Comptes Rendus
no. 4
High-frequency cell vibrations and spatial skin effect in thick cascade junction with heavy concentrated masses
[Vibrations cellulaires de haute fréquence et effet spatial « peau » dans une jonction cascade épaisse avec des masses lourdes concentrées]
Gregory A. Chechkin1  ; Taras A. Mel'nyk2
1 Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia
2 Department of Mathematical Physics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska st., 64/13, Kyiv 01601, Ukraine
Comptes Rendus. Mécanique, Volume 342 (2014) no. 4, pp. 221-228.

Nous étudions le comportement asymptotique des valeurs et fonctions propres du laplacien dans une jonction cascade épaisse bidimensionnelle, avec des masses lourdes concentrées. Si ε→0, nous présentons des approximations asymptotiques en deux termes pour les éléments propres dans les cas des masses concentrées « peu lourdes », « modérément lourdes » et « super-lourdes ». L'analyse asymptotique pour les vibrations à haute fréquence cellulaire est aussi trouvée.

We study the asymptotic behavior of eigenvalues and eigenfunctions of the Laplacian in a 2D thick cascade junction with heavy concentrated masses. We present two-term asymptotic approximations, as ε→0, for the eigenelements in the case of “slightly heavy”, “moderate heavy”, and “super heavy” concentrated masses. Asymptotics of high-frequency cell-vibrations are found as well.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2014.01.002
Keywords: Asymptotic expansion, Spectral problem, Thick cascade junction, Domain with highly oscillating boundary, Concentrated masses
Mots-clés : Développement asymptotique, Problème spectral, Jonction cascade épaisse, Domaine avec oscillation forte du bord, Masses concentrées

Gregory A. Chechkin 1 ; Taras A. Mel'nyk 2

1 Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia
2 Department of Mathematical Physics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska st., 64/13, Kyiv 01601, Ukraine 
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Gregory A. Chechkin; Taras A. Mel'nyk. High-frequency cell vibrations and spatial skin effect in thick cascade junction with heavy concentrated masses. Comptes Rendus. Mécanique, Volume 342 (2014) no. 4, pp. 221-228. doi : 10.1016/j.crme.2014.01.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2014.01.002/

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  • Gregory A. Chechkin; Tatiana P. Chechkina Random Homogenization in a Domain with Light Concentrated Masses, Mathematics, Volume 8 (2020) no. 5, p. 788 | DOI:10.3390/math8050788
  • T A Mel'nyk; G A Chechkin Eigenvibrations of thick cascade junctions with `very heavy' concentrated masses, Izvestiya: Mathematics, Volume 79 (2015) no. 3, p. 467 | DOI:10.1070/im2015v079n03abeh002751
  • Taras Anatol'evich Mel'nik; Gregory Aleksandrovich Chechkin Собственные колебания густых каскадных соединений со "сверхтяжелыми" концентрированными массами, Известия Российской академии наук. Серия математическая, Volume 79 (2015) no. 3, p. 41 | DOI:10.4213/im8238

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