optimum desing methodology for DGBB bearings

Simon KabusVestas Turbines Research

and Development, Motion Systems,

Vestas Wind Systems,

8200 Aarhus, Denmark

e-mail: [email protected]

Claus B. W. PedersenFE-Design,

22765 Hamburg, Germany

e-mail: [email protected]

Optimal Bearing HousingDesigning Using TopologyOptimizationThe internal load distribution in rolling bearings has a high impact on the bearing fa-tigue life. This study presents a method to optimize roller bearing housing design in orderto maximize the bearing fatigue life by ensuring an optimal internal load distribution. AnFE-model of a cylindrical roller bearing utilizing nonlinear springs in the roller model-ing is presented, which is capable of simulating the bearing load distribution efficiently.The optimal load distribution is achieved by specifying the desired internal load distribu-tion as design constraints in a topology optimization of the bearing housing design. Thesuperiority of the method is clearly demonstrated through case studies involving a cylin-drical roller bearing, where it is shown that the fatigue life is increased and the bearinghousing mass and roller contact misalignment is reduced. [DOI: 10.1115/1.4005951]

1 Introduction

The Lundberg/Palmgren rolling bearing fatigue life theoryadopted in Ref. [1] gives well defined methods for evaluating therating life of roller bearings based on the internal load distributionamong the rollers. Given a specific bearing design, the explicitrelationship allows for optimization of the internal load distribu-tion with the purpose of maximizing the bearing fatigue life. Thishas been investigated in Ref. [2] where the effect on bearing fa-tigue life of the outer ring section and clearance in a planet gearbearing is examined, in Ref. [3] where the maximum roller load isreduced by making the inner ring elliptical, and in Ref. [4] wherethe ring dimensions are optimized to maximize the fatigue life. Inthese studies, the bearing ring deflections are calculated by analyt-ical relationships and do not support an arbitrary stiffness of thesupporting structure, which can have a significant influence on theload distribution, as it will be demonstrated in this study.

In Refs. [5,6], simplified FE-models of a cylindrical roller bear-ing are presented, where each rolling element is represented by anumber of nonlinear springs representing the Hertzian contactstiffness between the roller and the raceways. The current paperemploys an equivalent model where the spring load from eachroller is used for bearing fatigue life evaluation. Topology optimi-zation is employed on the bearing housing design in order to max-imize the bearing life while reducing the housing mass. This isachieved by formulating design constraints for each roller load toensure an optimal load distribution in the bearing.

There are several reasons for applying the design force con-straints on each roller load instead of directly optimizing using thebearing fatigue life as an objective function. Firstly, the force con-straints on each roller should not yield different optimizationresults compared to directly applying the bearing life function.Secondly, to the author’s knowledge no optimization tools existsfor optimizing directly on the bearing life function using rigor-ously calculated sensitivities in topology optimization. Thirdly,using force constraints on each roller also allows one to apply amodified or different function for calculating the bearing life with-out considerable reprogramming.

It is the aim of this paper to present a new methodology forbearing housing designing. For ease of communication, the stud-ied cases involves a simple single bearing housing setup with oneload case. Except for an increased computational demand, nothingkeeps the method from being implemented on an actual bearingpedestal in a complete system taking into account more load

cases. The necessary steps in order to include multiple load casesare included.

This paper is organized with a summary of the bearing life cal-culation according to Ref. [1] followed by a subsequent deductionof the optimal load distribution in a cylindrical roller bearing withrespect to maximizing the fatigue life. Subsequently, the cylindri-cal roller bearing FE-model is presented with a description of theimplementation of load constraints in the model. A summary ofthe basic concepts of topology optimization, with emphasis onbearing housing optimization, follows before the optimal load dis-tribution and the bearing model are utilized in a topology optimi-zation where an optimal bearing housing is designed. Differenthousing designs are finally compared in two case studies, one withaligned contact and one misaligned. These studies are followed bythe concluding remarks.

2 Cylindrical Roller Bearing Fatigue Evaluation

By assuming aligned line contact in the roller/raceway contacts,the bearing fatigue calculations follow the procedure in Ref. [7]which is, given the assumptions, in accordance with Ref. [1]. It isassumed that the outer ring is stationary. The basic dynamiccapacity of a raceway contact, i.e., the load, which a roller/racewaycontact, for a 90% probability of survival, will endure for 1� 106

revolutions of a bearing ring, in a cylindrical roller bearing is

Qc;m ¼ bm � 552 � 1� cð Þ2927

16cð Þ14

� c29 � 2rrð Þ

2927 � l7

9 � Z�14 (1)

where m¼ 1 for the upper and m¼ 2 for the lower sign, whichrefers to an inner and outer raceway contact, respectively. Theconstant bm is a modification factor to accommodate for contem-porary improvements in bearing quality and is 1.1 for cylindricalroller bearings according to Refs. [7, 8].

