NewLifeTheoryRollingBearingsByLinkage-TT2012-55-5

See discussions, stats, and author proles for this publication at: /publication/254335605A New Life Theory for Rolling BearingsBy Linkage between Rolling Contact Fatigueand Structural FatigueArticle in Tribology Transactions September 2012DOI: 10.1080/10402004.2012.681342CITATIONS6READS1071 author:Shigeo ShimizuMeiji University34 PUBLICATIONS 162 CITATIONS SEE PROFILEAll content following this page was uploaded by Shigeo Shimizu on 19 March 2015.The user has requested enhancement of the downloaded le.This article was downloaded by: Shigeo ShimizuOn: 01 July 2012, At: 01:20Publisher: Taylor Bearing Steel; Fatigue Limit; Life The-ory; Minimum Life; Rating Load; Rolling Bearings; StressLifeCurve; Statistical Theory; Weibull DistributionINTRODUCTIONRecently, Shimizu (1) showed that the concept of minimumstrength of materials in a Weibull distribution function cannot beignored while reconsidering Weibulls theory (Weibull (2), (3)for static and dynamic strength of a material and the Lundberg-Palmgren (henceforth abbreviated as L-P) theory (Lundberg andPalmgren (4), (5) for rolling bearings. In other words, the com-petingmodelortheweakestlinkmodelforthenumberoflinks Kin a chain, where the reliability of the chain is expressed in termsof Weibull slope m; load/scale parameters 0and for the linkand the chain, respectively; and the location parameter as theManuscript received September 19, 2011Manuscript accepted March 26, 2012Review led by Michael Kotzalasminimum strength, is as follows:ln1R= Kparenleftbiggxn 0parenrightbiggm=parenleftbiggxn parenrightbiggm1Therefore, for any given probability of failure n%, thestrength of chain xnmay be expressed as shown by Eq. 2:xn= parenleftbiggln1Rparenrightbigg 1m+ , = K1m02Thus, for a very long chain, for example when K inEq. 2, 0 and so the statistical strength depending on thereliability vanishes. At this point the strength of the chain is onlyxn= , which is independent of the statistical strength.On the other hand, while using two-parameter Weibull distri-butionfunctionwith = 0,thestrengthofthechainwillapproachzero for K , which contradicts the real physical situation re-garding the strength of the chain or any other machine elements.A similar situation may be considered between the material fa-tigue and the rolling contact fatigue.In this article, the three-parameter Weibull distribution func-tion is used to develop a new theory for rolling bearings dealingwiththelinkagebetweentherollingcontactfatigueandstructuralfatigue of materials. To do this, the life characteristics in an al-ternating torsion fatigue probabilistic stresslife (P-S-N) test fortest samples made of same rolling bearing material and the lifecharacteristics for the rolling bearing races of different sizes andWeibull parameters are collected. Considering a linkage betweenthe two, a new theory for the load rating of rolling bearings is es-tablished, revealing a statistical strength expressing a 90% loadrating and a minimum strength expressing a 100% load rating.Further examination of life distributions of the two series of 61grease-lubricated #6206 deep-groove ball bearings revealed theexistence of linkage factors and a reduction factor for the loadratings.RELIABILITY FUNCTION AND LIFE FORMULABearing RacesFigure 1a shows the contact areas in the top view of the innerrace of the rolling bearing, and Fig. 1b is a sketch of the crosssection of the developed inner race contact. For a velocity veofthe rotating inner ring, the rolling element of diameter Dwwillroll and revolve with an orbital velocity vb.Further, Q is an imaginary equivalent rolling body load thatremains constant in its magnitude and direction and shows the558Downloaded by Shigeo Shimizu at 01:20 01 July 2012 A New Life Theory for Rolling Bearings 559NOMENCLATUREA0= Proportionality constant for specimenAprime0= Proportionality constant for raceA1= Material constant for ball bearinga = Semi-major axis of Hertz contact ellipsea1= (lnR/ln0.9)1/m= Reliability factorB1= Material constant for roller bearingb = Semi-minor axis of Hertz contact ellipseC = 90% Load rating/load rating of bearingCCAT= Basic dynamic load rating of catalogueCex= 90% Load rating obtained from life testCS= 90% stress rating of test specimenCprimeS= 100% stress rating of test specimenCSe= 90% Stress