第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
Null-field integral equation approach for free vibration analysis of
circular plates with multiple circular holes
Wei-Ming Lee1, Jeng-Tzong Chen2
Ya-Kuei Shiu1, Wei-Ting Tao1, Jyun-Chih Kao1
1
2
Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan
E-mail:wmlee@cc.chit.edu.tw jtchen@mail.ntou.edu.tw
Abstract
As quoted by Leissa [1]: "the free vibrations of
circular plates have been of practical and academic
interest for at least a century and a half", most research
work has focused on the free vibration analysis of
circular or annular plate [1-5]. However, only few
studies have conducted the problem of plate with an
eccentric hole [6].
In the past, some analytical solutions [2] for natural
frequencies of the circular or annular plates were solved.
Vera et al. [3] obtained analytical solutions by
implementing the same procedure as [2] in the Maple V
system and pointed out some inaccurate results in [7].
Recently some researchers intended to extend annular
plate [3,5] to the plate with an eccentric hole. Cheng et
al. [8] encountered difficulty and resorted to finite
element method to implement the vibration analysis of
annular-like plates due to the complicated expression
for this kind of plate. Laura et al. [9] determined the
natural frequencies of circular plate with an eccentric
hole by the Rayleigh-Ritz variational method. This
approximate analytic solution can provide the good
results, but some results are not accurate enough by
careful comparison.
The main gain is that the boundary element method
reduces the dimension of the original problem by one,
thus, the number of the introduced unknowns is much
less than that of the traditional domain type methods. In
addition, the domain mesh generation is not required,
which is generally the most difficult and time
consuming task. For the BEM applications to plate
problems, readers may consult with the review article
[10]. It is noted that improper integrals on the boundary
should be handled particularly when the BEM is used.
In the past, many researchers proposed several
regularization techniques to deal with the singularity
and hypersingularity. The determination of the Cauchy
In this paper, a semi-analytical approach for
eigenproblems of circular plate with multiple circular
holes is presented. Natural frequencies and natural
modes are determined by the null-field integral
formulation in conjunction with degenerate kernels,
tensor rotation and Fourier series. All the kernels in the
direct formulation are expanded into degenerate
(separable) form. By uniformly collocating points on
the boundary and taking finite terms of Fourier series, a
linear algebraic system can be constructed. The direct
searching approach is adopted to determine the natural
frequency through singular value decomposition (SVD).
The result of the annular plate, as a special case, is
compared with the analytical solution to verify the
validity of the present method. For the cases of circular
plates with an eccentric hole or multiple circular holes,
eigensolutions obtained by the present method are
compared with those of the existing approximate
analytical method or finite element method (ABAQUS).
Besides, the effect of eccentricity of the hole on the
natural frequencies and modes is also considered.
Moreover, the problem of spurious eigenvalue is
investigated and the SVD updating technique is
adopted to suppress the occurrence of spurious
eigenvalues. Excellent accuracy, fast rate of
convergence and high computational efficiency are the
main features of the present method due to the
semi-analytical procedure.
Keywords: boundary integral equation; null-field
integral equation, degenerate kernel,
vibration; spurious eigenvalue; SVD
updating technique
1. Introduction
1
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
principal value (CPV) and the Hadamard principal
value (HPV) in the singular and hypersingular integrals
are critical issues in BEM/BIEM [11]. For the plate
problem, it is more difficult to calculate the principle
values since the kernels are involved with
transcendental complex functions. In this paper, instead
of using the previous concepts, the kernel function is
recast into the degenerate kernel which is expanded into
