Existence and stability of the periodic orbits induced by grazing bifurcation in a cantilever beam system with single rigid impacting constraint

Abstract

Grazing, which can induce many nonclassical bifurcations, is a special dynamic phenomenon in some non-smooth dynamical systems such as vibro-impact systems with clearance. In this paper, the existence and stability of the periodic orbits induced by the grazing bifurcation in a cantilever beam system with impacts are uncovered. Firstly, the Poincaré mapping of the system is obtained by adopting the discontinuous mapping method. Then, by means of shooting method, the periodic orbits are determined, followed by the acquisition of Jacobian matrix in the case of non-impact is obtained subsequently. In addition, for various impacting patterns, a combination of inhomogeneous equations and inequations is obtained to determine the existence of period orbits after grazing, with the stability criterion of the grazing-induced periodic orbits given. Numerical results verify the effectiveness of theoretical analysis. What is more, we also give a conjecture about the relationship between eigenvalues and the type of periodic orbits when eigenvalues are imaginary numbers.