A Study on Non-linear Free Vibration of Cantilever Beams

Non-linear flexural vibration of beams with various boundary conditions such as hinged–hinged, clamped–clamped and clamped–hinged has been extensively studied using analytical approaches and numerical methods like FEM. A detailed review has been brought out by Rosenberg [1] and Satyamoorthy [2]. However, the investigation pertaining to the cantilever case, which is also of practical interest, is treated sparsely in the literature [3–9]. All these works are based on continuum approaches wherein either linear mode shape or truncation in series solution has been introduced for analysing the nonlinear vibration characteristics of the cantilever beams, except references [7–9]. This, in turn, may lead to underestimation of energy terms which may affect the actual prediction of type/degree of non-linearity. Furthermore, there is not much information available in the literature on how the changes in mode shapes at different amplitudes affect the type/degree of non-linearity. The FEM has been established as a powerful numerical tool that provides solutions to many complicated engineering problems and does not involve an a priori assumption of the actual nature of the mode shape. Analysis of large amplitude/non-linear vibration behaviour of cantilever beams by the finite element method appears to be lacking in the literature. Owing to the difficulty associated with obtaining accurate analytical solution to the cantilever problems due to the existence of non-linearity in both inertia and curvature terms, modelling and analysis of such a problem by the FEM become important. Hence, the study reported here has been carried out using the finite element approach.