A Comparative Analytical Study of 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
December, 2011
i
© Copyright 2011
by
Paul Wolfinger
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................... x
ABSTRACT ..................................................................................................................... xi
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
Derivation of Petroff’s Equation ............................................................ 9
COMSOL Model Description .......................................................................... 11
2.2.1
Plain Journal Bearing ........................................................................... 12
2.2.2
Elliptical Journal Bearing..................................................................... 13
2.2.3
Pressure Dam Journal Bearing ............................................................. 14
2.2.4
Offset/Lobe Journal Bearing ................................................................ 15
Conditions ........................................................................................................ 16
2.3.1
Case A: Concentric ............................................................................. 16
2.3.2
Case B: Effect to Eccentricity ............................................................. 17
2.3.3
Case C: Effect to Clearance ................................................................ 17
2.3.4
Case D: Effect to Rotational Speed ..................................................... 17
Assumptions ..................................................................................................... 17
3. Results / Discussion ................................................................................................... 20
3.1
Validation of COMSOL Model ....................................................................... 20
3.2
COMSOL Results – Bearing Geometry as a Function of Condition ............... 24
3.2.1
Plain Journal Bearing ........................................................................... 24
3.2.2
Elliptical Journal Bearing..................................................................... 32
iii
3.3
3.2.3
Pressure Dam Journal Bearing ............................................................. 40
3.2.4
Offset/Lobe Journal Bearing ................................................................ 48
COMSOL Results – Conditions as a Function of Bearing Geometry ............. 56
3.3.1
Case A: Concentric ............................................................................. 57
3.3.2
Case B: Effect to Eccentricity ............................................................. 58
3.3.3
Case C: Effect to Clearance ................................................................ 61
3.3.4
Case D: Effect to Rotational Speed ..................................................... 63
4. Conclusions................................................................................................................ 66
5. References.................................................................................................................. 69
6. Appendix A – Supplemental Information.................................................................. 70
iv
LIST OF TABLES
Table 1 Summary of Cases for Bearing Evaluation…………………………………….16
Table 2 Properties of Engine Oil .................................................................................... 18
Table 3 Summary of Plain Journal Bearing Results in Section 3.2.1 ............................. 30
Table 4 Summary of Elliptical Bearing Results in Section 3.2.2 ................................... 38
Table 5 Summary of Pressure Dam Bearing Results in Section 3.2.3 ........................... 46
Table 6 Summary of Offset/Lobe Bearing Results in Section 3.2.4 .............................. 54
Table 7 Summary of Cases for Bearing Evaluation ....................................................... 56
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.) [3] ...................... 4
Figure 2-3 Half of a Plain Journal Bearing (KN Series from Kingsbury, Inc.) [3] .......... 5
Figure 2-4 Plain Journal Bearing Cross Section Under Load [4] ..................................... 6
Figure 2-5 Plain Journal Bearing Clearance versus Theta [1] .......................................... 8
Figure 2-6 Sketches of the Four Bearing Types (Exaggerated) ..................................... 11
Figure 2-7 COMSOL Model - Plain Journal Bearing Section ....................................... 12
Figure 2-8 COMSOL Model – Elliptical Journal Bearing Section ................................ 13
Figure 2-9 COMSOL Model – Pressure Dam Journal Bearing Section ......................... 14
Figure 2-10 COMSOL Model – Offset/Lobe Journal Bearing Section .......................... 15
Figure 3-1 Plain Journal Bearing Validation Model – Pressure ..................................... 21
Figure 3-2 Validation Model – Pressure vs Arc Length ................................................. 21
Figure 3-3 Excel Plot – Pressure vs Theta ...................................................................... 22
Figure 3-4 Excel Plot – Pressure vs Arc Length............................................................. 23
Figure 3-5 Comparison Excel Plot – Pressure vs Theta ................................................. 23
Figure 3-6 Plain Journal Bearing Case A Pressure - COMSOL Results ........................ 25
Figure 3-9 Plain Journal Bearing Case B Temperature - COMSOL Results ................. 27
Figure 3-10 Plain Journal Bearing Case C Pressure - COMSOL Results ...................... 28
Figure 3-11 Plain Journal Bearing Case C Temperature - COMSOL Results ............... 28
Figure 3-12 Plain Journal Bearing Case D Pressure - COMSOL Results ...................... 29
Figure 3-13 Plain Journal Bearing Case D Temperature - COMSOL Results ............... 30
Figure 3-14 Plain Journal Bearing Cases – Pressure Results ......................................... 31
Figure 3-15 Plain Journal Bearing Cases – Temperature Results .................................. 31
Figure 3-16 Elliptical Bearing Case A Pressure - COMSOL Results ............................ 33
Figure 3-17 Elliptical Bearing Case A Temperature - COMSOL Results ..................... 33
Figure 3-19 Elliptical Bearing Case B Temperature - COMSOL Results...................... 35
Figure 3-20 Elliptical Bearing Case C Pressure - COMSOL Results ............................ 36
Figure 3-21 Elliptical Bearing Case C Temperature - COMSOL Results...................... 36
Figure 3-23 Elliptical Bearing Case D Temperature - COMSOL Results ..................... 38
