# Outlines by ewghwehws

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• pg 1
```									     M
S
V        BEM/MRM 31
Free vibration analysis of a circular
plate with multiple circular holes by
using addition theorem and direct
BIEM
Wei-Ming Lee1, Jeng-Tzong Chen2
1 Departmentof Mechanical Engineering, China University of Science and Technology, Taipei,
Taiwan
2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung,

Taiwan

Sep. 2-4, 2009 New Forest, UK                                     0
M   Outlines
S
V

1. Motivation
2. Methods of solution
3. Illustrated examples
4. Concluding remarks

1
M   Outlines
S
V

1. Motivation
2. Methods of solution
3. Illustrated examples
4. Concluding remarks

2
M     Overview of numerical methods
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Numerical method

Domain type                      Boundary type

FDM               FEM                                    BIEM
BEM
Meshless method

3
M          Motivation
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   Two questions of BIEM OR BEM
    The improper integral in the boundary integral
equation
   It is difficult to calculate the principal-value of plate problem
    High order derivative when field point and source
point are located on different circular boundaries
   Approaches to these problems
    The degenerate kernel , tensor transformation
Ref: Lee, W. M. & Chen, J. T., Null-field integral equation approach
for free vibration analysis of circular plates with multiple circular
holes. Computational Mechanics, 42, pp.733–747, 2008.

4
M     Motivation
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   The combination of tensor transformation and the
higher order derivative increases the difficulty in
computation and then affect the accuracy of its
solution.

   In addition, the method proposed by Lee & Chen
[10] belongs to point-matching approach and
requires more efforts for computation due to the
collocation of boundary nodes .

5
M        Motivation
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   Is it possible to have a method which needs not…

Tensor transformation

Collocation points

The answer is YES and please to see the next.

6
M   Outlines
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1. Motivation
2. Methods of solution
3. Illustrated examples
4. Concluding remarks

7
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Vibration of plate
V
u(x)
Governing Equation:
 u(x )   u(x ),
4            4
x 
 is  biharmonic operator frequency
 2the h
4      ω is the angle
4                ρ is the volume density
the
u is D lateral displacement thickness
h is the plates

D

the 3
is E hfrequency parameter rigidity
D is the flexural
E is the Young’s modulus
12(1   ) μ is thethin platesratio
is the domain of the Poisson’s

8
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Problem Statement
V

The eigenproblem of a circular plate with multiple circular holes
9
M       The integral representation for the plate problem
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u(x )   U (s,x ) v(s ) dB(s )    (s,x ) m(s ) dB(s )
B                        B

+  M (s,x )  (s ) dB(s)   V (s,x ) u(s) dB(s),   x 
B                         B

10
M     Kernel function
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The kernel function          is the fundamental
solution,

 1                    2           
U (s,x )= 2 Y0 ( r )  iJ 0 ( r )+ K 0 ( r )  ,
8 D                                  
which satisfies
 U (s,x )   U (s,x )= (s  x )
4           4

11
M     The slope, moment and effective shear operators
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()
K 
slope
n

moment                   2                 () 
2
K M   D   ()  (1   ) 2 
                  n 
effective shear
 2                    
KV   D   ()  (1   )   ()   
n
                 t   n  t  

12
M       Kernel functions
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In the polar coordinates of
U (s,x )
 (s,x )=K ,s (U (s,x ))=
R
M (s,x )=K M ,s (U (s,x ))
 2                      2U (s,x ) 
=  D   sU (s,x )+(1   )
                          R 2    
V ( s, x )  KV ,s (U ( s, x ))
                           1              1 U ( s, x )   
 D 
R
2U ( s, x )  (1   )  
s                     R  

 R  R    
              


13
M    Direct boundary integral equations
S
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displacement

with respect to the field point x
slope

with respect to the field point x
normal
moment

with respect to the field point x
effective
shear force

Among four equations, any two equations can be adopted to solve the problem.
14
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Null-field integral equations
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x 
C
By collocating the field point outside the domain, i.e.

