10
Chapter
I
lubrication theory
[6,
71.
Tower reported the results of a series of experi-
ments intended to determine the best methods to lubricate a railroad journal
bearing. Working with a partial journal bearing in an oil bath, he noticed
and later measured the pressure generated in the oil film.
Tower pointed out that without sufficient lubrication, the bearing oper-
ates in the boundary lubrication regime, whereas with adequate lubrication
the two surfaces are completely separated by an oil film. Petrov
[5]
also
conducted experiments to measure the frictional losses in bearings. He con-
cluded that friction in adequately lubricated bearings is due to the viscous
shearing of the fluid between the two surfaces and that viscosity is the most
important property of the fluid, and not density as previously assumed. He
also formulated the relationship for calculating the frictional resistance in
the fluid film as the product of viscosity, speed, and area, divided by the
thickness of the film.
The observations of Tower and Petrov proved to
be
the turning point in
the history of lubrication. Prior to their work, researchers had concentrated
their efforts on conducting friction drag tests on bearings. From Tower’s
experiments,
it
was realized that an understanding of the pressure generated
during the bearing operation is the key to perceive the mechanism of lubri-

thermal boundaries on the stationary and moving surfaces and includes a
pointwise variation of the film viscosity with temperature is generally
referred to as the thermohydrodynamic (THD) model. The THD analyses
in the past three decades have drawn considerable attention to the thermal
aspects of lubrication. Many experimental and theoretical studies have been
undertaken to shed some light on the influence of the energy generated in
the film, and the heat transfer within the film and to the surroundings, on
the generated pressure.
In 1929, McKee and McKee [39] performed a series of experiments on a
journal bearing. They observed that under conditions of high speed, the
viscosity diminished to
a
point where the product of viscosity and rotating
speed is a constant. Barber and Davenport [40] investigated friction in
several journal bearings. The journal center position with respect to the
bearing center was determined by a set of dial indicators. Information on
the load-carrying capacity and film pressure was presented.
In 1946, Fogg [41] found that parallel surface thrust bearings, contrary
to predictions by hydrodynamic theory, are capable of carrying a load. His
experiments demonstrated the ability
of
thrust bearings with parallel sur-
faces to carry loads of almost the same order of magnitude as can be
sustained by tilting pad thrust bearings with the same bearing area. This
observation, known as the Fogg effect, is explained by the concept of the
“thermal wedge,” where the expansion of the fluid as it heats up produces a
distortion of the velocity distribution curves similar to that produced by a
converging surface, developing a load-carrying capacity. Fogg also indi-
cated that this load-carrying ability does not depend on a round inlet
edge nor the thermal distortion

in. operating under
6000
rpm, failure due
to overheating of the bearing material (babbitt) occurred at about
340°F.
Booser et al.
[46]
observed a babbitt-limiting maximum temperature in the
range
of
266
to
392°F
for large steam turbine journal bearings. They also
formulated a one-dimensional analysis for estimating the maximum
temperature under both laminar and turbulent conditions.
In a study of heat effects in journal bearings, Dowson et al.
[47]
in 1966
conducted a major experimental investigation
of
temperature patterns and
heat balance of steadily loaded journal bearings. Their test apparatus was
capable
of
measuring the pressure distribution, load, speed, lubricant flow
rate, lubricant inlet and outlet temperatures, and temperature distribution
within the stationary bushing and rotating shaft. They found that the heat
flow patterns in the bushing are a combination of both radial flows and a
significant amount

of
the early applications of the energy equation to hydrodynamic lubrication
was made by Cope
[49]
in
1948.
His model was based on the assumptions
of negligible temperature variation across the film and negligible heat
conduction within the lubrication film as well as into the neighboring
solids. The consequence
of
the second assumption is that both the bearing
and the shaft are isothermal components, and thus all the generated heat is
carried out by the lubricant.
As
indicated in a review paper by Szeri
[50],
the belief, that the classical theory on one hand and Cope’s adiabatic
model on the other, bracket bearing performance in lubrication analysis,
was widely accepted for a while.
A
thermohydrodynamic hypothesis was
Introduction
13
later introduced by Seireg and Ezzat [51] to rationalize their experimental
findings.
An empirical procedure for prediction of the thermohydrodynamic
behavior of the fluid film was proposed in 1973 by Seireg and Ezzat. This
report presented results on the load-carrying capacity of the film from
extensive tests. These tests covered eccentricity ratios ranging from 0.6 to

