Piezoelectric Multilayer Beam Bending Actuators

Static and Dynamic Behavior and Aspects of Sensor Integration
1. Auflage 2007
ISBN: 978-3-540-32642-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 358 Seiten

An important aim of the presented book is the explanation of the application of piezoelectric materials such as piezoceramics within the wide field of electromechanical actuators and sensor technology. The reader should be presented the physical and mechanical properties of piezoceramics in a distinct way. In a next step, the reader is introduced into the mechanical description of the static behavior of piezoelectric multilayer beam bending actuators.

The description of the dynamic behavior of piezoelectric multilayered bending actuators is effected on the basis of Lagrange‘s formalism and Hamilton‘s principle. The achieved insights are used for the systematic development of the electromechanical circuit representation within the scope of the network theory for any design of piezoelectric bending actuators.

The applications of piezoelectric multilayer beam bending actuators can be extended by means of special displacement sensors allowing for the compensation of effects such as hysteresis, creep and drift being typical for piezoelectric actuators. Within the scope of the presented book, two different sensor-actuator-systems are presented being based on an integrated capacitive and inductive displacement sensor, respectively.

Analytical simulations of the static and dynamic behavior are compared to real measurement results of a specially developed piezoelectric multilayer beam bender. Here, the suitability of the developed theoretical aspects is shown in an outstanding way.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;List of Symbols;15
4;Focus of the Book;24
4.1;1 Introduction;25
4.1.1;1.1 Application Areas of Piezoelectric Actuators;25
4.1.2;1.2 Motivation and Aim of the Book;26
4.1.3;1.3 State of the Scientific Research;28
4.1.4;1.4 Textual Focus of the Book;33
5;Theoretical Aspects and Closed Form Analysis;36
5.1;2 Piezoelectric Materials;37
5.1.1;2.1 Discovery of Piezoelectricity;37
5.1.2;2.2 Direct and Inverse Piezoelectric Effect;38
5.1.3;2.3 Piezoelectric Ceramics;39
5.1.4;2.4 Perovskit Structure of PZT;40
5.1.5;2.5 Domain and Reversion Processes of PZT;41
5.1.6;2.6 Electromechanical Behavior;44
5.1.7;2.7 Piezoelectric Beam Bending Actuators;46
5.2;3 Linear Theory of Piezoelectric Materials;51
5.2.1;3.1 Energy Density of the Elastic Deformation;51
5.2.2;3.2 Energy Density of the Electrostatic Field;55
5.2.3;3.3 Thermodynamics of Deformation;56
5.3;4 Theory of the Static Behavior of Piezoelectric Beam Bending Actuators;66
5.3.1;4.1 Sectional Quantities of a Bending Beam;66
5.3.2;4.2 Bernoulli Hypothesis of Beam Bending Theory;68
5.3.3;4.3 Neutral Axis Position of a Multilayered Beam Bender;70
5.3.4;4.4 Forces and Moments within a Multilayer System;73
5.3.5;4.5 Total Stored Energy within a Multilayer System;74
5.3.6;4.6 Canonical Conjugates and Coupling Matrix;77
5.3.7;4.7 Principle of Virtual Work;79
5.3.8;4.8 Theorem of Minimum Total Potential Energy;80
5.3.9;4.9 Derivation of the Coupling Matrix;81
5.3.10;4.10 The Constituent Equations;94
5.4;5 Piezoelectric Beam Bending Actuators and Hamilton’s Principle;96
5.4.1;5.1 Constraints and Generalized Coordinates;96
5.4.2;5.2 D’Alembert’s Principle;97
5.4.3;5.3 Lagrange’s Equations;99
5.4.4;5.4 Euler-Lagrange Differential Equation;102
5.4.5;5.5 Hamilton’s Principle;106
5.4.6;5.6 Consideration of Non-Conservative Forces;107
5.4.7;5.7 Lagrange Function of Piezoelectric Beam Bending Actuators;110
5.4.8;5.8 Mechanical Work Done by Extensive Quantities and Frictional Force;114
5.4.9;5.9 Variation of the Lagrange Function;117
5.4.10;5.10 Variation of the Mechanical Work;118
5.4.11;5.11 Differential Equations of a Piezoelectric Multilayer Beam Bender;119
5.5;6 Theory of the Dynamic Behavior of Piezoelectric Beam Bending Actuators;122
5.5.1;6.1 Eigenmodes of a Clamped-Free Beam Bender;122
5.5.2;6.2 Orthogonality of Eigenfunctions;126
5.5.3;6.3 Description of Flexural Vibrations with Respect to Time;128
5.5.4;6.4 The Free Damped Flexural Vibration;129
