Analysis and Control of Mathematical Models of Fluttering Plates

Oregon State University, Corvallis OR

Investigators

Abstract

One of the fundamental problems in the field of aeroelasticity is the prediction and control of the instability known as aeroelastic flutter. Flutter occurs in a flexible structure immersed in a gas flow when aerodynamic loading excites the natural oscillatory modes of the structure; the result is a potentially violent interaction between the displacements of the structure and perturbations in the gas flow field. This phenomenon may occur in a multitude of applications including: buildings and bridges in strong winds, flag-like structures, the human respiratory system, and panel and flap structures on air and land vehicles. In the context of aircraft, flutter is a key concern. If the magnitude of the structural displacements due to flutter is sufficiently large, structural failure can occur. Small oscillations sustained over long periods of time may also bring about costly and/or hazardous fatigue in the structure. Very recently, the idea of harnessing flutter (naturally occurring, or induced) has been suggested to provide an alternative source of energy via piezoelectric "harvesting". For these reasons there is great interest in producing mathematical models that describe the flutter phenomenon in order to gain insight into the mechanisms of flow-structure coupling and predict the dynamics of the system based on its physical parameters. This project comprises a control of partial differential equations (PDEs) analysis of the principal model associated to panel and flap flutter. Results derived from PDE analyses are valuable for a variety of reasons. They: (i) guide and streamline experimental and numerical flutter threshold determination, (ii) can improve cost-effectiveness of experimentation and cut-down on design time of prototypes, (iii) indicate what types and locations of damping will be effective for a given flow-plate configuration.  The proposed investigations are based upon very recent progress in aeroelasticity that has permitted extensions of a classical flow-plate model used over the last 50 years. Obtaining results for flow-structure systems is demanding, as problems arise in the mismatching regularity of dynamics at the interface and ill defined or unbounded trace terms in the coupling. Recent advances in hyperbolic trace regularity theory, abstract coupled models, PDEs with delay, and geometrically constrained damping make modern PDE analysis of flow-plate models tractable. This proposal centers on well-posedness and stability in the presence of feedback controls for a class of nonlinear flow-plate models which include partially free plate boundary conditions and dynamic flow boundary conditions near plate edges. Fully nonlinear models accounting for both in-plane and out-of-plane motion in the structure, as well as nonlinear fluids, are considered. Moreover, recent stability analyses will be extended to the intermediary "transonic" flow regime and the piston-theoretic, hypersonic regime. Beyond well-posedness of these models, time convergence properties (i.e., attractors) of the dynamics will be considered to determine the sensitivity of non-transient behavior of the system to the plate's boundary conditions and external loading. The current proposal can be viewed as an analysis of models arising in aeroelasticity by providing a comparison between qualitative properties of well-posed PDEs to experimentally observed and/or numerically approximated behaviors.