Analytical-numerical studies on the stability and bifurcations of periodic motion in the vibro-impact systems with clearances

Abstract

This paper investigates the stability and bifurcations of periodic solutions in three-degree-of-freedom vibro-impact systems based on the explicit critical criteria and the discontinuous mapping method. Firstly, a six-dimensional Poincaré map is established by taking the impact surface as the Poincaré section. The explicit criteria including eigenvalue assignment and transversality condition are applied to determine the bifurcation point of co-dimension one pitchfork bifurcation. The stability and direction of the bifurcation solution are then studied by using center manifold reduction theory and normal form approach. Secondly, the bifurcation points of co-dimension-two Hopf–Hopf interaction bifurcation and pitchfork–Neimark–Sacker bifurcation are determined by applying the explicit critical criteria, and the local dynamic behaviors are examined in the neighborhood of these co-dimension-two bifurcation points. Finally, a six-dimensional Poincaré map formed by choosing the constant phase angle as the Poincaré section is used to investigate the existence and stability of grazing bifurcation based on the piecewise compound normal form map. The causes of the discontinuous jump and the coexistence of attractors near the grazing periodic motion are explained for the three-degree-of-freedom vibro-impact system with a moving constraint.