Research 
Journal of 
Engineering 
Sciences _________________________
______
____
________
 
ISSN 2278 
–
 
9472
 
Vol. 
2
(
7
), 
6
-
9
, 
July 
(201
3
)
 
 
Res. 
J.
 
Engineering 
Sci.
 
 
 
International Science Congress Association
 
 
 
 
    
6
 
A Computational Model on: Vibration of Square Plate with Varying 
Thickness and Thermal effect in Two Directions
 
 
Sharma Subodh Kumar
1
 
and 
Sharma Ashish Kumar
2
 
1
Dept. of Mathematics, Govt. 
P.G College, Ambala Cantt., Haryana, INDIA
 
2
Dept. of Mathematics, Pacific University, Udaipur, Raj
a
sthan, INDIA
 
 
 
Available online at: 
www.isca.in
 
Received
 
28
th
 
May
 
201
3
, revised
 
27
th
 
June
 
201
3
, accepted
 
15
th
 
July 
201
3
 
 
 
Abstract
A computational model presented here is to study the vibration of visco
-
elastic isotropic square plate with thermal effect on 
two direction varying thickness parabolically
. Equation of 
frequency is derived by using Rayleigh
-
Ritz technique with a two
-
term deflection function. 
All the 
calcul
ation made for the first two modes of 
vibration, for various values of thermal gradients 
and taper constant.
 
 
Keywords:
Square 
p
late, 
f
requency, 
t
hickness, 
t
hermal 
e
ffect, 
t
aper 
c
onstant.
 
 
Introduction
 
Square plates have wide applications in ships, aircrafts, bridges, 
and so on. A thorough dynamic study of their behavior and 
characteri
stics 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 to their utility in aircraft 
wings. There are different kinds of visco
-
elastic plates o
f 
variable thickness such as rectangular plates, square plates, 
circular plates, parallelogramic plates.
 
 
Modern engineering structures are based on different design 
types, which involve various types of anisotropic and non
-
homogeneous materials in the for
m of their structural 
components. Depending upon the requirement, durability and 
reliability, materials are being developed so that they can be 
used to provide better strength and efficiency. The equipment 
used in air jets, communications, and in other sim
ilar 
technological industries take into consideration such materials, 
which not only reduce the weight and size but also are reliable 
in terms of efficiency, strength, and economy.
 
 
Further, the study of vibration behaviour in the presence of 
thermal gradi
ent of visco
-
elastic plates is required due to its 
practical importance in the field of engineering because 
Machines very repeatedly operate under diverse temperature 
conditions. In majority of cases the impact of temperature are 
ignored yet they need to b
e taken in to consideration. Most of 
engineering materials are found to have linear relationship 
between modulus of elasticity and temperature. Applications of 
such materials are due to lessening of weight and size, low 
operating cost and enhancement in ef
ficiency and strength.
 
 
The objective of the present study is the thermal effect on 
vibration of square plate of varying thickness parabolically. It is 
clamped supported on all the four edges.
 
 
Methodology
 
Differential equation of transverse motion of a vi
sco
-
elastic 
plate in Cartesian co
-
ordinates
1
:
 
2
2
+
2
2
+
2
2
=
ℎ
2
2
 
 
 
            
(1)
 
 
The expression for M
x
, M
y
, M
yx
 
are given by
 
=
−
 
1
 
2
2
+
2
2
 
 
=
−
 
1
ቀ
2
2
+
2
2
ቁ
 
 
=
−
 
1
(
1
−
)
 
2
    
 
 
 
          
(2)
 
 
where 
 
 
is visco
-
elastic operator.
 
On substitution the values M
x
, M
y
 
and M
yx
 
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:
 
൦
D
1
ቀ
∂
4
W
∂x
4
+
2
∂
4
W
∂x
2
∂y
2
+
∂
4
w
∂y
4
ቁ
+
2
∂D
1
∂x
ቀ
∂
3
W
∂x
3
+
∂
3
W
∂x∂y
2
ቁ
+
2
∂D
1
∂y
ቀ
∂
3
W
∂y
3
+
∂
3
W
∂x
2
∂y
ቁ
+
∂
2
D
1
∂x
2
ቀ
∂
2
W
∂x
2
+
ϑ
∂
2
W
∂y
2
ቁ
+
∂
2
D
1
∂y
2
ቀ
∂
2
W
∂y
2
+
ϑ
∂
2
W
∂x
2
ቁ
+
2
(
1
-
ϑ
)
∂
2
D
1
∂x∂y
∂
2
W
∂x∂y
 
