The Duffing Equation - Nonlinear Oscillators andtheir Behaviour
Buy Rights Online Buy Rights

Rights Contact Login For More Details

More About This Title The Duffing Equation - Nonlinear Oscillators andtheir Behaviour


The Duffing Equation: Nonlinear Oscillators and their Behaviour brings together the results of a wealth of disseminated research literature on the Duffing equation, a key engineering model with a vast number of applications in science and engineering, summarizing the findings of this research. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a different form of the equation, relating it to various oscillatory problems and clearly linking the problem with the mathematics that describe it. The editors and the contributors explain the mathematical techniques required to study nonlinear dynamics, helping the reader with little mathematical background to understand the text.

The Duffing Equation provides a reference text for postgraduate and students and researchers of mechanical engineering and vibration / nonlinear dynamics as well as a useful tool for practising mechanical engineers.

  • Includes a chapter devoted to historical background on Georg Duffing and the equation that was named after him.
  • Includes a chapter solely devoted to practical examples of systems whose dynamic behaviour is described by the Duffing equation.
  • Contains a comprehensive treatment of the various forms of the Duffing equation.
  • Uses experimental, analytical and numerical methods as well as concepts of nonlinear dynamics to treat the physical systems in a unified way.


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.


List of Contributors.


1 Background: On Georg Duffing and the Duffing Equation (Ivana Kovacic and Michael J. Brennan).

1.1 Introduction.

1.2 Historical perspective.

1.3 A brief biography of Georg Duffing.

1.4 The work of Georg Duffing.

1.5 Contents of Duffing's book.

1.6 Research inspired by Duffing’s work.

1.7 Some other books on nonlinear dynamics.

1.8 Overview of this book.


2 Examples of Physical Systems Described by the Duffing Equation (Michael J. Brennan and Ivana Kovacic).

2.1 Introduction.

2.2 Nonlinear stiffness.

2.3 The pendulum.

2.4 Example of geometrical nonlinearity.

2.5 A system consisting of the pendulum and nonlinear stiffness.

2.6 Snap-through mechanism.

2.7 Nonlinear isolator.

2.8 Large deflection of a beam with nonlinear stiffness.

2.9 Beam with nonlinear stiffness due to inplane tension.

2.10 Nonlinear cable vibrations.

2.11 Nonlinear electrical circuit.

2.12 Summary.


3 Free Vibration of a Duffing Oscillator with Viscous Damping (Hiroshi Yabuno).

3.1 Introduction.

3.2 Fixed points and their stability.

3.3 Local bifurcation analysis.

3.4 Global analysis for softening nonlinear stiffness (γ< 0).

3.5 Global analysis for hardening nonlinear stiffness (γ< 0).

3.6 Summary.



4 Analysis Techniques for the Various Forms of the Duffing Equation (Livija Cveticanin).

4.1 Introduction.

4.2 Exact solution for free oscillations of the Duffing equation with cubic nonlinearity.

4.3 The elliptic harmonic balance method.

4.4 The elliptic Galerkin method.

4.5 The straightforward expansion method.

4.6 The elliptic Lindstedt–Poincaré method.

4.7 Averaging methods.

4.8 Elliptic homotopy methods.

4.9 Summary.


Appendix AI: Jacob elliptic function and elliptic integrals.

Appendix 4AII: The best L2 norm approximation.

5 Forced Harmonic Vibration of a Duffing Oscillator with Linear Viscous Damping (Tamas Kalmar-Nagy and Balakumar Balachandran).

5.1 Introduction.

5.2 Free and forced responses of the linear oscillator.

5.3 Amplitude and phase responses of the Duffing oscillator.

5.4 Periodic solutions, Poincare sections, and bifurcations.

5.5 Global dynamics.

5.6 Summary.


6 Forced Harmonic Vibration of a Duffing Oscillator with Different Damping Mechanisms (Asok Kumar Mallik).

6.1 Introduction.

6.2 Classification of nonlinear characteristics.

6.3 Harmonically excited Duffing oscillator with generalised damping.

6.4 Viscous damping.

6.5 Nonlinear damping in a hardening system.

6.6 Nonlinear damping in a softening system.

6.7 Nonlinear damping in a double-well potential oscillator.

6.8 Summary.



7 Forced Harmonic Vibration in a Duffing Oscillator with Negative Linear Stiffness and Linear Viscous Damping (Stefano Lenci and Giuseppe Rega).

7.1 Introduction.

7.2 Literature survey.

7.3 Dynamics of conservative and nonconservative systems.

7.4 Nonlinear periodic oscillations.

7.5 Transition to complex response.

7.6 Nonclassical analyses.

7.7 Summary.


8 Forced Harmonic Vibration of an Asymmetric Duffing Oscillator (Ivana Kovacic and Michael J. Brennan).

8.1 Introduction.

8.2 Models of the systems under consideration.

8.3 Regular response of the pure cubic oscillator.

8.4 Regular response of the single-well Helmholtz–Duffing oscillator.

8.5 Chaotic response of the pure cubic oscillator.

8.6 Chaotic response of the single-well Helmholtz–Duffing oscillator.

8.7 Summary.


Appendix Translation of Sections from Duffing's Original Book (Keith Worden and Heather Worden).




"The book is a very well written and tightly edited exposition, not only of Duffing equations, but also of the general behavior of nonlinear oscillators. The book is likely to be of interest and use to students, engineers, and researchers in the ongoing studies of nonlinear phenomena. The book cites over 340 references." (Zentralblatt MATH, 2011)