The Pennsylvania State University

The Graduate School

College of Engineering

DEVELOPMENT OF AN ANALYTICAL

DESIGN TOOL FOR JOURNAL BEARINGS

A Thesis in

Mechanical Engineering

by

Richard K. Naffin

© 2009 Richard K. Naffin

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

August 2009

The thesis of Richard K. Naffin was reviewed and approved* by the following: Liming Chang Professor of Mechanical Engineering Thesis Adviser Gita Talmage Professor of Mechanical Engineering Karen A. Thole Professor of Mechanical Engineering Head of the Department of Mechanical Engineering *Signatures are on file in the Graduate School

iii

Abstract

Reliable and efficient journal bearings operate with sufficient load capacity, low

frictional loss, low heat generation, and a sufficient supply of lubricating oil. To design

such bearings, values of the bearing design parameters are selected and used to calculate

bearing performance variables. The performance variables are then used to describe the

operational state of the bearing, aiding the designer in deciding whether the selected

values of the design parameters are optimal or not. In the current design methods, these

performance variables are often solved by interpolating and extracting solutions from

design tables. This technique may become tedious and inconvenient, especially when

trying to compile a large number of results to generate solution curves of the performance

variables.

This research develops an improved design method for journal bearings. The

solution is a design tool which uses analytic design modules to calculate the different

design considerations of the bearing system. Each module focuses on a single design

aspect and calculates the associated performance variables using a series of analytic

equations. This analytic method is implemented into a computer program and forms a

basic CAD package capable of generating solution curves of the performance variables.

These curves aid the designer in selecting the best design parameters for the system.

Thus, this basic analytic design tool produces similar results to the current manual

approach but in a manner which is more modern, time-efficient, user-friendly, and cost-

effective.

iv

Table of Contents

List of Figures....................................................................................................................vi

List of Tables....................................................................................................................vii

Nomenclature..................................................................................................................viii

Acknowledgements...........................................................................................................xi

Chapter 1: Introduction....................................................................................................1

Chapter 2: The Load Capacity Module...........................................................................8

2.1 Description of the Journal Bearing....................................................................8

2.2 The Simplified Models....................................................................................11

2.2.1 The Long Bearing Model..................................................................11

2.2.2 The Short Bearing Model..................................................................13

2.2.3 The Finite Bearing Technique..........................................................14

2.3 Development of an Analytic Finite Bearing Model.........................................15

2.4 Evaluation of the Finite Bearing Model...........................................................23

2.5 Refinement of the Finite Bearing Model.........................................................25

2.6 Summary of the Load Capacity Module..........................................................30

Chapter 3: The Temperature Module............................................................................32

3.1 Current Temperature Rise Calculation............................................................32

3.2 Analytic Equation for the Friction Factor........................................................35

3.3 Analytic Equation for the Inflow Factor..........................................................38

3.4 Analytic Equation for the Side Leakage Factor...............................................41

v

3.5 Summary of the Temperature Module.............................................................44

Chapter 4: Development of an Analytic Design Tool...................................................46

4.1 Summary and Integration of the Individual Modules......................................46

4.2 Implementation of the Core Design Calculations............................................51

4.3 Packaging the Basic Design Tool....................................................................55

4.4 Demonstrations of the Design Tool.................................................................59

4.4.1 Example 1.........................................................................................59

4.4.2 Example 2.........................................................................................63

Chapter 5: Summary and Recommendations...............................................................71

Bibliography.....................................................................................................................75

Appendix A: The Numerical Method Design Tables....................................................76

Appendix B: Sample Viscosity-Temperature Chart.....................................................80

Appendix C: MATLAB Code for the Bearing CAD Tool............................................81

vi

List of Figures

Figure 1-1 Shape of a Journal Bearing..........................................................................1

Figure 2-1 Geometry of a Journal Bearing....................................................................8

Figure 2-2 Shape of the Bearing Pressure Distribution in both the a) Circumferential and b) Axial Directions..............................................................................10

