On a Convolution-Evolution Equation in Aeroelasticity

摘要

Flutter is an instability endemic to aircraft wings that occurs at high enough airspeed in subsonic flight, and sets a bound (flutter boundary) on attainable airspeed in the subsonic regime. As is to be expected we need to make many simplifying assumptions before we can obtain a mathematically tractable model. Here we limit ourselves to a long, slender, "high-aspect-ratio" wing (see Figure 1) so that we can simplify the wing model to a cantilever beam with two degrees of freedom - corresponding to plunge and pitch. The aerodynamics is assumed to be inviscid and incompressible, characterized by the small disturbance potential field equations. We simplify the problem further by invoking "typical section" theory, so that we can work with 2D aerodynamics (see Figure 2). In this way we finally arrive at a model describable in terms of a time-dependent boundary value problem for a set of partial differential (wave) equations. We show that the abstract version reduces to a convolution-semigroup equation in a Hilbert space, and the Flutter problem can then be characterized as the problem of determining system stability as a function of the airspeed, but determining the Flutter Boundary analytically is still an open problem, with room for further research. Numerical calculations on typical cases do however establish that the model does capture the known features of flutter. This material has already been reported, most recently in [1]. What is new in this paper is a self-contained mathematical development of the aerodynamic theory leading to the results quoted without proof in [1] in the calculation of the lift and moment. The structure model used derives from that of Goland [2] further elaborated in [3]. In the Aerodynamic part (which is the more difficult compared to the Structure Dynamics), key use is made of the results due to Sohngen and Tricomi [4] in their investigation of the airfoil equation. We begin in Section 2 with the stucture model. Section 3 is devoted to the aerodynamics, and the calculation of the aerodynamic lift and moment. Section 4 assembles the total aeroelastic system and develops the abstract convolution-semigroup equation.