Discrete Systems
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Modelling of Mechanical Systems
Discrete Systems Volume I
Francois Rxisa
London and Sterling, VA
First published in France in 2001 by Hermes Science entitled 'Modelisation des systemes mecaniques, Systemes discrets, Tome l'. First published in Great Britain and the United States in 2004 by Kogan Page Science, an imprint of Kogan Page Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned addresses: 120 Pentonville Road London N1 9JN UK www.koganpagescience.com
22883 Quicksilver Drive Sterling VA 20166-2012 USA
© Hermes Science Publishing Limited, 2001 © Kogan Page Limited, 2004 The right of Francois Axisa to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. ISBN 1 9039 9651 1
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data Axisa, Francois, 1941Discrete systems / Francois Axisa. p. cm. "First published in France in 2001 by Hermes Science entitled: 'Modelisation des systemes mecaniques, Systemes discrets, Tome 1 ISBN 1-903996-51-1 1. Mechanics, Applied. 2. Discrete-time systems. I. Title. TA350.A95 2003 620.1'01'183--dc21
2003014087
Typeset by Kogan Page Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn www. biddies. co. uk
Contents
Foreword by Professor D J Ewins Preface Introduction
xi xiii xv
Chapter 1. Mechanical systems and equilibrium of forces 1.1. Modelling of mechanical systems 1.1.1. Geometry and distribution of masses 1.1.2. Motion relative to a given space 1.1.3. Coordinates 1.1.3.1. Degrees of freedom and generalized coordinates 1.1.3.2. Coordinate transformation 1. .3.3. Changes of reference frame 1.1.4. Kinematical constraints 1.1.4.1. Holonomic constraints 1.1.4.2. Nonholonomic constraints 1.1.4.3. Example: a constrained rigid wheel 1.1.5. Forces formulated explicitly as material laws 1.1.6. Forces formulated as constraint conditions 1.2. Basic principles of Newtonian mechanics 1.2.1. Newton's laws 1.2.1.1. Law of inertia 1.2.1.2. Law of motion (basic principle of dynamics) 1.2.1.3. Law of action and reaction 1.2.2. D'Alembert's principle of dynamical equilibrium 1.2.3. Equations of motion in terms of moments 1.2.3.1. Moment of a force and angular momentum 1.2.3.2. Plane rotation of a particle 1.2.3.3. Centrifugal and Coriolis forces 1.2.3.4. Applications to a few basic systems 1.2.4. Inertia forces in an accelerated reference frame 1.2.5. Concluding comments
1 2 2 3 3 3 5 7 8 8 11 11 16 19 20 20 20 21 22 24 26 26 28 29 30 35 38
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Chapter 2. Principle of virtual work and Lagrange's equations 39 2.1. Introduction 40 2.2. Mechanical energy and exchange of it 41 2.2.1. Work and generalized forces 41 2.2.1.1. Work performed by a force 41 2.2.1.2. Generalized displacements and forces 42 2.2.2. Work of inertial forces and kinetic energy 43 2.2.2.1. Linear motion (translation) in an inertial frame of reference 43 2.2.2.2. Rigid body rotating in an inertial frame of reference 44 2.2.2.3. Change of reference frame frame 46 2.2.2.4. Generalized inertial forces in a rotating frame frame 48 2.2.2.5. Properties of Hermitian matrices 50 2.2.3. Work performed by forces deriving from a potential 53 2.2.3.1. Potential energy 53 2.2.3.2. Generalized displacements and forces 54 2.2.4. Mechanical energy and the exchange of it with external systems 56 2.2.4.1. Conservative systems 56 2.2.4.2. Nonconservative systems 57 2.2.5. Work performed by constraint reactions and perfect constraints 59 2.3. Virtual work and Lagrange's equations 59 2.3.1. Principle of virtual work 59 2.3.2. Lagrange's equations 61 2.3.3. The Lagrange function (Lagrangian) 64 2.3.4. Special form of Lagrange's equations in the linear case 65 2.3.5. Lagrangian and Newtonian formulations 67 2.3.6. Application to a building resting on elastic foundations 67 2.3.6.1. Generalized displacements 68 2.3.6.2. Potential energy and stiffness 69 2.3.6.3. Generalized external loading and solution of the forced problem 70 2.3.6.4. Response to a distributed loading 71 2.3.6.5. Stiffness coefficients for distributed elastic foundations 72 2.3.6.6. Stiffness and mass matrices for any displacement field 74 Chapter 3. Hamilton's principle and Lagrange's equations of unconstrained systems 3.1. Introduction 3.2. The calculus of variations: first principles 3.2.1. Stationary and extremum values of a function 3.2.2. Static stability 3.2.2.1. Criterion for stability
79 80 82 82 85 85
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3.2.2.2. Static stability of a pair of upside-down and coupled pendulums 3.2.2.3. Buckling of a system of two articulated rigid bars 3.2.3. Stationary value of a definite integral 3.3. Variational formulation of Lagrange's equations 3.3.1. Principle of virtual work and Hamilton's principle 3.3.2. General form of the Lagrange's equations 3.3.3. Free motions of conservative systems 3.3.4. Forced motions of conservative systems 3.3.5. Nonconservative systems
86 87 94 101 101 102 103 106 109