Figure 2-3 Dimensionless Load versus Slenderness Ratio Data for the Simplification Models When the Eccentricity Ratio is 0.5...............................................17

Figure 2-4 Generation of the Dimensionless Load versus Slenderness Ratio Curve-fit

When the Eccentricity Ratio is 0.5............................................................18 Figure 2-5 Dimensionless Load versus Slenderness Ratio Curve Family...................22

Figure 4-1 Schematic of Integrated Model..................................................................49

Figure 4-2 Schematic of Basic Design Tool................................................................56

Figure 4-3 Solution Curves for a) Minimum Film Thickness, b) Friction Power Loss, c) Lubricant Side Leakage, and d) Outlet Temperature with Respect to the Clearance Ratio..........................................................................................61

Figure 4-4 Design Solution Curves of the a) Eccentricity Ratio and b) Minimum Film

Thickness with Respect to the Inlet Viscosity...........................................66 Figure 4-5 Design Solution Curves of the a) Friction Coefficient and b) Power Loss

with Respect to the Inlet Viscosity............................................................67 Figure 4-6 Design Solution Curves of the Lubricant a) Flow and b) Temperature

Conditions with Respect to the Inlet Viscosity..........................................68 Figure B-1: A Common Viscosity-Temperature Chart. This particular chart may be

used for the SAE oil grades 10, 20, 30, 40, 50, 60, and 70........................80

vii

List of Tables

Table 2-1 Error Evaluation of Preliminary Analytic-Finite-Bearing Model.............23

Table 2-2 Error Evaluation of Analytic-Finite-Bearing Model with Adjusted Mesh Points.........................................................................................................26

Table 2-3 Error Evaluation of Corrected Analytic-Finite-Bearing Model.................28

Table 2-4 Error Evaluation of the Film-Thickness-Performance Factor...................29

Table 3-1 Error Evaluation of the Preliminary Friction-Factor Equation..................36

Table 3-2 Error Evaluation of the Corrected Friction-Factor Equation.....................37 Table 3-3 Error Evaluation of the Preliminary Inflow-Factor Equation....................39

Table 3-4 Error Evaluation of the Corrected Inflow-Factor Equation.......................40

Table 3-5 Error Evaluation of the Preliminary Side-Leakage-Factor Equation.........42

Table 3-6 Error Evaluation of the Corrected Side-Leakage-Factor Equation............44 Table 4-1 List of Input Parameters Used in the Clearance Ratio Verification

Test.............................................................................................................60 Table 4-2 List of Input Parameters Used in the Inlet Viscosity Verification

Test.............................................................................................................64 Table A-1 The Finite-Bearing Design Tables as Compiled in Booser [1]..................76

viii

Nomenclature

Symbol Description of Symbol Units

AL Long-Bearing-Eccentricity-Ratio Function in Log Scale Dimensionless

AS Short-Bearing-Eccentricity-Ratio Function in Log Scale Dimensionless

C0 0th Coefficient of Finite Analytic Equation Dimensionless

C1 1st Coefficient of Finite Analytic Equation Dimensionless

C2 2nd Coefficient of Finite Analytic Equation Dimensionless

C3 3rd Coefficient of Finite Analytic Equation Dimensionless

CS Short-Bearing-Model Correction Factor Dimensionless

CL Long-Bearing-Model Correction Factor Dimensionless

Cf Friction-Coefficient Correction Factor Dimensionless

Cin Inflow Correction Factor Dimensionless

Cleak Side-Leakage-Flow Correction Factor Dimensionless

c Radial Clearance of Journal Bearing m

ch Specific Heat of Selected Lubricant J/(kg/°C)

