Research Article
Improved non-dimensional dynamic
influence function method based on
tow-domain method for vibration
analysis of membranes
Advances in Mechanical Engineering
1–8
Ó The Author(s) 2015
DOI: 10.1177/1687814015571012
aime.sagepub.com
SW Kang1 and SN Atluri2
Abstract
This article introduces an improved non-dimensional dynamic influence function method using a sub-domain method for
efficiently extracting the eigenvalues and mode shapes of concave membranes with arbitrary shapes. The nondimensional dynamic influence function method (non-dimensional dynamic influence function method), which was developed by the authors in 1999, gives highly accurate eigenvalues for membranes, plates, and acoustic cavities, compared
with the finite element method. However, it needs the inefficient procedure of calculating the singularity of a system
matrix in the frequency range of interest for extracting eigenvalues and mode shapes. To overcome the inefficient procedure, this article proposes a practical approach to make the system matrix equation of the concave membrane of interest into a form of algebraic eigenvalue problem. It is shown by several case studies that the proposed method has a
good convergence characteristics and yields very accurate eigenvalues, compared with an exact method and finite element method (ANSYS).
Keywords
NDIF method, membrane vibration, eigenvalue, two-domain method
Date received: 8 October 2014; accepted: 26 December 2014
Academic Editor: Gongnan Xie
Introduction
The authors developed the non-dimensional dynamic
influence function (NDIF) method for free vibration
analysis of arbitrarily shaped membranes.1,2 Although
the NDIF method has the feature that it yields highly
accurate solutions compared with the finite element
method (FEM)3 and the boundary element method,4,5
the final system matrix equation of the NDIF method
does not have a form of the algebraic eigenvalue problem6 unlike the FEM and the boundary element
method. As a result, the NDIF method needs the inefficient procedure of searching values of the frequency
parameter that make the system matrix singular by
sweeping the frequency parameter in the range of
interest.
The authors introduced an improved NDIF method
to eliminate the above inefficient procedure by changing the final system matrix equation in the NDIF
method into a form of the algebraic eigenvalue problem.7,8 However, the improved NDIF method has the
weak point that it does not give good solutions for
1
Department of Mechanical Systems Engineering, Hansung University,
Seoul, Korea
2
Department of Mechanical and Aerospace Engineering, University of
California, Irvine, CA, USA
Corresponding author:
SW 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 (http://www.uk.sagepub.com/aboutus/
openaccess.htm).
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Advances in Mechanical Engineering
concave membranes with high concavity and multiconnected membranes with holes. This article employs
the sub-domain method of dividing the entire domain
of a membrane into two convex sub-domains.
Although the proposed method is similar to the
authors’ previous method2 in employing the subdomain approach, the two methods differ in whether
the final system matrix equation has a form of the algebraic eigenvalue problem.
A vast literature exists for obtaining analytical solutions of free vibration of membranes and plates having
no exact solution as surveyed in the authors’ previous
articles.1,2,7 However, a survey of the literature performed by the authors reveals that only a very few articles of them dealt with concavely shaped membranes
and plates.
Recently, many researchers have studied new numerical methods for more accurate solutions of concave
membranes and plates than the FEM. For instance,
Wu et al.9 developed the local radial basis function–
based quadrature method for the vibration analysis of
arbitrarily shaped membranes and solved highly
concave-shaped membranes without the use of any
domain decomposition. Shu et al.10 applied the twodimensional least-square-based finite difference method
for solving free vibration problems of arbitrarily shaped
plates with simply supported and clamped edges. It
should be noted that the above methods9,10 still have a
limitation in accuracy owing to a large amount of
numerical calculation. In this article, a simple and practical approach, which is applicable to arbitrary shapes
and offers a highly accurate solution, is proposed by
extending the authors’ previous research.2,7 Note that
the proposed method is occasionally ill-posed when too
many nodes are used. On the other hand, the proposed
method does not consider the problem of spurious
eigenvalues, which have been studied in many articles
related to plate vibrations,11–16 because the spurious
eigenvalues do not appear in the NDIF method for
fixed membranes.
