Asymptotic properties of the problem on eigenvibrations of the bar with attached load
A.A. Samsonov a∗ , S.I. Solov’ev a∗∗ , D.M. Korosteleva b∗∗∗
aKazan Federal University, Kazan, 420008 Russia
bKazan State Power Engineering University, Kazan, 420066 Russia
E-mail: ∗anton.samsonov.kpfu@mail.ru, ∗∗sergei.solovyev@kpfu.ru,
∗∗∗diana.korosteleva.kpfu@mail.ru
Received September 14, 2019
DOI: 10.26907/2541-7746.2020.1.52-65
For citation: Samsonov A.A., Solov’ev S.I., Korosteleva D.M. Asymptotic properties of / the problem on eigenvibrations of the bar with attached load. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 1, pp. 52–65. doi: 10.26907/2541-7746.2020.1.52-65. (In Russian)
Abstract
The ordinary second-order differential eigenvalue problem describing eigenvibrations of an elastic bar with a load attached to its end was investigated. The problem has an increasing sequence of positive simple eigenvalues with a limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. In the paper, the behavior of the solutions in dependence on the load mass was studied. More precisely, limit differential eigenvalue problems were formulated and the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity were proved. The original differential eigenvalue problem was approximated by the mesh scheme of the finite element method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions were established. The results of this paper can be generalized for the cases of more complicated and important applied problems on eigenvibrations of beams, plates, and shells with attached loads.
Keywords: eigenvibration of bar, eigenvalue, eigenfunction, eigenvalue problem, mesh approximation, finite element method
Acknowledgments. The study was supported by the Russian Foundation for Basic Research (project no. 19-31-90063).
References
- Babuˇska I., Osborn J.E. Eigenvalue problem. In: Handbook of Numerical Analysis. Vol. II: Finite element methods. Ciarlet P.G., Lions J.L. (Eds.). Amsterdam, North-Holland, 1991, pp. 642–787.
- Mikhailov V.P. Differentsial’nye uravneniya v chastnykh proizvodnykh [Partial Differential Equations]. Moscow, Nauka, 1983. 424 p. (In Russian)
- Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki [Equations of Mathematical Physics]. Moscow, Nauka, 1977. 736 p. (In Russian)
- Solov’ev S.I. Eigenvibrations of a bar with elastically attached load. Differ. Equations, 2017, vol. 53, no. 3, pp. 409–423. doi: 10.1134/S0012266117030119.
- Andreev L.V., Dyshko A.L., Pavlenko I.D. Dinamika plastin i obolochek s sosredotochennymi massami [Dynamics of Plates and Shells with Concentrated Masses]. Moscow, Mashinostroenie, 1988. 200 p. (In Russian)
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