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
Mot 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 
@article{CRMECA_2014__342_4_221_0,
     author = {Gregory A. Chechkin and Taras A. Mel'nyk},
     title = {High-frequency cell vibrations and spatial skin effect in thick cascade junction with heavy concentrated masses},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {221--228},
     publisher = {Elsevier},
     volume = {342},
     number = {4},
     year = {2014},
     doi = {10.1016/j.crme.2014.01.002},
     language = {en},
}
TY  - JOUR
AU  - Gregory A. Chechkin
AU  - Taras A. Mel'nyk
TI  - High-frequency cell vibrations and spatial skin effect in thick cascade junction with heavy concentrated masses
JO  - Comptes Rendus. Mécanique
PY  - 2014
SP  - 221
EP  - 228
VL  - 342
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crme.2014.01.002
LA  - en
ID  - CRMECA_2014__342_4_221_0
ER  - 
%0 Journal Article
%A Gregory A. Chechkin
%A Taras A. Mel'nyk
%T High-frequency cell vibrations and spatial skin effect in thick cascade junction with heavy concentrated masses
%J Comptes Rendus. Mécanique
%D 2014
%P 221-228
%V 342
%N 4
%I Elsevier
%R 10.1016/j.crme.2014.01.002
%G en
%F CRMECA_2014__342_4_221_0
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/

[1] E. Sánchez-Palencia Perturbation of eigenvalues in thermo-elasticity and vibration of systems with concentrated masses, Trends and Applications of Pure Mathematics to Mechanics, Lect. Notes Phys., vol. 195, Springer-Verlag, 1984, pp. 346-368

[2] G.A. Chechkin; T.A. Mel'nyk Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses, Appl. Anal., Volume 91 (2012) no. 6, pp. 1055-1095 | DOI

[3] G.A. Chechkin; T.A. Mel'nyk Spatial-skin effect for eigenvibrations of a thick cascade junction with “heavy” concentrated masses, Math. Methods Appl. Sci., Volume 37 (2014) no. 1, pp. 56-74 | DOI

[4] M. Lobo; M.E. Pérez A skin effect for systems with many concentrated masses, C. R. Acad. Sci. Paris, Ser. IIb, Volume 327 (1999), pp. 771-776

[5] M. Lobo; M.E. Pérez Local problems for vibrating systems with concentrated masses: a review, C. R. Acad. Sci. Paris, Ser. IIb, Volume 331 (2003), pp. 303-317

[6] C. Leal; E. Sánchez-Palencia Perturbation of the eigenvalues of a membrane with concentrated mass, Q. Appl. Math., Volume 47 (1989), pp. 93-103

[7] O.A. Oleinik; J. Sánchez-Hubert; G.A. Yosifian On vibration of a membrane with concentrated masses, Bull. Sci. Math., Volume 115 (1991) no. 2, pp. 1-27

[8] G.A. Chechkin Asymptotic expansion of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case, Izv. Math., Volume 69 (2005) no. 4, pp. 805-846 (transl. from: Izv. RAN Ser. Mat., 69, 4, 2005, pp. 161-204)

[9] T.A. Mel'nyk Vibrations of a thick periodic junction with concentrated masses, Math. Models Methods Appl. Sci., Volume 11 (2001) no. 6, pp. 1001-1029

[10] O.A. Oleinik; G.A. Yosifian; A.S. Shamaev Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992 (Russian edition: MGU, Moscow, 1990)

[11] M.E. Pérez; S.A. Nazarov New asymptotic effects for the spectrum of problems on concentrated masses near the boundary, C. R. Acad. Sci. Paris, Ser. IIb, Volume 337 (2009), pp. 585-590

[12] A.M. Il'in Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Transl. Math. Monogr., vol. 102, American Mathematical Society, Providence, RI, USA, 1992

[13] V.A. Kondrat'ev; O.A. Oleinik Boundary-value problems for partial differential equations in non-smooth domains, Russ. Math. Surv., Volume 38 (1983) no. 2, pp. 1-86 (transl. from: Usp. Mat. Nauk, 38, 2, 1983, pp. 3-76)

[14] T.A. Mel'nyk Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwend., Volume 18 (1999) no. 4, pp. 953-975

[15] T.A. Mel'nyk Scheme of investigation of the spectrum of a family of perturbed operators and its application to spectral problems in thick junctions, Nonlinear Oscil., Volume 6 (2003) no. 2, pp. 232-249

[16] T.A. Mel'nyk Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1, Math. Methods Appl. Sci., Volume 23 (2000), pp. 321-346

[17] M.I. Vishik; L.A. Lyusternik Regular degeneration and boundary layer for linear differential equations with small parameter, Transl. Amer. Math. Soc., Ser. 2, Volume 20 (1962) no. 5, pp. 239-364 transl. from: Usp. Mat. Nauk, 12, 1957, pp. 3-122 (in Russian)

Cité par Sources :

Commentaires - Politique