vi
Figure 3-24 Elliptical Journal Bearing Cases – Pressure Results ................................... 39
Figure 3-25 Elliptical Journal Bearing Cases – Temperature Results ............................ 39
Figure 3-26 Pressure Dam Bearing Case A Pressure - COMSOL Results..................... 41
Figure 3-27 Pressure Dam Bearing Case A Temperature - COMSOL Results .............. 41
Figure 3-28 Pressure Dam Bearing Case B Pressure - COMSOL Results ..................... 42
Figure 3-29 Pressure Dam Bearing Case B Temperature - COMSOL Results .............. 43
Figure 3-30 Pressure Dam Bearing Case C Pressure - COMSOL Results ..................... 44
Figure 3-31 Pressure Dam Bearing Case C Temperature - COMSOL Results .............. 44
Figure 3-32 Pressure Dam Bearing Case D Pressure - COMSOL Results..................... 45
Figure 3-33 Pressure Dam Bearing Case D Temperature - COMSOL Results .............. 46
Figure 3-34 Pressure Dam Journal Bearing Cases – Pressure Results ........................... 47
Figure 3-35 Pressure Dam Journal Bearing Cases – Temperature Results .................... 47
Figure 3-36 Offset/Lobe Bearing Case A Pressure - COMSOL Results........................ 49
Figure 3-37 Offset/Lobe Bearing Case A Temperature - COMSOL Results ................. 49
Figure 3-38 Offset/Lobe Bearing Case B Pressure - COMSOL Results ........................ 50
Figure 3-39 Offset/Lobe Bearing Case B Temperature - COMSOL Results ................. 51
Figure 3-40 Offset/Lobe Bearing Case C Pressure - COMSOL Results ........................ 52
Figure 3-41 Offset/Lobe Bearing Case C Temperature - COMSOL Results ................. 52
Figure 3-43 Offset/Lobe Bearing Case D Temperature - COMSOL Results ................. 54
Figure 3-44 Offset/Lobe Journal Bearing Cases – Pressure Results .............................. 55
Figure 3-45 Offset/Lobe Journal Bearing Cases – Temperature Results ....................... 55
Figure 3-46 Case A for Each Bearing – Pressure Results .............................................. 57
Figure 3-47 Case A for Each Bearing – Temperature Results ....................................... 57
Figure 3-48 Case B for Each Bearing – Pressure Results .............................................. 59
Figure 3-49 Case B for Each Bearing – Temperature Results........................................ 59
Figure 3-50 Case C for Each Bearing – Pressure Results .............................................. 61
Figure 3-51 Case C for Each Bearing – Temperature Results........................................ 62
Figure 3-52 Case D for Each Bearing – Pressure Results .............................................. 63
Figure 3-53 Case D for Each Bearing – Temperature Results ....................................... 64
vii
NOMENCLATURE
rpm
revolutions per minute
rev
revolutions
rad
radians
m
meter (m)
s
second (s)
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
clearance; bearing radius minus the journal radius (m)
e
eccentricity of the bearing and journal center at equilibrium (m)
ε
eccentricity ratio; eccentricity divided by 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)
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 (m2)
viii
l
length of bearing (m)
P
pressure (Pa) or Pa multiplied by 1,000 (kPa)
lbf
pounds force
lbm
pounds mass
ix
ACKNOWLEDGMENT
The author would 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 also like to thank his classmates, who have caused him to
think critically during the course of study. Finally, he would like to thank his wife,
Megan, who has been so supportive throughout the entire Masters of Engineering in
Mechanical Engineering curriculum.
x
ABSTRACT
This project analyzes 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 although 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.
xi
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.
In an inclusive design, 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). The design pressure at the
supply of each bearing in the system must be known in order to determine the size of the
oil pumps required to move oil at various temperatures through the resistance of the
system and to the bearings. In addition, the heat load from the bearings and pumping of
the oil must be known in order to size the heat exchangers in the system properly to
deliver the oil at the appropriate temperature. Generally, full lubricating oil systems
have the capability of filtering (i.e., removing particles or solids) and purifying (i.e.,
removing water) to maintain bearing and component life and operation. The filtering
quality is dictated by the minimum expected clearance of each of the bearings, which is
dependent on load (i.e., load varies with rating, rotor weight and orientation, reaction
forces, etc.).
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 crosssectional area of the various journal bearings where no oil flows in or out of the bearings. In addition, no heating or cooling of the oil exists excluding the viscous heating
that occurs due to shearing of the fluid. This project provides an analysis of various
bearing types and provides a relative comparison of pressures and temperatures of oil at
various conditions as 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 gap between the rotating shaft (rotational) and the outer sleeve
(bearing) is filled with oil. By introducing eccentricity between the journal and bearing,
a load supporting wedge is generated due to the dynamics of the fluid. The fluid generally adheres to each body and shears 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 role 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
3
full bearings, which completely surround the shaft journal. Various full bearing types
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 that 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 a 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.) [3]
4
Figure 2-3 Half of a Plain Journal Bearing (KN Series from Kingsbury, Inc.) [3]
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 (i.e., the gap is
always full of oil). 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 in
exaggerated form.
5
Figure 2-4 Plain Journal Bearing Cross Section Under Load [4]
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 thickness associated with maximum pressure,
and “h” represents the thickness as a function of radial position, theta (θ).
For the laminar flow of a Newtonian fluid in two-dimensional Cartesian coordinates, 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
v
v
v
1 p
 2v
v w 
 2
x
y
z
 y
y
[3]
w
w
w
1 p
2w
u
v
w

