(HAS A-TM-816 90) I??!ROPUCTIOU !l'O BALL 381- 18392 - ,- .

BEARIHGS (3ASA) 86 p BC IrOS/BP A01 .CSCL 131

Intrdductidn to Ball Bearings

Bernard J. Hamrock Lewis RewPrrh Center Cleveland, Ohio and Duncan Dowson The Uniwrsiy of Leedr L&&, England

*

February 1981

NASA

https://ntrs.nasa.gov/search.jsp?R=19810009867 2020-03-10T04:07:29+00:00Z

The purpose of a b a l l bea r ing is t o provide a r e l a t i v e post-

t i o n i n g and r o t a t i o n a l freedom whi le t r a n s m i t t i n g a load between

two s t r u c t u r e s , usua l ly a s h a f t and a housing. For h igh r o t a t i o n a l

speeds - as found, f o r example, i n gyroscope b a l l hea r ings - t h e

purposz can be expanded t o inc lude r o t a t i o n a l freedom with prac-

t i c a l l y no wear i n t h e bearing. This cond i t ion can be achieved by

s e p a r a t i q g t h e bear ing p a r t s wi th a coherent f i l m of f l u i d known

as an elastohydrodynamic f i lm. Denhard (1966) pointed ou t t h a t t h e

elastohydrodynamic f i l m can be maintained n o t only when t h e bea r ing

c a r r i e s t h e load on a s h a f t , but a l s o when t h e bea r ing is preloaded

t o p o s i t i o n t h e s h a f t t o w i t h i n micro- o r nano-inch accuracy and

s t a b i l i t y . T h i s chapter provides background information on b a l l

bear ings; elastohydrodynamic l u b r i c a t i o n theory is discussecl i n

Chapter 7.

B a l l bea r ings a r e a s t d i n many kinds of machines and d~:vices

wi th r o t a t i n g p a r t s , as pointed out i n t h e pr~2ceding chapter . The

des igner is o f t e n c ~ n f r o n t e d w i t h d e c i s i o n s on whether a b a l l o r

f l u i d - f i l m bear ing should be used i n a p a r t i c u l a r a p p l i c a t i o n . The

fo l l ewing ~ h a r a c t e r i s t i c s make b a l l bea r fngs more d e s i r a b l e than

f lu id-f i l m bea r ings In many s i t u a t i o n s :

(1) Low r t a r t i n g and good operat ing f r i c t i o n

(2) The a b i l i t y to aupport combined r a d i a l and t h r u r t loads

(3) l e s s s e n s i t i v i t y to interrupt ion8 in lubr ica t ion

(4) No self-excited i n s t a b i l i t i e s

(5) Good low-temperature s t a r t i n g

Within reasonable l i m i t s , changes i n load, speed, and operating

temperature have but l i t t l e e f f e c t on the sa t f s f ac to ry performance

of b a l l bearings.

The following cha rac t e r i s t i c s make b a l l bearings l e s s desir-

ab le than f luid-f i lm bearings: - (1) F i n i t e fat igue l i f e subject t o v ide f luc tua t ions

(2) Larger space required i n the r ad i a l d i r ec t ion

(3) Low damping capacity

(4) Higher noise l e v e l

(5) More severe alignment requirements

(6) Higher cos t

In the l i g h t of these cha rac t e r i s t i c s , piston engines normally use

f luid-f i lm bearfngs, whereas j e t engines use b a l l bearings almost

exclusively. Each type of tear ing has its par t icu lar s t rong points ,

and care should be taken i n choosing the most appropriate type of

bearing for a given appl icat ion. Useful guidance on the important

issue of bearing se lec t ion has been presented by the Engineering

Sciences Data Unit (ESDU 1965, 1967).

The Engineering Sciences Data Unit documents (1965, 1967) pro-

vide an exce l len t guide t o the se lec t ion of the type of journal o r

th rus t bearing that is most l i k e l y t o give the required performance

when c o n r i d e r i n g t h e load, ~ p e e d , and g e m t r y of t h e bear ing. The

types of b e a r i n g conr idered were

(1) Rubbing bear ings , where t h e two bear ing s u r f a c e s rub

t o g e t h e r (e.g., un lubr ica ted bushings made from m a t e r i a l s based

on nylon, p o l y t e t r a f luoroethylene (PTFE) , and carbon)

(2) Oil-impregnated porous metal bea r ings , where a porous

metal bushing is impregnated w i t h l u b r i c a n t and thus g i v e s a s e l f -

l u b r i c a t i n g e f f e c t (as i n s i n t e r e d - i r o n and s in tered-bronze

bea r ings )

(3) Rol l ing bear ings , where r e l a t i v e motion is f a c i l i t a t e d

by i n t e r p o s i n g r o l l i n g elements between s t a t i o n a r y and moving

components ( a s i n b a l l , r o l l e r , and needle bear ings)

(4 ) Hydrodynamic f i l m bear ings , where the s u r f a c e s i n r e l a -

t i v e motion a r e kept a p a r t by p ressures genera ted hydrodynamically

i n t h e l u b r i c a n t f i l a

Figure 2.1, reproduced from t h e Engineering Sciences Data

Unit p u b l i c a t i o n (1965), gives a guide t o the t y p i c d l load t h a t can

be c a r r i e d a t va r ious speeds, f o r a nominal l i f e of 10,000 hours

a t room temperature, by journa l bear ings of va r ious types on s h a f t s

of t h e d iameters quoted. The heavy curves i n d i c a t e t h e p r e f e r r e d

type of j o u r n a l bear ing f o r a p a r t i c u l a r load , speed, and diameter

acd t h u s d i v i d e t h e graph i n t o d i s t i n c t i v e regions . The app l i ed

load and speed a r e usua l ly known, and t h i s enables a pre l iminary

assessment t o be made of t h e type cf journa l b e a r i n e most l i k e l y

t o be s u i t a b l e f o r a p a r t i c u l a r a p p l i c a t i o n .

In m n y carer t h e rhaf t d iamete r will a l r e a d y have been de-

termined by o t h e r conr ider8t ion8, and F Q u r e 2.1 can be ured t o

f i n d the type of journal bear ing t h a t w i l l g i v e adequate load

c a p a c i t y a t t h e r e q u i r e d rpeed.

These curves a r e bmed on good engineer ing p r a c t i c e and com-

m r i c a l l y a v a i l a b l e p a r t s . Higher loads and speeds o r smal le r

r h a f t d iameters a r e p o s s i b l e wi th excep t iona l ly h igh engineer ing

r t andards o r r p e c i a l l y produced mate r i a l s . Except f o r r o l l i n 8

bea r ings t h e curves a r e drawn f o r bea r ings with a width equal t o

t h e G l a e e t e r . A medium mineral o i l l u b r i c a n t is assumed f o r t h e

hydrodynamic bear ings .

S i m i l a r l y Figure 2.2 reproduced from t h e Engineering Sciences

Data Unit pub l i ca t ion (1967). g ives a guide t o t h e t y p i c a l maximum

load t h a t can be c a r r i e d a t va r ious speeds, f o r a nominal l i f e of

10,000 hours a t room temperature, by t h r u s t bear ings of va r ious

types on s h a f t s of t h e d iameters quoted. The heavy curves aga in

i n d i c a t e t h e p re fe r red type of bea r ing f o r a p a r t i c u l a r load,

speed, and diameter and thus d i v i d e t h e graph i n t o major regions .

The remainder of t h i s chap te r is concerned wi th the def i n i -

t i o n of d i f f e i e n t types of b a l l bea r ings and t h e i r geometry and

kinematics. Bearing m a t e r i a l s a n d manufacturing processes a r e

a l s o d i scussed i n t h i s chap te r , a s we l l a s s e p a r a t o r s (sometimes

c a l l e d cages o r r e t a i n e r s ) . The f a c t o r t h a t remains constant i n

t h e s e important in t roduc to ry s e c t i o n s is t h a t i t is assumed, f o r

the purposes of a n a l y s i s , t h a t t h e bea r ing c a r r i e s no load. The

cons ide ra t ion of loaded bear ings s t a r t s i n Chapter 3, but t h e

arterial covered in t h e p r e r e n t chap te r i a v i t a l In laying t h e

foundations f o r tha remainder of t h t book.

2.1 Types of Ball Bearings

The essential p a r t s of a b a l l bear ing - t h e i n n e r and o u t e r

r i n g , t h e b a l l s , and t h e s e p a r a t o r - a r e shown in F igure 2.3.

The inner r i n g is mounted on a s h a f t and h a s a groove i n which the

b a l l s r i d e . The o u t e r r i n g is usua l ly t h e s t a t i o n a r y p a r t of t h e

bea r ing and a l s o c o n t a i n s a groove t o guide and suppor t t h e b a l l s .

The s e p a r a t o r prevents con tac t between t h e b a l l s and t h u s reduces

f r i c t i o n , wear, and n o i s e from the regions where s e v e r e s l i d i n g

cond i t ions would occur. I n a few a p p l i c a t i o n s where opera t ing

c o n d i t i o n s a r e mild, t h e r i n g s and s e p a r a t o r can be omit ted and

loose b a l l s in te rposed between t h e s h a f t and housing. Th i s type

of b e a r i n g is sometimes found i n b icvc les . The v a s t ma jo r i ty of

b a l l b e a r i n g s , however, a r e preassemljled u n i t s c o n s i s t i n g of four

elements. There a r e many types of bcar ings because of v a r i a t i o n s

i n the des ign of r i n g s and s e p a r a t o r s and i n the number of b a l l s .

They can be divided i n t o c l a s s e s according t o t h e i r func t ion :

those t h a t suppor t a r a d i a l load, those t h a t suppor t a t h r u s t load,

o r those t h a t support a combination of t h r u s t and r a d i a l loads .

The l a s t t y p e is termed "angular-contact bearings. "

Deep Groove

I n deep-groove b a l l b e a r i n g s , sometimes c a l l e d Conrad bear ing8

a f t e r t h e i r German d e s i g n e r , t h e r a c e s a r e approximate ly one - fou r th

re deep as t h e b a l l d iameter . A c r o s s s e c t i o n of t h e b a l l and

o u t e r ring o f a deep-groove b a l l be.rring i e shown i n P i ~ u r e 2.4.

