Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow

The present paper is devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of the University of California at Los Angeles. 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 self-straining actuators. The differential parts of these equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. 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, which are the main object of interest in the present paper. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-self-adjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it was shown that the spectrum consists of two branches, and the precise spectral asymptotics with respect to the eigenvalue number was derived. The asymptotic approximations for the mode shapes have also been obtained. Based on the asymptotic results, it has been proved that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of the aeroelastic modes is asymptotically close to the discrete spectrum of the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.