Discrete gstems
This Page Intentionally Left Blank
Modelling of Mechanical Sgstems
Discrete
gstem
Volume I
Franl}ois Axisa
KOGAN PAGE SCIENCE
London and Sterling, VA
First published in France in 2001 by Hermes Science entitled 'Mod~lisation des systdmes
mdcaniques, Systdmes discrets, Tome 1'. 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 NI 9JN UK www.koganpagescience.com
22883 Quicksilver Drive Sterling VA 201662012 USA
9 Hermes Science Publishing Limited, 2001 9 Kogan Page Limited, 2004 The right of Franqois 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 CataloguinginPublication Data A CIP record for this book is available from the British Library.
Library of Congress CataloginginPublication Data Axisa, Franqois, 194 lDiscrete systems / Franqois Axisa. p. cm. "First published in France in 2001 by Hermes Science entitled: 'Mod61isation des syst6mes m6caniques, Syst6mes discrets, Tome I'." ISBN 190399651  1 1. Mechanics, Applied. 2. Discretetime systems. I. Title. TA350.A95 2003 620.1'0 I'183dc21 2003014087
Typeset by Kogan Page Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn
www.biddles.co.uk
Contents
F o r e w o r d by Professor D J Ewins .................................................................... xi Preface ............................................................................................................... xiii Introduction ........................................................................................................ xv C h a p t e r 1. M e c h a n i c a l systems and e q u i l i b r i u m o f forces ............................. 1 I.I. Modelling o f mechanical systems ................................................................. 2 1.1.1. G e o m e t r y and distribution o f masses ................................................. 2 1.1.2. Motion relative to a given space ........................................................ 3 1.1.3. Coordinates ........................................................................................ 3 1.1.3.1. Degrees o f freedom and generalized coordinates ................ 3 1.1.3.2. Coordinate transformation ................................................... 5 1.1.3.3. Changes o f reference frame ................................................. 7 1.1.4. Kinematical constraints ...................................................................... 8 1.1.4.1. H o l o n o m i c constraints ......................................................... 8 1.1.4.2. N o n h o l o n o m i c constraints ................................................... 11 1.1.4.3. Example: a constrained rigid wheel ..................................... 11 1.1.5. Forces formulated explicitly as material laws .................................... 16 1.1.6. Forces formulated as constraint conditions ........................................ 19 1.2. Basic principles o f N e w t o n i a n mechanics ..................................................... 20 1.2.1. N e w t o n ' s laws .................................................................................... 20 1.2.1.1. Law o f inertia ....................................................................... 20 1.2.1.2. Law o f motion (basic principle o f d y n a m i c s ) ...................... 21 1.2.1.3. Law o f action and reaction ................................................... 22 1.2.2. D ' A l e m b e r t ' s principle o f dynamical equilibrium ............................. 24 1.2.3. Equations o f motion in terms o f m o m e n t s ......................................... 26 1.2.3.1. M o m e n t o f a force and angular m o m e n t u m ......................... 26 1.2.3.2. Plane rotation o f a particle ................................................... 28 1.2.3.3. Centrifugal and Coriolis forces ............................................ 29 1.2.3.4. Applications to a few basic systems .................................... 30 1.2.4. Inertia forces in an accelerated reference frame ................................. 35 1.2.5. C o n c l u d i n g c o m m e n t s ........................................................................ 38
vi
Discrete systems
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 ................................................... 46 2.2.2.4. Generalized inertial forces in a rotating 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. i. 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. I. 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 ....................................................................................... 79 ~. 1. Introduction ................................................................................................... 80 3.2. The calculus of variations: first principles ..................................................... 82 3.2.1. Stationary and e• values of a function ................................... 82 3.2.2. Static stability ..................................................................................... 85 3.2.2.1. Criterion for stability ........................................................... 85
Contents 3.2.2.2. Static stability of a pair of upsidedown 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 ...................................................................
vii
86 87 94
101 101 102 103 106 109
Chapter 4. Constrained systems and Lagrange's undetermined multipliers .......................................................................................................... 111 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 ..............
112 112 112 116 116 127 127 130
Chapter 5. Autonomous oscillators ................................................................... 139 5.1. Linear oscillators ........................................................................................... 5.1.1. Mechanical oscillators ........................................................................ 5.1.2. Free vibration of conservative oscillators .......................................... 5.1.2.1. Timehistories 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. Timehistories 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. Selfsustaining oscillators .................................................................. 5.3. Numerical integration o f 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 timestep for stability .......................... 5.3. !.4. Accuracy o f the algorithm ...................................................
140 140 142 142 144 146 148 148 151 152 153 154 154 164 167 169 170 170
171 172 174
viii
Discrete systems 5.3.2. Application to a parametrically excited linear oscillator .................... 176 5.3.3. Application to an oscillator impacting against an elastic stop ........... 179 5.3.3.1. Impact force model .............................................................. 179 5.3.3.2. Constrained model ............................................................... 185 5.3.4. Newmark's implicit algorithm ........................................................... 188
Chapter 6. Natural modes of vibration of multi degree of freedom systems .................................................................................................. 191 6.1. Introduction ................................................................................................... 192 6.2. Vibratory equations of conservative systems ................................................ 193 6.2.1. Linearization of the equations of motion ........................................... 193 6.2.2. Solution of forced problems in statics ................................................ 194 6.3. Modal analysis of linear and conservative systems ....................................... 196 6.3.1. Coupling and uncoupling of the degrees of freedom ......................... 196 6.3.2. Natural modes of vibration ................................................................ 199 6.3.2.1. Basic principle of the modal analysis .................................. 199 6.3.2.2. Basic properties of the natural modes of vibration .............. 199 6.3.2.3. Modal analysis of 2DOF systems ....................................... 203 6.3.2.4. Natural modes of vibration as standing waves ..................... 209 6.4. A few extensions of the modal concept ......................................................... 215 6.4.1. Natural modes of vibration of constrained systems ........................... 215 6.4.2. Free modes of rigid body ................................................................... 218 6.4.3. Prestressed systems and buckling modes ........................................... 219 6.4.4. Rotating systems and whirling modes of vibration ............................ 227 6.4.4.1. Particle tied to a rotating wheel through springs .................. 228 6.4.4.2. Flywheel on flexible supports ............................................. 234
Chapter 7. Forced vibrations: response to transient excitations .................... 239 7.1. Introduction ................................................................................................... 240 7.2. Deterministic transient excitation signals ...................................................... 241 7.2.1. Locally integrable functions and regular distributions ....................... 241 7.2.2. Signals suited to describe transient excitations .................................. 242 7.2.3. Impulsive excitations" Dirac delta distribution .................................. 244 7.2.4. Excitations of infinite duration and finite energy ............................... 247 7.3. Forced response and Laplace transformation ................................................ 247 7.3.1. Laplace and inverse Laplace transformations .................................... 248 7.3.2. Transfer functions of the harmonic oscillator .................................... 249 7.3.3. External loads equivalent to nonzero initial conditions ..................... 250 7.3.3.1. Initial velocity and impulsive loading .................................. 250 7.3.3.2. Initial displacement and relaxation of a step load ................ 251 7.3.4. Timehistory o f the response to a transient excitation ....................... 252 7.3.4.1. Response to a rectangular pulse ........................................... 253 7.3.4.2. Response to a trapezoidal transient ...................................... 258 7.3.4.3. Response to a truncated sine function .................................. 260
Contents
7.4. Impulsive response and Green's function ...................................................... 267 7.4.1. Green's function of a harmonic oscillator .......................................... 267 7.4.2. Green's function and forced response to any transient ...................... 268 7.5. Response of MDOF linear systems ............................................................... 269 7.5.1. Transfer function matrix of a conservative system ............................ 269 7.5.2. Uncoupling by projection on the modal basis .................................... 271 7.5.2.1. Principle of the method ........................................................ 271 7.5.2.2. Modal expansion of the transfer and Green's functions ...... 272 7.5.3. Viscous damping ................................................................................ 280 7.5.3.1. Model of viscous and proportional damping ....................... 281 7.5.3.2. Non proportional viscous damping ...................................... 283 7.5.3.3. Implicit Newmark algorithm ................................................ 285
Chapter 8. Spectral analysis of deterministic time signals .............................. 295 8.1. Introduction ................................................................................................... 296 8.2. Basic principles of spectral analysis .............................................................. 298 8.2.1. Fourier series ...................................................................................... 298 8.2.2. Hiibert space of the functional vectors of period T ............................ 300 8.2.3. Application: propagation of nondispersive ID waves ...................... 309 8.2.4. Fourier transformation ....................................................................... 313 8.2.4.1. Definitions ........................................................................... 313 8.2.4.2. Properties of Fourier transforms .......................................... 313 8.2.4.3. PlancherelParseval theorem (product theorem) .................. 314 8.2.4.4. Fourier transform in the sense of distributions and Fourier series .............................................................................. 314 8.2.5. Spectral content of time signals ......................................................... 315 8.2.5.1. Spectral density o f energy of a transient signal ................... 315 8.2.5.2. Power spectral density of periodical functions .................... 318 8.2.5.3. Mutual or crossspectra ........................................................ 321 8.2.5.4. Spectra and correlation functions ......................................... 322 8.2.5.5. Coefficients of correlation ................................................... 323 8.2.5.6. Correlation of periodic signals ............................................. 326 8.2.5.7. Functions approximated by truncated Fourier series ........... 326 8.3. Digital signal processing ............................................................................... 328 8.3.1. Sampling of a time signal ................................................................... 328 8.3.2. The Shannon sampling theorem ......................................................... 329 8.3.3. Fourier transforms of the original and o f the truncated signals .......... 333 8.3.4. Discretization of the Fourier transform .............................................. 335 8.3.4.1. Discrete finite Fourier transform and Fourier series ............ 335 8.3.4.2. Definition and properties of the discrete Fourier transform ........................................................................................... 335 8.3.4.3. Illustrative example .............................................................. 336
ix
x
Discrete systems
Chapter 9. Spectral analysis of forced vibrations ............................................ 341 9.1. Introduction ................................................................................................... 342 9.2. Linear (harmonic) oscillator .......................................................................... 342 9.2.1. Spectra of excitation and response ..................................................... 342 9.2.2. Spectral properties of transfer functions ............................................ 343 9.2.2. I. General features of the displacement/force transfer function ................................................................................ 343 9.2.2.2. Spectral ranges of the oscillator response ............................ 346 9.3. MDOF linear systems .................................................................................... 352 9.3. I. Excitation and response spectra ......................................................... 352 9.3.2. Interesting features of the transfer functions ...................................... 352 9.3.3. Basic principles of the measurement of transfer functions ................ 358 9.3.4. Response spectra resulting from an MDOF excitation ...................... 359 9.3.5. Vibration absorber using antiresonant coupling ................................. 360 9.3.6. Shock absorber of a car suspension ................................................... 365 9.4. Forced vibrations of Duffing's oscillator ...................................................... 368 9.4. I. Periodic solutions and nonlinear resonances ...................................... 368 9.4.1. I. Ritz Galerkin method ........................................................... 368 9.4.1.2. Relationship between pulsation and amplitude of the oscillatory response .......................................................................... 370 9.4.1.3. Nonlinear resonance peak .................................................... 374 9.4.1.4. Hysteresis effect ................................................................... 374 9.4.2. Numerical simulations and chaotic vibrations ................................... 376 9.4.2. I. Periodic motions .................................................................. 376 9.4.2.2. Chaotic motions ................................................................... 379 Appendices .......................................................................................................... 383 Appendix 1: Vector spaces ................................................................................... 383 Appendix 2: Vector and multiple products of vectors .......................................... 389 Appendix 3: Euler's angles and kinetic energy of rotating bodies ....................... 390 Appendix 4" Hermitian and symmetrical matrices ............................................... 394 Appendix 5: Crout's and Choleski's decomposition of a matrix .......................... 398 Appendix 6" Some basic notions about distributions ........................................... 402 Appendix 7" Laplace transformation .................................................................... 409 Appendix 8: Modal computation by an inverse iteration method ......................... 414 Bibliography ....................................................................................................... 419 Index ................................................................................................................. 425 Series synopsis: modelling of mechanical systems .......................................... 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 Franqois 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 ingdnieur chevronn~ (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 'AngloSaxon' 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: singleDOF systems (oscillators); Chapter 6: modal analysis of conservative multiDOF systems; and Chapter 7: timedomain 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
xii
Discretesystems
concludes in Chapter 9 with what is very much the modem perspective of harmonic (transfer function) response analysis as applied to periodic, transient and random excitations, not only for linear systems but also nonlinear 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 Franqois Axisa will reward those who study it with new and unusual insights into the fascinating and notoriously difficulttomaster 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 skilfuily 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 selfcontained textbook for engineering and physical science students who are studying structural mechanics and fluidstructure 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), Fluidstructure Interaction in the absence of permanent flow, and finally, Flowinduced 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
xiv
Discretesystems
experience gained as a research engineer in nuclear engineering at the French Atomic Commission, and on course notes offered to 2 "d and 3 rd year engineer students from ECOLE NATIONALE SUPERIEURE DES TECHNIQUES AVANCEES, Paris and to graduate students of Paris V! 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 selfcontained manner, introductory material on the mathematical theory, which can be understood even by students without indepth 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, l 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 S a c l a y  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 Pa'fdoussis 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 Fran~;oise 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 . . . .
xvi
Discretesystems
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
Introduction xvii 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. Frangois Axisa August 2003
This Page Intentionally Left Blank
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.
2
Discretesystems
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.
Coordinates used to define its position versus time.
4.
Kinematical constraints connecting the relative motion of distinct parts of the system.
5.
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 masspoint, or particle. 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 = 1,2,.. 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)/9 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 threedimensional space (E), in short notation a 3Dspace. 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 masspoint can be specified by a vector belonging to an Euclidean and 3D vector space.
z.
v(t)
r
O'
. . . . . . . . . . .
3c
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 l) 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 (El), or to (E2). Clearly, the timehistory (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 o f 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).
4
Discretesystems
Z Z
Z x x " \ ,,,,
...... 9. .......
"i
r....,.......
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 3Dspace (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),O(t),tp(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 3Dspace, 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 3Dspace constitutes a 3NDOF 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 NDdimensional 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 qk(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 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: The reference frame is tied to the space and can be used to characterize the motion of this space with respect to another space. .
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. 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 pointtopoint mapping of the configuration space on itself. In this space, a point P(qk ) has indeed to be in a onetoone correspondence with a point P'(q~ ). The mapping is formulated by using a set of functions labelled f,.j 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: Xij =
fij(ql,q2
.....
qND) where/= 1,2,3 and j = 1,2 ..... N
[1.1]
Here, the subscript i indicates the axis of the Cartesian coordinate and the subscript j indicates the particle. The functions f~j 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:
fCij = Z O~qk ilk r fCiJ  O~qk qk k=l
[1.2]
6
Discretesystems
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: [q]* =[q,,q2 ..... qND] [ X ] T = [X1, YI,Z 1..... 9
XN,YN,ZN]
......
[ ] denotes a matrix and [ l T 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 [/] are rewritten as:
Xlj = Xj; X2j =Yj; X3j =Zj; flj = fxj; f2j = fYj[ f3j = fzj Moreover, omission of the coordinate axes to which the functions f refer (f,.j >fj ) 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 rewritten as: IX] : [f([q])] [J~]: [J(f([q]))] [q]
[1.31
ayj Jjk = o~qk In equations [1.3], j and k vary from 1 to ND and [J] is the Jacobian matrix of [/], also called the gradient transformation matrix. When the Jacobian determinant ( det [J ]) becomes zero, the transformation of coordinates ceases to be a onetoone correspondence. Geometrically,
[det[J I is the ratio of the infinitesimal
volume in
the [q]coordinate system over the infinitesimal volume in the [X]coordinate system.
Mechanical systems and equilibrium of forces
7
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 = rsinq~cos0; Y = rsin~osin0; Z = rcos(o Assuming that the differentiations are carried out in the order r,O,q~, one is led to the result that det[ J] =  r ~sin (,0. It remains negative, or equal to zero, in the whole domain r > 0, 0 < 0 < 2Ir, 0 < q~ < ~r. A zero value reflects a degeneration of the transformation on the polar axis (p = k~r, k = 0,1 .....
1.1.3.3 Changes of reference frame Let us consider two distinct spaces in relative motion with respect to the other. For instance (E 2) is assumed to be moving and (E l) is assumed to be at rest. Hence, I
the transformation of coordinates q j, as defined in a frame at rest in (E l ) , and q j as defined in a frame at rest in (E 2), depends explicitly upon time: I
qj = f j(q,,q2 ..... qND;t);
J = 1,2 ..... ND
Thus, the velocity components are transformed as: .,
o3fj+
qj = o~t
qk,
k = 1 , 2 ..... ND
O~qk
Or, in matrix notation:
[ql = [f([q])] I[~']: 57 +[J(f([q]))] [ZT]
[1.4]
[cgf/c)t] is the so called transport velocity. It can be easily identified with the velocity of the moving frame, with respect to the frame that is assumed to remain at rest. The second component is the relative velocity, which is easily identified with the velocity of the system as defined in the moving frame. Clearly, relation [1.4] reduces to relation [1.3] when the transport velocity becomes zero.
8
Discretesystems
EXAMPLE. A pendulum in a wagon Returning to Figure 1.1, in the Cartesian frame tied to (E l) one obtains:
X(t) = X 0 +
V ( r ) d r +Lsin 0; Z ( t ) = Z 0  Lcos0
X ( t ) = V ( t ) + LOcosO; Z(t) = LOsinO
Where X0,Z 0 are the initial coordinates of the fixed point of the pendulum. Incidentally, such a result indicates that a change of reference frame does not modify the number of degrees of freedom of a system, except if the transport motion is not prescribed, as illustrated later in Chapter 4, example 3 of subsection 4.3.2.
1.1.4 Kinematical constraints
In many mechanical systems, some generalized coordinates are connected to others through conditions which restrict the possible motions. In modelling real material systems, one is often led to formulate constraints of many kinds. To begin with, a distinction is made between two broad classes of conditions, namely those of holonomic constraints and those of nonholonomic constraints. Holonomic constraints reduce the number of degrees of freedom of the unconstrained system, while the nonholonomic constraints leave it unchanged. Though basically correct, this statement will be revised in Chapter 4, when formulation of constraints using Lagrange's multipliers will be introduced.
1.1.4.1 Holonomic constraints
To grasp what a holonomic constraint means, the simplest way is to start with a specific example. Consider a particle which is constrained to lay on the surface of a sphere of radius R, the origin of the frame being located at the centre of the sphere. This condition implies that the Cartesian coordinates of the point X, Y, Z obey the relationship: X2 +
y2
+ Z2 =
R2
[1.5]
Equation [1.5] reduces to two the number of degrees of freedom, as may be immediately checked by using spherical coordinates. The position of the point is then determined by the two independent variables ~p and 0. Though ND is clearly independent of the coordinate system used, it is also worth studying the problem when it is formulated in terms of Cartesian coordinates. By using equation [ 1.5] to calculate Z as a function of X and Y, an ambiguity arises concerning the sign of Z. From a geometrical point of view, this reflects symmetry in the problem. Fortunately, ambiguity disappears as soon as the geometrical aspect of the problem is completed by the kinematical one. The latter implies that Z(t) is continuous and therefore the following initial condition must be verified:
Mechanical systems and equilibrium of forces
9
li~[Z(t)]=Z(O)
Since the configuration of the system is assumed to be known at t = 0, the initial condition fixes the relevant sign of Z(t). Thus, the present example suffices to emphasize the kinematical nature of the concept of degrees of freedom. Generalizing this example, constraints are said to be holonomic (from the Greek o~.ov entire, as a whole) if they can be expressed as relations between the coordinates of a mechanical system, which take the following form:
~(q,,,,q,, .... ;t) = 0
[1.6]
The coordinates that are connected through [1.6] may refer to a single or to several distinct particles. Whichever the case actually is, the major point remains that equation [1.6] can be used to calculate (at least implicitly) one coordinate in terms of the others, with the condition that kinematical continuity removes any ambiguity due to a possible nonuniqueness of the roots of [1.6]. Hence, the NDDOF unconstrained system becomes a ( N D  1)DOF system, when constrained. On the other hand, amongst the holonomic constraints, it is convenient to draw a distinction between those that change continuously with time, and those that do not. The first are called rheonomic constraints (the radical rheo comes from the ancient Greek verb 9e0~ which means to pour, to run) and the second ones are called scleronomic constraints (the radical sclero comes from the ancient Greek adjective o~:~,rlgov which means stiff, set). For instance, depending upon whether R is time dependent or not, relation [1.5] is a rheonomic, or a scleronomic condition. In both cases, the particle becomes a 3  1 = 2DOF system. Such a result can be generalized to the case of motions constrained by several holonomic conditions according to the following rule:
A set of N particles moving in a 3Dspace constitutes a ( 3 N  L)DOF system if motion is constrained by L holonomic conditions which are mutually independent. Mutual independence of the constraint conditions implies that there is no redundancy in the set of equations which are used to formulate the constraints applied to the system. L linear equations are mutually independent, provided that no one of them can be obtained as a linear superposition of the others (see Appendix 1). In contrast, relations such as qn
=
0 and (qn)2 = 0 clearly form a redundant set
of conditions. EXAMPLE. Degrees of freedom of a rigid body Two particles P/, Pj are said to be rigidly connected to each other if their relative distance
R 0 remains constant. This scleronomic constraint may be
formulated as the following (9~0) condition:
10
Discretesystems

= t i 2 + r2 2(//,. i j ) = 
_
.
R 2.
is the radius vector used to specify the position of the ith particle. A possible set of generalized coordinates to describe this 5DOF system may be defined as follows: ql = X i; q2 = Yi ; q3 = Zi ; q4 = Oj ; q5 =(Pj
The first three coordinates set the position of P~ and the two remaining that of Pj, which lies on a sphere with centre P/ and radius R i j . It is of interest to go further by considering an arbitrarily large number of particles, thus modelling a rigid body. Let us start with a system of three particles. Five degrees of freedom are associated with two of them and the remaining particle is necessarily located on the circle defined by the intersection of two spheres of known centres P~ ,P2 and radii 17,13, R23. In this case, only one additional coordinate, for instance a curvilinear abscissa along the circle, is sufficient to determine unambiguously the position of this 6DOF system. In the case of four particles, generally forming a tetrahedron, one is led to the same result. Indeed, six degrees of freedom are still associated with three particles and the remaining particle is located at the intersection of three known spheres. The geometrical problem yields two possible positions. One is eliminated by applying the condition of kinematical continuity. In the case of five particles, four radius vectors are known and three additional (~j)relations are associated with the fifth particle. They are formulated in terms of the following scalar products: ~ . ( ~  F2) = a~; ~ . ( ~  F3) = a2; Fs.( ~  ~ ) = a 3
The known coefficients a l , a 2 , a 3 are the coordinates of P5 in the oblique frame ( ~  F 2 ) , ( ~  ~ ) , ( ~ 74). So, it can be easily verified that a further relation would merely be a linear superposition of the preceding relations. For instance, it is not difficult to check that ~.(F3  ~ ) = a 4 = a ~  a 2 , a relation which takes care of the compatibility of the redundant set of constraint relations. Generally, when formulating the rigidity of N particles systems, it could be thought necessary to write down C 2 = N ( N 
1)/2 ( ~ j ) relations. Fortunately, as
soon as N > 4 this becomes unnecessary, because the number of mutually independent relations is reduced to L = 3(N2). 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
Mechanical systems and equilibrium of forces
11
product of a translation and a rotation. In a 3Dspace, each of these elementary transformations is described by three independent parameters. As a final comment on the subject, the 5DOF 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:
4 X 2 + y2 + Z 2 < R
[ 1.7]
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 timedifferential 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:
[9~(qk,~]j .... ;t)O I
[1.8]
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 ._,,
..~
..,.
(Oxyplane), 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.
12
Discretesystems
1.
Unconstrained wheel
/ X
e
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 X G, Y~,Z~, and to choose three angular parameters to describe the rotation of the wheel about G.
Z
X 0
C x
Figure 1.4. Angular displacements of the wheel Figure 1.4 shows a possible choice where tp is the angle of rotation of the wheel about its axle, taken as the GXaxis. 0 is the angle of GX with the vertical Ozaxis of the reference frame. Finally, ~u is the angle between the Oxaxis and the intersection of the wheel plane (P) with the floor plane.
Mechanical systems and equilibrium of forces
13
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 Z c  R sin 0 ; then ND = 5, see Figure 1.5.
zl
"o
p
0
c
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 0 = 7r/2 holds and ND = 4. Motion can be described by using the horizontal translation of G, the rotation ~0 about the wheel axle and finally the angle ~u between the Oxaxis and the tangent to the trajectory of C, or even better, the direct angle ~ = n  ~u, as shown in Figures 1.6 and 1.7. The displacement of any point B at the disk periphery, which makes an angle ~0 with the vertical direction, counted starting from point C, is given by: X B = Xc + R sin q~cos 0" YB = Yc + R sin q~sin r ; Z s = R (1  cos (O) where Xc = Xc; Yc = Yc. ~Z
x
......"/)
~~
(p>O Y .................."'~Y
xZ...........
e
Figure 1.6. Vertical wheel in contact with the floor
14
Discrete systems
i i ..........
......::::i13...........
..............x.,
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 I , 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:
CB = R(odt ; C C ' =  C B =R(odt
= cc__Z"j =Red dt ,Yc = Rcp cos r ; l;"c = Rcp sin This yields the nonholonomic conditions: Xc + R~bcos r = 0 ; I;"c + R~bsin r = 0 . 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 C 1 = C is tied to the wheel and C 2 = C is tied to the plane. The velocities of the unfolded points are:
_.,,
,,
Vc2 = 0 However, rolling without sliding implies that C l and C 2 have the same velocity, which thus turns out to be zero. This yields: I7c + R~bJ = 0 ~
A'c + R~bc~162= 0 ; l;'c + R~bsinr = 0
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.
Mechanical systems and equilibrium of forces
15
N O T E .  Parallel parking Any set of values X c, Yc,(P,r can be obtained through the kinematics of rolling without sliding, but not by using an arbitrary path. The existence of such a restriction can be checked in practice, for instance when trying to park a car, as illustrated in Figure 1.8.
...........iJl i................................
!
Figure 1.8. Parallel parking: the trajectory indicated is not always feasible! 5.
The preceding conditions hold and the path of the contact point is prescribed Let the path of the contact point C be given, for instance by the explicit equation
Yc = f ( X c ) . It is expected that the wheel is restricted to a single degree of freedom system (in short, a SDOF system), which can be conveniently described by using either the variable (p, or X c . Indeed, starting from equation of the trajectory, the following calculation is performed:
Yc = f ( X c ) =~ f ' ( X c ) = tanr ~ r = h ( X c ) . Since f ' ( X c ) dYc / dX c = )'c / ~ c , the two following conditions are found to be equivalent to each other: I;"c + R~b sin 0 = 0 r
,~c + RO cos 0 = 0
Moreover, they can be integrated. This yields for instance:
1 r = h(X) =~ cos 0 = g(X) =~ q9 = k(X) = ~
It
f( g(X) dt = ~1
R
lt
dX g(X)
where, at this step, the subscript C has been dropped out to simplify notation. The simplest path that can be conceived is a straight line. Then, X = R(p (Figure 1.9).
16
Discretesystems
X
X. //'///////////////////////////////////d
Figure 1.9. Wheel rolling without sliding on a straight line
1.1.5 Forces formulated explicitly as material laws Various kinds of forces and moments may arise in a mechanical system. As it will be emphasized in subsection 1.2.2, it is convenient to begin by making a clear distinction between external and internal forces, or moments. By definition, the first do not depend upon the dynamical state of the system, while the second are related to it. The laws of mechanical behaviour, widely termed material laws, or constitutive laws, formulate the internal forces which are induced in the system (moving or not) as explicit analytical expressions. One is led first to make a distinction between static laws which involve only generalized displacements, (or their spatial derivatives in the case of continuous systems, cf. Volume 2) and dynamic laws which also involve time derivatives, or even primitives, of displacements. On the other hand, it is also useful to classify the laws, according to the linear or nonlinear nature of their analytical expression. EXAMPLE I.  Linear spring
~F
XI
F (PI )
XZ
F (Pz ) Figure 1.10. Linear spring acting in translation
Mechanical systems and equilibrium of forces
17
Figure 1.10 represents a spring that acts in translation along the Oxaxis. X 1 and X 2 denoting the displacements of the end points P/ and
P2, the spring exerts the
restoring forces:
F(PI) = K(XI  X2); F(P2) = K(X2  X1)
[1.9]
where the positive quantity K is the stiffness coefficient of the spring, expressed in Newtons per meter (N/m) using the S.I. units, which are used in this book with few exceptions. Expression [1.9] is a particular case of the law of linear elasticity, which is the simplest that can be conceived. EXAMPLE 2. 
Springs provided with gaps
I , I
X J
o!ii~ +J ~ g
c'~"
Figure 1.11. Model of an elastic ball bouncing between two fixed walls Figure 1.11a shows a simple system presenting nonlinear elastic behaviour. It consists of an elastic spherical ball, bouncing back and forth between two parallel rigid walls, which are separated from each other by the distance 2L. The ball of radius R, is slightly deformed elastically when in contact with one of the walls. As a first approximation, the following linear elastic contact law can be assumed:
F~ =  K~6R where
K c is the equivalent stiffness of the ball when deformed in the radial
direction by an amount 5R/R of its surface at the contact point (here, the rather unrealistic assumption is tacitly made that contact is restricted to a single point of the deformed boundary of the body). Furthermore, the contact force Fc is assumed to be in the radial direction. The global behaviour of the system is then governed by the nonlinear stiffness coefficient:
18
Discretesystems K=0ifIX[<J;
K = K , if IX]>J; w i t h J = L  R
where X is the displacement of the ball centre, with respect to the middle point at distance L from each wail. The gap J = LR of each contact spring is equal to half the total length of "free flight". Thus, the system reduces to a particle whose onedimensional motion is restricted by two springs of stiffness coefficient K~., which are actuated as soon as the displacement becomes greater than the gap, see Figure I. 1 lb. The corresponding material law is expressed as:
[  K c ( X  J ) if X > J F (X) = J Kc (X + J) if  X > J [
0 otherwise
N O T E .  Nonlinear contact spring Actually, the ball is not deformed according to a pinpoint contact with the rigid wall, but according to a surface whose area varies during impact. This can be taken into account by using the more sophisticated Hertzian model of elastic contact, which results in the nonlinear elastic law F,.
=
K'(dR) 3/2 , where tc depends upon the
elasticity constants of the materials and on the geometry of the contacting bodies, see for instance [TIM 70], [LAN 86], [YOU 01]. EXAMPLE 3 .  Viscous damping
C
2tl
v2
Figure 1.12. Conventional representation of a viscous damper According to the considerations which shall be developed in Chapter 2 concerning mechanical energy, any real system interacts with others in such a way that it can either lose or gain mechanical energy. Many kinds of physical processes can be responsible for such interactions, for instance those which are described in
Mechanical systems and equilibrium of forces
19
Volumes 3 and 4, devoted to fluidstructure coupled systems. From the mathematical viewpoint, viscous damping is the simplest model that can be conceived for taking into account such irreversible exchanges of energy. Figure 1.12 is a conventional representation of a viscous damper, which acts in translation along the Oxaxis. It induces the dynamic forces:
F( P, ) =  C ( :;:,  2~ ); F(P.,) = +C(X,2,_)
[1.10]
C is the viscous damping coefficient, expressed in Newton seconds per meter (Ns/m). Energy is dissipated, or gained, according to the positive or negative sign of
C, as discussed further in Chapters 2 and 5.
1.1.6 Forces f o r m u l a t e d as constraint conditions From the considerations discussed in the two last subsections, it is clear that the motion of a mechanical system can be restrained either by prescribing some constraint conditions, or by formulating explicitly some internal forces. For instance, in the example of the ball impacting against a wall, we could have opted for a model of unilateral constraint instead of the nonlinear elastic model (spring provided with gaps). In Chapter 5 subsection 5.3.2.2, this problem will be studied by using the two models successively. As an interesting result, it will be shown that discrepancies between the contact force and the unilateral condition models arise during the stages of contact, which concern the detailed description of the shocks (or impacts). Furthermore, the differences are found to vanish progressively as the impact stiffness coefficient K c is increased. As a matter of fact, kinematical constraints formulate implicitly internal forcing terms, which can be identified precisely with the forces, or moments, that are necessary to enforce the prescribed conditions. Such forces and moments are termed constraint reactions. In Chapter 4, we shall see how to calculate conveniently such reactions. EXAMPLE. Particle tied to a rigid string P
........... .~176176162
Figure 1.13. Particle tied to a rigid string and rupture of the string
20
Discretesystems
Let a rigid body be connected to a fixed point O by a nonextensible string. When the body is rotating, it is the centripetal reaction induced by the condition of nonextensibility that maintains the body at a constant distance from O. To be fully convinced of that, cut the string and observe the motion of the particle, which becomes rectilinear, as shown in Figure 1.13. When modelling internal forces arising in a mechanical system, it is largely a matter of convenience to choose between an implicit model (constraint conditions) and an explicit model (material laws). For instance, in terms of degrees of freedom, it is far more convenient to model a solid, of which deformation is negligible, as a rigid body, instead of using a model of internal forces which would take care of the approximate rigidity of the body on the atomic scale. On the other hand, the distinction between constraints and material laws is of major importance to the manner in which a problem in mechanics may be formulated, as will be further developed in Chapters 2 to 4. Based on considerations about work, it will be found suitable to model only a specific class of internal forces as constraint conditions, namely that of the so called perfect constraints, which, by definition, do not perform any work.
1.2. Basic principles of Newtonian mechanics 1.2.1 Newton's laws In order to predict the changes in the configuration of a mechanical system over time, it is necessary to establish the equations of motion (also called the equations of dynamical equilibrium) which govern the time evolution of the coordinates (or displacements), of the system. The formulation of these equations rests on the three laws postulated by Newton (1687), which are taken as the foundation of Newtonian mechanics. It is worthwhile, for clarity at least, to review them briefly and to point out some consequences of major interest. As a preliminary, it is also worth recalling that Newtonian laws introduce force as the primordial physical concept of mechanics. Since force is a vector quantity, the Newtonian approach results in a vector formulation of the equations of motion.
1.2.1.1 Law of inertia A particle not experiencing any force either remains at rest, or is in uniform rectilinear motion. The law of inertia is restating in terms of forces the older inertial principle of Galileo, according to which an isolated and undisturbed body keeps a constant velocity vector. As a corollary, the reference frames, which are in uniform rectilinear motion in relation to each other, are equivalent from the point of view of dynamics. This is simply because the forces are the same in all such reference frames, which are therefore called Galilean or inertial frames.
Mechanical systems and equilibrium of forces
21
According to formula [1.4], position and velocity of a particle are transformed from one Galilean frame to another one, as follows: F'=F+Vtt"
~7'V+V t
[1.11]
F and V are the vectors of position and velocity in the unprimed frame, F' and V' denote the transformed vectors in the primed frame. V, is the transport velocity of the unprimed frame with respect to the primed frame, which is assumed to be fixed. Thus the inertial law of Galileo postulates the invariance of the dynamic equations with respect to the Galilean transformations [1.11 ]. Accordingly, it can be interpreted as a principle of relativity, which implies that in any inertial frame: 1.
Time is the same.
2.
Mass of a particle is the same.
3.
Forces exerted on a particle are the same.
Another important consequence of such a principle is the nonexistence of a unique frame of reference that would be an absolute frame which should be adopted to express the equations of motion in a privileged form. Moreover, even the idea of fixed frame has to be interpreted in a relative meaning.
1.2.1.2 Law of motion (basic principle of dynamics) A particle experiencing a force is prompted in an accelerated motion, so that the acceleration multiplied by the mass of the particle is equal to the force. Hence, the second law generalizes the first one, as it connects the acceleration to the force exerted on the particle, giving thus the following equation of motion: ,.
mX = F(t)
[1.12]
X(t) is the displacement (or position) vector of the particle expressed in an inertial frame provided with a Cartesian coordinate system. F(t) designates the force, or the sum vector (the resultant) of the individual forces exerted on the particle. Such forces are generally time dependent. Equation [1.12] may also be rewritten in the following more general form: [ p ( t )  /~ ]
[1.13]
Equation [1.13] introduces the linear momentum ~ of a particle as a new physical quantity which is defined as the product of the mass times the vector velocity of the particle. Incidentally, it may be noted that form [1.13] holds, even if m is time dependent.
22
Discrete systems
Finally, as we shall see on several occasions in this book, it is also of major interest to define the action of a physical quantity over a time interval t~ to t2 , as the integral of it over this interval. As a first example, from equation [1.13] it can be stated that ~ measures the mechanical action of the resultant of the individual forces exerted on the particle. Indeed, it follows immediately from equation [1.13] that: f l2 _,.
F dt = p ( t 2 )  p ( t ~ )
[1.141
II
The preceding considerations can be extended to systems of several particles. However, some care is required to deal in a suitable manner with the mechanical interactions between the particles. First let Fj be the resultant of forces exerted on the jth particle. The corresponding equation of motion is:
m i X j = Fj
[1.15]
Now, Fj is generally the sum of internal and external forces. Furthermore, some internal forces, termed interaction forces, depend upon variables (coordinates, velocities etc.) which refer to distinct particles. Interaction forces are governed by a third law referred to as the law of action and reaction.
1.2.1.3 Law of action and reaction
I
//
I I I
2
I
I I
,'
I
F12~
i
~
''"
F31
Figure 1.14. Strong law of action and reaction in a system of particles The forces two particles exert on each other are equal and opposite to each other. According to this principle, when a particle P~ is acting on another particle Pj by exerting a force F/j then Py is reacting on P/ by exerting the force Fji =F,.j.
Mechanical systems and equilibrium of forces
23
Furthermore, when dealing with purely mechanical applications, as it will be always the case in this book, a more restrictive form of this law, broadly referred to as the strong law of action and reaction, is adopted by adding to it the following postulate:
The action and reaction forces are central; i.e. they lie along the line joining the particles. Thus, in a mechanical system comprising N interacting particles, for each pair one can define forces of mutual interaction which are central and exactly opposed to each other, as illustrated in Figure 1.14. EXAMPLE 1 .  Two bodies interacting by gravitation Let P~ and P2 be a pair of particles with masses m r and m 2 , which are attracted to each other by gravity. Newton's law of gravitation gives the interaction force:
r2
r
G designates the constant of universal gravitation (G = 6.6710 11 N.m 2 / kg 2) and r is the relative distance of the two particles ( r = P~P2)" A particular case of interest is that of a body with mass m orbiting round the Earth, which is modelled as a spherical rigid body with radius R = 6 3 7 0 k m and mass M =5.98• kg. However, in so far as r remains sufficiently near to R and m remains sufficiently less than M, the force of interaction can be drastically simplified, being reduced as a first approximation to: I,~ A t i
Fg  f mk r
= gmk
Fg is the gravity force and g designates the acceleration of Earth's gravity (g = 9.81ms2). f~ is the unit vector along the local vertical, which is oriented from the orbiting body toward the Earth's centre. The important point worth emphasising is that according to the above approximation the actual interaction force is replaced by an external force exerted on the orbiting body, while the motion of Earth is implicitly assumed to remain unaffected by the motion of the body. EXAMPLE 2 .  Motion of the centreofmass of a system of particles Let be a set of N interacting particles, subjected also to external Equation [1.15] can be written as:
m j X j = Fj = Fij + [~Je) ;V j  1,2..... N
forces
/~,~e).
