1984 Linkoping Roller Pressure

8/4/2019 1984 Linkoping Roller Pressure

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P r e s s u r e d i s tr i b u t io n in c r o w n e d r o l le r c o n t a c ts

B O T O R S T E N F E L TLink6ping Institute o f Technology, Departm ent o f Mechanical Engineering, Division o f SolidMechanics and Strength o f Materials, S-581 83 Link6ping, Sweden

B I L L Y F R E D R I K S S O NSAAB -SCAN IA AB , Aircraft Division, Stress Departm ent, S-581 88 Link6ping, SwedenThe fa t i gue l i f e o f a ro l l e r bea r ing i s heav i ly i n f luenced by the c rowning p ro f i l e o f t he ro l l e r s . Thepressure d i s t r i bu t ion fo r d i f f e ren t t y pes o f c rowning has been s tud ied . For so lving th i s t h ree -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 a n u m e r i c a l p r o c e d u r e f o r a n a l y s i s o f g e n e r a l e l a s t o - s t a t i c c o n t a c tp r o b l e m s h a s ' b e e n u s e d . T h e m e t h o d i s b a s e d o n a n i n c r e m e n t a l a n d i t e r a ti v e a l g o ri t h m a p p l i edto a se t o f l i nea r equa t ions e s t ab l i shed wi th f 'm i t e e l ement t echn ique . The con tac t su r faces a rea s s u m e d t o b e p e r f e c t l y s m o o t h , d r y a n d f r ic t io n l e s s. T h e p r e ss u r e d i s t ri b u t i o n b e t w e e n t h e b o d i e sh a s b e e n c o m p a r e d w i t h r e s u l ts o b t a i n e d f r o m o t h e r m e t h o d s . T h e i n f l u e n c e o n t h e p r e s su r e d i st r ib u -t i on by the f r ee bounda ry a t t he end o f t he f i n i t e cy l inde r s has a l so been inves t iga t ed . I t i s a l sos h o w n t h a t i t i s p o s si b le t o u s e t h e s a m e f 'm i te e l e m e n t m o d e l t o s t u d y d i f f e r e n t t y p e s o f c r o w n i n g ,t h u s m a k i n g i t e f f i ci e n t t o p e r f o r m p a r a m e t e r s u r v e y s. A m e t h o d o f o b t a i n in g r e q u i r e d o r ' o p t i m a l 'p re ssu re d i s t r i bu t ion i s sugges t ed .Ke y Words : p re ssu re d i s t r i bu t ion , ro l l e r bea r ings , c row ned ro l l e r s , c rown ing , p re ssu re con t ro l , op t ima lp re ssure

I N T R O D U C T I O NThe des ign o f cy l indr i ca l ro l l i ng de m en t s i n ro l l e r bea r ingsi s c r i t i c a l. Th e fa t i gue l i f e o f a bea r ing i s heav i ly i n f luencedb y t h e c r o w n i n g p r o f i l e o f th e r o l l e rs . T h e c o n t a c t p r e s s u red i s t r i b u t i o n a n d t h e l o c a t i o n o f s t r e s s c o n c e n t r a t i o n s a r en o n l i n e a f l y d e p e n d e n t o n t h e l o a d l ev e l. T h u s , t h e o p t i m a lc rowning p ro f i l e va r i e s wi th t he app l i ed load l eve l . Theg loba l s t i f fness o f a ro l l e r bea r ing i s a l so e f fec t ed by thec rowning p ro f i l e . S ince Her t z p re sen ted h i s c l a ss i ca l worko n c o n t a c t b e t w e e n e l as t ic b o d i e s g e o m e t r ic a l l y e x p r e s s e db y q u a d r a t ic f u n c t i o n s m a n y r e s e a r ch e r s h a v e p u b l i s h e dw o r k d o n e i n t h i s f i e l d . W h e n s t u d y i n g c r o w n e d r o l l e r sHe r t z ' so lu t ion i s no longe r accura t e and do es no t g ive t hep o s s i b il it y t o s t u d y t h e i m p o r t a n t a r e a at t h e e n d o f t h ero l l e r . The co n tac t a rea depa r t s f ro m the Her t z i an e l li pse .S e v e r a l a u t h o r s h a v e p u t a t t e n t i o n t o t h e n o n - H e r t z i a neases occur r ing in ro l l e r bea r ings . One o f t he f i r s t workspre sen ted in t h i s f i e ld was by Lundb e rg . 1 He so lved thei n t e g r a l e q u a t i o n u n d e r t h e a s s u m p t i o n s t h a t t h e c o n t a c tp re ssure d i s t r i bu t ion wi l l be cons t a n t a long the ro l l e r andac t ing on a r ec t angu la r con ta c t a rea . The s t i f fness was t akenf r o m t h e h a l f s p a e e . T h e m a i n r e s u l t w a s t h e w e U - k n o w nL u n d b e r g c r o w n i n g p r o f i l e .Equ ll .b r ium and c on t inu i ty equa t ions fo r t h ree -d imen-s iona l con tac t p rob lems cou ld be s t a t ed a s fo l lows:

f P(~, n ) d ~ d n - - F = 0 ( 1 )~2e

Received May 1983 . Written discussion closes May 1984.

an dw l ( x , y ) - w 2 (x ,y ) + z l ( x , y ) - - z2 ( x ,y ) = A on ~2e /Jl ( x , y ) - - w 2 ( x , y ) + z l ( x , y ) z 2 ( x , y ) > A ou t s ide ~2e(2)

wh ere F i s t he fo rce , p t he p re ssu re , wt and w2 the de fo r -m a t i o n o f b o d i e s 1 a n d 2 r e s p e ct i ve l y , z t a n d z 2 d e f i n e t h eb o u n d a r y o f t h e b o d i e s . A is th e g l o b a l a p p r o a c h o f t h e t w obod ie s and (x ,y ) ; (~ , *7) a re ca r t e s i an coord ina t e s i n t hecon tac t su r face ~2e .Man y re sea rche r s 1 -7 u t i l i se t he Bouss inesq fo rc e -d i sp l acement r e l a t i onsh ip a s a cons t i t u t i ve r e l a t i on fo r acy l inde r wi th a f i n i t e l eng th :1 - - v 2 / ' p ( ~ , ~ ) d ~ d r/w(x,y)= J (3)

