INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 2, No 2, 2011
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1 Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN Analysis of free virations of VISCO elastic square plate of variale thickness with temperature effect Anupam Khanna 1, Ashish Kumar Sharma 1 Asst. Professor., Dept. Of Mathematics, MMEC, MMU (MULLANA), AMBALA, INDIA Dept. Of Mathematics, MMEC, MMU (MULLANA), AMBALA, INDIA anupam_rajie@yahoo.co.in ABSTRACT A simple model presented here is to study the effect of linear thickness variation on viration of visco elastic square plate having clamped oundary condition on all the four edges. Two dimensional thermal effects on frequency of free virations of a visco elastic square plate of variale thickness are considered. An approximate ut quite convenient frequency equation is derived for a square plate (clamped at all the edges) y using Rayleigh Ritz technique with a two term deflection function. Both the modes of the frequency are calculated y the latest computational technique, MATLAB, for the various values of taper parameters and temperature gradient. All the results are presented in the graphs. Keywords: Visco elastic, Square Plate, Viration, Thermal gradient, Frequency 1. Introduction During the past four decades, viration of plates has ecome an important suject in engineering applications. There are several papers aout plate virations in open technical literature. Square plates have many engineering applications. These are commonly found in spacecrafts, missiles, land ase vehicles, off shore platforms, and underwater acoustic transducers. In the engineering we cannot move without considering the effect of viration ecause almost all machines and engineering structures experiences virations. As technology develops new discoveries have intensified the need for solution of various prolems of virations of plates with elastic or visco elastic medium. 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. Plates with thickness variaility are of great importance in a wide variety of engineering applications. Many researchers (A. Khanna, Ashish Kumar Sharma, 1) have analyzed the free viration of visco elastic plates with variale thickness for many years. The aim of present investigation is to study the two dimensional thermal effect on the viration of visco elastic square plate. It is also considered that the temperature varies linearly in oth the directions and thickness of square plate varies linearly in one direction and paraolic in other direction. It is assumed that the plate is clamped on all the four edges. Due to temperature variation, we assume that non homogeneity occurs in Modulus of Elasticity (E). 31
2 Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN For various numerical values of thermal gradient and taper constants; frequency for the first two modes of viration are calculated with the help of MATLAB. All results are shown in Graphs.. Equation of Motion Differential equation of motion for visco elastic square plate of variale thickness in Cartesian coordinate is given y equation (.1) [3]: ( ) ( ) ( ) [D W, + W, + W, + D, W, + W, + D W, + W, + 1 xxxx xxyy yyyy 1 x xxx xyy 1,y yyy yxx D 1,xx (W, xx + νw, yy ) + D 1,yy (W, yy + νw, xx ) + (1 ν )D1,xy W, xy ] ρ hp W = (.1) which is a differential equation of transverse motion for non homogeneous plate of variale thickness. Here, D 1 is the flexural rigidity of plate i.e. D = Eh v (.) 3 1 / 1(1 ) and corresponding two term deflection function is taken as [7] W = x a y a x a y a A + A x a y a x a y a (.3) [( / )( / )(1 / )(1 / )] [ 1 ( / )( / )(1 / )(1 / )] Assuming that the square plate of engineering material has a steady two dimensional temperature distriution i.e. ( 1 x/ a)( 1 y/a ) τ = τ (.4) where, τ denotes the temperature excess aove the reference temperature at any point on the plate and τ denotes the temperature at any point on the oundary of plate and a is the length of a side of square plate. The temperature dependence of the modulus of elasticity for most of engineering materials can e expressed in this form, E ( 1 γτ ) E = (.5) where, E is the value of the Young's modulus at reference temperature i.e. τ = and γ is the slope of the variation of E with τ. The modulus variation (.5) ecome E = E {(1 α )(1 x / a)(1 y / a )} (.6) where, α = γτ ( α < 1) thermal gradient. It is assumed that thickness varies linearly in one direction and paraolic in other direction as shown elow: h= h + x a + y a (.7) (1 β1 / )(1 β / ) where, β 1 & β are taper parameters in x & y directions respectively and h=h at x=y=. Put the value of E & h from equation (.6) & (.7) in the equation (.), one otain, 313
