Rolling-Element Fatigue Testing and Data Analysis—A … (ISO) and American National Standards...

Tribology Transactions, 54: 523-541, 2011ISSN: 1040-2004 print / 1547-397X onlineDOI: 10.1080/10402004.2011.568673

Rolling-Element Fatigue Testing and Data Analysis—A Tutorial

BRIAN L. VLCEK1 and ERWIN V. ZARETSKY2

1Georgia Southern UniversityStatesboro, Georgia

2NASA Glenn Research CenterCleveland, Ohio

In order to rank bearing materials, lubricants and other de-

sign variables using rolling-element bench-type fatigue testing

of bearing components and full-scale rolling-element bearing

tests, the investigator needs to be cognizant of the variables

that affect rolling-element fatigue life and be able to maintain

and control them within an acceptable experimental tolerance.

Once these variables are controlled, the number of tests and

the test conditions must be specified to assure reasonable statis-

tical certainty of the final results. There is a reasonable corre-

lation between the results from elemental test rigs with those

results obtained with full-scale bearings. Using the statistical

methods of Weibull and Johnson, the minimum number of tests

required can be determined. This article brings together and

discusses the technical aspects of rolling-element fatigue test-

ing and data analysis and makes recommendations to assure

quality and reliable testing of rolling-element specimens and

full-scale rolling-element bearings.

KEY WORDS

Rolling-Element Fatigue; Bearing Fatigue Testing; Bench-Type Testing; Probabilistic Analysis of Fatigue Data; WeibullAnalysis

INTRODUCTION

Though there can be multiple failure modes of rolling-elementbearings, the ultimate failure mode limiting bearing life is con-tact (rolling-element) surface fatigue of one or more of the run-ning tracks of the bearing components. Rolling-element fatigueis extremely variable but is statistically predictable depending onthe material (steel) type, the processing, the manufacturing, andoperating conditions (Zaretsky (1)). Sadeghi, et al. (2) providedan excellent review of this failure mode. Alley and Neu (3) pro-vided a recent attempt at modeling rolling-element fatigue. Withimproved manufacturing and material processing, the potential

This article is not subject to US copyright law.Manuscript received February 21, 2011

Manuscript accepted March 1, 2011Review led by Daniel Nelias

improvement in bearing life can be as much as 80 times that at-tainable in the late 1950s or as much as 400 times that attainablein 1940 (Zaretsky (4)).

In Germany in 1896, Stribeck (5) began fatigue testing full-scale rolling-element bearings. Rolling-element fatigue life anal-ysis is based on the initiation or first evidence of fatigue spallingon a loaded, contacting surface of a bearing. This spalling phe-nomenon is load cycle dependent. Generally, the spall begins inthe region of maximum shearing stresses, located below the con-tact surface, and propagates into a crack network. Failures otherthan that caused by classical rolling-element fatigue are consid-ered avoidable if the component is designed, handled, and in-stalled properly and is not overloaded (Zaretsky (1)). However,under low elastohydrodynamic (EHD) lubricant film thicknessconditions, rolling-element fatigue can be surface or near-surfaceinitiated with the spall propagating into the region of maximumshearing stresses. C. A Moyer and E. V. Zaretsky (1) discussed“failure modes related to bearing life” in detail.

The database for ball and roller bearing fatigue testing is ex-tensive. A concern that arises from these data and their analysis isthe variation between life calculations and the actual endurancecharacteristics of these components. Experience has shown thatendurance tests of groups of identical bearings under identicalconditions can produce a variation in life at a 90% probability ofsurvival (L10 life) from group to group. If a number of apparentlyidentical bearings are tested to fatigue at a specific load, there is awide dispersion of life among these bearings. For a group of 30 ormore bearings, the ratio of the longest to the shortest life may be20 or more (Zaretsky (1)). This variation can exceed reasonableengineering expectations.

Variables affecting rolling-element fatigue have been experi-mentally studied and documented by many investigators for over100 years. In 1963, Moult (6) discussed the then-known vari-ables that affect rolling-element fatigue life. He concluded that,“Qualitative comparisons of bearing performance require testbearing and test conditions to be statistically equivalent at allstages of processing, including the material, except for the vari-able being compared. Quantitative comparisons have the addi-tional requirement that the results must be related to the perfor-mance of a control bearing sample in combination with a controllubricant.”

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524 B. L. VLCEK AND E. V. ZARETSKY

NOMENCLATURE

A = Area due to plastic deformation and wear (m2, in2)C = Confidence number (fractional percentage or

percentage)e = Weibull slope or Weibull modulusF = Probability of failure (1 −S) (fractional percentage)H = Depth of stressed surface due to plastic deformation and

wear, maximum distance from chord to contact surface(m, in)

h′ = Distance from chord to original surface (m, in)j = Sequential failure number where j = 1, 2, 3, . . . , r

L = Life (revolutions, stress cycles, or hours)L10 = 10% Life or life at which 90% of population survives

(revolutions, stress cycles, or hours)L50 = Median life or life at which 50% of a population fails

(revolutions, stress cycles, or hours)Lβ = Characteristic life or life at which 63.2% of population

fails (revolutions, stress cycles, or hours)Lµ = Location parameter or life below which no failures will

occur (revolutions, stress cycles, or hours)

.

l = 1/2 Chord length (m, in)m = Total number of bearings or specimens in a subgroup,

number of bearing testersn = Number of data sets, test series, or samplesPr = Radial load (N, lb)Pt = Thrust load (N, lb)R = Original radius or curvature (m, in)Rp = Profile radius (m, in)r = Number of failures or number of subgroupsS = Probability of survival (1 −F) (fractional percentage)S90 = 90% Probability of survival (fractional percentage)ω = Rotational speed (rpm)

Subscripts

ir = Inner raceor = Outer racere = Rolling elementx = Probability of survival

A compilation of those variables that effect rolling-elementfatigue life were codified in Bamberger, et al. (7) as AmericanSociety of Mechanical Engineers’ (ASME) life-modifying factorsto the then-existing International Organization for Standardiza-tion (ISO) and American National Standards Institute/AmericanBearing Manufacturers Association (ANSI/ABMA) rolling bear-ing life prediction standards. Later these variables were expandedby the Society of Tribologists and Lubrication Engineers (STLE)to become more inclusive (Zaretsky (1)). The life-modifying fac-tors were updated from those of Bamberger, et al. (7) to reflectthe then-existing database (Zaretsky (1)).

In order to rank bearing materials, lubricants and design vari-ables using rolling-element bench-type fatigue testing and full-scale rolling-element bearing tests, the investigator needs to beaware of the variables that affect rolling-element fatigue life andbe able to maintain and control them within an acceptable exper-imental tolerance. Once these variables are controlled, the num-ber of tests and the test conditions need to be specified to assurereasonable statistical certainty of the final results. Using the sta-tistical methods of Weibull (8), (9) and Johnson (10), the mini-mum number of tests required can be determined.

There is reasonable correlation between the results from ele-mental test rigs with those results obtained with full-scale bear-ings. It is the objective of the work reported herein to bring to-gether and discuss the technical aspects of rolling-element fatiguetesting and make recommendations to assure quality and reli-able testing of rolling-element specimens and full-scale rolling-element bearings.

ROLLING-ELEMENT FATIGUE TESTING

Bench-Type Fatigue Testers

The determination of the rolling-element fatigue life of a bear-ing made from a particular steel or the effect of a specific lubri-

cant on fatigue life is an expensive undertaking. Very high scatteror dispersion in fatigue life makes it necessary to test a large num-ber of specimens. The cost is high in both money and time. This isborne out by the fact that a high-quality aircraft bearing can costupwards of several thousand dollars and run for several thousandhours. Bearing companies and various research laboratories havepioneered the use of bench-type rolling-element fatigue testersthat can simulate to various degrees the conditions found underfull-scale bearing operation. Generally, these test rigs performaccelerated rolling-element fatigue testing of element specimenssuch as a ball or a roller at maximum Hertz (compressive) stresslevels beyond 4.14 GPa (600 ksi).

The results obtained with the bench-type testers have beenused to indicate trends and to rank materials and lubricants. Thevast majority of published data have been obtained in bench-type fatigue testers. The results from these testers qualitatively,although not necessarily quantitatively, compare with test resultsfrom full-scale bearing tests (Zaretsky, et al. (11)).

The best compendium of rolling-element fatigue testing andtypes of bench-type rolling-element fatigue testers was that com-piled by Hoo (12) for the American Society for and Testing Ma-terials (ASTM). The more commonly used test rigs discussed inHoo (12) are as follows:

1. Two-disk machine or ring-to-ring rolling-contact fatigue tester2. Barwell four-ball fatigue tester3. NASA spin rig4. Pratt-Whitney one-ball fatigue tester (Fig. 1)5. General Electric rolling-contact (R-C) tester (Fig. 2)6. NASA five-ball fatigue tester (Fig. 3)7. Unisteel flat washer fatigue tester8. Federal-Mogul ball-rod rolling-contact fatigue (RCF) tester9. NTN cylinder-to-ball RC fatigue tester

10. NTN cylinder-to-cylinder RC fatigue tester

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Rolling-Element Fatigue Testing and Data Analysis 525

Fig. 1—P&W one-ball fatigue tester (Hoo (12) ): (a) test apparatus and (b)test-specimen assembly.

These test rigs employ an automatic failure detection andshutdown system.

The lubrication mode for the test specimen depends on thetype of test rig and the operating conditions. The lubricationmode can be by oil mist, oil jet (recirculating oil system), oil drip,or oil bath. Once-through, oil-mist lubrication for these types oftests appears to be the most efficient lubrication mode. Aftereach test the oil should be discarded and not reused. However, asample of oil from each test should be saved and cataloged forlater laboratory analysis, if necessary.