~2O t h e r a u t h o r s , s '9 h a v e p e r f o r m e d v a r i o u s m o d i f i c a ti o n s o fthe Bouss inesq fo rce -d i sp l acement r e l a t i onsh ip .A n o t h e r a s s u m p t i o n m o s t c o m m o n l y u se d is t o a s s u m etha t t he p re ssu re d i s t r i bu t ion pe rpend icu la r t o t he d i rec t ionof t he ro l l e r is an e l li pse :

p(w, y) = p(x , 0) (1 - - (y2/b2(x)))l/2 (4 )S e v e r a l s o l u t i o n m e t h o d s , f o r i n s t a n c e m a t h e m a t i c a l p r o -g ramming t echn iques , have been used to so lve the i n t eg ra lequa t ion p rob lem . The Bouss inesq f l ex ib i l i t y r e l a t i on seemsto have caused i l l - cond i t i on ing o f t he l i nea r sys t em ofequa t ion s a r is ing . T he re i s a lso a s ingu la r i t y p rob lem whe nca lcu la t ing the i n f luence o f t he d i sp l acement i n a po in tf rom a fo rce i n t he same po in t . The p re ssure has t o be i n t e -g ra t ed o ve r a su r round ing a rea and the d i sp l acement i s t hen

0264-682X/84/010032-08 $02.003 2 EngineeringAnalysis, 1984, Vol. 1, No. 1 1984 CM L Publications


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P r e s s u r e d i s t r i b u t i o n i n c r o w n e d r o l le r c o n t a c t s : B . T o r s t e n f e l t a n d B . F r e d r i k s s o nobta ine d , s The as su mpt ion of us ing the Bouss inesq con-s t it u t iv e r e l a t io n i s a p p r o p r i a t e i n t h e c e n t r e o f t h e c o n t a c ta rea . I t w i l l , however , overes t imate the s t i f fnes s a t the endo f t h e r o l l e r a n d t h e m e t h o d f a i l s w h e n t h e p r o b l e mbecomes in te res t ing .Thi s s evere res t r i c t ion has been rem oved by , for ins t ance ,Reusner 9 and Kalker . s R eusner i s t rea t ing the ro l l e rs asro l l ers wi th f in i t e l ength . The s t res s bo un dar y condi t iona t the end of the ro l l e rs i s approximate ly fu l f i l l ed througha super impos ing t echnique . Ka lker present s a so lu t ionwhich seems to g ive good resu l t s . I t i s based on the bas icas sumpt ion tha t the contac t a rea mus t be s l ender . I t has ther e q u i r e m e n t s t h a t t h e c r o w n i n g ra d i u s o f t h e r o l l e r m u s t b esevera l t imes as l a rge as the loca l rad ius o f the ro l l e r andc u t - o f f o f t h e a r e a i s n o t a l l o w e d . T hi s m e a n s t h e n t h a t t h eme tho d fa il s a t the end of the ro l l e r where edge e f fec t s a reinteres t ing.I n t h e p r e se n t w o r k n o n e o f t h e a b o v e m e n t i o n e da s s u m p t io n s a r e m a d e . A c o m b i n e d i n c r e m e n t a l i t er a t iv ea l g o r it h m b a s e d o n t h e m i n i m i s at i o n o f t h e p o t e n t i a lenergy , w hich i s e s t ab l i shed by us ing the f in i t e e l ement d i s-p l a c e m e n t m e t h o d . T h e e d g e e f f e c ts c o u l d t h e n b e s tu d i e d .T h e s o l u t i o n i s a p p r o x i m a t e d u e t o t h e a p p r o x i m a t e f u lf il -m e n t o f t h e s t r es s b o u n d a r y c o n d i t i o n s b y t h e f i n i te e l e.m e n t d i s p la c e m e n t m e t h o d . T h e f i n i t e e l e m e n t m o d e l isa u t o m a t i c a l l y g e n e r a t e d a n d t h e g r i d s i z e m a y b e c h o s e narb i t ra r i ly in d i f fe rent pa r t s of the co ntac t a rea .

S O L U T I O N M E T H O DT h e m e t h o d u s e d i n t h i s w o r k i s p r e s e n t e d b y F r e d r i k ss o n 13and Tor s tenfe l t 14 and i t w i l l on ly b e br i e f ly rev iewed.The bas ic equi l ibr ium, cont inui ty and cons t i tu t ive e l as -t i c i ty re l a t ions for smal l s tra ins and d i sp lacements :

Oi l , /+ Xi = 0 (5)e l l = (ui ,/ + u i , i ) (6 )

e l i = h i ik lOkl (7 )should be so lved t ak ing sur face t rac t ion and d i sp lacemente q u a l i t y b o u n d a r y c o n d i t i o n s i n t o a c c o u n t . F o r c o n t a c tproblem s i t is fur the r requi red to fu l f i l t he ine qua l i tyc o n d i t i o n s d e s c r i b e d b e l o w .Bodies A and B a re in contac t a t a poin t i , t he contac tpressure is P i and the in i t ia l gap o r in te r fe renc e i s 8 i . Wethus requi re : A p i - B pi = 0 (8 )

A U - B U - 8 i = 0 (9)Ap < 0 (10 )

I n t h e g e n e r al c a se w i t h a n u n k n o w n c o n t a c t a r ea t h e s ol u -t i o n o f t h e p r o b l e m h a s t o b e d o n e i n a n i t e r a ti v e m a n n e r .T h e p r e s e n t m e t h o d i s b a s e d o n t h e f i n i t e e l e m e n tm e t h o d . F o r m i n g t h e t o t a l p o t e n t i a l e n e r g y I I f o r a p r o b -l e m w i t h o u t f r i c t i o n w e o b t a in :

