Research Article
Improved non-dimensional dynamic
influence function method for vibration
analysis of arbitrarily shaped plates
with clamped edges
Advances in Mechanical Engineering
2016, Vol. 8(3) 1–8
Ó The Author(s) 2016
DOI: 10.1177/1687814016638586
aime.sagepub.com
Sang-Wook Kang1 and Satya N Atluri2
Abstract
A new formulation for the non-dimensional dynamic influence function method, which was developed by the authors, is
proposed to efficiently extract eigenvalues and mode shapes of clamped plates with arbitrary shapes. Compared with
the finite element and boundary element methods, the non-dimensional dynamic influence function method yields highly
accurate solutions in eigenvalue analysis problems of plates and membranes including acoustic cavities. However, the
non-dimensional dynamic influence function method requires the uneconomic procedure of calculating the singularity of
a system matrix in the frequency range of interest for extracting eigenvalues because it produces a non-algebraic eigenvalue problem. This article describes a new approach that reduces the problem of free vibrations of clamped plates to
an algebraic eigenvalue problem, the solution of which is straightforward. The validity and efficiency of the proposed
method are illustrated through several numerical examples.
Keywords
Non-dimensional dynamic influence function method, free vibration, arbitrarily shaped plate, algebraic eigenvalue problem, natural frequency, mode shape
Date received: 19 October 2015; accepted: 12 February 2016
Academic Editor: Farzad Ebrahimi
Introduction
For vibration problems of plates with arbitrary shapes,
numerical approximate methods such as the finite element method (FEM) and boundary element method
(BEM) are usually applied.1–5 However, these methods
cannot be expected to give very accurate results because
a large number of numerical calculations are required
as the number of nodes increases. To obtain highly
accurate results, the non-dimensional dynamic influence function (NDIF) method for free vibration analysis of arbitrarily shaped membranes has been
introduced6,7 by the authors and extended to arbitrarily
shaped acoustic cavities8 and clamped plates.9 It has
then been applied to arbitrarily shaped plates with a
mixed boundary composed of simply supported and
clamped edges10 and to arbitrarily shaped plates with
free edges.11
Recently, many researchers have studied various
meshless methods, which discretize the entire region of
a plate unlike the NDIF method. For instance, Bert
and Malik12 developed a methodology for the extension of the differential quadrature method to
1
Department of Mechanical Systems Engineering, Hansung University,
Seoul, Korea
2
Department of Mechanical and Aerospace Engineering, University of
California—Irvine, Irvine, CA, USA
Corresponding author:
Sang-Wook Kang, Department of Mechanical Systems Engineering,
Hansung University, Seoul 136-792, Korea.
Email: swkang@hansung.ac.kr
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License
(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
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Advances in Mechanical Engineering
irregularly shaped plates using the mapping concept of
a square region to a general curvilinear region. Misra13
proposed the radial basis function (RBF) method using
multiple linear regression analysis to obtain eigenvalues
of simply supported and clamped plates, which were
compared with the eigenvalues presented by the
authors’ previous research.9 Krowiak14 applied the
RBF-pseudospectral method to the free vibration analysis of two-dimensional structures, which combines the
meshless feature of the RBF method and the simplicity
of the pseudospectral method. He also used eigenvalues6 presented by the authors to show the accuracy of
his method. Bui and Nguyen15 developed a moving
Kriging-interpolation-based meshfree method having
Kronecker’s delta property. Finally, Rodrigues et al.16
presented a meshless technique based on the RBF-QR
algorithm for the static and vibration analyses of
plates. The above methods12–16 still have a limitation in
accuracy owing to the large number of numerical calculations compared with the NDIF method.9
Although the NDIF method yields highly accurate
results compared with FEM, BEM, and other numerical methods,9 its final system matrix equation does not
have a form of an algebraic eigenvalue problem. As a
result, the NDIF method needs the inefficient procedure of searching frequency values that make the system matrix singular in the frequency range of interest.