For line contact in roller bearings, Ref. [9] determined that aquartic mean roller load is appropriate for the rotating raceway,

Qe;1 ¼1

Z

XZ

j¼1

Q4j

!14

(2)

Equivalently, the mean load on the stationary raceway is

Qe;2 ¼1

Z

XZ

j¼1

Q4ej

! 14e

(3)Contributed by the Tribology Division of ASME for publication in the JOURNAL

OF TRIBOLOGY. Manuscript received October 28, 2011; final manuscript receivedJanuary 27, 2012; published online April 11, 2012. Assoc. Editor: Xiaolan Ai.

Journal of Tribology APRIL 2012, Vol. 134 / 021102-1Copyright VC 2012 by ASME

Downloaded From: http://tribology.asmedigitalcollection.asme.org/ on 04/28/2013 Terms of Use: http://asme.org/terms

where e is the Weibull slope approximated as 9/8 according toRef. [9]. The statistical combined life of the two raceways is cal-culated from

L10 ¼Qc;1

Qe;1

� ��4e

þ Qc;2

Qe;2

� ��4e !�1

e

(4)

From a contact point of view, the validity of Eq. (4) requires thatthe rollers are at line contact and that only negligible contact mis-alignment is present.

2.1 Optimal Roller Load Distribution. The explicit rela-tionship between roller loads and bearing fatigue life in Eq. (4)allows for optimization of the internal load distribution in thebearing with respect to bearing life. This means that using a spe-cific bearing, the life is optimized by ensuring an optimal internalload distribution among the rollers. The optimization is expressedas

Maximize: L10 Qð Þsubject to:

(5)

Equilibrium:XZ

j¼1

cos hj

� �� Qj ¼ F (6)

Equilibrium:XZ

j¼1

sin hj

� �� Qj ¼ 0 (7)

where hj is the angle from the radial load to the position of rollerj. Since a pure radial load is applied, only the in plane force com-ponents are considered in the equilibrium conditions in Eqs. (6)and (7). The optimal roller loads Qopt from the optimization inEq. (5) represent the load distribution yielding the highest possiblefatigue life according to Eq. (4). This load distribution is visual-ized in Fig. 1 for F¼ 1 N and Z¼ 52 along with the equivalentload distribution of a rigid bearing support. The bearing fatiguelife of the optimal load distribution exceeds the fatigue life of therigid support by approximately 24% according to Eq. (4) despitesupporting an identical external load.

It should be noted that the optimal load distribution yields asteep load increase when the rollers enter the loaded zone, whichcan be detrimental to the bearing because of the risk of smearing.Firstly, this solution might not be obtainable, but if the finaldesign yields an undesirable steep load increase, this can bechanged by modifying the Qopt values. However, any change toQopt yields reduced fatigue life, but other bearing factors shouldbe considered, if necessary.

In Sec. 3.2.1, it is described how Qopt is used for specifyingdesign constraints in the topology optimization of the bearinghousing.

3 FE-Modeling of Bearing System

Rolling element bearings constitute a significant FE-modelingchallenge due to the high number of contacts between the rollersand raceways. In Ref. [10], a single roller/raceway contact is mod-eled in FE using 50� 200¼ 10,000 contact elements in order tomodel the contact pressure with acceptable precision. In a com-plete bearing model with 50 rollers, this mesh density correspondsto 1� 106 contact elements in addition to the solid elements in themodel. Obviously, simplifications are needed in order to allow forefficient analysis of complete rolling element bearings in FE. Thisis addressed in Ref. [11] where a cylindrical roller bearing is mod-eled in 2D using contact elements. In many applications, the bear-ing housing is inadequately approximated as a 2D structure.References [5,6] present a more general approach, where eachroller is approximated as a number of axially distributed nonlinear

springs connecting the bearing inner and outer raceway. Apartfrom the high generality of this method, it does not involve con-tact elements, which again allows for relative coarse modelingmesh, a fast solution time, and good convergence performance.Obviously, this method does not allow for detailed contact evalua-tion including roller crowning, cage interference, contact pressureevaluation, and etc., but such level of details only have limitedinfluence with respect to simulating the load supported by eachroller, necessary in this study in order to optimize the bearing fa-tigue life according to Eq. (4). It is recognized that a final fatiguelife evaluation must include the actual pressure profile on eachcontact, including crowning, misalignment, etc. and comply withRef. [1], but for optimization purposes, the importance lies in thefourth power load/life relationship.

Since the spring modeling allows for direct access to the con-tact forces used when evaluating the bearing fatigue life and thepossibility of optimizing the load distribution, as discussed in thispaper, it is indeed an appropriate approximation.

