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IIUM Engineering Journal, Vol. 2, No. 1, 2001  A. A. Rizk and A. S. Ashour 
 

On leave from Faculty of Engineering, Alexandria University, Alexandria, Egypt. 
Corresponding Author 
 23

FREE VIBRATION OF VARIABLE THICKNESS PLATES USING 
CHARACTERISTIC ORTHOGONAL POLYNOMIAL STRIP FUNCTIONS 

SUBJECTED TO DIFFERENT COMBINATIONS OF BOUNDARY 
CONDITIONS 

A. A. Rizk and A. S. Ashour 

Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, 
53100 Kuala Lumpur, Malaysia 

Email:, ashour@iiu.edu.my 

 
ABSTRACT: A set of characteristic orthogonal 
polynomial strip functions was used to study the 
flexural vibration of variable thickness plates in one 
direction. The starting function in the set, which is 
satisfying the geometry and natural conditions, is 
developed from the solution of a static beam subjected 
to an arbitrary load that expanded into a Taylor series. 
The Gram-Schmidt orthogonalization process is used to 
generate the reminder functions of the set. The natural 
frequencies of the plat are calculated by using the finite 
strip method together with the transition matrix. 
Numerical results are obtained for different combination 
of boundary conditions, tapered ratio and aspect ratio. It 
is demonstrated that the method as well as the 
orthogonal strip functions yield good accuracy and fast 
convergence by comparisons with available results in 
the literature. This technique can also be used to solve 
some other complicated problems in the engineering 
applications.    

1. INTRODUCTION: 

Plates of variables thickness are commonly used as 
structural elements in many engineering applications 
such as aerospace, civil, and ocean engineering systems. 
The natural frequencies and mode shapes of the plate 
are useful in their operation environments. In the 
literature, there are many publications concerning the 
flexural vibration of uniform and non-uniform plates, 
see for examples Lissisa [1-5]. In last decades, many of 
the researchers are used different admissible functions 
to obtain the natural frequencies of rectangular and 
skew plates of variable thickness. Ashour [6] used the 
eigen functions derived from the solution of a classical 
beam vibration as basis functions. Cheung and Zhou [7] 
used a set of admissible functions that are developed 
from static solution of beam under arbitrary Taylor 
series load in Rayleigh-Ritz method. Zhou [8] used a 
combination of sine series and polynomials as the basis 
functions in Rayleigh-Ritz method. Grossi and Bhat [9] 
used boundary characteristic orthogonal polynomials in 
Rayleigh-Ritz method and applied Rayleigh-Schmidt 

method in their analysis. Singh and Saxena [10-12] used 
boundary characteristic orthogonal polynomials in two 
variables in Rayleigh-Ritz method. Also Lewi and lam 
[13], and Bhat [14] used two-dimensional orthogonal 
polynomials. Malhotra [15] used the conventional beam 
functions as the basis functions. Bhat [16] used a set of 
orthogonal polynomial functions in Rayleigh-Ritz 
method. 
 
In this paper, a set of orthogonal polynomial shape 
functions in x-direction is developed to study the free 
vibration of variable thickness plate in y-direction. The 
first strip function in the set that satisfies the geometry 
and natural boundary conditions is the first term in the 
series solution obtained by Cheung and Zhou [7] as a 
new set of admissible functions that are the static 
solution of the beam under an arbitrary static load 
expanded into Taylor series. The rest of the orthogonal 
polynomial strip functions are generated through Gram-
Schmidt orthogonalization process, which are also 
satisfy the geometry and natural boundary conditions. It 
is shown that, this set is the same as developed by Bhat 
[16]. The finite strip method in conjunction with 
transition matrix [17-18] is used to calculate the natural 
frequencies of the plate for different boundary 
conditions, tapered ratio and aspect ratio. Good 
accuracy and fast convergence for all cases presented in 
this paper are demonstrated when compared with 
available results in the literature.  

