Analytical Routes to Chaos in Nonlinear Engineering E-Book

Albert C. J. Luo

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 163

Veröffentlichungsjahr: 2014

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Analytical Methods

1.2 Book Layout

Chapter 2: Bifurcation Trees in Duffing Oscillators

2.1 Analytical Solutions

2.2 Period-1 Motions to Chaos

2.3 Period-3 Motions to Chaos

Chapter 3: Self-Excited Nonlinear Oscillators

3.1 van del Pol Oscillators

3.2 van del Pol-Duffing Oscillators

Chapter 4: Parametric Nonlinear Oscillators

4.1 Parametric, Quadratic Nonlinear Oscillators

4.2 Parametric Duffing Oscillators

Chapter 5: Nonlinear Jeffcott Rotor Systems

5.1 Analytical Periodic Motions

5.2 Frequency-Amplitude Characteristics

5.3 Numerical Simulations

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Chapter 1: Introduction

List of Illustrations

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.43

Figure 2.44

Figure 2.45

Figure 2.46

Figure 2.17

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.12

Figure 3.10

Figure 3.11

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

Figure 3.18

Figure 4.1

Figure 4.3

Figure 4.2

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.24

Figure 4.26

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Figure 4.18

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

List of Tables

Table 2.1

Table 3.1

Table 4.1

Table 5.1

Analytical Routes to Chaos in Nonlinear Engineering

Albert C. J. Luo

Southern Illinois University, USA

This edition first published 2014

© 2014 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data applied for.

Luo, Albert C. J.

Analytical routes to chaos in nonlinear engineering / Albert C.J. Luo.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-88394-5 (cloth)

1. Systems engineering. 2. Chaotic behavior in systems. 3. Nonlinear systems. 4. Nonlinear control theory.

I. Title.

TA168.L86 2014

629.8′36 – dc23

2014001974

A catalogue record for this book is available from the British Library.

1 2014

Preface

Periodic motions in nonlinear dynamical systems extensively exist in engineering and such periodic motions are paramount in engineering application. Since 1788, Lagrange used the method of averaging to investigate the gravitational three-body problem through a two-body problem with a perturbation. In the nineteenth century, Poincare developed the perturbation method to investigate the periodic motion of the three-body problem. In 1920, van der Pol used the averaging method to determine the periodic motions of self-excited systems in circuits. In 1945, Cartwright and Littlewood discussed the periodic motions of the van der Pol equation and proved the existence of periodic motions. In 1948, Levinson used a piecewise linear model to describe the van der Pol equation and determined the existence of periodic motions. In 1949, Levinson further developed the structures of periodic solutions in such a second order differential equation through the piecewise linear model, and discovered that infinite periodic solutions exist in such a piecewise linear model. On the other hand, in 1928, Fatou provided the first proof of asymptotic validity of the method of averaging through the existence of solutions of differential equations. In 1935, Krylov and Bogolyubov developed systematically the method of averaging. Thus, the perturbation method becomes a main analytical tool to investigate periodic motions of nonlinear oscillators in engineering. For example, in 1973 Nayfeh used the multiple-scale method to investigate Duffing oscillators for nonlinear structural dynamics. In the 1980s, since chaotic motions are observed in nonlinear vibrations, one tried to use the perturbation theory to describe chaotic motions. From the idea of the Lagrange standard form, the normal forms of nonlinear dynamical systems at equilibrium cannot be used for periodic motions and chaos in the original nonlinear dynamical systems. In 2012, the author systematically developed an analytical method to determine period-m flows in nonlinear dynamical systems. Thus this book will employ the analytical method to determine the analytical routes of periodic motions to chaos in nonlinear engineering.

This book presents analytical routes to chaos in a few typical engineering nonlinear dynamical systems through the recently developed analytical method. This book consists of five chapters. Chapter 1 gives a literature survey of analytical methods in nonlinear dynamical systems, including, the Lagrange standard form, the method of averaging, the Poincare perturbation method, and the generalized harmonic balance method. These analytical methods will be presented through theorems. In Chapter 2, the analytical bifurcation trees of period-m motion to chaos for the Duffing oscillator will be presented since the Duffing oscillator is extensively applied in structural dynamics. In Chapter 3, the period-m motion in the periodically forced, van der Pol oscillator will be presented analytically, and the analytical bifurcation trees of periodic motions to chaos in the van del Pol-Duffing oscillator will be discussed. In Chapter 4, the analytical solutions of period-m motions in parametric nonlinear oscillators will be presented through both a parametric, quadratic nonlinear oscillator and a parametric Duffing oscillator as two sampled problems. In Chapter 5, the bifurcation tree of periodic motions to chaos in a nonlinear Jeffcott rotor dynamical system will be presented, and the periodic motions to quasi-periodic motion will be discussed. All materials presented in this book will help one better understand nonlinear phenomena in nonlinear engineering.

