PIEZOELECTRIC SHELLS

SOLID MECHANICS AND ITS APPLICATIONS Volume 19

Series Editor: G.M.L. GLADWELL Solid Mechanics Division, Faculty of Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3G 1

Aims and Scope o/the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative research­ers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shens and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

Piezoelectric Shells Distributed Sensing and Control of Continua

by

H.S. TZOU Department of Engineering. University of Kentucky. Lexington. US.A.

... " SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

ISBN 978-94-010-4784-5 ISBN 978-94-011-1783-8 (eBook) DOI 10.1007/978-94-011-1783-8

Printed on acid-free pa per

All Rights Reserved © 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

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CONTENTS

Page

• PREFACE xvii

• ACKNO~EDGEMENT xxi

• CHAPTER 1

• INTRODUCTION 1

1.1 PIEZOELECTRIC CONTINUA 3

1.2 DISTRIBUTED SENSING AND VIBRATION CONTROLS 4

1.3 REMARKS 7

• CHAPTER 2 • PIEZOELECTRIC SHELL VIBRATION ThEORY

2.1 FUNDAMENTALS

§ 2.1.1 Definition of a Generic Piezoelectric

Shell Continuum

§ 2.1.2 Physical Laws

2.2 THICK PIEZOELECTRIC SHELL CONTINUUM

§ 2.2.1 Variation of Kinetic Energy

13

16

17

18 23

24

[ viii ] • Contents.

§ 2.2.2 Variation of Electric Enthalpy 24 § 2.2.3 Variation of Traction Force Energy 31

§ 2.2.4 Variation of Electric Potential Energy 32

§ 2.2.5 Electromechanical Equations by Hamilton's Principle 32

§ 2.2.6 Electromechanical Equations of Mechanical Motion 34 § 2.2.7 Mechanical Boundary Conditions 35 § 2.2.8 Charge Equation of Electrostatics 36 § 2.2.9 Electric Boundary Conditions 37

2.3 THICK PIEZOELECTRIC SHELL 38 § 2.3.1 Assumptions 38

§ 2.3.2 Electromechanical Equations in Principal Directions 40

§ 2.3.3 Electromechanical Equations in Rotations 41

§ 2.3.4 Definition of Resultant Forces and Moments 42 § 2.3.5 Mechanical Boundary Conditions 44

§ 2.3.6 Electrostatic Charge Equation and Boundary Conditions 47

2.4 THIN PIEZOELECTRIC SHELL 48

§ 2.4.1 Assumptions and Fundamentals 48 § 2.4.2 Electromechanical Equations

§ 2.4.3 Mechanical Boundary Conditions § 2.4.4 Electrostatic Charge Equation and

Electric Boundary Conditions

2.5 THIN ELASTIC SHELL 2.6 SUMMARY

• CHAPTER 3

• COMMON PIEZOELECTRIC CONTINUA AND

ACTIVE PIEZOELECTRIC STRUCTURES

3.1 REDUCTION PROCEDURES

3.2 PIEZOELECTRIC PLATES

§ 3.2.1 Thick Piezoelectric Rectangular Plate

51 52

55 56 58

63

64

66

67

• Contents. [ ix ]

§ 3.2.2 Thin Piezoelectric Plate 69 3.3 THIN PIEZOELECTRIC SHELL OF REVOLUTION 71

§ 3.3.1 Applications (Thin Piezoelectric Shell of Revolution) 74

3.4 ACTIVE PIEZOELECTRIC SHELL STRUCTURES 78 § 3.4.1 Active Piezoelastic Shell Equations 79 § 3.4.2 Simplified Representation and Structural Damping 81

3.5 CONTROL OF ACTIVE PIEZOELECTRIC STRUCTURES 83

§ 3.5.1 The Modal Expansion Method 84

§ 3.5.2 Steady-State Harmonic Response 87 § 3.5.3 Feedback Algorithms and Distributed Controls 89

3.6 DISTRIBUTED CONVOLVING ACTUATORS

AND CONTROLS 93 3.7 FREQUENCY CONTROL OF A PIEZOELECTRIC

BIMORPH BEAM 96 § 3.7.1 Free Vibration Analysis 99

§ 3.7.2 Finite Difference Results 101

§ 3.7.3 Laboratory Validation 103 3.8 DAMPING CONTROL OF PIEZOELECTRIC CYLINDRICAL

SHELLS 105

§ 3.8.1 Modeling and Analysis 106 § 3.S.2 Results and Discussion IDS

3.9 SUMMARY 111 3.10 APPENDIX 114

• CHAPTER 4

.. DISTRIBUTED SENSING AND CONTROL OF ELASTIC SHELLS 115

4.1 SYSTEM DEFINITIONS

§ 4.1.1 Membrane Forces

§ 4.1.2 Bending Moments

§ 4.1.3 Modal Expansion Method

118

121

121

122

[ x ] • Contents.

