 Open Access
 Authors : Ola Ragb , M. S. Matbuly
 Paper ID : IJERTV8IS120057
 Volume & Issue : Volume 08, Issue 12 (December 2019)
 Published (First Online): 11122019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Efficient Quadrature Solution for Composite Plates with Variable Thickness Resting on NonUniform and Nonlinear Elastic Foundation
Ola Ragb*, M. S. Matbuly
Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University,
P.O. 44519, Zagazig, Egypt
AbstractTwo Different schemes are examined for vibration analysis of composite plate with variable thickness resting on nonuniform and nonlinear elastic foundation problems. On the basis of first order transverse shear theory a basic equation of vibration is derived. Investigations are made over nonuniform and nonlinear Winkler and uniform Pasternak foundation model. Examined schemes are based on discrete singular convolution and moving least square differential quadrature method. Also, the obtained nonlinear algebraic system is solved by using iterative quadrature technique. This problem is solved for different boundary conditions, different shear correction factor and varying thickness in one and two directions. Numerical analysis is applied to investigate influence of different computational characteristics on convergence and accuracy of the obtained results. The obtained results agreed with the previous analytical and numerical ones. Further a parametric study is introduced to explore the influence of elastic and geometric characteristics of the vibrated plate, on results.
KeywordsComposite; Vibration; Shear Correction Factor; Nonuniform Elastic Foundation; Nonlinear Winkler Foundation; Variable Thickness; Discrete Singular Convolution; Moving Least Square.

INTRODUCTION
Nonuniform elastic plates with varying thickness are commonly used in ship and offshore structures, pavement of roads, footing of buildings and bases of machines. Therefore, the vibration analysis of like plates is of great importance for practical design and structural components.
Depending upon the requirement, durability and reliability, materials are being developed so that these may provide better strength, efficiency and economy. Therefore a study of character and behavior of these plates is required so that the full potential of these plates may be used. So, these plates have been studied analytically and numerically. In most cases, their closed form solutions are extremely difficult to establish. But, for some special cases of variable thickness of rectangular plate, investigations have been made and solutions have been obtained [14].The most commonly used numerical methods for such applications are Spline function approximation technique [56], RayleighRitz
method [78], Meshfree method[9], Mixed boundary node method [10], Finite strip method [11] and Finite element method [12], have been widely applied for the plate resting on uniform elastic foundation. Most of these studies are computationally expensive.
In seeking a more efficient numerical method is differential quadrature method (DQM) which requires fewer grid points yet achieves acceptable accuracy [1315]. By applying DQM, the free vibration problem of plate is translated into the eigenvalue problem. According to the selection of basis functions and influence domain for each point, there are more than versions of DQM. Discrete singular convolution differential quadrature method (DSCDQM) [1618], Moving least square differential quadrature method (MLSDQM) [1921] are the most reliable versions. DSCDQ method is that they exhibit exponential convergence of spectral methods while having the flexibility of local methods for complex boundary conditions. MLSDQ exploits the merits of both the DQ and meshless method.
The main aim of present work to apply two different schemes,( DSCDQM and MLSDQM) to solve vibration problems of composite plates with varying thickness. These plates are resting on nonuniform and nonlinear Winkler and uniform Pasternak elastic foundation. Based on a transverse shear theory, the governing equations of the problem are formulated. The unknown field quantities and their derivatives are approximated using DQ approximations. Then the obtained nonlinear algebraic system is solved by using iterative quadrature technique. The program of MATLAB is designed to solve the reduced eignvalue problem. Numerical analysis is implemented to investigate convergence and efficiency of each scheme. Accuracy of the obtained results is compared with the existing previous results. Further a parametric study is introduced to investigate the influence of elastic and geometric characteristics on natural frequencies and mode shapes.

FORMULATION OF THE PROBLEM
Consider a composite consisting of n plates interfacialy bonded with variable thickness h(x,y) and resting on nonuniform and nonlinear elastic foundation of Winkler and uniform Pasternak type , as shown in Fig.(1).
D (x , y ) E h 3 (x , y ) / [12(1 2 )]
is the flexural rigidity
of the plate. G, E and v are shear modulus, Youngs modulus and Poisson's ratio of the plate. k is the shear correction factor [2527].
Assuming harmonic behavior of the problem, the field quantities can be written as:
x (x , y ,t ) x e i t ,
Fig.1 Composite Plate with Variable Thickness resting on Nonuniform and Nonlinear Winkler Pasternak Foundation.
y (x , y ,t ) y e i t ,
w (x , y ,t ) W e i t
(5)
Each plate occupies (ai 1 x ai , 0 y b, i 1,n) ,
Where is the natural frequency of the plate and
where a and b are width and length of the composite.
Based on a firstorder shear deformation theory, the equations of motion for each plate can be written as [20, 22]:
i 1 .
x , y ,W are the amplitudes for x , y , and w ,
2 (1 ) 2 (1 ) 2 y
respectively.
D (x , y ) x x
x 2 2 y 2 2 x y
(1)
Substituting from Eq.(5) into (14), One can reduce the
kGh (x , y ) w
2
x I1 x ,
problem to:
2 2 2
x
t 2
x
(1 ) x
(1 ) y
D(x, y)
2 2 x2
2 y2 2 xy
y
y
D (x , y )
(1 ) y
(1 ) 2x
(6)
y 2
2 x 2
2 x y
(2)
W 2
kGh(x, y) x
x
I1x ,
y 1
y 1
kGh (x , y ) w
y
I
2 y
,
t 2
2
(1 ) 2
(1 ) 2
2w
2w
D(x, y) y y
x
kGh (x , y )
x
y
y2 2 x2
2 xy
x 2
y 2 x
y
(3)
(7)
3 2w 2w 2w
kGh(x, y) W
2 I ,
2 0
2 0
K1 (x )w

