Integrated Non-linear FE Modulefor Rolling Bearing Analysis

Dipl.-Ing. Hermann Golbach

INA reprint “Proceedings ofNAFEMS WORLD CONGRESS ’99 onEffective Engineering Analysis”, Volume 2Newport, Rhode Island, USA25 –28 April, 1999

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Integrated Non-linear FE Modulefor Rolling Bearing AnalysisDipl.-Ing. Hermann Golbach

Rolling bearings are commonlyused machine elements which theengineer encounters in the designof all kinds of machine systems.Due to their properties as non-linear, statically indeterminatesystems, rolling bearings placehigh demands on calculationmethods. A model for representingthe non-linear mechanicalbehaviour of rolling elements androlling bearings has been devel-oped for static finite elementanalyses and converted into theform of a user-defined element foruse in the ABAQUS/Standardsystem. This user element deter-mines, as a kind of structural

element, the non-linear contactstiffness in the Hertzian contactarea between the rolling elementand raceway on the basis ofanalytical geometrical and elasticconsiderations. Unlike the repre-sentation of the rolling elementusing conventional continuumelements and the resultingunavoidable fine mesh, this userelement manages with the mini-mum degrees of freedom. At thesame time, the accuracy achievedis very satisfactory, as has beenindicated by a comparison with theresults of a continuum finiteelement model of the particularrolling element type. In practical

application, only the essentialgeometric parameters of the rollingelement or rolling bearing arerequired for element definition. The user element, integrated as atype of module into the FE struc-ture of a machine system, enablesrealistic analyses both of the loaddistribution of the rolling bearingunder the influence of the elasticenvironment and of the stress anddeformation of the componentsadjacent to the bearing.

Figure 1 Needle roller bearing and linear guidance system

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1 IntroductionINA Wälzlager Schaeffler oHG is aninternational manufacturer of rollingbearings and a supplier to the automotiveindustry, employing over 24 000 people.The central calculation department at thecompany headquarters in Herzogenaurach(Germany) develops calculation methodsand tools for various products, particularlyrolling bearings.Rolling bearings are some of the mostimportant basic components in machine,vehicle and plant construction. They areused wherever machine elements movingin relation to each other require mutualsupport. As their name suggests, the rolling elements roll between theraceways of the components moving inrelation to each other. Among otherthings, rolling bearings are differentiatedaccording to the type of movementrelations in rotating and linear motionbearings and the shape of the rollingelement in ball and cylindrical rollerbearings. Figure 1 shows two examplesfrom INA’s extensive range of products: a rotating bearing with needle rollers asthe rolling elements and a linear motionbearing with balls as the rolling elements.

The external load is transmitted from oneraceway to the other, distributed overseveral rolling elements. There is a veryhigh local stress created at the contactpoint between the rolling elements andthe raceways, which occurs cyclicallywhen the bearing is in motion. In order toensure that this extreme stress can beendured reliably and for the requiredfatigue life, a rolling bearing made fromexcellent quality, designed under consid-eration of various environmental condi-tions, is essential. Analysis of the loaddistribution of the bearing is a key task,and is the first step in determining thefatigue life of the bearing.

2 Calculation of the loaddistribution in rolling bearings

Even the best rolling bearing will fail if itssystem behaviour is not adapted to themachine into which it is integrated. It istherefore necessary to analyse anddetermine in advance the intensive inter-actions between bearings and adjacentcomponents. For example, in a systemcomprising of bearings, shaft and hous-ing, the bending of the shaft and thedeformation of the housing influence the

bearing reactions and therefore the loaddistribution within the bearing. In turn thestiffness of the bearing influences theelastic line of the shaft (Figure 2).In mechanical terms, the system compo-nents, bearing, shaft and housing,represent spring elements which form a statically indeterminate spring system.The bearing, which generally has severalload-transmitting rolling elements, is itselfalso a high-grade statically indeterminatespring system, which is also characterisedby the strongly non-linear spring behaviourof the rolling elements:• rolling elements behave in a unidirec-

tional fashion, i. e. they only transmitcompressive and not tensile forces

• the compressive load-deflectionrelationship within the rolling contact is non-linear

• the behaviour of ball bearings(compared to roller bearings) is morecomplicated due to the fact the point-shaped contact areas may shift underload and the direction of force trans-mission also changes under load.