Although deep-groove b a l l b e a r i n g s are des igned t o c a r r y a r a d i a l

l oad , t h e y perform w e l l under a combined r a d i a l and t h r u s t load .

For t h i s reason t h i s is t h e most widely used t y p e of b a l l bea r ing .

The assembly of a deep-groove h e a r i n g is shown i n ti f ou r - s t age

i l l u s t r a t i o n i n F igure 2.5. The i nne r r i n g i s d i s p l a c e d w i t h i n t h e

o u t e r r i n g , l e a v i n g a c rescent -shaped space ( F i g u r e 2.5 (a ) ) . Balls

are i n s e r t e d i n t o t h e r e s u l t a n t wide c r e s c e n t opening ( F i s u r e

2 . 5 ( b ) ) . B v t h i s method on ly s l i ~ h t l y morc th,ln h a l f t h e annu la r

space between t h e i n n e r and o u t e r p.~thw:iy can he f i l l e d w i t h trillls.

The i n n e r r i n g is then moved back tt> be c o n c e n t r i c w i t h thv o u t c r

r i n g , and t h e h.111~ a r e spaced e q u a l l v around t h e b e a r i n g (Fig-

u r e 2 . 5 ( c ) ) . A s e p a r a t o r is then i n s t a l l e d (F igu re .? .S(d)) t o

ma in ta in equa l spac ing of t h e b a l l s .

Filled Notch

The most e f f e c t i v e way of inc reas ing bearing load capac i ty i a

t o i n c r e a s e t h e b a l l complement. In a f i l l ed -no tch b a l l bear ing,

a s shown i n Figure 2.6, t h i s is accomplished by c u t t i i g a notch i n

t h e r a c e s o f a conventional bear ing, thereby permi t t ing t h e inse r -

t i o n of the b a l l s . The notch need not be a s deep a s t h e grooves;

o r i t may be c u t only i n t h e o u t e r r ace , i n which case i t w i l l be

deeper than t h e o u t e r groove. Once t h e b a l l a have been i n s e r t e d

i n t o t h e bear ing through t h e notch, t h e notch is f i l l e d by an

i n s e r t . The i n s e r t should be kept on t h e unloaded s i d e of the

bear ing, f o r i t weakens t h e race . The increased b a l l complement

makes p o s s i b l e a 20 t o 40 percent inc rease i n radia l - load capaci ty

over t h a t o f t h e normal deep-groove bear ing, depending on bear ing

s i z e . Thrust-load capac i ty is s a c r i f i c e d , however, because t h e

f i l l e d notch is cu t almost t o t h e bottom of the race groove. The

bear ing can s u s t a i n l e s s t5an one-third of the t h r u s t load of the

deep-groove bear ing before t h e ba l l - race contact e l l i p s e begins

t o con tac t t h e edge of t h e f i l l e d notch.

Other Rad ia l Bearings

There a r e a number of o the r forms of r a d i a l b a l l bear ings f o r

s p e c i f i c a p p l i c a t i o n s , such a s instrument bear ings , counterbored

bear ings , o r a i r c r a f t c o n t r o l bear ings . The d e t a i l s of t h e spe-

c i f i c dea ign of t h e r e eprc ia l -app l i ca t ion bearinga can be found i n

moat bea r ing company cata logs .

Furthermore t h e r e a r e a p p l i c a t i o n s where i t is advantageous

t o have a double- r a t h e r than a single-row b a l l bear ing. A double-

row bear ing has g r e a t e r load capac i ty than a single-row bear ing.

However, t h e app l i ed load t h a t can be c a r r i e d by a double-row b a l l

bea r ing is less than twice t h a t of its single-row coun te rpa r t be-

cause of t h e p r a c t i c a l d i f f i c u l t y of providing f o r both rows of

b a l l s t o s h a r e t h e load equa l ly . Proper s h a r i n g of t h e app l i ed

load between t h e rows is a func t ion of t h e geometr ica l accuracy

of t h e grooves. Otherwise t h e s e bea r ings behave l i k e a s i n g l e -

row b a l l bear ing.

2.1.2 Angular-Contact Bearings

U n i d i r e c t i o n a l Thrust

A u n i d i r e c t i o n a l t h r u s t bear ing, which is the most popular

form of angular-contact bear ing, is shown i n Figure 2.7. It is

made w i t h one shoulder of t h e o u t e r r i n g counterbored almost t o

t h e bottom of t h e r a c e groove. The o u t e r r i n g is u s u a l l y heated

dur ing assembly so t h a t t h e inner r i n g and t h e ball-and-separator

a s s a b l y can be snapped i n t o p lace . This type of bear ing is de-

s igned t o suppor t combined r a d i a l and u n i d i r e c t i o n a l t h r u s t loads .

The amount of t h r u s t load t h a t t h i s bear ing can suppor t is a func t ion

of t h e c o n t a c t angle . A bear ing wi th a l a r g e c o n t a c t ang le , l i k e

t h a t e h o m i n l i l u r o 2.7 (b) , can ruppor t heavier t h r u s t l o a d s than

the smallor-contact-anale bear ing ehom i n Figure 2.7(a). The con-

tact ang le do08 not u s u a l l y exceed 40' s i n c e high c o n t a c t anglvv

r e s u l t i n coneiderable b a l l rp inn ing and heat generat ion. A con-

t a c t a n a l e of zero corresponds t o a r a d i a l bearing; a ' c o n t a c t ang le

of 90' corresponds t o a t h r u s t bear ing.

Self-Aligning Bearings

Se l f -a l ign ing b a l l bear ings normally have two rows of b a l l s

t h a t r o l l i n a comnon s p h e r i c a l r ace i n t h e ou te r r i n g , as shown

in Figure 2.8. Because of t h i s des i4n t h e inner r i n g , wi th the

b a l l complement, can a l i g n i t s e l f f r e e l y around the a x i s of the

s h a f t . The se l f -a l ign ing f e a t u r e of t h i s bear ing a l lows i t t o

a d j u s t t o small degrees of misalignment between s h a f t and housing

without malfunct ' -.. Even when t h e s h a f t bends under load, t h e

bear ing w i l l fol low the d e f l e c t i o n of the s h a f t without r e s i s t a n c e .

Self-alignment a l s o c o n t r i b u t e s t o smoeth running by n e u t r a l i z i n g

the e f f e c t of the b a l l s wobbling i n the grooves. This bearing is

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

f i c u l t t o ob ta in exact p a r a l l e l i s m between t h e s h a f t and housing

bores.

Two c h a r a c t e r i s t i c s l i m i t t h i s bear ing 's usefulness : It can-

no t suppor t moments; and because t h e b a l l s do not conform wel l t o

t h e o u t e r r a c e , which is not grooved, i t has redu,:ed load-carrying

capac i ty . Th i s l a t t e r l i m i t a t i o n is compensated f a r somewhat by

t h a ura of a v a r y large ba l l complamerir, which minimixor t h e l o a d

c a r r i e d by each b a l l .

Duplex

Because most a p p l i c a t i o n s r e q u i r e t h rus t - load suppor t i n two

d i r e c t i o n s , u n i d i r e c t i o n a l - t h r u s t angu la r - con tac t b e a r i n g s a r e

u s u a l l y s u p p l i e d i n matched p a i r s f o r duplex mounting. Back-to-

back and face- to- face mounted bearing^ make i t p o s s i b l e t o c a r r y

t h r u s t l oad i n e i t h e r d i r e c t i o n . Such b e a r i n g s a r e menufac t t~red

as matched p a i r s such t h a t , when they a r e mated and t h e r aces made

f l u s h , each b e a r i n g is s l i g h t l y pre loaded. The purpose of a pre-

load is t o keep t h e b e a r i n g loaded a t a l l times, even when t h e

loads a r e low and t h e speeds a r e h igh . T h i s p r e l o a d i n g of t h e

components a g a i n s t each o t h e r s t i f f e n s t h e assembly i n t h e a x i a l

d i r e c t i o n and h e l p s t o p reven t b a l l sk idd ing w i t h a c c e l e r a t i o n a t

l i g h t l oads .

Duplex back-to-back mounts (F igure 2 .9 (a ) ) and face- to- face

mounts (F igu re 2 .9(b) ) a r e i d e n t i c a l i n a l l t h e i r c h a r a c t e r i s t i c s

except r e s i s t a n c e t o moments. For t h e back-to-back arrangement

t h e l i n e s of c o n t a c t i n t e r s e c t o u t s i d e t h e b e a r i n g envelope . T h i s

t ype of mount is t h e r e f o r e more r e s i s t a n t t o moments and s h a f t

bend in g . Tan lemmounted angu la r - con tac t bea r ings (F igu re 2 . 9 ( c ) ) a r e

used i n a p p l i c a t i o n s r e q u i r i n g very h igh , u n i d i r e c t i o n a l - t h r u s t -

l oad c a p a c i t y . With c a r e f u l manufac ture and i n s t a l l a t i o n a tandem

bear ing p a i r can have a t h w r t capac i ty about 80 percen t g r e a t e r

than t h e capac i ty of a s i n g l e bear ing. Duplex bearing, a r c rup-

p l i e d in m t c h e d pairs and they should never b e repara ted u r uoed

wi th s i n g l e bea r ings from o t h e r p a i r s .

S p l i t Inner Ring

A i r c r a f t t u r b i n e engines o p e r a t i n g i n t h e speed range of

p r e a t e r than 2 m i l l i o n dbN (bear ing bore i n m i l l i m e t e r s times s h a f t

speed i n revo lu t ions pe r minute) has brought about t h e development

of t h e s p l i t - i n n e r - r i n g bea r ing shown i n Figure 2.10. The inner- r ing

r a c e is normally ground wi th a shim between t h e inner - r ing halves .

The shim is removed before assembly, eo t h a t t h e r e s u l t i n g inner-

r i n g groove i a shaped l i k e a g o t h i c arch. This provides a much

lower r a t i o of a x i a l - t o - r a d i a l play than when hoth r a c e grooves a r e

c i r c u l a r . This is Important where thermal g r a d i e n t s cause an appre-

c i a b l e l o s s i n i n i t i a l r a d i a l play ns t h e bea r ing comes up t o tem-

peratrlre. S u f f i c i e n t r a d i a l p lay must be b u i l t i n t o t h e bear ing

without a l lowing excess ive a x i a l play. Excessive a x i a l p lay r e s u l t s

i n lower accuracy of s h a f t l o c a t i o n .

