Extended homotopy analysis method for multi-degree-of-freedom non-autonomous nonlinear dynamical systems and its application

In normal circumstances, numerous practical engineering problems are multi-degree-of-freedom (MDOF) nonlinear non-autonomous dynamical systems. Generally, exact solutions for MDOF dynamical systems are hardly obtained; thus, the development of analytical approximations becomes a robust and appealing avenue for an analysis of these systems. The homotopy analysis method (HAM) is one of the analytical methods, which can overcome the foregoing barriers of conventional asymptotic techniques. It has been widely used for solving various nonlinear problems in physical science and engineering. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for MDOF weakly damped non-autonomous dynamical systems. In terms of its flexibility and applicability, the EHAM is also applied to derive the approximate solutions of parametrically and externally excited thin plate systems. Besides, comparisons are performed between the results obtained by the EHAM and the numerical integration (i.e. Runge–Kutta) method. The present findings show that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions.