8/4/2019 1979 Johnson Slender Contact

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I n t . Y . M e c h . ScL, Vol. 21, pp. 237-247. Pergamon Press 1979 . Printed in Grea t Britain

P R E S S U R E B E T W E E N E L A S T IC B O D I E S H A V I N G A

S L E N D E R A R E A O F C O N T A C T A N D A R B I T R A R Y

P R O F I L E S

L . N A Y A K

Re g i o n a l En g i n e e r i n g Co l l e g e , Ro u r k e l a , Or i s s a , I n d i a

a n d

K . L . J O HN S O N

U n i v e r s i t y E n g i n e e ri n g L a b o r a t o r y , C a m b r i d g e , E n g l a n d

(Received 16 A u g u s t 1 9 7 8 ; in revised form 26 O c t o b e r 1978)

S u m m a r y - - A n u m e r i c a l m e t h o d i s p r e s e n t e d f o r c a l c u l a t i n g t h e p r e s s u r e d i s t r i b u t io n a n d c o n t a c ta r e a s h a p e b e t w e e n t w o e l a s t i c b o d i e s o f a r b i tr a r y p r o f il e w h i c h m a k e c o n t a c t o v e r a s l e n d e rc o n t a c t a r e a , i . e . wh e r e t h e r e l a t i v e c u r v a t u r e o f t h e t wo p r o f i l e s i s m u c h s m a l l e r i n t h el o n g i t u d i n a l d i r e c t i o n t h a n i n t h e t r a n s v e r s e . Th e p r e s s u r e d i s t r i b u t i o n i s a s s u m e d t o b ep i e c e w i s e - l i n e a r in t h e l o n g i t u d i n a l d i r e c t i o n a n d s e m i - e l l ip t i c a l i n t h e t r a n s v e r s e . No a pr ior ir e l a t i o n s h i p i s a s s u m e d b e t we e n t h e s h a p e o f t h e c o n t a c t a r e a a n d t h e l o n g i t u d i n a l v a r i a t i o n i np r e s s u r e ; t h e y a r e f o u n d s i m u l t a n e o u s l y f r o m d u a l i n t e g r a l e q u a t i o n s f o r t h e c o m p a t i b i l i ty o f (a )t h e n o r m a l d i s p l a c e m e n t a n d ( b ) t h e t r a n s v e r s e c u r v a t u r e a l o n g t h e l o n g i t u d i n a l a x i s o f t h e

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

b e y o n d , t h e e n d s o f t h e c o n t a c t a r e a , t h e m e t h o d g i v e s a v e r y r e l i ab l e m e a s u r e o f t h e c o n t a c tp r e s s u r e d i s t r ib u t i o n . W h e r e d i s c o n t i n u i t ie s i n p r o f il e a re p r e s e n t , a t r o l l e r e n d s f o r e x a m p l e ,s t r e s s s i n g u l a r i t i e s a r e t o b e e x p e c t e d a n d l i k e a n y n u m e r i c a l m e t h o d , o n l y a p p r o x i m a t e v a l u e so f t h e s t r e s s c o n c e n t r a t i o n c a n b e f o u n d . I n t h e c a s e s s t u d i e d , t h e c o n c e n t r a t i o n o f p r e s s u r ea s s o c i a te d w i t h a " s h a r p " e d g e o f c o n t a c t i s f o u n d t o b e v e r y l o c al .

Th e m e t h o d h a s b e e n a p p l i e d t o b o t h c y l i n d r i c a l a n d v a r i o u s l y " c r o wn e d " r o l l e r s , a l s o t o ab al l " o v e r - r i d i n g " t h e e d g e o f a c l o s e l y c o n f o r m i n g g r o o v e .

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s e m i - l e n g t h o f c o n t a c t a r e as e m i - b r e a d t h o f c o n t a c t a r e ac o n t a c t p r e s s u r em a x H e r t z p r e s s u r e

! !4-~-~., = r e la t ive curv a ture in long i tudina l d i rec t ion

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1 16 ~-~ . = r e l a t i v e c u r v a t u r e i n t r a n s v e r s e d i r e c t i o n

Rv,x /a

b /ac o m b i n e d c o m p r e s s i o n o f b o t h b o d i e s a t x = y = 0

n o r m a l e l a s t i c d i s p l a c e m e n ts u r f a c e p r o f i l e o f b o d y r e l a t i v e t o t a n g e n t p l a n e a t o r i g i ne c c e n t r i c i t y o f m i d - p o i n t o f c o n t a c t a r e a f r o m o r i g i n

I - v , 2 + I - v : 2 " ~ 'E l - - ~ 2 / = c o m b i n e d e l a s ti c m o d u l u s

p i E 't o t a l c o n t a c t f o r c ec o m p l e t e e l l i p t i c i n t e g r a l o f a r g u m e n t k o f 1 s t a n d 2 n d k i n d s

1 . I N T R O D U C T I O N

C a l c u l a t i o n o f th e c o n t a c t p r e s s u r e b e t w e e n t w o e l a s t i c b o d i e s w h o s e i n i t i a l s e p a r a -

t i o n c a n n o t b e e x p r e s s e d a d e q u a t e l y b y a q u a d r a t i c f u n c t i o n c o n t i n u e s t o p r e s e n t

d i f f i c u l ti e s , in s p i t e o f m a n y s t u d i e s a n d i ts t e c h n o l o g i c a l i m p o r t a n c e . T h e f u n d a m e n -

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

H e r t z i a n e l l i p s e a n d i t s s h a p e i s n o t k n o w n a t t h e o u t s e t . I n t h i s p a p e r w e r e s t r i c t o u ra t t e n t i o n t o t h o s e c a s e s i n w h i c h t h e c o n t a c t p l a n f o r m h a s a l a r g e a s p e c t r a t i o , i . e . i ts

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

e f f e c t i v e l y c i r c u l a r a r c s . I n t h e s e c o r c u m s t a n c e s i t i s r e a s o n a b l e t o a s s u m e t h a t t h e

p r e s s u r e d i s t r i b u t i o n p e r p e n d i c u l a r t o t h e m a j o r a x i s o f t h e c o n t a c t a r e a i s H e r t z i a n ,

23 7

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238

a s s h o w n b y G r u b i n , 1 i .e .

