research article fatigue life analysis of rolling bearings based on quasistatic modeling ·...

Research ArticleFatigue Life Analysis of Rolling BearingsBased on Quasistatic Modeling

Wei Guo Hongrui Cao Zhengjia He and Laihao Yang

State Key Laboratory for Manufacturing Systems Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Hongrui Cao chrmailxjtueducn

Received 27 February 2015 Accepted 28 April 2015

Academic Editor Changjun Zheng

Copyright copy 2015 Wei Guo et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Rolling bearings are widely used in aeroengine machine tool spindles locomotive wheelset and so forth Rolling bearings areusually the weakest components that influence the remaining life of the whole machine In this paper a fatigue life predictionmethod is proposed based on quasistatic modeling of rolling bearings With consideration of radial centrifugal expansion andthermal deformations on the geometric displacement in the bearings the Jonesrsquo bearing model is updated which can predict thecontact angle deformation and load between rolling elements and bearing raceways more accurately Based on Hertz contacttheory and contact mechanics the contact stress field between rolling elements and raceways is calculated A coupling model offatigue life and damage for rolling bearings is given and verified through accelerated life test Afterwards the variation of bearinglife is investigated under different working conditions that is axial load radial load and rotational speed The results suggestedthat the working condition had a great influence on fatigue life of bearing parts and the order in which the damage appears onbearing parts

1 Introduction

Rolling bearings are widely used in rotating machinerysystem such as aeroengine rotors system machine toolspindles and train wheelset However they are the weaklinks of mechanical because their mechanical propertiesand operating state have significant impact on precisionreliability and life of the whole system The failure of rollingbearings not only affects the performance of mechanicalequipment but also causes serious accident Thus fatiguelife prediction for rolling bearing has important theoreticalsignificance and practical value

The dynamics behavior of rolling bearings is observablyaffected by factors such as centrifugal force gyroscopicmoment friction thermal deformation and external loadon unconventional conditions like high speed and highaccelerated velocity In this case the movement inside rollingbearing becomes very complex Besides pure rolling contactbetween rolling elements and raceways motions like spin-ning and skidding are accompanied The failure mechanismanalysis fault diagnosis and life prediction for rolling bear-ings become more difficult because of the significant change

of contact angle and contact load distribution Incipient fail-ure diagnosis is of great significance for keymajor equipmentHowever the response signals reflect incipient failures ofrolling bearings are not obvious and easy to be disturbed bypathways and working noises which bring new challengesto fault diagnosis technology based on signal processingand feature extraction On fault diagnosis of rolling bearingresearchers developed fault detect and diagnostic techniquesbased on vibration and signal processing [1ndash4] Patil et al [5]reviewed the research status of fault diagnosis on rolling bear-ings Research on fault modeling theory and failure mecha-nism for rolling bearings is particularly inadequate comparedwith fault detecting techniques In recent years studieson rolling bearings fault modeling gain more and moreattention from researchers Cao and Xiao [6] establishedcomplicated dynamicalmodel for self-aligning roller bearingSurface damage pretension and radial clearance problemswere studied Sawalhi and Randall [7] integrated nonlinearbearing model with model of gear and simulated spallingdamage of bearings Rafsanjani et al [8] developed a non-linear dynamical model for rolling bearings and provided

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 982350 10 pageshttpdxdoiorg1011552015982350

2 Shock and Vibration

mathematical description for roller inner raceway and outerraceway Patel et al [9] established a dynamical model fordeep groove ball bearing to study the vibration responsewhensingle point or multipoint faults existing on inner and outerraceways Above researches for dynamic simulation of rollingbearings are helpful to understand the failure mechanismand characteristic of bearings and provide fundamental basisfor monitoring and fault diagnosis of rolling bearings tosome extent However most of the models do not considerthe influence of parameters such as rotational speed oper-ation temperature rise and external load Considering thecentrifugal expansion and thermal deformation when innerraceway rotating Cao et al [10 11] improved Jonesrsquo rollingbearing model [12] A mechanical model for high speedrolling bearing was developed and verified by experimentsContact loads and contact positions inside rolling bearingsunder static load dynamic load and high rotational speedwere studied

The rolling bearing failure is typical rolling contactfatigue [13] Because fatigue spall of material originatingfrom subsurface caused by rolling contact is the main failureform of rolling bearing [14] researchers put forward a lotof rolling contact fatigue models to predict rolling bearinglife in past decades The deterministic research modelsfor rolling contact fatigue (RCF) as an important class ofbearing life models consider complete stress-strain behaviorinformation of materials under contact loading [15] Taraf etal [16] studied the modeling of the rolling contact fatigueinitiation life which was simulated with moving Hertziancontact pressure It was found that the size and shape ofdefects in material played an important role in fatigue crackinitiation Deshpande and Chandra Kishen [17] proposeda method for rolling contact fatigue crack propagationanalysis with the concepts of Hertzian contact mechanicsassociated with fracture mechanics This algorithm could beused to determine whether the bearing failure caused bycrack propagation Using the method of equivalent initialflaw size Liu and Mahadevan [18] applied the propagationmodel to calculate initiation fatigue life Then the modelfor crack initiation and growth was obtained Liu and Choi[19] developed a method to model the RCF life of finishhard machined surfaces and proved by experiment The RCFmodel was based on both the crack initiation life and thecrack propagation life

Calculation of contact stress field is essential for theprediction of bearing fatigue life and many studies havebeen conducted on solving contact stress field and RCFproblems Hertz created elasticity contact theory providingfoundation of contact stress analysis and calculation Hertztheory provided formula of surface stress field in correspond-ing contact area Lundberg and Palmgren [20] simplifiedHertz contact problem Corresponding contact parameterscould be obtained just by querying contact coefficient tablesLots of tests proved rolling contact fatigue often initiated insubsurface therefore the analysis for subsurface stress fieldof contact area is important to research failure mechanismand life prediction of contact fatigue [13] Johnson [21] gavea formula to calculate the principal stress at any depth in thesubsurface of contact area Sadeghirsquos group [22ndash25] developed

a Voronoi finite element model to simulate the microstruc-ture of material and calculate the stress field of contact areaThe effect of microstructure of material on contact stress fieldand rolling contact fatigue life was discussed

Traditional researches were mostly based on models forcontact between single roller and raceway but these modelshave not developedwholemechanicalmodel to analyze stressof a whole bearing Meanwhile the influences of parameterslike bearing structure and operating condition on bearinglife were not considered Based on modified Jonesrsquo model[12] this paper analyzes mechanics principle of a bearingand solves contact loads and contact angles under high-speedconditions On the basis of the above model stress field ofsurface and subsurface of bearing is calculated by dichotomyThen a couplingmodel of life and damage which consideringthe mechanical property of a bearing is established to predictthe fatigue initiation life of bearing parts and qualitativelyverified through accelerated life test