K 3w
K 2 x 2
I
y
t 2
y y 1 y
Where Io and I1 are mass moments of inertias [23]:
h /2
2W
kGh(x, y)
2W
x
y 3
K1(x)W K3W
0 x2
y2 x
y
(8)
I0 , I1
(1 , z2 ) dz,
(4)
h0 /2
2W
K2 x2
2W
y2
2 I0W
is the plate mass density. K
1 (x ) K1
(1 x / a)
is the nonuniform Winkler foundation stiffness which linearly varying along xdirection. is variation parameter of stiffness of foundaion. K2 is shear modulus of foundation
Boundary conditions can be expressed as follows:

Clamped edge:
reaction.
K 3 is the nonlinear Winkler foundation. t: time.
W 0,
nx y

ny x
0,
x (x , y ,t ),y (x , y ,t ),w (x , y ,t ) are normal strain
n n 0
(9)
rotations and transverse deflection [24].
The thickness variation function of plate is
x x y y
h(x , y ) h0 (1 x / a)(1 y / b). h0 is the constant
reference thickness value and , are variation parameter of thickness.


Simply Supporting of the first kind: SS1
(x ,b )
W 0,
W (x ,b
i ) W (x ,bi ),

D ( x i
x
(x ,b )
(x ,b )
(x ,b )
(n 2 n 2 ) x
(1 )n n
x
y i ) D ( x i y i ),
x y x
x y y
1
y
x (x ,bi )
x y
y (x ,bi )
(13)
x y
x y
( n 2 n 2 ) y
y
(1 )nx ny
y 0,
x
(10)
2
1
( y
(x ,b )
x )
(x ,b )
1
D ((n 2 n 2 ) x 2n n
x
( x i
2 y
y i ), (i 1, n )
x
2 x y y
x y x
(n 2 n 2 ) y x y x
2n n y ) 0
x y y




SOLUTION OF THE PROBLEM
Two different differential quadrature techniques are employed to reduce the governing equations into nonlinear eigenvalue problem, as follows [1621]:

Simply Supporting of the second kind: SS2
W 0, nx y ny x 0,

Discrete Singular Convolution Differential Quadrature Method (DSCDQM)
In this technique, regularized Shannon kernel (RSK) may be used as a shape function such that the unknown u(x)
(n 2 n 2 ) x (1 )n n x
(11)
and its derivatives can be approximated over a narrow
x y
( n 2 n 2 )
x
y
x y
(1 )n n
y
x 0
bandwidth
( x x M , x x M ) as [1618]:
2
x y x y
M sin[ (x x ) / h ]
( (x i x j ) )
y x
u (x )
i j x e
2 2
u (x
), (14)
i (x x ) / h j

Free edge:
j M i j x
(i N , N ),
kG h (n
W n
W n

n
) 0,
x x
y y
x x y y
where hx is the step size, 2M+1 is the effective
(n 2 n 2 ) x
(1 )n n
x
computational band width, is regularization parameter,
x y
x y
x y
( n 2 n 2 )
x
y
y
x y
(1 )nx ny
y
y 0,
x
(12)
= r hx and r is a computational parameter.
Derivatives of u, can be approximated as a weighted linear sum of ui , (i=N,N) as [1618]:
1 D ((n 2 n 2 ) x

2n n
x
u M x
2 x y y
x y x
x x x
i j M
Cij u (x j ),
(15)
2 2 y
y
y
(nx n ) x
y

2nx ny y ) 0
2u
x 2
x x
M
ij j
ij j
C xx u (x ),
i N , N ,
nx and ny are the directional cosines at a point on the
Where
i j M
boundary edge.
Along the interface between ith plate and (i+1) th one, the
(1)i j
hx (
e
(i j )2
2 2 ) ,
i j
continuity boundary conditions can be described as:
hx (i j )
C
C
x
ij
0
0
,
i j
(16)
i j 1
2 (i j )2
x by minimizing the following weighted quadratic for
( 2(1)
hx 2 (i j )2
xx
1 )e
2
hx ( 2 2 ) ,

j

N 1
h 2
C ij
1 2
(a) (x – xi )(u
i 1
(xi ) ui )
(20)
1
1
i j
2
3hx 2
N
(x – xi )(PT (xi )a(x) ui )2
i 1
Similarly, one can approximateu y ,u yy and calculated
C
C
, C
, C
y yy
ij ij .