L. NAYAKand K. L. JOHNSON

p ( x , y ) = p ( x , O) 'k / ( l - yZ/b2(x)) ( l )

w h e r e b ( x ) i s t h e s e m i - b r e a d t h o f t h e c o n t a c t a r e a a t l o c a t i o n x a l o n g i t s m a j o r a x i s

( t a k e n p a r a l l e l t o x - a x i s ) . F o r b o d i e s o f g i v e n p r o f i l e i t i s r e q u i r e d t o f i n d t h e

d i s t ri b u t i o n o f c o n t a c t w i d t h b ( x ) a n d p r e s s u r e a l o n g a m a j o r a x i s p ( x , 0 ) . P r a c t i c a la p p l i c a t i o n s o f s u c h p r o b l e m s a r i s e i n f i n d in g th e c o n t a c t p r e s s u r e d i s t r i b u t i o n in a

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

w h e r e t h e r a c e p r o f i le d e p a r t s f r o m a c ir c u l a r a rc . W e h a v e a l s o b e e n a s k e d t o

i n v e s t i g a te t h e s t r e s s c o n c e n t r a t i o n w h i c h a r i s e s w h e n t h e e ll ip s e o f c o n t a c t o f a ba ll

in a n a n g u l a r c o n t a c t b e a r i n g " o v e r - r i d e s " t h e e d g e o f th e r a c e w a y .

P r e v i o u s s t u d i e s h a v e g e n e r a l l y f o l lo w e d t w o a l t e r n a t iv e a p p r o a c h e s t o th e p r o b -

l em o f th e u n k n o w n p l a n f o r m . O n e a p p r o a c h h a s b e e n t o m a k e d i re c t e x p e r im e n t a l

o b s e r v a t i o n s o f t h e c o n t a c t a r e a u s in g s o m e t e c h n i q u e f o r m a k i n g t h e " f o o t p r i n t "

v i s i b l e ( G o o d e i l e et al . 2 a n d O r lo v 3) . T h e s e c o n d a p p r o a c h a s s u m e s t h a t th e l o c a l

s e m i - c o n t a c t b r e a d t h b ( x ) is r e l a t e d t o t h e c o r r e s p o n d i n g p r e s s u r e p ( x , 0 ) b y t h e

t w o - d i m e n s i o n a l ( p l a n e s t r a i n ) H e r t z r e l a t i o n s h i p :

b ( x ) = ( 2 / E ' ) R y ( x ) p ( x , O ) (2 )

w h e r e R y is t h e r e l a ti v e r a d i u s o f c u r v a t u r e in th e t r a n s v e r s e d i r e c ti o n . T h i s a s s u m p -

t io n h a s b e e n j u st if i ed b y G r u b i n ~ f o r c o n t a c t s w i t h t r a n s v e r s e c u r v a t u r e w h i c h v a r i e s

s l o w l y w i t h x . I t h a s b e e n a p p l i e d t o a x i a ll y p r o f il e d r o l l e rs b y K a l k e r 4 a n d K a n n e l )

W h i l s t t h e a s s u m p t i o n e m b o d i e d i n e q u a t i o n ( 1 ) i s l i k e l y t o b e a g o o d a p p r o x i m a t i o n

w h e n t h e p r e s s u r e v a r i e s s l o w l y w i t h x , i t i s n o t s o e a s y t o j u s t i f y i n r e g i o n s o f s t r e s s

c o n c e n t r a t i o n s u c h a s o c c u r a t t h e e n d s o f c y li n d r ic a l r o l le r s .

I n t h is p a p e r w e p r e s e n t a m e t h o d o f a n a l y s i s in w h i c h n o p r e s u p p o s i t i o n s a r e

m a d e a b o u t t h e v a r i a ti o n o f c o n t a c t w i d th . I n c o m m o n w i t h all t r e a t m e n t s o f th ep r o b l e m a n i n t e g r al e q u a t i o n f o r t h e p r e s s u r e v a r i a t i o n a l o n g t h e a x i s p ( x , 0 ) is

f o r m u l a t e d b y m a k i n g t h e e l a s ti c d i s p l a c e m e n t w Lz(x, 0 ) a l o n g t h e a x i s c o m p a t i b l e

w i t h t h e a x i a l p r o f i l e s z l , : ( x ) , i . e .

w~(x, O) + w2(x, O) = a - zj (x ) - z2(x) (3)

w h e r e a i s t h e c o m b i n e d c o m p r e s s i o n o f th e t w o b o d i e s a t x = y = 0 . T h i s e q u a t i o n

c o n t a i n s t h e u n k n o w n c o n t a c t w i d th d i s tr i b u t io n b ( x ) . T h e a s s u m p t i o n o f a n e l l i p t i c a l

d i s t ri b u t io n o f p r e s s u r e in t he t r a n s v e r s e d i r e c t i o n a c c o r d i n g t o ( l ) i m p l ie s a q u a d r a t ic

v a r i a t i o n i n d i s p l a c e m e n t w ( y ) . T h u s a s e c o n d i n te g r a l e q u a t i o n f o r p ( x , 0 ) c a n b e

e s t a b l i s h e d b y m a t c h i n g t h e t r a n s v e r s e c u r v a t u r e o f t h e e l a s t i c d i s p l a c e m e n t s t o t h er e l a ti v e t r a n s v e r s e c u r v a t u r e o f t h e tw o b o d i e s , i .e .

[~y {w t (x , y ) + w2(x, y )}]y=o - 1 - [ ~ + ~ ] . (4)R y ( x )

T h e d i s p l a c e m e n t e q u a t i o n ( 3) a n d t h e c u r v a t u r e e q u a t i o n ( 4 ) p r o v i d e t w o s i m u l -

t a n e o u s e q u a t i o n s f o r t h e a x ia l d i s t ri b u t io n o f p r e s s u r e p ( x , 0 ) a n d c o n t a c t w i d t h b ( x ) .

T h e d u a l i n t e g ra l e q u a t i o n s a r e f o r m u l a t e d i n S e c t i o n 2 a n d t h e n u m e r i c a l m e t h o d

u s e d f o r th e i r s o l u t i o n i s o u t l i n e d i n S e c t i o n 3 . F i n a l l y t h e r e s u l t s a r e p r e s e n t e d o f

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

2 . FORMULATION OF THE EQUATIONS

It will be assumed from the outset that the elastic deformation of both bodies is that of an elastichalf-space, as in the He rtz theory. This assumption mu st be kept in mind wh en applying the analysis topractical cases such as the stress concentration at roller ends. The bodies will be assumed to be frictionless,but the influence of tangential forces on the norm al pressure is always sm all and vanishes entirely w hen the

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P r e s s u r e b e t w e e n e l a s ti c b o d i e s h a v i n g a s l e n d e r a r e a o f c o n t a c t 2 3 9

e l a s ti c c o n s t a n t s o f t h e t w o b o d i e s a r e e q u a l . t W e a r e c o n c e r n e d , t h e n , o n l y w i th t h e n o r m a l d i s p l a c e m e n t

a t a p o i n t ( x. y ) d u e t o a n o r m a l p r e s s u r e d i s t r i b u t i o n p ( x ' , y ' ) , w h i c h i s g i v e n b y

y ) ! f p ( x ' , y ' ) d x ' d y 'w (x , = ~ - ; JA {(x - x ' ) ' + (y - y,)2},~2 (5)

w h e r e t h e i n t e g r a t i o n i s o v e r t h e c o n t a c t a r e a , A .

