Lee_Estimation Magnetic Pressure in Tube Expansion_JMPT_1996

8/11/2019 Lee_Estimation Magnetic Pressure in Tube Expansion_JMPT_1996

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E L S E V I E R Jo u rn a l o f M a t e r i a l s P ro c e ss i n g Te c h n o l o g y 5 7 1 9 96 ) 3 1 1 -3 1 5

JourMl of

M a t e r i a l s

P r o c e s s i n g

T e c h n o l o g y

st imation of the magnetic pressure in tube expansion by

electromagnetic forming

Sung H o Lee a, D on g Ny un g Lee b,

a Agen cy for Defense Development Taejon 305-600 South Korea

b Depar:ment o f Metallurgical Engineering Seoul National University Seoul 157-742 S outh Korea

Receive d 11 Nov em ber 1994

Industrial summary

In the analysis of electromagnetic forming processes it is essential to be able to estimate the magnetic pressure acting on the

workpiece. For this purpose the analysis of the equivalent electric circuit has been used widely. In this approach, it was assumed

that the workpiece and the forming co il are so long that the end ef fect can be neglected , which is far f rom actual pract ice.

Recently, the finite-element analysis of the electromagnetic f ield has been used by the present authors to obtain a more realistic

pressure d is tribu tion . In th is work , the analysis has been ex tended to investigate the ef fects of the geome try of the electromagnetic

forming system and the resistivity of the workpiece on the magnetic pressure in tube expansion.

The magnetic pressure var ies with the geometr ic parameters of the electromagnetic forming system as pred icted by circu it

analysis, but the magnitude of the variation depends on the resistivity of the workpiece. The magnetic pressure decreases as the

tube len gth increases. How ever, there is a limiting leng th above which the pressure no lo nger decreases. This length is almost equa l

to the length of the forming co il .

I I n t r o d u c t i o n

T h e a n a l y s i s o f e l e c t r o m a g n e t i c f o r m i n g p r o c e s s e s

c o n s i s t s o f t h e a n a l y s i s o f t h e e l e c t r i c c i r c u i t a n d o f t h e

d y n a m i c d e f o r m a t i o n o f th e w o r k p i e c e , t h e se a n a l y s e s

b e i n g c o u p l e d t o e a c h o t l he r . T h e p u r p o s e o f t h e e l e c t r ic

c i r c u i t a n a l y s i s i s t o c a l c u l a t e t h e m a g n e t i c p r e s s u r e a n d

t o a p p l y i t t o t h e d e f o r m a t i o n a n a l y s i s . V a r i o u s m e t h -

o d s [ 1 ,2 ] h av e b een u sed in th e e lec t r ic c i r cu i t an a ly s i s .

H o w e v e r , i n t h e a b o v e a p p r o a c h e s , i t w a s a s s u m e d t h a t

t h e t u b u l a r w o r k p i e c e a n d t h e c o i l w e r e s o l o n g t h a t t h e

e n d e f fe c t c o u l d b e n e g l e ct e d . T h u s , t h e d e f o r m a t i o n

b e h a v i o r o f t h e w o r k p i e c e c o u l d b e k n o w n o n l y a t t h e

c e n t e r o f t h e t u b e .

S u z u k i e t a l . [3 ] p e r f o r m e d t h e f i n i t e - e l e m e n t a n a l y s i s

o f tu b e e x p a n s i o n u n d e r a n a s s u s m e d m a g n e t i c p r e ss u r e

d i s t r i b u t i o n , b u t t h e r e e x i s t e d a d i f fe r e n c e b e t w e e n t h e

c a l c u l a t e d a n d e x p e r i m e n t a l r e s u l ts , t h e d i f f e r e n c e a r i s-

i n g m a i n l y f r o m t h e a s s u m p t i o n o f t h e p r e s s u re d i s t ri -

b u t i o n .

