Galerkin finite element method for non-linear beam vibrations

This paper develops a Galerid Finite Element to study13; non-linear vibrations of beams under the class of moderately13; large bending theory 'using the transverse displacement term13; 'w' alone whereas several previous attempts to do the same with a Rits - 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 IQ 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, till the required convergence is obtained. The second approach, assumes that the raodeshape does not change with the amplitude and by a virtual work type approach using the linear modeshape as the weighting vector reduces the problem13; to that of a single degree of freedom system* The eigenvalue13; corresponding to the chosen mode Is determined in a manner13; similar to but subtly different from the Reyleigh'a method.13; The accuracy and applicability of this approximate method la critically examined. The paper demonstrates with numerical results that the governing differential equations of the problem does not admit variable separable solutions (time and space) in the case of clamped-clamped, and simply supported-clamped case