Transverse Vibrations of Visco – Elastic Square Plate with Thickness and Temperature Variation

Transverse Vibrations of Visco – Elastic Square Plate with Thickness and Temperature Variation

Subodh Kumar Sharma, Ashish Kumar Sharma, Preeti Prashar

Associate Prof. Dept. Of Mathematics, P.G Degree College, Ambala Cantt. Haryana, India Research Scholar Dept. of Mathematics, Pacific University, Udaipur, Rajasthan, India Research Scholar Dept. of Mathematics, Pacific University, Udaipur, Rajasthan, India subodhamb@rediffmail.com, ashishk482@gmail.com,preetiprashar21@yahoo.in

Abstract

A mathematical model is developed with an aim that scientists and design engineers can make a use of it with a practical approach, for the welfare of the human beings. In which effect of thermal gradient is studied on vibration of homogeneous isotropic visco-elastic square plate having parabolic varying thickness. Using Rayleigh- Ritz procedure, frequency function is calculated for first mode of vibration for different values of thermal gradient and taper constants Parameter.

homogeneity occurs in Modulus of Elasticity. For various numerical values of thermal gradient and taper constants, frequencies 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

The governing differential equation of transverse motion of a visco-elastic plate of variable thickness in Cartesian co- ordinates, as below:-

1. Introduction

2M

x

2

2M

2

yx

2M

y

2

h

2 w

2

(1)

x xy y t

In the course of time, the study of vibration of plates has

The expression for Mx, My, Myx are given by

acquired great importance in the field of research, engineering and space technology. The visco-elastic behaviors of some materials invigorated scientists for modern designs and analysis techniques and their application to many practical problems. As technology

~ 2 w M x DD1 x 2

~ 2 w M y DD1 y 2

2 w

y 2

x 2

2 w

(2)

develops new discoveries have intensified the need for

~ 2 w

solution of various problems of vibrations of plates with

M yx

DD1 1

xy

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. Tapered plates are

On substitution the values Mx, My and Myx from equation

(2) in (1) and taking w, as a product of two function, equal to w(x,y,t)=W(x,y)T(t), equation (1) become:

4

4 W 4 W 4 W D 3W 3W D 3W 3W

generally used to model the structures. Plates with thickness

[D1 ( x

2

x2y2

) 2 1 (

y4 x

x3

xy

) 2 1 (

2 y

y3

x2y)

variability are of great importance in a wide variety of

2D

2 W

2 W

2D

2 W

2 W

2D 2 W T

1 (

) 1 (

) 2(1 ) 1 ] / hW –

engineering applications.

Plates of various geometries are commonly used as

x2

x2

y2

y2

y2

x2

xy xy D T

(3)

structural elements in various fields of engineering such as civil, naval and mechanical. In particular, rectangular plates are widely used in ocean structures and aerospace industry.

Here dot denote differentiation with respect to t. taking both sides of equation (3) are equal to a constant p2, we have

2

4 W 4 W 4 W D 3W 3W D 3W 3W

Plates with varying thickness possess a number of attractive

D1 2 2

4

2 1

4

3

2 1

2

3

2

x x y

y x x

xy y y x y

features such as material saving, weight reduction, stiffness

2D 2 W

2 W

2D 2 W

2 W

2D 2 W

enhancing, high strength and also meet the desirability of

1

2

2

1

2 2

2(1 ) 1 hp2 W 0

2

2

economy. A thorough dynamic study of their behavior and

x x

y

y y

x

xy xy

(4)

characteristics is essential to assess and use the full potentials of plates. In the aeronautical field, analysis of plates with variable thickness has been of great interest due

which is a differential equation of transverse motion for visco-elastic square plate of variable thickness. Here, D1 is the flexural rigidity of plate i.e.

to their utility in aircraft wings.

The aim of present investigation is to study one dimensional

D Ep / 12(1 v2 )

(5)

1

parabolic thermal effect on the vibration of visco-elastic square plate with varying thickness parabolic in one

and corresponding two-term deflection function is taken as [5]

direction. It is assumed that the plate is clamped on all the four edges and its temperature varies parabolic in one direction. Due to temperature variation, we assume that non

W [(x / a)( y / a)(1 x / a)(1 y / a)]2

[ A1 A2 (x / a)( y / a)(1 x / a)(1 y / a)]

(6)

Assuming that the square plate of engineering material has a steady one dimensional parabolic temperature distribution

T **

0 0

1 1[(1 X 2 )W 2 ]dYdX

(19)

0 0 1

i.e.

Here, 2 12(1 v2 )a2 / E h 2

is a frequency parameter.

0

(1 x2 / a2 )

(7)

Equation (19) consists two unknown constants i.e. A1 & A2

arising due to the substitution of W. These two constants are

where, denotes the temperature excess above the reference temperature at any point on the plate and 0 denotes the temperature at any point on the boundary of plate and a is the length of a side of square plate. The

to be determined as follows

n

(V ** 2T ** ) / A

On simplifying (2.17), one gets

, n = 1, 2 (20)

temperature dependence of the modulus of elasticity for

most of engineering materials can be expressed in this

E E0 1 – (8)

bn1 A1 bn2 A2 0 , n =1, 2 (21)

where, bn1, bn2 (n=1,2) involve parametric constant and the frequency parameter.

where, E0 is the value of the Young's modulus at reference For a non-trivial solution, the determinant of the coefficient

temperature i.e. 0 and is the slope of the variation of E with . The modulus variation (8) become

of equation (21) must be zero. So one gets, the frequency equation as

0

E E [1 (1 x2 / a2 )] (9)

b11b12 0

where,

0 (0 1) thermal gradient. It is

b21b22

(22)

assumed that thickness also varies one dimensional parabolic as shown below:

0 1

h h (1 x2 / a2 ) (10)

where, 1 is taper parameter in x direction respectively and h=h0 at x=y=0.

