Ultrasonic Motors: Technologies and Applications

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o

o

o 00

Sman airplane

MngJev 1I".un

Satell ite

Mars prober

Cell pnncture

Automobile

Optical III icroscope

Imegrated USM

Camera

For semiconductor manufacture

Mobi le phone

Walch

Color copier

Micro robot

piezoelectric act uator

Morphing \\;ng(by PZT actuator)

For aSlronom icaltelescope

Color figure 1

Aerial robot

Ultrasonic motors in applications

Space manipulator

Non-magnetic USM

V shape linear US

Ponable gasoline generator

USM with encoder

Non-contact USM

BUllerfty s hape linear USM

Surve il lance camera platfonn

Color figure 2

Three DOr USM

x- Y slage driven by rotarY USMs

Vacuum cleaning robot

Bar-type USM

Mode conversion type USMs

x-Y stage driven by line..."Ir

Joilll robot

Some ultrasonic motors developed by PDLab at NUAA and their applications

Active Il utter suppression system

MRI syringe

Fig. 1.1 I ( b )

Fig, 1.1 I ( a )

Fig,I,6

Fig. I,25

Fig. 1.1 9

Fig.1.8(b)

Fig. l.ll (d)

Fig. I.II (c)

Fig, 1.28

Fig. 1.26

Fig.2.15

- I

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Fig.5.2

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Fig,5. 19(a )

Color figure 3

Some colored figures in the book

Fig.6. 15(a)

Fig,6 ,15(b)

Fig.6.19

Fig.I I.25(b)

Fig. IO.2?

Fig.6.22

Fig.6.20

Fig. l l.29

Fig. 11 .20

Fig.14.1

Fig.12.35

Fig.11.24(b)

Fig.14 .6

e Fig. 14.?

Fig. 14. 17(b)

Fig. 14. 14

Color figure 4

Some colored figures in the book

Fig. 14.20

Fig. 15.32

Chunsheng Zhao

Ultrasonic Motors Technologies and Applications

Chunsheng Zhao


Wi th 564 figures, 14 of them in eolor

e;e Science Press Beijing .dl

'.£l Springer

Author

Chunsheng Zhao Precision Driving La bora tory Nanjing University of Aeronautics and Astronautics Nanjing 210016, China Email: [email protected]

ISBN 978-7-03-029018-9 Science Press, Beijing Springer ISBN 978-3-642-15304-4 e-ISBN 978-3-642-15305-1 Springer Heidelberg Dordrech t London New Y or k Library of Congress Control Number: 2010932502

© Science Press Beijing and Springer- Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights arc reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The usc of general descriptive names, registered names, trademarks, etc. in this publication docs not imply, even in the absence of a specific statement, that such names arc exempt from the relevant protective laws and regulations and therefore free for general use.

Cover design: Frido Sleinen-Broo, EStudio Calamar, Spain Printed on acid- free paper Springer is part of Springer Science+Business MediaC www.springer.com)

Foreword by Jorg Wallaschek Piezoelectric Ultrasonic Motors are fascinating actuators. They combine fast dynamics and high force, can be adopted to a wide range of applications, and offer many advantages in comparison to electromagnetic and other motors. As a consequence, research in the field of piezoelectric ultrasonic motors has attracted scientists from all over the world, and many valuable contributions have been made during the past decades, while some open research questions still wait for an answer. Professor Chunsheng Zhao is a pioneer in the research of piezoelectric ultrasonic motors. His contributions to the scientific progress in this field inelude theoretical and experimental works, whose results arc documented in a large number of scientific papers and patents, and he has also developed many different prototypes of piezoelectric ultrasonic motors. Professor Chunsheng Zhao was the organizer of important international conferences and he also was the founder and director of a very successful research lab which has been recognized world wide. The present book summarizes not only the results of Prof. Zhao's work, but also provides an excellent survey on the state of the art of piezoelectric ultrasonic motors, which can be used as a textbook for teaching as well as a reference for experts in the field. I sincerely wish that it will be of practical usc to a hopefully very large number of readers and that it will also stimulate further research in the field. January 26th, 2010 Prof. Dr. - lng. habil. J brg Wallaschek

Leibniz Universitiit Hannover

Foreword by Bangchun Wen Vibratory utilization engineering is a new branch of vibration science which has been developing since the second half of the 20th century. Due to the academic significance and important applications of the field, the Vibratory Utilization Engineering Specialty Commission (VUESC) was founded in affiliation with the Chinese Association of Vibration Engineering in the last few years. It is aimed at promoting wider use and further development of the discipline through regular conferences and communication. Prof. Chunsheng Zhao, a well-known expert in vibration engineering, has researched vibration and vibration utilization engineering for more than 40 years, and has achieved fruitful results in both theory and engineering applications of vibration. For the past 15 years, Prof. Zhao has specialized in ultrasonic motors. He and his research team have developed more than 30 new types of the ultrasonic motors and corresponding drivers with proprietary intellectual rights, 71 invention patents awarded and pending in China and published more than 500 papers. The project of "Research on Ultrasonic Motor" was awarded multiple national awards and international recognition. Based on the author and his team's research on ultrasonic motors over the last 15 years, this book has summarized their achievements as follows: Firstly, the author explains the systematic theory and design methods in using vibration and wave theory, ineluding motion mechanism, electromechanical coupling model, optimal design of structural parameters, driving, control techniques, etc. Secondly, the author creatively applies advanced analytical methods and techniques into research on the ultrasonic motors, ineluding dynamic substructure, structure dynamic modification, modal identification and separation techniques, etc. Thirdly, this book introduces many key techniques on the ultrasonic motors, ineluding an effective frequency auto-tracking technique solved by the author's team, which has been the bottleneck for application of the ultrasonic motor. The new concept of "anti-resonance/ constant current" put forward by the author is applied to the traveling wave ultrasonic motor, which can promote comprehensive performances of ultrasonic motors. Fourthly, the book shows a series of testing devices developed by the author independently or cooperatively and a series of testing methods provided by the author, which are used for various tests of parts and completed motors. Finally, this book integrates theory with application. It not only ineludes sys-

V111

Ultrasonic Motors Technologies and Ap plicalions

tematic theories and methods, but also introduces many engineering and industrial applications, such as in robots, the active flutter suppression of a two-dimensional wing, an injector for nuclear magnetic resonance, a target recognition/ tracking system, etc. In addition, the author is meticulous and precise in writing. His formula deducing, experimental data and figures are also very convincing. In summary, this book comprehensivcly and systematically describes the technologies of ultrasonic motors and their applications. It will certainly make contributions to this area. I believe the publication of this book will promote the worldwide development and practical applications of ultrasonic motors. I greatly appreciate the effort of my close friend, Prof. Zhao, in writing this wonderful book. Here I cite a Chinese poem as our mutual encouragement: "Although the stabled steed is old, he dreams to run a thousand miles". Mar 1, 2010 Bangchun Wen

Academician of Chinese Academy of Sciences Professor of :'\Iortheastern University

Preface As a new type of micro-motor, the Ultrasonic Motor CUSM) has gained rapid dcvelopmcnt and wide applications sincc thc 1980's. Unlike traditional motors with electromagnetic effect, USM is driven by ultrasonic vibration and piezoelectric effcct. This new typc of motor covcrs a wide rangc of subjects, ineluding mechanical vibration, tribology, matcrials scicncc, mechanical design, elcctronics, automatic control, super-prccision process, etc. Ultrasonic motors havc many cxccllcnt pcrformances and fcaturcs, such as simplc construction, high torquc dcnsity at low specd, dircct drive without spccd reduction gears, quick rcsponsc, better elcctromagnetic compatibility, high holding torque while power off, quict running, efficiency insensitivc to thc sizc, ctc. They have been applicd to robots, precise facilities, medical instruments, etc. With the dcvelopmcnt of new matcrials, advanced technologies, and ncw structural types, the construction and performance of ultrasonic motors will be improvcd, and their applications will bc broadencd to encompass a wider arca ineluding space vehieles, MEMS, semiconductor manufacturing, life sciences, etc. During my visit at MIT from 1992 to 1991, I started research on ultrasonic motors. I came back to China in 1991 and continued my research at Nanjing University of Aeronautics and Astronautics CNUAA). I built a research group in 1995. My group designed and manufactured a traveling wave rotary ultrasonic motor with integrated construction that operated properly by the end of that year. In 1997, I founded the Ultrasonic Motors Rescarch Center CUMRC) in NUAA. In 1999 I organised the First Chinese Workshop on Ultrasonic Motor TechnologiesCCWUMT) with the support of National Natural Sciences Foundation of China C:'\JSFC). The research and development in this area has rapidly advanced since then, and our Research Center was further promoted to be the Ultrasonic Motors Enginecring Rcsearch Center of Jiangsu Provincc in 2001. Fivc years later, the Research Center was renamed the Precision Driving Laboratory CPDLab). The 4th International Workshop on Piczoelectric Materials and Applications in Actuators(IWPMA1) was held at :'\JUAA on September 2007. For the past 15 ycars, our rcscarch team has systcmatically studied ultrasonic motors in depth and obtained considerable achievements, ineluding motion mechanism, electromcchanical coupling modcl, optimal design of structural parametcrs, driving/control techniqucs, ctc. We havc developed more than 30 typcs of ncw ultrasonic motors with indcpendent intellcctual property rights and corresponding drivcrs. Wc have 71 invention patcnts cither awarded or pending in China and more than 500 papers published in journals and conferences. Our projcct of "Rescarch on Ultrasonic Motors" was awarded multiple national awards.

x


The achievements of our team can be coneluded as follows:

1. In Theory On the basis of dynamic substructure theory, a comparatively well designed electromechanical coupling model of the traveling wave type rotary ultrasonic motor is built. A new friction interface model which takes the stator teeth and the radial sliding between the stator and rotor into consideration is proposed, and this model can precisely predict the output performances of the type of ultrasonic motors. Instead of the traditional concept of "resonance point/ constant voltage", a new concept of "anti-resonance point/constant current", which is more effective for improving the efficiency and stability of the traveling wave ultrasonic motor, is put forward. An effective frequency automatic tracking method which can lower the instability of ultrasonic motor's speed Cwi thin 5 %) is found, and this method succeeds in solving the bottleneck of the ultrasonic motor Cthe speed is down while the temperature is up). A method on solving the mode mixture in the ncar frequency of the ring stator or circular plate stator is obtained, and this method can improve the stability of the ultrasonic motor. The elliptical motion equation of a bar-type traveling wave ultrasonic motor is derived, and the concept of the effective ellipse orbit which provides a theoretical basis to the optimization of the bar-type ultrasonic motors is proposed. 2. Design Methods An optimal design of structure parameters for the ultrasonic motor put forward and the corresponding software is developed. By applying the sensitivity analysis of structure parameters and structural dynamic modification technique to the design of the ultrasonic motors, an effective method which can adjust the stator's two-phase or multi-phase modal frequencies to be the same. To propose that the design of piezoelectric ceramic components used for the ultrasonic motors should be in accordance with the strain mode of stator instead of its displacement mode; To put forward a method which simultaneously utilizes different types Cextension-contraction, bending and torsion) of vibration modes in-/out-of-planes for designing all types of ultrasonic motor; To point out that the design of the flexible rotor is very important, and to present some design methods for it; To provide the concept and design principles of the step ultrasonic motors. 3. Testing Techniques A series of test devices is developed independently or cooperatively. Some effective test methods have been proposed, ineluding modal tests with nm amplitude in ultrasonic frequency area, load characteristics tests in low speed and ultrasonic frequency area, response time tests at power on/off of the ultrasonic motors, measurement devices and methods of the dynamic friction between the stator and rotor, life test equipment and methods for the ultrasonic motor, test methods of the ultrasonic motor under an extreme environmentC vacuum, high/low temperatures), and performance measurement methods and preparation devices of new friction materials.

Preface

Xl

4. Applications Two series of the ultrasonie motors (TRUM and BTRUM) have been independently developed, and some of them are applied to industry, medical and precision instruments. Moreover, we have also provided prototypes of the ultrasonic motors to some companies. This creates favorable conditions for realizing the ultrasonic motor industrialization in China. At the same time, we have investigated some precision position and constant speed control systems with multi-variable (speed, frequency and phase) using the ultrasonic motors as actuators, ineluding a position control system used for suppressing a two-dimensional wing's flutter, a constant speed control system used for injector of nuelear magnetic resonance, a composite control system based on FN:'\J and Fuzzy control strategies which is used for drivel control a robot, a control system for automatically tracking targets based on vision, a fuzzy control system applied to portable gasoline generators, etc. In addition to the achievements and innovations mentioned above, this book also fully absorbs the most advanced and important results in this area over all world in order to enrich the content. There are 15 chapters in the book. Chapter 1 is an introduction, which describes the history, elassification, characteristics and applications of ultrasonic motors. Chapter 2 describes the fundamentals of piezoelectricity and piezoelectric materials used for ultrasonic motors, and emphasizes the influence of piezoelectric materials on the performance of ultrasonic motors. The knowledge on how to select the piezoelectric materials used for USM is also introduced. Chapter 3 introduces the fundamentals of tribology and tribomaterials used for ultrasonic motors. Some tribomaterials for ultrasonic motors are proposed. In addition, the components and produce process of two new kinds of friction materials are provided. Chapter 4 introduces the fundamentals of vibration and wave applied to ultrasonic motors. It expounds the displacement and strain modes of elastic bodies such as a common rectangular, circular, ring plates, and a cylindrical shell which are used for the stator of ultrasonic motors. The strain mode is a basis of the piezoelectric component polarization division for effectively exciting the stator. Moreover, some important concepts are analyzed, such as the relation between standing wave and traveling wave, mode superposition, mode separation, and wave propagation in elastic bodies. Chapters 5-11 describe the motion mechanism, electromechanical coupling model, optimal design of structure parameters, and testing for different types' ultrasonic motors, ineluding the disk- and bar-type traveling wave ultrasonic motors, the longitudinal-torsion hybrid type ultrasonic motor, the linear ultrasonic motor, the step ultrasonic motor, the non-contact ultrasonic motor, the surface wave ultrasonic motor, etc. These chapters are the most important as they represent our academic achievements and innovations. Chapters 12-13 describe the driving and control techniques of the ultrasonic

Xli


motors. Chapter 13 introduces the drive principles and design methods of the drivers in detail, and provides an actual driver circuit which is in use at PDLab. Chapter 14 introduces various tests of the ultrasonic motors, ineluding testing principles, methods, equipment, and the analysis of testing results. Chapter 15 summarizes the practical applications of ultrasonic motors and looks to the future of this area. This book is a comprehensive tutorial for practicing engineers and researchers developing the ultrasonic motor technologies and applications. It is also an up-todate reference for graduates taking a course on ultrasonic motor technologies. Finally, I tell my readers that I will greatly appreciate your comments and s ugges tions. June 5, 2010 Chunsheng Zhao

)JUAA, Nanjing, China

Acknowledgements First, I would like to thank the ='Jational ='Jatural Seiences Foundation of China, 863 High-Tech Projects, and provincial and ministerial funding projects. Many achievements described in this book are credited to these funding sources. I also express my gratitude to my colleagues who have contributed to this book. The followings people are especially thanked for writing and translating some chapters of my manuscripts: H uafeng Li, Chao Chen, Zhiyuan Yao, H ua Zhu, Ying Yang, Zhijun Sun, Weiqing Huang, Jiamei Jin, Yunlai Shi , Lin Yang, Junhui Hu, Jianhui Zhang, Qingjun Ding, Shengqiang Zhou, Yiping Wang, Yi Ding, Congyun Shi, Yubao Li, Jiantao Zhang, Wei Zheng, Hanming Peng, Xiangdong Zhao, Guiqin Wang, Wei Hu, Jian Liu, Dan Lu, Yucong Yin, Qi Chen, Ping Wang, et al. I am grateful to Prof. Bangchun Wen, Prof. Shizhu Wen, Prof. Jue Zhong, Prof. Zhiyun Shen, Prof. Liding Wang, Prof. Haiyan H u, Prof. Tieying Zhou, Prof. Chenglin Gu, Prof. Zhigang Yang, Prof. Fengquan Wang, Prof. Jinhao Qiu, Prof. Zhendong Dai, Prof. Zexiang Li, Prof. Haosu Luo, Prof. Baoku Li, Prof. Shuxiang Dong, Prof. Fei Zhou, Prof. Xiangtao Fan, Prof. Yuhong Liu, Dr. Chunning Zhang, Prof. Zhong You, et al. for their helpful comments and s ugges tions. I also want to express my heartfelt thanks to Prof. Jbrg Wallaschek, Prof. Kenji Uchino, Prof. Piotr Vasiljev, Prof. Scok-Jing Yoon, Prof. Yo shiro Tomikawa, Prof. Minoru Kuribayashi Kurosawa, Prof. Takhiro Takano, Prof. Takshi Maeno, Prof. Aydin Dogan, Prof. J ian S Dai. Dr. Toshiiku Sashida, Dr. Ichiro Ohumura, Dr. David Henderson, and Dr. Ryan Lee. They have provided the book with some papers, data, and photos via the conferences or the lectures in our PDLab. I especially thank Prof. J brg Wallaschek for writing a Foreword in the book. Last but certainly not least, I am grateful to my wife Fengying Wang, my da ugh ter Dr. Ying Zhao, my son-in-law Dr. Charles Zhou, and my grandsons Derek and J esse for their understanding and support to my work, for their care to my life, and for their encouragement to my soul.

Contents 1

Introduction··· ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 1

1. 1 History of Ultrasonic Motors ................................................ 2 1. 2 Characteristics of Ultrasonic Motors and Their Classification'" ...... 7 1. 2. 1

Characteristics of Ultrasonic Motors

1. 2. 2

Classification of Ultrasonic Motors'" ... ...... ... ...... ... ...... ... ... 8

...... ... ... ... ... ... ... ... ... ... 7

1. 3 Comparison with Electromagnetic Motors

.............................. 12

1. 3. 1

Load Characteristics ................................................... 13

1.3.2 1.3.3

Transient Response Characteristics

Energy Transform of Motors and Their Micromation

............ 13

................................. 14

1. 4 Applications and Development Trends of Ultrasonic Motors 1.4. 1

15

Applications .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · .. · 15

1.4.2 Development Trends ................................................... 15 References ........................................................................... 18

2

Fundamentals of Piezoelectricity and Piezoelectric Materials for Ultrasonic Motors ............................................................... 21

2. 1 Development and Classification of Piezoelectric Materials ............ 21 2.1. 1

Historical Development of Piezoelectric Materials .................. 21

2. 1. 2

Classification of Piezoelectric Materials'" ...... ... ...... ... ...... ... 22

2. 2 Electrical Properties of Piezoelectric Materials

... ... ...... ... ...... ... 23

2. 2. 1

Dielectric Properties and Dielectric Loss

...... ... ...... ... ...... ... 23

2. 2. 2

Ferroelectric Properties and Polarization

...... ... ...... ... ...... ... 25

2. 3 Properties and Constitutive Equations of Piezoelectric Materials

27 ........................... 27

2. 3. 1

Elastic Properties and Their Codficients

2. 3. 2

Piezoelectric Properties and Piezoelectric Equations

............... 28

2. 4 Vibration Types of Piezoelectric Vibrators .............................. 31 ............ 31

2.1. 1

Piezoelectric Vibrators and Their Equivalent Circuits

2. 4. 2

Characteristic Frequencies of Piezoelectric Vibrators ............... 33

2. 4. 3

Coupling Coefficient and Quality Factor

...... ... ...... ... ...... ... 35

2. 5 Applications of Piezoelectric Materials to Ultrasonic Motors 2.5. 1

Piezoelectric Ceramics Used for Ultrasonic Motors

2.5.2

Applications of Piezoelectric Materials to Other Actuators

38

............... 39 41

2. 6 Advances in Novel Piezoelectric Materials .............................. 45


XVI

References

3

........................................................................... 17

Fundamentals of Tribology and Tribomaterials for Ultrasonic Motors ........................................................................... 50

3. 1 Basic Tribology

............................................................... 51

3. 1. 1

Surface of Tribomatcrials ............................................. 51

3.1.2

Friction and Its Classification

3. 1. 3

Friction Mechanism ................................................... 53

3.1.4

Wear Mechanism ...................................................... 56

3. 1. 5

Wear Surface for Stator and Rotor of Ultrasonic Motors

....................................... 52

3.2 Tribomaterials Used for Ultrasonic Motors

58

........................... 58

3.2. 1

Basic Requirement, Classification and Selection Principle ......... 58

3. 2. 2

Influence of Composition on Tribological Properties ............... 60

3. 2. 3

Preparation of Tribomaterial .......................................... 62

3. 3 Influence of Tribomaterials on Performance of USM .................. 67 3.3. 1

Influence of Elastic Modulus and Hardness

3. 3. 2

Influence of Friction Coefficient

3. 3. 3

Influence of Anisotropy

........................ 67

...... ... ...... ... ...... ... ...... ... 69

...... ... ...... ... ...... ... ...... ... ...... ... 70

3. 4 Friction Testing for Tribomaterials ....................................... 72 3.1. 1

Quasi-static Friction Testing .......................................... 72

3.4.2

Dynamic Friction Testing ............................................. 73

References

4

........................................................................... 71

Fundamentals of Vibration for Ultrasonic Motors ........................ 76

1. 1 Natural Vibration of Elastic Body··· .................................... 76 4. 1. 1

Longitudinal Vibration of Bars ....................................... 77

1. 1. 2

Characteristics of Natural Modes .................................... 78

4. 1. 3

Torsional Vibration of Shafts

4.1.4

Bending Vibration of Beam

.......................................... 81

... ...... ... ...... ... ...... ... ...... ... 80

1. 1. 5

:"Iatural Vibration of Plates

.......................................... 82

4.1. 6

:"Iatural Vibration of Cylindrical Shells .............................. 92

4. 2 Forced Vibration of Elastic Body··· ....................................... 95 1. 2. 1

Response of PZT Bar to Distributed Electric Field

... ... ...... ... 96

4.2.2

Metallic Bar Excited by Single or Multiple PZT Pieces ............ 99

1.2.3

Response of Beam to Constant Electric Field Intensity

4.2.4

Excitation of Simply Supported Beam by PZT Pieces ............ 101

1.2. 5

Response of Thin Plate to PZT Piece Excitation .................. 106

101

1. 3 Wave Propagation in Elastic Body··· .................................... 107 ............................................. 107

4.3.1

Basic Concept of Wave

1.3.2

Waves in Elastic Body··· .......................................... 108

4.3.3

Superposition of Waves

............................................. 110

Contents

XVll

1. 3. 1

Formation of Traveling Waves

4.3. 5

Formation of Elliptical Trajectory··· .............................. ll4

... ...... ... ...... ... ...... ... ...... III

References ........................................................................... 115

5

Operating Mechanism and Modeling of Traveling Wave Rotary Ultrasonic Motor ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 118 ....................................... ll9

5. 1 Operating Mechanism of TRUM

.................. ll9

5.1.1

Structure and Operation Mechanism of TRUM

5.1.2

Formation of Traveling Wave in Stator

5. 1. 3

Elliptical Motion Trajectory of Surface Points on Stator ......... 123

5. 1. 1

EHect of Amplitude and Phase on Elliptical Motion··· ...... ... ...... 121

........................... 121

5. 1. 5

Polarization Pattern of Piezoelectric Ceramic Components

5. 1. 6

Three-dimensional Motion Analysis of Points on Stator ......... 129

128

5. 2 Semi-Analytical Electromechanical Coupling Model of Stator ...... 132 5. 2. 1

Substructure Division of Stator

.................................... 133

5. 2. 2

Characteristic Matrix of Substructures a and b

5.2.3

Characteristic Matrix of Substructure c··· ........................ 139

... ...... ... ...... 133

5.2.4

Electromechanical Coupling Model of Stator

5.2. 5

Computation Example of Dynamic Characteristics of Stator ...... 141

..................... 110

5. 3 Contact Model Between Stator and Rotor .............................. 111 5.3. 1

Interface Assumptions

............................................. 145

5.3.2

Interface Force and Power Transmission ........................... 146

5.3.3

Interface Energy Loss and Power Transmission Efficiency··· ... 118

5.3.4

Contact Model Between Stator and Rotor

5.3. 5

Contact InterLace Simulation

........................ 149

....................................... 150

5.4 Electromechanical Coupling Model of TRUM and Its Simulation ...... 152 5.1. 1

Electromechanical Coupling Model of TRUM ........................ 152

5.4.2 Performance Simulation of Ultrasonic Motor ..................... 151 References ........................................................................... 158

6

Design and Manufacture of Traveling Wave Rotary Ultrasonic Motors ........................................................................... 161

6. 1 General Design Process of TRUMs

.................................... 161

6.1. 1

Structure Sizes of the Stator

6. 1. 2

Design of Rotor Size

....................................... 162

6.1.3

Choice of Materials ................................................... 166

................................................ 165

6. 2 Operating Modes of Stator and Polarization of PZT Ring 6.2. 1 Design of Modal Frequency .0

6.2.2

••

0

••

0

••

0

••

0

••

0

••

••

0

••

0

••

0

••

0

••

0

••

0.

167 168

Polarization of Piezoelectric Ceramics .............................. 168

6. 3 Structure Form of Stator and Its Modal Analysis 6. 3. 1

0

Structure Form of Stator

... ...... ... ...... 170

...... ... ...... ... ...... ... ...... ... ...... 170


XV111

6. 3. 2

Modal Analysis of Stator

...... ... ...... ... ...... ... ...... ... ...... 170

6.4 Sensitivity Analysis and Avoiding of Mode Mixture of Stator ...... 172 6.1. 1

Principle of Sensitivity Analysis .................................... 172

6.4.2

Sensitivity Analysis of Stator for TRUM-60

6. 1. 3

Mode Separation of Stator for TRUM-60

..................... 173 ...... ... ...... ... ...... 174

6. 5 Optimal Design of Stator ................................................... 176 6.5. 1

Optimal Model of Stator ............................................. 176

6. 5. 2

Example of Optimal Design of Stator .............................. 177

6. 6 Adjustment of Two Phase Modal Frequencies of Stator ............ 179 6. 6. 1

Method of Adjusting of Two Phase Modal Frequencies

180

6. 6. 2 Example of Adjusting of Two Phase Modal Frequencies 182 6. 7 Analysis of Flexible Rotor ................................................ 183 6.7. 1

Importance of Rotor's Flexibility for Performance of Motor

185

6.7.2

Comparison of Contact Area of Rigid and Flexible Rotor

188

6.7.3

EHect of Rigid and Flexible Rotors on Mechanical Characteristics

189

6.7.4

Design and Manufacture of Flexible Stator ........................ 189

6.8 Manufacturing Techniques of TRUM ................................. 191 References ........................................................................... 193

7

Bar-type Traveling Wave Rotary Ultrasonic Motors ..................... 195

7. 1 Review of Bar-type Ultrasonic Motor

................................. 195

7. 2 Construction and Motion Mechanism of SDOF Motor ............... 196 7.2. 1 Construction ......................................................... 196 7.2.2

Motion Mechanism ................................................... 197

7. 3 Optimal Design for SDOF Motor

....................................... 201

7.3. 1

Design Principle ...................................................... 201

7.3.2

Dynamic Model

7.3.3

Sensitivity Analysis

7.3.4

Objective Function ................................................... 208

7.3. 5

Optimal Algorithm and Results

...................................................... 203 ................................................ 207 .................................... 208

7.3.6

Modal Frequency Modification of Stator ........................... 210

7. 3. 7

Design of Flexible Rotor ............................................. 213

7. 1 Performance Simulation for SDOF Motor .............................. 211 7. 4. 1

Dynamic Model

...................................................... 214

7.1.2

Contact Analysis

7.4.3

Performance Simulation ............................................. 216

................................................... 215

7.5 Motion Mechanism of 3-DOF Motor .................................... 219 7.5.1

Construction and Operating Modes ................................. 219

7.5.2


7. 6 Optimal Design of Stator of 3-DOF Motor 7.6. 1

Construction and Objective Function

........................... 221 .............................. 221

Contents 7.6.2

XIX


.................................... 223

7. 7 Performance Measurement of 3-DOF Motor ........................... 225 7.7. 1

Testing Equipment and Results .................................... 225

7.7.2

Effect of Pre-pressure on Mechanical Performance ............... 226

7. 8 Driving and Control Techniques of 3-DOF Motor

.................. 227

.............................. 227

7.8. 1

Configuration of the Control System

7.8.2

Control for Trajectory Tracking .................................... 227

References······ ... ...... ... ...... ... ...... ...... ... ...... ... ...... ... ...... ... ...... 229

8

Ultrasonic Motor Using Longitudinal-Torsional Hybrid Vibration

8. 1 Current Research of LTUM 8. 2 Multi-mode Type LTUM

... 232 ... ...... ... ...... ... ...... ... ...... ... ...... 232

...... ...... ... ...... ... ...... ... ...... ... ...... 235

8. 2. 1


8.2.2

Structure Design of Multi-mode Type LTUM ..................... 239

8. 3 Contact Model between Stator and Rotor .............................. 213 8. 3. 1

Modeling of Contact Interface··· ... ...... ... ...... ... ...... ... ...... 243

8. 3. 2

Friction Loss on Interface and Efficiency of LTUM

8.3.3

Simulation of Performance of LTUM .............................. 219

... ... ...... 218

8. 4 Mode Conversion Type Ultrasonic Motor .............................. 255 .................................... 255

8.1. 1

Structure and Operation Modes

8.4.2

Principle of Mode Conversion ....................................... 256

8.1.3 8.4.4

Design of Mode Conversion Type LTUM with Holes ............ 258 Testing ............................................................... 259

References ........................................................................... 262

9

Linear Ultrasonic Motors

.................................................. . 265

9. 1 State of the Art of Linear Ultrasonic Motors

266 9. 2 Linear Ultrasonic Motors Based on d'l Effect ........................ 270 9.2. 1

Linear Ultrasonic Motor with Double Driving Feet ............... 270

9.2.2

Linear Ultrasonic Motor with Single Driving Foot ............... 278

9. 3 Linear Ultrasonic Motors Based on d" Effect ........................ 281 9. 3. 1

Linear Ultrasonic Motor with Butterfly Shaped Stator

9. 3. 2

Linear Ultrasonic Motor with Wheel Shaped Stator

9.4 Contact Model of Standing Wave Type LUSMs 9.1. 1

Steady State Characteristics

282 ... ... ...... 288

..................... 292

....................................... 292

9.4.2

Transient Responses

................................................ 293

9.1.3

Simulation Examples

................................................ 294

9. 5 Synergetic Operation Technique of LUSMs ........................... 296 References ........................................................................... 298


xx

10

Step Ultrasonic Motors ...................................................... 300

10. 1 Step Control of USM ...................................................... 301 10. 1. 1 10.1.2

Startup and Shutdown Characteristics of USM ............... 301 Step Control for USM ............................................. 303

10. 1. 3 Factors Impacting on Single-step Positioning Accuracy 308 10. 2 Step USM with Fixed Step length ....................................... 309 10.2. 1 10.2.2

Standing Wave USM Used for Constructing Step USM 309 Modal Rotary Type Step USM ................................. 312

10.2.3

Self-correction Peak Type Step USM

........................... 322

References······ ... ...... ... ...... ... ...... ...... ... ...... ... ...... ... ...... ... ...... 325

11

Other Ultrasonic Motors

................................................... 327

11. 1 )Jon-Contact Type Ultrasonic Motors ................................. 327 11. 1. 1

Classification and Development

11. 1. 2

Operating Principle ................................................ 331

...... ... ...... ... ...... ... ...... 328

11. 1. 3

Design of Non-contact USM

11. 1. 1

Performance Measurement of Non-contact USM ............... 336

... ...... ... ...... ... ...... ... ...... 335

11.1. 5

Design of Non-contact Type USM with Disk Stator

11. 1. 6

Testing of :'\fon-contact USM with Disk Stator

11. 2 Linear Surface Acoustic Wave Motor

337

...... ... ...... 338

................................. 310

11. 2. 1

State of the Art ................................................... 340

11. 2. 2

Surface Acoustic Wave and Its Generation

11.2.3

Operation Mechanism ............................................. 316

..................... 343

References······ ... ...... ... ...... ... ...... ...... ... ...... ... ...... ... ...... ... ...... 349

12

Driving Techniques for Ultrasonic Motors .............................. 351

12.1 Design Requirements for Drivers ....................................... 351 12. 2 Signal Generator ............................................................ 353 12.2. 1

RC Multivibrator

................................................ 353

12.2.2

555 Multivibrator

................................................ 351

12. 2. 3

Voltage Controlled Oscillator .................................... 355

12. 3 Frequency Divider and Phase Splitter ................................. 356 12.3. 1

FDPS Composed by Shift Register .............................. 356

12.3.2

FDPS Composed by CPLD ....................................... 358

12.1 Power Amplifier Techniques ............................................. 359 12. 5 Electrical Characteristics of Ultrasonic Motors

..................... 362

12.5. 1

Experimental Results and System Description .................. 362

12.5.2 12.5.3

Analysis of Vibration States and Driving Method ............ 361 Experimental Results ............................................. 364

12. 6 Influence of Matching Circuit on Performance of Driver 367 12.6. 1 Influence of Matching Capacitor ................................. 368

Contents

XX!

12.6.2

Influence of Matching Inductors ................................. 371

12.6.3

Influence of USM on Driver

.................................... 373

12.7 :'\Jon-transformer Driver with Resonance Voltage Step-up ......... 376 References······ ... ...... ... ...... ... ...... ... ...... ... ...... ...... ... ...... ... ...... 383

13

Control Techniques for Ultrasonic Motors

.............................. 385

13. 1 Classification of Control for Ultrasonic Motors ..................... 385 13. 2 Speed Adjusting Mechanism and Control Methods of USM ...... 387 13.2. 1 Voltage Amplitude Adjusting .................................... 387 13.2.2

Frequency Adjusting

13.2.3

Phase Difference Adjusting ....................................... 389

............................................. 388

13. 3 Stability Control Techniques for Ultrasonic Motor 13.3. 1

............... 390

Principle of Frequency Automatic Tracking ..................... 390 .......................................... 392

13.3.2

Detection of Amplitude

13.3.3

Implementation of FAT System ................................. 392

13.4 Ultrasonic Motors Used as Servo Motors 13.1. 1

Ideal Servo Actuator- USM

........................... 393

.................................... 393 .................. 391

13.4.2

Requirements of Servo Control Using USM

13.4.3

Servo Control System Using USM .............................. 395

13.1.1

PID Controller Using USM ....................................... 397

13.4. 5

Adaptive Controller Using USM ................................. 404

13.1. 6 Fuzzy Controller Using USM .................................... 411 References ........................................................................... 117

14

Testing Techniques for Ultrasonic Motors

... ... ...... ... ...... ... ...... 119

14. 1 Modal Testing for Parts and Assemblies .............................. 419 11. 2 Measurements of Pre-pressure .......................................... 123 11. 3 Measurement of Transient Characteristics ........................... 121 14. 3. 1

Testing Principle··· ...... ...... ... ...... ... ...... ... ...... ... ...... 424

11. 3. 2

Transient Characteristics of USMs .............................. 126

14.4 Measurement of Load Characteristics ................................. 427 11. 1. 1

Measurement System for Load Characteristics··· ...... ... ...... 427

11. 1. 2

Measured Results for TRUM-60 Load Characteristics ......... 129

14. 5 Environmental Testing for Ultrasonic Motors

..................... 430

14.5.2

High/Low Temperature Environmental Testing ............... 130 Vacuum Environment Testing .................................... 132

11.5.3

Load Characteristics of USMs in Vibration Environment······ 434

11.5. 1

11.5.1

Load Characteristics of USMs under Strong Shock ............ 137

14. 5. 5

Test and Analysis of :'\roise from Ultrasonic Motors

438

11.5. 6 Testing of USMs in Hygrothermal Environment ............... 110 11. 6 Life Testing for USMs ................................................... 112


XXll

11.6. 1

Design 01 Life Testing System

................................. 112

Life Testing Results and Analysis for TRUM .................. 444 References ........................................................................... 115

14. 6. 2

15

Applications of Ultrasonic Motors in Engineering ..................... 448

15. 1 Applications in Domestic Engineering ................................. 119 15.1. 1

Application in Camera ............................................. 449

15.1.2

Application in Cell Phone

15.1.3

Application in Watch

....................................... 151

............................................. 151

15. 2 Applications in Industrial Engineering ................................. 452 .............................. 152

15.2. 1

Application in Gasoline Generator

15.2.2

Applications in Automobile ....................................... 153

15.2.3

Applications in Robot ............................................. 156

15.2.1

Application in Surveillance Camera PlatIorm

15.2. 5

Applications in Precision Positioning Stage ..................... 159

15. 3 Applications in Biological and Medical Engineering

.................. 458 ............... 461

15.3. 1

Applications in Medical Facility··· .............................. 161

15.3.2

Applications in Biomedical Engineering

15.4 Applications in Aerospace Engineering 15.1. 1

Applications in Aircraft

15.4.2

Applications in Aerospace

........................ 162

.............................. 464

.......................................... 161 ....................................... 466

References ........................................................................... 168

Index ....................................................................................... 469 Appendix A

Natural Vibration Frequencies and Mode Shape Functions of Bars Shafts, Beams, and Plates ... ... ... ... ... ... ... ... ... ... ... 477

Appendix B

Natural Vibration Mode Shapes of Bars, Shafts, and Beams .............................................................................. 179

Appendix C

Natural Vibration Displacement and Strain Mode Shapes of Plates, and Their Nephogram .................................... 187

Symbols The following symbols arc commonly used with the attached definitions. unless otherwise specified in the text. .1':.y.z

Spatial coordinates in a global system

u,v,w

Displacements in

.1':.

U o .Vo .Wo

Ampli tudes in

y. z directions

U o ,Va ,Wo

Displacements of neutral layer

u,v,w

V eloci ties in

u,v,w

Accelerations in

.1':.

.1':.

y. z directions

y. z directions .1':.

y. z directions

Tangential velocity Rotating speed (Speed) Length Width

b

h

Height. thickness

r(r, )

Radius

D(d)

Diameter

m

Mass

Mi

i th modal mass

Ki

i th modal stiffness

Fi

i th modal force

Ci

i th modal darning

S

Area

P E

Density of material

/1

Poisson ratio

G

Shear modulus of elasticity

lUx .Iy .Ie .Ip)

Inertia moment

A k

Wave length

Young's modulus

V clocity of wave propagation Wave number

/1 (/1d •/1, )

Friction coefficient

u(ai • Til )

Stress matrix

(Ci )

F( Fn.F,)

Strain matrix External force

Ultrasonic Motors Technolof{ies and Ap plicalions

XXIV

Friction force

Fr f(x) ,fey) ,fez)

Distributed forces in x, y, z directions

P(P o )

Pre-pressure (Preload)

M(M" Mx,

MT

M~)

Bending moment Torque Temporal variable Voltage sign function of the polarization of phase A Voltage sign function of the polarization of phase B Code of the tooth cell

e

Shape function of annular cell Mass matrix of annular cell /j"

Displacement column matrix of substructure a

M"

Mass matrix of substructure a

K"

Stiffness matrix of substructure a

M'

Mass matrix of tooth e

a'

Displacement column matrix of nodes of tooth e Displacement column matrix of inner nodes of tooth e

r

Condensed matrix of tooth e

K'

Condensed stiffness matrix of tooth e

T

Kinetic energy

V

Potential energy

f

Frequency (Friction force) Angular frequency

W

¢n (x)

,

¢n

nth mode shape

fn (W n ) c;p

nth mode frequency

cp( cpn)

Phase angle

q(t)

Mode shape matrix Modal coordinate Bending mode Mechanical quality factor Force coefficient matrix Shape function matrix Variable for structure design Sensitivity with respect to p; Relative sensitivity with respect to Pi Radial shape function matrix Node column matrix of annular cell Stiffness matrix of annular cell Displacement column of substructure b Mass matrix of substructure b Stiffness matrix of substructure b

Symbols

xxv

K'

Stiffness matrix of tooth e

aj

Displacement column matrix of boundary nodes of tooth e Static condensed matrix of tooth e Condensed mass matrix of tooth e

M'

Dielectric constant matrix under

E=

constant

Lagrange function Variational Work Charge on electrode Generalized coordinate column matrix K,

Generalized stiffness matrix

e,

Radial unit vector Axial unit vector Circumferential unit vector

PCP)

Polariza tion in tensi ty vector

E( Ei )

Electric field intensity vector

D( D i

Electric displacement vector

)

S(5,)

Strain tensor

TCTi)

Stress tensor

(Sij)

S

Flexibility coefficient matrix

c (e i})

Stiffness coefficient matrix

k

Electromechanical coupling coefficient

dedi} )

Piezoelectric constant matrix

im in i, i, i,

Minimum impedance frequency Series resonance frequency

i

p

Parallel resonance frequency

(Ei)

Dielectric constant matrix

Maximum impedance frequency Resonance frequency Anti-resonance frequency

v

Voltage

V pp

Peak-peak value of voltage

Vo VA,VB

Voltage amplitude

I (i. io )

Current matrix

Vol tage of phase A or B

I

Unit matrix

R(R l .Rd .Rm)

Resistance

C(Ca ,Cl .Cm ) L(L .Lm .Lp .L,)

Capacitance

Y

Admittance

j

Inductance

XXVI

Ultrasonic Motors Technolof{ies and Ap plicalions

Z(Zn,)

Impedance (Mechanical impendance)

W(W k )

Work (electric potential energy)

D

Duty cycle

1']

Integral time constant

Til

Differential time constant

K]

Integral coefficient

Kn Kp

Differential coefficient

VI

Isolated electrode voltage

Scale factor

Chapter 1

Introduction Traditional motors based on the electromagnetic principle have been invented and developed for more than one hundred years. As actuators and power sources, the motors have been widely used in many fields all over the world and have made a great contribution to our society. Over the years, the theories, design methods and manufacturing technologies of traditional motors have been developed so successfully that little improvement can be made to them.

However, due to ad-

vanced science and technology, especially in hi-tech products such as spaceships, satellites, launch vehicles, various electronic equipment, and precision instruments, many new requirements for motors have been raised, including a small size, light weight, low noise, no electromagnetic interference, etc. Due to limitations on the principle and structure, traditional motors are difficult to meet these requirements. Many countries in the world strive to explore various new, small, and special motors such as electrostatic motors, ultrasonic motors (USMs) , bionic motors, photo-thermal motors, shape memory alloy motors, microwave motors, etc. As integrated hi-tech products, the new, small, and special motors apply a variety of new technologies, including computers, automatic control, precision machinery, new material and modern manufacturing. They are increasingly becoming indispensable devices not only in the development of aerospace equipment, but also in the achievement of industrial automation, office automation and home automation. The ultrasonic motor is relatively mature among these new, small, and special motors. USMs have been developed as a new concept of motors since the 1980's. It utilizes the vibration of the elastic body (stator) in the ultrasonic frequency band and the reverse piezoelectric effect of piezoelectric materials. The mechanical movement and torque are obtained by means of the frictional contact force between the stator and rotor or slider. USMs can meet many new requirements for small and special motors because of advantages such as small size, light weight, compact structure, fast response, low noise, and no electromagnetic interference. )Jowadays, this kind of motors is being developed very quickly and applied to more and more fields. As an introduction to the book, this chapter summarizes the history, features, classification and applications of ultrasonic motors.

