Theory of Elasticity
Report at the end of term
Student number ：M96520007
Name：YI-JHOU LIN
Life-time Distinguished Professor：Jeng-Tzong Chen
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
Brief introduction:
Inverse Theory ?
Simple example:
P
△
1.Straight Computation Problem
P known
△ unknown
2. Inverse Computation Problem
P unknown
△ known
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
The question description:
1.Polar coordinate system r θ
ω
2.Constant angular velocity ω
3.Concentric circles rim L and L1
with the radii R and R1
4.The tightness function g(θ) is
unknown
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
Boundary conditions:
1. r  R1 ;  r  p1 (  ) ;  r  f 1 (  )
b
b
2. r  R ;    ;  r   ,  u  u0   i  v  v0   g(  )
Symbol:
1.  r ,  , r :stresses
:displacements
2. u ,v
r
3. i2 = -1
4.  *
r
:const should be determined upon solution
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
For solution of boundary value problen
(１)we mentally separate the disk and the shaft. We obtain the
following boundary conditions for the disk:
 r = R1  r  i r  p1    if 1   ,
disk  
 r = R1  r  i r  pc    if c   .
shaft  r = R  rb  i rb  pc    if c   .
 1
 2
pc   : normal ; f c   : tan gential
the normal pc   and fc   tangential contact stresses are unknown and
will be determined upon solution of the problem.
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
Without loss of generality, expanded in the Fourier series


' ik
on L1 ,
 p1    if 1     Ak e

k 



' ik
 p    if   
Ak e
on L , g     AkH e ik on L .

c
c

k 
k 

 3
planar theory of elasticity equations of volumetric forces, Let us represent the stressed
0
1
0
1
0
1
state in the rotating circular disk in the fo  r   r   r ,        , r   r   r
b
b0
b1
b
b0
b1
b
b0
b1
Similarly, the shaft  r   r   r ,        , r   r   r (these stresses
are known [1])
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
According to [2], boundary conditions (1) and (2), taking into account (3), can be
represented in the form
  ' ik
  Ak e
 k 
  z     z   e 2i  z   z    z     
' ik

A
ke
 k

0  z   0  z   e
2i
 z 0  z   0  z   
for r  R1 ,
for r  R ,

 Ae 
k 
  z 
d
k 
k
z
k
,  z  

cz
k 
k
k
, 0  z  

a
k 
5 
for r  R .
' ik
k
The complex potentials   z  ,  z  disk are   z  ,  z  shaft

4
k
z
k
, 0  z  

bz
k 
k
k
.
6 
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
For determination of the unknown coefficients Ak , we use boundary condition for
displacements.
    0  pc    4 Re   z    0  pc  

 2  R k   d k  d k  cos k  i  d k  d k  sin k  ,
k  
where γ is the weight of a unit volume of the disk; g is the acceleration of gravity;

0
3     2  2 1  


R 
pc   
4g



1
A0' R12  A0 R 2

A1 R
R  ; d0 
;
d

,
0
2
2
3 
1

k
2
R

R

0
1
2
  k cos k   k cos k  , d 1 
k  


B1
2 A1 R1

R14  R 4  1  k0  R12  R 2


Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
dk 
 1  k   R12  R 2  B k  B  k  R12 k  2  R 2 k  2 
 1  k  R
2
2
1
R
2
R
2k2
1
R
2k2
 R
2 k  2
1
R
2 k  2

,  k  2,  3, ... ,
B k  A'k R12 k  2  Ak R12 k  2 ,  1  k  d 0  A0  A0*  2GA0H ,
 1  k  d k R k  A k  A* k  2GAHk ,  1  k  d k R  k  Ak  Ak*  2GAkH
,
A0* 
G
G
G
 k0 a0  a0  b2 R 2  , A1* 
 k0 a1 R 1  b2 R 1  , A2* 
 k0 a 2 R 2  b0  ,
G0
G0
G0
Ak* 
G
k 0 a0 R  k
G0
 k   , A* k 
G 
 k  2
k 
b
R

1

k
a
R


k

2
k

G0 
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
optimal design, is provided by the minimization criterion
2
M
      
i
i 1
M
*
  min ,
U      i    * 
2
i 1
U
U
U
0 ,

0
,
0
 *
 kH
 kH
 k  0 ,1,2, ....
Inverse Theory of Elasticity Problem
of Mounting a Disk on a Rotating Shaft
REFERENCES:
[1]. Timoshenko, S. P., Soprotivlenie materialov (Mechanics of
Materials), Moscow: Nauka, 1965.
[2]. Muskhelishvili, N. I., Nekotorye osnovnye zadaci
matematicheskoi teorii uprugosti (Some Basic Problems of
the Mathematical Elasticity Theory), Moscow, Nauka: 1966.
[3]. Mirsalimov, V. M. and Allahyarov, E. A., The Breaking
Crack Build-Up in Perforated Planes by Uniform Ring
Switching, Int. Journ. of Fracture, 1996, vol. 79. no. 1. pp.
17–21.
Thanks
end