I f f fl 2 ( o i i e i i ) d V - (Xiui) dV -- (qiui) dS ( 1 1 )v v SqU t il is in g t h e f i n i te e l e m e n t m e t h o d f o r c a lc u l a t io n o f t h e s ein tegra ls we a r r ive a t equa t io n .(12) in ma t r ix no ta t ions :

M i n ( I I ( u ) = u t K u - - P t u l A u - - 8 / > 0 } ( 1 2 )

T h i s m i n im i s a t io n o f t h e p o t e n t i a l e n e r g y u n d e r i n e q u a l i t yc o n s t r a i n t s h a s t o b e p e r f o r m e d i n a n i t e r a t i v e w a y . T h ef 'mi te e l ement d i sc re t i s a t ion i s such tha t each of the nodesi n t h e c a n d i d a t e c o n t a c t a r e a h a s a n e ig h b o u r i n t h e o t h e rb o d y . T h e s e n o d e p a i r s a r e t r e a t e d a s c o n t a c t n o d e p a i r st h r o u g h o u t t h e i t er a t iv e p r o c e d u r e . T h e i n e q u a l i t y c o n -s t ra in t s co r respond ing to a nod e pa i r ins ide the co ntac t a reaar is e as e qua l i ty an d impl ies tha t the s e t of l inea r equa t ion si s o v e r d e t e r m i n e d . A l i n e ar t r a n s f o r m a t i o n is p e r f o r m e d i ne a c h i t e r a t io n a i m e d a t r e d u c i n g t h e n u m b e r o f e q u a t i o n sa n d m a k i n g t h e s y s t e m d e t e r m i n e d i n a c e r t a i n i t e r a t i o n .I n t r o d u c i n g t h e t r a n s f o r m a t i o n m a t r i x M w e o b t a i n :

M ( i ) t K o M ( i ) u ( O = M(i)t (Po -- Ko~ (i)) (13 )Af te r each i t e ra t ion inac t ive cons ta in t s a re checked andnod e pa i r s inside the co ntac t a rea a re checked for t ens i l ef o r c e s . T h e t r a n s f o r m a t i o n m a t r i x M h a s a n a t t r a c t i v efea ture . I t does inc lude only ze ros and uni t i e s . Thi s fac tm a k e s i t p o s s i b le t o p e r f o r m t h i s t r a n s f o r m a t i o n f a s te rt h a n a g e n e r a l l i n e a r t r a n s f o r m a t i o n b y m a k i n g u s e o fboolean opera t ions . I t i s ra t iona l to u t i l i s e a s t andards t if fnes s m at r ix as semb ly rout ine .The sys tem of equa t ions so lved g ives the cons i s t en tn o d a l c o n t a c t f o r c e s w h i c h s h o u l d b e t r a n s f o r m e d t ocon tac t pres sure . Thi s is don e in the fo l lowing way. Con-s id e r t h e c o n s i s te n t c o n t a c t f o r c e v e c t o r F e f o r a c o n t a c tsur face e l ement . Us ing the sur face e l eme nt in te rp ola t ionf u n c t i o n N e w e c a n w r i t e :

= ( N t e p ( x , y ) d ~ ( 1 4 )e12 e

Using the s ame in te rpola t ion func t ion for the pres surep ( x , y ) we obta in :F e = [ f NteNed~2] p e = C e P e ( 1 5 )

F~eS u m m a t i o n o f a ll c o n t a c t e l e m e n t f o r c e s m e a n s a nas semblage accord ing "to sur face e l em ent top olo gy and weo b t a i n :

R = C p ( 1 6 )w h e r e

R = E R e , p = E p e a n d C = C eT h e m a t r i x C i s r e c o g n i se d b ei n g t h e ' m a s s m a t r i x ' o f t h esur face e l ement wi th uni t dens i ty . C i s pos i t ive de f in i t eand wel l -condi t ioned and the pres sure i s thus obta ined bys o l u t io n o f e q u a t i o n ( 1 6 ) . I t h a s s h o w n t o g i ve g o o dr e s u l t s w h e n u s i n g t h e ' l u m p e d ' m a s s m a t r i x i n e q u a t i o n( 1 5 ) w h i c h a l so m e a n s a v e r y f a s t so l u t i o n o f e q u a t i o n ( 1 6 ) .T h e m e t h o d p r e s e n t e d i n t h i s p a p e r is i m p l e m e n t e d int h e g e n e r a l p u r p o s e f i n i t e e l e m e n t c o m p u t e r r o u t i n el ibra ry ASKA . I s Thi s rout in e l ibra ry , working wi th a d a tabase, is a u seful a nd f le xible tool to ut i l ise as a bas ic to olf o r w r i t in g a d d i t io n a l c o m p u t e r r o u t i n e s . T h e p r o c e d u r e f o re l a s t o s t a t i c c o n t a c t p r o b l e m s s h a l l b e l o o k e d u p o n a s aspec ia l so lver in the t 'mi t e e l emen t sys tem . F or each loadi n c r e m e n t i t e ra t i o n s a r e p e r f o r m e d u n t i l c o n v e r g e n ce i so b t a i n e d . T h e s o l u t i o n i n e a c h i t e r a t i o n i s o b t a in e d b y t h edi rec t e l imina t ion equa t ion so lver . An i t e ra t ive so lver forso lv ing the l inea r sys tem of equa t ions would b e an e f fec t ivee x t e n s i o n o f t h e p r o c e d u r e . A f t e r t h e f ir s t s o lu t i o n t h e r ewi l l be a good s t a r t vec tor ava i l ab le for th e fur th e r it e ra -t ions .

