Mechanical Engineering-Tribology In Machine Design Episode 11 pot

Friction, lubrication and wear in higher kinematic pairs 237 Energy dissipated due t o plastic defbrmations In the majority of cases, resistance to rolling is dominated by plastic defo

Friction, lubrication and wear in higher kinematic pairs 237

Energy dissipated due t o plastic defbrmations

In the majority of cases, resistance to rolling is dominated by plastic deformation of one or both contacting bodies In this case the energy is dissipated within the solids, at a depth corresponding to the maximum shear component of the contact stresses, rather than at the interface With materials having poor thermal conductivity the release of energy beneath the surface can lead to high internal temperatures and failure by thermal stress Generally metals behave differently than non-metals The inelastic properties of metals, and to some extent hard crystalline non-metallic solids, are governed by the movement of dislocations which, at normal temperatures, is not significantly influenced either by temperature or by the rate of deformation

The rolling friction characteristics of a material which has an elastic range of stress, followed by rate-independent plastic flow above a sharply defined yield stress, follow a typical pattern At low loads the deformation is predominantly elastic and the rolling resistance is given by the elastic hysteresis 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., the condition of full plasticity is reached, the rolling resistance may be estimated by the rigid-plastic theory The onset of full plasticity cannot be precisely defined but, from the knowledge of the static indentation

behaviour, where full plasticity is reached when W 1 2 a z 2 6 and

E a / Y R z 100, it follows that G W / k R z 300, where k is the yield stress in

shear of the solid

Energy dissipated due t o surface roughness

It is quite obvious that resistance to the rolling of a wheel is greqter on a rough surface than on a smooth one, but this aspect of the subject has received little analytical attention The surface irregularities influence the rolling friction in two ways First, they intensify the real contact pressure so that some local plastic deformation will occur even if the bulk stress level is within the elastic limit If the mating surface is hard and smooth the asperities will be deformed plastically on the first traversal but their deformation will become progressively more elastic with repeated traver- sals A decreasing rolling resistance with repeated rolling contact has been observed experimentally The second way in which roughness influences resistance 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 therefore unsteady Measurements of the resistance force show very large, high- frequency fluctuations Energy is dissipated in the rapid succession of small impacts between the surface irregularities Because the dissipation is by impact, the resistance due to this cause increases with the rolling speed

238 Tribology in machine design

cylinders of engineering surfaces in sliding contact Even surfaces in nominal rolling

contact, such as ball-bearings, normally experience some micro-slip, which necessitates lubrication if surface damage and wear are to be avoided A lubricating fluid acts in two ways First, it provides a thin adsorbed film t o the solid surfaces, preventing the adhesion which would otherwise take place and reducing friction through an interfacial layer of low shear strength This is the action known as boundary lubrication The film is generally very thin and its behaviour is very dependent upon the physical and chemical properties of both the lubricant and the solid surfaces The lubricant may act in a quite different way A relatively thick coherent film is drawn in between the surfaces and sufficient pressure is developed in the film to support the normal load without solid contact This action is known

as hydrodynamic lubrication It depends only upon the geometry of the contact and the viscous flow properties of the fluid The way in which a load-carrying film is generated between two cylinders in rolling and sliding contact is described in this section The theory can be applied to the lubrication of gear teeth, for example, which experience a relative motion

which, as shown in Section 6.2, is instantaneously equivalent to the

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

two solid surfaces moving with velocities V 1 and V 2 is shown in Fig 6.3

With thin, nearly parallel films, velocity components perpendicular to the film are negligible so that the pressure is uniform across the thickness At a

x low Reynolds number, for the case of a thin film and a viscous fluid, the inertia forces are negligible Then, for two-dimensional steady flow,

Figure 6.3 equilibrium of the fluid element gives

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

be integrated with respect to z Putting v = V 2 and V , at z = O 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 dpldx is zero

Friction, lubrication and wear in higher kinematic pairs 239 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 to the books on lubrication listed at the end of Chapter 5

