,1i

NASA TechnicalMemorandum 87017 1

' Fatigue Criterion to System Design, ",Life and Reliability

(J_SA°T_-87_I?) _A2TGUE C_TZ_ZON TO SYSTEM N85-27226D_SIGI, L_F_ AN5 _LIABILZ_¥ (H_Si} 22 pHC a_ 2/SF A_ 1 CSCL I_D

UnclasG3/37 21263

Erwin V. Zaretsky __

Lewis Research Center

_ Cleveland, Ohio _

Prepared for theTwenty-first Joint Propulsion Conference

' cosponsored by the AIAA, SAE, and ASMEMonterey, California, July 8-10, 1985

A

....... .... 198501891

https://ntrs.nasa.gov/search.jsp?R=19850018915 2018-06-27T03:16:27+00:00Z

Id

FAlI&UECRITERIONTO SYSTEM DESIGN,LIFE AND RELIABILITY

Erwln V. ZaretskyNationalAeronauticsand Space kdmlnlstratlon

Lewis ResearchCenterCleveland,Ohlo 44135

SUMMARY

_- A generalizedmethodologyto structuralllfe prediction,design,and, rellabllltybased upon a fatiguecriterionIs advanced. The llfe prediction

._ methodologyIs based In part on work of W. Welbulland 6. Lundbergand A.Palmgren. The approachincorporatesthe computedllfe of elementalstress

_ volumesof a complexmachineelementto predictsystemllfe. The resultsof''. N coupon fatiguetestingcan be incorporatedInto the analyslsa11owlngfor llfe

_.- _ predictionand componentor structuralrenewalrateswlth reasonablestatls-: tlc_l certainty. _

':_ INTRODUCTION

_ _ Designof machineelementsIs based for the most part on yield stresses:-_ and fatiguelimiting_tresses. In additionto the componentsmaterialprop.- ,_;"_' ertles,properconsiderationmust be given to the effectsof notches,surface !

: conaltlon,componentsize, residualstress,temperature,duty cycle and envl- i•_ ronmentalfactorssuchas corrosiveor chemicalexposure. For most machine

e_ements,Indlvldualsand organizationsusuallydevelopdesignmethodology"- based upon engineeringfundamentalsfound In most machinedesign texts and ;_ factorsbased upcn their corporateexperienceand test data. As a result,It

Is not too unusualf_r differentorganizationsor individuals,startingwlththe same or similardesign requirementsto reach dissimilarconclusionsordesignswhlle seeminglyapplyingthe same fundamentalengineeringprincipalsto the problem.

Settingaside the subjectivecreativeaspectsof design,there appearstobe nonunlformltyof data from which numbersand design factorsare selectedaswell as differencesin the computercodes and boundaryconditionsused In thedesign process. To compoundthese difficulties,fatiguedata which are usedto establishfatiguelimitsare usuallyof a limitednaturewlth the conditionsunder which the data were obtainednot adequatelydefinedor reported. Suchitems as temperature,humidity,number of specimens,specimensize and volume,heat treatment,hardness,surfacefinish and llfedistributionare not given.

The establishedfatigue11mlt for much of the reporteddata Is a meanvalue. From a statisticalviewpoint,the median value Is equal to or less thanthe mean. This can be interpretedas meaning that beforea fatiguelimit }s

• determinedthere Is a probabilitythat 50 percentor more of the specimenshadfalled. In other words, even at the fatiguelimitingstress,11fe Is finite,Of course,experienceddesign engineershave recognizedthls for years. They

i have added safetyfactorsto their design procedureusuallybased on exper-. lence. While these proceduresare generallyadequatethey can resultIn over-

designed,oversized,overweightand overcaststructures.

'i

A fundamental principle of good design Is to recognize that any structure •can fail. However, should It fail, it should be designed not to cause personalinjury or secondary damage. In other words, if the structure fails, it failsin a safe or benign manner. Once a structure Is designed to fall In a benl_nmanner, It then can be designed for finite life whereby the overall size,weight, and cost can be reduced and still meet the reliability requirements ofthe application.

In view of the aforementlon_ tt becomes the objective of the work reportedherein to advance a generalized methodology for structural llfe prediction,dcsltln, and reliability based upon a fatigue criterion. The life predictionmethodology ls based In part on the work of W. Weibull (refs. 1 and 2) and G.Lundberg and A. Palmgren (refs. 3 to b).

