Nonlinear Dynamics (eBook)

Title Page 2
Preface 6
Contents 8
Introduction 13
Brief Literature Overview 13
Asymptotic Meaning of the Approach 16
Two Simple Limits of Lyapunov Oscillator 16
Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis 18
Quick ‘Tutorial’ 21
Remarks on the Basic Functions 21
Viscous Dynamics under the Sawtooth Forcing 21
The Rectangular Cosine Input 23
Oscillatory Pipe Flow Model 24
Periodic Impulsive Loading 27
Strongly Nonlinear Oscillator 27
Geometrical Views on Nonlinearity 29
Geometrical Example 29
Nonlinear Equations and Nonlinear Phenomena 31
Rigid-Body Motions and Linear Systems 33
Remarks on the Multi-dimensional Case 35
Elementary Nonlinearities 36
Example of Simplification in Nonsmooth Limit 37
Non-smooth Time Arguments 38
Further Examples and Discussion 40
Differential Equations of Motion and Distributions 42
Non-smooth Coordinate Transformations 45
Caratheodory Substitution 45
Transformation of Positional Variables 45
Transformation of State Variables 47
Smooth Oscillating Processes 49
Linear and Weakly Non-linear Approaches 49
A Brief Overview of Smooth Methods 50
Periodic Motions of Quasi Linear Systems 50
The Idea of Averaging 51
Averaging Algorithm for Essentially Nonlinear Systems 53
Averaging in Complex Variables 55
Lie Group Approaches 56
Nonsmooth Processes as Asymptotic Limits 62
Lyapunov’ Oscillator 62
Nonlinear Oscillators Solvable in Elementary Functions 65
Hardening Case 67
Localized Damping 70
Softening Case 71
Nonsmoothness Hiden in Smooth Processes 72
Nonlinear Beats Model 73
Nonlinear Beat Dynamics: The Standard Averaging Approach 75
Asymptotic of Equipartition 80
Asymptotic of Dominants 82
Necessary Condition of Energy Trapping 84
Sufficient Condition of Energy Trapping 85
Transition from Normal to Local Modes 85
System Description 85
Normal and Local Mode Coordinates 87
Local Mode Interaction Dynamics 92
Auto-localized Modes in Nonlinear Coupled Oscillators 96
Nonsmooth Temporal Transformations (NSTT) 103
Non-smooth Time Transformations 103
Positive Time 104
‘Single-Tooth’ Substitution 106
‘Broken Time’ Substitution 106
Sawtooth Sine Transformation 107
Links between NSTT and Matrix Algebras 111
Differentiation and Integration Rules 112
NSTT Averaging 113
Generalizations on Asymmetrical Sawtooth Wave 115
Multiple Frequency Case 117
Idempotent Basis Generated by the Triangular Sine-Wave 119
Definitions and Algebraic Rules 119
Time Derivatives in the Idempotent Basis 121
Idempotent Basis Generated by Asymmetric Triangular Wave 122
Definition and Algebraic Properties 122
Differentiation Rules 124
Oscillators in the Idempotent Basis 125
Integration in the Idempotent Basis 126
Discussions, Remarks and Justifications 127
Remarks on Nonsmooth Solutions in the Classical Dynamics 128
Caratheodory Equation 129
Other Versions of Periodic Time Substitutions 132
General Case of Non-invertible Time and Its Physical Meaning 135
NSTT and Cnoidal Waves 135
Sawtooth Power Series 140
Manipulations with the Series 140
Smoothing Procedures 140
Sawtooth Series for Normal Modes 144
Periodic Version of Lie Series 144
Lie Series of Transformed Systems 147
Second-Order Non-autonomous Systems 147
NSTT of Lagrangian and Hamiltonian Equations 150
Remark on Multiple Argument Cases 153
NSTT for Linear and Piecewise-Linear Systems 154
Free Harmonic Oscillator: Temporal Quantization of Solutions 154
Non-autonomous Case 156
Standard Basis 156
Idempotent Basis 157
Systems under Periodic Pulsed Excitation 158
Regular Periodic Impulses 158
Harmonic Oscillator under the Periodic Impulsive Loading 160
Periodic Impulses with a Temporal ‘Dipole’ Shift 164
Parametric Excitation 166
Piecewise-Constant Excitation 166
Parametric Impulsive Excitation 168
General Case of Periodic Parametric Excitation 170
Input-Output Systems 172
Piecewise-Linear Oscillators with Asymmetric Characteristics 174
Amplitude-Phase Equations 175
Amplitude Solution 176
Phase Solution 177
Remarks on Generalized Taylor Expansions 181
Multiple Degrees-of-Freedom Case 182
The Amplitude-Phase Problem in the Idempotent Basis 186
Periodic and Transient Nonlinear Dynamics under Discontinuous Loading 188
Nonsmooth Two Variables Method 188
Resonances in the Duffing’s Oscillator under Impulsive Loading 191
Strongly Nonlinear Oscillator under Periodic Pulses 194
Impact Oscillators under Impulsive Loading 198
Strongly Nonlinear Vibrations 203
Periodic Solutions for First Order Dynamical Systems 203
Second Order Dynamical Systems 204
Periodic Solutions of Conservative Systems 206
The Vibroimpact Approximation 206
One Degree-of-Freedom General Conservative Oscillator 210
A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes 213
Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero 215
A Generalized Case of Odd Characteristics 217
Periodic Motions Close to Separatrix Loop 219
Self-excited Oscillator 222
Strongly Nonlinear Oscillator with Viscous Damping 226
Remark on NSTT Combined with Two Variables Expansion 230
Oscillator with Two Nonsmooth Limits 233
Bouncing Ball 238
The Kicked Rotor Model 242
Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics 243
Strongly Nonlinear Waves 248
Wave Processes in One-Dimensional Systems 248
Klein-Gordon Equation 249
Impact Modes and Parameter Variations 252
An Introductory Example 252
Parameter Variation and Averaging 256
A Two-Degrees-of-Freedom Model 259
Averaging in the 2DOF System 260
Impact Modes in Multiple Degrees of Freedom Systems 263
A Double-Pendulum with Amplitude Limiters 265
A Mass-Spring Chain under Constraint Conditions 267
Systems with Multiple Impacting Particles 269
Principal Trajectories of Forced Vibrations 272
Introductory Remarks 272
Principal Directions of Linear Forced Systems 274
Definition for Principal Trajectories of Nonlinear Discrete Systems 275
Asymptotic Expansions for Principal Trajectories 276
Definition for Principal Modes of Continuous Systems 278
NSTT and Shooting Method for Periodic Motions 281
Introductory Remarks 281
Problem Formulation 283
Sample Problems and Discussion 285
Smooth Loading 285
Step-Wise Discontinuous Input 292
Impulsive Loading 292
Other Applications 296
Periodic Solutions of the Period - n 296
Two-Degrees-of-Freedom Systems 299
The Autonomous Case 300
Essentially Non-periodic Processes 301
Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles 301
Impulsively Loaded Dynamical Systems 304
Harmonic Oscillator under Sequential Impulses 307
Random Suppression of Chaos 309
Spatially-Oscillating Structures 310
Periodic Nonsmooth Structures 310
Averaging for One-Dimensional Periodic Structures 317
Two Variable Expansions 318
Second Order Equations 320
Acoustic Waves from Non-smooth Periodic Boundary Sources 324
Spatio-temporal Periodicity 328
Membrane on a Two-Dimensional Periodic Foundation 331
The Idempotent Basis for Two-Dimensional Structures 337
References 343
APPENDIX 1 354
APPENDIX 2 356
APPENDIX 3 359
APPENDIX 4 363

