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Desjardins-Project-Final.pdf
MANE-6906- Friction, Wear and Lubrication of Materials
The Effect Eccentricity and
Temperature on Plain and
Tri-Lobed Journal Bearings
Marc Desjardins
12
Table of Contents
Table of Figures ............................................................................................................................................. 2
Table of Tables .............................................................................................................................................. 2
Purpose: ........................................................................................................................................................ 3
Background: .................................................................................................................................................. 3
Discussion:..................................................................................................................................................... 4
Plain Journal Bearing: .................................................................................................................................... 5
Tri-Lobed Journal Bearing: ............................................................................................................................ 9
Conclusion: .................................................................................................................................................. 13
References:.................................................................................................................................................. 14
1
Table of Figures
Figure 1: Journal Bearing and Force Balance ............................................................................................... 3
Figure 2: COMSOL 2D Pressure Distribution .............................................................................................. 5
Figure 3: Pressure Around Shaft Circumference ......................................................................................... 6
Figure 4: Comparison Between Maximum Pressure Results ....................................................................... 7
Figure 5: Engine Oil Viscosity Vs. Temperature ......................................................................................... 7
Figure 6: Comparison Between COMSOL And Analytical Results For A Varied Temperature ................ 9
Figure 7: COMSOL Model of Tri-Lobed Bearing ....................................................................................... 9
Figure 8: Maximum Fluid Pressure Vs Temperature For a Tri-Lobed Journal Bearing ............................ 10
Figure 9: Maximum Fluid Pressure Vs Eccentricity For Tri-Lobed Bearing ............................................ 11
Figure 10: COMSOL 2D Pressure Distribution Comparison ..................................................................... 12
Figure 11: Pressure Along Shaft Circumference of Tri-lobed Journal Bearing ......................................... 12
Table of Tables
Table 1: Constant Parameters With Varying Eccentricity ............................................................................ 5
Table 2: Maximum Fluid Pressure For COMSOL and Analytical Results .................................................. 6
Table 3: Parameter Constants As Temperature is Varied ............................................................................ 8
Table 4: COMSOL and Analytical Results For The Maximum Fluid Pressure With A Varied
Temperature .................................................................................................................................................. 8
Table 5: COMSOL Model Constants ......................................................................................................... 10
Table 6: Maximum Fluid Pressure For A Varying Temperature ............................................................... 10
Table 7: COMSOL Constants for a Varying Tri-Lobed Bearing Eccentricity .......................................... 11
Table 8: Maximum Pressure For a Varying Eccentricity In Two Separate Directions .............................. 11
2
Purpose:
The purpose of this paper is to investigate the effects of eccentricity and temperature on the pressure
developed between the journal shaft and shaft sleeve in both a plain journal bearing and tri-lobed journal
bearing. The influence of eccentricity is evaluated analytically and using computational fluid dynamic
software, while the effect of temperature is seen solely through the use of CFD software. Since very little
analytical expressions have been derived for tri-lobed journal bearings, the use of CFD is employed to
analyze pressure distributions and to vary the geometrical features and temperatures of the hydrodynamic
system.
Background:
Hydrodynamic journal bearings operate under the principal of building up a hydrodynamic wedge
forming between two solid parts sliding relative to one another. In a journal bearing, the gap between the
rotating shaft (journal) and the outer sleeve is filled with oil. By introducing eccentricity (centers not
aligned) between the journal and sleeve, a load supporting wedge is generated due to the dynamics of the
fluid. The load is used to offset an external load on the journal shaft such that there is a force equilibrium
between the fluid force and the external force on the shaft allowing the journal to be stable and rotate
freely with no solid-solid contact. Figure 1 below, although exaggerated, shows the force balance that
takes place in a journal bearing.
Figure 1: Journal Bearing and Force Balance
The plain journal bearing above is the most common type of journal bearing due to its simplicity, high
load capacity and ease of manufacture. However, as the design speeds of machines are increased, it’s
been found that the plain journal bearing has limitations due to oil whirl. If the shaft receives a disturbing
force or external shock, it can momentarily increase the eccentricity from its equilibrium position. When
this occurs, additional oil is immediately pumped into the space vacated by the shaft. This results in an
increased pressure of the load-carrying film, creating additional force between the oil film and shaft. In
this case, the oil film can actually drive the shaft ahead of it in a forward circular motion and into a
whirling path around the bearing within the bearing clearance. If there is sufficient damping within the
system, the shaft can be returned to its normal position and stability. Otherwise, the shaft will continue in
its whirling motion, which may become violent. Oil whirl is very undesirable because of high vibration
amplitudes, forces, and cyclic stresses that are imposed on the shaft, bearings and machine. If the shaft is
operating at very high rotational speeds and the oil whirl frequency coincides with the system’s natural
frequency oil whip can occur, which can potential lead to catastrophic failure in a relatively short amount
of time. To eliminate oil whirl, a variety of bearings, which are modifications to the profile of the plain
3
journal bearing, have been used. Variations include the lemon bore, pressure dam, and lobed bearings.
