Engineering Tribology Episode 1 Part 9 pps

The assumed value must lie between the inlet temperature ‘T inlet’ and the maximum temperature ‘T max’ of the bearing material, i.e.: T inlet < T eff,s1 < T max where: T eff,s1 is the ef

· Conducted/Convected Heat Ratio

From the above equations the ratio of conducted to convected heat can be calculated todetermine which of the two mechanisms of heat removal from the bearing is the moresignificant Combining (4.124) and (4.125) the ratio is given by:

K/ρσ = χ is the thermal diffusivity of the fluid [m2/s]

Typical values of density, specific heat, thermal conductivity and thermal diffusivity areshown in Table 2.7, Chapter 2

EXAMPLE

Find the ratio of conducted to convected heat in a journal bearing of diameter D = 0.1[m] and length L = 0.157 [m] which operates at 3000 [rpm] The hydrodynamic filmthickness is h = 0.0001 [m] The bearing is lubricated by mineral oil of thermal diffusivity

is discussed in the chapter on ‘Elastohydrodynamic Lubrication’ The significance of the ratio

of conducted to convected heat for hydrodynamic bearings is that convection must beincluded in the equations of heat transfer in a hydrodynamic film This condition rendersthe numerical analysis of the heat transfer equations much more complicated than wouldotherwise be the case, as is demonstrated in the next chapter

Another ramification of the above result is that conductive heat transfer is still significantalthough it is often the smaller component of overall heat transfer Most hydrodynamicbearings operate under a condition between adiabatic and isothermal heat transfer Adiabaticheat transfer can be modelled by a perfectly insulating shaft and bush In this case, all the heat

is transferred by the lubricant as convection Isothermal heat transfer represents a bearingmade of perfectly conductive material which maximizes heat transfer by conduction in thelubricant The adiabatic model gives the lowest load capacity since the highest possiblelubricant temperatures are predicted The isothermal model conversely predicts themaximum attainable load capacity Combined solution of the two models provides validupper and lower limits of load capacity If a more accurate estimate of load capacity isrequired then it is necessary to estimate heat transfer coefficients to the surrounding bearingstructure This is a very complex task and is still under investigation [35] This topic isdiscussed further in the chapter on ‘Computational Hydrodynamics’

As is probably very clear here, exact analysis of thermal effects in bearings is a demandingtask and most designers of bearings have used the ‘effective viscosity’ methods even thoughthey are at least partly based on supposition

Isoviscous Thermal Analysis of Bearings

A simple method of estimating the loss in load capacity due to frictional heat dissipation is toadopt an isoviscous model It is assumed that the lubricant viscosity is lowered by frictionalheating to a uniform value over the whole film Viscosity may vary with time duringoperation of the bearing but its value remains uniform throughout the lubricating film An

‘effective temperature’ is introduced with a corresponding ‘effective viscosity’ which is used

to calculate load capacity Two methods are available to find the effective temperature andviscosity, an ‘iterative method’ which requires computation and a ‘constant flow method’which can be executed on a pocket calculator These are discussed below

· Iterative Method

The iterative method is effective and accurate in finding the value of effective viscosity Thestandard procedure is conducted in the following stages:

· An effective bearing temperature is initially assumed for the purposes of iteration

The assumed value must lie between the inlet temperature ‘T inlet’ and the

maximum temperature ‘T max’ of the bearing material, i.e.:

T inlet < T eff,s1 < T max

where:

T eff,s1 is the effective temperature at the start ‘s’ of iteration, first cycle [°C].

The maximum temperature is usually set by the manufacturer and for most

bearing materials ‘T max’ is about 120 [°C] For computing purposes, the initial value

for ‘T eff,s1’ is usually assumed as equal to the inlet temperature

· the corresponding effective viscosity ‘ηeff,s1 ’ is found from ‘T eff,s1’ using the ASTMviscosity chart or applying the appropriate viscosity-temperature law, e.g Vogelequation

· for a given film geometry and effective viscosity, bearing parameters such as

friction force ‘F’ and lubricant flow rate ‘Q’ can now be calculated.

