Variational iteration method for free vibration analysis of a Timoshenko beam under various boundary conditions

In this paper, a relatively new method, namely variational iteration method (VIM), is developed for free vibration analysis of a Timoshenko beam with different boundary conditions. In the VIM, an appropriate Lagrange multiplier is first chosen according to order of the governing differential equation of the boundary value problem, and then an iteration process is used till the desired accuracy is achieved. Solution of VIM for natural frequencies and mode shapes of a Timoshenko beam is compared to the available exact closed-form solution and numerical results of differential quadrature method (DQM). The accuracy of VIM is approximately the same as exact solution and much better than the DQM for solving the free vibration of a Timoshenko beam. Also, convergence speed and simplicity of this method is more than the other two methods because it works with polynomial at the first iteration. Thus, VIM can be used for solving the complicate engineering problems which do not have analytical solution.