study of ball bearing torque under elastohyrodynamic lubrication

8/14/2019 study of ball bearing torque under elastohyrodynamic lubrication

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N 73 2 7 4 1 4

NASA TECHNICALMEMORANDUM N A S A TM X- 68271

CMOOX


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S TUDY O F B A L L BEARING T ORQUE UND E RE L A S TO H YDR O DYN A MIC LUBRICATION

by D. P. Townsend ,*C. W. Allen,** and E. V. Za re t sky*Lewis Res earch Cen ter

National Aeronaut ics and Space AdministrationCleveland, Ohio 44135

ABST RACTSpinning and rolling torques w ere mea sured in an ang ular-contact

bal l bear ing with andwithouta cage under several lubrication regimesin a modified NASA spinning torqu e apparatus. T wolubricants wereused--a di-2 ethylhexylsebacateand a synthetic paraffinic oil , at shaf tspeeds of 1000, 2000, and 3000 rpm and bearing loads f rom 45newtons(10 Ibs) to 403 newtons (90Ibs) . A n analytical model was developedf r o m previous spinning fr ictio n models to include rolling with spinningunder lubr ication regimes f r o m thin fi lm to flooded conditions. T h ebearing torque values have a wide variation, und er any conditiono fspeed and load, depen dingon the amount of lubr icant present in thebear ing . T he analytical m odel com pared favorab ly with experimentalresults unde r several lubr ica t ionregimes.

NASA-Lewis Research Center , M emb er A S M E .Univers i ty of Cal i fornia , Chico , M ember A S M E .


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N O M E N C L A T U R E2 2A . area of cage land, m ( i n . )

a ma j o rsemiaxis of contactellipse, m ( in . )b mi n o r semiaxis o f contact ellipse, m ( i n . )b ' semiwidth o f contact ellipse at y, m ( i n . )C radial cage clearance, m ( i n . )CD drag coefficient 2E materials properties facto r , N /m (psi)2E - , 2 modulus of elasticity, N /m (psi)e exponente un it vecto r along bearing axisF lubr icant factorF cage force , N( I b )F D f lu id -dy nam ic d rag , N( I b )F T T f r ic t ion forc e due to hysteresis, N( I b )F R friction for ce due torollingdrag, N( I b )Fg f r ic t ion force due tomicroslip, N( I b )f coefficient of frictionh film thickness, m ( i n . )h minim um dis tance be tween ba l l andgroove , m ( i n . )>.> , , * .i , j ,k uni t vector in x, y, and z directionK constant defining outer boundaryo f integrationM total spinning torq ue atball/race i n te rf ace , N -m ( I b - in . )oM .. total spinning torqu e in H ertzian ellipse, N - m ( I b - in . )M 2 spinning torq uedue toviscous drag outside H ertzian ellipse,

N -m ( I b - i n . )


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N number ofballsP normal load, N (Ib)R radius ofball, m(in.)R radius ofequivalent cylinder, m(in.)CR~ radius ofgroove, m(in.)Rj cage land radius, m(in.)R pitch radius, m (in.)RD radius ofrace, m(in.)r polar coordinate, m (in.)r valueof r at outerboundary ofHertzian ellipse, m(in.)oS contact stress, N/m (psi)T total bearing torque, N-m (Ib-in.)T cage torque, N-m (Ib-in.)\~ tT.. bearing torquedue toball spin torque, N-m (Ib-in.)T ^ bearing torquedue to rolling drag, N-m (Ib-in.)TV bearing torquedue toaerodynamic drag, N-m (Ib-in.)T. torquedue tohysteresis, N-m (Ib-in.)TV bearingtorquedue tocage drag, N-m (Ib-in.)Ug surface velocityofball relative tomovingcoordinate system,

m/sec (in./sec)U linear velocity ofball center, m/sec (in./sec)U D surface velocity of race relative to movingcoordinate system,

m/sec (in./sec)U g relative slipvelocity betweenballandrace, m/sec (in./sec)W load per un i t wid th , N/m (Ib/in.)x, y, z m o v i n g coordinate system, m( in.)

2-1 -1a pressure-viscosity exponent,(N/m ) (psi )


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| 3 anglewhich bal l angular velocityvector makeswithbearing axis,deg

0 loaded con tact angle, deg2 2/ j . absolute viscosity, N-sec/m (Ib-sec/in. )o 9I . L ambient viscosity, N-sec/m (Ib-sec/in. )

v Poisson's ratio3 2 4p densityof fluid, kg/m (Ib-sec /in. )oT shearstress, N/m (psi) 2r, transition shearstress, N/m (psi)

0 polar coordinatein the x-yplanefi angular velocity ofballcenter, rad/secfi- angular velocity of inner race, rad/secfi angularvelocity of outer race, rad/sec< j t angular velocityofballwithrespect to rotating coordinate system,

rad/seco > - relative angular velocity ofinner racewithrespecttomoving

coordinate system, rad/seco > relative angular velocityofouter racewith respect tomoving

coordinate system, rad/secO J angular velocity of rolling, rad/seca) angular velocityofspinning, rad/secsSubscripts:1 denotes inner raceo denotes outer race indicates vectors