NDIF method reviewed
NDIF
The NDIF primarily satisfies the governing equation of
the eigen-field of interest and physically describes the
displacement response of a point in an infinite domain
due to a unit displacement excited at another point.1,7
In the case of an infinite membrane (see Figure 1), the
NDIF between the excitation point Pk and the response
point P is given by the Bessel function of the first kind
of order zero as follows1
NDIF = J0 ðLjr rk jÞ
ð1Þ
y
infinite membrane
Pi
ri
P
r
P3
P2
P1
rk
x
PN
Pk
PN – 1
Figure 1. Infinite membrane with harmonic excitation points
that are distributed along the fictitious contour (dotted line)
with the same shape as the finite-sized membrane of interest.
in which L represents a frequency parameter; r and rk
denote the position vectors for Pk and P, respectively;
and jr rk j denotes the distance between Pk and P. The
NDIF satisfies the governing equation (Helmholtz
equation)
r2 W (r) + L2 W (r) = 0
ð2Þ
where W (r) is the transverse displacement of the finitesized membrane
pffiffiffiffiffiffiffiffiffi depicted as the dotted line in Figure 1,
L = v= T =r denotes the wavenumber in terms of the
angular frequency v, the uniform tension per unit
length T , and the mass per unit area r. Detailed illustrations on the NDIF are given in the authors’ previous
article.1
NDIF method
For free vibration analysis of an arbitrarily shaped
membrane, of which the boundary is illustrated by the
dotted line in Figure 1, N boundary points are first distributed at points P1 , P2 , . . . , PN along the boundary
depicted in an infinite membrane. Assuming that harmonic displacements of amplitudes A1 , A2 , . . . , AN are,
respectively, generated at points P1 , P2 , . . . , PN , the
total displacement response at the point P may be
obtained by the sum of responses (the linear combination of NDIFs given in equation (1)) that have resulted
from each boundary point, that is
W (r) =
N
X
Ak J0 ðLjr rk jÞ
ð3Þ
k=1
where Ak may be called a contribution coefficient.
Equation (3) is employed as an approximate solution
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Kang and Atluri
3
for the eigen-field of the finite-sized membrane. Note
that the approximate solution also satisfies the governing equation because the NDIF satisfies the governing
equation.
Next, the boundary condition given for the membrane is discretized at boundary points P1 , P2 , . . . , PN
as follows
W (ri ) = Ui ,
i = 1, 2, . . . , N
ð4Þ
P2(1)
P3(1)
P1(1) P1( 2 )
PN(11)+ Nc
where Ui denotes a boundary displacement given in
boundary point Pi . Then, applying the discrete boundary condition (equation (4)) to the approximate solution (equation (3)) gives
W (ri ) =
N
X
Ak J0 ðLjri rk jÞ = Ui ,
i = 1, 2, . . . , N
PN(11+) 2
PN( 22+) 2
PN(11+) 1
PN( 22+) 1
PN(11−) 1
P3( 2 )
PN( 22)+ Nc
Γc
DI
Γ1
P2( 2 )
PN(11) PN( 22)
DII
Γ2
PN( 22−) 1
Figure 2. Concavely shaped membrane subdivided into two
domains, DI and DII .
k=1
ð5Þ
Finally, equation (5) may be written in a simple
matrix form
SM(L)A = U
ð6Þ
where elements of the N 3 N system matrix SM(L) are
given by SMik = J0 (Ljri rk j), the contribution vector
A represents the contribution strength of the NDIFs
defined at each boundary point, and the displacement
vector U represents the discrete boundary condition. In
the case of the fixed boundary condition (U = 0), equation (6) leads to
SM(L)A = 0
ð7Þ
which is the system matrix equation of the membrane.