To overcome this limitation, we have recently developed an improved NDIF method for membranes17,18
and acoustic cavities19 with arbitrary shapes. In this
article, the basic concept of the improved NDIF
method17–19 is extended to arbitrarily shaped plates
with clamped edges. The proposed method is validated
by comparing its results with those of other methods
such as the NDIF method, FEM (ANSYS), and the
exact method. It is revealed that the proposed method
gives as accurate results as the NDIF method and
needs much shorter calculation time than the NDIF
method. Note that the computational speed of the
improved NDIF method was reported in detail in the
authors’ previous paper.17
NDIF method review
The equation of motion for free flexural vibration of a
uniform thin plate is written as20
D r4 w + r s
∂2 w
=0
∂t2
ð1Þ
where w = w(r, t) is the transverse deflection, rs is the
surface density, and D is the flexural rigidity expressed
as D = Eh3 =12(1 n2 ) in terms of Young’s modulus E,
Poisson’s ratio n, and the plate thickness h. Assuming a
harmonic motion w(r, t) = W (r) ejvt where v is the circular frequency, equation (1) becomes
Figure 1. Infinite plate including an arbitrarily shaped plate
(dashed line).
r4 W L4 W = 0
ð2Þ
where the frequency parameter L is
r v2
L= s
D
1=4
ð3Þ
Considering an infinite plate with a point source
located at point Ps as shown in Figure 1, the deflection
form of the plate depends on the scalar distance j r rs j
from the source point Ps to the field point P. If the
deflection at the source point is assumed to be regular
in value (no singularity exists in the entire domain), a
general solution of equation (2) is given by
W (r) = AJ0 ðLjr rs jÞ + BI0 ðLjr rs jÞ
ð4Þ
where J0 and I0 denote the Bessel function of the first
kind and the modified Bessel function of the first kind,
respectively.21 The general solution is termed the NDIF
of the infinite plate. It is a wave-type function that propagates omni-directionally from one point to infinity.
For free vibration analysis of an arbitrarily shaped
plate whose boundary is illustrated by the dotted line in
Figure. 1, N nodes are first distributed at points P1 ,
P2 ,., PN along the boundary. Assuming that harmonic
displacements of amplitudes A1 , A2 ,., AN are generated at points P1 , P2 ,., PN , respectively, the total displacement response at point P may be obtained by the
sum of responses (linear combination of NDIFs given
in equation (4)) that have resulted from each boundary
point as follows
W (r) =
N
X
Ak J0 ðLjr rk jÞ + Bk I0 ðLjr rk jÞ
ð5Þ
k =1
Equation (5) is employed as an approximate solution
for the free vibration of the finite-sized plate depicted
by the dotted line in Figure 1. Note that the
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Kang and Atluri
3
approximate solution also satisfies the governing equation (equation (2)) because the NDIF satisfies the governing equation.
If a clamped plate is considered, the boundary condition defined continuously along the dotted line in
Figure 1 is discretized so as to be satisfied only at the N
nodes as follows20
ð6Þ
W (ri ) = 0
∂W (ri )
= 0,
∂ni
N
X
Ak J0 ðLjri rk jÞ + Bk I0 ðLjri rk jÞ = 0,
k =1
ð8Þ
i = 1, 2, . . . , N
N
X
∂W (ri )
∂
=
Ak
J0 ðLjri rk jÞ
∂ni
∂ni
k =1
+ Bk
∂
I0 ðLjri rk jÞ = 0,
∂ni
i = 1, 2, . . . , N
ð9Þ
Equation (9) may be rewritten as
N
X
Ak LJ1 ðLjri rk jÞ
k =1
Bk LI1 ðLjri rk jÞ
∂
j ri rk j
∂ni
∂
jri rk j = 0,
∂ni
SM(J ) SM(I)
)
SM(J
SM(I)
n
n
A
C=
B
i = 1, 2, . . . , N
ð10Þ
ð14Þ
ð15Þ
The condition that the system matrix equation
(equation (13)) has a non-trivial solution gives that the
determinant of the system matrix SM(L) is zero as
follows
ð7Þ
i = 1, 2, . . . , N
where ni denotes the normal direction from the boundary at point Pi in Figure 1. Substituting equation (5)
into equations (6) and (7) gives
W (ri ) =
SM(L) =
det½SM(L) = 0
ð16Þ
Eigenvalues of the plate can be obtained from the
values of the frequency parameter (L) that satisfy equation (16). The mode shapes corresponding to the eigenvalues can be produced by substituting Ak and Bk
obtained from equation (13) into equation (5). Note
that a matrix manipulation process to reduce the system matrix size in equation (13) has not been described
in detail here.
