Periodic motions with impact chatters in an impact Duffing oscillator.

The periodic motions of discontinuous nonlinear dynamical systems are very difficult problems to solve in engineering and physics. Until now, except for numerical studies, one cannot find a better way to solve such problems. In fact, one still has difficulty obtaining periodic motions in continuous nonlinear dynamical systems. In this paper, a method is presented systematically for periodic motions in discontinuous nonlinear dynamical systems. The stability and grazing bifurcations of such periodic motions are studied. Such a method is presented through discussion on a periodically forced, impact Duffing oscillator. Thus, periodic motions with impact chatters in a periodically forced Duffing oscillator with one-sidewall constraint are studied. The analytical conditions for motion grazing at the boundary are developed from discontinuous dynamical systems. The impact Duffing oscillator is discretized to generate subimplicit mappings. With impact, the mapping structures are employed to construct specific impact periodic motions for an impact Duffing oscillator. The bifurcation trees of impact chatter periodic motions are achieved semi-analytically. The grazing and period-doubling bifurcations are obtained, and the grazing bifurcations are for the appearing and disappearance for an impact chatter periodic motion. The impact chatter periodic motions with and without grazing are presented for illustration of impact periodic motion complexity in the impact Duffing oscillator.