Nonlinear Dynamics: Between Linear and Impact Limits

Cover
......Page 1
Lecture Notes in Applied
and Computational Mechanics
Volume 52......Page 2
Nonlinear Dynamics......Page 4
ISBN: 9783642127984......Page 5
Preface......Page 6
Contents......Page 8
Brief Literature Overview......Page 13
Two Simple Limits of Lyapunov Oscillator......Page 16
Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis......Page 18
Viscous Dynamics under the Sawtooth Forcing......Page 21
The Rectangular Cosine Input......Page 23
Oscillatory Pipe Flow Model......Page 24
Strongly Nonlinear Oscillator......Page 27
Geometrical Example......Page 29
Nonlinear Equations and Nonlinear Phenomena......Page 31
Rigid-Body Motions and Linear Systems......Page 33
Remarks on the Multi-dimensional Case......Page 35
Elementary Nonlinearities......Page 36
Example of Simplification in Nonsmooth Limit......Page 37
Non-smooth Time Arguments......Page 38
Further Examples and Discussion......Page 40
Differential Equations of Motion and Distributions......Page 42
Transformation of Positional Variables......Page 45
Transformation of State Variables......Page 47
Linear and Weakly Non-linear Approaches......Page 49
Periodic Motions of Quasi Linear Systems......Page 50
The Idea of Averaging......Page 51
Averaging Algorithm for Essentially Nonlinear Systems......Page 53
Averaging in Complex Variables......Page 55
Lie Group Approaches......Page 56
Lyapunov’ Oscillator......Page 62
Nonlinear Oscillators Solvable in Elementary Functions......Page 65
Hardening Case......Page 67
Localized Damping......Page 70
Softening Case......Page 71
Nonsmoothness Hiden in Smooth Processes......Page 72
Nonlinear Beats Model......Page 73
Nonlinear Beat Dynamics: The Standard Averaging Approach......Page 75
Asymptotic of Equipartition......Page 80
Asymptotic of Dominants......Page 82
Necessary Condition of Energy Trapping......Page 84
System Description......Page 85
Normal and Local Mode Coordinates......Page 87
Local Mode Interaction Dynamics......Page 92
Auto-localized Modes in Nonlinear Coupled Oscillators......Page 96
Non-smooth Time Transformations......Page 103
Positive Time......Page 104
‘Broken Time’ Substitution......Page 106
Sawtooth Sine Transformation......Page 107
Links between NSTT and Matrix Algebras......Page 111
Differentiation and Integration Rules......Page 112
NSTT Averaging......Page 113
Generalizations on Asymmetrical Sawtooth Wave......Page 115
Multiple Frequency Case......Page 117
Definitions and Algebraic Rules......Page 119
Time Derivatives in the Idempotent Basis......Page 121
Definition and Algebraic Properties......Page 122
Differentiation Rules......Page 124
Oscillators in the Idempotent Basis......Page 125
Integration in the Idempotent Basis......Page 126
Discussions, Remarks and Justifications......Page 127
Remarks on Nonsmooth Solutions in the Classical Dynamics......Page 128
Caratheodory Equation......Page 129
Other Versions of Periodic Time Substitutions......Page 132
NSTT and Cnoidal Waves......Page 135
Smoothing Procedures......Page 140
Periodic Version of Lie Series......Page 144
Second-Order Non-autonomous Systems......Page 147
NSTT of Lagrangian and Hamiltonian Equations......Page 150
Remark on Multiple Argument Cases......Page 153
Free Harmonic Oscillator: Temporal Quantization of Solutions......Page 154
Standard Basis......Page 156
Idempotent Basis......Page 157
Regular Periodic Impulses......Page 158
Harmonic Oscillator under the Periodic Impulsive Loading......Page 160
Periodic Impulses with a Temporal ‘Dipole’ Shift......Page 164
Piecewise-Constant Excitation......Page 166
Parametric Impulsive Excitation......Page 168
General Case of Periodic Parametric Excitation......Page 170
Input-Output Systems......Page 172
Piecewise-Linear Oscillators with Asymmetric Characteristics......Page 174
Amplitude-Phase Equations......Page 175
Amplitude Solution......Page 176
Phase Solution......Page 177
Remarks on Generalized Taylor Expansions......Page 181
Multiple Degrees-of-Freedom Case......Page 182
The Amplitude-Phase Problem in the Idempotent Basis......Page 186
Nonsmooth Two Variables Method......Page 188
Resonances in the Duffing’s Oscillator under Impulsive Loading......Page 191
Strongly Nonlinear Oscillator under Periodic Pulses......Page 194
Impact Oscillators under Impulsive Loading......Page 198
Periodic Solutions for First Order Dynamical Systems......Page 203
Second Order Dynamical Systems......Page 204
The Vibroimpact Approximation......Page 206
One Degree-of-Freedom General Conservative Oscillator......Page 210
A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes......Page 213
Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero......Page 215
A Generalized Case of Odd Characteristics......Page 217
Periodic Motions Close to Separatrix Loop......Page 219
Self-excited Oscillator......Page 222
Strongly Nonlinear Oscillator with Viscous Damping......Page 226
Remark on NSTT Combined with Two Variables Expansion......Page 230
Oscillator with Two Nonsmooth Limits......Page 233
Bouncing Ball......Page 238
The Kicked Rotor Model......Page 242
Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics......Page 243
Wave Processes in One-Dimensional Systems......Page 248
Klein-Gordon Equation......Page 249
An Introductory Example......Page 252
Parameter Variation and Averaging......Page 256
A Two-Degrees-of-Freedom Model......Page 259
Averaging in the 2DOF System......Page 260
Impact Modes in Multiple Degrees of Freedom Systems......Page 263
A Double-Pendulum with Amplitude Limiters......Page 265
A Mass-Spring Chain under Constraint Conditions......Page 267
Systems with Multiple Impacting Particles......Page 269
Introductory Remarks......Page 272
Principal Directions of Linear Forced Systems......Page 274
Definition for Principal Trajectories of Nonlinear Discrete Systems......Page 275
Asymptotic Expansions for Principal Trajectories......Page 276
Definition for Principal Modes of Continuous Systems......Page 278
Introductory Remarks......Page 281
Problem Formulation......Page 283
Smooth Loading......Page 285
Impulsive Loading......Page 292
Periodic Solutions of the Period - n......Page 296
Two-Degrees-of-Freedom Systems......Page 299
The Autonomous Case......Page 300
Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles......Page 301
Impulsively Loaded Dynamical Systems......Page 304
Harmonic Oscillator under Sequential Impulses......Page 307
Random Suppression of Chaos......Page 309
Periodic Nonsmooth Structures......Page 310
Averaging for One-Dimensional Periodic Structures......Page 317
Two Variable Expansions......Page 318
Second Order Equations......Page 320
Acoustic Waves from Non-smooth Periodic Boundary Sources......Page 324
Spatio-temporal Periodicity......Page 328
Membrane on a Two-Dimensional Periodic Foundation......Page 331
The Idempotent Basis for Two-Dimensional Structures......Page 337
References......Page 343
Appendix......Page 354