 Open Access
 Authors : Kwanghun Kim , Hyeyong Pak , Suchol O , Chol Nam, Jonggil K O
 Paper ID : IJERTV9IS060076
 Volume & Issue : Volume 09, Issue 06 (June 2020)
 Published (First Online): 06062020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Application of Haar Wavelet Discretization Method for Free Vibration Analysis of Rectangular Plate
Kwanghun Kim1, Hyeyong Pak2
1,2department of Engineering Machine, Pyongyang University of Mechanical Engineering, Pyongyang, Democratic Peoples Republic of Korea
Suchol O3
3Institute of Science, Chongjin Mine & Metal University,
Chongjin, Democratic Peoples Republic of Korea
Chol Nam4
4Department of Resource Development Machine, Pyongyang University of Mechanical Engineering, Pyongyang, Democratic Peoples Republic of Korea
Jonggil K O5
5Department Of Machine Manufacturing, Pyongyang University of Mechanical Engineering, Pyongyang, Democratic Peoples Republic of Korea
AbstractThis paper focuses on the Haar wavelet discretization method (HWDM) for solving the problem of free vibration behaviour of rectangular plate. The displacements are expressed as Haar wavelet series and their integral. Since the constants generated during the integration process are decided as the setting of boundary conditions, therefore, the equations of motion of the entire system including boundary conditions are expressed as a series of algebraic equations. By solving the eigenvalue problem of these algebraic equation, the natural frequencies of the rectangular plate can be obtained. The accuracy and convergence of HWDM is verified against the results of the previous data, and the comparison results agree well. The effects of several geometric dimensions on the natural frequencies of rectangular plate under various boundary conditions are investigated. The numerical results for rectangular plate obtained by Haar wavelet discretization method may be served as benchmark solutions for future research.
KeywordsRectangular plate; Haar wavelet discretization method; Free vibration; Numerical analysis; Eigenvalue problem

INTRODUCTION
Plates are widely used in various engineering fields, and specially, in the case of ship and ocean structures they can be considered as one of the fundamental structural elements. It is very important to investigate the vibration behavior of plates which are widely used in various engineering fields. A lot of scholars have published a number of papers on the free vibration of rectangular plates [113]. Different methods such as RayleighRitz method, Galerkin method, finite element method and the method of separation of variables have been used to analyze the free vibration of rectangular plate. The results of the review show that although many studies have been conducted to analyze the free vibrations of rectangular plates, finding a reasonable approach to obtain the natural frequencies of rectangular plates is still an important problem. The Haar wavelet discretization method is a very simple and powerful method to solve the eigenvalue problem and the application in engineering of this method will be reviewed below.
In 1909, A. Haar proposed the Haar function, which made a great contribution to the emergence of wavelet theory. In 1981, J. Morlet [14] proposed the wavelet concept, which laid a good foundation for the formation of wavelet theory. Since then, the research of wavelet theory has entered a stage of rapid development. In 1981, Stromberg improved on the basis of the Haar function and found the first orthogonal wavelet. In 1982, expert Marr formed the "Mexican Hat" wavelet. In 1985, Meyer obtained a smooth wavelet orthonormal basis with a certain attenuation, namely Meyer base[15], which laid a good foundation for the further development of wavelet theory. In 1988, wavelet analyst Daubechies[16, 17] constructed an iterative method to construct a wavelet base that is nonzero only in a finite region (ie Daubechies base) and gave 10 presentations at a wavelet conference held in the United States. It caused a sensation, and then the book "Small Waves Ten" became a great book with great influence in contemporary mathematics, which pushed the development and practical application of wavelet theory to a climax. In 1989, Mallat[18, 19] proposed the famous "Mallat algorithm", which opened the development space for the engineering application of wavelet theory. After the 1990s, wavelet theory gradually matured. At present, research on wavelet theory has penetrated a wide variety of fields. Majak et al. [20, 21] developed Haar waveletbased discretization method for solving differential equations, discussed both, strong and weak formulations based apporaches, and introduced this method to solve solid mechanics problems. Recently, Hein and Feklistova[22, 23] applied HWDM for solving the vibration problems of functionally graded beams under some boundary conditions.
As can be seen from the literature review, the Haar method was used for vibration analysis of various types of structures, but twodimensional development is difficult, so there are very few examples applied to the vibration of plate structures. Therefore, the main purpose of this paper is to establish a solution method and system to apply conveniently and efficiently the Haar wavelet discretization method to the free vibration of a rectangular plate.