T o o b t a i n a n e x p r e s s i o n f o r t h e d i s p l a c e m e n t o n t h e x - a x i s , w (x , 0 ) t h e t r a n s v e r s e p r e s s u r e d i s t r ib u t i o n

g i v e n b y e q u a t i o n ( 1 ) i s s u b s t i t u t e d i n t o e q u a t i o n ( 5 ) a n d i n t e g r a t e d w i t h r e s p e c t t o y ' b e t w e e n t h e l i m i t s+_ b( x) . T h e r e s u l t w h e n s i m p li f ie d m a y b e w r i t te n

2 a ~ + l p ( ~ , )w (~ , 0) = ~rE-----3 -1 ~ - - - { K ( k ~ ) - E ( k~ ) } d ~ ' ( 6 )

w h e r e t h e l e n g t h o f t h e c o n t a c t i s 2 a ( w i t h t h e m i d - p o i n t c h o s e n a s o r i g in ) , ~ = x la . K ( k D a n d E ( k~ ) a r e

c o m p l e t e e l l ip t ic i n t e g r a ls o f t h e f i rs t a n d s e c o n d k i n d s , o f a r g u e m e n t k~, w h e r e

tg- J J '

g((;) =- b( O la

a n d

p( D = - p (x la , 0 ) .

S u b s t i t u ti n g t h e e l a s t ic d i s p l a c e m e n t s o f e q u a t i o n ( 6) i n to e q u a t i o n ( 3) g i v e s th e " d i s p l a c e m e n t e q u a t i o n " :

- 1 - ~ t . ~ E '

{ K (k ~ ) - ~ k ~ ) } d r ' = - ~ - { a - z ( x )} ( 7 )I

w h e r e z ( x ) r e p r e s e n t s t h e c o m b i n e d a x i al p r o f il e o f t h e t w o b o d i e s z l ( x ) + z 2 ( x ) . T h i s i s t h e e q u a t i o n

f o r m u l a t e d b y K a n n e l . -~ T h e v a r i a ti o n i n c o n t a c t b r e a d t h a p p e a r s i n th e a r g u e m e n t k ,, s o t h a t e v e n i f t h e

H e r t z r e l a t i o n s h i p b e t w e e n b ( x ) a n d p ( x ) ( i n e q u a t i o n 2 ) i s t a k e n t o h o l d , t h e i n t e g r a l e q u a t i o n ( 7 ) f o r p ( x )

i s n o n - l i n e a r .

W e s h a ll n o w p r o c e e d t o f o r m u l a t e t h e " c u r v a t u r e e q u a t i o n " . D i f f e r e n ti a t in g e q u a t i o n ( 5 ) t w i c e g i v e s

aZw(x,0) 1 f ° f0y 2 l r-E'

. b [ { ( X - -

X ')2+ y , 2 } - 3 / 2 - -

3y,2{(x _ x'): + y,2}-5:2]p ( x ' , y ' ) d x ' d y ' . ( 8 )

S u b s t i t u ti n g t h e t r a n s v e r s e p r e s s u r e d i s t r ib u t i o n f r o m ( 1) , in t e g r a ti n g w i t h r e s p e c t t o y ' a n d s i m p l i fy i n g ,

e n a b l e s e q u a t i o n ( 8) to b e e x p r e s s e d i n t e r m s o f c o m p l e t e e ll ip t ic i n te g r a l s o f s t a n d a r d f o r m . W h e n

s u b s t i t u t i n g i n t o e q u a t i o n ( 4 ) t h e c u r v a t u r e e q u a t i o n i s o b t a i n e d :

f _ P ( ~ ' ) ( 9 )~ k ~ { K ( k ~ ) - F _,(k e) d ~ :' - "n'E'a2 R r ( x ) "

O n c e a g a i n t h e c o n t a c t b r e a d t h v a r i a ti o n l i e s i n s i d e th e a r g u e m e n t k~ m a k i n g t h e e q u a t i o n n o n - l i n e a r.

E q u a t i o n s ( 7 ) a n d ( 8 ) a r e t h e d u a l i n t e g r a l e q u a t i o n s f o r t h e c o n t a c t b r e a d t h v a r i a t i o n g (~ :) a n d t h e

p r e s s u r e v a r i a t i o n p ( s r ) , g i v e n t h e a x i a l p r o f i l e z ( x ) a n d t h e t r a n s v e r s e c u r v a t u r e l i R a ( x ) . I n v i e w o f t h e i r

n o n - l i n e a r f o r m , t h e m e t h o d a d o p t e d f o r t h e i r s o l u t i o n h a s b e e n t o m a k e a n i n t e ll e g e n t f i rs t g u e s s a s t o t h e

s h a p e o f t h e p l a n f o r m . T h u s g ( f ) i s s p e c i f i e d a n d b o t h e q u a t i o n s ( 7 ) a n d ( 9 ) a r e l i n e a r i n p ( f ) . T h e y a r e

s o l v e d n u m e r i c a l l y a s d e s c r i b e d i n t h e n e x t s e c t i o n t o o b t a i n t w o i n d e p e n d e n t p r e s s u r e d i s t r i b u t i o n s p a ( s )

a n d P c (~ :) r e s p e c t i v e l y . I f , a t a n y p o i n t a l o n g t h e l e n g t h , a v a l u e o f t h e b r e a d t h h a s b e e n c h o s e n w h i c h i s t o os m a l l p a ( s ) w i ll e x c e e d P c ( ~) . A n a p p r o p r i a t e c o r r e c t i o n c a n t h e n b e m a d e t o g ( s ) a t t h a t p o i n t . A f e w

i t e r a ti o n s l e a d s t o a p l a n f o r m s h a p e w h i c h s a t is f i e s b o t h e q u a t i o n s ( 7 ) a n d ( 9 ) w i t h s e n s i b l y t h e s a m ep r e s s u r e d i s t r ib u t i o n ,