SYMBOLS

C dynamic load capacity _,

c stress-life exponent i

e Weibullslope, II¢=

F probability of failureI1

f failureprobabilitydensityfunction !

h exponent

L llfe ,_

LA adjusted life at a 90..percent probability ofsurvival

I.m mtssion life i#

LlO lO-percent life, life at which 90 percent of a population will survive

n load..lifeexponent

P appliedload

P(x) probabilityfor occurrenceof extent x

R cumulativerenewalfunction

r renewaldensityfunction

S probability of survival

S1 system probability of survival ¢

1 dummy variableof integration

t time or time function

_dll_" _,_ =r_ _ _llfj_J _

1985018915-TSA04

V stressed volume

X _lme function or stress function

XB characteristic strength

X_ stress below which no specimens fall

z depth to critical maximumshearing stress

shape parameter

B characteristic llfe

y mean tlme to failure

n stresscycles to failure

u iocatlot_parameteror tlme below which no failureIs expectedto occur

_u fatiguelimit

-icriticalstress

STAIISTICALMETHOD

In the late 1930's (circa1937) W. Welbullin Swedenattemptedto graph-ically llnearlzevarioustypes of statisticaldata distributionsfor smallsamplesizes. By trial and error,Welbullfound that by having the Inln I/Sas the ordinateand In X as the abscissa,where S Is the probabilityofsurvivalor statisticalpercentof samplessurvivingand X Is a time func-tion, most englneerln.qdata distributionswill plot on a straightllne(ref. 7). Hence, It becamepossiblef_r small amountsof data to estimateageneralizedpopulationdistributionfor a populationof infinitesize. Havingempiricallydeterminedthis, Welbulldevelopeda theoreticalbasis for whatwas to become known as the "Welbulldistribution"or "Weibullplot" which waspublishedIn 1939 (ref. l). Welbull(ref. 8) definesthe distributionfunctionas "an adequateexpressionfor a large class of phenomenawhich have the prop-erty that the probabilityof nonoccurrenceof an event Is equal to the productof the elementaryprobabilities,"

S . exp{- [(X - u)//13]I/"} (1)

The functioninvolvesthree parameters,

. a the shape parameter13 the scale parameter

the locationparameter

Where a the shape parameteris equal to I. 0.5 and 0.28, the respectived_strlbutlonsapproximatedare exponentlal,Raylelgh,and normal (Gausslan)_A typicalWelbullplot for rollingelementbearingfatiguedata Is shown infigure I. The slope of the llne "e" which Is calledthe "Welbullslope" is

3

1985018915-TSA05

I,L ,: _i

i : equal to 1/=. For most rolling-element bearing dat_ e equals 1.1. Thei location parameter _ Is a finite time under which there would be a zero

probability of failure to occur. If failure can occur In one stress cyclet

_, then the location parameter _ would be assumedto be zero.

For fatigue d_ta, If n stress cycles to failure ls substituted for X_; In equation (1) and e for 1/=, and where F = 1 - S, equation (1) can be,_.. written

T.. : e

_. , The characteristic life B Is the 63.2 percent failure llfe of the pop-_i_ ulatton-dtstrtbutton.

_2_:_i_ Weibull stated that the probability of survival S could be expressed as

__ In g ~ _CneV (3)F= 4

?

,_ _ where V Is the volume representation of stress concentration referred to_ _ herein as stressed volume, • ts the critical shearing stress and c is an_;t exponent denoting a stress-life relation where

n ~ _-c (4)i

_: _ lhe values of c and e can be determined experimentally.i ,i

.t

_ The effect of stressed volume can be Illustrated whereln two specimens ofL stress volumes V1 and VE respectively are subjected to equal stress

., _. If nl ts determined at a probability of survtval S1, then the:i probability of survival S2 for V2 for the ltfe Is given by the fol-

lowing expression

V /V2 1

S = S (S)2 1

i ,

_ where V2 > Vl

For the same probabilityof survival,the specimenswlth the largeri stressedvolume wlll have the lower llfe. Thls principlewas appliedto suc-

cessfullynormalizerolllng-elementfatiguedata by Carter (ref. g) andZaretsky(ref. lO). Thls principlewas appliedby Grlsaffe(ref. II) in aWelbullanalysisof shear bond strengthof plasma-sprayedaluminacoatingsonstainlesssteel (fig. 2). Grlsaffeshowedthat the calculatedmean bondstrengthdecreasedwlth increasingtest area In accordancewlth equation(5)(fig. 3). Further,the stressat which no specimensfailedcould be determinedby restatingequation(1) as follows:

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1985018915-TSA06

Fx = 1 - exp X_

where the parameters are now defined as

Fx statistical percent of specimens which when tested at one set ofconditions, fail at given stress or lower

X stress

Xg stress below which no specimens failed

XB characteristic strength

e Welbull slope• i

LUNDBERG - PALMGREN THEORY

Lundberg and Palmgren were contemporaries of Welbull. A problem existedin the roli_ng-element bearing industry to establish the lives of these bear-ings short of extensive testing. Taking the Welbull approach a step further,Lundberg and Palmgren (ref. 3) took equation (3) and added another element,the depth to the critical maximum shearing stress, z where

1

In _ ~ zh - (7)

