Communicated by CashThe scope of this paper is evaluating an oscillation system with nonlinearities, using a periodic solution called amplitudefrequency formulation, such as the motion of a rigid rod rocking back. The approach proposes a choice to overcome the difficulty of computing the periodic behavior of the oscillation problems in engineering. We are to compare the solutions results of this method with the exact ones in order to validate the approach and assess the accuracy of the solutions. This method has a distinguished feature, which makes it simple to use and agree with the exact solutions for various parameters. Moreover, it is perceived that with one-step iteration high accuracy of the solution will be achieved. We may apply the results of the solution to explain some of the practical physical problems.

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Single degree of freedom (SDOF) Two degrees of freedom (TDOF) Max-Min approach (MMA) a b s t r a c tMax-Min Approach (MMA) is applied to obtain an approximate solution of three practical cases in terms of a nonlinear oscillation system. After finding maximal and minimal solution thresholds of a nonlinear problem, an approximate solution of the nonlinear equation can be easily achieved using He Chengtian's interpolation. Numerical results indicate the effectiveness of the proposed method both in respect of the whole range of involved parameters as well as the excellent agreement with the approximate frequencies and periodic solutions with the exact ones. It is predicted that MMA can be found widely applicable in engineering.

This paper applies He's Energy balance method (EBM) to study periodic solutions of strongly nonlinear systems such as nonlinear vibrations and oscillations. The method is applied to two nonlinear differential equations. Some examples are given to illustrate the effectiveness and convenience of the method. The results are compared with the exact solution and the comparison showed a proper accuracy of this method. The method can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other science.

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Abstract-This paper deals with Approximate Analytical Solutions to nonlinear oscillations of a conservative, non-natural, single-degreeof-freedom system with odd nonlinearity. By extending the Variational approach proposed by He, we established approximate analytical formulas for the period and periodic solution.To illustrate the applicability and accuracy of the method, two examples are presented: (i) the motion of a rigid rod rocking back and forth on the circular surface without slipping, and (ii) Cubic-Quintic Duffing Oscillators. Comparison of the result which is obtained by this method with the obtained result by the Exact solution reveals that the He's Variational approach is very effective and convenient and can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other sciences.

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