Tribology in Machine Design 2009 Part 9 ppsx

The energy lost per unit time, per unit area of contact, should bethe product of the unit load, coefficient of friction and sliding velocity.. Finally, contact problems in rolling elemen

This fabric may then be bonded to a steel backing Such bearings are limited

by their adhesives Though they are relatively insensitive to high pressures,glues may give way to environmental chemicals or high temperatures -such

as those generated by high sliding speeds

B-3 Bonded plastic-based layer Thermoplastic tapes, thermosettingphenolic or polyamides filled with PTFE are bonded to a steel strip Theplastic layer may be moulded around wires or bondable cloth to facilitatewelding or glueing to the backing

B-4 Unbonded liners Cylinders of moulded nylon (filled or unfilled),acetal or reinforced PTFE are easily installed and replaced in metal sleeves.They generally cannot take as much load or speed as bonded liners, thoughsetting a reinforcing fabric into the polymer helps to improve the situation

C Homogeneous non-metallic composites

C-l Unfilled base resins are usually nylons, acetals, polyethylene cially high density), polyamides and PTFE Each of these has its ownspecial advantages, though they can only support relatively low loads.C-2 Single lubricant fillers These are made of nylons, acetals, poly-ethylenes, polyimides, PTFE, phenolics and polyphenylene sulfides withlubricant fillers - MoS2, PTFE or graphite Additions of silicone or oil arenot very popular

(espe-C-3 Single reinforcing fillers, typically fibreglass, in proportion from 10

to 30 per cent, can increase compressive strength, cohesion and temperatureresistance

C-4 Multiple fillers Various combinations of the materials alreadymentioned are used, plus bronze powder, metal oxides and sometimescarbon fibres

C-5 Fabric-and-filler composites are usually compression moulded fromphenolic resins filled with PTFE or MoS2 onto an open-weave reinforcingfabric

D Filament wound

Manufacturers can make sleeves of glass or other fibre, using techniquesdeveloped for the fabrication of pressure vessels The sleeves are then lined

in the same way as the metal sleeves

D-l Fibre-lined A strand of bondable material is twisted together with astrand of lubricant polymer The resulting thread is wound on a mandreland encapsulated in epoxy

D-2 Bonded fabric is of the same construction as that described in B-2.D-3 Bonded tape is described in B-3

5.11.2 Design considerations

There are a number of situations where self-lubricating and pre-lubricatedbearings of some kind should be considered by a designer The need toreduce maintenance or to increase reliability is frequently encountered in

practice Inaccessible bearings on all types of equipment are readycandidates for self-lubrication, as are pieces of equipment in remote places.

By replacing metal bearings with self-lubricating ones, substantial savingscould be made Similarly, self-lubrication can improve the service life ofequipment bound to be neglected, as for instance, consumer appliances.Seldom used but critical bearings are also prominent candidates; thepivots on an elevator emergency brake might remain motionless for years,but if called upon, the joint must move easily There is no reason to preventthe designer from using a self-lubricating bearing as a hydrodynamicallylubricated one There is little point in this, however, if the bearing will beproperly and continuously lubricated, but if there is a chance that the oilflow could stop, a self-lubricating bearing could prevent serious damageand the need for protracted shut-down and repair

A significant improvement in bearing performance may often beobtained by conventional liquid lubrication, and like any well-madejournal bearing, the oil lubricated self-lubricating bearing should lastalmost forever There are, however, a number of subtle interfacialphenomena which are sometimes noted and some of which are deleterious

to good operation A particular type of problem arises when the fluidmigrates to a significant depth into the matrix of the polymer and causes apremature failure This problem of premature failure is especially acute atintense levels of energy dissipation within the contact area

Another reason for selecting self-lubricating bearings is the necessity tocope with hostile environments Self-lubricating bearings retain their load-carrying capacity at high temperatures They can operate where rolling-element bearings fail due to fatigue, and where conventional lubricantsoxidize rapidly Furthermore, many self-lubricating polymers resist corro-sion very well