Test Procedures for Bench Type Testers

When testing in a bench-type rig, various precautions mustbe taken in order to obtain valid comparison of the lubricant,material, and/or operating variables. The procedure as outlinedis that followed by the authors and other investigators who haveconducted a considerable number of rolling-element fatiguestudies. It is mandatory, first of all, to have all of the rolling-element specimens fabricated from a single heat of material andheat-treated simultaneously to the same hardness. All specimensmust have the same surface finish unless, of course, the surfacefinish is the parameter being studied. The steel microstructure

Fig. 2—GE rolling-contact (RC) fatigue tester (Hoo (12) ): (a) test appara-tus and (b) test-specimen assembly.

Fig. 3—NASA five-ball fatigue tester (Hoo (12) ): (a) test apparatus and(b) test-specimen assembly.

should be reported together with the retained austenite andresidual stress before and after testing. It is of prime importancethat the difference in hardness between the contacting testspecimens remains constant.

Prior to testing, all contacting elements should be cleanedthoroughly with a solvent and wiped dry with a clean cloth. Sub-sequently, the mating elements should be coated with the test lu-bricant and installed in the tester.

Lubricant flow into the test assembly should be monitored. Inaddition, the specimens should be loaded prior to startup. If noload is placed on the specimens prior to starting the test, dam-age can result due to skidding of the contacting surfaces. Speedand test temperature should be periodically monitored during thetesting. Subsequent to testing, all specimens should be examinedand their condition recorded. It has been the authors’ experiencethat 30 specimens should be run for a given series of tests antic-ipating that at least one third to one half of the specimens willfail. Care must be taken to make sure that, in any series of tests inwhich material parameters are compared, all tests are conductedwith the same lubricant formulation obtained from the same lu-bricant batch. The same lubricant obtained from different batchescan give significantly different results. In testers where mating el-ements are not replaced subsequent to test, the elements must beexamined to make sure that they have not been damaged duringprior operation.

Where elevated temperature tests are to be conducted it is ageneral procedure to heat the specimen housing prior to begin-ning a test. Before startup, the lubricant flow is begun to makesure that there is a sufficient amount of lubricant in the test sys-tem to prevent wear at startup.

It is generally a good idea, where material comparisons areto be made, that the lubricant has good storage stability. It hasbeen the authors’ experience that superrefined mineral oils of theparaffinic or naphthenic type and polyalphaolefin (PAO) are sta-ble over extremely long periods of time and can give consistentresults, making material comparisons possible. However, whenusing ester-based lubricants, long-term stability of the lubricantmust be considered. To avoid water absorption, the lubricant

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should not be exposed to the atmosphere for any length of time.During storage, the lubricant must be kept under an inert covergas.

For a given type of fatigue tester where there are multiple rigsand/or multiple test heads being used, it is prudent to assure thateach test rig and/or test head provides the same test results. Thisassures that there are no sources of random variability or system-atic differences between the test rigs and/or test heads (McCooland Valori (13)). Because wear-out of the test apparatus is a ran-dom variable, these comparisons need to be made on a continu-ous basis.

Deformation Effects on Contact Geometry and Stress

In order to obtain rolling-element fatigue data in a bench-type, rolling-element fatigue test rig within a reasonable time,the maximum Hertz test stresses that are used are generally at orabove the static load capacity of the hardened bearing steel sur-face at which plastic deformation will occur. In addition to plasticdeformation, experience has shown that fracture of the contactingsurfaces may occur at maximum Hertz stresses at or beyond 6.9GPa (1,000 ksi). For efficiency of testing in these element testers,maximum Hertz stresses between 4.83 and 5.52 GPa (700 to 800ksi) are recommended.

With maximum compressive stresses of less than 4.14 GPa(600 ksi), gross plastic deformation of rolling surfaces will gen-erally not occur for normal hardened bearing steels (Zaretsky,et al. (14)). However, Drutowski and Mikus (15) reported thatsmall amounts of plastic deformation can occur at maximumHertz stresses as low as 1.1 GPa (160 ksi). Drutowski (16) fur-ther reported that for AISI 52100 steel there was an inversion be-tween hardness and the maximum Hertz stress necessary for theinitiation of plastic deformation. For a Rockwell C hardness of62, the onset of plastic deformation occurs at a maximum Hertzstress of 1.92 GPa (278 ksi). For a Rockwell C hardness of 58,the onset occurs at a maximum Hertz stress of 2.6 GPa (377 ksi).Drutowski (16) concluded that the contact stress at which plasticdeformation is initiated is a function of the steel structure but isindependent of the diameter of the rolling ball (roller).

Usually, in normal bearing operation, gross plastic deforma-tion is of no concern. However, in laboratory fatigue tests ofrolling elements, stresses greater than 4.14 GPa (600 ksi) arethe rule rather than the exception. Due to the isotropic or kine-matic hardening, the gross plastic deformation is accompaniedby changes in the contact geometry, residual stresses, and yieldlimit. As the amount of plastic deformation increases, the contactstress is reduced from that calculated using the Hertz formulas. Itwas found that the resultant Hertz stress is not only affected bythe applied load, material hardness, and structure but also by theelastohydrodynamic film formed (Zaretsky, et al. (14)).

Figure 4a is a schematic diagram of the transverse sectionof a ball surface or crowned roller. The line of the true sphereor crown and the profile after plastic deformation and wear areshown. Figure 4b is a schematic diagram of a surface trace of atransverse section with deviations from that of the true spherehighly magnified (Zaretsky, et al. (17)).

Fig. 4—Cross section of stressed ball track (not to scale; Zaretsky, et al.(17)): (a) schematic diagram of transverse section of ball surfaceand (b) schematic diagram of surface trace of transverse sectionwith deviation from true sphere or roller crown highly magnified.

For many applications where gross plastic deformation occurs,it is necessary to calculate the effective Hertz stress or the contactstress after plastic deformation occurs (Zaretsky, et al. (18)). Thiscan be accomplished by deriving the radius of curvature Rp of thedeformed rolling element as shown in Appendix A. Referring toFig. 5a, for a convex surface such as a ball or a crowned roller,

Rp =( A

H

)2

2{

R −[R2 − ( A

H

)2]1/2

− H} [1]

The deformed radius Rp ′ (not shown) in the plane perpendic-ular to the plane of the profile shown in Fig. 5(a) is

R′p = R′ − H [2]

where R′ is the radius of the body in the perpendicular plane.Because H is extremely small relative to R;

R′p = R′ [3]

Substituting Rp into the Hertz equations for contact stress,the effective Hertz stress after gross plastic deformation of therolling-element surface can be calculated.

For a race groove surface having a concave or negative radiusR, as shown in Fig. 5b,

Rp =( A

H

)2

2{

R −[R2 − ( A

H

)2]1/2

+ H} [4]

The area of deformation (and wear) A and the track depth Hcan be measured directly from a surface contour trace. Becauseplastic deformation can hardly change the roller diameter, themodified radius of curvature Rp as derived should only be appliedto ball bearings, not to cylindrical roller bearing.

The effect of deformation and wear on the profile radiusand the resultant maximum Hertz stress is illustrated in Table1 for the NASA five-ball fatigue tester. The radius of curvature

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Fig. 5—Rolling element bearing surfaces after plastic deformation (Zaret-sky, et al. (17)). (a) Cross section of ball or crowned roller surface(see Fig. 4) and (b) cross section of race surface.

across the running track of the upper test ball is increased. Onlyconsidering deformation of the upper test ball, the resultantmaximum Hertz stress is decreased by approximately 5%.However, considering deformation of both the upper and lowerballs, the maximum Hertz stress is decreased by approximately10% (Zaretsky, et al. (18)). At an initial maximum Hertz stressof approximately 4.14 GPa (600 ksi) no plastic deformation wasmeasured. This stress reasonably correlates with those stressesreported in the ISO Standards (ISO (19)) of 4.2 (609 ksi) and 4.0GPa (580 ksi) related to the static load capacity of ball and rollerbearings, respectively.

The contact model is presented as an introduction to the topicand can be used with reasonable engineering certainty to deter-mine the reduction in the contact stress resulting from gross plas-tic deformation. However, for greater accuracy, recent additional

work in elasto-plastic point contacts was reported on by Chen,et al. (20). Nelias, et al. (21), (22) have reported on elastic–plasticsliding contacts, and Wang et al. (23) have modeled elastic–plasticcontacts between a sphere and a flat.

Full-Scale Bearing Testers

As discussed above, the use of rolling-element fatigue testersqualitatively compares with results of full-scale bearing testsand/or from field data. However, the variables once defined frombench-type tests must be subject to realistic operating conditionssuch as magnitude and type of load, speed, mode of lubrication,and temperature such as those found in actual bearing applica-tions (Galbato (24)). In general, for fatigue testing full-scale bear-ings it is recommended that maximum resultant Hertz stressesbe equal to or less than 2.41 GPa (350 ksi). However, maximumHertz stresses as high as 3.1 GPa (450 ksi) can be used for test-ing many types of bearings. These requirements were discussedby Galbato (24). He appropriately stated that “The combined in-fluence of important environmental conditions which simulate asclosely as possible the expected service environment and that asufficient number of tests be performed to evaluate the scatterof fatigue life as characterized by the Weibull distribution (func-tion).”

From Galbato (24), the various arrangements for arrangingtesting ball and roller bearings are illustrated in Fig. 6. “Thestandard bearing test rigs include a stationary frame, a movablehousing frame, and a single test shaft on which test bearings aremounted. The type and size of the bearing will determine theoverall size of the test rig and drive system. Operating conditionsinclude the magnitude and type of load, speed, mode of lubrica-tion and temperature.” These variables should be continuouslymonitored.

A four-bearing configuration for radial-only loading is shownin Fig. 6a (Galbato (24)). This arrangement permits simultaneoustesting of four bearings. The applied radial load (2P) acts on thetwo inboard bearings. The reactionary forces (P) act on the twoinboard bearings.

A three-bearing configuration for combined loading is shownin Fig. 6b (Galbato (24)). This arrangement provides for the com-bined loading of the two outboard bearings. The applied load(2P) acts on the center load radial bearing (B). The thrust load(T) is applied axially on one outboard bearing and reacted to bythe other. In this arrangement, the reactionary radial load (P), incombination with the applied and reactionary thrust load (T) act-ing on the outboard test bearings, provides a simultaneous com-bined loading on these two bearings (Galbato (24)).