3. M E T H O D O F S O L U T I O N

T h e n u m e r i c a l m e t h o d e m p l o y e d f o r t h e s o l u t i o n o f t h e in t e g r al e q u a t i o n s ( 7 ) a n d ( 9 ) i s b a s e d o n t h e

m e t h o d o f s o l u t i o n o f t w o - d i m e n s i o n a l c o n t a c t p r o b l e m s u s e d b y B e n t a l l a n d J o h n s o n . 6 I n t h is m e t h o d t h e

c o n t a c t a r e a , o f l e n g t h 2 a , i s d i v i d e d i n t o 2 n s t r ip s e a c h o f w i d t h c . A t r i a n g u l a r u n i t o f p r e s s u r e i s t a k e n t o

a c t o n e a c h p a i r o f a d j a c e n t s t r i p s h a v i n g a c e n t r a l o r d i n a t e p j a n d f a l li n g to z e r o a t a d i s t a n c e _+ c o n e i t h e r

s i d e. S u p e r p o s i t i o n o f 2 n - 1 o v e r l a p p i n g t r i a n g u l a r u n i t s r e s u lt s i n a p i e c e w i s e - l i n e a r d i s tr i b u t io n o fp r e s s u r e w h i c h f a l ls t o z e r o a t e a c h e n d o f t h e c o n t a c t a r e a . T h e i n t e g ra l e q u a t i o n f o r t h e p r e s s u r e

d i s t r i b u t i o n i s s a t is f i e d d i s c r e t e l y a t t h e e d g e o f e a c h e l e m e n t a l s t r i p t h e r e b y y i e l d i n g a l i n e a r m a t r i xe q u a t i o n w h i c h c a n b e s o l v e d b y s t r a i g h t f o r w a r d i n v e r s i o n t o f i nd {p i} .

I n t h e p r e s e n t p r o b l e m t h e c o n t a c t a r e a i s a g a in d i v i d e d i n t o s tr i p s p e r p e n d i c u l a r t o t h e m a j o r a x i s. E a c h

u n i t o f p r e s s u r e a c t s o n a r e c t a n g u l a r b a s e o f w i d t h 2 c a n d b r e a d t h 2 b , a s s h o w n i n F ig . ! . T h e p r e s s u r e i s

t C a l c u l at i o n s i n s u p p o r t o f t h is s t a t e m e n t h a v e b e e n p e r f o r m e d b y H . D . C o n w a y a n d P . A . E n g e l , In t . J .M e c h . S c i . 11, 709 (1969) .

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240 L. NAYAK and K. L. JomqsoN

• p

- y

x

Fic. I. Press ure units. Each unit acts on a rectangular base 2b × 2c. The press ure distributionis triangular in longitudinal section and Hertzian in transverse section. The elastic displace-

ment at I due to the press ure unit centred at J is w(i, ]).

distributed eiliptically according to equation (1) in the transve rse direction; in the longitudinal direction it istriangular:

p( ~, )= ,fpj{(1 -(~cj - s ~ , ) l c } , ~ i - , " < s ~ ' < ~ (10)I.pi{(1 - (~' - ~i )lc }, ~j <-- ~' ~ I~j+,"

The dis placement at the point ~ = ~ due to the unit of pressure cent red at ~ = ~, denote d by w(i,j), is foundby substituting the pressu re distribution of equation (10) into equation (6) and integrating over the interval

~_, to ~i+t (see App endix 1). During this integration for a simple press ure unit, the bre adth g( O is consta ntat g(~). Similarly, substituting the unit pressure distribution into equation (8) and integrating gives thetransve rse curvature at ~ = ~i due to the pres sure unit centre d at ~- -~ , denot ed by O 2 w ( i , j ) l O y 2. The

pressure unit was also proposed by Koshy and Gohar. 7In co mmon with all conta ct press ure calculations, a singularity arises in the integrals of equations (6) and

(9) when the displac ement or curvature are being found a t a p oint beneat h the load, i.e. when i -- j. With thetwo-dimensional pressu re units used by Bentali and Johnso n the integrals could be evaluated in closed form

so that the difficulty vanished but, with the three -dimensional pressure units used here, equ ation s (6) and (8)

must be integrated numerically. However, as shown in the Appendix, by recasting the elliptic integrals a

satisfactory method of integration has been found. "

If the displac ements w(i,j) are normalised with respect to a and the pressu res p j normalised withrespect to the effective modelus E' the displacement equation (7) can be written

A ~ P ~ = tr~-{(~la) - z ( e , 6)la} (ll)

where A~ is the matrix of displacemen t influence coefficients and e specifies the location of the mid-point of

the contact area,

The corres ponding curvature equation is

c f r a

A i i P i = ~ x ~gl- ~'y'~x" (12)

The evaluation of the influence coefficients A~ and A~ is discussed in the Appendix. Given the axial profilez(O and the transverse curvature Ry(O, assuming a planform g(O, equations (11) and (12) can he inverted

to find the independent distributions of contact pressure.In performing this inversion, two class es of problem mus t be distinguished. In the first class the profiles

are smooth and continuous at the ends of the contact, so that the mid-point of the contact is located at anunknown distance from some arbitrarily chosen origin. In the second class the length and position of thecontac t area is uniquely defined by discontinuities in the profile, such as a cylindrical roller of finite length.In the first case the pressu re falls to zero at the en ds of th e conta ct area so that PI = P2,+1 = 0. In equation(!1) i and j take integral values fro m i to (2n + 1) and the equat ion is solved for (2n - I) unknown values ofP; together with the unknown values of a / a and e/a. (Naturally in a symmetrical cont act e = 0.) In equation

(12), on the other hand, the curvature condition is not maintained at the ends of the contact where thebreadth b--,0; whereupon i and j take values from 2 to 2n.

In cases where a discontinuity of profile prescribes the contact length the pressure at the ends of thecontact P~ and P2,+a are no longer zero. The pressure units at the ends are now h a l f u n i t s where theinfluence coefficien ts A~.I and Ak2n+ mus t be adjusted acco rdingly. In e quation (11), the location o f thecentre of the contact given by • is known p r i o r i and the indentation at the centre a must be prescribed. Inboth equatio ns (11) and (12) i and j take values fro m ! to (2n + I) to determine (2n + I) values of press ure.