compre-
hensive review of thermal effects in hydrodynamic bearings is given by
Khonsari [53] and deals with both journal and slider bearings.
In 1975, Seireg and
Doshi
[54] studied nonsteady state behavior of the
journal bearing performance. The transient bushing temperature distribu-
tion in journal bearing appears to be similar to the steady-state temperature
distribution. It was also found that the maximum bushing surface tempera-
ture occurs in the vicinity of minimum film thickness. The temperature level
as well as the circumferential temperature variation were found to rise with
an increase of eccentricity ratio and bearing speed. Later, Seireg and
Dandage [55] proposed an empirical thermohydrodynamic procedure to
calculate a modified Sommerfeld number which can be utilized in the stan-
dard formula (based on the isoviscous theory) to calculate eccentricity ratio,
oil flow, frictional loss, and temperature rise, as well as stiffness and damp-
ing coefficients for full journal bearings.
In 1980, Barwell and Lingard [56] measured the temperature distribu-
tion of plain journal bearings, and found that the maximum bearing tem-
perature, which is encountered at the point of minimum film thickness, is the
appropriate value for an estimate of effective viscosity to be used in load
capacity calculation. Tonnesen and Hansen [57] performed an experiment
14
Chapter
1
on a cylindrical fluid film bearing to study the thermal effects on the bearing
performance. Their test bearings were cylindrical and oil was supplied
through either one or two holes or through two-axial grooves, 180" apart.
Experiments were conducted with three types of turbine oils. Both viscosity
and oil inlet geometry were found to have a significant effect on the operat-

due to the elasticity of the contacting bodies has attracted considerable
attention from tribologists over the last half century. Some of the studies
related to frictional resistance in this elastohydrodynamic (EHD) regime
are briefly reviewed in the following with emphasis on effect of viscosity and
temperature in the film.
Dyson [60] interpreted some of the friction results in terms
of
a model of
viscoelastic liquid. He divided the experimental curves of frictional traction
versus sliding speed into three regions: the linear region, the nonlinear
Introduction
15
(ascending) region, and the thermal (descending) region. At low sliding
speeds a linear relation exists, the slope of which defines a quasi-
Newtonian viscosity, and the behavior is isothermal. At high sliding speeds
the frictional force decreases as sliding speed increases, and this can be
attributed to some extent to the influence of temperature on viscosity. In
the transition region, thermal effects provide a totally inadequate explana-
tion because the observed frictional traction may be several orders of mag-
nitude lower than the calculated values even when temperature effects are
considered.
Because of the high variation
of
pressure and temperature, many para-
meters such as temperature, load, sliding speed, the ratio of sliding speed to
rolling speed, viscosity, and surface roughness have great effects on
frictional traction.
Thermal analysis in concentrated contacts by Crook
[61, 621,
Cheng

been found that the temperature on the entry side remains small, but
it
does
have
a
very marked influence upon the temperatures within the pressure
zone, for instance, the introduction of
400
cm/sec sliding causes the effective
viscosity to fall in relation to its value in pure rolling by a factor of
50.
It has
also been shown that at high sliding speeds the effective viscosity is largely
independent of the viscosity
of
oil at entry conditions. This fact carries the
important implication that if an oil of higher viscosity is used to give the
surfaces greater protection by virtue
of
a thicker oil film, then there is little
penalty to be paid by way of greater frictional heating, and in fact at high
sliding speeds the frictional traction may be lower with the thicker film. It has
also been found that frictional tractions pass through a maximum as the
sliding is increased. This implies that if the disks were used as a friction
drive and the slip was allowed to exceed that at which the maximum traction
occurs, then a demand for a greater output torque, which would lead to even
greater sliding, would reduce the torque the drive can deliver.
Crook
[62]
conducted an experiment to prove his theory, and found that

