American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2013, 1(5), 103-109
DOI: 10.12691/ajams-1-5-5
Open AccessArticle

Vibration of Visco-Elastic Parallelogram Plate with Thickness Variation Linearly in one Direction and Parabolic in another Direction

Arun Kumar Gupta1, and Anuj Kumar1

1Department of Mathematics, M. S. College, Saharanpur, U.P., India

Pub. Date: November 15, 2013

Cite this paper:
Arun Kumar Gupta and Anuj Kumar. Vibration of Visco-Elastic Parallelogram Plate with Thickness Variation Linearly in one Direction and Parabolic in another Direction. American Journal of Applied Mathematics and Statistics. 2013; 1(5):103-109. doi: 10.12691/ajams-1-5-5


The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies bi-directionally. It is assumed that the plate is clamped on all the four edges and that the thickness varies linearly in one direction and parabolically in another direction. Using the separation of variables method and Rayleigh-Ritz technique with a two-term deflection function, the governing differential equation has been solved for vibration of visco-elastic parallelogram plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection is made. Visco-elastic of the plate is taken of the “Kelvin Type”. Time period and deflection function at different point for the first two modes of vibration are calculated for various values of taper constant, aspect ratio and skew angle and results are presented in tabular form. Alloy “Duralumin” is considered for all the material constants used in numerical calculations.

vibration parallelogram plate visco-elastic mechanics linear thickness variation parabolic thickness variation both directions

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[1]  Leissa A. W., Vibration of Plate, NASA SP-160, (1969).
[2]  Leissa A. W., Plate Vibration Research, 1976-1980, Classical Theory, The Shock and Vibration Digest, 13(9), (1981), 11-22.
[3]  Leissa A. W., Recent Studies in Plate Vibrations: Part I, Classical Theory, The Shock and Vibration Digest, 19(2), (1987),11-18.
[4]  Sobotka Z., Analysis of thick visco-elastic plates, Theoretical and Applied Mechanics, Proceeding Vol. I. Publishing House of the Bulgarian Academy of Sciences, Sofia., (1971),379-386.
[5]  Bland D.R., The theory of linear visco-elasticity, Pergamon Press, (1960).
[6]  Ilanko S., Comments on the historical bases of the Rayleigh and Ritz methods, J. Sound and Vibration, 319( 1-2),(2009), 731-733.
[7]  Eslami M. R., Shakeri M., Ohadi A.R. and Shiari B., Coupled thermo elasticity of shell of revolution effect of normal stress and coupling, AIAA Journal, 37(4), (1999), 496-512.
[8]  Leissa A. W. and Narita Y., Vibration studies for simply supported symmetrically laminated rectangular plates, Composite Structures, 12,(1989), 113-132.
[9]  Lekhnitski S.G., Anistropic plates, Ist Ed.English trans., Am.Iron and Steel Inst. (New York N.Y.) ( 1956).
[10]  Zhang L. and Zu J.W., Non-linear vibrations of visco-elastic moving belts part-I: force vibration analysis, J. Sound and Vibration, 216(1),(1998),75-91.
[11]  Zhang L and Zu J.W., Non linear vibration of parametrically excited visco-elastic moving belts part-II stability analysis, J. Appl.Mach., Trans. ASME, 66(2), (1999), 403-409.
[12]  Mivhel R., A periodic problem in visco-elasticity with variable coefficients, Int. J. of Engg. Sci., 19, (1981), 1145-1168.
[13]  Garrick I.E., Survey of Aero-thermo-elasticity, J. Aerospace Engg.,22,(1963), 140-147.
[14]  Gnossi R. O. and Laura P. A. A., Transverse vibrations of rectangular orthotropic plates with one or two free edges while the remaining are elastically restrained against rotation, Ocean Engg.,6(5),(1979),527-539.
[15]  Park J. and Mongeau Luc., Vibration and radiation of visco-elastically supported mindlin plates, J. Sound and Vibration, 318(4-5),(2008),1230-1249.
[16]  Gupta A.K., Kumar A. and Gupta Y.K., Vibration study of visco-elastic parallelogram plate of linearly varying thickness, International Journal of Engineering and Interdisciplinary Mathematics, 2(1),(2010),1-9.
[17]  Nagaya K., Vibrations and dynamic response of visco-elastic plates on non-periodic elastic supports, J. Engg. for Industry, 99, (1977),404-409.