The Duffing Equation Nonlinear Oscillators

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.

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.

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
Table of Contents provided by Publisher. All Rights Reserved.

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The Duffing Equation Nonlinear Oscillators and their Behaviour

Summary

Author Biography

Table of Contents

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