R e c e n t l y , t h e p r e s e n t a u t h o r s [ 4 ] o b t a i n e d , a m o r e

r e a l i s t i c p r e s s u r e d i s t r i b u t i o n b y t h e f i n i t e - e l e m e n t a n a l -

y s is o f t h e e l e c t r o m a g n e t i c f i e ld a n d s u c c e e d e d i n p r e -

d i c ti n g t h e d e f o r m a t i o n b e h a v i o r o f t h e t u b e b e i n g

e x p a n d e d .

* Corresponding author.

0924-0136/96/ 15.00 1996 Elsevier Science S.A. All rights reserved

S S D I 0924-0136(95)02086-2

I n t h i s w o r k , t h e e l e c t r o m a g n e t i c f i e l d a n a l y s i s h a s

b e e n e x t e n d e d t o i n v e s t i g a t e t h e e f f e c ts o f t h e g e o m e t r y

o f t h e e l e c t r o m a g n e t i c f o r m i n g s y s t e m a n d t h e r e s i st i v-

i t y o f t h e w o r k p i e c e o n t h e m a g n e t i c p r e s s u r e i n t u b e

e x p a n s i o n .

2 E l e c t r o m a g n e t i c fi e ld a n a l y s i s

I n t h e e l e c t r o m a g n e t i c f ie l d a n a l y s i s, t h e f o l l o w i n g

a s s u m p t i o n s a r e m a d e : ( i ) t h e p r i m a r y c u r r e n t i s r e p r e -

s e n t e d b y a c u r r e n t d e n s i t y d i s t r i b u t i o n i n t h e c r o s s

s e c t i o n o f t h e f o r m i n g c o i l ; ( i i ) t h e p e r m e a b i l i t y a n d

e l e c tr i ca l c o n d u c t i v i t y o f t he m a t e r i a l a r e c o n s t a n t a n d

i s o t r o p i c ; ( i i i ) t h e d i s p l a c e m e n t c u r r e n t d e n s i t y i s n e -

g l e c t e d ; a n d ( i v ) c y l i n d r i c a l c o o r d i n a t e s a r e u s e d a n d

o n l y t h e 0 c o m p o n e n t o f t he c u r r e n t d e n s i t y a n d t h e

v e c t o r p o t e n t i a l e x i s t ,

U n d e r t h e a b o v e a s s u m p t i o n s , M a x w e l l ' s e q u a t i o n s

m a y b e e x p r e s s e d a s :

~ B

V x E = - ~---}- (1 )

V x H = J (2)

v . B = o 3 )


2/5

312 Sung H o Lee , Dong Nytmg Le e/Jo urn al o1 Ma terials Processing Technology 57 (1996) 311 315

wh ere E i s the e lec t r i c f ie ld (V m ~ ), B i s the m agn e t ic

f l u x d e n s i t y ( W b m - 2 ) , H i s t h e m a g n e t i c f i e l d i n t e n s i t y

A m-I),

J i s t h e c u r r e n t d e n s i t y ( A m 2 ) a n d t i s t i m e .

T h e a b o v e f i el d v e c t o r s a r e n o t i n d e p e n d e n t , s i n c e

t h e y a r e r e l a t e d b y t h e f o l l o w i n g e q u a t i o n s :

B = / z H ( 4)

J = a E

(5)

w h e r e /~ is t h e p e r m e a b i l i t y ( H m ) a n d a is t h e

c o n d u c t i v i t y ( U m ~).

I t i s o f t e n c o n v e n i e n t t o c h o o s e a v e c t o r p o t e n t i a l A

a s a s y s t e m v a r i a b l e i n t h e e l e c t r o m a g n e t i c f ie l d a n a l y s i s

s u c h t h a t :

B = V x A (6)

0A

E = (7)

~ t

S u b s t i t u t i o n o f E q s . ( 4 ) a n d ( 5 ) i n t o E q . ( 2 ) w i t h E q s .

(6 ) and (7 ) g ive :

A

V ( v V

A ) = - - a ~ - + J (8 )

w h e r e v i s c a l le d t h e r e l u c t i v i ty a n d r e c i p r o c a l o f / t .

