Bifurcations and transition phenomena in an impact oscillator

The motion of mechanical systems with impacts is strongly nonlinear. Many different types of periodic and chaotic impact motions exist even for simple systems with external periodic excitation forces. The group of fundamental periodic motions is characterized by the different number of impacts in one motion period, which equals the excitation force period. Every motion has a region in the space of system parameters in which the solution can exist and is stable. There exist transition regions, so-named hysteresis regions and beat motion regions, which lie between the zones of neighbouring fundamental impact motions. Transition regions are determined by the boundaries of existence which correspond to grazing bifurcations and by the boundaries of stability corresponding to the period-doubling and saddle-node bifurcations. The transition between neighbouring periodic impact motions is never continuous, with the exception of singular points, where the existence boundaries and stability boundaries intersect. Jump phenomena appear on the hysteresis region boundaries and subharmonic and chaotic motions exist in the transition beat motion region. This paper presents results of theoretical analysis, analogue simulations and numerical solutions of a particular but typical impacting system.