Qualitative Analysis of Nonsmooth Dynamics: A Simple Discrete System with Unilateral Contact and Coulomb Friction

By Alain Léger and Elaine Pratt

Qualitative Analysis of Nonsmooth Dynamics: A Simple Discrete System with Unilateral Contact and Coulomb Friction explores the effects of small and large deformations to understand how shocks, sliding, and stick phases affect the trajectories of mechanical systems. By analyzing these non-regularities successively this work explores the set of equilibria and properties of periodic solutions of elementary mechanical systems, where no classical results issued from the theory of ordinary differential equations are readily available, such as stability, continuation or approximation of solutions. The authors focus on unilateral contact in presence of Coulomb friction and show, in particular, how any regularization would greatly simplify the mathematics but lead to unacceptable physical responses.

• Explores the effects of small and large deformations to understand how shocks, sliding, and stick phases affect the trajectories of mechanical systems
• Includes theoretical results concerning the full investigation of the behavior under constant or oscillating loadings, even in the case of the simplest mechanical systems
• Provides a focus on unilateral contact in presence of Coulomb friction
• Helps you gain an accurate understanding of how the transition occurs to ensure the safe use of any machine involving rotating or sliding mechanisms

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 26, 2016
• ISBN: 9780081012017

Alain Léger

Alain Leger received his PhDs in Mathematics and Sciences from the University Pierre et Marie Curie, Paris. He held positions as research engineer and senior research at EDF and CNRS respectively. He is the co-manager of the International Scientific Coordination Network of 'Wave Propagation in Complex Media'. His research areas lie in the mechanics of solids and structure, unilateral problems within the dynamics of discrete systems, bifurication theory and wave propagation in complex media.

Book preview

Qualitative Analysis of Nonsmooth Dynamics

A Simple Discrete System with Unilateral Contact and Coulomb Friction

Alain Léger

Elaine Pratt

Series Editor

Noël Challamel

Table of Contents

Cover image

Title page

Copyright

Preface

Introduction: The Mechanics of Unilateral Systems

1: The Model

Abstract:

1.1 About contact and friction conditions: nonsmooth nonlinearities

1.2 A class of discrete systems

1.3 Study of the restoring force

1.4 The dynamical problem

2: Mathematical Formulation

Abstract:

2.1 The complete mathematical statement

2.2 Computational methods

3: The Equilibrium States

Abstract:

3.1 In the linearized case

3.2 Equilibria at large strains

4: Stability

Abstract:

4.1 Stability of the equilibrium states in the classical sense

4.2 Introducing a new notion of stability

4.3 Analysis of simple models

4.4 A slightly more complicated mass–spring system

4.5 Numerical experiments on a finite element discretization of an elastic body

5: Exploring the Case of the Linear Restoring Force

Abstract:

5.1 Equilibrium states in the {Rt, Rn} plane under an oscillating loading

5.2 When no equilibrium solutions exist

5.3 When a single equilibrium solution exists

5.4 When infinitely many equilibria exist

5.5 Toward a more general excitation

6: The Case of the Nonlinear Restoring Force

Abstract:

6.1 Introduction

6.2 A particular case: Fn > 0 large enough and |Ft| large enough

6.3 Qualitative study of the set of equilibria

6.4 Qualitative dynamics when ε

6.5 Periodic solutions when ε

7: Open Problems and Challenges

Abstract:

7.1 Complementary calculations

7.2 Toward more challenging problems

Bibliography

Index

Copyright

First published 2016 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

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© ISTE Press Ltd 2016

The rights of Alain Léger and Elaine Pratt to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-094-2

Printed and bound in the UK and US

Preface

At the beginning of the 20th Century nonlinear dynamics was enriched by many fundamental findings, and from then on was widely studied within the framework of ordinary differential equations. Over the past 60 years, a second-order differential equation with a polynomial nonlinearity and a harmonic forcing, the so-called Duffing equation, was taken as an archetypal model of nonlinear dynamics in discrete systems. Its investigation produced predictive diagrams which are currently used for the occurrence of instability or the transition to chaos.