Chapter 4. Constrained systems and Lagrange's undetermined multipliers 4.1. Introduction 4.2. Constraints and Lagrange multipliers 4.2.1. Stationary value of a constrained function 4.2.2. Nonholonomic differential constraints 4.2.3. Lagrange's equations of a constrained system 4.3. Prescribed motions and transformation of reference frames 4.3.1. Prescribed displacements treated as rheonomic constraints 4.3.2. Prescribed motions and transformations of reference frame
111 112 112 112 116 116 127 127 130
Chapter 5. Autonomous oscillators 5.1. Linear oscillators 5.1.1. Mechanical oscillators 5.1.2. Free vibration of conservative oscillators 5.1.2.1. Time-histories of displacement 5.1.2.2. Phase portrait 5.1.2.3. Modal analysis 5.1.3. Free vibration of nonconservative linear oscillators 5.1.3.1. Time-histories of displacement 5.1.3.2. Phase portrait 5.1.3.3. Modal analysis 5.1.4. Static instability (divergence or buckling) 5.2. Nonlinear oscillators 5.2.1. Conservative oscillators 5.2.2. Damped oscillators 5.2.3. Self-sustaining oscillators 5.3. Numerical integration of the equation of motion 5.3.1. Explicit scheme of central differences of second order 5.3.1.1. Recursive process 5.3.1.2. Initialisation of the algorithm 5.3.1.3. Critical value of the time-step for stability 5.3.1.4. Accuracy of the algorithm
139 140 140 142 142 144 146 148 148 151 152 153 154 154 164 167 169 170 170 171 172 174
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5.3.2. Application to a parametrically excited linear oscillator 5.3.3. Application to an oscillator impacting against an elastic stop 5.3.3.1. Impact force model 5.3.3.2. Constrained model 5.3.4. Newmark's implicit algorithm
176 179 179 185 188
Chapter 6. Natural modes of vibration of multi degree of freedom systems 6.1. Introduction 6.2. Vibratory equations of conservative systems 6.2.1. Linearization of the equations of motion 6.2.2. Solution of forced problems in statics 6.3. Modal analysis of linear and conservative systems 6.3.1. Coupling and uncoupling of the degrees of freedom 6.3.2. Natural modes of vibration 6.3.2.1. Basic principle of the modal analysis 6.3.2.2. Basic properties of the natural modes of vibration 6.3.2.3. Modal analysis of 2-DOF systems 6.3.2.4. Natural modes of vibration as standing waves 6.4. A few extensions of the modal concept 6.4.1. Natural modes of vibration of constrained systems 6.4.2. Free modes of rigid body 6.4.3. Prestressed systems and buckling modes 6.4.4. Rotating systems and whirling modes of vibration 6.4.4.1. Particle tied to a rotating wheel through springs 6.4.4.2. Fly-wheel on flexible supports
191 192 193 193 194 196 196 199 199 199 203 209 215 215 218 219 227 228 234
Chapter 7. Forced vibrations: response to transient excitations 7.1. Introduction 7.2. Deterministic transient excitation signals 7.2.1. Locally integrable functions and regular distributions 7.2.2. Signals suited to describe transient excitations 7.2.3. Impulsive excitations: Dirac delta distribution 7.2.4. Excitations of infinite duration and finite energy 7.3. Forced response and Laplace transformation 7.3.1. Laplace and inverse Laplace transformations 7.3.2. Transfer functions of the harmonic oscillator 7.3.3. External loads equivalent to nonzero initial conditions 7.3.3.1. Initial velocity and impulsive loading 7.3.3.2. Initial displacement and relaxation of a step load 7.3.4. Time-history of the response to a transient excitation 7.3.4.1. Response to a rectangular pulse 7.3.4.2. Response to a trapezoidal transient 7.3.4.3. Response to a truncated sine function
239 240 241 241 242 244 247 247 248 249 250 250 251 252 253 258 260
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7.4. Impulsive response and Green's function 7.4.1. Green's function of a harmonic oscillator 7.4.2. Green's function and forced response to any transient 7.5. Response of MDOF linear systems 7.5.1. Transfer function matrix of a conservative system 7.5.2. Uncoupling by projection on the modal basis 7.5.2.1. Principle of the method 7.5.2.2. Modal expansion of the transfer and Green's functions 7.5.3. Viscous damping 7.5.3.1. Model of viscous and proportional damping 7.5.3.2. Non proportional viscous damping 7.5.3.3. Implicit Newmark algorithm
267 267 268 269 269 271 271 272 280 281 283 285
Chapter 8. Spectral analysis of deterministic time signals 8.1. Introduction 8.2. Basic principles of spectral analysis 8.2.1. Fourier series 8.2.2. Hilbert space of the functional vectors of period T 8.2.3. Application: propagation of nondispersive 1-D waves 8.2.4. Fourier transformation 8.2.4.1. Definitions 8.2.4.2. Properties of Fourier transforms 8.2.4.3. Plancherel-Parseval theorem (product theorem) 8.2.4.4. Fourier transform in the sense of distributions and Fourier series 8.2.5. Spectral content of time signals 8.2.5.1. Spectral density of energy of a transient signal 8.2.5.2. Power spectral density of periodical functions 8.2.5.3. Mutual or cross-spectra 8.2.5.4. Spectra and correlation functions 8.2.5.5. Coefficients of correlation 8.2.5.6. Correlation of periodic signals 8.2.5.7. Functions approximated by truncated Fourier series 8.3. Digital signal processing 8.3.1. Sampling of a time signal 8.3.2. The Shannon sampling theorem 8.3.3. Fourier transforms of the original and of the truncated signals 8.3.4. Discretization of the Fourier transform 8.3.4.1. Discrete finite Fourier transform and Fourier series 8.3.4.2. Definition and properties of the discrete Fourier transform 8.3.4.3. Illustrative example
