Mechanics amp Industry 16 604 (2015)ccopy AFM EDP Sciences 2015DOI 101051meca2015032wwwmechanics-industryorg

MechanicsampIndustry

Optimal design of paired tapered roller bearings undercentred radial and axial static loads

Eugenio Dragonia

Dipartimento di Scienze e Metodi dellrsquoIngegneria Universita degli Studi di Modena e Reggio Emilia Via Amendola 242122 Reggio Emilia Italy

Received 24 September 2014 Accepted 17 November 2014

Abstract ndash Based on the relationships from the ISO 76 standard this paper optimizes the internal dimen-sions of tapered roller bearings for maximum static load capacity A bearing system formed by two identicalbearings is assumed subjected to whatever combination of centred radial and axial forces It is shown thatthe static capacity increases linearly with the roller infill with the ratio of roller length to roller diameterand with the square of the pitch diameter of the roller set Further given the ratio of axial to radial forcean optimal contact angle exists which maximizes the static capacity of the bearing pair regardless of theactual bearing size and ratio of roller diameter to pitch diameter The optimization procedure can eitherbe used to design custom-made bearings or to pick from manufacturersrsquo catalogues the bearing with thebest contact angle for any assigned loading condition

Key words Radial rolling bearings tapered rollers static loading optimization design

1 Introduction

Ordinarily rolling bearings are not designed and builtin-house but are chosen by the designer from the cat-alogue of specialized manufactures The high degree ofspecialization has fostered the development of standard-ized high-quality products readily available off-the-shelfin a wide range of shape and dimensions at affordableprices Under particular design circumstances like verylarge bearings or tight mounting spaces the need can arisefor non-standard bearings which the regular market cansatisfy only at a considerable cost of time and money Insuch instances rolling bearings of simple geometry (aswith cylindrical or tapered rollers) can be manufacturedby the end user itself to meet the specific requirements ata fraction of the costs and delivery time requested by thespecialized suppliers

When tackling the construction of custom bearingsthe designer has the control of all the variables and thedesign is conveniently conducted according to optimiza-tion methods Unlike conventional machine elements forwhich a wealth of optimization criteria have been devel-oped since long [1 2] the category of rolling bearingshas received so far relatively little attention May be dueto the aforementioned passive design approach (selectionfrom a catalogue) towards these components until the

a Corresponding authoreugeniodragoniunimoreit

turn of the century the technical literature has been lim-ited to optimal bearing selectors [3] and simulators ofbearing kinematics [4] Papers dealing with the optimiza-tion of bearing features have appeared only lately aimedat maximizing one or several performance properties ofball bearings [5ndash11] cylindrical roller bearings [1213] andtapered roller bearings [14ndash19]

As for ball bearings Choi and Yoon [5] optimized anautomotive wheel-bearing unit with double-row angular-contact architecture using a genetic algorithm Theyshowed that the system life can be improved over thestandard design without any constraint violations Kalitaet al [6] adopted a multi-objective optimisation approachfor the design of ball bearings with enhanced dynamic ca-pacity static capacity and lubricant film thickness Thework uses both deterministic methods (penalty functions)and stochastic algorithms (simulated annealing and ge-netic search) The genetic approach was further exploredby Chakraborty et al [7] for ball bearings and its meritswere compared to conventional techniques Also based ongenetic algorithms a non-linear optimization procedurewas developed by Rao and Tiwari [8] for the design ofball bearings with maximum fatigue life under kinematicconstraints The optimized bearing yielded better fatiguelife as compared to standard catalogues This study wasfurther evolved [9] to achieve multi-objective optimizationof the bearing in terms of static load capacity fatigue lifeand elastohydrodynamic film thickness under inner and

Article published by EDP Sciences

E Dragoni Mechanics amp Industry 16 604 (2015)

Nomenclature

b Width of outer ring of bearingC0 Static radial load rating of the bearing

d Roller diameter (measured at midlength of roller)di Inside diameter of bearingdo Outside diameter of bearingdlowast

δ Optimum value of d for δ le δlim

D Pitch diameter of the roller setDlowast

δ Optimum value of D for δ le δlim

Fa Axial load acting on the most loaded bearing of the pair

Fr External radial load applied to the bearing pairFrs Radial load acting on the single bearing of the pair (=05 Fr)k Ratio of radial to axial external loads (=KaFr)Ka External axial load applied to the bearing pair

L Roller lengthLlowast

δ Optimum value of L for δ le δlim

P0 Static equivalent radial load acting on the most loaded bearing of the pairS0 Static safety factor of the bearing pair

s0 Intrinsic static safety factor of the bearing pairslowast0 Global optimum of s0

slowast0δ Optimum value of slowast0 for δ le δlim

X0 Y0 Y Load coefficients of the bearingZ Number of rollersZlowast

δ Optimum value of Z for δ le δlim

α Contact angle of bearing

αlowast Global optimum of ααlowast

δ Optimum value of α for δ le δlim

δ Pitch ratio of bearing (=dD)

δlowast Global optimum of δζ Filling ratio of the roller set (=ZdπD)λ Aspect ratio of the rollers (=LD)ξ Auxiliary variable (=δ cos α)

φ Component of the static safety factor (=ζλD2Fr)φlowast

δ Optimum value of φ for δ le δlim

outer geometric constraints Angular-contact ball bear-ings were optimized by Savsani et al [10] for maximumstatic and dynamic loading ratings and minimum filmthickness using a modified particle swarm optimizationtechnique Both single and multi-objective optimizationtasks were considered claiming advantages with respectto former mainly genetic numerical procedures Wei andChengzu [11] improved on the computational side of theproblem introducing non-dominated sorting genetic algo-rithms to optimize high-speed angular-contact ball bear-ings for fatigue life and frictional power losses

The optimization of cylindrical roller bearings wasfirst tackled by Kumar et al [12] for maximum dynamiccapacity using real-coded genetic algorithms and adopt-ing as design variables the diameter of the rollers theroller pitch diameter the roller length and the number ofrollers This work was then expanded by adding the rollercrowning (non linear profile) among the design parame-ters [13] and performing a Monte Carlo sensitivity analy-sis to investigate changes in the fatigue life of the bearingThe results showed that the multiplier of the logarithmicprofile deviation parameter has more effect on the fatiguelife as compared with other geometric parameters

A pioneering instance of tapered roller bearing op-timization can be traced back to the paper by Parkeret al [14] in which the performance of several large-bore(about 120 mm) tapered roller bearings was simulatedand tested at shaft speeds up to 20 000 rpm under com-bined thrust and radial load The computer-optimizedbearing design proved superior to equal-sized standardbearings tested for comparison Chaturbhuj et al [15] op-timized tapered roller bearings using genetic algorithmsand demonstrated that the fatigue life of the bearingimproved marginally compared with respect to standardbearings However some authors [1213] have pointed outthat some optimization constraints introduced in this pa-per were unrealistic A method for optimizing the geom-etry of tapered roller bearings at high speeds was devel-oped by Walker [16] with the main aim of determining thecup and cone angles which minimize the contact stressesunder a specific ratio of axial to radial load Walker foundthat at low speeds the optimum cup angle is 40 degreeswhereas the optimum value decreases to 10 degrees forthe highest speeds Wang et al [17] presented a mathe-matical model for optimizing the design of four-columnbearings with tapered rollers subject to several geometric

604-page 2

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 1 Longitudinal section of a two-stage bevelspur gearbox with ldquoOrdquo and ldquoXrdquo arranged tapered roller bearings susceptibleof customization andor optimization

constraints By acting on a rearrangement of the classi-cal variables (roller diameter and length pitch diameternumber of rollers and cup angle) they improved the dy-namic load rating by 22 and the life expectancy by 85over a commercial competitor of like dimensions Thoughinteresting for the potential of improvements it showsWang et alrsquos paper [17] does not provide details on theoptimization method behind the model and little can betaken away from the published results apart from thespecific example presented Two comprehensive contribu-tions to the optimal design of tapered roller bearings haverecently been published by Tiwari et al [1819] These twopapers contain also excellent reviews of the technical lit-erature on rolling bearings and the various optimizationmethods applied so far to bearing design

In general the above papers are focused on themethodological approach to the optimization problemand pay much less attention to the engineering value ofthe optimization results If most of the numerical algo-rithms referenced above can be beneficial to the special-istrsquos work they are of little use for the general-purposemechanical designer confronted with the task of design-ing simple custom bearings As outcome of an appliedresearch for a small Italian manufacturer of planetarygear drives the present author has recently published anoptimization procedure for radial cylindrical roller bear-ings [20] which overcomes these limitations Relying oneasy step-by-step calculations and with no need for spe-cific optimization backgrounds that procedure gives themacro-geometry of the bearing (roller diameter rollerlength pitch diameter of roller set number of rollers)which maximizes the static and the dynamic load ratingsunder realistic size constraints