2. THE CHARACTERISTIC 
ORTHOGONAL POLYNOMIAL STRIP 
FUNCTIONS: 

One of the admissible functions that are generated by 
Cheung and Zhou [7] is adopted in the present work. 
Consider a beam of length equal l and unit width under 
an arbitrary static load ( )p x . If the width and the depth 
of the beam are small compared to its length, the 
classical beam theory is valid. The non-dimensional 



 IIUM Engineering Journal, Vol. 2, No. 1, 2001  A. A. Rizk and A. S. Ashour 

 24

differential equation that is governing the static 
deflection of the beam may be put in the form 
 
 

4

4

d
( )

d
y

P x
x

=  (1) 

where 
x
l

x =  ,
4

( ) ( )
l

P p l
EI

x x=  and E I  is the 

flexural rigidity of the beam. The arbitrary static load 
( )P x  can be expanded into Taylor series as 

'

0

( ) ii
i

P Px x
¥

=

= å  (2) 

where 'iP  are the constants that can be determined if 
( )P x  is a given function. Using equation (2) into 

equation (1), the general solution of the differential 
equation can be put in the form  

2 3 4
0 1 2 3

0

( ) [ ]i i i i ii
i

y P b b b bx x x x x
¥

+

=

= + + + +å  (3) 

The first term in the series solution will be taken as the 
first strip function in the set of orthogonal polynomial 
strip functions, i.e. 

2 3 4
1 2 3( )o oy a a a ax x x x x= + + + +  (4) 

Where ( 0, 1, 2, 3)ia i =  are arbitrary constants to be 
determined from the boundary conditions. For clamped 
ends 

(0) (1)
(0) (1) 0.0o oo o

dy dy
y y

d dx x
= = = =  (5) 

Then the strip function (4) becomes 
2 3 4( ) 2oy x x x x= - +  (6) 

Equation (6) is the same as the strip function developed 
by Bhat [16]. The subsequent orthogonal polynomial 
functions are generated from Gram-Schmidt process 
[19]. The first orthogonal function 1( )y x  is obtained 
from 

1 1( ) ( ) ( )oy B yx x x= -  

where
1 1

2 2
1

0 0

/o oB y d y dx x x= ò ò . The higher 
terms in the set can be obtained by the recurrence 
relation between three successive members of the 
orthogonal polynomials set, 
i.e. 1 2( ) ( ) , 2k k k k ky B y C y kx x - -= - - ³   where  

1 1
2 2

1 1
0 0

/k k kB y d y dx x x- -= ò ò ,
1 1

2
1 2 2

0 0

/k k k kC y y d y dx x x- - -= ò ò . 

All the orthogonal polynomials that generated are 
satisfied the geometry and natural boundary conditions 
of the beam. 

3. PROBLEM FORMULATION: 

Assume a homogenous, isotropic, and linearly elastic 
rectangular plate in (x-y) plane as depicted in Fig. 1. 
The plate is of uniform in x-direction and varying 
thickness h(y) in y-direction.  Under the assumption of 
the classical deformation theory, the partial differential 
equation governing the vibration of such plate is given 
by  

22 2

22 2( )
W W

D y
x x y

¶ ¶ ¶
n

¶ ¶ ¶

é ùæ ö÷çê ú÷ç ÷+ê úç ÷ç ÷ê úç ÷÷çè øê úë û
22 2

2 2 2
( )

W W
D y

y xy

¶ ¶ ¶
n

¶ ¶¶

é ùæ ö÷çê ú÷ç ÷+ +ê úç ÷ç ÷ê úç ÷÷çè øê úë û
2 2 2

2
2 ( ) (1 ) ( )

W W
D y h y

x y x y t

¶ ¶ ¶
n r

¶ ¶ ¶ ¶ ¶

é ù
ê ú+ - = -ê ú
ê úë û

 (7) 

where 
3

2

( )
( )

