Toward a mathematical theory of aeroelasticity

Abstract

This paper initiates a mathematical theory of aeroelasticity centered on the canonical problem of the flutter boundary - an instability endemic to aircraft that limits attainable speed in the subsonic regime. We develop a continuum mathematical model that exhibits the known flutter phenomena and yet is amenable to analysis - non-numeric theory. Thus we model the wing as a cantilever beam and limit the aerodynamics to irrotational, isentropic so that we work with the quasilinear Transonic Small Disturbance Equations with the attached flow and Kutta-Joukowsky boundary conditions. We can obtain a Volterra expansion for the solution showing in particular that the stability is determined by the linearized model consistent with the Hopf Bifurcation Theory. Specializing to linear aerodynamics, the time domain version of the aeroelastic problem is shown to be a convolution-evolution equation in a Hilbert space. The aeroelastic modes are shown to be the eigenvalues of the infinitesimal generator of a semigroup, which models the combined aerostructure state space dynamics. We are also able to define flutter boundary in terms of the "root locus" - the modes as a function of the air speed U. We are able to track the dependence of the flutter boundary on the Mach number - a crucial problem in aeroelasticity - but many problems remain for Mach numbers close to one. The model and theory developed should open the way to Control Design for flutter boundary expansion. © 2005 by International Federation for Infonnation Processing. All rights reserved.