Analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol-Duffing oscillators by extended homotopy analysis method

In this study, the extended homotopy analysis method (EHAM) is applied to derive the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled oscillators. The present paper not only introduces the rationale for the EHAM for MDOF oscillators, but also strengthens the availability of the conventional homotopy analysis method (HAM) in solving complex MDOF dynamical systems. Employing the EHAM for the two-degree-of-freedom (TDOF) coupled van der Pol-Duffing oscillator, the explicit analytical solutions of frequency ω and displacements x1(t) and x2(t) are formulated for various initial conditions and physical parameters. To verify the accuracy and correctness of this approach, a number of comparisons are conducted between the results of the EHAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the third-order analytical solutions of the EHAM agree well with the numerical integration solutions, even if the time variable t progresses to a comparatively large domain in the time history responses.