Chapter 12: Hydrodynamic and
Hydrostatic Bearings
A cup of tea, standing in
a dry saucer, is apt to slip
about in an awkward
manner, for which a
remedy is found in
introduction of a few
drops of water, or tea,
wetting the parts in
contact.
Lord Rayleigh (1918)
A Kingsbury Bearing.
Hamrock • Fundamentals of Machine Elements
Density Wedge and Stretch
p
p
ρ
ub
ub(x)
Figure 12.1 Density wedge
mechanism.
Figure 12.2 Stretch mechanism.
Hamrock • Fundamentals of Machine Elements
Physical Wedge and Normal Squeeze
p
p
wa
ub
wb
Figure 12.3 Physical wedge
mechanism.
Figure 12.4 Normal squeexe
mechanism.
Hamrock • Fundamentals of Machine Elements
Translation Squeeze and Local Expansion
p
p
ua
ub
Figure 12.5 Translation squeeze
mechanism.
Heat
Figure 12.6 Local expansion
mechanism.
Hamrock • Fundamentals of Machine Elements
ua
Aa
Moving
ua
Da
Ca
E
Velocity Profiles in
Slider Bearings
Ba
B
A Stationary
A
(a)
Aa
Ba
B
A
(b)
N
ua
A′
B′
z
ua
Aa
h
A
Wz
L
ua
Ka Ba
ua
Da
H
B
x
I
B
Stationary
Figure 12.7 Velocity profiles in a
parallel-surface slider bearing.
Ca Ja
M
J
A
z
A
(c)
Figure 12.8 Flow within a fixed-incline
slider bearing (a) Couette flow; (b)
Poiseuille flow; (c) resulting velocity
profile.
Hamrock • Fundamentals of Machine Elements
Thrust Slider Bearing
ω
Bearing pad
A
Wt
A′
ro
ri
Thrust bearing
Lubricant
Bearing pad
Figure 12.9 Thrust slider bearing geometry.
Hamrock • Fundamentals of Machine Elements
Force and Pressure Profiles for Slider Bearing
Film pressure
distribution
&
sh
Wz
Wa
ub
Wxa
ho
q
z
Wza
Wzb
Fa
q qs
sh
ho
Fb
qs (both sides)
l
x
ub
Figure 12.10 Force components and oil
film geometry in a hydrodynamically
lubricated thrust slider bearing.
Figure 12.11 Side view of fixedincline slider bearing.
Hamrock • Fundamentals of Machine Elements
Design Procedure for Fixed-Incline Thrust Bearing
1.
Choose a pad length-to-width ratio. A square pad (λ = 1) is generally thought to give good
performance. If it is known whether maximum load or minimum power loss is more
important in a particular application, the outlet film thickness ratio Ho can be determined
2.
from Fig. 12.13.
Once λ and Ho are known, Fig. 12.14 can be used to obtain the bearing number Bt.
3.
From Fig. 12.15 determine the temperature rise due to shear heating for a given λ and Bt.
The volumetric specific heat Cs = ρCp, which is the dimensionless temperature rise
4.
6
2
parameter, is relatively constant for mineral oils and is equivalent to 1.36 × 10 N/(m °C).
Determine lubricant temperature. Mean temperature can be expressed as
!tm
t˜m = tmi +
2
where tmi = inlet temperature, °C. The inlet temperature is usually known beforehand.
Once the mean temperature tm is known, it can be used in Fig. 8.17 to determine the
viscosity of SAE oils, or Fig. 8.16 or Table 8.4 can be used. In using Table 8.4 if the
temperature is different from the three temperatures given, a linear interpolation can be
used.
(continued)
Hamrock • Fundamentals of Machine Elements
Design Procedure for Fixed-Incline Thrust Bearing
5. Make use of Eqs. (12.34) and (12.68) to get the outlet (minimum) film thickness h0 as
h0 = Hol
!
!0ubwt
WzBt
Once the outlet film thickness is known, the shoulder height sh can be directly obtained from sh
= ho/Ho. If in some applications the outlet film thickness is specified and either the velocity ub
or the normal applied load Wz is not known, Eq. (12.72) can be rewritten to establish ub or Wz.
6. Check Table 12.1 to see if the outlet (minimum) film thickness is sufficient for the pressurized
surface finish. If ho from Eq. (12.72) ≥ ho from Table 12.1, go to step 7. If ho from Eq. (12.72)
< ho from Table 12.1, consider one or both of the following steps:
a.
b.
Increase the bearing speed.
Decrease the load, the surface finish, or the inlet temperature. Upon making this change
return to step 3.
7. Evaluate the other performance parameters. Once an adequate minimum film thickness and a
proper lubricant temperature have been determined, the performance parameters can be
evaluated. Specifically, from Fig. 12.16 the power loss, the coefficient of friction, and the total
and side flows can be determined.