E n g i n e e r i n g A n a l y s i s , 1 9 8 4 , Vot 1 , No . 1 3 3


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Pressure distribution in crow ned roller contacts: B. To rstenfe lt and B. FredrikssonTHE FINITE ELEM ENT MODELThe con tac t p rob lem s analysed in th i s s tudy i s def ined byFig. 1 . A crow ned ro l ler w i th a cer ta in leng th and r ad ius i sp ressed be twee n two o ther iden t ica l non-crow ned ro l lerswi th para lle l cen t r e l ines . Because o f t r ip le- symm etry one-e igh th o f the ar r angement i s model led wi th f in i te e lements( see F ig . 2 ) . The su r f ace on the sym me try p lane betw eenthe tw o b ig cy l indr ica l ro l le rs th roug h the crow ned ro lleri s fo rced to r emain f la t when the load P i s app l ied . Themo del i s genera ted sem i-au tomat ica l ly . The r ef ine me nt o ft h e m es h i n t h e v i c i n it y o f t h e co n t ac t a r ea i s o b t a i n ed in ar epet i t ive manner . Three d i f f er en t ' un i t cubes ' a r e used fo rthis purpose (see Fig. 3) . These 'unit cubes ' are pi led intoeach o ther in th ree levels in the smal l ro l le r and in fourlevels in the b ig ro l ler . A computer rou t ine has beendeveloped w i th th e purpose o f genera t ing a l l r equ i r edinpu t da ta fo r d i f f er en t meshes des igned under cer ta inr eq u i r em en t s . T h e m es h p a t t e r n i n t h e can d i d a t e co n t ac tsu r f ace can be r a ther arb i t r ar i ly chosen wi th r ef inementsin d i f f er en t a r eas o f in ter es t . The d i scre t i sa t ion in thero i ling d i r ec t ion o f the con ta c t a r ea are f ixed to four o r t ennode pair s f rom the cen t r e l ine to the b order l ine o f thecand ida te con tac t zone ( see F ig . 4 ) . So l id e lemen ts w i th al inear or bi l inear shape function are used in the discret isa-t i o n o f t h e b o d i es . A h i d d en l i n e m es h o f t h e s m a l l ro l le ri s shown in F ig . 5 . The to ta l numbe r o f degrees o f f r eedomhas var ied between 400 0-8000 in the d i f f er en t app l ica t ionsan d t h e n u m b er o f can d i d a t e co n t ac t f r eed o m s i n v o lv ed i nthe i t e r a t ive p rocess var ied betw een 400-600 approx im ate ly .

Figure 2.

z

Idealisat ion o f the geom etry

Figure 3. The 'uni t cube s '

Figure 1. Three aligned rollers

RESULTS FRO M EXAMPLES STUDIEDThe f ' t r s t geomet ry inves t iga ted was tha t g iven by theLund berg crowning p ro f i le :

l - - v 2 P 2h (x ) = - - I n ( 1 7 )IrE Lo 1 - - ( x / Lo ) 2

34 Engineering Analysis, 1984, Vo l. 1, No. 1


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I I I. = t I I~ I

tqgure 4.

I

I

Figure 5.

Pressure distribution in crowned roller contacts: B. Torstenfelt and B. Fredriksson

V-4 -i I I I

I

I

I I I I I I I I f I I I I I I I m l l l ! I I I I[ / l I i ] [ i i i i i i i i i i l l l l [ i [ [ ii F I I Ii i I I I[ I , , I I I

A part o f the surface of a small roller mesh

w i t h c = 1 0 o r 2 0 w e t h e n o b t a i n f o r g eo m e t r i e s s t u d i ed i nt h i s w o r k a p p r o x i m a t e l y :X XL--~ ( c = 1 0 ) < 0 . 9 8 7 4 o r - - ( c = 2 0 ) < 0 . 9 9 7 4 6 ( 2 1 )L o

If then the load ing i s such tha t we wi l l have contac t to thev e r y e n d ( x = L o ) o f t h e r o l l e r w e o b s e r v e d t h a t t h e r e w i llb e a p r e s s u re p e a k a t x = L o . N o d e s w i t h x - c o o r d i n a t e sgrea te r tha n the l imi t s g iven by eq ua t io n (20) wi l l g ive ve rylow or even ze ro pres sure . I t i s sugges ted to add an in te r -p o l a t i o n f o r t h e g a p b e t w e e n t h e l i m i t f o r e q u a t i o n ( 1 7 )g i ve n b y e q u a t i o n ( 2 0 ) a n d t h e v a l u e g iv e n b y e q u a t i o n ( 1 9 )a t x = L o .I n t h e f i n i t e e l e m e n t c a l c u l a t i o n t h e t w o b o d i e s w e r et rea ted as cy l inders wi th a f in i t e l ength . The ra t io be tw eent h e l e n g t h o f t h e s m a l le r c r o w n e d r o l le r a n d t h e l e n g t h o fthe b igger cy l inder s imula t ing the inner r ing was chosen inaccordance wi th prac t i ca l bear ing appl i ca t ions . The resu l tof the ca lcu la t ions i s presented in F ig . 6 and compared tothe Lundberg so lu t ion . In th i s d iagram the pres sure prof i l e sa re p lo t t e d b oth a long the ro l l e r and in the ro l ling d i rec -t i o n . T h e f i n it e e l e m e n t s o l u t i o n s h o w s a s e x p e c t e d aw e a k e r b e h a v i o u r a t th e e n d o f th e c o n t a c t a r ea c o m p a r e dt o t h e L u n d b e r g s o l u t i o n . B y e q u i l i b ri u m r ea s o n s t h e m a x i -m u m p r e s su r e h a s t o i n c re a s e to w a r d s t h e c e n t r e o f t h ec o n t a c t a r e a . F o r x - v a l u e s c l o s e t o z e r o a n d a t 0 . 7 L oosc i l l a t ions a re observed . Thi s has been s tudied in de ta i land i t has been confh 'med tha t th i s i s due to the t rans i t ionf rom a f 'me to a coarse me sh in the f in i t e e l emen t mo de las s een in F ig . 4 . I t w i l l no t in f luence the resu l t i f a meanva lue i s used . Th e var i a t ion of the s em i -contac t wid th isa l so p l o t t e d a n d n o r m a l i s e d t o t h e c o n t a c t w i d t h c a l cu l a te da c c o r d i n g t o t h e L u n d b e r g t h e o r y . T h i s c u r v e i s e x tr a - a n di n t e r p o l a t e d t h r o u g h t h e c a n d i d a t e c o n t a c t n o d e p a i rs . T h eresul t shows which pa i r s a re in contac t and which a re not .P res sure va lues a t nodes in contac t a re used to ex t rapola teto ze ro w hich g ives the con tac t wid th a t the x-va lue . Thec o n t a c t b o u n d a r y c u r v e i s t h e n i n t e r p o l a t e d i n t h e x,y-

