Intelligent Motorized Spindle Technology (Springer Tracts in Mechanical Engineering) 9789811533273, 981153327X

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Springer Tracts in Mechanical Engineering

Yuhou Wu Lixiu Zhang

Intelligent Motorized Spindle Technology

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA

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Yuhou Wu Lixiu Zhang •

Intelligent Motorized Spindle Technology

123

Yuhou Wu School of Mechanical Engineering Shenyang Jianzhu University Shenyang, China

Lixiu Zhang School of Mechanical Engineering Shenyang Jianzhu University Shenyang, China

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-981-15-3327-3 ISBN 978-981-15-3328-0 (eBook) https://doi.org/10.1007/978-981-15-3328-0 © 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

Equipment manufacturing industry is the “backbone” of national economy. Numerical control (NC) machine tool is the “working machine tool” in modern equipment manufacturing industry and an important fundamental equipment for achieving the modernization of manufacturing technology and equipment. Intelligent NC machine tool is important for improving machining precision and efficiency as well as optimization of production process. Spindle system is a key functional part of NC machine tool, and intelligent machine tool should be equipped with intelligent spindle system. An intelligent motorized spindle should have the following features: it can automatically detect and optimize its own operational status; it can evaluate the quality of its own output; it has the ability to learn and make improvements. The key technologies involved are the automatic recognition of motor parameters, vibration inhibition, reduction of thermal deformation, interference prevention, automatic adjustment of lubricating oil amount, reduction of noise, etc. The use of intelligent motorized spindle can help improve the machining accuracy and efficiency of NC machine tool. Since 1995, the author of this book has been working on the key technologies of motorized spindle and has won many awards such as China National Science and Technology Progress Award, Science and Technology Invention Award, National Excellent Patent Award, National Patent Gold Award, and so on. This book is a summary of research outcomes of related fields and also refers to the latest academic literatures in these fields. The contents include three research hotspots (automatic recognition of motor parameters, vibration inhibition, and reduction of thermal deformation), research frontiers and future direction of motorized spindle development, and also include the theory basis and technical application of motorized spindle. This book discusses intelligent motorized spindle from four aspects. The first aspect is about the working principles, key technologies, and operational characteristics of motorized spindle. The second aspect is about the fundamentals of motorized spindle, including the basic theories and methods of spindle motion control, heat generation and heat transfer in spindle, and dynamic balancing of spindle. The third aspect is about the technologies of motorized spindle, including the application of intelligent algorithms to determination of stator v

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Preface

resistance, prediction of spindle temperature field, and online dynamic balancing of spindle. The fourth aspect is about future prospects, including the analysis of the performance and development of intelligent motorized spindle and high precision ceramic motorized spindle from the perspectives of intelligentization and integration, as well as the proposal of the industrialization of intelligent numerical control machine tool. Our work is financially supported by National Natural Science Foundation of China (Grant Nos. 59375228, 50975182, and 51375317), National High-tech R&D Program of China (863 Program, 2006AA03Z533), National Science & Technology Pillar Program during the Eleventh Five-year Plan Period (2006BAJ12B07), International Science and Technology Cooperation Project (2008DFA70330), Programme of Introducing Talents of Discipline to Universities (D18017), and Changjiang Scholar Innovation Team CNC Machine Tool Spindle System Project of Ministry of Education (IRT1160). In terms of writing style, this book strives to present the systematicity, theoreticity, advancement, and practicality of its contents in easy-to-understand language. This book can provide guidance for researchers and developers working on numerical control, industrial automation, electric drive technology, and key technologies of motorized spindle in research institutes, colleges, and corporations. This book can also provide guidance for advanced undergraduate and graduate students of related majors in higher education institutions. During writing, the author received help from the experts in Dalian University of Technology (China), Tsinghua University (China), Luoyang Bearing Research Institute Company (China), Shenyang Machine Tool Group (China), Shenyang Jianzhu University (China), and so on. The author also got support from the professors in University of Michigan (USA) and Transilvania University of Braşov (Romania). Here, the author would like to thank these people for their help and support. Due to the limited knowledge and expertise of the author, there are inevitably shortcomings in the book and valuable comments from readers are appreciated. Shenyang, China

Yuhou Wu Lixiu Zhang

About This Book

This book aims to improve the dynamic performance of motorized spindle and make it intelligent. Focusing on the thermal properties, driving characteristics, and dynamic balance of motorized spindle, this book discusses the basic theories and key technologies for intelligentizing its thermal control, motion control, vibration control, and fault diagnosis. By revealing the mechanism of motorized spindle’s nonlinear motions in coupled multi-fields (force field, flow field, temperature field, etc.), we study the thermal deformation of its stator and rotor, analyze the influence of changes of its internal structure and electromagnetic force on spindle’s dynamic performance, and develop methods to accurately predict spindle temperature rise as well as strategies to control its vibration. In addition, we propose methods for intelligent determination of stator resistance, which provides a basis for precise motion control of motorized spindle. Further, deep learning is applied to fault diagnosis of motorized spindle. Finally, taking intelligent motorized spindle as the future direction of motorized spindle development, this book presents several hot issues in its research and development. The integration of artificial intelligence and control technology into motorized spindle means that computer is employed to predict spindle temperature rise and thermal deformation, determine motor stator resistance, and control spindle vibration. This can greatly improve the operational performance and intelligence of motorized spindle and make numerical control (NC) equipment present high quality, good flexibility, high efficiency, and greenness. In this book, artificial intelligence is applied to control of spindle temperature rise, prediction of spindle thermal deformation, dynamic balancing of spindle, recognition of motor parameters, etc. These contents can be found in journal papers, theses, and research reports and summarized, for the first time, in this book. This book can provide guidance for researchers and developers working on numerical control, automation, electric drive technology, and key technologies of motorized spindle in research institutes, colleges, and corporations. This book can also provide guidance for advanced undergraduate and graduate students of related majors in higher education institutions.

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About This Book

This book discusses intelligent motorized spindle from four aspects. The first aspect is about the working principles, key technologies, and operational characteristics of motorized spindle. The second aspect is about the fundamentals of motorized spindle, including the basic theories and methods of spindle motion control, heat generation and heat transfer in spindle, and dynamic balancing of spindle. The third aspect is about the technologies of motorized spindle, including the application of intelligent algorithms to determination of stator resistance, prediction of spindle temperature field, and online dynamic balancing of spindle. The fourth aspect is about future prospects, including the analysis of the performance and development of intelligent motorized spindle and high precision ceramic motorized spindle from the perspectives of intelligentization and integration, as well as the proposal of the industrialization of intelligent numerical control machine tools.

Contents

1 Key Technologies and Performance of Motorized Spindle . . . . 1.1 The Structure and Working Principles of Motorized Spindle . 1.1.1 Structure and Classification of Motorized Spindles . . 1.1.2 Working Principles of Motorized Spindle Motor . . . . 1.1.3 Technical Parameters of Motorized Spindle . . . . . . . . 1.1.4 The Trend of Motorized Spindle Technology . . . . . . 1.2 Common Key Technologies for Motorized Spindle . . . . . . . . 1.2.1 Spindle Bearing Technology . . . . . . . . . . . . . . . . . . . 1.2.2 Motor and Control Technology . . . . . . . . . . . . . . . . 1.2.3 Lubrication and Cooling Technology . . . . . . . . . . . . 1.2.4 Dynamic Balance Technology . . . . . . . . . . . . . . . . . 1.2.5 Tool Interface Technology . . . . . . . . . . . . . . . . . . . . 1.3 Static and Dynamic Performance of Motorized Spindle . . . . . 1.3.1 Precision and Static Rigidity . . . . . . . . . . . . . . . . . . 1.3.2 Critical Rotational Speed . . . . . . . . . . . . . . . . . . . . . 1.3.3 Residual Dynamic Imbalance and Vibration Velocity for Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Noise and Sleeve Temperature Rise . . . . . . . . . . . . . 1.3.5 Pulling Force of Tensioning Tool . . . . . . . . . . . . . . . 1.3.6 Service Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.7 Other Servo Performances . . . . . . . . . . . . . . . . . . . . 1.4 Application of Intelligent Motorized Spindle to NC Machine Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Intelligent NC Machine Tool . . . . . . . . . . . . . . . . . . 1.4.2 Development of Motorized Spindle Towards High Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Motorized Spindle Drive Mode and Its Basic Theory . . . . . . . 2.1 Constant Voltage-to-Frequency Ratio Control . . . . . . . . . . . 2.1.1 Constant Voltage-to-Frequency Ratio Control . . . . . 2.1.2 Mechanical Characteristics of Motorized Spindle Under Voltage-Frequency Control . . . . . . . . . . . . . 2.1.3 Modeling and Simulation Analysis . . . . . . . . . . . . . 2.2 Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Coordinate Transformation . . . . . . . . . . . . . . . . . . . 2.2.2 Dynamic Mathematical Model of Motorized Spindle 2.2.3 Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Modeling and Simulation Analysis of Speed Sensorless Vector Control System . . . . . . . . . . . . . 2.3 Direct Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Principles of Direct Torque Control . . . . . . . . . 2.3.2 Mathematical Model of Inverter and Space Voltage Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Model of Direct Torque Control System for Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Simulation of Motorized Spindle Direct Torque Control System and Result Analysis . . . . . . . . . . . . 2.4 The Advantages and Disadvantages of Direct Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Intrinsic Connection Between Direct Torque Control and Vector Control . . . . . . . . . . . . . . . . . . 2.4.2 The Advantages and Disadvantages of Direct Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

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3 Heat Generation and Transfer of Motorized Spindle . . . . . . . . 3.1 Loss Analysis and Heat Generation Calculation of Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Heat Generation Analysis and Calculation of Motor . 3.1.2 Analysis and Calculation of Bearing Heat Generation 3.2 Heat Transfer in Motorized Spindle and Calculation of Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Basic Theory of Heat Transfer . . . . . . . . . . . . . . . . . 3.2.2 Heat Conduction in Motorized Spindle . . . . . . . . . . . 3.2.3 Heat Convection Between Motorized Spindle and External Medium . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Finite Element Formula of Temperature Field of Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Basic Theory of Thermoelasticity . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

4 Basic Theory and Method of Spindle Dynamic Balance . . 4.1 Dynamic Modeling of Rigid Rotor . . . . . . . . . . . . . . . 4.2 The Expression of Imbalance Amount . . . . . . . . . . . . . 4.3 Imbalance Classification . . . . . . . . . . . . . . . . . . . . . . . 4.4 Imbalance Tolerance Specification . . . . . . . . . . . . . . . . 4.5 Correction Plane Selection . . . . . . . . . . . . . . . . . . . . . 4.6 On-Site Dynamic Balance of Rotor System . . . . . . . . . 4.6.1 Dynamic Balance Principle of Rigid Rotor . . . . 4.6.2 Dynamic Balance Method of Rigid Rotor . . . . . 4.7 Dynamic Balance of Flexible Rotor . . . . . . . . . . . . . . . 4.8 Vibration Signal Extraction Algorithm . . . . . . . . . . . . . 4.8.1 Vibration Signal Smooth Processing Algorithm . 4.8.2 Signal Preprocessing Based on Time Domain Averaging and FIR Filtering . . . . . . . . . . . . . . . 4.8.3 Signal Extraction Method . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Intelligent Identification Technology of Stator Resistance of Motorized Spindle Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Effect of Stator Resistance on Direct Torque Control Performance . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Analysis of Stator Resistance Characteristics . . . . . . . 5.3 RBF Neural Network for Identification of Stator Resistance . 5.4 Hybrid Intelligent Identification of Stator Resistance . . . . . . . 5.4.1 Stator Resistance Identification Strategy Based on ANN-CBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Hybrid Intelligent Algorithm for Identifying Stator Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Simulation of Hybrid Intelligent Identification of Stator Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Experimental Verification . . . . . . . . . . . . . . . . . . . . . 5.5 Improvement of BP Neural Network Based on Biogeography-Based Optimization Algorithm . . . . . . . . . 5.5.1 Biogeography-Based Optimization Algorithm . . . . . . 5.5.2 Biogeography-Based Optimization Algorithm Based on Random Disturbance . . . . . . . . . . . . . . . . . 5.5.3 Application of BP Neural Network in the Estimation of the Stator Resistance of Motorized Spindle . . . . . .

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5.5.4 Stator Resistance Estimation Based on Improved BP Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.5.5 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6 Thermal Performance Prediction of Motorized Spindle . . . . . 6.1 Finite Element Model of Motorized Spindle Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calculation of Heat Generation of Motorized Spindle Based on Loss Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Temperature Field Prediction Model for Motorized Spindle Based on Heat Transfer Coefficient Optimization . . . . . . . . 6.3.1 Motorized Spindle Temperature Prediction Model Based on Genetic Algorithm . . . . . . . . . . . . . . . . . 6.3.2 Motorized Spindle Temperature Prediction Model Based on Least Squares Method . . . . . . . . . . . . . . . 6.4 Accuracy Analysis of Prediction Model . . . . . . . . . . . . . . . 6.4.1 Accuracy of Temperature Prediction Model Based on Least Squares Method . . . . . . . . . . . . . . . . . . . . 6.4.2 Accuracy of Temperature Prediction Model Based on Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Comparision of Accuracy Between the Two Temperature Prediction Models . . . . . . . . . . . . . . . 6.5 Loss Sensitivity Analysis of Temperature Prediction Model 6.5.1 Parameter Local Sensitivity Analysis . . . . . . . . . . . 6.5.2 Prediction Model Loss Sensitivity Analysis . . . . . . . 6.6 Thermal Deformation Prediction of Motorized Spindle . . . . 6.6.1 Finite Element Model for Motorized Spindle Thermal Deformation . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Thermal Deformation Prediction Model Based on Heat Transfer Coefficient Optimization . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Automatic Suppression of Motorized Spindle Vibration . . . . . . 7.1 Components and Working Principles of Dynamic Balancing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Balancing Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Software Design of Dynamic Balancing System . . . . . . . . . . 7.2.1 Overall Structure of Software System . . . . . . . . . . . . 7.2.2 Main Interface Design . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Signal Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Acquisition of the Phase Position of Counterweight Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.2.5 Calculation of Imbalance Amount and Influence Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Movement of Correction Mass . . . . . . . . . . . . . . . . 7.3 Characteristic Analysis of Dynamic Balancing System . . . . 7.3.1 The Influence of Rotational Speed on the Balancing Effect of Dynamic Balancing System . . . . . . . . . . . 7.3.2 The Influence of the Angle of Test Weight on the Balancing Effect of Dynamic Balancing System and Influence Coefficient . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Motorized Spindle Fault Diagnosis Technology Based on Deep Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Fault Analysis of Motorized Spindle . . . . . . . . . . . . . . . . . . 8.2 Deep Learning and Motorized Spindle . . . . . . . . . . . . . . . . . 8.3 Application of Deep Learning to Fault Diagnosis of Bearing . 8.4 Application of Deep Learning to Fault Diagnosis of Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Application of Deep Learning to Motorized Spindle State Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Development of Intelligent Ceramic Motorized Spindle . . . . . 9.1 Electromagnetic Properties of Ceramic Motorized Spindle . . 9.1.1 The Operational Characteristics of Power Supply Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Magnetic Leakage of Ceramic Motorized Spindle . . 9.2 Vibration and Noise Characteristics of Ceramic Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Vibration Characteristics of Ceramic Motorized Spindle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Noise Characteristics of Ceramic Motorized Spindle 9.3 Development of Intelligent Ceramic Motorized Spindle . . .

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Chapter 1

Key Technologies and Performance of Motorized Spindle

High-speed machining technology can solve many difficulties in manufacturing of mechanical products, such as obtaining special machining accuracy and surface quality. Therefore, it has been more and more widely applied to various equipment manufacturing industries and made high-speed NC machine tool become a fundamental tool in equipment manufacturing industry. Motorized spindle is a key functional part of NC machine tool and one of the major carriers of high-speed cutting technology. During high-speed machining, motorized spindle is the optimal choice to accurately move tool/workpiece and transfer energy required for cutting. This chapter introduces the working principles, key technologies and operational performance of motorized spindle.

1.1 The Structure and Working Principles of Motorized Spindle With the rapid development and improvement of variable frequency control technology (motor vector control, direct torque control, etc.), the mechanical structure of the main drive of high-speed NC machine tool has been greatly simplified, without using belt drive and gear drive any more. The spindle of machine tool is directly driven by a built-in motor, thus the length of the main drive chain is shortened to zero and “zero transmission” is realized. The integration of motor and machine tool spindle enables the spindle componment to become relatively independent from the transmission system and overall structure of machine tool so that it can be made into a “spindle unit”, commonly known as “motorized spindle” [1].

© Springer Nature Singapore Pte Ltd. 2020 Y. Wu and L. Zhang, Intelligent Motorized Spindle Technology, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3328-0_1

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1 Key Technologies and Performance of Motorized Spindle

1.1.1 Structure and Classification of Motorized Spindles Generally, a motorized spindle consists of front cover, back cover, shaft, front bearing, rear bearing, bearing preload, water jacket, shell, stator and rotor (Fig. 1.1). The stator of motorized spindle is made by repeated pressing of high-quality silicon steel sheets with high magnetic permeability, and the formed stator has punched grooves for wire embedding in its inner cavity. Rotor often consists of shaft, iron core and squirrel cage. There is a gap, called air gap, between rotor and stator. It provides a path for converting the electromagnetic field energy of stator into mechanical energy. The rotor is mounted onto shaft through interference fit method and located between front and rear bearings. The transmission of high torque is achieved through the utilization of friction force generated due to interference fit. Since there is a large interference between the inner hole of rotor and the matching surface of shaft, the rotor must be heated to about 200 °C in oil bath during assembly and then hot press assembly is quickly performed. The stator of motorized spindle is fixed in the spindle housing by a cooling jacket. In this way, the motor rotor is the main shaft of machine tool,

Fig. 1.1 Section of motorized spindle used for machining center (From Edel Maschinenbau GmbH company. http://www.edelgmbh.de/englisch/index.htm)

1.1 The Structure and Working Principles of Motorized Spindle

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Table 1.1 Classification of motorized spindles Classification criteria

Types

Support bearing

Rolling bearing, magnetic suspension bearing, fluid dynamic (static) bearing

Lubrication method

Grease, oil mist, oil/air

Cooling method

Water cooling, air cooling, self cooling

Application field

Turning, milling, grinding, drilling, rolling, centrifuging

Motor type

Asynchronous motorized spindle, permanent-magnet synchronous motorized spindle

Bearing type

Metal spindle, ceramic bearing spindle, ceramic spindle

and the sleeve of motorized spindle is the motor seat. Typically, a toothed disc is mounted at the rear of the spindle as an inductive encoder to achieve full closed-loop control of the motor. The inner tapered bore and end surface of the extruded part of the front end of spindle are used for mounting and fixing replaceable shank. High-speed motorized spindle units are usually classified according to support bearing types, lubrication methods, cooling methods, application fields and motor types. The classification of motorized spindles is shown in Table 1.1. According to application field, motorized spindles can be classified into spindles used for turning, milling, grinding, drilling, rolling, and centrifuging, respectively. (1) Motorized spindle used for grinding. The motorized spindle used for grinding is currently the most important and also the earliest type of motorized spindle studied and developed in various countries. It is mainly used to achieve highspeed grinding and high quality of grinding surface. This type of spindle is characterized by high speed, high precision and large output power and typically applied to bearing grinding machines, internal grinding machines, cylindrical grinding machines, etc. (2) Motorized spindle used for turning. High-speed motorized spindle used for turning enables good machining accuracy and surface roughness, and is especially suitable for machining of non-ferrous metal parts such as aluminum and copper. In addition to transmitting motion and torque, the spindle used in the turning center also drives the workpiece to rotate and directly bear cutting force. At a certain load and speed, the spindle should ensure that workpiece rotates accurately and steadily around its axis and maintains this performance under both dynamic and thermal conditions. (3) Motorized spindle used for milling. This type of motorized spindle is matched with NC milling, engraving and milling machine and machining center for high-speed milling and engraving. It is suitable for machining conventional parts, toolings and wood workpieces. There are two kinds of tool changes: automatic tool change and manual tool change. The spindle with automatic tool change has an automatic clamping and releasing system, which makes tool change convenient and quick. The spindle using manual tool change has a simple structure and is of low cost. It is suitable for machine tools that do not require

4

1 Key Technologies and Performance of Motorized Spindle

frequent tool change. The speed of motorized spindle used for engraving and milling is high, generally above 24,000 r/min. ER spring chuck is usually used to clamp the tool. The motor output can be classified into constant power output and constant torque output. Since motorized spindle used for large-scale NC milling does not have a tool magazine and needs not to change the tool, open-loop control is often used. The motorized spindle of the machining center usually employs closed-loop coding control. If low-speed and high-torque output is required, then it is necessary to provide the speed range of the spindle and the speed at the start of constant power stage, and the spindle should have exact stop function. Grease lubrication or oil/air lubrication is often used for this kind of spindle to reduce the pollution of oil mist to the environment. (4) Motorized spindle used for drilling The motorized spindle used for drilling mainly refers to that used for high-speed hole drilling of PCB board. Its speed can be divided into six grades: 60,000, 80,000, 90,000, 105,000, 120,000, and 180,000 r/min. Motorized spindle supported by grease-lubricated rolling bearings correspond to the first three grades of speed, with machining range of 0.2–0.7 mm. Motorized spindle supported by aerostatic bearings correspond to the last three grades of speed and can be used for drilling holes of 0.1–0.15 mm. (5) Other motorized spindles. The motorized spindle for high-speed centrifuge is widely used in various fields such as separation, crushing, atomization, and testing. The motorized spindle for high-speed roller is used to machine the internally threaded copper pipe of air conditioning equipment. There are also some special motorized spindles used for driving, testing, cutting, etc. According to the types of bearing materials, motorized spindle can be classified into metal spindle, ceramic bearing spindle and ceramic spindle. In metal spindle, the bearings are made of metal. In ceramic bearing spindle, some parts (e.g. rolling bodies or inner/outer ring) of bearings are made of ceramic. In ceramic spindle, bearings and other components such as shaft are all made of ceramic.

1.1.2 Working Principles of Motorized Spindle Motor 1. Three-phase asynchronous motorized spindle Three-phase symmetrical current is passed into the stator of three-phase asynchronous motorized spindle. Circular rotating magnetic flux is formed inside the motorized spindle. The formed magnetic field continuously rotates in space with the alternating current, that is, rotating fundamental magnetic field is formed. Figure 1.2 shows the simulated magnetic field in motorized spindle at different time. As can be seen, the magnetic field in motorized spindle is circular and the magnetic flux density rotates counterclockwise. If the rotor does not rotate, the rotor squirrel cage guide bar and the rotating magnetic flux density move in relative to each other, and then induced electromotive force is generated in the guide bar. Its direction is determined by the right-hand rule. The rotational speed of the rotating magnetic field is

1.1 The Structure and Working Principles of Motorized Spindle

5

Fig. 1.2 The simulation of rotating magnetic field in motorized spindle at different time

n s = 60

fs p

(1.1)

where p is the number of pole pairs, f s is power source frequency (Hz), n s is the speed of rotating magnetic field, also called synchronous speed (r/min). The rotor guide bars are short-circuited at the ends and a current i is generated in them. If we do not consider the phase position difference between electromotive force and the current, the current direction should be consistent with that of the electromotive force. The electromagnetic force that guide bars bear in the rotating magnetic field is F, resulting in the formation of electromagnetic torque T. The rotor circuit cuts the magnetic lines of force and the rotation direction is consistent with that of the rotating magnetomotive force, thus causing the rotor to rotate in this direction. Assume that the rotor speed is n r . When n s < n r , it indicates that the rotor guide bar and the magnetic field move in relative to each other. The directions of the generated electromotive force, current and force are the same as those when the rotor is not rotating. The direction of electromagnetic torque T is counterclockwise. The rotor continues to rotate and operates stably. When the rotational speed of rotor is equal to the synchronous speed n s , there is no relative motion between rotor and the rotating magnetic field. Therefore, rotor guide bars do not cut the rotating magnetic field, no induced electromotive force, rotor current and electromagnetic torque are formed, and the rotor will then not rotate. Therefore, the rotational speed of rotor of asynchronous motorized spindle is often lower than the synchronous speed of power source [2]. The difference between rotor rotational speed nr and synchronous speed n s is defined as slip ratio s, i.e. s=

ns − nr ns

(1.2)

The smaller the slip ratio is, the closer the rotor rotational speed with the synchronous rotational speed and the higher the efficiency of motorized spindle is.

6

1 Key Technologies and Performance of Motorized Spindle

Rs

nr Us

Rr

sX r

Xs

is

Es

.

ir

Er

fs

fr

Fig. 1.3 Schematic diagram of coupled stator and rotor circuit

(1) Stator voltage equation Figure 1.3 shows the diagram of coupled stator and rotor circuit. The frequency of stator is f s and the frequency of rotor is f r . The stator circuit and the rotating rotor circuit are coupled by the rotating magnetic field (main magnetic field) in the air gap. The magnetic field (main magnetic field) in air gap that rotates at synchronous rotational speed will induce a symmetrical three-phase electromotive force in the •

three-phase windings of the stator E s . According to Kirchhoff’s law, the supply •

voltage Us applied to each phase of stator should be equal to the sum of the negative •

•

value of electromotive force − E s and the leakage impedance voltage drop i s (Rs + jX sσ ) generated by the stator current. Since the three phases are symmetrical, we only need to analyze one of them (e.g. phase A). Therefore, the voltage equation for stator is •

•

•

Us = i s (Rs + j X sσ ) − E s

(1.3)

where Rs and X sσ are the resistance and leakage reactance of each phase of stator, •

•

respectively, and E s = − i m Z m . (2) Rotor voltage equation •

The main magnetic field in air gap not only induces an electromotive force E s with a •

frequency of f s in stator windings, but also induces an electromotive force E rs with •

slip frequency of f r = s f s in the rotating rotor windings. The effective value of E rs (E rs ) is E rs = 4.44s f s Nr kwr m

(1.4)

When the rotor stops rotating (s = 1), the induced electromotive force of each phase of the rotor is E r

1.1 The Structure and Working Principles of Motorized Spindle

7

E r = 4.44 f s Nr kwr m

(1.5)

E rs = sE r

(1.6)

This means that the induced electromotive force of rotor is positively proportional to the slip rate s. The higher the s, the higher the relative speed at which the main magnetic field “cut” the rotor windings and the greater the E rs . The windings of each phase of rotor also have resistance and leakage reactance. Since the frequency of rotor is f r = s f s , the leakage reactance X rσ s of rotor windings should be X rσ s = 2π f r L rσ = 2π s f s L rσ = s X rσ

(1.7)

where X rσ is the leakage reactance when rotor frequency equals f s (i.e., rotor stops rotating). The rotor windings of induction motor are usually short-circuited, i.e., terminal voltage Ur = 0. According to Kirchhoff’s second law, the voltage equation of one phase of rotor windings is written as •

•

E rs e jωr t = Irs e jωr t (Rr + js X rσ )

(1.8)

or •

•

E rs = i rs (Rr + js X rσ ) •

(1.9)

where i rs is rotor current and Rr is the resistance of each phase of rotor. Since stator and rotor are different in frequency, phase number and number of effective turns, they cannot be coupled together. In order to obtain an equivalent circuit that integrates both stator and rotor, the rotor frequency needs to be converted into stator frequency, and the numbers of phases and effective turns of rotor are converted into those of stator, respectively. In other words, frequency reduction and winding reduction are performed. Figure 1.4 shows the circuit diagram of stator and rotor of induction motor after frequency and winding reduction. After reduction, the voltage equation and magnetomotive force equation of the induction motor are ⎧ • • • ⎪ ⎪ U = i + j X E − (R ) s s s sσ s ⎪ ⎪ •  ⎪ • ⎪ ⎨ E  = i  Rr + j X  r r s rσ (1.10) • • • ⎪ ⎪ ⎪ E s = E r = − i m Z m ⎪ ⎪ • ⎪ ⎩ • • is + ir = im

8

1 Key Technologies and Performance of Motorized Spindle

X 2a

X 1a

R1

I2

I1

E2

E1

U1

E1

R2 1 s R2 s

Fig. 1.4 Circuit diagram of stator and rotor of induction motor after frequency and winding reduction X2

X1

R1

I1

R2 I2

Rm

U1

Im

E2

E1

1 s R s 2

Xm

Fig. 1.5 The equivalent T circuit of induction motor

According to Eq. (1.10), we can obtain the equivalent T circuit of the induction motor (Fig. 1.5). When the induction motor operates under speed of rotor is   no load, the rotational Rr close to the synchronous rotational speed s ≈ 0, s → ∞ and the rotor is equivalent to an open circuit. At this time, the rotor current is close to zero and the stator current is basically excitation current. When the motor is loaded, slip ratio increases R and sr decreases, leading to increase in rotor and stator currents. Under load, due to increase in stator current and leakage impedance voltage drop, E 1 and corresponding value of main magnetic flux are smaller than those under no load. At start-up, R s = 1, sr = R2 and the rotor and stator currents are very high. Since the leakage impedance voltage drop of stator is large, E s and value of main magnetic flux will be significantly reduced, only about 50–60% of those under no load. 2. Permanent-magnet synchronous motorized spindle The stator of permanent-magnet synchronous motorized spindle is basically the same as that of three-phase asynchronous motorized spindle and presents three-phase symmetrical structure. The rotor is a magnetic pole and can be classified into two types according to the rotor structure: salient pole and hidden pole. Salient pole is suitable for low-speed operation, whereas hidden pole is suitable for high-speed operation.

1.1 The Structure and Working Principles of Motorized Spindle

9

Therefore, the hidden pole is mainly used for motorized spindle, with permanent magnet excitation mode. When three-phase symmetrical current is passed into the stator windings of threephase synchronous motorized spindle, a rotating fundamental magnetic field is generated in the stator windings and the synchronous rotational speed is the same as Eq. (1.1). The rotor is made of permanent magnet, characterized by no excitation loss and high efficiency. Under the interaction of stator magnetic field and rotor permanent magnet, the rotor is pulled by the rotating fundamental magnetic field of stator to rotate at synchronous rotational speed. The rotational speed of rotor n 2 = n 1 , meaning that the rotor rotates at synchronous rotational speed.

1.1.3 Technical Parameters of Motorized Spindle The technical parameters of different motorized spindles are different. The technical parameters of motorized spindle used for grinding and cutting mainly include outer diameter of installation, maximum rotational speed, output power at maximum rotational speed, and lubrication method. The technical parameters of motorized spindle used for machining center mainly include outer diameter of installation, maximum rotational speed, calculated rotational speed, speed range at constant power, rated power, calculated torque, Dm · n value, etc. These technical parameters are defined as follows: (1) The outer diameter of installation refers to the diameter of the outermost sleeve of motorized spindle, i.e., the diameter of the outer casing of motorized spindle. (2) Calculated rotational speed is also called rated rotational speed or base speed. From a perspective of motor design, it refers to the rotational speed at the turning point of constant torque and constant power when the motor is working under continuous conditions (Figs. 1.6 and 1.7). It corresponds to point A in Fig. 1.7. When rotational speed is lower than the calculated value, the drive mode is a constant-torque mode. When rotational speed is higher than the calculated Fig. 1.6 Power and torque curves

10

1 Key Technologies and Performance of Motorized Spindle

Fig. 1.7 Power-rotational characteristics

(3) (4)

(5)

(6)

(7)

value, the drive mode is a constant-power mode. At this turning point, both the torque and power of motor are maximized and the optimal working efficiency is achieved. From a perspective of use, this point is preferably near the working rotational speed of motorized spindle. Calculated torque refers to the torque when rotational speed is no higher than the calculated rotational speed. Maximum rotational speed refers to the highest rotational speed that motorized spindle can achieve and is the limit of rotational speed at which the motorized spindle can still maintain a normal operation. At this rotational speed, the load capacity and working efficiency of motorized spindle are both low. Therefore, motorized spindle should not operate at this speed for a long time. Rated power refers to the working capacity of motorized spindle, denoted as P. It usually varies with power source frequency and voltage (except during speed adjustment at constant power). The nominal power on the nameplate of motorized spindle is the full-load output power at nominal voltage and rotational speed. The output power of motorized spindle generally decreases with decreasing rotational speed. This should be taken into account during selection of motorized spindle. Output torque indicates the output force of motorized spindle, denoted as M. The torque indicators of motorized spindle include maximum torque and rated torque. The maximum torque indicates the overload capacity of motorized spindle and rated torque indicates the load capacity of motorized spindle. If the torque borne by motorized spindle exceeds the maximum value, its rotational speed will decrease dramatically, even to zero. The maximum torque of motorized spindle is about two times of its rated torque. One should note that the instantaneous maximum torque under load cannot exceed the maximum torque of motorized spindle and its working torque should be slightly lower than its rated torque during spindle selection and use. Dm · n (where Dm is the medium diameter of bearing and n is the working rotational speed of motorized spindle) is an important characteristic parameter that reflects the power and rotational speed of motorized spindle. The power and rotational speed of motorized spindle are limited by its volume and bearing.

1.1 The Structure and Working Principles of Motorized Spindle

11

The greater the Dm · n value, the higher the performance requirement of motorized spindle. For regular motorized spindle, Dm · n = 1 × 106 and oil-air or oil-mist lubrication is used. Under a given value of Dm · n, the higher the n, the smaller the Dm , the lower the power and the poorer the rigidity. Therefore, one should not take high rotational speed as the only indicator when selecting motorized spindle. Table 1.2 lists the types and main specifications of some motorized spindles used for machining center and milling machine by a company.

1.1.4 The Trend of Motorized Spindle Technology In recent years, higher and higher requirements have been put forward for the performance of NC machine tool spindles in practical application with the development of machine tool technology, NC technology and high-speed cutting technology, as well as the solution of key technical challenges that restrict the development of motorized spindle. The trend of motorized spindle technology is mainly reflected in the following aspects: (1) High speed and high rigidity NC machine tool spindle has been developing towards high speed with the development of spindle bearing and lubrication technology, precision machining technology, dynamic balance technology, high-speed tool and tool interface technology, etc. The power and rotational speed of motorized spindle are limitted by its volume and bearings. Dm · n is an important comprehensive parameter that reflects the rigidity and rotational speed of motorized spindle. The greater the value of Dm · n, the higher the performance of motorized spindle. Therefore, when a high rotational speed is ensured, increasing the diameter and rigidity of motorized spindle is also one of the directions of motorized spindle development. (2) High speed high power or low speed high torque Modern NC machine tool should meet the requirements of both heavy cutting (lowspeed rough machining) and fine cutting (high-speed fine machining). Consequently, motorized spindle for NC machine tool should have low-speed high-torque performance or high-speed high-power performance. High-speed high-power motorized spindle has already been a direction of development of global machine tool industry. In recent years, the development of high-power semiconductor devices has been very rapid and the power levels of motorized spindle required in various applications can be satisfied, which lays a foundation for the development of high-speed motorized spindle towards high power. Table 1.3 lists the main parameters of motorized spindles of NC machine tools used in different countries.

180

185

200

HCS180-30000/16

HCS185g-8000/11

HCS200-18000/15

18,000

8000

30,000

20,000

15,000

40,000

24,000

18,000

60,000

50,000

42,000

Maximum rotational speed (r/min)

15

11

16

18

15

60

27

9

5.5

11

11

Rated power (KW)

1800

2130

15,000

12,000

6000

40,000

18,000

7500

60,000

30,000

30,000

Calculated rotational speed (r/min)

80

53

10

14

24

14

14

11

0.9

3.5

3.5

Calculated torque (N m)

OL

G

OL

G

G

OL

OL

G

OL

OL

OL

Lubrication

HSK-A63

HSK-A63

HSK-A50/E50

HSK-F63

HSK-A63

HSK-A50/E50

HSK-A63

HSK-A50

HSK-E25

HSK-E25

SK30

Tool interface

Note HCS-vector drive; OL-oil/air lubrication; G-permanent grease lubrication; SK-IS0-taper degree. All motorized spindles listed in this table adopt ceramic ball bearings

170

HCS170g-20000/18

HCS170-24000/27

170

170

HCS150g-18000/9

170

150

HC120-60000/5.5

HCS170g-15000/15

120

HC120-50000/11

HC170-40000/60

120

120

HC120-4200/11

Installation diameter (mm)

Type

Table 1.2 Types and main specifications of some motorized spindles used for machining center by a company

12 1 Key Technologies and Performance of Motorized Spindle

1.1 The Structure and Working Principles of Motorized Spindle

13

Table 1.3 The main parameters of motorized spindles of NC machine tools used in different countries Parameters company

Rotational speed (10,000 rpm)

Motor type

Luoyang Bearing Research Institute Co., Ltd. (China)

1.5

Induction motor

IBAG (Switzerland)

6

GMN (Germany)

Maximum power (KW)

Maximum torque (N m)

Spindle bearing

Lubrication method

22

200

Ceramic/steel

Grease/oil-mist

80

320

Ceramic/steel

Oil-air

>4.6

150

1250

Ceramic/steel

Oil-air

GAMFIOR (Italy)

>7.5

68

573

Ceramic/steel

Oil-air

Fisher (Switzerland)

6

20

450

Ceramic/steel

Oil-air

(3) Diversification of motor types and control methods Induction motor is frequently used for motorized spindle. However, limited by its structure and characteristics, induction motor can hardly be operated with optimum efficiency as the operating status changes, thus resulting in low power factor and low efficiency. Although the efficiency of motor system is improved by variable frequency speed regulation, vector control, power factor compensation, etc., the improvement is limited due to the working principles of induction motor. It sometimes cannot meet the requirements of systems when applied to high-precision high-speed motorized spindle systems with high requirements for position and speed. Therefore, using permanent magnet motor with low moment of inertia, high torque density and high control precision to replace induction motor is also an important direction of motorized spindle development. Vector control has been applied to most high-speed motorized spindles manufactured. In addition, adaptive control, direct torque control and stator optimization are also adopted to improve the performance of motorized spindle using induction motor. Constant torque control is often used for permanent magnet synchronous motor in heavy cutting (low-speed rough machining), whereas constant power control is used for fine cutting (high-speed precision machining). In addition, expanding the weak magnetic field of permanent magnet motor and at the same time improving its stability will also be a hot issue in high-speed motorized spindle research. (4) Intelligentization Intelligent machine tool requires intelligent motorized spindle. Currently, the development of intelligent motorized spindle mainly focuses on thermal deformation control, dynamic balance and fault diagnosis. Sensors are employed to detect the temperature and vibration of motorized spindle. Besides, intelligent algorithms are

14

1 Key Technologies and Performance of Motorized Spindle

utilized to determine spindle state, present early warning of faults, detect thermal deformation and provide online dynamic balancing solution.

1.2 Common Key Technologies for Motorized Spindle High-speed motorized spindle unit is a highly mechatronic system, consisting of mechanical system, control system, drive system and detection system (Fig. 1.8). The high operating speed leads to strict requirements for the design, manufacture and control of motorized spindle and brings about many technical challenges. The key technologies for high-speed motorized spindle include (1) high-speed high-precision ceramic bearing technology, (2) oil/air lubrication and cooling technology, (3) highspeed motor and drive technology, (4) engineering ceramic machining technology, as well as (5) precision manufacturing and assembly technology. In addition, motorized spindle also relies on supporting technologies such as (1) built-in pulse coder, (2) automatic tool change, (3) online automatic dynamic balancing, (4) axial positioning and accurate compensation, (5) temperature rise and vibration monitoring, (6) gas sealing of spindle ends, (7) cone purge flow systems, (8) fault monitoring and diagnosis, and (9) other security technologies.

Fig. 1.8 Schematic diagram of the whole motorized spindle system

1.2 Common Key Technologies for Motorized Spindle

15

1.2.1 Spindle Bearing Technology High-speed high-precision bearing is the core support component of high-speed motorized spindle. The design and selection of bearing is crucial in order to meet the high-speed and high-rigidity requirements. Notably, the maximum rotational speed of motorized spindle depends on the size, layout and lubrication methods of bearings. Therefore, for high-speed high-precision motorized spindle, bearings need to have good high-speed performance, high dynamic load capacity, good lubrication performance and low heat generation. Currently, the bearings used for motorized spindle mainly include rolling bearing, hydrostatic bearing and magnetic suspension bearing. 1. Support type of motorized spindle (1) Rolling bearing support Angular contact ball bearing is a type of rolling bearing that is most suitable for high-speed operation. It has many advantages such as low frictional resistance, low power consumption, low cost, easy serialization and standardization and has thus been widely applied to machining center, NC milling machine, NC turning machine, NC internal (cylindrical) grinding machine, and high-speed engraving and milling machine. Although rolling bearing has good high-speed performance, it can produce high centrifugal force and gyro moment during high-speed rotation and dynamic load finally exceeds cutting load. In the design of high-speed rolling bearing, hollow rolling body or engineering ceramics can be used to reduce the friction loss and inertia force of bearing during high-speed rotation and improve its dynamic performance. Currently, hybrid ceramic ball bearing is the most frequently used high-speed rolling bearing. In such type of bearing, the rolling bodies are hot-pressed Si3 N4 ceramic balls and bearing rings are steel rings. Since ceramic ball has a small expansion coefficient and its elastic modulus is 1.5 times of that of steel, the raceway gap changes little during operation and heat generation. Compared with steel bearing with the same specifications and precision, ceramic ball bearing enables higher rotational speed, longer service life and lower spindle temperature. Compared with steel ball bearing, hybrid ceramic ball bearing specifically has the following advantages: • The hybrid bearing made of ceramic and steel has good friction performance and the stress between materials and lubricant is reduced. • Since ceramic material has a low density, the generated centrifugal force is low during bearing operation. • The relatively small thermal expansion coefficient of ceramic material enables a low thermal sensitivity of bearing, small deformation of bearing, small changes of preload and good stability of rigidity during high-speed rotation. • The relatively high elastic modulus of ceramic material can improve the rigidity of bearing. Notably, if the inner and outer rings of bearing are also made of ceramic material, the centrifugal expansion of them or raceway, the change of preload, the contact area

16

1 Key Technologies and Performance of Motorized Spindle

between the balls and the raceway, heat generation and thermal expansion will all decrease. This kind of ceramic bearing enables higher rotational speed than ceramic ball bearing. During design, however, it is necessary to ensure that the stress between ceramic material and metal material cannot exceed the tensile strength of ceramic material when they are fitted together. (2) Magnetic suspension bearing support Magnetic suspension bearing is also called magnetic bearing. It relies on multiple pairs of electromagnets or magnetic poles that are placed oppositely on a circumference at a central angle of 180° to generate suction (or repulsion) forces to suspend the shaft in the air. The top of the shaft does not contact with the bearing and the radial clearance is about 1 mm. When load is applied, the position of shaft changes slightly and the change value is measured by position sensor. Through the electronic automatic control and feedback device, the suction (or repulsion) force of corresponding magnetic pole is changed so that the shaft can quickly move to the original position. Therefore, the shaft always rotates at high speed around its inertia axis. This kind of bearing is also called active magnetic bearing and its working principles are shown in Fig. 1.9. Magnetic suspension bearing is a new high-performance bearing that uses controllable electromagnetic force to suspend shaft in the air and requires no mechanical contact between the shaft and the stator. Since there is no mechanical contact, the shaft can achieve extremely high rotational speed. It has many advantages such as low mechanical abrasion, low energy consumption, small noise, no need for lubrication, no oil pollution, etc. Besides, magnetic suspension bearings are controllable bearings and the rigidity and damping of shaft are adjustable, which are conducive to machining.

