两自由度振动系统的斜碰撞分析

研究斜碰撞振动系统动力学的一个关键问题是对系统在碰撞前后的状态进行合理描述和正确计算．针对两弹性体斜碰撞问题，基于瞬间碰撞假设，提出了采用步进冲量来分析和求解斜碰撞前后的状态关系；并以弹簧摆和振子组成的两自由度斜碰撞振动系统为例，具体介绍了该算法如何实现．用解析方法讨论了该系统在斜碰撞过程中可能出现的各种力学现象，将冲量步进算法得到的数值解与解析结果进行对比，取得了完全一致的结果．该数值方法能适应多种斜碰撞问题的计算．

Analysis of oblique impact of a vibrating system of two degrees of freedom

A crucial issue in studying the dynamics of an oblique-impact vibrating system is to establish a simple, but reasonable impact model and a right computational method for the relation between pre-impact state and post-impact state. In this paper, an attempt is made to propose a uniform numerical method so as to describe the oblique-impact process of two elastic bodies in a system of multiple degrees of freedom and to lay the groundwork for the dynamic analysis of the oblique-impact vibrating system. When the instantaneous impact is assumed and the friction between two contact surfaces is taken into account, there may exist the reverse of relative rnicro-slip or its trend in tangential direction of the contact when oblique-impact occurs. Thus, the relation between pre-impact state and post-impact state cannot be described by using any simple impact laws, such as Newton's law or Poisson's law. To attack this problem, a numerical method, referred to as the incremental impulse method, is developed in this study. The method enables one to judge the direction of tangential micro-slip and to determine the state of the system at each incremental step of both normal impulse and tangential frictional impulse such that the possible reverse of relative micro-slip due to friction can be properly determined. An oblique-impact vibrating system of two degrees of freedom, composed of a spring-pendulum and a mass-spring oscillator, is used to illustrate the numerical method and some technical details. To verify the numerical method, the relation between pre-impact state and post-impact state of this system is analyzed in closed form and all possible cases of the tangential impact motion are discussed in detail. The analytical relation between pre-impact state and post-impact state is derived in three cases when the reverse of micro-slip occurs within the normal approach phase or within the restitution phase in impact process, or it does not show up in the impact process. A comparison between the numerical results and the analytical results indicates very good agreement. Compared with the analytical expressions applicable to a few of special vibro-impacting systems, the numerical method provides a widely feasible way for solving the oblique-impact problems of various dynamic systems.