Analyse asymptotique des modes de hautes fréquences dans les poutres minces

Irago, H.; Kerdid, Nabil; Viaño, J. M.

doi:10.1016/s0764-4442(98)80238-3

Cited by 7 publications

(1 citation statement)

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“…In fact, we get results for eigenvalues and eigenfunctions of (1.1) which are of interest in terms of the associated evolution problems since, from (1.8), we can construct standing waves which approach time-dependent solutions for long times, and these times can be precisely computed in terms of bounds for discrepancies such as that in (1.5) (see [41] and [34] for an abstract framework as well as for applications to very different vibrating systems). It should be emphasized that it seems to be a common fact to many mechanical systems arising in thin structures that the low frequencies give rise to longitudinal vibrations while for other kinds of vibrations such as torsional or stretching vibrations one must look among those associated to the high frequencies: see, for instance, [8], [23], [24], [35] and references therein. Also, we emphasize that, to our knowledge, this is the first work in the literature addressing the asymptotics of the high frequencies for a thin T-like shaped structure.…”

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confidence: 99%

Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure

Gaudiello

Gómez

Pérez‐Martínez

2020

Journal de Mathématiques Pures et Appliquées

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We consider a spectral problem for the Laplacian operator in a planar T-like shaped thin structure Ω ε , where ε denotes the transversal thickness of both branches. We assume the homogeneous Dirichlet boundary condition on the ends of the branches and the homogeneous Neumann boundary condition on the remaining part of the boundary of Ω ε . We study the asymptotic behavior, as ε tends to zero, of the high frequencies of such a problem. Unlike the asymptotic behavior of the low frequencies where the limit problem involves only longitudinal vibrations along each branch of the T-like shaped thin structure (i.e. 1D limit spectral problems), we obtain a two dimensional limit spectral problem which allows us to capture other kinds of vibrations. We also give a characterization of the asymptotic form of the eigenfunctions originating these vibrations.Résumé: On considère le problème spectral pour le Laplacien, dans une structure mince bidimensionnelle Ω ε en forme de T, où ε désigne l'épaisseur des deux branches du T. Les conditions aux limites sont du type Neumann homogène sur tout le bord sauf aux extrémités des branches où une condition de Dirichlet homogène est imposée. Onétudie le comportement asymptotique des hautes fréquences lorsque ε tend vers zéro. Contrairement au comportement asymptotique des basses fréquences, pour lesquelles le problème limite ne fait apparaître que des vibrations longitudinales le long de chaque branche de la structure (c'est-à-dire, des problèmes spectraux limites 1D ), on obtientà la limite un problème spectral bidimensionnel, qui nous permet de capter d'autres types de vibrations. On donneégalement une caractérisation de la forme asymptotique des fonctions propres qui sontà l'origine de ces vibrations.

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