C. Zhao, Ultrasonic Motors © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

2

1.1


History of Ultrasonic Motors L1J

In 1940s, scientists discovered the ceramic material of BaTi0 3

•

which is easy to

be processed and can be made into piezoelectric elements of special shape and can be poled in arbitrary direction. This discovery greatly promoted the development of piezoelectric actuator technology. As early as in 1948, Williams and Brown applied for the first patent of "piezomotor" in history-2-, whose structure is shown in Fig. 1. 1. Their invention reveals the basic tenets and creates a new period of ultrasonic motor. However, ultrasonic motors were not rapidly developed because of the limitations of materials and processing technology at that time. Man's attempt to use vibration of elastic bodies to obtain the power began with the clock and watch industry. In 1960, Swiss Bulova watch Co., Ltd. used the reciprocating displacement of a metal tuning fork to drive watch gears: 3 : , as shown in Fig. 1. 2. This clock's operating frequency is 360Hz. This watch has an error of one minute per month. which created a record at that time. About ten years later, roughly in 1970-1972, Siemens and Matsushita Electrical Industries developed a kind of linear actuator and step motor, which used the key component such as piezoelectric vibrator. Because the piezoelectric vibrator's resonant frequency is as high as tens of thousands hertz and vibration amplitude is very small. the motor can not obtain a large torque or power. Drive coil

Drive coil Vibration direction

Fig. 1. 1

Sta tor of the first ultrasonic motor

Fig. 1. 2 Driving mechanism of tuning fork watch

In 1965. Lavrinenko from the Soviet Union designed a kind of ultrasonic motor, as shown in Fig. 1. 3, which used the vibration of piezoelectric plate to drive the rotor. He was granted an invention patent: 4J and summed up the characteristics of the ultrasonic motor: simple structure, low cost, low speed, high torque, large energy density, high precision, and high energy conversion efficiency. In 1973, Barth in IBM proposed a structural design scheme with the principle of modern ultrasonic motor[3: , as shown in Fig. 1. 4. He used two piezoelectric actuators to produce longitudinal vibration of horns. The contact friction between the rotor surface and horns end drives the rotor. In 1975, Vishnevsky also proposed a design scheme similar to Barth' S[6 J • They used a spring to press the edge

Chapter 1

Introduction

3

of a rectangular piezoelectric composite stator, which excited a longitudinal bra tion modc to drive the rotor, as shown in Fig. 1. 5.

Vl-

Rotor Output shaft

P iezoelectTi c

actuator 2

Piezoelectric

actu ator I

Fig. 1. 3

Lavrinenko's USM

Fig. 1. 4

Barth's design of USM

Excitat ion i b'l131

Fig. 1. 5

Vishnevsky's design of USM

In 1981, Lithuanian Vasiliev successfully developed an ultrasonic motor with the ability of driving larger loads L7J , as shown in Fig. 1. 6. The stator of this motor is a Langevin vibrator, which excites the longitudinal vibration mode of the metal shect contacting with thc rotor that is driven by thc friction force bctwcen the sheet and rotor. This structure of the motor can decrease the operating frequency and amplify thc vibration amplitude at the samc time. Thc usc of such motors to thc wheel of gramophonc becamc the first practical application of piczoelectric actuator in that cra. Aftcr Vasiljev's rescarch findings, Sashida in 1982 dcsigncd and made a standing wavc ultrasonic motor:"], as shown in Fig. 1. 7. This motor uscd a Langcvin vibrator. Its driving frequency was 27. 8kHz, input electric power 90W, output mechanical power SOW, output torque O. 5:'\J-m, output rotational speed 2800 r/ min, and the efficiency 60 %. It can be said that this piezoelectric ultrasonic motor met the performance requirements for actual applications at the first time. Because this motor's metal film and the rotor were fixed at the same position, serious wears existed on the contact surfaces. To solve this problem, in 1983 Sashida designed and manufactured another traveling wave ultrasonic motor, and in 1985 was granted a patent: 9: in USA, as shown in Fig. 1. 8. This motor realizcd the rotor rotation through thc traveling wavc instcad of the standing wavc. The formcr drivcs thc rotor continuously, while the lattcr drives the rotor discontinuously, so thc abrasion on the contact


4

surface is decreased uSing the traveling wave. The successful development of such motors paved the way towards practical applications of ultrasonic motors. In the same year, Sashida put forward two design schemes of traveling wave linear ultrasonic motors(LUSMs) based on the same principle llo -: one is straight beam type, as shown in Fig. 1. 9, and the other is ring beam type, as shown in Fig. 1. 10. Piezoccralllic Metallic

Rotor Beari ng

Flexible sheet

Fig. 1. 6

Vasiliev's structure of USM

Fig. 1. 7

Sashida's standing wave USM

Rotor

,

: ~ Moving direction of rotor

)- - - - ~ ~...-,.-;..- - - - - - - - - - - - -: ;;;>.,.,;-~ L I Mode

Fig. 1. 12 Tomikawa's plate type longitudinal/bending motor

Fig. 1. 13 stator

IT

shape linear motor

Cylindrical-ball type ultrasonic motor with multi degrees of freedom also uses a composite mode of the longitudinal-bending vibration of the cylindrical stator and its operating mechanism will be described in detail in Chap. 7.

3. Based on composite mode of longitudinal-torsional vibration As shown in Fig. 1. 11, Kurosawa in 1991 developed longitudinal-torsional hybrid motor-19-20J. The unique characteristic of this motor was that stacked piezoelectric vibrator produced longitudinal vibration. This vibration could possess larger amplitude in conditions of low-voltage and non-resonance. The rotor's diameter was 50mm and total length was 82mm. The motor's no-load rotational speed measured was 100r/min, the maximum torque was O. 7)J'm, and the maximum efficiency was 33% when pre-pressure of 90N was applied on the rotor, and voltage 31 V,rn, imposed on torsional vibrator. Fig. 1. 15 shows a longitudinaltorsional hybrid ultrasonic motor with a brush developed by PDLab L21 -. Within the motor longitudinal and torsional piezoelectric ceramics were placed in the stator and rotor, respectively. This design could adjust structural parameters of the stator and rotor individually to keep the modal frequencies of the longitudinal and torsional vibrations as close as possible. The rational design of the structure of the motor could increase the pre-pressure on the contact surfaces in the operating process, which could improve output performance and operating efficiency. Because the torsional vibration piezoelectric ceramics were placed in the rotor, the brush was used for supplying electric power to the rotor. The motor was called the brush type longitudinal-torsional hybrid ultrasonic motor. The motor's diameter was 45mm, length was 210mm and maximum output torque was 2. 5N'm

Chapter 1

Introduction

11

when the operating frequency was 25kHz. @ - Nut

Stacked

Shaft

piezoceramic

Torsiona1 vibrator Piezoceramic -i§J~~

Kurosawa's longitudinal-torsional motor

Fig. 1. 14

Brush type longitudinal-torsional motor developed by PDLab

Fig. 1. 15

4. Based on bending vibration According to the structure of stator, ultrasonic motors based on bending vibration mode can be divided into three categories: bar-type, ring-type, and disktype, which all belong to traveling wave ultrasonic motors. In recent years, USMs with bar-type stator based on modes of the out-of-plane bending vibration have became a hotspot in the research area of micro actuator because of their advantages of simple structure, manufacturing convenience, and low cost. Some ultrasonic motors with bar-type stator have been applied to the micro-lens focusing system and medical endoscopy system. In 1988, Kurosawa designed a bar-type traveling wave ultrasonic motor with dual rotors: 22 ] , as shown in Fig. 1. 16. The stator also used Langevin vibrator and piezoelectric ceramics divided into four areas to excite two orthogonal bending modes. This motor's diameter was 20mm, maximum output torque was O. lS)J°m, and no-load maximum speed was 300r/min. This motor was very suitable for automated production, and had been widely applied in the lens focusing system of EOS camera of Canon Co., Ltd. In 1998, Morita used the method of hydrothermal deposition in the metal surface to obtain the piezoelectric thin film and successfully developed a kind of micro high-performance bar-type ultrasonic motor[20 "] shown in Fig. 1. 17.

The

6,um-thick PZT thin films were deposited in the surface of the titanium tube, on the external surface of which four electrodes were formed. The motor started to operate when alternating voltages with a phase difference of 71:/2 were imposed on electrodes and the middle part of the stator was connected to earth. Based on the d 31 effect, two pairs of the piezoelectric components in opposite position excite the corresponding bending modes of metal cylinder. The rotor's diameter was 2. 1mm, length was 10mm, maximum rotational speed was 880r/min under the excitation voltage 15Vpp

,

and the maximum torque was 7. 6,u)Jom. Two-axis me-

chanical arm driven by this motor could achieve step movement and lift an object with weight 109. This simple novel structure of the motor opened a new way for the development of bar-type micro USM['s:.


12

Fig. 1. 16

Kurosawa's bar-type motor

Micro USM based on piezoelectric film

Fig. 1. 17

5. Based on in-plane vibration In-plane vibration has three types: extension-eontraetion, bending, and torsion. In 1989, Takano used in-plane extension-contraction and bending vibration mode for developing an ultrasonic motor, as shown in Fig. 1. 18 L26 -. When alternating voltage signals with phase difference rr/2 were respectively imposed on circular piezoelectric ceramics with two areas, radial and tangential movement of drive point A(A') synthesized the elliptical motion, which drove the rotor. The rotor's diameter was 10mm, thickness was 2mm, operating frequency was 13. 3kHz, maximum output torque was 40mN 'm, and efficiency was 3. 5 %. In 2008, Tieying Zhou developed a linear ultrasonic motor, which used the inplane extension-contraction vibration modes of a hollow cylinder L27 -, as shown in Fig. 1. 19.

Pre-pressure

Fig. 1. 18

Pre-pressure

USM bascd on in-plane modes

Linear USM based on in-plane extension-contraction mode

Fig. 1. 19

In addition, there are some ultrasonic motors based on other modes such as a composite mode of the torsional and bending vibrations[2R:, a composite mode of the longitudinal and shear vibrations L29 -.

1.3

Comparison with Electromagnetic Motors

As mentioned above, ultrasonic motors are new concept motors that can directly drive loads devices and its operating principle is totally different from the electromagnetic motor. In the following sections we will compare the ultrasonic motor with the traditional DC motor in three aspects.

Chapter 1

1. 3. 1

13

Introduction

Load Characteristics

Figure 1. 20 shows the comparison of the load characteristics (measured) of DC motor with USM. It can be seen that when DC motor approaches the no-load speed. its efficiency is maximum and output torque is smaller. On the contrary. the ultrasonic motor's efficiency is maximum in conditions of the lower speed and higher torque. Therefore, USM is suitable for operating at low speed and high torque. and can directly drive loads. PowerfW

Efficiency!"!. Speedl(r/m in)

10

100

8

80

6

60

4

40

2

20

0

0

Maximum efficiency at high speed and low torque / power

20000

15000

10 000

5000

0

0

14

21

28

35

42

49

Out]JlIl torque !(N·m) (a) DC Molor PowerfW

Efficiency!"!. Speedl(r/min)

8.0

100

200

6.4

80

160

4.8

60

120

3.2

40

80

1.6

20

40

0

0

0

Maximum efficiency at low speed and high lorque

0

0.24

0.48

072

0.96

1.20

Output lorque I(N 'm) (b) Ullrasonic motor

Fig. 1. 20

1. 3. 2

Comparison of the measured load characteristics of DC motor with USM's

Energy Transform of Motors and Their Micromation

Electromagnetic motors convert the electromagnetic energy into mechanical energy based on the electromagnetic principle. The magnetic field is produced by the current applied to energize winding on a stator or a permanent magnet stator and there are many coils around the rotor. When the current is applied to drive coils

14


on a stator, the magnetic field drives a rotor to rotate. Therefore, the electromagnetic motor often consists of the stator and rotor, and a gap exists between them, which do not contact each other. Ultrasonic motors use the inverse piezoelectric effect of piezoelectric material to achieve the conversion of electrical energy to mechanical energy, and then ultrasonic vibrations of the stator arc transformed into the macro one-direction movement of the rotor through the friction between the stator and rotor. Except non-contact type ultrasonic motor (Chap. 11), the stator and rotor are in contact. Figure 1. 21 shows the comparison of efficiency vs. size of electromagnetic motors with ultrasonic motors:]:. It can be seen that the efficiency of an electromagnetic motor sharply deelines as the motor's diameter decreases to below 10 mm, while the efficiency of ultrasonic motor smoothly changes. At the same time, electromagnetic motor is more difficult to achieve miniaturization because its rotor must be rounded with coils and its structure is complex. As a contrast, the structure design of ultrasonic motor is more simple. Particularly bar-type traveling wave ultrasonic motor developed since 1990s is very suitable for miniaturization because of its simple structure, machining convenience, and low cost. New Scale Technology, Inc. uses a bar-type ultrasonic motor for the auto-focus system of a mobile phone camera. The motor's cross section is 1. 5mm XL 5mm, operating trip is 30mm, maximum speed is 8mm/ s and maximum output force is 1:'\1. In addition, Konica-Minolta , ]ohnson-:'\Janomotion, and Sam sung Electro Mechanics arc all using ultrasonic micro-motor for the camera lens.

1. 3. 3

Transient Response Characteristics

A comparison of transient response characteristics of the ultrasonic actuators with electromagnetic actuators' in welding system c,,: is shown in Fig. 1. 22. Obviously, the ultrasonic actuator responds much faster than the electromagnetic actuators. From zero speed to the stable speed, it only costs several milliseconds, and the stopping time to zero speed is even shorter. For the ultrasonic and electromagnetic motors , there are the similar rules. See Chap. 11 in detail. 40.---------------------, _... ->l >

40 50

-

The requirements in properties of piezoelectric materials have to be determined according to their specific purposes of the devices. These used in ultra-high-frequency (UHF) and high-frequency devices require the material to have low permittivity and small high-frequency dielectric loss. For energy transducer application, the coupling coefficient and acoustic impedance of the material arc often stressed. Materials with excellent frequency stability and high Qm values can be used as standard frequency oscillators. To satisfy the application of delay line, the materials have to be stable in frequency, and the velocity of sound in the materials should also be considered. Ceramics used in the electro-acoustic field should have a large permittivity, high kp value and high elastic compliance coefficient, and their dielectric loss doesn't matter too much to the devices. For hydroacoustic transducer applications, if used as receivers, it is necessary that the material has a large piezoelectric coefficient of g33 or g31 , large permittivity, high kp value and high compliance constant, but its Qrn value is not seriously required; if

Chapter 2

Fundamentals of Piezoelectricity and Piezoelectric'"

39

used as high-power emitters, it is important that the material has a low dielectric loss tanO' and a high Qrn undcr a rong elcctric field, additionally with a largc diclectric constant, high kp value and large piezoelectric constant. For materials uscd as filtcrs, thcy arc cxpcctcd to bc not only with cxccllcnt duration stability and temperature stability, but also with high Qm and low tanO'; the requirement in kp valuc dcpcnds on thc bandwidth of filtcrs. High-voltagc gcncrators and igniters require the materials to have large values of g33 and k33' a large permittivity, a moderately high Qm' as well as a low tan 0'. :'\Jowadays, by means of doping and substituting, thc propcrtics of piczoelcctric ccramics can bc adjustcd in a wide range to meet diverse application occasions.

2.5. 1

Piezoelectric Ceramics Used for Ultrasonic Motors

1. Piezoelectric ceramics for ultrasonic motors So far, piezoelectric ceramics, instead of piezoelectric single crystals, are mainly thc functional matcrials uscd in ultrasonic motors. Among thcm, Pb(Zrx TilE) 0, (PZT) bascd systcm is thc most important onc duc to its cxtraordinary propcrtics and is currcntly thc first choicc for ultrasonic motors. The preparation of PZT ceramics follows a standard ceramic process ineluding stcps of powdcr prcparing, forming and sintcring, succccdcd by a poling proccss which is requisite for piezoelectric ceramics to get piezoelectricity. Ulltrasonic motors are typical high-power devices, so that the used piezoelectric ccramics should bc with a low dielcctric loss tani) and a high mcchanical quality factor Qm under a strong electric field, as well as a reasonable piezoelectric constant d 33 and an elcctromcchanical coupling factor k p • Unfortunately, for PZT-based piezoelectric ceramics, in most cases the improvement in Qrn simultaneously induces degradations in d" and k p • Up to now, researchers from all over thc world havc conductcd a grcat dcal of fruitful work, which aim cd at cnhancing the piezoelectric constant and electromechanical coupling coefficient but without impairing the mechanical quality factor. Multi-constituent doped PZT ceramics havc bccn dcvelopcd for high powcr piczoelcctric dcviccs, and thc improvcmcnt in properties still continues. Ultrasonic motors now have been used in tremendous occasions to satisfy many applications. For different motors with specific purposes, piezoelectric elements wi th particular properties, dimensions and structures are required. The piezoelcctric elcmcnts thcn havc to bc shapcd into spccific dimcnsions and forms. Fig. 2. 10 (a) shows somc PZT piczoelcctric elcmcnts from industrial companics, and Fig. 2. 10 (b) displays PZT components used for ultrasonic motors designed by our PDLab at :'\JUAA. 2. Stability of Piezoelectric ceramics used in ultrasonic motors[1-2, 11-l2J During thc scrvicc of ultrasonic motors, thc uscd piczoelcctric matcrials havc bccn found fluctuant in thcir physical propcrtics with elapsc of durations and fluctuation of tcmpcraturcs. Somctimcs thc variation in propcrtics can bc pro-

40


(a) PiezoeleCTric ceramic elemCIlIs(Xinchang Silver River Electronic Co. Lid in China)

Fig. 2. 10

(b) PZT components lISed in ultrasonic molors developed by PDLab

PZT piezoelectric ceramic components

nounced enough to cause failure of the whole devices. Therefore, the stability of piczoelectric ccramics is utterly significant for thcir applications. (1) Aging stability Stabilities of piezoelectric ceramics with duration elapse and temperature fluctuation are called aging stability and tcmperaturc stability, rcspectively. Variations of physical propertics induccd by aging effcct will accumulatc in thc polarizcd ccramics at a gradually slowing-down ratc. Thc accumulation is irrevcrsible unlcss thc ccramics cxperience a ncw cxcitation such as a rcpolarization. Generally, as a result of aging, samples present decreases in dielectric constant, dielectric loss, piezoelcctric coefficicnt and elastic compliance and increases in mechanical quality factor and frequcncy factor. It is also found that such changes are roughly proportional to the logarithm of the duration. The aging curvc of the pol cd piezoelcctric ccramics may bc interfercd by environmental factors. To deal with possible disturbances, a prior artificial aging treatment is usually employed to stabilize the poled ceramics so that the ceramics won't fluctuate in properties with the environmental interferences. In practical manufactures, the poled piezoelectric ceramics are aged by heat treatment or heat cyeles. The prior heat treatment can help the ceramics to stabilize from other heat excitements. This stabilization is attributed to the mechanism that domain motions of the ceramic are enhanced and its internal stress is largely released during the artificial aging processing. Other artificial aging methods such as elcctric aging, mechanical aging and exposure to y-ray radiation of Co", can also achicvc similar cffects. (2) Thermal depolarization Thermal depolarization will occur during heating the piezoelectric ceramIcs. Dipolcs get in disorder gradually with the tempcraturc elevating, dcteriorating the piezoelectric performance simultaneously. Once the temperature reaches abovc the Curic point, whcre thc piczoelectricity disappcars thoroughly, the ultrasonic motor is irreversiblc dcstroyed. Consequcntly, it is ncccssary that thc divices operatc at tcmpcraturcs far below the Curie point. Thc tcmpcraturc limitation whcrc piezoelectric ccramics can safely work without rcmarkablc reduction in piczoelectric activity is approximately set at half of thc Curie point.

Chapter 2

Fundamentals of Piezoelectricity and Piezoelectric···

41

(3) Electric depolarization Electric dcpolarization happcns if a strong invcrse electric field is applicd on a polcd ccramic. Whcther the elcctric field will cause scrious dcpolarization in thc material depends on the material itself and the applying duration of the electric field. as well as the temperature that the material stays. For a direct current electric field. thc strcngth thrcshold causing dcpolarization is around 5001 OOOv Imm. Thc readers have also to notice that thc material can bc depolarizcd during the other half period when applying an altering current field to drive ultrasonic motors. (4) Mcchanical dcpolarization Excessive mechanical strcss can disarrange dipolcs in piczoelectric ceramIcs. leading to a failure in piczoelectric pcrformancc. This process is referrcd to as mechanical depolarization. Piczoelectric ceramics with differcnt compositions may allow different limitations of the safe mechanical stress. Reliable data and information have to be referred for reasonable application of materials.

2. 5. 2

Applications of Piezoelectric Materials to Other Actuators

Besides ultrasonic motors. other piezoelectric actuators based on different forms of piezoelectric materials havc also been widely applied in many fields. Considcring their application importancc and their distinct opcrating mcchanisms in contrast with ultrasonic motors, we will then launch a brief introduction for these piezoelectric actuators.

1. Piezoelectric stack actuators In conventional piezoelectric actuators. the displacements of single layer piezoelcctric actuators are found to bc too small to fulfilllargc strokc driving. Thc idca thcn comes naturally to stack several piezoelcctric ceramic pieccs together to form a piezoelectric stack actuator. As shown in Fig. 2. 11, the piezoelectric stack actuators are fabricated by agglutinating piezoelectric ceramic pieces in serics. Thcse picces are electrically parallel but mcchanically scrial. When a voltage IS applicd along the poling direction. each singlc picce produces a displacement, and all displacements sum up to the total output of the stack actuator.

Fig. 2. 11

Piezoelectric stack actuators [PI (Physik Instrument) L. P. ]

Current concerns for piczoelectric stack actuators are mainly on to compensatc thc hysteresis characteristic of the dcviccs. This bchavior refcrring to thc nonlin-

42


ear hysteresis between the input voltages and the output displacements, is an intrinsic trait of piezoelectric materials. Therefore, some compensation methods have to be adopted to improve the positioning precision of the actuators.

2. Piezoelectric bimorph actuators Piezoelectric bimorph was firstly invented by Baldwin Sawyer in 1931. Now it has been frequently used in piezoelectric elements for acoustic detections, USMs, laser beam deflectors, filters, accelerometers, optical choppers, etc l3J • There are four structures commonly used in piezoelectric bimorphs, whose schematic diagrams arc presented in Fig. 2.12. In diagrams (a) and (b) two identical piezoelectric plates arc bonded to each other, with their poling directions oppositely arranged. Electrodes are coated on both sides of the bimorph. These two structures arc therefore called antiparallel-type piezoelectric bimorphs or continuous-type piezoelectric bimorphs. The bimorph in Fig. 2. 12 (c) contains an extra electrode between two plates, and both plates are poled along the direction of the driving voltage. Contrastively, this structure is named as parallel-type piezoelectric bimorph. The actuator in Fig. 2. 12 (d) consists of a non-piezoelectric plate and a piezoelectric vibrator coated with electrodes.

{jJ U ~

(a)

f~

(e)

Fig. 2. 12

{jJ f[

t~ (b)

ff' (d)

Structures of piezoelectric bimorphs

When applied with an electric field, owing to the converse piezoelectric effect, the bimorphs in (a), (b) and (c) start to bend since one vibrator inside the structures extends and the other contracts. The working voltage of the bimorph in (c) is twice as large as those of the bimorphs in (a) and (b), so the bending deforma tion of the bimorph in (c) is also twice as large as those of the bimorphs in (a) and (b). The behavior of the fourth one in (d) is similar with that of the former three ones, but its motion and output is tunable by varying thickness ratios between the piezoelectric plate and the non-piezoelectric one or by changing the elasticity modulus of the non-piezoelectric material.

3. Functionally graded piezoelectric actuators For functionally graded piezoelectric materials, the composItions and structures are controlled to change gradually from one side to the other side. Correspondingly, the properties and functions of the materials change gradually too. A structural comparison between the bimorph and the functionally graded device is shown in Fig. 2. 13. In bimorph structure, the sharp interface is easy to concen-

Chapter 2


43

trate stress and induce interface cracks. However, these disadvantages can be effectively avoided in the gradient interface of the functionally graded material. + v --~

________________________~

-v --~

________________________~ (a) Piezoelectric bimorph

(b) Functionally graded piezoelectric material

Fig. 2. 13 A comparison between the structures of piezoelectric bimorph and functionally graded piezoelectric material

Figure 2. 14 shows the working mechanisms of a multi-layer functionally graded piezoelectric actuator[14:. Four layers with different compositions and properties are sintered together to form a structure without evident interfaces. The piezoelectric coefficients and dielectric constants vary from bottom to top in a positive and negative gradient. respectively. Under a DC voltage in thickness direction. the layers of smaller dielectric constants will have larger electric-field intensity distributions. As a result, each layer deforms in the way as the Fig. 2. l1(b) shows. The deformations then integrate into a uniform deformation in the whole device. as shown in Fig. 2. 14(c). Apparently, the internal stresses are greatly depressed in the device owing to the structural gradient. Furthermore, higher mechanical strength can be obtained for the absence of adhesive between the interfaces.

(a) Schematic structllre ofa functionally graded piezoelectric actuator with four piezoelectric layers

(b) The defomlation of each layer under applied voltage

(c) Total defomlation of the fu nctionally graded piezoelectric aCllmtor

Fig. 2. 14

Deformation mechanism of the functionally graded piezoelectric actuator

4. Piezoelectric fiber actuators The brittleness of piezoelectric ceramics restrains their applications in nonplanar devices. For this reason, active fiber composites (AFe) and macro fiber compos-


44

ites (MFC) were designed by American scientists from MIT and NASA in the 1990' s. These composites arc generally called piezoelectric fiber compositesL 15J . In AFC structures, arrays of piezoelectric fibers with round cross sections arc embedded in epoxy resin matrix. Interdigital electrodes are arranged perpendicularly to the axial direction of the fibers. The fabrication of MFC is analogical to that of AFC except that the piezoelectric fibers inside arc with rectangular cross sections. Large stains can be obtained in AFC and MFC by utilizing their axial d" piezoelectric characteristics. Comparing with piezoelectric ceramics, piezoelectric fiber composites possess better flexibility so that they can satisfy the applications in bending planes(Fig. 2.15).

Piezoelectric active fiber composites (left) and macro fiber composites (right)

Fig. 2. 15

Recently. metal core piezoelectric fibers (MPF) are developed to fabricate new piezoelectric sensors and actuatorS[16 17J. A typical structure of MPF is shown in Fig. 2. 16, where the piezoelectric fiber with diameter O. 15-0. 25mm is coated with a layer of metal electrode on the surface. In the center of the piezoelectric fiber, a metal core of O. 05mm in diameter, usually platinum. acts as another electrode. This metal core can also act as a medium that enhances the strength of the fiber. Metal COre Piezoelectric cCl'1lmic

T/

..... -

I

I

I

1 - .....

J-_I_

~-J

'1

~

Fig. 2. 16

Piezoelectric ceramic fiber with a metal core inside

We refer the fiber to as the full-electrode piezoelectric fiber if its surface is entirely coated with metal electrode. The piezoelectric fiber is polarized along the radial direction, so that the fiber will produce a radial extension vibration type under an applied electric field. A half-plated fiber is called the half-electrode piezoelectric fiber. Bending distortions will be produced in the half-electrode piezoelectric fiber after applying electric field. These tiny dimensional MPF can be con-

Chapter 2


45

veniently embedded into composites to drive deformations in the composites.

2.6

Advances in Novel Piezoelectric Materials

With the fast-growing application of ultrasonic motors and other piezoelectric devices, the demand for new piezoelectric materials is getting more and more imperative. Here we present some progress in the research on piezoelectric materials that are promising for future potential applications.

1. Multi-constituent doped PbCZr x Ti 1- x )()3CPZT) ceramics[1820: Multi-constituent doped PZT ceramics have been intensively investigated for long duration since its electrical and mechanical properties of the ceramics can be greatly varied by doping PZT with acceptors, donors, or isovalent dopant. It was verified that doping of Mn' I in PZT could improve the mechanical quality factor Qm; and its electromechanical coupling coefficient k" could be improved by Sb 5 - doping. So that PZT based ceramics doped with both Mn'- and Sb'- have been constantly attended. By these efforts some trinary and quaternary PZTbased materials with excellent sintcring property and repeatability have been found. Materials applied for high-power piezoelectric devices have been reported based on these systems. For example, in Mn'· doped quaternary PS:'\J-PZ:'\J-PZT system, experiments reveal that pure perovskitc phase can be formed within a wide range of Mn' I additives. Proper amounts of Mn'· additives can optimize the piezoelectric properties of the PS:'\J-PZ:'\J-PZT quaternary system for improving the mechanical quality factor Qm and decreasing the dielectric loss tanB. The ceramics with o. 5 % (mass fraction) dopant possess the best electromechanical properties that satisfy the requirement for USM and transformer applications. Recently a new series of quaternary PZT-based ceramics doped with Ba" and Sr'· have been developed with properties greatly improved. Among them, d 33 = 406pC/:'\J, kp

=

o. 55,

E=

2183, Qm = 1077, and tani) = 2. 7 %, respectively.

2. Relaxor ferroelectric single crystals[5] Relaxor ferroelcctrics such as lead magnesium niobate-lcad titanate (PM:'\J-PT) and lead zinc niobate-lead titanate (PZ:'\J-PT) single crystals exhibit much higher piezoelectric coefficients and electromechanical coupling coefficients than conventional piezoelectric ceramics- 5-. For example, the strain of these crystals reaches to 1. 7 %, almost an order larger in magnitude of the conventional piezoelectric ceramics. Furthermore, it has been verified that even temperature down to -200'C, the property of PMN-PT and PZN-PT single crystals are still comparable with the room-temperature property of PZT ceramics['l 24:. This nature enables PMN-PT and PZN-PT suitable for USM running at extreme-temperature conditions. Relaxor ferroelectric single crystals are promising to replace conventional piezoelectric ceramics in many devices, such as acoustic detectors, ultrasonic imaging devices, high-strain actuator, and ultrasonic motors used in extreme circumstances- 25 - 31J •

3. Lead- free piezoelectric materials[35] The lead-containing piezoelectric ceramics may cause serious hazard to the envi-

46


ronment and human health during their manufacturing, serving and disposing after failure. Therefore the development of environment-friendly lead-free piezoelectric ceramics is indispensable from the perspective of the global environment protection. Recent research on lead-free piezoelectric materials has been focused mainly on two promising systems: perovskite structural piezoelectric ceramics and bismuthlayered structural piezoelectric ceramics. The former family ineludes the solid solutions of Ko 5 :'\lao. 5 :'\Ib03 - LiTa0 3 , BaTi03 - Bio 5 Ko. 5 Ti0 3 , and Bio. 5 Nao. 5 Ti0 3 Bio. 5 Ko. 5 TiO, , with their compositions near the morpho tropic phase boundary. In optimized compositions the piezoelectric constant d 33 of these systems can reach 300pC/:'\I, which is elose to the value of PZT ceramics. The bismuth-layered structural solid solutions such as the donor-doped Bi4 Ti3 0 12 or Bi, TiTaO g systems are featured with high Curie point and relatively large piezoelectric coefficient, as well as the less temperature dependence of their resonant frequencies. These traits make them suitable for sensor and resonator applications.

4. Piezoelectric composites[6, 18: Piezoelectric composites have been developed since the late 70s of last century. The preparation of piezoelectric composites involves to incorporate piezoelectric ceramics and piezoelectric polymers with designed connectivity, mass/volume ratios and spatial distributions to form certain microstructures. Piezoelectric ceramics have disadvantages such as high density, extreme brittleness, easy fracture to mechanical impacts and poor capability to form complex shapes. On the other hand, piezoelectric polymers possess properties of excellent flexibility, low density and great machinability but poor temperature endurance and high cost. However, the properties of piezoelectric composites can be remarkable improved by taking advantage of the composition effect elaborately, so that piezoelectric composites keep the merits of both components of piezoelectric ceramic and polymer and overcome their disadvantages, offering excellent piezoelectric performance and mechanical flexibility. The manner of each phase connects with itself in composites is known as the "connectivity" of the composites, which is proposed by Newnham et al. in 1978. Fig. 2. 17 lists all ten types of connectivity of piezoelectric composites. These connectivity types are: 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, and 3-3. The first number in the expression represents the connecting dimension of the piezoelectric phase and the second number is the connecting dimension of the polymer phase. Different connectivity types mean different spatial distributions of the ceramic phase and the polymer phase and correspondingly different dielectric and piezoelectric properties in the composites.

5. Piezoelectric thin jilms[36 37J The progress in thin film deposition methods has provided the possibility of application thin films in almost all fields of modern science and technology. Now bunch of techniques have been employed to prepare piezoelectric materials from high-quality epitaxial films to large-area polycrystalline films. Among them,

Chapter 2


---__ ---0-0

0- I

0-2

0-3

I- I

1-2

2 -2

2 -3

1-3

Fig. 2. 17

47

3-3 (Two views)

Ten connectivity types of piezoelectric composites

sputter deposition, sol-gel, chemical vapor deposition (CVD), molecular-beam epitaxy (MBE) and pulsed laser deposition (PLD) are well established for piezoelectric films preparation. Piezoelectric films could have many important applications due to their versatile properties. For example, in surface acoustic wave ( SAW) devices. piezoelectric films have been widely used as the functional parts. The combination of micro sensors and actuators onto the surfaces of semiconductor integrated circuits creates a new research highlight of piezoelectric film micro mechanic-electric systems. Devices based on bulk piezoelectric materials usually operate with operating frequencies no more than hundreds hertz due to their dimension restrictions. On the other hand. devices based on piezoelectric films offer much higher operating frequency, extra flexibility in designing and shaping the device dimensions, as well as additional advantage in device miniaturization and integration. In various applications, piezoelectric films can replace their single crystal or ceramic counterparts, to provide similar functions with considerable satisfaction.

References [ 1] [2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9]

Zhiwcn Yin. Physics of Dielectrics (Second Edition). Bcijing: Prcss, 2005:1-8. (in Chincsc) B Jaffe, W R Cook, H Jaffe. Piezoelectric Ceramics. "few York: Academic Press, 1971: 1-5. Duan Fcng, Changxu Shi, Zhiguo Liu. Introduction to Material Science-An Integrated Approach. Beijing: Chemical Industry Press, 2002:324-350. (in Chinese) B Jaffe, R S Roth, S Marzullo. Piezoelectric properties of lead zirconate-Iead titanate solid solution ccramies. J. Appl. Phys., 1951,25: 809-100. R F Service. Shape-changing crystals get shifter. Science, 1997, 275: 1878. Changxu Shi, Hengde Li, Lian Zhou. Handbook of Materials Science and Engineering. Beijing: Chcmieal Industry Prcss, 2006: 7-76. (in Chincse) Shenghe Lin, Zhibi Ye, Yubin Wang. Piezoelectric Ceramics. Beijing: Defense Industry Press, 1980: 17-40. (in Chinese) Statc burcau of technical supcrvision. National Standards of the People's Republic of China CElT 3389. 1-1996. Bcijing: Standards Prcss of China, 1997: 2-3. Daorcn Song, Mingshan Xiao. Piezoelectricity and Its Application. Bcijing: Popular Sciencc


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[IOJ [l1J [I2J [13J [I1J [I5J [16J

[17J [I8J

[I9J

[20J [21J [22J [23J [21J [25J [26J [27J

[28J [29J [30J

Press, 1980:19-38. (in Chinese) Yuan Li, Zikai Qin, Zhigang Zhou. Measurement for Piezoelectric and Ferroelectric Materials. Beijing: Science Press, 2001:19-21. (in Chinese) Yuhuan Xu. Ferroelectric and Piezoelectric Materials. Beijing: Science Press, 1978: 118202. (in Chinese) Weilie Zhong. Physics of Ferroelectrics. Beijing: Science Press, 1996: 310-311. (in Chinese) J G Smits, S I Dalke, T K Cooney. The constituent equations of piezoelectric bimorphs. Sensors and Actuators A, 1991, 28: 41-61. J Qiu, J Tani, Ueno, et al. Fabrication and high durability of functionally graded piezoelectric bending actuators. Journal of Smart Materials and Structures, 2003, 12: 115-12l. R B Williams, G Park, D J Inman. An overview of composite actuators with piezoelectric fibers. Proc. of SP IE- The International Society of Optical Structures, 2002, 1753: 121-127. J Qiu, N Yamada, J Tani, et al. Fabrication of piezoelectric fibers with metal core. Pmc. of SP IE's 10th International Symposium on Smart Structures and Materials. San Diego, CA. , Active Materials: Behavior and Mechanics. DC Lagoudas, Ed., 2003, 5053: 175-183. G Sebald, J H Qiu, D Guyomar. Modeling the lateral resonance mode of piezoelectric fibers with metal core. Journal of Physics D, 2005, 38: 3733-3710. Qian Li, Ying Yang, Dandan Wan, et al. Microstructural characteristics and electrical properties of x Pb(Mg 1n Ta'/3)O,-(0. 1-.T)Pb(Mnl/3Sb2/')O,-0. 9Pb(Zr0.5zTio.48)03 high power piezoelectric ceramics. Materials Science and Engineering B, 2009, 163: 115-150. (in Chinese) J Ryu, D SPark, D Y Jeong. Effect of LazO, doping on the piezoelectric properties of PbZr03-PbTi03-Pb(Zn1l3 :'-Ib2/3) 03-Pb(Snl/3 Nbz/3) 03-yMn03 ceramics for high-power applications. Journal of Ceramic Processing Research, 2009, 10: 386-390. (in Chinese) G H Hacrtling. Ferroelectric ceramics: history and technology. Journal of the American Ceramic Society, 1999, 82: 797-818. (in Chinese) Fuxue Zhang, Likun Wang. Modern Piezoelectricity (Volume 1, Second Edition). Beijing: Science Press, 2003: 97-98. (in Chinese) J Van Randeraat, R B Setterington. Piezoelectric Ceramics. Mullard Limited, 1971: 15-16. S E Park, T R Shrout. Ultrahigh strain and piezoelectric behavior in relaxor based ferroelectric single crystals. Journal of Applied Physics, 1997, 82: 1801-1811. G Roger, D Busch. A survey of micro-actuator technologies for future spacecraft missions. [2009-05-26]. http://www.robotstore.eom/support. asp. S E Park, W Hackenberger. High performance single crystal piezoeleetries: applications and issues. Current Opinion in Solid State &. Materials Science, 2002, 6: 11-18. M Levy, S Ghimire, A K Bandyopadhyay, et al. PZ:'-I-PT single-crystal thin film monomorph actuator. Ferroelectrics Letters Section, 2002, 29 (3-4): 29-40. K S Moon, M Levy, Y K Hong, et al. Axial displacement measurement of a single-crystal actuator using phase-shift interferometry. IEEE Transactions on Industrial Electronics, 2005, 52 (4): 953-959. M Yang, M Zhu, C Robert, et al. Design and evaluation of linear ultrasonic motors for a cardiac compression assist device. Sensors and Actuators A, 2005, 119: 214-220. S Dong, L Yan, :'-I Wang. A small, linear, piezoelectric ultrasonic eryomotor. Applied Physics Letters, 86: 200505350l. Z Y Feng, T H He, H Q Xu, et al. High eleetrie-field-indueed strain of Pb(Mg 1/ 3Nbz/3)O,PbTi0 3 crystals in multilayer actuators. Solid State Communications, 2001, 130 (8): 557-562.

[31J [32J

S E Park, T R Shrout. Relaxor based ferroelectric single crystals for electro-mechanical actuators. Materials Research Innovations, 1997, 1 (1): 20-25. S Genti, D Damjanovie, :'-I Setter. Pb(Mg 1/3Nbz/3)O, and (1-x) Pb(Mg1n:'-lb'n)O,-.T PbTi03 relaxor ferroelectric thick films: processing and electrical characterization. Journal of Electroceramics, 2004,12 (3): 151-16l.

[:l3J

V Y Topolov. Orientation relationships between electromechanical properties of monoelinic

Chapter 2


O. 91Pb (Znl!' Nbz/,) 0,-0. 09PbTi0 3 single crystals.

[31J

[35J [36J [37J

49

Sensors and Actuators A-Physical.

2005.121 (1):118-155. S C Woody. S T Smith. X N Jiang, ct al. Pcrformancc of single-crystal Pb(Mgl/ 3 Nb z/3 ()3)32%PbTi0 3 stacked actuators with application to adaptive structures. Review of Scientific Instruments, 2005, 76 (7): 075112(1-8). T Takenaka, H :'-Iagata. Current status and prospects of lead-free piezoelectric ccramics, J. Euro. Ceramic. Society, 2005, 25: 2693-2700. Zhiwen Yin. Physics of Dielectrics (Second Edition). Beijing: Science Press, 2005: 778-831. C P Araujo, J F Scott, G W Taylor. Ferroelectric Thin Films: Synthesis and Basic Properties. Amsterdam: Gordon and Breach Scicnce Publishcrs, 1996: 1-8.

Chapter 3

Fundamentals of Tribology and Tribomaterials for Ultrasonic Motors Tribology is defined as "the sCIence and technology of phenomena occurnng at the contact interface between objects", and its main topics are friction and wear of materials. Ultrasonic motor is a new tribological actuator, which uses the friction at a contact area between a stator and rotor to convert the ultrasonic vibration of the stator into the linear or rotational motion of the rotor. It is evident that the ultrasonic motor with friction drive possesses features such as self-lock without power and a high self-lock torque. As far as the locking property of USM is concerned, the self-lock torque is higher than its stall torque, while the rotor's inertia is low. This indicates that the ultrasonic motor has rapid (millisecond scale) response property. Because USM transmits the power via friction at the contact area between the stator and rotor, stable and relatively high friction force at the contact interface is required. Since sliding wear between the rotor and stator is inevitable,

high wear resistance of tribomaterials in USMs (stator and rotor) is then essential to maintain precision control, because their wear causes changes in the contact condition between the stator and rotor and leads to a decrease in control accuracy. Generally, friction characteristics include output power property of friction surfaces, the microstructure of wear surface and the tribological property at contact surfaces, whereas the phys-chemical properties of tribomaterials include an elastic modulus, wear resistance, anisotropy, dependence to environments, etc.: 1 RJ Therefore, how to match the friction and wear characteristics between stator and rotor pairs is a key to guaranteeing the performance of USMs. Currently, the tribological behaviors of tribosystems without vibrations have been investigated. If the vertical and tangential high-frequency vibration components along .1':, y, and z axial directions are superimposed on the tribosystem, the friction and wear behaviors become sophisticated. Thus, it is imperative to study the friction and wear behaviors of tribomaterials under ultrasonic vibration. To improve the reliability and stability of USM, an advanced functional surface technology and nanotechnology must be used to adjust and enhance the tribological properties between the stator and rotor in adverse circumstances. Furthermore, it is important to devclop experimental methods for estimating tribomaterial life and efficiency because the life and the efficiency of USM largely depend on the tribological properties of the stator and rotor pairs. It is obvious


Chapter 3

Fundamentals of Tribology and Tribomaterials···

51

that friction interfaces for different ultrasonic motors exhibit various action fashions. However. due to the rotor or mover being impelled to move according to the elliptical motion at the contact points between the stator and rotor. the influences of the different action fashions on the stator and rotor are identical[9-15]. Therefore. it is of vital importance to analyze the tribological effect of the stator and rotor pair if we want to prolong the life of USM. In this chapter. the basic tribology will be first introduced. and then the preparation of tribomaterials in USM will be addressed. The influence of the phys-ehemieal properties of the tribomaterials on tribologic characteristics will be analyzed. After that. the experimental methods to determine the tribomaterial performance will be discussed.

