21

Synthesis of the research activity

1rst Part:

Rolling Bearing Modeling

Three families of rolling bearings have been modeled: ball, cylindrical roller and taperedroller bearings. There are two essential points in this work. The first one consists in writingthe adequate equilibrium and geometrical equations for each bearing element. The secondone concerns the lubrication, i.e., the calculation of the lubricant film thicknesses and frictionforces and torques at the different interactions between the rolling bearing elements.

This chapter is divided in four sections. After a short introduction in the first section, a briefdescription of the kinematics and mechanical equilibrium of each studied bearing kind isgiven in the second section. The third is relative to the lubrication, including a study of therheological properties of the lubricant as well as a review of the theories to calculate the EHLfilm thickness or to estimate the friction forces or torques. Finally, a few examples ofapplication are presented.

22

23

1. Introduction

To fulfill the requirements of the users of rotating machines in term of performances, cost,reliability and tolerances to malfunctions (surface defect, lubrication defect, overloading,misalignment, etc.), it is necessary to develop bearings of better quality while reducing theirsize and mass. This objective can be achieved by the improvement of the design and thelubrication of the bearing or by the elaboration and the use of new materials.

On the one hand, the development of aeronautical turbine engines requires higher operatingspeeds. Among the limitations encountered figure that imposed by the thrust rolling bearingsthat maintain the high-pressure compressors and turbines. A common criterion to evaluate theseverity of the operating conditions of a bearing is the DN product (product of the shaft speedin rpm by the pitch diameter of the bearing in mm). In future turboshaft engines this valueshould exceed 3.5 million, which poses some problem in terms of reliability and dissipatedpower. In addition, the increasing demand in term of performances leads to new concepts ofengines, in which the number of rolling bearing, their dimension, and their loading increase.The power dissipated by the mainshaft rolling bearings then starts to affect significantly thefuel consumption of the turbine engine. In this context, the knowledge of the power lossesbecomes a major concern for the designer.

Each bearing element must be dimensioned in an optimal way according to the material used.This concerns in particular the cage, which undergoes dynamic solicitations with the rollingelements and centrifugal hoop stress when the speed of rotation is significant. Except theexperience of bearing designers and sometimes that of users, there do not exist tools to designan optimized cage. New cages are obtained by homothety of existing and satisfactory triedsolutions. Thus, to prevent possible ruptures and to optimize the cage geometry, it is necessaryto know the dynamic loading experienced by the retainer. This is why we recently developed adynamic model. We initially chose to study cylindrical roller bearings, because of their simplegeometry.

Results concerning the internal kinematics and the power dissipated in cylindrical roller andball bearings will be presented. They show for example that the traditional assumption of aconstant friction coefficient and a control of the ball by one of the inner or outer rings does notexplain the real behavior of a high-speed ball bearing. An application to the detection of non-visible defect by the numerical simulation of acoustic emission will be also presented. Finallysome dynamic results describing the shocks between the rollers and the cage or the transienteffects at the contacts between the rollers and the rings will be given.

The body of this 1rst part includes three aspects. First, it relates to the kinematics and to thefundamental equation of dynamics that describe the rolling bearing equilibrium. Then theeffects of lubrication will be depicted, i.e. the rheological properties of the lubricants, and thetheories allowing the calculation of the EHL oil film thickness or the estimation of the frictionforces and torques. Finally, we will illustrate the performances of the models developed by acertain number of case studies.

24

2. Rolling bearing modeling

2.1 Principal notations

α° Bearing free contact angleα Contact angle (ball bearing) or half-cup angle (tapered roller bearing)ψ Angular position of the rolling elementθ Misalignment angle between the inner and outer ringsδ Elastic deformation at the contact between the raceway and the rolling

elementδa Axial displacement of the inner ringδr Radial displacement of the inner ringν Poisson's coefficientω Rotation speedωm Rotation speed of the cageωr Rotation speed of the rolling elementCC Friction torque between the cage and the ring guiding land (journal bearing)D Diameter of the rolling element (ball and cylindrical roller)DC Diameter of the cage pocketDm Pitch diameter of the rolling bearingDw Diameter of the rolling element (conical roller)E Young's modulusf Raceway conformityFa Applied axial loadFB Friction force between the rolling element and the ringFC Centrifugal force acting on a rolling elementFC1,2 Friction force between the rolling element and the cage (1=front; 2=rear)FOL Oleodynamic force acting on a rolling elementFr Applied radial loadh EHL lubricant film thickness between the rolling element and the ringH1,2 Lubricant film between the rolling element and the cage (1=front; 2=rear)JD Diametral clearancel Active lengthlw Roller lengthMg Gyroscopic momentN Number of rolling elementsQ Normal load between the rolling element and the ringRR Radius of the cylindrical rollerRs Curvature radiusWC1,2 or QC1,2 Normal load between the rolling element and the cage (1=front; 2=rear)WFC or QC Normal load between the cage and the ring guiding land (journal bearing)

Subscripts1 Refers to the front contact between the cage and the rolling element2 Refers to the rear contact between the cage and the rolling elementf Refers to the contact between the roller flange and the guiding shoulderi Refers to the contact with the inner ringj Refers to the contact rolling element n°jo Refers to the contact with the outer ring

25

2.2 Ball bearings

Assumptions

A ball bearing is a complicated mechanism and the resolution of its equilibrium requires somesimplifications. The analysis is quasi-static, only the centrifugal forces and the gyroscopicmoment are taken into account in this model. A quasi-static model seems well adapted to thecase of a bearing operating under a purely axial or purely radial load, which produces anidentical speed of rotation for all balls. The approach should be prudently extrapolated foroperating conditions that combine a slight misalignment, or a combination of radial and axialloads [37, 95, 179]. This analysis allows a good prediction of the stabilized behavior of thebearing (internal kinematics, load distribution, oil film thickness, power loss, etc.). However,such a model does not allow to estimate in a realistic way the dynamic efforts between theballs and the cage [87, 90].

A single row ball bearing is studied here, for operating conditions that correspond to a generalapplied load and a high rotation speed. The geometry of each bearing element are assumedperfect, i.e. balls are spherical, rings cylindrical, and raceway torical. The geometrical centerand the center of inertia are identical for each ball. The outer ring is fixed. The cage could beguided either by the inner or outer ring.

Interactions between the various bearing elements

The various models used to calculate the lubricant film thickness, as well as the friction forcesand torques are recapitulated in table 1. From most of the available literature, it can beestablished that contact forces resulting from ball-cage pocket and cage-ring pilot surfacecontacts are small compared with the forces at the ball-raceway contacts. Furthermore, due toa lubricating film at the contacts between the ball and the cage pocket, hydrodynamic modelscan be considered to avoid the limitation due to a Coulomb friction coefficient, and elasticdeformations are neglected in steady-state operating conditions.

Interactions between the cage and the ringsThe interaction between cage and outer- or inner-ring pilot surface is assumed purelyhydrodynamic. In a rolling bearing, the ratio of pilot surface width to cage diameter is alwayslower than 1/6. Therefore, the cage/ring contact is simply modeled by the well-known shortjournal bearing solution in laminar or turbulent flow. In this model, the attitude angle of thecage denotes the angle between the line center direction and the external load direction [80].The hydrodynamic load and friction torque acting on the cage are linked to the rotation speedand the eccentricity of the cage.

Interactions between the ball and the cage pocketConcerning the ball-cage pocket interface, the problem can be solved in two directionsindependently, and the two solutions are then superimposed as shown by Gupta [91]. Thus,for the ωr cosβ component of the ball rotation speed, the problem is similar to that of a longjournal bearing. Integrating the Reynolds equation over the contact region, the normal loadcontribution of the speed component is obtained directly from an algebraic relation, while thetraction force is obtained by a numerical integration. For the ωr sinβ component of the ballrotation, the model is similar to that of a short journal bearing, as described earlier for thecage-ring contact.

26

Interactions the ball and the ringThe ball-ring raceway contact is well described by the Hamrock and Dowson relation [102]for the EHL film thickness of oil film, corrected by a thermal reduction factor given by Guptaet al. [92], and by the Johnson and Tevaarwerk non-Newtonian model for the traction force[116].

Figure 2 – Interactions between the various ball bearing elements

CONTACT ASSUMPTIONS MODELS RESULTS

1. Cage-RingPilot Surfaces

- low load

- rigid surfaces

Hydrodynamic

- short journal bearing [85]

- normal load

- friction torque

- attitude angle

as function of the eccentricity

2. Ball-CagePocket

- low load

- rigid surfaces

Hydrodynamic

- short journal bearing [85]

- long journal bearing [91]

- normal load

- friction torque

as function of the ball position

3. Ball-Raceway

- from none up to heavyload (inner ring)

- elastic deformation

Elastohydrodynamic

- Hamrock and Dowsonrelation [102]

- Gupta et al. factor [92]

- lubricant film thickness

- thermal reduction factor

- rolling, sliding andspinning speeds

Elastohydrodynamic

- Johnson and Tevaarwerktheory[116]

- friction force

- spinning moment

Table 1 – Ball bearing element interactions

3.

1.

2.

27

Kinematics

The different coordinate frames used to determine the internal kinematics of the ball bearingare shown in figure 3. The RG coordinate frame, with its origin at the bearing center, is fixedin space, and is referred to as the inertial coordinate frame. The applied loads on the bearingare prescribed in these coordinates, the thrust load on the bearing being along the xG axis. Theinner ring is rotating along the xG axis. The R5 coordinate frame, shown in figure 3, is fixed inthe rolling element. This frame is used for formulating the rotation motion of the ball. This isa steady-state analysis therefore, transient effects are neglected.

y1, y2, y3y4 β

α

α

ββ'

β'

o

i

Y

X

C

o

o

o

XY

Ci

ii

O2

x4, x5

x3

x2

ω

mω

r

x1

O1

z1

z3, z4

x , x , xo iG

z2, Z , Zo i

Figure 3 – Definition of the reference frames used

The Cartesian coordinate frames commonly used for vector transformations are defined inappendix 2, as well as transformation matrices. These different coordinate systems are usefulto determine the movement of the ball surface relatively to the inner and outer raceway.

The absolute angular velocity vectors of the Ro frame (fixed in the outer ring) and of the Ri

frame (fixed in the inner ring) are given respectively in the following equations& &Ωo

Go o= ω .x (1) and

& &ΩiG

i i= ω .x (2)

28

The absolute angular velocity vectors of the R5 frame (fixed in the ball) is given by thefollowing relation:

& & & & & & & & & & &Ω Ω Ω Ω Ω Ω5G

1G

21

32

43

54

m 1 r 5= + + + + = + + + +ω ω. .x x0 0 0 (3)

Ball – inner ring contactSome simplifying assumptions are made in order to establish the dynam Tc,5/i of the relativemovement of the coordinate frame R5 fixed in the ball and the frame Ri fixed in the inner ring.Elastic deflection at each point of the contact ellipse is neglected in comparison with the balldiameter D. The component

& &δb y. 2 of the ball displacement vector is neglected in comparisonwith the pitch radius 1/2 dm. Then, the dynam Tc,5/i can be reduced at the contact ellipse centerCi as follows:

( )

−=

+−=Ω−Ω=Ω

)(C)(C)(C

.. = )(C Tc

iG

iiG

5ii

5

4,5ri,1,2o,G,imGi

G5

i5

i5/iVVV

xx&&&

&&&&&ωωω

(4)

with the velocity vectors written as (a subscript on a vector refers to the coordinate frame inwhich the vector components are expressed:

( )( )( )( )

−−+−+−+−−

=Ω'sin.cos.

cos.sin.sin.'cos.cos.

sin.sin.cos.'cos.cos.

'

i5

ββωαβωαββωωωαβωαββωωω

r

iririm

iririm

Ri

& (5)

( ) ( )

−−++−

+

=

im

imiir

r

R

i

DdD

D

CV

i

αωωβαββαω

ββω

cos.22.sin.sin'cos.cos.cos.2.

0

'sin.cos.2.

)(

'

i5

& (6)

respectively the rotation and the velocity vectors, Ri' being the reference frame linked to theball-inner ring contact ellipse.

Ball – outer ring contactSimilarly to the ball-inner ring contact, the dynam Tc,5/o of the relative movement of thecoordinate frame R5 fixed in the ball and the frame Ro fixed in the outer ring can be reduced atthe contact ellipse center Co as follows:

( )

−=

+−=Ω−Ω=Ω

)(C)(C)(C

.. = )(C Tc

oG

ooG

5oo

5

4,5ri,1,2o,G,omGo

G5

o5

o5/oVVV

xx&&&

&&&&&ωωω

(7)

with :( )( )( )( )

−++−−++−+

=Ω'sin.cos.

cos.sin.sin.'cos.cos.

sin.sin.cos.'cos.cos.

'

o5

ββωαβωαββωωωαβωαββωωω

r

ororom

ororom

Ro

& (8)

( ) ( )

+−++

=

om

omoor

r

R

o

DdD

D

CV

o

αωωβαββαω

ββω

cos.22.sin.sin'cos.cos.cos.2.

0

'sin.cos.2.

)(

'

o5

& (9)

respectively the rotation and the velocity vectors, Ro' being the reference frame linked to theball-outer ring contact ellipse.

29

Common ball-bearing analyses use the following angles:

α+

α=β

om

oo

cosd

Dsin

arctg (10) and

α+−

α=β

im

ii

cosd

Dsin

arctg (11)

These angles βo and βi are respectively related to an outer race control (0o5

&&=Ω ), and to an

inner race control ( 0i5

&&=Ω ), if the lateral slip along the CoXo and CiXi axes due to the

gyroscopic moment does not occur. The condition of no lateral slip due to gyroscopic effect isequivalent to setting the β' angle equal to zero.

Note that this assumption of full control of the ball by one of the two rings [106] is notphysically satisfactory since that induced a discontinuity in the kinematics.

Gyroscopic MomentIt can usually be assumed with minimal loss of calculation accuracy that pivotal motion due togyroscopic moment is negligible; then the angle β' is zero. The gyroscopic moment as definedabove is therefore resisted successfully by friction forces at the bearing raceways. Thefollowing expression may be obtain for ball bearings:

βωωπρ= sinD60

1M mr

5g (13)

If the bearing speed of rotation is sufficient that the outer raceway control is satisfied, then itcan be assumed with little effect on calculation accuracy that the ball gyroscopic moment isresisted entirely by frictional force at the ball-outer raceway contact. Otherwise, it is safe toassume that the ball gyroscopic moment is resisted equally at the ball-inner and ball-outerraceway contacts [106]. Therefore, in figure 4(a) and in equation 15, λij=0 and λoj=2 for outer

raceway control; otherwise λij=λoj=1.