24
Discretesystems
Now, by summing the above equations over j, one is led to the following result, of remarkable simplicity, which is a direct consequence of the law of action and reaction (either in the weak or in the strong form)" N
N
= E~(e)=~.(e)
Emjj~j
j=l
[1.16]
j=l
It is then convenient to reduce the lefthand side of equation [ 1.16] by defining the centreofmass of the system of particles. The position of G is given by: 
1 N
X G =~EmjXj,
N where M = E m j
j =1
[1.17]
jI
Using this definition, equation [ 1.16] becomes: MX~ =/~(e)
[1.18]
Hence, whatever the motion of the individual particles may be, the centreofmass behaves like a single particle with the total mass M of the system, which is subjected to the sum vector of the external forces. It is also convenient to define the resulting linear momentum P as:
N
= MXG = E PJ
[1.19]
j=l
~j is the linear momentum of the jth particle. When /~(e)= 0, /~ becomes constant and the motion of G is rectilinear and uniform.
1.2.2 D'Alembert's principle of dynamical equilibrium The concept of force of inertia  hereafter denoted F ~ )  which was introduced by d'Alembert (1743), allows us to unify the formulation of statics and dynamics. Force of inertia is clearly an internal force because it depends upon the dynamic state of the system. By making use of F~), equation [1.12] takes the form of an equilibrated balance of forces" /~ =  m s
[1.20]
Equation [ 1.20] extends to dynamic systems the same balance of forces as that utilized in statics. This balance of forces is contained in Newton's first law as a condition for mechanical equilibrium. Finally, it may be worth emphasizing that
Mechanical systems and equilibrium of forces
25
statics is only an idealized asymptotic case of dynamics in which motion is so slow that the time derivatives of displacement, and even the time evolution of forces, become negligible. In this book, a unified presentation of dynamic and static systems will be used, starting from the dynamical models. On the other hand, most systems are subjected to both internal and external forces at the same time. So, for sake of clarity, it is convenient to collect the internal forces on the lefthand side and the external forces on the righthand side of the dynamic equations. Accordingly, equilibrium is written as the following balance of forces: interna! : ~ ~ externall
[1.21]
Hereafter, this convention will be systematically respected when writing down the equations in their final form. EXAMPLE.  The linear damped massspring system The equation of motion of the oscillator, shown schematically in Figure 1.15, results from the force balance written as: gx
JrCX ] M X = F (e)
(t )
[1.22]
C
position of static equilibrium for F (et = 0 Figure 1.15. Linear damped massspring system KX
is the stiffness (elastic) force, C,~ is the viscous damping force and
M,~ is the inertia force. Finally, F (e) (t) denotes the external force, which excites the oscillator. Stress is laid on the point that the displacement X(t) is counted starting from the position O of the static equilibrium of the unloaded system F (e)  O . The equation corresponding to the static problem is KX = F (e) . Finally, when the time evolution of F(e)(t) is sufficiently slow, the inertia and damping forces remain negligible with respect to the stiffness force. Accordingly, the problem is governed by the equation KX(t)= F(e)(t), which is said to be quasistatic. The range of validity of such an approximation will be further discussed in Chapter 9,
26
Discrete systems
subsection 9.2.2, in relation to the frequency content of the excitation and response signals.
1.2.3 Equations of motion in terms of moments 1.2.3.1 Moment of a force and angular momentum
o ........................ P
/ O
1
mX
................
P
t . . . . ""~
Figure 1.16. Moment of a force and angular momentum of a particle One is often led to express equations of motion in terms of angular variables. In these cases, it is generally found convenient to formulate Newton's second law in terms of moments (or torques). It is recalled first that the moment of a force about a given point O is defined as the cross product: I.~e,0 = F x / ~ I
[1.23]
As shown in Figure 1.16, F is the radius vector from the point O to the particle P, which is subjected to the force F . It is also useful to define the moment about a
given axis of unit vector k , as the mixed product: .71,f~,,~= k.(F x F)
[ 1.241
Substitution of equation [ 1.12] into equation [ 1.23], yields: mFxX = FxF
[1.25]
At this stage, it is useful to define the following vector quantity: 7
Le, o(F,t) = F x m X = F x b
[ 1.26]
Lp/0 is the angular momentum of the particle P about the point O.
Now, when O is a fixed point, it is found that: 
X=~
_.
[1.271
Mechanical systems and equilibrium of forces
27
In this case, equation [1.25] takes the remarkably simple form:
[1.28]
; x i~  L e / o  M e / o ( F )
Equation [1.28] formulates the dynamical equilibrium of the moments about a fixed point and indicates also that the angular momentum measures the action of a moment. If a component of the latter is zero, the corresponding component of the angular momentum is an invariant of motion. EXAMPLE. Kepler's second law of planetary motion Even though the scope of the present book does not consider celestial mechanics, it would be a pity not to mention here one of the most famous occasions where astronomy provided the starting impulse to the development of Newtonian mechanics, which may be considered a direct application of equation [ 1.28]. Based on the careful measurements by Tycho Brahe, Kepler found by trial and error his wellknown law about planetary motion, according to which the radius vector from the Sun to a planet sweeps out equal areas in equal times. It was a major achievement of Newton to be able to demonstrate that such a statement simply proceeds from the fact that the force of gravitation attracting two particles to each other is directed towards the centreofmass of the system (the centre of the Sun as a first approximation). Indeed, on one hand it is realized that the area swept per unit time is given precisely b y a  I I ~ •
On the other hand, as the force is always directed
towards a fixed point, the angular momentum is constant and so is A, since mass is constant too. As pointed out in the true enlightening lecture by R.P. Feynman on the relation of Mathematics to Physics [FEY 65], the proof given by Newton goes along the lines just outlined, but by using a quite ingenious and simple geometrical reasoning, instead of the vector calculus.
z
G
"y //......~...*'"~
Xj
0 Y
Figure 1.17. Angular momentum about the centreofmass of a set of N particles More generally, equation [1.28] can be used to uncouple from each other the motion of the centreofmass G of a system and the motion of the system about G. To achieve this, O has to be taken at the centreofmass of the mechanical system, see
28
Discretesystems
Figure 1.17. Indeed, the total angular momentum of a system of N particles about a fixed point can be written as the sum: LS, O :
~ (iG+ ij ) x mj (rG + ~ )
[1.29]
J
Since G is assumed to be the centreofmass of the system, one obtains:
~ mj~ = O =:~ ~ mj~j = O J
J
And so, by developing the righthand side of [1.29], the remarkably simple result is established: =
[1.30]
§
L~/0 is the angular momentum about O of the material point G where the mass of
mj is concentrated.
the system M = ~ J
Such a simplification is very convenient particularly for analysing the motion of rigid bodies. Indeed, in this case, motion of the system about G reduces to a 3Drotation, as further discussed in Chapter 2, subsection 2.2.2.
1.2.3.2 Plane rotation of a particle 7,
Y
.,~
o
" ""'"" 9
"1
O /
..........j
. " "',..
X
Figure 1.18. Polar coordinate system
Mechanical systems and equilibrium of forces
29
Polar coordinates ( r , 0 ) are often used to describe motion taking place in a plane; see Figure 1.18. The following results are immediately established: 7 = rt~; .fi = m(/'t~ + rt)/~1) [1.31] with k = ti x/~1 where ~ and Ka are the unit vectors in the radial and in the tangential directions, respectively. I = mr 2 is the moment o f inertia of the particle about the axis Oz, perpendicular to the plane Oxy. On substituting expression [1.31] of L in equation [ 1.28], it can be seen that if r is constant, 0 is governed by the following equation:
Now, comparison of equations [1.12], or [1.13], with equations [1.32], or [1.28], shows that Newton's equation of motion keeps the same form, whether a linear or an angular displacement is used. Such a result will be extended in Chapters 2 and 3 to the case of generalized forces and displacements of any kind. On the other hand, the second equation [ 1.31] shows that if the particle is rotating about the Ozaxis with the angular velocity ~ = .62/~, the linear velocity can be written as: ...,,
V s
l  ,(2x~
[1.33]
Since equation [1.33] is expressed in terms of vectors, its validity does not depend ...,,
upon the direction of ,62. 1.2.3.3 Centrifugal and Coriolis forces
Referring back to Figure 1.18, let us consider now a particle sliding freely along the radial axis of unit vector K at constant velocity V = / ' . Furthermore, the radial ...,,
axis is assumed to rotate at the constant angular velocity Ok about the fixed point O. The inertia force impressed to the particle is readily found to be: d2 (r~)  m ~ dt 2
d(VK+rOK,) = m
dt
=  2 m V , Q ~ 1 + mrs
[ 1.34]
The expression [1.34] makes apparent two new kinds of inertial forces which are not proportional to an acceleration, in contrast to that arising in Newton's second law of motion [ 1.12] or [ 1.32]. The first term on the righthand side of [ 1.34] is known as a Coriolis force, which acts in the tangential direction, and the second term is known as a centrifugal force which acts radially in the outward direction from the centre of rotation. Such forces will be met again in subsection 1.2.4, in relation with transformations of reference frames.
30
Discretesystems
1.2.3.4 Applications to a few basic systems EXAMPLE 1 .  The simple pendulum .... ~,
X
Z Figure 1.19. Simple pendulum The simple pendulum is shown schematically in Figure 1.19. It consists of a rigid body tied to a fixed point O by a nonextensible string, which is supposed to move in a vertical plane. The dimensions of the massive body are assumed to be so small with respect to the length R of the string that it can be modelled as a particle P with mass M. The most convenient way of deriving the dynamic equation of this SDOF system is to take the angle 0 as the variable of displacement and to work out the balance of moments about O. This gives:  ~ / ~ ( i n t e r n a l ) : ~ / v (external)
:=} F x (  X F (internal)): F x ~ / 7 (external)
Indeed, making use of the moments instead of the forces allows one to eliminate directly the  a priori unknown  reaction of the constraint condition OP = R, which identifies with the tension T of the string. 0 being counted from the lowest position on the vertical axis, it is readily shown that:
MV = MRO F~," L= MR2b lr '
RF~• Fg =  M g R s i n O k
which gives the equation of motion:
MR20 +MgR sin/9 = 0
[1.35]
Equation [ 1.35] sums two distinct terms. The first one is a linear dynamical term produced by the moment of the inertia force and the second is a nonlinear stiffness term produced by the moment of the weight. In terms of balance of forces, the formulation is a bit more heavy. Adopting the Cartesian coordinates ( which depend upon 0 ), we obtain:
Mechanical systems and equilibrium of forces X=Rsin0;
)~=RScos0;
Z=Rcos8;
Z=RSsin0;
31
,~ = Rt~ cos 0  Rt92 sin 0 = (R8 sin 0 + R82 cos0)
The internal forces comprise the force of inertia and the tension of the string:  M X =  M R ( ' S c o s O  8 2sin O) ;  M Z = MR(Ssin O + b 2cos0) Tx = TsinO; Tz =  T c o s 0 ;
The external force is the weight /~(e)= Mg. The horizontal and vertical balancing of forces give the two following equations:  T sin 0  M R ( ' S c o s O  82sin 0) = 0  T cos0 + MR(0sin 0 + 82cos 0) + Mg = 0 In this elementary example, the string tension is very easily eliminated to obtain a unique equation in terms of the single independent parameter 8. As expected, this produces the same equation as that obtained by the balance of moments. Moreover, as an interesting byproduct, the balance of forces method provides us with the reaction of constraint, which is found to be: T = Mg cos 0 + MRS"
T comprises a nonlinear stiffness force induced by the weight and a centrifugal force, which is induced by the rotation of P around O. To conclude on this basic example, we must emphasize that elimination of the reactions of constraint in the case of systems having several degrees of freedom is generally much less simple than in the case of SDOF systems. We shall describe in Chapter 4 the very clever method devised by Lagrange to deal with such constrained systems in order to determine at the same time the equations of motion and the reactions of constraints. EXAMPLE 2 .  Wheel connected to a linear spring Let us consider the rigid wheel already described in subsection 1.1.4.3. The track of it on the floor is now assumed to be a straight line, taken as the 0xaxis. The Oxzplane is vertical and ~ designates the acceleration of gravity. The wheel centre is connected to a fixed and rigid wall by a spring (stiffness coefficient K) acting in translation, see Figure 1.20. M is the wheel mass, supposed to be uniformly distributed over the circular disk (radius R). The system is then moved away from its position of static equilibrium by imparting the horizontal displacement X0 to G. The exercise consists in formulating the equation of motion successively in the two following contrasted cases.
32
Discrete systems
S o e ic~ z
x
o v///////////////A
c

Figure 1.20. Wheel connected to a wall by a spring 1.
The wheel slides perfectly on the floor
The system is described by using the two independent variables X and r In the presence of gravity, the condition of contact with the floor Z = R implies a reaction from the floor T~ = M g k that balances the weight of the wheel. Clearly, this reaction does not enter into the equations of motion. X is governed by the equation of equilibrium of the horizontal forces KX + MX = 0. On the other hand, r is governed ..
by the equilibrium equation of moments i, = 0. The angular momentum L is easily determined as follows. One starts by calculating the velocity of a current point of the disk. Next, the angular momentum of an infinitesimal area of the disk is obtained, which is then integrated over the whole disk area. The mass per unit area of the disk being denoted p (kgm2), this gives: V = r(ou I
dL = pr3drdO(oj => L = p ( o j
r3drdO = J ( o j
where J is the moment o f inertia of the disk about the axis of rotation. From the ~.R 4
MR 2
above integral, it is found that in the present case J = p ~
=~ . Therefore, the 2 2 equation of motion reduces to JO = 0. Going a little bit further, it could be easily shown that, provided an initial velocity k (0), r (0) is imparted to the wheel, the response is a horizontal oscillation about the fLxed point of the spring, of frequency fl = ~ x
, linearly superposed on a rotation at constant angular velocity (o(0)
about the wheel axle. However, in the context of this book, mathematical techniques used to solve this kind of differential equations fully deserve an extensive presentation, which is postponed until Chapter 5.
Mechanical systems and equilibrium of tbrces 2.
33
The wheel is rolling without sliding on the floor
As already shown, the wheel is now a SDOF system. Furthermore, the condition X = Rip implies the existence of a horizontal reaction of the floor Tx = Txi. The equations of equilibrium become:
Kx+
= rx
Elimination of Tx yields:
Jk Tx=
R
R2
2
A single equation of motion is thus produced, expressed here in terms of X: .9 K X + M eX
=
0
J 3M 1 1 2K where M e = M +   = ~ ~ f = R2 2 ~ 3M
N O T E .  Inertia o f the rolling wheel As indicated by the last result, when the wheel is rolling without sliding, it oscillates at a lower frequency than when it slides without rolling. The reason for this is that rolling motion adds inertia to the system while stiffness remains unchanged. Accordingly, it can be advantageous to minimize the mass of rotating mechanical components. For instance, during acceleration, the legs of a cyclist are solicited "twice" by the mass of tires. Indeed, provided most of the mass is at the periphery of the wheel, it is found that J = MR', hence M eX" = 2 M X . EXAMPLE 3 .  Changing direction o f a spinning wheel
f'2
J=MR 2
/
!
Figure 1.21. Changing the direction of the axle of a spinning wheel
34
Discrete systems
Let us consider the simple experiment sketched in Figure 1.21, which consists of holding a spinning wheel by its axle and changing the direction of the axle. The spinning angular velocity s is assumed to be much larger than the angular velocity ~) imparted to the axle by the experimenter. If so, the angular momentum of the system remains practically collinear to the axis of spin. The mass M of the wheel is assumed to be essentially uniformly distributed at the periphery and so J = MR 2. We want to know the force, or torque, which has to be exerted on the axle to change its direction from horizontal to vertical.
.O
Figure 1.22.
Rotation of the spin axis
As indicated in Figure 1.22, 0 is the angle of the spin axis counted from the initial configuration, which is horizontal. Assuming that s for sake of simplicity, the following results hold, which are broadly known as the gyroscopic approximation: L = Js
I = cos0i + sin0k
L = J$2i = JK20 (sin0i" + cosO/~) = .71~F = 2el" x /~ * 0 Where .MF = 2gl x F is the torque imparted by the experimenter: .71~cr = 2el x/~ = 2e(cosOi" + sinO/~)x/~
F= Frj +Fzk Identifying component by component the rate of change of the angular
ja2~J momentum with the external torque, it is found that Fx = F z = 0"F,. = + '
"
. As 2s
shown in subsection 1.2.3.3, the force thus defined can be identified with a Coriolis force. Here, 0 is taken as negative, because the rotation considered in Figure 1.22 is ...,,
,..,
from i to k , i.e. in the direction opposite to that of a direct frame. Therefore, the
Mechanical systems and equilibrium of forces
35
experimenter must exert a vertical moment by pulling with the right hand and pushing with the left hand on the wheel axle. The law of action and reaction as applied to the two connected subsystems (the wheel on one side and the experimenter on the other), implies that the experimenter is subjected to a torque equal and opposite to L. Hence, if he stands on a swivel chair, he will turn to the left. Application of such a result is very familiar in cycling. Provided the bicycle runs forwards, s is negative. When the cyclist imparts a horizontal push with the right hand, the bike turns on the left and get tilted towards the inside of the bend. Conversely, by imparting a tilt to the bike, the cyclist can turn without activating the handlebar. However, such a way of driving is not advisable at low speed. Indeed, when the direction of L significantly departs from that of the axle of the wheels, it becomes harder to keep control of the motion!
1.2.4 Inertia forces in an accelerated reference frame It is not always convenient to formulate the equations of motion of a system by using inertial coordinates. Therefore, it is of interest to specify how the acceleration of a masspoint is transformed when noninertial coordinates are used. Here, for sake of simplicity, the problem is restricted to the case of motions of a particle taking place in a plane. The general case of 3D motion is discussed in Chapter 2. Finally, a general formula can be written, going along the same line as that used in subsection 1.1.3.3 to derive the general formula [ 1.4] for the transformation of velocity. Y
_
_. / X
~ (t} ~ ~
~
0
J
o
x
Figure 1.23. Change of Cartesian referenceframes As shown in Figure 1.23, we define an inertial Cartesian frame (~)
(axes
ox, oy and unit vectors t, i ) and an accelerated Cartesian frame ( ~ ) (axes OX,OY and unit vectors f, i ). The vector position of a particle is then given by:
36
Discrete systems ..__> ._...> o M = xi + yj ; OM = XI + YJ
[1.36]
Since (~) can be transformed into ( ~ ) by the product of a translation and a plane rotation, the coordinates are transformed as follows:
L
c
v'
where C = cos 0, S = sin 0
[1.37]
Incidentally, in accordance with the formula [ 1.4] the transformation [ 1.37] may also be expressed as: x = f~ ( t ; X , Y ) ) = Tx (t)+ X c o s ( O ( t ) )  Y sin (O(t)) y = fy ( t ; X , Y ) ) = Tr (t)+ X sin (O(t))+ Y cos (O(t)) Or, by using the notations of formula [ 1.4]"
[q'l= [rl+ [:l ][q] where[q']r=[x
yl;[q]r=[X
Y ] ; [ T ] r = ~ T x Ty 3
[1.38]
aL aL aX
OY
[+1= air
air
Lax or
is~ :]
By differentiating [1.37] with respect to time, the velocity of the particle is expressed as
E~]:,y+O['xlI~c ~IE Is xu +I~
[1.39]
The last term of expression [ 1.39] is the relative velocity in the accelerated frame ( ~ ) , as expressed in the inertial frame ( ~ ) . The relative velocity as expressed in the frame (P,z) is obviously:
SlEs~:Jill: E~]
[1.40]
On the other hand, the sum of the first two terms in [ 1.39] is identified with the transport velocity which has a component of translation and a component of rotation. The last, as expressed in the frame (P2) is in the tangential direction. In vector notation, it takes the same form as relation [ 1.33]:
Mechanical systems and equilibrium of forces
I~, ~']E'~~l[xl,,E~ ~x]0~ ~o,~ wherek=ixj=IxJand
37
[1.41]
R=XI+YJ
Now, by differentiating again the relation [1.39] with respect to time, the acceleration of the particle is expressed as:
E;]:[~;]+o~[:~~][x]+o[~ :~][xl+~o[~ :,~][~]+I~:;[~] [1.42] The last term stands for the relative acceleration. As expressed in ( ~ ) , this component is radial: [1.43] In the penultimate term we recognize a Coriolis acceleration, in accordance with [1.34]. As expressed in ( ~ ) this component becomes:
[1.44] The sum of the remaining terms can be identified with the transport acceleration. It is of interest to express the two components of rotation in ( ~ ) . The first rotation component is identified with the centripetal acceleration: r
_Q2/~
[1.451
The second rotation component, called the Euler acceleration, is tangential: ,O
[:~ 1[~ ~]Ex] [o o][x]oo,~. S
C
=
S
Y
.
0
Y
.
[1.46]
Thus, it is found again that when expressed in terms of variables defined in a rotating frame, the acceleration of a particle may consist of three distinct components which are not second time derivatives of displacement variables. The corresponding inertia forces are the centrifugal force +ms
the Coriolis force 2m,(2/~ x/~ and
the Euler force  m ~ k x/~. Finally, for sake of completeness it is worth to quote the general formula for transforming the acceleration from one frame to another. Time differentiation of [ 1.4] produces the following formula:
38
Discretesystems
~)t 2
c3t
"~ [q]2 j[ttl+ [J 1[/t1
[1.47]
1.2.5 Concludingcomments
Basically, Newton's laws of motion introduce two vector quantities which stand for the fundamental concepts of Newtonian mechanics. In the case of linear displacements (i.e. translations), the primordial quantity is force, and linear momentum measures the action of the force. In the case of angular displacements, the primordial quantity is the moment (or torque), and angular momentum measures the action of the moment. Nevertheless, as already pointed out, Newton's equation of motion keeps the same form, whether a linear or an angular displacement is used. On the other hand, the traditional presentation adopted here, which starts from Newton's three laws, could have been replaced by a more contemporary and axiomatic one, which is based on only two founding principles, that of the existence of inertial frames and that of conservation of momentum. However, the author believes that by doing so little would have been gained for applications in the field of mechanical engineering. The reader interested in such more formal aspects of classical dynamics may be referred to [JOS 02]. On the other hand, in the vectorial approach to mechanics, Newton's law of motion is directly applied to produce the dynamic equations of a given system. As already emphasized, this method is traditionally adopted in elementary mechanics, for reasons of clarity and mathematical simplicity. However, it requires one to write down a detailed balance in which all the forces exerted on the system are formulated explicitly, including the reactions induced by the constraints. As illustrated in the next chapter, even for simple systems such as a double pendulum, the vectorial method becomes intricate and unduly tedious. Fortunately, it is possible to avoid this difficulty by adopting the so called analytical or variational approach, which introduces the work of the forces as the primordial quantity. By doing so, the reactions of constraints are automatically eliminated, provided they do no work. Thus it is this second approach that shall be largely favoured hereafter in this book. However, this does not mean that the vectorial approach has to be rejected. On the contrary, the vectorial method remains an efficient tool, at least in order to understand better the physical meaning of the equations of motion established by the analytical method. Indeed, to most of us the concept of force remains more familiar and is more intuitively understood than the concept of work, or energy. Finally, coming back to the comment concerning linear and angular variables, it will be shown that the analytical approach makes systematic use of generalized quantities, which can be defined independently from their physical nature. Accordingly, linear and angular displacements may be understood as two particular kinds of generalized displacements, while forces and moments are two particular kinds of generalized forces.
Chapter 2
Principle of virtual work and Lagrange's equations
To quote from Lanczos [LAN 70], "by founding the analytical mechanics, Lagrange added nothing fundamentally different to Newton's laws, but provided an immensely powerful weapon to solve any mechanical problem on the basis of pure calculation" and his book "M6canique Analytique", first published in 1788 "may be considered rightly as an extraordinary achievement, which opened an entirely new world to mathematical modelling in mechanics". Analytical mechanics is entirely formulated in terms of generalized coordinates, providing thus mathematical expressions which hold independently of the specific coordinate system chosen. It introduces, as prime concepts, scalar quantities such as kinetic and potential energies, instead of force. Then, a variational principle concerning energy is used to derive the equations of motion in a purely analytical way. Variational principles and methods are very appealing in various branches of theoretical physics, as they bring out the symmetrical nature of the fundamental laws of physics. Following the approach of many textbooks devoted to mechanical engineering, the formalism of Lagrange's equations will be introduced here starting from the principle of virtual work. Nevertheless, it will be restated in the next chapter, starting from the principle of least action, as formulated by Hamilton, which offers richer perspectives both for a deeper understanding of physics and for the development of mathematical methods to solve practical problems.
40
Discretesystems
2.1. Introduction Direct use of Newton's second law for establishing the dynamic equations of constrained multi degrees of freedom systems can often be a tedious and even an unfeasible task. Actually, the constraint reactions arise as supplementary unknowns, which have to be eliminated, because they cannot be considered as additional independent variables. In most instances the elimination process is not a straightforward task, even in the case of 2DOF systems. This is illustrated in the following example, easily extended to an arbitrary number of DOF, to stress with the difficulty of the vectorial treatment of mechanics. EXAMPLE.  Double pendulum
0
t
X
..i,
k
i
,Ml I ,M2
Figure 2.1.
Doublependulum
Consider the system shown in Figure 2.1 where the two particles P~ and P2, with masses M 1 and M s , are constrained to remain in the vertical plane Oxz. Since the connecting strings are assumed to be rigid, we have to deal with a 2DOF system. Its motion is described by using the angular displacements 0~ and ~2. In the balance of forces (or moments), it is necessary to include the tensions T~,T2 as new unknown vectors, which have then to be eliminated to produce the final form of the dynamical equations, expressed in terms of independent variables only. However, the elimination process is much less obvious than in the case of the simple pendulum. On the other hand, it can be noted that T~ and T2 are normal to the path of the particles and hence do not perform any work. Moreover, this would remain true in the case of any kinematically admissible motion, i.e. a motion that is consistent with the kinematical constraints imposed on the system. Therefore, the idea is to establish the equations of motion, starting from a balance of work performed by the forces instead of forces themselves. This is the aim of analytical mechanics of which first notions are introduced in section 2.3, after having reviewed
Principle of virtual work and Lagrange's equations
41
the concepts of work and of mechanical energy and having discussed some general implications concerning the motion of discrete systems.
2.2. Mechanical energy and exchange of it
Mechanical energy is a scalar and additive quantity, which depends upon space, but not upon the coordinate system used to describe the motion. A natural way to introduce this quantity is to perform a balance of the work done by the generalized forces exerted on the material system, between two arbitrarily given times t 1 and t 2 .
2.2.1
Work and generalized forces
2.2.1.1 Work performed by a force ...,.
Let F be a force, whose point of application is changed by an infinitesimal amount dX. It performs the infinitesimal work defined as the scalar product (also known as the inner product)"
d74: = F.dX
[2.1]
NOTE. Notation of the scalar product In relation [2.1], the scalar product is defined in an Euclidean space with a dimension at most equal to three. Accordingly, the usual vectorial notation of Euclidean geometry is used. However, having in mind further extension to Ndimensional and functional vector spaces, the scalar product is also written by using either the matrix or the functional vector notation . For instance:
F.d'X  [F ]T [dX ]  < F, dfi, > Work done between t~ to t2 , is given by the integral:
~47F(tl't2 )= :~(t~) F.dX
[2.2]
Such a result can be immediately extended to the case of any system of N particles, because it suffices to sum up the individual contributions of each particle forming the whole system. Let Fj be the force impressed on the jth particle that is displaced by Xj(t). In Cartesian coordinates, the work produced by the whole system between t 1 and t2 can be written as:
42
Discretesystems
N f2j~t~)_ I[X(t~)] F,.d,,~, = [F]"[dX]
(tl'te)= Z
j=l
gj(tl )
J[X(tl)]
[F] T
=[F~,,F,,,,Fz,....FxN,F,.N,Fm~
[x]
=[x,,v,,z,
[2.3]
.... x,,,r',,,z,,]
2.2.1.2 Generalized displacements and forces Since work is defined as a scalar product, it does not depend upon the coordinate system used to express the force and displacement vectors. For instance, shifting from Cartesian to generalized coordinates [q(t)], we obtain:
WF(t,,t2) =
[F]" [ d X ] : /[X (tI)]
[Q]"
[dq]
./[q(tI)]
[2.4]
where [Q]" = [Q1,Q2.... QNo1; [q]~ = [q,,q2 .... qNo ] [Q] denotes the vector of the generalized forces. When the scalar product [Q]T [dq] is written by using the index notation, the individual contribution of each pair of conjugate (dual) components of force Qk and displacement qk appear explicitly: [Q]T[dq]=Qkdqk;
k : l .... ND
[2.51
Now, using the transformation rules [1.3], we may shift from Cartesian to generalized components of forces as follows:
dWF(t)=[Q]T [dq]=[F]T [ tgf ] [dq]
[2.6]
And so, identification of the force components gives: [Q]T =[F]T
[,gf/aq]=[F]T [J]
[2.7]
where [J]is the Jacobian matrix already introduced in Chapter 1, subsection 1.1.3.1.
Principle of virtual work and Lagrange's equations 2.2.2
43
Work o f inertial f o r c e s a n d kinetic e n e r g y
2.2.2.1 L i n e a r m o t i o n (translation) in an inertial f r a m e o f reference
Let X ( t ) be the displacement of a particle, as defined in a fixed Cartesian frame. The work of the inertia force is:
W ( t 1, t 2 ) =
 m J( .dX = J X (fi)
 m X . X dt = 
m X
[2.8]
t~
This result is now rewritten as: W(t,,t 2 )=As
=s
)s
2)
[2.9]
where s is the kinetic energy of the particle, given by:
w ere Ilxll This scalar quantity is said to be positive definite as it becomes strictly positive as soon as the particle moves. Extension to a system of N particles proceeds simply by adding the individual contributions of each particle:
~e~(t) =  ~ Z m , V 1 N
[2.11]
2
j=l
From the mathematical viewpoint, expression [2.11] is a quadratic form of the particle velocities, which is symmetrical, positive definite. It is written in matrix notation as"
I
],
[M][,e]
[2.12]
[,~] is the velocity vector of the system and [M] is the m a s s matrix. It takes here the particular form of a diagonal matrix, in which the mass coefficients related to the same jth particle (velocities "~i' I?j,2j) are all equal to mj. Since expression [2.12] remains invariant with respect to any change of coordinate system, [M] is actually a secondrank tensor. Moreover, as expression [2.12] is also invariant with respect to transposition, [M ] is symmetrical, a property which holds in any coordinate system.
44
Discretesystems
2.2.2.2 Rigid body rotating in an inertial frame of reference Let us consider a rigid body free to rotate around its centreofmass G, which is assumed to be fixed. This 3DOF system is described in a direct Cartesian frame with origin G. s
denotes the angular velocity vector, defined by"
.0 = & i + & j + n~K = q'j + q', j + ~=~
[2.13]
where ~,7'y,~Fz are the angles of rotation about the coordinate axes, as seen in Figure 2.2.
kz ~ Z~ \ \ \ \
\
7 i
Wx S X
WY
~1111
Figure 2.2. Direct Cartesianframe The linear velocity of a current point of the body is then obtained by using relation [1.33]" V = g 2 x ? . Since d ~ = s is an infinitesimal rotation, s can also be thought as an instantaneous rotation vector. As shown in Appendix 2, the coordinate transformation of a point in the infinitesimal rotation is"
I,
Y(t + dt) = + d ~ Z(t +dt) d~y
1 +d~
1
d~x 11Y(t) 1 _][.Z(t)
[2.14]
NOTE.  Infinitesimal and finite rotations If a finite rotation ~ = ~ j
+ ~,,] +~z/~
is considered, the transformation
F'= ~F x F does not correspond to a rotation anymore. Indeed, it can be easily checked that the Cartesian frame Oxyz is thus transformed into an oblique frame:
Principle of virtual work and Lagrange's equations
I =~xi
=~,
~,.
J =~xj
=


45
2

z
In order to describe finite rotations, one often makes use of the Euler angles, see Appendix 3. The reader is also referred to [GOL 80], [JOS 02] for a detailed discussion of finite rotations. The elementary angular momentum is (cf. formula [A2.7]): dL = pd77(~ x ~ x ? ) =
pd72(r2~2(F.s
[2.15]
where p is the mass per unit volume of the body and dle the elementary volume. Let .._>
x, y, z be the Cartesian coordinates of a current point. The components of dL are found to be:
dL~ = pd72{(y 2 + zZ )~Qx  yXg2y  ZXg2z } dey = pdle {(x 2 +
z 2 )~'2y  x y ~'2x 
Zy X2~}
[2.16]
dL z = pdl~{(x2 + y2 )X2z  yZX2y  XZs } Finally, by integrating [2.16] over the volume (V), the result is the matrix relation:
I[L]=[/l[all
[2.17]
[/] is the matrix of inertia of the body, whose Cartesian components are:
lxx = II~v,O(Y 2 + z2 )dxdydz;
Ixy = lyx =  ~ v pxydxdydz
lyy = ff~V /9( x2 + z2 )d,xdydz;
Ixz = Izx =  ~ v pxzdxdydz
lzz = ~fJv P( x2 + y2 )dxdydz;
lyz = lzy =  ~ V pyzdxdydz
[2.18]
[/] can be reduced to a diagonal form by using the coordinate frame of the principal axes of inertia of the body, which are given by the eigenvectors of [/]. Since the matrix is symmetrical, the eigenvalues and eigenvectors are real and the latter are orthogonal to each other (cf. subsection 2.2.2.5 and Appendix 4). Now, the kinetic energy of an elementary material volume with mass p d t ~ is given by:
=
2
=
2

}
[2.19]
46
Discrete systems Finally, integration over the whole body gives:
1 L..O = l[a]T Ill[K2]
[2.20]
Here again, it is found that kinetic energy is a quadratic and symmetrical form of velocity, which is positive definite. Hence, shifting from a motion of translation to a motion of rotation, the linear velocity vector is replaced by the angular velocity vector [,62] and the mass matrix is replaced by the inertia matrix [I], both having the same basic mathematical properties in that, just like [M], [/] is a secondrank tensor.
2.2.2.3 Change of referenceframe Let us consider a mechanical system set both in translation and rotation. Since kinetic energy is an additive scalar quantity, it can be expressed as a single quadratic positive definite form. However, it is often desirable to separate the translational and the rotational parts of motion. This is made possible by considering two frames of reference, one of them being an inertial frame and the other being tied to the centreofmass G of the system. In most instances, G is in an accelerated motion with respect to the inertial frame. Then one is led to calculate the kinetic energy, referring to the inertial frame, but still using coordinates (or displacements) defined in the accelerated frame. More generally, let ce~ be the kinetic energy of a mechanical system in an inertial frame, as given by equation [2.12] in terms of the Cartesian velocity vector [)~l" Let [ql be the displacement vector as defined in an accelerated frame. Relation [1.4] gives:
[X(t)]=[f([q])];
,gf ] +[j ([ql)][ql IX]= 57
Substitution of this expression in equation [2.12] yields:
8.=
89 [_~t]]"[M] [[Jl[q]+ [o~f]] 1[[ ]" [Jl[q]+ ~ = i [q]'[J +LOtJ
[
[M] [Jl[q]+
This form separates kinetic energy into three distinct components, namely:
1. Transportenergy
E2"=i[aOtf jl" [MlIa~f
[2.21]
Clearly, this is the only nonvanishing term when the system is at rest in the accelerated frame.
Principle of virtual work and Lagrange's equations 2.
47
Relative energy
oec(r) = 1 7101" [M'l[q] ; where [ M ' ] : [J]" [M ][J]
[2.22]
[M'] is the relative mass matrix, which is symmetrical and positive definite like [M]. It has to be noted that [J] and thus [M'] can depend upon the relative displacement vector [q], in contrast to [ M]. Thus, relative energy is still a quadratic symmetrical and positive form of the relative velocity vector. However, it can also depend upon [q]. 3.
Mutual energy (or crossenergy)
d""=7[L a, j Mutual energy Er(m) gathers the last two terms present in the total kinetic energy oe,~. However, as each of them is related to the other by a transposition, they become identified to the same scalar. Hence, ,f(m) is rewritten as"
[2.23] Mutual energy couples linearly the relative and the transport velocity vectors. As a final comment, it can also be noted that generally, [J] and [M][J] are not necessarily symmetrical. EXAMPLE. Particle rotating about a moving point
Z

/P
P
.>,o" x
Figure 2.3. Particle rotating about a moving point
48
Discretesystems Let Xoi + YoJ + Zok be the position vector of a point O, as defined in an inertial
Cartesian frame with unit vectors i', j,/~. Consider the Cartesian frame relative to O, . . . . . . . . .
.,,
...
..
with unit vectors /, J, K parallel to i, j, k . In the last reference frame, the position of the particle P, which rotates round O, can be described by the spherical coordinates R,8,~p, see Figure 2.3. However, direct use of such a coordinate system is not straightforward. It is more convenient to calculate the kinetic energy using the Cartesian coordinates defined in the frame relative to O. The result is then expressed in terms of the independent variables O,r The Cartesian coordinates of P in the inertial frame are: X = X0(t)+~; Y = Y0(t) +r/; Z = Z0(t)+~'; ~: = Rsinr
r/= RsintpsinO; ~"= Rcosr
The kinetic energy of P in the inertial flame is found to be: m
z2 9
m
m (~,02 + yo2 + Z,o2)+ m (~,0~ + l;,of/ + Z0r
.....> __..> OPz =O/]1+ P~Pz =/,1 (sin 0~i"+ cos01/~)+ Lz (sin Ozi"+ cosOz/~) Whence: I72 = (L~t~ cos01 + LeOz cos0~ )i"  (L~t~1sin01 + L20z sin02 )fc And so, the kinetic and potential energies of Pz are found to be: ~
s =~M2{L2~Oz~+ ~0~ + 2L~L~t~t~2 cos(01 02) } s : gM2 (~ cos02 + L~cos01 ) Now, from these results Lagrange's equations are obtained:
( M, + M 2) L~O, + Mz L~L~{ O~ cos(O,  0 2) + 0:2 sin(O,  0 2)} + g(M~ + M2)L ~sinO, = 0
66
Discretesystems
M~E~eb~ + M2L, L20' c o s ( O ,  0 2 )  M,L,L~b,(b,  b ~ ) s i n ( O ,  0 : ) +
gM2L2 sin 02 = 0 It is worth noticing that each step of the above calculation is straightforward. On the other hand, when the problem is restricted to the case of small motions about the state of static equilibrium 0~1,0~2 = 0 , it may be anticipated that it is sufficient to expand the above equations to the first order with respect to 0 l,O2 . This results in the linear matrix system:
L
M2L,L2
M2E22
b2
0
gM2L2 O;
It is noted that the mass and stiffness matrices are symmetrical and that the mass matrix is full, coupling thus the two degrees of freedom of the system. The underlying theoretical background and physical interpretation of such results will be further described in Chapter 3. Moreover, Chapter 6 will present a mathematical formalism, which is especially well suited to analyse the small motions governed by this kind of linear systems. As we shall see, they correspond to small vibrations about the position of a state of equilibrium 0,1,0,2 = 0, which is statically stable. To conclude on this example, external moments P'J?I(e) (t),P372(e) (t) are applied to each pendulum. This loading may be considered as derived from the external potential:
(o,.o2;,)=
(,))
The corresponding Lagrange equations are:
M2LIL2
M2~
02
0
gMzL2 02
PY~(~)(t)J
The last three chapters of this volume are devoted to the description of the mathematical tools and methods which are especially well suited to analyse the linear motions of such forced systems, in a systematic and straightforward way. In this respect, attention of the reader is drawn to the fact that such a mathematical background constitutes the backbone of the methodologies used by the engineer to perform the calculations required to analyse the design of a very large variety of important structures, even when calculation must be conducted beyond the linear domain of response.
