Semigroup Theory in Aeroelasticity

Abstract

Aerolasticity mixes Structural Dynamics with Aerodynamics—a “Tale of Two Semigroups,” so to speak. A fundamental problem — determining the bending-pitching wing-flutter speed in subsonic compressible flow — formulates as the asymptotic stability of the initial value problem for a Convolution-Semigroup equation in a Hilbert space of the form:

$$\dot{Y}(t) = \mathcal{A}Y(t) + \int_{0}^{t} {L(t - \sigma )\dot{Y}(\sigma )d\sigma }$$

where A is the infinitesimal generator of a Co-semigroup with compact resolvent and L(t)linear bounded for each t,and strongly continuous in t ≥ 0,both to be deduced from the solution of a boundary value problem for subsonic compressible aerodynamics of interest in itself.The latter is the linearized version of the transonic inviscid compressible flow equation (we consider only the 2D version)for the velocity potential Ø(x,z,t):

$$a_{\infty }^{2}(1 - {{M}^{2}})\frac{{{{\partial }^{2}}\phi }}{{\partial {{t}^{2}}}} + a_{\infty }^{2}\frac{{{{\partial }^{2}}\phi }}{{\partial {{z}^{2}}}} = \frac{{{{\partial }^{2}}\phi }}{{\partial {{t}^{2}}}} + 2M{{a}_{\infty }}\frac{{{{\partial }^{2}}\phi }}{{\partial t\partial x}}$$

in the half-plane -∞ < x < ∞ and 0 < z < ∞ with the(main)boundary condition

$$\begin{array}{*{20}{c}} {\frac{{\partial \phi }}{{\partial z}}(x,0 + ,t) = {{w}_{a}}(t,x),} & {|x| < b < \infty } \\ \end{array}$$

where the function w_a(t, x) is specified,and

$$U = M{{a}_{\infty }}$$

Is the airspeed, M is the Mach number,assumed less than one.This is a mixed initial-value boundary-value problem in which the Semigroup solution of the homogeneous boundary value problem plays an essential role.We also show that the convolution-semigroup equation can be represented as a pure semigroup equation in the form

$$\begin{array}{*{20}{c}} {Y(t) = PZ(t)} \\ {\dot{Z}(t) = AZ(t).} \\ \end{array}$$