· the values of ‘F’ and ‘Q’ are used to calculate the new effective temperature This

will be different from the previous value unless they happen to coincide Theeffective temperature is calculated from [4]:

where:

k is an empirical constant with a value of 0.8 giving good agreement

between theory and experiment [4];

∆T is the frictionally induced temperature rise dependent on ‘F’ and ‘Q’

[°C]

The frictionally induced temperature rise is found from the following argument.The heat generated in the bearing is:

H = FU

At equilibrium, the heat generated by friction balances the heat removed byconvection assuming an adiabatic bearing, thus:

A new effective temperature is then calculated from (4.128)

· for the new effective temperature ‘T eff,n1 ’ (‘n’ denotes new) a corresponding

effective viscosity is then found from, for example, the ASTM chart or Vogelequation

· if the difference between the new effective viscosity and the former effectiveviscosity is less than a prescribed limit then the iteration is terminated If thedifference in viscosities is still too large the new viscosity value ‘ηeff,s2’ is assumedand the procedure is continued until the required convergence is achieved Arelaxation factor is usually incorporated at this stage (see program listed inAppendix) to prevent unstable iteration

The iteration procedure is summarized in a flow-chart shown in Figure 4.52 and a computer

program ‘SIMPLE’ written in Matlab to perform this analysis for narrow journal bearings is

listed in the Appendix

End

Find new effective viscosity ηeff,n1

No Has viscosity converged?

ηeff,s1= ηeff,n1

Yes

Calculate new effective temperature T eff,n1

Calculate friction force F and oil flow Q

Define bearing geometry h = f(x)

Find effective starting viscosity ηeff,s1

Start

Assume effective starting temperature T eff,s1

Assume new effective

starting viscosity

ηeff,sn (n = 1, 2, 3 )

FIGURE 4.52 Flow chart for the iterative method in isoviscous analysis

· Constant Flow Method

The constant flow method is simpler than the formal iterative method and does not require

a computer In journal bearings, lubricant flow remains approximately constant between

eccentricity ratios 0.6 < ε < 0.95 as can be seen in Figure 4.36 It was found experimentally that

in this range the lubricant flow can be approximated by the formula [3,4]:

into (4.129) which in turn produces a new value of the effective temperature ‘T eff,n1’ A newcorresponding viscosity ‘ηeff,n1’ is then found and (4.129) re-applied This procedure isrepeated until satisfactory convergence is obtained

Non-Isoviscous Thermal Analysis of Bearings With Locally Varying Viscosity

The assumption that lubricant viscosity is uniform across the film is in fact erroneous andinevitably causes inaccuracy This error can be quite large especially when the lubricantviscosity has been severely reduced by frictional heating Lubricant viscosity varies withposition in the film, both in the plane of sliding and normal to the direction of sliding Anexample of computed film temperatures inside a journal bearing operating at an eccentricity

ratio of 0.8 and L/D = 1 is shown in Figure 4.53 The figure shows a temperature field through

a radial section parallel to the load line and an axial section through the midplane of thebearing The temperature distribution was calculated for a bearing speed of 2,000 [rpm],lubricant inlet temperature of 33°C and ambient temperature of 23°C [36]

It can be seen from Figure 4.53 that the hottest part of the lubricant film is at the centre of thebearing close to the minimum film thickness The temperature at this location is 50°C which

is 27°C higher than ambient temperature and causes a large viscosity loss in most lubricatingoils On the other hand, the shaft tends to a uniform temperature with angular positionbecause it is rotating at a high speed Bearing temperatures, even at the coolest point, arehigher than the lubricant inlet temperature Frictional heat accumulates in the bearing andthere is a large temperature rise from the initial level to ensure sufficient dissipation of heat

by convection or conduction through the external bearing structure The uniformity of shaft

temperature ensures a temperature difference across the lubricant film since the temperature