295 296 298 298 300 309 313 313 313 314 314 315 315 318 321 322 323 326 326 328 328 329 333 335 335 335 336
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Chapter 9. Spectral analysis of forced vibrations 9.1. Introduction 9.2. Linear (harmonic) oscillator 9.2.1. Spectra of excitation and response 9.2.2. Spectral properties of transfer functions 9.2.2.1. General features of the displacement/force transfer function 9.2.2.2. Spectral ranges of the oscillator response 9.3. MDOF linear systems 9.3.1. Excitation and response spectra 9.3.2. Interesting features of the transfer functions 9.3.3. Basic principles of the measurement of transfer functions 9.3.4. Response spectra resulting from an MDOF excitation 9.3.5. Vibration absorber using antiresonant coupling 9.3.6. Shock absorber of a car suspension 9.4. Forced vibrations of Duffing's oscillator 9.4.1. Periodic solutions and nonlinear resonances 9.4.l.l. RitzGalerkin method 9.4.1.2. Relationship between pulsation and amplitude of the oscillatory response 9.4.1.3. Nonlinear resonance peak 9.4.1.4. Hysteresis effect 9.4.2. Numerical simulations and chaotic vibrations 9.4.2.1. Periodic motions 9.4.2.2. Chaotic motions
341 342 342 342 343 343 346 352 352 352 358 359 360 365 368 368 368 370 374 374 376 376 379
Appendices Appendix 1: Vector spaces Appendix 2: Vector and multiple products of vectors Appendix 3: Euler's angles and kinetic energy of rotating bodies Appendix 4: Hermitian and symmetrical matrices Appendix 5: Grout's and Choleski's decomposition of a matrix Appendix 6: Some basic notions about distributions Appendix 7: Laplace transformation Appendix 8: Modal computation by an inverse iteration method
383 383 389 390 394 398 402 409 414
Bibliography Index Series synopsis: modelling of mechanical systems
419 425 435
Foreword
It is axiomatic in the international community of vibration specialists that texts on the subject written by French authors tend to be rather more mathematical and analytical in style and content than the corresponding works in English which tend, in turn, to emphasise the practical and pragmatic aspects. Indeed, that same comment can be applied to much of the teaching of vibration in the two languages. Of course, there are always exceptions to such sweeping generalisations, but the trend is real and it must be said that both approaches have their advantages and the serious scholar of the subject will benefit from studying texts from both camps. This new book by Francois Axisa seeks to break this stereotypical classification by combining features from both cultures. Here we have a text on vibrations which is written taking due regard for its French pedagogic heritage - rigour and completeness in the development of the underlying theoretical principles plus the goal of providing a general formulation - but whose overall direction is guided by the exigencies of the practising engineer and scientist. The author is both enseignant and ingenieur chevronne (an evocative and apposite phrase in the opening Introduction) and this dual perspective results in the addition of an unusual but fascinating contribution to the heavily populated territory of textbooks on analytical methods for structural vibration. This first volume of the series lays out the fundamentals of the underlying mechanics in more detail than is usually provided in the 'Anglo-Saxon' equivalents. Diligent study of the first four chapters will certainly enhance the understanding of these fundamentals for many who are already very familiar with the ensuing chapters on vibrating systems. The next three chapters present what might be described as the core topics of all texts on vibrating systems: Chapter 5: single-DOF systems (oscillators); Chapter 6: modal analysis of conservative multi-DOF systems; and Chapter 7: time-domain response analysis for damped and undamped systems. The final two chapters reflect clearly the preoccupation of the practising engineer with how much and what form of vibration response will be experienced by realistic structures in operation or service. Somewhat unusually for a first volume, this one includes in Chapter 8 the basis of spectral (i.e. Fourier and Laplace) analysis, and
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concludes in Chapter 9 with what is very much the modern perspective of harmonic (transfer function) response analysis as applied to periodic, transient and random excitations, not only for linear systems but also non-linear ones. These last two chapters clearly reflect the importance to the practising engineer of vibration response characteristics, in contrast to the relative abstraction of the virtual properties of modes which receive greater attention in many of the more academic texts. I feel sure that this work by Francois Axisa will reward those who study it with new and unusual insights into the fascinating and notoriously difficult-to-master subject of predicting and controlling the vibration properties of the complex practical structures encountered across a wide range of engineering sectors.