A disadvantage of t h e s p l i t - r i n g bea r ing is t h a t i t must op-

e r a t e a t f a i r l y high r a t i o s of t h r u s t t o r a d i a l load. .Any appre-

c i a b l e r a d i a l load causes simultaneous b a l l c o n t a c t on both ha lves

of the i n n e r r i n g , and t h i s r e s u l t s i n severe b a l l sk idding and

bea r ing d i s t r e s s .

The a i m p l e e t form of b a l l t h r u s t b e a r i n g i e rhown i n Fig-

u r e 2.11. I n t h i s t y p e o f b e a r i n g a s i n g l e row o f b a l l s , set i n

a s e p a r a t o r , runs i n two s imil~r grooves formed i n t h e s t a t i o n a r y

and r e v o l v i n g r i n g s . The r e v o l v i n g r i n g is f i x e d t o a s h a f t .

These grooves are u s u a l l y s h a l l o w e r t han t h e groove i n a deep-

groove r a d i a l b a l l bea r ing . Thrus. b a l l b e a r i n g s a r e n o t s u i t e d

t o c a r r y any a p p r e c i a b l e r a d i a l l o a d . However, r a d i a l l o a d i n g of

each b a l l a r i s e s from i t s own c e n t r i f u g a l f o r c e when i t is revolv-

i n g i n i ts t r a c k . The re fo re a t h igh speeds t h e s e f o r c e s can be

c o n s i d e r a b l e r e l a t i v e t o t h e t h r u s t Load, and t h i s may r e s t r i c t

t h e speed of o p e r a t i o n . Such hear in!;^ a r e a l s o speed l i m i t e d

because of s p i n n i n g i n t h e b a l l - r a c e c o n t a c t .

Because t h e y cannot s u p p o r t a p p r e c i a b l e r a d i a l l o a d s , t h r u s t

b a l l b e a r i n g s mst be used i n conjunc-t ion wi th r a d i a l b e a r i n g s .

Even a s l i g h t misalignmer., can cause d r a s t i c load c o n c e n t r a t i o n s ,

s o e i t h e r a h igh deg ree of squa renes ; must be main ta ined between

t h e s t a t i o n a r y r a c e s e a t and t h e s h a f t a x i s o r a s e l f - a l i g n i n g

s e a ; must be used . Fur thermore i t i s impor tan t i n t h i s t y p e of

b e a r i n g t h a t a l l b a l l s be v e r y a c c u r a t e l y manufac tured , bo th i n

roundness and d i a m e t e r , s i n c e any b a l l l a r g e r t h a n t h e o t h e r s

would c a r r y a d i s p r o p o r t i o n a t e f r a c t i o n of t h e t o t a l load du r ing

t h e whcle r e v o l u t i o n .

2.2 Ceoaatry of Ball Bearingl

The opera t ing c h a r a c t e r i s t i c s of a b a l l bea r ing depend g r e a t l y

on t h e d iamet ra l c lea rance of t h e bearing. This c lea rance v a r i e s

f o r t h e d i l f e r e n t types of bear ings discussed i n s e c t i o n 2.1. I n

t h e p resen t s e c t i o n t h e p r i n c i p a l geometrical r e l a t i o n s h i p qovern-

i n g t h e opera t ion of unloaded b a l l b ~ a r i n g s a r e developed. This

information w i l l be of v i t a l i n t e r e s t when such q u a n t i t i e s a s

s t r e s s e s , d e f l e c t i o n s , load capac i ty , and l i f e a r e considered i n

subsequent chapters . Bearings r a r e l y opera te i n t h e unloaded

s t a t e , but understanding of t h i s s e c t i o n is v i t a l t o t h e apprecia-

t i o n of t h e remaining chapters .

2.2.1 Conformal and Nonconfornal Surfaces

Before t h e va r ious geometrical r e l a t i o n s h i p s f o r b a l l bear ings

a r e developed, i t is d e s i r a b l e t o de f ine conformal and nonconformal

su r faces . Hydrodynamic l u b r i c a t i o n is g e ~ e r a l l y charac te r ized by

su r faces t h a t a r e conformal. That is, t h e sur faces f i t snugly i n t o

each o t h e r with a high degree of geometrical conformity, s o t h a t

t h e load is c a r r i e d over a r e l a t i v e l y l a r g e a r e a . Furthermore t h e

load-carrying sur face a rea remains e s s e n t i a l l y cons tanr whi le t h e

load is Fncrzased. Fluid-f i lm j o u n a l and s l i d e r bear ings e x h i b i t

conformal su r faces . In journa l bearings t h e r a d i a l c lea rance

between t h e s h a f t and bear1r.g is t y p i c a l l y one-thousandth of the

rha f t diameter; i n r l i d e r bearingr t he inc l ina t ion of the bearing

surface t o the runner is typica l ly one par t i n a thousand.

Nany machine elementr have contacting surfaces t ha t do not

conform t o each other very well. The f u l l burden of the load must

then be car r ied by a very smell contact area. In general the con-

t a c t a reas between nonconformal surfaces enlarge considerably with

increasing load but a r e still small compared with the contact areas

between conform1 surfaces. Some examples of these nonconformal

sur faces a r e mat itrg gear tee th , cams and followers, and rol l ing-

element bear lngs . The load per u n i t a rea i n conformal bearings is r e l a t ive ly

2 2 low, t yp ica l ly only 1 NN/m and seldom over 7 UN/m . By cont ras t ,

the load per u n i t area In nonconfonnal contacts , such a s those

2 t ha t e x i s t i n b a l l bearings, w i l l generally exceed 700 HN/m , even

a t modest applied loads. These high pressures r e s u l t i n e l a s t i c

deformation of t he bearing mater ia l s such tha t the e l l i p t i c a l con-

t a c t areas a r e formed f o r o i l f i lm gs!neration and load support.

The s igni f icance of the high contact pressures is tha t thcy r e s u l t

in a considerable increase i n f l u i d viscosi ty . Inasmuch a s vis-

cos i ty is a measure of a f l u i d ' s res i s tance t o flow, t h i s increase

grea t ly enhances the lub r i can t ' s a b i l i t y t o support load without

being squeezed out of t he contact zone.

The nonconfomal surfaces of a b a l l bearing are shown in

Figure 2.12. The b a l l and race conft)m t o some degree in the sec-

t i on shorn i n Figure 2.12(a), but tht? sec t iona l view shown in

F i g u r e 2.12(b) c l e a r l y e x h i b i t e l i t t l e confoxmity. Thie book is

concerned o n l y w i t h nonconfoxmal c o n t a c t s .

2.2.2 P i t c h Diameter, Clearance , and Race Conformity

The c r o s s s e c t i o n through a r a d i a l , s ingle- row b a l l b e a r i n g

shown i n F i g u r e 2.13 d e p i c t s t h e r a d i a l c l e a r a n c e and v a r i o u s

d i ame te r s . The p i t c h d i ame te r de is t h e mean o f t h e inne r - and

o u t e r - r i n g r a c e c o n t a c t d i a m e t e r s and IF g iven by

Also , from F i g u r e 2.13, t h e d i a m e r r a l c l e a r a n c e denoted by Pd

can be w r i t t e n as

Pd = do - d l - 2d ( 2 . 2 )

Diametra l c l e a r a n l e may t h e r e f o r e be thought o f a s t he maximum d i s -

t a n c e t h a t one r a c e can move d i a m e t r . ~ l l y w i t h r e s p e c t t o t h e o t h e r

when no measurable f o r c e is a p p l i e d .and both r a c e s l i e i n t h e same

p lane . Although d i a m e t r a l c l e a r a n c e is g e n e r a l l y used i n connec-

t i o n w i t h s ingle- row r a d i a l bea r ings . equa t ion (2.2) is a l s o ap-

p l i c a b l e t o angu la r - con tac t bea r ings .

Race conformi ty is a measure of t h e geomet r i ca l conformi ty

of t h e r a c e and t h e b a l l i n a p l a n e p a s s i n g through t h e bea r ing

a x i s , which is a l i n e pas s ing t h r o u g l ~ t h e c e n t e r of t h e bea r ing

and perpendicular t o its plane, and t ransverse t o the race. Fig-

ure 2.14 i r a cross sec t ion of a b a l l bearing showing race con-

formity, expressed as

For pe r f ac t conformity, where the radius of t he race o r groove is

equal t o the b a l l radius, f is equal t o 112. The c lose r the race

conforms t o the ba l l , the g rea t e r the f r i c t i o n a l hea t within the con-

t ac t . On the other hand, openrace curvature and reduced geometrical

conformity, whichreduce f r i c t i o n , a l s o i n c r e a s e the maximum contact

s t r e s s e s and consequently reduce the bearing f a t igue l i f e . F o r t h i s

reason most b a l l bearings made today have race conformity r a t i o s i n

the range 0 . 5 1 1 f 2 0.56, with f - 0.52 being the most comon

value. Therace conformity r a t i o fo r the outer race is usually made

s l i g h t l y l a rge r than tha t f o r the inner race t o compensate for t he

c loser conformity i n t he plane of the bearing between the outer

race and b a l l than between the inner race and b a l l . Th i s tends t o

equal ize the contact s t r e s s e s a t the inner and outer race contacts.

The d i f fe rence i n race conformity r a t i o does not normally exceed0.02.

2.2.3 Contact Angle, Endplay, and Slloulder Height

Contact Angle

Radial bearings have some a x i a l play s ince they a r e generally

designed t o have a diametral clearance, a s shown i n Figure 2.lS(a).

This impl ies a f r e c c o n t a c t ang le d i f f e r e n t from zero. Angular-

c o n t a c t bear ings a r e e p e c i f i c a l l y designed t o o p e r a t e under t h r u s t

loads. T h e c l e a r a n c o b u i l t i n t o t h e unloaded b e a r t r g , a l o n g w i t h t h e

race conformity r a t i o , determines t h e bearing f ree-contact angle.

Figure 2.15(b) shows a r a d i a l b a l l bear ing wi th contact due t o t h e

a x i a l s h i f t of t h e inner anrl o u t e r r i n g s when no measurable fo rce

is app l ied .