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P r e s s u r e b e t w e e n e l a st ic b o d i e s h a v i n g a s l e n d e r a r e a o f c o n t a c t 2 41

F i n a l ly t h e t o ta l l o a d i s f o u n d f r o m t h e s u m m a t i o n o f t h e p r e s s u r e o v e r t h e m e a s u r e d c o n t a c t a r e a , i. e.

2 F 1"÷~ E ' j = 2 n + l

" ~= J p ( E . ) g ( E . ) d E = - - ~ ~ P ( j ) ." f r O " I ' =i.r.i

( 1 3 )

T o t e s t t h e te c h n i q u e t h e m e t h o d w a s a p p l i e d t o a H e r t z c o n t a c t , h a v i n g a n e ll ip t ic a l p l a n f o r m o f a s p e c t

r a t i o a l b o = 10. ( i .e. R y l R , = 0 . 0 2 7 3 7 . ) I n t h i s c a s e g (E .) = ( b o l a ) ( l - E . 2 ) ~ I 2 i s k n o w n i n a d v a n c e . T h e a x i a l

p r e s s u r e d i s t r i b u t i o n a l o n g t h e c e n t r e l in e i s s h o w n i n F ig . 2 f o r d i f f e r e n t n u m b e r s o f s t ri p s . F o r n = 2 0 t h e

p r e s s u r e d i s t r i b u t i o n s f o u n d b y t h e d i s p l a c e m e n t e q u a t i o n ( 1 1 ) a n d b y t h e c u r v a t u r e e q u a t i o n ( 1 2 ) a r e i ne x c e l l e n t a g r e e m e n t w i t h e a c h o t h e r a n d w i t h t h e H e r t z t h e o r y . T h e t o t a l l o a d F , t h e c e n t r a l p r e s s u r e p 0 a n d

t h e c e n tr a l c o m p r e s s i o n a a r e c o m p a r e d i n T a b l e 1 b e l o w .

TABLE 1.

H e r t z D i s p l a c e m e n t C u r v a t u r e

t h e o r y e q u a t i o n e q u a t i o n

2 p o R , J E ' a 3 . 6 9 4 5 3 . 6 9 5 4 3 . 6 9 5 5

2 F R x / TrE ' a ~ 0 . 4 9 2 6 0 " 4 9 0 9 0 . 4 9 2 6

2 o l R x / a ~ 1 . 3 6 5 3 1 . 3 6 0 6 - -

~ b O

- I O - O . 5 O 0 . 5 I O

F IG . 2 . V e r i f i c a t io n o f t h e n u m e r i c a l m e t h o d b y a p p l i c a t i o n t o t h e H e r t z p r o b l e m , a/b = I 0 ,Rr/Rx = 0 - 0 27 4 . F r o m d i s p l a c e m e n t e q u a t i o n : x n = 4 , O n = I 0 , + n = 2 0 . F r o m c u r v a t u r e

e q u a t i o n : F i n = 2 0 . F r o m H e r t z t h e o r y :

4 . P R O F I L E D R O L L E R S

T o a v o i d t h e c o n c e n t r a t i o n o f c o n t a c t p r e s s u r e w h i c h o c c u r s a t t h e e n d s o f p e r f e c t l y c y l i n d r i c a l r o l l e rs

g e n t l y c r o w n e d p r o f il e s a r e f r e q u e n t l y u s e d . I n a c la s s i c p a p e r L u n d b e r g ~ s h o w e d t h a t , in o r d e r t o o b t a i n a

u n i f o r m d i s t r i b u t i o n o f p r e s s u r e P 0 a l o n g t h e e n t i r e l e n g t h o f a r o l le r t h e p r o f il e s h o u l d b e c r o w n e d

a c c o r d i n g t o t h e r e l a t i o n .

- p o a _ E . 2 ) - I [ E . [ < 1 . ( 1 4 a )(E. ) - ~-~7 1n(I

T h i s e x p r e s s i o n b e c o m e s a p p r o x i m a t e a s E. ~ 1 , w h e n

z ( 1 ) = p o a r l4E , L . + 2 In ( 2 a l b ) ] , IE .I = 1 . ( 1 4 b )

T h e d i s p l a c e m e n t a t t h e o r ig i n i s g i v e n b y

= p o a [ ! ' 2 In ( 4 a l b ) ] . ( 1 5 )a 2 E " - t

T h e c o n t a c t a r e a i s r e c t a n g u l a r w i t h b = 2 R v p o l E ' .

A f u r t h e r c h e c k o n o u r m e t h o d h a s b e e n o b t a i n e d b y s u b s t i tu t i n g th e L u n d b e r g p r of il e, d e f in e d b y

e q u a t i o n s ( 1 4 ) a n d ( 1 5 ) i n t o t h e r . h . s , o f e q u a t i o n ( 1 1 ) a n d i n v e r t i n g t o f i n d t h e p r e s s u r e d i s t r i b u t i o n .

T h e r e s u l t i s p l o t t e d i n F i g . 4 , c u r v e ( a ). T h e p r e s s u r e i s f o u n d t o b e a l m o s t e x a c t l y u n i f o r m a l o n g t h e l e n g t h

e x c e p t f o r a s m a l l d e v i a ti o n c l o s e t o t h e e n d . A v e r y s i m i l a r r e s u lt w a s f o u n d b y K o s h y a n d G o h a r . 7 T h e

d e v i a t io n m a y b e d u e t o c o m p u t i n g e r r o r s , b u t s e e m s m o r e l ik e ly t o b e d u e t o th e a p p r o x i m a t e n a t u r e o fe q u a t i o n ( 1 4a ) w h e n s a p p r o a c h e s 1 .0 .

T h e L u n d b e r g p r o f il e i s d i ff ic u l t t o m a n u f a c t u r e a n d i s , i n a n y c a s e , o n l y c o r r e c t a t t h e d e s i g n l o a d . A

p r a c t i c a l a p p r o x i m a t i o n t o t h e L u n d b e r g p r o f i le m a y b e o b t a i n e d b y a c i r c u l a r a rc ( " d u b - o f f " r a d i u s )

t o w a r d s t h e e n d s o f t h e r o l l e rs r u n n i n g t a n g e n t i a l l y i n t o t h e c e n t r e s e c t i o n w h i c h i s c y li n d r i c a l , i. e.

z ( x ) = ~ f 0 , 0 ~ I 1 1 < t

[ ( x - 1 ) 2 1 2 R ~ , I < - I x l " < a " ( 1 6 )

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242 L. NAYAK an d K. L. JOHNSON

2o

1 . 5 ~ -

i . o I

0 . 5 ~ - -

1 1 1 1 1

I I J0 0 . 2 0 . 4 0 . 6 0 -8 ,

i

i

1.0

F ro . 3 . R o l l e r w i th co n t in u o u s p ro f i l e ( "d u b -o f f " en d s ) . P res su re d i s t r i b u t io n : I I n i t i a l ca l -cu l a t i o n ; I 1 s t i t e r a t i o n ; 4 th ( f i na l ) i t e r a t i o n ; - - - c o n t a c t p l a n f o rm blbo.

