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.
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