 2
x
y
z
 z
y
[4]
6
Reducing the momentum equations above using conventional thin film assumptions (i.e.,
there is no flow in the y-direction or z-direction, and the change in flow in the xdirection is small and therefore can be neglected) results in the following governing
equations:
p
 2u
 2
x
y
[5]
Neglecting change in viscosity, µ, across the film thickness, h, allows for integration
twice:
u
1 p 2
y  ay  b
2 x
[6]
To solve Equation 6 for constants, boundary conditions must be applied:
u = U at y = 0
u = 0 at y = h
Solving for the constants in Equation 6,
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 
[7]
By integrating equation 7 across the film thickness, the Reynold’s equation is obtained.
h p h3

2 x 2
[8]
dh d  p h3 

U
 
dx dx  x 6 
[9]
p h3
U h  hm  
x 6
[10]
h
 udy U
hm
Rearranging to get in terms of pressure.
p
hh 
 6 U  3 m 
x
 h 
7
[11]
Equation 11 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
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
[12]
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 12, the
minimum film thickness is:
hmin  c  e  c1   
[13]
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 13, the film thickness, “h”, can be shown as:
h  c1    cos  
[14]
Substituting Equation 14 into Equation 11,


p
1
hm
 60R 2 

2
3

 c1    cos   c1    cos   
8
[15]

p
1
hm
R 
 60   

2
3

 c   1    cos   1    cos   
2
[16]
And integrating with respect to theta,
R
p  60  
c
2

1

hm
  1    cos    1    cos   d  C
2
1
3
[17]
As shown in Reference 5,
1    cos  
1  2
1   cos 
[18]
Applying the Sommerfeld solution to the integrated equation, which is evaluated at the
boundary condition:
p  p0 at   0  2
[19]
Solving for the constant, the pressure is shown as
R
p  60  
c
2
   sin  2    cos  


2
 2   2 1   2




[20]
2.1.2 Derivation of Petroff’s Equation
Related to work done by Sommerfeld, Petroff’s Law was developed to explain the
phenomenon of bearing friction. Petroff's method of lubrication analysis assumed a
concentric shaft and bearing, and produced the so-called 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
[21]
The lubricant shear stress is shown as [1]:
 
u
r r 0
9
[22]
Assuming a constant rate of shear, the equation is integrated with respect to the radius:
 
