Improved Non-dimensional Dynamic Influence Function Method

Improved Non-dimensional Dynamic Influence Function Method
Based on Tow-domain Method for Vibration Analysis of Membranes
S. W. Kang
Department of Mechanical Systems Engineering, Hansung University, Seoul, 136-792, Korea
([email protected])
S. N. Atluri
Department of Mechanical and Aerospace Engineering, University of California, Irvine,
92697, USA ([email protected])
ABSTRACT
This paper introduces an improved NDIF method using a subdomain method for efficiently
extracting the eigenvalues and mode shapes of concave membranes with arbitrary shapes.
The NDIF 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 (FEM). 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 paper 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 FEM (ANSYS).
1. INTRODUCTION
The authors developed the non-dimensional dynamic influence function method (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 [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 problem [6] unlike the
finite element method and the boundary element method. As the 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 concave membranes with high concavity and
1
multi-connected membranes with holes. This paper 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 author’s previous method [2] in employing the sub-domain
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 papers [1, 2, 7].
However, a survey of the literature performed by the authors reveals that only a very few
papers of them dealt with concavely shaped membranes and plates. For example, Irie et al.
[9] calculated the natural frequencies of concavely shaped hexagonal and octagonal plates
with simply supported edges by using series-type solutions and later extended their work to
L- shaped and I-shaped plates with clamped edges [10]. The authors proposed a sub-domain
method for the free vibration analysis of a concave polygonal plate by superposing plane
waves [11]. Cortinez [12] employed a subdomain method in the same manner as in the
present work but dealt with only a nonhomogeneous rectangular membrane. It should be
noted that the above methods [9-12] have the limitation that it is applicable only to special
shapes.
Recently, many researchers have studied new numerical methods for more accurate
solutions of concave membranes and plates than the finite element method. For instance, Wu
et al. [13] 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. [14] applied the two-dimensional
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
methods [13, 14] still have a limitation in accuracy owing to a large amount of numerical
calculation. In this paper, 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 doesn’t consider the problem of spurious
eigenvalues, which have been studied in many papers related to plate vibrations [15-24],
because the spurious eigenvalues doesn’t appear in the NDIF method for fixed membranes.
The authors also presented a simple scheme free from the spurious eigenvalues for free
2
vibration analysis of plates using the NDIF method [17, 18]. Young introduced the method of
fundamental solutions (MFS) to avoid the spurious eigenvalues of plates by using the
complex-valued kernels including the imaginary-part kernels [21]. Accordingly, this paper
concentrates the practical technique for efficiently extracting eigenvalues by changing the
final system matrix equation in the NDIF method into a form of the algebraic eigenvalue
problem [7, 8]. Note that most of the previous papers [15-24] require the inefficient
procedure for extracting eigenvalues because the determinant search method is used.
2. NON-DIMENSIONAL DYNAMIC INFLUNCE FUNCTION METHOD
REVIEWED
2.1 Non-dimensional Dynamic Influence Function
y
infinite membrane
Pi
P3
P
ri
r
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.
The Non-dimensional Dynamic Influence Function (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
3
point Pk and the response point P is given by the Bessel function of the first kind of order
zero as follows [1].
NDIF = J 0 ( r  rk ) ,
(1)
in which  represents a frequency parameter, r and rk denote the position vectors for
Pk and P , respectively, and r  rk denotes the distance between Pk and P . The NDIF
satisfies the governing equation (Helmholtz equation),
 2W (r )  2W (r )  0 ,
(2)
where W (r ) is the transverse displacement of the finite-sized membrane depicted as the
dotted line in Figure 1,    / T /  denotes the wavenumber in terms of the angular
frequency  , the uniform tension per unit length T , and the mass per unit area  .
Detailed illustrations on the NDIF are given in the authors’ previous paper [1].
2.2 Non-dimensional Dynamic Influence Function Method (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 Eq. (1)) that have resulted from each
boundary point: i.e.,
N
W (r )   Ak J 0 ( r  rk ) ,
(3)
k 1
where Ak may be called a contribution coefficient. Eq. (3) is employed as an approximate
solution 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 )  U i , i  1, 2,  , N ,
(4)
where U i denotes a boundary displacement given in boundary point Pi . Then, applying the
discrete boundary condition (Eq. (4)) to the approximate solution (Eq. (3)) gives
4
N
W (ri )   Ak J 0 ( ri  rk )  U i , i  1, 2,  , N .
(5)
k 1
Finally, Eq. (5) may be written in a simple matrix form:
SM ( ) A  U ,
(6)
where elements of the N  N system matrix SM ( ) are given by SM ik  J 0 ( ri  rk ) ,
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 ), Eq. (6) leads to
SM ( ) A  0 ,
(7)
which is the system matrix equation of the membrane.
Since the system matrix SM ( ) in Eq. (7) depends on the frequency parameter  , 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].
3. IMPROVED FORMULATION
3.1 Assumed solution and boundary conditions
P2(1)
P3(1)
P1(1) P1( 2 )
PN(11) Nc
DI
(1)
N 11
P
P3( 2 )
PN( 22) Nc
c
1
P2( 2 )
PN(11) 2
PN( 22) 2
PN(11) 1
PN( 22) 1
PN(11) PN( 22)
DII
2
PN( 22) 1
Figure 2 Concavely shaped membrane subdivided into two domains, D I and DII .
5
A concave membrane, illustrated as the solid line in Figure 2, is divided into the two
domains, D I and DII , which have the common boundary c . The domain D I is
surrounded by the fixed boundary 1 and the common boundary c . The domain DII is
surrounded by the fixed boundary 2 and the common boundary c . If the NDIF method is
first applied for D I , an approximate solution in D I is obtained by the linear combination
of NDIFs defined at boundary points on 1 and c . As a result, the approximate solution
may be, in the same manner as Eq. (3),
WI (r ) 
N 1 Nc
A
k 1
(1)
k
J 0 ( r  rk(1) ) ,
(8)
where N1 and Nc denote the numbers of boundary points on 1 and c , respectively,
Ak(1) denotes contribution coefficient for domain D I , and rk(1) denotes a position vector for
the k -th boundary point for D I . The fixed boundary condition at boundary points P1(1) ,
P2(1) , , PN(11) on 1 may be expressed as
WI (ri(1) )  0 , i  1, 2,, N1 .
(9)
where ri(1) denotes the position vector for Pi (1) . Applying the boundary condition, Eq. (9), to
the assumed solution, Eq. (8), gives
N 1 Nc
A
k 1
(I )
k
J 0 (Ri(,1k) )  0 , i  1, 2, , N1 ,
(10)
(1)
(1)
where Ri(,1k)  ri(1)  rk(1) denotes the distance between boundary points, Pi and Pk .
In order to decouple  from the Bessel function in Eq. (10), it is expanded in a Taylor
series [25] having the number of terms of the series M as follows.
M
J 0 (R )   2 j  (j1) (i, k ) ,
(1)
i ,k
(11)
j 0
where function  (j1) (i, k ) is
 j(1) (i, k ) 
(1) j ( Ri(,1k) / 2) 2 j
[( j  1)]2
.
(12)
Substituting Eq. (11) into Eq. (10) yields
N 1 Nc