A two-bearing configuration for thrust loading only is shownin Fig. 6c. In this arrangement, the thrust load (T) is applied axi-ally to one bearing and is reacted to by the second bearing.

The lubrication mode for these types of tests is usually recir-culating oil jet lubrication. Oil filtration of 10 µm or better shouldbe provided and the filter replaced after each test. Other lubrica-tion modes that are used are under- and through-the-race lubri-cation and oil mist. Though oil bath lubrication can be used forlow speed tests, it is not recommended. The oil condition shouldbe monitored and the oil changed after each test. The used oilshould be discarded and not reused. A sample of oil from each

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TABLE 1—DEFORMATION AND WEAR AND THEIR EFFECT ON MAXIMUM HERTZ STRESS FOR AISI M–50 STEEL BALLS IN NASA FIVE-BALL

FATIGUE TESTERa

Effective MaximumHertz Stress, GPa (ksi)

LubricantType

LubricantDesignation

Original ProfileRadius,

R (m, in.)

Deformation and WearArea from SurfaceTrace, A (Fig. 4)

(m2, in.2)

Depth from OriginalProfile to

Deformed Profile, H,(Figs. 4 and 5a)

(m, in.)

Calculated ProfileRadius, Rp

(Fig. 5a),(from Eq. [1])

(m, in.)

NoDeformation

and Wearof Support

Balls Assumed

Deformationand Wearof Support

Balls AssumedEqual toTest Ball

Ester NA-XL-3 6.35 × 10−3 4.95 × 10−10 16.68 × 10−7 8.35 × 10−3 5.27 5.03(0.250) (7.67 × 10−7) (6.567 × 10−5) (0.329) (765) (730)

NA-XL-8 6.35 × 10−3 5.16 × 10−10 16.75 × 10−7 8.18 × 10−3 5.29 5.06(0.250) (8.00 × 10−7) (6.596 × 10−5) (0.322) (767) (734)

Mineral oil NA-XL-4 6.35 × 10−3 6.13 × 10−10 20.19 × 10−7 8.79 × 10−3 5.23 4.95(0.250) (9.50 × 10−7) (7.947 × 10−5) (0.346) (759) (718)

NA-XL-7 6.35 × 10−3 5.37 × 10−10 17.95 × 10−7 8.51 × 10−3 5.27 5.00(0.250) (8.33 × 10−7) (7.067 × 10−5) (0.335) (763) (725)

aBall dia., 12.7 × 10−3 m (0.500 in.); Rockwell C hardness, 62; initial maximum Hertz stress, 5.52 GPa (800 ksi); contact angle, 10◦; number ofstress cycles, 30 × 103. Data from Zaretsky, et al. (17).

Fig. 6—Basic configurations for standard rolling-element bearing testrigs including stationary frame movable housing frame and sin-gle test shaft for mounted bearings (from Galbato (24): (a) Four-bearing arrangement for radial loading only; (b) three-bearingarrangement for combined loading; and (c) two-bearing arrange-ment for thrust loading only.

test should be saved and cataloged for later analysis, if necessary.In either case, the same lubrication conditions have to be assuredin the most loaded areas for all four bearings, especially whenbath lubrication is used.

Automatic shut-off systems are used for each test rig basedon temperature rise and monitoring any vibration amplitude in-crease that indicates the occurrence of an incipient fatigue spall.The bearings’ oil in and out temperatures and the bearings’ outer-(and, if possible, inner-) ring temperatures are measured andmonitored. Lubricant flow is measured and monitored. Chip de-tectors in the lubricating system should be critically placed to de-tect a metallic chip of a significant size emerging from a spall.Time of operation should be monitored for each test machine andeach test bearing (Galbato (24)).

Procedures for Testing Full-Scale Bearings

It should again be emphasized that bench-type testing is usedto observe comparative trends due to changes in lubricant, mate-rial type, material processing, operating parameters, etc. Thoughgreat efforts are made to appropriately select and control testingtechnique and test parameters, the only way to obtain quantita-tive data is to test full-scale bearings. However, the cost of test-ing bearings can be more than 100 times greater than simplifiedrolling-element specimens. Similar caution and test proceduresshould be taken with full-scale bearings as with bench-type testersdiscussed above.

A common problem in testing full-scale bearings is misalign-ment of the bearing within the rig. Prior to testing, the bearingshould be measured for dimensional clearances and tolerances.In general, test stresses lower than 2.41 GPa (350 ksi) maximumHertz are suggested. However, maximum Hertz stresses as highas 3.1 GPa (450 ksi) on the inner race can be used for testing manytypes of bearings.

Another common problem associated with full-scale bearingtesting is the failure of investigators to consider or report the

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interference fits between the bearing bore and shaft and the bear-ing outer diameter and housing. The interference fit between thebearing bore and the shaft will induce tensile hoop stresses in thebearing inner race. These tensile stresses can increase the mag-nitude of shearing stresses below the contacting surface betweenthe rolling elements and the bearing inner race and reduce thebearing fatigue life (Coe and Zaretsky (25); Oswald, et al. (26),(27)).

The interference fit between the bearing bore and shaft incombination with the interference fit between the bearing outerdiameter and the housing will reduce the bearing’s internalclearance and affect the bearing fatigue life. Hence, the samebearing run under the same operating conditions can producesignificantly different life results because of interference fits.Accordingly, a “fit-up study” should be conducted and reportedregarding the effect of interference fit and bearing internalclearance. Detailed findings and insights regarding interferencefit can be found in Coe and Zaretsky (25) and Oswald, et al.(26), (27).

Another problem is that the static load capacity of the bear-ing should not be exceeded. This is the maximum load that abearing can be permitted to support when not rotating. It is ar-bitrarily defined as the load that will produce a permanent in-dentation of the race having a depth equal to 0.0001 times therolling-element diameter. Based upon ISO 76:2006 (ISO (19))for through-hardened bearing steel of hardness Rockwell C 58and above, this load correlates to a maximum Hertz stress of ap-proximately 4.0 GPa (580 ksi) for cylindrical roller bearings. Forball bearings this load correlates to a maximum Hertz stress ofapproximately 4.2 (609 ksi; ISO (19)). When permanent defor-mation exceeds this value, bearing vibration and noise can no-ticeably increase when the bearing is subsequently rotated underlesser loads.

Although it is not recommended, a bearing can be loadedabove the static load capacity as long as the load is applied whenthe bearing is rotated. For an angular-contact ball bearing, theplastic deformation that occurs during rotation will be distributedevenly around the periphery of the races and will not be harmfuluntil it becomes more extensive. However, for a radially loadedball or roller bearing, when the maximum Hertz stress on theouter race exceeds 4.0 GPa (580 ksi), the plastic deformation willbe concentrated at the position of the maximum loaded rolling el-ement. In this event, exceeding the bearing’s static load capacitywould be unacceptable.

In many applications where marginal lubrication is a factor,such as at elevated temperature, high surface tangential speedsare desirable in the bearings because of elastohydrodynamic ef-fects. This can make a difference between the bearing operatingfor relatively short time periods with wear and surface distress be-ing the failure mode or operating for extended time periods withrolling-element fatigue being the failure mode. Lower speeds willnot necessarily give longer lives in terms of inner-race revolu-tions or, conversely, higher speeds will not give shorter lives interms of actual running time. Temperatures should be monitoredon the inner race and the outer race and in the housing aroundthe bearing. Lubricant sump and ambient temperatures shouldalso be monitored.

Fig. 7—Relative dynamic load capacity of 7208-size bearings as a func-tion of relative dynamic load capacity of five-ball fatigue testerfor various lubricants at 149◦C (300◦F; Zaretsky, et al. (17)).

Posttest examination of the failed test bearings should recordthose components; that is, balls or rollers, inner and outer races,of each bearing that has failed and their respective failure mode.Based on the methods of Johnson (10), it is possible to not onlydetermine the statistical life and failure distribution of the bearinggroup but also the lives of the ball or roller set, the inner race, andthe outer race.

Correlation of Bench-Type Tests with Full-Scale BearingResults

Bench-type fatigue tests can qualitatively rank material, lubri-cant, or operating variables. This is illustrated in Fig. 7, which is acomparison of the relative dynamic load capacity obtained in theNASA five-ball fatigue tester with four different test lubricantsat 149◦C (300◦F) to fatigue tests with 7208-size deep-groove ballbearings with the same four test lubricants (Zaretsky, et al. (17)).Each test lubricant for the bench tests and the full-scale bearingtests came from a single batch. Though the operating conditions,bearing steels, and contact geometries between the bench test rigconfiguration and the full-scale bearings are different, both typeof tests ranked the fatigue lives with each of the four lubricants inthe same order. These data are summarized in Table 2.

The operating conditions of the bench-type tests typically dif-fer from the in-service operating conditions for full-scale rolling-element bearings. Most bench-type tests are accelerated testswith extreme conditions (higher speeds, higher loads, and/orhigher operating temperatures) compared to typical bearing op-erating conditions. Though there may be differences in the mag-nitude of the quantitative lives, the general trends are compara-ble; that is, there is agreement in relative rankings of materials,lubricants, operating conditions, etc.

Figure 8 compares the relative load capacity also obtained inthe NASA five-ball fatigue tester to the load capacity of 207-size deep-groove ball bearings where ball hardness is the variable

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Fig. 8—Relative dynamic load capacity of 207-size bearings as functionof relative dynamic load capacity of five-ball fatigue tester forvarious dynamic load ball hardness. Nominal hardness of upperball (for five-ball tests) and inner race (for bearings), 63 RockwellC (Zaretsky, et al. (28)).