It may be seen in equation (13) that elements of the
system matrix SM(L) depend on the frequency parameter L. Therefore, the inefficient procedure of searching for the frequency parameter that makes the system
matrix singular by sweeping the frequency parameter in
the range of interest is required to extract eigenvalues
in the NDIF method. The next section presents an
improved theoretical formulation for the NDIF
method that makes the system matrix independent of
the frequency parameter.
Improved theoretical formulation
The four Bessel functions J0 , I0 , J1 , and I1 in equations
(8) and (10) are expanded in a Taylor series21 as follows
Equations (8) and (10) can be rewritten as matrix forms
(J )
(I)
SM A + SM B = 0
)
(I)
SM(J
n A + SMn B = 0
ð11Þ
M
X
( 1)s ðLjri rk j=2Þ2s
½G(s + 1)2
s=0
ð12Þ
where system matrices SM(J ) and SM(I) are given by
SM (J ) (i, k) = J0 (Ljri rk j) and SM (I) (i,k)=I0 (Ljri rk j),
)
(I)
respectively, and system matrices SM(J
n and SMn are
(J )
and
given
by
SMn (i,k)=(∂=∂ni )J0 (Ljri rk j)
SMn(I) (i,k)=(∂=∂ni )I0 (Ljri rk j), respectively. Note that
the calculation method for the diagonal elements of the
above four system matrices for ri rk =0 (i.e. i=k) was
presented in the authors’ previous paper.9
Furthermore, equations (11) and (12) can be combined into a single equation
SM(L) C = 0
J0 ðLjri rk jÞ ’
ð13Þ
where the final system matrix SM(L) and the unknown
vector C are given by
I 0 ðL j ri rk j Þ ’
M
X
ðLjri rk j=2Þ2s
s=0
J1 ðLjri rk jÞ ’
½G(s + 1)2
M
X
( 1)s ðLjri rk j=2Þ1 + 2s
s=0
I1 ðLjri rk jÞ ’
G(s + 1)G(s + 2)
M
X
ðLjri rk j=2Þ1 + 2s
s=0
G(s + 1)G(s + 2)
ð17Þ
ð18Þ
ð19Þ
ð20Þ
where M denotes the number of terms of the series and
G( . . . ) represents the gamma function. For simplicity,
equations (17)–(20) may be rewritten as
J0 ðLjri rk jÞ ’
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M
X
s=0
L2s fJs (i, k)
ð21Þ
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Advances in Mechanical Engineering
M
X
I0 ðLjri rk jÞ ’
L
2s
fIs (i, k)
ð22Þ
s=0
J1 ðLjri rk jÞ ’
M
X
L
1 + 2s
uJs (i, k)
ð23Þ
M
X
M
X
L
!
N
X
2s
s=0
s=0
I1 ðLjri rk jÞ ’
Changing the order of
tions (29) and (31) leads to
Ak fJs (i,k)
+
PN
k =1
M
X
ð25Þ
½G(s + 1)2
ðjri rk j=2Þ2s
M
X
L2(1 + s)
N
X
L2(1 + s)
ð28Þ
l0
N
X
Bk
M
X
2s
L fIs (i,k) =0,
+l
ð29Þ
k =1
Bk L
l0
N
X
0
l
N
X
Bk fIM (i, k) = 0
Ak cJ0 (i, k) + l1
N
X
Ak cJ1 (i, k)
k=1
N
X
N
X
Ak cJM (i, k)
Bk cI0 (i, k) + l1
k =1
N
X
ð37Þ
Bk cI1 (i, k)
k =1
+ + lM
N
X
Bk cIM (i, k) = 0
k =1
!
