 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| L.D. Akulenko and S.V. Nesterov, "Anomalous Dependence of the Vibration Frequencies of a Rod in an Elastic Medium on the Rod Length," Mech. Solids. 45 (2), 257-263 (2010) |
| Year |
2010 |
Volume |
45 |
Number |
2 |
Pages |
257-263 |
| DOI |
10.3103/S0025654410020111 |
| Title |
Anomalous Dependence of the Vibration Frequencies of a Rod in an Elastic Medium on the Rod Length |
| Author(s) |
L.D. Akulenko (Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, bolotnik@ipmnet.ru)
S.V. Nesterov (Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526 Russia, kumak@ipmnet.ru) |
| Abstract |
We use numerical-analytic methods to study the influence of the length of a thin inhomogeneous rod on its natural frequencies and the shapes of its plane transverse vibrations. We found that the existence of an external elastic medium described by the Winkler model can lead to an anomalous effect, i.e., to an increase in the natural frequencies of the vibration lower modes as the rod length increases continuously. We discovered rather subtle properties of this phenomenon in the case of variations in the length, the mode number, and the fixation method. We separately studied vibrations for the standard boundary conditions: fixation, hinged fixation, tangential fixation, and free end. We calculated several simple examples illustrating the anomalous dependence of the frequency of the rod natural vibrations in a strongly inhomogeneous elastic medium with different boundary conditions. |
| Keywords |
inhomogeneous rod, elastic medium, boundary conditions, natural frequencies and natural shapes, length, anomalous dependence |
| References |
| 1. | S. Timoshenko, D. H. Young, and W. Wear, Jr.,
Vibration Problems in Engineering, 4th Ed.
(Wiley, New York, 1974; Mashinostroenie, Moscow, 1985). |
| 2. | R. Courant and D. Hilbert,
Methods of Mathematical Physics
(Wiley, New York, 1951; Gostekhizdat, Moscow-Leningrad, 1951). |
| 3. | S. H. Gould,
Variational Methods for Eigenvalue Problems
(Univ. Press, Toronto-London-Oxford, 1966; Mir, Moscow, 1970). |
| 4. | L. D. Akulenko and S. V. Nesterov,
High-Precision Methods in Eigenvalue Problems and Their Applications
(CRC Press, Boca Raton, 2005). |
| 5. | L. D. Akulenko and S. V. Nesterov,
"Solution of Self-Adjoint Boundary-Value Problems of Order Four,"
Russ. J. Math. Phys.
10 (2), 121-133 (2003). |
| 6. | L. D. Akulenko, G. V. Kostin, and S. V. Nesterov,
"Numerical-Analytical Method for Studying Natural Vibrations of Nonuniform Rods,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 5, 180-191 (1995)
[Mech. Solids (Engl. Transl.)
30 (5), 173-182 (1995)]. |
| 7. | L. D. Akulenko and S. V. Nesterov, "A Frequency-Parametric
Analysis of Natural Vibrations of Non-Uniform Rods," Prikl. Mat.
Mekh. 67 (4), 588-602 (2003) [J. Appl. Math. Mech. (Engl.
Transl.)
67 (4), 525-537 (2003)]. |
| 8. | M. V. Ostrogradskii,
Differential Equations of the Problem of Isoperimeters,
ed. by L. S. Polak
(Fizmatlit, Moscow, 1959)
[in Russian]. |
| 9. | L. D. Akulenko and S. V. Nesterov,
"The Dependence of the Natural Frequencies of a One-Dimensional Elastic System on Its Length,"
Prikl. Mat. Mekh.
70 (3), 450-457 (2006)
[J. Appl. Math. Mech. (Engl. Transl.)
70 (3), 408-415 (2006)]. |
| 10. | G. A. Korn and T. M. Korn,
Mathematical Handbook for Scientists and Engineers
(Dover, New York, 2000; Nauka, Moscow, 1970). |
|
| Received |
27 October 2009 |
| Link to Fulltext |
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