However, be fore t h e f r e e contact is discussed, i t is import-

a n t t o d e f i n e t h e d i s t a n c e between t h e c e n t e r s of curva tu re of t h e

two r a c e s i n l i n e wi th t h e c e n t e r of the b a l l i n both Figures

2.15(a) and (b) . This d i s t a n c e - denoted by x i n Figure 2.15(a)

*ad by D i n Figure 2.15(b) - depends on race r a d i i and b a l l

diameter. Denoting q u a n t i t i e s r e f e r r e d t o t h e inner and ou te r

r aces by s u b s c r i p t s i and o , r e s p e c t i v e l y , we s e e from Fig-

u r e s 2.15(a) and (b) t h a t

0;

and

From these equat ions we can w r i t e t h a t

Thir d i r t a n c e , rhown in Figure 2.1S(b), w i l l be u r e f u l i n d e f i n i n g

t h e c o n t a c t angle .

By us ing equat ion (2.3), we can write equat ion (2.5) a 8

where

The q u a n t i t y B i n equat ion (2.8) is known as t h e t o t a l conformity

r a t i o and is a measure of t h e combined conformity of bo th t h e o u t e r

and i n n e r r a c e s t o t h e b a l l . It w i l l be seen t h a t c a l c u l a t i o n s of

bea r ing d e f l e c t i o n i n l a t e r chap te r s depend on the q u a n t i t y B.

The f ree-contact angle Bf (Figure 2.15(b)) is def ined as t h e

ang le mede by a l i n e through t h e p o i n t s of contact of t h e b a l l and

both races w i t h a plane perpendicular t o t h e bea r ing a x i s of ro ta -

t i o n when no measurable f o r c e is app l i ed . Note t h a t t h e c e n t e r s

of c u r v a t u r e of both the o u t e r and i n n e r r a c e s l i e on t h e l i n e

d e f i n i n g the f ree-contact angle . From Figure 2.15(b) t h e expres-

s i o n f o r t h e f ree-contact ang le can be w r i t t e n a s

D - Pd/2 cos B f = D

By us ing equa t ions (2.2) and (2.5) . we can w r i t e equat ion (2.9) a s

Equation (2.10) shows t h a t i f t h e s i z e o f t h e b a l l s is increased

and everything e l s e remain8 constant , t h e f ree-contact ang le is

decreased. S i m i l a r l y , i f t h e b a l l s i z e is decreased, t h e frce-

con tac t ang le is increased.

Prom equat ion (2.9) t h e d iamet ra l c lea rance Pd can be

writ :en as

This is an a l t e r n a t i v e way of d e f i n i n g the d iamet ra l c lea rance

from t h a t given in equat ion (2.2).

Endplay

Free endplay P, i s t h e maximm a x i a l movement of the inner

race with respec t t o t h e o u t e r , when both races a r e c o a x i a l l y

centered and no measurable fo rce is appl ied. Free endplay depends

on t o t a l curva tu re and contact ang le , a s shown i n Figure 2.15(b),

and can be w r i t t e n as

P = 2D s i n Bf e (2.12)

The v a r i a t i o n of f ree-contact ang le and endplay w i t h t h e r a t i o

P /2d is shown i n Figure 2.16 f o r four va lues of t o t a l conformity d

normally found i n single-row b a l l bear ings . El iminat ing Bf i n

equat ions (2.11) and (2.12) enables t h e fol lowing r e l a t i o n s h i p s

between free endplry and d h m e t r a l c lea rance to be es tab l i shed :

Shoulder Height

The shoulder h e i g h t found i n bal 1 bearings is i l l u s t r a t e d i n

Figure 2.17. Shoulder h e i g h t , o r r ace depth , is t h e depth of the

race groove measured from t h e shoulder t o t h e bottom of t h e groove

and is denoted by s In Figure 2.17. From t h i s f i g u r e t h e equa-

t i o n d e f i n i n g the shoulder height can be w r i t t e n as

s = r ( 1 - cos 0 ) (2.15)

The maximm poss ib le d iamet ra l c learance f o r complete r e t e n t i o n of

t h e ba l l - race con tac t wi th in t h e race under zero t h r u s t load is

given by

I n Chapter 3 t h e shoulder height requirement under a t h r u s t load

is discussed.

2.2.4 Curvature Sm and Diffc:ence

The undeformed geometry of c o n t a c t i n g a o l i d e i n a b a l l bear-

ing can be represented by two e l l i p s o i d s . The two s o l i d e with

d i f f e r e n t r a d i i of cu rva tu re i n a p a i r of p r i n c i p a l p lanes (x

and y) pass ing through t h e con tac t between t h e s o l i d s make con tac t

at a s i n g l e po in t under t h e cond i t ion of ze ro a p p l i e d load. Such a

cond i t ion is c a l l e d po in t contact and is shown i n Figure 2.18,

where t h e r a d i i of cu rva tu re a r e denoted by r's. It is assumed

throughout t h e book t h a t convex s u r f a c e s , a s shown i r Figure 2.18,

e x h i b i t p o s i t i v e curva tu re and concave s u r f a c e s , nega t ive curva-

t u r e . There fo re , i f t h e c e n t e r of cu rva tu re l i e s w i t h i n t h e s o l i d ,

t h e r a d i u s of cu rva tu re is p o s i t i v e ; i f the c e n t e r of cu rva tu re lies

o u t s i d e t h e s o l i d , t h e r a d i u s of cu rva tu re is nega t ive . It i e

important t o no te t h a t i f coord ina tes x and y a r e chosen such

t h a t

1 - 1 + - 1 1 > - + - 'ax 'bx ray 'by

coordinate x then determines the d i r e c t i o n of t h e semiminor a x i s

of t h e c o n t a c t a r e a when a load is app l i ed and y, t h e d i r e c t i o n

of the semimajor a x i s . I n t h i s chap te r we a r e concerned only with

unloaded c o n t a c t s , but the no ta t ion has been chosen such t h a t it

is a p p l i c a b l e f o r a l l t h e remaining chap te r s .

A cross s e c t i o n of a thrus t - loaded b a l l bea r ing opera t ing a t

a c o n t a c t angle p i a shown i n Figure 2.19. Equivalent r a d i i of

curvature f o r both inner- and outer-race contacts i n , and normal

t o , t h e d i r ec t ion of r o l l i n g can bc calculated from t h i s f igure.

The r a d i i of curvature f o r the b a l l - inner-race contact a r e

de - d cos B I

'bx 2 cos 6

fby ' ' f id ( 2 . 2 0 )

The r a d i i of curvature for the b a l l - ou: race contact a r e

Note t h a t the b a l l - inner-race and ' , a l l - outer-race contact

inequal i ty (2 .17 ) is s a t is£ led and tllat the s ign convent ion men-

tioned e a r l i e r has been adopted. Xn equations ( 2 .19 ) and ( 2 .22 )

0 is used instead of Bf s ince these equations a r e a l s o va l id

when a load is applied t o the contact.

The curvature sum and d i f fe rence , which a r e quan t i t i e s of

some importance i n the analysis of contact s t r e s se s and deforrna-

t i ons , a r e

where

Equation6 (2.26) and (2.27) e f f e c t i v e l y r e d e f i n e t h e problem of two

e l l i p s o i d a l s o l i d s approaching one another i n terms of an equiva-

l e n t e l l i p s o i d a l s o l i d of r a d i i Rx and Ry approaching a plane

i n t h e manner shown i n Figure 2.20. From the radius-of-curvature

express ions , t h e r a d i i Rx and. R f o r the con tac t example d i s - Y

cussed e a r l i e r can be w r i t t e n f o r the b a l l - inner- race contact a s

d(de - d cos 0) % ' (2 .28 )

2%

and f o r t h e b a l l - outer- race c o n t a c t as

d(de + d cos 0)

5( ' 2d

2.2.5 Geometrical Separa t ion of E l l i p s o i d a l So l id8

The geometr ica l eepara t ion (Sax + Sbx) o r (Say + Sby) betveen

two e l l i p s o i d a l s o l i d e is t h u s made equ iva len t t o t h a t between a

s i n g l e e l l i p s o i d a l s o l i d and a plane i n t h e manner uhbm i n

Figure 2.20. The geometr ica l requirement is simply t h a t a t any

va lue of x and y (Figure 2.20(a)) t h e geometr ica l separa t ion

between t h e two e l l i p s o i d s must be t h e same a s t h e s e p a r a t i o n

between t h e equ iva len t e l l i p s o i d and a p lane a t t h e same va lues

of x and y (Figure 2.20(b)) . From Figure 2.20(a) t h e follow-

ing can be w r i t t e n :

But 2rax .> Sax, s o equat ion (2.33) becomes

This is t h e well-known parabo l i c approximation t o t h e c i r c u l a r

s e c t ion of the s o l i d acd is v a l i d a s long a s the s e p a r a t i o n is

much s m a l l e r than t h e rad ius of curvature . S i m i l a r express ions

f o r Say, Sbx, and Sby c&n be w r i t t e n , and t h e express ion f o r

t h e t o t a l s e p a r a t i o n of an e l l i p s o i d a l s o l i d and a plane

(Figure 2.20(b)) can thus he w r i t t e n a s

9 .)

Dowron and HQginron (1966) have rhown i n a rimilar way that

t h e geomet r i ca l r e p a r a t i o n betweentvocylinderscanbe adequate ly

described by an equ iva len t c y l i n d e r near a p lane w i t h t h e geome-

t r i c a l s e p a r a t i o n

The d i f f e r e n c e between equa t ions (2 .35) and (2.36) i e t h a t the

r a d i i of c u r v a t u r e in t h e y d i r e c t i o n r and r - .! in- *Y b :

f i n i t e f o r t h e case of two c y l i n d e r s and thereby imply 1 . 1 1 ~ ~

is zero.