2 Quasistatic Modeling for Rolling Bearings

The movement inside an angular contact ball bearingbecomes complicated at high speed The contact load andthe contact angle are significantly changed because of cen-trifugal forces and gyroscopicmomentsMoreover operatingtemperature will increase with the run time growing sothat thermal deformation occurs on bearings These causesignificant change on contact stress field and then affectthe fatigue life of rolling bearing Jonesrsquo bearing model isa more complete bearing mechanics mode however it hasnot considered the dilatational strain caused by rapid risingtemperature and centrifugal force In this paper the Jonesrsquobearing model is improved with the consideration of effectof centrifugal force and gyroscopic couple and can take theinfluences of radial thermal expansion and centrifugationexpansion of inner raceway on geometric displacement insidethe bearing into account So the contact load and the contactangle under the influences of these factors can be calculated

21 Geometrical Properties of anAngular Contact Ball BearingThe typical geometric construction of an angular contactball bearing is shown in Figure 1 In the figure 120593119896 is theazimuth angle of the 119896th roller Based on finite element ideaa rolling bearing can be viewed as an element comprisedof an inner raceway node and an outer raceway node Themotion of each node contains 5 degrees of freedom (DOFs)In order to facilitate the analysis the outer raceway is fixedSet the relative displacements of these 5 DOFs between innerraceway and outer as Δ120575119909 Δ120575119910 Δ120575119911 Δ120574119910 and Δ120574119911

As shown in Figure 2 when rolling bearings operatedthe relative positions of inner raceway rolling element andouter raceway will be changedThe inner raceway and rollingelement havemoved to new locations respectively after bear-ings reach equilibrium states Then the distances betweencurvature centre of raceways and roller are as follows

Δ 119894119896 = 119903119894 minus 05119863+120575119894119896 = (119891119894 minus 05)119863+ 120575119894119896

Δ 119900119896 = 119903119900 minus 05119863+120575119900119896 = (119891119900 minus 05)119863+ 120575119900119896

(1)

Shock and Vibration 3

Outerraceway

BallInnerraceway

120574z

120574z

Dm

120575y

120575x

120575z

120579

1205930

120593k

y y

z

k

ox

k = 1 Δ120593

Figure 1 Geometric drawing of rolling bearings

Outer racewaycurvature center

Ball centerinitial position

Ball centerfinal position

Final positioninner racewaycurvature center

Initial positioninner racewaycurvature center

BD

Δ ok

Δik

Δicu Δicv

120579ok

120579ik

Uk

Uik

Vk

Vik120579

Figure 2 Geometric relationship of bearing inner raceway outerraceway and rolling elements

where subscript 119894 and 119900 indicate inner raceway and outerraceway respectively 119903 is curvature radius119863 is the diameterof rollers 119891 is ratio of diameter of roller to curvatureradius of raceways and 120575119894119896 and 120575119900119896 are contact deformationdisplacements of rollers on inner raceway and outer racewayrespectively

The relative displacement variation of inner racewaycurvature center is as follows

Δ 119894119888119906 = Δ120575119909 minusΔ120574119911119903119894119888 cos120593119896 +Δ120574119910119903119894119888 sin120593119896

Δ 119894119888V = Δ120575119910 cos120593119896 +Δ120575119911 cos120593119896 + 120576119894119903 +119906119894119903 minus 120576119900119903(2)

where 120576119894119903 and 120576119900119903 are radial thermal expansion of inner race-way and outer raceway respectively which can be obtained byfinite element heat analysis [26] 119906119894119903 is the expansion of innerraceway under the action of centrifugal force

Using the Pythagorean Theorem it can be seen fromFigure 2 that the displacement of bearing internal structurein working state is

(119880119894119896 minus119880119896)2+ (119881119894119896 minus119881119896)

2minusΔ

2119894119896= 0

1198802119896+119881

2119896minusΔ

2119900119896= 0

(3)

Accordingly the trigonometry function of contact anglebetween bearing raceways and rolling elements can bedescribed as follows

sin 120579119894119896 =119880119894119896 minus 119880119896

Δ 119894119896

cos 120579119894119896 =119881119894119896 minus 119881119896

Δ 119894119896

sin 120579119900119896 =119880119896

Δ 119900119896

cos 120579119900119896 =119881119896

Δ 119900119896

(4)

22 Force Balance Analysis When angular contact ball bear-ings operate at high speeds the contact between rolling ele-ment and raceways is not pure rolling contact but along withmotions such as spinning and skidding To simplify mattersin practice assume that pure rolling only occurs between balland inner raceway or outer raceway and both spinning andskidding exist on the other raceway Considering centrifugalforce and gyroscopic couple on balls the force of the 119896th ballis analyzed on the plane constructed by bearing axis and ballcenter as shown in Figure 3


Fck

120579ok

120579ikMgk

Qok

Qik

120582ikMgkD

120582okMgkD

120579ik ball inner raceway contact angle ( ∘)

Qok ball inner raceway contact load (N)

Fck centrifugal force for balls (N)

120579ok ball outer raceway contact angle (∘)

Qok ball outer raceway contact load (N)

Mgk gyroscopic moment for balls (Nmiddotm)

Figure 3 Force analysis of rolling balls

From Figure 3 considering the equilibrium of forces inthe horizontal and vertical directions

119876119900119896 cos 120579119900119896 minus119876119894119896 cos 120579119894119896 minus119872119892119896

119863(sin 120579119900119896 minus sin 120579119894119896)

minus 119865119888119896 = 0

119876119900119896 sin 120579119900119896 minus119876119894119896 sin 120579119894119896 +119872119892119896

119863(cos 120579119900119896 minus cos 120579119894119896) = 0

(5)

where 120579119894119896 and 120579119900119896 are contact angle of inner raceway and outerraceway 119865119888119896 and 119872119892119896 are centrifugal force and gyroscopicmoment for rollers respectively 119876119894119896 and 119876119900119896 are ball-innerand ball-outer raceway contact load respectively 120582119894119896 and 120582119900119896are corrected parameters of raceway control mode for theouter raceway controlling case 120582119894119896 = 0 and 120582119900119896 = 2 while120582119894119896 = 120582119900119896 = 1 in any other cases This will not cause muchinfluence to computational accuracy [27]

119876119894119896 = 11987011989412057532119894119896

119876119900119896 = 11987011990012057532119900119896

(6)

where 119870119894 and 119870119900 are respectively load deflection constantsof ball-inner and ball-outer raceway contact [27 28] Undera joint result of static load and thermal deformation theball-inner raceway contact deformation 120575119894119896 and ball-outerraceway contact deformation 120575119900119896 are

120575119894119896 = Δ 119894119896 + 120576119887 minus (119891119894 minus 05)119863

120575119900119896 = Δ 119900119896 + 120576119887 minus (119891119900 minus 05)119863(7)

z

b

a

y

y

xo

x

120590max120590

Figure 4 Ellipsoidal surface compressive stress distribution of pointcontact

where 120576119887 is thermal expansion deformation which can beobtained by finite element heat analysis

