Non Smooth Deterministic or Stochastic Discrete Dynamical Systems: Applications to Models with Friction or Impact Book

Introduction xi

Chapter 1. Some Simple Examples 1

1.1. Introduction 1

1.2. Frictions 1

1.2.1. Coulomb’s law 1

1.2.2. Differential equation with univalued operator and usual sign 3

1.2.3. Differential equation with multivalued term: differential inclusion 11

1.2.4. Other friction laws 12

1.3. Impact 16

1.3.1. Difficulties with writing the differential equation 16

1.3.2. Ill-posed problems 19

1.4. Probabilistic context 22

Chapter 2. Theoretical Deterministic Context 27

2.1. Introduction 27

2.2. Maximal monotone operators and first result on differential inclusions (in R) 27

2.2.1. Graphs (operators) definitions 28

2.2.2. Maximal monotone operators 29

2.2.3. Convex function, subdifferentials and operators 33

2.2.4. Resolvent and regularization 38

2.2.5. Taking the limit 40

2.2.6. First result of existence and uniqueness for a differential inclusion 40

2.3. Extension to any Hilbert space 45

2.4. Existence and uniqueness results in Hilbert space 57

2.5. Numerical scheme in a Hilbert space 59

2.5.1. The numerical scheme 59

2.5.2. State of the art summary and results shown in this publication 60

2.5.3. Convergence (general results and order 1/2) 61

2.5.4. Convergence (order one) 67

2.5.5. Change of scalar product 72

2.5.6. Resolvent calculation 74

2.5.7. More regular schemes 76

Chapter 3. Stochastic Theoretical Context 79

3.1. Introduction 79

3.2. Stochastic integral 79

3.2.1. The stochastic processes background 80

3.2.2. Stochastic integral 84

3.3. Stochastic differential equations 90

3.3.1. Existence and uniqueness of strong solution 91

3.3.2. Existence and uniqueness of weak solution 92

3.3.3. Kolmogorov and Fokker–Planck equations 95

3.4. Multivalued stochastic differential equations 101

3.4.1. Problem statement 101

3.4.2. Uniqueness and existence results 103

3.5. Numerical scheme 104

3.5.1. Which convergence: weak or strong? 106

3.5.2. Strong convergence results 108

3.5.3. Weak convergence results 122

Chapter 4. Riemannian Theoretical Context 129

4.1. Introduction 129

4.2. First or second order 129

4.3. Differential geometry 131

4.3.1. Sphere case 131

4.3.2. General case 132

4.4. Dynamics of the mechanical systems 139

4.4.1. Definition of mechanical system 139

4.4.2. Equation of the dynamics 141

4.5. Connection, covariant derivative, geodesics and parallel transport 144

4.6. Maximal monotone term 148

4.7. Stochastic term 149

4.8. Results on the existence and uniqueness of a solution 151

Chapter 5. Systems with Friction 155

5.1. Introduction 155

5.2. Examples of frictional systems with a finite number of degrees of freedom 155

5.2.1. General framework 155

5.2.2. Two elementary models 156

5.2.3. Assembly and results in finite dimensions 165

5.2.4. Conclusion 193

5.2.5. Examples of numerical simulation 194

5.2.6. Identification of the generalized Prandtl model (principles and simulation) 205

5.3. Another example: the case of a pendulum with friction 215

5.3.1. Formulation of the problem, existence and uniqueness 215

5.3.2. Numerical scheme 218

5.3.3. Numerical estimation of the order 219

5.3.4. Example of numerical simulations 221

5.3.5. Free oscillations 221

5.3.6. Forced oscillations 221

5.3.7. Transition matrix and calculation of the Lyapunov exponents 222

5.3.8. Melnikov’s method, transitory chaos and Lyapunov exponents 230

5.4. Elastoplastic oscillator under a stochastic forcing 231

5.4.1. Introduction 231

5.4.2. Modeling 232

5.4.3. Numerical scheme 236

5.4.4. Numerical results 238

5.5. Spherical pendulum under a stochastic external force 243

5.5.1. Establishment of the model 243

5.5.2. Numerical aspects 248

5.6. Gephyroidal model 255

5.6.1. Introduction 255

5.6.2. Description and transformation of the model 256

5.6.3. Quasi-static problems 263

5.6.4. Numerical simulations 265

5.6.5. Conclusion 267

5.7. Chain 268

5.7.1. Introduction 268

5.7.2. Description of the model 270

