Ball Bearings Mechanics-NASA Report

7/26/2019 Ball Bearings Mechanics-NASA Report

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,/

NASA Technical Memorandum 81691

b

Ball earing Mechanics

N

A

L A .

N A S A )

1 6 5 ; dC; A ~ u / f i k A d 1

Bernard J Hamrock

ewis Research Center

Clmland

Ohio

and

Duncan

Dowson

The University

o

ccdr

h d s England

June 98

3 CS

C S C L

3


2/105

CH PTER

B LL HE RING MECH NICS

The loads ca r r i e d by b a l l bea r ings a re t ransm i t t ed th rough

the ba l l s f rom one r i n g to the o the r . The magn itude o f th e l oad

ca r r i ed by an i nd i v idu a l b a l l depends on the i n te rn a l geome try

of t h e b e a r in g and t h e l o c a t i o n a f t h e b a l l a t any i n s t a n t .

Having determfned how a bea ring c a r r i e s load we can determine

how i t i s d i s t r i b u t e d among t h e b a l l s .

To da this

we

must

f i r s t d ev el op lo ad -d ef l e c t i o n r e a t i o n s h i p s f o r t h e b a l l - ra c e

contact . These r e l a t i o n s h i p s a re de ve lo pe d i n S e c t i on 3.1 f o r

any t y p e o f e l i p t i c a l c o nt a ct such as th os e f ou nd i n a b a l l

bearing.

The

de fo r na t i on w i t h i n the co n ta c t i s among o t he r

th lngs a fun c t i o n

o f

t he e l l i p t i c i t y param eter and t h e e l l i p t i c

i n t eg ra l s o f the f i r s t and second k inds .

Simp1 e d exp ress ions

th a t a1 ow qu ick ca l cu la t i o ns o f the de fo rma t ion t o be made

simply f rom a knowledge o f t he applSed load th e m at er ia l prop-

e r t es and

t h e

geometry of the contact ing e lements are presen-

t e d i n S e c tio n 3.2.

taost b a l l bea r ing app l i ca t i an s i n vo l ve s teady -s la te ro ta -

t i o n o f e i t h e r t i r e i n n e r o r o u t e r si\ g, o r bot h. However th e

ro ta t i ona l speeds are u s u a l l y not so g re a t as t o cause cen t r i f r r -

ga l fo rces or gyroscop ic moments o f s ig n i f i c a n t magni tude to ac t


3/105

on the b a l l . Consequen tl y t hese e f f e c t s a re igno red i n ana ly -

z i n g t he d i s t r i b u t i o n o f r a d i a l , t h r u s t , and com bined b a l l l o a ds

i n Sect ion

3.3.

I n h igh-speed b a l l bea rings t he ce t i t ri f uga l f o r c e ac t ing on

t h e i n d i v i d u a l b a l l s c an be s i g n i f i c a n t compared w i t h t h e

app l ied f o rces ac t in g on t he bear ing . I n h igh-speed bearings

b a l l gy ro sc op ic moments can a l s o be o f s i g ~ ~ i f i c a n tagnitude,

depending on th e conta ct angles, such t h a t the inner-race

con tac t ang les t end ~ inc rease and the outer - race con tac t

a ng le s t e n d t o decrea se. I n b ea ri ng s i n w hic h d r y f r i c t i o n o r

b ou nd ary l u b r i c a t i o n o cc u rs i n t h e c o n j u n c t i o n betw een t h e b a l l s

and races, t h i s can cause a s h i f t of co n t r o l between races and,

i n some cases, un sta ble be ar in g operat ion. Th is does n ot occur,

however, i h e c o n j u n c t o n s e xp e ri en c e f u 11 elastohydrodynamic

lub r i c a t ion . P rocedures f o r eva lu a t ing t he per fo tmance o f

h igh-speed b a l l bea r ings a re deve loped i n Sec t ion

3 4 .

E las to -

h yd ro dy na mic l u b r i c a t i o n o f b a l l - r a c e c o n t a c t s i s n o t c o n si de r ed

i n t h i s c h ap te r b u t w l l be t rea ted i n Chapter 8.

No r o l l

i

g-e lement bea r ing can g iv e u n l im i t ed 1 f e because

o f t h e pr ob a bi

l

t y o f f a t igue . Any s t ru c t u r a l m a t e r ia l subjec -

t e d

t o an un l im i t ed suecession of repeated or reversed s t resses

w l l u l t i m a t e l y f a i l . T he refo re a l l b a l l b ea rin gs e v e n t u a l l y

succumb t o f a t ig u e , w h ic h i s ma n if es te d b y s u rf a c e d i s t r e s s i n

t h e form o f f l a k i n g o f m e t a l l i c p a r tS c le s .

I n marry cases f l a k -

g may beg in as a crack below the su r f ace t h a t i s p ropaga ted t o


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th e surface, where

i t

even tua l

l y

fern is

a y

o r spa l 1. Fa t ig ue

i s assumed t o have oc c u r re d when t h e f i r s t c ra ck o r s p a l i s

observed on a load-carry ing sur face , A d es ig n c r i t e r i o n f o r t h e

f a t i g u e l i f e of b a l l b e a ri ng s i s de ve lo pe d i n S ec t io n 3.5.

B a l l b e a r i n g s c a n be l u b r i c a t e d s a t i s f a c t o r i l y w i t h a smal l

amount o f l u b r i c a n t s up p li ed t o t h e r i g h t a re a w i t h i n t h e

bearing,

S ec t i o n 3.6 c o n s id e r s t h e s e l e c t i o n o f a s a t i s f a ct o ry

lub r ic an t , as we11 as de scr ib i ng systems t h a t p rov ide a constan t

f lo w o f l u b r i c a n t t o t h e c on ta ct .

When an e l a s t i c s o l i d i s s u bj ec t ed t o a lo ad, s tr e ss e s ar e

produced t h a t increas e as t i re load i s increased . These s t re sses

a re assoc ia ted w i t h defo rma tions , wh ich a re de f i ne d by s t r a i n s ,

U niq ue re l a t i o n s h i p s e x i s t be tw ee n s t re s s e s and t h e i r c o r re -

sponding

s t r a i n s .

F o r e l a s t i c s o l i d s t h e s t re s se s a re

l

n e a r l y

related t o t h e s t r a i n s , w i t h t h e co ns ta nt o f p r o p o r t i o n a l i t y

b e in g an e l a s t i c c o ns ta nt t h a t a dop ts d i f f e r e n t v al ue s f o r d i f -

f e r e n t m a t e ri a ls ,

Thus

a

s i n ~ p l e e n s i l e l o a d a p p l ie d

t o

a ba r

p roduces a s t ress

u

a n d a s t r a i n

rl

where

1 =

Load

= St re ss i n a x i a l d i r e c t i o n

Cross-sect io t ia l area

3 1

l lange

n

l ngth

S t ra i n i n a x i a l d i r e c t i o n

O r i g i n a l l e n g t h

3 4


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and

E = E l a s t i c c o ns t an t o r modulus o f e l a s t i c i t y

C 3 . 3 )

A1 h oug h no s t r e s s a c t s t r a ns v e r s e ly t o t h e a x i a l d i r e c t i o n

t h e r e

w

11 never the less

be

d im e ns io na l changes i n t h a t d i r e c t i o n

su ch t h a t a s a b a r e x te n d s a x i a l l y t c o n t r a c t s t r a ns v e r s e ly .

T h e t r a n s v e r s e s t r a i n s c a re r e l a t ed t o t h e a x i a l s t r a i n s

r l

by

P o i s s o n ~ s a t i o such t h a t

w he re t h e n e g a t i v e s i g n means t h a t t h e t r a n s v e r s e s t r a i n

w l l

be

of t h e o p p o s i t e s i g n t o t h e a x i a l s t r a i n The m od ulus o f e l a s -

t i c i t y and P o i s s on l s r a t i o a r e tw o i m p or ta n t p ara me te rs u sed t o

d es c ri be t h e m a t e r i a l i n t h e a n al y si s

o f

c o n t a c t i n g s o l i d s .

As t h e st r e s s es i n c r e a s e w i t h i n t h e i n a t e r i a l e l a s t i c be-

h a v io r i s r ep la c ed by p l a s t i c f lo w i n w hich t h e m a t e r i a l i s

pemianent ly deformed.

The s t r e s s s t a t e a t w h ic h t h e t r a n s i t i o n

f r o m e l a s t i c t o p l a s t i c b eh a v io r o cc urs known as t h e y i e l d

s t r e s s h as a d e f i n i t e v a lu e

f o r

a g iv e n m a t e r i a l a t

a

g i v e n

temperature.

I n t h i s b ook e l a s t i c b e ha v io r a lo n e i s c o ns id er ed .

3.1.1 Surface Str ess es and De for n~ at i on

When two e l a s t i c s o l i d s a re b rought toge t her under a load

a

con tac t a rea deve lops the shape and s i z e o f wh ich depend on


6/105

t he app l i ed load, the e la s t i c p ro per t i es o f t he mate r i a l s , and

the cur va ture s of th e surfaces. When th e two s o l i d s shown i n

F i g u r e 2.18 have a nomlal load ap pl i ed t o them, th e shape o f th e

c o n t a c t a r e a i s e l l i p t i c a l , w i t h a b e i ng t h e s em im ajo r and b

the semiminor axis. I t has been c m o n t o r e f e r t o e l l l p t i c 6 1

con tac ts as po in t con tac ts , bu t s i nce t h i s book dea l s ma in l y

w i t h lo ad ed co n ta c ts , t h e t er m e l l i p t i c a l c o n t a c t i s a dopted.

For the special case where

r = r

and

rb

rby

aY

t he r e s u l t i n g c o n ta c t i s a c j r c l e r a t h e r t ha n an e l l i p s e .

Where r and r are b oth i n f i n i t e , t he i n i t i a l l i n e

aY by

con tac t develops i n t o a re c tang le when l oad i s app li ed.

The c o nt a ct e l l i p s e s o b ta in e d w i t h e i t h e r a r a d i a l o r a

t h ru s t l o ad f o r t he b a l l

-

i nner - race and b a l l ou te r - race

c o nt ac ts i n a b a l l b e a ri n g a r e shown i n F i g u r e 3.1. T h i s

book

i s concerned w i t h the con junc t i ons between so l i d s - w i t h c o n ta c t

a re as r a n g i ng fr o m c i r c u l a r t o r e c t a n g u l a r - and wi th the

ana lys i s o f con ta c ts i n a b a l l bear ing . Inasmuch as t h e s i z e

and shape of these contact areas are h i g h ly s i g n i f i c a n t t o t he

successfu l operat io n o f b a l l bear ings ,

i t

i s i m po rta nt t o

u n d e r s t a n d t h e i r c h a r a c t e r i s t i c s .