Since the system matrix SM(L) in equation (7)
depends on the frequency parameter L, the inefficient
procedure of calculating the singularity of the system
matrix in the frequency range of interest is required to
obtain the eigenvalues of the membrane. In the following section, an improved theoretical formulation for
the NDIF method is presented to make the system
matrix independent of the frequency parameter. The
improved formulation is applicable to not only convex
membranes but also concave one unlike the authors’
previous method.7
Improved formulation
Assumed solution and boundary conditions
A concave membrane, illustrated as the solid line in
Figure 2, is divided into the two domains, DI and DII ,
which have the common boundary Gc . The domain DI
is surrounded by the fixed boundary G1 and the common boundary Gc . The domain DII is surrounded by
the fixed boundary G2 and the common boundary Gc .
If the NDIF method is first applied for DI , an approximate solution in DI is obtained by the linear combination of NDIFs defined at boundary points on G1 and
Gc . As a result, the approximate solution may be in the
same manner as equation (3)
WI (r) =
N 1X
+ Nc
k =1
(1) A(1)
J
L
r
r
k 0
k ð8Þ
where N 1 and Nc denote the numbers of boundary
points on G1 and Gc , respectively; A(1)
k denotes contribution coefficient for domain DI ; and r(1)
k denotes a position vector for the kth boundary point for DI . The
fixed boundary condition at boundary points
(1)
(1)
P(1)
1 , P2 , . . . , PN 1 on G1 may be expressed as
WI (r(1)
i ) = 0,
i = 1, 2, . . . , N 1
r(1)
i
ð9Þ
P(1)
i .
where
denotes the position vector for
Applying
the boundary condition, equation (9), to the assumed
solution, equation (8), gives
N 1X
+ Nc
(1)
A(I)
k J0 (LRi, k ) = 0,
i = 1, 2, . . . , N 1
ð10Þ
k=1
(1)
(1) where R(1)
i, k = ri rk denotes the distance between
(1)
boundary points P(1)
i and Pk .
In order to decouple L from the Bessel function in
equation (10), it is expanded in a Taylor series17 having
the number of terms of the series M as follows
M
X
J0 (LR(1)
i, k ) ’
L2j f(1)
j (i, k)
ð11Þ
j=0
where function f(1)
j (i, k) is
f(1)
j (i, k) =
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2j
( 1)j (R(1)
i, k =2)
½G(j + 1)2
ð12Þ
4
Advances in Mechanical Engineering
Substituting equation (11) into equation (10) yields
N1X
+ Nc
A(1)
k
M
X
L2j u(1)
j (i, k) = 0,
i = 1, 2, . . . , N 1
j=0
k=1
ð13Þ
which is rearranged as follows
M
X
L
N 1X
+ Nc
2j
j=0
i = 1, 2, . . . , Nc
(1)
A(1)
k uj (i, k) = 0,
i = 1, 2, . . . , N 1
k=1
ð14Þ
In the same manner as equation (8), an approximate
solution in DII is assumed as
WII (r) =
N2X
+ Nc
A(2)
k J0
k =1
Lr r(2)
k ð15Þ
i = 1, 2, . . . , N 2
ð16Þ
is applied to equation (15), one can obtain
L
(1)
(1) A(1)
J
L
r
r
0
N1 + i
k
k N 2X
+ Nc
(2)
A(2)
k J0 (LRi, k ) = 0,
i = 1, 2, . . . , N 2
N 2X
+ Nc
(2)
A(2)
k uj (i, k) = 0,
Expanding the two Bessel functions in equation (22)
in a Taylor series and rearranging yield
M
X
N 1X
+ Nc
L2j
j=0
L2j
i = 1, 2, . . . , N 2
(2)
A(2)
k uj (N 2 + i, k),
i = 1, 2, . . . , Nc
k=1
slopes at boundary points P(1)
N 1 + 1,
(1)
. . . , PN 1 + Nc are the same as slopes at boundary
(2)
(2)
points P(2)
N 2 + 1 , PN 2 + 2 , . . . , PN 2 + Nc , respectively, one
can obtain, by differentiating equation (22)
Since
P(1)
N 1 + 2,
A(1)
k
k=1
=
N 2X
+ Nc
∂
∂n(1)
N1 + i
A(2)
k
k=1
(1) J0 Lr(1)
r
N1 + i
k ∂
J0
∂n(2)
N2 + i
(2) Lr(2)
N 2 + i rk ,
i = 1, 2, . . . , Nc
ð24Þ
2j
( 1)j (R(2)
i, k =2)
½G(j + 1)2
ð19Þ
Next, continuity conditions in displacement and
slope are considered on the common boundary Gc .
