Tribology International 40 (2007) 459–469

Calculation of Stribeck curves for (water) lubricated journal bearings

Alex de Krakera,�, Ron A.J. van Ostayena, Daniel J. Rixenb

aLaboratory of Tribology, Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2,

2628 CD Delft, The NetherlandsbEngineering Dynamics, Department of Precision and Microsystems Engineering, Faculty of 3mE, Delft University of Technology, Mekelweg 2,

2628 CD Delft, The Netherlands

Received 10 January 2006; received in revised form 10 April 2006; accepted 26 April 2006

Available online 14 June 2006

Abstract

This paper describes a mixed elastohydrodynamic lubrication (EHL) model for finite length elastic journal bearings. The finite element

method was employed to discretise the coupled system of 2D–3D Reynolds-structure equations and to compute Stribeck curves at

constant load. As underrelaxation strategies have been found to be insufficient for an iterative solution of this problem, artificial

dynamics have been added to the numerical structure equations in order to solve for stationary solutions of the fluid–structure problem.

An ideal plastic asperity contact model together with an effective film thickness formulation according to Chengwei and Linqing was

employed in order to compute the contact pressure in mixed lubrication. The method presented in this paper is applied to a typical water

lubricated journal bearing problem. Computed Stribeck curves are presented and the numerical performance of the method is evaluated.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Mixed lubrication; Soft-EHL; EHL; Journal bearing; Stribeck curve calculation; Fluid–structure interaction

1. Introduction

Many elastohydrodynamic (EHL) journal bearing mod-els and algorithms solving the coupled system of equationsarising from the interaction between the lubricating fluidand the deformation of the bearing have been published [1–5]. Early EHL journal bearing models, such as presented byHigginson [1], apply a 1D thin layer elastic deformationmodel in order to calculate the bearing deflection and showhow the load capacity for a given eccentricity ratio isaffected by the bearing flexibility. Oh and Huebner [2]developed a 2D–3D finite element approach to the EHLjournal bearing problem and employed a staggerediterative algorithm to solve for the fluid–structure equili-brium. It was concluded in their work that an elasticbearing is certainly not inferior compared to the rigidbearing, but has the ability to distribute the load over alarger bearing surface area and that for the same minimumfilm thickness and peak pressure a higher load capacity canbe obtained. With respect to their numerical solution

method, they remarked that it did not converge when thebearing deformation was of the same order of magnitude asthe film height. Even with various underrelaxationstrategies, convergence of their solution was not ensured.Potential asperity contact at high-eccentricity ratios is

left out of consideration in these early publications. Morerecently, the paper of Wang et al. [5] incorporates asperitycontact by the elastic–plastic asperity contact model of Leeand Ren in order to study mixed lubrication (ML)phenomena. Three major factors affecting lubricationperformance were studied: elastic deformation of thebearing, surface roughness effect on lubrication andasperity contact pressure. Bearing deflection was evaluatedby reduction of the full 3D FEM stiffness matrix into a 2Dflexibility matrix. The average flow model of Patir andCheng (P&C) was employed to account for roughnesseffects on lubrication.A most useful tool for the design of journal bearings is

the Stribeck curve as it clearly points out the criticaljournal velocity at which the transition from EHL to MLtakes place or at which velocity a certain acceptablecoefficient of friction is exceeded. For soft-EHL problems,such as arising from polymer bearings, it became clear that

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�Corresponding author. Tel.: +3115 27 83726; fax:+3115 27 87980.

E-mail address: A.deKraker@3mE.TUDelft.NL (A. de Kraker).

solving the coupled fluid–structure equations is notstraightforward [2]. The papers cited above provide usefulinsight into ML phenomena but cannot be used as a toolfor journal bearing design optimisation. In this paper, wereport about a computational method solving the mixedsoft-EHL problem which can be used in optimisation of thejournal bearing design.

2. Problem formulation and equations

In ideal smooth gaps that have small heights and heightvariations and a low Reynolds number, the flow isdominated by viscous forces and inertia effects can beneglected. The flow in such a gap is a combination ofCouette and Poisseuille flow and can be described by theReynolds equation

qqx�rh3

12Zqpf

qxþ

U1 þU2ð Þrh

2

� �þ

qqy�rh3

12Zqpf

qy

� �¼ 0,

(1)

with h the film thickness and r the fluid density. Since themagnitude of the pressure in a conformal contact asencountered in journal bearings remains limited, both thefluid density and viscosity Z are assumed constantthroughout this paper. The surface velocities are denotedby U1 and U2, respectively. Furthermore, pf represents thehydrodynamic or fluid pressure. Real surfaces are not idealsmooth and the film thickness is generally given by

hT ¼ hþ d1 þ d2, (2)

where h is the compliance or nominal film thickness and d1and d2 denote the roughness amplitudes of the surfaces.With s1 and s2 the standard deviations of d1 and d2,

respectively, the composite rms surface roughness—or Sq

value in engineering terms—for a pair of rough surfaces is

s ¼ Sq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis21 þ s22

q. (3)