2.3 Kinematics

The motion of the b a l l s , r i n g s , arrd s e p a r a t o r of a b a l l bear-

ing is important t o t h e friction, temperature r i s e , l i f e , and

l u b r i c a t i o n of the bearing. In t h i s s e c t i o n i t is assumed t h s t

t h e r e is no s l i p between t h e b a l l s and races dur ing r o t a t i o n of

the b a l l bea r ing , and a s i n t h e r e s t of the chap te r i t is a l s o

assumed t h a t t h e r e a r e no d i s t o r t ions. The i n f l u e n c e of high

speeds, where c e n t r i f u g a l e f f e c t s cause e l a s t i c d i s t o r t i o n s and

a divergence of the inner- and outer- race con tac t ang les from

t h e i r zero-speed value , is not considered a t t h i s s t age . I f

t h e s e assumptions were t o be considered in t h e subsequent theory,

a complex a n a l y s i s and computer program would be requ i red t o

o b t a i n a n approximate s o l u t i o n . Useful r e l a t i o n s can, however,

be de r ived w i t h i n theoe a r r u m ~ t i o n s , which a r e r e l a t i v e l y easy t o

app ly and which can be ured f o r the bulk of bea r ing a p p l i c a t i o n s ,

where modest speeds p r e v a i l ,

Attempts t o develop a parameter t h a t would be an accura te

gauaa of t h e b u r i n g opera t ing l i m i t have met with l i m i t e d mucceas.

The c r i t e r i o n most f r equen t ly used t o desc r ibe the speed capa-

b i l i t i e a o f a ball bear in8 is t h e dbN va lue (bore diameter i n

millimeters times r o t a t i v e speed i n revo lu t ions per minute). I t

should only be used a s a rough gu ide l ine f o r determining the

l l m i t i n g e ~ e e + of a bear ing because i t does no t t a k e proper ac-

count of t h e s i z e and c e n t r i f u g a l e f f e c t s . The v a r i a t i o n of l i m i t -

ing speed N wi th bore diameter db based on p resen t p r a c t i c e an t

6 6 f o r l i m i t i n g v a l u e s of dbN of 2x10 and 3x10 is shown i n Figure

2.21 f o r b a l l bear ings of varying bore diameter, frqm Bamberger,

e t al . (1971).

The s u r f a c e v e l o c i t i e s of the s o l i d n e n t e r i n g t h e conjunct ion

between t h e b a l l and rat. a r e c a l c u l a t e d f o r t h e genera l case i n

which both the inner and o u t e r races r o t a t e , even though i n most

p r a c t i c a l s i t u a t i o n s only one would be r o t a t i n g . T h e s impl i f i ed

s o l u t i o n can t h e r e f o r e be r e a d i l y deduced from the genera l r e s u l t s .

The var ious r o l l i n g speeds and dimensions t o be found i n a b a l l

bea r ing a r e shown i n F i g u ~ e 2.22. Frcm t h i s f i g u r e and t h e assump-

t i o c s mentioned e a r l i e r , t h e v e l o c i t i e s a t the inner- and ou te r -

r a c e c o n t a c t s , a s w e l l a s t h e v e l o c i t y a t t h e b a l l c e n t e r , can be

vr1.t t e n r e s p e c t i v e l y a s

de - d coe 0 - 2 w i

de + d cos 0 u = 0 2 '"0

Knowing these ve loc i t i e s , we can write two equations t h a t def ine

the sur face v e l o c i t i e s of the b a l l a t the ball-race contact,

namely,

Inner : d cos 0 de - d cos 0 - de

U~ = 2 '"B = 2 "i - 2 "'c

Outer:

d cos 6 d e de + d cos 6 U~

L

2 wg = 2 wc - 2 '"0

Solving f o r % and wc i n these two equations gives

1 te - O S " de + d cos B

"B ' d cos 0 "'i - 2 ( 2 . 4 2 )

1 Id. - d cos B uc =

The ve loc i t i e s of the sur faces enter ing the bal l - race contact

can be expressed by l e t t i n g the coordinate system r o t a t e about the

bearing center with veloci ty wC. This f i xes the bal l - race geom-

e t ry r e l a t i v e t o the observer. The anguiar v e l o c i t i e s of the

inner and o u t e r race8 m d t h e b a l l cen te r a f ter t h e t ranefonnat ion

are

O f r - "'1 ' '"c (2.44)

Subscr ip t r r e f e r s t o t h e r o t a t i n g coordinate system. Using equa-

t i o n s (2.42) and (2.43) we can w r i t e equat ions (2.44). (2.45). and

(2.46) as

de + d cos 6 "'if = (uJi - uo)

2%

d, - d cos 0 "'or

- 2%

(u0 - wi) (2.48)

2 2 2 d, - d cos B

U ~ r - 2d d, cos 0 (wi - w0) (2.49)

The s u r f a c e v e l o c i t i e s of the s o l i d s e n t e r i n g the b a l l - inner-race con tac t (where subscr ip t a represen t s t h e inner race

and s u b s c r i p t b t h e b a l l ) are then

d, - d cos 2 2 2

d, - d cos 0 "a i ' - 2 ' - 4de

(wi - do) (2.50)

d cos 8 3 2 2 d: - d cas 8

Since uai - u b i , t h e c o n d i t i o n f o r p u r e r o l l i n g i r s a t i s f i e d .

Likewise , f o r t h e b a l l - o u t e r - r a c e c o n t a c t

Note t h a t t h e c o n d i t i o n f o r p u r e r o l l i n g is a g a i n s a t i s f i e d a t t h e

b a l l - o u t e r - r a c e c o n t a c t .

2.4 Haterials and Manufac tur ing P r o c e s s e s

The good performance of b a l l b e a r i n g s is as dependent on t h e

p rope r c h o i c e and h igh q u a l i t y o f t h e i r m a t e r i a l s a s i t is on t h e i r

d e s i g n and p r e c i s e manufac tur ing . n ~ e o p e r a t i n g c o n d i t i o n s t o

which b a l l b e a r i n g s a r e exposed c a l l f o r t h e u s e o f m a t e r i a l s cap-

able o f w i t h s t a n d i n g h igh compress ive s t r e s s e s ove r m i l l i o n s of

stress c y c l e s w i thou t s i g n i f i c a n t wear. The m a t e r i a l must be ha rd

enough t o p r e v e n t e x c e s s i v e p l a s t i c de fo rma t ion under t h e s e h igh

compress ive s t r e s s e s and y e t n o t be s o b r i t t l e a s t o cause t h e

b e a r i n g t o f a i l when a shock load is a p p l i e d . In a d d i t i o n t o

t h e s e h i g h compressive s t r e s s e s , b e a r i w s must w i th s t and sub-

s u r f a c e s h e a r s t r e s s e s f r e q u e n t l y i n e x c e s s of 1.4 C N / m 2

(200,uOO l b f / i n 2 ) . Because t h e s t r e s s at P g iven p o i n t may

reach h i g h v a l u e s and t h e n be reduced t o z e r o at l e a s t once dur -

i n g each r e v o l u t i o n of t h e s h a f t , i t is neces sa ry t o u s e me ta l s

w i t h a h i g h f a t i g u e s t r e n g t h .

Before t h i s century t h e developmet-: of the b a l l bea r ing was

g r e a t l y r e ta rded by t h e l a c k of s u i t a b l e mate r i a l s . T h i s si:.ua-

t i o n was somewhat a l l e v i a t e d wi th t h e advent of caee-hardened

o t e e l s . However, u s i n g ouch materials makes i t d i f f i c u l t t o pro-

v i d e a c a s e of g r e a t homogeneity. A s i n g l e s o f t s p o t on a b a l l

due t o a s l i g h t de f i c iency of absorbed carbon is enough t o s i g -

n i f i c a n t l y reduce t h e l i f e of an otherwise well-made bearing. To

combat t h i s shortcoming, t h e through-hardening process was devel-

oped. I n t h i s process t h e hardness is achieved by co ld working

o r by heat t r e a t i n g , r a t h e r than by a s u r f a c e modi f i ca t ion such

a s c a r b u r i z i n g o r n i t r i d i n g .

The prime b a l l bear ing m a t e r i a l s i n c e about 1920 has been

SAE 52100 ( i n Europe e s s e n t i a l l y the same m a t e r i a l was c a l l e d

EN 31 and is now designated 534A99 o r 535A99), and even today

probably 90 percent of a l l b a l l bear ings a r e made from t h i s

m a t e r i a l . Tho 52100 m a t e r i a l is a through-hardened, high-carbon

chromium s t e e l t h a t a l s o con ta ins smal l amounts of manganese,

s i l i c o n , n i c k e l , copper, and molybdenum.

With t h e advent of machines l i k e t h e gas- turbine j e t engine ,

inc reased thermal demands have been placed on t h e m a t e r i a l s used

i n b a l l bea r ings . Th i s has r e s u l t e d i n t,le a d d i t i o n o f such

e lements a s molybdenum, tungs ten , chromium, and vanadiom t o pro-

mote t h e r e t e n t i o n of hardness a t high temperatures. Table 2 .1

l i s ts t h e chemical compositions of a nunibrr of bea r ing s t e e l s .

Of t h e s e s t e e l s M-53 is widely used by gas-turbine manufacturers

i n t h e United S t a t e s , and 18-4-1 is widely used i n Europe. Both

m a t e r i a l s (M-50 and 18-4-1) c o n t ~ i n elements t o promote high hard-

ness and good hardness r e t e n t i o n a t t h e h igh temperatures exper i -

enced by bea r ings i n gas- turbine engines . Parker and Zaretsky

(1978) found t h e ro l l ing-e lement Fat igue l i v e s of t h e s e two s t e e l s

t o have no s i g n i f i c a n t d i f fe rence .

Johnson (1964, 1965) developed t h e AMS 5749 s t e e l l i s t e d i n

Table 2.1. This m a t e r i a l combines t h e tempering, hot -hardness,

and hardness-re tent ion charact r i s t i c s of AISI M-50 s t e e l wi th t h e

c o r r o s i o n and oxidat ion r e s i v tance of h ISI 44GC sruFr.less s tee :

The impetus f o r t h e i n t r o d u c t i o n of t h i s s t e e l , as pointed out by

V a l o r i (1978), was t h a t e s t i m a t e s showed t h a t t h e United S t a t e s

Navy yea r ly r e j e c t e d M-50 engine bea r ings worth over $1 m i l l i o n

because of co r ros ion . AMS 5749 c o n t a i n s a h igher percentage of

carbon and chromium than AISI M-50 f o r improved c o r r o s i o n and wear

r es1 , t ance . The hot hardness and hardness r e t e n t i o n of AMS 5749

a r e b e t t e r than those o f AlSI 440C and s i m i l a r t o those of AISI

M-50 (Johnson, 1964). Parker and Hodder (1978) exper imenta l ly

found t h e ro l l ing-e lement f a t i g u e l i f e of AMS 5749 t o be 6 t o 12

t imes g r e a t e r than t h a t of AISI M-50.