a series form on each side (interior and exterior) of the
boundary by employing the addition theorem since the
double layer potential is discontinuous across the
boundary. In reality, addition theorems are expansion
formulae for the special functions (e.g. Bessel function,
spherical harmonics, etc.) in a selected coordinate
system [12]. Based on the direct boundary integral
formulation, Chen et al. [13] recently proposed
null-field integral equations in conjunction with
degenerate kernels and Fourier series to solve boundary
value problems with circular boundaries. Some
applications were done in the static stress calculations
of plate problems. The introduction of degenerate
kernel in companion with Fourier series was proved to
yield the exponential convergence [14] instead of the
linear algebraic convergence in BEM.
This paper presents a semi-analytical approach to
solve the natural frequencies and natural modes of
circular plate with multiple circular holes by using the
null-field integral formulation in conjunction with
degenerate kernels and Fourier series. A linear algebraic
system is constructed by taking finite terms of Fourier
series and uniformly locating the collocation points on
the boundary. By matching the boundary conditions,
the determinant of the matrix is zero to obtain the
nontrivial eigensolution. The direct searching approach
[15] is adopted to determine the natural frequency by
using singular value decomposition (SVD). After
determining the Fourier coefficients, the corresponding
mode shape of the circular plate with multiple circular
holes is obtained by using the boundary integral
equations for the domain point. For the plate problem,
the slope (bending angle) and moment in the normal
and tangential directions for the non-concentric domain
are determined with care under the adaptive observer
system. Therefore, the operator of transformation
matrix for the slope and moment is adopted to deal with
the problem for the non-concentric plate. Finally, the
analysis result of the annular plate, as our special case,
is compared with the analytical solution [3, 7] to verify
the validity of the present method. The results of the
circular plate with eccentric circular hole and multiple
circular holes are compared with those of approximate
analytical solution [9] and FEM using ABAQUS [16] to
demonstrate the validity of the proposed method. The
effect of various eccentricity of the hole on the
vibration characteristics of such plates under several
boundary conditions is also addressed. Moreover, the
problem of spurious eigenvalue is investigated and the
SVD updating technique [17] is employed to suppress
spurious eigenvalues.
2. Problem statement and boundary
integral formulation
2.1 Problem statement of plate eigenproblem
The governing equation for the free flexural
vibration of a uniform thin plate with randomly
distributed circular holes as shown in Figure 1 is
written as follows:
(1)
Ñ 4u( x ) = λ4u( x ),
xÎ Ω
where u is the lateral displacement, λ4 = aω2 ρ0 h/D , 
the frequency parameter, a the radius of the circular
plate,  the circular frequency, ρ0 the volume
density, D the flexural rigidity expressed as
D = Eh 3 /12(1-v 2 ) in terms of the Young’s modulus E ,
the Poisson ratio v and the plate thickness h, and Ω
the domain of the thin plate.
2.2 Boundary integral equation for the collocation
point in the domain
The integral representation for the plate
eigenproblem can be derived from the Rayleigh-Green
identity [15] as follows:
u( x ) = -
ò U ( s, x )v ( s)dB( s ) + ò  ( s, x )m( s )dB( s )
B
-
B
ò M ( s, x ) ( s)dB( s ) + ò V ( s, x )u( s )dB( s )
B
 ( x) = -
ò U ( s, x )v ( s)dB( s) +
(2)
B
ò  ( s, x )m( s)dB( s)
B
B
-
ò M ( s, x ) ( s)dB( s) + ò V ( s, x )u( s)dB( s)
B
(3)
B
m( x )    U m ( s, x )v ( s )dB( s )    m ( s, x )m( s )dB( s )
B
B
  M m ( s, x ) ( s )dB( s )   Vm ( s, x )u( s )dB( s )
B
2
B
(4)
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
v ( x )    U v ( s, x )v ( s )dB( s )    v ( s, x )m( s )dB( s )
B
The expressions for  ( x ), m( x ) and v ( x ) in
Eqs.(3)-(5), obtained by applying the operators in
Eqs.(8)-(10) to u(x) in Eq. (2) with respect to the field
point x(  , ), are
 u( x ) ,
 ( x )  K , x (u( x )) 
(14)

