Analysis on Nonlinear Coupled Vibrations of a Clamped Beam with a Tip-Mass Including Axial Inertia

Abstract

Analytical results are presented on nonlinear coupled vibrations of a clamped beam with a tip-mass constrained by an axial spring. First, governing equations of the thin beam are derived including both effects of the axial inertia force and the geometrical nonlinearity of the beam. The modified Galerkin procedure is applied to the governing equation by introducing a coordinate function for the axial displacement considering the quadratic nonlinear coupling with the deflection. Nonlinear periodic responses are calculated with the harmonic balance method. Frequency responses of the principal resonance of the fundamental vibration mode are compared by changing the mass attached to the beam end. When the mass is not attached to the beam, the frequency response agrees well with the result neglecting the axial inertia. As the mass is increased, the response curve in comparatively large amplitude shifts to the lower frequency range, owing to the axial inertia of the tip-mass. The analytical result of the frequency response, taking account of the axial inertia, agrees well with the relevant experimental result of a post-buckled beam formerly presented, which verifies our analytical results.