From (3) and (5) the unknown parameters 119880119896 119881119896 120575119900119896and 120575119894119896 will be solved with Newton iteration method thusthe contact angle and the contact load are obtained

3 Life Prediction Model ofRolling Contact Bearing

31 Contact Stress Field Modeling and Numerical Solutioninside Angular Contact Ball Bearing The contact stress fieldbetween rolling element and raceway is altered because of thesignificant changing of contact angle and contact load underhigh speed rotation which will affect the fatigue life of rollingbearingsTherefore it is very important to established contactstress model of relationship between rolling element andraceway and to find the high efficiency numerical solutionmethod

311 Ball-Raceway Contact Mechanics Model In angularcontact ball bearings the contact zone between a ball and araceway is elliptical based on Hertz contact theory Surfacestress distribution inside the contact area has a semiellipsoidas shown in Figure 4 In the figure 119886 and 119887 are the semimajorand semiminor axes of the elliptical area of contact respec-tively

When the contact load is119876 the normal stress at any point(119909 119910) in the contact area surface can be expressed as follows[13]

120590 = 120590max [1minus(119909

119886)

2minus(

119910

119887)

2]

12 (8)

where 120590max is the maximum contact stress at contact ellipsecenter From Hertz contact theory the semimajor axis 119886


semiminor axis 119887 and elastic contact deformation 120575 areshown as follows [13]

119886 = 119886lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

119887 = 119887lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

120575 = 120575lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

23sum120588

2

(9)

where 119864119894 (119894 = 1 2) is Youngrsquos modulus (MPa) 120585119894 (119894 = 1 2) isPoissonrsquos ratio and 120581 is a supplementary parameter 120581 = 119886119887Thus parameters 119886lowast 119887lowast and 120575lowast can be represented as

119886lowast= (

21205812119864120587

)

13

119887lowast= (

2119864120587120581

)

13

120575lowast=2119870120587

(120587

21205812119864)

13

(10)

where 119870 and 119864 are the complete elliptic integrals of the firstand second kind respectively [29]

According to (10) the key to the solution of Hertz contactproblem is to obtain the value of supplementary parameter 120581

312 Numerical Solution of Contact Model Harris andKotzalas [13] and Lundberg and Palmgren [20] suggestedsimplified computational methods to calculate Hertz contactstress However these methods all make approximation tomodel parameters which result in certain errors It will notonly affect the calculation accuracy of contact area stress fieldbut also cause some error to the calculation of rolling contactfatigue life Therefore dichotomy is used to simulate theHertz point contact problem to reduce the error of contactstress analysis Figure 5 shows the program chart

In order to determine the contact ellipse parameter 120581firstly the eccentricity of contact ellipse 119890 should be con-firmed while 119890 can be obtained from the following equation[27]

(2 minus 1198902) 119864 (119898) minus 2 (1 minus 119890

2)119870 (119898)

1198902119864 (119898)= 119865 (120588) (11)

where119898 is a supplementary parameter with the value of119898 =

1198902

119890 = radic1 minus 1205812 (12)

119870(119898) and 119864(119898) are the complete elliptic integrals of thefirst and the second kinds respectively 119865(120588) is a function ofprincipal curvature of contact bodies

119865 (120588) =

10038161003816100381610038161205881Ι minus 1205881ΙΙ1003816100381610038161003816 +

10038161003816100381610038161205882Ι minus 1205882ΙΙ1003816100381610038161003816

sum 120588 (13)

G(e) = F(120588) minus(2 minus e2)E(e2) minus 2(1 minus e2)K(e2)

e2E(e2)

x1 = 00001 x2 = 09999

m =x1 + x2

2

G(e) = 0

G(x1) middot G(m) lt 0

e = m

Output e

|x1 minus x2| lt 120598

Start

Initial value

Defining

Yes

Yes

Finish

Yes

No

No

No

x2 = m

x1 = m

Figure 5 Program chart of dichotomy

where 120588119894119895 (119894 = 1 2 119895 = I II) is the principal curvature ofcontact body sum120588 is the sum of principal curvature [13]

Function 119866(119890) is defined for the convenience of solving

119866 (119890) = 119865 (120588) minus(2 minus 119890

2) 119864 (119890

2) minus 2 (1 minus 119890

2)119870 (119890

2)

1198902119864 (119898) (14)

Then the solution of (11) can be equivalent to obtainthe zero match of function 119866(119890) while function 119866(119890) =

0 has unique solution in interval 119890 isin (0 1) the idea ofdichotomy is used to solve (11) numerically The value of119890 119864(1198902) and 119870(119890

2) can be obtained simultaneously when

solving (11) Initial values 1199091 and 1199092 as shown are taken in thesolution because the range of ellipse eccentricity is 119890 isin (0 1)Thus the more accurate value of contact ellipse parameter120581 is calculated if the error contact value is selected as small


Table 1 Parameters of 7311B angular contact ball bearing

Projects ValueInner raceway diametermm 55Outer raceway diametermm 120Number of rolling elements 12Ball nominal diametermm 20638Poissonrsquos ratio of ball and raceways 03Youngrsquos modulus of ball and racewaysNsdotmminus2 20811986411

Table 2 Contact area size

ProjectsContact betweenball and inner

raceway

Contact betweenball and outer

raceway119886mm 119887mm 119886mm 119887mm

Lundbergrsquos algorithm [27] 36860 03220 29102 04670This method 36893 03258 29059 04638Error 00894 1167 0148 0685

Table 3 Maximum contact stress 120590max of contact area

Projects Lundberg[27]MPa

ThismethodMPa Error

Inner raceway 24132 23828 126Ball whilecontact withinner raceway

24132 23828 126

Outer raceway 21082 21261 084Ball whilecontact withouter raceway

21082 21261 084

as possible and then contact parameters 119886lowast 119887lowast and 120575lowast are

solved numericallyThe above numerical method is used to solve and analyze

the stress field of 7311B angular contact ball bearing Table 1shows parameters of the bearing The results are comparedwith results of Lundbergrsquos simple algorithm [27] as shown inTables 2 and 3

From Table 2 some errors between contact area size gotthrough numerical method and Lundbergrsquos simple algorithmexist but all less than 15 So the two methods can beverified with each other The obtained maximum contactstresses are shown in Table 3 The maximum contact stresserror at inner raceway is about 126 and about 084 atouter raceway Thus it can be seen that Lundbergrsquos simplealgorithm can roughly estimate the contact problem whencalculation accuracy is not high

32 Coupling Model of Fatigue Life and Damage for RollingBearing Suppose contact fatigue damage of rolling bearingsis isotropic thus the damage variable can be expressed as ascalar119863 which means damage variable [30]

119863 =120575119878119863

120575119904 (15)

where 120575119878119863 is damaged area on the section and 120575119878 is a sectionarea of infinitesimal