D Journal Diameter m

e Eccentricity Between Bearing/Journal Centers m

f Coefficient of Friction Dimensionless

fL(ε) Long-Bearing-Eccentricity-Ratio Function Dimensionless

fS(ε) Short-Bearing-Eccentricity-Ratio Function Dimensionless

h Film Thickness m

h Film-Thickness-Performance Factor Dimensionless

ix

hmin Minimum Film Thickness m

L Bearing Length m

N Journal Rotational Speed rev/s or rpm

Ploss Frictional Power Loss Watts or Horsepower

p Bearing Pressure Distribution Pa

Qavg Average Lubricant Flow of Bearing m3/s

Qin Lubricant Inflow of Bearing Loaded Region m3/s

Qleak Lubricant Side Leakage Flow of Bearing m3/s

Qx Lubricant Circumferential Flow of Bearing m3/s

inQ Loaded-Region-Inflow-Performance Factor Dimensionless

leakQ Side-Leakage-Flow-Performance Factor Dimensionless

R Journal Radius m

R1 Bearing Radius m

R2 Journal Radius m

(R/c)f Frictional-Loss-Performance Factor Dimensionless

S Sommerfeld Number Dimensionless

Tavg Lubricant Average Temperature °C

Tin Lubricant Temperature at Bearing Inlet °C

Tout Maximum Lubricant Temperature °C

U Tangential Surface Velocity of Journal m/s

W Applied Load N

W Dimensionless Load Dimensionless

x

X Slenderness Ratio in Log Scale Dimensionless

XL Long Bearing Mesh Point in Log Scale Dimensionless

XS Short Bearing Mesh Point in Log Scale Dimensionless

x Circumferential Coordinate of Journal Bearing m

Y Dimensionless Load in Log Scale Dimensionless

z Axial Coordinate of Journal Bearing m

α Circumferential Angle of Pressure Termination Point degree, °

β Viscosity-Temperature Coefficient 1/°C

ΔT Lubricant Temperature Rise °C

Δε Eccentricity-Ratio Error Tolerance Dimensionless

ε Eccentricity Ratio Dimensionless

εhigh Maximum Eccentricity Ratio Dimensionless

εlow Minimum Eccentricity Ratio Dimensionless

θ Circumferential Angle of Journal Bearing degree, °

μ Lubricant Viscosity Pa-s

μavg Average Lubricant Viscosity Pa-s

μin Inlet Lubricant Viscosity Pa-s

π 3.14 Dimensionless

ρ Lubricant Density kg/m3

ω Journal Rotational Speed rad/s

xi

Acknowledgements

I would like to sincerely thank my academic advisor at the Pennsylvania State

University, Dr. Liming Chang, for granting me the opportunity to pursue this research.

His honest guidance went beyond simply providing assistance with this research, as his

advice helped strengthen my personal weaknesses as well. He is a very patient advisor

who strives to see his students continue to develop their skills and succeed after college.

I owe him a debt of gratitude.

I would also like to thank Dr. Gita Talmage, who was willing to review this

thesis. She provided many suggestions which enhanced the value of this work. I also

appreciate the advice she provided in enhancing my presentation skills.

1

Chapter 1: Introduction

A journal bearing is a very useful component when it comes to supporting load in

rotating machinery. As illustrated in Figure 1-1.a, a cylindrical bearing encases a

rotating shaft known as the journal. These two parts are kept separated from each other

through the process of hydrodynamic lubrication.

Figure 1-1: Shape of a Journal Bearing Source: Cameron, Alastair. Basic Lubrication Theory. 3rd ed. New York: A. Cameron/Ellis Horwood

Ltd., 1981. p. 128

2

Consider a non-rotating journal that is subjected to a downward applied load as in Figure

1-1.b. When the journal begins to rotate clockwise, the supplied lubricant is dragged into

the journal bearing in the direction of the motion. The flow continuity of the lubricant

induces a pressure that is generated in the converging portion of the bearing system

which is labeled with an ‘X’ in Figure 1-1.b. This pressure is known as the

hydrodynamic pressure. It reduces the inflow and increases the outflow of the lubricant

in that converging portion of the journal bearing, so that the flow continuity is

maintained. If the pressure is sufficiently high, it causes the journal to lift away from the

bearing surface as illustrated in Figure 1-1.c. This is the basis of hydrodynamic

lubrication in a journal bearing. If designed correctly, the hydrodynamic pressure keeps

the journal and bearing surfaces sufficiently separated during normal operation.

