Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

Abstract

A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.