APPLICATION OF HWDM
f = f (1) f (2)
f (n )T
Haar wavelet which is a group of square waves that has
a= a a a T ,
size of +1 and 1 in some intervals and has zeros in elsewhere is one of the simplest compactly supported orthogonal
1 2 n
d 3 f (0) d 2 f (0)
df (0) T
wavelet among the wavelet families. The details of Haar
wavelet series and their integrals can be found in the following references [2528].
b=
d 3
d d
f (0)
3 3
HWDM are used to discretize the derivatives in entire governing equations including boundary conditions. Since the
p4,1(1)
p4,2 (1)
p4,n (1)
1 1
6 2
1 1
3 3
Haar wavelet series is defined in the interval [0, 1], firstly, in
P p4,1(2 )
p4,2 (2 )
p4,n (2 )
2 2
2 1
order to apply the HWDM, a linear transformation statute 1
6 2
may be introduced for coordinate conversion from length
interval [0, L]of the rectangular plate to the interval[0,1]of the
3 2
Haar wavelet series, that is,
p4,1(n )
p4,2 (n )
p4,n (n )
n n n 1 6 2
In the HWDM, highest order derivatives of the displacements are expressed by Haar wavelet series, the lower order derivatives can be obtained by integrating Haar wavelet series.
The transverse vibration differential equation of the plate is expressed as
In practical process applications, the clamped boundary condition, simplysupported boundary condition and the free boundary condition are widely used, these boundary conditions can be written as follows in equation form.
x x1 x x1 , = y y1 y y1
x x L y y L
2 1 x 2 1 y
(1)
x x1 x x1 , = y y1 y y1
x x L y y L
2 1 x 2 1 y
(1)
w( ) 0, dw( ) 0, 0, 1 ,
d
w() 0, dw() 0, 0, 1
d
(7)
w( ) 0, dw( ) 0, 0, 1 ,
d
w() 0, dw() 0, 0, 1
d
(7)
Clamped boundary condition
4 w
4 w
4 w
4
4
x4
d 2w( )
w( ) 0, 0, 0, 1 ,
d 2
d 2w()
w() 0, 2 0, 0, 1
d
(8)
d 2w( )
w( ) 0, 0, 0, 1 ,
d 2
d 2w()
w() 0, 2 0, 0, 1
d
(8)
h
2 x2 y2
y4
k w 0
(2)
Simplysupported boundary condition
where k 4
2 , D is the flexural stiffness of the plate,
D
1
1
is defined as D E p / 12 1 2 .
Substituting equation (2) into equation (1), the vibration equation of the plate at local coordinates is defined as:
d 2w( ) d 3w( )
0, 0, 0, 1
d 2 d 3
d 2w() d 3w()
0, 0, 0, 1
d 2 d3
(9)
d 2w( ) d 3w( )
0, 0, 0, 1
d 2 d 3
d 2w() d 3w()
0, 0, 0, 1
d 2 d3
(9)
Free boundary condition
1 2 w
1 1 4 w
1 2 w
L4 2
2 L4 L4 2 2
k 4 w 0
L4 2
(3)
x x y y
d 4 f ( ) 2 m
ah
d 4 i1 i i
(4)
d 4 f ( ) 2 m
ah
d 4 i1 i i
(4)
By taking n=2m, f()=w(, ) (when is fixed, w is a function containing only the variable ), and the highest order derivative can be approximated by Haar wavelet along parallel to the axis:
By integrating the Eq. (4), the following derivative terms can be obtained.
Four boundary condition equations can be obtained by introducing the boundary condition, and can be written as follows in matrix form:
f = P a
b 2 b
(10)
d 3 f ( ) 2m d 3 f (0)
3 ai p1,i 3
d i1 d
(5,a)
d 2 f ( ) 2m d 3 f (0) d 2 f (0)
d 2 i 2,i d 3 d 2
a p
i1
(5,b)
df ( ) 2m 2 d 3 f (0) d 2 f (0) df (0)
ai p3,i 3 2
d i1 2 d d d
(5,c)
2m 3 d 3 f (0) 2 d 2 f (0) df (0)
f ( ) ai p4,i 6 d 3 2 d 2 d f (0)
i1
(5,d)
d 3 f ( ) 2m d 3 f (0)
3 ai p1,i 3
d i1 d
(5,a)
d 2 f ( ) 2m d 3 f (0) d 2 f (0)
d 2 i 2,i d 3 d 2
a p
i1
(5,b)
df ( ) 2m 2 d 3 f (0) d 2 f (0) df (0)
ai p3,i 3 2
d i1 2 d d d
(5,c)
2m 3 d 3 f (0) 2 d 2 f (0) df (0)
f ( ) ai p4,i 6 d 3 2 d 2 d f (0)
i1
(5,d)
where, for the clamped boundary condition
T
f = f (0)
f (1)
df (0)
df (1) ,
b d d
p4,1(0)
p4,2 (0)
p4,n (0) 0 0 0 1
1 1
p4,1(1) p4,2 (1) p4,n (1) 1 1
P2
6 2
p1,1(0) p1,2 (0) p1,n (0) 0 0 1 0
p1,1(1)
p1,2 (1)
p1,n (1)
1 1 1 0
2
for the simply supported boundary condition,
d 2 f (0)
d 2 f (1) T
The displacement function v() can be written in the form of a matrix as following:
f = P a
1 b
(6)
where
fb = f (0)
f (1)
d 2 d 2 ,
p4,1(0) p4,2 (0) p4,n (0) 0 0 0 1 d 2g n
1 1 2 cihi
p4,1(1) p4,2 (1) p4,n (1) 1 1 d i1
(17)
p4,1(0) p4,2 (0) p4,n (0) 0 0 0 1 d 2g n
1 1 2 cihi
p4,1(1) p4,2 (1) p4,n (1) 1 1 d i1
(17)
P2
6 2
By integrating second derivative of k in turn, the following
p2,1(0) p2,2 (0) p2,n (0) 0 1 0 0
expressions can be obtained.
p2,1(1)
p2,2 (1)
p2,n (1) 1 1 0 0
dg n
dg 0
and, for the free boundary condition
d ci p1,i d ,
d 2 f (0) d 2 f (1) d 3 f (0)
d 3 f (1) T
i1
fb =
2 2 3 3 ,
n dg 0
d d
d d
g ci p2,i d
dg 0
p2,1(0)
p2,1(1)
p2,2 (0)
p2,2 (1)
p2,n (0) 0 1 0 0
p2,n (1) 1 1 0 0
i1
g() can be written in matrix form as
p2,1(1) p2,2 (1) p2,n (1) 1 1
p ( ) p ( ) p ( ) 1 c c g 2,1 2 2,2 2 2,n 2 2 R1
d d
p2,1(n ) p2,2 (n ) p2,n (n ) n 1
(18)
p2,1(1) p2,2 (1) p2,n (1) 1 1
p ( ) p ( ) p ( ) 1 c c g 2,1 2 2,2 2 2,n 2 2 R1
d d
p2,1(n ) p2,2 (n ) p2,n (n ) n 1
(18)
P2
p3,1(0) p3,2 (0) p3,n (0) 1 0 0 0
p (1) p (1) p (1) 1 0 0 0
3,1 3,2 3,n
f P1 a a
= Q1
fb P2 b b
(11)
f P1 a a
= Q1
fb P2 b b
(11)
By combining Eq. (6) and Eq. (10) the following equation can be obtained.
where, g = g(1)
g(2)
g(n )T
as:
From Eq. (11), an unknown coefficient matrix is defined
a 1 f
Q1
b fb
(12)
a 1 f
Q1
b fb
(12)
Therefore, the fourth order derivative of displacement f
Two boundary condition equations can be obtained by introducing the boundary condition, and can be written as follows in matrix form (in this example the boundary condition is set simply supported boundary condition, in simply supported boundary condition g()= 2w/2=0, (=0, =1)):
g p2,1(0) p2,2 (0) p2,n (0) 0 1 c R c
b p (1) p (1) p (1) 1 1 d 2 d
2,1 2,2 2,n
T
where c= c c c T , d dg(0) g (0)
(19)
g p2,1(0) p2,2 (0) p2,n (0) 0 1 c R c
b p (1) p (1) p (1) 1 1 d 2 d
2,1 2,2 2,n
T
where c= c c c T , d dg(0) g (0)
(19)
can be expanded into the following as:
d 4 f 1 f
H1Q1 L1 f + L2 fb d 4 fb
(13)
d 4 f 1 f
H1Q1 L1 f + L2 fb d 4 fb
(13)
1 2
n d
1
1
Where L1 and L2 are the first n columns and the last four columns of the atrix H1Q 1 .
g R1 c c
= Q2
gb R2 d d
(20)
g R1 c c
= Q2
gb R2 d d
(20)
By combining Eq. (18) and Eq. (19) the following equation can be obtained.
d 4 f
d 4 f
d 4 f
d 4 f
T
1 , 2 , , n ,
d 4
d 4
d 4
d 4
p(1)
p(1)
p (1)
p (2 )
hn (1) 0 0 0 0
hn (2 ) 0 0 0 0
as:
From Eq. (20), an unknown coefficient matrix is defined
P1
P1
c 1 g
d Q2 g
b
(21)
c 1 g
d Q2 g
b
(21)
h ( ) h ( ) h ( ) 0 0 0 0
1 n 2 n n n
By using the tensor multiplying, d4f/d4can be extend to two dimensions
4w
L I w + L I f = K w M
4 1 y 2 y x x
(14)
Considering the boundary conditions, it is easy to know that f is a zero vector with a length equal to 2n, and it will be omitted. Iy is the unit matrix.
Similarly,
Therefore, the second order derivative of displacement g
can be expanded into the following as:
p(1) p (1) hn (1) 0 0
d 2 g p(2 ) p (2 ) hn (2 ) 0 0 c
2
d d
h ( ) h ( ) h ( ) 0 0
1 n 2 n n n
H Q1 g N g + N g
2 2 gb 1 2 b
(22)
Where N1 and N2 are the first n columns and the last two
4w
4 Ix L1 w = K yw
(15)
4w
4 Ix L1 w = K yw
(15)
1
columns of the matrix H 2Q2 .
p2,1(1) p2,2 (1) p2,n (1) 1 1
p2,1(2 ) p2,2 (2 ) p2,n (2 ) 2 1 m
l
n
p ( ) p ( ) p ( )
2,1 n 2,2 n 2,n n n 1
S m
1 n
(23)
p2,1(1) p2,2 (1) p2,n (1) 1 1
p2,1(2 ) p2,2 (2 ) p2,n (2 ) 2 1 m
l
n
p ( ) p ( ) p ( )
2,1 n 2,2 n 2,n n n 1
S m
1 n
(23)
The following expression will be used to calculate the Haar wavelet expression of the fourthorder mixed partial derivative of w.
4w 2 2w 2 2w
2 2 2 2
22 2
(16)
If g()= 2w/2, the second derivative of g() along the parallel axis can be approximated by Haar wavelet.
Similarly, for k()=w(, )
