Analysis of Various Parameters Associated with Oil
Lubricated Journal Bearings
by
Paul Wolfinger
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Guitierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Ct
November, 2011
i
© Copyright 2011
by
Paul Wolfinger
All Rights Reserved
ii
CONTENTS
Analysis of Various Parameters Associated with Oil ......................................................... i
Lubricated Journal Bearings ............................................................................................... i
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ACKNOWLEDGMENT .................................................................................................. xi
ABSTRACT .................................................................................................................... xii
1. Introduction / Background ........................................................................................... 1
2. Theory/Methodology ................................................................................................... 3
2.1
2.2
2.3
2.4
Bearing Fundamentals ........................................................................................ 3
2.1.1
Derivation of the Sommerfeld Equation ................................................ 5
2.1.2
Validation of COMSOL Model ............................................................. 9
2.1.3
Derivation of Petroff’s Equation .......................................................... 14
Model Description ............................................................................................ 15
2.2.1
Plain Journal Bearing ........................................................................... 16
2.2.2
Elliptical Journal Bearing..................................................................... 17
2.2.3
Pressure Dam Journal Bearing ............................................................. 18
2.2.4
Offset/Lobe Journal Bearing ................................................................ 20
Conditions ........................................................................................................ 21
2.3.1
Case A .................................................................................................. 21
2.3.2
Case B .................................................................................................. 22
2.3.3
Case C .................................................................................................. 22
2.3.4
Case D .................................................................................................. 22
Assumptions ..................................................................................................... 23
3. Results / Discussion ................................................................................................... 25
3.1
COMSOL Results – Bearing Geometry as a Function of Condition ............... 25
3.1.1
Plain Journal Bearing ........................................................................... 25
iii
3.2
3.1.2
Elliptical Journal Bearing..................................................................... 33
3.1.3
Pressure Dam Journal Bearing ............................................................. 41
3.1.4
Offset/Lobe Journal Bearing ................................................................ 49
COMSOL Results – Conditions as a Function of Bearing Geometry ............. 57
3.2.1
Case A .................................................................................................. 57
3.2.2
Case B .................................................................................................. 59
3.2.3
Case C .................................................................................................. 61
3.2.4
Case D .................................................................................................. 63
4. Conclusions................................................................................................................ 66
5. References.................................................................................................................. 69
6. Appendix A – Supplemental Excel Information ....................................................... 70
iv
LIST OF TABLES
Table 1 Summary of Cases for Bearing Evaluation ....................................................... 21
Table 2 Properties of Engine Oil .................................................................................... 23
Table 3 Summary of Plain Journal Bearing Results in Section 3.1.1 ............................. 31
Table 4 Summary of Elliptical Bearing Results in Section 3.1.2 ................................... 39
Table 5 Summary of Pressure Dam Bearing Results in Section 3.1.3 ........................... 47
Table 6 Summary of Offset/Lobe Bearing Results in Section 3.1.4 .............................. 55
Table 7 Summary of Cases for Bearing Evaluation ....................................................... 57
Table 8 Summary of Case A Results .............................................................................. 57
Table 9 Summary of Case B Results .............................................................................. 59
Table 10 Summary of Case C Results ............................................................................ 61
Table 11 Summary of Case D Results ............................................................................ 63
v
LIST OF FIGURES
Figure 1-1 Marine Lubricating Oil System Schematic ..................................................... 1
Figure 2-1 Newton’s concentric cylinders ........................................................................ 3
Figure 2-2 Plain Journal Bearing (KNG Model from Kingsbury, Inc.) ........................... 4
Figure 2-3 Half of a Plain Journal Bearing (KN Series from Kingsbury, Inc.) ................ 5
Figure 2-4 Plain Journal Bearing Cross Section Under Load [3] ..................................... 6
Figure 2-5 Plain Journal Bearing Clearance versus Theta [1] .......................................... 8
Figure 2-6 Plain Journal Bearing Validation Model – Pressure ..................................... 10
Figure 2-7 Validation Model – Pressure vs Arc Length ................................................. 10
Figure 2-8 Plain Journal Bearing Validation Model - Temperature ............................... 11
Figure 2-9 Excel Plot – Pressure vs Theta ...................................................................... 12
Figure 2-10 Excel Plot – Pressure vs Arc Length........................................................... 12
Figure 2-11 Comparison Excel Plot – Pressure vs Theta ............................................... 13
Figure 2-12 Comparison Excel Plot – Pressure vs Theta ............................................... 13
Figure 2-13 Sketch of Plain Journal Bearing Section ..................................................... 16
Figure 2-14 COMSOL Model - Plain Journal Bearing Section ..................................... 16
Figure 2-15 Sketch of Elliptical Journal Bearing Section .............................................. 17
Figure 2-16 COMSOL Model – Elliptical Journal Bearing Section .............................. 18
Figure 2-17 Sketch of Pressure Dam Journal Bearing Section....................................... 19
Figure 2-18 COMSOL Model – Pressure Dam Journal Bearing Section ....................... 19
Figure 2-19 Sketch of Offset/Lobe Journal Bearing Section.......................................... 20
Figure 2-20 COMSOL Model – Offset/Lobe Journal Bearing Section .......................... 20
Figure 3-1 Plain Journal Bearing Case A Pressure - COMSOL Results ........................ 25
Figure 3-2 Plain Journal Bearing Case A Temperature - COMSOL Results ................. 26
Figure 3-3 Plain Journal Bearing Case B Pressure - COMSOL Results ........................ 27
Figure 3-4 Plain Journal Bearing Case B Temperature - COMSOL Results ................. 27
Figure 3-5 Plain Journal Bearing Case C Pressure - COMSOL Results ........................ 28
Figure 3-6 Plain Journal Bearing Case C Temperature - COMSOL Results ................. 29
Figure 3-7 Plain Journal Bearing Case D Pressure - COMSOL Results ........................ 30
Figure 3-8 Plain Journal Bearing Case D Temperature - COMSOL Results ................. 30
Figure 3-9 Plain Journal Bearing Cases – Pressure Results ........................................... 31
vi
Figure 3-10 Plain Journal Bearing Cases – Temperature Results .................................. 32
Figure 3-11 Elliptical Bearing Case A Pressure - COMSOL Results ............................ 33
Figure 3-12 Elliptical Bearing Case A Temperature - COMSOL Results ..................... 34
Figure 3-13 Elliptical Bearing Case B Pressure - COMSOL Results ............................ 35
Figure 3-14 Elliptical Bearing Case B Temperature - COMSOL Results...................... 35
Figure 3-15 Elliptical Bearing Case C Pressure - COMSOL Results ............................ 36
Figure 3-16 Elliptical Bearing Case C Temperature - COMSOL Results...................... 37
Figure 3-17 Elliptical Bearing Case D Pressure - COMSOL Results ............................ 38
Figure 3-18 Elliptical Bearing Case D Temperature - COMSOL Results ..................... 38
Figure 3-19 Elliptical Journal Bearing Cases – Pressure Results ................................... 39
Figure 3-20 Elliptical Journal Bearing Cases – Temperature Results ............................ 40
Figure 3-21 Pressure Dam Bearing Case A Pressure - COMSOL Results..................... 41
Figure 3-22 Pressure Dam Bearing Case A Temperature - COMSOL Results .............. 42
Figure 3-23 Pressure Dam Bearing Case B Pressure - COMSOL Results ..................... 43
Figure 3-24 Pressure Dam Bearing Case B Temperature - COMSOL Results .............. 43
Figure 3-25 Pressure Dam Bearing Case C Pressure - COMSOL Results ..................... 44
Figure 3-26 Pressure Dam Bearing Case C Temperature - COMSOL Results .............. 45
Figure 3-27 Pressure Dam Bearing Case D Pressure - COMSOL Results..................... 46
Figure 3-28 Pressure Dam Bearing Case D Temperature - COMSOL Results .............. 46
Figure 3-29 Pressure Dam Journal Bearing Cases – Pressure Results ........................... 47
Figure 3-30 Pressure Dam Journal Bearing Cases – Temperature Results .................... 48
Figure 3-31 Offset/Lobe Bearing Case A Pressure - COMSOL Results........................ 49
Figure 3-32 Offset/Lobe Bearing Case A Temperature - COMSOL Results ................. 50
Figure 3-33 Offset/Lobe Bearing Case B Pressure - COMSOL Results ........................ 51
Figure 3-34 Offset/Lobe Bearing Case B Temperature - COMSOL Results ................. 51
Figure 3-35 Offset/Lobe Bearing Case C Pressure - COMSOL Results ........................ 52
Figure 3-36 Offset/Lobe Bearing Case C Temperature - COMSOL Results ................. 53
Figure 3-37 Offset/Lobe Bearing Case D Pressure - COMSOL Results........................ 54
Figure 3-38 Offset/Lobe Bearing Case D Temperature - COMSOL Results ................. 54
Figure 3-39 Offset/Lobe Journal Bearing Cases – Pressure Results .............................. 55
Figure 3-40 Offset/Lobe Journal Bearing Cases – Temperature Results ....................... 56
vii
Figure 3-41 Case A for Each Bearing – Pressure Results .............................................. 58
Figure 3-42 Case A for Each Bearing – Temperature Results ....................................... 58
Figure 3-43 Case B for Each Bearing – Pressure Results .............................................. 60
Figure 3-44 Case B for Each Bearing – Temperature Results........................................ 60
Figure 3-45 Case C for Each Bearing – Pressure Results .............................................. 62
Figure 3-46 Case C for Each Bearing – Temperature Results........................................ 62
Figure 3-47 Case D for Each Bearing – Pressure Results .............................................. 64
Figure 3-48 Case D for Each Bearing – Temperature Results ....................................... 64
viii
NOMENCLATURE
rpm
revolutions per minute
rev
revolutions
rad
radians
m
meter(s)
s
second
U
surface velocity (m/s)
ω
angular velocity (rad/s)
π
pi – constant (≈ 3.14159)
r
radius
o
center of the bearing
o’
center of the journal (shaft)
c
maximum clearance between the bearing and journal center (m)
e
eccentricity of the bearing and journal center - equilibrium (m)
ε
eccentricity ratio – eccentricity over clearance (e / c)
θ
radial position (rad)
h
film thickness as a function of radial position (m)
hm
film thickness at location of maximum pressure (m)
hmin
minimum film thickness (c – e) (m)
γ
Sommerfeld variable
Ri
inner radius (i.e., the radius of the journal (shaft))
Ro
outer radius (i.e., the radius of the bearing)
Nrev
rotational speed (rev/s)
Nrad
rotational speed (rad/s)
τ
shear stress (Pa)
µ
absolute viscosity (Pa*s)
T
torque (N*m)
F
force (N)
d
distance (m)
A
rotor surface area in bearing (m2)
ix
l
length of bearing (m)
P
pressure (Pa) or Pa times 1,000 (kPa)
lbf
pounds force
lbm
pounds mass
x
ACKNOWLEDGMENT
The author would like to thank his wife, who has been so supportive throughout the
entire Masters of Engineering in Mechanical Engineering curriculum. He would also
like to thank the faculty of Rensselaer Polytechnic Institute in Hartford, for sharing their
expertise throughout the curriculum; specifically for this project, he would like to thank
Professor Ernesto Guitierrez-Miravete for serving as project advisor and Professor Craig
Wagner for providing his expertise in fluid dynamics and boundary layers. He would
finally like to thank his classmates, who have caused him to think critically during the
course of study.
xi
ABSTRACT
This project analyzes various bearing types in order to develop relationships of various
parameters associated with oil lubricated journal bearings. In addition, it presents the
resulting pressure and temperature changes of oil inside each journal bearing.
The
model results show trends of four different types of journal bearings: plain, elliptical,
pressure dam, and offset/lobe journal bearings. Each bearing type is evaluated at four
conditions where eccentricity, geometry, and rotational speed are varied individually.
The power loss, in friction horsepower, was expected to directly correlate to the temperature rise of oil in the bearing. However, the temperature rise in the bearing from
case to case was largely dependent on the velocity gradient in the radial direction. Each
bearing type performed differently under the various cases. Each bearing model resulted
in pressure and temperature magnitudes and locations that were expected and explainable. Because pressures and temperatures did not vary significantly, it is concluded that
the overall trade-off between bearing types in industry is cost, simplicity, and manufacturability versus stability, variations in speed and load, and specific application.
xii
1. Introduction / Background
In marine propulsion plants, oil systems service bearings on main machinery and the
shafting for propulsion. Oil is contained in a sump or drainage tank, where service
pumps draw suction. Positive displacement type lubricating oil service pumps provide
pressure and constant flow to each bearing supply in order to lubricate bearing surfaces
and keep material temperatures within design limits. Lubricating oil coolers are provided to remove the heat transferred to the oil by the bearings.
Dependent on oil
characteristics for density, viscosity, and heat capacity, an oil temperature range on the
outlet of the coolers is maintained in order for ideal flow and heat removal. Figure 1-1
shows a common marine lubricating oil system schematically.
Figure 1-1 Marine Lubricating Oil System Schematic
1
Of the generic lubricating oil system described above, this project evaluates how the
fluid behaves in oil lubricated bearings and the effects of different journal bearing types.
Theoretically, the determination of the pressure drop and temperature change across the
bearing would be considered in designing the lubricating oil system (i.e., the pump
rating, heat exchanger capacity, piping size, etc). However, the scope of this project
does not cover the system design following the effects of bearing size and type on the
oil. The oil characteristics are analyzed inside a cross-sectional area of the various
journal bearings and are discussed below.
2
2. Theory/Methodology
2.1 Bearing Fundamentals
Fluid film bearings describe the bearing type that is designed to operate under the
principal of building up a hydrodynamic wedge between parts in relative sliding motion.
In bearing lubrication, the rotating shaft inside the bearing is in the presence of oil,
which generates a wedge between the two due to the dynamics of the fluid. The fluid
generally adheres to each body and must shear in order to accommodate the relative
motion [1]. Using basic principals such as Newton’s concentric cylinders, Figure 2-1, a
fluid film thickness between the bearing surface and rotor provides lubrication, of which
one can determine a velocity and temperature profile. Many factors play a roll in the
dependent parameters of oil pressure and temperature in bearings, such as rotor weight,
forces generated during operation, bearing geometry, fluid properties, rotational speed
(RPM), and bearing type.
All these elements must be considered when designing
bearings for any application.
Figure 2-1 Newton’s concentric cylinders
Journal bearings installed in industrial machinery generally fall into two categories; the
first is full bearings and the second is partial arc bearings. This project evaluates only
full bearings, which completely surround the shaft journal. Various full bearing types
3
are available in industry to support rotors of all sizes and specific applications (i.e.,
thrust and reaction loads, increased horizontal or vertical stability, etc) such as lobed,
pressure dam, and elliptical configurations. Different bearings are commonly manufactured in two halves and mated at a split line. The inner surface of the bearing where the
oil film exists is generally lined with a softer material than that of the bearing. This
material, called babbit, is a tin or lead based alloy and can vary in thickness from 1 to
100 mils depending upon the bearing diameter. In general, as bearing size increases, the
shaft RPM tends to decrease due to inertia. A babbit lining provides a surface which
will not mar or gouge the shaft if contact is made and to allow particles in the lubricant
to be imbedded in the liner without damaging the shaft [2].
This project evaluates several bearing styles of varying diameters and lengths, including
the plain journal bearing. A plain journal bearing, shown in Figure 2-2, contains an
even, symmetric surface in the full 360 degree arc with no grooves, pads, or pressure
dams. Although the plain journal bearing offers simplicity in design and manufacturing,
varying bearing styles in practical application results in superior rotor stability and even
bearing lubrication. An isometric view of half of an actual plain journal bearing, also
courtesy of Kingsbury, Inc., is provided in Figure 2-3.
Figure 2-2 Plain Journal Bearing (KNG Model from Kingsbury, Inc.)
4
Figure 2-3 Half of a Plain Journal Bearing (KN Series from Kingsbury, Inc.)
This pressure at the inlet of the bearing allows the oil to flow and flood the bearing,
creating a fluid film in the clearance gap between the rotor and bearing surface. For this
project, it is assumed that oil is not flowing in or out of the bearings. Grooves or ports to
deliver oil into and out of the bearing have been disregarded for simplicity.
2.1.1 Derivation of the Sommerfeld Equation
As mentioned above, if a load is applied to a journal as it rotates, it will be displaced
from the center of the bearing. With the lubricating film building up pressure to support
the load, the displaced shaft reaches an equilibrium position, as shown in Figure 2-4.
5
Figure 2-4 Plain Journal Bearing Cross Section Under Load [3]
To summarize Figure 2-4, “r” is the radius of the shaft, omega (ω) is the angular velocity
of the shaft, “ o’ ” is the center of the bearing, “o” is the center of the shaft or journal,
and “e” represents the difference between the two (eccentricity). In addition, “hmin” is
the minimum film thickness, “hm” is the approximate thickness associated with maximum pressure, and “h” represents the thickness as a function of radial position, theta (θ).
The continuity and momentum equations are:
u v w
 
