Abstract
In this paper, we study the problem of the boundary accumulation of a discrete spectrum, which is essential for a boundary-value problem of fourth order arising in the theory of small transverse vibrations in an inhomogeneous viscoelastic rod (a Kelvin—Voigt body). We establish conditions for such an accumulation and its asymptotics, which are expressed in terms of the coefficients defining the problem posed by the differential expression. The results obtained are illustrated by numerical computation data.
Similar content being viewed by others
References
V. N. Pivovarchik, “A boundary-value problem related to the vibrations of the rod with internal and external friction,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], (1987), no. 3, 68–71.
P. Lancaster and A. Shkalikov, “Damped vibration of beams and related spectral problems,” Canad. Appl. Math. Quat., 2 (1994), no. 1, 45–90.
R. O. Griniv and A. A. Shkalikov, “On the operator pencil appearing in the problem of the vibrations of the rod with internal friction,” Mat. Zametki [Math. Notes], 56 (1994), no. 2, 114–131.
A. A. Vladimirov, “On the accumulation of the eigenvalues of the differential operator functions,” Uspekhi Mat. Nauk [Russian Math. Surveys], 57 (2002), no. 1, 151–152.
R. Mennicken and H. Schmid, and A. A. Shkalikov, “On the eigenvalue accumulation of Sturm—Liouville problem depending nonlinearly on the spectral parameter,” Math. Nachr., 189 (1998), 157–170.
A. A. Vladimirov, R. O. Griniv, and A. A. Shkalikov, “Spectral analysis of the periodic differential matrices of mixed order,” Trudy Moskov. Mat. Obshch. [Trans. Moscow Math. Soc.], 63 (2002), 45–86.
F. S. Rofe-Beketov and A. M. Khol’kin, Spectral Analysis of Differential Operators: Relation between Spectral and Oscillatory Properties [in Russian] Mariupol, 2001.
M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow, 1969.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 79, no. 3, 2006, pp. 369–383.
Original Russian Text Copyright ©2006 by A. A. Vladimirov.
Rights and permissions
About this article
Cite this article
Vladimirov, A.A. On the accumulation of eigenvalues of operator pencils connected with the problem of vibrations in a viscoelastic rod. Math Notes 79, 342–355 (2006). https://doi.org/10.1007/s11006-006-0039-1
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0039-1