Periodic motions and bifurcations of a vibrating impact system

An inertial shaker is considered. Dynamics of the shaker is studied with special attention to periodic-impact stability and bifurcations. Dynamics of the shaker are analyzed by use of a mapping derived from the equations of motion. Each iterate of the mapping corresponds to the vibro-bench colliding with the cast each other once. The mapping is four dimensional in elastic impact case, three dimensional in inelastic impact case. The mapping, in inelastic impact case, is of piecewise property due to synchronous and non-synchronous motion of shaker and cast immediately after the impact, and singularities caused by the grazing contact motions of shaker and cast, so the inertial shaker exhibits two types of single-impact periodic motions. The influence of the piecewise property and singularities on global bifurcations and transitions to chaos is elucidated by numerical analyses. The single-impact periodic motion usually undergoes period doubling bifurcation with decrease in the forcing frequency. However, the period doubling cascades of the motion are discontinuous. Grazing contact of shaker and cast are likely to be a major cause of the discontinuous cascades of period doubling.