H e r t z

1881)

cons idered the s t resse s and deformat ions i n

two p e r f e c t l y smooth, e l l i p s o i d a l , c o n t a c ti n g e l a s t i c s o l i d s

much l i k e those shown i n F igure 2.18.

H i s a p p l i c a t i o n of t h e

c l a s s i c a l t he or y o f e l a s t i c i t y t o t h i s pro ble m fo rm s th e b a s i s

o f s t re ss ca l cu la t i o n f o r mach ine e lements such as b a l l and rol


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l e r bear ings, gears, sedls, and

cants.

The fo 1 o wing assumptions

were made

by

H e r t z 1881)

1)

The

ma ter i a l s a re homogeneous and t he y i e l d s t r ess i s

not exceeded.

(2 )

No tangent ia l forces are induced between the s o l i d s .

3 ) C on ta ct i s l i m i t e d t o a s ma ll p o r t i o n o f

the

surface,

such t ha t t he d i mens i ons o f t he con tac t r eg i on a re sma l l cow

pare d w i t h t h e r a d i i

of the

e l l i p s o i d s .

( 4 ) The s o l i d s a r e a t r e s t and i n e q u i l i b r i u m ( st e ad y

s ta te )

Making use o f these assumptigns, H er tz 1881) was ab l e t o

o b t a i n t h e f o l l o w i n g e x p re ss io n f o r

the

p re ss ur e w i t h i n t h e e l -

l i ps o i d a l con tac t shown i n F igure 3.2:

I f t he p ressure i s i n t eg ra ted over t he con ta c t a rea,

i t

i s fo un d

t h a t

Equat ion (3.5 ) de tenn lnes t he d i s t r i b u t i o n

o f

pressure o r

com-

press ive s t ress on the cornor

i z t r f

3ce;

i t

i s c l e a r l y a maximum

a t the cente r o f th e con tac t and decreases

t o zero a t

t h e p e r i -

phery.

The

e l l i p t i c i t y param eter

k

can be w r i t t e n i n t er ms o f

the

remimajor and semiminor axes o f t h e con tac t e l 1 pse as


8/105

Har r i s

(1966)

has shown tha t the e l l i p t j c i t y parameter can be

used t o r e l a te th e curvature d i f ferenc e expressed i n equat ion

(2.25)

and the e l 1 p t c i n te g r a ls o f t he f t r s t and second

kinds as

where

one-point i t e r a t i o n method th a t was adopted by Hamrock and

Anderson (1973) can be used t o ob ta in the e l l i p t i c i t y parameter,

where

= J kn)

3.11)

The

i t e r a t i o n process i s n orm ally co ntin ue d u n t i l kn+ d i f -

fe rs f rom tn by less than 1x10-~. Note t h a t t he e l l i p t i c -

i t y pa ra meter i s a f u n c t io n o f t h e r a d i i o f c u rv a tu re o f the

so l i ds on ly :

k

=

f(rax.rbx. ay. by)


9/105

That

i s

as the

l oad

increases, th e semi~najorand semiminor axes

of t h e co nt a c t e l l i p s e i nc rea se p ro p o r t i o n a t e l y t o each o th e r,

so th e e l

1

i p t i c i t y pa rameter rema ins cons tan t.

T h e e l l i p t i c i t y

param eter and e l l i p t i c i n t e g r a l s o f

th

f i r s t and second k i ~ d s re shown i n F igu re

3.3

f o r

a

range o f

t h e c u rv a tu re r a t i o R us ua l ly encountered i n concen-

Y x

t ra ted con tac ts.

Whbn the e l l i p i c i t y parameter

k, the normal app l ied load

F,

Po isson l s ra t i o

V

and the modulus o f e l a s t i c i t y

E

of the

co nta cti ng s o l i d s are known, th e semimajor and semiminor axes o f

t h e c o nt ac t e l 1 pse and t he maximum de fa m at io n a t the c en ter o f

t h e co n ta c t ca n b e w r i t t e n f r o m t h e a n a l ys i s

of

Her tz 1881) as

where

I n these equat ions a and b a re p ro por t i ona l t o

F~~~

and

213

a

i s p r op o rt io n a l t o

F

Knowing the deformat ion at the center of tne contact and

the natu ra l geometr ica l separat ion between the so l ids, equat ion

-

,~

,


10/105

2.35), we can w r i te the de fonnr ti on a t any po in t w i t h i n the d ry

Hc r tz ia n contac t as

Th i s e q u at io n i s used i n l a t e r cha pte rs t o d e fi ne t h e

i l m

th ickness wi th in the con junct ion .

3.1.2 Subsurface St resses

Fa t igue cracks usua l l y s t a r t a t a ce r ta in depth below the

su rface i n p lanes pa ra l l e l t o ,the d i re c t io n o f ro l l i n g . Because

o f th i s , spec ia l a t ten t io n must be g i ven t o the shear s t re ss

amp1 tu de oc cu rr in g i n t h i s plane. Furthermore a maximum shear

st ress i s reached

a t a

c e rt a in depth below the sur face. The

analysis used by Lundberg and Palmgren (1947) w i l l be used t o

d e f i n e t h i s s tr ess .

The stresse s are re fe rre d t o a rectangular co ordinate sys-

tem w it h i t s o r i g i n a t th e ce nt er

o f

t he c on ta ct , i t s

z

a x i s

co inc id ing w i th the in te r io r no rma l

of

the body considered, i t s

a x is i n t he d i r e c t i o n o f r o l l i n g , and i t s y a x i s i n th e d i re c -

t i o n pe rp e nd icu la r t o t h e r o l l i n g d ir e c t io n .

I n the ana lys i s

t h a t f o l l o w s t

i s assumed th a t y

= 0.

From Lundberg and Palmgren (1947) the following equations

can be wr i t ten:

2

3F cos 9 s i n 9 s i n

Y

T .

(a2tan2v b2cos2 )


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z r tan

y

ns

The maximum shear s tres s rmplttu de i s defined as

The amplitude of the shear stress rj0

i s

obtained fmm

or the point of

max i n ~ um

shear stress

tan

4 = La

2

t a n t -

The posit ion of the maximum point i s determined by

where


12/105

furthermore the magni tude of

the

uxirmnshear s tr es s i s g fv e n by

3.2 Simp1i e d S o lu t io n f o r

Elliptical-Contact

Deformation

The c l as s i c a l H er tz ian so lu t i on p resen ted i n the p r ev i 4s

s ec t ion requ ires t he c a l c u l a t i on o f t he e l l i p t i c i t y parameter

k

and t he c omple te e l l i p t i c i n t eg ra l s o f t he f i r s t and secand

kinds nd 4 T h is e nt aS ls f i n d i n g a s o l u t io n t o a t rans-

cendental equat ion relat ing, k , r nd

t o

the

geometry o f

the con tac t ing so l ids, as expressed i n equat ion

3.8).

T his i s

us ua lly accompl ished by some i t e r a t i v e numerical procedure, as

described By amrsck and Anderson 1973), o r w i t h t h e a i d o f

char ts, as shown y Jones

1946).

Brewe

and Hamrock

1977)

used a l inear regress ion

by

the

method o f l eas t squa res t o ob ta i n s i mp l i f i ed equat ions f o r k

nd 8 That

i s ,

f o r g i ven se ts o f p a i rs o f data, [kj,

R~/R,)J], . 1 2. ...,

a p o w e r f i t u s i n g a l in ea r

regression by the method of leas t squares res u l te d i n the f o l -

lowing

equation:


13/105

The asymptotic behavior of

and was sug ges tive o f th e

fu nc t io na l dependence th at and might ex hib i t . As a re -

s u l t

r

l o g a r i hmic and an inverse curve i t were t r i e d f o r

and

,

res pe ct iv ely . The fo l l ow ing expressions f rom Brtwe and

Hamrock

(1977)

prov ide an ex ce l len t approx imat ion t o the relr

t ionships between I and

Ry/R :

0.5968

J .0003 aTR (3.29)

Y X

Values o f

ji,

and re presented i n Tab le

3.1

and compared

w i t h th e Hamrock and Anderson (1973) ~ i r m e r ic a l l y etermined va l -

ues of k, 4 and

S

The agreement i s good.

Us ing these s imp l i f i ed express ions fo r r and and

equat ion (3.15) g iv es the deformat ion a t th e center o f th e con-

t a c t

where

Note t h a t t he l o a d -d e f le c t i o n co ns t an t i s a f u n c t i o n o f t he

ba l 1-race geometry and t he ma te ri a1 pr op er t es.

The re s u lt s of comparing 7 w i t h 6 are also shown i n

Table 3.1. The agreement i s again q u i t e good. Therefore the

de format ion a t the cen te r o f the con tac t can be ob ta ined d i -


14/105

rec t y f rom equat ions 3.28) t o 3.32). Th is valu able approxima-

t i o n e l i m i n a t e s t h e need t o use c ur ve f i t t i n g , c ha rt s, o r n u m r-

i c a l methods.

F i g u r e 3.4 shgws th ree d i f f e re n t degrees o f b a l - con tac t

conformi ty :

a b a l l on b a l l , a b a l l on a p la ne , and a b a l l

-

ou te r r i n g con tact . Table 3.2 uses t h i s f i g u r e

t o

show how the

degree o f con form i ty a f f ec ts the con tac t parameters. The ta b l e

shows that

k

i s n o t e x a c t ly eq ual t o u n i t y f o r t he b all-o n-

b a l l and b a l -on-p lane s i tu a t io n s because o f th e approximat on

represented by equat ion

3.28).

The d ia m e te r o f t h e b a l l s i s

t he same th roughout , and the ma te r i a l o f t he s o l i d s i s s tee l .

The ba l l

-

s u t e r- r in g c o n ta c t i s re p r e s e nt a t iv e o f a

209

r a d i a l

b a l l b e a ri ng .