An important issue related to the operation of machines is the protection

of the environment from contamination Sliding bearings do not make asmuch noise as rolling-contact bearings, and the plastic liners can act asdampers absorbing some vibration energy At the same time, many self-lubricating bearings are completely oil free, so that they cannot con-taminate their surroundings with a hydrocarbon mist - a point especiallyimportant to designers of medical equipment, food processing equipmentand business machines However, it should be pointed out that some self-lubricating materials, like the various lead-filled polymers, may emitcontaminants of their own It is known that fatigue limits the service life ofrolling-contact bearings, while wear constitutes the main limitation to thelife of self-lubricating bearings So it is not surprising that dry bearingsshould perform much better in applications that defeat rolling-elementbearings Oscillating motions of the order of a few degrees, for example,greatly accelerate needle bearing fatigue The rolling elements do notcirculate in and out of the load zone but instead, a single roller or a couple ofrollers will rock in and out of the zone always under load Under theseconditions rolling elements undergo accelerated fatigue and fail quickly.Oscillating motions pose even bigger problems for hydrodynamic bearings;

there is seldom enough time for an oil film to form as the shaft starts, stops,reverses itself and starts again Some metal-metal contact is inevitable Forthat matter, any equipment that is stopped and started frequently - even ifthe rotation is unidirectional will have problems with metallic contact atlow operating speeds The solution to this problem is offered by a self-lubricating bearing and actually there are a number of situations whereself-lubricating bearings are run with additional external lubrication Theself-lubrication is there for start-up, shut-down, and emergencies Lubri-cation also increases a bearing's load-carrying capacity Self-lubricatingbearings do not stick on start-up For instance, a bronze bushing and babbitshave a start-up coefficient of friction of around 0.3 while a PTFE basedbearing is only 0.05.

Traditionally, the performance of unlubricated bearings is measured in

terms of PV, the product of the bearing's unit loading and the relative

sliding velocity of the bearing and the mating surface The dimensions of

PV— (N/m2) x (m/s) are the dimensions of the energy flux This is to beexpected as the energy lost to friction and subsequently dissipated as heatshould be proportional to the frictional force multiplied by the slidingdistance The energy lost per unit time, per unit area of contact, should bethe product of the unit load, coefficient of friction and sliding velocity As

such, PV should give a good indication of the heat produced in a bearing If the bearing dissipates heat at a constant rate, then PV should be the

measure of the pressure and sliding velocity that the bearing can tolerate.The only complication is that the coefficient of friction is not constant butchanges with speed and the contact pressure Therefore it is justifiable to

take the PV values with some reserve.

The most important problem for the designer intending to utilize a lubricating bearing is to estimate its service life Unfortunately there is nouniversally accepted, comprehensive design and service life formulae, butinstead each manufacturer of self-lubricating bearings has its own andusually different method of projecting wear life This situation is partlyjustified by the fact that all calculation methods are based exclusively onexperimental results For general design purposes, ESDU Item No 76029 -'A guide on the design and selection of dry rubbing bearings' can berecommended Also, there are a number of so-called 'designers' handbooks'produced by manufacturers giving detailed information on the selectionand wear-life projection of self-lubricating bearings

self-References to Chapter 5 1 D F Wilcock and E R Booser Bearing Design and Application New York:

McGraw-Hill, 1957.

2 P R Trumpler Design of Film Bearings New York: The Macmillan Co., 1966.

3 F T Harwell Bearing Systems, Principles and Practice Oxford: Oxford

6 G B DuBois and F W Ocvirk The short bearing approximation for plain

journal bearings Trans ASME, 77 (1955), 1173-8.

7 W Gross Gas Film Lubrication New York: Wiley, 1962.

8 J Campbell, P P Love, F A Martin and S O Rafique Bearings forreciprocating machinery A review of the present state of theoretical, experi-

mental and service knowledge Proc Instn Mech Engrs, 182 (3A) (1967), 14-21.