Basic Equations

In this model, the kinematics of the cage and of each ball are unknown. They are determinedby solving the force and moment balance of the different bearing elements.

Equilibrium equationsThe traction forces and moments acting on the ball n°j are schematically shown in figure 4(a)and (b). The equilibrium equations for the balls are defined as follows:

( ) jjjojojijijgj

ojojijij FCFC2FC1-sinsinD

McosQcosQ0 ++− αλ−αλ+αα= (14)

( )ojojijijgj

ojojijij coscosD

MsinQsinQ0 αλ−αλ−αα= − (15)

0 = WC1j – WC2j + FBoj – FBij + Fol (16)

0 = FBoj cos αoj + FBij cos αij – FC1j – FC2j (17)

30

a)

αij

αoj

Qoj

ij

FCj

Q

Mgj

Mgjij D

λ

Mgjoj D

λ

b)Figure 4 – Ball loading at angular position ψj including gyroscopic moment effect

Equilibrium equations for the cage, figure 5, are written as:

[ ] χ++ψ−+ψ−= ∑=

sin)WFCWFC(sin)1FC2FC(cos)1WC2WC(0 oi

N

1jjjjj (18)

[ ] χ++ψ−−ψ−= ∑=

cos)WFCWFC(cos)1FC2FC(sin)1WC2WC(0 oi

N

1jjjjj (19)

[ ] oi

N

1jjmj CCCC2/D)2FC1FC(d)2WC1WC(0 −++−−= ∑

=

(20)

Figure 5 – Cage, force and moment equilibrium Figure 6 – Inner ring, force and moment equilibrium

The quasi-static equilibrium of the inner ring, shown in figure 6, can be reduced to twoequations defined as follows:

∑=

αλ−α=

N

1jij

gjijijija cos

D

MsinQF (21)

)cos(WFCcossinD

McosQF ij

N

1jij

gjijijijr χ−ψ

αλ+α= ∑

=

(22)

with χ = φ + ξ, φ being the attitude angle of the cage (short journal bearing effect) and ξ theangular location of the cage center.

31

The outer ring is stationary and mounted in its housing. The mounting forces ensure outer ringequilibrium without effect on bearing behavior.

Hydrodynamic and elastohydrodynamic models are used to determine the friction force andtorque at the different interactions, depending on the rheological properties of the lubricantand on the internal kinematics of the bearing.

Geometrical relationsFrom Harris [106], the geometrical relations can be determined by considering the ball centerdisplacement due to a generalized loading system including axial and radial loads, and anangular misalignment. Under zero load, the centers of the raceway groove curvature radii areseparated by a distance (fo+f i-1)D. Subjected to an applied static load, the distance between

centers increases by the amount of the contact deformations δi, δo less the lubricant filmthickness hi and ho, as shown in figure 7. Note that the contact exists only for a positive value

of δ.

Figure 7 – Ball center location with and without applied load,with Ri = (0,5 dm + (fi-0,5) D cos α°), from Harris [106]

Finally, two geometrical relations (equations 23 and 24) are obtained by comparing the initialand the final position of the ball center:

jro

io

ijijijiojojojo

coscos.D)1ff(

cos]hD)5.0f[(cos]hD)5.0f[(

ψδ+α−+=

α−δ+−+α−δ+−(23)

jmo

iao

io

ijijijiojojojo

cos)d5.0cos.D)5.0f((sin.D)1ff(

sin]hD)5.0f[(sin]hD)5.0f[(

ψ+α−θ+δ+α−+=

α−δ+−+α−δ+−(24)

Assessment

32

The set of non-linear equations (14)-(22) and (23)-(24) describes the quasi-static equilibriumof the ball bearing. The 6N+5 unknowns corresponding to the 6N+5 equations are recalled intable 2.

H1j Lubricant film thickness at the ball-cage pocket Nωrj Ball rotation speed (ball n°j) NQij Normal load at the ball n°j / inner ring contact NQoj Normal load at the ball n°j / outer ring contact Nαij Contact angle at the ball n°j / inner ring contact Nαoj Contact angle at the ball n°j / outer ring contact Nε Eccentricity of the cage 1ξ Angular location of the cage center 1

ωm Cage rotation speed 1δr Radial deflection of the inner ring 1δa Axial deflection of the inner ring 1

Table 2 – Unknowns in the set of equations to solve

Other relationsAccording to Harris [106], the normal loads are related to the normal contact deformation asfollows:

5.1.KQ δ= (25)

When the ball moves through the loaded zone, its circumferential position in the pocket ismodified. In this model, this position is an unknown. Normal and tangential load acting oneach ball– WC1,2 et FC1,2 – depend on the lubricant film thickness between the ball and thecage pocket H1,2 , and the surface speed. From figure 8, a geometrical relation is given by:

DC = D + H1j + H2j (26)

H1 H2D

DC

Figure 8 – Ball location in cage pocket

33

2.3 Cylindrical roller bearings

Two lubricated models have been developed. Both are 2D models (plan) that do not allow todescribe the bearing misalignment. The first is a quasi-static model; inertial effects exceptcentrifugal force are neglected. This basic model allows to obtain i) the internal kinematics ofthe bearing including the phenomenon of skidding [17, 24, 181, 211], ii) the load distributionon the rolling elements, and (iii) the power dissipated by the bearing. The second model isdynamic. It allows to estimate the dynamic efforts at the contacts between the rollers and thecage. It allows also to describe the internal kinematics and the load distribution in transientregime.

Assumptions

Solving bearing equilibrium requires a few simplifying assumptions. The geometry isassumed to be perfect, i.e., rings, cage and rollers are cylindrical. As it is a plane model, ringmisalignment and roller skewing are not considered. The outer ring is stationary. Though, it ispossible to take into account an outer or inner-ring guided cage, an outer or inner-ring guidedroller, and race flanges, which are guiding rollers, with or without flange angle.

A 2D model can not take into account the ring misalignment and thus the phenomenon ofroller skewing. However, a 2D dynamic model is sufficient to describe the phenomenon ofroller tilting in the cage pocket [105, 106], as shown in figure 9.

Figure 9 – Tilting and skewing movement of the roller in its cage pocket.

Roller skewingTilting

34

Roller bearing element interactions

Existing hydrodynamic and elasto-hydrodynamic models are used to determine the normal andtraction forces corresponding to the geometrical interactions between the different elements ofa roller bearing. This paragraph is limited to the discussion of the models and formulationsused in the two computer programs presented here.

The basic geometry of a roller bearing is presented in figure 10. The different formulationsrelating cage-ring pilot surface forces and moments to the cage rotation speed andeccentricity; tangential and normal cage forces to the roller position in the cage pocket; andthose defining traction forces at the roller-raceway interactions are summarized in table 3.From most of the available literature, it can be established that contact forces resulting fromroller-cage pocket and cage-ring pilot surface contacts are small in comparison with the forcesat the roller-raceway contacts. Furthermore, due to a lubricant film at the interfaces,hydrodynamic models can be considered, and elastic deformations are neglected in steady-state operating conditions. The computer program shows that rollers moving through theloaded zone, are located in the front of the cage pocket and consequently are driving the cage.While the opposite rollers, in the unloaded zone, are located at the rear of the cage pocket andact on the cage as a braking system.

The interaction between cage and outer or inner ring pilot surface is simply described by thewell-known "short journal bearing" solution as for ball bearings [85]. The hydrodynamic loadas well as the friction torque are function of the cage speed and eccentricity.

Resulting from a lubricant film at the interface, low load and conform surfaces, the interactionbetween roller and cage pocket is considered being purely hydrodynamic. This contact isdescribed by the analytical solution of Martin for isoviscous fluid. The hydrodynamic load andfriction force are function of the roller position relatively to the cage. However, an EHL modelis used when the shocks between the roller and the cage are significant.

Concerning the roller end-cage pocket edge interaction, assuming there is no roller skewing inthe cage pocket and no normal load between roller ends and cage pocket edges, the model issimilar to the hydrodynamic solution of Couette flow [160, 172].

The roller-ring raceway contact is described by Cheng [38, 39] for the EHL lubricant filmthickness for line contact, including a thermal corrector factor given by Gupta, Cheng et al.[92], and by Johnson and Tevaarwerk for the traction force [116].

Interactions between the roller end and the race guiding flange have been studied by numerousauthors, including Brown and Poon [29], Korrrenn [126], Warda et al. [127, 233], Li and Wen[135], Prisacaru et al. [184] or Zhou and Hoeprich [248] for tapered roller bearings. Weassume that there is no normal load and that the roller is rolling centered between the twoguiding shoulders without skew angle. Then we can use the well-known Couette flow model[160, 172]. The model can consider a race flange angle.

35

3

4

1

2

5

6

z

z

z

z

z

z

z

Figure 10 – Roller bearing element interactions

CONTACT ASSUMPTIONS MODELS RESULTS

c

WINDAGEAir-oil mixture generatesoleodynamic drag

Bearing tests

- experimental results - windage torque

d

CAGE / RING- low load- rigid surfaces- laminar or turbulent flow

Hydrodynamic

- short journal bearing

- load- torque- attitude angleas function of the eccentricity

e

ROLLER

/

- from none up to heavy load(inner ring)

Elastohydrodynamic

- theory of Cheng- Gupta, Cheng et al. factor

- lubricant film thickness- thermal reduction factor

RING RACEWAY- elastic deformations- rolling and sliding speeds

Elastohydrodynamic

- theory of Johnson andTevaarwerk

- friction force

f

ROLLER / CAGEPOCKET

- low load- rigid surfaces

Hydrodynamic

- theory of Martin

or Elastohydrodynamic

- normal load- friction forceas function of the roller position

g

ROLLER EDGES / POCKET

EDGES

- no load- no skewing

Hydrodynamic

- Couette flow - friction force

h

ROLLER EDGES / RING

RIDING

- no load- no skewing- with race flange angle

Hydrodynamic

- Couette flow- friction force- friction torque

Table 3 – Modeling of roller bearing element interactions

36

Basic equations

Equilibrium equations

In this model, the kinematics of the cage and of each roller are unknown, thus sliding at theroller-raceway may occur. They are determined by the force and moment balance of thedifferent bearing elements. They depend mainly on the effective friction coefficients resultingfrom the use of EHL or hydrodynamic models at the different interfaces.

The set of algebraic and differential equations that describes the dynamic behavior of theroller bearing is given here. Note that the set of differential equations, constitutes ofequilibrium equations and geometrical relations, is sufficient to obtain the steady-statesolution.

¾ Fundamental principle of dynamics applied to roller n° J in (Oj ; nj ; tj )

Figure 11 displays the internal geometry of the bearing in the case of roller guided by theouter ring. The interactions around a roller may be represented by traction forces andmoments, as shown in figure 12. The equilibrium of each roller in the reference frame (Oj ; nj ;t j ) is given by the 3 algebraic equations (27) to (29). An additional differential equation (30)which relates the roller position to its rotation speed, see figures 13 and 14, should be added toget the dynamic solution.

Bague Extérieure

Bague Intérieure

Cage

Q

F

FC2

QC2

Q

FC1

QC1

FE

FC

FOL O

CE

CC

oj

oj

j

j

j

oj

oj

j

j

ij

ij

j

j

F

Figure 11 – Internal geometry in the case of rollerguided by the outer ring

Figure 12 – Roller balance, forces and moments

Figure 13 – Kinematic of the roller bearing Figure 14 – Position of the roller in the cage pocket

Oc = cage centerOo = inner ring centerOj = center of roller n°JOo Oj = DMOo Oc = EX

37

0 = Qij – Qoj – FC1j + FC2j + FC (27)

roller · DM/2 · dWj / dt = QC2j – QC1j – FOL + Fij – Foj + FEoj + FEij (28)

roller· [dWj / dt + dWRj / dt] = RR · ( FC1j + FC2j – Fij – Foj ) + CEij + CEoj + CCj (29)

With dΨj / dt = Wj (30)

¾ Fundamental principle of dynamics applied to the cage in (Oc ; X0 ; Y0 )

Efforts applied to the cage are represented in figure 15. Finally, the cage equilibrium in theplan (Oc ; X0 ; Y0 ) is described by the 3 algebraic equations (31) to (33). Three additionaldifferential equations (34) to (36) which relates the cage position to its kinematics, see figures13 and 14, should be added to get the dynamic solution.

Figure 15 – Cage balance, forces and moments Figure 16 – Inner ring balance, forces and moments

mcage·dVxc/dt = ∑=

N

j 1

[(QC2j–QC1j)·cos Ψj + (FC2j+FC1j)·sin Ψj ] + (QCi+QCO )·sinχ (31)

mcage·dVyc/dt = ∑=

N

j 1

[(QC2j–QC1j)·sin Ψj + (FC1j–FC2j)·cos Ψj ] – (QCi +QCO )·cosχ (32)

Jcage·dWc/dt = ∑=

N

j 1

[(QC1j–QC2j)·DM/2 – (FC2j+FC1j)·RR – CCj] + CCi – CCo – EX·QCi·sinχ (33)

With dΨc / dt = Wc (34) dXc / dt = Vxc (35) and dYc / dt = Vyc (36)

¾ Inner ring equilibrium

The quasi-static equilibrium of the inner ring, figure 16, can be reduced to one equationdefined as follows:

∑=

N

j 1

[ Qij . cos Ψj ] – FR = 0 (37)

if the hydrodynamic load QCi at the cage-inner ring contact is neglected compared to thenormal loads Qij at the roller-raceway interface and the applied radial load FR.

The outer ring is stationary and mounted in its housing. The mounting forces ensure the outerring equilibrium without influencing the bearing behavior.