A 4.45-N l - l b f ) normal load has been cons idered

f o r each s i t u a t i o n ,

The maximum pr es su re decreases s i g n i f

ca n t l y as the curva tu re o f th e mat ing sur face approaches th a t o f

the ba l l . Tab le 3.2 shows th a t the curva tu re o f th e mat ing

s urfa ce s i s v e ry i m p or ta n t i n r e l a t i o n t o t h e ma gn itu de o f

th

maximum pressure or surface stress produced. A b a l l and r i n g o f

h i g h c o n f o r n i t y a r e t h us d e s i r a b l e f r o m t h e s t a n dp o in t o f

min in i izs ing he s t ress.

fab le

3.2

a l so shows th a t t he a rea o f

th

contact nab

inc reases w i t h t he con fo rm i t y

o f

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

A l

though t h i s e f f e c t m in im izes con tac t s t resses ,

t

can have an

u nd es ir ab le e f f e c t on t h e f o r c e o f f r i c t i o n , s in ce f r i c t i o n

fo rce i nc reases as t he con tac t a rea and he rce t he a rea o f t he


15/105

sheared lu b ri ca nt crease

r r

a beartng operat g under

elastohydrodynamic con ditio ns. The cur vatu res

of

the bear ing

races are the refo re genera l

ly

compromises that take into

cons iderat ion the s t ress , load capaci ty , and f r i c t i o n

character4 s t i es

o f

the bearing.

n

equations 3.24) t o

3.27)

the locat ion and magnjtude of

the maximum subsurface s k a r str es s are w r i t t e n as funct ions of

t ap n aux

i l i a ry

paraneter . Fur t l leno re i n equat ion

3.23)

th e e l l i p t i c i t y param eter

i s wrl t ten as

a f unc t i on o f

t .

the range for

l / k

i s

0 I l k

1

and the corresponding range

f o r t i s 1- t

1

4

A

l inear regress ion by

the method o f le a s t squares was used t o ob ta in

a

s i m p l i f i e d

formula f o r tl i n t erm s o f k, t he e l l i p t i c i t y param eter.

That is, f o r g iven s e t s o f p a i r s o f data [ I -

I l k )

J

-1

2,

..., n , a

power

f i t us ing a l i n e a r

regression

by

the method of least squares resul ted i n the

fa1 lowing equation:

The agreement between t h i s approximate eq uat ion and t he e xac t

s o lu t i o n i s w i t h i n 2 percent, The use of equation 3 , 3 3 )

great ly s imp l t f ie s the determination

o f t h e

values f o r t he

lo ca t on and magnitude o f the

maxl num

subsurf ace shear st re ss

expressed i n equat ions 3 . 24 )

t o

3 . 2 7 ) ,


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3.3 S t a t i c Load D i s t r i bu t i o n

Now that a s imp le a na ly t i c a l express ion f o r the de formation

i n ter ns o f the load has been detenii ined, t i s pos s i b l e t o con-

s ider how the bear ing load i s d is t r ib ut ed among the b a l l s w i t h in

a b a l l bear ing. Most b a l l bear ing app l icat ion s in vo lv e s teady-

s t a t e r o t a t i o n o f e i t h e r t h e in n e r o r o ut e r r i n g , o r both. I n

analyz ing the l oad d i s t r i b u t i o n on th e b a l l s ,

t

i s u s u a l l y s a t-

i s f a c to ry t o ignore t hese e f f e c t s i n most app l ic a ti ons. I n t h i s

sec t i on the rad ia l , th rus t , and combined load d i s t r i b u t i o ns o f

s ta t i c a l l y loaded b a l l bearings are i nves ti ga ted .

Fo r a g i ven ba l l - r ac e c on tac t t he l oad de f l ec t i on re l a t i on -

s h ip g i v en i n equat ion

3.31) can be rswr i t ten as

The t o t a l normal approach between two ra ces separated by a b a l l

i s the sum of the de formations under loa d between the bat 1 and

both races. There fore

where


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S u b s t i t u t i n g e q ua t io n s

(3.35)

t o (3.37) i n t o equa t i on J.34) and

s o l v i n g f o r K g i *

R e c a l l t h a t Ki and KO aye de f ined by eq ua tio n (3.32) and

th a t they a re a func t i on of ba l l - race geome try and ma te r ia l

p rope r t i es a lone .

The ana lys i s o f de fo rma t ion and l oad d i s t r i b u t i o n p resented

i n the fo l l o w ing th ree sec t i ons i s based on the work o f Jones

(

1946).

3.3.1 R ad ia l Load

r a d i a l l y lo ad ed b a l l b e a r in g w i t h r a d i a l c l e ar a nc e

Pd

i s shown i n F ig ur e 3.5.

I n t he c o n c e n tr i c p o s i t i o n shown i n

F ig ure 3.5 (a ) a un i fo rm r a d ia l c learance between the b a l l s and

t h e r i n g s o f P d/2 i s e v id e nt . l he a p p l i c a t i o n o f a s m a l l

r a d i a l l o a d t o t he sh a f t causes the i nne r r i n g t o move a d i s -

tance

Pd 2

befo re con tact i s made between a b a l l l o c a t e d on

t he l oad l i n e and the i nne r and ou te r t racks .

t

any angle

t h e r e w i l l s t i l l be a sma ll ra d ia l c l earance c tha t , i Pd

i s sma l l compared w i t h th e ra d ius

of

the tra ck s, can be ex-

pressed with adequate accuracy by


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On the load line, where

r =

0, the c learance i s zero,

b u t

when

=

90 t h e c le ar an ce r e t a i n s i t s i n i t i a l va lu e o f Pd/2.

The

app l i c a t i on of f u r t h e r l oad

w i l l

cause e ias t ic defoma-

t i o n

o f

some of the b a l l s and the e l im ina t io n o f c learance

around an arc

2 . f

he in te r fe rence o r to ta l e las t i c com-

p res sion on t he l oad l i n e i s

am

the corresponding e las t ic

compression

ay

a long a rad ius a t ang le t o t he l oad l i n e

w i l l

be

g i v en

by

Pd

6 1 =

( ax cos

y -

c )

=

ama,

* +)

cos

-

Now t i s c le ar from F igure 3 .5 (c) t h a t (a+ Pd / 2

r ep resents t he t o t a l r ad ia l d is placement o f t he i nne r r i n g o r

sha f t f rom the concent r i c pos i t ion

a.

Hence

The relat ionship between load and the elast ic compression along

the rad ius a t ang le

t o t h e lo ad v e ct or i s g iv en

y

equat ion

(3.34) as

312

Fy

=

K a *

Sub s t i t u t ing equation (3.39) in to t h i s equation g ives


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For s ta t i c equ i l i b r i um the app l ied ra d i a l l oad must equa l

t h e sum of th e components o f

the

b a l l lo ad s p a r a l l e l t o

the

d i r e c t i o n

o f t he

app l ied load

r

=

q

OS

Theref ore

The angular extent of the bearing arc

2qk

i n which t he b a l l s

are

loaded i s obtained by se t t i ng the ro o t exp ression i n 3.42)

equal to zero and so lv in g f o r .

The summation i n eq uat ion 3.42) app l ies on ly t o th e angu-

l a r e xt en t

o f

the loaded region, This equa t ion can be w r i t te n

i n i n teg ra l fo rm as

The i n t eg ra l i n th i s equat ion can be reduced to a s tandard e l -

l i p t i c i n te g ra l by the hypergeometr ic ser ies and the be ta func-

t i on .

f

t h e i n t e g r a l i s n u me ri ca ll y e v al ua ted d i r e c t l y , t h e

fo l lowing approx imate express ion

i s

der ived:


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(3.45)

Th is approximate express ion f i t s the exact numer ica l so lu t ion t o

w i t h i n *2 percent f o r s complete rnnge o f Pd/2 .

The load ca r r ie d by the most he av i ly loaded b a l l i s ob-

ta ined by su bs t i t u t i ng JI

=

0. i n eq uat ion (3.42) and dropp ing

the sumnation sign.

Di vid ing the maximum b a l l load (equa t ion

3.46) )

b y t h e t o t a l

a p p li e d r a d i a l l oa d o f t h e b e ar ln g ( eq u at io n ( 3 4 4 re

ar ra ng in g terms, and making use

o f

equa tion (3.45) giv e

where

When the diametral clearance

d

i s zero , the value o f

be-

comes 4.37. This i s the value der ived by Str ibeck (1901) f o r


21/105

bearing s o f zer o d ia me tra l c learance. The approach used by

S t ri b ec k was t o e v al u at e t h e f i n i t e s u m a t i o n c o r 5 / 2 0 f o r

var ious numbers o f ba l ls . He then de r iv ed th e ce le bra ted S t r i

beck eq u a t io n f o r s t a t i c l o a d -c a rr y in g c a p a c i t y by w r i t i n g t h e

more conse rva t i ve va lue o f 5 f o r t h e t h e o r e t i c a l v a lu e o f

4.37:

I n us ing equa t ion 3 .49)

t

should be remembered t h a t i s

cons ide red t o be

a c o n st a nt and t h a t t h e e f f e c t s of c learance

and a p p l i e d l o a d on l oa d d i s t r i b u t i o n a re n o t t a k en i n t o ac-

coun t. These e f f e c t s a re , however, cons idered i n ob ta in in g

equat ion 3.47). N ote a l s o t h a t t h e a n a l y t i c a l e xp r es s io n f o r

i n e q ua t io n 3.48) e na bl es a s o l u t i o n t o b e o b t a i n e d w i t h o u t

the a i d o f th e cha r ts used by Jones 1946 ) and H a r r i s 1966) .

3.3.2 Thrust Load

The s t a t i c t h ru s t - l o a d c a p a c i t y o f a b a l l b e a r in g may

be

def ined as the maximum th ru s t load th a t the b ear ing can endure

before th e co nt ac t e l l i p s e approaches a rac e shoulder, as shown

i n F ig ur e 3.6, o r t he load a t which t he a l lo wa ble mean compres-

s i ve s t r es s i s reached, wh icheve r i s sma ll e r .

B oth t h e l i m i t i n g

shoulder h e ig ht and the mean compressive s tr es s must be ca lc u l a-

t e d t o f i n d t h e s t a t i c t h ru s t- lo a d c a pa ci ty .


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T h e c on ta ct e l l p s e i n a bear ing race under a th rus t l o rd

i s

shown

i n Figure 3.6, Each ba l l

i s

su bj ec te d t o a n i d e n t i ca l

t hr us t c ~ n p ~ n e n t t/n , where Ft i s the t o t a l th ru st load.

The i n i t i a l c o nt ac t a ng le b ef o re

the

a p p l i c a t i o n o f a t h r u s t

load i s denoted by

rt.

Because o f th e appl ied thrus t . the

contact angle becomes . The normal b a l l th ru st load Ft

a c t s a t t h i s co nt a ct a n gle and i s w r i t t e n as

F

t

n s i n t~

cross sec t io n through an angular-contact bea ring under a thr us t

l oad F t i s shown i n F igu re

3.7.

