The Duffing Equation Nonlinear Oscillators and their Behaviour

by Kovacic, Ivana; Brennan, Michael J.

• ISBN13: 9780470715499
• ISBN10: 0470715499
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-04-04
• Publisher: Wiley

Summary

The Duffing Equation: Non-Linear Oscillators and their Behaviour collates together the results of a wealth of disseminated research literature on the Duffing equation, summarizing the findings of this research and applying it to real engineering problems. Each chapter is commissioned from an expert contributor in the field of non-linear dynamics and addresses a different form of the equation, relating it to real oscillatory problems and clearly linking the problem with the mathematics that describe it. It includes a chapter devoted to the mathematical techniques required to study non-linear dynamics, enabling the reader with little mathematical background to understand the text. Another chapter is solely devoted to practical examples of systems whose dynamic behavior is described by Duffing's equation.

Author Biography

Michael J Brennan, Dynamics Group, Institute of Sound and Vibration Research (ISVR), University of Southampton, UK
Professor Michael Brennan holds a personal chair in Engineering Dynamics and is Chairman of the Dynamics Research in the ISVR at Southampton University. He joined Southampton in 1995 after a 23 year career as an engineer in the Royal Navy. Since 1995 Professor Brennan has worked on several aspects of sound and vibration, specialising in the use of smart structures for active vibration control, active control of structurally-radiated sound and the condition monitoring of gear boxes by the analysis of vibration data and rotor dynamics. Mike Brennan has edited 3 conference proceedings, 3 book chapters, and over 200 academic journal and conference papers.

Ivana Kovavic, Department of Mathematics, Faculty of Technical Sciences, University of Novi Sad, Serbia Ivana
Kovavic is an associate professor within the Department of Mathematics at the University of Novi Sad in Serbia. She has authored two books in the Polish language, 30 journal and conference papers and edited 1 conference proceedings.

Table of Contents

Biographies of the Contributors

Preface

Background: On Georg Duffing and the Duffing Equation

Introduction

Historical Perspective

A Brief Biography of Georg Duffing

The Work of Georg Duffing

Contents of Duffing's Book

Research Inspired by Duffing's Work

Some other Books on Nonlinear Dynamics

Overview of this Book

References

Examples of Physical Systems Described by the Duffing Equation

Introduction

Nonlinear Stiffness

The Pendulum

Example of Geometrical Nonlinearity

A System Consisting of the Pendulum and Nonlinear Stiffness

Snap-Through Mechanism

Nonlinear Isolator

Large Deflection of a Beam with Nonlinear Stiffness

Beam with Nonlinear Stiffness due to in-plane Tension

Nonlinear Cable Vibrations

Nonlinear Electrical Circuit

Summary

References

Free Vibration of a Duffing Oscillator with Viscous Damping

Introduction

Fixed Points and their Stability

Local Bifurcation Analysis

Global Analysis for Softening Nonlinear Stiffness ( )

Global Analysis for Hardening Nonlinear Stiffness ( )

Summary

Acknowledgments

References

Analysis Techniques for the Various Forms of the Duffing Equation

Introduction

Exact Solution for Free Oscillations of the Duffing Equation with Cubic Nonlinearity

The Elliptic Harmonic Balance Method

The Elliptic Galerkin Method

The Straightforward Expansion Method

The Elliptic Lindstedt-Poincaré Method

Averaging Methods

Elliptic Homotopy Methods

Summary

References

Jacobi Elliptic Functions and Elliptic Integrals

The Best L₂ Norm Approximation

Forced Harmonic Vibration of a Duffing Oscillator with Linear Viscous Damping

Introduction

Free and Forced Responses of the Linear Oscillator

Amplitude and Phase Responses of the Duffing Oscillator

Periodic Solutions, Poincaré Sections, and Bifurcations

Global Dynamics

Summary

References

Forced Harmonic Vibration of a Duffing oscillator with Different Damping Mechanisms

Introduction

Classification of Nonlinear Characteristics

Harmonically Excited Duffing Oscillator with Generalized Damping

Viscous Damping

Nonlinear Damping in a Hardening System

Nonlinear Damping in a Softening System

Nonlinear Damping in Double-Well Potential Oscillator

Summary

Acknowledgments

References

Forced Harmonic Vibration in a Duffing Oscillator with Negative Linear Stiffness and Linear Viscous Damping

Introduction

Literature Survey

Dynamics of Conservative and Non-Conservative Systems

Nonlinear Periodic Oscillations

Transition to Complex Response

Non-Classical Analyses

Summary

References

Forced Harmonic Vibration of an Asymmetric Duffing Oscillator

Introduction

Models of the Systems under Consideration

Regular Response of the Pure Cubic Oscillator

Regular Response of the Single-Well Helmholtz-Duffing Oscillator

Chaotic Response of the Pure Cubic Oscillator

Chaotic Response of the Single-Well Helmholtz-Duffing Oscillator

Summary

References

Appendix: Translation of Sections from Duffing's Original Book

Glossary

Index

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