Fig. 1.9 Schematic diagram of magnetic suspension bearing (only one electromagnet of a pair placed in opposite sides is shown here)

1.2 Common Key Technologies for Motorized Spindle

17

When motorized spindle using magnetic suspension bearings rotates in the air, its Dm · n value is about 1–4 times of that of rolling bearing spindle and the highest linear velocity reaches 200 m/s (whereas that of ceramic ball bearing spindle is 80 m/s). Magnetic suspension bearing makes it easy to achieve online monitoring and diagnosis of motorized spindle. Since the position of the axis of magnetic suspension bearing is automatically adjusted by electronic feedback control system, the rigidity can be arbitrarily set, and automatic dynamic balance and active control of damping can be achieved to reduce spindle vibration to a very low level. The magnetic suspension bearing spindle is characterized by low temperature rise, high rotational accuracy (even reaching 0.1 μm) and small changes of axial size of spindle, thus it is a very promising motorized spindle type. However, practical applications show that the electromagnetic measurement and control system of this kind of bearing is very complicated, the technology is not mature enough and the cost is also very high. As a result, magnetic suspension bearing is only used in certain cases. (3) Fluid bearing support Hydrodynamic or hydrostatic bearing spindle uses “oil film” as support and has significant “error averaging effect” and good damping. Its rotational accuracy is far higher than that of rolling bearing spindle. Due to its high rigidity, small wear loss and long service life, it has been widely applied to precision and ultra-precision machine tools. The technological challenge of fluid bearing spindle lies in the control of temperature rise and thermal deformation of spindle during high-speed rotation. Gas bearing spindle uses “gas film” as support and its rotational accuracy and maximum rotational speed are higher than those of hydrodynamic and hydrostatic bearing spindles as well as rolling bearing spindle. It also has good thermal stability and is an indispensable core component of ultra-precision machine tools and printed circuit board (PCD) drilling machines. Hybrid bearing is a new multi-wedge oil film bearing that integrates the advantages of both hydrostatic and hydrodynamic bearings, avoids the severe heat generation of hydrostatic bearing during high-speed rotation and the complexity of its oil supply system, and solves dry friction that may occur when hydrodynamic bearing starts and stops. It has good high-speed performance and wide speed range, and can be applied to both high-power rough machining and highspeed precision machining. However, such bearings must be specially designed and separately produced, with low degree of standardization and difficult maintenance. Gas bearing spindle has been widely applied to ultra-precision machine tools and high-speed machine tools, such as diamond lathe for processing large optical lenses used in the aerospace and national defense fields, ultra-precision grinding machine for processing small aspherical optical lenses used in the electronics industry, and high-speed drilling machine for drilling printed circuit boards. The rotational speed of aerostatic bearing spindle can reach 100,000–200,000 r/min. However, their disadvantages include: ➀ Low rigidity and low bearing capacity. The technology of high-efficiency highspeed motorized spindle is mainly limited by the critical rotational speed of system and speed-torque characteristics of motor. The critical rotational speed

18

1 Key Technologies and Performance of Motorized Spindle

of gas bearing spindle depends on the rigidity of gas film and mass distribution of rotor. The drag capacity of motor depends on the properties of electromagnetic materials and electromagnetic design technology. High critical rotational speed is a prerequisite for achieving the high speed of spindle. The drag capacity of motor indicates the ability of spindle to operate at high speed. Both of them are indispensable indicators. ➁ Temperature effect. Temperature effect refers to the decrease in bearing performance due to variation of temperature caused by heat source and cold source. Although the temperature variation caused by the heat and shock wave generated during viscous shearing is small, the resulted deformation is in the same order of magnitude as the thickness of gas film and thus cannot be ignored. The heat generated can be coupled together through transfer, which makes the temperature distribution of the motorized spindle system, especially the gas film, more complicated. It is thus necessary to construct an effective temperature field model based on multivariate index. ➂ Inertia effect. Under high gas supply pressure, the inertia of the airflow caused by the high-speed rotation of gas bearing spindle will cause changes in the bearing capacity of bearing. The influence varies with bearing structure and increases with the increase of rotational speed. ➃ Low rotational accuracy. The rotational characteristics of ultra-precision machine tool spindle are determined by the rotational characteristics of rotor and the bearing characteristics of bearing. Some research analyzes the influence of rotor’s rotational characteristics on rotational accuracy by experiments. Besides, classical fluid lubrication theory is used to analyze the influence of the geometric error and imbalance amount of bearing-rotor system on rotational accuracy. Also, the two-dimensional mechanical analysis of bearing-rotor system is performed to analyze the influencing factors of rotational accuracy. However, the research results can only partly explain the influences of various components on rotational accuracy. They cannot effectively explain the degree of influence of each component, nor can they explain the formation mechanism of the rotational accuracy of spindle system. Therefore, it remains difficult to control the rotational accuracy of spindle system. 2. Rolling bearing configuration and preloading According to load magnitude, form, rotational speed, etc., the bearing configurations shown in Fig. 1.10 are generally adopted. The configuration in Fig. 1.10a is only applicable to motorized spindle with low load capacity used for grinding. If motorized spindle uses angular contact ball bearings, then it must be preloaded in the axial direction in order to work properly. Preloading cannot only eliminate the axial clearance of bearing, but also improve bearing rigidity and spindle rotational accuracy and inhibit vibration and slippage during self rotation of steel balls. Generally, a higher preload is more conducive to improving rigidity and rotational accuracy. However, the higher the preload, the greater the temperature rise. This can possibly result in burn damage, reduction of service life and even failure to work. Therefore,

1.2 Common Key Technologies for Motorized Spindle

19

Fig. 1.10 Frequently used bearing configurations for motorized spindle

an optimal preload value should be specifically designed for bearing according to the rotational speed and load capacity of motorized spindle. Rigid preloading is often adopted for motorized spindle with relatively low rotational speed and relatively narrow speed range. Specifically, preload is applied by utilizing the difference in width between the inner and outer spacers or the inner and outer rings of bearing. Although this method is simple, the preload will change when the lengths of shaft parts change due to heat generation. For motorized spindle with high rotational speed and wide speed range, elastic preloading should be adopted to reduce the influence of temperature or speed on preload. Elastic preloading is applied through springs. When each of the above two methods is used, the value of preload cannot be changed or adjusted after the assembly of motorized spindle. For motorized spindle with high requirements for use performance and service life, adjustable preloading device is adopted (Fig. 1.11). At the highest rotational speed, preload is applied by

20

1 Key Technologies and Performance of Motorized Spindle

Fig. 1.11 Schematic illustration of adjustable preloading device

Fig. 1.12 Schematic illustration of preloading device

springs. At relatively low rotational speed, oil pressure or air pressure is applied onto the piston to increase or decrease the preload so as to achieve the optimal preload value for the corresponding rotational speed. Figure 1.12 shows the schematic of preloading device.

1.2.2 Motor and Control Technology The design of motor determines the highest power, torque and performance of motorized spindle. Selecting a proper motor type and designing the electromagnetic parameters of motor are of great significance for improving the power density of motorized spindle and decreasing its volume and moment of inertia. Asynchronous motor with constant frequency is designed according to a given value of frequency. The motor frequency is allowed to change only within a small range during actual use and the variation range generally does not exceed 1% of the rated frequency. Otherwise, the performance of the motor will change greatly.

1.2 Common Key Technologies for Motorized Spindle

21

According to working requirements, the motorized spindle motor should have a wide speed range. Frequency converter can be used to achieve wide speed range and high speed during operation. Therefore, the motorized spindle motor changes with the change of power supply frequency and its output characteristics can meet the requirements for use. This requires that the ratio of motor voltage to motor frequency remains constant. In other words, the magnetic density of electromagnetic field inside the motor should remain constant when the frequency changes [3, 4]. During operation of high-speed motorized spindle, the basic electrical loss increases with the increase of winding current and the alternating frequency of magnetic flux in iron core, and there is also additional high frequency loss. Especially, the windage loss on rotor surface and bearing loss caused by the high-speed rotation of rotor account for a high proportion in total loss. Such losses are closely related to the rotational speed and heat dissipation conditions of the motorized spindle. It is necessary to couple electromagnetic field, stress field and temperature field to analyze and calculate the high frequency and high speed loss of stator and rotor, and temperature rise distribution. Compared with those of ordinary motor, the power density and loss of motorized spindle motor increase, while its heat dissipation area decreases. These make it more difficult to design a proper heat dissipation and cooling method for motorized spindle. The differences in design method between high-speed motorized spindle motor and traditional constant frequency motor relying on sinusoidal power supply lie in that: (1) There is no need to consider the problem of starting during design. Constant torque speed regulation is performed during low-speed operation and sufficient torque can be output. (2) The rotor grooves should be designed as shallow grooves to reduce skin effect and reduce rotor copper loss. (3) Radial size should be increased and axial size should be decreased to improve efficiency, output power, output torque and the critical rotational speed of rotor. (4) Design benchmark is generally not at 50 Hz. (5) The influence of higher harmonics should be considered. (6) Traditional electromagnetic design of asynchronous motor can be used at a certain working point. When frequency changes, the steady-state operational characteristics of motor at different frequencies should be calculated. Generally, the performance of motorized spindle is comprehensively described using the curves of power (P), torque (M), voltage (U), and current (I) versus rotational speed (n) (Fig. 1.13). According to these curves, we can obtain accurate information about the working performance of motorized spindle. In fact, the performance of motorized spindle is not only related to motor, but also related to the control mode that drives the operation of motor. For AC asynchronous induction motor, the frequently used drive control methods include scalar control, vector control, and direct torque control.

22

1 Key Technologies and Performance of Motorized Spindle

Fig. 1.13 Characteristic curves of motorized spindle

1. Scalar control Ordinary frequency converter adopts scalar control, characterized by constant torque drive, and the output power is proportional to the rotational speed. The relationships of torque and power with rotational speed are shown in Figs. 1.14 and 1.15, respectively. The output power under this kind of drive is unstable, especially at low speed, which fails to meet the low-speed high-torque requirement. Moreover, it does not enable accurate stopping of spindle and C-axis function. However, the cost of it is low. It is generally applied to high-speed motorized spindle for grinding, small hole drilling, engraving and milling. Fig. 1.14 The relationship between the torque and rotational speed of motorized spindle under scalar control

1.2 Common Key Technologies for Motorized Spindle

23

Fig. 1.15 The relationship between the power and rotational speed of motorized spindle under scalar control

According to the definitions of ISO, S1 means that the load remains constant during the whole (100%) running time of motor, and S6 means that the load changes during the running time of motor. Specifically, in each 2-min cycle, the motor bears load for 60% of the running time and bears no load for the remaining 40% of the running time. This operation mode of motor is called S6 . When motorized spindle is applied to machine tool, the load it bears often changes (when positioning, returning, tool change, etc. are performed, the machining process will be paused for a short time), thus it is economical to select power and torque according to S6 . The S1 and S6 data and peak data of motorized spindle provided by its manufacture are used to indicate its load capacity and overload capacity. In order to improve the drive quality of motorized spindle, a new frequency converter using a control strategy of “voltage/frequency = constant” has been developed. It enables constant-power drive of motor at a speed higher than the calculated rotational speed. Below the calculated rotational speed, torque increases rapidly from zero to achieve constant-torque drive with increase in rotational speed (Fig. 1.16).

Fig. 1.16 The curves of torque and power versus rotational speed for the new frequency converter

24

1 Key Technologies and Performance of Motorized Spindle

2. Vector control In vector control, constant-torque drive is used at low speed, and constant-power drive is used at medium and high speed. The relationships of torque and power with rotational speed are shown in Figs. 1.17 and 1.18, respectively. In vector control, the power and torque may decrease slightly at high speed or the highest speed (Figs. 1.19 and 1.20). When the vector control drive just starts, the torque is high (Figs. 1.17 and 1.19). Besides, the moment of inertia of motorized spindle is low. These ensure that the torque can instantaneously reach the maximum value. The vector control drive can be classified into two types: open-loop type and closed-loop type. The latter is equipped with a position sensor on the shaft to provide position and speed feedback. It not only has better dynamic performance, but also can achieve accurate stop at a certain preset position and C-axis function. As a comparison, scalar control drive enables a slightly poorer dynamic performance and does not enable accurate stop and C-axis functions, but its cost is relatively low. The three curves in each of above figures correspond to S1 , S6 and peak value. This is because both motorized spindle and the drive have the ability to allow shorttime or instantaneous overload. This feature can be utilized to maximize the working potential of motor and drive in different situations and thus increase economic benefit. Fig. 1.17 The relationship between the torque and rotational speed of motorized spindle under vector control

Fig. 1.18 The relationship between the power and rotational speed of motorized spindle under vector control

1.2 Common Key Technologies for Motorized Spindle

25

Fig. 1.19 The relationship between the torque and rotational speed of motorized spindle under vector control

Fig. 1.20 The relationship between the power and rotational speed of motorized spindle under vector control

Since permanent magnet motor is limited by power, motorized spindle mostly used asynchronous motor in the past and adopted vector closed-loop control to meet the requirements of accurate stop and rigid tapping. With the continuous enhancement of the performance of permanent magnet motor, it currently enables higher control accuracy and wider speed range than asynchronous motor. Therefore, permanent magnet synchronous motors are being more and more widely applied to high-speed motorized spindles. Especially in machining center, permanent-magnet synchronous motor spindle shows advantages such as small volume and compact structure, which make it suitable for use in high-precision heavy machining. Compared with AC asynchronous motor, it has the following advantages: (1) The rotor is made of permanent magnet material and it does not generate heat during operation. When AC asynchronous motor is used, the stator can be cooled by water, but the rotor cannot be cooled sufficiently. (2) The power density is high. This means that a high power and torque can be obtained with a motor of small size, which is conducive to reducing the radial size of motorized spindle. (3) The rotational speed of rotor is strictly synchronized with the power source frequency, thus the power factor is high and the efficiency is high.

26

1 Key Technologies and Performance of Motorized Spindle

(4) Vector control can also be used and the circuit is simpler than that of asynchronous motor. The servo control of motorized spindle needs to meet the requirements of precise positioning and rapid acceleration/deceleration. Therefore, vector control algorithm is often used and the control system consists of a current inner loop, a speed loop and a position loop. The high-speed all-digital processor enables complex servo control algorithms to be implemented and ensures the fast response of motorized spindle and servo control accuracy. Currently, there is small difference in the technical level of low-power permanent magnet synchronous motor and its servo control among countries. However, there is large difference in the technical level of high-power high-speed permanent magnet motor among countries. 40 KW permanent magnet motor with speed higher than 10,000 r/min and its control technology remains a hot research topic. 3. Direct torque control Direct torque control is another new high-efficiency variable frequency speed regulation technology after the development of vector control technology. In the mid-1980s, Prof. M. Depenbrock in Ruhr-Universität Bochum of Germany and Prof. I. Takahashi from Japan proposed hexagonal direct torque control scheme and circular direct torque control scheme, respectively. In 1987, the theory of direct torque control was further applied to weak magnetic speed control. The direct torque control technology employs space vector analysis to calculate and control motor torque directly in stator coordinate system. The stator magnetic field is used for orientation and discrete two-point adjustment (Band-Band) is used to generate PWM wave signals to directly achieve optimal control of the switching state of inverter and obtain high dynamic performance of motor. This technology eliminates the complicated vector transformation procedures and the simplification procedures of the mathematical model of motor, and requires no PWM signal generator. It is characterized by novel control idea, simple control structure, direct control means, and clear physical meaning of signal processing. Even in open loop state, it can output 100% of the rated torque and enables a load balancing function during multi dragging operations. However, the major disadvantage of direct torque control is the pulsation of torque and flux linkage. See Chap. 2 for more details on the principles and drive characteristics of direct torque control.

1.2.3 Lubrication and Cooling Technology The temperature rise of high-speed motorized spindle will affect the symmetry and gradient of the temperature field of the spindle system about the axis. During temperature rise, the spindle itself will experience axial elongation, and the center positions of the front and rear supports of the spindle will change in the radial direction. These changes can lead to radial displacement of the working end of the spindle. However,

1.2 Common Key Technologies for Motorized Spindle

27

if the cooling system is unevenly distributed, the temperature field will also become uneven, thus leading to reduction of machining accuracy and permanent damage to bearings and motor. Especially, overheating can cause permanent demagnetization of the permanent magnet of permanent magnet motor, which directly affects the performance of motor and induces concentrated heating of stator and bearing in the motorized spindle. Therefore, a uniform and effective cooling system has a decisive effect on the machining accuracy of high-speed motorized spindle system. The design of cooling system is closely related to that of lubrication system. If the performance of lubrication system is high, then relatively little heat is generated, which reduces burden on the cooling system. The heating sites in motorized spindle mainly include motor stator and rotor/bearing. The cooling of motorized spindle motor and bearing is usually achieved by convection of cooling fluids. The cooling of motorized spindle stator is achieved by designing a cooling jacket outside the stator, and the cooling effect is good. The cooling of rotor is relatively difficult and normally achieved through the flow of bearing lubricating oil to take away some heat. When rolling bearing rotates at high speed, a proper lubrication method is extremely important. Without proper lubrication, the bearing will be burned out due to overheating. Currently, there are three major lubrication methods for motorized spindle: (1) Grease lubrication is a one-time permanent lubrication that does not require any additional equipment and special maintenance. However, the temperature rise is high and the allowed highest rotational speed is relatively low, with Dm · n value below 1.0 × 106 . When hybrid bearing is used, the Dm · n value can be improved by 25–35%. (2) Oil/air lubrication is a kind of new and idea lubrication method. It is academically called gas-liquid two-phase fluid cooling and lubrication technology. It uses distribution valve to regularly and quantitatively supply oil-air mixture to the various parts of bearing according to their actual needs. This cannot only ensure that the various parts of bearing are lubricated, but also prevent greater temperature rise due to excessive lubricant and minimize oil mist pollution. The Dm · n value can reach 1.9 × 106 . Figure 1.21 schematically shows the lubrication system. Following the preset working procedure, the oil pump is switched on. According to the oil demand of the system that needs to be lubricated, lubricating oil is accurately metered and distributed by a quantitative dispenser and then sent to the network connected to compressed air. Then, the lubricating oil is mixed with the compressed air to form an oil-gas flow, which flows into the oil and gas pipeline. The compressed air is processed by a compressed air treatment device. Due to the influence of compressed air, the lubricating oil moves forward in a wavelike manner along the inner wall of the pipeline and gradually forms a continuous thin oil film. As the oil-air mixture enters bearing, its various parts are lubricated. Notably, external debris and water are unable to enter the bearing because the compressed air entering it ensures a positive pressure on the lubricating sites. This suggests a good sealing effect. Figure 1.22 schematically shows the oil/air lubrication device.

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1 Key Technologies and Performance of Motorized Spindle

Fig. 1.21 Schematic of lubrication system

Fig. 1.22 Schematic of oil/air lubrication device

1.2 Common Key Technologies for Motorized Spindle

29

Fig. 1.23 The working principles of oil mist lubrication

(3) Oil mist lubrication. Oil mist lubrication and oil/air lubrication both belong to gas-liquid two-phase fluid cooling and lubrication technology. The working principles of oil mist lubrication system are shown in Fig. 1.23 [5]. After the compressed air enters the valve body 1 through air inlet, it enters the nozzle cavity along the air inlet hole of nozzle 3 and is ejected from venturi tube 4 into atomization chamber 5. Then, a negative pressure is generated in vacuum chamber 2. This negative pressure enables lubricating oil to enter the vacuum chamber 2 through oil filter 8 and oil injection tube 7, and then to drop into the venturi tube 4. Subsequently, the oil droplets are sprayed into uneven oil particles by the airflow and then enter the upper portion of oil reservoir 9 from the mist vent of spray cover 6. Large oil particles fall back into the lower portion of oil reservoir 9 due to gravity effect. Only particles smaller than 3 μm remain in the gas to form an oil mist, which is then transported to the lubrication site along with the compressed air. In order to deliver lubricating oil to friction site, the lubricating oil is first atomized into very fine oil particles in an atomizing device. The surface tension of the atomized lubricating oil particles is greater than the attractive force of lubricating oil particles. Therefore, the atomized lubricating oil is in a gaseous state. In this state, the lubricating oil can be transported by the atomizing device and dispenser to reach

30

1 Key Technologies and Performance of Motorized Spindle

various friction sites. However, the oil mist does not completely form oil droplets after reaching the friction sites and this is not conducive to the formation of oil film required for lubrication. Therefore, a condensation nozzle is required to condense the oil mist to form oil droplets. Notably, oil mist can be transported only at a low speed because it can remain stable only under laminar flow conditions. In a turbulent state, the lubricating oil particles will accumulate due to collision among them and then form a large droplet, thus returning to a liquid state. In this liquid state, lubricating oil flows back to the container. Since the pressure of the oil mist is very low, the oil pipeline inside the motorized spindle must have a large cross-sectional area in order to overcome the resistance of the oil mist flow. In the pipeline, the oil in mist form is mixed with compressed air and they are transported in the same speed. Therefore, the air discharged from the lubrication sites contains tiny oil particles, which can cause environmental pollution and seriously endanger human health. Table 1.4 shows the comparison of oil mist lubrication and oil/air lubrication.

1.2.4 Dynamic Balance Technology The dynamic characteristics of motorized spindle largely determine the machining quality and cutting ability of machine tool. In the case of high-speed rotation or cutting, any disturbance of the motorized spindle itself or the environment will cause vibration of motorized spindle. Excessive vibration can cause abrasion and damage to the motorized spindle, increase the dynamic load borne by it, reduce its lifetime and accuracy and affect the stability of its dynamic balance. Therefore, research on the dynamic balance of motorized spindle is also a hot spot in the field of high-speed motorized spindle system. During the design of motorized spindle, it is necessary to fully consider forced vibration and self-excited vibration. The various parts of designed spindle should have good resistance to both types of vibrations. This ensures the good operational accuracy and transmission capability of spindle during high speed cutting. At high rotational speed, even a small amount of imbalance can cause the vibration of motorized spindle, thus affecting machining quality and accuracy. The dynamic performance of motorized spindle is determined by the amount of imbalance and the square of spindle’s rotational speed. At the same time, the motor rotor is directly fixed on the shaft, which increases the rotational mass of spindle. Therefore, highspeed motorized spindle has strict requirements for dynamic balance accuracy to ensure a good dynamic performance and machining accuracy at high rotational speed. Generally, the dynamic balance accuracy of the high-speed motorized spindle should generally reach G0.1–G0.4 (G = eω, where e is the displacement, i.e., eccentricity, between the center of mass and the center of rotation; ω is the angular velocity). Therefore, the amount of imbalance should be minimized during both the design and manufacture of motorized spindle. Strict symmetric design should be adopted. Since key connection and threaded connection are the main causes of dynamic imbalance

1.2 Common Key Technologies for Motorized Spindle

31

Table 1.4 Comparison of oil mist lubrication and oil/air lubrication Items

Oil mist lubrication

Oil/air lubrication

Fluid form

General gas-liquid two-phase fluid

Typical gas-liquid two-phase fluid

Lubricant pressure

0.04–0.06 bar

2–10 bar

Gas flow rate

2–5 m/s (lubricant and air are mixed to form oil mist; gas flow rate = lubricant flow rate)

30–80 m/s (lubricant is not atomized and the gas flow rate is much higher than the lubricant flow rate), 150–200 m/s in special cases

Lubricant flow rate

2–5 m/s (lubricant and air are mixed to form oil mist; gas flow rate = lubricant flow rate)

2–5 cm/s (lubricant is not atomized and the gas flow rate is much higher than the lubricant flow rate)

Heating and condensation

Lubricant is heated and condensed

Lubricant is not heated and condensed

Adaptability to lubricant viscosity

Only applicable to low viscosity lubricant ( 2200, the cooling water is in a turbulent flow state, and the Nusselt number is calculated by empirical formula 

N u = 0.012 Re

0.87



− 280 Pr

0.4

h3 =

1+

N uλw D



D l

 23 

Pr Prw

0.11 (3.23) (3.24)

where h 3 is the convective heat transfer coefficient between the stator and cooling water (unit: W/(m2 °C)); Nu is the Nusselt number of the fluid; Pr is the Prandtl number of the fluid; when the difference in temperature of the fluid is not significant, μf μw ≈ 1.05; Prf Prw ≈ 1[22]; λw is the thermal conductivity of water (unit: W/(m °C)).

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3 Heat Generation and Transfer of Motorized Spindle

Fig. 3.4 Heat transfer of stator and rotor

4. The convective heat transfer coefficient between stator-rotor gap and compressed air For a motorized spindle with axial ventilation, the convective heat transfer between the rotor and stator consists of two parts: one is the convective heat transfer generated by the circumferential speed of rotor and the other is the convective heat transfer generated by the axial speed of stator-rotor gap. Figure 3.4 shows the stator speed and rotor speed of motorized spindle, and the average speed is 1 2  v = va2 + vr2 /

(3.25)

The convective heat transfer coefficient between stator-rotor gap and compressed air can be expressed as [23]: h4 =

N uλa H

(3.26)

where  0.25 δ Re0.5 N u = 0.239 r Re =

uH r

(3.27) (3.28)

where h 4 is the convective heat transfer coefficient between stator-rotor gap and compressed air (unit: W/(m2 ·°C)); λa is the thermal conductivity of air (unit: W/(m °C)); r is the radius of rotor outer surface (unit: m); δ is the gap between the stator and rotor (unit: m); H is the qualitative dimension of geometric characteristic of the air gap (unit: m); Re is the Reynolds coefficient; Nu is the Nusselt number. 5. The convective heat transfer coefficient between motorized spindle surface and surrounding air During the operation of motorized spindle, as the internal temperature of spindle increases, the temperature of its outer surface also increases to a certain extent. Then,

3.2 Heat Transfer in Motorized Spindle and Calculation …

111

temperature difference occurs between the outer surface of motorized spindle and the surrounding air, leading to heat convection between them. It is assumed that the heat transfer between them is natural heat convection, and the composite convective heat transfer coefficient is [24, 25]. h 5 = 9.7

(3.29)

where h 5 is the convective heat transfer coefficient between the motorized spindle surface and surrounding air (unit: W/(m2 °C)).

3.3 The Finite Element Formula of Temperature Field of Motorized Spindle The heat transfer process of motorized spindle is very complicated, and energy conservation law is followed during both heat generation and heat dissipation. The heat transfer of motorized spindle is mainly in the ways of heat conduction and convection, and the energy conservation equation can be expressed as [26] ρ1 C p1

∂T + ρ2 C p2 v · ∇T = ∇ · (k∇T ) + Q ∂t

(3.30)

where Q = Ptot V , Ptot is the heat amount of heat source (unit: W); V is the volume of heat source (unit: m3 ); Q is the heat transfer rate (unit: W/m3 ); ρ 1 is the density of solid (unit: kg/m3 ); ρ2 is the density of fluid (unit: kg/m3 ); C p1 and C p2 are heat capacities of solid and fluid at atmospheric pressure (unit: J/(kg °C)); T is the temperature of motorized spindle (unit: ° C); v is the speed of fluid (unit: m/s); ∇ is the Laplace operator; k is the thermal conductivity (unit: W/(m °C). According to the heat transfer theory, the relationship between the convective heat transfer coefficient and the temperature can be expressed as q = h · (Text − T )

(3.31)

where q is the convective heat transfer flux (unit: W/m2 ); h is the convective heat transfer coefficient (unit: W/(m2 °C)); Text is the temperature of fluid medium (unit: °C). According to the above heat transfer equations used in the finite element simulation, if the convective heat transfer coefficients of corresponding components and heat source of motorized spindle are determinded, the temperature field of motorized spindle can be simulated.

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3 Heat Generation and Transfer of Motorized Spindle

3.4 The Basic Theory of Thermoelasticity Motorized spindle is considered to be an elastomer. When the spindle is influenced by internal heat source and uneven temperature field is generated, the various components inside spindle will undergo different degrees of expansion, and the volume of spindle will change. However, the spindle is constrained by its shell and the mounting fixture, and there is difference in expansion ratio among various spindle components. Thus, the volume of spindle cannot be freely changed. In other words, when the temperature changes, spindle components cannot freely expand or contract due to external constraints and mutual restraint between them. As a result, thermal stress is generated and causes slight displacements of the internal components of spindle, which is called thermal deformation. Such thermal deformation can influence the machining accuracy of machine tool. The cause of thermal stress is temperature variation. The research on thermal stress involves thermodynamics and heat transfer theory in addition to the relationship between the physical properties of materials and temperature. 1. Basic assumptions of thermoelasticity To make the established model meet practical and engineering requirements and be processed effectively by mathematical methods, four basic assumptions are relied on [19]: (1) (2) (3) (4)

The continuity assumption; The uniformity assumption; The isotropic assumption; The small deformation assumption.

2. Basic equation of thermoelasticity Suppose that the spindle is in a Cartesian coordinate system and a micro unit is selected from the heat conductor. The heat conductor is assumed to be made of isotropic homogeneous material. Both the density and specific heat capacity of the material are constant, and there is a heat source inside the heat conductor. Figure 3.5 shows the heat conduction of the micro unit. In order to obtain the three-dimensional transient temperature field of motorized spindle, the following thermal equilibrium differential equation must be satisfied [20] ρc p

      ∂ ∂ ∂ ∂T ∂T ∂T ∂T = kx + ky + kz +Q ∂t ∂x ∂x ∂y ∂y ∂z ∂z

(3.32)

where ρ and c p are the density (kg/m3 ) and specific heat capacity J/(kg °C) of spindle component material, respectively; T is the temperature (°C) of spindle at position (x, y, z) in moment t; t is time; k x , k y , k z are thermal conductivities (W/(m °C)) in x, y, z directions, respectively; Q is heat source (W/m3 ) inside spindle. In the equation, the left side is about the heat required for the temperature rise of micro

3.4 The Basic Theory of Thermoelasticity

113

Fig. 3.5 The heat conduction of micro unit

unit, the first three terms on the right side are the heat transferred into the micro unit along the x, y and z directions, and the last term is the heat generated by the heat source inside the micro unit. The differential equation shows that the heat required for the temperature rise of the micro unit should be balanced with the heat generated by the heat source in the micro unit plus the heat transferred to the micro unit. It is known that the temperature rise of spindle is T and, without restraint, a positive strain of αT will be produced, where α is the coefficient of linear expansion, and in an isotropic material, it does not change with direction. This means that the positive strain is the same in all directions, thus it is not accompanied by any shear strain. Also, assume that α does not change with temperature. Therefore, the deformation component of each point in the elastomer is: εx x = ε yy = εzz = αT and εx y = ε yz = εzx = 0. Due to external constraints and mutual restraint between spindle components, these spindle components cannot freely expand or contract and thermal stress is thus generated. The thermal stress will cause additional deformation due to the elasticity of the components, and then the total deformation components are [27]. ⎧   1 ⎪ εx = σx − μ σ y + σz + αT ⎪ ⎪ ⎪ E ⎪ ⎨  1 εy = σ y − μ(σx + σz ) + αT ⎪ E ⎪ ⎪ ⎪ ⎪ ⎩ ε = 1 σ − μσ + σ  + αT z z y x E

(3.33)

where εx , ε y , εz are the deformation components (μm) in x, y, z directions, respectively; E is the elastic modulus of material;μ is Poisson’s ratio; α is the linear expansion coefficient of material; T is the temperature rise (°C) of spindle; σx , σ y , σz are the stress components in x, y, z directions, respectively. Combined with space geometry equations and space equilibrium differential equations, the above equations can be solved, and the stress distribution, strain and displacement of motorized spindle can be obtained.

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3 Heat Generation and Transfer of Motorized Spindle

Spatial equilibrium differential equations are as follows: ⎧ ∂σ x ⎪ ⎨ ∂x + ∂σ y + ∂y ⎪ ⎩ ∂σz + ∂z

∂τ yx ∂y ∂τzy ∂z ∂τx z ∂x

+ ∂τ∂zzx + f x = 0 ∂τ + ∂ xx y + f y = 0 ∂τ + ∂ yyz + f z = 0

(3.34)

Space geometry equations are as follows: ⎧ εx = ∂∂ux , ε y = ∂∂vy , εz = ⎪ ⎪ ⎪ ⎨ γ = ∂w + ∂v yz ∂y ∂z ∂u ∂w ⎪ γ zx = ∂z + ∂ x ⎪ ⎪ ⎩ γx y = ∂∂vx + ∂u ∂y

∂w ∂z

(3.35)

Theoretically, as long as the temperature distribution is given and the above differential equations are solved, the stress distribution, strain and displacement of motorized spindle can be obtained.

References 1. Gmyrek Z, Boglietti A, Cavagnino A et al (2010) Estimation of iron losses in induction motors: calculation method, results, and analysis. IEEE Trans Ind Electron 57(1):161–171 2. Zhao HS, Liu XF, Luo YL (2010) Loss characteristics of cage induction motors under voltage deviation conditions. J Electr Mach Control 14(5):13–19 3. Huang PL, Hu QS, Cui Y et al (2007) Analytical calculation of motor core loss under PWM inverter power supply. Chin Soc Electr Eng 27(12):19–23 4. Zeng LQ, Wu H, Li H (2011) The influence of output harmonics of PWM inverter on the loss of asynchronous motor. Micro-motor 4(44):68–71 5. Luo C, Wang XY, Ning PQ (2006) Rotor loss of 12-phase high speed asynchronous generator. J Tsinghua Univ (Natural Science Edition) 46(1):9–12 6. Cen ZQ (1996) Design problem of three phase variable frequency and speed regulation asynchronous motor. Motor Technol 2:23 7. Zhang LX, Wu YH, Wang LY (2011) Analysis on the influence of vibration performance of air-gap of ceramic motorized spindle. Trans Tech Publications, Chengdu, China 8. Zhang LX, Wu YH, Wang LY et al (2012) The effect of air gap eccentricity on the rotor system of ceramic electric spindle. Mech Sci Technol 31(9):1512–1515 + 1521 9. Ma P, Zhou B, Li DN et al (2011) Thermal analysis of high speed built-in spindle by finite element method. Adv Mater Res 188:596–601 10. Wu YH, Tian F, Albert JS, et al (2012) Design and experimental analysis of temperature detection module of all ceramic electric spindle based on LabVIEW. Mach Tool Hydraul 17:60–63 11. Holkup T, Cao H, Koláˇr P et al (2010) Thermo-mechanical model of spindles. CIRP Ann Manuf Technol 59(1):365–368 12. Ge XS (2007) Basic principles of heat transfer and mass transfer. Chemical Industry Press, Beijing, China 13. Li CP, Cai F, Cheng SK (2012) The effect of cooling water flow rate on the temperature rise of automotive water-cooled motor. J Electr Mach Control 16(9):1–8

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14. Wang YF, Sun QG, Lv HB (2014) Comparative study on temperature field of oil bearing lubrication and injection lubrication of rolling bearings. Lubr Seal 39(2):66–70 15. Li SX, Zhao B, Bao YP et al (2012) Simulation analysis of thermal characteristics of high-speed electric spindle. Tool Technol 4:64–32 16. Liu JX (2009) Virtual simulation analysis of temperature field of AD1130 electric spindle. Journal of Henan Mechanical and Electrical College 6(17):71–74 17. Yu L, Wei YG, Shang YQ (2006) Thermal engineering and fluid mechanics. China Electric Power Press, Beijing, China 18. Staton DA, Cavagnino A et al (2008) Convection heat transfer and flow calculations suitable for electric machines thermal models. IEEE Trans Ind Electron 55(10):3509–3516 19. Xu ZL (2001) A brief tutorial on elastic mechanics. Higher Education Press, Beijing, China 20. Li SX, Zhao B, Bao YP et al (2012) Simulation analysis of thermal characteristics of high-speed electric spindle. Tool Technol 46(8):17–20 21. Zhang LX, Liu XH (2012) Modeling and simulation analysis of motorized spindle vector control. International Conference on Mechatronics and Intelligent Materials (MIM), GuiLin, China 22. Zhang K, Tong J, Xu XH et al (2007) Modeling and simulation of high-speed electric spindle direct torque control system. J Shenyang Jianzhu Univ Nat Sci, 04:664–667 23. Zhang K, Xu XH, Wang LJ, Wu YH (2006) Design of direct torque control system for high speed electric spindle under PMAC2. J Shenyang Jianzhu Univ Nat Sci 04:691–695 24. Xie LH, Zhao XY (2012) Fluid analysis and simulation of Ansys CFX. Electronic Industry Press, Beijing, China 25. Uhlmann E, Hu J et al (2012) Thermal modelling of a hing speed motor spindle. In: 5th CIRP conference on high performance cutting 2012, pp 313–318 26. Zhang JW, Yang ZY, Zhang Z (2009) The foundation and application of numerical simulation of fluid flow and heat transfer process, vol 1. Chemical Industry Press, pp 13–36 27. Fei YT (2009) Mechanical thermal deformation theory and application. National Defense Industry Press, Beijing, China

Chapter 4

Basic Theory and Method of Spindle Dynamic Balance

Dynamic imbalance and vibration of high-speed spindle are important factors affecting machining accuracy. Equipment failures caused by vibration account for 60–70% of all equipment failures. The balance of machine tool spindle with high precision after static balancing may still be destroyed due to design, manufacturing, workpiece clamping, wear abrasion, load shock and other factors. Therefore, online dynamic balancing technology should be applied to eliminate the influence of such factors on the spindle. In this chapter, the basic theory and method of spindle dynamic balance is introduced.

4.1 Dynamic Modeling of Rigid Rotor According to the standard ISO 1925:2001, a rigid rotor is defined as a rotor whose deflection due to the distribution of the given imbalance amount is less than the allowable limit when it rotates at any speed up to the highest operating speed. Under some conditions, the rotor can be considered a rigid rotor, while under other conditions, it can not be considered a rigid rotor [1]. When lathe spindle is studied, it can be regarded as a rigid rotor. The state of rotor is determined according to three aspects: the imbalance associated with rotational speed; the type of imbalance to be corrected; and the ability to maintain or change the relative position of mass unit within operating speed range. Balancing is a process of inspecting and, if necessary, adjusting rotor mass distribution to ensure that the residual imbalance, journal vibration and forces acting on the support are below specified limits at the corresponding operating speed. Rotor imbalance can be caused by problems in the design, materials, manufacturing, and assembly of spindle. Even during the mass production, each rotor has a different mass distribution along its axial direction.

© Springer Nature Singapore Pte Ltd. 2020 Y. Wu and L. Zhang, Intelligent Motorized Spindle Technology, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3328-0_4

117

118

4 Basic Theory and Method of Spindle Dynamic Balance

In most cases, rotor imbalance does not change significantly with rotational speed. Mass unit is an effective way to describe rotor mass distribution and its possible change with rotational speed. A mass unit can be a finite element, part or component. The state of rotor is also influenced by the design, manufacture and assembly of spindle. The tolerance of rotor to the imbalance caused by the change of rotational speed and bearing support state is defined as imbalance tolerance. The speed range of rotor is from zero to the maximum speed, including overspeed as the margin of working load. The imbalance of rigid rotor does not change significantly with rotational speed, and the relative position of all mass units of the rotor remains sufficiently constant within the operating rotational speed range. The imbalance of rigid rotor can be corrected between any two planes selected. Dynamic balance is based on the mechanism of rotor imbalance response. Therefore, establishing a dynamic model of rotor system and studying relationships among rotor dynamic response, imbalance amount and rotor system parameters are important for the study of dynamic characteristics of spindle system and dynamic balance technology. By studying the relationship between the rotational speed and vibration response of rotor, parameters such as the natural frequency of rotor can be investigated, which provides a guide for avoiding the critical rotational speed. The vibration states of rotor under different working conditions can also be investigated to provide reference for rationally designing correction amount. In addition, dynamics modeling and simulation also provide useful information for selecting dynamic balancing methods and designing spindle system structures. The dynamics modeling and simulation in the following are primarily for rigid rotors. For rigid rotors, the deformation caused by gravity and imbalance is neglected, and their dynamic balance should meet the following conditions: The balance of rigid rotor is independent of rotational speed. In other words, after the rotor reaches a balancing state at a certain speed, its imbalance amount will not significantly exceed the imbalance tolerance when the rotational speed changes to other values. The typical structure of a rigid rotor system is shown in Fig. 4.1. Two coordinate systems (O-XYZ and o-xyz) are established. O-XYZ is a fixed coordinate system and o-xyz is a rotating coordinate system. Point o is located at l

Fig. 4.1 Rigid rotor model z

l1

l2 mu ■

m o k1

c1

x

y Z

X

O

k2 Y

c2

4.1 Dynamic Modeling of Rigid Rotor

119

the centroid of spindle and the two coordinate systems initially coincide. The spring stiffness and damping at the left end of rotor are k 1 and c1 , respectively; and those at the right end of rotor are k 2 and c2 , respectively. The mass of rotor is m. The imbalance mass is mu . The moments of inertia of spindle are J x , J y and J z along the x, y and z axes, respectively. The coordinate of the imbalance mass mu in o-xyz is (ux , uy , uz ). Commonly, the axial displacement of rotor is small and can be neglected, thus the state of rotor can be expressed using the position of the centroid (X, Z), the displacements of shaft ends (X 1 , Z 1 ) and (X 2 , Z 2 ), and the rotational angle between o-xyz and O-XYZ. According to the above assumptions and the dynamics of rotor, the spindle dynamics equation can be expressed as ⎧ m X¨ + c1 X˙ 1 + c2 X˙ 2 + k1 X 1 + k2 X 2 = m u ω2 (u x cos ωt + u z sin ωt) ⎪ ⎪ ⎪ ⎪ m Z¨ + c1 Z˙ 1 + c2 Z˙ 2 + k1 Z˙ 1 + k2 Z 2 = m u ω2 (u z cos ωt − u x sin ωt) ⎪ ⎪ ⎨ ¨ Jx θ − c1 X˙ 1 l1 + c2 X˙ 2 l2 − k1 X 1l1 + k2 X 2 l2 − Jy ωϕ˙ ⎪ = −m u ω2 (u z cos ωt − u x sin ωt)u y ⎪ ⎪ ⎪ ⎪ J ϕ¨ − c1 Z˙ 1l1 + c2 Z˙ 2 l2 − k1 Z 1l1 + k2 Z 2 l2 − Jy ωθ˙ ⎪ ⎩ z = m u ω2 (u x cos ωt + u z sin ωt)u y

(4.1)

where the moment of inertia of rotor around o-xyz is Jx = Jz =

 1 2 m l1 − l1l2 + l22 3

Jy =

1 2 mr 2

(4.2) (4.3)

Since l = l1 + l 2 , the rotor with small amplitude meets following: ⎧ X 1 = X − l1 sin θ ⎪ ⎪ ⎨ X 2 = X − l2 sin θ ⎪ Z 1 = Z − l1 sin ϕ ⎪ ⎩ Z 2 = Z − l2 sin ϕ

≈ X − l1 θ ≈ X − l2 θ ≈ Z − l1 ϕ ≈ Z − l2 ϕ

(4.4)

Thus, Eq. (4.1) can be rewritten as: ⎧ ¨ m X + (c1 + c2 ) X˙ + (k1 + k2 )X − (c1 l1 − c2 l2 )θ˙ − (k1 l1 − k2 l2 )θ ⎪ ⎪ ⎪ ⎪ = m u ω2 (u x cos ωt + u z sin ωt) ⎪ ⎪ ⎪ ¨ ⎪ ⎪ m Z + (c1 + c2 ) Z˙ + (k1 + k2 )Z − (c1 l1 − c2 l2 )ϕ˙ − (k1 l1 − k2 l2 )ϕ ⎪ ⎪ ⎨ = m u ω2 (u z cos ωt − u x sin ωt) ⎪ Jx θ¨ + (c2 l2 − c1 l1 ) X˙ + (k2 l2 − k1 l1 )X + (c1 l12 + c2 l22 )θ˙ + (k1 l12 + k2 l22 )θ − Jy ωϕ˙ ⎪ ⎪ ⎪ ⎪ = −m u ω2 (u z cos ωt − u x sin ωt)u y ⎪ ⎪ ⎪ ⎪ ⎪ Jz ϕ¨ + (c2 l2 − c1 l1 ) Z˙ + (k2 l2 − k1 l1 )Z + (c1 l12 + c2 l22 )ϕ˙ + (k1 l12 + k2 l22 )ϕ + Jy ωθ˙ ⎪ ⎩ = m u ω2 (u x cos ωt + u z sin ωt)u y

(4.5)

120

4 Basic Theory and Method of Spindle Dynamic Balance

To facilitate programming, we set f 0 = X , f 1 = X˙ , f 2 = θ , f 3 = θ˙ , f 4 = Z , f 5 = Z˙ , f 6 = ϕ, f 7 = ϕ. ˙ Then, the equation set can be transformed into a form of X˙ = F(X, t): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

f˙0 = f 1 f˙1 = m1 [−(k1 + k2 ) f 0 − (c1 + c2 ) f 1 + (k1l1 − k2 l2 ) f 2  +(c1 l1 − c2 l2 ) f 3 − m u ω2 (u x cos ωt + u z sin ωt) f˙2 = f 3   f˙3 = J1x (k1l1 − k2 l2 ) f 0 + (c1l1 − c2 l2 ) f 1 − k1l12 + k2 l22 f 2    − c1l12 + c2 l22 f 3 + Jy ω f 7 − m u ω2 (u z cos ωt − u x sin ωt)u y f˙4 = f 5 f˙5 = m1 [−(k1 + k2 ) f 4 − (c1 + c2 ) f 5 + (k1l1 − k2 l2 ) f 6  +(c1 l1 − c2 l2 ) f 7 + m u ω2 (u z cos ωt + u x sin ωt) f˙6 = f 7   f˙7 = J1z (k1l1 − k2 l2 ) f 4 + (c1l1 − c2 l2 ) f 5 − k1l12 + k2 l22 f 6  2   − c1l1 + c2 l22 f 7 + Jy ω f 3 − m u ω2 (u x cos ωt − u z sin ωt)u y

(4.6)

The dynamic model of rigid rotor system includes trigonometric function. Thus, its analytical solution can hardly be obtained by direct deriving. Rotor dynamics simulation can be performed based on the rotor model using commercial software (differential equation solution VI module). The block diagram of the program is shown in Fig. 4.2, and the simulation results are shown in Fig. 4.3.