D J Ewins Professor of Vibration Engineering Imperial College London, UK
Preface
In mechanical engineering, the needs for design analyses increase and diversify very fast. Our capacity for industrial renewal means we must face profound issues concerning efficiency, safety, reliability and life of mechanical components. At the same time, powerful software systems are now available to the designer for tackling incredibly complex problems using computers. As a consequence, computational mechanics is now a central tool for the practising engineer and is used at every step of the designing process. However, it cannot be emphasized enough that, to make proper use of the possibilities offered by computational mechanics, it is of crucial importance to gain first a thorough background in theoretical mechanics. As the computational process by itself has become largely an automatic task, the engineer, or scientist, must primarily concentrate on producing a tractable model of the physical problem to be analysed. The use of any software system either in a university laboratory, or in a research department of an industrial company, requires that meaningful results be produced. This is only the case if sufficient effort was devoted to building an appropriate model, based on a sound theoretical analysis of the problem at hand. This often proves to be an intellectually demanding task, in which theoretical and pragmatic knowledge must be skilfully interwoven. To be successful in modelling, it is essential to resort to physical reasoning, in close relationship with the information of practical relevance. This series of four volumes is written as a self-contained textbook for engineering and physical science students who are studying structural mechanics and fluid-structure coupled systems at graduate level. It should also appeal to engineers and researchers in applied mechanics. The four volumes, already available in French, deal respectively with Discrete Systems, Basic Structural Elements (beams, plates and shells), Fluid-structure Interaction in the absence of permanent flow, and finally, Flow-induced Vibration. The purpose of the series is to equip the reader with a good understanding of a large variety of mechanical systems, based on a unifying theoretical framework. As the subject is obviously too vast to cover in an exhaustive way, presentation is deliberately restricted to those fundamental physical aspects and to the basic mathematical methods which constitute the backbone of any large software system currently used in mechanical engineering. Based on the
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experience gained as a research engineer in nuclear engineering at the French Atomic Commission, and on course notes offered to 2nd and 3rd year engineer students from ECOLE NATIONALE SUPERIEURE DES TECHNIQUES AVANCEES, Paris and to graduate students of Paris VI University, the style of presentation is to convey the main physical ideas and mathematical tools, in a progressive and comprehensible manner. The necessary mathematics is treated as an invaluable tool, but not as an end in itself. Considerable effort has been devoted to include a large number of worked exercises, especially selected for their relative simplicity and practical interest. They are discussed in some depth as enlightening illustrations of the basic ideas and concepts conveyed in the book. In this way, the text incorporates, in a self-contained manner, introductory material on the mathematical theory, which can be understood even by students without in-depth mathematical training. Furthermore, many of the worked exercises are well suited for numerical simulations by using software like MATLAB, which was utilised by the author for the numerous calculations and figures incorporated in the text. Such exercises provide invaluable training to familiarize the reader with the task of modelling a physical problem and of interpreting the results of numerical simulations. Finally, though not exhaustive, the references included in the book are believed to be sufficient for directing the reader towards more specialized and advanced literature concerning the specific subjects introduced in the book. To complete this work, I largely benefited from the input and help of many people. Unfortunately, it is impossible to properly acknowledge here all of them individually. However, I wish to express my gratitude to Alain Hoffmann, Head of the Department of Mechanics and Technology at the Centre of Nuclear Studies of Saclay, and to Pierre Sintes, Director of ENSTA, who provided me with the opportunity to be Professor at ENSTA. A special word of thanks goes to my colleagues at ENSTA and at Saclay - Ziad Moumni, Laurent Rota, Emanuel de Langre, lanis Politopoulos and Alain Millard - who assisted me very efficiently in teaching mechanics to the ENSTA students and who contributed significantly to the present book with pertinent suggestions and long discussions. Acknowledgments also go to the students themselves whose comments were also very stimulating and useful. I am also especially grateful to Professor Michael Paidoussis from McGill University Montreal, who encouraged me to produce an English edition of my book, which I found a quite challenging task afterwards! Finally, without the loving support and constant encouragement of my wife Fran9oise this book would not have materialized. Francois Axisa August 2003
Introduction
As a common teaching experience, graduate students are often somewhat astonished, and even disappointed, when the teacher starts the lectures on structural dynamics, and related more advanced topics, by reviewing in some detail the fundamentals of vector and analytical mechanics, which deal with discrete and idealized systems, instead of tackling directly the study of real and complicated systems that would impress their imagination and satisfy their legitimate eagerness to deal with engineering problems. In this respect, the reaction of an experienced engineer is of course the opposite, as his/her satisfaction is great when he/she is successful in reducing the behaviour of a set of interacting complicated structures to that of a few harmonic oscillators, without losing the essential aspects of the real problem. Clearly, it is necessary to acquire first a proper perspective to fully understand that this reduction is the very essence of the modelling process and that models are necessary to analyse real systems of practical interest. Moreover, the discrete models constitute the cornerstone of mathematical modelling, providing the analytical framework and numerical methods used in resolving mechanical engineering problems which can be incredibly complex. It is to these concepts and methods that this first volume is dedicated, while concentrating principally on the linear problems. Chapter 1 reviews the fundamental concepts and results of Newtonian mechanics. Besides the three founding principles enunciated by Newton, the essential points are first the concept of degrees of freedom (in short DOF) and generalized coordinates and then the formulation of internal forces which can either be expressed explicitly as material laws, or implicitly as kinematical conditions. Chapters 2 and 3 introduce the methods of analytical mechanics that use the work of forces, and not the forces themselves, as the primordial concept. This leads to Lagrange's equations, which are established starting first from the principle of the virtual work and then, in Chapter 3, from the principle of least action, due to Hamilton. The calculus of variations involved in such principles enables a discussion of the states of static equilibrium.