(1)
Displacements at boundary points P(1)
N 1 + 1 , PN 1 + 2 , . . . ,
(1)
PN 1 + Nc on Gc may be obtained by equation (8) as
follows
k=1
N 2X
+ Nc
ð23Þ
where function f(2)
j (i, k) is
N 1X
+ Nc
(1)
A(1)
k uj (N 1 + i, k)
k=1
M
X
N 1X
+ Nc
ð18Þ
WI (r(1)
N1 + i ) =
i = 1, 2, . . . , Nc
ð17Þ
k=1
f(2)
j (i, k) =
(2)
(2) A(2)
J
L
r
r
N2 + i
k 0
k ,
ð22Þ
j=0
(2)
(2) where R(2)
=
r
r
i
i, k
k denotes the distance between
and P(2)
boundary points P(2)
i
k . Expanding the Bessel
function in equation (17) in a Taylor series and rearranging yield
j=0
=
=
k=1
2j
N 1X
+ Nc
k=1
WII (r(2)
i ) = 0,
M
X
(1)
(1)
Since displacements at P(1)
N 1 + 1 , PN 1 + 2 , . . . , PN 1 + Nc
are, respectively, the same as displacements at P(2)
N 2 + 1,
(2)
(2)
PN 2 + 2 , . . . , PN 2 + Nc , one can obtain, by using equations (20) and (21)
k=1
where N 2 denotes the number of boundary points on
G2 in Figure 2, A(2)
k denotes contribution coefficient for
domain DII , and r(2)
k denotes a position vector for the
kth boundary point for DII . If the fixed boundary con(2)
(2)
dition at boundary points P(2)
1 , P2 , . . . , PN 2 on G2
N 2X
+ Nc
(2)
Displacements at boundary points P(2)
N 2 + 1 , PN 2 + 2 ,
(2)
. . . , PN 2 + Nc on Gc may be obtained by equation (15) as
follows
N 2X
+ Nc
(2)
(2)
(2) WII (r(2)
)
=
A
J
L
r
r
N2 + i
N2 + i
k 0
k ,
ð21Þ
k =1
N 1X
+ Nc
(1)
A(1)
k LJ1 (LRN 1 + i, k )
k =1
=
N 2X
+ Nc
∂R(1)
N 1 + i, k
∂nN(1)1 + i
(2)
A(2)
k LJ1 (LRN 2 + i, k )
k =1
(1)
(1) A(1)
k J0 LrN1 + i rk ,
i = 1, 2, . . . , Nc
(2)
where n(1)
N 1 + i and nN 2 + i denote the normal directions
(1)
from points PN 1 + i and P(2)
N 2 + i on Gc , respectively.
Equation (24) is rewritten as
∂R(2)
N 2 + i, k
∂nN(2)2 + i
,
i = 1, 2, . . . , Nc
ð25Þ
ð20Þ
The Bessel functions in equation (25) are expanded
in a Taylor series as follows
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Kang and Atluri
5
J1 (LR(a)
Na + i, k ) ’
1 + 2j
M
X
( 1)j (LR(a)
Na + i, k =2)
j=0
G(j + 1)G(j + 2)
ð26Þ
where a = 1 or 2. Substituting equation (26) into equation (25) leads to
M
X
L2j
j=0
=
N 1X
+ Nc
(1)
A(1)
k cj (N 1 + i, k)
k=1
M
X
L2j
j=0
N2X
+ Nc
(2)
A(2)
k cj (N 2 + i, k),
i = 1, 2, . . . , Nc
k =1
ð27Þ
where
c(1)
j (N1 + i, k)
c(a)
j (Na + i, k) =
and
c(2)
j (N 2 + i, k)
are given by
1 + 2j
( 1)j (R(a)
∂R(a)
Na + i, k =2)
Na + i, k
G(j + 1)G(j + 2)
∂n(a)
Na + i
ð28Þ
System matrix equation and eigenvalues
Equations (14), (18), (23), and (27) may be expressed in
simple matrix equations, respectively
ð35Þ
SML B = lSMR B
where the system matrices SML and SMR are given,
using the diagonal matrix I, by
3
I
0
0
7
0
I
0
7
..