U
r1  r0
U
c
2rN rev
 
c
 
[23]
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 the shearing of the fluid 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
[24]
Where
F  P A
[25]
Therefore,
T  (  A)  r
[26]
T  (  A)  r
 2rN rev 
T 
  2r  l   r
c


2 3
4 R lN rev
T
c
[27]
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
[28]
[29]
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
10
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 COMSOL 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,
Figure 2-6, and are discussed below.
Plain Journal
Elliptical
Pressure Dam
Offset/Lobe
Figure 2-6 Sketches of the Four Bearing Types (Exaggerated)
11
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 3.1 and is a
good baseline to compare with the other bearing types. Figure 2-7 shows a plain journal
bearing geometry modeled in COMSOL.
Figure 2-7 COMSOL Model - Plain Journal Bearing Section
As previously mentioned, Figure 2-6 is exaggerated to show eccentricity (i.e., the rotor is
off-center inside the bearing) between the bearing and shaft as if the rotor was loaded.
Figure 2-7 is an example of the model used in COMSOL and 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
12
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 2-8 shows an elliptical
journal bearing geometry modeled in COMSOL.
Figure 2-8 COMSOL Model – Elliptical Journal Bearing Section
Figure 2-6 is exaggerated to show eccentricity and elliptical geometry. Figure 2-8 was
created in COMSOL to model 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
13
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 2-9 shows a
pressure dam journal bearing geometry modeled in COMSOL.
Figure 2-9 COMSOL Model – Pressure Dam Journal Bearing Section
Figure 2-6 is exaggerated to show eccentricity and the geometry of the pressure dam.
Figure 2-9 was created in COMSOL to model 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.
14
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 2-10 shows a
pressure dam journal bearing geometry modeled in COMSOL.
Figure 2-10 COMSOL Model – Offset/Lobe Journal Bearing Section
Figure 2-6 is exaggerated to show eccentricity and the geometry of the offset / bearing
lobes. Figure 2-10 was created in COMSOL to model a realistic bearing 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.
15
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 the Table 1 conditions, and are described below:

Case A: Concentric

Case B: Effect to Eccentricity

Case C: Effect to Clearance

Case D: Effect to Rotational Speed
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: Concentric
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
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.
16
2.3.2 Case B: Effect to Eccentricity
Case B is similar to Case A in that the shaft rotational speed and geometries of both the
bearing and rotor are the same. That is, a rotor radius and rotational speed of 0.24m and
1,000 rpm. 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
shifted 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 3.1.
2.3.3 Case C: Effect to Clearance
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: Effect to Rotational Speed
Case D uses the larger dimensions of Case C in the model. The only parameter changing
in Case D is the shaft speed inside the bearing. This is the only Case where the shaft
speed is varied, which is in order to obtain a relationship between a rotational speed (N)
from Section 2.1.2. The shaft rotational speed is doubled from Case A, B, and C to
2,000 rpm.
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:
17

Similar to the derivations above, only the cross section of the bearing is considered, that is the bearing has infinite depth such that effects due to losses can be
neglected.