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Chapter 4 is devoted to the Lagrange undetermined multipliers method, which allows one to deal with systems constrained by kinematical conditions, by avoiding the necessity of eliminating first the superfluous variables. Mathematical and physical aspects of the method are described and illustrated by a few examples. Chapter 5 deals with the dynamical behaviour of single degree of freedom (SDOF) systems in the absence of any external loading (the so called autonomous systems) . The case of the linear oscillator is considered first to introduce the basic concepts used in the theory of vibrating systems. Then, the more complicated case of nonlinear autonomous oscillators is discussed based on a few examples to illustrate the significant differences between linear and nonlinear systems. The chapter is concluded by presenting two numerical algorithms, which are widely used in structural dynamics to produce approximate solutions of the equations of motion. Chapter 6 is concerned with the study of multi degrees of freedom systems (MDOF) in the autonomous and conservative case. It concentrates on the concept of natural modes of vibration, which is of paramount importance for analysing a large variety of problems in mechanics. From a physical standpoint, when an autonomous system, even if lightly damped, is provided initially with some amount of mechanical energy, it vibrates according to such modes, or most often according to a linear superposition of such modes. From a mathematical standpoint, the modes constitute an orthogonal vector basis which allows the equations of motion of any conservative, or lightly damped, linear system to be uncoupled. A few extensions of the concept of natural modes of vibration are also considered at the end of the chapter. Chapter 7 deals with the forced responses of linear systems subjected to external excitation, of the deterministic and transient type. A general method based on the Laplace transform is presented for solving the equations of motion. This provides the opportunity to introduce the important concept of the transfer function and of the Green's function. From the physical standpoint, the Green's function gives the response of a given DOF to an impulse of unit magnitude, applied to the same or another DOF. The transfer function is the Laplace transform of the Green's function. Application to a few problems illustrates the major features of the dynamic responses of a harmonic oscillator to deterministic transients. The study is then extended to the MDOF systems where the modal vector basis of the conservative system is first used to uncouple the equations of motions. Uncoupling in the presence of damping is also discussed and illustrated by an example. Chapters 8 and 9 present the basis of spectral analysis which has influenced the study of vibrations in both its theoretical and practical aspects most profoundly, since the discovery of the acoustical importance of fundamental vibration and its overtone by Pythagoras. Spectral analysis is based on Fourier series and the Fourier transform, which are reviewed first, because they are used to shift the analysis domain from the time to the frequency. Then, the spectral quantities which are appropriate to analyse the frequency content of time signals are introduced. Finally, the appropriate processing of digitized data to perform the spectral analysis of either measured, or computed signals is considered. Chapter 9 illustrates various
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applications of spectral analysis which are now commonly used by the scientists and the practising engineers in both theoretical and experimental studies of structural dynamics. The content of the English version of the present volume is essentially the same as that of the first edition in French. However, it benefited from a few but significant improvements and corrections. I am especially grateful and indebted to Professor John Fitzpatrick from Trinity College Dublin, who read and corrected the English edition of this volume, providing me with invaluable help in improving the use of English and with pertinent comments and suggestions on the content. I should like also to express my gratitude to Professor David Ewins from Imperial College London, for the foreword to this volume which points to precisely la raison d'etre I assigned to myself when writing this book. Finally, a special word of thanks goes to Lindsey Hofmeister and to Philip Kogan of Kogan Page Science, who were in charge of the present edition, for checking and rechecking every part of the manuscript. Their professional attitude has contributed significantly to the quality of this book. Any remaining errors and inaccuracies are purely my own. Francois Axisa August 200 3
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Chapter 1
Mechanical systems and equilibrium of forces
This chapter is intended both as a review of the basic principles of the Newtonian mechanics and as an introduction to a few physical concepts and mathematical notations we shall use throughout the book. Here, equilibrium equations (dynamic or static) of discrete systems are still derived by using the vector mechanics, i.e. direct balancing of the forces, and/or moments acting in the system. This approach is certainly the most familiar one to the majority of students, who are assumed to be already well acquainted with the basic techniques of vector analysis of Newtonian mechanics. However, the notions emphasized here, which concern degrees of freedom, generalized coordinates and kinematical conditions, are the first necessary ingredients of the Lagrangian formalism which is the subject of the next three chapters. Although the physical content of analytical mechanics founded by Lagrange (1788) is the same as that found in the Principia of Newton (1687), the analytical approach is more appealing from the logical viewpoint than the vectorial one and, even if more abstract, quickly reveals itself as far easier to apply when dealing with most material systems encountered in theoretical physics and mechanical engineering.
