Research of nonlinear vibrations of orthotropic plates with a complex form

Kurpa, L. V.; Shmatko, T. V.; Onufrienko, O. G.

doi:https://doi.org/10.1155/MPE/2006/26081

Mathematical Problems in Engineering

On this page

Abstract References Copyright Related Articles

Special Issue

Nonlinear Dynamics and Their Applications to Engineering Sciences

View this Special Issue

Open Access

Volume 2006 | Article ID 026081 | https://doi.org/10.1155/MPE/2006/26081

Research of nonlinear vibrations of orthotropic plates with a complex form

L. V. Kurpa,¹T. V. Shmatko,²and O. G. Onufrienko¹

Received30 Nov 2004

Revised27 May 2005

Accepted07 Jul 2005

Published25 Apr 2006

Abstract

This paper deals with effects of large amplitude on the free and forced flexural vibrations of elastic orthotropic plates of arbitrary shape. R-function method (RFM) is applied to obtain the basis functions need for expansion of sought solution into Fourier series. The initial nonlinear system of differential equations with partial derivatives is reduced to system of ordinary nonlinear differential equations by Galerkin procedure. The solving-obtained system is carried out by Runge-Kutta or Galerkin methods. The numerical results for the plate of complex form and also rectangular form and different boundary conditions have been presented and compared with other known results.

References

S. A. Ambartsumyan, The General Theory of Anisotropic Shells, Izdat. “Nauka”, Moscow, 1974.
View at: Google Scholar | MathSciNet
C. Y. Chia, Nonlinear Analysis of Plates, McGraw-Hill, New York, 1980.
View at: Google Scholar
H.-N. Chu and G. Herrmann, “Influence of large amplitudes on free flexural vibrations of rectangular elastic plates,” Journal of Applied Mechanics and Technical Physics, vol. 23, pp. 532–540, 1956.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. C. Dumir and A. Bhaskar, “Nonlinear forced vibration of orthotropic thin rectangular plates,” International Journal of Mechanical Science, vol. 30, no. 5, pp. 371–380, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Ganapathi, T. K. Varadan, and B. S. Sarma, “Nonlinear flexural vibrations of laminated orthotropic plates,” Computers & Structures, vol. 39, no. 6, pp. 685–688, 1991.
View at: Publisher Site | Google Scholar
K. Kanaka Raju and E. Hinton, “Nonlinear vibrations of thick plates using Mindlin plate elements,” International Journal for Numerical Methods in Engineering., vol. 15, no. 2, pp. 249–257, 1980.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. V. Kurpa, V. L. Rvachev, and E. Ventsel, “The $R$ -function method for the free vibration analysis of thin orthotropic plates of arbitrary shape,” Journal of Sound and Vibration, vol. 261, no. 1, pp. 109–122, 2003.
View at: Publisher Site | Google Scholar
T. Manoj, M. Ayyappan, K. S. Krishnan, and B. Nageswara Rao, “Nonlinear vibration analysis of thin laminated rectangular plates on elastic foundations,” ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik. Journal of Applied Mathematics and Mechanics, vol. 80, no. 3, pp. 183–192, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. Mei and K. Decha-Umphai, “A finite element method for nonlinear forced vibrations of rectangular plates,” AIAA Journal, vol. 23, no. 7, pp. 1104–1110, 1985.
View at: Google Scholar | Zentralblatt MATH
S. G. Mikhlin, Variational Methods in Mathematical Physics, Izdat. “Nauka”, Moscow, 2nd edition, 1970.
View at: Google Scholar | MathSciNet
M. K. Prabhakara and C. Y. Chia, “Non-linear flexural vibrations of orthotropic rectangular plates,” Journal of Sound and Vibration, vol. 52, no. 4, pp. 511–518, 1977.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
V. L. Rvachev, The Theory of $R$ -Functions and Some of Its Applications, Naukova Dumka, Kiev, 1982.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. L. Rvachev and L. V. Kurpa, $R$ -Functions in Problems of the Theory of Plates, Naukova Dumka, Kiev, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. L. Rvachev and L. V. Kurpa, “Application of $R$ -functions theory to plates and shells of a complex form,” Journal of Mechanical Engineering, no. 1, pp. 33–53, 1998 (Russian).
View at: Google Scholar
V. L. Rvachev and T. I. Sheiko, “ $R$ -functions in boundary value problems in mechanics,” Applied Mechanical Review, vol. 48, no. 4, pp. 151–188, 1995.
View at: Google Scholar
V. L. Rvachev and A. N. Shevchenko, The Problem-Orientation Languages and Systems for Engineering Research, Technica, Kiev, 1988.
View at: Google Scholar
B. S. Sarma, Nonlinear free vibrations of beams, plates and nonlinear panel flutter, M.S. thesis, Department of Aerospace Engineering, I. I. T. Madras, Chennai, 1987.
View at: Google Scholar | MathSciNet
Y. Shi and C. Mei, “A finite element time domain modal formulation for large amplitude free vibrations of beams and plates,” Journal of Sound and Vibration, vol. 193, no. 2, pp. 453–464, 1996.
View at: Publisher Site | Google Scholar
A. S. Vol'mir, Flexible Plates and Shells, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956.
View at: Google Scholar | MathSciNet
A. S. Vol'mir, Nonlinear Dynamics of Plates and Shells, Izdat. “Nauka”, Moscow, 1972.
View at: Google Scholar

Copyright

Copyright © 2006 L. V. Kurpa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

277

Downloads

888

Citations