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					     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
S
    V

                          Numerical method




            Domain type                      Boundary type




    FDM               FEM                                    BIEM
                                             BEM
                                                         Meshless method

                                                                           3
    M          Motivation
S
     V
   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
S
     V
   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
S
        V
       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
S
    V



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



                                   7
    M
S
            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
    M
S
          Problem Statement
    V




    The eigenproblem of a circular plate with multiple circular holes
                                                                        9
     M       The integral representation for the plate problem
 S
     V


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
S
     V

    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
 S
       V

                       ()
                  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
S
      V

    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
          V
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
             M
                     Null-field integral equations
    S
             V

                                                                                              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)
   S
           V
                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

                         Addition theorem
                                                                             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
     S
             V
     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
       M           Addition Theorem
   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
     M
 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
    V
                      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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