There are many features to the journal bearing which makes it popular to use in

industrial applications. High toleration to dirt, low cost, small diameter, low running

friction, and low noise are a few benefits suggested by Engineers Edge [2], an online

engineering website. Such beneficial features allow the journal bearing to be widely used

in various rotating systems, such as steam turbines, engines, pumps, motors, gearboxes,

and milling systems. According to the bearing manufacturer STI, these engineering

systems typically involve high horsepower and load [3]. With such extreme operating

conditions, it is critical that the bearing operates efficiently and reliably.

Reliable journal bearings operate with sufficient load capacity, low frictional loss,

low heat generation, and a sufficient supply of lubricant oil. Designers satisfy each of

these requirements by selecting optimal values of the bearing design parameters of

bearing length and clearance ratio, as well as selecting the proper lubricant grade. The

3

other design parameters of applied load, journal diameter, and journal rotational speed are

typically governed by the engineering system for which the bearing is being designed.

All of these parameters are used to determine the bearing performance variables, which

are physical quantities used to describe the operational state of the bearing such as the

film thickness, lubricant temperature rise, friction, and oil flow.

Depending on the bearing geometry, the performance variables are solved by

calculating analytic equations or by extracting data off of bearing design tables. Journal

bearing geometry is characterized by the slenderness ratio L/D, the bearing length over

journal diameter [4]. Using this ratio, some bearings may be classified as either infinitely

short or long. For a bearing to be considered long, different sources suggest that the

slenderness ratio be larger than a value of L/D = 2 [5] to L/D = 4 [4]. Likewise, for a

bearing to be considered short, the slenderness ratio should be smaller than a value of

L/D = 1/4 [4] to L/D = 1/3 [6]. These long and short bearing geometries simplify the

governing equations of the system into analytic approximations, forming simplified

models. For the bearings in between, known as finite bearings, no widely accepted

analytic approximations exist. In practice, the slenderness ratio varies from about 0.125

in the narrowest of bearings to about 2 [4]. For the majority of bearings in this range of

slenderness ratios, use of either simplified model does not provide meaningful design

solutions. Instead, the governing equations are solved numerically by either a finite

difference method or a finite element method [7]. The resulting finite bearing solutions

are compiled as a series of dimensionless performance factors that have been plotted or

tabulated into charts or tables, such as the tables [1] referenced in Appendix A. As

illustrated in the appendix, these tables tabulate the performance factors used to calculate

4

the performance variables of the nominal film thickness, oil flow, and coefficient of

friction. The factors are tabulated with respect to the slenderness ratio and a variable

referred to as the Sommerfeld number, a dimensionless combination of the design

parameters. Once the Sommerfeld number and L/D of a system is determined, the

performance factors are extracted from the table and used to calculate the performance

variables.

These design tables are widely used in the current design methods for journal

bearings, which usually consider both the load capacity and temperature aspects of the

system. Many of these current calculation procedures obtain meaningful design solutions

by calculating the bearing performance variables using the average viscosity of the

lubricant, where the corresponding average temperature needs to be determined. Arnell

[8] describes one such technique. Arnell’s iterative process, which uses design charts,

may be modified to be used with the design tables [1] of Appendix A. An “intuitive”

average temperature of the lubricant is initially estimated by the designer. Its

corresponding average viscosity is used to determine the Sommerfeld number. Once the

Sommerfeld number is calculated, the corresponding values of the performance factors

for the coefficient of friction and oil flow are “looked up” in the table. These factors are

used in the supplemental equations provided by the design table to calculate both the

temperature rise and average temperature of the bearing lubricant. This calculated

average temperature is compared to the estimated value. If the difference between these

two temperatures is not within a selected error tolerance, a new average temperature is

estimated and new data is extracted from the table. This procedure is repeated until the

temperature difference is less than the required tolerance. The designer then uses the

5

corresponding viscosity of the final average temperature to obtain the final performance

factors for the friction coefficient and oil flow and subsequently calculate the respective

performance variables. Then, the load capacity is determined by finding the

corresponding dimensionless factor in the table and using it to calculate the minimum

film thickness.

