi ; xj

xi xj ¼ a exp b

ð1Þ

where a is the influence factor, and the normalized term b is expressed as: Pn x j¼1 xj b ¼ n

Pn where x ¼

j¼1 xj

n

ð2Þ

where n is the number of CRs in each group. To improve the performance for ﬁltering the outliers of groups, the similarity of xi to the other evaluations in the group is denoted as AðSi Þ, and it is presented as:

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AðSi Þ ¼

n X j¼1

xi xj a exp ; i ¼ 1; . . .; n b

ð3Þ

Based on the values of similarity evaluation, the outliers of CR evaluations in each group can be ﬁltered according to the Pauta criterion [18]. For the vagueness of human language, it is more appropriate to rate CRs using fuzzy numbers. Fuzzy Delphi method is a widely used method for the group decision. In fuzzy Delphi method, the expert team may be invited to evaluate the CRs many times for achieving the consensus condition. The fuzzy Delphi method proposed by Wang and Chen [19] is used to evaluate the importance weight of CR. Consumers involve in OD for capturing their desired product, and the CRs and its evaluation values that are collected in OD tend to stabilization with the communications among consumers. As the number of consumers who involve in OD is increasing, an opening degree needs to be deﬁned to determine the degree of consumer involvement. Consumers who involve in OD are organized into different groups on the basis of time series. The opening degree is calculated according to the mean value of the CRs in each group. The deviation between the average of two adjacent groups and the expected mean value of the group are applied to determine the opening degree in OD.

4 Evaluating the Design Requirement in OD In the product design process, it is crucial to identify critical DRs according to CRs for maximizing consumer satisfaction. Fuzzy QFD has been widely applied for converting CRs into DRs. To evaluate DRs, the importance weight of CR and the relation intensity between CR and DR are considered as the inputs of fuzzy QFD [20]. Although many previous studies have been proposed for rating DRs, they cannot deal with an excessive number of CRs and evaluate DRs on the basis of the language information of CRs in OD. Therefore, a machine learning approach based on fuzzy QFD is proposed for evaluating DRs in OD. 4.1

Using a Machine Learning Approach to Identify the Relationship Between CRs and DRs

For rating DRs, the relationship between DR and CR need to be identiﬁed by a machine learning approach. To improve the efﬁciency of identifying the relationship between CR and DR in OD, it is necessary to preprocess the context of CRs. Stop words removal and word stemming methods can be applied to preprocess the language expression of CRs in OD. In addition, the keywords of DRs are applied for identifying the relation intensity between CRs and DRs. According to the words on different sides of the keyword, the relationship between CR and DR can be identiﬁed. Normally the words on both sides of the keyword have various effects on identiﬁcation of the relationship between the words and keyword. Consequently, it is assumed that Wk represents the keyword and Wc is the word of the sentence. The left and right words of Wk are denoted by Wl and Wr respectively. The keywords of DR and the preprocessed

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CR are applied for identifying the relationship between DR and CR. The probability that the ith left word Wli is related to DRk is denoted as: P0 ðWli ; Dk Þ ¼

CðWli ; Dk Þ C ðDk Þ

ð4Þ

where CðWli ; Dk Þ is the number of CRs that include Wli and relate to DRk simultaneously, CðDk Þ is the number of CRs related to DRk . Obviously, if CðWli ; Dk Þ equals zero, P0 ðWli ; Dk Þ is zero and it will cause the model inapplicable. To address this problem, Dirichlet Priors smoothing method can be used in the process of identifying the probability that the ith left word Wli is related to DRk , and it is deﬁned as: PðWli ; Dk Þ ¼

CðWli ; Dk Þ þ lPðWli ; C Þ C ðD k Þ þ l

ð5Þ

where l is a constant that can be used to adjust the smoothing item, PðWli ; CÞ represents the occurrence probability of Wli in training corpus C. According to Eq. (5), the probability that the left word Wl is related to DRk is denoted as: PðWl ; Dk Þ ¼

Nl 1X PðWli ; Dk Þ Nl li¼1

ð6Þ

where Nl is the number of the words that are on the left of the keyword. The probability that the right word Wr is related to DRk can be obtained with the same method. The probability that CRi is related to DRk on the basis of the Nik th keyword is expressed as: Pik ¼

1 1 10Nikn 2 Nikn h i 1 X N N þ P Wl ; Dk ikj þ P Wr ; Dk ikj 4Nikn Nikj ¼1

ð7Þ

N

where DNk ik is the Nik th keyword that is related to CRi , Nikj is the sequence of Dkki in N CRi , Nikn represents the number of Dkki in CRi . When there is no keyword in CR, the word with the maximum probability for relating to a CR is deﬁned as the keyword. Based on Eq. (7), a threshold d is suggested to identify the relationship between CRi and DRk . Then the relation intensity between each keyword and DR needs to be deﬁned by designers. According to the value of probability and the relation intensity between the keyword and DR, the relation intensity between CR and DR can be identiﬁed on the basis of the language expression of CR. 4.2

Evaluating the Design Requirement in Fuzzy QFD

Based on the importance weight of CR and the relation intensity between DR and CR, the importance weight of the DR can be evaluated in QFD. In practice, DRs are not independent with each other, and therefore the relation intensity among DRs should be

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considered in the process of evaluating DRs. Let Rij be the relationship between CRi and DRj , and cij represents the relationship between DRi and DRj . Based on previous study [19], the relation intensity between CRi and DRj that has been normalized by considering the relationship among DRs is expressed as: 1þ R0ij

¼ Rs ¼

P k6¼j

! ckj Rij ; ckj 2 ½0; 1

Rs Xn Xm i¼1

j¼1

1þ

X k6¼j

where !

ckj Rij

ð8Þ

where n is the number of CRs and m is the number of DRs. The importance weights of DRs can be obtained as: WjDR ¼ WiCR R0ij

ð9Þ

where WiCR is the importance weight of CRi . For the fuzziness of human thought as well as language, fuzzy evaluation method is more appropriate to rate DRs in QFD. Triangular fuzzy number (TFN) has been widely used in the fuzzy evaluation method. In this section, TFN is applied to represent the language expression (see Table 1). Supposing A is a TFN and it is denoted as A ¼ ða; b; cÞ, where a b c. To deal with fuzzy numbers, Rs ¼ ðRs a; Rs b; Rs cÞ in Eq. (8) is deﬁned as Rs ¼ Rs a þ Rs3 b þ Rs c. lA ð xÞ represents the membership function of A, and it can be expressed as:

Table 1. Linguistic expression and fuzzy number Linguistic expression Very weak relationship Weak relationship Medium relationship Strong relationship Very strong relationship

Symbol VW W M S VS

8 < ðx aÞ=ðb aÞ; l A ð xÞ ¼ ðc xÞ=ðc bÞ; : 0

Fuzzy number (0, 0.1, 0.2) (0.2, 0.3, 0.4) (0.4, 0.5, 0.6) (0.6, 0.7, 0.8) (0.8, 0.9, 1)

a x\b bxc otherwise

ð10Þ

Some algebraic operations of Ai ¼ ðai ; bi ; ci Þ and Aj ¼ aj ; bj ; cj can be deﬁned as follows [20]:

Analysis and Extraction of Consumer Information

Ai þ Aj ¼ ai þ aj ; bi þ bj ; ci þ cj Ai Aj ¼ ai aj ; bi bj ; ci cj

349

ð11Þ ð12Þ

Ai Aj ¼ ai aj ; bi bj ; ci cj

ð13Þ

Ai Aj ¼ ai aj ; bi bj ; ci cj

ð14Þ

kAi ¼ ðkai ; kbi ; kci Þ

ð15Þ

Based on QFD, the fuzzy importance weights of DRs can be obtained via Eq. (9) and algebraic operations. To determine the importance priority of DRs, the fuzzy numbers need to be converted to the crisp value. Then the defuzziﬁcation for a TFN can be expressed as follows: Rc xlA ð xÞ xð AÞ ¼ Rac a lA ð xÞ

ð16Þ

5 Case Study In this section, a case study of analyzing the CRs for evaluating the DRs of a smartphone in OD is proposed for demonstrating the effectiveness of the proposed method. Based on the Internet, consumers who are interested in the product are able to involve in OD. 5.1

Collecting and Analyzing CRs in OD

To capture CRs, an OD platform is constructed for collecting CRs and their evaluations. Consumers can publish CRs and evaluate CRs by using fuzzy numbers. Due to the fact that there will be a large number of evaluation data in OD, the evaluation data is divided into different groups and ﬁltered using the GOA. For example, a requirement CR0 can be evaluated by any consumer who involves in OD. In this study, the scale 1– 9 is used for rating CRs. The numbers 1 and 9 represent unimportant and very important respectively, and number 5 represents important. The evaluation values of CR0 are divided into different groups and the outliers are ﬁltered using GOA, and it is shown in Fig. 2. The evaluations of CRs are divided into 4 groups, and each group includes 50 values. Based on the Pauta criterion, the deviation values that are greater than 0 are deﬁned as outliers. Then the evaluation values are used to evaluate the importance weights of CRs according to fuzzy Delphi method, and the importance weight of CR0 is 7.7198 using the proposed method.

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Fig. 2. The analysis of CRs using GOM in OD

5.2

Evaluating Design Requirements

The importance weight of CR and the relationship between DR and CR are applied as the inputs of QFD for evaluating DR. The relationship between CR and DR can be identiﬁed according to the text language of CRs using the machine learning approach. In this section, six CRs (see Fig. 2) of a smartphone are selected as an example. Based on the machine learning approach and the formulation of Table 1, the relationship between CR and DR can be identiﬁed. For example, “running memory” is the keyword of CR1 and it corresponds to the DR “random access memory”. The language expression of CR1 is preprocessed using stop words removal and word stemming methods. Based on the preprocessed CR1 and Eq. (7), the relationship between CR1 and the DR “running memory” is identiﬁed as VS. The relationship between other CR and DR is identiﬁed with the same way (see Table 2). The relationship among DRs are evaluated by designers using fuzzy Delphi method according to Table 1. The fuzzy relationship matrix for a smartphone can be identiﬁed (see Fig. 3). In OD, designers can select the CRs according to their importance weights. Based on the importance weights of CRs and the relation intensity between DRs and CRs, the fuzzy importance weights of DRs can be evaluated by using Eqs. (8) and (9). Then according to Eq. (16), the crisp value of the importance weights of DRs can be obtained, and it is shown in Table 3. It can be seen that DR1 has the maximum importance weight (2.0622), and DR2 has the minimum importance weight (0.9157). Therefore, designers should pay more attention to DR1 than to DR2 . Based on the importance weights of DRs, the resources can be distributed efﬁciently in the process of product design.

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Table 2. The relationship between CRs and DRs CR

Expression

Keyword

DR

CR1 I suggest improving the running memory of the mobile phone

Running memory

CR2 I can’t hear what the other person is saying while I’m on the phone. It will be better if the phone has the function of noise reduction CR3 I also suggest increasing the service life of battery. It is better to reduce the battery charging time CR4 I hope that the camera function of mobile phone is more professional CR5 I hope that the smart phone has multiple operating modes. For example, it can be set up to receive only one person’s phone CR6 I would like the screen size to be 6.0inch

Hear, Noise

Random access memory Call quality

Relation Weight intensity VS 8.9522

VS

8.8047

Battery

Battery capability

VS

8.7096

Camera, Photos Operating modes

Photography VS

8.4972

System

M

7.9634

Screen size

Screen size

VS

7.7198

Fig. 3. The fuzzy QFD relationship matrix for a smartphone

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S. Li et al. Table 3. The importance weight of DR Design requirements Fuzzy value Crisp value (1.6553, 2.0483, 2.4833) 2.0622 DR1 DR2 (0.8144, 0.9157, 1.0178) 0.9157 DR3 (1.4493, 1.8116, 2.2140) 1.8251 DR4 (1.0995, 1.3256, 1.5711) 1.3322 (0.7366, 1.0121, 1.3251) 1.0246 DR5 DR6 (0.9989, 1.2043, 1.4274) 1.2103

Rank 1 6 2 3 5 4

6 Conclusions The analysis of consumer information is critical in the product design process. The importance of DRs can be evaluated according to CRs. Along with the popularization of Internet, consumers can involve in product design process and express their requirements freely. Since the product design is open to all the consumers, there will be a large amount of information about the product. It is difﬁcult for designers to rate DRs on the basis of these CRs. Therefore, this paper aims to rate DRs by analyzing consumer information depending on consumer involvement in OD. To rate DRs in OD, ﬁrstly, an OD platform is constructed for prompting consumer participation in product design. An SFM is proposed to analyze the CRs and a fuzzy Delphi method is used to rate the importance weights of CRs. Secondly, a machine learning approach is proposed to determine the relationship between DRs and CRs in OD. The importance weights of DRs are evaluated according to the weights of CRs and the relation intensity between DRs and CRs in fuzzy QFD. Finally, an example of a smartphone design was provided to illustrate the application of the method. For future research, the number of the words and the distance between the word and the keyword will be considered in the process of analyzing DRs in OD.

References 1. Wang, C.H., Hsueh, O.Z.: A novel approach to incorporate customer preference and perception into product conﬁguration: a case study on smart pads. Comput. Stand. Interfaces 35(5), 549–556 (2013) 2. Li, Y., Tang, J., Luo, X., Xu, J.: An integrated method of rough set, Kano’s model and AHP for rating customer requirements’ ﬁnal importance. Exp. Syst. Appl. 36(3), 7045–7053 (2009) 3. Nahm, Y.E., Ishikawa, H., Inoue, M.: New rating methods to prioritize customer requirements in QFD with incomplete customer preferences. Int. J. Adv. Manuf. Technol. 65(9–12), 1587–1604 (2013) 4. Zheng, P., Xu, X., Xie, S.Q.: A weighted interval rough number based method to determine relative importance ratings of customer requirements in QFD product planning. J. Intell. Manuf. 30(1), 3–16 (2019) 5. Dou, R., Li, W., Nan, G.: An integrated approach for dynamic customer requirement identiﬁcation for product development. Enterp. Inf. Syst. 13(4), 448–466 (2019)

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6. Tan, L., Tang, D., Wang, Q., Yang, J.: Open design pattern, method, and its selforganization mechanism. Procedia CIRP 56, 34–39 (2016) 7. Lettl, C.: Learning from users for radical innovation. Int. J. Technol. Manag. 33(1), 25–45 (2004) 8. Song, W., Ming, X., Han, Y., Wu, Z.: A rough set approach for evaluating vague customer requirement of industrial product-service system. Int. J. Prod. Res. 51(22), 6681–6701 (2013) 9. Liu, A., Hu, H., Zhang, X., Lei, D.: Novel two-phase approach for process optimization of customer collaborative design based on fuzzy-QFD and DSM. IEEE Trans. Eng. Manag. 64 (2), 193–207 (2017) 10. Vanegas, L.V., Labib, A.W.: A fuzzy quality function deployment (FQFD) model for deriving optimum targets. Int. J. Prod. Res. 39(1), 99–120 (2001) 11. Chen, Y., Fung, R.Y., Tang, J.: Rating technical attributes in fuzzy QFD by integrating fuzzy weighted average method and fuzzy expected value operator. Eur. J. Oper. Res. 174(3), 1553–1566 (2006) 12. Chen, L.H., Weng, M.C.: An evaluation approach to engineering design in QFD processes using fuzzy goal programming models. Eur. J. Oper. Res. 172(1), 230–248 (2006) 13. Wang, Y.M., Chin, K.S.: Technical importance ratings in fuzzy QFD by integrating fuzzy normalization and fuzzy weighted average. Comput. Math Appl. 62(11), 4207–4221 (2011) 14. Wang, Y.M.: A fuzzy-normalisation-based group decision-making approach for prioritising engineering design requirements in QFD under uncertainty. Int. J. Prod. Res. 50(23), 6963– 6977 (2012) 15. Miao, Y., Liu, Y., Chen, Y., Zhou, J., Ji, P.: Two uncertain chance-constrained programming models to setting target levels of design attributes in quality function deployment. Inf. Sci. 415, 156–170 (2017) 16. Baldwin, C.Y., Clark, K.B.: The architecture of participation: does code architecture mitigate free riding in the open source development model? Manag. Sci. 52(7), 1116–1127 (2006) 17. Yang, M.S., Wu, K.L.: A similarity-based robust clustering method. IEEE Trans. Pattern Anal. Mach. Intell. 26(4), 434–448 (2004) 18. Shen, C., Bao, X., Tan, J., Liu, S., Liu, Z.: Two noise-robust axial scanning multi-image phase retrieval algorithms based on Pauta criterion and smoothness constraint. Opt. Express 25(14), 16235–16249 (2017) 19. Wang, C.H., Chen, J.N.: Using quality function deployment for collaborative product design and optimal selection of module mix. Comput. Ind. Eng. 63(4), 1030–1037 (2012) 20. Liu, H.T.: The extension of fuzzy QFD: from product planning to part deployment. Exp. Syst. Appl. 36(8), 11131–11144 (2009)

Reduction in Aerodynamic Resistance of High-Speed Train Nose Based on Kriging Model and Multi-objective Optimization Tian Li(&), Deng Qin, Le Zhang, Jiye Zhang, and Weihua Zhang State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China {litian2008,jyzhang,tpl}@home.swjtu.edu.cn, [email protected], [email protected]

Abstract. In order to minimum the aerodynamic resistance of both the head and tail cars, an approach for aerodynamic design of a high-speed train nose based on Kriging model and multi-objective optimization is proposed. The aerodynamic resistances of the head and tail cars are chosen the dependent variables. 5 independent variables are related to 5 typical curves, which are the vertical outline of the front nose, vertical outline of the window, lateral outline of the nose, the ﬁrst auxiliary outline and the second auxiliary outline. Optimal Latin-hypercube design (OLHD) is used to generate the design matrix of the independent variables of the train nose. The dependent variables for every set of independent variables are obtained using computational fluid dynamics. The multi-objective optimization is solved by Non-dominated Sorting genetic algorithm II (NSGA-II). Compared to the initial value which is calculated using the original train model, the aerodynamic resistance of the head car decreased by about 6.5%, and the resistance of the tail car decreased by 5.0%. The magnitude of the decrease in the resistance of the head car is larger than that of the tail one. The vertical outline of the window is the most sensitive to aerodynamic resistance for both head and tail cars. The ﬁrst auxiliary outline has a contradictory effect on the aerodynamic resistance of the head and tail car. Keywords: Kriging model Multi-objective optimization Aerodynamic resistance Train aerodynamics

1 Introduction As known to all, high-speed train has been developed rapidly all over the world, especially in China. As a major breakthrough in the construction of an innovative country and the iconic achievement of independent innovation, high-speed railway has become China’s new “diplomatic business card” and “image representative.”

This project is supported by National Natural Science Foundation of China (Grant No. 51605397), Sichuan Science and Technology Program (No. 2019YJ0227) and Self-determined Project of State Key Laboratory of Traction Power (2019TPL_T02). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 354–370, 2020. https://doi.org/10.1007/978-981-32-9941-2_30

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The increase of train aerodynamic drag seriously restricts the increase of train running speed and affects the economics of train operation. In recent years, the multi-objective aerodynamic optimization design of the train nose has been attracted more and more attention in order to improve the operation economy and running safety of trains. Kwon [1] used the response surface method to optimize the high-speed train head shape, and studied the influence of the train head shape on the tunnel pressure wave. Lee [2] aimed to reduce the micro-pressure wave generated when the train entered the tunnel. The Kriging model and the improved support vector machine technology were used to generate the approximate model, and the high-speed train head type was optimized. Krajnovic [3] proposed an optimization design method for train aerodynamic characteristics based on response surface method. The results show that the combination optimization is the best. Vytla et al. [4] based on the hybrid method of genetic algorithm and particle swarm optimization, combined with the Kriging approximation model, optimized the longitudinal two-dimensional contour of the high-speed train head to reduce the aerodynamic drag and aerodynamic noise of the train. Munoz-Paniagua et al. [5] optimized the aerodynamic drag and maximum pressure gradient when the high-speed train entered the tunnel, combined with genetic algorithm and perceptron neural network method, and carried out threedimensional head shape optimization design. Sun et al. [6] carried out a singleobjective optimization design on the nose shape of the CRH3 train and the height of the upper wall of the driver’s cab. The aerodynamic drag was used as a single optimization target to solve the optimization goal. Yao et al. [7] simpliﬁed the train model for the three groups of CRH380A, established a three-dimensional parametric model of the train head by local function method, and established the optimization target and optimization design variables based on Kriging method. Yao et al. [8] selected train aerodynamic drag and tail car aerodynamic lift as optimization targets, and optimized the high-speed train head shape based on particle swarm optimization. Li et al. [9] parameterized the three-dimensional head shape of high-speed train based on free-form deformation technology, combined with genetic algorithm and Kriging approximation model to optimize multi-objective design of high-speed train head shape to reduce the aerodynamic drag of high-speed trains. Yu et al. [10] and Zhang et al. [11] established an optimized design flow for high-speed train head shape, with the aerodynamic drag of the train running in the open air and the maximum wheel load shedding rate of the ﬁrst car or the aerodynamic lift of the tail car as the optimization goal, through genetics. The algorithm optimizes the design of high-speed train heads. Zhang et al. [12] established a parametric model of the high-speed train including the bogie area, based on the Kriging approximation model and combined Multi-objective aerodynamic optimization design of high-speed train head shape based on genetic algorithm. The above researches concerned about the aerodynamic drag force of the train, for example, the aerodynamic drag of the head car or the whole train. Few work studied the effect of train nose on the aerodynamic drag force of the tail car. Moreover, the train nose affects the aerodynamic drag force of both head and tail cars. The design in train nose could be more complex if the contradiction for drag force of head and tail cars. In this research, the aerodynamic resistances of the head and tail cars are chosen as the dependent variables, an approach for aerodynamic design of a high-speed train nose based on Kriging model and multi-objective optimization is proposed to minimum the

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aerodynamic resistance of both the head and tail cars. Section 2 describes the numerical approach, and results are shown in Sect. 3, followed by conclusions.

2 Numerical Approaches The flow chart for the aerodynamic design of a train nose based on Kriging model and multi-objective optimization in this study can be described as follows. STEP1. According to the concerned objective variables and the parameterized train nose, choose the design variables including the dependent variables (also referred to as response variables or output variables) and independent variables (also referred to as input variables or predictor variables). And also the lower and upper values of the independent variables should be given. STEP2. Obtain the design matrix for the independent variables using Optimal Latin-hypercube design (OLHD). STEP3. Based on the parameterized train nose and every set of the independent variables, generate the train geometry using software CATIA. Establish the numerical simulation model for the aerodynamic including generating the mesh, numerical iteration and the result analysis. Obtain the dependent variables for every set of independent variables. STEP4. Establish Kriging model based on the above samples. STEP5. Perform the multi-objective optimization of the train nose based on the Non-dominated Sorting genetic algorithm II (NSGA-II) and the above Kriging model. The pareto results are obtained. The above procedures are illustrated in Fig. 1.

Determine the design variables

Concerned objective variables

Obtain the design matrix using OLHD

Design of Experiment

Generate the train geometry using CATIA

Parameterized train nose

Obtain the dependent variables

Computational Fluid Dynamics

Establish Kriging model

Multi-objective optimization using NSGA-II

Fig. 1. Conﬁguration of the exoskeleton arm system

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Design Variables and Parameterized Train Nose

The aerodynamic resistance of a train is one of the most important factors for the shape design of a high-speed train. Therefore, the aerodynamic resistances of the head and tail cars are chosen the dependent variables. Figure 2 shows the nose of a conceptual high-speed train, which is made up of several characteristic curves. Five typical curves are the vertical outline of the front nose C1, vertical outline of the window C2, lateral outline of the nose C3, the ﬁrst auxiliary outline C4 and the second auxiliary outline C5.

C2

C1

C5 C4 C3 Fig. 2. Train nose and characteristic lines

All curves are B-spline ones generated by several key points, and then B-spline surfaces are generated by the above curves. The shape of the train nose is composited of several B-spline surfaces. The independent variables correspond to the characteristic curves, which are d1, d2, d3, s4, s5, respectively. Variables d1, d2, d3 are the maximum distance for the deformation of the corresponding curve. Variables s4 and s5 are the scale size for the deformation of the corresponding curve. The parametric method of the train nose is similar to the one used by Zhang et al. [12]. Table 1 gives the lower and upper values of design variables. Table 1. Lower and upper values of design variables d1/mm d2/mm d3/mm s4 s5 Lower value −100 −200 −30 −0.1 −0.15 Upper value 200 400 50 0.15 0.4

Figure 3 shows the variations of lower and upper values of design variables. For the upper value of variables d1 and d2, the height of the nose and the window increase as shown in Fig. 3(a). In the original model, the ﬁrst and second auxiliary outlines are

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slightly concave (Fig. 3(e)), and they are concave at a certain for the upper value of variables s4 and s5 (Fig. 3(f)), whereas, they are convex when variables s4 and s5 are at lower value as shown in Fig. 3(d).

(a) Upper value of d1 and d2

(b) Initial value of d1 and d2

(c) Lower value of d1 and d2

(d) Upper value of s4 and s5

(e) Initial value of s4 and s5

(f) Lower value of s4 and s5

Fig. 3. Variations of independent variables

2.2

Design of Experiment (DOE)

The design of experiments (DOE) for the train nose is a design that aims to describe or explain the variation of independent variables under certain conditions that are hypothesized to reflect the variation. Optimal Latin-hypercube design (OLHD) was

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proposed by Park [13] to minimize the integrated mean squared error or maximize entropy. In this paper, OLHD is used to generate the design matrix of the independent variables of the train nose. In order to ensure the regression accuracy and the calculation efﬁciency, the number of designs is chosen as 301. The design matrix of independent variables is given in Table 2. The number 0 in Table 2 represents the original train model. Table 2. Design matrix of variables d1/mm 0 −97.00 −34.00 28.00 81.00 −47.00 … 29.00 9.00 114.00

0 1 2 3 4 5 … 299 300 301

2.3

d2/mm 0 272.00 −154.00 82.00 −138.00 342.00 … 358.00 158.00 28.00

d3/mm 0 −14.27 20.40 −5.73 19.07 44.67 … 2.00 32.40 −0.93

s4 0 −0.04 −0.05 −0.09 −0.08 0.12 … 0.12 0.03 −0.02

s5 0 −0.03 0.38 0.07 0.33 0.06 … 0.14 0.07 0.02

Numerical Model for Aerodynamics

The computational domain is shown in Fig. 4. The characteristic length H, which is based on the train height, is about 3.6 m. The computational domain has a length of 65H, width of 30H and height of 15H. A uniform velocity u = (83.33 m/s, 0, 0) is speciﬁed at the inlet boundary. A traction-free condition is prescribed at the outlet boundary, a symmetry condition is speciﬁed at the top boundary and 2 side boundaries, and a non-slip condition is speciﬁed all walls and train surface. The boundary conditions are shown in Fig. 4.

Symmetry Outlet

15H

Symmetry

Inlet

30 H

y x z

Ground Symmetry

50H

15H

Fig. 4. Computational domain and boundary conditions

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The commercial software FLUENT is used to obtain the aerodynamic characteristics of the train model. The Finite Volume Method (FVM) is adopted for the discretization of the Navier-Stokes equations. The Semi-Implicit Method for PressureLinked Equations (SIMPLE) algorithm is chosen to obtain the pressure and flow ﬁeld. The SST k-w model and the Second Order spatial discretization schemes are chosen in this study [14, 15]. The mesh with about 15 million cells is generated using ANSYS ICEM. A boundary layer is considered in the mesh, which make the y+ on the surface of the train around 30–150. The number of iterations is 4000 and the average value over the latest 1000 iterations is calculated and chosen as the dependent variables. 2.4

Kriging Model

Kriging model is a type of interpolation technique. It is extremely flexible due to the wide range of correlation functions. It provides a minimum error-variance estimate of any un-sampled value, and it tends to smooth out the details and extreme values of the original data set. Matlab Kriging Tool box DACE is adopted to establish the Kriging model for the aerodynamic forces calculated using different independent variables. The usage of the tool box DACE is to construct a kriging approximation model based on the independent variables and corresponding dependent variables. In this paper, the second order polynomial is chosen for the regression model, and the Gaussian is chosen for the correlation model. 2.5

Multi-objective Optimization

Multi-objective optimization is an area of multiple criterion design making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many ﬁelds of science, including engineering, economics and logistics. Non-dominated Sorting genetic algorithm II (NSGA-II) is a solid multi-objective algorithm, widely used in many real-world applications. While today it can be considered as an outdated approach, NSGA-II has still a great value, if not as a solid benchmark to test against. NSGA-II generates off springs using a speciﬁc type of crossover and mutation and then selects the next generation according to nondominated-sorting and crowding distance comparison.

3 Results According to numerical approaches described in Sect. 2, the aerodynamic design of a high-speed train nose based on Kriging model and NSGA-II is studied in this section. The results include the train aerodynamics, Kriging model and the multi-objective optimization.

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Train Aerodynamics

The aerodynamic resistances of trains given in Table 3 are monitored during the iteration for every design. Symbols Fdh, Fdm, and Fdt are the aerodynamic resistances of the head, middle and tail cars, respectively. Meanwhile, F is the total aerodynamic resistance of the train. It can be seen that the resistance of the head car is larger than that of the tail car. Among all cases, the resistance of the head car is smallest for case 162, and also the resistance of the whole train is. Compared to the original model, the resistance of the head car for case 162 decreased by about 8.87%, and the resistance of the train decreased by 4.47%. Case 164 has the smallest resistance of the tail car, which decreased by 5.01% with comparison to that of the original tail car. The magnitude of the decrease in the resistance of the head car is larger than that of the tail one.

Table 3. Aerodynamic resistances 0 1 2 3 4 … 162 163 164 … 299 300 301

Fdh 6473.2 5998.4 6307.5 6184.5 6246.2 … 5899.2 6246.5 6132.3 … 6157.8 6375.1 6099.1

Fdm 4082.2 4045.0 4166.9 4110.9 4102.7 … 4062.4 4028.6 4114.0 … 4027.7 4146.4 4099.5

Fdt 5298.7 5131.3 5190.2 5154.4 5213.4 … 5184.0 5225.2 5033.4 … 5269.6 5183.0 5077.9

F 15854.1 15174.7 15664.6 15449.8 15562.3 … 15145.6 15500.3 15279.7 … 15455.1 15704.5 15276.5

In order to discriminate the difference in the aerodynamic forces, the pressure distribution on the head car and tail car are shown in Figs. 5 and 6, respectively. Three train models are chosen including the original one, case 162 with a minimum resistance of the head car and case 164 with a minimum resistance of the tail car. For the head car, the pressure on the head car is basically positive. The maximum pressure is located at the nose tip. Compared to the original head car, the pressure on the coupling cover is smaller for both cases 162 and 164. Meanwhile, the pressure on the window for case 162 is smaller than that on the corresponding position for case 164. Therefore, case 162 has the minimum aerodynamic resistance of the head car. For the tail car, the pressure on the tail car is basically negative except for the region of the front nose. Compared to the original tail car, the positive pressure on the

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(a) The original model

(b) Case 162

(c) Case 164 Fig. 5. Pressure distribution on the nose of the head car (Pa)

front nose is larger for both cases 162 and 164. Meanwhile, the area of the positive pressure for case 162 is smaller than that on that for case 164. Therefore, case 162 has the minimum aerodynamic resistance of the tail car.

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(a) The original model

(b) Case 162

(c) Case 164 Fig. 6. Pressure distribution on the nose of the tail car (Pa)

3.2

Approximation Model

There are 302 sets of independent variables and corresponding dependent variables including the original train model. Among them, 15 random sets of samples are chosen randomly as the validation samples, and the others are used to establish the Kriging model. Figure 7 shows the error in the drag forces of head and tail cars between predicted and numerical results. The maximum error in the drag force of the head car is almost less than 2% and the average error is 0.97%. For the tail car, the maximum and average

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errors are 1.85% and 0.73%, respectively. Results show that the Kriging model has a great ability to predict the aerodynamic drag forces for both head and tail cars, especially for the tail car.

(a) head car

(b) tail car Fig. 7. Error between predicted and numerical results

3.3

Multi-objective Optimization

The optimization process of the NSGA-II includes selection, crossover, and mutation. The manners of crossover and mutation of the NSGA-II are the same as those of the standard genetic operations. In present study, the population size is set to be 60, the evolution generation is set to be 40 and the crossover probability is 0.9.

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(a) independent variable d1

(b) independent variable d2

(c) independent variable d3

(d) independent variable s4

(e) independent variable s5

Fig. 8. Convergence history of ﬁve independent variables

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Figure 8 shows the convergence history of ﬁve independent variables during the process of the multi-objective optimization. The pentagrams in the Fig. 8 are the optimal results. Variable d1 is converged to be −100 mm, which means that the lower of the vertical outline of the front nose C1, the better aerodynamic forces are. The variations of d2 and s5 are similar to that of d1. It has lower aerodynamic force if the vertical outline of the window C2 is lower, the ﬁrst auxiliary outline C4 is convex and the second auxiliary outline C5 is concave. While the variable s4 is in the middle range, it has a better performance. Moreover, the convergence of the variables d1, d2, and s5, is better than that of d3 and s4. Figure 9 shows the convergence history of aerodynamic drag force of the head and tail cars during the process of the multi-objective optimization. The red pentagrams in the ﬁgure represent the optimal results. Compared to the initial value which is calculated using the original train model, the aerodynamic resistance of the head car decreased by about 6.5%, and the resistance of the tail car decreased by 5.0%. The magnitude of the decrease in the resistance of the head car is larger than that of the tail one.

Fig. 9. Optimal results of multi-objective optimization

The main effect of all independent variables on the aerodynamic resistances of the head and tail cars is shown in Fig. 10. The main effect graph shows the effect of the independent variable on the dependent variable by plotting the relationship as determined by regression analysis of the data set. For the head car, the aerodynamic resistance increase with the incremental of variables d1, d2, and s5 and decrement of variable s4. Whereas the aerodynamic resistance of the tail car almost increase with the incremental of all independent variables.

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(a) head car

(b) tail car Fig. 10. Main effect of independent variables on the aerodynamic resistances of the head and tail cars

Pareto plot for the aerodynamic resistance of the head and tail cars is shown in Fig. 11. The pareto plot is similar to the main effect. For the head car, variables d1, d2 and s5 have positive effects on the aerodynamic resistance, and variables d3 and s4 give negative effects. Meanwhile, all independent variables have positive effects on the aerodynamic resistance of the tail car.

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(a) Head car

(b) Tail car Fig. 11. Pareto plot for the aerodynamic resistance of the head and tail cars

4 Conclusions (1) The approach of the reduction in the aerodynamic resistance of a high-speed train nose based on Kriging model and multi-objective optimization is proposed. (2) Compared to the original model, the resistance of the head car for case 162 decreased by about 8.87%, and the resistance of the train decreased by 4.47%. Case 164 has the smallest resistance of the tail car, which decreased by 5.01% with comparison to that of the original tail car. The magnitude of the decrease in the resistance of the head car is larger than that of the tail one. (3) The Kriging model has a great ability to predict the aerodynamic drag forces for both head and tail cars, especially for the tail car. (4) Compared to the initial value which is calculated using the original train model, the aerodynamic resistance of the head car decreased by about 6.5%, and the resistance of the tail car decreased by 5.0%. The magnitude of the decrease in the resistance of the head car is larger than that of the tail one.

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(5) Variables d1, d2 and s5 have positive effects on the aerodynamic resistance. The vertical outline of the window C2 is the most sensitive to aerodynamic resistance for both head and tail cars. The ﬁrst auxiliary outline C4 has contradict effect on the aerodynamic resistance of the head and tail car.

Appendix Appendix and supplement both mean material added at the end of a book. An appendix gives useful additional information, but even without it the rest of the book is complete: In the appendix are forty detailed charts. A supplement, bound in the book or published separately, is given for comparison, as an enhancement, to provide corrections, to present later information, and the like: A yearly supplement is issue.

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Research and Simulation on Pilot Conﬁguration in Multi-antenna System Based on Kalman Filter Ying Li1(&), Lei Cui2, and Zhe Zhang1,3 1

2

Department of Electronics and Information Engineering, Lanzhou Institute of Technology, Lanzhou 730050, China [email protected], [email protected] Department of Society, Gansu Academy of Mechanical Sciences CO. LTD., Lanzhou 730030, China [email protected] 3 Gansu Provincial Key Laboratory of Wearable Computing, School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China

Abstract. CRH has become a distinctive business card in the transportation industry around the world. However, transmitted signals suffer multi-path effect in wireless channels. Especially, server Doppler shift causes energy loss to received signals, and the wireless communication quality declines. In this paper, architecture of a joint Kalman Filter and multi-antenna system is designed, based on Ricean fading channels. A comparative analysis of channel performance estimation is given between the estimator designed by Ref. [5] and that of our architecture by simulations. For the same velocity of the train and Ricean factor, DFO estimation error is less than the estimator mentioned before for the same number of received antennas. Then, we compare DFO estimation with pilots decreased by 10 in a frame according to previous assumptions of 50, simulation results show that, before the pilot number drops to 20, DFO estimation error has no obvious growth, when equals to 20, error increases signiﬁcantly, that is, the estimation performance referred in this paper gets worse. Another important contribution for this paper is that the framework we have established provides a method and basis for the pilot conﬁguration in a multiantenna system to a receiver. Keywords: High Speed Railway Multi-antenna Kalman ﬁlter

DFO Ricean fading channel Pilot

1 Introduction The construction of China Railway High-Speed (CRH) is the longest in the world, which has become a distinctive business card of contemporary China [1–3]. By the end of 2018, the operational mileage of CRH exceeded 29000 km, two-thirds of the total This project is supported by National Natural Science Foundation of China (No. 61841202)Scientiﬁc Research Projects of Universities in Gansu province of China (2017A-1022017A-106). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 371–379, 2020. https://doi.org/10.1007/978-981-32-9941-2_31

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mileage of the High Speed Railway (HSR) in the world [4]. However, transmitting signals experience reflection, refraction and scattered paths multiple times to receivers in wireless channels, leading to signal degradation. For instance, the modulated signals transmit in high mobility situations suffer severe multi-path fading caused by Line-ofsight (LOS) propagation, and penetration loss caused by signals pass through the train body. Furthermore, Additive White Gaussian Noise (AWGN) and random phase shift can also lead to signal energy loss. Besides, the most challenge in the HSR channel is the serious Doppler shift, due to fast movement of HSR [5]. All problems above result in the decline of wireless communication quality, and the passengers’ communication requirements cannot be met as well. Regarding the problems mentioned before, many researchers focused on HSR channel estimations and compensations to improve wireless communication quality. For the penetration loss, a two-hop channel model was proposed by a bunch of literatures [6–8], in the uplink of the model, passengers’ data transmit ﬁrstly to antennas installed outside the carriage, then, the gathered data is transferred to the base station, and the opposite function for the downlink transmission process. The advantage of this model is the direct information exchange can be avoided, problems of penetration loss and energy loss can be solved as well. So, this model is applied in our paper. For multipath problem, MIMO technology signiﬁcantly improves the reliability of the channel and reduces the transmission error rate, and the problem of the multi-path fading has also been solved [9]. [10–14] studied some methods to estimate the channel performance in MIMO system. [15, 16] discussed pilot designing in massive MIMO networks. In this paper, we present architecture for Doppler shift estimation based on Kalman ﬁltering in multi-antenna system, estimation results can be given in simulation work based on the architecture we have noted before. Main contributions for this paper are: (1) the transmit channel model of joint Kalman Filter and multi-antenna system is proposed to estimate channel performance. (2) Comparisons of channel estimation based on Rice fading model with that of the estimator proposed in [5] are also studied by simulations. (3) Simulation works of pilot conﬁguration in data frames are shown and illustrated, by estimating channel character. (4) The framework can offer a way and basis for pilots’ conﬁguration in a cooperative antenna system to a receiver. The rest of the paper is organized as follows, in Sect. 2, we describe the system model and channel model. Then, comparisons of channel character estimation for those two estimators are shown, and pilots’ conﬁguration for our architecture is also shown in Sect. 3 by simulations. Conclusions are given in Sect. 4.

2 System Model 2.1

System Model

A two-hop model is shown in Fig. 1, passengers’ communications to the base station are done through antennas installed on the carriages, instead of communicating to the base station directly. Suppose there’s one antenna ﬁxed in one carriage, if we know the distance from the ﬁrst antenna to the origin, the distance from the origin to all antennas

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can be calculated, because the length of carriage is a ﬁxed value, typically 10 m. The train usually has 16 carriages, so the train length can also be calculated. Besides, the distance from base station to the origin is also known for 50 m [5].

Fig. 1. A two-hop model

2.2

Channel Model

We consider a transmission channel model shown in Fig. 2. It shows clearly that after estimating all parameters, received signals will be decoded, and then the information is sent to user terminals after demodulation.

Fig. 2. A transmission channel model

In Fig. 2, there is a multi-antenna system with N antennas over a high speed train. Each antenna receives signals transmitted from the base station, and it also transmits signals to the base stations. Received signals suffer an Additive White Gaussian Noise W with a variance of 2d2 . Small-Scale Fading causes server Doppler shift and multipath effect in channel. In addition, Large-Scale Fading causes Shadow-fading and path loss, etc. Therefore, we let l to be the Large-Scale fading factor for simulations. Then, the channel model is given by Y ¼ HX þ W

ð1Þ

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Where N is antenna number, R is received symbols in a frame.Y 2 SNR is the received signals vector, and H denotes Ricean channel fading with Ricean factor k, which indicates the intensity of the direct component relative to the multipath component. By Eq. (1), received signals can be rewritten as yn ¼

R1 X

hnr xnr þ wn ; n ¼ 1; 2; . . .N

ð2Þ

r¼0

The average received SNR for each antenna can be given as n o n o RE jhnr j2 E jxr j2 n o SNR ¼ E jwn j2

2.3

ð3Þ

Pilots and Kalman Filter Conﬁguration

It was reported that Kalman approach is of the superiority in improving estimation accuracy in time-varying linear systems [17]. For high mobility of HSR, channel estimation became very difﬁcult, especially for limited pilots. Therefore, we introduce Kalman ﬁltering into multi-antenna system for better DFO estimations. Suppose there are B symbols in a data frame, while we choose C of them for pilots, that is, there are (C-B) symbols after each pilot. Estimation accuracy increases at the expense of transmission efﬁciency and estimation complexity. In our architecture, we use limited pilots less than C for estimation, so we replace the removed pilots with Kalman ﬁltering. If the channel performance estimation results are similar to that without Kalman ﬁltering, it can be proved that the architecture proposed in this paper is reasonable, especially, pilots can be saved effectively. Pilot and Kalman ﬁlter conﬁguration is shown in Fig. 3. The probability Density Function (PDF) can be given by Ref. [5] f ðx; v; hÞ ¼

N [ C [

f ðxnr ; v; hnr Þ

ð4Þ

n¼0 c¼1

The coherent time T can be calculated by T = c/fv, if the train velocity and carrier frequency are given. Suppose v = 150 m/s, f = 1.8 GHz, then coherent time T 1.1 ms, the frame interval can be set to 1 ms in simulations in Sect. 3, while the sampling time is 0.25 ls.

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Fig. 3. Pilots and Kalman ﬁlter conﬁguration

3 Simulation Ref. [5] derives the channel parameter estimation performance such as path loss, velocity random phase, and DFO estimation error by the Theoretical Rao bound (CRB). Besides, theoretical symbol error rate function is also derived. In this section, ﬁrst, we compare the DFO estimation error of the architecture proposed in this paper with that of the estimator designed in Ref. [5], for good estimation performance results in Ref. [5]. Second, we compare the channel performance estimations for different number of pilots by simulations. Finally, theoretical comparison and analysis are summarized. For comparison purposes, we choose the 9th received antenna, the same as Ref. [5], Assume the distance from the base station to the ﬁrst antenna is within 100 m [18], we stipulate the estimation error is set no more than 1%, if not, error range can exceed 5%. In addition, it should be noted that, when calculating the DFO estimation error, there will be some deviation, because the receiver is moving constantly but the frequency offset estimation is only once. 3.1

Estimation Performance Compared with Ref. [5]

The channel estimation performance compared with that in Ref. [5] by simulations shown in Fig. 4, let v = 100 m/s, while the average of k is about 8.25 [18]. The sampling time TS = 0.25 ls, symbols in a frame B = 4000 and pilots C = 50, the Ricean factor k = 5 for Doppler shift estimation. From Fig. 4, we can conclude that (1) Simulation results of DFO estimation error performance are basically consistent with the theoretical performance with the same v and k, which can prove the correctness of simulation results. (2) The estimation error increase as a increases, especially for a > 100 m. (3) Comparison of simulated results show that estimation performance proposed in this paper is better than the estimator in Ref. [5].

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Fig. 4. Error performance of DFO estimation for the 9th received antenna, when v = 100 m/s, k = 5, C = 50

We know that Doppler shift increases as the train speed increases, Fig. 5 shows simulations on DFO estimation error performance when the velocity v = 150 m/s.

Fig. 5. Error performance of DFO estimation for the 9th received antenna, when v = 150 m/s, k = 5, C = 50

We conclude from Fig. 5 that DFO estimation error has no obvious growth under the condition of the train speed increase. Simulation results show that the estimation performance is stable for the framework proposed by us.

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Estimation Performance Compared Different Pilot Conﬁguration

We choose the average value of Ricean factor k = 8.25 for simulation in this section. The previous simulations are under the condition of C = 50. In this section, we reduce the number of pilots in a frame, then, DFO estimation error performance is given in Fig. 6.

Fig. 6. Comparisons of estimation performance for the 9th received antenna, when v = 150 m/s, k = 8.25, with different pilot conﬁguration

We conclude that, in the situation of C = 40 and C = 30, the DFO estimation error performance is close to that of C = 50, nevertheless, when C drops to 20, the performance begins to be worse. Simulation results demonstrate that, the estimation accuracy of our framework is relatively stable when k is reduced to a certain extent, which means the number of pilots for estimation can be saved effectively. 3.3

Comparisons of BER Performance for Different Ricean Channels

Figure 7 shows the comparisons of bit error rate (BER) performance for different Ricean channels by 16QAM. When k = ∞, means the interference of AWGN is added, we can conclude that, (1) The bit error still exists for both framework, the nonlinear communication system for high mobility channel is one reason, another is the estimation method needs to be further improved. (2) For the same SNR, BER performance applying our framework is better than the estimators’ in Ref. [5]. The estimation framework proposed by us is more efﬁcient. (3) When k varies to 8.25, BER performance becomes better obviously, because channel fading drops. The increase of Ricean factor k reduces the noise and the channel fading, and the estimation accuracy is improved.

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Fig. 7. Comparisons of BER performance for different Ricean channels, when k = ∞, k = 8.25

(4) The larger the SNR, the better the bit error rate performance. Especially, when BER is 10−4, SNR performance for both estimators’ is less than 1 dB.

4 Conclusion In this paper, an architecture for DOF and channel performance estimation is devised, which consists a multi-antenna system installed outside the train carriages, and Kalman ﬁltering is also included. A comparative analysis of channel performance estimation is given between the estimator in [5] and that of our architecture in Ricean fading channel by simulations. Simulation results show that, (1) The theoretical DFO estimation error performance is basically the same as the simulation performance with the same velocity and Ricean factor k. (2) The estimation error increase as a increases, especially for a > 100 m, a denotes the distance from the base station to the ﬁrst antenna. (3) Estimation performance proposed in this paper is better than that of the estimator in Ref. [5]. (4) DFO estimation error has no obvious growth as the train speed increases. (5) The number of pilots for estimation can be saved effectively by our framework. The framework we have established provides a method and basis for the pilot conﬁguration in a multi-antenna system to a receiver. (6) Simulation results prove the efﬁciency of estimation method. Simulations on BER performance show there are still some bit errors in system, we need to improve the estimation algorithm and framework to increase the estimation accuracy in our future work.

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References 1. Cao, J., Liu, X.C., Wang, Y., Li, Q.: Accessibility impacts of China’s high-speed rail network. J. Transp. Geogr. 28, 12–21 (2013) 2. Shaw, S.-L., Fang, Z., Shiwei, L., Tao, R.: Impacts of high speed rail on railroad network accessibility in China. J. Transp. Geogr. 40, 112–122 (2014) 3. Lvhua, W., Yongxue, L., Liang, M., Sun, C.: Potential impacts of China 2030 high-speed rail network on ground transportation accessibility. Sustainability 10, 1–16 (2018) 4. January 2019. http://society.people.com.cn/n1/2019/0104/c1008-30502584.html 5. Yaoqing, Y.A.N.G., Pingyi, F.A.N.: Doppler frequency offset estimation and diversity reception scheme of high-speed railway with multiple antennas on separated carriage. IEEE J. Mod. Transp. 20(4), 227–233 (2012) 6. Zhang, C., Fan, P., Dong, Y., Xiong, K.: Service-based high-speed railway base station arrangement. Wirel. Commun. Mob. Comput. 15(13), 1681–1694 (2015) 7. Dong, Y., Fan, P., Letaief, K.B.: High speed railway wireless communications: efﬁciency versus fairness. IEEE Trans. Veh. Technol. 63, 925–930 (2014) 8. Li, Y., Zhang, C.: Analytic comparisons of the high-speed railway base station arrangement strategy in fading channels, Beijing, pp. 187–191, November 2014 9. Oestges, C., Clerckx, B.: MIMO Wireless Communications: From Real-World Propagation to Space-Time Code Design, vol. 312. China Machine Press, Beijing (2010) 10. Shen, L., Yao, Y.-D., Wang, H., Wang, H.: ICA based semi-blind decoding method for a multicell multiuser massive MIMO uplink system in Rician/Rayleigh fading channels. IEEE Trans. Wirel. Commun. 16(11), 7501–7511 (2017) 11. Yapıcı, Y., Güvenç, I., Kakishima, Y.: A MAP-based layered detection algorithm and outage analysis over MIMO channels. IEEE Trans. Wirel. Commun. 1–29 (2017) 12. Li, T., Wang, X., Fan, P., Riihonen, T.: Position-aided large-scale MIMO channel estimation for high-speed railway communication systems. IEEE Trans. Veh. Technol. 66(10), 8964– 8978 (2017) 13. Lu, J., Chen, Z., Fan, P., Letaief, K.B.: Subcarrier grouping for MIMO-OFDM systems over correlated double-selective fading channels. Wirel. Commun. Mob. Comput. 1–30 (2015) 14. Kim, H.K., Byun, Y.S., Lee, Y.H.: Estimation of MIMO channel with imperfect channel correlation information. Wirel. Pers. Commun. 95(3), 3377–3389 (2017) 15. Akbar, N., Yan, S., Yang, N., Yuan, J.: Mitigating pilot contamination through locationaware pilot assignment in massive MIMO networks. In: 2016 IEEE Globecom Workshops, Washington, DC USA, December 2016 16. Noh, S., Zoltowski, M.D., Sung, Y., Love, D.J.: Pilot beam pattern design for channel estimation in massive MIMO systems. IEEE J. Sel. Top. Sig. Process. 8(5), 1–15 (2013) 17. Komninakis, C., Fragouli, C., Sayed, A.H., Wesel, R.D.: Multi-input multi-output fading channel tracking and equalization using Kalman estimation. IEEE Trans. Sig. Process. 50(5), 1065–1075 (2010) 18. Liu, L., Tao, C., Qiu, J.H., Yu, L.: Position-based modeling for wireless channel on highspeed railway under a viaduct at 2.35 GHz. IEEE J. Sel. Areas Commun. 30(4), 834–845 (2012)

Development and Experiment of an XhYhZ Micro-motion Stage Based on a Straight-Beam Three-Quarter Round Type Flexure Hinge Junlang Liang1, Lanyu Zhang1,2(&), Jian Gao1,2, Gengjun Zhong1, Guangtong Zhao1, and Jindi Zhang1 1

Laboratory of Electronic Precision Manufacturing Equipment and Technology of Ministry of Education, Guangdong University of Technology, Guangzhou 510006, China [email protected] 2 State Key Laboratory of Precision Electronic Manufacturing Technology and Equipment, Guangdong University of Technology, Guangzhou 510006, China

Abstract. This paper proposes a type of straight-beam three-quater round type (STR) flexure hinge for a micro-actuation motion stage in the ﬁeld of precision micro-operation manufacturing. Based on the hinge, a three degree-of-freedom precision parallel micro-motion stage is designed. Firstly, the rigid and flexure structural design of the STR flexure hinge is presented, and theoretical analysis and simulation performance veriﬁcation are carried out. Based on the elastic statics and calculus theory, the flexibility expression of the flexure hinge is derived under different stresses. The flexibility of the hinge is veriﬁed by ANSYS software. The results show that the designed flexure hinge can effectively achieve complaisant deflection and rigid support with high stress. Moreover, a XhYhZ micro-motion stage based on the STR hinge is established, and the dynamic performance of the stage is simulated and veriﬁed by Matlab/Simulink. Finally, the experimental stage is set up. The experimental results are compared with the simulation results, and the stage can achieve about 3.7 times micro-actuation ampliﬁcation ratio and 68 lm displacement stroke. Keywords: Micro-motion stage Precision micromotion operation

Flexure hinge Parallel mechanism

1 Introduction With the rapid development of MENS technology, the precision micro-motion stage based on flexure hinge has been widely used in many precision manufacturing and operation ﬁelds because the stage has the characteristics of no mechanical friction, no gap, high motion sensitivity and good stability, etc. It has an vital position in many

This work is supported in part by the National Natural Science Foundation under Grant 51675106, and 51905108, and in part by the Guangdong Provincial R&D Key Projects under Grant 2015A030312008, 2016A030308016 and 17ZK0091. © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 380–393, 2020. https://doi.org/10.1007/978-981-32-9941-2_32

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precision equipment and devices, such as microelectronic precision manufacturing, micro-positioning robots, ultra-precision processing equipment, etc. Scholars from all over the world have carried out extensive researches on the micro-motion stage flexure hinge [1]. Wu et al. proposed an concise compliance and accurate flexure hinge formula for achieving a rapid analysis [2]. Lobontiu et al. derived the calculation formula of the flexibility of the angular straight beam type flexure hinge by Cartesian second comparison. The results showed that the precision performance of straight-circular flexure hinges can realize better performance in the common hinge types [3]. At present, the straight beam flexure hinge is widely used because its advantages of high precision motion and stable rigidity. However, its stroke range is still limited, which generally has lower degree of flexibility. The flexibility degree of arc flexure hinge is higher than common hinges, which can obtain a large deflection by comparing with the straight beam type hinge. But, its precision stability is lower than the straight beam type [4]. Therefore, in order to improve the precision and deflection performance of the flexure hinge, this paper proposes a straight-beam three-quater round type (STR) flexure hinge by combining of the straight beam characteristics and the arc beam. The proposed hinge is symmetrically arranged based on the central point with a short wide straight beam in the middle, and round edges are designed around the wide straight beam. The flexibility calculation is derived by elastic statics and calculus theory, and the performance of the hinge is veriﬁed by ANSYS software (Fig. 1).

