MxedLubricationTheory

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1

CHAPTER 7 INTRODUCTION TO EHL AND MIXED LUBRICATION THEORIES

7-1 EHL and EHL problems

The application of the Reynolds equation to components in conformal contact, such as the

journal bearing, has successfully solved many design problems. It is natural to apply theReynolds equation to elements in counterformal contact, such as gears, cam and followers, and

rolling-element bearings, and look for the same success. Martin (1916) studied gear lubricationby means of rigid cylinders, which we know were simulations of tooth bodies at contact based on

the radii of curvature at the contact position, as shown in Figure 7-1. Guess what he found? Thepredicted film thickness was much less than the surface roughness of the best-machined gear

surfaces, sometimes was even negative. If it were true, the gears tooth under such film thicknessshould worn out rapidly. However, in reality, such gears worked fine. This suggested that the

real film thickness is thicker (or much thicker) than what was predicted by the rigid-cylinderlubrication model.

We know from elasticity that when two elements are in a counterformal contact, the contactpressure is highly localized, causing the surfaces to deform. Such deformation opens the gap, orincreases the film thickness. The elastic deformations of surfaces contribute to the lubrication

film thickness. Therefore, the Reynolds equation should be solved with an elasticity component,and such lubrication theory is named the elastohydrodynamic lubrication, or EHL in short.

The objectives of an EHL analysis are to obtain the lubrication status and determine the

distributions of hydrodynamic pressure and film thickness for a given geometry and a set of operating conditions. Film thickness is an important design parameter. With the assistance of the

EHL analysis, an engineer can adjust his/her design to ensure that the designed product worksunder full-film lubrication. The pressure distribution information will be further used for stress

and strain analyses, and for surface strength evaluation.

Figure 7-1. Equivalent cylinders for gear tooth contact

Gearteeth



2

7-2 Elastic deformation and the film-thickness equation for plane problems

The line contact and lubrication of parallel cylinders, as well as a cylinder and a half plane, maybe treated as a plane problem. As a matter of fact it is a plane-strain problem. The hydrodynamic

and mechanical responses to loading and motion are the same in every cross sectionperpendicular to the axis of the cylinder. Let's study the film thickness in such lubrication

problems. A deformation component should be included; however, let's start from a simpleproblem, the lubrication of two rigid bodies.

1 The film-thickness equation for rigid bodies.

Figure 7-2 shows a pair formed by a rigid cylinder and a rigid plane and a pair by two parallelrigid cylinders. We can easily derive the corresponding film thickness (which is really the

geometric gap) for each of them.

Figure 7-2 The geometry for the film thickness in a pair formed by a rigid cylinderand a rigid plane and a pair by rigid cylinders

For the cylinder-plane pair, the film thickness at position x is

R

xh

R

x Rh

R

x Rhhhh

2)

21(111

2

02

2

02

2

00 +=

−−+≈

−−+=∆+= (7-1)

Here, we approximate ∆h by

R

xh

2

2

=∆ (7-2)

For the cylinder-cylinder pair, the film thickness at position x is

2

2

1

2

021022 R

x

R

xhhhhh ++=∆+∆+= (7-3)

x

xh0

h0

h

h

∆h

∆h1

∆h2

R

R1

R2



3

If we use the equivalent cylinder concept, 1/R =1/R1+1/R2, equations (7-1) and (7-3) become the

same.

The film-thickness equation for elastic bodies. Now we proceed to consider the elasticdeformations in bodies. Figure 7-3 illustrates the deformations caused by a concentrated force at

x = 0 and by a distributed load. Let's analyze the deformation. For the pressure distributed inregion Ω on the surface of a half plane, we need the Flamant solution (1) to describe the

pressure-displacement relation. Remember, this is for the pressure on the surface of an infinitelylarge body (a half plane for the domain of a plane strain problem).

Figure 7-3. 2D elastic deformation caused by a concentrated force at x = 0 and a distributed

load in region Ω.

The normal surface displacement, dδa, due to the point force at x = 0

C x

E

P xd a +

−−= ||ln

)1(2)(

π

νδ

(7-4)

The core of Equation 7-4 is a Green's function having a singularity at the point of the force

application where the displacement tends to be infinite. We need to pick up a reference point, xr ,which is sufficiently far away from the region of interest. Therefore, the displacement we get is

the deformation with respect to that reference point.

||ln)1(2

)()()(2

r

r

aa

x

x

E

P xd xd xd

π

νδδδ

−−=−=

(7-5)

The normal surface displacement due to the distributed load is the integration of Equation (7-5)

over the loading region, Ω.

∫ Ω −

−−−= ξξ

ξ

ξ

π

νδ d p

x

x

E x

r

)(||ln)1(2

)(2

(7-6)

ξ

x

xr

δ(x)x

P



4

If we use the equivalent elastic modulus, E', from2

2

2

1

2

1 11

'

2

E E E

ν ν −+

−=

,in equation (7-6), we

obtain the displacement equation for two bodies under the same load

∫ Ω −−

= ξξξ

ξ

πδ d p x

x

E xr

)(||ln'

4

)( (7-7)

With the deformation solve, we can write the film-thickness equation for the counterformal

problem of two cylinders by means of their equivalent radius of curvature, R = R 1 R2 /(R1 + R2).

