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).