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1.1. Modelling of mechanical systems Formulation of a mathematical model is the first step in the process of analysing the behaviour of any real system. However, to produce a useful model, one must first adopt a set of simplifying assumptions which have to be relevant in relation to the physical features of the system to be modelled and to the specific information one is interested in. Thus, the aim of modelling is to produce an idealized description of reality, which is both expressible in a tractable mathematical form and sufficiently close to reality as far as the physical mechanisms of interest are concerned. Mechanical systems are made up of material bodies, i.e. finite portions of media endowed with mass. When excited by forces, or by prescribed motions, their position and shape change progressively with time, eventually reaching a new permanent equilibrium. When modelling such systems, it is necessary to specify five items listed below (and then discussed in the following order): 1. Geometry and mass distribution of the material system. 2. Space in which the motion is studied. 3. 4. 5.
Coordinates used to define its position versus time. Kinematical constraints connecting the relative motion of distinct parts of the system. Laws of mechanical behaviour of the material, and/or of mechanical interaction between distinct parts of the system.
1.1.1 Geometry and distribution of masses In the first instance, it is necessary to define the shape of the material system and the mass distribution within it. The simplest system that may be conceived is the mass-point, or panicle. This concept refers to a material body whose geometrical dimensions are neglected when describing its motion. The total mass m of the real body is attributed to the point, m being a positive scalar quantity. More generally, a collection of particles Pj (finite or not) with masses mj j = l,2,.. is called a discrete system. At first sight, a body whose dimensions are not neglected must be modelled as a continuous system, which is described by using a continuous (hence uncountable) set of points. They are endowed with a mass per unit volume (mass density) p that may vary either continuously, or piecewise continuously, with position inside the body. However, provided the body can be considered as rigid, i.e. remaining undeformed during the motion, it is still reducible to a discrete system, as further discussed in subsection 1.1.4.1. To conclude this subsection it is useful to recall that: In Newtonian mechanics, the mass of a particle is an invariant quantity of motion and the mass density is governed by the principle of conservation of matter.
Mechanical systems and equilibrium of forces
3
1.1.2 Motion relative to a given space The geometry of real mechanical systems can be defined in three-dimensional space ( E ) , in short notation a 3D-space. It is worthwhile recalling that: In Newtonian mechanics, the space in which the motion is studied is Euclidean in nature and its dimension is an integer less than or equal to three. Clearly, the position of any mass-point can be specified by a vector belonging to an Euclidean and 3D vector space.
Figure 1.1. Spaces associated with a wagon rolling on a railway
Furthermore, the space to which the motion may be referred is not unique. The simple example of Figure 1.1 helps to solidify this fundamental point. It shows a wagon rolling at cruising speed V(t) on a railway. Distinction is drawn between the space(E 1 ) tied to the rails and the space (E 2 ) tied to the wagon. The motion of a mechanical system installed in the wagon, a pendulum for instance, may thus be described either by referring to (E1), or to (E 2 ) . Clearly, the time-history (i.e. position versus time t) of the motion is dependant upon the space which is considered. To conclude this subsection it is worth recalling that: In Newtonian mechanics, time is a universal quantity, i.e. it does not depend upon the space (E) used to describe the motion.
1.1.3 Coordinates 1.1.3.1 Degrees of freedom and generalized coordinates In order to describe the motion of a set of particles it is necessary to know their position in (E), at each time. This requires the definition of a coordinate system related to (E).