of the bush varies with angular position Both the temperatures of the shaft and bush vary inthe axial direction The temperature characteristic of a journal bearing is clearly non-uniformand all ‘effective viscosity’ methods are, at best, approximations to a complex problem.The solution to the problem of calculating the pressure field and load capacity with variableviscosity involves the simultaneous solution of a variable viscosity form of the Reynoldsequation and a heat transfer equation for the lubricant film This is clearly beyond the scope

of analytical solution and numerical methods are almost exclusively employed The solutionmethod which is generally referred to as ‘thermohydrodynamics’ is discussed in the nextchapter Thermal effects render invalid many of the predictions of classical ‘isoviscoushydrodynamics’, e.g that the load capacity is proportional to surface speed, and constitute theprime reason why the viscosity index of an oil is such an important property for themaintenance of viscosity at high localized operating temperatures, as was discussed inChapter 2

film Bush

Attitude angleβ

Multiple Regression in Bearing Analysis

The multiple regression method is very useful in finding the correlation between variables,and also in expressing one variable, selected as a dependent variable, in terms of all the othervariables which are independent variables Any variable present in a particular process can

be selected as a dependent variable and expressed in terms of the other variables Some form

of approximating function is assumed and polynomials or exponential functions are themost frequently used Correlation coefficients between the variables and coefficients of

approximation are usually computed The technique is used for analysing experimental dataand has also been applied to bearing analysis [39] Theoretical data from several hundredbearings was analysed by multiple regression resulting in a set of equations which directlyprovide information about the design and performance parameters of journal bearings Theequations are given in an exponential form, i.e.:

z = Cv 1 a v 2 b v 3c v n m

where:

z is the dependent variable;

C is the calculated regression constant;

v 1 , v 2 , , v n are the independent variables, n =1,2,3, ;

a , b , , m are the calculated exponents

The dependent and independent variables together with calculated constants and exponentsare shown in Table 4.5 [39]

For example, if the load capacity is required to be calculated for a specific journal bearing,then the following equation from Table 4.5, row 1, can be used:

W = 2.7861 × 101υ37.8 °C −1.1υ93.3 °C 2.46 L 2.515 D 0.563 N 0.528 c −1.09 T S −0.383( )ε

1 − ε2 1.385

or row 7:

W = 1.7575 × 104υ37.8 °C −0.793υ93.3 °C 2.033 L 2.596 D 1.042 N 0.884 c −1.51 T mean −2.02( )ε

1 − ε2 1.108

TABLE 4.5 Multiple regression relationships between journal bearing design and

performance parameters [39] (Note that the temperature is in degreesFahrenheit [°F], i.e T°F = 1.8 × T°C + 32)

-0.162

- -2.020 -1.470

One of the equations is expressed in terms of lubricant supply temperature and the other interms of mean lubricant temperature Both, however, should give a similar result Theresults predicted by these equations are acceptable and can be used in quick engineeringanalysis.

Bearing Inlet Temperature and Thermal Interaction Between Pads of a Michell Bearing

Frictional heat not only affects the load capacity of a bearing by directly influencing theviscosity but also leads to thermal interaction between adjacent bearings For example,Michell pads are used in a combination of several pads distributed around a circle to form alarger thrust bearing According to isoviscous theory a minimum of space should be leftbetween the pads to maximize load bearing area However, experimental measurementshave revealed an optimum pad coverage fraction for maximum load capacity [37] A certainamount of space is required between the pads for the hot lubricant discharged from one pad

to be replaced by cool lubricant before entrainment in the following pad In practice, thereplacement of lubricant is never perfect and a phenomenon known as ‘hot oil carry over’ isalmost inevitable This phenomenon is illustrated schematically in Figure 4.54

It was found from boundary layer theory that the lubricant inlet temperature can becalculated from [38]:

where:

T s is the lubricant supply temperature [°C];

m is the hot oil carry over coefficient

FIGURE 4.54 ‘Hot oil carry over’ in a multiple pad bearing

The hot oil carry over coefficient is a function of sliding speed and space between adjacent

pads For a small gap width of 5 [mm] between pads, ‘m’ has a value of 0.8 at 20 [m/s] and 0.7

at 40 [m/s] For a large gap width of 50 [mm] between pads, ‘m’ has a value of 0.55 at 20 [m/s] and 0.5 at 40 [m/s] The minimum value of hot oil carry over coefficient occurs at approximately 40 [m/s] with a sharp rise in ‘m’ beyond this speed [38].