In the wake of that fruitful effort this paper tacklesthe engineering optimization of tapered roller bearingsencountered in many industrial applications (see Fig 1for a typical instance) The optimization involves thestatic load capacity of bearing pairs mounted accord-ing to either ldquoOrdquo or ldquoXrdquo arrangements subjected to a

combination of radial and axial forces Each bearing isdefined by the number diameter and length of the rollersby the contact angle (cup angle) and by the pitch diame-ter of the roller set Following the equations provided bythe standard ISO 76 [21] the safety factor of the bearingsystem is expressed in terms of three parameters the ra-tio between roller diameter and pitch diameter the cupangle and the ratio of axial to radial force This simpleexpression gives engineering insight into the problem andshows that for each given load ratio an optimal contactangle exists which maximizes the static load capacity ofthe bearing regardless of it actual size and proportionsTables of optimal contact angles are provided for quickreference

The optimization method presented here gives its bestresults for custom-made bearings for which the design pa-rameters can be varied with the greatest freedom How-ever the results disclosed are useful also for identify-ing the best commercial bearings that can be selectedfrom the manufacturersrsquo catalogues to fit a particularapplication

2 Problem statement

Figure 2 shows the baseline configuration of the bear-ing system examined in this paper The two bearings B1

and B2 are assumed to be equal and subjected to the ra-dial force Fr which is applied at the centre point of thepair In addition to Fr an axial thrust Ka also acts onthe system through the centre shaft

Geometrically each bearing is defined by the follow-ing parameters roller length L mean roller diameter d(measured at midlength of L) number of rollers Z pitchdiameter of the roller set D contact angle α The ISO 76standard [21] specifies that this angle should measure theslope of the raceway without retaining ribs which is nor-mally the outer one (cup) as shown in Figure 2 Shouldthe retaining rib be provided by the cup the contact angleα should refer to the inner raceway (cone)

604-page 3

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 2 Reference configuration of the bearing system with applied loads (Fr and Ka) and characteristic dimensions of thebearings

With reference to Figure 2 the paper seeks the set ofbearing parameters d D L Z α which maximizes thestatic load capacity of the system for given forces Ka andFr The search for the optimum will be based on the equa-tions provided by the ISO 76 standard [21] The effect ofthe flanges is disregarded because the roller-flange con-tact though relevant for the wear damage of the bearingespecially for high-speed applications pays no role understatic loading Although obtained explicitly for the par-ticular bearing combination depicted in Figure 2 (O ar-rangement with bearings removed from each other) theoptimal solution presented will be applicable also to othercombinations such as those shown in Figure 3 (O and Xarrangements with removed or paired bearings)

3 Theory

31 Static radial load rating

Subject to the conditions clarified below the standardISO 76 [21] gives the following field-tested expression forthe static radial load rating C0 (N) of tapered rollerbearings

C0 = 44(

1 minus d

Dcosα

)Z L d cosα (1)

Equation (1) in which the lengths are expressed in mmholds true if the bearing is built and installed under thefollowing assumptions (a) use of bearing steels with hard-ness HRC ge 58 (b) manufacture according to regular tol-erances [22 23] to enhance pressure uniformity over theroller length (c) accurate guide of rollers with roundedends to avoid pressure spikes at the edges [24] (d) mount-ing onto stiff shafts and within rigid housings (e) workingtemperature not higher than 150 C (f) an angular loadzone of 180 (ie the circumferential extent of the set of

rollers that are in contact with both inner and outer race-ways) Significant deviations from these reference condi-tions can be accommodated either by applying correctionfactors available in the literature [25] or by resorting tosecond-order methods [25 26] and specialized numericaltools [27] available to bearing manufacturers The extentof the load zone f is particularly sensitive to the endplay (ie axial clearance) with which the bearing pair ismounted and to axial displacements produced by the load(see further comments in Sect 32)

Table 1 compares the static load ratings predicted byEquation (1) with the actual load ratings of a selectionof tapered roller bearings retrieved from the catalogue ofa leading manufacturer (INA) The internal geometry ofthe bearings in Table 1 (properties from α to Z) werecalculated starting from the catalogue properties (fromdi to Y0) using the method described in the Appendix

By defining the filling ratio of the bearing ζ the as-pect ratio of the rollers λ and the pitch ratio δ as follows

ζ =Zd

πD(2)

λ =L

d(3)

δ =d

D(4)

Equation (1) becomes

C0 = 44 π ζ λD2 (1 minus δ cosα) δ cosα (5)

Although the theoretical limits for the positive parame-ters ζ λ and δ are ζ le 1 λ ge 0 and δ lt 1 in practicethe following ranges are generally observed 05 le ζ le 105 le λ le 2 and δ le 02

32 Static equivalent radial load

Let Frs = 05Fr be the radial load on the single bearingof the system in Figure 2 (remember that Fr is assumed

604-page 4

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 3 Examples of bearing combinations to which the present theory is applicable X-arrangement (top) O-arrangement(bottom) not paired bearings (left) paired bearings (right)

Table 1 Comparison of predicted and catalogue static load ratings for a selection of commercial tapered roller bearings(INA [31])

Properties from INA catalogue Derived properties (see Appendix) C0 (kN)Bearing di do b Y0 α d D L Z INA Eq (1)

(mm) (mm) (mm) (minus) () (mm) (mm) (mm) (minus)30210-A 50 90 17 079 156 10 70 141 20 96 10330220-A 100 180 29 079 156 20 140 241 20 325 35230230-A 150 270 38 076 161 30 210 316 20 630 69230310-A 50 110 23 096 129 15 80 189 15 148 14930320-A 100 215 39 096 129 2875 1575 32 16 500 51930330-A 150 320 55 096 129 425 235 451 16 1 030 1 08431310-A 50 110 19 04 288 15 80 173 15 125 12631320-X 100 215 35 04 288 2875 1575 32 16 480 47631330-X 150 320 50 04 288 425 235 457 16 1 040 1 007T7FC050 50 105 22 038 301 1375 775 203 16 135 144T7FC070 70 140 27 038 301 175 105 25 17 237 242T7FC095 95 180 33 038 301 2125 1375 305 19 400 406

604-page 5

E Dragoni Mechanics amp Industry 16 604 (2015)

centred between B1 and B2) and Fa the actual axial loadon the most loaded of the two bearings (bearing B1 inFig 2) Following ISO 76 [21] the static equivalent radialload acting on the most loaded roller bearing of the pairis

P0 = MAX Frs X0Frs + Y0Fa= MAX 05Fr 05X0 Fr + Y0 Fa (6)

whereX0 = 05 Y0 = 022

cosα

sinα(7)

Equation (6) does not account for narrow load zones (seeSect 31) and uses a lower limit corresponding to a loadzone of 180 degrees (P0 = Frs) Another implicit assump-tion behind Equation (6) is the absence of end momentsacting on the shaft at the sections coupled with the bear-ings which is consistent with the assumption of rigidshaft For a flexible shaft the radial loads on the twobearings would not necessarily be the same due to interde-pendence between radial displacements tilt rotations andunequal load zones Handling of these situations wouldnecessarily call for the use of in-house tools developed bybearing manufacturers (eg [27])

For a rigid shaft under the centre radial loading inFigure 2 and assuming neither end play nor axial preloadon the system the maximum axial force Fa is given bythe following universally accepted equation [28]

Fa = Ka + 05(

Frs

Y

)= Ka + 025

Fr

Y(8)

with [28]

Y = 04cosα

sinα(9)

Using professional tools it can be shown that an axialassembly preload equal to FaFrs asymp 155 would slightlyimprove the load carrying capacity of the bearing sys-tem with respect to the assumption of no end play Bycontrast a positive end play (clearance) or the axial dis-placement induced by the load itself would dramaticallydecrease the load zone of the secondary bearing (ie thebearing which does not support directly the external ax-ial load B2 in Fig 2) so that it could become the el-ement of the pair which experiences the highest contactstresses For this reason the end play should always bestrictly controlled and positive values should be avoidedwhenever possible

Using (7)ndash(9) and introducing the load ratio k as

k =Ka

Fr(10)

Equation (6) becomes

P0 = Fr MAX05 025 + 022

(cosα

sinαk + 0625

)(11)

33 Static safety factor

ISO 76 [21] defines the static safety factor S0 as

S0 =C0

P0(12)

By means of (5) and (11) Equation (12) gives

S0 =ζλD2

Fr

44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

)(13)

which can be written as

S0 = φs0 (14)

with

φ =ζλD2

Fr(15)

and

s0 =44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

) (16)