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

Case ( i ) - Rigid cylinders The geometry of two rotating rigid cylinders in contact is schematically shown in Fig 6.4 An ample supply of lubricant is provided on the entry side Within the region of interest the thickness ofthe film can be expressed

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

240 Tribology in machine design

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

at that point When this condition, together with p = O at x = - oo is

imposed on eqn (6.21) it is found that 5 , =0.443, whence x , =0.475(2Rh0)* The pressure distribution is shown by the solid line curve in Fig 6.4 In this

case 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 ho T o secure effective lubrication, ho must be greater than the surface irregularities It is seen from eqn (6.23) that the load carrying capacity of the film is generated by a rolling action expressed by ( V , + V , ) 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

Case ( i i ) - Elastic cylinders

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

where u,, and u,, are the normal elastic displacements of the two surfaces

and 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 hmin In all cases h m i n z 0 8 h l The lubrication process in which elastic deformation of the solid surface plays a significant role is known as elastohydrodynamic lubrication

Case (iii) - Variable viscosity of the lubricant

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

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

Friction, lubrication and wear in higher kinematic pairs 24 1

surprising that an increase in the viscosity with pressure is also a significant factor in elastohydrodynamic lubrication When sliding is a prevailing motion in the contact, frictional heating causes a rise in the temperature in the film which reduces the viscosity of the film However, for reasons which will be explained later, it is possible to separate the effects of pressure and temperature

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

where po is the viscosity at ambient pressure and temperature and a is a constant pressure coefficient of viscosity This is a reasonable description of the observed variation in the viscosity of most lubricants Substituting this

relationship into the Reynolds eqn (6.18) gives

This modified Reynolds equation for the hydrodynamic pressure in the field

must be solved simultaneously with eqn (6.24) for the effect of elastic

deformation on the film shape The solution to this problem can be obtained numerically There are a number of changes in the contact behaviour introduced by the pressure-viscosity effect Over an appreciable fraction of the contact area the film is approximately parallel This results

from eqn (6.26) When the exponent a p exceeds unity, the left-hand side

becomes small, hence h - h , becomes small, i.e h z h , =constant The corresponding pressure distribution is basically that of Hertz for dry contact, but a sharp pressure peak occurs on the exit side, followed by a rapid drop in pressure and thinning of the film where the viscosity falls back

to its ambient value po The characteristic features of highly loaded elastohydrodynamic contacts, that is a roughly parallel film with a constriction at the exit and a pressure distribution which approximates to Hertz but has a sharp peak near the exit, are now well established and supported by experiments It is sufficiently accurate to assume that the minimum film thickness is about 75 per cent of the thickness in the parallel section The important practical problem is to decide under what conditions it is permissible to neglect elastic deformation and/or variable viscosity Some guidance in this matter can be obtained by examining the values of the two non-dimensional parameters, the viscosity parameter g,

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

242 Tribology in machine design

influence on the film thickness The highly viscous fluid passes through the parallel zone until the pressure and the viscosity get back to normal at the exit This means a decrease in thickness of the film The inlet and exit regions are effectively independent They meet at the end of the parallel zone with a discontinuityin the slope ofthe 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 the region and can be predicted by the formula proposed by Dowson and Higginson for isothermal conditions

where G =aE is the dimensionless material parameter, V = [ p o

(V, + V 2 ) ] / 2 E R is the dimensionless speed parameter, W = w / E R L is the

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

Friction, lubrication and wear in higher kinematic pairs 243

on the piezo-viscous relation p =poeZP and reflects the change of viscosity with pressure

po is the lubricant viscosity at inlet surface temperature, V1, V2 are the surface velocities relative to contact region

where the plus sign assumes external contact (both surfaces convex) and the minus sign denotes internal contact (the surface with the larger radius of curvature is concave), w is the total load on the contact and L is the length of the contact