Principle of virtual work and Lagrange's equations
67
2.3.5 Lagrangian and Newtonian formulations Clearly, a formal equivalence must exist between the variational (Lagrange) and the vectorial (Newton) formulations of the equilibrium equations of any given mechanical system. The equivalence is readily established as follows. To begin with, the first inertial term of Lagrange's equations can be interpreted as the rate of the time variation of a generalized momentum, defined as follows: P"  c9q,,
[2.75]
The resultant of the other terms is identified with a generalized force Q,,, that can involve three distinct components, arising from: 1.
A potential s (q,,).
2.
The partial derivative of kinetic energy s (qn, q~) with respect to q,.
3.
The remaining forces Qn, which are related neither to s nor to s Accordingly, the general form of Newton's equations are found to be: [ p ]  [ Q ] : [0]
[2.76]
Since the vector equation [2.76] is expressed in terms of generalized quantities, it generalizes those already obtained in terms of physical displacements in Chapter 1, section 1.2. As a final note, let us consider a system which is described solely by an internal Lagrangian. If the latter is independent from a variable qn, the corresponding generalized momentum is obviously an invariant of motion.
2.3.6 Application to a building resting on elastic foundations Let us consider a building modelled as a homogeneous cuboid, with edge lengths
a, b, H. F = xi + yj + zk is the position vector of a current point referenced in the Cartesian frame, the origin of which is at the centreofmass G and the axes are parallel to the edges of the cuboid. The comers of the base B~, B2, B3, B4 are resting on elastic foundations, see Figure 2.6. Our purpose is restricted here to analysing the response of the building to a static external load applied laterally, assuming small displacements and neglecting gravity.
68
Discrete systems
~z
/
IT
!
i
~ ~
~ ~ ~
v
H/~.
~ ~
~ ~
,
Y
l~176
F " 9
.." 
! !
H
i
~
.~149162
i
X
~176176176176176149
&
B
Figure 2.6. A cuboid resting on elastic foundations
2.3.6.1 Generalized displacements
G
i
H/2 '
a/2
C
:Y
B
W
B
Figure 2.7. Displacement of the base of the rigid cuboid First, we adopt the six variables of physical displacement, namely the translations X~,Yt,Z ~ of G about the position of static equilibrium in the unloaded configuration, and the rotations gx,~'y,~'~ about the axes of the fixed frame Gxyz. The external load is a force Foj applied to a point P0(0, +a / 2, z 0). Foundations are modelled as a set of linear springs located at the comers of the base. The springs acting along the axial direction Gz have a stiffness coefficient denoted K. Those acting along the transverse directions Gx, Gy have a stiffness coefficient denoted k.
Principle of virtual work and Lagrange's equations
69
Since the external load is a force acting along the direction Gy, the response can be restricted by the conditions Z~ = 0;X~ = 0 and ~r = q/z = 0. Hence, we consider only the two relevant variables Y~ and ~x. However, it is soon recognized that the elastic energy of foundations is more conveniently expressed in terms of the Cartesian displacements XBi,YBi,ZBi, i = 1 tO4 of the corners than in terms of Y~ and ~x. The displacement of the base of the building is shown schematically in Figure 2.7., C being the middle point of segment B~B2 (or B3B4 ), to the first order in q/x the displacements are found to be:
H Z a
ZB1 = ZB4 =  :
a
~brx , ZB2 = ZB3  : ~bCx
which are then rewritten in terms of Y2 = ~xH /2 as:
where a = a / H denotes the reciprocal of the slenderness ratio of the solid. The small displacement field of the Gzaxis is Y (z) = Y~ z~x.
2.3.6.2 Potential energy and stiffness matrix In so far as gravity is neglected, the potential energy of the system is entirely given by the elastic energy in the foundations. Summing the contributions of each spring, we obtain: 4 = 2kYl2 + 2(k +otzK)Y~ +4kY1Y2 We recognize immediately a quadratic form, conveniently written as:
And so, the stiffness matrix is found to be:
[k k ] i: 1]
[K]=4 k
k+o~2K = 4 k
where the stiffness ratio 7' = K / k factor.
l+o,2y
is introduced as a physically pertinent scaling
70
Discretesystems By deriving s
with respect to Y~,Y2, the generalized stiffness forces are
produced according to the canonical form IQ(S)I=[K][Y]. In the present case, they are:
Q(S2)~ 4k 1
l+a~r r~
2.3.6.3 Generalized external loading and solution of the forced problem The generalized forces relative to the external load are determined by calculating its work (or in an equivalent way its external potential) in terms of the generalized displacements Y~,Y~.This gives:
w = E~ ') = For 0 V [ a ] =~ P is a local minimum.
2 ~ 8~'> I F ] < 0 V [ct] =~ P is a local maximum.
[3.101
3 ~  otherwise P is not a local extremum. Now, let 3,n,n = 1,2 ..... N , be the eigenvalues of the matrix
E
According
to the results already quoted in Chapter 2 and proved in Appendix 4 paragraph A4.4, it can be stated that: 1~ If 3,, > 0 V n ~ P is a local minimum.
2 ~ If 2. < 0 V n =, P is a local maximum.
[3.11]
3 ~ If some 2, < O, other >_0 ~ P is a saddle point. NOTE. Stationary and extremum value When performing the preceding analysis, it has been tacitly assumed that P lies inside the space of configuration. This condition is necessary for the virtual displacements to be reversible. Indeed, the function could otherwise have an extremum value without being stationary. As a practical example, let us consider a ball rolling on a hilly ground. The gravity potential has local minimum at the bottom of the valleys. However, if the ball is restrained along the slope by a rigid wall at such a boundary, the potential will be minimum without being stationary and the virtual displacements will be irreversible, since starting from the wall, it is allowed to go up but not down the slope. On the other hand, [F~>] is undetermined at such a boundary. EXAMPLE.  Stationary points of z = sinxcosy The system 3 z / 3 x = cos x cos y = 0. ~)z / ~)y =  sin x sin y = 0 has two sets of roots x = / r / 2 + k/r, y = n/r and x = n/r, y = / r / 2 + k/r. Here, k, n are either positive or negative integers. The nature of these stationary points is analyzed by using the eigenvalues equation: det
[sinxcosy 2
cosxsin y]
cos x sin y  sin x cos y  2
: (sin xcos y + 2)2
_
(cos x sin y
which yields: 3a =  s i n ( x + y) ; ~ =  s i n ( x  y ) . At the stationary points, we obtain: For the first set of roots, x = ~ / 2 + k#, y = n #
~/~
= ( I), ~ = ( I)
)2
:0
Hamilton's principle and Lagrange's equations
85
If k and n have the same even/odd parity, A~ = 2 2 =  1, the point is thus a maximum. If parity of k, n differs ~ = 2 2 = + 1, the point is thus a minimum. ,
For the second set of roots, x = k/r, y = n"/ 2 + n~" =r
~ = (  1)(k+''+l) ; ,,~ = (_ 1)(k,,)
Now it is found that ~ =  1 ; 2 2 = + 1 if k, n are of the same even/odd parity, and that ~ = 1;2 2 =  1 if parity of k, n differs. Hence, any root of the second set corresponds to a saddle point.
3.2.2
Static stabili~
3.2.2.1 Criterion for stability The study of the first and second variations of a function may be rightly considered as the cornerstone of the analysis of static stability of mechanical systems. In statics, the Lagrangian reduces to a potential energy s with the sign changed. Since, the first derivatives of Z; = _~ep stand precisely for the forces acting in the system, a point P of the configuration space where 6(t)Is corresponds to a state of static equilibrium. It may be worth noticing that the equilibrium is relative to the unloaded or to the loaded system, depending on the absence or the presence of an external component in the potential. The next step of the analysis is to determine whether a given equilibrium state is stable, or not. The static stability of a mechanical system can be defined according to the following rules, which can be easily understood from the physical viewpoint: 1.
P is stable provided that a small and actual (instead of virtual) variation [d q] = e [a] of the configuration about P induces restoring forces that tend to bring the system back to P, whatever the direction defined by [ct] may be. This case corresponds clearly to a local minimum, such that 8 (2) Is (P)~ > 0.
,
P is unstable if at least in one direction repelling forces are induced that tend to remove further the system from P. This is the case either when P is a local maximum 6 (2) Es (P)I < 0 , o r when it is a saddle point.
~
Finally, the equilibrium point P is said to be indifferent if no force is induced, whichever the direction [ct] may be. This is the case when
8(,EE, (p)]ov
86
Discrete systems
It has to be stressed that the local analysis becomes insufficient to settle the actual behaviour of unstable or indifferent systems, since it deals with infinitesimal variations only. Actually, as soon as a real system is repelled sufficiently far from static equilibrium, its behaviour becomes governed by nonlinear forces. From all what precedes, it is possible to state the following mathematical criterion concerning the static stability of a mechanical system:
A necessary and sufficient condition for a point of static equilibrium P to be stable is that all the eigenvalues of the matrix 8 (~'E~p (P)] are positive.
3.2.2.2 Static stability of a pair of upsidedown and coupled pendulums M
M
L K
Figure 3.3. Coupled and upsidedown pendulums Let us consider the system shown schematically in Figure 3.3. Each pendulum is maintained in the upward direction by a spring acting in rotation and the two pendulums are coupled to each other by a third spring. To simplify the algebra of the problem, the three springs are assumed to have the same stiffness coefficient K. The potential is written as:
s = ~ + 4 = MgL(c~
+ c~
) +K/'02 2 ~ 1 +022 + ( e l  e 2 ) 2)
Stability of the equilibrium configuration O, = O, = 0 is discussed in terms of the sign of the eigenvalues of the matrix:
Hamilton's principle and Lagrange's equations
t9201
a Oltg02
= K [(21) (2tr
,90,,90,_
=3~
with t r
87
MgL K
a ~0,_ ,~=o==o
Thus, the system is stable only if ~c< 1. The similar problem referring to the non upsidedown pendulums is deduced from the former one simply by changing the sign of Ir As expected, this new configuration is always stable. It is also easy to verify that the matrix, whose elements are the second derivatives of the potential, can be identified with the stiffness matrix of the system, once linearized about the static equilibrium configuration of reference. Therefore, the stability of a configuration of static equilibrium is guaranteed, provided the related stiffness matrix is positive definite. Now, when the stiffness matrix depends upon a free parameter, such as Ir the threshold of static instability occurs when the matrix becomes singular, passing from positive to negative. This kind of instability is termed divergence. However, in the language of engineering, it is broadly known as buckling. The linear analysis just performed provides the critical value of buckling, (• c = 1 in the present example), but gives no information concerning the direction in which the unstable pair of pendulums will fall down (toward the left or toward the right side?). This point will be further discussed in the next example. Finally, going back briefly to the case of the building resting on elastic foundations already studied in Chapter 2 section 2.4, it is found that when the gravity effect is taken into account, buckling occurs if the stiffness of the foundations are insufficient, or the slenderness ratio of the building is too large. Indeed, referring to the results of subsection 2.4.2, it is not difficult to show that the stiffness matrix including the gravity effect can be written as:
4k
E' ' ] 1 1 + ct2y 1r
where x= pab__.__..gg k
So, the building collapses as soon
as o~'2y 
K"~ 0
3.2.2.3 Buckling of a system of two articulated rigid bars We consider the system shown schematically in Figure 3.4. The bars (AC) and (CB) are of the same length L and assumed to be rigid. The possible motions are restricted to the plane of the figure. In addition, the end A can slide freely along the Oxaxis whereas B is fixed (position B0). However, the bar ( CB ) can rotate freely
88
Discrete systems
about B0 . Such a constraint is termed articulation, or pinned support. Finally, the two bars are also articulated together at point C. The angle of the base of the isosceles triangle (AC B o ) is denoted a: At A, a force F = Foi directed from A to B0 is applied. (AoBoCo) is the non deformed triangle, taken as the configuration of reference. Our purpose is to analyze the static stability of the system in relation to the base angle a 0 .
Y
C
Co
........ Ao
Bo
Ao
"A
Bo
"
Figure 3.4. Two articulated bars put in compression It is found convenient to divide the study into the following steps. 1.
Kinematics o f the system
Figure 3.4 shows a deformed configuration of the system. The displacement of point A is found to be: AoA = AoB o + BoA = 2L(cos a' 0  c o s a ' ) = X A
Then the coordinates of point C are: x c = XA + L cos a = L(2 cos ct 0  cos a') Yc = L sin ct
Now, the displacement of C is given by: x ~ = x~  x~0 = L ( c o s a0  c o s a )
Yc = Yc  Yco = L (sin o :  sin a0 ) 2.
The Lagrangian o f the system
In this problem, the Lagrangian reduces to the work of the external force: ~5 = ~
= FoX a = 2 L Fo (cos a'0  c o s c t ) =  2 L F 0 c o s a
Hamilton's principle and Lagrange's equations
89
where the superfluous constant term was finally omitted. The external potential is s =  W e . The generalized force conjugated to a is given by:
Q~, 
3a
 2LF o sin cr
Q,~ is twice the moment of F 0 about C, with sign changed, as implied by shifting from the righthand side to the lefthand side in the equilibrium equation.
3.
Positions of static equilibrium and stability
Stable configuration a, = (2k + l)n"
Bo
C
A
Vo X
Unstable configuration ap =2kn"
Fo
A o
Co 9
Bo ...
.......
x 9
A
Figure 3.5. Configurations of static equilibrium The positions of static equilibrium are deduced from the roots of the equation:
ds
o~s 3 cr
c9 a'
a'p=2kzc a, = (2k + 1)zc
Two distinct sets of roots are found, which are denoted ap, Ctl according to whether either the integer multiple of ~r is even, or not. The stability of the states of equilibrium is provided by:
02 s l d a '2
~P ,~l
= I2 LF~c~
0
where Fo > 0
The solutions ap = 2ktr are thus found to be unstable. They correspond to a unique configuration in which the two bars are put in a line, with the bar extremities
90
Discrete systems
ordered as A, C,B o . The equilibrium solutions cr1 = (2k + 1)n: are stable. Again, the bars are put in a line, but their extremities are now ordered as B0 , C, A. Such results are immediately understood when looking at Figure 3.5.
4.
Stabilizing effect of a stiffened articulation Here, the articulation located at point C is provided with a linear spring acting on
the angle ( A C B o) = ft. Its coefficient of angular stiffness is denoted K. The restoring moment induced by the spring is .714"=  K ( f l  f l o ) =  2 K
(orat o), in such a way
that it is zero in the non deformed configuration, see Figure 3.6. The elastic energy is s = 2 K (oror 0
)' and the new Lagrangian becomes:
= (2 K(::0 )~ + 2LF0cosa) TI y
i i
Co
Ao
Bo
Figure 3.6. Angular stiffness of the articulation C
5.
Static equilibrium of the stiffened system: symmetrical case o~o = 0 The equilibrium states cre are given by:
3 Z~I = 2 K Cte _ L Fo sin ot = 0 =r'
I
ct~ _ LFo _ ~ sin ote 2 K
where 9 is the reduced load factor, which characterizes the relative magnitude of the destabilizing over the stabilizing moments. The second derivative of the potential is written in the following reduced form:
_zl a2 L, = 1 
la, 2 a~
9 cos a, = 1  a, cotga~
Hamilton's principle and Lagrange's equations
91
It is realized that an equilibrium state a e ~ 0 can arise only if 9 > 1. If ~ < 1, the only possible state of equilibrium is a 1 = 0. Its stability is easily checked since 32~~p __ l_(I) > O.
When ~ > 1, two additional states of equilibrium occur, which are symmetrical to each other with respect to the O x  a x i s : a 2 =a~ ~ 0;~z 3 =  a e . Thus it suffices to restrict the analysis within the interval [0,~r]. Owing to its definition, a 2 has to verify the equilibrium equation: a 2~sina 2 =0
r
!
!
!
/ lO
9 = o.1 2 150
~
x 100
Figure 3.7.
(degree) ~ 50
Potential
0
i 50
of the symmetrical
I00
system ( ~
150
= O)
Now, the stability of equilibrium can be checked by using the criterion: o~2 Cp I = 1  a~cotg a'~ > 0 3a2 a~
The above inequality is valid since in the subinterval 0 < cr~ L is discarded here and will be briefly discussed later; see example 2 of this subsection. At any time, the variations of the constraint conditions are: ~9(SR,)6q. = 0
; g = 1 , 2 .... , L
; n = 1 . 2 ..... N
[4.10]
Accordingly, the action of the constrained system is postulated to be of the following form:
:i
Zx,( ,ld,
~/'= ~/+
[4.11]
L e=l
It must be pointed out that Lagrange multipliers At(t)
are now as yet
undetermined timefunctions, instead of undetermined constants, as is the case of constrained functions. Indeed, the constraint conditions have to hold at any time (if not otherwise specified) during the motion of the mechanical system. Owing to equations [4.10], it is immediately recognized that the variation of the constrained system is identical to that of the free system: ,C dt+
6[J/' 1:6
/],, (~R,)dt
iI
/],t 6~(9~, )]dt
: 6[.//1+
II
: 6[~/1
I!
where the convention of implicit summation on the repeated index g is used and will be used also hereafter to simplify notation. [.//] is then expressed in terms of the variations 8 q. :
B,C o~Z; o~.,.0. +
d ~
6[.~/]=
6q. dt
I
Regrouping now in 8 [J/'] all the factors pertaining to a given 6 q. , we have:
=
i
d
O•
aq. +
0~5
+/],t "
cg(~Rt q,: ) 1 6 q"
Once more, L multipliers 2 t (t) are specified in a suitable way to cancel the factors of L coordinate variations 8 q , , no matter which. Since, the constrained
118
Discretesystems
system has ( N  L) DOF, the factors of the ( N  L ) remaining 8 q,, also vanish. Therefore, the following N Lagrange equations governing the constrained system are obtained:
dqS and these are completed by the L constraint conditions. Here again, the method consists in transforming a system of ( N  L) equations involving ( N  L) independent variables into another equivalent system of (N § L) equations involving (N + L) interrelated variables, namely the displacements q, (t) and the undetermined Lagrange multipliers A,t (t). Restated in another fashion, the Lagrangian of the constrained system is written in the following form: [,C'= s + 2, (9t,);
e= 1,2 ..... L I
[4.12]
A priori, the multipliers 2 t should not be considered as additional degrees of freedom, since the calculation of the variation of .,4' is performed with regard to the q, only, which are considered as being the only variables of the problem. The undetermined multipliers d,, are calculated afterwards by using the constraint conditions. However, as in the case of constrained functions, wemay perform again the variations as well with regard to the 2~ as with regard to the q,. Of course, the coefficients of fi,;t, reduce to the constraint conditions. Whichever is the viewpoint adopted, one is led to the same system of equations:
dt
 q"~ 0 
o~q.
5R, (qk .... ;t) = 0
[4.13]
n = l , 2 ..... N ; e = l , 2 ..... L The constrained Lagrangian Z;' can be written as the extended form:
where: ~'~ = A,t(9~,);
e=1,2 ..... L
The quantity 74~c , which arises as a direct consequence of the presence of constraint conditions, clearly stands for some kind of work. As it cannot be related to the work done by any force already acting in the free system, the only possibility left is to interpret /4~c as the work done by the constraint reactions. Furthermore, in so far as
Constrained systems
119
the conditions (9tt) are referring to holonomic constraints, W~ is a function of the generalized displacements and of time only. Hence, a quite natural idea is to let it correspond to a potential of constraints, which has to be added to the potential of the free system: 4~ =  2~ (9~t)
[4.14]
But now, as a natural consequence of the formalism involved in the method of the undetermined Lagrange multipliers, it becomes necessary to reconsider our first point of view concerning holonomic constraints, according to which they restrict the number of DOF of the system, as described in Chapter 1. In contrast, according to the present point of view it is now stated that:
The holonomic constraint conditions do not restrict the number of degrees of freedom of the unconstrained system, but they modify its potential. Going a little bit further, by differentiating the potential of constraints with regard to the displacement variables q,, generalized constraint reactions ,~, are produced. ~ is interpreted as being the resultant of the generalized reactions induced by all the individual constraint conditions, which are prescribed to the nth degree of freedom of the free system. Indeed, differentiation yields: ~,,, : _
t9s
:/it c9(~)
[4.15]
It may be argued that, from the physical point of view, the present interpretation of the holonomic constraints is preferable to the mathematical definition given in Chapter 1, subsection 1.1.4.1. Actually, it is easily recognized that no realizable constraint can be made perfectly rigid. Consider for instance the mathematical condition qi  q j 0. It can be conveniently modelled physically using an elastic =
spring connecting the displacements q~ and q j. The potential of constraint is identified with that of the spring:
cec~: ~ 1K ~ ( q , q~) 2 where K~ designates the stiffness coefficient of the spring. The two constraint reactions acting on the constrained degrees of freedom are immediately found to be:
fl~ =Kcc (qi  qj) and ~ = + K~ (q;  qj) Since any part of the system must be in equilibrium, the force balance cannot depend upon the value of Kcc. In particular, the reactions must be independent
120
Discrete systems
of Kcc. On the other hand, the larger is Kcc the smaller is [qi  q j I In other words, the elastic connection tends to an ideal rigid constraint as K , tends to infinity. To conclude, it may also be noted that the signs attributed to 2 and to the constraint conditions in the potential of constraint are arbitrary. This has no effect on the generalized and physical reactions, as illustrated in the next example. EXAMPLE 1.  Plane rotation o f a particle a b o u t a f i x e d p o i n t
[ i
f ...... t
/ /
Ro
2"
Figure 4.1. Circular motion of a particle
Let us consider a particle which is left at first to move freely in the plane ( J ' ) . Polar coordinates r, 0 are used as generalized displacements. The Lagrangian of the free system reduces to: s
1
2
1
2 +  M r~ O 2 2
Then, the particle is constrained to a circular path about O. Accordingly, the constrained Lagrangian is written as: s 2
M~2 + _ 1M 2
r2 EJ2 + , ; / , ( r  R )
Lagrange's equations are found to be: M J:  M r 0 2  /]. = O Mr20 rR=O
+ 2Mri, O =0
Constrained systems
121
Substituting now the condition r = R into the first equation, Lagrange's multiplier /~ ( t )   M
R 6j2is found. A priori, we could wonder about the physical nature of
the constraint work thus obtained, because in the present form 2 looks more like an inertia force (cf. Chapter 1 subsection 1.2.4) than as a force deriving from a potential. However, it may also be argued that the circular motion of the particle is uniform, since no external torque is acting on it. Indeed, the second equation provides the solution 0 = ~2, where $2 is constant. Hence, the reaction may still be interpreted as deriving from a potential of constraint, given here by:
s : +Mg22R(rR)~
~
. . OCcr . .  / l =  M R J2 2 Or
As calculated according to the above procedure, ~c is the generalized reaction exerted by the fixed point on the free system. Formally, ~c and r are a pair of conjugated quantities. In physical terms, as r is a linear displacement, ~ is a force. It is immediately identified with the centripetal force exerted physically by the fixed point to balance the centrifugal force caused by the acceleration of the particle. To conclude this elementary exercise, the following comments may be made:
1.
Signs of Lagrange multipliers and constraint conditions
It can be easily checked with the present example that the sign convention attributed to the multipliers or to the constraint conditions has no influence on the calculated reactions. Adopting 74~c =  2 ( r  R ) , the first Lagrange equation becomes:
Mi;MrO
2 +2=0
The two remaining equations are left unchanged, hence:
Mr~QZ +2=O The constraint reaction is now given by:

~)r 2.
+~ =2MRS2 2 ~r
Nature of the constraint work and general relativity
The short discussion above on whether the constraint work is kinetic or potential in nature in the present example, points towards an equivalence between inertial and potential forces, indicating the presence of a link between accelerated motion and
122
Discrete systems
gravity. Indeed, even in everyday life, spinning motion is often used to mock up gravity. On the other hand, recognition of such a natural link was the starting "happiest thought" of Einstein in building up the theory of general relativity. As clearly discussed in layman's terms in [GRE 99], based on force measurements only, any accelerated motion can be described as a gravitational motion, provided suitable gravitational (or potential) forces are defined. As a consequence, "all observers, regardless of their state of motion, may proclaim they are stationary and the rest of the world is moving, so long as they include a suitable gravitational field in the description of their own surroundings. As a corollary, from the mechanical viewpoint all the vantage points are on equal footing". 3.
Physical modelling of the constraint condition
Turning back to the concrete problem of realizing the constraint condition involved in the present system, the particle will be connected to the fixed point O by using a slightly deformable string with an equivalent stiffness coefficient K~ made as large as possible, but remaining finite in any case. To obtain radial equilibrium, the centrifugal force M $'22r has to be balanced by the stiffness force Kc~ ( r  R ) , where r is the length of the extended string. Accordingly, )],Sr is identified with the virtual variation of the work of the spring and the following results are easily obtained: 2r
= ~74~c = Kc~ ( r  R ) S r =:~ /], = Kcc ( r  R) =  M s ~r
Provided that Kcc is sufficiently large, the relation r = R holds in practice. EXAMPLE 2 .  Static and hyperstatic equilibrium of a rigid rod Let us consider a rigid and uniform rod loaded by its own weight, which is resting on rigid knife edge supports, located at x = /_~ and at x = x I, as shown in Figure 4.2. The exercise consists in determining the support reactions as a function of x 1, by using Lagrange multipliers. Considering first the unsupported rod, the analysis is restricted to small vertical displacements: Z ( x ) = Zo + xO . where x is the abscissa of a current point of the rod axis and where use is made of the two following independent variables: 1.
Z 0 is the vertical displacement of the centreofmass O, located at the centre of the middle crosssection of the rod.
2.
0 is the rotation angle about O, in the plane of the figure.
Constrained systems
123
z i i !
LI2 Aa
+LI2
... '.~ ~'i x ................. . ~> "el _ ..... i ~;;7  ............... ,
:~,~,:~.~.~........ ,~,~,~
Figure 4.2. Rigid rod resting on knife edge supports
In statics, the Lagrangian of the free rod subjected to gravity reduces to the term of potential energy of gravity:
~;=p.R',,I_~(Zo +xo~=
~o
a result which could be easily anticipated. Now, the rigid supports maintaining the rod in static equilibrium are equivalent to the constraint conditions:
z (x,)= zo + Ox,
= z , = o;
Z (  L / 2 ) = Z o   L O = Z 2 =0 2
Defining two undetermined Lagrange multipliers A1,A2, the Lagrangian of the constrained system is written as"
(L)
z'= a4gZo+ A (Zo + x,O)+4 Zoso
from which the following Lagrange equations are derived:
Mg(A +4)=o; Ax, +&k=o 2
whence: ~ = Mg~L ; 4 = 2xiMg L + 2x~
L + 2x~
Then, the generalized reactions exerted by the supports, which refer to the conjugate variables Z0 and 0, are given by"
124
Discretesystems
&

=Mg
~)~c ao
L
=o
Using these generalized quantities, it is possible to return to the physical reactions which are applied to the rod by each support. Indeed, the work of reactions is unchanged in a transformation of coordinates (or displacements) and its variation as well. Hence we have:
Writing Z 0 in terms of the "physical" displacements at the supports Z l and Z 2, we obtain: L ~ Z l { 2xlt~Z 2
_ _ ~t ._M. g
~9 = "~z,  L + 2x I
,
= 2xlMg
L + 2x 1
The following remarks are appropriate to check the relevance of the physical reactions thus obtained: 1.
Clearly, the condition of global equilibrium of the rod implies that the resultant of the physical reactions must balance the weight of the rod, whereas the resultant moment about O must be zero. Obviously, these conditions could have been used to determine the support reactions, without having to resort to the method of Lagrange multipliers.
2.
As expected a priori, if x~ = U2, the two reactions are equal, in agreement with the symmetry of the problem.
3.
If x~ = 0, ~ = M g and ~ = 0. Thus, the total weight is applied to the support located just below. This result could also be anticipated, since the centreofmass of the rod is located at midspan.
4.
If x~ < 0, the sign of .~ is reversed and in the case of unilateral supports, the rod becomes unstable according to a rocking mode.
Constrained systems
125
.~ = MgL 6
.~f = MgL ".,%
L/2
\i i !
I ..... i2/ .................... XI
~(,)_(L,)Mg _ MgL 6
6
Figure 4.3. Reaction torque tending to the distribution of a Dirac dipole
If x I    )  L I 2 , the two supports merge together and ~ , ,~
tend towards
infinity with opposite signs. To interpret the physical meaning of such a result, we proceed as follows. Consider the support located at x I =  L I 2 + e / 2 . We have:
.~ (e)=  ( L  e ) M g s
;..~ = Mg L E
As shown schematically in Figure 4.3, the resulting moment of the reactions L e MgL Mge about the middle point x,,c = 2 +4 is .MR ( e ) 2 4 . Hence, it is noted that if e tends to zero, the reactive torque tends to ~(R = ( M g L ) / 2 , i.e. the finite value which allows one to balance exactly the external moment of the weight resultant; this result will be revisited later in terms of singular distributions, cf. Volume 2. Thus, provided the supports are bilateral, the rod can be maintained in stable equilibrium even if the lever arm is arbitrarily small, in principle at least. Such a support condition, known as a clamp, combines a zero displacement and a zero rotation condition. Finally, let us consider the cases where the rod is maintained by more than two supports. As easily understood, it becomes impossible to determine the support reactions if keeping in the framework of the mechanics of rigid bodies. Actually, with such support conditions, the number of unknown quantities entering in the Lagrange's equations of the constrained system becomes larger than the number of DOF of the free system. As a consequence, the system of equations to be solved is underdetermined. To illustrate this important point, let us consider the case of three supports located at x l,x2,x 3 , see Figure 4.4.
126
Discrete systems
g
x,
.....*'."9.~.~.,.~.;~.;j~.~:,ii~i ,.:~," \
~~~~
x,
\
~ i i i~i%~i,~,
Figure 4.4. Rigid rod with hyperstatic support conditions The constrained Lagrangian is written as: A~'
[email protected] ( Z 0 +OXl)[~, 2 ( Z 0 +OX2)'[~, 3 ( Z 0 + O X 3 )
Lagrange's equations involving Lagrange multipliers are: Mg (21 + 22 + 23) = 0
/~lXl + ~2X2 + ~3X3 = O The first equation refers to the force balance. Once more, it teaches us that weight and resultant of support reactions have to be exactly balanced. The second equation refers to the moment balance. Obviously, it becomes mathematically impossible to determine the individual reactions induced by each support, as the number of equations is now less than the number of variables. More generally, any mechanical system constrained by a number of holonomic conditions which is larger than the number of degrees of freedom of the unconstrained system is known as a hyperstatic system.
As a final comment, it is worth emphasizing that according to the definition given just above, the concept of hyperstatics is merely a byproduct of modelling solids as perfectly rigid bodies. As already stressed, such a model is adopted because of its mathematical convenience. However, one has to be aware that hyperstatics does not fit within physical reality, so far as the number of DOF is concerned. Indeed, as real materials are always deformable, the degrees of freeMom a material body cannot be enumerated. Moreover, since deformations occur at the supports, tiny as they may be, the reactions at each support can be unambiguously determined, whatever the number of supports is. Clearly, this kind of calculation has to be performed in the framework of the mechanics of continuous (deformable) solids, which is the object of Volume 2.
Constrained systems
127
4.3. Prescribed motions and transformation of reference frames
In many instances, material systems are excited by prescribing the motions of some DOF. A typical example is provided by the case of a building subjected to a seismic shake. It is assumed that the basement accompanies the motion of the ground. Here, we will describe two distinct methods to deal with such dynamical problems, each one having its own interest. The first method consists of modelling the prescribed motions as rheonomic constraint conditions. This allows one to obtain the equations of motion in terms of generalized displacements referring to an inertial frame. As already shown, Lagrange multipliers are providing the constraint reactions. Here, they identify with the forces which induce the prescribed motions, when impressed to the unconstrained system. The second method consists in defining an accelerated reference frame tied to the prescribed motion. Then, the equations of motion are expressed in terms of the relative displacements, which refer to the accelerated frame. The generalized forces which are equivalent to the prescribed accelerations are now derived from the kinetic energy of the system, as determined in the inertial frame, but expressed in terms of the relative variables.
4.3.1. Prescribed displacements treated as rheonomic constraints
A known displacement
Dj (t)
impressed to a variable
qj
can be formulated as
the rheonomic constraint condition:
qj  Di(t)
=0
[4.16]
Since rheonomic constraints belong to the class of holonomic constraints, the problem can be studied by using the method of Lagrange multipliers. Considering thus a system with a free Lagrangian of the type s = Cc Cp, we define the potential of constraint
s
= 2j (qi  D j (t)). The Lagrangian of the constrained
system is:
Z,'= s Ep + 3.j (qj Dj(t))
[4.17]
Lagrange's equations can be written as:
d t~ce~l_ dE~ + ;
,) q,
d 0 4 ] o~cec ~ ~qj ) cg q, qj  Dj (t) = 0
I
d q,
=O;i~j
O qj  ~ =0
[4.18]
128
Discretesystems
The only new point with regard to the scleronomic case is that the nature of the generalized reactions may differ, depending whether the kinetic energy is an explicit function of q j, or not. This is illustrated in the two following examples. EXAMPLE 1 .  Seismic excitation o f a massspring system Let us consider a massspring system in which the spring connects the mass M to a point O. The displacement of O, as defined in a given inertial flame, is a known function D(t) of time. Referring to this frame, X designates the displacement of M along the Oxaxis, counted from the equilibrium position at rest, and Y is the relative displacement as defined in the flame tied to O, see Figure 4.5. K is the stiffness coefficient of the spring and L is the length of the unloaded spring, which is specified here only to help in visualizing X and Y in Figure 4.5. Let X 0 be the displacement of O in the unconstrained system. The constrained Lagrangian is written as: ,G' = 1 ( M X z _ K ( X  X o )z) + A,(X ~ _ D ( t ) ) 2
Lagrange's equations are: ~ 1 7 6
K ( X  X o) + M X = 0
~KX
+MJr
K(xx0) X o  D(t) = 0
K
M
L
Figure 4.5. Seismic excitation of a massspring system
The constraint reaction is found to be /l = K ( X  D(t))= KY(t). Hence the work of the constraint is found to be a potential and the constraint reaction is a stiffness force.
Constrained systems
129
EXAMPLE 2.  Rotating massspring system
Z
,"'77~. X
Figure 4.6. Rotating massspring system
Consider a massspring system, which rotates at angular speed ,62(t) about the Ozaxis. The mass slides without friction along the direction of the spring force, taken as the Oxaxis. L is the length of the unloaded spring, see Figure 4.6. The motion of the free system is described by using the rotation angle 0 of Ox about Oz and the elongation X of the spring. The Lagrangian of the constrained system is written as:
x5'= 114 (02 (X + L)2+ ,~2)_ KX2 + A, (/~ s 2
2
Lagrange's equations are found to be" M O ( X + L) 2 + 2 M O X (X + L ) + / l = 0 M J~ + K X  M O 2 ( X + L ) = O
0
O(t) = 0
Hence, in contrast to all the former examples, it is found here that the Lagrange's multiplier is governed by the differential equation:
A   {M a ( x
+ L) + 2 M
}(x + L)
Physically A, is the moment about Oz of the resultant of two distinct inertia forces, each one acting in the plane of rotation and perpendicularly to Ox, namely the Euler f o r c e  M ,(2 (X + L), and the Coriolis force 2 M ,Q ,Y. It can also be noted that 2, has the dimension of an angular momentum, which can be quite naturally interpreted as the action of the resulting moment of the Euler and Coriolis forces.
130
Discretesystems Turning now to the second equation, it can be rewritten as: MX +(KM~22)x
=M,(22L
This form shows a negative stiffness term, which corresponds to the centrifugal force, as already discussed in example 1 of subsection 4.2.3. Clearly, this component is oriented in such a way as to move the mass away from the centre of rotation, whereas the spring restoring force is tending to bring it back. Thus the resulting stiffness coefficient (K  M ~2) can be either positive or negative, depending on the value of s
in relation to the critical spin velocity s c =~]K/M. Finally,
assuming here that s is constant, the term reported on the righthand side of the equation is the permanent centrifugal force induced by the rotation of the mass, when located at the nondeformed state of the spring X = 0. As this component is merely a constant, it may be preferable to eliminate it by transforming the displacement variable: ~:=XX
0 ; withX 0 =
M .Q2 L K  ff22 M
X 0 is the spring elongation which allows one to balance the permanent centrifugal force. Accordingly, the force equation is rewritten as: M~ + (K  . Q : M) ~ = 0 As written in terms of X, the equation describes the axial motion of the mass referenced to an unstressed initial state (s = 0), whereas the equation written in terms of $ describes the same motion as referenced to a permanent state s r 0, which is initially stressed, or "prestressed", by the permanent centrifugal force. Finally, as in example 1 of subsection 4.2.3, the centrifugal force may again be interpreted as deriving from a potential.
4.3.2. Prescribed motions and transformations of reference frame It is also worthwhile analysing the problem of prescribed motions by adopting another point of view to that of the rheonomic constraints. It is also possible, and even far more usual, to consider two distinct reference frames, namely: 1.
The inertial frame (A"), in which the prescribed motion is defined.
2.
