microtechnology and mems

microtechnology and memsSeries Editor: H. Fujita D. Liepmann

The series Microtechnology and MEMS comprises text books, monographs, andstate-of-the-art reports in the very active field of microsystems and microtech-nology. Written by leading physicists and engineers, the books describe the basic

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Microtechnology and Mems

Arno LenkRüdiger G. BallasRoland WerthschützkyGünther Pfeifer

Electromechanical Systemsin Microtechnology andMechatronicsElectrical, Mechanical and AcousticNetworks, their Interactions andApplications

123

With 313 Figures

ISSN 1615-8326ISBN 978-3-642-10805-1 e-ISBN 978-3-642-10806-8DOI 10.1007/978-3-642-10806-8Springer Heidelberg Dordrecht London New York

c© Springer-Verlag Berlin Heidelberg 2011This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication ofthis publication or parts thereof is permitted only under the provisions of the German Copyright Lawof September 9, 1965, in its current version, and permission for use must always be obtained fromSpringer. Violations are liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant pro-tective laws and regulations and therefore free for general use.

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Library of Congress Control Number: 2010936071

Professor Dr.-Ing habil Arno LenkDresden University of TechnologyFaculty of Electrical and Computer EngineeringHelmholtzstraße 1001069 DresdenGermany

Dr.-Ing Rüdiger G. BallasKarl MayerTextile MachineryBruehlstr. 2563179 ObertshausenGermanyrballas@karlmayer.de

Professor Dr.-Ing habil Roland WerthschützkyDarmstadt University of TechnologyInstitute for Electromechanical DesignMerckstr. 2564283 DarmstadtGermanywerthschuetzky@emk.tu-darmstadt.de

Professor Dr.-Ing habil Günther PfeiferDresden University of TechnologyFaculty of Electrical andComputer EngineeringHelmholtzstraße 1001069 DresdenGermany

Preface

Within the wide field of technical information processing, electromechanicalsystems consisting of coupled electrical and mechanical functional elementshave a significant importance. Both the design of interfaces between humanand information processing mechanisms and the design of interfaces with thematerial process during metrological data acquisition and actuatory influenceof process variables is made possible by these electromechanical systems. Ex-amples for realization of electromechanical systems in the form of devices,assemblies or components are:

• peripheral devices of data processing systems like printers, scanners, diskdrives and data memories,

• electroacoustic devices like loudspeakers, microphones and ultrasonic trans-ducers,

• sensors for medicine, automotive and process measurement engineering,

• actuators in the form of small drives and precision positioning systems.

The list mentioned above is increasingly extended by direct coupled sensor-actuator-systems with integrated data processing. Thus, the result is a smoothtransition to more complex electromechanical systems of mechatronics.The production of electromechanical systems results from enhanced preci-sion engineering methods and modern technologies of microtechnology andmicrosystems technology. In addition, used materials like high-grade steels,ceramics, glasses, silicon and quartz, are subjected to continuous further de-velopment.

In the phase of industrial development of electromechanical systems, the de-sign process based on a solution concept provides a fundamental stage. Here,geometrical, electrical and technological system parameters are defined being

vi Preface

based on a physical model considering special design criteria and technologi-cal limitations. The closed dynamic design of the overall system is made moredifficult by different subsystems consisting of electronic, mechanical, acousticand fluid elements.

The main objective of this book is the obtaining of a physically clear designmethod for complex electromechanical systems. This design method is basedon the network theory, electrical engineers and engineers of information tech-nology are familiar with. The total electromechanical system is described inthe form of a common technical circuit representation of different subsystemsincluding their interactions by means of network theory. Clear physical func-tions are assigned to either lumped or distributed elements of the network.The advantages of this design method are the application of clear analyticalmethods of electrical networks, the possibility of the closed design of physicallydifferent subsystems and the use of existing circuit simulation software. Thestructuring of electromechanical systems according to electrical, mechanicaland acoustic elementary networks and the introduction of passive transducersas two-port networks which describe the loss-free linear interactions betweenthe subsystems, are the fundamental conditions for the application of networktheory.