Bearings made from M-series s t e e l s can c o s t a s much a s 50

pe rcen t more than those made from s tandard SAE 52100 because of

0 g r ind ing d i f f i c u l t i e s . A t temperatures below about 180 C

(350' F) t h e r e appears t o be nc t e c h n i c a l o r c o s t advantage of

H-series s t e e l s over SAE 52100.

2.4.1 Melt in8 P r a c t i c e r

One important caure of ro l l ing-e lement f a t i g u e l a nonmctal l ic

i n c l u s i o n s , a s pointed out by Anderson and Zaretsky (1973). T h e ~ e

i n c l u s i o n s c o n s i s t of s u l f i d e s , a l m i n a t e s , s i l i c a t e s , and globular

oxides and can a c t as a t r e r s r a i s e r s . I n c i p i e n t cracks can emanate

from t h e s e inc lus ions , a s shown i n Figure 2.23, and can en la rge and

propagate under repeated stresses t o form a network of c racks t h a t

i n due course genera te a f a t i g u e spa11 o r p i t a s shown i n

Figure 2.24.

The vacuum-melting process reduces o r e l i m i n a t e s t h e s e non-

m e t a l l i c i n c l u s i o n s , as well a s entrapped gases and t r a c e elements

p resen t i n t h e metal . Exposing t h e melt t o a vacuum permits deox-

i d a t i o n t~ be performed e f f e c t i v e l y by the carbons. The products

formed when carbon is used a s a deox id ize r a r e gaseous and thus

escape i n t o the vacuum, l eav ing no harmful r e s idue i n t h e meta l .

This process t h e r e f o r e r e s u l t s i n a s u b s t a n t i a l l y c l e a n e r ma te r i a l .

Four prime vacuum-melting processes a r e vacuum induc t ion melt ing,

consumable-electrode vacuum mel t ing, e l e c t r o s l a g m e l t i r ~ g , and

vacuum induc t ion mel t ing - vacuum a r c remelt ing.

I n vacuum induc t ion mel t ing (VIM), a cold charge is melted i n

a h igh-pur i ty r e f r a c t o r y c r u c i b l e i n an induc t ive furnace and sub-

sequent ly poured v h i l e the melt is under a vacuum.

Consumable-electrode vacuum mel t ing (CVM), a l s o known a s

vacuum a r c remel t ing (VAR), is a technique i n which e l e c t r o d e s

made by primary hea t ing i n a i r a r e remelted by an e l e c t r i c a r c

process . The product t h u r remelted s o l i d i f i e e i n 8 water-cooled

copper mold under vacuum. The CVM method produce8 c l e a n e r and

more segrega t ion-f ree m a t e r i a l than t h e VIM process.

E l e c t r o s l a g me1tf.w o r remel t ing (ESR), a l s o known.as e lec -

t r o f l u x remel t ing (EFR), has been used e x t e n s i v e l y in' Europe and

t h e U. S. S.R. f o r improving bear ing s t e e l e . The ESR procees is a

consumable-electrode remel t ing procedure i n which a molten f l u x

blanket is maintained over t h e molten meta l . The f l u x blanket

s e r v e s e s s e n t i a l l y t h e same purpose a s a vacuum atmosphere i n

p r o t e c t i n g t h e ingo t from contamination.

I f t h e primary hea t i n t h e VAR process is vacuum induc t ion

mel t ing, t h e process i s c a l l e d double-vacuum mel t ing, o r VIM-VAR.

The high-qual i ty VIM-VAR AISI M-50 m a t e r i a l had a much longer

ro l l inq-e lement f a t i g u e l i f e than VAR m a t e r i a l i n a c c e l e r a t s d

f a t i g u e l i f e t e s t s ( S c h l a t t e r , 1974; Bamberger, 1972) and i n b a l l

bear ing t e s t s (Bamberger, e t a l . , 1976). The s t e e l produced by

t h e EFR process , wi th V I M primary hea t ing , a l s o showed longer l i f e

than t h a t produced by the VAR process with AISI M-50 i n acce le r -

a t e d rol l ing-e lement f a t i g u e t e s t s ( S c h l a t t e r , 1974; Bamberger,

1972). but t h e improvement was not a s g r e a t a s t h a t w i t h VIM-VAR.

Furthermore Parker and Hodder (1978) show t h a t VIM-VAR AMS 5749

s t e e l had a ro l l ing-e lement f a t i g u e l i f e 14 t imes g r e a t e r than

t h a t obta ined wi th V I M AMS 5749.

I n t h e l a s t 20 yea r s vacuum-melting techniques have no doubt

been most i n f l u e n t i a l i n improving the f a t i g u e l i f e o f b a l l bear-

ings . General ly vacuus-melted s t e e l s a r e no more expensive than

~ t a v i o u r l y ured air-melted ~ t e e l ~ . Mort b a l l bear ing aunufac-

t u r e r s are now ulring vacuum-melted s tee l r , i n t h e i r b e a r i n ~ s .

Furthermore, i n a p p l i c a t i o n s where very high r e l i a b i l i t y i s re-

qu i red , double-vacuum melting i s of ten 8pecif led .

2.4.2 E la te r i a l Hardness

Moat b a l l bear ing components a r e through-hardened w i t h i n t h e

range 60 t o 65 Rockwell C. I n general , bear ing l i f e can be ex-

tended by inc reas ing m a t e r i a l hardness i n t h i s range, where t h e

m a t e r i a l mainta ins i t s d u c t i l i t y . Furthermore Zaretsky, e t e l .

(1967) found t h a t bear ings assembled wi th b a l l s and r a c e s of

va r ious hardnesses had an optimum hardness combination f o r b e s t

f a t i g u e l i f e . Kaximum l i f e was obtained when the b a l l s were 1

t o 2 p o i n t s ha rder than t h e races . The reason f o r t h i s is t h a t

t h e r e s i d u a l s t r e s s e s induced i n the r a c e s a r e a l s o a func t ion

of t h e hardness combination. Compressive r e s i d u a l s t r e s s e s

induced i n t h e m a t e r i a l reduce t h e magnitude of the maximum

shear s t r e s s e s .

2.4.3 F i b e r Or ien ta t ion

The r a c e s and b a l l s of most b a l l bear ings a r e formed by

fo rg ing and thus have a f i b e r p a t t e r n t h a t r e f l e c t s t h e flow of

metal dur ing t h e forming process. Fat igue research on b a l l s in-

d i c a t a d r r t r o n g r f f r c t of f i b e r o r i e n t a t i o n on f a t i g u r , Hmter ia l r

w i t h t h e f i b e r e p a r a l l e l t o the r t r e s s e d r u r f a c e were r t r o n g s r t ;

those wi th f i b e r s normal t o t h e etrrfece were weakest. Th i s e f f e c t

hae been a p p l i e d t o bear ings by using forged races . A s k e t c h

of t h e f i b e r f low i n a convent ional ( top) and a forged r a c e

(bottom) is shown i n Figure 2.25. Fat igue d a t a repor ted by

Anderson and Zaretsky (1968) show a t l e a s t a t e n f o l d i n c r e a s e i n

bearrng l i f e wi th forged races as compared wi th r a c e s c u t from

tubing. However, forged races a r e used only i n those a p p l i c a t i o n s

where h igh r e l i a b i l i t y is required because of t h e g r e a t e r manufac-

t u r i n g c o s t . 1

2-4.4 Powder Metals

Another source of ro l l ing-con tac t f a t i g u e f a i l u r e s is t h e

presence of l a r g e ca rb ides i n t h e metal matr ix . These a c t a s weak I

p o i n t s under t h e h ighly c y c l i c loading experienced i n bea r ing

o p e r a t i o n and u l t i m a t e l y become the o r i g i n of f i n e c racks , which

lead t o s p a l l i n g and bearing f a i l u r e . Conventional process ing

techniques have not been a b l e t o provide the necessary c o n t r o l of

carbon s i z e and d i s t r i b u t i o n .

The use of powder meta ls i s a p o s s i b l e way t o produce a l l o y s

wi th h igh ly r e f i n e d ca rb ides s i n c e t h e ca rb ide s i z e is l imi ted by

t h e s i z e of t h e powder p a r t i c l e s and does not change v i t h subse-

quent process ing. Likewise, a highly uniform d i s t r i b u t i o n of

carbidma rhould be obta ined i n t h e f i n a l product becaure of t h e

random mixing of pal: t icler . The s t a r t i n g m a t e r i a l s f o r powder-metal procera ing a r e con-

v e n t i o n a l l y processed i n g o t s prepared by e i t h e r vacuum induct ion

mel t ing o r vacuum induc t ion mel t ing - vacuum a r c remel t ing.

Powder-metal proceseing h a s t e e n desc r ibed by Brown and P o t t e

(1977) :

Conversion of t h e i n g o t s i n t o powder was accomplished

wi th equipment which is schemat ica l ly represented i n

t h e s k e t c h shown i n Flgure 7. (See Figure 2.26.)

The a tomizing system is enclosed and is evacuated t o

a p r e s s u r e l e v e l which removes 311 t r a c e s of atmos-

p h e r i c contamination p r i o r t o r emel t ing of t h e i n g o t .

Melt ing is accomplished by indu:tion hea t ing tech-

n iques i n a c r u c i b l e i n t h e u p p x s e c t i o n of t h e u n i t .

A f t e r the metal becomes molten L t i e poured through

an atomizing nozzle i n t o another vacuum chamber loca ted

below the remel t chamber. The nolten metal i s discharged

a s r e f i n e d d r o p l e t s from t h e noczle. High v e l o c i t y i n e r t

gas b led i n t o t h e chamber coo l s the d r o p l e t s a s they

emerge from the riozzle. Vacuum pumping a c t s continuously

t o mainta in the chamber p r e s s u r ~ ? we l l below atmospheric.

The cooled d r o p l e t s s o l i d i f y anal drop t o t h e bottom of

t h e lower chamber. Th i s powder now c o n t a i n s c a r b i d e s

t h a t a r e many t imes smal le r t h a ~ l those t h a t were p resen t

i n t h e o r i g i n r l l r r g r ingo t . The powders a r e t o be re-

consol idated i n t o new ingo t8 at temperatures w e l l below

a l l o y melt temperatures i n order t o ensure t h a t no coa-

l e rcence o r growth of t h e a l l o y ca rb ider w i l l occur. By

m a n e of t h i s proceea i.t is assured t h a t t h e f i n a l prod-

u c t w i l l e x h i b i t r e f ined ca rb ides .