4 Optimization

41 Free optimization

Equation (14) shows that the static safety factor S0is proportional to the functions φ and s0 From Equa-tion (15) we see that for given radial load Fr function φincreases linearly with the filling ratio ζ and the aspectratio λ and goes up quadratically with the pitch diam-eter D Function φ can be regarded as a control factorthrough which the safety factor S0 can easily be madelarge at will by increasing the pitch diameter Similarlyfunction s0 in Equation (14) can be interpreted as an in-trinsic safety factor of the bearing system obtained whenthe filling ratio the aspect ratio the pitch diameter andthe radial load assume unit value From Equation (16) wesee that s0 depends non-linearly on α δ and k as shownby the three-dimensional charts in Figure 4

Using s0 as objective function the free optimizationproblem can be stated as follows Maximize s0 (X) sub-jected to k = const (=KaFr) with X = (δ α) Max-imization of s0 implies minimization of the denominatorand maximization of the numerator in Equation (16) Forany given load ratio k the denominator of Equation (16)achieves the absolute minimum value 05 when the fol-lowing condition is met

025 + 022(cosα

sinαk + 0625

)= 05 (17)

from which the optimum contact angle αlowast is obtained as

tan αlowast =8845

k asymp 1956k (18)

604-page 6

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 4 Charts of the intrinsic safety factor s0 for a selection of load ratios k

Letting ξ = δ cosα the numerator of Equation (16) ismaximum when

ddξ

[(1 minus ξ) ξ] = 0 (19)

givingξ = δ cosα = 05 (20)

Combination of Equations (18) and (20) yields the opti-mal pitch ratio δlowast as

δlowast =05

cosarctan

(8845k

) (21)

Introducing Equations (18) and (21) into (16) gives thefollowing absolute maximum value slowast0 for the intrinsicsafety factor

slowast0 = 22 π asymp 69115 (22)

The optimal contact angles and pitch ratios given byEquations (18) and (21) are collected in Table 2 for theload ratio k ranging from 0 to 1 Figure 5 depicts theshape assumed by the optimal bearing for k = 0 k = 03and k = 06

604-page 7

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 1 Longitudinal section of a two-stage bevelspur gearbox with ldquoOrdquo and ldquoXrdquo arranged tapered roller bearings susceptibleof customization andor optimization

constraints By acting on a rearrangement of the classi-cal variables (roller diameter and length pitch diameternumber of rollers and cup angle) they improved the dy-namic load rating by 22 and the life expectancy by 85over a commercial competitor of like dimensions Thoughinteresting for the potential of improvements it showsWang et alrsquos paper [17] does not provide details on theoptimization method behind the model and little can betaken away from the published results apart from thespecific example presented Two comprehensive contribu-tions to the optimal design of tapered roller bearings haverecently been published by Tiwari et al [1819] These twopapers contain also excellent reviews of the technical lit-erature on rolling bearings and the various optimizationmethods applied so far to bearing design

In general the above papers are focused on themethodological approach to the optimization problemand pay much less attention to the engineering value ofthe optimization results If most of the numerical algo-rithms referenced above can be beneficial to the special-istrsquos work they are of little use for the general-purposemechanical designer confronted with the task of design-ing simple custom bearings As outcome of an appliedresearch for a small Italian manufacturer of planetarygear drives the present author has recently published anoptimization procedure for radial cylindrical roller bear-ings [20] which overcomes these limitations Relying oneasy step-by-step calculations and with no need for spe-cific optimization backgrounds that procedure gives themacro-geometry of the bearing (roller diameter rollerlength pitch diameter of roller set number of rollers)which maximizes the static and the dynamic load ratingsunder realistic size constraints

In the wake of that fruitful effort this paper tacklesthe engineering optimization of tapered roller bearingsencountered in many industrial applications (see Fig 1for a typical instance) The optimization involves thestatic load capacity of bearing pairs mounted accord-ing to either ldquoOrdquo or ldquoXrdquo arrangements subjected to a

combination of radial and axial forces Each bearing isdefined by the number diameter and length of the rollersby the contact angle (cup angle) and by the pitch diame-ter of the roller set Following the equations provided bythe standard ISO 76 [21] the safety factor of the bearingsystem is expressed in terms of three parameters the ra-tio between roller diameter and pitch diameter the cupangle and the ratio of axial to radial force This simpleexpression gives engineering insight into the problem andshows that for each given load ratio an optimal contactangle exists which maximizes the static load capacity ofthe bearing regardless of it actual size and proportionsTables of optimal contact angles are provided for quickreference

The optimization method presented here gives its bestresults for custom-made bearings for which the design pa-rameters can be varied with the greatest freedom How-ever the results disclosed are useful also for identify-ing the best commercial bearings that can be selectedfrom the manufacturersrsquo catalogues to fit a particularapplication

2 Problem statement

Figure 2 shows the baseline configuration of the bear-ing system examined in this paper The two bearings B1

and B2 are assumed to be equal and subjected to the ra-dial force Fr which is applied at the centre point of thepair In addition to Fr an axial thrust Ka also acts onthe system through the centre shaft

Geometrically each bearing is defined by the follow-ing parameters roller length L mean roller diameter d(measured at midlength of L) number of rollers Z pitchdiameter of the roller set D contact angle α The ISO 76standard [21] specifies that this angle should measure theslope of the raceway without retaining ribs which is nor-mally the outer one (cup) as shown in Figure 2 Shouldthe retaining rib be provided by the cup the contact angleα should refer to the inner raceway (cone)

604-page 3

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 2 Reference configuration of the bearing system with applied loads (Fr and Ka) and characteristic dimensions of thebearings

With reference to Figure 2 the paper seeks the set ofbearing parameters d D L Z α which maximizes thestatic load capacity of the system for given forces Ka andFr The search for the optimum will be based on the equa-tions provided by the ISO 76 standard [21] The effect ofthe flanges is disregarded because the roller-flange con-tact though relevant for the wear damage of the bearingespecially for high-speed applications pays no role understatic loading Although obtained explicitly for the par-ticular bearing combination depicted in Figure 2 (O ar-rangement with bearings removed from each other) theoptimal solution presented will be applicable also to othercombinations such as those shown in Figure 3 (O and Xarrangements with removed or paired bearings)

3 Theory

31 Static radial load rating

Subject to the conditions clarified below the standardISO 76 [21] gives the following field-tested expression forthe static radial load rating C0 (N) of tapered rollerbearings

C0 = 44(

1 minus d

Dcosα

)Z L d cosα (1)

Equation (1) in which the lengths are expressed in mmholds true if the bearing is built and installed under thefollowing assumptions (a) use of bearing steels with hard-ness HRC ge 58 (b) manufacture according to regular tol-erances [22 23] to enhance pressure uniformity over theroller length (c) accurate guide of rollers with roundedends to avoid pressure spikes at the edges [24] (d) mount-ing onto stiff shafts and within rigid housings (e) workingtemperature not higher than 150 C (f) an angular loadzone of 180 (ie the circumferential extent of the set of

rollers that are in contact with both inner and outer race-ways) Significant deviations from these reference condi-tions can be accommodated either by applying correctionfactors available in the literature [25] or by resorting tosecond-order methods [25 26] and specialized numericaltools [27] available to bearing manufacturers The extentof the load zone f is particularly sensitive to the endplay (ie axial clearance) with which the bearing pair ismounted and to axial displacements produced by the load(see further comments in Sect 32)

Table 1 compares the static load ratings predicted byEquation (1) with the actual load ratings of a selectionof tapered roller bearings retrieved from the catalogue ofa leading manufacturer (INA) The internal geometry ofthe bearings in Table 1 (properties from α to Z) werecalculated starting from the catalogue properties (fromdi to Y0) using the method described in the Appendix

By defining the filling ratio of the bearing ζ the as-pect ratio of the rollers λ and the pitch ratio δ as follows

ζ =Zd

πD(2)

λ =L

d(3)

δ =d

D(4)

Equation (1) becomes

C0 = 44 π ζ λD2 (1 minus δ cosα) δ cosα (5)

Although the theoretical limits for the positive parame-ters ζ λ and δ are ζ le 1 λ ge 0 and δ lt 1 in practicethe following ranges are generally observed 05 le ζ le 105 le λ le 2 and δ le 02

32 Static equivalent radial load

Let Frs = 05Fr be the radial load on the single bearingof the system in Figure 2 (remember that Fr is assumed

604-page 4

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 3 Examples of bearing combinations to which the present theory is applicable X-arrangement (top) O-arrangement(bottom) not paired bearings (left) paired bearings (right)

Table 1 Comparison of predicted and catalogue static load ratings for a selection of commercial tapered roller bearings(INA [31])

Properties from INA catalogue Derived properties (see Appendix) C0 (kN)Bearing di do b Y0 α d D L Z INA Eq (1)