Fig. 1. Model diagram of the hinge and parameter sketch

For the precision micro-manipulation motion stage, the stage must have the characteristics of high precision output [5–7]. Now, the piezoelectric (PZT) element is widely used as the actuation element in the micro-motion stage with the characteristics of high-precision, high-speed response, large thrust, etc. But the stroke of PZT is limited, which is usually used with a displacement flexure hinge ampliﬁcation mechanism in the micro-motion stage to realize a large-stroke high-precision micro-motion [8, 9]. In this paper, the STR flexure hinge is used as the rotating link in a designed two-stage ampliﬁcation mechanism, and the spatial position of the amplifying mechanism is analyzed. The natural frequency of the stage is analyzed by dynamic theoretical analysis, which is veriﬁed with simulation software. Based on the set up of the experimental stage, the performance of the stage is veriﬁed with experiments.

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2 STR Flexure Analysis 2.1

Flexibility Analysis of the STR Hinge

The designed STR flexure hinge can be divided into two parts. The ﬁrst part is the equivalent two round edges of the hinge; the second part is the straight beam width at the middle of the hinge. Figure 2 is a dimensional sketch of the Part 1 of the hinge, where r is the inscribed radius of the circle; t is the minimum thickness of hinge; H is the width of the X-axis symmetry; h is the central angle of the inscribed cylindrical surface. Figure 3 is a schematic diagram of a micro-element parallel to the Y-axis direction including a height dx, a length d, and a width b.

Fig. 2. Dimensional sketch of Part 1 of the hinge

Fig. 3. Micro-element diagram parallel to the Y-axis direction

To study the compliance characteristics of the hinge, the upper end is assumed as a ﬁxed end, the lower end is assumed as a free end (Fig. 2). Six-dimensional generalized force act on point O, relatively, which are displaced in six degrees of the freedom point O. According to the Euler-Bernoulli beam theory and small deformation hypothesis, T the setting six-dimensional generalized force are Qo ¼ Fx ; Fy ; Fz ; Mx ; My ; Mz ; the T deformation is uo ¼ Dx; Dy; Dz; ax ; ay ; az : uo ¼ ½C Qo

ð1Þ

In the equation, the symmetric matrix C is set to measure the relationship between the load and the deformation. The I represents deformation, j represents the load, and the compliance element is Cij. The designed STR flexure hinge is composed of two identical T-type bodies. Firstly, the flexibility properties characteristics of the T-type bodies are studied. We set the elastic modulus parameter of the material is E, the shear modulus parameter is G,

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and I(x) is the moment of inertia of the cross section for the micro-element dx facing the central axis. In the O-xyz coordinate system, the hinge is divided into two regions, such as shown in Fig. 4. Then the area of this hinge section can be calculated.

Fig. 4. Schematic diagram of the partition of the T-shaped section

1. For the I section, the section area can be obtained when p=2 h p=2: a1 ¼ 2r þ t 2r cos h; dx1 ¼ d ðr sin hÞ ¼ r cos hdh

ð2Þ

2. For the II section, the section area can be obtained when p=2 h p: a ¼ 2r þ p þ r cos h dx2 ¼ d ðr sin hÞ ¼ r cos dh

ð3Þ

At the same time, when make s ¼ r=t; m ¼ r=p, the ﬁrst intermediate variable h(h) and the second intermediate variable g(h) can be obtained:

2.2

hðhÞ ¼ s þ 1=2 s cos h

ð4Þ

gðhÞ ¼ m þ 1=2 þ m cos h

ð5Þ

Compliance Formula

This section studies the rotational stiffness performance of the STR flexure hinge around the Z-axis. It is an important performance indicator for flexure. The moment Mz is expressed the lead to a deflection angle around the Z-axis. az ¼ CaMz Mz

ð6Þ

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The flexure of the angular deformation az can be expressed as: Cðaz Mz Þ ¼ az=Mz Z r ¼ ðMz dxÞ=ðEIz ð xÞÞ ðr Þ

Z ¼

ðp=2Þ

ðp=2Þ Z p ðp=2Þ

r cos hdh=ð2t hðhÞÞ3 þ

ð7Þ

r cos hdh=ð2p gðhÞÞ3

¼ 3r=2Eb f1 =t3 þ f2 =p3 Among them, the intermediate variables are: Z f1 ¼

ðp=2Þ

Z f2 ¼

ðp=2Þ

ðp=2Þ

ðp=2Þ

cos h= h3 ðhÞ dh

ð8Þ

cos h= g3 ðhÞ dh

ð9Þ

The flexibility of the hinge (Fig. 5) around the Z-axis compliance equation can be obtained according to the mechanics of materials: C 0 a M ¼ az =Mz ð ðzÞ z Þ Z ðh=2Þ ¼ ðMz dxÞ=ðEIz ð xÞÞ ðh=2Þ

¼ 12= Ebð2r þ tÞ3

Fig. 5. Schematic diagram of the second hinge around the Z axis direction

ð10Þ

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385

Calculation of Overall Rotational Stiffness of Flexure Hinge

The total flexibility of the flexure hinge about the Z-axis can be obtained from the superposition formula of compliance: Cz ¼ CðaðzÞ Mz Þ þ Cð0 a M Þ ðzÞ z 3 ¼ 3r=Eb f1 =t þ f2 =p3 þ 12h= Ebð2r þ tÞ3

2.4

ð11Þ

Finite Element Veriﬁcation and Comparison

Through the ANSYS software, the calculated formula of the overall stiffness/flexure hinge is veriﬁed (Figs. 6 and 7). The material of the STR hinge is aerospace aluminum 7075, and the corresponding flexibility is calculated. By comparing the theoretical solution with FEA, the error between the simulation and the theoretical analysis is less than 5%.

Fig. 6. Flexure hinge ﬁnite element model

Fig. 7. Finite element deformation of flexure hinge

In order to further verify the performance of the designed STR flexure hinge, the flexibility performance of the STR hinge is compared with the straight circular flexure hinge which is widely used in many motion stages. The simulation results are shown in

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Table 1. The results show that the STR hinge has better mobility in terms of compliance performance. Under the size parameters of r = 1.75; b = 12; t = 1.25; p = 1.5; h = 1.5, the flexure movement of the STR hinge is 6.4410−4 (better than the 6.0410−4 of the biaxial straight circular flexure hinge).

3 XhYhZ Micro-motion Stage Based on the STR Hinge 3.1

Structural Composition

In the ﬁeld of micro-motion precision manufacturing, there is a high requirement for the linear precision motion performance of the micro-motion stage [10, 11]. For achieving a high precision position adjustment micro-driving, the PZT is used as the driving component. A lever-type ampliﬁcation mechanism is designed, and the center of mass is designed by the two-lever alignment method. The proposed STR hinge is used in the ampliﬁcation mechanism. Finally, a XhYhZ micro-motion stage is established with three ampliﬁcation mechanisms. The speciﬁc structure is shown in Fig. 8.

Fig. 8. XhYhZ Micro-motion stage structure Table 1. Geometric parameters and flexibility comparison 组别 r/mm b/mm t/mm p/mm h/mm The straight circular compliance FEA 1 2 3 4 5 6

0.75 1 1.5 1.75 2 2.25

7 10 10 12 12 14

1 1 1.25 1.25 2 2.5

1 1 1.5 1.5 2 2.5

0.5 1 1.25 1.25 1.5 2

7.25 10−4 6.0410−4 2.1110−4 1.5510−4 0.8910−4 0.5710−4

The proposed STR hinge compliance Theoretical solution 7.7310−4 6.2710−4 2.7610−4 1.6510−4 0.9810−4 0.7110−4

FEA 7.3210−4 6.4410−4 2.2710−4 1.6910−4 0.9610−4 0.6810−4

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Analysis of Kinetic Theory

The structural features of micro-motion dynamics model (Fig. 9) can be represented with Newton’s second law kinematics, and the motion dynamics differential equations can be expressed as follows: 8 < F1 k1 x1 þ k2 ðx3 x1 Þ Cx01 þ k2 ðx2 x1 Þ ¼ m1 x001 F2 k1 x2 þ k2 ðx3 x1 Þ Cx02 k2 ðx2 x1 Þ ¼ m2 x002 : F3 k1 x3 k2 ðx3 x1 Þ Cx03 k2 ðx2 x1 Þ ¼ m3 x003

ð12Þ

Where k1 is the stiffness of the spring element in the micro-dynamic structure; k2 is the stiffness of the precision motion stage; C is the damping coefﬁcient of the piezoelectric ceramic; m1, m2, m3 are the equivalent masses of the three enlarged structures respectively; F1, F2, F3 represents the driving force of the drive piezoelectric micromotion stage. x1, x2, x3 are the output displacements, respectively.

Fig. 9. Micro-motion dynamic theoretical model

In order to analyze the dynamic performance of the stage, the kinetic equations are expressed in matrix form: MX 00 þ CX 0 þ KX ¼ f

ð13Þ

2

3 m1 0 0 Here, M ¼ 4 0 m2 0 5 is the equivalent mass matrix; K ¼ 0 0 m3 2 3 k1 þ 2k2 k2 k2 4 k2 k1 þ 2k2 k2 5 is the equivalent stiffness matrix for the kinetic k2 k2 k1 þ 2k2 2 3 0 1 C 0 0 x1 model; C ¼ 4 0 C 0 5 is the damping matrix for the dynamic model; X ¼ @ x2 A 0 0 C x3 0

1 0 00 1 x01 x1 is a displacement vector; X 0 ¼ @ x02 A is the speed vector; X 00 ¼ @ x002 A is the x03 x003

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2

3 0 F1 0 acceleration vector; f ¼ 4 0 F2 0 5 is the equivalent force matrix acting on the 0 0 F3 micro-motion process. Accordingly, the general displacement solution of the system can be expressed as: x ¼ A sinðxt þ hÞ

ð14Þ

Similarly, the velocity equation can be expressed as: x0 ¼ Ax cosðxt þ hÞ

ð15Þ

x00 ¼ Ax2 sinðxt þ hÞ

ð16Þ

The acceleration equation is:

0

1 A1 Here, A ¼ @ A2 A is the amplitude coefﬁcient of the system, x is the angular A3 frequency, h is the phase angle. The free vibration equation of the system can be obtained by using the displacement and acceleration equations in the above equation: K x2 M X ¼ 0

ð17Þ

For this equation, the characteristic equations of the system can be obtained by solving the characteristic equations: K x2 M ¼ 0

ð18Þ

The solves of the above characteristic equations are: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 ðmk1 Þ=m < x1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ mðk1 þ 3k2 ÞÞ=mﬃ x2 ¼ pðﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : x3 ¼ ðmðk1 þ 3k2 ÞÞ=m

ð19Þ

The natural frequency of the stage can be obtained based on the kinetic model parameters of Table 2. 8 < x1 ¼ 149:35 Hz x ¼ 216:43 Hz ð20Þ : 2 x3 ¼ 216:43 Hz

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Table 2. Related parameters of the dynamic model Parameter M (kg) K1 (N/m) K2 (N/m) Value 0.269 6000 2200

3.3

Natural Frequency Simulation Analysis

In order to verify the correctness of the dynamics theoretical analysis, the natural frequency analysis of the object is performed by the ANSYS/model simulation.

Fig. 10. Finite element model of the micro-motion stage

(a) First-order mode

(b) Second-order mode

(c)Third-order mode Fig. 11. Vibration pattern analysis

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J. Liang et al. Table 3. ANSYS natural frequency values x2 x3 Inherent frequency x1 (Hz) (Hz) (Hz) Value 141.34 229.05 229.09

Figure 10 is a ﬁnite element model of the micro-motion stage, and the model is meshed ﬁrstly. Moreover, by setting the model parameters, the amplitude of the stage is analyzed, and the analysis result is shown in Fig. 11. Table 3 is the simulation results of the natural frequency. Based on the theoretical solution and the natural frequency obtained by ANSYS, it can be seen that the error of theoretical analysis and simulation analysis is less than 5%, which proves the correctness of the dynamic analysis of the stage. 3.4

Dynamics Simulation Analysis

In order to observe dynamic performance of the stage, the rapid motion positioning, positioning overshoot, and the positioning accuracy performance of the stage are tested with MATLAB/Simulink simulation. The results are shown in Fig. 12. It can be seen that the stage designed in this paper can effectively meet the high-precision performance requirements of precision motion operating on its motion stage.

Fig. 12. Dynamic simulation result of the micro-motion stage

4 Experimental Analysis The experimental stage is shown in Fig. 13. The micro-motion stage is set under the laser precision manufacturing device from Han’s Laser Ltd. A laser interferometer is used to verify the position performance of the stage.

Development and Experiment of an XhYhZ Micro-motion Stage

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Fig. 13. Experimental setup of the designed micro-motion stage

4.1

Natural Frequency Experiment Veriﬁcation

In the experiments, the stage is mounted on a vibration isolation platform, and the natural frequency of the stage is measured. The experimental result is shown in Fig. 14. By using the Fourier analysis, a natural frequency 115.32 Hz is obtained, which is has a 10% error with the theoretical analysis because the testing viewﬁnder on the stage increases the stage quality.

Fig. 14. Laser interferometer amplitude map

4.2

Motion Positioning Experiment Veriﬁcation

To test the stage displacement point positioning accuracy performance, different stage strokes are test in the experiments. By inputting the different strokes to the PZT, the output of the stage can be obtained. The results are shown in Figs. 15 and 16. It can be seen that the stage can reach stable 3.7 times magniﬁcation effect, while the displacement distance was 68 lm. Moreover, the output close-loop precision of the stage is stable. The average positioning accuracy of the stage is about 30 nm with different output strokes from 10 lm to 68 lm.

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Fig. 15. Input and output diagram

Fig. 16. Scale enlargement

5 Conclusion This paper proposed a STR flexure hinge and applied it to a designed XhYhZ three degree-of-freedom micro-motion stage. Firstly, by using the elastic static formula and the calculus theory, the flexibility formula of the flexure hinge was derived. Based on the ANSYS simulation, the flexibility of theoretical analysis was veriﬁed. At the same time, by comparing the STR flexure hinge with the ordinary two-axis straight circular flexure hinge, the rigidity of the STR flexure hinge was veriﬁed, and the results showed that the proposed STR hinge can achieve high compliance deformation performance, which can be effectively applied into the bending action of the precision motion stage. Moreover, based on the designed STR hinge mechanism, a XhYhZ micro-motion stage is designed. The dynamic analysis of the stage was carried out, and the theoretical natural frequency of the stage was veriﬁed by the ANSYS simulation. The error between the simulation and the theoretical analysis is less than 2%. Based on the dynamic simulation, the results showed that the stage can achieve a short response time, small overshoot, and high precision position performance. Finally, the experimental stage was set up and the experimental analysis was carried out. The results showed that the stage can be achieved 3.7 times ampliﬁcation ratio, 68 lm displacement stroke, and about 30 nm positioning accuracy.

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References 1. Paros, J.M., Weisbord, L.: How to design flexure hinges. Mach. Des. 37(27), 151–156 (1965) 2. Wu, Y.F., Zhou, Z.Y.: Design calculation for flexure hinges. Rev. Sci. Instrum. 73(8), 3103– 3106 (2002) 3. Lobontiu, N., Paine, J.S.N., Garcia, E., et al.: Corner-ﬁlleted flexure hinges. J. Mech. Des. 123(3), 346–352 (2001) 4. Nishiwaki, S., et al.: Topology optimization of compliant mechanisms using the homogenization method. Int. J. Numer. Methods Eng. 42(3), 535–559 (1998) 5. Wu, Z.H., Huang, N.E.: Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 1(1), 1–41 (2009) 6. Ma, Z.Q., Li, Y.C., Liu, Z., et al.: Rolling bearing fault feature extraction based on variational mode decomposition and Teager energy operator. J. Vib. Shock 35(13), 134–139 (2016) 7. Wang, Y., Markert, R., Xiang, J., et al.: Research on variational mode decomposition and its application in detecting rub-impact fault of the rotor system. Mech. Syst. Sig. Process 60 (61), 243–251 (2015) 8. Itagaki, H., Tsutsumi, M.: Design method for high-gain control of linear motor feed drive systems using virtual linear friction. Procedia CIRP 1, 244–249 (2012) 9. Lee, T.H., Tan, K.K.: Intelligent control of precision linear actuators. Eng. Appl. Artif. Intell. 13, 671–684 (2000) 10. Sharon, A., Hogan, N, Hardt, D.E.: High bandwidth force regulation and inertia reduction using a macro/micro manipulator system. In: Proceedings of the IEEE International Conference on Robotics and Automation, Philadelphia, PA, USA, pp. 126–132 (1985) 11. Yang, Y.: Development status of micromanipulator technology for biomedical application. Chin. J. Mech. Eng. 23, 1–13 (2011)

An Improved PC-Kriging Method for Efﬁcient Robust Design Optimization Qizhang Lin1, Chao Chen2, Fenfen Xiong1(&), Shishi Chen3, and Fenggang Wang1 1

3

School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China [email protected] 2 Science and Technology on Complex Aviation Systems Simulation Laboratory, Beijing 100076, China Beijing Electromechanical Engineering Institute, Beijing 100074, China

Abstract. The polynomial-chaos-kriging (PC-Kriging) method has been derived as a new uncertainty propagation approach and widely used for robust design optimization in a straightforward manner, of which the statistical moments would be estimated through directly conducting Monte Carlo simulation (MCS) on the PC-Kriging model. However, the computational cost still cannot be negligible because thousands of statistical moment estimations might be performed during robust optimization, especially for highly nonlinear and complicated engineering problems. An analytical statistical moment estimation method is derived for PC-Kriging in this work to reduce the computational cost rather than referring to MCS. Meanwhile, a sequential sampling strategy is applied for PC-Kriging model construction, in which the sample points are not generated all at once, but sequentially allocated in the region with the largest prediction uncertainty to improve the accuracy of PC-Kriging model as much as possible. Through testing on three mathematical examples and an airfoil robust optimization design problem, it is noticed that the improved PC-Kriging method with analytical statistical moment estimation and sequential sampling strategy is more efﬁcient than the traditional ones, demonstrating its effectiveness and advantage. Nomenclature

D F Ma N P PC PCK p RðÞ R a b

= = = = = = = = = = = =

dimension of random variables information matrix in PC model mach number of flight flow ﬁeld number of sample points number of coefﬁcients in PC model polynomial chaos polynomial chaos kriging order of PC model auto-correlation function lift-to-drag ratio flight angle of attack coefﬁcient of PC model

© Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 394–411, 2020. https://doi.org/10.1007/978-981-32-9941-2_33

An Improved PC-Kriging Method for Efﬁcient Robust Design Optimization

r2 uðÞ

395

= prior variance of the gaussian random process = multi-dimensional orthogonal polynomial

1 Introduction Robust design optimization aims at optimizing the product performance while minimizing the sensitivity of performance to uncertainties, which has been widely applied to engineering design problems, such as aerospace engineering [1], automobile engineering [2], civil engineering [3]. As is well known, a key component of robust design optimization is to estimate the mean and variance of the output response, given the uncertainty information of inputs [4]. For practical engineering design problem, it often involves complex and time-consuming high-ﬁdelity simulation analysis models, such as ﬁnite element analysis (FEA) for structural analysis and computational fluid dynamics (CFD) for aerodynamic analysis. Compared to the design optimization based on traditional empirical formula, the application of these high-ﬁdelity simulation analyses signiﬁcantly improves the design accuracy, which however greatly increases the computational cost. Therefore, it is of great importance to increase the efﬁciency of statistical moment estimation (mean and variance) to perform robust design optimization for real engineering applications. Literature has seen many approaches about statistical moment calculation, among which the polynomial chaos (PC) method has been widely studied as it has a solid mathematical foundation and fast convergence rate [5]. With PC, a random variable can be represented as a stochastic metamodel, i.e., a weighted summation of some orthogonal polynomials, based on which the mean and variance of the random variable can be analytically obtained. Therefore, the PC method is popularly applied to robust design optimization in different ﬁelds, such as aerodynamic optimization [6], ship optimization [7], structure shape optimization [8], and trajectory optimization in flight dynamics [9–11]. To further improve the accuracy of PC, Schobi et al. proposed to combine the PC method with the classic Kriging technique to construct the so-called PC-Kriging method (hereinafter referred to as PCK for short) [12], within which a PC model is employed to describe the global trend of the original random response, and a Gaussian random process is used to capture its local characteristics. Many works have demonstrated that PCK is more accurate than both PC and Kriging [13, 14]. The PCK method can be considered as a generalized Kriging model, and the difference is that PCK is constructed in the stochastic domain, while the classic Kriging is done in the deterministic one. Once the PCK model is constructed, the mean and variance of the random output response can be conveniently obtained by conducting Monte Carlo simulation (MCS) on the PCK model, and the computational cost taken by one uncertainty analysis is almost negligible compared to the original time-consuming response function evaluation. However, when it is applied to the practical robust design optimization problems, especially those involving coupled multidisciplinary system that may require hundreds of thousands of uncertainty analyses, the computational cost

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taken here cannot be ignored. The PC-based analytical statistical moment estimation [15, 16] has been derived and widely applied, which however is not applicable for PCK as the Gaussian random process is added on the basis of PC model. Therefore, the analytical formulae of statistical moments (mean and variance) calculation for PCK are derived in this work to improve the efﬁciency of PCK-based robust design optimization. On the other hand, a certain number of sample points are required to estimate the unknown metamodel parameters (aka hyper-parameters) for both PC and PCK. Therefore, the sampling strategy has a great influence on the accuracy and efﬁciency of uncertainty analysis [17]. Till now, researches on the sampling strategy for PC and PCK generally focus on the comparative study of various sampling approaches [18, 19], such as random sampling, Latin hypercube sampling, Hemmasy sampling, Gaussian quadrature sampling, in which the sample points are basically generated altogether at one time according to the order of PC term and the dimensionality of uncertain inputs. However, just as the construction of metamodel in the deterministic ﬁeld, the characteristics of the response function cannot be considered during the sampling process for the one-stage sampling scheme, which probably causes the waste of limited computational resource. Therefore, the sequential sampling strategy is further introduced into the construction of PCK model in this paper. Considering that the uncertainty of metamodel can be conveniently quantiﬁed as the Gaussian process theory is employed in PCK [20, 21], the sample point is sequentially generated in the region with the largest uncertainty to update the PCK model and improve its accuracy as far as possible. As the sample points are adaptively generated with the sequential sampling strategy, the computational cost of the uncertainty analysis can be reduced to some extent. The remainder of this paper is organized as follows. In Sect. 2, the PCK method is briefly reviewed at ﬁrst. In Sect. 3, the proposed improved PCK method with analytical statistical moment estimation and sequential sampling is presented in detail, followed by numerical examples in Sect. 4 to test the effectiveness of the two proposed strategies. In Sect. 5, the improved PCK method is applied to an airfoil robust design optimization problem to further verify its effectiveness and applicability. Conclusions are drawn in the last section.

2 Review of PC-Kriging Method As a generalized Kriging model constructed in the probabilistic space, the PCK method employs a weighted sum of a series of orthogonal polynomials (i.e., PC model) to represent the global trend of the random response, and a Gaussian random process to capture its local variation characteristics. With PCK, the random response y ¼ cðxÞ; ðx ¼ ½x1 ; . . .; xd Þ can be expressed as: y M ðPCKÞ ðxÞ ¼

P X i¼0

bi ui ðxðnÞÞ þ r2 Z ðxÞ

ð1Þ

An Improved PC-Kriging Method for Efﬁcient Robust Design Optimization

397

where ui ðxðnÞÞ represents the multi-dimensional orthogonal polynomial obtained by conducting tensor product on one-dimensional orthogonal polynomials; M ðPCKÞ is PCK model; n is a standard random vector transformed from the random vector x in the original stochastic space; b ¼ ½b0 ; b1 ; . . .; bp and r2 denote the PC coefﬁcient vector and the prior variance of the Gaussian random process, respectively. The number of orthogonal polynomials in the PC model is P þ 1 ¼ ðp þ d Þ!=ðd!p!Þ, where p is the order of the model. ZðxÞ is a static Gaussian random process with zero mean and unit variance, which depends on the auto-correlation function between the existing sample points Rðx; x0 Þ ¼ Rðjx x0 j; hÞ. h is the hyper-parameter vector to be estimated by the maximum likelihood estimation (MLE) method or the cross-validation method (CV) as below. ^hMLE ¼ arg min 1 ðy FbÞT R1 ðy FbÞðdet RÞN1 N h h i ^hCV ¼ arg min yT R1 diag R1 2 R1 y h

ð2Þ ð3Þ

where N is the number of sample points; y ¼ ½y1 ; . . .; yN T is the output response vector at the sample points; R the autocorrelation matrix composed of the autocorrelation functions with element as Rij ¼ RðjxðiÞ xðjÞ j; hÞ; F the information matrix with ele ment as Fij ¼ ui xðjÞ ðnÞ . Once h is estimated, b and r2 can be expressed as: 1 bðhÞ ¼ FT R1 F FR1 y r2 ðhÞ ¼

1 ðy FbÞT R1 ðy FbÞ N

ð4Þ ð5Þ

Once the PCK model is constructed, the predicted output response at any point x can be expressed as yðPCKÞ ðxÞ ¼

P X

bi ui ðxðnÞÞ þ rðxÞT R1 ðy FbÞ

ð6Þ

i¼0

where rðxÞT is the correlated vector composed of the correlation function values between the predicted point x and all the existing sample points. Based on Eq. (6), the mean and variance of the output response y can be obtained using MCS.

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3 The Proposed Improved PCK Method 3.1

Analytical Statistical Moment Estimation

In order to save the computational cost of statistical moment estimation by using MCS directly on the PCK model, the analytical statistical moment estimation formulae are derived. It can be seen from Eq. (1) that the PCK model can be regarded as the sum of the PC term (denoted as gðxðnÞÞ) and the Gaussian random process (denoted as f(x)). y M ðPCKÞ ðxÞ ¼

P X

bi ui ðxðnÞÞ þ r2 Z ðxÞ ¼ gðxðnÞÞ þ f ðxÞ

ð7Þ

i¼0

Thus, the mean ly and variance r2y of the random output response y can be obtained by the following two equations. ly ¼ E M ðPCKÞ ¼ EðgðxðnÞÞÞ þ Eðf ðxÞÞ

ð8Þ

r2y ¼ var M ðPCKÞ ¼ varðgðxðnÞÞÞ þ varðf ðxÞÞ þ 2covðgðxðnÞÞ; f ðxÞÞ

ð9Þ

In practical implementation, it is found that the value of the covariance term covðgðxðnÞÞ; f ðxÞÞ is insigniﬁcant compared to the value of the variance terms in Eq. (9), and thus the covariance term is assumed to be neglected here, which will be veriﬁed in the subsequent example tests. (a) Calculating EðgðxðnÞÞÞ and varðgðxðnÞÞÞ To calculate Eqs. (8) and (9), the mean and variance of gðxðnÞÞ and f ðxÞ should be ﬁrstly calculated. According to the orthogonality of orthogonal polynomials in PC, one has Eðg12...N Þ ¼

P X

bi Eðui ðxðnÞÞÞ ¼ 0

ð10Þ

i¼1

Then, EðgðxðnÞÞÞ can be calculated as EðgðxðnÞÞÞ ¼ Eðg0 Þ þ Eðg12...N Þ ¼ Eðg0 Þ ¼ b0

ð11Þ

In Eqs. (10) and (11), gi (i = 0, 1, …, P) corresponds to each item bi ui ðxðnÞÞ of PC model in sequence. For varðgðxðnÞÞÞ, one has varðg0 Þ ¼ 0 varðg12...N Þ ¼

P X i¼1

ðbi Þ2 varðui ðxðnÞÞÞ

ð12Þ ð13Þ

An Improved PC-Kriging Method for Efﬁcient Robust Design Optimization

399

Clearly, one has varðui ðxðnÞÞÞ ¼ E ðui ðxðnÞÞÞ2 ðEðui ðxðnÞÞÞÞ2 ¼ E ðui ðxðnÞÞÞ2

ð14Þ

Through substituting Eqs. (14) into (13), var ðgðxðnÞÞÞ can be computed as varðgðxðnÞÞÞ ¼ varðg12...N Þ ¼

P X

ðbi Þ2 E ðui ðxðnÞÞÞ2

ð15Þ

i¼1

(b) Calculating Eðf ðxÞÞ and varðf ðxÞÞ f ðxÞ is a Gaussian random process, which can be expressed as below. f ðxÞ ¼ r2 Z ðxÞ ¼ rð xÞT R1 ðy FbÞ ¼

N X

ri ð xÞci

ð16Þ

i¼1

where ri is the i-th element of vector r and ci is the i-th element of vector c. c ¼ R1 ðy FbÞ

ð17Þ

As the Gaussian exponential model Rðjx x0 j; hÞ ¼ exp hjx x0 j2 is simple, and has good smoothness and wide scope of application, it is employed as the autocorrelation functions in this work. Then, f ðxÞ can be written as: f ð xÞ ¼

N X

ci

i¼1

xj mj r2 P exp j 2n2j d

2 !!

2 ! ! pﬃﬃﬃﬃﬃﬃ 1 xj m j ¼r ci P nj 2p pﬃﬃﬃﬃﬃﬃ exp j 2n2j nj 2p i¼1 N X d d d ¼ r2 ð2pÞ2 P nj ci P PN xj jmj ; n2j 2

N X

d

j

¼k

N X i¼1

d

i¼1

ð18Þ

j

d ci P PN xj jmj ; n2j j

d

where k ¼ r2 ð2pÞ2 P nj , d is the dimensionality of random input variables x, N is the j

number of sample points, mj and nj are respectively the mean and standard deviation of the j-th random input variables xj, PN ðÞ denotes the probability density function (PDF) of the normal distribution.

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Eðf ðxÞÞ can be expressed as: Eðf ðxÞÞ ¼ k

Z N

X d ci P PN xj jmj ; n2j PXj xj dxj

ð19Þ

j

i¼1

where PXj xj is the PDF of random input variable xj . In this work, it is assumed that all random variables are subject to uniform distribution for the simpliﬁcation of computation, and thus 0 1 xuj Z N X Bd C 1 Eðf ðxÞÞ ¼ k ci B PN xj jmj ; n2j dxj C @P A u l j x x j j i¼1 xlj ð20Þ " !#!

N X d xuj mj xlj mj 1 ¼ ci P u U U j x xl nj nj j j i¼1 where UðÞ is the cumulative distribution function (CDF) of the standard normal distribution; xuj and xlj represent the upper and lower limits of the distribution for xj. It is noteworthy that if the random input variable does not follow the uniform distribution, it can be ﬁrstly transformed into a uniform distribution by the transformation approach [22]. According to the relationship between variance and expectation, one has varðf ðxÞÞ ¼ var k

N X

ci P PN

¼E

k

¼E k

d

ci P PN

i¼1 2

j

i¼1 N X

d

N X N X i1 ¼1 i2 ¼1

j

xj jmj ; n2j

xj jmj ; n2j

!

! k

N X

d

c i P PN j

i¼1 d

ci1 ci2 P PN j

xj jmj ; n2j

PN

xj jmj ; n2j

xj jmj ; n2j

!! ðE ðf ðx ÞÞÞ2

! ðEðf ðxÞÞÞ2

ð21Þ where PN

xj jmj ; n2j

PN

xj jmj ; n2j

!2 2 ! 2 ! xj mj xj mj 1 pﬃﬃﬃﬃﬃﬃ exp ¼ exp 2n2j 2n2j nj 2p !2 ! 2 2 xj mj x j mj 1 pﬃﬃﬃﬃﬃﬃ exp ¼ 2n2j 2n2j nj 2p ! ! n2j 1 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ ¼ PN xj jmj ; 2 2nj 2p

ð22Þ

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By substituting Eq. (22) into Eq. (21), one has ! !! n2j 1 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ varðf ðxÞÞ ¼ E k ci1 ci2 P PN xj jmj ; ðEðf ðxÞÞÞ2 j 2 2nj 2p i1 ¼1 i2 ¼1 ! !Z N X N X d n2j 1 PXj xj dxj ðEðf ðxÞÞÞ2 ¼ k2 ci1 ci2 P PN pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ PN xj jmj ; j 2 2p 2 n j i1 ¼1 i2 ¼1 2

N X N X

d

ð23Þ Similarly, it is assumed that the random variables are subject to uniform distribution, and one has varðf ðxÞÞ ¼k 2

N X N X i1 ¼1 i2 ¼1

¼k

2

N X N X i1 ¼1 i2 ¼1

d

ci1 ci2 P PN j

u ! ! Zxj n2j 1 1 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ u PN xj jmj ; dxj ðEðf ðxÞÞÞ2 2 2nj 2p xj xlj

!

d

ci1 ci2 P PN j

xlj

"

xuj mj 1 1 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ u U n l pjﬃﬃ 2nj 2p xj xj 2

!

xlj mj U n pjﬃﬃ

!# ðEðf ðxÞÞÞ2

2

ð24Þ

(c) Calculating ly and r2y Based on Eqs. (8), (9), (11), (20), (16) and (24), one has N X

ly ¼ g 0 þ

ci

i¼1

r2y ¼

P X

ð25Þ

ðbi Þ2 E ðui ðxðnÞÞÞ2

i¼1

þk

" !#!

xuj mj xlj mj 1 P u U U j x xl nj nj j j d

2

N X N X i1 ¼1 i2 ¼1

d

ci1 ci2 P PN j

! " ! !# xuj mj xlj mj 1 1 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ u U ðEðf ðxÞÞÞ2 U n n pjﬃﬃ pjﬃﬃ 2nj 2p xj xlj 2 2

ð26Þ Once the PCK model is constructed, the mean and variance of the output response y can be obtained according to Eqs. (25) and (26). It should be pointed out that the random input variables are assumed to follow uniform distribution to make the analytical calculation of integrals possible and convenient. Of course, the random input variables do not necessarily obey the uniform distribution in practice. When they follow other distribution types (such as normal and exponential distributions, etc.), the transformation approach can be applied to convert them into uniform distributed variables once the PCK model is built, based on which Eqs. (25) and (26) are applied.

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Sequential Sampling

Considering that the PCK model is actually a generalized Kriging model, the variance of the response prediction can be expressed as follows [23]. VyðPCK Þ ðxÞ ¼ r

2

1 uðxðnÞÞ

T

0 r ð xÞ F T

FT R

1

uðxðnÞÞ rðxÞ

! ð27Þ

where uðxðnÞÞ ¼ ½u0 ðxðnÞÞ; u1 ðxðnÞÞ; . . .; uP ðxðnÞÞT . The response variance shown in Eq. (27) represents the conﬁdence level of predicted response value at the predicted point for the PCK model. The larger the value is, the greater the uncertainty of the predicted response value at this point is. Taking advantage of the response variance, a sequential sampling strategy is developed to improve the accuracy of the PCK model by sequentially generating sample points in the region with the largest prediction uncertainty. A step-by-step description of the proposed sequential PCK method for uncertainty analysis is presented as follows. Step 1: Generate N0 initial sample points X ¼ ½x1 ; . . .. . .; xN0 T according to the distribution information of the random input variables at certain design point, and calculate the corresponding output response values Y ¼ ½y1 ; . . .. . .; yN0 T of y ¼ cðxÞ. Step 2: Based on the current sample points X and Y, construct a PCK model according to Eqs. (1)–(6), and perform uncertainty analysis based on the PCK model to obtain uncertainty information of the output response y, such as mean and variance. Step 3: According to the distribution information of random input variables at certain design point, generate a large number of input sample points, and calculate the prediction variance at each sample point using Eq. (27). Select the input position x* with the largest prediction variance, which is added into the existing sample points, i.e. X ¼ ½X; x ,Y ¼ ½Y; cðx Þ. Then, update the PCK model using the extended sample points, based on which uncertainty analysis is conducted. Step 4: Repeat Step 3 till the variation of mean (ly ) and variance (r2y ) of the output response function y with respect to the previously obtained values are less than the speciﬁed value e or the total number of sample points reaches the limit Nmax.

4 Mathematical Examples In this section, three mathematical examples (see Table 1) are employed to test the effectiveness of the proposed analytical statistical moment estimation (denoted as PCKAE) and the sequential sampling (denoted as SPCK) strategies for PCK. Considering the compromise between accuracy and efﬁciency, the order of the PC term in the PCK model is set as p = 4. The simulation is done on a personal computer with i5-4590 CPU and 8.00 GB memory.

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Table 1. Mathematical examples for test No. Function Distribution 2 xi U ð0; 1Þ 2 2 1 cðxÞ ¼ 100 x2 x1 þ ð1 x1 Þ 0:5ðx2 þ x3 Þ 2 x1 N 55:29; 0:07932 ; x2 N 22:86; 0:00432 cðxÞ ¼ arccos xx14 þ 0:5ðx2 þ x3 Þ x3 N 22:86; 0:00432 ; x4 N 101:60; 0:07932 3 cðxÞ ¼ x1 þ x22 þ x33 þ x44 þ x55 þ x66 xi U ð0; 1Þ

4.1

Test of the Statistical Moment Estimation Strategy

The proposed analytical statistical moment estimation based PCK method (PCK-AE) is compared to the existing PCK approach that employs MCS (106 runs) on the PCK model for statistical moment estimation (denoted as PCK-MCS). The results of direct MCS on the original response model (106 runs) are used as benchmark to verify the effectiveness of PCK-AE and PCK-MCS. Table 2 shows the results of the three functions, in which N represents the number of function evaluations and T the calculation time of moment estimation. It can be seen that the proposed PCK-AE method can produce results that are very close to those of the existing PCK-MCS, and both results show great agreements to those of the direct MCS. Meanwhile, the calculation time for one uncertainty analysis of PCK-AE is clearly shorter than that of PCK-MCS. These results demonstrate the effectiveness and efﬁciency of the proposed analytical statistical moment estimation strategy. Table 2. Results of statistical moment estimation Function 1 ly r2y

T(s)

N

Function 2 ly r2y

T(s)

N

Function 3 ly r2y

T(s)

N

PCK- 465.667 375915.330 1.963 16 0.1222 0.0001377 1.468 71 1.5338 0.4723 3.207 211 AE PCK- 465.661 375911.287 6.009 16 0.1221 0.0001384 6.801 71 1.5381 0.4701 25.742 211 MCS 106 1.5969 0.4417 / 106 MCS 454.469 366451.284 / 106 0.1219 0.0001408 /

4.2

Test of the Sequential Sampling Strategy

In order to verify the effectiveness and advantage of the proposed SPCK method for uncertainty analysis, the three examples in Table 1 are tested with the same PC order. The existing PCK with one-stage sampling strategy is also tested for comparison. The results obtained by direct MCS (106 runs) are used as benchmark to verify the effectiveness of the proposed method. Figures 1, 2 and 3 show the relative errors (%) of the mean and variance of output response compared to MCS with the increase of the number of sample points. It is observed that with the increase of sample points, the errors of the existing PCK with one-stage sampling strategy are slightly decreased, while they are signiﬁcantly reduced

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for the proposed SPCK method. The interpretation is that the sample points generated by the SPCK method can more effectively improve the accuracy of the PCK model. Meanwhile, with the same number of sample points, the results of the SPCK method are more accurate than the existing one-stage PCK method. That is to say, in order to achieve the same accuracy of statistical moment estimation, the SPCK method requires fewer sample points. Such reduction of sample points will greatly improve the efﬁciency of robust design optimization, which will be veriﬁed in the application of the engineering example in the next section.

5 Application to Airfoil Robust Optimization 5.1

Problem Description

The airfoil optimization problem adopted from Ref. [24] aims to maximize the lift-todrag ratio (R) of the airfoil by designing the geometry of the airfoil subject to its constraint of the maximum thickness tmax . The flight states considered here are a = 1.55° and Ma = 0.7. As the main component for providing the lift, the wing suffers from many uncertainties during the flight process, such as the Mach number (Ma) of the incoming flow. In this example, Ma is considered to be uncertain and follow normal distribution Ma* N ð0:7; 0:0152 Þ. The mathematical model for airfoil robust optimization is: min F ðyÞ ¼(lR þ krR subject to :

tmax 0:1

ð28Þ

yi 2 ½0:006; 0:006

where y is the design variable vector, i.e. the vertical locations of the control points after the parametric modeling of the airfoil; and k is the weight set as k = 3 in this example. Due to the good local control property and differentiability, the B-spline curve [25] with 10 control points is employed for the geometry parametrization of the airfoil, where the horizontal locations of the 10 control points are ﬁxed as x = [0.1 0.3 0.5 0.7 0.9 0.9 0.7 0.5 0.3 0.1] and the vertical locations y are free to move for optimization, as shown in Fig. 4. The corresponding vertical locations of the 10 control points of the original airfoil are y = [0.04698, 0.06000, 0.05216, 0.03667, 0.01450, −0.01450, −0.03667, −0.05216, −0.06000, −0.04698].

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During the optimization, the computational fluid dynamics (CFD) analysis using Fluent17.0 is performed to obtain the aerodynamic data, with the k-omega two-path turbulence model. The steady-state density solver with Roe-FDS is employed. Figure 5 shows the mesh grids used in the CFD analysis. The number of nodes in the remote ﬁeld of the grid is 100 and the number in the airfoil boundary is 300. The genetic algorithm based on the Pareto strategy [26, 27] is selected as the optimization algorithm, with the population size as 40 and the evolutionary generation as 25.

Fig. 5. Grids used in CFD analysis

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Optimal Results and Analysis

The proposed analytical statistical moment estimation based PCK method (PCK-AE) and sequential PCK-AE with both analytical statistical moment estimation and sequential sampling strategies (SPCK-AE), PCK-MCS using one-stage sampling, and PCK-MCS with sequential sampling (SPCK-MCS) are all employed to calculate the statistical moments at each iteration design point during the airfoil robust optimization. The PC order is set as p = 2, the number of runs of MCS is set as 100,000, and the number of initial sample points used to construct the sequential PCK model is set as 4. Meanwhile, the deterministic optimization (DO) without considering any uncertainty is also done. Once the optimization is completed, the conﬁrmed mean and standard deviation of the lift-to-drag ratio are calculated using MCS through respectively substituting the optimal design variables obtained by robust optimizations with the four statistical moment estimation schemes as well as the deterministic optimization, and the NACA0012 original airfoil parameters into the CFD analysis model, with the consideration of uncertainty in Ma. Table 3 shows the conﬁrmed results, the total number of CFD calls required during the optimization, and the optimization time cost. Table 3. Comparison of the conﬁrmed results SPCK-AE 26.3258 lR rR 2.7318 # of CFD runs 1627 Time (h) 81

PCK-AE 26.2661 2.9501 2237 112

SPCK-MCS 26.3277 2.7410 1620 94

PCK-MCS 26.2658 2.9760 2270 132

DO 29.4807 4.7264 328 17

NACA0012 23.8161 4.9224 / /

Several observations can be made from Table 3. Firstly, it is found that the deterministic optimization signiﬁcantly increases the mean of airfoil lift-to-drag ratio lR , while the decrease in the corresponding standard deviation rR relative to the original airfoil NACA0012 is small, which is clearly larger than that of robust optimization. For all the robust optimizations using the four statistical moment estimation schemes, the mean of airfoil lift-to-drag ratio is clearly increased, and the standard deviation is signiﬁcantly reduced as well, which evidently improves the robustness of the airfoil design. Secondly, by comparing the results in Table 3 (SPCK-AE vs. SPCKMCS, PCK-AE vs. PCK-MCS), it is found that the proposed analytical statistical moment estimation strategy can clearly reduce the computational time of robust optimization compared to the MCS-based PCK methods (with about 15% reduction in time). Thirdly, by comparing the results in Table 3 (SPCK-AE vs. PCK-AE, SPCKMCS vs. PCK-MCS), it is noticed that the proposed sequential sampling strategy can signiﬁcantly reduce the number of CFD calls compared to the one-stage PCK, resulting in about 28% reduction of optimization time. These results further demonstrate the effectiveness and advantages of the proposed improved PCK method with analytical statistical moment estimation and sequential sampling. Figure 6 shows the optimized airfoils by the SPCK-AE based robust optimization and the deterministic optimization, and the original NACA0012 airfoil. Considering

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that the optimal results of robust optimizations using the four statistical moment estimation schemes (SPCK-AE, PCK-AE, SPCK-MCS and PCK-MCS) are very close, only the results of robust optimization using SPCK-AE are shown here. It is noticed that compared to the original NACA0012 airfoil, the upper surface leading edge thickness and curvature of the airfoil for both the deterministic optimization and robust optimization are reduced, resulting in the increase of lift. This is the main reason for the increase of lift-to-drag ratio.

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Figure 7 shows the diagrams of the static pressure cloud of the three airfoils shown in Fig. 6 with Ma = 0.7, from which it is observed that the regions in dark blue on the upper surface of the airfoil for both deterministic optimization and robust optimization are larger than that of the original one, and the region in dark blue of the deterministic optimization is larger than that of robust optimization. It indicates that both deterministic optimization and robust optimization can reduce the static pressure of upper surface compared to that of the original airfoil, and deterministic optimization can reduce more than robust optimization. Meanwhile, the regions in green on the lower surface for both deterministic optimization and robust optimization are larger than that of the original airfoil, and it is larger for deterministic optimization. This indicates that both deterministic optimization and robust optimization can increase the static pressure of the lower surface of the airfoil, and the deterministic optimization can increase more. In summary, the lift of optimized airfoil is increased, and the deterministic optimization improves more than that of the robust optimization. Therefore, the lift-to-drag ratio is increased after optimization, which is more obvious for the deterministic optimization. These results show great agreement to those displayed in Table 3.

An Improved PC-Kriging Method for Efﬁcient Robust Design Optimization

(a) SPCK-AE

(b) DO

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Fig. 7. Comparison of static pressure cloud diagram

In order to illustrate the robustness of the robust design optimization, the lift-todrag ratio of the optimized (both robust and deterministic) and original airfoils corresponding to Ma = [0.64, 0.66, 0.68, 0.70, 0.72, 0.74, 0.76, 0.78] are respectively calculated, of which the curves are shown in Fig. 8.

34 32 30 28

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Fig. 8. Variation of lift-to-drag ratio at different Mach numbers

It can be seen from Fig. 8 that in the region with Ma > 0.72, the airfoil lift-to-drag ratio obtained by the robust optimization changes more gently with the variation of Ma compared to the deterministic optimization. Although the deterministic optimization can improve the performance of lift-to-drag ratio, it may vary greatly under uncertainty during certain flight condition, which shows the necessity and signiﬁcance to perform robust design optimization to reduce the sensitivity of airfoil to uncertainty.

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6 Conclusions The existing PC-Kriging method employs the Monte Carlo simulation directly on the PC-Kriging model to obtain the statistical moments during robust optimization, which certainly would take lots of computational time especially for highly nonlinear complex design problem requiring a large number of statistical moment estimations. Therefore, an analytical statistical moment estimation strategy is proposed for PC-Kriging in this work. Meanwhile, a sequential sampling strategy is established to further reduce the computational cost, with which the sample point in the region with the greatest prediction uncertainty is sequentially allocated to update the PC-Kriging model. The effectiveness and advantages of the proposed two strategies are tested and veriﬁed by three mathematical examples. The application on the NACA0012 airfoil robust design optimization problem further demonstrates the effectiveness and applicability of the improved PC-Kriging method with analytical statistical moment estimation and sequential sampling strategy in solving practical engineering problems. Acknowledgement. The grant support from Science Challenge Project (No. TZ2018001) and Hongjian Innovation Foundation (No. BQ203-HYJJ-Q2018002) is greatly acknowledged.

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11. Wang, F.G., Xiong, F.F., Jiang, H., Song, J.M.: An enhanced data-driven polynomial chaos method for uncertainty propagation. Eng. Optim. 50(2), 1–20 (2017). https://doi.org/10. 1080/0305215x.2017.1323890 12. Schobi, R., Sudret, B., Wiart, J.: Polynomial-chaos-based Kriging. Statistics 5(2), 55–63 (2015). https://doi.org/10.1615/int.j.uncertaintyquantiﬁcation.2015012467 13. Kersaudy, P., Sudret, B., Varsier, N., Wiart, J.: A new surrogate modeling technique combining Kriging and polynomial chaos expansions application to uncertainty analysis in computational dosimetry. J. Comput. Phys. 286(14), 103–117 (2015). https://doi.org/10. 1016/j.jcp.2015.01.034 14. Schobi, R., Sudret, B.: Imprecise structural reliability analysis using PC-Kriging. In: 25th European Safety and Reliability Conference (2015). https://doi.org/10.1201/b19094-549 15. Xiong, F.F., Yang, S.X., Liu, Y., Chen, S.S.: Analysis Method of Engineering Probability Uncertainty. Science Press, Beijing (2015) 16. Sepahvand, K., Marburg, S., Hardtke, H.J.: Uncertainty quantiﬁcation in stochastic systems using polynomial chaos expansion. Int. J. Appl. Mech. 2(2), 305–353 (2010). https://doi.org/ 10.1142/s1758825110000524 17. Xiong, F.F.: Weighted stochastic response surface method considering sample weights. Struct. Multidiscip. Optim. 43(6), 837–849 (2011). https://doi.org/10.1007/s00158-0110621-3 18. Hosder, S., Walters, R.W., Balch, M.: Efﬁcient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: AIAA Non-Deterministic Approaches Conference (2007). https://doi.org/10.2514/6.2007-1939 19. Hampton, J., Doostan, A.: Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280(12), 363–386 (2015). https://doi.org/10.1016/j.jcp.2014.09.019 20. An, D., Choi, J.H.: Efﬁcient reliability analysis based on Bayesian framework under input variable and metamodel uncertainties. Struct. Multidiscip. Optim. 46(4), 533–547 (2012). https://doi.org/10.1007/s00158-012-0776-6 21. Xiong, F.F.: Robust design optimization considering metamodel uncertainty. J. Mech. Eng. 50(19), 136–143 (2014, in Chinese). https://doi.org/10.3901/jme.2014.19.136 22. Miller, F.P., Vandome, A.F., Mcbrewster, J.: Inverse Transform Sampling. Alphascript Publishing, German (2010) 23. Lockwood, B.A., Anitescu, M.: Gradient-enhanced universal Kriging for uncertainty propagation. Nucl. Sci. Eng. 170(2), 168–195 (2012). https://doi.org/10.13182/nse10-86 24. Ganesh Ram, R.K., Cooper, Y.N., Bhatia, V.: Design optimization and analysis of NACA0012 airfoil using computational fluid dynamics and genetic algorithm. Appl. Mech. Mater. 664(22) 111–116 (2014). https://www.scientiﬁc.net/AMM.664.111 25. Liang, Y., Cheng, X.Q., Li, Z.N., Xiang, J.W.: Multi-objective robust airfoil optimization based on non-uniform rational B-spline (NURBS) representation. Sci. China Ser. E: Technol. Sci. 53(10), 2708–2717 (2010). https://doi.org/10.1007/s11431-010-4075-4 26. Chen, X., Agarwal, R.K.: Optimization of wind turbine blade airfoils using a multi-objective genetic algorithm. J. Aircr. 50(2), 519–527 (2013). https://doi.org/10.2514/1.c031910 27. Cheng, F.Y., Li, D.: Multiobjective optimization design with Pareto genetic algorithm. J. Struct. Eng. 123(9), 1252–1261 (1997). https://doi.org/10.1061/(asce)0733-9445(1997) 123:9(1252)

Electro-Mechanical Response of a Cracked Piezoelectric Cantilever Beam Chao Liu, Wenguang Liu(&), and Yaobin Wang School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang 330063, China [email protected], [email protected], [email protected]

Abstract. Piezoelectric ceramics are of importance to the ﬁeld of aerospace, electric and electronic. However, crack often happens on the surface of piezoelectric ceramics due to harsh working environment. The purpose of this article is to study the impacts of crack on the electro-mechanical response of a piezoelectric beam. Firstly, a ﬁnite element model of the piezoelectric cantilever beam was set up by using the software of ABAQUS. And subsequently, the mechanic and electric induced response of the cantilever beam was discussed. Thereafter, a cracked piezoelectric cantilever beam was developed. The effects of crack on the electro-mechanical response of a cantilever beam were studied. Results indicate that the electro-mechanical response is proportional to the input force and the input voltage. As crack propagates, the variation rule of the electro-mechanical response depends on the crack size. This conclusion is helpful to the prediction of crack size, and to provide a new idea for the prediction of the piezoelectrical structure life. Keywords: Piezoelectric ceramic

Crack Electro-mechanical response

1 Introduction Piezoelectric ceramic is a kind of material that has unique mechanical properties and electro-mechanical coupling properties. Its positive and converse piezoelectric effects is widely used in the ﬁeld of energy-transducing. However, the piezoelectric ceramic is brittle. It is prone to failure or damage under the action of external loads for the piezoelectric ceramic with the initial defects. This characteristic is easy to affect the performance and reliability of the piezoelectric structure. Thus, it is of importance to study the fracture problem of a piezoelectric structure under the positive and converse piezoelectric effect of piezoelectric ceramic. The piezoelectric effect was ﬁrst discovered in 1880 by the brothers of P. Curie and J. Curie [1]. In the process of utilizing the mechanic-electric conversion characteristics of piezoelectric ceramics, piezoelectric energy harvesters and ultrasonic motors are of the most representative. Among them, the piezoelectric energy harvester is a device that This project is supported by National Natural Science Foundation of China (Grant No. 51565039), the Jiangxi Provincial Nautral Science Foundation (Grant No. 20181BAB206023). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 412–423, 2020. https://doi.org/10.1007/978-981-32-9941-2_34

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converts mechanical energy into electrical energy by using the positive piezoelectric effect, and is widely used as a sensor in the aerospace equipment. The piezoelectric cantilever beam is a typical model for the piezoelectric energy harvesters. But it has a drawback of collecting the vibrational energy from only one direction of the ceramic. The vibrations in the actual environment may come from any directions in the threedimensional space. An accurate and detailed analysis of an efﬁcient piezoelectric energy harvester has attracted the attention of many researchers. F or example, Fan et al. [2, 3] proposed a cantilever-type two-dimensional vibration energy harvesting system in 2014, which was superior to the linear piezoelectric energy harvesting device. Chen et al. [4] presented a kind of multi-dimensional energy harvesting device likes a dandelion structure, which can effectively collect multi-directional vibration. A three-dimensional vibration energy harvesting device was designed by Su et al. [5, 6] in 2013, which can realize energy collection in three-dimensional direction and adapt to the vibration response of actual broadband. The converse piezoelectric effect of piezoelectric ceramics is used by ultrasonic motors to convert electrical energy into mechanical energy. The research on piezoelectric theory represented by ultrasonic motors is also widely focused. Since Williams et al. [7] invented the ﬁrst piezoelectric actuator in the world, the achievement of piezoelectric driving studied by researchers at the forefront all the time. In 2015, Tan [8] applied piezoelectric actuators designed by himself to medical procedures, and designed the composite controller to drive the ultrasonic motor for precise driving. In the same year, a French scientist Soﬁane Ghenna invented a multi-modal ultrasonic motor [9] to enable the motor to run smoothly in the case of interference. In 2016, Canadian scientists Tavallaei et al. [10] studied the influences of temperature and other uncertain factors on the accuracy and stability of piezoelectric actuator operation, and gave a new situation to study the operational stability of ultrasonic motor. By analyzing the reliability of ultrasonic motor in detail, a second-order model traveling wave was proposed by Kuhne et al. [11] in 2018. It has been shown that the application of ultrasonic motor is more widely promoted. However, the characteristic of piezoelectric ceramics is brittleness, and their fracture toughness is usually only about 1 MPa. In the process of manufacturing piezoelectric material which is subjected to various initial defects, such as cracks, holes and inclusions, are inevitably generated. These initial defects are prone to growth under external loads, resulting in failure of the structure [12, 13]. Therefore, it is of great signiﬁcance to analyze the fracture behaviour which was focused by a lot of people [14]. In order to solve this problem, numerous research results and literatures are employed to analyze these fracture problems. For example, a T-shaped crack in a ceramic material was successfully observed by Kida et al. [15] and they investigated the thermo-electro-mechanical fracture behavior of an inﬁnite piezoelectric material which also was subjected to T-shaped crack under thermo-electro-mechanical loadings [16]. Kwon et al. [17] used the method of integral transformation to transform the

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moving anti-plane crack problem in a strip-shaped piezoelectric material into Fredholm integral equation solved, and the results were the same as those of Hou et al. [18]. In order to solve crack problems, complex analysis method is also employed to obtain the analytical results [19, 20]. Above studies shown that many researches on the piezoelectric characteristics and fracture mechanics of piezoelectric ceramic materials have been carried out by the workers. So far, most of them used the theoretical or experimental ways to study the problems about piezoelectric ceramics. However, due to the coupling relationship of the piezoelectric structure and the complexity of the structural parameters, theoretical modeling is extremely difﬁcult to derive, which leads to a large difference between the experimental design and the theoretical value. At the same time, considering the lack of research on the life prediction of the reliability of ultrasonic motor and energy harvester. The purpose of this article is to use the ABAQUS ﬁnite element software to establish a numerical model of piezoelectric beam, and to study the effects of crack on the output response of the piezoelectric ceramic cantilever beam with two kinds of working modes, such as positive piezoelectric effect and converse piezoelectric effect. It is of great signiﬁcance for the further study of the response relationship between the crack and the resonant frequency. And it is also helpful to predict the residual life of the cracked piezoelectric cantilever beam.