The rationale for including the depth z was that the initiation of afatigue crack occurred at the depth z and that the distance the crack needsto travel to the surface until a spall (pit) occurs Is equivalent to a criticalcrack length. In other words the greater the distance the maximum shearingstress below the surface the longer it takes for a fatigue crack to propagateto the surface, the longer the fatigue llfe. Lundberg and Palmgren took thero111ng bearing geometry, kinematic, stress theory of Hertz (ref. 12) andThomas and Hoersch (ref. 13) and incorporated then Into equation (7) to obtaina series of equations which relate the rolllng-element fatigue llfe of a bear-ing to the applied load wherein the resultant life would be in millions ofinner race revolutions. Knowing the speed of the bearing the llfe in hourscan be determined. Lundberg atldPalmgren further refined their approach tobearing llfe prediction to include a fictitious load designated C, thedynamic load capacity. The dyna,nlcload Is the theoretical load which whenapplied to the bearing would result in a llfe of one million Inner-race revo-lutlons and where

n

C=P _ (B)

i

1985018915-TSA07

4'1

P appliedload l!l

LlO TO-percentllfe, llfe at a gO-percentprobabilityof survival,Inner-racerevolutions

n Ioad-llfeexponentusuallytaken as 3.

Equation(B) can be written:

C nLlo = (_) (g)

Hence, the predicted life of a bearing can be determined by knowing C and P.

The Lundberg-Palmgren theory is now an international standard used byevery manufacturer of bearings around the world (ref. 14). The theory hasbeen applied to other machine elements and mechanical transmission systems(refs. 15 to 28). Table 1 ls a ltst of applications of the Lundberg-Palmgrentheory to llfe prediction.

SYSTEM LIFE AND RELIABILITY

The llfe and rellbabllltyof a systemis based upon the lives and rell-abilitiesof all of its components. Havingdeterminedthe life of the %ndl-vldualcomponentsusing Lunberg-Palmgren,the probabilityof survivalof the i

1entiresystem Is as follows _,

S1 = Sl • S2 • S3 ... Sn (TO)

The system life-rellabilltyequationcan be writtenas follows

I/_ll/ el (_22/e2 (_3/e3 (_BB/en]In 1 = In I L + L + L + L (II)

where Ll'aL_kL3 " " " Ln are the lives of each componentof thesystemat =u percentprobabilityof survival. The system llfe L can bedeterminedat each systemprobabilityof survival ST. For a gO-percentprobabilityof survival

e

1= + + ... + T- (12)

MISSIONLIFE

A systemdoes not usuallyoperateat one constantload in actual service.Miner'sRule Is used to sum fatiguedamageof a missionprofileconsistingofloads and tlme-at-loads. For a given probabilityof survival,the missionllfe for the system Lm is

1985018915-TSA08

ill:' !

where t a, t b, t c and t d are the fraction of the total tlme at loadsPa, Pb, Pc and Pd, respectively, and La, Lb, Lc and Ld are the systemlives at a given probabilityof survivalat loads Pa, Pb, Pc and Pd,

Welbullplot or distributioncan be constructedusing thls

_ respectively.

_ method by determiningthe missionllfe at varyingprobabilitiesof survival.lhls methodwas used by Lewlcklto determinethe fatiguellfe of a turbopropreductiongearbox(ref. 28). The analysiswas comparedto actualfield data.The resultsof thls comparisonare shown In figures4 and 5.

COMPONENTREPLACEMENTRATE

The WelbullanalysisIs a valuabletool for predictingthe llfe of com-ponentsor systems. The analysiswould describethe failurerate In field ,serviceonly if all componentswere put into serviceat the same tlme and If

._: failed componentswere not replaced. But, a certainnumber of componentsmustbe kept In operation,failurerateswlth replacementsIs of interest. As anexample,If there are lO 000 bearingsin the field, the user or manufacturer

_:: must know how many spare bearingswlll be neededover a given perlod In orderto keep lO 000 bearingsin operationat all times. Over the intervalof ser-

r_ ; vice operationIt may take l_ 000 replacementsto keep all lO c)O0bearingsrunning The Welbullanalysiscan never exceed I00 percent,but field failures

,_ can and often do exceedI00 percent(ref. 19).

i o The cumulativeprobabll_tyof failurefor the first time, assumingcon- i

_ stant serviceconditions,is a functionof the lengthof tlme that the bearing _ihas been running. Failureis the complimentaryfunctionto survivalaccording

_ ..... t0 the followingrelation'

" F(t) = l - S(t) (14) ;

The probabilitydensityfunctionfor failureIs

f(t) - _t (15)_T

_ ' The instantaneousprobabilityof failurefor the first tlme In any time inter-

_ val from t to t , At iS then given by=-_-_ P(flrstfailureIn at interval)= f(t)at (16)

By using renewaltheory (refs.29 and 30), the probabilityof having to replace" a bearingIn the field is writtenas follows:

P(maklnga replacement)--r(t)at (17)

where r(t) is called the renewaldensity, lhe renewaldensityis calculatedfrom the followingsummation(ref. 29):

X

:.. r(t) = _ fk(t) (18).: k=l

- where fk(t) Is the k-fold convolutionof f with itselfand Is computed_. by the followingrecurrencerelation(ref. 29):

7

_ ,

...........-....<"........"_ ':"" :"" __'_":_".... _"'_f_'_--'_'__'_ .... " ..... "......" "_-"-'-:'_'-_ - - - .......m -"---'-_-- _-__-_ ---- _" _4--'_ . " .2_2_

1985018915-TSA09

t/,

= J0 fk(T)fl(t - Tldl (fl _ f) (19)fk+l (t)

The expression1 fk(t)at gives the probability of a kth failure occurringIn the time interval t to t + at. Since, when a failure occurs,it can be for any value of k, it follows that the renewal density functionshould be defined as the sum over all the fk's.