∫ Ω −

−−+=

++=+∆+=

ξξ

ξ

ξ

π

δδ

d p

x

x

E R

xh

R

xhhhh

r

)(||ln

'

4

2

2

2

2

0

2

2

00

(7-8)

The derivation above clearly shows the geometry and mechanics, In computation practice, the

arbitrary constants may be processed in one by directly using Equation (7-4).

2 Lubricants, their viscosity and density

Lubricating oils, either mineral or synthetic, are common lubricants for parts under EHL. Theycan enter the loaded conjunction easily and flush away contaminants and debris. The flow of oil

can also help transfer heat out of the highly loaded area. Lubricating greases are also seen,especially in rolling-element bearings. The use of grease can help simplify housing geometry

because no lubricant replenishment system is necessary for most of such designs.

We know that the viscosity, η, of a lubricant is usually a function of temperature and pressure.The effective viscosity of a lubricant may obey one of the following relations.

Viscosity-temperature relationship:

T B Aa lg))lg(lg( −=+ ν T is in K a = 0.6

t ke

βη −= t is in C

Viscosity-pressure relationship:

peαηη 0=n

cp)1(0 +=ηη

Viscosity-temperature-pressure relationship:

)]()(1[ 000 t t p pc −+−+= βηη



5

) / / / (

00 T rpT T p

e+++= ββαηη

The density of a lubricant may be described by a function of pressure. Following is a density-pressure relationship for mineral oils (Dowson-Higgison, 1966)

p

p

7.11

6.01

0 ++=

ρρ

7-3. EHL formulation

The formulation for an EHL problem describes the lubricant properties, the hydrodynamics of the lubricant flow, and the elasticity of the bodies involved in the problem.

1 Lubrication. The Reynolds equation is employed to express the relationship between the

film thickness and the hydrodynamic pressure. For a line-contact problem, we ignore the side

leakage. Because both surfaces may move, the Reynolds equation is slightly different from whatwe derived before, where only one surface is assumed moving. Here, U = (U1+U2)/2.

x

hU

x

ph

x ∂

ρ∂

∂

∂

η

ρ

∂

∂ )(12

3

=

(7-9)

2 Lubricant. We will use the exponential viscosity-pressure relationship for simplicity

peαηη 0= (7-10)

The density-pressure relationship to be used is

p

p

7.11

6.01

0+

+=ρ

ρ(7-11)

3 Film thickness. The film-thickness equation (7-8) that considers body surfacedeformation is used.

∫ Ω −

−−+= ξξ

ξ

ξ

πd p

x

x

E R

xhh

r

)(||ln'

2

2 2

2

0(7-12)

For point-contact (or elliptic contact ) problems we only need to add a proper film geometry termin the other direction and replace the displacement term in the film thickness equation by the

following Boussinesq equation

∫ Ω −+−

−=22 )()(

),('

4)(

ζξ

ζξζξ

πδ

y x

d d p

E x

(7-13)



6

4 Load equation. Integration of the pressure obtained from the Reynolds equation resultsin the load supported by the film.

∫ = pdxW (7-14)

Equations (7-9) through (7-14) can only been solved numerically. The commonly usednumerical methods are the relaxation iteration method, the Newton Raphson method, and the

multigrade method.

7-4 EHL solutions for line-contact problems

A large amount of numerical computations have been conducted in the last century. In EHL

problems, two film-thickness values are usually calculated, the minimum film thickness and thecentral film thickness, as illustrated in Figure 7-4. The former determines the worst lubrication

condition, while the latter indicates the film geometry at the central flatted region. The numericalresults obtained in different regimes of lubrication yield film-thickness formulas for different

application, which are classified based on the characteristics of body elasticity and lubricantproperties.

Figure 7-4 Typical EHL film thickness and pressure distribution.

1. Typical solutions and formulas for minimum film thickness

Typical line-contact EHL solutions for minimum film thickness are Martin’s rigid-isoviscousformulas, Blok’s Rigid-piezoviscous formulas, Herrebrugh's Elastic-isoviscous formulas, and

Dowson-Higgison’s Elastic-piezoviscous formulas. In the formulas, load W = PB is used, where

B is the length (width) of the contact).

(1) Martin’s solution: formulas for rigid-isoviscous (RI) lubrication. Martin's solution does

not consider the body elasticity and viscosity variation due to pressure rise. Therefore, it may beapplied to the case where the hydrodynamic pressure is not high enough to cause significant

deformation and viscosity change.

hc hmin

Film thicknessPressure distribution



7

=

URB

W h

0

min9.4

η

(7-15)

(2) Blok’s Solution: formulas for rigid-piezoviscous (RV) lubrication. Blok’s solution doesnot consider the body elasticity but includes viscosity variation due to pressure rise. It applies to

the case where the viscosity is sensitive to the change of the hydrodynamic pressure.

