Pilipchuk Nonlinear Dynamics

Between Linear and Impact Limits
2010
ISBN: 978-3-642-12799-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 52, 360 Seiten

Reihe: Lecture Notes in Applied and Computational Mechanics

Nonlinear Dynamics represents a wide interdisciplinary area of research dealing with a variety of 'unusual' physical phenomena by means of nonlinear differential equations, discrete mappings, and related mathematical algorithms. However, with no real substitute for the linear superposition principle, the methods of Nonlinear Dynamics appeared to be very diverse, individual and technically complicated. This book makes an attempt to find a common ground for nonlinear dynamic analyses based on the existence of strongly nonlinear but quite simple counterparts to the linear models and tools. It is shown that, since the subgroup of rotations, harmonic oscillators, and the conventional complex analysis generate linear and weakly nonlinear approaches, then translations and reflections, impact oscillators, and hyperbolic (Clifford's) algebras must give rise to some 'quasi impact' methodology. Such strongly nonlinear methods are developed in several chapters of this book based on the idea of non-smooth time substitutions. Although most of the illustrations are based on mechanical oscillators, the area of applications may include also electric, electro-mechanical, electrochemical and other physical models generating strongly anharmonic temporal signals or spatial distributions. Possible applications to periodic elastic structures with non-smooth or discontinuous characteristics are outlined in the final chapter of the book.

Autoren/Hrsg.

Pilipchuk, Valery N.