6 Friction, lubrication and wear in higher kinematic pairs

6.1 Introduction It is well known in the theory of machines that if the normals to three points

of restraint of any plane figure have a common point of intersection, motion

is reduced to turning about that point For a simple turning pair in whichthe profile is circular, the common point of interaction is fixed relatively toeither element, and continuous turning is possible A pair of elements inwhich the centre of turning changes its position at the completion of anindefinitely small rotation, i.e the new position is again the common point

of intersection of the normals at three new points of restraint For this to bepossible the profiles will, in general, have differing geometric forms, and arethen referred to as a higher pair of elements Again, since the elements donot cover each other completely as in lower pairing and are assumed to becylindrical surfaces represented by the profiles, contact will occur along aline or lines instead of over a surface Relative motion of the elements maynow be a combination of both sliding and rolling

In higher pairing, friction may be a necessary counterpart of the closingforce as in the case of two friction wheels in contact Here the force on thewheels not only holds the cylinders in contact but must be sufficient toprevent relative sliding between the circular elements if closure is to becomplete In certain cases it is essential that force closure of higher pairsshall do more than maintain contact of the functional surfaces Forexample, the ball-bearing functions as a lower pair or as an incompletehigher pair of elements, it is, however, usually regarded as being a higherpair

This chapter is designed to provide familiarization and perspective toreaders planning to pursue in more detail any of the various topics covered

by the collective name of higher kinematic pairs There are two pervadingobjectives:

(i) to develop an understanding of the basic concepts of concentratedcontacts;

(ii) to develop a facility with the analytical techniques for predicting andassessing the behaviour of concentrated contacts which are typical forhigher kinematic pairs

The information contained in this chapter can be used to solve a number ofproblems common for all higher kinematic pairs First, problems as-sociated with contact between two nonconforming surfaces are discussed.They include the force transmitted at a point of contact, surface tractions,

elastic hysteresis during rolling, rolling friction, and the lubrication ofrollers Next, film thickness under isothermal elastohydrodynamic con-ditions, inlet viscous heating, regimes of line contact lubrication arepresented Finally, contact problems in rolling element bearings, gears, andcam-follower systems are reviewed and equations to evaluate requiredminimum film thickness are discussed.

6.2 Loads acting on In this section loads acting on a contact area and the way they are contact area transmitted from one surface to another shall be considered The load on

the contact can be resolved into a normal force P acting along the common

normal and a tangential force T opposed by friction The relationship between W and T is given by

where / is the coefficient of limiting friction T can be resolved into

components Tx and Ty parallel to axes x and y In a purely sliding contact the

tangential force reaches its limiting value in a direction opposed to thesliding velocity The force transmitted at a normal point of contact has theeffect of compressing solids so that they make contact over an area of finitesize As a result it becomes possible for the contact to transmit a resultantmoment in addition to a force This is schematically shown in Fig 6.1 The

components of this moment M x and M y are called rolling moments andoppose a rolling motion but are small enough to be neglected The thirdcomponent Mz, acting about the common normal, arises from frictionwithin the contact area and is referred to as the spin moment When spinaccompanies rolling, the energy dissipated by the spin moment is combinedwith that dissipated by the rolling moments to make up the overall rollingresistance

Free rolling is defined as a rolling motion in which spin is absent and

where the tangential force T at the contact point is zero This is the

condition of the unpowered and unbraked wheels of a vehicle if the rollingresistance and the friction in the bearings are neglected It is in markedcontrast with the driving wheels or the braked wheels which transmitsizeable tangential forces at their points of contact with the road or rail

The forces and moments discussed above are transmitted across the contactinterface by surface tractions at the interface The normal traction(pressure) is denoted here by w and the tangential traction (due to friction)

by t, shown acting on the lower surface in Fig 6.1 For overall equilibrium

approximately in the x-y plane Therefore

and

When the bodies have closely conforming curved surfaces, as for example in

a deep-groove ball-bearing, the contact area is warped appreciably out of

the tangent plane and the expressions for M x and My, eqn (6.4), have to be

modified to include terms involving the shear tractions t x and t y

6.4 Hysteresis losses Some energy is always dissipated during a cycle of loading and unloading

even within the so-called elastic limit This is because no solid is perfectlyelastic The energy loss is usually expressed as a fraction a of the maximumelastic strain energy stored in the solid during the cycle where a is referred to

as the hysteresis loss factor For most metals, stressed within the elasticlimit, the value of a is very small, less than 1 per cent, but for polymers andrubber it may be much larger

