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Volume 1, Number 2, 2011, Pages 183-191                                                                DOI:10.11948/2011012
Resonances of the SD oscillator due to the discontinuous phase
Qingjie Cao,Yeping Xiong,Marian Wiercigroch
Keywords:Resonances, generating function, canonical transformation, discontinuous Hamilto-nian, SD oscillator
      Resonant phenomenon of a harmonically excited system with multiple well dynamics plays a very important role in nonlinear dynamics research. In this paper, we investigate resonant behaviors of a discontinuous forced SD system with snap-through buckling and double-well dynamics. Firstly, a discontinuous time dependent Hamiltonian is derived from the discontinuous stage of SD oscillator providing a Du±ng type nonlinearity with snap-through buckling and double-well dynamics. This system comprises two subsystems connected at x = 0, where the system is discontinuous. We construct a series of generating functions and canonical transformations to get the canonical form of the system to reveal the complicated resonant behavior of the system. Furthermore, we introduce a composed winding number to explorer the complicated resonant phenomena of the system.This formulation for resonant phenomena given in this paper generalizes the formulation of n!0 = m! in regular perturbation theory, where n and m are relative prime integers,!0 and ! are the natural frequency and external frequencies respectively. Understanding the resonant behaviors of the SD oscillator at discontinuous stage enables us to further reveal the vibrational energy transition mechanism of the smooth and discontinuous nonlinear dynamic system.
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