2 The Modeling of Piezoelectric Ceramic Beam 2.1

Description of Piezoelectric Ceramic

Most of the common piezoelectric materials, particularly piezoelectric ceramics, are mainly divided into transversely isotropic piezoelectric ceramics and orthogonal anisotropy piezoelectric ceramics. And a transversely isotropic piezoelectric materials is named as the 6 mm crystal system with its consisting of 10 independent material constants, among which the elastic constant is 5, the piezoelectric constant is 3, and the dielectric constant is 2. An orthogonal anisotropy piezoelectric material is named as the 2 mm crystal system with 17 independent material constants, having 9 elastic constants, 5 piezoelectric constants and 3 dielectric constants. In this article, the 6 mm crystal piezoelectric ceramic (PZT-5H) is employed as the research object. And the isotropic plane is assumed to be perpendicular to the direction of polarization and make the z-axis as its direction of polarization. Since there are four kinds of boundary conditions of crystals, the corresponding piezoelectric constitutive equations are also four ways. Piezoelectric ceramics (PZT-5H) adopts the second type of piezoelectric equation with strain and electric ﬁeld intensity as independent variables, and the piezoelectric equation can be simpliﬁed as following.

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3 2 s11 S1 6 S2 7 6 s12 6 7 6 6 S3 7 6 s13 7 6 S¼6 6 S4 7 ¼ 6 0 6 7 6 4 S5 5 4 0 0 S6 2

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In the Eqs. (1) and (2), the subscript 1 denotes the direction along the x-axis and 2 denotes the y direction and 3 denotes the z direction. 2.2

Finite Element Model of Piezoelectric Cantilever Beam

A piezoelectric cantilever beam is considered in the present study. The geometry and the dimension of the beam, where one end is clamped and a piezoelectric ceramic sheet is also attached at the ﬁxed end, shown as Fig. 1. It is assumed that L and l are the length of cantilever beam and piezoelectric ceramic sheet, hp and hs are the thickness of piezoelectric ceramic sheet and cantilever, bp are the width of cantilever and piezoelectric ceramic sheet. As mentioned above, the parameters of the using piezoelectric ceramic is assumed to be shown in Table 1.

Fig. 1. Finite element model

The ﬁnite element model of the piezoelectric cantilever beam is developed by using the software of ABAQUS. Due to the anisotropic characteristics of piezoelectric materials, when using ABAQUS ﬁnite element software to simulate, the input of material parameters must correspond to the constitutive relationship.

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Parameters

Ceramic material (PZT-5H)

Length (mm) Width (mm) Thickness (mm) Density (kg/m3) Elastic modulous (Gpa) Poisson’s ration Dielectric constant Stiffness coefﬁcient (Gpa)

20 10 0.5 7500 – – k11 = 1.505; k33 = 1.301; (10−8) C11 = 126, C12 = 79.5, C13 = 84.1, C33 = 117, C44 = 23, C66 = 0.5(C11 − C12) e31 = − 6.5, e15 = 17, e33 = 23.3

Piezoelectric constant (c/m2)

Phosphor bronze base 60 10 2.5 8900 117 0.373 – – –

3 Effects of Center Crack on the Mechanic Induced Response 3.1

Finite Element Model of the Cracked Piezoelectric Cantilever Beam

In order to analyze the relationship of crack size and mechanical response, a ﬁnite element model of a cantilever piezoelectric beam with central crack was developed. It is assumed that the crack length is 0.2 mm and its width is 0.1 mm and it is in the center of the piezoelectric sheet. Figure 2 shows the ﬁnite element model. The mesh at the crack tip is reﬁned for considering the calculation accuracy of the model. Due to the stress singularity generally occurs at the crack tip during the vibration process, the ﬁrst node of the second circle at the crack tip is selected as the stress observation point as shown in Fig. 3.

Fig. 2. Finite element Model with Cracks

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Fig. 3. Schematic of stress observation points

3.2

Effects of Crack Size on the Mechanical Response

In order to investigate the effect of crack size on the mechanical response, the impact of the occurrence of cracks and the length of center crack on the output response is discussed. Since the 0.2 mm crack is designed ahead of time, this is the initial state of the crack under actual working conditions, which has practical research signiﬁcance. A kind of sinusoidal excitation force with a size of 1 N is selected by we to study the output response and the length of center crack growth from 0.2 mm, each step increases by 0.2. By comparing the output response of the piezoelectric cantilever beam with the central crack under different sinusoidal excitation forces and different crack lengths, the output responses of displacement, stress and the voltage are obtained. The output response of the cracked piezoelectric cantilever beam is analyzed as shown in Fig. 4. It can be seen from Fig. 4(a) that the initial size of 0.2 mm crack with increasing sinusoidal excitation force has little effect on the displacement of output response. However, when the sinusoidal excitation force is ﬁxed, the end midpoint displacement increases with the size of center crack becoming longer. The variation of displacement changes in the form of exponential function. Figure 4(b) shows the stress output response in PZT-5H under different excitation forces and different length of central cracks. The value of output stress with crack-free beam is greater than that stress with 0.2 mm crack and the difference of output stress is increased with the increasing of excitation force. The effect of the propagation of the central crack on the crack tip stress is approximately linear. Figure 4(c) shows the variation of piezoelectric ceramic output voltage subjected to different excitation forces is similar to the displacement response. The behavior of output voltage under excitation force is opposite of those output voltage under different length of central cracks. The value of output voltage is decreased by increasing the length of central crack. As expected, Fig. 5 shows the increase of central crack length leads to the decrease of the ﬁrst-order resonant frequency of the piezoelectric cantilever. The decreasing slope of the output voltage in longer length is greater than those in shorter length.

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Fig. 4. Effects of crack on mechanical response of piezoelectric cantilever beam

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Fig. 5. 1st modal frequency

4 Effect of Crack on Electrical Induced Response The driving stator of ultrasonic motor is essentially a piezoelectric cantilever structure, and the actuator is driven by deformation under converse piezoelectric effect. Due to the harsh working environment of ultrasonic motors, the physical characteristics of piezoelectric ceramic materials varies and it is are prone to fracture and failure during work. Moreover, ultrasonic motors are widely used in scenes with high safety levels such as artiﬁcial satellites and aerospace devices. A large amount of data shows that the cracks of the piezoelectric material are generally generated in the center, and extended to both sides. Thus, the relationship between the central crack and the electrical responses of the piezoelectric cantilever beam is studied. So as to clarify the mechanism of structural damage and to provide a theoretical guidance for the application and promotion of ultrasonic motor. In order to further analyze the effect of crack on the electrical response of piezoelectric cantilever beam, the output response of the piezoelectric cantilever beam with the central crack under a sinusoidal alternating voltage with 14th frequency of 29586.3 Hz and different crack lengths is analyzed. On the one hand, a sinusoidal alternating voltage with 14th frequency of 29586.3 Hz is applied to the upper surface of the piezoelectric sheet, on the other hand, an initial crack of 0.2 mm at the center of the piezoelectric sheet was set, and then, the effect of increasing crack length of the piezoelectric cantilever beam subjected to ﬁxed alternating voltage of 250 V on the surface with 14th frequency of 29586.3 Hz was studied. The output response of the piezoelectric cantilever is shown in Fig. 6. It can be seen from Fig. 6(a) that the displacement response is completely different to certain extent. When a 0.2 mm small crack occurs on the piezoelectric cantilever piezoelectric sheet, with the increasing of external excitation voltage, the difference of crack on displacement response becomes larger and larger. Under the alternating voltage of 250 V, the displacement response of the end midpoint decreases all the time until the propagation of central crack to 0.6 mm, then the value of displacement

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Fig. 6. Effects of crack on electrical response of piezoelectric cantilever beam

increases with central crack length less than 1.3 mm, and decreases with the central crack length more than 1.3 mm. Figure 6(b) shows that the effect of different excitation voltages on the stress response is little. Under the ﬁxed alternating voltage of 250 V, the stress at the crack tip of the piezoelectric sheet increases with the propagation of the crack, and its variation trend can be regarded as linear relationship approximately. The effect with a 0.2 mm crack for the displacement response and stress response is obvious different. To more clearly describe this change, the rate of decline of output response is deﬁned as follows: the rate of decline ¼

crack free response cracked response crack free response

ð3Þ

The relationship between the excitation voltage and the rate of decline in the output response is shown in Fig. 7.

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Fig. 7. Relationship between excitation voltage and the rate of decline in the output response

The Fig. 8. shows that the 0.2 mm small crack has a relatively large influence on the displacement response of the piezoelectric cantilever beam, and when the value of excitation voltage is reached to a certain extent, keeping the rate of decline trend in the output response unchanged.

Fig. 8. Relationship between crack length and the 14th modal frequency

This trend indicates that the resonant frequency of the piezoelectric cantilever beam has changed when the crack propagates, which affects the magnitude of the resonant amplitude, resulting in unstable output response. Therefore, crack propagation has a signiﬁcant influence on the resonant frequency of the piezoelectric cantilever beam.

5 Conclusions From the analysis of the electro-mechanical response of a cracked piezoelectric cantilever beam, some conclusions are as follows: (1) The electro-mechanical response is proportional to the input force and the input voltage.

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(2) As crack propagates, the variation rule of the electro-mechanical response depends on the crack size. This conclusion is helpful to the prediction of crack size, and to provide a new idea for the prediction of the piezoelectrical structure life.

References 1. Zhao, C.S.: Ultrasonic Motors Technologies and Applications, pp. 128–129. Science Press, Beijing (2007). (in Chinese) 2. Fan, K.Q., Chao, F.B., Zhang, J.G., et al.: Design and experimental veriﬁcation of a bidirectional nonlinear piezoelectric energy harvester. Energy Convers. Manag. 86(10), 561– 567 (2014) 3. Fan, K.Q., Chang, J.W., Pedrycz, W., et al.: A nonlinear piezoelectric energy harvest for various mechanical motions. Appl. Phys. Lett. 106(22), 223902 (2015) 4. Chen, R.W., Ren, L., Xia, H.K., et al.: Energy harvesting performance of a dandelion-like multi-directional piezoelectric vibration energy harvest. Sens. Actuators: Phys. 230, 1–8 (2015) 5. Su, W.J., Zu, J.: An innovative tri-directional broadband piezoelectric energy harvester. Appl. Phys. Lett. 103(20), 203901–203901-4 (2013) 6. Su, W.J., Zu, J.W.: Design and development of a novel bi-directional piezoelectric energy harvester. Smart Mater. Struct. 23(9), 095012 (2014) 7. Kurosawa, M., Kodaira, O., Tsuchitoi, Y., et al.: Transducer for high speed and large thrust ultrasonic linear motor using two sandwich-type vibrators. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45(5), 1188–1195 (1998) 8. Tan, K., Liang, W.Y., Huang, S.N., et al.: Precision control of piezoelectric ultrasonic motor for myringotomy with tube insertion. J. Dyn. Syst. Meas. Control 137(6), 064504-2– 064504-4 (2015) 9. Ghenna, S., Amberg, M., Giraud-Audine, C., et al.: Modelling and control of a travelling wave in a ﬁnite beam, using multi-modal approach and vector control method. In: Joint Conference of the IEEE International Frequency Control Symposium and the European Frequency and Time Forum, villeneuve-d’ascq, franch (2015) 10. Tavallaei, M., Atashzar, S., Drangova, M.: Robust motion control of ultrasonic motors under temperature disturbance. IEEE Trans. Ind. Electron. 63(4), 2360–2368 (2016) 11. Kuhne, M., Rochin, R., Santiesteban, R., et al.: Modeling and two-input sliding mode control of rotary traveling wave ultrasonic motors. IEEE Trans. Ind. Electron. 65(9), 7149– 7159 (2018) 12. Jamia, N., El-Borgi, S., Rekik, M., et al.: Investigation of the behavior of a mixed-mode crack in a functionally graded magneto-electro-elastic material by use of the non-local theory. Theor. Appl. Fract. Mech. 74, 126–142 (2014) 13. Herrmann, K., Loboda, V., Khodanen, T.: An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial. Arch. Appl. Mech. 80(6), 651–670 (2010) 14. Li, Y.D., Xiong, T., Cai, Q.G.: Coupled interfacial imperfections and their effects on the fracture behavior of a layered multiferroic cylinder. Acta Mech. 226(4), 1183–1199 (2015) 15. Kida, K., Saito, M., Kitamura, K.: Flaking failure originating from a single surface crack in silicon nitride under rolling contact fatigue. Fatigue Fract. Eng. Mater. Struct. 28(12), 1087– 1097 (2005)

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16. Ueda, S., Hatano, H.: T-shaped crack in a piezoelectric material thermo-electro-mechanical loadings. J. Therm. Stress. 35(1–3), 12–29 (2012) 17. Kwon, J., Lee, K., Kwon, S.: Moving crack in a piezoelectric ceramic strip under anti-plane shear loading. Mech. Res. Commun. 27(3), 327–332 (2000) 18. Hou, M.S., Qian, X.Q., Bian, W.F.: Energy release rate and bifurcation angers of piezoelectric materials with antiplane moving crack. Int. J. Fract. 107(14), 297–306 (2001) 19. Jangid, K., Bhargava, R.: Complex variable-based analysis for two semi-permeable collinear cracks in a piezoelectro-magnetic media. Mech. Adv. Mater. Struct. 24(12), 1007–1016 (2017) 20. Wan, Y.P., Yue, Y.P., Zhong, Z.: A mode III crack crossing the magnetoelectroelastic bimaterial interface under concentrated magnetoelectromechanical loads. Int. J. Solids Struct. 49(21), 3008–3021 (2012)

A Time-Variant Reliability Analysis Method Considering Maintenance Jingfei Liu, Chao Jiang(&), and Xiangyun Long State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha City 410082, People’s Republic of China {liujingfei,jiangc,longxy}@hnu.edu.cn

Abstract. A time-variant reliability analysis method taking into account the maintenance is proposed in this paper. The effects of maintenance on structural resistance and resistance decay function are ﬁrst quantiﬁed by repair functions. A new time-variant reliability model that are capable of considering maintenance is then established by integrating the repair functions. A solution strategy based on stochastic process discretization is then formulated to calculate the time-variant reliability. A reliability-based optimization approach for designing maintenance strategy is further presented. Finally, three numerical examples are investigated to verify the validity of the proposed method. Keywords: Time-variant reliability Maintenance Stochastic process discretization Reliability-based optimization

1 Introduction Due to material degeneration, random dynamic loads or other time-variant uncertain factors, the reliability of many engineering structures will generally degenerate with time, leading to the so-called time-variant reliability problem. The time-variant reliability analysis computes the probability that a system or component fulﬁlls its intended function over a time interval of interest without failures. It is very important for the full lifecycle design of structures. Therefore, the time-variant reliability analysis problem has received widespread attention in recent years. The main research methods to solve the time-variant reliability problem include the outcrossing rate method, the extremum method and the stochastic process discretization-based method. Among the above-mentioned methods, the most dominating one is the outcrossing rate method, in which the outcrossing rate of performance function at each moment in the design cycle need to be computed to calculate the reliability index under the assumption of Poisson process, Markov process or their improved models. Rice [1, 2] studied the issue of crossings between a dynamic This work is supported by the National Science Fund for Distinguished Young Scholars (51725502), the National Key Research and Development Project of China (2016YFD0701105), and the Science Challenge Project (TZ2018007). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 424–446, 2020. https://doi.org/10.1007/978-981-32-9941-2_35

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response and an allowable threshold and proposed the well-known ﬁrst-passage formula, which established the foundation for dynamic reliability analysis. Engelund et al. [3], proposed approximations of ﬁrst-passage times for differentiable processes based on higher-order threshold crossings. Schall et al. [4], and Rackwitz et al. [5], deal with the reliability problem with time-variant load through the outcrossing rate method. Li and Kiureghian [6–8] investigated the reliability problems involving nonlinear dynamics and inelastic space-variant ﬁnite element analysis. Andrieu-Renaud et al. [9], proposed the PHI2 method, which simpliﬁes the evaluation of outcrossing rate. Hu and Du [10] employed the First Order Reliability Method (FORM) [11–13] for the estimation of outcrossing rate and then proposed the JUR/FORM method. In addition to the above-mentioned outcrossing rate methods, there seems to be a trend to develop some more effective and conceptually simple methods to deal with time-variant reliability problems without using outcrossing rate. Jiang et al. [14], proposed a timevariant reliability analysis method based on stochastic process discretization (TRPD). Wang and Wang [15], and Hu and Du [16] developed several extremum methods where the extreme value distribution of the performance function was modeled ﬁrst and then transformed into the time-variant reliability. Singh et al. [17], analyzed the timevariant reliability of random dynamic systems using the importance sampling approach. Hu and Du [18] combined the equivalent Gaussian process and the stochastic process simulation, thereby proposed a ﬁrst order reliability method for time-variant problems. Besides, various surrogate model-based methods have been proposed to solve the timevariant problems [19–23]. Although the above-mentioned reliability analysis methods have been applied in the engineering successfully, they did not consider the practical maintenance for the structures during its life cycle. However, for many important equipment or structures (e.g. airplanes, automobiles and bridges etc.), regular inspections are usually employed to monitor their health status, and corresponding maintenance strategies are adopted to repair them [25–27]. Since maintenance can change the resistance [28] and the resistance decay function [29], the reliability of the repaired structure will increase [30–32]. Without considering the maintenance, the initial design of the structure will be quite conservative to satisfy the reliability requirement during the whole life cycle, which will cause the waste of resources. Therefore, it is essential to take the maintenance into the consideration of time-variant reliability analysis and optimization design. A time-variant reliability analysis method considering maintenance is developed in this paper. The impact of maintenance on structural reliability can be quantiﬁed, and the maintenance strategy can be optimized to ensure the safety and reliability of structure. The remainder of this paper is organized as follows: Sect. 2 provides the problem statement; Sect. 3 details the speciﬁc formulation for time-variant reliability analysis method considering maintenance; Sect. 4 proposes an optimization design approach for maintenance strategy. Section 5 discusses several numerical examples; And Sect. 6 gives the conclusion of this study.

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2 Brief Introduction of Time-Variant Reliability Analysis Structural time-variant reliability refers to the probability that a structure can complete its intended function within a speciﬁc time frame under speciﬁc conditions while experiencing the effects of time-variant uncertainty. The reliability of a structure within a time interval ½0; T can be expressed as [9]: Ps ðT Þ ¼ PfgðXðtÞ; Y; tÞ [ 0; 8t 2 ½0; Tg

ð1Þ

in which Pfg stands for the probability calculation, gðÞ is the performance function, XðtÞ ¼ ðX1 ðtÞ; X2 ðtÞ; . . .; Xm ðtÞÞ is an m-dimensional stochastic process vector, and Y ¼ ðY1 ; Y2 ; . . .; Yn Þ is an n-dimensional random vector. The outcrossing rate method [5, 9] is currently the most important method to solve time-variant reliability problem. It ﬁrst calculates the outcrossing rate, which is the mean number of crossings of the performance function from safe state to unsafe state in unit time. The structural reliability or failure probability can then be derived from the outcrossing rate. Therefore, the calculation of outcrossing rate is the core of this type of methods. The outcrossing rate m þ ðtÞ can be deﬁned as [33]: m þ ðt Þ ¼

PfN þ ðt; t þ DtÞ ¼ 1g Dt!0;Dt [ 0 Dt lim

ð2Þ

where N þ ðt; t þ DtÞ stands for the number of crossings of performance function from safe state to unsafe state within the time interval ½t; t þ Dt. Because the numerator of Eq. (2) stands for the probability that the performance function crosses only one time during the interval ½t; t þ Dt, the outcrossing rate can also be given as follows [34]: T PfgðXðtÞ; Y; tÞ [ 0 gðXðt þ DtÞ; Y; t þ DtÞ 0g m þ ðt Þ ¼ ð3Þ lim Dt!0;Dt [ 0 Dt If the outcrossing events are rare and independent, the outcrossing events can be considered to obey Poisson distribution. Therefore, Eq. (1) can be approximated by the following formula [35]: Z Ps ðT Þ ¼ 1 Ps ð0Þ exp

T

m þ ðtÞdt

ð4Þ

0

in which Ps ð0Þ stands for the initial reliability of structure. However, when the outcrossing events are strongly dependent, evaluating time-variant reliability by Eq. (4) may lead to a large error, thus some approaches [10, 35] have been proposed to better model the outcrossing events considering their dependence under different model assumptions, which improves the accuracy.

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3 Time-Variant Reliability Modeling and Analysis Considering Maintenance Though the time-variant reliability analysis methods have made great progress, most of them have not taken the influence of practical maintenance into account. However, maintenance is widespread in practical structures and have important impact on the structural reliability. Taking maintenance into the analysis of time-variant reliability can avoid the conservative initial design, thus to further reduce the waste of resources. It is necessary to develop a method that can quantify the impact of maintenance on the structural reliability and solve the time-variant reliability problem considering maintenance. Therefore, in this paper, for quantifying the impact of maintenance on structural reliability, a time-variant reliability model considering maintenance is ﬁrst proposed. The impacts of maintenance on structural resistance and resistance decay function are expressed by repair functions explicitly, and those repair functions are integrated into the performance function to further express the impact of maintenance on the structural reliability. Finally, an efﬁcient time-variant reliability analysis method iTRPD is employed to solve the model. 3.1

Reliability Modeling

According to the time-variant reliability analysis model [9, 10], the structural performance function before maintenance can be given by: gðtÞ ¼ gðXðtÞ; Y; tÞ

ð5Þ

Assume that XðtÞ¼½Xr ðtÞ; Xs ðtÞ, Xr ðtÞ ¼ ðX1 ðtÞ; X2 ðtÞ; . . .; Xa ðtÞÞ is a stochastic process vector representing the structural resistance, and Xs ðtÞ ¼ ðXa þ 1 ðtÞ; Xa þ 2 ðtÞ . . .; Xm ðtÞÞ is a stochastic process vector representing the structural load; Y¼½Yr ; Ys , Yr ¼ ðY1 ; Y2 ; . . .; Yb Þ is the random vector representing the structural resistance, and Ys ¼ ðYb þ 1 ; Yb þ 2 . . .; Yn Þ is the random vector representing the structural load. When the maintenance is considered, the performance function Eq. (5) can be further given as follows [36–39]: 8 < gðtÞ ¼ gðRðtÞ; SðtÞ; tÞ RðtÞ ¼ R0 uðXr ðtÞ; Yr ; tÞ ð6Þ : SðtÞ ¼ SðXs ðtÞ; Ys ; tÞ where RðtÞ represents the structural resistance, SðtÞ represents the structural load, R0 is the initial resistance of structure, and uðXr ðtÞ; Yr ; tÞ is the structural resistance decay function, whose speciﬁc forms are determined by material properties, service environment and other factors [39].

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Since maintenance generally has no effect on the external load, the structural load after i-th maintenance is assumed to be not changed. Therefore, the performance function of structure after i-th (i = 1, 2, …, n) maintenance is deﬁned as follows: gi ðt; si Þ ¼ gi Ri ðt; si Þ; Sðt; si Þ; t

ð7Þ

where gi ðÞ and Ri ðt; si Þ are the performance function and structural resistance after i-th maintenance, respectively, and si represents the time to take i-th maintenance. The effects of maintenance on structural resistance includes the effect on structural resistance [28] and the effect on structural resistance decay function [29]. Combining the speciﬁc maintenance strategy [25–27, 30–32], the structural resistance after i-th maintenance can be expressed as follows: Ri ðt; si Þ ¼ Ri0 ðR0 ; si Þui ðXr ðtÞ; Yr ; t; si Þ

ð8Þ

where Ri0 ðR0 ; si Þ ¼ R0 As eBs si and ui ðXr ðtÞ; Yr ; t; si Þ ¼ u0 ðXr ðtÞ; Yr ; tÞCs eDs si represent the structural resistance and resistance decay function after i-th maintenance, respectively, where As and Bs represent the maintenance control parameters of structural resistance, Cs and Ds represent the maintenance control parameters of structural resistance decay function. In this paper, the control parameters including As , Bs , Cs , Ds are employed to establish the repair functions, which are determined by practical maintenance strategy that includes maintenance time, means, etc. Different maintenance strategies will cause different impacts on the structure, and different impacts will be further represented by different control parameters. Actually, many structures need to be repaired in their design cycle according to their health statuses. If we can obtain those maintenance data, including the maintenance strategies and the corresponding impacts of maintenance on structures, then regression analysis or maximum likelihood estimation can be utilized to calculate the control parameters. Therefore, the impacts of maintenance on structural resistance and resistance decay function can be expressed by the repair functions explicitly. In turn, we can optimize the maintenance strategy by tuning those control parameters. For simplicity, a piecewise performance function is employed to represent the effect of maintenance on the structure during its design cycle. The piecewise performance function can be given as follows: 8 0 g ðRðtÞ; SðtÞ; tÞ; t 2 ½0; s1 Þ > > > < g1 ðR1 ðt; s1 Þ; Sðt; s1 Þ; t; s1 Þ; t 2 ½s1 ; s2 Þ GðXðtÞ; Y; tÞ ¼ .. > > . > : i i g ðR ðt; si Þ; Sðt; si Þ; t; si Þ; t 2 ½si ; T

ð9Þ

where g0 ðÞ stands for the performance function without maintenance, gi ðÞ is the performance function after i-th maintenance.

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Combining Eqs. (1) and (9), the reliability of the structure with i times maintenance during its design cycle ½0; T can be expressed as follows: Pis ð0; T Þ ¼ P g0 ðRðtÞ; SðtÞ; tÞ [ 0; 8t 2 ½0; s1 \ . . . \ g1 R1 ðtÞ; S1 ðtÞ; t; s1 [ 0; 8t 2 ½s1 ; s2 \ . . . \ gi Ri ðtÞ; Si ðtÞ; t; si [ 0; 8t 2 ½si ; T

ð10Þ

The time-variant reliability considering maintenance can be further calculated by solving the model in Eq. (10). 3.2

Reliability Solution

In order to calculate the overall safe probability of the structure with i times maintenance during its whole lifecycle ½0; T, the recently proposed improved time-variant reliability analysis method based on stochastic process discretization (iTRPD) [24] is introduced to solve this problem. The primary schedule of iTRPD can be summarized as follows. Firstly, discretize the time-variant performance function into a certain number of time-invariant performance function, thus the time-variant reliability problem is transformed into a timeinvariant system reliability problem. Secondly, perform time-invariant reliability analysis for the discretized performance functions at all the time points by FORM to calculate the reliability index vector, and a corresponding approach is further given to calculate the correlation coefﬁcient matrix q of all the components’ performance functions. Finally, the time-variant reliability can be approximately calculated based on the reliability index vector and the correlation coefﬁcient matrix. iTRPD avoids the calculation of outcrossing rate, greatly simplifying the process of solving time-variant reliability problems. Combining Eqs. (7–9), a time-variant performance function with i times maintenance GðXðtÞ; Y; tÞ is discretized over ½0; T at a certain number p of nodes with a time step Dt ¼ T=p, the time-variant reliability problem is changed to an a p-component time-invariant series system reliability problem, whose performance functions can be expressed as G Xj ðtÞ; Y; tj ; tj ¼ j 12 Dt; Xj ¼ Xr;j ; Xs;j ; j ¼ 1; 2; . . .; p, where tj represents the midpoint of the j-th discretization length. Hence, the reliability of the problem over the whole design cycle can be expressed as: (

Pis ð0; T Þ

p \ 1 T ¼P G Xj ; Y; tj [ 0; tj ¼ j Dt; Dt ¼ 2 p j¼1

ð11Þ

As the time is ﬁxed at tj , the stochastic processes XðtÞ degrade into random parameters Xj whose mean values and variances are determined by the mean value and

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covariance functions of the stochastic processes. Employing the Nataf transformation [40] to transform Xj and Y into standard Gaussian vectors X0j and Y0 , respectively: (

X0i ; COV X0i ¼ Nataf ðXi ; COV ðXi ÞÞ ðY0 ; COV ðY0 ÞÞ ¼ Nataf ðY; COV ðYÞÞ

ð12Þ

in which Nataf ðÞ represents the Nataf transformation. X0j and Y0 can be further transformed into standard independent Gaussian vectors by orthogonal transformation, respectively: (

U ¼ ATx X0j

ð13Þ

V ¼ ATy Y0

where ATx and ATy represent orthogonal transformation matrix of X0j and Y0 , after the above transformation, Eq. (11) can be represented as follows: (

Pis ð0; T Þ

p \ 1 T 0 ¼P G ðUj ; V; tj Þ; tj ¼ j Dt; Dt ¼ 2 p i¼1

ð14Þ

Therefore, the reliability index bj and MPP for each discretized performance function G0 ðUj ; V; tj Þ; j ¼ 1; 2; . . .; p can be calculated by FORM [11–13]. By linearizing G0 ðUj ; V; tj Þ; j ¼ 1; 2; . . .; p at their MPPs, Eq. (14) can be approximately expressed as: ( Pis ð0; T Þ

¼P

p \

) bj þ aU;j UTj

þ aV;j V [ 0 T

ð15Þ

i¼1

in which aU;j ; aV;j represents the unit gradient vector of the j-th performance function with respect to U; Vp at the MPP. Let Lj ¼ bj þ aU;j UTj þ aV;j VT , Eq. (15) can be written as: ( Pis ðT Þ

¼P

p \

) Lj [ 0

ð16Þ

j

Obviously, keeping the structure safe during the whole design cycle, every component corresponds to the time-invariant performance function G0 ðUj ; V; tj Þ; j ¼ 1; 2; . . .; p at arbitrary time tj needs to be positive. According to the ﬁrst-order method for series system reliability [32], Eq. (16) can be calculated by: Pis ð0; T Þ ¼ Up ðb; qÞ

ð17Þ

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where Up represents the p-dimensional standard Gaussian cumulative distribution function, b ¼ ðb1 ; b2 ; . . .; bp Þ represents the reliability index vector and q represents the correlation coefﬁcient matrix of the linearized performance functions. Assume all the stochastic processes in XðtÞ and random parameters in Y are mutually independent for simplicity. Each component of q, qj;k ; j 2 1; 2; . . .; p, is the correlation coefﬁcient of Lj and Lk : qj;k ¼

m X

au;j;l au;j;l q Uj;l ; Uk;l þ aTV;j aV;k

ð18Þ

l¼1

The calculation details of q can be referred to [24, 48]. Besides, the calculation of multi-dimensional Gaussian distribution can be referred to Refs. [35, 40–44]. The framework of the newly proposed method is summarized in Fig. 1.

Reliability modeling considering the maintenance

Discrete the stochastic process into p segments

Implement Nataf transformation and orthogonal transformation

Calculate the time invariant reliability index by FORM

Calculate the autocorrelation and intercorrelation of stochastic processes

The time variant reliability is obtained by calculating a multi-dimensional Gaussian integral

Fig. 1. Framework of the proposed method

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4 Optimization of the Maintenance Strategy Structural time-variant reliability refers to the probability that a structure can complete its intended function within a speciﬁc time frame under speciﬁc conditions while experiencing the effects of time-variant uncertainty. Thus, the time-variant reliability of a structure within a time interval ½ta ; tb can be expressed as: Ps ðta ; tb Þ ¼ PfgðXðtÞ; Y; tÞ [ 0; 8t 2 ½ta ; tb g

ð19Þ

The reliability indexes corresponding to the above time-variant reliability can be calculated as follows: bðta ; tb Þ ¼ U1 ðPs ðta ; tb ÞÞ

ð20Þ

Accumulative Reliability index

For better understanding of the impact of different maintenance strategies on structural reliability, a schematic diagram is given in Fig. 2. As shown in Fig. 2, bi ð0; si Þ represents the reliability index of structure within the time interval ½0; si , bH represents the structural reliability index during the whole design cycle ½0; T without maintenance, b1H and b2H correspond to the reliability indexes during the whole design cycle ½0; T with once and twice maintenance, respectively, and bt is the structural target reliability within the design cycle. As is seen from Fig. 2, when bH is smaller than bt , it indicates that the structure is not safe enough to satisfy the reliability requirement without maintenance during its design cycle. After being repaired once at time s1 , bH is increased to b1H , thus the repaired structure can satisfy the reliability requirement. Furthermore, if the second maintenance is employed at time s2 , then b1H is increased to b2H . Since different maintenance strategies will cause different influences on the structural reliability, it is essential to develop an optimization model for maintenance strategy, by which the optimal maintenance strategy can be obtained and the reliability of the structure can be ensured.

β 1 (0,τ 1 )

β 2 (0,τ 2 )

β H2 β H1

βt

βH

τ1

τ2

Design lifetime

Fig. 2. Schematic diagram of structural reliability index curve

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In the optimization of the maintenance strategy, the maintenance parameters and maintenance time are taken as design variables, and the objective is to maximize the structural time-variant reliability during its design cycle. Thus, the optimization model of the maintenance strategy can be expressed as follows: max

As ;Bs ;Cs ; Ds ;i;si s:t: biH bt

Pis ð0; T Þ

As ; Bs ; Cs ; Ds 2 X 0 i n; 0 si T

ð21Þ

where biH corresponds to the reliability index during the design cycle ½0; T with i ¼ 1; 2; . . .; n times maintenance, bt is the structural target reliability index within the whole design cycle, As , Bs , Cs , Ds are the parameters determined by maintenance strategy [25–27]. All of those parameters are collectively called maintenance strategy optimization parameters in the rest of this paper. In order to solve the optimization model in Eq. (21), the adaptive response surface method (ARSM) [49] is employed in this paper. ARSM ﬁrst uses some initial points to construct an initial response surface, which is then utilized to adaptively choose the next experimental points to update the response surface. This sequential procedure terminates when the convergence criterion is satisﬁed. Then, this easy to call response surface is optimized to obtain the optimal result. Combining Eq. (21) and Eqs. (11– 20), the optimal maintenance strategy can be easily obtained by ARSM.

5 Numerical Examples and Discussions In this section, three numerical examples are used to demonstrate the effectiveness of the proposed method. In all examples, the time-variant reliability analysis considering maintenance of the structure and the maintenance strategy optimization are both performed. Besides, only the maintenance time is optimized while the other maintenance strategy optimization parameters are given in each example. 5.1

A Ten-Bar Truss

In this example, a ten-bar truss [45] as shown in Fig. 3 is investigated. The length L of the horizontal and vertical bars is 9.1 m. Ai ; i ¼ 1; 2; . . .; 10 denote the cross-sectional areas of the bars. The Young’s modulus E of the truss is 68948 Mpa, and the density q is 2768 kg/m3. Joint 4 is subjected to a vertical load F1 , and joint 2 is subjected to a vertical load F2 and a horizontal load F3 ðtÞ. The vertical displacement dy ðtÞ at joint 2 is required to be no larger than the allowable displacement d ðtÞ ¼ d0 ð1 0:02tÞ, where d0 is the initial value of d ðtÞ. All of the involved random parameters are listed in

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Table 1, and the maintenance strategy optimization parameters are listed in Table 2. The performance function with and without maintenance are given as follows: 8 ðtÞ dy ðtÞ; t 2 ½0; T < gðtÞ ¼ d 6 0 10 0 pﬃﬃﬃ P P Ni ðtÞNi ðtÞ Ni ðtÞNi ðtÞ L d ð t Þ ¼ þ 2 : y E Ai Ai i¼1

ð22Þ

i¼7

8 < d ðtÞ dy ðtÞ; t 2 ½0; s1 Þ GðXðtÞ; Y; tÞ ¼ d 1 ðtÞ dy ðtÞ; t 2 ½s1 ; s2 Þ : 2 d ðtÞ dy ðtÞ; t 2 ½s2 ; T

ð23Þ

Where 8 i < d ðt; si Þ ¼ d0 As eBs si ð1 0:02tÞCs eDs si 6 0 10 0 pﬃﬃﬃ P P Ni ðtÞNi ðtÞ Ni ðtÞNi ðtÞ L þ 2 : dy ðt; si Þ ¼ E Ai Ai i¼1

ð24Þ

i¼7

pﬃﬃﬃ pﬃﬃﬃ ¼ F ðtÞ 2 2N8 ; N2 ¼ 2 2N10 pﬃﬃﬃ ¼ F1 2F ðtÞ þ F2 2 2N8 pﬃﬃﬃ ¼ F ðtÞ þ F2 2 2N10 pﬃﬃﬃ pﬃﬃﬃ 2 2N10 5 ¼ F ðt Þ 2 2N8 pﬃﬃﬃ N ¼ 2 2N 6 10 > pﬃﬃﬃ > > > N ¼ 2 ð F þ F ðtÞÞ þ N8 > 7 1 > > > N ¼ ða b a b2 Þ=ða11 a22 a12 a21 Þ > 8 22 1 12 pﬃﬃﬃ > > > N ¼ 2 F ð t Þ þ N > 9 10 : N10 ¼ ða11 b2 a21 b1 Þ=ða11 a22 a12 a21 Þ

ð25Þ

pﬃﬃﬃ 8 a11 ¼a22 ¼ 3=X1 þ 4 2 X2 L=ð2EðtÞÞ > > < a12 ¼a21 ¼ L=ð2X1 EðtÞÞ pﬃﬃﬃ pﬃﬃﬃ b ¼ ð F þ F ð t Þ Þ=X 2 2 ð F þ F ð t Þ Þ X2 2L=ð2EðtÞÞ > 1 1 1 1 > pﬃﬃﬃ : b2 ¼ 2ðF2 2F ðtÞÞ X1 4F ðtÞ=X2 L=ð2EðtÞÞ

ð26Þ

8 N1 > > > > > N 3 > > > > N4 > > > >

< BðtÞ rmax ðqðtÞ; E; q; tÞ; t 2 ½0; s1 Þ em As eBs s1 e0:01t Cs eDs s1 eðq; E; QðtÞÞ; t 2 ½0; s1 Þ > : em As eBs s2 e0:01t Cs eDs s2 eðq; E; QðtÞÞ; t 2 ½s2 ; T

ð35Þ

In this automobile frame example, the condition for structural reliability is that the limit strain satisfy the design requirement in the design cycle. The above-mentioned distributions of all the random variables and random process are listed in Table 13 and the maintenance strategy optimization parameters are listed in Table 14. The optimization model for automobile frame is given as: max Pis ð0; T Þ si

s:t:biH bt As ¼ 1; Bs ¼ 0:015 Cs ¼ 1; Ds ¼ 0:02 i¼f2g; 0 si T

ð36Þ

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Fig. 8. An automobile frame structure [48]

Table 13. Distributions of the random parameters for the automobile frame Parameter

Type of distribution Limit strain Type I extreme Young’s Modulus E Normal Density q Lognormal Load QðtÞ Gaussian process

Mean

Coefﬁcient of variation (%) 4.0 10 206.86 MPa 10 7700 kg/m3 10 500000 N 10

Autocorrelation coefﬁcient function NA NA NA exp½ð3s2 Þ

Table 14. Values of the maintenance strategy optimization parameters for the automobile frame Parameter As Bs Cs Ds bt Value 1 −0.015 1 0.02 1.8

In this example, we consider twice maintenance during the design cycle of the automobile frame. The reliability indexes of the automobile frame without maintenance are listed in Table 15. The optimal maintenance time is listed in Table 16, and the corresponding reliability indexes are shown in Table 17. It is clear from the tables that the automobile frame cannot satisfy the reliability requirement bt ¼ 2:0 in the whole

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design cycle without maintenance, and the reliability index b1H is increased by 16.92% after the automobile frame is repaired at time s1 ¼ 3:14 and s2 ¼ 6:05. The results for the automobile frame with twice maintenance are shown in Fig. 9. Table 15. The reliability indexes for the automobile frame without maintenance t (Years) b t (Years) b

1 2.59 6 2.03

2 2.34 7 1.97

3 2.26 8 1.92

4 2.17 9 1.87

5 2.09 10 1.83

Table 16. The optimal time for twice maintenance of the automobile frame s2

s1

maxb1H DbH ð%Þ

3.14 6.05 2.05

+16.92

Table 17. The reliability indexes for the automobile frame with twice optimal maintenance t (Years) b t (Years) b

2.7

1 2.56 6 2.13

2 2.32 7 2.11

3 2.15 8 2.10

4 2.21 9 2.08

5 2.17 10 2.05

No maintenance Once maintenance Best maintenance

2.6

Accumulative reliability index

2.5 2.4 2.3 2.2 2.1 2

1.9

βt

1.8 1.7 1

2

3

4 5 6 7 Design lifetime(years)

8

9

10

Fig. 9. The curves indicating the reliability indexes for the automobile frame with twice optimal maintenance

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6 Conclusions In this paper, a time-variant reliability method considering maintenance is proposed, which provides an effective tool for the problem of time-variant maintenance reliability analysis. A new reliability model is ﬁrst presented, by which the impact of maintenance on structural reliability can be quantiﬁed. Besides, an optimization model is also developed to determine the optimal maintenance strategy. Several numerical examples illustrated the effectiveness of the proposed method, the results of which indicate that maintenance can influence structural reliability, and reasonable maintenance strategy is necessary for structure to satisfy the reliability requirement during its design cycle. Thus, it is essential to take maintenance into account during the process of reliability analysis. In the future, this method can be extended to reliability-based multi-objective design optimization. Acknowledgements. This work is supported by the National Science Fund for Distinguished Young Scholars (51725502), the National Key Research and Development Project of China (2016YFD0701105), and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51621004).

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34. Hagen, Ø., Tvedt, L.: Vector process out-crossing as parallel system sensitivity measure. J. Eng. Mech. 117(10), 2201–2220 (1991) 35. Madsen, P.H., Krenk, S.: Integral equation method for the ﬁrst-passage problem in random vibration. J. Appl. Mech. 51(3), 674–679 (1984) 36. Enright, M.P., Frangopol, D.M.: Probabilistic analysis of resistance degradation of reinforced concrete bridge beams under corrosion. Eng. Struct. 20(11), 960–971 (1998) 37. Geidl, V., Saunders, S.: Calculation of reliability for time-varying loads and resistances. Struct. Saf. 4(4), 285–292 (1987) 38. Kameda, H., Koike, T.: Reliability theory of deteriorating structures. J. Struct. Div. 101(1), 295–310 (1975) 39. Mori, Y., Ellingwood, B.R.: Time-dependent system reliability analysis by adaptive importance sampling. Struct. Saf. 12(1), 59–73 (1993) 40. Liu, P.-L., Der Kiureghian, A.: Multivariate distribution models with prescribed marginals and covariances. Probabilistic Eng. Mech. 1(2), 105–112 (1986) 41. Hohenbichler, M., Rackwitz, R.: First-order concepts in system reliability. Struct. Saf. 1(3), 177–188 (1982) 42. Billinton, R., Jain, A.V.: The effect of rapid start and hot reserve units in spinning reserve studies. IEEE Trans. Power Appar. Syst. 2, 511–516 (1972) 43. Lian, G., Billinton, R.: Operating reserve risk assessment in composite power systems. IEEE Trans. Power Syst. 9(3), 1270–1276 (1994) 44. Low, B., Zhang, J., Tang, W.H.: Efﬁcient system reliability analysis illustrated for a retaining wall and a soil slope. Comput. Geotech. 38(2), 196–204 (2011) 45. Elishakoff, I., Haftka, R., Fang, J.: Structural design under bounded uncertainty— optimization with anti-optimization. Comput. Struct. 53(6), 1401–1405 (1994) 46. Guan, F.J., Han, X., Jiang, C.: Uncertain optimization of the engine crankshaft using interval methods. Eng. Mech. 25(9), 198–202 (2008). (in Chinese) 47. Wang, L.G., Hu, D.B.: FEM analysis on 368Q crankshaft fatigue strength and some discuss on relative problems. CSICE 18(3), 270–274 (2000) 48. Jiang, C., Han, X., Li, W.X., et al.: A hybrid reliability approach based on probability and interval for uncertain structures. J. Mech. Des. 134(3), 031001 (2012) 49. Wang, G.G.: Adaptive response surface method using inherited Latin hypercube design points. J. Mech. Des. 125(2), 210–220 (2003)

Research on Tooth Proﬁle Error of Non-circular Gears Based on Complex Surface Theory Yongping Liu, Fulin Liao, and Changbin Dong(&) School of Mechanical and Electronical Engineering, Lanzhou University of Technology, Lanzhou 730050, China [email protected], [email protected], [email protected]

Abstract. In view of the complex shape and structure of non-circular gears, it is difﬁcult to measure the machining errors of non-circular gears, and the complex surface theory is proposed to measure the errors. Firstly, the Euclidean transformation matrix is used to establish the error evaluation function, and the scanner is used to digitally scan the non-circular gears, collect the coordinates of the data points on the tooth surface. Then, the non-uniform rational B-spline function is used to accurately describe the tooth surface and reconstruct the tooth surface. Finally, the theoretical data are matched with the reconstructed tooth surface by coordinate transformation to calculate the tooth surface error. The feasibility of the proposed algorithm is proved by a calculation example, which provides a reference for further accurate detection of the non-circular gear tooth surface error. Keywords: Non-circular gear Tooth surface error Reconstruct tooth surface Match

1 Introduction The non-circular gears is a gear mechanism with variable transmission ratio, which has the advantages of large transmission power range, high transmission efﬁciency, accurate transmission ratio, long service life, strong carrying capacity and reliable operation [1]. At present, with the improvement of the transmission accuracy of non-circular gears, higher requirements are put forward for the processing accuracy and processing accuracy detection of non-circular gears. However, due to the complex shape and structure of non-circular gears, a wide variety of types, and many measurement parameters, so far, there is little research on their geometric accuracy, and there is a lack of effective measurement means [2]. To solve this problem, many scholars have done a lot of research. Among them, Wang mentioned the influence of NC hobbing processing

This project is supported by National Natural Science Foundation of China (Grant No. 51765032). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 447–455, 2020. https://doi.org/10.1007/978-981-32-9941-2_36

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errors and process errors on the accuracy of non-circular gears, and studied the errors of non-circular gears through interpolation algorithm and error compensation [3]. Lu and Wang put forward the use of polar coordinate method to measure the individual errors of eccentric circular gears in CNC gear measuring center [4]. Lin et al. proposed measuring and evaluating the tooth proﬁle error of super large gear based on NURBS surface ﬁtting [5]. Xie et al. put forward the method of NURBS model reconstruction and two-step matching to evaluate the surface error of non-bevel gears in variable ratio differential [6]. On the basis of the above research, aiming at the fact that the left and right sides of each tooth proﬁle of non-circular gear are different and the detection is difﬁcult, and in order to obtain accurate tooth surface error and make the transmission of gear pair more stable, this paper proposes to study the tooth proﬁle error of non-circular gear by using complex surface theory.

2 Mathematical Model for Evaluating Tooth Surface Error Before evaluating the tooth proﬁle error, it is necessary to reconstruct the tooth surface with actual data, and match the reconstructed tooth surface with the theoretical data point. As a rigid body, the reconstructed tooth surface is equivalent to the theoretical data of the tooth surface. The coordinate transformation makes the reconstructed tooth surface contain the theoretical data as much as possible, and minimizes the distance error between the theoretical data and the tooth surface [7]. So the key of matching is to solve the Euclidean transformation matrix. M, and ensure the theoretical data Pi ¼ ðxi ; yi ; zi Þði ¼ 0; 1; 2; . . .; nÞ were contained as much as possible by the recon0 0 0 0 structed tooth surface. Assume transformed data is Pi ¼ ðxi ; yi ; zi Þ, the relative between the theoretical data and the transformed data as follows: 0

0

0

ðxi ; yi ; zi ; 1Þ ¼ ðxi ; yi ; zi ; 1Þ Mði ¼ 0; 1; 2; . . .; nÞ M¼

R T

0 1

ð1Þ

ð2Þ

Where T ¼ ½ T x T y T z , Note that T is displacement matrix of theoretical data relative to reconstructed tooth surface, and T x , T y , T z is displacement vector that theoretical data along the x, y, and z axes of the coordinate axis respectively, and R is the rotation matrix of theoretical data relative to the reconstructed tooth surface, R as follows:

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R ¼ rij 33

449

ð3Þ

8 < r11 ¼ cos a cos b r ¼ cos b sin c : 12 r13 ¼ sin b

ð4Þ

8 < r21 ¼ sin a sin b cos c cos a sin c r ¼ sin a sin b sin c cos a cos c : 22 r23 ¼ sin a cos b

ð5Þ

8 < r31 ¼ cos a sin b cos c þ sin a sin c r ¼ cos a sin b sin c sin a cos c : 32 r33 ¼ cos a cos b

ð6Þ

Note that a, b, c is the rotation angle of theoretical data rotate around the x, y, and z axes respectively. According to the principle of least squares, the evaluation functions follows: F¼

n X P0 Qi 2 i

ð7Þ

i¼1 0

Note that the closest point of point Pi is the point Qi on CAD tooth surface model.

3 Data Acquisition of Non-circular Gear Tooth Surface The purpose of data acquisition is to obtain the three-dimensional point cloud of noncircular gears, which is the process of digitizing the shape of gears. At present, there are two main methods to obtain three-dimensional point cloud data of gears: stereo vision method and laser scanning measurement method. According to the geometric characteristics of the selected research object, a non-contact active laser ranging system is used to obtain the three-dimensional point cloud of the gear. Creaform Metrascan scanner is selected to digitize non-circular gears. Figure 1 is the scanning process, and Fig. 2 is the computer interface of the scanning process

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Fig. 1. The process of scanning

Fig. 2. Computer interface for scanning process.

In order to collect the actual and theoretical 3D coordinates of non-circular gears, the 3D point cloud and non-circular gear CAD model are imported into reverse engineering software. The command [analysis/point coordinates] is selected to collect the actual 3D coordinates of the point cloud, and the command [analysis/creation annotations] is selected to collect the 3D coordinates of the theoretical tooth surface of non-circular gear CAD model. In order to study the tooth surface errors of different teeth, the actual and theoretical three-dimensional coordinates of the teeth No. 1, No. 7, No. 14, No. 20, No. 27, and No. 33 are collected respectively. Table 1 shows the actual three-dimensional coordinates of the left tooth surface of No. 1.

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Table 1. The actual 3D coordinates of left tooth surface of the 1st tooth. Number 3D coordinates x y 1 63.906 −3.362 2 62.733 −2.705 3 61.416 −2.143 4 59.987 −1.706 5 63.860 −3.324 6 62.839 −2.761 … … … 59 61.326 −2.184 60 60.089 −1.837 61 63.953 −3.443 62 62.974 −2.879 63 61.327 −2.169 64 59.997 −1.777

z 1.722 1.688 1.705 1.669 3.003 2.948 … 22.180 22.254 23.771 23.717 23.830 23.823

4 Tooth Surface Reconstruction In order to calculate the distance between the theoretical data points and the reconstructed tooth surface, the Non-Uniform Rational B-Spline curve method was used to accurately describe the reconstructed tooth surface [8]. The rational form of p q times NURBS tooth surface as follows: m P n P

Sðu; vÞ ¼

xi; j Pi; j Ni;p ðuÞNj;q ðvÞ

i¼0 j¼0 m P n P

xi; j Ni;p ðuÞNj;q ðvÞ

ð8Þ

i¼0 j¼0

Note that Pi; j is control grid in direction of u and v, 0 u, v 1, xi; j is weight factor, Ni;p ðuÞ and Nj;q ðvÞ is Non-Uniform Rational B-Spline basis function, which was deﬁned on node vector U and V respectively. The node vector as follows: 9 8 8 > > > = < > > > > U ¼ 0; ; 0 ; u ; ; u ; 1; ; 1 > p þ 1 rp1 > |ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} > > > ; :|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} < pþ1 pþ1 9 8 ð9Þ > > > > = < > > > > V ¼ 0; ; 0; vq þ 1 ; ; vsq1 ; 1; ; 1 > > |ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} > > : ; :|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} qþ1

Where r ¼ n þ p þ 1,s ¼ m þ q þ 1.

qþ1

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The actual 3D coordinates of the collected tooth surface are imported into the numerical calculation software, and the reconstructed tooth surface can be obtained by ﬁtting and interpolating the above method. Figure 3 shows the reconstructed left and right tooth surfaces of No. 1 tooth.

-1.5 30 -2

z/mm

20 -2.5

10 0 59

-3 60

61

y/mm

62

63

-3.5

64

x/mm

65

(a) Left tooth surface of No. 1 tooth

30

-6.8

z/mm

20 -7 10 -7.2 0 58

-7.4 59

60

61

x/mm

y/mm 62

63

-7.6

(b) Right tooth surface of No. 1tooth Fig. 3. The reconstructed tooth surface of the No. 1 tooth.