Next, equations (36) and (37) may be expressed in
matrix equations as follows
L1 + 2s cJs (i, k)
!
L1 + 2s cIs (i, k)
and
ð36Þ
Bk fI1 (i, k)
k =1
= 0,
i = 1, 2, . . . , N
ð31Þ
where
N
X
k =1
+ + lM
ð30Þ
c(I)
s (i, k)
Ak fJM (i, k)
Bk fI0 (i, k) + l1
k =1
s=0
)
c(J
s (i, k)
N
X
k =1
L1 + 2s uJs (i, k)
M
X
0
N
X
+ + lM
!
s=0
N
X
Ak fJ1 (i, k)
k =1
k =1
Equation (30) is simplified further as
k =1
N
X
k =1
!
i = 1, 2, . . . , N
Ak L
Ak fJ0 (i, k) + l1
+ + lM
s=0
k =1
M
X
N
X
k =1
∂
jri rk j
∂ni
s=0
k =1
!
N
M
X
X
∂
1 + 2s I
Bk L
L
us (i, k)
jri rk j = 0,
∂n
i
s=0
k =1
N
X
i = 1, 2, . . . , N
ðjri rk j=2Þ1 + 2s
G(s + 1)G(s + 2)
Similarly, substituting equations (23) and (24) into
equation (10) yields
Ak L
Bk cIs (i, k) = 0,
k =1
ð27Þ
L fJs (i,k) +
M
X
!
( 1) ðjri rk j=2Þ
G(s + 1)G(s + 2)
1 + 2s
i=1,2, ...,N
N
X
N
X
After defining l = L2 , equations (34) and (35) are
rewritten as the forms of polynomial equations with
respect to l as follows
!
s=0
Ak cJs (i, k)
k =1
M
X
Substituting equations (21) and (22) into equation
(8) yields
Ak
!
ð35Þ
uIs (i, k) =
k =1
=0
k =1
ð26Þ
½G(s + 1)2
s
2s
!
ð34Þ
s=0
( 1)s ðjri rk j=2Þ2s
fIs (i, k) =
M
X
in equa-
Bk fIs (i,k)
ð24Þ
s=0
N
X
s=0
i =1, 2, ... ,N
L1 + 2s uIs (i, k)
where fJs (i, k), fIs (i, k), uJs (i, k), and uIs (i, k) are given by
uJs (i, k) =
PM
N
X
2s
s =0
k =1
s=0
fJs (i, k) =
L
and
are
l0 FJ0 + l1 FJ1 + + lM FJM A
+ l0 FI0 + l1 FI1 + + lM FIM B = 0
0 J
l C0 + l1 CJ1 + + lM CJM A
l0 CI0 + l1 CI1 + + lM CIM B = 0
ð38Þ
ð39Þ
)
(J )
c(J
s (i, k) = us (i, k)
∂
j ri rk j
∂ni
ð32Þ
where the elements of matrices F(s ) and C(s ) are given
by
(I)
c(I)
s (i, k) = us (i, k)
∂
j ri rk j
∂ni
ð33Þ
F(s ) (i, k) = f(j ) (i, k)
s
s
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s
s
ð40Þ
Kang and Atluri
5
C(s ) (i, k) = c(j ) (i, k)
ð41Þ
s
s
and unknown vectors A and B are
A = f A1
A2
AN gT
ð42Þ
B = f B1
B2
BN gT
ð43Þ
Furthermore, equations (38) and (39) may lead to a
single matrix equation
"
l
0
FJ0
FI0
CJ0
CI0
+ + lM
#
"
A
B
FJM
CJM
"
# A
FJ1
FI1
+l
J
I
B
C1 C1
# ð44Þ
A
0
FIM
=
B
0
CIM
1
which is expressed as a simple form as follows
l0 SM0 C + l1 SM1 C + + lM SMM C = 0 ð45Þ
where C = f A BgT , and SMs for s = 0, 1, . . . , M is
given by
FJs
SMs =
CJs
FIs
CIs
ð46Þ
Finally, equation (45) may be changed into the algebraic eigenvalue problem22 as follows
ð47Þ
SML D = l SMR D
where the system matrices SML and SMR are given
using the diagonal matrix I by
2
0
6 0
6 .
