Grazing and Hopf Bifurcations of a Periodic Forced System with Soft Impacts

Based on the research of a periodic forced system with soft impacts, the piecewise properties of the softimpacts system, such as asymmetric motion and singularity, were analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and grazing bifurcation, were investigated exactly. In the large constraint stiffness case, the Hopf bifurcation exists in the periodic forced system with soft impacts. The clearances of the system are the main reasons for influencing the chaotic motion. For small clearances, the grazing bifurcations bring about asymmetric motion and singularity. The steady 1-1-1 period orbits will exist within a wideband frequency range when appropriate system parameters are chosen.