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Rahnejat, Homer, Ramin Rahmani, Mahdi Mohammadpour, and P.M. Johns-Rahnejat. 2019. “Tribology ofPower Train Systems”. figshare. https://hdl.handle.net/2134/33106.

Tribology of Power Train SystemsH. Rahnejat, R. Rahmani, M. Mohammadpour, and P.M. Johns-RahnejatLoughborough University, United Kingdom

Introduction

Tribology is the study of friction, lubrication,and wear. It is a multidisciplinary subject cov-ering the mechanics of contacting surfaces,their roughness characteristics, lubrication, andmaterial behavior under normal load as wellas in traction. A main parameter of interest inany tribological study is the calculation oflubricant film thickness; an insufficient lubri-cant film can lead to the direct contact of mat-ing surfaces, which would lead to increasedfriction, heat generation, and many mechanismsof failure, such as wear, surface scoring, scuff-ing, and so on. The other parameter of impor-tance is the generated contact pressures,causing elastic deformation of the surfaces.Issues of component reliability, service life

(longevity), and operational efficiency havebecome progressively more important underpervading global competition. There is a signif-icant volume of literature on a plethora of tribo-logical issues, most of which requires a goodknowledge of specialist skills in numericalanalysis and/or the use of specialist methodsof measurement. The diversity of the subjectand the multitude of issues preclude the possi-bility of including everything in this article.In dealing with some of the issues in the area

of engine and power train tribology, this articleis confined to well-established and widelyaccepted analytical methods and design andanalysis charts. This includes coverage onlubricant rheology and prediction of lubricatingfilm thickness. Better lubrication reduces fric-tion, resulting in fewer incidents of wear andthereby improving component reliability andlife. Combined with improved contact mechan-ics, it also reduces stresses and lowers thechance of contact fatigue.In engine and power trains, as in many other

systems, friction as a source of energy loss is ofparamount concern for energy efficiency andreduced emissions (see also the preceding arti-cle, “Friction, Lubrication, and Wear of InternalCombustion Engine Parts,” in this Volume).Frictional losses in an internal combustionengine account for 15 to 20% of all its losses.The main contributor to the frictional power

loss is the piston-cylinder system, accountingfor 40 to 50% of all these losses, from pistonskirt- cylinder liner and piston ring pack to cyl-inder liner contacts.Engine journal bearings, comprising big-end

(connecting rod) bearings and crankshaft sup-port bearings, account for 20 to 30% of the fric-tional power loss, with the valve train contacts,particularly the cam-follower pair, contributingto 5 to 10% of these losses. The remaining fric-tional losses are from other conjunctions, suchas cam gears, camshaft bearings, and small-end (wrist-pin) bearings (sometimes referredto as the gudgeon pin), as well as due to pump-ing losses.

Contact Configuration

A primary concern is to choose the correctmethod for a given problem. The most appro-priate analytical methods to use for tribologicalanalyses tend to vary according to the type ofcontact (contact configuration). The classifica-tion depends on the degree of conformity ofthe mating surfaces. When the two surface geo-metries conform closely to one another (snugclosely, as shown in Fig. 1a), the configurationis referred to as conformal or conforming. Arelatively large contact area results, and gener-ally pressures of the order of a few to tens ofmegapascals are generated in gaps of severalto tens of micrometers. Good examples arejournal bearings supporting the crankshaft and

camshaft, and the connecting rod (big-end)bearing between the connecting rod and thecrankshaft. Note that the journal closely con-forms to the surface of the bearing bushing orshell. In other words, the convex radius of thejournal, R1, is close to the concave radius ofthe bushing, R2 (of the order 95% or closer).Conversely, the contact of a perfectly sphericalball bearing residing on a perfectly flat surfaceis just a single point (Fig. 1c). The convexradius of the sphere, R1, does not conform atall to the infinite radius of an ideal flat surface.This form of contact is nonconformal, some-times referred to as counterformal, such as thecontact between a pair of touching cylindershaving their axes perpendicular to each other.With a normal applied load, the surface of thesphere at the point of contact flattens (its defor-mation follows Hertzian contact theory, as dis-cussed later) if the flat counterface surface isassumed to be ideally rigid. The footprint shapebecomes a small circle of a few tenths of amillimeter radius subjected to high pressures,usually in the range of several hundred mega-pascals to a few gigapascals. Because the pres-sures are generated over such a small contactarea, this class of contact is often referred toas concentrated counterformal circular pointcontact.In an actual ball bearing, the balls are

retained in raceway grooves (Fig. 1b), althoughthere is some degree of conformity (the radiusof the ball can be as much as 80 to 85% ofthe raceway groove). The case is still termed

ASM Handbook, Volume 18, Friction, Lubrication, and Wear TechnologyGeorge E. Totten, editor

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All rights reservedwww.asminternational.org

R1

(a) (b)

ϕ ϕ ϕ ϕ

(c) (d)

R2R2

R2

R1 R1R1

Fig. 1 Various contact configurations

nonconformal, with the resulting contact foot-print being elliptical in shape (i.e., concentratedcounterformal elliptical point contact). The ballcontact with the raceway groove can be viewedin the zy-plane of contact (Fig. 2), where theball rotational velocity vector is perpendicularto the plane of paper (in the x-direction, shownin Fig. 1d). Therefore, a counterformal contactalso occurs in the zx-plane, a two-dimensionalfootprint of elliptical shape.There is a larger class of contacts that is nei-

ther conformal nor counterformal, for example,a piston compression ring ideally conformingcircumferentially to the inner surface of anideal right circular cylinder but with a paraboliccontact face profile not conforming to the cylin-der wall surface in the direction of sliding alongthe cylinder axis. The contact radius of an idealcylinder along its axis is considered to be infi-nite, although in practice this is not the case;usually there is a local contact radius of tensof meters. This case and many other similarcontacts are referred to as partially conforming(Fig. 1b). For piston compression rings, filmthickness of a few tenths to a few micrometerswould be expected at contact pressures of afew to several megapascals.Mechanics of contact depends on contact con-

formity as well as many other factors. Therefore,care should be taken to use the appropriate analyt-ical method of prediction (as highlighted later)for the cases considered in this article.

Contact Mechanics—FootprintShape and Elastic Deformation

The first step in any analysis is to determinethe contact area, where the applied load is car-ried. This is the prelude to calculating the gen-erated pressures as well as any localized contactdeformation, leading to the determination of thelubricant film thickness. In the field of dynam-ics/vibration, the general approach of any ana-lytical solution is to reduce the problem to aseries of masses, stiffness and damping ele-ments, and any applied forces. Then, for anygiven mass, Newton’s second law of motion isapplied to obtain the unknown acceleration:

a ¼ 1

m

XF

where SF is the net force acting on the mass, m,inclusive of the restraining forces of stiffnesselements, dissipative damping forces, and anyapplied forces. This simple approach leads tothe calculation of the unknown kinematic quan-tities: acceleration, a; velocity, v ¼ R a dt; anddisplacement, x ¼ R v dt. A similarly simpleyet profound model is required for contactmechanics. Hertz provided such a model, wherethe contact configurations in Fig. 1 are repre-sented by a rigid ellipsoidal solid of revolution(Fig. 2) contacting an elastic half-space of infi-nite dimensions compared with the dimensionsof the rigid ellipsoid, and with equivalent elas-tic properties to the original contacting pair.The ellipsoidal solid has two principal radii ofcontact in the planes zx and zy, which representthe curvatures of the original contacting pair atthe point or line of their touching contact. Thetwo principal radii of the equivalent ellipsoidalsolid, Rzx and Rzy (Fig. 2), become:

1

Rzx¼ 1

Rx1þ 1

Rx2and

1

Rzy¼ 1

Ry1þ 1

Ry2(Eq 1)

For the case shown in Fig. 2, because theradii of the original contacting solids differ inthe planes of contact, zx and zy, the equivalentsolid becomes an ellipsoid, as also shown inthe same figure. Note that concave radii, suchas that of the raceway groove, Ry2, are consid-ered to be negative (a concave curvature). Itcan be seen that when a concave radius exists,the equivalent radius of the ellipsoidal solidwill be larger than that of either of the originalbodies. Such a radius is often referred to as anequivalent increased radius (Rzy in Fig. 2).Conversely, with a pair of convex radii, theequivalent radius of the ellipsoidal solid willbe smaller than both the original contactingsolids, referred to as the decreased equivalentradius (Rzx in Fig. 2). A special case is a sphereon a plane (Fig. 1c), where for the sphere, Rx1 =Ry1 = R1, and for the flat plane, Rx2 = Ry2 = 1.Thus, the equivalent radii of the ellipsoidalsolid are Rzx = Rzy = R1, which is the sphericalball itself. Similarly, for a roller on a flat plane,Rzx = R1 and Rzy ! 1; therefore, a roller ofnominally infinite length results, which, in prac-tice, has finite length (the length of the roller).Referring to Fig. 1(a), assuming that R1 =

0.95R2, and using Eq 1, it can be seen that theequivalent increased radius becomes Rzx =20R2, which yields a large equivalent cylinderwith its length equal to the length of the bear-ing into the plane of paper along the zy-planefor a journal bearing. It can be shown thatthe resulting gap at the center of contact of thisequivalent cylinder with a flat surface is notrepresentative of the original clearancebetween the journal and the bearing bushing.Therefore, it is important to note that the Hert-zian methodology described in this sectiondoes not apply to conforming contacts of thetype depicted by Fig. 1(a). This point is rarelyunderstood, resulting in a large volume ofreported literature applying Hertzian contactmechanics analysis to journal bearings, ball-in-socket joints of a high degree of confor-mity, and even hip joint prostheses, which isquite erroneous. Furthermore, in wear analysisof counterface materials for such applications,pin-on-disc tribometers with pins of fairlysmall radii are often used, which is unrepre-sentative of the contact conditions. The mainreason for such misconception is that Hertziancontact mechanics deals with the localizednature of deformation (small elastic strainassumption). Of course, this is not the casefor any deformation of larger contact domains,such as journal bearing shells, rings, and anybody where the extent of deformation mayresult in significant changes to its overallgeometry. These forms of deformation maybe considered as global rather than local. Theappropriate theories for these were originallyset forth by St. Venant, culminating in specificanalytical and numerical methods, includingfinite-element analysis.As the equivalent rigidsolid of any shape (roller/cylindrical, sphere,or ellipsoidal shape) is pressed onto an elasticplane of equivalent elastic properties to thoseof the original contacting pairs, the resultingcontact footprint shape can be visualized. Fora sphere this is a circle, for a cylinder it is anarrow rectangular band, and for a generalizedellipsoid it is an elliptical shape. For a ball in araceway groove, this is an ellipse of semi-half-widths a and b (Fig. 3a); for a sphere on a flat,this is a circle, where a = b; and for a cylinder,a narrow rectangular band results (Fig. 3b).The pressure distribution on the elliptical

Z

y

yEquivalentsolid

Flat plane

Z

x

x

Z

Ry2

Ry1Rx2

Rx1

Rzx

Rzy

Fig. 2 Principal planes of contact and the equivalent ellipsoidal solid

Tribology of Power Train Systems / 917

contact footprint is an ellipsoid, and that overthe elastic line contact (rectangular band) iselliptical in cross section of an infinite lengthaccording to classical Hertzian theory(Ref 1). However, when a rigid cylinder offinite length is pressed onto an elastic half-space, the contact footprint spreads out at theedges due to the profile discontinuity (sharpedges) of the roller. Therefore, a dog-bone-or dumbbell-shaped footprint of finite lengthresults, as shown by Johns and Gohar (Ref 2)(Fig. 3c). The sharp edges of the roller yieldhigh generated pressures that can lead to wear

and fatigue. This is the reason for blending theedges of roller bearings; they are usuallyrelieved by the introduction of dub-off radiior through crowning. The contact relationshipsprovided in this section disregard these subtle-ties; otherwise, a numerical analysis would berequired. The ratio of the elliptical footprintsemi-half-widths a and b is known as the ellip-ticity ratio (sometimes referred to as the aspectratio). This is determined as:

e�p ¼a

be Rzy

Rzx

� �2=3

(Eq 2)

where, for a circular point contact, a = b; thus,e�p ¼ 1. Recall that for the idealized Hertzian

infinite line contact, Rzy ! 1; thus, e�p ¼ 1.

However, in reality, the length of a line contactis finite. It is reasonable to assume a line con-tact for e�p > 10 for a narrow rectangular band

of width b. The effective or equivalent modulusof the elastic half-space is:

2

E� ¼1� W21E1

þ 1� W22E2

(Eq 3)

where E1 and E2 are the Young’s elastic moduliof the contacting bodies, and W1 and W2 are theirrespective Poisson’s ratios. When the contact-ing surfaces are made of the same material,then the equivalent elastic modulus becomes:

E� ¼ E

1� W2(Eq 4)

which is usually termed the plane-strain elasticmodulus.Ignoring the magnitude of the edge pressure

peaks (sometimes referred to as pressure spikesor “pips”), whose determination requiresnumerical analysis, Table 1 provides the con-tact dimensions, localized elastic deformation,and generated pressures for different concen-trated counterformal contact types.Within the context of this article, the Hert-

zian relations in Table 1 apply to the contactof gearing systems in vehicular transmissions,differentials, cam-followers, and a host of otherpower train subsystems, where use is made ofball or rolling-element bearings. In all of thesecases, the contact geometry and applied loadmust be determined before these relations canbe used. Other key parameters are not used inclassical Hertzian theory, including operationalspeed and temperature. This is because Hert-zian theory is essentially for elastostatic fric-tionless contacts. Nevertheless, in many casesthe theory provides a good estimate of the pre-vailing conditions for generated pressures andcontact deflection that closely estimate the realconditions for lubricated concentrated counter-formal contacts. The frictionless assumption ofHertzian theory is often relaxed, and a coeffi-cient of friction is used to estimate friction.This approach is very empirical and should beavoided. Later, appropriate methods are out-lined for calculating friction.