I N T R O D U C T I O NIn bearing andgear applications considerable powerlossescan

occur even w h e ngood lubrication is present. In rolling-element bearings


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these power losses result in heat generation and increased temperatureof the lubricantand the bearing components. These losses occurdue toanumberoffactors; shearing of the lubricantin the bearing cavities,rubbingof the balls andcage (separator), cage drag, spinningand rollingof the balls in the raceways andchurningof the lubricant. Ball bearingkinematics are also affected bythese losses. Early analytical work [1-3]onpower losses in ball bearings wererestricted to the use ofCoulombfriction at the sliding contacts. More recently it hasbecome evidentthat the Coulombfriction modelis inadequatetocompletely describe allthe conditions in arealbearing [4, 5 |. In the later analysis [5] the EHDlubricant f i lmand the rheological properties of the lubricant were used todetermine bearing friction andkinematics. The methodof [5]uses anexponential modelfor the lubricant pressure-viscositycharacteristicswhi ch may predict powerlosseshigher than thosewhich generally occurin practice.

The torqueof aball spinningin agroovewith d i fferent conformitiesandseveral lubricants was measured [6-8]in the N A S Aspinning torqueapparatus. Later, an analytical modelwas developed[9, 10]which pre-dicted theextent that elastohydrodynamicf i lm contributedto the e f fec t iveball/race separation andspinning torque. Theanalysesshowed thatconventionalelastohydrodynamic lubricationis possible in thecaseof agroove having conformitiesup to 60percent.

Indevelopingthe analysisof [9] it was found that the assumption of aNewtonion fluidwith anexponentialpressure-viscosity relationship gave

Numbers in brackets designate references at end ofpaper.


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impossibly high values oftorque. It wasnecessary therefore to intro-duce a cutoff point at which th epressure-viscosity exponentdecreasedto a much lower value. T he model of |9| yielded satisfactory results

/>when th e calculated shear rate was of theorder of 10 reciprocal sec-onds. Comparison with data ofother researchers [11] showedthecutoff to occur at much higherpressures when the film thicknesses weremuchgreater than tho se of [9]. In other word s, a lower shear rateresulted in a higher cutoff pressure.

T heresearch reported herein, wh ichis based on the work reportedinitially in [12, 13], was undertakento investigate th e losses which occurin an angular-contact ball bearing under spinning an d rolling motionofth e balls. T he objectives were: (a)M odify th e composite viscositylubricant modelto take into account theshear rate dependencyeffects;(b) M easure experimental lythe torque in a thrust- loaded ball bearingwith andwithout a cage; (c)Extendthe previously-developed analyticaltechniques for determining torque of a ball spinning in a nonconforminggroove to thecase of a bearing operating with com bin ed spinning androll-ing; and (d)D etermine analy tically th e effect of lubricant viscosity andf luid-dynamic drag on bearing torque.

APPARATUS, SPECIMENS, AN DPROCEDURESpinning Torque Apparatus

A spinning torqu e apparatus (see figs. l(a) and (b))described pre-viously in |6. 7] wasalso used for thetests reported herein . T he apparatusessentially consists of a turb ine drive , a pneumatic load device, anupperand lowertest specimen, a lower test-housing assembly incorporating a


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hydrostatic airbearing, and a torque-measuring system. Anangular-contact ball bearingcan besubstituted for theupper andlower testspecimen in the apparatus. In operation, the bearing inner race ispneumatically loaded against the balls of the bearing throughthe driveshaft. As the drive shaft is rotated, the inner racerotateswithrespectto the outer race in the bearing. This causes anangular deflectionofthe outer race housing. This angular movement is sensed optically bythe torque-measuring systemand isconverted intoatorque value.During atest, thetorqueis continuously recorded on astrip chart.

Test BearingsT wotypes oftest bearings were used. Both types were conventional

204-size angular-contact bearingswithall butthreeballs removed. Onebearinghad a 26 contact anglewith a52-percent conformity at innerandouter races. The second bearing had a 17 contact anglewith53-and 54-percent conformity at the inner andouter races, respectively.Specifications of the bearings are given inTable 1.

Test ProcedureTestswere conducted in the spinning torque apparatus at room tem-

perature using the twotest bearings bothwith andwi thout acage. Testconditions were 1000, 2000, and3000 rpm andvarying loads from44 .5 N (10 lb) to 403 N (90lb). The tests were conductedwith twolubri-cants, adi-2-ethylhexyl sebacate and asynthetic paraffinic oil.

The bearings were first runwithoutacage andusing oil jet lubrica-tion at arate of 8cc/min. A f t e r the initial tests thebearing was


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8cleanedand a fewdrops oflubricant whichhadbeen dilutedby a 5 to 1addition ofhexanewas applied. Thehexane then evaporated leaving onlyathin f i lm oflubricanton the bearing surfaces. The bearings were thenrun again butonly longenough (usually5 to 15sec)at each speed and loadconditionto reach equilibrium conditions. Finally, the bearings were runwith aninner-race riding cage withjet lubrication.

