Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions

Yuri Golovaty; Delfina Gómez; Miguel Lobo; Eugenia Pérez

doi:10.1016/S1631-0721(02)01531-0

Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions

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

Comptes Rendus. Mécanique, Volume 330 (2002) no. 11, pp. 777-782.

Abstract
Résumé

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 $ℝ^{2}$ . The inclusion occupies a layer-like domain $ω_{ϵ} \subset Ω$ 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 (λ^ε,u^ε) of the associated spectral problem. We provide complete asymptotic series for the low frequencies λ^ε=O(ε²), the medium frequencies λ^ε=O(ε) and the corresponding eigenfunctions u^ε.

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 $Ω \subset ℝ^{2}$ , tandis que l'inclusion occupe une couche $ω_{ϵ} \subset Ω$ 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 (λ^ε,u^ε) du problème spectral associé. En fait, nous obtenons les séries asymptotiques complètes pour les basses fréquences λ^ε=O(ε²) et les moyennes fréquences λ^ε=O(ε), ainsi que les fonctions propres correspondantes u^ε.

Received: 2002-08-27
Revised: 2002-10-01
Published online: 2002-10-24

DOI: 10.1016/S1631-0721(02)01531-0

Keywords: vibrations, concentrated masses, spectral analysis, low frequencies, medium frequencies
Mots-clés : vibrations, masses concentrées, analyse spectrale, basses fréquences, moyennes fréquences

Author's affiliations:

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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     author = {Yuri Golovaty and Delfina G\'omez and Miguel Lobo and Eugenia P\'erez},
     title = {Asymptotics for the eigenelements of vibrating membranes with very heavy thin inclusions},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {777--782},
     publisher = {Elsevier},
     volume = {330},
     number = {11},
     year = {2002},
     doi = {10.1016/S1631-0721(02)01531-0},
     language = {en},
}

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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/

References
Cited by

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[2] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, North-Holland, London, 1992

[3] J. Sanchez-Hubert; E. Sanchez-Palencia Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer-Verlag, Heidelberg, 1989

[4] Yu.D. Golovaty; A.S. Lavrenyuk Asymptotic expansions of local eigenvibrations for plate with density perturbed in neighbourhood of one-dimensional manifold, Mat. Studii, Volume 13 (2000) no. 1, pp. 51-62

[5] H. Tchatat, Perturbations Spectrales pour des Systèmes avec Masses Concentrées, Thése 3eme cycle, Université Pierre et Marie Curie, Paris VI, Paris, 1984

[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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