Weight Minimization for a Thin Straight Wing with a Divergence Speed Constraint

Abstract

For a thin straight wing satisfying a given constraint on the divergence speed (i.e., the speed above which the twist of the wing leads to its failure), the problem of determining an optimal skin thickness distribution that minimizes the skin mass is considered. The mathematical formulation of the problem is as follows: minimize a linear functional over a set of essentially bounded measurable functions for which the smallest eigenvalue of a Sturm–Liouville problem is no less than a preset value. It is proved that this problem has a unique solution. Since only piecewise smooth thickness distributions satisfy the requirements for applications, the regularity of the optimal solution is analyzed. It turns out that the optimal solution is a Lipschitz continuous function. Additionally, it is shown that the solution depends continuously on a parameter determining the lowest possible divergence speed, i.e., the considered problem is well-posed in the sense of Hadamard. Finally, an iteration method for constructing minimizing sequences converging to an optimal solution in Holder spaces is proposed and numerical results are presented and discussed.