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
[Comportement asymptotique des éléments propres pour des membranes vibrantes avec des couches très minces et très lourdes]

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 (VO)
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.

Reçu le : 2002-08-27
Révisé le : 2002-10-01
Publié le : 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

Affiliations des auteurs :

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},
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     year = {2002},
     doi = {10.1016/S1631-0721(02)01531-0},
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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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Yuriy Golovaty; Delfina Gómez; Maria-Eugenia Pérez-Martínez On eigenvibrations of branched structures with heterogeneous mass density, Journal of Mathematical Analysis and Applications, Volume 549 (2025) no. 2, p. 26 (Id/No 129586) | DOI:10.1016/j.jmaa.2025.129586 | Zbl:8042390
Yuriy Golovaty Membranes with thin and heavy inclusions: asymptotics of spectra, Asymptotic Analysis, Volume 130 (2022) no. 1-2, pp. 23-51 | DOI:10.3233/asy-211743 | Zbl:1498.35529
Yu. D. Holovatyi; V. M. Hut Vibrating systems with rigid light-weight inclusions: asymptotics of the spectrum and eigenspaces, Ukrainian Mathematical Journal, Volume 64 (2013) no. 10, pp. 1495-1513 | DOI:10.1007/s11253-013-0731-8 | Zbl:1302.35153
D. Gömez; M. Lobo; M. E. Pérez High-Frequency Vibrations of Systems with Concentrated Masses Along Planes, Integral Methods in Science and Engineering, Volume 1 (2010), p. 149 | DOI:10.1007/978-0-8176-4899-2_15
Natalia Babych; Yuri Golovaty Low and high frequency approximations to eigenvibrations of string with double contrasts, Journal of Computational and Applied Mathematics, Volume 234 (2010) no. 6, pp. 1860-1867 | DOI:10.1016/j.cam.2009.08.037 | Zbl:1379.34082
D. Gómez; M. Lobo; S. A. Nazarov; E. Pérez Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 85 (2006) no. 4, pp. 598-632 | DOI:10.1016/j.matpur.2005.10.013 | Zbl:1168.35310
YU. D. GOLOVATY; D. GÓMEZ; M. LOBO; E. PÉREZ ON VIBRATING MEMBRANES WITH VERY HEAVY THIN INCLUSIONS, Mathematical Models and Methods in Applied Sciences, Volume 14 (2004) no. 07, p. 987 | DOI:10.1142/s0218202504003520
Miguel Lobo; Eugenia Pérez Local problems for vibrating systems with concentrated masses: a review, Comptes Rendus. Mécanique, Volume 331 (2003) no. 4, p. 303 | DOI:10.1016/s1631-0721(03)00058-5

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