12(1 )
Eh y

D y
n

=
-

  is the flexure rigidity of the 

plate,   is the Poisson’s ratio, E the Young’s modulus, 
r is the material density, and  W  is the deflection of 
the plate.  By normalizing the coordinate system with 
respect to the plate dimensions as follows:

x
a

x =  and 

y
b

h =   and the variable thickness ( )h y  with respect to 

the thickness of the plate oh  at  0h =  as
0

( )
( )

h
f

h
h

h = ,  

equation (7) can be written as  

{ }2 4 2 212 2 ( )W W W f W Wxxxx xxhh hhhh hhh xxhb b b h b+ + + + +
{ }2 2 22 3( ) ( ) 0f W W f Whh xxb h b n l h+ + - = , 

 (8) 

where 2l =
4 2

0

oh a
D

r w
 is the square of the natural 

frequency of the plate , oD is the flexural rigidity of the 

plate at 0h = , 
a
b

b =   is the aspect ratio of the plate 

and 
3

1 3

2 3

2 3 2

3 2

( )1
( )

( )

( )1
( )

( )

1
( )

( )

df
f

f d

d f
f

f d

f
f

h
h

h h

h
h

h h

h
h

üïï= ïïïïïïïïï= ýïïïïïï= ïïïïþï

 (9) 



IIUM Engineering Journal, Vol. 2, No. 1, 2001  A. A. Rizk and A. S. Ashour 
 

25 

 

Fig. 1  Geometry of rectangular plate of variable thickness 

For a plate striped in the x-direction as shown in Fig.1, 
the shape function ( , )W x h  may be assumed in the form  

0

( , ) ( ) ( )
N

i i
i

W X Yx h x h
=

= å  (10)  

where ( )iY h  are the unknown functions in  h -direction 
and ( )iX x  are the strip functions in x -direction that are 
developed in the previous section. Substituting equation 
(10), in equation  (8), multiplying both sides by ( )jX x , 
and integrating from 0 to 1, we may obtain 

{ }2 '' 4 2 2 ''' '1
0

2 2
N

iv
ij i ij i ij i ij i ij i

i

q Y n Y m Y f m Y n Yb b b b
=

+ + + +å
{ }2 2 '' 22 3 0ij i ij i ij if m Y n Y f m Yb b n l+ + - =  

              j = 0, 1, 2,.., N  (11) 

 where 
1

0

( ) ( )ij i jm X X dx x x= ò , 

1 2

2
0

( )
( )iij j

d X
n X d

d
x

x x
x

= ò ,   (12) 

1 4

4
0

( )
( )iij j

d X
q X d

d
x

x x
x

= ò  
From the orthogonality properties, 0ij ijm q= =  
for i j¹ , this is true for clamped-clamped boundary 
conditions in the x-direction. Therefore, equation (11) 
becomes  

4

4

jd Y

d h
'

1 22
0

1
2 2

N
ij

i i i
i jj

n
Y f Y f Y

m
n

b =

ì üï ïï ïé ù¢¢+ + +í ýê úë ûï ïï ïî þ
å   

''' ''
1 22 j jf Y f Y+ +  

2
34

1
0.0jj j

jj

q
f Y

m
l

b

é ù
ê ú+ - =ê ú
ê úë û

 

j = 0, 1, 2, ..N (13) 

The above system of fourth order differential equations 
can be rewritten as 4-N first order 
differential equations at any strip k as: 

{ }k k k
d

A
d h

=
Y

Y  (14) 

where the coefficients of the matrix {A}k , in general, 
are functions of h and the eigenvalue parameter l .  The 
vector kY is given by  

1 2 .. ..k i NY Y Y Yé ù= ê úë û
Y  (15) 

where 
' '' '''

i i i i iY Y Y Y Yé ù= ë û (16) 

Solving the above system of first order differential 
equations using the transition matrix technique [20] at 
any strip k , with boundaries 1k -  and  k  yield 

[ ] 1k kk -=Y B Y  (17) 

x

y

z

ho hb

a

b

h(y)



 IIUM Engineering Journal, Vol. 2, No. 1, 2001  A. A. Rizk and A. S. Ashour 

 26

where [ ]kB is called the transition matrix of the strip k  
and 1,k k -Y Y  are the nodal vectors of the 

boundaries k , 1k - . 

