多自由度含间隙振动系统周期运动的Hopf-pitchfork余维二分岔

建立了多自由度含间隙振动系统对称型周期碰撞运动及Poincaré映射的解析表达式,讨论了该映射不动点的稳定性与局部分岔。应用映射的中心流形和范式方法,研究了映射在Hopf-pitchfork余维二分岔点附近的参数开折,揭示了含间隙振动系统在余维二分岔点附近的动力学行为。在该类余维二分岔点附近,不仅存在对称型周期碰撞运动、Hopf分岔和叉式分岔,还存在非对称型周期碰撞运动及其Hopf分岔。通过数值仿真研究了余维二分岔点附近含间隙振动系统对称型周期碰撞运动经叉式分岔和Hopf分岔向混沌的转迁过程。

CODIMENSION 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 Poincare 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.