rating of raceCprimeSe= 100% Stress rating of raceC = 1= Theoretical 90% load rating for = 1Cprime= 1= Theoretical 100% load rating for prime= 1C= 100% Load rating of bearingCex= 100% Load rating obtained from life testcos = Ratio of curvature difference to curvature sumDe= Diameter of race or test specimenDpw= Pitch diameter of rolling bodiesDw= Diameter of rolling bodyE = Modulus of elasticityE0= 2.32 105N/mm2for steelF = Bearing loadf = Conformity factorfc= Factor for 90% load rating of bearingfprimec= Factor for 100% load rating of bearingJ1= Coefficient of rolling body load for rotating ringJ2= Coefficient of rolling body load for stationaryringJR= Coefficient of rolling body load for inner ringrotationJr= Coefficient of rolling body load for radial loadJS= Coefficient of rolling body load for outer ringrotationj = 1, 2, 3, ., ZK = Number of product law of probabilitiesk = Hertzs loaddisplacement exponentL10= 90% Rating life/rating lifeLwe= Effective length of rollerle= Stressed length of race or test specimen Dem = Weibull slopes, m = 10/9 (point contactbearings), m = 27/20 (roller bearings), andm = 3/2 (hourglass-shaped test specimens)m0= Poisons constant, 1/m0 1/3 for steelNn= Number of stress cycles to failure for n%n = (1 R) 100%, probability of failure %p = Loadlife exponentpprime= Loadlife exponent for minimum lifeQ = Rolling body loadQc= Rolling body load ratingQeq= Equivalent rolling body loadQmax= Maximum rolling body loadq = Stresslife exponentqprime= Stresslife exponent for minimum lifeR = 1 n/100 = ReliabilityR0= Reliability for link or small elementT = zy/maxT0= 0.25 (Line contact b/a 0)T1= 0.2139 (Point contact b/a 1)Tq= Torsional momentu = Number of stress cycles in one rotationvb= Orbital velocity of rolling bodyve= Velocity of rotating ringx = Random variable/strength of material/lifex10= Strength/life for 10% probability of failureZ = Number of rolling bodies per rowz0= Initiation depth of 0 = Contact angle = Linkage factor for of 90% stress ratingprime= Linkage factor for of 100% stress rating = Minimum life or location parameterN= Number of minimum stress cycles to failure = Exponent of Dwor DwLwe = Mutual approach between inner and outerraces = L63 = Load/scale parameter0= (N10 N)ln(1/0.9)1/m= Load/scaleparameter for link/small elementcorresponding to K = 1 = Reduction factor for Sigma1 = Curvature summax= Maximum Hertz stress0= Orthogonal shear stress amplitudemax= Alternating torsion shear stress amplitude = Geometrical coefficient for point contactj= Angular position of rolling body = Dwcos /Dpw = Geometrical coefficient for line contact = (1 cos )/(1 + cos) = Coefficient in Phi10= Coefficient in 0Subscriptse = e i (inner ring), e o (outer ring)h = h R (rotating ring), h S (stationaryring)same life as the raceways under various types of load distribu-tions. For an equivalent load Q at a depth z = z0, a maximumorthogonal shear stress amplitude 0occurs parallel to the race,and so a flaking failure has to occur somewhere at subsurface z0under an ideal stress distribution (Lundberg and Palmgren (4).For a point contact bearing, the locus of z0henceforth becomesthe equal stress line le,andle De, whereas for a line contactbearing, it is the equal stress area leLwe DeLwe(Shimizu (1).For a line contact bearing (b/a 0), Fig. 2 shows the fluctu-ation in the ratio of shear stress zyoccurring at a depth z = z0in accordance with the roller revolution and the maximum Hertzstress max;thatis,T = zy/max. In the case of a roller race, themaximum value occurs at y =0.87b, z0= b/2, and the stressratio is T = T0= 0/max= 0.25.The reliability function ln 1/R0for an element dleof the racecorresponding to K = 1with0is expressed by Eq. 3 for n%Downloaded by Shigeo Shimizu at 01:20 01 July 2012 560 S. SHIMIZUle= De=KdleR02adleStressed line/area for2bQxyzz0LwePoint C.Line C.Dia.DiR0zyxQDia.DwLwe2bvbvevbDia. Dw(b) Cross sections and developement of race(a) Contact areas and small race elementsQdle0Fig. 1Schematic of contacts for rolling bearings.failurelife Nn,Weibullslope m,andscale/loadparameter 0.Fur-thermore, by using a minimum life Nfor no failure to dle,withR0= 0.9 and n = 10%, 0is given by Eq. 4:ln1R0=parenleftbiggNn N0parenrightbiggm30= (N10 N)parenleftbiggln10.9parenrightbigg1m4In the stresslife (S-N) curve, according to Shimizu, et al. (6),generally the life (N10 N) is inversely proportional to 0to thepower of the stresslife exponent q resulting in Eq. 5, which isassumedasthevalueobtainedbytheP-S-Ntestforthesametypeof stress and the same kind of material.N10 N q05The reliability function for the element dlederived from Eqs.35 for point contact bearings is given by Eq. 6.ln1R0braceleftbigq0(Nn N)bracerightbigmdleln10.96Because each element of the inner or outer ring may not failat the same time, the probability Rebased on the multiplicationy /bT00.330.3030y+T0000+max00/=T0+T= zy/maxFig. 