含间隙振动系统周期振动的多样性和转迁特征

Diversity and transition characteristics of periodic vibration of a vibro-impact system with a clearance

以两自由度含间隙碰撞振动系统为研究对象,辨识周期振动的模式类型及其在双参平面内的发生区域和分布规律,揭示低频区域无冲击、基本冲击、颤碰和亚谐冲击等周期振动模式类型的多样性和转迁特征,以及擦边分岔点附近鞍结分岔的存在与位移振幅的变化形式之间的关系。在无冲击和基本冲击振动的边界线上存在若干具有自相似分形特征的舌形域。舌形域内亚谐振动的模式类型和分布具有规律性。由于擦边分岔的不可逆性,擦边和鞍结分岔线在相邻周期振动的发生区域之间形成迟滞域,并在舌形域的边界形成一个迟滞域群。相邻迟滞域边界线的横截相交点是奇异点,只有在奇异点,位移振幅连续变化,擦边分岔连续可逆。揭示了奇异点的二重擦边和倍化-鞍结余维二分岔特征。

A two-degree-of-freedom vibro-impact system with a clearance is considered.Diversity,existence and stability domains,and distribution regularities of periodic motions are obtained numerically in the double-parameter plane by constructing two Poincarémaps.Bifurcations of periodic motions in the low frequency region,such as impactless motion,fundamental impact,chattering and subharmonic impact motions,are analyzed.Diversity and transition characteristics of pattern types of periodic motions of the system,and relation between the existence of saddle-node bifurcation in the vicinity of the grazing bifurcation and the variation of displacement amplitude are revealed.Some small tongue-shaped regions that have self-similarity and fractal characteristic appear on the boundary of impactless motions and fundamental impact motions.Pattern type and distribution of subharmonic impact motions in the tongue-shaped regions show regularity.Given the irreversibility of grazing bifurcation of impactless motions,fundamental impact motions,and subharmonic impact motions in tongue-shaped regions,the hysteresis domain forms between the existence regions of adjacent periodic motions,and a group of hysteresis domains appear on the boundary of the tongue-shaped regions.Two boundary curves of each hysteresis zone intersect at two singular points.The grazing bifurcation is continuous and reversible and saddle-node bifurcation is absent only at these singular points.The displacement amplitude of the impact oscillator varies continuously just after them.Each singular point is the junction of adjacent hysteresis domains,and also a point of double grazing bifurcation flip-fold codimension-2 bifurcation.