Hidden line finite element idealisation

T h i s f o r m u l a w a s o b t a i n e d f r o m t h e a s s u m p t i o n o f a c o n -s tan t pres sure prof i l e a long the ro l l e r . Thi s means tha t thec o n t a c t a r e a i s a p e r f e c t r e c t a n g l e w i t h a c o n t a c t w i d t hd e t e r m i n e d b y t h e t w o - d i m e n s i o n a l H e r tz i a n r e l a t io nb e t w e e n t h e m a x i m u m c o n t a c t p r e ss u r e P o a n d t h e c o n t a c tw i d t h b e .be = CoPe ( 1 8 )

E q u a t i o n ( 1 7 ) i s n o t v a l id f o r x = L o . L u n d b e r g t h e n g iv es :h ( L o ) = E "~ o 1 . 19 3 2 + I n ( 1 9 )

T h e r e i s t h e n a n a r e a a t t h e e n d o f t h e r o l l e r w h e r e t h ec r o w n i n g i s n o t p r o p e r l y d e f m e d . I n t r o d u c i n g a n a r b i t r a r yc o n s t a n t c t h e l i m i t o f a p p l ic a b i li t y o f e q u a t i o n 0 7 ) m a yb e w r i t te n :( 2 0 )Lo \Lol

/ yFigure 6. The pressureLundberg crowning distribution obtained with

Engineering Analysis, 1984, Vol. 1, No. 1 3 5


5/8

Pressure distribution in crow ned roller contacts: B. Tor stenfelt and B. Fredrikssonp l a n e . T h i s h a s s h o w n t o b e a n a c c u r a te e n o u g h m e t h o dbecause in a rea s wi th h igh g rad ien t s we use a f ine mesh andin o the r a rea s i t is a ccep tab l e t o use a coa r se r mesh . F igure7 s h o w s t h e c a n d i d a t e c o n t a c t n o d e p a i r p a t t e r n a n d w h i c hnod e pa i r s a re i ns ide o r ou t s ide t he co n tac t a rea .I n o r d e r t o i n v e st ig a t e t h e i n f lu e n c e o f t h e f r e e b o u n -d a r y a t t h e e n d o f t h e r o l le r s o n t h e p r e s s u r e d is t r i b u t io na n u m b e r o f c a l c u la t io n s w i t h d i f f e r e n t l e n g t h s o f t h ero l l e r s has been pe r fo rmed . In F ig . 8 t he r e su l t s a re p lo t t edfor an add i t i ona l t h ree ca ses . The re su l t s ha s been norma l -i sed in t he same manne r a s i n F ig . 6 . In t he ca se wi th tworo l l e r s o f equa l l eng th t he p re ssure p ro f i l e and the semi -c o n t a c t l e n g t h s h o w a d e c r e a s in g t e n d e n c y a t t h e e n d o f t h er o l le r . T h e b o u n d a r y c o n d i t i o n s a t t h e e n d o f th e r o l l e rs h o w a n i m p o r t a n t i n f l u e n c e a t l e a st f o r v a l u e s LolL1 > 0 .8 .A s e c o n d s t u d y o n t h e i n f l u e n c e o f t h e f r e e b o u n d a r i e sh a s b e e n p e r f o r m e d . I n t h i s c a s e t h e s m a l l er c r o w n e d r o l le rh a s b e e n e x t e n d e d t o t h e s a m e l e n g t h a s t h e b i g r o l le r .N o t h i n g e ls e h a s b e e n c h a n g e d c o m p a r e d t o t h e f t rs t c a lc u -

h/bo| . ~

0 0 0 0 0 0 0 0 0 0 0 0 O 0 0 0 0 I ~ I I D

,

I , o n , . .n O C O 0 | | t ~, , i0 I .x/Io

l~gure 7. The candidate contact no de pair pat tern andcontact w id th for Lundberg crowning

h/bo/yFigure 8 . The p ressure d is~ ib i t ion ob ta ined w i t hLundberg crowning and wi th d i f f eren t l eng th o f the b igroller

P/Po [ z

/

/

!

1,

Figure 9. The pressure distribution obtain ed with Lu nd-berg crown ing and with an ex tende d small roller

P/Po, 2 _ : .

.5

" " !I "

k FEM1 ''" 31 L ]/~n~R 32 ,,i, , . . . . - ]

.2 5

I

O . 2 . 4 . 6 . 8x/LoFigure 10. A pressure distribution comparison along thesemi-contact length

l a t io n , F i g . 6 . S t i f fn e s s h a s b e e n a d d e d t o t h e e n d o f t h ec rowned ro l l e r . The re su l t s a re shown in F ig . 9 . Th i s com-p a r i s o n d e m o n s t r a t e s a b e t t e r a g r e e m e n t w i t h L u n d b e r g .T h e a s s u m p t i o n s m a d e b y L u n d b e r g a r e h e r e b e t t e r f u l f il le db e c a u s e o f t h e m a t e r ia l a d d e d .A c o m p a r i s o n w i t h t h e R e u s n e r m e t h o d 9 h a s a ls o b e e np e r f o r m e d . R e u s n e r d o e s t a k e t h e f r e e e n d b o u n d a r y c o n -d i t i o n s a p p r o x i m a t e l y i n t o a c c o u n t . T h e c r o w n i n g p r o f i l eused fo r t h i s compar i son i s de f ined in F ig . 10 . I t cou ld bes e e n : t h a t t h e a g r e e m e n t is r a t h e r g o o d b u t t h e f i n it e

