Long-time behavior of a model of extensible beams with nonlinear boundary dissipations

Fu, To; Narciso, Vando; Pelicer, Maurício Luciano

doi:10.1016/j.jmaa.2012.07.004

Cited by 23 publications

(31 citation statements)

References 34 publications

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“…In the case of nonlinear clamped and hinged beams, the well-posedness has been known for sometime; consult the abstract theory in [12] and note the classical references (for both 1-D Krieger and 2-D Berger nonlinearities) [2,3,5,15,21]. In the case of the cantilevered beam CF with α ≥ 0, see the recent work [26] (which includes discussion of other cantilevered beams [33]).…”

Section: Well-posednessmentioning

confidence: 99%

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

Howell¹,

Huneycutt²,

Webster³

et al. 2019

Mathematics in Engineering

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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.

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Section: Well-posednessmentioning

confidence: 99%

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

Howell¹,

Huneycutt²,

Webster³

et al. 2019

Mathematics in Engineering

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“…In the case of the beam, we have demonstrated that simplification, which yields the beam model here in (1.5). Generally speaking, this model has been historically referred to in the literature as a Krieger or Krieger-Woinowsky beam [WK50,MNP12], for instance. In the case of the 2-D full von Kármán system, when one takes in-plane accelerations to be negligible, the so-called scalar von Kármán equation [Lag89,CL10] are obtained.…”

Section: Nonlinear Beam and Plate Modelsmentioning

confidence: 99%

“…In [Ma03] the author considers the static version of the problem in [Ma01], with the associated nonlinear free boundary condition; that paper includes a very brief numerical study with specific polynomial nonlinear structure. The subsequent work [MNP12] is our primary motivating reference here, and addresses the dynamic equations with a general Berger-like nonlinearity from the point of view of attractors, following from [Ma03,MN10]. The latter analysis utilizes Lyapunov methods to demonstrate the dissipativity of the dynamical system, as well as a version of Theorem 8.3 to demonstrate asymptotic compactness.…”

Section: Berger Model For Cantilevered Beamsmentioning

confidence: 99%

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

Howell¹,

Toundykov²,

Webster³

2018

SIAM J. Math. Anal.

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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 non-dissipative RHS. 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 rotational inertia effects in the beam 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 existence of weak, possibly non-unique, solutions. For the former, inertial model, we prove that the associated non-gradient dynamical system has a compact global attractor. Finally, we study stability regimes and post-flutter dynamics (non-stationary end behaviors) using numerical methods for models with, and without, the rotational inertia terms.

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“…[9]- [11], [19], [20] and references therein), and several kinds of plate equations with memory (see e.g. [8], [14], [22], [24]- [27]. Therefore, we point out that our abstract results can be applied on a large variety of partial differential equations in which memory effects are considered.…”

Section: Introductionmentioning

confidence: 92%

“…Plate equations with or without memory effects have been attracted many researchers with various purposes. See [14]- [18], [22], [24], [26], [29], [31], [32] and references therein. Precisely, we present results concerning to the existence, uniqueness, regularity, unique continuation, blow-up alternative and continuous dependence of mild solutions to the strongly damped plate equation with memory (1.1)-(1.3).…”

Section: Introductionmentioning

confidence: 99%