ρ
hW
 
=
-
T
 
D
 
T
 
   
(3)
 
Here dot denote differentiation with respect to t, taking both 
sides of equation (3) are equal to a constant p
2
 
(square of
 
frequency), we have
 
 
[D
1
(W
,xxx
+2W
,xxyy
)
-
2D
1,x
(W
,xxx
+W
,xyy
)+2D
1,y
(W
,yyy
+W
,yxx
) 
 
+D
1,xx
(W
,xx
+W
,yy
)+D
1,yy
(W
,yy
+W
,xx
)+2(1
-
)D1
,xy
W
,xy
]
-
php
2
W=0
        
(4)
 
 
 
Research 
Journal of 
Engineering 
Sciences
___________
_________
_____
__________________
______
____
___
 
ISSN 2278 
–
 
9472 
 
Vol. 
2
(
7
),
 
6
-
9
, 
July 
(201
3
)
 
Res. 
J.
 
Engineering 
Sci.
 
 
 
International Science Congress Association
 
            
7
 
is a differential equation of transverse motion for non
-
homogeneous plate of variable thickness. Here, D
1
 
is the 
flexural rigidity of plate i.e.
 
1
=
ℎ
3
12
(
1
−
2
)
 
 
 
 
 
 
            
(5)
 
and corresponding two
-
term deflection function is taken as
2
 
 
=
ቂ
ቀ
ቁ
ቀ
ቁ
ቀ
1
−
ቁ
 
1
−
/
 
ቃ
2
ቂ
1
+
2
ቀ
ቁ
ቀ
ቁ
ቀ
1
−
ቁ
 
1
−
/
 
ቃ
 
 
   
(6)
 
 
In the above equation A
1
 
and A
2
 
are constants satisfy boundary 
conditions. Also, it is assumed that temperature varies 
parabolically in two directions i.e.
 
=
0
 
1
−
2
/
2
 
 
1
−
2
/
2
 
 
 
          
(7)
 
 
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 oundary of plate and “a” is the 
length of a side of square plate. The temperature dependence of 
the modulus of el
asticity for most of engineering materials can 
be expressed in this form
3
:
 
=
0
 
1
−
 
 
 
 
 
 
            
(8)
 
 
where,
 
E
0
 
is the value of the Young's modulus at reference 
temperature i.e. 
=
0
 
and  
 
is the slope of the variation of E 
with
 
. The modulus va
riation (5) become
 
=
0
ሾ
1
−
 
1
−
2
/
2
 
 
1
−
2
/
2
 
ሿ
 
          
(9)
 
where 
=
0
(
0
≤

1
)
, thermal gradient.
 
 
It is assumed that thickness also varies parabolic in x and y 
directions as shown below:
 
ℎ
=
ℎ
0
 
1
+
1
2
/
2
 
 
1
+
2
2
/
2
 
 
 
 
          
(10)
 
 
where 

1
 
is taper parameters in x
-
 
directions respectively and 
h=h
0
 
at
 
x=y=0.
 
 
 
Figure
-
1
 
Plate with parabolic varying thickness
 
 
Put the value of E 
and
 
h from equation (9) 
and
 
(10) in the 
equation (5), one obtain
 
1
=
 
0
 
1
−
 
1
−
2
/
2
 
 
1
−
2
/
2
 
൧
ℎ
0
 
1
+
1
2
/
2
 
 
1
+
2
2
/
2
 
൧
12
(
1
−
2
)
 
      
(11)
 
 
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
4
. So it is 
necessary for the problem under consideration that
 
 
∗
−
∗
 
=
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:
 
=
,
=
0
,
=
0
,
 
=
,
=
0
,
=
0
,
  
 
 
 
        
(13)
 
 
Now assuming the non
-
dimensional variables as
 
=
,
=
,
 
 
 
 
 
=
,
 
 
 
ℎ
 
=
ℎ
 
 
 
        
(14)
 
 
The kinetic energy K* and strain energy S* are 
5
 
∗
=
ቀ
1
2
ቁ
2
ℎ
0
 
 
 
5
 
 
ሾ
 
1
+
1
2
/
2
 
 
1
+
2
2
/
2
 
 
 
 
 
 
2
 
 
ሿ
1
0
1
0
 
 
 
    
 
