Modern Mechanical Engineering, 2011, 1, 1-5
doi:10.4236/mme.2011.11001 Published Online August 2011 (http://www.SciRP.org/journal/mme)
Analysis of Plate Vibration under Exponentially
Varying Thermal Condition
Tripti Johri, Indu Johri
Department of Applied Sciences & Humanities, J. P. Institute of Engineering & Technology, Meerut, India
Received July 23, 2011; revised August 6, 2011; accepted August 15, 2011
Abstract
A mathematical model is developed to assist the design engineers by analyzing the vibration response of
non-homogeneous orthotropic rectangular plate under exponentially varying thermal condition. Plate thick-
ness varies parabolically in both directions. Using Rayleigh Ritz approach, frequency parameter and two
term deflection function is calculated for diverse values of taper constants. For the best comprehension of the
vibration analysis, results are depicted graphically.
Keywords: Vibration, Non-Homogeneous, Plate, Thickness, Deflection Function, Taper Constants
1. Introduction
Study of vibration responses of plate has always been a
principal concern for design engineers. These plates are
used in numerous industries for the construction of in-
numerable vital structures and devices, such as space
shuttles, rockets, air craft’s, shafts, plate heat exchanger
and many more. Satellite antenna booms are used in
space as electric field or gravity-gradient probes. Heated
by sunlight, a temperature gradient is built up across the
cross-section. The thermal strain produces bending and
torsion of the boom. Due to the twist the direction of the
sunlight changes with respect to the cross-section and
another temperature distribution is obtained which again
causes another deformation. Similarly, a serious problem
in mechanical design of heat exchanger is flow induced
vibration.
Certain structures are less vulnerable against vibration
impacts whereas certain other are more vulnerable. As
we all know that vibration effects are now cannot be ne-
glected, as our day to day life is affected by them; From
kitchen to exercise centre, vibration effect are experi-
enced. The only thing is we do not put an eye on them
deliberately. Juicer, Mixer, massager, WholeBody Vi-
bration Plate (for fitness), all encompasses vibration
effect. Those were few inevitable and positive aspects of
vibration. Controlled vibrations are utilized in health
industry, paper industry, structural engineering, and
aeronautical engineering and in many more industries.
But uncontrolled vibration causes devastation. Occur-
rences of Tsunami, earthquake, collapse of structures are
few such most common devastating effects of vibration.
Thus the study of vibration responses in advance is of
immense importance for sustainable and positive effects
of vibrations for the well being of humans.
Monograph written by Leissa [1] is ample source of
knowledge in the field of plate vibration. Leissa [2] pro-
vides abundant knowledge about the numerous compli-
cating effects that can be introduced in the analysis of
plate vibration. Tomar and Gupta [3] evaluated the ex-
ponential thermal gradient effect on the vibration of
orthotropic rectangular plate with variation in thickness.
Rahimi & Davoodinik [4] studied the thermal behavior
of functionally graded plates under the exponential and
hyperbolic temperature conditions. They concluded that
temperature distribution profile plays vital role in ther-
mal resultant distribution of stresses and strains for FGP.
Shang, Wang and Li [5] analyzed the deformation of
laminated plates under exponential distributions of tem-
perature through the thickness. The plate under consid-
eration is simply supported. Javaheri & Eslami [6] used
classical plate theory for the buckling analysis of func-
tionally graded plates under four different types of ther-
mal load.Gupta, Johri & Vats [7] studied thermally in-
duced vibrations of an orthotropic rectangular plate using
Rayleigh Ritz approach. Gupta, Johri & Vats [8] calcu-
lated deflection function and frequency parameter for a
rectangular plate under the effect of linear temperature
distribution where thickness of plate was varied para-
bolically in both directions.

T. JOHRI ET AL.
2
In the present scenario design engineers are indulge in
making more efficient, vibration deficient and light
weight structures.
Present study is truly devoted for design engineers
utilizing rectangular plates for construction of devices or
structures. The effect of exponentially varying tempera-
ture distribution is analyzed for a non- homogeneous
orthotropic rectangular plate whose thickness varies
bi-directionally in parabolic manner. The non-homoge-
neity is assumed to arise due to the variation in the den-
sity of the plate material in linear manner along the
length of the plate. The frequencies and deflection func-
tion for first mode of vibration are calculated using
Rayleigh Ritz technique, for clamped plate, for diverse
values of non-homogeneity constant, taper constants and
temperature gradient. Results are demonstrated graphi-
cally.
Authentication of work is done by comparing the re-
sults for a uniform unheated homogeneous orthotropic
clamped rectangular plate with the results published by
the authors [3]. Results are found to be in good agree-
ment with those of published by Tomar & Gupta [3].
2. Methodology
Consider an orthotropic rectangular plate. Let us assume
that complicating effects are introduced in the plate by
density, thickness and thermal conditions.
Let the plate be subjected to an exponential thermal
variation along X-axis only, i.e.
0111
xa
ee
TT ee


 






