Handbook of Lubrication Episode 1 Part 7 pptx

Film pressure measurements by means of a vapor-deposited manganin strip confirmed analytical predictions on effect of load on location of the pressure spike.18,24,25 Film Thickness Chart

The exit construction was first seen experimentally in the circumferential film profile measured by the X-ray technique.22This was followed by optical interferometry.14,22While the measured nominal film shows good correlation with EHL theories, the measured ratio

of minimum film to nominal film appears to be considerably smaller than 0.7 to 0.75, as predicted analytically Film pressure measurements by means of a vapor-deposited manganin strip confirmed analytical predictions on effect of load on location of the pressure spike.18,24,25

Film Thickness Chart

Minimum film thickness of EHL line contacts may be determined from the Moes diagram26

of Figure 7 Film thickness parameter hmin/R, speed parameter ηou/E′R, load parameter, and lubricant parameter α*E′ are regrouped to form an implicit relation among only three independent parameters Only one family of curves is needed to relate the film thickness parameter with the other two parameters over a wide range of loads, speeds, and lubricant parameters Martin1results are an asymptote for the rigid/isoviscous case, and Herrebrugh27 for the elastic/isoviscous case

Point Contacts

Film Thickness

Figure 8 shows a point contact which is characterized by principal radii Rx1, Ry1for body

1 and Rx2, Ry2 for body 2 In general, the principal planes containing Rx1and Rx2 may not coincide: however, for most EHL contacts such as rolling bearings and gears, principal radii

Rx1 and Rx2 do lie in the same plane These surfaces can be convex, concave, or saddle-shape, depending on whether Rxand Ryare both positive, both negative, or mixed

Ertel-Grubin type of analysis can also be carried out for spherical contacts with a circular conjection Archard and Cowking28 solved the two-dimensional Reynolds equation outside the circular conjunction region for a film thickness distribution compatible to the Hertzian solution for an unlubricated contact An Ertel-Grubin type boundary condition, q = 1/α, around the circumference of the circular conjunction gave:

FIGURE 7 Survey diagram for incompressible and isothermal EHL.

Full computer solutions for elliptical contacts were made for flooded as well as for starved contacts.30Film thickness formula for the flooded contacts appear as:

Hc,F= 2.69 U0.67G0.53W–0.067(1 – 0.61e–0.73k) (12)

Hmin,F= 3.63 U0.68G0.49W–0.073(1 – e–0.68k) (13) where Hc,F = hc,F/Rx, hc,F= central film thickness for flooded contacts, Hmin,F= hmin,F/Rx,

hmin,F = minimum film thickness for flooded contacts, k = elliptical parameter, k = 1.03 (Ry/Rx)0.64, Rx – RxlRx2/(Rxl + Rx2), Ry = RylRy2/(Ryl + Ry2), W = w/ER2

x, w = total load, U = µo(u1+ u2)/2ERx, and G = αE

For the starved contacts, the formulas are

(14)

(15)

where subscript s refers to starved contacts, m is the distance of the inlet meniscus from the center of the contact, and m* is the inlet distance required for achieving flooded con-ditions; m* can be expressed as

(16)

where b is the semiminor axis of the elliptical conjunction in the rolling direction

In most starvation analyses, location of the inlet meniscus is not known beforehand and

is dependent on lubricant supply rate and system configuration Reduction in film thickness due to starvation in most ball bearings is considerably greater than from inlet healing.31,32

Optical techniques enabled extensive point-contact film thickness measurements and cor-relations with analysis.33-36Film thickness data by X-ray transmission37with crowned rollers showed a much stronger load dependence than that predicted by EHL analysis at maximum Hertzian pressures beyond 1 GPa (145,000 psi) This disagreement was explained by Gentle

et al.38as possibly due to a combination of thermal and surface roughness effects The point contact EHL film theory at extreme pressures [up to a maximum Hertzian pressure of 2 GPa (290,000 psi)l was validated by using a sapphire disk and tungsten carbide ball.38 Film thickness measurements using tungsten carbide disks confirm EHL film theories for loads

as high as 2.5 GPa (362,500 psi).39

146 CRC Handbook of Lubrication

Table 1 VALUES OF C, n 1 , AND n 2 FOR EQUATION 11

2 1.560 0.736 – 0.209

1 1.415 0.725 – 0.174 0.5 1.145 0.688 – 0.066

Point Contact Film Shape and Pressure Distribution

Film shape in a circular point contact was experimentally revealed by the interferometric map between a highly polished steel ball and a transparent plate.23,34,35,36,40 By identifying successive fringes, constant thickness contours can be mapped The effect of speed on film shaped constriction which is very narrow and very close to the trailing edge As speed increases or load decreases, the constriction becomes wider and less distinctive Except for extremely low loads and high speeds, the minimum film thickness is found at the two sides rather than at the center of the trailing edge, and it is more sensitive to load variation than the minimum film thickness for a line contact Analytical confirmation of the horseshoe-shaped constriction in a numerical solution of two-dimensional EHL equations41with a solid-like lubricant was followed by a series of full EHL solutions for circular and elliptical contacts30,42with a Newtonian lubricant

