Non-smooth Deterministic or Stochastic Discrete Dynamical ... · differential equations will allow...

Non-smooth Deterministic or Stochastic Discrete Dynamical Systems

Dedication

Some of Michelle SCHATZMAN’S works are much present in this publication. Along collaboration had brought us together. She had co-directed Jérôme BASTIEN andFrédéric BERNARDIN’S PhD Theses. We would have wished to see her associatedwith this publication again.

Her premature leave made it decide otherwise.

We pay tribute to her here.

Jérôme BASTIENFrédéric BERNARDIN

Claude-Henri Lamarque

Non-smooth Deterministic or Stochastic Discrete Dynamical Systems

Applications to Models with Friction or Impact

Jérôme Bastien Frédéric Bernardin

Claude-Henri Lamarque

Series Editor Noël Challamel

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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 licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA

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© ISTE Ltd 2013 The rights of Jérôme Bastien, Frédéric Bernardin and Claude-Henri Lamarque to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2012955112 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-525-2

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1. Some Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1. Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2. Differential equation with univalued operator and usual sign . . . 31.2.3. Differential equation with multivalued term:

differential inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.4. Other friction laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3. Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1. Difficulties with writing the differential equation . . . . . . . . . . 161.3.2. Ill-posed problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4. Probabilistic context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 2. Theoretical Deterministic Context . . . . . . . . . . . . . . . . . 272.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2. Maximal monotone operators and first result on differential

inclusions (in R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1. Graphs (operators) definitions . . . . . . . . . . . . . . . . . . . . . 282.2.2. Maximal monotone operators . . . . . . . . . . . . . . . . . . . . . 292.2.3. Convex function, subdifferentials and operators . . . . . . . . . . 332.2.4. Resolvent and regularization . . . . . . . . . . . . . . . . . . . . . 382.2.5. Taking the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.6. First result of existence and uniqueness for a

differential inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3. Extension to any Hilbert space . . . . . . . . . . . . . . . . . . . . . . . 452.4. Existence and uniqueness results in Hilbert space . . . . . . . . . . . . . 57

vi Non-Smooth Discrete Dynamic Systems

2.5. Numerical scheme in a Hilbert space . . . . . . . . . . . . . . . . . . . . 592.5.1. The numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . 592.5.2. State of the art summary and results shown in this publication . . 602.5.3. Convergence (general results and order 1/2) . . . . . . . . . . . . 612.5.4. Convergence (order one) . . . . . . . . . . . . . . . . . . . . . . . . 672.5.5. Change of scalar product . . . . . . . . . . . . . . . . . . . . . . . 722.5.6. Resolvent calculation . . . . . . . . . . . . . . . . . . . . . . . . . 742.5.7. More regular schemes . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 3. Stochastic Theoretical Context . . . . . . . . . . . . . . . . . . . 793.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2. Stochastic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2.1. The stochastic processes background . . . . . . . . . . . . . . . . 803.2.2. Stochastic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3. Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . 903.3.1. Existence and uniqueness of strong solution . . . . . . . . . . . . 913.3.2. Existence and uniqueness of weak solution . . . . . . . . . . . . . 923.3.3. Kolmogorov and Fokker–Planck equations . . . . . . . . . . . . . 95

3.4. Multivalued stochastic differential equations . . . . . . . . . . . . . . . 1013.4.1. Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.4.2. Uniqueness and existence results . . . . . . . . . . . . . . . . . . . 103

3.5. Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.5.1. Which convergence: weak or strong? . . . . . . . . . . . . . . . . 1063.5.2. Strong convergence results . . . . . . . . . . . . . . . . . . . . . . 1083.5.3. Weak convergence results . . . . . . . . . . . . . . . . . . . . . . . 122

Chapter 4. Riemannian Theoretical Context . . . . . . . . . . . . . . . . . . 1294.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2. First or second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3. Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.3.1. Sphere case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.3.2. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.4. Dynamics of the mechanical systems . . . . . . . . . . . . . . . . . . . . 1394.4.1. Definition of mechanical system . . . . . . . . . . . . . . . . . . . 1394.4.2. Equation of the dynamics . . . . . . . . . . . . . . . . . . . . . . . 141

4.5. Connection, covariant derivative, geodesics andparallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.6. Maximal monotone term . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.7. Stochastic term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.8. Results on the existence and uniqueness of a solution . . . . . . . . . . 151

Table of Contents vii

Chapter 5. Systems with Friction . . . . . . . . . . . . . . . . . . . . . . . . . 1555.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2. Examples of frictional systems with a finite number of

degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2.1. General framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2.2. Two elementary models . . . . . . . . . . . . . . . . . . . . . . . . 1565.2.3. Assembly and results in finite dimensions . . . . . . . . . . . . . . 1655.2.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.2.5. Examples of numerical simulation . . . . . . . . . . . . . . . . . . 1945.2.6. Identification of the generalized Prandtl model

(principles and simulation) . . . . . . . . . . . . . . . . . . . . . . 2055.3. Another example: the case of a pendulum with friction . . . . . . . . . 215

