Mechatronics
Mechatronics
SOLID MECHANICS AND ITS APPLICATIONS Volume 136 Series Editor: GML GLADWELL Aims and Scope of the Series The aim of this s es relates to solids n answering these questons on the chanics on computational mechanics chao sthe theo es of elastici plasticity an mechanics,biomechanics and machine design
SOLID MECHANICS AND ITS APPLICATIONS Volume 136 Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of 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, shells 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
Mechatronics Dynamics of Electromechanical and Piezoelectric Systems 4 A.PREUMONT ULB Active Structures Laboratory. Brussels,Belgium Springer
Mechatronics Dynamics of Electromechanical and Piezoelectric Systems by A. PREUMONT ULB Active Structures Laboratory, Brussels, Belgium
AC.I.P.Catalogue record for this book is available from the Library of Congress 1S070206952He oDodrcht.The Netrlads www.springer.com Printed on acid-free paper All Rights Reserved
A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4695-2 (HB) ISBN-13 978-1-4020-4695-7 (HB) ISBN-10 1-4020-4696-0 (e-book) ISBN-13 978-1-4020-4696-4 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands
"Tenez,mon ami,si vous y pensez bien vous trouverez qu'en tout, notre veritable sentiment n'est pas celui dans lequel nous n'avons jamais vacille; mais celui auquel nous sommes le plus habituellement revenus." Diderot (Entretien entre D'Alembert et Diderot)
” Tenez, mon ami, si vous y pensez bien, vous trouverez qu’en tout, notre v´eritable sentiment n’est pas celui dans lequel nous n’avons jamais vacill´e; mais celui auquel nous sommes le plus habituellement revenus.” Diderot, (Entretien entre D’Alembert et Diderot)
Contents Preface. 1 Lagr 11 12 Kinetic state functions 1.3 Generalized coordinates,kinematic constraints. 1.3.1 Virtual displacements. m8 1.4 The principle of virtual work 168 ple 1.7 of a line stem 19 172 Dissination functio 1.7.3 Example 1:Pendulum with a sliding mass 20 1.7.4 Example 2:Rotating pendulum. 2 1.7.5 Example 3:Rotating spring mass system .23 1.7.6 Example 4:Gyros opic effects.24 aws 。 din 103F ple:The ical 1.10 More on continuou 3 1.10.1 Rayleigh-Ritz method. 32 1.10.2 General continuous system. 34 1,l03 Green strain tens0r,.,.,.,.,.,34 1.10.4 Geometric strain energy due to prestress. 1.10.5 Lateral I vibration of a beam with axial loads . vii
Contents 1 Lagrangian dynamics of mechanical systems . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Kinetic state functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Generalized coordinates, kinematic constraints . . . . . . . . . . . 4 1.3.1 Virtual displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 The principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 D’Alembert’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.1 Lateral vibration of a beam . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7.1 Vibration of a linear, non-gyroscopic, discrete system 19 1.7.2 Dissipation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.7.3 Example 1: Pendulum with a sliding mass . . . . . . . . . 20 1.7.4 Example 2: Rotating pendulum. . . . . . . . . . . . . . . . . . . 22 1.7.5 Example 3: Rotating spring mass system . . . . . . . . . . 23 1.7.6 Example 4: Gyroscopic effects . . . . . . . . . . . . . . . . . . . . 24 1.8 Lagrange’s equations with constraints . . . . . . . . . . . . . . . . . . . 27 1.9 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.9.1 Jacobi integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.9.2 Ignorable coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9.3 Example: The spherical pendulum . . . . . . . . . . . . . . . . 32 1.10 More on continuous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.10.1 Rayleigh-Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.10.2 General continuous system . . . . . . . . . . . . . . . . . . . . . . . 34 1.10.3 Green strain tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.10.4 Geometric strain energy due to prestress. . . . . . . . . . . 35 1.10.5 Lateral vibration of a beam with axial loads . . . . . . . 37 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii vii
Contents B06 Example:Simply supported beam in compression. 2 Dynamics of electrical networks. 2.1 Introduction. Tonfor circuit clement 2.2 The Capacitor 2.3 Kirchho current sources 2.4 Hamilton' nciple for electrical r 2.4.1 Hamilton's principle.charge formulation 48 2.4.2 Hamilton's principle,flux linkage formulation. 49 2.4.3 Discussion. 1 2.5 Lagrange's equations. 5 2 Lagrange's equations,charge formulation 23 agrange's equations,fx linkage formulation 254E 2.6 Refer Electromechanical systems. Introduction ns for transducers. 3.2.3 Moving-coil transduce 68 3.3 Hamilton's principle 71 3.3.1 Displacement and charge formulation. 71 3.3.2 Displacement and flux linkage formulation. 3.4 Lagrange's equations 3.4 formulation flux linkage formulation . 3.5 eaker 3.5.3 Capacitive microphone 3.5.4 Proof-mass actuator 82 3.5.5 Electrodynamic isolator 84
