Lifting-Line Analysis of Roll Control and Variable Twist

Abstract

A more practical form of an analytical solution that can be used to predict the roll response for a wing of arbitrary planform with arbitrary spanwise variation of control surface deflection and wing twist is presented. This infinite series solution is based on Prandtl 's classical lifting-line theory and the Fourier coefficients are presented in a form that only depends on wing geometry. The solution can be used to predict rolling and yawing moments as well as the lift and induced drag, which result from control surface deflection, rolling rate, and wing twist. The analytical solution can be applied to wings with conventional ailerons or to wings utilizing wing-warping control. The method is also applied to full-span twisting control surfaces, named twisterons, which can be simultaneously used to provide roll control, high-lift, and minimum induced drag. A closed-form solution for optimum twist in a wing with linear taper is also presented. Nomenclature An = coefficients in the infinite series solution to the lifting-line equation an = planform contribution to the coefficients in the infinite series solution to the lifting-line equation b =w ingspan bn = twist contribution to the coefficients in the infinite series solution to the lifting-line equation Di C = induced drag coefficient L C = lift coefficient α , L C = wing lift slope α , ~ L C = airfoil section lift slope f L C δ , = change in wing lift coefficient with respect to flap deflection t L C δ , = change in wing lift coefficient with respect to twisteron deflection C = rolling moment coefficient p C , = change in rolling moment coefficient with respect to dimensionless rolling rate δ , C = change in rolling moment coefficient with respect to control surface deflection m