Tribology data handbook

VIII

Component Performanceand Design Data

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58 Fundamentals ofElastohydrodynamic Lubrication

Michael M. Khonsari and D. Y. HuaCONTENTS

Nomenclature.................................................................................................................................611Geometry of Contact....................................................................................................................613Dry Contact....................................................................................................................................614Elastohydrodynamic Line Contact..............................................................................................616Elastohydrodynamic Elliptical Contact.......................................................................................621Starvation........................................................................................................................................625Thermal Correction.......................................................................................................................625Partial-Film EHL............................................................................................................................627Traction............................................................................................................................................627Examples.........................................................................................................................................630References.......................................................................................................................................636

NOMENCLATURE

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612 Tribology Data Handbook



Fundamentals of Elastohydrodynamic Lubrication 613

GEOMETRY OF CONTACT

A general Hertzian contact between two bodies is shown in Figure 1.1 Two principal planes areused to characterize the geometry at the point of contact. R

xl, R

yl, and R

x2, R

y2are principal radii

for body 1 and body 2, respectively. In general, the principal planes of body 1 and body 2 maynot coincide. However, for most engineering machine elements, the principal radii R

xland R

x2,

as well as Ry1, and R

y2lie in the same plane. In this chapter, the following equivalent radii and

equivalent modulus of elasticity are introduced.

FIGURE 1 Geometry of elliptical contact.1

The equivalent radius in x direction is

and the equivalent radius in y direction is




where “+” and “-” represent convex and concave of the surface 2, respectively. Then, the curvaturesum in x and y direction is defined as

The equivalent elastic modulus is

The above equations are valid for the general case of an elliptical contact as formed between two ellip-soids with aligned principal axes, two crowned cylinders, or two cylinders that cross at right angle. Theelliptical contact can be reduced to two special cases:

Circular contact — when Rx1

= Ryl

= R1and R

x2= R

y2= R

2, i.e., contact between two spheres.

In this case, R = 1/(1/R1+ 1/R

2).

Line contact — both Ryl

and Ry2

are infinity. Then, Ry→ ∞ and the curvature sum R = R

x.

(cf. Figure 2).

FIGURE 2 Line contact: (a) nonconformal; (b) conformal; (c) equivalent elastic cylinder and rigidsurface.

DRY CONTACTLINE CONTACT

Two cylinders pressed against one another under a normal load will produce a plane rectangular con-tact area. If the cylinders are unequal, the contact area is not truly rectangular. Nevertheless, the planecontact is a reasonable assumption. Under a normal load, w, the “contact patch” will have width of2b. In the absence of lubricant, the normal load is parabolically distributed over this area. The half-width of contact and the maximum Hertzian contact pressure are functions of the load per unitlength, the equivalent curvature radius, R, and the equivalent elastic modulus, E. The

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Hertzian predictions of mean pressure, the magnitude and location of the maximum shearstress, as well as the normal approach of the centers are listed in Table 1.

CIRCULAR CONTACT

The contact between two spheres forms a circular region whose diameter is 2a. The radius ofthe contact and the maximum pressure in terms of the load, radii of the spheres, and elastici-ty modulus are given in Table 1 along with mean pressure, maximum shear stress, maximumtensile stress, and the normal approach of the center.

ELLIPTICAL CONTACT

The geometry of an elliptical contact is shown in Figure 1. The elliptic parameter k is definedas the ratio of the ellipse semimajor axis a to that of semiminor axis b. In general, the ellipti-cal parameter requires solving the first and the second elliptical integrals. The approximationof the elliptical parameter and the integrals can be used to simplify the expression which isrelated to the radius ratio.2 The definition and the approximation equations are listed in Table2. These approximations are valid for the range of 1 ≤ R

y/R

x≤ 100, or 1 ≤ κ ≤ 18.




The radii of the elliptical contact, a and b, as well as the maximum Hertzian pressure, pH, are

functions of several parameters such as load, equivalent radius of the bodies, and the elasticity mod-ulus, as well as the elliptic parameter and the elliptic integral. The appropriate equations are listed inTable 1. The contact deformation at the center of the contact is also provided in Table 1.