: The total number of replacements made during the first t units of timei ts obtain by integrating the renewal density function as follows:

R(t) = r(T)dT (20)

For a group of bearings that has been operating for some time with failuresoccurring and replacements being made, it is also important to know the MTBF

: (mean time expected between failures). According to the renewal density. theorem (ref. 30), _

-.i llm r(t) = 1 _ ! (21)

i t-= O_= tf(t)dt

_I Therefore, the mean time between failures is calculated as

_i ]

(22) j_1 MTBF : total number of bearings ii

_:' Reference lg reports the use of a computer program to evaluate .equation (1B) for rolling-element bearings. A Weibull slope of 1.1 wasassumed. Figure 6 shows plotted results, which glve the renewal density func- itlo_ for a case of where the failed bearings are removed from service andreplaced with new or restored bearings. For compari_._n,the probability den-s_.# for failures with no replacement (Welbull density function) is plotted

- also. The area under the curves represents the probability of failure.

.i I

The functions plotted in _Igure 6 were numerically integrated and areshown plotted In figure 7. These are the cumulative functions for renewal orfailure. The cumulative renewal functions indicate lO0 percent replacementbearings will have been needed by the time 8 LlO intervals have elapsed.By comparison, at the time of 8 LlO the Welbull distribution shows only65 percent of the original bearings will have failed. The difference In totalfailures would be due to replacement bearings failing.

The mean time between failure (MTBF) froraequation 22 can be obtainedfrom figure 6. The MIBF is the inverse of the probability density for failure.As an example a probability density for failure of 0.12 would give a MTBF value

of B.3 LIO.

FAGIGUE LIFE MODELING

The Welbull and Lundberg-Palmgren analyses have been primarily applied tohigh cycle fatigue with the material subjected to a Hertzlan stress field.However, as indicated in reference II, the Welbull analysis can be applied to

B

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1985018915-TSAI0

other types of durabilityproblems. What Is importantIs that a materialele_nentunly knows the state of stress It Is subjectedto and not whether ItIs in a bearing,gear, shaft,compressoror turbine. The crack propagationtime In hlgh cycle fatigueis generallya small fractiono_ the total time tofailure. In low cycle fatigue,the crack propagationtlme is generallya 51g-nlflcantfractionof the time to failure. In either case, the end result Istotal fractureof the componentmaking It no 'longeruseful for its intendedpurpose. If the failuredistributionis within standardstatisticalranges,then It can be representedby the Welbullanalysis. Hence,the analysisshouldbe blind to high or low cycle fatigue. This was recognizedby Ioannldesand

. tlarrls(ref. 31). Using Welbulland Lundberg-Palmgren,they Intruduceda"fatlguelimitingstress"and integratedthe computed _Ife of elementalstressvoldmesto predictllfe. This approachleads to a method of applylngLundberg-Palmgrenllfe predictiontechniquesto other componentsbesidesthose subjectto Hertzlanloading. It a|so allows an investigatorto use the resultsofcoupon testingto predictthe llfe of complexshapedcomponentssubjectedtononHertzlancyclic stressing. Ioannldesand Harris (ref. 31) appliedtheiranalyslssuccessfullyto rotatingbeam fatigue(flg. B), flat beams In reversedbending(fig. 9) and beams In reversetorsion. Based upon their approachor amodificationthereof,design proceduresfor structuressubjectedto fatigueloadingcan be formulatedwhich allowsfor finite llfe determinationIn theinitialdesign stage with reasonablestatisticalcertalnlty.

DESIGN OF COMPLEXSTRUCTURES

In recentyears, finiteelementstressanalysisof complexstructuressubjectedto thermaland mechanicalloadinghas reacha hlgh degreeof sophis-ticationand reliability. Now computersperformcalculationsin mere secondsand minutes for problemswhich only a short time ago would take hours or beimpossibleto perform. An exampleof a turbineblade analyzedusing finiteelementanalysisis shown in figure lO. Each elementof materialhas asso-ciatedwith it a stress. For design purposes,thls stress is within certainestablishedstressor strain limits. The ultimatellfeof the componentIsusuallybased upon empiricalcalculationsor extrapolatlonfrom field exper-ience. The resultsare at best highly speculative. By subjectingthe com-ponentto expensiveproductimprovementprograms(PIP)and by "makeand break"techniques,componentlives over a productsllfe time can usuallybe extendedto the useful llfe of the product.

