Accurate approximate analytical solutions for multi-degree-of-freedom coupled van der Pol-Duffing oscillators by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) nonlinear coupled oscillators. The periodic solutions for the three-degree-of-freedom (3DOF) coupled van der Pol-Duffing oscillators are applied to illustrate the validity and great potential of this method. For given physical parameters of nonlinear systems and with different initial conditions, the frequency ω , displacements x 1 ( t ) , x 2 ( t ) and x 3 ( t ) can be explicitly obtained. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge–Kutta) method. It is shown that the analytical solutions of the HAM are in excellent agreement with respect to the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses. Finally, the homotopy Pade technique is used to accelerate the convergence of the solutions.