5 Matching of Theoretical Data with Reconstructed Tooth Surface Matching is a process of making theoretical 3D coordinates and reconstructed tooth surface in the same coordinate system by coordinate transformation. Firstly, select 3 corner points Pi ði ¼ 0; 1; 2Þ from theoretical 3D coordinates, and select 3 corresponding corner points Qi ði ¼ 0; 1; 2Þ from reconstructed tooth surface, and then construct two sets of unit vectors with corner points from theoretical 3D coordinates and reconstructed tooth surface respectively, two sets of unit vectors as follows:

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8 P1 P0 > < e1 ¼ jP1 P0 j e2 ¼ e3 e1 > : e3 ¼ e1 P2 P0 jP2 P0 j

ð10Þ

8 0 > e ¼ Q1 Q0 > < 1 jQ1 Q0 j 0 0 0 e2 ¼ e3 e1 > > : e0 ¼ e0 Q2 Q0 3 1 jQ2 Q0 j

ð11Þ

And construct two local coordinate systems through taking point P0 and point Q0 as coordinate respectively, and take unit vector group ½ e1 e2 e3 and unit vector 0 origin 0 0 group e1 e2 e3 as axes respectively. Assume the two local coordinate systems are completely coincident through coordinate transformation of Eq (1). Therefore, the relation between the two local coordinates as follows:

0

e1

0

e2

0

e3

T

¼ ½ e1

e2

e3 T

ð12Þ

R ¼ rij 33

ð13Þ

The rotation angle of theoretical 3D coordinates around the x, y, and z axes were obtained by consisting of Eqs. (12) and (13), the rotation angle around the x, y, and z axes respectively as follows: 8 a ¼ arctan rr23 > 33 > < r13 b ¼ arctan pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ r2 r23 33 > > : c ¼ arctan r12 r11

ð14Þ

The translation vector T x , T y , T z in the translation matrix T were determined by T, T as follows: T ¼ ½P0 Q0 R

ð15Þ

According to the three-order non-circular gears studied and collected 3D coordinate of tooth surface, the Euclidean transformation matrix M is solved by the above method. Now the transformation matrix M of left tooth surface of 1st tooth was calculated by substituting 3D coordinate into Eqs. (10)–(15), the transformation matrix M of left tooth surface of 1st tooth as follows: 2

0:8256 0:0095 6 0:0767 0:0032 M¼6 4 0:2131 0:0802 0:06235 0:8326

0:0026 0:0003 0:0413 0:5325

3 0 07 7 05 1

ð16Þ

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6 Example Calculation Taking the third-order non-circular gear as an example, the speciﬁc parameters are shown in Table 2. The collected tooth data is calculated by the above method, and the calculation result is drawn into a graph. Figure 4 shows the tooth surface error of the left and right tooth faces of each tooth. Table 2. Experimental gear parameters Modulus m Number of teeth Z Center distance a Addendum coefﬁcient h*a Top clearance coefﬁcient C* Tooth width B Eccentricity e Pitch curve equation r

3 39 135 1 0.25 30 0.0773 57:5544 r ¼ 10:0773 cos 3u

Fig. 4. Error of tooth surface

It can be obtained from Fig. 4 that the tooth surface error of No. 1, No. 7, No. 14, No. 20, No. 27, and No. 33 is small, the error of six tooth surface is close, and the error of the left and right tooth surfaces of each tooth is close, and the phase difference is small.

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7 Conclusion From the above analysis, the following conclusions are drawn: (1) The error of tooth surface of No. 1, No. 7, No. 14, No. 20, No. 27, and No. 33 tooth is small and no more than 0.06 mm. In addition, error of 6 teeth surface and error of left and right tooth surface of every tooth are very close. (2) The error evaluation function constructed in this paper, the method of tooth surface reconstruction and the matching of theoretical data and reconstructed teeth surface are effective and feasible. (3) This paper ﬁnds a new way to measure the tooth surface error of non-circular gears.

References 1. Dooner, D.B.: Function generation utilizing an eight-link mechanism and optimized noncircular gear elements with application to automotive steering. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 13(215), 847–856 (1995) 2. Litvin, F.L., Gonzalez-Perez, I., Fuentes, A., et al.: Tandem design of mechanisms for function generation and output speed variation. Comput. Methods Appl. Mech. Eng. 198(5), 860–876 (2009) 3. Wang, Y.Z.: Study on Errors Analyse of CNC Hobbing Non-Circular Gears. Lanzhou University of Technology, Lanzhou (2013) 4. Lu, C.X., Wang, J.H.: Measuring technique for non-circular gears. J. Xi’an Inst. Technol. 1 (19), 58–61 (1999) 5. Lin, J.C., Shi, Z.Y., Pan, C.G., et al.: Proﬁle error evaluation of large gears based on NURBS surface ﬁtting. J. Instrum. 3(37), 533–539 (2016) 6. Xie, X., Zhang, X.B., Jia, J.M.: Surface error evaluation of non-circular bevel gear of variable ratio differential. Mech. Transm. 4(41), 48–52 (2017) 7. Liu, Y.P., Liu, J.X., Zhang, L.Y., et al.: Research on point cloud to the complex surface best ﬁtting. China Mech. Eng. 12(16), 1080–1082 (2005) 8. He, G.Y., Liu, X., Liu, P.P., et al.: Evaluating of complex surface proﬁle error based on subdivision and sphere approximation method. Comput. Integr. Manuf. Syst. 3(19), 474–479 (2013)

Topology Optimization Design of the Monocoque Bus Body Structure Zhuli Liu1(&), Xiangyu Huo1, Hao Zhou1, and Xiaoguang Wu2 1

2

School of Mechanical Engineering, Zhengzhou University, Zhengzhou 450001, China [email protected], [email protected], [email protected] Zhengzhou Yutong Bus Co., Ltd., Zhengzhou 450016, China

Abstract. Utilizing the monocoque bus body structure of low floor bus as the research object and the stress as constraint conditions, this paper developed the topology optimization model with the objective function of minimum structural mass under the horizontal bending conditions. Based on the results of topology optimization, a new bus body structure has been designed. This paper makes the ﬁnite element static analysis on the new body structure, which shows that its strength and stiffness meet the design requirements and that the topology is feasible. Further optimization of the body’s speciﬁc parameters can be put into effect to achieve the goal of the optimization in the design stage. Keywords: Monocoque body structure SIMP RAMP OptiStruct

Topology optimization design

1 Introduction With the Monocoque bus body structure is composed of proﬁle steel, the whole body structure plays a role in carrying. It is obviously better than separate frame construction with the advantages of structural safety, stability, ride comfort, energy efﬁciency and environmental protection. For this reason, the Monocoque bus body structure is the mainstream form of high-class coach [1]. Due to the complexity of the monocoque body structure, the traditional design method is to modify the basic model based on design requirements of the new model. This method not only requires design experience, but also has a certain blindness, and the ﬁnal design may not be reasonable enough. This paper takes the minimum mass of the monocoque bus body structure as the objective function, the strength and stiffness of body as constraint, and adopts the topology optimization to optimize the body structure, in order to ﬁnd the force transferring path, to realize the conceptual design of the body structure, and to provide reference for subsequent design.

This project is supported by National Key R&D Program of China (Grant No. 2018YFB0106204-03). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 456–465, 2020. https://doi.org/10.1007/978-981-32-9941-2_37

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2 Topology Optimization Theory and Mathematical Optimization Model of Body Topology optimization is an innovative technology in structural optimization. Within the given design space, material is distributed according to the force transferring path of the structure, while the mechanical properties of the structure and lightweight can be considered in the conceptual design phase. Topological optimization can be divided into topology optimization of discrete structure (truss, steel frame, etc.) and topology optimization of continuum structure (2d shell, 3d solid, etc.). Discrete topology optimization originated from the truss theory, which was proposed by Michell in 1904. The ground structure method of Dorn, Gomory and Greenberg, and Schmit’s transformation of structural problems into mathematical programming lay the foundation for the topology optimization of discrete structure. After Bendsoe and Kikuchi proposed the homogenization method to solve the structural topology optimization in 1988, the continuum topology optimization has been developed rapidly in theory. At present, the main methods of continuum topology optimization are homogenization method, variable density method, variable thickness method and topological function description method, of which homogenization method and variable density method are the most representative [2–4]. Homogenization method has strict mechanics and mathematics theory foundation, but it has many design variables, long solution time, and the calculation result is prone to checkerboard phenomenon. Therefore, it is usually used in topology optimization theory research. The variable density method assumes that the material density is variable, taking the element relative density as the design variable. In the calculation, when the element relative density is 0, the material in the element can be removed; when the element relative density is 1, the element reserves. There are two kinds of common interpolation models in variable density method: Solid isotropic material with penalization (SIMP) and Rational approximation of material properties (RAMP) [5, 6]. Variable density method has high computational efﬁciency, simple program implementation and wide applied range. In the conceptual design, the basic topological form of the structure can be determined only by the constraints and the loads of the structure. SIMP and RAMP are as follows: Ee Ee

SIMP

RAMP

¼

¼ ðxe Þp E0

xe E0 1 þ pð1 xe Þ

ð1Þ ð2Þ

In the formula, xe is element relative density, xe ¼ qe =q0 , qe is hypothetical element density, q0 is the real density of materials, Ee is hypothetical elasticity modulus of element, E0 is material elasticity modulus, p is penalty factor.

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SIMP and RAMP are shown in Figs. 1 and 2, SIMP has two drawbacks: SIMP cannot ensure that the penalty function is a concave function and cannot obtain the global optimal solution; when the value of p is greater than a certain value, the result is completely misleading. By contrast, the above situation does not occur in RAMP, so RAMP is more stable than SIMP [7].

Fig. 1. SIMP

Fig. 2. RAMP

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With the structural stress and displacement of ﬁnite element as the constraint condition, the body mass can be transformed into material volume based on the assumption that the body frame uses the same material. The topology optimization mathematical model with the minimum volume as the objective function is as follows: N X

ve

ð3Þ

s:t KU ¼ F

ð4Þ

ðre ÞVM xpe r

ð5Þ

min

e¼1

0 \ xmin xe 1

e ¼ 1; ; N

ð6Þ

In the formula, ve is element volume; K is stiffness matrix; U is displacement matrix; F is load; xe is element relative density; ðre ÞVM is the equivalent stress of the element; r is stress requirements under working conditions; p is penalty factor; xmin is the lower limit of element relative density.

3 Topology Optimization of Body Structure The general topology optimization software mainly includes TOSCA (Germany), OptiShape (Japan), OptiStruct (USA) and Genesis (USA) [8]. In this paper, the OptiStruct of HyperWorks software is used to optimize the structure of the body. 3.1

Determine the Topology Optimization Space

According to low-entry hybrid bus of yutong, the basic parameters of monocoque bus body structure are shown in Table 1.

Table 1. The basic design parameters Name Bus overall length Bus overall height Bus overall width Wheelbase Front/rear overhang Front/rear tread Front suspension Rear suspension

Parameter/mm 12000 3150 2550 6100 2670/3220 2096/1836 Airbag Bearing spring

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Topology optimization space is the largest area of material distribution. According to the basic parameters of the model, the topology optimization space for the bus can be determined, in which the front and middle parts of the bus adapt single floor, and the rear part adopts a double floor to store the battery and engine. The topology optimization space is shown in Fig. 3.

Fig. 3. Design space of topology optimization

Body material is Q345, elasticity modulus is 2.1E5 MPa, the Poisson’s ratio is 0.3, the density is 7.8E-9 T/mm3, the element type is shell element. Since the topology optimization design is a conceptual design, in order to reduce the calculation time, the element size is taken as 25 mm, and free meshing is performed in the hypermesh, and 203,728 elements are ﬁnally obtained. 3.2

Apply Constraints and Loads

Refer to Yutong bus, the fulcrum of rear overhang is connected to the body by bolts, and the airbag is connected to the front wheel house. In the topology optimization, the rigid region is selected in the connection position between the overhang and the body, and the RBE2 rigid node is used to replace the overhang fulcrum, so as to establish the coupling relationship between the RBE2 rigid node and the rigid region node. The simulation of rear overhang and front overhang fulcrum are shown in Figs. 4 and 5.

Fig. 4. Simulation of rear overhang

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Fig. 5. Simulation of front overhang

The constraints and loads are as follows in horizontal bending condition: Constrain the Y direction of the fulcrum of right airbag, the Y, Z directions of the left airbag, the X, Z directions of the fulcrum of right suspension, and the X, Y, Z directions of the left suspension; The X, Y, Z directions represent the direction of bus overall length, bus overall height and bus overall width. Body accessories such as passengers, engine, gearbox, clutch, air-conditioner are loaded at the corresponding positions according to the quality point. Due to the large difference between the design space and the actual material distribution space, the quality of the design space is not considered in the topology optimization design stage in order to avoid affecting the body structure 3.3

Topology Optimization Analysis

The design variable for topology optimization is element relative density. Considering strength safety requirement, the maximum stress in the design area is not more than 250 MPa, and the objective function is minimum volume. After 72 iterations, the density threshold is set to 0.09, which means that when the material density value is less than 0.09, the material at that location can be removed. The results of the topology optimization of body structure are shown in Fig. 6, and the reserved material shows the force transferring path of body structure under horizontal bending. In this circumstance, the convergence curve of the objective function value with the iteration number is shown in Fig. 7, the convergence is good.

Fig. 6. Results of topology optimization

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Fig. 7. Convergence curve of objective function

3.4

Topology Optimization Results

Figure 6 shows that the topology optimization result is an irregular pattern, which is different from the actual model and proﬁle shape. According to the manufacturing process and proﬁle selection, the body structure is mostly welded by rectangular steel pipe. Topology optimization results are interpreted as follows: (1) The red area in Fig. 6 is the main force transferring path, which has a large load, and material must be distributed on the path. (2) Considering the design of the body structure (such as structural symmetry, ensuring the closed loop structure after welding) and the bus operating conditions, increase the necessary distribution of materials. (3) Align the side wall pillar with the roof beam and floor beam. (4) In order to improve the anti-rollover ability of bus, the body shape should be square. (5) Take the lightweight design of the body as the criterion to minimize the distribution of materials. The force transferring paths and material distribution of the main structure such as the lower floor, upper floor, the connection of the left and right body side and floor are shown in Fig. 8. With the optimization results of the left body side as an example to make a brief description, it can be seen from Fig. 8(c) that the middle and front parts are the main transferring paths, and the side wall pillars are preferred. The path of the upper part of the body side is triangular, and the oblique brace is added at the upper waist beam. Similarly, oblique brace is added at the middle part of the body side and the lower waist beam. The right side is referred to the left side, which can be taken as symmetrical structure.

Topology Optimization Design of the Monocoque Bus Body Structure

(a) Lower floor result

(b) Upper floor result

(c) Left side result

(d) Floor connection result Fig. 8. Results of main structural topology optimization interpretation

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4 Veriﬁcation of Finite Element Analysis According to the topological optimization results and the cross-section size of the body frame, the new body structure is welded by Q345 steel pipe with cross-section size of 50 50 3 mm. The body frame is analyzed under horizontal bending condition. Stress and body displacement clouds are shown in Figs. 9 and 10. The maximum displacement is 8.6 mm, which occurs in the upper roof and lower floor of the middle part of the body. The maximum stress is 111 MPa, mainly located at the floor beam and the connection of upper/lower floor. The strength and stiffness of the body meet the design requirements.

Fig. 9. The displacement diagram of bus

Fig. 10. The stress diagram of bus

5 Conclusions Based on the variable density method, the topology optimization of the monocoque bus body is carried out, the load transferring path is clariﬁed, materials are distributed on the load transferring path, and a body frame is redesigned. The static analysis of the new body structure is carried out under bending condition, and the strength and stiffness of the body structure meet the design requirements, which shows that the topology optimization structure is feasible. In the future, based on the topology

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optimization results, the section size of rectangular steel pipe can be optimized, the speciﬁc parameters of body structure can be determined, and the optimization objective can be achieved in the design stage.

References 1. Liu, K.: Bus Body Design, pp. 24–27. Mechanical Industry Press, Beijing (2012). (in Chinese) 2. Zuo, K.: Research of Theory and Application about Topology Optimization of Continuum Structure. Huazhong University of Science and Technology (2004). (in Chinese) 3. Bendsoe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988) 4. Sigmund, O., Maute, K.: Topology Optimization Approaches, pp. 1031–1055. Springer, Berlin (2013) 5. Kongtian, Z., Liping, C., Fangyi, Z.: New theory and algorithm research about topology optimization based on artiﬁcial material density. Chin. J. Mech. Eng. 4(12), 31–37 (2004). (in Chinese) 6. Qiao, Z., Weihong, Z., Jihong, Z.: Topology optimization of structures under dynamic response constriants. Chin. J. Mech. Eng. 46(15), 45–50 (2010). (in Chinese) 7. Xiang, C., Xinjun, L.: Solving topology optimization problems based on RAMP method combined with guide-weight method. Chin. J. Mech. Eng. 48(1), 135–140 (2012). (in Chinese) 8. Weihong, Z., Min, W.: Applications of topology optimization in the automobilc industry. J. Kunming Univ. Sci. Technol. (Sci. Technol.) 3, 77–81 (2005). (in Chinese)

Analysis of Wind Vibration Response of Transmission Tower Zhuli Liu1(&), Beibei Liu1, Zhuan You1, and Mengli Li2 1

School of Mechanical Engineering, Zhengzhou University, Zhengzhou 450001, China [email protected], [email protected], [email protected] 2 Henan EPRI Gaoke Group Co. Ltd., Zhengzhou 450052, China [email protected]

Abstract. Angular tension-resistant transmission tower is taken as research object in this paper. The Davenport pulsating-wind speed spectrum was used to simulate wind speed time series and the wind power spectrum at different height nodes of transmission tower in MATLAB with the help of the linear ﬁltering method (AR). The reliability of the fluctuating wind speed time series simulation was conﬁrmed by comparing the wind power spectrum with Davenport power spectrum. Reﬁned ﬁnite element model of the tower was established by APDL in ANSYS. The wind-load was calculated by the simulation of wind speed time series, and the wind-load time-history curve was generated at the same time. Finally, the dynamic characteristics and the wind-induced dynamic response of transmission tower in hurricane ware studied. The results show that the top of the tower was prone to large deformation and acceleration under strong wind excitation. In other words, there was a signiﬁcant whipping effect. The research results will prove a useful reference for designing transmission tower and safeguarding the operation of transmission lines. Keywords: Transmission tower Pulsating wind simulation Dynamic characteristics Wind-induced dynamic response

1 Introduction The High-voltage transmission tower-line system is a key link of transmission line. If it fails, the entire line will not operate properly. Therefore, it’s great signiﬁcance to ensure it to operate safely. Generally, the high-voltage transmission tower-line system has a large span, and the tower body is flexible and tall. Hence, it has strong nonlinear characteristics and its mechanical properties is complex. When there are strong winds, it’s easy to cause complex vibration and collapse the tower [1]. In order to ensure the stability of tower in the strong wind, the dynamic characteristics and wind-induced dynamic response of transmission tower were studied in this paper.

This project is supported by state grid henan power company (Grant No. 5217021350HQ). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 466–476, 2020. https://doi.org/10.1007/978-981-32-9941-2_38

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2 Wind-Load Theory and Wind Speed Simulation 2.1

Wind-Load Theory and Characteristics

Air horizontal pressure gradient force is the cause of wind formation. According to the statistical data of wind speed, wind can be divided into mean wind and fluctuating wind [2]. (1) Mean wind. The mean wind is invariable for a certain period and it will not cause structural vibration. The mean wind can be treated as static force in mechanical analysis. The mean wind speed varies with height within a certain range of height. We can calculate it according to the exponential law [3] from Load code for the design of building structure [4]:

vðzÞ vðzr Þ

¼

a z zr

ð1Þ

Where VðZr Þ denotes the average wind speed at the reference height of Zr, a denotes the ground roughness index, V(z) denotes the mean wind speed at any height. (2) Fluctuating wind Davenport pulsating-wind speed spectrum [5] has been used widely to reach pulsating wind by domestic and foreign scholars and it is also the wind speed spectrum used in the relevant design speciﬁcations of China. The formula is as follows: Sv ðnÞ ¼ 4Kv210

X¼

X2 f ð1 þ X 2 Þ4=3

1200 f v10

ð2Þ

ð3Þ

Where f donates the pulse frequency (Hz), V10 donates the mean wind speed at 10 m (m/s); K is the surface roughness coefﬁcient, taken as 0.005. Wind speed can be regarded as the superposition of mean wind and fluctuating wind, expressed as: þ vðx; y; z; tÞ Vðx; y; z; tÞ ¼ VðzÞ

ð4Þ

Where V(z) is the mean wind speed at height z; V is the fluctuating wind speed at height Z; t denotes time.

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The Simulation of Tower’s Wind Speed Time Series

The tension-resistant transmission tower in a certain area is selected as the object of study. Selection of Class B Ground Roughness Types (suburbs, towns with sparse ﬁelds and houses, villages, jungles, hills and houses), the corresponding a = 0.16; the design mean wind speed V(zr) is 30 m/s at the reference height 10 m. The transmission tower is divided into 10 sections. Select the wind speed simulation points in each section (in Fig. 1). The mean wind speed V(z) of the wind speed simulation point at the height Z which is calculated by Eq. 1 was listed in Table 1.

Fig. 1. Location of the wind speed simulation points

Table 1. Mean wind speed at each point Note 1 2 3 4 5 6 7 8 9 10

Height (m) Area (m2) 9.0 4.856 12.0 1.520 17.0 2.527 22.0 3.490 28.0 1.952 30.0 1.921 36.1 2.146 38.1 2.974 42.1 0.880 44.1 1.274

Mean wind speed (m/s) 29.50 30.89 32.66 34.03 35.37 35.77 36.84 37.16 37.76 38.04

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Linear ﬁltering method (AR) is widely used to simulate wind speed of time-history, because of its fast calculation speed and small amount of calculation. In this method, the response values of the present time are represented by the linear combination of the response values of the past time and the white noise [6]. The mathematical expression of AR method as follows: V ðX; Y; Z; tÞ ¼

p X

uk V ðX; Y; Z; t kDtÞ þ N ðtÞ

ð5Þ

k¼1

Where uk denotes the autoregressive coefﬁcient matrix; P denotes the order of the model; N(t) denotes the independent random process vector; Dt denotes the time step of wind speed time series. The mean wind speed at simulated points of wind speed in each section is superimposed with the fluctuating wind speed obtained by the linear ﬁltering method. The wind speed time series of each point is simulated in MATLAB, the simulation time is 180 s. The wind speed time series curves of four simulation points are listed in Fig. 2.

Fig. 2. Wind speed time-history curve of simulated points

Wind power spectrum at 22 m and 38.1 m nodes are compared with Davenport power spectrum. The results are shown in Fig. 3. The simulated wind power spectrum

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at two selected nodes fluctuates around Davenport power spectrum, and the two power spectra have high ﬁtting degree. It proves the feasibility of using AR method to simulate fluctuating wind.

Fig. 3. Comparison diagram of power-spectrum

2.3

Numerical Calculation of Wind Load

Wind loads on transmission towers can be calculated by the following formulas [7]: Ftower ¼ us Af V 2 =1:6

ð6Þ

Where us denotes the shape coefﬁcient of transmission tower, according to Technical Regulation of Design for Tower and Pole Structures of Overhead Transmission Line us = 2.5; V denotes the node wind speed time series; Af denotes the wind-pressure projection area in simulated area. Equation 6 can convert the wind speed time series into the wind load time-history of a certain area at the simulation point. It can provide load conditions for subsequent analysis of wind-induced vibration response of tower.

3 Analysis of Dynamic Characteristics of Transmission Towers 3.1

Establishment of Finite Element Model of Transmission Towers

With the geometric center of the tower leg as the origin, the extension direction of the cross-arm as the X-axis, the vertical direction as the Z-axis, and the transmission traverse direction as the Y-axis, in ANSYS. BEAM188 element is selected to simulate the various poles in the tower by endowing the beam element with different crosssection shapes and parameters and accurately setting the orientation of the angle iron. The reﬁned beam-frame ﬁnite element model of the angular tension-resistant transmission tower is completed (as Fig. 4 shows).

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Fig. 4. Finite element model of tensile tower

3.2

Analysis of Dynamic Characteristics of Transition Towers

Block Lanczos method [8] is used for modal analysis of the tower in ANSYS. The natural frequencies and modes of each mode are obtained. The frequencies and modes of ten are listed in Table 2. Table 2. Natural frequency and vibration mode of tower Order 1 2 3 4 5 6 7 8 9 10

Frequency (Hz) Mode of vibration 2.4642 Swing along the Y axis 3.2359 Swing along the X axis 3.3292 Torsion around Z-axis 5.2201 Local deformation 5.6687 Swing along the Y axis & Torsion around Z-axis 5.8947 Swing along the X axis & Torsion around Z-axis 6.0398 Torsion around Z-axis & Local deformation 7.1057 Swing along the X axis & Torsion around Z-axis 7.1069 Local deformation 7.9560 Local deformation

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According to Table 2 and Fig. 5, the top ten natural frequencies of the tower are densely distributed, and the vibration modes are complex, it including lateral and longitudinal oscillation and torsion. At the same time, there are many local modes and spatial coupling modes. The local mode shapes appear many times at the legs and crossbars of the tower in the mode diagram (Fig. 5). It indicates that this part is the weak part of the tower structure which needs to be strengthened, and ensure the stability of the structure.

(a)First-order mode shape

(c)Third-order mode shape

(b)Second-order mode shape

(d)Fourth-order mode shape

Fig. 5. Primary vibration mode of tower

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4 Analysis of Wind Vibration Response of Transmission Tower 4.1

Time Domain Analysis Based on Newmark-b Method

Vibration of transmission tower under wind load belongs to forced vibration of damped structure, and its motion equation can be expressed as follows: ½Mf€ug þ [C]fu_ g þ [K]fug ¼ fFðtÞg

ð7Þ

Where [M] donates the mass matrix; [C] donates the damping matrix; [K] donates _ donates the velocity array; the stiffness matrix; {u} donates the displacement array; {u} {ü} donates the acceleration array; {F(t)} donates the wind load on the transmission tower. Newmark-b method is used to solve the forced vibration response of structures, and the integral solution of Eq. 7 can be solved directly. 4.2

Determining the Structural Damping of the Tower

Damping is an inherent property of structural dynamic characteristics. It is related to material and friction coefﬁcient of structure, and it’s an effective means to weaken vibration. The accuracy of simulation results can be ensured by correct calculation of damping. Since the accurate damping matrix needs to be further studied, the simpliﬁed actual damping matrix is used in this paper. The Rayleigh damping hypothesis [9] is used to assist the calculation. ½ C ¼ a ½ M þ b½ K

ð8Þ

Where [M] donates the mass matrix; [C] donates the damping matrix; [K] donates the stiffness matrix; a donates the mass damping coefﬁcient; b donates the stiffness damping coefﬁcient. a and b can be obtained by the following formula: 2wi wj ni wj nj wi 2 nj wj ni wi b¼ a¼ w2j w2i w2j w2i

ð9Þ

Where wi and wj represent the i-th and j-th vibration frequencies. They are usually taken for the ﬁrst and second order vibration frequencies; ni and nj represent the corresponding damping ratios of the i-th and j-th modes. For steel structures, they range from 0.01 to 0.02. 4.3

Generating Wind Load Time-History Curve

Based on the wind load numerical calculation method of tower described in Sect. 2.3, the wind load time-history of representative nodes of transmission tower can be

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calculated according to its geometric parameters and the simulation results of wind speed time series in Sect. 2.2 (as Fig. 6 shows).

Fig. 6. Time-history curve of wind-load at classic nodes

4.4

Analysis of Wind Vibration Response of Transmission Tower

The node displacement of transmission tower under strong wind is mainly affected by the wind load on the tower [10]. According to the wind speed time history simulation calculation method, the wind speed time history was transformed into the wind load time history at each node. The wind load was added to the tower model by APDL cyclic command flow, and then the wind vibration response was analyzed. Finally, the time-history curves of displacement, velocity and acceleration at the top and bottom of the tower were extracted (in Fig. 7). According to the extracted response curves of displacement, velocity and acceleration of tower at bottom and top, and combined with Table 3 found that: the displacement, velocity and acceleration increase with the increase of height of tower. The response peaks of displacement, velocity and acceleration at the top tower is 33.9 mm, 0.168 m/s and 3.89 m/s2 respectively. It was found that the top of the tower was prone to large deformation and acceleration under strong wind excitation. That is to say, there was a signiﬁcant whipping effect.

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Fig. 7. Response curves of displacement, velocity and acceleration at the bottom and top of tower Table 3. The maximum displacement, velocity and acceleration of the tower at the bottom and top Node Bottom 74 Top 518

Maximum displacement (mm) 1.81

Maximum velocity (m/s) 0.0034

Maximum acceleration (m/s2) 0.115

33.90

0.1680

3.890

5 Conclusions In this paper, a reﬁned ﬁnite element model of transmission tower was built based on the research object of the turret type tension-resistant tower. The dynamic characteristics of the tower model were analyzed. The tower model was loaded with simulated wind load, and the wind-induced time-history response was analyzed. The conclusions are as follows: (1) The most widely used Davenport wind speed spectrum is adopted to simulate the wind speed time history of transmission tower by using the linear ﬁltering method, and the simulated power spectrum is compared with Davenport power spectrum to verify the accuracy and applicability of the simulated spectrum.

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(2) The modal analysis of the tower model shows that the natural frequencies and mode shapes of each order clearly show that the weak parts of the transmission tower structure are the tower legs and the transverse arms prone to local deformation. (3) The wind-induced vibration response of the tower was analyzed, and the timehistory curves of displacement, velocity and acceleration at the representative nodes were extracted. It was found that for the single tower model, the displacement, velocity and acceleration also increased with the increase of height, and large deformation and acceleration occurred near the tower top, namely whipping effect. Based on the above conclusions, the structural strength of the weak parts, such as the tower leg and the cross arm, which are prone to local deformation, should be fully guaranteed to ensure the stability of the structure. For the high-rise transmission tower structure, the stiffness at the top should be increased to limit its wind-induced vibration response and prevent the occurrence of large displacement and acceleration at the top. The research results in this paper have important reference value for the subsequent design and structural improvement of the transmission tower.

References 1. Chen, J.: Research of Wind-Induced Vibration of Tower-Line System of Power Transmission Tower. Guangxi University, Nanning (2013). (in Chinese) 2. Wang, G., Zhao, P.: Analysis of Anti-overturning of a telecommunication tower under strong wind. J. Zhengzhou Univ. (Eng. Sci.) 33(02), 76–80 (2012). (in Chinese) 3. Ministry of housing and urban-rural development of the People’s Republic of China; General administration of quality supervision, inspection and quarantine of the People’s Republic of China. GB50009—2012 Load code for the design of building structures. China Building Industry Press, Beijing (2012). (in Chinese) 4. Walker, E.K.: The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum, effects of environment and complex load history on fatigue life, ASTM, 1970. STP462, 1–14 (1970) 5. Davenport, A.G.: Gust loading factors. J. Struct. Div. ASCE 93, 11–34 (1967) 6. He, W.: Studies on wind-induced dynamic response of high-rise lattice tower. Hunan University, Changsha (2009). (in Chinese) 7. Zhang, L.: Research on wind stochastic ﬁeld and dynamic reliability for high-rise building with wind loading. Tongji University, Shanghai (2006). (in Chinese) 8. Li, T., Sun, Y.: Analysis of the dynamic properties of smoke desulphurization absorber tower structure. J. Zhengzhou Univ. (Eng. Sci.) 02, 64–67 (2007). (in Chinese) 9. Wang, Z., Ma, R., et al.: Tower Structure. Science Press, Beijing (2004). (in Chinese) 10. Holmes, J.D.: Along-wind response of lattice tower: derivation of expressions for gust response factors. Eng. Struct. 16(4), 287–292 (1994)

Design of a Stabilize Device for Heavy Oil Transportation in Water Ring Haoran Lu, Fan Jiang(&), Yuliang Chen, Xiaolong Qi, and Zhongmin Xiao School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510000, China [email protected]

Abstract. During the long distance transportation of heavy oil by water ring, instability will occur under the action of gravity. In order to meet the demand of stable transportation of water-ring heavy oil, an innovative design method integrating TRIZ and Extenics is proposed. By analyzing the contradictions in innovative design and implementing effective extension transformation, a waterring heavy oil transportation stabilize device is designed. According to the principle of divergence analysis of a characteristic multi-value, two schemes are designed, in which the bottom liquid outlet is located at the entrance or exit of the stabilizing device. The N-S equation and the turbulence model have been used to describe the fluid flow. Taking the V0F model and the CSF model to track the oil-water interface, the numerical equation and analytical model have been established. The simulation results show that when the bottom liquid outlet is set at the entrance section of the stabilize device, the pipeline can better form the shape of heavy oil wrapped by water ring. Keywords: Core-annular flow Heavy oil transportation TRIZ Extenics Stabilize device Numerical simulation

1 Introduction The world’s heavy oil resources are abundant. With the large-scale exploitation of conventional crude oil on land, people have turned more attention to heavy oil. The viscosity of the heavy oil is too large, and the shear stress of the oil and the pipe wall is too large to be transported. The use of low-viscosity fluids to encircle the oil phase in a ring-shaped manner (forming an annular flow) for oil transport will greatly reduce the energy consumption of heavy oil transportation, and is one of the effective modes for heavy oil pipeline transportation. Many scholars have studied and discussed the theory of heavy oil-water core-annular flow viscosity, annular flow pattern and unstable surface tension of annular flow [1–5]. Pan et al. [6], ﬁrst proposed in the patent to use water to lubricate pipelines for viscous fluid transportation. This method can effectively This project is supported by Natural Science Foundation of Guangdong Province (Grant No. 2016A030313653); Science and Technology Program of Guangzhou (Grant No. 201504291436202). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 477–492, 2020. https://doi.org/10.1007/978-981-32-9941-2_39

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reduce friction loss and is suitable for pipeline transportation of high viscous crude oil. In 1994, Ho and Li [7] prepared a oil-in-water emulsion according to the water-oil ratio of 7–11, and successfully applied water ﬁlm transport. The thicker the oil, the greater the viscosity, the more suitable for liquid ring transport. Li et al. [8], studied the stability of the water ring through local distortion of the pipeline, hydrophilization of the pipe wall and gel adsorption layer on the i walnnerl of the pipe. China’s Shell Oil Company and Shengli Oilﬁeld Qinghe Oil Production Zone also adopted oil-water annular flow technology [9]. The main problem of water ring transportation technology at present: with the increase of transportation distance, the annular water ﬁlm of lowviscosity fluid (water) wrapped around high-viscosity fluid (heavy oil) will gradually disappear, that is, water ring instability. Jiang et al. [10], designed a new type of water ring generator, which can effectively improve the stable length of water ring and reduce transportation energy consumption, but there are still unstable problems. Aiming at the current problem, a design of water ring heavy oil transportation stabilize device based on TRIZ and extension is proposed, and a set of stabilize device is added for each distance to realize long-distance transportation of water ring.

2 Invention Design of Stabilize Device 2.1

Product Innovation Optimization Design Method Based on TRIZ and Extenics

2.1.1 TRIZ Theory and Extenics TRIZ theory was created by the Soviet scientist Genrich S. Altshuler in the middle of the 20th century. It is literally translated as the “inventive problem solving theory” and is an innovative problem-solving theory. Its main purpose is to study human beings inventing and the scientiﬁc principles and rules followed in the process of solving technical problems. The theory has been widely used in various engineering ﬁelds [11– 13]. Extenics is an original transversal subject put forward by Chinese scholar Cai Wen in 1983. It uses a formal model to explore the possibilities of things expansion and open up new laws and methods, and to solve contradictions. The basic theory of extenics is extension theory, which consists of three parts: elementary theory, extension set theory and extension logic. In innovation application, it mainly through extension modeling, expansion, transformation, and goodness evaluation to get ideas [14]. At present, extenics has entered some research areas and achieved a series of results. 2.1.2 Product Innovation Optimization Design Method Based on TRIZ and Extenics Innovative design will encounter many problems. The combination of TRIZ theory and extension innovation method will reduce the difﬁculty of solving. The key point is contradiction recognition and contradiction solving. Here, the TRIZ tool and the extension innovation method are integrated, and the innovative method is established as shown in Fig. 1.

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Need innovative products

No

Conflicts that hinder product

TRIZ theory

Method evaluation

Extenics

Method selection

Practice test

mechanism

Solved?

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Adjust the crossover idea

Evaluation of program excellence

Fig. 1. The cross fusion model of TRIZ and extenics

2.2

Stabilize Device Innovation Based on TRIZ and Extenics

2.2.1 Problem Recognition Based on Primitive Model Transportation of heavy oil by water-ring heavy oil can greatly improve transportation efﬁciency and reduce energy consumption. However, the density of heavy oil is smaller than that of water. With the increase of transportation distance, the stratiﬁcation of heavy oil and water will occur under the action of gravity. The upper layer is heavy oil and the lower layer is water. The contact between heavy oil and pipe wall in the upper layer produces greater friction resistance, which leads to the decrease of heavy oil transportation speed and efﬁciency, and also requires a large amount of energy consumption. Through these existing phenomena, the main identiﬁcation problem is the stratiﬁcation of oil and water.

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2.2.2 Problem Analysis Based on Primitive Model According to the formal description of Extenics, the ultimate goal G of establishing the water ring heavy oil conveying and stabilizing device, and the target event element is G¼

recovery

Control object requirements

Oil and water phases Water ring heavy oil

After long distance transportation, the stratiﬁcation of oil and water occurs. According to the formal description of extenics, the current conditional matter element L is 2 L¼4

Oil and waterphases

form flowstate

3 Upper and lower stratification of oilphase and waterphase 5 Slowoilspeed

Obviously, this constitutes an incompatibility problem: to improve efﬁciency requires the realization of water ring heavy oil, water ring heavy oil needs to be added again, the water content in the pipe increases, and the transportation efﬁciency is reduced. which is P ¼ G L. According to the incompatibility problem model, the expansion analysis is carried out to extract a contradiction: to improve the transport efﬁciency, it is necessary to improve the flow speed of heavy oil, and to improve the speed, the annular flow of oil and water should be formed. The formation of water ring needs to add water again, and the water content in the pipeline increases, so the transport efﬁciency decreases. The application of standard engineering technical parameters is described in Table 1. Table 1. Standardized description Improving parameter Deteriorating parameter

Contradiction Circular flow pattern

Standardized description of contradiction Parameter 12—shape

Increase energy consumption

Parameter 19—energy consumption of a moving object

2.2.3 Feasibility Solution Prediction Based on TRIZ Innovation Principle Looking for the contradiction matrix under the known technical parameters, getting the four kinds of recommended inventive principles [15], the corresponding serial number is: 02, 06, 14, 34, those are as follows: Principle 02 is extraction: (1) extracting negatively affected parts or attributes from an object; (2) extracting necessary parts from objects. Principle 06 is multi-purpose: (1) to make an object have multiple functions; (2) if the function of an object is replaced, the object can be cut. Principle 14 is curved surface: (1) from straight line to curve, from plane to sphere, from regular hexahedron to spherical structure; (2) using rod, sphere and helix; (3) from straight line to rotary motion, using centrifugal force. Principle 34 is abandonment and repair: (1) the use of dissolution, evaporation and other means to abandon the completed function of parts,

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or directly modify them in the process of system operation; (2) in the process of work quickly supplement system or object in the part of consumption. According to the above principles, it can be seen that the extraction principle, the multi-purpose principle and the curved surface principle are not suitable for fluid pipeline transportation, and they are not considered for the time being. At the same time, we hope that the annular flow pattern can be restored without increasing the water content. According to the suggestion of abandonment principle and repair principle, the original water phase in the pipeline can be abandoned by the replacement transformation in the basic extension transformation method, and new water phase can be added again to replace the initial water phase, so as to realize the recovery of the form of water-ring heavy oil. 2.2.4 Element Transformation of Feasible Solution According to the solution of TRIZ principle, it is necessary to remove part of the water accumulated at the bottom of the pipeline. Firstly, transform T1 so that T1 L1 ¼ L1 L11 . Among them, 2 T1 ¼ 4 2 L11 ¼ 4

Add

3 Actuator object L1 Dominant object L11 5 Transformation result L1 L11

Water phase outlet of pipeline

Shape Number Position

3 Rectangle 5 1 Bottom

At the same time, the location characteristics of the outlet at the bottom of the pipeline can be reﬁned and diverged. The divergence of matter elements (one eigenmultiple value) has L11 jfL111 ; L112 g. Among them, 2

L111

Water phase 6 outlet of pipeline ¼6 4 2

L112

Water phase 6 outlet of pipeline ¼6 4

Shape Number Position

Rectangle

3

7 7 5 1 Pipeline entrance section

Shape

Rectangle

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1 Pipeline outlet section

3 7 7 5

In order to restore the oil-water annular flow pattern of the fluid in the pipeline, so it is necessary to transform t so that T2 L1 ¼ L1 L12 . Among them,

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2 T2 ¼ 4

Add

3 Actuator object L1 5 Dominant object L12 Transformation result L1 L12

2

L12

Water phase 6 entrance of pipeline 6 ¼6 6 4

Shape

Rectangle

Number Position

3 Circumferential uniform distribution

3 7 7 7 7 5

Meanwhile, the characteristics of water phase inflow direction in pipeline can be reﬁned and diverged. The divergence of matter elements (one eigenmultiple value) has L12 jfL121 ; L122 g. Among them, 2

L121

Water phase 6 entrance of pipeline 6 ¼6 6 4 2

L122

2.3

Water phase 6 entrance of pipeline 6 ¼6 6 4

Inflow direction Number Position

Inflow direction Number Position

Vertical pipeline axis

3

7 7 7 3 7 5 Circumferential uniform distribution 3 Adherent spiral 7 encirclement 7 7 3 7 5 Circumferential uniform distribution

Final Program Objectives

According to the principle of divergence analysis of “one characteristic multi-value” [16]. According to the divergence analysis of the expansion type of the pipeline entrance, two schemes are obtained: (1) when water enters the pipeline, the flow direction is perpendicular to the pipeline axis; (2) when water enters the pipeline, there is an angle between the flow direction and the pipeline axis, that is, the flow enters the pipeline spirally. Comparing comprehensively, the water flow enters in the form of helix, which can better adhere to the inner wall of the pipeline under the action of centrifugal force. According to the divergence analysis of the pipeline outlet expansion type, the following two schemes are obtained: (1) the bottom liquid outlet is located at the inlet section of the pipeline of the stabilization device; (2) the bottom liquid outlet is located at the outlet section of the pipeline of the stabilization device. The ﬁnal scheme selection needs to be analyzed by simulation results. The overall design of the stabilizing device is innovative in the common conveying pipeline. Its main feature is to add a new water phase inlet and a water phase outlet, and replace the water phase of the inlet and outlet to restore the shape of the oil-water annular flow. Speciﬁcally, after the two-phase flow enters the stabilizing device, the water phase sinking at the bottom can flow into the liquid storage tank through the

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bottom outlet recovery pipe for storage, and the transfer pump continuously pumps water into the pipe body, and the water enters the pipe under the guidance of the connecting pipe. In the body, at this time, the water entering the pipe body is conveyed forward along the spiral groove around the inner body of the pipe body under the condition that the conveying pump is pressurized, thereby forming a water ring heavy oil in the pipe body; The water in the tank can be recycled to achieve recycling of the unit. As shown in Fig. 2;

1 flange, 2 pipe body, 3 spiral grooves, 4 connecting pipes, 5 recovery pipes, 6 conveyor pumps, 7 fixing plates, 8 storage tanks

Fig. 2. Transport stabilize device

3 Numerical Simulation of Stabilize Device 3.1

Control Equation

The two-phase fluid obeys the law of mass and momentum conservation. The standard k e model is chosen for turbulence model, and the continuous surface force CSF model is used for oil-water interaction. The continuity equation is: @q þ r ð~ vqÞ ¼ 0 @t

ð1Þ

q ¼ ao qo þ ð1 ao Þqw

ð2Þ

Formula: q is fluid density; t is time; r is divergence symbol; ~ v is fluid velocity; qo is oil phase density; ao is volume fraction of oil phase; qw is density of water phase. The momentum conservation equation is: @ ðq~ vÞ þ r ðq~ v~ vÞ ¼ rp þ rs þ q~ gþ~ F @t

ð3Þ

2 s ¼ l½ðr~ v þ r~ vT Þ r ð~ vIÞ 3

ð4Þ

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Formula: p is the pressure, s is the shear stress tensor, g is the acceleration of gravity, ~ F is the external volume force, l is the fluid viscosity, I is the unit tensor. The standard k e model is: @ðqkÞ @ðqk~ vÞ @ þ ¼ @t @xi @xj @ðqeÞ @t

@ðqe~ vÞ @xi 2 C2e q ek

þ

l @k lþ t þ Gk qe rk @xj

¼ @x@ j þ

h i @e C1e ke Gk l þ rlet @x j

lt ¼ qCl

k2 e

ð5Þ

ð6Þ

ð7Þ

Formula: k is the turbulent kinetic energy, e is the dissipation rate, Gk is the turbulent kinetic energy generation term caused by the average velocity gradient, C1e ; C2e ; Cl is the constant, and its values are 1.44, 1.92 and 0.09, respectively; rk is the turbulent Plante number of turbulent kinetic energy k and its value is 1.0; re is the turbulent Plante number of dissipation rate e and its value is 1.3; xi is the i coordinate direction; xj T is the j coordinate direction; lt is the turbulent viscosity. The surface tension model of CSF is: Fr ¼ row

ao qo jraw þ aw qw jrao 0:5ðqo þ qw Þ n j¼r j nj n ¼ rao

ð8Þ ð9Þ ð10Þ

Formula: Fr is the oil-water interfacial tension, which can be substituted by external volume force (3); row is the oil-water interfacial tension coefﬁcient; qo ; qw is the density of oil phase and water phase; ao ; aw is the volume fraction of oil phase and water phase; j is the curvature of interface; n is the unit normal direction of the volume fraction of oil phase. 3.2

Computational Model

The internal structure of the transport stabilize device is simpliﬁed, and the threedimensional model of the transport stabilizate device runner is created by using Solidworks software. The diameter of the runner is 20 mm, the length of the inlet pipeline is 10 mm, the length of the stabilizate device is 120 mm, the length of the outlet pipeline is 70 mm, the depth of the spiral depression of the inner wall of the pipeline is 1 mm, and the ratio of water to oil inlet area is about 2:8 [10]. The water phase is in the lower layer, the oil phase is in the upper layer, and the flow area model is shown in Fig. 3. The model is divided by using the mesh module in ANSYS

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Workbench. The maximum distortion of the mesh is 0.79, and the minimum orthogonal mass is 0.32, which meets the requirements of numerical simulation.

Fig. 3. Flow area model

The physical parameters of the fluids are shown in Table 2. The boundary conditions are set as follows: the inlet is a velocity inlet, the oil phase velocity is 1.2 m/s, the water phase velocity in the pipe is 1.02 m/s, the water phase velocity in the spiral inlet is 1.5 m/s, the outlet is a pressure outlet, and the rest are wall surfaces. Because the Reynolds number exceeds the critical value, turbulence occurs in the flow region. The standard turbulence model is adopted. SIMPLE algorithm is used to solve the flow equation. The second-order upwind scheme is chosen for all the discrete schemes in the equation, and the two-precision solution is obtained. Table 2. Physical properties of various fluids Name Density (kg m−3) Viscosity (ps s) Surface tension (N m−1) Oil 960.0 0.220000 0.039 Water 998.2 0.001003 –

4 The Analysis and Discussion of the Results 4.1

Scheme Simulation and Comparison

According to the thinking method of characteristic multi-value, we set the bottom liquid outlet at the entrance and exit of the pipeline of the conveying and stabilizing device respectively, and analyze it through numerical simulation. Comparing Figs. 4 and 5 (red area indicates that the area is full of oil, blue area indicates that the area is full of water, and other colors indicate oil-water mixing), when the bottom liquid outlet is set at the entrance section, there is no oil phase on the wall of the whole pipeline, all of them are water phase, and a good oil-water annular flow pattern is formed at the outlet of the pipeline, and the oil phase is wrapped uniformly by the water phase. When the bottom liquid outlet is set at the exit section, some oil phases will appear on the wall of the pipeline, and a better oil-water annular flow pattern has not been formed at the outlet of the pipeline, and the distribution of oil and water phases is not uniform. The main reason is that the water phase of the lower layer flows out from the bottom outlet and the oil level drops. At this time, the liquid of the spiral pipeline has entered the stabilize device pipeline. The oil phase and the water phase leaving the spiral pipeline can not form a good wrapping form. Under the impact of the water phase from the spiral pipeline, the oil phase adheres to the wall.

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Fig. 4. The volume fraction contours of oil phase when the bottom outlet is set at the entrance of pipeline

Fig. 5. The volume fraction contours of oil phase when the bottom outlet is set at the exit of pipeline

Comparing Figs. 6 and 7 synthetically, when observing the volume fraction contours of oil phase in each section, the two schemes have something in common: when the lower water phase passes through the spiral pipeline, a part of the water phase will be wrapped in the oil phase, and the two phases will move forward in a spiral way. At the same time, the water phase wrapped in the oil phase will gradually migrate to the pipeline wall, with the trend of leaving the oil phase wrapped. The difference is that when the bottom outlet is set at the entrance section of the pipeline, the stabilize device can form a good oil-water annular flow pattern; when the bottom outlet is set at the exit section of the pipeline, the intact flow pattern can not be formed. Therefore, the flow pattern in the pipeline varies with the location of the bottom liquid outlet, and the volume fraction of the oil phase varies with the cross-section. The speciﬁc relationship values are shown in Fig. 8. The volume fraction of oil phase decreases ﬁrst and then increases. The main reason is that there is a large amount of water flowing into the pipeline through the spiral pipeline, which leads to the decrease of the volume fraction of oil phase. After a long distance transportation, the two-phase flow is stable, and the volume fraction of oil phase increases gradually when the velocity of water phase is less than the velocity of oil phase, and ﬁnally approaches the volume fraction at the entrance. Comparatively, the recovery effect of oil-water annular flow with bottom outlet at the entrance of pipeline is higher than that with bottom outlet at the outlet of pipeline.

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Fig. 6. The volume fraction contours of oil phase of each section when the bottom outlet is set at the entrance section of the pipeline.

Fig. 7. The volume fraction contours of oil phase of each section when the bottom outlet is set at the exit section of the pipeline.

Fig. 8. Oil volume fraction comparison for different locations at the bottom

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The Formation of Two Phase Flow and Its Flow Field Analysis

As shown in Fig. 9 at the initial moment (0–0.06 s), the oil has a downward movement trend. The main reason is that the bottom liquid outlet is set at the entrance section of the pipeline of the transport stabilize device. At this time, part of the water is discharged from the pipeline, and the oil begins to move downward under the action of gravity. In the mid-term moment (0.06–0.12 s), some water enters the stabilize device through the spiral pipeline. At this time, the velocity and direction of the new fluid entering the pipeline are different from that of the original one, so some turbulent phenomena will occur in the contact area. At the same time, the new liquid ﬁlls the gap of the pipeline, and the liquid entering the spiral pipeline is better to contact the wall under the action of centrifugal force, thus blocking the oil phase and the wall contact. At the later stage (0.12–0.18 s), the oil is obviously wrapped in the middle, and there is obvious annular flow at the outlet of the pipeline. The water in the upper layer is slightly more than that in the lower layer, which is more conducive to maintaining the stability of the annular flow. At the later moment (0.12–0.18 s), the oil is obviously wrapped in the middle, and there is obvious annular flow at the outlet of the pipeline. The water in the upper layer is slightly more than that in the lower layer, which is more conducive to maintaining the stability of the annular flow. At the end moment (0.18– 0.8 s), it is obvious that a stable annular flow pattern has been formed in the pipeline. At this time, there is a water phase vortex in the middle of the pipeline, mainly because at the outlet of the spiral pipeline, the jets produced by the three pipelines converge to produce the vortex, which has no effect on the stability of the later transportation of the annular flow. As shown in Fig. 10, the flow velocity in the middle of the whole pipeline is faster than that near the pipe wall. The main reason is the influence of the liquid sticky wall to reduce the flow velocity. During pipeline transportation, there are two peaks of velocity increase along the central axis of the channel as shown in Fig. 11. The ﬁrst wave peak occurs at the end of the entrance of the spiral pipe. The reason is that there is a large amount of water flowing into the spiral pipe, and the high-speed flow around the center of the water squeezes the fluid to increase the flow velocity. The second wave peak occurs at the end of the helical pipe. The main reason is that the cross section of the main channel decreases. The fluid needs to leave the helical wall and enter the pipe. According to Bernoulli equation, the flow velocity increases with the decrease of the area through which the fluid flows, and then the flow velocity tends to be stable. As shown in Figs. 12 and 13, the total pressure between the inlet of the pipeline and the inlet of the helical pipeline increases slightly, and reaches the maximum at the inlet of the helical pipeline. Then the total pressure from this position to the outlet shows a signiﬁcant downward trend, and the pressure in the center of the pipeline is stronger than that on the wall of the pipeline. The main reason is that the fluid is squeezed at the entrance of the spiral pipe with the newly added fluid, therefore, the total pressure will rise, and then the fluid movement will be balanced, and the squeeze in the center will gradually decrease. However, the central extrusion strength is still greater than the extrusion strength of the fluid on the pipe wall, therefore, the total pressure in the middle of the channel decreases gradually but is still higher than that on the wall of the pipeline.

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0.02s

0.04s

0.06s

0.08s

0.1s

0.12s

0.14s

0.16s

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Fig. 9. Contours of volume fraction between heavy oil and water at different time

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Fig. 10. The velocity contours of stabilize device

Fig. 11. The velocity plot in the direction of an axis

Fig. 12. The total pressure contours of stabilize device

Fig. 13. The total pressure plot in the direction of an axis

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Evaluation of Program Excellence

According to the feasibility solution analysis of TRIZ innovation principle, we choose the combination of abandonment principle and repair principle. Through the replacement transformation of extension transformation method, we adopt the new method of adding water phase to replace the initial water phase. There is a certain angle between the direction of water phase and the axis of pipeline, that is, the water flow enters the channel in a spiral way. We have designed two different schemes for the initial water phase discharge. Through numerical simulation analysis, it can be concluded that both schemes have a certain degree of recovery effect. By comparing the volume fraction of oil phase and the recovery of water-ring heavy oil, we can ﬁnd that the effect of the bottom liquid outlet at the entrance section of the pipeline of the stabilization device is better than that of the bottom liquid outlet at the stable installation. The outlet section of the pipeline is located, so the bottom liquid outlet is selected for the inlet section of the pipeline of the stabilization device.

5 Conclusions and Recommendations (1) Innovative design approach combining TRIZ and extenics. Firstly, the contradiction problem in the innovative design is analyzed by using the extenics contradiction method, then the contradictory parameters are standardized, the TRIZ contradiction matrix is searched, the combination of abandonment principle and repair principle is selected to establish the feasible solution, implements the replacement transformation in extenics, and the innovative design scheme is obtained. (2) According to the principle of divergence analysis of a characteristic multi-value, two schemes are designed, in which the bottom liquid outlet is located at the entrance or exit of the stabilizing device. Through numerical simulation, according to the goodness evaluation method, the scheme of setting the bottom liquid outlet at the entrance section of the pipeline of the transport stabilize device is optimized. (3) Spiral pipeline is the core part of the whole stabilize device. The helical elevation angle, the number of helical coils and the wall thickness of the helical pipeline need to be further studied and discussed.