The above iterative procedure does contain some weaknesses, despite its

usefulness. The extraction process of the performance factors from the design tables is

time-consuming, tedious, and inconvenient. For each iteration step involved in

determining the average temperature, the value of the Sommerfeld number changes.

Therefore, for each step, the designer is required to manually extract new performance

factors from the table to aid in calculating the temperature rise. The data provided in the

design tables are also discrete in nature, resulting in the designer often interpolating

between the provided data points to obtain and extract the performance factor values.

Furthermore, this manual design method is not very practical to use when generating

solution curves of the bearing performance variables. This task requires running the

above design method several times before enough data points are compiled to make

meaningful curves. Consequently, the current design method is antiquated. Today, many

engineering design calculations are being developed or implemented into computer

programs. Such programs run the tedious calculations for the designer. In addition, the

solutions are saved and transferred to other programs for future work or reference.

It is desirable to modernize and improve the efficiency of the current design

procedure by developing a design method which is analytic in nature. This proposed

method would use continuous, analytic equations to calculate each performance factor

6

instead of extracting data from design tables. These analytic equations would then be

used to develop a bearing design model which would include both the load capacity and

temperature considerations of the bearing system. The continuous nature of the model

equations would allow coverage of the entire bearing geometry range, removing the need

to interpolate data. In addition, the analytic nature of this design model would allow the

development of a computer-aided design (CAD) tool. CAD is the application of

computer technology and software to planning, performance, and implementing the

design process [9]. This CAD package would be capable of generating solution curves of

the performance variables used to locate optimal design parameters. The curves would

be generated in an automatic fashion, cutting down the amount of time needed in the

design stage of engineering. The output would also be formulated into computer files

which would transfer the solutions to other software packages, allowing the CAD

package to interface with other design programs. This same transfer file could even be

sent through the internet to other computers, allowing the designer to communicate with

other designers on a global level. In summary, the aim of this project is to produce a

bearing design tool which will deliver results similar to the current manual approach, but

in a much more convenient and timely manner.

The focus of this research, which is the development of an analytic bearing design

tool, is broken into the following chapters. In Chapter 2, an analytic load-capacity design

module is developed to calculate film thickness. In Chapter 3, an analytic temperature

module is developed to calculate the lubricant temperature rise of the bearing. These two

modules are then integrated together in Chapter 4 to develop a basic design model. This

model is implemented as a computer program and further expanded into a CAD tool

7

capable of generating solution curves for multiple bearing performance variables.

Finally, Chapter 5 summarizes the bearing design package and its contributions to the

future of engineering design.

8

Chapter 2: The Load Capacity Module

2.1 Description of the Journal Bearing

This chapter focuses on the development of an analytic module which will aid in

the load-capacity design aspect of the journal bearing. Load capacity is the ability of a

bearing to support load and may be measured by calculating the nominal film thickness

required to separate the surfaces of the journal and bearing. The film thickness may be

calculated using nominal values of the bearing design parameters, which include the

bearing length, journal diameter, radial clearance, inlet viscosity of the lubricant, applied

load, and journal rotational speed. The geometric parameters are illustrated in

Figure 2-1.

Figure 2-1: Geometry of a Journal Bearing

Source: Arnell, R.D., et al. Tribology: Principles and Design Applications. 1st ed. London: MacMillan Education Ltd, 1991. p. 162

9

In Figure 2-1, R1 represents the radius of the bearing, while R2 is the radius of the

journal. The nominal difference between R1 and R2 is the radial clearance c. The offset

between the journal center C and bearing center O is known as the eccentricity e. From

this geometry, an equation for the film thickness in terms of θ is determined [8]:

cos1cos

c

ecech (2.1.1)

In equation (2.1.1), θ is measured from the point of maximum clearance. The minimum

film thickness occurs at point F, where θ = 180º, giving:

1min ch (2.1.2)

In equation (2.1.2), ε = e/c and is known as the eccentricity ratio. This ratio is considered

the dimensionless performance factor for load capacity.

