摘要
Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1 ： 4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1：4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1：4 resonance, are studied. Near the bifurcation point for 1：4 strong resonance there exist a Neimark-Sacker bifurcation of period one single-impact motion and a tangent （fold） bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems： namely, the ＂heteroclinic＂ circle formed by coinciding stable and unstable separatrices of saddles, Tin, Ton and Tout type tangent （fold） bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasiperiodic impact motions.
Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1 ： 4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1：4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1：4 resonance, are studied. Near the bifurcation point for 1：4 strong resonance there exist a Neimark-Sacker bifurcation of period one single-impact motion and a tangent （fold） bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems： namely, the ＂heteroclinic＂ circle formed by coinciding stable and unstable separatrices of saddles, Tin, Ton and Tout type tangent （fold） bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasiperiodic impact motions.
基金
The project supported by National Natural Science Foundation of China (50475109,10572055)
Natural Science Foundation of Gansu Province Government of China(3ZS061-A25-043(key item))