Embracing all surface details in a deterministic mannerwas—and still is—not feasible from a numerical point ofview in a coupled 2D–3D EHL problem. Hence, an averagerough Reynolds equation was derived by Patir and Cheng[6] correcting the Reynolds equation for the film heightvariations resulting from a randomly distributed surfaceprofile:

qqx�f0px

h3

12Zqpf

qxþðU1 þU2ÞhT

2þ

f0sðU1 �U2ÞSq

2

� �

þqqy�f0py

h3

12Zqpf

qy

� �¼ 0, ð4Þ

with h the nominal film thickness and hT the mean filmthickness. For a Gaussian distributed surface roughness,leaving roughness deformation out of consideration, hT isequal to h. The correction factors fpx and fpy are pressureflow factors and fs is a shear flow factor, correcting for thefluid transport by the roughness valleys. The shear flowfactor, however, is equal to zero if the surfaces have thesame Sq roughness [6], as is assumed throughout thispaper.Roughness deformation can be the result of local

hydrodynamic pressure build up or it can result fromcontact between the surfaces. Due to piezoviscous effects,roughness deformation due to local hydrodynamic pressurebuild up plays an important role in concentrated EHLcontacts—such as that occur in ball bearings—, but is ofpractically no importance in a journal bearing system. In

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Nomenclature

c radial clearance (m)C damping matrix (N s/m)D bearing diameter (m)e eccentricity (m)E Young’s modulus (Pa)f bearing coefficient of friction (–)f c coefficient of friction in dry contact (–)F c nodal contact force (N)F f nodal fluid force (N)h nominal film thickness (m)ht effective film thickness (m)hT mean film thickness (m)H hardness (Pa)K stiffness matrix (N/m)L bearing length (m)W bearing load (N)W t target load (N)p total pressure (Pa)pf hydrodynamic pressure (Pa)

pc contact pressure (Pa)R journal radius (m)S scaling factor (–)Sq surface roughness parameter (m)t bearing thickness (m)u vector with nodal displacements (m)_u discrete time derivative of u (m/s)U ;U1;U2 surface velocity (m/s)x; y; z local cartesian coordinate system (m)X ;Y ;Z global cartesian coordinate system (m)a numerical damping coefficient (s)Dt time step (s)Z dynamic viscosity (Pa s)n Poisson ratio (–)o journal frequency (rpm)f circumferential coordinate (rad)fpx;f

0px;fpy;f

0py pressure flow factor (–)

fs;f0s shear flow factor (–)

r fluid density ðkg=m3Þ

s0 projected bearing pressure (Pa)y load angle (�)

A. de Kraker et al. / Tribology International 40 (2007) 459–469460

case of contact between the surfaces, the highest asperitiesare flattened, altering the surface height distributionfunction. The nominal film thickness no longer representsthe mean film gap and can even become negative for largecontact fractions and the P&C average Reynolds equationbecomes insoluble. Wilson and Marsault [7] slightlymodified the P&C average Reynolds equation by replacingthe nominal film height h by an effective film height ht,representing the mean film height (volume between thesurfaces divided by the total area). The W&M averageReynolds equation reads

qqx�fpx

h3t

12Zqpf

qxþðU1 þU2Þht

2

� �

þqqy�fpy

h3t

12Zqpf

qy

� �¼ 0, ð5Þ

where fpx and fpy are alternative pressure flow factors.These pressure flow factors can be derived by comparingthe numerical solutions for a smooth and a rough controlvolume, provided the roughness satisfies the Reynoldsassumptions, similar to the method of Patir and Cheng [6].The numerical method presented in this paper to solve thecoupled Reynolds-structure deformation equations doesnot depend on the value of the pressure flow factors. Forsimplicity, fpx and fpy are set to 1.

2.1. Effective film height

To account for the change in the mean film heightbetween the surfaces in case of contact, we use an effective

(mean) film thickness formulation according to Chengweiand Linqing [8]. The effective film thickness ht representsthe fluid volume between the rough surfaces divided by thetotal surface area. The model of Chengwei and Linqing(C&L) includes no contact mechanics, but is based ongeometrical properties of the surface roughness only. Allsurface heights that are larger than the distance betweenthe mean planes of the surfaces are flattened to that surfacedistance h. In fact, one simply assumes a contactingasperity to be flattened, while surrounding asperitiesremain unaffected. This is illustrated in Fig. 1. Forsimplicity, only one rough surface is shown here.