0
x y z
[1]
u
u
u
u
1 p
 2u
v w  
 2
x
y
z
 x
y
[2]
u
w
w
w
1 p
2w
v
w

 2
x
y
z
 z
y
[3]
Reducing the momentum equation above using conventional thin film assumptions
results in the following governing equations:
6
p
 2u
 2
x
y
[4]
Neglecting change in viscosity, µ, across the film thickness, h, allows for integration
twice:
u
1 p 2
y  ay  b
2 x
[5]
To solve Equation 5 for constants, boundary conditions must be applied:
u = U at y = 0
u = 0 at y = h
Solving for the constants in Equation 5,
b U
a
1 p
U
h
2 x
h
Therefore, the velocity profile is:
y  1 p 
y

u  U 1   
yh1  
 h  2 x  h 
[6]
By integrating equation 6 across the film thickness, the Reynold’s equation is obtained.
h p h3
 udy U 2  x 2
hm
[7]
dh d  p h3 

 
dx dx  x 6 
[8]
h
U
U h  hm  
p h3
x 6
[9]
Rearranging to get in terms of pressure.
p
hh 
 6 U  3 m 
x
 h 
[10]
Equation 10 above is the general equation used to calculate the differential pressure in a
simplified journal bearing assuming concentric cylinders. In order to account for shaft
eccentricity, the film thickness must vary with theta. The distance between the centers
of the journal and bearing is known as the eccentricity, “e” (i.e., when the two are
7
concentric, e is zero). If the shaft were to move fully to be in contact with the bearing
surface, “e” will equal “c”, or the radial clearance of the bearing. Therefore,
e

c
[11]
Equation 11 represents the eccentricity ratio of any bearing due to loading. In this case,
the eccentricity ratio must be 0 ≤ ε ≤ 1. Because of the eccentricity of the shaft, there is
a location where the thickness of the film is at a minimum. Based on Equation 11, the
minimum film thickness is:
hmin  c  e  c1   
[12]
In order to show the bearing clearance with respect to theta, the bearing can be “rolled
out” to a two dimensional problem more like Cartesian coordinates, as in Figure 2-5.
Figure 2-5 Plain Journal Bearing Clearance versus Theta [1]
In Figure 2-5, hmin is shown at θ = π. To represent the line from 0 ≤ θ ≤ 2π based on
Equation 12, the film thickness, “h”, can be shown as:
h  c1    cos  
[13]
Substituting Equation 13 into Equation 10,


p
1
hm
 60R 2 

2
3

 c1    cos   c1    cos   
[14]

p
1
hm
R 
 60   

2
3

 c   1    cos   1    cos   
[15]
2
And integrating with respect to theta,
8
R
p  60  
c
2

1

hm
  1    cos    1    cos   d  C
2
1
3
[16]
As shown in Reference 4,
1    cos  
1  2
1   cos 
[17]
Applying the Sommerfeld solution to the integrated equation, which is evaluated at the
boundary condition:
p  p0 at   0  2
[18]
Solving for the constant, the pressure is shown as
R
p  60  
c
2
   sin  2    cos  