Notwithstanding t h e l imi ted euccees (Brown and P o t t s , 1977) i n

us ing t h i s process t o d a t e , i t seems t o o f f e r promise f o r app l i -

c a t i o n s where roll ing-element f a t i g u e i s a c r i t i c a l problem and

extreme r e l i a b i l i t y is des i red .

A thermomechanical process termed "aus f oming" has been

developed i n t h e hope of extending bear ing f a t i g u e l i f e . Ausform-

ing c o n s i s t s of re-forming a bear ing s t e e l by us ing a h o t - r o l l i n g

procedure whi le t h e m a t e r i a l is i n a metastable a u s t e n i t i c condi-

t i o n . Bamberger (1967) was *he f i r s t t o apply t h e p rocsss t o

rol l ing-element bear ings . To apply t h e ausforming process t o

bear ing s t e e l s , t h e m a t e r i a l must have a s lugg ish m a r t e n s i t i c

t ransformat ion behavior. The M-type s t e e l s e x h i b i t such behavior

and can be ausformed, whereas t h e SAE 52100 m a t e r i a l cannot be

t r e a t e d i n t h i s way because of i ts austeni te- to-martensi te t r a n s -

formation c h a r a c t e r i s t i c s . Bamberger ' s (1967) t e s t s ind ica ted

t h a t bear ings made from ausformed AISI M-50 were s u p e r i o r t o those

of n o r u l l y procerred &SO. Addit ional ly a r e l a t i o n r h i p war rhown

t o ucirt betwean t h e amount of deformation s u f f e r e d dur ina aurform-

ing and thcr f a t i g u e l i f e of t h e material. A 75 t o 80 percent work,

o r r educ t ion of arm, war requ i red t o r e l a i r e t h e m8ximum b e n e f i t .

2.5 Separa to rs

Ball bear ing s e p a r a t o r s , sometimes c a l l e d cages o r r e t a i n e r s ,

a r e bear ing components t h a t , al though never c a r r y i n g load, a r e

n e v e r t h e l e s s capable of e x e r t i n g a v i t a l i n f l u e n c e on t h e e f f i -

c iency of t h e bearing. I n a bear ing without a s e p a r a t o r t h e b a l l s

con tac t each ~ t h e r dur ing opera t ion an-' i n s o doing exper ience

severe s l i d i n g and f r i c t i o n . The separa to r t h e r e f o r e mainta ins

t h e proper d i s t a n c e between the b a l l s t o ensure proper load

d i s t r i b u t i o n and t o prevent s l i d i n g contact between t h e b a l l r

and thus permits s a f e operat ion of the bear ing a t a l l h igher

speed ranges. Furthermore a s e p a r a t o r is necessary f o r s e v e r a l

types of bear ings t o prevent t h e b a l l s from f a l l i n g out of t h e

bear ing dur ing handling.

The m a t e r i a i s used f o r s e p a r a t o r s vary according t o the type

of bear ing and t h e a p p l i c a t i o n . The mot' ccmon type of separa to r

used i n b a l l bear ings is made from two s t r i p s of carbon s t e e l t h a t

a r e pressed and r i v e t e d together . Called ribbon s e p a r a t o r s , they

a r e t h e l e a s t expensive t o manufacture and a r e e n t i r e l y s u i t a b l e

f o r many app l ica t ions .

Era d e r m and con8 t ruc t i on of the an&~lar-contac t bearing

allow the use of 8 one-piace reparator . The s impl ic i ty and

inherent s t rength of one-piece separators permit t h e i r fabr ic r -

t i on from many des i rab le mater ials . Reinforced phenolic and

bronze a r e the two m a t commonly used materials. Brohze separ-

a to r s o f f e r s t rength and l o r f r i c t i o n cha rac t e r i s t i c s and can be

operated a t temperatures t o 230' C (650' F). Reinforced phenolic

se;?ratore combine the advantages of l i g h t weight, s t rength , and

nongalling propert ies; they a re used f s r such high-speed applica-

t ions a s gyro bearings. Phenolic separatorR are , however, l imited

t o a maximum operating temperature of about 135' C (275' F).

The temperaturelirniteof the separator mater ia l s aiscuseed

i n t h i s sec t ion a r e shown in Figure 2.27. The bars represent the

temperature ranges throughout which these mater ials can be

expected t o operate s a t i s f a c t o r i l y fo r extended periods of time.

Where high speeds and high temperatures a t e cncountered, separ-

a t o r s of phosphor-bronze or i ron-si1ico~-bronze a r e sometimes

u9ed. These mater ials have excel lent m t i f r i c t i o , ~ propert ies

and favorable strength-to-weight r a t i o s .

At temperatures above 320° C (600' F) some suc e s s has been

obtained with nickel-based a l loys . Among the nickel-based a l loys

the most widely used is Monel. S-Monrl has been used successful ly

for r e t a ine r s i n bearings a t temperatiires of 540' C ( 1 0 0 0 ~ F) , as

shown i n Figure 2 .27 . This f igure a l s o shows tha t a l loy and tool

s t e e l s , su i tab ly plated for low f r i c t i o n , perform well a t temper-

a tu re s t o 540' C ( 1 0 0 0 ~ F) .

2.6 Clorurr

In t h i s chap te r t h e most popular and b a s i c type8 of b a l l

bear ing8 have been descr ibed. The geometric and k i n e m t i c r e l a -

t i o n s h i p s of the b a l l bear ing under a cond i t ion of no load have

a l s o been discussed. These t o p i c s a r e e x t r m e l y important i n

l a y i n g t h e foundat ion f o r the rena in ing chap te r s . The f i n a l p a r t

of t h e chap te r d e a l t wi th bear ing m a t e r i a l s and manufacturing

processes , a s w e l l a s t h e s e p a r a t o r s used i n b a l l bear ings . With

t h i s backgraund we now proceed t o d i s c u s s loaded c o n t a c t s .

A*, B*, C*,

D*, L*, M* 1 SYMBOLS

constant used i n equat ion (3.113)

r e l a x a t i o n c o e f f i c i e n t s

drag area o f b a l l , mL

semimajor ax i s of contact e l l i p s e , m

t o t a l conformi ty o f bear ing

semimincr ax i s of contact e l l i p s e , m

dynamic load capacity, N

drag c o e f f i c i e n t

constants

19,609 ~ / cm ' (28,440 l b f / i n 2 )

number o f equal d i v i s i o n s o f semimajor ax i s

d is tance between race curvature centers, m

ma te r i a l f ac to r

def ined by equat ion (5.63;

Deborah number

b a l l diameter, m

numbrr o f d i v i s i o n s i n semiminor ax i s

a v e r a l l diameter o f bear ing (F igure 2.13), m

bore diameter, m

p i t c h diameter, m

p i t c h diameter a f t e r dynamic e f f e c t s have acted on b a l l , m

inner-race diameter, m

outer-race diameter, m

4 1

modulus o f e l a s t i c i t y , N/m 2

/t ;:' , ~ / m ~ e f f e c t i v e e l a s t i c modulus, 2 - + - 2 2 in te rna l energy, m 1s

processing fac to r

- Hmin)/HminI x 100 2 112

e l l i p t i c i n teg ra l o f second k ind w i t h modulus ( 1 - l / k )

approximate e l 1 i p t i c in tegra l o f second k ind

dispersion exponent

normal applied load, N

normal appl ied load per u n i t length, Nlm

lub r i ca t i on fac to r

integrated normal appl ied load, N

cent r i fuga l force, N

maximum normal applied load ( a t J, = 0), N

a p p l ~ e d r a d i a l load, N

appl ied t h r u s t load, N

normal dpplied load a t angle +, N

e l l i p t i c i n teg ra l o f f i r s t k ind w i t n modulus (1 - l i t 2 ) 112

approximate e l l i p t i c i n teg ra l of f i r s t k ind

race conformity r a t i o

m s s?ri-face f i n i s h o f b a l i , m

nns surface f i n i s h o f race, m

dimensionless material; parameter, a€

f l u i d shear modulus, Nlm 2

haraness fac to r

g rav i t a t i ona l constant, mls 2

42

dimensionless elasticity parameter, u ~ ~ ~ ~ u ~ 3 2 dimensionless viscos parameter, GW IU

dimensionless film thickness, h/Rx

2 2 2 3 dimensionless film thickness, H(u/u)* - F h/u sORx

dimensionless central film thickness, hc/Ri

dimensionless central f i lm thickness for starved

lubrication condition

frictional heat, N m/s

dimensionless minimum film thickness obtained from EHL

el 1 i p t ical-contact theory

dimensionless minimum film thickness for a rectangular

contact

dimensionless minimum f i l m thickness for starved

lubrication condition

dimensionless central f l l m thickness obtained from

least-squares fit of data

dimensionless minimum film thickness obtained from

least-squares fit of data

dimensionless central-filmthickness - speed parameter, I i c ~ - O e 5

dimensionless minimuwfilm-thickness - speed parameter, Hmi -'

new estimate of constant in film thickness equation

film thickness, m

central film thickness, m

inlet film thickness, m

f 1 l m thickness a t po ln t o f marlmum pressure, where

dpldx = 0, m

minimum f i l m thickness, m

constant, m

diametral interference, m

b a l l mass moment o f i ne r t i a , m N s 2

i n teg ra l defined by equation (3.76)

i n teg ra l defined by equation (3.75)

funct ion o f k defined by equation (3 .8)

mechanical equivalent o f heat

po la r moment o f i ne r t i a , m N s 2

load-def l e c t i o n constant

e l l i p t i c i t y parameter, a/b

approximate e l 1 i p t i c i t y parameter

thermal conduct iv i ty, Nls 'C

l ub r i can t thermal conduct iv i ty , N/s 'C

f a t i gue l i f e

adjusted fa t igue l i f e

reduced hydrooynamic 1 i f t , from equation (6.21)

lenaths defined i n Figure 3.11, m

f a t i gue l i f e where 90 percent o f bearing populat ion w i 11

endure

fa t i gue l i f e where 50 percent o f bearing populat ion w i 11

endure

bearing length, m

constant used t o determine width o f side-leakage region

moment, Nrn 44

gyroscopic moment, Nm

dimensionless load-speed parameter, NU -0.75

torque requ i red t o produce spin, N m

mass o f b a l l , N s2/m

dimensionless i n l e t d is tance a t boundary between f u l l y

f l ooded and s tarved cond i t i ons

dimensionless i n l e t d is tance (Figures 7.1 and 9.1)

number o f d i v i s i o n s o f semimajor o r semiminor a x i s

dimensionless i n l e t d is tance boundary as obta ined f rom

Uedeven, e t a l . (1971)

r o t a t iona 1 speed, rpm

number o f b a l l s

r e f r a c t i ve index

constant used t o determine leng tn o f b u t l e t r eg i on

dimensionless pressure

dimensionless pressure d i f f e rence

d iamet ra l clearance, m

f r e e endplay, m

dimensionless He r t z i an pressure, N/m 2

pressure, N/m 2

maximum pressure w i t h i n contact , 3F/2nab, Nlm 2

i sov i scous asymptotic pressure, NI~'

s o l u t i o n t o homogeneous Reynolds equat ion

thermal load ing parameter

dimensionless dass f l o w r a t e per u n i t width, qn E 1 R 2 0 0 reduced pressure parameter

