《机电一体化技术》课程教学资源（书籍文献）Mechatronics，Dynamics of Electromechanical and Piezoelectric Systems，A Preumont

Preface The objective of my previous book.Vibration Control of Active Strue tures was to cross the hridge hetween Structural dynamics and auto matic Control.To insist on important control-structure interaction issues the book often relied on "ad-hoc"models and intuition (e.g.a thermal analogy for piezoelectric loads),and was seriously lacking in accuracy and depth on transduction and energy convers was in re ed it appe dthat this topic on its c This short book attempts to offer a systematic and unified way of ana- lyzing electromechanical and piezoelectric systems,following a Hamilton- Lagrange formulation.The transduction mechanisms and the Hamilton- Lagrange analysis of classical electromechanical systems ar systematic treatment s are devoted to the analysis of mechanical sys- tems electrical networks and classical electromechanical s tems rest tively:Hamilton's principle is extended to electromechanical systems fol- lowing two dual formulations.Except for a few examples,this part of the book closely follows the existing literature.The last three chapters are de- voted to piezoelectric systems.Chapter 4 analyzes discrete plezoelectr ndtheir troducti n into a structure;the approac ch parallel previ r ch e ar te en on pie lavers interact with the rting struetun sion from the perspective of active and passive damping:a unified approach is pro- posed,leading to a meaningful comparison of various active and passive techniques,and design guidelines for maximizing energy conversion. This b ook is intended for mechanical engineers (researchers and grad uate studer ts)w wish to et some tral transd the the dary c

Preface The objective of my previous book, Vibration Control of Active Struc￾tures, was to cross the bridge between Structural Dynamics and Auto￾matic Control. To insist on important control-structure interaction issues, the book often relied on “ad-hoc” models and intuition (e.g. a thermal analogy for piezoelectric loads), and was seriously lacking in accuracy and depth on transduction and energy conversion mechanisms which are essential in active structures. The present book project was initiated in preparation for a new edition, with the intention of redressing the imbal￾ance, by including a more serious treatment of the subject. As the work developed, it appeared that this topic was broad enough to justify a book on its own. This short book attempts to offer a systematic and unified way of ana￾lyzing electromechanical and piezoelectric systems, following a Hamilton￾Lagrange formulation. The transduction mechanisms and the Hamilton￾Lagrange analysis of classical electromechanical systems have been ad￾dressed in a few excellent textbooks (e.g. Dynamics of Mechanical and Electromechanical Systems by Crandall et al. in 1968), but to the author’s knowledge, there has been no similar systematic treatment of piezoelectric systems. The first three chapters are devoted to the analysis of mechanical sys￾tems, electrical networks and classical electromechanical systems, respec￾tively; Hamilton’s principle is extended to electromechanical systems fol￾lowing two dual formulations. Except for a few examples, this part of the book closely follows the existing literature. The last three chapters are de￾voted to piezoelectric systems. Chapter 4 analyzes discrete piezoelectric transducers and their introduction into a structure; the approach parallels that of the previous chapter with the appropriate energy and coenergy functions. Chapter 5 analyzes distributed systems, and focuses on piezo￾electric beams and laminates, with particular attention to the way the piezoelectric layers interact with the supporting structure (piezoelectric loads, modal filters, etc.). Chapter 6 examines energy conversion from the perspective of active and passive damping; a unified approach is pro￾posed, leading to a meaningful comparison of various active and passive techniques, and design guidelines for maximizing energy conversion. This book is intended for mechanical engineers (researchers and grad￾uate students) who wish to get some training in electromechanical and piezoelectric transducers, and improve their understanding of the sub￾tle interplay between mechanical response and electrical boundary condi￾xiii

