Global Bifurcation and Multipulse Type Chaotic Dynamics in the Nonlinear Nonplanar Oscillations of a Cantilever Beam

Abstract

The global bifurcation and multipulse type chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam are studied using the extended Melnikov method. The cantilever beam studied is subjected to a harmonic axial excitation and transverse excitations at the free end. After the governing nonlinear equations of nonplanar motion with parametric and external excitations are given, the Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 1:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance. By using the method of multiple scales the averaged equations are obtained. After transforming the averaged equation in a suitable form, the extended Melnikov method is employed to show the existence of chaotic dynamics by identifying Shilnikov-type multipulse orbits in the perturbed phase space. We are able to obtain the explicit restriction conditions on the damping, forcing, and the detuning parameters, under which multipulse-type chaotic dynamics is to be expected. Numerical simulations indicate that there exist different forms of the chaotic responses and jumping phenomena in the nonlinear nonplanar oscillations of the cantilever beam.