where, l = l(1) l(2)
T
l(n )T
dl(0) T
Table 1 lists the first four order natural frequency parameters of square thin plates with constant thickness under different boundary conditions using Haar wavelet, where
m= m1 m2 mn , n d l(0)
In a similar way to equation (19)
Lx=Ly=1m, h=0.1m and material is steel (E=210Gpa, =0.3, =7800kg/m3). It can be seen that a small number of matching points can achieve good numerical accuracy, and with the increase of the scaling factor J, the numerical results
l p2,1(0) p2,2 (0) p2,n (0) 0 1 m S m
b p2,1(1) p2,2 (1) p2,n (1) 1 1 n 2 n
(24)
l p2,1(0) p2,2 (0) p2,n (0) 0 1 m S m
b p2,1(1) p2,2 (1) p2,n (1) 1 1 n 2 n
(24)
24
are getting closer and closer to the literature , thus proving
By combining Eq. (23) and Eq. (24) the following equation can be obtained.
l S1 m m
l = S n Q3 n
b 2
(25)
Therefore, the second order derivative of displacement l
can be expanded into the following as:
p(1) p (1) hn (1) 0 0
d 2l p(2 ) p (2 ) hn (2 ) 0 0 m
2
d n
h ( ) h ( ) h ( ) 0 0
1 n 2 n n n
H Q1 l T l + T l
3 3 1 2 b
lb
(26)
3
3
Where T1 and T2 are the first n columns and the last two columns of the matrix H3Q 1 .
From above equations, we can obtain as following expression.
2 2w 2w
N1 I y N1 I y Ix T1 Kxyw
2 2 2
(27)
In this way, we can get an expression similar to Eq. (27)
that the structure is solved discretely using Haar wavelets in both directions. The natural frequency is feasible, and the boundary conditions can be accurately applied, which proves the feasibility of the method and provides a basis for the use of this method in more complex structures.
Table 2 to 5 shows the natural frequencies of the plates for different thicknesses under several boundary conditions using presented method.
TABLE2. NATURAL FREQUENCIES OF PLATE WITH CCCC BOUNDARY
Mode
h
0.02
0.04
0.06
0.08
0.1
0.2
1
179.9086
359.8172
539.7258
1468.621
899.543
1799.086
2
367.1552
734.3104
1101.466
1468.621
1835.776
3671.552
3
367.1552
734.3104
1101.466
2166.072
1835.776
3671.552
4
541.518
1083.036
1624.554
2635.765
2707.59
5415.18
5
658.9412
1317.882
1976.824
2648.261
3294.706
6589.412
6
662.0652
1324.13
1986.196
719.6344
3310.326
6620.652
Mode
h
0.02
0.04
0.06
0.08
0.1
0.2
1
179.9086
359.8172
539.7258
1468.621
899.543
1799.086
2
367.1552
734.3104
1101.466
1468.621
1835.776
3671.552
3
367.1552
734.3104
1101.466
2166.072
1835.776
3671.552
4
541.518
1083.036
1624.554
2635.765
2707.59
5415.18
5
658.9412
1317.882
1976.824
2648.261
3294.706
6589.412
6
662.0652
1324.13
1986.196
719.6344
3310.326
6620.652
CONDITION
TABLE NO 3. NATURAL FREQUENCIES OF PLATE WITH CSCS BOUNDARY
CONDITION
for
2 2w
Mode
h
0.02
0.04
0.06
0.08
0.1
0.2
1
144.7407
289.4814
434.2221
578.9627
723.7034
1447.407
2
273.82
547.64
821.46
1095.28
1369.1
2738.2
3
346.829
693.6581
1040.487
1387.316
1734.145
3468.29
4
473.3095
946.619
1419.929
1893.238
2366.548
4733.095
5
511.7808
1023.562
1535.342
2047.123
2558.904
5117.808
6
646.5281
1293.056
1939.584
2586.113
3232.641
6465.281
Mode
h
0.02
0.04
0.06
0.08
0.1
0.2
1
144.7407
289.4814
434.2221
578.9627
723.7034
1447.407
2
273.82
547.64
821.46
1095.28
1369.1
2738.2
3
346.829
693.6581
1040.487
1387.316
1734.145
3468.29
4
473.3095
946.619
1419.929
1893.238
2366.548
4733.095
5
511.7808
1023.562
1535.342
2047.123
2558.904
5117.808
6
646.5281
1293.056
1939.584
2586.113
3232.641
6465.281
.
2 2
Kx 2Kxy + K y z = k4 Ix I y1z
(28)
Kx 2Kxy + K y z = k4 Ix I y1z
(28)
Therefore, a governing equation for an orthogonal isotropic plate expressed by Haar wavelet and its integral can be obtained as:
Therefore, the natural frequency of the plate can be easily obtained according to equation (28).