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Figure 1.2. Cartesian (rectangular) and spherical coordinates
Let us start with the case of a single particle, the position of which is specified by using a Cartesian frame at rest in the 3D-space (E). In order to determine the position of the particle at each time t, it is necessary and sufficient to define three functions X ( t ) , Y ( t ) , Z ( t ) , which are the Cartesian components of the position vector of the particle in this frame. However, it is also clear that other coordinate systems could be used, for instance the spherical coordinates r(t),6(t),p(t), see Figure 1.2. In both cases, definitions for all three time functions are needed for determining the position of the particle. This number would remain unchanged if any other possible coordinate system were used. Accordingly, it is stated that a particle, which is free to move in a 3D-space, is a three degrees of freedom system (in short notation, a 3DOF system). This basic result can be generalized as follows: The number of degrees of freedom ND of a mechanical system is the number of time functions qk(t), k = 1,2,...,ND necessary and sufficient to specify the position of the whole set of its particles. Thus, a set of N particles free to move in a 3D-space constitutes a 3N-DOF system. On the other hand, the degrees of freedom of a deformable continuous medium cannot be enumerated. The functions denoted qk(t) form a set of ND independent parameters, or variables, which constitute the generalized coordinates of the system. Their physical nature may be varied at will, depending on the convenience found in using one specific quantity, or another, as a generalized coordinate. On the other hand, qk (t) can be interpreted as the coordinates of a single point which moves in an abstract M)-dimensional space, called the configuration space. The coordinates qk(t) are termed generalized displacements if they refer to the initial configuration in such a way that q k (0) = 0 k = 1,2,...ND. Thus, at any time, the configuration of the system may be described by a displacement vector denoted [q] which belongs to an Euclidean vector space with ND dimensions.
Mechanical systems and equilibrium of forces
5
On the other hand, it is also important to make a clear distinction between the concept of reference frame and that of coordinate system. Indeed, by a r reference frame we mean a frame tied to a given space, independently of the particular coordinate system that is used to define the position of the particles in this space. It can thus be pointed out that: 1.
The reference frame is tied to the space and can be used to characterize the motion of this space with respect to another space.
2.
The coordinate system specifies the way of defining the variables, which are used as coordinates (or displacements) to describe the motion of the mechanical system in a given space.
3.
The motion depends upon the reference frame considered but not upon the coordinate system.
4.
Any motion refers to a given space, or equivalently to a given reference frame.
Accordingly, in this book, distinction is made between the operation of coordinate transformation, which assumes implicitly that the space remains the same, and that of reference frame transformation, which assumes tacitly that the space is also changed. 1.1.3.2 Coordinate transformation Since the position of a mechanical system does not depend upon the coordinate system used to describe it, provided the space remains the same, changing from one coordinate system to another one is a point-to-point mapping of the configuration space on itself. In this space, a point P(qk) has indeed to be in a one-to-one correspondence with a point P'(q'k ). The mapping is formulated by using a set of functions labelled fij which specify the correspondence between the coordinates in the two systems. They must be finite, single valued, continuous and differentiable. For instance, any set of generalized coordinates qk of N particles can be transformed into a set of Cartesian coordinates by using relations of the following form:
Here, the subscript i indicates the axis of the Cartesian coordinate and the subscript j) indicates the particle. The functions fij are differentiable with respect to any variable qk , which is independent of all the others. As a consequence, the velocities in each system are connected by the following set of relations:
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Discrete systems
where the dot on a variable denotes a differentiation with respect to time. Thus, the quantities represented by qk are the generalized velocities of the system. On the other hand, in the second expression, use is made of Einstein's convention of implicit summation on the repeated indices, which is adopted for the rest of the book, unless otherwise explicitly indicated. As it can be easily anticipated, matrix calculus is an extremely powerful tool for analysing discrete mechanical systems in the linear domain. The necessary elements of theory will be introduced when needed in the context of our applications. For a thorough presentation, the reader can be referred for instance to [ANG 61], [BIS 65], [BAR 70], [JEN 77], [COL 87]. Then it will hereafter be often preferred to shift from the index notation to the matrix notation. Vectors are thus written as single column matrices, or in the transposed form, as single row matrices:
[ ] denotes a matrix and [ ]r denotes its transpose. Here, the axes Ox, Oy, Oz are labelled by a letter X, Y, Z, instead of the subscript i. Taking the particle labelled by j for instance, the components of [X] and [f] are rewritten as:
Moreover, omission of the coordinate axes to which the functions / refer ( f i j — f j ) provides further convenience for writing matrix relations. Accordingly, the subscript j is used now to individualize the degrees of freedom of the system instead of the particles. The relations [1.1] and [1.2] are then re-written as:
In equations [1.3], j and k vary from 1 to ND and [J] is the Jacobian matrix of [f], also called the gradient transformation matrix. When the Jacobian determinant (det[J]) becomes zero, the transformation of coordinates ceases to be a one-to-one correspondence. Geometrically, |det [J]| is the ratio of the infinitesimal volume in the [q] -coordinate system over the infinitesimal volume in the [X]-coordinate system.