k 1
M
Ak(1)  2 j  (j1) (i, k )  0 , i  1, 2,, N1 ,
j 0
which is rearranged as follows.
6
(13)
N 1 Nc
M
  A
2j
j 0
 j(1) (i, k )  0 , i  1, 2,, N1 .
(1)
k
k 1
(14)
In the same manner as Eq. (8), an approximate solution in DII is assumed as
N 2  Nc
A
WII (r ) 
k 1
( 2)
k
J 0 ( r  rk( 2) ) ,
(15)
where N 2 denotes the number of boundary points on 2 in Figure 2, Ak( 2 ) denotes
contribution coefficient for domain DII , and rk( 2 ) denotes a position vector for the k -th
boundary point for DII . If the fixed boundary condition at boundary points P1( 2) , P2( 2) , ,
PN( 22) on 2 ,
WII (ri( 2 ) )  0 , i  1, 2, , N 2 ,
(16)
is applied to Eq. (15), one can obtain
N 2  Nc
A
( 2)
k
k 1
J 0 (Ri(,2k) )  0 , i  1, 2, , N 2 ,
(17)
( 2)
( 2)
and Pk .
where Ri(,2k)  ri( 2)  rk( 2) denotes the distance between boundary points, Pi
Expanding the Bessel function in Eq. (17) in a Taylor series and rearranging yield
M
 2 j
N 2  Nc
j 0
A
k 1
( 2)
k
 (j 2) (i, k )  0 , i  1, 2,, N 2 ,
(18)
where function  (j 2) (i, k ) is
 (i, k ) 
( 2)
j
(1) j ( Ri(,2k) / 2) 2 j
[( j  1)]2
.
(19)
Next, continuity conditions in displacement and slope are considered on the common
(1)
(1)
boundary c . Displacements at boundary points PN11 , PN12 , , PN(11) Nc on c may be
obtained by Eq. (8) as follows.
WI (r
(1)
N 1 i
)
N 1 Nc
A
k 1
(1)
k
J 0 ( rN(11)i  rk(1) ) , i  1, 2,, Nc .
(20)
( 2)
( 2)
Displacements at boundary points PN 21 , PN 2 2 , , PN(22) Nc on c may be obtained by Eq.
(15) as follows.
WII (rN( 22)i ) 
N 2  Nc
A
k 1
( 2)
k
J 0 ( rN( 22)i  rk( 2) ) , i  1, 2,, Nc .
(21)
(1)
(1)
Since displacements at PN11 , PN12 ,  , PN(11) Nc are, respectively, the same as
7
( 2)
( 2)
displacements at PN 21 , PN 2 2 , , PN(22) Nc , one can obtain, by using Eqs. (20, 21),
N 1 Nc
A
(1)
k
k 1
J 0 ( r
(1)
N 1 i
r
(1)
k
)
N 2  Nc
A
( 2)
k
k 1
J 0 ( rN( 22)i  rk( 2) ) , i  1, 2,, Nc .
(22)
Expanding the two Bessel functions in Eq. (22) in a Taylor series and rearranging yield
M
 2 j
N 1 Nc
M
 Ak(1) (j1) ( N1  i, k )   2 j
j 0
k 1
j 0
N 2  Nc
A
k 1
( 2)
k
 (j 2 ) ( N 2  i, k ) ,
i  1, 2,, Nc .
(23)
(1)
(1)
Since slopes at boundary points PN11 , PN12 , , PN(11) Nc are, respectively, the same as
( 2)
( 2)
slopes at boundary points PN 21 , PN 2 2 , , PN( 22) Nc , one can obtain, by differentiating Eq.
(22),
N 2  Nc