38

Geometrical relations

The radial loading of the bearing produces a radial deflection of the inner ring, noted δr, asshown in figure 17. The knowledge of the elastic deformations and lubricant film thickness atthe inner and outer ring contacts gives for each roller the following algebraic equation, withJD the diametral bearing clearance:

( JD / 2 + δij + δoj – hij – hoj ) – δr · cos Ψj = 0 (38)

Figure 17 – Inner ring radial deflection

Assessment

The set of equations (27) to (38) describes the dynamic behavior of the roller bearing. The5N+7 unknowns corresponding to the set of 5N+7 equations are given in the table 4 below:

Ψj Angular location of roller n°j NWj Precession speed of roller n°j N

WRj Own rotation speed of roller n°j NQij Normal load at the roller n°j / inner ring contact NQoj Normal load at the roller n°j / outer ring contact NXc Horizontal position of the cage center 1Yc Vertical position of the cage center 1Ψc Angular location of the cage center 1Vxc Horizontal speed of the cage center 1Vyc Vertical speed of the cage center 1Wc Rotation speed of the cage 1δr Displacement of the inner ring along Yo 1

Table 4 – Unknowns in the set of equations to solve

39

Other relations

It is possible to reduce the size of the system of equations to resolve simultaneously to 4N+7relations and unknowns by subtracting the N equations (27) and corresponding unknowns Qoj,the load normal at the roller n°j / inner ring contact being deducted from the followingrelation:

Qoj = Qij – FC1j + FC2j + FC (39)

The passage of the roller in loaded zone produced a tilting movement of the roller in the cagepocket. In this model, the position of the roller in the pocket is unknown. According tofigure 14, a geometrical relation is given by the following expression:

H1j = DM/2 · (Ψcj – RC

1[ Xc · cos Ψcj + Yc · sin Ψcj ] – Ψj) – atan (2 · RR / DM) (40)

With Ψcj = Ψc + 2 · π / N · (j – 1)

For line contact, the relation between the normal load Q and the elastic deformation δ is givenby Palmgren [176]:

)/ .(])/E-4(1+)/E-[4(1 0.39 = 0.80.90.92

221

21 Q "ννδ (41)

where ν1, E1, ν2 and E2 are the Poisson's coefficient and the Young's modulus of bodies 1 and2 in contact, respectively, and " the contact length.

40

2.4 Tapered roller bearings

This paragraph presents the methodology to determine the internal load and displacementdistributions in a tapered roller bearing supporting a combination of axial and radial loads.

Assumptions

To simplify the calculation model the following assumptions are made. The inner ring (I.R.)rotates with the shaft at a constant angular velocity while the outer race (O.R.) is stationary.The bearing elements are considered rigid excepting the local zones of concentrated contactswhere the material is ideally elastic. The profiles of the roller and raceway are rectilinear (withno correction); the edge concentrations of pressure for line contacts are neglected. Underloading, the contact angles suffer small deviations and these are neglected. Finally, theskewing motion and bearing internal friction are ignored.

It should be mentioned that these assumptions are suitable for an estimation of the bearinglife. However, a more sophisticated model is required for more complex problems such as theprediction of heat dissipation, or for deeper investigations on abnormal bearing failures suchas the failure of cage.

Kinematics

Coordinate systems

Figure 18 – Tapered roller bearing componentsand Cartesian coordinate systems

Figure 19 – Steady-state equilibrium of a taperedroller

A typical tapered roller bearing is presented schematically in figure 18 containing also thethree principal Cartesian coordinate systems. Thus, it has been introduced an inertial frame(Xi , Yi , Zi) attached to the symmetry axis of the outer ring (or cup). The rotating or azimuthframe (XA , YA , ZA) is linked to the inner ring (or cone). The angular location of each roller isgiven by ψ (ψ = 0o corresponds to the direction of the Xi axis, and ψ = 90o to the Yi axis). Foreach tapered roller was defined a coordinate frame (Xw , Yw , Zw) with its origin fixed at thecenter of the big end of the roller. With respect to the inertial frame (Xi , Yi , Zi) the origin of

41

the roller frame (Xw , Yw , Zw) is defined by the vector components rb = X b , Yb , 0T. Thiscomponents are:

b bw w

i o

w w iX YD

2-

l2

+D

2-

l

2= −

cos sin tan

sin cosγ α α

γ α (42) and bw

Y = D .

2.

sin

tan

γε

(43)

where 2

+ = and

2

- = oiio αα

γαα

ε are the roller and bearing semi-cone angles, respectively.

Basic equations

Displacements and contact deformations

When the inner ring is loaded by the force vector F = Fx , 0 , FzT and is rotated with the

speed Ni (see figure 18), the inner ring suffers a displacement defined by the vector components ∆ = ∆x , 0 , ∆z

T. Consequently, the center of the big end of each roller is displaced in theXwZw- plane and tilted around the Yw-axis through the vector components u = ux , ϕy , uz

T.The problem consists to determine the bearing equilibrium under the external load above-mentioned. In this purpose the expressions for the elastic deformations (to contacts cone, cupand flange) will be defined. The elastic deformations were found as differences between thenormal components, on the contact area, of the corresponding final position vector:

δk = r *k - r

*wk . n k where k = i, o, or f (44)

The vector r* locates in the inertial frame the final position of the raceway surface point thatcomes into contact under load. It will be expressed in terms of the initial position vector r andof the inner ring displacement vector ∆. The vector r*w locates in the same frame the finalposition of the homologue points on the roller surface:

r *k = r k + ∆ where k = i, o, or f (45)

r *wk = r b + ρ *

wk where k = i, o, or f (46)

in which the vector ρ*w defines the distance components between the (Xw , Yw , Zw) - frame

origin and the roller surface point corresponding to the final position in the inertial frame. Thevector ρ*

w will be expressed by adding to the nominal (initial) position vector ρwk of thecontact point the roller displacement u and roller rotation vector ϕ corresponding to eachHertzian conjunction (roller-cone, roller-cup and roller- flange):

ρ*wk = ρwk + u + ϕ × ρwk where k = i, o, or f (47)

Substituting equations (45) and (46) into equation (44) the expressions of the contactdeformations can now be written as:

• for inner and outer line contact (k = i, o):

δi o( ψ, xw ) = fi o + xw⋅gi o with 0 ≤ xw ≤ lx (48)

where the quantities f and g depend both on the bearing internal geometry, the displacements ofthe bearing components (roller and cone) and the angular position of the roller:

fi = ux⋅Ai – uz⋅Bi + ϕy⋅Ci + ∆x⋅Di + ∆z⋅Ei (49a) gi = – ϕi (49b)fo = ux⋅Ao + uz⋅Bo + ϕy⋅Co (49c) go = ϕy (49d)

• for flange - roller end contact (k = f)

δf( ψ ) = – ux⋅Af – uz⋅Df – ϕy⋅Cf – ∆x⋅Df + ∆z⋅Ef (50)

The quantities Ak , Bk , Ck , Dk and Ek ( k = i, o, f ) depend only on the internal bearing geometry(Dw , Rs , lw , αi , αo , αf ).

42

Relation between the normal load and the elastic deformation

From the Palmgren's load-contact deformation relationship the contact load per unit lengthbecomes:

qk( ψ , xw ) = 0,35⋅E'⋅lw0,11⋅δk1,11( ψ, xw ) = Kk⋅δk

1,11( ψ , xw ) k = i , 0 (51)

that becomes by integration [56] :

( )[ ] [ ],0)(-)l,(K19

9 =f-glfK

19

9=)(Q 19/9

kw19/9kk

19/9kkwk

19/9kk ψδψδ⋅⋅⋅+⋅⋅ψ (52)

The negative values for δk( ψ, 0) and δk( ψ, lw) will indicate no contact between the roller andthe raceway. The flange - roller end contact is considered as a classical Hertzian point contact.The normal load at this contact was determined by the following relationship:

δf ( ψ ) = 27,8⋅10-5⋅δ*⋅Qf2/3⋅( Σρ )1/3 (53)

in which Qf (in Newtons) is the contact normal load, δ* is the dimensionless contact

deformation, and )sincos-sin(R+Y

sin-

R

2 =

sbs γλθθρΣ (in mm-1) is the curvature sum at the initial

point of the contact area, where Yb is given by equation (43) and λ = arcsin( Dw/2Rs ).

Roller equilibrium

The tapered roller equilibrium is schematically illustrated in figure 19 The quasi-dynamicequilibrium of a tapered roller is given by the following system of equations which are needed todetermine the unknowns ux , ϕy and uz for each roller:

γαγε

αγλ

γαγε

Σ

Σ

Σ

0

0

0

=

sinF-)-(cosQ+sin)Q+Q(-

M-)-(sin)l-cosR(Q+M-M

cosF+)-(sinQ+cos)Q-Q(

=

Q

M

Q

cffoi

gfcsfoi

cffoi

z

y

x

(54)

where Fc and Mg are the centrifugal force and gyroscopic moment, respectively. The quantitiesMi and Mo are the total contact moment for inner and outer raceway, respectively.

oi,=k, dX.X).X,(q = M wwwk

l

0

k

w

ψ∫ (55)

The non-linear system of equations (54) is solved by the Newton-Raphson iterative method.

Bearing equilibrium

In the XI and YI directions of the inertial frame, the external forces Fa and Fr must balance theinternal contact loads Qi and Qf on the inner ring. The following equations which define theinner ring equilibrium are needed to calculate the displacement vector ∆= ∆x, 0, ∆z

T. Thisnon-linear system is also solved numerically by the Newton-Raphson iterative scheme.

∫∫ππ

ψψπα

−ψψπα

Σ2

0

ff

i

2

0

iax d)(Q

2

cosd)(Q

2

sin-F=F (56)

∫∫ππ

ψψψπα

+ψψψπα

Σ2

0

ff

i

2

0

irz dcos)(Q

2

sindcos)(Q

2

cos-F=F (57)

43

2.5 Numerical procedure

Steady-state equilibrium

The sets of equations 0][&&&

=XH that describe the ball, cylindrical roller, and tapered rollerbearing equilibrium are constituted of non-linear algebraic equations for which there is noanalytical solution. Each system of equations is solved using the Newton-Raphson iterativemethod. By a first order limited development of each equation it comes:

0 = Hi(…, xj, …) = Hi(…, xj0, …) + ∑=

N

j 1

(xj – xj0) j

ji

x

xH

∂∂ ,...)(..., 0

where xj0 are the components of the initial vector 0X&

, and xj those of the final vector X&

.

Finally, the set of equations to solve becomes:

)X(HXx

)X(H0

j

0i&&&

&

−=∆•

∂

∂

where 0XXX&&&

−=∆ is the increment vector to determine. Thus, the vector XXX&&&

∆+= 0

becomes the new initial vector. The procedure is repeated and the process converges towardsthe solution whereas the norm of the error vector )( XH

&&

tends to 0. The numerical solution isreached when the maximal difference between to successive vector components is inferior tothe required precision ε.

The used method converges if the jacobian determinant calculated at each iteration is differentfrom zero. This case is always verified in practice for the 3 systems of equations definedpreviously for ball, cylindrical roller and tapered roller bearings. This quite robust methodconverges quickly when the initial solution is judiciously chosen. Typically, the convergenceis reached after 20 to 80 iterations for a maximum relative error of 10-6, depending on thenumber of rolling elements, the initial solution and on the choice of the relaxation coefficient.

Dynamic equilibrium

The system obtained to describe the dynamic behavior of cylindrical roller bearings is a non-linear algebro-differential system. It consists of 4N+7 equations with 4N+7 unknowns (seetable 4 in paragraph 2.2).

The system is thus of the form:

Differential equations ]t,)t(X[Gdt

)t(Yd &&

&

=

Algebraic equations 0])t(X[H&&&

=

where the unknowns are the components of the vector )(tX&

.

Equations are first discretized by an implicit Euler type method, which was selected for itsstability [96, 97]. Finally, a system of non-linear equations whose unknowns are given by the

)( ttX ∆+&

components is obtained and solved by the iterative method of Newton-Raphson.

44

3. Elastohydrodynamic lubrication (EHL)

3.1 Lubricant film thickness

Isothermal minimum and central film thickness hmin et hcen in fully flooded conditions

Under isothermal, steady-state and fully flooded conditions, the Hamrock and Dowsonformulae can be written for elliptical point contact geometry [99, 100, 101, 102, 103] :

)e -(1 * W*G * U2.266 = *H -0.68k-0.0730.490.68m (58)

)e 0.61-(1 * W*G * U1.691 = *H -0.73k-0.0670.530.67o (59)

where 1 )/R(R 1.03 =k 0.64xz ≥ (60)

With H*, U*, G* and W* the following dimensionless parameters:

x

iso

R

h*H = the lubricant film thickness parameter,

where hiso is the isothermal film thickness,

and2x1x

2x1xx RR

R.RR

+= the radius in the rolling direction,

UU

E Ro R

x

*.

.= µ the speed parameter,

where UR = U1 + U2 is the rolling speed,1

2

22

1

21

E

1

E

1.2E

−

υ−+υ−= the reduced elastic modulus,

and µo the inlet contact viscosity,

WQ

E Rx

*.

= 2the load parameter,

where Q is the normal load,

and G* = α.E the material parameter.

It is also possible to use Nijenbanning et al.'s formulae [174] for the calculation of theisothermal central film thickness. They present the advantage of extending the validity of thecalculation from EHL (piezo-viscous elastic regime) to hydrodynamic lubrication condition(piezo-viscous and isoviscous rigid regimes).

Thermal reduction factor

The viscosity of oil decreases very quickly when temperature increases. Thus, for large rollingand sliding speeds, energy due to lubricant shearing have significant effects on film thickness.Film thickness h can be calculated from the isothermal one hiso given above and with athermal factor φt such as:

h = φt . hiso (61)

45

Cheng [40], and Wilson, Sheu and Murch [153, 237] deduced a thermal reduction factor fromtheir numerical solution of the thermo-elastohydrodynamic lubrication of rolling-slidingcontacts. From these works, Gupta et al. [92] suggested the following reduction factor:

( ) 640.083.0

42.0o

LS23.21213.01

LE

p2.131

t ++

−

=φ (62)

where L is a thermal dimensionless coefficient,

( )2

f

21

K2

UU

TL+

∂

∂µ−= (63)

and S the slide dimensionless coefficient,

SU U

U U= −

+2

1 2

1 2(64)

The effects of the normal load are given by the term containing the hertzian pressure po.Thermal reduction coefficient variations with L and S parameters, calculated from equation(62), are shown in figure 20.

Figure 20 – Film thickness thermalreduction coefficient

There is also another formula, more particularly adapted to line contacts, which is due to Hsuand Lee [113]:

( ) 1S.875.0527.0447.0*

L687.0

T eLWG0766.01−

⋅+=φ (65)

where WL* is the dimensionless load parameter for line contact.

3.2 Rheological model for the calculation of friction force and torque

It is commonly known that lubrication in rolling bearings prevents metallic contacts, reducesfriction and wear of interacting elements, and serves as a coolant for the bearing. Further, thetractive behavior of the lubricant and the substantial quantity of oil present in the bearingcavity play a significant role in bearing performance. The magnitude of the frictioncoefficients found in the bearing and the air-oil mixture ratio greatly influences cage speedand power loss. In view of the scope of the subject, this section is limited to the discussion ofthe isothermal and thermal models and formulations effectively used in our computer codes.Note that the contact dimensions and the pressure distribution are given by the theory of Hertz[108] for line or point contacts.