ELASTOHYDRODYNAMIC LINE CONTACT

FILM SHAPE AND PRESSURE DISTRIBUTION

A typical film shape and pressure distribution of elastohydrodynamic lubrication (EHL) is shownin Figure 3. Generally, EHL pressure distribution closely resembles the dry Hertzian contact withthe major exception of a pressure build-up in the inlet region and a pressure spike in the exit region.Existence of the sharp pressure spike accompanied by a film constriction at the exit region areimportant characteristics of the elastohydrodynamic lubrication regime.

Several trends in EHL may be noted. First, increasing speed or decreasing load tend to increasethe magnitude of pressure spike and move its location towards the inlet region. Under very heavyloading, the pressure spike tends to decrease and eventually vanish, i.e., the pressure profileapproaches that of the dry Hertzian. In EHL applications, both the maximum Hertzian contactpressure and the pressure spike are important parameters. Although the pressure spike is very nar-row, its occurrence is very important since it may produce high subsurface stresses that directlyaffect the rolling element bearing fatigue life.

The minimum film thickness at the film constriction compared to surface roughness dictateswhether the lubrication film is thick enough to protect the surfaces. The central film thickness(essentially the parallel central region) is also a useful parameter in engineering design. The filmthickness is reduced by starvation of the lubricant and by inlet heating as discussed in sections on“Starvation” and “Thermal Correction.”

The appropriate EHL equations can be conveniently grouped in terms of the following dimen-sionless parameters:



Fundamentals of

Elastohydrodynam

ic Lubrication 617

FIGURE 3 Film shape and pressure distribution of line contact.




where the viscosity–pressure coefficient is defined as

In nonconformal contacts, it is important to include the variation of viscosity with pressure.There are two general relationship. The Barus viscosity–pressure relation is

The typical values of viscosity-pressure coefficient a for several lubricants are listed in Table3.3 The other relation due to Roelands4 is given below:

The typical value for z is 0.6, S0is 1.1 and a is 5.1 × 10-9

The EHL formulae reported in this chapter are based on Barus’ equation unless otherwise spec-ified.

REGIMES OF FLUID FILM LUBRICATION

Many expressions for evaluating EHL film thickness are available in the literature. These areobtained using curve fitting techniques to the numerical solutions of the governing equations thatinvolve the Reynolds equation coupled with surface deformation. These expressions, however, onlyapply to a particular range of operation conditions and cannot be extrapolated into different regimes.It is, therefore, necessary to define the regimes for appropriate usage of the film thickness expressions.Referring to Figure 4, the following regimes may be defined:5

• Rigid-isoviscous, load is not high enough to produce either an appreciable viscosity change or elas-tic deformation of contact surfaces

• Rigid-viscous, significant viscosity increase occurs due to high pressure but the elastic deformationof contact surfaces is negligible

• Elastic-isoviscous, elastic deformation of contact surfaces is quite large compared to the film thick-ness but the viscosity change due to pressure is negligible

• Elastic-viscous,6 viscosity changes due to pressure and elastic deformation of contact surfacesplay important roles. This is the regime of “full” EHL

FILM THICKNESS FORMULAE

The following dimensionless groups conveniently categorize the appropriate regime:




Film thickness formulae for the above-mentioned regimes are listed in Table 4.

PRESSURE SPIKE FORMULAE

Pressure spike amplitude and its locations are also determined by curve fitting the results of numeri-cal simulations. Data which were used in curve fitting covered a wide range of operating parameterswith dimensionless load W varying from 0.2045 × 10-4, dimensionless speed U varying from 0.1 × 10-11

tp 5.0 × 10-11, and values of dimensionless materials parameter G of 2504, 5007, and 7511. Onemust check to make certain that these restrictions are satisfied for a given application.

The pressure spike magnitude and its location are determined from the following expressions,7




FIGURE 4 Lubrication regimes of line contact.4 (From Roelands, D.J.A., CorrelationalAspects of the Velocity-Temperature-Pressure Relationship of Lubrication Oils, Druk, V.R.B.,Groningen, Netherlands, 1966.)