The main features of this book are based on the structure-oriented theoryof electromechanical systems developed by Arno Lenk in the 60s to 90s. Theresults were summarized in the books

”Elektromechanische Systeme — Systeme

mit konzentrierten Parametern” [1],”Elektromechanische Systeme — Systeme

mit verteilten Parametern” [2] and”Elektromechanische Systeme — Systeme

mit Hilfsenergie” [3] which were published in the 70s in the Verlag Technik.

The book is suitable for students of information technology, measurement andautomation engineering, mechatronics, technical acoustics as well as microsys-tems technology and precision engineering. The book enables the electricalengineer being familiar with network theory to get started quickly with thesolution of many dynamic problems concerning the design of coupled elec-trical, mechanical, acoustic and fluid systems. In addition, this book is alsosuitable for mechanical engineers in order to get started with the efficient andpractice-oriented design method for mechatronic systems. The necessary basicknowledge of network theory is summarized in an extra chapter.

We gratefully acknowledge Stephan Sindlinger, Stefan Leschka, Eric Starkeand Uwe Marschner, whose current research results are presented in the sec-tions concerning the finite network elements (Leschka, Sect. 6.3.1 and Sin-dlinger, Sect. 6.3.2), the combination of FEA and network theory (Starke,Sect. 6.4) and the application of electrodynamic and piezomagnetic actuators(Marschner, Sect. 8.1.2, Sect. 8.3.3 and Sect. 8.3.4).

Preface vii

Finally, our thanks go to Eva Hestermann-Beyerle and Birgit Kollmar-Thoniof Springer-Verlag, who offered an excellent cooperation and continuous sup-port while we were writing this book.

Dresden and Darmstadt, July 2010

Arno Lenk, Rudiger G. Ballas, Roland Werthschutzky, Gunther Pfeifer

Contents

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Part I Focus of the Book

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Focus of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Fields of Application and Examples for Electromechanical

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Design of Electromechanical Systems . . . . . . . . . . . . . . . . . . . . . . 91.4 Simulation Methods for Electromechanical Systems . . . . . . . . . 10

1.4.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Design Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Electromechanical Networks and Interactions . . . . . . . . . . . . . 152.1 Signal Description and Signal Transmission in Linear

Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.1 The Circular Function as Basic Module for Time

Functions of Linear Networks . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Fourier Expansion of Time Functions . . . . . . . . . . . . . . . . 202.1.3 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.4 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Electrical Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Mechanical Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 Mechanical Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.2 Electromechanical Interactions . . . . . . . . . . . . . . . . . . . . . 46

2.5 Structured Network Representation of Linear DynamicSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Basic Equations of Linear Networks . . . . . . . . . . . . . . . . . . . . . . . 58

x Contents

Part II Network Representation of Systems with Lumped andDistributed Parameters

3 Mechanical and Acoustic Networks with LumpedParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1 Mechanical Networks for Translational Motions . . . . . . . . . . . . . 62

3.1.1 Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1.2 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.1.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.1.4 Rules of Interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.1.5 Isomorphism between Mechanical and Electrical

Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.1.6 Representation of Transient Characteristics of Mass

Point Systems in the Frequency Domain (BODEdiagram) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.1.7 Network Representation of Mass Point Systems . . . . . . . 853.1.8 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.2 Mechanical Networks for Rotational Motions . . . . . . . . . . . . . . . 993.2.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.2.2 Components and System Equations . . . . . . . . . . . . . . . . . 1013.2.3 Isomorphism between Mechanical and Electrical

Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.2.4 Sample Application for a Rotational Network . . . . . . . . 106

3.3 Acoustic Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.3.2 Acoustic Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.3.3 Network Representation of Acoustic Systems . . . . . . . . . . 1103.3.4 Real Acoustic Components . . . . . . . . . . . . . . . . . . . . . . . . . 1143.3.5 Isomorphism between Acoustic and Electrical Circuits . 1203.3.6 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4 Abstract Linear Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.3 Nodal and Loop Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4 Characteristics of the Abstract Linear Network . . . . . . . . . . . . . . 132

