Dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation

A multi-degree-of-freedom vibratory system having symmetrically placed rigid stops is considered. The system consists of linear components, but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints. Such models play an important role in the studies of mechanical systems with clearances or gaps. Local codimension two bifurcation of maps, involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously, is analyzed by using the center manifold theorem technique and normal form method for maps. Symmetrical double-impact periodic motion and Poincaré map of the system are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with the codimension two bifurcation is obtained. Local behaviors of the vibratory systems with symmetrical rigid stops, near the points of codimension two bifurcations, are reported by the presentation of results for a two-degree-of-freedom vibratory system with symmetrical stops. The existence and stability of symmetrical double-impact periodic motion are analyzed explicitly. Also, local bifurcations at the points of change in stability, are analyzed. Near the point of codimension two bifurcation, there exists not only Hopf bifurcation of period-one double-impact motion, but also pitchfork bifurcation of the motion. Pitchfork bifurcation of period-one double-impact symmetrical motion results in the period-one double-impact unsymmetrical motion. The unsymmetrical double-impact motion is of two antisymmetrical forms due to different initial conditions and symmetrical stops. With change of the forcing frequency, the unsymmetrical double-impact periodic motion will undergo Hopf bifurcation. Moreover the period-one double-impact symmetrical motion will undergo Hopf bifurcation directly as the forcing frequency is changed in the contrary direction. The routes of quasi-periodic impact motions to chaos are observed by results from simulation.