For an infinitesimal small approach dh of the surfacesthe following relation holds for the change in fluid volume

between the surfaces:

dht ¼ ð1� acÞdh, (6)

with ac the surface area fraction that is in contact. Theright-hand side of Eq. (6) denotes the reduction of the fluidvolume between the surfaces due to the approach dh. It hasto be equal to the change in the mean volume between thesurfaces, denoted at the left-hand side of the equation.Eq. (6) can be rewritten to give

dht

dh¼ 1� ac ¼

Z h

�1

Fðh0Þdh0, (7)

where Fðh0Þ is the height distribution function of thecomposite surface roughness. For a Gaussian surfaceheight distribution function, this integral can be calculated:

dht

dh¼

1

21þ erf

1ffiffiffi2p

h

Sq

� �� �. (8)

The solution of this equation is

ht

Sq¼

1

2

h

Sq1þ erf

1ffiffiffi2p

h

Sq

� �� �þ

1ffiffiffiffiffiffi2pp e�ð1=2Þðh=SqÞ

2

. (9)

For large film heights (h=Sq43), the effective film height ht

is approximately equal to h. When h drops below 3Sq, thehighest asperities come into contact and an effective filmgap slightly larger than h remains. The other asymptote isfound for large negative values for h=Sq, which makes ht

approach zero. The effective film height function is shownin Fig. 2. The contact area fraction ac can then be definedas [8,9]:

ac ¼ 1�

Z h

�1

Fðh0Þdh0 ¼1

21� erf

1ffiffiffi2p

h

Sq

� �� �. (10)

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ac

h ht

Fig. 1. Schematic asperity contact according to the model of Chengwei

and Linqing.

-3 -2 -1 0 1 2 30

1

2

3

h/Sq

h t/S

q

Fig. 2. Effective film height as a function of the nominal nondimensional

film height h=Sq.

A. de Kraker et al. / Tribology International 40 (2007) 459–469 461

2.2. Contact pressure

Bowden and Tabor [10] suggested a fully plastic mode ofdeformation in asperity contacts and derived a linearrelation between contact force and asperity contactfraction, fulfilling Amontons first law of friction. Stochas-tic and purely elastic models were presented by Greenwoodand Williamson [11] and many other researchers, alsoobeying the first law of friction. The actual asperitydeformation mode will however probably be a combina-tion of elastic and plastic deformation. Though much efforthas been invested in developing a theoretical asperitycontact model, no universal closed-form expression de-scribing the elastic–plastic asperity contact pressure in arough contact has been presented. Presumably, it is thefractal nature of surface roughness that cannot beadequately represented by a limited number of surfaceparameters. Lee and Ren [12] numerically generated arandom rough surface and numerically solved the elastic–plastic deformation problem. They presented expressionsfor the contact area fraction and effective film height thatwere function fits to their numerical solutions. In [13] adetailed study, comparing the plastic model based onChengwei and Linqing with the numerical elastic–plasticmodel of Lee and Ren, is presented. Only small differenceswere found between both models. Furthermore, in case ofpolymers, one cannot easily distinguish between elastic andplastic behaviour, also due to the time-dependent proper-ties of polymers. Therefore, it is not clear which of themodels will give the best approximation. Further discus-sion with respect to the details of the asperity contactprocess is beyond the scope of this paper and will be leftout of consideration. Throughout the rest of this paper, wewill use the plastic asperity model of Bowden and Tabor.Future work will include validation of the model withexperiments. Accordingly, the asperity contact pressure isequal to the hardness H of the bearing material, such thatthe average contact pressure is given by

pc ¼ Hac ¼H

21� erf

1ffiffiffi2p

h

Sq

� �� �, (11)

with H three times the yield stress of the bearing material.See Fig. 3.

3. Application to the soft journal bearing problem

A schematic overview of a plain finite length journalbearing is depicted in Fig. 4. The bearing, having athickness t, is connected to the inner side of the bearinghousing. The inner radius and length of the bearing aredenoted by R and L, respectively. The radial clearance c isdetermined by the difference in radius between the journaland the bearing. A fluid film with viscosity Z is assumed tobe present in the gap. The displacement of the journalcentre is assumed to be parallel to the bearing centre, i.e.the bearing system is perfectly aligned, and is denoted by e.The angle between the line of centres and the load line,

usually referred to as load angle, is represented by y. Thebearing-to-journal stiffness ratio for polymer bearings issuch that both the journal deformation and bearinghousing deformation can be neglected. The nominal filmheight for the elastic journal bearing is defined as thedistance between the mean planes of the surfaces in thedeformed state:

h ¼ cþ e cosðfÞ þ uz, (12)

with uz the normal deformation of the bearing surface.Inplane displacements are assumed to be small comparedto the displacements normal to the bearing surface. Stress–strain relations for linear isotropic material according toHooke’s law were used to solve the standard 3D structuredeformation equations for the displacement uz of thebearing surface. Numerical solution by finite elements isdiscussed in the next section. The effective film height ht

follows from substitution of relation (12) into Eq. (9).For a journal bearing with stationary bearing and

journal surface velocity U, Eq. (5) gives

qqx�

h3t

12Zqpf

qxþ

Uht

2

� �þ

qqy�

h3t

12Zqpf

qy

� �¼ 0. (13)

ARTICLE IN PRESS

-3 -2 -1 0 1 2 30

H

h/Sq

p c

Fig. 3. Contact pressure as a function of the nominal nondimensional film

height h=Sq.

housingY

Tc

Tf

���

W

Wf

Wc

ZX

ZX

bearing

Fig. 4. Journal bearing geometry (left and centre) and bearing forces

(right).