B oth r ac e r a re a s s w d t o be

r i g id l y mounted, th a t i s , i ncapab le o f ra d i a l

defamation

From

th i s f i g u r e the con tac t ang le a f t e r the th ru s t l oad has been

appl ied can be w r i t t e n as

The

i n i t i a l con tac t ang le was g iven

i n

equat ion

2.9).

Using

that equat ion and rearranging terms

i n

equat ion

3.51)

g ive

From equa tion

3.34)

we can w r i t e

where


23/105

rn fi I

and re give n y equr t l ons

3.2U). 3.2Y), dnd

(3.30). respect ively.

From equation s (3.50) and (3.53)

Eq uat ion (3.55) can be solved numerically by the Newton-Raphson

method. The i t e ra t i v e e q u at io n t o be sa t i s f i e d i s

Th i s e q u at io n i s sa t i s f i e d when

l

-

6

i s e s s e n t i a l l y zero.

When a th r us t l oad i s app li ed , the shou lde r he igh t i s l i m i

t e d t o t h e d i st an c e

y

which the pressure-contact e l l i p s e can

approach th e shou lder. As long as the fo l lo wi ng ine qu al i ty i s

s a t i s i f e d , t h e p re ssure-co nt ac t e l l i p s e

w i

11 not exceed the

shoulder height 1 m i t :

e

> s i n

I G)

From Fig ur e 2.17 and equation (2.15) t he angle used t o de f i ne

the shou lder he ight e can be wr i t ten as


24/105

From Figu re

3.7

th e a x i a l d e f l e c t i o n

at

corresponding

t o a t h r us t load can be w r i t t e n as

6 t

m

0 6 ) s i n a - D s i n

i

3. 9)

Su bs t i t u t i ng equat ion 3 .52) i n t o equat ion 3.59) g iv es

D

s i n @-

c f )

t

COS 6

Having determined 6 i n equat ion 3.56) and 6f i n e qu at io n

2.9). we can ea s i l y eva lua te the re la t io ns h i p f o r

at.

3.3.3 Combined Load

For a combined r a d ia l and ax ia l load on a b a l l bea r ing we

cons ider the r e la t i v e d isp lacements

of

the inner and ou ter

r ings. We assume t h a t ne g l i g i b l e misal ignment of the bear i ng

can occur.

The

d isp lacements a re there fo re l i m i t ed to an ax ia l

displacement a t and a radia l d isp lacement

6

The races

a re the re fo re cons t ra ined t o re la t i v e movement i n para1 e l

planes,

The end re s u l t o f t h i s combined load ing i s shown i n

F igure 3 8 Note the d i f fe rence be tween th is f igure and F igure

3.7, wh ich re pr es en ts a x i a l lo ad in g alone. As was found when

d e a l i ng w i t h a p u r e l y r a d ia l load, t h e r a d i a l d is pla ce me nt i s a

f u n c t i o n o f t h e b a l l p o s i t io f t r e l a t f v e t o t h e a p p li e d load.

From Figure 3.8

2

D

6 )2

=

D cos

ef

a cos 1 ( D s i n

ef

at)

3.61)


25/105

Su b s t i t u t i n g t h i s e q u at io n i n t o e qu at io n 3.34) g i ve s

3.63)

where

K

i s efine i n e qu at io n 3.54). Also from Figure 3.8

t

s i n

fit

s i n

6

= 3.64)

r

r f

cos

r

cos 0os

i

cos

0

= 112 3.65)

The normal b a l l load F, which acts a t the contact ang le

0

(a long the 0 a

l i n e i n F ig ure

3.81

can

be

r eso lved i n t o two

components. One i s the th ru st fo rc e Ft p a ra l l e l

t o

t h e

bear ing axis, and th e other

is

t he r a d i a l f o r c e

Fr.

The

thrust component Ft can be w r i t t e n as

F t = s i n

0

3

b6)

ly

using equrtions 3.63) and

3.64)

th ls mla t fonshfp becaw


26/105

The r r d f r l component of load crlr be wrft ten as

r F cos ot

r

3.68)

From equat ions 3.63 ) and

3.65)

t h is expression can be

wr

t t e n

as

For the bearing to be n equilibrium after displacement,

the following conditions

must

be sat is f ied:


27/105

The extent of the load zone

1

i s ob ta ined by s e t t i n g the

numera to r i n these equa t ions t o zero o r

Under c e r t a i n c o n d i t i o n s o f a x i a l p r e lo a d and r a d i a l d i sp l a ce -

ment the va lu e o f cos li as determined y equat ion 3.72) w l l

be less than

1.

T h i s ind ica tes tha t the loaded zone ex tends

c o mpl ete ly aro un d t h e p i t c h c i r c l e . I n such c as es t h e l i m i t i n g

va lue

J R

i s t a ke n as W

Eq ua tio ns 3.71) and

3.72)

can

be

genera l ized t o i n c l u d e

any number o f ba l l s

by

the fo l l ow ing :


28/105

Note tha t t hese in teg ra ls a re func t ions o f t he th ree pa ramete rs

B t 6t/D,

and

sr /D

T h e se i n t e g r a l s a r e h yp e r e l l i p t i c

in te gr a l s th a t cannot be reduced t o s tandard fo rm t o pe nn i t so-

l u t i o n i n t erm s

o f

e l l i p t i c f u n c t i o n s and must th e r e f o r e be

eva lua ted num er ica l ly on a d i g i t a l computer.

Having detennined

at/D and

a /D

from eq ua tio ns S.73) and 3.74). we ca n

ob ta in the norma l ba l l l oad and ope ra t ing con tac t ang le a t any

b a l l p o s i t i o n JI f rom equat ions

3.63)

and

3.64)

._


29/105

Fo r b a l l b e a ri ng s t k t o p n r a t e a t

modest

speeds, as

csn

s i d e re d i n t h e p r ec e dj ng se c ti on , t h e c e n t r i f u g a l f o r c e o n th e

b a l l i s so n e g l i g i b l e t h a t t h e o n l y f or ce s t h a t keep t he b a l l i n

e q u i 1 b r i u m a r e t h e t wo c o n t a c t f o r c e s r e s u l t i n g f ro m t h e e x t e r -

n a l l y a p p l i e d l oa d. F o r such c o n d i t i o n s t h e c o n t a c t f o r c e s a r e

equ al and opp osi te, and the in ne r- and outer-race co nt ac t angles

ar e approx imate ly equal . The present se c t i on dea ls w i t h

h i gh-speed bear ings , where th e cen t r i fu ga l fo rc e developed on

the b a l l s becomes s ig n i f i c a n t and the i nner - and ou te r - race

con tac t ang les are no longe r equal . An angular -contac t be ar in g

i s ana lyzed s ince t he equa t i ons developed can be ap p l i e d t o

o t h e r t y p es o f b a l l b e ar in g s. combined r a d i a l nd a x i a l l o a d

i s c on sid er ed , b u t n ~ is a l ig n m e n t f t h e i n n e r and o u t e r r i n g s i s

exc luded . The n la te r i a l i n t h i s sec t i on was f i r s t developed by

Jones (

1956).

When a b a l l bea r ing operates a t h ig h speed, th e body for ce s

re su l t i n g from the b a l l s mot i on become s ig n i f i ca n t and mus t be

cons idered i n any a na l ys i s .

F igu re 3.9 shows the forces and

moments acting on

a

b a l l i n a hig h-s pe ed b a l l b e a r in g . The op-

e r a t i n g c o n ta c t a n gle a t t h e o u t er c o n ta c t i s l e s s t h an t h a t a t

the i nner con tac t because o f apprec iab le cen t r i f uga l f o rce and

gyroscopic moment.

I n t h i s fi gu r e, as w i t h t h e r e s t o f t h e


30/105

book su bscr ip t re fer s t o the inner race

arid

subsc r i p t o

t o t he ou te r race .

An exaggerated view i n Fig ure

3.10

shows the

b a l l

f i x e d i n

the p lane o f the paper and ro ta t i n g about i t s own center w i t h an

angu la r ve loc i t y o d i rec te d a t an angle t o the beari ng

cen ter l in e. The inne r and ou ter races r o ta te about the bearing

a x l s w i t h t angu la r ve loc i t i es oi and o0 r e l a t i ve t o

thq separato t . Fo r the l i ne ar ve loc i t y o f the races t o be equal

t o t he b a l l v e l o c i t y a t th e c on ta ct t h e f o l l o w i n g re -

l a t i o nsh ips must be sa t i s f i e d :

a -....-.

I f

t he o u t e r ra ce i s s t at io n a ry t he b a l l w i l l o r b i t the bear ing

ax i s w i th an angu la r ve loc i t y

uc

where

Then the abso lu te angula r ve loc i t y o f the i nner race i s

nip

where

There fo re fo r a s ta t i onary ou te r race and a ro ta t i ng i nner race

the fo l l ow ing can be wr i t ten :

=,F;;

4

]

-

d COS i

COS Bo


31/105

S i m i l a r l y f o r a s t a t i o n a r y i n n e r ra ce and a r o t a t i n g o u t e r r a ce

n o

- d

cos pi

r d COS

B

For s imu l taneous ro ta t i on

o f

th e ou ter and in ner races

F or a n a r b i t r a r y c h oi ce of t h e b a l l

w i l l

s p in r e l a t i v e

t o

the ra ce about th e normal a t t he center o f the contac t a rea .

t

i s c l e a r f ro m t h i s a n a ly s is t h a t t h e s p i n o f t h e

b a l l

may be d i f f e re n t r e l a t i v e t o each race , and t h i s p romp ted Jones

1 95 6) t o i n t r o d u c e t h e c o nc ep t o f r a c e c o n t r o l . f Coulomb

f r i c t i o n o r b ou nd ary l u b r i c a t i o n p re v ai 1s i n t h e c o n ju n ct io n s

between the b a l l and the i nne r and ou te r races , t he con junc t i on

s u b je c te d t o t h e l e a s t t o r q ue

w i l l

be prevented f rom spinning

y


32/105

f r i c t i o n w hi le the o t he r con junc t ion experiences sp in . The con-

t a c t a t whic h no s p i n occ urs i s c a l l ed t he c o n t ro l l i n g race .

f a l u b r i c a t i n g

i l m

ex is ts between the ba l l and each of

th e races, each o f th e conjunct ions can experience spin, and th e

r e l a t i v e n ~ o to n between the b a l l and the races i s detenn ined by

the eq u i l i b r i um o f t h e t orques r es u l t i n g f rom vi sc ous t r a c t i o ns

w i t h i n th e l u b r i c a n t .

