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E-grāmata: Duffing Equation: Nonlinear Oscillators and their Behaviour

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(Southampton University, UK), (University of Novi Sad, Russia)
  • Formāts: PDF+DRM
  • Izdošanas datums: 11-Feb-2011
  • Izdevniecība: John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9780470977866
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Duffing Equation: Nonlinear Oscillators and their BehaviourPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 11-Feb-2011
  • Izdevniecība: John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9780470977866
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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.

Recenzijas

"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)  

List of Contributors
xi
Preface xv
1 Background: On Georg Duffing and the Duffing Equation
1(24)
Ivana Kovacic
Michael J. Brennan
1.1 Introduction
1(1)
1.2 Historical perspective
2(3)
1.3 A brief biography of Georg Duffing
5(2)
1.4 The work of Georg Duffing
7(2)
1.5 Contents of Duffing's book
9(4)
1.5.1 Description of Duffing's book
9(3)
1.5.2 Reviews of Duffing's book
12(1)
1.6 Research inspired by Duffing's work
13(5)
1.6.1 1918-1952
13(2)
1.6.2 1962 to the present day
15(3)
1.7 Some other books on nonlinear dynamics
18(1)
1.8 Overview of this book
18(3)
References
21(4)
2 Examples of Physical Systems Described by the Duffing Equation
25(30)
Michael J. Brennan
Ivana Kovacic
2.1 Introduction
25(1)
2.2 Nonlinear stiffness
26(2)
2.3 The pendulum
28(1)
2.4 Example of geometrical nonlinearity
29(2)
2.5 A system consisting of the pendulum and nonlinear stiffness
31(1)
2.6 Snap-through mechanism
32(2)
2.7 Nonlinear isolator
34(3)
2.7.1 Quasi-zero stiffness isolator
35(2)
2.8 Large deflection of a beam with nonlinear stiffness
37(3)
2.9 Beam with nonlinear stiffness due to inplane tension
40(3)
2.10 Nonlinear cable vibrations
43(7)
2.11 Nonlinear electrical circuit
50(2)
2.11.1 The electrical circuit studied by Ueda
51(1)
2.12 Summary
52(1)
References
53(2)
3 Free Vibration of a Duffing Oscillator with Viscous Damping
55(26)
Hiroshi Yabuno
3.1 Introduction
55(1)
3.2 Fixed points and their stability
56(6)
3.2.1 Case when the nontrivial fixed points do not exist (αγ > 0)
58(1)
3.2.2 Case when the nontrivial fixed points exist (αγ < 0)
59(3)
3.2.3 Variation of phase portraits depending on linear stiffness and linear damping
62(1)
3.3 Local bifurcation analysis
62(6)
3.3.1 Bifurcation from trivial fixed points
62(5)
3.3.2 Bifurcation from nontrivial fixed points
67(1)
3.4 Global analysis for softening nonlinear stiffness (γ < 0)
68(4)
3.4.1 Phase portraits
68(1)
3.4.2 Global bifurcation analysis
69(3)
3.5 Global analysis for hardening nonlinear stiffness (γ > 0)
72(7)
3.5.1 Phase portraits
72(1)
3.5.2 Global bifurcation analysis
73(6)
3.6 Summary
79(1)
Acknowledgments
80(1)
References
80(1)
4 Analysis Techniques for the Various Forms of the Duffing Equation
81(58)
Livija Cveticanin
4.1 Introduction
81(2)
4.2 Exact solution for free oscillations of the Duffing equation with cubic nonlinearity
83(6)
4.2.1 The frequency and period of free oscillations of the Duffing oscillator
85(2)
4.2.2 Discussion
87(2)
4.3 The elliptic harmonic balance method
89(11)
4.3.1 The Duffing equation with a strong quadratic term
90(1)
4.3.2 The Duffing equation with damping
91(2)
4.3.3 The harmonically excited Duffing oscillator
93(5)
4.3.4 The harmonically excited pure cubic Duffing equation
98(2)
4.4 The elliptic Galerkin method
100(6)
4.4.1 Duffing oscillator with a strong excitation force of elliptic type
103(3)
4.5 The straightforward expansion method
106(4)
4.5.1 The Duffing equation with a small quadratic term
109(1)
4.6 The elliptic Lindstedt-Poincare method
110(5)
4.6.1 The Duffing equation with a small quadratic term
114(1)
4.7 Averaging methods
115(8)
4.7.1 The generalised elliptic averaging method
117(3)
4.7.2 Elliptic Krylov-Bogolubov (EKB) method for the pure cubic Duffing oscillator
120(3)
4.8 Elliptic homotopy methods
123(4)
4.8.1 The elliptic homotopy perturbation method
123(3)