Pressure spike location is

The center of pressure (the location of the center of pressure indicates the position atwhich the resulting force acts) is given by:

Another form of minimum film thickness expression is also available,7

In dimensional form where w is the load-per-width, minimum film thickness is




The central film thickness is

ELASTOHYDRODYNAMIC ELLIPTICAL CONTACT

The characteristic film shape and pressure distribution of an elliptical EHL is similar to that ofthe line contact. Some typical pressure and film thickness profiles predicted by the EHL theo-ry are shown in Figure 5.8 The maximum Hertzian contact pressure, pressure spike, and mini-mum film thickness, as well as central film thickness are of interest.

FIGURE 5 Typical contour plot of film thickness (left) and pressure profile (right) for a cir-cular contact.8

In order to show the different regimes of lubrication problems, the dimensionless parametersdefined in Equations 5 to 9 are used. The four regimes of rigid-isoviscous, rigid-viscous, elastic-iso-viscous and elastic-viscous are illustrated in Figure 6.9

FILM THICKNESS FORMULAE

To determine the appropriate regime, the following dimensionless parameter groups are defined as:

Film thickness formulae in these different regimes are summarized in Table 4 and Table 5.The minimum film thickness for more general consideration of the velocity vector is:14




FIGURE 6 Lubrication regimes of elliptical contact.8 (a) k = 111; (b) k = 1; (c) k = 3; (d) k = 6.

where




FIGURE 6 (Continued)

and




where, u and v are mean velocities in x and y direction, respectively; θ = tan-1 (u/ν). If purerolling or pure sliding exists, θ = 0 and ν = 0.

STARVATION

Reduction of film thickness due to starvation for a line contact is shown in Figure 7. Forstarved circular contacts, the film thickness formula is:15,16

where subscript s refers to starved boundary condition; subscript F denotes flooded contact mis the dimensionless distance of the inlet meniscus from the center of the contact; m* is thedimensionless inlet distance required for achieving flooded conditions:

D, n, and c for different regimes are listed in Table 6.

FIGURE 7 Influence of starvation on film thickness predicted by numerical simulation.Parameters h

starvedand h

floodedrefer to the starved film thicknesses, respectively. The distance from

the inlet meniscus to the edge of Hertzian boundary is denoted by Xj.19

THERMAL CORRECTION

For a line contact, film thickness reduction due to viscous heating of the lubricant at the con-junction inlet can be estimated by a thermal correction factor as




where the thermal correction factor Ct is17

where

uris rolling velocity, m/s; S is slide-roll ratio; K

fis the thermal conductivity of the lubricant,

W/(m ⋅ K).Reduction of film thickness due to inlet shear heating can be estimated from Figure 8,18

which is based on the following empirical viscosity–temperature relation.

FIGURE 8 Thermal correction factor. Parameter µo denotes the viscosity under the ambient con-dition and K

fis the lubricant thermal conductivity. With a known temperature-viscosity coefficient, β,

the dimensionless thermal parameter, Lm, and the thermal reduction factor, φ

f, are easily evaluated.18,19

Parameter L* is simply




PARTIAL-FILM EHL

Figure 919 illustrates full-film and partial-film elastohydrodynamic lubrication. Partial-film EHLis the regime where average film thickness becomes less than three times the composite sur-face roughness, h < 3σ. For determining partial-film EHL performance, surface roughnessparameters required for each surface include: (1) σ, root mean square of surface roughness; (2)surface roughness height distribution function; (3) λ

0.5x, λ

0.5y, 50% correlation lengths of sur-

face roughness in x and y directions; (4) autocorrelation function of roughness.

FIGURE 9 Full-film and partial-film lubrication.19

Typical contact area patterns for oriented rough surfaces are shown in Figure 10.20

Parameter γ is used to describe the surface pattern of the roughness.

where λ0.5x

and λ0.5y

are correlation lengths at which the autocorrelation function of the profileis 50% of the value at the origin. The autocorrelation function is a measure of the wave lengthstructure of a surface profile, defined as follows:

where λ is the correlation length; δ is the height function along the x direction; and Rx(λ) is

the autocorrelation function in the x direction.The surface roughness correction factor is defined as

Effect of surface roughness on the average film thickness of EHL contacts under purerolling condition is shown in Figure 1121 where Λ is film parameter, Λ = h

smooth/σ.