3. 1 3. 1. 1

Basic Tribology Surface of Tribomaterials

In a viewpoint of tribology. the friction between two sliding surfaces is largely governed by physical conditions and chemical interactions between sliding surfaces and environments. Physically. rough surfaces could create higher friction coefficient. while chemical interactions between two sliding surfaces and the environment can play an important role in the friction and wear behavior of tribocouples in different environments. Usually. the practical solid surfaces arc formed by cutting. grinding and polishing. These surfaces look smooth and arc sometimes called mirrors. But even a mirror surface is still rough microscopically. Since the solid surface has microasperities. the surface roughness is estimated by peaks and valleys with various amplitude and space. These microasperities with long wavelengths on the surface arc formed owing to the vibration of the work piece or tools during processing. The micro asperities arc distributed directionally or isotropically on the surface depending on processing methods. When the solid surfaces are processed via turning. milling and planning. the asperities arc distributed directionally. whereas the asperities arc distributed isotropic ally or equiprobably as the surfaces arc processed via electropolishing and lapping. The surface roughness is determined by R, (centerline average roughness). Ry (average roughness at ten points) andR,(maximum altitude roughness). respectively. Normally. R, is more often used to show the roughness value of surfaces. For the stator and the rotor in USM. their surface roughness R, is lower than 0. 51l-m. In order to measure the roughness value for such solid surface. a surface profilometer is popularly used. and the cross-sectional profile of the surface is measured as shown in Fig. 3. 1. Furthermore. the surface microstructure of metal will be changed during cutting process. The surface of metals usually consists of several layers which arc formed during their machining processes[16] • as shown in Fig. 3. 2. From the metal matrix. the first layer is the deformation layer(also known as the strain hardening layer) with about l0ll-m thickness. and then the Bielby layer is formed on the deformation layer due to the surface melting. flowing and quenching. Thus. the microstructure of Bielby layer is amorphous or microcrystal. The oxide layer is formed by the chemical reaction between metal

52


surface and oxygen in air. The outside layer is the absorption layer or contamination layer formed by the absorption of gas or liquid polar molecule on the solid surface in environments. It is obvious that the surface roughness and microstructure of tribomaterials have more influenees on the tribologieal properties of tribomaterials.

o

o

Dusl panicle wilh dia meter of I ~m Oxide laye r

-I

0. I

0.2 0 .3 0.4 0.5

0.6 0 .7

0 .8

x/ mm

Fig. 3. 1

3. 1. 2

Surfacc morphology of a stator

Fig. 3. 2

Typical surface of metal

Friction and Its Classification

The importance of friction may be seen in daily life. To decrease energy consumption in overcoming friction during sliding, the reduction of friction is extremcly important in modern technologies. However, when people walk or car moves. they need sufficiently high friction to push their body to move forward. Thus, it is imperative to control friction in our modern life. If friction effect exits on the contact surface between two sliding objects. which are called as a tribopair. Fig. 3. 3 shows a tribosystem constituted by objects A and B. It is clear that the object A is pressed firmly on the object B at a normal load of P. When object A is pushed by an external force F. a rclative motion or motion trend occurs at the contact surface between A and B. At this moment, A phenomenon to impede the rclative motion appears at the contact area.

This phenomenon is

called as friction. The force impeding the object motion at the contact surface is called as friction force, marked as Fr. The magnitude of Fr is related to the normal load. the contact surface status and the tribopairs.

Fig. 3. 3

Tribosystem

The following Coulomb's friction law:]6] is the most classical equation to

Chapter 3


53

describe the above tribosystem (3.1)

where /1 is a friction coefficient. In the viewpoint of the macro-motion state, frictions can be classified into static friction and dynamic friction. As seen in Fig. 3. 3, thc two objects A and B arc initially kcpt in relativcly stationary statc at thc normal prcssure P. When thc extcrnal force F is zcro, therc is no friction between A and B. When F increases gradually in a certain rang, the relative micro-motion will occur between A and B. In this case, the friction force Fr is callcd as a static friction force, which corresponds to a static friction cocfficicnt /1,. When F is higher than the maximal friction forcc, the rela ti ve macro-movc bctwccn A and B will occur. Thcn this friction force is called as a dynamic friction force, which corrcspond to a dynamic friction cocfficicnt /1d. According to the relative motion of the tribopair, the friction is classified as sliding friction, rolling friction, rolling and sliding friction: (1) Sliding friction is thc friction state that thc relativc spccd on thc contact surfacc of the tribopairs are not cqual to zcro. (2) Rolling friction is thc friction statc that the relativc specd at somc actual contact points are zero. (3) Rolling and sliding friction is the friction state that combines sliding friction and rolling friction. According to thc friction states of thc contact surface of tribopairs, thc friction can be classified into dry, boundary, fluid and mixed frictions: (1) Dry friction is the friction condition that there is no lubricant between two objccts; usually the object surfacc absorbs gas, aqucous vapors and so on. (2) Boundary friction is thc friction condition that thc cxtremely thin film of the lubrication oil separates the sliding surfaces of two objects. (3) Fluid friction is the friction condition of which the surface of two objects is completely separated by fluid film and the friction characteristics is decided by the fluid viscosity. (1) Mixed friction is the mixed conditions including dry friction, boundary friction and fluid friction.

3. 1. 3

Friction Mechanism

To cxplain thc occurrcncc of friction, Amontons put forward the mechanical junction thcory in 1699, while Tomlinson and Hardy put forward the molecular attraction theory in 1929 and 1936, respectively. Since 1938, Bowden had done detailed works about tribology. For example, Bowden and Tabor have distinguished the large difference bctwcen cffectivc contact arca and true contact arca in 1942. Finally, the adhcsion thcory had becn propos cd by Bowdcn in 1950 to elucidate the friction mechanism. Now, the adhesion theory and its development are reviewed here- 16 -.

1. Simple adhesion theory Even in the case of a mirror surface, the surface displays rough asperities

54


microscopically. When one solid contacts with the other, the surface mlcroasperisties will weld together due to the adhesive between a tribopair. When one of the solids moves on the other, the microweld junctions will break. So the entire friction process changes from the adhesion of asperities to the shear fracture of junctions alternatively. At the normal load pressure of p, the contact stresses at some contact asperities are so high that the plastic deformation occurs at contact zone, and then the contact area increases to the all area bearing load P. Suppose the yield strength of an ideal elastic-plastic material is rr, and the true contact area between two solids is S" then it has P=rr,S,. Once an object slides relatively against the other, the adhesive junctions arc fractured via shearing. This indicates that the shear force of the junctions is the main part of the friction force F. So the friction Fr force can be expressed as FI

S,T;,

=

=

(Plrr,)r;,

(3. 2)

and the dynamic friction coefficient (friction coefficient) is calculated as fl.d

=

FrlP

=

T:/rr,

(3. 3)

where T;' is the shear strength of the adhesive junction. It is obvious from Eq. (3. 3) that the friction coefficient is independent of the effective contact area, but is directly proportional to the shear strength of adhesive junction and inversely proportional to the yield strength of the ideal elastic-plastic material. If we consider the strain-hardening effect of tribomaterials, the shear strength Tb of the softer material is used to replace T;, in Eq. (3.3). Then (3. 1)

according to Eq. (3. 4), the friction coefficients of o. 2 for most metals arc acquired, but experimental results are mostly between O. 2-0. 5. This difference indicates that the simple adhesion theory has to be modified L16_

2. Modified adhesion theory In the situation of static friction, the true contact area S, is directly proportional to the normal load P. When the two objects in the tribosystem slide relatively, the true contact area S, will increase. Assuming the adhesive junction's yield is related to the composite stress of the compressive stress given by the normal load P and the shear stress given by tangential force, as shown in Fig. 3. 1, an empIrical equation for the modified model is introduced as (3. 5)

where a is an experimental constant. When a natural contamination film is formed on the contact surface and its shear strength is TI' there is (3. 6)

where Tb is the shear strength of the softer material in tribosystem. f3 is less than 1. When the ratio of Fr I S, is lower than Tr' with P increasing the contact area increases, while the contact area stops increasing as the ratio of FilS, equals to Tr. If the adhesive junction is sheared, the tribopairs start to slide over each other. If T in Eq. (3. 5) is replaced by Tr' the sliding criterion of the tribopair can be

Chapter 3


55

s,

Fig. 3. 4

Compressive stress and shear stress at junction

expressed as (3.7) it is known from Eq. (3. 7) that the number of adhesive junctions will increase if F is very high. Then aT~ ""=' a; or a = a; / T~' so a; = ari / f3 2. Combining those with Eq. (3. 7), the friction coefficient can be derived as /1d

f3 /aCl-f32)

(3. 8)

If the contact surfaces are well cleaned and the contact interface has good adhesion, Tr is close to a, of the softer material in contact and f3 is close to unit. It is clear from Eq. (3. 8) that /1d becomes infinite when f3 is equal to 1. When f3 decreases from one, /1d decreases quickly. Due to the shear strength of the contamination film lower than that of metal and the cease of junction growth, f3 is close to zero. Thus, Eq. (3. 8) can be represented as /1d = Tr / a" which agrees with the simple adhesive theory. Although the modified adhesion theory can explain the tribologieal phenomena of metals, it has been criticized because of the following questions such as: CDthe agreement between the theoretical calculation and the experimental results of the friction coefficient is not good; (2)the effect of surface roughness on friction is not considered in this theory; Gil there is a lack of evidence that strength is necessary for the junction formed. To overcome the shortcomings of the adhesion theory, Kragclskii proposed the molecular-mechanical theory based on the adhesion theory and the molecule attraction theory in 1939. Under very high pressure, the mieroasperities on the real contact surface for a tribopair arc mutual chimerism, and the micro asperities of the harder object arc impressed into the softer one. Moreover, the molecular attraction force is existed at the contact zone. Because the motion process is to overcome the mechanical chimerism of mieroasperities, ploughing and the molecular attraction force in the tribosystem, thus, the friction force is a sum of all tangential stresses induced by the mechanical chimerism of mieroasperities, ploughing and the molecule attraction of the contact junction, and expressed as (3. 9)


56

where a and yare related to phys-mechanical properties of contact surface. Combining Eqs.(3. 9) and (3. 1). the friction coefficient can be derived as

y+ aS,1 P

{1d =

(3. 10)

where y is the constant friction coefficient obtained from the mechanical chimerism theory. while ,I P is the variable of y after considering the influence of molecular attraction. This theory considers each factor comprehensively. and is usef ul not only to elucidate the mechanism of dry friction and boundary friction. but also to explain the tribology of metal and polymer material. The experimental friction coefficients arc listed in Table 3. 1. which shows the static and dynamic friction coefficients of tribomaterials with smooth surface.

as

Table 3. 1

Friction coefficient for normal tribopairs Static friction coefficients

Dynamic friction coefficients

Materials "fo lubricant

0.15

Steel-steel

o.

Lubricant

No lubricant

10-0. 12

o. o. o. o.

Steel-soft steel Steel-cast iron

0.30

Steel-bronze

0.15

Sol t steel-cast iron

0.20

Sol t steel-bronze

0.20

o.

18

Cast iron-bronze

o.

Bronze-bronze

Ebonite-steel

o.

0.05-0.10

20

o.

18

0.05-0.15

15

0.05-0.15

10

o. o. o. o.

18

0.07-0. 15

15

0.07-0. 12

15-0.20

0.07-0. 15

20

0.07-0. 10

10

0.01

Pure aluminum-brass hardened

o. o. o.

Steel-polycarbonate hardened

0.30

Steel-polyformaldehyde powder

0.16

Metallurgy-steel powder

o. 10 o. 10

Pure aluminum-steel

Bronze-bakelite

Metallurgy-cast iron

3.1.4

10-0.20

0.05-0. 15

o.

Cast iron-cast iron

10-0. 15

Lubricant

15

17 24 27

Wear Mechanism 1l718 -

Wear is the successive removal of surface materials by repeated friction and is mainly caused by microscopic mechanical fracture. Even when surface has some chemical reaction products. such as oxides. the volume loss from surface occurs mechanically in many cases. Although the various wear mechanisms have been proposed. it is difficult to predict the wear loss. It is elear that the wear process involves fatigue. fracture. corrosion and plastic deformation of tribomaterials. The wear mechanism is elassified by Burwell. Main wear mechanisms arc ad-

Chapter 3

Fundamentals of Tribology and Tribomatcrials···

57

hesive wear, abrasive wear, fatigue wear and corrosIve wear, respectively, which arc elucidated as follows.

1. Adhesive wear Adhesive wear is a form of wear which occurs when two smooth surfaces arc slid against each other, and the fragments arc pulled off from one surface to adhere to the other. Adhesive wear always arises from the formation and shear fracture of the junction. When the adhesive junction strength is lower than that of tribomaterials, shear fracture occurs at the joint interface, and the transfer of material is not obvious and wear rate is low. When the adhesive strength is higher than the yield strength of softer material in tribosystem, fracture takes place in the subsurface of softer metal ncar joint, and then wear become mild. When the junction strength is higher than those of tribomaterials, shear failure mainly occurs in the subsurface of soft metal. The fragments adhered to the hard metal make the softer surface scratched. If the junction strength is much higher than the shear strength of tribomaterials, shear fracture occurs at the deeper position of one or two metals, and then wear become severe. 2. Abrasive wear Abrasive wear is the form of wear which occurs when a rough hard surface slides on a softer surface, and ploughs a series grooves. The material originally in the grooves is normally removed in the form of loose fragments. Abrasive wear can also arise when hard and abrasive particles are introduced between sliding surface. In this situation, the abrasive grains adhere temporarily to one of the sliding surfaces, or else arc embedded into it, and plow out grooves in the other. The form of wear is generally called as the three-body abrasive wear. Usually there is the extremely high stress at contact area between abrasive grain and sliding surfaces, which makes the tribomaterials deform plastically and fatigue or fragment. If abrasive wear is caused by hard and rough surface, the form of wear is referred to as the two-body abrasive wear. When the motion direction of particles is parallel to the solid surface, the stress at contact zone between particle and smooth surface is low, which is characterized by the scratch line and shallow grooves on the surface. If the motion direction of particles is vertical to the solid surface, the collision contact stress at interface between particles and surfaces is high, which is characterized by the deeper groove on the surface and large size particles peeled off.

3. Fatigue wear Fatigue wear is observed during repeated sliding or rolling over a track. The repeated loading-unloading cyeles may induce the formation of surface or subsurface cracks, which eventually results in the break-up of the surface with the formation of large fragments, leaving large pits in the surface. Fatigue failure depends on the amplitude of the cycle shear stress. If the shear stress exceeds the endurance strength of materials during rolling, the wear particles arc generated by the initiation and propagation of crack. For the rolling contact, cracks arc usually initiated in subsurface. If the contact condition is the mixture of rolling

58


and a little sliding, the damage will occur close to surface. 4. Corrosive wear The mechanical-chemical reaction occurs at a sliding contact zone in the corrosive environment. and the corrosion elements are observed on the sliding friction surface. During sliding friction, the corrosion clements on the sliding surface arc worn away so that the corrosive attack can continue. This indicates that the corrosion and friction are promoted mutually in corrosive wear.

3. 1. 5

Wear Surface for Stator and Rotor of Ultrasonic Motors

Generally, wear occurs as a result of friction. For TRUM-60, its rotor surface is covered with the tribomaterial, and its stator is made of copper. Fig. 3. S shows the microstructure of wear track on a stator and rotor. As seen in Fig. 3. 5. it is clear that the plough grooves are formed on the worn surface owing to adding hard minerals such as alumina into the friction material as reinforced phase. Fig. 3. S (a) shows the optical microscope of a copper stator. The plough grooves arc generated on the worn surface of the stator owing to friction, and their direction is identical to the rotation direction of the rotor. If the hardness of tribomaterials (base materials) was lower than that of substrate metal, the plough grooves are formed on the rotor's surface. Thus, the wear mechanism is abrasive wear. It is obvious from Fig. 3. 5 (b) that the plough grooves arc formed on the rotor's surface owing to the friction effect, and the grooves' direction is the same as that of the rotor's rotation. Fig. 3. S(c) shows SEM image of polytetrafluoroethylene-based tribomaterials for rotor. As seen in Fig. 3. S (c), besides the plough grooves. the fragments arc observed to be pulled off the surface. This indicates that the wear mechanism is the mix wear including abrasive wear, fatigue wear and adhesive wear for the tribomaterials in USM.

(a) 0pl icalmi eroscopic ofstalor

(b) 0pliealmieroscopie ofrolor

Fig. 3. 5

3.2 3. 2. 1

(e) Scanning e el ctron microscopic ofrolor

Microscopic image of wear track

Tribomaterials Used for Ultrasonic Motors Basic Requirement, Classification and Selection Principle

In order to increase the mechanical characteristics and running life of ultrasonic

Chapter 3


59

motor, the surface of a stator or rotor is usually coated with tribomaterial or modificd using othcr surface processing methods. Currcntly, therc arc two elcments in the criteria to evaluate the ultrasonic motor's performances: CDthe ultrasonic motor posscsscs thc good output pcrformancc and running stability; @thc ultrasonic motor has the excellcnt reliability and running lifc. It is realized that the factors to influence the energy transmission are the situation of contact sur face (roughness, contact area), tangent friction force, longitudinal vibration velocity; but onc of main factors to induce thc ultrasonic motor running unstably and its life shortening is its tribology of its tribopairs.

1. Basic requirement and selection principle of tribomaterials in ultrasonic motors According to before-mentioned designing requirements, tribomaterials for USM should meet the following basic conditions: CD appropriatc static friction coefficient (0.15-0.3 for TRUM, greater than 0.2 for LUSM) , the coefficients of dynamic friction close to that of static friction, and thcre is no creep or crawl at low velocity; @good wcar-rcsistant pcrformancc and lcss wcar ratc for tribopairs surfaces; ®low frictional noise «15dB); @good surfaces hardness match of tribopairs; Ql) stablc phys-chcmical propcrties at room, high and low tempcraturc, low or high temperature tolerance; ® good vibration resistance and impact resistance propertics. The selection of friction coefficient will depend on the designing requirement of ultrasonic motors. For the motor of short-stroke, discontinuous working and short lifc, it is suitable to selcct thc tribomaterials with high cocfficicnts. But for the motor of long-stroke, continuous working and long life, it is suitable to select the tribomaterials with lower coefficients. Furthcrmore, thc tribomaterials' lifc is one of thc important factors deciding the running-life of ultrasonic motors, so it is especially important for the tribomaterials to have high wear-resistance ability. Due to the generation of frictional hcat, tcmpcraturc on thc friction surface will incrcasc with the running timc, and finally approach to a balance temperature higher than 100'C. Thus, it is more important to guarantee ultrasonic motors running stably if the tribomaterials havc good tcmperature stability of phys-chemical propcrties. 2. Classification of tribomaterials Tribomaterials for USM are often composites, which consist of matrix, relllforced filler and friction regulator. Matrix is used to form the main-body of tribomaterials, whilc rcinforced fillcr is used to cnhancc the mechanical charactcristics, and friction regulator is used to regulate the friction coefficients so as to enhance thc output torque and cfficicncy of USM. Morcovcr metal coating is also uscd as the tribomatcrials of TRUM to keep it running discontinuously. For cxample, the elinvar coating is used in the TRUM of camera. For thc tribomaterials uscd in USM, wc must consider how to match thcir friction cocfficient and wear rcsistance. Although the addition of hard assist materials could enhance the hardness of tribomaterial, the high content of hard assist materials leads to thc mating pair's surface becoming worn. If a little of

60


friction regulator is added into the matrix, the wear resistance and stability of tribomaterials would be enhanced. Then no stick-slip occurs, and vibration and noisy will be reduced. Based on the above-mentioned guiding theory, the tribomaterials used for USM could be classified as: CD polymer matrix; @ ceramic coatings; ® powder metallurgy; @metal coatings. There arc many kinds of tribomaterials for USM in the world. In Japan, Endo and Sasakp9: have reported a tribomaterial mainly made of neoprene, while in Germany, Rehbein and Wallasehek[2°:have devcloped a PTFE-based tribomaterial consisting of PTFE, polyimide, carbon fiber and steel. And in China, according to adhesive method, Baoku rjC21: has developed a tribomaterial with the main components of bisphenol type epoxy resin, phenol-formaldehyde epoxy resin, modified imidazole curing, frictional coefficient regulator, KH500 silane coupling agent and hardness regulator. Xuejun Liu, Tongsheng Li, et aI L22 - have developed an aromatic polyamide-based tribomaterial, and its main components are aromatic polyamide, cuprous chloride, graphite and carbon fiber. lianjun QUL23-21_ has reported a tribomaterials with the modified PTFE or nano PTFE. Recently, Zhiyuan Yao, Qingjun Ding and the author"-26 J have developed a series of tribomaterials used for the rotor and the stator. For coatings used in USM, Seok-Jin Yoon in Korea has indicated that the TiAI)J, Ti)J, DLC and Si-DLC coatings could be used on the stator- 27 -.

3. 2. 2

Influence of Composition on Tribological Properties

For TRUM, the matrix materials arc epoxy resin, phenol-formaldehyde reSin, PTFE, polyimide, neoprene, acrylonitrile-butadiene rubber (NBR) , etc., while reinforced fillers are aramid fiber, carbon fiber, wear resistance powder (mineral), wollastonite, calcium carbonate, alumina, etc. The proper addition of reinforced fiber will change the elastic modulus and the anisotropy of materials, and increase frictional coefficient as well as enhance the locked torque of motors. Due to the aramid fiber having high tensile strength as well as good thermal resistance and its friction coefficient higher than carbon fiber, so that aramid fiber is an ideal reinforced fiber. If the anti-wear powder is added into the tribomaterials made of phenol-formaldehyde resin modified by nitrile-butadiene rubber, the tribomaterials exhibit the high friction coefficient and then the locked torque of ultrasonic motor increases:"]. However, after the ultrasonic motor run for some time, its locked torque will decrease. This indicated that the frictional properties of the stator and rotor tribopair are not stable and the anti-wear ability of the stator/rotor tribopair is poor. In order to improve the anti-wear property of the material, the frictional regulator is added into the tribomaterial. At present, frictional regulators arc PTFE, copper oxide, molybdenum disulfide, graphite and copper powder etc., which can adjust the frictional coefficient. They arc absorbed on the surface of the anti-wear powder and distributed into the soft adhesive matrix to form the particle with specific function, and then could regulate the macroscopical friction property of tribomaterials. Table 3. 2 shows related properties of

Chapter 3

Fundamentals of Tribology and Tribomaterials'"

61

some raw materials [or tribomaterials based on polymers. Table 3. 2 Related properties of some raw materials for tribomaterials based on polymers Raw materials

Related properties

Epoxy resin

Middle friction coefficient, brittle. good wear resist anee with fillers, bad temperature stability, high polarity, easy adbesion

Phenolic resin

Middle friction coefficient, brittle, middle wear resistanee with fillers, bad temperature stability, high polarity, easy adhesion

Polytetrafluoroethylene

perature stability, tiny surface tension, small com-

Low friction coefficient, high self-lubricity, good tempression modulus, friction regulator

Matrix

Polyimide

Chloroprene rubber

Butadiene-arylonitrile rubber

Polyphenyl ester Highdensity polyethylene Aramid fiber

Reinforcing filler

Carbon fiber

Alumina

Middle friction coefficient, brittle, good wear resist ance, good temperature stability

Good toughness, bad high temperature stability, good wear resistance with fillers, low efficiency

Good toughness, bad bigb temperature stability, good wear resistance with fillers, low efficiency

Tiny wear loss, tiny creep deformation, good radiation resistance, tiny injury to coupled part, brittle

Low friction coefficient, bad temperature stability

High tensile strength, high friction coeffeient, good temperature stability High tensile strength, low friction coeffeient, good radiation resistance, good temperature stability

high hardness, good wear resistance, good temperature stability

Molybdenum

Solid lubricant, low friction eoeffcient, high friction coefficient at high temperature

Graphite

Solid lubricant, low friction coefficient, good temperature stability, good chemistry stability, conducting

Friction regulator

Copper oxide

High friction coefficient, good temperature stability

Ceramic composites and metal coatings arc used for LUSM. Moreover ceramic composites arc mainly alumina-, titania-, chromium oxide-based composites, etc. As is known, alumina has high hardness, high brittleness, and low wear

62


loss. If the alumina ceramics contains a certain amount titania, the alumina composites have good toughness, low wear loss and excellent heat insulation performance. The chromium oxide ceramics has low friction coefficient, good polishing performance and low wear loss. Recently, metal coatings are often used as tribomaterial for TRUM and LUSM, and their main components are nickel, chrome or their alloy. But short life is their shortcoming, so discontinuous condition is suitable for this technology. As above-mentioned, the composition of tribomaterials will have major influence on their tribological properties. The best composition of tribomaterial should be determined by using orthogonal matrix design with a few experiments.

3. 2. 3

Preparation of Tribomaterial

According to the type of tribomaterials, their preparation process method and the process equipment are different. The main equipment to prepare the epoxy resinbased tribomaterial is ordinary oven, while the main apparatuses to prepare PTFE-based tribomaterial arc hydraulic pressure machine bclow 20 ton and a high temperature sintering furnace above 100 'C. For the ceramic coatings, the main equipment is a plasma spraying device. Now, the preparations of the PTFE-based tribomaterial, epoxy resin-based tribomaterial and alumina-based tribomaterial arc introduced in detail here.

1. PTFE-based composite tribomaterial on rotor PTFE-based tribomaterials consist of the PTFE matrix, the reinforced agent of nano diamond powder and the regulator of copper powder. Its common compositions (molar percentage) are: CDPTFE matrix(60%-70%); @reinforcing agent 0%-25%); (3)regulator(5%-30%). It is clear that the matrix content is up to 60%. If the filler content is too high, the increase of hardness for tribomaterials will cause the abrasion wear of tribopair. There are three procedures to prepare the PTFE-based tribomaterial: CD three kinds of raw powders according to their molar ratios are mixed, stirred uniformly and dried up; @ the above mixture is filled into the mold, and pressed by the hydraulic machine to form an embryo of tribomaterial, then kept it at 40-60'C for 21-18h; C]the molding product is sintered at 370-380'C. As the temperature increases from 20 to 330'C, the heat speed is 40'C/h, while that is 30'C/h when the temperature rise from 330 to 380'C. When the temperature approaches to 380'C, it is kept for 4h. Figure 3. 6 shows the PTFE-based tribomaterial developed by PDLab. The PTFE-based tribomaterial is often adhered to a rotor. Firstly, the rotor is made via machining aluminum alloy. Then the sinter cd tribomaterial is cut into sheet (with the thickness of o. 2-0. 3mm) and adhibited on the rotor (as seen in Fig. 3. 7). 2. Epoxy resin-based tribomaterial on stator Epoxy resin-based tribomaterial consists of epoxy resin matrix, nano alumina reinforcing agent and PTFE regulator. Its common composition (molar percentage) are: CDmatrix of epoxy resin: 50%-60%; @reinforcing agent: 20%-35%;

Chapter 3


Tribomaterials based on polymers

Fig. 3. 6

Stators of TRUM with tribomaterials

Fig. 3. 8

63

Rotors with tribomaterials based on polymers

Fig. 3. 7

Rotors of BTRUM with tribomaterials

Fig. 3. 9

@regulator: 5%-30%. These raw materials including epoxy resin, PTFE, alumina, carbon fiber and curing agent arc cohered on the stator's surface and rotor's surface after being mixed up, and then lathe processes them to the required size after the composite is cured in heat at 80"C for 2 hours, as shown in Fig. 3. 8 and Fig. 3. 9. The frequency response experiments of stator arc performed with PSV-300F vibration measurement system using laser Doppler. A rotary tribometer designed by PDLab is used to test the wear and the friction of the samples. After running-in period of 10 hours, friction coefficients are tested under two different conditions: CD20"C, preload 100-250)J; @20"C ,prcload 100-250)J , voltage imposed on single face 15 V, frequency 37. 4kHz. The apparatus is located in a clean room with the relative humidity of 25%-50%. The rotor rotates at 12r/min, and the stator is fixed to the tribometer. Prior to testing, the eounterfaees are cleaned with ethanol, and dried. The normal load is continuously monitored and controlled with computer via using an eleetropneumatie valve. The data of normal load and friction force arc collected instantaneously. Table 3. 3 shows the friction coefficient of tribomaterials against the anodized aluminum rotor. It is obvious that the friction coefficient is not a constant value at different preload, and the higher the preload is, the higher the friction coefficient is. It indicates that the contact area increases with an increase of the preload due to the tribomaterial's toughness.


64

Table 3.3

Friction coefficient between tribomaterial and aluminum rotor anodized Friction coefficient

Preload/)!

Ordinary state

Under ultrasonic vibration

Variation ratc/ %

100

o.

150 7

O. 119 4

-20.7

150

0.153 1

0.123 2

-19.7

180

0.151 3

0.131 6

-11.7

200

O. 158 1

O. 140 7

-11. 0

250

O. 160 7

O. 113 8

-10.5

The friction coefficient decreases 10 %-20 % under ultrasonic vibration in comparison to that of the ordinary state- 10J • Due to ultrasonic vibration and impact, the contact area between tribomaterial and rotor under ultrasonic vibration is less than that of the ordinary state, and the preload decreases, thus the friction force and the apparent friction coefficient all decrease. As seen in Table 3. 3, it is evident that the more the preload is, the less the decrease degree is. The rotor is impacted by the stator coated with tribomaterial. When the pre-pressure increases, the effects of impact to contact area makes reduce. This indicates that the decrease degree of the friction coefficient is little. Figure 3. 10 shows that SEM images of friction surface of epoxy resin-based tribomaterial on stator. There arc some microcracks after running for 200 hours, while there is obvious delamination after running for 600 hours. However, there are no ploughing grooves. It could be coneluded that the wear mechanism of epoxy resin composite are fatigue wear and adhesive wear. Therefore, the antifatigue performance and high cohesion energy density of the tribomaterials should be investigated when they are used in ultrasonic motors. In addition, the hardness, thermal stability and friction coefficient of the tribomaterials are usually considered as key factors of effecting or wear resistance of the tribomaterials.

(a) 200h

Fig. 3. 10

(b) 600h

SEM images of friction surface

Because the variation of temperature results in the migration of frequency response, thus, one of key factors affecting the stability and adjustability of ultra-

Chapter 3


65

sonic motor is the width of frequency response. With increasing the width of frequency response, the stability and adjustability of ultrasonic motor increase. Fig. 3. 11 shows frequency responses of the stators. It is obvious that there is no migration for the frequency response when the stator( Fig. 3. 11 (b)) is covered with epoxy resin composite and the half-peak width of frequency response increases about 2 times of that of the stator (Fig. 3.11 (a)) without the composite. This indicates that the stability and adjustability of ultrasonic motor increase significantly via coating tribomaterial on the stator.

rL 1 1 1 f L Ll1 20

20

30

30

40

40

50

FrequencylkHz

50

Trioomatcria ll iner

l(b'

Frequency/kHz

Fig. 3. 11

Frequency response for stator

The experimental results show that the half-power bandwidth of working mode responding curve is widened if the tribomaterial is adhered on the surface of stator (Fig. 3. 11 (b)). This causes the ultrasonic motor having more stable rotation speed. Besides PTFE type the other tribomaterials developed by PDLab can meet the requirement for all kinds of ultrasonic motors (Fig. 3. 9). 3. Alumina composite as friction material on stator of LUSM Compared with traditional electromagnetic motor, one of the advantages of ultrasonic motors is excellent transient property, mainly in rapid response, self-locking performance and precise positioning. At present, linear ultrasonic motors have been used in rapid response unit, high-grade instruments, and precision control devices. One of the key factors effecting on transient property is hardness of tribomaterial. Tribomaterials based on polymers will delay the response time because of the polymers' toughness and heat deformation. So inorganic composites are often used as tribomaterials in linear ultrasonic motors. Because it is difficult to bond inorganic composites to the stator of ultrasonic motor without any tackiness agent. The plasma spray process is basically the spraying of molten or heat softened material onto a surface to provide a coating. Material in the form of powder is inj ected into a very high temperature plasma flame, where it is rapidly heated and accelerated to a high velocity. The hot material impacts on the substrate surface and rapidly cools forming a coating. This

66


plasma spray process carried out correctly is called a "cold process" (relative to the substrate material being coated) as the substrate temperature can be kept low during processing avoiding damage, metallurgical changes and distortion to the substrate material. Plasma spraying has the advantage that it can spray very high melting point materials such as refractory metals like tungsten, ceramics and zirconia unlike combustion processes. Plasma sprayed coatings arc generally much denser, stronger and cleaner than the other thermal spray processes with the exception of HVOF and detonation processes. Plasma spray coatings probably account for the widest range of thermal spray coatings and applications and makes this process the most versatile. The tribomaterial based on alumina is composition of the matrix of alumina and the regulator agent of titanium dioxide, etc. Its common composition (molar percentage) is: (1) matrix of alumina: 55%-80%; (2) the regulator agent of titanium dioxide: 10 %-10 %; (3) others: 0%-10%. Although in the prescription above, the ratio of matrix is up to 55%, however, the study shows that the ratio of the regulator agent of titanium dioxide has great effect on the friction properties of tribomaterial. It is clear that the high speed, torque, output efficiency and the efficiency of interface dynamical transmission would be acquired when adjusting the content of titanium dioxide in a certain range. The commercially available AI,O, powders with an average particle size of 40~70fLm arc used as a feedstock in the present study. The raw feedstock has the purity >99. Owt% of Al,0 3 component. And titanium dioxide powders with an average particle size of 50fLm are used as an additive to prepare the other feedstock of AI, 0, -TiO, composite. The composite powders with a content of around 20 % TiO, arc mechanically mixed in a rotary-vibrationmill, alcohol being used as a binder, and then suffered sieving and drying prior to the spraying.

Fig. 3. 12

AI, 0 3 - Ti0 2 tribomaterial

The DH-2080 atmospheric plasma spraying equipment made by Shanghai Dahao :'\Ianomaterials &. Thermal Spray Co., Ltd. in China is applied to prepare AI 2 0 3 -TiO, compositc coatings. Thc fcedstock powdcrs arc fed with a Twin-Systcm 1O-C. A mixturc of argon and hydrogen is uscd as plasma gas. During spra-

Chapter 3


67

ymg. the substrates and coatings are cooled using compressed air. Stainless steel coupons arc uscd as substrates. Beforc spraying. the substratcs are dcgrcascd ultrasonically in acctonc and grit blasted with corundum. In addition. the plasma torch is utilized to spray powders onto the unheated quartz substrate in order to observe the spreading and flattening morphology of impacted droplets. Disadvantagcs of the plasma spray process are relativc high cost and complcxity of process.

3.3 3. 3. 1

Influence of Tribomaterials on Performance of USM Influence of Elastic Modulus and Hardness

Elastic modulus E and hardness H are two essential parameters of materials. Elastic modulus is relatcd to the material atom composition. whilc hardncss has relevancc to the organization structurc of materials. It is elcar that hcat trcatment has no influcncc on the elastic modulus. but has grcat cffect on hardness. espccially for mctal alloy. Gencrally. hardness depends on the local elastic-plastic deformation of solid material during indentation loading. and the elastic modulus can be calculated from unloading process. Based on the conventional depthsensing indentation method proposed by Oliver and Pharr. Chinese researchers dcrivcd an analytical relationship between the reduccd modulus and hardncss for solid materials. It is found that the hardness and thc elastic modulus are interrelated to each other through the recovery resistance of materials. Experimental results show two important features: CD the reduced modulus predicted by the new E'- H relationship is the same as that obtained by the conventional method; CZ) the elastic modulus and hardness determined by the simple set of procedures arc comparable to thosc obtaincd by using thc convcntional method: 2"J.

1. Influence of elastic modulus on USM The elastic modulus of tribomaterials is one of the major physical parameters determining the friction characteristics. The results show that the elastic modulus affects the no-load speed. output torque. output power and start-stop characteristic of ultrasonic motors. Thc variation of opcrating spccd with the elastic modulus is not simply lincar relationship. The prescnt theory :29J indicates that undcr the condition of no-load and thc certain prc-prcssure in the range(250-300N). it is available that thc contact area between a stator and rotor would decrease as the elastic modulus of material increases in the range( o. 2-1. 5GPa) • which induces the average tangential velocity of thc rotor increasing at thc contact arca. With thc incrcase of thc avcrage tangential velocity and the decrcase in elasticity sliding motion. thc noload speed of ultrasonic motors would incrcascs. If the elastic modulus excecds thc above-mcntioned rangc. the averagc speed at contact area increascs and thc elastic sliding decreases. However. the high elastic modulus makes the interface area between the stator and rotor decreased. which causes a friction drive force and cnergy convcrsion rate bcing lower. In this casc. thc no load speed of ultra-

68


sonic motors decreases. The author29 J analyzes the influence of vanous elastic moduli on the output performance of ultrasonic motors using the simulation software for the traveling wave ultrasonic motor, and the results indicate that the contact width of the stator and rotor pair in a wavelcngth and the deformation strain of tribolaycr bccomc wide and large, with decreasing the elastic modulus of tribomaterials. On the contrary, the contact width between the stator and rotor will decrease with the increase of the elastic modulus or the contact stiffness. When the elastic modulus varies in thc rangc of o. 1-0. 5GPa, the no-load spced incrcases with mcreasmg the elastic modulus. The elastic modulus of tribomaterials also affects the locked torque and the output cfficicncy of USMs. According to thc contact models, if the tribolayer is soft and the contact area extend to the area beyond the points with the same circumferetial spccd of stator and rotor, the contact area includcs impcding area which weakcns thc stator's driving effect on rotor. Whcn the contact stiffncss of tribolayer is high, the contact arca would decrcase and becomc thc driving zone, and then thc locked torque incrcascs obviously. In meanwhilc, thc radial componcnt of thc contact forcc on this zone will bc low, and thc sliding loss on the intcrfacc will also decrcase. Thereforc, the output cfficiency of ultrasonic motors bccomcs high. From the above analysis, it is clcar that thc high rotational spccd, torque, output efficiency and thc cfficiency of dynamical transmission on thc intcrfacc would be acquircd whcn the elastic modulus of tribolayer properly incrcascs in a ccrtain rangc.

2. Influence of hardness The hardness of tribomaterials affects not only running speed and output torque, but also frictional noise. The influence of tribomaterials' hardness on the opcrating performance of somc linear ultrasonic motors has bcen rcported by Endo[19:. Thc author prepared several tribomatcrials, and studied the influence of hardness on the performance of USM. PTFE compositc, cpoxy composite, phcnolic composites, hard aluminum alloy, ccmented carbide as tribomatcrials on rotors are respectively paired to the stator made of phosphor bronze and piezoelectric ceramic. By controlling fillers and roughness, their friction coefficient can be adjusted to a similar valuc. Table 3. 4 shows the Vickcrs hardncss of fivc kinds of tribomaterials. It can bc seen that the rank of hardness is arranged from low to high: PTFE composite< epoxy composites < phenolic compositcs < anodizcd aluminum alloy < cemcnted carbidc. Table 3. 4 Tribomaterial

PTFE

Vickers hardness (HV)

11

Vickers hardness of tribomaterials Epoxy

Phenolic

Anodized

reSln

reSln

aluminum

Cemented carbide

39

80

453

1 120

Chapter 3


69

Figure 3. 13 shows the curves o[ speed vs. torque [or the ultrasonic motor with five kinds of tribomaterials. It is evident that with an increase of the hardness of tribomaterials, the no-load speed increases, while the stalling torque decreases. Moreover, the difference o[ the no-load speed also decreases. In other words, the hard tribomaterials could be applied to the ultrasonic motors with high speed, while the soft tribomaterials could be used in the ultrasonic motors with high stalling torque. Because the ultrasonic motors with hard tribomaterials often run with high noise, thus polymer composites are a main kind o[ tribomaterials especially used [or traveling wave rotary ultrasonic motors.

220 200

"'-.-,

.. .:.......

' :":':-- ...."

180 160

'2 140 120 0::

"'!l" c.

Vl

, ....

~

Anodized al uminu m Cemented carbibe

..... . '.;: : ....

.~ ~ .

-E ."

.

PTF E Epoxy resin Phenolic resi n

100

.:- ....~~.... ,.. .... .

., ... . . .....

'" .. , ... ~

80 60 40 20 0 00

0_2

0.4

0.6

0_8

1.0

L2

IA

Output torqu (N -m)

Fig. 3. 13

3. 3. 2

Mechanical characteristics for an ultrasonic motor

Influence of Friction Coefficient

The output torque of ultrasonic motors will increase with the friction coefficient in a certain range. The increase o[ output torque will stop as the friction coefficient increases to a certain value. If the friction coefficient becomes higher. the torque could not increase obviously. In this situation. the wear rate of tribolayer becomes high, and the noise of ultrasonic motors become aloud. Thus, the running life o[ USMs becomes short. In the viewpoint o[ tribology. there are primarily two friction mechanisms: the first is the sliding resistance caused by the mechanical chimerism of asperities for tribopair. This is a mechanical component in friction force. The second is the shearing resistance caused by the adhesion [unction o[ molecules at contact area. It is a molecular component in friction force. In order to enhance the friction force of the stator and rotor pair at a certain pre-pressure, the friction coefficient of the stator and rotor tribopair should be high. The simple method to raise the friction coefficient is the addition o[ hard particles into the tribomaterials. The hard particles can increase the sliding resistance caused by the mechanical chimerism of asperities. When the surface with hard micro-asperities is pressed to the

70


soft surface, the friction force is formed owmg to the ploughing resistance. In this casc, thcrc arc many ploughing groovcs and thc powdcr loss on thc surfacc of tribopairs in USM. The friction coefficient is increased using the second method, which decreases the surface roughness of tribopairs and augments the adhesion force between the molecules. In this case, the powder loss becomes slight. The above analysis indicates that the high-speed, high-torque, high-output power and interfacial dynamic transmission efficiency can be gained as the elastic modulus of the tribolayer increases in a certain range, and the output torque, efficiency, rotation speed, and power of ultrasonic motors can be enhanced by increasing the friction coefficient of tribomaterials.

