Dynamics of vibro-impact mechanical systems with large dissipation

An n-degree-of-freedom system having placed single stop and subjected to periodic excitation is considered. Based on the analysis of dynamics of the vibratory system with plastic impacts, we introduce a (2n−1)-dimensional map with dynamical variables defined at the impact instants. The nonlinear 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 the impact mass immediately after the impact, and the singularity of the map is generated via the grazing contact of both the impact mass and the rigid stop and corresponding instability of periodic-impact 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. The single-impact periodic motions and disturbed map, associated with free flight motion of the system, are derived analytically. Stability, sliding and period-doubling bifurcations of the single-impact periodic motions are analyzed by the presentation of results for a three-degree-of-freedom plastic impact oscillator. Finally two actual examples, the impact-forming machine and inertial shaker, are considered to further analyze periodic-impact motions and bifurcations of plastic impact oscillators. The free flight and sticking solutions of two impact machines are analyzed numerically, and regions of existence and stability of different periodic-impact motions are therefore presented. The influence of non-standard bifurcations and system parameters on dynamics of the vibro-impact machines is elucidated accordingly.