The viscosity of the lubricant at the temperature of the surface of the solid

in the contact inlet region is the effective viscosity for determining the film thickness This temperature may be considerably higher than the lubricant supply temperature and therefore the inlet viscosity may be substantially lower than anticipated, when based on the supply temperature Usually the inlet surface temperature is an unknown quantity in design analyses The solution to this problem is to use the lubricant system outlet temperature or

an average of the inlet and the outlet temperatures to obtain an estimate of the film thickness It should be pointed out that the predicted film thickness may be too large when the system supply temperature is used and the flow

of lubricant is not sufficient to keep the parts close to the lubricant inlet temperature Input data characterizing the lubricant, that is its viscosity, pressure-viscosity coefficient and temperature-viscosity coefficient are usually available from catalogues of lubricant manufacturers

The best way to illustrate the practical application ofeqn (6.27) is to solve

a numerical problem Two steel rollers of equal radius R1 = R2 = 100 mm and length L = 100 mm form an external contact Using the following input data estimate the thickness of the lubricating film

244 Tribology in machine design

Equivalent Young modulus

of the lubricant in the inlet zone and the second is the conduction of thermal energy accumulated in the bulk of the contacting solids to the inlet zone This second mechanism is probably only important in pure sliding where the conduction can occur through the stationary solid The heating at the inlet zone is significant only at high surface velocities and can be subjected

to certain simplified analysis Under conditions of high surface velocity or high lubricant viscosity the effect of inlet heating due to shear on effective viscosity ought t o be considered The engineering approach to this problem

is to use a thermal reduction factor, TI, which can be multiplied by the isothermal film thickness, ho, to give a better estimate of the actual film thickness

The thermal correction factor is a weak function of load and material parameters As a first approximation, the following expression may be used

to determine the thermal correction factor for line contacts

In eqn (6.28) the thermal loading factor, Tl, is defined as

Friction, lubrication and wear in higher kinematic pairs 245

6 is the lubricant viscosity-temperature coefficient, [=(l/p)/Ap/AT);

"C- '1 and p is the lubricant thermal conductivity

The viscosity-temperature coefficient, 6, can be adequately estimated from a temperature viscosity chart of the lubricant It is necessary to select two temperatures about 20" apart near the assumed inlet temperature and divide the viscosity difference by the average viscosity and temperature difference corresponding to the viscosity difference

Thermal conductivity, p, is relatively constant for classes of lubricants based on chemical composition For mineral oils, suitable values for these calculations are 0.124.15 W/mK The lower range applies for lower viscosity either resulting from lower molecular weight or higher tempera- ture To illustrate the procedure outlined above the numerical example solved earlier is used Thus, assuming that the lubricant is SAElO mineral oil, inlet temperature is 55 "C, corresponding 6 value is 0.045 "C-' and lubricant thermal conductivity is 0.12 W/m K, eqn (6.29) gives

Then, using eqn (6.28)

T , =0.857 -0.0234(2.32) +0.000168(2.32)2 =0.8

Finally, the thermally corrected film thickness is

his' means a 20 per cent reduction of the film thickness as a result of the heating at the inlet zone

In contrast with the heavily loaded line contacts which have been investigated very fully, the understanding of point contact lubrication is less advanced Any analysis of the problem naturally relies to a considerable extent on a knowledge of the local shape ofthe contact, which usually is not known in detail The foundation of the theoretical solution to the problem was laid down by Grubin He proposed that:

(i) the pressure distribution under lubricated conditions was almost Hertzian;

(ii) the shape of the entry gap was determined by the Hertz pressure alone, i.e the fluid pressure at the entry to the contact zone had negligible effect

The application of the Grubin approximation is quite simple in the case of line contacts The point contact case is far more complex due to side leakage effects Figure 6.6 shows the geometry of the point contact The various formulae for load, peak pressure, contact radius and surface deformation can be easily found in any standard textbook on elasticity (see Chapter 3 for more details) The relationships needed here are as follows:

246 Tribology in machine design

(i) the applied load PI/ producing a circular contact of radius a:

where p,,, is the peak pressure at the centre;

(ii) the contact radius a is found from the relationship

where R is the equivalent radius defined by 1 / R = l / R l + 1/R2 and E is

the equivalent materials modulus

where vdenotes the Poisson ratio and the subscripts 1 and 2 refer to the two materials in contact;