3 6 Engineering Analysis, 1984, Vo l. 1, No. 1


6/8

Pressure d is tr ibu t ion in cro wn ed roller contac ts: B. To rstenfe l t and B. Fredrikssone l e m e n t r e s u l t i n d i c a t e s a w e a k e r m o d e l t h a n t h e R e u s n e rmo de l . As we a re using the d i sp lacem ent f in i t e e l eme ntf o r m u l a t i o n i t c o u l d w i t h r a t h e r g o o d c o n f i d e n c e b e s t a te dt h a t e v e n th o u g h R e u s n e r ta k e s t h e e n d b o u n d a r y c o n -di t ion in to a ccou nt h i s mo de l i s s t il l a li t t l e b i t t oo s t i f f a tt h e v e r y e n d . T h i s i n d i c a te s t h e n t h a t t h e R e u s n e r p r e s su r ep e a k m i g h t b e n o n - c o n s e rv a t iv e .

GEOMETRY INFLUENCE O N THE PRESSUREDISTRIBUTIONN u m e r i c al m e t h o d s l i k e th e f i n it e e l e m e n t m e t h o d a r e t o o l swi th which i t i s theore t i ca l ly poss ib le to pe r form ana lys i so n t h is c o m p l e x t h r e e - d i m e n s io n a l p r o b le m s . I t i s, h o w e v e r ,ve ry expens ive to pe r form three -d imens iona l ans lys i s . I twould then be very benef i c i a l i f i t were poss ib le to use thesame mo de l to s tud y d i f fe ren t c rownings . The c lass ica lH e r t z p r o b l e m a n d t h e L u n d b e r g s o l u t i o n to c r o w n e d r o l le rcontac t s both as sume inf in i t e bodies wi th ' a r t i f i c i a l ' curvedb o u n d a r i e s a p p r o a c h i n g e a c h o t h e r . T h e F E M m o d e l sa p p r o x i m a t e t h e r e a l g e o m e t r y . I f w e h av e s o m e d e v ia t io n sf r o m t h e e x a c t g e o m e t r y t h is is , o f c o u r s e , w i t h t h e s a m ereasons as for o the r me thod s , acceptable . I t is no t neces sa ryt o c h a n g e t h e f 'm i te e l e m e n t m o d e l w h e n t h e c r o w n i n g isc h a n g e d . I t i s e n o u g h t o i n p u t t h e n e w g a p d u e t o t h e n e wc r o w n i n g . T h i s c o u l d b e c o n c l u d e d e i t h e r b y s t u d y i n g as imple one-d imens iona l example or by s tudying the bas icf in i t e e l ement equa t ions .Assume a on e-d imens iona l ba r w i th c ros s -sec t iona l a reaA , Y o u n g ' s m o d u l u s E a n d w i t h l e n g t h L . W h e n th e b a r iscompres sed a d i s t ance A a contac t force :

E AP = A (22)Lwill arise.A s s u m e n o w t h a t t h e b a r h a s b e e n s h o r t e n e d c a us in g a ni n i t i a l g a p e L b e f o r e t h e c o m p r e s s i o n . T h e c o n t a c t f o r c ew o u l d t h e n b e :

E A*' = (Z~ - eL ) (2 3)L ( 1 - - e )o r

E AP = - - ( A - - e L ) ( 1 + e + 0 ( e 2 ) ) ( 2 4 )Li f e < 1 w e o b t a i n a p p r o x i m a t e l y :

E AP - - - (ZX - - eL) (25)Lwhich i s the s ame resu l t a s obta ined when not changingg e o m e t r y b u t o n l y i n t r o d u c i n g t h e n e w g ap .In a genera l f in i t e e l ement appl i ca t ion th i s i s s een f romthe fo l lowing. Assume a presc r ibed d i sp lacement 8 / . In t ro-duc ing the e l ement s t i f fnes s mat r ix :

f BIoBidVv~

t h e f o l lo w i n g f o r c e v e c t o r P i arises:( 2 6 )

Ve V

I n t r o d u c e n o w a p r e s c r i b e d d i s p l a c e m e n t 8 - t - h / w h i c h isc a u s e d b y e h a h g i n g t h e g e o m e t r y o f t h e s o l i d s t r u c t u r es t u d i e d . T h e v o l u m e i s c h an g e d f r o m V to V + A V . W et h e n o b t a i n t h e n e w l o a d v e c t o r :P , + A P / = ( f ( B t D B j ) d V ) ( S j + h j ) (2 7)

V + A V

A s l o n g a s A V , ~ V w e m a y w r i t e :P i + A P , ~ ( f B ~ D B] d V ) ( S j - t - h j ) ( 2 8 )

V

a n d w e c a n t h u s u s e t h e s a m e f i n i t e d e m e n t m o d e l a sb e f o r e b y c h a n g i ng t h e g e o m e t r y .F o r L u n d b e r g a n d c i r c u l a r c r o w n i n g w e p e r f o r m e dc a l c u la t i o n s b o t h w i t h m o d e l l i n g t h e c o r r e c t g e o m e t r y a n dw i t h o u t t a k i n g c h a n g e i n g e o m e t r y d u e t o th e c r o w n i n gin to account . The resu l t i s shown in F ig . 11 . C i rcu la rc rown ing i s de f ine d by F ig . 12 . The d i f fe rences a re ve rysmal l . I t i s on ly a t the ve ry end the devia t ion exceeds 1%.Fo r the c i rcu la r c rowning i t i s a l so a ve ry h igh pres sure -peaka t t h e e n d . T h e p r e s s ur e a lo n g t h e a s y m m e t r y l in e ( y = 0 )i s shown in F ig . 12 . I t i s thus fu l ly acceptable to pe r form aparam ete r survey us ing the s ame FE-m ode l . I f t he in te res t -ing resu l t i s to s tudy the pres sure th i s is e spec ia l ly e f f ic i en tb e c a u s e y o u o n l y n e e d to d o t h e r e d u c t i o n t o c o n t a c td e g r ee s o f f r e e d o m o n c e .