Put the value of E & h from equation (9) & (10) in the equation (5), one obtain

With the help of equation (22), one can obtains a quadratic equation in 2 from which the two values of 2 can found. These two values represent the two modes of vibration of frequency i.e. 1 (Mode1) & 2 (Mode2) for different values of taper onstant and thermal gradient for a clamped plate.

4. Result and Discussion

1 0 0 1

D [E [1(1 x2 / a2 )]h 3 (1 x2 / a2 )3 ] /12(1 v2 )

3. Solution of Equation of Motion

(11)

The frequency Equation (22), one can find a quadratic

equation in 2 from which two roots can be determined. The frequency parameter corresponding to the first two modes

Rayleigh-Ritz technique is applied to solve the frequency equation. In this method, one requires maximum strain energy must be equal to the maximum kinetic energy. So it is necessary for the problem under consideration that

of vibration of clamped square plate have been computed for various values of temperature gradient () and taper constant 1. All calculations are carried out with the help of latest Matrix Laboratory computer software

(V * T * ) 0

(12)

for arbitrary variations of W satisfying relevant geometrical boundary conditions. Since the plate is assumed as clamped at all the four edges, so the boundary conditions are

W=W, x =0, x=0,a

In Fig. 1: – It is clearly seen that value of frequency decreases as value of thermal gradient increases from 0.0 to

1.0 for 1=0.0, 1=0.4 and 1=0.8 for both modes of vibrations.

In Fig. 2 :- Increasing value of frequency for both of the

W=W,y =0, y=0,a

(13)

modes of vibration is shown for increasing value of taper constant 1 from 0.0 to 1.0 for =0.2 , =0.4 and =0.8

Now assuming the non-dimensional variables as

X x / a,Y y / a,W W / a, h h / a (2.14)

The kinetic energy T* and strain energy V* are [2]

respectively. Note that in this case the value of frequency increased.

T * (1/ 2) p2 h a5

1 1[(1 X 2 )W 2 ]dYdX

(15)

and

0 0 0 1

0 0

XX YY XY

V * Q 1 1

2 2 3 2 2

[1 (1 X )](1 1 X ) {(W ,XX ) (W ,YY )

2vW , W , 2(1 v)(W , )2}dYdX

(16)

where,

Q E h 3a3 / 24(1 v2 )

0 0

Using equations (15) & (16) in equation (12), one get

1 1

where,

(V ** 2T ** ) 0

(17)

XX YY XY

V **

2 2 3 2 2

0 0

[1 (1 X )](1 1 X ) {(W ,XX ) (W ,YY )

2vW , W , 2(1 v)(W , )2}dYdX

(18) and

Frequency Vs Taper Constant

210

180

Mode 1 (=0.2)

150

120

0 0.2 0.4 0.6 0.8 1

1

30

0

Mode 3 (=0.8)

90

60

Mode 2 (=0.4)

Fig. 1:- Frequency vs. Thermal Gradient

Frequency Vs Thermal Gradient

200

Mode 1

180 (1=0.0)

160

140

120

100

Mode 2 (1=0.4)

80

60 Mode 3

(1=0.8)

40

20

0

0 0.2 0.4 0.6 0.8 1

Fig. 2:- Frequency vs. Taper Constant

1. Conclusion

   Motive is to provide such kind of a mathematical design so that scientist can perceive their potential in mechanical engineering field & increase strength, durability and efficiency of mechanical design and structuring with a practical approach .Actually this is the need of the hour to develop more but authentic mathematical model for the help of mechanical engineers/researchers/practitioners.

2. References

1. A. W. Leissa, Vibration of plates, NASA SP 160 ,1969.

2. Subodh Kumar Sharma and Ashish Kumar Sharma , Free Transverse Vibration of Visco-elastic C-C-C-C square plate with Thickness and Temperature Variation , Research Journal of Mathematical and Statistical Sciences, Vol. 1(7), pp. 1-11, 2013

3. A. W. Leissa, The free vibration of rectangular plates,

   Journal of Sound and Vibration, Vol. 31, 257-293,1973.

4. Subodh Kumar Sharma and Ashish Kumar Sharma , A Computational Model on : Vibration of square plate with varying Thickness and Thermal effect in two Directions , Research Journal of Engineering Sciences, Vol. 2(7), pp. 6-9, 2013.

5. A. Khanna, Ashish Kumar Sharma. Study of free Vibration of Visco-Elastic Square Plate of Variable Thickness with Thermal Effect, Innovative System Design and Engineering,

   U.S.A. Vol.2. No. 4, 2011.

6. A. Khanna, Ashish Kumar Sharma, Vibration Analysis of Visco-Elastic Square Plate of Variable Thickness with Thermal Gradient, International Journal of Engineering and Applied Sciences, Turkey, Vol. 3 Issue4, pp.1-6, 2011.