5.3.1. Formulation of the problem, existence and uniqueness . . . . . . . 2155.3.2. Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2185.3.3. Numerical estimation of the order . . . . . . . . . . . . . . . . . . 2195.3.4. Example of numerical simulations . . . . . . . . . . . . . . . . . . 2215.3.5. Free oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.3.6. Forced oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.3.7. Transition matrix and calculation of the Lyapunov exponents . . . 2225.3.8. Melnikov’s method, transitory chaos and Lyapunov exponents . . 230

5.4. Elastoplastic oscillator under a stochastic forcing . . . . . . . . . . . . . 2315.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.4.2. Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2325.4.3. Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2365.4.4. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.5. Spherical pendulum under a stochastic external force . . . . . . . . . . 2435.5.1. Establishment of the model . . . . . . . . . . . . . . . . . . . . . . 2435.5.2. Numerical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

5.6. Gephyroidal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.6.2. Description and transformation of the model . . . . . . . . . . . . 2565.6.3. Quasi-static problems . . . . . . . . . . . . . . . . . . . . . . . . . 2635.6.4. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 2655.6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

5.7. Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685.7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685.7.2. Description of the model . . . . . . . . . . . . . . . . . . . . . . . 2705.7.3. Transformation of the equations . . . . . . . . . . . . . . . . . . . 2715.7.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

5.8. An infinity of internal variables: continuous generalizedPrandtl model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835.8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835.8.2. Description of the continuous model . . . . . . . . . . . . . . . . . 284

viii Non-Smooth Discrete Dynamic Systems

5.8.3. Existence, uniqueness and regularity results . . . . . . . . . . . . . 2875.8.4. Application to the discrete case, and convergence of the

discrete model to the continuous model . . . . . . . . . . . . . . . 2895.8.5. Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2915.8.6. Study of hysteresis loops . . . . . . . . . . . . . . . . . . . . . . . 2935.8.7. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 301

5.9. Locally Lipschitz continuous spring . . . . . . . . . . . . . . . . . . . . 3015.9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.9.2. The studied model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.9.3. Results for the existence and uniqueness of the solutions . . . . . 3035.9.4. Convergence results for the numerical schemes . . . . . . . . . . . 3115.9.5. The locally Lipschitz continuous case . . . . . . . . . . . . . . . . 3135.9.6. Identification of the parameters from the hysteresis loops . . . . . 3145.9.7. Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 320

Chapter 6. Impact Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3256.1. Existence and uniqueness for simple problems

(one degree of freedom) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3266.1.1. The work of Schatzman–Paoli . . . . . . . . . . . . . . . . . . . . 3266.1.2. Simple case with one degree of freedom, forcing and impact:

piecewise analytical solutions . . . . . . . . . . . . . . . . . . . . . 3276.1.3. Adaptation of some classical methods . . . . . . . . . . . . . . . . 3296.1.4. Movement with the accumulation of impacts and a

sticking phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3336.1.5. Behavior of the numerical methods . . . . . . . . . . . . . . . . . . 3376.1.6. Convergence and order of one-step numerical methods applied to

non-smooth differential systems . . . . . . . . . . . . . . . . . . . 3386.1.7. Results of numerical experiments . . . . . . . . . . . . . . . . . . . 343

6.2. A particular behavior: grazing bifurcation . . . . . . . . . . . . . . . . . 3486.2.1. Approximation of the map in the general case . . . . . . . . . . . 3496.2.2. Particular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3506.2.3. Stability of the non-differentiable fixed point . . . . . . . . . . . . 3516.2.4. Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Chapter 7. Applications–Extensions . . . . . . . . . . . . . . . . . . . . . . . 3557.1. Oscillators with piecewise linear coupling and passive control . . . . . 355

7.1.1. Description of the model . . . . . . . . . . . . . . . . . . . . . . . 3567.1.2. Free oscillations of the system . . . . . . . . . . . . . . . . . . . . 3567.1.3. Order 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3627.1.4. Case of periodic forcing . . . . . . . . . . . . . . . . . . . . . . . . 3667.1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

7.2. Friction and passive control . . . . . . . . . . . . . . . . . . . . . . . . . 3787.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Table of Contents ix

7.2.2. Introduction to the models: smooth and non-smooth systems . . . 3797.3. The billiard ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3867.3.1. Maximal monotone framework . . . . . . . . . . . . . . . . . . . . 3867.3.2. More realistic but non-maximal monotone framework . . . . . . . 389

7.4. An industrial application: the case of a belt tensioner . . . . . . . . . . 3907.4.1. The theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.4.2. The tensioner used . . . . . . . . . . . . . . . . . . . . . . . . . . . 3927.4.3. Identification of the parameters . . . . . . . . . . . . . . . . . . . . 3927.4.4. Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

7.5. Problems with delay and memory . . . . . . . . . . . . . . . . . . . . . 3967.5.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3967.5.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

7.6. Other friction forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4007.6.1. More general forms (variable dynamical coefficient) . . . . . . . . 4017.6.2. With a variable static coefficient . . . . . . . . . . . . . . . . . . . 4197.6.3. With variable static and dynamical coefficients . . . . . . . . . . . 421