4.2 DISTRIBUTED SENSING OF ELASTIC SHELLS 122

§ 4.2.1 Sensor Output Signal 123 § 4.2.2 Local Point Voltage 127 § 4.2.3 Modal Identification - Modal Voltage 128

4.3 DISTRIBUTED VIBRATION CONTROL OF SHELLS 130

§ 4.3.1 Control Forces/Moments and System Equations 132 § 4.3.2 Direct Proportional Feedback Control 133

§ 4.3.3 Negative Velocity Feedback Control 134

§ 4.3.4 Lyapunov Feedback Control 135

§ 4.3.5 Modal Feedback Gains 136 4.4 STATE EQUATION 138 4.5 EXAMPLES 140

§ 4.5.1 Case 1: A Hemispheric Shell 141

§ 4.5.2 Case 2: A Rectangular Plate 143 § 4.5.3 Case 3: An Euler-Bernoulli Beam 146

4.6 SUMMARY 147

4.7 APPENDIX 153 § 4.7.1 Piezoelectricity Theory 153 § 4.7.2 Piezoelectric Matrix of Polyvinylidene Fluoride

(PVDF) 154

• CHAPTER 5 • MULTI-LAYERED SHELL ACTUATORS 155

5.1 A MULTI-LAYERED SHELL WITH DISTRIBUTED

ACTUATORS 156

§ 5.1.1 Assumptions 157

§ 5.1.2 Analysis of the i-th Distributed Actuator Layer 159

§ 5.1.3 Resultant Normal Forces and Moments of the Layered Shell 167

§ 5.1.4 Dynamic Equations of the Multi-Layered Shell 171

5.2 A MULTI-LAYERED PLATE ACTUATOR 175

• Contents. [ xi ]

§ 5.2.1 A Rectangular Plate 178

§ 5.2.2 A Circular Plate 178 § 5.2.3 An Elliptical Plate 178

5.3 A PIEZOELECTRIC/ELASTIC BEAM 178

§ 5.3.1 Fiberglass Layer (i = 1) 180

§ 5.3.2 Piezoelectric Control Layer (i = 2) 181

§ 5.3.3 Governing Equation of Motion for

the Entire Composite Beam

5.4 SUMMARY

• CHAPTER 6

• BOUNDARY CONTROL OF BEAMS

6.1 SYSTEM DEFINITION:

182

184

187

BEAM WITH SENSOR AND ACTUATOR 188

6.2 DISPLACEMENT FEEDBACK CONTROL 190

§ 6.2.1 Free Vibration Analysis 193

§ 6.2.2 Control Effectiveness: Displacement Feedback 198

6.3 VELOCITY FEEDBACK CONTROL 202

§ 6.3.1 Free Vibration Analysis 203 § 6.3.2 Control Effectiveness: Velocity Feedback 213

6.4 SUMMARY 226

• CHAPTER 7 • DISTRIBUTED CONTROL OF PLATES

WITH SEGMENTED SENSORS AND ACTUATORS

7.1 MATHEMATICAL MODELING

§ 7.1.1 Plate with Distributed Sensor and Actuator

§ 7.1.2 Modal Expansion and Vibration Controls

227

229

230

233

[ xii ] • Contents.

7.2 ONE-PIECE SYMMETRICALLY DISTRIBUTED

SENSOR/ACTUATOR 238 § 7.2.1 Distributed Sensing 238 § 7.2.2 Distributed Vibration Control 239 § 7.2.3 Equivalent Line Moment 240

§ 7.2.4 Modal Feedback Factor and Modal Equations 242 § 7.2.5 Modal Controllability and Observability 244

7.3 SEGMENTATION OF DISTRIBUTED SENSORS AND ACTUATORS 245

§ 7.3.1 Segmented Distributed Sensors 246

§ 7.3.2 Segmented Distributed Actuators 247 7.4 TRANSIENT RESPONSES 251 7.5 STEADY-STATE RESPONSES 253 7.6 EVALUATION OF CONTROLLED RESPONSES 254

§ 7.6.1 Control Effectiveness (Proportional Velocity

Feedback) 255

§ 7.6.2 Control Algorithms 255 7.7 NUMERICAL SOLUTIONS 257

§ 7.7.1 Free Vibration Analysis 260 § 7.7.2 Parametric Studies 260 § 7.7.3 Time-History Analyses (Negative Velocity