The prob lem of pred ic t ing v iscous t rac-

t i o ns i n e lastohy drody namic f i l m s s t i l l r equ i res f u r t he r work,

but the recent deve lopment o f the unders tand ing o f lubr icant

rheo logy i n

EHL

conjun c t ions out1 ned I n Chapter 10, together

w i t h the a b i l i t y t o p r e d ic t

i l m

t hic kn e ss o u t l i n e d i n t h i s

te x t , i nd i ca tes th a t a comp le te so lu t i o n to the p roblem may no t

be f a r away.

Spin i n the con junc t ions between a b a l l and the races o f a

bear ing i s impor tant f rom the po in t o f v iew o f energy losses and

hea t genera t ion . For t h i s reason the race-con t ro l theory o r i g i -

nated by Jones w i 11 be ou t l i n ed here, a l thoug h i t must be re-

c a l l e d t h a t i t was developed fo r dry f r i c t i o n o r boundary lub r i -

ca t on cond i t i ons be fo re so lu t i ons t o the e lastohydrodynamic

lu b r ic a ti o n problem became av ai la ble . The elastohydrodynamic

l u b r i c a t i o n o f b a l l b e ar in gs

w i

1.1 be co ns ide red i n Chap ter 8,

Sec t ion

8.9.

From F i gu re 3.10 t he ba l l s p i n ro t a t i on a l v e l o c i t i e s a t t he

inner and outer races can be wri t ten as

o

s i n f i is

uB

s in g i

-

5


33/105

= -

w s i n 6

o s in (eo - c

SO

0

The race-control concept of Jones

1956)

assumes th a t a1 1 the

sp in occurs at one conta ct and none a t the other. the conta ct

a t w hich no sp in o ccurs i s ca l l e d t he co n t r o l l i n g r ace. L i g h t l y

loaded bearing s may depart somewhat from t h i s si tu at io n.

f uSi

ahd

us

are made zero

i n

equations

3.87)

and 3.88). respec t ive ly . t he fo l lo w in g w i 11 resu l t :

I nne r - race con t ro l

d s i n

ei


34/105

where

v

=

c oe ff ic ie n t o f s l i d i n g f r i c t i o n

F =

c o n t a c t l o a d

a

=

semimajo r a x i s o f con tac t e l l i p s e ob ta ined f rom equa t ion

3 . 1 3 )

s

e l l i p t i c i n t e g r a l o f second k i n d o b t a in e d f ro m e q ua t io n

3.29)

Equat ion

3.92)

can be w r i t t en f o r b o th the ou te r- and i nne r -

race con tacts . Outer -race co n t ro l w i l l e x i s t i f

Mso

Msi.

I nne r - r ace con t ro l w i l l e x i s t i f

MSU < MSi.

n a g iv e n b a l l

bear ing t h a t opera tes under a g ive n speed and load, r o l l i n g

w i

t a k e p l a c e a t one r ac e and s pi n n in g a t t h e o t he r. R o l l i n g w i l l

the re fo re take p lace where MS i s g r e a t e r because o f t h e

g r e at e r g r i p p i n g a c t i o n .

The p o s i t i o n s o f t h e b a l l c e n t e r a nd t h e r a c e c u r v a t u r e

c e n t er s a t a n gu la r p o s i t i o n a r e shown i n F i g u re 3.11 w i t h

and w i t ho u t an app l i e d combined l oad . I n t h i s f i g u r e the ou te r -

race curv a tu re i s f i x ed . When speeds a re h igh and th e ce n t r i fu -

ga l fo r c e i s app rec iable , the i nne r and ou te r- r ace con tac t ang les

become d i ss im i l a r . Th i s r es u l t s i n the ou te r- r ace c on tac t ang le

B

b ein g l e s s th an t h e i n i t i a l c on t a c t an gle

ef

as shown

i n F ig ur e 3.11.

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

inne r and ou te r r i ngs

at

t h e a x i a l d i s t a n c e b e t w e e n t h e l o c i

o f inner - and ou te r - race curva tu re cen te rs i s

L1

D

s i n

f + d t

(3.93)


35/105

Furthermore i n accordance w i t h a r e la t i ve ra d i a l d isplacement o f

t h e r i n g c e n te rs

6

t he r a d ia l di sp lacement between th e l o c i

o f t he r ac e c u r v a t u r e c en te r a t each b a l l l o c a t i o n i s

L2

= D cos + dr cos

3.94)

where

J l

2 r ( j

-

1)

J

=

ls2,me.Sfl

n

(3.95)

and n i s the number o f ba l ls . From F ig ure 3.11 th e fo l lo wi ng

equat ions can be wr i t ten:

s i n B~

=

L1

- L3

( f i

0.5

6 i

The fo l lowing re la t ionships can thus be w r i t t e n w i t h r ef er en ce

t o F ig u re 3 11:

2

2

L4

+ L3 - [ d ( f o

-

0.5)

* do]*

= 0

(3.100)

2

( 0 cos

pf +

a cos

L4) +

( 0 s i n

~ + a t L3)

2

The

fo rc es and moments ac t i ng on th e b a l l a re shown i n F ig -


36/105

ur e 3.9. The norma l fo rc es shown i n t h i s f i g u re can be w r i t t e n

from

eq ua t io n (3.34) as

E q u i l i b r iu m o f f o r c e s i n t h e h o r i z o n t a l and v e r t i c a l d i r e c t i o n s

r e q u i r e s t h a t

M

F, s i o f1 -

F~

s i n si cos so - ( 1 - r lcos f l i ] = o

M

Fo cos

0 -

F . cos B~ s i n f1 - (1 sin fliJ - Fc = O

1

where

A - 1 f o r o u te r- ra ce c o n t r o l

A

=

0 f o r in n er -r ac e c o n t r o l

The ce n t r i f u g a l f o rce i n equa t i on (3.105) can

be

w r i t t e n a s

1

Fc = md u

e c

(3.106)

where

d = d

2L4

- 2 d f

- 0.5)cas flf

e e

o

(3.107)

and m i s t he mass o f t he ba l l . A j so th e gy roscop ic moment i n

equa t ions (3.104) and (3.105) can be w r i t t e n as

C Mg =

p ~ s u c

i n 5 (3.108)


37/105

where

1

i s mass momerrt

o inert

i a o f Ltle b i ~I

F ~ * ~ I I I

ttese

P

r e l a t i o n s h i p s , e q u a t i o n s 3.104) and (J.105) can be wr i t t en as

312

Ko6

L3 -

~ ~ 6 3 ( 0 i n fif + 6,

-

L3)

O

d f o -

0 g

a0

d ( f i

- 0.5 + ai

1

-

A) O cos 0f

+

a

cos

J

-

d(f i

-

0.5 ) + ai

(3.109)

2 1 u ~

i n c 1

-

r ) D

cos

sf

+

a t

-

a

P B C

-

d ( f

-

0 . 5 )

di

Equ ations 3.100), 3.101). 3.109), and 3.110) can be so lve d

s i ~ u l t a n e o u s l ~o r L j

L4

aO

and ai a t each b a l l

l o c a t i on once the va lues o f at and

6

a re assumed. The

Newton-Raphson method i s ge ne ra l ly used t o so lve these s imul-

taneous nonl near equat ions.

To f i n d how good th e i n i t i a l guess o f t h e v a lu e s o f

a

and at i s, a c o n d i t i o n o f e q u i l i b r iu m a p p l i e d t o t h e e n t i r e

b e a r i n g i s u sed

n

- s i n t l i j -

2 1

- A A ) M

t

d

cos

sij]

=

3.111)

j = l . . .

2 1 - A ~ M


38/105

q s i n

j

os j

o

j=l

.

Having computed values f o r L3,

L 4 ,

ci, and

do

a t each b a l l

p o s i t io n and knowing Ft and Fr as in p u t con di t io ns, we can

o b t a i n t h e v a lu e s o f at and a fro m equ atio ns 3.111) and

3 . 1 1 2 ) A f t e r o b t a in i n g th es e v alu es f o r

st

and ar i t i s

ne ce ssa ry t o r e p e a t t h e c a l c u l a t i o n s f o r

L3 L4

and

6o

a t each b a l l p o s i t i o n u n t i l t h e assumed v al ue s o f at and

r

agree w i t h these values found f rom equ at lons 3.111) and

3.112).

3.5 ~ a t i ~ u ei f e

B a l l bear ings can f a i l f rom numerous causes, i nc lud ing

f a u l t y h a n d l i n g and f i t t i n g , wear a s s o ci a te d w i t h d i r t , damage

t o th races o r separators, and fat igu e,

However,

i

they sur -

v i v e a1 1 th e o t h e r h az ard s, b a l l b e a r in g s e ve n tu a l l y f a i l be-

ca use o f f a t i g u e o f t h e b e a r in g m a t e r i a l . F o r t h i s re as on t h e

s u b j e ct of f a t i g u e c a l l s f o r s p ec i a l c o n si d er a t i on . F a ti gu e i s

caused by th e repeated s t resses developed i n the c ont ac t areas

between the b a l l and the , races and man i fes ts i t s e l f as a fa t i gu e

c r ac k s t a r t i n g a t o r b e lo w t h e s u rf ac e .

The fa t i g u e crack pro-

p a g a t e s u n t i l a p i e c e o f the r ac e o r b a l l m a t e r i a l s p a l l s o u t

and produces the fa i lu re . t y p i ca l fa t ig ue spa11 i s shown i n


39/105

Figure 2.24. On a micr osca le we can surmise t h a t th er e

w i l l

be

a w ide d ispers ion i n ma te r ia l s t rength o r res is tance t o f a-

t igue because o f Inhomogeneit ies

i n

th e m at er ial . Bearing ma-

t e r i a l s are complex a1 oy s and are thu s n ei th e r homogeneous nor

e q u a ll y r e s i s t a n t t o f a i l u r e a t a l l p o in t s. T he re fo re t h e f a -

t i gu e process can be expected t o be one i n which a group o f ap-

parent

l y

i d e n t i c a l b a l l b ea rin g s s ub je c te d t o i d e n t i c a l lo ads

speeds lu br ic at io n and environmental co nd i t io ns e xh ib i t wide

v a r i a t i o n s i n f a i l u r e t im es. For t h i s re aso n t h e f a t i g u e p ro -

cess m ust be t r e a t e d s t a t i s t i c a l l y . Tha t i s t h e f a t ig u e l i f e

of a b ea rin g i s n o rm a lly d ef in ed i n term s o f i t s s t a t i s t i c a l

a b i l i t y t o s urv iv e f o r a ce r t a i n pe r iod o f t ime .

