Friction and Lubrication in Mechanical Design Episode 2 Part 3 docx

7.7.4 Effective Viscosity Using the notation: Th = absolute bulk disk temperature e.g., Tb = 273.16 + "C A Ts = temperature rise for steel-steel contact ATc = temperature rise from Eq.

7.7.4 Effective Viscosity

Using the notation:

Th = absolute bulk disk temperature (e.g., Tb = 273.16 + "C)

A Ts = temperature rise for steel-steel contact

ATc = temperature rise from Eq (7.27) using the material properties of

A T = A Tc - A Ts = temperature rise difference between the steel-coating

AT, = effective temperature rise difference between the steel-coating con-

the contacting surfaces for steel-coating contact

contact and the steel-steel contact

tact and the steel-steel contact

Then:

where B is the coating thickness factor from the previous section

Then T', = Tb + AT, is used to calculate the viscosity for that coating

conditions, and the viscosity is then substituted into Eq (7.24) to calculate the corresponding coefficient of friction The viscosity of 10W30 oil is

calculated by the ASTM equation [27]:

lOg(cS + 0.6) = a - b log T , (7.32a)

therefore

(7.32 b) ( 7 3 2 ~ )

where T, is the absolute temperature ( K or R), cS is the kinematic viscosity (centistokes) a = 7.827 b = 3.045 for 10W30 oil For some commonly used

oil, a and 6 values are given in Table 7.3

7.7.5

For the reasons mentioned before, the effective modulus of elasticity, Et,,

for coated surface is desirable Using the well-known Hertz equation, one calculates the Hertz contact width for two cylinder contact as [27]:

Coating Thickness Effects on Modulus of Elasticity

(7.33)

I 1.398 10.43 1 10.303 10.293

4.64 18 4.5495 4.4367 4.4385 4.03 19 3.9705 3.9567

E' and U are the modulus of elasticity and Poisson's ratio

Coating material properties are used for E2 and u2 because coating

thickness is an order greater than the deformation depth (this can be seen later) Therefore, the deformation depth is calculated by (Fig 7.18):

Eh = modulus of elasticity of base material

E, = modulus of elasticity of coating material

E , = modulus of elasticity of coated surface

h, = coating film thickness

hd = elastic deformation depth

r = constant (it is found that r = 13 best fits the test data)

Figure 7.18 The contact of the shaft and the coated disk

(7.3 5 b)

For most metals used in engineering, the variation of U is small, and conse- quently, the variation in 1 - v2 is smaller Therefore, no significant error is expected from using the Poisson’s ratio of the base material or the coating material

Figures 7.21 and 7.22 show the comparison of the calculated coefficient

of friction in pure rolling conditions with the test results for chromium and steel Figure 7.23 shows the calculated coefficient of friction in the thermal regime compared with test results for tin, steel, chromium, and copper Figure 7.24 shows the calculated coefficient of friction in the thermal regime

compared with the results from Drozdov’s [ 171, Cameron’s [ 181, Kelley’s

[ 191, and Misharin’s [20] experiments Figures 7.25-7.28 show sample com- parisons of the experimental results with the curves, which are constructed

by using the calculated f,,f,, andf,, and appropriate curves against slide/roll ratios Figure 7.29 shows the comparison of Plint’s test data with prediction

It can be seen that the correlations are excellent

Rolling} Sliding Contacts

0.08

0.06

285

0 U=0.303m/r U=1.44mlfs

0.04

Ic

O*02!