3.1 Spring Modeling Approach. Many different theoreticaland empirical expressions exists for the roller/raceway contactdeflection. See Ref. [12] for an overview of several commonlyused relationships. In a parameter study in Ref. [6], it was foundthat for the spring modeling approach, the best suitable expressionis the one specified in Ref. [13], in which the roller/raceway con-tact deflection for steel/steel contact, identical for both raceways,is

di ¼ do ¼Q

910

l8

10

� 3:84 � 10�5 (8)

where Q is in N and l in mm. Equation (8) is reordered to beexpressed by a stiffness constant,

Q ¼ l8

10

3:84 � 10�5

!109

� d109

i ¼ ki � d109

i (9)

Interpreting the roller/inner race and roller/outer race contactdeflection as two springs in series, each roller can be representedby an effective spring stiffness substituting both contact stiffness.Noticing that the inner and outer race contact stiffness are equalaccording to Eq. (8) and that centrifugal forces are neglected, i.e.,d/2¼ di¼ do, the effective contact stiffness is

kr � d109 ¼ ki � d

109

i ¼ ki �d2

� �109

) kr ¼ki

2109

(10)

In this study, three springs are used to model each roller, assketched in Fig. 2(a), because it has been found sufficient to cap-ture the load distribution in the bearing and allows for modelingof misalignment, as shown in Fig. 2(b). To justify the use of threesprings, Fig. 3 shows a convergence study using the cylindricalbearing and setup presented in Sec. 5 and the V2 bearing housingshown in Fig. 8(a). Figure 3(a) shows a load distribution

Fig. 1 Optimal and rigidly supported load distributions

021102-2 / Vol. 134, APRIL 2012 Transactions of the ASME


comparison using four different numbers of springs, where Q1 is asingle spring and Q30 is thirty springs, which is considered to befully converged and is the basis of the error calculation. The rela-tive error is only calculated for Q30> 1 kN. Fig. 3(b) shows twodifferent convergence plots for the maximum force error andUsum, which is the total deflection error of the load node in themodel. It is evident that even a single spring ensures a reasonableapproximation of the converged load distribution. Although thethree spring results are not converged, the force error is< 1%, andconsidering the high computational performance, it is consideredto be appropriate. The presented method is, however, valid forany number of springs.

It is assumed that the rollers are at line contact, which will bethe case for the majority of the loaded rollers, and as each springrepresents an equal length of the contact, l/3, the spring stiffnessis the total roller stiffness kr in Eq. (10) divided by the number ofsprings representing each roller,

kjk ¼kr

3; j 2 1;Z½ �; k 2 1; 3½ � (11)

Since rollers only transfer forces in compression, the force deflec-tion relationship for roller j spring k is

qjk ¼ kjk � djk � c=2� �10

9 if djk > c=2

0 if djk � c=2

( )(12)

The diametral clearance c is assumed zero in the rest of the paper,but the procedure is general and will not require modifications forc = 0.

The relative raceway misalignment, see Fig. 2(b), at the rollerposition is derived from the spring deflections

aj ¼ tan�1 dj3 � dj1

� �� 3

2 � l

� �(13)

It should be noted that the analytical contact stiffness in Eq. (8)takes the compression of the raceway material into consideration.This introduces a potential redundancy since the stiffness of theraceways is also included in the solid elements of the FE-model.To eliminate this error, it is thus recommended to use a relativecoarse meshing of the raceways in order to limit the local racewaycompression at the spring attachment nodes and/or increase thespring stiffness in Eq. (12) to compensate for the raceway flexibil-ity. This modification can be established by applying unit loads onthe spring attachment nodes of the raceways in an FE-model onlyconsisting of the bearing rings, and in this way find the redundantlinear stiffness of the raceway attachment nodes. The final springstiffness is now found by considering the FE-spring to be the mid-dle of three springs in series: the inner raceway flexibility, the FE-spring, and the outer raceway flexibility.

3.2 Spring Deflection Constraints. In order for the topologyoptimization to yield a load distribution that complies with the op-timum load distribution calculated in Eq. (5), the maximumallowed load on each roller is implemented as design constraintsin Sec. 3.2.1. The optimum load distribution is derived for forces,which are furthermore converted to spring deflections in order tocomply with the requirements of the topology optimization. Addi-tionally, misalignment constraints defined in Sec. 3.2.2 ensure thatthe raceway misalignments are of acceptable size.

3.2.1 Force Constraints. In most cases, the optimal load dis-tribution derived in Sec. 2.1 is impossible to match in practice,and a scaling e is added to the optimal values

Qcon;j ¼ Qopt;j � 1þ eð Þ; e � 0 (14)

The load constraint equations are formulated as the sum of theload on the springs

X3

k¼1

qjk � Qcon;j ¼ Qopt;j � 1þ eð Þ (15)

Since the rollers are assumed to be at line contact, the springs rep-resenting each roller should be uniformly loaded, and Eq. (15) canbe expressed in deflections using Eq. (12)

Fig. 2 Modeling roller j using three springs

Fig. 3 Convergence study using different number of springs

Journal of Tribology APRIL 2012, Vol. 134 / 021102-3


X3

k¼1

djk

3� Qopt;j

kr� 1þ eð Þ

� � 910

þ c

2¼ dj (16)

where dj is the reference deflection value used to define the loadconstraints in Sec. 4.1. dj is thus the limit on the mean relativeapproach of the end nodes of the three springs of roller j along therespective spring axes.