Figure 2 Deformed system comprising of bearings, shaft and housing

Housing:– undeformed– deformed

External force

Load reactions

Ball bearing Shaft Roller bearing Contact pressure

In the past, the calculation methods forthese demanding non-linear, staticallyindeterminate mechanisms were devel-oped and improved in stages. They wereusually based on analytical foundations[1], [2]. At INA, these approaches havebeen converted into a sophisticatedcalculation program for rolling bearingdesign known as BEARINX®. In the case ofcomplex machine systems, BEARINX®

determines the equilibrium of the staticallyindeterminate spring systems in a closedcalculation (up to the equilibrium of asingle rolling element), taking preciselyinto account the interaction between theindividual components. However, if thegeometrical structure of the bearingsupport is complicated, BEARINX® mustassume that the bearing rings are rigidlysupported for simplification purposes. Inthe vast majority of cases, this assumptionis sufficient, but due to the increasingtrend towards lightweight designs (in theautomotive industry in particular), it isbecoming more necessary to take thedeflection of the bearing rings intoaccount from the beginning. In suchcases, the load distribution of the bearingmay only be calculated accurately on thebasis of numerical approaches, such asthe finite-element method (FEM). In the finite-element analysis (FEA) ofrolling bearings, the modelling of therolling contact is a core problem. A modelusing conventional continuum elementsvery quickly reaches the limits of thecomputing power of today's computers.In order to represent the curvature of the

rolling element, the non-linearity of theHertzian contact with the changing extentof the contact surface and the localstresses and deformations, very finemeshing is required, as can be seen inFigure 3 with the typical stress anddeformation state in the linear contactwith a cylindrical roller.The FEA of one or more rolling bearingsintegrated into a complex machinesystem using continuum elements forrolling element representation can there-fore no longer be realised with justifiableexpenditure. It is necessary to find amodel of the rolling element to representbehaviour with drastically reduceddegrees of freedom.

3 Model for representingrolling elements and rolling bearings

When developing a mathematical modelfor the description of the behaviour of therolling element, the mechanical charac-teristics of the rolling element has certainadvantages: it only transmits normalcompressive forces, and these aredistributed over a very small area incomparison with the usual geometricdimensions. One-dimensional springstiffnesses can therefore be used todescribe the local non-linear contactdeflection. However, existing springelements are not sufficiently suited, as will be shown later. The user elementsprovided for users by ABAQUS [3] offer amore suitable option for describing rolling

element behaviour. As the user elementsalso include the proportion of racewaysat the contact deflection, the localdeformation is independent from theoverall raceway deflection and thebearing rings can be modelled usingconventional elements.

3.1 Fundamentals of the user-defined element (ABAQUS)

The mechanical behaviour of theABAQUS user elements, which isgenerally non-linear, must be defined in a user subroutine. When calling thesubroutine, ABAQUS provides thesolution dependent nodal variables uM,the nodal coordinates and the elementproperties to be defined in the input fileas the essential information. Using thisinformation, the contribution of theelement at the residual vector of thestructure FN is defined in the subroutine:

and the contribution of the element at theJacobian matrix KNM:

Once the mechanical behaviour has beencoded in the user subroutine, only thenodes and the element properties needto be defined in the practical applicationof the user element as for conventionalelements.

KNM = – δFN(2)δuM

FN = FN (uM, Coord., Prop.) (1)

Figure 3 Stress and deformation state in linear contact

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Roller (undeformed)drawn)

Half space(deformed drawn)

F: Roller forceD: Roller diameterb: Semi-contact-width

b = b (F),b << D

Contact pressure

Lines of equal Von Mises Stress

3.2 Model of the ball

3.2.1 Geometric relationsAlthough ball bearings appear to besimple machine elements, their internalgeometry and therefore their motion andstress relations are quite complex. The most important geometric parametersare shown in Figure 4, with a profilerepresentation of an angular-contact ball bearing (under thrust load).Two of these parameters are the balldiameter D and the pitch diameter dm, on which the balls rotate around thebearing axis. The raceway groovecurvature radii of the inner and outer ring,ri and ro, are slightly larger than the radiusof the ball. The difference in curvature,which has been greatly magnified inFigure 4 for illustration purposes, ischaracterised by the osculation κ:

or

In the zero clearance and unloaded state,the ball makes contact with the inner andouter raceway at a single point. Underloading, the contact point increases to anelliptical contact surface. Under staticloading (without considering centrifugalforces), the connection line of the contact