Contact Fatigue

With calculated pressures for any of thecases highlighted in Table 1, the integrity ofthe contact under the prevailing condition canbe ascertained. One mode of failure is fatigueof the contacting surfaces. Fatigue occurs asthe result of inelastic deformation due to gener-ated stresses beneath the contacting surfacereaching a critical value. There are variousfatigue failure criteria. One is the maximumsubsurface shear stress criterion, known as

2

1.5

1

0.5

Axial position, m

Footprint shape [×10–3]

Pre

ssur

e, P

a

2a

p0

b

a

(a) (b)

(c)

xy

PL

0.2

0.2

–6 –4 –2 0 2 4 6

–6 –4 –2 0 2 4 6

[×103]

[×10–3]

2

1.5

1

0.5

Axial position, m

Footprint shape [×10–3]

Pre

ssur

e, P

a

2a

p0pp

b

a

(a) (b)

(c)

xy

PL

0.2

0.2

–6 –4 –2 0 2 4 6

–6 –4 –2 0 2 4 6

[×103]

[×10–3]

Fig. 3 Contact footprint and pressure distribution. (a) Elliptical point contact. (b) Long line contact. (c) Finite line contact

Table 1 Relationships for Hertzian contacts

Variable Elastic line contact Circular contact Elliptical contact

Contact half-width or radiusa ¼ 4WR

pLE�� �1=2

a ¼ 3WR4E�� �1=3 ffiffiffiffiffi

abp ¼ 3W

ffiffiffiffiffiffiffiffiffiRzxRzy

p4E�

� �1=3Maximum and mean contact pressures

p0 ¼ 4p pm ¼ WE�

pRL

� �1=2p0 ¼ 3

2pm ¼ 6WE�2

p3R2

� �1=3

p0 ¼ 3pm2

¼ 6WE�2p3RzxRzy

� �1=3Contact load W ¼ 2aLpm W ¼ pa2pm W ¼ pabpmContact center deflection d ¼ W

pLE� ln L3pE�2RW

� �þ 1

h id ¼ pp0a

2E� ¼ 9W2

16E�2R

� �1=3d ¼ 1

29W2

2E�2 ffiffiffiffiffiffiffiffiffiRzxRzy

p� �1=3

918 / Friction and Wear of Machine Components

Tresca. In a two-dimensional analysis, the max-imum shear stress is obtained as:

tmax ¼ 1

2s1 � s2j j (Eq 5)

where s1 and s2 are the principal stresses. Asthe applied load increases, so do the maximumHertzian pressure, p0, and the principal contactstress difference. Therefore, the maximum sub-surface shear stress increases. The onset ofinelastic deformation occurs when:

tmax ¼ sY

2¼ 0:3p0 (Eq 6)

where sY is the yield stress of the softer of thetwo counterface materials. Therefore, for agiven applied load, the onset of plastic defor-mation due to the Tresca yield criterion occursat the maximum Hertzian pressure of:

p0 ¼ 1:67sY ¼ 0:6H (Eq 7)

where H is the material indentation hardness(this is often provided in Vickers or Rockwellscale, but the units used in Eq 7 are in pascals).Hence, for a given load, such as the contactload on gear teeth or a flat tappet, and for givencontacting materials, if the maximum Hertziancontact pressure exceeds that given in Eq 7,then onset of inelastic deformation occurs at adepth z beneath the contact footprint center inthe softer of the two counterface surfaces, aftersome degree of work hardening, of course. Thechance of failure is increased with the existenceof any fault such as a crack, pore, void, and soon within the bulk of the material. Commonly,a crack grows to the contact surface, and asmall piece of material is removed. The phe-nomenon is referred to as fatigue spalling(ISO 6336). The depth at which the maximumshear stress occurs is z = 0.78a, where a is thehalf-width of the narrow band in the long linecontact (Table 1). Clearly, with the finite linecontact of a roller against a raceway or a camagainst a flat tappet, the edge pressure spikes(Fig. 2) induce subsurface stress fields of theirown. Because these edge stresses are usuallyhigher than the maximum Hertzian pressure,larger maximum subsurface shear stresses occurthere nearer to the contact surface, resulting in agreater chance of fatigue failure (Ref 3).The Tresca criterion is one of a number of

proposed yield criteria and is particularly suit-able for relatively hard-wearing but brittle con-tacting surfaces, such as many coatings of highhardness. Teodorescu et al. (Ref 4) provide

numerically obtained subsurface stress fieldsfor different coated surfaces, which show thatthe coincidence of maximum shear stress withthe interfacial layer between a coated layerand the original substrate material can lead totheir exfoliation at given maximum pressuresor tangential traction.Using the Tresca yield criterion for spherical

surfaces, the maximum Hertzian pressure (forelliptical and circular point contacts) at theonset of yield becomes:

tmax ¼ 0:31p0 ¼ sY

2(Eq 8)

Thus, the maximum Hertzian pressure at theonset of yield for an elliptical point contactbecomes:

p0 ¼ 1:60sY ¼ 0:6H (Eq 9)

The depth at which the maximum shear stressoccurs is z = 0.47a.

There are other failure criteria, including thedistortion energy hypothesis, also known as thevon Mises criterion. This criterion is moresuited to ductile substrates, hypothesizing theonset of yielding to be when the defined equiv-alent stress, se, reaches a certain limit (Ref 5, 6).The equivalent stress is defined in terms of thesubsurface stress field, se ¼ ffsxx;syy;szz; txy; tzx; tzyg (Ref 3). Various yield hypoth-eses are used with the equivalent stress. Onethat is progressively favored for bearings, gears,and cam-follower pairs is the alternating shearstress hypothesis (Ref 3, 7, 8). This is basedon the shear stresses tzx, tzy, which occur onorthogonal planes beneath the contact footprintin a cyclic manner, alternately stretching (ten-sioning) and compressing the bulk material(Fig. 4). Although these shear stresses havelower magnitudes than the maximum subsur-face shear stress, their repetitive cyclic rever-sals are often responsible for limiting theuseful life of the bearing (Ref 7, 8). Using thealternating shear stress hypothesis with the dis-tortion energy criterion, the equivalent stress atthe onset of yield becomes:

se ¼ 2jtzxmax=p0j (Eq 10)

where the alternating subsurface shear stress dis-tribution is given by Johnson (Ref 9), requiringnumerical evaluation, and its maximum doubleamplitude can be found as shown in Eq 10.The contact mechanics approach highlighted

here is applicable to many counterformal

concentrated contacts, including cam-followers,transmission gears, rolling-element, and ballbearing supports in some engines, transmissions,driveline, and axle components. In other words,the theory is generic and widely applicable forestimating applied pressures, contact stresses,and localized surface deformation and for pre-dicting the onset of fatigue. However, surfacetopographical information, lubricant properties,and an estimate of lubricant film thickness arestill required to predict friction, power loss, andgenerated contact temperature, all of which arecritical in the assessment of wear and system effi-ciency. Therefore, it is necessary to determine theprevailing regime of lubrication.

Regimes of Lubrication

Lubrication is an important component of tri-bology. Its main purpose is twofold. First, itmust provide a thin low-shear-strength layerof lubricant (termed the lubricant film thick-ness) to ideally completely separate the matingcontacting surfaces and carry the applied con-tact load. Secondly, the flow of lubricant shouldcool the contacting surfaces by carrying a pro-portion of the generated heat away through con-vection cooling.Real engineering surfaces are rough (not

smooth and frictionless, as assumed in classi-cal Hertzian theory). Boundary friction iscaused by the interaction of roughness peakson opposing contacting surfaces. (There is alsoviscous friction arising from shear of a thinlubricant film formed in the contact.) Thismeans that in boundary friction the relativemotion of contacting surfaces is resisted bytheir roughness features, which must besheared for continuance of motion. A muchgreater effort is usually needed for this thanfor shearing of a fluid (a lubricant film, i.e.,viscous friction). Viscous friction can becomesignificant at high loads when non-Newtonianlubricant behavior occurs, and lubricant shearstress tends to be its limiting value. Themechanisms underlying friction are discussedlater in the section “Friction and Power Loss”in this article. It suffices to state that, depend-ing on the thickness of the lubricant film, vari-ous degrees of direct interaction of roughsurface topography occur. The regime of lubri-cation depends on the extent of these interac-tions. Stribeck (Ref 10) devised a convenientway of relating roughness and film thicknessby a parameter, now called the oil film ratioor Stribeck’s oil film parameter:

l ¼ h

ss

(Eq 11)

where h is the lubricant film thickness, and

ss ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2s1 þ s2

s2

pis the root-mean-square

roughness of the two surfaces (the compositesurface roughness); ss1 and ss2 are the individ-ual surface roughnesses (usually the meanroughness heights of each contacting surface,

U

(a) (b) (c) (d) (e)

Fig. 4 Material subjected to reversing orthogonal shear stress. U, direction of rolling

Tribology of Power Train Systems / 919

although in many cases a better statisticalparameter would be more appropriate, such asthat discussed in the section “Surface Topogra-phy” in this article).Stribeck proposed a curve relating the coeffi-

cient of friction to his oil film parameters(Fig. 5). The coefficient of friction is m ¼ f

w,where f is friction, and W is the applied contactload. The Stribeck curve is only instructive, notquantitative. For a pair of surfaces and a partic-ular lubricant type, such curves are obtainedusing tribometers (devices that measure theresistance to motion under various operatingconditions: applied load, sliding speed, andbulk surface temperature). There are manytypes of tribometers, including pin-on-disc,roller-on-disc, and sliding strip-on-plate, eachrepresenting a different contact configuration.Depending on the operating conditions, it isnow more usual to construct the Stribeck curvewith its abscissa being the Hersey parameter,ZUp , where for a line contact, p ¼ w

L (load per

unit length), making the Hersey number nondi-mensional. Changing the contact speed, U, for agiven lubricant dynamic viscosity,Z, and agiven load and bulk temperature, the lubricationcondition alters as shown by the Stribeck curve.For all other operating conditions remaining thesame, counterface surfaces of different compos-ite root-mean-square roughness yield a Stribeckcurve as well (Fig. 5).The coefficient of friction of unity on the ver-

tical axis is quite arbitrary but represents veryclean, smooth counterface surfaces, whereadhesion would take place. When the abscissais the Hersey number, the intercept with thevertical axis represents the static coefficient offriction, where the contact is dry because nolubricant film is present when there is no rela-tive motion of the surfaces. In fact, it is oftenquite difficult to measure friction at very lowspeeds, because stick-slip motion occurs underthese conditions.The region between A and B is termed the

boundary regime of lubrication and has rela-tively high friction, because the lubricant filmis insufficient, being less than the averageroughness of the counterface surfaces. Forground steel counterfaces, the measured coeffi-cient of friction in this region is often reportedto be 0.2 to 0.3. Because an oxide film isformed on the surfaces in normal atmosphere,the coefficient of friction is reduced (surfaceoxides have lower shear strength than the par-ent surface material). With an increasing speed,a progressively thicker lubricant film is formed,and the value of l increases; thus, a lower num-ber of contact roughness heights (asperities)interact. Therefore, the generated friction inthe region B to C is due to the viscous shearof a lubricant film and a decreasing level ofdirect solid boundary friction. The lubricationin this region is termed a mixed regime of lubri-cation (a mix of fluid film and boundary lubri-cation, formed oxides, or, in some cases,lubricity of surface coatings).

