Applied Mathematics, 2010, 1, 357-365
doi:10.4236/am.2010.15047 Published Online November 2010 (http://www.SciRP.org/journal/am)
Study the Thermal Gradient Effect on Frequencies of a
Trapezoidal Plate of Linearly Varying Thickness
Arun Kumar Gupta, Pragati Sharma
Department of Mathematics, M. S. College, Saharanpur, India
Received August 12, 2010; revised August 24, 2010; accepted August 27, 2010
Abstract
In this paper, effect of thermal gradient on vibration of trapezoidal plate of varying thickness is studied.
Thermal effect and thickness variation is taken as linearly in x-direction. Rayleigh Ritz technique is used to
calculate the fundamental frequencies. The frequencies corresponding to the first two modes of vibrations are
obtained for a trapezoidal plate for different values of taper constant, thermal gradient and aspect ratio. Re-
sults are presented in graphical form.
Keywords: Frequencies, Trapezoidal Plate, Linearly Thickness, Thermal Gradient
1. Introduction
With the advancement of technology, plates of variable
thickness are being extensively used in civil, electronic,
mechanical, aerospace and marine engineering applica-
tions. It becomes very necessary, now a day, to study the
vibration behaviour of plates to avoid resonance excited
by internal or external forces. A number of researchers
have worked on free vibration analysis of plates of dif-
ferent shapes and variable thickness. Trapezoidal plates
are widely used in various structures; however, they have
been poorly studied, unlike other plates. Trapezoidal
plates find various applications in the construction of
modern high speed air craft. The vibration characteristics
of such plates are of interest to the designer. Liew and
Lam [1] worked on the vibrational response of symmet-
rically laminated trapezoidal composite plates with point
constraints. Chopra and Durvasula [2] worked on the
vibration of simply supported trapezoidal plates. Ortho-
tropic plates with clamped boundary conditions were
studied by Narita, Maruyama and Sonoda [3]. Liew and
Lim [4] have studied the free transverse vibrational
analysis of symmetric trapezoidal plates with linearly
varying thickness. Qatu, Jaber and Leissa [5] have work-
ed on the natural frequencies of trapezoidal plates with
completely free boundaries. Qatu [6] presents the natural
frequencies for laminated composite angle-ply triangular
and trapezoidal plates with completely free boundaries.
Bambill, Laura and Rossi [7] have studied the transverse
vibrations of rectangular, trapezoidal and triangular
orthotropic cantilever plates. Chen, Kitipornchai, Lim
and Liew [8] have worked on the free vibration of canti-
levered symmetrically laminated thick trapezoidal plates.
Saliba [9-10] discussed the free vibration analysis of
simply supported symmetrical trapezoidal plates as well
as the transverse free vibration of fully clamped symmet-
rical trapezoidal plates. Krishnan and Deshpande [11]
have studied the free vibration analysis of the trapezoidal
plates. Gupta et al. [12-15] have worked on the vibration
analysis of visco-elastic rectangular plates of variable
thickness and studied the thermal gradient effect and
non-homogeneity effect on the free vibrations of the
plate. Huang, Hsu and Lin [16] studied the experimental
and numerical investigations for the free vibration of
cantilever trapezoidal plates. Tomar and Gupta [17-18]
studied the effect of thermal gradient on frequencies of
orthotropic rectangular plate of variable thickness in one
and two direction.
As till now no authors discussed the thermally induced
vibration of trapezoidal plate of variable thickness. So, in
this paper the temperature effect on vibration of trape-
zoidal plate of linearly varying thickness is studied. Free
transverse vibrations of trapezoidal plates of varying
thickness with two opposite simply supported edges have
been studied on the basis of classical plate theory. In
order to calculate natural frequencies for first and second
mode of vibration, Rayleigh Ritz method is used. The
frequencies for the first and second mode of vibration is
calculated for the trapezoidal plate having C-S-C-S edges
for the different values of taper constant , thermal gradi-

A. K. GUPTA ET AL. 358
ent and aspect ratio and presented in graphically form.
2. Method of Analysis & Equation of Motion
A symmetric trapezoidal plate, as shown in Figure 1, of
varying thickness is taken into consideration. The thick-
ness of the plate h(

) is linear in x direction and is of
the form [4]: 0
1
( )[1(1)()]
2
hh    (1)
where 0
h is the maximum plate thickness occurs at
left edge & 0 h

is the minimum plate thickness occur-
ring at the right edge,

is the taper constant.
For most of the elastic materials, modulus of elasticity
(as a function of temperature) is described [19] as
0(1 )EE

  (2)
where 0 E is the value of young’s modulus along the
reference temperature, i.e., at

=0 and

is the slope
of the variation of E with

.
Assuming that the temperature of the trapezoidal
plates varies linearly in x direction only and if

and
0

denote the increase in temperature above the refer-
ence at any point at distance a
x
 
and at the
end
1
2
  respectively,
then

can be expressed as
0
1
2






(3)
where a
x
 
On substituting value of

from (3) into (2), one get
0
1
1( )
2
EE






(4)
where 0



(0 1)


, known as thermal gradient.
The expression for kinetic energy V and strain energy
T as given by [4] are:
2
22
22 22
22
22 22
2
2
11
1
()
2
2(1 )
1
A
ww
ab
ab ww
VD dA
ab
w
ab









 




































 (5)
and T=
A
dAwh
ab 22 )(
2
 (6)
b/ 2
b/ 2
h
0
a/ 2 a/
2
c/2
c/2
x,

x,

z
y,

O
)]5.0)(1(1[)(0  hh
Figure 1. Geometry of trapezoidal plate with variable thickness.