A N A L Y S I SGeneralized Rheological Model

The composite viscosity lubricant model [9] was modified to take intoaccount the shear ratedependencye f fec t s of a lubricant [12]. The newTheologicalmodel incorporates four parameters: (a)Ambient viscosity u ,(b ) Pressure-viscosity coeff icient or, (c) a lubricant factor F, and(d) transition shear stress, r . These parameters may berepresentedbythe fol lowing relations

c yT = p. e ojy/h and r < r (la)*- Vx

T = p. ea S u> y /h and r < r < FS (Ib)\J \jT= FS and ^ e coy/h> FS (Ic)

The transition shear stress r is introducedtoallowfor large ratios\s

T/'S to be present w h e n thepressure is low. The introduction of alubri-cant factor. F serves to limit theshear stress athighpressures andshearrates to afractionof the normalstress. Inorder to visualize the behaviorof the f luidunder theforegoing conditions.Fig. 2shows aplotofshear


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10properties andgeneralized model were usedin the analysispresented inthis paper.

Origin of Friction TorqueIna lubricated ball bearing there is anelastohydrodynamic f i lm

betweeneachrace andball. There is also some lubricant surroundingthe high-pressure elastohydrodynamic region. Amongthe possible sourcesof bearing f r ict ion in a thrust bearing of the type under investigation, arethe fol lowing:

1. Spinning frict ion arising within the elastohydrodynamic region.2. Spinningfriction due to the lubricant outside o f this region.3 . Rollingresistance due to the lubricant being squeezedout in f ront

of the rolling ball.4. Friction due totranslationalsliding in the elastohydrodynamic region.5. Fluid-dynamic drag of the balls as they orbit about th e center of the

bearing.6. Hysteresis lossesdue toelastic deformationofsteel during rolling.7. Cage viscous drag.T he contributionto the bearing torqueofeach o fthese effects will b e

considered independentlyandthenthe total bearing torque obtained b y thesummat ion of the individual contributions. Fo r the first ordercomputation,coupling betweenthe various effects will b eneglected.

Spinning TorqueAnalysis of thespinning torque is anextension of thework undertaken

previous ly for the spinning of aball in a nonconforming groove [9,10].


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11The coordinate system for aballbearing system is showninFig. 4. Thecontact ellipse for the ball/race interface is shown in Fig. 4(b). As inthe caseof thepure spinning reported in [9, 10]several simplifying assump-tions are made as fol lows:

(a) Thestress distribution is Hertzian.(b) The major axisof the ellipse is assumed to be considerably

greater than the minor axis. For a 54 percent curvature, theratioof themajor tominor axis is 5.2. For curvature lessthan54percent thisratio is larger.

(c) The major axisof the contact ellipse is considerably lessthantheball radius so thattheellipsemay beapproximatedby aplane ellipselying in the x-y plane.

(d) The significant velocities, as far as f i lm thickness and torque areconcerned, are those in the x direction. This fol lows f r o m the secondassumption.

(e)The nonsymmetryof the film thickness in the positive andnegativey directions is assumed to besmallso that momentscan beassumedto bebalanced about the z axis.

f ) The surface roughness is assumed small incomparison withthethickness of the elastohydrodynamic f i lm.

(g) Frictionalresistanceis entirely due toviscousshear,(h) Side leakage is neglected.( i) The f i lm thickness in the x direction is constant. j ) The surfaces are isothermal.O nthe basis of the preceding assumptions, the systemcan be reduced


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12to a n u m b e r ofelementalrollers ofwidth dy inrolling and sliding motionrelative to thegroove (Fig. 4 ).

T he equationfor the resulting spin torque aboutthe z axis is derivedin Appendix I b yintegrating the elemental moments over thewhole contactellipse as:

= / y j (3)In additionto the effect onspinning torque of the lubricant within the

Her tz iancontact region, the effect of the lubricant outside of this regionmust also b econsidered. T heexpression fo r this momentis given in [8]

> 7 r / 2

M s2 drRR + h -CO S

R \2I\/~t \ nG ] 21 r1/2 2-(R2- r2)1/2

(4)This ma y beintegrated numerically over the region outside the contactellipse. T he total moment M is therefore th e sum of the moments givenSb y th e Eqs. (3) and (4)

M = M , + M s si s2 (5a)Spinning torques are computedforboth inner andouter ball/race con-

tacts. For equilibriumit is necessary for the netmoment vector on theball in the z direction to bezero. Therefore,

Msi Mso (5b)


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M . + MSI SOM .si + M so

13

The valuesof the spinning torqueat the inner andouter ball/race con-tacts are dependentupontherelative spin velocitieswhichare inturndependent upon the angle /3 of the angular velocity vector. Inorder toobtain the correct value of /3, two extreme values are assumed and thenet moment on the ball computed for each and an interpolation procedure isused to arrive at that valuefor wh i ch the net momenton the ball is closetozero. The criterion usedin the numerical computationis


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14

A simple forceanalysisshows that for equilibrium:F0. - F0 - F_..+R, = 0 (8)Si So Ri Ro v '

andby taking momentsabout the centerof the ballR(Fgi- FRi) + R(FSo-FRJ=0 (9a)

It may be seen thatFSi =FRi


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15

Fluid Dynamic DragA stheballsorbit within thebearing there is adrag force present

which is approximately equivalentto the drag of asphere submergedina f luid. A napproximateanalysis ofthis can bemade using th e followingassumptions:

1.Each ballis assumed tobehaveas it wouldwhile moving in asteady stream of fluid.

