Comptes Rendus
Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions
Comptes Rendus. Mécanique, Volume 330 (2002) no. 11, pp. 777-782.

We consider the vibrations of a membrane that contains a very thin and heavy inclusion around a curve γ. We assume that the membrane occupies a domain Ω of 2. The inclusion occupies a layer-like domain ωϵΩ of width 2ε and it has a density of order O(ε−3). The density is of order O(1) outside this inclusion, the concentrated mass around the curve γ. ε is a positive parameter, ε∈(0,1). By means of asymptotic expansions, we describe the behaviour, as ε→0, of the eigenelements (λε,uε) of the associated spectral problem. We provide complete asymptotic series for the low frequencies λε=O(ε2), the medium frequencies λε=O(ε) and the corresponding eigenfunctions uε.

On considère les vibrations d'une membrane qui contient une très mince et très lourde inclusion placée autour d'une courbe γ. On suppose que la membrane occupe un domaine Ω2, tandis que l'inclusion occupe une couche ωϵΩ de largeur 2ε, la densité étant d'ordre O(ε−3). La densité est d'ordre O(1) en dehors de la petite inclusion : la masse est concentrée autour de γ. ε est un petit paramètre, ε∈(0,1). À l'aide des développements asymptotiques, nous décrivons le comportement, pour ε→0, des éléments propres (λε,uε) du problème spectral associé. En fait, nous obtenons les séries asymptotiques complètes pour les basses fréquences λε=O(ε2) et les moyennes fréquences λε=O(ε), ainsi que les fonctions propres correspondantes uε.

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Revised:
Published online:
DOI: 10.1016/S1631-0721(02)01531-0
Keywords: vibrations, concentrated masses, spectral analysis, low frequencies, medium frequencies
Mot clés : vibrations, masses concentrées, analyse spectrale, basses fréquences, moyennes fréquences

Yuri Golovaty 1; Delfina Gómez 2; Miguel Lobo 2; Eugenia Pérez 3

1 Department of Mechanics and Mathematics, Franko Lviv National University, Lviv, Ukraine
2 Departamento de Matemáticas, Estadı́stica y Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain
3 Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain
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     title = {Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions},
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Yuri Golovaty; Delfina Gómez; Miguel Lobo; Eugenia Pérez. Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions. Comptes Rendus. Mécanique, Volume 330 (2002) no. 11, pp. 777-782. doi : 10.1016/S1631-0721(02)01531-0. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01531-0/

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[6] Yu.D. Golovaty, D. Gómez, M. Lobo, E. Pérez, Vibrating membranes with very thin heavy inclusions around curves, in preparation

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