1.0

. Z

0 . 5 -

IC

A

o.

0 0 . 2 0 . 4 0 . 6 0 . 8 I - 0

~=~ , o

( a )

I L ' 7 L

J , : 7 cL - " I L AL . _ ; !

B : - I

L•--.-L - .

I p t I io o . 2 o 4 o . 6 o . s ,. o

(b )

F l o . 4 . ( a ) P r e s s u r e d i s t r i b u t i o n s w i t h p r of i l e d r o l l e rs : A L u n d b e r g p r of i le ; B p a r t l y c r o w n e d w i t hs l o p e d i s c o n t i n u i t y a t x = I ; C c r o w n e d w i t h " d u b - o f f " e n d s . C o) C o n t a c t p l a n f o r m s .

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Pres sure betwee n elastic bodies 'havi ng a slender area of contact 243

Equat ions (11) and (12) have been solved f or a roller of t his profile taking a = 5-5 mm, l = 3-6m m,

Ry = 6.35 mm, Rx = 104 mm and P0 = 1.93 KN/m m~. As a first approximation the contact area was taken to

be rectangu lar of width b0 wher e the roller is cyli ndrical (b0 = 2 R v P o l E ' ) and elliptical in the region of

"dub-off". The pressure distributions found from the displacement and curvature equations are shown in

Fig. 3. In places they deverge widely. The contact width at each cross section is then modified by theiteration formula:

b,.+, = b..{1 + h ( P d - p c ) 2Ry} (17)

where p a and Pc are the non-dim ensional press ures found from (11) and (12) respectively, and h is aconstant whose optimum value was found by experience to be about 0.5.

Equations (11) and (12) were then re-solved. The result of the 1st and 4th iterations are also shown inFig. 3. By the 4th iteration the two pressure distributions agreed to within 1.0%. A smooth continuouspressure distribution falling to zero at the ends of the contact is obtained. The final shape of contact

planform is also shown. It is evident that the shape of the planform closely follows the pressure distribution

in accordance with equation (2). This result suggests that in cases like the present one, where the profilevaries slowly and smoothly with x, the planform can be found from equation (2) as assumed by Kannel? In

such cases the curvature equation can be dispensed with, but the non-linear nature of the displacement

equation makes it desirable to still use an interative procedure between the pressure distribution and theplanform.

5. STRESS CONCENTRATIONS

In the previous section we considered bodies whose profiles are smooth and continuous. Many practicalproblems arise, however, in which there are real or effective discontinuities in profile which lead to stress.concent ration s having implication for plastic flow and fatigue failure.

Slight discontinuities in s l o p e of the profile can be present within the contact zone, i.e. contact is

maintained on either side of discontinuity. More severe discontinuities are associated with the edge ofcontact; a roller with a sharp end, for example.

In plane strain contacts of two-dimensional bodies a discontinuity of profile slope w i t h i n the contactleads to a "logarithmic singularity" in the cont act pressure. Thus the pr essure at a small distance x fr om thediscontinuity varies as In ]x[. Singularities which arise at the edges of con tact have been studied by

Dundurs9 who has shown that in frictionless contacts a "power singularity" would be expected i.e. thecontac t press ure at a small distance x from the edge would vary as x -m, where m var ies from 0.5 to 1

depending on the angle of discontinuity in the profiles and the relative elastic constants of the two bodies.We are c oncer ned in this paper with three-dimensi onal contac ts of narrow planfor m. Sufficiently close

to the ends, or to a discontinuity of axial profile, however, the stress distribution migllt be expected to

approach the two-dimensional situation. It is reasonable therefore to expect similar singularities in the

situation examined in this pape r whe re t here are discontinuities of axial profile or profile slope.At first sight it would appear that the existence of stress singularities would invalidate a numerical

method based upon a piecewise linear distribution of pressure acting over finite intervals of length. Inpractice, however, sharp discontinuities seldom arise: small discontinuities of slope at blend points arerounded off either by a finishing operation or by local pl~tstic deformation. It was considered useful,

therefore, to perse vere with the method presented in this pap ere ven in cases where nominal discontinuitiesexist. A few examples will be discussed below.

(a) C r o w n e d r o l l e r s

A further simplification in the manuf acturing proc ess for profiling rollers is achieved if the crown radius

is ground from a centr e on t he c entreline of the roller. The centre section of the roller, [xl-< I, remainscylindrical, so that the roller profile is then given by:

{ 0 0 - < [ x [ - lZ(X)

( x 2 - 1 2 ) 1 2 R ~ , l <- Ixl < a.

This profile has a slope discontinuity of//Rx at x = l. It has been used to find the pressure distribution in a

case comparab le with the previous e xample, taking I = 3.6 mm and Rx = 508 mm. The pressur e distributionis shown in Fig. 4 where it is compar ed with that for a smoo thly profiled roller.

It appears that the region of high stress in the vicinity of the discontinuity of profile slope is extremelylocal; only one computed point contributes to the peak. The effect of such a local peak pressure on thesubsurface stresses in the bodies is likely to be very small. In any case, slight rounding of the profile will

cause the singularity to disappear resulting in a pressure distribution much like the computed one.

(b) C y l i n d r i c a l r o l l e r s

A roller of finite length in contact with a flat race which extends b eyond the end of the roller has a stressconcentration at the end. The profile in this case is given by

z ( x ) = O, I x I < - a .