The key to cost effectivedesign is to be able to maximumlzecomponentlifeand minimizecost at the completionof the final design stagewithouttheneed for extensivetestingor field experience. It is proposedthat thisobjectivecan be accompllshedas follows:

(1) Having determinedthe stressof each elementalvolumeof a componentusing finite-elementanalysis,It Is possibleto establisha llfe of theelementalstressedvolumeusing eitherequation(3) or (7).

(2) Using equation(12) it Is then possibleto establishthe llfe of theentlre component.

(3) Carryingthe processa step further,applyingthe principleofequation(B) a "dynamic"capacityof the componentcan be determined.

9

W�s

1985018915-TSA11- '

,i 'ii.... (4) Once the dynamic capacity ls determined, the mission llfe of a com.-.

ponent or systemcan be establishedusing equation(13).i

(5) Using renewaltheory (eqs. (18) and (22)),the number of replacementsrequiredover a systemsuseful llfe and the mean time betweenfailures(MTBF)can be determined.

_,:i- By design iteration design llfe can be maximizedand cost minimizedl '

Further,by knowingfailurerates,productionguidelinescan be set prior to a_ productentlrlngthe market place._ ,

._ In the area of bearingtechnology,vast technologicalimprovements_ occurredbetweeninitialpublicationof Lundberg-Palmgren(1947)and contem-

_!: porarydevelopment. These technologicalimprovementswere factoredInto the_ analysisby the use of llfe adjustmentfactors(ref. 32) whereinthe adjusted

':__! llfe LA at a 90-percentpr°bablllty°f falluri_isn• LA = (D)(E)(F)(G)(H) (23)

_: where the llfe adjustmentfactorsare

_L_ D MaterialsFactor_i_I E ProcessingFactor_I F LubricationFactor_ G Speed Effect

H MisallgnmentFacto¢!

A similarapproachin the design of power--transmlttlngshaftswas taken

=_ by Loewenthal(ref. 33). The Loe_enthalapproachIs based on a fatigue limit-_ Ing stressbased on combinedtorsion,bending,and axial loading. The fatigue_°_ limitingstress Is modifiedby ten factorswhich in principleare In part_! incorporatedIn the Welbulland Lundberg.Palmgrenanalyses. Those _actors

outsidethe analysiscan be based upon experimentalcouponfatiguedata.

Most machineelementsare subjectedto their maximumdesign load only for

i a very small factionof their mission llfe cycle. Hence, by designingto allfe criterionand a mean cubic load ratherthan a fatiguelimitingstress it!is possibleto reducethe size of the machineelementwithouta reductionin

: system reliability

CONCLUSION

A generalizedmethodologyto structuralllfe prediction,design,andreliabilitybased upon a fatiguecriterionIs advanced. The llfe prediction

' methodologyIs based In part on work of W. Welbulland G. Lundbergand A.Palmgren. The approachincorporatesthe computedllfe of elementalstressvolumesof a complexmachineelementto predictsystem llfe. The rcsultsofcoupon fatiguetestingcan be incorporatedInto the analysisallowingfor llfepredictionand Componentor structuralrenewalrateswlth reasonablestaLls-tIcal certainty.

1985018915-TSA12

REFERENCES

I. Welbull,W., "The Phenomenonof Rupturein Solids,"Tng. VetanskapsAkad.-Hanal.,No. 153, 1939.

2. Welbull,W., "A StatisticalDistributionof Wide Applicability,"Trans.ASME, J. Appl. Mech., vol. 18, 1951, pp. 293-297.

3. Lundberg,G., and Palmgren,A., "DynamicCapacityof RollingBearings"Ing. VetanskapsAkad.-Handl.no. 196, 1947.

4. Lundberg,G., and Palnigren,"DynamicCapacityof RollingBearings,"AetaPolytechnlca,MechanicalEngineeringSeries,vol. l, no. 3, 1947.

5. Lundberg,G., and Palmgren,A.o "DynamicCapacityof RollingBearings,"Trans. ASME, J. Appl. Mech.,vol. 16, no. 2, 1949, pp. 165-172.

6. Lundberg,G. ana Palm@ren,A., "DynamicCapacityof RollerBearings,"Acta PolytechnlcaMechanicalEngineeringSeries,vol. 2, no. 3, 1952.

7. Welbull,W., PersonalCommunication,Jan. 23, Ig64.

8. Welbull,W., "EfficientMethodsfor EstimatingFatigueLife Distributionsof Roller Bearings,"Bidwell,3.B., ed., RollingContactPhenomena,

__ Elsevier,N.Y., N.Y., lg62, pp. 252-265.

g. Carter,T.L., "PreliminaryStudiesuf Rolllng-ContactFatigueLife ofHigi1-TemperatureBearingMaterials,"NASA RME57K12,April."IgSB.

i_ I0. Zaretsky,E.V., Anderson,W.J., and Parker,R.J., "The Effectof ContactAngle on Rolllng-ContactFatigueand BearingLoad Capacity,"ASLE Trans.,vol. 5, May 1962, pp. 210-219.