3 / 2

2 / 1

0

2 / 3

0

min

)(66.1

=

RU

W

VRB

Wh

η

α

η(7-16)

(3) Herrebrugh's Solution: formulas for elastic-isoviscous (EI) lubrication. Herrebrugh's

solution does consider the body elasticity but ignores the viscosity variation due to pressure rise.It applies to the case where the viscosity is not sensitive to the change of the hydrodynamic

pressure.

8.0

2 / 1

0

min

)'(0.3

=

URE B

W

VRB

Wh

η(7-17)

(4) Dowson-Higgison’s solution: formulas for elastic-piezoviscous (EV) lubrication.Dowson-Higgison’s solution considers both the body elasticity and the viscosity variation due to

pressure change. It is more general and applies to the case where both viscosity-pressure anddeformation-pressure relationships need to be taken into account.

06.0

2 / 1

54.0

2 / 1

0

2 / 3

0

min

)'()(65.2

=

URE B

W

RU

W

URB

Wh

ηα

η(7-18)

2 Solution generalization and EHL design for line contacts

The expressions of the formulas mentioned above may be significantly simplified and writteninto a generalized form by non-dimensionalization and parameter grouping.

(1) Dimensionless groups. Following dimensionless groups in Equation (7-19) are defined

for film-thickness formulas.

Dimensionless film thickness: Rh H minˆ =

Dimensionless load: RB E

W W

'ˆ =

(7-19)



8

Dimensionless speed: R E

U U

'ˆ 0η=

Dimensionless material parameter: 'ˆ E G α=

Here, U = (U1+U2)/2, R = R1 R2 /( R1 + R2), and2

2

2

1

2

1 11

'

2

E E E

ν ν −+

−=

.

(2) Minimum-film-thickness formulas. The dimensionless parameter groups (Equation (7-

19)) are further grouped into three g parameters shown in Equation (7-20). A generalizedformula for the minimum film thickness is then written with g parameters (Equation (7-21),

which is the same as Equation (7-22) written with dimensional parameters. Constants, Z, m, andn are given in Table 7-1.

H U

W gh ˆ)ˆ

ˆ

(= GU

W gv ˆ)ˆ

ˆ

( 2 / 1

2 / 3

= )ˆ

ˆ

( 2 / 1U

W ge = (7-20)

n

e

m

vh g Zgg = (7-21)

or

nm

URE B

W

RU

W Z

URB

Wh

=

2 / 12 / 1

0

2 / 3

0

min

)'()(η

α

η(7-22)

Table 7-1 Minimum-film-thickness formula for line-contact EHL

Z m n

Rigid-isoviscous 4.9 0 0

Rigid-piezoviscous 1.66 2/3 0

Elastic-isoviscous 3.0 0 0.8

Elastic-piezoviscous 2.65 0.54 0.06

In application, we should first determine the region of lubrication by means of g parameters from

Figure 7-5, and then select a proper formula for the analysis. The shapes of the pressuredistributions and film thickness corresponding to different regions are plotted in Figure 7-6.

(3) Central-film-thickness formula for line contacts. Similarly, a central-film-thicknessformulas for line contact is obtained by Dowson and Toyota (1979)

10.056.069.0 ˆˆˆ06.3ˆ −= W GU H c (7-23)



9

Figure 7-5 Regions of EHL formulas applications (Winer and Cheng 1980).Here, 1 = v and 3 = e.



10

Pressure distributions

Film thickness

Figure 7-6 Pressure and film shapes.



11

Example 7-1 A cylindrical roller in a roller bearing is 32 mm in diameter and 35 mm in width.

It interacts with the outer race of the bearing that has a concave radius of 103 mm. The roller andthe outer race are under a pure-rolling motion, and the outer race rotates at 100 rpm. The load is

3 KN, and the contact is lubricated with an oil of viscosity 0.003Pa-S and pressure-viscositycoefficient 2.3x10

-7m

2 /N. Calculate the minimum film thickness in the EHL contact.

Solution

R = R1 R2 /( R1 + R2)=16(-103)/(16-103) = 18.9mm

2

2

2

2

2

2

1

2

1 12

11

'

2

E E E E

ν ν ν −=

−+

−=

E' = E/(1-v2) = 208(10

9)/(1-0.3

2) = 228GPa

U1 = πDn/60 = 3.1416 (2x103.5)(100(10-3

)/60= 1.08m/s

The ball and the race are under a pure-rolling motion, U1 = U2

U = (U1 + U2)/2 = (1.08+1.08)/2 = 1.08 m/s

Non-dimensional parameters:

RB E

W W 'ˆ = =3000/(228x109x0.0189x0.035) = 1.99 x10-5

R E

U U

'ˆ 0η= = 0.003(1.08)/ /(228x10

9x0.0189) = 7.52 x10

-13

'ˆ E G α= = 2.3 x10-7

(228x109) = 524 x10

2

Then, g parameters:

GU W gv ˆ)ˆˆ(

2 / 1

2 / 3

= = (1.99 x10-5)3/2(524 x102)/(7.52 x10-13)1/2 = 3264

)ˆ

ˆ(

2 / 1U

W ge = = (1.99 x10

-5)/(7.52 x10

-13)

1/2= 22.9

This case is in the R-V region, Z = 1.66, m = 2/3, n = 0



12

n

e

m

vh g Zgg = = 1.66 (5366)2/3

=339.2

From

H

U

W gh

ˆ)

ˆ

ˆ(=

We have

hgW

U H )

ˆ

ˆ(ˆ = = (7.52 x10

-13)(339.2)/(1.99 x10

-5) = 1282 x10

-8

From

R

h H minˆ =

We have

R H h )

=min =(1282 x10-8

)x 0.0189 = 0.24µm.



13

7-5 Elliptical-contact problems (Dowson-Hamrock)

If the 1D Reynolds equation (7-9) is replaced by the 2D Reynolds equation and the Flamant

solution in Equation (7-12) by the Boussinesq formula (Equation 7-13)), the formulation forpoint-contact EHL, or elliptical-contact EHL, which is more general, is constructed. The

geometry of a typical elliptical-contact EHL is shown in Figure 7-7.

Figure 7-7 Geometry for elliptical-contact EHL.

The Non-dimensional parameters for EHL of elliptical contacts are as follows.

Dimensionless film thickness: x R

h H =ˆ , where

2

111

x x r R R+=

2'

ˆ x R E

W W = , whereb

b

a

a

E E E

22

11'

2 ν ν −+−=

(7-24)

Dimensionless speed: x R E

U U

'ˆ 0η=

Dimensionless material parameter: 'ˆ E G α=

The minimum film thickness formula (EV) is given in Equation (7-25).

)1(ˆˆˆ36.3ˆ 68.0073.049.068.0

min

k eW GU H −− −= (7-25)

where

64.0

03.1 /

≈=

x

y

R

Rbak is the elliptic ratio defined in Figure 7-7.

Ry2Rx2

Rx1Ry1

ab



14

21

111

x x x r R R+=

21

111

y y y r R R+=

The corresponding central film thickness equation has the following form.

)61.01(ˆˆˆ69.2ˆ 73.0067.053.067.0 k

c eW GU H −− −= (7-26)

The formulas for elliptical-contact EHL can also be generalized by means of the non-

dimensional parameters (Equation (7-24)) and the following g parameters. However, the regionsof application have to be determined from charts similar to that shown in Figure 7-5 but plotted

at different elliptical ratios.

H U

W

ghˆ

)ˆ

ˆ

(

2

= , GU

W

gv

ˆ)ˆ

ˆ

( 2

3

= , )ˆ

ˆ

( 2

3 / 8

U

W

ge = (7-27)

(1) Rigid-isoviscous

212 ]68.12

tan13.0[128 +

Φ= − β

βhg (7-28)

where64.0 / 1

03.1

≈= k

R

R

x

yβ

1

3

21

−

+=Φ

β

(2) Rigid-piezoviscous

)1(66.168.03 / 2 k

vh egg−−= (7-29)

(3) Elastic-isoviscous

)85.01(70.8 31.067.0 k

eh egg−−= (7-30)

(4) Elastic-piezoviscous

)1(45.368.017.049.0 k

evh eggg −−= (7-31)



15

7-6 Milestones of EHL



16

7-7 Introduction to mixed lubrication

A state of ideal hydrodynamic lubrication would be such a situation where a fluid completely

separates the surfaces of a pair of mechanical elements. That seems to be great! No solid-solidinteraction should happen. No wear, no damage should occur if surfaces could not contact.

Therefore, the parts should enjoy, theoretically, an infinitely long life. But we know that it isimpossible to have such great life and that some kinds of surface interactions are inevitable.

Then, we have to think about engineering surfaces. They are not smooth. They are rough, moreor less, and have all scales of irregularities. We have to consider the real surfaces, not the

idealized smooth surfaces, in lubrication problems.

Questions come. When should we begin to think that surfaces are rough? This one is easy toanswer. When the film thickness falls into the same order of magnitude as that of the surface

roughness, saying, h ∼ 3 σ, or the film thickness is about three times the RMS roughness of thesurface, we need consider the rough-surface effect. If the film thickness is larger, a plenty

amount fluid should flow through the interface. The main stream of the fluid does not “feel” the

surface undulation in a thick-film lubrication.

How does the surface roughness affect the fluid flow? This one is tough to answer. Fortunately,

Patir and Cheng (P & C, 1978) have a famous average Reynolds equation that describes theinfluence of roughness through a set of flow factors. How do the asperities contact and how do

the fluid and asperities interact? This question pair is tougher to answer. We need to analyze themechanics of asperity contact and the balance of fluid flow and asperity contact.