The region C to D and beyond corresponds tol � 3, which for a Gaussian distribution of sur-face asperity heights is shown to be subject to afluid film regime of lubrication. Generated fric-tion in this region is entirely due to the shear ofa lubricant film, which would completely sepa-rate the rough counterface surfaces. There arevarious forms (modes) of fluid film regimes oflubrication. The two main forms are hydrody-namics and elastohydrodynamics. The formerrefers to the formation of a coherent lubricantfilm between the contacting surfaces withoutany localized contact deformation. This occursat low-to-medium applied loads, with pressuresnot exceeding several to tens of megapascalsand with surfaces of high elastic modulus. Thefilm thickness increases with contact velocity,thus increasing the l value (for a given surfaceroughness). The coefficient of friction is muchlower than that with mixed and boundaryregimes of lubrication (a representative valueis shown in Fig. 5). If the coefficient of frictionis conceived to be the fractional energy loss in acontact, then for a boundary regime of lubrica-tion a value of m = 0.25 represents 25% energyloss or, in other words, a contact of 75% effi-ciency, while under a fluid film regime of lubri-cation with m = 0.005 the energy loss would bemerely 0.5%. This simple example shows theimportance of the principle of lubrication. Thedip in region C to D occurs in cases where loca-lized deformation of surfaces occurs (elastohy-drodynamic lubrication, or EHL; surfacedeformation and piezo-viscous action of thelubricant, described in the section “LubricantRheology” in this article), increasing the gapfilled with a film of lubricant and reducing thechance of direct contact of surfaces (greatervalues of l). Hence, the coefficient of frictionis reduced.As the contact velocity increases beyond

point D in Fig. 5, the lubricant film is enhanced,but the generated heat due to increased shearlowers the lubricant viscosity, Z, and thus thegenerated pressures, thereby reducing the con-tact deformation. Hydrodynamic conditionswith increased friction and drag would ensue.Not all types of contact, including those in

power train systems, undergo all the aforemen-tioned regimes of lubrication under normally

specified operating conditions. For instance,crankshaft journal bearing supports often oper-ate under a hydrodynamic regime of lubrica-tion, apart from under start-up conditions orunder intermittent stop-start in congested traf-fic, where entrainment of the lubricant into thecontact is interrupted. During such periodsthere may be insufficient film thickness, andsome degree of direct surface interactionsoccurs. To mitigate the resulting excess frictionand wear under these conditions, the surface ofthe bearing bushing is often coated with a thinwear-resistant layer, for example, with bismuthor indium (Ref 11). Sometimes the bushing sur-face has several soft and hard layers; thesoft layers allow some deformation (overlaybearings) and enhance the gap and improve filmretention, while hard top layers resist wear(Ref 8, 11). With pressures of the order of tensof megapascals, the soft layers such as alumi-num, copper, or tin deform, but the lubricantacts under iso-viscous conditions. This regimeof lubrication is known as iso-viscous-elastic(soft EHL) as opposed to elastohydrodynamics(or hard EHL), where high pressures increasethe viscosity of the lubricant as well (piezo-vis-cous effect). Therefore, the fluid film lubrica-tion after point D in Fig. 5 can be classified as(Fig. 6):

� Iso-viscous rigid: Under this condition, thegenerated pressures are insufficient to bringabout viscous action of the lubricant (seethe section “Lubricant Rheology” in thisarticle) or cause appreciable deformation ofcontacting surfaces. This regime of lubrica-tion is termed hydrodynamics in the tradi-tional sense. Applications include thick-shell journal bearings, often used in largeturbomachinery, as well as nominally lightlyloaded but interacting gears, such as the rat-tling unselected gears of vehicular transmis-sions (the gear rattle phenomenon) (Ref 12).Note that the first word, iso-viscous, refers tothe state of the lubricant, while the second,rigid, refers to the state of deformation ofthe contacting surfaces. The same adopteddefinition is true for the other followingcategories as well.

DCB

0.005

1

1 30

μ

A

λ

DCB

5

1 30

A

Fig. 5 Stribeck curve (not to scale)

Iso-viscous rigid Iso-viscous elastic

Dimensionless elasticity parameter, (Ge)

Dim

ensi

onle

ss v

isco

sity

par

amet

er, (

Gv)

Viscous rigid

108

107

106

105

104

103

102

10–2

10–3

10–4 10–3 10–2 102 103 104 10510–1 1 10

10–1

10

1

Viscous elastic

Iso-viscous rigid Iso-viscous elastic

Viscous rigid

8

7

6

5

4

3

2

2

3

4 3 2 2 3 4 51

0

Viscous elastic

Fig. 6 Fluid film regimes of lubrication

920 / Friction and Wear of Machine Components

� Viscous rigid: The generated pressures alterthe lubricant viscosity, but the contiguoussurfaces remain undeformed. These condi-tions are also referred to as hydrodynamic.They dominate in applications with hard sur-face materials, such as in some journal bear-ings, for example, the piston pin-borebearing, often hardened or coated with mate-rials of high elastic modulus.

� Iso-viscous elastic: The lubricant viscosityremains unaltered, but the contiguous sur-faces undergo elastic deformation. This con-dition is sometimes referred to as soft EHL,such as in the case of thin-shell or overlayjournal bearings.

� Viscous elastic: In addition to the lubricantviscosity changing, the contiguous solid sur-faces deform. This condition is often referredto as hard EHL. This is the main regime oflubrication for loaded gears, rolling-elementbearings, and cam-follower pairs.

The abscissa and the ordinate for Fig. 6 aredefined as:

Elastic parameters: Ge ¼ W�8=3

U�2 (Eq 12)

Viscous parameters: Gv ¼ G�W�3

U�2 (Eq 13)

Both parameters are nondimensional anddepend on the dimensionless groups G*, W*,and U*, which depend on the operating contactconditions. These parameters depend on thecontact configuration (see the sections “ContactConfiguration” and “Contact Mechanics—Foot-print Shape and Elastic Deformation” in thisarticle) such as the materials parameter:

G� ¼ E�a (Eq 14)

where E*is given by Eq 3, and a is the pressure-viscosity coefficient of the lubricant (see thesection “Lubricant Rheology” in this article).The load parameter is:

W� ¼ W

E�RzxRzy(Eq 15)

where Eq 15 is for the general case of an ellip-tical contact footprint. For a circular point con-tact, Rzx ¼ Rzy ¼ R, and for a finite line contactgeometry (see the section “Contact Fatigue” inthis article), Rzy ¼ L:The speed (or rolling viscosity) parameter is:

U� ¼ UZE�Rzx

(Eq 16)

where U is the speed of entraining motion ofthe lubricant into the contact; U ¼ u1þu2

2, which

is the average of the surface speeds of the con-tacting bodies; Z is the dynamic viscosity of thelubricant (see the section “Lubricant Rheology”in this article); and Rzx is the equivalent princi-pal radius of the ellipsoidal solid (see the sec-tion “Contact Fatigue” in this article) in the

direction of rolling/sliding. This depends onthe contact geometry. In some cases, the lubri-cant entrainment is at an angle to the principalradii of contact, such as in hypoid gears. Thedimensionless parameters U*, G*, and W* areoften used in film thickness equations foundthrough an extensive series of numerical predic-tions and formed through regression analysesfor nonconforming contacts. As can be seen,the dimensionless parameters depend on somelubricant properties (see the section “LubricantRheology” in this article). All the expressionsstated with regard to Fig. 6 are valid for non-conforming or partially conforming contactsonly. For conformal contacts, such as journalbearings, a different approach is used (see thesection “Piston-Cylinder Conjunctions” in thisarticle).

Lubricant Rheology

Lubricants used in internal combustionengines comprise a base oil and a multitude ofadditives to improve its performance. Theseinclude viscosity modifiers to improve load-carrying capacity, antiwear agents to reducewear of surfaces, antioxidants to inhibit lubri-cant thermal degradation, friction modifiers toreduce the effect of boundary friction, andothers. The additives are organic or inorganicmolecules and compounds at certain small per-centages (or parts per million) added to the baseoil to enhance certain desired measures ofperformance.The main properties of the lubricant in terms

of load-carrying capacity correspond to the for-mation of a thicker microscale film, due to itseffective viscosity as well as its density atlow-to-medium pressures (below its solidifica-tion pressure, usually approximately 200 to400 MPa, or 29 to 58 ksi). Lubricant densityplays a role of lesser importance than its viscos-ity. Above solidification pressure, the lubricantbecomes incompressible (under hard EHL),resembling an amorphous solid. Bulk rheologi-cal properties of lubricants are discussed here,not the actions of the various additives.Shear Characteristics of Lubricants. The

most important properties of a lubricant are itscapability to carry the applied contact loadand its viscous shear characteristics, affectingcontact friction. In these regards, viscosity isthe most prominent of lubricant properties. Vis-cosity is a measure of the resistance to flow of afluid. Newton described it as the internal fric-tion of layers of fluid in flow. In 1673, Newtonstated that “The shear stress between adjacentfluid layers is proportional to the negative valueof the velocity gradient between the twolayers.” Thus, viscosity was defined as:

Z ¼ t_g

(Eq 17)

where t is the shear stress, and _g ¼ Uh is the shear

strain rate, where U is the contact velocity,and h is the film thickness. Therefore, for a

Newtonian fluid (according to the previous def-inition), Z is invariable with respect to shearrate and is the slope of the (t � U) characteris-tics, as shown in Fig. 7. However, in practice,the viscosity of most fluids alters with shearrate; some exhibit shear thinning (pseudoplas-tics), while others become more viscous (dila-tants), as shown in Fig. 8. Both of thesecharacteristics deviate from the Newtonian def-inition of viscosity. Therefore, they are oftenreferred to as non-Newtonian fluids. However,note that non-Newtonian characteristics occurafter a certain shear rate, which varies accord-ing to the fluid.Most lubricants used in engine and power

train systems act with non-Newtonian shearthinning behavior. In some applications, suchas viscolock systems and limited slip differen-tials (Ref 13, 14), silicone-based fluids shearthin with increased slip of contacting surfacesto reduce friction induced by sliding. Anotherapplication in power train systems is the dis-persed silica particles in a polyethylene glycolsolution, which flocculate (particles come outof solution with increasing shear and raise itsviscosity, thus friction). They are used in fric-tion materials, for example, in brake pads.Most lubricants used in other power train

contacts act in a non-Newtonian shear thinningmanner. This occurs with increased shear rateand particularly with thin films in highlyloaded, concentrated counterformal contacts,as in gears and cam-follower pairs. When theapplied shear stress is removed, lubricant

t

g×

Slope, h

Fig. 7 Shear characteristics of a Newtonian fluid

Newtonian

Pseudoplastics

Dilatants

γ⋅

η

Fig. 8 Newtonian and non-Newtonian shear

Tribology of Power Train Systems / 921

viscosity returns almost instantaneously to itsoriginal state, such as when it exits the contact.Sometimes a finite period of time (relaxationtime) is required for this process to occur. Insuch cases, the lubricant behavior is referredto as thixotropic. For various lubricants, shearthinning is a function of their composition andoccurs when the linearity in ðt� _gÞ characteris-tics is lost (rather similar to the stress-strainrelationship after the Hooke’s elastic limit forsolids at sY). In the case of lubricants, thispoint is defined as t = t0, termed as the Eyringshear stress (after Eyring, Ref 15). In terms ofshear strain rate, a good guide for the onset of

non-Newtonian behavior is _g � 106 s�1 formost engine lubricants and transmission fluids,and t0 = 2 to 10 MPa (0.3 to 1.5 ksi). Of course,these values vary with contact pressure andtemperature.Cross (Ref 16) provided a flow curve for

dynamic viscosity of pseudoplastics as a func-tion of shear rate:

Z ¼ Z1 þ Z� Z1

1þ _g_gc

� �n (Eq 18)

whereZ is the low-shear viscosity of the lubricantat a particular temperature and pressure, and Z1is its limiting high-shear viscosity. For most min-eral-based and synthetic engine lubricants, thevalue of Z1 ¼ 6:31� 10�5N � s=m2. _gc is theshear rate at which the viscosity is midwaybetween the values Z0 and Z1. Also:

_gc ¼1

w(Eq 19)

where w is the relaxation time of the fluid. In thecase of mineral oils, this is approximately 1 ms.The exponent n in Eq 18 can be set to unity.Variation of Lubricant Viscosity with

Pressure and Temperature. Lubricant viscos-ity alters with pressure. The first expressionfor this variation was provided by Barus(Ref 17), which is widely used in literature:

Z ¼Z0eap (Eq 20)

where a is the pressure-viscosity coefficient inm2/N (sometimes called piezo-viscosity). Thisis a measure of lubricant response to appliedpressure. As Eq 20 shows, it affects the lubri-cant viscosity in an exponential manner; thus,it is critically responsible for the piezo-viscousbehavior (pressurization) of the lubricant. Thepressure-viscosity coefficient is itself a functionof temperature and the lubricant molecularstructure. Its value for most lubricants is in therange of 1.5 to 3.5 � 10�8 Pa�1 (for mostengine oils and transmission fluids) at 30 C(85 F) and 1.2 to 2 � 10�8 Pa�1 at 100 C(210 F). The Barus law (Eq 20) yields viscos-ity values that are fairly accurate for pressuresup to the lower levels of solidification pressureof fluids (which is generally in the range of 200to 400 MPa, or 29 to 58 ksi). It can be seen thatthe exponent ap � 1 for contact pressures of theorder of 100 MPa (14.5 ksi), where viscosity

nearly triples from its atmospheric value (usingEq 20), Z � 2.718Z (Note: e � 2.718). At400 MPa (58 ksi), the viscosity is increased

by a factor of ZZo e e4 � 2981, which is clearly

impractical. Therefore, for pressures of200 MPa (29 ksi) and above, alternative rela-tionships should be used. In practice, viscosityof the lubricant increases with pressure by2 orders of magnitude at the utmost (~200 to300 times), but at the same time it reducesbecause of the accompanying generated contactheat due to shear. The Barus law is suitable forconforming and most partially conforming con-tacts, such as journal bearing supports, big-endbearings, and piston rings, where pressures arein lower megapascals, indicating that usuallyap 1, where the conditions may be consid-ered as iso-viscous (meaning that lubricantpiezo-viscous action is not induced, which isreally an unfavorable outcome). On the otherhand, in counterformal contacts, generatedpressures are of the order of gigapascals, andit can be seen that ap � 1, which means thatthe lubricant would usually be solidified to anamorphous, incompressible solid (elastohydro-dynamic conditions). Any further rise in pres-sure simply deforms the solid surfaces andextends the footprint dimensions. Typical ball-to-race pressures in main shaft bearings of aero-engines can reach pressures of the order of 2.5to 3 GPa (0.36 to 0.44 � 106 psi) in take-offconditions, similar to hypoid gears of vehiculardifferentials of trucks at high loads of the orderof 6 to 10 kN (corresponding to contact pres-sures of the order of 1.5 to 2 GPa, or 0.22 to0.29 � 106 psi). Higher pressures can also beexperienced under impacting conditions. There-fore, piezo-viscous action of the lubricantshould be enhanced for some applications, withpressure-viscosity coefficients of the order of10�9 Pa�1. These are high-pressure oils.Roelands (Ref 18) provided an alternative

expression for lubricant-pressure dependence,suitable for medium-to-high pressures experi-enced in elastohydrodynamic contacts:

Z ¼ Z0eA 1þB p�p0ð Þ½ �Z�1f gð Þ (Eq 21)

where A ¼ lnZ0 þ 9:67, B ¼ 5:1� 10�9; and Zis a lubricant-specificconstant, sometimesreferredto as a viscosity-pressure index or exponent:

Z ¼ aCp

lnZ0 � lnZ1ð Þ (Eq 22)

where Cp is a pressure-viscosity coefficient,usually with a value of ~1.96 � 108 N/m2.Houpert (Ref 19) extended the expression

provided by Roelands to include the effect oftemperature:

Z ¼ Z0eA y�C

y0�C

� ��S0

1þBpð ÞZ�1

� �(Eq 23)

where A, B, and Z were provided previously, C= 138 K, y is the contact temperature, and y0 isthe reference or ambient temperature, both indegrees Kelvin. The exponent S0 is given as:

S0 ¼ b y0 � Cð ÞA

(Eq 24)

where b is the temperature-viscosity coefficient,usually in the range of 0.005 to 0.05 per degreeKelvin.For viscosity variation with temperature

alone, the originally proposed relationship wasprovided by Reynolds:

Z ¼ Z0e�b�y (Eq 25)

where the exponent b is the same as that definedpreviously, Dy = y � y0, and Z0 is the dynamicviscosity at the reference temperature, y0.As for Eq 20, the result of Eq 25 is quite lim-

iting. The most commonly used equation forviscosity variation with temperature is due toVogel (Ref 20):

Z ¼ aeb

y�cð Þ (Eq 26)

where a, b, and c must be obtained for eachlubricant from three sets of supplied data bythe lubricant manufacturer. The temperature,y, is usually in degrees Kelvin. However, if thisis used in degrees Celsius, then b as the inher-ent viscosity-temperature dependence of oil isusually in the range of 500 to ~2000, c = 50to ~150, and a is approximately the density ofthe lubricant. Alternative expressions for vis-cosity variation with temperature are also avail-able, such as that by Walther with twoparameters. The viscosity of lubricants changeswith blending and with the addition ofthickening additives, such as polymers and vis-cosity modifiers. A discussion of these issues inprovided in Ref 8.Variation of Lubricant Density with Pres-

sure and Temperature. Dowson and Higgin-son (Ref 21) provided an expression for thevariation of lubricant density with pressure.This is the most widely used expression fordensity-pressure dependence:

r ¼ r0 1þ 0:6� 10�9 p� p0ð Þ1þ 1:7� 10�9 p� p0ð Þ

(Eq 27)

This expression was subsequently expanded totake into account the effect of contact tempera-ture (Ref 22):

r ¼ r0 1þ 0:6� 10�9 p� p0ð Þ1þ 1:7� 10�9 p� p0ð Þ � c y� y0ð Þ

(Eq 28)

where c is the thermal expansivity of thelubricant, usually in the range of 6.4 to 7.3 �10�4 K�1.Variation of Lubricant Thermal Conduc-

tivity with Pressure and Temperature.Another important lubricant rheological prop-erty is its thermal conductivity, because thisaffects the ability of the lubricant to convectsome of the generated heat away from the con-tact. Larsson et al. (Ref 23) provided a relation-ship for the variation of thermal conductivitywith pressure:

922 / Friction and Wear of Machine Components

K ¼ a 1þ bp

1þ cp

� �(Eq 29)

where the constants are a = 0.137, b = 1.72, andc = 0.54.The variation of lubricant thermal conductiv-

ity with temperature can be stated as:

K ¼ a

s1� byð Þ (Eq 30)

where, in this case, a = 0.12, b = 1.667 � 10�4,and s is the specific gravity at 15.6 C (60.1 F)for temperature y, in degrees Celsius.Variation of Lubricant Specific Heat

Capacity and Thermal Diffusivity withTemperature. The specific heat capacity atany pressure, p, also varies with temperature.In its simplest form, it is roughly estimated as:

cp ¼ aþ bysn

(Eq 31)

where, in this case, a = 1630, b = 3.4, n = 0.5,and s is the specific gravity at 15.6 C(60.1 F) for temperature y, in degrees Celsius.For thermal diffusivity:

a ¼ K

rcp(Eq 32)

Predicting Lubricant Film Thickness

The first step in determining the prevailingconditions is to predict the lubricant filmthickness. The usual way is to simultaneouslysolve the Reynolds equation, the gap shape,and the lubricant rheological state equationsfor density and viscosity. The gap shapeincludes any nominal clearance, the unde-formed profile of contacting surfaces, andany deformation of contacting surfaces usingthe elasticity potential equation (Ref 8). How-ever, the method of solution, particularly forcounterformal contacts, involves detailednumerical analysis and does not lend itself toa readily accessible outcome to everyone. For-tunately, the results of multiple solutions forvarious nonconforming contact configurations(an ellipsoidal rigid solid loaded onto a semi-infinite elastic half-space) have been used inregression analyses to provide film thicknessformulas for a broad range of operating condi-tions, based on the dimensionless parametersG*, U*, and W* (see the section “Regimes ofLubrication” in this article).Figure 9 shows a micrograph of a lubricated

line contact of a roller against a flat plane(Ref 24). This image shows half of the footprintshape from the contact center of the roller, withthe central lubricant film thickness along thecenterline of the contact in the y-direction, h0;x is the direction of lubricant entrainment intothe contact from the inlet to the outlet, beyondwhich the exit flow breaks into cavitation.Figure 10 shows a typical film thickness varia-tion in the x-direction. The minimum film

thickness actually occurs in the vicinity of theoutlet.For counterformal long line contact, the cen-

tral lubricant film thickness can be predicted as(for rollers, spur gear teeth pairs) (Ref 21):

h�0 ¼ 1:93U�0:69G�0:56W��0:1 (Eq 33)

where:

h�0 ¼h0Rzx

A few important points should be noted.First, the lubricant film thickness is almostindependent of load, w (low exponent of W*).This is a feature of the hard EHL condition(recall the incompressible nature of the lubri-cant as an amorphous solid). On the contrary,in the case of hydrodynamics, h / w�1. Sec-ondly, the power indexes for U* and G* showthat the film thickness is crucially dependenton contact speed and the material combination.Thirdly, all of these equations are obtained withthe assumption of a fully flooded (or drowned)inlet condition. Finally, the regime of lubrica-tion is hard elastohydrodynamics. Thus, theseequations do not apply to the other modes offluid film lubrication in Fig. 6.For a finite line contact, the absolute mini-

mum film thickness at the edges of the contact,hsc, referred to as the side constriction filmthickness, should be obtained through numeri-cal analysis. However, the central lubricant filmthickness is given as, not considering any lubri-cant film squeeze (Ref 8, 25):

h�0 ¼ 1:67G�0:421U�0:541W�0:059 (Eq 34)

Similar observations can be made, as in thecase of Eq 33.For the general case of elliptical point con-

tact, as experienced by ball-to-raceway contactin ball bearings, hypoid and bevel gears, andcam-roller follower contact (Ref 26):

h�0 ¼ 4:31U�0:68G�0:49W��0:073

1� e�1:23RzyRzxð Þ2=3

� �(Eq 35)

where, in this case, the dimensionless para-meters are:

W� ¼ pW2E�R2

zx

;U� ¼ pZ0U

4E�Rzx;Ge ¼ 2

pðE�aÞ (Eq 36)

Note that for the case of a circular point con-tact, Rzy ¼ Rzx ¼ R.Therefore, the lubricant film thickness for

various counterformal contacts under EHL canbe estimated using Eq 33 to 35. Once the lubri-cant film thickness is obtained, the Stribeck oilfilm parameter, l, can be calculated (Eq 11), ifthe root-mean-square composite surface rough-ness is known. For conforming contact of jour-nal bearings, the appropriate lubricant filmthickness equation is provided in the section“Piston-Cylinder Conjunctions” in this article.

Surface Topography

Surface roughness is usually measuredthrough the use of various forms of profilometeror spectrometer, either using a contacting stylusrunning over the rough surface or throughimaging of the same. In all cases, the resultsare magnified and analytical calculations aremade to represent the surface by a series of sta-tistical parameters. Traditionally, arithmeticaverage roughness (Ra) has been used in indus-try to quantify the roughness of a surface:

Ra ¼ 1

L

Z L

0

z xð Þj jdx ¼ 1

n

Xn

i¼0zij j (Eq 37)

where, in continuous description, L is the lengthof the measured surface, and z(x) is themeasured profile as a function of length at anypoint, x. In discrete description, n is the numberof measured data points, and zi is the measuredsurface height at any given point along the sur-face. However, the arithmetic average or cen-terline average is not a good representation ofthe actual surface roughness. Advances in sur-face engineering have resulted in the use of var-ious more-pertinent parameters, describingvarious features of a rough surface. It is betterto use the geometric average roughness (Rq, or

Inlet

Outlet

Cavitation bubbles

hschm

h0

y

x

Fig. 9 Micrograph of lubricant film contour for a rolleragainst a flat. Source: Ref 24

–1.6 –1.2 –0.8 –0.4 0.4 0.8 1.20

X

Central film thickness

Minimum film thickness

Fig. 10 Lubricant film shape in the direction of rolling

Tribology of Power Train Systems / 923

root-mean-square roughness), which is moresensitive to the variations in roughness peaksand valleys:

Rq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

L

Z L

0

z2 xð Þdxs

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xn

i¼0z2i

r(Eq 38)

As a rule of thumb, for a given surface, usuallyRq should be 11% higher than its Ra value.An improved description of the surface rough-

ness for various engineering applicationsincludes parameters such as core roughness, Rk,reduced peak height, Rpk, and reduced valleyheight, Rvk. These parameters are obtainedthrough Abbott-Firestone or bearing area curvesand describe the surface texture or roughnessstructure.With the use of these sets of parameters,one can differentiate between the core roughness(the plateau), the peaks above the plateau, and thevalleys between the plateaus, such as for the caseof cross-hatched honed surfaces of cylinder liners(Fig. 11). These parameters are suitable for mon-itoring of running-in wear process (Ref 27, 28).During the process, Rpk usually changes dramati-cally until the roughness profile settles. In addi-tion, it is desirable to reduce the Rpk of thesurface with harder materials to reduce abrasivefriction. TheRvk is an estimate of the valleys withthe potential to act as reservoirs of trappedlubricant.Other roughness parameters are also used to

determine the nature of the surface roughnessdistribution, such as skewness and kurtosis.Skewness determines the symmetry of rough-

ness distribution. Positive values of skewnessindicate that the peaks are predominant, whilenegative values indicate dominance of thevalleys:

Rsk ¼ 1

Rq3

1

n

Xn

i¼0zi3

� �(Eq 39)

Kurtosis is a measure of deviation of theroughness profile from an ideal Gaussian distri-bution. Kurtosis values higher than 3 indicatethat there are excessive high peaks or valleysin the measured profile, while values lower than3 show a lack of these:

Rku ¼ 1

Rq4

1

n

Xn

i¼0zi4

� �(Eq 40)

A surface with Gaussian symmetric roughnessdistribution has a skewness of 0 and a kurtosisvalue of 3.Mathematical models such as the Greenwood

and Tripp model (Ref 29) used to estimateboundary friction in mixed or boundary regimesof lubrication are based on the contact ofopposing rough surfaces with a Gaussian distri-bution of roughness peaks. These models makeuse of other topographical parameters, such asthe average asperity peak radius, b, and the areadensity of asperity peaks at certain separationsof the contacting surfaces, k, as well as thecomposite surface roughness, ss (Eq 11). Forss, the arithmetic average or plateau coreroughness, Rq or Rk, may be used for run-in sur-faces, and during running-in wear with new sur-faces, Rpk is found to be the most appropriateparameter (Ref 28).The parameters b and k are evaluated using

spectral moments of the measured roughnessprofile (Ref 30). For anisotropic surfaces, thespectral moments should be calculated in twopreferably perpendicular directions and thencombined to provide a representative spectralmoment for the surface, based on which valuesof b and k can be determined (Ref 31). For iso-tropic surfaces, the area density of the asperitypeaks is calculated as (Ref 32):

k ¼ 1

32:65

Pni¼0 d2z=dx2ð Þ2Pni¼0 dz=dx2ð Þ2 (Eq 41)

and the average asperity radius as (Ref 33):

b ¼ 0:665ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Pni¼0 d2z=dx2ð Þ2

q (Eq 42)

All of these parameters can readily be obtainedfrom most postprocessing software installed onmodern surface-measuring optical metrologydevices. Thus, for two rough contacting sur-faces, the composite value becomes:

kc ¼ 1

2k1 þ k2ð Þ (Eq 43)

and

1

bc¼ 1

b1þ 1

b2(Eq 44)

In practice, the roughness of surfaces is depen-dent on the material and the processes carriedout, such as turning, grinding, honing, and soon. Representative values are provided in Ref 8.