1.10.6 Example: Simply supported beam in compression . . . 38 1.11 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 Dynamics of electrical networks . . . . . . . . . . . . . . . . . . . . . . . . 41 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Constitutive equations for circuit elements. . . . . . . . . . . . . . . 42 2.2.1 The Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.2 The Inductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.3 Voltage and current sources . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Kirchhoff’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 Hamilton’s principle for electrical networks . . . . . . . . . . . . . . 47 2.4.1 Hamilton’s principle, charge formulation . . . . . . . . . . . 48 2.4.2 Hamilton’s principle, flux linkage formulation . . . . . . 49 2.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5 Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.5.1 Lagrange’s equations, charge formulation . . . . . . . . . . 53 2.5.2 Lagrange’s equations, flux linkage formulation . . . . . . 54 2.5.3 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.4 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Electromechanical ystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Constitutive relations for transducers . . . . . . . . . . . . . . . . . . . 61 3.2.1 Movable-plate capacitor . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.2 Movable-core inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.3 Moving-coil transducer . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 Hamilton’s rinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.1 Displacement and charge formulation. . . . . . . . . . . . . . 71 3.3.2 Displacement and flux linkage formulation . . . . . . . . . 72 3.4 Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.1 Displacement and charge formulation. . . . . . . . . . . . . . 73 3.4.2 Displacement and flux linkage formulation . . . . . . . . . 73 3.4.3 Dissipation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5.1 Electromagnetic plunger . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5.2 Electromagnetic loudspeaker . . . . . . . . . . . . . . . . . . . . . 77 3.5.3 Capacitive microphone . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.5.4 Proof-mass actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.5.5 Electrodynamic isolator . . . . . . . . . . . . . . . . . . . . . . . . . 84 viii Contents s p
Contents ix 3.5.6 The Sky-hook damper.86 3.5.7 Geophone .87 3.5.8 One-axis magnetic suspension.89 3.6 General electromechanical transducer. .92 3.6.1 Constitutive equations. 2 3.6.2 Self-sensing. 93 3.7 References. 94 4 Piezoelectric systems 95 4.1 Introductio ”4”。”4”4”。”。”4。”4” 5% Contuve ratin f discrete trausducer 4.3.1 Interpretation of k2.103 4.4 Structure with a discrete piezoelectric transducer.105 4.4.1 Voltage source. .107 4.4.2 Current source 107 Admittanee of the piezoelectric transducer 10 4.4.4 Prestressed transducer 4.4.5 Active enhancement of the electromechanical coupling111 4.5 Multiple transducer systems.113 4.6 General piezoelectric structure.114 4.7 Piezoelectric material. .116 4.7.1 Constitutive relations . 116 A700 gy density function 118 4.8 Hamilton's principle. 121 4.9 Rosen's piezoelectric transformer. 124 4.10 References.130 5 Piezoelectric laminates.13 5.1 Piezoelectric beam actuator .131 5.1.1 Hamilton's principle .131 5.1.2 Piezoelectric loads. .133 5.2 Laminar sensor .136 5.2.1 Current and charge amplifiers 136 5.2.2 Distributeds ns0r0 utput. 5.3 Spatial modal filters.139 5.3.1 Modal actuator.139 5.3.2 Modal sensor.140
3.5.6 The Sky-hook damper . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.5.7 Geophone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.5.8 One-axis agnetic suspension . . . . . . . . . . . . . . . . . . . . 89 3.6 General electromechanical transducer . . . . . . . . . . . . . . . . . . . 92 3.6.1 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.6.2 Self-sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4 Piezoelectric ystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Piezoelectric transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Constitutive relations of a discrete transducer . . . . . . . . . . . . 99 4.3.1 Interpretation of k2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4 Structure with a discrete piezoelectric transducer . . . . . . . . . 105 4.4.1 Voltage source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.2 Current source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.3 Admittance of the piezoelectric transducer . . . . . . . . . 108 4.4.4 Prestressed transducer . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.4.5 Active enhancement of the electromechanical coupling111 4.5 Multiple transducer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.6 General piezoelectric structure . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 Piezoelectric material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.7.1 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.7.2 Coenergy density function . . . . . . . . . . . . . . . . . . . . . . . 