Inhaltsverzeichnis

1;Title Page;2
2;Preface;6
3;Contents;8
4;Introduction;13
4.1;Brief Literature Overview;13
4.2;Asymptotic Meaning of the Approach;16
4.2.1;Two Simple Limits of Lyapunov Oscillator;16
4.2.2;Oscillating Time and Hyperbolic Numbers, Standard and Idempotent Basis;18
4.3;Quick ‘Tutorial’;21
4.3.1;Remarks on the Basic Functions;21
4.3.2;Viscous Dynamics under the Sawtooth Forcing;21
4.3.3;The Rectangular Cosine Input;23
4.3.4;Oscillatory Pipe Flow Model;24
4.3.5;Periodic Impulsive Loading;27
4.3.6;Strongly Nonlinear Oscillator;27
4.4;Geometrical Views on Nonlinearity;29
4.4.1;Geometrical Example;29
4.4.2;Nonlinear Equations and Nonlinear Phenomena;31
4.4.3;Rigid-Body Motions and Linear Systems;33
4.4.4;Remarks on the Multi-dimensional Case;35
4.4.5;Elementary Nonlinearities;36
4.4.6;Example of Simplification in Nonsmooth Limit;37
4.4.7;Non-smooth Time Arguments;38
4.4.8;Further Examples and Discussion;40
4.4.9;Differential Equations of Motion and Distributions;42
4.5;Non-smooth Coordinate Transformations;45
4.5.1;Caratheodory Substitution;45
4.5.2;Transformation of Positional Variables;45
4.5.3;Transformation of State Variables;47
5;Smooth Oscillating Processes;49
5.1;Linear and Weakly Non-linear Approaches;49
5.2;A Brief Overview of Smooth Methods;50
5.2.1;Periodic Motions of Quasi Linear Systems;50
5.2.2;The Idea of Averaging;51
5.2.3;Averaging Algorithm for Essentially Nonlinear Systems;53
5.2.4;Averaging in Complex Variables;55
5.2.5;Lie Group Approaches;56
6;Nonsmooth Processes as Asymptotic Limits;62
6.1;Lyapunov’ Oscillator;62
6.2;Nonlinear Oscillators Solvable in Elementary Functions;65
6.2.1;Hardening Case;67
6.2.2;Localized Damping;70
6.2.3;Softening Case;71
6.3;Nonsmoothness Hiden in Smooth Processes;72
6.3.1;Nonlinear Beats Model;73
6.4;Nonlinear Beat Dynamics: The Standard Averaging Approach;75
6.4.1;Asymptotic of Equipartition;80
6.4.2;Asymptotic of Dominants;82
6.4.3;Necessary Condition of Energy Trapping;84
6.4.4;Sufficient Condition of Energy Trapping;85
6.5;Transition from Normal to Local Modes;85
6.6;System Description;85
6.7;Normal and Local Mode Coordinates;87
6.8;Local Mode Interaction Dynamics;92
6.9;Auto-localized Modes in Nonlinear Coupled Oscillators;96
7;Nonsmooth Temporal Transformations (NSTT);103
7.1;Non-smooth Time Transformations;103
7.1.1;Positive Time;104
7.1.2;‘Single-Tooth’ Substitution;106
7.1.3;‘Broken Time’ Substitution;106
7.1.4;Sawtooth Sine Transformation;107
7.1.5;Links between NSTT and Matrix Algebras;111
7.1.6;Differentiation and Integration Rules;112
7.1.7;NSTT Averaging;113
7.1.8;Generalizations on Asymmetrical Sawtooth Wave;115
7.1.9;Multiple Frequency Case;117
7.2;Idempotent Basis Generated by the Triangular Sine-Wave;119
7.2.1;Definitions and Algebraic Rules;119
7.2.2;Time Derivatives in the Idempotent Basis;121
7.3;Idempotent Basis Generated by Asymmetric Triangular Wave;122
7.3.1;Definition and Algebraic Properties;122
7.3.2;Differentiation Rules;124
7.3.3;Oscillators in the Idempotent Basis;125
7.3.4;Integration in the Idempotent Basis;126
7.4;Discussions, Remarks and Justifications;127
7.4.1;Remarks on Nonsmooth Solutions in the Classical Dynamics;128
7.4.2;Caratheodory Equation;129
7.4.3;Other Versions of Periodic Time Substitutions;132
7.4.4;General Case of Non-invertible Time and Its Physical Meaning;135
7.4.5;NSTT and Cnoidal Waves;135
8;Sawtooth Power Series;140
8.1;Manipulations with the Series;140
8.1.1;Smoothing Procedures;140
8.2;Sawtooth Series for Normal Modes;144
8.2.1;Periodic Version of Lie Series;144
8.3;Lie Series of Transformed Systems;147
8.3.1;Second-Order Non-autonomous Systems;147
8.3.2;NSTT of Lagrangian and Hamiltonian Equations;150
8.3.3;Remark on Multiple Argument Cases;153
9;NSTT for Linear and Piecewise-Linear Systems;154
9.1;Free Harmonic Oscillator: Temporal Quantization of Solutions;154
9.2;Non-autonomous Case;156
9.2.1;Standard Basis;156
9.2.2;Idempotent Basis;157
9.3;Systems under Periodic Pulsed Excitation;158