(mm) (mm) (mm) (minus) () (mm) (mm) (mm) (minus)30210-A 50 90 17 079 156 10 70 141 20 96 10330220-A 100 180 29 079 156 20 140 241 20 325 35230230-A 150 270 38 076 161 30 210 316 20 630 69230310-A 50 110 23 096 129 15 80 189 15 148 14930320-A 100 215 39 096 129 2875 1575 32 16 500 51930330-A 150 320 55 096 129 425 235 451 16 1 030 1 08431310-A 50 110 19 04 288 15 80 173 15 125 12631320-X 100 215 35 04 288 2875 1575 32 16 480 47631330-X 150 320 50 04 288 425 235 457 16 1 040 1 007T7FC050 50 105 22 038 301 1375 775 203 16 135 144T7FC070 70 140 27 038 301 175 105 25 17 237 242T7FC095 95 180 33 038 301 2125 1375 305 19 400 406

604-page 5

E Dragoni Mechanics amp Industry 16 604 (2015)

centred between B1 and B2) and Fa the actual axial loadon the most loaded of the two bearings (bearing B1 inFig 2) Following ISO 76 [21] the static equivalent radialload acting on the most loaded roller bearing of the pairis

P0 = MAX Frs X0Frs + Y0Fa= MAX 05Fr 05X0 Fr + Y0 Fa (6)

whereX0 = 05 Y0 = 022

cosα

sinα(7)

Equation (6) does not account for narrow load zones (seeSect 31) and uses a lower limit corresponding to a loadzone of 180 degrees (P0 = Frs) Another implicit assump-tion behind Equation (6) is the absence of end momentsacting on the shaft at the sections coupled with the bear-ings which is consistent with the assumption of rigidshaft For a flexible shaft the radial loads on the twobearings would not necessarily be the same due to interde-pendence between radial displacements tilt rotations andunequal load zones Handling of these situations wouldnecessarily call for the use of in-house tools developed bybearing manufacturers (eg [27])

For a rigid shaft under the centre radial loading inFigure 2 and assuming neither end play nor axial preloadon the system the maximum axial force Fa is given bythe following universally accepted equation [28]

Fa = Ka + 05(

Frs

Y

)= Ka + 025

Fr

Y(8)

with [28]

Y = 04cosα

sinα(9)

Using professional tools it can be shown that an axialassembly preload equal to FaFrs asymp 155 would slightlyimprove the load carrying capacity of the bearing sys-tem with respect to the assumption of no end play Bycontrast a positive end play (clearance) or the axial dis-placement induced by the load itself would dramaticallydecrease the load zone of the secondary bearing (ie thebearing which does not support directly the external ax-ial load B2 in Fig 2) so that it could become the el-ement of the pair which experiences the highest contactstresses For this reason the end play should always bestrictly controlled and positive values should be avoidedwhenever possible

Using (7)ndash(9) and introducing the load ratio k as

k =Ka

Fr(10)

Equation (6) becomes

P0 = Fr MAX05 025 + 022

(cosα

sinαk + 0625

)(11)

33 Static safety factor

ISO 76 [21] defines the static safety factor S0 as

S0 =C0

P0(12)

By means of (5) and (11) Equation (12) gives

S0 =ζλD2

Fr

44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

)(13)

which can be written as

S0 = φs0 (14)

with

φ =ζλD2

Fr(15)

and

s0 =44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

) (16)

4 Optimization

41 Free optimization

Equation (14) shows that the static safety factor S0is proportional to the functions φ and s0 From Equa-tion (15) we see that for given radial load Fr function φincreases linearly with the filling ratio ζ and the aspectratio λ and goes up quadratically with the pitch diam-eter D Function φ can be regarded as a control factorthrough which the safety factor S0 can easily be madelarge at will by increasing the pitch diameter Similarlyfunction s0 in Equation (14) can be interpreted as an in-trinsic safety factor of the bearing system obtained whenthe filling ratio the aspect ratio the pitch diameter andthe radial load assume unit value From Equation (16) wesee that s0 depends non-linearly on α δ and k as shownby the three-dimensional charts in Figure 4

Using s0 as objective function the free optimizationproblem can be stated as follows Maximize s0 (X) sub-jected to k = const (=KaFr) with X = (δ α) Max-imization of s0 implies minimization of the denominatorand maximization of the numerator in Equation (16) Forany given load ratio k the denominator of Equation (16)achieves the absolute minimum value 05 when the fol-lowing condition is met

025 + 022(cosα

sinαk + 0625

)= 05 (17)

from which the optimum contact angle αlowast is obtained as

tan αlowast =8845

k asymp 1956k (18)

604-page 6

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 4 Charts of the intrinsic safety factor s0 for a selection of load ratios k

Letting ξ = δ cosα the numerator of Equation (16) ismaximum when

ddξ

[(1 minus ξ) ξ] = 0 (19)

givingξ = δ cosα = 05 (20)

Combination of Equations (18) and (20) yields the opti-mal pitch ratio δlowast as

δlowast =05

cosarctan

(8845k

) (21)

Introducing Equations (18) and (21) into (16) gives thefollowing absolute maximum value slowast0 for the intrinsicsafety factor

slowast0 = 22 π asymp 69115 (22)

The optimal contact angles and pitch ratios given byEquations (18) and (21) are collected in Table 2 for theload ratio k ranging from 0 to 1 Figure 5 depicts theshape assumed by the optimal bearing for k = 0 k = 03and k = 06

604-page 7

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 2 Reference configuration of the bearing system with applied loads (Fr and Ka) and characteristic dimensions of thebearings

With reference to Figure 2 the paper seeks the set ofbearing parameters d D L Z α which maximizes thestatic load capacity of the system for given forces Ka andFr The search for the optimum will be based on the equa-tions provided by the ISO 76 standard [21] The effect ofthe flanges is disregarded because the roller-flange con-tact though relevant for the wear damage of the bearingespecially for high-speed applications pays no role understatic loading Although obtained explicitly for the par-ticular bearing combination depicted in Figure 2 (O ar-rangement with bearings removed from each other) theoptimal solution presented will be applicable also to othercombinations such as those shown in Figure 3 (O and Xarrangements with removed or paired bearings)

3 Theory

31 Static radial load rating

Subject to the conditions clarified below the standardISO 76 [21] gives the following field-tested expression forthe static radial load rating C0 (N) of tapered rollerbearings

C0 = 44(

1 minus d

Dcosα

)Z L d cosα (1)

Equation (1) in which the lengths are expressed in mmholds true if the bearing is built and installed under thefollowing assumptions (a) use of bearing steels with hard-ness HRC ge 58 (b) manufacture according to regular tol-erances [22 23] to enhance pressure uniformity over theroller length (c) accurate guide of rollers with roundedends to avoid pressure spikes at the edges [24] (d) mount-ing onto stiff shafts and within rigid housings (e) workingtemperature not higher than 150 C (f) an angular loadzone of 180 (ie the circumferential extent of the set of

rollers that are in contact with both inner and outer race-ways) Significant deviations from these reference condi-tions can be accommodated either by applying correctionfactors available in the literature [25] or by resorting tosecond-order methods [25 26] and specialized numericaltools [27] available to bearing manufacturers The extentof the load zone f is particularly sensitive to the endplay (ie axial clearance) with which the bearing pair ismounted and to axial displacements produced by the load(see further comments in Sect 32)

Table 1 compares the static load ratings predicted byEquation (1) with the actual load ratings of a selectionof tapered roller bearings retrieved from the catalogue ofa leading manufacturer (INA) The internal geometry ofthe bearings in Table 1 (properties from α to Z) werecalculated starting from the catalogue properties (fromdi to Y0) using the method described in the Appendix

By defining the filling ratio of the bearing ζ the as-pect ratio of the rollers λ and the pitch ratio δ as follows

ζ =Zd

πD(2)

λ =L

d(3)

δ =d

D(4)

Equation (1) becomes

C0 = 44 π ζ λD2 (1 minus δ cosα) δ cosα (5)

Although the theoretical limits for the positive parame-ters ζ λ and δ are ζ le 1 λ ge 0 and δ lt 1 in practicethe following ranges are generally observed 05 le ζ le 105 le λ le 2 and δ le 02

32 Static equivalent radial load

Let Frs = 05Fr be the radial load on the single bearingof the system in Figure 2 (remember that Fr is assumed

604-page 4

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 3 Examples of bearing combinations to which the present theory is applicable X-arrangement (top) O-arrangement(bottom) not paired bearings (left) paired bearings (right)

Table 1 Comparison of predicted and catalogue static load ratings for a selection of commercial tapered roller bearings(INA [31])

Properties from INA catalogue Derived properties (see Appendix) C0 (kN)Bearing di do b Y0 α d D L Z INA Eq (1)

