线性碰振系统周期解擦边分岔的一类映射分析方法

擦边分岔是碰振机械系统的一种重要分岔行为. 以固定相位面作为Poincaré截面， 建立了线性碰振系统单碰周期$n$运动的Poincaré映射. 通过分析该映射，得到了系统 发生擦边分岔的条件和分岔方程，并以单自由度碰振系统为实例验证了分析结果的正确性. 该方法不仅可以计算线性碰振系统擦边分岔的参数值，还可以计算系统的任意周 期$n$解的分岔参数值.

A MAP METHOD FOR GRAZING BIFURCATION IN LINEAR VIBRO-IMPACT SYSTEM

Grazing bifurcation is an important dynamical behavior of a vibro-impact system and is usually analyzed by choosing the impact plane as the Poincare section. However, this plane sometimes does not meet the transverse intersection condition of Poincare section, especially while grazing motion or chaos take place. Moreover, the bifurcation of impact number instead of period of the motion is considered in former cases. The bifurcations with time evolution are more attractive for a vibro-impact system. In this paper, the Poincare map of period-n motions with single-impact is set up for a linear vibro-impact system by using a fixed phase plane as the Poincare section here. Based on analysis of the Poincare map, the grazing bifurcation conditions and bifurcation equations are determined for the vibro-impact system, and a vibro-impact system with single DOF is used as an example to testify the obtained analytical result. A numerical simulation is carried out for the bifurcation diagram of the vibro-impact system, which agrees with analytical results very well. This method can be used to calculate not only the parameters of grazing bifurcation, but also those of any period-n motions, for a linear vibro-impact system.