3.2.2 Misalignment Constraints. To limit the relative race-way misalignments an additional constraint is added for eachloaded roller, using the misalignment angle calculated in Eq. (13)

aj

�� amax;j (17)

tan�1 dj3 � dj1

� �� 3

2 � l

� �� amax;j (18)

dj3 � dj1

�� tan amax;j

� �� 2 � l

3¼ da;j (19)

The maximum allowed misalignment angle for roller j, amax,j,can be chosen based on bearing manufacturer recommendationsand evaluation of the life impact according to Ref. [1] by simulat-ing the contact pressure at different misalignment angles usingdedicated FE contact simulations or elastic half space simulations.Both methods are demonstrated in Ref. [10], and the latter usedfor post-processing in Sec. 5.2.1. In all cases, the robustness of thedesign is improved if the allowed misalignment angle is chosen aslow as possible without making the design domain infeasible,causing unacceptable convergence performance or yielding excessmaterial usage. The variable da;j is used to formulate misalign-

ment constraints using spring deflections.In the final case study, an identical misalignment constraint is

applied for all rollers, but as indicated using subscript j, this canbe defined individually. This option can be utilized to allow alarger misalignment on low loaded contacts for which the impacton the total bearing life is small.

4 Conceptual Bearing Housing Designing Using

Topology Optimization

Topology optimization is frequently used in the early designphase producing new design suggestions where the optimizationtargets could be to obtain desirable stiffness properties and fulfillmass constraints. Usually, topology optimization provides the abilityto yield correct decisions in the early mechanical design phase. Theaim of the present paper is not to give a full theoretical backgroundof topology optimization. A comprehensive theoretical survey of to-pology optimization is found in Ref. [14] and the references therein.The intention is to give a brief introduction to topology optimizationand, specifically, address how topology optimization can be appliedfor obtaining conceptual designs of bearing houses.

4.1 Topology Optimization for Bearing HousingDesigning. Figure 4 shows the concept behind topology optimi-zation. The elemental relative density qe of each individual struc-

tural finite element in the CAE model is defined as designvariable, which can be solid (relative density at upper limitqe¼ 1) or void (relative density at minimum limit qe¼ qmin 0).The so-called SIMP-model is applied for modeling the elementalstiffness as being proportional to a power-law of the relative ele-mental density {q}, a vector containing the design variables of alldesign elements. Thereby, the optimization formulation for theoptimal bearing housing design yields

Minimize: Global compliance u qf gð Þf g; qf gð Þsubject to:

(20)

Equilibrium: R u qf gð Þf g; qf gð Þf gl ¼ 0f gfor l 2 1;max load case½ �

(21)

Mass: m qf gð Þ � cm � mini (22)

Deflections:X3

k¼1

d u qf gð Þf g; qf gð Þj;k;l3

� dj;l

for j 2 1; Z½ �; k 2 1; 3½ �(23)

Misalignments: d u qf gð Þf g; qf gð Þj;3;l � d u qf gð Þf g; qf gð Þj;1;l��

� da;j;l (24)

As an objective in Eq. (20), we want to maximize the stiffness.However, instead the total compliance is minimized, which isequivalent to maximizing the total stiffness. The force equilibrium{R}l in Eq. (21) of the entire bearing housing for the given loadcase l is solved using an implicit finite element solver [15] solvingdirectly for the displacements {u} of the entire bearing housing.The total mass of the bearing housing is constrained to fulfill acertain weight target described in Eq. (22), where mini is the massof the initial design space. As earlier described, the aim is toobtain an increased fatigue life of the bearing by designing againstan optimal roller deflection distribution. Thus, the roller deflec-tions dj,l are constraint against the constraint roller deflection dis-tribution dj;l from Eq. (23).

In general, more than one load case will be considered for abearing housing optimization, and each load case can be weightedin the objective function, as a starting point according to the per-centage of use at that condition. The roller load and misalignmentconstraints are more complicated to define since the optimum loaddistribution cannot be achieved for all load cases. A general rec-ommendation is to define the closest to optimum constraint valuesfor load cases representing the majority of use. For the remainingload cases, focus should initially be on reducing misalignmentssince contact misalignments in general are more severe than anon-optimum load distribution. After the topology optimizationthe results can be evaluated and if necessary the constraint valuescan be adjusted and the optimization rerun.

4.2 Adding Topology Optimization in Existing CAE andCAD Design Processes. The optimization formulation inEqs. (20)–(24) is solved using an iterative design process scheme

Fig. 4 Schematic overview of topology optimization workflow



as shown in the workflow of Fig. 5. In the final case studies, theiterative design process requires approximately 50 automatedoptimization runs. The iterative design process scheme can beimplemented in a predefined CAE and CAD workflow as in Ref.[16]. In this study, Refs. [15] and [17] have been applied.

A finite element design space model is created for the topologyoptimization as shown in Fig. 4(a). Afterwards, the design spacemodel is applied in an iterative design process (see Fig. 5). Foreach design iteration the nonlinear equilibrium of the finite ele-ment model are determined using Ref. [15]. Afterwards, the sensi-tivities with respect to the elemental relative densities arerigorously computed for the objective function and for the con-straints by means of the adjoint method (see, e.g., Refs.[14,16,18]). Then, the elemental relative densities are updatedusing mathematical programming, and a new finite element modelis generated. If the optimization has not converged, a new optimi-zation cycle is started.