(15)
 
and
 
∗
=
 
 
 
ሾ
1
−
 
1
−
2
 
 
1
−
2
 
ሿ
ሾ
 
1
+
1
2
/
2
 
 
1
+
2
2
/
1
0
1
0
2
    
 
3
 
,
2
+
,
2
+2
,
 
,
+2(1
−
)
,
2
 
(16)
 
 
where, 
=
0
ℎ
0
3
3
24
 
1
−
2
 
 
Using equations (15) 
and
 
(16) in equation (12), one get
 
 
 
∗
∗
−
2
∗
∗
 
=
0
 
 
 
 
 
          
(17)
 
 
where,
 
∗
∗
=
 
 
 
ሾ
1
−
 
1
−
2
 
 
1
−
2
 
ሿ
ሾ
 
1
+
1
0
1
0
1
2
/
2
 
 
1
+
2
2
/
2
 
 
 
 
 
 
ሿ
3
 
ቄ
 
 
,
 
2
+
 
 
,
 
2
+
2
 
,
 
 
 
 
 
,
+
2
(
1
−
)
 
 
,
 
2
 
൨
 
                                                                                         
 
 
 
and
 
  
∗
∗
=
 
 
 
 
1
+
1
2
/
2
 
 
1
+
2
2
/
2
 
 
 
 
 
 
 
2
 
 
 
 
 
൧
1
0
1
0
 
 
         
(19)
 
 
Here,
2
=
12
2
 
1
−
2
 
2
0
ℎ
0
2
 
is a frequency parameter.
 
Equation (19) consists two unknown constants i.e. A
1
 
and
 
A
2
 
arising due to the substitution of W. These two constants are to 
be determined as follows
6
:
 
  
 
∗
∗
−
2
∗
∗
 
,
=
0
, n=1, 
2
        
 
 
 
          
(20)
 
 
On simplifying (20), we get
 
1
1
+
2
2
=
0
, n=1, 2 
      
(21)
 
 
where bn
1
, bn
2 
(n=1,2) involve parametric constant and the 
frequency parameter.
 
 
For a non
-
trivial solution, the determinant of the coefficient of 
equation (21)
 
must be zero. So one gets, the frequency equation 
as
 
 
 
11
12
21
22
 
  
 
 
 
 
 
          
(22)
 
 
With the help of equation (22), one can obtains a quadratic 
equation in λ
2 
from which the two values of λ
 
2 
can found. These 
two values represent the two modes of
 
vibration of frequency 
i.e. λ
1 
(Mode1) 
and
 
λ
2 
(Mode2) for different values of taper 
constant and thermal gradient for a clamped plate.
 
(18)
 
Research 
Journal of 
Engineering 
Sciences
___________
_________
_____
__________________
______
____
___
 
ISSN 2278 
–
 
9472 
 
Vol. 
2
(
7
),
 
6
-
9
, 
July 
(201
3
)
 
Res. 
J.
 
Engineering 
Sci.
 
 
 
International Science Congress Association
 
            
8
 
Results and Discussion
 
Calculation of the frequency parameter are carried out with the 
help of computer software i.e. MATLAB. Computation has 
been done to obtain first two modes of frequency of square plate 
of variable thickness for different values of taper constants (

1
 
and 

2
), thermal gradient (α) at different points.
 
 
Table
-
1
 
Frequency vs. Thermal Gradient
 
α=0
 
β
1
= β
2
=0
 
β
1
= β
2
=0.6
 
0
 
0.2
 
0.4
 
0.6
 
0.8
 
1
 
140.88
 
134.28
 
127.34
 
119.99
 
112.18
 
103.78
 
35.99
 
34.31
 
32.52
 
30.63
 
28.60
 
26.39
 
273.42
 
264.27
 
254.81
 
244.98
 
234.77
 
224.12
 
70.88
 
68.46
 
65.91
 
63.21
 
60.31
 
57.17
 
It is clearly seen that value of frequency decreases as value of thermal gradient 
α
 
increases from 0.0 to 1.0 for 
β
1
 
= β
2
 
= 0.0 
and 
β
1
 
= 
β
2
 
=0.6 
for both modes of vibrations.
 