(iii) the gap outside the main contact regon at any radial distance r is (see Fig 6.6)

The expression in square brackets can be approximated over the region

a < r < 2 a by

By denoting that E = (1 1.43 W)/4aEho, eqn (6.32) becomes

By writing a non-dimensional version of the Reynolds equation in polar coordinates, and considering the quadrant of the annulus ADCBA, only the final differential equation can be derived Obviously, the equation does not have an exact analytical solution and it is usually solved numerically This has been done by many workers and the solution in a simplified form is as follows

where po is the oil viscosity at atmospheric pressure, cc is the pressure- viscosity coefficient and V is the surface velocity Equation (6.34) has been verified experimentally and it is now clear that despite an unfavourable geometric configuration, an elastohydrodynamic lubrication film exists at the nominal point contacts over a very wide range of conditions

6.10 Cam-follower The schematic geometry of a cam-flat-follower nose is shown in Fig 6.7

system The film parameter 3, for a cam-follower system can be calculated by the

Friction, lubrication and wear in higher kinematic pairs 247

In general, the value of ;l in cam systems is well below one In this regime, elastohydrodynamic lubrication is not very effective and we must rely heavily on surface film or boundary lubrication to protect the surfaces against scuffing and excessive wear

References to Chapter 6 1 F F Ling Surface Mechanics New York: Wiley, 1973

2 A W Crook The lubrication of rollers, I Phil Trans., A250 (1958), 18-34

3 A W Crook The lubrication of rollers, I1 and 111 Phil Trans., A254 (1961),

141 -50

4 A W Crook The lubrication of rollers, IV Phil Trans., A255 (1963), 261 -9

5 B Jacobson On the lubrication of heavily loaded cylindrical surfaces consider- ing surface deformations and solidification of the lubricant Trans ASME, J Lub Technol., 95 (1973), 176-84

6 D Dowson and G R Higginson Elastohydrodynamic Lubrication New York: Pergamon Press, 1966

7.1 Introduction

7 Rolling-contact bearings

In contrast with hydrodynamically lubricated journal bearings, which, for their low friction characteristics depend on a fluid film between the journal and the bearing surfaces, rolling-contact bearings employ a number of balls and rollers that roll, nominally, in an annular space T o some extent, these rolling elements help t o avoid gross sliding and the high coefficient of friction that is associated with sliding The mechanism of rolling friction is, therefore, discussed first followed by the review of the factors affecting the frictional losses

As a matter offact, the contact between the rolling elements and the races

or rings consists more of sliding than of actual rolling The condition of no interfacial slip is seldom maintained because of material elasticity and geometric factors It is natural then that the contact stress and the kinematics of the rolling-element bearing are presented in some detail in order t o stress their importance in the service life of this type ofbearing The advantages and disadvantages of rolling-contact bearings when they are compared with hydrodynamic bearings are well known and shall not be discussed here Instead, more attention is given t o the lubrication techniques and the function of the lubricant in bearing operation Finally, vibration and acoustic emission in rolling-element bearings are discussed as they are inherently associated with the running of the bearing Some methods t o combat the excessive noise emission are also suggested

7.2 Analysis of friction The resistance t o relative motion in rolling-contact bearings is due to many

in rolling-contact factors, the basic one being rolling friction This was long assumed t o be the

bearings only resistance to motion in this type of bearing It was established,

however, that the contribution of rolling friction is small though its effect on wear and tear and operating temperature is important These factors are especially important for miniature instrument rolling-contact bearings operating in the very accurate mechanisms of servo-systems, magnetic recorder mechanisms and other precision parts of instruments

Previously, rolling friction was treated as a mechanical process, i.e the interaction of rough surfaces of absolutely rigid bodies In 1876 Reynolds put forward the hypothesis according to which the frictional force due t o the rolling motion of a perfectly elastic body along a perfectly elastic substrate was a result of relative slip between the contacting surfaces resulting directly from their deformations Figure 7.1 provides a graphic