Journal of Mathematical Sciences and Applications
ISSN (Print): 2333-8784 ISSN (Online): 2333-8792 Website: http://www.sciepub.com/journal/jmsa Editor-in-chief: Prof. (Dr.) Vishwa Nath Maurya, Cenap ozel
Open Access
Journal Browser
Go
Journal of Mathematical Sciences and Applications. 2013, 1(3), 43-49
DOI: 10.12691/jmsa-1-3-2
Open AccessArticle

Effect of Thermal Gradient on Vibration of Non-Homogeneous Parallelogram Plate of Linearly Varying Thickness in Both Directions

Arun Kumar Gupta1, , Kumud Rana2 and Dharma Veer Gupta3

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

2Department of Mathematics, Maharaja Agarsain Institute of Technology, Ghaziabad, U.P., India

3Department of Mathematics, College of Engineering Roorkee, Roorkee, U.A., India

Pub. Date: December 06, 2013

Cite this paper:
Arun Kumar Gupta, Kumud Rana and Dharma Veer Gupta. Effect of Thermal Gradient on Vibration of Non-Homogeneous Parallelogram Plate of Linearly Varying Thickness in Both Directions. Journal of Mathematical Sciences and Applications. 2013; 1(3):43-49. doi: 10.12691/jmsa-1-3-2

Abstract

The paper presented here is to study the effect of thermal gradient on vibration of non-homogeneous parallelogram plate of linearly varying thickness in both directions. Thermal induced vibration of non-homogeneous parallelogram plate has been taken as one dimensional temperature distribution in linear from only. For non-homogeneity of the plate material, density is assumed to vary linearly. Using the method of separation of variables; the governing differential equation is solved. An approximate but, quite convenient frequency equation is derived by Rayleigh-Ritz technique with two terms deflection function. The frequencies corresponding to the first two modes of vibration has been computed for a clamped parallelogram plate for different values of non -homogeneity constant, aspect ratio, thermal constant, thickness variation constant and skew angle.

Keywords:
thermal vibration non-homogeneous parallelogram plate linearlyvarying thickness

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Gutierrez, R.H. and Laura P.A.A. ‘Fundamental frequency of transverse vibration of rectangular, anisotropic plate of discontinuously varying thickness’, J. Sound and Vibration, Vol. 248, pp. 573-577, 2001. Hearmen, R.F.S.
 
[2]  ‘The frequency of a flexural vibration of rectangular orthotropic plates with clamped or supported edges’, J. Appl. Mach., Trans. ASME, Vol. 26,No 4, pp. 537-540, 1959.
 
[3]  Nair, P.S. and Durvasula, S.‘Vibration of skew plates’, J. Sound and Vibration, Vol. 26, pp. 1-20, 1973.
 
[4]  Liew, K. M. and Lim, M. K. ‘Transverse vibration of trapezoidal plates of variable thickness: symmetric trapezoids,’ J. Sound and Vibration, Vol. 165, No. 1, pp. 45-67, 1993.
 
[5]  Gupta, A.K. and Kumar S. ‘Thermal effect on vibration of orthotropic rectangular plate with thickness variation is linearly & parabolically in x- and y- directions respectively, Journal of Mathematical Sciences and Applications , Vol. 1, No. 2, pp.32-35, 2013.
 
[6]  Kumud and Gupta, D.V. ’Thermal effect on vibration of non-homogeneous parallelogram plate of linearly varying thickness” , Journal of Experimental & Applied Mechanics, Vol. 4, No. 1, pp. 8-14, 2013.
 
[7]  Gupta, A.K. and Kumud ‘Thermal effect on vibration of non-homogeneous parallelogram plate of parabolically varying thickness’ Asian Journal of Applied Sciences, Vol. 1, No. 1, pp. 50-58, 2013.
 
[8]  Gupta, A.K., Panwar, V. and Vats, R.P. ‘Thermal gradient effect on vibrations of non-homogeneous rectangular plate having continuously parabolically varying thickness in both directions, Journal of Experimental & Applied Mechanics (2011), 2(1), 38-46(India).
 
[9]  Gupta, A.K., Kumar, M., Kumar, S. and Khanna, A.‘Thermal effect on vibration of parallelogram plate of bi-directional linearly varying thickness, Applied Mathematics (2011), 2(1), 33-38(USA).
 
[10]  Leissa, A.W. ‘Vibration of plate’, NASA SP-160, U.S. Govt. Printing office, 1969.
 
[11]  Leissa, A.W. ‘Recent studies in plate vibration 1981-1985: Part II, complicating effects’, The shock and Vibration Digest., Vol. 19, pp.10-24, 1987.
 
[12]  Olsson, U. ‘On free vibrations at temperature dependent material properties and transient temperature fields’, J. Appl. Mach., Trans. ASME, Vol. 39, 723-726, 1972.