(mm) (mm) (mm) (minus) () (mm) (mm) (mm) (minus)30210-A 50 90 17 079 156 10 70 141 20 96 10330220-A 100 180 29 079 156 20 140 241 20 325 35230230-A 150 270 38 076 161 30 210 316 20 630 69230310-A 50 110 23 096 129 15 80 189 15 148 14930320-A 100 215 39 096 129 2875 1575 32 16 500 51930330-A 150 320 55 096 129 425 235 451 16 1 030 1 08431310-A 50 110 19 04 288 15 80 173 15 125 12631320-X 100 215 35 04 288 2875 1575 32 16 480 47631330-X 150 320 50 04 288 425 235 457 16 1 040 1 007T7FC050 50 105 22 038 301 1375 775 203 16 135 144T7FC070 70 140 27 038 301 175 105 25 17 237 242T7FC095 95 180 33 038 301 2125 1375 305 19 400 406

604-page 5

E Dragoni Mechanics amp Industry 16 604 (2015)

centred between B1 and B2) and Fa the actual axial loadon the most loaded of the two bearings (bearing B1 inFig 2) Following ISO 76 [21] the static equivalent radialload acting on the most loaded roller bearing of the pairis

P0 = MAX Frs X0Frs + Y0Fa= MAX 05Fr 05X0 Fr + Y0 Fa (6)

whereX0 = 05 Y0 = 022

cosα

sinα(7)

Equation (6) does not account for narrow load zones (seeSect 31) and uses a lower limit corresponding to a loadzone of 180 degrees (P0 = Frs) Another implicit assump-tion behind Equation (6) is the absence of end momentsacting on the shaft at the sections coupled with the bear-ings which is consistent with the assumption of rigidshaft For a flexible shaft the radial loads on the twobearings would not necessarily be the same due to interde-pendence between radial displacements tilt rotations andunequal load zones Handling of these situations wouldnecessarily call for the use of in-house tools developed bybearing manufacturers (eg [27])

For a rigid shaft under the centre radial loading inFigure 2 and assuming neither end play nor axial preloadon the system the maximum axial force Fa is given bythe following universally accepted equation [28]

Fa = Ka + 05(

Frs

Y

)= Ka + 025

Fr

Y(8)

with [28]

Y = 04cosα

sinα(9)

Using professional tools it can be shown that an axialassembly preload equal to FaFrs asymp 155 would slightlyimprove the load carrying capacity of the bearing sys-tem with respect to the assumption of no end play Bycontrast a positive end play (clearance) or the axial dis-placement induced by the load itself would dramaticallydecrease the load zone of the secondary bearing (ie thebearing which does not support directly the external ax-ial load B2 in Fig 2) so that it could become the el-ement of the pair which experiences the highest contactstresses For this reason the end play should always bestrictly controlled and positive values should be avoidedwhenever possible

Using (7)ndash(9) and introducing the load ratio k as

k =Ka

Fr(10)

Equation (6) becomes

P0 = Fr MAX05 025 + 022

(cosα

sinαk + 0625

)(11)

33 Static safety factor

ISO 76 [21] defines the static safety factor S0 as

S0 =C0

P0(12)

By means of (5) and (11) Equation (12) gives

S0 =ζλD2

Fr

44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

)(13)

which can be written as

S0 = φs0 (14)

with

φ =ζλD2

Fr(15)

and

s0 =44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

) (16)

4 Optimization

41 Free optimization

Equation (14) shows that the static safety factor S0is proportional to the functions φ and s0 From Equa-tion (15) we see that for given radial load Fr function φincreases linearly with the filling ratio ζ and the aspectratio λ and goes up quadratically with the pitch diam-eter D Function φ can be regarded as a control factorthrough which the safety factor S0 can easily be madelarge at will by increasing the pitch diameter Similarlyfunction s0 in Equation (14) can be interpreted as an in-trinsic safety factor of the bearing system obtained whenthe filling ratio the aspect ratio the pitch diameter andthe radial load assume unit value From Equation (16) wesee that s0 depends non-linearly on α δ and k as shownby the three-dimensional charts in Figure 4

Using s0 as objective function the free optimizationproblem can be stated as follows Maximize s0 (X) sub-jected to k = const (=KaFr) with X = (δ α) Max-imization of s0 implies minimization of the denominatorand maximization of the numerator in Equation (16) Forany given load ratio k the denominator of Equation (16)achieves the absolute minimum value 05 when the fol-lowing condition is met

025 + 022(cosα

sinαk + 0625

)= 05 (17)

from which the optimum contact angle αlowast is obtained as

tan αlowast =8845

k asymp 1956k (18)

604-page 6

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 4 Charts of the intrinsic safety factor s0 for a selection of load ratios k

Letting ξ = δ cosα the numerator of Equation (16) ismaximum when

ddξ

[(1 minus ξ) ξ] = 0 (19)

givingξ = δ cosα = 05 (20)

Combination of Equations (18) and (20) yields the opti-mal pitch ratio δlowast as

δlowast =05

cosarctan

(8845k

) (21)

Introducing Equations (18) and (21) into (16) gives thefollowing absolute maximum value slowast0 for the intrinsicsafety factor

slowast0 = 22 π asymp 69115 (22)

The optimal contact angles and pitch ratios given byEquations (18) and (21) are collected in Table 2 for theload ratio k ranging from 0 to 1 Figure 5 depicts theshape assumed by the optimal bearing for k = 0 k = 03and k = 06

604-page 7

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 3 Examples of bearing combinations to which the present theory is applicable X-arrangement (top) O-arrangement(bottom) not paired bearings (left) paired bearings (right)

Table 1 Comparison of predicted and catalogue static load ratings for a selection of commercial tapered roller bearings(INA [31])

Properties from INA catalogue Derived properties (see Appendix) C0 (kN)Bearing di do b Y0 α d D L Z INA Eq (1)

(mm) (mm) (mm) (minus) () (mm) (mm) (mm) (minus)30210-A 50 90 17 079 156 10 70 141 20 96 10330220-A 100 180 29 079 156 20 140 241 20 325 35230230-A 150 270 38 076 161 30 210 316 20 630 69230310-A 50 110 23 096 129 15 80 189 15 148 14930320-A 100 215 39 096 129 2875 1575 32 16 500 51930330-A 150 320 55 096 129 425 235 451 16 1 030 1 08431310-A 50 110 19 04 288 15 80 173 15 125 12631320-X 100 215 35 04 288 2875 1575 32 16 480 47631330-X 150 320 50 04 288 425 235 457 16 1 040 1 007T7FC050 50 105 22 038 301 1375 775 203 16 135 144T7FC070 70 140 27 038 301 175 105 25 17 237 242T7FC095 95 180 33 038 301 2125 1375 305 19 400 406

604-page 5

E Dragoni Mechanics amp Industry 16 604 (2015)

centred between B1 and B2) and Fa the actual axial loadon the most loaded of the two bearings (bearing B1 inFig 2) Following ISO 76 [21] the static equivalent radialload acting on the most loaded roller bearing of the pairis

P0 = MAX Frs X0Frs + Y0Fa= MAX 05Fr 05X0 Fr + Y0 Fa (6)

whereX0 = 05 Y0 = 022

cosα

sinα(7)

Equation (6) does not account for narrow load zones (seeSect 31) and uses a lower limit corresponding to a loadzone of 180 degrees (P0 = Frs) Another implicit assump-tion behind Equation (6) is the absence of end momentsacting on the shaft at the sections coupled with the bear-ings which is consistent with the assumption of rigidshaft For a flexible shaft the radial loads on the twobearings would not necessarily be the same due to interde-pendence between radial displacements tilt rotations andunequal load zones Handling of these situations wouldnecessarily call for the use of in-house tools developed bybearing manufacturers (eg [27])

For a rigid shaft under the centre radial loading inFigure 2 and assuming neither end play nor axial preloadon the system the maximum axial force Fa is given bythe following universally accepted equation [28]

Fa = Ka + 05(

Frs

Y

)= Ka + 025

Fr

Y(8)

with [28]

Y = 04cosα

sinα(9)

Using professional tools it can be shown that an axialassembly preload equal to FaFrs asymp 155 would slightlyimprove the load carrying capacity of the bearing sys-tem with respect to the assumption of no end play Bycontrast a positive end play (clearance) or the axial dis-placement induced by the load itself would dramaticallydecrease the load zone of the secondary bearing (ie thebearing which does not support directly the external ax-ial load B2 in Fig 2) so that it could become the el-ement of the pair which experiences the highest contactstresses For this reason the end play should always bestrictly controlled and positive values should be avoidedwhenever possible

Using (7)ndash(9) and introducing the load ratio k as

k =Ka

Fr(10)

Equation (6) becomes

P0 = Fr MAX05 025 + 022

(cosα

sinαk + 0625

)(11)

33 Static safety factor

ISO 76 [21] defines the static safety factor S0 as

S0 =C0

P0(12)