A. K. GUPTA ET AL. 359
in which ()D

is the flexural rigidity of the plate, which
is given by

3
0
1
()1 12 DD




 


(7)
where
3
0
02
,,
12(1 )
Eh xy
D
ab


 
 (8)
is the flexural rigidity of the plate,

is the Poisson
ratio, A is the area of the plate,

is the mass density
per unit area of the plate and

is the angular fre-
quency of vibration.
Using (8) and (4) in (7)

3
00
2
3
1
() [112
12(1 )
1
1]
2
Eh
D






 





 


(9)
Using (9) in (5)

3
00
3
11
[1 11
2
222
12(1 )
2
22
11
22 22
]
2
22 2
11
2(1) 22 22
A
V
Eh ab
ww
ab
dA
ww w
ab
ab








 
 
 

 






 







 
 


 




 



(10)
Using (1) in (6)
22
0
1
1(1 )()
22
A
ab
Th wdA  




 (11)
3. Solution and Frequency Equations
Here Rayleigh Ritz technique is used for finding the so-
lution. According to this the maximum strain energy be
equal to the maximum kinetic energy i.e.
()0VT  
(12)
For the plate considered here boundaries are defined
by four straight lines
1
4242
1
4242
1
2
1
2
cc
bb
cc
bb









 


(13)
The two term deflection function taken as,
2
1
3
2
22
11
222 4
11
24 22
24 24
bc bc
wA
bcbc A
bc bcbc bc
 
 
 

 
 
 

 
 
 
 


 


 
 
 



 
 

 
 
 

(14)
where A1 and A2 are constants.
Equation (14) is taken to satisfy boundary condition
and provides a good estimation to the frequency.
Two edges of the plates are clamped and two are sim-
ply supported i.e. all the four degree of freedoms of the
nodes to the side faces of the plates are constrained.
Using (13) in (10) and (11)
V =

3
00
1
13
24 42 21
11
2
22
112(1 )
1
224
42
1
12
2
22
11
22 22
2
22 2
11
2(1) 22 22
c
c
bb
Eh
ab
c
c
bb
ww
ab
d
ww w
ab
ab



 

 


 

 
 








 











 





 

 
 




 




d 
(15)
1 1
4242 2
22
0
11
24242
1
[1 (1)()]
22
cc
bb
cc
bb
ab
Thwdd








(16)
Now Equation (12) becomes
2
11
()0VT 

 (17)

A. K. GUPTA ET AL. 360
where

1 1
3
4242
2
1
11
24242
2
22
22 22
2
22 2
22 22
1
11
22
1
12
11
11
2(1 )
cc
bb
cc
bb
ab
V
ww
ab
dd
ww w
ab ab





















 





 








 

  










(18)
1 1
4242 2
2
10
11
24242
1
[1 (1)()
2
cc
bb
cc
bb
Th wdd








(19)
In Equation (17),


2
00
252
2112
hE
a 


 is a frequen-
cy parameter.
Equation (17) involves the unknown, A1 and A2 arising
due to the substitution of w from Equation (14). These
unknowns are to be determined from Equation (17), for
which
2
11
1
2
11
2
()0
()0
VT
A
VT
A











(20)
On solving (20) we have
11220 mm
bAbA (21)
where 12
,( 1,2)
mm
bb m involves parametric const. and
frequency parameter. The determinant of the coefficient
of (21) must vanish for a non-zero solution.
Therefore the frequency equation comes out to be
11 12
21 22
0
bb
bb
 (22)
From Equation (22), a quadratic equation in 2

is
obtained which gives two values of 2

.
4. Result and Discussion
Frequency (22) is quadratic in2

, so it will gives two
roots. The frequency is calculated for the first two mode
of vibration for a c-s-c-s, trapezoidal plate with linearly
varying thickness, for various values ofa/b, c/b,

