Asymptotic, spectral, and numerical analysis of an aircraft wing model in subsonic airflow
Martin, Cynthia A
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In this dissertation, we carry out asymptotic, spectral and numerical analysis of an aircraft wing model in subsonic air flow. The wing is modeled as a finite length beam, which can bend and twist. The model is governed by a system of two coupled partial integro-differential equations and a two parameter family of physically meaningful boundary conditions modeling the action of the self-straining actuators. The unknown functions depend on time and one spatial variable. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type and represent the generalized forces and moments exerted on the wing due to the air flow. The system of equations of motion is equivalent to a single operator evolution convolution equation in the state space of the system, which is a Hilbert space equipped with the so-called energy metric. The Laplace transform of the solution can be represented in terms of the generalized resolvent operator, which is an analytic operator-valued function of the spectral parameter. This operator is a finite-meromorphic function with infinitely many poles, which is defined on the complex plane having the branch cut along the negative real semi-axis. The poles of the generalized resolvent are called the aeroelastic modes and they are precisely the quantities that can be measured experimentally. The residues at these poles are the projectors on the generalized eigenspaces. In this dissertation, our main object of interest is the dynamics generator of the differential part of the system. It is a nonselfadjoint operator in the state space with a pure discrete spectrum. We show that the spectrum consists of two branches, and derive the precise spectral asymptotics of those branches. The importance of the information on the spectrum of the differential part is related to the fact that the two sets of complex points, i.e.. the set of the aeroelastic modes and the set of the eigenvalues of the differential part of the model, are asymptotically close. This means that we have derived the asymptotic distribution of the aeroelatic modes, which is of importance for aircraft engineers.