【作者】 张睿；

【导师】 姜泽辉；

【作者基本信息】 哈尔滨工业大学 ， 光学， 2015， 硕士

【摘要】 蹦球指在竖直简谐振动台面上蹦跳的球,是一种最简单的非线性动力学系统,球与台面之间的碰撞恢复系数e(??10 e)决定了蹦球的运动性质。本文考虑一种极端情况,球与台面间碰撞恢复系数为0,二者进行完全非弹性碰撞。随着台面振动强度的增大,完全非弹性蹦球每一跳的飞行时间和相对台面的着陆速度会依次经历一系列倍周期分岔过程。在2n阶分岔点之前,存在一个“平台”区,平台区内蹦球进行稳定n倍周期运动;当n?1时,2n阶分岔点之后存在蹦跳密集区。在密集区内蹦球的运动非常复杂,存在大量倍周期分岔现象且分岔点非常密集,蹦球运动敏感地依赖于振动强度的变化,分岔相图中还存在着自相似结构。本文研究了完全非弹性蹦球密集区起点处和终点处的运动状况,发现在密集区的尾部有一段很短的区域,在这段区域内蹦球运动的周期由若干次跳跃组成,永远不会落入吸收区,但具有明显的周期性;在第二个密集区中间部位有一段仅由两次跳跃便落入吸收区的区间,这段区间之前的密集区部分也是永不落入吸收区的周期性运动。在完全非弹性蹦球中分别引入大小恒定的力和大小随蹦球相对振台速度变化的粘滞阻力,发现在这两种性质的阻力作用下,密集区尾部仍可观察到永不落入吸收区的稳定倍周期运动。将蹦球每一跳的飞行时间看做一维映射,引入Lyapunov指数,根据Lyapunov指数的正负判断蹦球的运动是否混沌,发现在不考虑阻力时蹦球运动不会出现混沌。加入大小恒定的阻力之后,蹦球的运动仍始终是倍周期的;在此基础上,加入一个与蹦球和振台的相对速度方向相反、大小成正比的粘滞阻力,蹦球的运动会出现混沌,且粘滞阻力系数越大,混沌出现时对应的振台振动强度越小。当粘滞阻力系数足够大时,第一密集区与第二个平台出现交叠,导致蹦球在第一密集区后面部分的运动就已经是混沌的,不再出现稳定的周期运动;某些特定粘滞阻力系数条件下,混沌运动中存在周期窗口,周期窗口中可观察到明显的倍周期分岔现象。更多还原

【Abstract】 A ball bouncing upon a vertically vibrated surface is one of the simplest nonlinear dynamical system, the collision recovery coefficient e( ￡￡10 e) between the ball and the table surface determines the nature of the bouncing ball movement. A extreme case is the bouncing ball is completely inelastic( e=0). With the increase of the vibration intensity, the ball’s flying time and the relative landing velocity between the ball and the table will experience a series of period doubling bifurcation. Before the 2nth bifurcation point, there is a “platform” area, in which the bouncing ball’s movement is periodic and its period is three times of the table’s period; when n>1, after the 2nth bifurcation point is a clumping zone. The movement of the ball in clumping zones is very complex, a lot of period doubling bifurcations exist and the bifurcation points are extremely dense, the movement of the bouncing ball depends sensitively on the variation of the vibration intensity, self similar structure is also existed in the phase diagram of the bifurcation.The completely inelastic bouncing ball’s movements in the start and end of the clumping zone were investigated, and we found at the end of the clumping zone, there is a short area in which the bouncing ball will never fall into the absorbing region, but its movement is still periodic. In the second clumping zone, there will be a short region in which the bouncing ball fall into the absorbing region after two jumps, in the clumping zone before this region the bouncing ball will also never fall into the absorbing region,while it move with periodicity. Adding two forms of resistance to completely inelastic bouncing ball: one is a value-constant resistance while another one is viscous resistance whose value is proportionate to the relative velocity between the ball and the table, and we can also found the ball’s periodic movement even though it will never fall into the absorbing region at the end of the clumping zones.Regard the flying time of every jump as one-dimensional mapping to get the Lyapunov exponent, and whether the ball’s movement is chaotic could be judged by the Lyapunov exponent is positive or negative. There will be no chaos when any form of resistance not be considered. When the value-constant resistance added to the completely inelastic bouncing ball singly, the movement of the ball is also periodic. On this basis,add viscous resistance to the system, chaos appears, and the larger the viscous resistance coefficient is, the earlier when the chaos appears. When the viscous resistance coefficient is large enough, there is a overlap between the first clumping zone and the second‘platform’, where the movement of the bouncing ball is chaotic. Under the condition of certain value of viscous resistance coefficient, periodic windows exists in chaos, and in the periodic windows bifurcation phenomenon can be obviously observed.更多还原