(1)
(1)
J
(

r

r
)

Ak( 2) ( 2) J 0 ( rN( 22)i  rk( 2 ) ) ,

0
N 1 i
k
(1)
n N 1i
n N 2i
k 1
N 1 Nc

k 1
Ak(1)
(24)
i  1, 2,, Nc ,
(1)
( 2)
(1)
( 2)
where nN1i and nN 2i denote the normal directions from points PN1i and PN 2i on c ,
respectively. Eq. (24) is rewritten as
N 1 Nc
A
k 1
(1)
k
 J 1 (R
(1)
N 1 i , k
)
R N(11)i ,k
n N(11) i

N 2  Nc
A
k 1
( 2)
k
 J 1 (R
( 2)
N 2i ,k
)
R N( 22)i ,k
n N( 22) i
,
(25)
i  1, 2,, Nc .
The Bessel functions in Eq. (25) are expanded in a Taylor series as follow.
J 1 ( R
M
(a)
Na  i , k
)
j 0
(a)
1 2 j
(1) j ( R Na
 i , k / 2)
( j  1)( j  2)
,
(26)
where a  1 or 2 . Substituting Eq. (26) into Eq. (25) leads to
N 1 Nc
M
  A
2j
j 0
k 1
M
 (j1) ( N1  i, k )   2 j
(1)
k
N 2  Nc
j 0
A
k 1
 (j 2 ) ( N 2  i, k ) ,
( 2)
k
(27)
i  1, 2,, Nc ,
where  (j1) ( N1  i, k ) and  (j 2 ) ( N 2  i, k ) are given by
 (j a ) ( Na  i, k ) 
(a)
1 2 j
(a)
R Na
(1) j ( R Na
 i , k / 2)
i ,k
( j  1)( j  2)
(a)
n Na
i
.
(28)
3.2 System matrix equation and eigenvalues
Eqs. (14, 18, 23, 27) may be expressed in simple matrix equations, respectively:
(Φ (01)   Φ1(1)  2 Φ (21)    M Φ (M1) ) A (1)  0 ,
8
(29)
(Φ (02 )   Φ1( 2 )  2 Φ (22)    M Φ (M2 ) ) A ( 2)  0 ,
(30)
ˆ (1)   Φ
ˆ (1)  2 Φ
ˆ (1)    M Φ
ˆ (1) ) A (1)
(Φ
0
1
2
M
,
( 2)
( 2)
2 ˆ ( 2)
ˆ
ˆ
ˆ ( 2) ) A ( 2)
 (Φ 0   Φ1   Φ 2     M Φ
M
(31)
(Ψ (01)   Ψ 1(1)  2 Ψ (21)    M Ψ (M1) ) A (1)
.
 (Ψ (02 )   Ψ 1( 2)  2 Ψ (22 )     M Ψ (M2 ) ) A ( 2)
(32)
ˆ (1) ,
In Eqs. (29-32), the i -th row and k -th column element of matrices Φ (j1) , Φ (j2 ) , Φ
j
ˆ ( 2) , Ψ (1) , and Ψ ( 2 ) are given by  (1) (i, k ) ,  ( 2) (i, k ) ,  (1) ( N1  i, k ) ,  ( 2) ( N 2  i, k ) ,
Φ
j
j
j
j
j
j
j
 (j1) ( N1  i, k ) and  (j 2 ) ( N 2  i, k ) , respectively. Eqs. (29-32) may be merged in a single
matrix equation as follows.
 A (1) 
( Q 0   Q1   Q 2     Q M )  ( 2)   0 ,
A 
(33)
Φ (j1) 0 