The loading conditions of the journal bearing are governed by the Reynolds

equation. During steady-state operation, the equation is in the following form [10]:

dx

dhU

z

ph

zx

ph

x633

(2.1.3)

In deriving the Reynolds equation, it is assumed that the flow is laminar, the fluid is

incompressible, the fluid viscosity remains constant, and no slip occurs between the fluid

and solid surfaces [7]. In equation (2.1.3), the film thickness h is given by equation

(2.1.1). The coordinate x is related to the circumferential coordinate θ and journal radius

R by:

Rx (2.1.4)

The tangential surface velocity U and angular velocity ω of the journal are related by:

RU (2.1.5)

10

The shape of the hydrodynamic pressure distribution p of equation (2.1.3) is illustrated in

Figure 2-2, where p(x,z) = 0 is the gage pressure.

a. b.

Figure 2-2: Shape of the Bearing Pressure Distribution in both the a) Circumferential and b) Axial Directions

The circumferential boundary conditions for equation (2.1.3) are given as [11]:

0,0 zp (2.1.6)

and

0,,

zx

pzp (2.1.7)

These two boundary equations form the Reynolds boundary condition, which contains an

unknown boundary point α that needs to be determined in the solution process [7].

11

Meanwhile, the axial boundary conditions for equation (2.1.3) are given as [11]:

02

,

L

xp (2.1.8)

and

00,

xz

p (2.1.9)

With these boundary conditions considered, no analytic solution exists for the Reynolds

equation in the above form. The system needs to be solved using numerical methods.

Under certain geometric conditions, however, equation (2.1.3) may be simplified to yield

analytic approximations. These approximations are known as either the long or short

bearing models and are described in the next section.

2.2 The Simplified Models

The geometry of the journal bearing is described by the slenderness ratio L/D, the

bearing length over journal diameter [4]. Using this ratio, journal bearings may be

categorized as either long, short, or finite bearings.

2.2.1 The Long Bearing Model

When the slenderness ratio L/D is sufficiently large, the magnitude of the axial

pressure variation becomes negligible compared to the pressure variation in the

circumferential direction. Therefore, the following assumption may be made:

z

p

x

p

(2.2.1)

12

Furthermore, it may be assumed that the oil film pressure does not change in the axial

direction [7]. Thus, equation (2.1.3) simplifies to [7]:

dx

dhU

dx

dph

dx

d 63

(2.2.2)

This is the long bearing model. However, a closed-form solution for this model cannot

be obtained using the Reynolds boundary condition because of the unknown boundary

point, α. By assuming that the pressure distribution terminates at the minimum film

thickness instead, which occurs at θ = π, α becomes zero. This simplification is known as

the half-Sommerfeld boundary condition and is written as [7]:

00 pp (2.2.3)

It is now possible to analytically solve equation (2.2.2) with the boundary condition

(2.2.3). The solution yields the following pressure distribution in terms of θ [5]:

222

2

cos12

cos2sin6

c

Rp 0 (2.2.4)

0p 2 (2.2.5)

Equation (2.2.4) is then integrated over the surface area that the pressure distribution acts

on to find the following hydrodynamic pressure load equation [5]:

22

2

12222

2

3

12

4

4

3

c

LDW (2.2.6)

Equation (2.2.6) may then be used to solve the bearing eccentricity ratio ε and

subsequently calculate hmin. When using the half-Sommerfeld boundary condition,

though, the calculated load may be lower than that obtained using the Reynolds boundary

condition for a wide range of operating conditions. Therefore, the long bearing model is

13

considered to provide conservative solutions for any journal bearing of a slenderness ratio

of L/D ≥ 3 [6].

2.2.2 The Short Bearing Model

When L/D is sufficiently small, the magnitude of the circumferential pressure

variation becomes very small compared to the pressure variation in the axial direction.