When the optimization has converged, then the new conceptualdesign for the bearing housing is still a finite element with non-smooth surfaces as show in Fig. 4(b). An automated smoothingprocess is applied, which smoothes the surfaces but keeps the top-ological layout as shown in Fig. 4(c). A new finite element modelis created for verification of the smoothed design. The smoothedoptimization result is transferred into a final CAD model if theverification model shows to fulfill the desired design targets. Allthe previous optimization steps illustrated in Fig. 5 are integratedin the software package TOSCA Structure [16].

5 Applying Topology Optimization on Bearing

Housing Design

The bearing housing topology optimization is demonstrated inan example using a type N28/1400 cylindrical roller bearing withbasic data shown in Table 1.

For ease of communication, the bearing is mounted in a simplebearing housing with a uniform thickness equal to the bearingwidth of 175 mm, shown in Fig. 6. For the finite element modelingnonlinear spring elements are applied, and the continuum modelconsists of eight-node hexahedral elements. The bearing outerring is connected to the bearing housing elements using bondedcontact formulation. Numerical tests have proven that the bonded

contact has no impact on the load distribution compared to match-ing the meshes at the interface and merging the coincident nodesor applying different types of frictional contacts. In all cases, it isassumed that the bearing rings are properly mounted withoutclearance, misalignment, etc.

In the following aligned study, the housing is meshed usingseven elements in the depth and 35,000 solid elements in total. Inthe misalignment study 15 elements are used in the depth and64,000 solid elements in total.

All FE-nodes on the bearing feet bottom surface are fixed in alldegrees of freedom, and the bearing inner ring is loaded by a ver-tical force, F¼ 1 MN, applied at a load node connected to theinner ring in a rigid MPC element spider web. Except in the verti-cal direction, the load node is fixed in all degrees of freedom. Theshaft can easily be included, but this has no principal influence onthe method discussed and only minor influence on the overall loaddistribution, and is thus excluded from the model. The used bear-ing housing material properties are Eh¼ 170 GPa, �h¼ 0.29 andqh¼ 7000 kg/m3 corresponding to cast iron, and the bearing ringsteel has Eb¼ 200 GPa and �b¼ 0.3.

The design space of the topology optimization is highlightedwith light colored elements in the V1 design while the upper sup-porting structure and the feet are excluded. Additionally the mate-rial located � 4 � rr beneath the bearing mounting surface is alsoexcluded in order to improve the robustness of the design by mak-ing the load distribution less sensible to variations in the designdomain and ensure that the roller load at intermediate positions,between the locations in the FE-model, are comparable.

5.1 Aligned Contact Study. A total of five different resultsare compared in a model with perfect alignment constrainedbetween the bearing and housing:

• rigid support/analytical distribution• V1 - solid housing (cm¼ 1)• V2 - arch housing• V3 - only mass constraint (cm¼ 0.3)• V4 - mass and load constraints (cm¼ 0.3, e¼ 0.05)

First, a rigid support is included to allow for comparison ofload distributions with the analytical one not taking the bearing

Fig. 5 Topological iterative design process in a present CAEand CAD workflow using TOSCA [16]

Table 1 Basic Data of the N28/1400 Cylindrical Roller Bearing

d[mm] D[mm] B[mm] Z rr[mm] rp[mm] l[mm]

1400 1700 175 52 40 778.5 105

Fig. 6 Initial V1 design with general model descriptions



housing flexibility into consideration. V1 is an analysis of the com-plete design volume, and V2 is an intuitive, geometrically simple,and lighter design, shown in Fig. 8(a). To underline the impor-tance of including roller load constraints in the optimization twooptimizations are included: V3 without load constraints and V4

including. Due to the boundary conditions, only very limited mis-alignment is present and the misalignment constraints in Eq. (19)are thus not considered in the aligned study. A remaining massfraction of cm¼ 0.3 is used in both optimizations, and in V4 a tol-erance of e¼ 0.05 is used since it has been found to be close tothe minimum obtainable limit.

The total roller load distributions of the considered designs areshown in Fig. 7, where each marker represents one roller. Figures8(b)–8(e) show the optimized relative elemental stiffness distribu-tion and smoothed V3 and V4 design, corresponding to Figs. 4(b)and 4(c). The convergence of the objective function and con-straints of the V4 design are shown in Fig. 8(f) where the horizon-tal lines indicate the constraint values. Due to symmetry only halfof the roller positions are included.

From the comparison, it is obvious that the supporting structurefor the studied cylindrical roller bearing has a significant influenceon the load distribution in the bearing. Even the solid V1 designhas a significant deviation from the rigid housing. The high flexi-bility in the center region of the V2 and V3 design clearly demon-strate the potential of applying the presented topologyoptimization method, where the soft spot across the feet gap canbe compensated, as shown for the V4 design.

