Analysis to grazing bifurcation in linear vibro-impact system with N-dimensions

Grazing bifurcation is an important dynamical behavior ofa vibro-impact system and is usually analyzed by choosing the impact plane as thePoincar\'e section. However, this plane sometimes does not meet thetransverse intersection condition of Poincar\'e section, especially whilegrazing motion or chaos take place. Moreover, the bifurcation of impactnumber instead of period of the motion is considered in former cases. Thebifurcations with time evolution are more attractive for a vibro-impactsystem.In this paper, the Poincar\'e map of period-n motions with single-impactis set up for a linear vibro-impact system by using a fixed phase plane asthe Poincar\'e section here. Based on analysis of the Poincar\'e map,the grazing bifurcation conditions and bifurcation equations are determinedfor the vibro-impact system, and a vibro-impact system with single DOF isused as an example to testify the obtained analytical result. A numericalsimulation is carried out for the bifurcation diagram of the vibro-impactsystem, which agrees with analytical results very well. This method canbe used tocalculate not only the parameters of grazing bifurcation, but also those ofany period-n motions, for a linear vibro-impact system.