×

zbMATH — the first resource for mathematics

The eigenvalue problem for networks of beams. (English) Zbl 0979.74026
Summary: We perform spectral analysis of different models of networks of Euler-Bernoulli beams. First, we derive the characteristic equation for the spectrum. Secondly, in some particular situations, we show that the spectrum depends only on the structure of the graph. Thirdly, we investigate the asymptotic behaviour of eigenvalues by proving the so-called Weyl’s formula.

MSC:
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Mehmeti, F.Ali, A characteristion of generalized \(C\^{}\{∞\}\) notion on nets, Integral equation and operator theory, 9, 753-766, (1986) · Zbl 0617.35022
[2] Mehmeti, F.Ali, Regular solutions of transmission and interaction problems for wave equations, Math. methods appl. sci., 11, 665-685, (1989) · Zbl 0722.35062
[3] Angad-Gaur, H.; Gaveau, B.; Okada, M., Explicit heat kernel on generalized cones, SIAM J. math. anal., 25, 1562-1576, (1994) · Zbl 0810.35142
[4] von Below, J., A characteristic equation associated to an eigenvalue problem on \(c\^{}\{2\}\)-networks, Linear algebra appl., 71, 309-325, (1985) · Zbl 0617.34010
[5] von Below, J., Classical solvability of linear parabolic equations on networks, J. differential equations, 72, 316-337, (1988) · Zbl 0674.35039
[6] von Below, J., Sturm – liouville eigenvalue problems on networks, Math. methods appl. sci., 10, 383-395, (1988) · Zbl 0652.34025
[7] von Below, J., Kirchhoff laws and diffusion on networks, Linear algebra appl., 121, 692-697, (1989)
[8] J. von Below, Parabolic network equations, Habilitation Thesis, Eberhard-Karls-Universität Tübingen, 1993 · Zbl 0833.35072
[9] von Below, J.; Nicaise, S., Dynamical interface transition with diffusion in ramified media, Commun. partial differential equations, 21, 255-279, (1996) · Zbl 0852.35057
[10] Borovskikh, A.; Mustafokulov, R.; Lazarev, K.; Pokornyi, Yu., A class of fourth-order differential equations on a spatial net, Doklady math., 52, 433-435, (1995) · Zbl 0891.34018
[11] Chen, G.; Delfour, M.; Krall, A.; Payre, G., Modeling, stabilization and control of serially connected beams, SIAM J. control opt., 25, 526-546, (1987) · Zbl 0621.93053
[12] Chen, G.; Krantz, S.; Russell, D.; Wayne, C.; West, H.; Coleman, M., Analysis, design and behavior of dissipative joints for coupled beams, SIAM J. appl. math., 49, 1665-1693, (1989) · Zbl 0685.73046
[13] R. Courant, D. Hilbert, Methods of Mathematical Physics I, Interscience, New-York, 1962 · Zbl 0099.29504
[14] B. Gaveau, M. Okada, T. Okada, Explicit heat kernels on graphs and spectral analysis: several complex variables, Princeton University Press, Math. Notes 38 (1993) 360-384 · Zbl 0803.35062
[15] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1980 · Zbl 0435.47001
[16] Lagnese, J.E.; Leugering, G.; Schmidt, E.J.P.G., Control of planar networks of Timoshenko beams, SIAM J. control optim., 31, 780-811, (1993) · Zbl 0775.93107
[17] J.E. Lagnese, G. Leugering, E.J.P.G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Birkhäuser, Boston, 1994 · Zbl 0810.73004
[18] H. Le Dret, Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications, RMA 19, Masson, Paris, 1991
[19] G. Leugering, Reverberation analysis and control of networks of elastic strings, in: Control of PDE and Appl., Lecture Notes in Pure and Applied Mathematics, vol. 174, Marcel Dekker, New York, 1996, 193-206 · Zbl 0867.73055
[20] G. Lumer, Connecting of local operators and evolution equation on network, Lecture Notes in Mathematics, vol. 787, Springer, Berlin, 1980, 219-234
[21] G. Lumer, Espaces ramifiés et diffusions sur les réseaux topologiques, C. R. Acad. Sc. Paris, Série A, 291 (1980) 627-630 · Zbl 0449.35110
[22] S. Nicaise, Some results on spectral theory over networks applied to nerve impulse transmission, Lecture Notes in Mathematics, vol. 1171, Springer, Berlin, 1985, 532-541 · Zbl 0586.35071
[23] S. Nicaise, Diffusion sur les espaces ramifiés, Thesis, Université de Mons, 1986
[24] S. Nicaise, Spectre des réseaux topologiques finis, Bull. Sci. Math., 2ème série 111 (1987) 401-413 · Zbl 0644.35076
[25] Pokornyi, Yu.; Penkin, O., Sturm theorems for equations on graphs, Soviet math. dokl., 40, 640-642, (1990) · Zbl 0708.34025
[26] Pokornyi, Yu.; Karelina, I., On the Green function ot the Dirichlet problem on a graph, Soviet math. dokl., 43, 732-734, (1991) · Zbl 0773.34020
[27] J.P. Roth, Le spectre du laplacien sur un graphe, Lecture Notes in Math. 1096, Springer, Berlin, 1984, 521-539 · Zbl 0557.58023
[28] R.E. Showalter, Hilbert space methods for partial differential equations, Monographs and Studies in Mathematics, 1, Pitman, Boston, 1977 · Zbl 0364.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.