×
Sabatini, Luca

Spectral estimates of vibration frequencies of anisotropic beams. (English) Zbl 07655737

Appl. Math., Praha 68, No. 1, 15-33 (2023).
Summary: The use of one theorem of spectral analysis proved by Bordoni on a model of linear anisotropic beam proposed by the author allows the determination of the variation range of vibration frequencies of a beam in two typical restraint conditions. The proposed method is very general and allows its use on a very wide set of problems of engineering practice and mathematical physics.

MSC:

74B05 Classical linear elasticity
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35P15 Estimates of eigenvalues in context of PDEs
47A75 Eigenvalue problems for linear operators

Keywords:

theory of beams; deformation of cross section; spectral geometry; comparison of spectra
PDFBibTeX XMLCite
\textit{L. Sabatini}, Appl. Math., Praha 68, No. 1, 15--33 (2023; Zbl 07655737)
Full Text: DOI

References:

[1] Bordoni, M., An estimate for finite sums of eigenvalues of fiber spaces, C. R. Acad. Sci., Paris Sér. I 315 (1992), 1079-1083 · Zbl 0761.53019
[2] Bordoni, M., Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds, Math. Ann. 298 (1994), 693-718 · Zbl 0791.58094 · doi:10.1007/BF01459757
[3] Bordoni, M., Spectral comparison between Dirac and Schrödinger operators, Rend. Mat. Appl., VII. Ser. 18 (1998), 181-196 · Zbl 0919.58064
[4] Brézis, H., Analyse fonctionnelle. Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1983), French · Zbl 0511.46001
[5] Picone, M.; Fichera, G., Trattato di analisi matematica, Tumminelli, Roma (1954), Italian · Zbl 0058.03803
[6] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. Vol. I: Functional Analysis, Academic Press, New York (1972) · Zbl 0242.46001 · doi:10.1016/b978-0-12-585001-8.x5001-6
[7] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, New York (1975) · Zbl 0308.47002
[8] Reed, M.; Simon, B., Methods of Modern Mathematical Physics. Vol. IV: Analysis of Operators, Academic Press, New York (1978) · Zbl 0401.47001
[9] Sabatini, L., Estimation of vibration frequencies of linear elastic membranes, Appl. Math., Praha 63 (2018), 37-53 · Zbl 1458.74092 · doi:10.21136/AM.2018.0316-16
[10] Sabatini, L., Estimates of the Laplacian spectrum and bounds of topological invariants for Riemannian manifolds with boundary, An. Ştiinţ. Univ. “Ovidius” Constanţa, Ser. Mat. 27 (2019), 179-211 · Zbl 1488.35405 · doi:10.2478/auom-2019-0027
[11] Sabatini, L., Estimates of the Laplacian spectrum and bounds of topological invariants for Riemannian manifolds with boundary II, An. Ştiinţ. Univ. “Ovidius” Constanţa, Ser. Mat. 28 (2020), 165-179 · Zbl 1488.35406 · doi:10.2478/auom-2020-0012
[12] Sabatini, L., A linear theory of beams with deformable cross section, J. Math. Model. 9 (2021), 465-483 · Zbl 1499.74006 · doi:10.22124/jmm.2021.17932.1548
[13] Tikhonov, A. N.; Samarskij, A. A., Equazioni della fisica matematica, Mir, Roma (1981), Italian · Zbl 0489.35001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.