(Zaretsky, et al. (28)). As can be seen for the lubricant and hard-ness variables, the correlation of the five-ball fatigue tester resultswith those of full-scale bearing tests on a relative basis is excel-lent. For both sizes of bearings, the maximum stress level was ap-proximately 2.41 GPa (350 ksi) at the inner race-ball contact incomparison with 5.52 GPa (800 ksi) for the five-ball tester.

A third example of the correlation between the NASA five-ball fatigue tester and full-scale bearings is shown in Fig. 9(Parker and Zaretsky (29); Bamberger and Zaretsky (30)). Therelative L10 lives of bearings made from three steels are comparedto rolling-element fatigue data from the NASA five-ball fatiguetester. These data show the effect of total percentage weight ofalloying element on bearing life. The bearings were thrust loaded120-mm bore angular-contact ball bearings made from AISI M-1, AISI M-50, and WB-49 steels (Bamberger and Zaretsky (30)).The AISI M-42 steel run in the five-ball tester had a similar mi-crostructure to WB-49. Both steels contained relative high per-centages of cobalt. The bearings were run at a maximum Hertzstress at the inner race of 2.23 GPa (323 ksi) and a temperatureat the outer ring of 316◦C (600◦F). The five-ball tests were run

Fig. 9—Comparison of results from NASA five-ball fatigue tester and 120-mm bore angular-contact ball bearings showing effect of totalweight percentage of alloying elements on fatigue life (Parkerand Zaretsky (29)).

at a nominal temperature of 66◦C (150◦F). Again, there is excel-lent correlation between the NASA five-ball fatigue tester fatiguedata with the full-scale bearing fatigue data.

These comparisons show that bench-type fatigue testers canreliably identify qualitative effects of many variables on rolling-element fatigue life. By benchmarking these life results to analready existing database, it is possible to develop bearing life-modifying factors with the Lundberg-Palmgren theory (Lundbergand Palmgren (31)) to predict bearing life with reasonable engi-neering certainty (Zaretsky (1)).

Testing Methodologies That Reduce Total Test Time

Experimentally determining rolling-element bearing life is acomplex, time-consuming, and costly task. In addition to the driv-ing need to find methods to reduce testing time (and cost), not allbearing and/or specimens that are tested will or can be expectedto fail in a reasonably prescribed time limit. Thus, methods to re-duce testing time are of great importance. An early explanationof three methodologies for reducing testing time can be found inthe classic work of Leonard G. Johnson (10). In the first method,more specimens are run simultaneously than are intended to fail.

TABLE 2—COMPARISON OF ROLLING-ELEMENT FATIGUE LIVES OBTAINED WITH FOUR TEST LUBRICANTS IN NASA FIVE-BALL FATIGUE

TESTER AND 7208-SIZE ANGULAR-CONTACT BALL BEARINGS (ZARETSKY, ET AL. (17))

Five-Ball Fatigue 7208-Size Angular-ContactTester2 Ball Bearingsb

Lubricant Lubricant L10 Life, Millions of Ratio of L10 Life L10 Life, Millions of Ratio of L10 LifeType Designation Stress Cycles to L10 Life with NA-XL-7 Millions of Inner-Race Revolutions to L10 Life with NA-XL-7

Ester NA-XL-3 13.7 0.31 1.1 0.33NA-XL-8 14.9 0.34 1.4 0.42

Mineral oil NA-XL-4 42.5 0.96 2.5 0.76NA-XL-7 44.5 1.00 3.3 1.00

aTemperature, 149◦C (300◦F); maximum Hertz stress, 5.52 GPa (800 ksi); material, vacuum processed (VP) AISI M-1 steel; steel RockwellC hardness, 63; speed, 10,000 rpm.bTemperature, 149◦C (300◦F); thrust load, 20,461 N (4,600 lb); maximum Hertz stress, 2.41 GPa (350 ksi); materials (assumed to be)air-melt processed (AM) AISI 52100 steel; hardness, not reported; speed, 3,450 rpm

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For example, for a Weibull slope of 1.0, the median time to fail10 out of 20 samples is 76% less than the median time requiredto fail 10 out of 10 samples. This is assuming that there are noreplacements of failed specimens, all testers are the same, and allspecimens are run simultaneously. Because the width of the con-fidence band is determined by the number of items failed and notthe total number of specimens in a test (Johnson (10)), the num-ber failed is 10 in both cases. As a result, the only difference is thatthe 10 items having the lowest lives in a lot of 20 are plotted ona Weibull plot instead of all 10 specimens in a lot of 10 (Johnson(10)). The first scenario will take 24% of the time it takes for thesecond, and yet the necessary life information can be obtained bythe first case.

Sequential analysis is a second methodology for shorteningtotal test time described by Johnson (10). Test specimens arefailed sequentially one after the other. The total number of spec-imens needed to fail is not known in advance. The investigatordecides after each failure whether additional testing is required.In this manner, the bare minimum number of runs needed todemonstrate an improvement or worsening of life is conducted. Amethod similar to this has been used extensively by E. V. Zaret-sky and colleagues since the late 1950s. An estimate of the 50%life (L50) is made and test bearings are then run on identicaltesters until this life is reached. As samples fail or are suspendedat the L50 life, they are removed, and new samples are mountedand evaluated to the estimated L50. In this manner, at least 50%of the samples are typically failed out of the entire available pop-ulation. In some cases, the L50 target may have to be adjustedas dictated by the number of failures encountered. The exper-imenter decides after each failure whether additional testing isrequired. In this manner, the minimum number of runs needed todemonstrate an improvement or worsening of life is conducted.

Sudden death testing (SDT) is the third method of reducingtesting time described by Johnson (10). The total accumulatedtest time is reduced by not running all specimens to failure. Thetotal number of specimens to be evaluated n is divided into equal-sized subgroups according to the number of available experimen-tal testers. Thus, there are m specimens in each equal-sized sub-group, and there are a total of r subgroups. The total numberof specimens n equals m times r. The specimens in each sub-group are fatigue tested identically and simultaneously on dif-ferent testers. The first subgroup of specimens is run until thefirst failure occurs. At this point, the surviving specimens are sus-pended and removed from testing. An equal set of new specimensnumbering m samples is next tested until the first failure in thatsubpopulation. This process is repeated until one failure is gener-ated for each of the subgroups. In the end, r failures are generatedwhile (m − 1) × r samples are suspended. Thus, the total accumu-lative test time is the time to fail r specimens times the number ofsamples concurrently tested m, not the time for n failures. Withproper corrections and analysis (Johnson (10)), reliability of thelife predicted by this sudden death methodology from r failures iscomparable to that obtained when failing the entire population.A detailed description of SDT is given in Appendix B.

Proving that any of these methodologies reduces total testtime while predicting accurate bearing fatigue lives would re-quire the generation of a significant and unreasonable quantity of

experimental data. Vlcek, et al. (32) have shown that computermodeling of bearing life based upon Weibull-Johnson MonteCarlo simulations results in reasonable engineering predictionsof bearing life that are relatively easy to determine. Vlcek, et al.(33) also performed Monte Carlo simulations combined with sud-den death testing in order to compare resultant bearing lives tocalculated bearing life and the cumulative test time and calendartime relative to sequential and censored sequential testing. Re-ductions up to 40% in bearing test time and calendar time can beachieved by testing to failure or the L50 life and terminating alltesting when the last of the predetermined bearing failures hasoccurred. Vlcek, et al. (33) found that sudden death testing is nota more efficient method to reduce bearing test time or calendartime when compared to censored sequential testing.

FATIGUE DATA ANALYSIS

Weibull Distribution Function

In 1939, Weibull (8), (9) developed a method and an equa-tion for statistically evaluating the fracture strength of materialsbased upon small population sizes. This method can be and hasbeen applied to analyze, determine, and predict the cumulativestatistical distribution of fatigue failure or any other phenomenonor physical characteristic that manifests a statistical distribution.The dispersion in life for a group of homogeneous test specimenscan be expressed by

ln ln1S

= e ln(

L − Lµ

Lβ − Lµ

)where 0 < L < ∞; 0 < S < 1 [5]

where S is the probability of survival as a fraction (0 ≤ S ≤ 1); eis the slope of the Weibull plot; L is the life cycle (stress cycles);Lµ is the location parameter, or the time (cycles) below which nofailure occurs; and Lβ is the characteristic life (stress cycles). Thecharacteristic life is that time at which 63.2% of a population willfail or 36.8% will survive.

The format of Eq. [5] is referred to as a three-parameterWeibull analysis. For most—if not all—failure phenomenon,there is a finite time period under operating conditions when nofailure will occur. In other words, there is zero probability of fail-ure, or a 100% probability of survival, for a period of time dur-ing which the probability density function is nonnegative. Thisvalue is represented by the location parameter Lµ. Without a sig-nificantly large database, this value is difficult to determine withreasonable engineering or statistical certainty. As a result, Lµ isusually assumed to be zero and Eq. [5] can be written as

ln ln1S

= e ln(

LLβ

)where 0 < L < ∞; 0 < S < 1 [6]

This format is referred to as the two-parameter Weibull distri-bution function. The estimated values of the Weibull slope e andLβ for the two-parameter Weibull analysis may not be equal tothose of the three-parameter analysis. As a result, for a given sur-vivability value S, the corresponding value of life L will be similarbut not necessarily the same in each analysis.

By plotting the ordinate scale as ln ln(1/S) and the abscissascale as ln L, a Weibull cumulative distribution will plot as astraight line, which is called a Weibull plot. Usually, the ordinateis graduated in statistical percentage of specimens failed F where

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Fig. 10—Weibull plot where (Weibull) slope or tangent of line is e; proba-bility of survival, Sβ, is 36.8%, at which L − Lβ or L/Lβ = 1: (a)schematic and (b) rolling-element bearing fatigue data.

F = [(1 − S) × 100]. Figure 10a is a generic Weibull plot withsome of the values of interest indicated. Figure 10b is a Weibullplot of actual bearing fatigue data.

The Weibull plot can be used to evaluate any phenomenonthat results in a statistical distribution. The tangent of the result-ing plot, called the Weibull slope and designated by e, definesthe statistical distribution. Weibull slopes of 1, 2, and 3.57 repre-sent exponential, Rayleigh, and Gaussian (normal) distributions,respectively.