Figure 6.2

In free rolling, the material of the bodies in contact undergoes a cycle ofloading and unloading as it flows through the region of contact deform-ation (Fig 6.2) The strain energy of material elements increases up to thecentre-plane due to the work of compression done by the contact pressureacting on the front half of the contact area After the centre-plane the strainenergy decreases and work is done against the contact pressures at the back

of the contact Neglecting any interfacial friction the strain energy of thematerial arriving at the centre-plane in time dt can be found from the workdone by the pressure on the leading half of the contact For a cylindricalcontact of unit width

where CD = V/R is the angular velocity of the roller Taking p(x) to be given

by the Hertz theory

where Wis the contact load If a small fraction a of this strain energy is now

assumed to be dissipated by hysteresis, the resultant moment required tomaintain the motion is given by equating the net work done to the energydissipated, then

where /r is defined as the coefficient of the rolling resistance Thus theresistance to rolling of bodies of imperfectly elastic materials can beexpressed in terms of their hysteresis loss factor This simple theory ofrolling friction is due to Tabor Using the same calculation for an ellipticalcontact area given the result

where a is the half-width of the contact ellipse in the direction of rolling For

There are basically two problems with this simple theory First, thehysteresis loss factor a is not usually a material constant In the case of

metals it increases with strain (a/R), particularly as the elastic limit of the

material is approached Second, the hysteresis loss factor in rolling cannot

be identified with the loss factor in a simple tension or compression cycle.The deformation cycle in the rolling contact, illustrated in Fig 6.2, involvesrotation of the principal axes of strain between points 2, 3 and 4, with verylittle change in total strain energy The hysteresis loss in such circumstancescannot be predicted from uniaxial stress data

The same deformation cycle in the surface would be produced by a rigidsphere rolling on an inelastic deformable plane surface as by a frictionlesssphere sliding along the surface In spite of the absence of interfacial frictionthe sliding sphere would be opposed by a resistance to motion due tohysteresis in the deformable body This resistance has been termed thedeformation component of friction Its value is the same as the rolling

resistance F r given by eqn (6.9)

6.5 Rolling friction Rolling motion is quite common in higher kinematic pairs Ideally it should

not cause much power loss, but in reality energy is dissipated in variousways giving rise to rolling friction The various sources of energy dissipation

in rolling may be classified into:

(i) those which arise through micro-slip and friction at the contactinterface;

(ii) those which are due to the inelastic properties of the material;(iii) those due to the roughness of the rolling surfaces

Free rolling has been defined as a motion in the absence of a resultant

tangential force Resistance to rolling is then manifested by a couple M y

which is demanded by the asymmetry of the pressure distribution, that is, byhigher pressures on the front half of the contact than on the rear The

trailing wheels of a vehicle, however, rotate in bearings assumed to be

frictionless and the rolling resistance is overcome by a tangential force T x

applied at the bearing and resisted at the contact interface Provided that

the rolling resistance is small (T x <^ W) these two situations are the same

within the usual approximations of small strain contact stress theory, i.e to

first order in (a/R) It is then convenient to write the rolling resistance as a non-dimensional coefficient f r expressed in terms of the rate of energydissipation P, thus

The quantity P/V is the energy dissipated per unit distance travelled Energy dissipated due to micro-slip

Energy dissipation due to micro-slip occurs at the interface when the rollingbodies have dissimilar elastic contacts The resistance from this causedepends upon the difference of the elastic constants expressed by theparameter /? (defined by eqn (6.11)) and the coefficient of sliding friction/

The resistance to rolling reaches a maximum value of

when fi/fx 5 Since, for typical combinations of materials, /? rarely exceeds

0.2, the rolling resistance due to micro-slip is extremely small It has beensuggested that micro-slip will also arise if the curvatures of two bodies aredifferent It is quite easy to see that the difference in strain between two such

surfaces will be second-order in (a/R) and hence negligible in any small

strain analysis A special case is when a ball rolls in a closely conforminggroove The maximum rolling resistance is given by

The shape of the contact ellipse (b/a) is a function of the conformity of the

ball and the groove; where the conformity is close, as in a deep groove

ball-bearing, b $> a and the rolling resistance from this cause becomes significant.