(1)
where T is the temperature excess above the reference
temperature at a distance
x
a and 0 T is the tempera-
ture excess above the reference temperature at the end of
the plate i.e. at x = a, where a is length of plate.
Thickness h of the plate is assumed to be varying
parabolically in both directions, i.e.
22
01 2
22
11
x
y
hh ab

 
 
 
 
(2)
where, 0
00
x
y
hh


 and 1

& 2

are two taper con-
stants.
Non- homogeneity or variation in density

is as-
sumed to varying linearly along X-axis, i.e.,

01
1
x
a    (3)
where, 00
x


 & 1

is the non-homogeneity pa-
rameter.
For most orthotropic materials, moduli of elasticity (as
a function of temperature) are defined as [8],

 
 
 
1
2
0
1
1
1
x
y
xy
ET ET
ET ET
GT GT






(4)
where,
x
Eand
y
Eare Young’s moduli in x- and y-di-
rections respectively and
x
y
Gis shear modulus,

is
Slope of variation of moduli with temperature. Using
Equation (1) in Equation (4), one has,



1
2
0
11 11
11 11
11 11
xa
x
xa
y
xa
xy
ee
ET Eee
ee
ET Eee
ee
GT Gee














 

 






 
 

 





 









 

 

 


 



(5)
The governing differential equation of transverse mo-
tion of an orthotropic rectangular plate of variable thick-
ness in Cartesian coordinate [3], is;
44 222
44 222
2 2332
23322
22
22 222 2
11
22 22 22
2
2
22
222
4
0
x
xy
yy
xx
yxy
H w wwwww
DD Hxx
xy xyy
HD
DD wwww w
yyx y
x
xyxx
DD
DD www w
x
yxy yy xy yx
w
ht 
  
 

 



 

 



  

    



(6)
where w is transverse deflection of plate, at the point (x,
y),

is mass density per unit volume of the plate ma-
terial, t is time, h is thickness of the plate at the point (x,
y),
x
D & y
D are flexural rigidities in x- and y-direc-
tions respectively and
x
y
D is torsional rigidity [3],

3
12 1
x
x
x
y
Eh
Dvv ,

3
12 1
y
y
x
y
Eh
Dvv ,
3
12
xy
xy
Gh
D
(7)
1()
x
yyx
D
DD



 and 12
x
y
H
DD , where
x

&
y

are Poisson’s ratio.
Assuming time harmonic motion, solution of Equation
(6), may be written as,
(, ,)(, )it wxytW xye

 (8)
where, w is frequency in radian and W(x, y) is a two term
deflection function.
For Clamped rectangular plate two term deflection
function is expressed as,

T. JOHRI ET AL.
3

22 22
1
33 33
2
11
,
11
xy xy
cab a b
Wxy
xy xy
cab a b

 


 
 


 


 

 

(9)
where, 1
c and 2 c are constants to be evaluated.
For a clamped plate, boundary conditions are,
0
W
Wx


 at 0,
x
a
0
W
Wy


 at 0,yb
In order to calculate the frequency

, Rayleigh Ritz
Technique is employed which states that maximum strain
energy must be equal to maximum kinetic energy, i.e.

0UT   (10)
where, U is strain energy and T is kinetic energy for a
plate executing transverse vibrations of mode shape W(x,
y), and are written as [3], respectively,
22
22
22
2
00 22 2
122
1
2
24
xy
ab
xy
WW
DD
xy
Udydx
WW W
DD
xy xy

 

 
 


 

 
 


 

 
 


(11)
22
00
1dd
2
qb
Tp hWyx   (12)
Using Equations (2), (3), (6), (7) and (9) in Equations
(11) and (12) and then putting these values of U & T in
Equation (10), one has,

2
11
0UT   (13)
where,

42
0
2
2
10
12 1
x
y
avv
Eh
 
 is the frequency para-
meter. Equation (13) contains two unknown constants
1
c and 2 c to be evaluated. Employing the following
method, these constants may be evaluated:

2
11
0
k
UT
c
 
 (14)
where k = 1,2
On simplifying Equation (14), we get following form,
11220 kk
rcrc
(15)
where, 1k
r & 2k r involves the parametric constants
and the frequency parameter.
For a non-zero solution, determinant of coefficients of
Equation (15) must vanish. In this way frequency equa-
tion comes out to be, as below,
11 12
21 22
0
rr
rr
 (16)
3. Result
Frequency Equation (16) provides the value of fre-
quency parameter and deflection function for the first
two modes of vibration for different values of taper
constants, thermal gradient parameter and non-ho-
mogeneity constant, for a clamped plate with linear
variation in thickness in both directions. Limitation of
method used lies in the consideration of only first
mode of vibration [3].
The parameter for orthotropic material has been
taken as [8],
2
1
0.32
E
E, 2
1
0.04
x
E
vE,