E q . ( 8 ) c a n b e e x p r e s s e d i n a f u n c t i o n w i t h o n e

c o m p o n e n t v e c t o r p o t e n t i a l a s [ 5 ] :

I - I = ; v [ ~ V B ' B + ( a O A T f - - J J ) A ] d V

(9 )

C o n v e n t i o n a l f u n c t i o n a l m i n i m i z a t i o n a n d f i n i t e -

e l e m e n t d i s c r e t i z a t io n o f E q . ( 9) g iv e s t h e m a t r i x e q u a -

t i o n o f v e c t o r p o t e n t i a l a n d i t s t i m e d e r i v a t i v e [ 5 , 6 ] .

S o l v i n g t h i s e q u a t i o n w i t h p r o p e r i n it ia l a n d b o u n d a r y

c o n d i t i o n s , t h e v e c t o r p o t e n t i a l A a t e v e r y n o d a l p o i n t

i s o b t a i n e d a n d t h e m a g n e t i c i n d u c t i o n c a n b e c a l c u -

l a t e d b y E q . ( 6 ) .

T h e m a g n e t i c p r e s s u r e e x e r t e d o n t h e i n s i d e w a l l o f

t h e t u b e i s c a l c u l a t e d f r o m t h e f o l l o w i n g e q u a t i o n [ 7 ] :

1 [B(B n - B 2 n ] 1 0 )

Bo

w h e r e n is t h e o u t w a r d n o r m a l u n i t v e c t o r o n t h e

s u r f a c e s u b j e c t e d t o p r e s s u r e a n d I i s M a x w e l l s s t r e s s

t e n s o r w h i c h i s e q u i v a l e n t t o t h e m a g n e t i c p r e s s u r e .

T h e l a y o u t o f t h e t u b e e x p a n s i o n s y s t e m is il l u s tr a t e d

i n F ig . l , w h i l s t t h e b a s i c d i m e n s i o n s o f t h e e l e c t r o m a g -

n e t ic f o r m i n g s y s t e m a r e g i v e n i n T a b l e 1 . A q u a r t e r o f

t h i s s y s t e m w a s c o n s i d e r e d i n t h e a n a l y s i s b e c a u s e o f i t s

s y m m e t r y . T h e m o d e l f o r t h e e l e c t r o m a g n e t i c f i el d an a l -

y s i s i s s h o w n i n F i g . 2 , a l o n g w i t h t h e b o u n d a r y

c o n d i t i o n s . A l t h o u g h t h e b o u n d a r y i s o p e n s p a c e , t h e

r e g i o n o f a n a l y s is i s l i m i te d b y t h e c o m p u t a t i o n c a p a -

b i l i ty a n d e f f ic i e n c y . I n t h e s e c a l c u l a t i o n s , t h e d i m e n -

s i o n s , th e d i m e n s i o n o f th e e n t i r e r e g i o n o f a n a l y s i s w a s

a b o u t f iv e t i m e s t h a t o f t h e c o il a n d t h e w o r k p i e c e .

F i n e r e l e m e n t s w e r e u s e d t o d e s c r i b e t h e r e g i o n o f t h e

WORKPIECE

Fig. 1. The forming coil and the tu bular w orkpiece n electromagnetic

forming.

c o i l a n d t h e w o r k p i e c e , a s s h o w n i n F ig . 3 . A x i s y m m e t -

r i c q u a d r i l a t e r a l e l e m e n t s w i t h f o u r n o d e s w e r e u s e d .

T h e m a t e r i a l p r o p e r t i e s u s e d i n t h e a n a l y s i s a r e g i v en

i n T a b l e 2 . T h e p e r m e a b i l i t y o f t h e c o il a n d t h e t u b e

m a t e r i a l w a s a s s u m e d t o b e e q u a l t o t h a t o f f r e e sp a c e .