Lobed bearings usually have two, three or four lobes. Lobed bearings are usually used in machinery with
very high rotational speeds and lighter loads. However, more and more systems are including variable
speed control. Therefore, the following discussion section analyzes the pressure created by a plain
journal bearing at 100 rotations per minute and a tri-lobed journal bearing at the same rotational speed.
Discussion:
In order to analytically analyze the effects of temperature and eccentricity of a plain journal bearing the
equations of continuity and motion must be combined, under the assumptions of lubrication theory, to
solve for the pressure distribution of the plain journal bearing. The key assumptions of lubrication theory
are: (1) laminar flow, (2) constant pressure across the film thickness, (3) steady state conditions, (4)
Newtonian fluid, (5) constant fluid density, (6) no-slip condition at the walls, (7) rigid and smooth walls,
(8) constant fluid viscosity, and (9) negligible body and inertial forces.
In this case we consider the case of an infinitely long journal bearing. This case is approximated when
the width is at least four times as large as the radius. The bearing radius rb is slightly larger than that of
the journal rj. The clearance is given by the following equation:
(Equation 1)
(Equation 2)
Under rotating and loaded conditions the centers of the journal and the bearing become separated by a
distance (e) called the eccentricity. Therefore, the distance between the journal and the bearing varies
over the circumference shown in Figure 1. The minimum film thickness (hmin) over the circumference of
the journal bearing is given by Equation 2:
Where: = e/c
In this case, the following integrated Reynolds equation can be derived by combining the governing
equations for continuity and fluid motion (Navier-Stokes).
(Equation 3)
"
!
Applying the Sommerfeld periodic solution to the integration of Equation 3 where p=p0 at theta equal to 0
and 2 yields the following equation for the pressure of the fluid between the journal (shaft) and the
sleeve.
/0 , # .
2
, 1 (Equation 4)
This equation is only accurate for positive pressures (i.e. not when cavitation occurs downstream of the
tightest clearance). The maximum pressure is calculated using Equation 5 below:
'() * + ) + #
$% &
) , (Equation 5)
4
Plain Journal Bearing:
The plain journal bearing maximum pressure is analyzed for its variation with eccentricity. The fluid
used in the analysis is regular engine oil. The analysis varies the eccentricity while keeping the following
parameters constant:
Table 1: Constant Parameters With Varying Eccentricity
Parameter
Variable
Value
Journal Diameter
rJ
0.19 m
Bearing (sleeve) Diameter
rB
0.2 m
Clearance
c
0.01 m
Rotational Speed
100 rpm or 10.47 rad/s
Temperature
T
298 K
0.55 Pa-s
Viscosity
For example, if the eccentricity of the system is 0.001 meter then epsilon will equal 0.1 meters (i.e e/c).
Substituting epsilon and the variables in Table 1 into Equation 5 gives:
#
$%
346 '4() *4 4+ ) 4+ 34**34)5 " &
- ,*34*7#8
) ,4 4 343
This exact geometry is modeled into the COMSOL software to compare the results from a simple CFD
analysis with the analytical study provided above. The pressure distribution for the above scenario is
shown in Figure 2 and Figure 3 below.
Figure 2: COMSOL 2D Pressure Distribution
5
Figure 3: Pressure Around Shaft Circumference
The maximum fluid pressures for the analytical and CFD analyses, for a varying eccentricity, are
provided in Table 2 and Figure 4 below.
