PERIODIC MOTIONS AND GLOBAL BIFURCATIONS OF A TWO-DEGREE-OF-FREEDOM SYSTEM WITH PLASTIC VIBRO-IMPACT

A two-degree-of-freedom (d.o.f.) impact system with proportional damping is considered. The maximum displacement of one of the masses is limited to a threshold value by a rigid wall, which gives rise to a non-linearity in the system. A limiting case of a dynamical problem arising in the mechanical systems with amplitude constraints is investigated. For the perfectly plastic vibro-impact case, dynamics of a two-degree-of-freedom vibratory system contacting a single stop is represented by a three-dimensional map. Existence and stability of period n single-impact motions are analyzed by analytical and numerical methods. It is shown that the vibro-impact system may exhibit two different types of periodic motions due to the piecewise property of the map. Transitions of two types of period n single-impact motions are demonstrated. The singularities of the Poincaré map, caused by grazing boundary motion of the impact oscillator, are considered. Due to the piecewise discontinuities and singularities of the map, the vibro-impact system is shown to undergo period-doubling bifurcations followed by complex sequence of transitions, in which the period-doubling cascades do not occur and extremely long-periodic and chaotic motions are generated directly with the motions with grazing boundary occurring. Finally, the influence of system parameters on periodic impacts and global bifurcations is discussed.