Application of the Homotopy Analysis Method for Determining the free Vibrations of Beam

In this paper we present the application of the homotopy analysis method for determining the free vibrations of the simply supported beam. The linear and nonlinear cases are considered.

The homotopy analysis gives the possibility to search for the solution of a wide range of problems described by means of the operator equations. The numerical examples are presented to confirm the exactness and fast convergence of the introduced method. The presented computational examples confirm the precision and the fast convergence of investigated method. In the linear case we knew the exact solutions, so we could compare with them the solutions obtained with the aid of homotopy analysis method. Differences between the obtained solutions were slight. However, the advantage of the examined method is that we receive here the approximate solution in the form of continuous function which can be used then in a further analysis or to perform various simulations.