κo = ro / D (> 0.5) (4)

κ i = ri / D (> 0.5) (3)

points or the centres of the contact sur-faces passes through the ball centre Mwand the centres of curvature of theraceway grooves of the inner and outerring, Mi and Mo. The external load istransmitted from one bearing ring to theother along this line in the form of acompressive force. The angle betweenthis line and the radial plane is called thecontact angle α of the bearing. Thedistance a between the curvature centresis the criterion for whether the ball isloaded or unloaded. In the zero clearanceand unloaded state, it is exactly a0:

In the loaded state, an elastic deflection δi or δo arises in the contact points fromthe ball to the raceway. The distance ofthe curvature centres from the racewayradii a is now greater than in theunloaded state. The relation between the sum of the two deflections and thedistances a and a0 is as follows:

3.2.2 Elasticity relationsFrom a purely geometric point of view,the direction of the ball force can bederived from the positions of the bearingrings in relation to each other. The theoryof elasticity is required to determine theextent of the force. The classical solution

δ = δi + δo = a – a0 (6)

a0 = ri + ro – D = (κ i + κo – 1) · D (5)

for the local stress and deformation oftwo non-conforming elastic bodies whichare pressed together was established byHertz [5]. For steel bodies, the mutualapproach of both bodies δ (in mm) at thecontact point is dependent on the force F(in N):

with ∑ρ : sum of curvatures in the principal curvature planes

δ* : Hertzian parameter (function of curvatures to be taken fromtable work)

3.2.3 User element ballThe position of the curvature centres ofthe raceway grooves and therefore theball force F and the contact angle α canbe determined from the position of twopoints located on the raceway grooves,provided that curvature is constant. To represent the ball behaviour, a 4-nodeelement with two nodes on each of theraceway grooves of the inner or outer ringis defined (Figure 5). Each of the nodeshas three translatory degrees of freedom(in the global coordinate system). In addition to the geometric parametersdescribed above, the user must alsodefine the bearing axis as an elementproperty. The ball force is divided intotwo statically-equivalent proportions onthe two nodes of each raceway groove.

δ = 2.79 · 10–4 · δ* · 3√∑ρ · F2 (7)

Figure 4 Geometric parameters of the unloaded and loaded ball Figure 5 User element ball

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3.2.4 VerificationFor verification of the user element ball, a very finely-meshed continuum FEmodel of a sector of a ball bearing wasgenerated. For loading, the inner andouter rings of the bearing are displacedaxially in relation to each other. The ballforce and the contact angle are indicatedby the axial displacement. A uniquematching of the characteristic curves ofthe user element and continuum FEmodel can be recognised (Figure 6).

3.2.5 Limits of the modelApplication of the user element is onlypermitted if the requirement of a constantraceway groove curvature is met. This requirement is not met in the case of thin-walled, insufficiently supportedbearing rings.

3.3 Model of the rollerThe following explanations are limited tothe special case of cylindrical rollingelements that are loaded in the trans-verse direction only, and which are notsubject to axial forces.

3.3.1 Geometric relationsDue the cylindrical shape of the rollers,the contact angle of radial bearingsremains constant at 0°. This considerablysimplifies the geometric description of theroller bearing in comparison to the ballbearing, particularly under load. Figure 7illustrates the most important rollerbearing parameters for determining theload distribution.The effective roller length leff is equal tothe total length lw minus the two edgeradii R. Although the rollers appear to beperfectly cylindrical macroscopically(except for the edge radii), the contour isprofiled slightly (on the order of a fewµms) to avoid edge contact pressure on the roller which would considerably

reduce the fatigue life of the bearing. The profile of real rollers can usually beapproximated by special functionsdependent on a small number ofparameters. Using the example of anuntilted loaded roller, Figure 8 illustratesthe effect profiling has on the contactpressure near the roller edge.