Proportional Feedback) 269 § 7.7.4 Control Algorithms 272

7.8 SUMMARY AND CONCLUSIONS 274

7.9 APPENDIX 279

• CHAPTER 8

.. CONVOLVING SHELL SENSORS AND ACTUATORS

APPLIED TO RINGS 283

8.1 SENSOR MECHANICS 285

§ 8.1.1 Thin Distributed Piezoelectric Shell- Continua 285

• Contents. [ xiii ]

§ 8.1.2 Spatially Distributed Piezoelectric

Convolving Sensor

8.2 CIRCULAR RINGS

§ 8.2.1 Electromechanical Equations of

a Piezoelectric Ring

§ 8.2.2 Natural Frequencies and Mode Shapes

§ 8.2.3 Numerical Example

8.3 DISTRIBUTED CONVOLVING RING SENSORS

8.4 EVALUATION OF DISTRIBUTED RING SENSORS

8.5 DISTRIBUTED VIBRATION CONTROLS

8.6 DISTRIBUTED CONVOLVING ACTUATORS

291

294

294

300

303

305

309

317

AND CONTROLS 321

8.7 DISTRIBUTED CONVOLVING RING ACTUATORS 324

§ 8.7.1 Spatially Convolving Piezoelectric Ring Actuators 325

8.8 EVALUATION OF CONVOLVING RING ACTUATORS 329

§ 8.8.1 Comparison of Bending and Membrane Effects 329

§ 8.8.2 Modal Damping Controls 330

8.9 SUMMARY AND CONCLUSIONS 334

• CHAPTER 9 • SENSING AND CONTROL OF CYLINDRICAL SHELLS 337

9.1 CYLINDRICAL SHELL 338

§ 9.1.1 Governing Piezoelectric Equations 339 § 9.1.2 Simplified Governing Equations and

Free Vibration Analysis 340

§ 9.1.3 Natural Frequencies and Modes of

a Laminated Cylindrical Shell 350

9.2 DISTRIBUTED SENSORS FOR CYLINDRICAL SHELLS 358

§ 9.2.1 Estimation of Sensor Signals 358

§ 9.2.2 Fully Qistributed Shell Sensor 360

§ 9.2.3 Diagonal Stripe Shell Sensors 364

[ xiv ] • Contents.

9.3 EVALUATION OF DISTRIBUTED CYLINDRICAL

SHELL SENSORS 368

§ 9.3.1 Fully Distributed Shell Sensor 368

§ 9.3.2 Diagonal Stripe Shell Sensors 374

9.4 TWO-DIMENSIONAL DISTRIBUTED ACTUATORS

FOR CYLINDRICAL SHELLS 380

§ 9.4.1 Fully Distributed Actuator 381

§ 9.4.2 Distributed Diagonal Stripe Actuator 385

9.5 EVALUATION OF DISTRIBUTED ACTUATORS 388

§ 9.5.1 Evaluation of Bending Effects 388

§ 9.5.2 Evaluation of the Fully Distributed Actuator 389

§ 9.5.3 Evaluation of the Diagonal Stripe Actuator 400

9.6 SUMMARY AND CONCLUSIONS 403

• CHAPTER 10

• FINITE ELEMENT FORMULATION AND ANALYSES 405

10.1 PIEZOELECTRIC FINITE ELEMENT FORMULATION 406

§ iO.1.1 Variational Equation and Piezoelectric Element 407

§ 10.1.2 New Piezoelectric Element with Internal DOF's 410

§ 10.1.3 Condensation of Internal DOF's 413

§ 10.1.4 System Equations 415

10.2 EIGENVALUE ANALYSIS 416

10.3 STRUCTURAL SENSING AND CONTROL 417

§ 10.3.1 Sensor Signal 420

§ 10.3.2 Distributed Actuators and Vibration Controls 420

§ 10.3.2.1 Constant Amplitude Negative Velocity

Feedback Control 421

§ 10.3.2.2 Constant Gain Negative Velocity

Feedback Control 422

§ 10.3.3 Time-Domain Integration

- The Modified Wilson-O Method 423

• Contents. [ XV ]

lOA STATE VARIABLE TRANSFORMATION 424 10.5 MATRIX CONDENSATION AND REDUCTION 427 10.6 SYSTEM AGGREGATION FOR CONTROL OF

LARGE-SCALE SYSTEMS 430

10.7 CASE STUDIES 433

§ 10.7.1 A Piezoelectric Bimorph Beam 434

§ 10.7.1.1 Micro-Displacement Actuation 436 § 10.7.1.2 Distributed Structural Identification 438

