Periodic motions and bifurcations of a vibro-impact system

Based on the analysis of a two-degree-of-freedom plastic impact oscillator, we introduce a three-dimensional map with dynamical variables defined at the impact instants. The non-linear dynamics of the vibro-impact system is analyzed by using the Poincaré map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of map is generated via the grazing contact of two masses and corresponding instability of periodic motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. Simulations of the free flight and sticking solutions are carried out, and regions of existence and stability of different impact motions are therefore presented in (δ, ω) plane of dimensionless clearance δ and frequency ω. The influence of non-standard bifurcations on dynamics of the vibro-impact system is elucidated accordingly.