B
  M v ( s, x ) ( s )dB( s )   Vv ( s, x )u( s )dB( s )
B
(5)
B
x 
where B is the boundary of the domain  , u(x),  ( x) ,
m(x) and v ( x ) are the displacement, slope, moment
and shear force. s and x mean the source and field
points, respectively. The kernel functions U ( s, x ) ,
 ( s, x ) , M ( s, x ) , V ( s, x ) , U (s, x) ,  (s, x) ,
M (s, x) , V (s, x) , Um (s, x) , m (s, x) , M m (s, x) , Vm (s, x) ,
Uv (s, x) , v (s, x) , Mv (s, x) and Vv (s, x) in Eqs.(2)-(5)
can be expanded to degenerate kernels by separating
the source and field points and will be elaborated on
later. The kernel function U ( s, x ) in Eq.(2) is the
fundamental solution which satisfies
Ñ 4U ( s, x ) - λ4U ( s, x ) = δ( s - x)
(6)
m( x )  K M , x (u( x ))  2u( x )  (1   )
v ( x)  KV , x (u ( x))

4
ù
1 é
2
êY0 ( λr ) + iJ 0 ( λr ) + ( K0 ( λr ) + iI 0 ( λr )) ú,
ú
8 λ 2 êë
π
û
(7)
B
2
¶ (×)
¶n
KV 
 ()
 n2
B
M ( s, x ) = K M,s (U ( s, x )) = νÑ 2sU ( s, x ) + (1 - ν)
B


(18)
B
B
  M m ( s, x ) ( s )dB( s )   Vm ( s, x )u( s )dB( s )
B
(19)
B
0    U v ( s, x )v ( s )dB( s )    v ( s, x )m( s )dB( s )
B
B
  M v ( s, x ) ( s )dB( s )   Vv ( s, x )u( s )dB( s ),
B
(20)
B
x   C  B,
where C is the complementary domain of  and
appropriate degenerate kernels are selected such that
x   C  B . Since the four equations of Eqs.(17)-(20)
in the plate formulation are provided, there are 6 ( C 24 )
options for choosing any two equations to solve the
problems. For simplicity and treatment of the spurious
eigenvalue, the Eqs.(17)-(19) are used to analyze the
plate problems. In the real implementation, the
(11)

 1      1 U ( s, x )  
 2s U ( s, x )  (1  )  

 
R
 R    R  R   
B
B
(12)
V ( s, x )  KV ,s (U ( s, x ))

(17)
0    U m ( s, x )v ( s )dB( s )    m ( s, x )m( s )dB( s )
(10)
¶ 2U ( s, x )
¶ R2
(16)
B
  M  ( s, x ) ( s )dB( s )   V ( s, x )u( s )dB( s )
(9)
¶ U ( s, x )
¶R
,
0    U ( s, x )v ( s )dB( s )    ( s, x )m( s )dB( s )
to the kernel U s, x  with respect to the source point,
where ¶ / ¶ n and ¶ / ¶ t are the normal and tangential
derivatives, respectively,  2 means the Laplacian
operator. In the polar coordinate of ( R, ), the three
kernel functions can be rewritten as:
Θ ( s, x ) = KΘ ,s (U ( s, x )) =
B
B
2
 2
     
 ()  (1   )   () 
n
 t   n   t 

  M ( s, x ) ( s )dB( s )   V ( s, x )u( s )dB( s )
(8)
K M   2 ()  (1  )

2.3 Null-field integral equations
The null-field integral equations derived by
collocating the field point outside the domain
(including the boundary point if exterior degenerate
kernels are adopted) are shown as follows:
0    U ( s, x )v ( s )dB( s )    ( s, x )m( s )dB( s )
where r  s  x and i   1 . The other three
kernels  ( s, x ) , M ( s, x ) and V ( s, x) in Eq.(2) can
be obtained by applying the following slope, moment
and effective shear operators defined by
KΘ =
 1      1 u ( x)  

 2s u ( x)  (1  )    