Φ (j2 ) 
0

Q j  ˆ (1)
ˆ ( 2)  .
Φ j  Φ
j
 (1)
( 2) 
Ψ j  Ψ j 
(34)
M
2
where matrix Q j is
Note that Eq. (33) is called the higher order polynomial eigenvalue problem [26]. Eq. (33)
may be changed into a linear matrix equation as follows.
SML B   SMR B ,
(35)
where the system metrics SML and SMR are given, using the diagonal matrix I , by
 0
 0

SM L   

 
 Q 0
I
0

0
I







 Q1
 Q2 

I

0
0 

  , SM R  0


I 

0
 Q M 1 
0
0 
I 0  0 
0    

  I
0 
0  0 Q M 
0
0

(36, 37)
and the vector B is given by

where A  A (1)
A ( 2 ) .
B  A A 2 A   M 1 A.
(38)
Finally, multiplying Eq. (35) by the inverse matrix of SMR yields
SM R1 SM L B   B ,
which may be expressed as the algebraic eigenvalue problem
9
(39)
SM B   B ,
(40)
SM  SM R1 SM L .
(41)
where the system matrix is given by
Note that the system matrix SM in Eq. (40) is independent of the frequency parameter  .
As the result, the eigenvalues of the membrane of interest can simply be extracted from Eq.
(40) without the inefficient procedure required in the original NDIF method [2].
4. 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).
4.1 Circular membrane
The proposed method is first applied to a circular membrane of unit radius where the exact
solution [27] is known. As shown in Figure 3, the membrane is intentionally divided into two
domains D I and DII for the proposed method. The boundary of each domain is
discretized with 12 boundary point and as the result 24 boundary points are used for the
membrane.
(1)
( 2)
P2(1) P1 P1
P5(1)
DI
P12(1)
P12( 2 )
P11(1)
P11( 2 )
P10(1)
P10( 2 )
P2( 2)
DII
P5( 2 )
( 2)
P8(1) P (1) P ( 2 ) P8
9
9
Figure 3 Circular membrane divided two domains using 24 boundary points.
10
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).
Proposed method (24 points)
FEM (ANSYS)
M  12
M  16
Exact
3668
M  20 method [27]
nodes
2378
nodes
1024
nodes
1
2.405
(0.00)
2.405
(0.00)
2.405
(0.00)
2.405
2.405
(0.00)
2.406
(0.05)
2.417
(0.50)
2
3.832
(0.00)
3.832
(0.00)
3.832
(0.00)
3.832
3.833
(0.04)
3.835
(0.07)
3.851
(0.50)
3
None
5.135
(0.02)
5.136
(0.00)
5.136
5.139
(0.05)
5.141
(0.10)
5.174
(0.74)
4
None
5.520
(0.00)
5.520
(0.00)
5.520
5.528
(0.14)
5.535
(0.27)
5.552
(0.58)
5
None
6.390
(0.16)
6.380
(0.00)
6.380
6.386
(0.10)
6.390
(0.16)
6.461
(1.27)
6
None
None
7.016
(0.00)
7.016
7.029
(0.19)
7.040
(0.35)
7.059
(0.61)
No.
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 method [27] 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 i th natural
frequency f i of the membrane can be calculated from  i  2 f i  / T where  i is the
i th eigenvalue in Table 1,  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 paper. If the plotted mode shape
doesn’t satisfy exactly the given boundary condition (the fixed boundary condition), it may
be said that the eigenvalue isn’t accurate and larger number of nodes and series functions are
required to improve its accuracy.
11
P1(1) P1(2)
P2(1)
P7(1)
DI
P16(1)
P16(2)
P15(1)
P15(2)
P14(1)
P14(2)
P12(1)
( 2)
P13(1) P13
P2(2)
DII
P7(2)
P12(2)
Figure 4 Rectangular membrane divided two domains using 32 boundary points.
4.2 Rectangular membrane
As shown in Figure 4, a rectangular membrane with dimensions 1.2m  0.9m is divided
into two domains of the boundaries are discretized with 16 points, respectively. In Table 2,
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).