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4. Jiang, F., Wang, K., Skote, M., et al.: Simulation of non-Newtonian oil-water core annular flow through return bends. Heat Mass Transf. 54, 37–48 (2017) 5. Jiang, F., Wang, Y., Ou, J.J., et al.: Numerical simulation of oil-water core annular flow in a U-bend based on the Eulerian model. Chem. Eng. Technol. 37(4), 659–666 (2014) 6. Pan, D.L., Tu, D.Y.: Experimental study on high viscosity crude oil transported by liquid loop. Oil Gas Storage Transp. 1(5), 1–8 (1982) 7. Ho, W.S., Li, N.N.: Core-annular flow of liquid membrane emulsion. AIChE J. 40(12), 1961–1968 (1994) 8. Li, J.Y., Liang, C.Q., Zhang, D.M., et al.: Stability of oil-water annular flow transport. Oil Gas Storage Transp. 7(6), 22–29 (1988) 9. Zhao, H.Y., Jing, J.Q.: Research status and progress of annular flow. Petrochem. Technol. 2, 124–126 (2015) 10. Jiang, W.M., Du, S.L., Liu, Y., et al.: Study on stability characteristics and structural optimization of a new type of core-annular flow generator with high viscosity oil. J. Hunan Univ. (Nat. Sci.) (8), 86–90 (2018) 11. Jiang, F., Sun, H., Hu, Y.D., et al.: The plan research on the mechanical foundation Innovative experiment Teaching system combined with TRIZ theory. Equip. Manuf. Technol. 2010(2), 190–192 (2010) 12. Jiang, F., Chen, W.P., Wang, Y.J., et al.: Collection Mode optimization of casting dust based on TRIZ. Adv. Mater. Res. 97–101, 2695–2698 (2010) 13. Robles, G.C., Negny, S., Lann, J.M.L.: Case-based reasoning and TRIZ: a coupling for innovative conception in chemical engineering. Chem. Eng. Process. Process. Intensif. 48 (1), 239–249 (2009) 14. Yang, C.Y., Cai, W.: Generating creative ideas for production based on extenics. J. Guangdong Univ. Technol. 33(1), 12–16 (2016) 15. Jiang, F., Li, S.J.: Did You Innovate Today – TRIZ Innovation Story. Intellectual Property Press, Beijing (2017) 16. Du, Y.S., Liu, Y.P.: The variation of features based on extenics. J. Guangdong Univ. Technol. 34(6), 9–14 (2017)

Design of Spatial Lissajous Trajectory Vibrating Screen Zhipeng Lyu(&) and Sizhu Zhou School of Mechanical Engineering, Yangtze University, Jingzhou 434023, China {Lvzhp,zhsz}@yangtzeu.edu.cn

Abstract. According to the formation principle of two-dimensional Lissajous curve, it is determined that the rotational speed of two linear exciting vibration motors is twice of the rotational speed of circular exciting vibration motors, the spatial Lissajous vibration trajectory can be realized. The relevant vibration equation is established and solved. At last, this paper innovatively divides the conventional single sieve box into two parallel symmetrical arrangements, and put forward a parallel double sieve box structure, then the spatial Lissajous trajectory vibrating screen is designed. Keywords: Lissajous spatial trajectory

Vibrating screen Design

1 Introduction Vibrating screen is the key equipment for material separation and is widely used in construction, mining, agriculture and petroleum [1]. For the planar motion of multibody dual-motor and multi-motor elemental-body vibrating screen, scholars have already conducted in-depth research and formed a certain theoretical basis [2–4]. At present, the vibrating trajectory of vibrating screen used in the ﬁeld is generally in the form of a circle, a straight line and a moving ellipse [5–8]. This paper proposes a new spatial Lissajous vibration trajectory which allows the surface of the screen to move in the X, Y and Z directions, making the solid-phase particles subjected to more uniform forces and this contributes to relieve the screen paste problem.

2 Formation Principle of Two-Dimensional Lissajous Curve The motion synthesized by two simple harmonic vibrations which are perpendicular and have different frequencies is complicated. However, when the frequency of the two partial vibrations is a simple integer ratio, a regular, stable, closed trajectory, the Lissajous curve, can be obtained. Jules Antoine Lissajous conducted a detailed research on this family of curves in 1857 [8]. Set the two simple harmonic vibration equations along the x and y directions as follows: This project is supported by the ﬁnancial support from the Hubei Provincial Department of education in China (Grant No. D20171303). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 493–498, 2020. https://doi.org/10.1007/978-981-32-9941-2_40

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(

x ¼ A1 cosðx1 t þ u1 Þ y ¼ A2 cosðx2 t þ u2 Þ

ð1Þ

Then, when the vibration frequency ratio x1:x2 is 2 and 3 and the initial phase difference x1:x2 is 0 and p/2, the two-dimensional Lissajous curve family is shown in Fig. 1.

Fig. 1. Two-dimensional Lissajous curves

Fig. 2. Spatial Lissajous trajectory

3 Formation Method of Spatial Lissajous Vibration Trajectory 3.1

Theoretical Equation for Spatial Lissajous Vibration Trajectory

The synthetic trajectory of three mutually perpendicular harmonic vibrations when the frequency of the partial vibration is a simple integer ratio is discussed. Set the expressions of the simple harmonic vibration parameter equations along the x, y and z directions as follows: 8 > < x ¼ A1 sinðx1 t þ u1 Þ y ¼ A2 cosðx2 t þ u2 Þ > : z ¼ A1 cosðx1 t þ u1 Þ

ð2Þ

It can be seen from analytical formula (2) that the synthetic trajectory is circular in the xz plane, and the trajectory is a reciprocating straight line in the y direction. When x1:x2 = 2, a stable and closed spatial vibration trajectory curve can be obtained, as shown in Fig. 2. This curve is similar to one of the two-dimensional Lissajous curves, which is referred as the spatial Lissajous trajectory in this paper.

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Fig. 3. The realization principle diagram of spatial Lissajous vibrating trajectory

3.2

Realization Principle of Spatial Lissajous Vibration Trajectory

The screen box, excitation motor and spring structure are simpliﬁed and the corresponding coordinate system is established, as shown in Fig. 3. According to the balance principle [9], if the object is to maintain the translational state, the vibration direction of the reciprocating straight line needs to be directed to its center of mass. Assuming that there is no phase difference between the two linear vibration motors, according to the Darren Bell principle, the equation of motion of the screen box along x, y and z direction is: 8 2 > < M€x þ 4kx x ¼ mc rc xc sin xc t M€y þ 4ky y ¼ 2ml rl x2l cos xl t > : M€z þ 4kz z ¼ mc rc x2c cos xc t

ð3Þ

In the formula: M - Screen box quality (including screen frame, vibration motor, drilling fluid and the reduced mass of rock debris), kg; mc - the quality of the eccentric block of the circular vibration motor, kg; ml - the quality of the eccentric block of the linear vibration motor, kg;

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rc - eccentric distance of the eccentric block of the circular vibration motor, m; rl - eccentric distance of the eccentric block of the linear vibration motor, m; kx, ky, kz - the stiffness equivalent of a single support spring in the x, y and z directions, N/m; xc - circular vibration motor speed, rad/s; xl - linear vibration motor speed, rad/s. According to the solution method of the ternary quadratic differential equation, the steady-state solution of Eq. (3) can be expressed as: 8 > < x ¼ Ax sinðxc tÞ y ¼ Ay cosðxl tÞ > : z ¼ Az cosðxc tÞ

ð4Þ

Equation (4) is the vibration trajectory equation of the model shown in Fig. 3 and Ax, Ay, Az is the amplitude along x, y and z directions, which are 8 mc rc x2c > > > Ax ¼ > > 4kx Mx2c > > > < 2ml rl x2l Ay ¼ > 4ky Mx2l > > > > > mc rc x2c > > : Az ¼ 4kz Mx2c

ð5Þ

Assume kx = kz, then Ax = Az. Formula (4) can be transformed into 8 > < x ¼ Ax sinðxc tÞ y ¼ Ay cosðxl tÞ > : z ¼ Ax cosðxc tÞ

ð6Þ

Comparing formulas (6) and (2), as long as xl = 2xc, a spatial Lissajous vibration trajectory shown in Fig. 2 can be realized. 3.3

Realization Principle of Spatial Lissajous Vibration Trajectory

In order to meet the speciﬁc installation requirements of the three motors to realize the spatial Lissajous trajectory, this paper innovatively divides the conventional single sieve box into two parallel symmetrical arrangements, and designs a parallel double sieve box structure, as shown in Fig. 4.

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Fig. 4. Parallel double screen box structure

4 Conclusions (1) According to the formation principle of two-dimensional Lissajous curve, it is determined that the rotational speed of two linear exciting vibration motors is twice of the rotational speed of circular exciting vibration motors, the spatial Lissajous vibration trajectory can be realized. (2) A parallel double sieve box structure can meet the requirements to realize the spatial Lissajous trajectory vibrating screen.

References 1. Jiang, H.S., Zhao, Y.M., Duan, C.L., Yang, X.L., Liu, C.S., Qiao, J.P., Diao, H.R.: Kinematics of variable-amplitude screen and analysis of particle behavior during the process of coal screening. Powder Technol. 306, 88–95 (2017) 2. Yin, Z.J., Zhang, H., Han, T.: Simulation of particle flow on an elliptical vibrating screen using the discrete element method. Powder Technol. 332, 432–454 (2016) 3. Liu, C.S., Wang, H., Zhao, Y.M., Zhao, L.L., DEM Dong, H.L.: Simulation of particle flow on a single deck banana screen. Int. J. Min. Sci. Technol. 23, 273–277 (2013) 4. Kichkar, I.Y.: Analysis of an assigned oscillatory trajectory of a vibratory drilling screen. Chem. Pet. Eng. 46(1–2), 69–71 (2010) 5. Kichkar, I.Y.: Positioning of shale-shaker drive with an assigned vibration trajectory. Chem. Pet. Eng. 46(3–4), 137–141 (2010) 6. Wang, Z.M., Wang, R.H., Xiao, W.S.: Screening mechanism of balanced elliptical motion shale shaker excited by backward elliptical vibrating forces. J. China Univ. Pet. 32(2), 104– 109 (2010)

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7. Wang, Z.M., Wang, R.H., Shi, L., Huang, D.H.: Development and application of a selfsynchronization and balanced elliptical motion shale shaker excited by backward elliptical vibrating forces. Oil Drill. Prod. Technol. 32(2), 89–92 (2010) 8. Taylor, C.A.: The Art and Science of Lecture Demonstration, pp. 50–54. CRC Press, Boca Raton (1988) 9. Zhao, C.Y., Wen, B.C., Zhang, X.L.: Synchronization of the four identical unbalanced rotors in a vibrating system of plane motion. Sci. China Technol. Sci. 53(2), 405–422 (2010)

Part Retrieval Technology Based on Geometric Shape and Topological Correlation Ning Ma, Bo Yang(&), and Junzhi Shang School of Mechanical Engineering, University of Jinan, Jinan 250022, China [email protected], [email protected], [email protected]

Abstract. Part retrieval technology plays an important role in the product structure reconstruction design. Because the basic functional and structural information of the part is represented by the shape and topological association of the geometric elements. A part retrieval technology based on geometric shape and topological correlation is proposed in this paper. The part retrieval method is constructed from the two aspects, the semantic level and the geometric topological correlation. Firstly, at the semantic level, the fuzzy query of the transplanted parts is realized by characterizing the key semantic description of the transplanted parts. Then, at the level of geometric topological relationship, the similarity calculation is performed on the geometric shape information and the topological feature information in the variant parts library and transplanted parts library. Through the similarity retrieval algorithm and the setting of the retrieval threshold, the transplanted parts which similar to the variant parts are retrieved. Finally, the feasibility of the technology is veriﬁed by an example. Keywords: Transplanted parts retrieval Similarity calculation Description semantics Geometric shape correlation Topological feature correlation

1 Introduction In the product design process, more than 75% of the engineering design is by improving the existing similar parts [1]. It can shorten the designing period and improve the efﬁciency. Therefore, the 3D CAD part retrieval technology plays an important role in the product structure reconstruction design and has become a research hot-spot in recent years. At present, the retrieval methods of 3D models mainly include: (1) Content-based retrieval methods. The method mainly uses the geometry of the extracted model or the topological association to describe the model. Based on this, the similarity between geometry and topology is calculated to complete the model retrieval [2]. For example, Gao et al. proposed a mechanical part retrieval method based on typical surface matching. They extract the geometrical topological information and estimate the

This project is supported by National Natural Science Foundation of China (Grant No. 51775239, 50975124). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 499–507, 2020. https://doi.org/10.1007/978-981-32-9941-2_41

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discriminant degree of the surface, ﬁnally apply the greedy algorithm to realize the part retrieval [3]. Zhu et al. proposed a 3D model retrieval method of mechanical parts based on skeleton tree [4]. (2) A retrieval method based on high-level semantics. The method deﬁnes the characteristics of model as semantic information that meets the user’s needs and extracts the semantic information to achieve similarity retrieval. For example, Xu et al. realized the region segmentation of the 3D model of the part by deﬁning the edge classiﬁcation speciﬁcation and used the region structure code to retrieve the similar structural parts [5]. The above methods effectively solved some basic problems of three-dimensional retrieval, but failed to realize the organic combination of global retrieval and local retrieval. Therefore, based on the above methods, this paper proposed a part retrieval method based on geometric shape and topological correlation. The method ﬁrstly performed a fuzzy query by characterizing the key semantics of the transplanted part. Then extracted the variant parts information and calculated the similarity between them and the geometric topology information of the transplanted parts. Finally, the transplanted parts similar to the variant parts are retrieved by setting the similarity retrieval algorithm and the retrieval threshold. This method achieved the uniﬁcation of local retrieval and global retrieval and made 3D retrieval more efﬁcient and accurate.

2 Fuzzy Retrieval Method Based on Part Description Semantics The method of part retrieval based on descriptive semantics is to use the descriptive semantics of the geometric model as the condition for geometric model retrieval. Then the data based fuzzy query method is used to retrieve the transplanted parts. 2.1

The Type of Describe Semantics

Geometric model descriptive semantics includes two aspects: functional descriptive semantics and physical attribute descriptive semantics. Functional descriptive semantics includes support unit, connection unit, transfer motion unit; physical attribute descriptive semantics including part name, part number, part type (classifying feature units: block part, cylinder part, cone parts, sphere parts, boss parts, etc.), geometric matching surface types (plane, cylindrical surface, spherical surface, regular surface, etc.) [6]. 2.2

Fuzzy Retrieval Process Based on Semantic

The 3D model retrieval technology based on descriptive semantics mainly includes the following process: (l) extracting and storing descriptive semantics; (2) matching descriptive semantics; (3) querying model matching; (4) outputting display query results. The process is shown in Fig. 1.

Part Retrieval Technology Based on Geometric Shape Descriptive semantics

Porting feature library

501

Data sheet Extract and index information

off line on line Query input Search User interface Visualization result

Descriptive semantics

Matching semantic

Search result

Matching geometric model

Fig. 1. Basic process of transplant unit retrieval based on descriptive semantics

With the descriptive semantics of physical attributes as the query keywords, the query high-level description semantics are divided into three levels: The ﬁrst level is the type of lap surface; The second level is target matching surface; The third level is plane. In the ﬁrst level semantics, the lap joints are divided into plane lap joints, cylindrical lap joints and other regular lap joints according to the different lap surfaces. In the second level semantics, the target matching surface is the geometric matching surface on the external transplanted part that needs to be associated with geometric constraints. Then, in the third level query semantics, the plane is taken as the target matching surface and ﬁnally the parts are found in the transplanted part library. The ﬁnal query results are shown in Fig. 2.

Input First level query semantics Query conditions

Mating type Input

Second level query semantics

Target matching surface Input

Third level query semantics

Plane

Query result

Target geometry matching plane - plane

Fig. 2. Part query based on geometric matching face type description semantics

3 Similarity Calculation and Part Retrieval Based on Geometric Shape and Topological Information In order to obtain the transplanted parts similar to the shape and topology of the variation model, the part retrieval method based on geometric model shape and topological information is selected. The retrieval of the transplanted features and the

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variation design of the parts can be realized quickly and effectively by setting the shape weight factor, topology weight factor and retrieval threshold. 3.1

Information Description Based on B-Rep

The B-Rep data structure records the geometric information of all geometric elements and their interconnected topological relations according to the level of “body-surfaceloop-edge-point” [7]. The transplanted structure is a B-Rep data structure and its topology elements include: point, edge, coedge, wire, loop, surface, sub-shell, shell, lump and body. The B-Rep information is extracted from the part model and the CAD part model is convert into the form of a geometric element adjacency graph. The CAD entity model is shown in Fig. 3(a), and its corresponding geometric elements attribute adjacent graph is shown in Fig. 3(b). F8

F4 F3

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(a) CAD solid model F8 e1

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(b) Attribute adjacency graph Fig. 3. CAD model and geometric elements attribute adjacent graph

A mathematical model of the geometric element attribute adjacency graph of the CAD solid model is created: g = [v, e], where v = [v1, v2, …, vi] represents the set of points in the geometric element attribute adjacent graph and vi represents the i-th

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surface in the B-Rep model; e = [e1, e2, …, ei] represents the set of edges and ei represents the i-th edge. The boundary expression method (B-Rep) can perform various geometric operations and judgments based on the surface and edge normal vectors for the dimensional geometric information G. The two adjacent faces are found by using the B-Rep information and the geometric relationship between the dimensional geometry information and the normal vector of the surface is judged, then to transform it into the correct surface [8]. The size conversion idea based on B-Rep is shown in the Fig. 4.

C onv ersion al gorit hm B-R ep inform ati on

Edge a dj ac ent inform ati on Siz e inform ati on mode l

Geo met ric rel ati onshi p Dime nsion informat ion

Dime nsion informa ti on

Fig. 4. The size conversion idea based on B-Rep

3.2

Similarity Calculation

The geometric shape information and the topological feature information values of the unit model are extracted. The similarity calculation is performed on the shape information description set of the two part models and similar part geometric models are retrieved. 3.2.1 Similarity Calculation of Geometric Shapes The geometric shape feature information includes six attribute values, which are stored in the initialized feature vector gi[k]. Where i = 1, 2, …… represents the number of the unit geometric model; k = 0, 1, 2, …, 5 represents the surface area, volume, body length, face length, shape ratio, and compact ratio of the unit geometry model i. The similarity degree of the two geometric models is calculated by the following formula: 5 P

similarityDegreeðg1 ; g2 Þ ¼

SimilarityDegreeðg1 ½i; g2 ½iÞ

0

ð1Þ

6

Where, ( SimilarityDegreeðg1 ½i; g2 ½iÞ ¼

0

jg1 ½ig2 ½ij maxfjg1 ½i;g2 ½ij;jg1 ½ig2 ½ijg

jg1 ½i g2 ½ij ¼ 0 jg1 ½i g2 ½ij 6¼ 0

ð2Þ

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The similarity value of the two elements models is calculated as: SimilarityValuegeomtry ¼

1:0 1:0 þ SimilarityDegreeðg1 ; g2 Þ

ð3Þ

3.2.2 Similarity Calculation of Topological Features The topological feature information includes 12 feature attribute values, which are stored in the initialized feature vector ti[k]. Where i = 1, 2, …… represents the number of the unit geometric model; k = 0, 1, 2, …, 11 represents the number of faces, the number of sides, the number of vertices, the number of loops, the number of features, the number of planes, the number of cylinders, the number of cone faces, the number of spheres, the number of inner loops, and the number of outer loops of the unit geometric model i. The similarity degree of the two element geometric models is calculated by the following formula: 11 P

SimilarityDegreeðt1 ; t2 Þ ¼

SimilarityDegreeðt1 ½i; t2 ½iÞ

0

ð4Þ

12

Where, ( SimilarityDegreeðt1 ½i; t2 ½iÞ ¼

0

jt1 ½it2 ½ij maxfjt1 ½i;t2 ½ij;jt1 ½it2 ½ijg

jt1 ½i t2 ½ij ¼ 0 jt1 ½i t2 ½ij 6¼ 0

ð5Þ

The topological similarity value of the two element models is calculated as: SimilarityValuetopology ¼

1:0 1:0 þ SimilarityDegreeðt1 ; t2 Þ

ð6Þ

3.2.3 Similarity Calculation of the Overall Model The overall similarity calculation values of the two element geometric models include two parts: the geometric similarity calculation value and the topological similarity calculation value. The overall similarity calculation value is: SimilarityValuemodel ¼

w1 SimilarityValuegeomtry þ w2 SimilarityValuetopolopy w1 þ w2

ð7Þ

Where, w1 þ w2 ¼ 1:0, w1 , w2 respectively represents the weight values of the similarity and topological structure of the unit model. When the user has higher requirements on the shape similarity, the value of w1 is larger. When the user has higher requirements on the topology similarity, the value of w2 is larger.

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Transplant Part Retrieval Process Based on Geometry and Topology Information

The 3D model retrieval technology based on geometric shape and topology information mainly includes the following parts: (l) extracting and storing the description information of shape and topology; (2) calculating model similarity; (3) matching model; (4) outputting query result. The basic process of part retrieval is as follows: The description information of shape and topology is extracted which stored in the transplanted parts library. And then the information set is stored as a ﬁeld in the database table. Before the retrieval, the shape and topology description information of the variant part are extracted. And then the model information similarity calculation is performed on the description information of the variant part and each model in the transplant part library. The similarity calculation value is obtained by setting the shape weighting factor, the topological weighting factor and the retrieval threshold. The basic process of retrieval is shown in the Fig. 5. Porting library Off-line

extract

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save

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extract

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Fig. 5. Basic process of part retrieval based on shape and topology information

4 Example As shown in the Fig. 6, the hinge holder is taken as a geometric model of the reference parts. Then the transplanted part model similar to its geometry and topology structure is retrieved in the transplanted part library. Reference geometry model

Target geometry model

1.00

0.9147

0.9618

0.8977

0.9321

0.8035

Fig. 6. Search results of the geometric model 1

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Firstly, shape description information and topology description information of the reference model are extracted. Then their similarity is calculated with the shape and topology description information of all part models in the transplanted part library. In this process, w1 is the geometric weight value, w2 is the topology weight value, the weight value is set: w1 ¼ 0:6, w2 ¼ 0:4 and the similarity value range is greater than or equal to 0.5. The search result is shown in the Fig. 6. w1 ¼ 0:7, w2 ¼ 0:3 and the similarity value range is greater than or equal to 0.5. The search result is shown in the Fig. 7.

Reference geometry model

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Fig. 7. Search results of the geometric model 2

5 Conclusions (1) In order to realize the organic combination of global retrieval and local retrieval, this paper proposes a part retrieval technology based on geometric shape and topological relationship. (2) The method performs fuzzy query retrieval on the parts by characterizing the key semantics of the transplanted parts. (3) The similarity between the geometrical and topological relationships of the mutated part and the transplanted part is calculated. (4) The transplanted parts similar to the variant parts are retrieved through the similarity search algorithm and the search threshold setting. The method can greatly shorten the designing period and improve product design efﬁciency.

References 1. Liu, Y.Y., Zhang, X., Yang, D.: Three-dimensional model retrieval of machine parts based on distance transformation. Agric. Equipment Veh. Eng. 56(05), 65–68 (2018) 2. Zhou, C., Yang, B., Yao, K., et al.: Research on generalized axis similarity retrieval method. Mech. Eng. 06, 4–8 (2016)

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3. Gao, Y., Wang, B., Hu, K.M., et al.: Mechanical parts retrieval based on typical face matching. J. Comput.-Aided Des. Comput. Graph. 23(04), 640–648 + 655 (2011) 4. Zhu, W.B., Geng, G.Q., Liu, Y.Y., et al.: 3D model retrieval method of mechanical parts based on skeleton tree. J. Mech. Eng. 52(13), 204–212 (2016) 5. Xu, J., Dong, Y.: 3D part model retrieval method based on surface region decomposition. J. Comput.-Aided Des. Comput. Graph. 29(05), 929–938 (2017) 6. Yao, K.: Research on the Rapid Variant Design Based on the Transplantation of the Unit. University of Jinan (2014) 7. Xu, J.H.: Variation Design Oriented Technology of Transplantation Element Retrieval and Fusion Process Evolution. Zhejiang University (2009) 8. Chen, Z.N., Lin, J., Xu, T.M.: Creation of dimension information model for three-dimensional machining process planning system. Mod. Manuf. Eng. 01, 33–38 (2016)

Analysis of Tool Wear and Wear Mechanism in Dry Forming Milling for Rail Milling Train Chao Pan1(&), Xiaoshan Gu1, Mulan Wang2,3, and Baosheng Wang2,3 1

China Railway Shanghai Administration Group Nanjing EMU Department, Nanjing 210012, China [email protected], [email protected] 2 Jiangsu Key Laboratory of Advanced Numerical Control Technology, Nanjing 211167, Jiangsu, China [email protected], [email protected] 3 Jiangsu Provincial Engineering Laboratory of Intelligent Manufacturing Equipment, Nanjing 211167, China

Abstract. Milling tools of the rail milling train have to face severe weariness due to the uneven surface hardness, inconsistent cutting rate and uncertain status of the rail itself as well as the vibration caused by the suspension system of the rail milling train and its characteristic hunting motion. This study performed a milling tool wearing status analysis and conducted relative research on wearing mechanisms with electron scanning microscope based on the typical milling tools and speciﬁc working conditions of rail milling train. Results indicate that the wearing status of rail milling train consists mainly of crater wear, notch wear, cutting edge chipping, surface exfoliation, milling sintering and tool break. The tool wearing conditions of rail milling train, compared with ordinary milling process, are signiﬁcantly severe, and associated with multiple types of wearing status of which certain relevance exists. The research is of great signiﬁcance for the improvement of the milling tool, the optimization of the process and the improvement of the mobile milling quality of the rail. Keywords: Rail

Forming dry milling Tool wear Wear mechanism

1 Introduction At present, there are mainly two mobile rail repairing methods, namely, rail grinding and rail milling. The method of rail grinding ﬁxes grinding materials such as motordriven grinding wheels at different points of railhead to grind surface metal through the rotation, translation or rolling motions of grinding materials, aiming at recovering the cross section and longitudinal proﬁle of the rail. Rail milling is a kind of new mobile rail maintenance technology, which makes rail maintenance more efﬁcient and more cost-effective. Rail milling train is a new-type and efﬁcient mobile maintenance This project is supported by National Natural Science Foundation of China (Grant No. 51405210), University Science Research Project of Jiangsu Province, China (Grant No. 18KJA460004). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 508–517, 2020. https://doi.org/10.1007/978-981-32-9941-2_42

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equipment of rail, it has milling and grinding as its two procedures during its operation process. Milling procedure mainly adopts peripheral milling technology. The milling cutter on the whirling head mills top surface of rail, gauge corner and gauge line by rotating whirling head around the longitudinal axis perpendicular to the rail. Rail milling train can be applied to most cases of track maintenance. The rail maintenance technology of rail milling train can reduce roughness on the rail surface, remove railhead surface defects, and mills railhead proﬁle into a targeted one. It has advantages of improving wheel-rail relation, reducing vibration and noises, elevating running stability of train and comfort of passengers and extending service life of rail [1–4]. Since rail should be made from difﬁcult-to-cut material, it is normally made from hot-rolled or quenched high-carbon steel and low carbon microalloyed steel. At present, the 60 kg rail materials in China are mainly PD3, U71Mn and BNbRE with harness between 280–320 HB. The surface of rail may be harded unevenly after repeated rolling of wheels. During mobile milling, in order to avoid influence of cutting fluid on environment and training running, dry milling is generally adopted, but the temperature at milling area may rise rapidly and cause cutter wearing [5]; besides, as there are both straight line and curve line milling demand, and there might be abnormal cases of uneven rail, surface defects and large rail verge which would directly affect the quality of rail processing and service life of cutters; compared with traditional machinery processing method, rail milling train has hunting motion and vibration of suspension system during working on rail, thus equipment status and processing object are always changing, and localization datum is also in constant change, and the stability of equipment and processing object may directly influence service life of cutters. Besides, tool wear is different in milling complicated proﬁle of rail compared with traditional processing method. Therefore, studying milling tool wear forms and tool wear mechanism may be signiﬁcant to optimization of procedure, improving tool and rail processing quality of rail milling train.

2 Mobile Rail Milling Technology The SF03-FFS rail milling train produced by Linsinger Company in Austria weighs 120 tons. It has two sets of procedures: milling and grinding. The train is ﬁxed with 2 milling units and 1 grinding unit per side. The milling cutters are used for cutting rail, grinding plate is used to improve the smoothness of rail surface. The milling procedure mainly adopts peripheral milling technology. The milling cutter on the whirling head mills rail top surface, gauge corner and gauge line by rotating whirling head around the longitudinal axis perpendicular to the rail and remove the damages on rail surface. Different from rail grinding train which needs many grinding wheels to ﬁnish rail milling, rail milling train adopts peripheral milling technology and grinds rail top surface by circumference surface of grinding plate and completely remove the minor concaves made by milling and improve the smoothness of railhead [6]. The milling cutter of rail milling train is mainly made up of blades, milling cutters and ﬁxed bolts. The working status of milling cutter is shown in Fig. 1. The diameter of disk milling cutter is 600 mm and made by wholistic processing. The blades were ﬁxed on the circumference surface of disk milling cutter in the radial direction, with two

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blade groups arranged laterally, each blade group containing eight blades. Moreover, the blades in each group are arranged in dislocation mode in the peripheral direction, forming an integral cutting edge with the same targeted proﬁle of the rail along the axle direction. Rail forming milling cutter is only adopted at gauge corner, where radius of curvature is small. The curve of concave circular of the forming milling cutter matches with that of gauge corner, and other blades have straight cutting edges. The appropriate blade pattern can mill rail into targeted proﬁle. Blades are made from hard alloy, coated with TiAlN on the surface. They can mill to a depth of 0.1–3 mm, and maximum depth of 5 mm at gauge corner. Their working speed can reach up to 420–1200 m/h. Milling range starts from the milling angle of 70° and 14 mm below rail surface of acting side to 40° outside the railhead. Rail section accuracy can reach 0.3 mm, and Accuracy of longitudinal proﬁles within the measuring range of 300 mm can reach 0.03 mm [7].

Fig. 1. Milling cutter of rail milling train

3 Analysis of Tool Wear in Forming Milling of Rail In order to study the tool wear status and mechanism of the rail milling train, after initial analysis, this study studied three typical blades after wearing, as is shown in Fig. 2. In the three blades, blades 1 and 2 have straight cutting edge, used for ﬁnish milling and rough milling of rail proﬁle respectively, blade 3 is in arc-shape, used for the forming cutting of rail gauge corner. Electron scanning microscope was applied to observe the front and rear face of the cutters and to analyze the wearing status and wearing mechanism.

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Fig. 2. Milling blades of rail milling train

Blade 1 as is shown in Fig. 3 is mainly used for ﬁnish milling of rail milling train. The blade is an indexable blade with four straight-line cutting edges. It is used for milling the circular arc of the rail proﬁle. The front face of the cutters shows that, crater wear is one of the main wear status. With expanding area of crater wear, there will be exfoliation on surface on the blade. In addition, cutting edge chipping also appears on all blades. With higher degree of wearing, milling sintering may emerge on the blade material.

1-1

1-3

1-2

1-4

Fig. 3. Wearing status of blade 1

Blade 2 as is show in Fig. 4 is mainly used for rough milling of rail. The wear status of front and rear faces of the blades showed that, the notch wear on rear face of blade 2 was more serious than traditional milling processing with larger wear area and greater depth, besides, in terms of milling depth, the wear on rear face was in the shape of triangle, with serious wear near the cutting edge, less wear area with increase of distance from the cutting edge. The front face of blades also had obvious crater wear.

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2-1 2-1

Front

face

of

2-2 2-2 Front face of cutter

Fig. 4. Wearing status of blade 2

Blade 3 is the key cutter in mobile forming milling. It mainly mills the small radius arc surface at gauge corner. The precision and performance of the blade directly influence the processing quality of gauge corner. Figure 5 shows that, the wear status of blade 3 has crater wear, notch wear at the rear face and milling sintering, as well as more serious cutting edge chipping, damages and surface exfoliation.

3-1

3-1 Front face of cutter

3-2

3-2 Front face of cutter

Fig. 5. Wearing status of blade 3

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4 Analysis of Wear Mechanism in Forming Milling of Rail An analysis on wear mechanism was done by considering the wear status of milling cutters. (1) Crater wear: crater wear at the front face of cutter is one of the main wear types for blades in rail milling train. Because of ﬁerce thermal shocks and mechanical shocks on the rail milling train during operation, only the method of dry milling should be adopted to avoid the influences by cutting fluid on the friction coefﬁcient of train wheels. In order to study the mechanism of crater wears on the front face of cutters, rail milling process was simulated by the software AdvantEdge FEM and the tool temperature and pressure distribution were analyzed. The parameters in milling process simulation and milling procedure are shown in Tables 1 and 2, and simulation results are shown in Fig. 6. Figure 6 shows that, in the process of milling, cutting heat takes place mainly at the contact area between front face of blades and metal chips, and the temperature rises sharply by friction at the contact area. Heat accumulates at the front surface of blades near the cutting edge, and there was ﬁerce frictions generated by constant contacts between front surface and metal chips. Affected by diffusion and bond-friction between front surface of blades and metal chips, the chips brought away the tool materials of the front surface of blades, thus resulting in crater wears on the front surface [8–11]. Table 1. Parameters of blades Material Hard alloy Surface coating TiAlN Thickness of surface coating l 5 Diameter, mm 60 Teeth number 2 Rake angle, deg 11 Relief angle, deg 1 Edge Radius, mm 0.02

Table 2. Parameters of milling process simulation Milling method Down milling Rotating speed of principal axis r/min 4000 Feed per tooth mm 2 Milling depth mm 3 Initial temperature °C 20

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(1) Temperature distribution

(2) Pressure distribution Fig. 6. Milling temperature and pressure results of simulation

(2) Milling sintering: in the milling process, wear and crumbling on cutting edges made the surface coating fall off and the cutting edge become round. To study the reasons for milling sintering, the milling process was simulated. The corner radius of cutting edge increased to 0.2 mm and there was no surface coating on the tool surface, simulation results are shown in Fig. 7. During the process of milling, milling cutters had ﬁerce frictions with metal chips, the corner radius of the cutting edge increased, and simulation temperature increased from 983 °C under normal conditions to 1042 °C, with evident increase in high-temperature area near the round corner of the cutting edge. Over-temperature brought about oxidizing reaction in the hard alloy and generated soft oxides. This is the phenomenon of milling sintering of the cutter materials [12]. In addition, one milling unit of the rail milling train were ﬁxed with 88 blades. When a blade breaks or chipped during milling and can’t be replaced with a new one immediately, continuous operation may aggravate milling sintering.

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(1) Temperature distribution

(2) Pressure distribution Fig. 7. Temperature and pressure distribution after increase of corner radius

(3) Notch wear: in terms of wear status, the blades of rail milling train has more serious notch wear than traditional milling with larger wear area and greater wear depth, moreover, the notch wear was in the shape of triangle at the rear surface of the blades in the depth direction of milling, and the wear near the concave circular near the cutting edge had the most serious wear. With increased distance from the cutting edge, the notch wear became less serious. In traditional milling, the wear on rear surface of blades is mainly elastic deformation of materials because of low back engagement and ﬁerce friction between the cutting surface and rear surface of the blades. During the mobile milling process, because of suspended rail bottom and the total elasticity of rail, the rail will have elastic bending under pressure and cutter relieving may occur during milling. After rotation of cutting edge, the rail recovers its height and has ﬁerce friction with rear surface of the blades and produces serious notch wear at the rear surface. In addition, since the rail proﬁle is arc-shaped, and the cutting edge of milling cutter is straight, there is friction between a flat surface and an arc surface, resulting deep friction depth at rear surface of the blades in the shape of triangle. (4) Cutting edge chipping: the breakage on milling cutter in mobile milling is mainly caused by chipping of cutting edge. As interrupted milling on the cutter ﬁerce shocks on rail, and high temperature at milling area reduces the strength of hard alloyed cutter, especially the strength of the area near cutting edge after wears produced on the front and rear surface of the blades, resulting in collapse of cutting edge under violent shocks. By studying the working conditions of forming

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milling, it can be found that, the cutting edge contacts the arc-shaped processing surface at a point, which enlarges constantly to a larger area, thus the shocks normally take place at one point which is the most vulnerable point in tool wear, therefore, cutting edge chipping may take place on blades 1 and 2. Blade 3 has arc-shaped cutting edge which milling the gauge corner of rail. Under the same feed conditions, blade 3 has higher cutting amount than blades 1 and 2 and makes the most serious deformation on this part of rail, thus the shocks on blade 3 was more serious and had most serious cutting edge chipping on blade 3. (5) Surface exfoliation: compared with tool steel and high-speed steel, the hard alloy has higher hardness and heat resistance, thus the blade is not easily bent. However, with lower tenacity and uneven organizational structure, surface exfoliation may easily occur in face of unstable processing equipment and ﬁerce shocks on the surface of cutters. (6) Fracture of cutter: because of the characteristics of hard alloy and its inner defects, under very bad processing conditions, the shocks may exceed the bearing capacity of the cutters and fracture the cutters.

5 Conclusions (1) The rail milling train have to face severe weariness due to the uneven surface hardness, inconsistent cutting rate and uncertain status of the rail itself as well as the vibration caused by the suspension system of the rail milling train and its characteristic hunting motion. (2) The wearing status of rail milling train consists mainly of crater wear, notch wear at the rear face of cutter, cutting edge chipping, surface exfoliation, milling sintering and tool break. There are multiple types of wearing status of which certain relevance exists. (3) Serious notch wear on rear surface of blades is caused because of suspended rail bottom and the total elasticity of rail and a triangle area of wearing is formed.

References 1. Liu, Y., Li, J., Cai, Y., et al.: Current state and development trend of rail grinding technology. Chin. Railway Sci. 35(4), 29–37 (2014). (in Chinese) 2. Jin, X., Du, X., Guo, J., et al.: States of arts of research on rail grinding. J. Southwest Jiaotong Univ. 45(1), 1–11 (2010). (in Chinese) 3. Zhou, Q., Tian, C., Zhang, Y., et al.: Research on key rail grinding technology of high-speed railway. Chin. Railway Sci. 33(2), 66–70 (2012). (in Chinese) 4. Liu, Z., Hu, J., Wang, Q., et al.: Research on the internal model control of the constant milling force of the rail milling train. J. Railway Sci. Eng. 10(1), 118–122 (2013). (in Chinese) 5. Yu, J., Lin, Y., Lin, H.: Review of tool wear in high-speed milling. Tool Eng. 49(8), 3–6 (2015). (in Chinese)

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6. Hu, J., Jiang, Y., Fang, J., et al.: Generalized predictive control of the grinding force of rail milling train. Chin. Railway Sci. 35(6), 84–90 (2014). (in Chinese) 7. Lima, F.F., Sales, W.F., Costa, E.S., et al.: Application of online rail maintenance technology for rail surface defects. Urban Mass Trans. 43(1), 677–685 (2017). (in Chinese) 8. Correa, J.G., Schroeter, R.B., Machado, A.R.: Tool life and wear mechanism analysis of carbide tools used in the machining of martensitic and supermartensitic stainless steels. Tribol. Int. 105, 102–117 (2017) 9. Maruda, R.W., Krolczykb, G.M., Nieslony, P., et al.: The influence of the cooling conditions on the cutting tool wear and the chip formation mechanism. J. Manuf. Process. 24, 107–115 (2016) 10. Sartori, S., Moro, L., Ghiotti, A., Bruschi, S.: On the tool wear mechanisms in dry and cryogenic turning additive manufactured titanium alloys. Tribol. Int. 105, 264–273 (2017) 11. Malakizadi, A., Gruber, H., Sadik, I., Nyborg, L.: An FEM-based approach for tool wear estimation in machining. Wear 368–369, 10–24 (2016) 12. Lima, F.F., Sales, W.F., Costa, E.S., et al.: Wear of ceramic tools when machining Inconel 751 using argon and oxygen as lubri-cooling atmospheres. Ceremics Int. 43, 677–685 (2017)

Wear Numerical Simulation and Life Prediction of Microstructure Surface Unfolding Wheel for Steel Ball Inspection Chengyi Pan(&), Xia Li, Hepeng Wang, Yanling Zhao, Jingzhong Xiang, Sihai Cui, and Yudong Bao Harbin University of Science and Technology, Harbin 150080, China [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

Abstract. Wear is the main failure form of unfolding wheel for bearing steel ball. When the wheel wear reaches a certain level, the unfolding effect to the surface of the steel ball is invalid. Aiming at the life prediction problem for the unfolding wheel, according to analysis the wear proﬁle and wear mechanism of the unfolding wheel drive surface, the main wear type of the unfolding wheel drive surface is determined as adhesive wear. A discrete Archard adhesion wear model is established based on the classical Archard adhesion wear model combined with the ﬁnite element method. Using the ANSYS software to simulate the wear of smooth surface unfolding wheel and microstructure surface unfolding wheel respectively, obtaining the relationship between wear depth and number of wear. Then establish the critical failure condition of the unfolding wheel according to the unfolding wheel structure and the unfolding principle. And based on the above theory, establish the wear life prediction model of the unfolding wheel. Finally, the effectiveness of the life prediction model is veriﬁed by an example. It is also found that the wear life of the microstructure surface unfolding wheel is about 10% higher than that of the smooth surface unfolding wheel. It provides a theoretical basis for the practical application of the surface microstructure of the unfolding wheel for steel ball inspection. Keywords: Steel ball unfolding wheel Wear type Wear model Microstructure Finite element Numerical simulation Life prediction

1 Introduction The unfolding wheel is the core component of the wheel type unfolding mechanism using for steel ball inspection. As shown in Fig. 1, the steel ball rotates under the movement of the driving wheel and is supported by the supporting wheel. A certain

This project is supported by Heilongjiang Provincial Natural Science Foundation of China (Grant No. E2017052). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 518–530, 2020. https://doi.org/10.1007/978-981-32-9941-2_43

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angle of deflection occurs under the friction of the unfolding wheel’s asymmetrical conical surface, thereby realizing the spiral unfolding of the steel ball [1].

Unfolding wheel Steel ball Support wheel

Drive wheel

Fig. 1. Contact state diagram of various parts of steel ball unfolding mechanism

However, the problem is that the slipping in the process of unfolding the steel ball is objective, it reduce the efﬁciency of steel ball detection. Moreover, the unfolding wheel is easy to fail due to wear. The unfolding wheel need to be replaced every 50000 times steel ball test [2], it increase the cost of steel ball detection. So it’s necessarily to conduct the analysis of the wear characteristics and life prediction for the unfolding wheel. In recent years, the microstructure surface technology combined with bionics provides a new research idea for improving the wear life of materials. German scholar Feldman [3] applies the microstructure surface to the cylinder surface and tested it on the AVL F528 engine, the results show that the wear of the cylinder and piston ring is reduced by 50%. Otero Nerea [4] applies the microstructure surface to tool surfaces, the surface-textured micro-processed sample was 10 times longer than the untreated sample in the life test. Since the 1970s, with the introduction of ﬁnite element theory and the rapid development of computer science, numerical simulation technology has played an increasingly important role in the study of friction and wear. Sun [5] used ANSYS software to model and simulate the non-smooth surface brake disc system, and obtained the wear amount and friction coefﬁcient through friction and wear test. Contrast test and numerical simulation results, it veriﬁed the credibility of numerical simulation results. Bortoleto [6] used ABAQUS software to numerically simulate the friction and wear test of the pin-plate under unlubricated conditions, Obtained stress ﬁeld distribution and surface topography change of wear trajectory. Compared with the wear test, the results were verify the consistency of the numerical simulation results under slight wear. Therefore, the author uses the wear numerical simulation method to study unfolding wheel microstructure drive surface increase friction and reduce wear characteristics, and microstructure surface unfolding wheel conduct the wear life prediction.

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2 Analysis of Unfolding Wheel Drive Surface Wear Type and Establishment of Discrete Archard Adhesive Wear Model 2.1

Analysis of Unfolding Wheel Drive Surface Wear Type

According to the mechanism and characteristics of friction surface damage, wear can be divided into abrasive wear, adhesive wear, fatigue wear and corrosion wear. In general, Wear failure of mechanical equipment components is the result of a combination of multiple wear type, but not all wear type dominate the wear process. For speciﬁc research objects, the type of wear that has less impact on the entire wear process can be ignored in order to study the wear mechanism more deeply and establish the wear model as well as predict the wear life. As shown in Table 1, the friction surface wear proﬁle characteristics of different wear types are different. Observing the deployed wheel with an ultra depth microscope, the wear proﬁle of the unfolding wheel drive surface after magnifying 200 times is shown in Fig. 2. Table 1. Friction surface wear proﬁle characteristics of different wear types Wear type Abrasive wear Adhesive wear Fatigue wear Corrosion wear

Wear proﬁle characteristics Scratches, grooves Pits, scratches, grooves Crack, peeling, Pits Reactant

a) Pits, scratches, grooves

b) Pits, scratches

Fig. 2. Wear proﬁle of the unfolding wheel drive surface after magnifying 200 times

From the analysis of the unfolding wheel drive surface wear proﬁle, the unfolding wheel drive surface has clear pits distribution and the scratches are also very obvious, there are many small pitting points. It can be roughly judged that the main type of wear is adhesive wear.

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From the analysis of the unfolding wheel drive surface wear mechanism, there is no chemical or electrochemical reaction occur during the process of unfolding the steel ball, and there are also no obvious alternating load, so corrosion wear and fatigue wear can be clearly excluded. For abrasive wear, the surface of the unfolding wheel and the steel ball has small roughness, and it is not easy to produce two-body wear. There is no abrasive grains between the unfolding wheel and the steel ball at the initial stage of wear, so there is no produce three-body wear. Although the drive surface material may fall off and formed the abrasive grains as the wear progresses, however, three-body wear is not the primary type of wear during the initial wear of the wheel drive surface. Adhesive wear is caused by friction surfaces in contact with each other due to shearing and metal adhesion. The reason why the steel ball can be spiral unfolded is the steel ball generating a certain angle of deflection while rotates, and the deflection of the steel ball is caused by the asymmetric conical surface on both sides drive the steel ball by friction. Therefore, the wear of the unfolding wheel drive surface is in line with the wear mechanism of the adhesive wear. In summary, from the two aspects of the unfolding wheel drive surface proﬁle characteristics and the wear mechanism, the main wear type of the unfolding wheel drive surface is adhesive wear. 2.2

Discrete Archard Adhesive Wear Model Based on Finite Element

The general formula for the classic Archard adhesive wear model is expressed as [7]: WV ¼ Km

PL H

ð1Þ

Among them, WV is the wear volume, P is the normal load, L is the slip distance, H is the Brinell hardness, and Km is the wear coefﬁcient. The differential of Eq. (1) can be expressed as: dWV ¼ Km

dP dL H

ð2Þ

The differential of the wear volume WV, the normal load P, and the slip distance L can be expressed as: 8 < dWV ¼ dh dA dP ¼ r dA : dL ¼ v dt

ð3Þ

Among them, h is the wear depth, A is the wear area, r is the contact stress of the friction pair contact point, v is the relative sliding speed between the two objects, and t is the wear time.

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The expression for the wear depth per unit time obtained by Eqs. (2) and (3) is: dh ¼ Km

vr dt H

ð4Þ

Select a contact node j on the unfolding wheel surface, assuming that the wear depth of the node in a very short period of time Dti is Dhi,j, where i represents the number of wear and j represents the node number, then: Dhi;j ¼ Km

vr Dti H

ð5Þ

For this junction, the wear depth of n wears is: hn;j ¼

n X i¼1

Dhi;j ¼

n X i¼1

km

vr Dti H

ð6Þ

Equation (6) is the wear depth of the j node after n wears in the whole time history. It is a discretized Archard adhesion wear model based on ﬁnite element numerical simulation. It is also the basis for the subsequent wear life prediction model.

3 Wear Numerical Simulation of the Micro-structure Unfolding Wheel First select the unfolding wheel drive surface microstructure type is pit type, the shape is diamond shape [8], the depth is 0.05 mm. The geometric parameters are shown in Fig. 3. Then simplify the steel ball wheel type unfolding mechanism and build the numerical simulation model of the steel ball-unfolding wheel.

Fig. 3. Unfolding wheel diamond surface microstructure geometric parameter map

The most effective way to reduce the numerical simulation calculation time and improve the numerical simulation accuracy is to reduce the number of meshes and improve the mesh quality. Before the meshing, the suppression function of the ANSYS

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Workbench drawing tool Design Modeler was used to suppress a part of the model, this part did not participate in the numerical simulation calculation. The model after suppression and meshing is shown in Fig. 4.

Fig. 4. Steel ball-unfolding wheel ﬁnite element mesh model

The boundary conditions are set using the relevant experimental data of the research group [2, 9]. The wear ﬁnite element numerical simulation belongs to the contact nonlinear analysis, and the calculation process is complicated and the result is not easy to converge. Therefore, it is unrealistic to simulate the entire life cycle of the unfolding wheel from the beginning of work to the wear failure by numerical simulation. Therefore, a short period of time is taken to repeatedly wear the same position of the unfolding wheel drive surface. The micro-segment time of the interception process is 0.02 s, and the wear process of the whole micro-segment is divided into four load steps. In order to ensure the ﬁnal convergence of the iterative calculation, the corresponding number of substeps can be set in each load step. In theory, the more substeps, the more iterative operations, the easier the operation to converge; However, the more substeps, the larger the amount of computation, and the longer the numerical simulation consumes. After repeated attempts, set the maximum number of substeps of the four load steps to 100. The minimum number of substeps is set to 10, The initial substeps are set to 10. Turn on the Auto time stepping option. The unfolding wheel drive surface experienced one wear during the 0.02 s period, and repeated 9-time wear numerical simulation at the same position of the unfolding wheel drive surface. Use ANSYS APDL to read the cumulative wear depth after each wear, the wear depth of the smooth surface unfolding wheel drive surface is shown in Fig. 5. The wear depth of the microstructure surface unfolding wheel drive surface is shown in Fig. 6, and the wear depth unit is mm.

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Fig. 5. Smooth surface unfolding wheel drive surface wear depth map

The maximum wear depth accumulated after 1 to 9 times wear numerical simulations was read from Figs. 5 and 6. The relationship between wear depth and the number of wear of the unfolding wheel is shown in Table 2. Fit the data in the Table 2 with software MATLAB, Obtained the relationship equation of smooth surface unfolding wheel wear depth h1 and number of wear n1 is: h1 ¼ 2:8212 107 n1 0:21345 107

ð7Þ

The relationship equation of microstructure surface unfolding wheel wear depth h2 and number of wear n2 is: h2 ¼ 2:5492 107 n2 0:22618 107

ð8Þ

It can be seen from the Eqs. (7) and (8) that the relationship between the wear depth and number of wear is approximately linear. The relationship equation of the unfolding wheel wear depth and number of wear obtained by the wear numerical simulation and provides a mathematical model basis for the establishment of the subsequent unfolding wheel wear life model.

Wear Numerical Simulation and Life Prediction

Fig. 6. Microstructure surface unfolding wheel drive surface wear depth map Table 2. Cumulative wear depth and number of wear of the unfolding wheel mm Number of Smooth surface wear n unfolding wheel Wear depth 1 2:93 107 2 5:56 107 3 8:23 107 4 1:08 106 5 1:38 106 6 1:59 106 7 1:86 106 8 2:31 106 9 2:59 106

Microstructure surface unfolding wheel Wear depth 2:72 107 5:21 107 7:09 107 9:33 107 1:21 106 1:48 106 1:72 106 2:07 106 2:32 106

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4 Establishment of Critical Failure Condition and Wear Life Model of Unfolding Wheel 4.1

Establishment of Unfolding Wheel Critical Failure Condition

As shown in Fig. 7, The deviation of the asymmetrical conical surface on both sides of the wheel is e, usually e take 1°. If unfolding wheel asymmetrical cone surface create wear, it will cause the deviation e decrease. When the e decrease to 0°, unfolding wheel asymmetric conical surface will lose the ability to provide deflection friction to steel ball. The steel ball will not be able to unfold along the spiral, and the wear depth at this time is the maximum wear depth hmax. As shown in Fig. 7, using the unfolding wheel axis as the x-axis, The perpendicular goes through the ball’s center and perpendicular to the x-axis as the y-axis. O is the origin of the coordinates. Build the Cartesian coordinate system. Q1 point is the intersection of the conical axis of the unfolding wheel and the horizontal rotary shaft. A1 is the contact point between the contour of the unfolding wheel and the steel ball at a certain moment. A2 is the intersection of the vertical line of the A1 point and the contour line below the unfolding wheel. M2 is the vertex of the conical surface on the side of the unfolding wheel. Pass M2 build the vertical line of A1A2, and the foot is M1, M1M2 is parallel to the horizontal rotation axis of the unfolding wheel. O1 is steel ball center, M2A1 is tangent to the steel ball, O1A1 is perpendicular to M2A1. According to the geometric principle knowledge, ∠Q1M2M is also e. Known by the unfolding wheel structure, ∠A1M2Q1 is 45°, ∠A1M2A2 is 90°

Fig. 7. Unfolding wheel wear failure geometry diagram

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Based on the above known geometric conditions, the lengths of line A1N1 and M1M2 can be deduced as:

lA1 N1 ¼ lM1 M2 ¼ r sinð45 þ eÞ

ð9Þ

According to the triangular function relation, obtained:

lM1 A2 ¼ lM1 M2 tanð45 eÞ

¼ r sinð45 þ eÞ tanð45 eÞ lA1 M2 lM 1 M 2 ¼ sinð45 þ eÞ sinð45 þ eÞ sinð45 eÞ r sinð45 þ eÞ r ¼ ¼ sinð45 þ eÞ sinð45 eÞ sinð45 eÞ

ð10Þ

lA1 A2 ¼

ð11Þ

According to the unfolding principle of the unfolding wheel, when e = 0°, the unfolding wheel can not provide asymmetric friction to make the steel ball unfold. The maximum wear depth is: hmax ¼ lA1 A2 2lM1 A2 r 2r sinð45 þ eÞ tanð45 eÞ ¼ sinð45 eÞ

ð12Þ

Equation (12) is the critical failure condition for the steel ball to detect the unfolding wheel wear. When the wear depth reaches or exceeds hmax, the unfolding wheel fails. 4.2

Unfolding Wheel Life Prediction Model

For a node j of the unfolding wheel drive surface, the total time that the steel ball rubs against it is Tj, then: Tj ¼

n X

Dti

ð13Þ

i¼1

According to Eqs. (6) and (13), the relationship between the maximum wear depth and wear time of a node can be obtained: hmax ¼ Km

vr Tmax H

ð14Þ

Where Tmax represents the wear time corresponding to the maximum wear depth, which is wear life.

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Convert Eq. (14) into the relationship between wear time and wear depth: Tmax ¼

H hmax vrKm

ð15Þ

In a very small period of time, the steel ball wear number to a node of the unfolding wheel drive surface is recorded as one time. For a node of the unfolding wheel drive surface, using number of wear as a measure of wear life is more convenient in practical calculations, so the relationship between the number of wear and the wear depth can be expressed as: Nmax ¼

H hmax þ B vrKm

ð16Þ

Where Nmax represents the maximum number of wear corresponding to the maximum wear depth, which is wear life; B is the linear compensation value. H Set Ks ¼ vrK , then the wear life of unfolding wheel can be expressed as: m Nmax ¼ Ks hmax þ B

ð17Þ

5 Example of Unfolding Wheel Life Prediction Take the bearing steel ball with a diameter of 16.6688 mm and the unfolding wheel matched with it as an example. The radius of the steel ball is r = 8.3344 mm, Take the unfolding angle of the unfolding cone as e = 1, take into Eq. (12), obtained the maximum wear depth of unfolding wheel failure hmax = 0.41872 mm. 5.1

Example of Smooth Surface Unfolding Wheel Life Prediction

For the smooth surface unfolding wheel, according to the Eq. (7), the relationship between the maximum wear depth and the maximum number of wear can be expressed as: hmax ¼ 2:8212 107 Nmax1 0:21345 107

ð18Þ

According to Eq. (17), the Eq. (18) is converted to the wear life prediction model as: Nmax1 ¼ 3:5446 106 hmax þ 0:07566

ð19Þ

Take hmax = 0.41872 mm into the Eq. (19), take the integer value, obtained Nmaxl = 1484195. Usually, every time a steel ball is tested, the steel ball needs to be turned 30 times on the unfolding wheel [10]. For a node of the unfolding wheel, it is

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rubbing 30 times. Therefore, the unfolding wheel wear life can be expressed by the total number of steel balls tested: 0 Nmax1 ¼

Nmax1 30

ð20Þ

0 Take the integer value, Nmax1 ¼ 49473, this means the unfolding wheel detection 54752 steel ball wills cause failure. This result is consistent with the fact that every 50,000 steel balls test will cause unfolding wheel fail due to wear and need to be replaced. It prove the effectiveness of the unfolding wheel life prediction model.