7.7. With the viscous dissipation term . . . . . . . . . . . . . . . . . . . . . . 4237.8. Ill-posed problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4247.8.1. First model: limit of a well-posed friction law . . . . . . . . . . . 4267.8.2. Second model: a differential inclusion without uniqueness . . . . 4277.8.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Appendix 1. Mathematical Reminders . . . . . . . . . . . . . . . . . . . . . . 431A1.1. Two Gronwall’s lemmas . . . . . . . . . . . . . . . . . . . . . . . . 431A1.2. Norms, scalar products, normed vector space, Banach and

Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432A1.2.1. Scalar products, norms . . . . . . . . . . . . . . . . . . . . . 432A1.2.2. Banach and Hilbert space, separable space . . . . . . . . . . 433

A1.3. Symmetric positive definite matrices . . . . . . . . . . . . . . . . . 435A1.4. Differentiable function . . . . . . . . . . . . . . . . . . . . . . . . . 435A1.5. Weak limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436A1.6. Continuous function spaces . . . . . . . . . . . . . . . . . . . . . . 436A1.7. Lp space of integrable functions . . . . . . . . . . . . . . . . . . . 437A1.7.1. Lp(Ω) space . . . . . . . . . . . . . . . . . . . . . . . . . . . 437A1.7.2. Lp(Ω,Rq) space . . . . . . . . . . . . . . . . . . . . . . . . . 438A1.7.3. Lp(Ω;H) spaces . . . . . . . . . . . . . . . . . . . . . . . . . 438

A1.8. Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439A1.8.1. Real values distributions . . . . . . . . . . . . . . . . . . . . . 439A1.8.2. Distributions with values in Rq . . . . . . . . . . . . . . . . . 440A1.8.3. Distributions with values in Hilbert space . . . . . . . . . . . 440

A1.9. Sobolev space definition . . . . . . . . . . . . . . . . . . . . . . . . 441A1.9.1. Functions with real values . . . . . . . . . . . . . . . . . . . . 441A1.9.2. Functions with values in Hilbert space . . . . . . . . . . . . . 441

x Non-Smooth Discrete Dynamic Systems

Appendix 2. Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 443A2.1. Functions defined on R . . . . . . . . . . . . . . . . . . . . . . . . 443A2.2. Functions defined on Hilbert space . . . . . . . . . . . . . . . . . . 446

A2.2.1. Any Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 446A2.2.2. Particular case of the finite dimension . . . . . . . . . . . . . 446

Appendix 3. Proof of Theorem 2.20 . . . . . . . . . . . . . . . . . . . . . . . . 447

Appendix 4. Proof of Theorem 3.18 . . . . . . . . . . . . . . . . . . . . . . . . 455

Appendix 5. Research of Convex Potential . . . . . . . . . . . . . . . . . . . 467A5.1. Method used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467A5.2. Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468A5.3. Lemma 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473A5.4. Lemma 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Introduction

I.1. The type of problem that is treated

This book discusses non-smooth mathematical problems in the context ofmodeling discrete dynamical systems. These non-smooth terms are necessary tomodel the physics of the problems considered in this book. Nonlinear particulardifferential equations will allow us to take into account these non smooth terms.A geometrical approach to the mechanics is sometimes necessary, leading to aRiemannian vision. External solicitations to the mechanical systems will be discussedin a deterministic or stochastic context.

This provides a short description of the problems from the mathematical point ofview.

From the physics point of view, the phenomena discussed are related todry friction, impacts and models combining various types of friction, leading tomacroscopic descriptions of various rheologies.

I.2. Different modeling and tools

I.2.1.What is poorly or not addressed

In this book, we have not covered all the approaches used to study thenon-smooth problem. For example, the “Moreau” approach [ACA 08, BRO 96],the complementary linear or nonlinear problem formulations [ACA 08, BRO 96],the “Ivanov” [BRO 96] approach, Frémond’s theory and his modeling of rigid solidcollisions [FRE 95], approaches concerning capable differential problems and thestability in the sense of Lyapounov [LEI 08] will not be discussed.

We will not analyze general Lagrangian mechanics in detail, that is with massmatrices dependent on displacements and velocities.

xii Non-Smooth Discrete Dynamic Systems

I.2.2.What will be used here

I.2.2.1. Deterministic context of friction: maximal monotone differential inclusionswith sub-differential nonlinearities

In this book, we will present models of friction by using “maximal monotonedifferential inclusion with sub-differential nonlinearities”. Each of these notions willbe presented in a simple manner in Chapter 1 and will be sufficient for the reader tothen go directly to Chapter 5. The more mathematically experienced readers can referto Chapter 2 for more details.

In this section, let us briefly justify each of these terms.

We will see in sections 1.2.2 and 1.2.3 that the friction force intervening in motionmust take several possible values, especially in the no slip phase of friction. If wewant to write differential equations correctly with a mechanical sense, usual functionssuch as the sign are not adapted and must be generalized and replaced by multivaluedoperators, which are a different approach but give a perfect framework to describethe motion of a solid subject to friction force. In this case, the differential equations,which bring multivalued operators, then contain terms that can be equal, not to aunique value, but to a set of possible values. The other terms (univalued) of thedifferential equation belong to this set and the differential equations are then replacedby differential inclusions.

The notion of monotonicity, which is valid for functions, is extended to thesemultivalued operators. From the mechanical point of view, the power dissipated fromthese friction forces is always negative: in the case of Coulomb’s force, friction alwaysopposes velocity. This is a result of the monotone aspect of the operators studied.