46

Isothermal model

Various studies have shown that under high pressure or short time of solicitation, the lubricantbehavior can not be described by a simple Newtonian model. More complex rheologicalmodels have been suggested and relied on transient viscosity and viscoelasticity to explaindiscrepancies between Newtonian theory and experiments. Johnson and Tevaarwerkdemonstrated that a simple non-linear model, such as the Maxwell model, could be used topredict with good accuracy traction results obtained on a two-disk machine. In thisviscoelastic model the vector rate of deformation γ

& has two constituents, one elastic, Eγ

& , and

the other one viscous, Vγ& :

ττ

µτ

ττ+τ=γ+γ=γ

ref

refVE F

G

&

&

&&&

&

& (66)

where τ& is the shear stress vector, G the elastic shear modulus, µ the dynamic viscosity, τref acharacteristic shear stress of the lubricant and F(x) a non-linear function.

For a Newtonian fluid, i.e. for low shear stress, the viscous law is linear:

µτ=γ&

& V

(67)

For a non-Newtonian fluid, or for high shear stress, different non-linear forms of the viscousterm have been proposed by Ree and Eyring [187, 188], Bair and Winer [6, 7], Gecim andWiner [86], and by Elsharkawy and Hamrock [77]. The non-linear function takes thefollowing form depending on the model used:

• ( ) ( )xsinhxF = (Ree & Eyring, 1955)

• ( ) ( ) ( )x1lnxxxF −−= (Bair & Winer, 1979)

• ( ) ( )xtanhxF 1−= (Gecim & Winer, 1980)

• ( ) ( ) n1nx1xxF −= (Elsharkawy & Hamrock, 1991)

The reference shear stress τref becomes a characteristic stress τo in the Ree and Eyring's modelor a limiting stress τL in other models. Note that for each fluid the viscosity µ, the shearmodulus G, and the characteristic or limiting stress τo or τL are assumed to be only relating topressure and temperature.

For high rolling speed along the (Ox) axis, with U the rolling speed, equation (66) becomes:

ττ

µτ

ττ+

∂τ∂=γ

ref

ref FxG

U&

&

&&& (68)

For the Ree-Eyring form of the viscous term, when sliding occurs in both x and z directions,equation (68) may be replaced by the set of equations (69) :

τ

τµτ

ττ

+τ

=γ

τ

τµτ

ττ

+τ

=γ

o

o

o

o

eq

eq

zzz

eq

eq

xxx

shdx

d

G

U

shdx

d

G

U

(69)

Where 2z

2xeq τ+τ=τ ,

h

.zUx

∆Ω+∆=γ andh

.xWz

∆Ω−∆=γ

γ&

47

Figure 21 presents an example of results obtained with unidirectional (line contact or 1D) orbi-directional (point contact 2D) models.

Figure 21 – Comparison of theoreticaltraction curves obtained with different

models (1D or 2D) or meshing.

Values of rheological parameters G and το, which vary with pressure and temperature, areobtained by curve fitting of experimental results. Figures 22 and 23 present the effects of thevariation of the elastic shear modulus G and of the characteristic stress το on the theoreticaltraction curves. As one can see, the shear modulus relates to the slope of the traction curvearound the origin whereas the characteristic stress is associated to the maximum frictioncoefficient.

Figure 22 – Theoretical traction curvesvs. the elastic shear modulus of the lubricant

Figure 23 – Theoretical traction curvesvs. the characteristic stress of the lubricant

The model can also take into account the tangential elastic deformation of the surfaces by themean of a compliance coefficient, L1, which is function of the contact geometry and elasticproperties of the material [134]. Equation (66) becomes then:

( )c1L

L

hLG1

GGihw ,F

xG

U

+=′

ττ

µτ

ττ+

∂τ∂

′=γ

&

&

&&& (70)

48

Thermal model

Thermal effects due to lubricant shearing have been included in the traction modeling byusing a simplified approach similar to those proposed by Houpert et al. [19, 109, 110, 112],itself based on the work of Johnson and Tevaarwerk [116, 215]. Moreover the thermalreduction factor of the EHL film thickness, φT, is used to determine the temperature rise in theinlet zone, ∆T, with the assumption that the isothermal EHL film thickness at T0 + ∆T is equalto the thermally corrected EHL film thickness at T0, i.e. hiso(T0 + ∆T)= φT x hiso(T0).

To take into account the lubricant heating by shearing, the rheological behavior of thelubricant describes by equation (66) is coupled to the energy equation (77), integrated alongthe film thickness:

x

T

k

cv

k

q

z

T2

2

δδρ

+−=δδ (71)

and to equation (72) [110], which gives the surface temperature rise (assuming that bothsurfaces are of identical material and have the same temperature):

( )( )∫−

ξξ−

−

πρ=−=∆

x

ya

surface

sss0ss d

x

dzdTk

Ukc

1TTT (72)

where q = ⋅& &γ τ represents the heat flux per volume unit dissipated in the lubricant film

thickness, k, ρ and c are the lubricant thermal conductivity, mass density, and mass capacity,respectively, and U is the average surface speed in the contact. The subscript s refers to thesurface properties.

Finally, the set of equations is integrated by the procedure proposed by Houpert [110],modified to accelerate the convergence.

3.3 Rheological parameters

The lubricant behavior is represented by three parameters, which are considered to vary withpressure and temperature: the dynamic viscosity µ(p,T), the shear modulus G(p,T) and acharacteristic shear stress τo(p,T) or limiting shear stress τL(p,T). Note that the knowledge ofthe lubricant viscosity versus pressure and temperature is sufficient for the calculation of theEHL film thickness. However, the evaluation of the traction forces requires in addition theknowledge of the shear modulus, and of the characteristic or limit shear stress according to theviscoelastic model used.

Viscosity

Several laws have been proposed to describe the viscosity variation with temperature andpressure. Among them are the well known Barus' law, the Roelands law, and also the WLFmodel. It appears that the WLF model modified by Yasutomi et al. [241], gives the bestcorrelation with the experimental results obtained for the fluid studied in the laboratory withP. Vergne, while covering a wider domain of pressure and temperature. Moreover, this modelis based on the physical concept of free volume, which presents the advantage of nicelyrelating the mechanical properties of the lubricant, such as its viscosity, to its physical state asfor example the free volume. This physical base confers to the WLF model the possibility toreproduce faithfully the variations of viscosity with pressure and temperature on almost all the

49

domain where the lubricant is liquid. It is also possible to use it far beyond the domain ofcharacterization, what can not be claimed by other models used by tribologists that are mostlyempirical.

For memory the Barus' law is written as [11]:p.e.)p( 0

αµ=µ (73)

which, generalized in a domain of pressure and temperature, gives from Gupta et al. [93]:

( )

−β+α

µ=µ 00

T

1

T

1.P.

e.T,p (74)

where α and β are the pressure- and temperature-viscosity coefficients, respectively, and To

and µo the reference temperature and viscosity. It should be noted that in equations (73) and(74) the exponential form of the viscosity variation with pressure overestimate the lubricantviscosity when the pressure exceeds 0.5 GPa, which makes these relations inaccurate forrolling bearing applications.

It is why today most tribologists prefer the Roelands' law [192]. This law is based on physicalconsiderations, at the opposite of the Barus law that is only empiric. It is particularly adaptedto mineral lubricants for which it was developed. After rewriting, the Roelands' law becomes:

[ ] ( )

−

−−++µ

µ=µ

−− 1

138T

138T101.5167.9)ln(

e)T,p(

0S

0

Z90

.0 (75)

where Z and So are the Roelands' parameters, assumed independent of pressure andtemperature, and that should be determined for each lubricant.

The WLF model modified by Yasutomi et al. [241] is given by the following expression:

( ) ( )( ) )p(F)p(TTC

)p(F)p(TTClogp,Tlog

g2

g1g1010 ×−+

×−×+µ=µ (76)

with ( ) ( ) ( ) ( )pA1lnATpTet pB1lnB1pF 210gg21 ++=+−=

where Tg(p) and F(p) represent the variation with pressure of the glassy state transitiontemperature and of the thermal expansion coefficient, respectively. Note that this modelrequires the determination of 6 constants for each lubricant: A1, A2, B1, B2, C1 et C2.

Pressure-viscosity coefficient

As discussed before, the pressure-viscosity coefficient is useful to calculate the EHL filmthickness. To take into account the effect of the pressure on the viscosity of the fluid westudied (Mobil Jet II, Pennzane SHF X2000, Fomblin Z25, Nye 186 A), it has been shownthat a simple Barus' law is inappropriate because unable to reproduce the curvature effectobserved on experimental curves, as seen in figures 25, 26 and 31 to 33.

The "inverse isoviscous asymptotic pressure", called α*, has been introduced to avoid thisdifficulty. It is defined as:

50

( ) ( )∫∞

µµ=

α0

* p,T

dp0,T

1(77)

It was shown by Bair [8] that this parameter was the most relevant for EHL film thicknesscalculation. Furthermore, its variation with temperature is easy to put in equation.

Shear modulus and characteristic or limit shear stress

The shear modulus G and the characteristic or limit shear stress, τo or τL, are assumed to takethe following forms [18, 84, 92, 93, 110] :

( )G T p G pT T

G G, exp= + −

0

0

1 1α β (78)

( )τ τ α βτ τL LT p pT T

, exp= + −

0

0

1 1 (79)

The hypothesis according to which µ, G and τL reach instantaneously their thermodynamicequilibrium value authorizes a fast calculation of traction forces. In practice, that limits thevalidity of the model to rolling speeds nearby that used for the traction force measurement.For a wider domain of validity, G and τL should also be functions of the time of passage in thecontact (tc = 2a/U).

Figure 24 – Experimental traction curvesFriction coefficient vs. the sliding rate, for different Hertz pressure.

Lubricant : Oil Mobil Jet II ; Specification : MIL-L-23699 ; Temperature : 373 K ; Speed U1+U2 : 42,35 m/s.

Direct rheological measurement of G and το or τL are possible but very delicate to implement.It is easier to obtain these parameters from experiments on two-disk machines, by measuringthe friction force versus the slide-to-roll ratio for different rolling speed, temperature andcontact pressure. An example of such experiments is given in figure 24. Parameters G and τοor τL are determined by minimizing the error between experimental and theoretical results.The model used in figure 24 is the isothermal model of Johnson and Tevaarwerk with a Ree-Eyring form of the viscous behavior.

51

Let introduce the Deborah number D and let study the fluid behavior according to thisnumber. The Deborah number compares the relaxation time of the fluid to the time of passagein the contact:

Dt

= τ(80)

where Gµ=τ and t

a

U= (µ and G are mean values in the contact).

then DU

G a= µ.

.(81)

There are different types of fluid behavior according to the Deborah number D:

For small D number, i.e. at low pressure and viscosity, (low rolling speed, hightemperature, high shear modulus), the fluid behavior is purely viscous non-linear thenequation (66) becomes ( Eγ << Vγ ):

ττ

µτ=γ≅γ

o

o

shV (82)

For large D number, i.e. at high pressure, viscosity or pressure-viscosity coefficient, thefluid behavior is elastic at low shear rate and viscous non-linear at high shear rate, then:

- if τeq << τo, the fluid behavior is purely elastic ( F(τeq) = 0 ).

- if τeq = τ o, the fluid behavior is viscous non-linear therefore 0=Eγ . The set of equations(69) is then reduced to:

ττ

µτ

ττγ

ττ

µτ

ττγ

=

=

o

eo

e

zz

o

eo

e

xx

sinh

sinh

(83)

then xeeq

x τ=τγγ

and ze

eq

z τ=τγγ

where

ττ

µτ=γ+γ=γ

o

eo2z

2xeq sinh

⇒ =

−τ τ

γ µτ

e oeq

osinh

1

Note that when o

eq

τµγ

is large, then

τ

µγτ≈τ

ooe

eq2

ln

For small Deborah number and shear stress rate, i.e. for small slide-to-roll ratio, the fluidbehavior is essentially elastic. Thus, by neglecting the viscous term in the set of equations(69), and by taking an average shear modulus, it can be shown that the slope of the tractioncurves around the origin is given by the following expressions:

( ) o

2

2121 p.h

a.3G

Q.h

c.a..4G

UUUU

cf =π=+−∂

∂(elliptical contact) (84a)

( ) o

2

2121 p.h.

a.16G

Q.h

l.a.8G

UUUU

cf

π==

+−∂∂

(line contact) (84b)

52

where cf is the friction coefficient. Therefore, it exists a linear relation between the shearmodulus and the slope at the origin of the traction curves, as previously illustrated in figure22.

When the slide-to-roll ratio increases, the viscous term becomes significant. Finally, at highsliding rate, the fluid behavior is mostly viscous. The friction coefficient then increases veryslowly with the slide-to-roll ratio if thermal effects are neglected. This lead to the

establishment of equations 85(a) and (b) relating the characteristic shear stress τO to thefriction coefficient cf.

τ

τ=

τπ

τ=

µ

∆

o

o

o

o

oo

o

o .3

p.cf.2sh.

p.c.a.

Q.cfsh.

pp.h

.U.

U

U(85a)

for elliptical point contact, or:

τπ

τ=

τ

τ=

µ

∆

o

o

o

o

oo

o

o .4

p.cf.sh.

p.l.a.2

Q.cfsh.

pp.h

.U.

U

U(85b)

for line contact.

The value of the characteristic shear stress το may be obtained by resolving the aboveequations.

Equations (84) and (85) allow to determine preliminary values of G and το, for a giventraction curve. Finally, the final values, which are pressure and temperature dependent, areobtained by curve-fitting the experimental and theoretical curves in the studied domain.

3.4 Hydrodynamic torque resistant to rolling

For EHL point contacts, the dimensions of the contact area are given by the Hertz theory. Thecontact is elliptical, with a the semi-minor axis in the rolling direction and c the semi-majoraxis in the transverse direction. The pressure distribution is approximated by the ellipticalhertzian solution and the contacting surfaces are separated by a lubricant film thicknessassumed uniform for the computation of traction force and torque. In fact in the reality, theEHL film thickness geometry and the pressure distribution differ slightly from the Hertzian(or dry) solution because a slight hydrodynamic pressure is generated before the contact inletand a restriction of the lubricant film thickness located near the contact outlet produces a peakof pressure. Consequently, the resultant pressure force is located lightly before the geometriccontact center. This distance δ', which depends on the normal load, surface speed and oilviscosity, is given by Hamrock in references [111, 215] relating to the dimensionlessparameters U* , W* and G* previously defined.