0 W-94703Wm W=l69408)(lm

Rolling/ Sliding Con tact s

A copper

Tin Chromium Copper

0.303 mjsec): (a) experimental; (b) calculated

Coefficient of friction vs slide/roll ratio ( W = 94,703 N/m, U =

RollinglSliding Con facts 293

v Steel " " " " l " ' " " "

A

A

Figure 7.26 Coefficient of friction vs slide/roll ratio ( W = 94,703 N/m, U =

0.303 m/sec): (a) experimental; (b) calculated

Figure 7.28 Coefficient of friction vs slidelroll ratio ( W = 378,812 N/m, U =

1.44 m/sec): (a) experimental; (b) calculated

Figure 7.29 Comparison of Plint's test data with prediction

All the test data used so far are obtained from ground or rougher surface contacts The chemical layer on the contact surfaces resulting from the manufacturing process and during operation is ignored because it wears off relatively quickly on a rough surface, especially at the real areas of contact where high shear stress is expected On the other hand, the surface chemical layer can be expected to play an important role in smooth surface contacts The properties of the contact surfaces are affected by this chemical layer because it can remain on the surfaces more easily than on a rough

surface In this case, Eq (7.27) can be used to account for this effect Cheng [7], Hirst [8], and Johnson [9] conducted experimental investiga-

tions on very smooth surface contacts All the contact surfaces were super- finished The A values are roughly between 80 and 100 for Cheng's test, 50

and 60 for Hirst's test, and 40 and 60 for Johnson's test (A = ho/Sc, and

Sc = ,/:fs:, where S1 and S2 are surface roughness of contact surfaces,

CLA.)

Because the surface chemical layers usually are very thin, they are assumed to have little effect on the elastic properties of the surfaces

Rolling/ Sliding Contacts 297

However, they affect the temperature rise in the contact zone significantly

In order to use Eq (7.27) to account for this effect, the thermal-physical

properties and thickness of the chemical layer are needed Because these values are not known, the following thermal-physical properties are used

TheS, values from the above tests are used to find the thickness of the corresponding chemical layers inversely The result is shown in Fig 7.30

It is found that the thickness decreases as the load increases as expected because the higher the load, the higher the shear stress in the lubricant, which results in a thinner chemical layer In Figs 7.31-7.33, the test data are compared with prediction It can be seen that the chemical layer makes a

Rolling} Sliding Contacts 299

Figure 7.33 Comparison of Johnson’s experimental data with prediction

great difference in the coefficient of friction for conditions with large A ratios

(>40) in the thermal regime Load can have a significant effect on the chemical layer thickness and Fig 7.30 can be used for evaluating the thick- ness as a function of normal load

The empirical formulas were checked for different regimes of lubrication, surface roughness, load, speed, and surface coating The formulas were used for evaluating rolling friction, and traction forces in the isothermal, non- linear, and thermal regimes of elastohydrodynamic lubrication Because of the current interest in surface coating, the formulas were also applied for determining the coefficient of friction for cylinders with surface layers of any arbitrary thickness and physical and thermal properties

It can be seen from the empirical formulas that:

1 It appears that in general, the slide/roll ratio has little direct effect

on the coefficient of friction in the thermal region (slidelroll

> 0.27)

2 The surface roughness effect is treated in this study as a function

of the surface generating process rather than the traditional surface roughness measurements

The oil film thickness is found to be better represented for friction calculation in a nondimensional form by normalizing it to the effective radius rather than the commonly used film thickness to roughness ratio A

Coating has a significant effect on the temperature rise in the contact zone This is represented by a factor B, as shown in Eq

(7.3 0)

Coating has an effect on the modulus of elasticity as shown in Eq

(7.35) This is represented by using an effective modulus of elas-

ticity for the coated surface For tin (whose modulus of elasticity differs from that of the base material most significantly among the three coating materials used), this correction produces a 50%

increase in the effective modulus of elasticity

7.8.1 Unlayered Steel-Steel Contact Surfaces

1 Given: contact surface radii r l , 1-2 (m, in.)

Surface velocities, U1, U, (mlsec, in./sec)

Dynamic viscosity of lubricant oil at entry condition qo (Pa-sec,

reyn)

Load F (n, lbf)

Surface roughness S1, S2 (m CLA, in CLA) or manufacturing processes

Density of lubricant p (kg/m3, (lb/in.3) x 0.0026)

Modulus of steel E (Pa, psi)

Contact width of surfaces y (m, in.)