1. P & C's Average Reynolds equation

In deriving the Average Reynolds equation, we assume that the roughness is a Reynoldsroughness, i.e., the asperities have small slopes, which is about a few degrees. Let’s first define

a few terms based Figure 7-8. Note we exaggerated the irregularities. If there is no contact anddeformation the compliance and the average gap are the same. The compliance and the average

gap are not the same once deformation is involved. The concept of the average gap is defined onthe separation of the centerlines of deformed surfaces.

Centerline

Centerline

h = T h

U2

U1

hT

δ2

δ1

hhT

T h

U1, U2: Surface velocities

δ1(x,y),δ2(x,y):Surface deviations from thecenterlines

hT(x,y): Gap between two surfaces

T h (x,y): Average gap

h(x,y): Compliance (film thickness, by

P&C), defined as the separationbetween two original

centerlines.

x

“Deformed” centerline

Figure 7-8 Gap and compliance.



17

(1) The average Reynolds equation

The Reynolds equation is valid to describe the flow between two surfaces with a Reynoldsroughness, and the corresponding film thickness is the gap at a position of interest.

t h

xhU U

y ph

y x ph

x

T T T T

∂∂+

∂∂+=

∂∂

∂∂+

∂∂

∂∂ 12)(6 21

33

ηη (7-32)

The corresponding flow rates are:

T

T

x hU U

x

phq

212

21

3 ++

∂∂

−=η (7-33)

y

phq T

y ∂∂

−=η12

3

(7-34)

If they are expressed over the average gap, we should have the terms of average flows:

sqT

T

x x RU U

hU U

x

phq ϕ

ηϕ

2212

2121

3 −+

++

∂∂

−=(7-35)

y

phq T

y y ∂∂

−=η

ϕ12

3

(7-36)

Here, flow factors are introduced and an additional term is added to the flow in the x direction.

However, the original P & C model defines the flows slightly differently because the term“compliance” is usually referred to as the film thickness.

sqT x x RU U

hU U

x

phq φ

ηφ

2212

21213 −

++

+∂∂

−=(7-37)

y

phq y y ∂

∂−=

ηφ

12

3

(7-38)

Let’s follow P & C’s flow equations. Here, φx and φy are pressure flow factors, and φs is a shear-

flow factor. The additional term, (U1- U2)Rq φs /2, shows the flow due to surface roughness. Forsmooth surfaces, Rq is 0 and so is the additional flow. U1or U2 has different influence to the flow.

When U1= U2, no additional flow enters the region of interest.

Let’s defined a differential control volume, ∆x ∆yT h , where

T h is the average gap of this

volume, and draw the flow equilibrium (Note, no flow along z):



18

The flow balance is the summation of the flows in and out of the control volume:

0)()( =∆∆∂

∂+∆−∆

∂

∂+∆+∆−∆

∂∂

+∆ T

T

yT

y

yT xT

x

xT h y x

t

hq xh y

y

qq xhq yh x

x

qq yh

t

h

y

q

x

qT y x

∂

∂−=

∂

∂+

∂

∂

(7-39)

Bringing in Equations (7-37) and (7-38) we have the average Reynolds equation:

( )t

h

x RU U

x

hU U

y

ph

y x

ph

x

T s

q

T

y x ∂∂

+∂

∂−+

∂∂

+=

∂∂

∂∂

+

∂∂

∂∂

126)(6 2121

33 φ

ηφ

ηφ

(7-40)

If U2 = 0 and U1 =U, Equation (7-40) becomes:

t h

x RU

xhU

y ph

y x ph

x

T s

q

T

y x ∂∂+

∂∂+

∂∂=

∂∂

∂∂+

∂∂

∂∂ 1266

33

φη

φη

φ(7-40’)

(2) Pressure flow factors

Equation (7-40) has defined the characteristics of the pressure flow factors. They may be very

large or very small (very influential) when h/ σ becomes very small, but tend to be unity when thefilm thickness is much larger than the height of surface asperities, or:

φx, φy → 1 as h/ Rq → ∞ (h → ∞ or Rq → 0)

Because x and y are orthogonal directions, we only need to define one of these two flow factors.

The other can be determined with the same method. Let's define φx first and only consider the

pressure-driven flow.

φx = Pressure-driven flow between rough surfaces/Pressure-driven flow between smooth surfaces

=)(

)(

pq

pq

x

x (7-41)

x x

qq x x ∆

∂∂+

xq

yq x∆

∆

T hFigure 7-9 A control volume

and flows.



