Functional Analysis and Applications Seminar 

15/07/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
A. V. Balakrishnan, The Flight Systems Research Center, University of California, LosAngeles, USA

Mathematical theory of aeroelasticity

The central problem of aeroelasticity involves an endemic safety issue - the determination of the 'Flutter Boundary' - the speed at which the wing structure becomes unstable at any given altitude. Currently all the theoretical work is computational - wedding the Lagrangian FEM structure codes to the Euler CFD codes to produce a 'time-marching' solution. While they can handle 'real life' nonlinear - complex geometry - structures and viscous flows, they are based on approximation by ordinary differential equations, and limited to specific numerical parameters. In turn this limits the generality of the results and understanding of phenomena involved; and of course inadequate for control design for possible stabilization ('Flutter Suppression'). In this presentation we show that the problem can be formulated retaining the full continuum models without approximation, as a boundary value problem for coupled nonlinear partial differential equations. The flutter speed can then be characterized as a Hopf bifurcation point for a nonlinear convolution-evolution equation in the time-domain, which - and this is the crucial point - is then determined completely by the linearized equations - linearized about the equilibrium state. A key step in this approach is a singular integral equation with a difference kernel, discovered by Camillo Possio in 1938, and bearing his name, linking the aerodynamics to the structure dynamics. A challenge here is to choose models which are amenable to analysis, taking advantage of recent advances in boundary value problems, and yet can display the phenomena of interest. The presentation will emphasize problem formulation but will include recent results both analytical and experimental (flight-tests).