4. BOUNDARY CONDITIONS:  

In this paper any combination of the classical boundary 
conditions such as simply supported, clamped, or free 
boundary conditions at either y = 0 or y = b is 
considered. After normalizing the boundary conditions 
and using equation (10) , we may have 

2

2
0

jd Y

d h
=  , 0 ( 0, 1)jY h h= = = ,  (18) 

For simply supported edges,   

0
jd Y

d h
=  ,  0 ( 0, 1)jY h h= = = . (19) 

For camped edges, and 

1

2

2

2
0

3

0

0

0
3

N
j

jj f ij i
i

N
j i

jj f ij
i

d Y
m c n Y

d
d Y dY

m c n
d d

h

h h

=

=

+ =

+ =

å

å
( 0, 1)h h= =  (20) 

where 
1 2f

c
n
b

=   and 
2 2

(2 )
fc

n
b
-

=  for free edges. 

5. NUMERICAL RESULTS AND 
CONCLUSION: 

To illustrate the validity of the proposed set of 
orthogonal polynomial strip functions in finite strip 
method, the natural frequencies for the free vibration 
plate of variable thickness in y-direction are calculated 

and compared with the available results obtained by 
different numerical methods. Numerical results are 
obtained for different combinations of classical 
boundary conditions, aspect ratio and tapered ratio. 

A linearly tapered plate is assumed with non-
dimensional variable thickness function defined by 

( ) (1 )f h dh= +  (21) 

where d  is the taper ratio given by 0
0

bh h
h

d
-

= , where 

bh is the thickness at 1h = .  
To investigate the accuracy and the convergence of the 
proposed admissible functions, the first four natural 
frequencies are calculated for different two cases, CSCS 
with taper ratio 0.5d = , and CCCC with taper 
ratio 0.4d = as shown in table 1. Different terms of the 
series solution in equation (10) ( )N  are considered. 
Some of the results that are obtained by using classical 
beam functions are calculated and presented to compare 
the convergent between the two admissible functions. 
Also,  the results compared with that obtained from 
Grossi and Bhat [9] and Singh and Saxena [11]. For the 
purpose of brevity, the symbolism CFCS for example, 
means a plate with varying thickness in the y-direction 
having clamped, free, clamped and simple supported 
edges at the boundaries, x= 0, y = b, x =a, and y = 0, 
respectively (see Fig. 1). In all calculations, a Poisson's 
ratio  = 0.3 was used. 
Table 2 demonstrates the natural frequencies for several 
combinations of boundary conditions, for two taper 
ratio d  (0,0.5), and two aspect ratio b  (1,2). A 
comparison is made with some results obtained from 
other numerical methods. Good accuracy has been seen 
for all the cases considered in this paper. 

Table 1: Convergence of the first four frequencies of square plate of variable thickness and comparison with 
data from different techniques (CSCS and CCCC plates) 

 N   1 2 3 
Classic 
CSCS 

1 1 .5 36.0213 68.3958  
2   36.0213 68.3958 85.2728 
3   35.9716 68.0789 85.2728 
4   35.9716 68.0789 85.2411 

       
Polynomial 

CSCS 
1   35.9672 68.1654 86.2478 
3   35.9227 67.8702 86.2478 
5   35.9173 67.8270 86.2303 
8   35.9165 67.8188 86.2229 

Singh and Saxena [11] 35.9640 68.033 85.2000 
    

Polynomial 1 1 .4 42.9255 87.6976  
CCCC 3   42.8647 87.4200            88.2030 

 5   42.8547 87.3755 88.1805 
Singh and Saxena [11] 42.9130 87.2900 87.5300 

 



 IIUM Engineering Journal, Vol. 2, No. 1, 2001  A. A. Rizk and A. S. Ashour 

 27

Table 2: The First four frequencies i (i = 1,2,3,4) of tapered plate in y-direction for different boundary 
conditions 