In tractive rolling, when large forces and moments are transmitted

between the bodies, it is meaningless to express rolling resistance as T x or

M y /R Nevertheless, energy is still dissipated in micro-slip and, for

comparison with free rolling, it is useful to define the effective rolling

resistance coefficient f r = P/VW This gives a measure of the loss of

efficiency of a tractive drive such as a belt, a driving wheel or a continuouslyvariable speed gear

Energy dissipated due to plastic deformations

In the majority of cases, resistance to rolling is dominated by plasticdeformation of one or both contacting bodies In this case the energy isdissipated within the solids, at a depth corresponding to the maximumshear component of the contact stresses, rather than at the interface Withmaterials having poor thermal conductivity the release of energy beneaththe surface can lead to high internal temperatures and failure by thermalstress Generally metals behave differently than non-metals The inelasticproperties of metals, and to some extent hard crystalline non-metallicsolids, are governed by the movement of dislocations which, at normaltemperatures, is not significantly influenced either by temperature or by therate of deformation

The rolling friction characteristics of a material which has an elasticrange of stress, followed by rate-independent plastic flow above a sharplydefined yield stress, follow a typical pattern At low loads the deformation ispredominantly elastic and the rolling resistance is given by the elastichysteresis equation (6.8) The hysteresis loss factor as found by experiment

is generally of the order of a few per cent

At high loads, when the plastic zone is no longer contained, i.e., thecondition of full plasticity is reached, the rolling resistance may beestimated by the rigid-plastic theory The onset of full plasticity cannot beprecisely defined but, from the knowledge of the static indentation

behaviour, where full plasticity is reached when W/2a&2.6 and Ea/YRx 100, it follows that GW/kR&3QQ, where k is the yield stress in

shear of the solid

Energy dissipated due to surface roughness

It is quite obvious that resistance to the rolling of a wheel is greater on arough surface than on a smooth one, but this aspect of the subject hasreceived little analytical attention The surface irregularities influence therolling friction in two ways First, they intensify the real contact pressure sothat some local plastic deformation will occur even if the bulk stress level iswithin the elastic limit If the mating surface is hard and smooth theasperities will be deformed plastically on the first traversal but theirdeformation will become progressively more elastic with repeated traver-sals A decreasing rolling resistance with repeated rolling contact has beenobserved experimentally The second way in which roughness influencesresistance is through the energy expended in climbing up the irregularities

It is significant with hard rough surfaces at light loads The centre-of-mass

of the roller moves up and down in its forward motion which is thereforeunsteady Measurements of the resistance force show very large, high-frequency fluctuations Energy is dissipated in the rapid succession of smallimpacts between the surface irregularities Because the dissipation is byimpact, the resistance due to this cause increases with the rolling speed

6.6 Lubrication of It is generally necessary to use a lubricant to ensure satisfactory operation cylinders of engineering surfaces in sliding contact Even surfaces in nominal rolling

contact, such as ball-bearings, normally experience some micro-slip, whichnecessitates lubrication if surface damage and wear are to be avoided Alubricating fluid acts in two ways First, it provides a thin adsorbed film tothe solid surfaces, preventing the adhesion which would otherwise takeplace and reducing friction through an interfacial layer of low shearstrength This is the action known as boundary lubrication The film isgenerally very thin and its behaviour is very dependent upon the physicaland chemical properties of both the lubricant and the solid surfaces Thelubricant may act in a quite different way A relatively thick coherent film isdrawn in between the surfaces and sufficient pressure is developed in thefilm to support the normal load without solid contact This action is known

as hydrodynamic lubrication It depends only upon the geometry of thecontact and the viscous flow properties of the fluid The way in which aload-carrying film is generated between two cylinders in rolling and slidingcontact is described in this section The theory can be applied to the