5.2

Example of Microstructure Surface Unfolding Wheel Life Prediction

For the microstructure surface unfolding wheel, according to the Eq. (8), the relationship between the maximum wear depth and the maximum number of wear can expressed as: hmax ¼ 2:5492 107 Nmax2 0:22618 107

ð21Þ

According to Eq. (17), the Eq. (21) is converted to the wear life prediction model as: Nmax2 ¼ 3:9228 106 hmax þ 0:08873

ð22Þ

Taking hmax = 0.41872 mm into the Eq. (17), we obtain Nmax2 = 1642555. Turn into the number of steel balls. 0 Nmax2 ¼

Nmax2 30

ð23Þ

Take the integer value, N′max2 = 54752, this means the unfolding wheel detect 54752 steel balls will cause failure. Comparing the wear life of the microstructure surface unfolding wheel and smooth surface unfolding wheel, the improvement rate of the wear life is calculated as: gm ¼

0 0 Nmax2 Nmax1 100% 0 Nmax1

ð24Þ

Substituting data for calculation, gm ¼ 10:67%, that means the wear life of microstructure surface unfolding wheel is about 10.68% higher than that of the smooth surface unfolding wheel.

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6 Conclusion (1) By observing the wear proﬁle and analysis wear mechanism of the unfolding wheel drive surface, the main wear type of the unfolding wheel drive surface is adhesive wear. Based on the classical Archard adhesion wear model and ﬁnite element theory, the discretized Archard adhesion wear model is established. (2) Through the numerical simulation of wear on the smooth surface unfolding wheel and the microstructure surface unfolding wheel, the cumulative wear amount of the single node is obtained, and the relationship equation between the number of wear and the wear depth is established by data ﬁtting. (3) According to the unfolding wheel structure and the unfolding principle, the critical failure conditions of the unfolding wheel are established, and based on the above theory, the wear life prediction model of the unfolding wheel is established. Finally, the effectiveness of the unfolding wheel wear life prediction model is veriﬁed by an example. It is found that the wear life of the microstructure surface unfolding wheel is about 10% higher than that of the smooth surface unfolding wheel.

References 1. Zhao, Y.L., Zhao, Z.Q., Bao, Y.D.: Principle and method of full surface development of steel balls. Chin. J. Mech. Eng. 52(17), 205–212 (2016) 2. Sun, M.M.: Analysis of Friction and Wear Characteristics of Steel Ball Unfolding Wheel Based on Finite Element Method, pp. 1+29–40. Harbin University of Science and Technology (2017) 3. Feldman, Y., Kligerman, Y., Etsion, I.: A hydrostatic laser surface textured gas seal. Tribol. Lett. 22(1), 21–28 (2006) 4. Otero, N., Romero, P., Gonzalez, A.: Surface texturing with laser micro-cladding to improve tribological properties. J. Laser Micro Nenoengineering 7(2), 152–154 (2012) 5. Sun, S.N., Xie, L.Y., Zhang, Y.Z.: Finite element analysis of friction and wear performance of non-smooth surface brake disc. J. Northeastern Univ. (Nat. Sci. Edn.) 35(11), 1597–1601 (2014) 6. Bortoleto, E.M., Rovani, A.C., Seriacopi, V., et al.: Experimental and numerical analysis of dry contact in the pin on disc test. Wear 301(1–2), 19–26 (2013) 7. Archard, J.F., Hirst, W.: An examination of a mild wear process. Proc. Roy. Soc.: Math. Phys. Eng. 238(1215), 515–528 (1957) 8. Zhao, Y.L., Geng, W., Bao, Y.D.: Analysis of friction and wear properties of microstructure on the unfolding wheel used for steel ball inspection. J. Tribol. 37(3), 348–356 (2017) 9. Kong, Y.L.: Study on Micro-structure Dry Friction Characteristics of Steel Ball Detecting Surface, pp. 22–57. Harbin University of Science and Technology (2016) 10. Xuan, J.P.: Research on Key Parts Structure of Steel Ball Full Surface Unfolding Device, pp. 23–34. Harbin University of Science and Technology (2014)

Design of Passive Gravity Balance Mechanism for Wearable Exoskeleton Suit Chenxi Qu1, Peng Yin2(&), Xiaohua Zhao3, and Liang Yang3 1

College of Automotive Engineering, Jilin University, Changchun 130025, China [email protected] 2 School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China [email protected] 3 Intelligent Technology R&D Center, Guangzhou Hyetone Mechanical and Electrical Equipment Co., Ltd., Guangzhou 510640, Guangdong, China {zhaoxiaohua,yangliang}@hyetone.com

Abstract. The wearable exoskeleton suit is a man-machine system which can combine the human limbs with machinery and environment. The key point of wearable exoskeleton suit is the design of passive gravity balance mechanism to achieve passive gravity balance in the system. Based on the principle of passive gravity balance technology, this paper deduces the theoretical equation of passive gravity balance. Then, the equations were applied to the passive gravity balance for single parallelogram mechanism and double parallelogram mechanism. Simultaneously, the analysis was also conducted using MATLAB optimization tool. The results show that the parallelogram passive gravity balance mechanism can achieve any position to hover within a certain range of spring tension. It can make the operating force at low level when doing the up and down movement. Consequently, with the help of this system, the vibration damage in some work conditions to human musculoskeletal can be signiﬁcantly reduced, it can be introduced to many work conditions, such as pneumatic impact drilling. Keywords: Exoskeleton Mechanical structure design Optimization design Passive gravity balance

1 Introduction In recent years, exoskeleton technology has become a new research hotspot in robotics, automatic control technology, artiﬁcial intelligence and other disciplines, and has been widely used in scientiﬁc research, industrial production, medical rehabilitation [1]. The wearable exoskeleton suit is an industrial exoskeleton mechanism that transmits the load of the tool held on the wearer’s hand to the ground under the premise of ensuring This project is supported by the 2017 Guangzhou Industrial Technology Major Research Projects (No 201802010067). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 531–545, 2020. https://doi.org/10.1007/978-981-32-9941-2_44

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the flexibility of the wearer, saving the wearer’s physical strength and preventing musculoskeletal diseases. The basic principle is the principle of leverage, that is, the hand tool is connected to the hip joint of the wearable exoskeleton suit through the parallelogram passive gravity balance mechanism, with the hip joint as the fulcrum, and the two mechanical legs of the wearable exoskeleton suit serve as the gravity transmission rod, passing through the lower limb The way the rear end of the exoskeleton increases the weight, so that the front end hand tool balances the moment between the hip joint and the rear end weight on the hip joint, so that the wearer can almost bear zero load. The design of passive gravity balance mechanism is an important part of exoskeleton design. The so-called passive gravity balance aimed at the system designed to be able to maintain the balance state within the range of activities without external force interference [2]. The basic principle is as follows: the center of mass of the whole system is always at a constant position at ﬁrst, then, the total mechanical energy of the system is always be a constant [3, 4]. The design of the passive gravity balance mechanism can take the following two measures: one is to ensure the relative position of the center of mass of the system under the motion condition is constant, and the other is that the total value of the mechanical energy at any position of the system does not change, and the gravity balance design of the system can be completed. In the realization of the gravity balance of the system, there are generally the following practices: First, the center of mass of the system is kept at a ﬁxed position; second, the spring is used reasonably, keeping the sum of the elastic potential energy and the gravitational potential energy of the system unchanged; the third is performing centroid positioning When using a mechanical structure of the parallelogram type. In the design process of the gravity balance system, this paper mainly adopts the second method, using the spring with the appropriate elastic coefﬁcient, so that the mechanical energy of the whole system is always constant. The passive balancing mechanism is used to connect the wearable exoskeleton suit with the hand tool while achieving the following functions: (1) Ensuring that the hand tool can move flexibly in the horizontal space; (2) Offsetting the heavy torque of the hand tool and enable the hand tool to stay at the comfortable working surface height of the human body; (3) Can be applied to handheld tools of different weights; (4) Applying a small force to the human hand, the hand tool can be moved up and down in a vertical direction within a certain height range, and can be hovered at any height position; (5) Reducing hand-held tools with continuous impact, such as pneumatic impact drills, to damage the human muscles and bones. In order to achieve the above functions, this paper uses a spring with a proper elastic coefﬁcient combined with a parallelogram structure to achieve the passive gravity balance of the design mechanism. One end is connected to the hip joint of the exoskeleton suit, and one end is connected with a hand tool, which is mainly used to balance the hand tool. The weight, while having a shock absorbing effect, both reduces the operator’s muscle fatigue and reduces the vibration damage to the operator’s joint muscles.

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2 Passive Gravity Equilibrium Theory System The static and dynamic stability of the equipment largely depends on whether the structure of the system can reach a balance of gravity [5]. At present, there are many studies on the passive gravity balance in the ﬁeld of industrial application, and the theory is relatively mature. Its basic means focus on the mutual offset of torques at mechanical joints [6], which greatly improves the stability of equipment. The use of a parallelogram to achieve gravity balance is based on the special properties of the parallelogram [7], the prototype of which has been around since the last century, SunilK and Agrawal reported the property of this type of parallel unit mechanism in the relevant published papers. The application of this parallel unit in low speed and low precision of the simple situations, such as lamp ﬁxtures [8]. After entering the 90s generation, the characteristics of this type of mechanism has caused a certain degree of attention, the design can guarantee the structure mechanical node torque offset between each other, making equipment to enhance the stability, and promoting the industrial applicability of this design [9]. The model and principle of passive balance technology are shown in Fig. 1.

Fig. 1. Schematic diagram of passive equilibrium

In order to ensure the structure designed to achieve static equilibrium (Friction was neglected here), We can know from the Eq. (1) that the system has zero kinetic energy at condition of rest when this system only consider the gravity factor but except the external force and the torque disturbance: s¼

d @T @T @V þ ¼0 dt @ q_ @q @q

ð1Þ

Due to the kinetic energy is unrelated with generalized coordinates in this system, then Eq. (1) can be simpliﬁed as: @V ¼0 @q

ð2Þ

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Hence, if the mechanical energy of the system is to be kept constant, V should be kept constant. As shown in Fig. 1, two values need to be calculated to determine the ﬁnal spring stiffness coefﬁcient and ensure the passive balance of gravity of the system: the elastic potential energy of the spring, and the gravitational potential energy of the bar. According to the model of bar (Fig. 1), one end of the bar is connected with a rotating hinge, and the other end is connected with a stretching spring. The horizontal plane where the rotating hinge in had selected be the zero potential energy surface [10]. Assuming that the system is static, the mechanical energy of the system is the sum of gravitational potential energy and elastic potential energy: E ¼ EPG þ EPF

ð3Þ

EPG ¼ Mglc sin h

ð4Þ

1 EPF ¼ kðx x0 Þ2 2

ð5Þ

And:

Among them: EPG ——System gravity potential; EPF ——Systemic potential energy; M——Mass of the rod; E——System mechanical energy; LC ——The distance from the center of mass of the rod to the hinge; x0 ——Initial length of the tension spring; k——Elastic coefﬁcient of tension spring; x——The length of the spring when the angle between the rod and the horizontal plane is h. Substituting Eqs. (4) and (5) into Eq. (3), hence: E ¼ Mglc sin h þ

1 k ð x x0 Þ 2 2

ð6Þ

Based on geometrical relationship: x2 ¼ d 2 þ l2 2dl sin h

ð7Þ

And: d——Distance between the mounting point on the left side of the spring and the rotating hinge; l——Length of the rod.

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Then the mechanical energy of the system can be rewritten as: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2dl sin h 2 2 2 2 2 E ¼ Mglc sin h þ k d þ l 2dl sin h þ x0 2x0 d þ l 1 2 2 2 d þl Hence, Eq. (9) is then established: 2dl sin h d 2 þ l2 1

ð8Þ

ð9Þ

The Taylor series was used to get simpliﬁcation in Eq. (8), the approximation is obtained as: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2dl sin h E ¼ Mglc sin h þ k d 2 þ l2 2dl sin h þ x20 2x0 d 2 þ l2 1 2 2 ð10Þ 2 d þl In order to make the system statically balanced, then: @E ¼0 @h

ð11Þ

At this time, the mechanical energy of the system is a constant, that is, does not change with the change of the rotation angle of the member, and when the Eq. (12) satisﬁes the Eq. (11): 2kdlx0 Mglc kdl þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0 d 2 þ l2

ð12Þ

Then, the spring coefﬁcient of the spring is: k¼

Mglc 2x0 ﬃ dl 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 d þl

ð13Þ

In this paper, the above steps have been veriﬁed the feasibility of the parallelogram passive gravity balance mechanism which used in terms of passive balance, and it always provides a theoretical calculation Equation. Using the theoretical calculation equations, structural parameters and spring elastic coefﬁcients can be determined by calculation results to achieve the passive balance of the parallelogram mechanism [11].

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3 Characteristic Analysis of Passive Gravity Balance Mechanism of Parallelogram Mechanism Figure 2 shows a spring parallelogram mechanism. The reference plane is X0 during analysis, and the initial length of the spring is X0. The upper and lower long rod are the same rods, and the total mechanical energy of the mechanism can be represented as Eq. (14): E ¼ m1 glc sin h þ m1 gðd þ lc sin hÞ þ m2 gðdc þ l sin hÞ þ

1 k ð x x0 Þ 2 2

1 ¼ ð2m1 lc þ m2 lÞg sin h þ m1 gd þ m2 gdc þ kðx x0 Þ2 2

ð14Þ

Among them: m1 ——The mass of long rod; m2 ——The mass of the short rod; x0 ——The initial length of the spring; dc ——The distance from the center of the short rod to the right lower end of the hinge; lc ——The distance from the center of the long rod to the left end of the hinge; l——The length of the long rod; d——The length of the short rod; h——The angle of rotation of the parallelogram; k——The spring constant of the tension spring; x——The length of the spring when the parallelogram rotation angle is h;

Fig. 2. Schematic diagram of a single-section spring parallelogram mechanism.

From the geometrically related: x2 ¼ l2 þ r 2 2lr sin h

ð15Þ

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In the equation, r——The distance from the left side of the spring mounting point to the left end of the hinge Then the mechanical energy of the system can be rewritten as: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 k l þ r2 2lr sin h þ x20 2x0 l2 þ r 2 2lr sin h 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2lr sin h ¼ ð2m1 lc þ m2 lÞg sin h þ m1 gd þ m2 gdc þ k l2 þ r 2 2lr sin h þ x20 2x0 l2 þ r2 1 2 2 l þ r2

E ¼ ð2m1 lc þ m2 lÞg sin h þ m1 gd þ m2 gdc þ

ð16Þ Hence, Eq. (17) is established: 2lr sin h d 2 þ l2 1

ð17Þ

Using the Taylor series reduction (16), the approximation has been obtained: E ¼ ð2m1 lc þ m2 lÞg sin h þ m1 gd þ m2 gdc pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2lr sin h þ k l2 þ r 2 2lr sin h þ x20 2x0 l2 þ r 2 1 2 2 l þ r2

ð18Þ

In order to make the system statically balanced, then: @E ¼0 @h

ð19Þ

At this time, the mechanical energy of the system is a constant, that is, it does not change with the change of the rotation angle of the member, when the Eq. (20) satisﬁes the Eq. (19): 2klrx0 ð2m1 lc þ m2 lÞg klr þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0 l2 þ r 2

ð20Þ

The elastic potential energy of the spring: ð2m1 lc þ m2 lÞg k¼ 2x0 lr 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 l þr

ð21Þ

It can be seen from Eq. (21) that k is a constant independent of h, the mechanical energy in this case is also unchanged, and the mechanism can perform the balance of any angle at 180°. If the structure is modiﬁed into two parallel four sides in series, and a weight of m3 is placed at the end of one of them, and the rotational degree of rotation in the vertical direction is increased, the structure can still achieve the balance at any angle, as shown in the Fig. 3.

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Fig. 3. Schematic diagram of two-section parallelogram passive gravity balance system.

Taking the X0 axis as the reference plane and performing static analysis on the mechanism of Fig. 3, the mechanical energy of the system is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2lr sin h1 k1 l2 þ r2 2lr sin h1 þ x20 2x0 l2 þ r 2 1 2 2 l þ r2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2lr sin h2 þ ð2m1 lc þ m2 lÞg sin h2 þ m1 gd þ m2 gdc þ k2 l2 þ r 2 2lr sin h2 þ x20 2x0 l2 þ r2 1 2 2 2 l þr

E ¼ð2m1 lc þ m2 lÞg sin h1 þ m1 gd þ m2 gdc þ

þ ð2m1 þ m2 Þgl sin h1 þ ðl sin h1 þ l sin h2 þ d H Þm3 g

ð22Þ Among them, m1 ——The mass of the long rod; m2 ——The mass of the short rod; m3 ——The mass of the weight suspended at the end of the parallelogram; k1 ——The elastic coefﬁcient of the tension spring of the ﬁrst parallel parallelogram mechanism; k2 ——The elastic coefﬁcient of the tension spring of the second parallel parallelogram mechanism; h1 ——The angle of rotation of the ﬁrst parallelogram mechanism; h2 ——The angle of rotation of the second parallelogram mechanism; dc ——The distance from the center of the short rod to the right lower end of the hinge; x0 ——The initial length of the spring; lc ——The distance from the center of the long rod to the left end of the hinge; H——The distance from the center of mass of the weight to the suspension point; x——The length of the spring when the parallelogram rotation angle is h; l——Length of the long rod; r——The distance from the left side of the spring mounting point to the left end of the hinge; d——The length of the short rod. Since the mechanism can maintain equilibrium at any position, it is known from the conservation of energy that the mechanical energy of the mechanism is constant, that is:

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@E ¼0 @h

ð23Þ

When the Eq. (22) satisﬁes the Eq. (23), it means: @E 2k1 lrx0 p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ¼ ð2m1 þ 2m2 þ m3 Þgl þ 2m1 lc g k1 lr þ cos h1 ¼ 0 @h1 l2 þ r 2 " # @E 2k2 lrx0 ¼ ð2m1 lc þ m2 l þ m3 lÞg k2 lr þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos h2 ¼ 0 @h2 l2 þ r 2

ð24Þ

ð25Þ

Constantly established, then: 2k1 lrx0 ð2m1 þ 2m2 þ m3 Þgl þ 2m1 lc g k1 lr þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0 l2 þ r 2

ð26Þ

2k2 lrx0 ð2m1 lc þ m2 l þ m3 lÞg k2 lr þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0 l2 þ r 2

ð27Þ

Then the spring coefﬁcient of the spring is: k1 ¼

ð2m1 þ 2m2 þ m3 Þgl þ 2m1 lc g 2x0 lr 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 l þr

ð28Þ

ð2m1 lc þ m2 l þ m3 lÞg 2x0 lr 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 l þr

ð29Þ

k2 ¼

In the manufacturing process of the spring, the spring’s elastic coefﬁcient has a property of uncertainty. Therefore, when designing the parallelogram passive balancing mechanism, the left side mounting position of the spring can be made adjustable, and its speciﬁc structure is shown in Fig. 4.

Fig. 4. Spring mounting position adjustment mechanism and spring installation position physical photo.

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4 Optimization Design of Parallelogram Passive Gravity Balance Mechanism When the suspension tool of the parallelogram passive gravity balance mechanism moves up and down in the vertical direction under the action of external force, the movement of the system is relatively slow, the change of the kinetic energy of the system is neglected, the elastic potential energy and the gravity potential energy of the system are transformed into each other. When the rate of change and the gravitational potential energy are inﬁnitely close to the rate of change of the angle, the work done by the external force will reach a minimum, that is, the operating force of the human hand reaches a minimum value [12]. 4.1

Optimization Design Model

Taking the single-parallel quadrilateral passive gravity balance mechanism as the optimization object, the mass of the end suspension tool of the mechanism is equivalent to the short rod of the mechanism. According to Eq. (16), the gravitational potential energy and elastic potential energy of the system at a certain position are: EPG ¼ ð2m1 lc þ m2 lÞg sin h þ m1 gd þ m2 gdc

ð30Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 EPF ¼ k l2 þ r 2 2lr sin h þ x20 2x0 l2 þ r2 2lr sin h 2

ð31Þ

Then the change rate of the system’s gravitational potential energy and elastic potential energy with angle h is: dEPG ¼ ð2m1 lc þ m2 lÞg cos h dh

ð32Þ

dEPF kx0 lr cos h ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ klr cos h dh l2 þ r 2 2lr sin h

ð33Þ

According to the design theory of the parallelogram passive gravity balance mechanism, combined with the reasonable range of the human working space, the parallelogram mechanism is designed, as shown in Fig. 5. The determined parameter values are as follows: the length l of the long rod is 200 mm, and the length d of the short rod is 70 mm, the distance lc of the long rod centroid to the left end rotating hinge is 100 mm, the distance dc of the short rod centroid to the right lower end rotating hinge is 35 mm, and the distance r of the left mounting point of the spring from the lower left end rotating hinge is 0–70 mm.

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(a)

541

(b)

Fig. 5. (a) Three-dimensional model of the parallelogram mechanism; (b) Physical photo of the parallelogram mechanism.

The 6061 aluminum alloy in the T4 heat treatment status was selected as the body material for the parallelogram mechanism. According to the material density and the volume of the component design, the long rod mass m1 is calculated as 0.1 kg, the short rod mass is about 0.1 kg, and the deﬁned tool mass is 8 kg. Equivalent to the short rod of the parallelogram mechanism, m2 is 8.1 kg. Substituting the parameter values into Eqs. 32 and 33 that: dEPG ¼ 16 cos h dh

ð34Þ

dEPF kx0 r cos h 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kr cos h 2 dh 1 þ 25r 10r sin h 5

ð35Þ

Among them, h——The angle of rotation of the parallelogram mechanism, variable; x0 ——The original length of the spring, design variable; k——Spring modulus of spring, design variable; r——The distance between the left side of the spring mounting point and the left end of the rotating hinge, design variable. According to the characteristics of the spring and the size of the parallelogram mechanism and the range of the movable space, the following constraints have been designed: G1 ¼ k 103

ð36Þ

G2 ¼ k 104

ð37Þ

G3 ¼ r [ 0

ð38Þ

G4 ¼ r\0:07

ð39Þ

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G5 ¼ x0 [ 0:05

ð40Þ

G6 ¼ x0 \0:2

ð41Þ

G7 ¼ h 45

ð42Þ

G8 ¼ h 45

ð43Þ

The difference between the rate of change of the system gravitational potential energy with the angle h and the rate of change of the elastic potential energy with the angle h is the objective function: Mingð xÞ ¼ max½absðDMi Þ

ðð44ÞÞ

Among them, DMi ——The value of the difference between the rate of change of the system’s gravitational potential energy with the angle h and the rate of change of the elastic potential energy with the angle h at different positions, i = 1, 2, 3 …n; gðxÞMMi ði ¼ 1; 2; 3. . .nÞ is the target function. The result of the optimized design is to minimize g (x), which means that the work done by the external force will reach a minimum, that is, the operating force of the human hand reaches a minimum value [13]. 4.2

Optimization Design Calculation and Result Analysis

Run the optimization main program in MATLAB just like Fig. 6. The optimization results are as follows:

Fig. 6. MATLAB optimization main program diagram.

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Spring modulus: k = 5508.241047; The distance between the left mounting point of the spring and the left end turning hinge: r = 0.066899; Original length of the spring: x0 = 0.197499; Rotation angle of the parallelogram mechanism: h = 42.690315; Objective function optimization value: minfgðhÞg = 0.000001065387323. It can be seen from the optimization results that when the appropriate tension spring is selected, the objective function optimization value is close to 0, indicating that the parallelogram passive gravity balance mechanism can well realize the functions of hovering at any position and moving up and down.

5 Passive Gravity Balance Mechanism Design In the design of the speciﬁc scheme, the spring with the appropriate elastic coefﬁcient is combined with the parallelogram structure, one end is connected to the hip joint of the wearable exoskeleton suit [14], and one end is connected with the hand tool, which is mainly used for balancing the weight of the hand tool and has the shock absorption effect [15]. It not only reduces the operator’s muscle fatigue, but also reduces the vibration damage to the operator’s joint muscles. The assembling drawing is shown in Fig. 7. The physical photo of the parallelogram passive gravity balancing mechanism is presented in is presented in Fig. 8. It would be a beneﬁcial tool for workers in many work conditions.

Fig. 7. Parallelogram passive gravity balancing mechanism.

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Fig. 8. Physical photo of the parallelogram passive gravity balancing mechanism.

6 Conclusions In this paper, the theoretical equations of passive gravity balance system is derived. Then, the mechanical energy of the whole system is always constant to achieve passive gravity balance by using the spring with the appropriate elastic coefﬁcient. The passive balance mechanism is used to connect the wearable exoskeleton suit with the hand tool. Consequently some functions have been realized as following: (1) The handheld tool can move flexibly in a horizontal space; (2) The heavy torque in the hand tool has been counteracted, so that the hand tool can stay at the comfortable height of the human body from the working surface; (3) The handheld tools can be applied to of different weights; (4) Applying a small force from the human hand, the hand tool can be moved up and down in a vertical direction within a certain height range, and can be hovered at any height position; (5) The vibration damage in some work conditions to human musculoskeletal can be signiﬁcantly reduced, it can be introduced to many work conditions, such as pneumatic impact drilling.

References 1. Rosen, J., Brand, M., Fuchs, M.B., et al.: A myosignal-based powered exoskeleton system. IEEE Trans. Syst. Man Cybern.-Part A Syst. Humans 31(3), 210–222 (2001) 2. Gang, H., Ditzinger, T., Ning, C.Z., et al.: Stochastic resonance without external periodic force. Phys. Rev. Lett. 71(6), 807 (1993) 3. Nathan, R.H.: A constant force generation mechanism. J. Mech. Transmissions Autom. Des. 107(4), 508–512 (1985)

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4. Sunil, K.A., Abbas, F.: Gravity-balancing of spatial robotic manipulators. Mech. Mach. Theor. 39, 1331–1344 (2004) 5. Mahrt, L.: Momentum balance of gravity flows. J. Atmos. Sci. 39(12), 2701–2711 (1982) 6. Wang, G.S.: Analysing the onset of multiple site damage at mechanical joints. Int. J. Fract. 105(3), 209–241 (2000) 7. Banala, S.K., Agrawal, S.K., Fattah, A., et al.: Gravity-balancing leg orthosis and its performance evaluation. IEEE Trans. Robot. 22(6), 1228–1239 (2006) 8. Latassa, F.M., Ray, J.G.: Compact fluorescent lamp assembly: U.S. Patent 4,347,460, 31 August 1982 9. Sunil, K.A., Abbas, F.: Theory and design of an orthotic device for full or partial gravitybalancing of a human leg during motion. IEEE Trans. Neural Syst. Rehabil. Eng. 12(2), 157–165 (2004) 10. Bernardi, F., Olivucci, M., Robb, M.A.: Potential energy surface crossings in organic photochemistry. Chem. Soc. Rev. 25(5), 321–328 (1996) 11. Morita, T., Kuribara, F., Shiozawa, Y., et al.: A novel mechanism design for gravity compensation in three dimensional space. In: Proceedings 2003 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM 2003), vol. 1, pp. 163–168. IEEE (2003) 12. Kruskal, J.B.: Nonmetric multidimensional scaling: a numerical method. Psychometrika 29 (2), 115–129 (1964) 13. Esch, F., Baird, A., Ling, N., et al.: Primary structure of bovine pituitary basic ﬁbroblast growth factor (FGF) and comparison with the amino-terminal sequence of bovine brain acidic FGF. Proc. National Acad. Sci. U.S.A. 82(19), 6507 (1985) 14. Low, K.H.: Robot-assisted gait rehabilitation: from exoskeletons to gait systems. In: 2011 Defense Science Research Conference and Expo (DSR), pp. 1–10. IEEE (2011) 15. Chiu, H.T., Shiang, T.Y.: Effects of insoles and additional shock absorption foam on the cushioning properties of sport shoes. J. Appl. Biomech. 23(2), 119–127 (2007)

Design and Analysis of Underwater Drag Reduction Property of Biomimetic Surface with Micro-nano Composite Structure Xuezhuang Ren1, Lijun Yang1, Chen Li1(&), Guanghua Cheng2,3, and Nan Liu1

2

1 School of Mechanical and Electrical Engineering, Shaanxi University of Science and Technology, Xi’an 710021, China [email protected], {yanglijun,lichen}@sust.edu.cn, [email protected], [email protected] Xi’an Institute of Optics and Precision Mechanics of the Chinese Academy of Sciences, Xi’an 710021, China [email protected] 3 School of Electronics, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

Abstract. Underwater drag reduction has an important impact on the speed and energy consumption of underwater vehicles. This is directly related to whether it can improve the operating efﬁciency of the underwater vehicle and whether it plays a role in energy-conservation and emission-reduction. A new type of bionic surface with micro-nano composite structure, which is designed to achieve drag reduction of the underwater vehicle is presented. And it is applied to a ﬁsh dart. The design source of the structure comes from dolphins’ ridge skin and mosquitoes’ mouthparts. The design of the structure is based on the method of bionics. A fluid mechanics method is taken to simulate the drag reduction effect of the micro-nano composite structure. According to the results of simulation optimization, the drag reduction mechanism of the composite structure is analyzed. From the analysis, the optimal structural parameters can be obtained. The simulation results show that the underwater drag reduction rate can reach 89.49% in the optimal structural parameters. Keywords: Bionic

Micro-nano composite structure Drag reduction

1 Introduction The rapid development of the marine industry relies on advanced marine engineering equipment and materials. The running speed and energy consumption of ships, vessels, torpedoes, and other marine vehicles are important indicators to evaluate their This project is supported by (1) Special Research Project of Shaanxi Provincial Department of Education (18JK0101) (2) Open Foundation of Chinese key laboratory of transient optics and photonic technology (SKLST201708) (3) National Natural Science Foundation of China (61705124) (4) Doctoral Scientiﬁc Research Foundation of Shaanxi University of Science and Technology (2016BJ-78). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 546–559, 2020. https://doi.org/10.1007/978-981-32-9941-2_45

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performance, and frictional resistance has an important impact on the speed and energy consumption of the vehicle. Through research on underwater vehicles, it can be found that the ratio of the frictional resistance of the surface of the underwater vehicle to the total resistance can reach 70% [1]. If the frictional resistance is reduced by 50%, the speed can be increased by 26%. Therefore, the design and mechanism of structure about underwater drag reduction have been given attention for a long time. The structural design of the flow drag reduction and the mechanism of flow drag reduction have been studied in the past sixty years. Early in the 1960s, Kramer [2] pioneered the study of dolphins skin and made bionic dolphin skin which achieved a 60% drag reduction effect when the bionic dolphin skin was put on the surface of a model. Huang [3] of the Institute of Marine Chemical Industry imitated the size of the dermal ridge of the dolphin skin and carried on numerical simulations to prove that the dermal ridge of the dolphins act as a traveling wave when they were cruising. Germany’s Scholle [4, 5] theoretically analyzed the drag reduction of different shapes and sizes with traveling waves, and he discussed the theoretical analysis of applying ridge structures which are perpendicular to the direction of flow (transverse grooves) to lowspeed creep flow for drag reduction. Ridgway [6] tested the sensitivity of dolphins’ skin, suggesting that the dolphins’ skin is sensitive enough to fell the surface vibrations and changes of pressure, which can cause small fluctuations in muscle contraction and reduce surface pressure. Grüneberger [7] investigated the drag reduction performance of longitudinal grooves and transverse grooves with small spacing. The resistance of longitudinal grooves increased by about 20% relative to that of transverse grooves. At present, the Northwest Institute of Technology, Tsinghua University, and other university are carrying on the research work. The study shows that the corrugated surface of the accompanying wave will achieve drag reduction under a certain Reynolds number. By systematic analysis, Yang [8] obtained that the wavy surface of the traveling wave has a better drag reduction effect at a certain speed. Zhang [9] compared the drag reduction effects of transverse grooves with longitudinal grooves detailedly and obtained that the two grooves have different drag reduction effects in different parts. However, most of the previous research work focused on the periodic arrangement of single grooved drag reduction elements. Cheng [10] studied the drag reduction of single-scale groove, designed the second-order grooved surface. And the calculation results show that the drag reduction effect is good. But this is also limited to the periodic arrangement of the periodic drag reduction elements. There are few reports on the application of bionic micro-nano composite structures to underwater drag reduction. As an underwater high-speed navigation instrument, the underwater high-speed ﬁsh dart has the characteristics of simple structure which is convenient to be researched. So a bionic drag reduction micro-nano structure is prepared on the surface of the underwater high-speed ﬁsh dart.

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2 Structural Design In the process of long-term evolution and natural selection, natural organisms have formed various functional features which provide rich structural resources for human creation and design. As a swimming expert among ﬁsh, dolphins swim freely in the water at speeds of 40–48 km per hour. Most of the body of the toothed cetacean, such as the dolphin, is coated by corrugated regular gullies which are called skin ridge [11], as shown in Fig. 1. It shows a sketch map of the microscopic structure of the skin ridge. After observing the structure of the skin ridge, Babenko [12] considered that the skin ridge on the epidermis of the living dolphins is relatively small, and skin ridge is more similar to the p and U shape than the sinusoidal shape.

Fig. 1. Skin ridge of dolphin and sketch map of the skin ridge

The mouthparts of mosquitoes are stinging and sucking, and Fig. 2 shows a sketch map of the structure of the mosquitoes’ mouthparts [13]. From the ﬁgure, we can see that the mouthparts consist of six needles, which are relatively long and can penetrate into the host’s capillaries smoothly when feeding. And Fig. 2 also shows an SEM photograph of a mosquito’s mandible, and it can be seen that the side of the mandible has a distinct sawtooth structure.

Fig. 2. Sketch map of the mouthparts of mosquitoes and SEM of a mosquito’s mandible [13]

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Mosquitoes can be undetected when biting people because the mouthparts produce rolling friction during the process of penetrating into the organism to reduce the resistance [14]. In view of the dolphin skin ridge, mosquito mouth device can play a role in drag reduction, the designed composite structure of them is applied to the surface of the ﬁsh dart. Figure 3(a) gives out the actual ﬁsh dart, used to shoot ﬁsh in ﬁshery. Figure 3(b) gives out the ﬁsh gun, used to launch the ﬁsh dart. In the study, the front end of the ﬁsh dart has a groove. In order to avoid the influence of the grooves on the underwater resistance in the study, the model shown in Fig. 3(c) is adopted.

(a)

(b)

(c)

Fig. 3. (a) The actual ﬁsh dart (b) ﬁsh gun (c) the model of ﬁsh dart

A kind of grooved composite structure is designed by referring to the micro-nano structure of dolphins skin ridge and mosquito mouth device, and its structural unit is Ushaped groove and sawtooth structure, as shown in Fig. 4(a) (b). The designed composite structure starts with a different arrangement of the grooves and a combination of different micro-nano structures. In order to utilize the best of the U-shaped structure’s characteristics, the following aspects are arranged: the periodic order is arranged longitudinally and transversely (P1 = P2 = P3 = … = Pn); the progressive order is arranged transversely (P1 < P2 < P3 < … < Pn); the quasi-periodic order [15, 16] is arranged longitudinally (Q1 * Qn are arranged according to the Fibonacci sequence, with default values, Q1 = 0.3 mm, and Q2 = 0.4854 mm). The sawtooth structure is formed by arranging the longitudinal U-shaped structures in a quasi-periodic order based on the transverse U-shaped structures arranged in a progressive order, as shown in Fig. 4(b). The U-shaped structure and the combined structure of the different arrangements described above are planned and laid out to obtain the composite structure shown in Fig. 4(c). And mark 1 in Fig. 4(c) shows a Ushaped structure arranged longitudinally (U-L) in a periodic order. Mark 2 in Fig. 4(c) is shown as a U-shaped structure arranged transversely (U-T) in a periodic order. Mark 3 in Fig. 4(c) is U-T in a progressive order, and mark 4 in Fig. 4(c) shows a sawtooth structure.

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spanwise P1

P2

direction P3

P4

P5

(a)

flow direction

P1 P2

P3 P4

P5

Q1

Q2

Q3

(b)

4

2

1

3

(c) Fig. 4. (a) U-shaped groove (b) sawtooth structure (c) composite structure

3 Simulation and Optimization 3.1

Conditions of Fluid Mechanics Calculation

The standard k-e model was selected in the Viscous mode using the hydrodynamics simulation software. The standard model is a semi-empirical Equation based on the turbulent flow energy and dissipation rate. The turbulent flow energy equation is an exact equation, and the turbulent dissipation rate equation is an empirical equation. The speciﬁc equations are as follows: The turbulent flow energy equation: @ @ @ l @k ðqkÞ þ ðqkui Þ ¼ ðl þ i Þ @t @xi @xj rk @xj þ Gk þ Gh qe YM þ Sk

ð1Þ

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The turbulent dissipation rate equation: @ @ @ l @e ðqeÞ þ ðqeui Þ ¼ ðl þ i Þ @t @xi @xj re @xj e e2 þ Gle ðGk þ G3e Gh Þ G2e q þ Se k k

ð2Þ

In the above equations, k is the turbulent flow energy, e is the turbulent dissipation function, t is the time, q is the fluid density (the fluid in this paper is water, and the density of water is 103 kg/m3); xi and ui are the coordinate components and the timeaveraged velocity component respectively in the i-coordinate direction; li is the turbulent motion viscosity coefﬁcient, Gk and Gh are the turbulent flow energy generated by the laminar velocity gradient and buoyancy respectively; YM is the diffusion fluctuation, Sk and Se are user-deﬁned parameters; rk and re represent the k equation and the e equation respectively, the Prandtl constants are rk = 1.0 and re= 1.3; G1e, G2e, and G3e are constants, respectively G1e= 1.44, G2e = 1.92, G3e = 1.0. Figure 5 shows the three-dimensional simulation model by the grooved structure of the composite arrangement. The other different arrangement of the grooved structures is arranged in this computational domain. In the computational domain, flow direction x length is 160 mm, the spanwise y width is 80 mm, and the vertical z height is 80 mm.

flow direction

z x y

1:1.5

Fig. 5. Sketch map of the computational domain

For convenient for comparing, differently arranged grooved structures are placed in the same computational domain for simulation. At the same time, in order to simplify the calculation process and improve the calculation accuracy, the top, bottom, left and right boundary of the computational domain are all in the condition of slip boundary which can be considered as boundary conditions that are not constrained by other physical surfaces. The grooved structural face is set to a wall function boundary condition, that means no-slip boundary condition. For the analysis of the turbulent flow state, in addition to setting the speed of 10 mm/s at the inlet, the turbulence intensity

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I = 0.04, the turbulent energy k = 0.271856, and the turbulent dissipation rate e = 83.18247 were also speciﬁed. For the outlet, the pressure P = 0 and the nonviscous stress are chosen to eliminate the viscous stress limitation of the Dirichlet boundary condition. In order to study the fluid drag reduction characteristics of the boundary turbulent state, the fluid dynamics mode is selected, and the ﬁnite volume method is used to discretely generate the mesh. However, for the corner angle of the primitive, it is difﬁcult to adopt a high-quality structured mesh. In order to accommodate the complex structure, an automatic generation of the mesh is employed and the denser process is performed in the grooved region. After repeated attempts, the size of the mesh is determined as follows: The outer domain of the grooved structure is sparse, and the length from the two sides to the center grid is gradually reduced by 0.87. The mesh is densely divided inside the groove (curvature resolution is 0.6, and the resolution of the narrow domain is 0.7). 3.2

Calculation Results and Analysis

3.2.1 Optimization of Composite Structure Parameters Referring to the structural dimensions of the actual dolphin skin ridge, a U-shaped structure is a better choice. In order to determine the optimal drag reduction parameters of the U-shaped structure, the width or depth is changed while ensuring the same depth or width. Ridgway [17] showed that the skin ridge peak spacing of the whales is 0.41– 2.35 mm and the peak-to-valley depth is 7–114 lm. Accordingly, the width variation range of the U-shaped structure is set to 0.25–2 mm, and the depth variation range is set to 0.01–2 mm. For the convenience of calculation, the width of the U-shaped structure in this paper is a chord of a circle, and the depth is the height of the arc corresponding to the chord. Through simulation, it is found that not all sizes of grooves can reduce drag. As shown in Fig. 6, at a width of 1.40 mm and a depth of 1.80 mm, the ﬁsh dart having the grooved surface of the size acts to increase the resistance. Under the size parameter of 0.38 mm in width and 0.10 mm in depth, the U-shaped structure has a better effect on drag reduction. The sawtooth structure is composed of a progressive U-T and a quasi-periodic U-L, and which is applied to the front end of the ﬁsh dart model, as shown in Fig. 7. Since the front end of the ﬁsh dart is tapered, in order to avoid the situation that the U-shaped grooves are deeper and deeper at the front of the model when the depth and width of the grooves are kept constant, a progressive arrangement is adopted. So at the front end of the tapered portion of the ﬁsh dart, the depth and width of the U-shaped groove are smaller, and the magnitudes of the depth and width is increased until the U-shaped groove size parameter is set as the U-groove size parameters described below and continue to align the transverse U-shaped grooves which are set as the U-groove size parameters described above (a depth is 0.1 mm and a width is 0.38 mm) in the tapered portion. Since the bursting of the water flow generated by the movement of the taper in the water is different from the flat’s, the dimensionless measurement is deﬁned according to the idea and method of studying the plane groove by Bhushan [18].

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Fig. 6. Drag reduction rate of different scales (a) same depth (b) same width

Q1 Q2 Q3 Q4 Q5 P1

P2

progressive U-T

P3

P4 P5

Fig. 7. The sawtooth structure

quasi-periodic U-L

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Dimensionless Equation: sþ ¼

sVs v

Vs ¼ ðs0 =qÞ0:5 s0 ¼ 0:0592

qU 2 1=5 R 2 e

ð3Þ ð4Þ ð5Þ

s+ is the dimensionless grooved width, s is the true grooved width dimension, m is the kinematic viscosity, Vs is the wall shear rate, q is the fluid density, s0 is the wall shear stress, U is the inflow velocity, Re is the Reynolds number. m is set to 1.308 10−6 m2/s, q is 103 kg/m3, U is set to 10 m/s, Re is 39808. Referring to the experimental results of Walsh [19], it shows by calculation that the minimum width of the U-T arranged in tapered position in a progressive order is 0.1 mm, and the depth is 0.025 mm; the maximum width is 0.2 mm, and the depth is 0.05 mm. 3.2.2 Flow Field on the Surface of the Structure During the analysis of flow ﬁeld, simulate the movement of water, the ﬁsh dart stays in a stationary state. In order to intuitively reflect the influence of different grooved arrangements to the movement of ﬁsh dart, Fig. 8 shows the distribution of velocity contour at a cross-section (z = 40) on a smooth surface, a U-L aligned on the tail, a U-T arrangement combined with a U-L arrangement (U-T&U-L), and composite structure (the structure combines U-L, U-T, and sawtooth structure). As we can see from Fig. 8, the arrangement of the groove structure has a signiﬁcant influence on the water flow velocity. Whether in the grooved domain or at the outlet, there are fluids of lower flow velocity in the vicinity of the grooves arranged in different structures. By comparing Fig. 8(a) and (b–d), it is known that the large flow velocity ﬁeld is concentrated on the surface of the smooth ﬁsh dart, so that the flow velocity through the surface is relatively high while the surface velocity of ﬁsh darts with grooves is lower, which can bring less resistance to the movement of ﬁsh darts. Comparing Fig. 8(b–d), it can be concluded that the U-T can enlarge the low-speed fluid domain on the surface of the ﬁsh dart; the U-L of the tail can extend the low-speed fluid domain at the outlet; the sawtooth structure of the tip of the ﬁsh dart can make the speed of the water flow of the ﬁsh darts decrease during the forward movement, and the resistance, of the water flow to it, is reduced. Because of these structures, the effect of drag reduction is achieved. Through the above analysis, it can be concluded that the composite structure has the best drag reduction effect. 3.2.3 Numerical Analysis of Drag Reduction Characteristics In order to study the drag reduction properties of differently arranged grooved structures, the frictional resistance of the structural surfaces was analyzed, and the drag reduction effect of the groove was studied to obtain the surface friction coefﬁcient. The Equation for calculating the surface friction coefﬁcient Cf is as follows:

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(a)

555

(b)

(c)

(d)

Fig. 8. Velocity distribution of fluid in different grooves (a) smooth surface (b) U-T (c) U-T & U-L (d) composite structure

sw 2 2 qref vref

Cf ¼ 1

ð6Þ

sw is the wall shear stress, qref and vref is the deﬁned reference density and velocity. The numerical calculation results of the surface friction coefﬁcient are shown in Fig. 9 (a). It can be seen from Fig. 9(a) and Eq. (6) that the magnitudes of the shear stress of the differently arranged grooved structures are smaller than the smooth faces’, and the magnitudes of the grooved shear stress arranged by the composite structure are the smallest. In order to have a detailed comparison of the drag reduction effect of different structures, calculate the surface drag reduction rate. The Equation is as follows: g¼

Cf ðsmoothÞ Cf ðgrooveÞ 100% Cf ðsmoothÞ

ð7Þ

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0.00373

surface friction coefficient

0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 4.96276E-4

0.0005

3.1208E-4

2.41893E-4

U-L & U-T

composite

0.0000 smooth

U-T

(a)

drag reduction rate

1.0

0.5

0.0 smooth

U-T

U-L & U-T

composite

(b) Fig. 9. (a) Surface friction coefﬁcient of different grooves (b) Drag reduction rate of different grooves

η is the drag reduction rate and Cf is the friction coefﬁcient. The results are given in Fig. 9(b). Compared with the single structural groove, the composite arrangement of the grooved structure has a better drag reduction effect, and the drag reduction rate can reach 89.49%.

4 Drag Reduction Mechanism To study the drag reduction mechanism of the grooved composite structure, we can start from the two aspects of grooved action and arrangement. Choi [20] believes that the longitudinal groove limits the spanwise motion of the flow vortex, causing the wall to burst to weaken, resulting in a reduction in wall frictional resistance. Bhushan [12] believes that the drag reduction of the rib structure can hinder the conversion of eddy currents and enhance the area of high-speed flow away from the surface.

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(a)

(b) Fig. 10. Wall shear stress (a) the composite structure (b) the smooth plane

The disturbance generated in the fluid is successive, that means, the disturbance is successively superimposed after the interaction of the fluid with grooves. Therefore, after the fluid passes through various ordered grooves, the shear stress is redistributed on the surface of the grooves due to coherent superposition. So when the actual fluid passes through the different ordered arrangement of the grooved structure, the direction and fluidity of the fluid change due to the shear stress. An important idea of this paper is to analyze the drag reduction mechanism of the composite structure by studying the wall shear stress. Figure 10(a) and (b) respectively show the wall shear stress distribution of the composite structure and the smooth surface. It can be seen from the ﬁgure that due to the existence of the composite structure, the wall shear stress exhibits a relatively large change. In the middle of the ﬁsh dart model, the wall shear stress of the composite structure is smaller than the smooth wall surface, and sometimes even larger than the smooth wall surface, that means, a viscous thrust is generated. The shear stress in the transverse U-shaped structure can reduce the turbulent burst and also make the free flow not directly in contact with the model wall, which is similar to “hydraulic bearings”, thereby reducing drag. By comparing the Fig. 10(a) and (b), the shear stress of the smooth plane at the front end of the ﬁsh dart is in the range of 5.466 102 − 1.083 103 Pa, which is about 1.5 times of the composite structure. Especially in the tip, middle and the tail of the ﬁsh dart, after the wall shear stress is

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modulated by the sawtooth structure and the U-L, the wall shear stress is no longer parallel to the fluid, and its direction is inclined to the flow direction at a certain angle, so the shear stress of the component acting on the model is greatly reduced, and the low-velocity fluid remains in the grooves. The modulation effect of the composite structure on the wall shear stress is 43% relative to the smooth plane, thereby the composite structure can achieve a better result of the drag reduction.

5 Conclusion (1) A bionic micro-nano composite structure was designed to achieve underwater drag reduction with the structure of the dolphin skin ridge and the mosquito mouth device being the reference. The fluid mechanic’s simulation method was used to simulate and optimize the size parameters of the composite structure to obtain the optimal structural parameters. (2) Applying the composite structure to the ﬁsh dart, when it moves in the water, the grooved structure can expand the area of the low-speed watershed, so that the high-speed watershed is far away from the surface, reducing the resistance of the water flow. The wall shear stress on the composite structure of the ﬁsh dart is greatly reduced by the action of the groove, and the direction of some wall shear stress is no longer parallel to the surface, so that the surface friction coefﬁcient is reduced, thereby achieving the purpose of drag reduction. (3) It is found that, by means of simulation, the micro-nano composite structure which is suitable for the surface of the ﬁsh dart achieve the effect of a drag reduction rate is 89.49% when moving in water. It is proved that the micro-nano composite structure can greatly improve the operating efﬁciency of the underwater vehicle and reduce the underwater navigation resistance.

References 1. Li, H.: Numerical simulation and mechanism analysis of drag reduction technology for underwater vehicles. Harbin Institute of Technology (2006) 2. Kramer, M.R., Cooper, H.L., Drews-Botsch, C.D., et al.: Metropolitan isolation segregation and Black-White disparities in very preterm birth: a test of mediating pathways and variance explained. Soc. Sci. Med. 71(12), 2108–2116 (2010) 3. Huang, W., Wang, B., et al.: Simulation numerical calculation of geometry of bionic baffle reducing material with accompanying wave. J. Ship Mech. 9(1), 14–17 (2005) 4. Scholle, M., Rund, A., Aksel, N.: Drag reduction and improvement of material transport in creeping ﬁlms. Arch. Appl. Mech. 75, 93–112 (2006) 5. Scholle, M.: Hydrodynamical modeling of lubricant friction between rough surfaces. Tribol. Int. 40, 1004–1011 (2007) 6. Ridgway, S., Samuelson, D., Van, A.K., et al.: On doing two things at once: dolphin brain and nose coordinate sonar clicks, buzzes, and emotional squeals with social sounds during ﬁsh capture. J. Exp. Biol. 218(Pt 24), 3987 (2015) 7. Grüneberger, R., Hage, W.: Drag characteristics of longitudinal and transverse riblets at low dimensionless spacings. Exp. Fluids 50(2), 363–373 (2011)

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8. Yang, Y.: Design and mechanism analysis of underwater drag reduction components of river raft appearance. Jiangsu University of Science and Technology (2017) 9. Zhang, F.: Study on drag reduction performance of non-smooth surface of bionic robot ﬁsh. Shandong University (2017) 10. Cheng, P., Jiang, C., Wu, C.: Numerical simulation of drag reduction characteristics of bionic secondary micro-grooves. Chin. J. Sci. Tech. Pap. 9(8), 940–943 (2014) 11. Yu, J., Liu, Z., Wu, W.: Progress in anti-noise and noise reduction of dolphin skin. In: The 30th Anniversary of the Establishment of the Underwater Noise Group of the Chinese Shipbuilding Engineering Society Academic Seminar (2015) 12. Babenko, V.V.: Boundary layer flow over elastic surfaces. Boundary Layer Flow over Elastic Surfaces (2012) 13. Gu, S.: Research on bionic syringe based on insect sucking mouth device. Jilin University (2008) 14. Qi, X.: Research on biomimetic low-resistance and antibacterial medical injection needle based on insect mouthpart. Jilin University (2014) 15. Wang, X., Qi, X., Qu, D.: Study on fluid drag reduction characteristics of one-dimensional periodic and quasi-periodic groove structures. Acta Phys. Sin. 62(5), 00075–82 (2013) 16. Zhang, M., Yan, X., Zhang, Y., et al.: Drag reduction effect and simulation analysis of onedimensional short-groove composite quasi-crystal structure. Acta Phys. Sin. 61(19), 000287–293 (2012) 17. Shoemaker, P.A., Ridgway, S.H.: Cutaneous Ridges in Odontocetes. Mar. Mammal Sci. 7 (1), 66–74 (1991) 18. Bhushan, B.: Biomimetics (2012) 19. Wood, C.M., Walsh, P.J., Kajimura, M., et al.: The influence of feeding and fasting on plasma metabolites in the dogﬁsh shark (Squalus acanthias). Comp. Biochem. Physiol. Part A Mol. Integr. Physiol. 155(4), 435–444 (2010) 20. Kim, W., Choi, H.: Effect of the spanwise computational domain size on the flow over a twodimensional bluff body with spanwise periodic perturbations at low Reynolds number. Comput. Fluids 183, 102–106 (2019)

A Method of Parametric Stability Region Determination for Non-linear Gear Transmission System Dongping Sheng1(&), Xiaozhen Li1, and Rupeng Zhu2 1

Changzhou Institute of Technology, Changzhou 213032, China [email protected] 2 College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Abstract. A general method of parametric stability region calculation for a non-linear transvers-torsional coupled gear train’s vibration model with multiple-clearances is studied. There are six steps in the method, and the ﬁrst step is strength and fatigue life threshold determination, and the second step was deciding numerical integration time according to the motion state of the system when the parameters discussed changed in their test range, the third step is calculating the maximum displacement of the system when the parameters discussed changed in their range by using nested loop algorithm, the forth step is calculating the value which equals the maximum displacement subtracted by minimum displacement by the same method with third step, the ﬁfth step is the stability judgment by comparing the value which is calculated in step four and ﬁve with threshold which is set in second step, and the last step is make the parametric stability region map of the system. As an example, the stability region calculation of a coupling transverse-torsional gear train vibration model with multiple-clearances is studied under the parameter of the backlash, mesh damp ratio, input rotation speed and the bearing clearance of the driving wheel and driven wheel, and the stable region of input rotation speed, backlash, meshing damp ratio and the bearing clearances are calculated respectively. Keywords: Transverse-torsional coupling Non-linear vibration Multiple-clearances Parametric stable region

Gear transmission system has the characteristics of high power and speed, harsh work conditions, small shape and low weight as well as high design objective. Gear system has been more and more extensive use in the ﬁeld of aviation, watercraft, automobile and heavy machine, and many scholars has been attracted into the research area of gear dynamics and stability demotic and overseas [1, 2]. Not only resonance would happen when excitation frequency close to natural frequency, but also close to multi-fold frequency in nonlinear system, which could cause instability. So ﬁnding out the stable and unstable region could guide the design and generate practical signiﬁcance. System parametric stable region includes the motion stability, vibration strength stability and fatigue life stability, motion stability means the motion condition does not This project is supported by Changzhou Institute of Technology. © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 560–569, 2020. https://doi.org/10.1007/978-981-32-9941-2_46

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change under small disturbance; strength stability means system vibration amplitude does not mutation under small disturbance and not cause gear crack under instantaneous peak stress; Fatigue life stability means gear would be disabled because of periodic or random stress Dr, the life of the system could be predicted by S-N curve. When under the periodic one motion, the amplitude of the system will change periodically under the periodic change of the meshing stiffness. Normally, the amplitude of the system will not surpass the stress limit except the severe overload condition. When system under the multi-periodic or chaos motion, Unilateral or bilateral impact would happen corresponding. The amplitude of vibration would not surpass stress limit, but the stress change range could cause failure with the time. In other words, those parametric region which bring failures called fatigue life instability. Lin [7] studied the parametric instability region caused by time-varying meshing stiffness, but ignored the influence generated by backlash. Li [8] studied the stability region of planetary gear transmission system, and the parameters including input speed, backlash and damping ratio are considered. From the available literature on gear system stability, the main research work focuses on the stability of the periodic motion state of a single pair of gears [4, 6], but it is rare to calculate the stability domain of multiple clearance and bending-torsion coupled nonlinear vibration system based on the concept of the stability of static vibration strength and fatigue strength. In this paper, the stability domain based on vibration strength and fatigue life of multiple clearances of nonlinear vibration system has been researched, and the parameters including input speed, backlash and mesh damping has been considered. The stable domain has been analyzed under the above parameters in different combinations.

