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The Duffing Equation: Nonlinear Oscillators and their Behaviour

The Duffing Equation: Nonlinear Oscillators and their Behaviour

Ivana Kovacic, Michael J. Brennan

ISBN: 978-0-470-97785-9

Mar 2011

392 pages


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.
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)