The property of maximality for these operators corresponds to the physicalfollowing notion: we will study Coulomb’s problem to a degree of freedom wherethe dynamic coefficient of friction is equal to the static coefficient of friction μS . Inthis case, the friction force in the static phase loses all of its values between −μS

and μS . Also the velocity can take all the values in the dynamic phase. The operatorconsidered is maximal: there is no hole.

We will see that the friction forces are equal to the sub-differential (a notion thatwill be defined later) of a convex function. This notion of sub-differential generalizesthe notion of a derivative or a gradient. The nonlinear term behaves in an analogousway to the gradient of a potential from where the friction forces are derived. Be carefulthat this field is non-conservative because it is also dependent on the velocities.

Thus, let us note that in the deterministic case, all of these models are formallydescribed by a “differential inclusion” of the type

u(t) +Au(t) f(t, u(t)). [I.1]

Introduction xiii

where A is a multivalued operator, equal to the “sub-differential” of a convex function.The symbol , which is turned around, means that the right-hand side1 f(t, u(t))belongs to the set of values on the left-hand side u(t) + Au(t). In this differentialinclusion:

– u is the vector Rp of internal and external displacements of models with theirpossible derivatives;

– f is regular and contains terms originally linear elastic, the eventual dampingterms (regular) and the external conditions;

– Au is a nonlinear multivalued term that provides friction forces.

I.2.2.2. Impact frameworkIn the impact framework, we will demonstrate in section 1.3 that, similar to [I.1],

this time we will have differential inclusions of the typeu(t) +Au(t) f (t, u(t), u(t)), [I.2]

with an impact law, whether conservative or not. The operator A will also be equal tothe simple sub-differential of the convex function, null in part of the phase space. Thisterm will, in fact, signify that the studied mass is “stuck” in part of the space and theforce that it is subject to will also be multivalued: null when the solid does not touchthe edges, it becomes equal to the reaction of the edges when the solid is in contactwith the edges: it can take a set of possible values in this case.

We also note the difference between [I.1] and [I.2]; this second inclusion isfundamentally of the second order when the first is of the first order; in [I.2], thesolutions are not necessarily functions but less regular objects2, contrary to [I.1] wherethe solutions are functions3.I.2.2.3. Stochastic and Riemannian context

The differential inclusion [I.1] introduced in the previous section makes externalforces appear in its regular part acting on the mechanical system considered. It can beuseful, sometimes necessary, to resort to a random model of external forces. Examplesthat come to mind are earthquakes, the wind or even swell [KRE 83].

Another source of hazard can originate from the mechanical system’s parameterswhere the characteristics are measured with inherent imprecision to all measured

1 This presentation was preferred to the following presentationf(t, u(t)) ∈ u(t) + Au(t),

probably out of habit of writing differential equations beginning with the derived term y (t) =f(t, u(t)).2 Radon measurements.3 Integrable and piecewise differential.

xiv Non-Smooth Discrete Dynamic Systems

systems. As long as the model is very sensitive to input values, consideration of thevariability in the model is an essential step itself with regard to obtaining simulationresults, accountable for the phenomenology studied.

In this book, we will not discuss the last case where the parameters are random,with the associated differentiable equations formalism being in the deterministic case.When time and hazard are coupled, which is the case for the external force, theformalism cannot be reduced to the deterministic case.

A particular theory must be used, which is for stochastic processes. Chapter 3presents this theory which comes in the introduction of processes themselves and ofstochastic integral and stochastic differential equations. In this chapter, multivaluedstochastic differential equations are discussed

dXt +A(Xt)dt b(t,Xt)dt+ σ(t,Xt)dWt, [I.3]

which is an extension of the differential inclusion [I.1] with the introduction of theBrownian motion W .

Examples of such models are given in Chapters 1 and 5 while considering rubbingmechanical systems which are solicited by white Gaussian noise and are eventuallyfiltered.

We will also present the Fokker–Planck equation in Chapter 3 governingprobability laws of a stochastic differential equation solution. We will also analyzewhy it is useful to establish the order of convergence of numerical methods.

Chapter 4 presents a generalization of equation [I.3] to the Riemannian manifolds.This is done to consider systems presenting a geometry that is reduced to the Euclideancase, for example those making intervening angles in their parameterization. We givean example in Chapter 5.

A reminder on differential and Riemannian geometry is presented in this chapter todefine a mechanical system in the context of analytical mechanics. In this formalism,the friction modeling by multivalued operators and stresses in a stochastic frameworkis also discussed.

I.3. References used

All the results presented in this book have been incurred in three PhD Thesesdefended at the University of Lyon I and codirected by Claude-Henri Lamarque andMichelle Schatzman: that of Jérôme Bastien in 2000 [BAS 00c], that of Olivier Janin4

4 Directed by Claude-Henri Lamarque only.

Introduction xv

in 2001 [JAN 01b] and that of Frédéric Bernardin in 2004 [BER 04c]. The first thesisrelates mainly to friction problems in the deterministic context, the second to impactproblems and the third to friction problems in the stochastic or Riemannian context.Generally, they are based on many existing results.

