Periodic solutions of multi-degree-of-freedom strongly nonlinear coupled van der Pol oscillators by homotopy analysis method

Abstract

The mathematical models of multi-degree-of-freedom (MDOF) strongly nonlinear dynamical systems are described by coupled second-order differential equations. In general, the exact solutions of MDOF strongly nonlinear dynamical systems are frequently unavailable. Therefore, efforts have been mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic tool for solving strongly nonlinear dynamical systems, and it provides a simple way to ensure the convergence of solution series by means of a convergence-control parameter \({\hbar}\). Unlike the classical perturbation techniques, this method is independent of the presence of small parameters in the governing equations of motion. In this paper, the HAM is applied to formulate the analytical approximate periodic solutions of MDOF strongly nonlinear coupled van der Pol oscillators. Within this research framework, the frequency and the displacements of two-degree-of-freedom (2-DOF) strongly nonlinear systems can be explicitly obtained. For authentication, comparisons are carried out between the results obtained by the homotopy analysis and numerical integration methods. It is shown that the fourth-order or eighth-order solutions of the present method provide excellent accuracy. Illustrative examples of three-degree-of-freedom (3-DOF) strongly nonlinear coupled van der Pol oscillators are also presented and discussed. Finally, the optimal HAM approach is used to accelerate the convergence of the solutions.