Global Bifurcations and Multipulse-Type Chaotic Dynamics in the Interactions of Two Flexural Modes of a Cantilever Beam

Abstract

Global bifurcations and multipulse-type chaotic dynamics in the interactions of two flexural modes 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’s procedure based on the first flexural mode in each direction is applied to the partial differential governing equation to obtain a two-degree-of-freedom non-autonomous nonlinear system. The resonant case considered here is one-to-one internal resonance, principal parametric resonance-1/2 subharmonic resonance. The method of multiple scales is used to derive four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of two interacting modes. After transforming the modulations equations into a suitable form, the extended Melnikov method is employed to show the existence of chaotic dynamics by identifying Silnikov-type multipulse jumping orbits in the perturbed phase space. We are able to obtain the explicit restrictions on the damping, forcing, and the detuning parameters, under which multipulse-type chaotic dynamics is to be expected. Physically, such jumping means sudden, large-amplitude departures of the beam from its planar oscillations. Numerical simulations indicate that there chaotic responses and jumping phenomenon in the nonlinear nonplanar oscillations of the cantilever beam.