9.3.1;Regular Periodic Impulses;158
9.3.2;Harmonic Oscillator under the Periodic Impulsive Loading;160
9.3.3;Periodic Impulses with a Temporal ‘Dipole’ Shift;164
9.4;Parametric Excitation;166
9.4.1;Piecewise-Constant Excitation;166
9.4.2;Parametric Impulsive Excitation;168
9.4.3;General Case of Periodic Parametric Excitation;170
9.5;Input-Output Systems;172
9.6;Piecewise-Linear Oscillators with Asymmetric Characteristics;174
9.6.1;Amplitude-Phase Equations;175
9.6.2;Amplitude Solution;176
9.6.3;Phase Solution;177
9.6.4;Remarks on Generalized Taylor Expansions;181
9.7;Multiple Degrees-of-Freedom Case;182
9.8;The Amplitude-Phase Problem in the Idempotent Basis;186
10;Periodic and Transient Nonlinear Dynamics under Discontinuous Loading;188
10.1;Nonsmooth Two Variables Method;188
10.2;Resonances in the Duffing’s Oscillator under Impulsive Loading;191
10.3;Strongly Nonlinear Oscillator under Periodic Pulses;194
10.4;Impact Oscillators under Impulsive Loading;198
11;Strongly Nonlinear Vibrations;203
11.1;Periodic Solutions for First Order Dynamical Systems;203
11.2;Second Order Dynamical Systems;204
11.3;Periodic Solutions of Conservative Systems;206
11.3.1;The Vibroimpact Approximation;206
11.3.2;One Degree-of-Freedom General Conservative Oscillator;210
11.3.3;A Nonlinear Mass-Spring Model That Becomes Linear at High Amplitudes;213
11.3.4;Strongly Non-linear Characteristic with a Step-Wise Discontinuity at Zero;215
11.3.5;A Generalized Case of Odd Characteristics;217
11.4;Periodic Motions Close to Separatrix Loop;219
11.5;Self-excited Oscillator;222
11.6;Strongly Nonlinear Oscillator with Viscous Damping;226
11.6.1;Remark on NSTT Combined with Two Variables Expansion;230
11.6.2;Oscillator with Two Nonsmooth Limits;233
11.7;Bouncing Ball;238
11.8;The Kicked Rotor Model;242
11.9;Oscillators with Piece-Wise Nonlinear Restoring Force Characteristics;243
12;Strongly Nonlinear Waves;248
12.1;Wave Processes in One-Dimensional Systems;248
12.2;Klein-Gordon Equation;249
13;Impact Modes and Parameter Variations;252
13.1;An Introductory Example;252
13.2;Parameter Variation and Averaging;256
13.3;A Two-Degrees-of-Freedom Model;259
13.4;Averaging in the 2DOF System;260
13.5;Impact Modes in Multiple Degrees of Freedom Systems;263
13.5.1;A Double-Pendulum with Amplitude Limiters;265
13.5.2;A Mass-Spring Chain under Constraint Conditions;267
13.6;Systems with Multiple Impacting Particles;269
14;Principal Trajectories of Forced Vibrations;272
14.1;Introductory Remarks;272
14.2;Principal Directions of Linear Forced Systems;274
14.3;Definition for Principal Trajectories of Nonlinear Discrete Systems;275
14.4;Asymptotic Expansions for Principal Trajectories;276
14.5;Definition for Principal Modes of Continuous Systems;278
15;NSTT and Shooting Method for Periodic Motions;281
15.1;Introductory Remarks;281
15.2;Problem Formulation;283
15.3;Sample Problems and Discussion;285
15.3.1;Smooth Loading;285
15.3.2;Step-Wise Discontinuous Input;292
15.3.3;Impulsive Loading;292
15.4;Other Applications;296
15.4.1;Periodic Solutions of the Period - n;296
15.4.2;Two-Degrees-of-Freedom Systems;299
15.4.3;The Autonomous Case;300
16;Essentially Non-periodic Processes;301
16.1;Nonsmooth Time Decomposition and Pulse Propagation in a Chain of Particles;301
16.2;Impulsively Loaded Dynamical Systems;304
16.2.1;Harmonic Oscillator under Sequential Impulses;307
16.2.2;Random Suppression of Chaos;309
17;Spatially-Oscillating Structures;310
17.1;Periodic Nonsmooth Structures;310
17.2;Averaging for One-Dimensional Periodic Structures;317
17.3;Two Variable Expansions;318
17.4;Second Order Equations;320
17.5;Acoustic Waves from Non-smooth Periodic Boundary Sources;324
17.6;Spatio-temporal Periodicity;328
17.7;Membrane on a Two-Dimensional Periodic Foundation;331
17.8;The Idempotent Basis for Two-Dimensional Structures;337
18;References;343
19;APPENDIX 1;354
20;APPENDIX 2;356
21;APPENDIX 3;359
22;APPENDIX 4;363

Leseproben

"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."