3 Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN ( )( ) ( ) ( ) 3 ( ) D1 = E 1 α 1 x/a 1 y /a h 1+ β1 x/a 1+ β y /a ]/1 1 ν (.8) Rayleigh Ritz technique is applied to solve the frequency equation. In this method, one requires maximum strain energy must e equal to the maximum kinetic energy. So it is necessary for the prolem under consideration that ** ** δ ( V T ) = (.9) for aritrary variations of W satisfying relevant geometrical oundary conditions. Since the plate is assumed as clamped at all the four edges, so the oundary conditions are W = W, = y x W = W, =,, x =, a The kinetic energy T** and strain energy V** are [] where, ** ** ( V λ T ) y =, a (.1) = (.11) 1 1 ** 3 3 = [1 α(1 )(1 )](1 + β1 ) (1 + β ) {(, XX ) + (, YY ) (.1) V X Y X Y W W + vw W + v W dydx, XX, YY (1 )(, XY ) } and 1 1 ** = + β + β 1 (.13) T [(1 X )(1 Y ) W ] dydx Here, λ = 1 ρ (1 v ) a / E h is a frequency parameter. Equation (.1) consists two unknown constants i.e. A 1 & A arising due to the sustitution of W. These two constants are to e determined as follows, n = 1, (.14) ** ** ( V λ T )/ A n On simplifying (.19), one gets n A 1 1 n A + =, n =1, (.15) where, n 1, n (n=1,) involve parametric constant and the frequency parameter. For a non trivial solution, the determinant of the coefficient of equation (.15) must e zero. So one gets, the frequency equation as = (.16) 314
4 Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN With the help of equation (.16), one can otains a quadratic equation in λ from which the two values of λ can found. These two values represent the two modes of viration of frequency i.e. λ 1 (Mode1) & λ (Mode) for different values of taper constant and thermal gradient for a clamped plate. 3. Result and Discussion Computation has een done for frequency of visco elastic square plate for different values of taper constants β 1 and β, thermal gradient α, at different points for first two modes of virations have een calculated numerically. In Figure 1 It is clearly seen that value of frequency decreases as value of thermal gradient increases from. to 1. for β 1 = β =., β 1 = β =.6 and β 1 = β =.8 for oth modes of virations. In Figure : It is evident that frequency increases continuously as increasing value of taper constant β 1 from. to 1. and i. β =., α=. ii. β =.6, α=.4 and iii. β =.8, α=.6 respectively. Frequency Vs Thermal gradient λ α Mode (β1 = β =.8 ) Mode (β1 = β =.6 ) Mode (β1 = β =. ) Figure 1: Frequency Vs Thermal gradient 315
5 Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN Frequency Vs Taper constant λ Mode (α =.6, β =.8 ) Mode (α =.4, β =.6 ) β1 Mode (α =., β =. ) Figure : Frequency Vs Tapper parameter 4. Conclusion So, main aim for our research is to develop a theoretical mathematical model for scientists and design engineers so that they can make a use of it with a practical approach, for the welfare of the human eings as well as for the advancement of technology. Acknowledgement It is with a feeling of great pleasure that I would like to express my most sincere heartfelt gratitude to Chancellor of MM University, Sh. Tarsam Garg and Head of the Mathematics Dept. Dr Deepak Gupta for providing me the necessary facilities in the department. I also very thankful to my Guide Dr Anupam Khanna, Asst Professor Dept. of Mathematics, MMU, Mullana, Amala for giving guidance, inspiration and lessings. 5. References 1. A.K. Gupta and Lalit Kumar. (8), Thermal effects on viration of nonhomogeneous visco elastic rectangular plate of linearly varying thickness in two directions, Meccanica, 43, pp
6 Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN A.K. Gupta and Anupam Khanna (1), Thermal Effect On Virations Of Parallelogram Plate Of Linearly Varying Thickness, Advanced Studies Of Theoretical Physics, 4(17), pp ). 3. A.K.Gupta and Harvinder Kaur. (8), Study of the effect of thermal gradient on free viration of clamped visco elastic rectangular plates with linearly thickness variations in oth directions, Meccanica, 43(4), pp A. Khanna, Ashish Kumar Sharma and Meenu Bhatia (11), Viration of Nonhomogenous Visco Elastic Square Plate of Variale Thickness in Both Directions, Innovative System Design and Engineering, U.S.A, (3). 5. A. Khanna, A. Kumar and M. Bhatia (11), A Computational Prediction on Two Dimensional Thermal Effect on Viration of Visco elastic Square Plate of Variale Thickness, Presented and Pulished in Proceeding of CONIAPS XIII, held in UPES, Deharadun. 6. A. Khanna, Ashish Kumar Sharma. (11), Study of free Viration of Visco Elastic Square Plate of Variale Thickness with Thermal Effect, Innovative System Design and Engineering, U.S.A, (3). 317
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