The accelerated frame (A') which accompanies the prescribed motion. Let us consider a system describeA by the displacement vector [q], as referred to
(A'). In (A"), the system is described by the displacement vector [q ']. Our purpose
Constrained systems
131
is to formulate Lagrange's equations in terms of [q] and its derivatives. Once more, it is stressed that though such equations are written in terms of relative variables defined in the accelerated frame, they still govern the dynamic equilibrium in the inertial frame (~"). The key is to start from the Lagrangian referring to the inertial frame and written in terms of the inertial variables [q'],[q']. Then, a suitable transformation of coordinates of the type [q'] ~ [q] is carried out to obtain the same Lagrangian, but written in terms of the relative variables [q],[c~]. It is immediately recognized that only the inertial terms have to be transformed because the others can be written directly as a function of the relative variables [q]. Starting from the general form of Lagrange's equations: d(c9 s /
~163 a , , = o
[4.19]
Q,, is the nth component of the resultant of the forces other than inertia forces. Kinetic energy is the sum of three distinct components C~(') , s
and s
which, in the general case, are dependent upon the variables qn, as shown in Chapter 2 subsection 2.2.2.3. It is also recalled that transport energy is independent from c~n, whereas mutual and relative energies are a linear and a quadratic function of c~n, respectively. The inertial part of the equation [4.19] is then written as:
O~~"(r) )
O~ (Gt~t)
+ cgqn ff~qn
+4
(m)
+ s
(r)
)
[4.201
Developing this expression and inverting the sign of it, the following expression of the generalized inertia force is obtained:
n
 ~
o3'qn + ~ 7 ~
+ c~j cgqj dc~n
t~ (G_C,(I)_~..G.~(m)_~ Gs
[4.21]
dqn
It is of interest to gather the various terms of this expression in such a way as to sort out the three following components, already identified several times in the context of rotating motions:
132
1.
Discretesystems
Relative component "  dq, ~
o~0,
[4.221
This component can be separated from the others by assuming transformation laws fq which do not depend explicitly upon time, hence describing a change of coordinates and not a change of reference frame.
2.
Transport component o,,, _
""
a'
dq,
m)
[4.23]
cgtdit,
This component can be separated from the others by assuming that the system is at rest in the accelerated frame q. = 0,'q n.
3.
Mutual component
""
o~q,
[4.24]
tg qj o~it,
This component gathers the remaining terms. It can be rewritten in a skewsymmetrical form and it can immediately be checked that s
q, ds
)"
Whence: O~2 E(m) [Q(m)lr = [q]r [(7]; where Go = t~ qi t~ t]j
d2 s [4.25]
tg qj tg iti
G j~ =  G o are known as the gyroscopic coefficients. [G] is the gyroscopic matrix which is obviously skewsymmetricaL It has to be pointed out that gyroscopic forces are conservative since they do no work. Thus, we have:
j
, 12 lI
[ql" [Qtm,~ dt =
f t2 l!
[q] r [G]r [t~] dt = 0
[4.261
Constrained systems
133
EXAMPLE 1 .  Seismic excitation of a massspring system
K
l/I;!
M
F(t) o L + Y(t) ///~!
Figure 4.7. Seismic excitation of a massspring system
The example already analysed in subsection 4.3.1 is revisited here by using the method of the reference frame transformation. Now the point O is subjected to a prescribed acceleration F ( t ) in the Ox direction, see Figure 4.7. Writing the equation of motion in terms of the variables Y and }; which refer to the accelerated frame, we obtain: K Y + M ~; =  M I ' ( t )
Thus, it is found that prescribing the acceleration F (t) to the relative frame is equivalent to forcing the oscillator by the external inertia force  M F (t). EXAMPLE 2.  Rotating massspring systems We consider once more the example 2 of subsection 4.3.1 where angular velocity = X2 is assumed to be constant. The Cartesian coordinates referred to the inertial frame are found to be: X ' = (L + X ) cosX2t ; Y'= (L + X) sin~2t
Kinetic energy in the inertial frame can be calculated directly or by using the general transformation formulas [2.21, 2.22, 2.23]. Whichever method is used, the two following terms can be identified: 9
Transport energy: s
= 1 M ~2 (L + X) 2 2
9
Relative energy: s
9
Mutual energy is zero.
= _1 M ,~ 2 2
The Lagrangian is: ~5 
2
(,~2 + ~22 X (2 L + X)) 
_r X 2 2
134
Discretesystems
Of course, the axial equation of motion is the same as that already found. EXAMPLE 3.  Pendulum attached to a spring ""' 9
X
~0 ! i
..,
,~:
M~ u
Z~ Figure 4.8. Pendulum attached to a spring
Let us consider a pendulum in which mass is connected to the fixed point through a spring, as shown in Figure 4.8. L is the length of the unstrained spring. Restricting the study to motion taking place in a vertical plane, we have to deal with a 2DOF system, which may be described by the angular variable 0 , defined in the inertial frame Ox, Oz and by the translation variable X, defined in the rotating frame, which goes along with the massspring system. Incidentally, it should be noted that in this problem we are making use of an accelerated frame, of which motion is not prescribed since 0 is unknown (cf. Chapter 1 subsection 1.1.3.3.). Potential and kinetic energies are found to be:
E, = ~ x ~ + Mg(L + x ) (1 cos0~ 2
4 7
~t~ § (c + x
0~
4o
(m~
with: ~e(r) _ M'~2,. s 2
= ~ ; ~(/_B)2 ecM 2
(,n) = MX (L + 2 X )02 2
where transport and mutual energies are defined in the same way as though 0 (t) were a prescribed function. The Lagrangian can be written as:
Constrained systems
135
= M_M_(fr2 + 02(L + X ) z ) _ K X Z _ Mg(L + X)(1cosO ) 2
2
Thus the following equations of motion are obtained: MX  M (L + X)6J 2 + KX  MgcosO =  M g
M (L + X)O + 2MOfC + mgsinO = 0 Once more, the permanent term in the right handside of the force equation could be eliminated. On the other hand, in the moment equation, the cross term is recognized as a Coriolis force, which couples nonlinearly the angular and the translation velocities of M. EXAMPLE 4.  Mass tied to a wheel through springs
"Y,,
I y' K XI
0
f2 Figure 4.9. Mass
tied to a rotating wheel through springs
As shown in Figure 4.9, the circular wheel rotates about its axle at constant angular speed s The pointmass M is connected to the wheel by linear springs K acting radially along two orthogonal directions defining a Cartesian frame in rotation with the wheel, of which the unit vectors are i, j , k . When the system is at rest, M coincides with the centre O of the circle and elongation of the springs vanishes. X, Y designate the coordinates of M referring to the rotating frame. We consider also the inertial frame Ox'y" and the X' Y' coordinates. (X' Y' ) and (X, Y) are related to each other through the rotation matrix:
[x:l Ec ] Ex]
where C = cos (,(2 t) ; S = sin (g2 t)
136
Discretesystems
Incidentally, it can be checked that the transformation matrix may be identified with the Jacobian matrix of the coordinate transformation. There is no difficulty in calculating the kinetic energy of M using the general expressions [2.21, 2.22, 2.23]. Indeed after some elementary algebra, it is found that:
M
=~
,,[o ~
4m)a[X rlM 0 However, it is still simpler to perform a direct calculation by determining the velocity of M in the inertial frame, in terms of the relative coordinates. This can be done either by using the vectorial relation: I7=$7r + ~ x ~ =
Xi" + l;'] + , . Q k x ( X  [
+ Yj)=(X ~2Y)[ +(Y + ~2X)j
or by making use of the coordinate transformation written in the matrix form, to obtain by differentiation:
In the inertial frame, the kinetic energy of M is thus found to be: M
2
The potential energy is: K
2
Finally, the two following dynamic equations are obtained: M ](  2 M ,.(2 }" + ( K  M s 2 ) X = 0
MI7 + 2 M s
+(KMS'22)Y =0
As expected from [4.25], the gyroscopic forces couple the two equations in a skewsymmetrical way. On the other hand, the rotation induces a centrifugal force on each DOF. The dynamical behaviour of such rotating systems is rather intriguing and will be discussed later, in Chapter 6 subsection 6.4.4 of this volume and again in Volume 4. As a final remark it is also of interest to realize that if the equations of motion are expressed in terms of the Cartesian displacements X', Y' defined in the O x ' y ' inertial frame, they reduce to those of two uncoupled harmonic oscillators:
Constrained systems
137
KX'+MX'=O KY'+M~;'=O Hence this system may serve as nice example to illustrate that motion may be modified drastically according to the reference frame adopted to describe it.
This Page Intentionally Left Blank
Chapter 5
Autonomous oscillators
In the previous four chapters of this book we have been concerned with the mathematical modelling of mechanical systems which results in a discrete set of timedifferential equations. The object of the following five chapters will be to describe the mathematical procedures available to solve such analytical models and to develop the generic features of the dynamical responses, which are of major interest for the mechanical engineer. Because of the theoretical and practical importance of this field, broadly referred to as vibration theory, the literature on the subject is particularly abundant. Amongst the engineering oriented textbooks devoted mainly to linear vibrations, let us quote in particular [DEN 56], [ROC 71 ], [CLO 75], [MEI 67], [ME170], [BIS 60], [GIB 88], [DEN 89], [WEA 90], [GER 97]. Most material systems, initially at rest in a given state of equilibrium, are likely to vibrate as soon as they depart from it. Vibrations are said to be free whenever they are triggered solely by delivering initially some amount of energy to at least one degree of freedom of the system. Such free vibrations reveal themselves as extremely useful for investigating the response properties of the material system to any kinds of excitations, as will be illustrated in specific examples in the following chapters of this book. This chapter is devoted to the analysis of the free vibrations of SDOF systems, starting with the socalled linear or harmonic oscillator which is the simplest vibrating system. It serves as a convenient introduction to several basic concepts, among them the concept of natural modes of vibration, which will be further extended in the next chapter to the case of MDOF systems. Furthermore, because of their relative simplicity, SDOF systems give us a good opportunity to make a first incursion into the nonlinear domain of dynamics. Indeed, even a limited study is enough to illustrate significant differences between the dynamical behaviour of linear and nonlinear systems.
140
Discrete systems
$.1. Linear oscillators 5.1.1 Mechanical oscillators A mechanical system is said to vibrate, or oscillate, when it is swinging back and forth about a permanent state of equilibrium, static or not. This state, which will be used as a reference to describe the vibration, is assumed to be stable. Indeed, vibration occurs as the result of the conflicting action of the inertial forces, which cause the resulting motion to overshoot the permanent state of equilibrium, and that of the restoring stiffness forces, which are tending to bring it back. Mechanical oscillators are the simplest systems of this kind. They correspond to devices which can be reduced to a masspoint and a spring. The back and forth motion takes place along a geometrical line which is prescribed a priori. Hence, according to the definition given in Chapter 1, a mechanical oscillator is a SDOF system. Clearly, such a concept arises as a mathematical idealization of a few real, but particularly simple, devices. Two archetypes, already introduced in the preceding chapters, are the massspring system and the simple pendulum. The basic difference between them is in the nature of the potential energy, and thus of the stiffness force, involved in the system. In masspringlike systems, restoring forces are induced by the elastic deformation of a body, while in pendulumlike systems, restoring forces arise as the consequence of a coupling between the motion and a permanent force field, the weight for instance. More generally, the permanent force field is said to prestress the system and the pendulum may be considered as a prestressed, or initially stressed
oscillator. The linear, or harmonic, oscillator is the linearized version of such systems. Its motion is governed by a linear timedifferential equation of second order and constant coefficients, which is written in the following canonical form:
IKq + cr + Mi~=a(e)(t) [
[5.1]
It is recalled that K is the stiffness coefficient, C is the viscous damping coefficient and M is the mass, or inertial, coefficient of the oscillator. In this chapter, study is restricted to the case of autonomous oscillators. Therefore, there is no external excitation, except that motion is triggered by imparting initially some amount of mechanical energy to the system: K (0)) 2 + M s ( 0 ) = ~(q ~ ( q (0)) 2
[5.2]
The motions of an autonomous system are said to be free in order to draw a clear distinction with the forced motions, which are induced by an external fluctuating (timevarying) excitation. Motions of the autonomous harmonic oscillator are thus governed by the homogeneous equation:
Autonomous oscillators
141
[5.3]
Kq + Co + Mi~ = O
This equation is provided with initial conditions of displacement and velocity: q0 = q(0); q0 = q ( 0 )
[5.4]
Referring now to the notions developed in the first three chapters of this volume, it can be stated that solutions are oscillatory in nature, so far as the coefficients K, C, M of equation [5.4] comply with the following conditions:
I.
Stiffness coefficient K > 0 The stiffness force Kq is derived from a potential ~e (0), which is written first
in terms of a generalized coordinate denoted ~. However, when the aim is to study the small motions about a position q0 of static equilibrium, one is naturally led to expand the potential in Taylor series with the point of expansion q0"
4%1 )
* (~176 dO'
dg]
+...
[5.5]
Now, by definition (cf. Chapter 3, subsection 3.2.2) a state of static and stable equilibrium is such that: =0"
d~q Oo
>0
'
[5.6]
dO20o
As pointed out just above, the motion can remain confined to an arbitrary small neighbourhood of the static equilibrium q0 only if it is stable. Within such an infinitesimal domain, potential energy can be approximated by the quadratic and positive definite form:
1
~ep(q) = "2 KqZ where q = 0  qo and K=
2.
d2~l 
>0
dO' IOo
[5.7]
Mass coefficient M > 0
Similar considerations hold concerning inertial forces. Kinetic energy can be written directly as a quadratic positive definite form: 1 dZE,r ] s (q) = ~ Mq 2 where M = >0
dit2 0o
[5.8]
142 3.
Discretesystems Viscous damping coefficient
Icl
1.
~176II 0.05
q#)(1,0
:
o.o4l/t ::
q(O) = 0 m
Cl ~I.05
,.,,,..
m l = lOO k g
9
/7 I OH:
0.03
0,02
0.0 !
O0
0,05
0.1
0,15
0.2
0.25
Figure 5.3. Free vibration of an overdamped oscillator (overcritical damping) Incidentally, the timehistory related to critical damping is obtained by using the trial solution: q(t) = ~(t)e At
[5.41]
Autonomous oscillators
151
Substitution of the function [5.41 ] into equation [5.31], leads one to: [5.42] Solution is produced by cancelling out every term of the above expression: 3,=~1;
cl=a+bt [5.43]
q(') : {q0 + (q0 re.q0 )t} eoh'
Hence, it is found that the free motion of a critically damped oscillator is of the same nonoscillating nature as that of the overcritically damped oscillator. Turning now to the case of negative damping, the analytical results established above are still valid, provided the sign of ~'l is changed. Clearly, the oscillator is dynamically unstable since its mechanical energy is steadily increasing with time, in an exponential way. The motion ceases to be an oscillation as soon as ~'l <  1 . Considering for instance the critical negative damping ~'c =  1 , the motion is still given by the law [5.43], provided the sign of 3, =~oI has been changed: q(') = {qo + (qo + ~
[5.44]
)t} e+ah,
It is of interest to note that the most important feature arising from the presence of nonconservative forces in a mechanical system is the loss of symmetry with regards to time which is thus introduced. In any real free motion, time elapses irreversibly, starting from a given origin, defined as the time at which energy is imparted to the system. In rather picturesque words, it may be said that time is now provided with an arrow. In Volumes 2 and 4, we will revisit the consequences of a lack of symmetry concerning the properties of the mathematical operators of mechanical systems. 5.1.3.2 Phase portrait 15
"'
,
~(0 !o
r .. o.ar f
..
o
!o[
~f;~ ,'.2 o.;~ o2, O.;~ ;
Figure
o.k o'., ~
025
0~
"i125
O.2 .4115
0 1
2 .... 0
5.4. Phase portraits of a subcritical oscillator
0 05
01
0 15
8.25
152
Discretesystems
,
,

1
9
50
0 50 5O
.
.
.
.
.
.
.
.
.
0 50 5O 0 50
t
,
i
Figure 5.5. Phase portrait of an overcritical linear oscillator
Unlike the case of conservative oscillators, the phase portrait of a damped oscillator cannot be obtained directly, but only as a byproduct of the timehistories. In particular, the phase trajectories become dependent on the initial conditions, in contrast to the undamped case. Nevertheless, for a given oscillator, all the phase trajectories are featured the same way. In the subcritical range < l, they are shaped as spirals which are either converging to, or diverging from, the origin (i.e. the state of static equilibrium), depending on the sign of ~'~, see Figure 5.4. Indeext, if damping is positive, the trajectories run from the representative point of initial conditions towards the representative point of static equilibrium. When damping is negative, they run in the reverse direction, even if the representative point of the initial conditions is "almost" coincident with the origin. Then, the origin of the phase plane is said to be either an attracting or a repulsing point, according to whether damping is positive or negative. Figure 5.5 displays the phase portrait of the overcritical oscillator. Of course, in this domain also, the origin of the phase plane is either an attracting or a repulsing point, according to the sign of damping coefficient. 5.1.3.3 Modal analysis
We proceed along the same steps as in the conservative case. The characteristic equation now produces two distinct complex roots: toc)
r.o,(+~/1
~'?
i~'t)
[5.451
As in the conservative case, it is sufficient to retain only the root of which the real part is positive, defining thus the natural pulsation of the vibratory component of
Autonomous oscillators
153
motion, also called natural pseudopulsation of the oscillation. The imaginary part of the complex pulsation [5.45] characterizes either the exponential decay, or increase in the magnitude of the oscillation, (depending on the sign of g l): ~o~c) = ~o. ( ~ 1  ~ ' :  i~',) Im(w~ c, ) a) = Re (a)~c));
~',
[5.461
~/(Re (w~c)))5 + (Im (a)~c)))2
A more detailed discussion of complex modes is postponed to Volumes 3 and 4 of this book, where a few mechanisms of exchange of energy between a fluid and a vibrating structure are described. 5.1.4 Static instability (divergence or buckling)
As could be expected, if the stiffness coefficient becomes negative, the oscillator becomes unstable. Indeed, the roots of the characteristic equation are now: [5.47] As a consequence, the masspoint is brought away from the rest position in a monotonical way, whatever the rate of dissipation due to damping may be. Damping only controls the speed of removal, see Figure 5.6. This may explain why buckling instability, which is static in nature, is also termed divergence.
10 5
10 *
10 ~
10'
0.2
0.4
0.6
0.8
1
1.2
Figure 5.6. Timehistories of a linear buckled oscillator (negative stiffness)
154
Discretesystems
5.2. Nonlinear oscillators
Though the present book is essentially devoted to the study of linear mechanical systems, it is nevertheless worthwhile to make a few incursions into the nonlinear domain, so far as only limited theoretical developments and calculation procedures are required. Indeed, description of a few major features of the dynamical behaviour of nonlinear systems is sufficient to clearly mark the profound differences which exist between linear and nonlinear systems. This also gives us a good opportunity to emphasise that even if one has to deal with a nonlinear system, it is still advisable to start by analysing first its behaviour in the linear domain, before embarking on the analysis in the nonlinear domain.
5.2.1
Conservative oscillators
The analysis of the dynamical response of autonomous conservative SDOF systems is straightforward, even in the nonlinear domain. As already pointed out in subsection 5.1.2.2, the problem can be solved analytically by using the first integral of energy. As in the linear case, the phase portrait is given by the levellines of constant mechanical energy ~m (q,q)= ~" As a general rule, according to whether they are closed or open, such lines characterize either periodic or non periodic motions. Moreover, the phase trajectories have no multiple points and do not cross each other. Consequently, under fixed initial conditions, there corresponds only one possible motion. As an exception worthy of note, a few specific trajectories, termed separatrices, may have multiple points. It has to be stressed now that occurrence of multiple points does not lead to any uncertainty concerning the motion of the system, since they can be reached only after an infinite amount of time. Thus, even if the separatrices are closed paths they do not stand for periodic motions. On the other hand, let us recall that the timehistories of motion, periodic or not, can be obtained implicitly in the form of the definite integral:
t=
g
q(,) [5.481
~(4_~ep(q) ) o)
In most instances, potential energy least one extremum value exists. Let
s (q) does not vary monotonically, thus at
qe be is a position of stable static equilibrium.
By definition, qe is such that the following conditions hold:
I dEvI
dEp =0; dq qe
dqE q,
>0
[5.49]
Autonomous oscillators
155
Thus if 4 lies within a certain neighbourhood of 4 (qe), equation 4 (q) = 4 has two roots denoted qmm and q~x, which bracket qe. In contrast with the linear case, the corresponding interval of energy is finite instead of infinitesimal. Within such an interval, motion is necessarily periodic in nature, since the particle is caught inside a sink of potential. The period is given by the definite integral: r'q~
@
rqmin
[5.50]
4 ~  E p (q)
Since ~ = s (qm~), the kernel of the integral has at least one singular point. As a consequence, before carrying out the numerical (or analytical) integration, it is suitable to make a variable transformation to remove the singularity. In contrast with the linear case, the period is a function of the magnitude of vibration. In addition it has to be noted that the integral [5.50] is not always convergent. This occurs in particular, along the separatrices. EXAMPLE 1.  Masspoint tied to a tensioned string M
z~o) / Lo= fl ~= Z/~0) Figure 5.7. Masspoint
tied to a tensioned string
The system studied here is schematized in Figure 5.7. The equivalent stiffness coefficient of the string is denoted K and its mass is assumed to be negligible in comparison with the mass M of the particle, which is located at midspan of the string. Z designates its transverse displacement in the plane of the Figure. 2L0 is the length of the unstretched string. L ( Z ) is the actual halflength in the deformed state. It is assumed that L (0)> L0 in such a way that the string is prestressed in tension: TO: K ( L ( 0 )  ~ ) :
KL0 ( f l  1 )
where fl : L ( O ) / L o : 1 + e o > 1
e 0 is the string deformation associated with the initial stress.
156
Discretesystems Making use of the dimensionless displacement ~: = Z / L(0), we have: L(~:) = L(O)41 + ~:2 fl~41 + ~:2 =
The nonlinearity of the problem is thus geometrical in nature, arising as a direct consequence of Pythagoras' theorem. The linear approximation consists in neglecting any variation of the string length with ~:. Elastic potential energy may be written as: ,s (~:;fl)= K { ( L ( ~ )  / ~ ) ~ (L(0)/.t))z }= K/~ {(fl~r + ~ 2  I ) a  ( f l  l ) 2 } where the constant is adjusted in such a way that s (0; fl)= 0.
9 10 le 6
reduced potential
,e:
1. 1.25
0 2
1.5
1
0.5
0
0.5
1
1.5
2
Figure 5.8. Elastic potential of the pretensioned string This potential is plotted in Figure 5.8 for three values of the stretching coefficient ,6 > 1. Provided ~: is still sufficiently small, the potential can be expanded as a Taylor's series truncated to the fourth order:
4 where r/0 = K ~
Autonomous oscillators
157
Hence, in the absence of permanent tension fl = 1, the elastic potential reduces to the quartic form:
Thus, it appears clearly that oscillations can be nearly linear provided the string is sufficiently tensioned and vibration amplitude is small enough, in such a way that the quadratic term of the potential dominates largely the terms of higher order. The quartic term is obviously responsible for the nonlinear behaviour of the oscillator. It dominates the quadratic term, as soon as vibration amplitude is larger than 2 4 e o , a value which can be much less than one. Let us consider for instance a steel wire of section area S = l m m 2 and length L0 = l m . It can be shown that the equivalent stiffness of the wire in tension is K = ES I ~ = 2105 Nm 1 ,where E = 2.1011 Pa is the Young's modulus of steel. The wire being tensioned at l k N , it is found that e o = 510 3 and 2 ~ 0 __0.14.
NOTE.  Elastic potential and work o f the tensioning (axial)force Instead of calculating the strain potential of the wire, it is also relevant to calculate the work done by the elastic forces in a transverse displacement Z. The tensioning force is: T ( ~ ) =  K ( L ( ~ : )  ~ ) =  K ~ (f141 + ~2 _ 1) The work resulting from the infinitesimal displacement dZ = f l ~ d ~ is: d W = 2 flLoT (~ )sin Od~
where sin 8 = x/1 + ~2
Hence the work done to change from configuration ~ = 0 to configuration ~: = ~ is found to be:
W(~:~)=  2 K ~
f12
~/1~d~j + ~:2
~ d ~  fl
= r/~ {fl2~:2  2fl
1 + ~2
1)}
0
The potential, obtained by changing the sign of the work, is identical to the elastic potential, as expected a priori. However, the interest in performing the present calculation is not only to check the self consistency of the mathematical formalism, but also to clarify the origin of the quadratic term present in the potential. Indeed,
158
Discretesystems
even in the approximation L ( ~ ) : L(0), the tension T(O)is doing a certain amount of work when M is displaced by Z. This work is found to be: d~ffL = 2flLoT (0)sin Od~
where sin 0 _0 = ~:
( ~ : , ) :  2 K ~ f l ( f l  1 ) f r ~:d~ ~ s
: 7?ofl(fl1)~ :2
J0
Thus, it is found that the potential is the sum of two quite distinct terms, namely: A prestress term, induced by the permanent tension of the wire. It reduces to a quadratic form, characterizing thus the linearized system. ,
An elastic term, induced by the change in length of the wire associated with the change of configuration. Approximated here as a quartic term, it governs the nonlinear behaviour of the oscillator. 0.4
,"I ..... ,io: s
. . . . p.  t.oJ .
0.3
1
,I, i
0.2
0.1 0
.
.
.
.
.
.
.
.
.
0.1 0.2
i
0.3
I !
0.4
I
i
~.~ _0., _0'.~ ~'.~ _,i,
I
0
,i,
i
,~
t
,~
,14 ,'.~
Figure 5.9. Phase portrait of the masspoint on a tensioned string
As shown in Figure 5.8, the resulting potential s has a single extremum, which is a minimum located at ~ = 0. The phase portrait is made up of closed lines and all the possible motions are periodic in nature, see Figure 5.9. Referring to the Rayleigh quotient [5.30], it can be anticipated that the natural frequency of the oscillator increases with vibration amplitude ~:~, since Ee (~:l) is increasing faster than a parabola when departing from ~: = 0. The Lagrangian of the system is written as:
Autonomous oscillators
~4
1 i~ 2
159
where I = M~
Accordingly, the equation of motion, broadly known as Duffing's equation, includes a cubic stiffness term:
I ~ + rlofl {2 ( fl 1) ~ + ~3 }= O The stiffness of the linearized oscillator is provided by stretching the string initially, hence it vanishes for the critical value tic = 1. The corresponding natural frequency is:
1 ~2fl(fl1)K M
f L = "~~
The nonlinear frequency fsL is obtained by using relation [5.48], which takes here the form: 1
T (~:,.)= ~ flK Jo ~/(1 y2 ){fl_ 1}+0.25~2 (1 + yZ)
where ~x,, is the magnitude of the oscillatory displacement and y = ~ / ~:m"
reducedfre~lUency.
...........
0.01
.
.
.
.
.
.
!~
~............
0.02 0.03 0.04 ~05
0.06 0.07 0.08 0,09
0.1
Figure 5.10. Nonlinear resonancefrequency of Duffing's oscillator
160
Discretesystems Then, the change of variable u = ~]1 y is used to avoid singularity of the kernel
at y = 1. Accordingly, the integral is finally written as: I
T ( ~ : ) = 4 2/2/2~
I
Jo
11.,+
du
(l.(,.'I'I}
In Figure 5.10 the reduced frequency f = f~./f~, is plotted versus ~:m for several values of the tensioning parameter ft. The required integrals have been calculated numerically by making use of the software MATLAB. It can be verified that the increase of the natural frequency with vibration amplitude becomes barely noticeable, at least in the scale of the figure, as soon as e o = f l  1 becomes larger than 5%. This because if fl is sufficiently large, the prestress force largely prevails on the elastic force. At the opposite, if e0 is less than 0.05%, the frequency becomes very sensitive to the vibration amplitude. Incidentally, such results indicate that when playing a string instrument, it is advisable to provide the strings with a fairly high tension. Piano strings are made of high strength steel wire tensioned up to around 10 9 Pa
( l k N / m m 2 )leading to a typical value of e 0 =0.5%. Finally, in the absence
of initial stress (e 0 = 0), the static equilibrium position ~e = 0 is indifferent, from the linear viewpoint; the system is, however, stabilized by the geometrical nonlinearity. EXAMPLE 2 .  Articulated bars, prestressed in compression M
p.
~= z / t(o)
~o) / L o fl
Figure 5.11. Masspoint linked to two rigid bars preset in compression The system is schematized in Figure 5.11. The two rigid bars are linked together by an articulation whose mass M is much larger than their own mass. On the other
Autonomous oscillators
161
hand, each bar is connected to a fixed point through a linear spring of stiffness coefficient K, acting in the axial direction of the bars. In the unstretched state, the total length of a bar including its connected spring is ~ . When the bars are set in coaxial configuration, the springs are compressed and the length becomes L < ~ . The problem is clearly the same as the preceding one, except that now the stretching parameter fl is less than one. As it could be expected, this is the cause of drastic changes in the dynamical behaviour of the system.

,
,[ '
1
9
,
'1
Potential
Ii
(aimensiontess)
J 0.8) 9
~l
!
~

' 0.4~176
oA i 0.2 i
0 10
8
6
4
2
0
2
4
6
(dimensionless)
t
0.8
,o,
/
i
0.6
0.4
C
B
0.2
0 10
1,
Figure 5.12. Potential of the compressed bars
8
10
162
Discretesystems
The potential is displayed in Figure 5.12, for a few values of ft. The presence of three extrema, hereafter denoted (A),(B),(C), is clearly conspicuous. (A) refers to the static equilibrium position ~A0, which is unstable. (B) and (C) refer to the symmetric positions of static equilibrium, which are stable: = ~ ' ,
~ c =  ~
Figure 5.13 is the phase portrait of the system. It comprises two distinct families of closed trajectories, of which the physical meaning is rather obvious. Indeed, if the imparted energy 4 is less than the potential barrier cep(0), the particle remains trapped in one of the potential sinks which are centered at (B) and at (C). Hence, the phase trajectories are made of a pair of closed orbits, each one surrounding (B), or (C). The specific orbit selected by the particle depends upon the initial conditions of motion.
~,
,
';
.
~
i
.
,
~ 1.5[ ~ .,o
"
;
i
o
;
1o
Figure 5.13. Phase portrait of the compressed bars On the other hand, if ~ is larger than cep(0), the pair of closed paths merge together to produce a single closed orbit surrounding the three positions of static equilibrium. Clearly the particle is now provided with a sufficient amount of energy to jump the potential barrier centered at (A). Finally, when ~ is exactly equal to s (0), the corresponding trajectory has a double point at (A). This particular trajectory is the separatrix, which marks the boundary between the two families described just above. Calculation of the period of motion occurring along the separatrix, would result in an infinite value. As a matter of fact, the repulsing point (A) can never been reached in a finite time, whatever the actual initial conditions may be.
Autonomous oscillators
163
EXAMPLE 3.  The plane pendulum The potential can be written as ~ee = MgL(1cosO). Accordingly, it vanishes at the lowest position (0 = 0). Provided the cyclic nature of the angular variable 0 is disregarded, it can be stated that the pendulum has the following positions of static equilibrium: I stable equilibrium :0s = 2nzc n = 1,2 .... [unstable equilibrium :01 = (2n + 1)~r
2
!
i
i
i
i
i
i
I
!
#
0"i_
(.
~ 10
I
1
I
I
1
1
1
I
I
8
6
4
2
0
2
4
6
8
10
Figure 5.14. Phase portrait of the plane pendulum The phase portrait displayed in Figure 5.14 comprises two distinct families of paths. The family of the closed orbits is related to energy levels ~ less than 2LMg .They stand for periodic oscillations about the lowest position. Calculation would show that the natural frequency is a decreasing function of the oscillation amplitude. This is simply because [sin 01 is less than 101 Energy levels higher than 2LMg result in open paths, which stand for nonperiodic motions, in which the pendulum is revolving around the fixed point. The separatrix has an infinity of double points which correspond to the angular positions of unstable static equilibrium. As a final remark on this system, it is noted that the cyclic nature of 0 can be accounted for simply by rolling the phase portrait of Figure 5.14 on a circular
164
Discretesystems
cylinder of unit radius. In this way, all the phase trajectories are suitably transformed into closed paths and the periodic nature of all the possible pendulum motions is recovered.
5.2.2 Damped oscillators In contrast with the conservative case, no general methods of establishing exact analytical solutions for phase trajectories exist in the nonconservative case, since energy is no more invariant. In so far as nonlinearities remain sufficiently weak, it is still possible to obtain approximate solutions analytically by using suitable perturbation methods. However, description of such methods, which are well documented in particular in [NAY 73], [NAY 79], is beyond the scope of this book. On the other hand, nowadays, it is also possible to perform numerical investigations on the computer, by integrating step by step in time the nonlinear equations of motion. Two typical direct integration methods are further described in section 5.3, namely central difference and Newmark's algorithms. Applications provided in this book were obtained by implementing them in the MATLAB software. EXAMPLE. Damped Duffing's oscillator Duffing's oscillator was already introduced in subsection 5.2.1. Here, the purpose is to investigate its behaviour in the presence of viscous damping. The equation of motion is thus written as:
co~q + eq 3 + 2co~9~it+ ii = 0 It is recalled that the position of static equilibrium q = 0 is unstable. However, the cubic stiffness term prevents the system going to infinity. Without making any calculation, it can be anticipated that the stable equilibrium positions (B) and (C) are now attracting points, since viscous damping is steadily dissipating mechanical energy of the system. The presence of two distinct attractors provides the dynamical behaviour of the nonlinear system with a somewhat intriguing feature, which contrasts with the case of the linear oscillator, namely a high sensitivity to the initial conditions of motion, at least in a certain range of values. Indeed, provided the mechanical energy initially imparted to the system is not too large in comparison with the potential energy at point (A), tiny variations in initial displacement and velocity, performed at constant energy, can be sufficient to change the final destination of the particle, from (B) to (C) or the reverse, as illustrated by the plots in Figures 5.15 and 5.16.
Autonomousoscillators 165 15
1
!
!
v
v
~)
lO
05 .i : :.:~ ::" " "
"=4.361 "": .....
300

v.
.
.
.
.
v

v
'1


v 

\'
4 q?o
5 10 I
5
0
.... 5i
10i
Figure5.15.Timehistoryandphaseportraitof Duffing'soscillator( m=I, o9=0.2Jr) case 1: theparticleis attractedby (B)
166
Discretesystems
15
,
i.
.
.
.
.
q(o)=7.1s
10 [ q(t)
(o.o2 t
q(o)~.s9e9
e 0.01 5
h= O.02he o
(0 5
IO
o
,3
....
i
I 50
i I0O
i
[
150
200
w
..
!
300
250
''
i i
"~
I0
i
5
i
I
0
5
.~
L
10
Figure 5.16. Timehistory and phase portrait of Duffing's oscillator ( re=l, o9= 0.2~ ) case 2: the particle is attracted by (C)
Furthermore, when a quantity such as the initial displacement is parametrically varied, the mechanical energy remaining constant, the particle is alternatively attracted by the point (B), or by the point (C), in successive intervals of q(0) values, which can be fairly narrow. A striking analogy with the game of roulette is worthy of
Autonomous oscillators
167
mention. Fundamentally, the pits in the rolling track of the game behave as attractors (sink of potential) and tiny variations in the initial conditions are sufficient to modify into which pit the ball will stop. More generally, the high sensitivity of nonlinear systems also occur in relation to changes in quantities other than the initial conditions. Since it is impossible in practice to control the exact value of any parameter of a real system, the sensitivity actually introduces the idea of chance or randomness into the behaviour of otherwise deterministic systems. Finally, when numerical investigations are carried out, one has to be conscious that, even if tiny, the errors which are unavoidably introduced in the computation process can have major consequences on the final result, as further discussed in section 5.3, in relation to the present example. These considerations are sufficient to point out that the task of analysing the response of dynamical systems is generally much more arduous in the nonlinear than in the linear domain. This will be further illustrated in Chapter 9, in connection with the forced motions of Duffing's oscillator, see subsection 9.4.2.
5.2.3 Selfsustaining oscillators
A selfsustaining oscillator is an oscillator which is dynamically unstable from the linear standpoint, for instance because its viscous damping coefficient is negative, but which is stabilized by the presence of nonlinear dissipative forces. The resulting oscillations, of which the magnitude is stabilized to a finite value by nonlinear forces, are then termed steady selfsustained oscillations, or vibrations. An archetype of such systems is the Van der Pol oscillator, which is governed by the equation: (.02q + o~(q2 1)q+/~ = 0 This equation is characterized by the presence of a nonconservative force comprising a linear and a nonlinear component, both of them being proportional to the velocity q (t). As a consequence of nonlinearity, the nonconservative force is found to dissipate, or alternatively, to produce mechanical energy depending upon the magnitude of the oscillation with respect to the critical value
Iq
1. we are
interested here in discussing the case of positive values of the parameter o~. Indeed, it is easily verified that this condition is necessary for obtaining steady selfsustained oscillations: 1.
If motion is initiated with Iq01 1 can be discussed qualitatively in the same way. Now the oscillator starts by losing energy. However, as soon as the amplitude of the vibration becomes sufficiently small, the same regime of steady sustained oscillations as above takes place, in which the energy balance over a cycle of vibration is exactly zero.
cycle
a = 0.126
q(t) "34
I
i
i
1
1
i
i
I
3
2
1
0
1
2
3
4
Figure 5.17. Phase portrait of the Van der Pol oscillator The steady oscillation qualitatively inferred from the above considerations defines a closed limit cycle in the phase plane. This cycle is immediately recognised as an attractor, on which all the possible phase trajectories are finally converging. The numerical integration of the Van der Pol equation confmns such a behaviour, see Figure 5.17. Furthermore, according to the computed timehistories it is found that provided the mechanical energy ~ of the limit cycle is large enough, within each period the motion may be split into two well contrasted stages of relatively slow and relatively fast motion, which occur alternately, see Figure 5.18. Clearly, such a pattern corresponds to the distinct stages of motion, within the periodic cycle, during which energy is gained and then lost by the selfexcited oscillator. Such sustained oscillations are often termed relaxation oscillations.
Autonomous oscillators
~c=
0~01 6
at
=
169
1.26
4 2 o
2 4 6 t
t
Figure 5.18. Timehistory of a relaxation oscillation (displacement and velocity)
5.3. Numerical integration of the equation of motion Methods for producing approximate solutions of the equations of motion have been researched for a long time, A plethora of papers and many books have been devoted to the subject. The approach here is restricted to two basic algorithms which are widely used in structural dynamics. The reader interested in the topic is referred to [BAT 76], [BEL 83], [PRE 89], [ARG 91 ], [GER 97]. The starting point of any numerical integration scheme is to discretize the time derivatives of any quantity, replacing it by a suitable finite difference approximation:
dq
Aq
dt
At
;
dit
A[t
dt
At
[5.51]
Therefore, q(t) is computed only at a discrete and finite sequence of successive timesteps. The various integration schemes available can be distinguished from each others by the two major following features: 1.
The value of the discretized timestep at which the equilibrium equations are verified.
2.
The order of the truncated Taylor series which are used to discretize the derivatives.
The first point is of paramount importance since it leads one to the existence of two quite distinct classes of algorithms, namely the explicit algorithm and the implicit algorithm. Hereafter, an example of each of these is described. As we shall see best in Chapter 7 in relation to MDOF systems, the general advantages of the explicit algorithms are that they are easier to programme and are, in many cases, more efficient than the implicit algorithms, especially for solving nonlinear problems. Nevertheless,
170
Discrete systems
they are also marked by conditional stability. Fundamentally, the timestep, hereafter denoted h, must be less than a certain critical value hc , otherwise the magnitude of motion is found to increase exponentially with time, independently of the physical model to be treated. For certain applications, in either the linear or the nonlinear domain, conditional stability becomes a major drawback because computational efficiency of explicit algorithms may be compromised by the necessity to select a timestep much smaller than the physical timescales of the problem. On the other hand, it is possible to establish implicit algorithms which are unconditionally stable, at least in the linear domain, for which computational efficiency can be optimised by selecting the value of h in accordance with the physical timescales of the problem. However, in nonlinear applications, the computational efficiency decreases as a consequence of the need to balance the system at each timestep by using an iterative process. Such preliminary remarks serve to emphasize that, the choice of a specific algorithm has to be made in accordance with the particularities of the problem to be solved.
