Nonlinear Elastic Plate in a Flow of Gas: Recent Results and Conjectures

Abstract

We give a survey of recent results on flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation (PDE) treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies.

A thorough look at the (in)stability of piston-theoretic beams

“…For more general studies of fluttering beams and plates (with extensive references), we point to [19]. In the discussion of piston theory, we mention the modern engineering references [41,42], classic engineering references [1,7], and the recent mathematical-minded survey [10]. Other mathematical references addressing piston-theoretic models include [25,11,6,12].…”

“…In this treatment we provide a multifaceted numerical study-from the dynamical systems point of view-for a simplified class of 1-D models representing the physical phenomenon known as aeroelastic flutter. Flutter is fluid-structure (feedback) instability that occurs between elastic displacements of a structure and responsive aerodynamic pressure changes at the flow-structure interface [7,19,10]. The onset of flutter, for particular flow parameters, represents a bifurcation in the linearization of the system [23], and the qualitative properties of the post-flutter dynamics can be analyzed from an infinite dimensional/control-theoretic point of view [24,10,14].…”

“…Beam (or plate) flutter is a topic of great interest in the engineering literature [7,16,1,34,40,45,41,42], as well as, more recently, the mathematical literature [10,25,31,32] (and many references therein). Though there is a vast (mostly engineering-oriented) literature on flow-structure instabilities, we provide a straight-forward analysis utilizing a 1-D structural model in multiple configurations of interest.…”

“…Indeed, if one can model the dynamic load across the structure, the issue is to determine under what conditions the non-conservative flow loading destabilizes structural (eigen)modes. However, to study the qualitative properties of the dynamics in the post-flutter regime, analysis will require some physical nonlinear restoring force to keep trajectories bounded in time [16,14,10]. Indeed, for the post-flutter dynamics of a given system, the theory of nonlinear dynamical systems is often used, e.g., compact global attractors [12,10] as well as exponential attractors [25,9,12].…”

A Cantilevered Extensible Beam in Axial Flow: Semigroup Well-posedness and Postflutter Regimes

“…For more general studies of fluttering beams and plates, we point to the classical references [Bol63,Dow15]. Specifically, in the discussions of piston theory, we should mention the modern engineering references [Ved13,Ved12], classic engineering references [Lig53, AZ56,Bol63], and the mathematical surveys [CDLW16a,CDLW16b]. Other mathematical references addressing piston-theoretic models include [HLW16, CL08, BT12, CL10].…”