On the vibration of a partially fastened membrane with many ‘light’ concentrated masses on the boundary

Gregory A. Chechkin

doi:10.1016/j.crme.2004.08.002

On the vibration of a partially fastened membrane with many ‘light’ concentrated masses on the boundary

Presented by: Évariste Sanchez-Palencia

Gregory A. Chechkin ¹

¹ Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia

Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 949-954.

Abstract
Résumé

We consider a partially fastened membrane with many concentrated masses near the boundary. Masses have the diameter $(a ɛ)$ ; the density is $O (1)$ outside the masses and $O ({(a ɛ)}^{- m})$ , $0 < m < 2$ , in the masses. We assume that the distance between masses is $O (ɛ)$ and a is fixed. We obtain the leading terms of the asymptotic expansion of eigenvalues and eigenfunctions of the respective spectral problems for the Laplacian in such a domain.

Nous considérons une membrane partiellement attachée avec plusieurs masses concentrées prés de la frontière. Le diamètre des masses est ègal à $(a ɛ)$ ; la densité est d'ordre $O (1)$ en dehors des masses et la densité des masses d'ordre $O ({(a ɛ)}^{- m})$ , $0 < m < 2$ . Nous supposons que la distance entre les masses est d'ordre $O (ɛ)$ et que a est fixé. Nous obtenons les termes principaux du développement asymptotique des valeurs propres et des fonctions propres du Laplacian dans un domaine de ce type.

Received: 2003-05-26
Accepted: 2004-09-08
Published online: 2004-11-05

DOI: 10.1016/j.crme.2004.08.002

Keywords: Vibrations, Homogenization, Small parameter, Eigenvalues, Asymptotics
Mots-clés : Vibrations, Homogénéisation, Petit paramètre, Valeur propre, Asymptotique

Author's affiliations:

Gregory A. Chechkin ¹

¹ Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia

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Gregory A. Chechkin. On the vibration of a partially fastened membrane with many ‘light’ concentrated masses on the boundary. Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 949-954. doi : 10.1016/j.crme.2004.08.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.08.002/

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