碰撞-渐进振动系统的周期振动与分岔

Periodic motions and bifurcations of a vibro-impact system with progressive motions

建立了碰撞-渐进振动系统的力学模型。分析了在相邻两次冲击之间,系统可能呈现的运动状态,给出了每种状态的判断条件和运动微分方程。采用数值计算的方法分析了单碰周期振动和p/1(p≥1)类基本碰撞振动的分岔特点,以及系统最佳渐进率对应的周期振动类型。结果表明:系统的最佳渐进效果发生在1/1周期振动时,质块M1冲击缓冲垫的速度峰值附近。由于碰撞振动系统特有的擦碰奇异性,使得在1/n(n≥2)单碰亚谐-渐进振动向混沌的转迁过程中,以及相邻p/1(p≥1)类基本碰撞-渐进振动之间的相互转迁过程中存在实擦边或虚擦边分岔和鞍结分岔等非光滑分岔。

The mechanical model of a vibro-impact system with progressive motions was established.The probable motion states presented by the system between two consecutive impacts were analyzed,and their judgement conditions as well as motion equations were put forward.Numerical simulations were used to examine the bifurcation characteristics of the system with two types of periodic motions,including single-impact periodic motions and p/1(p≥1)fundamental motions,and the periodic motion forms corresponding to the best progression.The results indicate that the best progression occurs in1/1motion,near the peak value of the impact velocity of the mass M1.Due to its own specific grazing singularity,the system presents two forms of non-smooth bifurcations,namely,the real-grazing or bare-grazing bifurcation and saddle-node bifurcation,in the process of transition from1/n(n≥2)single-impact subharmonic motion with progression to chaos,and in the process of mutual transition between adjacent p/1(p≥1)fundamental motions with progression.