2Orthogonal shear stress for line contact, b/a 0.law of probabilities of each element will be as follows:Re= RK0or ln1Re= Kln1R07So the integrated number of the products of the power repre-sentedbyK forpointandlinecontactsonracesreceivingacriticalstress 0is given by Eqs. 8 and 9:K 1dleleintegraldisplay0dle=ledleDedle8K 1dledLweleintegraldisplay0dleLwe/2integraldisplayLwe/2dLwe=leLwedledLweDeLwedledLwe9Thus, the reliability functions derived from the relationshipsin Eqs. 69 may be given by Eqs. 10 and 11:ln1Re mq0(Nn N)meDeln10.910ln1Re mq0(Nn N)meDeLweln10.911In addition, if u is the number of stress cycles on the race forone rotation of the rotating ring, then the relationship betweenthe number of stress cycles, Nn, until the n% failure and the lifeLnin units of 106revolutions may be shown by Eq. 12:(Nn N)e= u(Ln )e; Nn= uLn,N= u 12Now if the power of 1/mq is taken on both sides of the Eqs.10 and 11, and Aprime0is introduced as a proportionality constant,then by substituting from Eq. 12 into Eqs. 10 and 11, the re-arranged Eqs. 13 and 14 are obtained:0(Ln )1qe= Aprime0a1q1(umDe)1mq130(Ln )1qe= Aprime0a1q1(umDeLwe)1mq14Here a1is the reliability factor given by Eq. 15:a1=parenleftbigglnReln0.9parenrightbigg 1m15With n = 10% and so Re= 0.9, a1= 1andforL10 = 1,0= CSeis achieved; therefore, considering a 90% stress rating ofthe inner/outer ring, the next set of equations, Eqs. 16 and 17,is obtained.CSe= Aprime0(umDe)1mq16CSe= Aprime0(umDeLwe)1mq17Furthermore, it is possible to obtain a life equation, Eq. 18,involving the minimum life for any given value of reliability byderiving from the relationships in Eqs. 1317:(Ln )e= a1parenleftbiggCSe0parenrightbiggq18In addition, with a1= 1, the 90% rating life of the inner/outerring and the minimum life, when CprimeSeand qprimeare taken as the100% stress rating and stresslife exponent, respectively, will begiven by Eq. 19:(L10 )e=parenleftbiggCSe0parenrightbiggq,=parenleftbiggCprimeSe0parenrightbiggqprime19Downloaded by Shigeo Shimizu at 01:20 01 July 2012 A New Life Theory for Rolling Bearings 561le= DeRDeTqStressed lineeeDlK maxFig. 3Specimen under alternating torsion life test.P-S-N Test SpecimenFigure 3 shows the shape of the alternating torsion test spec-imen and the equal stress line lereceiving the maximum shearstress amplitude maxas a result of torque Tq. Due to the max-imum shear stress occurring on the circumference of diameterDeat the minimum cross section, the shearing cracks will initi-ate from points on the equal stress line in the circumferential andaxial directionsaswellasintheradial direction.Thecriticalmax-imum shear stress is given by Eq. 20 at the minimum cross sec-tion:max=16TqD3e20Thereafter, as shown in Fig. 4, the fatigue crack propagatesas a result of the conjugate tensile and compressive stresses 1=maxin a 45direction (in case of hardened materials), finallyleading to fracture. This result is for a specimen with an equalstress line similar to the point contact bearing as shown in Fig.1, and so the number of product law of probabilities K may beconsidered to follow as K De(Shimizu (1).Following a similar method as in Eq. 13 and assuming thespecimens Weibull slope as mSand taking A0as the proportion-ality constant, the life equation may be derived as given by Eq.21:max(Nn N)1q= A0a1q1D1mSqe21Now with n = 10% and a1= 1, for an alternating cycle N10N= 1, Eq. 22 for the stress rating of the specimen max= CSisobtained:CS= A0D1mSqe22Therefore, by following the same guidelines as for the lifeequation for the bearing race and applying Eq. 22 into Eq.21, and further assuming a 100% stress rating CprimeSrelated to N,the life equation for the specimen in Fig. 3that is, the P-S-NDe0maxmaxmaxmax+max+max+11Fig. 4Crackinitiationandpropagationbysurfaceshearstressandcon-jugate tensile stress.TABLE 1ESTIMATIONS OBTAINED BY ALTERNATING TORSIONP-S-NTEST, De0 R = 8 30 MM FOR TEST SPECIMENS IN FIG.3(GPA)N10 N=(CS/max)qN= (CprimeS/max)qprimemSqCSqprimeCprimeS3/2 31/3 2.27 9.92.27Data from Shimizu (1) and Shimizu, et al. (6).curvemay be expressed by Eqs. 23 and 24:Nn N= a1parenleftbiggCSmaxparenrightbiggq23N10 N=parenleftbiggCSmaxparenrightbiggq,N=parenleftbiggCprimeSmaxparenrightbiggqprime24Table1showstheestimatedvalueoftheWeibullslopemSandthe stresslife exponent q in the alternative torsion P-S-N test forthespecimeninFig.3(Shimizu,etal.