0   U (s,x ) v(s ) dB(s )    (s,x ) m(s ) dB(s )+  M (s,x )  (s ) dB(s )  V (s,x ) u(s ) dB(s )
B                           B                          B                             B

0   U (s,x ) v(s ) dB(s )    (s,x ) m(s ) dB(s )+  M  (s,x )  (s ) dB(s )  V (s,x ) u(s ) dB(s )
B                           B                           B                            B

0   U M (s,x ) v(s) dB(s)    M (s,x ) m(s) dB(s)+  M M (s,x )  (s) dB(s)  VM (s,x ) u(s) dB(s)
B                           B                          B                            B

0   UV (s,x ) v(s ) dB(s )   V (s,x ) m(s ) dB(s )+  M V (s,x )  (s ) dB(s )  VV (s,x ) u (s ) dB (s )
B                          B                        B                           B

x B
C
If degenerate kernel functions are used
15
M          Degenerate kernel (Separated kernel)
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1                         2          
U (s, x )=        Y0 ( r )  iJ 0 ( r )+ K0 ( r )                                                   s(R, )

8 2 D 
                                  
r

x( , )
where r =|x - s|
UI       O

UE               x( , )


1                                                      2
U ( s, x ) 
I

8 2   D

m -
{J m ( )[Ym ( R )  iJ m ( R )] 

I m ( ) K m ( R )}eim (  )


1                                                     2
U ( s, x ) 
E

8 2 D m  - 
{J m ( R )[Ym ( )  iJ m ( )] 

I m ( R ) K m ( )}eim (  )

16
M    Complex Fourier series expansions of boundary data
S
V

Displacement

u k ( sk )     
n 
an eink ,
k
sk  Bk ,   k  0,..., H

Bending slope

 k ( sk )      
n 
bn eink ,
k
sk  Bk ,    k  0,..., H

Bending moment

mk ( s )       
n 
cn eink ,
k
sk  Bk ,   k  0,..., H
shear force

v (s) 
k

n 
d n eink ,
k
sk  Bk ,   k  0,..., H
17
M
Analytical eigensolution for a circular plate
S             with multiple circular holes
V
Considering a clamped circular plate with H circular holes
and the null field near the circular boundary B                            0

Bp
B1   B0

BH

0   U E ( s0 , x0 ) v 0 ( s0 )dB0 ( s0 )    E ( s0 , x0 ) m 0 ( s0 )dB0 ( s0 )
B0                                     B0

H                                                                          
    M ( sk , xk )  ( sk )dBk ( sk )   V ( sk , xk ) u ( sk )dBk ( sk ) 
E          k                      E             k

 k 1 Bk
                                        Bk                                 


18
M
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 R0                                                2
0  2  {J m ( R0 )[Ym ( 0 )  iJ m ( 0 )]  I m ( R0 ) K m ( 0 )}d m eim 0           0

4 D m                                             
 R0                                                2
        
4 D m 
{J m ( R0 )[Ym ( 0 )  iJ m ( 0 )]  I m ( R0 ) K m ( 0 )}cm eim


                       0             0

H
  Rk                                          2
    2  {J m ( k )[ m ( Rk )  i m ( Rk )]  I m ( k ) m ( Rk )}bm eimk
Y               J                         K           k

k 1  4 m                                        
 Rk                                          2                                 
+ 2  {J m ( k )[  m ( Rk )  i  m ( Rk )]  I m ( k )  m ( Rk )}ame imk 
Y               J                          K          k

4 m                                                                         


         X m ( ) 
                 m2                 

The moment operator :                   ( )  D (1   )           (1   ) 2    2  X m ( ) 
X

               
m

                                               

 2                       
2 X m ( )              X m ( ) 
The shear operator :              ( )  D  m (1   )  ( ) 
X         
                    2        m 1   
2                   

m

                                                3    
19
S
V                                                                     


im p                                  i ( m n ) pk
x   J m ( p )e                            J mn (  rpk )e                    J n ( k )eink
