Piezoelectric Actuator Design via Multiobjective Optimization Methods

176 Pages·2005·1.66 MB·English

Piezoelectric Actuator Design via
Multiobjective Optimization Methods

zur Erlangung des akademischen Grades eines
DOKTORS DER INGENIEURWISSENSCHAFTEN (Dr.-Ing.)
der Fakultät für Maschinenbau
der Universität Paderborn

genehmigte
DISSERTATION

von
M. Eng. Bo Fu
aus Sichuan, VR China

Tag des Kolloquiums: 09.06.2005
Referent: Prof. Dr.-Ing. Jörg Wallaschek
Korreferent: Prof. Dr. Michael Dellnitz

Foreword
Piezoelectric actuators find widespread applications in almost all fields of engineering.
Ultrasonic welding, traveling wave motors or ultrasonic scalers are examples of systems
operated in resonance. Diesel injection valves, optical scanners and atomic force microscope
are examples of systems operating in a quasistatic mode.
The performance of piezoelectric actuators in a given application is mainly determined from
the tuning of the actuator characteristics to the load characteristics. Except for some very
limited special cases, no simple general design guidelines can be given today. Mathematical
methods for the optimization of piezoelectric actuators are therefore a very important tool for
the system designer.
In most engineering design tasks, multiple optimization criteria must be met. Despite
tremendous progress in the mathematical sciences, knowledge on methods and algorithms for
multicriteria optimization problems is limited in the engineering community.
Bo Fu was a scholar at the Paderborn Institute for Scientific Computation (PaSCo) and his
thesis work was funded within the DFG-Graduiertenkolleg “Scientific Computation:
Application-oriented Modeling and Development of Algorithms”. His thesis concentrates on
the study of multicriteria optimization problems arising in the design of piezoelectric
actuators and on mathematical methods which were developed for this class of optimization
problems. Classical and novel methods are studied in detail and applied to various typical
design problems of piezoelectric transducers. The results are of interest for all engineers
working on the design of piezoelectric actuators, which are interested in mathematical
background of multicriteria optimization, as well as for all mathematicians working in
multicriteria optimization, which are interested in engineering applications.

Paderborn, September 2005

(Prof. Dr.-Ing. Jörg Wallaschek)

Acknowledgments
First of all, I would like to express my sincere gratitude and appreciation to my thesis advisor,
Prof. Dr.-Ing. Jörg Wallaschek, for his continuous encouragement, support and patience
throughout my Ph.D. study. His observations and comments helped me to establish the overall
direction of my research and to move forward with investigation in depth, and his technical
and editorial advice was essential to the completion of this dissertation. I thank him for giving
me the opportunity to work with him.
I would also like to thank my co-advisor, Prof. Dr. Michael Dellnitz, for providing many
valuable comments and suggestions during the work. I am furthermore grateful to the other
members of my examining committee, Prof. Dr.-Ing. Roland Span and Prof. Dr.-Ing. Ansgar
Trächtler, for their valuable comments.
I would additionally like to thank all members of the Mechatronics and Dynamics research
group at the Heinz Nixdorf Institute for their help and support over the last few years.
Particular thanks go to Dr.-Ing. Tobias Hemsel for the valuable comments that were very
helpful in improving the presentation and contents of this dissertation. Thanks also to Dipl.-
Ing. Reinhard Böer, Dr.-Ing. Thomas Sattel, Dipl.-Ing. Michael Brökelmann, Dipl.-Ing. Maik
Mracek, Dipl.-Ing. Rafal Krol und Dipl.-Ing. Christian Potthast for their discussions and
support. Special thanks to Mrs. Marina Kassühlke and Mrs. Kerstin Hille for their help and
support throughout my Ph.D. study.
All the members of the Graduiertenkolleg of the Paderborn Institute for Scientific
Computation (PaSCo) are also due many thanks for the pleasant environment, assistance and
friendship. Particular thanks to M.Sc. Fang Wang for several disscussions and Dipl.-Ing.
Nicolai Neumann for proof-reading some of the chapters.
I gratefully acknowledge the China Scholarship Council, the Graduiertenkolleg of the PaSCo
and the Heinz Nixdorf Institute for providing scholarships to pursue doctoral studies in
Germany.
Finally, last, but not least, I would like to thank my family, my wife Ling and son Yiheng, for
their love, encouragement, patience and understanding during the past few years.
Thank you all.

