The Lifting Line Equation for a Curved Wing in Oscillatory Motion

An unsteady linear lifting line method for the determination of the circulation and lift distribution along the span of a curved wing subject to harmonic small amplitude oscillations is presented. The method relies on the Pistolesi-Weissinger 3/4-chord steady lifting line theory and couples it to the unsteady theory developed by Possio for the motion of lifting surfaces. It leads to an integro-differential equation of a modified Prandtl's type, where the unknown is the circulation. This equation has been carefully analysed in order to evidence all the singularities and to treat them in the most convenient way. The numerical procedure consists of a gaussian quadrature technique based on Chebyshev's polynomial approximation of the unknown function. The method has been appraised through the comparison of a number of solutions, pertaining to different wing configurations, with existing solutions based on lifting surface theory