5.5.5;6.5 Excitation by a Harmonic Force;131
5.5.6;6.6 Excitation by a Harmonic Moment;133
5.5.7;6.7 Excitation by a Harmonic Uniform Pressure Load;135
5.5.8;6.8 Excitation by a Harmonic Driving Voltage;136
5.5.9;6.9 Electrical Charge Generated by Harmonic Extensive Parameters;137
5.5.10;6.10 Dynamic Admittance Matrix;140
5.6;7 Network Representation of Piezoelectric Multilayered Bending Actuators;142
5.6.1;7.1 The Ideal Rod as Transducer for Translatory and Rotatory Quantities;143
5.6.2;7.2 Bending of a Differential Beam Segment;145
5.6.3;7.3 The Differential Beam Segment and Corresponding Correlations;148
5.6.4;7.4 Solution Approach to the Complex Equation of Flexural Vibrations;152
5.6.5;7.5 General Solution of the Equation for Flexural Vibrations;154
5.6.6;7.6 Solution of the Equation of Flexural Vibrations by Means of Reference Values;156
5.6.7;7.7 Admittance Matrix of a Beam Bender;156
5.6.8;7.8 Transition to the Piezoelectric Multilayer Beam Bending Actuator;161
5.6.9;7.9 The Clamped-Free Piezoelectric Multimorph;168
6;Measurement Setup and Validation of Theoretical Aspects;179
6.1;8 Measurement Setup for Piezoelectric Beam Bending Actuators;180
6.1.1;8.1 Measurement Setup;180
6.1.2;8.2 Automation of Measurement Setup;184
6.2;9 Measurements and Analytical Calculations;189
6.2.1;9.1 Used Multilayer Beam Bending Structure for Experimental Investigations;189
6.2.2;9.2 Static and Quasi-static Measurements;191
6.2.3;9.3 Dynamic Measurements;200
7;Sensor Integration for Tip Deflection Measurements;213
7.1;10 Piezoelectric Beam Bending Actuator with Integrated Sensor;214
7.1.1;10.1 Smart Pneumatic Micro Valve;215
7.1.2;10.2 Sensor Requirements;216
7.2;11 Tip Deflection Measurement - Capacitive Sensor Principle;218
7.2.1;11.1 Sensor Positioning;218
7.2.2;11.2 Sensor Electronics for Capacitive Strain Sensors;221
7.3;12 Tip Deflection Measurement - Inductive Sensor Principle;232
7.3.1;12.1 Measurement Setup and Basic Structure of the Inductive Proximity Sensor;232
7.3.2;12.2 Functioning of the Inductive Proximity Sensor;234
7.3.3;12.3 Equivalent Network Representation;238
7.3.4;12.4 Inductance of a Circular Loop Influenced by a Conductive Layer;242
7.3.5;12.5 Measurement Results;249
7.4;13 Conclusion;264
7.4.1;13.1 Summary and Results;264
7.4.2;13.2 Outlook;268
8;Appendix;270
8.1;A Work Done by Stresses Acting on an Infinitesimal Volume Element;271
8.2;B Derivation of the Coupling Matrix Elements;274
8.2.1;B.1 Multilayer Beam Bender Subjected to an External Static Moment;274
8.2.2;B.2 Multilayer Beam Bender Subjected to an External Static Force;277
8.2.3;B.3 Multilayer Beam Bender Subjected to a Uniform Pressure Load;279
8.2.4;B.4 Electrical Charge Generated by the Extensive Parameters;281
8.3;C Mechanical Potential and Kinetic Energy;291
8.4;D Derivation of the Electrical Enthalpy;293
8.5;E Correlation Between Material Parameters;295
8.6;F Work Done by Extensive Dynamic Quantities;297
8.6.1;F.1 Work Done by a Force;297
8.6.2;F.2 Work Done by a Moment;298
8.6.3;F.3 Work Done by a Driving Voltage;299
8.7;G On the Variation of the Lagrange Function;301
8.8;H On the Variation of the Work Done by Extensive Quantities;306
8.9;I On the Excitation by a Periodic Force;308
8.10;J Particular Solution of the Differential Equation for Flexural Vibrations;310
8.11;K Transition to the Differential Equations in Complex Form;312
8.12;L Orthogonality of Different Boundary Conditions;315
8.13;M Logarithmic Decrement;318
8.14;N Favored Sensor Principles and Sensor Signal Estimation;320
8.14.1;N.1 Resisitive Distance Sensors;321
8.14.2;N.2 Capacitive Distance Sensors;325
8.14.3;N.3 Piezoelectric Distance Sensor;330
8.14.4;N.4 Inductive Distance Sensor;332
8.15;O Methods of Measuring Small Capacitances with High Resolution;336
8.15.1;O.1 Direct Method;336
8.15.2;O.2 Self-balancing Capacitance Bridge;337
8.15.3;O.3 Charge Measurement;339
8.15.4;O.4 Measurement of the Integration Time;340
8.15.5;O.5 Oscillator Method;340
8.16;P To the Output Signal of the Instrumentation Amplifier;342
8.17;Q Alternating Magnetic Field Within a Conductive Layer;344
8.18;R Magnetic Field Calculation of a Circular Loop;346
9;References;350
10;Index;362