Nonsmooth Mechanics (eBook)

Bernard Brogliato is Senior Researcher at INRIA Grenoble, France, where he founded and leads the team BIPOP. He published more than 70 journal articles in the fields of systems and Control, Solid Mechanics, and Applied Mathematics, as well as 5 monographs. His research interests are in non-smooth dynamical systems (mechanical systems with constraints, impacts, friction, electrical circuits with non-smooth components, sliding-mode control, optimal control with state constraints), and dissipative systems. He was an Associate Editor for Automatica, and the chairman of two Euromech Colloquia dedicated to Impact Mechanics. He coordinated the FP5 European project SICONOS (2 million euros fundings, 13 partners), and two projects funded by the French National Agency for Scientific Research, on multiple impacts and discrete-time sliding mode control.

Preface 8
Acknowledgments 1
Contents 14
Notation 21
1 Impulsive Dynamics and Measure Differential Equations 23
1.1 Impulsive Forces 23
1.2 Measure Differential Equations (MDEs) 29
1.2.1 A First Class of MDEs 30
1.2.2 A Second Class of MDEs: ODEs Driven by Measure Inputs 33
1.2.3 Further Reading 37
1.2.4 A Third Class of MDEs: ODEs with State Jump Mappings 38
1.2.5 Further Reading 40
1.3 Systems Subject to Unilateral Constraints 41
1.3.1 General Considerations 41
1.3.2 Flows with Collisions (Vibro-Impact Systems) 48
1.3.3 Unilaterally Constrained Systems: A Geometric Approach 56
1.3.4 Bilaterally Constrained Mechanical Systems and Impulsive Dynamics 60
1.4 Changes of Coordinates in MDEs 61
1.4.1 From Measure to Carathéodory Systems 61
1.4.2 Decoupling of the Impulsive Effects (Commutativity Conditions) 64
1.4.3 From Unilaterally Constrained Mechanical Systems to Filippov's Differential Inclusions: the Zhuravlev--Ivanov Method 66
2 Viscoelastic Contact/Impact Rheological Models 72
2.1 Simple Examples 73
2.1.1 From Elastic to Hard Impact 73
2.1.2 From Damped to Plastic Impact 76
2.1.3 The General Case 77
2.2 Viscoelastic Contact Models and Restitution Coefficients 87
2.2.1 Linear Spring-Dashpot 87
2.2.2 Nonlinear Elasticity and Viscous Friction: Simon-Hunt-Crossley and Kuwabara-Kono Dissipations 89
2.2.3 Conclusions 98
2.3 Viscoelastic Models with Dry Friction Elements: Viscoelasto-Plastic Models 99
2.3.1 Conclusions and Further Reading 103
2.4 Penalizing Functions in Mathematical Analysis 104
2.4.1 The Elastic Rebound Case 104
2.4.2 The Case with Dissipation (Linear Viscous Friction) 105
2.4.3 Uniqueness of Solutions 110
2.4.4 Further Existence and Uniqueness Results 113
2.5 Some Comments on Compliant Models 114
3 Variational Principles 115
3.1 Virtual Displacements, Velocities, and Accelerations Principles 115
3.1.1 The ``Classical'' Presentation 115
3.1.2 Using Variational and Quasi-Variational Inequalities Formalisms 118
3.2 A Coordinate Invariance Principle 122
3.2.1 Perfect Constraints 123
3.3 Gauss' Principle 124
3.3.1 Further Reading 125
3.4 Lagrange Dynamics 127
3.4.1 External Impulsive Forces 127
3.4.2 Example: Flexible Joint Manipulators 128
3.5 Hamilton's Principle and Unilateral Constraints 130
3.5.1 Hamilton's Principle Without Impacts 130
3.5.2 Hamilton's Principle With Impacts 131
3.5.3 Modified Set of Curves 135
3.5.4 Modified Lagrangian Function 139
3.5.5 Additional Comments and Studies 142
4 Two Rigid Bodies Colliding 146
4.1 Dynamical Equations of Two Rigid Bodies Colliding 146
4.1.1 General Considerations 146
4.1.2 The Local Kinematics 148
4.1.3 The Gap Function 151
4.1.4 The Two-Body System Dynamics 154