Marine journal bearings are usually supplied with oil from a ship’s lubricating
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 20 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
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
18
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 1801 elements, depending on bearing geometry.
The
COMSOL stationary solver criterion is set for convergence tolerance of 0.002%
error in less than 100 iterations.
19
3. Results / Discussion
3.1 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 20, which is a
function of theta and includes geometric and fluid parameters. Because Equation 20 was
derived under the assumption that the fluid viscosity is constant, a single viscosity of
0.338 Pa*s 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 to Equation 20, 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. In addition, the derivation of Equation 20 assumes
that the initial pressure (PO) is the average pressure at the maximum film thickness, hmax,
consistent with Reference 6. In order to simplify the calculation, the bearing geometry
in COMSOL was offset only in the x-direction to determine the position of hmax without
calculation. The results canError! Reference source not found. be related to Case B of
the plain journal bearing by simply rotating the geometry 45 degrees counter-clockwise.
BError! Reference source not found.elow, Figure 3-1 provides the COMSOL two
dimensional results showing pressure in the plain journal bearing. Graphically, Error!
Reference source not found. Figure 3-2 shows the pressure results as a function of arc
length.
20
Figure 3-1 Plain Journal Bearing Validation Model – Pressure
Figure 3-2 Validation Model – Pressure vs Arc Length
21
As shown in Figure 3-1Error! Reference source not found. and Figure 3-2Error!
Reference source not found., the maximum and minimum pressure determined by
COMSOL for the plain journal bearing are 50.5 kPa and -109.4 kPa, respectively. Using
equation 20 above, the maximum and minimum pressure differentials calculated in
Excel were ±58.3 kPa. Consistent with Reference 6, the maximum and minimum
pressures were found in the third and second quadrants, assuming theta equals zero at the
point of hmax. By calculating the pressure generated (i.e., subtracting out the reference
pressure of PO = -7.15 kPa at hmax), a maximum and minimum pressure of 51.2 kPa and 65.5 kPa were determined using Equation 20. The results of using the Sommerfeld
equation to calculate pressures are shown graphically in Error! Reference source not
found.Figure 3-3Error! Reference source not found. and Figure 3-4Error! Reference
source not found.. First, Figure 3-3Error! Reference source not found. shows the
correlation of pressure to what is described in Reference 4. However, because bearing
rotation is in the clockwise direction, the graph is inverted. Second, Error! Reference
source not found.Figure 3-4 shifts the graph to compare it to Error! Reference source
not found. generated in COMSOL.
Figure 3-3 Excel Plot – Pressure vs Theta
22
Figure 3-4 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 arc length is shown
below in Figure 3-5Error! Reference source not found..
23
Figure 3-5 Comparison Excel Plot – Pressure vs Theta
As shown above, the calculated maximum and minimum pressures are approximately
1.4% and 40.2% off from the COMSOL solution, respectively. 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. A large difference between
the Sommerfeld solution and the COMSOL results is likely because of assumptions
made using the Sommerfeld solution. That is, there is no flow in the y-direction and the
change in flow in the x-direction is small and therefore can be neglected. However,
COMSOL solves the model using Navier-Stokes equations, considers velocities in the x
and y-direction, and accounts for inertia. Because the pressures calculated correlate to
the values generated in COMSOL, especially the maximum pressure, the method was
assumed to be correct. Then, the same COMSOL model was modified for each condition and bearing type described below. Therefore, despite slight differences shown in
Figure 3-5Error!
Reference source not found., the data generated in COMSOL provides
a valuable comparison of the cases considered relative to each other.
3.2 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.
3.2.1 Plain Journal Bearing
The bearing and rotor radius for this Case A (Concentric) 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 3-6. Figure 3-7
provides the maximum and minimum temperatures, which are 305.64 K and 305 K.
24
Figure 3-6 Plain Journal Bearing Case A Pressure - COMSOL Results
Figure 3-7 Plain Journal Bearing Case A Temperature - COMSOL Results
25
In Case B (Effects to Eccentricity), 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
shifted 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 3-8. Figure 3-9 provides the maximum and minimum temperatures,
which are 306.69 K and 305 K.
Figure 3-8 Plain Journal Bearing Case B Pressure - COMSOL Results
26
Figure 3-9 Plain Journal Bearing Case B Temperature - COMSOL Results
For Case C (Effects to Geometry), 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 3-10. Figure 3-11 provides the maximum and minimum temperatures, which
are 305.29 K and 305 K.
27
Figure 3-10 Plain Journal Bearing Case C Pressure - COMSOL Results
Figure 3-11 Plain Journal Bearing Case C Temperature - COMSOL Results
28
Case D (Effects to RPM) uses the larger rotor and bearing radii of 0.48m and 0.50m,
similar to Case C. Also, 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 3-12. Figure 3-13 provides the maximum and
