Application of the variational iteration method for nonlinear free vibration of conservative oscillators

In this paper, an analytical solution for the nonlinear free vibration of a conservative oscillator is presented. The nonlinear governing equation is solved by employing the Variational Iteration Method (VIM). This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a function. Obtained results reveal that the proposed method is very effective, simple and exact. In the present investigation, the results of this method are compared with those of the Homotopy Analysis Method (HAM), as well as those predicted by the Runge–Kutta method. The excellent accuracy of the obtained results is demonstrated by comparing them with available analytical and numerical results available in the literature. Furthermore, the numerical results for different particular cases of the problem are presented. The effects of different parameters on the ratio of nonlinear to linear natural frequency of the system are also studied. Consequently, the proposed analytical solution can be used as an efficient tool to study the effects of the material or geometrical parameters in the modeling of devices consisting of nonlinear conservative oscillators for their design and optimization, which requires a large number of simulations.