Poisson’s ratio for steel v (dimensionless)

Pressure-viscosity coefficient of lubricant a! ( 1 /Pa, 1 /psi)

2 Calculate:

E Effective modulus of elasticity E’ = -

Rolling/Sliding Con lac ts 30 I

Calculate coefficient of friction in the nonlinear region and its location.f, is calculated from Eq (7.22), z* is calculated from Eq

1 Given: contact surface radii r l , rl (m, in.)

surface velocities U 1 , U2 (m/sec, in./sec)

load F (N, lbf)

surface roughness S , , S2 (m CLA, in CLA) or manufacturing processes

density of lubricant p (kg/m3, (lb/in.3) x 0.0026)

modulus of steel E (Pa, psi)

contact width of surface y (m, in.)

Poisson's ratio for steel U (dimensionless)

Lubricant oil properties:

Thermal conductivity KO (W/m-'C), (BTU/(sec-in.-OF)) x 9338) Specific heat CO (J/(kg-"C), (BTU/lb-OF) x 3,604,437)

Pressure-viscosity coefficient a (I/Pa, l/psi)

Temperature-viscosity coefficient /3 (1 /"C, 1 / O F )

Dynamic viscosity at entry condition qo (Pa-sec, reyn)

Disk 1 base material properties:

Modulus of elasticity Ehl (Pa, psi)

Poisson's ratio uhl (dimensionless)

Thermal conductivity Kbl (W/(m-OC), (BTU/(sec-in-OF) x 9338) Specific heat cbl (J/(kg-"C), (BTU/(lb-OF) x 3,604,437)

Density pbl (kg/m3, (lb/in.3) x 0.0026)

Disk 1 surface material properties:

Modulus of elasticity ECl (Pa, psi)

Poisson's ratio ucl (dimensionless)

Thermal conductivity Kcl (W/(m-"C), (BTU/(sec-in.-OF) x 9338)

Specific heat ccl (J/(kg-"C), (BTU/(lb-OF) x 3,604,437)

Density pcl (kg/m3, (lb/in.3) x 0.0026)

Thickness hCl (m, in.)

Disk 2 base material properties:

Modulus of elasticity Eb2 (Pa, psi)

Poisson's ratio ub2 (dimensionless)

Thermal conductivity Kb2 (W/(m-OC), (BTU/-sec-in.-"F) x 9338)

Specific heat cb2 (J/kg-"C), (BTU/(lb-OF) x 3,604,437)

Density pb2 (kg/m3, (Ib/in.3) x 0.0026)

Disk 2 surface material properties:

Modulus of elasticity Ec2 (Pa, psi)

Poisson's ratio uc (dimensionless)

Thermal conductivity Kc2 (W/(m-OC), (BTU/(sec-in,-"F) x 9338)

Specific heat cc2 (J/(kg-"C), (BTU/(lb-OF) x 3,604,437)

Density pc2 (kg/m3, (lb/in.3) x 0.0026)

Thickness hc2 (m, in.)

Steel properties:

Modulus of elasticity Es (Pa, psi)

Poisson's ratio vs (dimensionless)

Thermal conductivity Ks (W/(m-OC), (BTU/(sec-in.-OF) x 9338)

Specific heat cs (J/(kg-"C), (BTU/(lb-OF) x 3,604,437)

Density ps (kg/m3, (lb/in.3) x 0.0026)

Bulk temperature Tb (K, R) (K = "C + 273.16)

Use previous section to calculate fr for steel-steel contact sur-

faces Substitute ATs, K s , cs, ps, and Es for AT, K1, c1, p1, El

and K2, c2, p2, E2 in Eqs (7.27) and (7.28), where f is replaced byf,, L is replaced by L = 3 2 J ' m

Rolling/Sliding Contacts 303

F Load per unit width W =-

5 Calculate hd by using Eq (7.34)

6 Substitute Ebl , Ecl , hcl for Eb, E,, h, in Eq (7.35) to calculate E e l

Substitute Eb2, Ec2, hc2 for Eb, Ec, h, in Eq (7.35) to calculate Ee2

7 Calculate the effective modulus of elasticity of the layered surfaces by:

For S < 0.05 pm take Se = 0.05 pm Then S - ec = , / S x

Calculate dimensionless U, q, W , S from Eqs (7.17)-(7.20)

except that E' is replaced by E,&

Calculate coefficient of rolling friction fr from Eq (7.2 1) Calculate coefficient of friction in the nonlinear region and its location fn is calculated from Eq (7.22), z* is calculated from