perature and contact pressure did not appear to affect the friction
coefficient. Below the ceiling the friction coefficient increased with pressure
and decreased with increasing rolling speed and temperature.
Dowson and Whitaker
[65]
developed a numerical procedure to solve
the EHD problem of rolling and sliding contacts lubricated by a Newtonian
fluid. It was found that sliding caused an increase in the film temperatures
within the zone, and the temperature rise was roughly proportional to the
square
of
the sliding velocity. Thermal effects restrained the coefficient
of
friction from reaching the high values which would occur in sliding contacts
under isothermal conditions.
Plint
[66]
proposed a formula for spherical contacts which relates
the coefficient of friction with the temperature on the central plane of the
contact zone and the radius
of
the contact zone.
There are other parameters which were investigated for their influence
on the frictional resistance in the EHD regime by many tribologists
[67-851.
Such parameters include load, rolling speed, shear rate, surface roughness,
etc. The results of some of these investigations are utilized in Chapter
7
for
developing generalized emperical relationships for predicting the coefficient

de Mathematique et de Physique, Paris, 1699.
Coulomb, C. A., Memoires de Mathematique et de Physique de 1’Academie
Royale des Sciences, Paris, 1785.
Reynolds,
O.,
“On the Theory
of
Lubrication and Its Application to Mr.
Beauchamp Tower’s Experiments Including an Experimental Determination
of
Olive Oil,” Phil. Trans.,
1886,
Vol.
177(i), pp.
157-234.
Petrov,
N.
P. “Friction in Machines and the Effect of the Lubricant,” Inzh.
Zh., St. Petersburg, Russia, 1883, Vol.
1,
pp. 71-140;
Vol.
2, pp. 227-279;
Vol.
3,
pp. 377436; Vol. 4, pp. 435-464 (in Russian).
Tower, B., “First Report on Friction Experiments (Friction of Lubricated
Bearings),” Proc. Inst. Mech. Engrs, 1883, pp. 632-659.
Tower, B., “Second Report on Friction Experiments (Experiments on Oil
Pressure in Bearings),” Proc. Inst. Mech. Engrs, 1885, pp. 58-70.

S.,
Matsuura, T., Uchizawa, M., Yura,
S.,
Shibata,
H.,
and Fujita,
H.,
“Friction and Wear Studies
on
Lubricants and Materials Applicable MEMS,”
Proc. of the IEEE Workshop
on
MicroElectro Mechanical Systems (MEMS),
Nara, Japan, Feb. 1991.
Ghodssi, R., Denton, D. D., Seireg, A. A., and B. Howland, “Rolling Friction
in Linear Microactuators,” JVSA, Aug. 1993.
Moore, A.J., Principles and Applications
of
Tribology, Pergamon Press, New
York, NY, 1975.
Rabinowicz, E., Friction and Wear of Materials, John Wiley
&
Sons, New
York, NY, 1965.
Stevens, J.
S.,
“Molecular Contact,” Phys. Rev., 1899, Vol. 8, pp. 49-56.
Rankin, J.
S.,
“The Elastic Range

40.
41.
42.
Seireg, A., and Weiter,
E.
J., “Viscoelastic Behavior of Frictional Hertzian
Contacts Under Ramp-Type Loads,” Proc. Inst. Mech. Engrs, 1966-67, Vol.
181, Pt. 30, pp. 200-206.
Seireg, A. and Weiter, E. J., “Frictional Interface Behavior Under Dynamic
Excitation,’’ Wear, 1963, Vol. 6, pp. 6677.
Seireg, A. and Weiter, E.
J.,
“Behavior of Frictional Hertzian Contacts Under
Impulsive Loading,” Wear, 1965, Vol.
8,
pp. 208-219.
“Designing for Zero Wear
-
Or
a Predictable Minimum,” Prod. Eng., August
Rabinowicz, E., “Variation
of
Friction and Wear
of
Solid Lubricant Films
with Film Thickness,” ASLE Trans., Vol. 10, n.1, 1967, pp. 1-7.
Reynolds,
O.,
Phil. Trans., 1875, p. 166.
Palmgren, A., “Ball and Roller Bearing Engineering,” S.H. Burbank,