By means of (5) and (11) Equation (12) gives

S0 =ζλD2

Fr

44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

)(13)

which can be written as

S0 = φs0 (14)

with

φ =ζλD2

Fr(15)

and

s0 =44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

) (16)

4 Optimization

41 Free optimization

Equation (14) shows that the static safety factor S0is proportional to the functions φ and s0 From Equa-tion (15) we see that for given radial load Fr function φincreases linearly with the filling ratio ζ and the aspectratio λ and goes up quadratically with the pitch diam-eter D Function φ can be regarded as a control factorthrough which the safety factor S0 can easily be madelarge at will by increasing the pitch diameter Similarlyfunction s0 in Equation (14) can be interpreted as an in-trinsic safety factor of the bearing system obtained whenthe filling ratio the aspect ratio the pitch diameter andthe radial load assume unit value From Equation (16) wesee that s0 depends non-linearly on α δ and k as shownby the three-dimensional charts in Figure 4

Using s0 as objective function the free optimizationproblem can be stated as follows Maximize s0 (X) sub-jected to k = const (=KaFr) with X = (δ α) Max-imization of s0 implies minimization of the denominatorand maximization of the numerator in Equation (16) Forany given load ratio k the denominator of Equation (16)achieves the absolute minimum value 05 when the fol-lowing condition is met

025 + 022(cosα

sinαk + 0625

)= 05 (17)

from which the optimum contact angle αlowast is obtained as

tan αlowast =8845

k asymp 1956k (18)

604-page 6

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 4 Charts of the intrinsic safety factor s0 for a selection of load ratios k

Letting ξ = δ cosα the numerator of Equation (16) ismaximum when

ddξ

[(1 minus ξ) ξ] = 0 (19)

givingξ = δ cosα = 05 (20)

Combination of Equations (18) and (20) yields the opti-mal pitch ratio δlowast as

δlowast =05

cosarctan

(8845k

) (21)

Introducing Equations (18) and (21) into (16) gives thefollowing absolute maximum value slowast0 for the intrinsicsafety factor

slowast0 = 22 π asymp 69115 (22)

The optimal contact angles and pitch ratios given byEquations (18) and (21) are collected in Table 2 for theload ratio k ranging from 0 to 1 Figure 5 depicts theshape assumed by the optimal bearing for k = 0 k = 03and k = 06

604-page 7

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

centred between B1 and B2) and Fa the actual axial loadon the most loaded of the two bearings (bearing B1 inFig 2) Following ISO 76 [21] the static equivalent radialload acting on the most loaded roller bearing of the pairis

P0 = MAX Frs X0Frs + Y0Fa= MAX 05Fr 05X0 Fr + Y0 Fa (6)

whereX0 = 05 Y0 = 022

cosα

sinα(7)

Equation (6) does not account for narrow load zones (seeSect 31) and uses a lower limit corresponding to a loadzone of 180 degrees (P0 = Frs) Another implicit assump-tion behind Equation (6) is the absence of end momentsacting on the shaft at the sections coupled with the bear-ings which is consistent with the assumption of rigidshaft For a flexible shaft the radial loads on the twobearings would not necessarily be the same due to interde-pendence between radial displacements tilt rotations andunequal load zones Handling of these situations wouldnecessarily call for the use of in-house tools developed bybearing manufacturers (eg [27])

For a rigid shaft under the centre radial loading inFigure 2 and assuming neither end play nor axial preloadon the system the maximum axial force Fa is given bythe following universally accepted equation [28]

Fa = Ka + 05(

Frs

Y

)= Ka + 025

Fr

Y(8)

with [28]

Y = 04cosα

sinα(9)

Using professional tools it can be shown that an axialassembly preload equal to FaFrs asymp 155 would slightlyimprove the load carrying capacity of the bearing sys-tem with respect to the assumption of no end play Bycontrast a positive end play (clearance) or the axial dis-placement induced by the load itself would dramaticallydecrease the load zone of the secondary bearing (ie thebearing which does not support directly the external ax-ial load B2 in Fig 2) so that it could become the el-ement of the pair which experiences the highest contactstresses For this reason the end play should always bestrictly controlled and positive values should be avoidedwhenever possible

Using (7)ndash(9) and introducing the load ratio k as

k =Ka

Fr(10)

Equation (6) becomes

P0 = Fr MAX05 025 + 022

(cosα

sinαk + 0625

)(11)

33 Static safety factor

ISO 76 [21] defines the static safety factor S0 as

S0 =C0

P0(12)

By means of (5) and (11) Equation (12) gives

S0 =ζλD2

Fr

44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

)(13)

which can be written as

S0 = φs0 (14)

with

φ =ζλD2

Fr(15)

and

s0 =44 π (1 minus δ cosα) δ cosα

MAX05 025 + 022

(cos αsin α k + 0625

) (16)

4 Optimization

41 Free optimization

Equation (14) shows that the static safety factor S0is proportional to the functions φ and s0 From Equa-tion (15) we see that for given radial load Fr function φincreases linearly with the filling ratio ζ and the aspectratio λ and goes up quadratically with the pitch diam-eter D Function φ can be regarded as a control factorthrough which the safety factor S0 can easily be madelarge at will by increasing the pitch diameter Similarlyfunction s0 in Equation (14) can be interpreted as an in-trinsic safety factor of the bearing system obtained whenthe filling ratio the aspect ratio the pitch diameter andthe radial load assume unit value From Equation (16) wesee that s0 depends non-linearly on α δ and k as shownby the three-dimensional charts in Figure 4

Using s0 as objective function the free optimizationproblem can be stated as follows Maximize s0 (X) sub-jected to k = const (=KaFr) with X = (δ α) Max-imization of s0 implies minimization of the denominatorand maximization of the numerator in Equation (16) Forany given load ratio k the denominator of Equation (16)achieves the absolute minimum value 05 when the fol-lowing condition is met

025 + 022(cosα

sinαk + 0625

)= 05 (17)

from which the optimum contact angle αlowast is obtained as

tan αlowast =8845

k asymp 1956k (18)

604-page 6

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 4 Charts of the intrinsic safety factor s0 for a selection of load ratios k

Letting ξ = δ cosα the numerator of Equation (16) ismaximum when

ddξ

[(1 minus ξ) ξ] = 0 (19)

givingξ = δ cosα = 05 (20)

Combination of Equations (18) and (20) yields the opti-mal pitch ratio δlowast as

δlowast =05

cosarctan

(8845k

) (21)

Introducing Equations (18) and (21) into (16) gives thefollowing absolute maximum value slowast0 for the intrinsicsafety factor

slowast0 = 22 π asymp 69115 (22)

The optimal contact angles and pitch ratios given byEquations (18) and (21) are collected in Table 2 for theload ratio k ranging from 0 to 1 Figure 5 depicts theshape assumed by the optimal bearing for k = 0 k = 03and k = 06

604-page 7

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 4 Charts of the intrinsic safety factor s0 for a selection of load ratios k

Letting ξ = δ cosα the numerator of Equation (16) ismaximum when

ddξ

[(1 minus ξ) ξ] = 0 (19)

givingξ = δ cosα = 05 (20)

Combination of Equations (18) and (20) yields the opti-mal pitch ratio δlowast as

δlowast =05

cosarctan

(8845k

) (21)

Introducing Equations (18) and (21) into (16) gives thefollowing absolute maximum value slowast0 for the intrinsicsafety factor

slowast0 = 22 π asymp 69115 (22)

The optimal contact angles and pitch ratios given byEquations (18) and (21) are collected in Table 2 for theload ratio k ranging from 0 to 1 Figure 5 depicts theshape assumed by the optimal bearing for k = 0 k = 03and k = 06

604-page 7

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 5 Optimal bearings for different load ratios k and constant pitch diameter D when no limits are put to the pitch ratioδ = dD

Table 2 Absolute optimum values of contact angle (αlowast) andpitch ratio (δlowast) for a range of load ratios (k)

k tan αlowast αlowast () δlowast

0 0 0 05001 0196 111 05102 0391 214 05403 0587 304 05804 0782 380 06405 0978 444 07006 1173 496 07707 1389 538 08508 1564 574 09309 1760 604 10110 1956 629 110

42 Constrained optimization

The proportions of the optimal bearings in Figure 5are too cumbersome especially for k gt 03 to be of prac-tical use A more fruitful approach is obtained by statingthe optimization problem with a constraint on the pitchratio as follows Maximize s0 (X) subjected to δ le δlim

and k = const (=KaFr) with X = (δ α) This problemis plotted for several k in Figure 6 with the contour linesof s0 drawn as a function of the contact angle α andthe pitch ratio δ The centre curve mn in Figure 6 corre-sponds to the global optimum safety factor s0 = slowast0 givenby Equation (22) For any chosen load ratio k the coor-dinates of point m in Figure 6 provide the global optimalvalues αlowast and δlowast listed in Table 2

