Limit cycle analysis of nonsmooth aeroelastic system of an airfoil by extended variational iteration method

The variational iteration method (VIM) was extended to solve the aeroelastic system of an airfoil with both cubic and nonsmooth stiffnesses (a piecewise freeplay), expressed by integral-differential equations. Limit cycle (LC) frequency and amplitude are unknown to be updated at every iteration stage. If the VIM is routinely applied, the LC solution needs to be embedded into both the integrals and the freeplay, which can result in two intractable problems: (1) the integrals lead to coupled terms between frequency and amplitude and (2) the Fourier expansion of the freeplay cannot be realized. In order to tackle these problems, the solution obtained at the previous iteration rather than the current unknown one is applied to the freeplay and the integrals. By this means, the frequency and amplitude are decoupled, and the freeplay can be expanded as Fourier series. Numerical examples show that very accurate LC solutions can be obtained by the presented method. This method is suitable for solving LCs of nonlinear dynamical systems with nonsmooth nonlinearities.