The effect of ‘hot oil carry over’ in a multipad thrust bearing was found to be sufficientlystrong to ensure that individual pad temperatures were reduced when the number of padswas reduced from 8 to 3 [37] This reduction in temperature occurred in spite of the greater

concentration of frictional power dissipation per pad at constant load and speed Even whenthe ratio of combined pad area to area swept by the pads was lowered to less than 35% thebearing still functioned efficiently It is therefore unnecessary to fit a large number of closelyspaced pads in a high-speed thrust bearing because this merely allows hot lubricant torecirculate almost indefinitely The number of pads can be reduced for the same load capacitythus achieving considerable economies in the manufacture of the bearing This is illustrated

in Figure 4.55

It can be seen that bearing temperature is reduced especially at high loads when four pads areused instead of eight The reduction in bearing temperature coincides with improved loadcapacity Removal of pads can raise load capacity under certain conditions, as shown inFigure 4.55, and this is the result of a diminished loss of lubricant operating viscosity

50 60 70 80 90

a)

0 12

0b)

Maximum pad temperature

3 000 [rpm]

5 000 [rpm]

FIGURE 4.55 Effect of pad number on the performance of a thrust bearing, a) bearing

temperature, b) power loss (adapted from [37] and [64])

As has been implied throughout this chapter, hydrodynamic lubrication is only effectivewhen an appreciable sliding velocity exists A sliding velocity of 1 [m/s] is typical of manybearings As the sliding velocity is reduced the film thickness also declines to maintain thepressure field This process is very effective as pressure magnitudes are proportional to thesquare of the reciprocal of film thickness Eventually though the film thickness will havediminished to such a level that the small high points or asperities on each surface will comeinto contact Contact between asperities causes wear and elevated friction This condition,where the hydrodynamic film still supports most of the load but cannot prevent somecontact between the opposing surfaces, is known as ‘partial hydrodynamic lubrication’ When

the sliding speed is reduced still further the hydrodynamic lubrication fails completely andsolid contact occurs A lubricant may still, however, influence the coefficient of friction andwear rate to some degree, as is discussed in subsequent chapters Original research into thelimits of hydrodynamic lubrication was performed early in the 20th century by Stribeck [40]and Gumbel [41] The limits of hydrodynamic lubrication are summarized in a graph shown

in Figure 4.56

When the friction measurements from a journal bearing were plotted on a graph against acontrolling parameter defined as ‘ηU/W’ it was found that, for all but very small sliding

speeds, friction ‘µ’ was proportional to the above parameter which is known as the ‘Stribeck

number’ When a critically low value of this parameter was reached, the friction rose fromvalues of about 0.01 to much higher levels of 0.1 or more The rapid change in the coefficient

of friction represents the termination of hydrodynamic lubrication Later work revealed thathydrodynamic lubrication persists until the largest asperities are separated by only a fewnanometres of fluid It was found that a minimum film thickness of more than twice thecombined roughness of the opposing surfaces ensures full hydrodynamic lubrication ofperfectly flat surfaces [3] With the level of surface roughness attainable today on machinedsurfaces, a minimum film thickness of a few micrometres could thus be acceptable In fact forsmall bearings, e.g a journal bearing of 80 [mm] diameter, it is possible to use twice thecombined roughness as a minimum limit for film thickness On the other hand for largehydrodynamic bearings, larger clearances are usually selected because of the great difficulty inensuring that such a small minimum film thickness is maintained over the entire bearingsurface Even if the bearing surfaces are machined accurately, elastic or thermal deflectionwould almost certainly cause contact between the bearing surfaces If contact between slidingsurfaces occurs then, particularly at high speeds of e.g 10 [m/s], the dramatic increase infrictional power dissipation can cause overheating of the lubricant and possibly seizure of thebearing Most hydrodynamic bearings, particularly the larger bearings, are designed to operate

at film thicknesses well above the estimated transition point between fully hydrodynamiclubrication and wearing contact because:

· the transition loads and speeds are difficult to specify accurately,

· bearing failure is almost inevitable if hydrodynamic lubrication is allowed to faileven momentarily

Increasing coefficient of friction caused

by partial contact between

shaft and bush

Rise in friction due to high eccentricity

at limit of hydrodynamic lubrication

Friction coefficient determined by hydrodynamic theory



ηU

W

µ

Zero friction coefficient at zero sliding speed according to Petroff theory

FIGURE 4.56 Schematic diagram of changes in friction coefficient at the limits of

hydrodynamic lubrication

4.8 HYDRODYNAMIC LUBRICATION WITH NON-NEWTONIAN FLUIDS

Most fluids in use as lubricants either have a rheology that cannot be described as Newtonian

or are modified by additives to cause deviations from Newtonian behaviour All fluids have

a non-zero density and therefore the hydrodynamic equations should ideally include theeffect of inertia and at high bearing speeds, turbulent flow can also occur Some of these floweffects are deleterious to bearing performance but others can be beneficial The Reynoldsequation presented so far does not include the characteristics of complex fluids and therefinement of hydrodynamic lubrication theory to model complex fluids is a subject of manycurrent research projects Some of the problems associated with complex fluids inhydrodynamics are outlined in this section

Turbulence and Hydrodynamic Lubrication

In most bearings operating at moderate speeds, laminar flow of lubricant prevails but at highspeed, the lubricant flow becomes turbulent and this affects the load capacity and particularlythe friction coefficient of the bearing Turbulent flow in a hydrodynamic bearing can bemodelled by introducing ‘turbulence coefficients’ into the Reynolds equation An example of

a modified Reynolds equation used in the analysis of Michell pad bearings is in the form [24]:

R is the radius of a bearing [m];

h is the film thickness [m];

U is the surface velocity [m/s];

L is the length of the bearing [m];

ρ is the oil density [kg/m3];

η is the oil viscosity [Pas]

The solution to this equation is usually obtained by computation since the ‘turbulencecoefficients’ are functions of film thickness At the high speeds where turbulence occurs,heating effects are quite significant and must be incorporated in the solution

It was found that the onset of turbulence in the bearing results in a slight increase in loadcapacity and marginal effect on stiffness coefficients (i.e the variation in hydrodynamic loadwith film thickness change), which can slightly alter the limiting speed before bearing

vibration occurs (i.e vibration stability threshold) The main effect of turbulence is a large

increase in bearing friction The friction coefficient in a turbulent bearing is about 80% largerthan if laminar flow prevails [24,42] To suppress turbulent flow additives in the form of oilsoluble macromolecules, e.g polymethylmethacrylate, are used, but they are eventuallydegraded by prolonged shearing [42].

Hydrodynamic Lubrication With Non-Newtonian Lubricants

Common examples of non-Newtonian lubricants are the multigrade oils with polymerviscosity index improvers described in Chapter 3 There are two forms of deviation fromideal Newtonian rheology which have been studied in terms of the effect on hydrodynamiclubrication The most common features of lubricating oil rheology are ‘shear-thinning’ and

‘viscoelasticity’