19

The pressure-driven flow between rough surfaces means that the flow has a pressure, p, at the

position of hT , while the pressure-driven flow between smooth surfaces is for the flow of the filmthickness defined by the compliance corresponding to hT and an average pressure, p . Recall

T T

x hU U x phq

212

21

3

++∂∂−=

η (7-42)

and

sT x x

U U h

U U

x

phq σφ

ηφ

2212

2121

3 −+

++

∂∂

−=(7-43)

Utilize the flow averaged across a calculation length, Ly, and for U1 =U2:

dy x

ph

L

dyhU U

x

ph

Lh

U U qh

U U

x

ph

y

T

y

T

L

y

T

L

y

T xT x

)12

(1

)212

(1

2212

0

3

21

0

3

2121

3

∫

∫

∂∂−=

++

∂∂

−+

=−+

=∂∂

η

ηηφ

(7-44)

Therefore, the pressure flow factor is defined as:

x

ph

dy x

ph

L

y

T

L

y x

∂∂

∂∂

=∫

η

ηφ

12

)12

(1

30

3

(7-45)

Obviously, the pressure flow factor corresponding to the average flows defined by the average

gap (Equations (7-35 and 7-36) is

x

ph

dy x

ph

L

yT

L

y x

∂∂

∂∂

=∫

η

ηϕ

12

)12

(1

30

3

(7-45')

7-8. Flow factors for the surface with a Gaussian asperity-height distribution

The asperity height distributions of lot of engineering surfaces may be simply expressed by aGaussian distribution.

2

2

2

)(

2

1)( q R

z

q

e R

z

µ

πφ

−−

=(7-46)



20

We may further simplify the problem by assuming that one of the mating surfaces is ideallysmooth while the other possesses the combined roughness of both, and use the combined

roughness value (Equation (4-16) in Equation (4-16).

22

21 qqq R R R += (7-47)

W need to recall the orientation parameter, or the aspect ratio, λ, defined as the ratio of the

correlation lengths in two orthogonal directions:

*

*

y

x

γ

γ λ =

(7-48)

And remember the following classifications:

λ = ∝ (9, P & C 1978, 6, Lee and Ren 1996) Longitudinal surface

λ = 1 Isotropic surface

λ = 0 (1/9, P & C 1978, 1/6, Lee and Ren 1996) Transverse surface

Patir and Cheng (1978) published a set of empirical flow-factor equations for the surface with a

Gaussian asperity-height distribution.

1.

Pressure flow factors

The pressure flow factor, φx, is found to be a function of a non-dimensional film thickness, h/ σ,

and the roughness orientation, λ.

),( λφφq

x x R

h=

(7-49)

) / (1 q Rhr

xCe

−−=φfor λ ≤1 (Isotropic and transverse) (7-50)

r

q

x R

hC −+= )(1φ

for λ >1 (Longitudinal) (7-51)

Coefficients, c and r, are given in Table 7-2.



21

Table 7-2 Coefficients c and r

λ c r Range

1/9 1.48 0.42 h/Rq > 1

1/6 1.38 0.42 h/Rq > 1

1/3 1.18 0.42 h/Rq > 0.75

1 0.9 0.56 h/Rq > 0.53 0.225 1.5 h/Rq > 0.5

6 0.52 1.5 h/Rq > 0.5

9 0.87 1.5 h/Rq > 0.5

Flow factors in two orthogonal directions are related through the following equations.

)1

,(),(λ

φλφφq

x

q

y y R

h

R

h==

(7-52)

2. Shear flow factors

The shear flow factor, φs, is found to be a function of the non-dimensional film thickness, h/ Rq,

and the roughness orientation, λ, as well as a shear flow factor for a single surface, Φ.

Obviously, φs is for the combined effect of the two mating rough surfaces.

),(),( 2

2

21

2

1 λσ

λφq

s

qq

s

q

q

s R

h

R R

h

R

RΦ

−Φ

=

(7-53)

Physically, if one of the surfaces is smooth, saying, Rq 1 = 0, we have

s

q

s

q

q

s R

h

R

RΦ−=Φ

−= ),( 2

2

2 λφ(7-54)

Recall the term for the additional flow,

sq R

U U q φ

2

21 −=

(7-55)

If U2 > U1, the rough surface is moving faster and there is an increase in the flow. On the other

hand if U2 < U1, the smooth surface is moving faster and there is a decrease in the flow.

Or for Rq2 = 0, we have

s

q

s

q

q

s R

h

R

RΦ=Φ

= ),( 2

2

2 λφ(7-56)



22

Here, if U1 > U2, the rough surface is moving faster and there is an increase in the flow. On the

other hand, if U1 < U2, the smooth surface is moving faster and there is a decrease in the flow. Ingeneral, there is an increase in the flow if the rough surface moves. Figure 7-10 illustrates this

observation.

For Rq1 = Rq2 = 0, we have φs = 0. No additional flow appears.

U U U

(a) (b) (c)

We need the following empirical relations to calculate the shear flow factor, φs:

2

32

1)(1

+

−

=Φ qq R

h

R

h

q

se

R

h A

ααα

for h/Rq

≤ 5 (7-57)

−

=Φ q R

h

s e A25.0

2

for h/ Rq > 5 (7-58)

The coefficients, A1, A2, α1, α2, α3, are in Table 7-3.

