RESONANCES OF THE SD OSCILLATOR DUE TO THE DISCONTINUOUS PHASE

Qingjie Cao; Yeping Xiong; Marian Wiercigroch

doi:10.11948/2011012

2011 Volume 1 Issue 2

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Qingjie Cao, Yeping Xiong, Marian Wiercigroch. RESONANCES OF THE SD OSCILLATOR DUE TO THE DISCONTINUOUS PHASE[J]. Journal of Applied Analysis & Computation, 2011, 1(2): 183-191. doi: 10.11948/2011012

Citation:

Qingjie Cao, Yeping Xiong, Marian Wiercigroch. RESONANCES OF THE SD OSCILLATOR DUE TO THE DISCONTINUOUS PHASE[J]. Journal of Applied Analysis & Computation, 2011, 1(2): 183-191. doi: 10.11948/2011012

RESONANCES OF THE SD OSCILLATOR DUE TO THE DISCONTINUOUS PHASE

1 School of Astronautics, Harbin Institute of Technology, Harbin 150001, China;
2 School of Engineering, University of Southampton, Southampton SO17 1BJ, UK;
3 Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK

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Abstract

Abstract

Resonance phenomena of a harmonically excited system with multiple potential well play an important role in nonlinear dynamics research. In this paper, we investigate the resonant behaviours of a discontinuous dynamical system with double well potential derived from the SD oscillator to gain better understanding of the transition of resonance mechanism. Firstly, the time dependent Hamiltonian is obtained for a Duffing type discontinuous system modelling snap-through buckling. This system comprises two subsystems connected at x=0, for which the system is discontinuous. We construct a series of generating functions and canonical transformations to obtain the canonical form of the system to investigate the complex resonant behaviours of the system. Furthermore, we introduce a composed winding number to explore complex resonant phenomena. The formulation for resonant phenomena given in this paper generalizes the formulation of nω₀=mω used in the regular perturbation theory, where n and m are relative prime integers, ω₀ and ω are the natural frequency and external frequencies respectively. Understanding the resonant behaviour of the SD oscillator at the discontinuous phase enables us to further reveal the vibrational energy transfer mechanism between smooth and discontinuous nonlinear dynamical systems.
- SD oscillator /
- resonance /
- generating function /
- canonical transformation /
- stochastic web /
- discontinuous Hamiltonian
MSC: 34C15;34A36;34F15

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