118 4.8 Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.9 Rosen’s piezoelectric transformer . . . . . . . . . . . . . . . . . . . . . . . 124 4.10 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5 Piezoelectric laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1 Piezoelectric beam actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1.1 Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1.2 Piezoelectric loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.2 Laminar sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.1 Current and charge amplifiers . . . . . . . . . . . . . . . . . . . . 136 5.2.2 Distributed sensor output. . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.3 Charge amplifier dynamics . . . . . . . . . . . . . . . . . . . . . . . 138 5.3 Spatial modal filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.3.1 Modal actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.3.2 Modal sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Contents ix s m
Contents 5.4 Active beam with collocated actuator-sensor.141 5.4.1 Frequency response function.142 5.4.2 Pole-zero pattern.143 5.4.3 Modal truncation. ,.145 5.5 Piezoelectric laminate 147 5.5.1 Two dimensional constitutive equations 148 5.5.2 Kirchhoff theory. 146 5.5.3 ess matrix of a multi-layer elastic laminate.149 5.5.4 Multi-laver laminate with a piezoelectric laver .151 5.5.5 Equivalent piezoelectric loads.152 5.5.6 Sensor output. .153 5.5.7 Remarks. 154 5 6 Reference 156 6 Active and passive damping with piezoelectric 6 159 tion 159 ”。”。”。”。” Active strut,open-loop FRF 6 6.3 Active damping via IFF.165 6.3.1 Voltage control.165 6.3.2 Modal coordinates. .167 6.3.3 Current control. 169 170 65 mp g via resistive shunting 172 6.5. Damping enhancement via negative capacitance shunting. 6.5.2 Generalized electromechanical coupling factor.176 6.6 Inductive shunting. .176 6.6.1 Alternative formulation. 181 6.7 Decentralized control 183 6 8 Gene piezoelectric structure. 184 6.9 Self-ser 6.9.1 Force sensing. 6.9.2 Displacement sensing.187 6.9.3 Transfer function.187 6.10 Other active damping strategies. .191 6.10.1 Lead control 101 6.10.2 Positive Position Feedback (PPF) 192
5.4.1 Frequency response function . . . . . . . . . . . . . . . . . . . . . 142 5.4.2 Pole-zero pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.4.3 Modal truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.5 Piezoelectric laminate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.5.1 Two dimensional constitutive equations . . . . . . . . . . . 148 5.5.2 Kirchhoff theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.5.3 Stiffness matrix of a multi-layer elastic laminate . . . . 149 5.5.4 Multi-layer laminate with a piezoelectric layer . . . . . . 151 5.5.5 Equivalent piezoelectric loads . . . . . . . . . . . . . . . . . . . . 152 5.5.6 Sensor output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.5.7 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Active strut, open-loop FRF . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.3 Active damping via IFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.3.1 Voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.3.2 Modal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.3.3 Current control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.4 Admittance of the piezoelectric transducer . . . . . . . . . . . . . . 170 6.5 Damping via resistive shunting . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.5.1 Damping enhancement via negative capacitance shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.5.2 Generalized electromechanical coupling factor . . . . . . 176 6.6 Inductive shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.6.1 Alternative formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.7 Decentralized control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.8 General piezoelectric structure . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.9 Self-sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.9.1 Force sensing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.9.2 Displacement sensing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.9.3 Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.10 Other active damping strategies . . . . . . . . . . . . . . . . . . . . . . . . 191 6.10.1 Lead control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.10.2 Positive Position Feedback (PPF). . . . . . . . . . . . . . . . . 192 x Contents 6 Active and passive damping with piezoelectric transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.4 Active beam with collocated actuator-sensor . . . . . . . . . . . . . 141
6.11 Remark 195 6 12 Refe Bibliography.199 Index.205
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Contents xi 6.11 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.12 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195