§ 10.7.2 Vibration Control of a Laminated Steel/PZT Beam 439 § 10.7.3 Distributed Vibration Identification and

Control of a Plate

§ 10.7.3.1 Distributed Structural Identification:

Modal Voltages

§ 10.7.3.2 Distributed Vibration Controls 10.8 SUMMARY AND CONCLUSIONS

10.9 APPENDIX

• APPENDIX

• LINEAR PIEZOELECTRICITY THEORY

Apx.1 DEFINITIONS

Apx.2 LINEAR PIEZOELECTRICITY RELATIONS Apx.3 ELASTIC, PIEZOELECTRIC,

AND DIELECTRIC RELATIONS

.. INDEX

443

444

447

451

455

457

457

458

460

463

_______________________________________________ [x~i]

PREFACE

Exploiting new advanced structures and electromechanical systems, e.g.,

adaptive structures, high-precision systems, micro electromechanical systems,

distributed sensors/actuators, precision manipulation and controls, etc., has been

becoming one of the mainstream research and development activities (structure & motion) in recent years. These new systems and devices could bring a new technological revolution in modern industries and further, directly or indirectly,

impact human life. In the search for and research in innovative technologies, it is proved that piezoelectric materials are very versatile in both sensor and actuator

applications. Consequently, piezoelectric technology has been widely applied to a

large number of industrial applications and devices, varying from thin-film micro

sensors/actuators to large space structures in addition to those relatively

conventional applications, e.g., sensors, actuators, hydrophones, precision manipulators, mobile robots, micro motors, etc. There have been a few books on piezoelectricity published in the past; however, a unified presentation of piezoelectric shells and distributed senSing/control applications is still lacking.

This book is intended to fill the gap and to pro~de practising engineers and researchers with an introduction to advanced piezoelectric shell theories and distributed sensor/actuator technologies in structural identification and control.

This book represents a collection of the author's recent research and

development on piezoelectric shells and related applications to distributed

measurement and control of continuaj it reflects six best-paper awards, including

[ xviii] • Contents.

two ASME Best-Paper Awards in recent years. To provide an "umbrella" theory

to most common piezoelectric continua, a generic piezoelectric deep shell defined in a tri-orthogonal curvilinear coordinate system is used as the fundamental

continuum to derive the piezoelasticity and distributed sensing/control theories. There are a number of shell continua discussed in the book: 1) three piezoelectric shell continua, 2) one elastic shell continuum laminated with distributed shell

layers, 3) one multi-layered shell actuator, and 4) one thin piezoelectric shell

layer. All of these shells are based on the generic deep shell continuum. The

major advantage of the generic shell theories is their generality. All theories developed based on the generic shell continuum can be easily simplified to account

for a large class of shell and non-shell continua (piezoelectric or elastic), such as shells of revolution, spherical shells, cylindrical shells, conical shells, plates (circular, elliptical, or rectangular), arches, rings, beams, rods, etc., with a variety of boundary conditions. Therefore, the important aspects of the book are to

introduce 1) the generic approach of piezoelectric shells and 2) the fundamental

electromechanics of distributed piezoelectric sensors/actuators in structural controls.

This book is divided into two major parts. The first part focuses on piezoelectric shell continua; the second part on distributed sensing and control of

elastic continua. To introduce the essential background in the area, Chapter 1 provides a general introduction to piezoelectric phenomena, piezoelectric materials, piezoelectric continua, and distributed sensing and control of continua. Detailed derivations of three piezoelastic shell theories are presented in Chapter 2. Applications of the the generic piezoelastic shell theories to common piezoelectric

continua based on a reduction procedure are described in Chapter 3. Distributed

convolving sensor and actuator theories in distributed identification and control

are also discussed. In addition, frequency and damping controls of piezoelectric

structures are demonstrated in Chapter 3. Chapter 4 gives generic distributed

shell sensor/actuator theories for elastic shell continua laminated with distributed

piezoelectric thin shell layers. Applications of the generic theories to other

common geometries are also demonstrated in case studies. Chapter 5 presents a

• Contents. [ xix ]

multi-layered shell actuator theory based on equivalent induced control actions. Distributed measurements (or identifications) and controls of elastic continua are studied in Chapters 6 to 10. Boundary controls of an elastic cantilever beam are investigated in Chapter 6. Distributed sensing and control of plates with fully distributed and segmented sensors/actuators are studied in Chapter 7. Detailed parametric studies of segmented sensors/actuators are emphasized. In Chapter 8, distributed shell sensor electromechanics of a generic piezoelectric shell layer are

first presented and followed by generic theories of distributed shaped and

convolved sensors/actuators. These theories are then applied to an elastic circular

ring. Distributed convolving modal sensors/actuators are designed; and their

performances evaluated. Following the same procedures, distributed sensing and control of cylindrical shells are presented in Chapter 9. It is well known that theoretical models and solutions are limited to relatively ideal geometries and boundary conditions. In order to broaden the practical applications of

piezoelectric technology, piezoelectric finite elements, especially a "thin" solid

element, and senSing/control algorithms are formulated in Chapter 10. Validations and applications of the piezoelectric thin finite element to micro

manipulations, distributed identification (modal voltages), and distributed control of continua are presented in case studies.