91.035.0

2

3/8022.0

2

3

a

c

)2/*U(

*W

)2/*U(

*W.*Ga25.4'

=δ

−

(86)

Then the resistant momentum can be expressed by:

M Qroll = δ'. (87)

53

3.5 Oleodynamic drag

The bearing is filled with a mixture of air and oil. The drag force which is imposed on therolling elements when they translate through a fluid can be estimated using the followingrelation (here for ball bearings):

Fol = 1/2 ρ Cd S V2 (88)

Where ρ = [Xol.ρoil+(100-Xol).ρair]/100 is the equivalent oil-air mass density of the fluidpresent in the bearing cavity, with Xol the amount oflubricant in the bearing cavity (in percent),

S = π.D2/4 – Lc.D is the ball frontal area to the flow direction with Lcthe cage thickness,

and V = 1/2 dm.ω m is the orbital velocity of the ball.

The drag coefficient Cd was determined by Schlichting [197] as function of the Reynoldsnumber defined as ℜe = ρVD/µ, where µ is the dynamic viscosity of the fluid. For example,for a ball and a Reynolds number varying between 103 and 2.105, the Cd value varies between0.4 and 0.5.

Parker [177] showed that for different speeds and ball bearing geometries, and for alubrication through the inner ring or by an external jet, the amount of lubricant in the bearingcavity may be estimated by the following equation:

1,7

0,37710

md.Ni

QhXol = (89)

where Xol is the amount of lubricant in the bearing cavity (in percent),

Qh is the oil flow rate (in cm3/min),

Ni is the shaft rotation speed (in rpm),

and dm the bearing pitch diameter (in mm).

54

4. Main Results

4.1 Rheological and tribological properties of some liquid lubricants used in turbine enginesand in aerospace

Experimental devices

Rheometers operating at ambient pressureIt concerns the measurement of the dynamic viscosity for temperatures ranging from -50 to+200°C and shear rates up to 10+5 s-1. To cover the domain of temperature, several devices areused: i.e., a cone / plan rheometer for low temperatures and another one with coaxial cylindersfor high temperatures as well as for measurement at high shear rate.

High pressure rheometerThe rheometer used for high pressure measurement was developed in the Laboratory ofMechanics of Contacts (LMC) [190, 221]. It is the principle of a falling body viscometer.Under the effect of the gravity, a cylindrical body moves inside a vertical cylinder. The fallingspeed ratio obtained under various conditions of pressure and temperature is inverselyproportional to the variation of the dynamic viscosity. The falling speed is measured by meansof an ultrasonic sensor calibrated before each test. Measurement can be made in a range oftemperatures between 20 and 164°C, for a pressure up to 700 MPa and a viscosity varyingbetween 10-3 and 10+3 Pa.s.

Two-disk machineTraction tests have been carried out on the high-speed two-disk machine available in thelaboratory [82, 83]. The values of the EHL parameters simulating the real operating conditionsare reproduced at the nominal scale and in permanent regime. Contact is realized between twotest disks of parallel axes independently driven. Mechanical parameters imposed whichgovern the EHL contact are the macro and the micro-geometry of the disk raceway, thevelocity of the contacting surfaces, the normal load as well as the flow and temperature of theoil jet. This device allows, among others, to measure the friction coefficient continuouslyduring any variation of the sliding rate (U1-U2)/(U1+U2) from -10 to +10 %, for various rollingspeeds, lubricant temperatures, and contact pressures.

Lubricant for aeronautic applications (Oil Mobil Jet II, MIL-L-23699 specifications)

Tested lubricantThe lubricant tested is a tetraester of 5 cst viscosity at 100°C, qualified for use in gas turbineengine lubrication systems under the MIL-L-23699 specifications. This lubricant has beenstudied by Houpert [110], Gupta et al.[93], and more recently by Vergne and Nélias [220].

Thermal conductivity KfThe lubricant thermal conductivity Kf is given in N/s/K by the following relation [93]:

( )67,366T1022753,2148676,0K 4f −⋅−= − (90)

55

Specific mass ρ(T)The specific mass has been measured [220] at ambient pressure in a domain of temperatureranging from -20 to 170°C. It varies linearly versus the temperature, with a negative slope of-0.752 g/ml/K and a reference value of 1013.4 g/ml at 0°C:

( ) ( )15,273T752,04,1013T −⋅−=ρ (91)

Dynamic viscosity µ(p,T)Measurement performed at ambient pressure did not exhibit a Newtonian behavior for shearrate up to 3 104 s-1. However, the dynamic viscosity varies very strongly according to thetemperature: it is divided by more than 100 between -5 and +170°C. A dynamic viscosity of157, 17, 4.7, 2.1, and 1.2 mPa.s was found at 0, 50, 100, 150, and 200°C, respectively.

Measures versus pressure up to 600 MPa were performed at 0, 40, 82, 124 and 164°C. Resultswere analyzed by means of the modified WLF model as proposed by Yasutomi et al.[241].Figure 25 presents a comparison between the experimental and theoretical results that weobtained [220]. An excellent agreement was observed. Figure 26 shows also results obtainedby other authors for the same lubricant. For the same oil Gupta et al. [93] used the Barus' law,equation (74), whereas Houpert [110] attempted to deduct the value of the Roelandsparameters Z and So, equation (75), and while Prat et al. [182] used a power law. The valuethat we obtained for the 7 parameters of the WLF modified model are indicated in table 5 thatfollows.

Pressure (MPa)0 200 400 600 800 1000

Vis

cosi

ty (

Pa

.s)

10-3

10-2

10-1

100

101

102

103

Exp. 0°C Exp. 40°C Exp. 82°C Exp. 124°C Exp. 164°C WLF model 0°C WLF model 40°C WLF model 82°C WLF model 124°C WLF model 164°C

Pressure (MPa)0 200 400 600 800 1000

Vis

cosi

ty (

Pa.

s)

10-3

10-2

10-1

100

101

102

103

104

105

Exp. 40°C Exp. 164°C Gupta 40°C WLF 40°C Roelands 40°C Power law 40°C Gupta 164°C WLF 164°C Roelands 164°C Power law 164°C

Figure 25 – Comparison between the dynamic viscositymeasured and the theoretical one given by the modified

WLF model (Oil Mobil Jet II).

Figure 26 – Comparison between different models:Barus' law [93], Roelands' law [110], power law [182],

and modified WLF model [220] (Oil Mobil Jet II).

A1 (°C) A2 (GPa-1) B1 B2 (GPa-1) C1 C2 (°C) Tg(0) (°C)171,96 0,4294 0,1961 17,434 16,342 29,406 -107,4

Table 5 – Value of the parameters of the modified WLF model (Oil Mobil Jet II).

56

Inverse isoviscous asymptotic pressure α*(T)The values of α* obtained at different temperatures are given in table 6, as well as thepressure-viscosity coefficient commonly used and noted α50 as it is deduced from ameasurement at 50 MPa through the Barus' law. Here α* comes from a numerical integrationof equation (77) in a large domain of pressure in order to reduce the numerical error.

T (°C) α* (GPa-1) α50 (GPa-1)0 20,2 20,140 14,8 15,782 11,5 13,0124 9,15 12,0164 7,60 11,0

Table 6 – Comparison between α* and α50 (Oil Mobil Jet II)

From direct integration of equation (76) in equation (77), it comes the following relationshipwith the temperature expressed in Kelvin:

244

2

6

10599.14726.066.16510626.410292.6

* TTTT

−++

+−+−=α (92)

Only one paper by Present et al. [183] has been found in the literature to compare with ourresults for pressure up to 600 MPa. Among 10 fluids studied by Present et al., 4 are syntheticfluids qualified for use in turbine engine under the MIL-L-23699 specifications. The result ofour investigation is compared with published data of Present et al. [183] in table 7. Results arein agreement although our results show a stronger variation of the viscosity with pressure atlow temperature (40°C). This difference may result from a numerical integration error, ameasurement error, or from a difference in the chemical composition of lubricants. However,a comparison with ASME data for a di(2-ethylhexyl)-sebacate fluid shows a maximum errorof 15 % for the viscosity measurement [222]. Consequently, it seems that the observeddifference is related to the nature of tested fluids.

Fluid C[183]

Fluid D[183]

Fluid I[183]

Fluid J[183]

Mobil Jet II[220]

40°C 11,6 12,4 11,9 11,9 14,8100°C 9,85 9,72 8,95 8,95 10,4150°C 8,41 8,23 7,88 7,85 8,11

Table 7 – Value of the “inverse isoviscous asymptotic pressure” α* (in GPa-1),comparison between our results [220] and the ones of Present et al. [183].

A comparison between the results obtained by Gupta et al. [93] (with a Barus' law), byHoupert [110] (Roelands' law), and our current investigations (experiments and modifiedWLF model) is given in figure 27. A significant difference is found between results of eachmodel. This may have rather important consequences, as for example on the evaluation of theoil film thickness. Figure 28 gives the central film thickness calculated with the EHL formulaproposed by Nijenbanning et al. [174], for a spherical point contact (Rx=Rz=0,02 m), amaximum contact pressure of 1,5 GPa and surface velocities U1=U2=40 m/s. As shown infigure 28 the error in the evaluation of the lubricant film thickness can reach 20 %.

57

Temperature (°C)

0 50 100 150 200 250

α* (P

a-1

)

5x10-9

6x10-9

7x10-9

8x10-9

9x10-9

2x10-8

3x10-8

10-8

ExperimentWLF modelGupta modelRoelands model

Temperature (°C)

50 100 150 200

Hce

ntre

2

3

4

5

6

WLF model Gupta model Roelands model

Figure 27 – Value of α*:comparison between our results [220] (with the modifiedWLF model) and those of Gupta et al. [93] (Barus' law)

and Houpert [110] (Roelands' law)(Oil Mobil Jet II).

Figure 28 – Central lubricant film thickness calculatedat 1.05 GPa with the formula of Nijenbanning et al.

[174] for different models: modified WLF model [220],Barus' law [93] and Roelands' law [110]

(Oil Mobil Jet II).

Lubricant elastic shear modulus G(p,T) and characteristic shear stress το(p,T)The value of various coefficients used in the calculation of G and το (see equations 78 and 79)which is deducted from experimental traction curves obtained on a two-disk machine, is givenin table 8. These results are valid only for the studied lubricant (Mobil Jet II), in the envelopeof experimental operating conditions used for testing, and for the type of model used in thecalculation (here an isothermal model of Maxwell, with a Ree-Eyring viscous term). They donot correspond to intrinsic values of the lubricant. Finally, a comparison of experimental andtheoretical curves is given in figure 29.

To 300 °K To 300 °K

Go 4,82 104 Pa το 1,56 105 Pa

αG 6,11 10-9 Pa-1 ατ 2,41 10-9 Pa-1

βG 2,45 103 K βτ -7,86 102 K

Table 8 – Parameters of the elastic shear modulus and characteristic shear stress obtained forthe Mobil Jet II oil qualified under the MIL-L-23699 specifications

58

Experiments Theory (isothermal model)G = 4,82 10 4 exp(6,11 10 -9 P + 2450 (1/T-1/300))τ = 1,56 10 5 exp(2,41 10 -9 P + 786 (1/T-1/300))

Figure 29 – Comparison between experiments (left column) and theory (right column)for 2 different rolling speeds (42.35 and 121.93 m/s) and temperatures (297 and 378 K).

Oil Mobil Jet II (MIL-L-23699)

59

Lubricants for aerospace applications (Pennzane SHF X2000, Nye 186 A, Fomblin Z25)

Some fluid lubricants (oil or grease) were developed specifically for spatial applications. Theysupplant in some mechanisms solid lubricants such as lead, PTFE or MoS2. Indeed, solidlubrication turns out badly adapted for mechanisms operating at high contact pressure, highsliding velocity, or requiring high fatigue life or a constant coefficient of friction. Thus, solidlubrication remains inappropriate for rolling bearings.

Three oils developed for spatial applications (Pennzane SHF X2000, Fomblin Z25 and Nye186 A) were studied. If the first two oils are used for applications under secondary vacuum foralready several years, the last one is new and presents an interest for mechanisms workingunder partial pressure. Let us note that Pennzane SHF X2000 and Fomblin Z25 were alreadythe object of rheological characterizations [182, 204]. However, results will be recalled toallow a comparison with the Nye 186 A oil.

Tested lubricantsLiquid lubricants developed for aerospace applications should have specific properties. Thevapor pressure is low in order to minimize losses by evaporation and to limit pollution due tothe degassing. Their viscosity index is high; so, their viscosity is hardly sensitive totemperature variations as those found by mechanisms in orbit (from -30°C to +80°C). Theirinertia and chemical stability are generally compatible with a 10-year life duration as required.

The Pennzane SHF X2000 oil is a non-linear synthetic hydrocarbon of typical chemicalformula C65H130.

The Fomblin Z25 oil is a perfluoropolyether (PFPE) of typical chemical formula:

CF3-[(O-CF2CF2)p-(O-CF2)q]n-O-CF3 where p/q=0,6-0,7 and m≈13000 gr/mol

The Nye 186 A oil belongs to the family of synthetic hydrocarbons and can be related to thefamily of poly-alpha-olefines. It is a poly-1-decane: this fluid should be considered as apolymer with mass spread over a wide specter.

Some interesting properties of the fluid lubricant studied are reported in table 9 below. Theseproperties, which come from documents supplied by the manufacturers, are useful for a spaceusage.

Pennzane SHF X2000 Fomblin Z25 Nye 186 A

Vapor pressure 1.4 10-12 mbar at 20°C 4 10-12 mbar at 20°C 7 10-8 mbar at 26°C

Flow point < -55°C -70°C < -48°C

Kinematic viscosity 110 cst at 40°C14.6 cst at 100°C

154 cst at 38°C47 cst at 99°C

103.2 cst at 40°C14.6 cst at 100°C

Viscosity index 137 355 146

Specific mass 0.85 gr/ml at 15.6°C 1.85 gr/ml at 15.6°C

Evaporation 0.06 %(23h at 125°C, 10-5 mbar) 0.03% (22h at 149°C) 0,9% (22h at 149°C)

Table 9 – Properties of the fluid lubricant studied

60

ViscosityAs discussed previously, the modified WLF model proposed by Yasutomi et al. [241] givesthe best results compared with viscosity measurement. The parameters of the model obtainedfor the three fluid lubricants studied are given in table 10 below. For numerical reasons, Tg0,the temperature of glassy-state transition at the atmospheric pressure, was determine for atransition viscosity µg of 10+7 Pa.s, instead of 10+12 in theory.