Taking a prescribed value of ,- given by equation (15) with b d a = 10, equations (11) and (12) have to besolved to find the pressure distribution. Assuming the cut-off at the ends of the rollers to be sharp, h a l f

units of pressure are taken to act at the roller ends. The pressure distribution is shown in Fig. 5. The stressconcentr ation at the end is se en to be a ver y local effect. In the first instance the problem was solved takingn = 20 which gave rise to a stress concentr ation factor at the roller end of 4.56. Halving the width of thestrips at the outer half of roller, i.e. effectively changing to n = 40, has the effect of increasing the stress

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244 L. NAYAK and K. L. JoHNSON

6. 0

5- 0

~ 4.C

o. 3'C

2(3

IC

0

ir

x - . K - K - x - ~ , - K - -K , - - K - ~

I I

o !2 o!4 o6 o . e ,.o

FIG. 5. Cylindr ical roller with "square" ends. x n = 20; ©n = 40.

concentr ation factor at the roller end to 5.38, while making neglig/ble difference to the pre ssure elsewhe re.It is not surprising that the stress concentration rises as the width of the pressure elements is decreased

since, for a roller with a s harp cut-off, the pressur e rises theoretically to infinity. In practice roller ends donot terminate sharply but have a small end radius of magnitude - R,IIO. If the end radius blends smoothly

with the cylindrical barrel the en d s tress- concentr ation is then finite. Only a three -dimensiona l analysistaking the end radius into account could determine the end pressure with certainty but it is felt that a stress

concentration factor of about 5 is probably about right.The spread in the breadth of the contact area at its ends is shown enlarged in Fig. 6. Also shown in a

plot of p(x)lPo, which by equation (2) should repre sent b(x)lbo where P0 and b0 are the contact pressure and

contact breadth in the centre of the roller. It is clear that the assumption that the contact breadth and

contact pressure are related by the two-dimensional Hertz equation (equation 2) is well founded except atthe point of stress concentration, where the increase in pressure is very much greater than the increase in

contact breadth.As a simple experimental check on the overall validity of the theory, the shape of the contact area was

measured by observing the footprint made in a very thin copper plate on the flat surface of a hard steel

roller end .by the cylindrical surface of a contacting roller. The edges of the flat surface were chamfered at5, 15 and 25 ° but no significant difference with ch amfe r angle was obser ved in the f ootprint shape. Thenumerical method employs pressure elements of rectangular planform in which the pressure falls to zero at theedges. The final theoretical planform of the contact area is therefore stepped as shown in Fig. 6. Exact

comparison with experiment is therefore not possible, but the general agreement is very satisfactory andprovides good support for the numerical results.

A fu rther exper iment was perf orme d with a cylindrical roller (38 mm dia.) in contac t with th e flatend of a roller which has an edge radius of 1.5 ram. The footprint measurements are also shown in Fig. 6.Excep t for the value at the edge itself the footp rint breadth is not significantly affected by whe ther the edge

is radiused or chamfered.

(c) Over-ride in ball bearingsWhen the ratio of thrust load to radial load in an angular contact bearing becomes excessive the contact

//

- - - - ~ J e I I I I

o . , o . s 0 . 6 o . ~ 0 : 8 o. 0

FIG. 6. Pla r~or m of cylindri cal roller; Calcula ted; Q measured (avere4~e of 3and 25° c h a m f e r ) ; x measured (radiused end); -- - pressure distribution

1"25

1.20

q - 1 5

b (x )

b o

- H O

t.0 5

PO

tests with 5 , 1 5

p ( x ) l p o .

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Pressure between elastic bodies having a slender area of contact 245

1 2

IO

,~ 8

O.

I

0.2 0.4 0.6 0"8 ~

~=Xla

F ]~ . 7. Over-ride in an angular contact ball-bearing. Ball diameter 7/16 in:, track radius 0.226 in.,nominal contact shape alb = 10.82. Contact pressures for 0.1, 0.2, 0.35 and 0-5 over-ride,

A - ll2{(lIRxl) + (l/Rx~)}.

angle increases sufficiently for the elliptical contact area of the ball to over-ride the edge of the race. Edgeloading conditions then arise which are qualitatively similar to the end load in a cylindrical roller. Thisproblem has been analysed by the method presented here for a bearing having the following geometry andloading: ball dia. = 1 . i 125 mm (7/16 in.), track radius = 0.226 in., nominal contact shape a/b = 10.82. Theresulting contact pressure distributions for varying degrees of over-ride are shown in Fig. 7. The high stressconcentration at the edge of the contact, with very little change in contact pressure elsewhere, is equallyapparent in this problem.

6. CONCLUSIONS

A method has been presented for finding the pressure distribution and contact area

planform for bodies in approximate "line contact", i.e. where the contact area is long

and narrow. Their transverse profiles are effectively circular arcs but the longitudinal

profiles are arbitrary. This method differs from previous approaches in that it makes

no presupposition about the longitudinal variation in the breadth of the contact area.

A pair of integral equations have been formulated for the longitudinal variation of

contact pressure p ( x ) and contact breadth b(x): they are solved numerically by a

procedure which iterates the contact breadth b ( x ) . Convergence is very rapid and

satisfactory results have been obtained with about 4 iterations.

In the cases examined it has been shown that the usual assumption, that the

breadth of the contact area is related to the pressure at any point by the two-

dimensional Hertz equation (equation 2), is a good approximation provided the

pressure gradient is small, but that it does not hold close to a point of stress

concentration. There the breadth is appreciably less than the value given by equation

( 2 ) .

In circumstances where the profiles of the bodies are smooth and continuous to

the edge of the contact area and beyond, the method gives contact pressure and area

distributions which are considered to be extremely reliable. The area predictions have

been supported by footprint measurements. Smoothly blended rollers, with or without

angular misalignment, have been exami ned. Ball bearings in confor ming but non-

circular profile tracks, and taper roller bearings or bevel gears which have continuous

variations of transverse radius, provide further practical examples to which the

method would give satisfactory results.

In cases where there are discontinuities in the axial profile some difficulty of

interpretation of the pressure distribution arises, in common with any numerical

analysis of a situation in which there are stress singularities. In those cases where

there is a small discontinuity in s lo p e of the profile wi th in the axial length of thecontact, the logarithmic singularity appears to be extremely local and the numerical

method is considered to give good results at all parts except very close to the

singularity. In mos t practical case s the slope discontinuity is "roun ded off" by finishing

procedures, whereupon its effect becomes negligible, "End effects" are not negligible

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246 L. NAYAK and K. L. JOHNSON

h o w e v e r . A s h a r p e n d l e a d s t o a " p o w e r " s i n g u l ar i ty a n d e v e n w e l l r a d i u s e d e n d s g i v e

r is e to a h ig h s t r e ss c o n c e n t r a t i o n w h i c h i s r e v e a l e d b y t h e n u m e r i c a l m e t h o d . T h e

s t re s s c o n c e n t r a t io n f a c t o r f o r r a d i u s ed e n d s g i v e n b y t h e m e t h o d c a n b e r e g a r d e d a

r e a s o n a b l e f i rs t a p p r o x i m a t i o n t o t h e t r u e v a l u e , w h i c h c o u l d o n l y b e f o u n d b y a f ul l

t h r e e - d i m e n s i o n a l a n a l y s i s .

