Homotopy Analysis Method for Limit Cycle Oscillations of an Airfoil with Cubic Nonlinearities

An analytical approximate technique for nonlinear problems, namely the homotopy analysis method, is employed to propose an approach for the aeroelastic system of a two-dimensional airfoil with a cubic nonlinearity. The frequency and amplitude of the limit cycle oscillation are expanded as power series of an embedding parameter. A series of algebraic equations governing the coefficients of the series are then derived. All the equations are linear except the first one. This provides us with a simple iteration scheme to seek high-order approximations. The frequency and amplitude of the limit cycle oscillation are obtained with a high degree of accuracy. It turns out that the frequency is independent of the coefficient of the cubic nonlinearity, and that the amplitude is in inverse proportion to the square root of this coefficient.