Improved exponential time series approximation of unsteady aerodynamic operators

A procedure has been developed for approximating unsteady aerodynamic operators as truncated exponential series in the time domain. The approximation is accomplished using a least squares minimization fit to aerodynamic data in the frequency domain. The procedure extends previous methods by including the pole locations as unknown parameters of the least squares minimization. In addition, the error associated with both the real and imaginary parts of the Fourier transform of the approximation is minimized. A Newton-Raphson search algorithm is used to find the minimum of the weighted square error in the parameter space of the approximation while constraining the poles to be in the left half-plane. By freely varying the poles of the approximation during the numerical least squares minimization, the representation of the unsteady aerodynamics is improved and is comparable to existing higher-order Fade approximations. Hence, the method offers the aeroelastic designer a more direct method of finding approximate aerodynamic states. However, because the minima of the square error in the cost function found are not necessarily global and depend on the number of poles in the approximation, the initial trial minimum, and the details of the cost minimization algorithm, the poles found in the search do not necessarily correspond to the theoretical poles of the aerodynamic transfer function. Example exponential time series approximations of the Theodorsen function are presented and compared with a Fade approximation and other exponential time series approximations. Nomenclature A = curvature matrix of the cost function J A =true aerodynamic impulse response spectrum A' — approximate aerodynamic impulse response spectrum a - coefficient defined in Eq. (14) an = coefficients of the approximation bn - pole locations of the approximation c = cost minimization step length C = Theodorsen circulation function d - cost minimization search direction vector F = real part of A F' = real part of A' g = gradient of the cost function / G = imaginary part of A G' = imaginary part of A' J = weighted square error cost function k = reduced frequency ^max - frequency corresponding to the peak in IGI M = number of data points to be fitted N — number of terms in the series t = npndimensional time s — Laplace domain variable x — parameter vector of the coefficients and poles am = weighting factors for the real part of the approximation error 0m = weighting factors for the imaginary part of the approximation error = true step response ' = approximate step response