Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer

Geredeli, Pelin G.; Lasiecka, Irena; Webster, Justin T.

doi:10.1016/j.jde.2012.10.016

Cited by 23 publications

(19 citation statements)

References 49 publications

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“…• If the damping coefficient is large, OR L ≡ 0, then the quasi-stability estimate can be shown on the compact global attractor to get finite dimensionality and smoothness of the attractor. The technique from [19,13] (showcased here) circumvents the need for backward-in-time-smallness, the key ingredient utilized in [10] which requires the dynamics to be gradient (i.e., L ≡ 0).…”

Section: Past Results In Relation To Work Hereinmentioning

confidence: 99%

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

Howell¹,

Lasiecka²,

Webster³

2016

EECT

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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 ...

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Section: Past Results In Relation To Work Hereinmentioning

confidence: 99%

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

Howell¹,

Lasiecka²,

Webster³

2016

EECT

Self Cite

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“…F 0 represents in-plane forces within the plate, F x is nonlinear transverse force and Lx is a first order differential operator modeling a nonconservative force. In the last fifteen years have appeared papers which study the existence of a global attractor which captures the long-time behavior of solutions to (1) and properties of this attractor, such as smoothness and finite dimensionality (see [12] and reference therein). without rotational forces.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we investigate (1) with α = 0 i.e. [5,12,13] and reference therein). The model without rotational forces displays less regularity, hence the analysis of the nonlinear effects is more subtle i.e.…”

Section: Introductionmentioning

confidence: 99%

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Solvability and stability of von Karman model without rotational inertia with nonlinear forces

Nowakowski

Rogowski

2015

Nonlinear Analysis: Real World Applications

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“…given by 26) and depends continuously on the initial data. In particular, if y 0 ∈ D(A ) then the solution is strong.…”

Section: Wellposedness/regularity Resultsmentioning

confidence: 99%

Long-time dynamics of two classes of beam and plate equations

Monteiro¹

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Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer

Cited by 23 publications

References 49 publications

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

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

Solvability and stability of von Karman model without rotational inertia with nonlinear forces

Long-time dynamics of two classes of beam and plate equations

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