The references of published works during or pursued after these theses (allcowritten by the authors) are all cited under their corresponding chapters. Thereferences concerning both the mathematical context and its applications and themechanical aspects are the following:– for friction problems in the deterministic context [BAS 00a, BAS 00b, BAS 02a,

BAS 02b, BAS 05, BAS 07a, BAS 07b, BAS 08a, BAS 08b, BAS 09, BAS 12,LAM 00, LAM 03, LAM 05a, SCH 99, SCH 10], or even in a deterministic andstochastic context [LAM 05b];– for piecewise continuous problems [GOU 05, LAM 11];– for friction problems in a Euclidean stochastic context [BER 03, BER 09];– for friction problems in a Riemannian stochastic context [BER 04a, BER 04b];– for impact problems [JAN 01a, JAN 02].

Most of the results of these works presented in this book are discussed.Nevertheless, the reader may need to refer to these references for theoretical ortechnical results.

I.4. Reading guide

A simple example of one degree of freedom is given in Chapter 1, demonstratingthe different aspects of the three problems studied (friction, impact and stochasticcontext). The multivalued derivative inclusion notions are simplified and summarized.Simple simulations also show the numerical method used.

The mathematical novice reader can then consult Chapter 2, which demonstratesfundamentals on maximal multivalued operators and recalls the existing results ofexistence and uniqueness, and the numerical method used for derivative inclusionsthat allow friction modeling in a deterministic context. These results are also basedon the knowledge of notions that are supposed to be known (integration, distributionand Hilbert space) which are briefly summarized in Appendix 1. Several notionson convex functions are also recalled in Appendix 2. Any interested reader whowishes to read the complete text on the proof of the convergence of first order ofthe numerical method in the deterministic case should refer to Appendix 3 (recentlypublished result).

xvi Non-Smooth Discrete Dynamic Systems

Both Chapters 3 and 4 present the notions used5 in the stochastic and Riemanniancontext. Any reader wishing to read the complete text on the proof of theweak convergence of the numerical method in the stochastic context is referred toAppendix 4. For these chapters, a number of prerequisites, notably the fundamentalson measuring and the functions of bounded variations (BV), are not recalled in thebody of the text.

Any reader wishing to study friction problems in the deterministic context directlycan refer to Chapter 5, where a number of examples are proposed on discretemechanical systems; prior reading of Chapter 1 will let the reader understandmultivalued derivative inclusions and the numerical methods used. For all problemsgiven in the deterministic context, the numerical methods are completely explicit andare thus exploitable. In these deterministic problems, several technical calculationsare presented in Appendix 5, which is optional. In this chapter, we will presenta continuous model (in section 5.8), which remains considered as only discrete tothe extent that the internal variables are infinite. In this chapter, a model will alsobe presented in the stochastic context in section 5.4 and will then almost certainlyrecall the more theoretical Chapter 3. Also, a model with a cubic term is presented insection 5.9 in the deterministic context and is then accessible to the non-probabilisticreader; however, it is also given in the stochastic context. Finally, a model isdemonstrated in section 5.5 in the Riemannian context in Chapter 4.

Chapter 6 presents impact models. Section 6.1 presents a simple case with onedegree of freedom and simulations with completely explicit numerical methods thatare ready to be used. Section 6.2 is more delicate and discusses the problem of grazingbifurcation.

Chapter 7 presents models with new frictions that are less academic than thosepresented in Chapter 5. Some of them (sections 7.1 and 7.2) give their applicationsin passive control. Section 7.4 presents an industrial application of friction modelsfor the study of a belt tensioner distribution. Section 7.6.1 proposes more general andrealistic friction laws than Coulomb’s friction.

5 Even more mathematically difficult than in Chapter 2.

Chapter 1

Some Simple Examples

1.1. Introduction

In this chapter, we give three simple examples of mechanical models in the threedomains that are concerned in this book: friction problems (section 1.2), impactproblems (section 1.3) and friction problems in the stochastic context (section 1.4).Each of them treats a simple mechanical system, a mass with one degree of freedomresting on a support plane. This chapter constitutes an informal introduction to the restof the chapters where the theory will be developed with different application examplesof the work.

1.2. Frictions

1.2.1. Coulomb’s law

We recall Coulomb’s friction law: we consider a solid resting on a rough groundplane, which exerts a reaction R on that solid. This reaction decomposes itself on anormal component N , perpendicular to the ground, and a tangent component T (seeFigure 1.1). As long as the ratio T/N does not exceed a certain limit f0, there isadherence and the solid remains at rest. Once the value is reached, there is a slip, wehave T/N = f0 and the tangential force is opposed to the relative velocity between thesolid and the ground. We note the adherence condition being that the vectorR remainsin a cone. The value f0 depends on the nature of the ground and the solid. We assumehere that the dynamic friction coefficient (in slip phase or dynamic phase) is equal tothe static friction coefficient (in adherence phase or static phase). Occasionally, thisstatic coefficient of friction is supposed to be larger than the dynamic, which can benumerically dangerous (see section 7.8). For more details, we can consult section 8.5of [GIE 85].