1 Modeling The multiple clearances and bending-torsional nonlinear gear transmission system is consist of a pair of gear, shaft and bearing, the gear is assumed to be spur gear and without considering friction. This mathematical model is shown in Fig. 1. In Fig. 1, the angle displacement of active and passive gear are denoted by hd and hp , base radius are symbolized by rd and rp , ed and ep are the eccentric errors, ud and up are phase errors. The active gear’s supporting stiffness, backlash and damping are described by kd , bd and cd , the passive gear’s stiffness, backlash and damping are denoted by kp , bp and cp . The comprehensive meshing error, time-varying meshing stiffness, backlash and meshing damping are denoted by eðtÞ, kðtÞ, b and cm , ed and ep are active and passive gear’s eccentric error, ud and up are the active and passive gear’s initial phase. According to the characteristics of meshing stiffness of the spur gear, it could be assumed to be square wave. Periodical curve could be expressed by Fourier series, the ﬁrst item is adopt and the time-varying meshing stiffness could be expressed as follows kðtÞ ¼ km þ ka sinðxt þ uÞ

ð1Þ

Where, km is the average value of meshing stiffness, ka is alternating stiffness, u is initial phase, and x is meshing frequency.

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Fig. 1. Multiple-clearance transverse-torsional coupled model

The system has four degrees, and it’s coordinate could be expressed by ðhd ; Xd ; hp ; Xp ÞT , where hd and Xd are the active gear’s rotating and longitudinal freedom, hp and Xp are the passive gear’s rotating and longitudinal freedom. 1.1

Dynamic Meshing Force

For single pair external meshed gear, the relative displacement Xr on the meshing line and comprehensive error could be expressed as 8 < Xr ¼ xd xp eðtÞ eðtÞ ¼ e cos xt þ ed cosðxd t þ ud Þ ð2Þ : þ ep cosðxp t þ up Þ Where, eðtÞ is the project on the meshing line of the comprehensive error, xd and xp are the displacement on the meshing line of active and passive. The nonlinear function of backlash and supporting clearances could be expressed as 8 > < Xr b Xr [ b f ðXr ; bÞ ¼ 0 jXr j b > : Xr þ b Xr \ b 8 > < Xd bd Xd [ bd fd ðXd ; bd Þ ¼ 0 jXd j bd ð3Þ > : Xd þ bd Xd \ bd 8 X p [ bp Xp bp > > < Xp bp fp ðXp ; bp Þ ¼ 0 > > : X p þ bp Xp \ bp

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The dynamic force is consist of elastic restoring force and damping force, and could be expressed as F ¼ kðtÞf ðXr ; bÞ þ cX_ r

ð4Þ

Damping coefﬁcient is expressed as cm ¼ 2f

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ km =ð1=md þ 1=mp Þ

ð5Þ

Where, f is the damping ratio, md is the mass of active gear, mp is the mass of passive gear. 1.2

Controlling Equation

Under the effect of torque Td and load Tp , the differential equation for the system shown in Fig. 1 could be expressed as 8 €d þ cd X_ d þ kd fd ðXd ; bd Þ md X > > > > ¼ cm X_ km f ðX; bÞ > > > > € m X þ cp X_ d þ kp fp ðXp ; bp Þ > > < p p ¼ cm X_ þ km f ðXr ; bÞ € > Id hd þ rd cm ðrd h_ d rp h_ p þ X_ d X_ p eðtÞÞ > > > > þ ðkm kc cosðxtÞÞf ðXr ; bÞ ¼ Td > > > > Ip €hp þ rp cm ðrp h_ p rd h_ d X_ d þ X_ p þ eðtÞÞ > : ðkm kc cosðxtÞÞf ðXr ; bÞ ¼ Tp

ð6Þ

In order to eliminate the rigid displacement and simplify the equation, under the condition of without affecting the solution of equation, the relative coordinates is introduced as follows X ¼ xd xp þ Xd Xp eðtÞ

ð7Þ

Where X is the superposition of the relative displacement on the meshing line, which has the same dynamic characteristics with xd and xp . The dimensionless time is introduced, which could be expressed as s ¼ xn t, where pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xn ¼ km =me , km is the average value of meshing stiffness, me ¼ md mp =ðmd þ mp Þ, the nominal displacement scale bc is introduced as well. Based on the above parameters, the dimensionless displacement, velocity and acceleration could be expressed as € x2 , b ¼ bb , X ¼ x=x , X ¼ x =x , X ¼ x =x c , X_ ¼ Xb _ c xn , X € ¼ Xb X ¼ Xb c n c n d d n p p n respectively. Put all the above parameters into Eq. (6), and simplify it into matric form, the dimensionless equation could be expressed as

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1 2 30 1 € _ n11 n12 n13 X X 7B _ C 6 7B € C 6 4 0 1 0 5@ X d A þ 4 n21 n22 0 5@ X d A € _ p 0 0 1 n31 0 n33 X X p 2 30 1 0 1 bÞ 1 k11 k12 k13 f ðX; F 6 7B C B C þ 4 k21 k22 0 5@ fd ðXd ; bd Þ A ¼ @ 0 A p ; bp Þ k31 0 k33 f p ðX 0 2

1

0

0

30

ð8Þ

Where n11 ¼ cm =me xn þ cm =md xn þ cm mp xn , n12 ¼ cd =md xn , n13 ¼ cp mp xn , n21 ¼ cm =md xn , n22 ¼ cd =md xn , n31 ¼ cm mp xn , n33 ¼ cp mp xn , k12 ¼ kd md x2n , k13 ¼ kp mp x2n , k11 ¼ ðkm ka cosðXtÞÞ ð1 me x2n þ 1 md x2n þ 1 mp x2n Þ, k21 ¼ ðkm ka cosðXtÞÞ md x2n , k22 ¼ kd md x2n , k31 ¼ ðkm ka cosðXtÞÞ mp x2n , k33 ¼ kp 1 ¼ F bc me x2n þ ex2 cosðXtÞ bc x2n þ ed x2d cosðXd tÞ bc x2n þ ep x2p cosðXp tÞ mp x2 , F . n bc x2n .

2 Calculation Method of Parametric Region Refer to connotation of the static strength and fatigue life stability region, the calculation method of stable region of multiple clearances and bending-torsional coupled gear system could obey the following steps: 1. In order to analyze the stable and unstable region quantitatively, two index have been deﬁned. According to the bending strength and fatigue life, a vibration static unstable displacement threshold value X and fatigue life unstable threshold value DX have established. 2. Choose appropriate Poincare section R ¼ fðs; X Þ 2 R Rn js ¼ mod ð2p=xÞ; xi ¼ maxðabsðxi0 ! xiT Þg; where xi means the maximum displacement under a certain periodic excitation force of the ith state variable, x is the meshing frequency. 3. Choose appropriate Poincare section R ¼ fðs; XÞ 2 R Rn js ¼ mod ð2p=xÞ; Dxi ¼ absðmaxðxi0 ! xiT Þ minðxi0 ! xiT ÞÞg; Dxi ¼ absðmaxðxi0 ! xiT Þ minðxi0 ! xiT ÞÞ means the absolute value of the difference for the maximum and minimum value under a certain periodic excitation force of the ith state variable. 4. By the method of numerical integration and in a certain time region, system’s maximum and minimum stable displacements are obtained, and absolute value of the subtraction is gained as well [8]. 5. Use the method of loop nesting, calculate system’s maximum and minimum displacement under different parameter combination, and compare the subtraction with the unstable threshold value. Evaluate whether the system motion is in unstable condition, and put “1” into the matrix if the motion is unstable. 6. System’s stable and unstable region could be obtained by outputting the data in the matrix graphically.

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3 Stable Region Calculation Taking the multiple clearances and bending-torsional coupled gear transmission system as study object, and select a group of parameters, including m = 3 mm, a = 20°, zd = 40, zp = 80, ed = 10 lm, ep = 10 lm, bc = 10 lm, bd = 10 lm, bp = 10 lm, kd ¼ 0:2 GN=m, kp ¼ 0:35 GN=m, input power P = 200 kw. Choose input speed n, backlash b, damping ratio n, active and passive gear’s supporting clearances bd and bp as research parameters. Set the ﬁve times of backlash as the strength threshold, and ten times of backlash as the fatigue life threshold. 3.1

Single Parameter Stable Region

There are ﬁve main parameters in multiple clearances and bending-torsional coupled gear transmission system, including input speed, backlash, meshing damping, active and passive gear’s supporting clearances. In order to reduce the computing scale, only taking input speed as single parameter to study the stable region. When backlash b ¼ 4 105 m, damping ratio n = 0.05, input speed is in between 1000–10000 r/min, the curve of strength threshold X versus input speed is obtained, as shown in Fig. 2. As shown in Fig. 2, the speed n is mostly located in between 5300– 5400 r/min and 6200–7600 r/min when maximum displacement surpass the 5b. In this region, the maximum displacement exceed the 5b, and could be regarded as the unstable region, the other speed range could be considered as stable region.

Fig. 2. System motion static strength threshold-rotation speed

Under the same backlash and damping ration, when input speed is in between 1000–10000 r/min, the curve of fatigue life threshold DX versus input speed is obtained as well, as shown in Fig. 3, where the speed n is mostly located in between 4800–5200 r/min and 5600–9000 r/min when maximum displacement surpass the 10b, which means the speed in this range could cause system unstable. In this region, the maximum displacement exceed the 5b, and could be regarded as the unstable region, the other speed range could be considered as stable region.

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Fig. 3. System motion fatigue strength threshold-rotation speed

Think Figs. 2 and 3 together, the single parameter stable region based on the static strength and fatigue life could be obtained. When fatigue life threshold becomes bigger, the stable region will be compressed. Both considering static strength and fatigue life threshold, the unstable region is located in between 4800–5400 r/min and 5600– 9000 r/min. Additionally, the ﬁrst critical speed of the system could be obtained from Figs. 2 and 3, which almost located in between the range of 6000–8000 r/min, by theoretical calculation, the critical speed of this system is 6837 r/min. 3.2

Double Parameters Stable Region

Because the system has many kinds of double parameter combinations, now take input speed and backlash for example to study the stable region. Assume the damping ratio equals 0.05, active gears supporting clearance is 2 10−5 m, and study the stable region when speed located in between 1000-10000r/min and backlash in between 1 10−5–10 10−5 m. In Figs. 4, 5 and 6, “” means stable region. Figure 4 is the stable region calculated by strength threshold value, Fig. 5 is the stable region calculated by fatigue value threshold value, and Fig. 6 is the double parameter stable region calculated by static strength threshold and fatigue life threshold value. By comparing the Figs. 4, 5 and 6, it could be conclude that the double parameter stable region is the common region of the Figs. 4 and 5, which is mostly located in the range of 5000– 9000 r/min.

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Fig. 4. Parametric stable region of rotation speed-backlash (Static strength threshold equals 5b)

Fig. 5. Parametric stable region of rotation speed-backlash (Fatigue strength threshold equals 10b)

Fig. 6. Parametric stable region of rotation speed-backlash (Static and fatigue strength threshold equal 5b and 10b)

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Triple Parameters Stable Region

Figure 7 is the triple stable region of speed versus backlash versus damping ration diagram by the method above, where the range of damping ratio is in between 0–0.4, backlash is in between 1 10−5 and 8 10−5m, the speed range is from 1000 to 8000 r/min, and “” means the stable region. As shown in Fig., it could be seen that the unstable region mostly located in the range of big backlash, low speed and low damping. Figure 8 is the triple parameter stable region of backlash versus active gear’s supporting clearances versus passive gear’s supporting clearance by the same method. From Figs. 7 and 8, the stable region and appropriate parameters group could be obtained and guide the detail design. By the same method, the four or ﬁve parameter stable region could be calculated as well.

Fig. 7. Stable region of rotation speed-backlash-damp ratio

Fig. 8. Stable region of backlash-drive bearing clearance-driven bearing clearance

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4 Conclusion 1. A kind of stable region calculation method have been proposed for the multiple clearances and bending-torsional coupled gear transmission system. By choosing appropriate static strength and fatigue life threshold, calculate the related parameter range, and the stable region could be achieved correspondingly. 2. A four degree nonlinear gear transmission system has been established, and the parameters including backlash, supporting clearance, input speed and damping ratio has been considered to calculate the stable region. By the calculate method, the single parameter, double parameter and triple parameter stable region was obtained which could guide the design of gear transmission system.

References 1. Zhu, F., Li, W., Maropoulos, P.G.: Tolerance analysis of a planetary gear reducer under cad circumstance. Chin. J. Mech. Eng. 18(3), 342–345 (2005) 2. Sun, D., Qin, D., Wang, H.: Control strategy of a parallel hybrid car with a metal beltplanetary gear continuously variable transmission system. Chin. J. Mech. Eng. 15(3), 199– 203 (2002) 3. Sun, Z., Ji, H., Shen, Y.: Nonliear dynamics of 2 K-H planetary gear train. J. Tsing-hua Univ. Sci. Technol. 43(5), 636–639 (2003) 4. Gao, Z., Shen, Y., Li, S.: Research on the periodic solution structure and its stability off nonlinear system with clearance. Chin. J. Mech. Eng. 40(5), 17–22 (2004) 5. Chen, A., Luo, S., Wang, W.: Numerical investigations on dynamic transmission error and stability of a geared rotor-bearing system. J. Mech. Eng. 40(4), 21–25 (2004) 6. Yu, Y., Yu, L., Liu, H.: Stability and bifurcation of nonlinear bearing-rotor system. J. Mech. Eng. 40(10), 62–67 (2004) 7. Lin, J., Parker, R.G.: Planetary gear parametric instability caused by mesh stiffness variation. J. Sound Vib. 1, 129–145 (2002) 8. Li, T., Zhu, R., Bao, H.: Method of stability region determination for planetary gear train’s parameters based on nonlinear vibration model. J. Aerosp. Power 27(6), 1416–1423 (2012) 9. Zhang, W.: The application of fatigue damage theory for gear fatigue calculation under nonstable and varying load. 15(supp.), 52–57 (1987) 10. Wang, S., Shen, Y., Dong, H.: Chaos and bifurcation analysis of a spur gear pair with combine friction and clearance. Chin. J. Mech. Eng. 38(9), 8–11 (2002) 11. Li, T., Zhu, R., Bao, H.: Nonlinear torsional vibration modeling and bifurcation characteristic study of a planetary gear train. Chin. J. Mech. Eng. 47(21), 76–83 (2011) 12. Sun, Z., Shen, Y., Wang, S., Li, H.: Bifurcation and chaos of star gear system. Chin. J. Mech. Eng. 37(12), 11–15 (2001) 13. Ahmadian, M., DeGuilio, A.P.: Recent advances in the use of piezoceramics for vibration suppression. Shock Vibr. Dig. 33(1), 15–22 (2001) 14. Gu, H.C., Song, G.B.: Active vibration suppression of a flexible beam with piezoceramic patches using robust model reference control. Smart Mater. Struct. 16(4), 1453–1459 (2007) 15. Lamarque, C.H., Bastien, J.: Numerical study of a forced pendulum with friction. Nonlinear Dyn. 23(4), 335–352 (2000) 16. di Bernardo, M., Kowalczyk, P., Nordmark, A.: Bifurcations of dynamical systems with sliding: derivation of normal form mapping. Phys. D 170(3–4), 175–205 (2002) 17. Louroza, M.A., Roitman, N., Magluta, C.: Vibration reduction using passive absorption system with Coulomb damping. Mech. Syst. Sig. Process. 19(3), 537–549 (2005)

Performance and Parameter Sensitivity Analysis of Finger Seal with Radial Clearance Hua Su(&) School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China [email protected]

Abstract. Finger seal is an advanced compliant seal which can be utilized to separate high pressure (HP) and low pressure (LP) zones in high speed rotating shaft environment. The work concerns leakage simulation of a multi-layer ﬁnger seal with radial clearance and the effects of structural and operating parameters on ﬁnger seal performance. Leakage, viscosity friction and load capacity of ﬁnger seal under different conditions are calculated by computational fluid dynamics method. Using single factor method six input parameters influence rules to sealing behavior are investigated. Leakage and friction increase with increasing radial clearance or ﬁnger number. When lengthening the flow path the leakage decrease sharply but friction and load increase slightly. Leakage, friction and load all increase approximate linearly with pressure increase. Decreases of leakage are observed when raising gas temperature or rotor rotating speed. Six input parameters sensitivity analysis to ﬁnger seal performance are conducted combined with orthogonal design and fuzzy satisfactory function evaluation. It is concluded that radial clearance is the most sensitive factor of all. Pressure and radial clearance and rotating speed are the top three factors impact on friction. Flow length and pressure have signiﬁcant roles on load capacity. Finger number has the least effect on all performance. The work presented makes sense to study the ﬁnger seal performance accurately and economically. It also provides useful insights to design the ﬁnger seal with better performance. Keywords: Finger seal Load-carry capacity

Radial clearance Leakage Friction

1 Introduction Finger seal is a new compliant seal presented after brush seal which is regarded as a revolutionary new technology in air-to-air sealing for secondary flow control and gas path sealing in gas turbine engines [1, 2]. A ﬁnger seal is composed of multiple staggered laminates with each having a large number of flexible ﬁngers around the rotor surface. The interface between the free end of the ﬁnger named as ﬁnger foot pad and the rotor surface is the main sealing path. However, the flow path is not evenly around the circumferential direction due to the gaps between ﬁnger element sticks. The leakage This project is supported by National Natural Science Foundation of China (Grant No. 51575445). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 570–591, 2020. https://doi.org/10.1007/978-981-32-9941-2_47

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clearance is influenced by the ﬁnger seal installation status and operating conditions [3– 5]. Sometimes in order to reduce heat and wear of the seal, an initial clearance is set between the ﬁnger foot pad and rotor surface. In addition, the hysteresis of the ﬁnger seal [6] and heat deformation may change the clearance between ﬁnger pad and rotor. This radial clearance is variable for different working conditions which affect the leakage of the ﬁnger seal signiﬁcantly. It is helpful to understand the leakage and other flow behavior such as sealing force and viscosity friction of ﬁnger seal in different radial clearance and working conditions. And this work is fundamental to design and analyze the ﬁnger seal with high performance. The radial clearance between the ﬁnger foot pad and rotor surface can be derived by theoretical analysis or experimental method. However, how to calculate the leakage with a given radial clearance exactly has not been studied in detail. In previous work the radial clearance of ﬁnger seal is often assumed as simple annular gap and the gaps between the ﬁnger sticks are neglected [7, 8]. Some researchers use the annular incompressible flow to compute the leakage of the ﬁnger seal. Many researchers use relative clearance to compare the performance of gas seal based on the annular gap assumption [5, 9]. In the annular leakage formula only the diameter of the gap is considered then others structural parameters affecting the clearance, such as the number of the ﬁngers and the length of the axial flow path, are not included yet. And the rotating speed is not considered in the procedure of leakage estimation. Furthermore, the gas flow tribology behavior such as viscosity friction and gas ﬁlm load capacity are also important for study the performance and dynamic stability of the seal-rotor system, which are seldom studied in the ﬁeld of contacting ﬁnger seal applications. All these work must be established on understanding the ﬁnger seal flow state more accurately. In addition, the ﬁnger seal performance is affected by many structural and operating parameters. The problem how these parameters effect the seal leakage has always received extensive attention by many researchers [2–5, 10]. However, most are qualitative analysis, or study the leakage under structural and operating conditions separately. The difference and importance of every input parameter is not speciﬁc. Also, how the parameters effect on the ﬁlm viscosity friction and load capacity of contacting ﬁnger seal has not been concerned yet. Therefore, the quantitative or the sensitivity analysis of the structural and operating parameters on sealing properties is inadequate by now. Whereas, this information is an indispensable part for seal design. The 3D numerical model of the ﬁnger seal radial clearance flow ﬁeld is established in the presented work. The numerical analysis in this study is carried out using the computational fluid dynamics (CFD) module of COMSOL software. The leakage, viscosity friction and load capacity are calculated under different structural and operating conditions. The gaps between ﬁnger sticks are included in the model. Thus, a more accurate set of results could be expected from the current analysis. Furthermore, in order to ascertain the parameters importance to the performance of ﬁnger seal, the effects of structural and working conditions on ﬁnger seal performance are investigated by single factor analysis ﬁrstly. Then based on orthogonal design the simulation scheme is presented and fuzzy satisfactory function is established to estimate the satisfactory degree of the performance of ﬁnger seal. Through the range analysis of the satisfactory function, the sensitivity of the structural and operating

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parameters to seal leakage, friction and load are determined respectively. And the sequence of influence degree to seal performance from big to small is obtained consequently.

2 Geometry of Finger Seal The structure of ﬁnger seal studied in this work is shown in Fig. 1. The gap between ﬁnger foot pad (see Fig. 1(b)) and rotor surface is very thin that is magniﬁed in Fig. 1 (d). The repetitive pattern of the seal in circumferential direction allows the application of periodic symmetric boundary conditions on a sector of the seal. Therefore, the flow ﬁeld of the seal corresponding one ﬁnger section constitutes the computational domain, which is shown in Fig. 2 (the middle part). The main parameters of the ﬁnger seal which affect the flow path geometry include the following parameters: ﬁnger slice numbers, ns, means the number of ﬁnger slices; ﬁnger number, s, means the number of ﬁngers around the circumference of a “ﬁnger slice”; gap between ﬁnger sticks, d; thickness of a ﬁnger slice, tf; ﬁnger foot height, x, and radial clearance between ﬁnger pad and rotor, h, shown in Fig. 1.

(a) One finger slice of seal face

(b) Details of finger foot

(c) Configuration of finger seal

(d) Details of radial clearance

Fig. 1. Structure of ﬁnger seal

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3 Simulation Method The radial clearance between the ﬁnger foot pad and rotor is very thin from a few microns to a dozens of microns and the flow in axial direction is very short (about 1 to 3 mm). In this thin and short flow ﬁeld the flow distribution will vary greatly. In order to describe the gas flow reasonably and actually, parts of inlet and outlet cavity are integrated in the simulation model according to the structure of the seal system. The ﬁnger seal flow ﬁeld model with radial clearance is established based on commercial software COMSOL (5.1). Figure 2 shows the mesh model of the computational domain. The raised parts in the middle of the model represent the gaps between ﬁnger sticks. 3.1

Governing Equations

For compressible and iso-viscous fluid, the governing Navier-Stokes momentum and continuity equations in the absence of any body force such as gravity can be written in the following vector form in the steady state flow conditions [11]: ~ ðq~ r VÞ ¼ 0

ð1Þ

~~ ~ þ gr ~2 ~ qð~ V rÞ V ¼ rp V

ð2Þ

Where, q is gas density, p is pressure, η is dynamic viscosity of gas. In cylindrical coordinates the gradient, convective and vector Laplacian and ordinary Laplace operators take the following form, respectively: @ ~ ð @ þ 1Þ^r þ 1 @ ^ h þ ^z r @r r r @h @z

ð3Þ

~ Vr @ þ Vh 1 @ þ VZ @ ~ V r @r r @h @z

ð4Þ

1 2 @Vh 1 2 @Vr ^ ~2 ~ r V ðr2 Vr 2 Vr 2 Þ^r þ ðr2 Vh 2 Vh þ 2 Þh þ r2 Vz^z r r @h r r @h

ð5Þ

r2

1@ @ 1 @2 @2 ðr Þ þ 2 2 þ 2 r @r @r r @h @z

ð6Þ

In addition, the density of the gas (air) is replaced from the ideal gas law: q¼

p RT

Where, R is speciﬁc gas constant, T is temperature.

ð7Þ

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For the numerical analysis in the current study, the circumferential Reynolds number and axial Reynolds number can be determined by Eqs. (8) and (9) respectively [12]. If the axial Reynolds numbers of the ﬁnger seal are less than 2100 the flow may be regarded as laminar flow. Reh ¼

qrxh g

ð8Þ

m_ gpr

ð9Þ

Rez ¼

For gas flow the Knudsen numbers Kn is commonly used to decide if the gas rarefaction effects need to be considered. According to Eq. (10), if Kn < 0.01, continuum flow with no slip boundary conditions can be employed. k g Kn ¼ ¼ h ph

rﬃﬃﬃﬃﬃﬃﬃﬃﬃ pRT 2

ð10Þ

Where, r is average rotor radius, x is angular velocity, h is clearance, m_ is mass flow, k is molecular mean free path, η is the dynamic viscosity of the gas. Besides, the surface of ﬁnger seal and rotor are assumed as smooth and the asperities are not considered in this paper. 3.2

Boundary Conditions

The boundary conditions of flow simulation are shown in Fig. 2. The pressure boundary condition over the inlet is set to given pressure drop and the outlet is considered to be constant and given by 1 atmosphere. The surface contacting with the rotor is set to be the moving wall and the rotating speed is given on it. Two cutting surfaces are set periodic boundary conditions. And the others are set to ﬁxed walls. The fluid is regarded as ideal gas with a given temperature. The computational domain shown in Fig. 2 is covered by a mesh of quadrilateral and tetrahedral elements. Because the ﬁlm thickness grid has a great influence on the results, in order to verify the grid independence, the grid in the ﬁlm thickness direction takes 5–10 layers, and the other grid sizes are changed accordingly to ensure the grid quality. It is found that the leakage and friction increases 1.66% and 2.17% respectively, and the load decrease 0.29% when the number of mesh elements increase from 17,646 to 63,360. Considering the computation time and accuracy, the next results are achieved by using the element number of 36,624 with 6 layers of grids in the ﬁlm thickness direction.

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(a) Mesh of flow field and boundary conditions

(b) Radial clearance meshed layers Fig. 2. Finger seal flow ﬁeld with radial clearance

3.3

Performance Parameters

The gas ﬁlm load capacity is calculated as ZZ Fload ¼

pdA

ð11Þ

qvdAout

ð12Þ

Where, Aout is the area of outlet surface. The viscosity friction is calculated as ZZ sf dA Ffriction ¼

ð13Þ

A

Where, A is the area of load surface. The mass flow leakage is calculated as ZZ m_ ¼ AOUT

A

Where, sf is sheer stress.

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4 Simulation Results and Discussions The ﬁnger seal flow behavior with variations of ﬁnger seal geometry and working conditions are analyzed by single factor method, that is, the ﬁnger seal performances are determined by sequentially varying each of the input factors and by ﬁxing all other factors to constant values. The seal geometry of this and operating parameters are provided in Table 1.

Table 1. Geometrical and operational data for ﬁnger seal Parameter Finger number, s Gap between ﬁnger, d Finger slice number, ns Thickness of a ﬁnger slice, tf Flow path length, l = ns tf Finger foot height, x Radial clearance, h Radius of rotor, rr Pressure drop, Dp Rotating speed, n Operating temperature, t

4.1

Value 30–45 0.4 8 0.15–0.3 1.2–2.4 0.53 5–25 33 0.1–0.5 0–15000 293–873

Unit – mm – mm mm mm lm mm MPa rpm K

Compare with Compressible and Incompressible Flow

Figure 3 shows the ﬁnger seal leakage with different radial clearance in compressible and incompressible conditions. From the graph it can be seen that the leakage increase with the radial clearance for compressible or incompressible condition. However, the leakages for compressible flow are obviously greater than that of the incompressible

Fig. 3. Leakage for different radial clearance in compressible/incompressible conditions

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condition at the same clearance, even if the Mach number in the presented conditions are less than 0.3 which is often considered as incompressible flow. So the compressibility effect should be considered when calculate the leakage of ﬁnger seal. Figure 4 shows the viscosity friction and load capacity of ﬁnger seal with different radial clearance in compressible and incompressible conditions. These results are obtained under the following conditions: ﬁnger number is 33, flow length is 1.6 mm (8 slices, tf is 0.2 mm), pressure drop is 0.1 MPa, rotation speed is 5000 rpm, and gas temperature is 293 K. It is found that the load decreases and the friction increases with the radial clearance increasing. When radial clearance increases, the flow velocity increases obviously and varies sharply. These can be found from the Fig. 5(a) and (b) of velocity contour under the radial clearance of 5 lm and 25 lm respectively. The velocity variation gradient along the ﬁlm thickness increases, which results in friction increase. Compared with the compressible and incompressible model, the load capacity is bigger and the friction is lesser in compressible condition.

Fig. 4. Friction and load with different radial clearance in compressible/incompressible conditions

In order to get more accurate flow results of ﬁnger seal, the following analysis results are all obtained in consideration the gas compressibility effect. In order to express clearly, the flow performance parameters such as leakage, load and friction are expressed as relative values with relative variations of conditions. The relative parameters are deﬁned as: relative ﬁnger number, S = s/s0; relative flow length, L = l/ l0; relative pressure, P = Dp/p0; relative speed, N = n/n0; relative temperature, T = t/t0. Here, the basic parameters are set to: s0 = 33, l0 = 1.6 mm, p0 = 0.1 MPa, n0 = 5000 rpm, t0 = 293 K. The relative leakage, relative friction and relative load are _ m_ 0 , FF = Ffriction/Ffriction0, FL = Fload/Fload0. Here, m_ 0 , Ffriction0, and deﬁned as: M ¼ m= Fload0 are obtained when the geometrical and operating parameters are set to the basic values.

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(a) Gas flow velocity field when h=5μm in compressible conditions

V(m/s)

(b) Gas flow velocity field when h=25μm in compressible conditions Fig. 5. Gas flow velocity ﬁeld in different radial clearance

4.2

Change Finger Numbers

Figure 6 show the ﬁnger seal relative leakage, friction and load vary with relative ﬁnger numbers. It is found that the leakage increase with the ﬁnger number increasing while the gap between the adjacent ﬁnger stick is constant (=10 lm). When adding the ﬁnger numbers, the width of ﬁnger stick decreases which is similar to widen the leakage path. This result could not be observed by using the approximation circular ring formula to calculate the leakage of ﬁnger seal. It is also seen that the ﬁnger number has small effect on flow friction and load capacity. The friction increase slightly with the ﬁnger number increasing, while contrary to the load capacity. Actually, when the ﬁnger number increases, the flow path becomes more complicated adding the flow resistance a bit. According to the calculation the friction just increases about 0.9% for ﬁnger number changing from 30 to 45.

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Fig. 6. Relative leakage, friction and load with different ﬁnger number

4.3

Change Flow Length

The variations of ﬁnger seal leakage, friction and load with different flow length are shown in Fig. 7. If the number of ﬁnger slice is constant (here it is 8) and every slice has the same thickness, when changing the thickness of a ﬁnger slice we can get different flow length of the seal. Figure 7 shows the performance results when a slice thickness is set from 0.15 mm to 0.30 mm respectively. It can be seen that the leakage decrease obviously with the flow length increasing, while the friction and load increase slightly.

Fig. 7. Relative leakage, friction and load with different flow length

4.4

Change Pressure Differential

Figure 8 shows the ﬁnger seal leakage, friction and load varying with the pressure differential between HP and LP. It is found that the leakage, friction and load all increase obviously with increasing pressure and almost maintain linear relationships respectively. This means pressure flow works remarkably in ﬁnger seal flow.

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Fig. 8. Relative leakage, friction and load with different pressure

4.5

Change Rotating Speed

The ﬁnger seal leakage, friction and load capacity variation with the rotor rotating speed can be found in Fig. 9. It shows that the leakage declines slightly with the speed increasing. It is because that the flow direction changes due to the rotation and results in more flow keeping in the clearance which reduces the leakage. This can be demonstrated from the gas flow velocity distributions shown in Figs. 10 and 11 with zero rotating speed and 15000 rpm respectively.

Fig. 9. Relative leakage, friction and load with different rotating speed V(m/s)

Fig. 10. Gas flow velocity ﬁeld when n = 0 rpm

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V(m/s)

Fig. 11. Velocity ﬁeld when n = 15000 rpm

It is also seen that the flow viscosity friction increase slightly with the rotating speed and the load capacity is almost keep constant in the varying range. Figures 12 and 13 show the gas pressure distributions with the rotating speed of zero and 15000 rpm respectively. It is found that the pressure distribution is almost same under different velocity. As a whole, the rotor’s rotating speed has small effects on radial clearance gas flow of ﬁnger seal. The results also prove that pressure flow hold a dominant position for ﬁnger seal.

Fig. 12. Pressure distribution of ﬁnger bottom surface when n = 0

Fig. 13. Pressure distribution of ﬁnger bottom surface when n = 15000 rpm

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Change Temperature

From Fig. 14 it is found that the gas operating temperature may influence the leakage and viscosity friction obviously. When the gas operating temperature rises, the leakage decreases and load increases mainly due to an increase in the gas viscosity. And we can ﬁnd that the flow friction increase with the temperature increasing.

Fig. 14. Relative leakage, friction and load with different gas temperature

4.7

Validate Simulation Results

In order to validate the simulation method and results presented in this paper, we test the ﬁnger seal leakage using the seal test rig shown in Reference [13]. The test hardware, apparatus, and experimental procedures were described in Reference [13]. With the limitations of test device, only leakage rate were tested at room temperature (293 K). The testing ﬁnger seal geometry are as follows: ﬁnger number, s, is 33; the thickness of a ﬁnger slice, tf, is 0.2 mm; the designed initial built radial clearance between ﬁnger pad and rotor, h, is 30 lm. The other parameters are shown in Table 1. Figure 15 shows a comparison of leakage varying with pressure differential in static condition gained with experiments and simulations. It is found that the leakages increase with the increasing pressure differential referring to the test or simulation results. Similar results could be found in Reference [3].

Fig. 15. Static leakage comparison between simulation and experiment results

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Here, only the static results are compared because the radial gap leakage of the ﬁnger seal is affected by hysteresis, frictional heat, and circumferential flow caused by rotor rotation under dynamic conditions. And it is currently impossible to measure the radial gap under dynamic conditions, so the accurate result of the leakage cannot be calculated. The presented results demonstrate that the CFD simulation results are consistent with those of experiments, although there are some errors between them. It is shown that the leakage experiment results are slightly larger than that of the simulation results. Because the simulation is the amount of radial gap leakage, in fact, in addition to the radial gap leakage, the surface roughness effect and unevenness between the ﬁnger slices will also cause certain leakage from circumferential gap. This is also pointed out in the literature [9]. In addition, there are some measurement errors in the test.

5 Parametersensitivity Analysis 5.1

Parameter Sensitivity Analysis Method

In above analysis, the impacts of the structural and operating parameters on the ﬁnger seal performance are analyzed through the single factor method. In order to identify parameters of ﬁnger seal that contribute to most to the variability of leakage, viscosity friction or ﬁlm load, the parameters sensitivity analysis (SA) are conducted in this study. The SA is based on the relationship between the input factors and the outputs considerations. For ﬁnger seal it is difﬁcult to establish the mathematical functions to describe the relationship of so much input structural and working parameters and output performances, including leakage, friction and load capacity. Thus we use the CFD model discussed above to establish the implicit functions of ﬁnger seal performance. Another issue that deserves concern is the large computational demand in SA. In order to get the reasonable and accurate results large samples must be used in the procedure. However, the use of more strategic, efﬁcient, and effective sampling approaches, can signiﬁcantly reduce calculation amount. Therefore, this research integrates orthogonal design into the SA to provide a ‘sufﬁcient’ number of inputoutput samples. Further study needs to perform the quantitative measure of output caused by each input factor, that is, a measure of sensitivity. Since the input factors for ﬁnger seal include structural and working parameters and there are multi output factors need attention such as seal leakage, friction and load, the satisfactory functions of the seal performance are presented based on fuzzy evaluation [14, 15]. This method could unify the parameters with different dimension and physical interpretation by a uniﬁed

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satisfactory function. According to range analysis of the satisfactory function, we can understand the effects of individual input factors on outputs. 5.2

Orthogonal Design and Simulation Results

The simulation program of ﬁnger seal for SA is created by orthogonal method. The 25 simulation tests totally shown in Table 2 are arranged with 6 factors and 5 levels [16]. Table 3 refers to the corresponding factors and levels of ﬁnger seal. Literatures indicate that proper assignment of input ranges is more influential on SA results [17]. Different variation range of the levels for each factor generate different SA results. In order to avoid the assessment deviation due to the different variation range selected, the same relative increment is set to every level for each factor. In this case, the relative change rate of the level for each factor is set as 15 percent, shown in Table 4. And all the levels should meet the operating and manufacturing demands of ﬁnger seal. By using the CFD method mentioned above, the 25 simulation results of the combination with different factors and level inputs are obtained, which are shown in Table 4.

Table 2. Simulation experiments layout_Factors No. Factors h s l 1 1 1 1 2 1 2 2 3 1 3 3 4 1 4 4 5 1 5 5 6 2 1 2 7 2 2 3 8 2 3 4 9 2 4 5 10 2 5 1 11 3 1 3 12 3 2 4 13 3 3 5

Dp 1 2 3 4 5 3 4 5 1 2 5 1 2

n 1 2 3 4 5 4 5 1 2 3 2 3 4

t 1 2 3 4 5 5 1 2 3 4 4 5 1

No. Factors h s l 14 3 4 1 15 3 5 2 16 4 1 4 17 4 2 5 18 4 3 1 19 4 4 2 20 4 5 3 21 5 1 5 22 5 2 1 23 5 3 2 24 5 4 3 25 5 5 4

Dp 3 4 2 3 4 5 1 4 5 1 2 3

n 5 1 5 1 2 3 4 3 4 5 1 2

t 2 3 3 4 5 1 2 2 3 4 5 1

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Table 3. Levels and factors of ﬁnger seal Level Factors h(lm) s 1 14 29 2 16.1 33 3 18.5 38 4 21.3 44 5 24.5 51

l(mm) 1.6 1.84 2.08 2.4 2.8

Dp(MP) 0.1 0.115 0.132 0.152 0.175

n(rpm) 5000 5750 6612.5 7604 8745

t(K) 293 337 387.5 445.6 512.5

Table 4. Simulation experiments layout_Results Results No. m(g/s) 1 0.2481 2 0.2137 3 0.1866 4 0.1595 5 0.1397 6 0.1923 7 0.4988 8 0.4352 9 0.1517 10 0.2264 11 0.4556 12 0.1599 13 0.4067 14 0.5804 15 0.5114 16 0.4289 17 0.3703 18 0.5370 19 1.1699 20 0.5065 21 0.9121 22 1.2009 23 0.5176 24 0.4555 25 1.0334

5.3

_ f ðmÞ 0.8064 0.8524 0.8853 0.9149 0.9340 0.8787 0.4189 0.5157 0.9226 0.8360 0.4839 0.9145 0.5608 0.3080 0.4008 0.5255 0.6191 0.3648 0.0083 0.4078 0.0545 0.0065 0.3919 0.4841 0.0239

Ffriction(N) 0.0969 0.1153 0.1357 0.1591 0.1886 0.1536 0.1637 0.1957 0.1227 0.1279 0.2203 0.1391 0.1509 0.1510 0.1793 0.1714 0.2032 0.2036 0.1894 0.1411 0.2321 0.2157 0.1596 0.1876 0.1840

f(Ff) 0.7760 0.6984 0.6086 0.5053 0.3831 0.5289 0.4854 0.3556 0.6660 0.6431 0.2700 0.5934 0.5408 0.5406 0.4200 0.4526 0.3281 0.3268 0.3797 0.5844 0.2338 0.2850 0.5029 0.3870 0.4010

Fload(N) 51.5852 62.9112 75.6010 93.3152 117.1623 66.9610 79.5531 99.0014 90.7940 54.7536 86.0140 77.6820 95.0988 56.8674 70.7982 81.2058 100.9833 61.0584 74.0344 66.2388 105.7949 63.3105 58.5352 70.5320 84.0990

f(Fl) 0.2411 0.3365 0.4471 0.5945 0.7590 0.3718 0.4811 0.6380 0.5745 0.2671 0.5356 0.4650 0.6084 0.2848 0.4053 0.4952 0.6526 0.3206 0.4335 0.3655 0.6866 0.3400 0.2990 0.4029 0.5196

f(z) 0.6277 0.6514 0.6708 0.6959 0.7162 0.6217 0.4575 0.5043 0.7412 0.6074 0.4352 0.6833 0.5691 0.3708 0.4079 0.4946 0.5419 0.3401 0.2473 0.4481 0.2979 0.1901 0.3973 0.4306 0.2857

Sensitivity Measurement Based on Fuzzy Evaluation

It is found that the output factors, such as leakage, friction and load of ﬁnger seal, have different values and dimensions in simulations. It is difﬁcult to evaluate the sensitivity

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and importance of each input parameter to the output performance of ﬁnger seal. Hence, a satisfactory function based on fuzzy evaluation is presented. For leakage and friction, which are expected as small as possible, the satisfactory function of leakage and friction are deﬁned as follows respectively: _ ¼ ekm m_ ; f ðmÞ

km [ 0

ð14Þ

f ðFf Þ ¼ ekf Ff ;

kf [ 0

ð15Þ

2

2

Load capacity, the bigger the better, whose satisfactory function is set as f ðFl Þ ¼ 1 ekl Fl ; 2

kl [ 0

ð16Þ

Where, km, kf, kl is the coefﬁcient of leakage, friction and load, respectively, which relates the variation and acceptance level of each performance indicator. In this study the extreme values of the simulation results are used to determine the coefﬁcients ki, (i = m, f, l). For example, the max satisfactory function of leakage is set to 0.9 for the minimum leakage among the all results, and the min satisfactory function is set to 0.1 for the maximum leakage. Then we can get the range of km when the max and min satisfactory function is substituted into Eq. (14) respectively. The average is taken as the coefﬁcient km. Under the analysis conditions in this paper, km is calculated as 3.4967. By the same way, we can get the coefﬁcient kf as 26.9841. For load coefﬁcient kl, the max satisfactory function of load is set to 0.9 for the maximum load, and the min satisfactory function is set to 0.1 for the minimum load, then the load coefﬁcient kl, 1.0366 10−4, is obtained similarly. _ f(Ff), When the coefﬁcients, km, kf, kl are determined, the satisfactory function f ðmÞ, f(Fl), for every simulation set, could be calculated by Eqs. (14), (15) and (16) respectively. The satisfactory function results for every simulation sample are shown in Table 3. Then the range analysis of uniﬁed satisfactory function are used as quantitative indicators of each input variable’s importance, that is measure of sensitivity. For _ f(Ff) or f(Fl), the range of all samples for each input different output function, f ðmÞ, parameter is calculated. The relative importance of each of these input factors could be judged according to the order of the range value from the perspective of different seal performance. If one wants to know the influence of parameters on the integrated performance of ﬁnger seal, a multi-index synthesis satisfactory function, f(z), could be deﬁned by weighted arithmetic average of every single satisfactory function of seal performance, shown in Eq. (17). Deﬁnitely, the weight coefﬁcients should be determined in advance according to the importance of every indicator. f ðzÞ ¼

n X i¼1

wi f ðxi Þ

ð17Þ

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5.4

587

Sensitivity Analysis Results

Through the orthogonal table, the range analysis results for leakage, friction and load are shown in Tables 5, 6 and 7, respectively. Where, Ki is the average of satisfactory function for level i, i = 1–5. The range value is helpful to intuitively indicate how the input parameters impact the sealing performance. From Table 5, it is found that the order of the influence of parameters on leakage from big to small is: radial clearance (h), temperature(t), pressure drop(Dp), flow length(l), rotating speed(n), and ﬁnger number(s). Among these parameters, radial clearance gain the highest range value among the others. It means that the radial clearance has a high influence on the leakage of ﬁnger seal as we expected. Next, temperature, pressure drop and flow length also have important effects on leakage changes. Both rotating speed and ﬁnger number have least impacts on leakage relatively. As a result, radial clearance should be determined accurately which would influence the result of leakage sensitively. As we know, the radial clearance is varying in operating conditions, which will changed due to rotor run out, thermal deformation, hysteresis and blow down effect. So determination the accurate radial clearance of ﬁnger seal is challenging and important work whether in theory calculation or practice test. In addition, the flow length should be paid more attention in leakage analysis, however, in previous researches it was not considered in contacting ﬁnger seal leakage analysis.

Table 5. Range analysis of satisfactory function for leakage K1 K2 K3 K4 K5 Range Order

h 0.8786 0.7144 0.5336 0.3851 0.1922 0.6864 1

s 0.5498 0.5623 0.5437 0.5276 0.5205 0.0418 6

l 0.4643 0.5064 0.5360 0.5789 0.6182 0.1539 4

Dp 0.6886 0.6518 0.5430 0.4308 0.3897 0.2990 3

n 0.5652 0.5295 0.5397 0.5537 0.5157 0.0495 5

t 0.3637 0.4277 0.5481 0.6491 0.7152 0.3515 2

Table 6. Range analysis of satisfactory function for friction h 0.5943 K1 0.5358 K2 0.4730 K3 0.4143 K4 0.3619 K5 Range 0.2323 Order 2

s 0.4523 0.4781 0.4670 0.4957 0.4863 0.0434 6

l 0.5143 0.5060 0.4671 0.4616 0.4304 0.0839 4

Dp 0.6245 0.5444 0.4815 0.3943 0.3347 0.2898 1

n 2.2668 2.3623 2.4587 2.4444 2.3646 0.1919 3

t 0.5166 0.4826 0.4865 0.4499 0.4439 0.0727 5

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H. Su Table 7. Range analysis of satisfactory function for load K1 K2 K3 K4 K5 Range Order

h 0.4756 0.4665 0.4598 0.4535 0.4496 0.0260 3

s 0.4660 0.4551 0.4626 0.4581 0.4633 0.0110 6

l 0.2907 0.3692 0.4464 0.5425 0.6562 0.3655 1

Dp 0.3890 0.4220 0.4552 0.4976 0.5412 0.1522 2

n 0.4680 0.4574 0.4599 0.4560 0.4638 0.0119 5

t 0.4567 0.4623 0.4524 0.4698 0.4639 0.0173 4

In Table 6, we may rank the parameters according to the range value of friction satisfactory function from big to small as: pressure drop,radial clearance, rotating speed, flow length, temperature, and ﬁnger number. The ﬁrst three factors have similar effects on viscosity friction. It indicates that these parameters play important roles in thermal power loss consideration. Relatively, the impacts of flow length, temperature, and ﬁnger number on friction are small. From Table 7, it is seen that the influence of parameters on load from big to small is: flow length, pressure drop, radial clearance, temperature, speed, and ﬁnger number. The ﬁrst two parameters, especially flow length, play more signiﬁcant roles than others. Therefore, adding the size of axial flow length is the most efﬁcient to improve the ﬁlm load capacity. Such structure can be found in non-contacting ﬁnger seal with an extended pad along axial in LP. It is concluded that for different performance the relative importance of the input parameters are different. Among the 6 input factors studied here, radial clearance keeps in the top three, which is very important to ﬁnger seal performance. However, the influence of ﬁnger number is the least one relatively, which is always in the last place. While, it is worthy to note that ﬁnger number would have influence on ﬁnger stiffness which result in variation of hysteresis clearance. It may be considered in study the real radial clearance change of ﬁnger seal. Moreover, not only the structural parameters, but also the operating conditions would influence the performance of ﬁnger seal distinctively, sometimes the operating parameters are more sensitive to the performance of ﬁnger seal than structural factors. If we hope to know the importance of parameters to the whole performance of ﬁnger seal, the synthesis satisfactory function may be calculated by Eq. (17). Similarly, the range analysis is used to judge the order of importance of parameters to the integrate performance of ﬁnger seal. For example, the weigh coefﬁcient for leakage, friction and load is set to 0.4, 0.3 and 0.3, respectively. The satisfactory function, f(z), is listed in Table 4 and the range analysis results are shown in Table 8. The sequence of influence of parameters on the integrate performance from large to small is: radial clearance, pressure drop, flow length, temperature, rotating speed, and ﬁnger number. The results indicate that the radial clearance is a signiﬁcant parameter in ﬁnger seal, then pressure drop, flow length, temperature, are also closely related to the whole performance of ﬁnger seal. While rotating speed and ﬁnger number have little impacts on the ranking of integrate performance.

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Table 8. Range analysis of satisfactory function for integrate performance K1 K2 K3 K4 K5 Range Order

h 0.6724 0.5864 0.4933 0.4144 0.3203 0.3521 1

s 0.4954 0.5048 0.4963 0.4972 0.4931 0.0118 6

l 0.4272 0.4651 0.4885 0.5328 0.5733 0.1460 3

Dp 0.5795 0.5506 0.4982 0.4399 0.4186 0.1609 2

n 0.5025 0.4908 0.5014 0.5050 0.4873 0.0177 5

t 0.4375 0.4545 0.5009 0.5355 0.5584 0.1209 4

6 Conclusions With the same sealing diameter and manufacturing conditions, radial clearance, flow length and ﬁnger number can influence the geometry of leakage path of ﬁnger seal, which result in the variation of sealing performance combined with different operating conditions such as pressure drop, rotating speed and temperature. CFD simulation is an effective method to investigate the ﬁnger seal flow behavior with complicated and detailed geometry. The leakage, viscosity friction and ﬁlm load of ﬁnger seal are calculated by different combination of input parameters. The flow behavior is quite different whether the gas compressibility is considered even as the Mach number is less than 0.3. Then it is suggested that compressibility should be taken into account in course of numerical simulation of ﬁnger seal. Single factor method has advantage to get the influence rule of any parameter on the performance of ﬁnger seal qualitatively. The leakage and friction increase but the load decrease with increasing radial clearance. When the sealing diameter and radial clearance are constant, changing the ﬁnger number may influence the leakage and friction and load capacity, which cannot be shown by former simpliﬁed annular flow model. When adding the ﬁnger numbers the leakage and friction increase and the load decreases slightly. When the thickness of ﬁnger slice increases, the flow path is lengthened which results in the leakage decline and the friction and load capacity increase. Operating conditions such as pressure drop, rotating speed and temperature influence the ﬁnger seal flow behavior in different ways. The leakage, friction and load are found to be proportional to the pressure drop approximately. When temperature increase the leakage decreases, the friction and load increase, in contrast to what would be expected of liquids. The rotating speed affects the flow direction in gas ﬁlm which leads to the variation of leakage and friction at different speed. Parameters SA is necessary to judge the proportional impacts of multi input parameters on output indicators. A procedure to perform the SA combined with CFD, orthogonal design and fuzzy evaluation has been proposed. Using orthogonal design is helpful to select reasonable and uniform samples to deduce the computation amount. A kind of fuzzy evaluation function is presented to measure the sensitivity of parameters to different outputs. It is found that the same parameter has different sensitivity to leakage, friction or load performance, respectively. Leakage is most signiﬁcantly

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influenced by the radial clearance. While, temperature, pressure and flow length are also important and sensitive parameters to leakage secondly. Rotor speed and ﬁnger number have relatively small impact on leakage. In addition, the parameters SA to the friction and load capacity are analyzed in this research. It is found that pressure drop, radial clearance and rotating speed have more signiﬁcant impact on friction and the flow length influence the load capacity mostly. The parameter SA of the comprehensive performance of ﬁnger seal is conducted using integrated satisfactory function according to the expected weight coefﬁcient deﬁned to every output indicator. In the case presented, radial clearance is the most worthy concerned input parameter and signiﬁcant to overall ﬁnger seal performance. Among the six inputs parameters discussed here, the ﬁnger number has the least impact on ﬁnger seal performance. SA results allows analyzing quantitatively the contribution of each input parameter to the output performance variance, providing important insights to design ﬁnger seal effectively.

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14. Su, H., Lang, D.X., Qi, F.: Structural parameters sensitivity analysis of hydrodynamic ﬁnger seal with herringbone-grooved rotor based on fuzzy valuation. Lubr. Eng. 37(5), 7–12 (2012) 15. Su, H.H., Yao, Z.J.: Fuzzy analysis method for multi-orthogonal test. J. Nanjing Univ. Aeronaut. Astronaut. 36(2), 29–33 (2004) 16. Chen, K.: Design and Analysis of Experiments. Tsinghua University Press, Beijing (2005). (in Chinese) 17. Muleta, M.K., Nicklow, J.W.: Sensitivity and uncertainty analysis coupled with automatic calibration for a distributed watershed model. J. Hydrol. 306, 127–145 (2005)

Calculation and Experimental Study on Comprehensive Stiffness of Angular Contact Ball Bearings Peng Sun, Weifang Chen(&), Chuan Su, and Yusu Shen College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China [email protected], [email protected], [email protected], [email protected]

Abstract. The stiffness of the rolling bearing has a great influence on the dynamic characteristics of motorized spindle. In order to obtain the stiffness of angular contact ball bearing under oil-jet lubrication, the theoretical calculation model and experiment method are proposed in this paper. Based on hertz contact theory and elastohydrodynamic lubrication theory, the quasi-static analysis model of angular contact ball bearing is established. The factors including ball centrifugal force, gyro moment, preload and oil ﬁlm thickness are taken into consideration and theoretically analyzed. The test platform on comprehensive stiffness is designed and the stiffness is experimentally veriﬁed under different factors. The results show that the comprehensive stiffness increases with the increase of axial, radial load and preload but decreases with the increase of rotational speed. Therefore, the adjustment of the working condition of angular ball bearing can make it work in a better mechanical state and thus improve the bearing life. Keywords: Angular contact ball bearings Oil ﬁlm stiffness Contact stiffness

Comprehensive stiffness

1 Introduction Angular contact ball bearings are important driving and supporting parts in numerical control machines. In the process of supporting loads, the boundary conditions of bearings are constantly changing, so the description of the stiffness matrix is more complicated. Besides, as an elastomer, the mechanical properties of bearings play a key role in the dynamic characteristics of a motorized spindle system and can signiﬁcantly influence the dynamic performance of the motorized spindle system. Many scholars had done some research on this. Chen [1] regarded angular contact ball bearing as the research object and derived ﬁve-DOF expression of stiffness without considering factors such as oil ﬁlm or centrifugal force. Yu [2] established a calculation model of angular contact ball bearings according to the principle of elastohydrodynamic This project is supported by National Natural Science Foundation of China (Grant No. 51775277). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 592–601, 2020. https://doi.org/10.1007/978-981-32-9941-2_48

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lubrication and the result showed that the stiffness of bearing tends to decrease when considering the lubricating oil. Tang [3] built a quasi-dynamics analysis model with centrifugal force, gyroscopic moment and lubricating oil ﬁlm, which demonstrated that the contact stiffness and oil ﬁlm stiffness are not much different when the rotational speed of bearings reaches a certain level. Xiong [4] developed a calculation model of bearing considering lubricating oil, the deformation of ring, the effects of temperature and the centrifugal force. Other factors were also studied under different preload and speed condition. Shi [5] set up the test platform on axial stiffness of bearing, concluding that the hertz analytical solution is rather consistent with the measured value when the load value is small. However, as the load increases, the error will increase. Li [6] obtained the dynamic stiffness of the angular contact ball bearing by making use of the synchronous rotational radial load with unbalanced mass. However, the rotational speed and the radial load have an inter-related relationship. Besides, the test platform has certain limitations and can’t comprehensively analyzes the influencing factors. Fang [7] established the deformation expression of angular contact ball bearing with the coordinate transformation method and designed the relevant test platform. However, what the test platform measured is the absolute displacement of the bearing and can’t fully meet the exact requirement. Ali [8] and Lambert [9] obtained the radial stiffness of bearing by measuring the mass, acceleration and relative displacement of the inner and outer ring, which indirectly expresses the contact force between the shaft and the inner ring by the product of acceleration and mass of shaft and provides a new idea for solving the stiffness of bearing through the reasonable matching of sensor position and data processing method. At present, the existing mechanical model of bearings can realize the approximate calculation of stiffness, but the comprehensive factors have not been fully analyzed. It has not yet reached the precise level in the test part of stiffness. Due to the poor repeatability of the test results, it can not play a good guiding signiﬁcance. There are not many related references about the test part of the rolling bearing comprehensive stiffness. Thus, a perfect test plan has not yet been formed. Based on above investigation and survey, angular contact ball bearing will be carefully studied with an improved quasi-static model. Furthermore, a complete experimental scheme is proposed to verify the theoretical analysis.