When incorporating non-Newtonian lubricant behaviour into hydrodynamics it is necessary

to derive a controlling equation from a basic rheological formula relating shear stress toshear rate The introduction of special coefficients into Reynolds equation to allow for thiseffect, in a similar manner to that conducted when considering turbulent flow, is inaccurate

as an analytical method [43] This is because of the variation of the effective viscosity withshear rate which depends on the position in the film A detailed description of the analysisand examples of the shear-thinning effect on standard lubricating oils is provided in [43] Ashear-thinning non-Newtonian oil provides only a reduced load capacity as compared to aNewtonian oil with the same apparent viscosity at near zero shear rate An example of theeffect of shear-thinning on bearing load capacity is shown in Figure 4.57

It can be seen from Figure 4.57 that the effect of shear-thinning is most pronounced at higheccentricity ratios where shear rates are greatest For example the load capacity of theNewtonian lubricant at an eccentricity ratio of 0.6 is only equalled by the shear-thinninglubricant at an eccentricity ratio of about 0.7 and there is an even greater difference at highereccentricities This means that a Newtonian lubricant may provide a sharper increase in loadcapacity with eccentricity ratio than non-Newtonian lubricants At extremes of eccentricity,i.e high loads, the reduced load capacity compared to an equivalent Newtonian fluid may becritical to bearing survival

FIGURE 4.57 Comparison of load capacity versus eccentricity ratio for a shear-thinning

lubricating oil and an equivalent Newtonian lubricating oil [43]

In contrast, viscoelasticity, which is the combination of elasticity and viscosity in a fluid,generally has a positive effect on load capacity It has often been reported that the addition ofviscoelastic additives to lubricating oils raises the load capacity of journal bearings andreduces any wear that might occur A review of the work published on the effects ofviscoelastic lubricants in hydrodynamics is summarized in [44] The strongest effect is in theload capacity improvement at high eccentricity ratios which tends to prevent bearing failureunder extremes of load [45] Viscoelastic lubricants can also raise the threshold of vibrationinstability in hydrodynamic bearings with stiff shafts [44] The substitution of a viscoelasticlubricant may be a simple solution to bearing vibration.

Inertial Effects in Hydrodynamics

Fluid inertia, like turbulence, becomes significant at high bearing speeds and alters manybearing performance characteristics It is found that although cavitation is suppressed by fluidinertia [46-50] in journal bearings, the effect on load capacity is quite small for higheccentricity ratios, ε > 0.55 [50] Inertia effects become significant when the reduced Reynolds

number is greater than unity, i.e Re* > 1 The reduced Reynolds number is defined as:

Re* = (ρUh 0 /η)(h 0 /R) = ρ Uh 0

2

where:

h 0 is the minimum film thickness [m];

R is the radius of the bearing [m]

For the analysis of bearings with significant inertia effects a modified form of the Reynoldsequation is used The one-dimensional Reynolds equation which includes an inertia term is

H is a parameter allowing for inertia effects [49]

‘H(x)’ is defined entirely in terms of higher powers of ‘h’, ‘dh/dx’, ‘∂p/∂x’ and ‘∂ 2 p/ ∂x 2’obtained from the standard Reynolds equation which ignores inertia effects The full form of

the expression for ‘H(x)’ can be found in [49].

The effect of inertia in high speed turbulent journal bearings is to raise the vibration stabilitythreshold until it is almost identical to that predicted by laminar theory [51]

The influence of inertia is fundamental to the operation of ‘squeeze-film dampers’ Asqueeze film damper is a journal bearing which is fitted around a rolling contact bearing.These systems are used in aircraft engines where rolling bearings prevent bearing failure inthe event of interruption to the lubricant supply The journal bearing or ‘squeeze filmdamper’ is used to provide damping of the shaft vibrations which are completely unaffected(not suppressed) by the presence of the rolling bearings A schematic diagram of a squeezefilm damper is shown in Figure 4.58

At the high speeds which are typical of the operation of squeeze dampers, the journal bearingcreates higher friction than the rolling bearing so that it does not usually rotate There arealways, however, imbalance forces and these form a rotating load vector with a high angularvelocity The rolling bearing housing is therefore prevented from making contact with the