          
(15)
By the same way, the kernel functions U (s, x) ,
 (s, x) , M (s, x) , V (s, x) , Um (s, x) , m (s, x) ,
M m (s, x) , Vm (s, x) , Uv (s, x) , v (s, x) , Mv (s, x) and
Vv (s, x) can be obtained by applying the operators in
Eqs.(8)-(10) respectively to the kernel functions U,  ,
M and V with respect to the field point x(  , ).
where  is the biharmonic operator and  ( s - x ) is
the Dirac-delta function, respectively. Considering the
two singular solutions [18], Y0 ( λr ) and K0 ( λr ) ,and
two regular solutions, J 0 ( λr ) and I0 ( λr ) , in the
fundamental solution, we have the complex-valued
kernel,
U ( s, x ) =
 2u( x ) ,
 2
(13)
3
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
M
m( s )  a0   (an cos n  bn sin n ), s  B ,
collocation points in the null-field integral equation can
be exactly located on the real boundary, B, while kernel
functions are expressed in proper degenerate forms.
Novelly, all the improper integrals disappear and
transform to series sum in the BIEs since the potential
across the boundary can be explicitly described in each
side by using degenerate kernel. Successful experiences
on biharmonic problems can be found in [13].
M
v ( s )  p0   ( pn cos n  qn sin n ), s  B ,
1 ¥
å εm{J m ( λρ)[Ym ( λR) + iJ m ( λR)]
8 λ2 m= 0
2
+ I m ( λρ )[ K m ( λR ) + i(- 1)m I m ( λR )]}
π
cos éëm (θ - φ)ùû , ρ < R
U E ( s, x ) =
1 ¥
å εm{J m ( λR)[Ym ( λρ) + iJ m ( λρ)]
8 λ2 m= 0
2
+ I m ( λR )[ K m ( λρ) + i(- 1)m I m ( λρ)]}
π
cos éëm (θ - φ)ùû , ρ ³ R
where a0, an, bn, c0, cn, dn, g0, gn, hn, p0, pn and qn are the
Fourier coefficients and M is the number of Fourier
series terms.
3 Adaptive observer system and
transformation of tensor components
3.1 Adaptive observer system
Consider a plate problem with circular boundaries as
shown in Figure 1. Since the direct boundary integral
equations are frame indifferent (i.e. rule of objectivity),
the origin of the observer system can be adaptively
located on the center of the corresponding boundary
contour under integration. Adaptive observer system is
chosen to fully employ the circular property by
expanding the kernels into degenerate forms. Figure 3
shows the boundary integration for the circular
boundaries in the adaptive observer system. The
dummy variable in the circular contour integration is
the angle (θ) instead of radial coordinate (R). By using
the adaptive system, all the boundary integrals can be
determined analytically free of principal value senses.
(21)
where  m is the Neumann factor (  m =1, m=0 ;
 m =2, m=1,2, L ,∞) and the superscripts "I" and "E"
denote the interior and exterior cases for the degenerate
kernel U ( s, x ) to distinguish r < R and r > R ,
respectively as shown in Figure 2. The degenerate
kernels  ( s, x ) , M ( s, x ) and V ( s, x ) in the
null-field boundary integral equations can be obtained
by applying the operators of Eqs.(11)-(13) to the
degenerate kernel U ( s, x) of Eq.(21).
In order to fully utilize the geometry of circular
boundary, the displacement u(s), slope  ( s ) , moment
m(s) and shear force v ( s ) along the circular
boundaries in the null-field integral equations can be
expanded in terms of Fourier series respectively as
shown below:
3.2 Transformation of tensor components
Since the slope, moment and effective shear force are
calculated in the plate problem, potential gradient or
higher-order gradient need to be manipulated with care.
For the non-concentric case, special treatment for the
potential gradient should be taken care as the source
and field points locate on different circular boundaries.
As shown in Figure 3, the angle i of the collocation
point xi is described in the center of the circle under
integration and the angle c is described in the center
of the circle on which the collocation point is located.
According to the transformation of the component of
the vector Eq.(26) and the tensor Eq.(27)[19],
()n   cos(δ ) sin(δ )   () r 
 ()   -sin(δ ) cos(δ )  ()  ,
 θ
 t 
M
u ( s )  c0   (cn cos n  d n sin n ), s  B ,
n 1
n 1
(26)
.
(22)
é(×) nn ù
ê ú=
êë(×) nt úû
M
 ( s )  g 0   ( g n cos n  hn sin n ), s  B ,
(25)
n 1
2.4 Degenerate kernels and Fourier series for
boundary densities
In the polar coordinate, the field point and source
point can be expressed as (  , ) and ( R, ),
respectively. By employing the separation technique for
the source and field points, the kernel functions
U ( s, x ) are expanded in the series form as follows:
U I ( s, x ) =
(24)
n 1
(23)
é
cos 2 ( )
ê
ê - sin( )cos( )
ë
sin ( )
sin( )cos( )
2
é(×) rr ù
2sin( )cos( ) ùê ú
úê(×) ú
ú
cos 2 ( ) - sin 2 ( )úê
ûê(×) ú
ë r û
(27)
, the operators in Eqs.(14) and (15) can be transformed
as follows:
4
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
KΘR  cos  δ 
 ()
 ()
 sin  δ 
n
t
 2 ()
K MR   v  (1  )sin 2 ( )   2 ()  cos(2 )(1  )
 n2
 sin(2 )(1  )
   () 


n t 
Fourier series are described. By considering the outer
clamped circular boundary and inner free circular
boundary as an example, a linear algebraic system can
be written due to orthogonal property as follows:
(28)
(29)
 U 11