Friction and Power Loss

Viscous Friction. Friction in lubricated con-tacts is generated due to the shear of a thinlubricant film. The shear stress is obtained as:

t ¼ h

2

dp

dx Z�U

h(Eq 45)

where the first term is shear caused throughapplication of the pressure gradient (lubricantentering into the contact wedge). This is calledPoiseuille shear, after the French physiologistwho first defined it. This component of shearis dominant at the inlet to a contact andbecomes negligible relative to the second term,which is the shear of the lubricant film due tothe relative motion of the contiguous surfaces,termed Couette shear, after the French physicistwho first defined it. DU is the sliding speed (therelative speed of the contacting surfaces,�U ¼ u1 � u2ð Þj j). The shear stress in Eq 45is most appropriate for Newtonian behavior of

the lubricant. As the shear rate, _g ¼ �Uh ,

increases, a limit is reached where any furtherincrease in shear rate has no effect on the shearstress. At this limit, the proportional linearitybetween shear stress and shear strain rate islost. The shear stress at this limit is defined asthe Eyring shear stress (Ref 15) (see the section“Shear Characteristics of Lubricants” in thisarticle). The shear behavior of the lubricantthereafter is referred to as non-Newtonian(Ref 8). Under non-Newtonian shear, the behav-ior of the lubricant is visco-elastic, meaning thatshear stress is not only dependent on the shearstrain rate but also on the shear strain itself (note:for elasticmaterial in shear, t/ g, and for viscousmedia, t / _g). Thus, t / ðg; _gÞ. Thereafter,with further shear the lubricant enters into avisco-plastic behavior, with a diminutive sur-face-adsorbed film (Fig. 12). Evans and John-son (Ref 34) provide charts for shear behavior

Valleys

Roughnesspeaks

X = 0.712 mmY = 0.54 mmZ = 6.68 μm

μm

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0

Plateau

Fig. 11 Typical image of a cross-hatched honed specimen surface using white light interferometry

924 / Friction and Wear of Machine Components

of some lubricants. Because such films are notcoherent (continuous in nature), a limiting shearstress is reached as:

tL ¼ tL0 � epm (Eq 46)

where clearly the limiting shear stress is a functionof the average or mean pressure, pm (Pascal pres-sure as contact load over the area); e is the pres-sure-induced shear coefficient, usually in therange of 0.05 to 0.3; and tL0 is the limiting shearstress of the lubricant at atmospheric pressure.The mathematical definition of friction is:

f ¼ZZ

tdxdy (Eq 47)

where the contact occurs in the xy-plane. There-fore, for Newtonian shear, viscous frictionbecomes (Eq 45):

fv ¼ZZ

h

2

dp

dx Z�U

h

� �dxdy (Eq 48)

Note that in the contact:

h

2

dp

dx Z�U

h

With the film thickness, h, obtained from theappropriate equation in the section “PredictingLubricant Film Thickness” in this article, andthe viscosity of the lubricant determined at agiven temperature and pressure from the section“Lubricant Rheology,” the viscous friction canbe obtained if the contact area is known. Forthe case of nonconforming contacts, these arereadily available in Table 1, where for acircular point contact footprint, A = pa2; foran elliptical contact footprint, A = pab; andfor a finite line contact as a rectangular band,A = 2aL. For journal bearings, see the section“Piston-Cylinder Conjunctions” in this article.For the non-Newtonian fluid film regime of

lubrication in the visco-elastic (Eyring) region,an analytical approach is reported by Evansand Johnson (Ref 34), where the coefficient offriction, m, is given as:

m ¼ 0:87at0 þ 1:74t0pm

ln1:2

t0h0

2KZ0

1þ 9:6x

� �12

" #(Eq 49)

where values for the pressure-viscosity coeffi-cient, a, lubricant thermal conductivity, K, andviscosity, Z, for the correct operating condi-tions can be calculated using the expressionsin the section “Regimes of Lubrication” in thisarticle. Also:

x ¼ 4

pK

h0=Rzx

pmE�RzxK0r0c0U

� �1=2

(Eq 50)

which is in terms of lubricant thermal conduc-tivity and that of solid surfaces, where r0 isthe density, c0 is the specific heat capacity,and K0 is the thermal conductivity.For the visco-plastic region, where interac-

tions occur in the glassy amorphous state:

m ¼ tLpm

(Eq 51)

Therefore, the viscous component of frictionbecomes:

fv ¼ mW (Eq 52)

where W is the contact load.Prior to any calculation of friction, it is

important to establish the regime of traction(shear behavior of the lubricant: Newtonian,non-Newtonian visco-elastic Eyring, or non-Newtonian visco-plastic). For this purpose,Reiner (Ref 35) defined a number, calledDeborah’s number, De, which is the ratio ofrelaxation time for a material to adjust to anapplied state of stress to a characteristic timeof the investigation. In rheology/lubrication,this characteristic time is the time of passageof the lubricant through a given contact con-junction. The relaxation time of a lubricant inshear is Z

G, where G is the shear modulus ofthe lubricant, which only becomes significantat high pressures (typical of visco-elastic andvisco-plastic traction), experienced in elastohy-drodynamic contacts (pressures in the gigapas-cal range). The value of G depends on themaximum generated pressure (Table 1) as(Ref 8):

G ¼ 0:1þ 3p0 (Eq 53)

The unit for this equation is gigapascals, andthe value of 0.1 GPa represents the onset oflubricant solidification to an incompressibleamorphous solid, taken to be 100 MPa(14.5 ksi) in this case for most engine lubri-cants. p0 is the maximum Hertzian pressure(Table 1).The time of passage of the lubricant through

a contact depends on the contact footprintdimension in the direction of flow, usually 2a(Table 1), and the speed of entrainment of thelubricant into the contact, U ¼ u1 þ u2, or, inother words, 2aU . Therefore, the Deborah number

becomes:

De ¼ ZU2aG

(Eq 54)

Viscosity should be adjusted for pressure, usingthe Roelands equation (Ref 18) (see the section“Lubricant Rheology” in this article). It can beseen that at high viscosities or high lubricantentrainment speeds, a large Deborah numberwould result, indicating visco-elastic behavioras the lubricant elastically deforms/shears inflow through the contact. Conversely, at lowerviscosities and entraining speeds, the Deborahnumber is reduced, such as in Newtonian shear,when De � 1. With more excessive pressures,the lubricant acts as a glassy solid and, depend-ing on the transient value of G, undergoes plas-tic deformation as well. Like a solid, the stress-strain relationship governs the conditions ratherthan the stress-strain rate, as predominates forliquids. The limiting shear stress becomes:

tL ¼ gG (Eq 55)

which can be compared with Eq 46 to get anidea of shear strain under any prevailing condi-tions, using the data already provided and thevalue of tL0 for a particular lubricant from itsmanufacturer.It should be noted that as either of the con-

tact pressures or the speed of entrainingmotion increases, the generated temperaturesreduce the lubricant viscosity; thus, the lubri-cant shear stress falls in all the stated regimesof traction.Boundary Friction. When the thickness of

the lubricant film is comparable to the rough-ness of the contacting surfaces, interactionsoccur between the asperities of the roughcounterface surfaces. The regime of lubrica-tion is no longer fluid film lubrication butmixed or boundary regimes of lubrication(see the section “Regimes of Lubrication” inthis article). The result is increased frictionthrough boundary interactions (i.e., boundaryfriction). There are two main mechanisms forthis form of friction. One is welding of theopposing asperities under pressure into junc-tures that must be broken for the continuanceof relative motion of the contacting surfaces.The resisting mechanism is called adhesivefriction. In the process, globules of the softerof the two surfaces may be broken and stickto the harder one or contaminate the lubricant,resulting in adhesive wear. There is also fric-tion due to tribochemical reaction layersformed on the contiguous surfaces in contact.The other mechanism of boundary friction is theoblique contact of opposing asperities with thoseof the harder surface, deforming those of thesofter counterface. The resistance introduced bythese asperities in deformation (elastically orplastically) accounts for a mechanism referredto as deformation or plowing friction. This canlead to a host of wear mechanisms, mainly abra-sive wear and scuffing. Similar counterface mate-rials should be avoided due to solid solubility ofsimilar materials and bonding under pressure.As a guideline, a mixed regime of lubricationoccurs when 1<l<3, and a boundary regime oflubrication occurs when l � 1.

Visco-plasticNon-Newtonian

New

toni

an

Shear rate

She

ar s

tres

s

τ0

τL

Fig. 12 Typical shear rate characteristics of lubricants

Tribology of Power Train Systems / 925

When mixed or boundary regimes of lubrica-tion occur, the Greenwood and Tripp (Ref 29)model is often used to calculate boundary fric-tion. The method assumes a Gaussian distribu-tion of surface asperity heights that introducessome limitations along with real rough engi-neering surfaces (see the section “SurfaceTopography” in this article) (Ref 36).Under a mixed regime of lubrication, a por-

tion of the contact load is carried by the oppos-ing asperities of the counterface surfaces as afunction of their separation (Ref 29):

Wa ¼ 16ffiffiffi2

p

15p kbssð Þ2

ffiffiffiffiffiss

b

rE�AF5=2 lð Þ (Eq 56)

where b is the average asperity tip radius, ss isthe composite root-mean-square surface rough-ness, and k is the asperity distribution per unitscanned/measured area. The product kbss isknown as the roughness parameter, usually inthe range of 0.03 to 0.07, and s/b = 10�4 to10�2 (a measure of asperity slope) for manyengineering surfaces. This parameter affectsthe adhesion of surfaces, as described by Fullerand Tabor (Ref 37). F5/2(l), a statistical function,can be represented by a polynomial fit for ease ofapplication:(see Eq 57 at bottom of page)

The load-carrying capacity of the lubricantfilm is obtained as Wh ¼

RRpdxdy, which is

usually obtained through numerical solution ofthe Reynolds equation to find the pressure dis-tribution, p(x ,y). Analytical solutions are avail-able for simple cases, many of which areprovided in Ref 8. Therefore, the applied load,W, is carried by the lubricant film as well as aportion (usually small) of directly contactingopposing asperities; hence, W ¼ Wh þWa.Similarly, the asperity contact area at any

given separation of surfaces, represented bythe Stribeck’s oil film parameter, l, is (Ref 29):

Aa ¼ p2 xbsð Þ2AF2 lð Þ (Eq 58)

The statistical function F2ðlÞ is:(see Eq 59 at bottom of page)

Boundary friction can be evaluated as:

fb ¼ tLAa þ BWa (Eq 60)

where B is analogous to the asperity-scale adhe-sive coefficient of friction. It is also assumedthat pockets of lubricant trapped in the valleysbetween asperity peaks and any ultrathinadsorbed film at the summit of asperities wouldshear at the limiting shear stress. Therefore,with measured values of roughness parameterand asperity slope, as well as using an

appropriate film thickness equation (see the sec-tion “Predicting Lubricant Film Thickness” inthis article), boundary friction can be predicted.Total Friction and Power Loss. The total

generated contact friction is the result of combinedviscous and boundary friction components, thus:

f ¼ fv þ fb (Eq 61)

The frictional power loss is obtained as:

Pl ¼ f�U (Eq 62)

where DU is the sliding velocity (relative speedof the contacting surfaces).It is important to evaluate the frictional

power loss to ascertain the efficiency of thepower train, such as engine subsystems, enginebearings, the piston-cylinder system, and thevalve train system.