From the radial displacement δrad and thetilting ψ of the bearing rings towards eachother, it is possible to specify thedeflection of the roller δ depending on the axial coordinate x as follows, takinginto account the bearing clearance sradand the profiling p(x):

δ(x) = δrad – srad / 2 – ψ · x – 2 · p(x) (8)

Figure 6 Continuum FE model of the ball and comparison of results

Figure 7 Geometric parameters of the unloaded and loaded roller

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Cont. mod. ball force

User El. ball force

Cont. mod. contact angle

User El. contact angle

Axial displacement [mm]

Bal

l fo

rce

F [k

N]

Co

ntac

t an

gle

α[°

]

0 0,04 0,08 0,12 0,16

8

6

4

2

0

40

30

20

10

0

Bild 8 Roller profile and influence on loaddistribution

Pro

file

dep

th [

µm]

Roller length

50

40

30

20

10

0

–leff/2 0 leff/2

Pre

ssur

e [N

/mm

2 ]

Roller length

Profiled roller

Unprofiled roller

5 000

4 000

3 000

2 000

1 000

0–leff/2 0 leff/2

3.3.2 Elasticity relationsThe literature provides different load-deflection relationships for the mutualapproach δ (in mm) of a cylindrical rollerof the finite length l (in mm) that ispressed against the infinite half spacewith the force F (in N). Some of these arespecified on an analytical basis, andothers are specified on an empiricalbasis. In an extensive parameter studyusing the FE model of an (unprofiled)roller of different lengths and diametersintroduced in Section 3.3.4, the load-deflection relation specified in [2] provedto be the most suitable:

3.3.3 User element rollerThe roller is discretized into a limitednumber of n laminae parallel to the radialplane of the bearing (Figure 9). Each ofthese has 1/n of the contact stiffness ofthe whole roller and is connected withone node to the raceway of the inner andouter ring. The total number of nodes on

δ = 3.84 · 10–5 · F0.9(9)l0.8

the user element roller is therefore 2n.The deflection of the laminae can bedetermined from the radial displacementsof the related nodes (taking into accountradial clearance and profiling). Each of thelaminae can only transmit normal forcesin the direction of their original connectionline, i. e. in the local coordinate system ofthe roller, the nodes in the directions xand z are free of forces. The load-transmitting length l that influences thedeflection relation (8) is determined fromthe number of loaded laminae.

In cases where the roller shows anunsymmetrical pressure distribution ofthe two line contacts with regard to itsaxis (due, for example, to an additionalprofiling of one of the two raceways), the model must be modified slightly: thelaminae which are independent of eachother must be interlinked on the middleline by means of beam elements in orderto represent both contacts of a laminaseparately. Because of the highlyincreased computational effort use of thesecond model is only recommendedwhen absolutely necessary.

3.3.4 VerificationTo verify the user element of the roller, a half-symmetrical continuum FE modelof the roller and of the infinite half spacewas created and loaded in a tilted way.The load was deliberately selected to beso extremely high that the profile was nolonger sufficient to avoid edge contactpressure. With the exception of the edgecontact pressure, the pressures are wellmatched (Figure 10).

3.3.5 Limits of the modelThe above comparison calculation showsthe limits of the laminae model: any edgecontact pressures that occur cannot bedetermined directly. However, as theseare limited to a very small area, theeffects on the global load distributionwithin the bearing remain insignificant.Together with the roller force andmoment which were sufficiently preciselydetermined in the analysis of the wholebearing, the continuum FE model of asingle rolling element can then be used toexamine rolling elements which could becritically loaded in relation to edgecontact pressure.