The scatter in the data is inversely proportional to the Weibullslope; that is, the lower the value of the Weibull slope, the largerthe scatter in the data and vice versa. The Weibull slope is also

liable to statistical variation depending on the sample size (database) making up the distribution (Johnson (10)). As the samplesize becomes smaller, there is a greater statistical variation in theWeibull slope.

Whether the data are generated experimentally or simulatedanalytically, the ultimate goal is to determine the life and char-acteristics of a larger population from a limited amount of data.There are various statistical methods for determining bearing lifeestimates from fatigue data that for practical engineering pur-poses give similar results. These methods differ significantly, how-ever, in their level of complexity to apply and limitations to theirapplication. For its relative ease of use and engineering applica-tion with comparable reliability, we have selected the Weibull-based Leonard G. Johnson approach (10), which utilizes a linearregression least-squares-fit method over the significantly morecomplicated maximum likelihood estimation method (Lieblein(34)). The authors’ experience has been that the least-squares-fitmethod of Johnson (10) gives similar results to that of the maxi-mum likelihood estimation methods (Lieblein (34); Cohen (35)).However, the maximum likelihood estimation method is moresensitive to early failures, biasing the Weibull slope to indicatemore scatter than actually exists in the data. There are computerprograms that are commercially available that can perform theseWeibull analyses.

Bearing Component Lives

Where there are design, manufacturing, material, and metal-lurgical variables among the inner race, outer race, and rolling-element (ball or roller) set, it is important to distinguish the ef-fects of these variables on the resultant life of a rolling-elementbearing.

Lundberg and Palmgren (31) presented bearing life as only afunction of the lives of the inner and outer races as follows:(

1L10

)e

=(

1Lir

)e

+(

1Lor

)e

[7a]

They do not directly calculate the life of the rolling-element(ball or roller) set of the bearing. However, through benchmark-ing of the equations with bearing life data by use of a material-geometry factor, the life of the rolling-element set is implicitlyincluded in the bearing life calculation. The material-geometryfactor does not differentiate the effects of these variables but em-pirically blends them together. Accordingly, Lundberg and Palm-gren (31) should have written their equation relating bearing lifeto the individual components lives based on the Weibull (8), (9)equation as follows:(

1L10

)e

=(

1Lir

)e

+(

1Lre

)e

+(

1Lor

)e

[7b]

where the Weibull slope e is the same for each of the componentsas well as for the bearing as a system. The value of the L10 lifeis the same as that in Eq. [7a]. However, the values for the innerand outer race live will be greater than those of Eq. [7a]. Equa-tions [7a] and [7b] are based on strict series reliability derived inAppendix C. Equation [7b] can be rewritten as

1 = +[

L10

Lre

]e

+[

L10

Lir

]e

+[

L10

Lor

]e

[7c]

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Though the experimental bearing L10 life may equal or greaterthan that calculated according to Lundberg and Palmgren (31),the experimental values of the inner and outer race lives shouldalways be higher than that calculated.

From Johnson (10), the fraction of failures due to the failureof a bearing component is

Fraction of inner-race failures =[

L10

Lir

]e

[8a]

Fraction of rolling-element failures =[

L10

Lre

]e

[8b]

Fraction of outer-race failures =[

L10

Lor

]e

[8c]

From Eqs. [8a] to [8c], if the life of the bearing and the frac-tions of the total failures represented by the inner race, the outerrace, and the rolling-element set are known, the life of each ofthese components can be calculated. Hence, by observation, it ispossible to determine the life of each of the bearing componentswith respect to the life of the bearing.

Bearing Life Variation

Vlcek, et al. (32) randomly assembled and tested 340 virtualbearing sets totaling 31,400 radially loaded and thrust-loadedrolling-element bearings. It was assumed that each bearing wasassembled from three separate bins of components, with one bincontaining 1,000 inner rings, one with 1,000 rolling element sets,and one with 1,000 outer rings. The median ranks of the individ-ual components were assigned and then virtual bearing assem-blies were created using a Monte Carlo technique.

Vlcek, et al. (32) determined the L10 maximum limit andL10 minimum limit for the number of bearings failed, r, using aWeibull-based Monte Carlo method. By fitting the resultant livesfor different size populations of failed bearings (Fig. 11), equa-tions were determined for both of these limits:

Maximum variation L10 life

= calculated L10 life (1 + 6r−0.6) [9a]

Minimum variation L10 life

= calculated L10 life (1−1.5r−0.33) where r > 3 [9b]

Minimum L10 life = 0 where r ≤ 3 [9c]

These curves compared favorably with the 90% confidencelimits of Johnson (10) at a Weibull slope of 1.5 (Fig. 12; Vlcek,et al. (32)).

Moult (6) published a series of successive fatigue tests withcylindrical roller bearings made from a single heat of air-melted(AM) and subsequently consumable-electrode vacuum remelted(CVM) AISI 8620 steel conducted over a 5-year period. Thesebearings were lubricated with a single batch of Mil-L-7808 diesterlubricant and three individual types of SAE 30 mineral oil lubri-cants. The base stocks of these mineral oils were a superrefinednaphthenic, a naphthenic, and a paraffinic. Moult’s (6) databasecomprised approximately 106 roller bearing fatigue failures. In-dividual test series ranged from 10 to 14 bearings with as manyas 39 bearings making up a composite group used for purposes of

Fig. 11—Maximum and minimum variation of L 10 lives as percentage ofcalculated L 10 for groups of r bearings and 90% confidencelimits based on Weibull slope and number of bearings failedr (Vlcek, et al. (32)): (a) 50-mm bore deep-groove ball bearing;(b) 50-mm bore angular-contact ball bearing; and (c) 90% confi-dence limits.

comparison. The variation in Moult’s (6) data base varied in ac-cordance with Eqs. [9a] to [9c]. For an example, the individual L10

lives for the AM AISI 8620 steel lubricated with the Mil-L-7808lubricant varied from 85 to 1900% from the composite bearingL10 life value with this lubricant. Further, when comparing thevariation in bearing L10 lives for the AM AISI 8620 steel lubri-cated with the Mil-L-7808 lubricant with the composite L10 bear-ing life obtained with the AM AISI 8620 steel bearings lubricatedwith the SAE 30 mineral oil, the bearing L10 lives with the Mil-L-7808 lubricant ranged from 7 to 115% of that obtained with theSAE 30 mineral oil (Moult (6)).

Rules can be implied from these results to compare and distin-guish resultant lives of identical bearings either from two or moresources or made using different manufacturing methods. The

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Fig. 12—Rules for comparing bearing life results to calculated life (Vlcek, et al. (32)): (a) bearing sets A and B are acceptable; (b) bearing set A isacceptable, bearing set B is not acceptable; (c) bearing sets A and B are acceptable; (d) bearing sets A and B are not acceptable; (e) bearingsets A and B are acceptable; and (f) bearing set A is acceptable, bearing set B is not acceptable.

following rules are suggested to determine whether the bearingsare acceptable for their intended application or whether there aresignificant differences between the two groups of bearings.

1. If the L10 lives of both bearing tests are between the maximumand minimum L10 life variations, there can be no conclusionthat there is a significant difference between the two sets ofbearings regardless of the ratio of the L10 lives. The bearingsets are acceptable for their intended application (Fig. 12a).

2. If the L10 life of one set of bearings is greater than the max-imum variation and the second set is less than the minimumvalue, there exists a significant difference between the bearingsets. Only one bearing set is acceptable for its intended appli-cation (Fig. 12b).

3. If the L10 lives of both sets of bearings exceed the maximumvariation, the bearing life differences may or may not be signif-icant and should be evaluated based upon calculation of confi-dence numbers according to the method of Johnson (10). Bothsets of bearings are acceptable for their intended application(Fig. 12c).

4. If the L10 lives of both sets of bearings are less than the mini-mum variation, the bearing life differences may or may not be

significant. However, neither set of bearings is acceptable forits intended application (Fig. 12d).

5. If the L10 life of one set of bearings exceeds the maximumvariation and the other set is between the maximum and min-imum variation, the bearing life differences may or may notbe significant and should be evaluated based upon calculationof confidence numbers according to the method of Johnson(10). Both sets of bearings are acceptable for their intendedapplication (Fig. 12e).

6. If the L10 life of one set of bearings is less than the minimumvariation and the other set is between the maximum and min-imum variation, there exists a significant difference betweenthe bearing sets. Only one set of bearings is acceptable for itsintended application (Fig. 12f).

WEIBULL SLOPE VARIATION

Johnson (10) analyzed the probable variation of the Weibullslope as a function of the number of bearings tested to failure.Based on the Johnson analysis, in 90% of all possible cases theresultant Weibull slope will be within the limits shown in Fig. 13based upon a Weibull slope of 1.11. Based on Johnson (10), the

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Fig. 13—Extremes of Weibull slope from Monte Carlo testing for eachgroup of 10 bearing trials of r bearings compared with predicted90% probable error. Data from Vlcek, et al. (32).

approximate relation for the number of bearings failed r and thelimits of the value of Weibull slope e equal to 1.11 are as follows:

Maximum Weibull slope = 1.11 + 1.31r−0.5 [10a]

Minimum Weibull slope = 1.11 − 1.31r−0.5 [10b]

The results of the extremes in the Weibull slopes for eachgroup of the 10 bearing trials of r bearings from the Monte Carlosimulation of Vlcek, et al. (33) are compared with the Johnsonanalysis (10) in Fig. 13.