Hamrock • Fundamentals of Machine Elements
Slider Bearings:
Configuration and Film
Thickness
Sliding surface
or runner
ro
ri
Na
Dimensionless minimum film thickness, Ho = ho /sh
1.0
.8
Maximum
normal
load
.6
.4
.2
0
l
Minimum power
consumed
1
2
3
4
Length-to-width ratio, λ = l/wt
Figure 12.13 Chart for determining
Pads
minimum film thickness
corresponding to maximum load or
Figure 12.12 Configuration of
minimum power loss for various
multiple fixed-incline thrust slider
pad proportions in fixed-incline
bearing
bearings.
Hamrock • Fundamentals of Machine Elements
Film Thickness for Hydrodynamic Bearings
Surface finish
(centerline average), Ra
µm
µin.
0.1-0.2
4-8
Examples of
manufacturing
methods
Grind, lap,
and superfinish
Grind and lap
Approximate
relative
costs
17-20
Allowable outlet (minimum)
film thicknessa , ho
µm
µin.
2.5
100
Description of surface
Mirror-like surface
without toolmarks;
close tolerances
.2-.4
8-16
Smooth surface with17-20
6.2
250
out scratches; close
tolerances
.4-.8
16-32
Smooth surfaces; close
Grind, file,
10
12.5
500
tolderances
and lap
.8-1.6
32-63
Accurate bearing surGrind, precision
7
25
1000
face without toolmarks
mill, and file
1.6-3.2
63-125
Smooth surface withShape, mill,
5
50
2000
out objectionable toolgrind and
marks; moderate
turn
tolerances
a The values of film thickness are given only for guidance. They indicate the film thickness required to avoid metal-to-metal
contact under clean oil conditions with no misalignment. It may be necessary to take a larger film thickness than that
indicated (e.g., to obtain an acceptable temperature rise). It has been assumed that the average surface finish of the pads
is the same as that of the runner.
Table 12.1 Allowable outlet (minimum) film thickness for a given
surface finish.
Hamrock • Fundamentals of Machine Elements
Thrust Bearing - Film Thickness
10
Dimensionless minimum film thickness, Ho = ho /sh
6
Length-towidth ratio,
λ
4
0
2
1/2
1
1
2
3
4
.6
.4
.2
.1
0
1
2
4
6
10
20
40
60
100
200
400
1000
Bearing number, Bt
Figure 12.14 Chart for determining minimum film thickness for
fixed-incline thrust bearings.
Hamrock • Fundamentals of Machine Elements
Thrust Bearings - Temperature Rise
1000
Dimensionless temperature rise, 0.9 Cs wt l ∆t m /Wz
600
400
Length-towidth ratio,
λ
200
100
4
3
60
2
40
1
20
1/2
0
10
6
4
0
1
2
4
6
10
20
40
60
100
200
400
1000
Bearing number, Bt
Figure 12.14 Chart for determining dimensionless temperature rise due
to viscous shear heating of lubricant for fixed-incline thrust bearings.
Hamrock • Fundamentals of Machine Elements
Thrust Bearings - Friction Coefficient
1000
Dimensionless coefficient of friction, µl / sh
600
Length-towidth ratio,
λ
400
200
100
4
60
40
1/2
1
3
2
0
20
10
6
4
2
1
0
1
2
4
6
10
20
40
60
100
200
400
1000
Bearing number, Bt
Figure 12.16a Chart for determining friction coefficient for fixedincline thrust bearings.
Hamrock • Fundamentals of Machine Elements
Thrust Bearings - Power Loss
1000
Power loss variable, 1.5 hpl / Wz ub sh, W-s/N-m
600
Length-towidth ratio,
L
400
200
4
100
1
60
40
3
2
1/2
0
20
10
6
4
2
1
0
1
2
4
6
10
20
40 60
100
200
400
1000
Bearing number, Bt
(b)
Figure 12.16b Chart for determining power loss for fixed-incline
thrust bearings.
Hamrock • Fundamentals of Machine Elements
Thrust Bearings - Lubricant Flow
4
q
q – qs
0
Dimensionless flow, q/wt ub sh
qs (both sides)
Length-towidth ratio,
L
1/2
1
3
2
4
3
1
0
2
1
2
4
6
10
20
40
60
100
200
400
1000
Bearing number, Bt
(c)
Figure 12.16c Chart for determining
lubricant flow for fixedincline thrust bearings.
Hamrock • Fundamentals of Machine Elements
Thrust Bearings - Side Flow
1.0
Volumetric flow ratio, qs /q
.8
Length-towidth ratio,
λ
.6
4
3
.4
2
.2
1
1/2
0
0
1
2
4
6
10
20
40
60
100
200
400
1000
Bearing number, Bt
Figure 12.16d Chart for determining lubricant side flow for fixedincline thrust bearings.
Hamrock • Fundamentals of Machine Elements
Wr
Journal Bearing Pressure Distribution
e
0a
0
Wb
hmin
F0
&
Film pressure, p
Fmax
Figure 12.17 Pressure
distribution around a journal
bearing.
pmax
Hamrock • Fundamentals of Machine Elements
Concentric Journal Bearing
u
Na
Figure 12.18 Concentric
Journal Bearing
r
c
wt
2πr
u
u
Bearing
h=c
Journal
Figure 12.19 Developed journal bearing surfaces for a concentric
journal bearing.