2 Non-Smooth Discrete Dynamic Systems

R

T

N

Figure 1.1. Coulomb’s friction

We now assume that the solid moves along an axis and we note x the abscissaof the solid. We can then assume the normal of the force N to be constant. In thiscase, the cone transforms into an interval: the tangential force is named g and thevelocity of the solid is x. We introduce the function sign defined by:

sign(x) =

⎧⎪⎨⎪⎩−1 if x < 0,

1 if x > 0,

0 if x = 0.

[1.1]

Then there is a number α > 0 such that:

if x = 0, g ∈ [−α, α], [1.2a]if x = 0, g = −αsign (x). [1.2b]

We remark that [1.2] is not equivalent to g = −αsign (x), which implies that in thestatic phase, we have g = 0! (see section 1.2.2). Nevertheless, we can rewrite [1.2]under a more condensed form, by introducing a multivalued operator, meaning anapplication of R in the set of R parts. Let us then define the operator σ by:

σ(x) =

⎧⎪⎨⎪⎩−1 if x < 0,

1 if x > 0,

[−1, 1] if x = 0.

[1.3]

We can also see it as part of R2, represented in Figure 1.2. We then have:g ∈ −ασ (x). [1.4]

We next identify the graph of R2 notions and of multivalued operators on R. We willthen discuss the σ graph defined by [1.3]. We note that in the dynamic phase (when xis non-zero), the force g is equal to −αsign (x) and that in the static phase (when x isnull), g belongs to [−α, α]. We will return to the graphs in more detail in Chapter 2.

Let us assume now that the studied solid has a mass m, an external force F actingon it and that we are given two initial conditions:

x(0) = x0, x(0) = x0. [1.5a]

Some Simple Examples 3

1

-1

x

y

σ

Figure 1.2. The operator σ

The fundamental principle of dynamics gives:mx = F + g. [1.5b]

1.2.2. Differential equation with univalued operator and usual signBefore studying a differential inclusion with Coulomb’s law friction force in

section 1.2.3, let us now look at what happens if we choose to define the functiong, thanks to the “usual” sign defined by [1.1], meaning:

g(t) = −α sign (x(t)). [1.6]We will see that the problem formed by equations [1.5] and [1.6] equivalent to:

x(0) = x0, x(0) = x0, [1.7a]mx(t) + α sign (x(t)) = F (t), [1.7b]

is ill-posed and that the analytical resolution of this simple problem does not providethe solution! We note here that by setting u0 = x(0) and u = x, [1.7] is equivalent to:

u(0) = u0, [1.8a]

∀t ∈ [0, T ], u(t) +α

msign (u(t)) =

1

mF (t). [1.8b]

To simplify, we assume thatm = α = 1. [1.9]

We then have the differential equation:u(0) = u0, [1.10a]∀t ∈ [0, T ], u(t) + sign (u(t)) = F (t). [1.10b]


Let us now consider a special case:

u0 = 0, F (t) = t. [1.11]1) We will first assume – which is physically well conceived – that the mass is at

rest for a certain amount of time, which is necessary for obtaining an external forcelarge enough to move it. Let us assume then:

∃T > 0: ∀t ∈]0, T ], u(t) = 0, [1.12]Hence, u = t in [0, T ] and then u(t) = t2/2 > 0, as soon as t > 0. This iscontradictory to our starting hypothesis [1.12] (and against intuition).

2) Let us then try the other hypothesis:∀T > 0, ∃t0 ∈]0, T ]: u(t0) = 0. [1.13]

We note:ε = sign(u(t0)) ∈ {−1, 1}. [1.14]

We can set T such that:

T <1

2. [1.15]

We define t1, the smallest real number of [0, t0[ such that u is the sign of u(t0) on]t1, t0]. By continuity,

u(t1) = 0. [1.16]We therefore have by integration of [1.10b] on [t1, t0]:

u(t0)− u(t1) + ε(t0 − t1) =1

2t20 − t21 ,

Then,

u(t0) =1

2t20 − t21 − ε(t0 − t1) = (t0 − t1)

1

2(t0 + t1)− ε ,

and therefore,

sign(u(t0)) =1

2(t0 + t1)− ε.

We still have two cases: either ε = 1 and from [1.15]

sign(u(t0)) =1

2(t0 + t1)− 1 < t0 − 1 <

1

2− 1 = −1

2,


which contradicts [1.14]. Thus, ε = −1 and

sign(u(t0)) =1

2(t0 + t1) + 1 > 1,

which contradicts [1.14].

In conclusion, none of these hypotheses are compatible with the model, yet theycorrespond to what we expect as possible answers to the problem. The explanationcan be found in the expression of an incorrect sign. Besides, the calculation of thefriction force with this “usual” expression of the sign is not correct from a physicalpoint of view: for a mass at rest, the friction force would be null!

Another question arises: is it possible to define for the usual sign simple numericalschemes that would provide reasonable solutions, at least from a physical point ofview? Let us use the same example, the same expression of the sign and test theconstruction of numerical schemes.