T h e e f f e c t o f s u c h s t r e s s c o n c e n t r a t i o n s u p o n t h e e l a st i c li m i t o r f a t ig u e s t r e n g t h

o f t h e c o m p o n e n t s d e p e n d s u p o n t h e s t r e s s fi el d i n s id e t h e b o d i e s d i e to t h e c o n t a c tp r e s s u r e . T h e i n t e r n a l s t r e s s f i el d d u e t o a n e n d s t r e s s c o n c e n t r a t i o n i s s ti ll o p e n t o

i n v e s t i g a t io n a n d g o e s b e y o n d t h e s c o p e o f t hi s p a p e r .

A c k n o w l e d g e m e n t - - T h i s work was carried out with the support of a grant from Messrs. R.H.P. Ltd.,Chelmsford, England, to whom we are grateful for permission to publish this paper.

REFERENCES

1. A. N. GRUBIN, Inves tigation of t he con tact of machine componen ts. C en t ra l Sc i en t i f i c Res . I n s t . f o r

T e ch . a n d M e c h . E n g n g B o o k 30, p. 1. Moscow (1948) (DSIR Translation) .

2. R. A. GOODELLE, W. J. DERNER and L. E. ROOT, T r an s . A S L E 13, 268 (1970).3. A. V. ORLOV,M a c h i n o v e d e n i e , A N S S R , No. 2 (1973).

4. J. J. KALKER, T r a n s . A S M E ( S e r. E ) , J . A p p l . M e c h . 39, 1125 (1972).

5. J W. KANNEL, T r a n s . A S M E ( S er . F ) , J. L u b . T e c h . 96, 508 (1974).6. R. H. BENTALL and K. L. JOHNSON,In t . J . M ech . Sc i . 9, 389 (1967).

7 . M . K O S H Y a n d R . G O n A R , P r o c . 18th M a c h i n e T o o l Res. and Design Conf., I m p e r i a l C o l l e g e , L o n d o n

(Sept. 1977).8. G. LUNDBERG,F o r s c h u n g a u f D i e n G e b i e t e d e s l n g e n i e u r - w e s e n s 10, 201 (1939).9. J. DUNDURS,J. Elas t . 2, 109 (1972).

10. P. F. BYRD and M. D. FRIEDMAN, H a n d b o o k o f E l l i p t i c I n t e g r a l s f o r E n g i n e e r s a n d P h y s i c i s ts .

Springer-Verlag, Berlin (1954).

APPENDIX

I n f l u e n c e c o e n ic i e n t s f o r d i s p l a c e m e n t a n d c u r v a t u r e d u e t o a u n i t o f p r e s s u r e

The influence coefficient for a com plete pre ssure unit is split into coefficients for two half-units asfollows:

For displacements

where

and

Similarly, for c urvature

where

and

A ~ = ~ i d ( j - - I , i ) + ~ p l d ( j , J + I )

' f ~ J ( 1 ~ - ~ / ~ - ' { K ( / ~ ) - E ( / ~ ) } d s'p i d ( j - l , j ) = .r~j_, -- ) ( A I )

( A 2 )

A ~ j = d ~ [ ( j - l , j ) + ~ b : f j , j + I )

~ : ( j - l , j ) = t g (~ j ) J J~,_,( A 3 )

d : : ( j , j + I ) = t g - ' ~ j ) J J ~

These integrals present no difficulty except when i = j , w h e r e u p o n k = 1 a n d K ( k ~ ) b e c o m e s i n f i n it e . T Ohandle the singularity when i = k we write equations (AI) and .(A2) in the for m

@ i d ( j - - I , j ) = d p l a ( j , j + I ) = 1 1 + 1 2 + 1 3 (A 5 )

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P r e s s u r e b e t w e e n e l a s ti c b o d i e s h a v i n g a s l e n d e r a r e a o f c o n t a c t 2 4 7

w h e r e

I3 = - f ,"_, ( I - ~ ) k , - l E ( / q ) d , ' ,

a n d w h e r e ~ i s a s u i t a b l y c h o s e n s m a l l n u m b e r . T h e s i n g u l a r i t y i s n o w c o n f i n e d t o / 2 , w h i c h c a n b e

r e - w r i t t e n :

12 = g ( ( " ) f t' { 1 k ' g ( , , ) ~ d k- - ~ -- -- -~ - j K ( k ) ~ ( A 6 )

w h er e k' = (1 - k2) 1/2 an d k~ = {1 + (~/g(~i))2} t2.P r o v i d e d ¢ i s s m a l l , k . ~ ! , a n d I2 i s a p p r o x i m a t e l y g i v e n b y

i _ = g ( 6 , ) f ' K ( k ) g 2(~ i) f l K ( k ) A b

w h e r e k s = ~ (1 + k . ) . B y u s i n g T a b l e s o f E l l i p t i c F u n c t i o n s b y B y r d a n d F r i e d m a n I° w e o b t a i n

== g(~/~) 2 g K (k )12 '- '~ " ~ K ( I /' V ' 2 ).. . , o - ~ ` ' ' 1 ' '~ ¢¢ ')~ ,' - - C t ~ - ~ ' ) ~ - ~ " ~ " ' ~ " ' . ~ ' ,. ) + ¢1 ' , : )' ¢ ' , . ) ~ ] . - . ~ , ¢ A 7 )

E q u a t i o n ( A T ) c o n t a i n s n o s i n g u l a r i t y .

T h e s a m e p r o c e d u r e i s f o l l o w e d t o e v a l u a t e t h e i n f lu e n c e c o e f fi c ie n t o f c u r v a t u r e w h e n i = j. W e w r i te

cbi'(j - i ) = ~b[ (j , j + 1) = /4 + I s + 16 (A8)

w h e r e

a n d

2 6 1 - ,

1 6 _ g - 2 ( ~ i ) { ~(1_~i-~'k~]k~E(k~)d~,.j ~ , _ , c ]

A g a i n I s c o n t a i n s t h e s i n g u la r it y . U s i n g t h e s a m e a p p r o x i m a t i o n a s b e f o r e I s m a y b e w r i t t e n

~" K( k) 1

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

( A g )