Damage variable 119863 ranges from 0 to 1 When 119863 is 0 itmeans the section has not been injured When119863 is 1 it indi-cates the section is full of damage and the bearing material isdestroyed entirely In fact real material is destroyed before119863reaches 1 Use 119863119888 to express the critical damage threshold ofmaterial experiments show that 119863119888 has a value between 02and 08 for metal material

Based on damage mechanics the rate of damage evolu-tion of high-cycle fatigue links with damage variable 119863 andstress level 120590 which expressed by a nonlinear equation asfollows [30]

d119863d119873

= 119891 (120590119863) (16)

Based on the nonlinear equation a two-parameter lifemodel coupled with damage was proposed by Chaboche andLesne [31] and Xiao et al [32] which was widely used topredict the fatigue life of crack initiation

d119863d119873

= [Δ120590

120590119903 (1 minus 119863)]

119898

(17)

where Δ120590 is the maximum variation range of stress in astress cycle 120590119903 and 119898 are two temperature-related materialconstants

For rolling contact note that the subsurface normalstresses on all the contact areas between raceway androllers are compressive and prevent crack propagation someassumptions are made [33] (1) damage accumulation cannotbe caused by subsurface pressure stress acting on cell nodes(2) the shear stress acting on cell nodes help to damageaccumulation and subsurface micro-crack propagation Thisis consistent with the view of subsurface crack in rollingbearing only propagating in mode II (sliding mode crack)Thus only shear stress amplitude Δ120591 can be used to predictthe fatigue life The damage evolution equation is expressedas follows

d119863d119873

= [Δ120591

120590119903 (1 minus 119863)]

119898

(18)

In view of the rolling contact fatigue damage is causedonly by the shear behavior of material the rolling contactfatigue is assumed to equivalent to torsional fatigue Soparameters 120590119903 and 119898 can be obtained from torsional fatiguecurve of a material

119873119891 =1

(119898 + 1)[120590119903

Δ120591]

119898

= [2120591119891Δ120591

]

119861

(19)

where 120591119891 is the stress intensity factor and 119861 is the fatigueintensity index In the torsional fatigue graph minus1119861 is theslope of S-N curve and 120591119891 is the vertical axis intercept of S-N curve


Outer racewayInner racewayBall

0 2000

15

20

25

30

35

40

45

4000 6000 8000Rotational speed (rmin)

Fatig

ue li

fe (m

illio

n cy

cles)

Figure 6 The change of fatigue life with rotational speed 119899

From the equivalence of rolling contact fatigue failuremechanism and torsional fatigue failure mechanism thestress parameters in both cases are assumed to be the same

119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)

33 Simulation of Fatigue Life of Bearing Parts under Dif-ferent Loads and Rotational Speeds According to the abovequasistatic model contact angles and contact loads of abearing change with different operation conditions whichhave large effects on contact stress field and fatigue life ofbearing parts Ignore the impact of friction lubrication andmaterial inclusion on fatigue life of bearing parts to sametype rolling bearing the fatigue life of bearing parts is mainlyaffected by rotational speed 119899 axial load 119865119886 and radial load119865119903 This paper discusses the influences of these factors underthe circumstance of fixed outer raceway and rotating innerraceway

The contact angle and the contact load distribution of7311B angular contact ball bearing are calculated by the abovemodified Jonesrsquo model under different operating conditionsSubstitute these calculated load data in life prediction modeland then the fatigue life of bearing parts is obtained

331 Effect of Rotational Speed on Fatigue Life of Bearing PartsWhile external load is invariable contact angles and contactloads of a bearing will be significantly changed because of theincrease of centrifugal force with rotational speed increasingIt will cause the life of bearing parts to be different Settingaxial load 119865119886 = 50 kN and radial load 119865119903 = 0 changing thespindle rotational speed then the changes of fatigue life ofbearing parts with rotational speed can be predicted as shownin Figure 6


Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60

Figure 7 The change of fatigue life with axial load 119865119886

As can be seen from Figure 6 the fatigue life of outerraceway is shortened gradually with the increasing of therotational speed and the fatigue life of inner raceway andball tend to be larger This is mainly due to the change of themaximum orthogonal shear stress at contact area caused bythe variation of centrifugal force The maximum orthogonalshear stress of the outer raceway will increase with theincrease of centrifugal force but the orthogonal shear stressof inner raceway decreases From the simulation results thefatigue life of outer raceway is the longest ball is the secondand inner raceway is the lowest

332 Effect of Axial Load on Fatigue Life of Bearing Parts Inthis case only the axial load is changed while the rotationalspeed and the radial load are constant This situation affectsthe stress field at contact area and further affects the fatiguelife of a bearing Figure 7 shows the variation trend of fatiguelife of bearing parts with axial load 119865119886 under the case radialload 119865119903 = 0 and rotational speed 119899 = 1400 rmin

As shown in Figure 7 increasing of axial load causesthe decrease of fatigue life of ball outer raceway and innerraceway This is because the maximum orthogonal shearstress increases with the axial load increasing However dueto the different increase rate of orthogonal shear stress fatiguelife of the three parts has different decline rates The fatiguelife of inner raceway falls at the fastest rate ball second andouter raceway the slowest We also see from the figure thatwhen axial load less than 29 kN the inner raceway has thelongest life of the three parts But when axial load bigger than29 kN the outer raceway life becomes the longest though itdoes not appear to be much different among fatigue life ofthree parts

333 Effect of Radial Load on Fatigue Life of Bearing PartsThe fatigue life of three bearing parts is also influenced by


Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30


Figure 8 The change of fatigue life with radial load 119865119903

Figure 9 T20-60nF bearing fatigue life tester

adjusting radial load and keeping axial load and rotationalspeedThe fatigue life trend changes with radial load119865119903 underthe case axial load 119865119886 = 50 kN and rotational speed 119899 =

1400 rmin is shown in Figure 8It can be found from Figure 8 that the fatigue life of outer

raceway and ball decreases with the increasing radial loadwhile the fatigue life of inner raceway increases slowly Atfirst the inner raceway life is the shortest among three partsHowever it exceeds ball life when radial load increases toabout 5 kN and exceeds outer raceway life when the loadat 17 kN or so The increase of radial load leads to contactangle and contact load changing which causes themaximumorthogonal shear stress of outer raceway and ball increase andinner raceway decreaseThis situation results in the variationof bearing parts life

34 Accelerated Life Test of Bearing To verify the accuracyof the model an accelerated life test was performed Theexperimental work carried out on the T20-60nF bearingfatigue life tester as shown in Figure 9 7311B angular contactball bearing was used in the test under one simulationworking condition as 119899 = 1400 rmin 119865119886 = 50 kN and