3. 3. 3

Influence of Anisotropy

The tribolayer on stators or rotors with a certain thickness

IS

distributed on the

annular area. This area is r 2 ~ r~ r3 , as shown in Fig. 3. 14. The sand {} denote arc-length and angel respectively. The contact model between the stators and rotors of ultrasonic motors is very complicated (see Chap. 5). One simple model is that the tribolayers arc supposed as the axial and circumferential independent springs. If the elastic coefficients of axial and circumferential springs are k n and k., respectively, the dynamic friction coefficient is !1d and the deformation of the friction layer is 0, the pre-load of Po is equal to kno under static status, as shown in Fig. 3. 15. Tribomaterial

r

Fig. 3. 14 Rotor of traveling wanc USM

Fig. 3. 15 Deformation status of rotor and stator

If the tribomaterial is pasted on the rotor, the free surface of the tribolayer is against the surface of the stator. In the situation of the ultrasonic motor operating, the stator affects the rotor through tribolayer. Assuming that the axial(normal) pressure f.(r,{},t) and circumferential shear force f.(r,{},t) have influences on the rotor in the friction area, they arc respectively expressed as

f ( r, {} ,t) -- {kn (w + 0) , n

f, (r, {}, t)

0,

=

w+O>o w+o~O

sign(V" - V) !1dn (r,{), t)

(3. 11) (3. 12)

where w is the displacement (z direction) of points on the surface of the stator,

Chapter 3


71

V" is the corresponding circumferential velocity, V, is the circumference speed at the contact point between the rotor and stator. In the above-mentioned model, k n and /1d have different effects on the interaction between stators and rotor. Eq. (3.11) shows that the value of k n influences the contact state of the stator and rotor pair. When thc stator and rotor contact mutually, Eq. (3. 11) is exprcsscd as fn(r,{),t)=knw+k,J'j, where the constant forcc of k,J'j is cqual to Po as prcload, while k n w is alternating force, which represents the interaction between the stator and rotor during operating, and

o. 5k

n

w' is the work done by alterna-

ting force in axial direction. When k n is high, k nwand o. 5k nw' become high. Due to ineffcctive work done by thc ultrasonic motor along axial dircction, the ultrasonic motor's energy would lose. When the stator and rotor contact mutually, Eq. (3. 12) is changed as f,(r,{),t)

=

sign(V" - V,)/1dkn (w+ 0)

(3. 13)

It is clcar from Eq. (3. 13) that the valuc of /1dkn affects the transmission of thc energy from the stator to the rotor. With an increase in the values of /1dkn' the tangential force between the stator and rotor increases at the proper pre-load. Actually, with a decrease in the value of kn' the deformation amount of the tribolayer increascs and the contact width bctween thc stator and rotor in a wavelength enlarges gradually. This indicated that the value of k n should vary in a suitable rangc. If Lt is a contact time in which the point G on the stator contacts with the rotor in one period, the work done by this point to the rotor is

(3. 14) wherc hi is thc distance from thc stator surfacc to thc neutral layer. It is indicated that the stator transmits the cffectivc encrgy to the rotor though thc tribolaycr along circumfcrential dircction. In herc, the tribolayer in circumfercntial direction is considered as the spring with the elastic coefficient of k" which decidcs the output cfficiency of thc ultrasonic motors. Thc output efficiency usually incrcascs with an increasc in thc value of k,. The above-mentioned analysis indicates that in order to increase the operating efficiency of ultrasonic motors, it is necessary to make the anisotropic tribomaterials with low vertical elastic modulus. But in order to obtain the high output torque, the friction coefficient and the circumferential elastic coefficient k, for tribomaterials should be high. The anisotropic tribomaterials prepared in this way are beneficial to improving the output characteristics of ultrasonic motors. Based on the preparation of isotropic tribomaterials, the anisotropic tribomaterials can be prepared increasing the circumferential elastic modulus k,. After glass or carbon fibers are added into the isotropic tribomaterials, the fibers are distributed and stirred circumferentially, and then the anisotropic tribomaterials arc acquired.


72

3.4

Friction Testing for Tribomaterials

Currently, there are two methods to measure the friction coefficients of tribomaterials: The first method is to determine the traditional static friction coefficients. The second method is to measure the dynamic friction coefficients based on the operating principle of USM.

3. 4. 1

Quasi-static Friction Testing

1. Summary Quasi-static tribometer, as shown in Fig. 3. 16, is used to measure friction coefficients of tribomaterials at low speed. The friction coefficient measured by the method is called a quasi-static friction coefficient.

.. ~

Sensor

,-~!!!!!!~II'

Signal amplifier

Data acquisition card

Motion control card

Motion loading: vert ical, el vel , rotational;

Fig. 3. 16

Data acquisition : adhesive force, friction force(moment), friction coefficient

Schematic diagram of quasi-static tribometer

The tribometer is controlled by a computer, whose software system IS wmdows interface in Chinese, and operated easily. Its data analysis software can accomplish the data acquisition and storage, and translate the test data into Word, Excel or other general software. Data record system adopts 12bits AID converter, the record speed can reach to 1000kHz as the experimental curve is shown and the dynamic saving disk works. According to the configurations of different sensors, the tribometer can accomplish the adhesion, friction and wear experiments.

2. Operating principle The tribometer includes hardware and software systems. The hardware system consists of 5 parts: the level moving part, the vertical moving part, the rotation part, the force sensor, and the control box of the motor. Meanwhile the attachments to the tribometer include the motion control card of motors, the signal amplified card of sensors, the data acquisition card, computer, and so on. The motion compartments such as level motion, vertical motion and rotation parts all arc driven by step motors. Software system consists of the drive and control system of step motors, the data acquisition and the data analysis software.

Chapter 3

Fundamentals 01 Tribology and Tribomaterials···

73

As seen in Fig. 3. 16, the relative movement between tribopairs is generated via moving parts, and a ccrtain prc-prcssurc is imposcd to thc stator. Thc rcaltimc data collcction and storagc of prc-prcssurc and friction forcc arc carricd out by using sensor, signal amplifier, and data collection card. The control system includes computer, motion control card, and control program, and controls the motion dircction and spccd of motion parts and thc prc-prcssurc bctwccn tribopairs.

3.4.2

Dynamic Friction Testing

1. Summary The dynamic friction test is used to measure the dynamic friction coefficients of tribomaterials during friction. For the running USM, there are macroscopic and microscopic rclativc motion at thc contact arca of thc stator and rotor tribopairs simultancously. Thc microscopic relativc motion shows two aspccts: CD thc stators and rotors are in the contact state with pulsation variation, which makes the contact stress of the stator and rotor tribopair to change periodically; @there is altcrnating rclativc motion along circumfcrcntial dircction. This motion statc causcs thc intcraction bctwccn thc stators and rotors bcing complicatcd, and thcn thc uniquc friction charactcristics arc cxhibitcd. To analyzc friction cocfficicnt of tribomaterials during running, a dynamic friction test machine is made and provided by Harbin Institute of technology. This machine can simulate the motion at thc contact point of thc stator and rotor pair for ultrasonic motors, and thcn mcasurc thc dynamic friction cocfficicnts for ultrasonic motors. 2. Work mechanism Thc dynamic tribomctcr utilizcs thc front cnd of bar with longitudinal and flcxural vibration modes to simulate the elliptical motion of a surface point on a stator for ultrasonic motor. When the bar excites a composite ultrasonic vibration made of a longitudinal and a flcxural vibration, thcn thc bar front cnd producc a highfrcqucncy microscopic clliptical motion, and thcn thc clliptical functions arc simulated. The dynamic tribometer consists of mechanical system, signal output, and data collcction and transfcr systcm, ctc. Thc mcchanical systcm is a vcrtical structurc. It is convcnicnt for loading prc-prcssurc and thc amplificd output of thc instant friction driving force as the ultrasonic micro-tribo test is done. As seen in Fig. 3. 17, the tribometer includes the ultrasonic vibration parts to simulate the high-frcqucncy clliptical motion of thc point on thc stator for thc traveling wavc ultrasonic motors, thc prc-tightcning structurc to adjust thc prc-load and thc position of instant kinetic positive pressure sensor, the pre-tightening part to regulate the position of output and sensor, the supporting and position structures of transmission output axis and cxpcrimcntal tablc. Thc signal transfcr and output parts includc thc piczoelcctric scnsor to mcasurc thc instant dynamic driving force, and corresponding electric charge amplifier.

74


IISp"cinl~~~z"" I~~tnicsen sor I sensor 2 Operation simulation equipment for TRUM

Fig. 3. 17

Data acquisition card

Computer

Charge amplifier

Schematic diagrams 01 dynamic tribomctcr

References [ 1

J

[ 2

[3

J J

[ 4

J

[ 5

J

[ 6

J

[ 7

J

[ 8

J

[ 9

J

[IOJ [llJ

[I2J

[13J

[14J

[15J

Shizhu Wen. Existing state and development of tribology research in China. Chinese Journal of Mechanical Engineering, 2004, 40 (11): 1-6. Zhongrong Zhou, Leo Vincent. Fretting Wear. Beijing: Scicncc Prcss, 2002. (in Chines c) T Ishii, S Ueha, K "Iakamura. Wear properties and life prediction of frictional material for ultrasonic motor. Japanese J oumal of Applied Physics, 1995, 34: 2765-2770. H Storck, W Littmann, J Wallasehek. The effect of friction reduction in presence of ultrasonic vibration and its relevance to traveling wave ultrasonic molors. [lltrasonics, 2002, 40: 379-383. T Yamaguchi, K Adachi, Y Ishimine, et a1. Wear mode control of drive tip of ultrasonic motor for prccision positioning. Wear, 2001, 256: 115-152. M Kurosawa. Efficiency of traveling wave type ultrasonic motors. J. Acoust. Soc. J pn, 1988,11(1): 10-16. N M Hagood, A J McFarland. Modeling of piezoelectric rotary ultrasonic motor. IEEE Trans. Ultrason., Ferroelee!., Freq. Contr., 1995, 42(2): 210-224. P Hagcdorn, T Sattel, D Spcziari, ct a1. The importance of motor flcxibility in traveling wave ultrasonic motors. Smart Mater. Struct., 1998, 7: 352-368. Hcming Sun, Chunshcng Zhao, Xiaodong Zhu. Simulation on friction characteristic of ultrasonic motor using longitudinal and torsional modc. Journal of Southeast University (Natural Science Edition), 2002, 32(1): 621-626. (in Chincse) Heming Sun, Hui Guo. Thc relation of preprcssure and output-torquc of longitudinal and torsional ultrasonic motor. Tribology, 2001, 21( 1): 52-54. (in Chinese) Hui Guo, Taizhc Tan, Xinbao "ling. Moving track of the surfacc particlc and torquc for thc ultrasonic motor using thc traveling wavc in the plane. Tribology, 2002, 22(5): 386-390. (in Chincse) Xiangdong Zhao, Changqing Liu, Hcming Sun, ct a1. Output characteristics of thc frictional interface of traveling wave type ultrasonic motors. Small & Special Machines, 2000, 21 (3): 21-22. (in Chinese) Xiangdong Zhao, Bo Chen, Chunsheng Zhao. "Ionlinearly frictional interface model of rotated traveling wave typc ultrasonic motor. Journal of Nanjing University of Aeronautics & Astronautics, 2003, 35(6): 629-633. (in Chinese) Hai Xu, Chunsheng Zhao. Contact process and friction analysis of linear ultrasonic motor. Journal of Nanjing University of Aeronautics & Astronautics, 2005, 37(2): 144-149. (in Chinese) Qianwci Chcn, Wciqing Huang, Chunshcng Zhao. Mcasurcmcnt of scrviec lifc of ultrasonic

Chapter 3

[l6J [17J [l8J [19J [20J [21J [22J [23J

[24J [25J

[26J

[27J

[28J

[29J

Fundamentals 01 Tribology and Tribomaterials'"

75

motors. Journal of Vibration, Measurement & Diagnosis, 2004, 24 (l): 19-22. (in Chinese) J Halling. Principles of Tribology. Beijing: China Machine Press, 1981. (in Chinese) Shizhu Wen. Principles of Tribology. Beijing: Tsinghua University Press, 1990. (in Chinese) Zhendong Dai, Min Wang, Qunji Xue. Introduction to the Thermodynamics of Friction Systems. Beijing: National Defense Industry Press, 2002. A Endo, N Sasaki. Investigation o[ [rietional material [or ultrasonic motor. Japanese Journal of Applied Physics, 1987, 26: 197-199. P Rhbein, J Wallasehek. Friction and wear behavior of polymer/steel and alumina/ alumina under high-fretting conditions. Wear, 1998, 216(2): 97-105. Baoku Li. Preparation [or new [rietion material. Technology on Adhesion & Sealing, 2001, 22(3): 7-8. (in Chinese) Xujun Liu, Tongsheng Li, Tian Nong, et al. Manufacture and application of aromatic polyamide based [rietional material. China Plastics Industry, 1999, 27(3): 25-26. (in Chinese) Jianjun Qu. Friction Driving Mechanism and Friction Material Research on Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. Harbin: Harbin Institute of Technology, 1998. (in Chinese) Jianjun Qu. The Contact Model and the Properties of the Friction Materials for Ultrasonic Motors. Post-doctoral Report. Beijing: Tsinghua University, 2001. (in Chinese) Zhiyuan Yao, Qingjun Ding, Chunsheng Zhao. Preparation and auxiliary tools of thermoset resin-based friction material and friction layer of ultrasonic motors. Chinese Invention Patent, CN200610040708. 5, 2006. Zhiyuan Yao, Qingjun Ding, Chunsheng Zhao. Using PTFE-based filled with carbon fiber as friction material of ultrasonic motors and its fabrication. Chinese Invention Patent, CN200610010709. X, 2006. H-P Ko, SKim, J-S Kim, et al. Wear and dynamic properties o[ piezoelectric ultrasonic motor with [rietional materials coated stator. Materials Chemistry and Physics, 2005 (90): 391395. Y W Bao, W Wang, Y C Zhou. Investigation of the relationship between elastic modulus and hardness based on depth-sensing indentation measurements. Acta Material, 2004, 52 (18) : 5397-5404. Chao Chen. The Research on Theory Model for the Rotary Driveling Wave Ultrasonic Motor. Dissertation for the Degree of Doctor of Philosophy. Nanjing: :'-Ianjing University of Aeronautics and Astronautics, 2005. (in Chinese)

Chapter 4

Fundamentals of Vibration for Ultrasonic Motors An ultrasonic motor is onc of thc most typical cxamplcs of utilizing vibration. In order to understand its motion mechanisms and design principles, it is necessary to start with clastic body vibrations. Vibration is a classic topic in mcchanical cngineering and many references can be found 11-2J. In this chapter the vibration of clastic bodics is discusscd. It providcs thc ncccssary thcorctical foundation for the subsequent chapters of this book and for readers who are interested in ultrasonic motor technologies, but have little exposure to mechanical vibration. In general, structures (including an ultrasonic motor's structure) are made of simplc componcnts such as bcams, platcs, and shclls. Thcy havc a continuous distribution of mass and stiffness, called a continuous system (elastic body), that has an infinitc numbcr of natural modcs (natural frcqucncics and corrcsponding mode shapes). Analytical solutions to an elastic body vibration equation arc limitcd to only simplc gcomctrics with spccific boundary conditions. In most other cases, numerical methods are used instead to obtain approximate solutions. Thc Finitc Elcmcnt Mcthod (FEM) is thc most cffcctivc onc, of which somc highly sophisticated software, such as NASTRA:'\J, A:'\JSYS, ATILA, etc., is bascd on. Most of thc numcrical analysis in this book wcrc donc by ANSYS, which is a powerful package capable of static and dynamic analysis, modal analysis, timc domain analysis of structurcs, ctc. Ultrasonic motors utilize the inverse piezoelectric effect of piezoelectric ceramic clcmcnts to gcncratc strcss or strain, which cxcitcs a stator (clastic body) to produce forced vibration response. The response is converted into the rotational or lincar motion of a rotor or slidcr by thc friction bctwccn thc stator and rotor. Thcreforc, in ordcr to dcsign ultrasonic motors, pcoplc must also mastcr thc forccd vibration of clastic bodics c, 4J.

4. 1

Natural Vibration of Elastic Body

In the section, we will introduce the natural vibration of elastic body, including bars (shafts, beams), plates, shells, etc. A straight elastic strut can undergo longitudinal, torsional, and latcral vibration. If x dcnotcs thc longitudinal (ccn-


Chapter 1

Fundamentals of Vibration for Ultrasonic Motors

77

troidaD aXIs, and y and z represent the directions of principal axes of a cross section, the longitudinal vibrations take place in the x direction, the torsional vibrations occur about the x axis, and the lateral vibrations involve motion in either the

.Ly

plane or the

.LZ

plane. The strut subjected to longitudinal vibration is of-

ten called a bar. We consider first the longitudinal vibration of a uniform bar using a simple theory.

4, 1. 1

Longitudinal Vibration of Bars

In the condition of no external force and no damping, we can obtain the equation governing the natural vibration of the bar in the longitudinal direction:

psa2u at'

=

~(ESaU)

a.L

ax

(4. 1)

where u(x, t) is the displacement function of the bar in axial direction; S(x) , E(x) , and P(.L) are the cross section area, elastic modulus of material, and mass density of the bar, respectively. For a uniform bar, Eq. (4. 1) can be simplified as

au at' 2

E a'u

(1. 2)

p a.L'

The solution of the above equation can be obtained through the separation of variables. Assume that the solution can be expressed as (1. 3)

Substituting Eq. (4.3) into Eq. (4.2) and using the method of separation of variables can yield

d2Xt~t) d'KL) dx'

+w'q(t)

0

=

+ WE'I'·L '13...--I.( )

(1. 1)

0

(1. 5)

+ Bcoswt

(1. 6)

+ Dcosw J"f-.L

(1. 7)

=

From Eqs. (1.1) and (1.5), we can obtain

q(t)

~(.L)

=

=

Asinwt

Csinw J"f-.L

The complete solution of Eq. (1. 2) becomes

U(.L, t)

=

(Asinwt

+ Bcoswt) (Csinw J"f-.L + Dcosw J"f-.L)

(1. 8)

where w denotes the frequency of vibration, the function ~(.L) represents the mode shape, the constants C and D can be evaluated from the boundary conditions, the function q (t) indicates harmonic motion, and the constants A and B can be determined from the initial conditions of the bar. The general solution of Eq. (4.2) becomes


78

11-]

According to boundary conditions of bars, we obtain the natural frequency of vibrationwn(n = 1,2,3,···) and corresponding modc shapes ~n(x) , which arc summarizcd in Tablc A. 1 and Fig. B. 1 in Appcndixes A and B, respcctively.

4. 1. 2

Characteristics of Natural Modes

All of ultrasonic motors make use of thc "mode" of elastic bodics. Thc word "mode" is used to describe either the natural mode of vibration ( W n ' ~n ) or the mode shape ~n. In other words, the nth mode refers to the nth natural frequency and corresponding mode shape, or refers only to the mode shape ~n. It has been noted that usc of word "modc" has becn very loosc in litcratures. From Tablc A. 1 and Fig. B. 1, it can be observcd that the mode is in fact a wavc in spacc whosc amplitude ratio of various points along axis direction of the uniform bar holds a constant for all time. The certain points (called nodes) on the bar undergo zero amplitude, whereas other points (called antinodes) attain maximum amplitude. The nodes and antinodes occur at regular spaces along the bar and remain the fixcd positions for all timc. This form of vibration is callcd a standing wavc, which is widely utilized in dcsign of ultrasonic motors. The modcs posscss thc following important characteristics:

1. Infinite number of natural modes An elastic body is a continuous system with an infinitc numbcr of dcgrces of frcedom. The system possesses an infinite number of natural frequencies (modal frequencics) and modc shapcs, i. c., wc ha vc modal parameters n

(w n , ~n)'

=

1, 2,3, ...

In general, each natural frequency corresponds with one mode shape.

2. Dependence of modal parameters Gcnerally, modal paramctcrs dcpend on mass, stiffness distribution of thc elastic body, and its boundary conditions. 3. Orthogonality of mode shapes Whcn a bar vibratcs longitudinally, from Eq. (4. 5) any of the modcs must satisfy

-.! (E S dx Therefore for modes

(Wi'

~i)

and (w j

d¢) d.L '

p Sw 2 ~

= -

~j)'

~ =

~(x)

(1. 10)

there are

d d.L

(E 'd.L S d¢i)

d d.L

(ES d¢j )S dx - - P

=-

P

S

2-1.

(4. 11)

2-1.

(4. 12)

Wi't'i

Wj 't'j

Chapter 1


79

The two end positions of the bar are denoted by 0 and l, respectively. Multiplying Eqs. (1. 11) and (1. 12) by ~j and ~i , respectively, and carrying out integration yields

I' I' o

M)

d ( E5 -d' d.r = ~J -d

o ~i

w;

I' I'

p5~i~Jd.r

(1. 13)

d ( E5 MJ - - WJ2 oP5~J~id.r dx dx ) d.r -

(1.11)

X.T

0

Applying integration by parts to Eqs. (4. 13) and (4. 14), respectively, and using the free boundary condition ( E5M/d.r = 0) or the fixed boundary condition (~ = 0) of the bar, the following results can be obtained:

t

Mi -d MJd x - - Wi'It P5-1.'l'i'l'j -I. d - I E5 -d X o .T.T 0

(4. 15)

t Mi -d Mjdx - - Wj'It P5-1.'l'i'l'j -I. d - I E5 -d X o .T.T 0

(4. 16)

Subtracting Eq. (1. 16) from Eq. (1.15), the remainder is

(w~ -w;)I>5~i~Jd.r

0

=

(1. 17)

Because Wi cFW J ' we have (1. 18a) For a uniform bar we obtain (4. 18b) Comparing Eq. (1. 18a) with Eq. (1.16) gives

MJ dx I to E 5 Mi dx dx

0

=

(4. 19a)

For a uniform bar, there is

MJdx I to Mi d.r d.r

=

0

(4. 19b)

Eq. (1.18) or (1. 19) is the orthogonal condition of mode shapes. More precisely speaking, Eq. (4.18) is the orthogonal condition of displacement mode shapes. Similarly, multiplying both sides of Eq. (1.11) by ~i and then integrating from 0 to l results in

(E5 dd~i )d.r I 'o ~i ddX .T

=-

W;I' p5~; d.r

(1.20)

0

Integrating it by parts and then applying the boundary conditions, there is

Ki Mi

(1.21)


80

Ki

I' ,

(dcPi) 2 oES dx d.1':

=

(1. 22) (4. 23)

where Ki and Mi are the ith (order) modal stiffness and modal mass of the bar. respeetively. Sometimes they are also ealled as the ith (order) generalized stiffness and generalized mass of the bar. From Eqs. (1.22) and (1.23). it can be observed that each natural frequency Wi corresponds to both the modal stiffness Ki and modal mass Mi. The three modal properties given above arc universal to the vibration system. not only for bars. shafts. and beams. but also for plates. shells. and more complex vibration systems. 4. Normalization of modes The mode describes the amplitude distribution of an elastic body at corresponding natural frequency. It shows that the amplitudes of all points on the elastic body are not independent. with being proportional to each other. The process used to select the specific ratio or multiples is called normalization. Currently there arc three major methods for normalization L5J : (1) The maximum amplitude of a mode shape is regulated to 1. (2) The modal mass is taken as 1, that is. Mi (3)

J: ¢;

d.1':

=

=

I>S¢;

dx

=

1.

1.

5. Strain modes The stress-strain relation of a bar is (J

=

F

-

S

=

au

E-

a.1':

=

EE

(4. 24)

For the nth (order) mode shape function of the bar. the corresponding strain function of the bar can be deduced as En =

au: a.1'

=

dcPn(X)qn(t) d.1':

=

¢'()

()

n .1': qn t

(1. 25)

¢:

(X) is defined as the nth (order) strain mode shape function of longitudinal vibration or the nth strain mode. 'Table A.2 represents the natural frequencies and corresponding strain mode shape functions of the bar with three boundary conditions. Fig. B. 2 denotes the first four (order) mode shapes of the bar with the boundary conditions. From Eq. (4. 19) the strain modes also possess orthogonality. The modal characteristics mentioned above exist in the natural vibration of bars. as well as in those of shafts. beams. plates. and shells. but which possess different expressions in their displacement and strain mode functions.

4. 1. 3

Torsional Vibration of Shafts

The strut subjected to torsional vibration is often called a shaft. The equation

Chapter 1


81

governing the natural vibration of the shaft about its axis can be described by (4. 26)

where e(x, t) = e is the twist angle of the shaft around x axis, Ie (x) = Ie indicates the mass moment of inertia of the unit length of the shaft around :r axis, Ger) = G and J Cd = J represent the shear modulus of the material and the aera moment of inertia of the shaft about :r axis, respectively. For a uniform shaft, Eq. (1.26) reduces to I a'e Oat'

=

GJ a'e

(1.27)

a.x'

When the shaft has a circular section, Io=pJ , Eq. (4.27) becomes

G a'e

a'e at

(4. 28)

p a.x'

Note that Eqs. (4.28) and (4.2) arc mathematically the same. So the characteristics of the torsional and longitudinal vibrations of a bar behave in the same form. Hence detailed discussion of the torsional vibration of the shaft is omitted and only the final results are given in Table A. 3. :'\Iote that the e.xpressions of natural frequencies are only suitable for shafts with a circular section.

4. 1. 4

Bending Vibration of Beam

The strut subjected to bending vibration is often called a beam. We consider the thin beam for which the length is much large than depth (at least 10 times) and the deflections arc small compared to the depth. Then, the rotation of cross sections of the beam is neglected compared to the translation, and the angular distortion due to shear is negligible compared to the bending deformation. Applying Euler-Bernoulli theory, the equation governing the natural vibration of the beam in its lateral direction can be described by

~ (E1 ax'

a'w)+ 5 a 2 w a.x' Pat'

=

0

(1. 29)

where w(.x, t) = w is the lateral vibration displacement of the beam, I (.x) = 1 denotes the area moment of inertia of the beam's cross section about the neutral axis, S(.x), E(.x), and p(.x) represent the cross section area, the elastic modulus of the material, and the mass density of the beam, respectively. For a uniform beam, we have (4. 30)

Letting w(x,t)

=

cp(x)q(t)

(4. 31)

Substituting Eq. (1. 31) into Eq. (1. 30) and using the method of separation of variables, we can obtain the general solution of Eq. (4. 30)

82


W(.T, t)

where the constants An and En can be determined from the initial conditions, the constants C and Dn can be evaluated from the boundary conditions, from which we can obtain following characteristic equation, natural frequencies, and corresponding mode shapes for a uniform beam simply supported, respectively: sinX n Wn

=

_ X~ l'

-

0, n

=

(1. 33)

1,2,3'"

{IT '\j ps

(4. 34)

Fnsinpnx

(4. 35)

S¢i(xH;(x)dx.

Taking the actual damping into account, the consumption function can be expressed as

Chapter 1


D where C i}

=

C}i

=

J:

103

~ ~ ~ Ci}ej,(t)q} (t)

=

I-J

(4. 99)

)-1

C¢i (x) ¢j (x) dx.

Substituting Eqs. (4.95) - (4.99) into the Lagrange Eq. (4.76) produces Mij(t) +Cq(t) +Kq(t)

=

(1.100)

F(t)

where M = [m i} J, C= [e i} J, and K = [k i} ] are the generalized mass, damping and stiffness matrices of the beam, respectively. kij = k;; kt, F(t) = b hh"

. [J:

+

e3l E3 rfl: (.T) d.TJ is the column matrix of the generalized force.

According to the orthogonality conditions of displacement and bending strain modal shapes of the beam: 1 '] , we can obtain

or (4.101) where M n , C , and Kn are the nth modal mass, damping, and stiffness of the beam, respectively, and its modal force is

Fn (t) = bhh p

f:

e31 E3 rfl}n (.T) d.T = bhe 31 e iwt

where

Fn

=

bhe 3 1

J:

J:

Vo rfl: (.T) d.T = Fn e iwt

Vo rfl~ (x) dx

(1.102)

(4.103)

Subjected to the excitation of the PZT strip, the steady-state bending vibration response of the beam can be expressed as w(x,t) =

2.:Fn¢n(X)/[Kn

Jo-w:) + (2Sn wj

]eiCwt-.n)

11=]

(4.104) 11-]

where (4.105) Comparison of Eqs. (1.105) and (1.82) reveals that they are identical in formation, and thus the same eonelusions can be drawn, which are not to be repeated here. It is important to note the followings: (1) The content of the modal force amplitude Fn of Eq. (1.102) is different from that of Eq. (1.80).

If Vo

=

EN'~ (x) =

Vex) n

=

m

n

=F

m

(1.106)

104


Then, the mth pure modal response of the beam can be excited, and the response is proportional to the width of the PZT strip b, the eonstant e31 , and the height of beam h. (2) It is known from Eq. (1. 106) that in order to obtain the "pure" mode of a uniform beam, distribution of V(.L) must be the same as the strain distribution. It ean be learnt by comparing Figs. B. 3 with B. 4 that among four typical boundary eonditions, the displacement mode and strain modes are identical only in the simple supported beam, whereas in the other three conditions the two kind of modes are different. (3) In reality, it is difficult to impose a distributed voltage identically to strain modal function of a beam. In genera), a number of PZT strips (pieces) are affixed on a beam to approximately achieve "pure" mode excitation.

4. 2. 4

Excitation of Simply Supported Beam by PZT Pieces

As shown in Fig. 4. 17, the simply supported beam is excited by one PZT strip (piece). Let a,b, and hI' denote the length, width, and thickness of the PZT piece, respectively, and the characteristic coordinate of the excitation PZT is .L o • The excitation effect of the PZT piece at point .Lo is expressed as "distributed foree" by the unit pulse function with variable (x- x o ). For a simply supported beam, we can obtain from Eq. (1. 102): Fn(t)

=

bhe3IVoeiw'J:B(x-xo)1«x)dx 2

=-

nIT) . nIT iw' b h eol V 0 ( T sIn Txoe

(4.107)

xo

h

0

xCI)

g"""

,d; ~ _.

;1 ~

~E3 I

Fig. 4. 17

A metallic beam excited by single PZT piece

Thus, the steady-state bending vibration response of the beam to one PZT piece excitation is 2

ex:

W(.L, t)

=

~

-

b h e 31 Vo (nt) sin nlIT.Lo ( sin nlITx)

n=l

I[K n ~(l-w~) =

~ AnB nei(w'-.n)

+

(2l;"n wj

Jei(wt-~n) (4.108)

Chapter 1

Fundamentals 01 Vibration Ior Ultrasonic Motors

105

where

{

An

=-

En -

bh

e VO (T) 3l

(,jCl - iLI~)

2

sin

TXo sin nZ'I[x

+ (2Sn iLlY)

(1.109)

The form of Eq. (4.89) is similar to that of Eq. (4.109) about the longitudinal vibration response to one single point excitation, and hence related conclusions are not repeated here. The displacement and strain modes possess the same shape in the simply supported beam. Therefore, one PZT piece is placed wi thin the range of the half wavelength of its mode, the polarization direction of the PZT matches with the "+" and" - " of the mode, and PZT piece's width beam's width and its length a ,1./2. as shown in Fig. 1. 18.

b
has a positive angle a with

dinate plane formed bye, and eo, and the latter

148


eo. fn and f, can be calculated by Eq. (5.83). In Fig. 5. 25 .f~ and f:. are forces (reaction forces of fn and fJ acting on the rotor along the axial and tangent directions. respectively. Thus. the distributivc interface force endured by the stator tceth can be written as (5.86) where f~e and f~, are the absolute values of the circumferential and radial componets of f~. the negative sign ahead .I;, shows that the axial pressure endured by stator is along thc ncgative coordinate axis of z. sgn(V,J. and sgn (V,o - V,) arc sign functions. which can bc dctermincd by thc relativc velocity betwecn the stator and rotor on the contact interface: sgn(V,J

=

1, {+ -1,

sgn(V,o - VJ

=

(5. 87a)

1, {+ -1,

(5. 87b)

The values of f,eand f"for components can be determined by the following formula: { f,o : I f, ~~sa I fer - I f,sma I

(5. 88)

In this formula a is defined as the friction angle which can be dctcrmined by thc relative velocity of the contact point 8 between the stator and rotor

I I tana I = I VI Vcr - V, ,0

(5.89)

Thc friction angle dcscribcs the relativc sliding trcnd along thc radial on thc point 8 bctwcen thc stator and rotor. its rangc is from -180° to 180°. Some references simply considcr that points on the interfacc havc only two-dimcnsional traj ectory formed by the circumferential and the axial motions. i. e. f~ is along direction 8. They is to overlook the radial slip. and the driving cffect is cnlarged.

5. 3. 3

Interface Energy Loss and Power Transmission Efficiency

The frictional power loss on the contact interface can be calculated by the following formula: (5.90) wherc P d, and P dO arc thc power losses causcd by thc radial and circumfercntial slip:

t II f~

~, [1 (

I Vcr I dS) dt

(5.91a)

e-l See)

(5. 91b)

Chapter 5

Operating Mechanism and Modeling of Traveling···

149

Regarding the average energy in a cycle. t is the certain moment when the motor arrives a stable operating. and T is the cycle of the drive voltage imposed on the piezoclectric ceramic components. The transmission efficiency of the contact interface between the stator and rotor can be defined as (5.92) when

Pout

is the mechanical energy output from the motor Pout

1 fr-T . T , M T (3 dt

=

(5.93)

(3 is the rotation speed of the rotor and MT is the load moment endured by the rotor. The transmission efficiency of the contact interface is the indicator which can measure the performance of the interface between the stator and rotor.

5. 3. 4

Contact Model Between Stator and Rotor

Stator teeth and the friction layer are in the state of either contact or isolation. During the course of the contact there are two conditions as shown in Fig. 5. 26: the first is a quasi-contact condition of tooth e. the second is a full contact state of tooth e 1. The radial width 6.r of the contact area is very narrow. and its average radial location is r,.. At this average radius. along the circle. we can sclect a number of auxiliary interpolation points to determine the contact condition between the rotor and teeth.

+

Rotor substrate

e+l

e,

~o o

o Element nodes of tooth • Interpolative nodes of tooth

Tooth e Ca) FE discretization of tooth e and interpolative nodes

Fig. 5. 26

Cb) Contact states of two teeth

Contact state of stator tooth

Fe is a modal force acting on the on stator caused by the distributed contact force of tooth e. After the superposition of the modal forces of n teeth. we can get the total modal force acting Fe in Eq. (5.82) on the stator:

Fe

N

=

~

Fe

(5. 91)

e-j

We can get the axial force acting on the rotor through the integration and su-


150

perposition of the axial forces located on the top of each tooth in the contact area. Fi

=

i= Ilfn

(5.95)

dS

e-l SCe)

in which S (e) is the integration on the top of each tooth Similarly, the driving torque acting on the rotor is: MTi

=

t If

In

the contact area.

(5.96)

sgn(V,e - VJrf,e dS

e-l See)

in which the sign function is defined as Eq. (5. 87).

5.3.5

Contact Interface Simulation

TRUM-60 stator, with a diameter of 60 mm, Bog operating mode and made of phosphor bronze, is selected as an example for the following analysis. The rotor is made of duralumin, the excitation voltage and frequency arc 120VI'I' and 37. 98kHz, respectively, and the pre-pressure is 140)J. Fig. 5. 27 shows the axial displacement of the stator and the friction layer compression. Fig. 5. 28 shows the radial or circumferential displacement of the points on the stator surface and the circumferential speed of the corresponding points on the rotor. In this case, (ad' a,2) denote the actual contact area in one wavelength, a01 and a02 arc points with the same circumferential speed of the points on the stator and rotor, and a p corresponds to the wave crest of the traveling wave in the stator. X 10-6

~

S-

~

0

u

~

2

~

0

'0

E

..,"0

Friction layer

J1"

]

-I

.

.

,

'6

]

0

§

jUp

aclj

0

,

.,

10

15

.,

.OJ

Axial

.

i Contact area:

""'" -2 ]. -3 -5

0.2

~

-0

20

" 6

0.6

U

0.8 -5

e"

jac2

25

30

Angle coordinatel(")

Fig. 5. 27 Actual contact area in one wavelength

35

40

,

;I

0.2 0.4

~ ~

,

Contact area

0.4

OJ

'6

0

= " S

~;.

.,,

0.8 0.6

Circumferential

0

10

15

20

25

30

35

40

Angle coordinatel(")

Fig. 5. 28 Velocity components at contact interface in one wavelength

Figure 5. 29 shows the distribution of contact angle in the contact area in one wavelength. Moreover the radial and circumferential contact pressure distributions arc shown in Fig. 5. 30, which denotes that the radial and circumferential components of the contact pressure possess the same order in magnitude. Therefore it can be concluded that the power losses from the radial and circumferential friction also possess the same order in magnitude. It is found that there arc four different ranges of the contact area.

Chapter 5 (1) In (ad'

aOl)


151

the points on the stator surface have slower circumferential

velocities than those on the rotor in the negative direction of eo in absolute value. Meanwhile the radial velocities of the particles are along in the positive direction of er • so the contact angle ranges form -180 to - 90 The friction force fro in Eq. (5. 88) is along the reverse direction against the rotary direction of the ro0

0

•

tor. In this case the stator prevents the rotor rotating. I

.

.

Contact area

I

180

7' E

~

~ ?

Q

~

90

.§

"5

or>

.~

0

'6

'" g

U

~

.

Contaci area

I

4 3 2

I

~a - I g -2

- 90

-1 80 -5

.

7 X:..: F I-" O_ ' --~----!----------, 6 - - Circum ferential componenl - Radial compollclll

g -3

a ~l

0

10

15

20

0

25

30

35

40

u -4

-5

0

5

Angle coordinate/CO)

Contact angle distribution in one wavelength

Fig. 5. 29

(2) In (aOl • a p

)

10

15

20

25

30

35

40

Ang le coordinate/CO)

Distribution of contact pressure applied to stator

Fig. 5. 30

the contact angle ranges form -90 0 to 0 0. In this case the ro-

tor is propelled along in the reverse direction against the propagating wave. At the point aOl the stator and rotor possess the same circumferential speed. then they have no relative motion are the eo direction. So the friction pressure is along the radial direction absolutely. and the corresponding serve abrasion happens. (3) In (a p

•

a02) the contact angle ranges from 0 0 to 90 0

•

which denotes that

the radial velocity component of the points on the stator surface become positive. In this case. the circumferential velocities of the points on the stator surface are higher than those on the rotor. So the rotor is still propelled along the reverse direction of the propagating wave. The radial forcing component acting on the stator becomes positive. Point gential contact pressure

t'

aDZ

is the same as point

aOl.

This indicates that tan-

is absolutely along the radial direction. and the serve

abrasion exists in radial one. (4) In (a02 • a c2 ) the points on the stator surface have slower circumferential velocity than those of the rotor in absolute value. and the radial components of the points on the stator are in the negative direction of er • And the contact angle is between 90 0 and 180 0 • In this case the rotor will be prevented by the stator. Figure 5. 31 shows the mechanical power losses at the interface and the output power for various torque levels. In Fig. 5. 31 it is found that the radial friction force causes more dissipated power while no load is applied to TRUM. With the increasing load. the power losses from the circumferential friction become a main part. The power loss of

152


30

- - - TOlallosses allhe imerface

25

- -I n the circumferenfia l directi on

-

20

~ 30

"-

~ ; ., ~/

In theradia l direcri on

.--'.- .- .-

15

...... .,"

10

~; /

~

.,.,"

35 30 ::R e::

5

25

".

>.

u

"3:0

4

~ 6

3

5 20 "u E 15

2

10

Co

-

7

6

"

~ 6

O~--~~~~--~--~--~

o

0.2

0.4

0_6

0.8

1.2

OutpUI lorque/(N "Ill )

OUlpUI torque/(N" m)

(a) Lo ses al inlerface under various loads

(b) Transmission efficiency of interface fl , a nd outpul power lmder various loads

Fig. 5. 31 Losses, transmission efficiency and power for TRUM under various loads

5W, which is 60 percent of total losses in the interface, happens under the noload condition. However the power loss is only 2W, which is 10 percent of total losses in the interface when the motor is stalled. Then it is obvious that the power losses from the radial friction cannot be neglected: 49 50J. With the increasing torque, the rotor speed decreases, and the relative motion in the circumferential direction between the stator and rotor at the interface becomes more and more violent. In this case the contact angle becomes small, i. e. the circumferential component of tangential force increases at the interface, and the circumferential friction abrasion is more and more. Meanwhile the small radial component results in the smaller power loss from the radial friction, as shown in Fig. 5. 31( a) .

5.4

5.4.1

Electromechanical Coupling Model of TR UM and Its Simulation Electromechanical Coupling Model of TRUM

1. Dynamic equation of rigid rotor Under the assumptions of the rigid rotor, the rotor moves only have rigid body movement along and around the axis. M, W

+ C~ W =

Fi - Po

(5. 97 a) (5. 97b)

where M, and] , are the mass and rotational inertia of the rotor, respectively, and C~ and C~ are the damping along and around the axis, respectively. They are mainly from the bearings, friction materials, rotor damping material. Fi and MTi are the axial force and driving torque acting on the rotor. Po and MT are the prepressure and load torque acting on rotor, respectively.

Chapter 5


153

2. Electromechanical coupling model of ultrasonic motor From the dynamic model of the semi-analytical for stator in Eq. (5. 78) , utilizing the dynamic equations of the rigid rotor Eq. (5. 97) and generalized forces acting on the interface of the stator and rotor Eqs. (5.91), (5.95), and (5.96), we can make the comprehensive electromechanical coupling model of TRUM

t If t II

M,q+C, 4+K,q Mrw+

C~W

=

KV+ Fe

=

fndS- Po

SCe)

Jt~ + c~ /J =

sgn(VrB

(5. 98)

V r ) fre rdS - MT

-

e-l SCe)

The alternating current equation of flowing through the piezoelectric ceramics can be derived from Eq. (5. 67): 1

bc;,V+ KT 4=I

(5.99)

)=3

where 1= [IA III T' is the current column matrix of flowing through the two groups of ceramic electrodes attached to the stator.

3. Energy conversion and output efficiency of ultrasonic motor system The entire electrical energy can be described by the following formula. (5. 100) where Pin is the input power, P oot and P w are the output power and the loss power, respectively: (5. 101) (5. 102)

where t is the certain moment when the motor achieves a stable speed, and Tis the cyele of the alternating voltage imposed on the piezoelectric ceramic compo-

/J

nents, is the rotor rotation speed. P,k is the power loss of the contact interface, and P d, and Pdt arc the power loss due to the damping of the stator and rotor, which arc given as follows, respectively: (5. 103)

1 T

ft.

T

•

c~ (3' dt

(5. 104)

1 ft. T T T t l Vdt

(5. 105)

t

The input power is Pin

=

The output efficiency of TRUM can be defined as (5.106)


154

5. 4. 2

Performance Simulation of Ultrasonic Motor

1. Mechanical properties Taking the TRUM-60 as an example, based on the proposed modcl the performanee simulation was conducted. All the material and structural parameters, the imposed voltage and pre-pressure are the same as in Section 5. 3. 5. The motor's characteristic curves simulated and tested are shown in Figs. 5. 32 and 5. 33. The signs, are the sampling points from different tests for the same motor, and rough line is the simulation result from dynamic model proposed by the author. The dotted line is simulation results based on simplified contact model ignoring the radial slips between the stator and rotor. As simplified model exaggerates the driving effect, the predicted performance is higher than the testing results[51: Figures 5. 32 and 5. 33 show that 3D contact theory simulation results are quite consistent with the testing results. When the output torque increases to about O. 5:'\J·m, the motor achieves the maximum efficiency, close to 30 percent.

"+"

200 r - - - - - - - - - - - - - - - - ,

- - - Simplified model Proposed model Measured results

175

?

C

125

ioo &

75

•

530

..

