Description
Book SynopsisNonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations.
Table of ContentsPreface ix
1 Introduction 1
1.1 Analytical Methods 1
1.1.1 Lagrange Standard Form 1
1.1.2 Perturbation Methods 2
1.1.3 Method of Averaging 5
1.1.4 Generalized Harmonic Balance 8
1.2 Book Layout 24
2 Bifurcation Trees in Duffing Oscillators 25
2.1 Analytical Solutions 25
2.2 Period-1 Motions to Chaos 32
2.2.1 Period-1 Motions 33
2.2.2 Period-1 to Period-4 Motions 35
2.2.3 Numerical Simulations 52
2.3 Period-3 Motions to Chaos 57
2.3.1 Independent, Symmetric Period-3 Motions 57
2.3.2 Asymmetric Period-3 Motions 64
2.3.3 Period-3 to Period-6 Motions 71
2.3.4 Numerical Illustrations 82
3 Self-Excited Nonlinear Oscillators 87
3.1 van del Pol Oscillators 87
3.1.1 Analytical Solutions 87
3.1.2 Frequency-Amplitude Characteristics 97
3.1.3 Numerical Illustrations 110
3.2 van del Pol-Duffing Oscillators 114
3.2.1 Finite Fourier Series Solutions 114
3.2.2 Analytical Predictions 130
3.2.3 Numerical Illustrations 143
4 Parametric Nonlinear Oscillators 151
4.1 Parametric, Quadratic Nonlinear Oscillators 151
4.1.1 Analytical Solutions 151
4.1.2 Analytical Routes to Chaos 156
4.1.3 Numerical Simulations 169
4.2 Parametric Duffing Oscillators 186
4.2.1 Formulations 186
4.2.2 Parametric Hardening Duffing Oscillators 194
5 Nonlinear Jeffcott Rotor Systems 209
5.1 Analytical Periodic Motions 209
5.2 Frequency-Amplitude Characteristics 225
5.2.1 Period-1 Motions 226
5.2.2 Analytical Bifurcation Trees 231
5.2.3 Independent Period-5 Motion 239
5.3 Numerical Simulations 246
References 261
Index 265