.T II. Grlsaffe,S.L., "Analysisof Shear Bond Strengthof Plasma-SprayedAluminaCoatingson StainlessSteel,"NASA TND-3113,1965.

12. Hertz,H., MiscellaneousPapers. Part V-The Contactof ElasticSolids.The MacMillanCo., (London),1896, pp. 146-162.

13. Thomas,H.R. ond Hoersch,V.A., "StressesDue to the Pressureof One: ElasticSolid Upon Another,"Univ. Ill., Eng. ExperimentStationBull.,; vol. 27, no. 46, July IS, 1930.

_.: 14. InternationalStandardsOrganization,"RollingBearlngs-DynamlcsLoadRatlngand Rating Llfe-Partl: CalculationMethods,"InternationalStandard28!/I-1977(E).

15. Cow., J.J., Townsend,D.P., and Zaretsky,E.V., "Analysisof DynamicCapacityof Low-Contac%-RatloSpur Gears Using Lundberg-PalmgrenTheory,"NASA TN D-BO2g,1975.

16. Coy, J.J, and Zaretsky,E.V., "Life Analysisof HelicalGear Sets using_,- Lunberg-PalmgrenTheory,NASA TN D-GO45,Ig?5.

III

_-_. _ J

1985018915-TSA13

I

17. Coy, J.3., Townsend, O.P., anQ Zaretsky, E.V., "Dynamic Lapaclty andSurface Fatigue Life for Spul and Helical Gears," ASMElrans, J. ofLubrication Teclmology, vol. gB, no. 2, pp. 267-276, 1976.

"i

18. Coy. 3.3., Loewenthal,S.H., and Zaretsky,E.V., "FatigueLife Analysisfor TractionDriveswlth Applicationto a ToroldalType Geometry,"NASATN D-B362,December1976.

19. Coy, 3.J., Zaretsky,E.V., and Cowgl!l,G.R., "FatigueLife AnalysisofRestoredand RefurbishedBearings,"NASA TN D-B486,May 1977. .

• I

20. Townsend,D.P., Coy, 3.J., and Zaretsky,E.V., "ExperimentalandAnalyticalLoad-LifeRealtlonfor AISI 9310 Steel Spur Gears,"ASMETrans. J. MechanicalDesign,vol. I00, no. l, pp. 54-60,JanuaryIg7B.

I

-_ 21. Rohn, D.A., Loewenthal,S.H., and Coy. 3.3., "SimplifiedFatigueLife" ASME Trans MechanicalDesign,Analysisfo_"TractionDrive Contacts, • (

vol. I03, no. 2, pp. 430-439,April IgBl. I. i

._: 22. Coy, J.J., Rohn, D.A., and Loewenthal S.H., "Life Analysisof a Nasvytls 1MultlrollerPlanetaryTractionDrlve,'_NASA TP-171O,April 1981.

23. Coy, J.O., 'townsend,O.P., and Zaretsky,E.V., 'tAnUpdateon the Life_i Analysisof Spur Gears,"AdvancedPower TransmissionTechnology,Fischer,

6.K, ed., NASA CP-2210,pp. 421-433,19B2.

_ 24. Rohn, D.A., Loewenthal,S.H., and Coy, J.J., "SizingCriteriafor: TractionDrives,"Power TransmissionTechnology,Fischer,G.K, ed., NASA= CP-2210,pp. 299-315,1982.

25. Savage,M., Parldon,C.A., and Coy, _.J., "ReliabilityModel forPlanetaryGear Trains,"ASME Trans.3. Mech. Trans. and Automationin

= Design,vol. I05, no. 3, Sept, 1983, pp. 291-397

26. Savage,M., Knorr, R.3., and Coy, J._., "Life and ReliabilityModels forHelicopterTransmissions,"AHS Paper AHS-RWP-16,Nov. IgB2.

27. Rohn, D,A., Loewenthal,S.H., and Coy. 3.3., "ShortCut for PredlclntTractlon-DrlveFatigueLife,"MachineDesign,vol. 55, no. 17, pp. 73-77,

._: 1983.

28. Lewlckl,D.G., Black,3.D., Savage,M. and Coy, 3.J., "FatigueLife- Analysisof a TurbopropR_ductlonGearbox,"NASA TM-BTOIi,1985., (Presentedat AIAA/SAE/ASMEJoint PropulsionConference,Cincinnati,June

11-13, lgB4.)

29. Bralow,R.E., Proschan,F., and Hunter,L.C., MathematlcalTheoryofReliability,John Wiley and Sons, Inc., 1965, pp. 48-61.

30. Lloyd,D.K., and Lipow,M., Reliability; ManagementMethodsandMathematics,Prentlce-Hall,Inc., 1962, pp. 271-27B.

• 31. loannldes,E., and Harris,T.A., "A New FatigueLife Model for RollingBearings,"ASME paper B4-Trlb-2B,1984 (to be publishedin ASME Trans.,

r _. Trlb.,IgB5).