CONTACT PRESSURE DISTRIBUTION CONTRO LIn a des ign s i tua t ion i t would be va luable to be ab le todes ign for a pres sure d i s t r ibu t ion tha t i s op t imal in somes e n s e , f o r i n s t a n c e t o m i n i m i s e t h e m a x i m u m c o n t a c t

. 0 0 5

, , . , 0g l.

O _

L u n d b e r g- - - - c i r c u l a rc r o w n i n g

' Ii q m l m m l u - - ~ ;

. 2 . 4 . 6

II

.

Ii/j J _ ~

F

J.8

X / L oFigure 11. The rela tive pressure deviat ion f i )r two cases .Po stands f o r va lues f ro m exac t geom etry and p f i~r va luesf r o m t h e a p p r o x i m a t i o n

Engineering Analysis, 1984, I io l . 1 , No. 1 3 7


7/8

Pressure distribution in crow ne d roller contacts: B. Tor stenfelt and B. Fredriksson1 .e

P / P ,L4 T1 .2

. 6 - - - - Z

. 4 . . . . L o

. 2 t ! . . . .

0 . 2 5 . 5 .7 5

iII p ~J

Figure l2. The pressure dis tr ibut ion forcrowning

1.x / L o

a circular

pressure . Opt imisa t ion p rob lems in e l a s t i c con tac t s haveb e e n f o r m u l a t e d a n d s o l v e d b y K i k u c h i a n d T a y l o r . x~ Asa n a p p l i c a t io n t h e m a x i m u m p r e s su r e i n a t w o - d i m e n s i o n a lc o n t a c t p r o b l e m i s m i n i m i s e d .W e w i ll h e r e p r e s e n t a n o t h e r , s i m p l e r m e t h o d o f o b t a in -ing p re ssure d i s t r ibu t ion con t ro l . Assume tha t a p re ssured i s t r ibu t ion Preq(X,y) i s r equ i red on ~2 . Apply ing th i sp r e s s u re o n b o t h t h e e l as t ic b o d i es i n c o n t a c t w e o b t a i n t h ed i sp lacement s Wl(X, y ) a n d w~(x , y ) re spec t ive ly . I f we n owc r o w n t h e b o d i e s i n s u c h a w a y t h a t t h e t o t a l g ap h ( x , y )is :

h ( x , y ) = w - z (x , y ) - - w 2 ( x , y ) - - A ( 2 9 )a n d a p p l y t h e e x t e r n a l l o a d :

P = f Preq(X,y) d[2 (30).1

~q ewe wi ll ob ta in the p re ssure d i s t r ibu t ion Preq(X,y) a s a r e su l to f t h e c o n t a c t s o l u t io n . T h i s re q u i r es , h o w e v e r , t h a t t h econ tac t su r face [2e i s know n a p r /o r / . I f t h i s i s no t t h e casew e c o u l d a p p l y t h e p r o c e d u r e it e r a ti v e l y . T h e p r o c e d u r ep r o p o s e d c o u l d b e c o n c l u d e d a n d f o r m u l a t e d f r o m t h er e c i p r o c i t y t h e o r e m o f e l a s ti c i ty p r o b l e m s .We have app l i ed th i s approach to a two- and a th ree -d i m e n s i o n a l p r o b l e m . I n t h e t w o - d i m e n s i o n a l c a s e w eappl ied a ser ies o f press ure prot~fles as show n in Fig. 13 to at 'mi t e punch and an in f in i t e ha l fp l ane . The d i sp l acement s

1 . 2P / P o

wx and w2 re spec t ive ly were then in t rod uced a s a gap ,e q u a t i o n ( 2 9 ) , a n d t h e c o n t a c t p r o b l e m s o l v ed w i t h t h ec o n t a c t a l g o r it h m . W e t h e n o b t a i n e d w i t h e x c e l l e n t a gr ee -m e n t t h e r e q u e s t e d p r e s s ur e d i s t r ib u t i o n a s s h o w n i nFig. 13.I n t h e t h r e e - d i m e n s i o n a l e x a m p l e w e u s e d t h e r o l l e rc o n t a c t p r o b l e m d e s c r ib e d p r e v io u s l y . W e a p p l ie d o n b o t h

t h e r o ll e rs t h e c o n s i s t e n t n o d a l f o r c e s o b t a i n e d f r o m t h eLun dberg p re ssure d i s t r ibu t ion , i . e . e ll i p t ic in y -d i rec t iona n d c o n s t a n t i n x - d i r e c ti o n . T h e c o n t a c t a r ea w a s r e c t-a n g u la r . A p p l y i n g t h e s e f o r c e s t o b o t h t h e r o l l er s w eobta ined the d i sp l acement s Wl and w2, t hese va lues werethe n g iv ing the gap on th e cen t re l ine o f t he ro l le r s . I t is no tposs ib l e t o i n t rod uce the com ple t e gap Wl (X, y ) - w2 (x , y )b e c a u s e t h e r e i s t h e r e q u i r e m e n t o n t h e r o l l e r s t o b ec y l i n d e r s . T h e g a p w a s t h e n c h a n g e d a c c o r d i n g l y . T h ec o n t a c t p r o b l e m w a s t h e n s o l v e d . W e o b t a i n e d t h e p r e s su r ed i s t r ibu t ion on the cen t re l ine shown in F ig . 14 , which i s

[ ]

P r s q / O ' o Co n t u ~ s o lu t io nf o r 2 D p r o b le m

~l 111 0+o

I i N0 . 2 5 . 5 . 7 5 1x / a o

Figure 13. Re quir ed and calculated pressures in a two-dimensional p unch pro blem

1.' @O * j s | I s * * * ~

. 8 . . . . . . j' P r e q / P o - - Co n t a c t s o lu t io o f o r3 0 r o l l e r p r o b le m

: I. 6 ] i l

0 . 2 5 . 5 3 5 1 .x / t oFigure 14. Re quir ed a nd calculated pressures along thesemi-contact length in the three-dimensional roller prob lem

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P r e s s u r e d i s t r i b u t i o n i n c r o w n e d r o l le r c o n t a c t s : B . T o r s t e n f e l t a n d B . F r e d r i k s s o n

c o n s i d e r e d t o b e a g o o d f u l f i l m e n t o f t h e r e q u e s t . T h eo s c i l la t i o n s s h o w n a r e s t il l d u e t o t h e t r a n s i t i o n f r o mc o a r s e t o f 'm e m e s h e s .

