Asymptotic approaches in nonlinear dynamics: new trends and applications

Cover......Page 1
Springer Series in Synergetics......Page 2
Title......Page 3
ISBN 3-540-63894-6......Page 4
Preface......Page 5
Contents......Page 9
1. Introduction: Some General Principlesof Asymptotologyl......Page 13
1.1 An Illustrative Example......Page 14
1.2 Reducing the Dimensionality of a System......Page 16
1.3 Continualization......Page 17
1.4 Averaging......Page 18
1.5 Renormalization......Page 19
1.7 Linearization......Page 20
1.8 Pade Approximants......Page 21
1.9 Modern Computers and Asymptotic Methods......Page 22
1.11 Problems and Perspectives......Page 23
2.1 The Classical Perturbation Technique: an Introduction......Page 25
2.2 Krylov-Bogolubov-Mitropolskij Metho......Page 31
2.3 Equivalent Linearization......Page 36
2.4.1 Introduction......Page 38
2.4.2 Nonresonance Oscillations......Page 39
2.4.3 Oscillations in the Neighbourhood of Resonance......Page 43
2.5 Nonstationary Nonlinear Systems......Page 54
2.6.1 Analysed System and Equation of Motion......Page 67
2.6.2 Transformation of the Equations of Motion to the Main Coordinates......Page 71
2.6.3 Zones of Instability of the First Order......Page 74
2.6.4 Calculation Examples......Page 86
2.7.1 One-Degree-of-Freedom System......Page 93
2.7.2.1 Introduction......Page 98
2.7.2.2 Autonomous System......Page 99
2.8 Hopf Bifurcation......Page 105
2.9.1 Introduction......Page 112
2.9.2 Control of Vibro-Impact Periodic Orbits......Page 113
2.9.3 Stability Control......Page 115
2.9.4 Simulation Results......Page 117
2.10.1 Definition......Page 118
2.10.2 Free Oscillations and Close Natural Frequencies!......Page 120
2.11.1 Choice of Asymptotic Expansion Parameters......Page 130
2.11.2 6-Expansions in Nonlinear Mechanics (49]......Page 133
2.11.3 Asymptotic Solutions for Nonlinear Systems with High Degrees of Nonlinearity......Page 140
2.11.4 Square-Well Problem of Quantum Theory......Page 142
2.12.1 .One-Point Pade Approximants: General Definitions and Properties......Page 144
2.12.2 Using One-Point Parle Approximants in Dynamics......Page 146
2.12.3 Matching Limit Expansions......Page 151
2.12.4 Matching Local Expansions in Nonlinear Dynamics!......Page 155
2.12.5 Generalizations and Problems......Page 160
3.1 Continuous Approximation for a Nonlinear Chain......Page 163
3.2.1 Nonhomogeneous Rod......Page 167
3.2.2 Stringer Plate......Page 170
3.2.3 Perforated Membrane......Page 174
3.2.4 Perforated Plate......Page 178
3.3.1 Berger and Berger-Like Equations for Plates and Shells......Page 183
3.3.2 \"Method of Freezing\" in the Nonlinear Theory of Viscoelasticity......Page 188
3.4.1 Straightforward Bolotin Approach......Page 189
3.4.2 Modified Bolotin Approach......Page 197
3.5.1 Circular Rings and Axisymmetric Cylindrical Shells......Page 203
3.5.2 Reinforced and Isotropic Cylindrical Shells......Page 209
3.5.3 Nonlinear Oscillations of a Cylindrical Panel......Page 226
3.5.4 Stability of Thin Spherical Shells Under Dynamic Loading1......Page 231
3.5.5 Asymptotic Investigation of the Nonlinear Dynamic Boundary Value Problem for a Rod......Page 240
3.6 One-Point Pade Approximants Using the Method of Boundary Condition Perturbation......Page 241
3.7 Two-Point Parle Approximants: A Plate on Nonlinear Support......Page 248
3.8 Solitons and Soliton-Like Approaches in the Case of Strong Nonlinearity......Page 252
3.9.1 Introduction......Page 259
3.9.2 Modified Envelope Equation......Page 260
4.1 Periodic Oscillations of Discrete-Continuous Systems with a Time Delay......Page 265
4.2 Simple Perturbation Technique......Page 279
4.3.1 Introduction......Page 292
4.3.2 Dynamics Equations......Page 293
4.3.3 Averaging......Page 294
4.3.4 Numerical Results......Page 297
General References......Page 303
Detailed References (d)......Page 311
Index......Page 315