Modified Exponential Series Approximation for the Theodorsen Function

An improved method is developed for the approximation of unsteady aerodynamics in the time domain by a series of decaying exponentials. The new method is different from the previous procedures in that it consistently accounts for the case when the optimum values of the lag parameters in the exponential series are close to one another. This is achieved by introducing a time-weighted exponential series for the repeated pole case. The method uses a nongradient optimizing procedure. Approximations are presented for Theodorsen's lift deficiency function and results are compared with those of a gradient-based method that was published recently. HE representation of the general motion of an aeroelastic structure requires the availability pf the unsteady aerody- namic forces in the time domain. An important feature of these forces is the lag associated with the circulatory wake. Theodorsen1 employed a lift deficiency function in the re- duced frequency domain to represent this effect for the oscilla- tory flow over an airfoil. Jones2 used a two-term series of decaying exponentials in the time domain to approximate the effect of circulation for the transient aeroelastic motion and solved for the linear coefficients of the series by using the Fourier transform to convert the transfer function into the frequency domain where it must equal Theodorsen's circula- tion function. This idea was extended by Do well3 who used an exponential series in the time domain to represent the unsteady aerodynamic transfer function and then transformed it into a rational function in the Laplace domain. In contrast with this approach is the conventional least-squares method4 that be- gins with a suitable approximation in the Laplace domain and then transforms it into the time domain using the inverse Laplace transform. It is to be noted that, although certain effects, such as the aerodynamic damping and the aerody- namic inertia, can be included in the rational function approx- imation in the Laplace domain, they are essentially left out in the exponential series approximation in the time domain. However, this difference between the two methods is not present when only the circulatory effect is being approxi- mated, such as the lift deficiency function of Theodorsen. The accuracy of an exponential series approximation de- pends crucially on the values of the nonlinear lag parameters that occur in the exponentials. Recently, Peter son and Craw- ley5 have shown .that, when these parameters are optimized to give a minimum squared error between the exact and the approximate values of the transfer function, the accuracy of the approximation is greatly enhanced. However, the objec- tive-function minima obtained by Peterson and Crawley5 may not be unique because the gradient-based optimizer is unable to escape the local minima. Also, the frequently encountered cases when the optimum values of two or more lag parameters are nearly the same are mistaken to indicate that the same fit accuracy can be achieved by reducing the number of lag states. Moreover, their inclusion of the numerator as well as the denominator coefficients of the transfer function in the Laplace domain as the free parameters of optimization ap- pears to be a less efficient procedure than the one in which the numerator coefficients are determined by a least-squares fit. The latter option is utilized in the present method, which uses a simplex nongradient optimizer to locate the absolute minima of the objective function. The most significant improvement is achieved by using a consistent optimizing procedure, intro- duced earlier by the authors,6 which correctly accounts for the case of repeated lag parameters by employing a new approxi- mation that contains time-weighted exponentials. The time- weighting functions are polynomials in time of one degree less than the multiplicity of the corresponding lag parameters. By using this series, the repeated pole case not only becomes meaningful but also allows an improvement in the fit accuracy with the frequency domain data. Gradient-Based Method as Compared to the Present Method The present example compares the gradient-based opti- mization scheme of Peterson and Crawley,5 who used the exponential time-series approximation of Dowell3 as the basic transfer function, with the nongradient optimized scheme of the present method employing the same transfer function. For the purpose of comparison, both methods approximate the Theodorsen circulation function.