minimum temperatures, which are 305.51 K and 305 K.
Figure 3-12 Plain Journal Bearing Case D Pressure - COMSOL Results
29
Figure 3-13 Plain Journal Bearing Case D Temperature - COMSOL Results
The results of the various Cases evaluated in this section with the plain journal bearing
models above are summarized in Table 3.
Table 3 Summary of Plain Journal Bearing Results in Section 3.2.1
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
30
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 3-14 and Figure 3-15, respectively.
Figure 3-14 Plain Journal Bearing Cases – Pressure Results
Figure 3-15 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 3-14. 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
31
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 3-15.
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).
3.2.2 Elliptical Journal Bearing
The bearing and rotor radius for this Case A (Concentric) 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 3-16. Figure 3-17 provides the maximum and minimum temperatures,
which are 305.59 K and 305 K.
32
Figure 3-16 Elliptical Bearing Case A Pressure - COMSOL Results
Figure 3-17 Elliptical Bearing Case A Temperature - COMSOL Results
33
In Case B (Effects to Eccentricity), 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
shifted 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 3-18. Figure 3-19 provides the maximum and minimum temperatures,
which are 305.73 K and 305 K.
Figure 3-18 Elliptical Bearing Case B Pressure - COMSOL Results
34
Figure 3-19 Elliptical Bearing Case B Temperature - COMSOL Results
For Case C (Effects to Geometry), 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 3-20. Figure 3-21 provides the maximum and minimum temperatures, which
are 305.24 K and 305 K.
35
Figure 3-20 Elliptical Bearing Case C Pressure - COMSOL Results
Figure 3-21 Elliptical Bearing Case C Temperature - COMSOL Results
36
Case D (Effects to RPM) 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 3-22. Figure 3-23 provides the maximum and
minimum temperatures, which are 305.3 K and 305 K.
Figure 3-22 Elliptical Bearing Case D Pressure - COMSOL Results
37
Figure 3-23 Elliptical Bearing Case D Temperature - COMSOL Results
The results of the various Cases evaluated in this sectionwith the elliptical bearing
models above are summarized in Table 4.
Table 4 Summary of Elliptical Bearing Results in Section 3.2.2
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
38
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 3-24 and Figure 3-25, respectively.
Figure 3-24 Elliptical Journal Bearing Cases – Pressure Results
Figure 3-25 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 3-24. 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
39
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 3.3.
The change in oil temperature varied with the different conditions in a similar fashion to
the plain journal bearing, and is provided in Figure 3-25. 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.
3.2.3 Pressure Dam Journal Bearing
Similar to the plain journal bearing, the bearing radius for Case A (Concentric) 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 3-26.
Figure 3-27 provides the maximum and minimum temperatures, which are 305.57 K and
305 K.
40
Figure 3-26 Pressure Dam Bearing Case A Pressure - COMSOL Results
Figure 3-27 Pressure Dam Bearing Case A Temperature - COMSOL Results
41
In Case B (Effects to Eccentricity), 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
shifted 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 3-28. Figure 3-29 provides the maximum and minimum temperatures,
which are 306.98 K and 305 K.
Figure 3-28 Pressure Dam Bearing Case B Pressure - COMSOL Results
42
Figure 3-29 Pressure Dam Bearing Case B Temperature - COMSOL Results
For Case C (Effects to Geometry), 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 3-30. Figure 3-31 provides the maximum and minimum temperatures, which
are 305.29 K and 305 K.
43
Figure 3-30 Pressure Dam Bearing Case C Pressure - COMSOL Results
Figure 3-31 Pressure Dam Bearing Case C Temperature - COMSOL Results
44
Case D (Effects to RPM) 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 3-32. Figure 3-33 provides the maximum and
minimum temperatures, which are 305.66 K and 305 K.
Figure 3-32 Pressure Dam Bearing Case D Pressure - COMSOL Results
45
Figure 3-33 Pressure Dam Bearing Case D Temperature - COMSOL Results
The results of the various Cases evaluated in this section with the pressure dam bearing
models above are summarized in Table 5.
Table 5 Summary of Pressure Dam Bearing Results in Section 3.2.3
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
46
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 3-34 and Figure 3-35, respectively.
Figure 3-34 Pressure Dam Journal Bearing Cases – Pressure Results
Figure 3-35 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 3-34. 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
47
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 3.3.
The change in oil temperature varied consistently with the previous bearing types, and is
provided in Figure 3-35. 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).
3.2.4 Offset/Lobe Journal Bearing
Similar to the plain journal bearing, the rotor radius for Case A (Concentric) 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 3-36. Figure 3-37 provides the maximum and minimum
temperatures, which are 305.73 K and 305 K.
48
Figure 3-36 Offset/Lobe Bearing Case A Pressure - COMSOL Results
Figure 3-37 Offset/Lobe Bearing Case A Temperature - COMSOL Results
49
In Case B (Effects to Eccentricity), 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
