The present paper deals with two different analytical methods to a system with inertia and static type cubic nonlinearities. The free vibration of a mass grounded by linear and nonlinear springs in series is modeled, and a comparison between two analytical techniques is studied to obtain the periodic solutions. These methods are: energy balance method and coupled homotopy variational approac. Moreover, higher order solutions are investigated for energy balance method and coupled homotopy variational approach. The periodic solutions are analytically verified, and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. Comparison of the present solutions is made with the exact solution and excellent agreement is noted. Consequently, the proposed analytical methods can be used as efficient tools 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.

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