5 Mechanical Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.1 Translational-Rotational Transducer . . . . . . . . . . . . . . . . . . . . . . 137

5.1.1 Rigid Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.1.2 Bending Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.2 Mechanical-Acoustic Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.2.1 Ideal and Real Mechanical-Acoustic Piston Transducers 1475.2.2 General Elastomechanical-Acoustic Plate Transducer . . . 149

Contents xi

5.3 Characteristics of Selected Mechanical-Acoustic Transducers . . 155

6 Mechanical and Acoustic Networks with DistributedParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.1 Representation of Mechanical Systems as one-dimensional

Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.1.1 Extensional Waves within a Rod . . . . . . . . . . . . . . . . . . . . 1666.1.2 Approximate Calculation of the Input Impedance . . . . . 1726.1.3 Approximate Representation of an Impedance at

Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.1.4 Approximated two-port Network Representation at

Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.1.5 Flexural Vibrations within a Rod . . . . . . . . . . . . . . . . . . . 183

6.2 Network Representation of Acoustic Systems as LinearWaveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.3 Modeling of Transducer Structures with Finite NetworkElements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.3.1 Ultrasonic Microactuator with Capacitive Diaphragm

Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.3.2 Fluid-filled Pressure Transmission System of a

Differential Pressure Sensor . . . . . . . . . . . . . . . . . . . . . . . . 1986.4 Combined Simulation with Network and Finite Element

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2026.4.1 Applied Combination of Network Methods and Finite

Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2046.4.2 Combined Simulation using the Example of a Dipole

Bass Loudspeaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2096.4.3 Combined Simulation using the Example of a

Microphone with Thin Acoustic Damping Fabric . . . . . . 216

Part III Electromechanical Transducers

7 Electromechanical Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.1 Classification of Electromechanical Interactions . . . . . . . . . . . . . . 2297.2 Network Representation of Electromechanical Interactions . . . 233

8 Magnetic Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2478.1 Electrodynamic Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8.1.1 Derivation of the Two-Port Transducer Network . . . . . . . 2478.1.2 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

8.2 Electromagnetic Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2678.2.1 Derivation of the Two-Port Transducer Network . . . . . . . 2688.2.2 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.3 Piezomagnetic Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

xii Contents

8.3.1 Derivation of the Two-Port Transducer Network . . . . . . 2868.3.2 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2968.3.3 Piezomagnetic Unimorph Bending Elements . . . . . . . . . . 3028.3.4 Example of a Parametric Magnetoelastic Bending

Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

9 Electrical Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3139.1 Electrostatic Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

9.1.1 Electrostatic Plate Transducer . . . . . . . . . . . . . . . . . . . . . . 3139.1.2 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3239.1.3 Electrostatic Diaphragm Transducer . . . . . . . . . . . . . . . . 3319.1.4 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3349.1.5 Electrostatic Solid Body Transducers . . . . . . . . . . . . . . . . 3399.1.6 Sample Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

9.2 Piezoelectric Transducers with Lumped Parameters . . . . . . . . . 3459.2.1 Model Representation of the Piezoelectric Effect . . . . . . . 3459.2.2 Piezoelectric Equations of State and Circuit Diagram

for Longitudinal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 3489.2.3 General Piezoelectric Equations of State . . . . . . . . . . . . . 3509.2.4 Piezoelectric Transducers and Corresponding

Equivalent Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3539.2.5 Piezoelectric Bending Bimorph Elements . . . . . . . . . . . . 3589.2.6 Piezoelectric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3609.2.7 Sample Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

9.3 Piezoelectric Transducer as one-dimensional Waveguide . . . . . . 3709.3.1 Transition from Lumped Parameters to the Waveguide

using the Example of an Accelerometer . . . . . . . . . . . . . . 3719.3.2 Piezoelectric Longitudinal Oscillator as Waveguide . . . . 3759.3.3 Piezoelectric Thickness Oscillator as Waveguide . . . . . . 3759.3.4 Sample Applications of Piezoelectric Longitudinal and