J.,
and Appeldoorn,
J.
K.,
“The Effect of Lubricant Viscosity on
Metallic Contact and Friction in a Sliding System,” ASLE Trans. 1962, Vol.
5,
Furey, M. J., “Surface Roughness on Metallic Contact and Friction,” ASLE
Trans., 1963, Vol. 6, pp. 49-59.
Eng, B. and Freeman, P., Lubrication and Friction, Pitman, New York,
NY,
1962.
McKee,
S.
A. and McKee, T. R., “Friction of Journal Bearing as Influenced by
Clearance and Length,” ASME Trans., 1929, Vol. 51, pp. 161-171.
Barber,
E.
and Davenport,
C.,
“Investigation of Journal Bearing
Performance,” Penn. State Coll. Eng. Exp. Stat. Bull., 1933, Vol. 27(42).
Fogg, A., “Fluid Film Lubrication of Parallel Thrust Surfaces,’’ Proc. Inst.
Mech. Engrs, 1946, Vol. 155, pp. 49-67.
Cameron, A., “Hydrodynamic Lubrication of Rotating Disk in Pure Sliding,
New Type
of
Oil Film Formation,” J Inst. Petrol., Vol. 37, p. 471.
15, 1966, pp. 41-50.
pp. 149-159.

Trans., Vol. 13, n.4, Oct. 1970, pp. 262-268. Also, “Maximum Temperature for
Hydrodynamic Bearings Under Steady Load,” Lubric. Eng., Vol. 26, n.7, July
1970, pp. 226-235.
Dowson, D., Hudson, J., Hunter, B., and March, C., “An Experimental
Investigation of the Thermal Equilibrium of Steadily Loaded Journal
Bearings,” Proc. Inst. Mech. Engrs, 1966-67, Vol. 101, 3B.
Cameron, A., The Principles of Lubrication, Longmans Green
&
Co., London,
England, 1966.
Cope, W., “The Hydrodynamic Theory of Film Lubrication,” Proc. Roy. Soc.,
Szeri, A.
Z.,
“Some Extensions of the Lubrication Theory of Osborne
Reynolds,” J. of Tribol., 1987, pp. 21-36.
Seireg, A. and Ewat,
H.,
“Thermohydrodynamic Phenomena
in
Fluid Film
Lubrication,” J. Lubr. Technol., 1973, pp. 187-194.
Wang, N.
Z.
and Seireg, A., Experimental Investigation
in
the Performance of
the Thermohydrodynamic Lubrication of Reciprocating Slider Bearing, ASLE
paper
No.
87-AM-3A-3, 1987.

pp. 422428.
Boncompain, R., Fillon, M., and Frene, J., “Analysis of Thermal Effects
in
Hydrodynamic Bearings,” J. Tribol., 1986, Vol.
108,
pp. 219-224.
1948, Vol. A197, pp. 201-216.
20
Chapter
I
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
Dyson, A., “Frictional Traction and Lubricant Rheology in
Elastohydrodynamic Lubrication,” Phil. Trans. Roy.
Soc.,

1967-68,
Vol.
182,
Pt.
1,
No.
14.
Dowson, D., and Whitaker, A. V.,
“A
Numerical Procedure for the Solution of
the Elastohydrodynamic Problem of Rolling and Sliding Contacts Lubricated
by a Newtonian Fluid,” Proc. Inst. Mech. Engrs,
196546,
Vol.
180,
Pt.
3B,
Plint, M.
A.,
“Traction in Elastohydrodynamic Contacts,” Proc. Inst. Mech.
Engrs,
1967-68,
Vol.
182,
Pt.
1,
No.
14,
p.
300.

and Moore, A.
J.,
“Non-Newtonian Behavior in Elasto-hydrody-
namic Lubrication,” Proc. Roy. Soc.,
1974,
Vol.
A337,
pp.
101-121.
Johnson,
K.
L.,
and Tevaarwerk, J. L., “Shear Behavior of
Elastohydrodynamic Oil Films,” Proc. Roy. Soc.,
1977,
Vol.
A356,
pp.
Conry, T. F., Johnson,
K.
L., and Owen, S., “Viscosity in the Thermal Regime
of Elastohydrodynamic Traction,” 6th Lubrication Symposium, Lyon, Sept.,
1979.
Trachman,
E.
G., and Cheng,
H.
S.,
“Thermal and Non-Newtonian Effects on
Traction in Elastohydrodynamic Contacts,” Elastohydrodynamic Lubrication,