..
..
..
7
.
.
.
.
7
7
..
..
..
5
.
I
.
.
Q1 Q2 QM1
3
2
I 0 0 0
60 I 0 0 7
6
.. 7
.. ..
7
6
SMR = 6 0 0
. 7
.
.
7
6. . .
.. I
4 .. ..
0 5
0 0 0 QM
2
0
6 0
6 .
6
SML = 6 ..
6 .
4 ..
Q0
ð36Þ
ð37Þ
and the vector B is given by
B= A
lA
l2 A
lM1 A
ð38Þ
where A = f A(1) A(2) g.
Finally, multiplying equation (35) by the inverse
matrix of SMR yields
(1)
(1)
2 (1)
M (1)
(F(1)
0 + lF1 + l F2 + + l FM )A = 0 ð29Þ
SM1
R SML B = l B
(2)
(2)
2 (2)
M (2)
(F(2)
0 + lF1 + l F2 + + l FM )A = 0 ð30Þ
which may be expressed as the algebraic eigenvalue
problem
(1)
(1)
(1)
(1)
^ + l2 F
^ + lF
^ + + lM F
^ )A(1)
(F
0
1
2
M
(2)
(2)
(2)
ð31Þ
(2)
^ + lF
^ + l2 F
^ + + lM F
^ )A(2)
= (F
0
1
2
M
(C(1)
0
2 (1)
+ lC(1)
1 + l C2 +
(2)
2 (2)
= (C(2)
0 + lC1 + l C2
(1)
+ l C(1)
M )A
(2)
+ + lM C(2)
M )A
given
by
f(2)
j (N2 + i, k),
j
j
j
j
(1)
(2)
fj (i, k), fj (i, k),
c(1)
and
j (N 1 + i, k),
where the system matrix is given by
j
j
(1)
fj (N 1 + i, k),
c(2)
j (N 2 + i, k),
A(1)
(Q0 + lQ1 + l Q2 + + l QM )
A(2)
M
= 0 ð33Þ
where matrix Qj is
3
0
F(1)
j
(2)
6 0 Fj 7
6
7
Qj = 6 ^ (1) ^ (2) 7
4 Fj Fj 5
(2)
C(1)
j Cj
SM = SM1
R SML
ð32Þ
respectively. Equations (29)–(32) may be merged in a
single matrix equation as follows
2
ð40Þ
M
In equations (29)–(32), the ith row and kth column
^ (1) , F
^ (2) , C(1) , and C(2)
element of matrices F(1) , F(2) , F
are
SM B = l B
ð39Þ
2
ð34Þ
Note that equation (33) is called the higher order
polynomial eigenvalue problem.18 Equation (33) may
be changed into a linear matrix equation as follows
ð41Þ
Note that the system matrix SM in equation (40) is
independent of the frequency parameter l. As a result,
the eigenvalues of the membrane of interest can simply
be extracted from equation (40) without the inefficient
procedure required in the original NDIF method.2
Case studies
The validity of the proposed method is shown through
numerical tests of circular, rectangular, and highly concave membranes. For each case, the eigenvalues
obtained by the proposed method are compared with
those computed by the exact solution and FEM
(ANSYS).