(6).Thespecimenmaterialwas JIS SUJ2/AISI 52100 (henceforth abbreviated as SUJ2) witha hardness range of HRC 5862. The Weibull slope was observedto recur as a constant value mS= 3/2 for the life distribution ofsix lots at stress amplitude values max= 0.51.0 GPa (Shimizu(1). Based on the P-S-N test carried out for a specimen diameterDe0= 8 mm, the stress rating CSmay be evaluated from Eq. 24,whereas the proportionality constant A0is determined from Eq.22, leading to Eq. 25:A0= CSD1mSqe025Furthermore, because no tests were conducted for acylindrical-shaped specimen (diameter length = De Lwe)foralternative torsion stress, maxat equal stress area, the P-S-N testdata are unavailable. By using the proportionality constant A0ofEq. 25 and by replacing stress line Dewith stress area DeLweinEq. 22, the stress rating may be written as Eq. 26:CS= A0(DeLwe)1mSq26In addition, because the Weibull slope mSmay possibly varysimilarly for point and line contact bearings, the failure mode forcylindrical specimens under the alternative torsion test must beconfirmed.Linkage between Races and P-S-N Test SpecimenUsing Eqs. 12, 19 and 24 for the life of the bearing raceandthetestspecimeninanalternatingtorsiontest,anequalstresscondition such as 0= maxmay be considered after putting to-gether the total number of stress cycle units. Furthermore, in or-der to consider the difference between the fatigue strength re-lated to the rolling contact fatigue in the subsurface material (0and life) and the structural failure on the surface material (maxand life), a linkage factor as a proportionality constant was in-troduced as shown in Eq. 27:CSe=parenleftbig106uparenrightbig1qCS27Downloaded by Shigeo Shimizu at 01:20 01 July 2012 562 S. SHIMIZUAt this point, if the diameter Decorresponding to the stressrating CSof the test specimen shown in Eqs. 22 and 26 is usedas De= Diand De= Dofor the inner and outer races in Eq. 27,the stress rating CSefor the race may be given by Eqs. 28 and29 for point contact and line contact bearings, respectively.CSe= 106qA0(umSDe)1mSq28CSe= 106qA0(umSDeLwe)1mSq29Because a similar idea is expressed by Eqs. 28 and 29 forstructural fatigue and by Eqs. 16 and 17 for rolling contact fa-tigue,theproportionalityconstant Aprime0fortheinner/outerbearingraceways may be determined by Eqs. 30 and 31 for point andline contact bearings, respectively.Aprime0 106qA0D1mqparenleftBig1mmSparenrightBige30Aprime0 106qA0(DeLwe)1mqparenleftBig1mmSparenrightBig31Shear Stress Amplitude and Life FormulaThe relationship between the maximum Hertz stress maxontheracewaycontactportionreceivingtherollingbodyloadQandthe maximum shear stress amplitude 0occurring at a depth z0belowthesurfaceinthepassage(refertoFig.2)maybeexpressedby Eq. 32:0= Tmax,max=3Q2ab32Here, the major and minor axes a and b of the contact ellipsemay be expressed in terms of the parameters , , the modulus ofelasticity E, Poissons constant 1/m0, and the summation of cur-vatures Sigma1 of two contacting bodies by Eq. 33:a = parenleftbigg3QE0Sigma1parenrightbigg13, b = parenleftbigg3QE0Sigma1parenrightbigg13, E0=m20Em20 133In addition, Eq. 32 can be expressed as Eq. 34:0= Tparenleftbigg32parenrightbiggparenleftbiggQab2parenrightbigg12parenleftbigg1aparenrightbigg12Q1234Now if the parameters in Eq. 33 and the rolling body diam-eter Dware introduced in Eq. 34, an effective Eq. 35 for thepoint and line contact bearings may be obtained.0= Tparenleftbigg32parenrightbiggparenleftbiggE0DwSigma132parenrightbigg12parenleftbiggDwaparenrightbigg12parenleftbiggQD2wparenrightbigg1235On the other hand, if Dwis introduced into Eqs. 13 and 14,Eqs.36and37 forbothpointandlinecontactbearingsmaybederived:0(Ln )1qe= a1q1Aprime0parenleftbiggumDeDwparenrightbigg1mqD1mqw360(Ln )1qe= a1q1Aprime0parenleftbiggumDelaD2wparenrightbigg1mqD2mqw37LOAD RATING AND LIF