Temperature

For a line contact, a detailed study of thermal effects required solution of the energy equation in the lubricant film considering heat generation by shearing and compressing the lubricant, heat convected away by the lubricant, and heat conducted into the solids.43,44

Film thickness level is influenced only by temperature rise in the inlet region discussed earlier; subsequent large temperature rise in the Hertzian conjunction has little influence The predicted temperature field within the Hertzian conjunction depends strongly on the lubricant rheological model used in evaluating the heat generated by sliding For a Newtonian model44 for steel contacts with a maximum Hertzian pressure up to 0.5 GPa, the principal feature of exit film constriction and pressure peaks are unaltered when thermal effects are included However, for loads higher than 0.5 GPa, the Newtonian lubricant model predicts

a sliding frictional coefficient almost an order of magnitude higher than measured Since practical sliding EHL contacts such as gears and cams involve pressures greater than 0.5 GPa, a Newtonian lubricant model is needed for the frictional heat Successful non-Newtonian models are discussed in the next section

Early measurement of surface temperature profiles were made with a platinium wire temperature transducer24at moderate loads Later, an improved transducer45,46with a titanium wire deposited over a silica layer gave good results at much higher loads While vapor-deposited probes are yet to perfected, a promising infrared technique was developed by Nagaraj et al.47 for measuring the surface as well as film temperature in circular contacts Figure 10 shows surface temperature along the center strip of the circular contact The measured temperature show good agreement with that predicted from the Jaeger-Archard48

formula

Friction

In rolling and sliding EHD contacts, frictional force has two components: one due to rolling and the other due to slip between the surfaces Except for nearly pure rolling con-ditions, sliding friction is always much larger than rolling friction Basic features of sliding friction are revealed in Figure 11 by data from a two-disk machine.49In the low-slip region, friction increases linearly with slip As slip increases, friction gradually tapers off in a nonlinear region where the stress is no longer governed by linear constitutive relations of the lubricant In the high-slip region, friction decreases with slip because of thermal influence

on lubricant properties at high-sliding speeds

Low-Slip Friction

For low-slip frictions, sliding friction can be predicted from the Maxwell viscoelastic model If equilibrium viscosity and shear modulus are used, however, predicted friction is much greater than measured This disagreement led to the argument that the viscosity, when

FIGURE 9 Effect of load and speed on film shape (a) W = 10 lb, U = μ°u/ER = 3 × 10 −11 ; (b) 10 lb, 1.8 × 10 −10 ; and (c) 5 lb, 1.5 × 10 −9

stress = (1/2τijτij)1/2, τO = representative stress, a fluid property, η = viscosity, and G = shear modulus τO, η, and G are fluid properties deduced from traction tests Limited values

are given for five oils.52A similar nonlinear viscous and plastic model introduced recently

by Bair and Winer58 used rheological constants from tests totally independent of any data from the EHL contact itself

The Bair and Winer58model in dimensional form can be written as

(19)

In dimensionless form, it is

(20) where

Only three primary physical properties are required: low shear stress viscosity, µo, limiting elastic shear modulus, Gx, and limiting yield shear stress, τL, all as functions of temperature and pressure The behavior of the dimensionless equation is shown in Figure 13 Agreement between theory and experiment is good

FIGURE 12 Nonlinear Maxwell fluid with zero-shear-rate viscosity and infinite-rate shear modulus G F(τ) denotes the nonlinear viscous function.

where λ0.5xis the correlation length at which the acf of the profile is 50% of the value at the origin; γ may be interpreted as the length-to-width ratio of a representative asperity contact Purely transverse, isotropic, and purely longitudinal roughness patterns have γ = 0,1,∞, respectively Surfaces with γ > 1 are longitudinally oriented

For determining partial EHL performance, surface roughness parameters required for each surface include: (1) σ — rms surface roughness, (2) height distribution function, (3)λ0.5x,

λ0.5y— 50% correlation lengths in x and y directions, and (4) acf (autocorrelation function)

Average Film Thickness

Pure longitudinal or transverse roughness was first explored by Johnson et al.64for pure rolling contact based on Christensen’s stochastic theory.65 They developed a Grubin type solution for σ <<h and concluded that the effect of roughness on average film thickness is minimal

For h/σ > 3 and for rolling and sliding contacts, Berthè66and Chow and Cheng67showed that:

1 For pure rolling contact with pure transverse roughness, average film thickness is higher than predicted by smooth surface EHL theory This effect is greatly enhanced

as h/σ approaches three For sliding contacts with one surface smoother than the other, the roughness effect is enhanced if the smoother surface is faster and retarded if the smoother surface is slower

2 For pure longitudinal roughness, average film thickness is lower than predicted by the smooth surface theory Superimposing of sliding on rolling has little influence on the roughness effect for pure longitudinal surfaces