Besides the load distribution of each design, Fig. 7 additionallyshows three different load distributions, namely the optimal loaddistribution Qopt according to Eq. (5), the force constraints Qcon

according to Eq. (15), the load distribution of the final optimiza-tion iteration Qfinal, which complies with all constraints, and therealized load distribution after the automated smooth processQsmooth. It should be noted that different smooth options are avail-able, and some iterations were needed in order to obtain the excel-lent correlation shown. The lower plot shows a verification of thefinal smoothed design where the bearing is additionally simulatedwith a rotation of 180�/Z to verify the roller load at the intermediatepositions. This plot shows that the maximum error between theroller loads and the constraints are <þ 2%. Furthermore, it is seenthat the deviation on forces at intermediate positions is negligible,which is due to the non-design space below the bearing mountingsurface distributing the forces from the supporting trusses.

Fig. 7 Load distribution evaluation

Fig. 8 Considered bearing housings in aligned study



Table 2 gives an overview of the fatigue life according toEq. (4) calculated for the optimal load distribution as well as eachof the five designs. dr is the radial deflection measured at the loadnode to give an impression of the overall stiffness influence of thebearing housing design.

This evaluation clearly demonstrates the potential of the pre-sented method for automated topology optimization of bearinghousings. Despite the simplicity of the considered problem, itwould have required numerous manual iterations to get even closeto the desired load distribution, which ensures a bearing fatiguelife superior to any of the considered designs.

Simultaneously, an optimum material usage is ensured.

5.2 Misaligned Contact Study. In the second study, a mis-alignment on the inner bearing ring is imposed by rotating theinner ring about the horizontal axis causing maximum roller mis-alignment angle at w = 180�. A misalignment of 1 mrad is appliedand the V5 design is optimized using the same constraint values asfor V4 but also including misalignment constraints:

• V5 - mass, load, and misalignment constraints (cm¼ 0.3,e¼ 0.05, amax¼ 0.02 mrad).

The chosen misalignment constraints ensure that only 1/50 ofthe externally applied misalignment is allowed as relative mis-alignment between the raceways at the roller positions.

Figure 9 shows the relative elemental stiffness distribution andsmoothed V5 design.

Figures 10(a) and 10(b) show the convergence of the objectiveand constraints of which all are complied with in the final optimi-zation iteration. Figure 10(c) shows the final load distributionincluding results of the V4 design in order to show the impact of

Table 2 Bearing Housing Performance Comparison

Design L10 L10/L10(Qopt) mh[kg] dr[mm]

Qopt 4786 1 – –Rigid 3881 0.81 – 0.047V1 3325 0.69 2769 0.096V2 3517 0.73 1977 0.110V3 2376 0.50 1438 0.150V4 4295 0.90 1438 0.158

Fig. 9 V5 bearing housing for misaligned study

Fig. 10 Optimization convergence and evaluation of load distributions and misalignments



the misalignment, which is quite significant compared to the rela-tively small misalignment imposed on the inner ring. Figure 10(d)shows the contact misalignment of all the designs presented inSec. 5.1, the final optimization iteration, and the realized resultsafter smoothening. Two different smoothed results are shown; oneusing 1st order 4-node tetrahedral elements and one using 2ndorder 10-node elements in the remesh after smoothening. Com-pared to the final optimization iteration, the 4-node model is stifferand thus results in larger misalignments, and the 10-node model ismore flexible and thus results in smaller misalignments. This rela-tively big difference is due to the fact that the misalignment con-straints are very tight and the realized misalignment is, seen from acontact point of view, negligible in practice, and the improvementscompared to V1-V4 are evident. The difference on the load distribu-tion in the bearing is, however, not significantly influenced by thechoice of element type, and thus the results are not illustrated.

The main differences compared to the V4 design are a change inthe truss design and that these are supporting the bearing in an off-set from the housing center plane in order to ensure that the bear-ing housing angular deflections match the imposed misalignmenton the bearing inner ring. The impact of this change is highlightedin Fig. 11 where the deflection of the V4 and the two V5 housingdesigns are compared, and the superiority of the V5 design againstthe imposed misalignment is evident.

5.2.1 Life Evaluation of Misaligned Contact Study. The va-lidity of the fatigue life calculation in Eq. (4) requires all rollers tobe at line contact with zero misalignment. To allow for a fatiguelife evaluation of the misaligned contacts, the more detailed pro-cedure in Ref. [1], taking the actual contact pressure along theroller/raceway contacts into consideration, should be followed.

Since the micro geometry is unknown, a full logarithmic rollercrowning profile as proposed by Ref. [19] is used with an assumeddesign load Qd¼ 150 kN, while the raceways are assumed cylin-drical. The crowning profile is thus

Cp xð Þ ¼ 2 � Qd � 1� v2ð Þp � l � Eb

� ln 1

1� 2 � xl

� �2

0B@

1CA (25)

Where Cp is the crowning profile, x is the coordinate along theroller length, zero at center, and Eb¼ 200 GPa and �¼ 0.3 areelastic properties of the bearing. Obviously the precision of thepressure calculations and the calculated fatigue life greatly dependon the geometry, but for the purpose of illustration it is sufficient.