Fig. 4.2 Rotor dynamics simulation program

4.1 Dynamic Modeling of Rigid Rotor

121

Fig. 4.3 Rotor dynamics simulation interface

The rotor parameters that can be set in the program include radius, rotational speed, centroid position, imbalance mass, relative position of imbalance mass, bearing stiffness and damping. The simulation results are shown as vibration waveforms in the x and z directions, as well as the variation in the position of centroid.

4.2 The Expression of Imbalance Amount According to the definition given in the standard GBT 6444-2008, the imbalance amount (unit: g mm) is expressed as the product of the imbalance mass and the distance between its centroid and the axis (radius). The imbalance phase angle represents the polar angle of the imbalance mass in the rotating coordinate system, which is in a plane perpendicular to the rotor axis and rotates with the rotor. A vector with the same magnitude of imbalance amount and along the direction of the imbalance phase angle is called the imbalanced vector [2]. The spindle is subjected to eccentric centrifugal force due to the imbalance mass during rotation. The force acts on it is shown in Fig. 4.4, and the centrifugal force can be expressed as

122

4 Basic Theory and Method of Spindle Dynamic Balance

Fig. 4.4 The centrifugal inertia force acting on the rotor

F = meω2 = me(

πn 2 ) 30

(4.7)

where F is the inertial centrifugal force (N); m is the imbalance mass (g); n is the spindle rotational speed (r/min); ω is the spindle angular velocity (rad/s); e is the eccentricity of spindle imbalanced mass (mm). According to Eq. (4.7), when the rotational speed of spindle remains constant, the centrifugal force is proportional to the product of spindle mass and the eccentricity. According to the theory of spindle dynamic balance, the product me is called the imbalance amount U, then F = Uω2 = U

π n 2

(4.8)

30

It can be seen that the inertial centrifugal force caused by the imbalance mass of the spindle is related to the rotational speed and imbalance mass. Based on the principle of leverage, the imbalance distributed on the rigid rotor can be treated equivalently. As shown in Fig. 4.5, the load V on the beam between the two support points is equivalent to loads distributed on two parallel planes, V 1 and V 2 . The equations are as follows Fig. 4.5 Load equivalent processing

V V1

V2

b

a L

4.2 The Expression of Imbalance Amount

123

b L a V2 = V × L V1 = V ×

(4.9) (4.10)

In this way, all imbalance vectors distributed along the rotor can be equivalently processed on two selected planes, and the obtained vector sum is called resultant imbalance. The resultant imbalance can be expressed as Ur =

K

Uk

(4.11)

k=1

where U r is the resultant imbalance vector (g mm); U k is the kth imbalance vector (k is from 1 to K). The vector sum of the moments of all imbalance vectors distributed along the rotor is called the resultant imbalance moment. The imbalance of the rigid rotor is fully described by the resultant imbalance together with the resultant imbalance moment. The resultant imbalance vector is independent of a particular radial plane, and the value of the resultant imbalance moment and the direction of phase angle depend on the axial position of the selected resultant imbalance. The resultant imbalance vector is the vector sum of the equivalent imbalanced vector of the dynamic imbalance. Resultant imbalance moment is typically represented as a pair of imbalance vectors with the same value and opposite direction on any two different radial planes. The resultant imbalance moment can be expressed as K   ZUr − Zk × U k Pr =

(4.12)

k=1

where P r is the resultant imbalance moment (g mm); U k is the kth imbalance vector (k is from 1 to K); Zk is the vector from the a reference point to axial position of the plane U k ; ZUr is the vector from the same reference point to axial position of resultant imbalance plane U r . An actual rotor can be considered to consist of an infinite number of disks with very thin thickness along the axial direction, as shown in Fig. 4.6a. If the mass distribution of each thin disk constituting the rotor is uniform, then the centroid of the rotor coincides with the center of rotation. When the rotor rotates, the resultant force of the centrifugal forces in all directions is zero, and the rotor is in a balance state. If the centroid of rotor does not coincide with the center of rotation, imbalance will occur in the rotor. As shown in Fig. 4.6b, all imbalances are equivalent to imbalances on two planes, U I and U II .

124

4 Basic Theory and Method of Spindle Dynamic Balance U5 U4

UII

U3 U2

UI

U1

(a) Spindle consisting of numerous thin disks

(b) Imbalances equivalent to imbalances on two planes

Fig. 4.6 Imbalance distribution of the rotor

4.3 Imbalance Classification According to the relationship among the centroid of rotor, the principal inertia axis (PIA) and the rotary axis, the imbalance of rotor can be classified into four types: (1) Static imbalance Static imbalance is a condition where the PIA of the rotor is displaced from but still parallel to the rotary axis. As shown in Fig. 4.7, the PIA of the rotor is parallel to the rotatory axis with a displacement of e. Static imbalance often occurs in rotors with a short axial length such as disk. U

r

u

e

m

Fig. 4.7 Static imbalance

4.3 Imbalance Classification

125

U

u

θ

e

X b

Fig. 4.8 Quasi-static imbalance

(2) Quasi-static imbalance The quasi-static imbalance is a condition where the PIA intersects the rotary shaft axis at a point other than the centroid. As shown in Fig. 4.8, the PIA of the rotor is not parallel to the rotary axis, and interests it at point X with an angle of θ . The displacement from the centroid to the rotary axis is e. (3) Couple imbalance The couple imbalance is a condition where the PIA of the rotor intersects the shaft axis at the centroid. As shown in Fig. 4.9, the PIA of the rotor intersects the shaft axis at the centroid and at an angle of θ . The magnitude of the couple imbalance can be expressed by the vector sum of the moment of the two dynamic imbalance vectors versus a reference point on the shaft axis. If the static imbalance of the rotor is corrected on any plane other than the plane containing

U

θ

-U b

Fig. 4.9 Couple imbalance

126

4 Basic Theory and Method of Spindle Dynamic Balance

Fig. 4.10 Dynamic imbalance

the reference point, then the couple imbalance will be changed. The unit of the couple imbalance is g mm2 , and the second length unit is the unit of the distance b between two test planes. (4) Dynamic imbalance The dynamic imbalance is a condition where the PIA of rotor is in an arbitrary position with respect to the shaft axis. As shown in Fig. 4.10, PIA is not parallel to the shaft axis or does not intersect it. Under some special conditions (i.e., the above three conditions), PIA can be parallel to or intersect the shaft axis. The dynamic imbalance can be expressed equivalently by two imbalance vectors, which fully represent the total imbalance of the rotor in two specified planes (perpendicular to the axis), as shown in Fig. 4.6b.

4.4 Imbalance Tolerance Specification In practice, the imbalance can be expressed by a single vector U (the imbalance vector). To ensure that the rotor operates well, the imbalance amount (residual imbalance Ur es ) should not be greater than the permissible imbalance U per . Ur es ≤ U per

(4.13)

U per is defined as the total imbalance tolerance of the centroid plane. For all rotors with double-side balance, the total tolerance should be assigned to the corresponding tolerance plane. As mentioned above, the permissible imbalance is proportional to the mass of the rotor Uper ∝ m

(4.14)

4.4 Imbalance Tolerance Specification

127

If permissible imbalance is proportional to the mass of the rotor, the permissible residual imbalance can be obtained by Eq. (4.9) e per =

U per m

(4.15)

where the international unit of e per is kg m/kg, and the practical unit of it is g mm/kg, which is equivalent to µm. In the case of resultant imbalance rotor (i.e., a disk perpendicular to the shaft axis), e per is the distance from the centroid to the shaft axis. For rotors with two types of imbalance, e per is an artificial quantity, which contains effects of the resultant imbalance and resultant imbalance moment, and therefore e per generally cannot be shown on the rotor. In practice, as long as the journal has high accuracy (roundness, straightness, etc.), a small e per can be obtained. In some cases, belt, air or self-propelled device may be used to balance the rotor based on its working support. In other cases, the rotor needs to be driven by itself on a fully assembled bearing seat under operating conditions and temperatures to achieve the balance. Experience has shown that for the same types of rotors, the permissive residual imbalance is usually inversely proportional to the rotor’s operating speed n eper ∝

1 n

(4.16)

For rotors with similar geometry and operating at the same speed, the stress-tobearing load ratios (generated by centrifugal force) within the rotor are the same. For rotors whose operating rotational speed is much lower than the designed maximum rotational speed, for example, some types of AC motors with a maximum rotational speed of 3000 r/min rotating at a speed of 1000 r/min (this similarity rule may be too strict), a great value of e per (with ratio of 3000/1000) can be used. According to the experience and similar conditions, a balance quality grade G has been established, which meets the classification requirements of the balance quality of typical mechanical types, as shown in Table 4.1. After the dynamic balance quality grade G of spindle is selected according to the spindle type, the permissive imbalance of the spindle can be obtained in combination with the mass of spindle: U per = 1000 ·

(e per · ω) · m ω

(4.17)

where U per is the permissive imbalance (g mm); (e per · ω) is the selected balance quality grade (mm/s); m is the rotor mass (kg); ω is the angular velocity of the rotor (rad/s) (Fig. 4.11). We can also obtain the e per from the Fig. 4.8, and then with m, the U per can be expressed

128

4 Basic Theory and Method of Spindle Dynamic Balance

Table 4.1 The guide for classification of the balance quality of rigid rotor [2, 3] Mechanical type: general

Balance quality grade G

Value

Crankshaft gear for large marine diesel engines with low-speed and inherent unbalance

G 4000

4000

Crankshaft gear for large marine diesel engines with low-speed and inherent balance

G 1600

1600

Resilient mounting crankshaft driving device with inherent unbalance

G 630

630

Rigid mounting crankshaft driving device with inherent unbalance

G 250

250

Reciprocating engine for cars, trucks and locomotives

G 100

100

Automobile wheel, hub, wheel assembly, drive shaft, resilient mounting crankshaft drive device with inherent unbalance

G 40

40

Agricultural machinery; rigid mounting crankshaft driving device with inherent unbalance; grinder; drive shaft (universal drive shaft, propeller shaft)

G 16

16

Aviation gas turbine; centrifuge (separator, decanter); motor and generator with a maximum rotational speed of 950 r/min (spindle center height is not less than 80 mm) Motor with spindle center height less than 80 mm; fan; gear; universal machinery; machine tool; paper machine; process industrial machine; pump; turbocharger

G 6.3

6.3

Compressor; computer drive unit; motor and generator with a maximum rotational speed greater than 950 r/min (spindle center height is not less than 1080 mm); gas turbine and steam turbine; machine tool drive device; textile machinery

G 2.5

2.5

Sound and image equipment; grinding machine drive

G1

1

Gyro; spindle and drive parts for high precision systems

G 0.4

0.4

Uper = epec · m = 18.6g mm

(4.18)

The maximum rotational speed of the spindle is 4500 r/min, the maximum diameter of the rotor is 165 mm, the length is 722 mm, and the weight is 23.32 kg. According to Table 4.1, the dynamic balance accuracy grade of this spindle is G 0.4. As shown in Fig. 4.8, the permissive residual unbalance e per is 0.8 g mm/kg. Then, U per can be obtained U per = e per · m = 18.6g mm

(4.19)

4.5 Correction Plane Selection

129

Fig. 4.11 U per determined by the balance quality grade G and operating rotational speed n [3–5]

4.5 Correction Plane Selection When determining the imbalance tolerance, it is best to use the specified reference planes. For these reference planes, the residual imbalance on each plane must be less than the respective tolerance, regardless of the position of phase angle. For rigid rotors, the imbalance tolerance always has two ideal planes. In most cases, these planes are near the support plane, and the purpose of balancing is to reduce the force and vibration transmitted by the rotor to the periphery through the support. Rotors with residual imbalance greater than the imbalance tolerance should be corrected. In some cases, these imbalances can not be corrected on the selected tolerance plane, and have to be corrected on the plane where the addition, removal or reconfiguration of the material are easy. The number of correction planes depends on the magnitude and distribution of the initial imbalance of the rotor, the design of the rotor, and the relative position from the correction plane to the tolerance plane. Commonly, to simplify the operation, the tolerance plane is selected on the correction plane. Some rotors have resultant imbalance that exceeds the tolerance, and the resultant imbalance moments are within the range of tolerance. The single-side balance can be performed as long as the rotor has very large support pitch, the very small axial runout during rotation, and the suitable calibration plane. These conditions can be studied separately in each case. After the single-side balance of the rotor is achieved, the measured maximum residual imbalance torque is divided by the bearing spacing to obtain a couple imbalance (a pair of imbalances). Even in the worst case, if the amount of imbalance obtained by this method is acceptable, then single-side balance is sufficient. For single-side balance, the rotor may not rotate. However, considering the sensitivity and accuracy, the rotor will be rotated in most cases, then the resultant imbalance can be measured.

130

4 Basic Theory and Method of Spindle Dynamic Balance

If the rigid rotor cannot meet requirements of single-side balance, then it is necessary to reduce the imbalance moment. Commonly, the resultant imbalance and the resultant imbalance moment together form the dynamic imbalance. The two imbalance vectors on the double planes are called equivalent imbalance vectors. For double-side dynamic balance, the rotor should be rotated, otherwise the imbalance moment cannot be detected. Although all rigid rotors can theoretically achieve balance on two planes, there are cases where more than two correction planes are used. For example, the resultant imbalance and the imbalance couple should be corrected separately; the resultant imbalance cannot be corrected on one plane or both planes; correction is performed along the axial direction of the rotor. In special cases, because the correction plane is limited (such as drilling holes on multiple counterweights to correct the crankshaft) or the function and strength of the rotor need to be ensured, it is necessary to correct imbalance along the axial direction of the rotor.

4.6 On-Site Dynamic Balance of Rotor System 4.6.1 Dynamic Balance Principle of Rigid Rotor The on-site dynamic balance of rotor, also known as the whole maneuver dynamic balance, is performed after the machine is installed at the work site. The final operating and vibration conditions of the machine are related to the rotational speed, bearing support, rotor stiffness, overall stiffness, drive conditions and machine load. The dynamic balance of rotor on working machine is directly detected using the vibration meter and the rotor is then balanced if the imbalance of rotor is found [6]. Compared with the balance on a special balance machine, on-site dynamic balance has some advantages. With the development of large-scale and high-speed machinery, more and more attention has been paid to the on-site dynamic balance. For the on-site dynamic balance, the measurement and elimination of the imbalance are two key aspects. The rotor can be divided into two types: rigid rotor and flexible rotor. There is difference in dynamic balance method for these two types of rotors. Commonly, the influence coefficient method is used for the on-site dynamic balance of the rigid rotor [7], and a combination of the modal balance method and the influence coefficient method is used for the flexible rotor. For rigid rotors, to achieve balance, it is necessary to eliminate the effects of imbalance vectors and imbalance moments. The imbalance of the rotor is distributed around the entire rotor, which means all imbalances along the axis can be simplified to a resultant force and resultant couple along the centroid. The rotor imbalance is decomposed into two components along directions x and z, as shown in Fig. 4.12. u(y) indicates the imbalance distribution function of rotor. u(y) is decomposed into u x (y) and u z (y), and both of them belong to planar force system u(y) = u x (y) + ju z (y)

(4.20)

4.6 On-Site Dynamic Balance of Rotor System z

131 z

z

u o

ux

uz

o

y

x

o

y

x

y

x

Fig. 4.12 The decomposition of imbalance

Equilibrium equations along x and z directions are established ⎧ N

 ⎪ ⎪ Q x,i = 0 ⎨ u x (y)dy + i=1

N

 ⎪ ⎪ ⎩ u x (y)ydy + Q x,i yi = 0

(4.21)

i=1

⎧ N

 ⎪ ⎪ Q z,i = 0 ⎨ u z (y)dy + i=1

N

 ⎪ ⎪ ⎩ u z (y)ydy + Q z,i yi = 0

(4.22)

i=1

where Q x,i and Q z,i are the compensation values along the x and z directions respectively; yi is the axial coordinate of the compensation value. When there are only two correction planes, which means that components of the compensation value along the same axial position are in the same plane, components can be combined.  Q 1 = Q x,1 + j Q z,1 (4.23) Q 2 = Q x,2 + j Q z,2 Thus, the dynamic balance equation of rigid rotor can be obtained by multiplying the Eq. (4.22) by j and adding it to the Eq. (4.21) ⎧ N

 ⎪ ⎪ Qi = 0 ⎨ u(y)dy + i=1

N

 ⎪ ⎪ ⎩ u(y)ydy + Q i yi = 0

(4.24)

i=1

When N = 2, the equation has a unique solution, and to perform the dynamic balance, only two compensation quantities are required. If the components of u(y) decomposed onto the correction planes I and II are U1 and U2 , then the compensation values on these two planes must meet 

U 1 + Q1 = 0 U 2 + Q2 = 0

(4.25)

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4 Basic Theory and Method of Spindle Dynamic Balance

Therefore, for the imbalance of rigid rotor, it is only necessary to compensate on the two calibration planes. In addition, the deformation of the rotor is small, which indicates that the imbalance distribution of the rotor does not change with the variation of rotational speed.

4.6.2 Dynamic Balance Method of Rigid Rotor There are usually two dynamic balance methods for rigid rotor: the balance machine method and the on-site balance method. The latter method has been more widely used. The traditional on-site dynamic balance methods mainly include the test weight shift method, three-point method, etc. These methods have many disadvantages, however, such as a large number of starting times, poor precision and serious damage to the machine. By the influence coefficient method, the magnitude and direction of the compensation value can be accurately calculated, and the number of starting times is small. The basic idea of the influence coefficient method is: rotor and bearing constitute a linear system, whose vibration response is a linear superposition of the vibration responses caused by imbalance, and the vibration response caused by unit imbalance on the balance plane is called the influence coefficient. 1. Single-side dynamic balance influence coefficient method When the rotor is subjected to single-side dynamic balance, only the imbalance vector of rotor is considered, and the imbalance moment is not considered. The specific steps of single-side balance influence coefficient method are as follows: First, the spindle starts without test weight, and rotates until to the rotational speed to be balanced. Then, the amplitude and phase position A0 of the original vibration on the corrected plane are measured. Then, the spindle stops, and the test weight P is added on correction plane of the rotor, the size of P is the product of the mass and radius of test weight (g mm). The test weight can be selected according to the empirical formula P = m ×r

(4.26)

where P is the value of the test weight vector (g mm); m is the mass of the test weight (g); r is the radius of the test weight position (mm); empirical formula m = M A0 , M is the rotor mass (kg); n is the rotational speed at which rotor is (10∼15)r (n/3000)2 balanced (r/min). Then, the spindle restarts, and rotates to the same speed. The amplitude and phase position of the vibration with the test weight is measured. The influence coefficient can be calculated by twice measured results and the test weight

4.6 On-Site Dynamic Balance of Rotor System

K =

133

A1 − A0 P

(4.27)

where K is the influence coefficient (µm/(g mm)), which represents the imbalance vibration response caused by the unit imbalance amount on the correction plane at the measurement point; A0 is the original vibration response, whose magnitude is the amplitude (µm); A1 is vibration response with the test weight, whose magnitude is the amplitude (µm). Then, the original imbalance amount (unit: g mm) is obtained U =−

A0 K

(4.28)

Q has the same value and opposite direction with U, thus Q = −U = −

A0 K

(4.29)

2. Double-side dynamic balance influence coefficient method In the case of double-side dynamic balance, it is necessary to consider not only the imbalance vector but also the imbalance moment. For the double-side dynamic balance influence coefficient method of rigid rotor, the influence coefficient and imbalance amount of the system (in this case both of them are the matrix) can be obtained by adding the test weight. Figure 4.13 shows double-side dynamic balance system model of the rotor, and planes I and II are rotor’s two correction planes. The process of double-side dynamic balance influence coefficient method is similar to that of dynamic balance influence coefficient method, and the difference is that there are test weight and measuring point for the former method. The process is as follows: First, the spindle starts and rotates to the balancing speed without the test weight. The amplitude and phase position of the original vibration on the calibration plane are A10 and A20 , respectively. Fig. 4.13 Doulbe-side dynamic system model

UII

UI

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4 Basic Theory and Method of Spindle Dynamic Balance

Then, the machine stops, and the first test weight P 1 is added on the correction plane I. Vibration responses of two positions are measured as A11 and A21 , respectively. Then, similarly, the test weight P 1 on the plane I is removed, and the second test weight P 2 is added on the plane II. Vibration responses of two positions are measured as A12 and A22 . The influence coefficient can be calculated by twice measured results and the test weight ⎧ K 11 ⎪ ⎪ ⎪ ⎨K 21 ⎪ K 12 ⎪ ⎪ ⎩ K 22

= = = =

A11 − A10 P1 A21 − A20 P1 A12 − A10 P2 A22 − A20 P2

(4.30)

where P j is the jth test weight; K i, j is the influence coefficient of the test weight j on the calibration plane i; Ai, j is the vibration response of the jth measurement on the i plane, whose magnitude is the amplitude. The relationship between the original vibration and the original imbalance is as follows  A10 = K 11 U 10 + K 12 U 20 (4.31) A20 = K 21 U 10 − K 22 U 20 Thus, the original imbalance U is 

U 10 = U 20 =

A12 K 22 − A22 K 12 K 11 K 22 −K 12 K 21 A22 K 11 − A12 K 21 K 11 K 22 −K 12 K 21

(4.32)

And the compensation is 

Q 1 = −U 10 Q 2 = −U 20

(4.33)

4.7 Dynamic Balance of Flexible Rotor Flexible rotor is a rotor whose flexibility, due to the centrifugal force caused by the imbalance, exceeds the allowable limit. During rotation, flexible rotor is subjected to centrifugal force, and its rotary shaft usually oscillates in a curve. The centrifugal force changes with rotational speed, and the flexibility of the rotary shaft also changes. Therefore, although the amplitude of vibration is within the permissive range at a

4.7 Dynamic Balance of Flexible Rotor

135

certain rotational speed, the balance state may not be maintained if the rotational speed changes. At present, there are many methods for flexible rotor to achieve dynamic balance. Among them, the influence coefficient method and the modal balance method are the most widely used. The influence coefficient method of flexible rotor is the influence coefficient balance method used with multiple planes and rotational speeds, which indicates the balance with multiple balance rotational speeds and correspondingly increasing number of correction planes. It is an improvement based on the influence coefficient balance method of rigid rotor. The modal balance method is based on the idea of stepwise balance and the principle of orthogonality between the main vibration harmonics of each order and the vibration flexibility of other orders. The two methods are described below. (1) Influence coefficient method To achieve the balance of flexible rotor with the influence coefficient method, the balance rotational speed and the number of correction planes should be increased. Similar to the one-side or double-side influence coefficient methods, the relationship between the correction amount and the influence coefficient is expressed as U0 + K Q = 0

(4.34)

where K is the influence coefficient matrix. ⎡

K 111 K 112 .. ⎢ .. ⎢ . . ⎢ n n K =⎢ K K ⎢ m1 m2 ⎢ . .. ⎣ .. . N N K M2 K M1

⎤ · · · K 11d . ⎥ .. . .. ⎥ ⎥ · · · K nmd ⎥ ⎥ . ⎥ .. . .. ⎦ N ··· KM D

(4.35)

where K nmd is the influence coefficient of the correction plane d(d = 1, 2, · · · , D) on the measuring point m(m = 1, 2, · · · , M) when the balance rotational speed is ωn (n = 1, 2, · · · , N ). U 0 is the initial vibration matrix of the system; U 0 (n, m) is the initial vibration of the measuring point when the rotational speed is ωn . U 0 is expressed as U 0 = [U 0 (1, 1), · · · , U 0 (n, m), · · · , U 0 (N , M)]T

(4.36)

Q is the amount of compensation required for achieve the balance of initial vibration, and it can be expressed as Q = [Q 1 , Q 2 , · · · , Q D ]T

(4.37)

When K is a non-singular square matrix, the equation has a unique solution, which means the multi-planes influence coefficient method of flexible rotor should meet

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4 Basic Theory and Method of Spindle Dynamic Balance

the following requirements: the number of calibration planes is the product of the number of measuring points and the number of balance rotational speed. Commonly, the rotor does not provide enough calibration plane, and the equation has no unique solution. Then, the ordinary used method is to abandon the requirement that the vibration response of each measuring point is zero, and adopt the least square method to minimize the residual vibration. (2) Modal balance method The imbalance state of the flexible rotor can be represented by modal imbalance, which is also called the mode of vibration imbalance. It is expressed based on the bending curve of each order mode of vibration. If the imbalance U k is determined by its ordinates, single mode of vibration imbalance can be obtained U n,k = U k × Φ(Z k )

(4.38)

where U n,k is the single mode of vibration imbalance; U k is the single imbalance vector; Φn (Z k ) is the nth-order mode of vibration function. All imbalances of a certain order mode of vibration describe the distribution of mode of vibration imbalance, and sum of these imbalances is the resultant modal unbalance U n,k (4.39) U n,r = The n-order resultant modal imbalance represents the distribution of imbalances at a certain mode of vibration. Each imbalance U k can be decomposed into the imbalance components of the main mode of vibration for each corresponding order. Due to the orthogonality of the main mode of vibration, the imbalance component of a certain mode of vibration can only cause the flexibility component of the order. Therefore, the imbalance component of each order can be eliminated by adding the compensation proportional to each mode of vibration to the rotor.

4.8 Vibration Signal Extraction Algorithm 4.8.1 Vibration Signal Smooth Processing Algorithm During the acquisition of on-site signals, due to the interference of the working environment, operation error, sampling error, etc., there will be abnormal data in the acquired signals. These results in that the fitted curve based on discrete data after the A/D conversion consists of broken line and appears many burrs, causing increased errors of data processing result and lower extraction accuracy of the signal amplitude and phase. Curve fitting can be used to eliminate singular points, and then appropriate points can be added to the original position according to the principle of

4.8 Vibration Signal Extraction Algorithm

137

data statistics, which reduces interference components and improves the smoothness of curve. Common numerical approximation fitting ways include linear interpolation, Hermite interpolation, least square interpolation, spline interpolation, and so on. Then, the cubic spline interpolation is introduced into the vibration signal algorithm. The principle is that the cubic spline interpolation function fits the acquired discrete data points to the cubic polynomial curve with many segments. The fitted curve maintains the variation characteristics of the original curve, and has less sampling errors, improving the extraction accuracy of the vibration amplitude to a certain extent. The cubic spline interpolation has higher stability and smoothness, and the fitted signal is close to the actual vibration signal. Thus, the practical value is high. There are n observed data (x i , pi ) i = 0, 1, 2, 3…, n acquired during a set of experiments. Where x is the sampling time, and p is the vibration value. When there is a cubic curve function relationship between variables x and p, the second derivative  of the function can be assumed i (i = 0, 1, 2, · · · , n). y(x) is a cubic as y (xi ) = M curve function in the interval xi , xi+1 , so y  (xi ) can be set as a linear function 

y (x) = Mi

xi+1 − x x − xi + Mi+1 αi αi

(4.40)

where αi = xi+1 − xi is interpolation basis function. y  (x) is integrated twice, and using y(xi ) = pi , y(xi+1 ) = pi+1 , we can obtain   Mi αi2 xi+1 − x (xi+1 − x)3 (x − xi )3 + Mi+1 + pi − y(x) = Mi 6αi 6αi 6 αi  2 Mi+1 αi x − xi + pi+1 − , i = 0, 1, · · · , n − 1 (4.41) 6 αi To obtain Mi (i = 0, 1, · · · , n), the derivative of y(x) is taken 

y (x) = −Mi

pi+1 − pi Mi+1 − Mi (xi+1 − x)2 (x − xi )2 + Mi+1 + − αi 2αi 2αi αi 6 (4.42) 

y (xi + 0) = −

αi αi pi+1 − pi Mi − Mi+1 + 3 6 αi

(4.43)

In the same way 

y (xi − 0) = 

αi−1 αi−1 pi − pi+1 Mi−1 + Mi + 6 3 αi−1

(4.44)



According to y (xi + 0) = y (xi − 0), the following equation can be obtained μi Mi−1 + 2Mi + λi Mi+1 = di , i = 1, 2, · · · , n − 1

(4.45)

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4 Basic Theory and Method of Spindle Dynamic Balance

p x ,x ]− p[xi−1 ,xi ] αi−1 i where μi = αi−1 , λi = αi−1α+α , di = 6 [ i i+1 . −αi αi−1 +αi i     According to the boundary conditions f (x0 ) = p0 , f (xn ) = pn , the following equations can be obtained

 6

 p[x0 , x1 ] − p0 α0  6  Mn−1 + 2Mn = pn − p xn−1 , xn αn−1 2M0 + M1 =

(4.46) (4.47)

Thus, the Mi can be calculated, and the cubic spline fitting function can be solved.

4.8.2 Signal Preprocessing Based on Time Domain Averaging and FIR Filtering The vibration signals acquired on site contain a high content of high-frequency noise signals. These noise signals are random signals and can be suppressed by timedomain averaging. The steps of time domain averaging algorithm are to first intercept N periods of the processed curve and then superimpose the N periodic signals to calculate the corresponding maximum amplitude and time. The time domain averaged signals show significantly reduced noise signals and improved signal-to-noise ratio. The specific time domain averaging process is as follows. The sampled signals can be expressed as S(t) = Y (t) + n(t)

(4.48)

where Y (t) is the periodic signals in the sampled signals; n(t) is the noise in the sampled signals. N periods of S(t) are obtained by intercepting the signals with the period, and are superimposed to obtain S(ti ) = NY (ti ) +



Nn (ti )

(4.49)

The average signal is obtained by dividing the sum of signal by N i) ¯ i ) = Y (ti ) + n(t S(t √ N

(4.50)

As shown in Eq. (4.50), the amplitude of noise signal after time domain averaging decreases to √1N of the original one. Thus, the more the superimposed signal periods, the better the effect of high frequency noise suppression. After the high-frequency component of the signals is effectively reduced, other inter-frequency components of the signals need to be processed, and only signals with

4.8 Vibration Signal Extraction Algorithm Fig. 4.14 The principle of filter

139

V(t)/V(n)

Q(t)/Q(n) H(t)/H(n)

the frequency close to rotating frequency are retained. Digital filter has the function of screening signals with different frequencies. When the rotor frequency is low, low-pass filter is used to ensure that the passband cutoff frequency is greater than the rotating frequency. When the rotor frequency is high, bandpass filter is used to ensure that the rotating frequency is greater than the low cutoff frequency of passband and lower than the high cutoff frequency of passband. Filter should have high stability and enable the phase of signal to experience linear offset. Therefore, the finite impulse response filter with good stability and high precision is chosen. As shown in Fig. 4.14, after the signal passes through the FIR filter, strict offset linear phase is produced. V (t) is the input signal of the filter. After the signal is processed by the filter, the output signal is Q(t). h(t) is the impulse response function of the filter, and after Z-transform of complex frequency domain, the discrete  system function of the FIR M h(n)z −n . V (n), h(n), digital filter can be obtained. The expression is H (z) = n=0 Q(n) are discrete values of signals mentioned above, respectively. V ( f ), h( f ), Q( f ) are spectrums of these signals, respectively. Then, Q(n) =

n

V (i)h(n − i)

(4.51)

i=0

Q( f ) = V ( f ) × h( f )

(4.52)

From the frequency domain, the process of signal filtering is actually to multiply the input signal by frequency domain transformed by the filter impulse response signal. By changing the spectrum h( f ) of the filter, certain spectrums of the input signal can be removed, weakened or strengthened, the desired output signal can be obtained. From the time domain, the signal filtering process is realized after the input signal is convolved with the impulse response of the filter, and the implementation of the digital filter function can be realized by a software algorithm program. Thus, the FIR digital filter has a high flexibility, resulting in that the required data can be obtained by changing parameters of the filter. The design methods of FIR filter include frequency sampling method, window function method and equiripple approximation method. As shown in Fig. 4.15, filtered signal is regular with phase of strict linear offset when frequency sampling method and equiripple approximation method are used to design the filter. When the filter is designed by the window function method, the energy of filtered signal leaks, and the amplitude is significantly reduced. As shown in Figs. 4.16 and 4.17, compared with the window function filter with the same order, the equiripple approximation filter has lower passband and stopband waviness (which has more uniform errors and is more stable), wider passband range, less transition band time, better stopband

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4 Basic Theory and Method of Spindle Dynamic Balance

Fig. 4.15 The comparison between waveforms of original signal and filter signal using a equiripple approximation method; b window function method

Fig. 4.16 Transfer function of equiripple approximation filter a amplitude versus frequency; b phase versus frequency

Fig. 4.17 Transfer function of window function a amplitude versus frequency; b phase versus frequency

attenuation characteristics and higher edge frequency. The equiripple approximation filter is the waveform conditioning of LabVIEW, and designed by the digital FIR filter module. The order of the filter, which influences the filter’s properties, should be considered during the process of design. It can be calculated according to the following equation. In addition, the passband boundary frequency of the filter should be determined by the actual working conditions, sampling rate, the order of filter, passband characteristics, and stopband attenuation characteristics.

4.8 Vibration Signal Extraction Algorithm

N=

141

  1 1 2 lg 3 10β1 β2 h

(4.53)

where β1 is the passband waviness; β2 is the stopband waviness; h is normalized transition bandwidth.

4.8.3 Signal Extraction Method Automatic tracking filter is used to extract fundamental frequency vibration signal, which can be expressed as s(t) = a0 + a1 sin(ω1 t + φ1 ) +

n

ai sin(ωi t + φi ) + n(t)

(4.54)

i=2

where a0 is a direct current component; a1 is the amplitude of fundamental frequency signal; φ1 is the phase of fundamental frequency signal; ai is the amplitude of interfrequency signal; ωi is the frequency of each signal; φi is the phase of inter-frequency signal; n(t) is interference signal. If the frequency of fundamental frequency rotational speed signal is ω1 , then the reference signal is y(t) = sin ω1 t

(4.55)

z(t) = cos ω1 t

(4.56)

The principle of automatic tracking filter is: first, multiply s(t) by y(t) and z(t), respectively. a1 a1 cos φ1 + a0 sin ω1 t + n(t) sin ω1 t − cos(2ω1 t + φ1 ) 2 2 n 1 ai (cos((ω1 + ωi )t + φi ) − cos((ω1 − ωi )t − φi )) − 2 i=2

s(t)y(t) =

a1 a1 sin φ1 + a0 cos ω1 t + n(t) cos ω1 t + sin(2ω1 t + φ1 ) 2 2 n 1 ai (sin((ω1 + ωi )t + φi ) − sin((ω1 − ωi )t − φi )) + 2 i=2

(4.57)

s(t)z(t) =

(4.58)

The first terms of Eqs. (4.57) and (4.58) represent direct current signal, and other terms represent alternative current signal. The amplitude and phase information of the vibration signal with the same frequency as the rotational speed are included in the direct current component. Therefore, only the useful direct current component should

142

4 Basic Theory and Method of Spindle Dynamic Balance

be extracted, which can be achieved by low-pass filtering of multiplied signals with very low cutoff frequency. Then, the filtered direct current component is multiplied by the reference sinusoidal signal and cosine signal, respectively. Subsequently, the signals are input into the adder. Finally, the fundamental frequency imbalance signal is obtained   1 1 s(t) = 2 a1 sin ω1 t cos φ1 + a1 cos ω1 t sin φ1 = a1 sin(ω1 t + φ1 ) (4.59) 2 2 The extracted fundamental frequency vibration signal is processed using the fast FFT transform. The amplitude and phase corresponding to the spectral line with the largest index energy can achieve the amplitude and phase extraction of fundamental frequency vibration signal. If there are N sequences of discrete sampling, then the DFT processing of discrete sequence is N −1

s(k) =

s(n)e− j

2π N

nk

(4.60)

n=0

When k =1, s(1) is the component of fundamental frequency, then N −1

s(1) =

s(n)e− j

2π N

n

(4.61)

n=0

The following can be obtained by expanding the formula 

a1 =

N −1

s(n) cos

n=0

N −1 2π n  2π n b1 = s(n) sin N N n=0

(4.62)

Thus, s(1) can be transformed to 



s(1) = a1 + jb1 = F1 ∠θ1

(4.63)

The periodic function s(t) can be expanded by Fourier series, and the Fourier series expansion of the periodic function with period of T1 is   2π s(t) = a0 + Ak sin k t + φk T1 k=1  Ak = ak2 + bk2 ∞

φk = arctan

ak bk

(4.64) (4.65) (4.66)

4.8 Vibration Signal Extraction Algorithm

143

The number of sampling points of the acquired signals is determined by the sampling period Ts , and the specific calculation formula is N = T0 /Ts . When the number of sampling points is sufficient in one period, the discretization equations of ak and bk are as follows   N −1 N −1 Ts 2 2π kn s(nTs ) cos(kω1 nTs ) = s(nTs ) cos T1 n−0 N n−0 N

(4.66)

  N −1 N −1 Ts 2 2π kn bk = 2 s(nTs ) sin(kω1 nTs ) = s(nTs ) sin T1 n−0 N n−0 N

(4.67)

ak = 2

By substituting s(t) into Eq. (4.66) and making k = 1, we can obtain    N −1  4π n A1 a1 = sin + φ1 + sin φ1 = A1 sin φ1 N n−0 N

(4.68)

Similarly, we can obtain b1 = A1 cos φ1 . Using values of a1 and b1 , the amplitude and phase of the vibration signal can be obtained according to the following equation A1 =

 a1 a12 + b12 , φ1 = arctan b1

(4.69) 

According to the comparison between formula a1 and formula a1 , and between  formula b1 and formula b1 , the following conclusion can be drawn: the amplitude of actual imbalance signal is the product of the amplitude obtained by Fourier transform and 2/N . Figure 4.18 shows the use of automatic tracking filter to extract fundamental frequency vibration signal.

Fig. 4.18 The principle of the fundamental frequency vibration signal extracted by automatic tracking filter

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4 Basic Theory and Method of Spindle Dynamic Balance

References 1. ISO 1925: 2001, Mechanical vibration, balanced vocabulary 2. GB/T 6444-2008. National Standards Writing Group of the People’s Republic of China. Mechanical vibration, balanced vocabulary (GB/T 6444-2008). Changchun Testing Machine Research Institute, Changchun 3. National Standards Writing Group of the People’s Republic of China. Mechanical vibration, balance quality requirements of constant state (rigid) rotor, Part 1: specification and balance tolerance test (GB/T 9239.1-2006). Changchun Testing Machine Research Institute, Changchun 4. Xu BG, Qu LS (2000) Holographic dynamic balancing technique for asymmetric rotors. J Xi’an Jiaotong Univ 34(3):60–65 5. An SL, Yang LM (2007) Rotor field dynamic balancing technology, 1st edn. National Defense Industry Press, Beijing, China 6. Bin GF, Li XJ, Shen YP et al (2016) Research on maneuver balance of multi-span rotor shafting without test reforming based on dynamic finite element model. J Mech Eng 52(21):78–86 7. Ouyang HB, Wang XX (2000) Chaos optimization of structural parameters of electromagnetic online automatic balancing head. China Mech Eng 11(5):557–559

Chapter 5

Intelligent Identification Technology of Stator Resistance of Motorized Spindle Motor

Problems such as non linearity, parametric changes, disturbance and noise exist in the drive system of high-performance motorized spindle. The solution of these problems is the key to improving the control precision and control performance of motorized spind drive system. To this end, researchers begin to use modern control theory and constantly seek and adopt new control methods and strategies to promote the control technology of motorized spindle to a higher level. In this chapter, we analyze the difficulties in the control of motorized spindle and explore the application of artificial intelligence to the parameter identification of motorized spindle motors.

5.1 Overview The operational characteristics of a motorized spindle are closely related to its control mode. Among the high-performance control modes of modern motors, direct torque control shows great application potential. Direct torque control requires the accurate estimation of stator flux linkage. When the stator flux linkage is estimated by voltage integration method, the only motor parameter used is stator resistance. Traditional direct torque control method ignores the change of stator resistance in the estimation of flux linkage and the stator resistance is considered a constant value. However, after operation for a period of time, the temperature of spindle increases and the stator resistance thus changes. The change of stator resistance causes error in flux linkage observation, leading to the oscillation of motorized spindle at low speed, reduction of load capacity, distortion of current and large harmonic component, which affect the speed regulation range of motor. This book focuses on direct torque control and introduces the application of intelligent control technology to direct torque control.

© Springer Nature Singapore Pte Ltd. 2020 Y. Wu and L. Zhang, Intelligent Motorized Spindle Technology, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3328-0_5

145

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5 Intelligent Identification Technology of Stator Resistance …

Motorized spindle, no matter it uses three-phase induction motor or permanent magnet motor, is a high-order, multivariate and strongly coupled nonlinear timevarying system. Therefore, the control of high-performance motorized spindle is very complicated and an effective control method is needed to achieve high-quality dynamic performance. As mentioned above, the starting point of vector control is to consider them equivalently as a separately excited DC motor. The basic principle is to use space vector theory and vector transformation to realize the decoupling control of torque and rotor flux linkage in a specific space coordinate system. At this time, in terms of control of torque, it has the same control characteristics as DC motor, thus the servo drive system can obtain good dynamic performance. However, vector control does not fundamentally change the nonlinear characteristics of system, as shown by the vector control equation (5.1). For a three-phase induction motor spindle, the stator and rotor voltage equations in MT coordinate system oriented by rotor magnetic field are ⎧ u sM = Rs i sM + pψsM − ωs ψsT ⎪ ⎪ ⎨ u sT = Rs i sT + ωs ψsM − pψsT ⎪ u rM = Rs i rM + pψrM ⎪ ⎩ u rT = Rs i rT + pψrM

(5.1)

The stator and rotor flux linkage equations are ⎧ sM = L s i sM + L r i rM ⎪ ⎪ ⎨ sT = L s i sT + L r i sT ⎪  = L m i sM + L r i rM ⎪ ⎩ rM ψrT = 0

(5.2)

Electromagnetic torque equation is Te = pn

Lm ψrM i T Lr

(5.3)

In the dq coordinate system oriented by rotor magnetic field, the stator and rotor voltage and flux linkage equations of permanent magnet motor spindle are 

u d = Rs i d + pψd − ωr ψd u q = Rs i q + pψq − ωr ψd  d = L d i d + ψf q = L q i q

Electromagnetic torque equation is

(5.4) (5.5)

5.1 Overview

147

Te = n p ψf i q

(5.6)

As can be seen from Eq. (5.3), the decoupling control of torque and flux linkage has been achieved within a specific reference coordinate system. Torque Te and rotor flux linkage ψr can be independently controlled by controlling the torque component i T and the excitation component i M of stator current. In Eq. (5.6), ψf can be considered constant, and the torque is positively proportional to the cross-axis current component i q . Equations (5.3) and (5.6) show that when the excitation remains constant, electromagnetic torque is positively proportional to torque current component, which is similar to the case of DC motor. However, the voltage equations of the two are still cross-coupled and are nonlinear. For the system, the mechanical equation must also be considered. The mechanical equations of three-phase induction motor spindle and three-phase permanent magnet synchronous motor spindle are the same. dθr = ωr dt

(5.7)

dωr = n p Te − R ωr − n p T1 dt

(5.8)

The motion state of the system is fully described by the voltage equation, flux linkage equation, torque equation and mechanical equation. It can be seen that this is a group of nonlinear equations. In addition, the nonlinearity of the equations also reflects in that many of the parameters are nonlinear. For example, stator and rotor inductances L s L r and magnetizing inductance L m vary with the degree of saturation of the main magnetic circuit. Stator and rotor resistance values Rs and Rr are related to the ambient temperature. The damping coefficient R in the mechanical equation is a nonlinear parameter related to friction and motion. Permanent magnet excitation magnetic field ψf is also a nonlinear function of temperature. All these parameters constantly change during the operation of motorized spindle. In addition, the basis of space vector theory is to assume that the magnetomotive force and magnetic field are sinusoidally distributed in space. There are also many other assumptions as premise. In fact, these assumptions are not completely consistent with the actual situation of motorized spindle. Harmonic torque is surely generated in electromagnetic torque, but further details are unknown. In actual control systems, it is usually regarded as a kind of disturbance, which cannot be suppressed by vector control itself but by the regulator of the control system. Therefore, some nonlinearity problems of the motorized spindle under vector control are still a great obstacle to further improve its steady-state and dynamic performances. In vector control systems, the position, velocity and current loops are generally controlled by conventional PI or PID controllers. These controllers are basically designed based on linear theory and can display good control performance only at a specific operating point or within a limited range. In order to obtain good dynamic performance and eliminate static difference, the parameter settings of PI controllers

148

5 Intelligent Identification Technology of Stator Resistance …

are very sensitive to disturbances and changes in system parameters. To solve this problem, various improvement measurements have been proposed, such as using a model reference adaptive controller to eliminate the effects of parameter changes. However, this kind of regulator has not only a very complicated structure, but also poor real-time performance. Its design is heavily dependent on accurate mathematical models and accurate identification of parameters. Although direct torque control is not a control method based on the mathematical model of motorized spindle, accurate parameters of motorized spindle are still necessary in the estimation of stator flux linkage and torque in order to compare their reference values with their actual values. The estimation accuracy and speed estimation range are greatly affected by the changes in the parameters of motorized spindle. Therefore, it is necessary to further solve the control problems of nonlinearity, parameter variation, disturbance and noise in high-performance servo drive system. By dong this, the control precision and control performance of servo drive system can be further improved and the modern control technology of motorized spindle can be promoted to a higher level. Therefore, people are using modern control theories to continuously seek and adopt more advanced control methods, strategies and techniques.