TRACTION

In EHL, as in all lubrication mechanisms, surface traction is present. In pure rolling, the rollingtraction is F

R. When sliding occurs, a sliding traction, F

S, will be present. The total traction force

on faster and slower surfaces will be

where “+” is for the faster surface and “-” is for the slower surface.




FIGURE 10 Contact pattern of oriented rough surfaces: left, transverse (γ < 1); center,isotropic (γ = 1); and right, longitudinal (γ = 1).19,20

FIGURE 11 Effect of surface roughness on film thickness.19,21 PH/E = 0.003; pure rolling; G

= 3333; σ/R = 1.8 × 10-5.

Typical traction curves measured experimentally at various mean contact pressures are shownin Figure 12. Rolling traction is much smaller than sliding traction, except for pure rolling. Inthe low-slip region, traction increases almost linearly as slip increases. If the lubricant is assumedto behave as a Newtonian fluid, this linear trend persists over large slips. However, experimentalmeasurements show that the traction curve rises linearly from pure rolling (zero traction) andreaches a plateau at a certain slip ratio in the so-called nonlinear isothermal region shown inFigure 12. In this region, the linearly viscous (Newtonian) constitutive equation for the lubricantis no longer valid. In the so-called thermal region, traction tends to drop with increasing slip. This trend can only be predicted if the model properly incorporates non-Newtonian effects withthermal consideration. One example of the traction coefficient predicted, using Bair-Winer’sconstitutive equation22 with its comparison to experimental data, is shown in Figure 13.23. Theinterested reader may refer to References 23 and 24 for the details of the formulation of thegoverning equations for generalized non-Newtonian formulation including thermal effects and




numerical solution technique. The effects of load, speed, and inlet temperature on tractioncoefficient curves are illustrated in Figure 14. These trends are important in predicting thetrend of traction under various operating conditions. For example, increasing the mean con-tact pressure tends to increase the traction coefficient, whereas increasing speed results in areduction of friction.

FIGURE 12 Experimental traction curve under various mean contact pressures, illustratingthe linear, nonlinear isothermal, and thermal traction regimes.19

FIGURE 13 Comparison of thermoelastohydrodynamic traction coefficient using the Bair-Winer’s constitutive equation and experimental results (W = 5.5185 × 10-5, U

1= 2.8 m/s, G =

5152, τo

= 1.4 × 107 N/m2, β = 0.05).23 The experimental results are taken from a researchreport published by Zhang et al. at the Twente University of Technology 1983. (FromKhonsari, M.M. and Hua, D.Y., J. Tricol., Trans. ASME, 116(1),37–46, 1994. With permission.)




FIGURE 14 Effects of load, speed, and inlet temperature on the traction curve.

EXAMPLES

LINE CONTACT

Consider a cylindrical roller of 40 mm diameter and 30 mm length contacting a cylinder of 120mm diameter which rotates at 1000 rpm. The load on the roller is 3000 N. The viscosity of thelubricant at ambient pressure and room temperature is 0.04 N ⋅ s/m2. The pressure viscositycoefficient is 2.1 × 10-8m2/N. The two surfaces are steel with an elastic modulus of 2.08 × 1011

N/m2 and Poisson ratio of 0.3.

Geometry of contactFrom Equation 1, the equivalent radius is

The equivalent elastic modulus is defined by Equation 4. As the material is the same for thetwo surfaces,

For pure rolling, the rolling velocity is




Dry contactFrom Table 1, the half-width of Hertzian contact is

Maximum Hertzian contact pressure is

Mean contact pressure is

Maximum shear stress is

The location of τmax

is at x = 0 and z = 1.02 × 10-4m (refer to Table 1).

