Riesz basis property of mode shapes for aircraft wing model (subsonic case)

Abstract

The present paper is devoted to the Riesz basis property of the mode shapes for an aircraft wing model in an inviscid subsonic airflow. The model has been developed in the Flight Systems Research Center of the University of California at Los Angeles in collaboration with NASA Dryden Flight Research Center. The model has been successfully tested in a series of flight experiments at Edwards Airforce Base, CA, and has been extensively studied numerically. 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 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 a finite—meromorphic operator—valued function of the spectral parameter. Its poles are precisely the aeroelastic modes. In the author's previous works, it has been shown that the set of aeroelastic modes asymptotically splits into two disjoint subsets called the β -branch and the δ -branch, and precise spectral asymptotics with respect to the eigenvalue number have been derived for both branches. The asymptotical approximations for the mode shapes have also been obtained. In the present work, the author proves that the set of the mode shapes forms an unconditional basis (the Riesz basis) in the Hilbert state space of the system. 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.