5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle Above analysis reveals that it is necessary to identify the stator resistance of motorized spindle in order to reduce the influence of stator resistance change on the control performance of direct torque control system. There are two main categories of online estimation schemes practically used for stator resistance identification in direct torque control. The first category involves the use of adaptive mechanisms for online identification of stator resistance, mainly including methods based on observers and methods based on model reference adaptive systems [1–8]. The second category involves the use of artificial intelligence techniques such as artificial neural network, fuzzy logic control, and neuro-fuzzy control in stator resistance identification process [9–14].

5.2.1 Effect of Stator Resistance on Direct Torque Control Performance In vector control and direct torque control, the change in stator resistance Rs has a great influence on the control performance [15, 16]. Stator resistance is a very critical parameter for direct torque control, no matter the motor used is induction motor or three-phase permanent magnet motor. The control of stator flux linkage and torque

5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle

149

both belong to Bang–Bang control, which is a control method that does not rely on the mathematical model of motor and thus should be independent of parameter variation. However, direct torque control requires the estimation of stator flux linkage and torque. If voltage integration method is used to estimate stator flux linkage, then the only motor parameter used in the whole control process is the stator resistance Rs . When stator frequency is low, the change of Rs can greatly affect the estimation of flux linkage and thus directly affects the accuracy of torque estimation. At present, online identification of stator resistance remains a technical problem to be solved. 1. Effect of stator resistance on flux linkage When u–i model is adopted as stator flux linkage estimation model, the only parameter of motorized spindle used in the calculation of stator flux linkage is stator resistance. Assume that the change amount of stator resistance under the influence of temperature or other factors is Rs , and the change amount of current due to stator resistance change is i s . Then, the actual value of stator flux linkage is ψs =

 [Us − (i s + i s )(Rs + Rs )]dt

(5.9)

At this time, the estimated value of stator flux linkage is ψs∗ =

 [Us − (i s + i s )Rs ]dt

(5.10)

From Eqs. (5.9) and (5.10), we can obtain the flux linkage deviation caused by stator resistance change  ψs (t) =

[Rs (i s + i s )]dt 

=

t

[Rs (i s + i s )]dt  t e jωt dt = Rs |i s + i s | 0

0

Rs |i s + i s | jωt = e +C jω

(5.11)

It can be known from Eq. (5.11) that the change of stator resistance will affect the observation of stator flux linkage. The error in flux linkage estimation caused by stator resistance error consists of non-time-varying part and part that varies sinusoidally with time. Therefore, stator resistance error surely affects direct torque control performance. 2. Effect of stator resistance on torque In direct torque control, the estimated value of torque is as shown in Eq. (5.12), where n p is the number of pole pairs of motor.

150

5 Intelligent Identification Technology of Stator Resistance …

Te =

3 n p |ψs × i s | 2

(5.12)

Based on the above equation, assume that the change amount of stator resistance under the influence of temperature or other factors is Rs , and the change amount of current due to stator resistance change is i s . The actual value of torque is Te =

3 n p |ψs × (i s + i s )| 2

(5.13)

At this time, the estimated value of torque is T∗e =

3 n p |ψs∗ × (i s + i s )| 2

(5.14)

By subtracting Eq. (5.14) from Eq. (5.13), we obtain Te = Te − T∗e 3 3 = n p |ψs × (i s + i s )| − n p |ψs∗ × (i s + i s )| 2 2

3  = n p ψs |(i s + i s )| sin α − ψs∗ |(i s + i s )| sin β 2 3 = n p |(i s + i s )|( ψs − ψs∗ )(sin α − sin β) 2

(5.15)

Equation (5.15) shows that the change in resistance causes a change in current and affects the estimation of flux linkage. Since there is coupling relationship between torque and flux linkage as well as current, the error in torque estimation is related to current amplitude and the amplitude of flux linkage error. As can be seen, stator resistance has a great influence on the control performance of direct torque control system. 3. Simulation analysis of the effect of stator resistance on direct torque control performance The influence of stator resistance on direct torque control performance is mainly reflected in its influence on the estimation of stator flux linkage and torque [17, 18]. In addition, changes in stator resistance also lead to changes in the harmonic wave of stator current during direct torque control. To further prove this conclusion, simulation experiment was carried out using the direct torque control simulation model. The PID settings in the rotational speed estimator were K p = 90, K I = 1990, and K d = 35. In the simulation process, the stator resistance was set to Rs = 1.13  for motorized spindle and to Rs = 1.13  and Rs = 1.2  for flux linkage estimation model, and load torque was set to TL = 3N · m. Then, the changes in stator flux linkage, output torque and stator current harmonics were estimated. Figure 5.1 shows the effect of stator resistance on flux linkage. As can be seen, when the resistance estimated by the flux linkage estimation model is the same as

5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle

151

Fig. 5.1 Effect of stator resistance on flux linkage

the actual value of stator resistance, the flux linkage is hexagonal. However, when the resistance estimated by the flux linkage estimation model is different from the actual value of stator resistance, the shape of flux linkage changes and is no longer hexagonal. Figure 5.2 shows the effect of stator resistance on output torque. As can be seen, when the resistance estimated by the flux linkage estimation model is the same as the actual value of stator resistance, the output torque of motorized spindle is close to the load torque, that is, the output torque fluctuates around 3N · m. However, when the resistance estimated by the flux linkage estimation model is different from the actual value of stator resistance, the output torque of motorized spindle is much higher than the load torque, and the torque is out of control. Figure 5.3 shows the effect of stator resistance on stator current harmonics. As can be seen, when the stator resistance estimated by the flux estimation model is no longer Fig. 5.2 Effect of stator resistance on output torque

152

5 Intelligent Identification Technology of Stator Resistance …

(a) Rs = 1.13 Ω

(b) Rs = 1.2 Ω

Fig. 5.3 The effect of stator resistance on stator current harmonics

accurate, the amplitude of stator current harmonics will increase significantly. As a result, the harmonic vibration and harmonic loss of motorized spindle also increase.

5.2.2 Analysis of Stator Resistance Characteristics 1. Analysis of factors affecting stator resistance The stator resistance of motorized spindle is directly related to its structural parameters and the actual operational conditions. (1) Influence of structural parameters of motorized spindle on stator resistance In order to analyze the influencing factors of stator resistance, the calculation method of stator resistance should be firstly analyzed. According to the literature, the resistance of each phase of induction motor stator winding can be calculated according to the following equation Rs = K F ρω

2N1lσ A σ 1 a1

(5.16)

where N1 is the number of turns connected in series in each phase; lσ is the average length of coil half turns; Aσ 1 is the cross-sectional area of conductor; a1 is the number of parallel branches of phase windings; ρω is the resistivity of conductor at the reference operating temperature; K F is the resistance increase factor. As revealed by Eq. (5.16), in addition to N1 , lσ , Aσ 1 , a1 , and ρω , stator frequency, stator current, operational time, and spindle temperature all affect stator resistance. (2) Influence of temperature on stator resistance

5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle

153

Fig. 5.4 Influence of temperature on stator resistance

Theoretically speaking, temperature is the main factor affecting stator resistance. The relationship between the stator resistance of a motor and temperature is Rs2 =

235 + t2 Rs1 235 + t1

(5.17)

where Rs1 is cold-state resistance at t1 ; Rs2 is cold-state resistance at t2 . Equation (5.17) indicates that stator resistance increases linearly with increasing temperature. Figure 5.4 shows a comparison of the measured and calculated values of the stator resistance of a certain type of motorized spindle at different temperatures. As can be seen, the general trend of stator resistance with temperature is close to that revealed by theoretical calculation, but the two are still different. This shows that stator resistance is not only determined by temperature, but also determined by other factors. (3) Influence of stator current on stator resistance When the current of motorized spindle increases, the copper loss of stator increases and finally the temperature of motorized spindle increases. In addition, increase in stator current also leads to increase in the magnetic flux leakage of motorized spindle as well as increase in stray loss. Therefore, the influence of stator current on stator resistance is in essence the influence of temperature on stator resistance. (4) Influence of frequency on stator resistance For asynchronous motors, the rotational speed is adjusted by changing power source frequency. When power source frequency is increased, the stator resistance may be changed due to skin effect. When the rotational speed of motor is in weak magnetic speed range, skin effect may also occur in the motor due to the saturation of magnetic circuit, which not only leads to changes in rotor resistance and rotor inductance, but also has a certain influence on stator resistance.

154

5 Intelligent Identification Technology of Stator Resistance …

Fig. 5.5 Device for measuring stator resistance

2. Experimental study on stator resistance characteristics The stator resistance of motorized spindle is often measured by LCR-821 tester with an accuracy of 0.1%. The experimental device is shown in Fig. 5.5. (1) Methods for measuring stator resistance ➀ Cold stator resistance In order to ensure that the test current of the tested winding does not exceed 10% of its normal operating current, the motorized spindle is energized for no more than 1 min. During meaurement, the rotor of motorized spindle remains still. The appearing end of stator winding is used for measurement. Each resistance is measured for three times, and the difference between each measurement and the average of three measurements is in the range about ±0.5% of the average. Then, the average of three measurements is taken as the actual value of resistance. Since the measured motorized spindle is star-connected, the resistance of the stator winding of each phase can be calculated as follows: ⎧ ⎨ Ra = Rmed − Rbc (5.18) R = Rmed − Rca ⎩ b Rc = Rmed − Rab After initial measurement, it is found that the difference between the wire end resistance of stator and the average of three wire end resistance is less than 2% of the average value, thus the final stator resistance is R=

1 Rav 2

Rav is the average resistance of the three line ends.

(5.19)

5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle

155

➁ Detection of hot stator resistance and temperature rise of motorized spindle In order to analyze the influence of spindle temperature rise, spindle frequency and stator current on stator resistance, it is necessary to detect the hot stator resistance of motorized spindle. The device used for measuring hot stator resistance is the same as that used for measuring the cold stator resistance of motorized spindle. The difference is that when hot stator resistance is measured, the motorized spindle is required to operate at the same frequency for 120 min and in this period it is stopped every 10 min. The stator resistance, the temperature of spindle shell, stator current and the corresponding time are measured and recorded at the moment when the power is cut off. In order to reduce the influence of temperature change caused by power cut-off on resistance readings, it is necessary to use extrapolation method to correct the results. Specifically, the curve of resistance Rs versus time t is plotted in semilogarithmic coordinate system. The point at which the extended curve intersects the vertical axis is taken the stator resistance at the moment of power cut-off. The value of stator current is read from the inverter, and the temperature of spindle housing is detected by the device shown in Fig. 5.6. The device is a multi-channel temperature tester with an accuracy of 0.5%. During the operation of motorized spindle, direct torque control is used and temperature is kept constant at 25 °C by water cooling. (2) Experimental results and analysis The cold stator resistance of a motorized spindle was measured to be 1.07965 . Table 5.1 shows experimental data on stator resistance and its influencing factors. The stator resistance data in the table are hot stator resistance data after calculation and processing. Figure 5.7 shows the variation of the stator resistance of motorized spindle with frequency under no load. As can be seen, the stator resistance of motorized spindle fluctuates and shows an overall rising trend with increasing frequency. This trend is related to the relationship between resistance and temperature rise.

Fig. 5.6 Temperature rise detection device

156

5 Intelligent Identification Technology of Stator Resistance …

Table 5.1 Experimental data on stator resistance and its influencing factors Code

Operating time/min

Frequency/Hz

Temperature of spindle housing/°C

1

10

Stator current/A

Stator resistance/

10

25.1

5.85

1.0860

2

15

25.1

5.79

1.0867

…

…

…

48

245

33.0

5.84

250

35.1

5.79

1.116

10

25.2

5.74

1.086

51

15

25.2

5.54

1.091

…

…

…

97

245

35.3

5.78

1.118

98

250

36.3

5.74

1.119

49 50

20

…

…

1.115

…

…

…

…

…

491

110

10

26.9

5.71

1.090

492

15

27.3

5.58

1.091

…

…

…

538

245

36.6

5.71

1.121

539

250

37.3

5.66

1.124

540

120

…

…

…

…

…

10

27.0

5.84

1.090

541

15

27.5

5.61

1.092

…

…

587

245

35.9

5.81

1.115

588

250

36.9

5.77

1.123

Fig. 5.7 Influence of frequency on stator resistance of motorized spindle

5.2 Analysis of Stator Resistance Characteristics of Motorized Spindle

157

Fig. 5.8 Variation of stator resistance of motorized spindle with operating time

Figure 5.8 shows the variation of the stator resistance of motorized spindle under no load with time. With the extension of operating time, the stator resistance of motorized spindle gradually increases, and the mean value of stator resistance is related to the operating frequency. The higher the operating frequency, the greater the stator resistance. Figure 5.9 shows the influence of the temperature of spindle housing on stator resistance under no load. The relationship between the temperature of spindle housing and stator resistance is plotted at 10 Hz, 20 Hz, 30 Hz, 140 Hz, 160 Hz, 240 Hz, and 250 Hz, respectively. At various operating frequencies, the stator resistance increases nonlinearly with increasing temperature. The temperature range of motorized spindle varies with operating frequency. At high speeds, the temperature of motorized spindle is high and thus the stator resistance is high. At low speeds, the temperature of motorized spindle is relatively low and thus the stator resistance is relatively low. Fig. 5.9 Influence of temperature on the stator resistance of motorized spindle

158

5 Intelligent Identification Technology of Stator Resistance …

Fig. 5.10 Influence of stator current on the stator resistance of motorized spindle

Figure 5.10 shows the influence of stator current on stator resistance under no load. The relationship between stator current and stator resistance is plotted at 10 Hz, 20 Hz, 30 Hz, 140 Hz, 160 Hz, 240 Hz, and 250 Hz, respectively. It can be seen that the stator resistance varies nonlinearly with increasing stator current at various operating frequencies. The relationship between stator current and stator resistance varies with operating frequency. Apparently, the influence of stator current on resistance is affected by operating frequency.

5.3 RBF Neural Network for Identification of Stator Resistance 1. RBF neural network design Neural network enables good nonlinear approximation and is used for analyzing the factors affecting stator resistance. First, RBF neural network is used to identify stator resistance. RBF neural network is a three-layer forward network. The first layer is the input layer and composed of signal source nodes. The second layer is the hidden layer. The number of hidden units depends on the problem described. The transformation function of hidden unit is a non-negative nonlinear function that is radially symmetric and attenuated to the center point. The third layer is the output layer, which responds to the input layer. The basic idea of RBF neural network is to use radial basis function as the “base” of hidden unit to construct the hidden layer space. The hidden layer transforms the input vector and specifically transforms the low-dimensional input data into the high-dimensional space, thus making linearly inseparable problem in low dimensional space become linearly separable in high dimensional space. In the RBF network used to identify stator resistance (Fig. 5.11), 3-n-1 network

5.3 RBF Neural Network for Identification of Stator Resistance

159

Fig. 5.11 Structure of RBF neural network

structure is adopted. The input of the network is the operating frequency f, stator current Is and temperature T of the motorized spindle. The number of input nodes is 3, the number of hidden layer nodes is n, and the output of the network is stator resistance Rs . Since Gaussian function is simple in form, radially symmetric, smooth and easy to resolve, it is used as the activation function, and the stator resistance Rs can be expressed as Y [X ; W ; c; σ ] = W T [X ; c; σ ]

(5.20)

Y = Rs

(5.21)

X = [ f, Is , T ]T ∈ R 3 × 1

(5.22)

W = [w1 , · · ·, wh ]T

(5.23)

c = [c1 , · · ·, ch ]T

(5.24)

σ = [σ1 , · · ·, σh ]T

(5.25)

[X (k); c; σ ] = [ 1 [X (k); c1 ; σ1 ], · · ·, h [X (k); ch ; σh ]]T  1 X (k) − ci 2 [X (k); ci ; σi ] = exp − 2 σi2

(5.26) (5.27)

where W is the estimated weight matrix, is the activation function, and c ∈ R 3 × h , σ ∈ R h is the estimated value and normalized parameter of hidden layer node center, respectively. Equation (5.26) shows the mapping relationship between the input and output of radial basis function network. Equation (5.27) is a Gaussian function. In the design of neural networks, the selection of the center c and number n of hidden layer nodes is a key issue. Among them, the number of hidden layer nodes generally satisfy

160

5 Intelligent Identification Technology of Stator Resistance …

n=

√

ni + n o + a

(5.28)

where n is the number of hidden layer nodes; n i is the number of input layer nodes; n o is the number of output layer nodes; a is a constant between 1 and 10. It can be obtained through Eq. (5.28) that the number of hidden layer nodes of the RBF neural network to be established is from 3 to 12. In MATLAB, the newrb function can be used to automatically increase the number of neurons in the hidden layer until the mean square error meets the requirement of precision error or the number of neurons reaches the maximum value. The block diagram of RBF is shown in Fig. 5.12.

Fig. 5.12 Block diagram of RBF algorithm

5.3 RBF Neural Network for Identification of Stator Resistance

161

Assume that the number h of hidden layer units in the RBF network (i.e., the number of basis functions) has been determined. Then, the key to determining network performance is the selection of the center ch of h basis functions. K-mean clustering method is a simple algorithm. 2. Acquisition of sample data The experimental data in Table 5.3 were used for RBF identification of stator resistance. The data were acquired as follows. The operating frequency was changed from 10 to 250 Hz with an interval of 5 Hz and at each frequency a group of data were collected. Each group of data included the temperature of spindle housing, stator current and stator resistance recorded every 10 min after the motorized spindle was first operated for 10 min at a certain frequency. Note that since stator is embedded inside the motorized spindle, its temperature can hardly be measured. The temperature of spindle housing was thus used here to reflect the temperature of stator. The motorized spindle was operated for 120 min at each frequency. 3. Use of RBF neural network to identify stator resistance The data in Table 5.1 were normalized and grouped. The 588 groups of experimental data were divided into two groups: the data listed in Table 5.2 was used for network training, and the data listed in Table 5.3 was used for testing the performance of RBF network. In the process of training, the optimal number of hidden layer nodes was determined by comparison and then the structure of the network was determined. As can be seen from Fig. 5.13, after three times of training, the error reaches the allowable range (the error is no greater than 0.01 ). At this time, the number of hidden layer nodes is three. In the experiment, the accuracy of the instrument used to measure stator resistance reached 0.01 , thus the error in measured data was no higher than 0.005 . If the difference between the response value of the network and the actual value Table 5.2 Sample data for training RBF neural network for identification of stator resistance Sample

Frequency/Hz

Current/A

1

10

5.85

25.1

1.0860

2

10

5.74

25.2

1.0864

3

10

5.70

26.5

…

…

195

130

5.46

31.2

1.1144

196

130

5.42

31.9

1.1154

197

130

5.47

31.5

…

…

390

250

5.68

37.4

1.1229

391

250

5.75

37.8

1.1226

392

250

5.66

37.3

1.1239

…

…

Temperature/°C

…

…

Resistance/

1.0890 …

1.1174 …

162

5 Intelligent Identification Technology of Stator Resistance …

Table 5.3 Data for RBF neural network identification of stator resistance Sample

Frequency/Hz

1

10

Current/A 5.75

26

1.0883

2

10

5.675

26.8

1.0897

3

10

5.7

26.8

…

…

97

130

5.405

30.2

98

130

5.46

32.067

1.1173

99

130

5.445

31.467

1.1162

…

…

195

250

5.685

37.533

1.1243

196

250

5.77

36.9

1.1228

…

…

Temperature/°C

…

…

Resistance/

1.0911 … 1.1117

…

Fig. 5.13 Error performance curve of training

was between −0.005 and 0.005 , then the network can be considered to meet the identification requirements. Figure 5.14 shows the network response based on training data. Figure 5.15 shows the error curve between the actual output value and the predicted value by the RBF network. As shown in Figs. 5.14 and 5.15, the training result reaches the expected error range. The remaining 1/3 of the data were used to test the network’s ability to identify stator resistance. Table 5.3 lists the data for testing the performance of the RBF neural network. Table 5.4 shows the testing results of the performance of RBF neural network in identifying stator resistance. Figure 5.16 shows the output curve of RBF neural network after the testing data were input. Clearly, the output values were close to the actual values of stator resistance. Figure 5.17 shows the error of RBF neural network after training in identifying stator resistance. As can be seen, the error of the model in identifying stator resistance was between −0.015 and 0.006 .

5.3 RBF Neural Network for Identification of Stator Resistance

163

Fig. 5.14 The output diagram of stator resistance network after training

Fig. 5.15 The output error of stator resistance network after training

Table 5.4 The performance of RBF neural network in identifying stator resistance Testing sample

Value output by the network/

Actual value of stator resistance/

Error/

1

1.0887

1.0883

0.00044

2

1.0911

1.0897

0.00144

3 …

1.0912 …

1.0911 …

0.00013 …

97

1.1114

1.1117

−0.00034

98

1.1149

1.1173

−0.00239

99

1.1139

1.1162

−0.00232

…

…

…

…

195

1.1217

1.1243

−0.00255

196

1.1217

1.1228

−0.00113

164

5 Intelligent Identification Technology of Stator Resistance …

Fig. 5.16 Output of RBF neural network after testing data were input

Fig. 5.17 Error of RBF neural network in identifying stator resistance

5.4 Hybrid Intelligent Identification of Stator Resistance 5.4.1 Stator Resistance Identification Strategy Based on ANN-CBR Due to the complexity of the operation process and factors influencing stator resistance, it is very difficult to effectively estimate the stator resistance of high-speed motorized spindle. Although stator resistance identification based on RBF neural network enables good accuracy, neural networks suffer from problems such as slow convergence and local optima in practical applications. Therefore, the latest research results in the field of artificial intelligence are continuously absorbed and used for the identification of stator resistance. Especially, neural network and other artificial intelligence technologies are combined in use to identify the stator resistance of motorized spindle, which is a problem with nonlinear characteristics. This helps enrich the

5.4 Hybrid Intelligent Identification of Stator Resistance

165

Fig. 5.18 The use of CBR to solve a problem

intelligent estimation methods for stator resistance and is of great significance to the development of intelligent control technologies for motorized spindles. Case-based reasoning (CBR) is an important branch of artificial intelligence. It aims to solve new problems by using existing experience and cases and obtains new solutions by referring to solutions to similar problems in the knowledge base. The use of CBR is as shown in Fig. 5.18. CBR is suitable for use in decision-making environment where there is no strong theoretical model and the knowledge is incomplete, difficult to define, or inconsistently defined, but rich experiences have been accumulated [19, 27]. There have been estensive research on CBR. CBR has currently been successfully applied to many systems and projects, including medical diagnostics, law, circuit or mechanical design, fault diagnosis, agriculture, meteorology, software engineering and so on [1–12]. For the problem of nonlinear parameter estimation, some researchers [16] combine artificial neural network (ANN) and CBR to form ANN-CBR model structure, which combines the advantages of neural network and case-based reasoning and realizes hybrid intelligent identification of stator resistance. The model structure is shown in Fig. 5.19. ANN is a module set before CBR. The input information is trained and indexed according to the attribute characteristics of the case. Case similar to the index and other working condition information established by the neural network is searched in the case library of CBR module so as to correct the output of the neural network. Finally, the revised result is saved for the new case. Therefore, CBR does not require a large amount of historical data to solve a new problem. Instead, it solves a new problem through referring to similar problems and using rules generated by the reasoning process, thus CBR can be easily adjusted and extended [20–26]. Stator resistance identification is a nonlinear parameter identification problem. According to the ANN-CBR model and existing research results, an intelligent identification method is established for stator resistance. There are two types of stator resistance of motorized spindle at a given frequency under two operating conditions. One is initial cold stator resistance Rs0 at power supply frequency f during the initial

166

5 Intelligent Identification Technology of Stator Resistance …

Fig. 5.19 The structure of ANN-CBR model

operation of motorized spindle. The other is hot stator resistance at power supply frequency f after operation time of t. In this stage, the stator current i s and temperature T of the motorized spindle change and thus hot stator resistance changes. The stator resistance Rs of motorized spindle can be estimated in two steps. First, stator frequency f is selected as the input quantity and stator resistance Rs as the output quantity. Then, neural network estimator is constructed and stator resistance j is estimated. The result is recorded as an initial value Rs0 . Second, the amount of j change Rs in the stator resistance is calculated based on the measured temperature j ∗ T j and measured current i ∗j at the preset frequency f and then the initial value Rs0 is automatically corrected. j

(5.29)

Rs0 = gcs ( f )

(5.30)

Rsj = gbs ( f, T j∗ , i ∗j )

(5.31)

Rsj = Rs0 + Rsj j

where gcs () represents an unknown nonlinear relationship; gbs () represents an j unknown nonlinear relationship of the amount of change Rs in stator resistance ∗ with the measured temperature T j , the preset frequency f, and the measured current i ∗j . The specific scheme for identifying stator resistance using the hybrid intelligent method is shown in Fig. 5.20. As long as there are enough intermediate hidden layer nodes, RBF network can approach any function with any size of error and thus RBF network has the advantages of simple training and high learning efficiency [20–26]. j Therefore, to estimate stator resistance, Rs0 should be first estimated and RBF neural network is used for estimation. Frequency f is as the input of RBF-based neural network, stator resistance Rs0 as the output, and the initial value of stator resistance Rs0 is obtained. Subsequently, CBR is used for dynamic correction of the initial value of stator resistance [20–34]. Experiment is conducted to obtain the relationship

5.4 Hybrid Intelligent Identification of Stator Resistance

167

Fig. 5.20 Hybrid intelligent identification of stator resistance

of the variation amount of stator resistance Rs with spindle temperature T j∗ and stator current i ∗j and thus a typical case library of stator resistance is established. Stator resistance correction is then achieved by means of retrieval, reuse, correction and save. The two-step estimation strategy makes full use of the data (temperature, current and resistance) collected at a regular time interval under different frequencies to ensure the accuracy of parameter estimation. After hybrid intelligent estimation, j the estimated stator resistance value Rs is substituted into the direct torque control model to estimate flux linkage and torque. The above stator resistance estimation strategy is used at the beginning of each speed regulation and in accordance with the operating frequency f i of motorized spindle. After the motorized spindle is started for operation, the stator resistance is automatically adjusted according to the measured temperature of spindle housing and the measured stator current of inverter. j

5.4.2 Hybrid Intelligent Algorithm for Identifying Stator Resistance 1. Estimation of initial value Rs0 based on RBF network technology Table 5.5 lists the data for constructing RBF network. Table 5.6 shows the output of RBF network trained by using the data listed in Table 5.5. Figure 5.21 shows the initial value of stator resistance output by RBF network and Fig. 5.22 shows the error in the initial stator resistance estimated by the RBF network. When frequency f is used alone as the input and stator resistance Rs0 is used as the output, there is large error in the estimation result of RBF neural network (Fig. 5.22), thus the output data need to be corrected. 2. Dynamic correction based on CBR j

After a rough estimate through the first step, the initial value Rs0 can be dynamically j corrected using CBR. CBR obtains the dynamic correction amount Rs according to

168

5 Intelligent Identification Technology of Stator Resistance …

Table 5.5 Data for constructing RBF network Sample

Frequency/Hz

1

10

Actual resistance (output of RBF network)/ 1.08930

2

15

1.09250

3

20

…

…

24

125

1.11410

25

130

1.11445

26

135

1.11610

…

…

47

240

1.11990

48

245

1.11920

49

250

1.12095

1.09305 …

…

Table 5.6 Error in the output of RBF network Sample

Output of RBF network (initial value)/

Actual output of RBF network (initial value)/

Error in the output of RBF network/

1

1.09116

1.08930

0.00186

2

1.09038

1.09250

−0.00212

3

1.09041

1.09305

−0.00264

…

…

…

…

24

1.11355

1.11410

−0.00055

25

1.11388

1.11445

−0.00057

1.11610

−0.00182

26 …

1.11428 …

…

…

48

1.12069

1.11920

0.00149

49

1.12087

1.12095

−0.00008

Fig. 5.21 Initial output of RBF network

5.4 Hybrid Intelligent Identification of Stator Resistance

169

Fig. 5.22 Error in the initial output of RBF network after training

three measured variables including the temperature of spindle housing, the operating frequency of motorized spindle and stator current. Through case retrieval and case j reuse, CBR finally obtains the dynamic correction amount Rs . First, the spindle ∗ housing temperature T j measured by temperature detection system, stator current i ∗j read from inverter and power supply frequency f j∗ of inverter are used as retrieval feature. Then, the case library is searched and the retrieved cases are reused to obtain j j j the correction amount Rs of stator resistance. Subsequently, Rs and Rs0 obtained j through the first step are summed to obtain the estimated value of stator resistance Rs under certain operating conditions. In order to verify the correctness of the estimated j ∗j value, Rs is compared with the measured stator resistance Rs for error analysis. If the error is within ±1%, then estimated value is considered valid and the case can be saved to the case library; if the error is not within the range, the value should be further corrected. In the following, we discuss in detail case representation, case retrieval and case reuse. j

➀ Case representation of parameter correction amount Rs based on structured framework Case representation is the basis of CBR and CBR technology depends to a large extent on the representation structure and content of collected cases. Case is a kind of expert experience knowledge, and knowledge is currently represented using various methods such as first-order predicate representation, production rule representation, framework expression, semantic network representation, script method, process representation, petri net method, object-oriented method, etc. [28–35]. The empirical knowledge about operating conditions in the case-based reasoning process is generally expressed in a structured way, which is a structured description of the corresponding domain. Therefore, the case needs to be represented based on framework structure. This means each case in the case library is composed of retrieval features and solution features. The retrieval features include f, i j and T, and the solution feature is Rs , on which basis a hierarchical framework is formed. In other words, case

170

5 Intelligent Identification Technology of Stator Resistance …

is a 2-tuple Case = (C, Y) represented by retrieval features “C” and solution feature “Y”. “C” stands for the description of the problem, including the measured spindle housing temperature T, measured stator current i and actual control frequency f. Similarly, the retrieval features of the case studied consist of T j , i j and f j , and the j solution feature Y is Rs . j

➁ Establishment of a case library for parameter Rs

The initial value of stator resistance is automatically adjusted by CBR technology. An j increment Rs is added to the initial value. This increment is determined according to the calculation result of Eq. (5.32). Specifically, it is measured as the difference between the average value of stator resistance measured at various time points and the actual value of stator resistance under a certain operating frequency of motorized spindle. Rsj = Rs∗ j − Rsn

(5.32)

∗j

where Rs is the measured value of stator resistance; Rsn is the average value of stator resistance measured at different time points under frequency n. The correction amount (increment), corresponding measured spindle housing temperature T j∗ , stator current i ∗j read from inverter, and the power supply frequency f j∗ of inverter constitute a case, which is saved in the case library, thereby establishing a case library for the stator resistance of motorized spindle. Table 5.7 shows data for establishing CBR. ➂ Case retrieval based on nearest neighbor method The powerful function of CBR system lies in that it can quickly and accurately retrieve relevant cases from its memory bank. Therefore, case retrieval is the most critical factor for the success of CBR method. Its objective is to make sure that the retrieved cases are as relevant or similar as possible to the current case. Table 5.7 Data for establishing CBR j

Temperature/°C

Correction amount of Rs /

5.85

25.1

−0.0033

10

5.75

26.0

−0.001

10

5.70

26.5

−0.00035

…

…

5.46

31.2

−0.0001

5.46

32.1

0.0028

5.47

31.5

0.00295

Sample

Frequency/Hz

Current/A

1

10

2 3 …

…

195

130

196

130

197

130

…

…

390

250

5.69

37.5

0.0033

391

250

5.75

37.8

0.00165

392

250

5.77

36.9

0.00185

…

…

…

…

5.4 Hybrid Intelligent Identification of Stator Resistance

171

When stator resistance is corrected using CBR, the case retrieval process can be divided into three stages: classification, selection and confirmation. In the classification stage, the case library is searched for cases related to the case studied and this search is based on the prominent features of the case studied. The case library of stator resistance takes input frequency as a prominent feature. In the selection stage, relationships and mappings are established between the case studied and cases selected in the first stage and detailed analysis is performed. Through comprehensive comparison of the case studied with cases initially selected, the case (or cases) with the highest similarity to the case studied is selected. In the selection stage, it is necessary to calculate the similarity of the characteristic attributes of selected cases to those of the case studied. In the confirmation stage, the similarity, reusability and modifiability of the selected case in the second stage are evaluated to make sure that the selected case is appropriate. There are three frequently used case retrieval techniques: nearest neighbor method, induction method (decision tree method), and knowledge guidance method. These three methods can also be used in combination. Nearest neighbor method is the most frequently used method for case retrieval in CBR system. It uses the feature weights of the input case in terms of matching with those in the case library to calculate the distance between two cases in the feature space and thus obtain the similarity between the two cases for retrieval. It is applicable to situation where the number of cases is small and the definition of the search target is not sufficient. The details are as follows: X is a point in the n-dimensional feature space D = {D1 , D2 , · · · , Dn }, and X i ∈ Di . The distance between X and Y in D is Dist(X, Y ) =



1/ r Wi

∗ D(X i , Yi )

r

(5.33)

1

We have ⎧ ⎨ |X i − Yi | If Di is continuous D(X, Y ) = 0 If Di is discrete and X i = Yi ⎩ 1 If Di is discrete and X i = Yi

(5.34)

If Di is continuous, when r is 1, Dist(X, Y ) is Manhattan distance; when r is 2, Dist(X, Y ) is Euclidean distance; when r > 0, Dist(X, Y ) is generally called Minkowski distance. According to the definition of distance, we can define the similarity between two cases as SIMx y = 1 − Dist(X, Y ) (when Dist(X, Y ) ∈ [0, 1]) 1 (when Dist(X, Y ) ∈ [0, ∞)) SIMx y = (1 + μDist(x,y)

(5.35)

The simplest nearest neighbor algorithm uses a weighted average method. After the similarities of all features are weighted and summed, the similarity of the two

172

5 Intelligent Identification Technology of Stator Resistance …

cases can be obtained. This book uses associated search method based on the nearest neighbor to find the deviation case in the case library that is close to the deviation of current actual working condition. Since the search features are all numeric, the similarity function is defined by the Euclidean distance dpq   2  2  2  21 d pq = w1 T p − T q + w2 Isp − Isq + w3 f p − f q

(5.36)

where T p and T q represent the temperature features of cases p and q, respectively; p q Is and Is represent stator current features of cases p and q, respectively; f p and f q represent the operating frequency features of p and q, respectively; w1 , w2 , and w3 represent the weighting coefficients of temperature, stator current, and operating frequency, respectively, to characterize their importance. Then, the similarity SIM pq of cases p and q is SIM pq =

1 1 + μd pq

(5.37)

where μ is a positive real number, which is selected according to the actual data distribution. The key of the nearest neighbor method is to determine the weighting coefficients of the retrieval features. The weighting coefficients of retrieval features characterize their importance to the case solution. The accuracy of the weighting coefficients of retrieval features will directly affect the final retrieval result. Usually, artificial neural network, rough set and so on can be used to determine the weights of features. This book uses the three-layer BP network model [84] shown in Fig. 5.23 to determine Fig. 5.23 BP network model for determining weights

RS

W

Zj

Zm

Z2

...

Z1 V

f

Is

T

5.4 Hybrid Intelligent Identification of Stator Resistance

173

Table 5.8 Input-to-hidden layer weights Hidden layer nodes

1

Frequency/Hz

2

3

−0.0745

−1.9261

−0.8822

3.5183

0.5828

−0.6679

0.0545

Current/A Temperature/°C

4

5

6

7

3.2169

−1.8802

−0.5654

−0.1197

−6.0962

−3.6421

−5.5567

−1.9398

5.7043

−0.7745

0.8291

−0.6725

0.1039

0.2681

Table 5.9 Hidden-to-output layer weights Hidden layer node 1

Hidden layer node 2

Hidden layer node 3

Hidden layer node 4

Hidden layer node 5

Hidden layer node 6

Hidden layer node 7

−0.2750

−0.1259

−0.9300

0.8748

−0.5268

−0.6446

−0.2516

the weights of features. In the three-layer BP network model, the input vector is X = {T, i s , f }, hidden layer output vector is Z = {z 1 , z 2 , z 3 }, and output layer output vector is Y = {Rs }. The weight matrix from input layer to hidden layer is represented by V, and V = {v1 , v2 , v3 }. The weight matrix from hidden layer to output layer is represented by W = {w1 , w2 , w3 }. The established BP network was used for training, and the number of nodes of the hidden layer after training is seven. After training, the obtained weight values from the input layer to the hidden layer are shown in Table 5.8. The weight values from the hidden layer to the output layer are shown in Table 5.9. The purpose of using BP network shown in Fig. 5.23 is to obtain the weights of influences of network input factors on the output decision, thus the following processing is required: First, the correlation coefficient ri j is calculated and let x = wik , then ri j =

p  k=1

vki

1 − e−x 1 + e−x

(5.38)

Let y = ri j , and the correlation coefficient is 1 − e−r Ri j = 1 + e−r

(5.39)

Then, the absolute influence coefficient of input factor is wij =

Ri j m 

(5.40)

Ri j

i=1

where i is the input unit of neural network, i = 1, 2, 3, · · · , m; j is the output unit of neural network, j = 1; k is the hidden unit of neural network, k = 1, 2, · · · , p;

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5 Intelligent Identification Technology of Stator Resistance …

vki is the weight coefficient between the input layer neuron i and the hidden layer neuron k; w jk is the weight coefficient between the output layer neuron j and the hidden layer neuron k. Finally, the influences of the three retrieval features including stator frequency, stator current and temperature on the correction value of CBR output are determined by BP network. The weights were calculated to be 0.3606, 0.3587 and 0.2807, respectively. The weights of the three retrieval features were substituted into Eq. (5.36) and the obtained d pq was substituted into Eq. (5.37). Finally, the similarity of cases was obtained. 3. Case reuse based on replacement method After case retrieval, case reuse is performed. Case reuse is a process of deciding how to solve the new case from the solution of the retrieved case based on the description of the features of the new case. Case reuse is a difficult task in the casebased reasoning process. In some simple systems, the solution of the retrieved case can be directly used as the solution to the new case. This method is suitable for problems where the reasoning process is complex but the solution is simple. In most cases, however, because there are no cases in the case library that exactly match the new case, the solution to the retrieved case needs to be adjusted to obtain a solution for the new case. Simple case adjustments only require simple replacement of some of the components of the solution to the retrieved case, whereas complex case adjustments may even require modification of the entire structure of the solution to the retrieved case. There are generally four methods for case reuse: substitution, conversion, specific target-driven method, and derived replay [36]. Since the problem studied here is problem solving type, the problem case has the same representation structure and descriptive attributes as historical case. Therefore, substitution method is used for case reuse. Suppose that a total of L cases with a similarity greater than 0.85 c1 , c2 , . . . , cL are retrieved [37], and the corresponding case solutions are i i i i , Rs,2 , Rs,3 , · · · , Rs,L . The similarities of the problem case to the retrived Rs,1 cases are SIM p,c1 , SIM p,c2 , SIM p,c3 , . . . , SIM p,cL . Then, the following formula is j used to obtain the solution Rs to the problem case under the current working conditions n 

Rsj

=

h=1

i SIM p,ch × Rs,h n 

(5.41) SIM p,ch

h=1

This means that the similarity between the retrieved case and the problem case is taken as weight and then all cases are summed. The solutions to the retrieved cases are adjusted and substituted to obtain the new value. Once the reuse case is j j obtained, it is sent as the appropriate Rs and the final Rs is obtained. The results obtained by using the data in Table 5.10 to search the built CBR case library are j shown in Table 5.11. It can be seen from the table that the error Rs obtained after case retrieval and reuse is small and the precision is high.

5.4 Hybrid Intelligent Identification of Stator Resistance

175

Table 5.10 CBR test data Sample

Frequency/Hz

Temperature/°C

Resistance correction value/

1

10

Current/A 5.74

25.2

−0.00285

2

10

5.69

26.7

−0.00010

3

10

5.69

26.7

…

…

97

130

98

130

99

130

…

…

195

250

5.68

37.4

0.00200

196

250

5.66

37.3

0.00295

…

0.00190

…

….

5.44

29.7

−0.00240

5.42

31.9

0.00095

5.42

31.4

−0.00035

…

…

…

Table 5.11 CBR output error Sample

CBR system output (correction value)/

CBR actual output (correction value)/

Error/

1

−0.00330

−0.00285

−0.00045

2

−0.00114

−0.00010

−0.00104

3

0.00038

0.00190

−0.00152

…

…

…

97

−0.00094

−0.00240

0.00146

98

0.00095

0.00095

0.00000

99

0.00096

−0.00035

0.00131

…

…

…

…

…

195

0.00176

0.00200

−0.00024

196

0.00277

0.00295

−0.00018

5.4.3 Simulation of Hybrid Intelligent Identification of Stator Resistance This book determines that the coefficient μ = 1 in Eq. (5.37) by experimental studies. Using the model combining neural network with case-based reasoning, we simulated hybrid intelligent estimation of stator resistance. The test data used in the experiment are listed in Table 5.12. Through the test, the system response results and corresponding errors can be obtained (Table 5.13). Figure 5.24 shows the comparison among the initial value, the overall system response and the actual resistance. The error between RBF output, i.e., Rs0 and the actual resistance is large. After CBR correction, the response value of the system is in good agreement with the actual value of resistance. The error between the system response and the actual value of stator resistance is shown in Fig. 5.25. As can be seen,

176

5 Intelligent Identification Technology of Stator Resistance …

Table 5.12 Test data for intelligent estimation of stator resistance Sample

Frequency/Hz

1

10

Current/A 5.74

Temperature/°C 25.2

1.0865

2

10

5.69

26.7

1.0892

3

10

5.69

26.7

…

…

97

130

5.44

29.7

1.1121

98

130

5.42

31.9

1.1154

99

130

5.42

31.4

1.1141

…

…

195

250

5.68

37.4

1.1230

196

250

5.66

37.3

1.1239

…

1.0912

…

…

Resistance/

…

…

…

Table 5.13 Hybrid intelligent estimation results of stator resistance Sample

RBF output (initial value)/

CBR output (corrected value)/

System output value/

Actual resistance/

Error/

1

1.09116

−0.00330

1.0879

1.0865

0.00141

2

1.09116

−0.00114

1.0900

1.0892

0.00081

3

1.09116

0.00038

1.0915

1.0912

0.00033

…

…

…

…

…

…

97

1.11388

−0.00094

1.1129

1.1121

0.00089

98

1.11388

0.00095

1.1148

1.1154

−0.00057

99 …

1.11388 …

0.00096 …

1.1148 …

1.1141 …

0.00074 …

195

1.12087

0.00176

1.1226

1.1230

−0.00032

196

1.12087

0.00277

1.1236

1.1239

−0.00026

Fig. 5.24 Output results of hybrid intelligent estimation system

5.4 Hybrid Intelligent Identification of Stator Resistance

177

Fig. 5.25 Hybrid intelligent estimation error

the error of the ANN-CBR hybrid intelligent estimation system for stator resistance is within ±0.002 . The estimation accuracy is doubled than when the RBF system is used alone for stator resistance estimation.