Proposed method (32 points)
FEM (ANSYS)
M  12
M  16
Exact
2806
M  20 method [27]
nodes
1813
nodes
1089
nodes
1
4.363
(0.00)
4.363
(0.00)
4.363
(0.00)
4.363
4.364
(0.03)
4.364
(0.03)
4.365
(0.05)
2
6.293
(0.00)
6.293
(0.00)
6.293
(0.00)
6.293
6.295
(0.03)
6.296
(0.06)
6.301
(0.13)
3
7.458
(0.03)
7.456
(0.00)
7.456
(0.00)
7.456
7.461
(0.07)
7.464
(0.11)
7.467
(0.15)
4
None
8.595
(0.00)
8.595
(0.00)
8.595
8.602
(0.08)
8.606
(0.13)
8.621
(0.30)
5
None
8.727
(0.00)
8.727
(0.00)
8.727
8.732
(0.06)
8.736
(0.10)
8.741
(0.16)
6
None
10.51
(0.00)
10.51
(0.00)
10.51
10.518
(0.07)
10.523
(0.13)
10.54
(0.29)
No.
12
eigenvalues obtained by the proposed method for M  12 , M  16 , and M  20 are
compared with eigenvalues obtained by the exact method [27] 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 obtained when M increases (for M  16 and M  20 )
and fully converge to eigenvalues given by the exact method.
0.2 m
P6(1)
P6( 2 )
P2( 2 )
P2(1)
DI
P10(1)
P1(1)
P1( 2 )
P16(1)
P16( 2 )
(1)
15
( 2)
15
P
P13(1)
0.45 m
DII
P10( 2 )
P
P14(1) P14( 2 )
P13( 2 )
Figure 5 Highly concave membrane divided two domains using 32 boundary points.
4.3 Highly concave membrane
Figure 5 shows a 1.2m  0.9m 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 has 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 this difference happens will be revealed in a future study.
13
Table 3 Eigenvalues of the highly concave membrane obtained by the proposed method
and FEM (parenthesized values denote differences (%) with respect to FEM).
Proposed method (32 points)
No.
1701
nodes
5.71
FEM (ANSYS)
976
451
nodes
nodes
5.72
5.74
M  12
M  16
M  20
1
5.89
(3.15)
5.89
(3.15)
5.89
(3.15)
2
6.41
(0.16)
6.41
(0.16)
6.41
(0.16)
6.42
6.43
6.44
3
8.19
(0.24)
8.19
(0.24)
8.19
(0.24)
8.17
8.18
8.21
4
None
8.88
(0.23)
8.88
(0.23)
8.89
8.90
8.92
5
None
9.95
(0.81)
9.95
(0.81)
9.87
9.89
9.94
6
None
11.23
(0.71)
11.23
(0.71)
11.31
11.34
11.43
5. CONCLUSION
A practical, improved NDIF method for the free vibration analysis of concave membranes
with arbitrary shapes was proposed in the paper. 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.
ACKNOWLEDGEMENT
This research was financially supported by Hansung University.
REFERENCES
[1] S. W. Kang, J. M. Lee, and Y. J. Kang 1999 Journal of Sound and Vibration 221, 117-132.
14
Vibration analysis of arbitrarily shaped membranes using non-dimensional dynamic influence
function.
[2] S. W. Kang and J. M. Lee 2000 Journal of Sound and Vibration 234(3), 455-470.
Application of free vibration analysis of membranes using the non-dimensional dynamic
influence function.
[3] T. J. R. Hughes 1987 The Finite Element Method. New Jersey: Prentice-Hall.
[4] D. Nardini and C. A. Brebbia 1982 Applied Mathematical Modeling 7(3), 157~162. A
new approach to free vibration analysis using boundary elements.
[5] C. A. Brebbia, J. C. F. Telles and L. C. Wrobel 1984 Boundary Element Techniques. New
York: Springer-Verlag.
[6] J. H. Wilkinson 1965 The Algebraic Eigenvalue Problem. New York: Oxford University
Press.
[7] S. W. Kang, S. H. Kim and S. Atluri 2012, Journal of Vibration and Acoustics,
Transactions of ASME 134, 1-8(041008). Application of the non-dimensional dynamic
influence function method for free vibration analysis of arbitrarily shaped membranes.