A. de Kraker et al. / Tribology International 40 (2007) 459–469462

The total pressure on the bearing surface is given by thesum of the hydrodynamic and contact pressure

p ¼ pc þ pf , (14)

where pc follows from Eq. (11) and pf from the numericalsolution of Eq. (13).

3.1. Boundary conditions

As the bearing liner is connected to a rigid housing andthe thickness of the bearing liner is relatively small, i.e. inthe order of 10% of the bearing diameter, the bearing canbe ‘unwrapped’ onto a local cartesian coordinate systemðx; y; zÞ. In this coordinate system, x corresponds to thecircumferential coordinate, y still denotes the axialcoordinate and z is the thickness coordinate of the bearing.Due to symmetry with respect to the plane y ¼ 0, thecomputational domain can be divided into half. Boundaryconditions applied to the structure deformation problemare zero displacement at the interface between the bearingand the housing (at z ¼ 0) and zero displacement in axialdirection on the symmetry surface (at y ¼ 0). Periodicalboundary conditions apply to the surface at x ¼ 0 and 2p.Boundary conditions for the Reynolds flow problem arezero pressure at the side end of the bearing (y ¼ L=2) andno-flow boundary condition in axial direction at thesymmetry line y ¼ 0. Again, periodical boundary condi-tions have to be prescribed at x ¼ 0 and 2p for a 360�

bearing without lubricant supply groove.

3.2. Cavitation

Mathematical solution of the Reynolds equation maypredict a negative value for the pressure in diverging parts ofthe film. In practice, a cavitated area is found due to dissolvedgasses that appear when the pressure drops below the vapourpressure. The Reynolds cavitation boundary condition withpcav equal to the vapour pressure has been applied

pfXpcav (15)

and

qpf

qx¼

qpf

qy¼ 0 (16)

at the cavitation boundary.

3.3. Friction

Shear forces are transferred through both the fluid andthe contacting asperities. The shear stress transferred bythe fluid is given in [1]:

tf ¼ZU

ht

�ht

2

qpf

qx. (17)

The traction introduced by asperity contact is written as alinear function of the contact pressure:

tc ¼ f cpc, (18)

with f c the coefficient of friction under pure boundarylubrication (BL) conditions, that has to be obtained fromexperiments. In the simulations presented in this paper, f c

has been set to a value of 0:12, which is representative forultra high molecular weight polyethylene (UHMWPE).The shear stresses acting on the bearing surface result in atorque as well as a force. The latter contributes to the loadcapacity of the bearing. The torque can be found byintegration of the shear stresses over the bearing surfacearea, multiplied by the bearing radius:

T ¼ T f þ Tc ¼ R

ZOtf dOþ R

ZOtc dO. (19)

Both the normal pressure and the traction at the bearingsurface contribute in carrying the load. The vector ~W c inFig. 4 is defined as the vectorial sum of the forces acting onthe shaft that are generated by the normal pressure andtraction due to contact:

~W c ¼W c;X

W c;Z

!¼

RO sinðxÞpc dOþ

RO cosðxÞtc dOR

O cosðxÞpc dO�RO sinðxÞtc dO

!.

(20)

Note that pc;f and tc;f are defined as the pressure and shearstress acting on the bearing surface. Similar to W c, the loadcarried by the fluid is given by

~W f ¼W f ;X

W f ;Z

!¼

RO sinðxÞpf dOþ

RO cosðxÞtf dOR

O cosðxÞpf dO�RO sinðxÞtf dO

!.

(21)

The total load vector ~W then follows from the vectorialsum of the fluid and contact component (Fig. 4). From theload vector, the load capacity W and load angle y can bederived. According to [1], the bearing coefficient of frictionis determined by

f ¼T

WR. (22)

4. Numerical method

The commercially available finite element code SE-PRAN has been used to discretise both the Reynoldsequation on a 2D grid and the structure deformationequations on a 3D grid.