2. Interactionof theballs witheach other is neglected.3. Any effect due to rotation of the balls is neglected.On th e basis of thepreceding assumptions th e drag FJ-. of a single

ball may becomputedby the following fo rmula[15].F D = C D 7 7 R 2 ( l /2pUJ ) (11)

A difficulty arises because th e densityof themedium must beknown.T he viscosity of the mediumalsomust b e knownbecause the drag coef-ficient Cj-vis dependent upon th eReynolds number [15]. T he annularspace, however , is notfilledwith ahomogeneousfluidbut amixtureofair andlubricant.

T he drag force may, however, b ebracketed b y computing onevaluebased upon air alone and another value based on the annular spacebeingfilled with lubricant.

As shown in Fig. 6, for equilibriumthe drag force must b ebalanced b yforces at the inner andouterball/race contact. These forces wouldbecaused bymicroslip at the ball/race contact andwouldeachbeequal to


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16

F,-y/2. T he result ing torqueon the inner race is then:T3i =RRi FD /2

ando n the outer race isT3o=RRoV2

T he torque at the inner race therefore adds to the friction torque seen bythe rotating inner ring. T he fluid drag torque on the outer raceacts inth e opposite direction to the other torque values previouslydiscussed (seeFig. 6). Hence, th e value of fluid drag torque shouldb e subtracted f romth e other torque values previously discussed.

HysteresisWhen a ballrolls on a plate or in agroove , elastic deformationof the

ball and gro ove will occu r. Th e application and relaxation of load as theball rolls along th eg roove willresult in acertain amount ofhysteresisloss within th estressed zones o f the b all and gr oo ve.

An experimental investigation of thehysteresis loss for balls rollingon a flat plate was rep or ted in [16]. Asemi-analytical relationship wasderived in [16] betweenth e s pecific d amp ing capacityof the material andtheresisting fo rc e. This relationship is:

/I 1 \ /P \2F w =0 .1315 / + W I x Specific damping capacity (13)1E1 E2JUJFor an AISI 52100steel ball (R.,60) a valueof the specific damping{jcapacity of0 .007is givenin [16].


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17Since the stressedregion is assumed to bethat enclosedwithinthe

Hertziancontact region, Eq. (13)may be modified for anelliptic contactregion similar to that present between aball andgroovebysubstitutingthe areaof anellipse for that of acircle. Equation (13) then becomes

2/ I 1\PFw = 0.1315/-?- + \ l xSpecific dampingcapacityH VE i vw (14)Thetorque on the outer racedue tohysteresis is then obtained as:

T = NFW (R +Rcos 9) (15)4o PCageDrag

Reference is madeto Fig. 7, which is adiagramof aninner raceriding cagewith radial clearance C and anominal rubbingarea A .

^_ >

If it is assumed that (a) the radial clearance C is constantand is muchlessthantheland radius R^; (b) theregion betweenthe cage andland isfilled with lubricant; and (c) the velocity gradient in the radial directionis linear, the Pretroff formula for drag of a concentric plane bearing maybe used to determine cage drag.

O nthe basisof the foregoing assumptions the shearstresswithinthelubricant is given by

(o - n.)RLT = M -l- (16)

The total cage torquewould begivenby:

- O.)AcRL2/C (17)


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18Inorder toconsider the effect ofthison the overall bearing torque,

thebearing ball cage system shown in Fig. 7mustbeconsidered. Ifinertia forces are neglected and the fr ic t ion between the ball and cage isalso neglected, thenforequilibrium

TBall/cagereaction = (18a)RPTcRace/ballreaction = (18b)2 R

PThe torqueon theouter race inducedby the cage isTRR~T --5-5 (19)5 2Rp

T heprecedinganalysis considers the kinematic conditions to remain thesame as for the cageless bearing. Also, the friction between ball andpocket is neglected. Byconsidering the equilibriumof the ball asshownin Fig. 7(c)with aconstant coeff icient of fr ic t ion f ,

Fco = - (1 + f) (20)co 2Thetorque on the outer racedue to the cage is therefore increased by afac tor of (1 -i- f). Fo r an ef fect ive coeff icient of friction, f = 0.2 the effectwouldbe a 20percentincrease in thetorquedue to the cage.


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19TotalBearing Torque

Neglectingcoupling betweenthe various effects considered individually,and combining Eqs. (7), ( l O b ) , (12b) , (15)and (19), thegrosstorque on theouter race can beobtained where

T =T l o + T 2 o - T 3 o + T 4 o + T 5 o


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20

torque for oil jet lubrication. The calculated values oftorque increasemo r e withspeed thando the experimental values. This result is mostlikely caused b ymore lubricant being centrifugally thrownout of the bear-in gas the speed increases for the experimental values.