In the case of partial differential equations the studies are more recent, not as clear and not as complete. It is well known that partial differential equations are classified into different types, namely elliptic, parabolic or hyperbolic, according to the respective orders of the derivative in the different directions, each type leading to very different behaviors. Elliptic partial differential equations are most commonly found to describe the equilibria of solids and structures. In dynamics, parabolic equations may describe the bending deformations of structures but are mainly used in fluids mechanics through the Navier–Stokes equation. Any other motions, whether membrane deformations of structures or dynamics of three-dimensional deformations of solids, are given by hyperbolic equations. Certainly, the nonlinear partial differential equation that has been the most extensively studied is the Navier–Stokes equation, but its behavior remains the topic of many very difficult and technical research studies. In the mechanics of structures, hyperbolic partial differential equations still involve unknowns even concerning basic stability properties.

But the equation of motion changes drastically when the boundary conditions involve conditions of contact or friction, although contact or friction are quite common physical situations. In the case of discrete systems, the equation of motion at first glance still looks like an ordinary differential equation, but a closer look reveals that the equation is far from being classical: the right-hand side involves terms that are not functions, and the equation itself should be understood in the sense of measures. By comparison with classical ordinary differential equations, this feature represents the transition from smooth to nonsmooth dynamics. Very few studies have been performed on this subject. This book book endeavors to investigate the behavior of very simple mechanical systems when the nonsmoothness arises from non-regularized unilateral contact and Coulomb friction.

This book results from a long-term collaboration between the authors. The authors have been working together for the past 15 years, and have also been interested in complementary subjects with respect to the topic of this book; one of the authors has been working for many years on bifurcation and stability problems arising from the mechanics of solids and structures, and the other has been working on nonlinear dynamics for ordinary differential equations.

Over all these years, the authors have been working in the Laboratory of Mechanics and Acoustics of the CNRS in Marseille as a part of the group on Contact Mechanics, whose interests range from theoretical aspects of statics or dynamics to industrial applications, in collaboration with many PhD students.

The authors are highly indebted to Jean-Jacques Moreau; his presence at Montpellier and his regular visits to Marseille were a strong support to the foundations of nonsmooth dynamics in our group. The authors also wish to thank Michel Jean, who built the numerical tools to compute nonsmooth dynamics and to whom the authors owe explicit contributions to the present work. Through an extended analysis of the dynamics of simple systems, this book intends to contribute to the understanding of nonsmooth dynamics. This book is dedicated to the memory of Jean-Jacques Moreau.

Alain Léger; Elaine Pratt, Laboratoire de Mécanique et d'Acoustique, CNRS Marseille

February 2016

Introduction: The Mechanics of Unilateral Systems

Abstract: A brief glance at the state of the art concerning a large class of problems of mechanics may help us to understand the objectives of the present book. From a very wide and very abstract point of view, the statement of a problem of solid mechanics requires a specific combination of concepts coming from physics. For example, does the physics require that the problem be modeled by a discrete system or by a continuous medium? Does it imply that the boundary conditions are unilateral or bilateral? Do these conditions involve contact and friction? Is the problem static or dynamical? The complexity of these alternatives may rapidly increase by considering subclasses also arising from physics, for example does the system undergo large deformations or small deformations, involve distance interactions or contact interactions.

Keywords: Coulomb friction, Discrete systems, Dynamical problem, Friction laws, Linearly elastic continuous media, Static problem, Unilateral systems

A brief glance at the state of the art concerning a large class of problems of mechanics may help us to understand the objectives of the present book. From a very wide and very abstract point of view, the statement of a problem of solid mechanics requires a specific combination of concepts coming from physics. For example, does the physics require that the problem be modeled by a discrete system or by a continuous medium? Does it imply that the boundary conditions are unilateral or bilateral? Do these conditions involve contact and friction? Is the problem static or dynamical? The complexity of these alternatives may rapidly increase by considering subclasses also arising from physics, for example does the system undergo large deformations or small deformations, involve distance interactions or contact interactions.

An initial outline of the present book would be that it considers a specific combination of the following alternatives: the model is discrete or continuous, the problem is static or dynamical, involves unilateral contact, with or without friction, and in each case, focus will be concentrated on the question of utmost physical interest, namely, the question of whether there exists a solution to the corresponding mathematical model, and moreover does there exist a single solution.

1) In the case of discrete systems:

i) The static problem of a discrete system with or without friction is now well understood, but it is important to bear in mind that although an equilibrium problem may look simple, the theoretical results have been established quite recently and even now justify an extended analysis.

without friction under strong restrictions on the regularity of the loading, by Patrick Ballard in 2000. The case with friction is precisely the subject of this book. The dynamical response is relatively well understood only for simple models, but remains intricate for more general problems, although well-posedness for large-size systems was obtained in 2014 under conditions on the loading similar to those concerning the case without friction.