12

" " .... 1985018915-TSA14

32. Bamberger, E.N., et al., Life Adjustment Factors for Ball and RollerBearlngs-AnEngineeringDesign G_ilde,ASME, N.Y., 1971.

33. Loewenthal,S.H., "Designof Power TransmittingShafts,"NASA RP-lI23,July IgB4.

TABLE i. - LUNDBERG-PALMGRENFATIGUELIFE ANALYSIS

LBasedupon 1939Weibulltheory.]

Analysis Year published

Bearings 1947Spur gears 1975Helicalgears 1976Restoredand refurbishedbearings 1977Load-liferelation 197bToroidaldrive (traction) 1976Optimizationof multirollertractiondrive 1980Simplifiedlife analysisfor tractiondrives 1980Nasvytisdrive 1981Spiralbevel gears 1982Planetaryassemblies 1982Transmissionassemblies 1983Rotary beam, torsionx tension-compression 1984

13

'!_ :_-'_--="- " _' "i_-_'-_.:t..',,_ .................... _r-- _ _........ ....._,,_..........._=_ ..................... ..

1985018915-TSB01

80- "

°iiC_

_n _1_1,1 I , I,a,l_l u , n,l,l,I22 5 10 20 50 100 200 500 1000

LIFE,hr

Figure1. - Weibulldistribution of bearingfatiguelife for140-.rambore-sizeangular-contactball bearing.MtL-L-?BOBlubricant_ thrust load, 9500Ibs; speed,10000rpm_temperature,121°C (250OF).

9,- lgO

!°o40

_ 2O i

F-.10 -.-

k,,..

6-

1 i I iI,l,I4 2 4 6 8 10xl02SHEARBOND STRENGTH,Ib/sqIn.

Figure2.oWeibulldistributionfor bond-shear-strengthforcoatingofspraypowderonstainlesssteel(ref. ll).

1985018915-TSB02

I ROOT-MEAN-SQUARE

1000 SURFACEROUGHNESS,pin,

280 115.=_' ..... MEDIANVALUECURVE

800_O CALCULATEDFOR50-PERCENT

_ PROBABILITYOF SURVIVALO _ SINGLEMEASUREDDETERMINATION

v,_ 600kk--O___C_-...-_F'-\_ MEDIANTIoNSVALUEOF10DETERMINA-

=

21)1)-- 13

.. I I I0 .2 .4 .6

... PROJECTEDTESTAREA,sqIn.; i._ Figure3.-Effecto_projectedtestareaon measuredandcalcu-

latedbondshearstrengthofcoatingsattwosurfacerough-:. nesses(ref.11).

-9,I"--FRONTPINIONBRG.

"" 80 SY _'e.-MAINDRIVEBRG.: . 60 F/'_PLANETBRG.,.J

40 r'"-MAINDRIVEGEAR,.,_ 20 '."-PLANETGEAR

_.=¢ ll _-PROPRADIALBRG.o I "-PROPTHRUSTBRG.

421 "-CARRIERSUPPORT

FIVEPLANET/ LSUN '_-PINION BRG.< BRG,SET./ GEAR GEAR0

=." I ,l,ldil I JlJl,hl I i lllllll100 1000 10000 1000O0

TIMETOREMOVAL,hr

Figure4. - Analyticallypredictedsystemandcomponentfatiguemissionlives,Combinedmaterialandmaterialprocessinglifeadjustmentfactorsusedforbearings.Gearmaterialisbaseline

i material-nofactorsneeded.NolubricationIli'eadjustmentfactorsused(ref.28).

: ..................,....: _,................................ ...,..,_,..,,.,,..,,___,.,.,,.,,.,,.,"I_OU]_lO /ODuo

/- ANALYTICAL-NOLIFE98 -- // ADJUSTMENTFACTORS

ANALYTICAL-LUbRICATION.// USEDLIFEADJUSTMENT// // / .1/FACTORSUSED-3 /-/ ,/- X/80

y'/ / -_//_LFIELD

_ 60 // / ,/ //" EXPERIENCE/ / / #'I_" 40 /" ,/ /t _. LANALYTICAL.J t / ,* J" MATERIALSLIFE

20 // / // / _,D.IIJSTMENT

_,. _/ ,// FACTORSUSED_' lo_ ,/

6 /i //,,,-,4 -- LANALYTICAL-MATERIALSANDO'_ LUBRICATIONLIFEADJUSTMENTa.

2 -- FACTORSUSED

I I lllllll I I I IIIlll I I I I1_111100 1000 1O000 100000

TIMETOREMOVAL,hr

Figure5. - Turbopropreductiongearboxfatiguelives (fieldexperience)comparedwithanalyticallypredictedsystemmissionlives. (Materialfactor impliescombinedmaterialandmaterialprocessingfactor.) (Ref. 28.)