“Fundamental Research on Gear
Lubrication,” Bull. JSME,
1961,
Vol.
4(14),
p.
382.
266(
1
170),
pp.
1-33.
p.
57.
2 15-236.
17(4),
pp.
271-279.
Introduction
21
77.
78.
79.
80.
81.
82.
83.
84.
85.
Sasaki, T. Okamura,

105,
p. 187.
Greenwood, J. A., and Tripp, J. H., “The Elastic Contact of Rough Spheres,”
J. Appl. Mech., March 1967, p. 153.
Lindberg, R. A., “Processes and Materials
of
Manufacture,” Allyn and Bacon,
1977, pp. 628-637.
“Wear Control Handbook”, ASME, 1980.
Szeri, A.
Z.,
Tribology: Friction, Lubrication and Wear, Hemisphere, New
York, NY, 1980.
2
The Contact Between Smooth Surfaces
2.1
INTRODUCTION
It is well known that no surface, natural or manufactured, is perfectly
smooth. Nonetheless the idealized case of elastic bodies with smooth sur-
faces is considered in this chapter as the theoretical reference for the contact
between rough surfaces. The latter will be discussed in Chapter
4
and used
as the basis for evaluating the frictional resistance.
The equations governing the pressure distribution due to normal loads
are given without detailed derivations. Readers interested in detailed deriva-
tions can find them in some
of
the books and publications given in the
references at the end of the chapter

2.1
Concentrated load on a semi-infinite elastic solid.
a,
=
horizontal stress at any point
I
=
P((1
2n
-
2v)[$
-
z
v2
(E
2n
~3(,2~2)-5/2)-”~]
-
3r2Z(r2
+
22)-5/2
az
=
vertical stress at any point
-
3p
23(,.2~2)-5/2
2n
rrz
=

=
horizontal displacement
and
w
=
vertical displacement
where
E
=
elastic modulus
At
the surface where
2
=
0,
the equations for the displacements become:
P(l
-
2)
(tt’)Z,0
=
____
(1
-
2u)(l
+
u)P
(&=0
=
-

u2)qa
-
JrE
(w)~=”
=
deflection at the center
of
the loaded circle
-
2(
1
-
u2)qa
-
E
(oZ)r=O
=
vertical stress at any point on the Z-axis
(a2
+
z*)3/2
z3
1
-4
-I+
-(
The Contact Between Smooth Surfaces
25
(I?
Figure

3:
Uniform Pressure over
a
Rectangular Area on the Surface of
a
Semi-Infinite
Solid
In this case (Fig. 2.3) the average deflection under the uniform pressure
q
is
calculated from:
W,,e
=
k(1
-
u2)
;
.JA
where
A
=
area
of
rectangle
k
=
factor dependent
on
the ratio
b/a

100
0.95
0.94
0.92
0.88
0.82
0.71
0.37
27
Case
4:
A
Rigid Circular Cylinder Pressed Against a Semi-Infinite Solid
In this case, which is shown in
Fig.
2.4,
the displacement
of
the rigid cylinder
is calculated from:
P(1
-
”*)
w=-
2aE
P
Figure
2.4
Rigid cylinder over a semi-infinite elastic solid.
28

5:
Two
Spherical Bodies in Contact
In this case (Fig.
2.5)
the area of contact
is
circular with radius
a
given by
a
=
0.88
/:
R,
Figure
2.5
Spherical bodies in contact.
The
Contact
Betwven
Smooth
Surfaces
assuming a Poisson’s ratio
U
=
0.3
and the pressure distribution over this
29
area

a rectangle with width
b
and
length equal to the length
of
the cylinders. The design relationships in this
case are:
q
=
pressure
on
the
area of contact
where
P’
=
load
per
unit length of the cylinders
111
R, RI R2
_
-
+-