An approach to calculate the natural frequencies of an elastic parallelepiped with different boundary conditions is proposed. The approach rationally combines the inverse-iteration method of successive approximations and the advanced Kantorovich–Vlasov method. The efficiency of the approach (the accuracy of the results and the number of approximating functions) is demonstrated against the Ritz method with different basis systems, including B-splines. The dependence of the lower frequencies of a three-dimensional cantilever beam on its cross-sectional dimensions is examined
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. A. Zhilin and T. P. Il’icheva, ”Spectra and modes of vibrations based on the three-dimensional theory of elasticity and plate theory,” Izv. AN USSR, Mekh. Tverd. Tela, No. 2, 94–103 (1980).
Von L. Collatz, Eigenvalue Problems with Engineering Applications [in German], Akad. Verlagsges., Leipzig (1963).
R. C. Barta, L. F. Qian, and L. M. Chen, ”Natural frequencies of thick square plates made of orthotropic, trigonal, monoclinic, hexagonal and triclinic materials,” J. Sound Vibr., 270, No. 4–5, 1074–1086 (2004).
E. I. Bespalova, ”Solving stationary problems for shallow shells by a generalized Kantorovich–Vlasov method,” Int.Appl. Mech., 44, No. 11, 1283–1293 (2008).
E. I. Bespalova and G. P. Urusova, ”Solving the torsion problem for anisotropic prism by the advanced Kantorovich–Vlasov method,” Int. Appl. Mech., 46, No. 2, 149–158 (2010).
Ya. M. Grigorenko, O. A. Avramenko, and S. N. Yaremchenko, ”Spline-approximation solution of two-dimensional problems of statics for orthotropic conical shells in a refined formulation,” Int. Appl. Mech., 43, No. 11, 1218–1227 (2007).
A. Ya. Grigorenko and T. L. Efimova, ”Using spline-approximation to solve problems of axisymmetric free vibration of thick-walled orthotropic cylinders,” Int. Appl. Mech., 44, No. 10, 1137–1147 (2008).
A. Ya. Grigorenko and S. A. Maltsev, ”Natural vibrations of thin conical panels of variable thickness,” Int. Appl. Mech., 45, No. 11, 1221–1231 (2009).
Ya. M. Grigorenko, N. N. Kryukov, and N. S. Yakovenko, ”Using spline functions to solve boundary-value problems for laminated orthotropic trapezoidal plates of variable thickness,” Int. Appl. Mech., 41, No. 4, 413–420 (2005).
A. W. Leissa and Z. Zhang, ”On the three-dimensional vibrations of cantilevered rectangular parallelepiped,” J. Acoust. Soc. Amer., 73, 2013–2021 (1983).
M. Levinson, ”Free vibrations of simply supported rectangular plates: An exact elasticity solution,” J. Sound Vibr., 98, 289–298 (1985).
K. M. Liew, K. C. Hung, and K. M. Lim, ”A continuum three-dimensional vibration analysis of thick rectangular plates,” Int. J. Solids Struct., 30, 3357–3379 (1993).
W. C. Lim, ”Three-dimensional vibration analysis of a cantilevered parallelepiped: exact and approximate solutions,” J. Acoust. Soc. Amer., 106, 3375–3381 (1999).
O. G. McGee and A. W. Leissa, ”Three-dimensional free vibrations of thick skewed cantilevered plate,” J. Sound Vibr., 144, 305–322 (1991).
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, No. 1–2, 329–353 (2008).
S. Srinivas, C. V. Joga Rao, and A. K. Rao, ”An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates,” J. Sound Vibr., 12, 187–199 (1970).
D. Zhou, Y. K. Cheung, F. T. K. Au, and S. H. Lo, ”Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method,” Int. J. Solids Struct., 39, 6339–6353 (2002).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 47, No. 4, pp. 76–88, July 2011.
Rights and permissions
About this article
Cite this article
Bespalova, E.I. Determining the natural frequencies of an elastic parallelepiped by the advanced Kantorovich–Vlasov method. Int Appl Mech 47, 410–421 (2011). https://doi.org/10.1007/s10778-011-0467-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-011-0467-9