xiv Preface aham s in 1875,a his ack of the electrical knowled oded to me his mochan difficulties.Henry simply replied:"Get it".The beauty of the Hamilton. Lagrange formulation is that,once the appropriate energy and coenergy functions are used,all the electromagnetic forces (electrostatic,Lorentz, reluctance forces,.)and the multi-physics constitutive equations are au- tomatically accounted for Acknowledgements hto my pr sent and fo hich have le P mit me in ring the mg and n ost of the figur os The comments of the Series Editor.Prof.Graham Gladwell,and of my friend Michel Geradin,have been very useful in improving this text.I am also in- debted to ESA/ESTEC,EU,FNRS and the IUAP program of the SSTC for their generous and continous support of the Active Structres Labo This book was partly written while I was visi ting professor Notation Notation is ays a sou which blend dis ted ficlds ith r hist ach own well established notation This book is noexcention to this rule since mechatronics mixes,analytical mechanics,structural mechanics,electrical networks,electromagnetism,piezoelectricity and automatic control,etc. The notation has been chosen according to the following rules:(i)We llow the IEBE Standard on Piezoelectricity as much as we can )When ere is no ambiguity,we t will not mak calars vectors and mat ne e clear from t ill be (e.g T wil denote the

tions, and vice versa. In so doing, we follow the famous advice given by Prof. Joseph Henry to Alexander Graham Bell, who had consulted him in connection with his telephone experiments in 1875, and lamented over his lack of the electrical knowledge needed to overcome his mechanical difficulties. Henry simply replied: “Get it”. The beauty of the Hamilton￾Lagrange formulation is that, once the appropriate energy and coenergy functions are used, all the electromagnetic forces (electrostatic, Lorentz, reluctance forces,.) and the multi-physics constitutive equations are au￾tomatically accounted for. Acknowledgements I am indebted to my present and former graduate students and cowork￾ers who, by their enthusiasm and curiosity, raised many of the questions which have led to this book. Particular thanks are due to Amit Kalyani, Bruno de Marneffe, More Avraam and Arnaud Deraemaeker who helped me in preparing the manuscript, and produced most of the figures. The comments of the Series Editor, Prof. Graham Gladwell, and of my friend Michel Geradin, have been very useful in improving this text. I am also in￾debted to ESA/ESTEC, EU, FNRS and the IUAP program of the SSTC for their generous and continuous support of the Active Structures Labo￾ratory of ULB. This book was partly written while I was visiting professor at Universit´e de Technologie de Compi`egne (Laboratoire Roberval). Notation Notation is always a source of problems when writing a book, and the difficulty is further magnified as one attempts to address interdisciplinary subjects, which blend disconnected fields with a long history, each with its own, well established notation. This book is no exception to this rule, since mechatronics mixes, analytical mechanics, structural mechanics, electrical networks, electromagnetism, piezoelectricity and automatic control, etc. The notation has been chosen according to the following rules: (i) We shall follow the IEEE Standard on Piezoelectricity as much as we can. (ii) When there is no ambiguity, we will not make explicit distinction between scalars, vectors and matrices; the meaning will be clear from the context. In some circumstances, when the distinction is felt necessary, col￾umn vectors will be made explicit by { } (e.g. {T} will denote the stress xiv Preface

Preface vector,while Tij denotes the stress tensor).(The partial derivative will be denoted ei r by tial deth choice of on otatio or the ther will be by and confo :11 summation on repeated indexes (Einstein's summation convention)will be assumed even when it is not explicitly mentioned. Andre Preumont Brussels,Decembre 2005

vector, while Tij denotes the stress tensor). (iii) The partial derivative will be denoted either by ∂/∂xi or by the subscript ,i (the index after the comma indicates the variable with respect to which the partial deriva￾tive is taken); the choice of one notation or the other will be guided by clarity, compactness and conformity to the classical literature. Similarly, summation on repeated indexes (Einstein’s summation convention) will be assumed even when it is not explicitly mentioned. Andr´e Preumont Brussels, Decembre 2005. Preface xv