shifted 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 3-38. Figure 3-39 provides the maximum and minimum temperatures,
which are 306.10 K and 305 K.
Figure 3-38 Offset/Lobe Bearing Case B Pressure - COMSOL Results
50
Figure 3-39 Offset/Lobe Bearing Case B Temperature - COMSOL Results
For Case C (Effects to Geometry), 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 3-40. Figure 3-41 provides the maximum and minimum temperatures, which
are 305.33 K and 305 K.
51
Figure 3-40 Offset/Lobe Bearing Case C Pressure - COMSOL Results
Figure 3-41 Offset/Lobe Bearing Case C Temperature - COMSOL Results
52
Case D (Effects to RPM) 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 3-42. Figure 3-43 provides the maximum and
minimum temperatures, which are 305.59 K and 305 K.
Figure 3-42 Offset/Lobe Bearing Case D Pressure - COMSOL Results
53
Figure 3-43 Offset/Lobe Bearing Case D Temperature - COMSOL Results
The results of the various Cases evaluated in this section with the pressure dam bearing
models above are summarized in Table 6.
Table 6 Summary of Offset/Lobe Bearing Results in Section 3.2.4
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
54
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 3-44 and Figure 3-45, respectively.
Figure 3-44 Offset/Lobe Journal Bearing Cases – Pressure Results
Figure 3-45 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 3-44. 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
55
areas are in the horizontal plane. A comparison of the various bearing types with the
results obtained in COMSOL is provided in Section 3.3.
The change in oil temperature varied consistently with the previous bearing types, and is
provided in Figure 3-45. 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).
3.3 COMSOL Results – Conditions as a Function of Bearing
Geometry
Using the results obtained in Section 3.2, 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
56
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
3.3.1 Case A: Concentric
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 3-46 and Figure 3-47, respectively.
Figure 3-46 Case A for Each Bearing – Pressure Results
Figure 3-47 Case A for Each Bearing – Temperature Results
57
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 3-46. 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.
The change in oil temperature varied independent of the pressure results, and is shown
graphically for each bearing type in Figure 3-47. 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) [Error! Reference
source not found.]. 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.
3.3.2 Case B: Effect to Eccentricity
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,
58
as discussed in Section 3.2. 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 3-48 and Figure 3-49, respectively.
Figure 3-48 Case B for Each Bearing – Pressure Results
Figure 3-49 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 3-48. Maximum pressures between the plain,
elliptical, and pressure dam type bearings were essentially the same in Case B. Howev59
er, 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
exists in this location, the downstream area after the lobe results in a slightly lower
pressure, as seen in Figure 3-38.
The change in oil temperature varied independent of the pressure results, and is shown
graphically for each bearing type in Figure 3-49. 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.
60
3.3.3 Case C: Effect to Clearance
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 3-50 and Figure 3-51, respectively.
Figure 3-50 Case C for Each Bearing – Pressure Results
61
Figure 3-51 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 3-50. 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
lobe at about -5 degrees, still in the fourth quadrant, as seen in Figure 3-40. 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 3-51. 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
62
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.
3.3.4 Case D: Effect to Rotational Speed
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 3-52 and Figure 3-53, respectively.
Figure 3-52 Case D for Each Bearing – Pressure Results
63
Figure 3-53 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 3-52. 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
the fourth quadrant, but at the edge of the lobe at about -5 degrees, as seen in Figure
3-42. 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 3-53. 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 tempera64
ture 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 COMSOL models show that
the results trend 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 varied 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 2.1.2 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 29 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
[29] from previous
[30]
Equation 30 shows the relation of Torque to parameters such as radius, rotational speed,
and radial clearance. It was found that using the relationships in Equation 30 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 
[31]
[32]
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.
http://www.kingsbury.com/pdf/catalog-FPJ.pdf, Kingsbury, Inc., July, 2007.
4.
B. Bhushan, Introduction to Tribology, John Wiley & Sons, 2002.
5.
J. Halling, Principles of Tribology, Scholium International, 1978.
6.
B. Hamrock, Fundamentals of Fluid Film Lubrication, Second Edition, Marcel
Dekker, Inc., 2004.
7.
Frank M. White, Viscous Fluid Flow, McGraw Hill, 3rd Edition, 2006.
69
6. Appendix A – Supplemental 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