Confidence Number

It is important to know the confidence in the conclusions thatcan be derived from one series of tests relative to a standard testseries. Johnson (10) developed a statistical method for compar-ing two populations and/or variables that he termed confidence

Fig. 14—Comparison of endurance test results for two material lots usingWeibull plots. Ratio of L 10 lives, 2; Weibull slope for each lot,1.7 (Johnson (10)).

numbers. For purposes of example, Weibull plots of lives of bear-ings made from two hypothetical bearing steels are compared inFig. 14. Lot 1 is made from a standard bearing steel containing7 failures. Lot 2 is made from an experimental bearing steel andcontains 10 failures. The L10 for Lots 1 and 2 are 38 and 76 h,respectively. Both lots have a Weibull slope of 1.7. The questionthat can be asked is “How much confidence can be placed in theapparent superiority of the experimental steel population (Lot 2)over the standard steel population (Lot 1)?”

In addition to the dependence upon the amount of separationbetween the two Weibull plots, or the ratio of the longer life tothe shorter life at one level [F(x)], the answer to the question ofconfidence depends upon the number of failures (number of datapoints), r1 and r2 in each population of Lots 1 and 2, respectively,and also upon the Weibull slopes of their populations. An impor-tant point to be aware of is that the degree of confidence of onelot over the other need not be constant from one level [F(x)] toanother. As an example, it is possible to have a significant im-provement at the 50% life, L50, without any improvement at theL10 life and vice versa. Because improvement in early life is ofprime importance and is used for comparison purposes, the con-fidence in the L10 life will be considered. An extensive treatmentwas presented by Johnson (10).

Johnson (10) created a series of graphs shown in Fig. 15 where1/(1 − C) is plotted against the L10 life ratio on logarithmic co-ordinates with the Weibull slope and [(r1− 1)(r2− 1)] held con-stant. The graphs shown in Fig. 15 are straight lines (or verynearly so). For convenience, the ordinate is graduated as the con-fidence number. Johnson (10) claimed no theoretical basis for thelinearity of the plots.

The quantity (r1− 1) is known as the number of degrees offreedom for Lot 1. There are (r1− 1) degrees of freedom becausefor a fixed value of failures r1, only (r1− 1) can be chosen arbi-trarily, the last one being determined from the remaining values.Similarly, (r2− 1) is the number of degrees of freedom in Lot 2.The product of [(r1− 1)(r2− 1)] is the total degrees of freedom ofthe pair of lots.

Referring to the Weibull plots of Fig. 14,

r1 = number of failures of Lot 1 = 7

r2 = number of failures of Lot 2 = 10

Thus, the total degrees of freedom = (7 − 1)(10 − 1) = 54.The Weibull slopes of both lots are approximately 1.7 and the

L10 life ratio equals 76/38 or 2. Interpolating between Figs. 15dand 15e for 54 degrees of freedom at Weibull slopes of 1.6 and1.8, the confidence numbers are 75 and 78%, respectively. At aWeibull slope of 1.7 the average confidence number is 76.5% [(75+ 78)/2]. This indicates that there are 765 chances out of 1,000that the L10 life in the population of Lot 2 is superior to the L10

life in the population of Lot 1. (A 95% confidence number isequivalent to 2σ confidence band and a 68% confidence numberis equivalent to a 1σ confidence band.) In general, it is desirable tohave a confidence number of 90 or greater to assure a statisticallysignificant difference between two populations.

If Lots 1 and 2 have unequal Weibull slopes, the confidencenumber can be determined by treating the two lots as if both had

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Fig. 15—Confidence numbers as a function of L 10 life ratio and total degrees of freedom (Johnson (10)).

the Weibull slope of Lot 1. Then obtain a second confidence num-ber by treating both lots as if both had a Weibull slope of Lot 2.To arrive at the true confidence number the two results are aver-aged together (Johnson (10)).

To determine the confidence number for successive compar-isons or successive tests, Johnson (10) provided the following for-mula:

C = 1 − 2n−1∏n

i=1(1 − Ci) [11]

As an example, suppose that there is a second set of tests tocompare Lots 1 and 2. For the second set of tests, the resultantconfidence number is 0.8. Then from Eq. [11], the combined con-fidence number is

C = 1 − 2(1 − C1)(1 − C2) = 1 − 2(1 − 0.765)(1 − 0.8) = 0.906

[12]

Assume that there is a third set of data that provides a confi-dence number of 0.65. The combined confidence number is

C = 1 − 2(3−1)(1 − C1)(1 − C2)(1 − C3)

= 1 − 4(1 − 0.765)(1 − 0.8)(1 − 0.65) = 0.934 [13]

From the above, it may be concluded with reasonable engi-neering and statistical certainty that Lot 2 will provide a longerrolling-element fatigue life than Lot 1 approximately 93% of thetime. However, there will be instances (approximately 7% of thetime) where the population of Lot 1 will give longer lives.

Determining Significance of Test Results

If the data being compared are from either full-scale bearingtests or bench-type tests, confidence numbers can be used. Thepreviously established rules based on variation in life, however,should only be applied to full-scale bearing testing where the livesof a bearing population with which the experimental or field dataare to be compared are known and/or predicted. However, thereare instances where there are test results where there are eitherinsufficient test failures and/or the ratio of the life results are notsufficient to conclude with any reasonable engineering and/or sta-tistical certainty that life differences in the variables being testedexist. This can occur where a single variable is being tested suchas material chemistry, additive and lubricant chemistry, materialhardness, etc.

For purposes of example, let us assume that there appears tobe a relation between the volume of an additive present in a lubri-cant and bearing fatigue life. In this example we will assume thatin four test series life increases with increases in additive con-tent where test series D with the highest percentage of additivegives the longest life and test series A with no additive presentgives the lowest life. Test series C has an additive content greaterthan test series B but less than test series D. The order of thetest results from longest life to shortest life are D, C, B, and A.However, the confidence number between test series D and Ais less than 90%. It may be possible to combine the confidencenumbers as discussed in the previous section to obtain a confi-dence number greater than 90% showing an effect of additivevolume on fatigue life. Alternatively, the following formula may

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be used.

Percentage probability of a relation existing

= [1 − 1/(n!)] × 100 [14]

where n is the number of test series or groups.In the example above, n = 4 and factorial (n!) = 4 × 3 × 2 ×

1 = 24. From Eq. [14] there is a 96% probability that there existsa relation between additive volume percentage and fatigue life.However, assume that there were only three test series whereby(n!) = 3 × 2 × 1 = 6. From Eq. [14] there is an 83% probabilitythat there exists a relation between additive volume percentageand fatigue life. This percentage may be insufficient to concludewith any statistical certainty that a relation does exist.

SUMMARY

In order to rank bearing materials, lubricants and design vari-ables using rolling-element bench-type fatigue testing and rolling-element bearing tests, the investigator needs to be cognizant ofthe variables that affect rolling-element fatigue life and be ableto maintain and control them within an acceptable experimen-tal tolerance. This article brings together and discusses the tech-nical aspects of rolling-element fatigue testing and data analysisand makes recommendations to assure quality and reliable test-ing of rolling-element specimens and full-scale rolling-elementbearings. These are as follows:

1. When testing to obtain valid comparison of the lubricant,material, and/or operating variables in bench-type rolling-element fatigue testers, all of the rolling-element specimensmust fabricated from a single heat of material, heat-treated si-multaneously to the same hardness. All specimens must havethe same surface finish unless the surface finish is the parame-ter being studied. The steel microstructure should be reportedtogether with the retained austenite and residual stress beforeand after testing. It is of prime importance that the differencein hardness between the contacting test specimens remainsconstant.

2. Where high reliability is desired for a given application it ispreferable to test a group of bearings to determine their ac-tual lives. In such cases, the same precautions should be takenas with a bench-type tester discussed above. In general, teststresses lower than 2.4 GPa (350 ksi) maximum Hertz are sug-gested. However, maximum Hertz stresses as high as 3.1 GPa(450 ksi) on the inner race can be used for testing many typesof bearings.

3. The interference fit between the bearing bore and the shaftneeds to be controlled and reported. The interference fit canchange the bearing’s internal diametrical clearance and willalso induce tensile hoop stresses in the bearing inner race.These tensile stresses can increase the magnitude of shearingstresses below the contacting surface between the rolling ele-ments and the bearing inner race and reduce the bearing fa-tigue life. It can also alter the Hertz stress-life relation.

4. Comparisons show that bench-type fatigue testers can reli-ably identify qualitative effects of many variables on rolling-element fatigue life. By benchmarking these life results to an

already existing database, it is possible to develop bearing life-modifying factors with the Lundberg-Palmgren theory to pre-dict bearing life with reasonable engineering certainty.

5. Reductions up to 40% in bearing test time and calendar timecan be achieved by testing half of the specimens to failure orto theL50 life and terminating all testing when the last of thepredetermined bearing failures have occurred. Sudden deathtesting is not a more efficient method to reduce bearing testtime or calendar time when compared to censored sequentialtesting.

6. Once variables are controlled, the number of tests and the testconditions need to be specified to assure reasonable statisticalcertainty of the final results. Using the statistical methods ofWeibull and Johnson, the minimum number of tests requiredcan be determined.

REFERENCES(1) Zaretsky, E. V. (Ed.). (1992), STLE Life Factors for Rolling Bearings,

STLE Paper No. SP-34, Society of Tribologists and Lubrication Engineers,Park Ridge, IL.

(2) Sadeghi, F., Jalalahmadi, B., Slack, T. S., Raje, N., and Arakere, N. K.(2009), “A Review of Rolling Contact Fatigue,” Journal of Tribology,131(4), Art. No. 041403.

(3) Alley, E. S. and Neu, R. W. (2010), “Microstructure-Sensitive Modeling ofRolling Contact Fatigue,” International Journal of Fatigue, 32(5), pp 841-850.

(4) Zaretsky, E. V. (Ed.). (1997), Tribology for Aerospace Applications, STLESP-37, Society of Tribologists and Lubrication Engineers, Park Ridge, IL.

(5) Stribeck, R. (1900), “Reports From the Central Laboratory for ScientificInvestigation,” Hess, H. (Trans.), 1907, Transactions of the ASME, 29, pp420-466.

(6) Moult, J. F. (1963) “Critical Aspects in Bearing Fatigue Testing,” Lubrica-tion Engineering, 19(12), pp 503-511.