Table 2: Maximum Fluid Pressure For COMSOL and Analytical Results
Max Pressure Using Analysis
Max Pressure using COMSOL
Eccentricity
(Pa)
(Pa)
0.0005
622
627
0.001
1,251
1,235
0.0015
1,891
1,856
0.002
2,552
2,503
0.0025
3,241
3186
0.003
3,972
3,920
0.0035
4,758
4,717
0.004
5,623
5,602
0.0045
6,595
6,602
0.005
7,718
7,766
0.0055
9,057
9,179
0.006
10,713
10,909
0.0065
12,852
13,134
0.007
15,768
16,161
6
Figure 4: Comparison Between Maximum Pressure Results
As shown in Figure 4 and Table 2 above, the COMSOL analysis correlates very well with Equation 5 and
the Sommerfeld analysis. At lower eccentricity values COMSOL predicts a slightly lower maximum
pressure while at the higher
er values of eccentricity it predicts a slightly higher maximum pressure.
The second portion of this investigation is to vary the temperature of the oil and calculate the maximum
pressure using both Equation 5 and the COMSOL software. To use Equation 5, the
he effect of temperature
is taken into account by varying the viscosity using Figure 5 and the parameters in Table 3.
Figure 5: Engine Oil Viscosity Vs. Temperature
7
Table 3: Parameter Constants As Temperature is Varied
Parameter
Journal Diameter
Bearing (sleeve) Diameter
Clearance
Rotational Speed
Eccentricity
Epsilon
Variable
rJ
rB
c
e
ε
Value
0.19 m
0.2 m
0.01 m
100 rpm or 10.47 rad/s
0.005 m
0.5
For example, if the temperature of the oil is 95°F and the viscosity is 0.28 Pa-s, Equation 5 becomes:
346 '4*() *4* 4*+ ) 4*+ #
$% 34,934)5 " &
- ':6*37#8
) ,4* 4* 343
The maximum fluid pressures for the analytical and CFD analyses, for a varying oil temperature, are
provided in Table 4 and Figure 6 below.
Table 4: COMSOL and Analytical Results For The Maximum Fluid Pressure With A Varied Temperature
Temperature
(°F)
32
50
68
86
104
122
140
158
176
194
212
230
Maximum Pressure
Using Analysis (Pa)
54,216
24,871
11,263
5,432
2,962
1,750
1,021
605
450
331
243
182
Maximum Pressure
Using COMSOL (Pa)
54,659
25,060
11,338
5,465
2,985
1,774
1,051
639
487
371
286
228
8
Figure 6: Comparison Between COMSOL And Analytical Results For A Varied Temperature
Once again the COMSOL results correlate well with Equation 5 and the Sommerfeld solution. The
maximum pressure varies linearly with viscosity but as shown in Figure 5,, viscosity does not vary linearly
with temperature. The COMSOL program predicts a sligh
slightly
tly higher maximum fluid pressure. At high
temperatures (>75°F) the maximum fluid pressure is far less than with colder oil. These results show why
cooling the lubricating oil is necessary to keep the maximum fluid pressure within a specified range and
the resulting force such that the journal does not contact the outer sleeve.
Tri-Lobed
Lobed Journal Bearing:
An exaggerated version of a tri-lobed
lobed journal bearing was modeled in COMSOL using three ellipses’ and
tangents evenly spaced around the bearing shown in Figure 7.. The effect of temperature on the tri-lobed
tri
bearing was done by varying the temperature and recording the maximum pressure while keeping the
parameters in Table 5 constant.
Figure 77: COMSOL Model Of A Tri-Lobed Bearing
9
Table 5: COMSOL Model Constants
Parameter
Variable
Value
Journal Diameter
rJ
0.19 m
Bearing (sleeve) Diameter
rB
0.2 m
Clearance
c
0.01 m
Rotational Speed
100 rpm or 10.47 rad/s
Eccentricity
e
0.002 m
The results of the study are shown in Table 6 and Figure 8 below. Similar to the plain journal bearing the
maximum fluid pressure decreases rapidly in from 32°F to approximately 120°F and then seems to flatten
as the temperature is further increased
increased.