5.4.4 Experimental Verification In order to further verify the hybrid intelligent estimation results, the results are input into the direct torque control model to verify the consistency of the estimated and actual stator resistance in flux linkage estimation, torque estimation and stator current harmonics estimation. The actual stator resistance in the simulation is Rs = 1.1239 , and the estimated value by the hybrid intelligent system is Rs = 1.1236 . The simulation results are in Figs. 5.26, 5.27, and 5.28. As can be seen, the estimated stator current harmonics, stator flux linkage and output torque based on estimated stator resistance are consistent with those based on the actual stator resistance. This further demonstrates the effectiveness of the hybrid intelligent estimation method proposed here in estimating stator resistance.

5.5 Improvement of BP Neural Network Based on Biogeography-Based Optimization Algorithm In recent years, BP neural network has been continuously optimized by using intelligent optimization algorithm and its accuracy is greatly improved compared with that of traditional BP neural network. However, there are still some problems. For example, the operating speed of optimization algorithm cannot meet requirements. When BP neural network is optimized by the optimization algorithm, the weight and threshold can easily fall into local optima. MLBBO-BP algorithm uses MLBBO to

178

5 Intelligent Identification Technology of Stator Resistance …

(a) Simulation results based on actual stator resistamce

(b) Simulation results based on estimateds tator resistance

Fig. 5.26 Comparison of stator current harmonics

Fig. 5.27 Comparison of stator flux linkage

optimize the weight and threshold of BP neural network, and then uses the trained BP neural network to estimate the stator resistance of motorized spindle. This method can solve the above-mentioned problems well and enables higher precision, faster operation and less local optima problems.

5.5.1 Biogeography-Based Optimization Algorithm Biogeography-based optimization algorithm (BBO) is a kind of algorithm that can simulate biogeographical features. It establishes independent habitat mathematical models and uses mathematical methods to simulate the migration, mutation and

5.5 Improvement of BP Neural Network …

179

Fig. 5.28 Comparison of output torque

death of biological species. The algorithm is mainly used for optimization of nonlinear systems and functions. Similar to any colony algorithm, bee colony algorithm and genetic algorithm, BBO algorithm is a type of swarm intelligence optimization algorithm. In BBO, the two most important concepts are suitability index variable (SIV) and habitat suitability index (HSI). SIV reflects the specific environmental variables of a certain habitat H and is usually represented as a set of n-dimensional vectors. In actual application, SIV is an independent variable and HSI is a solution set corresponding to SIV. Generally, an ecosystem H ps consists of ps habitats, which means the population size of BBO algorithm is ps. The migration operator, mutation operator and clearance operator are the three basic operators of BBO algorithm. They can effectively change the species quality of the habitat. The steps of BBO algorithm are as follows: (1) Initialization. The initial populations are randomly generated. In the algorithm, random numbers are usually used to generate the initial populations. (2) Calculate the solution and habitat suitability index (HSI) of each population. (3) If the termination condition is not met or the required number of iterations is not reached, all the solutions are arranged in ascending order of HSI. (4) Mobility is determined according to the number of populations. The migration algorithm is used to migrate the populations. The results of the migration are saved. This operation effectively updates the quality of the species, but does not produce new species. (5) Similarly, mutation probability is determined according to the population number, and then the populations are subjected to mutation operation according to the mutation probability. This operation can generate a new species with instability. (6) The same solution is eliminated by performing a cleanup operation of the population with the clearance operator.

180

5 Intelligent Identification Technology of Stator Resistance …

(7) Return to (2) and re-circulate until the termination condition is met. 1. Migration operator BBO algorithm uses migration operator to share information in the habitat, and such sharing can effectively improve the HSI of habitat. However, not all species will undergo migration operation. BBO adopts a concept of probability in the implementation of migration operation. The probabilities of migration and emigration are determined according to the number of known species. When one habitat meets immigration conditions and another habitat meets emigration conditions, then the species in the two habitats will be exchanged. The algorithm of the migration operator is as follows: Choose habitat Hi based on immigration probability λi . If Hi is chosen, Choose habitat H j based on emigration probability μ j . Generate a random number. If the generated random number is smaller than μ j , Hi (S I V ) = H j (S I V ). (5) Loop for ps times until the termination condition is met.

(1) (2) (3) (4)

As can be seen, the core idea of migration operator is to replace poorer solution with better ones and thus improve the quality of solution in the population. 2. Mutation operator Mutation plays an important role in the normal evolution of populations. Unlike migration operator that constantly changes the solution within the population, mutation operator produces a new solution to replace the original one. Generally speaking, the mutation probability in a habitat is inversely proportional to the number of populations in the habitat. This means that the smaller the number of biological species, the higher the mutation probability; the larger the number of biological species, the lower the mutation probability. The algorithm of mutation operator is as follows: (1) Let k = 1, then calculate the mutation probability pk by using the immigration rate λk and the emigration rate μk . (2) Generate a random number and select Hk based on the random number and the magnitude of pk . (3) If Hk is selected, then a randomly generated SIV is used to replace Hk (S I V ). (4) Return to step (1) and let k = k + 1. The process is ended until k = n.

5.5 Improvement of BP Neural Network …

181

3. Clearance operator The role of clearance operator is to remove similar solutions generated during migration operation. In an ecosystem, a large number of similar solutions are not conducive to the diversity of populations. In actual application, if two solutions are the same, then the two solutions are considered to be similar solutions and a new solution is randomly generated to replace one of the original solutions. This procedure ensures that the update of the solutions in the population is effective. The algorithm of clearance operator is as follows (1) The algorithm is initialized. Let i = 1 and j = 2. (2) Hi and H j are compared. If they are equal, replace the original H i (SIV ) with the randomly generated SIV. (3) Determine whether j = ps. If they are not equal, return to (1) and let j = j + 1; if they are equal, then perform (4). (4) Determine whether i = ps. If they are equal, then the process is finished; if not, return to (1) and let i = i + 1 and j = j + 2.

5.5.2 Biogeography-Based Optimization Algorithm Based on Random Disturbance Exploration and development capabilities are the basic capabilities of BBO algorithm. Exploration capability refers to the horizontal search ability of the algorithm. Strong exploration capability enables the algorithm to have the opportunity to search for more high-quality solutions, and also prevent the algorithm from falling into local optima. However, excessive exploration ability will limit the precision of solution. Development capability refers to the ability of the algorithm to use existing information in the population to develop vertically in the neighborhood of approximate optimal solution. The algorithm with strong development capability can not only search for more accurate solutions, but also show high convergence speed. However, excessive development capability may lead to local optima problems. Therefore, the two capabilities need to be balanced. If one of them is too high, it is not conducive to the optimization of the algorithm. Therefore, the key to algorithm design is to balance the two capabilities so that the algorithm not only has not only good accuracy and high convergence speed, but also suffers from less local optima problems. In BBO algorithm, the migration operator shares the solutions in the population so that the low- and high-quality solutions can be exchanged, which ensures high development capability of algorithm and improves the quality of solution. In addition, mutation operator can effectively improve the diversity of the solutions and can introduce new high-quality solutions on the basis of migration, which ensure high exploration capability of the algorithm. Further, clear operator can effectively prevent the homogenization of the solutions and prevent the algorithm from falling into local optima.

182

5 Intelligent Identification Technology of Stator Resistance …

In original BBO algorithm, migration operator is the core operator, but it only uses the SIV of high-quality solution to replace the SIV of low-quality solution. Therefore, there often occur homogenization, poor diversity and local optima problems of solutions in the population. Moreover, although the algorithm guarantees the normal evolution of low-quality solutions, it does not have a qualified mechanism to ensure that high-quality solutions can evolve into better solutions. This not only makes the algorithm fall into local optima, but also reduces the convergence speed of algorithm. The BBO algorithm based on random perturbation (MLBBO) can effectively solve this problem by using hybrid migration operator, Cauchy mutation operator and random perturbation operator. The algorithm introduces a new migration equation, improves the original migration operator, and introduces a random perturbation operator. These can effectively accelerate the convergence speed of algorithm, increase the diversity of populations, and prevent the algorithm from falling into local optima. Introduction to the three operators is as follows 1. Hybrid migration operator MLBBO adopts the mutation strategy of differential evolution algorithm (DE) [38]. This can not only maintain the development capability of migration operator, but also effectively improve its exploration ability. The equation is as shown in (5.42): Vi = X best + F(X r 1 − X r 2 ) + F(X r 3 − X r 4 )

(5.42)

where X best is the optimal solution vector; F is the scaling factor; i ∈ {1, … , ps} represents the ith solution vector. The original equation is improved and the improved migration equation is Hi (S I V ) = Hbest (S I V ) + F[Hr 1 (S I V ) − Hr 2 (S I V )] + F[Hr 3 (S I V ) − Hr 4 (S I V )]

(5.43) where H best is the optimal solution; H r1 , H r2 , H r3 and H r4 are four different solution vectors randomly selected. Compared with the original migration equation, this equation enables to obtain more information since its scaling factor F can balance the migration equation. According to Ma [39], sinusoidal migration curve has the best performance among the six curves introduced. In sum, the algorithm of the hybrid migration operator is as follows: (1) The algorithm is initialized, and let i = 1 and j = 1. (2) Choose H i based on immigration probability λi . (3) If H i is selected, four different positive integers r 1 , r 2 , r 3 and r 4 are randomly generated. (4) Choose H j based on emigration probability μj . (5) Generate a random number and determine whether the random number is smaller than μj . If so, let Hi (S I V ) = H j (S I V ); otherwise, let Hi (S I V ) = Hbest (S I V ) + F[Hr 1 (S I V ) − Hr 2 (S I V )] + F[Hr 3 (S I V ) − Hr 4 (S I V )].

5.5 Improvement of BP Neural Network …

183

(6) Determine whether j = ps. If so, then j = 1 and i = i + 1, and return to (2); if not, j = j + 1, and return to (3). (7) Determine whether i > ps. If yes, the algorithm ends, save result and exit; if not, then i = i + 1 and return to (2). 2. Cauchy mutation operator The original mutation operator produces a new solution to replace the original solution, which can effectively prevent the algorithm from easily falling into local optima. The mutation operator used in the MLBBO algorithm is the Cauchy mutation operator proposed by Gong [40]. It can effectively improve the convergence speed and accuracy of the solution. The probability density function of the Cauchy distribution is f t (x) =

t 1 2 π t + x2

(5.44)

where x ∈ R and t > 0. When t = 1, the Cauchy mutation equation is Hi (S I V ) = Hi (S I V ) + δ(1)

(5.45)

where δ(1) is the Cauchy distribution value of parameter t = 1. 3. Random disturbance operator In original BBO algorithm, migration operator can effectively improve the quality of solution in the population, but the traditional migration operator simply replaces lowquality solution in the population with high-quality solution. This can easily cause the homogenization of solutions. In order to solve this problem, MLBBO introduces a random disturbance operator on the basis of original BBO. This operator generates perturbation with a certain probability in the vicinity of the good solution. In general, for the sake of simplicity, the algorithm intends to perturb each solution in the first half of the population, and its perturbation formula is defined as  Hi (S I V ) =

Hi (S I V ) + a × [Hk (S I V ) − Hi (S I V )] rand(0, 1) < pl Hi (S I V ) else

(5.46)

where Hk is a randomly chosen solution; pl represents the disturbance frequency; a is the disturbance amplitude.

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5.5.3 Application of BP Neural Network in the Estimation of the Stator Resistance of Motorized Spindle With the continuous research on artificial neural networks, researchers at home and abroad have proposed more than 70 neural network models, including perceptron, multi-layer feedforward network, fuzzy neural network and so on. The multi-layer feedforward network is a multi-layer mapping network that reversely transfers and modifies errors. It can be applied to highly nonlinear models such as target recognition, adaptive control, complex pattern recognition, etc. BP neural network is one of the most widely used neural network models. However, traditional BP neural network models have the disadvantages of slow convergence and easily falling into local minimum, which limit their application. Therefore, there have been more and more algorithms for improving BP neural networks, which has become a hot trend in machine learning and deep learning fields. In order to test the performance of traditional BP neural network in estimating the stator resistance of motorized spindle, data are initialized and normalized in MATLAB and the BP neural network model shown in Fig. 5.29 is constructed. According to the magnitude of the stator resistance of motorized spindle under different working conditions, the input layer-hidden layer-output layer adopts the 4-4-1 network structure, that is, 4 input nodes, 4 hidden layer nodes and 1 output node. The four input layer nodes represent frequency, operating time, stator resistance temperature and current flowing through the stator resistance. The four hidden layers are determined empirically, and there is no theoretical support for determining the number of hidden layer nodes. The one output layer node represents the stator resistance of motorized spindle. LM function is used as the training function for BP neural network. This

Fig. 5.29 BP neural network model

5.5 Improvement of BP Neural Network …

185

Fig. 5.30 Traditional BP neural network for estimating stator resistance error

function is faster than other functions. Random number method is used to determine the weight and threshold of traditional BP neural network. Before the BP neural network model is built, it is necessary to use MATLAB to normalize the original data and then use the random function in MATLAB to randomly divide the processed data into two groups. Among them, one group including 480 data points are used as training data for BP neural network, and the other group including 108 data points are used as test data for BP neural network. After the data are processed, the training of BP neural network can be performed. After training of BP neural network, the test data are input into the already constructed BP neural network and the error of the stator resistance of motorized spindle can be estimated by using the traditional BP neural network (Fig. 5.30). As shown in Fig. 5.30, the error of the BP neural network in estimating the stator resistance of motorized spindle is between −0.03 and 0.02 . Since the stator resistance of motorized spindle is between 2.1 and 2.5 , the error of traditional BP neural network in estimating stator resistance can reach more than 1%, and this accuracy can not meet the requirements. Therefore, traditional BP neural network has great defects in estimating the stator resistance of motorized spindle.

5.5.4 Stator Resistance Estimation Based on Improved BP Neural Network MLBBO-BP algorithm is proposed to solve the defects of BP neural network. In this algorithm, MLBBO is used to optimize the weight and threshold of the BP neural network, thereby improving the accuracy of stator resistance estimation by the BP neural network [41]. The MLBBO algorithm has two main parameters: habitat suitability index (HSI) and suitability index variables (SIV). In the text, SIV is equivalent to the weight and threshold to be optimized. Prior to optimization, the weights and thresholds are initialized and normalized into the vector x (i.e., SIV), that is, x = [w1 , b1 , w2 , b2 ], and

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5 Intelligent Identification Technology of Stator Resistance …

assigned a set of initial random numbers. w1 and b1 are the weights and thresholds between the input layer and the hidden layer, respectively; w2 and b2 are the weights and thresholds between the hidden layer and the output layer. HSI is equivalent to an important indicator for evaluating the weights and thresholds. Here, the mean square error of training of BP neural network is used as HSI, see formula (5.47). MSE =

p m 1   (y − yi j )2 mp i=1 j=1 i j

(5.47)

where MSE is the mean square error of the network; m is the number of output nodes; p is the number of training samples; yi j is the expected output value of the network; yi j is the actual output value of the network. The magnitude of HSI corresponds to the performance of each set of weights and thresholds in the BP neural network. The BP neural network trained with good weights and thresholds has a smaller mean square error. In order to improve the quality of species, MLBBO introduces migration operator, mutation operator, clearance operator and random perturbation operator. Migration operator can effectively exchange the solution vectors in the SIV. The solution corresponding to high HSI shares information to the solution corresponding to low HSI with a certain emigration rate. The low HSI solution accepts many new features from the high HSI solution and these additional new features can effectively improve the quality of the solution. Ma et al. [42] proposed different migration models, among which cosine mobility model has thebest performance (see Fig. 5.31). In this model,  + 1 , where λs indicates the immigration the immigration rate is λs = 2I cos sπ n rate for s populations, μs is the emigration rate, s is the number of current species, n is the maximum number of populations, and I and E correspond to the maximum immigration rate and the maximum emigration rate, respectively. Fig. 5.31 BBO algorithm cosine mobility model

5.5 Improvement of BP Neural Network …

187

Mmutation operator can effectively enrich the diversity of the population. In the continuous turnover of population, when a certain probability is reached, mutation will occur in the population and the probability of such mutation is generally low. Assume that BBO algorithm mutates the characteristic variables of habitat according to the probability of the number of species Ps in each habitat. Then, the probability Pk that the number of species at the next moment is s is calculated as (5.48) ⎧ ⎪ ⎪ ⎨

−(λs + μs )Ps + μs+1 Ps+1 , s = 0 −(λs + μs )Ps + λs−1 Ps−1 + μs+1 Ps+1 , Ps = ⎪ 1≤s ≤n−1 ⎪ ⎩ −(λs + μs )Ps + λs−1 Ps−1 , s = n

(5.48)

Mutation plays an important role in the evolution of population. Generally speaking, the habitat with a larger number of populations is more stable and the mutation rate is lower, whereas the habitat with a smaller number of populations is less stable and the mutation rate is higher. This theory is also applicable to MLBBO algorithm. The corresponding function is shown in (5.49)  Pk m(xi ) = m max 1 − Pmax

(5.49)

The main role of clearance operator is to prevent migration operator from producing similar solutions, which can reduce the diversity of populations. Therefore, Simon proposed to use clearance operator to change the similar solutions. When two solutions are similar or identical, a random solution is used to replace one of the two solutions. The use of random perturbation operator can help solve the local optima problem of traditional BBO, increase the anti-interference ability of ecosystem and greatly improve the exploration ability of algorithm. The random perturbation operator is described as Eq. (5.50)  Hi (SIV) =

Hi (SIV) + a × (Hk (SIV) − Hi (SIV)) rand(0, 1) < pl Hi (SIV) else

(5.50)

where H k is a randomly chosen solution; pl is the perturbation frequency; a is the perturbation amplitude. The operator generates perturbation at a certain frequency in the vicinity of the good solution with a certain probability. The parameters used in the optimization of algorithm are as follows: the maximum emigration rate is I = E = 1, the number of evolutions is 500, the population size is 20, the mutation probability is 0.05, the perturbation amplitude is a = 0.8, and the perturbation frequency is pl = 0.2.

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5 Intelligent Identification Technology of Stator Resistance …

Fig. 5.32 Neural network training steps and mean square error

5.5.5 Experimental Study Before the construction of a neural network, the stator resistance data were first grouped and processed. A total of 480 random sets of experimental data were used to train the neural network model, and another 108 sets of data were used to test the neural network model. The data samples were then normalized and the neural network was constructed. The weights and thresholds of the neural network to be optimized were uniformly integrated into N p row vectors by random number method and added to the MLBBO optimization algorithm. After 500 loop iterations, the best set of vectors can be obtained, and this set of vectors were decomposed into the weights and thresholds required by the neural network. Then, the “optimal” neural network model was used for testing. The number of training steps and training mean square error are shown in Fig. 5.32. As shown in Fig. 5.32, when the number of training steps reached 10, the mean square errors of the training data and the testing data tended to be stable and the training ended. Therefore, in the estimation of stator resistance by the MLBBO-BP method, the training result is converged. In order to evaluate the stator resistance estimation results, the actual stator resiatnce corresponding to the test data were compared with the estimation results based on the test data (Fig. 5.33). The network output curve shows the output stator resistance estimated by the MLBBO-BP neural network, and the sample curve shows the actual stator resistance corresponding to the test data. Comparing the two curves, we can see the high fitting degree of the sample data and the network output data. The errors are within the allowed error range, and the two curves almost coincide. Therefore, the neural network now has the ability to estimate stator resistance. The error between the value output by MLBBO-BP optimization neural network and actual value is shown in Fig. 5.34. Clearly, the error of BP neural network trained with MLBBO in estimating stator resistance is within ±0.007 , and thus

5.5 Improvement of BP Neural Network …

189

Fig. 5.33 Network output resistance curve and actual resistance curve

Fig. 5.34 Error between network output and sample output

it can be used to accurately estimate stator resistance. By using the collected stator resistance data under different working conditions, the accuracy of the MLBBO-BP method in estimating stator resistance was calculated to be ±0.3%. It can be seen that MLBBO-BP has strong ability to estimate stator resistance.

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Fig. 5.35 Error between stator resistance estimated by the MLBBO-BP method and by the original BP neural network

Figure 5.35 shows the error between stator resistance obtained by MLBBO-BP method and by the traditional BP neural network. It can be seen that the stator resistance obtained by the MLBBO-BP method is closer to the actual value after the same number of iterations.

References 1. Foo GHB, Rahman MF (2010) Direct torque control of an IPM-synchronous motor drive at very low speed using a sliding-mode stator flux observer. In: Proceeding of 11th international conference on electrical machines and systems. Wuhan, China, pp 933–942 2. Barut M (2010) Bi input-extended Kalman filter based estimation technique for speedsensorless control of induction motors. Energy Convers Manag. 2010, 51(10):2032–2040 3. Soltani J, Abootorabi ZH, Arab MGR (2013) Stator-flux-oriented based encoderless direct torque control for synchronous reluctance machines using sliding mode approach. Proc World Acad Sci, Eng Technol (58):883–889 4. Soltani J, Markadeh GRA, Abjadi NR, et al (2007) A new adaptive Direct Torque Control (DTC) scheme based-on SVM for adjustable speed sensorless induction motor drive. In: Proceeding of 11th international conference on electrical machines and systems. Wuhan, China, pp 497–502 5. Bai HY, Liu J, Li YJ, et al (2007) Research on direct torque control of permanent magnet synchronous motor drive based on extended state flux linkages observer. In: 2007 8th international conference on electronic measurement & instruments. Xi’an, China, pp 4-718–4-721 6. Xiao D, Foo G, Rahman MF(2010) A new combined adaptive flux observer with HF signal injection for sensorless direct torque and flux control of matrix converter fed ipmsm over a wide speed range. In: 2010 IEEE energy conversion congress and exposition. Atlanta Georgia, USA, pp 1859–1866 7. Foo GHB, Rahman MF (2010) Sensorless direct torque and flux-controlled IPM synchronous motor drive at very low speed without signal injection. IEEE Trans Ind Electron 57(1):395–403 8. Abjadi NR, Markadeh GA, Soltani J (2010) Model following sliding-mode control of a sixphase induction motor drive. J Power Electron 10(6):694–701

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9. Zidani F, Diallo D, Benbouzid MEH, et al (2006) Direct torque control of induction motor with fuzzy stator resistance adaptation. IEEE Trans Energy Convers 21(2):619–621 10. Zaimeddine R, Berkouk EM, Refoufi L (2007) Two approaches for direct torque control using a three-level voltage source inverter with real time estimation of an induction motors stator resistance. Mediterranean J Meas Control 3(3):134–142 11. Sayouti Y, Abbou A, Akherraz M, et al (2009) On-line neural network stator resistance estimation in direct torque controlled induction motor drive. In: ISDA 2009—9th international conference on intelligent systems design and applications. University of PisaPisa, Italy, pp 988–992 12. Tlemcani A, Bouchhida O, Benmansour K, et al (2009) Direct torque control strategy (DTC) based on fuzzy logic controller for a permanent magnet synchronous machine drive. J Electr Eng Technol 4(1):66–78 13. Aktas M, Ibrahim OH (2010) Stator resistance estimation using ANN in DTC IM drives. Turk J Electr Eng Comput Sci 18(2):197–210 14. Draou A, Miloudi A (2010) A simplified speed controller for direct torque neuro fuzzy controlled induction machine drive based on a variable gain PI controller. In: PEOCO 2010—4th international power engineering and optimization conference, program and abstracts. TBDShah Alam, Selangor, Malaysia, pp 533–538 15. Zhang LX, Liu XH, Wu YH (2012) Analysis of stator resistance characteristics of electric spindle. Dev Innov Mech Electr Prod 25(05):162–164 16. Zhang LX (2012) Research on electromagnetic characteristics and control strategy improvement of electric spindle unit, Ph.D. thesis. Dalian University of Technology, Dalian 17. Zhang LX, Wu YH, Pian JX (2013) Intelligent identification method for stator resistance of electric spindle. J Shenyang Jianzhu Univ (Nat Sci Ed) 29(06):1098–1103 18. Zhang LX, Zhou YJ, Pian JX et al (2011) Stator resistance identification of electric spindle based on RBF network. Control Eng 18(S1):44–47 19. Zbigniew K, Maria MK, Stefan Z, et al (2005) CBR methodology application in an expert system for aided design ship’s engine room automation. Expert Syst Appl 29(2):256–263 20. Zhang YM, Qu SR, Wen KG (2007) Short-term traffic flow prediction based on chaotic and RBF neural networks. Syst Eng 25(11):26–30 21. Wang JS, Gao ZW (2008) Network traffic modeling and prediction based on RBF neural network. Comput Eng Appl (13):6–7+11 22. Zhu SX, Zhang RJ (2007) Comparison of BP and RBF neural networks in face recognition. J Instrum 28(2):375–379 23. Sum J, Leung C, Ho K (2006) Prediction error of a fault tolerant neural network. Neurocomputing 72(1–3):653–658 24. Zeng W, Wang Q (2015) Learning from adaptive neural network control of an under actuated rigid spacecraft. Neurocomputing 168(C):690–697 25. Lee CC, Chiang YC, Shih CY, et al. (2009) Noisy time series prediction using M-estimator based robust radial basis function neural networks with growing and pruning techniques. Expert Syst Appl 36(3):4717–4724 26. Zhang LX, Wu YH, Zhang K (2014) Hybrid intelligent method of identifying stator resistance of motorized spindle. Int J Smart Sens Intell Syst 7(2):781–797 27. Pian JX (2010) Research on modeling and control method of laminar cooling process for hot rolled strip. Ph.D. thesis. Northeastern University, Shenyang, China 28. Vong CM, Wong PK, Ip WF (2011) Case-based expert system using wavelet packet transform and kernel-based feature manipulation for engine ignition system diagnosis. Eng Appl Artif Intell 24(7):1281–1294 29. Olsson E, Funk P, Xiong N (2004) Fault diagnosis in industry using sensor readings and case-based reasoning. J Intell Fuzzy Syst 15(1):41–46 30. Stéphane N, Le L JM (2008) Case-based reasoning for chemical engineering design. Chem Eng Res Des 86(6(A)):648–658 31. Reguera Acevedo P, Fuertes Martinez JJ, Dominguez Gonzalez M, et al (2008) Case-based reasoning and system identification for control engineering learning. IEEE Trans Educ 51(2):271–281

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32. Tsai CY, Chiu CC (2007). A case-based reasoning system for PCB principal process parameter identification. Expert Syst Appl 32(4):1183–1193 33. Wu LJ, Zhang JY, Gao LX (2009) A review of intelligent diagnosis systems based on neural networks and case-based reasoning. Mech Des Manuf 3:261–263 34. Kolodner JL, Cox MT, Gonzalez-Calero PA. Case-based reasoning-inspired approaches to education. Knowl Eng Rev 20(3):299–303 35. Yang BS, Jeong SK, Oh YM, et al (2004) Case-based reasoning system with Petri nets for induction motor fault diagnosis. Expert Syst Appl 27(2):301–311 36. Shi ZZ (2010) Knowledge discovery, 2nd edn. Tsinghua University Press, Beijing, China 37. Bai M (2007) Application of artificial neural network in determining the weight of agricultural land classification factors, Master’s thesis. Chang’an University, Xi’an, China (in Chinese) 38. Feng SL (2014) Biogeography optimization algorithm and its application in biological sequence pattern discovery. University of Electronic Science and Technology, Chengdu, China 39. Ma HP (2010) An analysis of the equilibrium of migration models for biogeography-based optimization. Inf Sci 180(18):3444–3464 40. Gong WY, Cai ZH, Ling CX, et al (2010) A real-coded biogeography-based optimization with mutation. Appl Math Comput 216(9):2749–2758 41. Wu YH, Zhang YL, Zhang LX (2017) Stator resistance identification of high speed grinding electric spindle based on biogeographic optimization algorithm. J Shenyang Jianzhu Univ (Nat Sci Ed) 33(5):898–905 42. Ma HP, Li X, Lin SD (2009) Mobility model analysis of biogeographic optimization algorithm. J Southeast Univ (Nat Sci Ed) 39(S1):16–21

Chapter 6

Thermal Performance Prediction of Motorized Spindle

In addition to the requirements for the high speed and high power of motorized spindle, high-speed machining also requires the ability of spindle to control its own temperature rise and thermal deformation. This can help guarantee the machining accuracy of machine tool and is the prerequisite of intelligentization of machine tool. There are usually two ways to control the thermal deformation of motorized spindle. First, extremely high temperature can be avoided through the optimal structural design of motorized spindle. Second, the temperature rise of motorized spindle can be actively controlled through the use of cooling/lubrication system, the thermal deformation of motorized spindle is predicted and compensation is implemented. There are essential differences between the two methods, but both require the preestablishment of motorized spindle thermal model and the prediction of temperature field, i.e., thermal deformation. The commonly used thermal design and mechanical design methods for motorized spindle are finite element analysis. However, this analysis method has two problems. First, there is a lack of comprehensive consideration of factors. In the analysis, many factors such as rotational speed, oil/gas two-phase flow, preload after thermal deformation of bearing and so on are often neglected, and the accuracy of temperature field prediction is affected. The optimization design result of motorized spindle unit is thus affected. Second, the dynamic changes of the influencing factors are neglected. Specifically, the fluctuation of electromagnetic field, flow field, temperature field and stress field as well as their mutual coupling all can affect the temperature field prediction accuracy and optimization design result. The multi-field coupling condition increases the complexity of the thermal performance of motorized spindle. At present, temperature rise prediction and active control remains difficult for motorized spindle in academia and industry. Through the analysis of the heat generation and transfer mechanism of motorized spindle, the temperature field of motorized spindle can be obtained (Fig. 6.1). In motorized spindle, the heat generated by motor and bearing is transmitted to other

© Springer Nature Singapore Pte Ltd. 2020 Y. Wu and L. Zhang, Intelligent Motorized Spindle Technology, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3328-0_6

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6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.1 Schematic diagram of temperature field of motorized spindle system

parts through heat conduction. Since the conduction rate is mainly related to material properties and the temperature difference between various parts of the motorized spindle, it is not necessary to consider improving the prediction accuracy of temperature field from the perspective of heat conduction. When heat source remains unchanged, an accurate motorized spindle temperature field prediction model [1–7] can be established from the viewpoint of improving the calculation accuracy of heat transfer coefficient.

6.1 Finite Element Model of Motorized Spindle Temperature Field COMSOL Multiphysics, a multi-physics coupled finite element simulation software, was used to analyze the temperature field of motorized spindle. Its function is realized through six steps: geometric modeling, adding material properties, selecting physical fields and setting boundary conditions, meshing, solving and post processing. 1. Geometric modeling 3D modeling was selected and the unit of model size was set to mm. CAD or Solidworks geometry was imported into COMSOL finite element analysis software through model import function. Model can also be built directly in the finite element software. 170SD30-SY motorized spindle was taken as research object. The threedimensional models of motor and bearing were used for simulation analysis. Under the premise of ensuring calculation accuracy, a pair of angular contact ball bearings, rotor, stator, etc. were simplified and assembled on the main shaft. All the screws, vent holes, oil passage holes and some other small structures were ignored. Then, the three-dimensional model of the 170SD30-SY motorized spindle was constructed as shown in Fig. 6.2.

6.1 Finite Element Model of Motorized Spindle Temperature Field

195

Fig. 6.2 Finite element model of motorized spindle

2. Material properties The various components of motorized spindle were defined according to the material properties listed in Table 6.1. 3. Select physical field and set boundary conditions The solid heat transfer module in the commercial finite element software can meet the needs of establishment of motorized spindle temperature field prediction model. Motor stator, rotor and bearing were set as heat sources. There exists thermal convection between rotor end and surrounding air, between bearing and the compressed air, between stator and cooling water, between stator-rotor gap and compressed air, and between motorized spindle outer surface and surrounding air. These boundaries should be set as heat exchange boundaries through heat transfer coefficient settings. Table 6.1 Material properties for the model Component

Material

Density/[g/cm3 ]

Thermal conductivity/[W/(m °C)]

Specific heat capacity/[J/(kg °C)]

Stator windings

Copper

8.856

400

386

Iron core of stator

Silicon steel

7.852

35

535

Water jacket

45Cr

7.850

60.50

434

Rotor bar

Cast aluminum

2770

–

875

Shaft

45Cr

7.850

60.50

434

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6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.3 The meshing model of motorized spindle

4. Meshing Since the motorized spindle model is a revolving body, regular tetrahedral and diamond-shaped grids were used for meshing. The number of tetrahedral grids was 337,076 and the number of triangular grids is 61,172. The minimum mass of grid was 0.06701, the average size of grid was 0.7022 and the total mesh volume was 647,5000.0 mm3 . Figure 6.3 shows the meshing model of motorized spindle. 5. Solving and post processing Steady-state solver or transient solver can be selected according to actual needs. During calculation, physical field interfaces were selected at the same time, and relative tolerance, error estimation factor and maximum number of iterations were set. One can also use the default settings. After calculation, post processing can be performed according to needs. Different cloud maps such as temperature field cloud map, isotherm graph, heat flux cloud map, flow field map, etc. can be obtained by changing the physical quantity expressions in the graphing tool.

6.2 Calculation of Heat Generation of Motorized Spindle Based on Loss Test The main characteristics of motorized spindle in operation are inverter power supply, and frequent fluctuation of operational speed and load. Inverter power supply causes motor to experience harmonic loss due to electromagnetic harmonics, thus the heat generated by motor can not be ignored. The high-speed variable load operation adds to the complexity of factors affecting bearing heat generation due to friction. In order to obtain the accurate heat generated by motor and bearing, the motorized spindle

6.2 Calculation of Heat Generation of Motorized Spindle …

197

loading and performance testing system shown in Fig. 6.4 was used to test motorized spindle motor loss and bearing friction loss. The specific steps are as follows [8]: (1) The loading motorized spindle and the motorized spindle to be tested were connected by a coupling. The power supply to the motorized spindle to be tested was cut off and its rotation was driven by the loading motorized spindle, thus the two motorized spindles were synchronously rotated. The loss of the motorized spindle to be tested was in the form of friction loss and at this time, the input power of the loading motorized spindle can be measured as PJ1 . (2) The two spindles were disconnected. Then the loading motorized spindle was made to rotate at the same speed as in the first step. At this time, the input power of the loading motorized spindle can be measured as PJ2 . (3) The two spindles remained disconnected. The testing motorized spindle was made to rotate under no load at the same speed as in the first step, and the input voltage and current of the spindle can then be measured by electrical parameter measuring instrument. Then, the input power of the testing mtorized spindle can be obtained as Pin . Torque and rotational speed sensors can measure the output torque and rotational speed of the testing motorized spindle, and its output power Pout can also be obtained. The friction loss of the testing motorized spindle was Pf = PJ 1 − PJ 2 , and the motor loss of the testing motorized spindle was Pe = Pin − Pout − Pf . Using the above method and the automatic loading device shown in Fig. 6.5, we can measure the friction loss and motor loss of the 170SD30-SY motorized spindle. Under ideal conditions, loss is in the form of heat generation.

Fig. 6.4 Motorized spindle loading and performance testing system

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Fig. 6.5 Automatic loading device for motorized spindle

In general, loss is related to load. The higher the load, the greater the loss [9]. The loss is considered the main factor causing the temperature rise of motorized spindle, and the calculation results of heat generation due to loss are then loaded into the finite element model. Here, only loss data under no load were taken as a calculation example.

6.3 Temperature Field Prediction Model for Motorized Spindle Based on Heat Transfer Coefficient Optimization In traditional temperature field models, the heat transfer coefficient of each part of the motorized spindle is calculated by empirical formula. However, in actual application, there exist differences between different motorized spindles, and the heat transfer coefficient of motorized spindle is affected by many factors and presents dynamic changes [10–13]. Therefore, the heat transfer coefficient obtained by the empirical formula can lead to errors in the prediction model. In order to improve the prediction accuracy of the model and reduce the influence caused by error in the heat transfer coefficient, it is necessary to optimize the heat transfer coefficient obtained by theoretical and empirical formulas [14–29]. First, the temperature of the motorized spindle under certain working condition is obtained through experiments. Then, the theoretical and empirical formulas are used to calculate the heat transfer coefficient of each part of motorized spindle under the working condition. The obtained initial values of heat transfer coefficients are input

6.3 Temperature Field Prediction Model for Motorized Spindle …

199

Fig. 6.6 Flowchart of the temperature prediction model for motorized spindle

into the finite element model to obtain the temperature field of the motorized spindle. The experimental and simulating temperature data of corresponding positions are extracted separately. The optimal value of the heat transfer coefficient of each part is obtained by the optimization algorithm, and the accurate temperature field of motorized spindle is obtained. Figure 6.6 is a flowchart of the temperature prediction model for the motorized spindle.

6.3.1 Motorized Spindle Temperature Prediction Model Based on Genetic Algorithm In order to reduce the calculation amount and time and at the same time ensure the prediction accuracy of the model, this section proposes to establish an intelligent and accurate motorized spindle temperature prediction model based on the use of genetic algorithm to optimize the heat transfer coefficient. 1. Introduction to genetic algorithm Genetic algorithm is a computational model constructed by simulating natural selection based on Darwinian theory of biological evolution and genetic evolutionary process. It relies on natural selection process to search for the optimal solution. The initial search of the genetic algorithm starts from a population of candidate solutions

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6 Thermal Performance Prediction of Motorized Spindle

to a problem. A number of individuals constitute the population through gene encoding. In fact, each individual is an entity with chromosome features. Chromosome is a set of genes and the main carrier of genetic information. The combination of genes in the chromosome reflects the genotype, which determines the external appearance of individual. There are several characteristics that distinguish genetic algorithm from traditional optimization algorithms [1]. (1) The search process of genetic algorithm starts from an initial population and can be performed from several points and in several directions at the same time. It can avoid local optima and is thus a global optimization method. (2) Genetic algorithm relies only on the numerical information of objective function to search for the optimal solution and has no special requirements for the properties of optimization model. Therefore, it is a universally applicable algorithm. (3) Since genetic algorithm searches in the population, it is not sensitive to initial points and shows good robustness. The five elements of genetic algorithm are coding, the generation of initial population, the evaluation of the fitness of individual in the population, genetic operation, and the setting of algorithm parameters during operation (Fig. 6.7). During selection, crossover and mutation, random functions are employed to randomly simulate biological evolutionary process.

Fig. 6.7 Flowchart of genetic algorithm

6.3 Temperature Field Prediction Model for Motorized Spindle …

201

2. Optimization of heat transfer coefficient based on genetic algorithm (1) Parameter settings of heat transfer coefficient optimization based on genetic algorithm ➀ Use binary coding to discrete independent variables. The coding length is determined according to discrete precision. In this example, the discrete precision was set to 0.01. The range of heat transfer coefficient was h max −hmin +1 . [hmin , hmax ]. Then, the coding length was l = log2 0.01 ➁ Select individuals according to their fitness. Here, roulette was used for  selection. Let PPi = ij=1 Pi and PP0 = 0, where PPi is cumulative probability; Pi is the selection probability of individual and calculated according fitness(hi ) , where fitness(hi ) is the fitness of individual and NP is to Pi = NP i=1 fitness(hi ) the number of individuals in the population. In each rotation, a random number r was randomly generated between 0 and 1. When PPi−1 ≤ r ≤ PPi , individual i was selected. ➂ Crossover. In this example, single point crossover was adopted. The crossover probability pc = 0.8. ➃ Mutation. In mutation, the value at certain position of the next generation is reversed according to mutation probability. For instance, 0 might be changed to 1, with a very low mutation probability between 0 and 0.05. Here, the mutation probability pm = 0.05. (2) Fitness function The calculation formula of roulette selection probability shows that the greater the individual’s fitness value, the higher the selection probability. In order to optimize the heat transfer coefficient of motorized spindle, let f be the objective function of the optimization problem to be solved, and fit is its fitness function. 1  Tei − Tsi 31 i=1

(6.1)

1 1 + fi

(6.2)

31

f =

fit =

where Tei is the experimentally measured temperature of motorized spindle (°C); Tsi is the simulated temperature of motorized spindle (°C). The fitness value is set as fit ≤ 0.67. This means tha when fi ≤ 0.5, the iteration is terminated, and the optimal heat transfer coefficient and the optimal motorized spindle temperature field are output. (3) Steps of the genetic algorithm to optimize heat transfer coefficient According to the flowchart of genetic algorithm, the steps to optimize the heat transfer coefficient are as follows.

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6 Thermal Performance Prediction of Motorized Spindle

Step 1: According to the theory and empirical formula of heat transfer coefficient in Chap. 2 of this book, the heat transfer coefficients at five positions of the motorized spindle under working conditions are calculated. In addition, the temperatures at 31 measuring points of the motorized spindle under working conditions are monitored. The distribution of the measuring points is shown in Fig. 5.3. Step 2: Initialize each parameter in the genetic algorithm, the population number M = 80, the maximum number of iterations N = 100, and the numerical range of each heat transfer coefficient is set. Step 3: The obtained heat transfer coefficients are input into the motorized spindle temperature prediction model, and the initial temperature field is calculated. Step 4: The predicted temperatures of the 31 measuring points are extracted and compared with the corresponding measured temperatures to evaluate the initial heat transfer coefficients and obtain fitness value. The roulette strategy is used to determine the individual’s fitness and determine whether the termination condition is met. Step 5: If fit ≤ 0.67, then fi ≤ 0.5 is satisfied and the obtained heat transfer coefficient is the optimal value. The loop can then be terminated, and the optimal heat transfer coefficient and the temperature field prediction model are output. If fit ≥ 0.67, then fi ≤ 0.5 is not satisfied, and the program returns to step 6 and continue the loop. Step 6: The selection probability is calculated according to the fitness value and generated individuals are selected by roulette. The greater the fitness value, the higher the probability that the individual is selected. The individuals with low fitness values are gradually eliminated. Step 7: Generate a new individual according to the set crossover probability pc = 0.8 and using a single point crossover. Step 8: Generate a new individual using the binary variation method according to the set mutation probability pm = 0.05. Step 9: After the new generation of populations are generated by crossover and mutation, return to step 2. 3. Motorized spindle temperature prediction model based on genetic algorithm The following working conditions were assumed in calculation: (1) Environmental temperature T 0 = 12 °C; (2) Machinery oil (20#) was used as lubricating oil in oil/air lubrication system. The temperature of compressed air at inlet T a = 8 °C. The pressure of compressed air at inlet P = 0.365 MPa. (3) The inlet temperature of water cooling system T w = 12 °C, and flow rate Q = 0.25 m3 /h. (4) The rotational speed under no load was 10,000 r/min. According to the above conditions, the initial values of heat transfer coefficients for temperature field optimization were obtained. Through the loss test, the stator loss at a rotational speed of 10,000 r/min was obtained (Table 6.2). Table 6.2 Stator loss at a rotational speed of 10,000 r/min Parameters

Rotor heat generation H 1 /W

Stator heat generation H 2 /W

Bearing heat generation H 3 /W

Values

157

314

98

6.3 Temperature Field Prediction Model for Motorized Spindle …

203

Table 6.3 Initial values of heat transfer coefficient Parameters

Heat transfer coefficient between motorized spindle and external air h1 /[W/(m2 °C)]

Heat transfer coefficient between rotor and stator h2 /[W/(m2 °C)]

Heat transfer coefficient of shaft end h3 /[W/(m2 °C)]

Heat transfer coefficient between bearing and compressed air h4 /[W/(m2 °C)]

Heat transfer coefficient between stator and cooling water h5 /[W/(m2 °C)]

Calculated values

9.7

146.81

121.35

71.42

190.12

According to the above conditions, the initial values of heat transfer coefficients for temperature field optimization were obtained (Table 6.3). The initial values of heat transfer coefficients were input into the finite element model of motorized spindle, and the initial temperature field of motorized spindle before heat transfer coefficient optimization was simulated and obtained. The simulated temperature data were extracted. On the basis of the simulated and measured steady-state temperature data, genetic algorithm was used to perform optimization iteration for 100 generations. Then, the optimal heat transfer coefficients (Table 6.4) and accurate temperature prediction model were obtained. When the 25th, 50th and 100th generations were reached, the isotherm map of the motorized spindle was extracted, respectively. As the number of iterations increased, the temperature field of motorized spindle changed (Fig. 6.8). The experimentally monitored temperatures at certain points of shell, front bearing, stator and rear bearing were compared with the simulated steady-state temperatures at the same positions before and after optimization (Fig. 6.9). Before optimization, the average error reached 8 °C. After optimization, the average error decreased to 0.81 °C. 4. Experimental verification In order to validate the prediction results of the prediction model, the working conditions were changed into the following: (1) Environmental temperature T 0 = 12 °C; Table 6.4 Heat transfer coefficients after optimization Parameters

Heat transfer coefficient between motorized spindle and external air h1 /[W/(m2 °C)]

Heat transfer coefficient between rotor and stator h2 /[W/(m2 °C)]

Heat transfer coefficient of shaft end h3 /[W/(m2 °C)]

Heat transfer coefficient between bearing and compressed air h4 /[W/(m2 °C)]

Heat transfer coefficient between stator and cooling water h5 /[W/(m2 °C)]

Calculated values

19.99

188.42

188.20

127.71

500.29

204

6 Thermal Performance Prediction of Motorized Spindle

(a) The isotherm map of motorized spindle before the optimization of heat transfer coefficients

(c) The isotherm map of motorized spindle obtained after iterations for 50 times.

