Analysis of Nonlinear Oscillation Systems using He’s Variational Approach

This paper applies a new variational approach proposed by Ji-Huan He for nonlinear oscillators. Two examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy. The method can be easily extended to other strongly nonlinear oscillations and it can be found widely applicable in engineering and science. 1. Introduction Nonlinear oscillations systems are such phenomena that mostly occur nonlinearly; hence solving of governing equations have been one of the most time-consuming and difficult affairs among researchers of vibrations. Therefore, many researchers and scientist of both vibrations and mathematics have recently paid much attention to find and develop approximate solutions. If there is no small parameter in the equation, the traditional perturbation methods cannot be applied directly. Recently, considerable attention has been directed towards the analytical solutions for nonlinear equations without possible small parameters. The traditional perturbation methods have many shortcomings, and they are not valid for strongly nonlinear equations. To overcome the shortcomings, many new techniques have appeared in open literature [1―5]. Variational methods have been, and continue to be, popular tools for nonlinear analysis. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial-functions. Recently, some approximate variational methods, including approximate energy method [6, 7] and variational iteration method [8–12], to soliton solution, bifurcation, limit cycle and period solutions of nonlinear equations have been given much attention. The approximate energy approach [6] can be applied not only to weakly nonlinear equations, but also to strongly nonlinear ones. The so obtained results are valid for the whole solution domain [7]. Variational iteration method is based on a general Lagrange multiplier, and it can be applied to various nonlinear equations [13, 14]. In the present paper, we use a new variational method proposed by J.H. He for nonlinear oscillators [15].