A fast harmonic balance technique for periodic oscillations of an aeroelastic airfoil

The harmonic balance (HB) method is utilized to obtain the periodic solutions for the two-dimensional airfoil with cubic nonlinearity in pitch undergoing subsonic flow. In the course of formulating the HB algebraic system, the manipulation software Mathematica is employed to deal with the complex Fourier coefficients involved with the nonlinear term. In general, to solve the HB algebraic system, either a symbolic calculation or a numerical approximation of the Jacobian matrix is required in each iteration, which is computationally expensive. To remedy this drawback, the Jacobian matrix is explicitly derived in this paper. The effects of exploiting the explicit Jacobian matrix on the accuracy and efficiency of the HB method are investigated, through comparing with the case using a numerical Jacobian matrix calculated by a three-point difference technique. Moreover, the spectral analysis is applied to the periodic motions by the numerical method to provide insight into the distribution of the dominant frequencies, so as to provide a reasonable suggestion for the truncation of the Fourier series expansion in the HB method. In addition, a frequency modulation phenomenon is identified in the pitch motions via spectral analysis, whose effect on the accuracy of the HB method is examined both numerically and analytically. Finally, illustrative examples validate that the HB method with as many harmonics as the spectral analysis suggests can yield sufficiently accurate solutions.