,

and

= .33. All these results are presented in
graphical form. First mode of vibrations is presented in
Figure (a) and second mode of vibrations is presented in
Figure (b).
Figures 2 and 3: In these figures different values of
taper constant

and different aspect ratio c/b = 1.0,0.5
and two values of thermal gradient 0.0,0.4

 are taken.
Figures 4 and 5: In these figures different values
of thermal gradient

and different aspect ratio c/b =
1.0,0.5, a/b = 1.0 and two values of taper constant
0.0,0.4


are taken.
Figures 6 and 7: In these figures different aspect ratio
a/b = 1.0,0.75, c/b = 1.0,0.75,0.50,0.25 and four combi-
nation of thermal gradient

and taper constant

i.e.
0.0, 0.0;0.0, 0.4;
0.4, 0.0;0.4, 0.4






Figures 2 and 3 show that increase in taper constant

makes a increase in frequency for 0.0,0.4

. Also
with increase in aspect ratio c/b the frequency decreases
for first two mode of vibration.
0.0 0.2 0.4 0.6 0.8 1.0
20
22
24
26
28
30


First mode
c/b=1.0, a/b=1.0
=0.0
=0.4
(a)

A. K. GUPTA ET AL. 361
0.0 0.2 0.4 0.6 0.8 1.0
80
90
100
110
120
130
140
150


Second mode
c/b=1.0, a/b=1.0
 =0.0
 =0.4
(b)
Figure 2. (a) Frequency parameter λ vs. taper constant α; (b) Frequency parameter λ vs. taper constant α.
0.0 0.2 0.4 0.6 0.8 1.0
31
32
33
34
35
36
37
38
39


First mode
c/b=.5, a/b=1.0
=0.0
=0.4
(a)
0.0 0.2 0.4 0.6 0.8 1.0
140
150
160
170
180
190
200
210
220
Second mode
c/b=.5, a/b=1.0
=0.0
=0.4


(b)
Figure 3. (a) Frequency parameter λ vs. taper constant

; (b) Frequency parameter λ vs. taper constant

.

A. K. GUPTA ET AL. 362
0.0 0.2 0.4 0.6 0.8 1.0
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
22.0
22.5


First mode
c/b=1.0, a/b=1.0
 =0.0
 =0.4
(a)
0.00.20.40.60.81.0
76
78
80
82
84
86
88
90
92
94
96
98
100
102
104
106


Second mode
c/b=1.0, a/b=1.0
 =0.0
 =0.4
(b)
Figure 4. (a) Frequency parameter λ vs. thermal gradient β; (b) Frequency parameter λ vs. thermal gradient β.
0.00.20.40.60.81.0
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5


First mode
c/b=0.5, a/b=1.0
 =0.0
 =0.4
(a)

A. K. GUPTA ET AL. 363
0.00.20.40.60.81.0
125
130
135
140
145
150
155
160
165
170


Second mode
c/b=0.5, a/b=1.0
 =0.0
 =0.4
(b)
Figure 5. (a) Frequency parameter λ vs. thermal gradient β; (b) Frequency parameter λ vs. thermal gradient β.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
20
24
28
32
36
40
44

c/b
First mode
a/b=1.0
 =0.0, =0.0
 =0.0, =0.4
 =0.4, =0.0
 =0.4, =0.4
(a)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
80
100
120
140
160
180
200
220

c/b
Second mode
a/b=1.0
 =0.0, =0.0
 =0.0, =0.4
 =0.4, =0.0
 =0.4, =0.4
(b)
Figure 6. (a) Frequency parameter λ vs. c/b; (b) Frequency parameter λ vs. c/b.

A. K. GUPTA ET AL. 364
0.3 0.40.5 0.60.7 0.8 0.91.0
15
20
25
30
35
40

c/b
First mode
a/b=.75
 =0.0, =0.0
 =0.0, =0.4
 =0.4, =0.0
 =0.4, =0.4
(a)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
60
70
80
90
100
110
120
130
140
150
160

c/b
Second mode
a/b=.75
 =0.0, =0.0
 =0.0, =0.4
 =0.4, =0.0
 =0.4, =0.4
(b)
Figure 7. (a) Frequency parameter λ vs. c/b; (b) Frequency parameter λ vs. c/b.
Figures 4 and 5 show that increase in thermal gradient

the frequency decreases. The result is display- ed for
the following two cases of taper constant: 0.0

, 0.4 .
Also with increase in aspect ratio c/b the frequency de-
creases for first two mode of vibration.
Figures 6 and 7 show that with increase in aspect ratio
c/b the frequency decreases for first two mode of vibra-
tion. Also with increase in aspect ratio a/b the frequency
increase for first two mode of vibration. Also the differ-
ence be tween the lines of 0.0,0.0 



and 0.4


,
0.4

 is negligible.
5. Conclusions
The Rayliegh-Ritz method has been successfully used to
make a comprehensive study of the linear vibration
characteristics of trapezoidal plates. Accurate data has
obtained by varying the length ratios a/b and c/b.
The results for trapezoidal plates of varying thickness
are verified by the literature (4). It has been shown that
the method provides accurate results.
These results have been determined with considerable
accuracy in order that they may serve as future data for
researchers who desire to investigate the accuracy of
their new numerical methods.
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