4.1. Reynolds pressure field

In order to satisfy the Reynolds cavitation condition, thenumerical solution to the Reynolds equation must beconstrained by an additional boundary condition:

pfXpcav. (23)

Nodal points on the bearing surface that have a negativevalue for the pressure solution are identified as cavitationpoints and the essential boundary condition pf ¼ pcav isapplied to such a point. Since the magnitude of the fluid

ARTICLE IN PRESSA. de Kraker et al. / Tribology International 40 (2007) 459–469 463

pressure is large compared to the vapour pressure, pcav isset to zero. In order to satisfy continuity an extra conditionis needed at the cavitation boundary:

qpf

qxi

¼ 0. (24)

It has been shown in [14] that the continuity condition ofEq. (24) is satisfied automatically by successive iteration ofthe constrained (Eq. (23)) solution. In practice, the film isruptured—or starved—in the cavitation area while in ourmodel the film is fully flooded but at zero pressure level.Viscous forces in the cavitation area are set to zero. In oursolution the reformation boundary will therefore be founda little upstream. In spite of this, the error with respect tothe bearing load capacity will remain rather limited sincethe pressure in the vicinity of the reformation boundary isapproximately zero and the pressure gradient qp=qx issmall. The error with respect to the calculation of thefriction forces is small as well. At the reformationboundary, the pressure gradient is small and the viscousforce is dominated by the Couette flow component, givenby the first term in Eq. (17). Soft bearings operate at higheccentricity ratio and therefore, the film thickness isrelatively large at the reformation boundary. As a result,friction losses are small at the location of the reformationboundary.

4.2. Bearing deformation

The FEM discretisation of the structure force equili-brium resembles a set of algebraic equations:

Ku ¼ F, (25)

with K the stiffness matrix, u the nodal displacements of thestructure and F the sum of the nodal hydrodynamic andcontact forces that act on the bearing surface:

F ¼ Ff þ Fc. (26)

The hydrodynamic forces Ff are found by integration ofthe hydrodynamic pressure over the element surface area.The contact forces Fc follow from integration of thecontact pressure (Eq. (11)) over the element surface area.Note that all forces in Eq. (26) have only one component,normal to the bearing surface. We found that neglectinginplane forces acting on the bearing surface due to fluidand contact shear results in a very small error. Inplaneforces due to fluid and contact shear are therefore ignoredin the solutions that are presented in the next section.

The solution of the coupled problem is obtained by astaggering scheme between the Reynolds equation and thebearing deformation problem. It can be seen as a pseudo-dynamic stepping. In Eq. (25) no viscous dissipation ordamping is present since we are solving for stationaryoperating points. To prevent numerical instability of thestaggered scheme between fluid and structure, artificialdamping is added to the structure:

C _uþ Ku ¼ SF with C ¼ aK . (27)

The coefficient a is a damping constant and S is a scalingfactor that will be explained further on in this section. Thediscrete time derivative of u at time tþ 1 is given by

_utþ1 ¼utþ1 � ut

Dt. (28)

Substituting this in Eq. (27) gives

Kutþ1 ¼Dt

aþ Dt

� �SFt þ

aaþ Dt

� �Kut. (29)

The term Kut can be replaced by the right-hand side of theequilibrium equation at time step t. The factors betweenbrackets can be considered as underrelaxation factors

applied to the solution. Eq. (29) yields the bearingdeformation at time step tþ 1. As a criterion forconvergence we use C _u5Ku (see Eq. (27)). This gives

ak_utþ1k

kutþ1k¼

aDt

kutþ1 � utk

kutþ1kptol, (30)

with tol a small value in the order of 10�5. A suitable a=Dt

ratio has to be chosen, depending on the problem andoperating conditions. In full film conditions, the bearingdeformation is small compared to the film thickness andthe equilibrium can be found relatively easily compared tothe ML regime, where the contribution of the contactpressure appears. Choosing a single a=Dt ratio that ensuresconvergence in the ML region results in unnecessarilymany iterations in the EHL area. Therefore, a simpleheuristic time step searching rule is applied in order to finda suitable a=Dt so that larger time steps are used when theproblem allows it. When divergence of the solution isdetected, i.e. the value of expression (30) increases withrespect to the previous iteration, the time step is decreased.Otherwise, the time step is slowly increased, i.e. a=Dt isdecreased.To compute points on the Stribeck curve, one has to

iterate for a constant bearing load W t. Given the previoussolutions (e,W) this is a matter of zero finding. A Secantmethod was used in order to make a new estimate for thejournal eccentricity. Generally, about six iterations areneeded. The entire numerical scheme is shown in Fig. 5.The scaling factor S that was introduced in Eq. (27) is

set to

S ¼W

W t, (31)

with W t the target load and W the load computed in thecurrent iteration. As a consequence, the total load that isapplied to the structure problem by the contact andhydrodynamic pressure is equal to the target load,regardless of the eccentricity e. In fact, only the shape ofthe load distribution is passed to the structure deformationproblem. Hence, no large differences in the solution for thestructure deformation between successive iterations appearand we only iterate for the correct balance between contactand fluid pressure. As a result, faster convergence isobtained for the inner loop (see Fig. 5) where we solve forthe equilibrium between fluid and structure. Intermediate