From each FE-analysis, the roller load Qj and tilt angle aj areused to evaluate each roller/raceway contact (aj being split oninner and outer raceway contact) using an elastic half space simu-lation as proposed in Ref. [1] and demonstrated in, e.g., Refs. [10,20], which is efficiently able to calculate non-Hertzian contactpressures including roller edge effects, misalignment, etc. For anillustration of the calculated pressures, Fig. 12(a) shows the high-est loaded inner raceway contact of the V3 and V5 design, the for-ward nearly symmetric distribution being V5. The highest pressurealong the roller/raceway contact is extracted and Fig. 12(b) showsthe distribution for the highest loaded roller in each design, wherethe pressure peak is enlarged on the right picture. Figure 12(c)shows the unwrapped raceway pressure of the loaded zone of twodesigns, where the intermediate space between the rollers, visual-ized with vertical lines, is filled using linear interpolation of thecontact pressures. For each design, the bearing fatigue life is nowcalculated, see Ref. [21] for a comprehensive explanation of theprocedure, according to Ref. [1] with the results shown in Table 3.From this comparison, the superiority of the V5 design is evidentwith a life ratio ranging from approximately 2-7 compared to theother design. As discussed previously, the actual fatigue life hashigh dependence on the crowning profile, but it should be notedthat the life in Table 3 is inversely dependent to the misalignmentsshown in Fig. 12, except for V2/V3, which is due to the generalbetter performance of V2.

Fig. 11 Deflection sum comparison of V4 and V5 bearing crosssections at 3 300 magnification with misalignment indicationsat bottom roller position. Scale in mm

Fig. 12 Evaluation of non-Hertzian contact pressures for misaligned contact study

Table 3 Bearing Housing Performance Comparison

Design L10 L10/L10(Qopt) mh[kg] dr[mm]

Qopt 4786 1 – –Rigid 632 0.13 – 0.047V1 869 0.18 2769 0.096V2 1143 0.24 1977 0.110V3 1085 0.23 1438 0.150V4 2134 0.45 1438 0.158V5 4219 0.88 1438 0.165



6 Conclusions

The load distribution in rolling element bearings is highly influ-enced by the stiffness of the supporting structure. As demonstratedin this paper, this influence can yield an inferior as well as supe-rior bearing fatigue life compared to a perfectly rigid support,depending on the actual load distribution. Optimizing a bearinghousing is a complex challenge due to several factors. First of all,the bearing modeling is a complicated task, interpreting the FE-results is difficult, and finally the conventional workflow involv-ing CAD, FE-meshing, bearing modeling, simulation, result inves-tigation, and interpretation is highly inconvenient. This is furthercomplicated if different people are involved in the different steps.

This paper presents a new method for bearing housing design-ing where topology optimization is utilized to ensure an optimalhousing material usage and optimal load distribution. It is shownhow the optimal roller loads are explicitly included in the topol-ogy optimization since the optimization will otherwise yield anunfavorable load distribution. Besides ensuring an optimal loaddistribution, it is also demonstrated how contact misalignment canbe minimized using design constraints. This is a huge benefit sinceeven small contact misalignments can cause roller edge loads,which severely reduces the bearing fatigue life.

The final studies show the potential of this new method whereit is demonstrated that the topology optimization can be used in abearing housing design to yield optimal material usage, load dis-tribution, and minimized contact misalignment. That shaft angularmisalignments due to, e.g., shaft bending can be compensated forthrough housing design can yield a significant improvement inbearing fatigue life or have the additional benefit that it can poten-tially allow for usage of a cylindrical roller bearing in applicationsthat initially are disqualified due to excessive shaft misalignments.

The present study features a simple cylindrical roller bearinghousing due to simplicity of communication, but the presentedmethod can be implemented equivalently on complete systemsincluding more load cases and on other types of rolling elementbearings.

Nomenclaturebm ¼ rating factorB ¼ bearing widthc ¼ diametral clearance

Cp ¼ crowning profiled ¼ bearing bore diameterD ¼ bearing outer diametere ¼ Weibull slopeE ¼ Young’s modulus of elasticityF ¼ radial forcek ¼ stiffnessl ¼ effective roller contact length

L10 ¼ basic rating life, in million revolutionsm ¼ massq ¼ spring loadQ ¼ roller load

Qc ¼ basic dynamic capacity of a raceway contactQd ¼ crowning profile design loadQe ¼ equivalent roller load

r ¼ radiusu ¼ displacementsx ¼ coordinate along rollerZ ¼ total number of rollers in a bearinga ¼ relative misalignment between inner and outer racewayc ¼ auxiliary parameter, rr/rp

cm ¼ mass reduction fractiond ¼ total deflection of both raceway contacts of a rollere ¼ optimization constraint tolerance� ¼ Poisson’s ratio

hj ¼ angle from resultant radial force to roller jq ¼ density

qe ¼ relative elemental densityW ¼ bearing angular position

Subscriptsb ¼ bearingh ¼ housingi ¼ inner ring/racewayj ¼ roller indexk ¼ spring indexl ¼ load case index

m ¼ raceway indexo ¼ outer ring/racewayp ¼ pitch circler ¼ roller

con ¼ constraint valueini ¼ initial value

max ¼ maximum valueopt ¼ optimal value

Superscript* ¼ reference value for constraints

References[1] International Organization for Standardization, 2008, “ISO/TS 16281:2008

Rolling Bearings—Methods for Calculating the Modified Reference RatingLife for Universally Loaded Bearings,” Standard 1, Geneva, Switzerland.

