Global dynamics for impacting cantilever beam supported by oblique springs

Abstract

The chaos and subharmonic bifurcation of a cantilever beam supported by oblique springs under bilateral asymmetric rigid constraints are investigated in this paper. It is difficult to investigate analytically the chaos and subharmonic bifurcation of the system because the stiffness term of the oblique spring support structure is a transcendental function. Firstly, the stiffness term of the system is fitted by the approximation method, and the homoclinic orbit and its internal orbits of the approximate system are compared with the orbits of the original system. Secondly, the threshold conditions for subharmonic bifurcation and homoclinic chaos are presented by applying the Melnikov method to the non-smooth impacting cantilever beam system. Moreover, the stability of impacting orbits is analyzed by combining characteristic multipliers of smooth manifolds with impact function, and the relationship between subharmonic bifurcation and chaos is investigated. Finally, the effects of damping, excitation frequency, excitation amplitude and impact coefficient of restitution on chaos and subharmonic bifurcation are investigated based on threshold conditions, which further verifies the theoretical analysis.