ARTICLE IN PRESSA. de Kraker et al. / Tribology International 40 (2007) 459–469464

solutions are however incorrect with respect to theeccentricity ratio that was estimated. In the second loop,we search for the correct load by adjusting the eccentricity.Once this loop has converged, the total force equilibrium

between fluid and structure and the eccentricity ratio isobtained. The outermost loop in the numerical scheme hasbeen added to construct the Stribeck curve by reduction ofthe surface velocity U from a high initial value U start downto a very low value U stop (� 10�3U start). In the next section,we will show how the number of iterations to calculate apoint on the Stribeck curve is affected by the introductionof the scaling factor.

5. Results

In this section, some preliminary results with respect tothe water lubricated polymer journal bearing are shown.As discussed previously, the bearing was unwrapped onto acartesian coordinate system and divided into 52� 8� 2nonuniformly distributed elements. Locally a finer meshsize was used in the vicinity of the contact region, i.e. aty ¼ 180� and at the outer edge of the bearing. Furtherglobal mesh refinement did not improve the solution. Forthe structure deformation equations, 3D linear elementswere used. 2D rectangular linear elements were used for theReynolds equation. The nodes of the 2D and 3D elementscoincide at the bearing surface to ensure correct couplingbetween the fluid and structure problem. Table 1 lists thedesign and material parameters for an arbitrary butrepresentative reference bearing that is used in thesimulations. The values that were used for the Young’smodulus, hardness and Poisson ratio are representative forUHMWPE.A typical fluid and contact pressure distribution in ML

conditions is shown in Fig. 6. Due to the soft bearingmaterial, a relatively large area contributes in carrying theload. Therefore, the maximum contact and fluid pressureremains limited. Initial contact typically occurs at the sideends of the bearing since, because of side flow, the fluidpressure drops rapidly to the ambient pressure at thislocation. As one runs into BL, the maximum contactpressure is found at the centre of the bearing.In Fig. 7 computed Stribeck curves (left) and minimum

film height (right) are shown. In each set of figures, thesensitivity with respect to a single parameter is shown. Thereference case as listed in Table 1 is plotted in black in all

ARTICLE IN PRESS

Ustart(3)

(2)

(1)

estimate e

e

compute h

h

compute ht

compute Pc

ht

solve Reynolds equation

compute cavitation area

compute W

scale P

P

solve 3D structure equations

with artificial damping at t+1

u

C u << K u ?

no

no

no

yes

yes

yes

W = Wt ?

choose new U

U < Ustop ?

stop

Pc

(eq 29)

(eq 31)

(eq 19 + 20)

(eq 12)

(eq 10)

(eq 8)

(eq 11)

Fig. 5. Numerical scheme.

Table 1

Design parameters for the reference bearing

Description Parameter Value Dimension

Bearing radius R 25 mm

Bearing length L 100 mm

Bearing thickness t 10 mm

Radial clearance c 0.125 mm

Composite surface roughness Sq 0.424 mmYoung’s modulus E 1 GPa

Poisson ratio n 0.45 –

Hardness H 50 MPa

Coefficient of friction in BL f c 0.12 –

Fluid viscosity Z 0.001 Pa s

Load W 1000 N

A. de Kraker et al. / Tribology International 40 (2007) 459–469 465

graphs. From top to bottom, the radial clearance C ¼ c=R,the Sq surface roughness value, the load W, the Young’smodulus E and the material hardness H is varied.

From the first set of figures it is clear that for smallerradial clearance, a lower coefficient of friction and largerminimum film height is obtained for the same operatingconditions in ML. Reduction of the radial clearance by afactor two results into a reduction of the rotationalfrequency at the ML–EHL transition by approximately afactor two.

The sensitivity with respect to the combined surfaceroughness (see Eq. (3)) is shown in the second pair offigures. For smaller combined surface roughness the ML–EHL transition is shifted to the left and a lower coefficientof friction in ML is obtained for the same operating point.Clearly, the ML–EHL transition is strongly affected by thesurface roughness.

In the third set of figures, the load is varied with respectto the reference case. The parameter s0 ¼W=ðLDÞ denotesthe projected bearing pressure. As is to be expected, forincreasing load, the Stribeck curve slightly shifts to theright.