Piston-Cylinder Conjunctions

The ring pack usually uses three to fourrings, comprising one to two compression rings,an oil control ring, and a scraper ring. In gen-eral, the losses are nearly equally sharedbetween the piston skirt and the ring pack.Among the piston rings, the tribology of thetop compression ring is the most complex,because its main function is to seal the combus-tion chamber, thereby guarding against flow ofcombustion gases and soot into the crankcase.This phenomenon is called blow-by. Therefore,good sealing action of the compression ring(s)is required to mitigate blow-by and any lossof power. As a result, there is a diminishedgap between the compression ring contact faceand the liner or bore surface, promotingincreased friction. In an engine cycle compris-ing in-take, compression, combustion (orpower), and exhaust strokes (for a four-strokeengine), the compression ring is subjected to atransient regime of lubrication, which usuallyincludes boundary, mixed, and hydrodynamicconditions. Hydrodynamics often dominate inall the engine strokes, except at piston reversalsat the top and bottom dead centers of pistonmotion. Here, because of a change in the direc-tion of sliding velocity, there is momentary ces-sation of entraining motion of the lubricant, U =0; thus, no lubrication occurs, except for anyentrapped pockets of lubricant on the rough sur-face topography or lubrication through squeeze-film motion. Mixed or boundary regimes oflubrication are prevalent in these parts of thecycle and also near the top dead center in thecompression stroke and in a large part of thepower stroke because of the higher contact

pressures. It is rare that the contact pressures(usually of the order of several megapascals)would cause any local deformation of contactingsurfaces or significantly cause piezo-viscousaction of the lubricant. Thus, elastohydrody-namic conditions are not usually encountered,although some have claimed otherwise in theliterature.Due to the complexity of the ring-liner con-

junction, numerical solutions of the Reynoldsequation or even more general solutions of theNavier-Stokes equations are used. Readers canrefer to such solutions in Ref 38 to 42. However,there have been some analytical solutions that,although idealized to a certain extent, providegood estimates of prevailing conditions. Further-more, for a handbook with a broad readership,such solutions are more appropriate. The piston-skirt contact with the cylinder liner is usually sub-ject to an elastohydrodynamic regime of lubrica-tion, with thermoelastic deformation of the skirtbeing a function of skirt structure and contact pro-file. No accurate analytical solution is possible.Readers are referred to Ref 43 to 45.To obtain any analytical solution, certain

simplifying assumptions are made. In this case,it is assumed that the ring conforms perfectly tothe surface of the liner, which is considered tobe an ideal right circular cylinder. In practice,the ring is subject to modal deformation whensubjected to varying contact loads and frictionduring an engine cycle. It deforms in its radialplane (in-plane motion) as well as in the axialdirection of the piston (out-of-plane motions,including twist and flutter motions) (Ref 46–48).Lubricant film thickness is formed along thering contacting face in the axial direction ofmotion. The film thickness is generally quite thin(tenths of a micrometer to a few micrometers).Therefore, one can assume no side leakage ofthe lubricant from its axial flow into the periph-eral direction of the cylinder. These assumptionsmean that a one-dimensional solution of the Rey-nolds equation can be undertaken, which lendsitself to an analytical approach:

@

@xh3

@p

@x

� �¼ 6Z0U

@h

@xþ 12Z0

@h

@t(Eq 63)

whereZ0 is the dynamic viscosity of the lubricantat atmospheric pressure, assumed here because,as shown in the section “Variation of LubricantViscosity with Pressure and Temperature” in thisarticle, with low-to-medium hydrodynamic pres-sures,Z ~Z0. However,Z0 should be adjusted forthe contact temperature to better represent theprevailing conditions. In the case of ring-cylinderliner contact, Morris et al. (Ref 40) have shownthat liner temperature can be safely used for this

F5=2 ¼ �0:004l5 � 0:057l4 � 0:29l3 � 0:784l2 � 0:784l� 0:617 for l< 2:50; for l � 2:5

�ðEq 57Þ

F2ðlÞ ¼ �0:002l5 � 0:028l4 � 0:173l3 þ 0:526l2 � 0:804l� 0:500 for l < 2:50; for l � 2:5

�ðEq 59Þ

926 / Friction and Wear of Machine Components

purpose. The film thickness, h, is required for thesolution of the Reynolds equation (Eq 63), toobtain the pressure distribution, p. The film thick-ness is a function of the ring contact face profileand the nominal minimum gap (designed clear-ance), h0, as:

h ¼ h0 þ SðxÞ (Eq 64)

where S(x) is the axial profile of the ring, asshown in Fig. 13.A parabolic ring face profile is shown in

Fig. 13. However, in practice there are manyforms of ring face profile, some fairly flat withchamfered edges. In most cases, the profile canbe approximated with the following expression:

S xð Þ ¼ Cx o

b=2þ o

� �m

(Eq 65)

where m = 2 yields a parabolic profile, C is thering crown height, o is any crown offset fromthe center, with “+” shifting the vertex of theparabola toward its left side and “�” shiftingthe vertex of the parabola toward its right side,as shown in Fig. 13. The crown height and theparabolic shape alter with running-in and grad-ual wear, as shown by Rahmani et al. (Ref 39).The profile of a run-in ring was measured,showing a complex mix of various orders ofm. Morris et al. (Ref 27) provide analysis forrings with different values of m.For a parabolic contact face profile:

S xð Þ ¼ 1

2Rx oð Þ2 (Eq 66)

where the radius of curvature of the ringprofile is:

R ¼ 1

2Cb=2þ oð Þ2 (Eq 67)

It is now possible to solve the Reynolds equa-tion (Eq 63) analytically by integrating it twicewith respect to x. Two boundary conditions arerequired to find the integration constants. Theseboundary conditions are based on the assumedinlet and outlet of the contact. They also dependon the sense of motion of the ring: upstroketoward the top dead center or downstroketoward the bottom dead center. In the case ofthe former, the pressure at the inlet is the com-bustion chamber pressure, and in the case of the

latter, the inlet pressure is the interring pressure(between the compression and the oil controlrings). For simplicity, this is often taken to bethe crankcase pressure at approximately theatmospheric pressure.The lubricant is drawn into the contact in

the opposite sense to the ring sliding motion.This means that in the upstroke sense of thering, the inlet meniscus would be in the ringconverging profile adjacent to the combustionchamber, where one would expect starved con-ditions. In the downstroke sense of the ring,the inlet is fed by any volume of surface lubri-cant between the oil control ring and the lowerside of the compression ring. Most analyticalsolutions assume a fully flooded inlet. Thismeans that the inlet distance to the center ofthe contact can be assumed to be theoreticallyat infinity, xi ! �1; the negative sign isassigned to the inlet, where the center of thecontact is at x = 0. In practice, the fully floodedinlet is at a distance of 11.298 times the half-width of the sliding contact (in the case of aring of face width b, the half-width can be takenas b/2). The general case of rolling and slidingcontacts is shown by Birkhoff and Hays(Ref 49). Therefore, at least for most of the ringupstroke motion, the inlet to the contact wouldbe partially starved. Tipei (Ref 50) shows thateven with an assumed drowned inlet, some ofthe inward flow into the converging wedge ofthe contact is subject to counter and swirl flows(observed experimentally in Ref 49). Underthese conditions, the actual inlet occurs at thestagnation point, with no reverse and swirlflows thereafter, where:

@p

@x

x¼xi

¼ 2Z0U

h2i(Eq 68)

Integrating the Reynolds condition (Eq 63)once and using Eq 67, the position of theinlet meniscus can be determined. Tipei(Ref 50) favored Reynolds (Ref 51) or Swift(Ref 52)-Stieber (Ref 53) boundary conditions,based on the observation of the inlet flow fieldprior to the stagnation point, where, in additionto condition (Eq 68), p = pc. On the other hand,the zero reverse flow inlet boundary conditionsof Prandtl-Hopkins (Ref 54) require zero lubri-cant film velocity as well as its gradient todetermine the stagnation boundary. It is shownthat the pressure gradient at the actual inlet inthis case becomes (Ref 55):

@p

@x

x¼xi

¼ 4Z0U

h2i(Eq 69)

Using this condition in conjunction with theReynolds equation determines the inlet bound-ary to be at (Ref 55):

xi ¼ �2:623ffiffiffiffiffiffiffiffiRh0

p(Eq 70)

For outlet (exit) boundary conditions, the mostwidely used are the Swift-Stieber conditions(Ref 53), where:

p ¼ pc;@p

@x¼ 0 at x ¼ xc (Eq 71)

where xc is the lubricant film rupture point,beyond which cavitation occurs, and pc isthe cavitation vaporization pressure, assumedto be the atmospheric pressure in most appli-cations. The inlet rupture point is thenobtained as:

xc ¼ 0:672ffiffiffiffiffiffiffiffiRh0

p(Eq 72)

With solution of the Reynolds equation(Eq 63), using any assumed inlet and outletboundary conditions, the pressure distributionis obtained, together with the inlet and outletdistances from the center of the contact, xiand xc. Gohar (Ref 55) provides a solutionthat, with the use of Eq 72 and with zero inletpressure for a fully flooded inlet for a slidingcontact, becomes:

p ¼Z0�Ur0ffiffiffiR

p

h3=20

8:1724

1þ x2

2Rh0

� �24

þ 1:5133xffiffiffiffiffiffiffiffiRh0

p tan�1 xffiffiffiffiffiffiffiffiffiffi2Rh0

p� �

þ 3:3617

#(Eq 73)

where r0 is the bore radius. To account for theinlet and outlet pressures, the result of Eq 73can be supplemented by the average hydrostaticpressure caused by these pressures:

pav ¼ Pi þ Po

2(Eq 74)

The sliding speed of the ring is DU. This is afunction of the engine stroke/compression ratioand rotational speed of the crankshaft and itshigher harmonics (Ref 56). For contributionsup to second engine order (first harmonic ofengine speed):

�U � orc sinotþ rclsin2ot

� �(Eq 75)

where l is theconnecting rodlength, rc is thecrank-pin radius, ando is the engine speed in rad/s.The outlet boundary conditions determine the

position of lubricant film rupture (start of thecavitation zone). The most commonly usedone is the Reynolds or Swift-Stieber boundarycondition. Now, the hydrodynamic load-carrying capacity can be obtained as:

Wh ¼ 2pr0

Z xc

�xi

pdx (Eq 76)

Using a fully flooded inlet and Swift-Stieberexit boundary conditions, the hydrodynamicreaction becomes (Ref 55):

Wh ¼ 15:394Z0r0RU

h0(Eq 77)

This should equate the applied contact forcedue to gas pressure acting behind the inner

Positiveoffset

No offset

Negative offset S(x)

x

Fig. 13 Parabolic ring contact face profile

Tribology of Power Train Systems / 927

rim of the ring and the ring tension force, bothof which strive to push the ring against the sur-face of the liner. For a ring of contact facewidth b, the ring tension force is:

Ft ¼ per0b (Eq 78)

where the elastic pressure, pe, is:

pe ¼ GEJ

3pba4(Eq 79)

where a is the ring thickness, G is the free endgap of the ring, and EJ is its flexural rigidity.Note that the ring is incomplete with a gap,which in its free state (unfitted) is usually 2 to8 mm (0.08 to 0.32 in.). When fitted in situ,the end gap is of the order of a few tenths ofa millimeter, which, under applied gas pressure,reduces to a few micrometers.There are various empirical and analytical

methods for determining the gas pressure load-ing of the compression ring. Assuming that100% of the gas pressure acts behind the fullyconformed ring to the surface of an idealizedright circular cylindrical liner, the gas pressureloading becomes:

Fg ¼ 2pr0bPG (Eq 80)

where PG is the combustion chamber pressure,which is measured using a pressure sensorinserted into the chamber.Thus, any load carried by the direct interac-

tion of surfaces can be obtained, without theneed to determine the film thickness and theStribeck’s oil film thickness parameter:

Wa ¼ Ft þ Fg �Wh (Eq 81)

Gohar (Ref 55) also shows that:(see Eq 82 at bottom of page)

Therefore, knowing the value of xc, viscousfriction can be obtained easily. For starved con-ditions, the lower limit, �b/2, can be replacedby xi from Eq 70.Film thickness can be calculated using Eq 64.

Then, the procedure in the sections “BoundaryFriction” and “Total Friction and Power Loss”in this article can be followed to find the bound-ary friction contribution, fb, and thus the totalgenerated friction.The rather simple procedure described here

yields quite acceptable predictions. Gore et al.(Ref 57) measured the friction of compressionring-cylinder liner contact under motorizedcondition for a high-performance single-cylin-der motocross motorbike engine, using a float-ing liner. The predictions of their analyticalmodel, quite similar to the one reported here,

agreed well with the measurements, within anerror of 5 to 10%.