(b) The isotherm map of motorized spindle obtained after iterations for 25 times.

(d) The isotherm map of motorized spindle obtained after iterations for 100 times.

Fig. 6.8 The isotherm map of motorized spindle changing with iteration times of algorithm

(2) Machinery oil (20#) was used as lubricating oil in oil/air lubrication system. The inlet temperature of compressed air T a = 18 °C. The inlet pressure of compressed air P = 0.35 MPa. (3) The inlet temperature of water cooling system T w = 19 °C, and flow rate Q = 0.32 m3 /h. (4) The rotational speed under no load was 15,000 r/min. According to the above conditions, the stator loss was obtained (Table 6.5) and the initial values of heat transfer coefficients for temperature field simulation were obtained (Table 6.6). The initial values of heat transfer coefficients were input into the finite element model of motorized spindle. Then the initial temperature field before the optimization of heat transfer coefficients was simulated and obtained. The simulated temperature

6.3 Temperature Field Prediction Model for Motorized Spindle …

205

Fig. 6.9 The comparison between the experimentally monitored temperatures of various parts of motorized spindle and the simulated steady-state temperatures before and after optimization

Table 6.5 Motorized spindle loss at a rotational speed of 10,000 r/min Parameters

Rotor heat generation H 1 /W

Stator heat generation H 2 /W

Bearing heat generation H 3 /W

Values

226.33

452.67

151

Table 6.6 Initial values of heat transfer coefficient Parameters

Heat transfer coefficient between motorized spindle and external air h1 /[W/(m2 °C)]

Heat transfer coefficient between rotor and stator h2 /[W/(m2 °C)]

Heat transfer coefficient of shaft end h3 /[W/(m2 °C)]

Heat transfer coefficient between bearing and compressed air h4 /[W/(m2 °C)]

Heat transfer coefficient between stator and cooling water h5 /[W/(m2 °C)]

Calculated values

9.7

178.37

142.45

109.12

223.85

data were extracted. On the basis of the simulated and measured steady-state temperature data, genetic algorithm was used to perform optimization iteration for 100 generations. Then, the optimal heat transfer coefficients were obtained (Table 6.7) and input into the finite element model of motorized spindle to accurately predict temperature field. When the 100th generation was reached, the isotherm map of the motorized spindle was extracted. It can be seen from Fig. 6.10 that the temperature field of the motorized spindle changes after optimization of heat transfer coefficient. Four measuring points were randomly selected (Fig. 6.11). The measured and pre-

206

6 Thermal Performance Prediction of Motorized Spindle

Table 6.7 Heat transfer coefficients after optimization Parameters

Heat transfer coefficient between motorized spindle and external air h1 /[W/(m2 °C)]

Heat transfer coefficient between rotor and stator h2 /[W/(m2 °C)]

Heat transfer coefficient of shaft end h3 /[W/(m2 °C)]

Heat transfer coefficient between bearing and compressed air h4 /[W/(m2 °C)]

Heat transfer coefficient between stator and cooling water h5 /[W/(m2 °C)]

Calculated values

10.81

171.26

223.83

134.9

333.38

(a) Before optimization

(b) After optimization

Fig. 6.10 The isotherm map of motorized spindle before and after the optimization of heat transfer coefficients

Fig. 6.11 Four randomly selected points

6.3 Temperature Field Prediction Model for Motorized Spindle …

207

Fig. 6.12 Comparison of the measured and predicted temperatures of these four points after optimization

dicted temperatures of these four points after optimization were compared (Fig. 6.12). The average error of the temperature prediction model after optimization was only 0.75 °C.

6.3.2 Motorized Spindle Temperature Prediction Model Based on Least Squares Method In many research fields, least squares method is widely used as an effective data processing method. In practical applications, the least squares method is often used to process experimental data to obtain optimized or relatively ideal parameter values. Therefore, we use here the least squares method to optimize the heat transfer coefficients of five parts of the motorized spindle. 1. Heat transfer coefficient optimization based on least squares method During steady state operation, the heat exchange between various parts of motorized spindle follows the law of conservation of energy, and heat is transferred from heat sources including bearing, stator and rotor to other parts of the motorized spindle through heat conduction. The convective heat exchange between motorized spindle and external air as well as coolant also follows the law of conservation of energy. For heat conduction, according to Fourier’s law, the relationship between heat flow vector and temperature gradient is q = −λgradT

(6.3)

208

6 Thermal Performance Prediction of Motorized Spindle

where q is heat flow vector (W); λ is the thermal conductivity of material (W/(m °C)); gradT is the temperature gradient at a certain point on the motorized spindle; T is the simulated motorized spindle temperature field related to position and time (T = f (x, y, z, t)). For convective heat transfer, according to Newton’s law of cooling, the relationship of heat transfer coefficient with heat flow vector between motorized spindle surface and air as well as cooling water is [48]. q = h(T − T0 )

(6.4)

where h is heat transfer coefficient (W/(m °C)); T0 is ambient temperature (°C). According to the above basic theories, the heat flow vector at any point can be obtained by using the finite element algorithm, which can lay a foundation for the optimization of heat transfer coefficient. The sum of the squares of errors between simulated temperature rise Tk and  m 2 measured temperature rise Ti i=1 ri is used to measure the overall magnitude of error ri (i = 1, 2, …, m) (for the convenience of calculation, let m = 5). According to the Newton’s law of cooling in Eq. (6.4), the relationship between the simulated temperature rise of motorized spindle and heat flow vector can be obtained Tk = qhkk . Let ak = h1k (k = 1, 2, . . . , 5), and Imin =

m 

ri2 =

i=1

m 

(ak qk − Ti )2

(6.5)

i=1

Therefore, Eq. (6.5) is transformed into an extremum problem of multivariate function a1 , a2 , . . . , a5 . The necessary condition for finding the minimum value of multivariate function is m

m   ∂Imin =2 ak qk − Ti × qij = 0 ∂aj i=1 k=1

i.e., m n   k=1

i=1

qij ) ak =

m 

qij Ti

(6.6)

i=1

Equation (6.6) is a system of linear equations for a1 , a2 , . . . , an , and can be expressed as a matrix

6.3 Temperature Field Prediction Model for Motorized Spindle …

⎡

m  ⎢ m + 1 i=1 qi ⎢ m m ⎢  2 ⎢ ⎢ i=1 qi i=1 qi ⎢ ⎢ .. .. ⎢ . . ⎢ m m ⎣ n  qi qin+1 i=1

i=1

209

⎡ m ⎤ ⎤  qin ⎥ ⎡ ⎤ ⎢ Ti ⎥ i=1 ⎢ i=1 ⎥ ⎥ a1 m m ⎢  ⎥  n+1 ⎥ ⎢ a2 ⎥ ⎢ ⎥ ··· qi ⎥ q T i ⎢ ⎢ ⎥ ⎥ ⎥ i=1 ⎥ · ⎢ . ⎥ = ⎢ i=1 ⎥ ⎥ ⎣ .. ⎦ ⎢ ⎥ .. .. ⎢ ⎥ ⎥ ··· . . ⎢ ⎥ ⎥ an m m ⎣ ⎦ ⎦   2n n ··· qi qi T ···

m 

i=1

(6.7)

i=1

where Tk is the simulated temperature rise of a certain point on the surface of motorized spindle (°C) and Tk = (Tsimulated − T0 ), where Tsimulated is the temperature at a certain point on the surface of motorized spindle obtained by simulation (°C); qk is the convective heat between a point on the surface of motorized spindle and air (W); Ti is the measured temperature rise of a certain point on the surface of motorized spindle and Ti = (Tmeasured − T0 ), where Tmeasured is the temperature of randomly selected point on the surface of motorized spindle measured by the experimental device shown in Fig. 3.5 (°C); k, i, j, m, and n are constants, and k = i = j = m = n = 1, 2, . . . , 5. Equation (6.7) was solved to obtain the minimum value of ak . From hk = 1 − 1, 2, . . . , 5), we can obtain the optimal heat transfer coefficient hk of each (k ak part. 2. Temperature prediction model based on least squares method The following working conditions were assumed in calculation: (1) Environmental temperature T 0 = 23 °C; (2) Machinery oil (20#) was used as lubricating oil in oil/air lubrication system. The temperature of compressed air at inlet T a = 18 °C. The pressure of compressed air at inlet P = 0.365 MPa. (3) The inlet temperature of water cooling system T w = 20 °C, and flow rate Q = 0.32 m3 /h. (4) The rotational speed under no load was 12,000 r/min. According to the above conditions, the heat transfer coefficients for temperature field simulation were obtained (Table 6.8). At the same time, the temperatures of 31 measuring points on the motorized spindle under working conditions were measured by temperature measuring device, and temperature data at five of the measuring points were used as sample data for heat transfer coefficient optimization (Table 6.9). The data listed in Table 6.8 were input into the 1/4 finite element model of motorized spindle, and the conductive heat flux clouds in three directions x, y, and z can be obtained (Fig. 6.13), which serve as a basis for least squares optimization of heat transfer coefficients. In order to more intuitively observe the temperature field of motorized spindle and reduce calculation time, the optimized model also takes 1/4 of the 3D model. As shown in Fig. 6.14, the temperature of each part of the motorized spindle changed obviously after optimization. The optimization of heat transfer coefficient thus has a great influence on the accurate prediction of the temperature field of motorized spindle. The temperature data of other four measuring points under the same working conditions were selected to analyze the accuracy of the model. The data are listed in

Heat transfer coefficient between bearing and compressed air h1 /[W/(m2 °C)]

113

Parameters

Values

96

Heat transfer coefficient of shaft end h2 /[W/(m2 °C)]

156

Heat transfer coefficient between rotor and statorh3 /[W/(m2 °C)]

Table 6.8 Boundary conditions for temperature field simulation

169

Heat transfer coefficient between stator and cooling waterh4 /[W/(m2 °C)]

9.7

Heat transfer coefficient between motorized spindle and external air h5 /[W/(m2 °C)] 157

Rotor heat generation H 1 /W

314

Stator heat generation H 2 /W

98

Bearing heat generation H 3 /W

210 6 Thermal Performance Prediction of Motorized Spindle

6.3 Temperature Field Prediction Model for Motorized Spindle …

211

Table 6.9 Measured temperature data for heat transfer coefficient optimization Code

Time t/s

Measuring point 1 temperature T /°C

Measuring point 12 temperature T /°C

Measuring point 18 temperature T /°C

Measuring point 27 temperature T /°C

Measuring point 28 temperature T /°C

1

0

23.19

2

20

23.31

23.31

23.81

23.75

23.56

23.44

23.81

23.75

3

40

23.81

23.56

23.81

23.81

24.75

24.25 24.63

4

60

23.81

24.06

24.81

24.75

…

…

…

…

…

…

…

307

5940

35.69

39.81

40.06

39.81

41.75

308

5960

35.69

39.75

40.06

39.81

41.75

309

5980

35.69

39.81

40.06

39.81

41.75

310

6000

35.75

39.81

40.06

39.81

41.75

Table 6.10. The simulated temperature of the measuring points before and after optimization were compared with the corresponding measured temperature data (Fig. 6.15) to determine the accuracy of the prediction model. Figure 6.15 reveals that the average error of the temperature prediction model before optimization is 2.71 °C and the relative error is 7.12%, whereas the average error of the temperature prediction model after optimization is 0.89 °C and the relative error is 2.34%.

6.4 Accuracy Analysis of Prediction Model In order to further verify the validity of the two temperature field prediction models and compare their advantages and disadvantages, it is necessary to compare the measured and simulated temperatures of other measuring points that were not used for heat transfer coefficient optimization. The errors of the models were calculated to determine the accuracies of the two prediction models. Standard deviation of the error, variance and standard deviation were used to evaluate the accuracy of prediction model. The standard deviation of the error SDE of the prediction model is calculated as follows  SDE =

  n 2 1  e i=1 i (Ti −  Ti )2 = n n i=1

n

The variance S 2 of the prediction model is calculated as follows

(6.8)

212

6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.13 Motorized spindle heat flow vector cloud

(a) x direction

(b) y direction

(c) z direction

6.4 Accuracy Analysis of Prediction Model

(a) Temperature field before optimization

213

(b) Temperature field after optimization

Fig. 6.14 Motorized spindle temperature field

Table 6.10 Temperature data used to verify the accuracy of the prediction model Code

Time t/s

Measuring point 6 temperature T /°C

Measuring point 14 temperature T /°C

Measuring point 18 temperature T /°C

Measuring point 29 temperature T /°C

1

0

23.31

2

20

23.31

23.94

23.38

23.94

24.19

23.56

3

40

23.06

23.41

24.63

23.94

24.25 24.38

4

60

23.51

24.94

24.19

…

…

…

…

…

…

307

5940

35.83

39.50

39.50

39.88

308

5960

35.80

39.44

39.50

39.88

309

5980

35.81

39.50

39.44

39.88

310

6000

35.84

39.50

39.50

39.88

1 1 2 a (Te − Ts )2 = n i=1 n i=1 n

S2 =

n

(6.9)

The standard deviation s of the prediction model is calculated as follows    n  n 1  1  2  S= a2 (Te − Ts ) =  n i=1 n i=1

(6.10)

214

6 Thermal Performance Prediction of Motorized Spindle

(a) Front bearing

(b) Stator

(c) Shell

(d) Rear bearing

Fig. 6.15 Comparison of the simulated temperature of the measuring points before and after optimization with the corresponding measured temperature data

6.4.1 Accuracy of Temperature Prediction Model Based on Least Squares Method The temperature data of the 31 measuring points on motorized spindle were all used as sample data for heat transfer coefficient optimization by genetic algorithm. Therefore, these data cannot be reused for accuracy analysis. The temperature measuring system can monitor the temperature of the water inlet and outlet of the motorized spindle. Therefore, the cooling water outlet temperature can be used to evaluate the prediction accuracy. Table 6.11 shows the simulated temperature T * of motorized spindle water outlet. Figure 6.16 shows the comparison of the simulated and measured temperature at water outlet of motorized spindle at a rotational speed of 12,000 r/min. Clearly, the error of the prediction model is small and the prediction accuracy is high. According to Eqs. (6.8)–(6.10), the standard deviation of the error, variance and standard deviation of the temperature prediction model based on least squares method were 0.96%, 1.92 × 10−4 and 0.014, respectively.

Temperature

Time t/s

T * /°C

20

0

20.96

400

21.9

800

Table 6.11 Simulated water outlet temperature 22.70

1200 22.80

1600 22.92

2000 23.04

2400 23.16

2800

23.29

3200

23.41

3600

23.49

4000

23.53

4400

23.59

4800

6.4 Accuracy Analysis of Prediction Model 215

216

6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.16 The comparison of the simulated and measured temperature at water outlet of motorized spindle at a rotational speed n of 12,000 r/min

6.4.2 Accuracy of Temperature Prediction Model Based on Genetic Algorithm The temperature data of the 31 measuring points on motorized spindle were all used as sample data for heat transfer coefficient optimization by genetic algorithm. Therefore, these data cannot be reused for accuracy analysis. The temperature measuring system can monitor the temperature of the water inlet and outlet of the motorized spindle. Therefore, the water outlet temperature can be used to evaluate the prediction accuracy. Table 6.12 shows the simulated temperature T * of motorized spindle Table 6.12 Simulated water outlet temperature

Time t/s

Water outlet temperature T * /°C 10,000 r/min

15,000 r/min

0

12

18.76

400

14.19

19.62

800

15.03

20.03

1200

15.37

22.34

1600

15.43

23.21

2000

15.71

23.62

2400

15.68

23.84

2800

15.78

24

3200

15.87

24.09

3600

15.90

24.28

4000

15.93

24.43

4400

16.03

24.50

4800

16.05

24.61

6.4 Accuracy Analysis of Prediction Model

(a) n=10000r/min

217

(b) n=15000r/min

Fig. 6.17 The comparison of the simulated and measured water outlet temperature of motorized spindle

water outlet. Figure 6.17 shows the comparison of the simulated and measured water outlet temperature of motorized spindle at a rotational speed of 10,000 and 15,000 r/min. Clearly, the error of the prediction model is small and the prediction accuracy is high. According to Eqs. (6.8)–(6.10), the standard deviation of the error, variance and standard deviation of the temperature prediction model based on genetic algorithm were 1.5%, 2.8745 × 10−4 and 0.045, respectively.

6.4.3 Comparision of Accuracy Between the Two Temperature Prediction Models The accuracy of the two temperature prediction models after optimization were compared and the results are listed in Table 6.13. Table 6.13 suggests that the least squares method enables a higher prediction accuracy of temperature field than genetic algorithm. When genetic algorithm was adopted, the temperature data used were the steady state temperature data of the motorized spindle. It was assumed that the heat transfer coefficient of each part of the motorized spindle does not change with time and the temperature rise of the spindle. In other words, the heat transfer coefficients remained constant. In fact, however, Table 6.13 Comparision of accuracy between two temperature prediction models Standard deviation of the error SDE/% Least squares method

1.4125

Genetic algorithm

2.05

Variance/10−4

Standard deviation

7.93

0.024

22.87

0.031

218

6 Thermal Performance Prediction of Motorized Spindle

there is a complicated coupling relationship between heat transfer coefficient and temperature field. When least squares method was adopted, the temperature data used were transient temperature data. The heat transfer coefficients changed with time and the temperature rise of motorized spindle. In other words, the heat transfer coefficients were also transient, which is closer to the actual situation. Therefore, even if only temperature data of five measuring points were used, the least squares method still enables a higher prediction accuracy of motorized spindle temperature field.

6.5 Loss Sensitivity Analysis of Temperature Prediction Model In addition to the heat transfer coefficient, the heat loss of motorized spindle is also a parameter that affects the accuracy of the prediction model proposed here. In fact, there exists a certain error in the loss test. In the modeling process, it is assumed that 100% of motor loss and bearing loss are converted into heat and measured by the motorized spindle performance test system. Therefore, when heat source is set as the heat source boundary condition in the temperature prediction model, it is necessary to analyze the loss sensitivity of the prediction model.

6.5.1 Parameter Local Sensitivity Analysis Sensitivity analysis aims to qualitatively or quantitatively evaluate the influence of model parameter errors on model results, and is a tool used in model parameterization and model correction. Local sensitivity analysis generates small perturbations to the optimal estimate of a parameter, and calculates the rates of changes of the model output caused by the small perturbations under the condition that other parameters remain unchanged. The local sensitivity analysis method is simple, easy to implement and has small calculation amount. Therefore, it is chosen to study the loss sensitivity of the motorized spindle temperature prediction model. Morris method is a widely used sensitivity analysis method. It is also called a one-step-at-a-time method, meaning that in each run only one input parameter is given a new value. The parameter value can be randomly changed within the given range. The change of model output in relative to change of model input, i.e., rate of change, is used to represent the degree of parameter’s influence on model output. The modified Morris method uses independent variables to fix step size change and the average of Morris coefficients calculated by multiple disturbances is taken as the parameter sensitivity index. The calculation formula is

6.5 Loss Sensitivity Analysis of Temperature Prediction Model Table 6.14 Parameter sensitivity levels

Level

219

Morris coefficient

Sensitivity

I

0 ≤ |Si | < 0.05

Not sensitive

II

0.05 ≤ |Si | < 0.2

Medium sensitivity

III

0.2 ≤ |Si | < 1

Sensitive

IV

|Si | ≥ 1

Highly sensitive

  n−1  (Yi+1 − Yi ) Y0  S= (n − 1) (Pi+1 − Pi ) 100 i=0

(6.11)

where S is Morris coefficient; Yi is the model output after ith run; Yi+1 is the model output after (i + 1)th run; Y0 is the initial calculation result after parameter calibration; Pi is the percentage change of the parameter value in relative to the initial parameter value after parameter calibration in ith run; Pi+1 is the percentage change of the parameter value in relative to initial parameter value after parameter calibration in (i + 1)th tun; n is the running times of model. The greater the Morris coefficient S, the more sensitive the model is to the parameter. Table 6.14 lists four parameter sensitivity levels.

6.5.2 Prediction Model Loss Sensitivity Analysis For the convenience of calculation, the simulated steady-state temperature data of the four measuring points on motorized spindle shown in Fig. 6.11 were taken as the output value for sensitivity determination. S is the Morris coefficient of loss sensitivity. The predicted temperature is the output value of the model. The simulated data of the four measuring points on motorized spindle are the initial calculation result after parameter calibration. −10, −5, 5, and 10% are the percentage changes of parameter value in relative to initial parameter value after parameter calibration and after model running for i = 1, 2, 3 and 4 times. The total running times of model is 4. These data were substituted into Eq. (6.11). The Morris coefficients of loss sensitivity of different measuring points in the model were obtained (Table 6.15). The average Morris coefficient of loss sensitivity of the selected four measuring points is |S| = 0.262. According to Table 6.14, we can see that the loss sensitivity Table 6.15 The Morris coefficients of loss sensitivity of different measuring points in the model Measuring point

Measuring point 1

Measuring point 2

Measuring point 3

Measuring point 4

Morris coefficient

0.293

−0.449

0.004

−0.302

220

6 Thermal Performance Prediction of Motorized Spindle

of the motorized spindle temperature prediction model is of level III. Therefore, the accuracy of loss calculation has a great influence on the accuracy of motorized spindle temperature prediction model. Temperature prediction model after heat transfer coefficient optimization by genetic algorithm was taken as an example. It was assumed that the loss of motor and bearing measured during loss test at a rotational speed of 15,000 r/min were completely (100%) converted into heat. Then, the modified Morris method was used to analyze local sensitivity of the temperature prediction model to loss, and disturbances were performed at four levels (−10, −5, 5, and 10%). Figure 6.18 shows the comparison between predicted temperature and measuring temperature of the four measuring points under different disturbances. According to Eqs. (6.8)–(6.10) and Fig. 6.18, the SDE, variance and standard deviation of the temperature prediction model were 9.16%, 0.00847 and 0.0920, respectively, under −10% disturbance of loss; 5.4%, 0.00297 and 0.0545, respectively, under −5% disturbance of loss; 1.26%, 2.86 × 10−4 and 0.0169, respectively, under 5% disturbance of loss; and 4.51%, 0.00219 and 0.0468, respectively, under 10% disturbance of loss. In sum, the error between the predicted and measured temperature is the smallest under 5% disturbance of loss. This further proves the great influence of the accuracy of loss calculation on the accuracy of temperature prediction model.

(a) -10%

(c) 5%

(b) -5%

(d) 10%

Fig. 6.18 Comparison between predicted temperature and measuring temperature of the four measuring points under different disturbances

6.6 Thermal Deformation Prediction of Motorized Spindle

221

6.6 Thermal Deformation Prediction of Motorized Spindle At present, there are two methods for thermal deformation prediction of motorized spindle: experimental data modeling and finite element modeling. Since thermal deformation can hardly be measured in real time, finite element modeling is often used by Chinese scholars to predict the thermal deformation of motorized spindle. In order to improve the accuracy of thermal deformation prediction, the experimental data are used for finite element modeling and the heat transfer coefficient of motorized spindle is optimized using the biogeography-based optimization [30–32].

6.6.1 Finite Element Model for Motorized Spindle Thermal Deformation The temperature rise and thermal deformation of motorized spindle are two coupled and interactive processes. Finite element coupled field analysis can be divided into indirect coupling and direct coupling analysis. Compared with traditional finite element software, the multi-physics coupled finite element software used here directly couples the temperature field and structure field of motorized spindle. This coupling method shows high accuracy and high speed of calculation, and avoids errors caused by interaction among multiple fields. The operation process is divided into six steps: geometric modeling, adding material properties, selecting physical fields and setting boundary conditions, meshing, solving, and post-processing. 1. Geometric modeling 3D modeling was selected and the unit of model size was set to mm. CAD or Solidworks geometry was imported into COMSOL finite element analysis software through model import function. Model can also be directly built in the finite element software. 100MD60Y4 motorized spindle was taken as research object. On the basis of ensuring calculation accuracy, the main structures of motorized spindle including main shaft, stator, rotor, bearing, balance ring, water jacket and outer casing were considered, whereas some small structures such as screws, vent holes and oil passages were ignored. In order to reduce the amount of calculation, the spindle system was regarded as an axisymmetric structure. Then, a 1/4 three-dimensional model of the 100MD60Y4 motorized spindle was established. Figure 6.19 shows the threedimensional model of the 100MD60Y4 motorized spindle. Subsequently, the various components of motorized spindle were defined according to the material properties listed in Table 6.16. According to the need to establish a model of motorized spindle that couples temperature field and structure field, the multi-physics thermal stress module was selected and it can directly couple solid heat transfer module and solid mechanics module. The stator, rotor and bearing of motor were all set as heat sources. There exist different degrees of heat transfer between motorized spindle and external air,

222

6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.19 Three-dimensional geometric model of motorized spindle

Table 6.16 Material properties for the model Component

Material

Density/g/cm3

Linear expansion coefficient/10−6 °C−1

Thermal conductivity/[W/(m °C)]

Specific heat capacity/[J/(kg °C)]

Stator windings

Copper

8.856

9

400

386

Stator core

Silicon steel

7.852

13

35

535

Water jacket

40Cr

7.850

10

60.50

434

Rotor bar

Cast aluminum

2770

23.6

167

875

Shaft and housing

40Cr

7.850

10

60.50

434

between cooling gas and stator and rotor in the air gap, between shaft head and air, between bearing and the compressed air, between stator water jacket and cooling water, and between shaft end and air. Therefore, different heat transfer coefficients were set for these different positions as boundary conditions. Fixed support was also set.Considering that the model structure is a solid of revolution, it was meshed into regular tetrahedral and triangular grids. The numbers of tetrahedral and triangular grids were 25,480 and 8110, respectively. The minimum grid mass was 0.2024, the average grid mass was 0.6764, and the total grid volume is 348,300.0 mm3 . Figure 6.20 shows the meshing model of motorized spindle. 2. Boundary conditions The rotational speed of motorized spindle was set at 12,000 r/min. The ambient temperature was set at 15 °C. The gas pressure at the inlet of oil/air lubrication system was set at 0.20 MPa, the oil supply per time was 16.3 mm3 /min, and the oil supply interval was 3 min. A constant temperature of 18 °C was maintained by the

6.6 Thermal Deformation Prediction of Motorized Spindle

223

Fig. 6.20 The meshing model of motorized spindle [33, 34]

constant temperature water cooling system and the water flow rate was 10 L/min. The motorized spindle housing was set as a fixed support. The boundary conditions for simulation are listed in Table 6.17. The temperature field and thermal deformation cloud charts of the motorized spindle are shown in Fig. 6.21. As shown in Fig. 6.21a, when the motorized spindle reaches a steady state, a clear temperature distribution is presented inside it. The temperature at rotor is the highest and even reaches 52.9 °C. Although the heat generated by rotor only accounts for about 1/3 of the heat generated by motor, it cannot be well dissipated since the internal cavity of motorized spindle is very small. As a result, the temperature rise of rotor is the highest. Although the heat generated by stator accounts for about 2/3 of the heat generated by motor, the steady state temperature of stator reaches only about 32 °C. This is mainly because the cooling system can continuously cool the stator by taking away most of the heat generated by it [35]. The temperature rises of the front and rear bearings are also relatively high. The temperature at the front bearing is slightly higher than that at the rear bearing. This is because the front bearing is closer to the built-in motor and the heat tends to be transferred from the rotor to the front bearing. As shown in Fig. 6.21b, when the motorized spindle reaches a steady state, the thermal deformation occurs mainly in its front end and the maximum thermal deformation can even reach 75.8 µm. In actual machine tool, the front end of spindle is where the tool is installed. The large thermal deformation at this position means that during actual machining process, the tool will also undergo deformation and thus the machining accuracy of the machine tool will be reduced, which is a problem that cannot be ignored.

Sator heat generation/W

153.8

Parameters

Values

76.0

Rotor heat generation/W

Table 6.17 Boundary conditions for simulation

70

Bearing heat generation/W

206.5

Heat transfer coefficient between rotor and stator/[W/(m2 °C)]

9.7

Heat transfer coefficient between motorized spindle and external air/[W/(m2 °C)] 99.2

Heat transfer coefficient between bearing and compressed air/[W/(m2 °C)] 119.76

Heat transfer coefficient of shaft end/[W/(m2 °C)]

85.6

Heat transfer coefficient between shaft head and air/[W/(m2 °C)]

550

Heat transfer coefficient between cooling water and water jacket/[W/(m2 °C)]

224 6 Thermal Performance Prediction of Motorized Spindle

6.6 Thermal Deformation Prediction of Motorized Spindle

(a) Temperature field cloud

225

(b) Thermal deformation cloud

Fig. 6.21 The temperature field and thermal deformation cloud charts of motorized spindle

6.6.2 Thermal Deformation Prediction Model Based on Heat Transfer Coefficient Optimization As the boundary condition for finite element analysis, the magnitude of convective heat transfer coefficient reflects the intensity of convective heat transfer on the surface of each component inside the motorized spindle. The accuracy of heat transfer coefficient has the most significant influence on the accuracy of the finite element analysis of motorized spindle. At present, the commonly used method for determining the convective heat transfer coefficient is to first determine the flow state of fluid and then use corresponding empirical formula to calculate the heat transfer coefficient. Since the empirical formula is only approximate, there is a large difference between the calculated and the actual convective heat transfer coefficient values. Yang et al. [36] proposed to combine experimental data with neural network to calculate the convective heat transfer coefficient, and verified the accuracy of the method through experiments. Pian Jinxiang et al. [15] proposed to use an artificial bee colony optimization algorithm to optimize the heat transfer coefficient of machine tool spindle. Their results show that the method can improve the accuracy of convection heat transfer coefficient. Biogeography-based optimization (BBO) is a new type of swarm-based intelligent optimization algorithm. It achieves global optimization through cooperation and competition among population individuals. It is simple, easy to implement and suitable for solving high dimensional optimization problems [37]. Jain et al. used BBO to solve the overload scheduling problem of power plant generator [38]. Ren et al. applied BBO to neural network fault diagnosis method [39]. Ren et al. applied BBO to fault diagnosis of pumping unit [40]. Lu et al. used dual-mechanism BBO to

226

6 Thermal Performance Prediction of Motorized Spindle

evaluate cylindricity error [41]. These studies verify the feasibility, correctness and superiority of BBO in engineering applications [42]. 1. Technology roadmap The technology roadmap of the thermal deformation prediction model based on BBO algorithm for motorized spindle is as follows: (1) According to the motion parameters and electrical parameters of motorized spindle, the stator loss, rotor loss and bearing loss are calculated and all converted into heat. (2) A finite element model of the temperature field of motorized spindle is established. The amount of heat generated and heat transfer coefficients are taken as the boundary conditions of the model. Then, the temperature field  T is calculated. (3) The temperature data of the of motorized spindle are col⎤ ⎡ stationary component . 0 . 0 ⎢ T1 . Tn ⎥ ⎢ lected and extracted T = ⎣ · · · · · · · · · ⎥ ⎦, where n is the number of temperature . m m . T1 . Tn measuring points and m is the number of data acquisition times.   (4) The measured steady state temperature T = T1m · · · Tnm and the finite ele m  ment model are used to calculate temperature  T=  T1 · · ·  Tnm . Optimization algorithm is used to optimize the heat transfer coefficients and the optimization results are input into the finite element model. (5) A finite element model of the thermal deformation of motorized spindle is established and used to calculate the thermal deformation. The dynamic thermal deformation prediction model for the motorized spindle is shown in Fig. 6.22. Since motor is integrated into the spindle unit and the rotational speed of spindle is often high, a large amount of heat is generated during operation. The resultant temperature rise can decrease the thermal and dynamic performance of motorized spindle, thereby affecting its normal operation. Therefore, measures must be taken to control the temperature of motorized spindle. Forced water circulation is often used to cool the stator of motorized spindle. Specifically, the cooling water is forced to circulate outside the spindle stator and thus take away the heat generated during the high-speed rotation of spindle. In addition to the lubrication of spindle bearings, oil/air lubrication system can also cool the bearings through the flow of compressed air [43]. Due to the unique cooling and lubrication methods of motorized spindle, the heat exchange process inside it is complicated. In addition to heat conduction, heat transfer is in the form of convection and the convective heat transfer coefficient varies with different parts of motorized spindle. According to the structural characteristics of motorized spindle (Fig. 6.23), the heat transfer coefficients of motorized spindle can be divided into six types: the heat transfer coefficient h1 of natural convection between spindle housing and external air; the heat transfer coefficient h2 of forced convection between cooling gas and stator and rotor in air gap, the heat transfer

6.6 Thermal Deformation Prediction of Motorized Spindle

227

Fig. 6.22 Dynamic thermal deformation prediction model for the motorized spindle

h1: heat transfer coefficient of natural convection between spindle housing and external air; h2: the heat transfer coefficient of forced convection between cooling gas and stator and rotor in air gap; h3: the heat transfer coefficient of forced convection between shaft head and air; h4: the heat transfer coefficient of forced convection between bearing and compressed air; h5: the heat transfer coefficient of forced convection between stator water jacket and cooling water; h6: the heat transfer coefficient of forced convection between shaft end and air.

Fig. 6.23 The main structure of motorized spindle and the heat transfer form of each part

228

6 Thermal Performance Prediction of Motorized Spindle

coefficient h3 of forced convection between shaft head and air, the heat transfer coefficient h4 of forced convection between bearing and compressed air, the heat transfer coefficient h5 of forced convection between stator water jacket and cooling water, and the heat transfer coefficient h6 of forced convection between shaft end and air. Figure 6.23 shows the main structure of motorized spindle and the heat transfer form of each part. 2. Heat transfer coefficient optimization based on BBO The specific steps of BBO-based heat transfer coefficient optimization are as follows: Step 1: Initialize various parameters in BBO algorithm. Habitat number S = 50, evolution times N = 100, maximum species number n = 6, maximum immigration rate I = 1, maximum emigration rate E = 1, and maximum mutation rate mmax = 0.05. The numerical range of each heat transfer coefficient is also set. Step 2: According to the theory of heat transfer coefficient and empirical formula [15−20] and in combination with the specific working  conditions, the initial values of heat transfer coefficients h = h01 h02 h03 h04 h05 h06 are calculated. Step 3: The obtained heat transfer coefficients h are input into the motorized spindle temperature field prediction model, and the temperature field is calculated. The predicted temperatures of the n measuring points are extracted. Step 4: The fitness value is evaluated on the basis of the measured and predicted temperature of the n measuring points. Step 5: Determine whether the termination condition is met. The loop termination condition is: if fi ≤ 0.5 is satisfied, then the obtained heat transfer coefficient value is the optimal value, the loop is terminated, and the optimal heat transfer coefficient and the temperature field prediction model are output; if fi > 0.5, then return to step 6 and start the loop. Step 6: Calculate the suitability, immigration rate and emigration rate of each habitat. For the specific formula, see Chap. 5 [29]. Step 7: Perform migration. Step 8: Calculate the probability of mutation [20]. Step 9: Generate the next set of habitats and return to step 3 for loop. The flow chart for optimizing the heat transfer coefficient is shown in Fig. 6.24. 3. Motorized spindle temperature test When BBO is used to optimize heat transfer coefficient, it is necessary to collect the temperature data of motorized spindle. Due to the high speed rotation and structural precision requirements of motorized spindle, only the temperature data of some stationary components of motorized spindle can be collected. Considering the correlation between temperature measuring points and thermal deformation prediction, seven temperature measuring points were selected here. The distribution of the temperature measuring points on the motorized spindle is shown in Fig. 6.25. There are five sensors placed on motorized spindle. Sensors 1 and 2 measure the temperature of the outer surface of front bearing. Sensors 3 and 4 measure the temperature of

6.6 Thermal Deformation Prediction of Motorized Spindle

229

Fig. 6.24 The flow chart for heat transfer coefficient optimization

the outer surface of rear bearing. Sensor 5 is placed between sensors 1 and 3. In addition, two sensors (6 and 7) are built in the stator coil and outer ring of front bearing, respectively. This means that sensor 6 measures stator temperature and sensor 7 measures the temperature of front bearing. The test conditions are the same as the simulation conditions. Each test time was set at 3600 s. A set of data were collected

230

6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.25 Distribution of temperature measuring points on motorized spindle

every 20 s in the experiment and a total of 180 sets of data were collected. The test was carried out three times and the average of three measurements was taken. The average temperature of each position of the motorized spindle during operation is shown in Table 6.18. 4. Heat transfer coefficient optimization According to the distribution of temperature measuring points shown in Fig. 6.25,   T = T1180 . . . T6180 were selected from the 180 data sets in Table 6.18 for heat transfer coefficient optimization by BBO. Nodes were set in the corresponding positions of finite element model and the simulated steady-state temperature data   the  T =  T6180 of these positions were extracted. Then the simulated and T1180 · · ·    experimentally measured temperature data T = T1180 . . . T6180 at 3600th s were used to optimize the heat transfer coefficients according to BBO. Let the population number M = 50 and the maximum number of iterations N = 100. The objective function curve in the iterative process of BBO algorithm is shown in Fig. 6.26. After 90 iterations, the objective function value is almost close to 0.5, and when the num∼ h100 ber of iterations reaches 100, the convective heat transfer coefficients h100 1 6 are 10.54, 180.84, 65.03, 80.28, 455.13 and 119.76, respectively. At this point, the objective function f it = 0.4995 and satisfies the iteration termination condition. The temperature fields and thermal deformation fields after iterations for 25, 50 and 100 times were extracted. It can be seen from Fig. 6.27 that as the iterative process continues, the temperature field and thermal deformation field of the motorized spindle change constantly. 5. Verification of transient temperature prediction error In order to explain the effect of heat transfer coefficient optimization⎡on the prediction ⎤ .. 0 0 T . T 6 ⎥ ⎢ 1 ⎥ accuracy of temperature field, the measured temperature T = ⎢ ⎣ ··· ··· ··· ⎦ . T1180 .. T6180

Time/s

0

20

40

60

…

3540

3560

3580

3600

Temperature measuring points

1

2

3

4

…

177

178

179

180

33.40

33.40

33.40

33.39

…

14.01

13.39

13.24

12.96

Temperature measuring point 1

Temperature/°C

33.41

33.41

33.41

33.41

…

14

13.4

13.25

12.96

Temperature measuring point 2

32.5

32.5

32.5

32.48

…

13.6

13.33

13.15

12.97

Temperature measuring point 3

Table 6.18 Measured temperatures of different positions of motorized spindle

32.5

32.5

32.5

32.5

…

13.56

13.32

13.14

12.97

Temperature measuring point 4

25.7

25.7

25.7

25.7

…

13.2

13.11

13.03

12.96

Temperature measuring point 5

34.6

34.6

34.6

34.6

…

15.39

15.27

15.16

15

Temperature measuring point 6

38.5

38.5

38.5

38.5

…

16.17

15.95

15.84

15

Temperature measuring point 7

6.6 Thermal Deformation Prediction of Motorized Spindle 231

232

6 Thermal Performance Prediction of Motorized Spindle

Fig. 6.26 Objective function curve

⎤ .. 0 0   . T T 6 ⎥ ⎢ 1 ⎥ were compared with the simulated transient temperature  T =⎢ ⎣ · · · · · · · · · ⎦ of .  T 180 T 180 ..  ⎡

1

6

motorized spindle before and after heat transfer coefficient optimization. Figure 6.28 shows comparison of measured temperature data with simulated temperature data before and after heat transfer coefficient optimization. Clearly, after heat transfer coefficient optimization, the simulated temperature data are closer to the measured temperature data. The simulated temperature curve of spindle front bearing after heat transfer coefficient optimization is highly consistent with the measured temperature curve (Fig. 6.29). In the time range of 0–1800 s, the maximum temperature difference between the two curve is 3.6 °C. In 100–3600 s, the temperature difference between the two is even smaller. Therefore, the accuracy of the prediction model is high. Figure 6.30 shows comparison of measured thermal deformation with simulated thermal deformation after heat transfer coefficient optimization. Clearly, the simulated thermal deformation of motorized spindle after heat transfer coefficient optimization is highly consistent with the measured thermal deformation (Fig. 6.30). The error of thermal deformation prediction is small and the prediction accuracy is high.