Thickness Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3819.3.5 Piezoelectric Beam Bending Element as Waveguide . . . 3929.3.6 Sample Applications of Piezoelectric Beam Bending

Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

10 Reciprocity in Linear Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 41310.1 Reciprocity Relations in Networks with only One Physical

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41310.2 Reciprocity Relations in General Linear Two-Port Networks . . 41510.3 Electromechanical Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41710.4 Mechanical-Acoustic Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Contents xiii

A Characteristics of Selected Materials . . . . . . . . . . . . . . . . . . . . . . 425A.1 Material Characteristics of Crystalline Quartz . . . . . . . . . . . . . . . 425A.2 Piezoelectric Constants of Sensor Materials . . . . . . . . . . . . . . . . . 426A.3 Characteristics of Metallic Structural Materials . . . . . . . . . . . . . . 427A.4 Material Characteristics of Silicon and Passivation Layers . . . . 428

A.4.1 Comparison of Main Characteristics of Silicon, SiliconDioxide and Silicon Nitride Layers . . . . . . . . . . . . . . . . . . 428

A.4.2 Characteristics of Silicon Dioxide Layers . . . . . . . . . . . . . . 429A.4.3 Characteristics of Silicon Nitride Layers . . . . . . . . . . . . . . 430

A.5 Characteristics of Ceramic Structural Materials . . . . . . . . . . . . . 431A.6 Material Characteristics of Selected Polymers . . . . . . . . . . . . . . . 432A.7 Characteristics of Plastics as Structural Materials . . . . . . . . . . . 433A.8 Composition and Material Characteristics of Selected Glasses . 434A.9 Material Characteristics of Metallic Solders and Glass Solders . 435A.10 Sound Velocity and Characteristic Impedance . . . . . . . . . . . . . . . 436

B Signal Description and Transfer within Linear Networks . . 437B.1 Fourier Expansion of Time Functions . . . . . . . . . . . . . . . . . . . . . . 437

B.1.1 Estimate of Approximation Error with NumericalAnalyses of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 437

B.1.2 Sample Application for the Periodic Iteration ofSingular Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

B.2 Ideal Impulse and Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . 442B.2.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442B.2.2 Ideal Impulses and their System Response . . . . . . . . . . . . 443B.2.3 The Ideal Step Function and its System Response . . . . . 448

B.3 The Convolution Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Part IV Appendix

.

List of Symbols

Symbol Quantity Unit

A cross-sectional area, pole face m2

A two-port matrixAel electrically active area m2

AF cross-sectional area (femur) m2

Ai area, rod segment area m2

Aij two-port matrix elementsAK piston area m2

AP cross-sectional area (prosthesis)Amech mechanically active area m2

a (t) acceleration m s−2

a complex acceleration (excitation) m s−2

a (ω) , b (ω) coefficients (Fourier integral transform) sai, bi, ci Fourier coefficients sB vector of magnetic induction TB (p) complex transfer function

(Laplace transform)B (ω) complex transfer function

(Fourier transform)B complex amplitude of transfer functionB0 magnetic induction TBk component of magnetic induction TBmax maximum magnetic induction TBS saturation induction Tb width mC capacitance FC0 reference capacitance FCb capacitance (mechanically locked state) FCK cable capacitance FCnm capacitance matrix elements F

xvi List of Symbols

Symbol Quantity Unit

c (ω) spectral densities sc (ω) coefficient (Fourier integral transform) sc wave velocity m s−1

