Galerkin finite element method for non-linear beam vibrations

Abstract

A Galerkin finite element method is presented, for studying non-linear vibrations of beams describable in terms of moderately large bending theory. The transverse displacement term w alone is used, although several previous attempts to do the same with a Ritz element have failed. This, together with certain assumptions regarding the nature of the vibration, allows an eigenvalue-like quantity characteristic of non-linear vibration to be defined and computed for various amplitudes of vibration. The solution to the non-linear eigenvalue problem is effected in two ways. In one, the exact mode shape and the frequency corresponding to the reference amplitude of vibration are determined by solving iteratively a series of eigenvalue problems until the required convergence is obtained. In the second approach, one assumes that the mode shape does not change with the amplitude and, by a virtual work type approach in which the linear mode shape is used as the weighting vector, reduces the problem to that of a single degree of freedom system. The eigenvalue corresponding to the chosen mode is determined in a manner similar to but subtly different from Rayleigh's method. The accuracy and applicability of this approximate method is critically examined. Numerical results are presented to demonstrate that the governing differential equations of the problem do not admit variables separable solutions (time and space) in the clamped-clamped and simply supported-clamped cases.