N    1 2 3 4 
5 CCCS 1 0.0 31.7809 63.1909 71.8811 100.879 
5 0.5 39.6479 89.7691 77.9645  
5 2 0.0 73.2638 108.272 168.863 210.392 
5 0.5 88.0933 134.256 211.415  

Singh and Saxena [11] 88.2750 134.190 206.890  
5 CSCC 1 0.5 39.2255 78.7977 87.3291  

Singh and Saxena [11] 39.2820 78.9690 86.404  
5  2 0.5 93.6774 134.575 206.9541 93.6773 

Singh and Saxena [11] 93.8310 134.520 202.580  
5 CFCC 1 0.0 24.2741 40.2840 62.9298  
5 0.5 27.6150 49.8425 68.0844 95.9289 
5 2 0.0 31.8625 63.3429 103.4920 127.067 
5 0.5 39.5637 75.8020 133.582 144.134 
5 CFCF 1 0.0 22.3639 26.7401 43.8835 62.9323 
5 0.5 26.8121 34.3755 54.7061 68.0833 
5 2 0.0 22.3197 36.5568 62.9065 62.9389 
5 CFCS 1 0.5 27.1122 44.2948 68.0726 91.3042 
5 2 0.0 26.9892 62.9285 80.0701 108.2721 
5 0.5 32.0426 74.2904 101.6199 142.2618 
5 CSCS 1 0.0 28.9125 54.5913 70.1724 94.7093 
5 0.5 35.9173 67.8270 86.2303 126.5393 
5 2 0.0 54.5901 94.7081 158.9818 170.1827 
5 0.5 67.7910 117.7756 196.9454 211.1377 

 
 

6. CONCLUSION REMARKS: 

The finite strip transition matrix method (FSTM) by 
using new set of orthogonal polynomial strip functions 
presented in this paper is used to investigate the flexure 
vibration of plate of generally variable thickness with 
general classical boundary conditions.  The method is 
easily implemented in a computer program and yield a 
fast convergence and reliable results.  The results 
obtained by the proposed method are compared with 
known results from different numerical and approximate 
methods. Excellent agreement has been achieved for all 
plate configurations considered in this paper. The 
comparison results illustrate the high accuracy and 
efficiency of the present method. Other complexity 
effects such as orthotropic, edges restrained against 
translation and rotation, concentrated masses, etc., can 
be easily implemented in the present method. 

REFERENCES: 

[1] A. W. Leissa, Vibration of plates (NASA sp 160), 
Washington, D.C., U.S. Governments Printing Office, 
1969. 

[2] A. W. Leissa, The Free Vibration of Rectangular 
Plates, Journal of Sound and Vibration 31, pp. 257-293, 
1973. 

[3] A. W. Leissa,  Recent Research in Plate Vibration. 
1973- 1976Classical Theory, The Shock and Vibration 
Digest.10., (1977), pp.13-24, 1977. 

[4] A. W. Leissa,,Plate Vibration Research. 1976-1980: 
Classical Theory, The Shock and Vibration Digest.13, 
pp. 11-22, 1981. 

[5] A. W.  Leissa,., Recent  Studies in Plate Vibrations. 
1981-1985, Part I, Classical Theory, The Shock and 
Vibration Digest.19, pp. 11-18, 1987,. 

[6] A. S. Ashour, A Semi-Analytical Solution of The 
Flexural Vibration of Orthotropic Plates of Variable 
Thickness, Journal of Sound and Vibration,  in press, 
2000. 

[7] Y. K. Cheung and D. Zhou,  The free vibration of 
Tapered Rectangular Plates Using New Set of Beam 
Functions with The Rayleigh-Ritz Method, Journal of 
Sound and Vibration223, pp. 703-722, 1999. 

[8] D. Zhou,  Natural Frequencies of Rectangular Plates 
Using a Set Of Static Beam Function in Rayleigh-Ritz 
Method , Journal of Sound and Vibration 189, pp. 81- 
87, 1996. 

[9] R. O. Gross and R.B. Bhat, Natural Frequencies of Edge 
Restrained Tapered Rectangular Plates , Journal of 
Sound and Vibration 185, pp. 335-343, 1995. 