Figure 6.3

lubrication of gear teeth, for example, which experience a relative motionwhich, as shown in Section 6.2, is instantaneously equivalent to thecombined rolling and sliding contact of two cylinders

A thin film of an incompressible lubricating fluid, viscosity //, between

two solid surfaces moving with velocities V\ and V 2 is shown in Fig 6.3.With thin, nearly parallel films, velocity components perpendicular to thefilm are negligible so that the pressure is uniform across the thickness At alow Reynolds number, for the case of a thin film and a viscous fluid, theinertia forces are negligible Then, for two-dimensional steady flow,equilibrium of the fluid element gives

where v is the stream velocity Since dp/dx is independent of z, eqn (6.14) can

be integrated with respect to z Putting v = V 2 and V v at z =0 and h, gives a

parabolic velocity profile, as shown in Fig 6.3, expressed by

The volume flow rate Q across any section of the film is

For continuity of flow, Q is the same for all cross-sections, i.e.

where h^ is the film thickness at which the pressure gradient dp/dx is zero.

Eliminating Q gives

This is Reynolds equation for a steady two-dimensional flow in a thin

lubricating film Given the variation in thickness of the film h(x), it can be integrated to give pressure p(x) developed by hydrodynamic action For a

more complete discussion of the Reynolds equation the reader is referred tothe books on lubrication listed at the end of Chapter 5

Now, eqn (6.18) will be used to find the pressure developed in a filmbetween two rotating cylinders

Figure 6.4

The geometry of two rotating rigid cylinders in contact is schematicallyshown in Fig 6.4 An ample supply of lubricant is provided on the entryside Within the region of interest the thickness of the film can be expressedby

where l/R = 1/Rt + l/R 2 and h is the thickness at x=0 Substituting eqn

(6.19) into (6.18) gives

By making the substitution c=tan l [x/(2Rh)*] eqn (6.20) can be

in-tegrated to give an expression for the pressure distribution

where ^=tan i [xi/(2Rh 0 )*'] and xt is the value of x where h = h v and

dp/dx=0 The values of ^ and A are found from the end conditions.

At the start it is assumed that the pressure is zero at distant points at entryand exit, i.e p = 0 a t x = ± o o The resulting pressure distribution is shown

by the dotted line in Fig 6.4 It is positive in the converging zone at entry

and equally negative in the diverging zone at exit The total force W

supported by the film is clearly zero in this case However this solution isunrealistic since a region of large negative pressure cannot exist in normalambient conditions In practice the flow at the exit breaks down intostreamers separated by fingers of air penetrating from the rear The pressure

is approximately ambient in this region The precise point of filmbreakdown is determined by consideration of the three-dimensional flow in

the streamers and is influenced by surface tension forces However it hasbeen found that it can be located with reasonable accuracy by imposing thecondition

at that point When this condition, together with p=0 at x= — oo isimposed on eqn (6.21) it is found that ^ =0.443, whence xx =0.475(2/?/i0)*.The pressure distribution is shown by the solid line curve in Fig 6.4 In thiscase the total load supported by the film is given by

In most practical situations it is the load which is specified Then, eqn (6.23)

can be used to calculate the minimum film thickness h 0 To secure effective

lubrication, h 0 must be greater than the surface irregularities It is seen fromeqn (6.23) that the load carrying capacity of the film is generated by a rolling

action expressed by (V^ + V 2 ) If the cylinders rotate at the same peripheral

velocity in opposite directions, then (V^ + V 2 ] is zero, and no pressure is

developed in the film

Cose (ii) - Elastic cylinders

Under all engineering loads the cylinders deform elastically in the pressurezone so that the expression for the film profile becomes

where u zl and u z2 are the normal elastic displacements of the two surfacesand are given by the Hertz theory Thus

This equation and the Reynolds eqn (6.18) constitute a pair of simultaneous

equations for the film shape h(x) and the pressure p(x) They can be combined into a single integral equation for h(x) which can be solved

numerically The film shape obtained in that way is then substituted into

the Reynolds equation to find the pressure distribution p(x).