4.1.5 Dynamical Equations and Energy Loss at Collision Times 156
4.1.6 The Percussion Center 161
4.2 Restitution Laws 162
4.2.1 Elastoplastic Impacts and Restitution Coefficients 166
4.2.2 Adhesive Effects 177
4.2.3 Beyond Hertz: Conformal Contact Models 181
4.2.4 Conditions for Quasistatic Impacts 183
4.2.5 Incorporating Friction Effects 187
4.2.6 Conclusions 190
4.2.7 Material Parameters: Some Values 191
4.3 Impacts with Friction 192
4.3.1 Simple Examples 192
4.3.2 Kinematic CoR: Brach's Method 207
4.3.3 Additional Comments and Studies 212
4.3.4 Kinematic CoR: Frémond's approach 217
4.3.5 First Order Impact Dynamics: Darboux-Keller's Shock Equations 219
4.3.6 The Energetic Coefficient of Restitution 236
4.3.7 Examples 241
4.3.8 Other Energetical Coefficients 244
4.3.9 Additional Comments and Studies 244
4.3.10 Multiple Microcollisions Phenomenon: Toward a Global Coefficient 246
4.3.11 Conclusion 250
4.3.12 The Thomson-and-Tait Formula 251
4.3.13 Graphical Analysis of the Shock Dynamics 252
4.4 Impacts in Flexible Structures 255
4.4.1 Multimodal Modeling Approach 255
4.4.2 Infinite Dimensional System Approach 257
4.4.3 Further Reading 258
4.5 General Comments 258
5 Nonsmooth Lagrangian Systems 260
5.1 Lagrange Dynamics with Multiple Constraints 260
5.1.1 Frictionless Bilateral Constraints: The Contact Problem 263
5.1.2 Frictionless Unilateral Constraints: The Contact Problem 266
5.1.3 Mixed Bilateral/Unilateral Frictionless Constraints: The Contact Problem 270
5.1.4 Singular Mass Matrix: From Singular Lagrange's to Singular Hamilton's Dynamics 273
5.2 Moreau's Sweeping Process 275
5.2.1 First-Order Sweeping Process 275
5.2.2 Second-Order Sweeping Process: Frictionless Mechanical Systems 277
5.2.3 Well-Posedness Results 296
5.2.4 Continuous Dependence on Initial Data 303
5.3 Coulomb's Friction 304
5.3.1 Coulomb's Friction Model 305
5.3.2 Coulomb--Moreau's Disk 307
5.3.3 De Saxcé's Associated Formulation 309
5.3.4 Coulomb's Friction at the Acceleration Level 311
5.3.5 Further Comments on Friction Models 312
5.3.6 Sweeping Process with Friction 313
5.3.7 Additional Comments and Studies 316
5.4 Complementarity Formulations 316
5.4.1 Two Bodies: Signorini's Conditions 317
5.4.2 Linear Complementarity Problem (LCP) 318
5.4.3 Relationships with Quadratic Problems 321
5.4.4 Linear Complementarity Systems (LCS) 323
5.4.5 Controllability of LCS 343
5.4.6 Observability and Observers for LCS 346
5.4.7 Complementarity Systems and Hybrid Dynamical Systems 346
5.5 The Contact Problem with Coulomb's Friction 348
5.5.1 Introduction 348
5.5.2 Dissipativity of the Constrained Lagrange Dynamics 349
5.5.3 Extension of the Results of Sects.5.1.1, 5.1.2, 5.1.3? 350
5.5.4 The Contact Problem for a Planar Particle 351
5.5.5 A Second Simple Mechanism with Friction 354
5.5.6 Non-Uniqueness of the Contact Force 358
5.5.7 Comments 360
5.6 Painlevé's Paradoxes: Sliding Rod Example 361
5.6.1 The Dynamics of Painlevé's Example 361
5.6.2 The Contact LCP 363
5.6.3 Analysis of the Dynamical Singularities 366
5.6.4 Further Reading 371
5.6.5 Conclusions 374
5.7 Numerical Simulation 375
5.7.1 Event-Driven Algorithms 375
5.7.2 Compliant Contact/Impact Models 376
5.7.3 Time-Stepping (Event-Capturing) Numerical Algorithms 377
6 Generalized Impact Laws and Multiple Impacts 390
6.1 Particular Features of Multiple Impacts 390
6.1.1 Some Specific Features of Multiple Impacts 391
6.1.2 Han-Gilmore's and Binary Collisions Models 398
6.1.3 Penalization at Contacts (Compliance) 402
6.1.4 Multiplicity of Multiple Impacts 404