T h eexperimentalresults for the thin f i lm lubrication with th e syntheticparaff inic oil are shown inFig. 10. From these data it is seen thatthespeed effect is practically nonexistent for the conditions shown. T heexperi-mental trend of the data agreeclosely with the calculated values o ftorqueconsidering thespinningterm (eq. (7))andhysteresis loss (eq. (15)) only.

Using the synthetic paraffinic lubricant, th ebearing torque with je tlubr ication was anorder ofmagnitudegreater thanthe torque withthin filmlubr ication (Figs. 9 and10). However , using thedi-2-ethylhexyl sebacatelubr icant , thebearing torquew asessentially thesame with both jet andthin fi lm lubrication. This is becausethedi-2-ethylhexyl sebacate is muchless viscous than the syntheticparaff inic oil and was not fully retained inth ebearing with je t lubrication.

T hecomputed minimumandmaximum torques expected f rom thefluid-dynamicdrag are giveninTable2. Theminimum valueisdeterminedbyconsidering the annular space in thebearing to befilledwith air only. T odetermine th emaximum value o ffluid-dynamic drag, the annular space isconsidered to be completely filled withlubricant. On thebasis of air alone,thecontribution of fluid-dynamic drag to the total bearing torque is insignifi-cant. If it is assumed that the annular space is completelyfilled withthesynthe t ic paraff inic oil, amaximum valueoffluid-dynamic torqueo f_ Q8.6x10 newton-meters (0 .076Ib-in.) at 3000rpm is calculated.


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21

T he com puted bearing torques due tohysteresis effects only, areshown in F ig . 11. Th ese torques are dependentonly upon loadandcontact angle and are independentofspeed and lubricant. However, th ehysteresis effect alone is insignificantwhen comparedwith the otherfac tors considered.

T he agreement between the experimentalresults andcomputed valuesfo r th e bearings without a cage is general lygood, althoughthe computedtorque is, with fewexceptions, less thanthe corresponding experimentalvalue. In all cases, an extrapolation of the curves back to the zero loadpoint yields a finite torque at the no-load con dition. Th is is to beexpectedin the caseof the di-2-ethy lhexy l sebacate and the synthetic paraffiniclubricants with adequate lubricant supply be causethe rollingresistancethrough the lubricant willstill b epresent with zero load. However , for thecaseof the thin f i lm, th e only torques computed are thosedue to the ballspin andhysteresis effects . There would, however, b e asmall torquedueto rolling resistance with anyno nzero lubricant f i lm. Additionof the torquewouldraise the computed valueb yapproxim atelyth esame amount over thewhole load range, and bring th e computed andexperimental values intocloser agreement .

T he exper imentala nd calculated bearing torques which include cagedrag are shown inFigs. 12. Figures 12(a)and (b )show th eresults fo rthe twobearings with the di -2-ethy lhexy l sebacate lubr icant . With thislubricant , the calculateda ndexperimental torques are in fair agreement.Comparing Figs. 12(a)and (b)with Figs. 8(a)and (b), respectively, th e

Vcalculated values of cage drag were found to accountfor approximately


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2298percent of the total bearing torque at a thrust loadof 44. 5newtons (10 Ib)and approximately9 3percent at a thrust loadof 403newtons (90 Ib) and aspeed of 3000 rpm for the bearing with the 26 contact angle. For thebearing with the 17 contact angle under the same conditions, cage dragaccounted for approximately 97 and 92percent of the total calculated bearingtorque, respect ively . T he experimental values of cage drag were found tobe approximately 95 and 87percen t, resp ectively , for the bearing with th e20 contact angle. For the bearing with the 17 contact angle, th eexperi-mental values of cage drag were approximately95 and 85percent, respec-t ively, of the entire b earing to rqu e under thesameexp erimental conditions.

Figure 12(c)and (d) are theresults for the twobearings with th esynthetic paraffinic o il lubricant. It is apparent that except at the speedof 1000 rp m , the calculated torq ues are as much as approximately 100 per-centgreater than the experimen tal values. Th e most probab lereason fo rthis discrepancy at the higher speeds is that the more viscous syntheticparaffinic o il does no tcom pletelyfill th e bearing cavitybu t is either centri-fuged out of the bearing or may not enter th e cage-land areaof the innerrace inlarge quantit ies. A s aresult, less torque would occur thancalcu-lated.

Comparing th e torque data ofFigs. 12(c)and (d)withFigs. 9(a)and (b)fo r th e synthet icparaffinic oil, th e calculated values of cage drag at aspeed of 3000 rpm and a thrust loadof 403newtons (90Ibs) were approximately27 and 29percent of the total calculated bearing torque for the 17 and 20contact angle bearing, respectively. Under thesame conditions, theexperi-mental values of cage torque were approxim ately 2 and 10percent,respect ively , of the total experimental bearing torque.


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23Theseresults would clearly indicate that with a viscous fluid,

such as the synthet icparaffinic oil, cage drag is less of a factor intotal torq ue values than with aless viscous fluid, such as the di-2-ethylhexyl sebacate. F urth er, these results would tendto affirm theprevious speculation that the lubricant dues not enter the cage-land areaof the inner race in large quantities with the viscous lubricant.

