【作者】 陈良；

【导师】 李群宏；

【作者基本信息】 广西大学 ， 基础数学， 2005， 硕士

【摘要】 多自由度碰撞振动系统一直是非线性动力学研究中的一个热点之一。 本文主要研究了一类两自由度碰撞振动系统的周期碰撞运动,借助理论分析和数值仿真的方法对系统单碰周期n运动的存在性与共存性等性质进行了研究,并推导出能描述系统运动过程的不连续映射,然后讨论了系统周期运动的稳定性的情况。 首先,本文用非光滑动力系统理论对一类具有较复杂碰撞条件且碰撞位置不明确的两自由度碰撞振动系统进行了分析,经过大量的计算及详细的理论推导,给出了该系统在特定参数区域内单碰周期n运动的存在性定理和不存在性定理,得到了一种寻找多自由度碰撞振动系统中多个周期轨道共存的分析方法。由此可以细化了系统在各个参数区间内的运动情况,特别是对如何保证系统在单碰周期运动中不会发生其他碰撞的问题得到了较好的认识。同时,通过数值模拟也验证了理论结果。 随后,本文利用流形理论及复合映射的方法推导了系统的不连续映射,把运动流形的过程和碰撞过程结合起来。在一已知特定运动轨道(如一单碰周期n运动轨道)的基础上,通过系统的不连续映射即可知道该轨道上一点及其附近领域内各点往后的运动状态。 最后,本文利用扰动分析的方法,在结合系统不连续映射的基础上,建立以定相位面为Poincaré截面的Poincaré映射,讨论了系统周期运动的稳定性.更多还原

【Abstract】 Vibro-impact system is one of the focuses in the research of nonlinear dynamics.The periodic impact motions in a vibro-impact system of two degrees of freedom are presented in this dissertation.By using the analytic and numerical methods,existence and coexistence of single impact period-n responses in the above system are obtained and a discontinuity mapping which can describe the motions of the system is deduced.Furthermore the stability of the periodic motions of the system is considered.Firstly,a two-degree-of-freedom vibro-impact system with more complicated impacting conditions and uncertain impacting position is studied in this paper by the theory of non-smooth dynamical systems.The existence and non-existence theorems of single impact periood-n responses in some parameter regions are presented through a great deal of computation and detailed theoretical deducing,with this provides an analytic method to find out the coexistence of periodic trajectories in the multi-degree-of-freedom vibro-impact systems. In this way,dynamics in particular parameter regions of the systems can be known well and so is the problem of how to guarantee that other more impacts could not happen so that single impact periodic responses can occur. Subsequently simulation illustrates the theoretical results.After that,by using the theory of manifold and the method of composition mapping , the discontinuity mapping of the system is deduced and it composites the process of motion manifold and the process of impact.On the basis of a particular known motion trajectory (such as a trajectory of single impact period-n responses),the future motion states of a given point on the trajectory and other points in the proximity of it can be determined subsequently.Finally, in this paper the Poincare mapping with definite phase surface as Poincare section is established ,then stability of periodic motions of the system is investigated by using the deduced discontinuity mapping and the method of perturbation analysis.更多还原