Engine Bearings

Journal bearings are used in many areas ofengine and drive train systems. They are usedas connecting rod bearings, crankshaft andcamshaft support bearings, and in several otherapplications. Figure 14 shows a typical circularbore journal bearing. This is typical of crank-shaft support bearings. Connecting rod (big-end) bearings are not usually of circular config-uration. They mostly have an elliptical bore/bushing, commonly referred to as “lemon-shaped” bearings. In fact, bushing or shellshapes of higher-order out-of-roundness alsoexist. The reason for these out-of-round bush-ings can be appreciated by noting that theeccentric position of the journal center withinthe bushing, along the instantaneous line ofcenters (LOC), e, creates a pressurized wedge(region), shown by the film thickness, h, inFig. 14. Directly opposite to the high-pressureregion, a divergent gap emerges and thereforeis often unloaded with no coherent film of lubri-cant. This is the cavitation area, with a mix oflubricant, air, and vapor. The angle, j, ismeasured from the LOC. The minimum clear-ance occurs at j = 180 and the maximumclearance at j = 0. The film thickness aroundthe journal is:

h ¼ cþ ecosj (Eq 83)

Therefore, the minimum and maximum filmthickness values are:

hmin ¼ c� e and hmax ¼ cþ e (Eq 84)

where c is the nominal designed clearance and isusually a fraction of the journal radius (termed theclearance ratio), typically in the range of1

1000� c

R � 1100

, where R ¼ D2. The upper limit

corresponds to most commercial and mass-manufactured road vehicles, and the lower limitis that of high-performance racing vehicles,where higher pressures are generated in smallerclearances with higher applied loads.The clearance,c, is an important designparame-

ter,aswellas theeccentricity,e.Therefore, thefilmthickness can be expressed as h ¼ c 1þ ecosjð Þ,where the eccentricity ratio is:

e ¼ e

c(Eq 85)

The minimum film thickness becomeshmin ¼ cð1� eÞ, and when e ! 1, direct

contact of surfaces takes place. Therefore,determination of e in a design process is essen-tial. Conversely, when e ! 0, no high-pressurewedge would be formed. The loss of lubricantreaction means that under certain rotationalconditions the phenomenon known as journalwhirl would occur, leading to impact of thejournal with the bearing bushing.Referring to Fig. 14, if the journal was station-

ary with no applied load, it would fall under itsown weight, W, onto the bushing surface untilthe lubricant film reaction, Wz, equilibrates itsweight, with the LOC being vertical. The lubri-cant film would be sustained under pure squeezecondition, with the surfaces finally coming intocontact. With journal rotation, the LOC becomesinclined to the vertical, because any applied loadis supplemented by the effect of generated fric-tion torque. There are now two components oflubricant reaction:Wz along the LOC andWx per-pendicular to it, with W being their resultant,making an angle, c (attitude angle), with theLOC, which is also a function of e as:

c ¼ tan�1 p4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

p

e

!" #(Eq 86)

Therefore, it is important to determine e in anydesign or analysis. To do so in a generic man-ner, use should be made of a chart, known asthe Sommerfeld chart (Fig. 15).The Sommerfeld number is defined as:

� ¼ W

ZoRLc

R

� �2(Eq 87)

fv ¼

Z0�U

h0

2ffiffiffiffiffiffiffiffiffiffi2Rh0

ptan�1 xffiffiffiffiffiffiffiffiffiffi

2Rh0p� �

� 1:839ffiffiffiffiffiffiffiffiffiffi2Rh0

ptan�1 xffiffiffiffiffiffiffiffiffiffi

2Rh0p� �

�1:839 2h0Rð Þ x

2Rh0 þ x2ð Þ

26643775xc

�b=2

for: � b2� x < xc

Z0�U

h0

ffiffiffiffiffiffiffiffiffiffi2Rh0

ptan�1 xffiffiffiffiffiffiffiffiffiffi

2Rh0p� � b=2

xc

for: xc � x � b2

8>>>>>>><>>>>>>>:ðEq 82Þ

Fig. 14 Circular bore journal bearing. LOC, line ofcenter

928 / Friction and Wear of Machine Components

The Sommerfeld number is another impor-tant design parameter based on the intendedoperating conditions: load, W; clearance ratio;journal speed, o ¼ 2pN (N in rev/s); and lubri-cant dynamic viscosity, Z, which should beadjusted for temperature (see the section “Vari-ation of Lubricant Viscosity with Pressure andTemperature” in this article). L is the width ofthe bearing, sometimes referred to as its lengthin engine parlance, because it is along the axialdirection of the crankshaft.The various curves in Fig. 15 correspond to

the ratio L/D. When L/D � 0.5, the bearing istermed a short-width bearing, and long-widthbearing is when L/D � 2. Most crankshaftengine bearings fall between the two, and thosein the camshaft system are near enough toshort-width configuration. Thus, the Sommer-feld chart provided here suffices for the purposeof any engine bearing analysis. Therefore, forgiven operating conditions—load, speed, andtemperature—for a given bearing geometryand lubricant used, the Sommerfeld numbercan be calculated, and for a given L/D ratio,the eccentricity ratio is obtained using the chartin Fig. 15. Film thickness can then be obtainedusing Eq 85 as well as the attitude angle fromEq 86. Then, the following relationships pro-vide viscous friction, fv, friction torque, Tf,power loss, Pl, and minimum required flow rateto guard against starvation, Q, as (Ref 8):

fv ¼ ce2R

W sincþ 2pZoR2 L

c

� �1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1� e2p (Eq 88)

Tf ¼ fvR (Eq 89)

Pl ¼ Tfo (Eq 90)

Ql ¼ oRLce (Eq 91)

The negative signs in the aforementioned equa-tions correspond only to the direction of frictionon the opposing surfaces. A typical design eval-uation process would involve calculation of theSommerfeld number for a range of loads andengine speeds for a given bearing geometryand lubricant choice. In each case, the eccen-tricity ratio is obtained for various values ofthe Sommerfeld number. For good design, an

eccentricity ratio in the range of 0.65 � e �0.85 is sought. The film thickness is thencalculated using Eq 85, which ideally shouldbe l ¼ h

ss� 3 (Eq 11) to attain hydrodynamic

lubrication. If the film thickness falls belowthe stated condition, then some degree of sur-face interactions would occur, leading to theirwear. The section “Boundary Friction” in thisarticle deals with the general case of boundaryfriction. In the case of journal bearings, the sur-face roughness can either be regarded as isotro-pic/uniform, transversal (along the bearingcircumference), or longitudinal (along the bear-ing width). Then, the film thickness is expectedto comprise a nominal value, h (Eq 85), and astochastic contribution, hs, based on the surfaceroughness pattern. The Reynolds equation isthen solved numerically to include the effectof roughness (Ref 58). Solutions for ellipticbore big-end bearings are provided in Ref 58and 59. To mitigate direct contact of surfaces,and thus wear, it is important to enhance theload-carrying capacity of the contact. Oneapproach is to pump pressure into the contact,causing additional hydrostatic effect. Anotherapproach is to use bearings with elliptic orhigher-order bores (bushing), where instead ofone high-pressure zone, several high-pressurewedges are formed, where the film thicknessfor the usual elliptic bore connecting rod bear-ing is of the form:

h ¼ cð1þ Gcos2jþ ecosjÞ (Eq 92)

where G ¼ a�bc , where a and b are the elliptic

bore semimajor and semiminor half-widths,and 0 � G � 1 is the degree of noncircularity,with G ¼ 0 representing a circular bore bear-ing. The effect of G on improving film thick-ness can be seen in Fig. 16.Another approach is to use an overlay on the

bearing bushing that comprises low elasticmodulus layers (e.g., copper, Babbitt), whichwould locally elastically deform, thus creatingan additional gap for the lubricant to run intoas well as a thin, hard layer (e.g., indium, bis-muth) to resist wear when surfaces come into

contact (e.g., under start-up conditions).Theseissues are discussed in Ref 8 and 11.Now, returning to the issue of journal whirl,

the lubricant film acts like a mass-spring sys-tem. Its natural frequency is of the form(Ref 8):

o0 ¼ n

ffiffiffig

c

r(Eq 93)

where g is the gravitational acceleration(9.81 m/s2), and c is the bearing clearance.The constant n � 1.3 for 0.1 � e � 0.65 andn � 2 for e > 0.7. Clearly, when the speed ofcrankshaft rotation � ! o0; resonant condi-tions occur, known as synchronous whirl,where the center of the rotating journal withspeed O commences to rotate around the centerof the bushing at speed o0. This problem can beovercome by quickly speeding up or downthrough this condition. Another condition iswhen � � 2o0 for a horizontal bearing (suchas the crankshaft support bearings), whereuponthe lubricant entraining speed into the contactceases; thus, there is no lubricant reaction.The unloaded journal can spin and strike thebushing, causing catastrophic failure (Ref 8).To avoid this condition, a series of numericalanalyses are carried out to obtain a whirl stabil-ity chart, shown in Fig. 17. The ordinate in thefigure is the dimensionless stability factor g

c�2. It

can be seen that as the speed decreases, thevalue of the factor increases, affording betterstability for a given bearing clearance, c. Thearea under the curve includes all conditions thatlead to the diminution of lubricant entrainmentinto the convergent high-pressure zone, ascer-tained through numerical solution of the Rey-nolds equation. An increasing e corresponds toincreased loading, thus higher pressures andimproved load-carrying capacity. It is clear thatjournal bearings operate in a more stable man-ner at higher eccentricity ratios, e � 0:7; for abroader range of engine speeds, O (above andto the right of the demarcation line). However,as the eccentricity ratio increases, there is a ten-dency for thinner films, which can lead to directcontact of surfaces, causing wear, scuffing, andscoring due to excess heat generation. There-fore, for given operating conditions, the

1

Som

mer

feld

No.

(Δ)

Eccentricity ratio (ε)

102

10–1

10–2

10–3

10

1 0.5

0.25

0.125

0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1

Fig. 15 Sommerfeld chart

G=0.0

3

2.5

2

1.5

1

0.5

040 80 120 160 200 240 280 320 360

G=0.3G=0.5G=1.0

Circumferential location, deg

Film

thic

knes

s, μ

m

Fig. 16 Improved film thickness with elliptic borebearings. Source: Ref 58

STABLE

UNSTABLE

Increasing mass

0.2

0.1

0.06

Decreasing speed

0 0.2 0.6 0.8

ω0cΩ2

g

= 0.5Ω0.04

0.02

0.01

ε

Fig. 17 Journal bearing whirl stability chart. Source:Ref 8

Tribology of Power Train Systems / 929

eccentricity ratio can be obtained from theSommerfeld chart in Fig. 15 and used inFig. 17 to ascertain the state of system stability.

Cam-Tappet Contact

Unlike journal bearings (conforming) andring-liner contact (partially conforming), thecam-follower contact is nonconforming (coun-terformal) and subject to high loads and gener-ated pressures (see the section “ContactConfiguration” in this article). The regime oflubrication is mostly elastohydrodynamics (seethe section “Regimes of Lubrication”); thus,the section “Contact Fatigue” is directly appli-cable in this case. Furthermore, for all intentsand purposes, Hertzian conditions can be usedto obtain an estimate of generated pressures,contact footprint dimensions, and elastic deflec-tion of contacting surfaces in the medium-to-highly loaded parts of the cam-follower cycle(Table 1). For most cam-follower contacts, themedium-to-high contact loads occur betweenthe valve opening and closing points in cam tra-verse through the cam nose position. However,an elastohydrodynamic analysis is required todetermine the lubricant film thickness (Hertziantheory is for dry contacts), friction, and powerloss.Figure 18 shows a typical automotive poly-

nomial cam. It comprises a base circle with aslight clearance, so that the valve remainsclosed between the valve closing and its open-ing point, marked on the cam profile. The valvelift profile from the start of the flank to the camnose position is shown in the inset to the figure.The polynomial lift function, s, is of the form(Ref 8):

s ¼ smax þ C2y2 þ Cmy

m þ Cpyp þ Cqy

q þ Cryr;

y ¼ y=y

(Eq 94)

where smax is the maximum lift at the cam nose;y is the cam-lift angle, commencing at the start

of the flank; and y is the maximum cam-liftangle at the position of the cam nose. The expo-nents of y are the polynomial powers, shownfor three different polynomial cams in the keyto the inset in Fig. 18. The polynomial coeffi-cients are obtained as:(see Eq 95 at bottom of page)

where m, p, q, and r are the polynomial coeffi-cients in a polynomial of the form given inEq 94.The flank profile for valve closure in most

cases is quite similar to the opening one.However, this need not be the case. The profile

is essentially designed in line with enginethermodynamic requirements. Therefore, asfar as tribological assessment is concerned,the cam profile is a given, unless thepreliminary design, based on engine thermody-namics, results in excess friction and wearand thus poor reliability of the valve trainsystem.There are many types of valve train systems.

Here, the case of an overhead cam-tappet (a flatfollower) is considered. Figure 19 shows adirect overhead cam valve train system, withan automotive cam-flat tappet arrangement.The contact film thickness, h, is usually of

the order of a few tenths of a micrometer to afew micrometers (utmost), formed due to thelocalized elastic deformation of contacting sur-faces, d. The valve is subjected to lift, s, veloc-ity, v, and acceleration, a, as the cam rotates.The valve lift profile is so chosen to complywith the required engine breathing and, at thesame time, results in a continuous (nonjerky)acceleration profile. Figure 19 also shows thetypical variations of s, v, and a for a polynomialcam, where s is given by Eq 94 and 95 for thepolynomial cam. The valve velocity is:

v ¼ @s

@t¼ _s (Eq 96)

As the cam rotates with an angular velocity, o,there is an accelerative curvature change,termed geometrical acceleration:

jy ¼ @2s

@y2¼ d2s

dðotÞ2 ¼1

o2

d2s

dt2¼ 1

o2a (Eq 97)

Thus, valve acceleration becomes:

a ¼ o2jy (Eq 98)

C2 ¼ �smax � m � p � q � r= m� 2ð Þ p� 2ð Þ q� 2ð Þ r � 2ð Þ½ �Cm¼ �smax � 2 � p � q � r= 2� mð Þ p� mð Þ q� mð Þ r � mð Þ½ �Cp ¼ �smax � 2 � m � q � r= 2� pð Þ m� pð Þ q� pð Þ r � pð Þ½ �Cq ¼ �smax � 2 � m � p � r= 2� qð Þ m� qð Þ p� qð Þ r � qð Þ½ �Cr ¼ �smax � 2 � m � p � q= 2� rð Þ m� rð Þ p� rð Þ q� rð Þ½ �

8>>>><>>>>: ðEq 95Þ

Cam nose

Valveopening

Ramp

2–12–22–32–42

2–10–18–26–34

2–8–14–20–26

Clearance Valveclosing

Radius ofthe basecircle

10

8

6

4S (

mm

)

2

00 0.2

θ θˆ (.)