T h e e f f e c t s o f t h e g e o m e t r y o f t h e e l e c t r o m a g n e t i c

f o r m i n g s y s t e m a n d t h e r e s is t iv i t y o f t h e w o r k p i e c e o n

t h e m a g n e t i c p r e s s u r e w e r e i n v e s t i g a t e d b y v a r y i n g t h e

a b o v e p a r a m e t e r s , T a b l e s 1 a n d 2 .

T h e p r i m a r y c u r r e n t f l o w i n g i n t h e f o r m i n g c o i l w a s

c a l c u l a t e d u s i n g t h e f o l l o w i n g e q u a t i o n [ 3 ] :

i = - 1 0.5 e x p ( - 1 4 3 0 t ) + 1 .0 6 x 1 04 e x p ( - 3 2 4 0 t )

x s in(17 800 t + 9.90 x 10 4) (11)

3 . R e s u l t s a n d d i s c u s s i o n

3 .1. E f f e c t o f the f or m ing co i l and t he t ube geom e t ry

on t he magne t i c pre ssure

T h e m a g n e t i c p r e s s u r e , a s s u m i n g a u n i f o r m m a g n e t i c

f lu x d e n s it y d i s t r ib u t i o n a n d n o l e a k a g e o f th e m a g n e t i c

f i e l d i n s i d e t h e t u b e h a v e b e e n e x p r e s s e d a s [ 2 ] :

(12)

P = 2 - k P /

k a y

w h e r e i i s t h e p r i m a r y c u r r e n t , p i s t h e p i t c h o f t h e c o i l ,

b i s t h e e f f e c t iv e c o i l r a d i u s a n d a i s t h e e f f e c t i v e t u b e

r a d i u s .

Table 1

The b asic dimensions of the forming coil and the workpiece

Coil (mm) 76

Coil radius (effective) mm ) 42

Coil pitch (mm) 4

Wo rkpiece length (ram) 76

Wo rkpiece radius (inner) (mm) 44

Gap between coil 1

workpiece (mm)


3/5

Sung Ho Lee, Dong Nyung Lee/Journal of Materials Processing Technology 57 1996)311-315 313

A : : 0

A = 0

C 0 1 ;

IWORKFECE

zLll

-0

A=0

F i g . 2 . M o d e l f o r t h e e l e c t r o m a g n e t i c f ie l d a n a ly s i s .

T h e e f f ec t s o f t h e c o i l p i tc h , t h e t u b e a n d t h e c o i l

r a d i i o n t h e m a g n e t i c p r e s s u r e , c a l c u l a t e d f o r a t u b e

e q u a l t o t h e c o i l i n l e n g t h , a r e s h o w n i n F i g s . 4 - 6 ,

re spec t ive ly . In these f i~J res the do t t ed curves a re re -

l a t e d t o t h e m a g n e t i c p r e s s u r e c a l c u l a t e d f r o m E q . ( 1 2 ).

T h e m a g n e t i c p r e s s u r e s i n t h e f i g u r e s a r e t h e m a x i m u m

v a l u e s a t th e l o n g i t u d i n a l c e n t e r o f t h e t u b e . T h e m a g -

n e t i c p r e s s u r e s o b t a i n e d f r o m t h e p r e s e n t c a l c u l a t i o n s

s h o w r e a c t i o n s t o g e o m e t r i c p a r a m e t e r s o f t h e c o i l a n d

t h e t u b e s i m i l a r t o t h o s e o b t a i n e d f r o m t h e u s e o f

Eq. (12).

3.2. Effect of the resistivity of the workpiece of the

magnetic pressure

A s s h o w n i n F i g s . 4 - 6 , t h e m a g n i t u d e o f t h e m a g -

n e t i c p r e s s u r e v a r i e s w i t h r e s i s t i v i t y o f t h e w o r k p i e c e ,

t h u s i t c a n b e s u g g e s t e d t h a t E q . ( 1 2 ) b e r e w r i t t e n a s :

,_ _ .o iy 7

7 t~ \a

= - - K ( 1 3 )

m

w i t h

i W b V

K = . o t t z

50 mm

T a b l e 2

T h e m a t e r i a l p r o p e r t ie s o f t h e c o i l a n d t h e w o r k p i e c e