3.5.1 Load Factor

The predominant factor i n d et erm in in g t he f a t i g u e l i f e o f a

b a l l bear ing i s t he load f ac to r. The re la t ion sh ip between l i f e

and load developed here i s based on

a

we l l - l ub r ic at ed system and

a

bearing made o f a ir -m e lt e d m ate rials . To p re d ic t how lon g a

p a r t i c u l a r b e a r i n g w l l run under a spe c i f i c load two esse n t ia l

p ieces o f in format ion are requ i red:

1) An accurate w a n t a t ve es tima te o f t he 1 f e

d i spers ion or scat ter

(2 ) An express ion f o r the dynamic load capa c i ty o r


40/105

abt 1 y o the bear ing t o endure g iven load

f o r a s t iyu la ted number o f s t ress cyc les or

r e v o l u t i o n s

t y p i c a l

distribution

o f t he fa t i gu e l i f e o f i d e n t i c a l

b a l l b e a ri ng s op e r at in g un der n o m in a ll y i d e n t i c a l c o n d it i o n s i s

p resented i n F igu re 3-12 Th is f i g u re shows th a t the number o f

revo lu t ions that a bear ing can comple te w i th 1 percent

proba-

b i l i t y o f s urv iv al .

Ca i s

zero. A l te rn at iv e l y th e proba-

b i l l t y o f ny b e a ri n g i n t h e p o p u la t io n h a vi ng i n f i n i t e endur-

a

ance i s zero. F ai lu re i s normal ly assumed t o have occurred when

t h e f i r s t s p a l l i s observed on a load-carry ing sur face.

Bearing manufacturers

have

chosen

t o

use one or two points

n

the

curve i n Figure 3.12 t o descr ibe bear in g endurance:

1) The f a t ig u e l i f e t h a t 90 percent o f the bear ing

papu la t ion w i l l endure (LO)

( 2 )

The median l i fe , that is ,

t he

l i f e t h a t 50 p er ce nt

o f t h e p o p u l a t i o n

w i l l

endure LS0)

Bearing manufacturers lmost u n i v e rs a l ly r e f e r t o a N r a t i n g

1 i f e w as a measure of t he fat igue endurance of a given bear ing

opera t ing under g i v e n l o a d c on dit io ns . T h is r a t i n g l i f e i s

the e st {mated L10 fa t ig ue 1 f e

of

a l a rg e popu la t ion o f such

bearings operat g under the speci f ied loadi

ng

Fat igue 1 Xe i s genera l

y

s ta te d i n

rni

l ion s o f revo lu-


41/105

t ions. As an a l t er na t ive

i t

may be and f requ ent l y i s g iven i n

hours of successful operat ion a t a given speed.

Weibu ll ( 1949) has pos tu la ted th a t the fa t i g ue l i v e s o f a

homogeneous group o f b a l l bearing s are d i spersed accord g t o

th e f o l l o ~ i n g e l a t i o n :

where

i f e , m i l l i o n s o f r e v o lu t io n s

e

disp ersio n exponent (s lo pe of W eibul l p l o t ) o r measure of

s c a t te r i n b ea rin g l i v e s

ons tant, such t ha t e I n i s ve r t i ca l i n t e r cep t on

Weibul l plot when

=

1

The f a t ig u e l i f e

L

i n equat ion (3.113) i s the L10 l i fe ,

b u t i t i s s imply ref er re d t o here and throughout t h e remainder

o f t he book as f a t i g ue l i f e L.

The so-cal led Weibu ll d i s t r i b u t i o n g i ven i n equation

(3.113) re su l t s f rom a s ta t i s t i ca l theory o f s t rength based on

th e theory of pr ob ab il i ty , where th e dependence f strength on

volume i s expla ined by the d ispers ion i n ma ter ia l s t rength.

This i s the weakest l i n k M heory . Equat ion (3.113) i s used f o r

p l o t t i n g f a t i g u e f a i l u r e s t o d e te rm in e t h e L1 l i ves .

t y p i c a l U e i b u l l p l o t of b e ar in g f a ti g u e f a i l u r e s i s g iv e n i n

F igure 3.13. The ex pe r i i i n ta l res u l ts shown as c i r cu la r po in ts

i n t h i s f l g u r e c on fi rm t h a t b ea ri ng l i v e s conform w e l l w i t h t h e

W e ib u ll d i s t r i b u t i o n and t h a t t h e b e a r in g f a t i g u e d a t a w l l p l o t


42/105

as a s t r a i g h t l i n e .

W it h a te ch ni qu e f o r t r e a t i n g l i f e d i s p e r s i o n now a v a i l -

able , an exp ress ion f o r the dynamic load cap ac i t y t h a t

a

b e a r i n g

can ca r r y fo r a g i ven number o f s t ress cyc les w i th a g i ven p rob -

a b i l i t y o f s u r v i v a l m ust be d e ri ve d .

From the weakest- l ink

theo ry we ge t the r e l a t i o ns h i p be tween the l i f e o f an assemb ly

the bea r ing ) and i t s components th e inne r and ou t e r r i n gs ) :

F o r b a l l b e a r i n g s

e

=

1019. The fo l l o w in g expres s ion can be

w r i t t e n f o r th e fa t i g u e l i f e o f e l l i p t i c a l con tac ts

where

F = s t a t i c l o ad c a p a c it y

dynamic load cap ac i ty

U sin g t h i s e q ud t io n and ch ang in g t h e f a t i g u e l i f e f ro m m i l l i o n s

o f r e v o l u t i o n s t o h ou rs of s u cc e ss ful o p e r a t i o n a t a g i v e n

speed, we can w r i t e equat ion 3 .114) as

The

s ta t i c loads Fi and

Fo

can be ob ta ined f rom e i th e r


43/105

Sec t ion

3.3

or 3.4

or

' the appropriate load and speed condi-

t io n . I n equat ion (3 .116) N i s e xpre ssed i n r e vo lu t i o n s p e r

minute, and the fa t igue l i f e i s expressed i n hours of successfu l

opera t ion a t the g iven speed

N

From Lundberg and Palmgren

(1947)

the dynamic load ca pac ity

o f t h e in ne r r i n g can be wr i t ten as

where

f maximum orthogonal subsurface shear stress

ai =

r a t i o

of depth

of

maximum shear stress

o f

in ne r r i n g t o

semiminor ax is o f con tact e l l pse,

zO b

( k.1

ui number o f st re ss cy cl es p e r r e v o l u t i o n o f i nn er r i n g

With proper changing of subscr ipts f rom

t o

o, equation

(3.117) can represent t he dynamic load cap acity o f the o uter

r i n g

Co

The number

o f

s t ress cyc les pe r revo lu t ion

denotes

t he

number of bal ls that pass

a

given point (under load)

on

the race

of one piny whi le the other r ing I tas turned through one complete


44/105

revo lu t i on .

t he i n n e r r i n g p er u n i t o f t tm e i s

-

(d d cos

i

I n e qu at io n

(3,117)

the dianleters

o f

the

inner and outer

races are wr i t ten as

i

=

d -

d

CQS ei

(3.120)

do

= d cos B0

(3.121)

Hamrock and Anderson (1973) f ound tha t fo r mos t ba l l bear -

i n g

onfigorhtions the

v a r i a t i o n o f T and

@

i s such

th t

the

following approximation can be made:

Table

3.3

presents corresponding values for

l / k ,

T

and

@

as

we1

1

as va lues o f ( T ~ I T ) ~ ~ ( @ I @ ~ ) ~ * ~or corresPondin9

values of I l k , From these values the fol lowing simple formula

can

be

w r i t t e n :

Table

3.3 a

so

shows the

good accuracy

o f t t l i s

apyrox imale


45/105

formula.

I n equ at io n 3.117) th e curvatu re sum R can be obta ined

from equa t i on 2.24), and the e l l i p t i c i n te g r a l o f t he second

k i n d and t h e e l l i p t i c i t y param eter

k

can be obta ined f rom

equat ions 3.29) and 3.28). re sp ec t iv ely . y mak ing use o f the

s t a t i c l o a d s

Fi

and Fo obta ined f rom ei th e r Se ct ion 3.3

or

3.4

and equa t ions 3 .117 ) t o 3 .123 ), t he fa t i gue l i f e i n

ope ra t i ng hou rs o f t he b ea r i ng can be ob ta i ned from equa t i on

3.116).

The dynamic lo ad ca pa ci ty C j u s t deve loped can be used t o

d e te n ni ne th e r e l a t i v e im porta nce o f c e n t r i f u g a l e f f e ct s i n b a l l

bea ri ngs o f d i f f e re n t s izes . This

was

done by Hamrock and

Anderson 1973) y co mp arin g t h e r a t i o o f

d 3 ~ ? t o t h e dy-

namic load capac i ty

C

I n t he p revious chap te r

t

was noted

t h a t

db

i s t h e b o r e d ia m ete r i n m i l l i m e t e r s and N i s t h e

r o t a t i o n a l s peed

i n

re vo lu t io ns per minute. The fa c to r d3N2

i s pro po r t ion a l t o the ce nt r i f ug a l fo rce , and the dynamic capa-

c i t y i s a measure o f t he l oad capac i t y o f t h e bear ing . Fo r

e x t r a - l i g h t s e r ie s a ng ula r- co nt ac t b a l l b a r f ngs ope ra t i ng a t a

va lue of d,~

of

3

m i l l i o n , T a b l e

3.4

shows the ts a t io .o f d N

t o dynamic

capaci ty C

f o r fou r bore d iameters db. Ce ntr i -

f ug a l e f f ec t s a re shown to be re l a t i v e l y more seve re i n sma l l

be arin gs when d ~ s kep t constant .

The e f f e c t o f r a ce c o n f o r m i t y r a t i o

on f a t ig u e l i f e a t


46/105

h i gh o pera t ing speeds i s shown i n Fig ure 3.14. Th is f ig u r e was

obta ined f rom Winn, e t a l . (1974) f o r a 20-mn-bore b a l l bea r ing

o pe ra ti ng a t 120,000 rpm.

Note th a t an i nc rease i n ou te r - race

c u r va t u re b r i n g s a bo ut a s u b s t a n t i a l d ec re ase i n f a t i g u e l i f e .