Hamrock • Fundamentals of Machine Elements
Typical Radial Load for Journal Bearings
Application
Automotive engines:
Main bearings
Connecting rod bearing
Diesel engines:
Main bearings
Connecting rod bearing
Electric motors
Steam turbines
Gear reducers
Centrifugal pumps
Air compressors:
Main bearings
Crankpin
Centrifugal pumps
Average radial load per area, Wr∗
psi
MPa
600-750
1700-2300
4-5
10-15
900-1700
1150-2300
120-250
150-300
120-250
100-180
6-12
8-15
0.8-1.5
1.0-2.0
0.8-1.5
0.6-1.2
140-280
280-500
100-180
1-2
2-4
0.6-1.2
Table 12.2 Typical radial load per area Wr* in use for journal bearings.
Hamrock • Fundamentals of Machine Elements
Journal Bearing - Film Thickness
Figure 12.20 Effect of bearing number on minimum film
thickness for four diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Journal Bearings - Attitude Angle
Attitude angle, Φ, deg
100
Diameter-towidth ratio,
λj
80
0
1
60
2
4
40
20
0
10 –2
10 –1
100
101
Bearing number, Bj
Figure 12.21 Effect of bearing number on attitude angle for four
different diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Journal Bearing - Friction Coefficient
2 × 102
Dimensionless coefficient
of friction variable, rbµ /c
102
Diameter-towidth ratio,
λj
2
1
10
0
4
1
100
0
10 –2
10 –1
Bearing number, Bj
100
101
Figure 12.21 Effect of bearing number on coefficient of friction
for four different diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Journal Bearing - Flow Rate
Dimensionless volumetric
flow rate, Q = 2πq/rbcwtω b
8
Diameter-towidth ratio,
λj
6
4
2
1
4
0
2
0
10 – 2
10 –1
100
101
Bearing number, Bj
Figure 12.23 Effect of bearing number on dimensionless volume
flow rate for four different diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Journal Bearing - Side Flow
Side-leakage flow ratio, qs /q
1.0
4
.8
Diameter-towidth ratio,
λj
2
.6
.4
1
.2
0
10 –2
10 –1
100
101
Bearing number, Bj
Figure 12.21 Effect of bearing number on side-flow leakage for
four different diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Journal Bearing - Maximum Pressure
Dimensionless maximum film
pressure, Pmax = Wr /2rb wt pmax
1.0
.8
Diameter-towidth ratio,
λj
0
.6
1
.4
2
4
.2
0
10 – 2
10 –1
100
101
Bearing number, Bj
Figure 12.25 Effect of bearing number on dimensionless maximum
film pressure for four different diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Diameter-towidth ratio,
λj
φ0
φmax
100
25
0
1
2
4
80
20
4
60
2
40
15
10
1
20
5
0
0
0
10 – 2
10 –1
100
Location of maximum pressure, φmax, deg
Location of terminating pressure, φ0, deg
Journal Bearing - Maximum and
Terminating Pressure Location
101
Bearing number, Bj
Figure 12.21 Effect of bearing number on location of terminating
and maximum pressure for four different diameter-to-width ratios.
Hamrock • Fundamentals of Machine Elements
Minimum film thickness, h min; power loss, hp;
outlet temperature, tmo; volumetric flow rate, q
Journal Bearing - Effect of Radial Clearance
hmin
hp
t mo
q
0
.5
1.0
1.5
2.0
2.5
3.0 × 10–3
Radial clearance, c, in.
Figure 12.27 Effect of radial clearance on some performance
parameters for a particular case.
Hamrock • Fundamentals of Machine Elements
Squeeze Film Bearing
w = – ∂h
∂t
l
z
ho
Surface b
x
Surface a
Figure 12.21 Parallel-surface squeeze film bearing
Hamrock • Fundamentals of Machine Elements
Wz
Wz
Bearing runner
Bearing pad
Bearing
recess
Restrictor
Recess pressure,
pr = 0
pr > 0
Flow,
q=0
Supply pressure,
ps = 0
ps > 0
Manifold
Fluid Film in
Hydrostatic
Bearing
(b)
(a)
Wz
Wz
ho
q
p = pl
q=0
p = ps
p = pl
(d)
(c)
Wz + ∆Wz
Wz – ∆Wz
ho + ∆ ho
ho – ∆ho
q
q
p = pr + ∆ pr
p = ps
(e)
p = pr
p = pr – ∆ pr
p = ps
Figure 12.29 Formation of fluid
film in hydrostatic bearing
system. (a) Pump off; (b) pressure
buildup; (c) pressure times recess
area equals normal applied load;
(d) bearing operation; (e)
increased load; (f) decreased load.
(f)
Hamrock • Fundamentals of Machine Elements
Wz
Radial Flow
Hydrostatic Bearing
ro
ri
p=0
sh
pr
q
ho
Figure 12.30 Radial flow hydrostratic
thrust bearing with circular step pad.
Hamrock • Fundamentals of Machine Elements