一类双自由度碰振系统运动分析

基于Poincare映射方法对一类两自由度碰撞系统进行了分析。经过详细的理论演算得到单碰周期n的次谐运动的存在性判据和稳定性条件，给出计算Jacobi矩阵特征值的公式。数值模拟表明，该方法具有令人满意的结果。此外，还讨论了当不满足所提出的单碰周期n次谐运动的存在性条件时，可能会出现的运动形式。

ANALYSIS TO MOTIONS OF A TWO-DEGREE-OF- FREEDOM VIBRO-IMPACT SYSTEM

An undamped two-degree-of-freedom vibro-impact system with a harmonic excitation is under consideration based on the Poincar 'e map in this paper. In general case, due to the complexity of this kind of problems, it is difficult to give a satisfied theoretical analysis on the issues of periodic impact motions, bifurcations and chaos for multi-degree-of-freedom dynamical systems. So far, most of achievements are obtained for the one-degree-of-freedom vibro-impact systems with the property of piecewise linear by distinguished scholars, for instance, Shaw, Whiston and Nordmark, etc. In addition, a few cases have been studied for the piecewise linear vibro-impact systems with two degrees of freedom provided that there is a fixed rigid constraint boundary. Here we take into account a situation in which two oscillators move relatively under the harmonic excitation and the contact surface is rigid but unknown before impact. Our attention is paid to single impact period- n motions which have an impact occurred for every n times of the external excitation period. By a great deal of computation, we get an existent criterion of single impact period- n motions in the above piecewise linear impact system. The criterion is governed by a second order algebraic equation that is clearly solvable. This means that it is possible that there exist periodic impact motions even if the impact border is uncertain in the system with rather complicated impact law since two oscillators are involved in impact events. Moreover, the stability of periodic motions is studied and the corresponding formula is derived from the Poincar 'e map of the system discussed. The complex form of the derived stability criterion prevents us from developing more analytical results except the numerical simulation. Finally, it is shown that our conclusions are valid through the numerical simulation. And other motion modes are explored when the criterion for single impact period- n subharmonic motions is not available.