Singularities of aerodynamic transfer functions calculated on the basis of an unsteady lifting surface model in subsonic flow

Abstract

A direct method to calculate the Laplace transformed pressure distribution on subsonic lifting surfaces is considered. The kernel function is analytically continued in the entire p-plane (of the non-dimensional Laplace variable), and the discretizing procedure follows the lifting lines (or doublet-lattice) method developed for simple harmonic motion. The aerodynamic influence coefficient matrix is a function of Mach number M and the complex variable p. In the first part of the paper, some analytical properties of this matrix were investigated on the basis of numerical calculations performed for an aspect-ratio-3 rectangular wing. The main conclusion of this paper is that for M ≠ 0, there exist a large (probably infinite) set of latent roots of the matrix in the left half of the p-plane which (usually) reflect in poles of the transfer functions. For M → 1, all latent roots tend to the origin (p = 0). For M → 0, all latent roots move to infinity and probably, for M = 0 there are no roots in the finite part of the p-plane. The distribution of latent roots in the p-plane does not depend on the number of aerodynamic elements introduced by the discretization (within the limits of accuracy of the calculation method). The algebraic equations are well-conditioned in the right half of the p-plane and in a strip parallel to the imaginary axis in left half of the p-plane. The width of this strip depends on the Mach number. In the second part of the paper, an approximation to the aerodynamic transfer functions based on the identified singularities and calculated left and right-hand latent vectors of the aerodynamic influence coefficients matrix is developed. It avoids the ill-posed analytical continuation from the imaginary axis in the whole p-plane. The results clarify also some unexpected phenomena observed in Laplace-domain calculations, and described in the literature.