119865119903 = 0 Four acceleration sensors and an acoustic emissionsensor were installed to monitor the work status Because thecrack initiation life is difficult to determine in test the lifemodel cannot be quantitatively verified through comparingcalculated life with test value Because for bearing the failureof one of the parts means the failure of whole bearing if thefirst damage part of bearing in experiment is in agreementwith simulated result the model can be proved qualitativelyAccording to the simulation result the fatigue life is 137 times 106cycles 452times 106 cycles and 210times 106 cycles for inner racewayouter raceway and ball respectively It can be seen from theresult that the life of the inner raceway is the lowest that isthe inner raceway will be the most easily damaged part

35 Results Analysis There was no obvious damage thatcan be seen in outer raceway inner raceway and ball afterexperiment Small dots were found on the surface of innerraceway as shown in Figure 10(a) while not found on outerraceway (shown in Figure 10(b)) and ball under VMS-1510Gimagemeasurement instrument To further confirm the smalldots on inner raceway were fatigue damage the surfacecharacterization of the three bearing parts was studied underscanning electron microscope (SEM) as shown in Figure 11

Obvious pits about 25 120583m in size on the surface of innerraceway can be seen in Figure 11(a) which can be determinedtomicrospalling caused by contact fatigue inmorphologyOnthe other side there was no pit on surface of outer racewayand ball as can be seen in Figures 11(b) and 11(c)This suggeststhat the fatigue damage first appears on inner raceway ofbearing in accelerated life testThis result which is consistentwith the fatigue life model demonstrates qualitatively thecorrectness of coupling model of life and damage

4 Conclusion

In current study a quasistatic model considering mechanicalproperties of whole bearing was introduced into the fatiguelife calculation of angular contact ball bearing Then acouplingmodel of fatigue life and damage was established forrolling bearing The fatigue life of bearing parts was analyzedunder different rotational speed axial load and radial loadThe results have shown that different working condition has agreat influence on fatigue lives of bearing parts under settingconditions in this paper specifically as follows

(1) With the increasing of rotational speed the fatiguelife of inner raceway and rollers is up while of outerraceway decline The rotational speed does not muchaffect the order in which the damage appear onbearing parts

(2) The fatigue life of three parts decreases with the axialload increasing the fastest of which is the life ofinner raceway The rolling elements and raceways aredamaged in different order under different load valueThe inner raceway is easiest to emerge failure whenaxial load is higher than 29 kN

(3) As radial load increasing the fatigue life of outerraceway and rollers falls while of inner raceway rising


(a) Surface photo of inner raceway (b) Surface photo of outer raceway

Figure 10 Surface photo of raceways under VMS-1510G

Pits


(c) Surface photo of ball

Figure 11 SEM photos of bearing parts

a bit The rollers are easiest to damage when load islarger than 5 kN

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is jointly supported by the National NaturalScience Foundation of China (no 51421004) the NationalBasic Research Program of China (no 2011CB706606) andthe Fundamental Research Funds for the Central University(CXTD2014001)

References

[1] R B Randall and J Antoni ldquoRolling element bearingdiagnostics-a tutorialrdquo Mechanical Systems and Signal Process-ing vol 25 no 2 pp 485ndash520 2011

[2] L L Jiang Y L Liu X J Li and A Chen ldquoDegradationassessment and fault diagnosis for roller bearing based on ARmodel and fuzzy cluster analysisrdquo Shock and Vibration vol 18no 1-2 pp 127ndash137 2011

[3] Y Lei J Lin Z He and Y Zi ldquoApplication of an improvedkurtogram method for fault diagnosis of rolling element bear-ingsrdquo Mechanical Systems and Signal Processing vol 25 no 5pp 1738ndash1749 2011

[4] P Chang and B Lin ldquoVibration signal analysis of journalbearing supported rotor system by cyclostationarityrdquo Shock andVibration vol 2014 Article ID 952958 16 pages 2014


[5] M S Patil J Mathew and P K RajendraKumar ldquoBearingsignature analysis as a medium for fault detection a reviewrdquoJournal of TribologymdashTransactions of the ASME vol 130 no 1Article ID 014001 7 pages 2008

[6] M Cao and J Xiao ldquoA comprehensive dynamic model ofdouble-row spherical roller bearingmdashmodel development andcase studies on surface defects preloads and radial clearancerdquoMechanical Systems and Signal Processing vol 22 no 2 pp 467ndash489 2008

[7] N Sawalhi and R B Randall ldquoSimulating gear and bearinginteractions in the presence of faults Part I The combined gearbearing dynamic model and the simulation of localised bearingfaultsrdquoMechanical Systems and Signal Processing vol 22 no 8pp 1924ndash1951 2008

[8] A Rafsanjani S Abbasion A Farshidianfar andHMoeenfardldquoNonlinear dynamic modeling of surface defects in rolling ele-ment bearing systemsrdquo Journal of Sound and Vibration vol 319no 3ndash5 pp 1150ndash1174 2009

[9] V N Patel N Tandon and R K Pandey ldquoA dynamic model forvibration studies of deep groove ball bearings considering singleand multiple defects in racesrdquo Journal of Tribology vol 132 no4 Article ID 041101 10 pages 2010

[10] H-R Cao Z-J He and Y-Y Zi ldquoModeling of a high-speedrolling bearing and its damage mechanism analysisrdquo Journal ofVibration and Shock vol 31 no 19 pp 134ndash140 2012 (Chinese)

[11] H R Cao L K Niu and Z J He ldquoMethod for vibrationresponse simulation and sensor placement optimization of amachine tool spindle systemwith a bearing defectrdquo Sensors vol12 no 7 pp 8732ndash8754 2012

[12] A B Jones ldquoA general theory for elastically constrained balland radial roller bearings under arbitrary load and speedconditionsrdquo Journal of Fluids Engineering vol 82 no 2 pp 309ndash320 1960

[13] T A Harris and M N Kotzalas Essential Concepts of BearingTechnology CRCPress BocaRaton Fla USA 5th edition 2006

[14] A Grabulov R Petrov and H W Zandbergen ldquoEBSD inves-tigation of the crack initiation and TEMFIB analyses ofthe microstructural changes around the cracks formed underRollingContact Fatigue (RCF)rdquo International Journal of Fatiguevol 32 no 3 pp 576ndash583 2010

[15] F Sadeghi B Jalalahmadi T S Slack N Raje and N KArakere ldquoA review of rolling contact fatiguerdquo ASME Journal ofTribology vol 131 no 4 pp 1ndash15 2009

[16] M Taraf E H Zahaf O Oussouaddi and A ZeghloulldquoNumerical analysis for predicting the rolling contact fatiguecrack initiation in a railwaywheel steelrdquoTribology Internationalvol 43 no 3 pp 585ndash593 2010

[17] A S Deshpande and J M Chandra Kishen ldquoFatigue crackpropagation in rocker and roller-rocker bearings of railway steelbridgesrdquoEngineering FractureMechanics vol 77 no 9 pp 1454ndash1466 2010

