Advances in Physics Theories and Applications www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol 4, 2012 14 Vibrational Characteristics of C-C-C-C Visco-Elstic Plate with Varying Thickness Anupam Khanna* Dept. Of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana (INDIA) *E-mail of corresponding author: rajieanupam@gmail.com Abstract The main objective of the present study is to analyze vibrational characteristics of visco-elastic rectangular plate whose thickness varies in two directions. Two dimensional variation of thickness is considered as cubically in x-direction and linearly in y-direction. 4 Rayleigh- Ritz technique is used to get convenient frequency equation. Time period (K) and deflection (w) i.e. amplitude of vibrating mode at different instants of time for the first two modes of vibration are calculated for various values of taper constant and aspect ratio. Keywords: Visco-elastic, recatngular plate, variable thickness and vibration. 1. Introduction In these days’ researchers, scientists and technocrats are in search of material having less weight, size, low expenses, enhanced durability and strength. Now a days, plates of variable thickness commonly used in Modern technology such as in aerospace, nuclear plants, power plants etc. It may also used for construction of wings & tails of aero planes, rockets and missiles. There are different kinds of visco-elastic plates of variable thickness such as rectangular plates, square plates, circular plates, parallelogramic plates. Recent development in technology, such as aeronautical field, nuclear reactor etc., study of vibrations of visco elastic isotropic plates with one directional varying thickness has great importance. But a very little work have been done in this field of two dimensional varying thickness. Few papers are available on the vibrations of uniform visco-elastic isotropic beams and plates. It is assumed that rectangular visco-elastic plate is clamped support on all the four edges (C-C-C-C) and the visco-elastic properties of the plate are of the ‘Kelvin’ type. All the material constants used in numerical calculation have been taken for the alloy ‘DURALIUM’. Assuming that deflection is small so it is refered to amplitude of vibrating mode. Time period and deflection at different instant of time for first two modes of vibration for various values of aspect ratio (a/b) and taper constants (β 1 & β 2) are calculated. All the above results are also illustrated with graphs. 2. Analysis and Equation of Motion The equation of motion of a visco-elastic isotropic plate of variable thickness is [4]: [D 1 (∂ 4 W/∂x 4 +2∂ 4 W/∂x 2 ∂y 2 +∂ 4 W/∂y 4 )+2D 1 , x (∂ 3 W/∂x 3 +∂ 3 W/∂x∂y 2 )+2D 1,y ∂ 3 W/∂y 3 +∂ 3 W/∂x 2 ∂y)+D 1,xx (∂ 2 W/∂x 2 +ν∂ 2 W/∂y 2 )+D 1,yy (∂ 2 W/∂y 2 +ν∂ 2 W/∂x 2 ) +2(1-ν)D 1,xy ∂ 2 W/∂x∂y] - ρ h p 2 W = 0 ------(1) and .. ~ T+ p 2 DT =0 -------(2) where equations (1) and (2) are the differential equation of motion for isotropic plate of variable thickness and time function for visco-elastic plate for free vibration respectively with a constant p 2 . The expressions for Kinetic energy T and Strain energy V are [6]