Stability of periodic motion, bifurcations and chaos of a two-degree-of-freedom vibratory system with symmertical rigid stops

A two-degree-of-freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Such models play an important role in the studies of mechanical systems with clearances or gaps. The period-one double-impact symmetrical motion and its Poincaré map are derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motion are analyzed by the equation of Poincaré map. The routes from period-one double-impact symmetrical motion to chaos, via pitchfork bifurcations and period-doubling bifurcation, are studied by numerical simulation. Some non-typical routes to chaos, caused by grazing the stops and Hopf bifurcation of period two four-impact motion, are analyzed. Hopf bifurcations of period-one double-impact symmetrical and antisymmetrical motions are shown to exist in the two-degree-of-freedom vibratory system with two-sided stops. Interesting feature like the period-one four-impact symmetrical motion is also found, and its route to chaos is analyzed. It is of special interest to acquire an overall picture of the system dynamics for some extreme values of parameters, especially those which relate to the degenerated case of a single-degree-of-freedom system, and these analyses are presented here.