[18] Y Liu and S Mahadevan ldquoProbabilistic fatigue life predictionusing an equivalent initial flaw size distributionrdquo InternationalJournal of Fatigue vol 31 no 3 pp 476ndash487 2009

[19] C R Liu and Y Choi ldquoRolling contact fatigue life modelincorporating residual stress scatterrdquo International Journal ofMechanical Sciences vol 50 no 12 pp 1572ndash1577 2008

[20] G Lundberg and A Palmgren ldquoDynamic capacity of rollingbearingsrdquo Acta Polytechnica Mechanical Engineering Series vol1 no 3 p 196 1947

[21] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 9th edition 1987

[22] N Raje and F Sadeghi ldquoStatistical numerical modelling ofsub-surface initiated spalling in bearing contactsrdquo Proceedingsof the Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 223 no 6 pp 849ndash858 2009

[23] B Jalalahmadi A new voronoi finite element fatigue damagemodel [PhD thesis] Purdue University West Lafayette IndUSA 2010

[24] N Weinzapfel and F Sadeghi ldquoNumerical modeling of sub-surface initiated spalling in rolling contactsrdquo Tribology Interna-tional vol 59 pp 210ndash221 2013

[25] A Warhadpande F Sadeghi M N Kotzalas and G DollldquoEffects of plasticity on subsurface initiated spalling in rollingcontact fatiguerdquo International Journal of Fatigue vol 36 no 1pp 80ndash95 2012

[26] T Holkup H Cao P Kolar Y Altintas and J Zeleny ldquoThermo-mechanical model of spindlesrdquo CIRP AnnalsmdashManufacturingTechnology vol 59 no 1 pp 365ndash368 2010

[27] T A Harris Rolling Bearing Analysis John Wiley amp Sons NewYork NY USA 1991

[28] D E Brewe and B J Hamrock ldquoSimplified solution forelliptical-contact deformation between two elastic solidsrdquo Jour-nal of Lubrication Technology vol 99 no 4 pp 485ndash487 1977

[29] I N Bronshtein K A Semendiaev andK AHirschHandbookof Mathematics Van Nostrand Reinhold New York NY USA1985

[30] J Lemaitre A Course on Damage Mechanics Springer BerlinGermany 1992

[31] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988

[32] Y-C Xiao S Li and Z Gao ldquoA continuum damage mechanicsmodel for high cycle fatiguerdquo International Journal of Fatiguevol 20 no 7 pp 503ndash508 1998

[33] N R Nihar Statistical Numerical Modeling of Subsurface Initi-ated Spalling in Bearing Contacts Purdue University 2008

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



mathematical description for roller inner raceway and outerraceway Patel et al [9] established a dynamical model fordeep groove ball bearing to study the vibration responsewhensingle point or multipoint faults existing on inner and outerraceways Above researches for dynamic simulation of rollingbearings are helpful to understand the failure mechanismand characteristic of bearings and provide fundamental basisfor monitoring and fault diagnosis of rolling bearings tosome extent However most of the models do not considerthe influence of parameters such as rotational speed oper-ation temperature rise and external load Considering thecentrifugal expansion and thermal deformation when innerraceway rotating Cao et al [10 11] improved Jonesrsquo rollingbearing model [12] A mechanical model for high speedrolling bearing was developed and verified by experimentsContact loads and contact positions inside rolling bearingsunder static load dynamic load and high rotational speedwere studied

The rolling bearing failure is typical rolling contactfatigue [13] Because fatigue spall of material originatingfrom subsurface caused by rolling contact is the main failureform of rolling bearing [14] researchers put forward a lotof rolling contact fatigue models to predict rolling bearinglife in past decades The deterministic research modelsfor rolling contact fatigue (RCF) as an important class ofbearing life models consider complete stress-strain behaviorinformation of materials under contact loading [15] Taraf etal [16] studied the modeling of the rolling contact fatigueinitiation life which was simulated with moving Hertziancontact pressure It was found that the size and shape ofdefects in material played an important role in fatigue crackinitiation Deshpande and Chandra Kishen [17] proposeda method for rolling contact fatigue crack propagationanalysis with the concepts of Hertzian contact mechanicsassociated with fracture mechanics This algorithm could beused to determine whether the bearing failure caused bycrack propagation Using the method of equivalent initialflaw size Liu and Mahadevan [18] applied the propagationmodel to calculate initiation fatigue life Then the modelfor crack initiation and growth was obtained Liu and Choi[19] developed a method to model the RCF life of finishhard machined surfaces and proved by experiment The RCFmodel was based on both the crack initiation life and thecrack propagation life

Calculation of contact stress field is essential for theprediction of bearing fatigue life and many studies havebeen conducted on solving contact stress field and RCFproblems Hertz created elasticity contact theory providingfoundation of contact stress analysis and calculation Hertztheory provided formula of surface stress field in correspond-ing contact area Lundberg and Palmgren [20] simplifiedHertz contact problem Corresponding contact parameterscould be obtained just by querying contact coefficient tablesLots of tests proved rolling contact fatigue often initiated insubsurface therefore the analysis for subsurface stress fieldof contact area is important to research failure mechanismand life prediction of contact fatigue [13] Johnson [21] gavea formula to calculate the principal stress at any depth in thesubsurface of contact area Sadeghirsquos group [22ndash25] developed

a Voronoi finite element model to simulate the microstruc-ture of material and calculate the stress field of contact areaThe effect of microstructure of material on contact stress fieldand rolling contact fatigue life was discussed

Traditional researches were mostly based on models forcontact between single roller and raceway but these modelshave not developedwholemechanicalmodel to analyze stressof a whole bearing Meanwhile the influences of parameterslike bearing structure and operating condition on bearinglife were not considered Based on modified Jonesrsquo model[12] this paper analyzes mechanics principle of a bearingand solves contact loads and contact angles under high-speedconditions On the basis of the above model stress field ofsurface and subsurface of bearing is calculated by dichotomyThen a couplingmodel of life and damage which consideringthe mechanical property of a bearing is established to predictthe fatigue initiation life of bearing parts and qualitativelyverified through accelerated life test

2 Quasistatic Modeling for Rolling Bearings

The movement inside an angular contact ball bearingbecomes complicated at high speed The contact load andthe contact angle are significantly changed because of cen-trifugal forces and gyroscopicmomentsMoreover operatingtemperature will increase with the run time growing sothat thermal deformation occurs on bearings These causesignificant change on contact stress field and then affectthe fatigue life of rolling bearing Jonesrsquo bearing model isa more complete bearing mechanics mode however it hasnot considered the dilatational strain caused by rapid risingtemperature and centrifugal force In this paper the Jonesrsquobearing model is improved with the consideration of effectof centrifugal force and gyroscopic couple and can take theinfluences of radial thermal expansion and centrifugationexpansion of inner raceway on geometric displacement insidethe bearing into account So the contact load and the contactangle under the influences of these factors can be calculated

