Transition law of adjacent fundamental motions in vibro-impact system with progression

The vibro-impact phenomena which widely exists in power mechanical system will make the system exhibit complex dynamic response. So far the research on stability and bifurcation of p/1 fundamental motions of vibro-impact system is still rare, and most studies of vibro-impact dynamics are based on single-parameter bifurcation analysis. In this paper, taking a small vibro-impact driver as engineering background, a mechanical model of a vibro-impact system with progressive motions is established. Types of impact between the vibration exciter and the cushion, and conditions of progressive motions of the slider are analyzed. Judgment conditions and motion equations of four probable motion states presented by the system are put forward. Based on bifurcation analysis of two-dimensional parameters, existence regions and distribution laws of different types of periodic motions of the system are obtained in the (ω,l) parameter plane. Transition laws of adjacent p/1 fundamental motions are analyzed in detail. In the right region of the existence region of 5/1 fundamental motion, there exists a singular point Xp on the boundary between adjacent regions of p/1 fundamental motions, which is the critical point of bifurcation characteristics of adjacent p/1 fundamental motions. In the region with l less than lXp, adjacent p/1 fundamental motions are transited mutually by real-grazing bifurcation and saddle-node bifurcation. Two periodic attractors can coexist in the hysteresis region, which exists between real-grazing bifurcation boundary and saddle-node bifurcation boundary. In the region with l more than lXp, there exists a transition region between adjacent regions of p/1 fundamental motions. The system exhibits (2p+2)=2 and (2p+1)=2 motions in the transition region. In the left region of the existence region of 5/1 fundamental motion, p/1 fundamental motion transits to (p+1)=1 fundamental motion via multi-sliding bifurcation. © 2017, Editorial Office of Chinese Journal of Theoretical and Applied Mechanics. All right reserved.