Transverse Vibration Analysis of Euler-Bernoulli Beams Using Analytical Approximate Techniques

Abstract

The vibration problems of uniform and nonuniform Euler-Bernoulli beams have been solved analytically or approximately [1-5] for various end conditions. In order to calculate fundamental natural frequencies and related mode shapes, well known variational techniques such as Rayleigh_Ritz and Galerkin methods have been applied in the past. Besides these techniques, some discretized numerical methods were also applied to beam vibration analysis successfully. Recently, by the emergence of new and innovative semi analytical approximation methods, research on this subject has gained momentum. Among these studies, Liu and Gurram [6] used He’s Variational Iteration Method to analyze the free vibration of an Euler-Bernoulli beam under various supporting conditions. Similarly, Lai et al [7] used Adomian Decomposition Method (ADM) as an innovative eigenvalue solver for free vibration of Euler-Bernoulli beam again under various supporting conditions. By doing some mathematical elaborations on the method, the authors obtained ith natural frequencies and modes shapes one at a time. Hsu et al. [8] again used Modified Adomian Decomposition Method to solve free vibration of non-uniform Euler-Bernoulli beams with general elastically end conditions. Ozgumus and Kaya [9] used a new analytical approximation method namely Differential Transforms Method to analyze flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam. Hsu et al. [10] also used Modified Adomian Decomposition Method, a new analytical approximation method, to solve eigenvalue problem for free vibration of uniform Timoshenko beams. Ho and Chen [11] studied the problem of free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using Differential Transform Method. Another researcher, Register [12] found a general expression for the modal frequencies of a beam with symmetric spring boundary conditions. In addition, Wang [13] studied the dynamic analysis of generally supported beam. Yieh [14] determined the natural frequencies and natural