6.6 Thermal Deformation Prediction of Motorized Spindle

(a) Temperature field (ºC) after 25 iterations

(b) Axial thermal deformation field (μm) after 25 iterations

(c) Temperature field (ºC) after 50 iterations

(d) Axial thermal deformation field (μm) after 50 iterations

(e) Temperature field (ºC) after 100 iterations

(f) Axial thermal deformation field (μm) after 100 iterations

233

Fig. 6.27 The temperature fields and thermal deformation fields of motorized spindle after different number of iterations

234

6 Thermal Performance Prediction of Motorized Spindle

(a) Temperature measuring point 6

(b) Temperature measuring point 1

(c) Temperature measuring point 3

(d) Temperature measuring point 5

Fig. 6.28 Comparison of measured temperature data with simulated temperature data before and after heat transfer coefficient optimization Fig. 6.29 Comparison of measured temperature data with simulated temperature data of spindle front bearing after heat transfer coefficient optimization

6.6 Thermal Deformation Prediction of Motorized Spindle

235

Fig. 6.30 Comparison of measured thermal deformation with simulated thermal deformation after heat transfer coefficient optimization

References 1. Zhang LX, Li CQ, Wu Y et al (2017) Hybrid prediction model of the temperature field of a motorized spindle. Appl Sci 7(10):1091-1–14 2. Zhang LX, Li JP, Wu YH et al (2018) Prediction model and experimental validation for the thermal deformation of motorized spindle. Heat Mass Trans 54(2):1–14 3. Zhang LX, Li CQ, Li JP et al (2017) Temperature rise prediction model of high-speed and high-precision electric spindle. J Mech Eng 53(23):129–136 4. Zhang LX, Xia WL, Li JJ et al (2016) Prediction of surface temperature of electric spindle based on genetic neural network. Control Eng 23(2):243–248 5. Wu YH, Li SH (2018) Ceramic motorized spindle. ISCI Publishing Ltd., London 6. Zhang LX, Li JP, Li CQ et al (2018) Experimental study on temperature rise and thermal deformation of high-speed spindle of CNC machine tools. Mech Des Manuf 01:129–132 7. Shu YJ, He BM (2011) Investigation of the high speed rolling bearing temperature rise with oil-air lubrication. J Tribol Trans ASME 133(2):021101-1–021101-9 8. Zhang LX (2015) A test system and method of electric spindle load. China 201310522511. 5 9. Shi QH, Zhang LX, Wu YH et al (2015) Analysis of thermal deformation simulation of mechanical spindle of precision lathe based on thermal-structural coupling. Dev Innov Mech Electr Prod 02:120–122 10. Zhang K, Xu WZ, Zhang LX (2018) The effect of contact thermal resistance on the thermal characteristics of high-speed electric spindle. Combined machine tools and automated machining technology 04:23–28 11. Zhang LX, Yu SY (2018) The effect of oil mist generated by oil and gas lubrication of high-speed electric spindle on air quality. J Shenyang Jianzhu Univ (Natural Science Edition) 02:341–349 12. Zhang K, Wang XK, Zhang LX (2018) Modeling and air flow field analysis of 100MD60Y4 electric spindle oil and gas lubrication system. J Shenyang Jianzhu Univ (Natural Science Edition) 03:505–513 13. Zhang LX, Xia WL, Li JJ et al (2014) An intelligent prediction method for surface temperature of electric spindle of high-speed CNC machine tool. Dev Innov Mech Electr Prod 04:133–135 14. Pian JX, Liu MJ, Zhang LX et al (2015) Optimization of convective heat transfer coefficient of machine tool spindle based on bee colony algorithm. J Instrum 12:2706–2713

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15. Pian JX, Liu MJ, Liu JX et al (2016) Optimization of convective heat transfer coefficients of the mechanical spindle based on bee colony algorithm. Control Eng 23(9):1349–1355 16. Zhang LX, Li JP, Li CQ (2016) Experimental study on deviation and thermal deformation of 150MD24Z7.5. Electr Spindl (09):59–61 17. Wu YH, Yu WD, Zhang LX et al (2016) Thermal characteristics analysis of 150MD24Y20 high speed electric spindle. J Shenyang Jianzhu Univ (Natural Science Edition) 04:703–709 18. Zhang LX, Gong WJ (2017) Simulation analysis and research on the coupling relationship between air gap and thermal deformation of electric spindle. Micro Motor 11:34–39 19. Zhang LX, Gong WJ (2017) Experiment on the influence factors of thermal characteristics of 100MD60Y4 high speed electric spindle. J Shenyang Jianzhu Univ (Natural Science Edition) 04:703–712 20. Zhang LX, Li CQ, Li JP (2016) Experimental study on factors affecting the temperature rise of high-speed spindle. Comb Mach Tools Autom Mach Technol (06):75–77+91 21. Zhang LX, Liu T, Li CQ (2015) The influence of cooling water flow rate on the temperature rise of the electric spindle motor. Comb Mach Tools Autom Mach Technol (08):36–38+42 22. Zhang K, Chen N, Zhang LX et al (2015) The effect of cooling water channel width on the temperature rise of ceramic electric spindle. Mech Des Manuf 03:104–106 23. Zhang LX, Li CQ, Li JP et al (2017) The effect of high-speed electric spindle cooling water parameters on its temperature field. Mech Sci Technol 09:1414–1420 24. Zhang LX, Yan M, Wu YH et al (2015) Simulation analysis for two different materials motorized spindles with model coupled multi-physics. Mater Res Innov 19(1):S2-1–7 25. Zhang LX, Yan M, Wu YH (2014) Machine-electric-thermal-magnetic coupling model of highspeed electric spindle and its dynamic performance analysis. Comb Mach Tools Autom Mach Technol 11:35–38 26. Zhang LX, Yan M, Wu YH et al (2015) Machine tool and hydraulic (13):120–124 27. Zhang LX, Yan M, Wu YH et al (2016) Multi-field coupling model and dynamic performance prediction of 150MD24Z7.5 high speed electric spindle. Vib Shock 35(01):59–65 28. Wu YH, Yu WD, Zhang LX et al (2014) Temperature field analysis of electric spindle based on loss experiment. J Shenyang Jianzhu Univ (Natural Science Edition) 30(01):142–146 29. Wu YH, Zhang K, Deng HB et al A balancing device with built-in spindle mechanical online. China zl201510398608. 9, 2017. 05. 24 30. Wu YH, Zhang K, Deng HB et al. Mechanical online dynamic balancing system with built-in Spindle: Chinese patent, CN105021352A. 2015. 11. 04 31. Stephen H, Lane M. Rotating machine active balancer and method of dynamically balancing a rotating machine shaft with torsional vibrations. US 20060005623A1. 2006. 06. 12 32. Wang ZM (2009) Noise and vibration measurement. Science Press, Beijing, China, pp 134–135 33. Gong WJ, Zhang LX, Li JP et al (2016) A review of thermal deformation prediction models for electric spindles. Dev Innov Mech Electr Prod 29(03):125–128 34. Zhang LX, Gong WJ, Zhang K et al (2018) Thermal deformation prediction of high-speed motorized spindle based on biogeography optimization algorithm. Int J Adv Manuf Technol 97(5–8):3141–3151 35. Zhang LX, Liu T, Wu YH (2016) Design of the motorized spindle temperature control system with PID algorithm. Republic of:Springer Verlag, Seoul, Korea 36. Yang H, Yin GF, Fang H et al (2011) Research on calculation method of convective heat transfer coefficient in machine finite element thermal analysis. J Sichuan Univ (Engineering Science Edition) 43(04):241–248 37. Rahmati SHA, Zandieh M (2012) A new biogeography-based optimization (BBO) algorithm for the flexible job shop scheduling problem. Int J Adv Manuf Technol 58(9–12):1115–1129 38. Jain J, Singh R (2013) Biogeographic-based optimization algorithm for load dispatch in power system. Int J Emerg Technol Adv Eng 3(7):549–553 39. Ren WJ, Zhao YJ, Wang TR et al (2014) Research on neural network fault diagnosis method based on biogeographic optimization algorithm. Chem Autom Instrum 41(02):149–153+18 40. Ren WJ, Zhao YJ, Wang TR et al (2014) Research on fault diagnosis of pumping unit based on biogeographic optimization algorithm. J Syst Simul 26(06):1244–1250

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Chapter 7

Automatic Suppression of Motorized Spindle Vibration

Modern high-speed CNC machining center features high rotational speed of spindle, high operational precision and high machining efficiency. The improvement of speed and precision is based on high-precision dynamic balance. However, for motorized spindle, the existence of dynamic imbalance is inevitable due to factors such as manufacturing and installation errors as well as material unevenness. During high speed operation, the motorized spindle has higher requirements for control of imbalance than rotor. A slight imbalance may result in severe loss of the rotational accuracy of spindle and even the instability of bearing support system. Only by controlling the residual imbalance of motorized spindle within a certain range can the vibration of spindle during high-speed operation be suppressed and the machining accuracy be ensured. In order to reduce the imbalance of motorized spindle, asymmetric structure should be avoided during design and error should be minimized during the processing and assembly process. Even if these are done, spindle imbalance cannot be completely eliminated, thus a motorized spindle often undergoes initial dynamic balancing to reduce imbalance before leaving the factory. However, the slight asymmetry, abrasion or sticking of spindle tool can still destroy the original dynamic balance. Moreover, the spindle tool system is disturbed by complex factors such as cutting force excitation, thermal deformation and centrifugal force due to high-speed rotation, which also destroy the dynamic balance of motorized spindle, thereby reducing the stability of high-speed machine tool spindle system. When traditional offline dynamic balancing method is used to eliminate the imbalance of motorized spindle, the interruption of the machining process will lead to low efficiency. Therefore, research on dynamic imbalance of high-speed spindle and its online control technology is conducive to improving the performance of high-speed motorized spindle and ensuring the long-term stability and efficient operation of machine tools, thus improving the overall level of machine tool industry and manufacturing industry.

© Springer Nature Singapore Pte Ltd. 2020 Y. Wu and L. Zhang, Intelligent Motorized Spindle Technology, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3328-0_7

239

240

7 Automatic Suppression of Motorized Spindle Vibration

Online automatic balancing technology of rotor mainly includes passive dynamic balance technology and active dynamic balance technology. The working principle of passive dynamic balance technology is that when the flexible rotor works above the critical rotational speed, the relationship between its imbalance and vibration responses is obtuse, and the counterweight under the action of centrifugal force will automatically compensate for the original imbalance. Due to its limited accuracy, this technology is less used in industry. Active dynamic balance technology achieves automatic rotor balancing by means of external energy input. It is usually divided into two categories: one is direct active vibration control, which directly applies an external force on the rotating object to offset the centrifugal force caused by the imbalance. The applied external force is generally electromagnetic force, liquid impact force and so on. The other type is mass redistribution method, which balances the rotor with a balanced terminal that rotates with the rotor. The balanced terminal can eliminate imbalance by adjusting mass distribution [1]. Rotating vibration control technology is a typical fault self-healing technology and related research mainly focuses on imbalance vibration control. The core of active vibration control technology lies in the study of balancing device, control method and vibration signal acquisition and processing.

7.1 Components and Working Principles of Dynamic Balancing System Built-in online dynamic balancing system for motorized spindle mainly consists of balancing head, spindle, motor, eddy current sensor, base, controller, data collector, and computer [2–5]. The spindle is hollow and the balancing head is installed in the inner hole of the spindle. Several Hall sensors are also installed internally to measure phase position and rotational speed. The controller is directly connected to the balancing head for its drive control and signal transmission. The vibration is generally measured with eddy current displacement sensor. Figure 7.1 shows the block diagram of vibration test system.

7.1.1 Balancing Head Slip ring-based online dynamic balancing device is mainly composed of a static ring and a moving ring [6]. The static ring is an electromagnetic driver, which is mainly composed of a coil and an iron core. The moving ring is an actuator mainly composed of counterweight plates supported by bearings, and is a component that forms dynamic balance compensation vector. When the two counterweight plates in the device rotate to a certain angle, a vector is synthesized. When the magnitude of the vector is equal to that of the original imbalance and their directions are opposite,

7.1 Components and Working Principles of Dynamic Balancing …

241

Fig. 7.1 The block diagram of vibration test system

dynamic balance can be achieved. When the imbalance amounts produced by the two counterweight plates are at an angle of 180°, the device has no balancing effect. When the imbalance amounts generated by the two counterweight plates are completely overlapped, the balancing head has the maximum balancing ability. When it is detected that the spindle vibration exceeds the set threshold, the software system starts to process the vibration signals, calculate the positions where the counterweight plates should arrive, and then send an electric pulse to the static ring coil through the controller. Then, the coil generates an electromagnetic field, and the counterweight plates are driven to rotate to the target positions and compensate for the imbalance of the machine. The working principles of the slip ring are shown in Fig. 7.2. The static ring is made of material with high magnetic permeability. It is characterized by alternative arrangement of boss and groove. The lengths of the boss and the groove are both equal to the distance between the permanent magnets on the moving ring. The role of static ring is to stabilize the balance position and transmit the electromagnetic field generated by the coil. When the coil is excitated, the iron teeth interact with the permanent magnets on the counterweight plates to achieve stepping. When the coil excitation is completed, the counterweight plates remain in the next position and stop moving. In the stable position, the spatial relationship between the permanent magnets and the inner edge of the magnetic plate is shown in Fig. 7.3.

242

7 Automatic Suppression of Motorized Spindle Vibration

Permanent magnet

Magnetic line of force of permanent magnet

Permanent magnet

Driving force Moving ring Counterweight plate

Magnetic line of force of static ring

Fig. 7.2 The working principles of the slip ring

Fig. 7.3 Slip ring-based balancing head

When the next voltage pulse arrives, the electromagnetic force will push the counterweight plates to move to the next stable position, which means that they will move past unstable positions as shown in Fig. 7.4. Figure 7.5 shows the interaction between electromagnetic fields during the slipping process. Comparing Figs. 7.2 and 7.5, we can see that the magnetoresistance of the closed magnetic circuit of the permanent magnet in Fig. 7.2 is lower than that in Fig. 7.5, and the counterweight plate can stay in a stable position with a lower magnetic resistance.

7.1 Components and Working Principles of Dynamic Balancing …

243

Fig. 7.4 Sliding of counterweight plate from the current stable position to next stable position Magnetic line of force of magnetic line of force Permanent magnet

Permanent magnet

Driving force Moving ring Counterweight plate

Magnetic line of force of static ring

Fig. 7.5 The interaction between electromagnetic field and permanent magnetic field during the slipping process of counterweight plate

7.1.2 Sensor The online dynamic balancing system determines the characteristics of imbalance quantity according to the vibration information of rotor system and performs mass adjustment to achieve online dynamic balancing. The quality of measured vibration signals has a decisive influence on the performance of the dynamic balancing system. Ordinary vibration signal has three elements: vibration amplitude, vibration frequency and vibration phase position. In order to obtain these three elements, it is necessary to set a measurement starting point on the rotating shaft or the turntable as a reference phase position signal. Then, one can measure vibration signal starting from this point and phase signal generated by the reference point itself. The vibration amplitude, frequency and phase can be accurately calculated from the phase position signals and vibration signals.

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7 Automatic Suppression of Motorized Spindle Vibration

1. Phase position signals and speed measuring sensor The phase position signal is a rectangular wave signal with the same period as rotation, and the rising edge of the signal can be used as a measurement starting point of vibration signal. However, conventional rotor system generally does not have a signal generating device with the same rotational period as the system, and must be designed separately. Common signal generating devices include photoelectric sensors and Hall elements. Photoelectric sensor is designed based on the principle of light reflection and photoelectric effect. The working principles of a typical photoelectric sensor are as follows: Infrared light is emitted from the infrared emitting tube and reflected by semi-lens onto the spindle to be tested. A reflective sticker is attached to the corresponding position of the spindle. When the infrared light reaches the reflective sticker, it can be reflected back and directed through the semi-lens to the infrared receiving tube. When the infrared receiving tube receives infrared light with intensity higher than the specified intensity, a corresponding electrical signal change will be generated, and the signal can be output through a relevant circuit. The photoelectric sensor generates a waveform per rotation of the rotary shaft. Therefore, during the operation of rotor system, the photoelectric sensor outputs signal with a period the same as that of rotor rotation. Since this method can avoid the influence of system vibration on signals and the cost of the sensor is low, it is widely used in rotational speed measurement, dynamic balancing and so on. However, photoelectric sensor system is generally large in size and can hardly be built in spindle. Moreover, it is susceptible to ambient light and the influence of obstructions. Therefore, Hall element is used here as the signal generating device. Hall element is a magnetic field sensor fabricated according to Hall effect. Hall effect is a kind of magnetoelectric effect, which was discovered in 1879 by A. H. Hall (1855–1938) when he studied the conductive mechanism of metals. Later, this effect is also found in semiconductors, conductive fluids, etc., and the Hall effect of semiconductors is much stronger than that of metals. Various Hall elements made according to this phenomenon are widely used in industrial automation, detection, information processing, etc. Hall element has many advantages such as firm structure, small size, light weight, long service life, convenient installation, low power consumption, high frequency, vibration resistance, as well as dust/oil/water vapor/salt fog corrosion resistance. Linear Hall devices have high precision and good linearity. Hall switches have no contact, little abrasion loss, clear output waveform, no jitter, no rebound, and high position repeatability (reach micron level). Hall elements can be used in a wide temperature range from −55 to 150 °C. The schematic of Hall element is shown in Fig. 7.6a. A Hall semiconductor chip is integrated inside the element. When a constant current passes through the chip and the element is in a magnetic field, the electron flow is shifted to one side. This causes the chip to generate a potential difference perpendicular to the direction of the current, which is the so-called Hall voltage. The Hall voltage changes with magnetic field strength. The stronger the magnetic field, the higher the voltage. The weaker the

7.1 Components and Working Principles of Dynamic Balancing …

(a) Package appearance

245

(b) Internal block diagram

Fig. 7.6 Hall element

magnetic field, the lower the voltage. Generally, the Hall voltage is low and of only a few millivolts. After amplified by an amplifier in the integrated circuit, the voltage becomes high enough to output a strong signal. Figure 7.6b is the block diagram of the internal integrated amplifier circuit of Hall element. Pin 1 is the power line, pin 2 is the ground line, and pin 3 is the output line of the amplified signal. The Hall element is small in size (several millimeters) and has a simple peripheral circuit. Three Hall elements are built into the balancing head and fixed on stationary part. Magnets are fixed in the corresponding positions of the balancing head case and the two counterweight plates. The magnets rotate with the spindle. As the magnetic induction detected by the Hall element changes, the output voltage of Hall element also changes, thus the phase position information of the spindle and the counterweight plates can be obtained. 2. Vibration measuring sensor In order to measure vibration signals, various sensors are used to convert the vibration responses of machine, such as displacement, velocity variation or acceleration, into electrical signals. After being amplified by electronic circuit, these signals are sent to the corresponding signal analysis and processing instrument, and the three elements of vibration can be obtained. (1) Amplitude: It indicates the amplitude and energy level of machine vibration. In most cases, the quality of a machine is evaluated according to the magnitude of its vibration amplitude. (2) Frequency: It is defined as the number of vibrations per unit of time and used to further study the source of the excitation force of machine. (3) Phase position: Vibration is a vector. In order to accurately describe it, we should not only measure its magnitude, but also measure its direction. In the process of dynamic balancing, phase position is used to reflect the position of imbalance. There are three important physical quantities in vibration research, namely displacement, velocity and acceleration. These three physical quantities are closely related to each other. For general time average measurement, the phase relationship between these three physical quantities can be ignored. At a certain frequency,

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7 Automatic Suppression of Motorized Spindle Vibration

velocity can be obtained by dividing the acceleration by a coefficient proportional to frequency; displacement can be obtained by dividing acceleration by a coefficient proportional to the square of frequency. These calculations can be implemented in the measuring instrument by integral operations [7]. Vibration displacement X (t) = A sin ωt

(7.1)

π dX = A sin(ωt + ) X˙ (t) = dt 2

(7.2)

d2 X X¨ (t) = = ω2 A sin(ωt + π ) dt 2

(7.3)

Vibration velocity

Vibration acceleration

When the vibration signal is in different frequency ranges, the relationship of vibration intensity with acceleration, velocity and displacement is also different. In general, when the signal is in a high frequency range, the intensity of vibration is proportional to the acceleration. When the signal is in an intermediate frequency range, the intensity of vibration is proportional to velocity. When the signal is in a low frequency range, the intensity of vibration is proportional to displacement. Therefore, for different machines and different frequency ranges, different amplitude measurement parameters should be selected to accurately measure the intensity of vibration. The selection of vibration sensor is generally determined by factors such as measuring site, ambient temperature, ambient humidity, magnetic field, vibration frequency and amplitude range, and matching requirements of supporting instruments. There are three types of commonly used vibration sensors and matched amplifiers: eddy current displacement sensors and transmitters; electric speed sensors and amplifiers; piezoelectric acceleration sensors and amplifiers. Figure 7.7a shows the sensor

(a) Eddy current displacement sensor probe Fig. 7.7 Photograph of eddy current displacement sensor kit

(b) Preamplifier

7.1 Components and Working Principles of Dynamic Balancing System

247

probe and Fig. 7.7b shows the preamplifier for powering and transmitting signals. In addition to the slip-ring balancing head, the core component of the dynamic balancing system also includes a test system. The test system is responsible for rationally allocating the entire system resources so that the various components of system are in an orderly action and cooperate with each other. Finally, the counterweight plates are adjusted to achieve dynamic balance. The test system includes a controller, data collector and so on.

7.2 Software Design of Dynamic Balancing System Online dynamic balancing system software is written in the commonly used virtual instrument software LabVIEW to realize parameter input, signal acquisition, signal processing and dynamic balance control.

7.2.1 Overall Structure of Software System The ultimate goal of the built-in dynamic balancing system is to change the mass distribution of the rotating parts of balancing head and thus offset imbalance and eliminate the eccentricity of rotor. Finally, the vibration caused by the imbalance amount during rotor rotation can be reduced below the expected value, and smooth operation of rotor system is achieved. On this basis, the goal of software design is to output suitable adjusted signals to rotate the counterweight plates to specific positions, thereby changing the mass distribution of rotor system and achieving online dynamic balancing. In order to achieve this goal, various functional modules need to be designed and integrated to form the whole software. As shown in Fig. 7.8, the online dynamic balancing system software mainly includes the following parts: signal acquisition and data processing module, control module, correction module, and basic parameter input module. In addition, it includes the main interface panel and the influencing factor panel.

Fig. 7.8 Constituent modules of online dynamic balancing system sofware

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7 Automatic Suppression of Motorized Spindle Vibration

The signal acquisition and data processing module collects reference and vibration signals, acquires information about counterweight plate phase position and processes vibration signals. The correction module splits imbalance amount and corrects the movement of mass. The basic parameter input module mainly calculates influence coefficients and inputs spindle parameters. The control module coordinates the workflow of each module, make some basic decisions, display the operating status and results, etc. The flowchart of the software is shown in Fig. 7.9. After running, the software first evaluates the vibration data to determine whether the system should be balanced. If yes, then it calculates the imbalance amount and the compensation amount, adjust the positions of the counterweight plates according to the compensation amount, and finally decide whether to stop operation. The automatic tracking filter algorithm for vibration signal extraction based on time domain averaging and FIR filtering is shown in Fig. 7.10. This method first

Fig. 7.9 Flow chart of dynamic balancing system

7.2 Software Design of Dynamic Balancing System

Fig. 7.10 Flow chart of vibration signal measuring algorithm

249

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7 Automatic Suppression of Motorized Spindle Vibration

preprocesses the signals by resampling and interpolation, the purpose of which is to avoid spectrum leakage and fence effect and to ensure the accuracy of subsequent signal processing. The cubic spline curve fitting can be performed on the obtained vibration signals and high-order numerical approximation can effectively improve the precision of amplitude extraction. Then, the time-domain averaging method is used to eliminate high frequency noise signals from vibration signals and improve the signalto-noise ratio of signals. This step ensures that the automatic tracking filter works under weak noise. Signals at other frequencies are filtered out by the designed FIR filter. After filtering, only signal components with frequencies close to the rotational frequency of spindle are preserved. Finally, the amplitudes of the vibration signals with fundamental frequency are extracted by automatic tracking filter. In order to measure the amplitude and phase position of spindle vibration signals, the key is to obtain fundamental frequency to construct standard sine and cosine signals, then extract the DC component containing amplitude and phase position information according to related theories, and finally obtain the amplitude and phase position of fundamental frequency signals through calculations.

7.2.2 Main Interface Design The human-computer interaction interface is an important part of the system. It includes the display of the status of the entire system and experimental results, parameter display and settings, process flow of the device, etc. Nowadays, humancomputer interaction system is more dependent on software design. This is especially true in the field of virtual instruments. Virtual instruments use a series of standard hardware and design various instruments through software. Their human-computer interaction system relies on software design. The interface of the built-in dynamic balancing system software is shown in Fig. 7.11. It mainly includes the display of vibration information, the display of counterweight phase position, the display of imbalance phase position, the indication of coil excitation and the input of basic parameters.

7.2.3 Signal Acquisition Rotational speed, and amplitude and phase position of fundamental frequency signals are important physical quantities in the measurement of rotational mechanical vibrations. In order to measure rotational speed and phase position, it is necessary to set a reference site on the rotor so that a reference signal can be output every rotation of the shaft. The reference signals can not only be used to measure the rotational speed of rotor, but also serve as the basis for measuring the phase positions of vibration signals. In the same way, the phase position of the two counterweight plates of the built-in slip ring balancing head can be obtained.

7.2 Software Design of Dynamic Balancing System

251

Fig. 7.11 The interface of built-in dynamic balancing system software

The sensor output is analog signals and cannot be processed directly by computer that only recognizes digital signals. First, analog-to-digital conversion is performed to obtain the corresponding digital signals. This step is done by data acquisition card. Therefore, the software design only needs to include the signal acquisition program for the data acquisition card used. The signal acquisition system outputs a total of three kinds of phase position signals and one kind of vibration signals, as shown in Fig. 7.12. The signal acquisition includes three parts and four channels of acquisition tasks.

Fig. 7.12 Signal acquisition system

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7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.13 Schematic of phase position measurement

1. Acquisition of phase position and rotational speed signals The reference signals need to be measured by the Hall element. The working principles are shown in Fig. 7.13. Hall elements are built into the balancing head and fixed on stationary part. Magnets are fixed in the corresponding positions of the balancing head casing and the two counterweight plates. The magnets rotate with the spindle. When the permanent magnet on the casing (or counterweight plate) rotates to the same position as the Hall element, the magnetic induction detected by the Hall element changes and the output voltage of Hall element also changes. Therefore, rectangular wave signals that are completely synchronized with the rotational frequency of rotor are obtained, and the phase position information of the spindle and counterweight plates can be obtained. Phase position signals are rectangular wave signals with a period the same as that of rotation, and the rising edge of the signals can be used as a measurement starting point for vibration signals. 2. Vibration signal acquisition Phase position is of great significance in the process of dynamic balancing. Its determination enables to obtain the phase position of imbalance vector and the target position of counterweight plate during dynamic balancing. If the phase position measurement of imbalance signals is not accurate, no good balancing effect can be achieved regardless of the method used. In the measurement of rotational mechanical vibration, phase position has a special definition: the angle at which the vibration peak lags behind the reference phase position. As shown in Fig. 7.14, the rotational Fig. 7.14 Schematic of vibration signal phase position measurement

7.2 Software Design of Dynamic Balancing System

253

Fig. 7.15 Signal acquisition program

speed signal is a rectangular wave detected by Hall element after each rotation of the rotor. Based on the rising edge of the rotational speed signal, the vibration signal is collected by the vibration sensor and ϕ is the phase position of vibration signal according to the definition of phase position [8]. 3. Signal acquisition program The program uses DAQmx functions of LabVIEW to acquire signals on multiple channels simultaneously. For example, the 4-channel signal acquisition system includes 3 channels of Hall element signals and 1 channel of vibration sensor signals. After acquisition, the original signals of each channel are separated and proceed to the next step. In order to avoid occupying too much system resources, a certain cycle interval is set. The main block diagram is shown in Fig. 7.15.

7.2.4 Acquisition of the Phase Position of Counterweight Plate The “Extract Single Tone Information” function of LabVIEW is used to extract the frequency, amplitude and phase position information of fundamental frequency signals. First, the frequencies of the reference signals are extracted as the fundamental frequencies and converted into rotational speeds (rev/min). Then, the phase positions of the three channels of signals are extracted, and the reference phase position is compared with the phase positions of the counterweight plates to obtain the phase angles of the two counterweight plates with respect to the reference. The main block diagram is shown in Fig. 7.16. 1. Data processing The collected spindle vibration signals are mixed signals containing multiple frequency components. In the process of dynamic balancing, only the fundamental frequency signals of the same frequency as rotational speed are concerned. Therefore, it is necessary to analyze and process the vibration signals to extract the fundamental frequency signals. Full cycle sampling and cross-correlation filtering are used to extract the imbalance signals. The flowchart of the method is shown in Fig. 7.17. Full

254

7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.16 Phase position detection procedure

Fig. 7.17 Flowchart of vibration signal analysis and processing

cycle sampling enables to obtain ideal signal waveform. Full cycle vibration signals are sampled in accordance with the same reference signal to reduce spectrum leakage. Cross-correlation filtering enables accurate extraction of the vibration signals. The application of full cycle sampling and cross-correlation filtering improves the calculation accuracy of rotor dynamic balancing. 2. Composition of vibration signals Since the rotor system itself has some problems such as misalignment and basic vibrations, the vibration signals collected by the sensor in the actual dynamic balance test are even more complicated. In addition to fundamental frequency component, there are DC, different frequency harmonics, random noise of a certain bandwidth, etc. [9]. However, the fundamental frequency vibration is still the major vibration caused by imbalance. For the convenience of representation, the acquired vibration signal is generally expressed as y(t) = C + N (t) + A0 sin(ω0 t + ϕ0 ) +

n 

Ai sin(ωi t + ϕi )

(7.4)

i=1

where C is the DC component of the signal; N (t) is random noise; A0 sin(ω0 t +ϕ0 ) is the fundamental frequency signal; A0 is the amplitude of the fundamental frequency signal (mm); ϕ0 is the phase position of the fundamental frequency signal (rad); n  Ai sin(ωi t + ϕi ) is the superposition of harmonic signals; Ai is the amplitude of i=1

7.2 Software Design of Dynamic Balancing System

255

the harmonic signal (mm); ϕ i is the phase position of the harmonic signal (rad); ω0 is the angular velocity corresponding to the rotational frequency, (rad/s); ωi is the angular velocity corresponding to other frequency multiplications (rad/s). In order to achieve dynamic balance, the key is to accurately extract the amplitude A0 and phase ϕ0 of the fundamental frequency signal A0 sin(ω0 t +ϕ0 ). Through the analysis of the composition and spectrum of spindle vibration signals, the characteristics of the imbalance signals can be obtained: the time domain waveform is simple harmonic signal, there are multiple spectral lines in the spectrogram, and the frequency corresponding to the largest amplitude is the fundamental frequency. 3. Full cycle sampling of vibration signals In the dynamic balance test of rotor, the vibration signals need to be sampled in full cycles to ensure that the signal reference is the same for each acquisition, and the spectrum leakage and fence effect due to time domain truncation can be avoided [10, 11]. Traditionally, the triggering of signals is controlled by hardware circuit phaselocked frequency multiplier method to realize the full cycle sampling of signals [12, 13]. This method can automatically realize the real-time simultaneous sampling of multi channels of signals and show good synchronization performance, but it needs specialized hardware. The full cycle sampling is implemented based on the use of software algorithms. First, reference signals and vibration signals are simultaneously collected, and then the collected full cycle vibration signals are intercepted, thereby achieving full cycle sampling. The method can realize full cycle sampling by using the existing data acquisition card, and has good versatility. First, proper sampling parameters are set. The number of sampling points N should satisfy the equation: N = 2n . Under the limitation of hardware conditions, the sampling frequency f s should not only satisfy sampling theorem, but also be set as high as possible so that the waveform of the reference signal can be extracted more accurately. The sampling time is longer than the period of the signal to ensure the integrity of the reference signal. After the parameters are set, the reference signals and vibration signals are synchronously acquired, and then the collected vibration signals are processed. The two adjacent rising edges of reference signals are used as reference, and then the vibration signals are intercepted in full cycles. The intercepted vibration signals are full cycle signals and based on the same phase position of the rotor. Figure 7.18 shows the schematic of the full sampling of vibration signals. The first rising edge of the reference signal array is taken as the starting position for vibration signal acquisition. Then, the number of cycles for vibration signal acquisition is determined according to the sampling rate and the number of sampling points. Afterwards, the vibration signals are acquired in full cycles. The number of signal cycles is output and used for subsequent calculations. Figure 7.19 shows the block diagram of full cycle signal acquisition algorithm.

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7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.18 Schematic of full cycle sampling of vibration signals

Fig. 7.19 Block diagram of full cycle signal acquisition algorithm

4. Fundamental frequency signals extracted by correlation filtering method Currently, the methods used to extract base frequency vibration signals from mixed signals include analog and digital bandpass tracking filters [14, 15], integral digital filter [16], zero phase shift digital filter [17] and so on. The analog and digital bandpass tracking filters and the integral digital filter can generate phase position errors in output signals during the filtering process, thereby affecting the accuracy of subsequent dynamic balance calculation. Although zero phase shift digital filter realizes zero phase shift, it can lead to a certain degree of error in signal amplitudes. Correlation filtering method is used in this book to extract the imbalance signals [18]. It can effectively suppress interferences such as DC component and noise in the signals, and accurately extract imbalance signals. Correlation filtering can extract signals of a specific frequency from complex signals containing useful signals, DC bias, harmonic frequency component, random noise and so on. The working principles of correlation filtering are shown in Fig. 7.20. After full cycle vibration signals are acquired, the correlation filtering method is used to extract fundamental frequency signals [19, 20]. According to the fundamental frequency of reference signal, the signal generator function of LabVIEW is used

7.2 Software Design of Dynamic Balancing System

257

Fig. 7.20 Working principles of correlation filtering method

Fig. 7.21 Correlation filtering algorithm

to generate sine signals and cosine signals with fundamental frequency and 90° difference in phase position. Then, the correlation processing of vibration signals is performed, and finally the fundamental frequency signals are extracted. The block diagram is shown in Fig. 7.21. 5. Method for extracting amplitude and phase position of fundamental frequency signal After imbalance vibration signals are extracted by full cycle sampling and correlation filtering, the amplitude and phase position of the imbalance signals should be accurately calculated. There are usually three methods used to obtain the amplitude and phase position information of signals: fast Fourier transform (FFT), cross correlation, and discrete Fourier transform (DFT) [21]. FFT method has poor anti-interference ability, and due to the influence of time domain truncation, it suffers from problems such as energy leakage, amplitude decrease, precision decrease, and limited improvement of amplitude accuracy after windowing. The phase position extracted by cross correlation method is affected by the AD conversion accuracy, and the base frequency variation

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7 Automatic Suppression of Motorized Spindle Vibration

due to rotational speed fluctuation can cause decrease in the extraction accuracy of phase position. DFT method [22–25] can extract full cycle rotor vibration signals according to the rotational speed signals and the rotational period, and then perform DFT calculation on the full cycle signals. The amplitude and phase position of base frequency component are obtained, and finally the values are compared over multiple cycles to obtain a more accurate amplitude and phase position of base frequency vibration. Full cycle extraction of continuous signals can effectively solve the problem of spectrum leakage and fence effect in DFT processing. It is unnecessary to calculate the N point DFT, and one only need to calculate the k line of interest, which greatly reduce the calculation time. For ideal white noise, after the calculation of N point DFT, high frequency interference is basically suppressed. The basic procedures of the discrete Fourier transform are as follows. The finite sequence of single-cycle discrete vibration signals is {x n } (n = 1, 2, 3, …, N), and after discrete Fourier transform it becomes Xk =

N −1 

xn W nk , (k = 0, 1, 2, · · ·, N − 1)

(7.5)

n=0

If the rotation factor W n is Wn = e

− j∗2πn N

,

(7.6)

then the spectrum X 1 to be obtained is Xk =

N −1 

xn W nk =

n=0

N −1 

xn cos(

n=0

N −1  2π n 2π n )− j ) xn sin( N N n=0

(7.7)

Since W n is symmetrical, W n = −W N/2+n and then X 1 can be transformed into N /2−1

X1 =



(xn − xn+N /2 )W n

(7.8)

n=0

Let X 1 = R+Bi. Then the corresponding real and imaginary parts are as follows N /2−1

R=



(xn − xn+N /2 ) cos(

2π n ) N

(7.9)

(xn − xn+N /2 ) sin(

2π n ) N

(7.10)

n=0 N /2−1

B=

 n=0

7.2 Software Design of Dynamic Balancing System

259

Fig. 7.22 Block diagram of the algorithm for extracting DFT amplitude and phase position

Then the signal amplitude A is A=

2 2 R + B2 N

(7.11)

and the corresponding phase position is ϕ = arctan

B R

(7.12)

After fundamental frequency signals are extracted, the amplitude and phase position of the signals are extracted, and then the amplitudes and phase positions of most recent signals are averaged to increase the stability and accuracy of the result. The mean values are used as the final output result. Experiments show that the five most recent signals can be used to find the average. Figure 7.22 shows the block diagram of the algorithm for extracting the DFT amplitude and phase position.

7.2.5 Calculation of Imbalance Amount and Influence Coefficient The amplitude and phase position of vibration displacement signals extracted by above method are susceptible to interference and the accuracy can decrease. In order to improve the accuracy and stability of dynamic balance, signals over multiple cycles are averaged to obtain the amplitude and phase position information. Then, the dynamic balancing calculation is performed by using the influence coefficient method of single-sided balance. The specific calculation method is shown in Fig. 7.23 [26].

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7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.23 Flowchart of influence coefficient calculation

(1) First, the motorized spindle is started under no load to the rotational speed to be balanced, and the amplitude and phase position A0 of the original vibration at the correction plane position are measured. (2) Then the motorized spindle is stopped and the test weight P is added to the rotor correction plane. (3) The motorized spindle is restarted to the same rotational speed and the amplitude and phase position A1 of vibration under load are measured. (4) The influence coefficient is calculated as K = (A1 − A0 )/P. (5) Then the original imbalance amount is calculated as U = −A0 /K. (6) Since the compensation amount and the imbalance amount have the same magnitude but opposite direction, we have Q = −U. The block diagram for calculating the imbalance amount and the influence coefficient is shown in Fig. 7.24. Given the original vibration, the vibration in loading test and the vibration after loading test, the corresponding influence coefficients and compensation amounts can be calculated.

Fig. 7.24 Block diagram for calculating the imbalance amount and influence coefficient

7.2 Software Design of Dynamic Balancing System

261

Fig. 7.25 The front panel of the program for calculating imbalance amount and influence coefficient

The front panel of the program is shown in Fig. 7.25. It includes the input of initial vibration information, the input of loading test information, the input of vibration information after loading test, and the display of the result.

7.2.6 Movement of Correction Mass According to the calculation result of the imbalance amount, the compensation mass is split into two vectors, and then the counterweight plates are moved from their original positions to the designated positions according to the vector result. The motorized spindle rotates in clockwise direction, and the direction of positive angle difference is set to be counterclockwise. The specific procedures are shown in Fig. 7.26. After the program is started, the compensation amount is first divided into two vectors. Then, the movement paths of counterweight plates are calculated. Subsequently, excitation coil is driven to move the counterweight plates to the target positions. Figure 7.27 is the block diagram for calculating the required moving angles of counterweight plates. After compensation amount information is input, it is split into two vectors according to the cosine theorem, and the two vectors are then assigned to the two counterweight plates, and the angle at which each counterweight plate needs to move is calculated. To make the program simple, commonly used block diagram is edited as a sub VI (including calculating the difference between the two angles and panning the angle value to a range of 0–360°) for multiple calls. It is known that the counterweight plate cannot arbitrarily move in the circumferential direction, and the moving step is set at 10°. This means that the counterweight plate can reach a total of 36 positions (360°/10° = 36, numbered from 0 to 35) in the circumferential direction. This also determines the balance accuracy of the balancing

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7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.26 Flowchart of movement of correction mass

head. Then, according to the angle at which the counterweight plate needs to move, the number of moving steps can be determined. Based on the working principles of the slip ring balancing head, the magnetic pole directions of the circumferentially distributed magnets on counterweight plate are staggered, thus the direction and position of counterweight plate during movement should be considered. If the counterweight plate is in an even-number position, forward excitation current is given when forward movement is required, and reverse excitation current is given when reverse movement is required. If the counterweight

7.2 Software Design of Dynamic Balancing System

263

Fig. 7.27 Program for allocating compensation mass

is in an odd-number position, reverse excitation current is given when forward movement is required, and forward excitation current is given when reverse movement is required. Figure 7.28 shows the flowchart of controlling the excitation current to move the counterweight plate. Figure 7.29 shows the block diagram for moving the counterweight plate.

7.3 Characteristic Analysis of Dynamic Balancing System 7.3.1 The Influence of Rotational Speed on the Balancing Effect of Dynamic Balancing System When running at low speed, the motorized spindle can be regarded as a rigid system. After dynamic balancing, the balancing effect is relatively stable. When running at high speed, however, the motorized spindle becomes a flexible system. In this case, after dynamic balancing at a certain speed, the balancing effect cannot necessarily be maintained at other speeds. According to modal analysis and vibration measurement

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7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.28 Flowchart for moving counterweight plate

results, as well as general industrial standards, the machine tool spindle itself does not resonate during normal operation. However, the vibration of the entire experimental platform is greatly affected by the rotational speed of spindle, and the dynamic balancing effect may be affected by changes in rotational speed. The basic idea of the experiment is: after the spindle is dynamically balanced at a certain speed, the spindle speed is changed to investigate the stability of the balancing effect. The specific procedures are as follows. First, the vibration amounts of spindle at 3300, 3600, 3900 and 4200 r/min without dynamic balancing are measured. Second, the spindle is dynamically balanced at 3600 r/min and the vibration amount is measured. The speed is then changed to 3300, 3900 and 4200 r/min, and vibration amounts at these speeds are measured. Third, the spindle is dynamically balanced at 3900 r/min and the vibration amount is measured. The speed is then changed to 3300, 3600 and 4200 r/min, and the vibration amounts at these speeds are measured.

7.3 Characteristic Analysis of Dynamic Balancing System

265

Fig. 7.29 Program for moving counterweight plate

Finally, the spindle is dynamically balanced at 4200 r/min and the vibration amount is measured. The speed is then changed to 3300, 3600 and 3900 r/min, and the vibration amounts at these speeds are measured. The results are listed in Table 7.1. The experimental results are plotted with rotational speed as the horizontal axis and the vibration amplitude as the vertical axis (Fig. 7.30). Then the stability of dynamic balancing effects at different rotational speeds is analyzed. As can be seen, when the motorized spindle is imbalanced, the vibration is large at each of the rotational speeds investigated. After dynamic balancing at 3600 r/min, the vibration amplitude remains basically unchanged as speed decreases to 3300 r/min, and the vibration amplitude increases as the speed increases to 3900 and 4200 r/min. After dynamic balancing at 3900 r/min, the vibration amplitude increases as the Table 7.1 Dynamic balancing effect at different rotational speeds Rotational speed at which vibration is measured (r/min)

Vibration amplitude (μm)

Without dynamic balancing

3600

3900

4200

3300

4.9

0.5

3.4

3.6

3600

5.7

0.5

3.2

3.2

3900

4.7

1.8

0.8

1.5

4200

5.7

0.9

1.3

0.6

Rotational speed at which dynamic balancing is performed (r/min)

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Fig. 7.30 Comparison of the dynamic balancing effects of spindle at different speeds

speed increases or decreases. After dynamic balancing at 4200 r/min, the vibration amplitude increases as the speed decreases. It can be found that the trends of spindle vibration with rotational speed are similar after dynamic balancing at 3300 and 3600 r/min, and the trends of spindle vibration with rotational speed are also similar after dynamic balancing at 3900 and 4200 r/min. The results show that changing rotational speed has an effect on the dynamic balance of the experimental platform. Further analysis and research are needed to unravel the underlying reasons. It may be because the system becomes a flexible system as its speed exceeds the critical speed. It is necessary to perform dynamics and modal analysis of the whole system. The dynamic balancing method for rigid spindle should not be used in this condition and the dynamic balancing method for flexible spindle should be adopted instead [27].

7.3.2 The Influence of the Angle of Test Weight on the Balancing Effect of Dynamic Balancing System and Influence Coefficient In addition to rotational speed, the theoretical method used for dynamic balancing also has a direct impact on the dynamic balancing effect. According to influence coefficient method, in the balancing process, the position where the test weight is loaded and the magnitude of the test weight also have important influences on the dynamic balancing effect. If the test weight is too small, then it has no significant influence on spindle vibration. If the test weight is too large, however, it may significantly increase spindle vibration, causing damage to the equipment. The position of test weight has similar influence on spindle vibration. At present, some empirical

7.3 Characteristic Analysis of Dynamic Balancing System

267

formulas are used to calculate the test weight and there is no universally accepted value of the test weight. The system intends to use the influence coefficient [28] when the vibration amplitude and phase position change are large. In the experiment, the same test weights (8.1 g) were loaded at different angles of the spindle end face, and the vibrations before and after loading were measured. Several sets of amplitude differences and phase position differences corresponding to the test angles were obtained, and the influence coefficient of weight loading at different angles was obtained. The weight loading angles, vibration amplitude difference and phase position difference before and after weight loading, and corresponding influence coefficient are shown in Table 7.2. In order to visually reflect the relationship among the vibration amplitude difference and phase position difference before and after weight loading, weight loading angle, and influence coefficient, the results were plotted with weight loading angle as abscissa, vibration amplitude difference and phase position difference as the left and right ordinates, respectively (Fig. 7.31). The size of the symbol in the figure represents the magnitude of the influence coefficient, and the unit of value is 10−3 μm/(g mm). There is a “sinusoidal” relationship between the vibration amplitude difference and the weight loading angle. The relationship between the vibration phase position difference and weight loading angle is slightly linear, but fluctuates. The influence coefficient has a weak relationship with the weight loading angle, but is related to the vibration amplitude difference and phase position difference. The larger the vibration amplitude and phase position change, the greater the influence coefficient. However, at 15°, the vibration amplitude difference is small but the influence coefficient is large, which is probably due to the small vibration amplitude difference. Table 7.2 Results obatined at different weight loading angles Weight loading angle (°)

Amplitude difference (μm)

Phase position difference (°)

Amplitude of influence coefficient (μm/(g mm))

15

0.12

64.87

0.00757

45

1.36

−38.61

0.00382

75

1.76

−30.45

0.00374

105

1.95

−25.24

0.00374

135

1.74

−18.65

0.00313

165

1.29

−9.43

0.00234

195

0.68

0

0.00196

225

−0.34

−3.14

0.0012

255

−0.98

0.62

0.00213

315

−1.43

−38.16

0.00313

345

−0.78

−17.85

0.00142

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7 Automatic Suppression of Motorized Spindle Vibration

Fig. 7.31 Relationship among the vibration amplitude difference before and after weight loading, vibration phase position difference before and after weight loading, weight loading angle, and influence coefficient

According to Fig. 7.31, two points with large vibration amplitude difference and phase position difference (75° and 315°) were selected, and two points with small vibration amplitude difference or phase position difference (15° and 195°) were also selected. Then, the dynamic balancing experiment was performed. The results are listed in Table 7.3. The spindle vibrations before and after dynamic balancing were compared. Figure 7.32 shows the comparison of the dynamic balancing effects after weight loading at different angles. The experimental results show that weight loading at four different angles all reduced spindle vibration amplitude, but the reduction effects were different. When the test weight was loaded at 15°, 75° and 195°, the dynamic balancing effects were not good. When the test weight was loaded at 315°, the dynamic balancing effect was good. At a weight loading angle of 75°, the vibration cannot be reduced to a desired level because the vibration amount is too large, but the reduction effect is still better than that obtained at a weight loading anlgle of 15° and 195°. In sum, Table 7.3 Dynamic balancing effects under different weight loading angles Angle (°)

Vibration amplitude before balancing (μm)

Vibration amplitude after balancing (μm)

Percent decrease (%)

15

3.92

2.74

30.1

75

5.84

2.83

51.5

195

4.93

2.78

43.6

315

2.86

1.32

53.8

7.3 Characteristic Analysis of Dynamic Balancing System

269

Fig. 7.32 Comparison of the dynamic balancing effects after weight loading at different angles

the experimental results are consistent with expectations, that is, when the change of vibration amplitude and phase position is relatively large, the influence coefficient is relatively reliable and its magnitude is large.

