Reduction of Boundary Value Problem to Possio Integral Equation in Theoretical Aeroelasticity

A V Balakrishnan,M A Shubov

doi:10.1155/2008/846282

Abstract

The present paper is the first in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. We provide a rigorous reduction of the initial boundary value problem involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions to a singular integral equation, the Possio equation. The question of its solvability will be addressed in our forthcoming work.

Highlights

The present paper is the first in a series of three works devoted to a systematic study of a specific singular integral equation that plays a key role in aeroelasticity
In our forthcoming work, we will prove a unique solvability of 5.61 using a two-step procedure
We prove that as |λ|→∞, the right-hand side of the integral operator of 5.61 splits into two distinct parts: one does not get smaller and another one asymptotically goes to zero in a Banach space equipped with a specific norm

Summary

Introduction

The present paper is the first in a series of three works devoted to a systematic study of a specific singular integral equation that plays a key role in aeroelasticity. Summarizing all of the above, one can see that the entire aeroelastic problem structure and aerodynamics becomes in general a nonlinear convolution due to the expressions for the lift and moment and/or evolution equation in terms of the structure state variables, whose stability with respect to the parameter U is the “flutter problem” one has to resolve. Such a double transform allows us to give the first version of the Possio equation see 3.22. In our series of papers, we are addressing the problem of the unique solvability using rigorous analytical tools

Aerodynamic field equation

Mikhlin multipliers

Concluding remarks