Nonlinear dynamics of piecewise constant systems and implementation of piecewise constant arguments

Liming Dai9789812818508, 9812818502

Piecewise constant systems exist in widely expanded areas such as engineering, physics, and mathematics. Extraordinary and complex characteristics of piecewise constant systems have been reported in recent years. This book provides the methodologies for analyzing and assessing nonlinear piecewise constant systems on a theoretically and practically sound basis. Recently developed approaches for theoretically analyzing and numerically solving the nonlinear piecewise constant dynamic systems are reviewed. A new greatest integer argument with a piecewise constant function is utilized for nonlinear dynamic analyses and for establishing a novel criterion in diagnosing irregular and chaotic solutions from the regular solutions of a nonlinear dynamic system. The newly established piecewise constantization methodology and its implementation in analytically solving for nonlinear dynamic problems are also presented.
Contents: Fundamentals of Conventional and Piecewise Constant Systems; Preliminary Theorems and Techniques for Analysis of Nonlinear Piecewise Constant Systems; Piecewise Constant Dynamical Systems and Their Behavior; Analytical and Semi-Analytical Solution Development with Piecewise Constant Arguments; Numerical and Improved Semi-Analytical Approaches Implementing Piecewise Constant Arguments; Application of P-T Method on Multi-Degree-of-Freedom Nonlinear Dynamic Systems; Periodicity-Ratio and Its Application in Diagnosing Irregularities of Nonlinear Systems.

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Table of contents :
CONTENTS……Page 14
Preface……Page 8
1.1. Preliminary Remarks……Page 18
1.2. Remarks on the Development and Analyses of Piecewise Constant Systems in History……Page 19
Physical Model……Page 23
Mathematical Modeling……Page 25
Derivation of Governing Equations……Page 26
Solution Development……Page 28
Result Analysis and Interpretation……Page 30
1.4. Fundamentals of Dynamic System Modeling in Science and Engineering……Page 31
Discharging Capacitor……Page 32
Driven Froude Pendulum……Page 33
Workpiece-Cutter System……Page 35
1.5. Piecewise Constant Systems and Their Modeling……Page 39
1.5.1. Greatest Integer Functions……Page 40
Vertically Transmitted Diseases Among American Dog Ticks……Page 42
Geneva Wheel……Page 45
Flexible Support of Electrodynamic Shaker……Page 48
1.6. Implementing Piecewise Constant Arguments in Dynamic Problem Solving……Page 51
References……Page 55
2.2. Nonlinear Behaviors and Fundamental Analytical and Geometric Tools of Nonlinear Dynamics……Page 58
2.2.1. Periodic Responses of Linear and Nonlinear Dynamic Systems……Page 59
2.2.2. Poincare Map……Page 62
2.2.3. Quasiperiodic Response of Nonlinear Systems……Page 65
2.2.5. Bifurcation of Nonlinear Systems……Page 67
2.3. Lyapunov Exponent……Page 69
2.4. Characteristics of Numerical Solutions and Runge-Kutta Method……Page 73
References……Page 76
3.1. Introduction……Page 78
3.2. Governing Equations of Dynamic Systems with Piecewise Constant Variables……Page 81
3.3. Solution Development of Simple Dynamic Systems Subjected to Piecewise Constant Excitations……Page 83
3.4. Development of Analytical Solutions via Piecewise Constant Variables……Page 86
3.5. General Vibration Systems under Piecewise Constant Excitations……Page 89
3.6. Derivation and Characteristics of Approximate and Numerical Solutions of Dynamic Systems with Piecewise Constant Variables……Page 96
3.7. Extraordinary and Nonlinear Behavior of Linear Piecewise Constant Systems……Page 103
3.8. Oscillatory Properties of Dynamic Systems with Piecewise Constant Variables……Page 108
3.9. Approximate and Numerical Technique of Small Interval with Piecewise Constant Variable……Page 117
3.10. Characteristics of Approximate Results with Piecewise Constant Variable in Small Intervals……Page 123
References……Page 130
4.1. Introduction……Page 134
4.2. A New Piecewise Constant Argument [Nt]/N……Page 135
4.3. Solving for Dynamic Systems with Implementation of Piecewise Constant Arguments……Page 138
4.4. Analytical Solutions of Free Vibration Systems via Piecewise Constantization……Page 144
4.5. Analytical Solutions to Undamped Systems with Piecewise Constant Excitations……Page 150
4.6. Development of General Analytical Solutions for Linear Vibration Systems……Page 153
4.7. Semi-Analytical and Approximate Solutions for Nonlinear Piecewise Constant Dynamic Systems……Page 158
References……Page 162
5.1. Introduction……Page 164
5.2. Numerical Solutions for Linear Dynamic Systems via Piecewise Constant Procedure……Page 166
5.3. Numerical Solutions of Nonlinear Systems……Page 174
5.4. Chaotic Behavior of Numerical Solutions for Nonlinear Systems……Page 179
5.5. Development of P-T Method……Page 185
5.6. Analytical and Numerical Approaches and the Approaches Implementing P-T Method……Page 193
5.7. Numerical Solution Comparison between P-T and Runge-Kutta Methods……Page 199
5.8. Consistency Analysis of Numerical Solutions with Implementation of Piecewise Constant Arguments……Page 211
5.9. Step Size Control……Page 214
5.10. Characteristics of the P-T Method……Page 216
References……Page 218
6.1. Introduction……Page 220
6.2.1. Governing Equations and Solution Development of Linear MDOF Systems……Page 224
6.2.2. Solving for Nonlinear MDOF Systems……Page 227
6.3.1. Solving Nonlinear Systems Directly Implementing P-T Method……Page 228
6.3.2. Nonlinear Systems with Linear Coupling and Proportional Damping……Page 231
6.3.3. Nonlinear Systems with Linear Coupling and General Damping……Page 233
6.4. Numerical Solutions via Piecewise Constantization……Page 235
References……Page 248
7.1. Introduction……Page 250
7.2. Phase Trajectories of Periodic, Nonperiodic and Chaotic Behavior of Nonlinear Systems……Page 253
7.3. Poincare Maps and Their Relation with Piecewise Constant Dynamic Systems……Page 255
7.4. Bifurcation of Piecewise Constant Dynamic Systems……Page 259
7.5. Derivation of Periodicity-Ratio……Page 260
7.6. Distinction of Quasiperiodic Motion from Chaos……Page 275
7.7. Comparison of Periodicity-Ratio and Lyapunov-Exponent……Page 277
7.8. Characteristics of Periodicity-Ratio……Page 297
7.9. Implementation of Periodicity-Ratio in Analyzing Nonlinear Dynamic Problems……Page 299
References……Page 310
1. L’Hopital’s Rule for Indeterminate……Page 314
2. Proof of the Existence of for Following Relationship……Page 315
3. Derivation of the Limits Shown in Equations (3.22), (3.23) and (3.24) as w Approaches Zero……Page 316
4. Matrix Manipulations……Page 318
2. Transpose of Matrices……Page 320
4. Determinants……Page 321
6. Eigenvalues and Eigenvectors……Page 322
8. Diagonalization of Matrices……Page 323
Appendix C Computer Programs for Analyses of Dynamics……Page 326
Index……Page 342