Lagrangian dynamics of mechanical systems 1.1 Introduction This book considers the modelling of electromechanical systems in an unified way based on Hamilton's principle.This chapter starts with a review of the Lagrangian dynamics of mechanical systems,the next chap- ter proceeds with the Lagrangian dynamics electrical networks and apters address a wide class of electromechanical systems, motivated by the e substitution of scalar and work)for vo ntitios (fo m nomentum)in classical ve r dy ralized co ordinates are substituted for physical coordinates,which allows a formula tion independent of the reference frame.Systems are considered globally, rather than every component independently,with the advantage of elimi- nating the interaction forces(resulting from constraints)between the var entary parts of the system.The choice of generalized coordinates ation of the variational form of the equations of from its of virtual work tended to d ics th ks to d'Al leading eventually to Hamilton's principle and the Lagr ange equations for discrete systems. Hamilton's principle is an alternative to Newton's laws and it can be agah时 argued that, comprehen sible to the unexperienced reader and that its derivation fo a system of particles will ease its acceptance as an alternative formula

1 Lagrangian dynamics of mechanical systems 1.1 Introduction This book considers the modelling of electromechanical systems in an unified way based on Hamilton’s principle. This chapter starts with a review of the Lagrangian dynamics of mechanical systems, the next chap￾ter proceeds with the Lagrangian dynamics of electrical networks and the remaining chapters address a wide class of electromechanical systems, including piezoelectric structures. Lagrangian dynamics has been motivated by the substitution of scalar quantities (energy and work) for vector quantities (force, momentum, torque, angular momentum) in classical vector dynamics. Generalized co￾ordinates are substituted for physical coordinates, which allows a formula￾tion independent of the reference frame. Systems are considered globally, rather than every component independently, with the advantage of elimi￾nating the interaction forces (resulting from constraints) between the var￾ious elementary parts of the system. The choice of generalized coordinates is not unique. The derivation of the variational form of the equations of dynamics from its vector counterpart (Newton’s laws) is done through the principle of virtual work, extended to dynamics thanks to d’Alembert’s principle, leading eventually to Hamilton’s principle and the Lagrange equations for discrete systems. Hamilton’s principle is an alternative to Newton’s laws and it can be argued that, as such, it is a fundamental law of physics which cannot be derived. We believe, however, that its form may not be immediately comprehensible to the unexperienced reader and that its derivation for a system of particles will ease its acceptance as an alternative formula- 1

2 1 Lagrangian dynamics of mechanical systems tion of dynam equHamilton's principe is in fact more general han on's ed to dis It ic he nethods in dy g the finite method 1.2 Kinetic state functions Consider a partice traveling n the directiont of the aw, f= (1.1) The increment of work on the particle is （1.2) 70 the totlone切广in incrnn the momet T(p)= (1.3) mnction of the instantaneous momen- dl' =U (1.4) dp Up to now,no explicit relation between p and v has been assumed;the constitutive equation of Newtonian mechanics is 卫=ma (1.5) Substituting in Equ.(1.3),one gets (1.6) A complem state function can be defined as the kinetic (Fig1.1

2 1 Lagrangian dynamics of mechanical systems tion of dynamic equilibrium. Hamilton’s principle is in fact more general than Newton’s laws, because it can be generalized to distributed systems (governed by partial differential equations) and, as we shall see later, to electromechanical systems. It is also the starting point for the formula￾tion of many numerical methods in dynamics, including the finite element method. 1.2 Kinetic state functions Consider a particle travelling in the direction x with a linear momentum p. According to Newton’s law, the force acting on the particle equals the rate of change of the momentum: f = dp dt (1.1) The increment of work on the particle is fdx = dp dt dx = dp dt v dt = v dp (1.2) where v = dx/dt is the velocity of the particle. The kinetic energy function T(p) is defined as the total work done by f in increasing the momentum from 0 to p T(p) = Z p 0 v dp (1.3) According to this definition, T is a function of the instantaneous momen￾tum p, with derivative equal to the instantaneous velocity dT dp = v (1.4) Up to now, no explicit relation between p and v has been assumed; the constitutive equation of Newtonian mechanics is p = mv (1.5) Substituting in Equ.(1.3), one gets T(p) = p2 2m (1.6) A complementary kinetic state function can be defined as the kinetic coenergy function (Fig.1.1)