Impacts in Mechanical Systems: Analysis and Modelling (Lecture Notes in Physics)

Springer......Page 1
The Production Process......Page 2
Impacts in Mechanical Systems......Page 3
Preface......Page 5
Contents......Page 8
1 Introduction......Page 9
2 Some preliminaries from convex analysis and fucntional analysis......Page 11
3.1 The Lipschitz continuous sweeping process......Page 17
3.2 An application: Evolution of a quasi-static mechanical system......Page 24
3.3 The state-dependent sweping process......Page 31
3.4 Additional remarks......Page 37
4 Second-order problems......Page 41
5.1 Frictionless unilateral contact......Page 51
5.2 Unilateral contact with friction......Page 62
References......Page 65
Introduction......Page 69
Normal Contact Force – Penalized Method......Page 70
Tangential Contact Force – Penalized Method......Page 71
Open Systems......Page 74
Closed loop systems......Page 75
Method of Integration......Page 76
Runge-Kutta Methods......Page 77
Interpolation of the Results......Page 78
Integration of the Differential-Algebraic Systems by the Runge-Kutta Methods......Page 79
Methods of correction of the drift......Page 82
Calculation of the acceleration and of the Lagrange multipliers.......Page 83
Modelling of Frictional Contact......Page 84
Definition of a Unilateral Connection......Page 85
Contact Law......Page 87
A First Approach......Page 90
A Contact without Friction......Page 91
A Contact with Friction......Page 92
Generalization......Page 94
Formulation of a Problem of Optimization under Constraints......Page 96
Contact without Friction......Page 97
Contact with Friction......Page 98
Bibliographical Analysis......Page 100
Determination of our Strategy......Page 113
Contact without Friction......Page 115
Contact with friction......Page 116
Convergence......Page 117
Contact with Friction......Page 118
Structure of the Gauss-Seidel Procedure in Acceleration......Page 119
Bilateral Constraints......Page 121
Joint drivers......Page 122
Particular Cases......Page 123
Indeterminacy......Page 124
Inconsistency......Page 125
Formulation of the problem......Page 127
Contact Law......Page 129
Tangential Contact Law......Page 130
Resolution of the problem......Page 132
Definitions......Page 133
Definition......Page 136
Implementation......Page 137
Propagation of the Impulses......Page 138
Interpretation of the Impulses in Forces......Page 140
Problem of Capture......Page 143
Industrial Example - C60 Circuit Breaker......Page 144
Second Experiment......Page 146
Conclusion......Page 147
Acknowledgement......Page 148
References......Page 149
1 Introduction......Page 153
2.1 Unilateral Constraints......Page 154
2.2 Equations of Smooth Motion......Page 155
2.3 Impacts......Page 156
2.4 Multiple Impacts......Page 157
3.1 Concepts of Stability of Motion with Impacts......Page 159
3.2 Poincaré Maps......Page 161
3.3.2 Linearization......Page 164
3.3.3 Stability Conditions......Page 167
3.3.4 Bifurcations......Page 168
4.1 Appearance of Grazing......Page 169
4.2.1 One Unilateral Constraint......Page 171
4.2.2 Double Unilateral Contraint......Page 173
4.3 One-Degree-of-Freedom System with Periodic Forcing......Page 175
4.4 Multiple Degrees of Freedom......Page 180
5.1 Orthogonality Conditions......Page 181
5.2 Variation of Periodic Orbit......Page 183
5.3 Stability Conditions......Page 186
5.4 Bifurcations......Page 190
Acknowledgements......Page 192
References......Page 193
Contact Problems for Elasto-Plastic Impact in Multi-Body Systems......Page 196
1. Introduction......Page 197
2. Impact Process......Page 198
3. ‘Rigid’ Body Impact Theory for Smooth Hard Bodies......Page 199
4. Extended Hertz Theory for Elastic-Plastic Impact......Page 202
4.1 Elastic Indentation from Normal Contact Force......Page 203
4.2 Indentation at Yield of Elastic-Plastic Bodies......Page 205
4.3 Fully Plastic Indentation......Page 207
4.4 Elastic Unloading from Fully Plastic Indentation......Page 209
4.5 Energetic Coefficient of Restitution......Page 211
5. Effect of Tangential Compliance Between Colliding Bodies......Page 214
5.1 Normal Relative Velocity for Collinear Impact......Page 216
5.2 Tangential Relative Velocity for Collinear Impact and Dry Friction......Page 217
Tangential Velocity Changes During Slip......Page 218
Transition from Initial Slip to Intermediate Period of Stick......Page 219
5.3 Change of Relative Velocity for Different Slip Processes......Page 220
5.4 Example: Oblique Impact Of Sphere......Page 225
5.5 Maximum Force From Oblique Impact Of Sphere......Page 227
5.6 Discrete Modelling of Tangential Compliance......Page 228
6. Chain Reactions From Impact In Multi-Body System......Page 229
Wave Propagation In Linear Coaxial Periodic System......Page 230
6.2 Example: Impact Response Of Multi-body System With Graduated Properties......Page 232
7. CONCLUSION......Page 239
References......Page 240
1 Introduction......Page 242
2.1 Three ball chain......Page 244
2.2 Impulse Correlation Ratio......Page 245
2.3 The solution method......Page 247
2.4 Multiple impacts......Page 250
2.5 Post impact bouncing patterns of a three ball cradle......Page 252
2.6 Generalization of the three ball approach to N-balls......Page 255
2.7 I termittent collisions......Page 256
2.8 Special Examples......Page 257
2.9 Experiments......Page 261
2.10 Experimental verification of the Impulse Correlation Ratio......Page 262
2.11 Experimental observation of the small center ball behavior......Page 265
3.1 Problem Description......Page 266
3.2 Impulse Correlation Ratio......Page 267
3.3 Velocity impulse relationships......Page 269
3.4 The single impact case......Page 271
3.5 Multiple Sequences of Impacts......Page 275
3.6 Numerical analysis example......Page 276
References......Page 279