 | | 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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| S.O. Papkov, "Asymptotically Exact Solution of the Problem of Harmonic Vibrations of an Elastic Parallelepiped," Mech. Solids. 52 (6), 686-699 (2017) |
| Year |
2017 |
Volume |
52 |
Number |
6 |
Pages |
686-699 |
| DOI |
10.3103/S0025654417060085 |
| Title |
Asymptotically Exact Solution of the Problem of Harmonic Vibrations of an Elastic Parallelepiped |
| Author(s) |
S.O. Papkov (Sevastopol State University, ul. Universitetskaya 33, Sevastopol, 299053 Russia;, stanislav.papkov@gmail.com) |
| Abstract |
An asymptotically exact solution of the classical problem of elasticity about the steady-state forced vibrations of an elastic rectangular parallelepiped is constructed. The general solution of the vibration equations is constructed in the form of double Fourier series with undetermined coefficients, and an infinite system of linear algebraic equations is obtained for determining these coefficients. An analysis of the infinite system permits determining the asymptotics of the unknowns which are used to convolve the double series in both equations of the infinite systems and the displacement and stress components. The efficiency of this approach is illustrated by numerical examples and comparison with known solutions. The spectrum of the parallelepiped symmetric vibrations is studied for various ratios of its sides. |
| Keywords |
rectangular parallelepiped, infinite system of linear equations, asymptotics, free vibrations, natural frequencies |
| References |
| 1. | G. Lamé,
Leçons sur la théorie mathématique de l'Élasticité des corps solides
(Mallet-Bachelier, Paris, 1852). |
| 2. | E. Mathieu,
Théorie de l'élasticité des corps solides
(Gauthier-Villars, Paris, 1890). |
| 3. | S. P. Timoshenko and S. Woinowsky-Krieger,
Theory of Plates and Shells
(McGraw-Hill, New York, 1959; Nauka, Moscow, 1966). |
| 4. | S. Iguchi,
"Die Eigenschwingungen mit Klangfiguren der vierseitig freien recteckigen Platte,"
Ingenieur-Archiv
21 (5-6), 303-322 (1953). |
| 5. | D. J. Gorman,
"Free Vibration Analysis of the Completely Free Rectangular Plate by the Method of Superposition,"
J. Sound Vibr.
57 (3), 437-447 (1978). |
| 6. | D. J. Gorman,
"Accurate In-Plane Free Vibration Analysis of Rectangular Orthotropic Plates,"
J. Sound Vibr.
323 (1-2), 426-443 (2009). |
| 7. | S. O. Papkov and J. R. Banerjee,
"A New Method for Free Vibration and Buckling Analysis of Rectangular Orthotropic Plates,"
J. Sound Vibr.
339, 342-358 (2015). |
| 8. | R. D. Mindlin and E. A. Fox,
"Vibrations and Waves in Elastic Bars of Rectangular Cross-Section,"
Trans. ASME J. Appl. Mech.
27 (1), 152-158 (1960). |
| 9. | V. T. Grinchenko and V. V. Meleshko,
Harmonic Vibrations and Waves in Elastic Bodies
(Naukova Dumka, Kiev, 1981)
[in Russian]. |
| 10. | J. R. Hutchinson and S. D. Zillmer,
"Vibrations of a Free Rectangular Parallelepiped,"
J. Appl. Mech.
50 (1), 123-130 (1983). |
| 11. | Y. Qu, G. Yuan, S. Wu, and G. Meng,
"Three-Dimensional Elasticity Solution for Vibration Analysis
of Composite Rectangular Parallelepiped,"
Eur. J. Mech. A/Solids
42, 376-394 (2013). |
| 12. | H. Nagino, T. Mikami, and T. Mizusawa,
"Three-Dimensional Free Vibration Analysis of Isotropic Rectangular Plates Using
the B-Spline Ritz Method,"
J. Sound Vibr.
317 (1-2), 329-353 (2008). |
| 13. | C. S. Huang, O. G. McGee, and K. P. Wang,
"Three-Dimensional Vibrations of Cracked Rectangular Parallelepiped
of Functionally Graded Material,"
Int. J. Mech. Sci.
70, 1-25 (2013). |
| 14. | W. Nowacki,
Theory of Elasticity
(PWN, Warszawa, 1970; Mir, Moscow, 1975). |
| 15. | M. V. Fedoryuk,
Asymptotics: Integrals and Series
(Nauka, Moscow, 1987)
[in Russian]. |
| 16. | A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,
Integrals and Series: Elementary Functions
(Nauka, Moscow, 1981)
[in Russian]. |
|
| Received |
23 June 2015 |
| Link to Fulltext |
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