2
 2   2 1   2




[19]
2.1.2 Validation of COMSOL Model
In order to confirm that the results of the COMSOL model were valid, a model at one
condition was validated with the Sommerfeld Equation derived in Section 2.1.1. The
plain journal bearing model provided in Section 2.2.1 at the Case B condition described
in Section 2.3.1 was used for the validation (i.e., a plain journal bearing at 1000 rpm,
with an eccentricity ratio of 0.424).
Microsoft Excel was used to validate the results generated in COMSOL. Varying theta
from 0 to 2π in 36 increments, pressure was calculated using Equation 19, which is a
function of theta and includes geometric and fluid parameters. Because Equation 19 was
derived under the assumption that the fluid viscosity is constant, a single viscosity for
engine oil was used based on the specified temperature (305K). However, the COMSOL
model was created using laminar, non-isothermal physics with viscous heating. In order
to allow for comparison, viscous heating was disabled in the COMSOL model to focus
solely on pressure of the fluid in the bearing thereby keeping the fluid viscosity constant.
Figure 2-6 below provides the COMSOL two dimensional results showing pressure in
the plain journal bearing. Graphically, Figure 2-7 shows the pressure results as a func-
9
tion of arc length. For information only, the temperature of the fluid in the model is
represented in Figure 2-8.
Figure 2-6 Plain Journal Bearing Validation Model – Pressure
Figure 2-7 Validation Model – Pressure vs Arc Length
10
Figure 2-8 Plain Journal Bearing Validation Model - Temperature
As shown in Figure 2-6 and Figure 2-7, the maximum and minimum pressure determined by COMSOL for the plain journal bearing are 110.1 kPa and -115.6 kPa,
respectively. Using equation 19 above, the maximum and minimum pressures calculated
in Excel were ±62.0 kPa. The maximum and minimum pressures were found at theta
equal to 230 and 130 degrees from zero, assuming the zero location is along the parallel
from point o to o’ as shown in Figure 2-4. The results of using the equation to calculate
pressures are shown graphically in Figure 2-9 and Figure 2-10. First, Figure 2-9 shows
the correlation of pressure to what is described in Reference 3. However, because
bearing rotation is in the clockwise direction, the graph is inverted. Second, Figure 2-10
shifts the graph to compare it to Figure 2-7 generated in COMSOL.
11
Figure 2-9 Excel Plot – Pressure vs Theta
Figure 2-10 Excel Plot – Pressure vs Arc Length
Combining the results from COMSOL and Excel, and then plotting on the same graph
provides a comparison of the data. The pressure as a function of theta and arc length are
shown below in Figure 2-11 and Figure 2-12, respectively
12
Figure 2-11 Comparison Excel Plot – Pressure vs Theta
Figure 2-12 Comparison Excel Plot – Pressure vs Theta
As shown above, the calculated minimum and maximum pressures are approximately
45% less than COMSOL. The pressure variation as a function of position (arc length) is
consistent between both methods. Although the results show a difference in pressure,
the error is expected for several reasons. First, assumptions during the derivation of the
Reynolds equation / Sommerfeld solution such as no flow in the radial direction. Also,
the bearing is assumed to have an infinite length, when L/D ratios can affect the assumptions.
13
2.1.3 Derivation of Petroff’s Equation
Related to work done by Sommerfeld, the Petroff Law was developed to explain the
phenomenon of bearing friction. Petroff's method of lubrication analysis assumed a
concentric shaft and bearing and produces the Petroff equation. This equation, derived
below, defines groups of relevant dimensionless parameters and predicts a coefficient of
friction, even when the shaft is not concentric.
Considering a concentric shaft rotating inside a bearing, it can be assumed that the radial
clearance space is completely filled with lubricant, the bearing is subjected to a negligible load, the bearing is of sufficient length to neglect changes in flow at the ends, and
that leakage is negligible. The surface velocity of the shaft is:
U  2rN rev
[20]
The lubricant shear stress is shown as [1]:
 
u
r r 0
[21]
Assuming a constant rate of shear, the equation is integrated with respect to the radius:
 
U
r1  r0
U
c
2rN rev
 
c
 
[22]
where c is the gap clearance in the bearing. The amount of stress (i.e., pressure in
Pascals) required to shear a fluid of a known viscosity is dependent on how much of the
fluid is involved and how quickly it is occurring. Therefore, the circumference and
rotational speed are also necessary in the shear stress equation above. Knowing the
force required over an area to shear a fluid is required in order to determine the amount
of torque required to overcome this loss. Torque is defined as:
T  F d
[23]
Where
F  P A
[24]
Therefore,
T  (  A)  r
[25]
14
T  (  A)  r
 2rN rev 
T 
  2r  l   r
c


2 3
4 R lN rev
T
c
[26]
The torque required to move the shaft through the lubricating oil can be directly related
to power (kW).
Power 
Power 
2TN rev
60,000
2
 3 R 3lN rev
7,500  c
[27]
[28]
Because marine propulsion systems are rated for a large amount of power (e.g., 1 x 103
to 1 x 105 W), even a small percentage of the power lost can result in a significant
amount of heat being generated and transferred to the lubricating oil. Using the above
equation, power lost in the bearing can be calculated based on known geometry and
operating conditions at the bearing (e.g., the bearing RPM, oil density, bearing film
thickness). Therefore, Section 2.3 varies several of these important bearing design
parameters to confirm correlations developed above. As bearing size and shaft speed
increases, the losses to fluid shear should also increase; on the other hand, an increase in
bearing clearance should decrease the amount of power lost, or heat, to the oil.
2.2 Model Description
Bearing models were generated in the COMSOL Multiphysics program, which allows
the user to manipulate geometry, material properties, mesh sizing, and overall conditions
to evaluate flow and heat transfer among other parameters. Several bearing types were
chosen to trade-off the effects of geometry with resulting oil pressure and temperature
with the benefits of bearing stability. All of the journal bearings below are assumed to be
made from the same material. In general, journal bearings are manufactured with a
material that is durable and have low wear to the bearing and shaft, low friction, resistant
to elevated temperatures, and corrosion resistant. Four bearing types were considered
and are discussed below.
15
2.2.1 Plain Journal Bearing
Journal bearings consisting of a plain bore are the simplest form of journal bearing used
in industry. They have a circular cross section and generally only a single lubricant feed
groove. Although relatively easy to manufacture, simple to analyze, and capable of
carrying a high amount of load, the plain journal bearing has a greater potential for
bearing-induced rotor instability [1]. Because the plain journal bearing is essentially a
rotating cylinder within a cylinder, it was chosen for validation in Section 2.1.2 and is a
good baseline to compare with the other bearing types. Figure 2-13 and Figure 2-14
show both a sketch of a plain journal bearing and the geometry modeled in COMSOL,
respectively.
Figure 2-13 Sketch of Plain Journal Bearing Section
Figure 2-14 COMSOL Model - Plain Journal Bearing Section
16
As shown above, Figure 2-13 is exaggerated to show eccentricity (i.e., the rotor is offcenter inside the bearing) between the bearing and shaft as if the rotor was loaded.
Figure 2-14 assumes a realistic eccentricity for the plain journal bearing, which is
evaluated in three of the four conditions for each bearing type. The four conditions for
each model are discussed at detail in Section 2.3. The plain journal bearing is modeled
to have a radius for the various conditions of either 0.25m or 0.50m with a rotor radius
of 0.24m or 0.48m, respectively. Plain journal bearings have good load carrying capability and excellent damping, however the design is at risk to rotor instability.
2.2.2 Elliptical Journal Bearing
The elliptical journal bearing is a variation to a plain journal bearing in that the circular
rotor is supported by a bearing that is ovular in shape. Therefore, the bearing clearance
is reduced in one plane. In order to manufacture the elliptical, or lemon bore bearing,
shells that were originally circular are intentionally put through large crush, shimmed
offset, or offset machining. This causes the bearings to have clearance that varies evenly
around the circumference each half-turn of the shaft [1]. Figure 2-15 and Figure 2-16
show both a sketch of an elliptical journal bearing and the geometry modeled in
COMSOL, respectively.
Figure 2-15 Sketch of Elliptical Journal Bearing Section
17
Figure 2-16 COMSOL Model – Elliptical Journal Bearing Section
As shown above, Figure 2-15 is exaggerated to show eccentricity and elliptical geometry. Figure 2-16 was created in COMSOL for a realistic bearing size for the elliptical
journal bearing. Similar to the plain journal bearing, the bearing radius for the various
conditions was kept consistent and assumed to be either 0.25m or 0.50m with a rotor
radius of 0.24m or 0.48m, respectively. In order to model the ovular shape of the
bearing, the horizontal (i.e., along the x-plane) distance was increased to 0.26m or
0.52m. Similar to the plain journal bearing, manufacturing and installation costs are
low. Elliptical bore bearings can have a good load carrying capability and are often used
when the load direction is well known (e.g., load is due to gravity). This offers the load
direction to be handled by the stiffest part of the bearing on the minor axis.
2.2.3 Pressure Dam Journal Bearing
A pressure dam bearing has a single wide circumferential groove of gradually increasing
depth. Generally, the groove is in the upper bearing shell and over approximately 135
degrees. The purpose of the pressure dam is to create a pocket that causes a static
pressure to build up, keeping the bearing slightly off-center to stiffen it. This can be
advantageous when significant downward loads can be expected [1]. Figure 2-17 and
18
Figure 2-18 show both a sketch of a pressure dam journal bearing and the geometry
modeled in COMSOL, respectively.
Figure 2-17 Sketch of Pressure Dam Journal Bearing Section
Figure 2-18 COMSOL Model – Pressure Dam Journal Bearing Section
As shown above, Figure 2-17 is exaggerated to show eccentricity and the geometry of
the pressure dam. Figure 2-18 was created in COMSOL for a realistic bearing size for
the pressure dam journal bearing. Similar to the plain journal bearing, the bearing radius
for the various conditions was kept consistent and assumed to be 0.25m with a rotor
radius of 0.24m. In order to model the pressure dam, a circle with radius of 0.26m was
used for approximately 90 degrees.
19
2.2.4 Offset/Lobe Journal Bearing
Similar to the elliptical bearing described in Section 2.2.2, offset bore journal bearings
can be manufactured by offset machining, which decreases the cross-coupled force the
oil wedge creates. In practical application, this tends to break up the consistent whirling
of the lubricant around the bearing that causes shaft whip [1]. Figure 2-19 and Figure
2-20 show both a sketch of a pressure dam journal bearing and the geometry modeled in
COMSOL, respectively.
Figure 2-19 Sketch of Offset/Lobe Journal Bearing Section
Figure 2-20 COMSOL Model – Offset/Lobe Journal Bearing Section
As shown above, Figure 2-19 is exaggerated to show eccentricity and the geometry of
the offset / bearing lobes. Figure 2-20 was created in COMSOL for a realistic bearing
20
size for the offset/lobe journal bearing. Similar to the plain journal bearing, the rotor
radius for the various conditions was kept consistent at 0.24m. The bearing radius
started at 0.25m, but was offset 0.01m. In order to model the lobes along the horizontal
axis, two circles with radii of 0.1m were centered on the offset section of the bearing.
The offset journal bearing is similar to a pressure dam style bearing, however there are
two pressure dams in the x-plane with lobes at the end of the dam.
2.3 Conditions
Each of the four bearing types discussed in Section 2.2 were evaluated at four different
conditions. The varying conditions allow for relationships to be developed between
bearing size, eccentricity, and shaft RPM. To summarize, each of the bearings were
evaluated in COMSOL according to Table 1.
Table 1 Summary of Cases for Bearing Evaluation
Case
Bearing Radius (m)
Rotor Radius (m)
Rotational Speed
(rad/sec)
Offset in X-dir (m)
Offset in Y-dir (m)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
0.25
0.24
B
0.25
0.24
C
0.50
0.48
D
0.50
0.48
104.7
104.7
104.7
209.4
0
0
0
0.01
0.000
0.003
0.003
0.003
0.003
0.003
0.003
0.00424 0.00424 0.00424
0.01
0.02
0.02
0.424
0.212
0.212
Where specific bearing geometries required additional modification from Case A and B
to Case C and D (i.e., the model size doubled in diameter), dimensions of areas such as
lobe size and pressure dam were scaled up accordingly. Each of the cases shown in
Table 1 are described below.
2.3.1 Case A
In order to establish a baseline, Case A was evaluated to determine that the model
created in COMSOL was constructed properly, and with correct values. The bearing and
21
rotor radius for this condition are 0.25m and 0.24m, respectively, and are considered
concentric to each other during shaft rotation. Shaft speed is assumed to 1,000 rpm.
2.3.2 Case B
Case B is similar to Case A in that the shaft rotational speed and geometries of both the
bearing and rotor are the same. However, a slight eccentricity was introduced in the
expected direction of normal machine operation (i.e., if the shaft is rotating in a clockwise manner, the center of the shaft will move down and to the left, as shown in Figure
2-4). The shaft was moved 0.003m in the –x and –y direction (i.e., left and down) to
provide eccentricity. Non-concentric shafts are common in real applications, even at
very low clearances. Because of this, Case B was considered for validation of the
COMSOL model, as mentioned in Section 2.1.2.
2.3.3 Case C
The purpose of Case C is to allow only the size of the bearing and rotor to change. That
is, the radii of both dimensions are doubled from Case A and B from 0.24m and 0.25m
to 0.48m and 0.50m. In this Case, the shaft offset of 0.003m in both the horizontal and
vertical are kept the same, decreasing the eccentricity ratio by half (i.e., eccentricity
stays the same but the radial clearance associated with the ratio is larger). Shaft rotational speed is the same as Case A and B at 1,000 rpm.
2.3.4 Case D
Case D uses the larger dimensions of Case C in the model. The only parameter changing
in Case D is the shaft speed in the bearing. This is the only Case where the shaft speed
is varied, and this is in order to obtain a relationship between a rotational speed (N) from
Section 2.1.3. The shaft rotational speed is doubled from Case A, B, and C to 2,000
rpm.
22
2.4 Assumptions
For modeling in COMSOL, several assumptions were made to evaluate each of the
bearings at the various cases. Below, the assumptions are listed:

Similar to the derivations above, only the cross section of the bearing is considered, and the bearing has infinite depth such that effects due to losses can be
neglected.

Marine journal bearings are usually supplied with lubricating oil from a ship’s
system, which also returns the oil back to a collection area via drain to cool it.
The interface with the system is not considered, and the bearing has no oil inlet
or outlet.

Oil properties are consistent with engine oil and change with temperature, with
the exception of the COMSOL validation model, which assumed a constant oil
density due to Equation 19 also holding density constant for oil at 305 K. The
engine oil properties are provided in Table 2.
Table 2 Properties of Engine Oil
Temp (K)
273
275
280
285
290
295
300
305
310
315
320
325
330
335
340
μ
3.85
3.30
2.24
1.51
1.01
0.69
0.47
0.34
0.25
0.19
0.15
0.11
0.09
0.06
0.05
Cp
1,797
1,805
1,826
1,846
1,867
1,888
1,908
1,929
1,950
1,971
1,992
2,013
2,035
2,056
2,077
23
ρ
899.6
898.4
895.4
892.5
889.5
886.5
883.5
880.5
877.6
874.6
871.6
868.7
865.7
862.8
859.8
k
0.147
0.147
0.147
0.146
0.145
0.145
0.144
0.144
0.143
0.143
0.142
0.141
0.141
0.140
0.140

The bearing and journal material are not specified for the model

Physics in COMSOL were set to steady state, non-isothermal laminar flow,
where there is a no slip condition at walls. For heat transfer, the temperature at
bearing surface is defined as 305 K and the journal surface insulated (i.e., no heat
transfer across the boundary). Therefore, the only source of heat for the oil is
from viscous heating.

