Feedback Stabilization of a Fluttering Panel in an Inviscid Subsonic Potential Flow

Abstract

We consider a cantilevered (clamped-free) beam in an axial potential flow. Certain flow velocities may bring about a bounded-response instability in the structure, termed flutter. As a preliminary analysis, we employ the theory of large deflections and utilize a piston-theoretic approximation of the flow for appropriate parameters, yielding a nonlinear (Berger/Woinowsky-Krieger) beam equation with a nondissipative right-hand side. As we obtain this structural model via a simplification, we arrive at a nonstandard nonlinear boundary condition that necessitates careful well-posedness analysis. We account for beam rotational inertia effects and discuss technical issues that necessitate this feature. We demonstrate nonlinear semigroup well-posedness of the model with the rotational inertia terms. For the case with no rotational inertia, we utilize a Galerkin approach to establish the existence of weak, possibly nonunique, solutions. For the former, inertial model, we prove that the associated nongradient dynamical system has a compact global attractor. Finally, we study stability regimes and postflutter dynamics (nonstationary end behaviors) using numerical methods for models with, and without, the rotational inertia terms.