Moving Least Squares Differential Quadrature Method (MLSDQM)
Where (xxi) is a positive weight function defined over the influence domain, (i, i=1,N)
The stationary value of (a) with respect to a(x) leads to a linear equation, such as:
In this technique, the influence domain, (i, i=1: N), for each node is determined as shown in Fig. (2). Over each
A(x)a(x) B(x)u
(21)
influence domain, the nodal unknowns can be approximated as [1921]:
from which
a(x) A1 (x)B(x)u , (22)
u (xi ) u h
N
1
1
(xi ) j (xi ) u (x j ),(i 1, N ),
(18)
where
T
j 1
A(x) P(xi ) i (x)P
(xi )
1
1
N
i i
i i
n
n
i 1
(x)P(x )PT
(xi ),
u=u1 u2
u T ,
B(x)=P(x) (x)
1 (x)P(x1)
2 (x)P(x2 )
n (x)P(xn )
.
Fig.2 Domain discretization for moving least
On suitable substitution from Eq. (22) into (19), uh(x) can then be expressed as:
u h (x) PT (x)a(x)
1
1
i
i
N
squares differential quadrature.
PT (x)A1 (x)B
xu=i (x)ui
(23)
Where the shape function j (xi
) can be obtained using
i 1
MLS approximation as follows:
Where the nodal shape function:
Let
m
i
i
u h (x) Pi (x)ai (x) PT
(x)a(x),
(19)
i (x) = PT (x)A1 (x) B
x
(24)
i 1
W here a(x) a (x),a (x), ,a
(x) T is a vector of
Determination of the shape function i (x) and its partial
1 2 m
unknown coefficients.
derivatives can be simplified as follows [1921]:
i x PT xA1 xBi x T xBi x
(25)
PT (x) p1 (x), p2 (x), , pm (x) is a complete set
of monomial basis. m is the number of basis terms. The coefficients aj (x),( j 1, m ) , can be obtained at any point
Since A(x) is a symmetric matrix, then obtained through
Ax x P x
x
can be
(26)
Therefore, the problem of determination of the shape function is reduced to solution of Eq.(26). This Equation can be solved using LU decomposition and backsubstitution, which requires fewer computations than the inversion of
A(x). Further, the first and second order partial derivatives of
W i 0,
n i n i 0, n i
n
n
y y
y y
j (xi ) can be determined as follows [1921]:
x y y x x x
Differentiate Eq. (26) with respect to L, K, (L, K =X,Y) such as:
i 0, (i 1, N )
(34)
A(x),L (x) P,L (x) A,L (x) (x),

Simply supporting of the first kind: SS1
(L X ,Y )
(27)
W i 0,
N (n 2 n 2 ) c x (1 )n n c y j
x y ij x y ij x
0,
A(x),LK (x) P,LK (x) A,LK (x)(x)
j 1 ( n 2 n 2 )c y
(1 )n n c x j
(35)
(28)
x y ij x y ij y
A,L (x),K (x) A,K (x),L (x), (L, K X ,Y )
N (n 2 n 2 ) c y 2n n c x j
1 D ij
x y ij x y ij x
0,
2 j 1 (n 2 n 2 ) c x 2n n c y j
The first and second order partial derivatives of the shape function can be described as:
x y ij x y ij y
(i 1, N )
(x ) c L (x ) T (x )B (x )


Simply supporting of the second kind: SS2
j ,L i j i j ,L i j i
T (x )B (x
), (L x , y ),
(29)
W i 0,
n i n i
0,
j i j ,L i
x y y x
N (n 2 n 2 )c x (1 )n n c y j
(x ) c LK (x ) T
(x )B (x )
x y ij x y ij x
0
j ,LK i j i j ,LK i j i
j 1 ( n 2 n 2 )c y (1 )n n c x j
(36)
T (x )B (x ) T (x )B (x ) T
(x )B (x ),
x y ij x y ij y



i j ,LK i j ,L i j ,K i j ,K i j ,L i
(L, K x , y )
(30)

Free edge:
(i 1, N )
On suitable substitution from Eqs.(1430) into (68), the problem can be reduced to the following nonlinear eigenvalue problem:
N
N
x ij y ij x x y y
x ij y ij x x y y
kG h ij ((n c x n c y )W j ) kG h ij (n i n i ) 0,
j 1
N (n 2 n 2 )c x (1 )n n c y j
kGhij c xW j D ij c xx 1 c yy kGh ij j
x y ij x y ij x
0,
N ij
ij
2 ij
x
j 1 ( n 2 n 2 )c y (1 )n n c x j
(37)
x y ij x y ij y
j 1 D ij 1 c x c y j
(31)
N (n 2 n 2 )c y 2n n c x j
2 ik kj y
1 D ij x y ij x y ij x
0
2
2 2 x y j
2 I j , (i , k 1, N )
j 1 (nx ny )cij 2nx ny cij y
1 x (i 1, N )
kGhij c yW
j D ij 1 c x c y
j
N ij
2 ik
kj x



NUMERICAL RESULTS
j 1 D ij c yy

1 c xx

kGhij j
(32)
This section presents numerical results that
ij
2 ij
y
demonstrate convergence and efficiency of each one of
the proposed schemes for vibration analysis of non
y
y
2 I1 j , (i , k
N
N
kGhij c xx
1, N )
c yy W
j c x j c y j
uniform and nonlinear Winkler and uniform Pasternak foundation of composite plate. This plate varying with thickness. For all results, the boundary conditions (3437) are augmented in the governing equations (3133). Then
j 1
ij ij
ij x
ij y
the obtained nonlinear algebraic system is solved by
K c xx
c yy
W j K iW
j K
(W 2 )W j
(33)
using
u
iterative quadrature
ar f
technique
[34]. Thee
2 ij ij 1 3
comp tational ch acteristics o each schem are adapted
2 I 0W j , (i 1, N )
to reach accurate results with error of order 1010. The obtained frequencies are normalized such as:
The boundary conditions (913) can also be approximated using DQMs as:

Clamped edge:
( I (20 ) where 0 is the fundamental
frequency of isotropic squared plate.
For DSCDQ scheme based on regularized Shannon kernel (RSK), the problem is also solved over a uniform grids ranging from 7*7 to 19*19. The bandwidth 2M+1
ranges from 5 to 15 and the regularization parameter = r hx ranges from 1.5hx to 3 hx , where hx =1/N1.
Table (1) shows convergence of the obtained
fundamental frequency to the exact and numerical ones [2831] over grid size 13*13, bandwidth 11 and regulization parameter = 2.82 hx.
TABLE. 1 Comparison between the fundamental frequency due to DSCDQMRSK with the bandwidth 2M+1
fundamental frequency
Number of grid points
9
11
13
15
17
regularization parameter
Band width
DSCDQM RSK
=1.8 hx
2M+1 =7
2.03
2.46
2.74
2.89
2.9545
2M+1 =9
2.23
2.69
2.82
2.93
2.9736
2M+1 =11
2.58
2.88
2.91
2.95
2.9828
2M+1 =13
2.79
2.92
2.96
2.99
2.9991
=2.4 hx
2M+1 =7
2.29
2.64
2.83
2.92
2.9683
2M+1 =9
2.52
2.72
2.88
2.96
2.9743
2M+1 =11
2.61
2.91
2.94
2.98
2.9915
2M+1 =13
2.86
2.95
2.99
2.99
2.9976
=2.62 hx
2M+1 =7
2.58
2.72
2.89
2.96
2.9697
2M+1 =9
2.70
2.85
2.94
2.99
2.9953
2M+1 =11
2.76
2.94
2.99
3.01
3.0224
2M+1 =13
2.89
2.98
3.01
3.02
3.0213
=2.82 hx
2M+1 =7
2.86
2.92
2.98
2.99
3.0026
2M+1 =9
2.89
2.98
2.99
3.005
3.0141
2M+1 =11
2.94
2.99
3.02
3.022
3.0215
2M+1 =13
2.97
3.01
3.02
3.022
3.0215
Exact results [28]
3.0215
Ritz method[29]
3.0214
Radial basis[30]
3.0216
Element free Galerkin[31] (15×15)
3.0225
fundamental frequency
Number of grid points
9
11
13
15
17
regularization parameter
Band width
DSCDQM RSK
=1.8 hx
2M+1 =7
2.03
2.46
2.74
2.89
2.9545
2M+1 =9
2.23
2.69
2.82
2.93
2.9736
2M+1 =11
2.58
2.88
2.91
2.95
2.9828
2M+1 =13
2.79
2.92
2.96
2.99
2.9991
=2.4 hx
2M+1 =7
2.29
2.64
2.83
2.92
29683
2M+1 =9
2.52
2.72
2.88
2.96
2.9743
2M+1 =11
2.61
2.91
2.94
2.98
2.9915
2M+1 =13
2.86
2.95
2.99
2.99
2.9976
=2.62 hx
2M+1 =7
2.58
2.72
2.89
2.96
2.9697
2M+1 =9
2.70
2.85
2.94
2.99
2.9953
2M+1 =11
2.76
2.94
2.99
3.01
3.0224
2M+1 =13
2.89
2.98
3.01
3.02
3.0213
=2.82 hx
2M+1 =7
2.86
2.92
2.98
2.99
3.0026
2M+1 =9
2.89
2.98
2.99
3.005
3.0141
2M+1 =11
2.94
2.99
3.02
3.022
3.0215
2M+1 =13
2.97
3.01
3.02
3.022
3.0215
Exact results [28]
3.0215
Ritz method[29]
3.0214
Radial basis[30]
3.0216
Element free Galerkin[31] (15×15)
3.0225
,regularization parameter , the grid points N and the previous results for a regular discretized isotropic simply supported squared plate:( h0/a=0.01,v=0.3, k=5/6,K1=500,=0,=0,=0, K2=0, K3=0).
For MLSDQ scheme, circular influence domain, (i, i=1,N), is considered as shown in Fig.2. Gaussian weight function is employed such as [1921]:
size 11*11 , completeness order 4 and raduis of support domain dmax = 5 .
TABLE.2 Comparison between the fundamental frequency due to MLSDQM with the radius of support domain dmax completeness order Nc , the grid points N and the previous results for a regular discretized isotropic simply supported squared plate: (h/a=0.01,v=0.3, k=5/6, K1=100,=0,=0,=0, K2=0, K3=0).
fundamental frequency
Number of grid points
7
9
11
13
completeness order
Radius of support domain
MLSDQM
Nc=2
dmax =5
2.64
2.516
2.446
2.4509
dmax =6
2.31
2.295
2.272
2.2612
dmax =7
2.26
2.254
2.251
2.2491
dmax =8
2.25
2.247
2.246
2.2436
Nc =3
dmax =5
2.25
2.245
2.245
2.2441
dmax =6
2.25
2.243
2.243
2.2421
dmax =7
2.24
2.242
2.242
2.2414
dmax =8
2.24
2.241
2.241
2.2413
Nc =4
dmax =5
2.24
2.241
2.2413
2.2413
dmax =6
2.24
2.241
2.2413
2.2413
dmax =7
2.24
2.241
2.2413
2.2413
dmax =8
2.24
2.241
2.2413
2.2413
Nc =5
dmax =5
2.74
2.244
2.2413
2.2413
dmax =6
2.25
2.249
2.2413
2.2413
dmax =7
2.26
2.241
2.2413
2.2413
dmax =8
2.26
2.241
2.2413
2.2413
Exact results [28]
2.2413
Ritz method[29]
2.2413
Radial basis[30]
2.2414
Element free Galerkin[31] (15×15)
2.2427
Table (3) shows that the best value of Shear factor correction is to be taken (5 /6v). Tables (3,4) show that the fundamental frequency decrease with increasing Poisson ratio and Shear factor correction. Also ,the value of shear factor correction helps to achieve more accurate results.
TABLE.3 Comparison between the fundamental frequency, Poisson , shear factor correction K and the previous results
exp((d
i
/ c )2 ) exp((r / c )2 )
Poisson ratio v
Discrete Green function [33]
=
= ( + ( ))
(( ))
= ( + ( ))
( ( ))
=
=
0.15
8.1424
8.1638
8.1639
8.164
8.163
0.23
8.1110
8.1110
8.1112
8.1112
8.111
8.111
0.3
8.0640
8.04586
8.0460
8.046
8.046
0.5
7.8090
7.7430
7.7432
7.743
7.744
Poisson ratio v
Discrete Green function [33]
=
= ( + ( ))
(( ))
= ( + ( ))
( ( ))
=
=
0.15
8.1424
8.1638
8.1639
8.164
8.163
0.23
8.1110
8.1110
8.1112
8.1112
8.111
8.111
0.3
8.0640
8.04586
8.0460
8.046
8.046
0.5
7.8090
7.7430
7.7432
7.743
7.744
2
di r
(38)
for CCCS squared plate: (h0/a=0.01, K1=0,=0.5,=0.5,=0, K =0 , K =0,E /E =2.45).
w i (x )
0
0
1 exp((r / c ) )
di r
2 3 1 2
Where di is the distance from a nodal point xi
to a field one
x located in the influence domain of xi. r is the radius of support domain and c is the dilation parameter. In the present work, the dilation parameter is selected such as: c=r/4.
The scheme is employed with varying completeness order Nc ranges from 2 to 5 and the raduis of support domain dmax = r/ hx ranges from 4 to 9. Also, the problem is also solved over a uniform grids ranging from 5*5 to 19*19. Table (2) shows convergence of the obtained fundamental frequency to the exact and numerical ones [2831] over grid
Boundary coditions
natural frequency
1
2
3
Execution time (sec)
variable thickness
results
SSSS
=0
DSCDQM RSK
4.902
7.253
8.374
1.780280
MLSDQM
4.902
7.253
8.374
1.839144
Element free Galerkin [31]
4.900
7.256
8.382
Discrete Green function [33]
4.902
7.253
8.374
=0.4
DSCDQM RSK
5.360
7.928
9.150
1.889903
MLSDQM
5.360
7.928
9.150
1.918841
Element free Galerkin [31]
5.356
7.931
9.159
Discrete Green function [33]
5.360
7.928
9.150
=0.8
DSCDQM RSK
5.770
8.525
9.831
1.900178
MLSDQM
5.770
8.525
9.831
1.928867
Element free Galerkin [31]
5.772
8.529
9.845
Discrete Green function [33]
5.770
8.525
9.831
CCCC
=0
DSCDQM RSK
6.780
8.953
10.29
1.632990
MLSDQM
6.780
8.953
10.29
1.646533
Element free Galerkin [31]
6.748
8.905
10.24
Discrete Green function [33]
6.780
8.953
10.29
=0.4
DSCDQM RSK
7.402
9.770
11.23
1.716354
MLSDQM
7.402
9.770
11.23
1.726170
Element free Galerkin [31]
7.371
9.723
11.18
Discrete Green function [33]
7.402
9.770
11.23
=0.8
DSCDQM RSK
7.945
10.48
12.05
1.731625
MLSDQM
7.945
10.48
12.05
1.742609
Element free Galerkin [31]
7.915
10.43
12.01
Discrete Green function [33]
7.945
10.48
12.05
Boundary coditions
