Comptes Rendus
Local problems for vibrating systems with concentrated masses: a review
Comptes Rendus. Mécanique, Volume 331 (2003) no. 4, pp. 303-317.

In this review we collect certain results obtained in the last decades on vibrating systems with concentrated masses. In particular, we show the connection of the eigenvalues and eigenfunctions of the local problem with the low and high frequency vibrations of the original problem.

Ce rapport-ci contient quelques resultats obtenus tout au long des denières décades sur les systèmes vibratoires avec masses concentrées. Notamment, on met en evidence la connexion entre les éléments propres du problème local et les vibrations de basses fréquences et d'hautes fréquences du problème original.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-0721(03)00058-5
Keywords: Vibrations, Spectral analysis, Concentrated masses, Low frequencies, High frequencies
Mot clés : Vibrations, Analyse spectrale, Masses concentrées, Fréquences basses, Fréquences hautes

Miguel Lobo 1; Eugenia Pérez 2

1 Departamento de Matemáticas, Estadı́stica y Computación, Universidad de Cantabria, Avenida de los Castros s/n. 39005 Santander, Spain
2 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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Miguel Lobo; Eugenia Pérez. Local problems for vibrating systems with concentrated masses: a review. Comptes Rendus. Mécanique, Volume 331 (2003) no. 4, pp. 303-317. doi : 10.1016/S1631-0721(03)00058-5. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00058-5/

[1] E. Sanchez-Palencia Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, Trend in Applications of Pure Mathematics to Mechanics, Lecture Notes in Phys., 195, Springer-Verlag, Berlin, 1984, pp. 346-368

[2] E. Sanchez-Palencia; H. Tchatat Vibration de systèmes élastiques avec des masses concentrées, Rend. Sem. Mat. Univ. Politec. Torino, Volume 42 (1984) no. 3, pp. 43-63

[3] O.A. Oleinik Homogenization problems in elasticity. Spectra of singularly perturbed operators (R.J. Knops; A.A. Lacey, eds.), Non-Classical Continuum Mechanics, Cambridge University Press, New York, 1987, pp. 81-95

[4] Yu.D. Golovaty; S.A. Nazarov; O.A. Oleinik; T.S. Soboleva Eigenoscillations of a string with an additional mass, Siberian Math. J., Volume 29 (1988) no. 5, pp. 744-760

[5] C. Leal; J. Sanchez-Hubert Perturbation of the eigenvalues of a membrane with a concentrated mass, Quart. Appl. Math., Volume XLVII (1989) no. 1, pp. 93-103

[6] J. Sanchez-Hubert Perturbation des valeurs propres pour des systèmes avec masse concentrée, C. R. Acad. Sci. Paris, Sér. II, Volume 309 (1989), pp. 507-510

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

[8] O.A. Oleinik; J. Sanchez-Hubert; G.A. Yosifian On vibrations of a membrane with concentrated masses, Bull. Sci. Math. Sér. 2, Volume 115 (1991), pp. 1-27

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

[10] S.A. Nazarov Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions, Math. Model Numer. Anal., Volume 27 (1993) no. 6, pp. 777-799

[11] Yu.D. Golovaty Spectral properties of oscillatory systems with adjoined masses, Trans. Moscow Math. Soc., Volume 54 (1993), pp. 23-59

[12] M. Lobo; E. Pérez On vibrations of a body with many concentrated masses near the boundary, Math. Models Methods Appl. Sci., Volume 3 (1993) no. 2, pp. 249-273

[13] M. Lobo; E. Pérez Vibrations of a body with many concentrated masses near the boundary: High frequency vibrations (E. Sanchez-Palencia, ed.), Spectral Analysis of Complex Structures, Hermann, Paris, 1995, pp. 85-101

[14] M. Lobo; E. Pérez Vibrations of a membrane with many concentrated masses near the boundary, Math. Models Methods Appl. Sci., Volume 5 (1995) no. 5, pp. 565-585

[15] M. Lobo; E. Pérez On the local vibrations for systems with many concentrated masses, C. R. Acad. Sci. Paris, Sér. IIb, Volume 324 (1997), pp. 323-329

[16] D. Gómez; M. Lobo; E. Pérez On the eigenfunctions associated with the high frequencies in systems with a concentrated mass, J. Math. Pures Appl., Volume 78 (1999), pp. 841-865

[17] M. Lobo; E. Pérez The skin effect in vibrating systems with many concentrated masses, Math. Methods Appl. Sci., Volume 24 (2001), pp. 59-80