NUMERICAL RESULTS
In this section, new numerical data that future researchers can use as benchmarks are presented along with parameter studies.
Boundary Condition 
Mode 
J=2 
J=3 
J=4 
J=5 
Ref. [24] 
Present 
Present 
Present 
Present 

CCCC 
1 
3.65939 
3.65014 
3.64713 
3.6463 
3.6467 
2 
7.52806 
7.46223 
7.44301 
7.438 
7.4416 

3 
7.52806 
7.46223 
7.44301 
7.438 
7.4416 

4 
11.1062 
11.0136 
10.9777 
10.968 
10.974 

SSSS 
1 
2.00959 
2.00241 
2.0006 
2.0002 
2.00 
2 
5.06696 
5.01684 
5.00422 
5.0011 
5.00 

3 
5.06696 
5.01684 
5.00422 
5.0011 
5.00 

4 
8.15074 
8.03836 
8.00963 
8.0024 
8.00 

CSCS 
1 
2.94584 
2.93673 
2.9342 
2.9336 
2.9336 
2 
5.60883 
5.56345 
5.55091 
5.5477 
5.5484 

3 
7.12473 
7.05063 
7.03096 
7.026 
7.0285 

4 
9.73665 
9.62799 
9.59499 
9.5864 
9.5888 
Boundary Condition 
Mode 
J=2 
J=3 
J=4 
J=5 
Ref. [24] 
Present 
Present 
Present 
Present 