Pennzane SHF X2000 Nye 186 A Fomblin Z25µg (Pa.s) 107 107 107

Tg0 (K) 185.691 188.04 152.455A1 (°C) 69.8093 53.9262 48.3241A2 (GPa-1) 1.67903 2.26829 2.96467B1 0.212452 0.223438 0.224572B2 (GPa-1) 11.8028 12.4888 23.8537C1 11.8362 11.5171 10.0745C2 (°C) 60.5908 53.979 54.4713Mean error // measures 4.95 % 3.43 % 5.44 %Maximum error // measures 16.94 % 8.00 % 13.38 %

Table 10 – Parameters of the modified WLF model for the 3 fluid lubricants studied

The experimental and theoretical evolution of the ambient pressure viscosity with temperatureis shown in figure 30 for the three studied oils. Note that tests have been performed in adomain of temperature typical of space applications. Two comment follow the observation ofresults presented. First, the viscosity of the Fomblin Z25 oil is more stable than that of the 2other lubricants (the Y axis is logarithmic). Second, the Pennzane SHF X2000 and the Nye186 A oils present very close value of viscosity and similar trends for the variations withtemperature, at such a point that the same model parameters could be used for the two,without loss of accuracy within the domain of measurement.

Figures 31 - 33 present for each tested oil the evolution of the viscosity according to pressureand temperature. Values cover more than three orders of magnitude, typically from 10-2 to20 Pa.s. Note that experimental results do not align themselves perfectly on a straight linethat, in our system of axes, would mean that viscosity follows an exponential increase.Therefore the classical pressure-viscosity coefficient - obtained by presupposing anexponential relation - varies appreciably with the pressure (for example from 16.4 to 9.8 GPa-1

for the Nye 186 A oil at 75°C) while it is assumed to be an intrinsic parameter, i.e., constant ata given temperature. Finally, a very good agreement between experimental data and valuesobtained with the modified WLF model is observed.

61

Comparaison des viscosités dynamiques des trois fluides dans la zone -25, +75 °C (domaine spatial)

0.01

0.1

1

10

-25 0 25 50 75

Température (°C)

Vis

cosi

té d

ynam

ique

(

Pa.

s)

Nye 186 A

Pennzane SHF X 2000

Fomblin Z 25

Variation de la viscosité avec la pression et la températureHuile Fomblin Z25

-24.0 °C-9.0 °C

13.0 °C40.0 °C

63.0 °C

0.01

0.1

1

10

100

1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pression P (GPa)

Vis

cosi

té d

ynam

ique

(

Pa.

s)

Mesures Fomblin Z25

Modèle WLF

Figure 30 – Variation of the viscosity with temperaturefor: Pennzane SHF X2000, Nye 186 A and Fomblin Z25

(line: modified WLF model; point: experiments)

Figure 31 – Variation of the viscosity with pressureat different temperatures for the Fomblin Z25 oil

Variation de la viscosité avec la pression et la températureHuile Pennzane SHF X2000

-24.0 °C

-9.0 °C

13.0 °C

40.0 °C

63.0 °C

0.01

0.1

1

10

100

1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pression P (GPa)

Vis

cosi

té d

ynam

ique

(

Pa.

s)

Mesures Pennzane SHF X2000

Modèle WLF

Variation de la viscosité avec la pression et la températureHuile Nye 186 A

25.0 °C49.7 °C

75.4 °C

100.2 °C

0.01

0.1

1

10

100

1000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pression (GPa)

Vis

cosi

té d

ynam

ique

(

Pa.

s)

Mesures Nye 186 A

Modèle WLF Nye

Modèle WLF Pennzane

Figure 32 – Variation of the viscosity with pressurefor the Pennzane SHF X2000 at different temperatures

Figure 33 – Variation of the viscosity with pressurefor the Nye 186 A oil at different temperatures

Inverse isoviscous asymptotic pressureThe calculation of the "inverse isoviscous asymptotic pressure" α* was done for the threelubricants studied. There are two ways to perform this calculation, either directly byintegrating the experimental value, or by first identifying WLF's model parameters and thencalculating numerically equation (77).

Results of both ways are reported in figure 34. Note again that the variation with temperatureof the "inverse isoviscous asymptotic pressure" α* is lower for the Fomblin Z25 oil, especiallyat high temperature. Moreover and once again, the Pennzane SHF X2000 and the Nye 186 Alubricants give very close and complementary results.

62

Evolution du coefficient de piézoviscosité α * avec la température

5

10

15

20

25

30

-30 0 30 60 90 120

Température (°C)

Coe

ffici

ent d

e pi

ézov

isco

sité

*

(GP

a-1

)

Modèle Pennzane SHF X2000

Modèle Nye 186 A

Modèle Fomblin Z25

Mesures Pennzane SHF X2000

Mesures Nye 186 A

Mesures Fomblin Z25

Figure 34 – "Inverse isoviscous asymptotic pressure"α* versus temperature for the 3 oils studiedPoints: values deduced from experiments

Line: calculated values from the modified WLF model

A good estimation of the α* coefficient calculated with the modified WLF model can beobtained on a wide domain of temperature (with a maximum error lower than 0.1 %between-30 and +300°C) with the following formula:

( ) ( )α α* = + + +

−+

−∞

A

T

A

T

A

T T

A

T Tc c

00 012

10

0

11

02

(93)

where: T T Cc g0 0 2= −

The corresponding values for the three lubricants studied are given in the following table:

Pennzane SHF X2000 Nye 186 A Fomblin Z25α∞ (GPa-1) 0 0 0A00 (K/GPa) -54,677 -1,7295 103 -1,3222 103

A01 (K2/GPa) 8,4243 105 4,9567 105 7,3974 105

Tc0 (K) 125,1002 134,061 97,948A10 (K/GPa) 1,4264 103 3,2079 103 4,0438 103

A11 (K2/GPa) 1,3876 104 -1,97 104 -1,3132 105

Table 11 – Parameters for the calculation of α* (GPa-1) by equation (93)

In conclusion, it should be noted that the modified WLF model, which is based on physicalconsiderations, is the most reliable to represent the variation of the viscosity on a wide domainof temperature and pressure.

63

Traction coefficientOnly results for the Pennzane SHF X2000 oil are reported are. The law of viscosity used inthe traction model for the Pennzane SHF X2000 is the modified WLF model. The coefficientof friction is calculated by the thermal model described before in paragraph 3.2.

The non-linear function relating the viscous shear rate to the shear stress is the one proposedby Elsharkawy & Hamrock [77], with an exponent n equal to 2 (cf. paragraph 3.2). This modelpresents the advantage to be able to describe a viscous or elastic behavior linear at low shearrate and plastic at high shear rate, with a transition between the various types of behaviordepending on the exponent n. Furthermore, the shear stress saturation encountered at highshear stress facilitates the identification of the limiting shear stress τL, that is not case with theJohnson & Tevaarwerk's model [116].

Parameters G and τL depend on pressure and temperature, according to equations (78) and(79), with pressure dependent coefficients αG and ατ that are constant. After identification, thevalues of parameters giving the best correlation with experiments are the following ones:

G0 (at T0 = 303.15 K) (MPa) 0.565138

αG (GPa-1) 3.41106

βG (K) 1426.14

τL0 (at T0 = 303.15 K) (MPa) 16.0444

ατ (GPa-1) 1.08853

βτ (K) -977.746

Table 12 – Parameters of the visco-elastic model for the Pennzane SHF X2000 oil(viscous term of Elsharkawy & Hamrock [77@, n = 2)

A comparison of experimental and theoretical results is illustrated by the graphs of figure 35.Considering the wide domain of temperature and pressure investigated during measurement,model gives satisfactory results. A good agreement is found for the mean values oftemperature or pressure, and degrades at the extremities of the domain of measure. Toimprove this point, it seems that coefficients αG and ατ should be temperature dependent (orβG and βτ pressure dependent).

Note that, for a mean Deborah number (D V Ga= µ ) ranging from 0.1 (at high temperature,low speed) to 30 (at low temperature, high speed) on the whole domain of measure, the fluidbehavior can not be correctly described by a visco-elastic model. Elastic effects are alwaysimportant, partially because of the strong viscosity of the Pennzane SHF X2000 oil,particularly at low temperature. Finally, thermal effects in the contact inlet are not negligible.They have an important consequence on the slope of the traction curves at low sliding rate.

Heating in the high-pressure zone is sometimes important, producing an increase oftemperature up to 80 K, but its effect on the traction coefficient remains rather low, thelimiting shear stress τL being hardly sensitive to temperature. In term of friction, the non-linearbehavior of the Pennzane lubricant appears strongly marked at high slide-to-roll ratio. Thus, itcan not be correctly described by a simple viscous linear model, even with thermalconsiderations.

64

-8

-6

-4

-2

0

2

4

6

8

-20-15-10-505101520

Taux de glissem ent ∆V/V (%)

Coe

ffici

ent d

e fr

otte

men

t (%

)

Modèle - 1.8 GPa

Mesures - 1.8 GPa

Modèle - 1.2 GPa

Mesures - 1.2 GPa

Modèle - 0.6 GPa

Mesures - 0.6 GPa

T = 273 K V = 3 m /s

-8

-6

-4

-2

0

2

4

6

8

-20-15-10-505101520

Taux de glissem ent ∆V/V (%)

Coe

ffici

ent d

e fr

otte

men

t (%

)

Modèle - 1.8 GPa

Mesures - 1.8 GPa

Modèle - 1.2 GPa

Mesures - 1.2 GPa

Modèle - 0.6 GPa

Mesures - 0.6 GPa

T = 273 K V = 6 m/s

-8

-6

-4

-2

0

2

4

6

8

-20-15-10-505101520

Taux de glissem ent ∆V/V (%)

Coe

ffici

ent d

e fr

otte

men

t (%

)

Modèle - 1.8 GPa

Mesures - 1.8 GPa

Modèle - 1.2 GPa

Mesures - 1.2 GPa

Modèle - 0.6 GPa

Mesures - 0.6 GPa

T = 313 K V = 3 m /s

-8

-6

-4

-2

0

2

4

6

8

-20-15-10-505101520

Taux de glissem ent ∆V/V (%)

Coe

ffici

ent d

e fr

otte

men

t (%

)

Modèle - 1.8 GPa

Mesures - 1.8 GPa

Modèle - 1.2 GPa

Mesures - 1.2 GPa

Modèle - 0.6 GPa

Mesures - 0.6 GPa

T = 313 K V = 6 m/s

-8

-6

-4

-2

0

2

4

6

8

-20-15-10-505101520

Taux de glissem ent ∆V/V (%)

Coe

ffici

ent d

e fr

otte

men

t (%

)

Modèle - 1.8 GPa

Mesures - 1.8 GPa

Modèle - 1.2 GPa

Mesures - 1.2 GPa

Modèle - 0.6 GPa

Mesures - 0.6 GPa

T = 353 K V = 3 m /s

-8

-6

-4

-2

0

2

4

6

8

-20-15-10-505101520

Taux de glissement ∆V/V (% )

Coe

ffici

ent d

e fr

otte

men

t (%

)

Modèle - 1.8 GPa

Mesures - 1.8 GPa

Modèle - 1.2 GPa

Mesures - 1.2 GPa

Modèle - 0.6 GPa

Mesures - 0.6 GPa

T = 353 K V = 6 m /s

Figure 35 – Comparison between experimental and theoretical (thermal model) tractioncurves for the Pennzane SHF X2000 lubricant

65

4.2 Power loss

Purpose of the studyPower loss in mainshaft and gearbox rolling bearings constitute a significant part of thespecific fuel consumption of gas turbine engines. The oil flow can reach hundreds of liters byhour and by bearing. However, only a small fraction of this debit serves effectively forlubricating the contacting surfaces, the rest serving for cooling in order to maintain areasonable temperature, that besides contributes to heat generation by churning. Moreover,throwing more lubricant at the bearing to achieve a few degrees temperature reductionincreases oil heating and leads to oversized and overpriced engine oil systems (oil mass, oiltank, exchangers, filters, pumps). It is clear from this analysis that it is advantageous to reducethe oil flow to the bearings, with associated benefits on oil system components. Therefore, anestimation of the rolling bearing power losses is necessary to refine lubrication techniques andto optimize machine component design.

The steady-state model described earlier in paragraph 2.2 allows to predict and to locatepower losses in a high-speed cylindrical roller bearing operating under purely radial load.Results presented here show the effects of operating conditions (speed, load, temperature andoil flow) and of internal geometry (internal clearances, type of cage guiding, type of rollerguiding) on the heat generation. This work has been published [160, 172]. Moreover, a similarstudy, not presented here, has been carried out for high-speed ball bearings [169, 170].

DataThe specifications of the roller bearing studied are listed in table 13. Both cage and rollers areguided by the inner-ring. Lubrication is provided through the inner race, therefore, the wholeinput flow rate goes through the bearing. The range of operating conditions is given intable 14. Experimental data comes from the SNECMA company.

ROLLER BEARING SPECIFICATIONS RANGE OF OPERATING CONDITIONS

External geometry (ID, OD, W) (mm) 119 x 164 x 40 Shaft speed (rpm) 0 to 20 000Pitch diameter (mm) 142 Radial load (daN) 0 to 5 000Number of roller 30 Lubricant flow (l/h) 0 to 300Roller diameter (mm) 12 Lubricant inlet temperature (°C) 60 to 200Roller length (mm) 14 Lubricant specification MIL-L-23699

Bearing diametral clearance (µm) 30Cage guidance type Inner ring Table 14 – Operating conditionsCage diametral guiding clearance (µm) 480Roller guidance type Inner ringRoller axial guiding clearance (µm) 30Race flange angle (degree) 0.375Race and ball material AISI M-50

Table 13 – Roller bearing specifications

Test-Model CorrelationThe calculated drive power is first compared with experimental results. It must be noted thatthe drive power is equal to the total power loss. The experimental work used to validate thecomputer program has been performed on several high-speed roller bearing test rigs. Thepredicted bearing heat generation agreed very well with the experimental data obtained fromdifferent sizes of roller bearings (35 to 142 mm pitch diameter), for a lubricant provided by jetor through the inner ring and over a speed range from 0.3 up to 3 million DN. An example of

66

results is presented in figure 36. It corresponds to a comparison of the measured andcalculated power loss versus the lubricant flow, for the roller bearing described in table 13.

Figure 36 – Predicted and experimental total power lossvs. lubricant flow and inner ring speed.

(Lubricant inlet temperature, 100°C ; radial load, 2500 daN)

Symbols=test results ; line=model

Figure 37 – Predicted total power lossvs. inner ring speed and radial load.

(Lubricant flow, 150 l/h ; lubricant inlet temperature, 100°C)

Figure 38 – Predicted total power lossvs. lubricant inlet temperature

and inner ring speed.

(Radial load, 2500 daN ; lubricant flow, 150 l/h)

Total power loss prediction versus operating conditionsFigure 37 shows the total power loss prediction versus the shaft speed for several radial loadsvarying between 500 and 4000 daN, a lubricant flow of 150 l/h and a lubricant inlettemperature of 100°C. Heat generation increases greatly with the shaft speed, whereas itseems independent of radial load (all plots seem as one). This last result is explained by theeffect of the rolling friction that has been initially omitted in the model.