Figure 6 shows that for reasonable values of the limitpitch ratio (ie δlim le 05) an optimal contact angleαlowast

δ always exists which depends on the limit pitch ratioitself Take for example the chart in Figure 6 for k = 04and assume δ le δlim = 03 meaning that only the regionbelow line ab is feasible The greatest value of s0 that canbe achieved in that region is obtained by moving on lineab and sweeping the contact angle from a to b until theoptimum point M is reached Point M is defined as thetangent point between ab itself and whichever contour lineoccurs to be touching the line ab The abscissa of point Mgives the optimal value αlowast

δ for the contact angle (αlowastδ asymp 35

in this example) The contour line of s0 passing from Mgives the corresponding intrinsic safety factor (slowast0δ asymp 50in the example)

This constrained optimization can be performed sys-tematically once and for all using Equation (16) for anycombination of load ratios k and pitch ratios δ For givenk and δ the intrinsic safety factor s0 in Equation (16)depends only on α and the optimum value αlowast

δ is easilyfound numerically Optimal values of αlowast

δ and slowast0δ obtainedin this way are reported in Table 3 for load ratios in therange 0 le k le 10 and pitch ratios 005 le δ le 025 (themost likely to occur in practice) Optimal bearings forδlim = 02 are shown in Figure 7 for k = 0 (αlowast

δ = 0)k = 03 (αlowast

δ = 30) and k = 06 (αlowastδ = 38) In marked

contrast with the bearings in Figure 6 awkward andhardly feasible the designs in Figure 7 are sleek and tech-nically viable

The use of Table 3 for the optimal design of taperedroller bearings is easily performed as clarified by the fol-lowing example Assume that the bearing system in Fig-ure 2 has to be designed for the loads Fr = 600 000 N andKa = 180 000 N with a safety factor S0 = 15 Calculatingfrom Equation (10) k = KaFr = 600 000180 000 = 03and assuming a limit pitch ratio δlim = 015 Table 3 givesthe optimal contact angle αlowast

δ = 30 and the optimal in-trinsic safety factor slowast0δ = 3114 From Equation (14) thevalue φlowast

δ = S0slowast0δ = 153114 asymp 00482 is calculatedwhich adopting a filling ratio ζ = 08 (Eq (2)) andan aspect ratio λ = 15 (Eq (3)) and using (15) givesDlowast

δ = (φlowastδFrζλ)05 = (00482 times 600 00008 times 15)05 asymp

155 mm From Equations (4) (3) and (2) the optimalmean roller diameter optimal roller length and optimalnumber of rollers are finally obtained as dlowastδ = δlimDlowast

δ =015 times 155 asymp 233 mm Llowast

δ = λdlowastδ = 15 times 233 asymp 35 mmZlowast

δ = πζ Dlowastδdlowastδ = π times 08 times 155233 asymp 17

5 Discussion

51 Review of the results

Table 1 shows that despite its simplicity Equation (1)predicts quite accurately the static load ratings of com-

604-page 8

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Fig 6 Contour lines of the intrinsic safety factor s0 for a selection of load ratios k

Fig 7 Optimal bearings for different load ratios k and limit pitch ratio δlim = 02

604-page 9

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Table 3 Optimum values of contact angle (αlowastδ) and corresponding intrinsic safety factors (slowast0δ) for given load ratios (k) and

limit pitch ratios (δlim)

δlim

005 01 015 02 025

k αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ () slowast0δ αlowast

δ () slowast0δ αlowastδ(

) slowast0δ

0 0 1313 0 2488 0 3525 0 4423 0 5184

01 110 1289 110 2444 11 3467 110 4356 110 5114

02 215 1226 215 2333 215 3320 215 4187 215 4935

03 295 1141 300 2179 300 3114 300 3945 300 4673

04 320 1062 325 2031 330 2907 340 3691 350 4387

05 340 998 345 1909 350 2736 360 3479 370 4141

06 355 943 360 1806 370 2590 380 3298 390 3931

07 370 895 375 1717 385 2465 395 3141 405 3748

08 385 854 390 1638 395 2354 405 3003 415 3587

09 395 817 400 1568 410 2255 415 2880 425 3443

10 405 784 410 1506 420 2167 425 2769 435 3314

mercial tapered roller bearings Within the wide range ofdimensions and contact angles considered the maximumerror in Table 1 (see Appendix) is less than 10 per cent(bearing 3023-A) while the average absolute error is justabove 4 per cent Errors of the same order of magnitudewere calculated by Dragoni [20] for one standard cylindri-cal roller bearing for which the internal dimensions wereknown exactly not simply estimated as in Table 1

For the case of purely radial load (k = 0) Table 2 givesa global optimum solution with contact angle αlowast = 0 andpitch ratio δlowast = 05 which means cylindrical rollers ofdiameter d equal to one half of the pitch diameter DThis result coincides with the global optimal proportionsreported by Dragoni [20] for the specific category of ra-dial cylindrical roller bearings For increasing axial loads(k gt 0) the global optima for the pitch ratio collectedin Table 2 become more and more unlikely for real-lifeapplications (see Fig 5) The reason why the theoreticaloptimization tends to these quite odd shapes is perhapsimputable to the fact that the empirical expression (1)was developed to fit the experimental behaviour of bear-ings with pitch ratios much lower than 05 as commonlyencountered in practice (for example the pitch ratios ofthe twelve commercial bearings in Tab 1 range from 014to 019)

Table 3 developed to take into account realistic ge-ometric constraint on δ (δ le δlim = 005 025) showsthat the optimum contact angle αlowast

δ depends both onthe load ratio k and the limit pitch ratio δlim Howeverwhile the effect of the load ratio is strong (with αlowast

δ increas-ing monotonically with k) the dependence on the pitchratio is weak (on passing from δlim = 005 to δlim = 025the maximum relative increase of the optimum contactangle is about 10 and occurs for k = 06) For k = 03the optimal contact angle in Table 3 is about 30 whichis the highest contact angle prescribed by the ISO 355standard [29] and offered by most bearing manufacturers(see Tab 1) For load ratios greater than 03 the optimal

contact angle exceeds 30 and reaches the optimum valueslightly above 40 for k = 1 In this range of operationspecial supplies [30] or custom constructions are neededto achieve the maximum load capacity

Walker [16] confirms that for low-speed roller bearingsloaded under prevailing axial thrusts the optimal coneangle is about 40 as predicted by Table 1 for k ge 08Conversely for k = 0 purely radial load Table 1 predictscorrectly αlowast = 0 and Equation (21) predicts δlowast = 05meaning cylindrical rollers with diameter one half of thepitch diameter This result coincides with the global op-timum reported by Dragoni [20] for the specific categoryof radial cylindrical roller bearings

Table 3 also shows that the optimal intrinsic safetyfactor slowast0δ rapidly increases with the limit pitch ratioδlim and decreases with the load ratio k The increase ofslowast0δ with δlim is a consequence of the marked gradient ofthe surfaces of s0 in Figure 4 (confirmed by the densityof the contour lines in Fig 6) for pitch ratios in the range0 le δ le 05 The decrease of slowast0δ with k is due to the factthat given the radial force Fr in Figure 2 an increase ofthe load ratio k implies a greater total force on the bear-ing system with respect to the condition of pure radialloading

With reference to the numerical example at the end ofthe Section Constrained optimization it is easily verifiedthat substituting the design data Fr = 600 kN and k =03 together with the optimal results αlowast

δ = 30 dlowastδ =23 mm Llowast

δ = 345 mm Zlowastδ = 17 for Fr k α d L and

Z in Equations (11) and (1) gives P0 = 300 kN andC0 = 4588 kN respectively These values imply a safetyfactor S0 = C0P0 = 4588300 asymp 153 slightly greaterthan the design value of 15 This small difference is dueto roundoff of the variables involved in the calculationsespecially as regards to the optimum number of rollers Zlowast

δ(the exact value 1672 was rounded to 17 in the example)

604-page 10

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

Table A1 Relationships between catalogue data (Y0 di do b) and internal bearing properties (α d D L Z)

Internal property Relationship Numerical values Source

α = arctan

(022

Y0

)minus Equation (7)

d = q1 (do minus di) q1 = 025 Textbook [32]

D = q2 (do + di) q2 = 05 Assumption

L = q3b

cos αq3 = 08 Assumption

Z = q4(do + di)

dq4 = 145 Textbook [32]