 U11
 21
 U
 21
 U

 L1
 U
 L1
 U
where   c - i . When the angle c equals to the
angle i or the angle difference  equals to zero,
Eqs.(28) and (29) are simplified to the Eqs.(14) and
(15). Considering non-concentric cases, the degenerate
kernels, U (s, x) ,  (s, x) , M (s, x) , V (s, x) Um (s, x) ,
m (s, x) and M m ( s, x ) can be obtained by applying the
operators of Eqs.(28)-(29) to the degenerate kernel
U ( s, x ) , ( s, x ) , M ( s, x ) and V ( s, x ) with respect
to the field point x.
Consider the plate problem with circular domain
containing H randomly distributed circular holes
centered at the position vector oj ( j =1, 2, L ,L), (L= 1+
H and o1 is the position vector of the outer circular
boundary for the plate), as shown in Figure 3 in which
Rj denotes the radius of the jth circular region, x j is
the collocation point and Bj is the boundary of the jth
circular hole. By uniformly collocating the N (=2M+1)
collocation points on each circular boundary in Eqs. (17)
- (19), we have
 M ( s, x ) ( s )  V ( s, x )u( s ) dB j ( s ), x  B,
(30)
j 1 B j
 M  ( s, x ) ( s )  V ( s, x )u( s ) dB j ( s ), x  B.
 M 1L
11
 M12
V12
 M1L

M
22
M 2 L
21 M 22 V 22
M 2 L
 L1 M L 2 V
L2
M LL
L1 M L 2 V L 2
M LL
21
22
V
V 1L   v 1 
   0 
V1L  m 1  0 
   
V 2 L   2  0 
   
V 2 L   u 2   0 
    

V LL   L  0 
   0 
V LL   u L 
(32)
The explicit expressions for the sub-matrices of [ U ij ],
[  ij ], [ M ij ], [ V ij ], [ U ij ], [ ij ], [ Mij ] and [V ij ] can
be obtained through replacing K in Eq.(34) by U , Θ ,
M , V , U , Θ , M  and V 
N
0    U ( s, x )v ( s )   ( s, x )m( s )
V 12
 c0i 
 p0i 
 g0i 
 a0i 








 c1i 
 p1i 
 g1i 
 a1i 
 i 
 i 
 i 
 i 
b 
d 
q 
h 
u i   1   i   1  mi   1  v i   1 
 





 (33)




i
 i 
 i 
i
aM 
 gM 
 cM 
pM 
i
b 
 hi 
d i 
q i 
 M
 M
 M
 M
N
j 1 B j
 M 12
where L denotes the number of circular boundaries
(including inner and outer circular boundaries). For
brevity, a unified form [ U ij ] ( i  1, 2,3, , L and
j  1, 2,3, , L ) denote the response of U ( s, x) kernel
on the ith circle due to the source on the jth circle.
Otherwise, the same definition is for [  ij ], [ M ij ],
[ V ij ], [ U ij ], [ ij ], [ Mij ] and [ V ij ] kernels. The
i
i
i
explicit expressions for sub-vectors [ u ]，[  ],[ m ]
i
and [ v ] can be described as follows:
4 Linear algebraic system
0    U ( s, x )v ( s )   ( s, x )m( s )
 11
(31)
N
0    U m ( s, x )v ( s )   m ( s, x )m( s )
j 1 B j
 M m ( s, x ) ( s )  Vm ( s, x )u( s ) dB j ( s ), x  B.
 K0ijC ( 1 ,1 )
K1ijC ( 1 ,1 )
K1ijS ( 1 ,1 )
 ij
ij
ij
K
(

,

)
K
(

,

)
K
1C
2 2
1S (  2 , 2 )
 0C 2 2
ij
K 


 K0ijC (  N , N ) K1ijC (  N , N ) K1ijS (  N , N )
(32)
The main gain by using degenerate kernel in the BIE
is that singular integrals due to the kernels can be
transformed to series sum free of facing principal
values. The selection of interior or exterior degenerate
kernel depends on   R or   R , respectively,
according to the observer system. Besides, the path is
counterclockwise for the outer circle; otherwise, it is
clockwise. In the B j integration, the origin of the
observer system is adaptively set to collocate at the
center o j from which the degenerate kernels and
ij
K MS
( 1 ,1 ) 

ij
K MS
(  2 , 2 ) 



ij
K MS
(  N , N )  N  N
(34)
where k and k (k=1,2,…,N) shown in figure 3 are
the angle and radius of the kth collocation point on each
boundary in the observer system and the element of the
sub-matrices can be integrated by
ij
KnC
( i ,i )  
2
0
5
K (R, ; i ,i ) cos(n )( Rd )
n=0,1,2, ,M,
(35)
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
ij
KnS
( i ,i )  
2
0
K (R, ; i ,i ) sin(n )( Rd )
n=1,2, ,M,
 U 11
 11
 U
 21
 U
 21
 U
 U 11