E l e c t r i c P e r m e a b i l i t y

c o n d u c t i v i ty 4 m - t )

( 0 m - )

C o i l O F H C ) c o p p e r 6 .8 5 10 7 1 .2 6 x 1 0 -

W or kp i e c e A 1 1050) 3 . 56 x l 07 1 . 26 x 10 -

w h e r e m i s a c o n s t a n t w h i c h v a r i e s w i t h t h e r e s i s t i v i t y

o f t h e w o r k p i e c e .

T h e r e l a t i o n s h i p b e t w e e n t h e m a g n e t i c p r e s s u r e a n d

K i s shown in F ig . 7 , da t a po in t s in the f igure be ing

f r o m F i g s . 4 t o 6 . T h e m a g n e t i c p r e s s u r e v a r i e s l i n e a r l y

w i t h K , w h i c h m e a n s t h a t t h e v a l u e o f m i s c o n s t a n t , a s

s u g g e s t e d . T h e c h a n g e i n m w i t h r e s i s t i v i t y i s s h o w n i n

F i g . 8 , t h e v a l u e o f m b e i n g a l i n e a r f u n c t i o n o f t h e

res i s t iv i ty of the workpiece .

3.3. Effect of the workpieee length on the magnetic

pressure

C h a n g e o f t h e m a g n e t i c p r e s s u r e w i t h w o r k p i e c e

length i s i l l us t ra t ed in F ig . 9 . The magne t i c p res sures in

t h e f i g u r e a r e t h e m a x i m u m v a l u e a t t h e l o n g i t u d i n a l

c e n t e r o f t h e t u b e . A s t h e l e n g t h o f t h e t u b e i n c r e a s e s,

t h e m a g n e t i c p r e s s u r e d e c r e a s e s , b u t a b o v e a p a r t i c u l a r

l i m i t o f th e l e n g t h i t r e m a i n s c o n s t a n t , t h i s l im i t i n g

l e n g t h o f t h e tu b e b e i n g a l m o s t e q u a l t o t h e l e n g t h o f

t h e f o r m i n g c o i l .

S i n c e t h e m a g n e t i c p r e s s u re , P , i s p r o p o r t i o n a l t o t h e

s q u a r e o f t h e p r i m a r y c u r r e n t, i l, a n d t h e p r i m a r y

c u r r e n t i s a l s o p r o p o r t i o n a l t o i2 /L w h e r e i2 is the

s e c o n d a r y c u r r e n t a n d L i s t h e l e n g t h o f t h e t u b e [ 2 ] ,

t h e m a g n e t i c p r e s s u r e c a n b e w r i t t e n a s f o l l o w s :

P = C , i 2 / L ) 2

14)

3 . 5

3 . 0

O

n 2 . 5

~E

L,.~2 . 0

n -

~ 1 . 5

b J

n -

~ 1 . 0

0 . 5

[ ] E ] B ~ E ]

2 . 8 m i c r o h r n c m

~\ CX:~CO 5

O , ~ . , ~ A 1 5

\

\

\

\

\

\

[]

\

O

8

I I

0 . 3 . '0 2 0 6 . ; , . 0 , . ' 0 9 . 0 , o . o

C O IL P I T C H , r n m

F i g . 4 . M a g n e t i c p r e s s u r e a s a f u n c t i o n o f t h e c o il p i t c h c o i l l e n g t h

F i g . 3 . F i n i t e - e l e m e n t m e s h f b r t h e e l e c t r o m a g n e t i c f i el d a n a l y s i s. 7 6 m m , c o i l r a d i u s 4 2 m m , tu b e l e n g t h 7 6 m m , tu b e r a d i u s 4 4 m m ) .