0
1
10.09
xy
Gvv
E
Results are displayed graphically. Figure 1 depicts
the variation of frequency parameter Ω with the ther-
mal gradient parameter ‘
 ’ for the following two cases:
α1 = 0.0, β1 = 0.0, β2= 0.0 and α1 = 0.0, β1= 0.2, β2 = 0.6.
In Figure 2, Variation in frequency parameter with
non-homogeneity of the plate material is taken into
consideration for the following two cases:
α = 0.0, β1 = 0.0, β2 = 0.0 and α = 0.0, β1 = 0.2, β2 = 0.6
Figures 3 and 4 display the variation of taper con-
stant ‘β1’ and ‘β2’ with frequency parameter ‘Ω’, re-
spectively, for the following cases:
α1 = 0.0, α = 0.0, β2 or β1 = 0.0
α1 = 0.0, α = 0.0, β2 or β1 = 0.6
α1 = 0.0, α = 0.4, β2 or β1 = 0.0
α1 = 0.0, α = 0.4, β2 or β1 = 0.6
α1 = 0.8, α = 0.0, β2 or β1 = 0.0
α1 = 0.8, α = 0.0, β2 or β1 = 0.6
α1 = 0.8, α = 0.4, β2 or β1 = 0.0
α1 = 0.8, α = 0.4, β2 or β1 = 0.6
Figure 5 displays the variation of deflection func-
tion W with X for the following cases:
α1 = 0.0, α = 0.0, β1 = 0.0, β2=0.0, a/b =1.5 for Y =
0.2 and 0.4
α1 = 0.8, α = 0.4, β1 = 0.2, β2 = 0.6, a/b = 1.5 for Y =
0.2 and 0.4
4. Conclusions
From the above results it is seen that the frequency of
vibration reduces on increasing thermal gradient and
non-homogeneity, whereas increase in taper constants
increases the frequency of vibration. A comparative
study was carried out for the plates regarding variation in
thickness under exponential temperature gradient i.e.

T. JOHRI ET AL.
4
Figure 1. Frequency parameter ‘λ’ Vs. ‘α’.
Figure 2. Frequency parameter ‘λ’ Vs. ‘α1’.
Figure 3. ‘λ’ Vs. taper constant ‘β1’.
Figure 4. ‘λ’ Vs. taper constant ‘β2’.
Figure 5. Deflection Vs. X (= x/a).
plates with linear and parabolic variations in thickness
were compared. It was found that vibration effects were
significantly less pronounced (lesser values of frequency
parameter) for plates having parabolic bi-directional va-
riation in thickness as compared to that of linear bi-di-
rectional variation in thickness. Hence it is concluded
that plates with parabolic variation in thickness are more
stable as compared to those of linearly varying thickness,
for bearing up of exponential thermal gradient effects.
Yet it was well thought-out that as compared to expo-
nential variation in thermal gradient, linear variation in
temperature is better. Hence parabolic bi-directional va-
riation in thickness under linear temperature distribution
is a nice combination of conditions for the bearing up of
vibration effects, till the further considerations.
5. References
[1] A. W. Leissa, “Vibrations of Plates,” NASA SP-160,

T. JOHRI ET AL.
5
1969.
[2] A. W. Leissa, “Recent Research in Plate Vibrations:
Complicating Effects,” The Shock and Vibration Digest,
Vol. 9, No. 11, 1977, pp. 1-35.
[3] J. S. Tomar and A. K. Gupta, “Effect of Exponential
Temperature Variation on Frequencies of an Orthotropic
Rectangular Plate of Exponentially Varying Thickness,”
Proceedings of Workshop on Computer Application in
Continuum Mechanics, Department of Mathematics,
University of Roorkee, Roorkee, 11-13 March 1985, pp.
47-52.
[4] G. H. Rahimi and A. R. Davoodinik, “Thermal Behavior
Analysis of the Functionally Graded Timoshenko’s
Beam,” IUST International Journal of Engineering Sci-
ence, Vol. 19, 2008, pp. 105-113.
[5] F. L. Shang, Z. K. Wang and Z. H. Li, “Analysis of
Thermally Induced Cylindrical Flexure of Laminated
Plates with Piezoelectric Layers,” Composites Part B:
Engineering, Vol. 28, No. 3, 1997, pp. 185-193.R. Java-
heri and M. R. Eslami, “Thermal Buckling of Function-
ally Graded Plates,” AIAA Journal, Vol. 40, No. 1, 2002,
pp. 162-169.
[7] A. K. Gupta, T. Johri and R. P. Vats, “Study of Thermal
Gradient Effect on a Non-Homogeneous Orthotropic
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Variation,” Meccanica, Vol. 45, No. 3, 2010, pp. 393-
400. doi:10.1007/s11012-009-9258-3
[8] A. K. Gupta, T. Johri and R. P. Vats, “Thermal Effect on
Vibration of Non-homogeneous Orthotropic Rectangular
Plate Having Bi-directional Parabolically Varying Thick-
ness,” Proceedings of World Congress on Engineering
and Computer Science, San Francisco, 2007, pp. 784-
787.