Lagrange-type formulation for finite element analysis of non-linear beam vibrations

Abstract

A Lagrange-type formulation for finite element analysis of non-linear vibrations of immovably supported beams is presented. Two equations of motion coupled in axial and transverse displacements are derived by using Lagrange's equations. By neglecting the in-plane inertial effects, these equations are written in terms of the transverse displacement alone. Upon defining certain properties for the non-linear oscillatory behaviour of the transverse displacement, the governing equation is reduced to an equation in space alone from which the eigenvalue-like quantity is computed. The governing equation is solved in two ways. A direct iteration technique is used in the first method to compute a numerically exact mode shape and the corresponding frequency. A Rayleigh quotient type of formulation, similar to linear vibration analysis, is used in the second approach to evaluate the frequency of vibration for a fundamental mode which is determined from a linear FEM model and is maintained constant at all amplitudes. Numerical results are compared with available results and they corroborate the observations of earlier research workers.