The modulus of elasticity has been varied in the fourthpair of figures. Due to larger bearing deformation for softermaterials, the contact pressure is distributed over a largerarea. As a result, the contact pressure is reduced and aneffective fluid film is preserved between the surfaces even atlow journal frequencies. Furthermore, it can be seen thatthe minimum film thickness obtained for the softer bearingis larger than that for the more rigid bearing in most cases.However, the minimum film height for the softest bearing

material ðE ¼ 0:01MPaÞ intersects the two other curves.This is probably due to the large bearing deformationtogether with the high Poisson ratio of 0:45. The nearlyincompressible behaviour causes the bearing material to bepressed upwards at the side ends of the bearing, where thehydrodynamic load is minimal, resulting into contact at theside ends only (Fig. 8). The apparent discontinuity—whichis not a real discontinuity but in fact a continuoustransition—in the minimum film height at approximately50 rpm appears at the moment that the contact areas at theside ends of the bearing are connected by a narrow contactzone through the centre of the bearing. This is illustrated inFig. 9, where a contour plot of the contact pressure isshown for o ¼ 54 and 33 rpm.The bottom pair in Fig. 7 shows how the material

hardness affects the bearing performance. Mathematically,the hardness parameter H represents some kind of rough-ness stiffness and affects the computation of the contactpressure (Eq. (11)). It strongly affects the coefficient offriction in ML and the minimum film height that isobtained in ML and BL.

5.1. Numerical performance

Fig. 10 shows the number of iterations per Stribeck point(iSp) that is needed to compute the point on the Stribeckcurve at the corresponding journal frequency for thereference bearing. For a fixed a=Dt value of 100, up to5000 iterations are necessary in the vicinity of the ML–EHL transition. The time step heuristic (TSH) provided asimple but useful method to choose a suitable a=Dt ratiofor a particular journal frequency. It was found that in theBL region the average a=Dt ratio is small. This is due to thefact that the problem is contact-dominated and the relationbetween film height and bearing deformation is approxi-mately linear. Similarly, a small a=Dt ratio is sufficient tosolve the problem very rapidly in the EHL region. In thiscase, however, the fluid pressure dominates the problemand the bearing deformation is small compared to the filmheight. Hence, the solution is not very sensitive to a changein film height. Close to the ML–EHL transition, theproblem is not dominated by fluid or contact pressure anda larger a=Dt ratio was found to be necessary. In practice,we have used 1oa=Dto105 for this ratio.From Fig. 10, it can also be concluded that the numerical

effort is further reduced by the application of the scalingfactor S (Eq. (31)). During successive iterations novariation in the total load that is applied to the structureappears. Therefore, the differences in bearing deformationand film thickness during successive iterations remainrather limited. In fact, we only solve for the correctdistribution between contact and hydrodynamic pressure inthe inner loop of the algorithm. The combination (blackline) of TSH and scaling results in a low and almostconstant number of iterations for the complete Stribeckcurve with an average number of iterations per Stribeckpoint of approximately 150.

ARTICLE IN PRESS

090

180270

360

-L/20

L/20

0.25

0.5

φ [degrees]

p f [M

Pa]

090

180270

360

-L/20

L/20

0.25

0.5

φ [degrees]

p c [M

Pa]

Fig. 6. Typical fluid and contact pressure distribution for a soft journal

bearing in mixed lubrication.

A. de Kraker et al. / Tribology International 40 (2007) 459–469466

ARTICLE IN PRESS

10 100 10000

0.04

0.08

0.12

ω [rpm]

f [-]

C = 0.00125C = 0.0025C = 0.005C = 0.01

10 100 10000

4

8

16

24

ω [rpm]

h t,m

in/S

q [-

]

C = 0.00125C = 0.0025C = 0.005C = 0.01

10 100 10000

0.04

0.08

0.12

ω [rpm]

f [-]

Sq = 0.21 µmSq = 0.42 µmSq = 0.84 µm

10 100 10000

4

8

16

24

ω [rpm]h t

,min

/Sq

[-]

Sq = 0.21 µmSq = 0.42 µmSq = 0.84 µm

10 100 10000

0.04

0.08

0.12

ω [rpm]

f [-]

σ0 = 0.1 MPaσ0 = 0.2 MPaσ0 = 0.4 MPaσ0 = 0.8 MPa

10 100 10000

4

8

16

24

ω [rpm]

h t,m

in/S

q [-

]

σ0 = 0.1 MPa

σ0 = 0.2 MPaσ0 = 0.4 MPaσ0 = 0.8 MPa

10 100 10000

0.04

0.08

0.12

ω [rpm]

f [-]

E = 1 GPaE = 0.1 GPaE = 0.01 GPa

10 100 10000

4

8

16

24

ω [rpm]

h t,m

in/S

q [-

]

E = 1 GPaE = 0.1 GPaE = 0.01 GPa

10 100 10000

0.04

0.08

0.12

ω [rpm]

f [-]

H = 5 MPaH = 50 MPaH = 500 MPa

10 100 10000

4

8

16

24

ω [rpm]

h t,m

in/S

q [-

]

H = 5 MPaH = 50 MPaH = 500 MPa

Fig. 7. Sensitivity of the computed Stribeck curve and minimum film height with respect to the design parameters C and Sq, the load W and material

parameters E and H.