In fact, a decrease in power loss can be observed for radial loads lower than 200 daN, asshown in figure 39(c). It must be emphasized that, as earlier noted for ball bearings [170],power loss prediction and location depend strongly on the roller bearing internal geometry,and more specifically on cage and roller guidance type, i.e. guided by inner- or outer-ring. For

67

example with this specific internal geometry, where both cage and rollers are guided by theinner ring, a small cage slip occurs for radial load lower than 200 daN. This phenomenon isimportant because changes in cage speed produce a strong variation in windage loss,according to equation (89) in paragraph 3.5.

Figure 38 shows the total power loss prediction versus the lubricant inlet temperature varyingfrom 60°C up to 200°C, over a speed range from 2500 to 14000 rpm, with a lubricant flow of150 l/h and a radial load of 2500 daN. It can be noted that the decrease in total power losswith increased lubricant inlet temperature is very marked. That is mostly due to the viscositydrop with temperature, the variation of the viscosity with the temperature being exponential.

The predicted trends of increased heat generation with increased shaft speed, increasedlubricant flow rate and decreased lubricant inlet temperature were experimentally verified.

Power loss location versus operating conditionsFigures 39(a) to (c) present the estimated distribution of the heat dissipated in the bearing,versus the lubricant flow rate (figure 39(a)), the inner ring speed (figure 39(b)) and the radialload (figure 39(c)). The cage contribution to the total power loss is important, mainly due tothe fluid windage, and also due to the short journal dissipation at the cage-inner ring pilotsurfaces and to the power dissipated at the roller-cage pocket interfaces.

The decrease of the total power loss with decreased lubricant flow rate is mainly due to thewindage loss reduction, as shown in figure 39(a). Predicted power loss location shows thatcage-ring pilot surface contribution to the total energy dissipated can be more important thanthe windage contribution at very high speeds (figure 39(b)). A shaft speed of 20000 rpm for a142 mm pitch diameter bearing corresponds to 2.84 million DN.

As it was mentioned above, figure 39(c) presents some interesting results on power lossdistribution under light radial load, typically for radial loads lower than 200 daN. The cageslip, which produces a decrease in windage and total power losses, also strongly modifies thepower loss distribution. Then, dissipation can reach 300 W at the roller-race flange interface,and 150 W by sliding at the roller-inner ring raceway contact. Note that both cage slipphenomenon and power loss distribution are strongly dependent on the roller bearing internalgeometries, and then results for other geometries can be completely different.

Power loss location versus internal geometryA parametric study was conducted in order to evaluate the effects of bearing internalclearances on power loss prediction and location. Power loss versus the bearing diametralclearance, the cage diametral guiding clearance, the roller axial guiding clearance and the raceflange angle value are presented respectively in figures 40(a), (b), (c), and (d). The basic rollerbearing geometry is presented in table 13. Operating conditions are the following; an innerring speed of 10500 rpm, a radial load of 2500 daN, a lubricant flow of 150 l/h and a lubricantinlet temperature of 100°C.

Figure 40(a) shows that no important change can be observed in the distribution of heatgenerated in the bearing versus the diametral clearance, as far as this clearance remainspositive. Increase of the total power loss with decreased cage diametral guiding clearance ismostly due to the cage-ring pilot surface contribution, as shown in figure 40(b). Local effectsof roller-race flange geometry are presented in figure 40(c) and (d). Decreasing both roller

68

axial guiding clearance and race flange angle value increases power loss generated at thisinterface, without significant effects on the total power loss.

(a) – Power loss location vs. lubricant flow.

(b) – Power loss location vs. inner ring speed.

(c) – Power loss location vs. radial load.

Figure 39 – Power loss location vs. operating conditions (regular view on the left, zoom on the right).Nominal operating conditions:

Inner ring speed, 10500 rpm; radial load, 2500 daN; lubricant flow, 150 l/h; lubricant inlet temperature, 100°C.

69

(a) – Power loss location vs. bearing diametral clearance.

(b) – Power loss location vs. cage diametral guiding clearance.

(c) - Power loss location vs. roller axial guiding clearance.

70

(d) – Power loss location vs. race flange angle.

Figure 40 – Power loss location vs. internal bearing geometry (regular view on the left, zoom on the right).Nominal operating conditions:

Inner ring speed, 10500 rpm; radial load, 2500 daN; lubricant flow, 150 l/h; lubricant inlet temperature, 100°C.

Concluding remarksThe roller bearing model that has been developed in order to evaluate the heat generation iscomprehensive and original. This model includes both hydrodynamic and elastohydrodynamicanalyses to describe interactions among the various rolling bearing elements, and an empiricalwindage torque model. Experimental results were compared to computer predictions. Finally,the influence of the operating conditions and internal geometry was studied.

The following major results were obtained:

1. Although it is a quasi-static model, a good correlation between theory and experiment wasobtained on power loss.

2. Results obtained are coherent with the literature, notably those of the NASA [50, 51, 52,124, 179, 198, 199, 200, 201, 202, 203], Cretu et al. [57], Rumbarger et al. [193], Warda etal. [127], and Chittenden et al. [46].

3. The contribution to the total energy dissipated of each lubricated contact is established, i.e.,interface between roller and raceway, roller and race flange, roller and cage pocket, cageand ring pilot surface, etc.

4. Results show that parameters affecting the power loss may be classified in descendingorder as follows:- for operating conditions; the rotation speed, the lubricant inlet temperature and the

lubricant flow, whereas the radial load effect is considered to be less important.- for internal geometry; the cage guiding clearance and its location on inner- or outer-ring,

the roller guiding clearance, its location and the value of its race flange angle. Theinfluence of the bearing diametral clearance can be considered as negligible.

Finally, it was point out that the knowledge of internal kinematics and contact loads in high-speed roller bearings is of great interest on heat generation, since power losses are related tosliding velocities and traction forces.

71

4.3 Progressive ball-raceway control change in lubricated ball bearings

Purpose of the studyRecent experimental studies on high-speed lubricated ball bearings tend to show that thisassumption of an inner or outer raceway control is no more valid. The energy dissipated byspinning of the balls seems well distributed towards the inner and the outer races. The steady-state model presented earlier in paragraph 2.2 is used here to study the progressive ball-raceway control change versus operating conditions.

DataThe specifications of the bearing studied are listed in table 15. The cage is guided by theinner-race. The lubricant used is the oil Mobil Jet II, qualified to the MIL-L-23699specifications, and described before in paragraph 4.1. Lubrication is provided under the innerrace, therefore the whole input flow rate goes through the bearing.

TYPE ANGULAR CONTACT BB

Number of ballsBall diameterBearing pitch diameterBearing contact angleInner race conformityOuter race conformityRace and ball material

2020.638 mm

153 mm30°

0.52250.51125

80 DCV 40 (M50)

Table 15 – Angular contact ball bearing specifications

The range of operating conditions is given below:

- Shaft speed: 5000 to 20000 rpm- Axial load: 500 to 5000 daN- Inlet oil temperature: 100 °C

The notion of a mixed control between raceways requires the definition of the parameter β% asfollowing, which corresponds to the percentage of control of the ball by the outer ring:

oi

i% .100

β−ββ−β

=β (94)

Effect of the Gyroscopic MomentThe gyroscopic moment about the rolling direction creates a component of the ball rotationalong the rolling direction. This ball rotation component produces a new gyroscopic momentabout the radial axis resulting in change of the ball rotation about the radial direction.Consequently, the gyroscopic moment resulting in transversal sliding at the ball-raceinterfaces works to modify the direction of the ball rotation. From Aramaki et al. [3], the βangle decreases in response to this behavior and a slight β' angle appears. A decrease in β wasobserved experimentally by Kawamura et al. [122].

Transversal sliding due to gyroscopic moment is taken into account in the present analysis.However, an exploitation of the results obtained shows that the power loss by friction due totransversal sliding never exceeds 2% of the total power loss for the operating conditions of thetypical example described here.

72

Friction coefficient versus the spin-to-roll ratioFigure 41 shows the friction coefficient by spinning versus the spin-to-roll ratio for severalHertz pressure ranging from 1.19 to 2.38 GPa. The rheological properties of the lubricant aregiven above in paragraph 4.1.

ωspin/ωroll

0.0 0.2 0.4 0.6 0.8 1.0

Fric

tion

Co

effic

ient

=

Spi

nni

ng T

orqu

e/(N

orm

al L

oad.

(a.c

)0

. 5)

0.00

0.05

0.10

0.15

0.20

Pmax. Hertz

1.19 GPa

1.50 GPa

1.89 GPa

2.38 GPa

Figure 41 – Friction coefficient by spinning versus spin-to-roll ratio (isothermal model)Lubricant: Oil Mobil Jet II (MIL-L-23699) ; ωroll=5337.27 rad/s ; a=2.605 mm (for Pmax Hertz=1.89 GPa)

a/c=8.1423, U*=4.5473 10-10, G*=2364.49, W*=4.5341 10-5, 9.0682 10-5, 1.1814 10-4, and 3.6273 10-4

Power loss by spinning at the ball / inner ring and ball / outer ring contactsCommon ball bearing analyses use the simplifying assumption that balls roll without spinningon one raceway and spin and roll on the other. Such an assumption has no sense physicallysince raceway control change produces a discontinuity of the ball kinematics, the ball attitudeangle β being only free to be equal to βi or βo.

Taking into account spinning and lubricant at both inner- and outer-raceway contacts allows todetermine the external loading acting on a ball. The notion of a mixed control betweenraceways comes from minimizing the energy dissipated around a ball. Using such an analysis,a value of the ball attitude angle β can be found different to βi and βo.

For the purpose of illustrate whether mixed raceway control occurs or not, the attitude angleminimizing the energy dissipated is calculated for the ball bearing example described earlierin table 15. The bearing is 4000 daN thrust loaded and the shaft is rotating at 12500 rpm.Figure 42 shows the power losses by spinning at the outer- and inner-raceway contacts versusthe ball attitude angle. The total power loss is the sum, per ball, of the power dissipated byspinning at both outer- and inner-ring contacts. If pure rolling occurs at the outer racewaycontact, which defines the outer raceway control, then the attitude angle β is equal toβo=0.45 rad. For this condition, the energy dissipated by spinning at the outer raceway contactis equal to zero. Similarly, if no spinning occurs at the inner raceway contact (inner racewaycontrol), the attitude angle β is equal to βi=0.74 rad. and there is no power loss by spinning atthis contact. For this specific example, the total energy dissipated by spinning is minimizedfor a value of β equals to 0.62 rad. (i.e., β%=43.22 %), which is an intermediate value betweenβi and βo.

73

β (rad)

0.5 0.6 0.7

0

50

100

150

200

250

300

β (rad)

0.5 0.6 0.7

Pow

er L

oss

(W)

0

50

100

150

200

250

300

βiβο

Ball / Outer-Ring Contact

Total Power Loss

Ball / Inner-Ring Contact

Figure 42 – Power loss by spinning and per ball vs. attitude angle β(Thrust load = 4000 daN ; Shaft speed = 12500 rpm)

This illustration shows that ball kinematics is mainly governed by the loading transmittedthrough the lubricant at the different interactions between the ball and the races.

Ball attitude angle ββ vs. thrust load and shaft speedSimilar analysis has been used for the angular contact ball bearing of the previous example,operating at speed ranging from 5000 up to 20000 rpm, under thrust load from 5000 up to50000 N. Results are presented in figure 43. Results on β% value versus shaft speed and thrustload show that a very smooth change of raceway control is produced. This control is mainlygoverned by the inner race at low speeds and high thrust loads, by the outer race at highrotation speeds and low loads, and is strongly mixed at intermediate speeds and loads. Thisresult is remarkable since the operating conditions that produce a mixed control correspond tocurrent operation of the gas turbine engine.

0

20

40

60

80

100

0

5000

10000

15000

20000

010000

2000030000

40000

β % (

in %

of

O.R

. con

trol

)

Ni

(rpm

)

FA (N)

O.R. control

I.R. control

Figure 43 – Ball attitude angle β vs. thrust load FA and shaft speed Ni

74

Results show that a very smooth change of raceway control is produced, function of operatingconditions. These results tend to show that this assumption of an inner or outer racewaycontrol is no more valid. The energy dissipated by spinning of the balls seems well distributedtowards the inner and the outer races. These results are coherent with those of Aramaki et al.[3] and Touma et al. [217, 122], that more particularly studied the performance of ballbearings with silicon nitride ceramic balls in high-speed spindles for machine tools,respectively from theoretical and experimental point of view.

Concluding remarksFrom energy balance considerations, a comprehensive method to determine the rotationattitude angle of a ball in lubricated ball bearings has been proposed. Taking into account therheological properties of the lubricant at contacts between ball and inner- or outer-racewayallows one to determine the external loading acting on a ball, including spinning effects.Spinning can occurs simultaneously at ball / inner raceway and ball / outer raceway contacts.The energy dissipated by ball spinning seems well distributed towards the inner and the outerraces to minimize the energy generated. Results show that a very smooth change of racewaycontrol is produced, function of operating conditions.

The main conclusions are the following:

1) Results tend to show that the common assumption of an inner or outer raceway controlis no more valid. The control of the ball kinematics is mainly governed by the inner raceat low rotation speed and high thrust load, by the outer race at high speed and low load,and it is mixed at intermediate speed and load.

2) The lubricant behavior is of great interest on the determination of the internal ballbearing kinematics and heat generation.

Finally, this new analysis makes clear that the power loss is lower than previously expected.This allows to explain why a ball bearing will be able to operate satisfactory at speed up to4 million DN, and to propose original technologies for future turbine engines.

75

4.4 Effort between roller and cage

Purpose of the studyExcept the experience of bearing designers and sometimes also that of users, it does not existin our knowledge a tool to conceive an optimal cage. New cage designs are obtained mostlyby homothety of experienced solutions. To foresee possible cage failure and so to optimize thecage design, it is necessary to know the mechanical solicitations applied on it in operation.These information can result from the use of sophisticated codes such as "Adore" by Gupta[90, 91] or "Beast" by SKF [206], who are a rather complicated software to implement for thefirst one or confidential for the second one. An alternative consists in using a simplifiedapproach as that presented in paragraph 2.2, easy to implement.

DataThis study is limited to cylindrical roller bearings, since their geometry is quite simple. Thedynamic code has been described earlier in paragraph 2.2. The internal geometry of the rollerbearing studied as well as its operating conditions are given in table 16 below.