The “fine” mesh for each bearing model was used in all cases, which results in
approximately 918 to 1801elements, depending on bearing geometry.
The
COMSOL stationary solver criterion is set for convergence tolerance of 0.002%
error in less than 100 iterations.
24
3. Results / Discussion
3.1 COMSOL Results – Bearing Geometry as a Function of Condition
Using the bearing geometries described in Section 2.2, results were obtained with
COMSOL Multiphysics. In this section, each bearing geometry is presented individually
at the various Cases discussed in Section 2.3 to compare the effects to oil pressure and
temperature.
3.1.1 Plain Journal Bearing
The bearing and rotor radius for this Case A are 0.25m and 0.24m, respectively, and are
considered concentric to each other during shaft rotation. Shaft speed is assumed to
1,000 rpm in the clockwise direction. The maximum and minimum pressures for Case A
are 4.36 kPa and -5.49 kPa, as shown in Figure 3-1. Figure 3-2 provides the maximum
and minimum temperatures, which are 305.64 K and 305 K.
Figure 3-1 Plain Journal Bearing Case A Pressure - COMSOL Results
25
Figure 3-2 Plain Journal Bearing Case A Temperature - COMSOL Results
In Case B, the shaft rotational speed and geometries of both the bearing and rotor are the
same as Case A. However, a slight eccentricity was introduced in the expected direction
of normal machine operation (i.e., if the shaft is rotating in a clockwise manner, the
center of the shaft will move down and to the left). The shaft was moved 0.003m in the
–x and –y direction (i.e., left and down) to provide eccentricity. The maximum and
minimum pressures for Case B are 104.98 kPa and -130.96 kPa, as shown in Figure 3-3.
Figure 3-4 provides the maximum and minimum temperatures, which are 306.69 K and
305 K.
26
Figure 3-3 Plain Journal Bearing Case B Pressure - COMSOL Results
Figure 3-4 Plain Journal Bearing Case B Temperature - COMSOL Results
27
For Case C, the radii of both the shaft and bearing dimensions are doubled when compared to Case A and B. That is, the rotor and bearing radii are 0.48m and 0.50m in Case
C, but the shaft offset of 0.003m in both the horizontal and vertical is maintained. Shaft
rotational speed is the same as Case A and B at 1,000 rpm. The maximum and minimum
pressures for Case C are 190.74 kPa and -209.30 kPa, as shown in Figure 3-5. Figure
3-6 provides the maximum and minimum temperatures, which are 305.29 K and 305 K.
Figure 3-5 Plain Journal Bearing Case C Pressure - COMSOL Results
28
Figure 3-6 Plain Journal Bearing Case C Temperature - COMSOL Results
Case D uses the larger rotor and bearing radii of 0.48m and 0.50m, similar to Case C.
The shaft offset of 0.003m in both the horizontal and vertical is maintained.
The shaft
rotational speed, however, is doubled from 1,000 rpm in Case A, B, and C to 2,000 rpm.
The maximum and minimum pressures for Case D are 701.24 kPa and -795.62 kPa, as
shown in Figure 3-7. Figure 3-8 provides the maximum and minimum temperatures,
which are 305.51 K and 305 K.
29
Figure 3-7 Plain Journal Bearing Case D Pressure - COMSOL Results
Figure 3-8 Plain Journal Bearing Case D Temperature - COMSOL Results
30
The results of the various Cases evaluated with the plain journal bearing models above
are summarized in Table 3.
Table 3 Summary of Plain Journal Bearing Results in Section 3.1.1
Case
Bearing Radius (m)
Rotor Radius (m)
Rotational Speed
(rad/sec)
Offset in X-dir (m)
Offset in Y-dir (m)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
A
0.25
0.24
B
0.25
0.24
C
0.50
0.48
D
0.50
0.48
104.7
104.7
104.7
209.4
0
0
0
0.01
0.000
4.36
-5.49
305.64
0.003
0.003
0.003
0.003
0.003
0.003
0.00424 0.00424 0.00424
0.01
0.02
0.02
0.424
0.212
0.212
104.98 190.74 701.24
-130.96 -209.30 -795.62
306.69 305.29 305.51
From Table 3, the pressure and temperature changes associated with the plain journal
bearing models are provided in Figure 3-9 and Figure 3-10, respectively.
Figure 3-9 Plain Journal Bearing Cases – Pressure Results
31
Figure 3-10 Plain Journal Bearing Cases – Temperature Results
Comparing the pressures in Table 3, the pressure generated increases from Case A to
Case D, as shown in Figure 3-9. Case A results in the smallest pressure generation due
to the bearing and shaft being concentric. Case B has a pressure approximately 24 times
greater than Case A because of the slight eccentricity introduced. When comparing Case
C to Case B, the pressure increases almost 82% due to the increase in bearing and shaft
diameter. Finally, from Case C to Case D, the pressure is approximately 3.7 times
greater. Case D not only uses the larger bearing geometry, but the shaft rotational speed
was doubled. The change in oil temperature also varied with respect to the various
Cases, and is provided in Figure 3-10.
The oil temperature in Case A increased 0.64 K due to viscous heating in the bearing. In
Case B, the change in temperature is approximately 2.6 times larger than Case A due to
the eccentricity of the shaft. Cases C and D result in the lower temperature changes,
about 45% and 80% of Case A, respectively. The lower change in temperature is
associated with the decrease in eccentricity ratio of the last two cases (i.e., from Case B
to Cases C and D, the bearing geometry doubled, but the eccentricity remained the same,
thereby reducing the eccentricity ratio by half). The maximum temperature in Case D is
larger than in Case C due to the increase in rotational speed of the shaft (i.e., the fluid is
shearing quicker).
32
3.1.2 Elliptical Journal Bearing
The bearing and rotor radius for this Case A are 0.25m and 0.24m, respectively, and are
considered concentric to each other during shaft rotation. In order to model the ovular
shape of the bearing, the horizontal (i.e., along the x-plane) distance was increased to
0.26m. Shaft speed is assumed to 1,000 rpm in the clockwise direction. The maximum
and minimum pressures for Case A are 36.73 kPa and -56.77 kPa, as shown in Figure
3-11. Figure 3-12 provides the maximum and minimum temperatures, which are 305.59
K and 305 K.
Figure 3-11 Elliptical Bearing Case A Pressure - COMSOL Results
33
Figure 3-12 Elliptical Bearing Case A Temperature - COMSOL Results
In Case B, the shaft rotational speed and geometries of both the bearing and rotor are the
same as Case A. However, a slight eccentricity was introduced in the expected direction
of normal machine operation (i.e., if the shaft is rotating in a clockwise manner, the
center of the shaft will move down and to the left). The shaft was moved 0.003m in the
–x and –y direction (i.e., left and down) to provide eccentricity. The maximum and
minimum pressures for Case B are 106.13 kPa and -154.52 kPa, as shown in Figure
3-13. Figure 3-14 provides the maximum and minimum temperatures, which are 305.73
K and 305 K.
34
Figure 3-13 Elliptical Bearing Case B Pressure - COMSOL Results
Figure 3-14 Elliptical Bearing Case B Temperature - COMSOL Results
35
For Case C, the radii of both the shaft and bearing dimensions are doubled when compared to Case A and B. That is, the rotor and bearing radii are 0.48m and 0.50m in Case
C, but the shaft offset of 0.003m in both the horizontal and vertical is maintained. Shaft
rotational speed is the same as Case A and B at 1,000 rpm. The maximum and minimum
pressures for Case C are 245.79 kPa and -376.15 kPa, as shown in Figure 3-15. Figure
3-16 provides the maximum and minimum temperatures, which are 305.24 K and 305 K.
Figure 3-15 Elliptical Bearing Case C Pressure - COMSOL Results
36
Figure 3-16 Elliptical Bearing Case C Temperature - COMSOL Results
Case D uses the larger rotor and bearing radii of 0.48m and 0.50m, similar to Case C.
The shaft offset of 0.003m in both the horizontal and vertical is maintained.
The shaft
rotational speed, however, is doubled from 1,000 rpm in Case A, B, and C to 2,000 rpm.
The maximum and minimum pressures for Case D are 1,217.2 kPa and -1,651.7 kPa, as
shown in Figure 3-17. Figure 3-18 provides the maximum and minimum temperatures,
which are 305.3 K and 305 K.
37
Figure 3-17 Elliptical Bearing Case D Pressure - COMSOL Results
Figure 3-18 Elliptical Bearing Case D Temperature - COMSOL Results
38
The results of the various Cases evaluated with the elliptical bearing models above are
summarized in Table 4.
Table 4 Summary of Elliptical Bearing Results in Section 3.1.2
Case
Bearing Radius (m)
Rotor Radius (m)
Rotational Speed
(rad/sec)
Offset in X-dir (m)
Offset in Y-dir (m)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
A
0.25
0.24
B
0.25
0.24
C
0.50
0.48
D
0.50
0.48
104.7
104.7
104.7
209.4
0
0
0
0.01
0.000
36.73
-56.77
305.59
0.003
0.003
0.003
0.003
0.003
0.003
0.00424 0.00424 0.00424
0.01
0.02
0.02
0.424
0.212
0.212
106.13 245.79 1,217.2
-154.52 -376.15 -1,651.7
305.73 305.24
305.30
From Table 4, the pressure and temperature changes associated with the plain journal
bearing models are provided in Figure 3-19 and Figure 3-20, respectively.
Figure 3-19 Elliptical Journal Bearing Cases – Pressure Results
39
Figure 3-20 Elliptical Journal Bearing Cases – Temperature Results
Comparing the pressures in Table 4, the pressure generated varies with respect to the
cases, as shown in Figure 3-19. Case A results in the smallest pressure generation due to
the bearing and shaft being concentric. Case B has a pressure approximately 2.9 times
greater than Case A because of the slight eccentricity introduced. When comparing Case
C to Case B, the pressure increases about 2.3 times due to the increase in bearing and
shaft diameter. Finally, from Case C to Case D, the pressure increases about 5 times.
Consistent with the plain journal bearing, Case D produced the largest pressure; however, the Case D elliptical bearing maximum pressure is 74% larger than the maximum
pressure of the plain journal bearing. A comparison of the various bearing types with
the results obtained in COMSOL is provided in Section 3.2.
The change in oil temperature varied with the different conditions in a similar fashion to
the plain journal bearing, and is provided in Figure 3-20. The oil temperature in Case A
increased 0.59 K due to viscous heating in the bearing. In Case B, the change in temperature increased almost 24% from Case A due to the eccentricity of the shaft. Cases C
and D result in the lower temperature changes, about 41% and 51% of Case A, respectively. The lower change in temperature is associated with the decrease in eccentricity
ratio of the last two cases.
40
3.1.3 Pressure Dam Journal Bearing
Similar to the plain journal bearing, the bearing radius for Case A was kept consistent
and assumed to be 0.25m with a rotor radius of 0.24m. In order to model the pressure
dam, a circle with radius of 0.26m was used for approximately 90 degrees. Shaft speed
is assumed to 1,000 rpm in the clockwise direction. The maximum and minimum
pressures for Case A are 48.76 kPa and -22.39 kPa, as shown in Figure 3-21. Figure
3-22 provides the maximum and minimum temperatures, which are 305.57 K and 305 K.
Figure 3-21 Pressure Dam Bearing Case A Pressure - COMSOL Results
41
Figure 3-22 Pressure Dam Bearing Case A Temperature - COMSOL Results
In Case B, the shaft rotational speed and geometries of both the bearing and rotor are the
same as Case A. However, a slight eccentricity was introduced in the expected direction
of normal machine operation (i.e., if the shaft is rotating in a clockwise manner, the
center of the shaft will move down and to the left). The shaft was moved 0.003m in the
–x and –y direction (i.e., left and down) to provide eccentricity. The maximum and
minimum pressures for Case B are 107.92 kPa and -114.06 kPa, as shown in Figure
3-23. Figure 3-24 provides the maximum and minimum temperatures, which are 306.98
K and 305 K.
42
Figure 3-23 Pressure Dam Bearing Case B Pressure - COMSOL Results
Figure 3-24 Pressure Dam Bearing Case B Temperature - COMSOL Results
43
For Case C, the radii of both the shaft and bearing dimensions are doubled when compared to Case A and B. That is, the rotor and bearing radii are 0.48m and 0.50m in Case
C, but the shaft offset of 0.003m in both the horizontal and vertical is maintained. Shaft
rotational speed is the same as Case A and B at 1,000 rpm. The maximum and minimum
pressures for Case C are 249.13 kPa and -224.20 kPa, as shown in Figure 3-25. Figure
3-26 provides the maximum and minimum temperatures, which are 305.29 K and 305 K.
Figure 3-25 Pressure Dam Bearing Case C Pressure - COMSOL Results