2 Theoretical Calculation Model According to the hertz elastic contact theory [10] and elastohydrodynamic lubrication theory, analysis of the stress condition of angular contact ball bearings has been conducted. As shown in Fig. 1, the position change of inner and outer ring curvature center, before and after the angular contact ball bearing is loaded, has been showed in the schematic diagram. Since the outer ring and the housing are ﬁxed, point E and E′ coincide.

P. Sun et al. Radial

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X aj The end position of inner ring curvature center j-h

0.5 (f i -

(f o

-0. 5)D w

+u

ojh

Y rj

oj

αij

αoj α 0

O'

O

)

ui Dw+

I'

ij

I

urcosΨ

ua+θRicosΨ The first position of inner ring curvature center The end position of ball center

The first position of the ball center The first(end) position of outer ring curvature center

E (E')

Axial

Fig. 1. The position change of inner and outer ring curvature center before and after loaded

2.1

Geometric Equation

According to Fig. 1, for the ball in any position j, the following formula holds: 8 < Xaj ¼ LIE sin a0 þ ua þ hRi cos wj Y ¼ LIE cos a0 þ ur cos wj : rj Ri ¼ 0:5Dm þ ðfi 0:5ÞDw cos a0

ð1Þ

where a0 is the initial contact angle; LIE is the distance between inner and outer ring curvature center when the bearing is not loaded; ua, ur and h are respectively axial and radial deformation and rotation angle after the bearing is loaded; Ri is the radius of the circle where the inner ring curvature center is located; Dm is the diameter of the bearing pitch; Dw is the diameter of the ball; fi(o) is the curvature of inner (outer) groove. Considering the axial and radial load, the centrifugal force of balls, the gyro moment and the lubricating oil ﬁlm, the contact angle of the inner ring (outer ring is the same) can be expressed as: 8 < sin aij ¼ Xaj ½ðfi 0:5ÞDw þ uoj hoj sin aoj ðfi 0:5ÞDw þ uij hij : cos aij ¼ Yrj ½ðfi 0:5ÞDw þ uoj hoj cos aoj

ð2Þ

ðfe 0:5ÞDw þ uij hij

where uij and uoj are the hertz deformation between the jth ball and the contact point of inner and outer ring; hij and hoj are the oil ﬁlm thickness between the jth ball and the contact point of inner and outer ring. 2.2

Force Balance Equation Between the Ball and the Inner and Outer Ring

When the angular contact ball bearing rotates in high speed, in addition to the contact force generated by external load, the combined effect of centrifugal force and gyro

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moment are also applied on the ball. The force on the ball and the inner ring is shown in Fig. 2

Qoj F cj

λijM ij/D b

λojM oj/D b αoj

αij

Qij M gj ΣQij

F a0

j=1

ΣλijM ij/D b j=1

Fig. 2. The force on the ball and the inner ring

The force balance relationship is determined by the following formulas: (

Mgj Mgj Db cos aij þ koj Db cos aoj M M Qoj cos aoj þ kij Dgjb sin aij koj Dgjb sin aoj

Qij sin aij Qoj sin aoj kij

¼0

Qij cos aij

þ Fcj ¼ 0

ð3Þ

where Qij and Qoj are the forces between the ball and the inner and outer ring, they could be expressed as: (

Qij ¼ Kij u1:5 ij Qoj ¼ Koj u1:5 oj

ð4Þ

When the system is statically balanced, the radial load and bearing load on the inner ring of the bearing can be expressed respectively as: 8 Z P Mgj > > > Fa þ Fa0 ðQij sin aij þ kij Db cos aij Þ ¼ 0 > > j¼1 > > < Z P M Fr ðQij cos aij kij Dgjb sin aij Þ cos wj ¼ 0 > j¼1 > > > Z > P > M M > : M ½ðQij sin aij þ kij Dgjb cos aij ÞRi ri kij Dgjb cos wj ¼ 0

ð5Þ

j¼1

where Z is the number of balls; Fa0 is the initial preload; ri is the radius inner ring curvature. Equations (1), (2), (3), (4) can be combined as the ﬁrst group of equations, and (5) as the second group of equations, then they can be solved jointly by Newton-Raphson

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iterative method. Under the condition, the axial and radial deformation and the angle of the bearing could be solved and then the comprehensive stiffness of the bearing with these parameters can be obtained.

3 Test and Analysis 3.1

Test Platform

In order to verify the correctness of the theoretical model, a test platform of bearings on comprehensive stiffness is designed in this paper. As shown in Fig. 3, the test platform consists of ﬁve parts: power output system, spindle transmission system, axial and radial loading system, lubrication system and data acquisition system.

Fig. 3. The test platform of bearing on comprehensive stiffness

The axial loading and testing part includes a loader, a piezoelectric force sensor and a displacement sensor. The axial force is applied on the outer ring of the bearing through a cylinder by the axial loader and can be measured by the force sensor located between the loader and the cylinder. The displacement sensor is mounted in the center hole of the cylinder for measuring the axial displacement of the end face of the shaft. The value measured by the axial displacement sensor represents the relative displacement of the inner and outer ring since the inner ring and the shaft, the outer ring and the cylinder have the same displacement in the axial direction. The radial loading and testing part includes a loader, a piezoelectric force sensor, a displacement sensor and the housing. Similar to the principle of axial testing, the inner ring and the shaft, the outer ring and the housing have the same displacement in the radial direction and the radial value measured by the sensor which is embedded in the housing represents the relative displacement of the inner and outer ring in the radial direction. The supporting assembly ﬁxed on the baseboard is used to support for the transmission shaft. Several oil and gas holes and rotating shaft seals are provided on the outer surface of the housing for bearing lubrication.

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597

Principle

The test principle of comprehensive stiffness is shown in Fig. 4. The slowly increasing load in the axial and radial direction will be applied on the bearing, and then the physical output signal of each sensor is converted into an identiﬁable analog signal through the transmitter. After ﬁltering and other calculation, the actual forcedisplacement data can be obtained. Finally the axial and radial stiffness of bearings are obtained by deriving the force-displacement curve which is obtained by data ﬁtting. Radial loader

Force sensor

Data acquisition system Displacement sensor

Cylinder transmitter

analog signal

filter and other calculation

data fitting

comprehensive stiffness

transducer

Axial loader

Motor water tank

Force sensor

Coupling Supporting assmebly

housing

Displacement sensor

Fig. 4. The test principle of comprehensive stiffness

3.3

Result and Analysis

The type of the angular contact ball bearing selected in this paper is S7212AC. The basic parameters are shown in Table 1. Table 1. The basic parameters of S7212AC Parameter Outer diameter Inner diameter Width Ball number Elasticity modulus Density Ball diameter Initial contact angle Curvature of outer groove Curvature of inner groove Poisson’s ratio material

Symbol D d B Z E P Dw a fe fi l /

Value 110 mm 60 mm 22 mm 16 2.07 1011 Pa 7800 kg/m3 12.5 mm 25° 0.53 0.52 0.3 9Cr18

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The Utel dynamic signal acquisition device is used for data acquisition. The acquired displacement and force data are polynomial ﬁtted to obtain a ﬁtting curve. The slope of the curve represents the stiffness in a certain working condition. Figure 5 shows the stiffness comparison of the test processing result and the theoretical result.

(a) The stiffness comparison in radial direction

(b) The stiffness comparison in axial direction

Fig. 5. The stiffness comparison of the test processing result and the theoretical result

When the radial load changes from 0 to 5000 N, the test of comprehensive stiffness on the angular contact ball bearing under operating conditions is carried out. When the bearing is running, the displacement data is periodically fluctuated. The mean value of the displacement data corresponding to the related load is ﬁtted to obtain the forcedisplacement relationship of the angular contact ball bearing and then the bearing stiffness is obtained.

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As shown in Fig. 6, comparative tests have been done under different working conditions. It can be seen that the stiffness generally decreases with the increase of the rotational speed. And the larger the radial load at the same speed is, the larger the radial comprehensive stiffness becomes.

(a) Radial stiffness under different working conditions

(b) Axial stiffness under different working conditions

Fig. 6. The comparative test under different working condition

The comparative test of the comprehensive stiffness of angular contact ball bearings subjected to different preload has been carried out. As the preload increases, the contact load between the ball and the inner and outer ring increases when other factors remain unchanged, which will lead to the increase of the contact stiffness and account for why the axial and radial comprehensive stiffness increase in Fig. 7.

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(a) Radial stiffness under different preload

(b) Axial stiffness under different preload

Fig. 7. The stiffness under different preload

4 Conclusions In this paper, the calculation model of the comprehensive stiffness of angular contact ball bearings was established. The effects of external load, centrifugal force, rotational speed, lubricating oil ﬁlm, preload and gyro moment were fully considered. The test platform of comprehensive stiffness was built and the influence under different working conditions was analyzed. The main conclusions are as follows: (1) Under the premise that other factors remain unchanged, the axial and radial comprehensive stiffness of angular contact ball bearings increase with the increase of axial and radial loads. The maximum error between the theoretical radial comprehensive stiffness and the measured radial stiffness is 6.5%, while the maximum error between the theoretical axial stiffness and the measured axial stiffness is 13% and the error decreases with the increase of the load; (2) The comprehensive stiffness of the angular contact ball bearing will decrease when the elastohydrodynamic lubrication effect is considered; (3) With the increase of the rotational speed, the axial comprehensive stiffness of the angular contact ball bearing will decrease, for the oil ﬁlm thickness of the lubricating oil is reduced, which leads to the increase of the oil ﬁlm stiffness. Different from the axial comprehensive stiffness, the radial comprehensive stiffness is almost unchanged;

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(4) The increase of the preload will make the axial and radial comprehensive stiffness of the angular contact ball bearing increase.

References 1. Chen, S.J., Zhang, C.Y.: Analytical derivation and computer solution of bearing stiffness matrix. Bearing 2, 1–4 (2006) 2. Yu, Y.J., Chen, G.D.: Study on stiffness characteristics of high speed angular contact ball bearings considering elastohydrodynamic lubrication. J. Northwestern Polytechnical Univ. 1, 125–131 (2016) 3. Tang, Y.B.: Research on Mechanical Characteristics of Aeroengine High Speed Rolling Bearings. Nanjing University of Aeronautics and Astronautics (2005) 4. Xiong, W.L., Zhao, Z.S.: Research on dynamic stiffness of ball bearing considering the influence of ferrule deformation and lubrication. Chin. Mech. Eng. 26(11), 1421–1428 (2015) 5. Shi, S.C., Wu, J.W.: Axial stiffness analysis and experiment of thin-wall angular contact ball bearings. J. Harbin Eng. Univ. 44(7), 32–37 (2012) 6. Li, C.J.: Experimental study on dynamic stiffness of angular contact ball bearings. J. Xi’an Jiaotong Univ. 47(7), 68–72 (2013) 7. Fang, B., Zhang, L.: Theoretical calculation and experiment of angular contact ball bearing stiffness. J. Jilin Univ. (Eng. Edn.) 42(04), 840–844 (2012) 8. Ali, N.J., García, J.M.: Experimental studies on the dynamic characteristics of rolling element bearings. ARCHIVE Proc. Inst. Mech. Eng. Part J J. Eng. Tribol. 224(7), 659–666 (2010). 1994–1996 (vol. 208–210) 9. Lambert, R.J., Pollard, A., Stone, B.J.: Some characteristics of rolling-element bearings under oscillating conditions. Part 1: theory and rig design. Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn. 220(1), 171–179 (2006) 10. Harris, T.A., Mindel, M.H.: Rolling element bearing dynamics. Wear 23(3), 311–337 (1973) 11. Huang, Z.Q.: Design and Calculation of Ball Beaing. Mechanical Industry Press (2003)

Design of a New Hydraulic Manipulator with Kinematic and Dynamic Analysis Yao Sun1, Yi Wan1(&), Xichang Liang1, Xin Huang1, and Ziruo Liu2 1

Key Laboratory of High-Efﬁciency and Clean Mechanical Manufacture, College of Mechanical Engineering, Shandong University, Jinan 250061, China [email protected], [email protected] 2 National Demonstration Center for Experimental Mechanical Engineering Education, Shandong University, Jinan 250061, China

Abstract. Since the traditional hydraulic manipulator is heavy and serious in stress concentration, a new 6-DOF hydraulic manipulator was presented with high deadweight ratio and its 3D solid model was constructed. The kinematics of the manipulator was analyzed according to the structural characteristic and the kinematic constraint. A forward kinematics model of the manipulator was conducted, and the complete analytical solution of the inverse kinematics was obtained. In addition, a dynamic analysis method based on Lagrangian equation is used to optimize the structure of hydraulic manipulator. The application of Lagrangian equation to dynamic modelling and force analysis leads to an elegant and geometrically meaningful formulation. The method is implemented for the hydraulic manipulator and simulation results are provided. All the results show that the designed manipulator have a better Kinematic characteristic and work performance. Keywords: Hydraulic

Manipulator Kinematic analysis Dynamic analysis

1 Introduction Manipulator is the most widely used mechanical device in the ﬁeld of robot at present. It is used to replace manual operation in automobile manufacturing, electronic and electric industries, which greatly reduces labor cost, and effectively improves product quality [1]. Most traditional mechanical arm are designed for a single application, used to complete a speciﬁc task, and with the method of motor and reducer, the carrying capacity is relatively small. Therefore, a new type of drive mode of manipulator which has a large carrying capacity and high reliability is developed. This research has designed a hydraulic mechanical arm with light weight, small inertia, large output and stiffness. In addition, the pressure of the hydraulic system is adjustable, which can improve the output characteristics, such as increase or decrease output power and deadweight ratio. In particular, it needs to be point out that carrying capacity of

This project is supported by the Fundamental Research Funds of Shandong University (2017JC041) and the Key Research Project of Shandong Province (2017CXGC0917). © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 602–614, 2020. https://doi.org/10.1007/978-981-32-9941-2_49

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manipulator, an important indicator, is one of important standards to judge the robot arm performance. The BigDog from Boston Dynamics company which use hydraulic system as driving power is mainly used for rugged handling work. The robot, used for military or engineering, has a outstanding sport ability. The hydraulic actuator of BigDog that together with four institutions and based on the two characteristics of hydraulic transmission is chosen as the joint driving element. Hydraulic load capacity of the hydraulic system depends on pressure and the size of actuator, and its speed hinge on flow of the hydraulic system [2–6]. In order to achieve automation of the hydraulic manipulators, the automation of excavators [7, 8], dumpers, cranes and telescopic handler [9, 10] have been developed. Dynamics and kinematics research are the hotspot of robot research [11–17]. The kinematic algorithm can meet the real-time requirements of general robot control system [18]. However, dynamic and kinematic analysis is an important research direction in the automation of hydraulic machines, especially for manipulator. Through the analysis of dynamic and kinematic, we can obtain the output characteristics including motion performance and working performance, which are important standards of hydraulic robot. In this paper, we propose a new 7-DOF redundant manipulator that has better flexibility and higher deadweight ratio. The double-manipulator rescue robot, which have two hydraulic mechanical arms, only weight 4 tons, but have a load capacity of 2 tons and 10 mm motion accuracy. In order to improve the velocity and force properties of the hydraulic manipulator, we need to analyze kinematic and dynamic of the mechanical arm. Then we carried out ADAMS analysis of the manipulator to verify the correctness of the dynamic analysis. The results of speed and force curves show that the hydraulic manipulator of the double-rescue robot has a good speed and force properties, which is suitable for rescue mission.

2 Kinematics 2.1

Base System Model

In order to meet the requirement of the mobile flexibility of the manipulator, the mechanical structure adopts the design idea of redundant degrees of freedom. Since human arm has good flexibility, mechanical arm is designed according to ergonomic principles. Human arm is a typical 7-DOF redundant mechanism that can achieve four basic forms of motion: flexion, extension, rotation and translation. Studying the structure of human arm is conducive to better design of related anthropomorphic robot arm. However, it should be noted that the bone structure of human body cannot be directly applied to the design of mechanical arm, which needs reasonable and practical improvement. Hydraulic cylinder is taken instead of arm muscle in hydraulic manipulator. Considering the feasibility of installing the hydraulic drive components, a new 7DOF

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Fig. 1. The double-manipulator rescue robot

Fig. 2. Geometric structure of the manipulator

Fig. 3. Geometric structure of swing freedom

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hydraulic manipulator, combined with the human arm conﬁguration, is proposed as Fig. 2, which can achieve basic movement of human arm. The design of the wrist joint adopts the double-cylinder collaborative mode, so that the wrist has the structure of abduction, with 2-DOF includes swing and rotation. The wrist of the hydraulic manipulator shows in Fig. 3. The control system of hydraulic manipulator is based on electro-hydraulic proportional control technology design, which can control the flow of hydraulic system through electrical signals and flow direction. The hydraulic system of the mechanical arm includes hydraulic pump, electromagnetic proportional valve, hydraulic cylinder, motor and other components. hydraulic cylinders and motors are driving parts, including six hydraulic cylinders and two motors. The six hydraulic cylinders include 5DOF and three types of action, that is rotation, pitch and swing. And the two hydraulic motors correspond to the rotational motion of the wrist joint and base joint, respectively. 2.2

D-H Coordinate System

The kinematic equations of manipulator have varieties of methods that like early space vector polygon method and Quaternion Algorithm. The current common methods include: homogeneous coordinate, Denavit-Hartenberg Matrix and screw theory. The forward kinematics equation is solved by Denavit-Hartenberg Matrix method. The double-manipulator rescue robot is shown in Fig. 1. Each manipulator of the robot has 7-DOF, and the two mechanical arms have 14 degrees of freedom in total. In order to obtain larger unit torque, the double-manipulator robot is driven by hydraulic. the hydraulic driven system, which includes motors and serve values, is inside the doublemanipulator rescue robot. In this paper, the mathematical model is extracted from the entity model of the manipulator, according to the method of D-H Matrix. The coordinate system of the connecting rod is shown in Fig. 4. The coordinate system {0} is the base coordinate, the coordinate system {7} is the end effector coordinate of the mechanical arm, and the coordinate system {1} to {5} is the link coordinate system.

Fig. 4. D-H coordinate system of the manipulator

Firstly, the rod and connection mode of the mechanical arm are deﬁned. Torsion angle a, rod length l, joint angle h and joint distance d were used to describe each link. Torsion angle a and bar length l are used to describe the characteristics of the ith link

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itself, and joint angle h and joint distance d are used to represent the relationship between the ith and i + 1 bar. 2.3

Kinematic Equation

The articular angles of the manipulator is given in Table 1, which includes the angle between the x-axis of the adjacent coordinate system. In this section, the end effector position of the manipulator in coordinate system relative to the base coordinate system is solved. Table 1 deﬁnes the parameters of solving the forward kinematics of the manipulator that the angle values of each joint are deﬁned or given.

Table 1. Notation di hi li ai qi M R P

Distance between the x axes in the adjacent coordinate system Angle between the x axis in the adjacent coordinate system Distance between the z axis in the adjacent coordinate system Angle between the z axis in the adjacent coordinate system Variable of the ith joint Position of the manipulator in the D-H coordinate system Position matrix of the end-effector coordinate system relative to the base coordinate system Position vector of the end-effector coordinate system relative to the base coordinate system

The math model of kinematic equation is given as follow: X M¼ f ð qi Þ

ð1Þ

The homogeneous transformation matrix from system {0} to system {1} is given as Eq. (2): M01 ¼ Rotðz; h1 Þ Transð0; 0; l1 Þ Rotðx; a1 Þ

ð2Þ

Where, M01 is the homogeneous transformation matrix. Rot(x, a1) represent {0} coordinate system rotates about the x0 axis by a1 degree. Trans(0, 0, l1) represent {0} frame shift along the z0 axis by l1. Similarly, M12 ¼ Rotðz; h2 Þ Transðl2 ; 0; 0Þ

ð3Þ

M23 ¼ Rotðz; h3 Þ Transðl3 ; 0; 0Þ

ð4Þ

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M34 ¼ Rotðz; h4 Þ Transðl4 ; 0; 0Þ

ð5Þ

M45 ¼ Rotðz; h5 Þ Transðl5 ; 0; 0Þ Rotðz; h6 Þ Rotðx; 90Þ

ð6Þ

M56 ¼ Rotðz; h7 Þ Transðl6 ; 0; 0Þ

ð7Þ

The coordinate of the terminal position of the hydraulic manipulator can be obtained by continuous multiplication of six homogeneous coordinate transformation matrices, so the homogeneous transformation matrix:

M06 ¼ M01 M12 M23 M34 M45 M56

X ¼ 0

P 1

ð8Þ

where P ¼ ½ px py pz T . Input the D-H parameters into Eq. (8), we can obtain kinematic equation of the hydraulic manipulator. Then, the transformation equation of the end effector coordinate system relative to the base coordinate system is obtained by further solving: nx ¼ cos h1 cos x6 cos h7 þ sin h1 sin h7

ð10Þ

ny ¼ sin h1 cos x6 cos h7 þ cos h1 sin h7

ð11Þ

nz ¼ sin x6 cos h7

ð12Þ

ox ¼ cos h1 cos x6 sin h7 þ sin h1 cos h7

ð13Þ

oy ¼ sin h1 cos x6 sin h7 cos h1 cos h7

ð14Þ

oz ¼ sin x6 sin h7

ð15Þ

ax ¼ cos h1 sin x6

ð16Þ

ay ¼ sin h1 sin x6

ð17Þ

az ¼ cos x6

ð18Þ

px ¼ cos h1 ðl6 cos x6 l5 cos x4 þ l4 cos x3 þ l3 cos h2 þ l2 Þ

ð19Þ

py ¼ sin h1 ðl6 cos x6 l5 cos x4 þ l4 cos x3 þ l3 cos h2 þ l2 Þ

ð20Þ

pz ¼ l6 sin x6 l5 sin x4 þ l4 sin x3 þ l3 sin h2 þ l1

ð21Þ

Where xi ¼

i P

hi .

j¼2

The matrix R and P can be solved according to the above transformation equation. Given a vector of joint angles hðh1 ; ; h7 Þ, we can obtain the end effector position vector of the manipulator

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T P ¼ px ; py ; pz

ð22Þ

and attitude matrix 2

nx X ¼ 4 ny nz

2.4

ox oy oz

3 ax ay 5 az

ð23Þ

Joint Variable Analysis

Different from the manipulator driven by motor, the hydraulic manipulator is driven by hydraulic cylinder, whose joint parameters cannot be given directly, but can be calculated by the stroke of the hydraulic cylinder and the length of the arm structure. Therefore, for hydraulic manipulator, joint parameters need to be analyzed and calculated. In order to calculate joint parameters, we take joint 2 as an example. Joint 2 model diagram and structure diagram are shown in Fig. 5.

Fig. 5. Structure of the 2th joint

According to the geometric relationship, the following equation can be obtained: /2 ¼ arccos

jOM j2 þ jON j2 jMN j2 jAM j þ arcsin 2jOM jjON j jOM j h2 ¼ /2 þ b2

p 2

ð24Þ ð25Þ

Where jMN j 2 ½h; h þ a. joint range can be obtained: "

jOM j2 þ jON j2 h2 jAM j jOM j2 þ jON j2 ðh þ aÞ2 jAM j ; arc þ arcsin þ arcsin q2 2 arccos 2jOM jjON j 2jOM jjON j jOM j jOM j

#

ð25Þ

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Where, OA, OB, ON, OM, b are known constants, MN is a variable parameter, the value range of u2 is the joint range q2 of the hydraulic manipulator. The relationship between the joint variable h2 and the hydraulic cylinder stroke a is also obtained. Similarly, the range of other joints variables qi and hi of the manipulator can be calculated.

3 Inverse Kinematic The purpose of inverse kinematic is to obtain the trajectory of each joint after determining expected trajectory of the end effector. In order to solve joint parameters, we should determine position vector of end effector ﬁrstly. In other words, the inverse motion analysis of the manipulator is to calculate the joint parameters hðh1 ; ; h7 Þ from the known end effector position vector P. according to Eq. (26) p px ; py ; pz f fh1 ; h2 ; h3 ; h4 ; h5 ; h6 ; h7 g ¼ 0

ð26Þ

The obtained hðh1 ; ; h7 Þ is the solution of the inverse motion calculation of the manipulator. However, in actual working environment on site, it is difﬁcult to set Eq. (26) to be exactly zero. The inverse solution obtained from Eq. (26) may have multiple solutions, and what we need is the optimal solution. We deﬁne G(h) as follows: GðhÞ ¼ min

7 X

j hi j

ð27Þ

i¼1

In order to obtain the optimal solution, we use penalty function method:

Fðx; MÞ ¼ min f ðhÞ þ M GðhÞ VðhÞ ¼ max f_ ðhÞ

ð28Þ

Where v(h) is the rotational velocity of the revolute joint. M is the penalty function playing the role of “punishment”. The penalty function can ﬁrst take the value of 1 to calculate the optimal solution x_ of F(x, M). If x_ does not meet the constraint conditions, the value of the penalty function M will be magniﬁed by 10 times and repeated in turn until the solution satisfying the constraint conditions is obtained.

4 Dynamic Analysis 4.1

Lagrangian Dynamics

Dynamic model is derived by Jacobian matrix. Lagrange equation is shown as Eq. (29).

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d dt

@L

!

@ hi

@L ¼ si @hi

ð29Þ

Where L¼KP

ð30Þ

K and P present the kinetic energy and potential energy of the system, respectively. hðh1 ; ; h7 Þ presents generalized coordinate system. According Eq. (29), we can obtain simpliﬁed formula as follows:

M ðhÞ€h þ H h; h_ þ GðhÞ ¼ s

ð31Þ

Where in Eq. (30), M is a symmetric positive deﬁnite matrix 2

M11 6 .. MðqÞ ¼ 4 . M71

3 M17 .. 7 . 5

ð32Þ

M77

_ presents generalized force vector of Coriolis force and centrifugal force. where, Hðh; hÞ GðhÞ ¼ ½P1 P6 T presents gravity vector matrix. Derived from the Jacobian matrix, we can obtain Eq. (32). R_ 7 ¼ J7 h_ R_ 7 ¼ P_ 7

x7

T

¼

x_ 7

y_ 7

ð33Þ z_ 7

a_ 7

b_ 7

c_ 7

ð34Þ

Equation (33) presents the terminal velocity and angular velocity of the manipulator. Differentiate Eq. (32), we can obtain Eq. (33): dR7 ¼ J7 dh

ð35Þ

dh is virtual joint displacement, dR7 is virtual displacement and virtual angular displacement of the manipulator at the terminal. According to the virtual displacement principle, the sum of the virtual work done by each joint is equal to the virtual work done by the end of manipulator. sT dh ¼ F7T dR7

ð36Þ

Where F7 ¼ ½ F7X F7Y F7Z s7X s7Y s7Z T . F7 is the force and torque of the terminal of manipulator. Input Eq. (34) into Eq. (35):

Design of a New Hydraulic Manipulator with Kinematic and Dynamic Analysis

sT dh ¼ F7T J7 dh

611

ð37Þ

According Eqs. (29) to (37), we can obtain the equation as follows: M ðqÞ€q þ H ðq; q_ Þ þ GðqÞ ¼ J7T F7

4.2

ð38Þ

ADAMS Simulation

ADAMS software is selected for dynamic simulation in this paper. Firstly, the virtual prototype model of manipulator is established. Then, add revolute pair and driven force on the model. In order to make the simulation more consistent with the actual situation, we add friction at the rotation (Figs. 6 and 10).

Fig. 6. The manipulator model in ADAMS

Figure 7 shows the displacement curves of the end effector of the doublemanipulator rescue robot, which includes three directions. Figures 8 and 9 shows velocity and acceleration curve of the end effector, respectively. In the ﬁrst 4 s of applying force, the speed of the manipulator increases steadily. It reaches maximum speed at the 4th second. The displacement of the manipulator steadily increases after 4 s, then it turns into decrease. As it shows in Figs. 7, 8 and 9, we can come to the conclusion that the driving force of each hydraulic cylinder is stable. And the torque increases or decreases steadily that is basically the same as the displacement curve during the movement of the manipulator. The above indicates that the hydraulic manipulator has a good force and velocity performance.

Fig. 7. Displacement curve

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Fig. 8. Velocity curve

Fig. 9. Acceleration curve

Fig. 10. Force curve of the end-effector

5 Conclusions After the above theoretical analysis and counter seismic calculation, the following conclusions are obtained: (1) The designed manipulator has 7-DOF, which can simulate the movement of human arm and keep the dynamic stability that proves the rationality of manipulator structure. (2) The D-H method was used to establish the kinematics model of the 7-DOF hydraulic manipulator, and the position matrix of the end-effector was obtained.

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Aiming at the motion planning problem of redundant hydraulic manipulator, the inverse kinematics solution is solved. On the basis of the commonly used methods at present, considering the joint angle limit of the manipulator, the avoidance of singular form and the accuracy of the solution, the inverse kinematics solution based on penalty function method is proposed. (3) Based on the Lagrangian dynamics analysis, the mathematical model of joint torque is derived, and the robot control system can be optimized according to the dynamics parameters, which provides a basis for the closed-loop servo control of hydraulic manipulator in the future. (4) Using the ADAMS software for the dynamic simulation analysis of hydraulic manipulator obtained the joint torque as well as the end-effector of velocity and force curve, from the point of the ﬁnal result, manipulator parameters meet the use requirement, although there are some error, but will not affect the realization of the kinematics and dynamics characteristics of manipulator.

References 1. Choi, J.: Robust position control of electro-hydrostatic actuator systems with radial basis function neural networks. J. Adv. Mech. Des. Syst. Manuf. 7(2), 257–267 (2013) 2. Nelson, G., Saunders, A., Neville, N., et al.: Petman: a humanoid robot for testing chemical protective clothing. J. Robot. Soc. Jpn. 30(4), 372–377 (2012) 3. Dedonato, M., Dimitrov, V., Du, R.X., et al.: Human-in-the-loop control of a humanoid robot for disaster response: a report from the DARPA robotics challenge trials. J. Field Robot. 32(2), 275–292 (2015) 4. Fallon, M., Kuindersma, S., Karumanchi, S., et al.: An architecture for online affordancebased perception and whole-body planning. J. Field Robot. 32(2), 229–254 (2015) 5. Ding, L.: Key technology analysis of big dog quadruped robot. J. Mech. Eng. 51(7), 1–23 (2015) 6. Wu, W.G.: Research progress of humanoid robots for mobile operation and artiﬁcial intelligence. J. Harbin Inst. Technol. 47(7), 1–19 (2015) 7. Kim, Y.B., Ha, J., Kang, H., et al.: Dynamically optimal trajectories for earthmoving excavators. Autom. Constr. 35, 568–578 (2013) 8. Kim, D., Kim, J., Lee, K., et al.: Excavator tele-operation system using a human arm. Autom. Constr. 18(2), 173–182 (2009) 9. Park, J.Y., Chang, P.H.: Vibration control of a telescopic handler using time delay control and commandless input shaping technique. Control Eng. Pract. 12(6), 769–780 (2004) 10. Činkelj, J., Kamnik, R., Čepon, P., et al.: Closed-loop control of hydraulic telescopic handler. Autom. Constr. 19(7), 954–963 (2010) 11. Zhao, Y.N., Gao, F., Hu, Y.: Novel method for six-legged robots turning valves based on force sensing. Mech. Mach. Theor. 133, 64–83 (2019) 12. Wu, G.L., Bai, S.P.: Design and kinematic analysis of a 3-RRR spherical parallel manipulator reconﬁgured with four–bar linkages. Robot. Comput.-Integr. Manuf. 56, 55–65 (2019) 13. Andrej, C., Olav, E.: Dynamic modelling and force analysis of a knuckle boom crane using screw theory. Mech. Mach. Theor. 133, 179–194 (2019)

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14. Zhang, D.S., Xu, Y.D., Yao, J.T., Zhao, Y.S.: Design of a novel 5-DOF hybrid serial-parallel manipulator and theoretical analysis of its parallel part. Robotics and Computer-Integrated Manufacturing 53, 228–239 (2018) 15. Liu, Q., Ge, W.M., Wang, X.F., Zhang, H.Y.: Dynamics modeling and simulation of constrained flexible load with manipulator operation. 42(3) (2018) 16. Han, Y., Wu, J.H., Liu, C., Xiong, Z.H.: Static model analysis and identiﬁcation for serial articulated manipulators. Robot. Comput.-Integr. Manuf. 57, 155–165 (2019) 17. Liang, X.C., Wan, Y., Zhang, C.R., Kou, Y.Y.: Design of position feedback system with data fusion technology for large hydraulic manipulator. In: Advances in Mechanical Design, pp. 349–361 (2017) 18. Wan, Y., Kou, Y.Y., Liang, X.C.: Closed-loop inverse kinematic analysis of redundant manipulators with joint limits. Mech. Mach. Sci. 55, 1241–1255 (2018)

Research on Electro-Hydraulic Servo System Based on BP-RBF Neural Network Tao Chen, Wenqun Zhang, and Jianggui Han(&) College of Power Engineering, Naval University of Engineering, Wuhan 430000, Hubei, China [email protected], [email protected], [email protected]

Abstract. Aiming at the accuracy of the severe interference control system commonly used in marine electro-hydraulic servo system, a nonlinear model of the control system is established. In order to control the excess force, a composite PID controller with BP neuron algorithm is proposed. The integration and setting of parameters were carried out by neurons. The tracking ability of RBF neurons was used to construct a PID composite tracker, and simulation experiments were carried out to show that the relevant parameters were self-corrected. The controller has high control stability and fast speed. The other characteristics signiﬁcantly suppress the excess force in the electro-hydraulic servo system, thus effectively optimizing the stability of the entire electro-hydraulic servo control system against external load force interference. Keywords: Electro-hydraulic servo system PID controller BP neural network

Redundant force control

1 Introduction The rudder electromechanical servo system has the characteristics of high precision, high speed and high power. It is widely used in the marine and aerospace sectors. One of the applications is the electro-hydraulic servo cooperative control system. Suppressing excess power is a difﬁcult problem between them. There are many ways to solve this problem, such as the single pendulum load suppression method with single pendulum load drive frequency and load frequency [1–3], the dual valve parallel compensation scheme to meet the double ten index of the loading system [4, 5], or the double differential structure decoupling. Compensation control signiﬁcantly reduces excess force [6]. There is also the problem of solving the problem that the nonlinear excess force is too large by the composite controller, which succeeds in reducing the original ±22N to about ±4N [7]. In this paper on the basis of establishing the nonlinear control problem of composite controller PID control, the idea of negative feedback correction is adopted. The PID composite control module is constructed by BP-RBF neural network unit to complete the optimization function of the electro-hydraulic servo system [8]. The simulation data shows that the nonlinear model monitored by the PID module shows good monitoring accuracy and successfully produces less than 10% of the given input, thus completing the effective suppression of the excess force in the electro-hydraulic © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 615–622, 2020. https://doi.org/10.1007/978-981-32-9941-2_50

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servo system. The suppression of the excess force of the servo system provides new help and method.

2 Mathematical Modeling of Electro-Hydraulic Servo System The rudder electromechanical servo system is mainly composed of oil source, ﬁlter (sparse ﬁlter, ﬁne ﬁlter), overflow valve, electro-hydraulic servo valve and oil cylinder. According to the needs of experimental simulation, therefore, the system structure is properly idealized, and only the mathematical model of the key parts is considered. Figure 1 is a schematic diagram of the principle of the rudder electromechanical servo system.

Fig. 1. Schematic diagram of electro-hydraulic system

In The flow equation for the nonlinearization of the electro-hydraulic servo valve is shown in the following formula. 8 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pS pL > > ; Xv 0 xx > C < v v q QL ¼ Kq ðpL ÞXv ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > pL pS > > ; Xv \0 C xx : v v q

ð1Þ

In the formula, the load flow is QL , the flow gain is kq , the load pressure drop is pL , the spool displacement is xv , the flow coefﬁcient is Cv , the area gradient is x, the oil supply pressure is PS , and the liquid density is q. The total flow of the hydraulic cylinder is composed of the flow required by the propulsion cylinder piston, the total compression flow, and the total leakage flow. The continuity equation of the total flow of the hydraulic cylinder:

Research on Electro-Hydraulic Servo System Based on BP-RBF Neural Network

QL ¼ Ap

dxt Vt dpL þ þ Ct pL dt 4be dt

617

ð2Þ

In the formula, the effective area of the piston is AP , the displacement of the piston is xt , the total volume of the two oil chambers is Vt , the comprehensive elastic modulus of the system is be , and the total leakage coefﬁcient of the hydraulic cylinder is Ct . The force balance equation is: Ap pL ¼ m

d 2 xP dxP þ FL þ kxP þ Bp dt2 dt

FL ¼ kP ðxP x0 Þ

ð3Þ ð4Þ

Among them, the total mass of the piston and the load is m, the viscous damping coefﬁcient of the piston and the load is BP , the spring stiffness of the load is k, and any external load force acting on the piston is FL . The second-order part is mainly composed of the leakage flow, the quality of the valve core, and the compressibility of the liquid. The natural frequency of the hydraulic pressure is large, so there is no influence on the dynamic performance of the valve. On the other hand, the open-loop gain coefﬁcient of the pressure feedback loop is the maximum. It is also much smaller than the minimum value of the open-loop gain coefﬁcient of the force feedback loop, so it can be ignored, so its transfer function is: GðsÞ ¼

Ksv 2 SðxS2 sv

þ

2nsv xsv

S þ 1Þ

ð5Þ

In this formula, the speed ampliﬁcation factor is Ksv , the natural frequency of the second-order factor of the closed-loop system is xsv , the damping ratio is nsv .

3 Basic Principles of BP and RBF Neural Network The error back propagation algorithm relies on the idea of least squares in mathematics. Through the gradient search scheme, the difference between the true value of the control system and the ideal value is averaged, and the mean square is minimized. The input value of the BP neural network system passes through the input segment, the recessive segment and the output segment at the same time, and the neuron mesh of each segment only controls the characteristics of the next segment. When the expected output of the output segment does not conform to the actual output error, then the reverse propagation is performed, and the correlation weight and the closed value of each layer of neurons are changed, so that the error function is reduced to the negative gradient side, thereby achieving the true output value and The mean square between the ideal output values is the smallest. The radial basis function, the neural network contains a single recessive layer of three-layer pre-feedback network, is a partially approaching neural network. The characteristic of the RBF network is that only a few connection weight segments in a small area of the input area determine the output of

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the network, and ﬁnally the local network has the advantages of fast learning speed and high precision.

4 Design of PID Controller Based on BP-RBF Identiﬁcation In order to obtain a high-speed, high-precision output load force for the load control system, a series of optimization measures are needed to overcome the excess force. In order to make the PID controller complete excellent and precise control functions, this paper uses BP neural network to integrate and set related parameters for the PID controller to achieve three parameters (xd , xi xp ) and dynamic self-stability. The role of the RBF neural network is to identify the state of the monitored person, thereby inputting its state to the control system to optimize the accuracy of the control system. Finally, after RBF neural network identiﬁcation is used to assist the optimization of the entire PID controller system, the control stability and accuracy of the system are signiﬁcantly enhanced [6]. The abbreviated algorithm of the PID controller control system based on BP neural network identiﬁcation based on RBF neural network is divided into the following steps: ① Set the basic structure of the BP neural network, that is, set the number of input segment points N and the number of recessive segment nodes R, and set the initial segment of each segment weighting coefﬁcient y1ij ð0Þ, Select the learning efﬁciency k and set the inertia coefﬁcient b to get x ¼ 1. ② Set the input segment node in the RBF network a, the number of recessive segments C, and set the vector of the recessive segment dj ð0Þ, the vector initial value of the base band cj ð0Þ, the weighting coefﬁcient yj ð0Þ, the learning efﬁciency k1 and the inertia coefﬁcient b1 to get x ¼ 1. ③ Extract the parameter to get rinðxÞ and youtðxÞ, the calculated instantaneous error is eðxÞ ¼ rinðxÞ youtðxÞ. ④ According to the formula, the output of each segment of the BP neural network is obtained. The result of the BP neural network output segment is the relevant basic parameters (xd , xi , xp ) of the PID controller. Calculate the result value of its PID controller is uðxÞ, at the same time, uðxÞ is output to the RBF recognition neural network and the electro-hydraulic control system, and the output value youtðx þ 1Þ of the control system is calculated. ⑤ According to the formula, the value of each segment of the neurons in the RBF recognition neural network is ymoutðx þ 1Þ. ⑥ The gradient reduction method after optimization is used to adjust the weighting coefﬁcient and learning efﬁciency of the neural network. ⑦

Fig. 2. PID block Diagram

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Network learning is adopted for the control system, and the weighting coefﬁcients y1ij ðxÞ and y2ij ðxÞ are optimized in real time, thereby completing the self-optimization function of the PID controller related data. Last reset, loop to ① (Fig. 2).

5 Simulation Analysis The composition of the RBF neural network is 3-6-1, and the BP is 3-4-3. At the same time, the given values are sum, and then the ideal curve of the comparison is compared with the theoretical curve. Figure 3 is a given three output curves. Comparing the nonlinear output curve with the linear output curve, it can be seen that when the system is at low frequency, the system is found to have a good tracking and monitoring function; at the same time, the comparison curve of Fig. 3 can be found that the nonlinear curve has a large difference from the linear curve. The change of the nonlinear curve is more stable, and the optimization of the theoretical output curve is good. In order to compare the optimization results of the electro-hydraulic control system with the non-linear control algorithm assisted by the composite controller PID, as shown in Fig. 4, the image of the PID module compared with the general electrohydraulic servo system is obtained.

Fig. 3. Nonlinear, theoretical, and linear output

According to the comparison output in the ﬁgure, with the help of the composite module PID, the optimized excess force is 1.5N, while the general linear electrohydraulic servo system has a peak force of 12N, which is found to be negative for the

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Fig. 4. Excess force simulation curve

excess force of the PID module. The feedback is higher than 87.5%, which has a more signiﬁcant optimization effect.

6 Experimental Veriﬁcation The experimental setup includes a loading test rig, a force object, an electrohydraulic servo system, a pump source, and corresponding mechanical and electrical connections [9]. The test oil source pressure is 17 MPa, and the test environment is room temperature. The motion track of the stressed object is SðtÞ ¼ 4 sin pt, and load instruction is 0N. When the electro-hydraulic system inputs a sinusoidal signal with a amplitude of 0.08 m and a frequency of 0.5 Hz, the results of the excess force experiment using the ordinary PID control method and the BP-RBF-based composite control method are shown in Figs. 5 and 6. It can be seen from the experimental results that the excess force is obviously reduced after the introduction of the composite controller, and we also found that when the system turns, the excess force will be jagged [10].

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Fig. 5. The experimental curve without BP-RBF

Fig. 6. The experimental curve with BP-RBF

7 Conclusions In this paper, the ship rudder electro-hydraulic servo system is taken as the research subject, and a high-precision nonlinear model is constructed. According to the experimental results, it can be seen that the system works in the nonlinear low

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frequency band and the characteristics in the linear low frequency band are basically the same, but high. The tracking effect during band operation is more accurate. When the sinusoidal signal is input through the BP-RBP composite PID controller, the actual error of the system with the general controller is small, and the control advantage with high precision is provided, indicating the effectiveness of the control system, and further on the ship rudder electro-hydraulic system. The simulation and characterization of the simulation have the meaning of throwing bricks.

References 1. Zhen, L.: Research on the method of suppressing the excess force of single pendulum load electro-hydraulic servo dynamic loading system. Lanzhou University of Technology (2013) 2. Yang, H., Hui, W., Yi, Z., Yang, P.: Excessive force analysis of electro-hydraulic servo loading system. Hydraulics Pneumatics 36(09), 55–58 (2016) 3. Wu, J.: Nonlinear dynamic system identiﬁcation method based on BP neural network. In: Proceedings of the 12th Annual Conference of Control and Application of Chinese Aeronautical Society, China Aviation Society Automatic Control Professional Branch, p. 6. Session: China Aviation Society (2006) 4. Xue, J.: Design of PLC-based electro hydraulic servo position closed loop system. In: Proceedings of 2016 2nd International Conference on Materials Engineering and Information Technology Applications (MEITA 2016), International Society for Information and Engineering: International Society for Computer Science and Electronic Technology (Computer Science and Electronic Technology International Society), p. 5 (2016) 5. Ma, C.: Simulation studies of CMAC-PID combined control for electro-hydraulic position servo system. In: Proceedings of 2017 2nd International Conference on Computer, Mechatronics and Electronic Engineering (CMEE 2017), Advanced Science and Industry Research Center: Science and Engineering Research Center, p. 5 (2017) 6. Liu, X.: Research on Residual Force Compensation and Digital Control Strategy of Electrohydraulic Servo System. Beijing Jiaotong University (2008) 7. Cheng, X., Jin, X., Zhang, M., Zheng, B.: Study on the excess force of passive electrohydraulic loading system. Mach. Des. Manuf (09), 71–75 (2018) 8. Zhao, S., Sun, S., Li, G.: A structural compensation method for the excess force of electrohydraulic load simulator. J. Henan Univ. Sci. Technol.: Nat. Sci. Edn. 36(04), 23–26+31+5 (2015) 9. Jin, Y., Yun, N.: Research on the mechanism of excessive force generation and its elimination method in passive loading system. Helicopter Technol. (01), 18–22 (2003) 10. Yang, F., Liu, C., Hao, Y., Ma, X.: Residual force suppression based on composite control. Mach. Tool Hydraulics 43(19), 82–85+90 (2015)

Collision Simulation of GQ70 Light Oil Tank Car at the Level Crossing Maopeng Tian, Ronghua Li, Xiujuan Zhang(&), and Miao Jin School of Mechanical Engineering, Dalian Jiaotong University, Dalian 116028, China [email protected], [email protected], [email protected], [email protected]

Abstract. The simulation and analysis of the collision process between GQ70 light oil tank car and truck at the level crossing are carried out by using the theory of explicit dynamics. The results are obtained which are the equivalent stress values of the tank, the vertical uplift and transverse displacement of each wheel set. Meanwhile, it is studied that is the influence of two collision parameters (mass and speed) on tank strength. It is concluded that the speed and mass of the collision vary with the intensity of the collision in the primary curve and the quadratic curve, respectively. The effect of the former on the collision performance is higher than that of the latter. The research in this paper provides a certain reference for the optimization design of tank car. Keywords: Railway tank car Collision mass Collision performance Explicit dynamics

Collision speed

1 Introduction Railway tank car is one of the main types of railway freight trafﬁc, which is widely used and plays an important role in railway freight trafﬁc. When the railway tank car passes through the level crossing and collides with the passing vehicle, it is easy to cause the derailment of the tank car, the damage of the tank body and so on. Therefore, it is very important to study the side crash-worthiness of railway tank car. Many scholars have simulated and analyzed the collision process of rail vehicles. Among them, Yang et al. [1], used the method of multi-rigid-body dynamics to simulate the rail vehicle collision. Lu [2] has studied the collision among different groups of trains under nonlinear conditions by using the method of multi-body dynamics, and derived the calculation formula of the collision energy allocation of the train. Dalapati et al. [3], proposed an agent-based solution to study the real-time collision process in train operation. Lu et al. [4], established a vertical vehicle collision model considering the track model by using the method of multi-body dynamics. Lei et al. [5], used the method of explicit dynamics to simulate and analyze train collision behavior. Deng [6] used the method of explicit dynamics to study the collision performance of a G70 modiﬁed light oil tank car at different speeds in front-line operation. Wang et al. [7], used the method of one-dimensional energy allocation and the method of threedimensional vehicle collision simulation to study the influence of empty travel of train © Springer Nature Singapore Pte Ltd. 2020 J. Tan (Ed.): ICMD 2019, MMS 77, pp. 623–633, 2020. https://doi.org/10.1007/978-981-32-9941-2_51

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energy absorption system on train crash performance. Wang et al. [8], used the method of ﬁnite element and multi-body dynamics to simulate the collision resistance of the track train, and optimized energy absorbing and climbing device for the train. Zhou et al. [9], used the method of ﬁnite element and multi-rigid body dynamics to simulate the collision scene of rail vehicle and studied the derailment risk of the train. The above researches mainly focus on the straight-line frontal collision of rail vehicles, but there is little research on the side collision of vehicles at the level crossing. Compared with the normal collision, the railway tank car is more prone to derailment and other safety accidents under side collision. In this paper, the theory of explicit dynamics is used to simulate the collision accident of GQ70 light oil tank car at the level crossing under the condition of empty vehicle. The maximum equivalent stresses of the tank during the collision are obtained. The influence of truck mass and collision speed on the maximum equivalent stress of the tank is also analyzed. The research result of this paper provides a certain reference for the optimization design of the tank car.

2 Basic Theory of Explicit Dynamics Compared with implicit algorithm, the best advantage of the explicit algorithm is that the calculation has good stability. For the Lagrangian formulations currently available in the explicit dynamics system and the mesh moves and distorts with the material variation. The conservation of mass is automatically satisﬁed. The density at any time can be determined from the current volume of the zone and its initial mass, and can be described as: q¼

p0 V0 m ¼ V V

ð1Þ

Where q is the element density at any time; q0 is element initial density; V0 is element initial volume; m is element mass; and V is element volume at any time. Represented by the following formula, the partial differential equations express the relation between the acceleration and the stress tensor rij . @rxx @rxy @rxz þ þ @x @y @z @ryx @ryy @ryz € y ¼ by þ þ q þ @x @y @z @rzx @rzy @rzz € z ¼ bz þ þ þ q @x @y @z € ¼ bx þ q

ð2Þ

Where q is element density at any time; €x; €y; €z are acceleration of elements in x, y and z directions, respectively; rij is stress tensor.

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3 Finite Element Analysis 3.1

Finite Element Model

The location of light oil tank car is uncertain when its collision is happened at level crossing. But the location is mainly divided into two kinds: one is the direct collision between the truck and the middle part of the tank, the other is the collision between the truck and the tractor frame of the tank car. In this paper, two kinds of collision models of GQ70 light oil tank car and truck at the level crossing are established by using CREO software as shown in Fig. 1. And the ﬁnite element simulation is carried out with ANSYS/Explicit Dynamics module. Where the tank car runs in a straight line at 20 km/h speed in the X direction. The truck moves in a straight line along the Z direction at the same speed and crashes into the middle part of the tank car and the tractor frame of the tank car, respectively. The total weight of GQ70 light oil tank car is 23.6t. Its material is Q345A, its density is 7850 kg/m3, its elastic modulus is 206GPa, its Poisson’s ratio is 0.28, and its yield limit is 345 MPa. When the constraints are established, the ground and rail are ﬁxed and the gravity acceleration is added to the whole system.

a) Collision in the middle part of tank car

b) Collision in the tractor frame Fig. 1. Collision of tank car at the level crossing

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Finite Element Analysis Results

Equivalent stress σ/MPa

3.2.1 Results of Collision in the Middle Part of Tank Car When the middle part of GQ70 light oil tank car is impacted by the side of 8t truck, its maximum stress variation is shown in Fig. 2. The collision time is 0.2 s and the maximum stress occurs at the 0.006 s. The equivalent stress cloud diagram of the tank is shown in Fig. 3. The value of the maximum equivalent stress is 344.54 MPa, which has reached the yield limit of the tank material. Subsequently, because of collision and separation between the tank car and the truck, the equivalent stress value decreases gradually, and its average value is approximately 100 MPa. 400 300 200 100 0 0

0.05

0.1 Time t/s

0.15

0.2

Fig. 2. Equivalent stress curve of tank

Fig. 3. Equivalent stress contour of tank at maximum stress time

During the collision process, a large vertical lift and lateral displacement occurred in the tank car. The four wheel sets are calibrated along the motion direction of the tank car as Wheel sets A, B, C, and D, where Wheel sets A and B are the wheel sets on the collided side as shown in Fig. 4. Figure 5 shows the curves of maximum vertical uplift over time for each wheel set. It can be seen that the vertical uplifts of Wheel sets A and B increase linearly within the time of 0.2 s, and the maximum vertical uplift reaches 357.4 mm. Meanwhile, the vertical lifts of Wheel sets C and D are smaller than those of Wheel sets A and B. The tank car has a obvious roll-over trend. Figure 6 illustrates

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the time-dependent curves of the transverse displacement of four wheel sets. The four wheel sets all have the large lateral displacements within the time of 0.2 s, and the maximum lateral displacement is up to 178.28 mm.

Vertical uplift d/mm

Fig. 4. Diagram of wheel sets

400 350 300 250 200 150 100 50 0 -50

A

0

0.05

B

0.1

C

D

0.15

0.2

Time t/s

Transverse displacement d/mm

Fig. 5. Vertical uplift of four wheel sets

200 180 160 140 120 100 80 60 40 20 0 -20

A

0

0.05

B

0.1

C

0.15

D

0.2

Time t/mm

Fig. 6. Transverse displacement of four wheel sets

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Equivalent stress σ/MPa

3.2.2 Collision Results of Tractor Frame of Tank Car When the tractor frame of GQ70 light oil tank car is collided by the side of 8t heavy truck, the maximum equivalent stress changes are shown in Fig. 7. It can be seen that the collision time is 0.2 s and the equivalent stress curve reaches the maximum equivalent stress at 0.004 s. Figure 8 shows the equivalent stress cloud of the tank at this time, and the maximum equivalent stress is 193.84 MPa. Subsequently, there is a collision and separation between the tanker and the truck. However, due to the coupling between the tank and the tractor frame, the maximum equivalent stress value of the tank is always in a large vibration. In this paper, it is only considered the maximum equivalent stress of the tank at the moment of collision, so the maximum equivalent stress for the ﬁrst collision is taken as the research object.

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Time t/s

Fig. 7. Equivalent stress curve of tank

Fig. 8. Stress contour of tank at maximum stress time

During the collision process, the tank car also has a large ver