Figure 9 User element roller (two variants)

Figure 10 Continuum FE model of roller and comparison of results

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Model for symmetricalpressure distribution

User elementBeam elements

Model for unsymmetricalpressure distribution

Pre

ssur

e [N

/mm

2 ]

Roller length

User ElementContinuum model

8 000

6 000

4 000

2 000

0–leff/2 0 leff/2

3.4 Entire ball and roller bearingIn the analysis of shaft systems withmultiple bearings, there may be only onebearing which is of particular interest.Also, due to the thick-walled design ofthe bearing's support structure, it may bepossible to neglect local deformations ofthat structure. For the efficient bearingmodelling of such cases, two userelements were developed to representthe behaviour of an entire ball or rollerbearing, assuming that the bearing ringsare rigid. To describe the positions ofboth bearing rings, a middle node with 3

translatorial and 3 rotational degrees offreedom were introduced for each ring. In the FE model, these are then coupledto the bearing support in the housing and the shaft using enforced constraints. After carrying out a few coordinatetransformations, it is possible to deter-mine the load distribution in the entirebearing from the displacements androtations of both nodes of the bearingelement using the relations (3) to (8)outlined above. In order to describe theelement properties, specificationsconcerning the number of rows i, number

of rolling elements per row z and thepitch distance between the rows t arerequired in addition to the geometricparameters of a single rolling element.

3.5 Overview of elements generated

Table 1 provides an overview of the fouruser elements developed, while Table 2summarises the element propertysymbols used.

4 Evaluation of the loaddistribution: the bearing fatigue life

Out of the load distribution of the relevantbearing, analytical relations can be usedin a subsequent calculation to work outthe distribution of Hertzian pressure onthe contact surface and the subsurfacestresses [5]. In a moving bearing, thisloading of the rolling element andraceway occurs cyclically. The resultingfatigue life, which is generally limited, can also be obtained by evaluating thebearing load distribution. The new fatiguelife theory according to Ioannides andHarris [2], [4] as an expansion of theclassical Lundberg-Palmgren-theory isthe latest state of research in this field. To carry out a fatigue life calculation, the lubrication and contamination state,as well as the material properties of thebearing, are required in addition to thebearing load distribution.

Figure 11 FE model of the gear and the bearing support (mesh unposted)

Bearing support Elastic Rigid

User element Ball Roller Ball bearing Roller bearing

Number of nodes 4 2 · n 2 2

Degrees of freedom ux, uy, uz ux, uy, uz ux, uy, uz, ux, uy, uzper node ϕx, ϕy, ϕz ϕx, ϕy, ϕz

Number of degrees of 12 6 · n 12 12freedom of the element

Element properties xj, j = 1,6 xj, j = 1,6 xj, j = 1,6 xj, j = 1,6D D i idm dm z zκ i lw t tκo leff D Ds aj, j = 1,3 dm dmα0 s κ i lw

κo leffs aj, j = 1,3α0 s

Output variables F Fj, j = 1, n Fj, j = 1, (i · z) Fj, j = 1, (i · z · n)α α j, j = 1, (i · z)

Common symbols Ball/ball bearing Roller/roller bearing

xj: Axis parameters κ i: Inner osculation n: No. of laminaei: No. of rows κo: Outer osculation lw: Overall roller lengthz: No. of rolling elements α0: Nominal contact angle leff: Effective roller lengtht: pitch distance α: Operational contact angle aj: Profile parametersD: Ball/roller

diameterdm: Pitch diameters: Bearing clearanceF: Ball/lamina force

Table 2 Symbols

Table 1 Overview of the four elements

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5 Application exampleThe application of the user element“roller” and the knowledge gained by itsuse with the influence of the elasticstructural support on the bearing loaddistribution are to be illustrated using theexample of a shaft bearing of a motor car gear. The FE model of the gearbox,which is made of a light metal alloy, wasprovided by the customer and expandedto include the bearing, a drawn cup rollerbearing (Figure 11).Two essential deformation proportions ofthe bearing supporting structure ariseunder load: an ovalisation in the circum-ferential direction and a conicity in theaxial direction (Figure 12).In comparison to rigid support, thedeformation has the following effects onload distribution: due to the ovalisation of

the hub, the load zone stretches over agreater circumferential angle area, thedistribution of the roller forces is morebalanced overall and the maximum rollerforce is reduced by 23% (Figure 13). This positive influence of deformation inthe circumferential direction is opposedby the negative influence of deformationin the axial direction: the lack of supportin the frontal area of the bearing supportcauses the rollers to be loaded in a moretilted way. These two deformationinfluences almost compensate for eachother with regard to the maximumoccurring Hertzian pressure: in compari-son to the calculation with rigid support,this is reduced only slightly by 5%, while fatigue life extends by 20%. Using this calculation, it was possible toshow firstly that the hub deformationregarded as a whole in this actual