Circular membrane
The proposed method is first applied to a circular
membrane of unit radius where the exact solution19 is
known. As shown in Figure 3, the membrane is intentionally divided into two domains DI and DII for the
proposed method. The boundary of each domain is
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Advances in Mechanical Engineering
Table 1. Eigenvalues of the circular membrane obtained by the proposed method, the exact method, and FEM (parenthesized values
denote errors (%) with respect to the exact method).
No.
1
2
3
4
5
6
Exact method19
Proposed method (24 points)
M = 12
M = 16
M = 20
2.405 (0.00)
3.832 (0.00)
None
None
None
None
2.405 (0.00)
3.832 (0.00)
5.135 (0.02)
5.520 (0.00)
6.390 (0.16)
None
2.405 (0.00)
3.832 (0.00)
5.136 (0.00)
5.520 (0.00)
6.380 (0.00)
7.016 (0.00)
2.405
3.832
5.136
5.520
6.380
7.016
FEM (ANSYS)
3668 nodes
2378 nodes
1024 nodes
2.405 (0.00)
3.833 (0.04)
5.139 (0.05)
5.528 (0.14)
6.386 (0.10)
7.029 (0.19)
2.406 (0.05)
3.835 (0.07)
5.141 (0.10)
5.535 (0.27)
6.390 (0.16)
7.040 (0.35)
2.417 (0.50)
3.851 (0.50)
5.174 (0.74)
5.552 (0.58)
6.461 (1.27)
7.059 (0.61)
FEM: finite element method.
Table 2. Eigenvalues of the rectangular membrane obtained by the proposed method, the exact method, and FEM (parenthesized
values denote errors (%) with respect to the exact method).
No.
1
2
3
4
5
6
Exact method19
Proposed method (32 points)
M = 12
M = 16
M = 20
4.363 (0.00)
6.293 (0.00)
7.458 (0.03)
None
None
None
4.363 (0.00)
6.293 (0.00)
7.456 (0.00)
8.595 (0.00)
8.727 (0.00)
10.51 (0.00)
4.363 (0.00)
6.293 (0.00)
7.456 (0.00)
8.595 (0.00)
8.727 (0.00)
10.51 (0.00)
4.363
6.293
7.456
8.595
8.727
10.51
FEM (ANSYS)
2806 nodes
1813 nodes
1089 nodes
4.364 (0.03)
6.295 (0.03)
7.461 (0.07)
8.602 (0.08)
8.732 (0.06)
10.518 (0.07)
4.364 (0.03)
6.296 (0.06)
7.464 (0.11)
8.606 (0.13)
8.736 (0.10)
10.523 (0.13)
4.365 (0.05)
6.301 (0.13)
7.467 (0.15)
8.621 (0.30)
8.741 (0.16)
10.54 (0.29)
FEM: finite element method.
(1)
2
P
P5(1)
DI
P1(1) P1( 2 )
P12(1)
P12( 2 )
P11(1)
P11( 2 )
P10(1)
P10( 2 )
P8(1) P (1) P ( 2 )
9
9
( 2)
2
P
DII
P5( 2 )
P8( 2 )
Figure 3. Circular membrane divided into two domains using
24 boundary points.
discretized with 12 boundary point and as a result 24
boundary points are used for the membrane.
Eigenvalues obtained by the proposed method using
M = 12, M = 16, and M = 20 are presented in Table 1,
which also shows the eigenvalues given by the exact
method19 and FEM (ANSYS). In Table 1, it may be
said that the eigenvalues by the proposed method in the
case of M = 20 converge accurately to those by the
exact method. Furthermore, it should be noted in
Table 1 that the proposed method using only 24 boundary points yields more accurate eigenvalues than FEM
(ANSYS) using 3668 nodes. In addition, the ith natural
of ffi the membrane can be calculated from
frequencypfiffiffiffiffiffiffiffiffi
Li = 2pfi r=T where Li is the ith eigenvalue in Table
1, r is the mass per unit area, and T is the uniform tension per unit length.
On the other hand, the accuracy of an eigenvalue
obtained by the proposed method can be verified by
plotting its mode shape, which is omitted in the article.