5.3.1 Explicit scheme o f central differences o f second order
5.3.1.1 Recursive process Let us start with the Taylor series, truncated to the second order: h2
q~+, = q~ + hit~ +  ~ i(l~ h2 qn_, = q,,  hit. +  :  ~
[5.521
2
where n = 1,2 .... is the nth timestep, of duration h. From relation [5.52], it is immediately found that: 2hqn = qn+l  q~i h2/1, = q~,l  2q~ + qnl
[5.53]
Now, let us consider the forced equation of a damped harmonic oscillator:
a~2q(t) + 2r
+/~(t)  Q(t) M
[5.54]
The reason for introducing an external force, already here, will become apparent in subsection 5.3.2 where the treatment of nonlinearities will be described. Then, the force balance is written at the nth timestep:
Autonomous oscillators o (Ol2q,, + 2(Ol~'lO,, + qn = ~""2"n M
171
[5.55]
Substitution of the discretized derivatives [5.53] into equation [5.55] gives: o)~q,, + o919'
q,,§  q,,i q,,+l  2q,, + q,,I = Q,, + h h2 M
[5.56]
which can be written as: q.+l (1 + ~1~'i)+ q. ( 0 " 2  2 ) + q,,_, (1~1~" 1) = h2Q" M
[5.57]
where the dimensionless frequency ~l = ht~ is used. Finally, relation [5.57] is written as a recursive sequence, producing the value of q,.~ expressly in terms of quantities defined at the foregoing timesteps n and nl" q,,+l = a q . + flq.i + YQ. 2  W2
~lql  1
;fl = ~ ; y = 1 + to'lg"~ 1 + nr191
where ct = ~
h2
[5.58]
M (1 + ~1ffl )
5.3.1.2 Initialisation o f the a l g o r i t h m
The actual motion of the oscillator is analysed starting from an initial dynamical state determined by the physical displacement X 0 and velocity Vo . However, in order to initialise the recursive sequence [5.58], it is necessary to define two fictitious displacements at timesteps h and 2h, the actual motion starting at time t = 0. One is thus led to express the physical initial conditions in terms of the recursive sequence: X o = qo = ctq1 + flq2
V0 = q0  ql  q1 2h
[5.59]
ql = CtXo + flq1
Using the expressions [5.59], the displacement values required to initialise the sequence are easily obtained. The following intermediate results are produced" V0 _ ql  q1 =~ ql = ql  2hV 0 2h q~ = CtXo + flq1 ~ q~ = CtXo + flq~  2hVo
substituting [5.60] into [5.59], the desired results are readily obtained:
[5.60]
172
Discretesystems ql = ctX~  2 h 1fl
Vo
[5.61]
X o  a'q_~ q  2 ~"
5.3.1.3 Critical value of the timestep for stability It must be realized that the fact of writing the force balance at the nth timestep (see equations [5.55] and [5.58]) leads one to compute the configuration of the system at the next timestep. Basically, the configuration at the step n is assumed to be already known and the sequence is used to extrapolate the configuration at the step n + 1. We may thus anticipate that if too large a timestep is chosen, such an extrapolation is likely to become unrealistic, inducing eventually a numerical instability, marked by the divergence of the sequence [5.58] whatever the external excitation may be. This can be checked analytically in the linear domain of response. The proof is worth considering, since a striking analogy exists between the behaviour of the recursive sequence and the dynamical behaviour of a linear oscillator. In the absence of any external excitation, the process [5.58] can be written in matrix notation, starting from n = 1:
[5.621
As the initial values q~,qo are known by using [5.61] (after suitable renumbering of the sequential index), it is found that:
([L]f [q]l = [q],+, =:~
/[ 1
0fl])n[q~]=[ qn+~] qo L q,
[5.63]
Now, the process [5.63] converges only if the modulus of the eigenvalues of the recursion matrix [L] is less than, or equal to one. Let us assume that [L] is a regular matrix with two distinct eigenvalues, it is then possible to transform it into a diagonal matrix by using the eigenvector matrix, hereafter denoted [~]. In geometrical terms, this is interpreted as a transformation of similarity defined by: [A]= [~]~ [L][~]
[5.64]
which leaves the eigenvalues and the angle between two vectors unchanged. Now, in the frame of the eigenvectors, the recursion process is written as:
Autonomous oscillators
~
l:Iu ] where ul IOllql Un_I
173
[5.651
Un
and the mapping [5.63] is transformed into:
u
[5.66]
It is immediately realized that convergence of the process requires that the moduli of ,,ll and 22 be less or equal to one. After some elementary algebra the following results are obtained"
,;t~~r
~ 2,.~=
or+~a,z +4fl
2  ~r~ + ~/~t4  4ar~ ( 1  ~'~ ) ,2 ~
[5.67]
2(1I ~T1~1)
The simplest case to investigate is the conservative case (~'~ = 0 ) where it is found that: If ar 1 < 2, IAI = I~1 = 1
If ar 1 > 2,
[5.68]
IAI < 1;1~1 >
Therefore, the following critical value of the timestep for stability of the recursive scheme is derived: 2 to"1 < 2 ~ hc = m co1
[5.69]
It can also be verified that damping does not modify this stability threshold, provided its value is positive, as shown in Figure 5.19. As could be expected, in the case of negative damping, the modulus of the two eigenvalues are found to be larger than one, even if the timestep is arbitrarily small. This is clearly a direct consequence of the instability of the physical system.
174
Discretesystems
m o d i ~ o.f~r eilenvalma
1.2
0.9F
k//~e = /
oo, /a2/
,,',41/
"\\~X
""....
O
'\ ,]5
/"
/,''"""
k/kc = 0.1
~
./J.2/
/
e.6
1.4
11.2
\
0.7
'%%~. ",%.,
%
O3
ff
, e.
I' N~S '
e_2 0:4 o~
o]1
1
1.2
,
,
1.4 1.6
~""%'. l.l
2
of tke , ~ ~
2
k/ke =1.01
'
l
I.I
/all> 1
easeA~ / 9
I
~..
~
= 0~1
' i
~ / 1 ~ i 1 > i
.,,"~176176176176 ...... .,,...,,~~""~
~2
t.S
t.6
L4
ta
I
MS
O.6
4L4
4.l
Figure 5.19. Modulus of the eigenvalues of matrix [L] versus reduced damping
5.3.1.4 Accuracy of the algorithm
q(O) = l m
r= 1 5 H z
. II 'i
0.5
.
i
."1 t. _
!
0.5
_
.. . i
.1.. Iv~o= o.t
~~
o:~
o12
oi~
oi,
oi~
t (s)
oi,
o17
0:8
o19
,
Figure 5.20. Computed timehistory of the undamped massspring system
o
Autonomous oscillators
175
0.2
signal of relative, error
0"15t 0.1 0.05
1
0.05
0.1s
51
t(s)
~
o11
I
.
"4
I
012 0.3 014 0.5 0.6 017 018 019. 1
Figure 5.21. Error signal It is first emphasized that stability and accuracy of a dynamical algorithm are two distinct properties which must not to be confused with each other. After having discussed the problem of stability, it is appropriate to investigate the errors introduced in relation to the timestep of the algorithm. In this respect, it turns out that computing the free vibration of an undamped linear oscillator is sufficient to provide us with the essential information about accuracy. As a typical example, consider the case of a massspring system, mass 4 kg and natural frequency 15 Hz, which is set in motion by the initial displacement X 0 = 1 m and the initial velocity V0 = 0 m/s. The computations have been performed using MATLAB. The timehistory of Figure 5.20 indicates that the central difference scheme provides the correct value of the vibration amplitude, without inducing any numerical attenuation or damping. Nevertheless, a systematic error is still made as shown in Figure 5.21, which is a plot of the error signal obtained by subtracting the analytical solution from the numerical one. The error signal can be described as a sinusoidal component at the natural frequency of the massspring system which is "modulated" by a linearly increasing envelope. The slope of the envelope signal is an increasing function of h. Actually, the error is caused by a small shift in time in the computed solution with respect to the theoretical solution. As a general result, it can be stated that no algorithm can be free from any error either in amplitude or in time. However, the errors can be made as small as desired, by diminishing the value of the timestep. Nevertheless, even tiny roundoff errors can be of importance when dealing with nonlinear systems. For instance, in the case of the damped Duffing's oscillator, numerical simulations performed by using the central difference scheme with the
176
Discretesystems
same initial conditions but with distinct timesteps may result in distinct trajectories ending either at (B) or at (C), as illustrated in Figure 5.22.
I0
(=0.02 ',
,
',
,
,
~I [ ~ ,' I
~ I
,
I
~
,
I I
I I
t I
! I
I
l!
~ !
I t
t I
I I
r I
i I
I
I I
,' , l
,' ' I:
', ' I
I I
!
[
' 9
I
i
1
i
i ~
I
o
5.22.
,
! I
,
I
,
i
t q(O)=O
I0
Figure
E = O.OI
0.05
I
o.~
I
,, ,,
"
;
.
,, :: i ! ~;
t
J
I
~
!
i
!
I
'
!
'
I
"
!
!'
i
I
!
!I" I I
J
'
,
~'
I I
i t ,,; ,' .ll .,.! ',I '!
:,
;I ~.' :;I
:
,
;,
: ;
I t
I
er
!1
I I
!I ~, lI
'. ;I
iJ
l
t
q(O)=lO .
h~e,
1
"
.
o.15.
o12
0.25.
Sensitivity of the numerical solution of the damped Duffing's equation to the timestep of the central difference algorithm
5.3.2 Application to a parametrically excited linear oscillator Linear oscillators are said to be parametric when at least one of their coefficients K,M, or C is timedependent. As an example, we consider a simple pendulum vibrating in a fixed plane with a small angle O about its support point O'. The latter is moving in the vertical direction about the fixed point O, according to the given oscillation Z 0 sin (COot), where the displacement Z 0 is assumed to be very small in comparison with the length R of the pendulum, see Figure 5.23. The Lagrangian of the system is:
s
"2 + 2RZo00 )  Mg Zo+ T ( R ' O ' + Zo
Whence the following equation of motion:



Autonomous oscillators
O+ ( 2 ~
R
) 0=0r
177
6/+ 092 (1 _ K.sin 090t)0 = 0
ZotO~
where 092 = g a n d / r R
g
I
..... Z"0..sin ......(root) ....................................................X...........
l
~
g
..~ 9 .......
i M z V
Figure 5.23. Pendulum
with a moving support
According to this equation, known as Mathieu's equation, the pendulum behaves as if the support would be fixed and the mass would be subjected to a gravity field oscillating sinusoidally at the prescribed angular frequency 090 . Solutions of more general parametric equations, known as Hill's equations, are extensively discussed from an analytical point of view in many textbooks, see in particular [WHI 44], [ANG 61], [MAG 66], [BER 84], [NAY 79]. As a major result, the presence of parametric instabilities (also termed parametric resonances) is shown to occur if 090 lies inside intervals centered at the resonance frequencies 090 / 09~ = 2 / n , n = 1,2... The width of such intervals is an increasing function of ~c. Without entering into a detailed discussion of such systems, it can be said in a qualitative way that the physical mechanism responsible for the parametric instability is clearly an energy transfer from the prescribed motion toward the pendulum. As would be expected, such a transfer can be globally positive (i.e. when averaged over a cycle of oscillation) only when the ratio of the response frequency of the pendulum to that of the prescribed oscillation of the support is sufficiently close to particular values. Width of the intervals where the so called parametric resonant responses can take place is expected to increase with the mechanical energy involved in the prescribed motion.
178
Discretesystems
0.5
0
0.4 t
,
,
10
20
mO/ml
r 9.81 ,
=2.1
0.3 03 0.1 0
0.1 03
0.4
.t 0
30
40
SO
60
70
80
o
30
I
_/ / II II
w/,,l =2.oJ
20 10
O tO 20
..30 t 40
0
10
20
30
40
50
60
70
Figure 5.24. Two distinct timehistories of the parametric pendulum The integration algorithms of the explicit type allow one to solve numerically the equation of the parametric pendulum in a quite convenient way. It is found suitable to deal with the parametrically varying stiffness force as an external force. In accordance with the recursive scheme [5.58], this force component is known explicitly at the required nth timestep. Figure 5.24 displays two samples of such simulated timehistories; one refers to a nonresonant (stable) response and the other refers to a resonant (unstable) response. In this kind of application, it is found that a timestep of the order of one tenth of the critical value for stability provides satisfactory results. In the nonresonant response, the difference between the pendulum and the excitation frequency results in a beating phenomenon marked by a low frequency modulation of the amplitude of the response. The beating phenomenon will be further discussed in Chapter 7.
Autonomous oscillators
179
5.3.3 Application to an oscillator impacting against an elastic stop
5.3.3.1 Impact force model
k L
180
Discrete systems
An analytical solution is easily obtained in the case of stiff impacts. The motion before the first impact is governed by the following linear equation:
m(r.o2X
+ ,~)=0
X o = 0 ; 2 o>0; X(t)_m The variable transformation q = X  L leads one to the autonomous system:
Kcq + miil = O q(to )= O ; dl(to )= X (to ) Thus, motion of the oscillator during the shock is described by the following timehistory: X (t) = L +
2
(to) sin ( ~
(t  to )) where r = ~~
Duration ~'c of contact between the oscillator and the obstacle, also termed the
shock duration, is inferred from the contact condition X (t o + rc ) = L. Whence we deduce that:
Autonomous oscillators =~'_1 rCco~2
181
T~
where Tc is the natural period of the "shockoscillator". Incidentally, it may be noted that shock duration does not depend upon the energy imparted initially to the oscillator, and thus is also independent of the strength of the impact. Moreover, it is also easy to check that, in agreement with the principle of conservation of energy during an elastic impact, the velocity of the oscillator at the end of the shock is precisely ,~ (t 0). On the other hand, the impact force is identified with the shockspring reaction, which is thus given by: (t
Fc(t)=K~
)
~,'0 sin (r c ( t  to )) t  t o ~ [0, z'c]
The interesting point, worth emphasizing and easily checked, is that the action of the impact force identifies with the change of linear momentum of the oscillator from start to end of the shock: (t ) ,o I sin (o)cv)dv =  2mX (t o) co~ Jo
kX 2
L) 2
2 ::
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
X ik.. /
"freefl~ht"
o.L............. , /
shock
Figure 5.26. Potential and phase portrait of the impacting oscillator
182
Discretesystems
Finally, the period of the nonlinear oscillation is calculated by using the invariance of energy. The procedure is readily understood by referring to Figure 5.26, which sketches the potential and the phase portrait of the system. The potential curve is made up of two distinct parabolic segments, one of them refers to the "freeflight" stage 9 of motion and the second, much stiffer than the first one, refers to the shockstage. Therefore, the phase portrait is made of two distinct elliptic arcs. They correspond to the phase portrait of two linear oscillators, distinct from each other by the value of the stiffness coefficient, namely k for the "freeflight" oscillator and K~ = k + K c  K~ for the shockoscillator. Duration of a cycle of "free flight" is given by the definite integrals of the type [5.21], calculated between the following boundaries:
Xmin =  ~  ~ ,
Xmax = L
So, in reduced quantities: ~, = 1 ;~2 =
~L
~:0
The period is thus given by: 2
i0
d2j
+ 2_.
1
'f0
d~j
(
T1
:
+2
2Arcsin
tXo))
In the case of violent impact, the above result takes the simpler form: T=
1+
+ . X0
As expected, the period of the impacting oscillator is found to be less than the period T~ of the non impacting oscillator. Moreover, if the shock stiffness tends to infinity, the shock duration tends to zero. However, it may be verified that important aspect of the motion such as the "free flight" stage and the action of the impact force remain unchanged. This interesting aspect of the problem will be discussed further in the next subsection. On the other hand, from the analysis made above, it may be readily understood that numerical simulations using explicit algorithms require a timestep the value of which is related to the shortest timescale of the problem, i.e. r c and not T~. More specifically, when using the central differences scheme, in formula [5.69], a~ has to be replaced by toc . As a consequence, in the presence of very stiff impacts, computational time can become excessive. Hence, from a practical point of view, it is advisable to determine first the minimum timescale which is needed to interpret
Autonomous oscillators
183
usefully the results of the numerical simulation and then to adjust the value of the impact stiffness accordingly. Let us consider for instance the crash of a car against a rigid wall. Such an event lasts typically about 100 ms. Crash tests clearly indicate that such a transient may be satisfactorily described by using a timestep of about one millisecond. Hence, to perform numerical simulations of the crash, a suitable value of Kc such that toc /2nr = lkHz is appropriate.
0.2 0.15 0.1 0.05
!
i
i
X (m)
i
i
1
I
i
i
k / K c = 0. O1 h / h c = 0. 02
A ::" ;T;:s
s .........
0.05 0.1 0.15 0.2 0.25
t(s) i
i
I
1
i
I
1
1
i
0.02
0.04
0.06
0.08
0 1
0.12
0.14
0.16
0.18
0.2
Figure 5.27. Displacement of the impacting oscillator ( M~ = 10Skg" f~ = IOHz'r =0; "~o = 15m/s ) Figures 5.27 to 5.30 illustrate the use of the central differences method to treat the present problem. The simulation refers to the case k / K c = 0.01 .The nonlinearity of the response and the elastic compression of the impacted spring during shocks are clearly detectable in Figure 5.27, which shows the timehistory of displacement. The velocity plot of Figure 5.28 makes the nonlinearity even more conspicuous, because velocity reverses abruptly during the shocks. As the system is conservative, maximum velocity remains exactly equal to the value imparted initially to the oscillator ( )~0 = 54 km/h). The timehistory of the impact force is plotted in Figure 5.29. It is shaped as a negative halfsine curve repeated periodically at the frequency of the oscillation. Force amplitude depends upon the stiffness coefficient of the shockspring but the action over a shock is precisely equal to the change of linear momentum 2MI,~ 0 . Anticipating the mathematical formalism, it can be said that in the limit of infinite stiffness, impact forces tend to a periodic series of Dirac 8
distributions (cf. Chapter 7 and Appendix 6), weighted by 2M1,~ 0 . Moreover, in the
184
Discretesystems
limit of infinite impactstiffness, the shockspring model could be advantageously replaced by a nonholonomic constrained model, from the analytical viewpoint at least, as discussed in the next subsection. However, so far as the dynamical response of the oscillator is concerned, not much is gained by letting the impact time be vanishingly small. Indeed, as soon as K~ is sufficiently large, the truly relevant quantity is the action of the impact force and not the particular force value at a given time.
!
2o~ X (m/s)
shock
10
9149
25
9149
9
0
9
0.02
0.04
Figure 5.28.
o,'1~
0,06
0.08
.
0.1
0.12
.
0.14
0.16
0.18
O,2
Velocityof the impactingoscillator
,
1 2
F
shocks'
"
9.s. i ~llv
'
:F 10
!
O.
w.:..........................,..........:
.............
t(s) i
I
o.o$
o,1
Figure 5.29. Impactforces
o,15
Autonomous oscillators
185
20
15
10
shock
10
i/

15 i
!
,
X
!
ZO.3
0.2
011
0
0.1
0~.2
0.3
Figure 5.30. Phase portrait
Finally, the phase portrait shown in Figure 5.30 is found to be in satisfactory agreement with the theoretical sketch of Figure 5.26. When plotted over a large number of cycles, it provides a good test to check the periodicity of the computed response. Indeed, any error with respect to periodicity results unavoidably in trajectories which superpose imperfectly from one cycle to the next.
5.3.3.2 Constrained model Starting from the last model, it is of interest to further investigate the asymptotic case K c ) oo analytically. For this purpose, impacts are now modelled according to the following conservative unilateral constraint condition:
X(t)o),o
andz'_  r  e
(,>o),o
Thus, r_ is the time "just before" an impact and r§ is the time "just after" the same impact, occurring at time r . Before the first impact, the motion triggered by the initial velocity "~0 is:
186
Discrete systems
Xl(t):
~ col
~',(t)=)~0cos(colt ) 0 < t < r l
The impact constraint implies that: L = "~0 sin (r_olr~)=, r 1 = 1 Arcsin(/_~. l /
Xo)
with the auxiliary condition for impact : )~o > Lm~ The first impact occurs at the incident velocity:
Thus, the impact constraint provides us with the necessary initial conditions to describe the "freeflight" motion after the first impact. It is found that: X 1(t')= Lcos(colt')Xcsin(colt col
")
21 (t')=/_zo~ sin (co~t') 2 c cos (wit') where t" = t  r~ Now, periodicity of the motion implies that:
where Tc is the still unknown period of the impacting system. Therefore, the condition of periodicity can be suitably written as the following mapping:
E']ICOS'O 'sin'o ]o :l[ '~c
a)~ sin(m~T~) cos(olT ~) J 2c
The above system can be solved directly, or as an eigenvalue problem. Let us outline the direct method first. From the first equation, it is found that:
Autonomous oscillators /.zo, _
L
187
sin (rolT~)
Xcos( o r )
The second equation provides another relation which turns out to be compatible with the first one: L(DI_
,~
1 + COS((DlTc)
sin (m~T~)
After a few trigonometric calculations, the above result may be rewritten as: Lm~_ "Yc c~176 2 / which is a form suitable for comparison with the result provided by the impact force model: Tl 1+ Arcsin ~ Xo) ) = lim KY~ =2n" Substituting this expression in the result of the constrained model, it is found that: a)lL /Arcsin/r~ m.lLl/l_/~ ) ~ 2c tan [~~0) ) = ( X0 )~ ~, X0 ) which is the correct result, since 2~ = 21 (r~)= 2 o 1 Lml
, as already shown.
Now solving the problem as an eigenvalue problem, the mapping is first changed into the equivalent form:
~c
o91 sin (r_o~Tc ) cos(oo, Tc
According to this new formulation, the required solution must correspond to the eigenvalue 2 = + 1 and the related eigenvector may be interpreted as an invariant or fixed point of the mapping. Actual resolution of the problem presents no difficulty. The eigenvalue equation is:
188
Discretesystems (cos ((.OiT~)  2 ) ( c o s (a)~Tc)+ 2 )  (sin (a),T~))2 = 22  1=0
The eigenvectors are conveniently written as:
,~
=1=,
 + I ~
UOl
l+cos(a, r )
"~c
sin (talc)
Lo)l _
sin (o)fc)

providing thus the same compatible equations as those obtained by using the direct method. As a final remark, the reader can check that the mapping:
obtained by discarding the finite jump condition of velocity at the impact, leads one 2~r to the linear period T~ = .
tot
5.3.4
Newmark's implicit algorithm
As seen in the last subsection, explicit algorithms use the equilibrium equation at the nth timestep in order to determine the dynamical state of the system at the next (n+ 1)th timestep. It may be noted, that in this procedure, the force balance written down refers to a known configuration and is used to extrapolate the configuration at the next timestep. As already pointed out, the drawback of such an explicit scheme is its conditional stability, which is governed by the smallest timescale of the numerical model. In contrast with this, the implicit schemes use the force balance written down at the (n+l)th timestep to determine the state of the system at the same (n+ 1)th timestep. Such a procedure allows one to build unconditionally stable algorithms. Though there exists a whole family of Newmark's algorithms, it is sufficient for our purpose to describe here the most broadly used version of them, which is implicit and free of numerical damping. Let us start from the Taylor's formulae:
f(t,,
+h)=
h2 h p f(p) f(t,)+hf'(t,)+~f'(t,)+...+(t,) + Rp p!
where the residue is given by:
[5.70]
Autonomous oscillators
Rp+l = ~ f (hP P*')(/,+hO) (p+l)!
189
0 1 degrees of freedom, (also called multi degrees of freedom systems, or briefly MDOF systems) is governed by a system of N linear equations, algebraic in the case of statics and differential with respect to time in the case of dynamics. In most instances, such equations couple together several generalized displacements. Therefore, it is extremely useful to find a systematic procedure allowing one to uncouple such systems of equations, in statics as well as in dynamics. Indeed, if suitably uncoupled, the system is reduced to a set of N oscillators independent of each other, which is very convenient for further analysis. The problem of uncoupling will thus serve as a guideline for most of the considerations which shall be made in this chapter. In the first instance, section 6.2 is concerned with the task of linearizing Lagrange's equations about a static state of stable equilibrium. In the conservative case, after linearization, the system is thus characterized by a stiffness matrix [K] and a mass matrix [M]. Both of them are symmetrical, [M] is positive definite and [K] is positive. Section 6.3 constitutes the core of the present chapter. It deals with autonomous conservative and linear systems which vibrate freely about a static state of stable equilibrium. We shall show that a judicious transformation of displacement variables allows one to uncouple the equations of motion. Moreover, the same uncoupling procedure applies also in the case of statics. The column vectors of the transformation matrix are formed by N linearly independent mode shapes of vibration of the system. As expressed on this modal basis the dynamic equations of the mechanical system reduce to a set of N uncoupled linear oscillators. The natural frequencies of such oscillators are the modal frequencies, of the system, whereas the masses and stiffness coefficients are called the modal mass and stiffness coefficients (or the generalized mass and stiffness coefficients). Clearly, the coupling between the variables on the physical basis (or in any non modal basis) is accounted for by the mode shapes which interrelate the displacement of each oscillator. Section 6.4 extends the concept of vibration modes to systems other than those already considered in section 6.2. Namely, the following items are discussed:
Natural modes of vibration of constrained systems. After linearization of the constraint conditions, the Lagrangian of a constrained system gives rise to a set of linear differential equations, mixing the variables of displacements and constraint reactions. In the conservative case, such systems can be characterized by a [K] and a [M] matrix, which are still symmetrical. However, as the variables of the problem now mix components of displacement and of forces, the physical meaning of stiffness and mass matrices [K] and [M] does not
Natural modes of vibration
193
hold anymore. Nevertheless, modal analysis of such systems can proceed in the same way as in the unconstrained case. .
.
.
The free modes of rigid bodies, of which the natural frequency is zero. They correspond to motion allowing the potential energy of the system to remain unchanged. The modes of elastic buckling, which may occur in many prestressed systems. Such modes are analysed successively from the static and dynamical point of views. The whirling modes, which take place in rotating systems as a consequence of gyroscopic coupling (cf. Chapter 4, subsection 4.3.2).
On the other hand, extension to nonconservative systems will be described in Volumes 3 and 4, in relation to fluidstructure coupled systems. Indeed, at this stage, we will be able to bring the physical meaning of such modes to light by discussing the behaviour of a few examples.
6.2. Vibratory equations of conservative systems 6.2.1. Linearization of the equations of motion Let us consider a conservative NDOF mechanical system, of which P is a static state of stable equilibrium. Motions around P are described by using N generalized displacements independent from each other, which are the components of the displacement vector [q]. In accordance with this definition, [q] is zero at P. The system is governed by Lagrange's equations: 3~ 3 [ q ]

3 [ q ] =[0] [6.1]
Z  E,r ( [ q ] ' [ q ] )  4 ([q]) Now, we restrict our interest to motions of small magnitude, in such a way that the equations can be linearized about the configuration P, which is taken as a reference. As already indicated in Chapter 2, subsection 2.3.4, the system [6.1] takes the canonical form: [K][q]+ [M ][/~] = [0]
[6.2]
Here, it is of interest to start by relating the stiffness matrix [K] and the mass matrix [M] to the quadratic approximation of the Lagrangian. With this aim in mind, it is appropriate to carry out an expansion of ,G in Taylor's series, limited to the
194
Discretesystems
quadratic terms in [q], or [q]. Referring to the forms [2.12] and [2.20] of Chapter 2, kinetic energy is written as:
1 _i
[M ][q] [6.3] r
d [q] o~[,)1 [+[ol
M jk = d Ojd qk [ql,M[ol
where only the constant terms are retained in [M ]. Clearly, such a mass matrix is symmetrical and positive definite, i.e. all the eigenvalues are positive, as already discussed in Chapter 3 subsection 3.2.1. On the other hand, it is recalled that the state of static equilibrium is defined by the conditions:
a qk q.o = 0 ~ La [ql_[t.H0~ = [01 [6.4] ~ q1c9qk qt=qj__0
tg[q]
[q]=[Ol
[K] is the stiffness matrix, which is thus found to be symmetrical. Moreover, provided the equilibrium state is stable, [K] is also positive definite, becoming eventually simply positive, if the case of indifferent equilibrium is included as a possibility. Of course, potential energy takes the quadratic form:
r = l[q]~ [Kl[q]
[6.51
To conclude this subsection, it is noted for further reference that the equations governing the linear system, when excited by an external force vector, take the canonical form: [K][q]+ [M ][/~1 =
IQ (e)(t)]
[6.6]
6.2.2. Solution of forced problems in statics 1.
Resolution by inversion of the [K] matrix
Adopting the static, or quasistatic approximation, the forced problem reduces to the following linear algebraic system:
Natural modes of vibration
[g][q]=[a (e)]
195
[6.7]
Provided the state of equilibrium taken as a reference is not indifferent, [K] is a regular matrix so the system [6.7] can be solved by using the inverse matrix [K] ~ , at least formally:
(e)]
However, it has to be stressed that such a solution is physically meaningful only if [K] is positive definite. Otherwise, the matrix would be singular or the material system would be unstable and its mechanical response to any external excitation would be time dependent and nonlinear in nature. EXAMPLE. 
Coupled pendulums
Going back to the pair of coupled pendulums already considered in Chapter 3, subsection 3.2.2, the forced problem takes the form: +l
1
=
AL +1
(a)
The roots of the determinant A = ( 2  x)5 _ 1 are x 1 = 1 and x 2 = 3. x 1 corresponds precisely to the threshold of static instability of the system. Hence, it is rather obvious that the displacements which may be calculated by using the above formula are physically meaningful in the domain Ir < 1r1 = 1 only. Indeed, when the system becomes statically unstable, the pendulums are attracted toward the position of stable equilibrium 0e, = 7r. As is well known in practice, it follows that large oscillations about 0e, occur instead of the solution given by (a).
2. Inversion and triangular decomposition of a matrix In the process of solving large systems of linear algebraic equations numerically, calculation of the inverse matrix is not advisable because is a costly and weakly accurate operation. It is much more preferable to perform a triangular decomposition, using Choleski's decomposition described in Appendix 5. Doing so, the transformed matrix takes the form: [ K ]  I T ] T [T]
[6.8]
where [T] is an upper (or right) triangular matrix, i.e. a matrix in which all the elements below (or to the left of) the main diagonal are zero. IT] T is the lower (or
196
Discretesystems
left) triangular matrix obtained by transposing [T]. Moreover if [K] is regular, [T] is regular too. Substituting [6.8] into [6.7], the system to be solved takes the convenient form: [T] T [T][q]=[Q(')I=, I [Tr
[q']=Ea")l [ [T][q]=[q']
[6.9]
Then, we can proceed easily to the solution of the two triangular systems involved in [6.9], without having to perform any matrix inversion, provided of course that no diagonal term of IT] vanishes, that is to say provided [K] is regular.
6.3. M o d a l analysis of linear and conservative systems
Obviously, the task of solving any system of the kind [6.6] is made much easier if both [K] and [M] are diagonal. Indeed, in such a case, one has to deal only with a system of N forced harmonic oscillators, which are independent from each other. Solution of any forced problem in statics is thus immediate. Resolution of any forced dynamical problem is reduced to that of a system of N uncoupled ordinary differential equations. Moreover, such equations are linear and their coefficients are constant. The mathematical techniques for solving analytically such equations form the subject of the last three chapters of this volume. In the present section it will be shown that conservative and linear dynamical systems can always be transformed into a diagonal form. This can be achieved by using a judicious linear transformation of coordinates.
6.3.1.
Couplingand uncouplingof the degreesoffreedom
In order to grasp better the problem of DOF coupling and uncoupling in a material system, it is found appropriate to analyse first the system of two harmonic oscillators, coupled together by a linear spring, as shown in Figure 6.1. The Lagrangian of the system is:
,~ I{(MIXI2 + M 2X2 )_ (KIX2 + K2X2)_ Kc(X 1_ X2 )2} 2 which gives the following system of dynamical equations, written in matrix form:
Kc
9
I [o1 M2 X21
K2 +Kc X2
+
=
[6.10]
Natural modes of vibration
197
This system is clearly coupled through the nondiagonal terms of the stiffness matrix. At first sight, such a result seems quite natural since the physical component connecting the two massive bodies is precisely a spring of stiffness coefficient K c .
!
\ /
i
Positions of..
static equilibrium
Figure 6.1. Two harmonic oscillators coupled together by a spring Nevertheless, this point is worthy of further consideration. Indeed, from the mathematical viewpoint, it is clear that the basis of the eigenvectors of [K] allows one to transform [K] into a similar diagonal matrix. Therefore, it is realized that such a transformation lets the stiffness coupling disappear. However, there is no reason why, through the same basis transformation, the mass matrix [M] would be changed into another diagonal matrix. As a consequence, the transformed equations are no longer coupled through stiffness but they are likely to be coupled through inertia! This clearly indicates that: The inertial, or the elastic nature, of the coupling is not an intrinsic property of a given mechanical system but merely a consequence of the choice made in defining the generalized displacements which serve to describe it.
The reasoning above leaves the possibility of the existence of a particular basis in which both the [K] and [M] matrices would take a diagonal form. Obviously, if it exists, such a basis is the key to the problem, as it will allow one to uncouple the system [6.6]. It can be easily shown that such a basis does exist for the system [6.10], provided the two oscillators are identical to each other. Indeed, in this case, any vector [ct flit is a possible eigenvector of [M]; thus, in particular, the eigenvectors of [K] can be selected. When normalized to a unit modulus, they take the transposed forms: [q~]T= 1 [1 1]"
[fp2]r= 1 [1 1]
The physical interpretation of such eigenvectors is clear: The symmetrical shape [r
is said to be "'inphase": the two masses move in
the same direction at any time, with the same amplitude. When the system
198
Discretesystems vibrates in accordance with the inphase eigenmode, the connecting spring does no work. [r is said to be "outofphase": the two masses, move in the opposite direction at any time, again with the same amplitude. When the system vibrates in accordance with the outofphase eigenmode, the connecting spring does work.
The transformation of displacement variables related to the uncoupling is of the kind:
q, = ot(X, + Xe ) ;q2 = ot(Xi  X2 ) where the scaling factor a is arbitrary. Such a result is rather intuitive and could indeed be obtained directly by the elementary procedure of variable elimination. Adopting o~ = 1/2, the transformed system is:
[0
K?2K~][qq~21+[O
O1[~~2]=[~ ]
Therefore, in terms of generalized quantities the "inphase" oscillator is characterized by the mass coefficient M and by the stiffness coefficient K, whereas the "outofphase" oscillator is characterized by the mass coefficient M and the stiffness coefficient K + 2K c . Moreover, the only difference between the generalized and the physical oscillators is the existence of a fixed phase relationship between the two physical displacements, which does not hold in the case of really uncoupled oscillators. It is also of interest to define the following functionals of potential and kinetic energy:
[o,,1~ [~1[~,,1 = KII[~,II
2, LO,er [KIt~,el=(K +2Kc)ll[~ll 2 [~, r [M ][~,] = M I1[0,112 ' [~ ]~ [M ][02] = g 11[~2II2
Such functionals are independent from the coordinate system, provided the basis vectors are normed in the same manner, for instance by the condition:
i1[~,112:11[~2ll~ =1 These very important results, which are introduced in this subsection by taking a particularly simple case, can be extended to any system of the type [6.6], by introducing the concept of natural modes of vibration. This is the object of the next subsections.
Natural modes of vibration
199
6.3.2. Natural modes of vibration
6.3.2.1. Basic principle of the modal analysis Here the embryo of modal analysis introduced in Chapter 5 subsection 5.1.2.3, is extended to the case of NDOF systems. With this object in mind, we search for non trivial solutions of the autonomous version of system [6.6] which are of the harmonic form [q]e i~ , independently from any particular initial conditions Of motion. We are thus led to formulate the problem as the following linear and algebraic system:
[M ]][q] = [0]
[6.11]
Since such a system is homogeneous, it is clear that non trivial solutions, denoted 2 which are the roots of [r ], can be obtained only if 092 takes the discrete values to n, the characteristic equation:
detl[Kl~o~ [M 1] = 0
[6.12]
It is immediately noticed that [6.12] is a polynomial equation of degree N with respect to the variable 2 = ~o2 . Consequently, it has N ,~ roots, some of them being eventually multiple. The natural modes of vibration are then defined as harmonic vibrations characterized by the following complex amplitude:
[[q~,l ei~"' where a), = + x ~ , I
[6.13]
co, is the natural pulsation of the mode indexed by the integer n (n = 1.... N), [q~, ] is the corresponding mode shape.
6.3.2.2. Basic properties of the natural modes of vibration Starting from the properties of symmetry and positive sign of the [K] and [M] matrices, which are inherent to any conservative and stable mechanical system, the following properties of paramount importance may be inferred:
1.
The modal pulsations are positive and the mode shapes are real. At first, we show that the ~,, are necessarily positive, if [K] and [M] are
positive definite. Indeed, letting [~p,,] be a vector solution related to a given 2 n, we can state that:
200
Discretesystems
[~0. ]~ [[K ].~. [M ]] [~0. ]  0 ~ K. .~,,M. : 0
[6.14]
where K. : [~0. ]~ [K][~o,, ] and M. : [rp.]T [M ][q~. ] Relations [6.14] are obtained by performing the scalar product by [fa,] of the terms of the modal equations. In the language of geometry, such a scalar product is interpreted as the projection of the modal equations onto a mode shape vector related to 2 n . Now, since [K] and [M] are assumed to be positive definite, K,, and M, and thus 2, are also positive definite. This basic result is conveniently written in the form of a Rayleigh quotient, which extends to NDOF systems the quotient already introduced in Chapter 5, (cf. relation [5.30]): ~o~
K. 
[~0.]~ [K][~0.] 
M.
<Ev. > =
~
[~0.]~ [M][~0.]
>
0
[6.15]
<E. >
According to the Rayleigh quotient, the modal (or generalized) stiffness can be identified with the functional of potential energy (d'p,) and the modal mass with the functional of kinetic energy (s
The modal pulsation (_o, = +~'~, is also taken as
positive. Its physical meaning is clear, since it is the angular frequency of the harmonic oscillation of the real system, vibrating freely in the nth mode. Incidentally, it also happens that one has to deal with a null stiffness matrix. This is in particular the case of free (i.e. not provided with any support) bodies which have six modes of rigid body at zero frequency. In subsection 6.4.1 we shall come back to this particular case, which is of practical importance in many instances. On the other hand, since 2. and the elements of the [K] and [M] matrices are real quantities, the mode shapes are necessarily real too. The vectors [r provide the relative algebraic amplitude of the displacements of the various degrees of freedom of the system. The term of "relative amplitude" is highly appropriate since [r ] is defined except for an arbitrary multiplicative constant. This is also precisely the reason why [r
is called a mode "shape". Usually a specific value of the
constant is used to define a norm for [r ]. Amongst the most widely used, let us quote the norm of the unit modulus: [[r generalized
displacement
max([r
1, that of the largest component of 1, i = 1,2 ..... N
and finally the mass
normalisation already mentioned in Chapter 5, according to which: [(~,]~ [M][(~,]=I, [(~,,]~ [ K ] [ ( ~ ] = ~ , ~
Natural modes of vibration 201 Finally, it may be also useful to mention that when the generalized displacements of two distinct degrees of freedom have the same sign they are said to be inphase, whereas if they are of opposite sign they are said to be outofphase. 2.