[10] B. Singh and V. Saxena,  Transverse Vibration of Skew 
Plates with Variable Thickness, Journal of Sound and 
Vibration 206, pp. 1-13, 1997. 

[11] B. Singh. and V. Saxena,  Transverse Vibration of 
Rectangular Plate with Bi-directional  Thickness 
Variation, Journal of Sound and Vibration 198, pp. 51- 
65, 1996. 

[12] B. Singh  and V. Saxena,  Flexural Vibration of Skew 
Plate Using Boundary Characteristic Orthogonal 
Polynomials in Two Variables, Journal of Sound and 
Vibration 173, pp. 157- 178, 1994. 



 IIUM Engineering Journal, Vol. 2, No. 1, 2001  A. A. Rizk and A. S. Ashour 

 28

[13] K. M. Liew and K. L. Lam, Application of Two 
Dimensional Orthogonal Plate Function to  Flexural 
Vibration of Skew Plates, Journal of Sound and 
Vibration 139, pp. 241-252, 1990. 

[14] R. B. Bhat, Flexura Vibration of Polygonal Plates Using 
Characteristic Orthogonal Polynomials in Two 
Variables, Journal of Sound and Vibration 114, pp. 65-
71, 1987. 

[15]  S. K. Malhotra, N. Ganesan and M.A. Veluswami, 
Vibration of Orthotropic Square Plate Having Variable 
Thickness (Parabolic Shape), Journal of Sound and 
Vibration 119, pp. 184-188, 1987. 

[16] R.B. Bhat, Natural Frequencies of Rectangular Plates 
Using Characteristic Orthogonal Polynomials in 
Rayleigh-Ritz Method, Journal of Sound and Vibration 
102, pp. 493-499, 1985. 

[17] A. M. Farag, Mathematical Analysis of Free and Forced 
Vibration of Rectangular Plate,  Ph.D. thesis, Faculty of 
Engineering , Alexandria University, Alexandria, Egypt, 
1994. 

[18] A.M. Farag and A.S. Ashour, Free Vibration of 
Orthotropic Skew Plates , Journal of Vibration and 
Acoustics Vol. 122,  pp. 313-317, 2000. 

[19] S.J. Leon, Linear Algebra with Applications. New Jersey, 
Prentic-Hall International, Inc. Saddle River, 1998. 

[20] R. Zurmuhl, Numerical Analysis for Engineers and 
Physicists. , New York, Springer-Verlag Berlin 
Heidelberg,  1976. 

BIOGRAPHIES 

Dr. Abd El- Fattah A. Rizk is currently Associate 
Professor at the faculty of Engineering, International 
Islamic University Malaysia, Malaysia, on leave from 
Faculty of Engineering, Alexandria University, 
Alexandria, Egypt.  He was born in Alexandria 1952. 
He obtained his B.Sc. in Mechanical Engineering in 
1975, Faculty of Engineering, Alexandria University, 
Egypt, and B.Sc. in Mathematics in 1979, Faculty of 
Science, Alexandria University, Egypt. He received his 
M.Sc. in 1984 and Ph.D. in 1988 from Faculty of 
Engineering, Lehigh University, USA. His area of 
interest is Solid and Fracture Mechanics. 
 
Ahmed A. S. Ashour was born in Alexandria, Egypt in 
1959. He received his bachelor's degree in 1982 in 
Telecommunications Engineering, his MSc. in 
dynamical system model for traffic noise in 1988 from 
Alexandria University.  He obtained his MSc in 
Engineering Mechanics in 1992 and his PhD in 
Acoustics in 1994 from The University of Texas at 
Austin, which was comprised of a study of the fracture 
impedance method.  His research interests include wave 
propagation in solids and fluid, structural vibration and 
noise, and mathematical modeling and simulation.  He 
is an Associate Professor at Alexandria University, 
Egypt.  Currently, He is Associate Professor and head, 
Department of Science in Engineering, International 
Islamic University Malaysia.