C ODIMENSION TWO HOPF-PITCHFORK BIFURCATION OF PERIODIC MOTION OF THE MULTI-DEGREE-OF-FREEDOM VIBRATORY SYSTEM WITH A CLEARANCE

A multi-degree-of-freedom vibratory system with a clearance 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.Such models play an important role in the study of mechanical systems with clearances or gaps.Local codimension two bifurcation of maps,associated with Hopf-pitchfork case,is analyzed using the center manifold theorem and normal form method of maps.The period-one double-impact symmetrical motion and Poincaré map of the vibratory system with symmetrical rigid stops are derived analytically.The existence and stability of period-one double-impact symmetrical motion are analyzed explicitly.Near the point of codimension two bifurcation there exists not only Hopf bifurcation of period-one double-impact symmetrical motion,but also pitchfork bifurcation of the motion,which results in the period-one double-impact unsymmetrical motion.With change of the forcing frequency,the unsymmetrical double-impact periodic motion will undergo Hopf bifurcation.The routes of quasi-periodic impact motions to chaos are observed from simulation results.