application example did not cause anyessential changes with reference to theconventional analysis assuming rigidlysupported bearing rings, and secondly,that there is still potential for optimisation(better support over the axial length). Generally speaking, it is not possible tosay in advance with certainty whether theelastic deformation of the bearing supportstructure has a significant influence onthe bearing load distribution, and there-fore on the fatigue life of the bearing, and whether this influence is positive ornegative overall. Conclusions can only bedrawn in individual cases using an FEanalysis with representation of thebearing supporting structure.

Figure 12 Deformation of the bearing supporting structure under load

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View Y

View X

Undeformed structure

View X View Y

Deformation incircumferential direction

Deformation in axial direction

Figure 13 Load distribution of the bearing

Direction of load

Elastic housingRigid housing

Distribution ofroller forces

Pressure distribution ofthe highest loaded roller

Roller length

Elastic housingRigid housing

–leff/2 0 leff/2

3 000

2 500

2 000

1 500

1 000

500

0

Pre

ssur

e [N

/mm

2 ]

1 000 N

6 ConclusionThe user element of the rolling elementdisplayed represents the mechanicalbehaviour of the contact between rollingelement and raceway with great accuracy,and simultaneously with the minimumdegrees of freedom. In comparison to amodel with conventional continuumelements, a huge amount of calculationtime is saved, and analyses of bearingload distribution in the elastic environmentare enabled using current computerpower. A more reliable design and fatiguelife prediction of the bearing where thereis significant local deformation of thebearing support structure can be carriedout using a FE analysis with the userelement. The element descriptionrequires little modelling expense incomparison to an explicit definition of thecharacteristic stiffness curves of therolling element, as can be carried outusing only a few geometric parameters. This user-friendly type of elementdefinition in the practical application alsoallows calculation engineers from non-bearing industries to integrate the user

element of the rolling bearing into theirmodels as a kind of black box, withouthaving to be familiar with the fundamentalsof rolling element behaviour. This allowsthem to make a more realistic analysis ofthe stress and deformation of theirproducts adjacent to the bearing.

The method outlined above is a consistentfurther development in the course ofintegral solutions in the CAE process andthe necessity of representing the systembehaviour of products as early as possiblein the development stages (keyword:virtual product). In the next step, it isplanned to support the user element inthe pre-processor environment also, inorder to improve handling and achievecomplete integration into the CAEprocess.

The advantages of the rolling elementmodel outlined can be summarised asfollows:

• improved quality of calculation anddesign

• saving of both manpower andcalculation time.

References[1] Eschmann/Hasbargen/Weigand:

Die Wälzlagerpraxis.Oldenbourg Verlag GmbH, 1978

[2] Harris, T. A.:Rolling Bearing Analysis. John Wiley & Sons, Inc., 1991

[3] Hibbit, Karlsson & Sorensen:ABAQUS Version 5.5Hibbit, Karlsson & Sorensen Ltd., 1995

[4] Ioannides, E./Beswick, J. M.: Moderne Wälzlagertechnik. Vogel Buchverlag, 1991

[5] Johnson, K. L.:Contact Mechanics.Cambridge University Press, 1985

About the author:Dipl.-Ing. Hermann Golbach works in thecentral CAE department of INA Wälzlager Schaeffler oHG,Herzogenaurach, Germany, specializing in complex, non-linear FE calculation.

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BEARINX®, INA’s well-know calculation pro-gram for bearing design is mentioned onpage 4 of this publication. Our qualifiedengineers always do their best to providedesigners with competent advice on-site indesigning bearing supports. You can findout more about “BEARINX®” in INA’s specialpublication “Calculation Service for RollingBearings”.

To request this publication, write to:INA Wälzlager Schaeffler oHGAbteilung IVS-MSSIndustriestraße 1–391074 Herzogenaurach (Germany)

Sac

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in G

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INA Wälzlager Schaeffler oHG

91072 Herzogenaurach (Germany)Telephone (+49 91 32) 82-0Fax (+49 91 32) 82-49 50http://www.ina.com