Vl

+

+

150

50r-----.-.~_~_-.----------, 45 ." - - - Simplified model .. .... Prorosed model ';f 40 ~. .. Measured results " i; 35 ,/ G

i:§

t

25

~20

8

50

15

,,

25

0.2

0.4

0.6

0.8

1.2

1.4

0.2

Output torque/(N . m)

Mechanical characteristics for TRUM-60

Fig. 5. 32

0.4

0.6

0.8

1.2

1.4

Output torque/(N . m)

Output efficiency characteristics for TRUM-60

Fig. 5. 33

2. The transient characteristics of the motor The transient characteristics of USM refer to the startup and shutdown performance. Its simulation can be done with Eq. (5. 98) , and the results are shown in Fig. 5. 31. The results show that the motor can arrive at a stable operating within only about O. 8ms from the startup. Due to the frictional brake of the interface, the motor's shutdown time is even less, just about O. 6ms to complete stop. The responses of various loads show that motor's transient response time changes a little, only the output speed decreases step by step. 3. Influence of pre-pressure on the output characteristics With the increasing pre-pressure, the contact area between the stator and rotor becomes larger, and then the interface energy loss and power transmission effi-

Chapter 5

Operating Mechanism and Modeling 01 Traveling···

155

ciency will also change. The contact area between the stator and rotor and the compression distribution of TRUM-60 in a wavelength are presented under different pre-pressure as shown in Fig. 5. 35. With the increasing pre-pressure, the contact region becomes larger. Fig. 5. 36

-€

ISO 160

I~

140 120

OAN·m

100

0.6N·m

SO 60

a-

~

is

g 0.6

O.SN-m

13 0.4 .s

0 u" 0.2

0.2 0.4

0.6

1.2 1.6

O.S

0 -5

1.S

lims

3.5

k, k uP1

=

0

(6.7)

Multiplying Eq. (6. 7) on the left by I/>Y leads to (6. 8)

The mass matrix and stiffness matrix is symmetry. so we can obtain

I/>Y(K-w;M) =I/>Y(K-w;M)T

[(K-w;M)l/>kJ T

=

=

0

(6. 9)

Therefore. Eq. (6.9) can be written as ".;r

'f"k

[dK, _ 2 dM, ] A, UP1 Wk UP1 'f"k

_

uWk, A,T A, 2Wk d P1'f"k M 'f"k

=

0

(6. 10)

The sensitivity of the kth modal frequency Wk to structure parameter P1 is defined as

UWk dP 1

=

1 A,T [dK , dM] A, 2Wkl/>l'Ml/>k 'f"k uP1 -Wk UP1 'f"k

(6. 11)

According to the second kind of mode normalized method mentioned in Chap. 1. we can obtain (6.12)

1 Therefore. Eq. (6.10) becomes

UWk __1_ A,T [dK _ 'Yk d P1 2Wk UP1

2

Wk

dM] A, 'Yk UP1

(6. 13)

The relative sensitivity of modal frequency to structure parameter is

S1< ,

"- P

=

P1UWk Wkdpj

=

AA,T 2 'Yk

2Wk

[dK _ '1 uP 1

, dM] A, uP1 'Yk

Wk '1

(6. 11)

Based on FE analysis. it is difficult to describe the mass matrix M and stiffness matrix K of discrete structure as a continuous function of design variables. So. Eq. (6.11) can be solved by using difference approximation and perturbation method. When parameter P1 has a perturbation LP1

•

the matrix M and K will

have perturbation LM and 6,K. respectively. Therefore. we can obtain (6. 15)

6. 4. 2

Sensitivity Analysis of Stator for TRUM-60

Table 6. 1 shows the structure parameters of some TRUM-60 stator developed by PDLab. which arc illustrated in Fig. 6. 4. The material of the stator is phosphor bronze.

174

Ultrasonic Motors Technologies and Ap plicalions Initial size of TRUM-60 stator(Unit: mm)

Table 6. 1

PI

P,

P3

p,

Pc

32

44

44

60

60

P7

ps

po

PIO

PI I (Tooth number)

0.7

0.6

1.5

0.5

72

Pc

Based on the strueture parameters in Table 6. 1. the stator FE modcl (3 168 nodes and 1 584 clements) is built. When the parameters have perturbations (increased by 1%), it can be obtained to calculate the sensitivities of the operating modes Bog and other adjacent modes B15 • which are out-of-plane bending modes with one nodal circle and five nodal radiuses, and E21 , an in-plane extension-contraction mode. as shown in Fig. 6. 10. Although the stator has many types of modes. for each type of modes its mode frequencies increases with the mode order. It means that the frequency sensitivity of the mode of some type to design variables can represent that of this type of modes. According to the sensitivities of different types of modes. it is possible to fulfill mode separation. 2

x 10"'

I

1.5 I

I

I

•

t

:

:

I

- - - - - ~ - - - - - -:- - - - -- +-- - - - -:- - - _. --i - - . 1/-+£&&,1~l , , ._ J ______ , ____ _ ----- 1. ------'-----"-, , , - -- , I ____

-----

x:

~

I

~

;-- n-----

f --- ' -----~ ~- --- · ----

:,

- 0.5 - - - - -

I

_ 1I

I _

_

_

..L _ _ _ _ _ I

I _ 1_

_

_

_

0.

_

I

I

_ __ _ '- _ _

_

__

J _ __ _ __

I L

_

_ _

_

_

, :

____- L_ _ _ _ p, p, p,

~

lia '

__= r__~~__rm~~Ub~~~~~~--~ I~I ~ ~ ~ : ~ :' :,L..J

1 :,, ;;;1 -1,1

O ~--~~_,

-1 L-__

_ 8.,. c::::::::J B"

I

_ L_ _ _ _~_ _ _ _L __ _ _ _L __ _~_ _ _ _~

p,

p,

p,

p"

Design variables

Fig. 6. 10 Sensitivity of Bog mode and its adjacent modes of TRUM-60 to structure parameters

Figure 6. 10 shows that the sensitivity of Bog to parameter PI is very small and that of B15 and E'l to P1 is relatively bigger. Therefore, it is possible to separate Bog from B 15 and E'I by adjusting PI. The sensitivities of E'I to structure parameters Pc , P8' and PlO are contrary to that of B09 and B15 • So, it is possible to separate in-plane extension contraction mode E21 from others.

6. 4. 3

Mode Separation of Stator for TRUM-60

In order to validate above method. the stator of TRUM-60 shown in Fig. 6. 9 is chosen as the example of the mode separation. The structural sizes of the stator are given in Table 6. 1. The frequency of the operating modes and other adjacent modes are calculated and are shown in Table 6. 2. It is noticed that the frequency

Chapter 6

175

Design and Manufacture of Traveling Wave···

of the operating modes Bog are close to those of other modes, i. e. 113Hz higher than that of B I5 and 812 Hz higher than that of E 21 • In this case the modal mixture exists. It is possible to separate the operating modes from the non-operating modes by reducing the frequencies. According to the sensitivity analysis in Fig. 6. 10, the decrease of PI can reduce the modal frequency of B15 and E21 , which results in the mode separation expected. When PI is 28mm. the calculated modal frequencies closing to Bog are shown in Table 6. 3. There is the difference of 2741Hz between Bog and B15 after PI is adjusted. In practice, it is found that the normal operating of an ultrasonic motor can be hardly affected by the interference modes with the frequency difference of 2kHz at least from operating modes. As a result, the modified stator can meet the design requirement. Table 6.4 shows the measured modal frequency of the stator of TRUM-60 before and after the mode separation. Table 6.2

Calculated modal frequency of stator for TRUM-60 before mode separation

Various types of modes Modal frequency/Hz

Table 6.3

38 243

38 612

39 055

44 325

Calculated modal frequency of stator for TRUM-60 after mode separation

Various types of modes Modal frequency/Hz

Table 6.4

35 784

36 229

38 970

42 764

Measured modal frequency of stator for TRUM-60 Various types of modes

pdmm E 21 /Hz

Bog/Hz

Before mode separation 32

38 167

38 321

39 023

13 856

After mode separation 28

35 413

35 824

38 660

41 367

10

~E ~

'u 0

5

~

0

36

38 j lkHz

40

Fig. 6. 11 Measured frequency response and mode shapes of TRUM-60 after mode separation

176


Based on the adjusted sizes. another stator is manufactured. The out-of-plane bending modes are observed by PSV-300F-B. The results are illustrated in Fig. 6. 11. which shows the peak and modal shape of out-of-plane bending mode B 15 • because only out-of-plane modes can be measured by PSV-300F-B.

6.5

Optimal Design of Stator

After determining the structure form of a stator. material properties and operating modes arc given. the parameter optimization of the stator structure is followed. In this section an optimization model is presented from the design requirements of the stator. taking into account the nature dynamic and dynamical response. Based on parametrieal finite element analysis in Section 6. 3 and sensitivity analyses in Section 6. 4. the sequential quadratic programming is adopted1l7 -

6. 5. 1

Optimal Model of Stator

Parameter optimization of a stator is discussed in this section. and design parameters are continuous in variation ranges. After the parametrieal finite element model in Section 6. 3 is applied. and the optimal design of the stator can be considered as the mathematical model of the corresponding optimization. In this way the optimization of the stator parameters is conducted with the given ultrasonic frequency. operating mode. appropriate difference between the operating mode and others and the special operating ranges. The optimal model is mathematically written as follows max s. t.

obj=Wov" p:h~Pi~P:'h

(6.16)

Lf~2kHz

fo~20kHz

where VB is velocity amplitude in the circumferential direction. p~h and p;'h are the upper and lower bounds of variation range of the ith structure parameter. respectively. Lf is the frequency difference between the operating modal frequency and the nearest interferential modal frequency. fa is the operating modal frequency. While some stator parameters are fixed. the others arc variable. namely design variables. Design variables arc constrained by factors of the structure form and processing technique. such as the minimum of the tooth groove param-

eter PlO is o. 3mm. Based on the above mentioned. the first elass constraints are formed. namely boundary constraints. The second elass constraints are formed by requirements of dynamic perform-

anees. namely performance constraints. The two constraints arc derived from enough large frequency differenee(Lf >2kHz) and operating modes in the ultrasonic frequency range. The relationships between stator parameters and dynamic performances obtained by finite element analysis are nonlinear implicit functions.

Chaptcr 6

Dcsign and Manufacturc of Traveling Wavc···

177

In Eq. (6.16) the dynamical response performances are regarded as the object. The limit speed of an ultrasonic motor is theoretically derived from the circumferential speed of the points on the stator surface- 18J . The speed factor wnWO (product of operating modal frequency and axial amplitude) is regarded as the evaluation index119-zoJ. According to the simulation analysis in Section 5. 1. 3, the output characteristics of an ultrasonic motor are greatly related to axial amplitude and circumferential velocity. The product of axial amplitude and circumferential velocity Wo v" can be regarded as another evaluation index: 6: , which can be transformed to wnW;. Compared with speed factor, the index much more emphasizes the contribution of axial amplitude to the output characteristics of USM. So the latter index is adopted here. The axial displacement and circumferential velocity can be gained from the harmonic response analysis under the conditions of the certain operating modal frequency fa and excitation voltage.

6. 5. 2

Example of Optimal Design of Stator

The exemplification of optimization design of TRUM-40 stator IS gIven here. The parameters of a piezoelectric ceramic ring are known, the inner and outer radiuses are 28mm and 40mm, respectively, and the width is o. 5mm. Firstly, the B07 modes are adopted as the operating modes based on trial calculation. According to the similar parameters of TRUM-60 stator and processing technique, the parameters of TRUM-40 stator are defined preliminarily. The five parameters Pz, P3' Pl' P5' and Pll are regarded as fixed and listed in Table 6. 5, the others as design variables. Table 6.5

pz 28

P3 28

The fixed parameters of stator for TRUM-40 (Unit:mm) Pl1 (Tooth number) Pc 40

40

56

The initial values of p, which reads [ Pl ,Pc, P7 ,p, ,P9 ,PlO J, is [20 o. 5 o. 5 5 2.0 o. 5J, and the upper and lower boundaries of variation range of the design variables plb = [11 o. 5 o. 5 o. 5 1. 0 o. 1J, pub = [21 2. 5 1. 0 2. 5 3. 0 o. 8].

o.

The mathematical form is as follows mm s. t.

obj=-Wov,,= f(pJ cj (pJ = Pi- p~b~O(i= 1,6,7,8,9,10; j= 1,2, ... ,6) C5

C ll C1

(pJ = p;'b - Pi~O (pJ = /:::,.f- 2~0 (Pi) = fa - 20~0

. }

C10

,(Pi)=50- fo~O (6. 17)

The optimization problem is solved by a numerical iterative search algorithm, namely sequential quadratic programming (SQP) algorithm. The sensitivity analysis method in Section 6. 3 is used to compute sensitivities of the operating modal frequency and frequency difference, and the forward difference method is used for the sensitivity of the evaluation index, the difference step is one percent

178


of current values. The search direction is solved by the quadratic programmmg sub-problem formed by the sensitivity analysis results. and the search step is solved along the solved direction by unconstrained problem, namely min obj1 =

-Wov" +r[2.:

1min{O.cJ(p)} IJ.

here the penalty factor r is 1 000. obj1 is

the sum of the object and penalty term. The convergence eriterions of dot pitch (2-norm distance) and the decrease of the objective function arc adopted, and the values of termination accuracy are 1. Oe-3. This algorithm converges within 31 steps, and Fig. 6. 12 shows the change process of the object and penalty term.

.,o ""

'vvA

1200

.

Q.

~

;:; 'E' o

800

'"

'0

,1L\ ,

\.

"

,

400

\ \.._~-

o o

5

10

20

15

25

30

Iteration teD

Fig. 6. 12

Change process of sum of object and penalty

The initial and optimal design parameters arc listed in Table 6. 6. Compared with the initial, the evaluation index of optimum is lower, but satisfied with the constraint of frequency difference. Restricted to the processing technique, the final design is obtained by retaining the first decimal place. Table 6. 6

Design variable

State variable

Initial and optimal design parameters of TRUM-40 stator Variable

Unit

Initial value

Optimal value

Final value

PI

mm

20. 0

17.635

17. 6

Pc

mm

0.5

0.506

0.5

P7

mm

0.5

0.694

0.7

ps

mm

0.5

0.501

0.5

pg

mm

2. 0

2.001

2.0

PlO

mm

0.5

O. 504

0.5

!::,f

kHz

0.987

2.096

2. 191

fo

kHz

37.010

10.356

10.321

Wov."

p'm ·m/s

6.639

5.276

5. 150

The finite element model of the final stator is established and solved to obtain and other modes shown in Fig. 6. 13.

B07

Chaptcr 6


179

FEM model ofTRUM-40 talor

B07(4032 IHz)

Interference rnode(38 130Hz)

Interference rnode{4 760Hz)

B..(31 27 1Hz)

B",,(5 1 00 I Hz)

Fig. 6. 13

B07

modc and ncarcst modcs of stator for TRUM-10

According to the final design, the stator manufactured is tested with PSV300F-B, the result is shown in Fig. 6. 11. The three peaks in Fig. 6. 11 are corresponding to B06 • B07 , and B08 modes. respectively. Exciting at the fixed frequency 40 143Hz. the axial amplitude of TRUM-40 stator is 3fLm. 20

~ ~

"""2

"§,

"0

15

40 143Hz

10

'" ::E

5

0

J\..... 30

Fig. 6. 14

6.6

Frequency/kHz

40

50

Frequency responses of stator for TRUM-40

Adjustment of Two Phase Modal Frequencies of Stator

It is important for the good mechanical characteristics and stable operating of USM to make the frequencies of two orthogonal modes of the stator to be the same. In reality, due to the fabrication error and heterogeneous materials, the two frequencies do not coincide with each other. Fig. 6. 15 shows the modal nephogram and the frequency response characteristics of a TRUM-60 stator. When the driving voltage of phase A is applied to the stator. if the frequencies of

180


the two modes coincide with each other, only one peak excited by A phase appears in the frequency response curve. corresponding to one modal shape; if their frequencies do not coincide with each other, peaks A and B excited by A phase appears in the curve. When the driving voltage of B phase is applied to it, the corresponding modal nephogram can be obtained. as shown in Fig. 6. 15(b). In this case the excitation frequency is elose to those of peaks A and B, then the response won't be satisfactory. Moreover, a rotary mode can also be excited by only one phase excitation. This is why a traveling wave can come into being in a stator with only one phase. Therefore. when the two phase excitations are made at the same frequency. the amplitudes of two standing waves arc different. According to Chap. 5. the traveling wave in the stator is distorted, which will result in the decrease of the performance and the unstable speed of the motor. This section presents a modification technique for the stator to make the modal frequencies of two phases coincide with each other. ~

2

E E

"=' 1.5

"

"0

.~

Q.

...

E

C

-g ~

0.5 0

34

(a) Mode for A phase

Fig. 6. 15

6.6.1

40

(b) Mode ror B pIJa e

Frequency response curve and modal nephogram of TRUM-60 stator

Method of Adjusting of Two Phase Modal Frequencies

The modification method for the stator is proposed based on structural perturbation theory. An appropriate mass or stiffness is added on the stator to modify its modal frequencies. A portion cut from the stator can be considered as the addition of the negative mass or stiffness. The structure modification is small, so it can be assumed that the modal shapes change little- 2 ]-. According to Thomson's Structural perturbation theory:22 23J , the perturbation mass or stiffness have an effect on kinetic energy and potential energy, respectively. Based on Lagrange equations, the dynamic equations for the modified stator can be written. The new modal frequency can be calculated by solving the dynamic equations. The modification method can be

Chaptcr 6


181

obtained by the analysis of the computation result. Assuming V" P,' and C, are the volume, density, and stiffness matrix of a stator, respectively. The distributing mass with density PI and volume VI' whose deformation is ignored, is added to the stator. Meanwhile some material with stiffness matrix C2 and volume V, , whose mass is ignored, is placed on the stator. The derivation of Eq. (5. 28) in Chap. 5 leads to the stator vibration displacement, velocity and strain vectors as follows

CPmq

(6. 18a)

U = CPmq

(6. 18b)

e=cp'mq

(6.18c)

u

=

where CPm is the shape function matrix, CP:n is the strain matrix. Eq. (6. 18c) depicts the transformation between strain tensors and modal coordinates. Considering the added mass and stiffness, strain energy and kinetic energy of the stator can be written as

f

~

=

~ L/,uTu dV + ~ L/IUTU dV

v, eTC,

e dV +

~

f

=

V

2

eT C2 e dV

(6. 19a) (6. 19b)

Inserting equations above into Lagrange equation without dissipation leads to

1,2,···

(6.20)

where CPmk is the kth column of the matrix CPm. From Eq. (6.20), the modal frequencies of the stator before and after modification can be calculated as follows

(6.21)

[L,

cp'mkTC, cp'mk dV

f

Vs

+~

CP~'k P, CPmk dV + :i= .i

L,

f

cp'm/'C, cp'm} dV I t (6. 22) cp~,} Pi CPmj dV

V1

The analysis of Eq. (6. 22) leads to the following conelusions: (1) If only the ametabolic distributing mass with density PI and volume VI added to the stator, the modal frequency becomes

IS

(6.23)

182


It is lower than the old modal frequency. (2) If some material with stiffness matrix C, and volume V, , whose mass is ig-

nored, is added to the stator, the new modal frequency is

(6.21)

It is higher than the old modal frequency. (3) If the hole is drilled somewhere with volume Vo in the stator, the modal frequency is

f

cr(k T C, cP'mk dV - ~

r LcP~'k v

f

1

cP'm, T C, cP'm; dV- ,-

v

~ ( cP~,; p, cPm;dV

p, cPmkdV -

(6.25)

According to weighted orthogonality of modal shape function, Eq. (6.25) can be induced into

Kk - K'k ----''--------';Mk -M'k where

Kk Mk .6

=

= =

W

iv, f

V,

cP:nk T C, cP:nk dV,

,

K'k

=

(6.26)

~w

;::::::::: Wk -

f cP:nk cP:nk dV f cP~k cPmk dV T

C,

(6. 27a)

Vo

cP~k p,

(K'k -

wi

Mk

cPmk dV , M'k)

M'k

=

p,

(6. 27b)

Vo

(6. 27c)

where Kk and Mk are the kth modal mass and modal stiffness of the original stator, respectively. K'k and M'k are the kth modal mass and modal stiffness after modifying the stator, respectively. Considering Eq. (6.26) , we can draw the conelusions as follows: If .6 w 0, the modified modal frequency is lower than the original one. Therefore, drilling hole at someplace can increase or decrease the modal frequency. The drilling hole is more feasible than the other ways. The following section will define the position of the hole.

6.6.2

Example of Adjusting of Two Phase Modal Frequencies

The finite element model of the modified stator is proposed using ANSYS. The K~ - wiM~ value of every element in the case of B09 mode can be calculated from the element table of A='JSYS software, which is shown in Fig. 6. 16. From Fig. 6. 16, it is shown that the (K:- w;M~) values of the tooth groove located at the wave crest or trough are big positive values. If the part of the

Chaptcr 6


- . 1 e-04

Fig. 6. 16

- .7c- 04

- Ae - 05

Distribution of

- .Se - 06

183

.3c - OS

K:- wrM: for Bog mode of stator

tooth groove at the wave crest or trough is cut, the modal frequency will decrease. On the other hand, the (K'k - wiM'k) values of tooth top located at the wave crest or trough arc big negative values. If the part of tooth top is cut, the modal frequency will increase. At the nodal diameter, whether the part of the tooth groove or tooth top is cut, the modal frequency changes very little. USM's operating depends on two orthogonal modes. So one point located at the wave crest of the modal shape of a stator, must lie at the node radius of another one, as shown in Fig. 6. 15. In order to make the two modal frequencies the same, we shall cut the tooth groove, which lies at the wave crest or trough of the mode with higher frequency. In this way the higher modal frequency will decrease while the other modal frequency changes very little, so the two modal frequencies can be equal to each other. An application example is presented·"·. Fig. 6. 15 shows that the two modal frequencies measured arc 36. 81 kHz and 37. 23kHz, respectively. And the difference between the two modal frequencies is o. 42kHz. Using the PSV-300F-B we can find the operating mode shape with higher frequency, and deepen O.1mm on the tooth groove at the wave crest of the mode, as shown in Figs. 6. 17 and 6. 18. The result of the modified stator is obtained by PSV-300F-B. It is noticed that the two modal frequencies are 36. 67kHz and 36. 63kHz, respectively, and that the difference of them is only o. 04kHz now. The difference between two modal frequencies decreases by ten times. Recently, a new hole-drilling device, which can automatically adjust the two modal frequencies to coincide with each other, is fabricated by PDLab. It is applicable for the modal frequency modification of traveling wave type rotary USMs and rod shape rotary USMs.

6.7

Analysis of Flexible Rotor

In the early development of an ultrasonic motor, the disk spnng was used between the stator and rotor to produce the pre-pressure and the contact pressure between the stator and rotor, as shown in Fig. 6. 19. The rotor is called rigid ro-


184

tor, whose deformation is considerable small and ignored.

6

~

6 7

f-

N

0..

8 h,

Fig. 7. 9

9

10

Structure and FE model of stator

The displacement field of the element j can be expressed by the node displacements as ~~Xl

IVs l 3x3 ] ~~n

=

Nrn ~~n

(7.22)

204


where l3x3 is the unit matrix, &.n = [UI VI WI Us Vs Ws T is the displacement vector of the element nodes, and N m is the matrix of the shape functions. Thc strain tcnsor E of thc elcment can bc cxpressed as (7.23) where Lm is the matrix of differentiation operators. By substituting Eq. (7.22) to Eq. (7.23), it can bc obtained that (7.21) To nodes of piczoelectric elemcnts, the electric potcntial vcctor has to be added. Because in practical application the piezoelectric ceramic ring is very thin and thc voltages are applicd to its surface, the electric potential function can bc approximated as (7.25) wherc qu and qd arc the potentials on the two surfaces of the ccramic ring. The electric field is simplified as a constant and only imposed to the ceramic ring in thc thickness direction. Thcreforc it can be exprcsscd as

o E

=

[E,

(7.26)

o

wherc hI' is the thickness of thc ring. The stress tensor ( j of a metallic element can be written as (j

=

(7.27)

C,E

where C, represents the stress stiffness coefficient matrix of the metallic element. If defining the poling direction of thc ring as z direction, from thc sccond piezoelectric cquation, the strcss tensor of the piezoelcctric elemcnt can bc written as (7.28) where C; and e are the stress stiffness coefficient matrix and the piezoelectric constant matrix of the piczoelectric matcrial, respcctively. Whether it is thc metal or the piczoelectric matcrial, thc kinetic encrgy of thc element j can be defined as Tj

=

~f ~JT ~JdV 2 vP

=

~ 2 ~JTM ~J m

mm

m

(7.29)

J

where p is its density and (7.30) For the metallic elcment, the potcntial energy can bc defined as

Chapter 7

Bar-type Traveling Wave Rotary Ultrasonic Motors

2If v

V~

T

e udV

2If v

=

T

e C, edV

=

IT

..

2~;" K:':m~;"

205

(7.31)

e

J

where (7.32) For the piezoelectric elements, the potential energy is expressed as (7.33) Substituting Eqs. (7.21) and (7. 26)into Eq. (7.33), it can be obtained that

V Jp -where

",jT KJmmUm "'J

21

Urn

jK:m ~ f JI'

21

-

BT CEB

-

m

"'JT KJme qe

Urn

dV

m

f Bm e B,dV Vj

K;", -

1

TI'

(7.34)

(7.35)

Vj

From the second piezoelectric equation, the electric potential energy of piezoelectric element j can be expressed as

Wi,

=

~

LET (ee +tE)dV

~ q~K~m~;n + ~ q~K;,q,

=

(7.36)

J

where

{ K~m K~,

=

=

K;",T

=

LB~eBmdV j

LB~t B,dV

(7.37)

J

where t is the dielectric constant matrix of the piezoelectric material. According to the Hamilton's principle, the energy equation for the stator divided into N elements can be represented as (7.38) where 1'1 , V j

,

and W j are the kinetic energy, potential energy, and electric en-

ergy of element j, respectively. oWr and oW" arc the virtual mechanical and electrical works done by the external force and the charge, respectively. Making the modal analysis of the stator, Wf=W,,=O. Substituting Eqs. (7.29), (7.31), (7.33), and (7.31) into Eq. (7.38) and performing variation of ~m and qc' the dynamical differential equation of the system can be obtained:

{

Mmm ~m K~,~m

+ Kmm

- K"q,

~m =

=

Qq

F, - Kmcqc

(7.39) (7.10)

where Mmm , Kmm' K m" and K" arc the mass, mechanical stiffness, piezoelectric

206


stiffness and dielectric stiffness matrices of the stator, respectively. ()m is the displacement vector of the element nodes, (Fe - Km,q,) is the generalized force vector applied to the stator and F,. will be discussed in Section 7. 4. 2. Qq is the electric charge vector applied to the ceramic surface. The natural vibration equation of the stator can be expressed as

(7.41) Solving this generalized eigenvalue problem can lead to n eigenvalues and the corresponding eigenvectors, i. e. the modal frequencies and mode shapes. Table 7. 1 shows the comparison of the results calculated by two different methods with the experimental result when the stator material is stainless steel (3Cr13), and the piezoelectric material is PZT8. The calculation is conducted by this dynamic model (Matlab environment) and ANSYS software. It can be seen that these three results have reached good agreement and this dynamic model can be used to do the sensitivity analysis and optimization of the stator. Table 7. 1

Comparison of results calculated with experimental one

Modal frequency of the first bending vibration/Hz Tolerance relative to experimental result/ %

Calculated result by the author

Calculated result by ANSYS software

Experimental result

34 537

33 701

33 250

3.87

1. 36

Because the motor always operates elose to resonance, without other interference modes, ()m is simplified as

cp, ] [ql (t) ]

(7.42)

q, (t)

where CPl and cP, are the two orthogonal bending mode shapes, and q = [ql q2 ] T is the modal coordinate column matrix. Inserting Eq. (7.12) into Eq. (7.39), then pre-multiplied by cpT at both sides of the equation, it can be obtained that (7.13) where Fq = cpT F, is the modal force column matrix corresponding to the contact force vector F,. under the modal coordinate column matrix q, and K= - CPTK"" is the force coefficient matrix which represents the conversion capacity of the piezoelectric material from the electrical energy input to the mechanical energy. If the eigenvectors cA and cP, are normalized by the modal mass, then the modal mass, modal stiffness, and force coefficient matrix can be expressed as

(7.44)

~ 1/>,') (cfi[1 1/>01) •

=

1,2.3.··· .16

(7.47)

where I/>'i is the column matrix of the modal shape calculated by FEM. 1/>01 is the column matrix of the normalized reference mode. which is the first bending mode of the stator. and the latter can be expressed as follows:

1/>01

[4. 29

=

2. 92

- o. 96

- 1. 0

-

1. 13

-

o. 72

-

o. 37 o. 54

-

o. 56

o.

0.02

50

r

Here. taking i = 16. the first 16 modal frequencies and the corresponding MAC values can be calculated. The first 6 modes are rigid motion modes and the corresponding frequency values are equal to zero. So Table 7. 2 only lists 7-16 order modes. The eighth mode with the largest MAC value is the first bending mode. while the seventh mode with the almost same MAC value as that of the eighth mode is the orthogonal mode of the eighth mode. Mode calculation and corresponding MAC value MAC value Order flHz MAC value

Table 7. 2

Order

flHz

7 8

31 31 31 50 50

10 11

o. o. o. o.

537 538 656 499 499

996 998 670

12

022 O. 044

15 16

58 59 59 62 68

13

11

0.611 o. 177 0.251

898 553 551 554 238

o. 884 o. 888

As shown m Fig. 7. 9. the relation between the numbers of physical parameters and dimensions can be shown in Table 7. 3. Parameters PI • P, • P,. P5' and P7 are selected as the optimal variables based on the sensitivity analysis method in Section 6. 3. 2. Table 7. 3 Parameters and numbers

Dimension of the structural parameters

Structural design parameters of the stator

PI

pz

P,

P,

P5

P6

P7

rz

r3

r6

hi

hz

h3

hs

208

7.3.4


Objective Function

The design requirements for the vibration mode of the BTRUM have been discussed in Section 7. 3. 1. Doing the optimal design, these requirements must be considered in the objective function F for the optimization algorithm, which ineludes: (1) The first bending modal frequency fbi and the target design frequency ft should be as elose as possible, that is FI

=

I

fbi - ft

(7.18)

I

(2) The amplitude of points on the driving surface should be as large as possible, while the amplitude of points on the lower mass should be as small as possible. If ¢tr and ¢tt x represent the first bending mode shape values in the x direction of Node 1 and :'\Jode 11 in Fig. 7. 9, respectively, this means

F

2

-I ¢i,i" 1

(7.49)

¢i,~

-

(3) Piezoelectric ceramics group should be placed on the antinode( with maximum strain) of the first bending mode of the stator, which means the position of Node 7 should be ncar the middle of the antinode, that is

(7.50) where ¢;,~ and ¢t~ represent the bending mode shape values in the 01': direction of Nodes 6 and 8 in Fig. 7. 9, respectively. (4) The difference between the bending mode frequency and the nearest interference one should be as large as possible, that is

F,

=

I

fO

1

(7.51)

fO

_ ill -

_ iut

Based on the above four requirements and taking into account differences of orders of magnitude, the partial objective functions arc multiplied by weighted coeffieients, and the global objective function for the stator design can be expressed as

(7.52) i=l

where Pi stands for the weighted coefficients. The mathematical model of the optimal design of the stator can be expressed as a minimum problem with boundary constraints:

,

min F

=

b PiFi

(7.53)

i-"j

j

=

1,2,1,5,7

where p;b and P't' represent the lower and upper boundaries of the optimal variables, respectively.

7.3.5


The pattern search algorithm in Matlab toolbox is used in the optimization mod-

Chapter 7


209

el L37 .• and the boundary of the design variables is defined as plb

=

[7

pub

=

[9. 5

1. 5

1

6. 5

6 1

1J 10

(7.54)

5J

The start values of the pattern search algorithm are p"

=

[7.75

6

2.7

8

(7.55)

3J

The vector in Eqs. (7.54) and (7.55) represents the dimensions of the design variables PI • P, • P4 , Pc , and P7 in mm. The stator modal frequency at the start point is 31 538Hz. the design goal frequency is 36 OOOHz and the weighted coefficients arc f31 = 1. f32 = 1. 5 X 10' • f33 = 50, and f3, = 1. 2 X 10 5. Figure 7. 10 shows the iterative process and the optimal results of the design variables. The values of the partial objective functions arc compared in Table 7. 4. where F represents frequency of first bending mode, A represents amplitude ratio of point on the driving surface to the one on the bottom of stator, D represents distance between Node 7 and the middle of the antinode of the bending mode, and T represents tolerance between the frequency of operating mode and interference mode. It can be the seen that after the optimization, the modal frequency of the stator is 36 003Hz, the piezoelectric ceramic group is closer to the center of the antinode of the bending mode, and the difference between the frequencies of the operating mode and the interference mode becomes larger. In addition, the amplitude ratio of the point on the driving surface to the one on the bottom of the s ta tor increases to 10. 79, as shown in Fig. 7. 11.

-§

7000

<E

5000

"

1000

g

Convergence of the objective function value is at I 486

:'" "5 3000

:0-

0

0

10

5

I5

20

25

30

35

40

45

50

Iterative times

~

0 .:;;

"E"

:0 -;;;

.;

C.

0

0.0 10 0.008 0.006 0.004 0.004 0.002

0.0077

0'1i 5 0.001 02 2

Fig. 7.10

Table 7. 4

0.0092

0.0065

n

5

3 4 De ign variables

Iterative process and optimal results

Values of the partial objective functions after the optimal design

F/Hz

A

D

T/Hz

Start values

34 538

7.65

40. 129

117.5

Optimal results

36 003

10.79

O. 101

1 522. 7

This numerical example shows that the optimal model can meet all the design

210


7

After optimal design

4

"

~ ~

"

:;:;:

•

Before optimal dcsi!:,'ll

5

"d 0

Model proposed

Ansys

6

by author

Fitting curve

*

---

D

_ ...

0

_---

.. ~

2

D· •••.••. ". 0

2

I 2

4

'.

~-~~.-.-.-.-.-.-.-.-.-.-.-.

5

6

7

8

9

10

11

Numbers of nodes

Fig. 7. 11

Comparison of mode shapes bdore and after optimal design

requirements for the stator and it provides the theoretical foundation for the optimal design of the BTRUM with SDOF.

7.3.6

Modal Frequency Modification of Stator

The stator of BTRUM is axially symmetrical. and has two orthogonal operating modes with the same shape and same frequency. However. in reality. the two operating frequencies usually don't coincide with each other because of the influence of heterogeneous materials. irregular elamping pressure of screws or a machining error. Thus. when the stator arc driven by the two phase voltages at the same frequency. the amplitudes generated are different. Then the elliptical motion of the points on stator is distorted. resulting in the unstable speed of the motor. The BTRUM presented in this section is composed of one stator and two rotors. The characteristics of the velocity response versus frequency for the stator is shown in Fig. 7. 12. Solid and dash-dot lines indicate frequency responses excited by phases A and B. respectively. The mode shapes of two orthogonal first bending modes arc shown in two small figures. Their two frequencies arc 26. 56kHz and 26. 73kHz. respectively. The difference between the two frequencies is 170 Hz. which doesn't satisfy the requirements of the design. Generally the difference should be no more than 100Hz. To solve this practical problem. a modification technique is proposed which can adjust the difference to no more than 100Hz in an assembled state. 1. Finite element analysis In Section 6. 6. according to the structural perturbation theory. forming a recess portion in a predetermined portion of the stator. or adding an appropriate mass can change the modal frequencies of the stator. From Eq. (6. 27 c). it is noted

Chapter 7


,,

12

W

10

~

8

c.

6

"" ,~ ~

211

- - - Phase A

_. -

PhaseB

?>

g

~

2

0 20

22

24

32

34

36

38

40

42

Freq llency/ kHz

Fig. 7.12

Velocity response versus frequency for stator

that the modal mass matrix M, is a positive number, hence the increase or decrease of the modified modal frequencies depends on the value of (K~ - wiM~) and modification ways. There are a number of ways to modify the modal frequencies. Compared with other adjustment ways. chamfering the stator is more convenient. In this section, how to define the position of chamfered portions is presented using finite element analyses. The finite element model of the stator is built using ANSYS software. The (K: - wiM:) value of the first bending mode is calculated utilizing the clement table, as shown in Fig. 7. 13.

Ant inode

- 243.068 - 167.20 1 -91334 - 15.467 - 205. 134 - 129.268 - 53.40 1 22.466

Fig. 7. 13

60.J

99.333

Distribution of ( K: - wkM:) value of first bending mode

Figure 7. 13 shows that the ( K:- wiM:) value of the area closed to the line passed through the antinode changes from positive to negative along the line from both ends to the middle. Therefore. it is effective to increase the modal frequency by removing the part of mass on both ends. It is also effective to decrease the modal frequency by removing the part of mass on the middle. However, the mid-

212


die of the stator is piezoelectric ceramIc nngs. We can't modify this part. In fact. when the mass elosed to the middle is removed. it is also helpful in decreasing the modal frequency . If the regions ncar the line passed through the node are concerned as shown in Fig. 7. 13, the ( K~-w;M~) value doesn't change significantly from both ends to the middle. In particular, the ( K; - w;M~) value of the region between two slots of the stator is elose to zero. If the mass of these regions is removed, the modal frequency docs not change significantly. BTRUM operates using two orthogonal bending modes. If one point is located in the antinode of one of modes, the point must be located in the node of the other mode. Thus. when the regions near the line, which passes through the antinode of one of modes. arc modified, and the modal frequency of the mode can be changed. Meanwhile, the other modal frequency changes very little.

2. Example of modal frequency modification Modal frequency modification will be carried out for the stator mentioned above. The modal frequency induced by phase B is higher than that by phase A. Therefore. in order to adjust the modal frequencies to achieve the same. it is necessary to reduce the modal frequency corresponding to phase B and keep the modal frequency corresponding to phase A unchanged as much as possible. Chamfered portions on the stator are shown in Fig. 7. 11. The chamfered planes are perpendicular to the nodal diameter of the mode corresponding to phase A. In this way, the modal frequency to phase B can be decreased and that to phase A changes slightly. Fig. 7. 15 shows the velocity response versus the frequency of the modified stator. The solid line and the dash-dot line indicate the frequency response curves to phase A and phase B. respectively. The two modal frequencies of the stator arc 26. 23kHz and 26. 30kHz, respectively, which have the difference of 70 Hz that the original value is 170 Hz. After the stator is modified, this meets the requirements for the design. The efficiency, running stability, and output performance of the motor can be improved. II

10 ~

E

9

~

8

'" .E

g.

6

C

'"

4

·0 0

~

- - PhaseA

7 5

_ . - PhaseB

3 2 I

0

Fig. 7. 14 on stator

Portions modified

28 30 32 FrcquencylkHz

34

36

38

40

Fig. 7. 15 Velocity response versus frequency after modified stator

Chapter 7

7.3.7


213

Design of Flexible Rotor

For BTRUM, the consumption of the input power can be divided into three parts L38 - : the first part is the damping loss in the conversion process from electrical energy to the mechanical energy. The second part is the friction loss between the stator and rotor, including their radial relative slip. This portion is 80 percent of the total energy 10ss L39- The third part is the inner loss of piezoelectric material. Thus, the key problem to improve the efficiency of BTRUM is how to avoid the relative slip between the stator and rotor. The two typical structure of rotors will be compared in the next section, and it can be proved that the rotor with the inner flange structure can effectively reduce the relative slip. The rotor with an outer flange structure is shown in Fig. 7. 16 (a). When the stator is in stationary state. the contact point on the stator and the point on rotor are assumed as points a and A. respectively. before the vibration of the stator is excited. In the xOz plane, when the stator is operating, the rotor is deformed by the action of the stator. In the condition, point a moves to a' and point A moves to A'. Because the directions of the displacements aa' and AA' are opposite in the x direction. the radial relative slip is inevitable.

I

zL' :

o

x

Enlarged local pari orthe map (a) Contact between outer HaJlge and the rotor

Rotor

zL "

O x .. ··•

Enl arged local pan oflhe map

(b) Contact between inner nange and the rotor

Fig. 7.16

Radial slip between stator and rotor

Reversely, for the rotor with an inner flange strueture. it ean be found from

214


Fig. 7. 16(b) that the directions of the displacements of aa ' and AA' are identical in the x direction. and the radial relative slip is reduced. This conelusion can be further verified by finite element analysis. Two dimensional :'\Jode-to-Surface contact model is used to simulate the contact behavior between the stator and rotor. The element type is contac18. and the bending deformation is simulated by the synthesis motion of the stator in the x and z directions. Fig. 7.17 shows the results calculated by FEM. From Fig. 7.17(a). we can see that for the outer flange rotor. the direction of the displacements of the stator and rotor in the contact area are exactly opposite to each other in the :x; axis direction. and the relative displacement increases with time. From Fig. 7. 17 (b). we can sec that for the inner flange rotor, the displacements of the stator and rotor in the contact area possess the same direction. so the relative slip between the stator and rotor is reduced: 13J • This simulation gets the same conclusion as the above theoretical analysis does. X 10'"

X 10-7

3

2

~

g

il '" ~

•

0

o

'"""

Displacemen. of J)Qint A Displacemelll of point a

"

;6;

Displacement of point A

-I

o

Displacement of point a

.tT,

-~-

Relationship between multistep

and mode frequency of stator

stator with multistep

Based on the analysis above, PDLab has developed one multi-mode L TUM with diameter of

o. 045m.

The difference between the first order longitudinal and

the second torsional frequeneies is

o. 3kHz.

This indieates that the stator with

multistep ean effectively make the first order longitudinal and the second order torsional frequencies elose.

4. Optimum design for amplitude of stator The amplitude of the stator is generally few micro, how to design the stator is of great significance to augment the amplitude of the end surface of the stator, then improve the mechanical performance of motors. According to the magnification theory of ultrasonic solid horn- 19J

,

the relationship between the stator's structure

and the amplitude of driving end is studied by Heming Sun and Chunsheng Zhao in detail under the first order longitudinal mode. Fig. 8. 17 (a) shows the finite element model of a stator. The material used for the part with larger section is the steel which has a higher acoustic impedance, whereas the material used for the part with smaller section is aluminum which has a lower acoustic impedance. Fig. 8. 17 (b) shows the first order longitudinal mode shape of the stator. From Fig. 8. 17 (b) , the displacement amplitude value at right end is much greater than that at zero point of coordinate

.T.

Thereby, the stator with multistep has really

the function to magnify amplitude. As a matter of fact, besides the stator with multistep, the shape of the stator can be made as exponential function type, conic type, catenary curve type, and so on.

In a word, the stator with variable cross-section and the material with different acoustic impedance can notably magnify the amplitude at the free end of stator.

Chapter 8

Steel

PZT

Ultrasonic Motor Using Longitudinal-Torsional···

Duralumin

~

243

I

I

...Y -30 (a) Finite element model

Fig. 8. 17

8.3

-15

o

15

30

x/mm

45

(b) Amplitude distribution along axial direction

Change of displacement amplitude with variation of cross-section

Contact Model between Stator and Rotor

Like the traveling wave ultrasonic motors, the contact model between a stator and rotor of L TUM is vcry important. From thc model, we can conduct the simulation to prcdict thc load charactcristics of L TUM. W c can also analysc that thc load charactcristics are influenced by the amplitudes of longitudinal and torsional vibration, the pre-pressure between the stator and rotor, the friction material, and geometrical and physical parameters. It is to be regretted that so far as it goes, we cannot find a satisfactory contact model, since Ueha and Tomikawa firstly proposed the contact model in 1993: 20J • Based on the model, Jifeng Guo, Shujuan Gong and Xiao Liu, et al. have developed a model- 21 - 22J which takes advantage of energy conservation principle in one period, and derives the expressions of the output torquc, contact duration and othcr parametcrs in 2003. Finally, they applied these expressions for simulation.

