Dynamics of a plastic-impact system with oscillatory and progressive motions

A mathematical model is developed to describe oscillatory and progressive motions in dynamics of a plastic impact oscillator with a frictional slider. Dynamics of the impact oscillator is analyzed by a five-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the Poincaré map. The piecewise property is caused by the transitions of free flight and sticking motions of impacting masses immediately after the impact, and the singularity of the map is generated via the grazing contact of impacting masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularities and various parameters on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams for before-impact velocity versus forcing frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-n single-impact motions of the plastic-impact oscillator are found to exhibit extensive and systematic characteristics.