Finally, I would like to appreciate my students (Jianzhe Huang, Arash Bagaei Laken, Bo Yu, and Dennis O'Connor) for applying the recently developed analytical method to nonlinear engineering systems and completing numerical computations. Herein, I would like to thank my wife (Sherry X. Huang) and my children (Yanyi Luo, Robin Ruo-Bing Luo, and Robert Zong-Yuan Luo) again for tolerance, patience, understanding, and continuous support.

Albert C.J. LuoEdwardsville, Illinois, USA

Chapter 1Introduction

In this chapter, analytical methods for approximate solutions of periodic motions to chaos in nonlinear dynamical systems will be presented briefly. The Lagrange stand form, perturbation method, method of averaging, harmonic balance, generalized harmonic balance will be discussed. A brief literature survey will be completed to present a main development skeleton of analytical methods for periodic motions in nonlinear dynamical systems. The weakness of current approximate, analytical methods will also be discussed in this chapter, and the significance of analytical methods in nonlinear engineering will be presented.

1.1 Analytical Methods

Since the appearance of Newton's mechanics, one has been interested in periodic motion. From the Fourier series theory, any periodic function can be expressed by a Fourier series expansion with different harmonics. The periodic motion in dynamical systems is a closed curve in state space in the prescribed period. In addition to simple oscillations in mechanical systems, one has been interested in motions of moon, earth, and sun in the three-body problem. The earliest approximation method is the method of averaging, and the idea of averaging originates from Lagrange (1788).

1.1.1 Lagrange Standard Form

Consider an initial value problem for and

1.1

where is an matrix and continuous with time is a —continuous vector function of and The unperturbed system is linear and such a linear system has independent basic solution to form a fundamental matrix . That is,

1.2

where is constant, determined by initial conditions. As in Luo (2012aa,b), a linear transformation is introduced as

1.3

Substitution of Equation (1.3) into Equation (1.1) gives

1.4

With , we obtain

1.5

The foregoing form is called the Lagrange standard form.

Consider a vibration problem as

1.6

From the basic solution of the unperturbed system, we have a transformation as

1.7

Using this transformation, Equation (1.6) becomes

1.8

where

1.9

If the function and is T-periodic with

1.10

where

1.11

1.1.2 Perturbation Methods

In the end of the nineteenth century, Poincare (1890) provided the qualitative analysis of dynamical systems to determine periodic solutions and stability, and developed the perturbation theory for periodic solution. In addition, Poincare (1899) discovered that the motion of a nonlinear coupled oscillator is sensitive to the initial condition, and qualitatively stated that the motion in the vicinity of unstable fixed points of nonlinear oscillation systems may be stochastic under regular applied forces. In the twentieth century, one followed Poincare's ideas to develop and apply the qualitative theory to investigate the complexity of motions in dynamical systems. With Poincare's influence, Birkhoff (1913) continued Poincare's work, and a proof of Poincare's geometric theorem was given. Birkhoff (1927) showed that both stable and unstable fixed points of nonlinear oscillation systems with two degrees of freedom must exist whenever their frequency ratio (also called resonance) is rational. The sub-resonances in periodic motions of such systems change the topological structures of phase trajectories, and the island chains are obtained when the dynamical systems are renormalized with fine scales. In such qualitative and quantitative analysis, the Taylor series expansion and the perturbation analysis play an important role. However, the Taylor series expansion analysis is valid in the small finite domain under certain convergent conditions, and the perturbation analysis based on the small parameters, as an approximate estimate, is only acceptable for a very small domain with a short time period. From Verhulst (1991), the perturbation solution of dynamical system can be stated as follows.

Theorem 1.1 Consider a dynamical system

1.12

with and is a —continuous vector function of and Assume can be expanded in a Taylor series with respect to as

1.13

with and is continuous in and with times continuously differentiable with , and is continuous in and and satisfies Lipschitz—continuous in Suppose there is a -series of as

1.14

Application of Equation (1.14) to Equation (1.12), using the Taylor series expansion of with respect to power of and equating coefficients with the initial condition

1.15

generates an approximate solution of with

1.16

on the time-scale 1.

Proof

The proof can be referred to Verhulst (1991).

Assume that in Equation (1.12) can be expanded in a convergent Taylor series with respect to and in a finite domain. Consider an unperturbed system in Equation (1.12) as

1.17

Using a transform

1.18

Equation (1.12) becomes

1.19

where and

1.20

Thus, the Poincare perturbation theory for nonlinear dynamical systems can also be stated as follows:

Theorem 1.2 (Poincare) Consider a dynamical system

1.21

with and . is a —continuous vector function of and If such a vector function can be expanded in a convergent power series with respect to and for and then can be expanded in a convergent power series with respect to and in the vicinity of and on time scale 1.