44
Figure 3-26 Pressure Dam Bearing Case C Temperature - COMSOL Results
Case D uses the larger rotor and bearing radii of 0.48m and 0.50m, similar to Case C.
The shaft offset of 0.003m in both the horizontal and vertical is maintained.
The shaft
rotational speed, however, is doubled from 1,000 rpm in Case A, B, and C to 2,000 rpm.
The maximum and minimum pressures for Case D are 763.42 kPa and -788.30 kPa, as
shown in Figure 3-27. Figure 3-28 provides the maximum and minimum temperatures,
which are 305.66 K and 305 K.
45
Figure 3-27 Pressure Dam Bearing Case D Pressure - COMSOL Results
Figure 3-28 Pressure Dam Bearing Case D Temperature - COMSOL Results
46
The results of the various Cases evaluated with the pressure dam bearing models above
are summarized in Table 5.
Table 5 Summary of Pressure Dam Bearing Results in Section 3.1.3
Case
Bearing Radius (m)
Rotor Radius (m)
Rotational Speed
(rad/sec)
Offset in X-dir (m)
Offset in Y-dir (m)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
A
0.25
0.24
B
0.25
0.24
C
0.50
0.48
D
0.50
0.48
104.7
104.7
104.7
209.4
0
0
0
0.01
0.000
48.76
-22.39
305.57
0.003
0.003
0.003
0.003
0.003
0.003
0.00424 0.00424 0.00424
0.01
0.02
0.02
0.424
0.212
0.212
107.92 249.13 763.42
-114.06 -224.20 -788.30
306.98 305.29 305.66
From Table 5, the pressure and temperature changes associated with the plain journal
bearing models are provided in Figure 3-29 and Figure 3-30, respectively.
Figure 3-29 Pressure Dam Journal Bearing Cases – Pressure Results
47
Figure 3-30 Pressure Dam Journal Bearing Cases – Temperature Results
Comparing the pressures in Table 5, the pressure generated increases from Case A to
Case D, as shown in Figure 3-29. Case A results in the smallest pressure generation due
to the bearing and shaft being concentric. Case B has a pressure approximately 2.2 times
greater than Case A because of the slight eccentricity introduced. When comparing Case
C to Case B, the pressure increases about 2.3 times due to the increase in bearing and
shaft diameter. Finally, from Case C to Case D, the pressure is approximately 3.1 times
greater. Case D not only uses the larger bearing geometry, but the shaft rotational speed
was doubled. This bearing type is similar to the plain journal bearing, and the pressures
generated from the model reflect that (i.e., with the exception of Case A, the pressure
dam in this bearing does not significantly impact the resulting pressure from the plain
journal type). A comparison of the various bearing types with the results obtained in
COMSOL is provided in Section 3.2.
The change in oil temperature varied consistently with the previous bearing types, and is
provided in Figure 3-30. The oil temperature in Case A increased 0.57 K due to viscous
heating in the bearing. In Case B, the change in temperature is approximately 3.5 times
larger than Case A due to the eccentricity of the shaft. Case C results in the lowest
temperature change of about 51% of Case A, and Case D is almost 16% larger than Case
A. The maximum temperature in Case D is larger than in Case C due to the increase in
rotational speed of the shaft (i.e., the fluid is shearing quicker).
48
3.1.4 Offset/Lobe Journal Bearing
Similar to the plain journal bearing, the rotor radius for Case A was kept consistent at
0.24m. The bearing radius started at 0.25m, but was offset 0.01m. In order to model the
lobes along the horizontal axis, two circles with radii of 0.1m were centered on the offset
section of the bearing. Shaft speed is assumed to 1,000 rpm in the clockwise direction.
The maximum and minimum pressures for Case A are 40.82 kPa and -30.94 kPa, as
shown in Figure 3-31. Figure 3-32 provides the maximum and minimum temperatures,
which are 305.73 K and 305 K.
Figure 3-31 Offset/Lobe Bearing Case A Pressure - COMSOL Results
49
Figure 3-32 Offset/Lobe Bearing Case A Temperature - COMSOL Results
In Case B, the shaft rotational speed and geometries of both the bearing and rotor are the
same as Case A. However, a slight eccentricity was introduced in the expected direction
of normal machine operation (i.e., if the shaft is rotating in a clockwise manner, the
center of the shaft will move down and to the left). The shaft was moved 0.003m in the
–x and –y direction (i.e., left and down) to provide eccentricity. The maximum and
minimum pressures for Case B are 66.24 kPa and -89.62 kPa, as shown in Figure 3-33.
Figure 3-34 provides the maximum and minimum temperatures, which are 306.10 K and
305 K.
50
Figure 3-33 Offset/Lobe Bearing Case B Pressure - COMSOL Results
Figure 3-34 Offset/Lobe Bearing Case B Temperature - COMSOL Results
51
For Case C, the radii of both the shaft and bearing dimensions are doubled when compared to Case A and B. That is, the rotor and bearing radii are 0.48m and 0.50m in Case
C, but the shaft offset of 0.003m in both the horizontal and vertical is maintained. Shaft
rotational speed is the same as Case A and B at 1,000 rpm. The maximum and minimum
pressures for Case C are 159.57 kPa and -197.65 kPa, as shown in Figure 3-35. Figure
3-36 provides the maximum and minimum temperatures, which are 305.33 K and 305 K.
Figure 3-35 Offset/Lobe Bearing Case C Pressure - COMSOL Results
52
Figure 3-36 Offset/Lobe Bearing Case C Temperature - COMSOL Results
Case D uses the larger rotor and bearing radii of 0.48m and 0.50m, similar to Case C.
The shaft offset of 0.003m in both the horizontal and vertical is maintained.
The shaft
rotational speed, however, is doubled from 1,000 rpm in Case A, B, and C to 2,000 rpm.
The maximum and minimum pressures for Case D are 575.86 kPa and -727.71 kPa, as
shown in Figure 3-37. Figure 3-38 provides the maximum and minimum temperatures,
which are 305.59 K and 305 K.
53
Figure 3-37 Offset/Lobe Bearing Case D Pressure - COMSOL Results
Figure 3-38 Offset/Lobe Bearing Case D Temperature - COMSOL Results
54
The results of the various Cases evaluated with the pressure dam bearing models above
are summarized in Table 6.
Table 6 Summary of Offset/Lobe Bearing Results in Section 3.1.4
Case
Bearing Radius (m)
Rotor Radius (m)
Rotational Speed
(rad/sec)
Offset in X-dir (m)
Offset in Y-dir (m)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
A
0.25
0.24
B
0.25
0.24
C
0.50
0.48
D
0.50
0.48
104.7
104.7
104.7
209.4
0
0
0
0.01
0.000
40.82
-30.94
305.73
0.003
0.003
0.003
0.003
0.003
0.003
0.00424 0.00424 0.00424
0.01
0.02
0.02
0.424
0.212
0.212
66.24
159.57 575.86
-89.62 -197.65 -727.71
306.10 305.33 305.59
From Table 6, the pressure and temperature changes associated with the plain journal
bearing models are provided in Figure 3-39 and Figure 3-40, respectively.
Figure 3-39 Offset/Lobe Journal Bearing Cases – Pressure Results
55
Figure 3-40 Offset/Lobe Journal Bearing Cases – Temperature Results
Comparing the pressures in Table 6, the pressure generated increases from Case A to
Case D, as shown in Figure 3-39. Case A results in the smallest pressure generation due
to the bearing and shaft being concentric. Case B has a pressure approximately 1.6 times
greater than Case A because of the slight eccentricity introduced. When comparing Case
C to Case B, the pressure increases about 2.4 times due to the increase in bearing and
shaft diameter. Finally, from Case C to Case D, the pressure is approximately 3.6 times
greater. Case D not only uses the larger bearing geometry, but the shaft rotational speed
was doubled. This bearing type has traits of both the pressure dam and elliptical journal
bearings in that there are areas around the shaft where oil can collect, however those
areas are in the horizontal plane. A comparison of the various bearing types with the
results obtained in COMSOL is provided in Section 3.2.
The change in oil temperature varied consistently with the previous bearing types, and is
provided in Figure 3-40. The oil temperature in Case A increased 0.73 K due to viscous
heating in the bearing. In Case B, the change in temperature is approximately 1.5 times
larger than Case A due to the eccentricity of the shaft. Case C results in the lowest
temperature change of about 45% of Case A, and Case D is almost 81% of Case A. The
maximum temperature in Case D is larger than in Case C due to the increase in rotational speed of the shaft (i.e., the fluid is shearing quicker).
56
3.2 COMSOL Results – Conditions as a Function of Bearing
Geometry
Using the results obtained in Section 3.1, presentation of the data is manipulated in this
section to compare the four bearing types. Below, each Case discussed in Section 2.3 is
presented individually with respect to the bearing geometries to evaluate the benefits to
various bearing types and formulate advantages and disadvantages. Table 7 is provided
to reiterate the cases summarized in Table 1.
Table 7 Summary of Cases for Bearing Evaluation
Case
Bearing Radius (m)
Rotor Radius (m)
Rotational Speed
(rad/sec)
Offset in X-dir (m)
Offset in Y-dir (m)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
0.25
0.24
B
0.25
0.24
C
0.50
0.48
D
0.50
0.48
104.7
104.7
104.7
209.4
0
0
0
0.01
0.000
0.003
0.003
0.003
0.003
0.003
0.003
0.00424 0.00424 0.00424
0.01
0.02
0.02
0.424
0.212
0.212
3.2.1 Case A
As discussed in Section 2.3.1, Case A assumes a concentric shaft of 0.24m radius inside
the 0.25m radius bearing. Although having no eccentricity is unrealistic, Case A provides a benchmark for the results and allows for comparison relative to bearings with
slightly offset shafts. The results of each bearing type for Case A are provided in Table
8.
Table 8 Summary of Case A Results
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
Plain
4.36
-5.49
305.64
Elliptical
36.73
-56.77
305.59
Pressure Dam
48.76
-22.39
305.57
Offset/Lobe
40.82
-30.94
305.73
From Table 8, the pressure and temperature changes associated with Case A are provided graphically in Figure 3-41 and Figure 3-42, respectively.
57
Figure 3-41 Case A for Each Bearing – Pressure Results
Figure 3-42 Case A for Each Bearing – Temperature Results
Comparing the pressures in Table 8, the pressure generated in each case is similar with
the exception of the plain journal, as shown in Figure 3-41. The plain journal bearing
results in the smallest pressure generation due to the concentricity. The elliptical bearing
has a pressure approximately 8.4 times greater than the plain journal. When comparing
the pressure dam and offset/lobe type bearings, the overall change in pressure is about
75% and 77% of the elliptical journal bearing. The elliptical bearing has the largest
change in pressure of all the bearing types, and is expected to (especially in the negative
pressure) due to the bearing type creating the most variation in radial clearance with
respect to theta. In theory, the elliptical journal bearing is behaving similar to the
eccentric shaft of Case B, but an order of magnitude less.
58
The change in oil temperature varied independent of the pressure results, and is shown
graphically for each bearing type in Figure 3-42. As shown in equation XX, larger
temperature changes due to viscous heating will occur where the velocity gradients are
also larger (i.e., where the change in velocity occurs at the fastest rate in the radial
direction). The oil temperature for the plain journal increased 0.64 K due to viscous
heating in the bearing. In the elliptical bearing, the change in temperature is approximately 7.9% less than the plain journal bearing. The pressure dam journal bearing
results in the lowest temperature change of about 89% of the plain journal, and the
offset/lobe bearing is about 14% larger than the plain journal. The maximum temperature delta in the offset/lobe bearing is larger than the other bearing types due to the lower
pressures generated coupled with the smaller clearances resulting from the offset bearing
geometry.
3.2.2 Case B
As discussed in Section 2.3.2, Case B assumes a slight eccentricity between a shaft of
0.24m radius inside the 0.25m radius bearing. Eccentricity of 0.003m in both the
negative ‘x’ and negative ‘y’ direction is assumed, thus moving the shaft at a 45 degree
angle. Case B provides a more realistic model for a shaft under some load that has
reached equilibrium due to the fluid film wedge created. Comparisons with Case A also
show how the pressure and temperature vary due to an off-center rotor inside a bearing,
as discussed in Section 3.1. The results of each bearing type for Case B are provided in
Table 9.
Table 9 Summary of Case B Results