natural frequency
1
2
3
Execution time (sec)
variable thickness
results
SSSS
=0
DSCDQM RSK
4.902
7.253
8.374
1.780280
MLSDQM
4.902
7.253
8.374
1.839144
Element free Galerkin [31]
4.900
7.256
8.382
Discrete Green function [33]
4.902
7.253
8.374
=0.4
DSCDQM RSK
5.360
7.928
9.150
1.889903
MLSDQM
5.360
7.928
9.150
1.918841
Element free Galerkin [31]
5.356
7.931
9.159
Discrete Green function [33]
5.360
7.928
9.150
=0.8
DSCDQM RSK
5.770
8.525
9.831
1.900178
MLSDQM
5.770
8.525
9.831
1.928867
Element free Galerkin [31]
5.772
8.529
9.845
Discrete Green function [33]
5.770
8.525
9.831
CCCC
=0
DSCDQM RSK
6.780
8.953
10.29
1.632990
MLSDQM
6.780
8.953
10.29
1.646533
Element free Galerkin [31]
6.748
8.905
10.24
Discrete Green function [33]
6.780
8.953
10.29
=0.4
DSCDQM RSK
7.402
9.770
11.23
1.716354
MLSDQM
7.402
9.770
11.23
1.726170
Element free Galerkin [31]
7.371
9.723
11.18
Discrete Green function [33]
7.402
9.770
11.23
=0.8
DSCDQM RSK
7.945
10.48
12.05
1.731625
MLSDQM
7.945
10.48
12.05
1.742609
Element free Galerkin [31]
7.915
10.43
12.01
Discrete Green function [33]
7.945
10.48
12.05
TABLE.4 Comparison between the fundamental frequency, Poisson , shear facor correction K and different boundary condition for squared plate: (h /a=0.01, K =100,=0,=0,=0,
0 1
K2=50 , K3=0,E1/E2=2.45).
Poisson ratio v
shear factor
correction
=
SSSS
CCCC
CSSS
CCSC
0.15
0.8547
3.9382
5.20912
4.2312
4.87028
0.23
0.8666
3.9207
5.18596
4.2123
4.84863
0.3
0.8772
3.8935
5.1501
4.1831
4.8151
0.5
0.9091
3.7429
4.95105
4.0213
4.62899
Tables (5, 6) show that execution time of DSCDQMRSK scheme is less than that of MLSDQM. Therefore, it is more efficient than MLSDQM for vibration analysis of variable thickness, nonuniform and nonlinear elastically supported composite plate.
TABLE. 5 Comparison between the obtained natural frequencies and the previous results for isotropic clamped plate: (h0/a=0.015,v=0.15 ,=0,=0,=0, K2=0,K3=0).
Subgrade reaction
K1=1390.2
K1=2780.4
Execution time (sec)
Results
2
3
1
2
3
MLSDQM (11×11),
Nc=4 and dmax =5
5.245
8.316
8.316
6.463
9.132
9.132
1.676119
DSCDQM RSK(13×13)
=2.82hx and M=5
5.245
8.316
8.316
6.463
9.132
9.132
1.656792
Mixed finite element [32]
5.245
8.316
8.316
6.463
9.132
9.132
Element free Galerkin[31]
5.267
8.391
8.392
6.477
9.202
9.203
Ritz method[29 ]
5.259
8.432
8.432
6.460
9.248
9.248
Radial basis[30 ]
5.244
8.313
8.313
6.463
9.130
9.130
Also, for different boundary conditions and varying thickness in one and two directions, tables (6,7) insist that DSCDQMRSK scheme is the best choice for such problem.
TABLE. 6 Comparison between the natural frequency due to DSCDQMRSK and MLSDQM with the variable thickness in one direction ,different boundary conditions and the previous results for a squared plate: (h/a=0.01,E1/E2=2.45 v=0.23, K1=0, =0, K2=0,K3=0).
TABLE. 7 Comparison between the natural frequency due to DSCDQMRSK and MLSDQM with the variable thickness , in two direction ,different boundary conditions and the previous results for a squared plate: (h0/a=0.01,E1/E2=2.45, v=0.23, K1=0, K2=0, K3=0).
Boundary coditions
natural frequency
variable thickness
results
SSSS
0.5
0.5
DSCDQM RSK
3.635
5.335
6.087
MLSDQM
3.635
5.335
6.087
Element free Galerkin [31]
3.633
5.3460
6.0957
Discrete Green function [33]
3.635
5.335
6.087
0.5
0.5
DSCDQM RSK
4.704
6.937
7.966
MLSDQM
4.704
6.937
7.966
Element free Galerkin [31]
4.707
6.9422
7.9751
Discrete Green function [33]
4.704
6.937
7.966
0.5
0.5
DSCDQM RSK
4.708
6.933
7.904
MLSDQM
4.708
6.933
7.904
Element free Galerkin [31]
4.708
6.9420
7.9146
Discrete Green function [33]
4.708
6.933
7.904
0.5
0.5
DSCDQM RSK
6.086
9.022
10.350
MLSDQM
6.086
9.022
10.350
Element free Galerkin [31]
6.099
9.013
10.359
Discrete Green function [33]
6.086
9.022
10.350
CCCC
0.5
0.5
DSCDQM RSK
4.955
6.548
7.440
MLSDQM
4.955
6.548
7.440
Element free Galerkin [31]
4.936
6.532
7.4167
Discrete Green function [33]
4.955
6.548
7.440
0.5
0.5
DSCDQM RSK
6.453
8.510
9.748
MLSDQM
6.453
8.510
9.748
Element free Galerkin [31]
6.427
8.481
9.715
Discrete Green function [33]
6.453
8.510
9.748
0.5
0.5
DSCDQM RSK
6.447
8.525
9.671
MLSDQM
6.447
8.525
9.671
Element free Galerkin [31]
6.420
8.501
9.639
Discrete Green function [33]
6.45
8.525
9.671
0.5
0.5
DSCDQM RSK
8.390
11.076
12.666
MLSDQM
8.390
11.076
12.666
Element free Galerkin [31]
8.36
11.040
12.632
Discrete Gree function [33]
8.390
11.076
12.666
Boundary coditions
natural frequency
variable thickness
results
SSSS
0.5
0.5
DSCDQM RSK
3.635
5.335
6.087
MLSDQM
3.635
5.335
6.087
Element free Galerkin [31]
3.633
5.3460
6.0957
Discrete Green function [33]
3.635
5.335
6.087
0.5
0.5
DSCDQM RSK
4.704
6.937
7.966
MLSDQM
4.704
6.937
7.966
Element free Galerkin [31]
4.707
6.9422
7.9751
Discrete Green function [33]
4.704
6.937
7.966
0.5
0.5
DSCDQM RSK
4.708
6.933
7.904
MLSDQM
4.708
6.933
7.904
Element free Galerkin [31]
4.708
6.9420
7.9146
Discrete Green function [33]
4.708
6.933
7.904
0.5
0.5
DSCDQM RSK
6.086
9.022
10.350
MLSDQM
6.086
9.022
10.350
Element free Galerkin [31]
6.099
9.013
10.359
Discrete Green function [33]
6.086
9.022
10.350
CCCC
0.5
0.5
DSCDQM RSK
4.955
6.548
7.440
MLSDQM
4.955
6.548
7.440
Element free Galerkin [31]
4.936
6.532
7.4167
Discrete Green function [33]
4.955
6.548
7.440
0.5
0.5
DSCDQM RSK
6.453
8.510
9.748
MLSDQM
6.453
8.510
9.748
Element free Galerkin [31]
6.427
8.481
9.715
Discrete Green function [33]
6.453
8.510
9.748
0.5
0.5
DSCDQM RSK
6.447
8.525
9.671
MLSDQM
6.447
8.525
9.671
Element free Galerkin [31]
6.420
8.501
9.639
Discrete Green function [33]
6.45
8.525
9.671
0.5
0.5
DSCDQM RSK
8.390
11.076
12.666
MLSDQM
8.390
11.076
12.666
Element free Galerkin [31]
8.36
11.040
12.632
Discrete Green function [33]
8.390
11.076
12.666
Furthermore, a parametric study is introduced to investigate the influence of elastic and geometric characteristics of the composite on the values of natural frequencies.
Figures (315) show that the natural frequencies decrease with increasing variable thickness (, ), Young's modulus gradation ratio and thickness ratio. As well as, Figs. (3 10,1213) show that the natural frequencies are increased with increasing variation parameter of foundation stiffness , Winkler and Pasternak foundation, shear modulus gradation ratio and aspect ratio a/b . The case of (E1=E2=E3=E4=E5, G1=G2=G3=G4=G5 and p=p=p=h4=p) is a limiting case of this study which was previously solved in [28] and [2933]. Figures(1215) show the first three mode shapes of the vibration waves along the interface.Furthermore, Figs.(14

