American Journal of Computational and Applied Mathematics 2012, 2(1): 34-36 DOI: 10.5923/j.ajcam.20120201.06 Effect of Thermal Gradient on Vibration of Visco-Elastic Plate with Thickness Variation Anupam Khanna 1,* , Ashish Kumar Sharma 2 1 Dept. of Mathematics, MMEC, MMU (Mullana), Ambala, India 2 Research Scholar, Dept. of Mathematics, MMEC, MMU (Mullana), Ambala, India Abstract Visco- Elastic Plates are being increasingly used in the aeronautical and aerospace industry as well as in other fields of modern technology. Plates with variable thickness are of great importance in a wide variety of engineering appli- cations i.e. nuclear reactor, aeronautical field, naval structure, submarine, earth-quake resistors etc. The analysis is pre- sented here is to study the two dimensional thermal effect on vibration of visco-elastic square plate of variable thickness. Temperature & thickness both vary linearly in one direction and parabolically in another direction. A frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Both the modes of the frequency are calcu- lated by the latest computational technique, MATLAB, for the various values of taper parameters and temperature gradient. Keywords Visco-Elastic, Square Plate, Vibration, Thermal Gradient, Taper Constant 1. Introduction Since new materials and alloys are in great use in the construction of technically designed structures therefore the application of visco-elasticity is the need of the hour. Ta- pered plates are generally used to model the structures. Plates with thickness variability are of great importance in a wide variety of engineering applications. With the advancement of technology, the requirement to know the effect of temperature on visco-elastic plates of variable thickness has become vital due to their applications in various engineering branches such as nuclear power plants, engineering, industries etc. Further in mechanical system where certain parts of machine have to operate under ele- vated temperature, its effect is far from negligible and ob- viously cause non-homogeneity in the plate material i.e. elastic constants (young modulus etc.) of the materials be- comes functions of space variables. In an up-date survey of literature, authors have come across various models to account for non-homogeneity of plate materials proposed by researchers dealing with vibra- tion but none of them consider non-homogeneity with thermal effect on visco-elastic plates. It also indicates that sufficient work on one dimensional temperature variation has been done but negligible work has been done in the field of two dimensional temperature variation. Recently, A.K. Gupta and Anupam Khanna[1], studied the Thermal Effect on Vibrations of Parallelogram Plate of * Corresponding author: anupam_rajie@yahoo.co.in (Anupam Khanna) Published online at http://journal.sapub.org/ajcam Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved Linearly Varying Thickness. A.K. Gupta and Anupam Khanna[2], studied the Vibration of clamped visco-elastic rectangular plate with parabolic thickness variations. A.K. Gupta and Anupam Khanna[3], has been studied on Free vibration of clamped visco-elastic rectangular plate having bi-direction exponentially thickness variations. A.K. Gupta and A. Khanna[4], also studied the, Vibration of Visco- elastic rectangular plate with linearly thickness variations in both directions. Anupam Khanna, Ashish Kumar Sharma[5], studied the Study of free Vibration of Visco-Elastic Square Plate of Variable Thickness with Thermal Effect. Anupam Khanna, Ashish Kumar Sharma[6], has been studied on Vibration Analysis of Visco-Elastic Square Plate of Variable Thickness with Thermal Gradient. We assume that non homogeneity occurs in Modulus of Elasticity. For various numerical values of thermal gradient and taper constants; frequency for the first two modes of vibration are calculated with the help of latest software. All results are shown in Graphs. 2. Equation of Motion and Analysis The governing differential equation of transverse motion of a visco-elastic plate of variable thickness in Cartesian co-ordinates, as : 2 2 2 2 yx y x 2 2 2 M M M w 2 ρh x xy y t ∂ ∂ ∂ ∂ + + = ∂ ∂∂ ∂ ∂ (2.1) The expression for M x , M y , M yx are given by