ci complex Fourier coefficient sc∗i complex conjugate Fourier coefficient s

c(Q)ij elastic coefficient for Q = const. Nm−1

cEij elastic coefficient for E = 0 Nm−2

cHij elastic coefficient fur H = 0 Nm−2

cl extensional wave velocity m s−1

cL sound propagation velocity (air) m s−1

cp specific heat capacity for p = const. J kg−1K−1

cV specific heat capacity for V = const. J kg−1K−1

cW sound propagation velocity (water) m s−1

D electric displacement Cm−2

D vector of electric displacement Cm−2

D0 electric reference displacement Cm−2

Del electrically generated electric displacement Cm−2

Dmech mechanically generated electric displacement Cm−2

Dn component of electric displacement Cm−2

dij piezoelectric coefficient mV−1

dji piezomagnetic coefficient mA−1

deff effective distance mE Young’s modulus Nm−2

electric field strength Vm−1

E complex Young’s modulus Nm−2

E vector of electric field Vm−1

E0 electric reference field strength Vm−1

EB33 Young’s modulus (magnetostrict. material) Nm−2

Ec coercive field strength Vm−1

EF Young’s modulus (femur) Nm−2

EP Young’s modulus (prosthesis) Nm−2

EPoly Young’s modulus (polymer) Nm−2

Em component of electric field vector Vm−1

Emax maximum electric field strength Vm−1

EI average bending stiffness Nm2

enj piezoelectric modulus NV−1

ejn piezomagnetic modulus NA−1m−1

ex, ey, ez unit vectors (Cartesian coordinates)F (t) force NF (p) Laplace transformF (ω) Fourier transformF, F i complex force N

F force amplitude NF 0 excitational force, source force N

List of Symbols xvii

Symbol Quantity Unit

FB bottom force NFd distal excitation force NFel Coulomb force NFel vector of electric force NFel,n vector of electric force at reference point n NFi vector of affecting force NFK short circuit force NFm inertia force NFmag magnetic force NFmag vector of magnetic force NFmag,n vector of magnetic force at reference point n NFmax maximum force NFmech mechanical force NFmech,n vector of mechanical force at reference point n NFn spring force NFn complex spring force NFn vector of force at reference point n N

Fn spring force amplitude NFr frictional force NF r complex frictional force N

Fr frictional force amplitude NFW transducer force NFx, Fy, Fz components of force vector Nf frequency Hzf (x) complex load per unit length Nm−1

f0 resonant frequency, assigned frequency Hzfg cut-off frequency HzfP, fp parallel-resonant frequency Hzfr resonant frequency HzfS, fs series-resonant frequency HzG conductance S

shear modulus Nm−2

Gi proportionality factorGmn leg conductance Sg (t) normalized impulse response 1

(weighting function)H enthalpy J

magnetic field strength Am−1

H admittance matrixHapp applied magnetic field Am−1

Hin magnetic field in material Am−1

Hm component of magnetic field Am−1

Hmax maximum magnetic field strength Am−1

h height m

xviii List of Symbols

Symbol Quantity Unit

h translational admittance m s−1N−1

h (ω) complex admittancehD wave admittance s kg−1

hi layer thickness (layer i) mhij admittance matrix elementshR rotational admittance J−1 s−1

I geometrical moment of inertia m4

integral of Dirac delta function sdirect current A

I0 supply current AIF moment of inertia (femur) m4

Im magnetic flux rate Wb s−1

IP moment of inertia (prosthesis) m4

i (t) current Aı current amplitude Ai complex current Ai summation indexiK short-circuit current AiW transducer current AJ magnetic polarization Tj imaginary unitK,M,N summation limitsKel,K

0el coefficients

Kmag,K0mag coefficients

Kmag,r rotational transduction coefficient NmWb−1

Ki constant 1

K(ξ)km reciprocal inductance oefficient for ξ = const. H−1

k coupling factor, transformation ratio 1kel electric coupling factor 1kmech mechanical coupling factor 1kmn,i coupling factor (layer i) 1L inductance HL0 reference inductance, air coil inductance HLAir inductance in air HLb inductance (mechanically locked state) HLm inductance in magnetic layer HL∞ inductance of planar coil Hl length, lever length, beam bender length ml0 reference length, neutral position mlel electrically active length mli rod segment length mlmech mechanically active length mM torsional moment NmM0 magnetostrictive generated moment Nm

List of Symbols xix

Symbol Quantity Unit

M0 excitational moment NmMi vector of torsional moment NmM i complex torsional moment NmMW transducer moment NmMa,Ma,Si,Mai acoustic masses kgm−4