Eq (7.23)

Calculate minimum oil film thickness ho from Eq (7.26), where

G = aE,I,

Calculate E by using Eq (7.28), where z = 0.27

Calculate A T, by Eq (7.27), where K l , c1, p1 , El are replaced by

L = 26,f =f! from step 2 for steel-steel contact surfaces Calculate D 1 by using Eq (7.29) where K, c, p, U are substituted

by K 1 , c1 , p l , El Use Eq (7.30) to calculate p1 where h, = hCl , Calculate D2 by using Eq (7.29) where K, c, p, U are substituted

by K2, c2, p2, E2 Use Eq (7.30) to calculate p2 where h, = hr2,

Calculate AT, by using Eq (7.31) where /? = (PI + P2)/2 Use Eq (7.32) to calculate a and 6 for theparticular lubricant

as follows Suppose that the viscosity is rnl at T1 and m2 at T2 (where T1 and T2 are absolute temperature, say,

K = 273.16 + "C, ml and m2 are kinematic viscosity in centi-

& I * cc19 Pcl9 & I * K2 9 c2 * P2 * E2 are replaced by Kc2 * cc2 * Pc2 , Ec2 *

D = 0 1

D = 02

a and 6 (If lubricant is SAE 10, SAE 20, SAE 30, SAE 40, SAE

Use Eq (7.32) to calculate the viscosity q at temperatufe

T,(q = pou where po is the lubricant density, U is the kinematic viscosity)

Use Eq (7.26) to find ho where qo = q, G = al&

Findfo from Fig 7.12 Find (Sc,r/R)', from Fig 7.13b Calculate

coefficient of thermal frictionf, from Eq (7.24)

If the difference betweenf, value in step 21 andf, value in step 14 does not satisfy your accuracy requirement, go back to step 14, replace .ft by fr value in step 21 and iterate until the accuracy requirement is satisfied

Use f;.fn, z*, and J to construct the coefficient of friction curve versus sliding/rolling ratio as in Fig 7.1 1, where sliding speed

= lU, - U21, and rolling speed = U

The following are some illustrative examples for the application of the developed empirical formulas in sample cases

Figure 7.34 shows calculated coefficient of friction versus sliding/rolling ratio for different rolling speeds, T = 26°C (78.8"F), steel-steel contact,

ground surfaces, S = 0.03 pm (1 2 pin.), W = 378,8 12 N/m (2 160 lbf/in.), 10W30 oil, R = 0.0234m (0.92 in.)

Figure 7.35 shows calculated coefficient of friction versus sliding/rolling

ratio for different normal loads, T = 26°C (78.8"F), steel-steel contact,

ground surfaces, S = 0.03 pm (12 pin.), U1 = 3.2 m/sec (126 in./sec), 10W30 oil, R = 0.0234 m (0.92 in.)

Figure 7.36 shows calculated coefficient of friction versus slidinglrolling ratio for different effective radii, T = 26°C (78.8"F), steel-steel contact ground surfaces, S = 0.03 pm (12 pin.), W = 378,8 12 N/m (2 160 lbf/in.), 10W30 oil, U1 = 3.2m/sec (126in./sec)

Figure 7.37 shows calculated coefficient of friction versus sliding/ rolling ratio for different viscosity, steel-steel contact, ground surfaces,

S = 0.03 pm (12pin.), W = 378,812N/m (21601bf/in.), 10W30 oil, U1 = 3.2 m/sec (126 in./sec), R = 0.0234 m (0.92 in.)

Figure 7.38 shows calculated coefficient of friction versus sliding/

rolling ratio for different materials, T = 26°C (78.8"F), ground surfaces,