Semi-analytical Solution for Nonlinear Transverse Vibration Analysis of an Euler-Bernoulli Beam with Multiple Concentrated Masses Using Variational Iteration Method

In this article, linear and nonlinear transverse vibration of Euler-Bernoulli beams with multiple concentrated masses have been investigated using variational iteration method (VIM), and the effects of concentrated mass on natural frequencies and mode shapes have been taken into account as well. VIM is a powerful method with high convergence which gives analytical solution to linear problems and is capable of being extended to present semi-analytical solutions to nonlinear ones. The proper choice of Lagrange's multiplier and Initial Function can increase the convergence speed. In this study, along with presenting the suitable trend for determining these two functions, the obtained frequencies in linear and nonlinear states are compared with those calculated from other methods, and the accuracy and convergence speed of this procedure are examined. It is concluded that with increasing the intensity of concentrated mass, linear natural frequency and the ratio of nonlinear frequency to linear one will decline.