 
Table
-
2
 
Frequency vs. Taper 
parameter
 
β
1
 
β
2
=0.2 and α =0.3
 
β
2
=0.2 and α =0.6
 
0
 
0.2
 
0.4
 
0.6
 
0.8
 
1
 
147.14
 
166.37
 
186.58
 
207.61
 
229.33
 
251.64
 
37.78
 
42.79
 
48.09
 
53.63
 
59.35
 
65.22
 
135.71
 
154.40
 
174.01
 
194.36
 
215.37
 
236.93
 
34.82
 
39.67
 
44.78
 
50.11
 
55.59
 
61.22
 
Also it is obvious to 
understand the increment in frequency as value of tapper constant β
1
 
from 0.0 to 1.0 for
 
i. 
β
2
=0.2 and α =0.3.
, 
ii. 
β
2
=0.2 and α =0.6
 
 
 
Figure
-
2
 
Frequency vs. Thermal Gradient (For β
1
 
= β
2
 
= 0.0 
and 
β
1
 
= β
2
 
=0.6.)
 
Research 
Journal of 
Engineering 
Sciences
___________
_________
_____
__________________
______
____
___
 
ISSN 2278 
–
 
9472 
 
Vol. 
2
(
7
),
 
6
-
9
, 
July 
(201
3
)
 
Res. 
J.
 
Engineering 
Sci.
 
 
 
International Science Congress Association
 
            
9
 
 
Figure
-
3
 
Frequency vs. Taper Parameter 
(For β
2
=0.2 and α =0.3and β
2
=0.2 and α =0.6)
 
 
Conclusion
 
Motive is to provide such kind of a mathematical design so that 
scientist can perceive their potential in mechanical engineering 
field 
and
 
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/practition
ers.
 
Therefore mechanical 
engineers and technocrats are advised to study and get the 
practical importance of the present paper and to provide much 
better structure and machines with more safety and economy.
 
 
References
 
1.
 
Leissa 
A.W.
,
 
Vibration of plates
,
 
NAS
A SP
-
160 
(1969)
 
2.
 
Tomar J.S. and Gupta A.K., Effect of thermal gradient on 
frequencies of an orthotropic rectangular plate whose 
thickness varies in two directions, 
Journal sound and 
vibration
, 
98(2)
, 257
-
262 
(1985)
 
3.
 
Gupta A.K. and Lalit Kumar, Thermal effect
s on vibration 
of non
-
homogeneous visco
-
elastic rectangular plate of 
linearly varying thickness in two directions
,
 
Meccanica
, 
43
, 
47
-
54 
(2008)
 
4.
 
Gupta A.K. and Khanna A., 
Vibration of visco
-
elastic 
rectangular plate with linearly thickness variations in both
 
directions, 
J. Sound and Vibration
, 
301
, 450
-
457 
(
2007
)
 
5.
 
Larrondo H.A., Avalos D.R., Laura P.A.A. and Rossi R.E., 
Vibration of simply supported rectangular plates with 
varying thickness and same aspect ratio cutouts , 
J. Sound 
and Vibration
, 
244
, No.
4
, 738
-
746 
(2001)
 
6.
 
Gupta A.K. and Anupam Khanna, Free vibration of clamped 
visco
-
elastic rectangular plate having bi
-
direction 
exponentially thickness variations, 
Journal of Theoretical 
and Applied Mechanics
,
47(2)
, 457
-
471 
(2009)
 
7.
 
Khanna A. and Ashish Kumar 
Sharma
,Thermally Induced 
Vibration of Non
-
 
Homogenous Visco
-
Elastic Plate of 
Variable thickness, 
Advances in Physics Theories and 
Applications
, 
1
, 1
-
5 
(2011)
 
8.
 
Khanna A. and Ashish Kumar Sharma
, Analysis of free 
Vibration of Visco
-
Elastic Square Plate of Var
iable 
Thickness with Temperature effect, 
International Journal 
o
f 
Applied Engineering Research, 
Dindigul
, 
2
,(
2)
, 312
-
317 
(
2011
)
 
9.
 
Khanna A. and Ashish Kumar Sharma
, Mechanical 
Vibration of Visco
-
Elastic Plate with Thickness Variation, 
International Journal o
f Applied Mathematical Research
,
 
1(2),
150
-
158 
(2012)
 
10.
 
Khanna A. and Ashish Kumar Sharma
, Vibration Analysis 
of Visco
-
Elastic Square Plate of Variable Thickness with 
Thermal Gradient, 
International Journal of Engineering 
and 
Applied Sciences
, 
3(4),
 
1
-
6 
(2011)
 
 
<end>