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Applications of He's Homotopy Perturbation Method to Obtain Second-order Approximations of the Coupled Two-Degree-of-Freedom Systems

In this paper we apply a Modified He's Homotopy Perturbation Method (MHHPM) as a strong method to solve accurately higher-order approximate analytical solutions of coupled two-degree-of-freedom (CTDOF), which is described with a system of two coupled strong non-linear differential equations. MHHPM is modified version of He's HPM by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution into the second-order linear differential equation. For the first approximation we achieve a high accuracy of solutions with a maximal relative error less than 3.14 % for small and large values of oscillation amplitude, while for the second iteration the fabulous relative error is 0.064 %. It is predicted that MHHPM can be found widely applicable in engineering which needs extremely accurate solutions.