Investigating the Solutions of Two Classical Nonlinear Oscillators by the AG Method

In this article a simple solution for analyzing some classical nonlinear oscillators are presented and discussed. The proposed method is named AG method which is of applicability to many nonlinear problems in physical and applied sciences. The problems under consideration include Gaylord's oscillator as well as an oscillating mass joined to two springs in series arrangement such that one of the springs is assumed to have an elastic constant with nonlinear property. The Gaylord's oscillator is composed of a rigid and relatively long rod vibrating on a circular surface with certain radius. The different methods such as variational iteration method, energy balance method, and parameterized perturbation method which have been utilized to find the approximate solutions of nonlinear problems are reviewed and investigated at first. Then the algorithm of the AG method which has not already been applied to this oscillator is described and the obtained results are compared with those produced by mentioned conventional methods. The solution of the problem is derived in a closed form and finally is validated by the exact and numerical solutions. Angular displacement and velocity curves in terms of time are calculated and illustrated for special cases of selected length of the oscillating rod and also the radius of circular surface. The other nonlinear problem includes a mass attached to two springs with series arrangement which is of many applications in analyzing mechanical structures with oscillatory nature. AG method is applied to find uniform solutions with desired accuracy to this oscillator, as well. It is clearly found that the employed method is very efficient and needs simple processes to find approximate solutions with acceptable accuracy to nonlinear differential equations. Using this method, we can avoid complicated solving procedures in addition to a huge amount of mathematical manipulations such as exhaustive integrations and successive mathematical operations. Finally, we will find that the AG method is of relative accuracy plus simplicity with respect to the other methods in analyzing the behavior of oscillators with strong nonlinearities in the terms including conservative restoring forces.