(7) Bamberger, E. N., Harris, T. A., Kacmarsky, W. M., Moyer, C. A., Parker,R. J., Sherlock, J. J., and Zaretsky, E. V. (1971), Life Adjustment Factorsfor Ball and Roller Bearings, an Engineering Design Guide, ASME: NewYork.

(8) Weibull, W. (1939), “A Statistical Theory of the Strength of Materials,” In-giniors Vetenskaps Adademien (Proc. Royal Swedish Academy of Engr.),151.

(9) Weibull, W. (1939), “The Phenomenon of Rupture,” Inginiors VetenskapsAdademien (Proc. Royal Swedish Academy of Engr.), 153.

(10) Johnson, L. G. (1964), The Statistical Treatment of Fatigue Experiments,Elsevier: Amsterdam.

(11) Zaretsky, E. V., Parker, R. J., and Anderson, W. J. (1982), “NASA Five-Ball Fatigue Tester—Over 20 Years of Research,” Rolling Contact FatigueTesting of Bearing Steels, Hoo, J. J. C. (Ed.), ASTM STP 771 AmericanSociety of Testing Materials: Philadelphia, pp 5-45.

(12) Hoo, J. J. C. (1982), Rolling Contact Fatigue Testing of Bearing Steels,ASTM STP 771, American Society of Testing Materials, Philadelphia, PA.

(13) McCool, J. I. and Valori, R. (2009), “Evaluating the Validity of RollingContact Fatigue Results,” Tribology Transactions, 52(2), pp 223-230.

(14) Zaretsky, E. V., Sibley, L. B., and Anderson, W. J. (1963), “The Role ofElastohydrodynamic Lubrication in Rolling-Contact Fatigue,” Journal ofBasic Engineering, Series D, 85, pp 439-450.

(15) Drutowski, R. C. and Mikus, E. B. (1960), “The Effect of Ball BearingSteel Structure on Rolling Friction and Plastic Deformation,” Journal ofBasic Engineering, Series D, 82, pp 302-308.

(16) Drutowski, R. C. (1962), “The Linear Dependence of Rolling Friction onStressed Volume,” Rolling Contact Phenomena, Bidwell, J. B. (Ed.), Else-vier: Amsterdam, pp 113-131.

(17) Zaretsky, E. V., Anderson, W. J., and Parker, R. J. (1962), “The Effectof Nine Lubricants on Rolling-Contact Fatigue Life,” NASA TN D-1404,National Aeronautics and Space Administration, Washington, DC.

(18) Zaretsky, E. V., Anderson, W. J., and Parker, R. J. (1962), “The Effect ofContact Angle on Rolling-Contact Fatigue and Bearing Load Capacity,”ASLE Transactions, 5, pp 210-219.

(19) International Organization for Standardization. (2006), Rolling Bearings—Static Load Ratings, International Organization for Standardization:Geneva, Switzerland.

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(20) Chen, W. W., Wang, Q. J., Wang, F., Keer, L. M., and Cao, J. (2008),“Three-Dimensional Repeated Elasto-Plastic Point Contacts, Rolling, andSliding,” Journal of Applied Mechanics, 75(2), Art. No. 021021.

(21) Nelias, D., Antaluca, E., Boucly, V., and Cretu, S. (2007), “A Three-Dimensional Semianalytical Model for Elastic-Plastic Sliding-Contacts,”Journal of Tribology, 129(4), pp 761-771.

(22) Nelias, D., Antaluca, E., and Boucly, V. (2007), “Rolling of an Elastic El-lipsoid upon an Elastic-Plastic Flat,” Journal of Tribology, 129(4), pp 791-800.

(23) Wang, F. S., Block, J. M., Chen, W. W., Martini, A., Zhou, K., Keer, L. M.,and Wang, Q. J. (2009), “A Multilevel Model for Elastic-Plastic Contactbetween a Sphere and a Flat Rough Surface,” Journal of Tribology, 131(2),Art. No. 021409.

(24) Galbato, A. T. (1982), “Methods of Testing for Rolling Contact Fatigue ofBearing Steels,” Rolling Contact Fatigue Testing of Bearing Steels, Hoo, J.J. C. (Ed.), American Society of Testing Materials: Philadelphia, pp 169-189.

(25) Coe, H. H. and Zaretsky, E. V. (1986), “Effect of Interference Fits onRoller Bearing Fatigue Life,” ASLE Transactions, 30(2), pp 131-140.

(26) Oswald, F. W., Zaretsky, E. V., and Poplawski, J. V. (2009), “InterferenceFit Life Factors for Roller Bearings,” Tribology Transactions, 52(3), pp415-426.

(27) Oswald, F. W., Zaretsky, E. V., and Poplawski, J. V. (2011), “InterferenceFit Life Factors for Ball Bearings,” Tribology Transactions, 54(1), pp 1-20.

(28) Zaretsky, E. V., Parker, R. J., and Anderson, W. J. (1967), “ComponentHardness Differences and Their Effect on Rolling-Element Fatigue Life,”Journal of Lubrication Technology, 89(1), pp 47-62.

(29) Parker, R. J. and Zaretsky, E. V. (1972), “Rolling-Element Fatigue Livesof through Hardened Bearing Materials,” Journal of Lubrication Technol-ogy, 94(2), pp 165-173.

(30) Bamberger, E. N. and Zaretsky, E. V. (1971), “Fatigue Lives at 600◦F of120-Millimeter-Bore Ball Bearings of AISI M-50, AISI M-1 and WB-49Steels,” NASA TN D-6156, National Aeronautics and Space Administra-tion, Washington, DC.

(31) Lundberg, G. and Palmgren, A. (1947), “Dynamic Capacity of RollingBearings,” Acta Polytechnica Mechanical Engineering Series, 1(3), Stock-holm, Sweden.

(32) Vlcek, B. L., Hendricks, R. C., and Zaretsky, E. V. (2003), “Determina-tion of Rolling-Element Fatigue Life from Computer Generated BearingTests,” Tribology Transactions, 46(3), pp 479-493.

(33) Vlcek, B. L., Hendricks, R. C., and Zaretsky, E. V. (2004), “Monte CarloSimulation of Sudden Death Bearing Testing,” Tribology Transactions,47(2), pp 188-199.

(34) Lieblein, J. (1954), “A New Method of Analyzing Extreme-Value Data,”NACA TN 3053, National Advisory Committee for Aeronautics, Wash-ington, DC.

(35) Cohen, C. (1965), “Maximum Likelihood Estimation in the Weibull Dis-tribution Based on Complete and on Censored Samples,” Technometrics,7(4), pp 579-588.

(36) McCool, J. (1982), “Analysis of Sets of Two-Parameter Weibull Data Aris-ing in Rolling Contact Endurance Testing,” Rolling Contact Fatigue Test-ing of Bearing Steels, Hoo, J. J. C. (Ed.), ASTM ST 771 American Societyof Testing Materials: Philadelphia, pp 293-319.

(37) Houpert, L. (2003), “An Engineering Approach to Confidence Intervalsand Endurance Test Strategies,” Tribology Transactions, 46(2), pp 248-259.

(38) Skinner, K., Keates, J., and Zimmer, W. (2001), “A Comparison of ThreeEstimators of the Weibull Parameters,” Quality and Reliability Engineer-ing International, 17, pp 249-256.

(39) Poplawski, J. V., Peters, S. M., and Zaretsky, E. V. (2001), “Effect of RollerProfile on Cylindrical Roller Bearing Life Prediction—Part I: Comparisonof Bearing Life Theories,” Tribology Transactions, 44(3), pp 339-350.

(40) Poplawski, J. V., Peters, S. M., and Zaretsky, E. V. (2001), “Effect of RollerProfile on Cylindrical Roller Bearing Life Prediction—Part II: Comparisonof Roller Profiles,” Tribology Transactions, 44(3), pp 417-427.

APPENDIX A—EFFECT OF GROSS PLASTICDEFORMATION ON CONTACT RADII

For many applications where gross plastic deformation oc-curs, it is necessary to calculate the effective Hertz stress or thestress after plastic deformation (Zaretsky, et al. (17)). This canbe accomplished by deriving the radius of curvature Rp of the de-formed rolling element as follows:

Referring to Fig. 5a let

A = 12

H (2l) [A1]

approximately, then

h′ = h + H or h = h′ − H [A2]

Then

l2 = R2p − (R − h)2

= 2Rph − h2 [A3]

and

Rp = l2 + h2

2h[A4]

If Rp = R and h = h′, Eq. [A3] becomes

l = 2Rh′ − (h′)2

and

h′ = R ± (R2 − x2)1/2

[A5]

Substituting Eq. [A5] into Eq. [A2] and solving for h in termsof A, R, and H results in

h = R ±[

R2 −(

AH

)2]1/2

− H [A6]

If H is substituted into Eq. [A4] and a negative sign selectedfor the radical in Eq. [A6],

Rp =( A

H

)2 +{

R −[R2 − ( A

H

)2]1/2

− H}2

2{

R −[R2 − ( A

H

)2]1/2

− H} [A7]

Because {R − [R2 − ( AH )2]1/2 − H}2 is small relative to ( A

H )2, it canbe neglected in the numerator; then

Rp =( A

H

)2

2{

R −[R2 − ( A

H

)2]1/2

− H} [A8]

The deformed radius R′p (not shown) in the plane perpendic-

ular to the plane of the profile shown in Fig. 5a is

R′p = R′ − H [A9]

where R′ is the radius of the body in the perpendicular plane.Because H is extremely small relative to R′,

R′p = R′ [A10]

Substituting Rp in the Hertzian equations will give the effec-tive compressive Hertz stress after gross plastic deformation ofthe rolling-element surface.

For a race groove surface having a concave or negative radiusR, as shown in Fig. 5b,

Rp =( A

H

)2

2{

R −[R2 − ( A

H

)2]1/2

+ H} [A11]

The area of deformation A and track depth H can be measureddirectly from a surface contour trace.