Table 7-3 Coefficients, A1, A2, α1, α2,

λ A1 α1 α2 α3 A2

1/9 2.046 1.12 0.78 0.03 1.856

1/6 1.962 1.08 0.77 0.03 1.754

1/3 1.858 1.01 0.76 0.03 1.561

1 1.899 0.98 0.92 0.05 1.126

3 1.56 0.85 1.13 0.08 0.5566 1.29 0.62 1.09 0.08 0.388

9 1.011 0.54 1.07 0.08 0.295

Figure 7-10 Variations of the flow (a) two smooth surfaces, (b) thesmooth surface is moving, and (c) the rough surface is moving.



23

3. Discussions on the roughness effects on lubrication

Roughness has a profound effect on fluid flows in both of the x and y directions when the

magnitudes of the film thickness and the roughness are compatible. The smaller the filmthickness, the stronger the influence of the roughness. Let’s take a journal bearing as an

example. For wider bearings, saying L/D is about or larger than unity, the flow is predominatelyalong the circumferential direction and side leakage may be negligibly small. Side leakage may

only affect the pressures at bearing edges. We can deduce, from the average Reynolds equation(see P & C’s original paper for flow factor charts), that a transverse roughness tends to increase

the load capacity because the flow factor in the circumferential direction, x, is small (φx, fortransverse). A restriction in this direction may lift the pressure. On the other hand, for narrower

bearings, saying L/D is much smaller than one, the flow is predominately along the widthdirection and the side leakage may be of the most importance. We can deduce, from the average

Reynolds equation (see P & C’s original paper for flow factor charts), that a longitudinalroughness tends to increase the load capacity because the flow factor in the width direction, y, is

small (φy, transverse). This means that a restriction in this y direction can confine the leakage

flows and lift the pressure.

4. Discussions on the average Reynolds equation

The average Reynolds equation derived by Patir and Cheng (1978) relates the roughness effectand fluid flow through a few flow factors and keeps the original form of the Reynolds equation.

The flow factors are functions of the film thickness and two statistical parameters, the RMSroughness and the orientation parameter (aspect ratio). Empirical formulas for flow factors for

surfaces with Gaussian asperity height distribution are given. This average Reynolds equation iseasy to be programmed and convenient to use.

However, we should realize that two statistical parameters might be insufficient to define anengineering surface. Also, because the flows are averaged across the film thickness, the averageReynolds equation does not allow cross-film property variations. Some strong assumptions

(Newtonian fluid, no surface deformation, and no flow cavitation) were made in the derivation.Most importantly, mixed film terms, the average gap and compliance, are used in the equation

(Equation 7-40)) and therefore, an additional relation is needed to convert the average gap to thecompliance, or vice versa.

( )t

h

x RU U

x

hU U

y

ph

y x

ph

x

T s

q

T

y x ∂∂

+∂

∂−+

∂∂

+=

∂∂

∂∂

+

∂∂

∂∂

126)(6 2121

33 φ

ηφ

ηφ

(7-59)

A reasonable solution to this problem may be defining the same term in the equation shown

above. Instead of using the compliance, we can employ the average gap in the pressure driventerms. We have already done the necessary derivation work, seeing Equations (7-35), (7-36),

and (7-45’):

sT

T

x x

U U h

U U

x

phq σϕ

ηϕ

2212

2121

3 −+

++

∂∂

−=(7-60)



24

y

phq T

y y ∂∂

−=η

ϕ12

3

(7-61)

x

ph

dy x

ph

L

T

L

y x

y

T

∂∂

∂∂= ∫

η

ηϕ

12

)12

(1

30

3

(7-62)

Those are all what we need for the gap-version of the average Reynolds equation. Wilson (1998)and She and Wang (1998) have done so for some cases.

7-9. Practice for mixed lubrication calculation

Our MOBI code has a simple component of mixed lubrication. This time we need to input thesurface properties in 4 in Table 7-3.

Table 7-3 The input screen

Are your input data in Metric System? (Enter Y for Metric System or N for English System)

Which group of data do you want to input or change?

1. Geometric parameters

2. Material properties of journal and bearing

3. Lubricant4. Surface roughness5. Operating conditions

6. None (EXIT from the input mode)

Note: If this is not your first run, probably you don't need to input all of the data. You may onlymake change based on your last run.

Each of the items in the input main menu leads to an input sheet, where detailed data are required

to define a case.

The code will give us the information about the composite roughness, asperity friction, and thefilm thickness/roughness ratio, which is call "lamda," as listed in the output data sheet in Table

7-4.