[2] Jones, A., and Harris, T., 1963, “Analysis of Rolling Element Idler Gear Bear-ing Having a Deformable Outer Race Structure,” J. Basic Eng., 85(2), pp.273–277.

[3] Harris, T., and Broschard, J., 1964, “Analysis of an Improved Planetary GearTransmission Bearing,” J. Basic Eng., 86(3), pp. 457–462.

[4] Harris, T., 1965, “Optimizing the Fatigue Life of Flexibly Mounted, RollingBearings,” Lubr. Eng, 21(10), pp. 420–428.

[5] Molnr, L., Vradi, K., Bdai, G., Zwierczyk, P., and Oroszvry, L., 2010,“Simplified Modeling for Needle Roller Bearings to Analyze EngineeringStructures by FEM,” Mech. Eng., 54(1), pp. 27–33.

[6] Golbach, H., 1999, “Integrated Non-Linear FE Module for Rolling BearingAnalysis,” Proceedings of NAFEMS World Congress ‘99 on Effective Engi-neering Analysis, NAFEMS, ed., Vol. 2.

[7] Harris, T. A., and Kotzales, M. N., 2007, Rolling Bearing Analysis - EssentialConcepts of Bearing Technology, Taylor & Francis Group, Boca Raton,Florida.

[8] International Organization for Standardization, 2007, “ISO 281:2007 RollingBearings-Dynamic Load Ratings and Rating Life,” Standard 2, Geneva,Switzerland.

[9] Lundberg, G., and Palmgren, A., 1952, “Dynamic Capacity of RollingBearings,” Acta Polytech. Mech. Eng., 2(4), pp. 5–32.

[10] Kabus, S., Hansen, M. R., and Mouritsen, O. Ø., 2011, “A New High Perform-ance Method for Evaluating Contacts in Time Domain Rolling Bearing Simu-lations,” J. Tribol., submitted.

[11] Demirhan, N., and Kanber, B., 2008, “Stress and Displacement Distributions onCylindrical Roller Bearing Rings Using FEM,” Mech. Based Des. Struct.Mach., 36(1), pp. 86–102.

[12] Teutsch, R., and Sauer, B., 2004, “An Alternative Slicing Technique to Con-sider Pressure Concentrations in Non-Hertzian Line Contacts,” J. Tribol.,126(3), pp. 436–442.

[13] Palmgren, A., 1959, Grundlagen der Walzlagertechnik, Franckh’sche Verlag-shandlung, Stuttgart, Germany.

[14] Bendsøe, M. P., and Sigmund, O., 2003, Topology Optimization: Theory, Meth-ods and Applications, Springer-Verlag, Berlin, Germany.

[15] ANSYSVR , 13.0, ANSYS, Inc., www.ansys.com.[16] TOSCAVR , 7.1, FE-Design GmbH., www.fe-design.com.[17] CreoVR , 1.0, PTC, www.ptc.com.[18] Buhl, T., Pedersen, C. B. W., and Sigmund, O., 2000, “Stiffness Design of Geo-

metrically Non-Linear Structures Using Topology Optimization,” Struct. Multi-discip. Optim., 19(2), pp. 93–104.

[19] Lundberg, G., 1939, “Elastische Beruhrung zweier Halbraume,” Forsch. Geb.Ingenieurwes., 10(5), pp. 201–211.

[20] Cretu, S., Antaluca, E., and Cretu, O., 2003, “The Study of Non-Hertzian Con-centrated Contacts by a GC-DFFT Technique,” National Tribology Conference.

[21] Kabus, S., Hansen, M. R., and Mouritsen, O.Ø., 2011, “A Cylindrical RollerBearing Model to Accurately Consider Non-Hertzian Contact Pressure in TimeDomain Simulations,” J. Tribol., submitted.



http://dx.doi.org/10.1115/1.3656576

http://dx.doi.org/10.1115/1.3653139

http://dx.doi.org/10.1080/15397730701842537

http://dx.doi.org/10.1080/15397730701842537

http://dx.doi.org/10.1115/1.1739244

www.ansys.com

www.fe-design.com

www.ptc.com

http://dx.doi.org/10.1007/s001580050089

http://dx.doi.org/10.1007/s001580050089

http://dx.doi.org/10.1007/BF02584950

http://dx.doi.org/10.1007/BF02584950

optimum desing methodology for DGBB bearings

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