S U M M A R YT heSpinning Torque Apparatuswas modified to measure the torque

on a thrust loaded204size (20-mm bore) ball bearings having contactangles of 17 and 26 with an d without a cage. Friction torquewas meas-ured for thrust loads varyingf rom 44. 5newtons (10Ibs) to 403newtons(90 Ibs) at speeds of 1000, 2000, and 3000 r p m . Tests were conductedwith either a di-2-e thy lhexy l sebacate and asyn thetic paraffinic oil as thelubricant. T he lubrication modew as either oil jet lubrication directed atth e bearing contacting surfaces of athin surface film of lubricant appliedon the bearingraces andballs. A n analytical mo delwhich included rollingresistance and a generalized rheological modelwas developedandextendedf rom previous modelfor spinning torque and lubricant rheology. T hefollowing results were obtained:

1. The calculated bearing torques using th e bearing analytical modeland lubrican t rheological mod el develo ped for determ ining the torques ina thrust-loaded ball bearing were in fair agreement withthe experimentalresults.

2. Cage drag was found to be primarily a function oflubricant viscosity.


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24

For the d i-2-ethy lhex y l sebacate, cage drag was found to account forapproximately 87 to 95percent oftotal experimental bearing torque. Ho w-ever , for the more viscous synthetic paraffinic oil, cage drag was found toaccount for approximately2 to 10percent of total experimental bearingtorque.

3. For ab earing without a cage with arelatively lo wviscosity fluidsuch as the di-2-ethy lhexy l sebacate thelargest con tribution to b earingtorquewas ball spin torque. For a more viscous oil suchas the syntheticparaffinic oil, thelargest contr ibutorto bearing torque is theresistanceto rolling throughthe lubricant. Theresistancetorollingis affectedbyth e amountof lubricant present.

4. With a low viscosity lubricant anexcess supplyoflubricant to thebearing has asmall affect on bearing torque. H owever, with a viscouslubricant, a nominalflow of oil to the bearing can result in a tenfold in -creasein bearing torque when compared withthe value obtained with onlyathin f i lm of lubricant present.


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25

APPENDI XDerivation ofS pinning T orque

Analysis of the spinning torque is anextension of the wo rk under takenpreviously for the spinning of a ball in a noncoriforming groove [9,10]. T hecontactellipse for theball/race interface is shownin Fig. 4. Thesystemcan be reduced to a number of elemental rollers of width dy inrolling andsliding motionrelative to the gro ov e (F ig. 4). R elative to the rotatingcoordina te sys tem the surface velocityof the e lemental roller is obtainedf r o m [13]for the inner race as:

UBi="(a)riR +^siy) (A la)andfor the outer race as:

U D = (eo R - a) y) ( A l b )Bo v ro soy' v 'T he magnitudeof the surface velocities of the inner andouter raceswithrespectto the rotating coordinate systems are:

IL,. = a; . R (A 2 a )Ri n v 'U D = w R ( A 2 b )Ro ro

T he slip velocityof theball withrespectto the inner race f rom [13] isobtained as:

USi= ""si y ( A 2 C )and for the outer race

U = -


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26

T he t rac t ive force on each element of the con tac t ellipse is then given as:/*b

f.J~ bdF = / rdx dy iFor a Newtonian fluid withalinear velocity gradient

r =

w h e r e the fi lm thickness h is given in [17]as

/ 1 . 6 a - 6 E - 0 3h = o /UB+ URW'0 . 1 3

and

E = 2 , 2, * " "2E,

RR

E,

R = R

W =

R + R R

0.75P

(A3)

(A4)

(A5a)

(A5b)

(A5c)

(A5d)

According to [12] the assumption of Newtonian behavior for the lubri-cant is notrealistic athighpressures andshear ratesand amoregeneralmodel is:


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27T= jLLoeaSUs/h and t < TC

T = /ioea?SUg/h and TC < r < FS

r = F S and fti> FS

(A6a)

(A6b)

(A6c)The resulting spin torque aboutthe z axis is then obtained byinte-

grating the elemental moments over thewhole contact ellipse as:M si - f d K l =f' (A7)

Thecomputationof the shearstress T proceeds as follows1. At a given value of y the filmthickness is computedusingEq. (A5a) ,2. For each valueof x at the given value of y the Hertzian contact

pressure is givenby

S = 1.5P?rab

i /yfla)

fxf] J Jf|l/2

(A8)

3. Theshearstress r is computedby Eq. (A6a)if its value islessthan the criticalshearstress T . If theshearstress is larger than TL* L,butlessthan FS it is computedby Eq. ( A 6 b ) . However, if the shearstressis larger than FS, Eq. (A6c)is used.


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2 8REFERENCES

1. Poritsky, H. Hewlet t , C. W., Jr. and Coleman, R. E., Jr. "SlidingFrictionofBall Bearingsof the Pivot Type," J. A ppl . M ech . ,Vol.14,N o. 4 , Dec. 1947, pp. 261-26 8.