C O N C L U ~ O N SD i f f e r e n t t y p e s o f c r o w n i n g w e r e s t u d i e d a n d c o m p a r e d t oo t h e r s o l u t io n s . T h e g e n e r a l p u r p o s e c o n t a c t a l g o r i t h m w a ss h o w n t o b e e f f i c i e n t t o u s e f o r t h i s s t u d y . I t w a s s h o w nt h a t t o s t u d y p r e s s u r e d i s t r i b u t i o n i t i s i m p o r t a n t t op r o p e r l y t a k e e n d b o u n d a r y c o n d i t io n s in t o a c c o u n t . T h eR e u s n e r s o l u t i o n s e e m s t o g i ve a t o o s t i f f m o d e l a t t h e e n da n d t h u s a n o n - c o n s e r v a t i v e p r e s s u r e i n t h e c e n t r e .

I t w a s s h o w n t h a t i t i s p o s s i b l e t o u s e t h e s a m e f 'm i t ee l e m e n t m o d e l t o s t u d y d i f f e r e n t t y p e s o f c r o w n i n g . It w asa l so s h o w n t h a t i t is p o s s i b l e t o c o n t r o l t h e p r e s s u r ed i s t r i b u t i o n b y a p p l y i n g f o r c e s c o m p a t i b l e w i t h t h e p r e s -s u r e d i s t r i b u t i o n r e q u i r e d o n b o t h t h e c o n t a c t i n g b o d i e ss e p a r a t e l y . T h e d i f f e r e n c e o f t h e d i s p l a c e m e n t s th u so b t a i n e d i s u s e d a s t h e i n i t i a l g a p .A C K N O W L E D G E M E N TT h i s w o r k w a s p a r t l y s u p p o r t e d b y S K F E n g i n e e r i n gR e s e a r c h C e n t r e , E R C , i n t h e N e t h e r l a n d s . M r J a n d e M u la t E R C i s g r e a t l y a c k n o w l e d g e d f o r h is s u p p o r t i n t h i ss t u d y .

REFERENCES1 Lu ndb erg, G. Elastisc he Bertihtung zweier Halbr~iume, For-schungs auf dem Gebiete des Ingenieurwesens 1939 , l 0 (5 ) ,2012 Singh, K . P. and Paul , B. St ress conc entra t ion in crownedrol lers , J . Engn. Ind. 1974, 3, 5

3 Oh, K . P. and Trachm an, E. G . A nume rical proced ure fordes igning prof i led rol l ing e lements , ASME, J. LubricationTechn. 1976, 00, 5474 Nay ak, L. and Johnson , K . L. Pressure betwe en e las t ic bodieshaving a s lender area of contact and arbi t rary prof i les , Int . ZMech. Sci. 1979, 21 , 23 75 N ayak , L . S ingu l a r i t y cons ide ra t ion in num er i ca l so lu t ion o fcontact s t ress problems, ASM E, J. Lubrication Techn. 1981,00, 16 Ha r tnet t , M. J . The analys is of contact s t resses in rol l inge l em en t bearings, ASM E, J. Lubrication Techn. 1979, 101, 1057 Ha r tnet t , M. J . and K annel , J . W. Con tact s tresses betw eenelas t ic cyl inders : a comprehensive theore t ica l and exper i -m en ta l app roach , ASM E, J. Lubrication Techn. 1981, 103, 408 K a lke r , J . J . A n a sym pto t i c m e thod fo r t he ca l cu la t i on o f t hecon tact s t ress in rol ler bear ings , 1975. Presented in connect io nwith a seminar on S KF European Research Centre , Jutphaa s ,The N ether lands , 19759 Reu sner, H. Druckfl~chenbelastung und Oberfli ichenverschie-bung im W~lzkon takt yon Rotat io nsk6 rpern , SKR Kngel lager-f ab r iken G m bH , 197710 Ahm adi , N . , Kcer , L. M. and Mtt ra , T. Non-her tz ian contacts t ress analysis for an e las t ic hal f sp ac e- norm al and s lid ingcon tact , Int . J . Solids Struct . 1983, 19 (4), 35711 Malmberg, P. Con tact s t resses in rol ler beat ing rol lers - com-

pa r i son be tw een t he K a lke r aysm pto t i c m e thod and som eother theor ies , SKR Engineer ing & Research Centre B .V. ,1977, NL76D002.12 K ikueh i , N . and Tay lo r , J . E . Shape op t im im t ion fo r un il a t e r a le l a s t i c con t ac t p rob l em s , Num . Meth. Coupled Prob l . , Proc.Int . Conf . held a t Univ. Col lege , Swansea, 7-11 September ,1981,430--44113 Fredr ikss on, B. On e las tos ta t ic problem s wi th f r ic t ion. D iss .No. 6 , Link~ping Stua ies in Science and Techn ology, Link6p ingIns t i tu te o f Technology, Link~ping, Sweden, 197614 Tors tenfe l t , B. Con tact problem s wi th f r ic t ion in generalpu rpose f i n i t e e l em en t com pu te r p rog ram s , Comp. & Struct .1983, 16 (1-4 ) , 48715 ASK A Par t I - Linear s ta t ic analys is , user ' s reference manu al ,A S K A U M 202 , l n s t i t u t t i l t S t a t i k und D ynam ik , U n ive rs i tyo f S tu t t ga r t , S tu t t ga r t , 1979

E n g i n e e r i n g A n a l y s i s , 1 9 8 4 , V o l . 1 , N o . 1 3 9

1984 Linkoping Roller Pressure

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