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Bernard Brogliato (Ed.)

Impactsin Mechanical SystemsAnalysis and Modelling

13

EditorBernard BrogliatoLaboratoire dAutomatique de GrenobleUMR, CNRS-INPG 5528, ENSIEGBP 46, Domaine Universitaire38402 Saint Martin dH`eres, France

Library of Congress Cataloging-in-Publication Data applied for.Die Deutsche Bibliothek - CIP-EinheitsaufnahmeImpacts in mechanical systems : analysis and modelling / BernardBrogliato (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; HongKong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 2000(Lecture notes in physics ; Vol. 551)(Physics and astronomy online library)ISBN 3-540-67523-X

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Preface

This volume contains extended versions of the ve plenary lectures given at theEuromech Colloquium 397 Impact in Mechanical Systems, at Grenoble, fromJune 30 until July 12, 1999, France. As the title indicates, it is devoted to thestudy of rigid multi-body mechanical systems subject to nonsmooth eects, suchas impacts, Coulomb friction, constraints addition and deletion. Actually, thisrepresents a large and important class of nonsmooth mechanical systems. Itsstudy can be traced back to the ancient Greeks. The 17th and 18th centuriesalso witnessed a great deal of (scientic) excitement about shock dynamics.Prestigious names like Descartes, Newton, Poisson, Gauss, Huygens, Bernoulli,to name a few, have long been attached to the study of collisional eects betweentwo rigid bodies. Later, scientists like Darboux, Routh, and Carnot, also contributed signicantly to the eld. The earliest studies on impact dynamics wereessentially motivated by fundamental scientic questions in physics (what is therole of hardness in the rebound phenomenon, is springiness necessary for a rebound to occur, use in light models), as well as more practical goals (calculationof bullet trajectories).Interest in such a class of dynamical systems today is certainly much morerelated to engineering, and in particular the development of simulation software for virtual prototyping, a topic of great importance in industry. However,it still possesses strong connections with physics: the study of granular matter,planetary rings, may benet from using the models described in this book (letus also recall that so-called billiards, which are a particular class of impactinglossless mechanical systems, have motivated intense mathematical studies). Inparticular, the study of Newtons cradle is closely related to what one needs toproperly understand and predict phenomena such as clusterization and uidization, which are well known in granular matter dynamics. In addition, numericalsimulations are quite important in these elds. Furthermore, scientic communities like computer science (virtual reality), robotics, systems and control, appliedmathematics and, evidently, applied mechanics nd various elds of investigationin multi-body systems. As the reader will see throughout this book, nonsmoothmechanical systems with unilateral (or inequality) constraints represent a veryinteresting class of dynamical systems. They are not a simple extension of systems with bilateral constraints, or of systems with impulses. To express it ina language that has recently become fashionable in the computer science andsystems and control communities, they constitute a class of hybrid dynamicalsystems; in other words, they merge continuous as well as discrete-event phe-

VI

nomena (roughly, their state space may be seen as the product of Rn with anite set of symbols). Contrary to some widely held opinion, their dynamics isquite complex.Many important problems associated with the dynamics of multi-body mechanical systems with unilateral constraints still remain open: mathematicalproblems (existence, uniqueness, continuous dependence on initial data, bifurcations, chaos), numerical analysis problems (how to discretize such a complexmixture of dierential equations and algebraic conditions), mechanics (some phenomena, such as multiple impacts, with or without friction, still require muchstudy on the modelling side), systems analysis (controllability, stabilizability).The ve chapters in this book contain contributions related to mathematics,modelling and numerical simulations. Mathematical Analysis The rst chapter, by M. Kunze (MathematicsDept., Cologne University) and M.D.P. Monteiro-Marques (MathematicsDept., Lisbon University), is devoted to presenting the so-called Moreaussweeping process. This evolution problem, invented by Moreau in the 1960s,applies to quasistatics as well as to dynamics. It was rst motivated by applications in uid mechanics and later on in nonsmooth mechanical systems.The focus of this chapter is on mathematical analysis. Numerical Analysis and Simulation The second chapter, by M. Abadie(Schneider Electric Research Center, Grenoble), is dedicated to numericalsimulation problems. It describes the work done at the company SchneiderElectric to improve the virtual prototyping of electrical devices. It also contains an overview of the existing tools for simulation of nonsmooth mechanical systems. As with other analyses, our nonsmooth systems require veryspecic numerical tools and cannot be accomodated by classical softwareand algorithms. The algorithms presented in this chapter have been inspiredby the discretization of the sweeping process as done by Moreau (see therst contribution ), with appropriate modications to comply with industrial needs (they are to be used directly by Schneiders engineers), whereasMoreaus scheme was primarily devoted to the simulation of granular matter. Stability and Bifurcations The third chapter by A. Ivanov (Universityof Moscow), deals with stability and bifurcation phenomena. It is a factthat systems with unilateral constraints possess specic sorts of bifurcationswhich are not encountered in smooth dynamics. They occur with grazingtrajectories and have therefore been called grazing bifurcations. Also, thestability of trajectories requires new analytical tools. Energetical Restitution Coecient The contribution of the fourth chapter by W.J. Stronge (Mechanical Engineering Dept., Cambridge University),focuses on collisions between two bodies using elasto-plastic models. It concentrates on the study of an energy coecient of restitution. It also containssome developments on multiple impacts (the so-called Newtons cradle). Multiple Impacts The nal chapter, by Y, Hurmuzlu and V. Ceanga (Mechanical Engineering Dept., Southern Methodist University), concentrates onmultiple impacts without friction. It proposes a completely new way to attack the multiple impact problem, by using a new set of physical coecients

VII

(including the energetical coecient presented in the previous chapter) todescribe the shock phenomenon. Newtons cradle and the rocking block areused to develop the theoretical analysis. Experiments conrm the analysis.In summary, this volume is an advanced introduction to the eld of analysis,modelling and numerical simulation of rigid body mechanical systems with unilateral constraints. It will be worthwhile reading for anybody interested in thistopic, be it at the mathematical, mechanical or numerical level. In fact, all theseelds of investigation feed one another and it is almost compulsory to have ageneral view of the problems in the other elds to be able to propose sound solutions in a particular domain. The book contains some established (although notalways very well known outside the nonsmooth dynamics community) results,as well as quite new ideas.I would like to express warm thanks to my colleagues who kindly accepted toprepare a plenary talk at the Euromech 397, and, most importantly, who madethe eort to write these chapters. They are gratefully acknowledged here. I wouldalso like to recall that the Euromech Colloquium 397 was organized within theframework of a European INTAS project coordinated by Bill Stronge. This bookrounds o this cooperation nicely.Saint-Martin dH`eres, June 2000

Bernard Brogliato

Contents

An Introduction to Moreaus Sweeping ProcessM. Kunze, M.D.P. Monteiro Marqu`es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Dynamic Simulation of Rigid Bodies:Modelling of Frictional ContactM. Abadie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Stability of Periodic Motions with ImpactsA.P. Ivanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Contact Problems for Elasto-plastic Impactin Multi-body SystemsW.J. Stronge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Impulse Correlation Ratioin Solving Multiple Impact ProblemsY. Hurmuzlu, V. Ceanga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

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