2. Jones, A. B. "Ball MotionandS liding Friction inBall Bearings," J.Basic Eng., Vol. 81, N o. 1, Mar. 1959, pp. 1-12.

3. Reichenbach, G. S. , "The Importanceo fS pinning Friction inThrust-CarryingBall Bearings," J. Basic Eng., Vol.82, No. 2, June 1960,pp.295-301.

4. Harris, T. A., "Ball MotioninThrust-Loaded Angular Contact Bearingswith Coulomb Friction, " J. Lub. Tech. Vol. 93, No. 1, Jan. 1971,pp. 32-38.

5. Harris, T. A., "AnAnalyticalM ethodto Prevent Skiddingin Thrust-Loaded AngularContact Ball Bearings," J. Lub. Tech., Vol.93,N o. 1, Jan. 1971, pp. 17-24.6. Miller, T., Parker, J., andZaretsky, E. V., "Apparatus for StudyingBall Spinning Friction," NASATND-2796, 1965.

7. Dietrich, M. W., Parker, R. J., andZaretsky, E. V., "Effec t ofBall-Race Conformity onS pinningFriction. " NASATND-4669, 1968.

8. Dietrich, M. W., Parker, R. J., Zaretsky, E .V., andAnderson, W. J ,"Contact Conformity Effects onSpinningTorqueandFriction," J.Lubr . Tech., Vol. 91, No. 2, Apr. 1969, pp. 308-313.

9. Allen, C. W., Townsend, D. P., andZaretsky, E. V., "Elastohydro-dynamic Lubricationof aS pinning Ballin a Nonconforming Groove , "J . Lubr . Tech., Vol. 92, No. 1, Jan. 1970, pp. 89-96.


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2910. Allen, C. W., Townsend, D. P., and Zaretsky, E. V., "Comparison o f

Conventional andMicroasperity Elastohydrodynamic Lubrication of aBall Spinningin a Nonconforming Groobe," NASATND-6761,April 1972.

11. Johnson, K. L., andCameron, R., "Shear Behaviour ofElastohydro-dynamicOilFilmsat HighRolling ContactPressures," Proc. Inst.M e c h . Eng., Vol. 182,N o. 14, 1967-68, pp. 307-330.

12. Allen, C. W., Townsend, D. P., andZaretsky, E. V., "New General-ized Rheological Modelfor Lubrication of aBall Spinningin aNon-conforming Groove. NASA T ND-7280, 1973.

13. Townsend, D. P., Allen, C. W., andZaretsky, E. V., "Friction Lossesin a Lubricated Thrust Loaded Cageless Angular Contact Bearing, "Proposed NASA Technical Note, 1973.

14. Wolveridge, P. E., Baglin, K. P., andArchard, J. F., "The StarvedLubrication ofCylindersin Line Contact," Proc. Inst. Mech. Engrs.1970-71, Vol. 185, 81/71, p. D428.

15. Eskinazi, Salamon," Principles ofFluid Mechanics" AllynandBacon,Boston, 1968, p. 438.

16. Drutowski, R. C., "Energy Lossesof Balls RollingonPlates," J. BasicEngr., Vol. 81, No. 1, June 1959, pp. 233-239.

17. Dawson, D., andHigginson, G . R. , Elastohydrodynamic Lubrication,Pergamon Press (1966).


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30

TABLE 1. - TEST B E A R I N G SPECIFICATIONS Type A j Type B

Inside diameter, mm (in.) 20 (0.7874)Outside diameter, mm (in.) 47 (1.8504)iWidth , mm (in.) 14 ( 0 . 5 5 1 2 )Pitchdiameter, mm(in.) . . 33.5 (1.319).Nominal contact angle, percent 26 17Inner race curvature, percent 52 i 53

Outer race curvature, percent 52 54Number ofballs 3Balldiameter, mm (in.) 7.15(0.281)Rockwell Chardness - inner race 62-64Rockwell Chardness - outer race 62-64Rockwell hardness - balls 62-64Surface finish, rms - races, jitm (^in.) 0.15 (6)Surface finish, rms - balls, p.m(p - in . ) 0.025 - 0.05 (1 - 2)


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31

T A B L E 2 . - C A L C U L A T E DV A L U E S O F F L U I D - D Y N A M I CDRAGB A S E D U P O N C O M P L E T E L Y F I L L I N G A NA N N U L A R

B E A R I N G C A V I T Y W I T HA H O M O G E N E O U S F LU I D[Bearing bore, 20 mm; number ofballs; 3, cage, none;

temperature, roomambient.]Fluid

Air

Di-2-ethylhexylsebacate

Syntheticparaff inicoil

Shaftspeed,rpm

1000

3000

1000

3000

1000

3000

Ballorbitalspeed,m/sec

(in./sec)0.7

(27 .5)2.1(84)0.7

( 27 . 5 )2.1(84)0.7

( 2 7 .5 )2.1(84)

Reynoldsnumber

327

990

287

880

10.7

33

Dragcoeffic ient ,

CD

0 . 55

0 . 4 2

0.7

0 . 4 5

4

2

Drag fo rc eper ball,newtons

( lb)