Ma,L acoustic mass of moved air kgm−4

Ma,M acoustic mass of a strip diaphragm kgm−4

Ma0 acoustic reference mass kgm−4

m∗ effective mass kgm mass kgm,n reference pointsm0, µ mass per unit length kgm−1

mers equivalent mass kgN transducer factor 1Na,Nai acoustic compliance (adiab. change of cond.) m5N−1

Na,iso acoustic compliance (isoth. change of cond.) m5N−1

Na,K acoustic short-circuit compliance m5N−1

Na,L acoustic open-circuit compliance m5N−1

Na,M acoustic compliance (diaphragm) m5N−1

Na,P acoustic compliance (plate) m5N−1

Na0 acoustic reference compliance m5N−1

Nd demagnetization factor 1n mechanical compliance mN−1

n0 compliance per unit length N−1

n0 translational compliance mN−1

nC field compliance mN−1

ners equivalent compliance mN−1

nK short-circuit compliance, mN−1

compliance of piezoceramics mN−1

nL compliance (electric open-circuit) mN−1

nmech mechanical compliance mN−1

n0R rotational compliance per unit length N−1m−2

nR rotational compliance N−1m−1

nRK rotational short-circuit compliance N−1m−1

nstat static compliance mN−1

P power WPa, Pak radiated acoustic power WPel electric power WPmech mechanical power WP vector of polarization Cm−2

Pi internal polarization Cm−2

Pr remanent polarization Cm−2

p complex frequency s−1

p (t) pressure Pa

xx List of Symbols

Symbol Quantity Unit

p pressure amplitude Pap0 reference pressure Papi

complex pressure Pa

pW

transducer pressure Pa

Q charge CQ0 reference charge CQ Q-factor 1Qel electrically generated charge CQm0 reference point charge CQmech mechanically generated charge CQn charge at reference point n Cq (t) volumetric flow m3 s−1

q0

excitational volumetric flow m3 s−1

qW

volumetric flow of transducer m3 s−1

R general gas constant Jmol−1K−1

R, r radius mR resistance ΩRi internal resistance ΩRmag magnetic resistance H−1

Rmn leg resistance Ωr friction impedance N sm−1

ra coefficient of friction per unit length N sm−2

ri position vector (reference point i) mrR torsional friction impedance NmsS mechancial strain 1SA area strain 1Si complex mechanical strain (layer i) 1Si component of mechanical strain 1Smax maximum mechanical strain 1Sr remanent mechanical strain 1SS saturation magnetostriction 1s (t) normalized step function 1s elastic constant m2N−1

sEij elastic compliance for E = 0 m2N−1

sHij elastic compliance for H = 0 m2N−1

T mechanical stress Nm−2

oscillation period sT0 fundamental oscillation period s

mechanical prestressing Nm−2

reference temperature KTE additional mechanical stress Nm−2

Tj component of mechanical stress Nm−2

T complex mechanical stress Nm−2

TM Maxwell stress Nm−2

List of Symbols xxi

Symbol Quantity Unit

Tmax maximum mechanical stress Nm−2

t time sU direct current voltage VU0 supply voltage Vu (t) electrical voltage Vu0 source voltage VuL complex open-circuit voltage VuW transducer voltage VV volume m3

V0 reference volume m3

VA armature volume m3

Vm magnetic voltage AVm,Air magnetic voltage (air) AVm,m magnetic voltage (ferromag. layer) Av (t) velocity m s−1

v complex velocity m s−1

vd distal velocity m s−1

vL open-circuit velocity m s−1

vS velocity (generator) m s−1

vW transducer velocity m s−1

W internal energy JWel electrical field energy JWkin kinetic energy JWmag magnetic field energy JWmech mechanical energy Jw number of turns 1

energy density Jm−3

wel electical field energy density Jm−3

wmag magnetic field energy density Jm−3

X transformer-like transducer constantXm eigenmode m 1x complex input variable 1x (t) , x (t) complex time functions 1x (t) periodic approximate function 1x∗ (t) complex conjugate time function 1x, y, z coordinate axesx amplitude of input variable 1x arithmetic mean 1x0 reference position mx1, x2, x3 coordinate axesY gyrator-like transducer constantY ∗ imaginary transducer constanty complex output variable 1y amplitude of output variable 1