References 1. Zhang Y, Mei XS (2013) High-speed spindle dynamic balance and its online control technology. Chin Eng Sci 15(01):87–92 2. Wu YH, Zhang K, Deng HB et al. Mechanical online dynamic balancing system with built-in Spindle: Chinese patent, CN105021352A.2015.11.04 3. Stephen H, Lane M. Rotating machine active balancer and method of dynamically balancing a rotating machine shaft with torsional vibrations. US 20060005623A1.2006.06.12 4. Wu YH, Zhang K, Deng HB et al. A test system and method of electric spindle load. China. zl201510398758.X.2017.10.20.156 5. Wu YH, Zhang K, Deng HB et al. Adjustment method of mechanical online dynamic balance system with built-in spindle: Chinese patent CN104990670A.2015.10.21.140 6. Wang ZM (2009) Noise and vibration measurement. Science Press, Beijing, China, pp 134–135 7. Bai ZG, Tang GJ (2012) A method for extracting the amplitude and phase characteristics of rotor dynamic unbalanced signals. Power Sci Eng 10(3):69–90 8. Yang JG, Xie DJ, Gao W (2003) Research on dynamic balance method based on multi-sensor data fusion. Power Eng 23(2):2275–2278 9. Lv Y, Zhu J, Tang B (2009) Parameter estimation of nonlinear FM signal based on DPI. J Electr Meas Instrum 23(6):63–67 10. Wen H, Teng ZS, Wang Y et al (2011) Research on measurement algorithm of spectrum leakage suppression and improved dielectric loss angle. J Instrum 32(9):2087–2094 11. Zhu LK, Mao LS, Tang M (2000) Using the single-chip microcomputer to achieve the whole cycle sampling method. Instrum Technol 2:34–35 12. Long HJ, Sun CF, Mo GL (2016) Research and implementation of general algorithm for helicopter vibration detection. Vib Test Diagn 03:524–528 13. Zhang BC, Yang XH, Wu D et al (2005) Detection of the imbalance of the brake drum of the car. Comput Measur Control 1:27–29 14. Tao LM, Li Y, Wen XS (2007) Signal tracking filtering method based on switched capacitor technology and its application in rotor dynamic balance. China Mech Eng 8(2):427–430

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15. Wu SB, Lu QH (2003) Integral digital tracking filtering method and its application in rotor balancing. Eng Mech (2):76–79 16. Ji YB, Qin SR, Tang BP (2000) Zero phase digital filter. J Chongqing Univ 6(11):4–7 17. Guo JH, Wu X, Liu XQ et al (2011) Research on the method of extracting the amplitude phase of vibration signal in rotor dynamic balance. Mach Electron 10:6–10 18. Luo DY (2000) Engineering signal analysis. Kunming: J Kunming Univ Sci Technol 19. Wu YH, Tian F, Shao M et al (2013) Research on analog parameter identification module based on LabVIEW. Control Eng 20(01):69–71+75 20. Wu YH, Tian F, Shao M et al (2011) Research on vibration signal preprocessing module of all ceramic electric spindle based on LabVIEW. J Shenyang Jianzhu Univ 27(06):1177–1182 21. Cai Y, Hao CP, Hou CH (2013) The design of DFT modulation filter bank with linear phase non-uniform bandwidth. J Instrum 10:2293–2299 22. Phoong SM, Chang YB, Chen CY (2005) DFT-modulated filterbank transceivers for multipath fading channels. IEEE Trans Signal Process 53(1):182–192 23. Yiu KFC, Grbic N, Nordholm S et al (2004) Multicriteria design of oversampled uniform DFT filter banks. IEEE Signal Process Lett 11(6):541–544 24. Bracewell RN (2005) Fourier transform and its application (Third Edition). Translated by Yin QY, Zhang JG. Xi’an Jiaotong University Press, Xi’an, China, pp 203–230 25. Wang ZH, Sun CY, Liu DR et al (2013) Vibration mode analysis of basic flexible rotor system. J Shenyang Jianzhu Univ 29(2):367–371 26. Pan X, Zhu QX, Wu HQ, Gao JJ (2018) The application of precise targeting method in the dynamic balance of machine tool spindle. Mach Tool Hydraul 46(7):1–4 27. Zhang K, Zhang CY, Zhang LX et al (2018) Characteristic analysis and experiment of online dynamic balance system with electromagnetic slip ring type. Vib Test Diagn 38(1):34–38+203 28. Wang Z, Zhu FL, Tu W (2018) Research on the extraction of vibration signal of electric spindle by cross-correlation method. Comb Mach Tools Autom Mach Technol (2):87–89+93

Chapter 8

Motorized Spindle Fault Diagnosis Technology Based on Deep Learning

Motorized spindle is a key functional component of machine tool. Its reliability and accuracy directly affect the operational reliability and machining accuracy of the entire CNC machine tool, and are even directly related to the production efficiency of the entire production line. The fault analysis of motorized spindle belongs to the category of reliability physics, and is the study of the fault phenomenon, mechanism, mode and hazard of motorized spindle. It is a necessary measure to realize the reliability growth of motorized spindle, and is a basic work ensuring the reliability of motorized spindle. The fault diagnosis of motorized spindle can provide a basis for subsequent reliability model establishment, reliability analysis and reliability design.

8.1 Fault Analysis of Motorized Spindle Faults can be caused by mistakes in design, manufacture, use and maintenance processes. Fault diagnosis is the assessment of the causes of the faults. Each fault mode may correspond to multiple causes, including the product’s own problems and other external causes. Internal causes include changes in physical, electrical, chemical, and mechanical stresses, and external causes include failures of other products, environmental factors, and human factors. The fault of motorized spindle can be roughly divided into two categories: mechanical fault and electrical fault. The main faults of motorized spindle include the following six aspects: excessive temperature rise; poor rigidity; large vibration; poor precision; difficulty in starting; and speed drop. (1) Excessive temperature rise means that the temperature rise of the hottest spot on the housing of motorized spindle using oil mist lubrication exceeds 25 °C, or the temperature rise of the hottest spot on the housing of motorized spindle using grease lubrication exceeds 30 °C. There are many reasons for the excessive temperature rise of motorized spindle, and these reasons mainly include

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assembly factors, lubrication factors, use factors, electrical factors and manufacturing factors. The assembly factors include the following: 1. Poor bearing quality and high temperature rise of bearing during high speed rotation. 2. Bearing installation error. 3. The assembly environment is unclean and debris enter the working area of bearing in assembly. 4. Too small clearance between spindle bearing and the inner hole of bearing seat. 5. Too large interference between bearing guard and the inner hole of bearing. 6. The inner and outer adjustment pads of bearing are not parallel and not vertical. 7. The outer diameter of the outer adjustment pad of rear bearing of spindle without guide bush is too large so that there is no gap between the pad and the seat hole, and the rear bearing cannot move freely in the rear seat hole. 8. The interbearing adjustment pad is not well ground, resulting in that only one set of bearings bear axial load. 9. The load applied to motorized spindle is too high. The lubrication factors include the following: 1. Wrong grease grand is selected and the grease lubrication performance is not good. 2. Insufficient grease is added and lubrication is thus insufficient. 3. Too much grease is added and the excess grease cannot be throwed off in a short time of operation. 4. The quality of grease in spindle is not checked when it is re-started after left aside for a long time. 5. After bearing is cleaned, grease is directly loaded into spindle before the bearing is dried. As a result, when the spindle is running at high speed, most of the grease is thrown out of the working area. 6. The oil mist lubrication system works abnormally and cannot guarantee stable and continuous oil supply. 7. Wrong oil is selected, the viscosity does not meet requirements, or the oil quality is inferior. 8. The oil mist lubrication system supplies too much oil to bearings, causing the steel balls in bearings to stir the oil and generate heat. 9. The exhaust passage of the motorized spindle is blocked. The manufacture factors include the following: 1. The clearance between rotary part and corresponding fixed part is too small or their coaxiality is poor, resulting in a slight rubbing between them. 2. The interference between rear bearing seat, steel ball and the rear seat hole of the housing is too large, and the bearing seat can hardly move freely in the seat hole of housing. 3. The rolling guide bush steel ball is riveted tightly and it cannot rotate freely, and the load applied by the spindle does not work. The electrical factors include the following: 1. The offline parameters of stator windings are incorrect, or wiring error causes the wire package to be severely heated. 2. Stator coil is damp and the insulation to the ground is decreased. 3. The rotor is broken. The use factors include the following: 1. The motor is not sufficiently cooled, the flow rate of cooling system is too low, the water tank is too small, or the cooling water pipe is blocked up. 2. The motorized spindle and the power supply do not match, or the voltage of the three-phase IF power supply is too high. 3. The output waveform of three-phase static variable frequency power supply is poor or the higher harmonic content is too high. 4. Motorized spindle has been working under overloading or overspeed conditions for a long time. 5. Spindle and bearing are damaged due to crashing accident or sudden stop of rotation of spindle.

8.1 Fault Analysis of Motorized Spindle

273

(2) The motorized spindle has poor rigidity. The poor rigidity of motorized spindle is mainly manifested as low load capacity. The main causes of the poor rigidity of motorized spindle include: 1. The load applied during spindle installation is low. 2. The clearance between bearing and spindle housing front seat hole is too large. 3. The front and rear lock nuts of spindle become loose. 4. The grinding wheel rod is too slender, and the clearance between it and the fitting part of the shaft is too large, resulting in poor contact rigidity of the connecting rod. 5. The adjustment pad of spindle bearing is not approporiate. 6. There is a large gap between the rolling guide sleeve or sliding sleeve at spindle rear end, seat hole, and bearing seat. (3) The vibration of motorized spindle is severe. Vibration signals are the main signals for diagnosing the failure of motorized spindle. The major causes of large vibration of motorized spindle include: 1. The quality of spindle bearing is inferior and its vibration is large, or the precision of bearing is decreased rapidly and its durability is poor. 2. After the spindle has been used for a long time, the shaft is deformed and the original balance accuracy is decreased. 3. The rotating parts installed on the motorized spindle have poor concentricity and the fitting clearance is too large. 4. Improper assembly results in decreased spindle bearing accuracy. 5. The load applied to spindle is too low and the built-in spring is broken and deformed. 6. The spindle grinding wheel rod is too slender. 7. The inner and outer diameters of stator motor are not concentric, resulting in uneven air gap between stator and rotor. 8. The rotor cage become loose, suffers from poor contact or is broken. (4) The motorized spindle has poor precision. In the course of service, the motorized spindle will gradually lose its original precision. The main causes of poor precision of motorized spindle include: 1. The quality of spindle bearing is low. 2. The shaft is severely deformed and loses its original manufacturing precision. 3. The precisions of spindle parts are poor and the assembly precision is also poor. 4. Bearing experiences unrecoverable deformation during assembly. 5. The grinding wheel rod has poor quality and the assembly precision is poor. 6. The seat hole of front bearing is too large and the fitting clearance is too large. 7. The preload applied to spindle is too low or preload cannot be applied to spindle because certain part is stuck. 8. The front and rear nuts of spindle are loose, causing serious axial runout. (5) It is hard to start the motorized spindle. The difficulty in starting the motorized spindle can be caused by electrical or mechanical reasons. The electrical reasons include: 1. The power supply capacity is low. 2. The power frequency grid voltage is too low. 3. The start protection of static frequency converter is not adjusted to the optimum value. 4. The air gap of motor is too small. 5. The wiring of spindle stator is wrong. 6. The spindle motor rotor is severely broken. 7. The spindle motor stator is disconnected. The mechanical reasons include: 1. The preload applied to motorized spindle is too high. 2. The grease used by the grease lubrication system is too thick and too much grease is input into the system. 3. There is friction between the rotating and fixed parts of spindle.

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(6) The speed drop of motorized spindle means that increasing the resistance torque of spindle will slightly lower the rotational speed of the spindle during normal operation. Generally, the slip should be controlled within 2–5%. If the rotational speed of motorized spindle drops significantly under loading conditions and even the stall phenomenon occurs, it will be regarded as speed drop phenomenon. The main causes of the above phenomenon include: 1. The spindle rotor cage is broken or the contact is poor. 2. The intermediate frequency power supply voltage is too low.

8.2 Deep Learning and Motorized Spindle In recent years, deep learning [1] has been developing rapidly in academia and industry. It enables significantly improved recognition rates in many traditional recognition tasks [2], showing its ability to deal with complex recognition tasks. Deep learning is derived from the study of neural networks and can be regarded as a deep neural network. Through it, we can obtain deep feature representation and avoid the complicated and cumbersome manual selection procedures and the dimensional disaster of high-dimensional data. Several well recognized deep learning models include deep belief network based on restricted Boltzmann machine [3, 4], stacked autoencoders based on autoencoder [5], convolutional neural networks [6], and recurrent neural networks [7]. Researchers begin to study the use of deep learning in the field of control. Current research mainly focuses on control target recognition, state feature extraction, system parameter identification, and control strategy calculation. In particular, the combination of deep learning and reinforcement learning has led to exciting research results. As shown in Fig. 8.1, deep learning has been applied to all aspects of the control system [8]. In the field of motorized spindle control, accurate motor parameters are essential to improve the quality of the control system. Motor parameter identification is unavoidable in both direct torque control and vector control. According to the basic idea of vector control in Chap. 2, the stator resistance, stator inductance, rotor resistance, rotor inductance and mutual inductance of motorized spindle all need to be identified. The motorized spindle is a complex nonlinear dynamic system whose motor parameters also show dynamic nonlinear characteristics. Neural networks have the ability to fit complex nonlinear functions and can be used for system identification. However, shallow neural networks are susceptible to local optima problems during training, and sometimes cannot accurately describe dynamic systems. Therefore, an identification mode of neuron network + CBR is proposed in Chap. 5. With the application of deep learning in control system, the motor parameter identification is hopefully achieved by deep learning. Some studies have investigated the use of deep learning methods for system identification. Since the system model is replaced

8.2 Deep Learning and Motorized Spindle

275

Fig. 8.1 Application of deep learning to all aspects of the control field

by a deep neural network, the system identification task is turned into parameter optimization of deep neural network.

8.3 Application of Deep Learning to Fault Diagnosis of Bearing The faults of CNC machine tool spindle are mainly due to faults of stator, rotor, bearing and broach mechanism. For fault diagnosis of motorized spindle, it is necessary to collect and process fault signals and extract fault features to identify the fault mode. At present, the commonly used fault diagnosis methods based on vibration signals mainly transform signals in the time-frequency domain, and include Fourier transform, wavelet analysis, empirical mode decomposition (EMD), Hilbert-Huang transform, S transform, support vector machine (SVM), BP neural network, etc. DBN is essentially a multilayer perceptron neural network that is composed of multiple RBM networks and a supervised BP network. The lower layer represents original data details and the higher layers represent data attribute categories or features. The layers are abstracted from lower to upper layers. It can gradually mine the deep features of data. The training process of DBN includes pre-training and finetuning. In pre-training, DBN uses a layer-by-layer training approach to treat each layer of RBM. The output of the lower RBM hidden layer is used as the input to the upper RBM visible layer. In fine-tuning, the BP network of the last layer is trained by supervised learning and the error obtained by comparing the actual output and the standard labeling information is propagated layer by layer to achieve fine tuning of the entire DBN weight and offset.

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Combining the characteristics of rolling bearing vibration signals and the advantages of deep learning, future fault diagnosis methods for bearing should combine unsupervised learning and supervised learning in DBN. At the same time, future fault diagnosis methods should be able to mine deep features under variable loading conditions and recognize the multi states of rolling bearing. The specific procedures are as follows: (1) The vibration signals of rolling bearing under certain load conditions are obtained and used training data set to perform EEMD decomposition to obtain several IMFs. Experiments are performed with time-saving and high accuracy principles and the first t IMFs with high fault sensitivity are selected. (2) The IMFs of vibration signals for the same state are made to undergo Hilbert transform and obtain an envelope spectrum. High-dimensional sample features with t envelope spectra are constructed and taken as the input of DBN. (3) The number of hidden layers N and the learning rate of DBN are set. The number of nodes of each hidden layer (m1, m2, …, mN) is determined by genetic algorithm (GA). Then each restricted Boltzmann machine is trained layer by layer by unsupervised learning until all RBM training is completed. (4) The error back propagation principle of back-propagation (BP) network is used to fine-tune the weight and offset, and the multi-state recognition model of rolling bearing under variable loading conditions is constructed. (5) The vibration signals of rolling bearing under loading conditions different from the conditions under which training data are obtained are used as test data. The data are processed according to the same EEMD-Hilbert envelope spectrum method. The fault state recognition model obtained in step (4) is used to realize the multi-state recognition of rolling bearing under different loading conditions.

8.4 Application of Deep Learning to Fault Diagnosis of Motorized Spindle The high-speed motorized spindle is a complex system coupling mechanical, electrical, thermal and magnetic fileds. Motors and mechanical devices are subjected to electromagnetic radiation and interference, and also affected by environmental changes such as temperature, humidity and mechanical stress changes. There often occur faults caused by pitting erosion of inner and outer rings, uneven abrasion, failure of sealing system, contamination of lubricating oil by impurities and so on. These faults are recurrent, but last for only a short time and can disappear without treatment. These faults are special faults different from permanent faults and called intermittent faults. Due to the particularity and complexity of intermittent faults, traditional diagnostic methods for permanent faults and transient faults cannot be directly applied to the monitoring or fault diagnosis of high-speed motorized spindle with intermittent faults.

8.4 Application of Deep Learning to Fault Diagnosis of Motorized …

277

The timing dynamic characteristics between the coupling fields of high-speed motorized spindle and between intermittent faults are common. For example, the rotor system of high-speed motorized spindle with deep transverse cracks on the shaft can generate static rotor collision due to excessive vibration. The loose spindle will also experience rubbing fault due to excessive vibration. In addition, when the temperature rise of spindle is too high, the spindle will undergo thermal deformation, which may cause the spindle system to vibrate, and finally result in collision and friction, misalignment, imbalance, asymmetric support, cracks, looseness, etc. At the same time, system vibration affects the optimal preload of spindle, which causes temperature rise, vibration and preload system to couple with each other, further affecting the machining accuracy of product. This makes the timing dynamic characteristics of faults of spindle system universal. Deep learning [9] has high ability to extract and characterize fault features, and to identify parameters and states. It automatically extracts features from a large amount of data, identifies the state of spindle (normal state, permanent fault state or intermittent fault state), and finally achieve parameter identification of mechanism model, model update and dynamic compensation. Thereby, establishment of hybrid model of high-speed motorized spindle and intelligent diagnosis intermittent faults are realized. The hybrid model cannot only link the information before and after the occurence of fault, but also deal with non-stationary and poorly reproducible signals. It can also reveal the complex mapping relationship between signal characteristics and the three states of spindle, identify the state of spindle and the type of intermittent fault, and find the signs of early development of failure. The identification of spindle state can be summarized as follows: First, the vibration signal spectrum and temperature rise signals of motorized spindle are obtained and used as training samples. Deep learning is used to perform unsupervised learning on the training samples, and the deep neural network (DNN) is pre-trained layer by layer so that it can efficiently mine the intermittent fault characteristics of training samples. Then, DNN is fine-tuned by supervised learning, and the output layer with classification function is added. According to the state type of the sample and the constructed corresponding fault mechanism model, the DNN parameters are fine-tuned, and the adaptive extraction of intermittent fault features based on deep learning and identification of state type can be achieved. Convolutional neural network (CNN) is a multi-layer neural network whose basic structure consists of input layer, convolutional layer (conv), subsampling layer (pooling), fully connected layer, and output layer (classifier). CNN has two main features: sparse connection and weight sharing. In each iterative process, a part of hidden layer nodes are randomly invalidated according to a certain proportion so that they do not participate in the training process of forward propagation. The weights of the failed hidden layer neurons are preserved during each training session, but will not be updated in the current iteration. The above operations are repeated in each iteration of training and in the final diagnosis all hidden layer neurons are required to participate in the calculation. This is similar to the combination of networks with different structures, called dropout process. Dropout is introduced into the training

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Fig. 8.2 Fault diagnosis model established using CNND

process of CNN program to prevent the model from over-fitting and improve its accuracy, i.e., forming a CNND model. Figure 8.2 shows the fault diagnosis model established using CNND for motorized spindle. CNND method is used to obtain the scatter plot of the main components of features. It can be seen that the same faults of motorized spindle can be well gathered so that different faults can be effectively separated. Only a small number of faults including misaligned bearings, loose bearing outer ring and bearing imbalance are gathered and a high diagnostic accuracy is obtained.

8.5 Application of Deep Learning to Motorized Spindle State Evaluation The dynamic performance of motorized spindle is affected by many factors. Considering the operating conditions of motorized spindle and the limitations of the machining process, the relevant indicators of the operating state of motorized spindle are analyzed from the common faults of the spindle and index system for the evaluation of motorized spindle system is constructed. Different from fault diagnosis, spindle state evaluation not only distinguish the normal state from the fault state of motorized spindle, but also classifies the state of motorized spindle under normal operation into several levels to make sure that it works in an optimal state. Therefore, it is required to comprehensively consider various relevant indices and achieve

8.5 Application of Deep Learning to Motorized Spindle State …

279

Fig. 8.3 Evaluation system of the operating state of motorized spindle

fine-tuning of motorized spindle system before failure to achieve long-term efficient work. Figure 8.3 shows the evaluation system of the operating state of motorized spindle [10]. Several main monitoring indicators of the operating state of motorized spindle include vibration, temperature rise and efficiency. Other indicators are all related to these three indicators. For example, the temperature of the cooling water and the flow rate of cooling water are related to the temperature rise of motorized spindle. Besides, the lubricating oil and compressed air can take away a small but not negligible part of the heat generated inside the spindle, and are thereby related to the temperature rise of motorized spindle. Therefore, the temperature rise of motorized spindle can be used to comprehensively characterize the complex changes inside the spindle and then deep level indicators are used for evaluating the later state of spindle. On this basis, the index system for the evaluation of motorized spindle is constructed as shown in Fig. 8.4. Figure 8.5 shows the basic process of evaluation and analysis of the operating state of motorized spindle system. The prerequisite of the evaluation of the operating state of motorized spindle system is that there must be a state level data model. Therefore, it is necessary to first identify and classify the state data from the offline data to determine the classification rules. Then, the data characteristic information included in the state level data is used to determine the state evaluation rules. After the state evaluation rules are determined, the state of spindle is evaluated according to the sample data collected online. If the operating state of the device is not optimal, it is necessary to find out the reason. The result is used as a reference for the maintenance and repair of the motorized spindle system.

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Fig. 8.4 Index system for the evaluation of motorized spindle

Fig. 8.5 The basic process of evaluation and analysis of the operating state of motorized spindle system

References 1. LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521:436–444 2. Krizhevsky A, Sutskever I, Hinton GE (2012) ImageNet classification with deep convolutional neural networks. In: Advances in Neural Information Processing Systems 25; 26th Annual Conference on Neural Information Processing Systems 2012, vol 2, pp 1097–1105 3. Hinton GE, Salakhutdinov RR (2006) Reducing the dimensionality of data with neural networks. Science 313(5786):504–507 4. Hinton GE, Osindero S, Teh YW (2006) A fast learning algorithm for deep belief nets. Neural Comput 18(7):1527–1554 5. Bengio Y, Lamblin P, Dan P et al (2007) Greedy layer-wise training of deep networks. In: Proceedings of the 2006 Conference on Advances in Neural Information Processing Systems 19, pp 153–160 6. Lecun Y, Bottou L, Bengio Y et al (1998) Gradient-based learning applied to document recognition. Proc IEEE 86(11):2278–2324 7. Sutskever I (2013) Training recurrent neural networks. University of Toronto, Canada

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8. Duan YJ, Lv YS, Zhang J et al (2016) Deep learning for control: the state of the art and prospects. Acta Automatica Sinica 42(5):643–654 9. Wang YJ,Na XD,Kang SQ et al (2017) State recognition method of a rolling bearing based on EEMD-Hilbert envelope spectrum and DBN under variable load. In: Zhongguo Dianji Gongcheng Xuebao/Proceedings of the Chinese society of electrical engineering, vol 37, no 23, pp 6943–6950 10. Fan LT, Chen JK, Zhang K et al (2017) Operating state evaluation model of motorized spindle based on composite index. Chongqing, China: Institute of Electrical and Electronics Engineers Inc.

Chapter 9

Development of Intelligent Ceramic Motorized Spindle

Ceramic motorized spindle refers to motorized spindle some components of which are made of ceramic materials. There are two types of ceramic motorized spindle, including ceramic bearing motorized spindle and full ceramic motorized spindle. New engineering materials such as ceramics have been used for machine tool spindles for many years. In 1989, Namba et al. in Japan invented a glass-ceramic spindle for ultra-precision surface grinding. The glass-ceramic spindle has zero thermal expansion and enables ultra-precision grinding of sub-micron and nano-scale surfaces of optical and electronic materials. The researchers in MIT used epoxy resin-graphite composite to make machine tool spindle and tested its performance. Compared with steel spindle, the spindle made of this composite material had increased cutting width (by 23%) and increased damping efficiency (by 20%). Even when the temperature difference between the spindle and the machine bed was large, the pre-tension remained almost constant because the longitudinal thermal expansion coefficient of the composite material is close to zero. In 1998, at the North American Metalworking and Manufacturing Exhibition held in Los Angeles, C series machining centers (450 MC, 600 MC and 800 MC) with ceramic spindles were presented. Due to the use of ceramic spindle, the C series machining centers not only have decreased weights, but also thermal expansion rates about 5% lower than those of general machining centers. Due to the combination with high-power output motor (maximum output power of 15 kW), the speed rise time of the spindle during operation is very short (6000 r/min, 0.3 s) and the spindle also shows high rigidity and high machining accuracy. In 2002, Kyung Geun Bang and Dai Gil Lee designed a composite air spindle with carbon fiber material. Compared with traditional steel spindle, it has the characteristics of small rotational inertia and high natural frequency. Due to these characteristics, the bending deformation and thermal deformation of the spindle system due to centrifugal force and temperature rise under the limit rotational speed are greatly reduced, and the speed and accuracy of the spindle system are improved. In 2003, Weck and Brecher et al. proposed the design of a multi-point angular contact (3P, 4P) ceramic © Springer Nature Singapore Pte Ltd. 2020 Y. Wu and L. Zhang, Intelligent Motorized Spindle Technology, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-3328-0_9

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Fig. 9.1 ZrO2 ceramic rotor

Fig. 9.2 Ceramic motorized spindle prototype

spindle bearing where the additional contact points are supposed to limit the positions of the bearing balls to prevent displacement and thereby increase rigidity. From 2005 to 2006, in order to improve the wear resistance of bearing, schaeffler et al. sprayed a wear-resistant ceramic coating on bearing groove. Results show that the spindle speed of the machine tool was increased, and the Dm · n value even reached 3.5 × 106 mm/min or higher. In 2008, supported by the National 863 Program of China, Yuhou Wu et al. from Shenyang Jianzhu University applied ceramic materials to bearing components and rotor shaft. They successfully developed a ceramic motorized spindle without inner ring and initially tested its operational characteristics (Figs. 9.1 and 9.2). Full ceramic motorized spindle is a new type of motorized spindle whose mechanical shaft and bearings are made of engineering ceramic materials. Its structure is basically the same as that of conventional motorized spindle. Due to the non-conducting and non-magnetic properties of ceramic material, the spindle and bearings can not only avoid extra electromagnetic loss caused by shaft current such as that in metal motorized spindle, but also reduce the extra electromagnetic loss caused by the higher harmonics in the output power supply of frequency converter. Due to the low density and low thermal expansion coefficient of ceramic material, the weight of motorized

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spindle and the centrifugal force generated during high speed rotation are reduced, and the spindle deformation caused by temperature rise is also reduced. These help improve the operating precision of motorized spindle. Therefore, the performance of ceramic motorized spindle is quite different from that of metal motorized spindle. The study of ceramic materials is of great significance for optimizing the performance of motorized spindle.

9.1 Electromagnetic Properties of Ceramic Motorized Spindle In order to meet high-speed and controllable requirements, motorized spindle is generally powered by an inverter. Motorized spindle driven by inverter power supply has many advantages, such as wide speed range and high torque within the wide speed range. However, the input voltage, frequency and waveform of variable frequency speed regulation device will change during the speed regulation process, and the changes have different effects on the various losses and efficiency of motorized spindle. This is mainly due to the fact that the output voltage of inverter contains a high content of higher harmonics, which cause additional losses in motor. The electromagnetic properties of ceramic motorized spindle under the control of inverter deserve more attention.

9.1.1 The Operational Characteristics of Power Supply Inverter The output voltage harmonics of inverter is the main cause of the electromagnetic vibration of motorized spindle. In order to correctly evaluate the magnetic field characteristics of ceramic motorized spindle, it is necessary to apply a power analyzer to test the inverter output voltage and current harmonics. Figure 9.3 shows the photograph of power analyzer. Figure 9.4 shows the voltage harmonic spectra of inverter at different frequencies. Figure 9.5 shows the current harmonic spectra of inverter at different frequencies. The presence of harmonics in the current or voltage waveform of power supply will affect the loss and vibration of motorized spindle. Figures 9.4 and 9.5 show that the 3th, 5th and 7th order harmonic peaks in voltage and current generated by the inverter are generally greater than the 2nd and 4th order harmonic peaks. This means that odd harmonics have more significant influences on the vibration of motorized spindle than even harmonics. Harmonic voltages and currents increase with increasing frequency, and harmonic power also increases. Comparing Figs. 9.4 with 9.5, it is found that the harmonic response of inverter increases with the increasing fundamental frequency of power supply, and under the excitation of the high-frequency harmonic voltage and

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Fig. 9.3 Power analyzer

current, the increase of the frequency will lead to more significant skin effect. The harmonic current and voltage waveforms are distorted. The low-frequency peak value and distribution density of the spectrum at high frequencies are much higher than those at medium or lower frequencies. The harmonic power of the inverter generates a large amount of 3rd order harmonic currents flowing through the conductor, which makes the conductor overheat and increases the harmonic loss. It is an important factor affecting the mechanical vibration and heat generation of motorized spindle. Figure 9.6 shows the relationship between the output electrical parameters of inverter and the rotational speed of motorized spindle. It can be seen that the output voltage of the inverter increases linearly with the increase of rotational speed. The voltage reaches a saturation state when the rotational speed of motorized spindle reaches the rated value. Notably, there is also a linear relationship between output power and rotational speed.

9.1.2 Magnetic Leakage of Ceramic Motorized Spindle The properties of ceramic materials and metal materials are different. Ceramic materials are selected as the materials for the shaft and bearing of motorized spindle, which is due to their good thermal properties, as well as non-magnetic and nonconductive properties. Here, magnetic field tester is used to measure the magnetic field surrounding the ceramic motorized spindle, and a comparison with that of metal motorized spindle is performed. Figure 9.7 is the photograph of ceramic motorized spindle to be tested, and Fig. 9.8 is the photograph of magnetic field tester. Figure 9.9 shows the magnetic leakage around the ceramic motorized spindle in the time domain. Figure 9.10 shows the comparison of the magnetic leakage around the ceramic motorized spindle and metal motorized spindle in the time domain. As shown in Fig. 9.9, there are irregular peaks in the time-domain diagram of the magnetic leakage of ceramic motorized spindle shell. This reflects that the harmonic effect has great influence on the electromagnetic field, and the distortion of harmonic current and

9.1 Electromagnetic Properties of Ceramic Motorized Spindle Fig. 9.4 Voltage harmonic spectra of inverter at different frequencies

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Fig. 9.5 Current harmonic spectra of inverter at different frequencies

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Fig. 9.8 Magnetic field tester

Fig. 9.9 Magnetic leakage arround ceramic motorized spindle

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Fig. 9.10 Comparison of magnetic leakage arround ceramic and metal motorized spindle

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voltage increases the magnetic leakage. As shown in Fig. 9.10, the magnetic leakage around the ceramic motorized spindle is much lower than that around the metal motorized spindle. This shows that the magnetic field density between the ceramic stator and rotor poles is much lower than that of metal motorized spindle. This is mainly because ceramic materials cannot be magnetized. The reverse magnetic material acts against eddy current effect, leading to demagnetizing effect. The magnetic flux density between the magnetic poles reaches a saturation state slowly, and the magnetic flux diffusion to the peripheral region of the magnetic poles is reduced. This has a certain inhibitory effect on magnetic leakage. In order to compare the magnetic flux leakage around ceramic motorized spindle under different loads, the magnetic flux leakage of ceramic motorized spindle and metal motorized spindle under no load, 5-N radial load and 10-N radial load were studied. Figure 9.11 shows the effect of load on the magnetic flux leakage of ceramic motorized spindle. Figure 9.12 shows the effect of load on the magnetic flux leakage of metal motorized spindle. Figures 9.11 and 9.12 show that the magnetic field around the motorized spindle is greatly affected by the applied load. In Fig. 9.11, as the load increases, the amplitudes of magnetic flux leakage at different frequencies around the motorized spindle increase rapidly, the proportion of fundamental decreases and the proportion of higher harmonics increases. Clearly, the applied load has a great influence on the distribution of the magnetic field of motorized spindle. This is because as the load increases, the torque output increases, the voltage and power also increase, the magnetic flux density and magnetic flux between the stator and rotor increase, and finally the harmonics increase. Comparing Figs. 9.11 and 9.12, we can see that the amplitude of magnetic flux leakage around the metal motorized spindle is significantly higher than that around the ceramic motorized spindle. Due to the diamagnetic effect of ceramic material, the diffusion of magnetic field is suppressed. Compared with that of metal motorized spindle, the magnetic flux leakage of ceramic electric spindle is more affected by loading.

9.2 Vibration and Noise Characteristics of Ceramic Motorized Spindle 9.2.1 Vibration Characteristics of Ceramic Motorized Spindle Polytec OFV-505 non-contact laser vibrometer is used to measure the vibration of the housing and shaft of ceramic and metal motorized spindles. Figures 9.13 and 9.14 shows the metal motorized spindle and ceramic motorized spindle to be tested, respectively. Figure 9.15 shows the vibration spectra of ceramic and metal motorized spindles at different rotational speeds. It can be seen that with the increase of rotational speed, the vibration of ceramic motorized spindle is intensified, and the vibration peaks regularly shift to the low frequency region, but the change amount is not large,

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(a) 18000 r, no load

(b) 18000 r, 5-N load

(c) 18000 r, 10-N load Fig. 9.11 Effect of load on the magnetic flux leakage of ceramic motorized spindle

9.2 Vibration and Noise Characteristics of Ceramic Motorized Spindle

(a) 18000 rpm, no load

(b) 18000 rpm, 5-N load

(c) 18000 rpm, 10-N load Fig. 9.12 Effect of load on the magnetic flux leakage of metal motorized spindle

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Fig. 9.13 Metal motorized spindle to be tested

Fig. 9.14 Ceramic motorized spindle to be tested

indicating that the dynamic performance of ceramic motorized spindle is good. In Fig. 9.15, the vibration frequencies of the two motorized spindles are very different. The number of vibration peaks of metal motorized spindle in the low frequency region is larger than that of ceramic motorized spindle in the low frequency region.

9.2 Vibration and Noise Characteristics of Ceramic Motorized Spindle

(a) Ceramic motorized spindle, 12000 rpm

(c) Ceramic motorized spindle, 15000 rpm

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(d) Metal motorized spindle, 15000 rpm

(f) Metal motorized spindle, 18000 rpm

Fig. 9.15 The vibration spectra of ceramic and metal motorized spindle at different rotational speeds

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Fig. 9.16 Comparison of the vibrations of metal and ceramic motorized spindles

For metal motorized spindle, multiple irregular peaks appear in the low frequency region of its vibration spetrum and the energy in the low frequency region is greater than that in the high frequency region. These results show that the vibration of metal motorized spindle is greatly affected by electromagnetic vibration and harmonic effects. Figure 9.16 shows the comparison of vibration between metal motorized spindle and ceramic motorized spindle. The vibration amplitudes of the two motorized spindles are both directly related to rotational speed. As the rotational speed increases, the vibration amplitude shows a rising trend. The vibration of motorized spindle mainly consists of two categories. One is the mechanical vibration of the internal structure of motorized spindle, and the other is the electromagnetic vibration caused by the high frequency variation of stator and rotor air gap. According to Fig. 9.10, the magnetic flux leakage of metal motorized spindle is greater than that of ceramic motorized spindle. In Fig. 9.16, the vibration of metal motorized spindle is slightly larger than that of ceramic motorized spindle. It can be obtained that the electromagnetic and harmonic vibrations are one of the important causes of the vibration of motorized spindle.

9.2.2 Noise Characteristics of Ceramic Motorized Spindle With the development of science and technology, high-speed machining has become the trend of modern manufacturing technology. Motorized spindle is the core component of high-end CNC machine tool, and it is a high-speed machining technology characterized by zero transmission. Noise is an important indicator of the performance of motorized spindle. The noise of motorized spindle mainly includes mechanical noise, magnetic field noise and pneumatic noise. Mechanical noise is mainly caused by friction and shock vibration of bearings supporting the main shaft

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Fig. 9.17 Noise waveform of full-ceramic bearing motirzed spindle at different rotational speeds

at high speed. Magnetic field noise is caused by uneven magnetic field air gap due to rotor motion. Pneumatic noise is mainly caused by the impact of input oil and gas of the lubrication system on high-speed rotor and bearing. Pneumatic noise is small compared with mechanical noise and magnetic field noise. This section conducts noise tests on three kinds of motorized spindles with oil/air lubrication system. We mainly discuss the mechanical noise and magnetic field noise of motorized spindle. Figures 9.17, 9.18, 9.19, 9.20, 9.21, 9.22, 9.23, 9.24 and 9.25 show the noise waveforms of full-ceramic bearing motirzed spindle, metal bearing motorized spindle and ceramic shaft motorized spindle without inner ring at different rotational speeds. Figure 9.17 suggests that with increase in rotational speed, the amplitude of the radiated noise of full ceramic bearing motorized spindle increases and the periodicity of the noise wave becomes more obvious. Figure 9.18 reveals that with increase in rotational speed, the amplitude of the radiated noise of steel bearing motorized spindle also increases, but the increase amount is smaller than that of the amplitude of the radiated noise of full ceramic bearing motorized spindle. Figure 9.19 shows that the amplitude of the radiated noise of ceramic shaft motorized spindle without inner ring is basically independent of rotational speed in the speed range of 10,000–18,000 rpm. The above results indicate that the radiated noise of ceramic shaft motorized spindle without inner ring is significantly higher than that of full ceramic bearing motorized spindle and steel bearing motorized spindle. Figure 9.20 shows a plot of radiated noise vs. rotational speed. The measured speed range is 10,000–18,000 rpm.

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Fig. 9.18 Noise waveform of metal bearing motirzed spindle at different rotational speeds

Fig. 9.19 Noise waveform of ceramic shaft motorized spindle without inner ring at different rotational speeds

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Fig. 9.20 Curve of radiated noise versus rotational speed

Fig. 9.21 Noise spectra of full ceramic bearing motorized spindle at different rotational speeds

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Fig. 9.22 Noise spectra of metal bearing motorized spindle at different rotational speeds

Fig. 9.23 Noise spectra of ceramic shaft motorized spindle without inner ring at different rotational speeds

9.2 Vibration and Noise Characteristics of Ceramic Motorized Spindle

Fig. 9.24 Block diagram of the intelligent control system of motorized spindle

Fig. 9.25 Intelligent temperature field prediction system for ceramic motorized spindle

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With increase in rotational speed, the radiated noises of full ceramic bearing and metal bearing motorized spindles increase, with the former changing greatly and the latter changing smoothly. At low speed, the radiated noise of full ceramic bearing motorized spindle is lower than that of metal bearing motorized spindle. At high speed, the radiated noise of full ceramic bearing motorized spindle is higher than that of metal bearing motorized spindle. Among the three kinds of motorized spindles, the ceramic shaft motorized spindle without inner ring shows the highest radiated noise, but the change of its noise with the change of rotational speed is the smallest. Its noise increases slightly and then decreases with rotational speed, and the sound pressure level of the noise is generally maintained between 88 and 90 dB. Since the bearing clearance of ceramic shaft motorized spindle without inner ring is relatively large, a large vibration noise is generated during its operation. With increase in rotational speed, the metal bearing motorized spindle shows relatively small vibration and the vibration changes smoothly. As a result, its radiated noise is relatively small and increases linearly with rotational speed. With increase in the rotational speed of full ceramic bearing motorized spindle, the friction between the balls and ring increases rapidly, resulting in relatively large friction noise. Therefore, compared with metal bearing motorized spindle, the full ceramic bearing motorized spindle shows great changes in noise and the noise increases nonlinearly with rotational speed. Figures 9.21, 9.22 and 9.23 show noise spectra of full ceramic bearing motorized spindle, metal bearing motorized spindle and ceramic shaft motorized spindle without inner ring at different rotational speeds. Figure 9.21 reveals that the radiated noise of full ceramic bearing motorized spindle is two times of the rotational frequency (two peaks appearing from low to high frequency) and impact and friction noise between the balls and ceramic occurs (frequency near 2000 Hz). At low speeds, the mechanical noise generated during rotation is large, whereas the magnetic field noise caused by magnetic field vibration (frequency is 6000–8000 Hz) is relatively small. As the rotational speed increases, the magnetic field vibration is intensified, causing the magnetic field noise to slowly exceed the mechanical noise (bearing vibration noise). With increase in rotational speed, the noise frequencies gradually shift to the right. Figure 9.22 shows that the radiated noise characteristics of metal bearing motorized spindle are similar to those of all ceramic bearing motorized spindle. However, at lower speeds, the friction and impact noise of the former are more severe than that of the latter. At 18,000 rpm, the magnetic field noise (around 5000 Hz) of metal bearing motorized spindle is slightly higher its mechanical noise. Figure 9.23 shows that the noise spectrum of ceramic shaft motorized spindle without inner ring is rather messy, but it displays the existence of mechanical vibration and magnetic field vibration frequencies. At lower speeds, the rotational frequency noise of ceramic shaft motorized spindle without inner ring is not significant, and the noise caused by friction and impact between the balls, ring and cage are obvious. At higher speeds (18,000 rpm), the rotational frequency noise become significant. With increase in rotational speed, the noise variation of magnetic field frequency is not obvious, and the change of the whole noise spectrum is not very large. It is probably because the ceramic shaft is not magnetically conductive

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compared with the steel shaft, which explains the large difference in noise spectrum between ceramic shaft and steel shaft motorized spindles. In sum, at low speeds, the noise of full ceramic bearing motorized spindle and metal bearing motorized spindle are mainly mechanical vibration noise. With increase in rotational speed, the main frequency is biased toward the vibration frequency of electromagnetic field so that the magnetic field noise at higher rotational speed is higher than mechanical noise. For ceramic shaft motorized spindle without inner ring, the noise spectrum is relatively complicated, and the change of its spectrum has little relationship with rotational speed.

9.3 Development of Intelligent Ceramic Motorized Spindle The above analysis reveals that the ceramic motorized spindle and metal motorized spindle show difference in vibration, noise and electromagnetic properties. The ceramic motorized spindle is essentially different from metal motorized spindle in terms of temperature field calculation, thermal deformation calculation and fault diagnosis. To develop intelligent ceramic motorized spindle, it is necessary to first collect a large amount of operational data on ceramic motorized spindle. Using the techniques in Chaps. 5–8 and combining with the large amount of data collected during the operation of ceramic motorized spindle, we develop an intelligent ceramic motorized spindle system. Figure 9.23 is a block diagram of the intelligent control system of motorized spindle. The system includes a temperature field prediction system, a thermal deformation prediction system, an online dynamic balancing device, and a failure early warning system. Through the installation of a large number of temperature, acceleration and flow rate sensors inside the motorized spindle and its subsystems, the internal temperatures and vibration changes of the spindle can be perceived. According to the collected data about the speed, load, flow rate of lubricating oil, temperature of cooling water and other parameters of motorized spindle, expert analysis and evaluation system based on intelligent algorithm is used to calculate the temperature filed, thermal deformation, imbalance amount and phase position of the spindle, evaluate its operating state and perform fault diagnosis. Figure 9.25 shows an intelligent system for temperature field prediction. The system is a subsystem of the intelligent control system of ceramic motorized spindle. After temperature field prediction, the thermal deformation is further predicted. The thermal deformation prediction system can achieve a prediction accuracy of 1.3% within 30 s.