x p  rpk  xk                                                                n 

k

p

im p                                   i ( m  n ) pk
k
I m (  p )e                           I mn (  rpk )e                      I n (  k )eink
n 
rpk          Ok
p
 pk
Οp                                                     

i ( m  n ) pk
Ymn (  rpk )e                 J n (  k )eink ,            k  rpk
im p     n 
Ym (  p )e                 

 J mn ( rpk )e pk Yn (k )eink ,
i ( m  n )
 k  rpk
 n 
xk   k , k                                              

x p    p , p                                 
      ( 1)n K mn (  rpk )e
i ( m n ) pk
I n (  k )eink ,               k  rpk
im p     n 
rpk   rpk ,  pk 
K m (  p )e            


i ( m n ) pk
( 1)mn I mn (  rpk )e                  K n (  k )eink ,             k  rpk
 n 

20
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S
V

 R0     
22
0  2  {J m ( R00)[Ym (( 00)) iJ mm(( 0)]  II m ( R0 ) K m  0 )}dd m imim0
{J m ( R )[Ym   iJ            0 )]  m ( R0 ) K m ( ( 0 )}m ee 
00       0

4 D m 
m                                                
 R0      
22
        0

4 D m 
{J m ( R )[Ym   iJ  0 )]  m( R0 ) K m ( ( 0 )}cm im
m( R00)[Ym (( 00)) iJ mm(( 0)]  II m ( R0 ) K m  0 )}cm ee im0
{J 

                   00          0

m  
H
H
   Rk                                                 2

    2 2  {[ m ( Rk )[ i m ( Rk )] m ( R n ( rk 0 )ei ((m  n )k k)0 m ( 0k))}bm eim
Rk
                       {J m  k )  m ( Rk )  i  J m k )]  I m   J K  R
Y           Y J              J                                     k               k

k 1  4  m                                                                n
k 1
4 m                                    n  

 Rk                                          
2                       k k 
+ 2  {J m ( k )[  m  Rk )Rk i  m ( m kn)] rk 0 )em( (kk ) nm( R)}ea0e im 
2 K

Y
( m (  )  I R (  I i m  n ) 0 I K (  0 k )}inm bm
J                                                    k

4 m                                    n                                            
 Rk                                          
 2  {[  m ( Rk )  i  m (  Rk )]  J m  n ( rk 0 )ei ( m  n )k 0 J n ( 0 )
Y               J

4 m                                     n 

2                 

        ( Rk )  I m  n ( rk 0 )ei ( m  n ) I n ( 0 )}ein am 
K                                    k0                0   k


m
n                                                      
21
M
The first and second null integral equation
S
V

  k      H     
k
0     e
m 
im0
Am (0 )d m  Bm (0 )cm     Amn (0 )bn   Bmn (0 )an 
0         0    0        0

k 1  n 
k

n 
k


where                                                                                                                         (24)
 R0                                         2
Am (0 ) 
0
{J m ( R0 )[Ym (0 )  iJ m (0 )]  I m ( R0 ) K m (0 )}
4 2 D                                        
 R0                                         2
Bm (0 )  
0
                                       
{J m ( R0 )[Ym (0 )  iJ m (0 )]  I m (  R0 ) K m ( 0 )}
4 D                                          
 Rk i ( n m )                                                             2
Amn (0 ) 
k
e           {J n m ( rk 0 )nJ ( Rk )[Ym ( 0 )  iJ m ( 0 )]  ( 1) n m I n m ( rk 0 ) nI ( Rk ) Km ( 0 )}
k0

4 2                                                                         
 Rk                                                                        2
Bmn (0 )   2 ei ( n  m ) {J n m ( rk 0 )  nJ ( Rk )[Ym ( 0 )  iJ m ( 0 )]  ( 1) n m I n m ( rk 0 )  nI ( Rk ) Km ( 0 )}
k                              k0

4                                                                         


  k            
H
k
0      e
m 
im0
C (0 )d  D (0 )c     Cmn (0 )bn   Dmn (0 )an 
0
m
0
m
0
m
k 1  n 
0
m
k

n 
k


(29)
22
M
S
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Bp
B1   B0

BH

Considering the null field near the
circular boundary B p, P=1,…,H

23
M
S
V

 M            H  
k
e               E (  p )d  F (  p )c +    Emn (  p )bn   Fmn (  p )an 