[8] S. W. Kang and S. Atluri 2014, Advances of Mechanical Engineering, 2014, Article ID
363570, 1-9. Application of Non-dimensional Dynamic Influence Function Method for
Eigenmode Analysis of Two-Dimensional Acoustic Cavities.
[9] T. Irie, G. Yamada and M. Tsujino 1981 Journal of the Acoustical Society of America
69(5), 1507-1509. Natural frequencies of concavely shaped polygonal plates with simply
supported edges.
[10] T. Iire, G. Yamada and Y. Narita 1988 Journal of Sound and Vibration 61(4), 571-583.
Free vibration of cross-shaped, I-shaped and L-shaped plates clamped at all edges.
[11] S. W. Kang and S. Atluri 2009 Journal of Sound and Vibration 327, 271-284. Free
vibration analysis of arbitrarily shaped polygonal plates with simply supported edges using a
sub-domain method.
[12] V. H. Cortinez and P. A. A. Laura 1991 Journal of Sound and Vibration 156(2), 217-225.
Vibrations of non-homogeneous rectangular membranes.
[13] W. X. Wu, C. Shu, and C.M. Wang 2007 Journal of Sound and Vibration 306, 252-270.
Vibration analysis of arbitrarily shaped membranes using local radial basis function-based
differential quadrature method.
[14] C. Shu,, W. X. Wu, H. Ding, and C.M. Wang 2007 Computer Methods in Applied
Mechanics and Engineering 196 1330–1343. Free vibration analysis of plates using least15
square-based finite difference method.
[15] J. R. Hutchinson 1988 Vibration of Plates, In Boundary elements X. Springer, Berlin
(see pp. 413-430).
[16] J. R. Hutchinson 1991 Analysis of Plates and Shells by boundary collocation. Springer,
Berlin (see pp. 314-368).
[17] S. W. Kang and J. M. Lee 2001 Journal of Sound and Vibration 242, 9-26. Free vibration
analysis of arbitrarily shaped plates with clamped edges using wave-type functions.
[18] S. W. Kang 2002 Journal of Sound and Vibration 256, 533-549. Free vibration analysis
of arbitrarily shaped plates with a mixed boundary condition using non-dimensional dynamic
influence functions.
[19] J. T. Chen et al. 2004 Computational Mechanics 34, 165-180. Mathematical analysis and
numerical study of true and spurious eigenequations for free vibration of plates using realpart BEM.
[20] C. W. Chen et al. 2005 CMES: Computer Modeling in Engineering and Sciences 8, 2944. Eigenanalysis for membranes with stringers using the methods of fundamental solutions
and domain decomposition.
[21] D. L. Young et al. 2006 CMC: Tech Science Press 4(1), 1-10. The method of
fundamental solutions for eigenfrequencies of plate vibrations.
[22] J. T. Chen, J. W. Lee and S. Y. Leu 2012 Meccanica 47(5), 1103-1117. Analytical and
numerical investigation for true and spurious eigensolutions of an elliptical membrane using
the real-part dual BIEM/BEM.
[23] L. Chen et al. 2013 International Journal of Computational Methods 10(2), 1-16. On the
true and spurious eigenvalues by using the real or the imaginary-part of the method of
fundamental solutions.
[24] J. T. Chen et al. 2003 Journal of Sound and Vibration 262, 370-378. Comments on “Free
vibration analysis of arbitrarily shaped plates with clamped edges using wave-type function”.
[25] M. R. Spiegel 1983 Advanced Mathematics. Singapore: McGraw-Hill.
[26] I. Gohberg, P. Lancaster, and L. Rodman 1982 Matrix Polynomials. New York:
Academic Press.
[27] R. D. Blevins 1979 Formulas for Natural Frequency and Mode shape. Litton
Educational, New York.
16