Patir and Cheng62developed an average flow model to handle roughnesses of an arbitrary surface pattern parameter γ and extended the results to h/σ below three where part of the load is shared by asperity contacts Figure 14 depicts the flow pattern for longitudinally oriented (γ > 1), trasversely oriented (γ < 1), and isotropic roughness (γ = 1) In Figure

15, the ratio of actual film thickness to the smooth surface film thickness is plotted against film parameters Λ = hsmooth/σ

Asperity Load to EHL Load Ratio

Average asperity contact pressure in partial EHL is a function of the ratio of compliance

to composite surface roughness h/σ Here, compliance is the distance between the two mean planes based on the underformed surfaces For Gaussian surfaces, Tallian59 has derived the asperity load as a function of h/σ for both plastically or elastically deformed asperities The load sharing ratios in circumferential ground EHL contacts (longitudinal roughness) can be obtained by a full numerical solution for disks with known surface roughness characteristics.68

Average Friction

Once the ratio of asperity load to fluid pressure load is determined, total friction force in partial EHL can be estimated by:

where F = total frictional force, µa, µEHL = coefficient of friction for asperity load and hydrodynamic load, respectively, and Qa,QEHL = asperity, hydrodynamic load For most partial EHL contacts, µa is believed to be between 0.1 and 0.2 The value of µEHL can be taken from frictional coefficients for full-film EHL contacts

sion, bending of pads, and thermal distortion can significantly affect performance of large, high-speed thrust bearings.75,76 Because deformation effects are sensitive to detailed pad geometry, they can only be determined by elaborate computer codes.77

APPLICATION TO MACHINE COMPONENTS

Based on EHL theories, effectiveness of lubrication in rolling element bearings,3,78-81

gears,3,78 and cams82 can be calculated through the film parameter Λ, the ratio of film thickness to the composite surface roughness In this section, formulas are taken mostly from an EHL guide book.78

Rolling Element Bearings

Roller bearings usually have line contacts and Equation 9 should be used to calculate film thickness For ball bearings, contacts are elliptical with semimajor axis normal to the direction

of rolling and Equations 10 through 13 should be used; to evaluate the speed and load parameter, rolling speed and contact dimensions must be determined from the geometry and kinematics of the system Reference 78 gives formulas for all common commercial rolling bearings A simplified film thickness formula, which does not involve detailed bearing geometry and yet gives an adequate prediction of film thickness, is given below:78

(23)

where ∧ = h/σ, D = bearing outside diameter, m or in., C = a constant given in Table

2, dimensionless, LP = µoα · 1011, sec, µo = viscosity, N-sec/m2 or lb-sec/in.2, α = pressure-viscosity coefficient, m2/N or in.2/lb, N = difference between the inner and outer race speeds, rpm, h = film thickness in microns if D is in meters or in microinches if D

is in inches, and σ = composite roughness, µm or µin Typical values of α for bearings are given in Table 3

An adequate ∧ for protecting bearing surfaces against early surface fatigue was shown

to be greater than 1.5 Typical values of lubricant parameter, LP, for motor oils can be found in Figure 16

Table 2 VALUES OF C FOR BEARING RACEWAYS

Bearing type Inner race Outer race

Spherical and cylindrical 8.37 × 10 –4 8.99 × 10 –4

Tapered and needle 8.01 × 10 −4 8.48 × 10 –4

Table 3 TYPICAL VALUES OFσ

FOR BEARINGS

Composite roughness

Spherical and cylindrical 0.356 14

Note: Where:

⎟⎟ = Absolute (positive) value Ng = gear wheel speed, rpm Ts = sun gear torque

C = Center distance NR = ring gear speed, rpm TR = ring gear torque

ED = reduced modulus (equation 2) Ns = sun gear speed, rpm γG = gear cone angle

F = face width RGm = midface pitch radius γP = pinion cone angle

mG = gear ratio RR = ring gear radius φn = normal pressure angle

n = Number of planets Rs = sun gear radius ψ = helix angle

Nc = Carrier speed, rpm TG = gear wheel torque ψm = midface spiral angle

Table 5 TYPICAL VALUES OF COMPOSITE

ROUGHNESS, 

(24)

where G = geometrical parameter from Table 4, LP = µoα · 1011, sec, N = gear rotational speed, rpm, Wτ/ = load per unit length of contact from Table 4, and σ; = composition roughness, see Table 5

158 CRC Handbook of Lubrication

Table 4 GEAR EQUATIONS

The critical value of Λ at which a 5% probability of surface distress is expected is an empirical function of pitch line velocity V as shown in Figure 17 Equations for V for different types of gears are given in Table 4

Cam-Follower Systems

The film parameter Λ for a cam-flat follower Figure 18 system can be calculated by Equation 25:

(25) FIGURE 17 Adjusted specific film thickness vs pitch line velocity (5% probability of distress).

FIGURE 18 Geometry of a cam-follower contact.