52 Limitations of the model and future work

The present optimization is built on the assumptionthat the critical bearing of the pair is the one that di-rectly supports the axial load (B1 in Fig 2) This is thenatural consequence of using Equation (6) for calculatingthe maximum equivalent load on the system Though thisapproach is coherent with the design formulae present inthe manufacturersrsquo catalogues it has limitations when thebearing pair is mounted with end play (axial clearance)or undergoes large axial deformations under load A largeaxial load induces a large axial displacement which causesa very narrow load zone in the second bearing (B2 inFig 2 not considered here) hence generating risk of largeroller-race load and pressure Under these unfavourableworking conditions the second bearing may thus becomethe critical element which is not optimized here In thiscase the full answer to the problem cannot be obtainedwith the present analytical model and requires numericaltools considering equilibrium and compatibility (deforma-tions) of bearings shaft and housing as a whole

In practical terms the optimal design described inthis paper strictly holds true when the bearing systemis assembled with a light preload that would compensatethe axial displacement induced in the secondary bearingby the externally applied load This is certainly a limita-tion but not a prohibitive one since preloading of taperedroller bearings is a common procedure for the many ad-vantages it brings about such as (a) increase of the stiff-ness (b) noise reduction (c) improved rotational preci-sion (d) wear compensation and (e) longer life

Spurred by the insightful results obtained for thestatic loading covered in this paper the next step of theresearch will focus on the optimization of the bearing sys-tem in Figure 2 for maximum dynamic load-carrying ca-pacity In the light of the comments on the sensitivity ofthe contact zone emphasized in this section an effort willbe made to incorporate the effects of the bearing defor-mation into the optimization procedure

6 Conclusion

Using the empirical relationships provided by theISO 76 standard the internal dimensions of tapered rollerbearings are optimized for maximum static load capacityThe bearing system investigated comprises two identicalbearings undergoing whatever combination of radial andaxial forces Assuming that the radial force is applied at

equal distance from the bearings of the pair and that nei-ther end play (axial clearance) nor abnormal preloadingaffect the assembly the closed-form optimization processleads to the following general results

ndash the static load capacity increases linearly with the fill-ing ratio (number of rollers divided by the maximumnumber which can fill the bearing) and the aspect ra-tio (ratio of roller length to mean roller diameter) andgoes up with the square of the pitch diameter of theroller set

ndash given the ratio of axial to radial force global optimaexist for the contact angle and the pitch ratio (ratioof roller diameter to pitch diameter) which maximizethe static load capacity

ndash the bearing proportions at the global optima are toosturdy (contact angles greater than 60 degrees pitchratios equal to or greater than 05) to be used in prac-tice

ndash if the pitch ratio is constrained to stay below reason-able limits (le025) an optimal contact angle existswhich maximizes the static load capacity regardlessof the actual size and proportions of the bearing

ndash the results of the optimization are conveniently sum-marized by a general table and a few simple equationsthat can be followed step-by-step to design the opti-mal bearing that suits any given application

ndash the optimization procedure can either be used to de-sign custom-made bearings (thus exploiting the geo-metrical freedom to the full) or to pick from the manu-facturersrsquo catalogues the bearing with the best contactangle for any assigned loading

Appendix Internal dimensions of taperedroller bearings

The internal dimensions of rolling bearings are pro-prietary data which the manufacturers do not provide intheir catalogues However starting from the external di-mensions of the bearings (di do b in Fig 2) and thestatic coefficient Y0 available from the catalogues theinternal properties (α d D L Z in Fig 2) can be calcu-lated within 10ndash15 per cent of the true value Table A1shows how the internal properties displayed in Table 1were obtained step-by-step from the catalogue data usingelementary geometrical considerations and characteristicproportions available from technical textbooks [32]

604-page 11

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12

E Dragoni Mechanics amp Industry 16 604 (2015)

References

[1] A Seireg A survey of optimisation of mechanical designJ Eng Ind T ASME 94 (1972) 495ndash499

[2] AA Seireg J Rodriguez Optimizing the shape of me-chanical elements and structures CRC Press ClevelandOH 1997

[3] J Ahluwalia SK Gupta VP Agrawal Computer-aidedoptimum selection of roller bearings Comput Aid Des25 (1993) 493ndash499

[4] H Aramaki Rolling bearing analysis program packageBRAIN NSK Technical Journal Motion Control 3 (1997)15ndash24

[5] D-H Choi K-C Yoon A Design Method of an auto-motive wheel-bearing unit with discrete design variablesusing genetic Algorithms J Tribol T ASME 123 (2001)181ndash187

[6] K Kalita R Tiwari SK Kakoty Multi-objective op-timisation in rolling element bearing system designIn Proceedings of the International Conference onOptimisation (SIGOPT 2002) Lambrecht Germany2002

[7] I Chakraborty V Kumar SB Nair R Tiwari Rollingelement bearing design through genetic algorithms EngOptimiz 35 (2003) 649ndash659

[8] BR Rao R Tiwari Optimum design of rolling elementbearings using genetic algorithms Mech Mach Theory42 (2007) 233ndash250

[9] S Gupta R Tiwari SB Nair Multi-objective designoptimisation of rolling bearings using genetic AlgorithmsMech Mach Theory 42 (2007) 1418ndash1443

[10] V Savsani RV Rao DP Vakharia Multi-objective de-sign optimization of ball bearings using a modified par-ticle swarm optimization technique Int J Des Eng 1(2008) 412ndash433

[11] Y Wei R Chengzu Optimal design of high speed an-gular contact ball bearing using a multiobjective evolu-tion algorithm In Proceedings of the IEEE Int Conf onComputing Control and Industrial Engineering (CCIE2010) Wuhan China 2010

[12] KS Kumar R Tiwari RS Reddy Development of anoptimum design methodology of cylindrical roller bear-ings using genetic algorithms Int J Comput Meth EngSci Mech 9 (2008) 321ndash341

[13] KS Kumar R Tiwari VVN Prasad An optimum de-sign of crowned cylindrical roller bearings using geneticAlgorithms J Mech Des T ASME 131 (2009) 051011-1ndash051011-14

[14] RJ Parker SI Pinel HR Signer Performance ofcomputer-optimized tapered-roller bearings to 24 MillionDN J Tribol T ASME 103 (1981) 13ndash 20

[15] N Chaturbhuj SB Nair R Tiwari Design optimizationfor tapered roller bearings using genetic algorithms InProceedings of the International Conference on ArtificialIntelligence (IC-AI 03 2003) CSREA Press Las VegasNevada 2003 Vol 1 pp 421ndash427

[16] B Walker High speed tapered roller bearing opti-mization MS Thesis Rensselaer Polytechnic InstituteHartford Connecticut 2008

[17] Z Wang L Meng H Wensi E Zhang Optimal design ofparameters for four column tapered roller bearing ApplMech Mater 63-64 (2011) 201ndash 204

[18] R Tiwari KK Sunil RS Reddy An optimal designmethodology of tapered roller bearings using geneticAlgorithms Int J Comp Meth Eng Sci Mech 13(2012) 108ndash127

[19] R Tiwari R Chandran Thermal based optimum designof tapered roller bearing through evolutionary AlgorithmIn Proceedings of the ASME 2013 Gas Turbine IndiaConference Bangalore Karnataka India 2013 (paperNo GTINDIA2013-3792)

[20] E Dragoni Optimal design of radial cylindrical rollerbearings for maximum load-carrying capacity ProcIMechE Part C J Mech Eng Sci 227 (2013) 2393ndash2401

[21] ISO 76 Rolling bearings ndash Static load ratings 2006[22] H Wiesner Rolling bearings TC4 meets GPS TC213

Evolution 19 (2012) 24ndash28[23] ISO 492 Rolling bearings ndash Radial bearings ndash Tolerances

2002[24] KL Johnson Contact Mechanics Cambridge University

Press Cambridge UK 1985[25] TA Harris Rolling Bearing Analysis 4th edn John

Wiley amp Sons New York 2000[26] NSK Bearing internal load distribution and dis-

placement httpwwwjpnskcomapp01enctrg

indexcgigr=dnamppno=nsk_cat_e728g_5 (access date2112014)

[27] INA BEARINX-online Shaft Calculation http

wwwinadecontentinadeenproducts_services

calculatingCalculation_and_testingjsp (accessdate 2112014)

[28] G Niemann H Winter B-R Hohn MaschinenelementeSpringer Vol I Berlin Germany 2005

[29] ISO 355 Rolling bearings ndash Tapered roller bearings ndashBoundary dimensions and series designations 2007

[30] TIMKEN Tapered Roller Bearing Catalogue http

wwwtimkencomen-usproductsPagesCatalogs

aspx (access date 2492014)[31] INA Tapered Roller Bearing Catalogue httpwww

inadecontentinadedeproducts_services

rotativ_productstapered_roller_bearings

tapered_roller_bearingsjsp (access date 2492014)[32] G Niemann Maschinenelemente Springer Berlin

Germany 1981 Vol I

604-page 12