Asymptotic analysis of aircraft wing model in subsonic air flow

Abstract

This paper is the first in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works of Balakrishnan. The model is governed by a system of two coupled integro‐differential equations and a two parameter family of boundary conditions modelling the action of the self‐straining actuators. The unknown functions (the bending and the torsion angle) 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 convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution type equation in the state space of the system equipped with the so‐called energy metric. The Laplace transform of the solution of this equation can be represented in terms of the so‐called generalized resolvent operator. The generalized resolvent operator is an operator‐valued function of the spectral parameter. This generalized resolvent operator is a finite meromorphic function defined on the complex plane having the branch cut along the negative real semi‐axis. The poles of the generalized resolvent are precisely the aeroelastic modes, and the residues at these poles are the projectors on the generalized eigenspaces. In this paper, our main object of interest is the dynamics generator of the differential parts of the system. It is a non‐selfadjoint operator in the state space with a pure discrete spectrum. In the present paper, we show that the spectrum consists of two branches, and we derive their precise spectral asymptotics. Based on these results, in the next paper we will derive the asymptotics of the aeroelastic modes and approximations for the mode shapes.