2 volume f l o w r a t e per u n i t w id tn i n x d i r e c t i o n , m I s

45

2 volume f l o w r a t e per u n i t w id th i n y d i r ec t i on , m Is

curva tu re sum, m

a r i t hme t i ca l mean dev ia t i on def ined i n equat ion (4.1) . m

opera t iona l hardness o f bear ing ma te r l a l

e f f ec t i ve r ad ius i n x d i r ec t i on , m

e f f ec t i ve r ad ius i n y d i rec t ion , m

race curva tu re radius, m

r a d i i o f curvature, m

c y l i n d r i c a l p o l a r coordinates

spher i ca l p o l a r coordinates

de f ined i n F igure 5.4

geometric separation, m

geometric separat ion f o r l i n e contact , m

emp i r i ca l constant

shoulder height, m

' ~!Pmax

tangen t ia l ( t r a c t i on ) force, N

temperature, 'C

b a l l surface temperature, 'C

average l ub r i can t temperature, 'C

b a l l surface temperature r i se , 'C

viscous drag force, N

time, s

aux i 1 i ary parameter

v e l o c i t y o f ba l l - race contact , m/s

46

u c v e l o c i t y o f b a l l center, mls

U dimensionless speed parameter, nOu/E1Rx

u sur face v e l o c i t y i n d i r e c t i o n o f motion. (ua + ub)/2, m/s

C

u number o f s t ress cyc les pe r r e v o l u t i o n

s l i d i n g ve loc i t y , ua - ub,

sur face v e l o c i t y i n t ransverse d i r ec t i on , m/s

W dimensionless load parameter, F / E ' R ~

w sur face v e l o c i t y i n d i r e c t i o n o f f i l m , mls

X dimensionless coordinate, x/R X

Y dimensionless coordinate, y/Rx

X t * I t dimensionless grouping f rom equat ion (6.14)

ex te rna l forces, N

Z constant def ined by equat ion (3.48)

1 v i s c o s i t y pressure index, a dimensionless constant

Y. 7, 7, 5 ) coord inate system

a pressure-vi scosi t y c o e f f i c i e n t o f l ub r i ca t i on , m ' / ~

r ad ius r a t i o , R /R Y X

con tac t angle, r ad

f r ee o r i n i t i a l contact angle, rad

i t e r a t e d value o f contact angle, rad

curvature d i f f e r e n c e

2 v iscous d iss ipa t ion , N l m s

t o t a l s t r a i n ra te , s'l

e l a s t i c s t r a i n ra te , s' 1

viscous s t r a i n ra te . s-l

f l o w angle, deg

t o t a l e l a s t i c defotmation, m

l u b r i c a n t v l s cos i t y temperature coe f f i c i en t , ot-1

e l a s t i c deformat ion due t o pressure di f ference, m

r a d i a l displacement, m

a x i a l displacement, m

displacement a t some l o c a t i o n x, in

approximate e l a s t i c deformation, m

e l a s t i c defotmat i on o f rec tangular area, m

c o e f f i c i e n t o f determinat ion

s t r a i n i n a x i a l d i r e c t i o n

s t r a i n i n t ransverse d i r e c t i o n

angle between b a l l r o t a t i o n a l a x i s and bear ing

c e n t e r l i n e (F igure 3.10)

probabi 1 i t y of su r v i va l

abso lu te v i s c o s i t y a t gauge pressure, N s/m 2

dimensionless v i s cos i t y , I-,/,, 0

v i s c o s i t y a t atmospheric pressure, N slm 2

6 . 3 1 ~ 1 0 - ~ N sln?(0.0631 cP)

angle used t o de f ine shoulder he igh t

f i l m parameter ( r a t i o o f f i l m th ickness t o composite

sur face roughness)

equals 1 f o r outer-race c o n t r o l ana 0 f o r inner-race

c o n t r o l

second c o e f f i c i e n t o f v i s c o s i t y

Arcnard-Cowking side-leakage fac to r , (1 + 213 a,)- 1

r e l a x a t i o n f a c t o r

48

c o e f f i c i e n t of s l i d i n g f r i c t i o n

Po isson 's r a t i o

d ivergence o f , v e l o c i t y vector , ( a u a x ) + (av /ay) + ( awlaz), s - I 1'' I

2 4 l u b r i c a n t d&s i ty , N s Im

dimensionless dens i ty , p / p o

2 4 d e n s i t y a t atmospheric pressure, N s /m

normal s t ress , ~ l m '

s t r e s s i n a x i a l d i r e c t i o n , Nlm 2

shear s t ress , Nlm 2

maximum subsurf ace shear s t ress , Nlm 2

shear s t ress , N/m 2

7 e q u i v a l e n t s t ress , N/m'

l i m i t i n g shear s t ress , Nlm 2

r a t i o o f deptn of maximum shear s t r e s s t o semiminor a x i s o f

c o n t a c t e l 1 i pse

a u x i l i a r y angle

tnermal reduc t i o n f a c t o r

angu ldr l o c a t i o n

l i m i t i n g va lue o f $

abso lu te angu lar v e l o c i t y o f i n n e r race, r a d l s

abso lu te angu lar v e l o c i t y o f o u t e r race, r a d l s

angu la r v e l o c i t y , r a d l s

angu lar v e l o c i t y o f b a l l - r a c e con tac t , r a d l s

angu lar v e l o c i t y of b a l l about i t s own center , r a a / s

49

u C

0 s Subscr ipts:

Superscr ip t :

( - 1

10

angular v e l o c i t y o f b a l l around shaft center, r a d l s

b a l l sp i n r o t a t lotla l ve l oc i t y , r ad / s

s o l i d a

s o l i d b

c e n t r a l

b a l l cen te r

i sovi scous-elastic regime

i sovi scous-r ig id regime

i nner race

Kapi t z a

minimum

i t e r a t i o n

outer race

p i ezov i scous-elast i c regime

p i e r c v i scous-*igid regime

f o r rectangular area

f o r s tarved cond i t i ons

coord inate system

approximate

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Table 2.1 Chemical Compositione of Various Bearing Steels

Bearing C Mn Si Mo W Cr V Co ateel Chemical Composition, wt%, Balance Iron

AISI 440C SAE 52100 AMS 5749 AISI M-50 18-4-1 A m M-10 AISI M-2 AISI M-1 WE49

- --- ---- Rubbing b r i n g s - - - Oil - impreg~td porous metal b r i n g s

Rolling b r i n g s - - -0- Hydrodynamic oil-film bearings

1 10 ld ld 104 ld 106 Frequency d rotallon, rpm

Figure 2.1. - General guide to journal bearing type. (Except for roller bearrngs, curv?; are drawn for bearings with width equal to d~ameter. A med~um-viscosity mineral 011 lubricant IS assumed for hydrodynamic bearings. 1 From Englneer~ng Sciences Data Unit (1965).

wd~ 'uollepl p huanbal j

$1 4" pol 8'' 10' 01 1 - -.--

I ' i I

t ! ~.--L.-..A.- i_~..i-_~ A PO' €01 101 01 I 1-01 PI

U (bl k) (dl

Figure 2.5. - krsrmbk 0( CmtM b c u t q .

Figure 2 6. - Filled-notch ball baring.

(a) Small angle. (b) large angle.

Figure 27. - Angularsontat! ball bring.

Figure 28. - Self-aligning brll Wring.

lab k k - b Q c k mountsd (bl face-to-face mounted. Ic) Tandem mounted

Figure 2.9. - Duplex prirs d mquhrzonbct ball bearings.

Figure 2.1Q - Split- inner-ring ball wring.

Figure 2.11. - lhrust ball bearing.

F gure 2.12. - Ball bearing components. Example d nonconformal surfaces.

Figure 2.14. - Cross section d tell and oubr rue. s h w i q rue conformity.

Figure 2 13. - Cross section through radbl, sinqle-rw ball bearings.

Axis d rotation - - -- Axis d rdrtion --- ra, mitial position. rbl Shiftd mitlon.

Fqure 2.15. - Cross section d radial ball bearing. sharing tell-rre contact due to axial shift d inner and outer rings.

figure 2 16. - Freeankt angle and endpky a function d Pd% for four values d tobl mformity.

Figure 2 ll. - Shoulder helpht in ball bring.

Figure 2 18. - Geometry d contacting elastic solids.

Figure 2 19. - Cross sction d ball Wring.

y * t Plans x - OPlme y 0 plane x - OPlane

lc ~ V J rr:twdnt ellipsoidal solids. (bl Equivtknt ellipsoibl wlM nur r plane.

Figure 2.20. - Equivalmt ellipsoibl solids.

...

Wring b e . nm

Figure 2.21. - Limiting N w l w for present prwtke and 3xld 8# for b l l k t r '? q s d wrious bore dkmhrs. Baud on information from Bambryr. et a1. 11971,.

Figure 2, W - Rolllnp spuds in b l l baring.

Flqure LR - F -track mina!,-T Irm inckuriwl Fm Anderson a a ,'. ., - v 11973).

Figure 2.24. -Typical fat@* sprll in Wring raze. From lnkrsm and Zaretsry 119731.