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IssuesArchive of Issues2017-6pp.686-699

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Total articles in the database: 10864
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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
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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).
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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
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