25

Table 7-4 The output file

****************************************************************************

INPUT DATA

1. GEOMETRIC PARAMETERS

Bearing diameter: 24.00000 mm

Bearing length: 12.00000 mm

Bearing radial clearance: 0.0150000 mm

Equivalent housing diameter: 89.91423 mm

Shaft length from the bearing center

to the free end of the shaft: 18.00012 mm

2. MATERIAL PROPERTIES OF BEARING AND JOURNAL

Bearing Journal Unit

Young's modulus 206.80000 206.84100 GPa

Poisson's ratio 0.30000 0.30000

Thermal expansion coef. 11.3940 11.3940 mm/mm*deg.C

3. LUBRICANT PROPERTIES

Name: SAE-5W30 with additives

Viscosity: 68.0000 cSt at 40.000 deg.C

10.2000 cSt at 100.000 deg.C

Density at room temperature: 0.8700000 g/cm**3

4. SURFACE ROUGHNESS

RMS roughness of bearing surface: 1.00000 micron

RMS roughness of journal surface: 1.00000 micron

5. OPERATING CONDITIONS

Bearing stationary, journal rotating

Static load, the value is: 2000.000 N

Rotational speed: 500.000 rpm

Working temperature: 40.000 deg.C

Initial excentricity: Ex = 0.00000 micron

Ey = 0.00000 micron



26

RESULTS OF CALCULATION

----------------------

Composite roughness: 1.41421 micron

Actual radial clearance: 0.0150000 mm

Actual dynamic viscosity: 0.0591600 Pa*s

Viscosity equation for SAE-5W30 with additives

lg ( lg ( v + 0.6)) = 8.44874 + ( -3.27949 * lg T )

where v is viscosity in cSt, T is temperature in deg.K.

No. of iteration: 1 Ex= -4.44692 Ey= -11.82549 Temp.= 1.6640

No. of iteration: 2 Ex= -4.44706 Ey= -11.82538 Temp.= 1.6638

No. of iteration: 3 Converged solution shown as follows:

THET LAMDA TH TA PMAX EX EY

degree N N GPa micron micron

360.00 1.6731 4.15949 94.91836 0.03022622 -4.44706 -11.82538

Temperature increase = 1.6638 deg.C

Here, the meaningful results are TH, the hydrodynamic viscous friction, PMAX, or Pmax, EX andEY, or ex and ey.

e = SQRT(ex2

+ ey2) = C – hmin

Lamda = – hmin /Rq.



27

References

Arnell, R. D, Davies, P. B., Halling, J. and Whomes, T. L., Tribology, Principles and Designs,

Springer-Verlag, 1991.

Cheng, H.S., 1965, "A Refined Solution to the Thermal-Elastohydrodynamic Lubrication of Rolling and Sliding Cylinders," ASLE Transactions, 8: pp. 397-410.

Dowson, D., and Higginson, G. R., 1961, “New Roller-Bearing Lubrication Formula,”

Engineering (London), 192, pp.158-159.

Dowson, D., and Higginson, G. R., 1966, “Elastohydrodynamic Lubrication,” Pergamon Press.Elrod, H. G., 1981, “A Cavitation Algorithm,” ASME Journal of Lubrication Technology, Vol.

103, pp.350-354.

Hamrock, B, "Fluid Film Lubrication,"

Hamrock, B. J., and Dowson, D., 1976, “Isothermal Elastohydrodynamic Lubrication of Point

Contacts, Part 1 Theoretical Formulation,” ASME Journal of Lubrication Technology, 98,

pp.223-229 (1976).


Contacts, Part 2 Ellipticity Parameter Results,” ASME Journal of Lubrication Technology, 98,pp.375-383 (1976).


Contacts, Part 3 Fully Flooded Results,” ASME Journal of Lubrication Technology, 99,

pp.264-276.


Contacts, Part 4 Starvation Results,” ASME Journal of Lubrication Technology, 99, pp.15-23.

Johnson, K. L., 1996, Contact Mechanics, Cambridge University Press.

Lee, Si C., and Cheng, H.S., 1992, “On the Relation of Load to Average Gap in the Contact

Between Surfaces with Longitudinal Roughness,” STLE Tribology Trans., 35, pp.523-529.

Lee, S. C. and Ren, N., 1996, "Behavior of Elastic-Plastic Rough Surface Contacts as Affected

by the Surface Topography, Load and Materials,” STLE Tribology Transactions, Vol. 39, pp.67-74.

Pinkus, O. and Sternlicht, B., Theory of Hydrodynamic Lubrication, McGraw-Hill, Inc., 1961.

Patir, N. and Cheng, H. S., 1978, “An Average Flow Model for Determine Effects of ThreeDimensional Roughness on Partial Hydrodynamic Lubrication ,” ASME Journal of Lubrication

Technology, Vol. 100, pp.12-17.



Shen, M., "A Computer Analysis of Lubrication of Dynamically Loaded Journal BearingsIncluding Effects of Asperity Contact," MS Thesis, Northwestern University, 1986.

Shi, F. and Wang, Q., 1998, "A Mixed-TEHD Model for Journal Bearing Conformal Contacts,

Part I: Model Formulation and Approximation of Heat Transfer Considering Asperity Contacts,"ASME Journal of Tribology, Vol. 120, pp. 198-205.

Winer, W. and Cheng, H. S., Wear Control Handbook , ASME, Edited by Peterson and Winner,

1980.

Zhu, D. and Wen, S., 1984, "A Full Numerical Solution for the ThermoelastohydrodynamicProblem in Elliptical Contacts," ASME Journal of Tribology, 106: 246-254.

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