If the plotted mode shape does not satisfy exactly the
given boundary condition (the fixed boundary condition), it may be said that the eigenvalue is not accurate
and larger number of nodes and series functions are
required to improve its accuracy.
Rectangular membrane
As shown in Figure 4, a rectangular membrane with
dimensions 1:2 m 3 0:9 m is divided into two domains
of the boundaries discretized with 16 points, respectively. In Table 2, eigenvalues obtained by the proposed
method for M = 12, M = 16, and M = 20 are compared with eigenvalues obtained by the exact method19
and FEM (ANSYS). In Table 2, it may be observed
that only the first three eigenvalues are obtained for
M = 12, but the first six eigenvalues are successfully
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Kang and Atluri
7
Table 3. Eigenvalues of the highly concave membrane obtained by the proposed method and FEM (parenthesized values denote
differences (%) with respect to FEM using 1701 nodes).
No.
Proposed method (32 points)
1
2
3
4
5
6
FEM (ANSYS)
M = 12
M = 16
M = 20
1701 nodes
976 nodes
451 nodes
5.89 (3.15)
6.41 (0.16)
8.19 (0.24)
None
None
None
5.89 (3.15)
6.41 (0.16)
8.19 (0.24)
8.88 (0.23)
9.95 (0.81)
11.23 (0.71)
5.89 (3.15)
6.41 (0.16)
8.19 (0.24)
8.88 (0.23)
9.95 (0.81)
11.23 (0.71)
5.71
6.42
8.17
8.89
9.87
11.31
5.72
6.43
8.18
8.90
9.89
11.34
5.74
6.44
8.21
8.92
9.94
11.43
FEM: finite element method.
P2(1)
P7(1)
DI
P1(1) P1( 2)
P16(1)
P16( 2)
P15(1)
P15( 2)
(1)
14
( 2)
14
0.2 m
P2( 2)
(1)
6
P6( 2 )
P
P2(1)
D II
P7( 2)
DI
P
P12(1)
P
( 2)
P13(1) P13
P10(1)
P12( 2)
P13(1)
P2( 2 )
P1(1)
P1( 2 )
P16(1)
P16( 2 )
P15(1)
P15( 2 )
P14(1) P14( 2 )
0.45 m
DII
P10( 2 )
P13( 2 )
Figure 4. Rectangular membrane divided two domains using
32 boundary points.
Figure 5. Highly concave membrane divided two domains
using 32 boundary points.
obtained when M increases (for M = 16 and M = 20)
and fully converge to eigenvalues given by the exact
method.
this difference happens will be revealed in a future
study.
Conclusion
Highly concave membrane
Figure 5 shows a 1:2 m 3 0:9 m rectangular membrane
with a V-shaped notch. The membrane is divided into
two domains to apply the proposed method. Each
domain is discretized with 16 boundary points as shown
in Figure 5. Eigenvalues obtained by the proposed
method and FEM (ANSYS) are summarized in Table 3
where it is shown that eigenvalues by the proposed
method fully converge for M = 16. Since the current
membrane has no exact solution, eigenvalues by the
proposed method are compared with those by FEM. It
may be observed in Table 3 that the five eigenvalues
except the first eigenvalue by the proposed method
have very small differences within 0.81% with respect
to FEM using 1701 nodes. Interestingly, it is noticed
that the first eigenvalues by the proposed method and
FEM have relatively large difference. The reason why
A practical, improved NDIF method for the free vibration analysis of concave membranes with arbitrary
shapes was proposed in the article. It was revealed that
the proposed method yields much more accurate eigenvalues, which converge to exact values for membranes
having exact solutions, than FEM using a large number
of nodes. On the other hand, the proposed method cannot be directly applied to a multiply connected membrane because it divides the region of the membrane of
interest into only two regions. In order to overcome
this weak point, an extended way of dividing the region
of the membrane into more than two regions will be
developed in future research.
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
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8
Advances in Mechanical Engineering
Funding
This research was financially supported by Hansung
University.
10.
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