Analytical Routes to Chaos in Nonlinear Engineering

Nonlinear 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. This works well up to some accuracy and some range for the input values, but some interesting phenomena such as chaos and singularities are hidden by linearization and perturbation analysis. It follows that some aspects of the behavior of a nonlinear system appear commonly to be chaotic, unpredictable or counterintuitive. Although such a chaotic behavior may resemble a random behavior, it is absolutely deterministic. Analytical Routes to Chaos in Nonlinear Engineering discusses analytical solutions of periodic motions to chaos or quasi–periodic motions in nonlinear dynamical systems in engineering and considers engineering applications, design, and control. It systematically discusses complex nonlinear phenomena in engineering nonlinear systems, including the periodically forced Duffing oscillator, nonlinear self–excited systems, nonlinear parametric systems and nonlinear rotor systems. Nonlinear models used in engineering are also presented and a brief history of the topic is provided. Key features: Considers engineering applications, design and control Presents analytical techniques to show how to find the periodic motions to chaos in nonlinear dynamical systems Systematically discusses complex nonlinear phenomena in engineering nonlinear systems Presents extensively used nonlinear models in engineering Analytical Routes to Chaos in Nonlinear Engineering is a practical reference for researchers and practitioners across engineering, mathematics and physics disciplines, and is also a useful source of information for graduate and senior undergraduate students in these areas.

Preface Chapter 1  Introduction 1 1.1 Analytical methods 1 1.1.1 Lagrange standard forms 1 1.1.2 Perturbation methods 3 1.1.3 Method of averaging 6 1.1.4 Gernalized harmonic balance 8 1.2 boook layout 25 Chapter 2  Bifurcation trees in Duffing Oscillators  27 2.1 Analyitcal period–m motions 27 2.2 Hardening Duffing oscillators 35 2.2.1 Period–1 motion to chaos 35 2.2.1.1 Period–1 motions 35 2.2.1.2 Period–1 to period–4 motions 40 2.2.1.3 Numerical simulations 59 2.2.2 Period–3 motion to chaos 69 2.2.2.1 Independent, symeertic period–3 motions 69 2.2.2.2 Asymmetric period–3 motions 72 2.2.2.3 Period–3 to period–6 motions 80 2.2.2.4 Illustrations 89 2.3 Softening Duffing oscilattors 95 2.3.1 Symmetric period–1 motions 95 2.3.2 Asymmtric period–1 motions to chaos 99 2.3.3 Illstrations for periodic motions 112 2.4 Twin–well Duffing oscilattors 126 2.4.1 Rough prediction of period–1 motions 126 2.4.2 Period–1 motions to chaos 131 2.4.3 Numerical illstrations 151 Chapter 3  Self–excited Nonlinear Oscillators 155 3.1 van der Pol oscillators 155 3.1.1 Analytical solutions 155 3.1.2 Frequecy–amplitude characteristics 167 3.1.3 Numerical illustratyions 179 3.2 van der Pol–Duffing oscillator 188 3.2.1 Finite Fourier series solutions 188 3.2.2 Analytical predictions 201 3.2.3 Numerical illustratyions 206 Chapter 4  Parametric Nonlinear Oscillators 223 4.1 Parametric, quadratic nonlinear oscillators 223 4.1.1 Analytical solutions 223 4.1.2 Analytical routes to chaos 228 4.1.3 Numerical simulations 251 4.2 Parametric Duffing oscillators 261 4.2.1 Formulations 261 4.2.2 Parametric hardening Duffing oscillators 269 Chapter 5 Nonlinear Jeffcott Rotor Systems 285 5.1 Analytical periodic motions 285 5.2 Frequency–amplitude characteristics 301 5.2.1 Period–1 motions 302 5.2.2 Analytical bifurcation trees 308 5.2.3 Independent, symeertric period–5 motions 318 5.3 Numerical simulations 323 References 341 Subject index 347

Professor Luo is currently a Distinguished Research Professor at Southern Illinois University Edwardsville. He is an international renowned figure in the area of nonlinear dynamics and mechanics. For about 30 years, Dr. Luo’s contributions on nonlinear dynamical systems and mechanics lie in (i) the local singularity theory for discontinuous dynamical systems, (ii) Dynamical systems synchronization, (iii) Analytical solutions of periodic and chaotic motions in nonlinear dynamical systems, (iv) The theory for stochastic and resonant layer in nonlinear Hamiltonian systems, (v) The full nonlinear theory for a deformable body. Such contributions have been scattered into 13 monographs and over 200 peer–reviewed journal and conference papers. His new research results are changing the traditional thinking in nonlinear physics and mathematics. Dr. Luo has served as an editor for the Journal “Communications in Nonlinear Science and Numerical simulation”, book series on Nonlinear Physical Science (HEP) and Nonlinear Systems and Complexity (Springer). Dr. Luo is the editorial member for two journals (i.e., IMeCh E Part K Journal of Multibody Dynamics and Journal of Vibration and Control). He also organized over 30 international symposiums and conferences on Dynamics and Control.