8. 3. 1

Modeling of Contact Interface

In this section, we citcd and improved the model developed by Refs. [21J and [22]. Whcn thc motor opcrates in a stcady state, an elliptical motion trajcctory of a point on the stator surface is shown in Fig. 8. 18. The point moves along A, B, C, and D. U means torsional displaccment of thc point, and U y mcans the longitudinal displaccmcnt of the point. Thcy can bc cxpressed as E

{

U

E : - UEsinwt Uzcoswt -

(8.7)

Uz

wherc U x and U z arc the torsional and longitudinal amplitudcs of thc point respcctively. In the steady state, the stator's vibration is transformed into the rotor by intermittcnt contacts bctwcen thc stator and rotor with a friction layer. The deformation of friction layer can be described by Fig. 8. 18, where point a indicates thc initial contact point between thc stator and rotor, and point b indicates the final contact point. The axial displacement of the initial contact point is U m , ({Ja and ({Jb indicate thc initial and final contact anglcs, and thc corresponding timcs are ta and t b , rcspcctively. Then the contact duration ({J is cqual to (({Jb - ({Ja)' in which the rotor is always in contact with the stator. The contact time during every vi-

244


B

A

wt

f)

Fig. 8. 18 Elliptic locus(lcIt) and contact conditionCright) of a point on stator surface

bration cyelc is Tc= cp/w. At this momcnt, thc relationship bctwccnCPa and contact duration cp is cpa = (rc - cp) /2. Hence (8. 8)

Rcgarding thc rotor as rigid body, thc forcc acting on thc rotor along axial direction is shown in Fig. 8. 19, where Po is the pre-pressure applied to the rotor by the spring, and P is the impact force acting on the rotor from the stator through thc friction laycr, thc rotor mass is m. and thc vibration pcriod of thc stator is T = 2rc/ w. Thc impact forcc P in onc vibration pcriod can bc expressed as t

E (0, t a )

t E (t a , t,, )

(8. 9)

t E (t b • 1')

where kr = ErS/ hr is the equivalent stiffness of the friction layer; E[O S, and hr are the Young's modulus. contact area. and thickness of the friction material, rcspcctively. C