5.7.3. Transformation of the equations 271

5.7.4. Conclusion 283

5.8. An infinity of internal variables: continuous generalized Prandtl model 283

5.8.1. Introduction 283

5.8.2. Description of the continuous model 284

5.8.3. Existence, uniqueness and regularity results 287

5.8.4. Application to the discrete case, and convergence of the discrete model to the continuous model 289

5.8.5. Numerical scheme 291

5.8.6. Study of hysteresis loops 293

5.8.7. Numerical simulations 301

5.9. Locally Lipschitz continuous spring 301

5.9.1. Introduction 301

5.9.2. The studied model 301

5.9.3. Results for the existence and uniqueness of the solutions 303

5.9.4. Convergence results for the numerical schemes 311

5.9.5. The locally Lipschitz continuous case 313

5.9.6. Identification of the parameters from the hysteresis loops 314

5.9.7. Numerical simulations 320

Chapter 6. Impact Systems 325

6.1. Existence and uniqueness for simple problems (one degree of freedom) 326

6.1.1. The work of Schatzman–Paoli 326

6.1.2. Simple case with one degree of freedom, forcing and impact: piecewise analytical solutions 327

6.1.3. Adaptation of some classical methods 329

6.1.4. Movement with the accumulation of impacts and a sticking phase 333

6.1.5. Behavior of the numerical methods 337

6.1.6. Convergence and order of one-step numerical methods applied to non-smooth differential systems 338

6.1.7. Results of numerical experiments 343

6.2. A particular behavior: grazing bifurcation 348

6.2.1. Approximation of the map in the general case 349

6.2.2. Particular case 350

6.2.3. Stability of the non-differentiable fixed point 351

6.2.4. Numerical example 353

Chapter 7. Applications–Extensions 355

7.1. Oscillators with piecewise linear coupling and passive control 355

7.1.1. Description of the model 356

7.1.2. Free oscillations of the system 356

7.1.3. Order ��1 362

7.1.4. Case of periodic forcing 366

7.1.5. Conclusion 377

7.2. Friction and passive control 378

7.2.1. Introduction 378

7.2.2. Introduction to the models: smooth and non-smooth systems 379

7.3. The billiard ball 386

7.3.1. Maximal monotone framework 386

7.3.2. More realistic but non-maximal monotone framework 389

7.4. An industrial application: the case of a belt tensioner 390

7.4.1. The theory 390

7.4.2. The tensioner used 392

7.4.3. Identification of the parameters 392

7.4.4. Validation 393

7.5. Problems with delay and memory 396

7.5.1. Theory 396

7.5.2. Applications 399

7.6. Other friction forces 400

7.6.1. More general forms (variable dynamical coefficient) 401

7.6.2. With a variable static coefficient 419

7.6.3. With variable static and dynamical coefficients 421

7.7. With the viscous dissipation term 423

7.8. Ill-posed problems 424

7.8.1. First model: limit of a well-posed friction law 426

7.8.2. Second model: a differential inclusion without uniqueness 427

7.8.3. Conclusion 429

Appendix 1. Mathematical Reminders 431

A1.1. Two Gronwall’s lemmas 431

A1.2. Norms, scalar products, normed vector space, Banach and Hilbert space 432

A1.2.1. Scalar products, norms 432

A1.2.2. Banach and Hilbert space, separable space 433

A1.3. Symmetric positive definite matrices 435

A1.4. Differentiable function 435

A1.5. Weak limit 436

A1.6. Continuous function spaces 436

A1.7. Lp space of integrable functions 437

A1.7.1. Lp(Ω) space 437

A1.7.2. Lp(Ω, Rq ) space 438

A1.7.3. Lp(Ω; H) spaces 438

A1.8. Distributions 439

A1.8.1. Real values distributions 439

A1.8.2. Distributions with values in Rq 440

A1.8.3. Distributions with values in Hilbert space 440

A1.9. Sobolev space definition 441

A1.9.1. Functions with real values 441

A1.9.2. Functions with values in Hilbert space 441

Appendix 2. Convex Functions 443

A2.1. Functions defined on R 443

A2.2. Functions defined on Hilbert space 446

A2.2.1. Any Hilbert space 446

A2.2.2. 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 467

A5.1. Method used 467

A5.2. Lemma 5.1 468

A5.3. Lemma 5.4 473

A5.4. Lemma 7.1 476

Bibliography 477

Index 495