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
Plain
104.98
-130.96
306.69
Elliptical
106.13
-154.52
305.73
Pressure Dam
107.92
-114.06
306.98
Offset/Lobe
66.24
-89.62
306.10
From Table 9, the pressure and temperature changes associated with Case B are provided graphically in Figure 3-43 and Figure 3-44, respectively.
59
Figure 3-43 Case B for Each Bearing – Pressure Results
Figure 3-44 Case B for Each Bearing – Temperature Results
Comparing the pressures in Table 9, the pressure generated with each bearing type is
similar in magnitude, as shown in Figure 3-43. Maximum pressures between the plain,
elliptical, and pressure dam type bearings were essentially the same in Case B. However, the total change in pressure varied between all bearings. The plain journal bearing
pressure change is approximately 236kPa. The elliptical bearing has a pressure delta
approximately 10.5% larger than the plain journal. When comparing the pressure dam
and offset/lobe type bearings, the overall change in pressure is about 94% and 66% of
the plain journal bearing. The offset/lobe journal bearing results in the smallest pressure
change. Interestingly, the maximum pressure for all the bearings occurs in the fourth
quadrant, between -45 and -60 degrees (assuming the zero degree position is horizontal
and to the right on the x-plane). On the other hand, the minimum pressure is generally in
the third quadrant, with the exception of the offset/lobe bearing; although a low pressure
60
exists in this location, the downstream area after the lobe results in a slightly lower
pressure, as seen in Figure 3-33.
The change in oil temperature varied independent of the pressure results, and is shown
graphically for each bearing type in Figure 3-44. The oil temperature for the plain
journal increased 1.69 K due to viscous heating in the bearing. In the elliptical bearing,
the change in temperature is approximately 43% less than the plain journal bearing. The
pressure dam journal bearing results in the highest temperature change of about 1.98 K.
The offset/lobe bearing is about 65% of the plain journal. The maximum temperature
delta in the pressure dam bearing is produced in the first quadrant at approximate 20
degrees. In this area of the bearing, a slight backflow in the opposite direction causes
the greatest velocity gradient and therefore larger temperature increase.
3.2.3 Case C
As discussed in Section 2.3.3, Case C assumes the same eccentricity as Case B between
a shaft of 0.48m radius inside the 0.50m radius bearing. Eccentricity of 0.003m in both
the negative ‘x’ and negative ‘y’ direction is assumed, thus moving the shaft at a 45
degree angle. Comparison of Case C with Case B shows how the pressure and temperature vary due to the increase in bearing and shaft dimensions, but decreasing the
eccentricity ratio by half due to the larger bearing geometry. The results of each bearing
type for Case C are provided in Table 10.
Table 10 Summary of Case C Results
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
Plain
190.74
-209.30
305.29
Elliptical
245.79
-376.15
305.24
Pressure Dam
249.13
-224.20
305.29
Offset/Lobe
159.57
-197.65
305.33
From Table 10, the pressure and temperature changes associated with Case C are provided graphically in Figure 3-45 and Figure 3-46, respectively.
61
Figure 3-45 Case C for Each Bearing – Pressure Results
Figure 3-46 Case C for Each Bearing – Temperature Results
Comparing the pressures in Table 10, the pressure generated with each bearing type
follows the same trend as in Case B, as shown in Figure 3-45. The maximum pressure
and total change in pressure for each bearing increased by 1.8 to 2.4 times that of Case
B. The plain journal bearing pressure change is approximately 400kPa. The elliptical
bearing has a pressure delta approximately 56% larger than the plain journal. When
comparing the pressure dam and offset/lobe type bearings, the overall change in pressure
is about 118% and 89% of the plain journal bearing. The offset/lobe journal bearing
results in the smallest pressure change. Interestingly, the maximum pressure for all the
bearings occurs in the fourth quadrant at about -45 degrees (assuming the zero degree
position is horizontal and to the right on the x-plane) with the exception of the offset/lobe type bearing. In this case, the maximum pressure is generated at the edge of the
62
lobe at about -5 degrees, still in the fourth quadrant, as seen in Figure 3-35. The approximate doubling of pressure from Case B to Case C is attributed to the doubling of the
bearing geometry.
The change in oil temperature varied independent of the pressure results, and is shown
graphically for each bearing type in Figure 3-46. The oil temperature for the plain
journal increased 0.29 K due to viscous heating in the bearing. In the elliptical bearing,
the change in temperature is approximately 17% less than the plain journal bearing. The
pressure dam bearing results in the same temperature increase as the plain journal. The
offset/lobe journal bearing results in the highest temperature change of 0.33 K, about
14% more than the plain journal bearing. The changes in temperature are 15% to 33%
of the Case B values, and can be attributed to the increase in bearing/shaft size that
reduced the eccentricity ratio.
3.2.4 Case D
As discussed in Section 2.3.4, Case D assumes the same geometry and eccentricity as
Case C. Eccentricity of 0.003m in both the negative ‘x’ and negative ‘y’ direction is
used, thus moving the shaft at a 45 degree angle. Comparison of Case D with Case C
shows how the pressure and temperature vary due to the increase only rotational speed
of the shaft. Rotational speed in this case is doubled to 2,000 rpm. The results of each
bearing type for Case D are provided in Table 11.
Table 11 Summary of Case D Results
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Max Temperature (K)
Plain
701.24
-795.62
305.51
Elliptical
1,217.2
-1,651.7
305.30
Pressure Dam
763.42
-788.30
305.66
Offset/Lobe
575.86
-727.71
305.59
From Table 11, the pressure and temperature changes associated with Case C are provided graphically in Figure 3-47 and Figure 3-48, respectively.
63
Figure 3-47 Case D for Each Bearing – Pressure Results
Figure 3-48 Case D for Each Bearing – Temperature Results
Comparing the pressures in Table 11, the pressure generated with each bearing type
follows similar trends as in Case B and C, as shown in Figure 3-47. The maximum
pressure and total change in pressure for each bearing increased by 3 to 5 times that of
Case C. The plain journal bearing pressure change is approximately 1,497kPa. The
elliptical bearing has a pressure delta approximately 1.9 times that of the plain journal.
When comparing the pressure dam and offset/lobe type bearings, the overall change in
pressure is about 104% and 87% of the plain journal bearing, which is similar in trend to
Case C. The offset/lobe journal bearing results in the smallest pressure change. Interestingly, the maximum pressure for all the bearings occurs in the fourth quadrant at about 45 degrees (assuming the zero degree position is horizontal and to the right on the xplane). The offset/lobe type bearing shows a maximum pressure also being generated in
64
the fourth quadrant, but at the edge of the lobe at about -5 degrees, as seen in Figure
3-37. For all bearings, the increase to the change in pressure is 3 to 5 times the Case C
value, which be attributed to the doubling of the journal rotational speed because as the
geometry was kept the same from Case C to D.
The change in oil temperature varied independent of the pressure results and in the same
way as Case B, and is also shown graphically for each bearing type in Figure 3-48. The
oil temperature for the plain journal increased 0.51 K due to viscous heating in the
bearing. In the elliptical bearing, the change in temperature is approximately 41% less
than the plain journal bearing. The pressure dam bearing results in the largest temperature increase of Case D, with a change in temperature 29% more than the plain journal.
The offset/lobe journal bearing results in the temperature change almost 16% larger than
the plan journal bearing. The Case D larger bearing dimensions result in temperature
changes that are roughly 2 to 3 times less than the Case B values. Higher rotational
speed increased the change in temperature 1.3 to 2.3 times the Case C values.
65
4. Conclusions
After analyzing the various bearing types in Section 2.2, the results show that the
COMSOL model trends as expected between the varying conditions of the four cases
discussed in Section 2.3. That is, for each bearing type the pressure range increased
from Case A to D. The maximum pressure increased when eccentricity was introduced,
when bearing geometry was doubled but eccentricity stayed the same, and again when
rotational speed doubled but geometry and eccentricity remained the same. This result
was observed for each of the bearing types, although the bearing types performed
differently in magnitude.
The plain journal bearing compared nicely with the other bearing types evaluated. For
Case A where the shaft was concentric with the bearing, the plain journal produced very
little change in oil pressure compared to the other bearings with geometrical changes. In
general, the plain journal bearing produced lower pressures than the other bearings over
the various cases. The data shows that lower bearing pressures tend to result in higher
changes in temperature. Minimum and maximum oil pressures in the plain journal
bearing were located in the third and fourth quadrant at approximately 190 degrees and
315 degrees from zero (i.e., the zero location assuming horizontal and to the right in the
figures presented above).
The elliptical bearing consistently produced the largest pressure differential over the four
conditions evaluated, due to the predominately lower pressure generated in the bearing.
Maximum oil pressures for the elliptical bearing were generated in the fourth quadrant
(at approximately 315 degrees assuming zero in the horizontal to the right). The minimum pressure for each elliptical bearing case was located just downstream of the
bottom-most point of the bearing in the third quadrant (approximate 260 degrees from
the zero point). Although the pressure was greater than the other bearings, the elliptical
bearing showed the very little increase in oil temperature. Aside from Case A where the
journal was concentric with the bearing, the elliptical bearing resulted in the lowest
temperature change.
66
The pressure dam and offset/lobe journal bearings performed similarly with regard to oil
pressure in the bearing. The pressure dam bearing was usually between the elliptical and
offset/lobe type bearing in pressure, and also producing the largest maximum pressure
for every case except Case D. For each case except Case A, the maximum and minimum pressures for the pressure dam bearing were found in the fourth and third quadrant,
respectively. Because Case A was a concentric journal, the largest pressure resulted at
the pressure dam.
The offset/lobe journal bearing resulted in the lowest pressures throughout the four
conditions considered, with the exception of Case A. Interestingly, the minimum and
maximum oil pressures were concentrated around the lobes of the bearing; specifically,
at the point on the lobe closest to the moving shaft. Because the pressures were lower
than the other bearing types, the change in temperature of oil inside the offset/lobe type
bearing tended to be larger.
A goal during this project was to consider the use of Petroff’s Equation, derived in
Section 2.1.3 as Equation 28, for rough approximations of bearing losses. That is, if the
losses and other parameters of a bearing type (e.g., speed, radius, radial clearance, etc.)
were known, scaling horsepower losses based on the known parameters could be accomplished. Equation 28 provides a relationship between various parameters and the power
loss in the bearing, and is repeated below:
FHP 
T
2
 3 R 3lN rev
7,500  c
4 2 R 3lN rev
c
[26] from previous
[29]
Equation 29 shows the relation of Torque to parameters such as radius, rotational speed,
and radial clearance. It was found that using the relationships in Equation 29 provided a
rough approximation for the pressure generated in the bearing. Specifically for the plain
journal bearing, using Case B to approximate Case C is as follows:
67
R
PC  PB   C
 RB