 Um11

 U 21
 21
 Um
(36)
in which the selection of interior or exterior degenerate
kernel depends on the position of collocation point with
respective to the center of circle under integration as
presented in figure 3. According to the direct-searching
scheme, the eigenvalues can be obtained by applying
the SVD technique to the matrix in Eq.(32). Once the
eigenvalues are obtained, the associated mode shape
can be obtained by substituting the corresponding
eigenvectors into the boundary integral equations for
the domain point.

 M

21
 M 22
12
21  M22

V 

V 22 

V22  4 N 4 N
12
 v1 
0 
 
,
0 
m 1 
 
= 
 2
 
0 
 2
0 
4 N 1
 u 4 N 1
 11  M 12 V 12 
 m11
21

12
M m
22
M
 m21  M m22

Vm12 
22 


Vm22  4 N 4 N
V
 v1 
0 
 
,
0 
m 1 
 
=
 2
 
 
0 
 2
0 
4 N 1
 u 4 N 1

 M

11
M
22
12
m11  M m12
 21  M 22
m21  M m22
22 
V 
V 12 

Vm12 

V 22 

Vm22 8 N 4 N
 v1 
0 
 
0 
m 1 
 
= 
 2
 
0 
 2
0 
4 N 1
 u 4 N 1
(39)
6.1.1 A circular plate with an eccentricity e/R1=0.2
When the center of circular hole is displaced by 0.2
m from the center of outer circle, the significant change
in natural frequency and mode shape are discussed.
From the minimum singular value of the influence
matrix versus the frequency parameter  , the
multiplicity is one only due to the lack of axial
symmetry. The former eight eigenmodes versus the
number of terms of Fourier series is shown in Figure 5.
The value of frequency parameters is relevant to the
number of the Fourier series. From the convergence
analysis, six terms of the Fourier series are required to
capture the former eight eigenmodes. Figure 6 presents
the former seven eigenvalues and eigenmodes of the
present method and FEM using ABAQUS and the
result also indicates good agreements in this eccentric
case.
(37)
Similarly, Eqs.(30) and (32) yield
 U 11

11
 U m
 21
 U
 21
 U m
21
6.1 A circular plate with one circular hole [12]
A circular plate with one circular hole where the
center is located along a radial axis from 0.0 to 0.5 is
considered to show the effect on spurious eigenvalues
as shown in Figure 4. The outer and inner radii are one
meter (R1 = 1m), and 0.4 meter (R2 = 0.4m),
respectively.
5.1 SVD updating technique
The approach to suppress the appearance of spurious
frequency is the criterion of satisfying all Eqs. (17)-(20)
at the same time. Considering the circular plate with an
eccentric hole subject to the outer clamped boundary
and inner free boundary, Eqs.(30) and (31) reduce to
11

 21  M 22 V 22 
Natural frequencies and modes for circular plates
with circular holes are determined by using the present
method and FEM using ABAQUS for comparison. In
all cases, the inner boundary is subject to the free
boundary condition. The thickness of plate is 0.002m
and the Poisson ratio  =1/3.
For the 2-D multiply-connected problem [20],
spurious eigenvalues occur even though the
complex-valued kernel function is employed to solve
the eigenproblem. This may cause the present method
fail. Therefore, SVD updating technique is adopted to
suppress the appearance of spurious eigenvalue. The
concept of this technique is to provide sufficient
constrains to overcome the rank deficiency of the
system.
 11  M 12 V 12 