On the o ther hand an inc rease i n inner - race curv a ture does no t

a f fec t the 1 f e t o any apprec iab le degree. The reason f o r t h i s

i s

t h a t a t h i g h speeds t h e c e n t r i f u g a l f o r c e a c t s a g a i n s t t h e

outer race.

I t i s t hu s im po rt an t i n o p ti m iz i ng t h e b ea ri ng l i f e

n h igh-speed appl i c a t ons th a t t he ou te r- race con fo r l n i t y r a t i o

sho uld remain as low as po ssi ble . Confo rmity expressed by a

c ur va tu re r a t i o o f

0.515

t o 0.520 represen ts the lowes t

th res ho ld o f p resent manuf ac tu r i ng p rac t i ce s .

The con ta c t ang le i n ba i

l

bear ings i s ex t reme ly import ant

inasmuch as i t c r i t i c a l l y a f f e c t s t h e b e ar in g s t i f f n e s s and

1 f e . T yp ic al v a r i a t i o n s o f f a t ig u e 1 f e w i t h i n i t i a l c or lta ct

ang le

Bf

f o r a med ium-size bear ing opera t ing a t a va lue o f dbN

o f 1 .5 m i l l i o n are shown i n F ig ure 3.15. The contac t -ang le range

sugges ted i n F i gu re 3.15 i s t y p i c a l of b ea ri ng s o pe r a ti n g a t h i g h

speed s.

I n r ec e n t y e ar s b e t t e r u n d e rs ta n di ng o f b a l l b e a r in g de-

s ign , m ater ia ls , p rocess ing , and lu b r i ca t i o n has pe rm i t te d an

improvement i n be ar in g performance.

T hi s r e f l e c t e d i n e i t h e r

h i ghe r bea r i ng

re

i a b i

l

t y o r l o n ge r e x pe cte d

1

ves than those

obta ined f rom equat ion (3.116) o r b a l l bear ing ca ta logs . s a

r e s u l t Bamberger, e t a l . (1971) a r r i v e d a t an e x p re s s io n f o r t ir e

a d ju s te d be a r in g f a t i g u e 1 f e


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.. .---

La EFGL

where

m a t e r i a i f a c t o r

E

p ro ce s s in g f a c t o r

= l u b r i c a t i o n f a c t o r

G ,= ha rdness fac to r

The nex t th ree sec t i ons dea l wi th t hese fac to rs .

3.5.2

L u b r i c a t i o n F a c t o r

I f

a b a l l b ea rin g i s adequate ly des igned and lu b r ic a te d ,

t h e r o l l i n g su r f a c e s c an be se p arat ed b y a l u b r i c a n t

f i l m

En-

durance te s t i n g o f bea rings , as repo r ted by T a l l an , e t a l

1965). has demonstra ted th a t when the lu b r ic an t

f i

m i s t h i c k

enough t o s e p ar a te t h e t wo c o n t a c ti n g b od ie s, f a t i g u e l i f e o f

the ~e a r i n g s g r ea t l y extended . Converse ly , when the

f i l m

i s

n o t t h i c k enough t o p ro v i d e f u l l s e p a ra t io n b etw een t n e a s p e ri -

t i e s i n t he c o nt a ct zone, t he l i f e of t h e b e a ri n g i s ad ve rs ely

a f f e c t e d by t h e h i g h sh ea r r e s u l t i n g f r o m d i r e c t m e ta l- to -me ta l

con ta c t . An exp ress ion f o r the f i m t hi ck n es s i n b a l l b e ar in g s

i s d ev elo pe d l a t e r ,

b u t i t i s conven ien t t o i l l u s t r a t e i t s

e f f e c t on fa t i gu e l i f e i n t h i s s ec t io n.

o e s t a b l i s h tihe e f f e c t o f finl th ickness on the lift f

any g i ven bea ri ng , we f i r s t ca l cu l a te t he f i l m paramete r A


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The re l a t i on sh ip between h and the

f i l m

th i ckness

h

i s

where

f r

=

rms s u r f ace f i n i s h o f r ac e

f

rms s u rf ac e f i n i s h o f b a l l

A m ore d e t a i l e d d is c u ss i o n o f s ur fa ce to po gr ap hy i s g iv e n i n

Sec t i on 4.1, and the rms i s de f ine d by equ at ion 4.2).

With the f i l m parameter A known, Figure 3.16 can be used

t o d ete rm in e t h e l u b r i c a t i o n f a c t o r

r.

N o t e f r o m t h i s f i g u r e

t h a t w h e i t h e f i l m parameter va lues f a l l be low approx imate ly

1.2, t he b e a r in g f a t i g u e l i f e i s a dv er se ly a f f e c t e d s in c e i s

l ess than

1

Conversely, when th e v.sllres o f ar e between 1.2

and 3, b e a r i n g f a t i g u e l i f e i s , a p p r e c i a b l y exten de d. F i l m pa-

rame ters h i ghe r t han

3

do n o t y i e l d a ny f u r t h e r im provem ent i n

t h e l u b r i c a t i o n f a c t o r m a i n ly because a t t he se v al ue s o f

A

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

f i l m

i s t h i ck enough t o separa te the ext reme peaks

o f t h e i n t e r a c t i n g s ur fa ce s.

3.5.3 M at er ia l Fac tor

Bamberger, e t a l. 1971) have shown th a t be ar ing ma te r ia ls

c a n s i g n i f i c a n t l y a f f e c t t h e u l t i m a t e pe rf or ma nc e o f a b e ar in g .

A s ment ioned i n Chapter 2 th e most f reque nt Py used

st l

f o r

b a l l b e ar in g s i s l S l

52100.

The dynamic loa d cap acity , as ca l-


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cu la ted f rom equat ion 3.117) o r any be ar ing manu fac turerss ca t -

a log, i s based on a i r -me l ted 52100 st e e l t h a t has been hardened

t o 58 Rockwel l

C R,).

Because o f improvements i n th e q u a l i t y

g a i r -m e l ted s tee ls , Bamberger, e t a l . 1971) suggested the

v a lu e o f t h e m a t e r i a l f a c t o r shown i n T a bl e

3.5.

Fac to rs

ta k i ng i n t o account vacuum remel t ing, hardness, and ot he r pro-

cess ing va r ia bl es are considered separa te ly . Many o f the mater-

i a l s i n t h i s t a b l e were d isc usse d i n S e ct io n 2.4, and the chemi-

c a l compos it ions o f many o f t hese s t ee l s a re g i ven i n Tah le 2.1.

3.5.4 Proc essing Fa ct or s

Improvements i n processing tec hnique s have als o extended

fa t i gu e l i f e . The va r ious me l t i ng p rac t i ces have been d i s -

cu sse d i n S e c t io n

2 4 1

Zare tsky, e t a l 1969) found tha t

consuma ble-electrode vacuum re m el tin g CVM) gave up t o 13 t imes

longer l i f e than a i r me l t ing . Hgwever, Bamberger, e t a l . 1971)

recommended that a processing factor f

o f 3 be used f o r

a l l

CVM

bea ring stee ls. This valu e may be somewhat con serv at ive ,

bu t t he con fi dence fac to r f o r ach ievi ng th i s l e ve i o f improve -

ment i s h igh .

A no th er p r o c es s in g f a c t o r t h a t s e r i o u s l y a f f e c t s b e a ri ng

fa t i g u e 1 f e i s m a te ri a l hardness. The minimum recomnended

hardness f o r b a l l b e a r ln g s t e e l s i s 8 RE A drop i n ha rdness

from t ha t va lue because o f e i t h e r poo r heat t rea tnwn t o r h i gh


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operat ing temperatures w i apprec iab l y sho r ten the bea r i ng fa -

t i g u e 1 i f e , as po in ted ou t by Bamberger, e t a l e 1971). To

enable an est im ate t o be made o f th e e f f e c ts o f hardness change

on bea r i ng l i f e , a hardness fa c t o r

i s d e f i n ed as

where

Rc

i s t he ope ra t i ona l hardness o f t h e bea r i ng mate r-

i a l , N ote t h a t t h e r e l a t i o n s h i p p re se nt ed y equa t i on 3.126)

i n d ic a t es t h a t b ea ri ng l i f e i s h i g h l y s e n s i t i v e t o changes i n

hardness. Thus, f o r example,

a

two-po in t d rop i n hardness t o

56

R

w i l l cause a

32

p e rc e nt d r op i n b e ar in g f a t i g u e l i f e .

Once the va r i ous fac to rs i n equa t i on 3.124) have

been

de-

f in e d , t h e ad j u st e d f a t i g u e l i f e L a c an be c a l c u l a t e d f r o m

t h a t e qu a ti on . T h i s e qu a t io n e n ab le s t h e d e si gn e r t o a r r i v e a t

a more r e a l i s t i c e s t i m a t e o f b e a r i n g f a t i g u e 1 f e .

3.6 Bear i ng Lub r i ca t i on

Wi thou t adequa te l u b r i c a t i o n o f t h e ba l l - race con junc t ion ,

various degrees of damage w i l l r e s u l t t o t h e r o l l i n g ele rre nts

r

the races, o r both. These inc lud e th e development o f scuf f ing ,

p l a s t i c f lo w , a nd p i t t i n g . The f a t i g u e

l i f e

o f t n e b a l l - r a c e

con tac t t he re fo re depends on t h i s con junc t ion hav ing

an

adequate

l u b r i c a n t f i l m as p o i n t e d o u t i n t h e p r e v i ou s s e ct io n ,


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For many years the op in ion preva i ed that th maximum con-

t a c t pr es s ur e i n t h e b a l l - r a c e c o n t a c t pr ec lu d ed

t he

poss ib f l i t y

o f a l u b r i c a n t

f i l m

e x i s t in g i n t he con junc t ion , However,

t

i s

now gen era l l y accepted no t on l y t h a t a lub r i c an t

f i l m

i s pre-

s en t, b u t a l s o t h a t t h e n a t u r e of t h e l u b r i c a n t

f i l m

has

an

im-

p o r t a n t i n f l u e n c e on t h e f a t i g u e l i f e o f t h e b e ar in g . B esid es

p r o v i d i n g a f i l m t h e l u b r i c a n t i n a b a l l b e a r in g m ust p r o v id e

cor ros ion p ro t ec t io n and ac t as a coo lan t .