21 Geometrical Properties of anAngular Contact Ball BearingThe typical geometric construction of an angular contactball bearing is shown in Figure 1 In the figure 120593119896 is theazimuth angle of the 119896th roller Based on finite element ideaa rolling bearing can be viewed as an element comprisedof an inner raceway node and an outer raceway node Themotion of each node contains 5 degrees of freedom (DOFs)In order to facilitate the analysis the outer raceway is fixedSet the relative displacements of these 5 DOFs between innerraceway and outer as Δ120575119909 Δ120575119910 Δ120575119911 Δ120574119910 and Δ120574119911

As shown in Figure 2 when rolling bearings operatedthe relative positions of inner raceway rolling element andouter raceway will be changedThe inner raceway and rollingelement havemoved to new locations respectively after bear-ings reach equilibrium states Then the distances betweencurvature centre of raceways and roller are as follows

Δ 119894119896 = 119903119894 minus 05119863+120575119894119896 = (119891119894 minus 05)119863+ 120575119894119896

Δ 119900119896 = 119903119900 minus 05119863+120575119900119896 = (119891119900 minus 05)119863+ 120575119900119896

(1)


Outerraceway

BallInnerraceway

120574z

120574z

Dm

120575y

120575x

120575z

120579

1205930

120593k

y y

z

k

ox

k = 1 Δ120593







BD

Δ ok

Δik

Δicu Δicv

120579ok

120579ik

Uk

Uik

Vk

Vik120579








(119880119894119896 minus119880119896)2+ (119881119894119896 minus119881119896)

2minusΔ

2119894119896= 0

1198802119896+119881

2119896minusΔ

2119900119896= 0

(3)


sin 120579119894119896 =119880119894119896 minus 119880119896

Δ 119894119896

cos 120579119894119896 =119881119894119896 minus 119881119896

Δ 119894119896

sin 120579119900119896 =119880119896

Δ 119900119896

cos 120579119900119896 =119881119896

Δ 119900119896

(4)



Fck

120579ok

120579ikMgk

Qok

Qik

120582ikMgkD

120582okMgkD










119863(sin 120579119900119896 minus sin 120579119894119896)

minus 119865119888119896 = 0

119876119900119896 sin 120579119900119896 minus119876119894119896 sin 120579119894119896 +119872119892119896

119863(cos 120579119900119896 minus cos 120579119894119896) = 0

(5)


119876119894119896 = 11987011989412057532119894119896

119876119900119896 = 11987011990012057532119900119896

(6)


120575119894119896 = Δ 119894119896 + 120576119887 minus (119891119894 minus 05)119863

120575119900119896 = Δ 119900119896 + 120576119887 minus (119891119900 minus 05)119863(7)

z

b

a

y

y

xo

x

120590max120590








120590 = 120590max [1minus(119909

119886)

2minus(

119910

119887)

2]

12 (8)




119886 = 119886lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

119887 = 119887lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

120575 = 120575lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

23sum120588

2

(9)


119886lowast= (

21205812119864120587

)

13

119887lowast= (

2119864120587120581

)

13

120575lowast=2119870120587

(120587

21205812119864)

13

(10)






2)119870 (119898)

1198902119864 (119898)= 119865 (120588) (11)


1198902

119890 = radic1 minus 1205812 (12)


119865 (120588) =

10038161003816100381610038161205881Ι minus 1205881ΙΙ1003816100381610038161003816 +

10038161003816100381610038161205882Ι minus 1205882ΙΙ1003816100381610038161003816

sum 120588 (13)


e2E(e2)

x1 = 00001 x2 = 09999

m =x1 + x2

2

G(e) = 0


e = m

Output e

|x1 minus x2| lt 120598

Start

Initial value

Defining

Yes

Yes

Finish

Yes

No

No

No

x2 = m

x1 = m




119866 (119890) = 119865 (120588) minus(2 minus 119890

2) 119864 (119890

2) minus 2 (1 minus 119890

2)119870 (119890

2)

1198902119864 (119898) (14)










raceway






ThismethodMPa Error


24132 23828 126


21082 21261 084






119863 =120575119878119863

120575119904 (15)




d119863d119873

= 119891 (120590119863) (16)


d119863d119873

= [Δ120590

120590119903 (1 minus 119863)]

119898

(17)



d119863d119873

= [Δ120591

120590119903 (1 minus 119863)]

119898

(18)


119873119891 =1

(119898 + 1)[120590119903

Δ120591]

119898

= [2120591119891Δ120591

]

119861

(19)




0 2000

15

20

25

30

35

40

45


Fatig

ue li

fe (m

illio

n cy

cles)



119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)





Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60







Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Outerraceway

BallInnerraceway

120574z

120574z

Dm

120575y

120575x

120575z

120579

1205930

120593k

y y

z

k

ox

k = 1 Δ120593







BD

Δ ok

Δik

Δicu Δicv

120579ok

120579ik

Uk

Uik

Vk

Vik120579








(119880119894119896 minus119880119896)2+ (119881119894119896 minus119881119896)

2minusΔ

2119894119896= 0

1198802119896+119881

2119896minusΔ

2119900119896= 0

(3)


sin 120579119894119896 =119880119894119896 minus 119880119896

Δ 119894119896

cos 120579119894119896 =119881119894119896 minus 119881119896

Δ 119894119896

sin 120579119900119896 =119880119896

Δ 119900119896

cos 120579119900119896 =119881119896

Δ 119900119896

(4)



Fck

120579ok

120579ikMgk

Qok

Qik

120582ikMgkD

120582okMgkD










119863(sin 120579119900119896 minus sin 120579119894119896)

minus 119865119888119896 = 0

119876119900119896 sin 120579119900119896 minus119876119894119896 sin 120579119894119896 +119872119892119896

119863(cos 120579119900119896 minus cos 120579119894119896) = 0

(5)


119876119894119896 = 11987011989412057532119894119896

119876119900119896 = 11987011990012057532119900119896

(6)


120575119894119896 = Δ 119894119896 + 120576119887 minus (119891119894 minus 05)119863

120575119900119896 = Δ 119900119896 + 120576119887 minus (119891119900 minus 05)119863(7)

z

b

a

y

y

xo

x

120590max120590








120590 = 120590max [1minus(119909

119886)

2minus(

119910

119887)

2]

12 (8)




119886 = 119886lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

119887 = 119887lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

120575 = 120575lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

23sum120588

2

(9)


119886lowast= (

21205812119864120587

)

13

119887lowast= (

2119864120587120581

)

13

120575lowast=2119870120587

(120587

21205812119864)

13

(10)






2)119870 (119898)

1198902119864 (119898)= 119865 (120588) (11)


1198902

119890 = radic1 minus 1205812 (12)


119865 (120588) =

10038161003816100381610038161205881Ι minus 1205881ΙΙ1003816100381610038161003816 +

