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

Zhu Xifeng*, 1, Gao Quanfu2
1 School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, P.R. China
2 Key Laboratory of System Dynamics and Reliability of Rail Transport Equipment of Gansu Province, Lanzhou, 730070, P.R. China

© 2014 Zhu and Gao

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: ( This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the School of Mechatronic Engineering, Lanzhou Jiaotong University, Box no.406, 88 West Anning Road, Lanzhou, Gansu Province, Postcard: 730070, P.R. China; Tel: 0086-931-4938143; E-mail:


Based on the research of a periodic forced system with soft impacts, the piecewise properties of the soft-impacts 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 thoroughly. In the case of large constraint stiffness, the Hopf bifurcation is observed 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.

Keywords: Grazing bifurcation, soft impacts, periodic motion, vibration.