0.4 0.6 0.8 1

Fig. 18 Automotive polynomial cam

Valveclosing

s, v, a

[m]

Valve lift

Valve velocity

Valveopening

Valveclosing

amax smax

Valveacceleration

Cam

nos

e

Cam angle

[m/s]

[m/s2]

0

Valveopening

ms,ks

mv

R

h

s

ω

δ

Fig. 19 Direct overhead cam-tappet valve trainsystem

930 / Friction and Wear of Machine Components

The applied load is the result of two forces. Oneis the valve spring elastic force, Fe, which isthe result of any applied preload, Fp, and theinstantaneous spring force, Fs, caused by thevalve lift, s. Thus:

Fe ¼ Fp þ Fs ¼ ksðsþ spÞ (Eq 99)

where sp is any precompression of the valvespring, often carried out to guard against anyloss of contact that may occur with a surge ininertial force, Fi (Fig. 20). Therefore, there isideally always a net contact force, W.The inertial force is:

Fi ¼ mv þ 1

3ms

� �a (Eq 100)

where mv is the mass of the valve, and ms isthe mass of the valve spring. The valve springis an unequal-pitched nonlinear spring of non-negligible mass. It can be shown that a third ofits mass contributes to the valve dynamics, asstated in Eq 100. In reality, its stiffness altersas a function of its compression. Therefore,the approach here, using a constant stiffness,is idealized. A better representation of valvespring stiffness is shown in Ref 60. The varia-tions of the elastic force, Fe, and the inertialforce, Fi, are shown in Fig. 20. The net resultis the contact force, W:

W ¼ Fe � Fi (Eq 101)

Clearly, if Fi � Fe, there would be no con-tact force. This leads to the valve spring surgeeffect, loss of contact, and subsequent cam-tap-pet impact. In some cases, the valve springsurge can lead to spring coil clash as well. Tomitigate these damaging effects, a suitable pre-load is necessary, as shown in the figure. Alsonote that a is a function of engine speed, o(Eq 98), and because the inertial force signifi-cantly alters post valve opening and prior tovalve closure, the contact force changesaccordingly.With the contact force determined, the max-

imum Hertzian contact pressure can beobtained by using Table 1 for the case of elas-tic line contact footprint geometry in the caseof the cam-flat tappet arrangement. For cam-roller configuration, the contact footprint areais elliptical when using the relations in Table 1.To find the lubricant film thickness, the speedof entraining motion of the lubricant into thecontact should be determined first. The tappetspins as the cam rotates. This is to avoid arepetitive path of sliding contact, which tendsto scuff the tappet surface. Therefore, thecam is designed to approach the tappet with apredefined eccentricity that encourages thetappet to spin, thus reducing the chance ofscuffing (Ref 61). Ignoring the effect of tappetspin, the relative sliding speed of the cam sur-face with respect to an assumed stationary tap-pet becomes:

vc ¼ or (Eq 102)

where vc is the instantaneous cam surfacevelocity, and r is the effective contact radiusof the cam at the point of contact:

r ¼ Rþ sþ jy (Eq 103)

where R is the base circle radius. Therefore, foran assumed stationary tappet, the speed ofentraining motion of the lubricant into the con-tact becomes (average speed of the two contact-ing surfaces):

u ¼ 1

2o Rþ sþ a

o2

� �(Eq 104)

Now, an extrapolated oil film thickness equa-tion, such as those in the section “PredictingLubricant Film Thickness” in this article, forthe line contact geometry can be used to obtainthe lubricant film thickness (Ref 8, 25). With aknown film thickness, viscous and boundaryfriction contributions can be obtained throughthe procedures highlighted in the section “Fric-tion and Power Loss.”Figure 21 shows that at either side of the

cam nose, the speed of entraining motionreverses. This means that the inlet to the con-tact conjunction reverses in position. There-fore, there is a short period of time with zeroentrainment velocity, and, as a result, thelubricant film diminishes. Any quite thin filmof lubricant is sustained by pure squeeze-filmmotion or through entrapment in the roughnessof the contiguous surfaces. Therefore, there isa chance of wear in these positions. Figure 21shows the film thickness at high and low camrotational speeds, with little change in the inletreversal positions. The figure also shows thefilm thickness in a polar representation in acam cycle.

Transmission and DifferentialGearing Systems

Aside from the engine conjunctions, thedrive train system also contributes to frictionalpower loss. This occurs mainly in the gear paircontacts in the transmission and the differentialunit. There are also losses due to bearing sup-ports (usually ball, rolling-element, and taperroller bearings) of gear shafts, such as the trans-mission input and output shafts, the pinion shaftof the differential, and the output axle half-shafts of the differential ring gear. These sys-tems are fairly efficient compared with journalbearings and piston-cylinder conjunctions,because they are generally subjected to the elas-tohydrodynamic regime of lubrication, with theleast generated friction (Fig. 5). The transmis-sion efficiency for helical gears of the usualautomotive transmission systems and hypoidor bevel gears of the differential unit is usuallyin the range of 93 to 98% per meshing pair.This means that the frictional losses, being themain source of inefficiency, account for 2 to7% of the input power. Nevertheless, anyreduction of these would improve fuel effi-ciency and, by implication, the emissions fromthe power train system. With elastohydrody-namic conditions, the main areas of concern ingearing systems as well as their bearing sup-ports are contact fatigue with excessive loadsand poor noise and vibration performance withloss of contact and/or any preload. There are alarge number of gear pair configurations. Here,the emphasis is on cylindrical gears—spur andhelical—because these are most commonly

Force on the valveElastic force

Springpreload force

Valveopening

Inertia force

Contact force

deg

deg

N

N

0

0

Valveclosing

High ω Low ω

Fig. 20 Forces acting in a direct overhead valve trainsystem

Ent

rain

men

t mot

ion,

m/s

4

3

2

1

0

–1–135 135–90 90–45 450

Crank angle degree

Oil film thicknessInlet reversal

Degree0

Valveclosing

Valveopening

μm

ω

Fig. 21 Speed of lubricant entrainment and filmthickness in a cam cycle

Tribology of Power Train Systems / 931

used in transmission systems. For spur gears,readers can also refer to Ref 62 and 63; for heli-cal gears of transmission systems, analyticaland numerical solutions are provided by De laCruz et al. in Ref 64 and 65; and for hypoidgear pairs, analytical solutions are provided inRef 66 and 67. More comprehensive numericalsolutions are reported in Ref 68 and 69.For all cases of gearing systems, in an analyt-

ical approach, lubricant film thickness can beobtained using the equations in the section“Predicting Lubricant Film Thickness” in thisarticle. Note: for spur and helical gears, usethe line contact equation; for spiral gear teeth,use any elliptical point contact equation; andfor bevel and hypoid gears, use an ellipticalpoint contact equation with angled flow lubri-cant entrainment, such as Eq 35 (Ref 26). Foran analytical solution, viscous friction can beobtained using Eq 49, 50, and 52, because gearsgenerally operate under non-Newtonian viscousshear of the lubricant. Boundary friction contri-bution can be obtained by using the procedureset out in the section “Boundary Friction” inthis article, if the Stribeck oil film parameter,l < 3 (in the section “Regimes of Lubrica-tion”). To ascertain the potential for fatigue ofcontacting surfaces, use the approach used inthe section “Contact Fatigue.” A more compre-hensive approach is highlighted by Paouriset al. (Ref 70) for the case of hypoid gears,which is applicable to all gear pairs.

Inlet Starvation

Most gear teeth contacts are usually partiallystarved. However, a sufficient inlet meniscus oflubricant exists to form a thin elastohydrody-namic film. As noted in the section “Piston-Cyl-inder Conjunctions” in this article, some of theinward flow into the contact is subject to swirland counter (or reverse) flow (Fig. 22). Thus,only a portion of lubricant available at theentrance to the contact is actually entrained intoit. This means that many contacts, includingthose of gear teeth, are subject to starvationwith thinner films than that predicted, usuallyassuming a fully flooded or drowned inlet.Based on numerical analyses, Hamrock and

Dowson (Ref 71) provided a dimensionlessparameter, m*, which defines the inlet boundaryat the onset of starvation as:

m� ¼ 1þ 3:06Rzx

b

� �2 h0Rzx

" #0:58(Eq 105)

where Rzx is the equivalent radius in the princi-pal plane of contact zx, with x being the direc-tion of entraining motion. For a pair of gearteeth, Rzx is the instantaneous equivalent radiusof the contacting teeth pair during meshing(each approximated by a cylinder of a givenradius), and b is the half-width of the rectangu-lar strip for a narrow band line contact or thesemiminor half-width of an elliptical contactfootprint (see the section “Contact

Configuration” in this article). The actual inletparameter, m, is a function of the surface speedof the contacting surfaces, k ¼ u1

u2. Therefore, m

is the inlet parameter for the condition of zeroreverse flow, at an angle yi; to the centerlineof the contact. This was obtained throughpotential flow analysis by Tipei (Ref 50). Theinlet distance to the zero reverse boundary is:

m ¼ xib¼ Rzx sin yi

b(Eq 106)

where xi is the inlet distance, and

yi ¼ cos�1ð1� h0;sRzx

coshWi � 1ð ÞÞ (Eq 107)

The inlet position angle, yi as a function ofcoshWi, varies with the value of k. For pure roll-ing condition, k ¼ 1, and the value of coshWi ¼11:28 (Ref 50). For other k values, refer toRef 50. h0 is the central film thickness, forexample, using Eq 35. The relationship betweenm and m* is given in Ref 71 as:

h0;Sh0

� �¼ m� 1

m� � 1

� �0:29

(Eq 108)

This relationship can be used to obtain thelubricant film thickness with contact starvation,h0;S, which is reduced from h0, thus affectingboth viscous friction as well as friction contri-bution due to direct boundary interactions.The approach here is validated by a combinednumerical and experimental study using opticalinterferometry as well as measurement of con-tact pressure distribution for the case of circularpoint contacts (Ref 72) and applied to hypoidgear pairs in Ref 73.The variables ye and xe mark the outlet posi-

tion from the conjunction (Ref 50).

Cylindrical Spur Gears

To use the procedures highlighted previ-ously, the teeth-pair contact load, speed oflubricant entraining motion, and the contactsliding velocity are required at any instant oftime during the meshing cycle of a pair of teeth,

all of which are functions of system dynamicsas well as contact geometry (gear teeth geome-try). During a meshing period, one, two, orsometimes three teeth pairs are in contact.Figure 23 shows the length of a meshing cycle,ga, where X is any instant of contact from thestart of a meshing cycle. Two teeth pairs(a leading and an intermediate pair) are insimultaneous contact. As the leading pairbegins to depart, a trailing pair enters into con-tact. Therefore, in the demonstrated case here,there are instances of two simultaneous teethpairs in contact, and for a proportion of a mesh-ing cycle, represented in the figure by the basepitch, pe, a single teeth pair carries all the load.The instantaneous equivalent radius of curva-

ture of contact for a pair of cylindrical spurgears at any instant during the meshing cyclebecomes:

Rzx Xð Þ ¼ rb1tanjþ Xð Þ rb2tanj� Xð Þrb1 þ rb2ð Þtanj (Eq 109)

where rb1 and rb2 are the base circle radii of thegear pairs, and j is the pressure angle. Forgears other than spur gears, an analyticalexpression such as that in Eq 109 is not easilyobtained. In these cases, tooth contact analysisis usually carried out, providing both the geo-metrical information, such as the principal radiiof contact, as well as contact kinematics anddeformation information (Ref 74).The transmitted torque determines the teeth

pair contact load, W � Trp, where the torque, T,

is the input torque, resident on the transmissioninput shaft. When more than one pair of teeth isin contact, one can assume that this torque isequally shared between them.The speed of entraining motion of the lubri-

cant into the contact is obtained as:

U Xð Þ ¼ orp Xcos jð Þ 1

rb1� 1

rb2

� �þ 2sin jð Þ

� �(Eq 110)

The sliding velocity is obtained as:

�U Xð Þ ¼ orpXcos jð Þ 1

rb1� 1

rb2

� �(Eq 111)

θi θe

Rzx

Xi Xe h0

B

A

Fig. 22 Inlet reverse and swirl flows causing contactstarvation

Base pitch, pe

Double contact Double contactSingle contact

X

A BBase pitch, pe

Length of the meshing cycle, ga

D E

Fig. 23 Meshing cycle with simultaneous teeth-paircontacts

932 / Friction and Wear of Machine Components

Now the relationships for prediction of filmthickness, friction, and power loss can be usedfor all the teeth pairs in simultaneous mesh dur-ing a meshing cycle, 0 � X � ga, and summedup for each value of X. Figure 24 shows thetotal power loss for a meshing cycle. Theabscissa value of zero indicates the beginningof a meshing cycle, and the value of unity indi-cates the end of the meshing cycle. For a spurgear pair, the sliding velocity diminishes atthe pitch point contact, as shown in the figure.

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60

50

40

30

20

10

00 0.2 0.4 0.6 0.8 1

Tot

al p

ower

loss

, W

Position along the meshing cycle in one cycle

Fig. 24 Power loss for one meshing cycle

Tribology of Power Train Systems / 933

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