Some mathematical methods in three-dimensional subsonic flutter-derivative theory

Abstract

In this report are given some of the methods used in three-dimensional subsonic flutter-derivative theory. It is hoped that it will provide a basis for the reading of specific papers on the subject. Two proofs are given of the basic integral equation: the first proof uses co-ordinates fixed in space, the second uses co-ordinates fixed in the wing. Analytical solutions of the integral equation can be found for very few planforms and in these particular cases the problem is more easily solved by starting from the differential equation. Nevertheless, for the sake of completeness and to be consistent with the rest of the report a method is given for obtaining an analytical solution of the integral equation. The rest of the report deals with the numerical solution of the integral equation. Since Gaussian integration is fundamental to the method of solution it is explained in some detail. Two methods of chordwise integration and two methods of spanwise integration are applied to the solution of the integral equation. Computational details are not given since they vary from author to author and a reader interested in a particular variant of the general method of solution is referred to the relevant papers. Several methods exist for eyaluating incompressible derivatives but these have now been superseded by the methods used for compressible flow and so no account of them is given. No attempt has been made to give an historical survey of the subject. The references give the places where results may conveniently be found; they are not necessarily original papers.