theory of elasticity report at the end of term student number ： m96520007 name ： yi-jhou lin...

Theory of ElasticityTheory of ElasticityReport at the end of termReport at the end of term

Student numberStudent number ：：M96520007M96520007

NameName：： YI-JHOU LIN YI-JHOU LIN

Life-time Distinguished ProfessorLife-time Distinguished Professor ：： Jeng-Tzong CheJeng-Tzong Chenn

Inverse Theory of Elasticity Problem Inverse Theory of Elasticity Problem of Mounting a Disk on a Rotating Shaftof 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


The question description:

ω

1.Polar coordinate system r θ

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

Boundary conditions:

1.

2.

Symbol:

1.

2.

3. i2 = -1

4.

r r

1 r 1 r 1

b br 0 0

r R ; p ( ) ; f ( )

r R ; ; , u u i v v g( )


r r, ,

u ,v :stresses

:displacements

* :const should be determined upon solution

For solution of boundary value problen( １ )we mentally separate the disk and the shaft. We obtain the following boundary conditions for the disk:

the normal and tangential contact stresses are unknown and will be determined upon solution of the problem.

1 r r 1 1

1 r r c c

b br r c c

c c

r = R i p if ,disk 1

r = R i p if .

shaft r = R i p if . 2

p : normal ; f : tan gential


cp cf

Without loss of generality, expanded in the Fourier series

planar theory of elasticity equations of volumetric forces, Let us represent the stressed

state in the rotating circular disk in the fo

Similarly, the shaft (these stresses

are known [1])


' ik1 1 k 1

k

' ik H ikc c k k

k k

p if A e on L ,

3

p if A e on L , g A e on L .

0 1 0 1 0 1r r r r r r, ,

b b0 b1 b b0 b1 b b0 b1r r r r r r, ,

According to [2], boundary conditions (1) and (2), taking into account (3), can be

represented in the form

The complex potentials disk are shaft


' ikk 1

k2i

' ikk

k

2i ' ik0 0 0 0 k

k

A e for r R ,

z z e z z z 4

A e for r R ,

z z e z z z A e for r R . 5

z , z z , z

k k k kk k 0 k 0 k

k k k k

z d z , z c z , z a z , z 6b .z

For determination of the unknown coefficients , we use boundary condition for

displacements.

where γ is the weight of a unit volume of the disk; g is the acceleration of gravity;


kA

0 0c c

kk k k k

k

p 4 Re z p

2 R d d cos k i d d sin k ,

2 ' 2 20 2 2 0 1 0 1

1 0 02 201

1 1 1c k k 1 4 4 2 2

k 1 0 1

3 A R A R A R1R R ; d ; d ,

4 g 3 1 k2 R R

B 2 A Rp cos k cos k , d

R R 1 k R R


2 2 2k 2 2k 21 k k 1

k 2 2 2 2k 2 2k 2 2k 2 2k 21 1 1

' 2k 2 2k 2 * H0k k 1 k 1 0 0 0

k * H k * Hk kk k k k k k

* 2 *0 0 0 0 2 1

0

1 k R R R Rd , k 2, 3, ... ,

1 k R R R R R R

A R A R , 1 k d A A 2GA ,

1 k d R A A 2GA , 1 k d R A A 2GA ,

GA k a a b R , A

G

B B

B

1 1 * 20 1 2 2 0 2 0

0 0

k 2* k * kk 0 0 k k 2 k

0 0

G Gk a R b R , A k a R b ,

G G

G GA k a R k , A b R 1 k a R

G G

optimal design, is provided by the minimization criterion


2M

i *i 1

2M

i *i 1

H H* k k

min ,

U

U U U0 , 0 , 0 k 0 ,1,2, ....

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

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theory of elasticity report at the end of term student number ： m96520007 name ： yi-jhou lin...

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