xxii List of Symbols

Symbol Quantity Unit

Z electrical impedance ΩZa acoustic impedance N sm−5

Za,L acoustic friction of moved air N sm−5

Za,r acoustic friction N sm−5

Za0 acoustic reference friction N sm−5

z mechanical impedance kg s−1

z0 reference wave impedance kg s−1

zW wave impedance kg s−1

α coefficient of thermal expansion K−1

α (ω) , β (ω) coefficients (Fourier integral transform) sα, β, γ abstract componentsαf temperature coefficient of resonant frequency K−1

αi, βi Fourier coefficients 1β wave number m−1

γ propagation constant m−1

abstract admittanceγmn

, ρmn

matrix elements

∆A surface segment m2

∆C capacitance change F∆Cb capacitance change F

(mechanically locked state)∆F,∆Fi force change N∆Fmag change of magnetic field force N∆fr resonant frequency change Hz∆i current change A∆l change in length m∆L inductance change H∆m mass element kg∆n spring element N−1m−1

∆nI interface compliance N−1m−1

∆nR rotational compliance (beam segment) N−1m−1

∆nRK rotational short-circuit compliance N−1m−1

(beam segment)∆p pressure change Pa∆P power change W∆Q charge change C∆r equivalent viscous damping N sm−1

(differential beam element)∆R resistance change Ω∆t time difference s∆T temperature change K∆u voltage change V∆V volume change m3

∆W energy change J

List of Symbols xxiii

Symbol Quantity Unit

∆Wel electric field energy change J∆x rod element length, position change m∆µr permeability change 1∆ϕ angle change rad∆Φ magnetic flux change Wb∆ω angular frequency change s−1

δ error of Fourier spectral density 1δ (t) Dirac delta function s−1

ε permittivityε0 electric constant Fm−1

εSmn permittivity for S = 0 Fm−1

εTmn permittivity for T = 0 Fm−1

εr dielectric constant 1η loss factor, efficiency factor 1θ moment of inertia kgm2

ϑ temperature CϑCurie Curie temperature Cκ adiabatic exponent 1λ heat conductivity Wm−1K−1

λ, λB, λD wavelength mλm abstract flow coordinate 1λ amplitude of abstract flow coordinate 1µ differential coordinate 1

permeability Hm−1

voltage integral V sviscosity Pa s

µ amplitude of abstract differential coordinate 1µ0 magnetic constant Hm−1

µHij permeability for H = 0 Hm−1

µair dynamic viscosity of air Pa sµn

differential coordinate 1

µTnm permeability for T = 0 Hm−1

µr relative permeability 1ν Poisson ratio 1ξ (t) deflection, displacement mξ complex displacement mξg limit deflection mξmax maximum deflection mρ density kgm−3

ρ abstract impedanceρ (x) charge density function Cm−3

unbalance amplitude mρ0 reference density kgm−3

ρF density (femur) kgm−3

xxiv List of Symbols

Symbol Quantity Unit

ρK density (piezoelectric ceramics) kgm−3

ρL density (air) kgm−3

ρP density (prosthesis) kgm−3

ρW density (water) kgm−3

σ conductance Sτ impulse time sΦ magnetic flux WbΦe flux in magnetic circuit WbΦ0 constant flux component Wbϕ phase angle radϕ0 reference angle radϕB (ω) phase angle of transfer function radϕi phase angle of harmonic rad

oscillating componentϕn phase angle of spring force radϕr phase angle of frictional force radϕv phase angle of velocity radϕx phase angle of input quantity radϕy phase angle of output quantity radχ porosity, magnetic susceptibility 1Ω angular velocity s−1

Ω complex angular velocity s−1

Ω0 excitational angular velocity s−1

ΩW transducer angular velocity s−1

ω angular frequency s−1

ω0 reference frequency s−1

ω0 resonant frequency s−1

ωg cut-off angular frequency s−1

ωi characteristic frequency, angular frequency s−1

of harmonic oscillating componentωk discrete angular frequency s−1