The mode shapes are orthogonal to each other with respect to [K] and [M]
By orthogonality of any pair of mode shapes with respect to the stiffness and the with the following m a s s matrices is meant that the vectors [tpj] and [q~k]c~ conditions:
E(pj ] T [K ][q~k] : {K 0 >otherwiseO if k = j [6.16] {Mo>Oifk=J Eq~,~ [M l[tpk ] = otherwise The most direct way to prove that the conditions [6.16] hold is to transform the modal system [6.11] in such a way as to turn it back to a standard eigenvalue and eigenvector problem of a symmetrical matrix. Indeed, it is recalled that the eigenvalues of a symmetrical matrix are real and the related eigenvectors are orthogonal with each other (for mathematical proof, see Appendix 4). The close analogy between the modal calculation presented just above and the standard eigenvalue problem of a matrix is rather obvious. Indeed, since [M ] is always regular, it is possible to multiply equation [6.11] by the inverse [M ]1, to obtain the standard eigenvalue equation: E[A]2, [l]][q~, ] = [0] where In]= [M ]1 [K] and [I] denotes the unity matrix
[6.17]
However, the range of this first idea remains too limited, since in most instances [A] is not symmetrical, even if both [K] and [M] are symmetrical matrices. Therefore, it is necessary to have recourse to a more subtle transformation, which preserves the symmetry of the modal equation. The key is to transform the massmatrix by using Choleski's decomposition, already invoked in subsection 6.2.2. Doing so, the transformed matrix takes the form: [M 1= IT] T [T] Since [M ] is regular, [T] is regular too. Equation [6.11] is thus rewritten as:
[6.18]
202
Discretesystems [[K]/],, IT]" [T]] [(p, ] : [0]
[6.19]
Then, we introduce the transformed modal vector:
[~,.]= [r][~.]
[6.20]
The transformed system is:
[t, 0, the system is compressed, which turns out to have a destabilizing effect. The system buckles according to the inphase mode at the critical load parameter: ';/.c~=o90 ~ Fc~ = 2 K I L . The outofphase mode is destabilized at the higher critical load parameter: 2c2 = ~t32~. The last result is not surprising since, for the same amount of destabilizing work WFo, the stabilizing springs do more work according to the outofphase mode than according to the inphase mode.
outofphase mode
inph ase mode
Y i I
~
e,
q 
cz
i
....
_x.,
A~A
Figure 6.15. Mode
i AyA
..............x.....,
q
shapes of the system of two articulated bars
Finally, if F0 < 0, the system is tensioned, which turns out to enhance the stiffness of the system. Therefore, the natural frequencies of the two modes increase with the magnitude
IF01 of tension.
EXAMPLE 3 .  Stretched chain o f c o u p l e d massspring systems ~A
i l
i t ....
,d
.......... 9
L
".,,
""
...fig "
.............. #'2" . . . . . . . . . .
direalon of propag aaon of the trat,etUng wa~e
)
.L ,L "r ;' 1/'ralqsp ~tT~ ~ e
Figure 6.16. Stretched oscillator chain and transverse waves
Let us consider once more the chain of N oscillators described in subsection 6.3.2.4. But here we assume that the chain has been stretched in such a way that the state of static equilibrium is now stressed. The length of the stretched chain is L = (N + 1)g, instead of ~ = (N + 1)g 0 which refers to the relaxed state. Further,
Natural modes of vibration
227
we are now interested in the small vibrations which take place in the plane Oxz of the Figure 6.16. The tension in the stretched chain is readily found to be TO= K ( g  g o ) . Denoting Z,, the small transverse displacement of the nth particle, the elastic energy of the first oscillator is calculated as follows:
K
c?e(1) = y(&  s
)2
where ~2 =
~2 + Z 2
The problem being restricted to small linear oscillations, it is left as an exercise for the reader to verify that the following approximation suffices, which is identified with the prestress potential:
~(1) 7~ 2 Applying the same procedure to the other oscillators, the total prestress potential is found to be:
_ r0& s
.
2e o [ 1 + +
z,,ly+ 

.
}
On the other hand, it is also easy to show that in the linear approximation the longitudinal oscillations are independent from the stressed state. Thus, it turns out that the transverse oscillations of the system are governed by the same equations as the longitudinal oscillations, provided the elastic coefficient K is replaced by the prestress coefficient:
The above result shows that the ratio of the natural frequencies of the transverse to the longitudinal modes is simply the square root of the stretching deformation, which is usually a small parameter.
6.4.4. Rotating systems and whirling modes of vibration Let us consider a mechanical system vibrating about a state of permanent rotation, characterized by the spin velocity ~ . [q] denotes the vector of generalized displacements defined in the rotating frame. In agreement with the results established in Chapter 4 subsection 4.3.2, the kinetic energy of vibration is the sum of the three following components, which are suitably expanded in Taylor's series up to the second order:
228
Discretesystems Relative energy is reduced to a quadratic form in [q], involving constant coefficients of mass. The latter constitute the elements of the mass matrix [M ], which is symmetrical and positive definite.
2.
Transport energy is reduced to a quadratic form in [q], involving constant coefficients of stiffness. The latter constitute the elements of the stiffnessmatrix [K0] of centrifugal prestress, which is usually symmetrical and negative definite.
3.
Mutual energy is reduced to a bilinear form in [q] and [q], the coefficients of which constitute the elements of the antisymmetrical gyroscopic matrix [G]. Therefore, the system of modal equations takes the following canonical form:
l[[K]+[Ko(.Q)]+,ia~[G(~)]a~2[M]][q]= iOii
[6.491
In this modal system, s is thus acting as a load factor. On the other hand, though system [6.49] involves imaginary terms, it still remains conservative as already shown in Chapter 4, relation [4.26]. Consequently, the natural frequencies of the system remain real and positive, in so far as the threshold for buckling instability is not yet crossed. On the other hand, the gyroscopic coupling induces mode shapes which are complex. It is worth examining further such a particularity by considering two distinct examples. 6.4.4.1 Particle tied to a rotating wheel through springs
Y
~ y' x K
0
Figure 6.17. Masspoint tied to a rotating wheel through linear springs
Natural modes of vibration 229 Returning to the system shown in Figure 6.17 which was already considered in Chapter 4, subsection 4.3.2, we recall that the equations of motion written by using the displacements X,Y defined in the rotating frame were found to be: MX  2M g21;"+ (K  .Q2M )X : 0 MY + 2 M Y 2 X + ( K  ~ 2 2 M ) Y : O
As a first step, it is of interest to isolate the gyroscopic coupling effect. We start thus by discarding the centrifugal terms. Accordingly, the system of modal equations reduces to:
+2i(o~
[6.50]
(oo (o2
The characteristic equation is:
)(0,0
+
0
[6.51]
which has the two following positive roots: [6.521 The reason for using the subscripts 'F' and 'B' will be made clear below by investigating the mode shapes related to the modal pulsations [6.52]. Substituting (OF,(OB in the system [6.50], the following complex mode shapes are produced: [6.53]
(oF ~ X F =iY F ;coB ~ X B =iYs
Such shapes are conventionally normalized here by the condition X F = X B " 1. On the other hand, the real vibration is obtained by retaining either the real part or the imaginary part of the complex amplitudes:
AF=E1U]eiWFt; AB=[[ +ijleiWst (a)
[6.54]
(b)
Figure 6.18. Forward mode: (a) Re [A F ], (b) Im [A F ]
230
Discrete systems
1. Forward mode The real part of
A e : Ii i] ]e it~ gives:
Re[Al: ]: Re[(ii])(costoFt * isintoet)l=:~
[6.55]
XF = COStort, YF = + sin toFt The imaginary part of Ae : I i"  / j ] e
i~~ gives"
Im[Ar]:ImI(iU)(costoFt+isintort)]~
[6.56]
X e = sin toFt; YF =  c o s toet As shown in Figure 6.18, plotting the motion of the masspoint given either by [6.55] or [6.56], results in a circular motion at angular speed toF, which is oriented in the same counter clockwise direction as the permanent rotation. For this reason, such a mode is known as a forward (or direct) mode. In the inertial frame, the angular speed is found to be toe + ~ = 4to 2 + s
.
2. Backward mode
a
Ty,
(a)
..................... Xl~
.re_X__,
Figure 6.19. Backward mode: (a) Re[A R], (b) Im[AR ]
The real part of
AB = [i +i l i e it~ gives:
Re[A B]= Re [(i" +i ])(cos toBt + i sin tost)] ~ X s = + cos tost; Ys =  sin tost The imaginary part of
A B = [i +i l i e iC~ gives:
[6.57]
Natural modes of vibration 231
Im[A B] : Im[([ + i ;)(cos ogBt+ i sin ~Bt)l =:>
[6.58]
X 8 = + sin coBt; YB = + cos oBt As shown in Figure 6.19, circular motions governed by [6.57] or [6.58] are oriented in the clockwise direction, opposite to that of the permanent rotation. For this reason, such a mode is known as a backward (or retrograde) mode. In the inertial frame, the angular speed is found to be opposite to that of the forward mode: ,.(2  (_hiB " ~/(.t) O "~" ~ 2 2
A straightforward calculation shows that the forward and the backward modes are orthogonal to each other:
Ill [Ov]r [(P,]" =[1
i]
i]
E1 ]
Eli i
Il ]
+i1 +i = 0
i =0;[q~B]r [cpF]* =[1
fifo
4
[6.59]
J
3.s
xx =A'y
2.5
9
21
~
forward
j# 0.5 ~
0! 0.5
0
.0/~o0 ~ 0.2
,
0.4
t 0.6
i 0.8
,
~
~
1
1.2
1.4
,
~
,
1.6
1.8
2
Figure 6.20. Campbell's diagram of the system in the presence of gyroscopic coupling and without centrifugal stiffness, case of symmetrical supports Ky = K z
In Figure 6.20, the natural frequencies of the system are plotted versus the speed of permanent rotation. Such a plot, broadly known as a Campbell diagram, is presented here in a dimensionless form, by using co0 as a pertinent scaling factor. It is noted that the backward frequency tends to infinity with s
0, whereas the
forward frequency tends asymptotically to zero. In this particular system, the
232
Discrete systems
gyroscopic coupling is found to increase the frequency of the backward mode and to diminish that of the forward mode. However, we shall see later that the trend is inverted in many other systems. On the other hand, even if either the backward or the forward natural frequency tends to zero when s tends to infinity, the gyroscopic effect does not induce any buckling instability of the rotating system. Taking now into account both the gyroscopic and the centrifugal effects, the modal system is written as:
ioo
I (.l)  .0 2  (1)2
(1)2 __,('~2 __0 2
+2ito$2
E0o]
y
[6.60]
The characteristic equation is:
(,002a
+) ( , O g  a '  , o '2 oa)=0
[6.61]
which has the four roots: [6.62]
a~] =to o  ~ 2 ;a~ =tOo,f2 ;to3 =a~o + Q ;to+ = O  t o 0
I
p/0
..~
oeeo~ e
44eO
4 eToO
2.5
~ ~y
.~
.~ ..~176
..~176
...o...."~~176 "*~"
.~..'~ ~176176 .
\ \
~ .~176176176176176 "
1.5
..~176
9backward modes
ooO ~176176
\
~
"''**'~176
0.5
"+.%
forward mode ..% ~ o
.,."" .. .~ ~
~.
.+.~
*..~176 " ..+
026
9
~176 9
",~
o18
,~ ,~ ~176
~ ~
,
~176"'
,12
1"//~0
i24
,26
ils
Figure 6.21. Campbell's diagram of the system in the presence of gyroscopic and centrifugal effects, case of symmetrical supports Ky = K z
Restricting the analysis to non negative frequencies, it is found that the forward and the backward modes have the natural pulsations:
Natural modes of vibration w~ = w o  s
233 [6.63]
;w B = ~ + ~ o o
In the range .Q < too, there is a forward and a backward mode and it is found that to~ increases linearly with s
whereas top decreases linearly with g2. The frequency
of the forward mode vanishes as .(2 = too. In the range .(2 > too, it is found that two backward modes are coexisting, their frequency increasing linearly with s see Figure 6.21. Because of the presence of gyroscopic coupling, analysis of stability is less straightforward than in the case of systems vibrating about a static state of equilibrium. However, it may be noted that the four roots [6.62] of the characteristic equation are real at any spin velocity. Therefore, in contrast to the systems analysed in subsection 6.4.3, there is no bifurcation, marking a change of sign of stiffness and consequently, no buckling. However, the analysis of the same system, performed in the case of asymmetrical supports (Ky ,: Kz), evidences the existence of a finite interval of angular speeds into which the system is buckled. This interval is found to be precisely equal to the gap between the natural frequencies of the uncoupled system at s = 0, see Figure 6.22.
f/fo
4.5
Ky =
4Kx
.................................. :
.~176176176176176176176176176176176 ~176176176
3.5
~~ .~ o..~176176176176176176176176176176176 ,,.~176176176176176176176176 ~ 2.5
backward
.................................... forward
1.5
sS ...... ..... ~.~
0.5
00
.......... ~
~oS
":
\
domainoJ
~I buckling ~176176176176 i .........~_
: ~: : I
..............
05
1
1.5
ii..........]I............
......
.O/toO
~176176176176176176 ,.~ i
2
2.5
3
Figure 6.22. Campbell's diagram of the system in the presence of gyroscopic and centrifugal
effects, case of asymmetrical supports Ky r K z
234
Discrete systems
6.4.4.2 Flywheel on flexible supports The system to be discussed here is sketched in Figure 6.23. It is made of a rigid disk locked at midspan to a rigid axle, the ends of which are supported by linear springs acting into two transverse directions (fixed unit vectors j , k ), orthogonal to each other. It is found convenient to define the inertial Cartesian frame Oxyz, of unit vectors i, j , k . Origin O coincides with the centreofmass of the disc.
~ 2 ~
*K"~ ~UyY ), R
x
~
Figure 6.23. Flywheel on flexible supports
The system spins about the axle at the angular speed s where 1 is the unit vector along the axle in its actual configuration. It can be defined as the transform of i through the small rotations ~uy, ~uz about the axis j , k which describe the rocking modes of vibration of the system about O. We are interested here in investigating how such modes are coupled by the permanent spin. Equations of small vibrations can be obtained either as a direct application of the theorem of angular momentum (cf. Chapter 1, subsection 1.2.3.2, example 3) or by using the formalism of Lagrange equations. In the present problem, the first method is the easiest one. Indeed, as we shall see a little later, calculation of kinetic energy is not so straightforward here, because the rate of rotation involves the large component $2I.
1.
Vector formulation: rate of change of angular momentum In the absence of vibration, the angular momentum of the disk is: =Js
where J 
MR 2 2
In the presence of a vibration according to the rocking modes, it becomes:
[6.64]
Natural modes of vibration 235
{_,= I (q2yj + 1/)'~/~)+Jg2(q/~j~r/~ ) where/ = ~J
[6.65]
The equations of motion are thus readily obtained as a direct consequence of the balance of moments (cf. equation [ 1.27]), which takes here the form: L = restoring moment exerted by the springs Therefore, the following equations are obtained:
Zy._  2 K g 2~y
[6.66]
L~. = 2Kg2~z Substituting the result [6.65] in [6.66], the equations take the form:
I ~r + 2 1 Q ~ t + 2 Kg21gy = 0 I(/2~  21,Q~y + 2KgZyt~  0
[6.67]
whence the modal system: 03o (0 2 2iw..(2
+ oolE,,yl:[Ool wl, reo  2Kg 2
eO~  r.O2
yt z
[6.68]
The system [6.68] is quite similar to [6.50], the only difference being the sign inversion in the gyroscopic coupling. Accordingly, the modal pulsations are: =
+a
;,o. =
+,Oo  a
[6.69]
The sign inversion of the coupling implies that of the evolution with s of the forward and backward modes is also inverted, see Figure 6.24. The physical displacement of the end A of the axle is given by:
Z(t)=gReEq/yei~ [6.70]
Y (t)= +gRe[q/zei~
+gsina)Ft
As expected, the vibration takes place in a plane passing through Ox, which rotates in the forward direction ,(2 at the angular speed toF . It is worth emphasizing that, in contrast with the former example, in the present case rotation of the modes is relative to the inertial frame.
236
Discrete systems
f(.O)/f(O) o O"~
forward
...'"" .""
mode
ooO~176 o ~176176176176176176176 ~ 9149176176 o~
~ ~149 o~176 ~176176
~ ~
~
o~176
backward
.....
o~176176149 9176176149176176176
made
a,~o) .......
0
0
O5
1
o. . . . . .
...................................
15
2
9. . . . . . .
2.5
oo
3
Figure 6.24. Campbell's diagram of the disk on flexible supports
2.
Lagrange's equations
The crucial point in formulating Lagrange's equations of this system is to produce a correct expression for the kinetic energy of rotation. Use is made of the noninertial Cartesian frame, Ox'y'z', which is defined in such a way as to keep the same notations as in Appendix 3. Accordingly, Oz" is the spin axis and Ox', Oy' are an orthogonal pair of principal axes of inertia of the disk. Therefore, kinetic energy of rotation can be written as:
s = 1{1 (s + Or2,)+ j~22}
[6.71]
where ,O' is the vector of rotation rate as defined in the frame Ox'y'z" which is corotative with the disc. The problem now is to determine the components of ,O' in terms of the angular velocities as defined in the inertial frame. Thus, it may be noted that the frame transformation required here is the reciprocal of that used in the first example. As a large rotation rate s is present in the problem, it is no longer possible to linearize the frame transformation. It may be worth noting that such a linear calculation would lead to wrong results, as spurious centrifugal forces which have no physical meaning would appear, since the system rotates about the centreofmass. In order to carry out the pertinent calculation use is made of Euler's angles. The procedure is detailed in Appendix 3.
Natural modes of vibration 237 In the frame used to define the Euler angles, the Lagrangian can be written as: ,~ = I (q)2 +q/~92 )+ J (~rq)~ _ ~ u 
7
)_ Kg2 (I//2 + ~ 2 ) 
[6.721

Lagrange's equations are:
lOx +21~y +2Ke2~x=~1
l~y 21~(trx +2KgZ~y
[6.73]
Of course, the equations [6.73] differ from [6.67] only by the different labelling of the coordinate axes. The following permutations: z ~ x, x ) y, y) z allow one to return to the definition used in [6.67]. Dynamical behaviour and stability of rotating systems will be further investigated in Volume 4, Chapter 2, including damping, see also for instance [GEN 95] and specialized books such as [LAL 90], [CHI 93], [KRA 93], [ADA 01].
This Page Intentionally Left Blank
Chapter 7
Forced vibrations" response to transient excitations
The study of the forced responses of material systems to various kinds of external excitation is usually the central object of modelling. Determination, or at least some global description, of such responses is clearly of major concern in many fields of physics and of mechanical engineering. In contrast to the vibration modes, which remain an abstract concept, forced responses are amenable to measurement. Furthermore, they control operating and mechanical integrity conditions of most material devices and structures. The problem of solving the equations of motion is often extremely difficult, requiring generally the use of advanced computational methods. This is mainly because of the presence of nonlinearities. However, linear Lagrange's equations may usually be solved analytically, provided that the coefficients are constant and that the excitation has a suitable analytical form. Here, we describe a general method for solving this kind of problems based on the Laplace transformation, which allows one to replace the time dependent differential equations by algebraic equations, expressed in terms of the Laplace variable. As will be shown, this method presents several advantages over the direct method of solving the differential equations in the time domain.
240
Discretesystems
7.1. Introduction Let us consider a discrete and linear mechanical system, which is excited by an external force vector, applied starting from time t = 0. As already seen in the preceding chapters, the motion is governed by a set of ordinary differential equations with constant coefficients and of second order with respect to time. The problem is fully determined if the equations are provided with a suitable set of initial conditions, which specify the dynamical state of the system at t = 0. Equations of motion are thus written as:
[K][q]+ [C][q]+ [M ][/~] = Ie (`) (t)] [7.1] [q(O)J=[qo];
Eq(O)~=[qo];
[Q(')(O)I=[O]
It has to be emphasized that the vector of external force [Q(')(t)l is assumed to be identically zero at any time in the range t < 0, i.e. before motion is studied. In this chapter, a general method is described for analysing the system [7.1]. Furthermore, by applying the method to a few simple cases for which analytical solutions are available in closed form, it is possible to highlight the major features of the dynamical responses to a broad class of excitation signals, known as transients. As a preliminary, section 7.2 describes the mathematical properties of the deterministic transients, which are physically relevant as excitation signals in mechanics. Section 7.3 deals with the forced responses of SDOF linear systems. Analysis of such problems is performed by using the Laplace transformation. For this mathematical tool, a brief review of the definition and the few properties that are necessary in our applications are presented. The reader, who would like to deepen or refresh his/her, theoretical background on the subject, is referred to standard textbooks such as [DETT 84], [ZEM 65]. In the domain of the Laplace variable (the so called image domain), response is conveniently expressed as the product of two distinct functions. One of them, known as the image of the external loading, characterizes the excitation signal and the initial conditions, whereas the other one, known as a transfer function, characterizes solely the response properties of the excited system. Thus, in terms of Laplace transforms, a very clear distinction is made between the intrinsic properties of the excited system and those of the external excitation, in such a way that the influence of each of these quantities in the response can be investigated separately. The inverse Laplace transformation is then used to obtain the response in the time domain. At this step, mathematical difficulties may arise, because the inverse Laplace transform is available analytically, or not, depending on the analytical form of excitation signal. Fortunately, in practice the study of a few typical transients leading to simple analytical solutions is sufficient to illustrate the main features of the response signals which are of major interest to the engineer.
Forced vibrations
241
In section 7.4, the harmonic oscillator is used to introduce the concept of Green's function also called unit impulsive response in the language of engineering which is defined as the inverse Laplace transform of a transfer function of the system considered, or in more physical terms, as a response to an impulse of unit magnitude, applied to a single degree of freedom. When extended to MDOF and continuous systems, Green's functions provide us with a powerful tool for analysing various dynamical problems, e.g. waves travelling in continuous media, as further discussed in Volumes 2 and 3. These are also often encountered in modal testing procedures, since excitation of the structures by an impact is an experimental technique which is widely used. In section 7.5, the formalism is extended to the case of MDOF linear systems. Their response properties are now entirely characterized by a transfer function matrix, which is generally full. Fortunately, it can be uncoupled by projecting it on a modal basis, provided a suitable model for damping can be adopted. Then, Newmark's implicit algorithm for numerical integration of the equations of motion, which was introduced in chapter 5 in the case of SDOF systems, is extended to MDOF systems. Indeed, implicit algorithms present a few interesting particularities worthy of mention when passing from SDOF to MDOF systems, in particular, the solution procedure becomes truly implicit. Application is made to a heavily damped system in modelling the shock absorber in a car suspension.
7.2. Deterministic transient excitation signals 7.2.1 Locally integrable functions and regular distributions Let us start with deterministic and real functions of time f (t). By definition, such functions connect to each other the real numbers t~ and f (t~), where t t lies within the range of definition o f f (t). Moreover, the mapping tI ~ f (t~) is deterministic, which means free of any uncertainty, or randomness. Restricting study to the physical context of mechanics, f (t) is assumed to be a continuous or a piecewise continuous function of finite magnitude: [ f (t)[ < B V t
[7.2]
Bounded functions with a finite number of jumps are locally integrable, which means that:
f
I f (t)[ dt does exist, V the finite interval r I,t
Such functions generate regular distributions defined by the integrals:
[7.3]
242
Discretesystems
(f ,I//):
f (t)I//(t)dt
[7.4]
where use is made of the functional vector notation of the scalar product. Accordingly, f(t) and Ip'(t) are interpreted here as functional vectors, as detailed in Appendix 1. On the other hand, ~ (t) designates an auxiliary function, termed test
function that complies with the following very restrictive conditions: 9
~ (t) is identically zero, outside a finite interval z. (t) can be differentiated up to any desired order.
N O T E .  Theory of distributions The concept of distributions, introduced first by Dirac, and then formalized mathematically by Schwartz, is a convenient tool for analysing mechanical systems. Appendix 6 may be used as a mathematical refresher concerning the few definitions and theorems which are necessary for our applications. To the reader interested in the subject, the book by Stakgold [STA 70], in which there is a clear and pragmatic presentation of the mathematical formalism, is recommended. Amongst many other references of interest, let us quote [SCH 50], [BRE 65], [ZEM 65].
7.2.2 Signals suited to describe transient excitations For mathematical convenience a signal suited to describe a transient excitation is first defined as a locally integrable function that complies further with the following conditions: 1.
f (t) ~: 0, only if 0< t < r
2.
f (t) is upperbounded. As a corollary, such transients are integrable in any time interval:
p=
I f (t)dt=
f (t)dt= f (t)dt=(f(t),I)
[7.5]
oo
According to this definition, p is the action of the excitation signal over its total duration r. As already stated in Chapter 1, ifj~t) stands for a force, p is a linear momentum, and if fit) stands for a moment, p is an angular momentum. On the other hand, when writing the integral of action in the functional form (i.e. as in the last expression of [7.5]), it is considered that the unit function can be considered to be a particular test function, since the integral is restricted here to the finite interval r.
Forced vibrations 3.
Furthermore it is also required that
E=
X ~
Considering
(f(t))z dt=
f(t)
I
0
f(t)
243
be square integrable:
(f(t))z dt=(f(t),f(t)) 0 only, that is during the free stage of the response. Therefore, during the forced stage, the timehistory of motion is given by the integral:
i a+i**
1
4n'2e ~
r (~') = ~t~ Ja_i.. S (4~2 + 4l't.fflS+ S2) .~,.~,@
ds = A~ + ~
+@
[7.39]
are the residues belonging to the three poles: [7.40]
The following intermediate results are then easily found:
4//.2 e~, r
4//.2 e#:
=
=
@r 2 
s,s
SiS2 = 4~2
; Sl _ S2 = _4~i41 _ ~.2 = 4~'izl
~l ( r ) = 1+
s2ei*'r_ S l eis2r 4~zi z~
r (r)'" 1 
e
2#~!r
2iz~
{(ffl +izl)e2i~z: +(gl +izl)e2i'~: }
Finally, the timehistory of response is written as:
~ (r) = l e2~C:{c~
r)+ ffL~sin [7.41]
where zt = 4 1  ~'12
Forced vibrations
255
If damping is sufficiently small, the last term of [7.41 ] can be neglected: ~:1( r ) = 1  e 2~'~ cos(2zcr)
r e [0,0]
[7.42]
Incidentally, the result [7.42] shows that the response of the oscillator to a step load is the sum of the response to the static load defined by the magnitude of the step and of a harmonic oscillation at the natural frequency of the oscillator, induced by the initial jump in the load. Provided damping is sufficiently small (typically 91 < 0.1 ), the conservative approximation is sufficient to assess a reasonable upper bound to the maximum amplitude of the response in relation to pulse duration, which is information of major interest to the mechanical design engineer. Therefore, in what follows we assume that ~'~ = 0. Motion during the free stage starting at r = 0 , can be deduced from the previous one simply by using the delay theorem. From the simplified result [7.42] it is found that:
r e
1.
z,~c,(,0)
cos
(2,(r

O)) e 2~'' cos(2n'r) 
r >O
[7.43]
Pulse of long duration: 0 > 0.5
The timehistory of displacement is plotted in Figure 7.5. During the forced stage, the response is a harmonic vibration of unit amplitude, at the natural frequency of the oscillator, which is centred about ~:s = I. During the free stage, a similar response is observed, except that the free oscillation is now centred about ~x= 0. The magnitude of displacement is always less, or equal to twice the static value. The first occurrence of a maximum (or peak value) is at time rl,,, = 0.5, i.e. the halfperiod of the free oscillation. The first maximum of the free motion is defined by: 1+20
[7.44]
Its value is thus less or equal to that already reached during the forced stage. Incidentally, it is noted that if the duration of the pulse is an exact multiple of the natural period of the oscillator, the free stage oscillation vanishes. Indeed, the oscillator is exactly put at rest at the end of the pulse and it will remain at rest for ever, if not excited again.
256
Discrete systems
2.5
~(r) ..........
ia = 2.1
~'a
1.5
0.5
0.$
L
I
I
I
Figure 7.5. Response to a pulse of long duration (dashed line: excitation, full line: response) 2.
Pulse of short duration: 0 < 0.5
Figure 7.6 shows a typical response to a short pulse. Its amplitude increases steadily during the forced phase up to:
~:, (0)= 1  c o s (2JrO) < 2
[7.45]
Maximum displacement occurs during the free stage:
~ = 2 sin (nrO) }~
,;
9 < f(t),g(t)>=~
,#
_atg(tldt"~
,X o
g(t) dt=.~ at g(t) dt
For At 4 0, one obtains:
> A,,O .im_' f~ g (t) dt = lim l ( z l t At .I~
lim
,g(t) =
A/,O
~0
g(e))
At
where At 0.5 ), the maximum displacement is now given by: Isin(~O~ ~:m = 1 +
~rO
[7.48]
Forced vibrations 259
Figure 7.8. Trapezoidal transient with a nonzero rise time
max(q(t)/qs)
2~ 1.5
0
8>0.5
0.5
1
1.5
2
2.5
3
3.5
Figure 7.9. Maximum displacement versus the rise time of the pulse Figure 7.9 is a plot of the function [7.48]. It shows in particular that the discrepancy between the static and dynamic magnitude of the response is less than 10%, as soon as O becomes greater than about four. From the foregoing analyses, the two following conclusions of practical interest arise, which hold for every transient presenting a single peak: 1.
2.
The response amplitude of an oscillator excited by a single peaked transient is at most twice the response to the static load of peak value. The longer the duration of the transient ( O > 0.5 ), the shorter the rise (or decay) time, the closer to this upper bound is the actual peak value of response. A damping ratio up to a few percent does not modify the above results very significantly.
260
Discretesystems
The second point stems from the fact that, as already shown in Chapter 5, the characteristic timescale for damping is: r a = l/(2r~')
[7.49]
Therefore, if ~" is sufficiently small, r a >> 0.5. Nevertheless, the above statements must be drastically modified if the transient excitation presents several peaks, acquiring thus an oscillatory nature. This point is evidenced in the next subsection, based on a simple analytical example. 7.3.4.3 Response to a truncated sine function
1 0.8 0.6 a = l.J
0.4 0.2 0 0.2 0.4
0 = 6/a=
4
0.6 0.8 1 f
o
I
1
~
~
,
i
~
,
Figure 7.10. Sine function truncated to six periods
Using the same scaling factors as those already defined in subsection 7.3.4, the analytical expression of the signal displayed in Figure 7.10 is: (r) = sin (2n'ctr){2/(r) 1./(r 0)}
[7.50]
where a is the reduced frequency of the signal. The oscillator is assumed to be conservative and initially at rest. In the image domain, the response during the forced stage is found to be: 8~30~
Forced vibrations
2
w

261

1.5t ~(r) A
~'=oa=l'5
1
0 ~6/a
O.SO
 4
free stage
....................
1.5[ "20
V
z 1
,
2
3
I

4
I
5
6
i
!
Figure 7.11. Response of the undamped oscillator to a truncated sine
2

,
,
i
1.s t ~(r) A 1
i
,
t

a ~. 1.5;(= a
o ,~
0"I_
4
5
6
7
8
Figure 7.12. Response of the undamped oscillator to a truncated sine Let us consider first the nonresonant case o: ~: 1. The inverse Laplace transform is~l (z')= ~zsin (2gz')sin (2mzr)
t7521
262
Discrete systems
This result shows that the total response is the sum of a free oscillation at the natural frequency of the oscillator, which is caused by the starting effect of excitation, and of a forced oscillation at the frequency of the excitation signal.
1
!
~
!
i
!
!
~(r) 1.~
84
i
i
a
i
0 4.33...
r
=1.5
1 0.5 0 0.5 1 1.5 2 0
i
l
2
4
~
t
t
,
l
6
8
10
12
14
Figure 7.13. Response of the damped oscillator to a truncated sine Figures 7.11 and 7.12 show two samples of timehistories, referring to a = 1.5. In the first case, the oscillator remains at rest during the free stage, since duration of the transient excitation is an exact multiple of the natural period of the oscillator. This peculiarity may be used to check errors induced in algorithms of numerical integration of the equation of motion. In the second case O = 3.66... so the free stage oscillation is clearly present. Figure 7.13 refers to a similar calculation for a damped oscillator (~" = 2%). The magnitude of the response during the forced stage is still very close to that calculated in the conservative case, and a damped oscillation is taking place during the free stage. Finally, Figure 7.14 refers to the same case as Figure 7.13, except that damping is greater and that duration of excitation is practically infinite (equal or longer than the duration of the computed response). As evidenced in Figure 7.14, motion during the forced stage tends progressively to a steady harmonic oscillation at the frequency of excitation. The underlying reason for such a result is the exponential decay of the free vibration.
Forced vibrations 263
21
';
'
1.5
.
I : I I I
.
.
a = 1.5; (=
.
4%
I
O > 40 "
1
_
0.5 0 0.5 ! I I I
ransient regime
1.5
! :I
steady regime
z
!
2
0
5
10
1
20
2
30
35
Figure 7.14. Response of the damped oscillator to a truncated sine, case 6 >oo Therefore it is useful, in practice, to separate the forced stage into two distinct regimes. The first one is a transient regime during which the presence of the free oscillation, induced by the starting effect of excitation, is still significant. The second regime is a steady vibration resulting from the sole persisting response component, namely the response forced by the sine. Of course, the larger the damping, the shorter is the transient regime. In terms of energy, the existence of steady vibration indicates that, when averaged over a cycle, the work produced by the excitation is exactly balanced by the energy dissipated by viscous damping. In this respect, the present situation is similar to that of the limit cycle of the autonomous Van der Pol oscillator (cf. Chapter 5, subsection 5.2.3).
It is then of particular interest to discuss the nearly resonant case a  ~ 1, as it provides us with the opportunity to introduce the beating phenomenon, which is encountered in many fields of physics. Basically, beats result from the linear superposition of two harmonic signals of similar amplitude and of similar but still distinct frequencies: sin (c~ * ~ )* sin (co2t*r AO)
where
)= 2sin (AoJ. A ~)sin (coc . ~ c )
(/)1 (/)2 ," A~r  l/fl ~r
2 o) = o)1 +co~. g~ 2
2 g~ +~'2 2
[7.53]
264
Discrete systems
10
i a=0.9;(=O
~(r
5
5
.10t
i
0
101 15
~
1
I
l
4
6
8
10
i
i
I
1

a=.0.9"(0"0>30
g(r)
s
~"
I
2
I
~ ,,
,/
\
,, I
,,
\
,,
5:
x..
10
envelope
"150
5
1o
./., 15
2o
25
30
Figure 7.15. Response to nearly resonant harmonic excitation The result [7.53] shows that the response resulting from the interference between two harmonic signals of close frequencies co1,co2 may be interpreted as an harmonic signal occurring at the central frequency a~c , which is modulated in amplitude at the much lower frequency Aco. Accordingly, the response involves two quite distinct timescales T~ = ~r / a~c, and T2 = ~" / A(t).
Forced vibrations
265
Coming back to the present problem, the general result [7.52] can be rewritten as: sin (~er)cos (2n'r) [7.54]
E
where a = 1 + e, IlelI T~). Figure 7.15 shows two typical responses to a nearly resonant excitation. The first plot corresponds to a relatively short duration transient T~ < 0 < T2 and the second corresponds to a long lasting transient O > T2 . In terms of energy, it can be noted that the mechanical energy of the oscillation, during the forced stage, varies according to the slow timescale T2 . Indeed, it is easy to show that:
/2
esin (n'er)
[7.55]
Thus, the work done by the nearly resonant excitation changes sign at the low modulating frequency. Finally, we consider the resonant case a = 1. The response of the undamped oscillator, in the image domain, is found to be: ~ (s) =
4n'2
[7.56]
Multiplicity of the poles s = +2ig equals two and the method of residues provides the following solution in the timedomain: ~:~( r ) = ~ { s i n ( 2 n ' r )  2~rr cos (2~rr)}
[7.57]
This result clearly shows a drastic change in the motion, when shifting from a non resonant or nearly resonant excitation to an exactly resonant one. The second term of solution [7.57] includes a linear envelope which modulates the amplitude of the harmonic oscillation. Accordingly, from one cycle at the natural frequency to the next, the magnitude of motion is found to increase proportionally to the elapsed time. In terms of energy, the work produced by the resonant force is now always positive, when averaged over the natural period, in contrast to the non resonant case, see Figure 7.16.
266
Discretesystems
100
80
,,~''~d'd''~a''p'~"~'~"
60
i
40 20 0
20 40 60
t~,,~
. . . .
'J..t !  I U i nl
80 100
I
I
0
Figure 7.16. Response to a long lasting resonant sine excitation 3O
20
10
0
10
20
30
h
'
Figure 7.17. Resonant response of the dissipative harmonic oscillator
The presence of dissipation modifies this unrealistic result. Indeed, in the presence of damping it is found that, provided 0 is sufficiently long, the response tends to a steady oscillatory regime in which energy produced by the external force is dissipated by the viscous damping force, see Figure 7.17. The response in the steady regime is a cosine, the magnitude of which is proportional to the reciprocal of damping: 4n~ ~I = (4~r2 + s 2 )(4~r2 + 4~6.1s +
s2)
cos (2n'r) =~ ~:f = 2ff1
[7.58]
Forced vibrations 267 The major conclusion of practical interest arising from the result [7.58] may be put in words as follows:
A resonant excitation applied to a harmonic oscillator during a time larger than the timescale of damping induces a steady oscillation, the magnitude of which is inversely proportional to the damping ratio. Thus, when the system is lightly damped, as it is often the case, the magnitude of the dynamical response is much larger than that of static response. This will be emphasized again in Chapter 9 by adopting a spectral standpoint. By itself, such a result suffices to motivate the dimensioning of structures against vibrational problems, when they are intended to withstand dynamical loads.