"Chapter 1 Introduction (p. 1-2)

Abstract. This chapter contains physical and mathematical preliminaries with di?erent introductory remarks. Although some of the statements are informal and rather intuitive, they nevertheless provide hints on selecting the generating models and corresponding analytical techniques. The idea is that simplicity of a mathematical formalism is caused by hidden links between the corresponding generating models and subgroups of rigid-body motions.

Such motions may be quali?ed indeed as elementary macro-dynamic phenomena developed in the physical space. For instance, since rigid-body rotations are associated with sine waves and therefore (smooth) harmonic analyses then translations and mirror-wise re?ections must reveal adequate tools for strongly unharmonic and nonsmooth approaches. This viewpoint is illustrated by physical examples, problem formulations and solutions.

1.1 Brief Literature Overview

Analytical methods of conventional nonlinear dynamics are based on the classical theory of di?erential equations dealing with smooth coordinate transformations, asymptotic integrations and averaging. The corresponding solutions often include quasi harmonic expansions as a generic feature that explicitly points to the physical basis of these methods namely - the harmonic oscillator. Generally speaking, some of the techniques are also applicable to dynamical systems close to integrable but not necessarily linear.

However, nonlinear generating solutions are seldom available in closed form [9]. As a result, strongly nonlinear methods usually target speci?c situations and are rather di?cult to use in other cases. Generating models for strongly nonlinear analytical tools with a wide range of applicability must obviously 1) capture the most common features of oscillating processes regardless their nonlinear speci?cs, 2) possess simple enough solutions in order to provide e?ciency of perturbation schemes, and 3) describe essentially nonlinear phenomena out of the scope of the weakly nonlinear methods.

So the key notion of the present work suggests possible recipes for selecting such models among so-called non-smooth systems while keeping the class of smooth motions still within the range of applicability. Note that di?erent non-smooth cases have been also considered for several decades by practical and theoretical reasons. On the physical point of view, this kind of modeling essentially employs the idea of perfect spatio-temporal localization of strong nonlinearities or impulsive loadings. For instance, sudden jumps of restoring force characteristics are represented by absolutely sti? constraints under the assumption that the dynamics in between the constraints is smooth and simple enough to describe. As a result, the system dynamics is discretized in terms of mappings and matchings di?erent pieces of solutions."