Regime of lubricationRefer to Table 4 and Equations 5 through 9. Calculating the dimensionless parameters yieldsthe following results:

Dimensionless velocity

Dimensionless material parameter

Dimensionless load

To determine the regime of lubrication, from Table 4 the dimensionless viscosity parameter is




The dimensionless elasticity parameter is

From Figure 4, this is within the regime of elastic-viscous and the dimensionless film thick-ness parameter is

Film thicknessFrom Table 4, the minimum film thickness is

In dimensional form, we get the film thickness as

If the alternative equation (20) is used, the minimum film thickness is

and from Equation 22, the central film thickness is

Starvation effectAssuming the distance from inlet oil meniscus to inlet edge of Hertzian boundary, x

iis 2b,

From Figure 7, the reduction of film thickness is about 0.8.

Pressure spikeFrom Equation 17, the dimensionless pressure spike amplitude is

The dimensional pressure spike is

Dimensionless distance of the spike from the center of Hertzian contact by Equation 18 is




The dimensional distance from the center of the pressure to the center of Hertzian contact is

Consider the same rolling velocity and load, but with slip of 0.15 between two surfaces.Estimate the thermal reduction in the film thickness. Assuming β = 0.05 and K

f= 0.12 W/(m

⋅ K), from Equation 37

Then using Equation 34, thermal correction factor Ctis

ELLIPTICAL CONTACT

Consider a steel roller of 40 mm diameter with a 50 mm crown radius (surface 1) contact with80 mm diameter steel cylinder (surface 2). Rotation speed of the roller is 1500 rpm and thecylinder is 1000 rpm. The load is 50 N. Viscosity of the lubricant is 0.028 N ⋅ s/m2. The visco-pressure parameter is 1.45 × 10-8 m2/N. Equivalent elastic modulus for steel is 2.3 × 1011 N/m2.

Geometry of contactRadii of the two surfaces are:

Velocities of the two surfaces are:

Rolling velocity is

From Equations 1 and 2, the equivalent radii are




From Equation 3, the curvature sum in x and y direction is,

From Table 2, the elliptic parameter is

Dry contactFrom Table 2, the second kind of elliptic integral is

From Table 1, the elliptic contact radius is:

From the definition of the elliptic parameter in Table 2

The maximum Hertzian contact pressure is

the mean pressure is

Regime of lubricationAppropriate dimensionless parameters are:

Dimensionless velocity

Dimensionless material parameter

Dimensionless load




To determine the regime of lubrication (cf. Table 4), the dimensionless viscosity parameter is

The dimensionless elasticity parameter is

From Figure 6 (d), it is in the elastic-viscous regime.

Film thicknessFrom Table 4, the dimensionless minimum film thickness parameter is

From Table 4, the dimensionless minimum film thickness is

In dimensional form the minimum film thickness is

The dimensionless central film thickness parameter is

The dimensionless central film thickness is

In dimensional form, the central film thickness is

Starvation effectFrom Equations 31 and 32 and Table 6, m* for minimum film thickness is




Assuming dimensionless inlet distance m = 1.5, the reduction of minimum film thickness is

m* for the central film thickness is

Reduction of the central film thickness for m = 1.5 is

REFERENCES

1. Hamrock, B.J. and Dowson, D., Minimum Film Thickness in Elliptical Contacts for DifferentRegimes of Fluid Film Lubricants, NASA Tech. Pap., No. 1342, 1978.

2. Brewe, D.E. and Hamrock, B.J., Simplified solution of elliptical contact deformation between twoelastic solids, J. Lubr. Technol. Trans. ASME, 99(4), 485–487, 1977.

3. Jones, W.R., Johnson, R.L., Sanborn, D.M., and Winer, W.O., Viscosity-pressure measurements ofseveral lubricants to 5.5 × 108 N/m2 (8 × 104 psi) and 149°C (300°F), Trans. ASLE, 18(4),249–262, 1975.

4. Roelands, D.J.A., Correlational aspects of the viscosity-temperature-pressure relationship of lubri-cating oils, Druk, V.R.B., Groningen, Netherlands, 1966.