show that the amplitudes of W increase with increasing linear and nonlinear elastic foundation parameters.Also, these figures show that the natural frequencies do not affect significantly by nonlinear elastic foundation parameter k3.
Normalized fundamental frequency
Normalized fundamental frequency
28
26 =1000
24
22 =500
20
18
16 =100
14
12
10 =0
8
6
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thickness variable
Normalized fundamental frequency
Normalized fundamental frequency
28
26 =1000
24
22 =500
20
18
16 =100
14
12
10 =0
8
6
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thickness variable

FIG. 3 Variation of the normalized fundamental frequency withthickness variable (, ) and nonuniform Winkler foundationfor squared simply supported plates. A: =0, B: =0.5(K1=1000, K2=0,K3=0).
Normalized fundamental frequency
Normalized fundamental frequency
Normalized fundamental frequency
Normalized fundamental frequency
44 42
42 40
=1000
40 38
38
38
=500
36
36 =100 34
34
32
32 =0 30
30
28
28
=1000
=500
=100
=0
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thikness variable
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thikness variable
Normalized fundamental frequency
Normalized fundamental frequency
42
40 =1000
38 =500
36 =100
34
FIG. 5 Variation of the normalized fundamental frequency with thickness variable (, )
,nonuniform ,nonlinear Winkler and Pasternak foundation for squared simply supported plates. A: =0, B: =0.5 (K1=1000, K2=10, K3=50)
Normalized fundamental frequency
Normalized fundamental frequency
32 =0 18
30
28 15
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
B Thikness variable
12
=0.5,0.5
9 =0.5,0.5
FIG. 4 Variation of the normalized fundamental frequency with thickness variable (, ) and nonuniform Winkler and uniform Pasternak foundation for squared simply
supported plates. A: =0, B: =0.5 (K1=1000, K2=10,K3=0)
Normalized fundamental frequency
Normalized fundamental frequency
44
=0.5,0.5
=0.5,0.5
6
3
0 2000 4000 6000 8000 10000
A
42
=1000
40
38 =500
36 =100
34
32 =0
30
28
27
Normalized fundamental frequency
Normalized fundamental fequency
24
21
18 =0.5,0.5
15 =0.5,0.5
12 =0.5,0.5
=0.5,0.5
9
6
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thikness variable
3
0 2000 4000 6000 8000 10000
FIG. 6 Variation of the normalized fundamental frequency with thickness variable (, ) and nonuniform Winkler foundation for squared simply supported plates . A: K1=10, B: K1=100 ( K2=0,K3=0)
Normalized fundamental frequency
Normalized fundamental frequency
30
28
26
24 =0.5,0.5
22 =0.5,0.5
=0.5,0.5
20
=0.5,0.5
18
16
0 2000 4000 6000 8000 10000

