含弹性约束碰撞振动系统的不连续分岔

Discontinuous bifurcations in an impact oscillator with elastic constraint

考虑单自由度弹性碰撞振动系统,构建不同模式周期运动的Poincaré映射,推导周期运动分岔分析的延拓打靶法。应用数值方法辨识系统在(ω,b)参数平面的稳定周期运动模式及其参数域。基于延拓打靶法和Floquet理论分析周期运动的稳定性与分岔,揭示两参数平面内迟滞域和亚谐包含域的形成机理以及亚临界周期倍化分岔引起迟滞现象的原因。弹性碰撞振动系统的擦边分岔是连续的。在p/1与(p+1)/1运动的转迁过程中,p/1运动的鞍结型擦边分岔产生迟滞域,p/1运动的周期倍化分岔或周期倍化型擦边分岔产生亚谐包含域。在亚临界周期倍化分岔的极小邻域内,鞍结分岔或鞍结型擦边分岔使系统响应产生迟滞现象。

A single-degree-of-freedom impact oscillator with elastic constraint was considered.Poincarémaps of different types of periodic motions were constructed,and a continuation shooting method was presented for the bifurcation analysis of periodic motions.The period motion modes and occurrence regions of the system were identified in the(ω,b)-parameter plane by numerical simulation.Based on the continuation shooting method and Floquet theory,the stability and bifurcations of periodic motions were analyzed.The formation mechanism of hysteresis and subharmonic inclusions regions in the two-parameter plane was revealed,as well as the hysteresis phenomenon caused by subcritical period-doubling bifurcation.It is found that the grazing bifurcation is continuous in the elastic impact system.In the transition between p/1 and(p+1)/1 motions,the hysteresis region is created by the SN-type grazing bifurcation of p/1 motion,and the subharmonic inclusions region is created by the period-doubling bifurcation or PD-type grazing bifurcation of p/1 motion.The saddle-node bifurcation or SN-type grazing bifurcation causes the hysteresis phenomenon of the system response in the minimal neighborhood of subcritical period-doubling bifurcation.