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

2016

EECT

Self Cite

We consider a nonlinear (Berger or Von Karman) clamped plate model with a piston-theoretic right hand side-which include non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of strong tools such as backward-in-time smallness of velocities and finiteness of the dissipation integral. Modern quasi-stability techniques are utilized to show the existence of compact global attractors and generalized fractal exponential attractors. Specific results depending on the size of the damping parameter and the nonlinearity in force. For the Berger plate, in the presence of large damping, the existence of a proper global attractor (whose fractal dimension is finite in the state space) is shown via a decomposition of the nonlinear dynamics. This leads to the construction of a compact set upon which quasi-stability theory can be implemented. Numerical investigations for appropriate 1-D models are presented which explore and support the abstract results presented herein.2 these two terms is the key driver of interest in this plate problem (and in the general flutter phenomenon). Such a non-dissipative, piston-theoretic model has been intensively studied in for different boundary conditions and resistive damping forces, see [5,9,10].In line with the aeroelasticity literature [16,17] (and references therein), we primarily focus on the theory of large deflections for the plate. Taking the boundary to be clamped (as we do) or hinged, von Karman theory is considered accurate [15,16,24]. Moreover, as we consider thin plates, we do not consider rotational inertia effects [24], nor in plane accelerations [25]. In addition, by making an ad hoc simplification to von Karman's scalar plate model, one arrives at a "friendlier" plate model, known as the Berger equation [3, 33, 39] 2 . Other work on flutter dynamics have considered Berger-like beam and plate models (see [9] and references therein), as well as the scalar von Karman equations [10,13,14,38]. Physically, when the plate has no portion of its boundary free, the Berger approximation of von Karman's dynamics is taken to be valid [20,34,37].Remark 1.1. In 1-D, the appropriate model (reflecting an appropriate singular limit of the full von Karman dynamics [25] as in-plane accelerations vanish) is the 1-D analog of the Berger model presented here (sometimes also referred to as a Timoshenko beam [32], or Krieger beam [30]). In Section 7 we consider some numerical results for this 1-D Berger beam model.Many of the results herein are simultaneously valid for both the von Karman and Berger plate dynamics. However, the principal results presented below in Section 1.4.1 on the existence and ...

Section: F = F Bmentioning

confidence: 99%

Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping

Howell¹,

Lasiecka²,

2016

EECT

Self Cite

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Howell¹,

Huneycutt²,

Mathematics in Engineering

et al. 2019

We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms provided through the flow effects. Three different configurations are considered: a clamped panel, a hinged panel, and a flag (a cantilever clamped at the leading edge, free at the trailing edge). After providing the functional framework for the dynamics, recent results on well-posedness and long-time behavior of the associated dynamical system for solutions are presented. Having provided this theoretical context, in-depth numerical stability analyses are provided, focusing both at the onset of flow-induced instability (flutter), and qualitative properties of the post-flutter dynamics across configurations. Modal approximations are utilized, as well as finite difference schemes.

“…In these variables, the pressure/density of the fluid has the form p = (∂ t + U · ∇)φ. Due to the impermeability assumption, in the case of the perfect fluid, we have only one Neumann-type boundary condition given above via the operator L. The (semigroup) well-posedness [23,42] and stability properties [22,24,36] of this model have been intensively studied.…”

Section: Discussion Of Main Results In Relation To the Literaturementioning

confidence: 99%

Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary

Avalos¹,

Geredeli²,

Discrete &Amp; Continuous Dynamical Systems - B

2018

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile U. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically [18]-the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a C0-semigroup e At t≥0 on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator A which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field U ∈ H 3 (O) has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.