5. Hooke, C.J., The elastohydrodynamic lubrication of heavily loaded contacts, J. Mech. Eng. Sci., 19(4),149–156, 1977.

6. Dowson, D. and Higginson, G.R., Elastohydrodynamic Lubrication, Pergamon Press, Oxford, 1977.7. Pan, P. and Hamrock, B.J., Simple formulae for performance parameters used in elastohydrody-

namically lubricated line contacts, J. Tribol., Trans. ASME, 111(2), 246–251, 1989.8. Venner, C.H., Multilevel Solution of the EHL Line and Point Contact Problems, Ph.D. thesis,

University of Twente, Enschede, Netherlands, ISBN 90-9003974-0, 1991.9. Esfahamian, M. and Hamrock, B.J., Fluid-film lubrication regimes revisited, STLE Tribol. Trans.,

34(4), 618–632, 1991.10. Brewe, D.E., Hamrock, B.J., and Taylor, C.M., Effects of geometry on hydrodynamic film thick-

ness, J. Lubr. Technol., Trans. ASME, 101(2), 231–239, 1979.11. Jeng, Y.R., Hamrock, B.J., and Brewe, D.E., Piezoviscous effects in nonconformal contacts lubri-

cated hydrodynamically, ASLE Trans., 30(4), 452–464, 1987.12. Hamrock, B.J. and Dowson, D., Elastohydrodynamic lubrication of elliptical contacts for materi-

als of low elastic modulus, I. Fully flooded conjunctions, J. Lubr. Technol., Trans. ASME, 100(2),236–245, 1978.

13. Hamrock, B.J. and Dowson, D., Isothermal elastohydrodynamic lubrication of point contacts, III.Fully flooded results, J. Lubr. Technol., Trans. ASME, 99(2), 264–276, 1977.

14. Chittenden, R.J. et al., Theoretical analysis of isothermal EHL concentrated contacts: I and II,Proc. R. Soc., London, Ser. A, 387, 245–294, 1985.



15. Hamrock, B.J. and Dowson, D., Isothermal elastohydrodynamic lubrication of point contacts, IV.Starvation results, J. Lubr. Technol., Trans. ASME, 99(1), 15–23, 1977.

16. Hamrock, B.J. and Dowson, D., Elastohydrodynamic lubrication of elliptical contacts for materi-als of low elastic modulus, II. Starved conjunctions, J. Lubr. Technol., Trans. ASME, 101(1),92–98,1979.

17. Gupta, P.K. et al., Visco-elastic effects in Mil-L-7808 type lubricant, I. Analytical formulation,STLE Tribol. Trans., 34(4), 608–617, 1991.

18. Cheng, H.S., Calculation of elastohydrodynamic film thickness in high-speed rolling and slidingcontacts, Rep. No. MTI-67TR24, Mechanical Technology, Latham, NY, 1967.

19. Cheng, H.S., Elastohydrodynamic lubrication, CRC Handbook of Lubrication, Vol. 2, CRC Press,1984, 139–162.

20. Patir, N. and Cheng, H.S., Effect of surface roughness on the central film thickness in EHD con-tacts, Elastohydrodynamic and Related Topics, Proc. 5th Leeds-Lyon Symp. Tribology, Institution ofMechanical Engineers, London, 1978, 15–21.

21. Patir, N. and Cheng, H.S., An average flow model for determining effects of three dimensionalroughness on partial hydrodynamic lubrication, J. Lubr. Technol., Trans. ASME, 100(1), 12–17,1978.

22. Bair, S. and Winer, W.O., A rheological model for EHL contacts based on primary laboratory data,J. Lubr. Technol., Trans. ASME, 101, 258–265, 1979.

23. Khonsari, M.M. and Hua, D.Y., Thermal elastohydrodynamic analysis using a generalized non-Newtonian formulation with application to Bair-Winer constitutive equation, J. Tribol., Trans.ASME, 116(1), 37–46, 1994.

24. Khonsari, M.M. and Hua, D. Y. Generalized non-Newtonian elastohydrodynamic lubrication,Tribol. Int., 26, 45–411, 1994.




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