1) Let us start with an explicit Euler numerical scheme (see, e.g. [BAS 03,CRO 84, SCH 01]) for problems [1.10]–[1.11]. By discretizing, we then introducethe sequences un u(tn), and tn = nh, for n ∈ {0, ..., N}, with h = T/N as thetime step (chosen constant) and u0 = u(0) = 0. We get u0 = 0 and

∀n ∈ {0, ..., N − 1}, un+1 − un

h+ sign(un) = nh, [1.17]

or finally∀n ∈ {0, ..., N − 1}, un+1 = un + h (hn− sign(un)). [1.18]

Figure 1.3 shows the results obtained for two values of h. Some oscillationsemerge; at each time value, the function approached changes sign and it can no longerremain at 0. We also note that the amplitude of the oscillations seem to decrease withh. This issue can be demonstrated in the following way: we generally assume that:

u0 ∈ [−2h, 2h] and ∀t ∈ [0, T ], |F (t)| ≤ 1. [1.19]Then, let (vn)0≤n≤N be defined by:

∀n ∈ {0, ..., N − 1}, vn =un

h. [1.20]

The scheme [1.17] is then written as:∀n ∈ {0, ..., N − 1}, vn+1 = − sign(vn) + vn + F (tn). [1.21]

Showing by induction on n ∈ {0, ..., N} that:∀n ∈ {0, ..., N}, vn ∈ [−2, 2]. [1.22]


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−3 Euler scheme

h=0.0002h=0.002

Figure 1.3. Two approximations of [1.10] and [1.11] given bythe explicit Euler scheme [1.18]

For n = 0, this comes from the initial condition [1.19]. Let us assume that [1.20] istrue for n. We can observe that the function x → − sign(x) + x maps the interval[−2, 2] in [−1, 1]. Thus, − sign(vn) + vn belongs to [−1, 1]; since F (tn) ∈ [−1, 1],we deduce from this that by summing, vn+1 ∈ [−2, 2]. We then deduce from [1.22]that:

∀n ∈ {0, ..., N}, |un| ≤ 2h. [1.23]Numerically, we have drawn for N describing a logarithmic interval

[10nmin, 10nmax ] curve (N,max0≤n≤N (|un/h|)) (see Figure 1.4(a)), whichcorroborates [1.23]. The oscillations are more difficult to demonstrate formally.We remark that u1 = 0; we have then drawn in Figure 1.4(b)) the curve(N,min2≤n≤N (|un/h|)), which shows that the minimum seems to be comprisedbetween two curves, one above corresponding to non-zero values (of the order of10−5).

Finally for each value ofN , we can calculate the number P (N) of sign changes ofun defined by the number of valuesn ∈ {2, . . . , N−1} such as unun+1 < 0. We haveP (N) ≤ N − 2. We have drawn the curve (N,P (N)) in Figure 1.5. On this curve,the points seem perfectly aligned: the correlation is equal to r = 0.9999999999074,for a slope a = 0.5000000232 and an ordinate at the origin b = −1.3748. We thenhave:

P (N) ≈ 0.5000000232N − 1.3748.


0 0.5 1 1.5 2 2.5 3 3.5x 105

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2maximum

a)max0≤n≤N (|un/h|) versus N

0 0.5 1 1.5 2 2.5 3 3.5x 105

−18

−16

−14

−12

−10

−8

−6

−4

−2log 10 of minimum

b) log10(min2≤n≤N (|un/h|)) versus N

Figure 1.4. Extreme values of |un/h| versus N for N ∈ [10nmin , 10nmax ]with nmin = 1 and nmax = 5 .5


0 0.5 1 1.5 2 2.5 3 3.5x 105

0

2

4

6

8

10

12

14

16x 104 Number of changing signs

Figure 1.5. Number P (N) of changing signs of un versus N

2) Let us continue with a numerical implicit Euler scheme for the same problem[1.10]–[1.11]. In this case, with the same notations, [1.17] must be replaced by:

∀n ∈ {0, ..., N − 1}, un+1 − un

h+ sign(un+1) = nh, [1.24]

with u0 = 0. Let us examine the possible states and demonstrate that un is not definedfor n ≥ 2. Let us assume that N is large enough so that:

h < 1. [1.25]For n = 0, [1.24] gives:

u1

h+ sign(u1) = 0. [1.26]

We can check that u1 = 0 is a solution of [1.26]. On the other hand, u1 > 0 isimpossible since [1.26] would give u1/h = −1. Likewise, u1 < 0 is impossible since[1.26] would give u1/h = 1. Thus, for n = 1, [1.24] gives:

u2

h+ sign(u2) = h. [1.27]

The u2 = 0 case is impossible since [1.27] would give 0 = h. Likewise, the u2 > 0case is impossible since [1.27] would give:

u2 = (h− 1)h < 0,


according to [1.25]. Finally, the u2 < 0 case is impossible because [1.27] would give:u2 = (h+ 1)h > 0.

3) We have also tested the resolution of [1.10] and [1.11] with two Matlab©solvers (Figure 1.6).

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−2.5

−2

−1.5

−1

−0.5

0

0.5

1x 10−5 approximate calculation with solvers

ode23ode45

Figure 1.6. Two approximations of [1.10] and [1.11]given by two Matlab © solvers

Oscillations still appear and the velocity is non-zero. For the more precise secondsolver (ode23), the amplitude of the oscillations is weaker.

