An application of Flax's variational principle to lifting-surface theory

Abstract

The loading on a harmonically oscillating wing is represented by a linear combination of given functions which satisfy the edge conditions. The coefficients in this linear combination are determined, by an application of the variational principle due to Flax, so that the required generalised airforces acting on the wing are obtained to the greatest accuracy possible. For the particular case of subsonic flow, it is shown that, when certain numerical integration techniques are used, the results reduce to those obtained from a normal collocation procedure for lifting-surface theory. The procedure using the variational principle is shown to be superior to one which obtains the coefficients in the loading expression by minimising the integral of the square of the difference between the actual and calculated upwashes on the wing surface. Illustrative examples in two-dimensional incompressible oscillatory flow are given.