Marianna Shubov
(Department of Mathematics,
Texas Tech University)
Asymptotic and Spectral Analysis of Aircraft Wing Model in
Subsonic Airflow
Abstract
The aircraft wing model, which will be discussed in this talk, has been developed
in the Flight Systems Research Center of UCLA in collaboration with NASA
Dryden Flight Systems Center. The mathematical formulation of this model
has been originally presented in the works by A.V. Balakrishnan. The model
has been recently tested in a series of flight experiments at Edwards Airforce
Base, CA. The experimental results have shown very good agreement with the
theoretical predictions of the model for at least several lowest aeroelastic
modes. The objective of the entire wing modeling project is to analyze the
flutter phenomenon in aircraft wing in order to control/suppress flutter
by using self—straining actuators.
In this talk, I will present the results of my six recent works and of a
joint paper with A.V.Balakrishnan devoted to the mathematical analysis of
the model. The model is governed by a system of two coupled linear integro
– differential equations and a two-parameter family of boundary conditions
modeling the action of self--straining actuators. The differential parts
of the equations of motion form a coupled linear hyperbolic system; the integral
parts are of the convolution type. The aforementioned system is equivalent
to a single operator evolution – convolution equation in the state space
equipped with the energy metric. The Laplace transform of the solution of
this equation can be represented in terms of the so-called generalized resolvent
operator, which is an operator – valued function of the spectral parameter.
This generalized resolvent operator is a finite – meromorphic function on
the complex plane having a branch – cut along the negative real semi – axis.
Its poles are precisely the aeroelastic modes and the residues at these poles
are the generalized projectors on the corresponding eigenspaces.
I will describe the following results:
- Asymptotics of the eigenvalues and eigenvectors of the
nonselfadjoint operator, which is a dynamics generator of the differential
part of the model.
- Riesz basis property of the generalized eigenvectors
of the above operator in the state space.
- Asymptotics of aeroelastic modes and mode shapes for
the entire model.
- Riesz basis property of the mode shapes.
- Application of the results of asymptotic and spectral
analysis to the representation of the solution to the main initial – boundary
value problem in the space – time domain.
- The problem of flutter control will be discussed.
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