We have drawn in Figure 1.7 the results given by explicit Euler and two Matlab©solvers, this time on [0, 2]. Oscillations still appear and the velocity is non-zero. Forthe more precise second solver (ode23), the amplitude of the oscillations is smaller.Beyond 1 (see Figure 1.7(b)), the oscillations disappear: we are in the case where thepoint material slips with a friction force equal to −1 and there are no more problems.COMMENT 1.1.– The astute reader will understand that even if the numerical solutiongiven by [1.10] and [1.11] approaches zero, it cannot tend toward a solution of [1.11],since the second member F (t) − sign(u) is not continuous in u and so the usualtheorems of convergence of numerical schemes [BAS 03, CRO 84, SCH 01] cannotbe applied.

Thus, it is clear that if we look at the consistency of the results provided by theseschemes with the usual sign, they are not capable of returning correct behavior from amathematical or physics point of view.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6approximate calculation with Euler scheme and solvers

ode23ode45Euler

a) on the whole interval

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.040

0.5

1

1.5

2

2.5

3

3.5

4x 10−3 approximate calculation with Euler scheme and solvers

ode23ode45Euler

b) Around t = 1

Figure 1.7. Three approximations of [1.10] and [1.11] given by Explicit Eulernumerical scheme and two Matlab© solvers on [0 , 2 ]

Let us take advantage of this simple example to demonstrate why the usual signis not adapted to good modeling. We consider mass m initially at rest on the supportsubjected to an external slowly growing force F (t) starting from zero. As long as theforce is not large enough to overcome the friction effect (let us say at an intervalof time [0, T ], T > 0), the mass will stay motionless. In [1.10] and [1.11], theterm −α sign(u(t)) corresponds to Coulomb’s friction force. Clearly on [0, T ], thisamounts to: 0 = F (t), which is not coherent with the hypothesis on f(t), exceptfor t = 0. It appears that we must develop the possibility for the Coulomb’s frictionmodel to not be univalued: as the mass is not set in motion, sign(u(t)) = sign(0)must take multiple values allowing F (t) to balance. This leads us to consider the


friction term to be modeled as a graph (the graph of a function then being a specialcase). The motion equation will then be described not by the previous equality but bya differential inclusion, as shown in section 1.2.3.

1.2.3. Differential equation with multivalued term: differential inclusionIn this section, we demonstrate that the problem is well set if the force g is defined

by [1.4]:g(t) ∈ −ασ (x(t)). [1.28]

We are therefore seeking two functions x and g checking systems [1.5] and [1.28]. Bysetting:

g(t) = mx(t)− F (t),

this system can also be written in the equivalent following form:x(0) = x0, x(0) = x0,mx(t)− F (t) ∈ −ασ (x),

then again by returning the symbol ∈:x(0) = x0, x(0) = x0, [1.29a]∀t ∈ [0, T ], mx(t) + ασ (x(t)) F (t). [1.29b]

This is not a differential equation but a differential inclusion. We will then take acloser look at this differential inclusion [1.29], where the data is F and the unknownfunction is x. As in section 1.2.2, by setting u0 = x(0) and u = x, we replace [1.29]by:

u(0) = u0,

∀t ∈ [0, T ], u(t) +α

mσ (u(t))

1

mF (t).

To simplify things, we will assume that [1.9] occurs. We therefore have a differentialinclusion:

u(0) = u0, [1.30a]∀t ∈ [0, T ], u(t) + σ (u(t)) F (t). [1.30b]

We will come back to this example, according to initial conditions and secondmembers in section 2.2.6.1.

As in section 1.2.2, we can present several simulations. By anticipatingsection 2.2.6.1, we note that the inclusions [1.11] and [1.30] are of the type [2.26]with the hypotheses [2.35] where ta = 1 and T = 2 are valid. The exact expression is


then given by [2.36], thus here:

∀t ∈ [0, T ], u(t) =

⎧⎪⎪⎨⎪⎪⎩0 if t ≤ 1,

1

2t2 − t+

1

2if t ≥ 1.

[1.31]

We note that in section 2.2.6.1, the ad hoc numerical implicit Euler scheme used (see[2.32]) provides the solution with an error in 0(h).

The results are given in Figure 1.8. In this figure, we note that in the static phase,on [0, 1], u is null and the numerical scheme gives a solution that is rigorously null,without oscillation.

To conclude, we therefore note that all is well here with a differential inclusion,contrary to the differential equation ill-posed in section 1.2.2.

COMMENT 1.2.– Let us note two important things here. The force g is continuous;otherwise said, the σ graph does not present a “hole”. Moreover, the power of theforce dissipated by friction is always negative because:

g(t)x(t) = 0, if x(t) = 0,

g(t)x(t) = −α sign (x(t)) x(t) = −α |x(t)| , if x(t) = 0.

We therefore have:

g(t)x(t) ≤ 0. [1.32]

The function sign is monotone; we deduce from it the monotonicity of the σ graphthat can be written as:

∀x, y ∈ R, u ∈ σ(x), v ∈ σ(y) =⇒ (v − u)(y − x) ≥ 0. [1.33]

Since α is strictly positive and 0 ∈ σ(0), [1.32] is only a consequence of [1.28] and[1.33].

1.2.4. Other friction laws

Other type of frictions can be considered, which are more realistic than idealCoulomb’s friction.

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