A. de Kraker et al. / Tribology International 40 (2007) 459–469 467

6. Conclusions

In this paper an iterative finite element algorithm for themixed-EHL journal bearing problem with soft surfaces hasbeen presented. Standard linear elastic structure equationswere used to describe the elastic behaviour of the bearing.The fluid pressure in the gap is calculated using theReynolds equation. To calculate the effective film heightand contact pressure, a stochastic ideal plastic contactmodel was used. Adding artificial damping to the structuredeformation equations provides a stable and robustmethod to find a solution to the fluid–structure problem,arising from the soft journal bearing that operates in ML.

The performance of soft mixed-EHL journal bearings interms of load capacity and friction has been found superiorwith respect to more rigid bearings. This was alreadysuggested by Huebner and Oh in 1973, but they were notable to solve the EHL problem for soft bearings.

The amount of computing effort was reduced by scalingthe load that is applied to the structure equations duringiterative solution of the fluid and structure equations.Intermediate solutions are incorrect since there is actuallyno equilibrium between the journal eccentricity e and theload that is carried. However, the equilibrium is satisfiedfor the final solution and convergence is obtained muchfaster. A TSH was applied to find a suitable ratio betweenthe time step and the damping constant, also reducing thecomputation time.Future work will include validation of the model with

laboratory experiments.

References

[1] Higginson GR. The theoretical effects of elastic deformation of the

bearing liner on journal bearing performance. Proceedings on the

elastohydrodynamic lubrication symposium, 21–23 September 1965.

London: The Institution of Mechanical Engineers; 1966.

[2] Oh KP, Huebner KH. Solution of the elastohydrodynamic finite

journal bearing problem. J Lubr Technol 1973;July:342–52.

[3] Wang QJ, Shi F, Lee SiC. A mixed-lubrication study of journal

bearing conformal contacts. J Tribol 1997;119:456–61.

[4] Shi F, Wang QJ. A mixed-TEHD model for journal-bearing

conformal contacts—Part I: model formulation and approximation

of heat transfer considering asperity contact. J Tribol 1998;120:

198–205.

[5] Wang QJ, Shi F, Lee SiC. A mixed-TEHD model for journal-bearing

conformal contacts—Part II: contact, film thickness, and perfor-

mance analyses. J Tribol 1998;120:206–13.

[6] Patir N, Cheng HS. Application of average flow model to lubrication

between rough sliding surfaces. J Lubr Technol 1978;78:1–10.

[7] Wilson WRD, Marsault N. Partial hydrodynamic lubrication with

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[8] Chengwei W, Linqing Z. An average Reynolds equation for partial

film lubrication with a contact factor. J Tribol 1989;111:188–91.

[9] Ostayen van RAJ, Beek van A, Ros M. A mathematical model of the

hydro-support, an elasto-hydrostatic thrust bearing with mixed

lubrication. Tribol Int 2004;37:607–16.

ARTICLE IN PRESS

0 90 180 270 360-L/2

0

L/2

φ [degrees]

Fig. 8. Contour plot of the contact pressure distribution for the reference

bearing geometry with E ¼ 0:01MPa at o ¼ 244 rpm. Contact only

occurs at the side ends of the bearing.

0 90 180 270 360-L/2

0

L/2

φ [degrees]

0 90 180 270 360-L/2

0

L/2

φ [degrees]

Fig. 9. Contour plot of the contact pressure distribution for the reference

bearing geometry with E ¼ 0:01MPa at o ¼ 54 rpm (upper figure) and

o ¼ 33 rpm (lower figure).

10 100 10000

2000

4000

6000

ω [rpm]

iSp

α/∆ t = 100

TSH

TSH + S

Fig. 10. Effect of scaling (S) and time step heuristic (TSH) on the total

number of iterations (iSp) needed to compute a point on the Stribeck

curve. The Stribeck curve is plotted to indicate lubrication regimes.

A. de Kraker et al. / Tribology International 40 (2007) 459–469468

[10] Bowden FP, Tabor D. Friction and lubrication of solids, revised ed.

Oxford: Clarendon; 2001 ISBN 0-19-850777-1.

[11] Greenwood JA, Williamson JBP. Contact of nominally flat surfaces.

Philos Trans R Soc London, Ser A 1966;295:300–19.

[12] Lee SiC, Ren N. Behavior of elastic–plastic rough surface contacts as

affected by surface topography, load, and material hardness. Tribol

Trans 1996;39:67–74.

[13] Ostayen van RAJ. The hydro-support: an elasto-hydrostatic thrust

bearing with mixed lubrication. PhD thesis, Delft University of

Technology; 2002.

[14] Christopherson DG. A new mathematical model for the solution of

film lubrication. Proc Int Mech Eng 1941;149:126–35.

ARTICLE IN PRESSA. de Kraker et al. / Tribology International 40 (2007) 459–469 469