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APPENDIX B—SUDDEN DEATH TESTING TECHNIQUES

As discussed by Vlcek, et al. (33), in sudden death testing, thetotal number of specimens to be evaluated is divided into equalsubgroups that can be evaluated simultaneously. The first sub-group of specimens is run simultaneously until the first failureoccurs. The surviving tests are terminated (i.e., suspended), andnew test specimens are mounted in the testers. This process is re-peated until all of the test specimens in the population have beenscreened.

A technique developed by Vlcek, et al. (32) was used to gener-ate virtual bearing sets that were analyzed as if they were suddendeath tested. Total populations with 36, 72, and 144 deep-groovebearings were generated. The populations were then sequentiallybroken into subgroups representing all possible combinations ofsudden death test series of (m × r), where m was the numberof bearing testers used simultaneously and r was the number ofsets of m bearings necessary to achieve the total number of bear-ings n in the total population. The total number of bearings nequals m times r. For example, if 36 bearings were to be evalu-ated, and there were 4 bearing fatigue testers available, the valueof m would be 4 and the value of r would be 9, whereby n equals36 (4 × 9). For this example, the first four bearings were run si-multaneously until the first failure occurred. The three survivingtests were suspended, and four new bearings were mounted. Thisprocess was repeated until nine failures occurred, one from eachof the sets of 4, with (m− 1) ×r or 27 suspensions.

These nine failures and their corresponding median ranks, onefor each subgroup of size m, were then plotted on a Weibull plot.This is illustrated as Step 1 in the schematic of a generic Weibullplot of the sudden death line (SDL) of Fig. B1. This SDL rep-resents the distribution of first failures in each subgroup. TheSDL of each series was next shifted on its respective Weibull plot,Steps 2 and 3 of Fig. B1, so that the curve represented the failuresof the total population, not just that of one out of r bearings.

Various methods exist for shifting the SDL line (Fig. B1) andfinding the life of a total population n based upon sudden deathtesting data where only r failures are considered. The slope andcharacteristic life of the subpopulation generated during suddendeath testing can be found from maximum likelihood estimators

Fig. B1—Schematic of generic Weibull plot of the sudden death lineshifted to total population line (Vlcek, et al. (33)).

of Cohen (35) that are obtained from an iterative process. Thislife for r samples must be corrected to represent the life of theoriginal population containing n samples. One way to achievethis, as reported by McCool (36), is to multiply the subpopulationlife estimator by the number of samples (m) in each equally sizedsubpopulation raised to the inverse of the slope estimator. Confi-dence limits can be placed upon these values for a limited numberof cases provided in tables in the open literature or by extensiveMonte Carlo simulations (McCool (36)). Houpert (37) also pro-posed a technique for determining the life of a larger populationbased upon a subpopulation determined from sudden death test-ing.

For its simplicity of application and relative engineering accu-racy, we prefer a technique presented in Johnson (10) for shift-ing the sudden death line so that the life and characteristics ofthe larger bearing population (n) can be projected. The genericWeibull plot of Fig. B1 accompanies the following steps and in-cludes many of the elements mentioned.

Step 1 Plotting the subgroup of r failures on Weibull paper. Toplot the failures, median ranks are assigned to the sequentiallyordered lives. Median ranks were defined using

median rank = (j − 0.3)/(r + 0.4) [B1]

where j = 1, 2, 3, . . . , r. For sudden death testing, the number ofsubsets r equal n. A discussion and comparison of median rankdefinitions is available in Houpert (37) and Skinner, et al. (38).The median ranks along with their corresponding lives are nextplotted on Weibull paper. The locus of points is fitted with a lin-ear curve. From the Weibull plot the Weibull slope, the L10 life,L50 life, and characteristic life, Lβ, are determined. This SDL rep-resents the distribution of first failures at the median rank for onefailure out of the number of samples (m) in each of the subgroups.

(Step 2) Determining by how much the SDL must be shifted to ac-curately estimate the L10 life for the entire population. The me-dian rank (Eq. [B1]) must be determined for the first failureout of the number of bearings simultaneously evaluated (m).In the above example, for an [(m = 4) × (r = 9)] test, the foursimultaneous testers are stopped after the first test failure oc-curs; thus the mean order number j equals 1 and the subgroupsize m equals 4 where it is assumed that in Eq. [B1] m equals r.The median rank for 1 out of 4, found using Eq. [B2] below, is0.1591. This is the value to which the SDL must be shifted (Fig.B1). In general,

First failure median rank (FFMR) = (1 − 0.3)/(m + 0.4) [B2]

(Step 3) Constructing the total population line (TPL) by shiftingthe SDL. At the L50 intersection of the SDL, a vertical line isdrawn down to the median rank value determined in Step 2.Through this point, a line is drawn parallel to the SDL createdin Step 1. The slopes of both lines are assumed to be equal.Figure B1 is a generic Weibull plot of the SDL and the shiftedTPL representative of this technique.

(Step 4) Determining lives from the shifted TPL. The Weibullslope and lives are read directly from the TPL generated in

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TABLE 3—COMPARISON BETWEEN LIVES OBTAINED FROM MONTE

CARLO SUDDEN DEATH TESTING AND THAT ADJUSTED FOR TOTAL

POPULATIONa

Bearing Life, hBearing PopulationSize L10 L50 Lβ Weibull Slope, e

Sudden death for 9 failures 2,354 14,569 20,772 1.033Adjusted for total population

from sudden death9,003 55,729 79,453 1.033

Calculatedb (actual) for totalpopulation

6,912 37,729 47,729 1.11

aNumber of failures, 9; number of bearings tested, 36; number of testrigs, 4; assumed Weibull slope, 1.11; resultant Weibull slope, 1.033;bearing size and type, 50-mm bore deep-groove ball bearing (Vlcek,et al. (33)).bLife based on Zaretsky’s rule and lubricant life factor (Zaretsky(1)).

Step 3. Table 3 contains the Weibull slope, L10 life, L50 life, andcharacteristic life, Lβ, for a typical [(m = 4) × (r = 9)] suddendeath test of a virtual deep groove bearing. The values obtainedfrom the SDL and the shifted TPL are provided.

APPENDIX C—DERIVATION OF STRICT SERIESRELIABILITY

As discussed and presented in Poplawski, et al. (39), (40), in1947, Lundberg and Palmgren (31), using the Weibull (8), (9)equation for rolling-element bearing life analysis, first derivedthe relationship between individual bearing component lives andbearing system L10 life. The following derivation is based on butis not identical to the Lundberg-Palmgren (31) analysis, which didnot include the rolling-element (ball or roller) set life.

Referring to Fig. 10a and Eq. [6], the Weibull equation can bewritten as

ln ln[

1Sx

]= e ln

[LLβ

][C1]

where L is the number of cycles to failure at a probability of sur-vival of Sx. The characteristic life Lβ is the life at a 36.8% prob-ability of survival or a 63.2 [(1 − 0.368) × 100 = 63.2] percentprobability of failure.

Figure C1 is a sketch of multiple Weibull plots where eachWeibull plot represents a cumulative distribution of the bearingand each component of the bearing system. The Weibull plot ofthe bearing represents the combined Weibull plots of the (1) in-ner race, (2) rolling-element set, and (3) outer race. All plots areassumed to have the same Weibull slope e. The slope e can bedefined as follows:

e =ln ln

[1Sx

]− ln ln

[1

S90

]ln L − ln L10

[C2a]

or

ln[

1Sx

]ln

[1

S90

] =[

LL10

]e

[C2b]

Fig. C1—Sketch of multiple Weibull plots where each numbered plot rep-resents the cumulative distribution of each component of thebearing Weibull plot represents combined distribution of plots1, 2, 3 (all plots are assumed to have the same Weibull slope e;Poplawski, et al. (39)).

From Eqs. [C1] and [C2b],

ln[

1Sx

]=

[ln

1S90

] [L

L10

]e

=[

LLβ

]e

[C3]

and

Sx = exp −[

LLβ

]e

[C4]

where Eq. [C4] is identical to Eq. [C1].For a given time or life L, each component in a system

will have a different reliability S. For a series reliability sys-tem the probability of survival of the system Sx at a giventime or life L is the product of the probabilities of survivalof each of the components making up the system where for abearing

Sx = S1 · S2 · S3 [C5]

Combining Eqs. [C4] and [C5] gives

exp −[

LLβ

]e

= exp −[

LLβ1

]e

× exp −[

LLβ2

]e

× exp −[

LLβ3

]e

[C6a]

exp −[

LLβ

]e

= exp −{[

LLβ1

]e

+[

LLβ2

]e

+[

LLβ3

]e}[C6b]

It is assumed that the Weibull slope e is the same for all com-ponents. From Eq. [C6b]

−[

LLβ

]e

= −{[

LLβ1

]e

+[

LLβ2

]e

+[

LLβ3

]e}[C7a]

Factoring out L from Eq. [C7a] gives[1Lβ

]e

=[

1Lβ1

]e

+[

1Lβ2

]e

+[

1Lβ3

]e

[C7b]

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From Eq. [C3] the characteristic lives Lβ1, Lβ2, Lβ3, etc., can be re-placed with the respective lives L1, L2, and L3, at S90 (or the livesof each component that have the same probability of survival S90)as follows:[

ln1

S90

] [1

L10

]e

=[

ln1

S90

] [1L1

]e

+[

ln1

S90

] [1L2

]e

+[

ln1

S90

] [1L3

]e

[C8]

where, in general, from Eq. [C3][1Lβ

]e

=[

ln1

S90

] [1

L10

]e

[C9a]

and [1

Lβ1

]e

=[

ln1

S90

] [1L1

]e

, etc. [C9b]

Factoring out[ln 1

S90

]from Eq. [C8] gives

[1

L10

]=

{[1L1

]e

+[

1L2

]e

+[

1L3

]e}1/e

[C10]

or rewriting Eq. [C10] results in(1

L10

)e

=(

1Lir

)e

+(

1Lre

)e

+(

1Lor

)e

[C11]

Equation [C11] is identical to Eq. [7b] of the text.

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