7.4. Impulsive response and Green's function 7.4.1 Green's function of a harmonic oscillator The response in the time domain of an oscillator to initial conditions is easily calculated by using the residue theorem. In the case of subcritical damping, which is of major interest for structural dimensioning against vibration problems, it is found that:
q(t)e~"{qoIcos(~Oat)+ ('/)a = O)l r
~',
)
sin(a~at ) +q~
t
 ~"12
which identifies of course with the result already established in Chapter 5 by solving directly the differential equation of motion (cf. subsection 5.1.3.1). We consider now the response to a unit impulse occurring at time to . This response is known as a Green's function of the oscillator. It is given by: m
G ( t  to) = f e
0
0 Tk
Figure 7.19. Transient signal decomposed as a sequence of successive pulses The Green's function allows us to express the response of a harmonic oscillator to any transient excitation Q(e)(t) as a convolution product. This is an immediate consequence of the convolution theorem (formula [A7.8] of Appendix 7) as applied to relation [7.23], which gives:
q(t)=Q(e)(t)
=
Q(O(r)G(tr) dr
[7.62]
Forced vibrations 269 Physical understanding of such a result can be gained by splitting up Qte) (t) into a sequence of impulses. Indeed, it may be noted that:
a(e)(t): f~a(e) (~,)~(tT) d~"
[7.63]
On the other hand, 8 ( t  v ) can be obtained as the limit of a sequence of the rectangular pulses displayed in Figure 7.18. Accordingly, we have:
Q(e) (t)
I~Q (e)(~')~( t  r )
d r = !irn( h k~Q (e) (z"k )8, (tz" k ))
[7.64]
The series [7.64], where time v~ is within the interval [tk , tk + h/n], is suitable to state the Riemann's integrability of
Q(e)(t). On the other hand, the same series
can also be interpreted as splitting up Q(e) (t) into a sequence of successive pulses, the number of which tends to infinity and width of which tends to zero as n tends to infinity, see Figure 7.19. Since the response of the oscillator to the impulse
Q(e) (z)8(t v) is Q(e)(v)G(t z), the result [7.62] appears as a mere consequence of the superposition principle, which holds for any linear system. According to this principle, the response can be built by summing the individual responses to partial excitations. In the present problem, the latter are made up of the successive impulses, which result in Q(e) when superposed. Before leaving the subject, it may be noted that the formulation [7.62] is more interesting as a nice theoretical result than as a convenient formula to calculate the response of the oscillator. Actually, in most instances the convolution integral is not easily calculated analytically, and its numerical computation is cumbersome. Nevertheless, a very simple application of [7.62] will serve us in Chapter 8 to introduce the spectral analysis of linear systems.
7.5. Response of MDOF linear systems 7.5.1 Transfer function matrix of a conservative system The Laplace transformation of the matrix system [7.1 ], without damping, results in: [7.65]
270
Discrete systems The relation [7.26] becomes: [7.661
In principle, the system [7.65] can be solved by inversion. One is thus led to define a transfer function matrix In(s)], in such a way that:
where
[[g]+ s'[M]l'
[7.67]
The matrix equation [7.67] is the natural extension to MDOF systems of the formalism already introduced in subsection 7.3.2, in connection with the harmonic oscillator. The transfer function matrix enables us to relate the response, a displacement in this case, of any individual degree of freedom to the excitations acting on the whole set of the degrees of freedom of the mechanical system. Clearly, the concept of the transfer function matrix can be extended to any linear system, mechanical or not.
Figure 7.20. Transfer box of a linear system with N inputs and N outputs
The transfer box shown in Figure 7.20 extends that of Figure 7.2 to linear systems provided with N inputs and N outputs. Every element Hi1 of the transfer function matrix can be interpreted as a transfer function connecting the input of index i to the output of index j. In the case of multiple inputs (excitations), the output of index j is the superposition of the partial responses of the jth DOF to the individual inputs.
Forced vibrations
271
On the other hand, in the case of conservative mechanical systems, the matrices [K] and [M] are symmetrical. Thus, the transfer function matrix is also symmetrical. A corollary of such a symmetry is known as Maxwell's theorem of reciprocity according to which:
The response of the jth DOF to a unit load applied to the ith DOF is the same as the response of the ith DOF to a unit load applied to the jth DOF.
Finally, as an immediate extension of SDOF systems, Hu(s ) is the Laplace transform of the response of the jth DOF, to a unit impulse applied to the ith DOF at time t = O: [7.68]
TL' [ Hij ( s ) ] = G 0 (t )
where Gij (t) is the Green's function relative to the ith input and the jth output.
7.5.2 Uncoupling by projection on the modal basis
7.5.2.1 Principle of the method Using directly formula [7.67] to calculate the response of a NDOF system would be very clumsy, at least as soon as N > 2. Of course, it is advisable to use the symmetry properties of the operators [K] and [M] of the conservative mechanics by projecting first the equations of motion on a modal basis. (cf. Chapter 6, subsection 6.3.2.2). Let us thus consider a mechanical system described by using physical coordinates (or displacements), which is written in the image domain as:
[M
(,)]
(,]]
[7.691
As already discussed in Chapter 6, the basis of the natural modes of vibrations is complete and the mode shapes [(ak] can be used to define the column vectors of the modal transformation matrix [~] (in short, the [~] basis). Therefore, the solution for any forced problem of the type [7.69] can be decomposed in a unique manner on the modal basis, by projecting it on [~]. As a result, the solution is expanded as the modal series:
[[ ] N =
k=)
][
[7.70]
272
Discretesystems qk (s) are the natural coordinates of the vector [,~ (s)]. Moreover, since [(I)] is
orthonormal with respect to [K] and [M l, the modal projection of [7.69] produces a diagonal system, the current line of which reduces to the equation of a forced harmonic oscillator: "'((o~+s2)~lk(S) O'k(s)" k = l , 2 .... N Mk '
[7.71]
where o k is the natural pulsation and M k is the generalized mass of the kth mode. Qk (s) is the generalized force applied to the kth mode, which is defined as the projection of the physical load I F (s)] on the kth mode shape: [7.72] Of course, the physical load can also be expanded as a modal series, since it is a vector pertaining to the same vector space as the response vectors. Hence, we have: N
[ P (s)] = Z 0 k (s)[fak ]
[7.73]
k=l
Accordingly, the generalized forces Ok (S) are simply the coordinates of IF(s)] on the It/i] basis. Thus, one is naturally led to the following statement: The concept of natural modes of vibration enables us to transform a forced and coupled system, of size N x N, into a forced system of N harmonic oscillators, which are uncoupled from each other. As we shall see on many occasions hereafter, such a result is of paramount importance, from the viewpoints both of physical modelling and computation. 7.5.2.2 Modal expansion of the transfer and Green's functions Let us consider a unit impulse applied to the ith DOF of a NDOF material system, which is assumed to be initially at rest. In the timedomain the (transposed) load vector is written as: [F(t)]r =[0
0 ... 0 ~ ( t ) . . .
O]
[7.74]
where 8 (t) is located at the a ith column. The Laplace transform of the load vector is: [F(s)]~=[O
0 ... 0 1 ... O]
[7.751
Forced vibrations 273 Projection on the modal basis produces the following vector of generalized load: [7.76]
[(2(s)] ~ = Eo , (i) q~2(i) ... ON (i)]
where Cpk(i) stands for the ith component of the kth mode shape, in agreement with the notation already adopted in Chapter 6, subsection 6.3.2.4, for describing the mode shapes of a chain of oscillators. Solution of system [7.71] is immediate. Then, the use of formula [7.70] produces the expansion in modal series of the transfer function connecting the displacement of the jth DOF to the force applied to the ith DOF, which is written as: N
[7.77]
H (i, j ; s ) = ~ (Pk (i)(Pk (J)
,s')
It is convenient to put the indices i and j as arguments of the transfer functions and mode shapes for future extension of the formalism to continuous systems. Indeed, it will suffice to replace the indices by the coordinates of a pair of points in a Euclidean space, namely the loaded point and the point at which response is calculated, or measured (see Volume 2). The inverse Laplace transformation of [7.77] gives the expansion in modal series of the impulsive response (or Green's function), which is immediately found to be:
G(i j;t) ~ (Pk(i)(Pk (J) '
k=l
sin(cOkt)
[7.78]
MkCO~
According to either relation [7.77] or [7.78], an impulse applied to the ith DOF is able to excite all the natural modes of the physical system, except those which are orthogonal to the impulse, i.e. the modes such that cpk (i)= 0. Furthermore, in order to get a nonzero contribution of mode [~0k] to the response of the jth DOF, it is also needed that (Pk (J) ~: 0. The response to a given mode is proportional to the product of the modal components of displacements referring to the degrees of freedom involved in the excitation and in the response. Again, the symmetry of the product is in agreement with Maxwell's theorem of reciprocity. More generally, if a load vector with several nonzero components is considered, all the excited modes are such that: [F] T[c,0k]~0;
where [ F (s )l T =[/71
F2 ... F~_~ F,
... F N] [7.79]
274
Discretesystems
EXAMPLE 1.  Impulsive responses of two identical oscillators in the presence of weak coupling The system shown in Figure 7.21 comprises two identical massspring systems connected together by a spring with a stiffness coefficient Kc much smaller than that of the oscillators. One is interested in determining the displacement response induced on the second oscillator by an impulsive excitation of the first one. By definition, the response in the time domain is the Green's function G(1,2;t), times the amplitude Po = MVo of the exciting impulse. The modal properties of the system are easily obtained by adapting to the present problem the results already established in Chapter 6, subsection 6.3.1. The modal system is written as:
(002(1 + K) (.0
m 2K (Oo2(1 + x )  ~ 2
_~2x
X1 = X2
where o0~ ~; x =   0 The physical meaning of the ESD can be made even clearer by considering the signal energy contained in the infinitesimal band of frequencies [f, f + df] which is of course"
dEx~( f , f + df) = Sxx ( f )df 2501
.
.
.
[8.26]
.
.
.
.
.
.
__[ESD.~tU/Hz)2
A =IOU
2oo
~=O.Is
I
\
~ ~
150
1i i ' ~
,oo
i!
o
0
, , 10
20
,J, 30
40
, 50
, 60
70
80
90
L00
Figure 8.13. Energy spectrum of a rectangularpulse of energy A 2z
316
Discretesystems
This is illustrated schematically in Figure 8.13, taking the example of the energy spectrum of a rectangular pulse of length r, measured in seconds. If U stands for the physical unit of the amplitude A, the spectral density of energy is expressed in
(U I Hz)2. Now, it is also interesting to calculate the spectrum of a rectangular pulse of variable length but of constant action (amplitude AIr ): ..
a foe_e,,,,/.,dt
X (/)= 7
=r ~ ( / ) =
A( 1 e2'~/'~) 2iyrfz"
((1ee"/~)(1e+2'#/')l Sxx(f)=2A 2
(2yrfz.)2
( asin (yr/r)) 2
z'=2
yrfz"
Most of the energy is contained in the frequency range [0,1/'r]. fc = 1/z is the cutoff frequency of the spectrum, see Figure 8.14. Incidentally, the product theorem implies that:
a" S:Sxx" : aeIiIasin' ' 'i2df yrf z" in agreement with the mathematical formula
sin x dx = yr. x
200 18~ 1
ESD(N/Hz) 2
160 140 120
if 10"
/ 10 e
10 z
10 2
10 5
Figure 8.14. Energyspectrumof a rectangularpulse in semi logarithmicscale
Spectral analysis of deterministic time signals
317
When r   + 0 , the pulse of action A approaches the impulse A8 (t)whose spectrum is fiat, with the constant density 2A 2 . A fiat spectrum is known as a white spectrum, using an expression borrowed from optics. Incidentally, it is also found that though energy of a Dirac impulse cannot be defined in the time domain (cf. NOTE in subsection 7.2.3), it would be found infinite if calculated in the spectral domain. In any case, the importance of such peculiarities is more formal than practical, as they merely indicate that a Dirac impulse is not physically feasible, though it can be satisfactorily mocked up by suitable transients, such as a rectangular pulse, provided the proper timescale is respected. As a general rule, the frequency range covered by the spectrum of a signal increases in proportion to the reciprocal of the timescale r of the signal, more specifically the spectrum extends up to the cutoff frequency fc "11 z. Figure 8.14 is a typical example, as the rectangular pulse lasts O.01s whereas the cutoff frequency is precisely found to be l OOHz. In other words, the size of the time and frequency ranges filled by the signal, denoted At and Af respectively, is inverted when shifting from one domain of description to the other, in such a way that the following invariant holds:
[At x Af = 1]
[8.27]
The similarity of relation [8.27] with the Eisenberg principle of uncertainty met in quantum mechanics is rather striking. N O T E .  Shock or impact signal As outlined just above, the spectrum of a rectangular pulse can be used to represent a shock, or impact signal produced by a shock hammer (cf. Chapter 5). By adjusting properly the stiffness of the impactor, it is possible to adjust the frequency range over which the spectrum is essentially white. However, the contact force induced by the impact of a harmonic oscillator against a rigid wall is shaped as a half sine and not as a rectangular pulse, as detailed in subsection 5.3.3.1. Then, as an exercise, it is of interest to compare the spectra of such transients suitably scaled to last the same time 7: and have the same unit action. Accordingly the halfsine signal is written as: F ( t ) = a'P~ s i n ( ~ ~3 The Fourier transform is given by: " Ii _ 1 F ( f ) = ~Pofs sin(2lgf, t)e2i~/tdt; where f,  27 After some straightforward algebra, we obtain:
318
Discretesystems
~(:r):
Po(l+e2i#f').where f r = f
//
2(~ f / ) '
The energy spectrum follows immediately:
s ~ (f~): e0~ ~+ cos ('~Ir)
9
It may be noted that the spectrum has no singularity, even at the reduced frequency fr = 1. As shown in Figure 8.15, the spectrum of the halfsine is found to be essentially the same as that of the rectangular pulse, provided the timesignals are properly scaled.
200
180
DSE (N/Hz) 2
160
\\/
Ii
A : 1ON
140
t=O.O1 s
120
100 80
t 1
rectangular ~ e
60
i
\',, 20
0 10 t
,
.
.
.
.
.
.
.
t
l0 t
.
.
.
.
.
.
.
.
i
10 !
10 z
10 3
Figure 8.15. Spectraof equivalentrectangularpulse and halfsinetransients
8.2.5.2. Power spectral density of periodical functions The theory of distributions enables us to extend the spectral concept to the case of periodic functions of finite power. Accordingly, the physical quantity of relevance here is the power spectral density, (PSD in short notation) expressed in U21Hz, where again U stands for the physical unit of the signal, which is now periodic. From the Fourier series [8.7], it is clear that we have to deal with a discrete spectrum, in which the signal power is entirely concentrated at the frequencies of the harmonic
Spectral analysis of deterministic time signals
319
contributions to the signal. Such contributions are naturally viewed as spectral lines of arbitrarily small width and finite power. Total power of the signal is thus: oo
Ovxx [Co] z + 2 ~ [ C , , ] /
[8.28]
tl =1
It is found appropriate to write this line spectrum as a series of Dirac's impulses
6(fnfl ) where f~ is the signal frequency (the so calledfundamental frequency)" oo
Sxx(f)=~x =[co]Zs(o)+2E[C,~8(f nf~)
[8.29]
n=l
Such a result is consistent with the Fourier transform [8.23]" n=.too
J~(f) = EXAMPLE 1 . 
ECnS(fnfl) Power spectrum of a sinusoid
The periodical signal which has the simplest spectrum is of course the sinusoid
X(t) = A sin(to: +q~). Its power spectrum, when restricted to the domain f > 0, comprises the single line:
S x x ( f ) = A 2 a ( f fo) 2 A sinusoid is said to be optics.
monochromatic using again an expression borrowed from
Power spectrum of a periodic sequence of rectangular pulses
EXAMPLE 2 . 
Let us start by expanding the signal shown in Figure 8.2 as a Fourier series in the complex domain:
X(t)=ZC"exp
T
tl  ...oo
For n 4:0 it is found that:
1
6'. = ~
X (t)exp 
2i~nt T
whence, for n varying from 0 to + ~ :
'
1 I ~r
gJo
X (t)exp
2i~nt 7'
dt
320
Discretesystems
[CO]~ : (erA)5 ; and if n > 0, 2 [6', ]2 : 2/
A sin (anzr)/~
Thus the power can be expanded as: 'Y'xx=[Co] 2 + 2 Z [ c . ] z= A2 a2 +2
n.
nI
/ ~
whereas the power spectrum is written as:
(f)+2~(
Sxx (f )= A21c~28
(n#)
6 ( f  nf~ ) ; where f~
=~
It consists of a sequence of lines spaced from each other by fl. The envelope is precisely the continuous spectrum of a single rectangular pulse, see Figure 8.16 where the spectrum is plotted using linear scales, in contrast with the loglog plot of Figure 8.15. The longer the duration of the pulse (or)1), the narrower the frequency range of the spectral lines with a significant amount of power. Besides this, it may be noted that to shift the signal by a certain delay, leads simply to add a phase angle to the spectral components of the Fourier transform, and of course, the spectrum remains unchanged.
'"
IX II
120
,~176
10i
f (H# ~...ar
. . . .
go
I~
l~
l~l
_
ltl

Figure 8.16. Powerspectrumof a periodicsequenceof rectangularpulses
On the other hand, it is also noted that as ~'xx = c~A2, we can infer that:
(a'n~'~lj) 0
[8.39]
Now, as the energy is essentially positive, so is the quadratic form in a, or b. The discriminant of the equation in a, or b, is thus necessarily negative and so:
{
b2 ( . ~ ("t') Exx ) < 0 Vb~O a2(..~,xx('t'),Sxx) 0 m where 0,, = min (01, 02 )
O~r ~
f
0
if r < 0m ifv>O,,
EXAMPLE 2.  Decreasing exponential signals Let us consider the signal X (t)= Ae'1821 (t). The function and the coefficient of autocorrelation are found to be:
326
Discretesystems
~,xx ( r ) : A2e ~,0
e 2,,0 d t : A2Oe,O~ 2
=~ Pxx (r) = e "o
Let Y and X be two exponentials with characteristic times of decay 0~ and 02 . The function and the coefficient of intercorrelation between Y and X are found to be: .gv,xr (3)= A~A2 0102 e_,,o~ ~ Pxr (3)= e ''~
o,+o,
(r): r
0'02 e"~ =* Prx (r)= e ''st
+o,
This time, the result depends upon the integration order as ~ x v ( Z ) , 5~rx(Z)
8.2.5.6. Correlation of periodic signals As already emphasized, periodic signals have an infinite energy whereas those which correspond to physical quantities have a finite power. As a consequence, the concept of functions and coefficients of correlation can be extended to such periodic signals, provided power is used instead of energy. In this manner we define the autoand crosscorrelation functions:
1 I +TI2
1f+r/2X(t)X (t+3) dt"
X (t)r (t + r) dt
[8.44]
8.2.5.7. Functions approximated by truncated Fourier series Let X(t) be a function of period T, or even non periodic but considered over a finite time interval T only. Furthermore, X(O is assumed to be bounded and to have, at most, a finite number of discontinuities and extrema on T. Such a function can always be approximated to by a trigonometric series of the type:
X (t)=__ SN ( t ) : a 0 + E
a, cos
2ntI T
+/,,sin
(2 nt)) 'T
[8.45]
n=l
The aim is now to determine the coefficients of the series which provide us with the best approximation. Of course, what we mean by "best approximation" has to be specified in accordance with a suitable criterion concerning the behaviour of an error function. In this respect, it may be argued that a criterion based on the energy of the error signal is well suited for applications in mechanics. Therefore, the problem is to
Spectral analysis of deterministic time signals 327 minimize, with respect to the unknown coefficients [a]N ,[b]N, the error signal defined as the power of the gap signal:
Ill
E([a]u,[blN)=~
(f(t)Su(t)f
[8.46]
dt
We already know that in such a problem one has first to make E stationary (cf. Chapter 3, subsection 3.2.1). The condition for optimizing the approximation [8.45] is thus: , "'~)SN t)E _ 2 i r ( f ( t )  SN It)) t)an dt = 0 i)a n T o [8.47] _
0b.
d,0
0
Performing the calculation prescribed by [8.47], we come precisely across the coefficients of the Fourier series: 2
f ( t )  a n cos
7
2n~rt
cos
2nrct
r
f ( t )  b n sin 2T~~t sin 2 n m o
7'
dt = 0 =:~ a n =
dt = 0 ~ bn =
2
T
f (t)cos
f (t)sin
2nm
dt
r 2n~t
dt
7'
Now, it can be understood without further calculation that the stationary solution corresponds in fact to a minimum. Indeed, the error function [8.46] may be interpreted as the equation of a paraboloid in a Ndimensional space which has a unique stationary point (equations [8.47] are linear) and this point cannot be a maximum since the error can be made arbitrarily large. Therefore, it may be concluded that the Fourier series is the trigonometric series which provides us with the best approximation over the interval T of the original function, in terms of power (or energy). To conclude on this subject, it is noted that the approximation method based on the minimization of the energy of the gap signal is also broadly known as the least squares method. On the other hand, the procedure of minimisation [8.47] would hold if the function were expanded on any other basis of orthogonal functions.
328
Discretesystems
8.3. Digital signal processing 8.3.1. Sampling of a time signal Generally, the timesignal originating from a measurement arises as a function which is continuous. Numerical recording and processing of such signals are necessarily preceded by an operation of discretization, which consists of sampling the original signal to retain only a finite sequence of values, taken at successive discrete times: X(t)~ {X(tn)},n = 1,2..... N. Ideally, the sampled signal is related to the original signal by the following transformation: N
(t)= X (t)~ 8 (tt. )
[8.48]
n=l
where the hat symbol (^) marks the distinction between the sampled and the original signal. Most often, the discrete values are evenly spaced in time, for instance t = ( n  1)z', where r, is the sampling period. Therefore, the ideal transformation [8.48] may be viewed as the multiplication of the original signal by a periodic sequence of Dirac's impulses, broadly known as a Dirac's comb, which is truncated to N consecutive teeth. On the other hand, time signals resulting from numerical simulation on a computer are directly available as sampled signals, where the sampling period identifies with the timestep of computation, provided the latter is kept constant. In fact, digital processing presents three peculiarities of practical importance, which will be described here, starting from the case of a measured signal to be digitized. Generally, the signal is first obtained in analogue form as the output X(t) of a sensor, typically a voltage proportional to the physical quantity we are interested in. X(t) is then processed through a device called an analoguetodigital converter (in short AD). The AD comprises a locked sampling circuit, which converts the analogue information into a sequence of discrete numerical values denoted X(t n), coded as binary numbers of fixed size. This operation calls for the three following remarks: 1. As the storage capacity of any electronic device is necessarily finite, the signal must be processed sequentially at successive time intervals, whose duration is denoted 20. This first necessary operation thus consists of transforming the original signal X(t) into the truncated signal X(t;20), defined as the product of X(t) by the timecentred rectangular window:
x (t ;20) = x (t){~ (t + 0 )  ~ (t 0)} X (t ;20) may be viewed as a sample of the original signal X(t).
[8.49]
Spectral analysis of deterministic time signals
329
2. Every individual AD conversion takes a finite time ,'~, during which the input of the AD converter is locked. Of course, rt cannot exceed the sampling period r s . In a similar way, a computed signal arising from a numerical simulation may be viewed as a signal sampled according to r e =rs = h where h is the timestep of the computation. Hence, to perform the sampling transformation [8.49], the ideal Dirac comb is replaced in practice by a comb made of rectangular pulses of duration rr, which are repeated at the sampling period r s . 3. The binary coding in words of fixed size leads to quantification of the numerical values of the sampled signal which can be made discernible. For instance, a word comprising 16 digits allows one to quantify the value of ,~ according to 214  1 = 16 383 distinct levels. Indeed, one digit is used to specify the sign of the number and another digit, called the digit of parity, is used to detect possible coding errors. The coding dynamic is defined as the ratio of the maximum value which can be coded over the elementary step of coding. Usually, the coding dynamics is expressed in decibels, that is by using the logarithmic scale: D =101og
/IX
[8.50]
where it is noted that the relative scaling of the signal is defined in terms of energy, or power. According to [8.50], the dynamics provided by a word of 16 digits is equal to 84 dB. Of course, this last remark holds also in the case of a signal arising from numerical simulations.
8.3.2. The Shannon sampling theorem In order to make a proper use of the storage and data processing capabilities of the digital instrumentation, or computer, it is not advisable to scan a signal at a sampling frequency fs = 1/~', much higher than the largest frequency of interest with respect to the physical content. Therefore, one is naturally faced with the problem of determining what minimum value of f, is to be adopted when the aim is to analyse the sampled
signal up to the highest frequency fro" In order to answer such a
question, it is appropriate to analyse how the Fourier transforms of the continuous original signal X(t) and that of the sampled signal A'(t) are connected to each other. Thus, let us consider a signal X(t) which is first truncated on 20 to produce the sample X(t;20), which in turn is sampled to produce a sequence of 2N+1 discrete values "~n (nrs) where  N < n < N and 20 = 2Nl's . Using the ideal sampling relation [8.48], we can write that:
330
Discretesystems '+O
rF I f( (t )]
= I ~( ( f )] 2N+I
= 2N+I
l
n=N ~ X(t)t~ ( t  n f , ) e2i#yt dt
[8.5~1
n=N
O
We thus obtain:
(f
)]
= ~_, ~2,e 2'~"/~"
2N+I
[8.52]
n=N
It is worth emphasizing that in relation [8.52] frequency can still be viewed as a continuous variable. Now, the point is to establish the mathematical connection between X(f) and X(f). With this object in mind, it is first noted that as the original and the sampled
signals are restricted to the finite interval 20 = 2Nr,, nothing
prevents us from assuming that the truncated functions X(t;20) and ~2(t;20) are ^
20 periodic. Therefore, X(t;20 ) can be written in the non truncated form [8.48]" n+~
X' (t;20)= ~ X ( t ; 2 0 ) 8 ( t  n ~ )
[8.53]
n.l~
The unlimited Dirac comb can be viewed as a periodic distribution, which can be expanded as a Fourier series" n = +**
k = +**

t.ke
n =o.
1 ' = 
k =o.
k = +**
[8.54]
~ ' s k =o.
If k denotes an integer index, it is easily found that:
e 2i~k~ 1 8 (tk~" s )e 2'~'r' dt  ~
1 f(k+l/2y~,
Ck
= 
r, JCkl~2)~,
r~
r~
[8.55]
The relation [8.54] enables us to rewrite the ideal sampling transformation [8.53] in the following equivalent form:
k~+~ ) e 2i~k''~" .~ (t; 20) : X (t; 20_________) Ts
[8.56]
k=**
Now, starting from the form [8.56], it is possible to calculate once more the Fourier transform of the sampled and 20 periodic signal, based on the convolution theorem (see formula [A7.9] in Appendix 7), according to which it is found that: ( f ; 2 0 ) = ~ (f;2O)  k~TFEe2,~u/" ] ~'s
k=**
[8.57]
Spectral analysis of deterministic time signals 331 Furthermore, relation [8.21] provides us with the following results:
r F [ e ' " " ] = a (f  kL )
[8.58]
and: +oo
(f;20)
kf~)=
i
X (f~;20)a(f  f~kf,)df~ = X ( f kf,,2O) [8.59]
whence the final result connecting the Fourier transforms of the continuous original signal and that of the sampled signal is deduced, as truncated in a time interval 20 : .
k=+**
( f ; 2 0 ) = fs ~ X ( f kf~,2O)
[8.601
k =~
The result [8.60] is of major importance in practice, as it shows that the discrete sampling operation modifies the Fourier transform of the original signal. Fortunately, starting from ,~ ( f ; 2 0 ) , it is however possible to recover the desired information contained in ,~ ( f ; 2 0 ) ,
provided suitable precautions are taken in the data
processing. A priori, we can be faced with two distinct cases, which control the rules specified in the Shannon sampling theorem.
First case: the spectral range of the original signal is finite The Fourier transform of the original, yet truncated, signal extends over a finite frequency range Af = [f~,+f~] where f~ stands for the cutoff frequency of the truncated signal. When this is the case, it becomes possible to recover ,~ ( f ; 2 0 ) without any ambiguity from ,~ ( f ; 2 0 ) , provided the so called Shannon sampling rule is fulfilled:
lf,;2f l
[8.611
This rule arises as a direct consequence of the formula [8.60]. Indeed, if fs < 2fc it is found that the sum of two consecutive components in the series [8.60] leads necessarily to a spectral overlapping, hence to an irreversible loss of information. In contrast, if the criterion [8.61] is fulfilled, no overlap takes place and the information contained in the original signal is preserved. Figure 8.17 shows schematically how overlap is avoided, or not, depending whether the sampling frequency is appropriate, or not.
332
Discrete systems
) f
L
o
+L L>2L
"2
X(f)
f
>
0
f~fc
f~+fc
A
X(f)
L < 2L
f f
~
L+f~
Figure 8.17. Problem of spectral overlapping of an undersampled signal


w


w
f= tO Hz ;fs = g Hz
,

N = IO
1
0.5
,
/
0.
,
t (~9
0
Figure 8.18.Phenomenon of aliasing (forexplanation see text)
Spectral analysis of deterministic time signals 333 The spectral overlap phenomenon is further illustrated in Figure 8.18, which illustrates what it is meant by "aliasing". Figure 8.18 shows that a sinusoid at 10 Hz (full line), sampled at 9 Hz over ls (dots) cannot be distinguished from a sinusoid at 1 Hz (dashed line). In agreement with [8.60], the aliased frequency fa is related to the actual frequency f b y fa = f  f s ,
where fs is the sampling frequency.
Second case: original signal of "infinite" bandwidth A priori, if the bandwidth of the signal is unlimited, the spectral overlapping is unavoidable, whatever the sampling frequency may be. Of course, no physically feasible signal is provided with an infinite bandwidth; furthermore, one is often led to diminish the spectral domain of a signal on purpose, by using a lowpass filter. Thus, the same sampling rule holds as in the former case, the cutoff frequency being that of the filter. To conclude on this point, it is worth emphasising that the Shannon criterion must be satisfied also when numerical simulations are performed, the cutoff frequency being identified with the reciprocal of the timestep of computation. Therefore, it is necessary to select a suitable timestep for describing satisfactorily the spectral content of the simulated signal, up to the highest frequency which is desired.
8.3.3. Fourier transforms of the original and of the truncated signals
2.5
26=1s
2
1.5
1
0.5
0
0.5 I0
l
t
l
!
I
.... t
Figure 8.19. Fourier transform of a rectangular pulse
Shannon's theorem provides us with a criterion for sampling a signal, in such a way that the Fourier transform X(f,20) of the sampled signal can reproduce exactly
334
Discretesystems
the same information as that contained in the Fourier transform X(f,20) of the original and truncated signal (such a Fourier transform is termed truncated or finite Fourier transform). However, the problem now is to find out the relation between the finite Fourier transform and the Fourier transform of the original signal, without any truncation. In order to investigate this let us return to the definition [8.49] of a truncated signal. The convolution theorem implies that: X ( f , 2 0 ) = X(f)
sin (2n'fO)
+ 0 )  2/(tO)}] = X(f)
~rf
[8.621
Figure 8.19 shows the shape of the Fourier transform of the rectangular pulse used for windowing the original signal, which is marked by a central peak plus a series of progressively decaying undulations, or ripples. Due to ,.such tipples, the ,. convolution induces some distortion of X(f,20) with respect to X(f). The shorter 0 is, the larger is the distortion. The truncation effect is illustrated in Figure 8.20, which refers to the windowing of a cosine function at fl = 1Hz. The convolution product is:
(f;2O) = )~(f) 
sin (2JrfO) r
A I+"
sin(2~r(f  fo )O)
7
de~ A lsin(2~r(ffl)0)
sin(2~r(f + fl)0)}
(Z+:,)
signalfrequency: f  IHz 2 6=30s
p_.___ 10
20 ~ 2
f (H~ 1,5
1
0.5
0
0.$
.L 1
1.$
2
Figure 8.20. Fourier transform of the truncated cosine signal This is precisely the same windowing effect which is conspicuous in the plots of the truncated Fourier series shown in Figure 8.9 and 8.10. As a final remark, it is possible to alleviate such defects by using other window shapes, smoother than a
Spectral analysis of deterministic time signals 335 rectangular pulse. However, such windows have their own drawbacks as described in the same references as those already quoted above.
8.3.4. Discretization of the Fourier transform
8.3.4.1. Discrete finite Fourier transform and Fourier series Returning to the original function, truncated over20. If periodicity 20 assumed, it can be expanded as a Fourier series: k+**
is
1
X (t;2O)= ~ Cke2"k<S*''" with 6 f =  ~=_.. ' 20
[8.63]
The coefficients are given by [8.8] and performing now a Fourier transform of the series [8.63], we obtain, in terms of distributions: ""
1 k=+o*
x (f;2o)= 7 E
C~a(fkaf)
[8.64]
k =~
This expression is a discrete version of the Fourier transform of the truncated signal. It indicates, in particular, that the smallest identifiable frequency separation contained in the truncated signal is precisely 8 f = 1/20, a result which is not surprising.
8.3.4.2. Definition and properties of the discrete Fourier transform When time data are processed at the sampling frequency fs, the only transform which is available is a finite sampled function ,g ( f ; 2 0 ) , and not ,~ ( f ; 2 0 ) as given by the infinite series [8.64]. Since the smallest frequency separation which can be accounted for by the sampled time signal ,~ (t; 20) is f~ = 1/20, this value may be used as a suitable frequencystep to sample ,~ ( f ; 2 0 ) . Of course, only the terms at frequencies less or equal to the cutoff value f, / 2 , are retained in agreement with the Shannon criterion. This produces 2N+ 1 values of the sampled discrete Fourier transform, which is finally written as:
(A;20
= ~ff ~
X,e ^ ~'~{"~'=~)
[8.651
rI=N
Relation [8.65] is obtained by substituting into formula [8.52] the current frequency f b y the sampled value kfs and by adopting the normalization factor l12N.
336
Discretesystems
On the other hand, since the time samples {,Y, } are real quantities, the 2N+1 values of the discrete Fourier transform are interrelated by the condition of complex conjugation:
L
_.
)
t8.661
Such a result shows that starting from 2N+1 sampled values of a time signal, formula [8.65] provides us with N+I sampled values of the Fourier transform, defined in the domain f > 0. It is thus possible to restrict the computation to the range f > 0, without any loss of information. This is to be expected, because the computation procedure [8.65] transforms 2N+1 real numbers into N complex numbers, plus the real number at zero frequency (k = 0). It may also be noted that if the case of an unbounded sequence of successive integer values k is considered, it can be concluded that the corresponding discrete Fourier transform becomes a sequence with periodicity 2N: X k = Xk+2N
[8.67]
This also is in full agreement with the considerations made in the last subsection about the assumed periodicity of the truncated time signal. On the other hand, the inverse Fourier transformation is found to be given by: {,~',}= ~ ,Yke§
[8.68]
k=N
A priori, examining the formulas [8.65] and [8.68], it could be concluded that the transformation requires a number of calculating operations which is proportional to (2N + 1)2. Fortunately, in 1965, Cooley and Tukey made available a fast Fourier transformation (FFT) algorithm, which reduces this number significantly (number of calculating operations proportional to N log 2 ( N ) , provided N is an integral multiple of 2 ( N = 2 m; m : integer ). The availability of this algorithm [COO 65] has greatly promoted the use of spectral analysis since the late sixties, and it is now a basic technique of signal processing, in common use in mechanical engineering in particular. Modified FFT algorithms to which the limitation of N = 2 mdoes not apply have also been developed, see for instance [BR174]. 8.3.4.3. Illustrative example It is worthwhile to conclude this chapter by applying the foregoing general notions to a specific example, at least in order to illustrate their practical relevance. Let us consider the following signal:
Spectral analysis of deterministic time signals
337
X (t) = 2 { 2 / ( t  8 . 0 4 2 )  2 / ( t  8.342)}+ 0.06 sin(10~t)(1 + 3 sin(16#t)) where time t is given in second. Hence, the highest frequency component of the signal is f3 = 13 Hz. It arises from the addition of the frequencies of the two sine functions present in the above analytical expression. Two other spectral lines are also expected, at 3 and 5 Hz respectively. Finally, the spectral continuum arising from the rectangular pulse is marked by a cutoff frequency of about 3.33 Hz.
2.5

;
!
i
)
w
X(t)
(
~'= st.2 Hz 2 N = 2048
26=4~ 1.5
0.5
lalalaW~h/lalalal~WWlataWtal~lalalalata~lal~laW~Wl~a~ia ~ I
.
0
5
.
. 10
. 15
.
. 20
.
t(s) 25
30
35
40
Figure 8.21. Timehistory of the signal to be processed 4
)
3.5
/~D
f
~" = 51.2 H z
f/= 3 Hz
2 N = 2048 3 2.5
F2=SIIz
/3=13Hz
f 1.5
0.5
f (Hz) 0
0
'I'o
"
i'~
2'o
25
Figure 8.22. Power spectrum of the signal suitably sampled
338
Discretesystems
2.5 X(t) 2
9
fs=12.SHz 2 N  512 2 6=40s
9 1.5
1
o
9:'. .~....
I
I.
...':. ~.$.
.:. ...'. .~..'...~.
.:.
..'.
...
.:..
.':.
:X,.~. ,.4~;. .',,.,~. ,.,..;...'.~..~. ,.o.,.:.;.:..z.,;...~,..,,:,,,:~.. :'~r~,."..a,~.r "."3 e ~'~e.
~?"
9
0.5" 0
"
9
.~.
.
9
otPO #~ r ..
"
e ~e
9 9
"8
Q o ;p o.
e''o ~. . 9
%,
e.
e ir'~o.
;.
9 .
'
'
'
'
'
'
,
5
10
15
20
25
30
55
9
o.
I 40
Figure 8.23. Timehistory of the undersampled signal
i
!
fl =3 H~
PSD
0.9
fs= l Z S Hz 2N = 512
/ f 3 =13 HZ folded back to O.2 Hz
0.$ #.7 0.6
#.J ~ = 5Hz
#.4 0.3 0.2 .~
O.I 0
f(
0
1
2
3
4
J
Figure 8.24. Power spectrum of the undersampled signal
The s a m p l e d f u n c t i o n X(t) is displayed in Figure 8.21, over 2 0 = 4 0 s. The sampling frequency fs is 51.2 H z providing thus 2 0 4 8 sampled values. This
sampling rate is more than compatible with the Shannon criterion. It has to be emphasized that the continuous line conspicuous in the plot is merely due to an
Spectral analysis of deterministic time signals 339 artefact of the plotting subroutine which has been activated on purpose, in order to help visualisation of the analytical function. Figure 8.22 is a plot of the PSD of the sampled signal. The frequency range of the discrete spectrum extends up to f~ / 2. The frequency separation between two successive components is 6 f = 1/20 = 12.5 mHz. Figure 8.23 displays the sampled time signal obtained by adopting fs = 12.8Hz, providing thus 512 sampled values which are "undersampled" with respect to the Shannon criterion. Here the plot has been performed by deactivating the optional continuous line display of the MATLAB subroutine. Doing so, it can be seen that the original signal is hardly recognisable based on the sampled signal, as soon as N becomes small enough, as it could be verified even when using a sampling frequency in agreement with the Shannon criterion. Figure 8.24 is a plot of the corresponding PSD with the continuous line display activated. The sampling frequency is still sufficient to describe the spectral content of the pulse and the two lines at 3 and 5 Hz. However as expected, it is found insufficient to describe the line at 13 Hz. Indeed, on the plot of Figure 8.24 the line at 13 Hz is folded back to the much lower frequency f3 = f 3  f s = 0.2 Hz, in full agreement with the principle schematically displayed in Figure 8.17. The above example is found suitable in order to illustrate the practical rules it is necessary to comply with when processing digital data. If the numerical signal originates from an analogue record, a necessary preliminary is to filter the analogue signal up to the cutoff frequency fc = fm before digitalisation, where fm stands for the maximum frequency of interest. Then the signal is sampled at the frequency fs = 2fro. For identifying spectral components separated in frequency by at least 8f,
the length of the sampled signal must be is selected as: Ts = 2 0 
1
[8.691
8f Starting now from the results provided by a numerical simulation performed on the computer, the problem can be restated as follows. The practical object is to compute a signal X(t), whose spectral content is bounded by the maximum frequency f,,,. Moreover, spectral components separated by at least 8 f must be identified. According to the foregoing considerations, it is necessary to perform the numerical simulation with the timestep of the dynamic integration algorithm set to: h
1
1
 ~ fs 2/',,
The duration of the computed signal is still given by [8.69].
