Bifurcation of nonlinear normal modes of a cantilever beam under harmonic excitation

Abstract

Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two degrees-of-freedom nonlinear model for the vibration in the transverse direction is obtained by the discretization of the governing equation using Galerkins method based on the eigenmodes in each direction. The method of multiple scales is used to derive two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of the dominant mode for the case of 1:1 internal resonance. The bifurcation diagrams are computed considering the frequency of excitation and the magnitude of the excitation as the control parameters. The stability of the fixed point is determined by examining the eigenvalues of the Jacobian matrix. The results show that a saddle-node-type bifurcation of the solution can occur under certain parameter conditions.