11  M12 V12 
6 Numerical results and discussions
5 Spurious eigenvalues in multiplyconnected plate eigenproblem
 U 11

 U11
 21
 U
 21
 U
 11  M 12 V 12 
(38)
To obtain an overdetermined system, we can
combine Eqs. (37) and (38) by using the SVD
technique of updating term as shown below:
6.1.2 The effect of eccentricity
The effect of eccentricity of e/R1 on the frequency
parameter is shown in Figure 7. The repeated
frequencies occurring in the annular case, i.e. mode (m,
6
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
n) with m  0 , are gradually separated into two
distinct values as the eccentricity increases due to initial
axial symmetry broken. By viewing this figure, the
spurious eigenvalue (7.9906) is always there and
independent of the eccentricity.
model are 7570 and 7296. After comparing with the
present method, the high efficiency can also be
observed for this case. Figure 12 shows the former six
natural frequency parameters and modes of FEM using
ABAQUS and the present method. Excellent agreement
between the results of the present method and those of
ABAQUS is observed.
6.1.3 Spurious eigenvalue and its remedy
The cases of eccentricity e/R1=0.2 and three different
boundary conditions ((a) clamped (b) simply supported
(c) free) subject to the outer boundary are considered
here. By using the U formulation, the minimum
singular value of the influence matrix versus the
frequency parameter  is shown in Figure 8. The
U formulation means that both Eqs. (30) and (31)
are used to construct the influence matrix in Eq. (32) in
the boundary integral formulation. Similarly, the UM
formulation is adopted using Eqs. (30) and (32). Figure
10 indicates that the spurious eigenvalue (7.9906) is
independent of the specified boundary condition and its
value happens to be the true eigenvalue of circular
clamped plate with a radius of 0.4 m. When UM
formulation is applied to solve the same problem,
Figure 8 shows similar results as Figure 9 except for the
different spurious eigenvalue (5.5811) which just
equals to the true eigenvalue of circular
simply-supported plate with a radius of 0.4 m. The
numerical result shows that the occurrence of spurious
eigenvalue depends on the size of the circular hole and
the formulation employed. The specified type of
boundary condition and the location of the center of the
hole can not change the spurious eigenvalues. Figure 10
shows that the SVD technique can successfully
suppress the spurious eigenvalue.
7 Concluding remarks
A semi-analytical approach for solving the natural
frequencies and natural modes for the circular plate
with multiple circular holes was proposed. Natural
frequencies and natural modes were determined by
employing the null-field integral formulation in
conjunction with degenerate kernels and Fourier series.
The improper integrals in the null-field integral
formulation were avoided by using the degenerate
kernels and were easily calculated through the series
sum. The potential across the circular boundary was
described explicitly by the interior and exterior
expressions of degenerate kernels, respectively. The
degenerate kernels for the displacement, slope, moment
and effective shear in the plate eigenproblem have been
derived. Furthermore, for the non-concentric case, the
degenerate kernels have also been derived on the
adaptive observer system. Once the Fourier coefficients
of boundary densities have been determined, the
corresponding mode shape can be obtained by using the
boundary integral equations for domain points. The
effect of eccentricity of the hole on the natural
frequencies is addressed. The natural frequencies and
mode shapes for the multiply-connected plate problems
with multiple circular holes have been solved easily and
efficiently by using the present method in comparison
with the available approximate analytic solutions and
FEM results using ABAQUS. Excellent agreement
between the results of the present method and those of
ABAQUS is observed. The occurrence of spurious
eigenvalue depends on the size of the circular hole and
the formulation. The specified type of boundary
condition and the location of the hole influence the true
eigenvalue. Finally, the SVD technique can
successfully suppress the spurious eigenvalue and the
present method can obtain very accurate semi-analytic
solutions. From the numerical results presented in this
paper, the present method provides more accurate
semi-analytical eigensolutions for the circular plate
6.2 A circular plate with three circular holes
In order to demonstrate the generality of the present
method, a circular plate with three holes is considered
as shown in Figure 11. The radii of holes are 0.4m,
0.2m and 0.2m and the coordinates of the center are
(0.5,0), (-0.3,0.4) and (-0.3,-0.4), respectively, in the
coordinate system with an origin at the center of outer