2) In the case of linearly elastic continuous media the situation is totally different, and way beyond the scope of this book:

i) The formulation of the equilibrium problem of a linearly elastic solid at small strains submitted to unilateral contact conditions without friction is due to Antonio Signorini [SIG 59] in the 1950's, and Gaetano Fichera [FIC 63] proved in the 1960's existence and uniqueness for this equilibrium problem. Later on in the 1960s, these results were generalized within the new framework of variational inequalities. Uniqueness is lost in general if the framework of linear elasticity at small strains is removed and replaced either by nonlinear elasticity, by large strains, or by any other nonlinear behavior. However, unilateral conditions at the boundary are not in general cause of this loss of uniqueness.

ii) No similar result exists if the conditions at the boundary involve Coulomb friction even in the static case where the problem remains open and certainly difficult.

iii) Almost all the problems related to the dynamics of a continuous medium in the presence of unilateral contact are open for the most part. There are no results at all in the presence of Coulomb friction. In the case of contact without friction, only two results can be viewed as first steps toward solving the problem. The first one is the work by G. Lebeau and M. Schatzman who obtained existence and uniqueness of an energy preserving solution to the wave equation, but their result was strictly restricted to a half space with unilateral contact at the boundary. The second one was obtained by J. U. Kim, as opposed to the result by G. Lebeau and M. Schatzman, it dealt with a bounded domain of any smooth enough shape. However, there was a strong restriction: it did not apply to elasticity but only to the harmonic operator, and moreover established existence but not uniqueness. Of course, an existence result can be seen as an important first step and is often the most difficult step, but here it was considered in the mechanical community as being of little physical interest as it dealt only with the Laplacian.

3) Are the investigation of the behavior and the computation of trajectories of solutions to continuous systems involving unilateral contact and Coulomb friction possible at short term?

i) Regularizing contact or friction laws: The contact and friction laws will be discussed in Chapter 2, but the essential point is that these laws are represented by multivalued mapping and not by functions. Regularizing these laws means changing these multivalued mappings into functions, but no convergence of the regularized solution toward the solution of the corresponding mechanical problem has been established and indeed no estimation of the error is available because there are as yet no results giving existence and uniqueness of the solution under unilateral contact and friction. Therefore, regularization does not seem to be an interesting path to follow.

ii) Discretizing continuous bodies: Another idea, currently used in commercial softwares, comes as a trivial consequence of points (1) and (2) above: in the case of discrete systems, exploring the set of equilibrium states or the dynamics has either already been performed or is possible, while almost all the problems whether static or dynamical are open in the case of continuous media. Then, the natural idea was to discretize the continuous media at hand, by a finite element method for example, and to perform the calculations using the corresponding discrete problem. The use of a finite element method or of any other discretization of a continuous medium requires the convergence of the solution, which means that the solution should tend to the solution for the continuous body when discretization step size, for example mesh size in the case of finite elements, tends to zero. Unfortunately, such a convergence result does not exist when the continuous body is submitted to unilateral contact and to Coulomb friction and, as in the case of regularization, establishing such a convergence result would lead to the same difficulties as those encountered when tackling the continuous problem itself.

iii) Restricting attention to discrete systems: Discrete systems involve one more equation than continuous bodies, essentially because the discrete model removes the possibility of waves propagating inside the body after an impact time, but discrete models, nevertheless, exhibit a very large range of behaviors that are far from being well known. Even if studying discrete models is a strong restriction with regard to the analysis of the mechanics of unilateral systems in general, they will certainly bring important insight on nonsmoothness, whether in statics or dynamics. The authors have decided in the present book to restrict their attention to discrete systems.

The book is organized as follows:

– Chapter 2 aims at describing the models. The description first focuses on the constitutive laws. The meaning of the unilateral contact and friction laws will be given in detail. Then, the finite degree of freedom mechanical devices that will be studied in the following chapters will be presented, either at small or at large deformations.

– Chapter 3 is concerned with theoretical results and numerical tools. After setting the mathematical foundations of the dynamical problem of a discrete system submitted, in addition to external forces, to unilateral contact and Coulomb friction with some rigid support, the current state of theoretical results for this problem is given. In particular, the necessary conditions insuring uniqueness are stated and counterexamples to uniqueness are given. Then, a number of numerical methods designed to harness the specificity of the dynamics are given. Specificity means that unilateral contact implies, for instance, that a trajectory may involve impacts and jumps; Coulomb friction may signify that a