This book can be used as a professional reference for scientists and

practising engineers who would like to understand the new piezoelectric technology in distributed sensing, identification, and control. The book can also be used in classroom teaching as a textbook for graduate students. Technical backgrounds in vibration and solid mechanics should be adequate to master the contents.

----------------------------------------------- [xxi ]

ACKNOWLEDGEMENT

This research was supported by a grant from the National Science Foundation (No. RII-8610671) and the Commonwealth of Kentucky (1986-1991). A grant from the Army Research Office (DAAL03-91-G-0065) (1991-1994), Technical Monitor: Dr. Gary L. Anderson, is gratefully acknowledged. (Contents of the information do not necessarily reflect the position or the policy of the government, nor should official endorsement be inferred.) Graduate assistantships (1987-1988 and 1992-1993) provided by the Center for Robotics and ManufacturiI).g Systems at the University of Kentucky are also gratefully acknowledged.

A fellowship supported by;>the Institute of Space and Astronautical Science

(ISAS), Japanese Ministry of Education, is gratefully acknowledged. (The first draft Of the first "seven chapters was finished during the eight-month stay at ISAS.) The author is particularly thankful to his hosts, Prof. M. C. Natori and Prof. K. Miura, in the Division of Spacecraft Engineering for their warm hospitality during that period.

The author ,would like to express his sincere gratitude to a number of people

who are (or were) influential to his career development: Prof. A. J. Schiff

(Stanford University), Prof. H. T.Y. Yang (Purdue University), Prof. R. A. Altenkirch (Mississippi State University), Prof. J. T.P. Yao (Texas A&M

[xxii] • Contents.

University), and Prof. D. J. Inman (Virginia Polytechnic Institute and State University). From these people, directly or indirectly, he acquired the knowledge and philosophy to compete in this challenging research community. The author is

especially grateful to Prof. W. Soedel (Purdue University) for his excellent teaching in shell vibrations which laid down the foundation for this book.

The author also would like to thank his friends and colleagues who are (or were) very supportive and stimulating during those long days and nights over the years (in alphabetical order): Dr. G. L. Anderson (U.S. Army Research Office), Prof. A. Baz (CatholiC University of America), Prof. V. Birman (University of Missouri), Prof. Y.-F. Chou (National Taiwan University, Taiwan, China), Prof.

P. Destuynder (Institute of Aerospace Engineering, France), Prof. T. Fukuda (Nagoya University, Japan), Dr. J. E. Hubbard, Jr. (Optron Systems, Inc.), Dr. C.-K. Lee, (IBM Almaden Research Laboratory), Prof. K.-M. Lee (Georgia Institute of Technology), Prof. F. Ma (University of California, Berkeley), Prof. M. C. Natori (Institute of Space and Astronautical Science, Japan), Prof. D. L. Polla (University of Minnesota), Prof. K. K. Tamma (University of Minnesota), Prof. K. W. Wang (Penn State University), Prof. B. Yang (University of Southern California), and all faculty in the Department of Mechanical Engineering at the University of Kentucky.

A special appreciation goes to Dr. N. Hollingworth, the Acquisition Editor of Kluwer Academic Publishers, for his persistent encouragement and continued interest. The author is particularly thankful to Professor G.M.L. Gladwell

(University of Waterloo) for his thorough review of the book manuscript. In

addition, the author would like to thank all his graduate students, post-doctoral

research associates, and visiting scholars who actually executed the research.

Lastly, it seems fascinating that the author accidentally returned to his "lost roots" as an unexpected total surprise. He was told that he is actually the

• Contents. [ xxiii ]

fourth academic generation from Professor R. D. Mindlin (Columbia University) who was conducting research on piezoelectric plates in the early 70's. In a search for lost roots, the author found that a large number of scientists and researchers

are actually IIrelatedll to the same lIacademicll family; a few of them are very

prominent in their respective areas. This turns out to be the greatest reward to his work on piezoelectric shells.

Horn-Sen Tzou

Lexington, Kentucky

December 24, 1992