To my mother, my father, my wife and my son

CONTENTS I

Contents
1 Introduction.......................................................................................................1
1.1 Motivation.............................................................................................................1
1.2 Objective...............................................................................................................2
1.3 Scope.....................................................................................................................2
2 Piezoelectric Actuators ....................................................................................5
2.1 Piezoelectric Effect................................................................................................5
2.2 Piezoelectric Actuators..........................................................................................8
2.3 Models of Piezoelectric Actuators........................................................................9
2.3.1 Nonparametric Models..............................................................................9
2.3.2 Continuum Models..................................................................................12
2.3.3 Finite Element Method............................................................................18
2.3.4 Lumped Parameter Models......................................................................18
2.4 Typical Design Goals..........................................................................................23
2.4.1 One Stroke Driving..................................................................................23
2.4.2 Resonant Driving.....................................................................................23
2.5 State of the Art of Optimization of Piezoelectric Actuators...............................24
3 Multiobjective Optimization Methods............................................................29
3.1 Basic Concepts of Multiobjective Optimization.................................................29
3.2 Traditional Multiobjective Optimization Methods..............................................31
3.2.1 Weighted Sum Method............................................................................32
3.2.2 ε-Constraint Method................................................................................33
3.2.3 Weighted Metric Methods.......................................................................33
3.2.4 Value Function Method...........................................................................34
3.3 Multiobjective Evolutionary Algorithms............................................................35
3.3.1 Basic Principles of Evolutionary Algorithms..........................................35
3.3.2 Fitness Assignment and Fitness Sharing.................................................39
3.3.3 General Procedures of Multi-objective Evolutionary Algorithms..........41
3.3.4 Multiobjective Genetic Algorithm .........................................................42
3.3.5 Non-dominated Sorting Genetic Algorithm ...........................................43
3.3.6 Strength Pareto Evolutionary Algorithm ................................................43
II CONTENTS
3.3.7 Elitist Non-dominated Sorting Genetic Algorithm..................................45
3.4 Constraint Handling in Multiobjective Evolutionary Algorithms.......................45
3.4.1 Methods based on Preserving Feasibility of Solutions............................46
3.4.2 Methods based on Penalty Functions......................................................46
3.4.3 Methods based on Feasibility and Domination of Solutions...................47
3.5 Performance Metrics for Evaluating MOEAs...................................................48
4 Multiobjective Optimization of Piezoelectric Transducers..........................51
4.1 Langevin Transducers.........................................................................................51
4.2 Performance Criteria...........................................................................................53
4.3 Determination of Optimal Prestress....................................................................57
4.3.1 Freely Vibrating Transducers..................................................................61
4.3.2 Transducers with a Mechanical Load......................................................62
4.4 Modeling of Langevin Transducers using Transfer Matrix Methods.................65
4.5 Optimization of Symmetrical Langevin-type Transducers.................................70
4.5.1 Derivation of the Whole Transfer Matrix................................................71
4.5.2 Problem Formulation of the Symmetrical Transducer without Loads....73
4.5.3 Implementation of the Optimization Process..........................................78
4.6 Optimization of Langevin-type Transducers with a Stepped Horn.....................91
4.6.1 Derivation of the Whole Transfer Matrix................................................91
4.6.2 Problem Formulation of Transducers without Loads..............................92
4.6.3 Implementation of Optimization.............................................................95
4.6.4 Problem Formulation of Transducers with a Mechanical Load............105
4.6.5 Implementation of Optimization...........................................................108
5 Results and Discussions..............................................................................113
5.1 Discussion of the Results for the Symmetrical Transducer.............................113
5.1.1 Analysis of the Results of the Optimization..........................................114
5.1.2 Determination of the Preferred Solution...............................................116
5.2 Discussion of the Results for Stepped-horn Transducers without Load...........122
5.2.1 Analysis of the Results of the Optimization..........................................124
5.2.2 Determination of the Preferred Solution...............................................125
5.3 Discussion of the Results for Stepped-horn Transducers with Load................132
5.3.1 Analysis of the Results of the Optimization..........................................134
5.3.2 Determination of the Preferred Solution...............................................135