circle. By only using the UM formulation and eight
terms of Fourier series (M=8), the spurious eigenvalue
5.5811 occurs due to 0.4 m radii of inner hole. The
spurious can be filtered out by using SVD updating
technique. The same problem is also solved by using
ABAQUS. The numbers of node and element of FEM
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第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
with an eccentric circular hole or multiple holes so far.
[12] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals,
series, and products, 5th edition, Academic Press,
1996.
[13] J. T. Chen, C. C. Hsiao, S. Y. Leu, Null-field
integral equation approach for plate problems with
circular holes, ASME Journal of Applied
Mechanics, 73 (2006) 679-693.
[14] R.
Kress.,
Linear
integral
equations,
Springer-Verlag, New York, 1989.
[15] M. Kitahara, Boundary Integral Equation Methods
in Eigenvalue Problems of Elastodynamics and
Thin Plates. Elsevier: Amsterdam, 1985.
[16] ABAQUS 6.5 Hibbitt, Karlsson and Sorensen, Inc.,
RI, 2004.
[17] J T Chen, S Y Lin, I L Chen, Y T Lee.
Mathematical analysis and numerical study to
free vibrations of annular plates using BIEM and
BEM. International Journal for Numerical
Methods in Engineering 2006; 65:236-263
[18] J R.Hutchinson, Analysis of plates and shells by
boundary collocation. In Boundary Elements
Analysis of Plates and Shells, Beskos DE (ed.).
Springer: Berlin, (1991) 314–368.
[19] J. N. Reddy, Mechanics of laminated composite
plates and shells: theory and analysis, CRC Press,
2004.
[20] J T Chen, J H Lin, S R Kuo, S W Chyuan,
Boundary element analysis for the Helmholtz
eigenvalue problems with a multiply connected
domain. Proceedings of the Royal Society of
London, Series A 2001; 457: 2521–2546.
References
[1] A.W. Leissa and Y. Narita, Natural frequencies of
simply supported circular plates, Journal of Sound
and Vibration 70 (1980) 221-229.
[2] S. M. Vogel, D.W. Skinner, Natural frequencies of
transversely vibrating uniform annular plates,
ASME Journal of Applied Mechanics 32 (1965)
926-931.
[3] D. A. Vega, S. A. Vera, M. D. Sanchez, P. A. A.
Laura, Transverse vibrations of circular, annular
plates with a free inner boundary, Journal of the
Acoustical Society of America 103 (1998)
1225-1226.
[4] J. T. Chen, S. Y. Lin, I. L. Chen, Y. T. Lee,
Mathematical analysis and numerical study to free
vibrations of annular plates using BIEM and BEM,
International Journal for Numerical Methods in
Engineering 65 ( 2006) 236–263.
[5] W.O. Wong, L.H. Yam, Y.Y.Li , L.Y.Law,
K.T.Chan, Vibration analysis of annular plates
using mode subtraction method. J. Sound Vibration,
2000; 232 (4) : 807–22.
[6] H. B. Khurasia, and S. Rawtani, 1978, Vibration
analysis of circular plates with eccentric hole.
ASME Journal of Applied Mechanics, Vol. 45, pp.
215-217.
[7] A.W. Leissa, Vibration of plates, NASA SP-160,
(1969).
[8] L. Cheng, Y. Y. Li, L. H. Yam, Vibration analysis
of annular-like plates, Journal of Sound and
Vibration 262 (2003) 1153-1170.
[9] P.A.A. Laura, U. Masia, D.R. Avalos, Small
amplitude, transverse vibrations of circular plates
elastically restrained against rotation with an
eccentric circular perforation with a free edge,
Journal of Sound and Vibration 292 (2006)
1004-1010.
[10] C. P. Providatis, D. E. Beskos, Dynamic analysis
of plates by boundary elements, ASME Applied
Mechanics Reviews 52(7) (1999) 213-236.
[11] J. T. Chen, H. K. Hong, Review of dual boundary
element methods with emphasis on hypersingular
integrals and divergent series. ASME Applied
Mechanics Reviews 52(1) (1999) 17-33.
Figure 1. Problem statement for an eigenproblem with
multiple circular holes
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第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
Figure 2. Collocation point and boundary contour
integration in the boundary integral equation in
the adaptive observer system
Figure 4. A circular plate with one circular hole
Figure 3. Transformation of tensor components
Figure 5. Natural frequency parameter versus the number of
terms of Fourier series for clamped-free plate
Figure 6. The former seven eigenvalues and eigenmodes of two methods for clamped-free plate
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第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
Figure 7. Effect of variation of e on the frequency
parameter for clamped-free plate
Figure 8. The minimum singular
value versus the frequency
parameter using U , 
formulation
Figure 11. A circular plate with three circular holes
Figure 9. The minimum singular
value versus the frequency
parameter using U , M
formulation
Figure 10. The minimum singular
value versus the frequency
parameter using the SVD
technique of updating term
Figure 12. The former six eigenvalues and eigenmodes of two methods for a circular plate with three circular holes
10
第十五屆中華民國振動與噪音工程學術研討會 中國文化大學 中華民國九十六年六月十六日
11