10038161003816100381610038161205882Ι minus 1205882ΙΙ1003816100381610038161003816

sum 120588 (13)


e2E(e2)

x1 = 00001 x2 = 09999

m =x1 + x2

2

G(e) = 0


e = m

Output e

|x1 minus x2| lt 120598

Start

Initial value

Defining

Yes

Yes

Finish

Yes

No

No

No

x2 = m

x1 = m




119866 (119890) = 119865 (120588) minus(2 minus 119890

2) 119864 (119890

2) minus 2 (1 minus 119890

2)119870 (119890

2)

1198902119864 (119898) (14)










raceway






ThismethodMPa Error


24132 23828 126


21082 21261 084






119863 =120575119878119863

120575119904 (15)




d119863d119873

= 119891 (120590119863) (16)


d119863d119873

= [Δ120590

120590119903 (1 minus 119863)]

119898

(17)



d119863d119873

= [Δ120591

120590119903 (1 minus 119863)]

119898

(18)


119873119891 =1

(119898 + 1)[120590119903

Δ120591]

119898

= [2120591119891Δ120591

]

119861

(19)




0 2000

15

20

25

30

35

40

45


Fatig

ue li

fe (m

illio

n cy

cles)



119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)





Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60







Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Fck

120579ok

120579ikMgk

Qok

Qik

120582ikMgkD

120582okMgkD










119863(sin 120579119900119896 minus sin 120579119894119896)

minus 119865119888119896 = 0

119876119900119896 sin 120579119900119896 minus119876119894119896 sin 120579119894119896 +119872119892119896

119863(cos 120579119900119896 minus cos 120579119894119896) = 0

(5)


119876119894119896 = 11987011989412057532119894119896

119876119900119896 = 11987011990012057532119900119896

(6)


120575119894119896 = Δ 119894119896 + 120576119887 minus (119891119894 minus 05)119863

120575119900119896 = Δ 119900119896 + 120576119887 minus (119891119900 minus 05)119863(7)

z

b

a

y

y

xo

x

120590max120590








120590 = 120590max [1minus(119909

119886)

2minus(

119910

119887)

2]

12 (8)




119886 = 119886lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

119887 = 119887lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

120575 = 120575lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

23sum120588

2

(9)


119886lowast= (

21205812119864120587

)

13

119887lowast= (

2119864120587120581

)

13

120575lowast=2119870120587

(120587

21205812119864)

13

(10)






2)119870 (119898)

1198902119864 (119898)= 119865 (120588) (11)


1198902

119890 = radic1 minus 1205812 (12)


119865 (120588) =

10038161003816100381610038161205881Ι minus 1205881ΙΙ1003816100381610038161003816 +

10038161003816100381610038161205882Ι minus 1205882ΙΙ1003816100381610038161003816

sum 120588 (13)


e2E(e2)

x1 = 00001 x2 = 09999

m =x1 + x2

2

G(e) = 0


e = m

Output e

|x1 minus x2| lt 120598

Start

Initial value

Defining

Yes

Yes

Finish

Yes

No

No

No

x2 = m

x1 = m




119866 (119890) = 119865 (120588) minus(2 minus 119890

2) 119864 (119890

2) minus 2 (1 minus 119890

2)119870 (119890

2)

1198902119864 (119898) (14)










raceway






ThismethodMPa Error


24132 23828 126


21082 21261 084






119863 =120575119878119863

120575119904 (15)




d119863d119873

= 119891 (120590119863) (16)


d119863d119873

= [Δ120590

120590119903 (1 minus 119863)]

119898

(17)



d119863d119873

= [Δ120591

120590119903 (1 minus 119863)]

119898

(18)


119873119891 =1

(119898 + 1)[120590119903

Δ120591]

119898

= [2120591119891Δ120591

]

119861

(19)




0 2000

15

20

25

30

35

40

45


Fatig

ue li

fe (m

illio

n cy

cles)



119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)





Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60







Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119886 = 119886lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

119887 = 119887lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

13

120575 = 120575lowast[

31198762sum120588

(1 minus 1205851

2

1198641

+1 minus 1205852

2

1198642

)]

23sum120588

2

(9)


119886lowast= (

21205812119864120587

)

13

119887lowast= (

2119864120587120581

)

13

120575lowast=2119870120587

(120587

21205812119864)

13

(10)






2)119870 (119898)

1198902119864 (119898)= 119865 (120588) (11)


1198902

119890 = radic1 minus 1205812 (12)


119865 (120588) =

10038161003816100381610038161205881Ι minus 1205881ΙΙ1003816100381610038161003816 +

10038161003816100381610038161205882Ι minus 1205882ΙΙ1003816100381610038161003816

sum 120588 (13)


e2E(e2)

x1 = 00001 x2 = 09999

m =x1 + x2

2

G(e) = 0


e = m

Output e

|x1 minus x2| lt 120598

Start

Initial value

Defining

Yes

Yes

Finish

Yes

No

No

No

x2 = m

x1 = m




119866 (119890) = 119865 (120588) minus(2 minus 119890

2) 119864 (119890

2) minus 2 (1 minus 119890

2)119870 (119890

2)

1198902119864 (119898) (14)










raceway






ThismethodMPa Error


24132 23828 126


21082 21261 084






119863 =120575119878119863

120575119904 (15)




d119863d119873

= 119891 (120590119863) (16)


d119863d119873

= [Δ120590

120590119903 (1 minus 119863)]

119898

(17)



d119863d119873

= [Δ120591

120590119903 (1 minus 119863)]

119898

(18)


119873119891 =1

(119898 + 1)[120590119903

Δ120591]

119898

= [2120591119891Δ120591

]

119861

(19)




0 2000

15

20

25

30

35

40

45


Fatig

ue li

fe (m

illio

n cy

cles)



119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)





Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60







Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














raceway






ThismethodMPa Error


24132 23828 126


21082 21261 084






119863 =120575119878119863

120575119904 (15)




d119863d119873

= 119891 (120590119863) (16)


d119863d119873

= [Δ120590

120590119903 (1 minus 119863)]

119898

(17)



d119863d119873

= [Δ120591

120590119903 (1 minus 119863)]

119898

(18)


119873119891 =1

(119898 + 1)[120590119903

Δ120591]

119898

= [2120591119891Δ120591

]

119861

(19)




0 2000

15

20

25

30

35

40

45


Fatig

ue li

fe (m

illio

n cy

cles)



119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)





Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60







Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0 2000

15

20

25

30

35

40

45


Fatig

ue li

fe (m

illio

n cy

cles)



119898 = 119861

120590119903 = 2120591119891 (119861 + 1)1119861

(20)





Axial load (kN)20

20

40

60

80

100

0

Fatig

ue li

fe (m

illio

n cy

cles)

30 40 50 60







Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Radial load (kN)0

0

1

2

3

4

5

Fatig

ue li

fe (m

illio

n cy

cles)

5 10 15 20 25 30











4 Conclusion








Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Pits







Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article fatigue life analysis of rolling bearings based on quasistatic modeling ·...

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