Minimum-Weight Design of an Orthotropic Shear Panel with Fixed Flutter Speed

Abstract

THIS paper deals with the weight minimization of rectangular flat panels placed in a high supersonic flowfield and subject to a flutter speed constraint. In the establishing of the structural operator a pure transverse shear plate model was used, which may, be considered as a complement of the Love-Kirchhoff type model. By using the theory of optimal control of distributed parameter systems, necessary conditions for the minimum-weight panel are derived. These are supplemented with a condition ensuring that the flutter speed of the optimal panel coincide with the prescribed one. It is shown that the optimal thickness distribution is symmetrical with respect to the panel midpoint. Numerical rough estimates obtained via Galerkin's method are presented. Contents The field of weight minimization of panels subjected to aeroelastic constraints has been investigated thoroughly during the past decade, as it may be inferred from the specialized literature. Throughout these investigations, whether dealing with one-dimensional (see Refs. 1-4) or twodimensional aeroelastic optimization problems,5-8 the appropriate structural operator was established on the basis of the Love-Kirchhoff type model. As it is known, this model involves the ab-initio disregard of transverse shear effects. In contrast to this approach, a somewhat opposite structural type model is used here, in which the rigidities in transverse shear are considered as finite, and in bending as negligible (such a panel will be termed a pure transverse shear panel). This model—first introduced by Armand 9—is practically motivated by the advent of new composite materials that enjoy exotic properties. A generalized form of this structural model, including transverse shear orthotropicity effects, will be used here for approaching the present aeroelastic optimization problem. The structure to be analyzed consists of an elastic, rectangular flat thin panel (axb) of nonuniform thickness h = h(x]yx2), where OxjX2 denotes the in-plane coordinate system (Ox2 is the stream wise coordinate, while Ox2—the span wise one—coincides with the panel leading edge). The panel is exposed to a high supersonic gas flow over its upper face. The aeroelastic optimization problem dealt with here consists of finding the thickness distribution which minimizes the panel weight, while maintaining the same flutter speed as that of a uniform-thickness reference panel. As usual, 8 the reference panel is defined as the panel of uniform thickness H0 having the same ofthbtropy characteristics and boundary conditions as its counterpart of nonuniform thickness.