6.2 Kinematic Multiple-Impact Law (Generalized Newton) 405
6.2.1 The Quasi-Lagrange Equations 405
6.2.2 The Kinetic Energy 409
6.2.3 The Contact Forces Power 411
6.2.4 Restitution Law for Frictionless Systems 413
6.2.5 Restitution Law with Tangential Effects 417
6.2.6 Tangential Restitution 421
6.2.7 Comments 421
6.3 Energetic-CoR Multiple-Impact Law 422
6.3.1 Presentation of the LZB Impact Dynamics 423
6.3.2 Applications and Validations 426
6.3.3 Comparison of Different Multiple Impact Mappings 431
6.4 Further Reading 432
6.4.1 Kinetic Restitution (Poisson) 432
6.4.2 Kinematic Restitution (Newton and Moreau) 433
6.4.3 Other Approaches 433
7 Stability of Nonsmooth Dynamical Systems 435
7.1 Stability of Measure Differential Equations 435
7.1.1 Stability of Impulsive ODEs 435
7.1.2 Stability of Measure Driven ODEs (MDEs) 437
7.1.3 Additional Comments and Studies 438
7.2 Stability of the Discrete Dynamic Equations 439
7.2.1 The Bouncing-Ball with Fixed Obstacle 440
7.2.2 Lyapunov Stability of Discrete-Time Systems 443
7.3 Impact Oscillators 444
7.3.1 Existence of Periodic Trajectories 444
7.3.2 Further Reading 448
7.3.3 Comments on the Poincaré Impact Map Stability Analysis 450
7.3.4 Other Studies on Stability 454
7.3.5 Bouncing-Ball with Moving Base 455
7.3.6 Additional Comments and Studies 456
7.4 Grazing or C-Bifurcations 458
7.4.1 The Stroboscopic Poincaré Map Discontinuities 460
7.4.2 The Stroboscopic Poincaré Map Around Grazing-Motions 463
7.4.3 Further Comments and Studies 465
7.5 Complementarity Lagrangian Systems: Stability of Fixed Points 466
7.5.1 The Dynamical System 467
7.5.2 The Stability Analysis 469
7.5.3 Dissipativity Properties 472
7.5.4 Further Reading and Comments 475
7.5.5 Global Finite-Time Stability via the Zhuravlev-Ivanov Transformation 478
7.6 Stabilization of Impacting Systems: From Compliant to Rigid Models 480
7.6.1 System's Dynamics 480
7.6.2 Lyapunov Stability Analysis 482
7.6.3 Analysis of Quadratic Stability Conditions for Large Stiffness Values 483
7.6.4 A Stiffness-Independent Convergence Analysis 487
7.7 Stability of Linear Complementarity Systems 491
7.8 Further Reading 493
8 Trajectory Tracking Feedback Control 495
8.1 Trajectory Tracking: Rigid-Joint Rigid-Body Systems 495
8.1.1 Basic Concepts 497
8.1.2 Controller Design 503
8.1.3 Tracking Control Framework 505
8.1.4 Design of the Desired Contact Force During Constraint Phases 508
8.1.5 Strategy for Takeoff at the End of Constraint Phases ?kJ 510
8.1.6 Closed-Loop Stability Analysis 512
8.1.7 Illustrative Examples 513
8.1.8 Proof of Lemma 8.1 517
8.1.9 Proof of Theorem 8.1 521
8.2 Short Bibliography 524
8.3 Trajectory Tracking: Flexible-Joint Rigid-Link Systems 526
8.3.1 Basic Concepts 527
8.3.2 Tracking Control Framework 530
8.3.3 Desired Contact Force During Constraint Phases 533
8.3.4 Strategy for Takeoff at the End of Constraint Phases ?2k+1Bk 535
8.3.5 Closed-Loop Stability Analysis 536
8.3.6 Illustrative Example 537
8.3.7 Proof of Proposition 8.7 541
8.3.8 Proof of Lemma 8.2 542
8.3.9 Proof of Lemma 8.3 542
8.3.10 Proof of Theorem 8.2 544
8.4 A Unified Point of View 547
8.5 Further Results 547
8.5.1 Experimental Control of the Transition Phase 547
8.5.2 Juggling Robots Analysis and Control 549
8.5.3 Mechanisms with Joint Clearance 550
8.5.4 Observability and State Observers 551
Erratum to: Nonsmooth Mechanics 553
References 562
Appendix A Distributions, Measures, Functionsof Bounded Variations 564
Appendix B Elements of Convex Analysis 575
References 591
Index 647