[8.70]
This Page Intentionally Left Blank
Chapter 9
Spectral analysis of forced vibrations
The methods of spectral analysis are of special interest in the treatment of dynamical systems, in particular, but not exclusively, when they are linear. We shall first show that by considering the spectral content of excitation and natural modes of vibration (in short the "modal spectrum") it becomes possible to identify those modes which have to be retained in the dynamical model and those which can be neglected. This aspect of modelling is of crucial importance since the modal sequence of real material systems is a priori infinite, as detailed in Volume 2. On the other hand, for many applications in mechanical engineering the pertinent information is contained in the spectral properties of the response signals and little would be gained by embarking on solving the problem in the timedomain. Finally, even if the dynamical system is far from linear, spectral analysis still provides quite valuable information concerning the existence and the nature of nonlinearities present in the system on one hand, and the periodic or the chaotic properties of the response to periodic excitation, on the other.
342
Discretesystems
9.1. Introduction The mathematical tools described in the preceding chapter are applied here to analyse the spectral properties of the response of mechanical systems vibrating according to steadystate regimes. Section 9.2 is devoted to the analysis of the damped harmonic oscillator. It is shown that the spectral properties of the response can be fully understood by starting from those of the transfer function and of the excitation signal. In section 9.3, the methods of spectral analysis are applied to the case of MDOF linear systems. The most prominent feature of the spectral properties of the transfer functions is the presence of three distinct frequency domains, which require distinct levels of accuracy in their description, namely the domain of quasistatic responses, resonant responses and finally the domain of quasiinertial responses. Clearly, the most demanding domain for modelling is that of resonant responses, in which the description of the vibration modes, including damping, is of crucial importance. General results are further discussed based on the analysis of two simple devices of practical interest. The first system is a vibration absorber which makes use of the principle of antiresonance. The second system is the shock absorber of a car suspension which was already described in Chapter 7, subsection 7.5.3. Finally, in section 9.4 another incursion into the nonlinear domain is considered, based on the classical example of the steady state responses of the Duffing oscillator to harmonic excitations. The study presented here demonstrates the profound differences which exist between the dynamical behaviour of linear and nonlinear systems. The most prominent feature, also encountered in many other nonlinear dynamical systems, is the possible occurrence of chaotic responses. For certain values of the internal coefficients of the oscillator and for certain values of the external harmonic forcing function, though the system is basically deterministic in nature, the response becomes so irregular that it is impossible to ascertain which dynamical state the system will have at a fixed time, as though the response were random. On the other hand, chaotic responses are also marked by a continuous spectrum, in contrast with the periodic, or pseudo periodic responses. Chaotic dynamics stems from the pioneering work of Poincar6. However, it was not before the second half of the XXth century and the advent of numerical computers that the practical and philosophical implications of deterministic chaos were fully investigated. The reader interested in an historical survey and a qualitative introduction to the subject, is referred in particular to [BER 84] and [GLE 88].
9.2. Linear (harmonic) oscillator 9.2.1. Spectra of excitation and response By transposing to the frequency domain the basic equation [7.23], which relates the response of a SDOF linear system to its excitation, we obtain in terms of Fourier transforms:
Spectral analysis of forced vibrations
Io
~ ((/))O(e)(CO)I
343 [9.1]
Starting from [9.1] the following spectral relationships follow immediately:
Sqq ( c o ) :
[9.2]
I n (co~2 SQQ (CO)
S;; (co) = H (co)See (co)]
[9.3]
As further investigated in the next subsection, these formulas show the filtering properties of the oscillator which acts as a narrow band filter, centred at the natural frequency, relative bandwidth being controlled by damping.
9.2.2. Spectral properties of transfer functions 9.2.2.1. General features of the displacement~force transfer function 10 6 _
HH* ~
10 4 _
C = 0. 003 (=o.oi ( = o.o~
10 2
10 0 _
10 z
............ 10 4 10 I
zTPfl ' 10 e
, ,,~., 101
Figure 9.1. Squared modulus of the displacement/force transferfunction of the damped harmonic oscillator in loglog scales Starting from the formula [7.24], the transfer function of the damped oscillator, relative to the output/input ratio "displacement/force" is now written in terms of the reduced frequency z, as follows: H(z)=
1 t,'1z' + 2izfl )" ; where z = f / fl = co/col K
[9.41
344
Discretesystems
f~ is the natural frequency and g l is the damping ratio of the linear oscillator. In accordance with the spectral relation [8.20], it is convenient to describe the complex quantity [9.4] by using its squared modulus and its phase: iH(z~2 =
1 K2 I(1z2 ? + (2z9",)2]
[9.5]
~ (z) = a r c t a n ~
[9.6]
The shape of IH (z "~2 is depicted in Figure 9.1 for several values of the damping ratio, by using logarithmic scales. If ~1 is sufficiently less than one, the curves are marked by a prominent and sharp peak centred at the resonant frequency z = 1. The peak value is given by: nl2x=
1 2g'~K
[9.7]
Provided again that ~1 is sufficiently small, the magnitude of the peak is reduced by a half when frequencies are shifted by Az = +g'~ from the resonant frequency: ~2 0.51H 2
[9.8]
=
Moreover, the area under the curve can be expressed analytically in terms of the oscillator coefficients:
ii , fl H(z)I 2dz : 2
I.(z)l
= 2~.1K2
I
[9.9]
When z is sufficiently smaller than one, it is found that the modulus of the transfer function is essentially constant and equal to IlK. But if z is sufficiently larger than one, IHI decays according to a z 2 law. The detailed shape of the resonance peak is better shown in Figure 9.2, where IH (z ~2 is plotted in the vicinity of resonance by using semi logarithmic scales.
Spectral analysis of forced vibrations
105
345

HH *
=0.003
10 4
=0.01 10 3
~
10 2
/~
=0.1
lO s
;
Z
!
i0 ~ 0.9
t
0.95
~ ,
1
,
1.05
1.1
1
1.15
Figure 9.2. Squared modulus of the displacement~force transfer function of the damped harmonic oscillator in the vicinity of resonance
According to the phase curve of H, if z is much less than one, the phase between the response and the excitation is essentially zero. Thus, the response is "inphase" with the excitation. At the opposite, when z is sufficiently larger than one, the phase is practically equal to ~r. Of course, as far as phasing is concerned, nr is equivalent to +It, both values corresponding to an opposition of phase between the response and the excitation. As shown in Figure 9.3, most of the variation of the phase angle from zero to +~r takes place in a narrow range of frequencies centred at resonance, where the slope of r
is proportional to ~~. At resonance the phase
lag is  n / 2 , a value which corresponds to a phase quadrature. Finally, it may be noted that the phase is varied from  n / 4 to 3n / 4 over the narrow frequency range Az = 1T~'l.Therefore, the most suitable way to display graphically the resonant portion of the transfer function in detail is to make an Argand (or Nyquist) plot, in which the imaginary part is plotted versus the real part, as shown in Figure 9.4. It is left to the reader as an exercise to show that the Argand plot of the transfer function [9.4] is a circle, from which the parameters of the oscillator can be conveniently extracted.
346
Discrete systems
o.s~
,,
I
o~_...~
Lj/
.,~..'~
'
Z.= O.OO3
',,,V~~,:o..
1.5 ;......... _ "2t
 ~r./21
7,
3.50 . 9
0.95
1
1.05
1 1
1.15
Figure 9.3. Phase function ~z) in the vicinity of resonance
1
1
,
i
"
'
"
"r ! ! .
.
.
.
.
.
.
.
.
.
.
.
.
.
2
I
4
~,.,i\
~~
\
'
i!
t;
10 12
!
14
8

4
2
0
2
4
6
8
Figure 9.4. Argand plot of the transferfunction
9.2.2.2. Spectral ranges of the oscillator response It is of interest to investigate the consequences of the response properties of the harmonic oscillator on the forced response to an excitation of which the spectrum is
Spectral analysis of forced vibrations
347
known. Starting from the first result [9.2], we can draw a few conclusions of major interest, based on the order of magnitude of the ratio f~/fc of the resonant frequency f~ over the cutoff frequency fc of the excitation. Let us examine first the case of a monochromatic excitation of pulsation toe and magnitude Q~e). As already shown in Chapter 7, the forced response is also harmonic, at pulsation toe" Thus the three frequency ranges identified in Figure 9.5 are worth discussing.
lO s
10 z
,' 1
~ . o.o4 A
lO t
i
/\i
~.
!
!~
i l
I
!
l
1
I0 ~
10 t
10 z
10 s
 0 4
I
'
'
[ re.s'ponse :
[ :
I
'
i rc,~'onant i
quam statw respor~e
'
I
i
I
&
,
I
i0 s
,
i
I
I
10 e
\
inertial r~ponse ~
\
I \
,
i
I
I
?
~
.
.
.
.
\4
.
I
lO t
Figure 9.5. Schematic identification of the spectral ranges of the oscillator response 1.
Quasistatic range o f response: toe (co)
[9.221
j=l
As a definition, the autocorrelation spectrum (ESD, or PSD) of the response is given by:
Sqq (i;o)): qi ((o)q~ (o)): s H (i, j;(o)O~ ~, ((1)) s
j=l
k=1
(i,k;a))O~ r (a))
[9.23]
360
Discretesystems
As a consequence of the quadratic nature of the above operation, it is found that the respollse spectrum is related to the auto and cross spectra of the excitation signals, through the following formula: N
N
Sqq (i;(o) = Z Z H(i, j;(o)H* (i,k;a))SQQ (j,k;o)) k=l
[9.24]
1=1
The actual calculation of such a formula may be rather tedious, in particular because the product of the transfer functions generates coupling terms between all the natural modes of vibration of the system. It is thus worth mentioning that such crossed terms can be generally neglected, provided the resonant frequencies are sufficiently apart from each other, the relevant criterion being given by the following inequality: fn+l  fn
>> 1
[9.25]
Thus, provided the condition [9.25] is fulfilled, each individual resonance peak of response is practically independent from the others. Accordingly, it becomes possible to calculate the energy, or the power, of the physical response simply by summing up the partial contributions of each individual resonance: N
~q : Es
(fn)
[9.26]
n"l
Or, in terms of standard deviations, or r.m.s, values: [9.27]
9.3.5. Vibration absorber using antiresonant coupling
In practice, it often occurs that monochromatic and resonant excitation of poorly damped systems is inevitable. If no specific action were taken, excessive vibration would occur, leading generally to a failure of the component, typically by fatigue. An efficient and elegant way to reduce the response to a safe level is to perturb the system by coupling it to a "small resonant oscillator", tuned at the natural frequency of the resonance to be cured. By "small" it is meant that the mass and the stiffness coefficients of the oscillator, though tuned to the proper ratio, are much less than the corresponding generalized quantities of the resonant mode to be perturbed. IndeX, as shown below, the effect of the perturbing oscillator is to replace the undesirable resonance of the initial system by a pair of closely spaced resonances, of which the contributions to the response cancel out each other over a fairly small frequency range centred at the natural frequency of the unperturbed system. In other words, the
Spectral analysis of forced vibrations
361
perturbed system has an antiresonance centred at the resonance of the initial system. This kind of vibration absorber is found to be particularly efficient precisely in the case of small damping. However, its range of application is restricted to the case of monochromatic, or fairly narrow bandwidth excitations.
X
M
k
Figure 9.17. Schematic picture of the antiresonant vibration absorber How it works can be understood by using a model restrained solely to the resonant mode of the initial system, plus the perturbing oscillator, as shown in Figure 9.17. As the excitation is assumed to be monochromatic and in resonance with a given mode of the initial system, the response of the other modes may be neglected. The perturbed model is governed by the two following equations: [(k + K )  t o 2 M ] X  k Y = FS(totoo)
There would be no difficulty to solve directly the above system by eliminating the displacement Y of the perturbing oscillator between the two equations. However, we prefer to use the modal method to illustrate the underlying physics of the problem. Qualitatively, the important point is that the two modes arising from the coupling of the two oscillators have quite similar properties. In particular, the two oscillators being individually tuned at the same angular frequency too, the modal frequencies of the perturbed system are found to bracket too in a fairly narrow interval [tot,to2]. The response of M to a monochromatic excitation of M at too is given by the following modal series: Frp2 (1)
+
Frp22 (1)
MI (0)2 _ 092 + 2itoltoogo) M2 (0)2 _ 0)2 + 2ito2toogo) In this step, to simplify the algebra, it is assumed here that the damping ratios of the two perturbed modes are the same and equal to that of the non perturbed mode, noted go. However, the physical mechanism of the antiresonance would be the same even if distinct damping ratios would be considered. Indeed, the major point is that
362
Discretesystems
the two modal contributions cancel each other in a frequency interval of the order of 28co =a~2 co, because the sign of the real part of the denominator is changed from one term to the next. The quantitative analysis is left to the reader as an exercise. Writing down the vibration equations in terms of the dimensionless parameter of perturbation: m fl =   = 2M
k 2K
0
[9.40]
This condition may be expressed as: a2 4G2,0
(1_~.2) > 3r/ _Tp
[9.41]
Incidentally, it can be noted that in the linear case, we already know that at resonance we have: ,~
=
o~2 2
[9.42]
4~"
and the above inequality is automatically verified, since it reduces
to
q2 ) 0.
As p is necessarily positive, the condition [9.41 ] can be transformed into: 4(1~ 2)
/9 2 + ~ p  ~
3r/
a2 >0 3r/q 2 
[9.43]
By calculating the roots of the expression [9.43] equated to zero, it is an easy task to show that the required inequality holds so far as: p <  ~1 {~2 ( 1  )~.2 +~4(1~
"2)2 +3r/a2/ ~.2}
[9.44]
The internal coefficients a,r/ of the oscillator being fLxed, the nonlinear resonance is def'med by the condition of maximum amplitude:
Spectral analysis of forced vibrations
,.
=
1,
=
373
~4(q2  1)2 + 3r/a2/q 2 [9.45]
+
37/
which in turn determines the frequency of the nonlinear resonance by using [9.39]. After some elementary algebra the following result is obtained: 3 7/~,,, 2 Ores = ~1 q2 + "4"
[9.46]
101
p(~)/p(o) 6"= 1 ~
lo0
10t
11 10
2
. . . . . . .
10 ~
Figure
9.24.
1
i0 s
. . . . . . . .
1
104
. . . . . . . .
1
103
. . . . . . . .
1
10 2
?
. . . . . . . . .
10t
i
i
.....
10~
Limitation o f the vibration amplitude due to the nonlinearity
Qualitatively, the major results arising from the formulas [9.45] and [9.46] may be summarized as follows: 1.
The magnitude of the nonlinear resonant response can be substantially less than in the linear case, as shown in Figure 9.24, where the ratio of p(r/) to p ( 0 ) is plotted versus r/. It can be noted that the nonlinear magnitude tends asymptotically to the linear one if 7/ tends to zero, as expected.
2.
The nonlinear resonance frequency varies with the amplitude of the excitation.
In the present example, .('2re$ is found to increase with a, a result which is not surprising since the effect of the nonlinearity present in equation [9.28] is to increase the stiffness of the oscillator, provided r/> 0, as already evidenced in Chapter 5, subsection 5.2.1.
374
Discrete systems
9. 4.1.3. Nonlinear resonance peak
16
r
14
12
T/=0.05
10
r
O0 iI
0:w
i
115
:2
. . . . . . . . .
r . . . . . . .
2.S
t
3
3.w
Figure 9.25. Peak o f nonlinear resonance
It is also of interest to investigate how the an~litude of the nonlinear response behaves when the pulsation ~ of the harmonic excitation is continuously varied, sweeping through the resonant range. A typical result is'shown in Figure 9.25, which is shaped as a "curved resonance peak". Excitation being maintained at a fixed amplitude ( a = 1 ), the response is a decreasing function of damping, as in the linear case. Then, in contrast to the linear case, the peak is bent, here to the fight became r/> 0. The practical consequences of such a feature will be further discussed in the next subsection. 9. 4.1.4. Hysteresis effect The hysteresis effect related to a bent peak of resonance is best evidenced when the nonlinear oscillator is excited by a sinusoidal signal, of which the frequency is progressively varied through the resonance peak. The results obtained in terms of timehistories are illustrated in Figures 9.29 and 9.30. They can be conveniently explained starting from the plots of Figure 9.26, in which the arrows indicate the direction of frequency sweeping. /
In the upper plot, frequency is progressively increased, at a sufficiently slow rate for observing at each frequency a steady vibration regime. Starting from a frequency below the resonance frequency, the magnitude of the response is
Spectral analysis of forced vibrations 375 increasing monotonically up to the top of the peak, (point A in the figure). When frequency is further increased, even by a very tiny increment, the magnitude of the vibration jumps down abruptly from point A to point B.
r/~ 0.05
J O0
~
015
1
B
1.5
2
2.5
3
3.5
3
2. ~
~ A
l'i i/ 2
D
....
~ 0
...... 0.5
1
1.5
2
2.5
.~ 3
3.5
4
Figure 9.26. Sweeping of the resonance peak
.
In the bottom plot, excitation frequency is progressively decreased starting from a value higher than the resonance frequency. Now, the magnitude of the vibration increases monotonically up to the point C, where the slope of the resonance curve goes to infinity. When the frequency is fiu'ther decreased, the magnitude of the vibration jumps abruptly from point C to point D.
376
Discretesystems
Therefore, inside a certain fi'equency range bracketing the nonlinear resonance, the magnitude of the response to a given excitation is found to be dependent on the previous history of the excitation signal, eventually by a large amount.
9.4.2. Numerical simulations and chaotic vibrations The numerical integration of Duffing's equation using an explicit algorithm, such as the method of the central differences, is quite straightforward. The main problem is the choice of an appropriate value of the timestep which has to be sufficiently less than the critical value [5.68] which was derived in Chapter 5 in the framework of linear oscillators in order to cope properly with the stiffening effect of the nonlinearity investigated here. 9. 4.2.1. Periodic motions
1
4
05
2
0
0
0.5 1
1
"2
o15
'
o
4
o15
6
2
0
2
2O 10
0
0 I0
a = 100
4
5
0
5
20
10
5
0
5
10
Figure 9.27. Phase portraits of periodic solutions When using numerical simulations, the first step is to check the existence of periodic responses, such as those predicted by the approximated analytical solutions of subsection 9.4.1. Then the various responses of the oscillator to sinusoidal forces
Spectral analysis of forced vibrations
377
of fixed frequency and variable amplitude can be computed. The results referring to the eventual steady states of the response are suitably visualized as phase portraits. Indeed, if motion is periodic, the phase portrait reduces to a closed curve, as soon as the length of the simulated timehistory is sufficient to be in a steady regime of response. Periodicity of the motion can also be checked by performing a spectral analysis of the response, the spectral signature of a periodic signal being a sequence of lines in harmonic progression, as seen in Chapter 8.
800
i

response
Ioor
/
6O0
(: w~ 400
r/! 200 a
force
= I00
0
,td I
200
400
6000
I
25
i
311
3
40
45
Figure 9.28. ,4 typical sample of timehistory of periodic motion (here amplitude of the displacementfull line was magnified by a factorl O0 to adapt the scale of the figure to the magnitude of the excitation force (dashed line)) A few phase portraits of periodic motions are shown in Figure 9.27. As indicated in the figures, damping ratio is fairly large q = 0.1, to save computing time, the coefficient of the cubic term is also fairly large 77= 1, to produce a sufficiently strong nonlinearity and finally the excitation frequency is the same as that of the linear resonance, for contrasting the behaviour of the linear and the nonlinear oscillators. From such plots, it is found that motions are indeed periodic, coinciding with the period of excitation, in agreement with the analytical calculation made above. Furthermore, it is found that the shape of the phase cycle is more complicated when excitation amplitude is increased. The secondary loops which become apparent in the phase portrait correspond to secondary oscillations within the global cycle at the driving frequency, as seen in the timehistory of Figure 9.28.
378
Discretesystems
,
s[ #(0
=F
1
,,idllh
.,,,,llilMiiigl,,
,,=2
:
= 5% cz~ ]
v
1.5 = a < X , Y > ....
.
.
.
.
....
..,,,
< X 1+ X z , Y > = < X ~ , Y > + < X z , Y > <X,X ....
< X,X
[A1.8]
> >_0 ....
_,.
....
>= O cz~ X = O
where the asterisk marks the complex conjugation. From the conditions [A1.8], it can be inferred that:
=a' and the Schwarz triangular inequality:
[A1.9]
Appendices
387
Vectors are said to be parallel to each other whenever: [AI.10] Vectors are said to be orthogonal (or perpendicular) to each other if" <X,Y>=0
[AI.ll]
whereas < X, X >r 0; < Y, Y >v=0
A.1.6.2. Orthogonal projection of a vector
It may be shown without difficulty that an orthogonal set of vectors is independent. On the other hand, the projection of a vector X on the line generated by a vector Y is given by:
f(p = <X,Y > ~
[Al.12]
Ilell~ __,,
_.~
_..
X and Y being a pair of known vectors, the following decomposition of X is unique"
X=Xp+Z 2e=<X,Y>~;~
=0
[Al.13]
Ilell~ where o~course, it is a~sume~,,,at Ilell*0 A1.6.3. Hilbert spaces
On the other hand, x/<X, 27 > can be interpreted as a norm, known as the natural norm of the scalar product space"
4 FII
[Al.141
A vector space provided with a scalar product and which is complete in the metric of the natural norm is called a Hilbert space.
388
Discrete systems
AI.7. Hilbert space of square integrable functions
funcfiomf(t)defmed on the domain a < t < b for ~lf (t)12dtexists, is a Hilbert space ~c)(a,b).
The set of all complexvalued which the Lebesgue integral
The scalar product is defined by: b
(x(~),Y(O)io.~l f. x(Or" (~)d~
[Al.15]
Appendices 389
Appendix 2: Vector and multiple products of vectors
The vector (or external) product of .(2 by F is noted: [A2.1]
V = .~ • i
The Cartesian components of the vector product are written as: ,
j
k
Vx =Z.('2yY.(2 z
ax ar a , =
Vy=Xa~Zax
X
V~ = Y . ( 2 x  X . ( 2 y
Y
Z
[A2.2]
relations which can be gathered in a matrix form:
ixl i0 _Oz~ v~= ~z Vz ~
0 ax
~ o
[A2.3]
They can be also written in tensor form, as expressed in indicial notation: V~= euk.Ojr, avec i, j, k = 1, 2, 3
[A2.4]
The permutation symbol 6#, is given by: f zuk = 0, whenever at least two indices are repeated. zuk = + 1, if i, j, k is an even permutation.
[A2.5]
6gk =  1, if i, j , k is an odd permutation. The notation [A2.4] is convenient for proving the following relations"
P ~.(~x~) ~.(~x,~) ~.(~x~)~,,,,~.~,.~,
[A2.6]
,~  ,~x(~x e) (,~.e)~_ (,~.~)e
[A2.7]
w, = .4jcjB,  . 4 j s F ,
_ (~• ~),(e• ~) (~, e)(~, ~) (,~,~)(~, e) s = ,4~ (c~D~  c,~) (A~c~)(~,~,,)(,4~ )(~, c,)
[A2.8]
390
Discretesystems
Appendix 3: Euler's angles and kinetic energy of rotating bodies Euler's angles p,O,~ enable us to transform a Cartesian direct frame Oxyz into another Cartesian direct frame Ox'y'z' by means of three successive rotations performed in a specific sequence:
z=~
x
x
0 '=~
Figure A3.1. The rotationsdefining the Eulerian angles
Starting from the frame Oxyz, the counter clock~se (direct) rotation by an angle p about the axis Oz produces a first intermediate frame OCr/~'. Then, similar counter clockwise rotations are carried out by an angle /9 about the axis 0r to produce the frame Or by an angle g about Off' to produce the final transformed frame Ox'y'z'.
A3.1. Application: kinetic energy of a rigid body of revolution Let us consider a rigid body of revolution, rotating about it centreofmass G, which is assumed to be fixed. Let (R) be an inertial Cartesian frame with axes
Gx, Gy, Gz and (R') an accelerated flame, rotating with the body. Moreover, the axes Gx', Gy', Gz' of the Cartesian coordinates in (R') are assumed to be a set of principal axes of inertia of the body, Gz' being taken as the axis of symmetry. The coefficient of inertia about Gz' is denoted J and I = J/2, about the two other axes. The kinetic energy of the rotating body is thus (cf. Chapter 2): , ~ = 1 {1 (.f2x=.+ .(2;.)+ Js
[A3.1]
Appendices 391
t~., O c o s ~, \!
i
.....'"'"
......~
 t)sin~
Xr
4' o ,o
~O~. z
y

!
zs.
\ 0
X
..'
t
~176176176
f qbx, : ~bsinOsin g/ r = ~bsin 0 cos
~r
0~, z
yF
: ~
cos 0
11t
xz_ Figure A3.2. Inertial and corotatingframes linked to the centreofmass of the rigid body In (R), the angular speed of the body is described by the components 0, (o, ~. Now, the problem is to determine the Cartesian components of .O in (R'). Referring to Figure A3.2, it is found that: 0 x, " 0COSlpt;
0sinv;
0y,
~bx,=Chsin0sin~,;
0z, = 0
~by,=cbsin0cosv;
r
~x,=0; ~y=0; r Collecting these partial results, the following components of .O in (R') are found:
392
Discretesystems
~2~ = ~psin0sin~, +0cos~, #2y = ~sin0cos~,bsin~,
[A3.2]
~, =~cos0+~ The kinetic energy of the rotating body is written as:
/l,k = iiz, ll~ = E i~, k
a, 9< v ~ , z, > = 0
[A4.5]
Appendices
395
a result which is simply impossible, since the norm of V, is strictly positive. As a consequence, to the N distinct eigenvalues N eigenvectors correspond, forming an orthogonal set. If N is the dimension of the Hilbert space on which the hermitian matrix is operating, the eigenvectors form an orthogonal basis of the space. We analyse next the case of multiple eigenvalues. It suffices to consider the case of multiplicity 2. Let W/be an eigenvector solution connected to Aa = 22 = 2"
=2 = 0 ; j = 3, 4,..., N
[A4.7]
Let W_, be a vector perpendicular to W~ and to the whole set of V,. It may be used to complete the orthogonal basis { ~ , if'2,/~,, l?/v}. The aim is to prove that
W: is another eigenvector related to 2.. With this object in mind, let us define the following vector: 3
U = n(/'~'2)=a,~
+ az I'V2 + E a j . l,Tj.
[A4.8]
j=l
Orthogonality of the basis implies:
= aj = < v j , n ff'~ > = .11 11 , ()
[A4.9]
whence we deduce that: aj = 0 , j ~ [ 3 , 4 , . . . , N ] =:> U=a~ W~ +a2W2
[A4.10]
As the same reasoning may be applied to W~, the desired result is reached, according to which: ....
....
_..
U = cr2 W2 = 2 W 2
[A4.111
Finally, as W~ and W2 are two distinct eigenvectors connected to the same eigenvalue 2 , any linear combination of the two vectors is also an eigenvector connected to 2 . Thus, it may be concluded that to an eigenvalue of multiplicity 2, a whole plane of eigenvectors corresponds, defined by the linear form a~ ~ + a 2 W2 . Moreover, this plane is perpendicular to the set {/7} of the eigenvectors which are
396
Discretesystems ,,
....
connected to all the other eigenvalues. ~ , W2 form an orthogonal basis of such a plane and it is pointed out that there exists an inf'mity of other such bases; they can be obtained simply by transforming ~ , W2 through a plane rotation of arbitrary angle. Clearly, the reasoning presented here in the particular case of a twofold eigenvalue may be extended to the case of any multiplicity m. The linear form al ~ + ... + a mWm generates an hyperplane of dimension m. To conclude this paragraph, it can thus be stated that it is always possible to build an orthogonal basis, and even an orthonormed basis, if desired, in the Ndimensional Hilbert space of dimension N, on which [H] is operating, which is formed by an orthogonal set of N eigenvectors.
A4.3. Transformation of coordinates It is first noted that:
z,
wl
= [r;]" w l [z.] :
IIr, II
[A4.12]
The eigenvector matrix [~], columns of which are N independent and normed eigenvectors, enables us to transform [ H ] into a similar matrix which is diagonal through the following formula: [A]
=
[~*]~[H] [~] ; ~.j
=
A,~
if j
=
i and 0 otherwise
[A4.13]
Transformation [A4.13] is known as a similarity transformation. Two matrices related to each other by a similarity transformation are said to be to be similar to each other. They have the same eigenvalues. Similarity can also be interpreted as a change of generalized coordinates: [q']~ [H] [q]
=
[q"]~ [~]r[H] [~][q~]
[A4.14]
whence the following formulas of coordinates transformation are deduced: [q] =[~][q~] r
[q q = [~]1 [q] = [~.]~ [q]
A4.4. Sign of the symmetric matrices A symmetric matrix [S] being given, we de/me the quadratic form:
[A4.15]
Appendices 397 [q]r [S][q] = cr
[A4.16]
[S] is said to be positively def'mite whenever cr is a definite and positive scalar, for all [q] ~ [0]. [S] is said to be negatively definite whenever cr is a definite and negative scalar, for all [q] ~ [0]. In the fi'ame of the eigenvectors of [S], [A4.16] becomes: N
[q]r [S][q] = [q']r [~]r [S][@][q'] = ~'~.iq; 2 = cr
tA4.17]
i=1
As the q; can take any arbitrary value, it is immediately inferred that: 9
The necessary and sufficient condition for [S] to be positive definite is that all the eigenvalues are positive.
9
The necessary and sufficient condition for [S] to be negative definite is that all the eigenvalues are negative.
Another possibility is that the eigenvalues are of the same sign, except at least one of them, which is zero. In such cases, the matrices are said to be non negative, or non positive, according to the sign of the non zero eigenvalues. Finally, when the eigenvalues are not all of the same sign, the sign of the matrix is not defined.
398
Discrete systems
Appendix 5: Crout's and Choleski's decomposition of a matrix A5.1. LUdecomposition of a non symmetric matrix
Any regular N x N matrix [A] can be written as the product of a righttriangular, or upper triangular, matrix [U] and a legtriangular, or lowtriangular matrix [L], based on the Gaussian method of elimination:
[A]=[L][U]
[A5.1]
For instance ifN = 4, [A5.1] is expressed as:
I A.. A.~_ A.~ A.. [Z~. 0 0 A~. A~_ 4~ A~4 =/L~. L._ 0
0 U.. U.~ U.~ U.. 0 0 U,2 U~3 U~4 [A5.2]
A3, A32 A33 ,434 /L3, L32 L33 0 00 A4..442 .443 A44 LL41 L42 L43 L44
0 0
U33 U34 0 U.
The decomposition is not unique.
A5.1.1. Crout 's decomposition The linear equations involved in the system [A5.2] are:
f! < j =j >j
Li,U,j + Li2U2j + .... + LiiU O.= A 0 Li,@j + Li2U2j + .... + LiiU ~. = ~.j
[A5.3]
Li,U,j + Li2U2j + .... + Lo.Uii = A O.
We have thus at our disposal N 2 linear equations involving N ( N + 1) unknowns. Such a system is clearly underdetermined and the N superfluous unknowns can be specified in an arbitrary way. One possible and rather natural choice is to let Li~ = 1. The remaining unknowns are then easily computed according to the following sequence of equations, which is repeated for every j: For i = 1, U~j = A~j
[A5.4] iI
For i = 2,..., j" U U = ,~.j  ~ LikUkj k=l
For i = j + l , j + 2 , . . . , N "
Lo
[A5.5]
Appendices 399 When applying such a procedure, it is verified that the numerical values of the elements Li,,Ukj, were made available before being used to compute those of Lu, U/j. On the other hand, it can be noted that the elements ,~j of the matrix to be transformed are used once only, hence the coefficients Lu,Uo of the transformed matrix can be conveniently stored at the place of ,~.j. Obviously, it is also useless to store the coefficients Li~ = 1. Finally, it is realized that whenever i = j , the equations [A5.5] and [A5.6] become identical to each other, except that [A5.6] involves a division by U~. As a consequence, the suitable choice of the pivot for the column can be made after having computed every possible candidate. Division of all coefficients is then performed at once. A5.1.2. Computation of the determinant and of the reciprocal matrix The LUdecomposition of[A] enables us to compute det[A] very easily, since: det [A] = (det [L])(det [U])
[A5.7]
Now, the determinant of a triangular matrix is obtained by multiplying all the diagonal elements with each other. Thus, det[L] = 1 and relation [A5.7] becomes: N
det [A] det [U] = II ujj
[A5.8]
j= l
However, to produce the proper sign of the determinant, it is necessary to take into account which parity arises from the row permutations involved in the optimised choice of the pivot. Actual computation of the reciprocal matrix [A] ~ can be carried out in the most suitable way by solving successively N forced systems of the type:
[A]IAf~I=[L][U]IAyl]=EIj]; j= 1,2,...N
[A5.91
where EA]'] stands for thejth column of [A]' and EIj] denotes thejth column of the identity matrix. More generally, any forced problem is solved according to two steps carried out successively, both of them involving the solution of a triangular system. The principle of the method is as follows: [A][X] = [F] r [L]IU][X] = [F] Letting [UI[X] = [Y], the two following systems are solved successively:
[A5.10]
400
Discretesystems
[L][Y]=[F] IV]Ix]Iv]
[A5.11]
A5.2. Choleski's decomposition of a symmetric matrix
Here it is shown that a regular and symmetric matrix [S], can be written as: [A5.12]
[S]=[U]r[U]
where [U] is again an upper triangular matrix. Indeed, starting from the decomposition [A5.1 ], [S] is written as:
[sl = [LI[D][0]
tAS. 131
EU] is obtained by normalizing the rows of [U] by the corresponding diagonal element
Uii.
Of course, as [S] is assumed to be regular, none is zero. [D] is the
diagonal matrix gathering the diagonal terms of [U]. Equations [A5.5] and [A.5.6] may be now written as: i1
O ,j = s ,,  Z [, ,, O ,j
kl iI
LO.'SijZLik~fkj
whenever/<j
[A5.14]
whenever i > j
k=l
The first relation [A5.14] is transposed, to obtain: j1
whenever j < i" Uji = sji  ~ / ' : U k i k=l i1
[A5.15]
whenever/> j" Oj~  Sij.  Z ZikOkj =Zig. = L o k=l
It is worth noting that the permutation of the indices i andj under the summation sign, when shifting from one formulation of 00. to the other, is indifferent since it is correctly reflected in the summation index. Thus, the following decomposition is reached:
Is] = I 0 ] ~
[D]IO]
[A5.16]
Appendices 401 Finally, to find the result [A5.12] it suffices to redef'me [U] (or equivalently [L]) as:
402
Discrete systems
Appendix 6" Some basic notions about distributions
A6.1. Functions described as functionals
Let y =fix) be a real function of the real variable x. We are used to consideringf as a mapping of the line Ox of the real numbers on itself. Indeed, it specifies a correspondence between a real number x, called the independent variable, and another number y, known as the value of the function, of the type Yi = f(xi), which can be put in a more concrete form by defining a table of correspondence of the type: X
X1
X2
X3
"'"
$
$
$
$
...
Y Yl
[A6.1]
Y2 Y3
Now, there exists another point of view for describing f, which is proving to be more general and often preferable for applications to physics than the classical one, recalled just above. Indeed, a physical device is required in any case for measuring the values taken by a function which describes a physical quantity. However, one has to realize that insmmaentation is unable to establish any pointtopoint correspondence between measured quantifies, but only a functional correspondence which express the action on the measurement system. In mathematical terms, one is led to consider the action of fix) on a set of auxiliary functions ~o(x), which are selected judiciously, as detailed in subsection A6.2. Assuming that f is defined for any real value of x, action is defined as:
f(x)go(x)dx
= ( f , go) =
[A6.2]
f i s then described according to the following table of correspondence: tp
~0~ r $ ~
~P3 "'" ~ ...
~t
~
~
~
...
A6.2. Vector subspace of test functions
For go to be a suitable test function defined in the real domain, it is required that: 1.
go is identically zero outside a bounded set of real values.
Appendices 2.
403
cp may be differentiated up to any arbitrary order. It may be easily checked that the set of all possible {cp } are forming a vector
subset (or linear manifold) (.~). Moreover, whenever cp ~ ( ~ ) , the derivatives to any order of cp belong also to ( ~ ) .
EXAMPLE.  ~0(x) =
f(1/exp x2 i
0
if
Ixl < 1
[A6.3]
otherwise
Furthermore, a restraining criterion for convergence in (~9) is stated, according to which a sequence {cpj}~ ( ~ ) is said to converge towards a function cp belonging to ( ~ ) if, and only if, the following conditions are fulfilled: 1.
All the (pj vanish outside a common f'mite interval, independently of j;
2.
cpj and their derivatives of any order k, converge uniformly to ~0 and to the corresponding derivatives, when j approaches inf'lnity.
N O T E .  Extension to several variables More generally, to accommodate the description of multivariable functions as functionals, the test functions can be defined over a space of real numbers with dimension n, denoted (R n ), where n is, of course, a positive integer. For the sake of brevity, they will be denoted cp(~), where the position vector f is defmed in a ndimensional Euclidean space.
A6.3. Distributions A distribution T is a continuous and linear functional, built on the vector space (~). This means that to any test function cp ~ ( 9 ) , T lets correspond a scalar (real,
or complex, number) denoted T(~), in accordance with the following properties:
If q)j>cp, then T ((pj)+ T (cp)
[A6.4]
The distributions also form a vector space, denoted (~'). In particular, the addition of two distributions, and the product of a distribution by a scalar are def'med by:
404
Discrete systems
r~ + r, = r, (~)+ r~ (~)} xr=~r(~)
[A6.51
A6.4. Regular distributions Let f(~) be a function of n real variables, which is locally integrable. It gives rise to a distribution through the scalar product: [A6.61 It can be proved that this linear functional is continuous and that two locally integrable functions, which are not equal, almost everywhere generate distinct distributions.
A6.5. Singular distributions All the distributions which are not generated by locally integrable functions are said to be singular. This is the case in particular of the Dirac distribution 8, which is def'med as follows:
(8 (~), r (~)} =