[18] E. Pérez On the whispering gallery modes on the interfaces of membranescomposed of two materials with very different densities, Math. Models Methods Appl. Sci., Volume 13 (2003) no. 1, pp. 75-98

[19] G.A. Chechkin; E. Pérez; E.I. Yablokova On eigenvibrations of a body with light concentrated masses on the surface, Russian Math. Surveys, Volume 57 (2002) no. 6, pp. 195-196

[20] D. Gómez; M. Lobo; E. Pérez On the vibrations of a plate with a concentrated mass and very small thickness, Math. Methods Appl. Sci., Volume 26 (2003), pp. 27-65

[21] E. Pérez, Vibrating systems with concentrated masses: on the local problem and the low frequencies, in: C. Constanda, A. Larguillier, M. Ahues (Eds.), Proceedings of the 7th International Conference on Integral Methods in Sciences and Engineering, Birkhäuser, 2003, to appear

[22] Yu.D. Golovaty; D. Gómez; M. Lobo; E. Pérez Asymptotics for the eigenelements of vibrating membranes with very heavy thing inclusions, C. R. Mécanique, Volume 330 (2002) no. 11, pp. 777-782

[23] T. Melnyk Vibrations of a thick periodic junction with concentrated masses, Math. Models Methods Appl. Sci., Volume 11 (2001) no. 6, pp. 1001-1027

[24] M. Lobo; E. Pérez High frequency vibrations in a stiff problem, Math. Models Methods Appl. Sci., Volume 7 (1997) no. 2, pp. 291-311

[25] M. Lobo; E. Pérez Asymptotic behavior of an elastic body with a surface having small stuck regions, RAIRO Modél. Math. Anal. Numér., Volume 22 (1988) no. 4, pp. 609-624

[26] A. Brillard; M. Lobo; E. Pérez Homogénéisation de frontieres par epi-convergence en élasticité linéaire, RAIRO Modél. Math. Anal. Numér., Volume 24 (1990) no. 1, pp. 5-26

[27] M. Lobo; E. Pérez Boundary homogenization of certain elliptic problems for cylindrical bodies, Bull. Sci. Math. Sér. IIb, Volume 116 (1992), pp. 399-426

[28] D. Cioranescu; F. Murat Un terme étrange venu d'ailleurs (H. Brezis; J.-L. Lions, eds.), Collège de France Séminar, Vol. II & III, Res. Notes Math., 60 & 70, Pitman, London, 1982, pp. 98-138 (154–178)

[29] M. Lobo; O.A. Oleinik; T.A. Shaposhnikova; E. Pérez On homogenization of solutions of boundary value problems in domains, perforated along manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 25 (1998), pp. 611-629

[30] T. Kato Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966

[31] E. Sanchez-Palencia Nonhomogeneous Media and Vibration Theory, Springer-Verlag, Berlin, 1980

[32] J. Rappaz; J. Sanchez-Hubert; E. Sanchez-Palencia; D. Vassiliev On spectral pollution in the finite element approximation of thin elastic “membrane” shells, Numer. Math., Volume 75 (1997) no. 4, pp. 473-500

[33] M. Lobo, S.A. Nazarov, E. Pérez, Eigenoscillations of contrastly non-homogeneous elastic body. Asymptotic and uniform estimates for the eigenvalues, in preparation

[34] H. Attouch Convergence for Functions and Operators, Pitman, London, 1984

[35] M.I. Visik; L.A. Lusternik Regular degeneration and boundary layer for linear differential equations with small parameter, Trans. Amer. Math. Soc. Ser. 2, Volume 20 (1957), pp. 239-364

[36] V.I. Arnold Modes and quasimodes, Funct. Anal. Appl., Volume 6 (1972) no. 2, pp. 94-101

[37] V.F. Lazutkin Semiclassical asymptotics of eigenfunctions (M.V. Fedoryuk, ed.), Partial Differential Equations V, Springer-Verlag, Heidelberg, 1999, pp. 133-171

[38] I.M. Guelfand; G.E. Chilov Les distributions, Tome III, Dunod, Paris, 1965

[39] S.A. Nazarov Asymptotic Theory of Thin Plates and Rods, Vol. 1. Dimension Reduction and Integral Estimates, Novosibirsk, Nauchnaya Kniga, 2002

[40] D. Leguillon; E. Sanchez-Palencia Computation of Singular Solutions in Elliptic Problems and Elasticity, Masson, Paris, 1987

[41] E.L. Beltrami; M.R. Wholers Distributions and the Boundary Values of Analytic Functions, Academic Press, New York, 1966

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