An important parameter from the point of view of the designer is the

minimum film thickness h min In all cases h min &Q.8hi The lubrication

process in which elastic deformation of the solid surface plays a significantrole is known as elastohydrodynamic lubrication

Case (Hi) - Variable viscosity of the lubricant

It is well known that the viscosity of most practical lubricants is verysensitive to changes in pressure and temperature In contacts characteristic

of higher kinematic pairs, the pressures tend to be high so that it is not

surprising that an increase in the viscosity with pressure is also a significantfactor in elastohydrodynamic lubrication When sliding is a prevailingmotion in the contact, frictional heating causes a rise in the temperature inthe film which reduces the viscosity of the film However, for reasons whichwill be explained later, it is possible to separate the effects of pressure andtemperature.

Let us consider an isothermal film in which variation in the viscosity withpressure is given by the equation

where /i0 is the viscosity at ambient pressure and temperature and a is aconstant pressure coefficient of viscosity This is a reasonable description ofthe observed variation in the viscosity of most lubricants Substituting thisrelationship into the Reynolds eqn (6.18) gives

This modified Reynolds equation for the hydrodynamic pressure in the fieldmust be solved simultaneously with eqn (6.24) for the effect of elasticdeformation on the film shape The solution to this problem can beobtained numerically There are a number of changes in the contactbehaviour introduced by the pressure-viscosity effect Over an appreciablefraction of the contact area the film is approximately parallel This resultsfrom eqn (6.26) When the exponent ap exceeds unity, the left-hand side

becomes small, hence h — h^ becomes small, i.e hxh t = constant Thecorresponding pressure distribution is basically that of Hertz for drycontact, but a sharp pressure peak occurs on the exit side, followed by arapid drop in pressure and thinning of the film where the viscosity falls back

to its ambient value /^0- The characteristic features of highly loadedelastohydrodynamic contacts, that is a roughly parallel film with aconstriction at the exit and a pressure distribution which approximates toHertz but has a sharp peak near the exit, are now well established andsupported by experiments It is sufficiently accurate to assume that theminimum film thickness is about 75 per cent of the thickness in the parallelsection The important practical problem is to decide under whatconditions it is permissible to neglect elastic deformation and/or variableviscosity Some guidance in this matter can be obtained by examining thevalues of the two non-dimensional parameters, the viscosity parameter 0V

and the elasticity parameter g e which are presented and discussed inChapter 2, Section 2.12.1 The mechanism of elastohydrodynamic lubri-cation with a pressure dependent lubricant is now clear The pressuredevelops by hydrodynamic action in the entry region with a simultaneousvery large increase in the viscosity The film thickness at the end of theconverging zone is limited by the necessity of maintaining a finite pressure.This requirement virtually determines the film thickness in terms of thespeed, roller radii and the viscous properties of the lubricant Increasing theload increases the elastic deformation of the rollers with only a minor

influence on the film thickness The highly viscous fluid passes through theparallel zone until the pressure and the viscosity get back to normal at theexit This means a decrease in thickness of the film The inlet and exitregions are effectively independent They meet at the end of the parallelzone with a discontinuity in the slope of the surface which is associated with

a sharp peak in the pressure

6.7 Analysis of line In this section line contact lubrication is presented in a way which can be contact lubrication directly utilized by the designer The geometry of a typical line contact is

shown in Fig 6.5 The minimum film thickness occurs at the exit of theregion and can be predicted by the formula proposed by Dowson andHigginson for isothermal conditions

where G = a£ is the dimensionless material parameter, K=[/i0

(V, + V 2 )~\/2ER is the dimensionless speed parameter, W=w/ERL is the

dimensionless load parameter, a is the pressure-viscosity coefficient based

Figure 6.5