Not o n l y t he b a l l - ra c e c o n ta c t b u t a l s o a l l t h e i n t e rf a c e s

between moving elements must be pr op er ly l ub r ic at ed . The ba l l -

separator arid race-separator con tacts expe r ience m os t ly impact

load ing and t he re fo re have g rea te r po s s i b i l i t i e s o f me ta l -t o -

metal co nta ct, even when the b ea ring has an adequate supply o f

l u b r ic a n t . For t h i

s

reason th e separator su r f aces are genera l l y

coa ted w i t h

a

l o w - f r i c t i o n m a t e r i a l .

The ba l l - r ac e con tac t s i n b a l l bear ings can gen era l l y be

s a t i s f a c t o r i y l u b r i c a t e d w i t h a s ma ll amount of a p p r op r ia t e

l u b r i c a n t s u p p l i e d t o t h e r i g h t a re a w i t h i n t h e b e a ri ng . The

m a jo r c o n s i d e ra t i o n s i n pr o pe r b a l l b e a r i n g l u b r i c a t i o n a re

1)

S e l ec t io n o f a s u i t a b l e l u b r i c a n t

2 ) S e l e c t i o n o f a s ystem t h a t w i l l provide an adequate

and co n st an t f l o w o f t h i s l u b r i c a n t t o t h e c o n ta c t

These two t op ics a re cons idered i n t he f o l l o w i ng sec tions.

3.6.1 L u b r i c a n t s


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Bo th o i l s and greases a re ex tens i ve l y used as l ub r i c an ts

f o r a l l t yp e s of b a l l b ea ri ng s o ve r a wide range of speeds and

opera t ing tempera tures. The cho ice i s f r eq ue nt ly de termined by

cons ide ra t i on s o t he r than l ub r i ca t i on requ iremen ts a lone .

Because of i t s f l u i d i t y

o i l

has a number o f advantages over

grease:

I t

c an e n t e r

tilt

o ad ed c o n ju n c t i o r i m os t r e a d i l y t o

f l u s h away contaminants such s wate r and d i r t and pa r t i c u -

l a r l y t o t r a n s f er heat rom heav i ly loaded bear ings.

I t i s a l s o

f re que n t l y ddvan tageous t o l u b r i c a t e bea rings from t c e n t ra l o i 1

system used fo r o t he r machine par ts.

Grease however

i extensive ly used because i t p e rm i t s

simp1i e d des igns

of

housings

and

bear ing enc losures wh ich

re qu ir e le s s maintenance and because i t

i s

more e f f e c t i v e i n

sea l i n g aga ins t d i r t and contam inan ts . I t a lso reauces poss ib le

damage t o the p rocess o r p roduct f rom o i l leakage.

i

1 L u b r i c a t i o n

E x c e p t f o r

a

few s pec ia l requ iremen ts pe t ro leum o i l s sa t -

i s f y mos t ope ra t i ng cond i t i ons . H igh -qua l i t y p roduc ts f r ee

f rom adu l te ran ts tha t can have an abras ive o r lapp ing act ion

are recomnended. Anima l a r vege tab le o i l s o r pe t ro leun i o i l s o f

p oo r q u a l i t y te n d t o o x i d i z e t o d ev el op a ci ds and t o f o rm

s lu dg e o r r e s i n l i k e d e p os i t s on the be ar ing surfaces. They thus

pena e bea ring performance o r endurance.


53/105

composite o f reconmended lub r i ca n t v i s co s i t i es a t 38 C

(100

F)

i s shown as Figu re

3.17.

I n

many b a l l bearSng applica-

6 2

t i o n s an o i l e q ui va le nt t o an

SAE-10

m otor o i l

(40x10

m /s , or

40

cS, a t 38 C (300 F ) ) o r a l i g h t t u r b i n e o i l i s t he most

frequent choice.

For a number o f m i l i t a r y app l ica t ion s where th e opera-

t i o n a l r e q u i r e ~ n e n t sspan the temperature range -54 t o

204

C

(-65

t o 400

F),

syn t h e t i c o i l s a r e used. Es t e r l u b r i ca n t s a re

most f reque nt ly employed i n t h i s temperature range.

I n a pp li ca -

tions where temperatures exceed 260

C

(500

F),

most syn the t ics

w i l l

qu ic k l y b reak down, and e i t h e r a so l i d l ub r i ca n t (e.g.,

nos2) o r a po lypheny l e the r i s recanmnded. more de ta i l ed

d iscus s ion of sy n th e t i c lu br i ca n ts can be found i n B isson and

Anderson (

1964).

Grease Lu br ica t io n

The s imple st method o f l u b r i c a t i n g a b e a r in g i s t o a p p l y

grease, because o f i t s r e l a t i v e l y n o n f l u i d ch a r a c t e r i s t i c s .

Danger o f leakage i s reduced, and th e housing and enclos ure de-

s ign can be s imp le r and l ess c os t l y t han those used w i t h o i 1.

Grease can be packed i n t o bear ings and re ta in ed wi th inexpensive

closures, b u t packing should no t be excessive and

the

manufac-

t u r e r ' s rocommendat ions should be closely adhered to.


54/105

The m a jo r l i m i t a t i o n o f g rease l u b r i c a t i o n i s t h a t i t i s

n o t p a r t i c u l a r l y u s ef u l

i n

high-speed appl icat ions.

I n

general

i t i s n o t employed f o r speed f a c t o r s ( d b ~ ,b ore i n

m i

1 i m t e r s

t imes speed i n rev o lu t io ns pe r m inu te ) over 200,000 al though

sel ect ed greases have been used su cc es sf ul ly f o r h ig he r speed

f a c t o r s w i t h s p e c i a l l y d esig ne d b a l l b e ar in gs .

Greases vary widely i n propert ies, depending on the type

and grade o r cons i s tency

.

Fo r t h i s re as on w spec i f i c recom-

mendat ions can be made. Greases used f o r most bear ing op er at ing

c o n d i t i o n s c nn s i s t o f p et ro le um , d i e s t e r , p o l y e s te r , o r s i i c o n e

o i l s t h i ckened w i t h sodium o r l i t h iu m soaps o r w i t h more recen t-

l y deve loped nonsoap th ickeners , Genera l ch ar ac te r i s t ic s of

greases are as fo l lows:

1)

Petroleum o i 1 greases are be st f o r general-purpose op-

e r a t i o n f r o m

-34

t o

149'

C

(-,30

o

300

F) .

( 2 ) Die s te r o i l g reases are designed fo r low-- temperature

ser v ice down t o -54

(-65'

F .

(3)

E ste r-b ase d g re as es a r e s i m i l a r t o d i e s t e r o i l g re ase s

b u t h a v e b e t t e r h igh-temperature cha rac ter i s t cs , cover in g a

range from -73' t o 177' (-100 t o 350

F .

( 4 ) S i l i c on e o i 1 greases a re used f o r b ot h high- and low-

temperature operat ion, over the widest temperature range o f a11

greases

(-73'

t o 232'

C -100*

t o

450'

F), b u t have t he d isad-

van t ige o f 1ow 1aad-carry in g c apacity.


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( 5 ) F lu o r o s i1 cone

4

1 greases have a11 t he des i rab le fea -

t u r e s

o f

s i cone 05 1 g reases p l us good load-car ry ing capa c i ty

and resis tance t o fuels, so lvents, and co rro si ve substances.

They have a very low v o l a t i l i t y i n vacuums down to

log

t o r r ,

which mzkises them u se fu l i n aerospace y p l c a t ons.

(6 ) Perf 1u o ri na te d o i 1 greases have a h ig h degree of chemi-

c a l ine rt ne ss and ar e completf ly nonflamnable. They have good

load-carry ing ca pa ci ty and can operate a t temperatures as h ig h

as

288' C

(550 F )

f o r l on g periods, ~ h i c h~ak es hem usefu l

i n

the chemical process and aerospace indu str i es, where h i gh re l i a -

b i l t y j u s t i f i e s th e ad d it io n a l c os t.

Grease consistency i s important s i nc e grease w i 11 slump

bad ly and churn excess ive ly when too s o f t and f a i l t o lub r i c a te

when to o hard. E i t he r co nd it io n causes improper lubr ic at i on ,

excessive temperature r is e, and poor performance and can shor ten

bear ing i fe .

A v al ua bl e g ui de t o t h e e s t i m a t io n o f t h e u s e fu l l i f e o f

grease i n ro l l ing-e leme nt bear ings has been publ ished by the

Engineering Sciences Data Unit (1978).

t has been demonstrated recently by Aihara and Dowson

(1979)

and by Wilson (1979) that the

f i l m

th ickness i n g rease-

lu br ic at ed components can be cal cu la ted w i t h adequate accuracy

by us ing the v is co s i t y o f t he base o i l i n t he e lastohyd rodynamic

equa t ions see Chapter 8). Ai ha ra and Dowson compared m

thickness measurements made by capacitance techniques

on a


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grease- lubr icated, two-d i sc machine w i t h the p r e d i c t i o n s of

e lastohydrodynamic theory.

i

son repor ted an ex tens ive and

impress ive range o f experSments on a g rease- lub r ica ted r o l e r

bear ing.

This work enables th e e lastohydrodynamic the ory de-

veloped

i n

t h i s t e x t t o be a p p l i e d w i t h c o nf id e n ce t o g rease-

l u b r i c a t e d b a l l b ea rin gs ,

3.6.2 L u b r ic a t io n Systems

The q u a n t i t y o f l u b r i c a n t r e q u i r e d t o m a i n t a i n ad eq ua te

l u b r i c a t i o n o f b a l l b e a ri ng s i s s m al l. D ata p re s en te d by

i cock and Booser

1957)

show t h a t f o r me dium-si ze, deep-groove

b a l l bea ri ngs ope ra t i ng a t modera te l oads and speeds 2 . 1 6 ~ 1 0 ~

d b ~ ) , h e q u a n ti ty o f o i l r eq ui r ed i s a bout

0.5

rnglhr. The

o i 1 re qu ir em en t i s d et ermin ed by t h e s e v e r i t y o f t h e o p e ra ti n g

cond i t i ons .

Some

o f

t h e t e c hn iq u es m ost f r e q u e n t l y used t o l u b r i c a t e a

b a l l b e a r i n g a re a e s c r i bed i n t n e f o l l o w i ng p ara gra ph s.

Fo rced Lub r i ca t i on

A lt ho ug h th e q u a n t i t y o f o i1 r e q u i r e d t o p ro v i d e a de qua te

l u b r f c a t i o n i s s ma ll,

i t

i s f r e q u en t ly d e s i r a b le i n h e a v i ly

loaded h igh-speed bea r ings t o use the o i l t o t

Ball Bearings Mechanics-NASA Report

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