4/5

3 1 4 Sung Ho Lee , Dong Nyung L ee /J ourn a l o f Mater ials Process ing Technology 57 1996) 311- 315

3 .5

3 .0

O

n 2 . 5

L S 2 . 0

n

~ 1 . 5

Wad

~- 1 . 0

0 . 5

[]

o []

o []

0 D

Eqn.(12)

D D D D D

2 .8 m ic rohm-cm

OOOOO5

AAAAA

15

i i i

02 .0 43 0 44 .0 45 .0 4 6 .0 47 0 48 0 49 .0

WORK P I E C E RADIUS mm

F i g . 5 . M a g n e t i c p r e s s u r e a s a f u n c t i o n o f t h e t u b e r a d i u s c o i l l e n g t h

7 6 m m , c o i l r a d i u s 4 2 m m , c o i l p i c h 4 m m , t u b e l e n g t h 7 6 m m ) .

wh e re C 1 i s a cons tan t .

F ro m t h e a n a l o g y o f th e p r i n c ip l e s o f t h e t ra n s fo rm e r

a n d t h e e l e c t ro m a g n e t i c fo rm i n g s y s t e m , i t c a n b e d e -

d u c e d t h a t t h e n u m b e r o f t u rn s o f t h e s e c o n d a ry c o i l i n

t h e t r a n s fo rm e r c o r r e s p o n d s t o t h e l e n g t h o f t h e t u b e in

the e lec t romagnet ic fo rming sys tem, wh ich resu l t s in the

fo l lowing equat ion :

i l N I = i 2 N 2 o c i 2 L

15)

wh e re N1 a n d N2 a r e t h e n u m b e r o f t u rn s o f t h e

p r i m a ry a n d t h e s e c o n d a ry c o i l i n t h e t r a n s fo rm e r ,

respect ively .

Co mb in ing Eqs . 14 ) and 15) in to one , bear ing in

m i n d t h a t i~N i s cons tan t :

= C 2 L -4 ,

16)

whe re C2 i s a cons tan t .

F ig . l0 shows tha t the above sugges t ion i s va l id ,

p rov ided th a t l eaka ge o f the magnet ic f i eld in s ide the

tube i s neg ligible . Fr om the ex i s tence o f the l imi t ing

l e n g th o f th e t u b e , a b o v e wh i c h t h e p r e s s u re n o l o n g e r

4 . 0

DDDDD

2.8 microhrn-crn

/

OOOOO5

13~

3 . 0 A A A A A

1 5

LJ

~ 2 . 0

w

Q _

1 . 0

0.0 ' ' ' '

0 . 0 1 . 0 2 . 0 3 .0 4 . 0 5 . 0 6 . 0

K MPa

F i g . 7 . R e l a t i o n s h i p b e t w e e n t h e m a g n e t i c p r e s s u r e a n d t h e g e o m e t r i c

f a c t o r , K i n E q . 13 ).

decreases , i t can be in fer red tha t the l eng th over wh ich

t h e s e c o n d a ry c u r r e n t i s i n d u c e d i n t h e t u b e d o e s n o t

i n c rea s e b e y o n d t h e l e n g th o f t h e fo rm i n g c o i l. T h i s

si tuat io n is sho wn in Fig . 11 in which L+ is the e ffect ive

l e n g t h wh e re t h e s e c o n d ry c u r r e n t f l o ws .

4 Conclus ions

T h e e f f e c t o f t h e g e o m e t ry o f t h e e l e c t ro m a g n e t i c

fo rm i n g s y s t e m a n d t h e r e s i st i vi t y o f th e w o rk p i e c e o n

t h e m a g n e t i c p r e s s u re h a v e b e e n i n v e s t i g a t e d i n t u b e

expans ion . The magnet ic p ressu re var ies wi th the geo-

m e t r i c p a ra m e t e r s o f th e e l e c t ro m a g n e t i c fo rm i n g s y s -

tem as p red ic ted by c i rcu i t ana lys i s , bu t i t s magn i tude

d e p e n d s o n t h e r e s is t iv i ty o f t h e wo rk p i e c e . T h e m a g -

ne t ic p ressu re decreases as the tube leng th increases .