CCCC 
1 
3.65939 
3.65014 
3.64713 
3.6463 
3.6467 
2 
7.52806 
7.46223 
7.44301 
7.438 
7.4416 

3 
7.52806 
7.46223 
7.44301 
7.438 
7.4416 

4 
11.1062 
11.0136 
10.9777 
10.968 
10.974 

SSSS 
1 
2.00959 
2.00241 
2.0006 
2.0002 
2.00 
2 
5.06696 
5.01684 
5.00422 
5.0011 
5.00 

3 
5.06696 
5.01684 
5.00422 
5.0011 
5.00 

4 
8.15074 
8.03836 
8.00963 
8.0024 
8.00 

CSCS 
1 
2.94584 
2.93673 
2.9342 
2.9336 
2.9336 
2 
5.60883 
5.56345 
5.55091 
5.5477 
5.5484 

3 
7.12473 
7.05063 
7.03096 
7.026 
7.0285 

4 
9.73665 
9.62799 
9.59499 
9.5864 
9.5888 
TEBLE I. FREQUENCY PARAMETERS OF PLATE WITH VARIOUS BOUNDARY CONDITIONS =L2/h(/(E1)1/2
TABLE NO 4. NATURAL FREQUENCIES OF PLATE WITH SSSS BOUNDARY
CONDITION
td>
0.06
Mode 
h 

0.02 
0.04 
0.08 
0.1 
0.2 

1 
98.68739 
197.3748 
296.0622 
394.7495 
493.4369 
986.8739 
2 
246.8521 
493.7043 
740.5564 
987.4085 
1234.261 
2468.521 
3 
246.8521 
493.7043 
740.5564 
987.4085 
1234.261 
2468.521 
4 
395.1056 
790.2112 
1185.317 
1580.422 
1975.528 
3951.056 
5 
494.19 
988.38 
1482.57 
1976.76 
2470.95 
4941.9 
6 
494.19 
988.38 
1482.57 
1976.76 
2470.95 
4941.9 
TABLE NO 5. NATURAL FREQUENCIES OF PLATE WITH CFFF BOUNDARY
CONDITION
Mode 
h 

0.02 
0.04 
0.06 
0.08 
0.1 
0.2 

1 
144.5568 
289.1136 
433.6705 
578.2273 
722.7841 
1445.568 
2 
267.354 
534.708 
802.0619 
1069.416 
1336.77 
2673.54 
3 
346.8095 
693.619 
1040.429 
1387.238 
1734.048 
3468.095 
4 
469.3112 
938.6224 
1407.934 
1877.245 
2346.556 
4693.112 
5 
472.3318 
944.6637 
1416.996 
1889.327 
2361.659 
4723.318 
6 
646.5246 
1293.049 
1939.574 
2586.098 
3232.623 
6465.246 
Fig. 1 shows the natural frequency change of the plate with increasing A under several boundary conditions. As shown in the Fig.1, the natural frequencies are decreased as Ly/Lx increases. In particular, it can be seen that the natural frequencies are decreased rapidly until Ly/Lx=1, then are gradually decreased.
Fig.2 The change of natural frequencies as increasing of thickness h IV. NUMERICAL RESULTS In this paper, a reasonable analysis method based on the Haar wavelet has been presented to obtain the natural frequencies of rectangular plate with several boundary conditions. The basic principles and formulas of the Haar wavelet collocation method are described, and its 
approximation to displacement functions using HWDM are discussed an indepth. The solution process of Haar wavelet used in free vibration analysis of rectangular plate is given in detail. The efficiency and accuracy of presented method are proved for natural frequencies of rectangular plate with several boundary conditions. Some conclusions obtained through numerical examples and free vibration analysis results for the rectangular plate are presented, these data may be may be served as benchmark solutions for future research. ACKNOWLEDGMENT The authors also gratefully acknowledge the supports from 
Fig.1 The change of natural frequencies as increasing of Ly/Lx Pyongyang University of Mechanical Engineering of DPRK. (a)CCCC, (b)CFCF, (c)CSCS, (d)SFSF 
Lastly, Fig. 2 shows the change in natural frequency of a plate with a completely clamped boundary with increasing thickness h. It can be clearly seen from the figure that as the thickness h of the plate increases, the natural frequency also increases.
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