4.8.2 The elliptic homotopy analysis method
126(1)
4.9 Summary
127(1)
References
128(3)
Appendix 4AI Jacobi elliptic functions and elliptic integrals
131(4)
Appendix 4AII The best L2 norm approximation
135(4)
5 Forced Harmonic Vibration of a Duffing Oscillator with Linear Viscous Damping
139(36)
Tamas Kalmar-Nagy
Balakumar Balachandran
5.1 Introduction
139(2)
5.2 Free and forced responses of the linear oscillator
141(3)
5.2.1 Free oscillations and timescales
141(1)
5.2.2 Forced oscillations
142(2)
5.3 Amplitude and phase responses of the Duffing oscillator
144(17)
5.3.1 Primary resonance
145(11)
5.3.2 Secondary resonances
156(5)
5.4 Periodic solutions, Poincare sections, and bifurcations
161(7)
5.4.1 Periodic solutions
161(1)
5.4.2 Poincare section and Poincare map
161(2)
5.4.3 The Ueda oscillator
163(1)
5.4.4 Bifurcations and chaos in the Duffing oscillator with a softening spring
163(5)
5.5 Global dynamics
168(5)
5.6 Summary
173(1)
References
173(2)
6 Forced Harmonic Vibration of a Duffing Oscillator with Different Damping Mechanisms
175(44)
Asok Kumar Mallik
6.1 Introduction
175(1)
6.2 Classification of nonlinear characteristics
176(2)
6.2.1 Stiffness force
176(1)
6.2.2 Damping force
176(1)
6.2.3 Equivalent viscous damping
177(1)
6.3 Harmonically excited Duffing oscillator with generalised damping
178(1)
6.4 Viscous damping
178(15)
6.4.1 Harmonic solution for a hardening system
178(8)
6.4.2 Harmonic solution for a softening system
186(1)
6.4.3 Superharmonic and subharmonic response
187(1)
6.4.4 Chaotic and other types of responses
188(1)
6.4.5 Experimental and numerical results
188(5)
6.5 Nonlinear damping in a hardening system
193(15)
6.5.1 Harmonic solution
193(6)
6.5.2 Stability analysis
199(1)
6.5.3 Chaotic motion
200(3)
6.5.4 Coulomb damping
203(5)
6.6 Nonlinear damping in a softening system
208(3)
6.7 Nonlinear damping in a double-well potential oscillator
211(4)
6.8 Summary
215(1)
Acknowledgments
215(1)
References
215(4)
7 Forced Harmonic Vibration in a Duffing Oscillator with Negative Linear Stiffness and Linear Viscous Damping
219(58)
Stefano Lenci
Giuseppe Rega
7.1 Introduction
219(1)
7.2 Literature survey
220(8)
7.2.1 Former numerical studies and approximate criteria for chaos
222(3)
7.2.2 Refined computational investigations
225(1)
7.2.3 Control of nonlinear dynamics
226(2)
7.3 Dynamics of conservative and nonconservative systems
228(7)
7.3.1 The conservative case
228(4)
7.3.2 The effect of damping
232(2)
7.3.3 The effect of the excitation
234(1)
7.4 Nonlinear periodic oscillations
235(5)
7.5 Transition to complex response
240(17)
7.5.1 Bifurcation diagrams, behaviour chart and basins of attraction
240(11)
7.5.2 Analytical prediction via the Melnikov method
251(6)
7.6 Nonclassical analyses
257(12)
7.6.1 Control of homoclinic bifurcation
257(7)
7.6.2 Dynamical integrity
264(5)
7.7 Summary
269(1)
References
270(7)
8 Forced Harmonic Vibration of an Asymmetric Duffing Oscillator
277(46)
Ivana Kovacic
Michael J. Brennan
8.1 Introduction
277(1)
8.2 Models of the systems under consideration
278(3)
8.3 Regular response of the pure cubic oscillator
281(16)
8.3.1 Primary resonance: transient solution
282(1)
8.3.2 Primary resonance: steady-state solution
283(13)
8.3.3 Some secondary resonance responses
296(1)
8.4 Regular response of the single-well Helmholtz-Duffing oscillator
297(11)
8.4.1 Primary resonance response via perturbation method
297(6)
8.4.2 Frequency-response curves
303(2)
8.4.3 Analysis of the steady-state response: coexisting attractors
305(2)
8.4.4 Some secondary resonance responses
307(1)
8.5 Chaotic response of the pure cubic oscillator
308(9)
8.5.1 A cascade of period-doubling bifurcations as a route to chaos: analytical considerations
309(5)
8.5.2 A cascade of period-doubling bifurcations: numerical simulations
314(3)
8.6 Chaotic response of the single-well Helmholtz-Duffing oscillator
317(3)
8.6.1 Routes to chaos
319(1)
8.7 Summary
320(1)
References
320(3)
Appendix Translation of Sections from Duffing's Original Book 323(32)
Keith Worden
Heather Worden
Glossary 355(10)
Index 365
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.
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