51
Normalized fundamental frequency
Normalized fundamental frequency
48
45
42
39 =0.5,0.5
36 =0.5,0.5
33
30 =0.5,0.5
27 =0.5,0.5
24
21
18
0 2000 4000 6000 8000 10000
Normalized fundamental frequency
Normalized fundamental frequency
51
48
45
42
39 =0.5,0.5
36 =0.5,0.5
33
30 =0.5,0.5
27 =0.5,0.5
24
21
18
FIG. 8 Variation of the normalized fundamental frequency with thickness variable (, ) and nonuniform, nonlinear Winkler and Pasternak foundation for squared simply supported plates. A: K1=10, B: K1=100 ( K2=10,K3=100)
60
54
Natural frequencies
Natural frequencies
48
42
0 2000 4000
B
6000 8000 10000
36
30 E3=E2=0.2E1
E5=0.2E4
FIG. 7 Variation of the normalized fundamental frequency 24
E3=E2=E1=E4=E5=1
E4=E5=2E1
with thickness variable (, ) and nonuniform 18
E =E =E =3E
Winkler and Pasternak foundation for squared
12 E3=0.3E2 E =E =E =5E
1 4 5 2
simply supported plates. A: K1=10, B: K1=100 ( K2=10,K3=0)
3 2 1 4
6
0
Normalized fundamental frequency
Normalized fundamental frequency
30
28
26
24 =0.5,0.5
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5
A Location along the interface.
60
54
Natural frequencies
Natural frequencies
48
42
22 =0.5,0.5
=0.5,0.5
20
=0.5,0.5
36
30 G1=G4=G5=1.5G2
24
G4=G5=2G1
G5=1.75G4 G3=G2=G1=G4=G5=1
18
16
0 2000 4000 6000 8000 10000
A
18 G3=0.75G2
12
6
0
G3=G2=G1=0.5G4 G3=G2=0.25G1
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5
B Location along the interface.
FIG. 9 Variation of the natural frequencies with Young's and Shear modulus gradation ratio of a squared simply supported composite (K1=100, K2=10,K3=50, h0/a=0.01,=1000, =0.5, =0.5, v1=v2=v3=v4=v5).
250
225
Natural frequencies
Natural frequencies
200
175
150
125
a1=a4=a5=3a2
a5=4a4
48
Natural frequencies
Natural frequencies
42
36
h =h =0.5h
100 a =a =2a
3 2 1
30
p=p=p=.7h4
h =h =h =h =h =1
4 5 1
75
p=.6p
24
3 2 1 4 5
50
3 2
3 2
a =(2/3)a
25 a =a =0.5a
a =a =a =.75a
a3=a2=a1=a4=a5=1
18 h =h =h =3h
h4=p=2p
3 2 1
0
3 2 1 4
1 4 5 2
p=4h4
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5
A Location along the interface.
130
120
110
Natural frequencies
Natural frequencies
100
90
12
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5
B Location along the interface.
FIG. 11 Variation of the natural frequencies with thickness of a squared elastically supported composite. A: Simply supported plates, B: Clamped plates
80
70 a1=a4=a5=3a2
60
50
40
30
a5=4a4
a4=a5=2a1
0.100
(K1=1000, K2=1000, K3=1000, =1000, =0.5, =0.5;E1=E2=E3=E4=E5, G1=G2=G3=G4=G5, v1=v2=v3=v4=v5).
20 a3=(2/3)a2
a =a =a =a =a =1
10 a3=a2=0.5a1 a3=a2=a1=.75a4
0
3 2 1 4 5
Lateral mode shape: w
Lateral mode shape: w
0.075
Fundamental mode
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5
B Location along the interface.
FIG. 10 Variation of the natural frequencies with aspect ratio (a/b) and thickness (h/a) for Clamped composite plates. A: h0/a=0.01, B: h0/a=0.1 (K1=500, K2=100, K3=100,=1000, =0.5, = 0.5;E1=E2=E3=E4=E5, G1=G2=G3=G4=G5,
v1=v2=v3=v4=v5).
0.050
0.025
0.000
0.025
Second mode Third mode
50 A
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time along the interface: (x, y=0)
45
Natural frequencies
Natural frequencies
40
35 h =h =0.5h
0.25
Lateral mode shape: w
Lateral mode shape: w
0.20
Fundamental mode
3 2 1
30
p=p=p=.7h4
h =h =h =h =h =1
0.15
25 p=.6p
3 2 1 4 5
20
p=h4=p=3p
15
10
h4=p=2p
p=4h4
0.10
0.05
0.00
Second mode
Third mode
0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5