im p
0                         p
m
p
m        m
p            p
m
k   k        k
(31)
m                                          k  0  n  M        n              
k p
where
 Rp                                               2
Em ( p ) 
p
{J m ( p )[ m ( R p )  i m ( R p )]  I m ( p ) m ( R p )}
Y               J                           K

4 2

 Rp                                             2
Fmp ( p )   2 {J m (  p )[  m ( R p )  i  m ( R p )]  I m ( p ) m ( R p )}
Y              J                          K

4                                               
  Rk i ( n  m )kp                                                                  2
 4 2 D e                {J n  m ( rkp ) J m ( p )[Yn ( Rk )  iJ n ( Rk )]  I n  m ( rkp ) I m (  p ) K n (  Rk )},       k =0
                                                                                     
Emn ( p )  
k

  Rk ei ( n  m )kp {J ( ) J ( R )[Y ( r )  iJ ( r )]  2 ( 1) m I (  ) I (  R ) K (  r )}, k  0,p
 4 2

m       p    n        k   n m     kp       n m    kp

m        p    n      k      n m   kp

      Rk i ( n  m )kp                                                              2
  4 D e                  {J n  m ( rkp ) J m ( p )[Yn( Rk )  iJ n ( Rk )]  I n m (  rkp ) I m (  p ) K n (  Rk )},
                                                           k 0
                                                                                      
Fmn ( p )  
k

   Rk ei ( n  m )kp {J ( )  J ( R )[Y ( r )  iJ ( r )]  2 ( 1) m I (  )  I (  R ) K (  r )}, k  0,p
 4 2

m       p    n        k    n m    kp       n m    kp

m         p    n      k      n m  kp


 M                 H
k
e               G ( p )d  H ( p )c +    Gmn (  p )bn   H mn (  p )an 
im p
0                          p
m
p
m
p
m
k  p
m
k        k
(36)
m                                        k  0  n  M        n               
k p

24
A couple infinite system of simultaneous linear
M
S
algebraic equations
V

(37)

If m=0, ±1, ±2,….±M, a truncated (H+1)(2M+1) system of equations is given.
25
M   Direct-searching scheme
S
V    The eigenvalue can be obtained by applying the SVD
technique to the system of Eq.(37).

3.196      4.487

26
M       The SVD updating technique
S
V
To provide sufficient constrains, UM formulation is considered.

(40)

By combing eqns. (37) and (40),
spurious eigenvalues can be suppressed.

27
M    The SVD updating technique
S
V
A circular plate with an eccentric hole
B1   B0
formulation

formulation

28
M   Outlines
S
V

1. Motivation
2. Methods of solution
3. Illustrated examples
4. Concluding remarks

29
M   A circular clamped plate with three circular
S       free holes
V

30
Natural frequency parameter versus the number of
M
S
terms of Fourier series
V
Fast rate of convergence with few terms of Fourier series

31
The minimum singular value versus the frequency
M
S
parameter by using three different methods
V

32
The former five natural frequency parameters and mode
M   shapes for a circular clamped plate with three circular free
S       holes by using the present method, semi-analytical method
V   and FEM

33
M   Outlines
S
V

1. Motivation
2. Methods of solution
3. Illustrated examples
4. Concluding remarks

34
M
S
Concluding remarks
V
1.   Natural frequencies and natural modes of a circular plate with multiple
circular holes have been solved theoretically.

2.   Based on the addition theorem, two critical problems of improper
integration and the higher derivative in the multiply-connected
domain problem were successively treated in a novel way.
3.   A couple infinite system of simultaneous equations has been
derived as an analytical model for the free vibration of a circular
plate with multiple circular holes .
4.   The SVD updating technique can successfully suppress the
appearance of spurious eigenvalues .
5.   Numerical results show good accuracy and fast rate of
convergence thanks to the analytical method.

35
M
S
V

The End
Thanks for your kind attention

36

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