6 . 2X10"6(1 .4X10"6)4 4 .5 X1 0 ~ 6

(10.0X10'6)

6 . 2X10" 3(1.4*10~ 3)35.6X10" 3(8 :OX10~ 3)35.6X10'3(8.0X10"3)

1 6 9 x l O ~ 3(38.0X10"3)

Totaldragtorque,N-m

( Ib - in . )

0 .32X10" 6(2 .8X10"6)

2 .3X10"6(20X10~ 6)

0 . 32X10" 3(2 .8X10~ 3 )

1.8X10'3(16X10"3)1.8X10"3(16X10'3)8.6X10~ 3(76X10"3)


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D R I V E T U R B I N EMAGNETICSPEEDP I C K U P - ,

T E S T B E A R I N G - 7iCONNECTING SPLINE- . /

LOAD CHAMBER ','P R E S S U R EREADOUT^

50x10

T U R B IN E D R I V E AIR-L O AD ING P ISTO N

L O A D | A I R | I NBAL L / \BUSHING-/.,,,CO UNTERWE IGHTSYSTEM--

i, 'rLOW ER/ / ( T E S T

D R I V E ' /,' H O U S I N GSHA /^,'/,'rRADIAL

,'AIR1 1 B E A R I N G

--PAD

/ < - A X I A LA I R/ B E A R I N G^ H O L L O W S H A F T

L O W E R T E S T -H O U S I N G A S S E M B L Y .M H R U S T G A S B E A R I N G P A D

O P T I C A L T O R Q U E M E T E RC A L I B R A T I N GLOAD P A N - TO R Q U E R O D M I R R O R^ - D A M P I N G S Y S T E M( a ) G E N E R A L C U T A W A Y V I E W . CD-10720-15

7000 r-

6000

5000


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oaOf

CLU

ace.

Ba:eUJrooUJo

(2o

eceu.

co

Oz

oo

S,O3

c1c

t eCD Ji- "S S ? 8.E

_01O1Z


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OIosu.LLJosg


36/38

_ _ Tsi-sgs*UJQ_S

1atiHU

o

1

o

2

-A

3

CALCULATD

111\

\\^ OOvO 00

1 1 1O CO sO^

111t 1vvv\A i gig11f111oj \aCONTCTANGLE, \i )i \A \ .i \ *i i 1 1 * \\ \ \ \\ \ \0 -c?12 AXIALLOAD,N 1Iii_ 8s o_eS23456 AXIALLOAD,LB bCONTCTANGLE,2ss s g s s W - N 'snoaoi^ r e s j ^ o o % o ^ c \ i o^4

Nh8i '3noyoi

Scco>&

*5o'cc:EroQO

e -ocnjQ

rsooccae

e

tooo

oash

e2mm;s

1

2

1Oo1o*'3=

ucO

%o>3B

IS

2enccSCO

ftoEEO>&OJ< yocw.SL3H

rW c -oV X3a39 C** = .003ft- .8 S1 acj?

-18

o< _ >sI I I I


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S P E E D ,RPM

a*TCO

oc

C A L C U L A T E D (ALL S P E E D S ). /

10 20 30 40 50 60A X I A L L O A D , L B

(b) C ON TA C T AN G LE,20

70 80 90

Figure 10. -Comparisonof experimental and calculated bearingtorqueas afunctionof loadfor asynthetic paraffinicoil.Bearingboresize,20 mm; speed, 1000,2000,and 3000rpm;numberofballs, 3;cage, none; temperature, roomambient;typeoflubrication,thin film.

.008

.006

.004

.002fl

.0008.0006S -0004

Uo .0002

I I

100 200 300A X I A L L O A D ,N

I I I I I I I

C O N T A C TA N G L E ,D E C17

400

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0AXIAL L O A D , L B

Figure1L - Calculated bearing torque due to hysteresis as afunctionofload. Bearing bore size, 20 mm; numberofballs, 3;cage, none; lubrication, none.


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.12

.08

.04

0.16

.12

.08

.04

01.2

.016

.012

.008

.0040

.016

.012

.008

.004r 0

E X P E R I M E N T A L ,RPM100020003000

_ --- C A L C U L A T E D

I N N E R R A C ES P E E D ,

RPM^3000 ~ t>^-02000^1000 o

I(a ) L U B R I C A N T , DI-2-ETHYLHEXYL

S E B A C A T E CO N T ACT ANG LE , 17

a (b) L U B R I C A N T , 01-2-ETHYLHEXYLg S E B A C A T E C O N T A C T A N G L E , 26

.4

01.6

1.2

3000

J ( c) L U B R I C A N T , S Y N T H E TI C P A R A F F I N I CO I L ; CO NT ACT ANG LE , 17

.16

.12

.08

.04

. _ - - 3000

2000

100 200L O A D , NI

300 400

20 40 60L O A D , L B

80 100

(d) LUBRICANT, SYNT HET IC PARAFFIN CO IL ; CO N T ACT ANG LE , 26

Figure 12. - Com parison of calculated and experimentalbearing torqueas afunctionofloadfor abearingwith

study of ball bearing torque under elastohyrodynamic lubrication

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