Location along the interface.
0.05
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)
FIG. 12 Variation of the lateral mode shapes with time and nonuniform Winkler foundation for squared clamped plate at =0, =0,K2=0,K3=0, =1000. A: K1=10, B: K1=100
0.150
Lateral mode shape: w
Lateral mode shape: w
0.125
0.100
0.075
0.050
0.025
0.000
0.025
0.050
Fundamental mode
Second mode Third mode
0.440
0.385
Lateral mode shape: w
Lateral mode shape: w
0.330
0.275
0.220
0.165
0.110
0.055
0.000
Fundamental mode
Second mode Third mode
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)
0.385
Fundamental mode
0.055
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)
FIG. 14 Variation of the lateral mode shapes with time and nonuniform and nonlinear Winkler
Lateral mode shape: w
Lateral mode shape: w
0.330
0.275
0.220
0.165
0.110
0.055
0.000
0.055
Second mode Third mode
0.200
0.175
Lateral mode shape: w
Lateral mode shape: w
0.150
0.125
foundation for squared clamped plate at =0.5, =0.5,K2=0, K3=100, =1000. A: K1=10, B: K1=500
Fundamental mode
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
B Time along the interface: (x, y=0)
FIG. 13 Variation of the lateral mode shapes with time and nonuniform Winkler and Pasternak foundation for squared clamped plate at =0.5, =0.5,K2=100, K3=0, =1000. A: K1=10, B: K1=500
0.100
0.075
0.050
0.025
0.000
0.025
0.050
Second mode Third mode
0.150
0.125
Fundamental mode
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A Time along the interface: (x, y=0)
0.585
Lateral mode shape: w
Lateral mode shape: w
0.100
0.075
0.050
0.025
0.000
0.025
0.050
Second mode Third mode
0.520
Lateral mode shape: w
Lateral mode shape: w
0.455
0.390
0.325
0.260
0.195
0.130
0.065
0.000
0.065
Fundamental mode
Second mode Third mode
A
0.150
Lateral mode shape: w
Lateral mode shape: w
0.125
0.100
0.075
0.050
0.0250.000
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time along the interface: (x, y=0)
Fundamental mode
Second mode Third mode
0.130
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
B Time along the interface: (x, y=0)
FIG. 15 Variation of the lateral mode shapes with time and nonuniform and nonlinear Winkler and Pasternak foundation for squared clamped plate at =0.5, =0.5,K2=100, K3=100, =1000. A: K1=10, B: K1=500
0.025
0.050
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time along the interface: (x, y=0)


CONCLUSION
Different Quadrature schemes have been successfully applied for vibration analysis of composite plate with variable thickness resting on nonuniform and nonlinear elastic foundation. Iterative quadrature technique is used to solve the nonlinear algebraic system. A matlab program is designed for each scheme such that the maximum error (comparing with the previous exact results) is 1010 . Also, Execution time for each scheme, is determined. It is concluded that discrete singular convolution differential quadrature method based on regularized Shannon kernel (DSCDQMRSK) with grid size 13*13 , bandwidth 2M+1
11 and regulization parameter = 2.82 hx leads to best accurate efficient results for the concerned problem. shear factor correction (K=5/6v )is considered for such problem. Based on this scheme, a parametric study is introduced to investigate the influence of elastic and geometric characteristics of the vibrated plate, on results. It is aimed that these results may be useful for design purposes of engineering fields.
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