Table 6: Maximum Fluid Pressure For A Varying Temperature
Temperature (°F)
Maximum Fluid Pressure (Pa)
32
29,084
50
13,336
68
6,038
86
2,946
104
1,653
122
1,023
140
645
158
432
176
365
194
314
212
276
Figure 8: Maximum Fluid Pressure Vs Temperature For a Tri
Tri-Lobed
Lobed Journal Bearing
Due to the perfectly circular nature of the plain journal bearing, the direction of eccentricity is
independent of the maximum fluid pressure developed. In a tri-lobed
lobed bearing, the maximum fluid
pressure developed is dependent on the direction of eccentricity. This analysis bounds the problem by
analyzing the maximum pressure developed when the eccentricity is directed at one of the lobes (i.e. the
top lobe) and when the eccentricity is directed between the lobes (i.e. directly to the right). The
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maximum pressures are recorded in Table 8,, for both scenarios, while keep the following parameters
constant:
Table 7: COMSOL Constants for a Varying Tri
Tri-Lobed
Lobed Bearing Eccentricity
Parameter
Variable
Value
Journal Diameter
rJ
0.19 m
Bearing (sleeve) Diameter
rB
0.2 m
Clearance
c
0.01 m
Rotational Speed
100 rpm or 10.47 rad/s
Temperature
T
86°F
Table 8: Maximum Pressure For a Varying Eccentricity In Two Separate Directions
Eccentricity Right (m) Maximum Pressure (Pa) Eccentricity Up (m)
Maximum Pressure (Pa)
0.001
1,615
0.001
1,056
0.002
2,946
0.002
1,582
0.003
4,484
0.003
2,132
0.004
6,309
0.004
2,823
0.005
8,590
0.005
3,697
0.006
11,671
0.006
4,920
0.007
16,044
0.007
6,797
0.008
23,082
0.008
10,170
0.009
37,900
0.009
18,628
Figure 9: Maximum Fluid Pressure Vs Eccentricity For Tri
Tri-Lobed Bearing
The graph above shows that the direction of eccentricity for a tri
tri-lobed
lobed bearing depends on the direction.
The increase in the maximum pressure also seems to be more severe when compared to the plain journal
bearing (Figure 4).
Figure 10 shows the
he 2D pressure plots from COMSOL for two separate computation
computations using COMSOL.
The figure on the right shows the results of a small eccentricity (0.002 m) to the right while the image on
11
the left shows a large eccentricity to the right (0.008 m). One can see how the left lobe in the left image
has a much lower pressure due to the increased clearance than the left lobe in the left image.
Figure 10: COMSOL 2D Pressure Distribution Comparison
Unlike the plain journal bearing (Figure 2) the tri-lobed bearing (right above) has more than one high
point pressure along the circumference of the journal bearing. This is due to geometry of the tri-lobed
bearing and how the fluid gap expands and contracts more than once around the circumference of the
journal shaft. A typical pressure distribution along the circumference of the journal shaft of the tri-lobed
journal bearing is shown in Figure 11. The two high points represent the contraction of two of the lobes
while the dip represents cavitation downstream of the smallest fluid gap in the flowpath.
Figure 11: Pressure Along Shaft Circumference of Tri-lobed Journal Bearing
12
Conclusion:
Based on the results of this study, one can conclude that the COMSOL program very closely
mimics Sommerfeld’s analytical solution for the maximum oil pressure using fluid dynamic and
lubrication theory analysis. It is obvious that the maximum oil pressure, of both the plain and tri-lobed
journal bearings, varies with temperature in the same manner that viscosity varies with temperature.
Varying the oil temperature also revealed an extreme delta in the maximum attainable pressure, and
therefore maximum load capacity of the journal bearing, as oil temperature is increased.
The tri-lobed journal bearing functions similarly, although lower in magnitude, in terms of the
maximum oil pressure, when compared to the plain journal bearing at high eccentricity values. However,
for the tri-lobed bearing, the direction of eccentricity can have a significant effect on the maximum
attainable pressure. At low eccentricity values, the pressure distribution of the tri-lobed bearing contains
more pressure fluctuations than the plain journal bearing due to the slightly more complex geometry.
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References:
a) Hydrodynamic Journal Bearing, Kopeliovich, Dr. Dmitri; SubsTech;
http://www.substech.com/dokuwiki/doku.php?id=hydrodynamic_journal_bearing&do=sh
ow
b) Multilobe Bearings, John Crane Corporation; dated October 2008;
http://www.kupujucpavky.cz/e-catalogue/DataSheets/Bearings/Multilobe_bearings.pdf
c) A Comparative Modeling Study of Journal Bearings Used in Marine Systems, Wolfinger,
Paul; dated 2011.
d) http://www.kingsbury.com/pdf/catalog-FPJ.pdf, Kingsbury, Inc., July, 2007.
e) Fluid Film Bearing Fundamentals and Failure Analysis, Zeidan F.Y., Herbage B.S;
http://www.bearingsplus.com/uploads/literature/turbomachinery/tms_vol2016_1991.pdf
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