Asymptotics of natural oscillations of a spatial junction of thin elastic rods
Comptes Rendus. Mécanique, Volume 344 (2016) no. 9, pp. 684-688.

A one-dimensional model of an elastic junction that contains hard- and readily-movable thin rods is derived, and asymptotic formulas for eigenvalues with rigorous estimates for remainders are given. In addition to vector functions satisfying classical ordinary differential equations, the model involves algebraic unknowns and algebraic relations corresponding to the longitudinal rigid motion of readily-movable rods.

Nous obtenons un modèle 1D d'une jonction élastique qui contient des barres dures fines facilement déplaçables, et nous donnons un développement asymptotique des valeurs propres justifié par des estimations d'erreur rigoureuses. En plus de fonctions vectorielles satisfaisant des équations différentielles ordinaires, le modèle met en jeu des inconnues et des relations algébriques correspondant au mouvement rigide longitudinal des barres.

Accepted:
Published online:
DOI: 10.1016/j.crme.2016.04.001
Keywords: Elastic junction of thin rods, Hard- and readily-movable rods, Natural oscillations, Asymptotics, Dimension reductions
Mot clés : Jonction élastique de barres fines, Barres dures déplaçables, Oscillations libres, Analyse asymptotique, Réduction de dimension

Sergei A. Nazarov 1, 2, 3; Andrey S. Slutskij 1, 3

1 Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia
2 Saint Petersburg State Polytechnical University, Polytechnicheskaya ul., 29, 195251, Saint Petersburg, Russia
3 Institute of Problems of Mechanical Engineering RAS, V.O., Bol'shoi pr., 61, 199178, Saint Petersburg, Russia
@article{CRMECA_2016__344_9_684_0,
author = {Sergei A. Nazarov and Andrey S. Slutskij},
title = {Asymptotics of natural oscillations of a spatial junction of thin elastic rods},
journal = {Comptes Rendus. M\'ecanique},
pages = {684--688},
publisher = {Elsevier},
volume = {344},
number = {9},
year = {2016},
doi = {10.1016/j.crme.2016.04.001},
language = {en},
}
TY  - JOUR
AU  - Sergei A. Nazarov
AU  - Andrey S. Slutskij
TI  - Asymptotics of natural oscillations of a spatial junction of thin elastic rods
JO  - Comptes Rendus. Mécanique
PY  - 2016
SP  - 684
EP  - 688
VL  - 344
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crme.2016.04.001
LA  - en
ID  - CRMECA_2016__344_9_684_0
ER  - 
%0 Journal Article
%A Sergei A. Nazarov
%A Andrey S. Slutskij
%T Asymptotics of natural oscillations of a spatial junction of thin elastic rods
%J Comptes Rendus. Mécanique
%D 2016
%P 684-688
%V 344
%N 9
%I Elsevier
%R 10.1016/j.crme.2016.04.001
%G en
%F CRMECA_2016__344_9_684_0
Sergei A. Nazarov; Andrey S. Slutskij. Asymptotics of natural oscillations of a spatial junction of thin elastic rods. Comptes Rendus. Mécanique, Volume 344 (2016) no. 9, pp. 684-688. doi : 10.1016/j.crme.2016.04.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.04.001/

[1] J. Sanchez-Hubert; É. Sanchez-Palencia C. R. Acad. Sci. Paris, Ser. II, 312 (1991) no. 4, pp. 337-344

[2] J. Sanchez-Hubert; É. Sanchez-Palencia Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris, 1992

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

[4] G.P. Panasenko Multi-Scale Modelling for Structures and Composites, Springer, 2005

[5] H. Le Dret J. Math. Pures Appl., 68 (1989), pp. 365-397

[6] S.A. Nazarov; A.S. Slutskii Proc. Steklov Inst. Math., 236 (2002) no. 1, pp. 222-249

[7] S.A. Nazarov; A.S. Slutskii Transl. Amer. Math. Soc., Ser. 2, 214 (2005), pp. 59-108

[8] W.G. Mazja; S.A. Nasarow; B.A. Plamenewski Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, Bd. 1, Akademie-Verlag, Berlin, 1991 (Bd. 2, 1991)

[9] V. Kozlov; V.G. Maz'ja; A.B. Movchan Asymptotic Analysis of Fields in Multi-Structures, Oxford University Press, Oxford, UK, 1999

[10] J.-L. Lions; E. Magenes Non-homogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972

[11] S.A. Nazarov; A.S. Slutskii Sib. Math. J., 43 (2002) no. 6, pp. 1069-1079

[12] S.A. Nazarov J. Math. Sci., 114 (2003) no. 5, pp. 1657-1725

[13] S.A. Nazarov C. R. Mecanique, 330 (2002), pp. 603-607

Cited by Sources:

The work was supported by St.-Petersburg University, grant 0.38.237.2014, and Russian Foundation of Basic Research, grant 15-01-02175.