3
 0.5 
PC  236  

 .25 
PC  472kPa
3
 NC

 NB
 C B

 CC



 104.7  0.01 



 104.7  0.02 
[30]
[31]
The COMSOL differential pressure was determined to be 400 kPa, and the scaled value
is 18% larger. Performing the same scaling from Case B to Case D for the plain journal
bearing was less accurate, resulting a pressure delta 37% less (944 kPa) than the
COMSOL value.
The power loss, in friction horsepower, was expected to directly correlate to the temperature rise of oil in the bearing. For every 1 horsepower lost in the bearing, 3,412 kW
(2,545 Btu/hr) of heat can be transferred to the oil due to the shearing of oil from the
moving journal. However, the temperature rise in the bearing from case to case was
largely dependent on the velocity gradient in the radial direction. As the radius of the
bearing increased, the gradient where the eccentricity created the minimum film thickness was not as large.
In general, Case B for each bearing resulted in the larger
temperature change. Case C and D are comparable in that only rotational speed changes
and bearing geometry does not vary. Excluding the elliptical bearing, when the rotational speed of the bearing doubled, the change in temperature of the oil also doubled,
approximately.
Each bearing type performed differently under the various cases. Areas where concentrated pressure increases developed are expected to give the journal stability. There is a
trade-off in heat and pressure generated, rotor stability, and internal clearances. Each
bearing model resulted in pressure and temperature magnitudes and locations that were
expected and explainable. Because pressures and temperatures did not vary a significant
amount, it is apparent that the overall trade-off between bearing types is cost, simplicity,
and manufacturability versus stability, variations in speed and load, and specific applications in industry.
68
5. References
1.
Everett C. Hunt, Modern Marine Engineer’s Manual, Third Edition, Cornell
Maritime Press, 1999.
2.
http://www.reliabilitydirect.com/appnotes/jb.html, Sales Technology, Inc., 2000.
3.
B. Bhushan, Introduction to Tribology, John Wiley & Sons, 2002.
4.
J. Halling, Principles of Tribology, Scholium International, 1978.
5.
Frank M. White, Viscous Fluid Flow, McGraw Hill, 3rd Edition, 2006.
6.
W. M. Kays, M. E. Crawford, and B. Weigand, Convective Heat and Mass
Transfer, 4th Edition, McGraw-Hill, New York 2005.
7.
Dudley D. Fuller, Theory and Practice of Lubrication for Engineers, Second
Edition, John Wiley & Sons, 1984.
8.
J. Tevaarwerk, P.E., Fluid Film Bearing Design and Hydrodynamic Lubrication,
Case Western Reserve University, Presentation Notes from August 9 – 11, 2004.
69
6. Appendix A – Supplemental Excel Information
Table of Contents
71. Validation
72. Plain Journal Bearing
73. Elliptical Journal Bearing
74. Pressure Dam Journal Bearing
75. Offset/Lobe Journal Bearing
76. Case Comparison of all Bearings
77. Fluid Properties of Engine Oil
70
Outer radius
Inner radius
Eccentricity (e)
Radial Clearance ( c)
Eccentricity Ratio (E)
Viscosity
Density
Omega
Max Press
Min Press
0.25
0.24
0.0042426
0.01
0.424264
0.338044676
880.5355937
-104.7
m
m
m
m
Pa*s
kg/m^3
rad/s
COMSOL
1.10E+05
-1.16E+05
Halling
6.083E+04
-6.08E+04
% Diff
44.75%
47.36%
Theta (rad)
0.00
0.39
0.79
1.18
1.57
1.96
2.36
2.75
3.14
3.53
3.93
4.32
4.71
5.11
5.50
5.89
6.28
P-Po
0
-11715
-23863
-36666
-49594
-60002
-60833
-41275
0
41275
60833
60002
49594
36666
23863
11715
0
COMSOL
-1.50E+04
-3.70E+04
-6.80E+04
-9.00E+04
-1.14E+05
-1.16E+05
-1.12E+05
-5.30E+04
3.00E+04
8.90E+04
1.10E+05
1.04E+05
8.50E+04
6.80E+04
4.30E+04
2.50E+04
7.00E+03
71
Oil Parameters
Oil Type
Inlet Temperature
Density
Viscosity at Inlet
Specific Heat
Ratio of Specific Heats
Engine Oil (SAE
30)
305 °K
880.5 kg/m^3
0.34 Pa*s
1929.3 J/(kg*K)
0.14
Bearing Parameters
Bearing Type
Bearing Size
Bearing Radius (m)
Rotor Radius (m)
Offset in X-dir (m)
Offset in Y-dir (m)
A
0.25
0.24
0
0
Plain Journal
B
C
0.25
0.50
0.24
0.48
0.003
0.003
0.003
0.003
D
0.50
0.48
0.003
0.003
Conditions
Rotational Speed (rad/sec)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
104.7
0.00000
0.01
0.000
B
104.7
0.00424
0.01
0.424
C
104.7
0.00424
0.02
0.212
D
209.4
0.00424
0.02
0.212
Results - Bearing Size A
Maximum Pressure (kPa)
Minimum Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
A
4.36
-5.49
0.64
305.64
305
B
104.98
-130.96
1.69
306.69
305
C
190.74
-209.30
0.29
305.29
305
D
701.24
-795.62
0.51
305.51
305
Results
72
Oil Parameters
Oil Type
Inlet Temperature
Density
Viscosity at Inlet
Specific Heat
Ratio of Specific Heats
Engine Oil (SAE
30)
305 °K
880.5 kg/m^3
0.34 Pa*s
1929.3 J/(kg*K)
0.14
Bearing Parameters
Bearing Type
Bearing Size
Bearing Radius (m)
Rotor Radius (m)
Offset in X-dir (m)
Offset in Y-dir (m)
Conditions
Rotational Speed (rad/sec)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
0.25
0.24
0
0
Elliptical Journal
B
C
0.25
0.50
0.24
0.48
0.003
0.003
0.003
0.003
D
0.50
0.48
0.003
0.003
A
104.7
0.00000
0.01
0.000
B
104.7
0.00424
0.01
0.424
C
104.7
0.00424
0.02
0.212
D
209.4
0.00424
0.02
0.212
Results - Bearing Size A
Maximum Pressure (kPa)
A
36.73
B
106.13
C
245.79
Minimum Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
-56.77
0.59
305.59
305
-154.52
0.73
305.73
305
-376.15
0.24
305.24
305
D
1,217.16
1,651.70
0.30
305.3
305
Results
73
Oil Parameters
Oil Type
Inlet Temperature
Density
Viscosity at Inlet
Specific Heat
Ratio of Specific Heats
Engine Oil (SAE
30)
305 °K
880.5 kg/m^3
0.34 Pa*s
1929.3 J/(kg*K)
0.14
Bearing Parameters
Bearing Type
Bearing Size
Bearing Radius (m)
Rotor Radius (m)
Offset in X-dir (m)
Offset in Y-dir (m)
A
0.25
0.24
0
0
Pressure Dam Journal
B
C
0.25
0.50
0.24
0.48
0.003
0.003
0.003
0.003
D
0.50
0.48
0.003
0.003
Conditions
Rotational Speed (rad/sec)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
104.7
0.00000
0.01
0.000
B
104.7
0.00424
0.01
0.424
C
104.7
0.00424
0.02
0.212
D
209.4
0.00424
0.02
0.212
Results - Bearing Size A
Maximum Pressure (kPa)
Minimum Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
A
48.76
-22.39
0.57
305.57
305
B
107.92
-114.06
1.98
306.98
305
C
249.13
-224.20
0.29
305.29
305
D
763.42
-788.30
0.66
305.66
305
Results
74
Oil Parameters
Oil Type
Inlet Temperature
Density
Viscosity at Inlet
Specific Heat
Ratio of Specific Heats
Engine Oil (SAE
30)
305 °K
880.5 kg/m^3
0.34 Pa*s
1929.3 J/(kg*K)
0.14
Bearing Parameters
Bearing Type
Bearing Size
Bearing Radius (m)
Rotor Radius (m)
Offset in X-dir (m)
Offset in Y-dir (m)
A
0.25
0.24
0
0
Offset Journal
B
C
0.25
0.50
0.24
0.48
0.003
0.003
0.003
0.003
D
0.50
0.48
0.003
0.003
Conditions
Rotational Speed (rad/sec)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
104.7
0.00000
0.01
0.000
B
104.7
0.00424
0.01
0.424
C
104.7
0.00424
0.02
0.212
D
209.4
0.00424
0.02
0.212
Results - Bearing Size A
Maximum Pressure (kPa)
Minimum Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
A
40.82
-30.94
0.73
305.73
305
B
66.24
-89.62
1.10
306.1
305
C
159.57
-197.65
0.33
305.33
305
D
575.86
-727.71
0.59
305.59
305
Results
75
Bearing Size
Bearing Radius (m)
Rotor Radius (m)
Offset in X-dir (m)
Offset in Y-dir (m)
A
0.25
0.24
0
0
B
0.25
0.24
0.003
0.003
C
0.50
0.48
0.003
0.003
D
0.50
0.48
0.003
0.003
Conditions
Rotational Speed (rad/sec)
Eccentricity (m)
Radial Clearance (m)
Eccentricity Ratio
A
104.7
0.00000
0.01
0.000
B
104.7
0.00424
0.01
0.424
C
104.7
0.00424
0.02
0.212
D
209.4
0.00424
0.02
0.212
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
Plain
4.36
-5.49
0.64
305.64
305
Elliptical
36.73
-56.77
0.59
305.59
305
Pressure
Dam
48.76
-22.39
0.57
305.57
305
Offset/Lobe
40.82
-30.94
0.73
305.73
305
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
Plain
104.98
-130.96
1.69
306.69
305
Elliptical
106.13
-154.52
0.73
305.73
305
Pressure
Dam
107.92
-114.06
1.98
306.98
305
Offset/Lobe
66.24
-89.62
1.10
306.10
305
Bearing
Max Pressure (kPa)
Min Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
Plain
190.74
-209.30
0.29
305.29
305
Elliptical
245.79
-376.15
0.24
305.24
305
Pressure
Dam
249.13
-224.20
0.29
305.29
305
Offset/Lobe
159.57
-197.65
0.33
305.33
305
Bearing
Max Pressure (kPa)
Plain
701.24
Pressure
Dam
763.42
Offset/Lobe
575.86
Min Pressure (kPa)
Delta Temperature (K)
Maximum Temperature (K)
Minimum Temperature (K)
-795.62
0.51
305.51
305
Elliptical
1,217.16
1,651.70
0.30
305.30
305
-788.30
0.66
305.66
305
-727.71
0.59
305.59
305
Case
A
Case
B
Case
C
Case
D
76
Viscosity
4.50
4.00
3.50
Viscosity
3.00
2.50
2.00
1.50
1.00
0.50
0.00
270
280
290
300
310
320
330
340
350
330
340
350
330
340
350
Tem perature (K)
Specific Heat
2,100
2,050
Specific Heat
2,000
1,950
1,900
1,850
1,800
1,750
270
280
290
300
310
320
Tem perature (K)
Density
905.0
900.0
895.0
Density
890.0
885.0
880.0
875.0
870.0
865.0
860.0
855.0
270
280
290
300
310
320
Tem perature (K)
77