~

ROlation axi s

Pre· pres lire /'0

Rotor

~~~~22222S:~- Friction layer 1111 pact force P

Fig. 8. 19

Axial force acting on rotor

In fact. the simplification model shown in Fig. 8. 19 is similar to a single de-

Chapter 8


245

gree freedom system which is made up of a rotor and a spring. The natural frequency of thc system is lower due to lower spring stiffncss k. and thc frequcncy of thc impact forcc acting on thc rotor from the stator is abovc 20kHz when thc motor operates in a steady state. therefore the rotor can be considered as immovablc obj ect approximately in axial dircction. That is. ignoring thc momcntum variation in the axial direction. the axial impulse to the rotor in one vibration period is

[cp - Po)dt

[mdv

=

=

(8. 10)

0

from which

f T Pdt

=

Po 2IT

(8. 11)

w

o

In one vibration pcriod. thc impact forcc is influcnccd by thc contact time ta to tb' namely by contact anglc rpa to rpb. Therefore Eq. (8. 11) can bc simplified as

f::Pd(wt)

=

2ITPo

(8.12)

Substituting Eqs. (8. 8) and (8. 9) into Eq. (8. 12). it can bc obtaincd

2ITP o

=

klU~

[2sin(rp/2) - rp cos(rp/2) ]

(8. 13)

Equation (8.13) indicatcs that thc contact duration rp is related to the prcpressurc. elasticity modulus. contact arca. thickncss of friction layer. and longitudinal amplitude value. If these parameters are given. the contact duration can be obtained. When rp=2IT. the stator contacts with the rotor during the whole vibration period. and the pre-pressure is kfU, all the time. When the pre-pressure is greater than kIU,. the stator will contact with the rotor always. klU, is a critical value between continuous and intermittent contact. so the critical pre-pressure is defincd as P,=kIU,. Therefore in Eq. (8. 9) the impact force can be revised as

Po < P, Po ;?: P,

= =

klUx klUx

(8. 14)

From Eq. (8. 7). thc tangcntial vibration velocity of point on stator surface is and V" is the velocity amplitude of torsional vibration. then we have

Ux '

(8. 15) At the same time. considering that the rotor has certain inertia moment and thc contact time is short. thc rotation spccd of thc rotor can bc supposcd to be a steady value V,. Fig. 8. 20 shows the relationship between the rotation speed and the torsional vibration velocity of the stator surface during one vibration period. It is observcd from Fig. 8. 20(a) that when the stator begin contact with thc rotor and thc torsional vibration velocity is highcr than thc rotary spccd in thc moment. thc friction forcc F m will do positive work. as curve abo It can be also observcd from Fig. 8. 20(b) that when thc torsional vibration velocity is lower than

246


the rotary speed, the friction force Fm will do negative work, as curve ac and db. A friction coefficient is relevant to relative velocity between the torsional vibration velocity and the rotary speed, which is shown in Fig. 8. 21 and can be expressed as[23:

(8. 16) where /1d is a sliding friction coefficient. /1, is a static friction coefficient and k[ is the coefficient of adhesion, as shown in Fig. 8. 21.

(~ I

(~I

(a)

Fig. 8. 20

(b)

Relationship between the torsional velocity and the rotary speed of LTUM

Fig. 8. 21

Coulomb model

The total friction force between the stator and the rotor can be expressed as

Fm

=

ff,dS s

=

f/1

~ dS =

/1P

(8. 17)

s

Ignoring the friction moment induced by bearing and the damping moment caused by structure on the rotor. the torque produced in one vibration period should be the same as the torque consumed by the load, and the average output torque in one vibration period is

(8. 18) where Rov is the mean radius of the friction layer. In the case of Po < kIU~, there arc two situations: (1) when Pc) (8. 25) When Po = pc. at critical contact duration rp= 2rr, and substituting it into the above equation, its stall torque is the same as that formed under condition of Po>

p,.. In which the stall torque is not related to the pre-pressure value. At this condi tion, the stall torque cannot be increased by enhancing pre-pressure.

8. 3. 2

Friction Loss on Interface and Efficiency of L TUM

Friction loss on the contact surfaces can be calculated as

(8. 26) where V; is the torsional velocity of the stator, V, is the linear velocity of contact point on the rotor. The efficiency between the stator and rotor can be expressed as P01lt

(8. 27)

where P oot is the output power

P OUI where

iJ is

=

1

T

IT MT pdt . 0

(8. 28)

the angular speed.

In the case of Po

rpa' the loss can be divided as driving and block parts:

(8. 29)

Chapter 8

=


!L)_ (cosm

3 m -sin3 -.lkU 1( I y {k 1 V'"[-.l(eos 3 Tl 2

(2kl vy" + kl

V;, cos f

+,ud

-sin!L)] 2

!

V,,) [ (T - 1( ~ p)-

+ (kl V; + ,ud V, + 2kl V,V"cos f -

Tl

249

+ ,udV" cos f ) (sin f

(sin2rpl - sinrp) ] - COSrpl )

(,udV,cosf+klV;cosf)(rpl-1(~P)} (8. 30)

PdD

~ kfU

=

y

~

{kl V;, ( COSrpl -

cos 3 rpl )

(2kl vy" + kl V;, cos f

- ,udV,,) (

+ (kl V; -,ud V, + 2kl V,V"cos f + (,udV, cos f

~

-

T+ ! sin2rpl)

- ,ud V"COS f ) COSrpl

- kl V; cos f ) ( ; - rpl ) } (8. 31)

The total loss on the contact surface can be expressed as (8. 32)

8.3.3

Simulation of Performance of LTUM

From Eq. (8.25). we can predict the load characteristics and analyze the influence of the pre-pressure. exciting voltage and friction layer on the performance of the motor. Various parameters adopted in the simulation are shown in Table 8.2. Table 8.2

Parameters adopted in the simulation

Parameters

)J umerical values

Rotor Quality m/kg

2.0c-2

Pre-pressure Po / N

1. 2c2

Operating frequency

f /KHz

5. 2el

Elasticity modulus Er/Pa

3.5e-1O

Friction layer area 5/m 2

8.2c-5

Effective contact radius R,"/m

6.6e-3

Thickncss of friction layer hr/m

o.

Longitudinal vibration amplitudc Uy/m

3.0c-6

Torsional vibration amplitude Ux/m

3.0e-6

3e-3

1. Contact pressure between the stator and rotor The most essential influence for the output torque is the contact pressure between the stator and rotor. which is relevant to the longitudinal amplitude. pre-


250

pressure and elasticity modulus and thickness of the friction layer, as shown m Figs. 8. 22-8. 25.

~::l ~

0.

~

8

550

400 ,-----------------------, :...... \ - - U,= 1e-6m /'"", .... _.__.... U,=2e-6m '1. \' 320 - - - U.=3e-6m .... ........ U,=4e-6m 240

/ ] \ ,. I /. '. \.

440

1

,,,~;./ ,:, ,,,,\~i ' q:

330

.~

;

,~

/. 160 , l,

~

' i\

13

220

.s

"0 U

,I .!

80 1/

- - P()=40N ........... P()=110N - - - P()=180N P()=250N

110

. ..... .

OL---~--~~~--~--~

o

0,2

0.6

0.4

0 ,8

0.2

0.4

Iff

Fig. 8. 22 Contact pressures vs. contact duration under various U z

i

~ 0.

;;

;:l o u

640

400

480

. ....

; , / ;.... ' \ \

;

:

.

320 ..... J .

...

:

-"r-0.3e-311l •.•..•..•.• " r=0,6e-311l

320

E,- 1.75e9

_ . _ E,=2.45e9 .

;/.- \ ,\. :,. .\ ':

,

- - E =0.35e9 --_._... E,= 1.05e9

....... 1'"\~ ''' : '' --;. -..'.

~

..

~ c-

..

:

- - - il r=0,ge-3m

240

o

;:l 160 0 u 80

0.2

0.8

Fig. 8. 23 Contact pressures vs. contact duration under various Po

800 .-------~--------~----,

./ .

0.6 tiT

0.6

0.4

0 0.8

""',.,',

0

..... .

0.2

111'

0.4

0.6

"., if

0,8

tiT

Fig. 8. 24 Contact pressures vs. contact duration under various Ef

Figures 8. 22-21 indicate that as U z

....

Fig. 8. 2S Con tact pressures vs. contact duration under various h f '

Po , and E f increase, the contact duration

will decrease and the amplitude of contact pressure will increase.

Fig. 8. 25

shows that as h f increases, contact area will increase and the amplitude of contact pressure will decrease.

2. Influence of pre-pressure on output characteristics Relationships of the stall torque and no-load speed versus the pre-pressure are shown in Figs. 8. 26 and 8. 27. The figure indicates that the stall torque will increase by the augmentation of pre-pressure, while the no-load speed will decrease. When the pre-pressure is higher than that at critical pre-pressure, the stall torque will no longer increase.

Chapter 8

0.6


1500

.---~-~--~-~------,

0.48

251

r--~--~-~--~----'

1300

I

~.

~ "go .9

~

0.36

gj f}

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1100 900

en

700

160

240

400

320

500~-~--~-~--~-~

o

120

Pre-pressureIN

Fig. 8. 26

240

360

600

480

Pre-pressure/N

Stall torque vs. pre-pressure

Fig. 8. 27

No-load speed vs. pre-pressure

Figure 8. 28 shows the load characteristics under various pre-pressures. It can be observed that when prc-pressure increases, the stall torque increases obviously, 1400

20r-----------------,

.. 1120 ~-...;..

. . :,. . ___ ;__ ----------

U,i~crease://'

320 . ___ [T,-3e-6m

,

960

400r--~-~-~-~-~-.

- - [,,"-le-6m: ----------- [,,"-2e-6m

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_. U,-4e-6m

-S

I

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640

-:.,.~ .. ~"- r::.~I.';~~~~. :

. ~

.,

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........... .. ..... .....

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r-

II

,-L-

According to I displacement direction I -'-

-'-

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'"~

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.S

:~

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··,···f··· .. ···.···,·· ·;····.···, ·· ·,· ···,· · ,

-9

'V::::::::::~~

~>-. FrequencylkHz (a) Synergetic operarion of line.,r ultrasonic motors

Fig. 9. 59

(b) Frequency response obtained by ingle piezoelectric vibrator

Synergetic operating of linear ultrasonic motors and frequency response

Chapter 9

297


the considerations for driving a single motor, Mracek, Hemsel, Wallaschek, et at. evolved four main driving strategies for a set of motors as shown in Fig. 9. 60:

8L3

8 1. 4

81.5

81.6

81.7

8 1. 8

81.9

8L3

8 1.4

Fig. 9. 60

8 1. 5

81.6

81.7

Frequency /kHz

(a) Individual exci lalion

(b) Single resonalll excitation

"I o

81.3

81.4

FrequencylkHz

81.5

81.6

81.7

81.8

81.9

·········i

81.8

81.9

........ j.......... j........ _-;.......... j..... .

80 80.2 80.4 80.6 80.8 81 81.2 81.4 81.6 81.8

Freq ucncyik Hz

Frequency /kHz

(c) Sweep excitation

(d) Single non·reSOnanl .,citalion

Driving method of synergetic operating of linear ultrasonic motors

(1) Individual excitation: as shown in Fig. 9. 60(a), every vibrator is excited by its own resonance frequency. A complex "beat" coupling phenomenon will be produced when the four stators work together on the slider with closed-spaced vibration frequency. The stator will also produce "beat" phenomenon because of the stator is excited by other stator through the slider in the adjacent driving frequency. This behavior will result in the instability of the running of the slider and lower efficiency of the motor. (2) Single resonant excitation: all vibrators will be driven at a single excitation frequency within the resonance area of all vibrators, as shown in Fig. 9. 60 (b). This approach can avoid the coupling phenomenon. However, it will aeeclerate the wear and also lower the efficiency of the motor for the driving velocity of each stator is different. (3) Sweep excitation: as is depicted in Fig. 9. 60(e), a bandwidth in the resonance area of all vibrators will be defined and the excitation signal will be swept up and down in frequency. Operating in this style, each stator has the same vibration period. there will be no "beat" phenomenon, and the contribution of each stator for driving the slider is similar. However, the energy diffusion problem has not been solved yet because the tangential velocity of each stator is different at random time. (1) Single non-resonant excitation: Equal to the "Single resonance excitation"

but in this case the excitation frequency will be chosen in the non-resonant area

298


of all frequency responses. as shown in Fig. 9. 60 (d). In contrast to individual excitation strategy. an excitation at a single frequency in the non-resonant area would be very simple and stable, but of low efficiency. Furthermore, the amplitude of each stator will decrease largely if the driving frequency departures from the resonant frequency too much. In conclusion. the main problem for synergetic operating technique of linear ultrasonic motors is to improve machining quality and ensure close dynamic characteristics of every vibrator. Now, the synergetic operating technique of linear ultrasonic motors has been used step by step. Fig. 9. 61 shows an application in a large-scale astronomical telescope system. in which three transducers developed by Kurosawa as shown in Fig. 9. 6 have been used.

Speed (no· load): 0.8 8m/s; Torque(max.): 8.3N· 1ll

Fig. 9. 61

Application of thc synergetic operating of linear ultrasonic motors

References [ 1J

[ 2J

Jian Liu. Study on Linear Ultrasonic Motor Based on In-plane Vibration Modes. Dissertation for the Degree of Master. "fanjing: Nanjing University of Aeronautics and Astronautics, 2001. (in Chinese) Chaodong Li. Research on Longitudinal and Bending Vibration Linear Ultrasonic Motor with Large Thrust. Dissertation for the Degree of Doctor of Philosophy. "fanjing: Nanjing

University of Aeronautics and Astronautics, 1999. (in Chinese) [ 3J

T Sashida, T Press, 2002.

[ 4J

T Takano, Y Tomikawa. Characteristics of the ultrasonic linear motor using radial and non-axisymmetric vibration modes of an annular plate. Jpn. J. Appl. Phys, 1995, 31Partl(9B): 5288-5291.

[ 5J

Kenjo. An Introduction to Ultrasonic Motors.

USA: Oxford University

W Wishnewskiy, S Kovalev, 0 Vyshnevskyy. New Ultrasonic Piezoelectric Actuator for Nanopositioning. Bremen, 2001.

[ 6J [ 7J

T Wakai, M K Kurosawa, T Higuchi. Transducer for an ultrasonic linear motor with flexible driving part. IEEE Ultrasonic Symposium, 1998(1) :683-686. Ultra High Vacuum Systcm for Chcmical Analysis and Thickncss Measurcment. [2007-0615]. hup: / /www.baysidemotion.com/web/BMGHome. nsf.

[ 8J [ 9J [10J

[2006-08-06]. hup: / /www. rockwell. eom/anorad/produets/airbearing systems/ntype/n250. html. Squiggle motor ovcrvicw. [2007-6-6]. http://www.ncwscaletcch.com/downloads.htm!. R Yoshita, Y Okamoto. Micro-piezoelectric actuator. Journal of Precision Engineering Society, 2002, 68(5): 615-618. (in Japanese)

Chapter 9 [llJ


299

M Kuwana, T Kanbara, M Tikami. Driving device using piezoelectric actuator. Proceedings of Spring Symposium of Precision Engineering Society, Japan, 1999: 311. (in Japanese)

[l2J

Weidong Liu, Sbiebun Di, Wansbeng Zbao, et al. Design and analysis of linear bipolar ultrasonic motor. Piezoelectrics & Acuustooptics, 1997, 19(4): 226-230. (in Cbinese)

[13J

Chenglin Gu, Gan Dong. Double IT type linear piezoelectric ultrasonic motor. Proceedings of the CSEE, 1998, 18(2): 226-330. (in Chinese) Chaodong Li, Hua Yao, Renqing Pei, et al. Small-sized bionic [Dot ultrasonic linear motor. Small & Special Machines, 2001, 11(6): 10-11. (in Chinese)

[14J [lSJ [16J

Wei Hu. Study on Linear traveling Wave Ultrasonic Motors. Dissertation for the Degree of Master. :'-Ianjing: :'-Ianjing University of Aeronautics and Astronautics, 1996. (in Chinese) Wei Hu, Chunsheng Zhao. Study on linear traveling wave ultrasonic motors. Journal of & Diagnosis, 1996, 16(3): 8-14. (in Chinese)

Vibration, Measurement

[l7J

Chaodong Li, Long Jin, Chunsheng Zhao. The characteristics of hybrid transducer type linear ultrasonic motor with large thrust and large stroke. Acta Acustica, 1999, 24 (6): 653-

[l8J

Chunsheng Zhao, Jian Liu. Linear ultrasonic motor and its vibrator. Patent, ZL9811128. 2, 1998-05-07. (in Chinese)

[19J

Jian Liu, Chunsheng Zhao. Study on the linear ultrasonic motor based on the vibration plane of the rectangular plate. Acta Acustica, 2003, 28 (1): 86-90. (in Chinese)

[20J

Jian Liu, Chunsheng Zhao. Design of the linear ultrasonic motor based on the vibration In plane o[ the rectangular plate. New Progress on Vibration and Wave Technology. Shenyang: Northeastern University Press, 2000: 255-259. (in Chinese) Chunsheng Zhao, Jian Liu. Linear ultrasonic motor based on the vibration in plane of the rectangular plate. Chinese Invention Patent, ZL01l27038. 1, 2001-07-27. (in Chinese)

651. (in Chinese)

[21J [22J

Chinese Invention In

Guoqing Huang. Research on a Longitudinal-bending Vibration Coupled Type Linear Ultrasunic Motor with Two Statur and Precision Stage. Dissertation [or the Degree of Master.

[23J

Nanjing: Nanjing University of Aeronautics and Astronautics, 2001. (in Chinese) Qunting Liu. Research on Longitudinal-bending Vibration Coupled Type Linear Ultrasonic Motor with Multi-stators. Dissertation [or the Degree o[ Master. Nanjing: Nanjing University

[21J

of Aeronautics and Astronautics, 2001. (in Chinese) Weiqing Huang, Qunting Liu, Chunsheng Zhao. Research on a working stage driven by linear ultrasonic motor. Small & Special Machines, 2004(3): 17-18. (in Chinese)

[25J [26J

[27J

Chunsheng Zhao, Jiamei Jin. Square plate type linear ultrasonic motor and its excited mode. Chinese Invention Patent, CN20071002096S. 7, 2007-01-0S. (in Chinese) Dong Yang. Research on V Shaped Linear Ultrasonic Motor with Double Amplitude Transformer. Dissertation [or the Degree o[ Master. Nanjing: Nanjing University o[ Aeronautics &. Astronautics, 2009. (in Chinese) Chunsheng Zhao, Yubao Li. A butterfly type linear ultrasonic motor and its excited mode. Chinese Invention Patent, CN200710021372.8. 2007-06-10. (in Chinese)

[28J

[29J [30J

[31] [32J

Yunlai Shi, Hanlei Zhang, Chunsheng Zhao, et al. Two DOF positioning stage using linear ultrasonic motors. Transactiuns of .1'Vanjing [lniversity oj Aeronautics & Astrunautics, 2008,25(3): 161-168. Shuxiang Dong, Li Yan, :'-Iaigang Wang, et al. A small, linear, piezoelectric ultrasonic eryomotor. Applied Physics Letters, 2005,86, 05350l. S Ueha, Y Tomikawa. Ultrasonic Motors Theory and Applications. Ox[ord: Ox[ord Science Publications, 1993. J C Piedboeuf, J D Carufel, R Hurteau. Friction and stick-slip in robots: simulation and experimentation. Multibody System Dynamics, 2000 (4): 341-354. M Mracek, T Hemsel, J Wallasehek. Synergetic operation of ultrasonic linear motors. The First International Workshop on Ultrasonic Motors and Actuators, 2005 (11): 23-21.

Chapter 10

Step Ultrasonic Motors Great research progress for step USMs has been achieved since their invention more than 10 years ago. In 1991, Kusakabe developed a standing wave and selfcorrection USM[1:, which made the USM succeed in step motion without feedback. Later, Miyazawa put forward a step USM[2 ': using a shifting standing wave mode based on lijima's standing wave USM in 1993. Furthermore, in 1999, the author and Guiqing Wang, et al. designed and fabricated a new selfcorrection type USM[1-5:. In the same year, the author and Long J in, et al. developed a mode rotary type step USM: 67J with 80, 120, and 168 steps in one eirele based on the theory proposed by Miyazawa. In 2000, Snitka designed an ultrasonic aetuator: 8: based on a linear USM using two modes, whose positioning accuracy reached nano-meter level. In 2003, Yong Jin, Jifeng Guo, et al. made a step USM L9 J by combining a longitudinal mode and a torsional mode. In the next year, Xiangeheng Chu made a shaking-head type step USM: lOJ . In 2005, the author and Jiamei Jin, et al. developed a self-correction type step USM using modal rotation- 11J , a mode alternation type step USM_ 12 -13 J , and a linear type step USM and a rotary type step USM using vibrator alternation- 11 - 15J . Several prototypes were fabricated in different sizes for each type. Compared to an electromagnetic step motor, the step USM has good characteristies, such as simpler structure, smaller size, better environmental adaptability, and no-electromagnetic interferenee1l6 -. It can be widely applied to optical devices, robotics, space shuttles, automatic control systems, military facilities, medical equipment and so on. The nano-meter step USM can also be used in electron beams, ion beams, X-ray, scanning electron microscope positioning, etc. A step USM can be described as an ultrasonic motor which can realize the step motion with a certain step size. USMs inelude two major types: a traveling wave and a standing wave, and these two types of USMs operate in two different ways. The traveling wave USM has good controllability, stability, and long life- 17 -22J , while the standing wave USM has different performance and can realize movement with different forms[20 3<J. Thus, many standing wave USMs arc made by different operating modes and different driving schemes. The step USM is a significant branch of USMs. The step USMs can be divided into the adjustable steplength type and the fixed steplength type. The adjustable steplength USM is driven step by step by switching onloff the driving signal in a carefully designed sequence. Its step length is decided by the


Chapter 10

Step Ultrasonic Motors

301

width of the driving pulse L35 -. When the USM starts or shuts down, fluctuations of the velocity will affect the uniformity of every step. When every step displacement is smaller, this fluctuation will be not negligible. In the condition of longstep and thc multi-stcp positioning, the fccdback loops havc to bc applied in order to reduce the position error. The fixed step length USMs possess some special structural forms to obtain step motion in the open-loop control c'6:. This kind of step USMs has no accumulated error in multi-step operating. The single-step operating error is produced mainly from the machine work and assembly error.

10. 1

Step Control of USM

An adjustable steplength USM is controlled by switching onloff the driving signal. Theoretically, the electromagnetic motors can also realize the step motion by similar operating, but the control is more difficult because the coil motor can not very quickly response to the control signal and can not make self-locking when its power is shut down. USMs possess characteristics of self-locking, rapid response and high position resolution. These naturally present advantages for the control of accurate step positioning. USM's self-locking feature comes from the operating principle of friction drive. Usually, in the operating state, the contact area between the stator and rotor (or silder) is smaller than that in the non-operating state. Besides when it operates, there is also local sliding on the frictional interface, which makes USM's self-locking torque (or force) greater than the stalling torque (or the maximum output force). In other words, with the power shutdown, USM's rotor (or slider) will be quickly locked at the position where it arrives. This advantage is beneficial to the control of precision positioning. The rapid response means the stator will soon achieve the steady state after power on. Namely, USM can reach its rating speed quickly. USM's high positioning resolution is due to its micro vibration amplitude and high operating frequency. Generally, displacement in the sub-micron or nanometer level can be achieved when the stator makes the rotor (or slider) move in a vibration eyele. Plenty of experiments show that large fluctuation of the positioning accuracy appears at the period of starting and ending the motor.

10.1.1

Startup and Shutdown Characteristics of USM

Under normal circumstances, in order to increase the amplitude and reduce the energy consumption, the driving frequency is designed to be elose to the resonance frequency of the stator. In this way, the dynamic characteristics of USM is a key factor in precision positioning. The process, from the beginning of the stator's vibration to the accumulation of the energy and finally to its stable operating state, is called as the starting state of the motor. On the other hand, the

302


process, from the moment when USM's power is off and the stator obtains its own initial condition of vibration to the momcnt whcn thc vibration of the stator stops completely and finally to states of shutdown the rotor (or slider), is called as statc of shutdown of the motor. The intcrmcdiate statc betwecn thcse two states is the steady opcrating statc of USM.

1. Stator's vibration response According to the vibration thcory dcscribcd in Chap. 4, the forccd vibration of thc stator can bc divided into thrcc stages: CDa startup(beginning of an cxcitation); CZ)a steady state(kccping thc excitation); @an attenuation(switching thc excitation). Here, stage CD and ® will be discussed because the vibration characteristics of thesc two stages directly rclate to USM's response speed and positioning resolution. Thc Ref. [37J has analyzed thc problem in detail and obtained thc conclusion that the time for USM's stator vibrating in the stage CD and ® is different. The diffcrcncc bctween thc two stagcs' time is mainly dctermincd by damping of the stator. The greater thc damping is, thc biggcr thc diffcrence will bc. Gcnerally, the damping of USM's stator is relatively small. So, the difference between the two stagcs' time is not large. It is not enough to only consider the viscosity damping of the stator. In actual circumstancc other damping factors still cxist. From thc point view of energy, it is valid that in the beginning of the forced vibration the damping prevents the system from accumulating energy. The greater the damping is, the longer the time t, to reach the steady-state will be. In the decaying stage, the damping will consume cncrgy. In other words, the grcater the damping is, the sooner thc decaying stage will bc, thc shortcr the timc t, to stop thc vibration will be. Consequently when the damping is bigger, the difference between the two stages' time 6.t = t, - t, will be greater. 2. The actual operating process of the motor Corresponding to the vibration of the stator, the actual operating process of USM can be also divided into three stages: CD startup stage; CZ) steady operating stage; @shutdown stagc. The characteristics of startup stage and shutdown stage for a typical traveling wavc USM arc shown in Fig. 14. 12 and Fig. 14. 13. In the startup stage of USM, there is some delay, from the beginning of the vibration of the stator to the starting of the output of the rotor's shaft, which is callcd mechanical hystercsis of the motor. This is due to thc wholc system's damping and stiffness of the shaft coupling. Whcn the rotor is asscmblcd, thc boundary conditions of the stator will change. It's equivalent to adding the constraint to the stator, which increases its natural frcquency and dccreascs its amplitudc. When the stator's amplitude dcclincs, thc pre-pressurc will increasc the contact arca on the contact interfacc between the stator and rotor at the same time. :'\Jot only the transmittability of thc tangential forcc on the contact interface is incrcascd, but also thc arca of sliding friction is increased, resulting in more energy dissipation. Thereby, the startup time increascs with thc increase of prc-prcssurc, whilc thc shutdown timc

Chapter 10


303

decreases with the increase of pre-pressure. Thc startup and shutdown characteristics for TRUM-4S under different prcpressurc arc shown in Fig. 10. 1. It illustrates that thc startup time incrcases with the pre-pressure going up while the shutdown time declines slightly with the pre-pressure nsmg up, as shown in Fig. 10. 1. 1)

r----------------------------------,

.

-

• Startup

•

II~

"0Il

__

~

___ L_ _

~

220

_ _ _ _L __ _

~

Pre-~lr~~sLJrl'

_ __ L_ _

"btl

2~1I

ShI11~hn\ fl

~

__

~

280

'-N

Fig. 10. 1 Startup and shutdown characteristics of TRUM-15 under different pre-pressure

10. 1. 2

Step Control for USM

Based on the preceding analysis, we can conclude that besides the load, other factors influencing the startup and shutdown time come from the motor itself. Among them thc most unstable factor is thc frictional intcrfacc betwecn thc stator and rotor. When the USM runs, the frictional properties at the interface will change from time to timc duc to wcar, hcating, and other rcasons, which lcads to fluctuation of the startup and shutdown time. These are the main reasons that makc USM instability whcn it starts up or shuts down. Figure 10. 2 shows test results regarding to the angular displacement versus thc powcron time for TRUM-45. Fig. 10. 2 (a) shows that the poweron time is 10ms, and the fluctuation of the displacement is large. It is primarily due to the startup timc of thc motor is about 8ms. Besides, therc is the ovcrswing, which needs about 30ms to make the speed relatively stable. The situation will be better when the poweron time more than 10ms, as shown in the Fig. 10. 2(b). The fluctuation reduces significantly and good linear relationship can be observed when thc powcron timc more than lOOms, as shown in Fig. 10. 2(c). Another test is regarding to the motor's repeatability. As shown in the Fig. 10. 3, the USM was rcpcatedly tcstcd 10 times. Thc maximal dcgrcc of decentralization based on the arithmetical mean value demonstrates that the shorter thc powcron timc is, the worse thc repeatability is. When it turns in thc counterclockwise direction, the degree of decentralization is lower than 10% if the timc is morc than ISms. Whcn it turns in the clockwise direction, the dcgrcc of decentralization is lower than 10 % if the time is more than 25ms. For this USM, the diffcrcncc of deccntralization in the two directions is mainly causcd by the speed difference in these two directions of the USM. Further, the degree of thc decentralization shows the position error of thc timing displacemcnt. Thc


304

longer the electrical power is applied. the longer the step of the USM is, and the smaller the position error is. 14,---------------------------, • Coulllerclockwise • Clockwise

>::'

"1l" E

'" ~ :;:; ~,;;, ~

~

"'

4

~ ~

2 01

2

4

3

~

0

...."

>::'

E

'" ~ :;:; );;

:;

2!'

-
min. Previous three control methods can use a control strategy called DCM (Differential Composite Motion) to get a smaller step angle. Tsinghua University Xiangcheng Chu. Zengping Xing. et al. described this strategyllo-. and fulfilled the step control in a shaking head type USM. whose diameter. length and operating frequency are ¢15mm. 42mm and 30-45kHz. respectively. It's no-load speed is 150r/min. stall torque 0.12N·m. startup time o. 24ms. and minimum step angle 12" . The step angle Bi can be obtained by the way of turning forward and backward. As shown in Fig. 10. 7. the every step contains two processes: one is starting forward. running and decaying; the other is starting backward. running and decaying. There are some advantages in this way: the smaller step angle and greater output torque can be obtained. The displacements in the two startup and deca-

308


ying processes can be subtracted with each other, which can reduce the error from instability statc. But thc way is forbiddcn whcn thc ovcrshot is not allowcd. Po it inof shutdown

~

80

~

60

§

'"

~

'is

TfJ( r) 0,

Fig. 10. 6 Stepping motion of USM using closed-loop control

10. 1. 3

40

O

~-L-L~L-~-L

500 I 000

__

I 500 2 000

I /ms

Fig. 10.7 DeM's locating strategy of stepping

Factors Impacting on Single-step Positioning Accuracy

For the step USM, the single-step displacement accuracy depends on its stability including the stability of stator's vibration and the force transferring to friction interface at different time and in different environment.

1. Stability of stator's vibration The main factor influencing the stability of stator's vibration is the change of connecting performance between the stator and piezoelectric ceramics. At present, there are three ways to connect a stator with a piezoelectric ceramics: compacting way. welding way and bonding way. (1) Compacting way Compacting way is conducted by a clamping force induced with connecting bolt, which makes both the modal frequency and amplitude sensitive to the clamping force. When USM is running, the heat from both the piezoelectric ceramic pieces and the friction interface will cause the stator's temperature rise. Due to thermal expansion the clamping force changes, then results in the fluctuation of the stator's modal frequencies and amplitudes. In addition, after long running, the motor's rated speed will change. Generally, thc piezoclcctric constant d" is bigger than its d 31 • and the clamping method is more suitable for these transducers that take the d" piczoelectric effcct, such as an ultrasonic motor using longitudinal-torsional bybrid vibration (scc Chap. 8). (2) Welding way Welding way is the mcthod that joins piczoclectric ccramic picces and a stator under a high tcmperature. which is bcneficial to more cffectively transfer piczoclectric ceramic deformation to the stator; further, it can provide high excitation cfficicncy along with dccreasing mcchanical hystcresis. In addition, the wclding possesses high rcliability and stability. and thc performances of USM arc preferable. However, in general, with the stator vibration. the strain of the piezoelectric ceramic picces on the welding surfacc and thc strain of the stator surface are

Chapter 10


309

not totally consistent. This leads to a loealized stress concentration when USM is running, and the welding layer will has fatigue failure. (3) Bonding way Bonding way is the method of which water glue cement and so on bond a piezoelectric ceramic pieces to a stator in certain temperatures and pressures. Currently, an epoxy resin and acrylic adhesive is uscd extcnsivcly for USM. The adhcsivc possesses sufficient strength, which can provide thc clastic link bctween the stator and piezoelectric ceramic pieces and release the stress concentration on the bonding surface. Mostly this approach can be just utilized for the d 31 piezoelectric effect. Beside the thickness of the adhesive layer has great effect on the excitation efficiency, tempcrature changc also affect the clasticity of thc layer. Comparatively, the welding way and bonding way can make USM operating stable, while compacting way can induce some unstability.

2. Stability

0

f friction interface

Good performancc of thc friction interface can cnhancc the torque or thc running velocity of USM. A moderate friction coefficient, high hardness and high wear ability are thc basic rcquiremcnts for thc friction interface. Thc stability of thc friction interface is affcctcd by thc factors including the charactcristics of friction materials, thc surfacc morphology of the intcrfacc and the changes of the tribopair to cnvironmcnt, such as tempcraturc, humidity, and vacuum (scc Chap. 3). Thc stability of friction interfacc is a kcy to achieve thc stablc step interval of the step USM. The wear and tear of the tribopair change operating condition. From analysis above, we can draw the following conclusions: applying the materials with a moderate friction coefficient, high hardness and high wearability is bcneficial for obtaining more stablc friction interface.

10. 2

Step USM with Fixed Step length

10.2.1

Standing Wave USM Used for Constructing Step USM

The standing wave USM discussed in this section is a type of rotational USM whosc stator is ring shapc, as shown in Fig. 10.8. Thcre arc 4 tccth distributcd equally on the ring stator. Thc rotor is prcssed on it by thc prc-prcssure providcd with a spring. The monodirectional polarized piezoelectric ceramic ring has 8 uniform electrodes, as shown in the Fig. 10. 9. Figure 10. 10 is a diagram expanded in the circumferential direction of the USM. Thc figurc shows that thc stator's modc can be excitcd by thc driving voltage. This mode is the standing wavc ¢({}). In the coordinate of Fig. 10. 10, the standing wave can be expressed as: ¢({}) =

Wo

+

sin(wt Wo

+ ({J) cos(k{} -

sink{}sinkasin(wt

ka )

+ ({J)

= =

Wo

cosk{}coska sin(wt

¢l ({})

+ ¢, ({})

+ ({J) (10. 5)


310

1

Pre-pressure

I'.

Fig. 10.9 Piezoelectric ceramic's subareas of uniform electrodes of rotary USM using standing wave

Fig. 10.8 Basic structure of rotation USM using standing wave

z

Stator mode __ a _ /

d

"'- I I

0

Fig. 10. 10

'"

I

"-

/1 1 ./

I

II "-

B

./

-Eosm (011) EUSlll

(rul)

Operating mode of standing wave type rotatory USM

=

0,

Mode ¢(B)

Mode I Mode 2

()

Fig. 10. 11

Where epj (B)

Decomposition of stator's operating mode

+

+

coskBcoska sin(wt rp), ep2 (8) = Wo sinkBsinka sin(wt rp) , k is the number of the standing wave in the stator, and here k = 2. ep] (B) and ep2 (B) are two standing waves whose phase difference is rr/2 decomposed from ep(B) in thc spacc. Moreovcr they are callcd as the mode 1 and mode 2, rcspecti vcly, as shown in Fig. 10. II. If tceth are takcn as rigid, tccth' s movcmen t will bc dctermined by thc rotational and translational movement of the points on the neutral layer of ring which is in thc bending vibration. In this figure, the tooth No.1 lies in the location of B = rr. The movement of the point on the neutral layer of the ring in the z direction is =

Wo

(10. 6)

The teeth's translational movement is determined by mode 1. That is to say, thc point's angle of rotation is

y

=

arctan ()ep(BB) I rd

a-IT

()ep2 (8) I arctan --Brd

a-IT

(10. 7)

Chapter 10


311

Teeth's rotary movement is determined by mode 2. Where r is the stator's outer diameter. It is valid to assume that the distancc between one point on the top of tooth No.1 and the neutral layer of the ring is h, and then the movement of this point in the () direction is

~siny """ ~

{)t =

r

r

CJcp, (()) I rd{)

= O-IT

~ kw o sin(wt + rp) sinka

(10. 8)

r

Movement of this point in the z direction is Wo

coskCi' sin(wt

+ rp) (10. 9)

From the deduction above, the mode 1 makes the tip of the tooth to move in thc z direction, while the mode 2 makes the tip of thc tooth to movc in thc () direction. Further, Zj and {)j correspond to the point's amplitude of the tip of thc stator tooth in the Z and () direction, respectively. Giving the pre-pressure, an excitation voltage and height of teeth, the angle Ci' bccome thc kcy indicator determining thc synthesis of displaccments. As the angle Ci' changcs, the wavc shapc form cd by the tip's movcment is decided. The angle Ci' can be determined from design requirement so that the polarized pattern of piezoelectric ceramic ring can be determined. Thc wave shapes of CPt ({)) and cP, (()) as thc Ci' changcs arc shown in Fig. 10. 12. 2

-;(0) ---- ¢ ,(O)

2

-;(0) ---- ;,(0)

--- ;,(0)

--- ;,(0)

0 -\

-\

-~

-2

-0

0

(0) a ; ;I[/ 12

2

2 (b) a ; I[/3

4

2

- \

-20~--------------~ 2 --------------~4

(c) a

~ I[/4

Fig. 10. 12

When 0 ~

Ci'

~

-20~--------------~--------------~4

(d)

cr ~ I[ /6

Decomposition of standing wave with different a

re/1 , the amplitude of horizontal movement is larger than the


312

one of vertical movement; when rr/ 1 ~ a ~ rr/2 ,the latter is larger. The shape of the locus is determined by h and a. When the designer needs bigger torque, it's better to make Zt larger. If the designer needs bigger speed, it's better to make at larger.

re=~=?f?? ~ ~ Jj=vsin(WI)~ ~/?74 ~ 3§s1l1«(~ L------(-a-)- - - - 1 - -E""" ,-:......:.J"'sl""' -; ·n"' (wc:./""")....

(a)

.

E- Vs in(w/)

Rotor direction

Rotor direction I (b)

(b)

Rotor direction

Rotor direction

I

(c)

(c)

Operating principle of rotor rotation in anticlockwise

Fig. 10. 13

Operating principle of rotor rotation in clockwise

Fig. 10. 14

Figure 10. 13(a) shows an operating shape produced by the annular plate type stator when a single phase voltage E excites the stator. Two of the four teeth of stator move in counterclockwise and contact with the rotor. The friction between the stator and rotor drives the rotor turning in counterclockwise. The other two teeth move in clockwise. Because they located close to the trough and do not contact with the rotor, then they cannot push the rotor. In the first half cycle of the stator's vibration. the teeth :'\10. 2 and No.1 contact with the rotor and drive the rotor, as shown in Fig. 10. 13(b). In the second half cycle of the stator's vibration, the teeth No.1 and No.3 contact with rotor and drive the rotor, as shown in Fig. 10. 13(c). Figure 10. 11 shows the driving voltage which makes the rotor turn reversely. The vibration mode is shown in Fig. 10. 11 (a). The moving decomposition of the teeth's tip and the rotor rotation are shown in Fig. 10. l1(b) and Fig. 10. 11(c).

10.2.2

Modal Rotary Type Step USM

1. Structure of USM The structure of a modal rotary type step USM is basically the same as the previous standing wave type USM. It is made of the annular stator, piezoelectric ceramic ring, spring and rotor. The difference is that the teeth arc on the rotor instead of on the stator, as shown in Fig. 10. 15.

2. Driving mechanism The stator has a mode

BOk

excited by the piezoelectric ceramic ring. According to

thin plate vibration theory, the displacement of point P on the stator's surface in

Chapter 10


313

1

Pre· pressure Po

Spri ng Rotor

ROlDr Icelh

Fig. 10. 15 Basic structure of modal rotary step type USM and electrodes of piezoelectric ceramic ring

the z direction can be expressed as wp(r,{),t) = R(r)sin(k{))sin(wt)

(10. 10)

The speed of point P along the () direction can be expressed as (10.11) wherc d is thc thickncss of the platc, k is the numbcr of thc wavc, w is thc driving angular frequency, r is the radial coordinate of the point, and R(r) is the Bessel function

Whcre J n (kr) and Y n (k r) arc the first and sccond type nth ordcr Bessel functions, respectively; In (kr) and Kn (kr) are the first and second type nth order corrected Bcssel functions. rcs pecti vely. The rotor motion is acquired by two actions from the contact of the stator with rotor. One is the motion of the points on the stator's surface, and the other is the circumfercntial component forcc at the contact points of the stator with rotor. From Eq. (10. 11), the spccd V, of point P along () dircction, which locates at thc stator's surfacc bctwcen thc wavc crest and nodal diameter, changcs within thc cyele of the stator's vibration. as shown in Fig. 10. 16. In the first quartcr ofthc cyele of the stator's vibration (namely, wt 16a) or thc fourth quarter (i. e. wt

=

=

0-rr/2, as shown in Fig. 10.

3rr/2-2rr. as shown in Fig. 10. 16(d)), thc

speed V , direction aims at the nodal diameter of the mode shape. In the second quarter (i. e. wt=rr/2-rr, as shown in Fig. 10. 16(b)) or in the third quarter (i. e. wt=rr-3rr/2, as shown in Fig. 10. 16(c)), the specd V, direction aims at thc crests (or trough) of the mode shape. If there is no relative sliding on the contacting surface between the stator and rotor, the rotor will acquire the momentum in the circumferential direction that depends on thc contacting period bctwcen thc rotor's tccth and stator.

314


z z

(WI =0- 1[/ 2)

Es in (w/) (w l = x - 3x/2) (c)

z

z

Es in(w!} (wl =3x / 2 - 21t )

Es in(w /) (w / =1[ / 2-x)

(d)

(b)

Fig. 10. 16

Speed component V, of contact point on stator

along circumferential direction

z

Es in (w l) (wl = 1[ -37t / 2) (e)

Esi n(wf) (w / =O-1[ / 2) (a)

z

Es in(WI) (wl =1[ / 2-1[) (b)

Fig. 10. 17

Esi n(W/) (WI =31[ / 2 - 2x ) (d)

Forcc component jd of contact point on

rotor along circumferential direction

The force fd in the circumferential direction is caused by the changing the slope on the contacting surface of the stator with rotor under the pre-pressure Po. The direction of the force fd also changes during the cycle of the stator's vibration, as shown in Fig. 10. 17. In the first quarter of the cycle of the stator's

Chapter 10


315

vibration G. e. wt=O-rr/Z. as shown in Fig. 10. 17(a» or in the second quarter (i. e. wt= rr/Z-rr, as shown in Fig. 10. 17 (b», the force Id direction aims at the nodal diametcr of the mode shapc. In thc third quartcr (i. c. wt = rr-3 rr/ Z. as shown in Fig. 10. 17(c» or the fourth quarter of the cycle of the stator's vibration (i. e. wt=3rr/Z-Zrr. as shown in Fig. 10. 17(d». the direction of the force Id aims at thc trough or crests of the vibration shapc. If thcre is rclative sliding on the contacting surfacc between thc stator and rotor. thc rotor will be driven by the circumferential force component Id whose direction depends on the contacting pcriod betwecn the rotor's tecth and thc stator. If the rotor only contacts with thc stator during the first quartcr of thc cycle of the stator's vibration, which can assure that the motion component V, has the samc dircction with thc forcc component Id. In fact, if the numbcr of teeth on thc rotor is as twicc as that of the nodal diamctcrs, the stator will contact with the rotor in the first quarter or the second quarter of the vibration, i. e. wt= O-rr/ Z or wt=rr/Z-rr, as shown in Fig. 10. 18.

()

()

£s in Q)/ (Q)/"O- rr)

Es in (v / (Q)t=rr-2rr) (b)

(a)

Contact points between the stator and rotor

Fig. 10. 18

8

Fig. 10. 19

(01

Force analysis of rotor's teeth

Fig. 10. 20 Relationship of V, and Id vs. time

Assuming that the rotor contacts with stator during the first 1/1 cycle of the vibration, the circumfercntial component Id induced by contacting thc stator with rotor teeth under the pre-pressure Po can be expressed as

Id tanB

=

=

Po cosB sinB

1 dz(r,{),t) r

d{)

=

=

~o sinn

R(r)

k --cos(k{)sin(wt) r

316


Considering that fj is small, there is an approximate relationship, as shown in Fig. 10. 19 sin2fj """ tan2fi

=

2 tanfj """ 2 tanfj 1 - tan'fj

Thus

fd """

Potanfj

=

R (r)

.

Pok --cos(k{})sm(wt) r

(l0. 12)

The changes of V, and fd versus time t are shown in Fig. 10. 20. In the period of the contact, the circumferential movement of the points on the stator's surface contributes to drive the rotor; while in the anaphase of the contact, the circumferential force component produced by contacting both the stator and rotor teeth under the pre-pressure will also drive the rotor more effectively. Thereby, the dcsign on the contact period of the stator with rotor is directly relatcd to the dcsign of frictional fcaturcs of thc contacting interfacc. The friction coefficient that is big enough can make the horizontal movement of the points on the stator to transmittcd reliably to thc rotor; the small friction coefficient makc thc stator and rotor teeth to possess relativc slide under the prcpressure, so that the circumferential force component Fd can come into better use for the driving force. It is impossible to design two friction materials with friction coefficients on the same surface, and only one of driving methods can be chosen to design the motor: (1) Usc horizontal movcment of the points on thc stator surface to drive thc rotor, which has large enough friction cocfficicnt. (2) Use the circumferential component force under pre-pressure between the stator and the rotor's teeth to drive the rotor, keeping the friction coefficient as small as possiblc.

3. Stepping principle Modal rotary type step USM relies on the vibration of the stator ring to make the rotor stepping. As shown in Fig. 10. 15, the piezoelectric ceramic ring is polarized in one way in the axial direction. The number of electrode is 20. Four teeth are distributed on the rotor uniformly, and B02 is the operating mode. On the purpose of convenient observation, the stator and the rotor are expanded along eireumferential. as shown in Fig. 10. 2l. The sinusoidal voltage is utilized for the electrodes of the piezoelectric ceramic ring, as shown in Fig. 10. 21(a). The bending standing wave Bo, of the stator is excited. The rotor's teeth move forward to the stator's nodal diameter under the effect of the stator, as shown in Figs. 10. 21(b) and (c). When the rotor's teeth locate at the nodal diameter, as shown in Fig. 10. 21 (d), the first step of the motor's movement will be over. When the step is finished, the power is supplied to the electrodes of the piezoelectric ceramic ring. From Fig. 10. 21(e) , the bending standing wave B02 of the stator is excited again, but at this time the standing wave has turned 21(/20 arcs around the motor's shaft. The rotor's teeth moves forward to the stator's nodal diameter, as shown in Figs. 10. 21 (0 and (g).

Chapter 10


317

--r·L__ TT--TTT--TTT--IU-' ___ boSll1(

Base signal

:1 n Ch 0 0

:1 / I £1 0 (a) Schematic diagram

Fig. 12. 10

12. 3. 2

r-I I

•I •I I

I

•I

(b) Waveforms ofnodes

Dead-zone circuit set up by D flip-flop method

FDPS Composed by CPLD 20 21

The above circuits can only generate driving signals with the dead-zone for fixedtime, i. e. the range of the dead-zone is irrelative with the signal frequency. In applications however, using driving signals with 40 % duty cycle for all frequencies to drive the USM can achieve higher efficiency. Utilizing the aforementioned discrete devices is difficult to fulfill this object, but a complex programmable logic device (CPLD) is competent for this task. CPLD has the advantages of high performance, high density of integration,

Chapter 12

and easy development.

359

Driving Techniques for Ultrasonic Motors

It

IS

designed and simulated with software MAX

+

PLUSII of ALTER A Co .. To develop the CPLD, graphies and programming language are used. Fig. 12. 11 is the simulation results of CPLD. In this figure, elk is a elock whose frequency is adjustable, cw and ccw are control signals for CW, CCW, and STOP (STOP when CW and CCW are inphase; CW or CCW when they are out of phase), and qo - q3 are driving signals with 10 % duty cyele to drive the power transistor. Fig. 12. 12 is the actual waveforms of driving signal and output voltage. 20.0 fls

D-- clk

0

D-- cw

0

40.0 JJS

60.0 fls

80.0 fls

100.OflS

120.0flS

D-- ccw

-aq3

0

-aq2 -aql

0

-aqo

0

Fig. 12. 11

Simulation waveforms of CPLD

e> >

> (5 ~

~

-

N

G

:;:

u

51-lsIDiv

Fig. 12. 12

12.4

Waveforms of driving signal and output voltage

Power Amplifier Techniques

Ultrasonic motors need the signals with a high freguency, high voltage, and certain power. Small signals generated from the oscillator have to be amplified. This means that the generated unipolar square wave should be transformed into bipolar square wave to drive high-frequency transformers. The power amplifier components are mainly power transistor, MOSFET, and 1GBT. Since USM usually has low power, and operates at relatively high frequency, MOSFET is suitable for such application. The MOSFET device has a high input impedance of up to 40MD. It is a volt-

360


age-drive type switching device. When the grid-source voltage is greater than the threshold, the MOSFET is on. Otherwise it is off. It is relative simple to drive MOSFET. In some occasions, CMOS and IC can drive it directly, which predigests the driving circuit and reduces cost. The grid-source part of MOSFET can be regarded as a capacitor. Commonly, the grid-driving circuit used ineludes TTL driving circuit. Fig. 12. 13 shows a TTL driving circuit with emitter follower. 5V

Fig. 12. 13

15V

TTL driving circuit with emitter follower

According to different topological structures, the power amplifier circuit made of MOSFET has three types- 22 - 23J •

1. Push-pull converter Figure 12. l1(a) is the main circuit of a push-pull converter, which is excited by the driving signals applied to the gates, two switching elements conduct alternately through the middle point of the primary side of the transformer. Fig. 12. 14(b) shows the waveforms on every node. We can sec from this figure that the unipolar square waves alternately applied to the gate arc converted to bipolar square waves by this circuit. Due to the very low conducting resistance and leak current, the loss of MOSFET is very small in a period from ON to OFF. Q, Grid voltage

nsLJ

LnJL 0, Drain voltage LnJL O, Gridvollage

Q2Drain voltage Outp ut ,'oltage

Cal Main circuit

nsLJ

oor uu

(b) Wavefonns of nodes

Fig. 12. 14

Push-pull converter

Chapter 12

361


In the push-pull circuit and the following full-bridge and half-bridge circuits, it is not allowed that the series of two switehing MOSFETs conduct simultaneously, whieh will eause damage of the MOSFETs. Due to this reason, we have to

use driving signals with the duty cycle of less than 50%. This means that the dead-zone is indispensable, see section 12. 3. Push-pull converter is the simplest structure for power amplifier, suitable for the driver using low-voltage DC supply.

2. Full bridge converter In push-pull converter, the rating voltage of the MOSFET is at least two times of DC supply. For safety design, the rating voltage of the MOSFET should be 3. 3 times of the supply. If DC power is supplied by rectification of AC network, the voltage on the MOSFET may be 1 OOOV in worst condition. At present, the switching MOSFET of rating voltage 1 OOOV with suitable switching speed is very expensive. Therefore, we seldom use push-pull circuit when the source is directly supplied by AC network.

II[

Vout

Vout

+--------,*

Fig. 12. 15

Full bridge converter

+-----------,11 [

t;fL Fig. 12. 16

Half bridge converter

However, a full-bridge converter can solve this problem. Fig. 12. 15 is the circuit of the full-bridge converter. The opposite MOSFETs QJ and Q" Q2 and ~conduct alternately, namely, the QJ and Q, conduct within the first half period, Q2and Q4 conduct within the second half period. We can see from this figure that the voltage on the MOSFET is half of that of push-pull converter. The reliability of the full-bridge is enhanced, but four MOSFETs are used. Therefore, the full-bridge converter is suitable for power supply with high voltage.

3. Half bridge converter One arm of the full-bridge can be substituted by two capacitors, as shown in Fig. 12. 16. This is a half-bridge converter, which is widely used in low power applications. The voltage of the middle point of capacitors is about V /2, so the primary voltage of the transformer is V /2, and that of full-bridge is V. This means that for same power, the primary current of the half-bridge is double of that of the full-bridge.

362

12. 5 12. 5. 1


Electrical Characteristics of Ultrasonic Motors Experimental Results and System Description

1. Experimental phenomena The stator made of a piezoelectric element and a metallic elastomer is the key part of ultrasonic motors. The vibration of the stator, the friction between the stator and rotor, temperature, etc. have influence on the piezoelectric element's performance, and USM's characteristics. The electrical admittance curve is almost symmetric near the resonant frequency when the stator of USM is driven by low power, and the electric and mechanical characteristics arc almost the same, which can be explained by linear theory of piezoelectric material. However, the drive voltage applied to TRUM is usually higher than 300Vpp and output power is more than 8W. In this case, some special phenomena are shown: temperature increases obviously, resonant frequency drifts seriously as shown in Fig. 12. 17, and some nonlinear phenomena happen, such as electrical admittance curve of stator from low to high frequencies does not coincide with that from high to low frequencies,

as shown

in Fig. 12. 18. 0.0 12 , - - - - - - - - - - - - - - - - , - - - Curve rrom low 10 high frequencies 0.010 Co I'Ve fI'ol1l high to

100 __- - - Increasing lemperal\lJe

90 80

:5 E

e 0.008

70

7

60

g

1l

~ 50

i

low frequencies

"

-6

40

4.16

O
0028 , . . . - - - - - - - - - - - - - - , -

ij

.c;

8 OIl

-= is. 8" 1l

-"- .... ... ... ...

500

450

I

400

0>

3

4

0.026 ---

I:E 0.024

to

2

6

Fit curve Measurtd dal8

•

4. 170

Inpu t powerrw (a) Shift of resona.1I frequency and

750

-

4. 174

~'~-- ---e----~----~

4. 15

367

7

iii

ear reSOnanl fre-

quency

ear anti -res-ooanl

frequency

0.022

~ /" ~ P

--"

•

/ , /r;('

0.020

0.0 18

/ , / p'

0.016 0.014

0.012

'L.._~_~_-'-_

o

2

3

_'__

4

_'__.J

5

In put powerrw

Input powerfW

(c) Mechanical resonance quality factor

(d) Electromechanical coupling coefficieru

6

Comparison of characteristics of stator near the resonant frequency with anti-resonant one

Fig. 12. 24

In fact, the contact force between the stator and rotor cannot be ignored. Moreover, the contact force is different with various loads, so the equivalent circuit cannot completely describe the response characteristic of the stator while the motor is operating. In this case, the parameters must be experimentally obtained with an operating USM. Therefore, there are certain difficulties to design USM based on equivalent circuit.

12.6

Influence of Matching Circuit on Performance of Driver

The functions of matching circuit are: (Dpower matching: the USM is a capacitive load. power matching can reduce reactive loss and increase the efficiency of the system; CZ) filtering: the output of the transformer is high voltage bipolar square wave, which comprises many harmonic components. A matching circuit could filter the unnecessary harmonic waves to achieve a necessary and basic wave. avoiding exciting non-operating modes of a stator- 3 • 26. 28J. To reduce the self-lose of the matching circuit, it is often composed of capacitor or inductor. The matching circuit inevitably influences the electrical and vi-

368


bration characteristics of USM. Therefore, study on the matching circuit is very important to design the driver. According to the simplified equivalent circuit in Fig 12.21 and the impedance expression of the stator, the equivalent impedance of the stator can be written as following when the dynamic resistor RHO and the parallel resistor Rd are omitted for simplicity

Z

=

iX

= _

i _1_ • 1 - ( f j f) 2 we o l-(fp/f)'

02. 7)

where Co is the static elamped capacitance of the motor, I, and II' are the series resonance and parallel resonance frequencies of the stator, respectively. Although there are many kinds of matching circuits, it is usually composed of inductors or capacitors. Here we will take the stator of a disk-type USM as an object to investigate, PSV-300F-B used as testing tool to study the influence of the matching inductor and capacitor on the electrical and vibration performance.

12. 6. 1

Influence of Matching Capacitor

1. Series capacitor When the stator is in series with eapaeitor C, the equivalent impedanee is

Z'

=-

i[---.L + _1_ • 1- ( f j f): ] we

we o

1 - (fe/f)

02. 8)

For electrical analysis, the matching capacitor is chosen as 30nF, o. lll-F, O.18Il-F, 1. 0ll-F, 3. 31l-F, l0ll-F, and 171l-F. Fig. 12. 25 shows the sweep results, where Fig. 12. 25(a) is for the former four eapacitors, and Fig. 12. 25(b) is for the latter four capacitors. It is shown from testing results that for the stator in series with capacitor, the resonance frequency of the stator I, (""" I,) rises, and the anti-resonance frequency I, (""" Ip) keeps the same. Besides, the less the series capacitor is, the higher the resonance frequency is, and the less the equivalent admittance is. The series capacitor translates the phase frequency curve of the motor near the resonance frequency. But when the series capacitor is greater than certain value 00ll-F in this case) , the amplitude and phase frequencies characteristic keeps the same, as shown in Fig. 12. 25(b). At the same time, when in series with capacitor the vibration characteristic of the stator is measured with PSV-300F-B, applying constant voltage value of 20Vpp • The vibration characteristic of the stator is elose to that of non-matching in series with larger capacitance. But with decreasing the capacitance, the vibration amplitude becomes smaller, and the resonance frequency is higher. Fig. 12.26 shows the vibration characteristic for the capacitances of 30nF and lOnF. It can be seen from this figure that the frequency change of vibration characteristic is coincident with that of electrical characteristic. Smaller capacitance makes the equivalent impedance larger, which reduces the driving current and then the amplitude.

Chapter 12

0.02

100

0,015

50

Non-matching

~

§

,S:

""~

0.01

10 ~I'

0

5:

.§ ..:

369

Driving Techniques 10r Ultrasonic Motors

- 50

0,005

0 4, 15 4,16

4,17

4,18

4, 19

42 , 4.21

4.22

1'Hz

x lO'

-100 L-_L-_'--_'--_'--_''_____-'-------' 4,154. 16 4.174, 184. 19 4 .2 4214.22 . x lO' 11Hz

(a) Freqllcncy sweep cllrve I

0,02 50

0.0 15 (/)

]

"'1;l"

~ 0,0 1

"§

0

~

c::

"0

..:

-50

0.005

o

-I OO ~_'___"_____"_____"_____-'----_~~

4. 15 4 . 16

4.17

4. 18 4. 19 [ 1Hz

4 .2

421 .

4,22 xlD'

4. 15

4. 16

4 .17 4,18 4 . 19 [fl-lz

4.2

421. 4.22 xlD'

(b) Frequency sweep curve 2

Fig. 12. 25

Testing characteristics of stator with series capacitor 1.8 ,------,,------,-----y--,,----,.----,.-----, 1.6

14

0.8 0.6

0.4 0.2 0

4,14

4. 12

4 ,16

4, 18

4,2

4.22

[fl-lz

Fig. 12.26

4.24 xlO'

Amplitude-frequency characteristics 01 the stator with series capacitor

2. Parallel capacitor When the stator is in parallel with capacitor C, the equivalent admittance is Y

I

.

=

{

wC

+ wC o •

1- (fJf)' 1 - ( f j f) 2

]

(12. 9)


370

,..

0.04 r----,.--r---r----,,----r--y---.--........

0.03 ~

__ - /

...

,

I

I I

I1

0.1!1F

I

,

.. _ _ - - - - -

"

!I ~0.02

i
-J

C/Q

:::;S'

...tJ

N

f-'

rt (J

"...

::r po

376


...f1..f'L 1__ Q,

M,

+v0 - - - -----+---c::-

50 kHz

'"'

....> " r;

"~ V,

volta ge

4

r oj

~ Outpu t

~

6

10

V

V

V 20

30

40 IIIl S

50

60

70

80

Fig. 12. 41

Waveforms of LC resonance voltage step-up circuit

Primary driving

circuit without transformer

Thc detail cd operating proccss is as follows: Suppose that the waveform of the switch K is shown in Fig. 12.42, whcn t= 0, the switch turns on; at t = to, the switch turns of[; and at t = t\ it turns on again. The on-of[ cycle T= t, - to, and the duty cycle D = (t, - t 1 ) IT. Stagc I (Swi tch turns on): At t = 0, thc switch turns on, the original circuit can bc transformed into Fig. 12. 13. The C is shorted by the switch, and the power E and L make up a loop through the switch. The current i increases linearly: i = io EtlL When thc currcnt is not continuous. i = EtlL Stagc II (Swi tch turns off): At t = to, the switch K turns of[, Land C forms a resonance circuit, the original circuitcan be transformed to Fig. 12.11. The corresponding state equations are

+

J ~~' 1I C

~

K

di dt

i-~

= =

E

-

(12.15)

u,

L

E ()

10

II

C

E

R

12

Fig. 12.42 Waveform of switch signal

Fig. 12.43

Equivalent circuit with closed switch

Fig. 12.44

Equivalent circuit with open switch

Chapter 12


379

When the current is discontinuous, the initial conditions of these equations are i(t = EDT/Land U, (t = 0, and assume the resistance R is very large, then the voltage u,(t) and the current i(t)at the resonant stage can be derived as j

)

j

{

h were w Let

~'

(t)

=

E [1 - cosw(t - to)

i(t)

=

ECw[sinw(t- to)

Uc

(1 _2rcD) l'

=

=

)

=

1. ;rc

IS

+ DTwsinw(t -

to)]

+ DTwcosw(t- to)]

02.16)

t h e resonance angu Iar f requency.

0, then the maximum value of u,. is

=

U,.m"x

[u,(tHn"x

= E[ 1 + /1 + (DTw)']

02. 17)

Thc above analyses are derived from the assumption that Land C only resonate one cycle at the OFF stage. In fact, we can choose different L so as to let them resonate k (k = 1 ,2,3 ... ) cycles at the OFF stage, and the inductance should be decided by

L=

O-D)'T' (2rcS)'C

02. 18)

2rcS -:-O-:---....:.D=-:-)=1'

02.19)

And the angular frequency is w -

Substitute Eq. 02. 19) to 02. 17)

02. 20) Then we can get the rclations between maximum voltage unn"x and duty cycle under different resonance cycles k, as shown in Fig. 12. 45. It can be seen from this figure that the maximum voltage increases with the increase of k under the 20 ····1 cycle - 2 cycle --- 3 cycle

18 16

~

OIl

lS "0 >

E E

" . ~

:2

I

14

I

I

I

I

I

I

I

I

I

///

12 10 8 6 4 .................... - .................... 2~

Q1

-- --

-----------//

......

__ __ __ __ __ __ ____ Q2 Q3 Q4 Q5 Q6 Q7 Q8 ~

~

~

~

~

~

~~

Q9

Duty cycle

Fig. 12. 45

Maximum voltage at different cycle wave


380

same duty cycle. Therefore, we should choose high k to get sufficient high output voltage. And at the same k, the U cmox increases with the increase of duty cycle, which is easy to understand. Larger duty cycle leads to more energy stored in the inductor, which causes a higher voltage during resonance duration. However, the voltage applied to USM is not an ideal sine wave, as shown in Fig. 12.41, which contains abundant harmonic waves. In order to drive USM efficiently, we should choose suitable harmonic wave. Therefore, it's necessary to analyze the output voltage using Fourier analyses. According to Fourier transformation, the output voltage can be rewritten as

k=]

let n

=

wT z; ,m =

ak =

k=]

DTw ,a

=

(1- D)21(, then the parameters in Eq. (12.21) are

~f2rr Uccoskx dx 0

1(

E{ 1 . k _sin(n+k)a_sin(n-k)a sm a 2(n+k) 2(n-k)

-; k

bk

+ m[1-cos(n+k)a+1-cos(n-k)a]} 2 n+k n-k 1 f,rr U c sinkx dx 1(

E 1(

{1- cosk a _ 1- cos(n+ k)a 0

k

1 - eos(n - k)a

2(n+k)

2(n-k)

+ m [Sin(n+k)a _ sin(n-k)a]} 2 n+k n-k

Therefore, the amplitudes of every harmonic wave under different duty cycle can be derived, as shown in Figs. 12.16 and 12. 17.

-c\ 5 4

- C2

Shutoff period is 1 cycle

.... C3 --- C4

_.- C,

2

/ I

""

I

I

I

I

I

I

I

.=:-;: .... ..:....~>

.o . . . .o •

.o

"

"S

>:'

.'"

0.

~

Fig. 13.5 USM's speed vs. frequency under different temperature

Supposing that P j is the operating point of USM which corresponds to the rotary speed n1' before the temperature increases, when the temperature increases, and the characteristic curve of the ultrasonic motor changes from 51 to 5,.

If the driving signal frequcncy is fixcd, thc operating point will be P" rotary spced of USM will decrcase to

n,.

and thc Thus, thc drivcr decreases automati-

cally thc driving signal frcquency, which makcs the operating point to movc to

P3

,

so as to keep the original rotary speed nj. This is frequency automatic track-

ing mcthod. When thc rotor of the ultrasonic motor only contacts with thc wave crcst of thc points of the stator surface, according to Eq. (13. 1), thc rotary spccd nand thc axial amplitudc of thc stator's bending vibration Wo havc thc following rclationship (13. 6) where a is a proportional constant. Therefore. the rotary speed of the motor is proportional to the stator's amplitude and the exciting frequency. Actually, the amplitude is influenced by many parameters, the most important one is the exciting frequency (13. 6) can be rewritten as n

=

awo W Ot Cwo) (

~

)

Wo'

Thus, the Eq.

(13. 7)

In Eq. (13.7). thcre is a t added to the suffix of the amplitude. which means that thc amplitudc is time variablc. That is to say, cvcn thc frequency is constant, the amplitude still changcs slowly. If therc is a minor variation L thc operating point

Wo ,

w

ncar

then

The reason that this formula can be approximated in calculation is Lw/ Wo

< 1%

gcnerally. According to Eq. (13. 8), if the exciting frequency can be adjusted