6
SML = 6 ..
6 .
4 ..
SM0
I
0
..
.
..
.
SM1
2
I
60
6
6
SMR = 6 0
6.
4 ..
0
0
I
0
..
.
0
..
.
..
.
0
I
..
.
..
.
SM2
0
0
..
.
..
.
3
0
0
..
.
I
SMM1
..
.
0
0
..
.
I
0
0
SMM
7
7
7
7 ð48Þ
7
5
lCT
l2 CT
lM1 CT
i
ð51Þ
The ith mode shape can be obtained by plotting
equation (5), where the unknown coefficients Ak and Bk
are given by CT (CT = f A Bg ) in equation (51).
Verification examples
We performed numerical tests for a clamped circular
plate having an exact solution and an arbitrarily shaped
plate with clamped edges to examine the validity and
accuracy of the proposed method. The eigenvalues and
mode shapes of the plates are obtained through the
numerical testes. Since the eigenvalues have the same
dimension as the frequency parameter in equation (2),
the natural frequencies of the plates can be calculated
from equation (3).
Clamped circular plate
The proposed method is first applied to a clamped circular plate of unit radius (Figure 2). The boundary of the
plate is discretized with 16 nodes for the proposed
method. The eigenvalues obtained by the proposed
method using M = 20 and M = 25 are presented in
Table 1, which also shows the eigenvalues given by the
exact method,24 NDIF method,9 and FEM (ANSYS). It
can be observed that the eigenvalues obtained by the
proposed method in the case of M = 25 converge rapidly
and accurately to those obtained by the exact method.
Also, the errors with respect to the exact method do not
decrease proportionally as M increases for M\20, and a
meaningful trend of eigenvalue change is not observed.
Note that the errors in Table 1 are calculated by
Proposed method exact method
3 100
Exact method
Furthermore, note that the proposed method using
only 16 nodes yields more accurate eigenvalues than
3
7
7
7
7
7
5
Di = C T
ð49Þ
and the vector D is
D = CT
lCT
l2 CT
lM1 CT
ð50Þ
Note that unlike SM(L) in equation (13), the system
matrices SML and SMR in equation (47) are independent of the frequency parameter. Therefore, eigenvalues
can simply be extracted from equation (47) without the
inefficient procedure required in the original NDIF
method.9 The ith eigenvector Di can be obtained from
equation (47) for mode shapes as follows23
Figure 2. Clamped circular plate discretized with 16 nodes.
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Advances in Mechanical Engineering
Table 1. Eigenvalues of the circular plate obtained by the proposed method, the exact method, the NDIF method, and FEM
(parenthesized values denote errors (%) with respect to the exact method).
No.
1
2
3
4
5
6
Proposed method (16 nodes)
M = 20
M = 25
3.196 (0.00)
4.611 (0.00)
5.906 (0.00)
6.306 (0.00)
7.144 (0.00)
7.820 (0.27)
3.196 (0.00)
4.611 (0.00)
5.906 (0.00)
6.306 (0.00)
7.144 (0.00)
7.799 (0.00)
Exact method24
NDIF method9 (16 nodes)
FEM (1835 nodes)
3.196
4.611
5.906
6.306
7.144
7.799
3.196 (0.00)
4.611 (0.00)
5.906 (0.00)
6.306 (0.00)
7.144 (0.00)
7.799 (0.00)
3.201 (0.16)
4.610 (0.02)
5.906 (0.00)
6.323 (0.27)
7.143 (0.01)
7.811 (0.15)
NDIF: non-dimensional dynamic influence function; FEM: finite element method.
