Homotopy analysis method for limit cycle flutter of airfoils

An analytical approximate method for non-linear problems, namely the homotopy analysis method, is for the first time employed to propose a new approach for the flutter system of a two-dimensional airfoil with a cubic structural nonlinearity. The frequencies and amplitudes of limit cycle flutters are described as Maclaurin series of an embedding parameter. A series of algebraic equations in the priori unknown coefficients of the series are derived. All the equations are linear except the first one. This provides us with a simple iteration scheme to seek the solutions to any desired accuracy. Numerical examples are presented to illustrate the validity and efficiency of the proposed approach. The approximations of the frequencies and amplitudes of the limit cycles are obtained very accurately. Furthermore, it is proved the frequencies are independent of the coefficient of the cubic nonlinearity, while the amplitudes are in inverse proportion to the square root of the coefficient.