We analytically solve the problem of forced steady-state vibrations of an orthotropic rectangle. The problem is reduced to a quasiregular infinite system of linear equations by the superposition method. The use of Koyalovich limitants enables us to obtain bilateral estimates for the entire infinite sequence of unknowns. Numerical examples of the realization of an algorithm are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. T. Grinchenko and V. V. Meleshko, Harmonic Vibrations and Waves in Elastic Bodies [in Russian], Naukova Dumka, Kiev (1981).
L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [in Russian], Fizmatgiz, Moscow–Leningrad (1962).
S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body [in Russian], Nauka, Moscow (1977).
V. V. Meleshko and S. O. Papkov, “Bending vibrations of elastic rectangular plates with free ends: from Chladni (1809) and Ritz (1909) up to the present day,” Akust. Visn., 12, No. 4, 34–51 (2009).
S. O. Papkov, “Steady-state forced vibrations of a prism for given displacements on the boundary,” Akust. Visn., 11, No. 4, 36–43 (2008).
S. O. Papkov and V. N. Chekhov, “On the localization of natural vibrations of a rectangular prism by elimination of unknowns in a quasiregular infinite system,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 57–62 (2004).
V. N. Chekhov and A. V. Pan, “On limiting expressions of Koyalovich limitants,” Dopov. Nats. Akad. Nauk Ukr., No. 3, 31–36 (2007).
J. R. Banerjee and W. D. Gunawardana, “Dynamic stiffness matrix development and free vibration analysis of a moving beam,” J. Sound Vibrat., 303, 135–143 (2007).
M. I. G. Bloor and M. J. Wilson, “An approximate analytic solution method for biharmonic problem,” Proc. R. Soc. Lond. A, 462, 1107–1121 (2006).
V. V. Meleshko, “Bending of an elastic rectangular clamped plate: exact versus ‘engineering’ solutions,” J. Elasticity, 48, 1–50 (1997).
V. P. Revenko, “Three-dimensional stress state of an orthotropic rectangular prism under a transverse force applied at its end,” Prikl. Mekh., 41, No. 4, 30–37 (2005); English translation: Int. Appl. Mech., 41, No. 4, 367–373 (2005).
Y. F. Xing and B. Liu, “New exact solution for free vibrations of thin orthotropic rectangular plates,” Compos. Struct., 89, 567–574 (2009).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 55, No. 2, pp. 177–185, April–June, 2012.
Rights and permissions
About this article
Cite this article
Papkov, S.О. Steady-state forced vibrations of a rectangular orthotropic plate. J Math Sci 192, 691–702 (2013). https://doi.org/10.1007/s10958-013-1426-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1426-2