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On the other hand, possible transformations are such that the sign of det [J ] remains constant, the latter depending upon the specific ordering of the variables of differentiation adopted, so that no zero crossing of det[J] occurs. EXAMPLE. - Transformation of spherical to Cartesian coordinates X = rsin(pcos0\ Y = rsin^7sin#; Z = rcos
4 this becomes unnecessary, because the number of mutually independent relations is reduced to L = 3(N-2). Such a reasoning can be extended to the case of a rigid body by considering N as being arbitrarily large. Hence, a rigid body can be described as a discrete system having at most 6 DOF. Of course, by using geometry, we could have derived such a result much more directly. Indeed, the displacement transformation of a geometrical figure can always be reduced to the
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product of a translation and a rotation. In a 3D-space, each of these elementary transformations is described by three independent parameters. As a final comment on the subject, the 5-DOF system of two particles is merely a particular case, arising because the particles having no dimension, any rotation about the P1P2 axis leaves the system unchanged.
1.1.4.2 Nonholonomic constraints The degrees of freedom of mechanical systems may be constrained by conditions which differ from the generic type [1.6]. Such conditions are called nonholonomic. As a first example, let us consider the case of a particle that is constrained to move inside a sphere, or at the surface of it. The Cartesian coordinates of the particle are thus governed by the condition:
Clearly, the condition [1.7] drastically restrains the space in which the motion can take place, but does not restrict the number of degrees of freedom of the particle. Relations of the type [1.7] are known as conditions of unilateral contact. During the contact, they reduce to equalities, and thus to holonomic conditions. Amongst the various kinds of nonholonomic constraints, the time-differential conditions are particularly worthy of mention because they are often used in rolling contact problems. They are expressed in terms of displacements and velocities of some points of the mechanical system:
It is also worth noting that [1.8] reduces to a holonomic condition when the differential equation can be integrated. An illustrative example is discussed in the next subsection. 1.1.4.3 Example: a constrained rigid wheel Let us consider a wheel modelled as a rigid circular disk of radius R, rolling on a horizontal floor, as shown in Figure 1.3. Oxyz is a Cartesian frame tied to the floor (Oxy-plane), with unit vectors i ,j,k . We are interested in determining the number of degrees of freedom and in defining convenient generalized displacements to describe the system, when subjected to various constraint conditions.
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Discrete systems
1.
Unconstrained wheel
Figure 1.3. Rigid wheel rolling on a horizontal floor
Since the wheel is assumed to be rigid, ND = 6. A natural idea for defining convenient generalized displacements is to make use of the displacement of the centre G of the wheel, which yields the parameters of translation XG,YG,ZG, and to choose three angular parameters to describe the rotation of the wheel about G.
Figure 1.4. Angular displacements of the wheel
Figure 1.4 shows a possible choice where cp is the angle of rotation of the wheel about its axle, taken as the GX-axis. 0 is the angle of GX with the vertical Oz-axis of the reference frame. Finally, ip is the angle between the Ox-axis and the intersection of the wheel plane (P) with the floor plane.
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2. The wheel is constrained to keep in contact with the floor The contact point of the wheel with the floor is denoted C. The constraint takes the form of the holonomic condition ZG = Rsin0; then ND = 5, see Figure 1.5.
Figure 1.5. Wheel in contact with the floor 3. The wheel keeps in contact with the floor and its axle keeps parallel to the floor Since the plane of the wheel remains perpendicular to the floor, the additional holonomic relation 6 = n/2 holds and ND = 4. Motion can be described by using the horizontal translation of G, the rotation (p about the wheel axle and finally the angle i// between the O.x-axis and the tangent to the trajectory of C, or even better, the direct angle 0 = n - ip , as shown in Figures 1.6 and 1.7. The displacement of any point B at the disk periphery, which makes an angle (p with the vertical direction, counted starting from point C, is given by:
where
Figure 1.6. Vertical wheel in contact with the floor
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Discrete systems
Figure 1.7. Condition of rolling without sliding
4.
Additional constraint of rolling without sliding
As shown in Figure 1.6, it is convenient to consider the direct Cartesian frame GXYZ with unit vectors J,J,K. One possible way to formulate the constraint condition is to balance the distances covered, during the same time, by the contact point along the disk periphery and along the path in the floor plane. Rolling without sliding implies that the length of arc CB is the same as the length of arc CC', see Figure 1.7. Accordingly, we have:
This yields the nonholonomic conditions: The second approach makes use of the concept of instantaneous centre of rotation, according to which the contact point is split conceptually into two distinct points, namely c1 = C is tied to the wheel and C2=C is tied to the plane. The velocities of the unfolded points are:
However, rolling without sliding implies that C1 and C2 have the same velocity, which thus turns out to be zero. This yields:
In the general case, the above equations cannot be integrated. Therefore, they are expressing nonholonomic conditions and the number of degrees of freedom remains unchanged.
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NOTE. - Parallel parking Any set of values Xc,Yc,