Number of roller 30 Pocket longitudinal clearance 300 µm

Roller diameter 12 mm Pocket axial clearance 350 µm

Total roller length 14 mm Bearing diametral clearance 30 µm

Cylindrical length of rollers 12.4 mm Cage diametral guiding clearance 240 µm

External diameter of the inner ring 135 mm Cage and roller guidance by the Inner Ring

Internal diameter of the outer ring 154 mm Cage made in bronze

Diameter of the inner ring raceway 130 mm Rings and balls made in AISI M-50 steel

Roller axial guiding clearance 30 µm Stationary radial load 25 000 N

Race flange angle 0.375 degree Rotation speed of the I.R. 14 000 rpm

Inner diameter of the cage 135 mm Inlet oil temperature 100°C

Outer diameter of the cage 146 mm Lubricant Mobil jet II

Ring-cage pilot surface width 3.98 mm

Table 16 – Internal geometry of the roller bearing studied and operating conditions

Results obtained without perturbation of the inner ring rotation speedThe displacement of the cage center from its initial position in the numerical procedure isshown in figure 44. Under the effect of a stationary radial load applied to the bearing, the cagereach an equilibrium position while behaving like a short journal bearing. Figure 45 presentsthe corresponding rotation speed of the cage that is quickly stabilized after a transient phase.

Figure 44 – Displacement of the cage center Figure 45 – Rotation speed of the cageThe normal effort between the 8th roller and the cage separator, WCR, versus the rollerangular position is given in figure 46. The normal load between the roller and the inner ring,

76

WBI, is also indicated. A shock appears each time the roller exists the loaded zone, defined asthe angular section when roller-I.R. load is positive (WBI > 0). More precisely, this shockoccurs between the roller and the front cage pocket separator (WCR > 0).

Figure 46 – Normal load between the roller and the cage separator (WCR)and load at the roller-inner ring contact (WBI) for roller number 8 (non-perturbed case).

Results of the modeling show that the rollers which are located in the angular zone where theradial load is applied to the bearing, are accelerated and thus become driving rollers for thecage. While the others, localized in the opposed area, tend to slow down the movement of thecage.

This tool allows to estimate the roller-cage efforts, but not only. It also permits to gaintransient information for other contacts such as roller - ring raceway contacts. Thus, figure 47below gives the sliding velocity and the maximum pressure of Hertz at the contact betweenthe same roller and the inner and outer ring raceways, BI, BE, PMAXI , and PMAXE, respectively.

Figure 47 – Sliding velocity and contact pressure at the roller – inner and outer ring contacts

77

Results obtained with perturbation of the inner ring rotation speedResults that follow were obtained with the same data that the previous case, except therotation speed of the shaft which is perturbed by a sinusoidal signal of frequency f (in Hz) andamplitude Ampli (in %):

WA = WANominal + Ampli × WANominal × sin (2π ×f×t)

A frequency of 40 Hz and an amplitude of 10 % of the nominal speed (14 000 rpm ) has beenchosen for the shaft speed perturbation. This perturbation is not realist from energy point ofview, however it approaches an extreme situation given by the turbine manufacturer, which,has been since revised to a 30 Hz frequency and a 5 % amplitude. Figure 48 shows theposition of the cage center. This one does not any more settle comfortably as in the non-perturbed case. The represents The corresponding shaft and cage orbital rotation speeds, WAand WC, respectively, are represented in figure 49. It appears that the cage speed oscillates ina sinusoidal way around its nominal value with however a slight phase shift and a relativeamplitude slightly reduced.

Figure 48 – Location of the cage center Figure 49 – Orbital speed of the cage and of the inner ring

The normal effort between the 8th roller and the cage separator, WCR, and the normal loadbetween the roller and the inner ring, WBI, are given in figure 50 versus the roller angularposition. This perturbed situation differs notably from the non-perturbed one (see figure 46)by the increase in the shock magnitude, i.e. up to 600 N instead of 120 N about.

Figure 50 – Normal load between the roller and the cage separator (WCR)and load at the roller-inner ring contact (WBI) for roller number 8 (perturbed case).

78

The sliding velocity and the maximum hertzian pressure are indicated in figure 51, for thecontact between the inner- and outer-ring raceways. It should be noted that a globalperturbation of the shaft speed locally increases the sliding velocities at the EHL contactsbetween rollers and rings, which therefore augments the risk of scuffing.

Figure 51 – Sliding velocity and contact pressure at the roller – inner and outer ring contacts.

Influence of the operating conditions (radial load and shaft speed)Lastly, the evolution of the average amplitude of the normal load between the roller and thecage is indicated in table 17 for various shaft speed and radial load.

10 000 14 000 Shaft speed WA in rpm15 000 70 N 100 N25 000 80 N 120 N

Radial load FR in NTable 17 – Average amplitude of the roller-cage effort

Concluding remarksThe dynamic model proposed allows to estimate the transient efforts between the rollers andthe cage for a given set of operating conditions. For non-perturbed regime, i.e. when the shaftspeed and the radial load vector are constant, a shock occurs between the roller and the frontcage pocket separator each time the roller exists from the loaded zone. Thus, every cagepocket separator sees two shocks by cage rotation. For perturbed regime, the most importantshocks do not occur when rollers go out of the loaded zone but rather in a chaotic way, andtheir amplitude is several times those obtained in the non-perturbed regime. This is typicallythe situation when the shaft speed is oscillating around a nominal value, for example due toturbine power control, or when the load is rotating (unbalance mass). Besides, the dynamiccode developed permits to describe the transient behavior of other contacts such as the EHLline contact between rollers and outer- or inner- ring raceways. It has been shown that thislater contact may become critical in term of scuffing. Finally, the proposed model is a firststep towards a more complete 3D model in which the ring misalignment as well as the rollerskewing could be taken into account.

79

4.5 Transient EHL lubricant film thickness

Purpose of the studyThis study aims to the estimation of the duration during which an lubricating film remainspresent at the interface between the ball and the inner ring in an axially loaded ball bearingwhen its rotation speed passes instantaneously from a nominal speed to zero. The applicationconcerns two angular contact ball bearings mounted in a X arrangement in a satellitemechanism. The bearings are lubricated by the release during operation of a space lubricantfrom porous retainer.

DataThe contact parameters come from the ball bearing code described earlier in paragraph 2.2. Atransient EHL code developed by Couhier [53], which is based on the previous work ofDowson and Higginson [61, 62, 63] and more recently of Lubrecht and Venner [143, 218,219], has been adapted and used. This code allows to calculate the lubricant film thickness forline contact and for transient operating conditions such as here the stop and go movement. Itshould be recalled that an EHL line contact model gives a good approximation of the ellipticcontact between balls and rings in ball bearing since the ellipticity ratio is close to 8.

Condition #1 #2

Axial load, FA (N) 250 500

Inner ring (I.R.) rotation speed, Ni (rpm) 10.5 10.5

Hertzian pressure at the ball/I.R. contact, Ph (GPa) 1.06 1.33

Semi-minor axis of the contact ellipse, a (mm) 0.046 0.057

Semi-major axis of the contact ellipse, c (mm) 0.314 0.393

Bal

l bea

ring

calc

ulat

ion

Rolling speed at the ball/ I.R. contact, (U1+U2)/2 (m/s) 0.02 0.02

Inlet oil temperature (°C) 20 40

Equivalent reduced radius, R’ (mm) 2.435 2.435

Equivalent Young modulus, E’ (MPa) 225275 225275

Equivalent load per length unit, F (N/mm) 76.31 120.14

Contact half-width, a (mm) 0.0458 0.0575

Hertzian pressure, Ph (GPa) 1.06 1.33

Central film thickness, Hcen (µm) 0.0511 0.0213

EH

L ca

lcul

atio

nat

the

ball/

inne

r rin

g co

ntac

t

Minimum film thickness, Hmin (µm) 0.0383 0.0159

Table 18 – Summary of ball bearing and EHL calculations performed

The initial EHL code has been modified to account for the rheological properties of thePennzane oil described before in paragraph 4.1, with a modified WLF model [241] for theviscosity [166]. Two transient calculations have been performed to simulate operatingconditions of two different qualification tests, a reference test and a severe life pre-validationtest. Data and stationary results corresponding to these calculations are reported in table 18.

80

A first result in table 18 is the effect of the temperature on the lubricant film thickness (centraland minimum). These results are strongly dependent on the rheological properties of thelubricant reported before. The results of the transient EHL calculations are presented infigures 52 and 53 in terms of rolling speed and central and minimum film thicknesses versustime. It should be noted that the minimum film thickness collapses at the edges of the linecontact (x = ±a) as soon as the rolling speed tends to zero, whereas the central film thickness(x = 0) remains approximately constant during 2 ms about for both calculations. This may beexplained by the local pressure in the center of the contact, which is so high that itsignificantly increases the oil viscosity and therefore slows down the oil flow. For the studiedoperating conditions that leads to a volume of lubricant entrapped in the contact center for aduration of approximately 2 ms. Finally, the central film thickness remains nearly constantduring this time duration. Note that result of these simulations are coherent with experimentalresults of Sugimura et al. [207].

Condition n°1 (T=20°C; Phertz=1060 MPa; a=0,046 mm; Penzanne Oil)

0,00E+00

1,00E-05

2,00E-05

3,00E-05

4,00E-05

5,00E-05

0 2 4 6 8 10

t (ms)

0

0,005

0,01

0,015

0,02

H en X=0 (mm) Hmin (mm) Umoy (m/s)

H, Hmin (mm) Umoy (m/s)

Figure 52 – Central and minimum film thicknesses vs. time at the ball / I.R. contact for condition #1.

Condition n°2 (T=40°C; Phertz=1330 MPa; a=0,058 mm; Penzanne Oil)

0,00E+00

5,00E-06

1,00E-05

1,50E-05

2,00E-05

2,50E-05

0 1 2 3 4 5 6 7 8

t (ms)

0

0,005

0,01

0,015

0,02

H en X=0 (mm) Hmin (mm) Umoy (m/s)

H, Hmin (mm) Umoy (m/s)

Figure 53 – Central and minimum film thicknesses vs. time at the ball / I.R. contact for condition #2.

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5. Conclusion

This first part synthesizes a 10-year work on the modeling of rolling bearings. It should berecalled that bearings are key elements in a multitude of equipment and machines,contributing to the reliability of airplanes and space launchers and making ventilatorsnoiseless and machine tools precise. More than 5 billion bearings are produced each year inthe world, of which nearly 40% are absorbed by the production of 50 million cars. Even if99.5% of the bearings finish their life without problem or are replaced during preventivemaintenance actions, the failure of the 0.5% remainders is at the origin of an intense activityof research and development on lubrication, design or materials. The bearings alone justifymore than half of current research tasks undertaken on elastohydrodynamic lubrication, whichemploys several hundreds of researchers in the world.

Study of rolling bearings is closely related to the knowledge of the lubrication mechanisms.Indeed, internal kinematics and, to a lesser extent, the load distribution on rolling elementsdepend on the nature of the lubricant and on its rheological and tribological properties. Theprincipal lubricant rheological property that interests bearing users and designers is theviscosity, which depends on the operating contact pressure and temperature. Its tribologicalbehavior results first in a friction coefficient itself related to temperature, pressure and rollingspeed, whose estimate is complex but fundamental.

A comprehensive steady-state model, eventually without cage, is enough to describe theinteractions between the rolling elements and the rings. This makes it possible the calculationof the fatigue life following the ISO 281 standard [10, 114], which however corresponds onlyto rolling contact fatigue of rolling bearing for almost ideal operating conditions, i.e. at theexception of other modes of failure (scuffing, cage breaking, abrasive wear, etc). On the otherhand, it is essential to explicitly introduce the cage into modeling to determine the energydissipated by the bearing or to predict effective sliding speeds encountered in operation.Finally, a more sophisticated dynamic model is necessary to estimate the shocks on the cage,or to explain some abnormal operations observed in transient state. This approach is thusnecessary for more critical applications in order to dimension judiciously the retainer or tooptimize the internal geometry of the bearing.

This research along with numerous discussions that we had with our industrial or scientificpartners have produced some notable results, more particularly for aeronautical applications.The most significant seems to be the dissociation of the lubrication of cooling functions. Thusit becomes now usual to feed rolling bearings with only a small quantity of oil that ensures thelubricating function, the major part of the oil flow being derived around the bearing throughthe shaft and/or through the bearing housing to cool it. The use of a inner ring cage guidancetype becomes also more common, while at the same time a lubrication through the inner ringis preferred which both act to reduce the risk of cage skidding in also decreasing power losses.In a more academic step, it can be noted that this work was at the origin of 6 M. Sc. Thesis (ofwhich 3 continued by a Ph.D. thesis on another subject). Finally this research resulted in3 publications in the "Journal of Engineering Tribology (IMEch)", 2 in "TribologyTransactions (STLE)", 1 in "Lubrication Science (Leaf Coppin)", 1 in "Tribology Series(Elsevier, Leeds-Lyon Symposium)", 2 in the "Revue Française de Mécanique" and 1 in"Mécanique Industrielle et Matériaux", plus 14 communications in congresses withproceedings.

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However a certain number of scientific questions remains. They should be the subject offurther attentions in the coming years. Except the problems of surface damages, which will bediscussed in the following chapter, let mention:

- The evaluation of the real temperatures and clearances in operating bearings, which aregenerally introduced as data for the numerical simulations whereas they should result fromthe operating conditions and the nominal geometry.

- The influence of the elastic deformation of both rings and housing, in particular when thinrings are assembled in a flexible housing. It is a fundamental point that modifies the loaddistribution among rolling elements and sometimes leads to rolling element take off,which affects the bearing life directly. It is for example the case of some helicoptergearbox ball bearings or of turbine intershaft roller bearings.

- The study of bearing operating in transient regime, what should explain a certain numberof failure observed in real situations, including some ruptures of cage.

- The study of starved lubrication, not only for grease lubricated bearings, but also for high-speed applications, and for rolling bearings with porous cage impregnated of lubricant.

- The identification of lubricant rheological behavior at high pressure, in particular withregard to the local limiting shear stress and the condition of adherence or slip at the walls(see the recent work of Ehret et al. [76]).

- The development of solid lubrication for high temperature applications.

- The study of lubricant of substitution, such as kerosene or water with or without additive.

- The prediction of oleodynamic drag acting on the cage, for which little is know due to thecomplexity of the problem.

- The optimization of the contacts between the roller end and the ring guiding flange, whichare sensitive to scuffing and therefore limit the maximum rotation speed and/or axial load.

- The study of ball bearings with 4 contact points, which have complex internal kinematicsand load distribution.

Finally, only a fine and rigorous modeling of rolling bearings allows to know their internalkinematics and load distribution. This is an essential stage to tackle the problems of surfacedamage insofar as the local operating conditions at the interactions between rolling bearingelements can move away notably from ideal conditions. It is the subject of the next part.