Figure 3. Mode shapes of the circular plate produced by the proposed method using 16 nodes for M = 25: (a) 1st mode, (b) 2nd
mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.
FEM (ANSYS) using 1835 nodes. Although the computational speed (CPU time) of the proposed method is
not presented here, it has improved vastly compared
with that of the NDIF method.
In addition, mode shapes produced by the proposed
method using 16 nodes for M = 25 are presented in
Figure 3, and they agree well with those given by the
exact method.24
The accuracy of an eigenvalue obtained by the proposed method can be verified by plotting its mode
shape. If the plotted mode shape does not exactly satisfy the given boundary condition (the clamped boundary condition), it may be said that the eigenvalue is not
accurate, and increasing the number of nodes or series
functions is required to improve its accuracy.
Arbitrarily shaped plate with clamped edges
An arbitrarily shaped plate whose boundary is composed of p
a ffiffisemicircle
of unit radius and two equilateral
ffi
edges of 2 m in length is shown in Figure 4. The normal directions at the three corner points P1 , P6 , and P16
are approximately determined by the sum of the two
Figure 4. Arbitrarily shaped plate discretized by 20 boundary
nodes (the arrows denote the normal directions of the
corresponding nodes).
normal vectors of the edges adjacent to each corner.
The eigenvalues obtained by the proposed method,
NDIF method,9 and FEM (ANSYS) are summarized in
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Kang and Atluri
7
Table 2. Eigenvalues of the arbitrarily shaped plate obtained by the proposed method, the NDIF method, and FEM (parenthesized
values denote errors (%) with respect to FEM using 961 nodes).
No.
1
2
3
4
5
6
Proposed method (20 nodes)
M = 20
M = 25
3.634 (0.03)
5.121 (0.04)
5.283 (0.08)
6.457 (0.16)
6.844 (0.10)
7.013 (0.21)
3.634 (0.03)
5.121 (0.04)
5.283 (0.08)
6.457 (0.16)
6.844 (0.10)
7.014 (0.20)
NDIF method9 (16 nodes)
3.634 (0.03)
5.121 (0.04)
5.283 (0.08)
6.457 (0.16)
6.844 (0.10)
7.014 (0.20)
FEM
961 nodes
576 nodes
3.633
5.119
5.279
6.447
6.837
7.028
3.631
5.116
5.274
6.439
6.830
7.019
NDIF: non-dimensional dynamic influence function; FEM: finite element method.
Figure 5. Mode shapes of the arbitrarily shaped plate obtained by the proposed method using 20 nodes for M = 25: (a) 1st mode,
(b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.
Figure 6. Mode shapes of the arbitrarily shaped plate obtained by FEM (ANSYS): (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th
mode, (e) 5th mode, and (f) 6th mode.
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Advances in Mechanical Engineering
Table 2. Since the current plate has no exact solution,
errors of the proposed method with respect to FEM
using 961 nodes are calculated. It may be observed that
the proposed method for M = 25 has very small errors
within 0.2% despite using only 20 nodes. Note that the
errors with respect to FEM (961 nodes) do not decrease
proportionally as M increases, and a meaningful trend
of eigenvalue change is not observed in the same manner
as the clamped circular plate.
Figure 5 shows the mode shapes obtained by the
proposed method, which are in good agreement with
those obtained by FEM (ANSYS), which are shown in
Figure 6.
Conclusion
An improved NDIF method is proposed to more efficiently extract eigenvalues and mode shapes of arbitrarily shaped plates with clamped edges. It is revealed
that the proposed method yields highly accurate eigenvalues, which converge to the exact solution. Owing to
its concise formulation, the proposed method gives
eigenvalues that are much more accurate than those
obtained by FEM, which uses a large number of nodes.
The proposed method should be extended to accurately
analyze concave plates because it gives accurate results
for only convex plates. To overcome this limitation, a
sub-domain method of dividing the concave region of
interest into several convex regions will be developed in
future research.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this
article: This research was supported by the Basic Science
Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of
Education (2014R1A1A2057076).
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