However , there i s a l imi t ing leng th , above which the

pressu re no longer decreases . Th is l eng th i s a lmos t

equal to the l eng th o f the fo rm ing co i l .

4 . 0

3 . 5

0 3 . 0

13

. 2 . 5

l ad

~ 2 . 0

o3

o3

,.~ 1.5

1 . 0

[] []

E3

zx / x A

Eqn.(12)

O O O O O

2.8 rn icrohm-cm

0 .5 O O O O 0 5

~ 15

0 0 0 . 0 3 5 .0 4 0 . 0 4 5 . 0 5 0 ~ . 0 5 5 . 0

C O I L R A D IU S , r n m

F i g . 6 . M a g n e t i c p r e s s u r e a s a f u n c t i o n o f t h e co i l r a d i u s c o i l l e n g t h

7 6 m m , c o i l ra d i u s 4 m m , g a p c o i l a n d t u b e 1 m m ) .

3 .2

2 .8

2 .4

E 2 . 0

1 . 6

1.2

0 .8

0 .0

[]

5.0 10.0 15.0 20.0 25.0

R E S I S T I V I T Y , m i c r o h m - c m

F i g . 8 . C h a n g e i n m i n E q . 1 3 ) w i t h t h e r e s i s ti v i t y o f t h e w o r k p i e c e .


5/5

S u n g H o

Lee,

Dong Nyung Lee/J ou rn al o f Materials Processing Technology 57 1996) 31 1-3 15

315

7,0

6 0

0

O

~E 5 . 0

uS

~4.0

w

n, 3.0

O _

2 0

O

I3

[]

0 [] 0

i I I

1 . 0 . 0 5 0 0 1 0 0 . 0 1 5 0 . 0 2 0 0 . 0

W O R K P I E C E L E N G T H , m m

Fig. 9. Change of the magnetic presure with the tube length (coil

length 76mm, coi l radius 42 mm, co il pitch 4mm, tube radius

44 mm).

A cknow ledgement s

The authors would like to express their gratitude to

Professor Song-Yop Hahn, Dept . of Electrical Engi-

neering, Seoul National Universi ty, for his valuable

comments on the electromagnetic field analysis.

R ef erences

[1] K Baines, J.L. Duncan and W. Johnson,

Proc. Inst. Mech. Engrs.,

180

(1965) 93.

[2] S.T.S. A1-Hassani,J.L. Dunc~mand W. Johnson,

Proc. Int. Conf.

Manuf. Technol . ,

ASTME, 1967, p. 853.

4.0

3.8

0

o.. 3.6

u13.4

Y

1~3.2

i I

m3.0

]

2 8

2 .0 2 0 1 0 , 0 1 0 8 0 .0 8 o ' . 0 , 0 0 . 0 , 2 0 . 0 x , o - .

W O R K P I E C E L E N G T H - 4 , t u r n - 4

Fig. 10. Relationship between the magnetic pressure and L -4, where

L is the length of the tube.

W O R K I ~ I E C E - -L p - - 1

x x ~ x x x

~

L c : ~ _ [ O I L

Fig. 11. Effective workpiece length, Le, and coi l length, Lc.

[3] H. Suzuki, H. Negishi, Y. Yokouchi and M. Murata,

J. Japanese

Soc. Tech. Plasticity, 27

(1986) 1254.

[4] Sung Ho Lee and Dong Nyung Lee,

A SM E J . Eng. Mate r .

TechnoL, 116

(1994) 250.

[5] S.C. Tandon, A.F. Armor and M.V.K. Chari,

1EEE Trans . , PAS

102

(1983) 1089.

[6] P.P. Silvester and R.L Ferrari,

Finite Elements for Electrical

Engineers,

Cambridge University Press, Cambridge, 1983.

[7] C.H. Durney and C.C. Johnson,

Introduction to Modern Electro-

magnetics,

McGraw-Hill, New York, 1969, p. 163.

Lee_Estimation Magnetic Pressure in Tube Expansion_JMPT_1996

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