392


+

online to make W Ot (wo 6.w) = const, we can guarantee the stable rotary speed of the rotor. This is the principle of the frequency automatic tracking method.

13. 3. 2

Detection of Amplitude

In order to make the rotary speed of the rotor stably, the vibration amplitude of the stator must be constant. Then, how to detect the vibration amplitude of the stator? Generally a method used is to install a piezoelectric ceramic transducer in the piezoelectric ceramic piece pasted on the stator. This transducer is the isolated electrode mentioned in Chap. 5. Under the action of the traveling wave, the isolated electrode will generate alternating voltage induced by the piezoelectric effect. According to Eq. (2. 16b) , and letting 1=0, the voltage is . V I = - : - - -K C WI

lw

(13. 9)

./0

where Co , K , and WI represent the elamped capacitor of the isolated electrode, the force coefficient of the piezoelectric ceramic piece, and the vibration speed of the isolated electrode, respectively. Since W Ot is a slowly variable signal WI

=

tt (Wotsinwt) """ wWOteoswt

(13. 10)

Substituting Eq. (13. 10) into Eq. (13. 9), we can obtain:

VI

= -

KWOt

--'------c coswt 1

(13. 11)

~o

Therefore, the alternating voltage of the isolated electrode is an alternating signal having the same frequency as the exciting frequency, its amplitude is proportional to the amplitude of the traveling wave in the stator, and the rotary speed is also proportional to the amplitude of the traveling wave in the stator. Therefore, it can be conel uded theoretically that if the transmission between the stator and rotor is ideal, the amplitude of the alternating voltage on the isolated electrode is proportional to the rotary speed of the motor. If the alternating voltage VI of the isolated electrode is rectified and filtered furthermore, the average voltage obtained is also proportional to the rotary speed of the motor.

13. 3. 3

Implementation of FAT System

Figure 13. 6 shows a elosed loop control system FAT. In the system, the feedback voltage comes from the isolated electrode, which is also called as an isolated voltage. Experimental result shows that the load not only influences the rotary speed, but also influences the isolated voltage. That is to say, under the same rotary speed, if the load is different, the isolated voltage is also different. There is still no reasonable explanation yet at present. Because the operating region of USM is often between the resonance and antiresonance points, therefore the operating frequency locates in the right side of the maximum rotary speed. In Fig. 13. 6, the isolated voltage is applied to the in-

Chapter 13


393

Given voltage ,...-------,

+

Block diagram of the feedback electrode voltage feedback control system

Fig. 13. 6

put terminal of PI controller. When the whole system is stable, the isolated voltage equals to the given voltage. The speed adjusting of USM can be obtained through changing the given voltage. If the given voltage is a constant, the USM's speed is also constant under constant load. Figure 13. 7 shows the relationship between time and the rotary speed under open loop and the temperature. At the same time, Fig. 13. 7 also shows the relationship between time, the rotary speed, and the frequency under elosed loop of the motor. From this figure, after the frequency automatic tracking control (closed loop) is used, the variety of the motor's rotary speed is kept within 5 %. As the temperature of the motor increases, its open loop speed drops dramatically. Under constant load, the frequency automatic tracking technique can fairly compensate the variation of the motor's speed caused by temperature change. 150

55

140 '-

~

~

""s;-

50 !-l

130

~

120

J:

45 [

'-,

~

~

Il O

C ~ 0 100

""

N

B

40

90 35

80 70 60

30 0

5

10

15 {/min

20

25

30

25

Fig. 13.7 Experimental result of rotary speed varies with temperature in TRUM-60

13. 4 13.4.1

Ultrasonic Motors Used as Servo Motors Ideal Servo Actuator-USM

As equipment to transform the electric energy into mechanical energy, as viewed from applications, an elcctromotor can be divided into two kinds: a power motor and servo motor. They have the same operating principle, but the different function. The significance of the power motor lies in exporting a large enough power

394


and generating a large enough torque (or force), so as to drive mechanical facilities. The usage of the servo motor lies in that it can change its operating state, speed, output torgue (or force), etc., to adapt quickly to the continuously variational operating condition. In addition, their operating state, performance, and requirements are also different. In general, the servo motor is used in the electromechanical servo system as an actuator, whereas the power motor is used for general mechanical motive power systems. According to the operating demand of servo systems, the following requirements arc put forward to the servo actuator: (1) The output torque and rotary speed can meet the requirement of load. (2) A relatively wide adjustable range of speed and torque, and controllability in low speed operating. (3) A rapid response, that is to say, a quick start and stop, or reverse, and a quick response to signal variation. (1) A large torque/mass ratio and small volume as far as possible. (5) A linear relationships between the rotary speed and torque, and between the torque and control variable. According to above requirements, and compared with the characteristics of an ultrasonic motor, it can be seen that the performance of the ultrasonic motor are good enough to make it an ideal and small servo motor. However, as a novel servo actuator, whether or not the ultrasonic motor can be widely used for the servo system and whether or not its excellent performance can be given are determined in a large extent by whether or not it is convenient to be controlled and whether or not it has good control performance. Therefore, profound and systematical research should be carried out on the servo control techniques using ultrasonic motors.

13.4.2

Requirements of Servo Control Using USM

The servo control using ultrasonic motors mainly means its position and speed controls, and sometimes also includes its output torque control. However, whether the position or the speed controls, the servo control system based on ultrasonic motors generally includes following components: transducer used to feedback the position or speed of USM; an ultrasonic motor used as an actuator to implement the motion, such as the mechanical objects driven by USM; the controller used to implement a control strategy and a control algorithm; a servo driver used to transform and amplify the control signal exported by a controller to make it meet the requirements of the format, energy, amplitude, power of driving, etc. In the design of the servo system based on ultrasonic motors, we must consider that: (1) The stability of operating, which is the foundation to make sure the position and speed of a motor to track a command. (2) The accuracy of tracking, which means the output position or speed of a system is very close to a target.

Chapter 13


395

(3) The rapidity of response, which means the system arrives at a target value within a time as short as possible. (4) The robustness of control, which means that under situations when the characteristics of USM change or there is external interference, the control performance of a system can be kept. Many strategies and plans have been applied to the accurate servo control of ultrasonic motors. Limited by the space, this chapter will only introduce several usual and already applied control strategies. For other control strategies, please refer to relative writings.

13. 4. 3

Servo Control System Using USM '

1. Control system based on computer The implementation of control strategies IS based on reliable detection and control equipment. For example, the speed control of ultrasonic motors need to detect precisely the instantaneous rotary speed of USMs, while the position control needs to measure precisely the current position of USMs. For this sake. in order to obtain a superior man-machine interface to implement different control strategies, the PDLab made the robot arm, which is based on ultrasonic motors, as shown in Fig. 13. 8. This system ineludes: three traveling wave rotary ultrasonic motors. each of which cones ponds to a driving and control circuits, a rotary photoelectric encoder, a frequency/voltage transformation circuit. a motion controller, and a computer used for all USMs, as shown in Fig. 13. 9. Because the following control testing completed are all based on the system, we can compare with the control quality of various control strategies.

Fig. 13. 8

Robot arm based on ultrasonic motors

In Fig. 13. 9. when the motor rotates, the rotary photoelectric encoder gives electric pulses. which can indicate the position of the motor. When the rotary speed of the motor varies, the frequency of the electric pulse from the encoder varies accordingly. The computer determines the instantaneous rotary speed of the motor according to the voltage sampled by the frequency/voltage transformation circuit. as shown in Fig. 13. 10. simultaneously obtains the position of the motor by reading the values from the counter, then determines the corresponding

396

I


J--f

USM

-;..:..::.:..:.---------

State

~

I

Driving and control circuit

Rotary photoelectric encoder

Mechanical load

I

" " ~~

--- --- -- -- --

~o

BlO

r

RlS CW/CCW

§

'" u

'-------

I

.. _-------Servo control signal

Fig. 13. 9

DAC

I

Timer interrupt request

] '8 U

ADC

l-

I

I I

-----t--------+--------+-----------Computer system (control algorithm) I

~

I I

~

'13

""'"

Ii

I I I I

IReversible counterl

~ "

I I I

!

~S

r-

GO~400b;;d-------------i

IQuad frequency processing circuit I

> "

.~

Frequency/voltage converter

-" "'" "'"

u

~;§

~~

T

--------_ ..

I

Composition of control system based on computer using ultrasonic motor

controlled variable using certain control algorithm, and finally sends out the control signal to driving and control circuit through the interface circuit. We use the voltage from the isolated electrode on the stator to control the rotary speed of USM. However, no matter which speed detection method is used, the feedback signal needs an AID transformation before they arc sent to the computer.

l'i ..

10

v'"' 8

c,

70

:r-!

:r--

r-"I

,,..-,

II :- -- -Expected valuel - Actual value

"::"

"

0;

.2 60 ;;; 0

"" tis

0.3

0.4

Fig. 13. 14 Response of PI control with 10ad(Kp = 20, K, = 2)

0.5

50 1.-

40

0

0. 1

,'-"

....... 0.)

0.2

,'--'

:t.-.

0.4

0.;

I/ S

Fig. 13. 15 Square wave tracking response(Kp =20, K,=2)

In the control experiments, through regulating PI coefficients continuously,


400

the effect of the control can be improved. However, because of the randomness of trying cocfficicnts, and espccially bccausc of the strong timc-variation and nonlinearity of USM, which are influenced by the increase of temperature, the interference of load, and also other factors, the parameters of the motor and its speed characteristic both vary. It is hard to select the ideal coefficients for the controller. If fixed gain is still used for PID coefficients to control the motor, the control quality of the system will be reduced. The results of PID control using the motor with a load are enough to prove this point. Under this condition, if we can implemcnt thc advantagc of PID control and obtain good control effect, it is bcttcr to dynamically tune thc gain of PID controllcr during thc opcrating of thc motor according to its operating statc, to compcnsatc the influence brought by the time-variation and nonlinearity of the motor. On-line PID coefficients tuning can be obtained through the fuzzy technique, the neural network technique, the gcnetic algorithm, etc. )Jext, the neural nctwork tcchnique will be introduccd to implement the on-line tuning of PID coefficients.

2. Neural network variable gain P ID control using U5Mr 4 , 22

23-

The gain of PID controller can cmbody the charactcristic of a control system ovcrall. Howcvcr, its robustncss and sclf-adapting are relatively poor. Thc ncutral network has relatively strong ability of approximation in the nonlinearity, self-adapting, and robustness. If thesc two tcchniques arc combincd, which means the coefficients of PID control are tuned on-line through the neural network, namely, whcn the motor is undcr differcnt operating conditions or its inner paramctcrs vary, and diffcrcnt coefficients of PID are selected on-linc to control the objcct, theoretically this method will obtain better control quality than thc fixcd gain PID control. In addition, this technique will omit the complicated proccss in tuning the coefficicnts of PID control. (1) Basic ideology of the ncural nctwork PID control In order to explain the essence of the neural network PID control, Eq. (13. 13) is modified as an increment equation Lu(k)

=

Kp[e(k) - e(k - 1) ]

+ KD[e(k) -

+ Kre(k) + e(k -

2e(k - 1)

(13. 11)

2) ]

Eq. (13. 14) is diffcrcnt from the fixcd gain PID. In the variablc gain PID control algorithm, the values of K p , K r , and Kn depend on the adjustable parameters of thc operating condition of thc system. Thus, abovc equation can bc writtcn as more general form Lu(k)

=

r3(K p ,Kr ,Kn ,e(k) ,e(k -1) ,u(k -1) ,e(k - 2»

Where r3(') is the nonlinear function relating to K p

,

K"

KD

,

(13.15)

e(k), e(k -

1) ,

u(k - 1), and e(k - 2). Lots of methods can be used to approximate r3('). If BP

(Back-Propagation) nctwork is uscd to approximatc r3( • ), the neural nctwork PID control is made up. (2) Structure of the neural nctwork PID control using ultrasonic motor Thc neural nctwork PID controller using the ultrasonic motor is shown In

Chaptcr 13

Control Tcchniqucs for Ultrasonic Motors

401

Fig. 13. 16. It includes two parts: CDThe PID controller controls directly the objcct, whosc cocfficicnts arc variablc; CZ)BP ncural nctwork adjusts on-linc thc cocfficicnts of PID controllcr according to thc opcrating condition of thc systcm to make certain performances of the system best. The output of this neural network corresponds to the three adjustable coefficients of PID controller.

Fig. 13. 16

:'\feural network PID control system using ultrasonic motor

Figure 13. 17 is the topological structure of the neural network used, which has one hidden layer. Theoretically, BPNN (Back-Propagation Neural Network) with onc hiddcn laycr can approximatc to any nonlincar function, and with thc incrcasc of thc nodcs in thc hiddcn laycr, thc approximatc accuracy of thc nctwork increases too. But the calculation amount of the network increases dramatically. Therefore, the nodes of the hidden layer are set to six. According to Eq. (13.15), thc input of thc nctwork is sct as e(k), e(k-l), and e(k - 2). Thc cxcitation function of thc ncuron in thc hiddcn laycr is dctcrmincd according to the following equation: tanh(.r)

(13.16)

e(k) e(k-l) e(k-2)

].F3

Fig. 13. 17

{F6

Structure of BPNN network used for the control of object

Because the coefficients of PID cannot be negative, the excitation function of thc ncuron in thc output laycr uscs thc nonncgativc Sigmoid function


402

[l+tanh(.:r:)] 2

17;,(.:r:) =

.

(13. 17)

In order to avoid saturation during the operating of the neural network, the input of BPNN can be normalized. Because the value of the output layer excitation function g, (x) is within (0,1) , the output of the neural network needs to multiply an appropriate proportional factor to become the coefficients of PID controller. (3) Forward calculation of BPNN The forward calculation is a process for BP:'\JN to determine the values of Kp , K J , and K D • From Fig. 13. 17, for the nodes in the input layers of the neural network, their input/ output relationships are (1)

{

OJ

_

-

ajY

e(k - j),

.Tk-j

=

I,

M= 3

j=O,1,2

(13. 18)

For the nodes in the hidden layer, their input/ output relationships are

b 3

net;2) (k) {

0;2)

=

(k) =

o~) (k)

aji) (k)

f[:~t;2) (k) ] , 1,

Q

i

=

0,1, ... ,Q - 1

(13.19)

6

=

where is the connection weight between the node j from the input layer and node i from the hidden layer; w;}} (= 8i ) is the threshold value, and the superscript (1), (2), and (3) represent the input layer, the hidden layer, and the output layer respectively. For those nodes in the output layer, their input/output relationships are 6

{

net Z3 ) (k) a;') (k)

=

=.

~ W~3) a;2) (k)

g, [net;') (k) ] ,

(13. 20) l

=

0,1,2

is the connection weight between the node i in the hidden layer and where the node l in the output layer; wg) is the threshold value; and wg) =8,. Thus, the coefficients of PID controller is

Kp (k)

=

Gpae') (k),

K, (k)

=

G,a;') (k),

KD (k)

=

GDoi') (k) (13. 21)

where G p , G" and G D are the proportion coefficients selected properly. (4) Adjusting of the weight in the BP:'\JN In order to adjust the weight of the network, the modifying function of the weight is set as (13. 22) The steepest descent method is used to modify the weight of the network, i. e. according to the criterion J to search and regulate under the direction of the neg-

Chapter 13


403

ative gradient, and plus an inertia term which makes the search rapid convergent to a global minimum value. Thus, the modified amplitude of the weight is

!:::"w~') (k + 1)

=-

"I

a~3) + a!:::"w~') (k)

(13. 23)

aW/i

where r;( > 0) is a learning rate; a (~ 0) is a smoothing factor; a!:::"w~') is a inertia term. While a]

a] ay(k

ay(k+1) duCk)

+ 1)

aw~3)

duCk) ao;3)(k)

ao;3)(k)

anet;3) (k)

anet;3) (k)

aw~3)

(13. 24) According to Eq. (13. 11), following equations can be obtained

J

du (k) (3) (

ao o

l

k

=

)

Gp [e(k) - e(k - 1) ]

du (k) , aO ;3) (k) = Gle(k) duCk) (3) (

ao,

k

=

)

(13. 25)

GD [e(k) - 2e(k - 1)

+ e(k -

2) ]

Therefore, the adjusting amounts of the connection weight in the output layer of BP='JN are !:::"W~3) (k

+ 1) =

r;B;3)

0;') (k) + a!:::"w~3) (k)

J B(3) (k + 1). ay (k + 1). duCk) • (k)] e duCk) Cb;') (k) g, ne I g: [net;') (k) ] g, [net;') (k) ] {I - g, [net;3) (k) ] } , l l=0,1,2 =

I

t (3 )

[

I

I

(13.26)

=

Similarly, according to the above predication method, the adjusting amounts of the connection weight in the hidden layer are

1

!:::,.w~') (k+1)= r;B;')o;]) (k)+a!:::"wt2)

B;2) =

j' [net;2) (k) ]

f' [net;2)

(k) ]

=

1-

~ [B;3)w/i (k) J, i

0,1, ... , Q-1 (13.27)

=

f [~et;2) (k) ]

The modification of the weight needs to calculate the partial derivative term

dy;~(t) 1) ,

which involves the model of USM. However, currently there is no

effective model of USM. In order to resolve the problem, generally, the symbol-

. f· [d y (k+1)]. db· dy(k+1) lC unctIOn sgn duCk) IS use to su stltute duCk) . Finally, the calculation formula of the weight value in the hidden layer and the output layer are coneluded, respectively: (k

+ 1) =

(k+ 1) =

(k)+!:::"w~')(k+1), l=0,1,2,3

(k)

+ !:::"wt2) (k + 1) ,

i

=

0,2, ... , Q

(13. 28)


404

(5) Control results Figure 13. IS shows experimental results of the neural network PID control using USM. In Fig. 13. IS (a), the neural network PID control not only obtains relatively high position control accuracy (stable state position error: -0. oS~+ o. OSO). but also keeps original control quality of the system even when the load is added. Fig. 13. IS (b) shows that USM possesses rapid and high accuracy servo position control performance. which modify PID controller's parameters online through neural network.

.

.

_ . 71.-0.0 N· III -

>::'

7i.- O.2 N ·III

>::'

'"

90

80

a

0;,

'"

60

«'"

"2

'"

40

0;,

a

.2 0;

0;

« 0 0

0.5

2

1.5

- - Aci llal val lie

. - - - -Expecled val ue

20 0

0

2

4

I/S

(a)

.,t

40

5 E

30

'"0. a 20

a:

10

I\... V'"

It"" ......,

. :

. J\ jL fl 1'- r~

I

10

2 ,------------r-----------,

~ ---~ It

m {l)

" i,

-K

-

'1 ..J

~

OK!)

~

~i·~_(~:L · X=~.--L:L_j~~:C~

~

8

(b)

50

~

6 I/S

~

-

)

i 'f

10

o o

n ,c)

"" 5

lIs

I/S

c)

(d)

10

Fig. 13. 18 Experimental results of the neural network prD controller using USM

13.4.5

Adaptive Controller Using USM L5 • 78J

It is well known that the performance of USM is influenced by temperature, load. pre-pressure, and its driver. etc., and it has strong time-variation and noncertainty. The good operating of PID controller can make a system obtain relatively high control accuracy within short operating time. The parameters of the system vary after operating for a long time, therefore, it is hard on PID control to keep the system in a superior control quality. Under this condition, a selfadapting control will be a good choice. The self-adapting is a control method adjusting the parameters of the controller continuously to compensate the variation in the characteristic of controlled system, and especially suitable to system with time-varying and nonlinearity. The self-adapting control mainly includes a self-

Chapter 13


405

tuning control and the model reference adapting control. The application of these two techniques in the control system using USM will be discussed as follows.

1. Self-tuning control using USMC3· 24 2S] In all kinds of the self-tuning control methods, the self-tuning controller usmg the generalized least square error control is most suitable to the inverse unstable system. Therefore, we introduce this algorithm into the servo position control system using USM. Fig. 13. 19 shows the block diagram of the self-tuning control system using USM. The kernel of the self-tuning control system is to identify dynamically the parameters of the USM through identifier to determine the control function of the controller, to keep the error of the system least.

I

'--""T""---,I-+-Y_(k..> Object

I

Block diagram of self-adapting control system using the ultrasonic motor

Fig. 13. 19

It is supposed that the mathematical model of the motor can be de script as the following difference equations

+

AY(k) = q-dBu(k) w(k) { A = l+alq-l +···+anq-n

B

=

bo +b1 q-l

+ ... +bmq-m

(13. 29)

where d is the delay of the control; q -1 is the reverse shift factor; w(k) is white noise; nand m are the order numbers of A and B, respectively. Since this system is a precise position control system, the controller accepts the expected position and the actual position as inputs, and chooses the phase difference as output. In the self-tuning controller, generally, the principle of the control is to make the following cost function 11 least (13. 30)

where Y and Yd are the actual position and the expected position, respectively; A is the control factor, which is to limit the very large input of the control, and improve the stability of the self-tuning closed loop system. In order to obtain the input which can make the 11 least, an auxiliary system is defined as r(k

+ d)

=

y(k

+ d)

- Yd

+ AU (k)

(13.31)

The corresponding object function is

12

=

E{[r(k+d)]'}

It can be proved that u can make the obj ect function

(13. 32)

12 of the auxiliary system


406

least. and can also make the original object function 1] least. The general optimal expectation model of the output for the auxiliary system is rO (k + d/ k)

yO (k + d/ k) - Yd + AU (k)

=

(13. 33)

whereyO(k+d/k) is the optimal expectation of the y(k+d) in time k+d. and composes of time k and the previous information. Let reek + d/ k) = O. and obtain directly the control law of the system u(k)

Yd -

=

yO(k

A

+ d/k)

(13. 31)

Because

=

yO(k+d/k)

(13. 35)

Gy(k) +BFu(k)

where G and F are determined by the equation of Diophantine

1

FA+q-dG

=

1 F. G

1 + flq-l + ... + fnfq-nJ f!;o + f!;l q-l + ... + f!;ngq-ng

= =

degF degG

=

Thus. we obtain the input of the control which makes (k)

Yd (k) - l'Y (k) BF+A

=

u

or

B

1

u(k)

=

ho

f'..

11 least

Yd (k) - l'Y (k) M+A

B

(13. 37a)

=

ng

+ A [Yd (k) -

(13. 36)

d-1 n-1

=

f!;iY (k -

i) -

miu (k -

(13. 37b)

i) ]

Because the parameters of the system a i and hi are unknown. and they are needed to be identified. we define the cost function 1, as N

13

(13.38)

L-pN-k{y(k) -(k)

[y(k -

=

{ (J =

From

;11;

=

1) .y(k -

2) .... . y(k - n).

u(k - d) • u(k - d [ -

a2 ••••• -

al • -

an

1) •...• u(k -

.ho .h 1

•••• •

d - m)

r

(13. 39)

hmJ T

O. we can obtain the estimation algorithm of the recursive least

mean square for the parameter

J

O(k)

=

O(k -

+

1 ! K(k)

=

P(k)

=

p

(J

1)

+ K(k) [y(k)

- cf>T (k) O(k -

P(k - 1) cf>(k) cfir (k) P(k - 1) cf>(k)

[P(k -

1) - K(k) cf>T (k) P(k -

1) ]

(13. 40) 1) ]

where p is the oblivion factor. whose range is [0.9. 0.99].

Chapter 13


407

Generally, the less the value of p, the stronger the ability of tracking time-varying parameters, and simultaneously the severer is the influence of the noise interference. Because the system with USM is a slow time-varying system, set p as o. 99. The closed loop equation of the system is

_ L!L Yd +;. +BFe(k)

y(k) -

A;. +B

A;. +B

(13.11)

When;' = 0, the closed loop characteristic equation of the system is B = o. If it has any zero pole residing outside of a unit circle, the system is an irreversible stable system, namely the closed loop system is unstable. Thus, selecting a suitable;' can control the reverse stable system. The block diagram of the system is shown in Fig. 13. 20.

Fig. 13. 20

Program block diagram of least square error control

2. Model reference adaptive control using USM L1J The model reference adaptive control (MRAC) is a kind of control method based on model. It adjusts the parameters of the controller to compensate the change of the parameters of the controlled object. This method does not have high requirement of the model of the controlled object. According to the operating characteristics of USM, this kind of method can be considered in the control system using USM. The MRAC position control system using USM is shown in Fig. 13. 21. In the MRAC system, a reference model G m (5) is set, its controller is composed of the pre-filter it, the feedback compensator F (t) and the self-tuning mechanism. ret) is the reference input of the system. When USM operates, the system regulates dynamically the parameters of the controller Ie and F(t) through the generalized error of the position em' which is obtained through comparing the actual position x, (t) of the motor with the output position x'" (t) of the reference model


408

G m (s) • to compensate the error introduced by the nonlinearity of the motor and the changes in the parameter of the motor. to make the actual position of the motor close to the reference modcl as much as possible. In the figure. the frequency modulation method is used to control USM which uses the variation of the frequency as the control variable. The value of this variable is added eventually to the reference frequency to form the driving frequency of the motor. In the control methods there arc three important problems:

f*(/)

Reference model

r (t)

x,(/)

USMmodel

'------I Self·adapting 1--_ _-' mechanism

MRAC control system using USM

Fig. 13. 21

(1) An approximate transfer function model of USM Because the speed response characteristic of USM possesses that of the first order inertial element. and the MRAC has low requirement of the model. therefore. the transform function of USM can be supposed as G( ) I

S

=

D,(s) = UI(s)

kUSM res",s+

1

(13. 12)

where D, (s) and U I (s) are the expressions of the motor rotary speed and the controlled variable of the frequency converter in the frequency domain. respectively; rUSM is the time constant of the motor; kUSM is the proportional gain in the model.

For a time-varying object. kus", contains certain amount of time-varying uncertainty. In the design of the MRAC. kus", can be considered as constant. According to Eq. (13.12). the nominal transfer function of the motor uSing the position as output is CPOS

=

(13.13)

s( res", + 1)

where @(s) is the expression of the angular displacement of the rotor in frequency domain. a o = l/rusM' bo = kesM/resM· The above model is written as the form of the state space equation X,(t) = A,x,(t) +B,UI(t)

Wherein

B,

=

[~J

(13. 11) (13. 15)


Chapter 13

409

According to Fig. 13. 21, the controlled variable of the output in the MRAC is (13. 46) (2) The design of the reference model In the design of the reference model, the speed characteristic of USM should be considered sufficiently, but the capacity of USM cannot be exceeded. The reference model should satisfy the requirements of rapid response, a little overshoot, and no steady state error. According to the model reference adaptive control theory, the order of the reference model should not exceed the order of the system itself. After these factors arc considered, using the standard second order link as the reference model, whose state equation can be described as

{ xm : Am Xm Ym - CmX

+ Bm r

(13. 17)

where

A

= m

[ -

0 ktnl

Cm

=

[l,OJ

According to the requirement of the optimal response characteristic of the second order system, the damping ratio of the reference model is S"= o. 707, and the step response is designed based on the rising time t, = o. 14s, then

Am

=

[_

~61

_216. 8].

Bm

=

[3~lJ

(13. 48)

(3) The self-adapting law of the MRAC controller In the model reference self-adapting control system, the position error, which indicates the difference between the actual output of controlled object and the model output, called the generalized state error, that is (13. 19) From Eq. (13.19) ,we can obtain

x,(t)

=

[A, -B,F(t)Jx,(t) +B,K(t)Uf(t)

(13. 50)

From Eqs. (13. 17) and (13. 50), the equation of the generalized state error vector is (13.51) When

Ku

and F(t) are adjusted to

Ku =

K: and F= F* , respectively, the ad-

justablesystemmatehesthemodel. Here, Am=A,+B,F* and Bm=B,K: ,soEq. (13. 51) can be written as (13. 52) whereF=F* -Fm;K=K: -K u • The kernel in the design of the MRAC control system is to determine the selfadapting principle of Ku and F (t). Thus, a Lyapunov function within a space composed by general state error is defined

410


(13. 53) where P, r-;1 and r;1 are the symmetric positive definite matrix. Obviously,

>

when em # O,L(t)

0

Differentiating two sides of the above equation to time, then Let)

=

1

~T

T··T

ZemPem+emPem+tr(F

r-;

1

-

-T

F+F

r-;

1 ~

~ e~ (PAm + A: P) em + tr(F T r-;1 F + x,e~

1 ~

---"

~

-1---"

F+Kr, K+Kr 2 K)

PBm K :

-1

F) (13. 54)

Because Am is set as a stable matrix during the design of the reference model, a symmetrical positive definite matrix Q can be chosen to establish PAm

+ AmP=

-Q. For arbitrary em #0, the first item of the above equation is negative definite. If choose

{F-

(B

---" = -

r

K

r, (BmK:

=-

Therefore, considering BmK: -1

=

1

B"

{F=

it

m

K'-l)Tp u

-1)T

emx,T

(13.55)

Pemur

then self-adapting law is

r

r

=

B: Pemx:

1

2

(13.56)

B: PemUr

Adjusting the parameters of the controller according to the above equation, the second and third items in Eq. (13.51) are both zero, and L(t) is negative definite. Obviously, the adapting theory designed by this way can guarantee the global asymptotic stability of the model reference adapting control system using USM. Thus, if t - CXJ, em (t) (1) The results of the control

90

r

"""

M

,...

....,

,...,

1""\

o.

,..,

- • - - OUlput of Ihe model OUlpul of Ihe molar

80

~60 ;;

g .'il 40 ~ 20

o o

10

15

20

25

lis (a) Response for asquare wave comm and

Fig. 13. 22

25 lis (b) Parameler F in square wave track ing "-

Results of MRAC using USM

Chapter 13


411

Figure 13.22 shows the results of MRAC control using USM. Where. Fig. 13. 22(a) is the result of the tracking control to a square wave when the reference input is 45 -90 It can be seen that the rotary speed of the motor tracks quickly the output of the reference model. and the tracking precision is relatively high. Fig. 13. 22 (b) gives the variation of the parameters of the self-adapting control. 0

13.4.6

0

•

Fuzzy Controller Using USM L2629 -

A fuzzy control is based on human's experience and knowledge. and employs the fuzzy reasoning as means. and makes decision through imitating the human's thinking manner to implement the technique of the computer intelligent control. The fuzzy control describes a system through some language variables. Its implementation does not need the precision mathematical model of a controlled obj ect. Therefore. control methods based on the fuzzy logic reasoning have become important control techniques using USM.

1. Fuzzy control law oj USM Figure 13. 23 shows the block diagram of a fuzzy position control system using USM. Its kernel is the fuzzy logic controller (FLC). which uses the two-dimension input structure. Generally. the two-dimension FLC takes the error of the controlled subject and the variance ratio of the error as the inputs. However. according to the operating feature of USM. in order to guarantee the high robustness of the control system when the parameters and the operating condition of the motor vary. the position error e (e = Yd - y) and the angular speed w, of the motor are taken as the inputs of FLC. The frequency adjusting method is used to regulate the speed of the motor. Therefore. a frequency base point f' is needed. which corresponds to the rated speed of the motor. In Fig. 13. 23. we use the increment of the operating frequency (frequency variable) 6.u as the output of FLC. and u as the control variable of the frequency converter. FLC

r-----------------------~

y.

+

S

e

AU

.~

-=

~

u(k-I)

I1r(k)

r f(k)

+

------USM Fig. 13. 23

Block diagram of fuzzy control system using USM

2. FuzziJication oj input/output variables Af ter the input/ output variables of FLC are determined. based on the requirement of the fuzzification. first of all. they need to be quantized. i. e.

412


transforming them from a basic domain into the fuzzy domain. The grade of the quantization influences dramatically the control quality of the system. Generally. the smaller the quantization grades of the error e. the larger the system's overshoot. and the longer the transient process. Too much quantization grades will increase the calculation amount and the memory occupation. The main objective that this book uses the rotary speedw, of the motor as the input variable of FLC is to reduce overshoot. Theoretically. the thinner the partition of the quantization grade for w,. the smaller the overshoot. However. too slim partition will influence the response speed of the system. As the output of the FLC. too little quantization grades of 6.u will make the system oscillate. and too much quantization grades will elongate the transient process of the system's dynamic response. After considering these factors synthetically. we take the basic domain [ - 1 ( . 1( ] of the error e as fifteen grades. Introduce a sign X into the fuzzy domain of the position error

X= {-7.-6 ... ·.0.· ... +6.+7} We take the basic domain [ -10.10 J rad/ s of the rotary speed w, and the basic domain [ - O. 5. O. SJ kHz of 6.u as eleven grades; Y and Z indicate the signs of the fuzzy domain.

Y= {-S.-4.···.0.···.+4.+5}. Z= {-5.-4.···.0.···.+4.+S} During operating. the rotary speed w, of the motor may fall outside of [ -10. 10Jrad/s. then the quantization grades are set as 5 (when positive error) or -S (when negative error). The quantization factors of the position error e and the rotary speed w, arc G, = 9. 92(l/rad) and Gw=O. S5[ l/(rad/s)J. respectively. The proportional factor of the frequency variable is G/',u = O. l( kHz/l). During the controlling process. in order to increase the response sensitivity of FLC to little error. while quantizing the minor-error input signal, we properly increase the quantization factor. The language variables corresponding to e. w, • and 6.u are J:;. ~. and 6. l.l • respectively. The control strategy of the fuzzy control system using USM embodies in the fuzzy control rules. According to the operating characteristics of USM. the following language values for J:;. ~. and ~ l.l are introduced. {)JB (negative big). )JM (negative middle). )JS (negative small). ZO (zero). PS (positive small). PM (positive middle). PB (positive big)} Defining the fuzzy of variables is an important content of fuzzification. and its essence is to determine the shape of the membership function curve in the fuzzy domain. The membership grade functions defined by variables J:;. ~. and 6.1.1 arc shown in Fig. 13. 24. where J:; and ~ usc the triangle distribution function. and 6.U uses the single-point fuzzy set. Once the membership function curves are defined and discreted in the fuzzy domain. the membership grade of points in the fuzzy domain will be obtained. and the fuzzy subset of fuzzy variables arc constructed.

Chapter 13

:c!


413

0.5

"-

- 2 -I (a) Displacemcnls

NB

4 5

2

0

(b) Angular speeds

NS

NM

PM

PS

ZO

PB

~I

"-

I

o -5

-4

I

-3

I

-2

-I

I

0

I

I

I

2

3

I

5

4

(c) Frequency variable

Fig. 13.24

Membership degree functions of the input/ output variables in the FLC

3. Establishment of the fuzzy control rule The rule of the fuzzy control is based on the experience from manual control, and the rationality of its constitution directly influences the control quality of FLC. The experiments of the manual control using USM show that if the current position of the rotor is far away from the objective position, the current rotary speed of the motor is relatively slow, and then in order to implcment a rapid positioning, the motor needs to speed up. Therefore, in order to decrease the driving frequency of the motor, FLC needs to export a negative frequency-vary value. On the contrary, if the current position of the motor is elose to the expected position, and the speed of the motor is relatively fast, the motor needs to decelerate. Thus, need to increase the driving frequency of the motor, FLC needs to export a positive frequency-vary value. According to these experiences obtained from the operating, the fuzzy control rules for USM are established as shown in Table 13. 2. Table 13. 2

Fuzzy control rule for USM E ~

fill

W

ZO

PS

PM

PB

NB

NM

)IS

NB

ZO

PS

PM

PB

PM

PS

ZO

NM

NS

ZO

PS

PM

PS

ZO

)IS

NS

NM

NS

ZO

PS

ZO

NS

)1M

ZO

NB

NM

)IS

PB

NS

NM

)lB

PS

NM

NS

ZO

PS

ZO

NS

)1M

PM

NS

ZO

PS

PM

PS

ZO

)IS

PB

ZO

PS

PM

PB

PM

PS

ZO

I

414


In order to carry out fuzzy reasoning, the fuzzy control rules from above table should be written as the fuzzy condition proposition, that is Rule 11 if I.:; =:'\JB and ~ = NB then L l.l = ZO; Rule kj Rule 77 where,1k

W=B

and if I.:;

,/.J) ,

=

then

~)

L l.l

=

!;;k) ;

then L l.l = ZOo PB represent the language values of I.:;, ~, and Ll.l in their do-

PB

and C)

~

and

~=

main, respeeti vcly. According to the fuzzy mathematical theory, every fuzzy condition sentence corresponds to a fuzzy relationship, that is

ISk) = ,1k X 12) X !;;k)'

k = 1, 2 , ... , 7; j = 1, 2 , ... , 7

(13. 57)

All of these control rules correspond to a general fuzzy relationship IS, which can be expressed as

IS

k=7,j=7

=

U ,1k

k=l,j=l

X

12j

X

!;;kj

(13. 58)

The membership function of IS can be calculated according to the following equation /1R (.:r:,y,z)

i=7,j=7

=.

V

I-l,)-l

/1A, (.:r:)

1\

/1B, (y)

1\

/1c, (z) ,

.:r:E X,yE Y,zE Z

(13.59)

where /1::t, (.:r:) is the membership grade of the fuzzy subset ~k at the position error x;/1':c (y) is the membership grade of the fuzzy subset ~j at the angular speed y; /1c.. (.:r:) is the membership grade of the fuzzy subset

fk) at the frequency

variable z. when the fuzzy values of the position error and the angular speed inputted to FLC are ~' and ~' respectively, according to the fuzzy composition rules, the fuzzy value of the frequency variable exported by FLC is (13. 60) where" 0" is the fuzzy synthesis operator. The above equation can be written as the form of the membership function: (13.61)

4. DeJuzziJication The frequency-variation value obtained through the fuzzy reasoning equation is a fuzzy vector, and cannot be directly used for the control using USM. So, we need to decide an explicit value from this vector, and carry out the dcfuzzifieation. In all kinds of dcfuzzifieation methods, a center of gravity (COA) method is simple, and easy to be implemented, and its compositive performance is good. Therefore, the explicit value of the frequency-variation variable using COA method is

Control Tcchniqucs for Ultrasonic Motors

Chaptcr 13

415

II

~ fl.;:,u (Zi) Zi LU

=

Zi

...:i----;I7-\- - - -

E Z

(13. 62)

~fl.;:,u (Zi) i-"j

where fl.;:,\[ (zJ is the membership grade of the frequency variable L l.l in its fuzzy domain. Theoretically. according to the value of e and w, gathered. using the Eq. (13.61) and Eq. (13. 62). the corresponding grade of frequency variable LU can be calculated online. However. in order to improve the real-time behavior of the control, we can calculate off-line the frequency variable LU according to all possible combination of the elements in X and Y. and then make a look-up table of the fuzzy control using USM. as shown in the Table 13. 3. and store this table in the system. When the fuzzy control using USM is needed. the computer looks up this table according to the position error and the angular speed of the motor. therefore. we can obtain the frequency-vary value needed within the current sampling period. Table 13. 3 t,U

w -4

-5 1

E

Fuzzy inquiry control table of position control using USM

-2

-3 1

1

-1 1

0 1

+1 1

+2 1

+3 1

+4 1

IH

-7

0

-1

-1

-2

-3

-5

-3

-2

-1

-1

0

-6

0

0

-1

-2

-3

-5

-2

-2

-1

0

0

-5

1

0

-1

-1

-2

-1

-2

-1

-1

0

1

-4

1

0

0

-1

-2

-4

-1

-1

0

0

1

-3

1

1

0

-1

-1

-3

-1

0

0

1

1

-2

2

1

1

0

0

-2

0

0

1

1

2

-1

3

2

1

1

0

-1

0

1

1

2

3

+0

5

1

3

2

1

5

1

2

3

1

5

+1

3

2

1

0

0

-1

0

0

1

2

3

+2

2

1

1

0

0

-2

0

0

1

1

2

+3

1

1

0

-1

-1

-3

-1

0

0

1

1

+1

1

0

0

-1

-2

-1

-2

-1

0

0

1

+5

1

0

-1

-1

-2

-4

-2

-1

-1

0

1

+6

0

0

-1

-2

-3

-5

-3

-2

-1

0

0

+7

0

-1

-1

-2

-3

-5

-3

-2

-1

-1

0

Actually. the value obtained from the fuzzy decision is only a grade in the output fuzzy domain. and is not the controlled value requested by USM. Therefore. it needs to be multiplied to a proportion factor G;:,u to become an accurate value. In addition. FLC output needs to add to the control value u(k - 1) obtained from

416


the last sampling period to become the control value of the current sampling period, because Fig. 13. 23 uses the increment structure for the controller. That is (13. 63)

5. Control results In order to inspect the control effect of FLC, we made the fuzzy control experiments of TRUM-60 on the control system established above. Set the reference frequency as 40kHz. Considering that the fuzzy control determines the control value by inquiring the table, the sample period is set to 4ms. Fig. 13. 25(a) shows the response curve of the system in 90° step input, its stable error is within ±O. 28°, the response time is around o. 18s, and the control precision is low. Fig. 13. 25 (b) shows the step response curve under load. It can be seen that the response time increases a little (about o. 2s) after load addcd, but the control error docs not change much. This indicates that the fuzzy control has relatively high robustness to load changes of the motor. 90 80

""

" 00

60

" .~

40

'"

20

:a 0

100

~

""

" 00 :a

50

" .~ 0

'"

0 0

0.5

1.5

0

2

/ 0 0.2

0.5

1.5

2

tis

tis

Ca) Step response without load

Cb) Step response with O.2N·m load

Fig. 13. 25

Step response of fuzzy position control

Figure 13. 26 shows the tracking results of the USM with 90° square wave input. It can be seen that USM oscillates within 0-90° repeatedly and carries out the position tracking rapidly when the target position varies within 0-90° according to the pattern of the square wave. This indicates that the fuzzy control

,

""a'"' 00

90 80

I

60

rr II

I

I

"

(5

'"

20 0

I

, 2

4

I

I

6

8

Actual value

_ . Ex pected

, I ,

I

o

-

,I

C 40

f:a

.n

I

I

I 10 12

value

~~ I

I

I

I 14

"

0"

12

'ED

.

~

,." ::

I

I

16 18

20

"2 "§

U

1 000

500

0 - 500 -1 000 ~-L~-L~~-L~LL-L~-L~LL~

o

2

4

6

8

~

(a) Square wave tracking

10

12

14

16

~

(b) Varial ion of cOl1lrol qualll ily

Fig. 13. 26 Square wave tracking results of fuzzy position control using USM

18 20

Chapter 13

Control Techniques 10r Ultrasonic Motors

417

can implement a rapid position tracking control using USM. It is worth to be noticed that the control value in Fig. 13.26 (b) is the digital format of a frequency control variable, and when its value is negative, "negative sign" only indicates the control effect which makes the motor to rotate reversely.

References [ 1

J

Chunsheng Zhao. Recent progress in ultrasonic motor techniques. ] oumal of Vibration, Measure&. Diagnosis, 2004, 24(1): 1-5. (in Chinese)

ment

[ 2

J

Jiakui Zu. Research on Driving and Control Techniques for Traveling- Wave Ultrasonic Motor Based on Its Electric Characteristics. Post-doctoral Report. 'lanjing: 'lanjing University

o[ Aeronautics and Astronautics, 2004. (in Chinese) [ 3

J

Huafcng Li. Study on Ultrasonic Motor and Its Precise Servo-control System. Dissertation for the Degree of Doctor of Philosophy. Wuhan: Huazhong University of Science and Technology, 2002. (in Chinese)

[ 1

J

[ 5

J

Honglin He. Research on the Ultrasonic Motor and Its Application in the Robot. Dissertation for the Degree of Doctor of Philosophy. 'lanjing: 'lanjing University of Aeronautics and Astronautics, 2007. (in Chinese) M W Spong. Robust and adaptive control o[ manipulators. IEEE Transactions on Automatic Control, 2001(3): 186-210.

[ 6

J

[ 7

J

S Furuya. Load-adaptive frequency tracking control implementation of two-phase resonant invert [Dr ultrasonic motor. IEEE Transactions on Power Electronics, 1992, 7(3): 542-550. T Seniyu. Adjustable speed control o[ ultrasonic motor by adaptive control. IEEE Transactions on Power Electronics, 1995, 10(5): 532-538.

[ 8

J

T Senjyu, T Kashiwagi, K Uezato. Position control of ultrasonic motors with adaptive deadzone compensation with fuzzy inference.

IEEE Transactiuns un Power Electrunics, 2002,

17(2): 265-272. [ 9

J

J Mass, T Schulte, "I Frohlcke. Model-based control for ultrasonic motors. IEEE/ ASME Transactions on Mechatronics, 2000,5(2): 165-180.

[10J

Sahin Yildirim. Design o[ adaptive robot control system using recurrent neural network. International ] oumal of Intelligent and Robotics System, 2005, 11(3): 217-26l.

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T Senjyu, H Miyazato, S Yokoda, et al. Position control of ultrasonic motors using neural

[12J

network. IEEE Transactions on Power Electronics, 1998, 13(3): 381-387. Faa-jeng Lin, Rong-J ong Wai, Rou- Yongi Duan. Neural-network controller [or parallel resonant ultrasonic motor drive. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1999(4): 494-501.

[13J

Faa-jeng Lin, L C Kuo. Identification and control o[ rotary traveling-wave type ultrasonic motor using neural-networks. IEEE Transactions on Control Systems Technology, 2001(4): 672-680.

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Y Izuno, R Takeda, M Nakaoka. 'lew fuzzy reasoning-based high-performance speed/position servo control schemes incorporating ultrasonic motor.

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G Bal. A digitally control drive system for traveling-wave ultrasonic motor. Turkey] ournal Electrical Engineering and Computer Sciences, 2003,11(3): 155-167.

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S W Chung. A thesis submitted in partial [ulfillment o[ the requirements [Dr the degree o[ master of philosophy. Motion Control of a Traveling-wave Ultrasonic Motor. Hong Kong: University of Hong Kong, 200l.

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S W Chung, K T Chau. Servo speed control o[ traveling wave ultrasonic motors using pulse width modulation. Electric Power Components and Systems, 2001, 29(8): 31-37.

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Faa-jeng Lin, Rong-Jong Wai, Hsin-Hai Yu. Adaptive fuzzy-neural-network controller for

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Ultrasonic Motors Technologies and Ap plicalions ultrasonic motor drive using LLCC resonant technique. IEEE Transactiuns un Ultrasunics,

Ferroelectrics, and Frequency Control, 1999(3): 715-727.

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Faa-jeng Lin, Rong-jong Wai, C C Lee. Fuzzy neural-network position controller for ultrasonic motor drive using push-pull DC-DC converter. IEEE Proceedings on Control Theory Application, 1998, 34(1): 363-368.

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Zhihua Chen. Research on the Control of Ultrasonic Motors. Post-doctoral Work Report. Nanjing: Nanjing University of Aeronautics and Astronautics, 2003. (in Chinese)

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Zhihua Chen, Chunsheng Zhao. The control of resonant tracking of ultrasonic motor. & Acoustooptics, 2003, 25 (2): 119-151. (in Chinese) Honglin He, Hua Zhu, Chunsheng Zhao. Position control of an ultrasonic motor using fuzzyPI technique. Mechanical Science and Technology, 2006, 25 (5): 603-607. (in Chinese) Piezoelectrics

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Shousong Hu. Theory of Automatic Control (4th Edition). Beijing: Science Press, 2001. (in Chinese) Huafcng Li, Chenglin Gu. Precise position control of ultrasonic motors using adaptive control. Piezoelectrics & Acoustooptics, 2003, 25(2): 155-158. (in Chinese)

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Chapter 14

Testing Techniques for Ultrasonic Motors In addition to theoretical research on ultrasonic motors, we also investigate their tcsting techniqucs. Testing can not only verify the theoretical method, but also propose solutions for some problems that can not be resolved by theoretical analysis. Furthermore, the performance of the products also requires e.Lperiments according to certain standard measurement methods. Therefore, much attention has been paid to the testing techniques of USMs around the world .1 " . With the broad applications of USMs in aerospace field, semiconductor manufacture field, etc., the USA and Japan conducted certain basic research on USMs under extreme environmental conditions in 1998. Russia and Germany conducted some explorations in this aspect as well. Unfortunately, these techniques and data have not been reported C' 7:. In 2004, the European Space Agency successfully developed USMs suitable for vacuum environments and conducted many experimental studies, ineluding mechanical characteristics, life testing, and vacuum thermal cycling experiments. At present, research institutes in China have also been carrying out some preliminary research on testing techniques for USMS·8-11 .. USM testing techniques mainly include modal testing of parts and assemblies, measurement of pre-pressure, transient characteristics, mechanical performance, adaptivity to extreme environment conditions·]5J, their lives, etc. USMs run at a high frequency above 20kHz but low speed, contrastively, traditional electromagnetic motors(EMs) usually run at a low frequency of 50Hz/60Hz but high speed. Thus, many measuring devices and methods for traditional electromagnetic motors cannot be directly used for USMs. The mechanical characterization measurement system for EMs, including the hardware and software, must be updated. This chapter mainly describes the purposes, requirements, methods, equipment and results of the testing for USMs.

14. 1

Modal Testing for Parts and Assemblies

An ultrasonic motor USM relies on the converse piezoelectric effect of piezoelectric ceramics to excite the resonance of stator in ultrasonic frequency range and realize its rotation. Therefore, the natural modes of the stator are critical to the performance of the USM. It is always e.Lpected that one of the modes of the manufactured stator can be consistent with pre-calculated design, and greater response can be obtained under a certain e.Lcitation voltage. Furthermore, in order to confirm the design of the stator and check the manufacturing precision,



420

the correctness of the piezoelectric ceramic bonding process and to guarantee the assembly quality, the modal test for USM is necessary. Traditional modal tests employ some special excitations (such as hammering) to excite a structure and get its modal response. The response is then converted into electrical signals by using contact-type vibration sensors. After the signal amplification and data processing. the modal parameters of the structure are acquired 116_. Considering that the USM's stator is a structure with small size and light weight, the use of the general contact-type sensors will have non-negligible effect on the modal characteristics of the stator. Therefore, in the initial stage of the author's study, due to the test equipment restriction, we used dynamic signal analyzer HP3562 to measure admittance curve of the stator and obtain its modal frequencies. With the aid of laser holography, we can measure the nodal pattern of the stator. Then, SS330 type electronic speckle laser vibrometer is used for measuring modal frequencies, and node patterns can be measured at the same time 1l7 -. The above apparatus and measurement methods have played an important role and helped us to complete a series of experimental studies 1l8- 19 -. However, electronic speckle laser vibrometer is only suitable for measuring the node pattern of out-plane vibration, and is not capable of measuring longitudinal vibrations and in-plane vibrations of bar-type stator as well as the quantitatively measuring amplitudes. Meanwhile, its adjustment and operating are more troublesome, and it is susceptible to the external environment interference. These disadvantages confine the application to electronic speckle laser vibrometer in the USM studies. In 2002, we adopted a more advanced device PSV-300F-B type Doppler laser vibration measurement system (PSV-300F-B) :20 J , as shown in Fig. 14. 1. The system has some advantages, such as wide test band (0-100kHz), a high accuracy (displacement can be measured to nm level) and non-contact measurement. etc. It can achieve a rapid multi-point measurement through the definition of scanning grid on the measured stator, and then obtain quantitatively amplitude vs. frequency curve, phase vs. frequency curve. and the mode shape of the stator.

Fig. 14. 1

PSV-300F-B type Doppler laser vibration measurement system

Chapter 11


421

The block diagram of PSV-300F-B is shown in Fig. 11. 2, and the operating principle of the system is shown in Fig. 14. 3. It can be seen that the system consists of two major components hardware and data-processing software. The high-precision laser interferometer is the core of the hardware. In the system, the computer scanning module produces a digital signal, which is converted into an analog signal by D/ A converter in connecting box. The analog signal is amplified into an excitation voltage by a power amplifier. The voltage is applied to a target tested, such as a stator, finally then induces the target to vibrate. At the same time, a laser beam. which comes from the laser interferometer in the scanning laser head and has a certain frequency. irradiates the surface of the measured vibrating target, and the reflected scattering laser from the measured target is collected. The reflecting light beam induces the certain frequency to change due to the Doppler effect and interferes with the laser beam (as reference) in scanning laser head, then generates the Doppler frequency shift which is proportional to the vibration velocity of the measured target. Photoelectric detector records the interference signals and outputs an analog voltage signal through the decoder processing. This signal is processed by the computer via a high-speed A/ D conversion in the connecting box, and finally mode shapes and amplitude fre-

Output module Animation module Driver module

Scanning module Analysis module -----------------------------------------~

Computer control and date processing system

Fig. 14.2

System hardware

Components of PSV-300F-B type Doppler laser vibration measurement system

Controller

-

..... - -----

synchronillion conncclor -

~.~ Vu ,•. ~~I---"' U M slator

Fig. 14.3

-- --

-

1'VV

. -~

1

Power amplifi_

................ -

Schematics of PSV-300F-B type Doppler laser vibration measurement system


422

quency response curves can be recorded and displayed on the computer monitor. In this system, the scanning and measurement of the stator surface are completed through the couple of high-speed swing lens in the front of the laser interferometer. Software and computers arc used for the automatic control, data quality assessment, modal analysis, and display of the full system. The modal testing of TRUM-60 stator shown in Fig. 11.1 is carried out by PSV-300F-B and its measurement results are shown in Figs. 11. 5-11. 7. The stator frequency response characteristic curve is shown in Fig. 14. 5. While using PSV-300F-B the measuring beam must be directed to the target surface (such as the upper surface of the stator), generally it is used for the measuring of frequency response of the stator without the rotor. Figs. 14. 6 and 14. 7 show B09 mode shape and nephogram of the stator, respectively. From the two figures we can get the relative and absolute velocities, amplitude and node line position. It can be seen from Figs. 14. 6 and 14. 7 that the stator's mode B09 possesses 9 nodal diameters and its velocity amplitude reaches O. 8m/ sunder 38. 2kHz. The test results arc in good agreement with the calculated results through the theoretical model in Chap. 5.

~

..,~"

15

.~ 10

Q.

"OJ

.g

5

~

0

l"'-_. . . ~~_. . . .;: : .J:. . ,.: '"-~

+--__ W

~

~\..~

__

50

~

J IkHz

Fig. 14.4

Fig. 14. 5 Frequency response of stator of TRUM-60

Stator of TRUM-60

Fig. 14. 6 Modc shapc stator of TRUM-60

B09

of

Fig. 14.7 Mode nephogram of TRUM-60

B09

of stator

Some types of USMs operate based on the in-plane mode of a stator. For ex-

Chapter 11


423

ample. a longitudinal-torsional type USM is designed by utilizing the longitudinal and torsional modes of the stator. and the torsional mode is exactly an in-plane mode. The measurement principles of the in-plane and out-plane modes are similar, although the measuring devices are different. The more details can be obtained in Refs. [21-26].

14.2

Measurement of Pre-pressure

Pre-pressure refers to the pressure between the stator and rotor in assembled products. It has an important influence on the dynamic characteristics of the stator, the contact characteristics between the stator and rotor, and whole USM mechanical performance. The appropriate pre-pressure can effectively reduce the abrasion and noise. and ensure that the whole system has good output performance. The relationship between USM performance and pre-pressure can be obtained through pre-pressure testing. so we need to design the pre-pressure range of USM and ensure that USM has excellent performance and a longer service life. The pre-pressure testing mainly includes measurement of pre-pressure, the relation between USM's performance and the pre-pressure, etc. Since there are no commercially available test devices, PDLab designed and manufactured several sets of the pre-pressure measurement devices for TRUM series products. A new pre-pressure test device was developed in order to investigate the influence of the pre-pressure on the USM's performance[28:. The schematic picture and measurement system of the pre-pressure are shown in Figs. 11. 8 and 11. 9, respectively. In Fig. 14. 8. the pre-pressure between the stator and rator of USM is applied with the load rod of pressure sensor. USM

Fig. 14. 8

Pressure sensor

Pre-pressure regulator

Pre-pressure adjusting device

Fig. 14.9

Pre-pressure measurement system

The pre-pressure regulator supporting plate is designed to "II" shape, which avoids the radial slide of the sensor and ensures that the sensor has only one axial degree of freedom. This can also guarantee the pre-pressure measurement accuracy. Moreover, clearance fit is adopted between the supporting plate and the inner hole of the shell, and their contact areas and sliding friction are reduced because of the "II" shape design. The regulator with fine-pitch threads can make pre-pressure to adjust precisely. The motor and the sensor are assembled together, and the latter is used for picking up the pre-pressure, which is sent to the measurement system. The pre-pressure value can be automatically adjusted and

424


displayed by the system. Figure 14. 10 is the measurement results of USM using the new pre-pressure regulator. It proves that pre-pressure has great influenee on the mechanical characteristics of the USM. When the input voltage and excitation frequency remain the same, different no-load speed and stall torque are obtained under different pre-pressure respectively. From Fig. 14. 10, there is an optimized range of the pre-pressure, which can make no-load speed and stall torque to elose simultaneously to the maximum value. 1.4 r - - - - - - - - - - - - - - - - - - --,

250r-----~----~----~----~----__,

L2

200

EI

c:

~

150

""!:l

100

>::' Q.

~0.8

g

0.6

.- +70'C -.- +80'C - 0 - +25 'CCAfter test)

0

-*- +60'C

-20 0,0

0,2

0.4

0,6

0,8

1.0

Output torque/(N' m)

Fig. 14. 19 Measured mechanical characteristics for TRUM-60 under high temperature environment

become smaller under low temperature environment. In high temperature environment, the torque increases while speed becomes lower. After finishing tests in the high/ low temperature, we measured again its mechanical characteristics at room temperature. Compared with the results before the high/ low testing, the speed and torque at room temperature changes little.

14.5.2

Vacuum Environment Testing

According to thc rcquircmcnts of USMs uscd for spacc aircraft and scmiconductor preparation, it is necessary to study its load characteristics under vacuum condition. Specifically, we need to develop testing methods in vacuum environmcnt, cstablish relatcd tcsting cquipmcnts, cxplorc thc load charactcristics of USMs as function of thc vacuum lcvel, ctc. In thc study, PDLab dcvelopcd a set of test devices consisting of an optical encoder, hysteretic machine, torque sensor and flexible coupling, as shown in Figs. 14. 20 and 14. 21.

Chapter 11

Fig. 14. 20

Fig. 14. 21


433

Vacuum environment testing system for USM

Block diagram of vacuum environment testing system for USM

When the motor is operating. its speed ean be ealculated by using the number of pulse signals generated by the encoder. A load can be applied to the motor by the hysteretic machine, and can be changed from 0 to the stall torque of the motor by varying the current imposed on the hysteretic machine. The value of the load can be measured by the torque sensor, which is selected as a strain type since it is not sensitive to the vacuum environment. The torque sensor can convert the measured torque signal to frequency one. The signals obtained by the torque sensor and optical encoder are transmitted to the external apparatuses by binding posts inside the vacuum chamber. Using the three timing-counter of an A T89S52 type micro controller , we can measure and process the pulse signals from the optical encoder and torque sensor simultaneously, and the results are displayed in real-time. Because the signals are contaminated by interference noise in the process of transmission. the signal has to be filtered before they are sent to the microcontroller. The driver is placed outside the vacuum chamber and the driving current can be sent to the motor by the binding posts, so there is no need to consider the influence of the vacuum environment on the driver. In the testing, the load is gradually increased until the motor is stalled. The


434

mechanical characteristics of the TRUM-60 under the normal and vacuum envIronment arc shown in Fig. 14. 22. respectively. from which we can sec that the mechanical characteristics of the TRUM-60 changes little under low vacuum 0010- 3 Pa) environment.

Atmosphere -.- 10 Pa __ 10-3 Pa

100

~

i

OJ OJ

75

50

Ultrasonic Motors: Technologies and Applications

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Enzyme Technologies for Pharmaceutical and Biotechnological Applications

VANET Vehicular Applications and Inter-Networking Technologies

Computational Collective Intelligence. Technologies and Applications

Advanced Array Systems, Applications and RF Technologies

Commerce in Space: Infrastructures, Technologies, and Applications

Ultrasonic Motors: Technologies and Applications

Cyberinfrastructure Technologies and Applications

Ultrasonic Processes and Machines: Dynamics, Control and Applications

Bio-MEMS Technologies and Applications

Advanced Web Technologies and Applications

Intelligent Databases: Technologies and Applications

Bio-MEMS: Technologies and Applications

Light Sources: Technologies and Applications

Smart Card Technologies and Applications

Bio-MEMS: Technologies and Applications

Men, Motors, and Markets

Electric Motors and Drives: Fundamentals, Types and Applications

Combinatorial Chemistry and Technologies: Methods and Applications

Molecular Motors

Gene Therapy Technologies, Applications and Regulations

Gene Therapy Technologies, Applications and Regulations

Rough Computing: Theories, Technologies and Applications

Rough Computing: Theories, Technologies and Applications

Multimedia Technologies: Concepts, Methodologies, Tools, and Applications

Database Technologies: Concepts, Methodologies, Tools, and Applications

Plant Proteomics. Technologies, Strategies, and Applications

Single Cell Analysis: Technologies and Applications

Database Technologies: Concepts, Methodologies, Tools, and Applications

Alignment Technologies and Applications of Liquid Crystals

Encyclopedia Of Database Technologies And Applications

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Enzyme Technologies for Pharmaceutical and Biotechnological Applications

VANET Vehicular Applications and Inter-Networking Technologies

Computational Collective Intelligence. Technologies and Applications

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