"----- REPLACEMENTSADDEDTO

POPULATION.... NOREPLACEMENTS

.16>-I,--

_:g .08

,'L

O 2 4 6 8 10 12 14 16 18 20

RUNNINGTIME,t • rllLl0

Figdre6. - Renewaldensityfunctionscomparedforrolling-elemontbearings.Alsoshownisproba-bilitydensityforfailureofnewbearingassumingnoreplacements.Areaundercurvesrepresentsprobebilltyo! failure.

...... 1985018915-TSB04

320--CUMULATIVEREPLACEMENTS

280 -- ------- CUMULATIVEFAILURES

240-- (NOREPLACEMENTS)

160

12o

_" 80 ._..._.........-.-'--...... "_6o _

;i0 2 4 6 8 I0 12 14 16 18 20

RUNNINGTIME,t•rilL10

Figure7.-Cumulativerenewalandfailureforrolling-elemeqtbearings.

1985018915-TSB05

i_

i

I FATIGUEi_ l LIMIT,

I %,' L Nlmm2

! , z2oo o

, 550 O EXPERIMENTAL_ _:..-_'_'_,,_\ _ SUSPENDED_ II00 _ "_'_ _k TEST

_'_'_ \ "--"-- THEORETICAL_ 7_o_,_-..,.c-,,',,o\

__" " zooo_ 65o-Y' _ 900

u')

_; _-_°_o5 lo_ _0_ _o_• _ Ll0LIFE,CYCLES

_ _ (a)Martensiticsteel.

____ _ 15o

_', _ 275-- /- 0

. 200J 250_'__0,, _

250--

_/.. 2_)o4 I I j10p 106 107• Llo LIFE,CYCLES

_- (b)Softannealedsteel.

L. Figure8. - Comparisonbetweenexperimentalandpredictedstress-liferelationforAISI 52100steelrotatingbeamspecimensforcalculatedvaluesoffatiguelimit ou. (Ref.31.)

• . _ .. , ........ . ..... ...... , .........

!

• I

I

z

""- Rc

_,

(a)Flatbeamconfiguration.

FATIGUE iLIMIT, 0 EXPERIMENTAL iou, -- THEORETICAL J

Nlmm2 _ SUSPENDEDTEST i!N

looo L55°''"_b_ _¢-

_, I I I I Iz 600103 104 105 106 107 I@

LSOLIFE,CYCLES

(b)Stress-liferelationshipforcalculatedvaldesof fatiguelimitou.

Figure9. - Comparisonbetweenexperimentalandpredictedstress-liferelationfor flat beamspecimensmadefrombearingsteelandtestedin reversebending(ref. 31).

1985018915-TSB07

OF POOf_ Qu,:_if_

Figure10. - Finiteelementanalysisof turbineenginefan blade.

• ° *4

1985018915-TSBO8

t, Report No, 2, GovernmentAccession No, 3, Recipient'= Catalog No.

NASA TM-87017• °

4. Title and Subtitle 5, Report Date

FatigueCriterionto System Design,Life andReI iab i 1i ty e.PerformingOrganizationCode

505-37-7Z

,:. 7, Author(=) 8. Performlng Organization Report No,

, Erwin V. Zaretsky E-2562:._ , 10. Work Unit No.

_= 9.Pedormlng Organization Name and A_was11. Contract or Grant No.

_ NationalAeronauticsand Space Administration5; _ Lewis ResearchCenter

Cleveland, Ohio 44135 13.TypeofReportandPerlodOovere¢|_='; .

'_: 12. Sponsoring Agency Name and Address Technic a 1 Memorandum_ National Aeronautics and Space Administration _4.SponsoringAgencyCf_de-_ Washington,D.C. 20546

" 15. Supplementary Notes

'- Preparedfor the Twenty-firstJoint PropulsionConference,cosponsoredby theAIAA, SAE, and ASME, Monterey,California,July8-I0, 1985.

i _ ,_. 16, Abstract

A generalized methodology to structural life prediction, design, and reliability_. based upon a fatiguecriterionis advanced. The lifepredictionmethodologyis: _ based in part on work of W. Weibulland G. Lundbergand A. Paimgren. Thei_ approachincorporatesthe computed lifeof elementalstress volumesof a complex

machineelementto preaictsystem life. The resultsof couponfatiguetestingcan be incorporatedinto the analysisallowingfor lifepredictionand component

_,L_! or structuralrenewalrates with reasonablestatisticalcertainty.

11, Key Words (Suggested by Authorlsl) 18, Distribution Statement

Life prediction Unclassified - unlimiteaFatigue STAR Category37

• ReliabilityDesign

: lg, Security Classlf, (9f this report) 20. Security Clssslf. (of this page) 21, No, of pages 22, Price", Unclassified Unclassified

tL

,: _ *For sale by the National Technical Information Service, Springfield, Virginia 22161 _t_ _j`,`,_!

1985018915-TSB09