Therefore, the following assumption may be made [10]:

z

p

x

p

(2.2.7)

Using this assumption, equation (2.1.3) simplifies to:

dx

dhU

z

ph

z63

(2.2.8)

This is the short bearing model. Analytically solving equation (2.2.8) with the axial

boundary conditions:

02

,

L

xp (2.1.8)

and

00,

xz

p (2.1.9)

yields the following pressure distribution equation [10]:

dx

dhLz

h

Uzxp

4

3,

22

3

LzL

2

1

2

1 (2.2.9)

Equation (2.2.9) is then rewritten in terms of θ [10]:

2

2

32 4cos1

sin3, z

L

czp

0 and LzL

2

1

2

1 (2.2.10)

14

0, zp 2 and LzL2

1

2

1 (2.2.11)

Like the long bearing model, equation (2.2.10) is integrated over the surface area the

pressure distribution acts on to produce the following load equation [10]:

2

12

222

3

162.018

c

LDW (2.2.12)

Equation (2.2.12) may then be used to solve the bearing eccentricity ratio ε and

subsequently calculate hmin. The short bearing model is considered to provide reasonable

solutions for any journal bearing of a slenderness ratio of L/D ≤ 1/4 [4].

2.2.3 The Finite Bearing Technique

Finite bearings, which fall in the geometric range ¼ < L/D < 3, lack an analytic

model to aid in their load capacity calculations. Instead, these bearings are designed

using solutions based on numerical methods [7] which are tabulated as dimensionless

performance factors. These factors, which are used to calculate the different performance

variables, are dependent on the Sommerfeld number for a range of slenderness ratios, as

presented in the design tables of Appendix A [1]. The Sommerfeld number is defined as

[1]:

W

NDL

c

RS

2

(A1.1)

where N is the journal rotational speed in rev/s.

The current load-capacity design process for finite bearings involves extracting

the eccentricity ratio ε from the design tables. For example, consider a bearing in which

L/D = 1/2 and has a Sommerfeld number equal to S = 0.55. After locating the L/D = ½

15

table, the designer scrolls down the Sommerfeld number column. Because S falls in

between the table values of S = 0.5093 and S = 0.6337, the variable ε may be obtained

using a linear interpolation technique. The resulting value is then input into equation

(2.1.2) to calculate hmin. This design process may be tedious, time-consuming and

inconvenient for the designer due to the iterative techniques that are often involved when

designing journal bearings. It is necessary to develop a calculation process for finite

bearings which is completely analytic in nature. The proposed design model is to consist

of continuous equations which relate W and ε, similar to what equations (2.2.6) and

(2.2.12) do of the long and short bearing models.

2.3 Development of an Analytic Finite Bearing Model

A dimensionless load parameter may first be defined as:

WD

cW

4

2

(2.3.1)

Substituting equation (2.3.1) into equation (2.2.6) yields a dimensionless load equation

for the long bearing model:

D

LW

22

2

12222

124

43

L/D ≥ 3.0 (2.3.2)

Equation (2.3.2) may be written as:

D

LfW L L/D ≥ 3.0 (2.3.3)

16

where fL(ε) is a function dependent on the eccentricity ratio. Similarly, substituting

equation (2.3.1) into equation (2.2.12) yields a dimensionless load equation for the short

bearing model:

3

2

12

22162.0

18

D

LW

L/D < 1/4 (2.3.4)

which may be written as:

3

D

LfW S L/D < 1/4 (2.3.5)

where fS(ε) is a function dependent on the eccentricity ratio. After taking the logarithm of

both sides, both equations (2.3.3) and (2.3.5) respectively become:

D

LfW L loglog)log( L/D ≥ 3.0 (2.3.6)

and

D

LfW S log3log)log( L/D < 1/4 (2.3.7)

Equations (2.3.6) and (2.3.7) now relate )log(W to log(L/D) by two straight-line

segments, which are illustrated in Figure 2-3 for ε = 0.5.

17

Figure 2-3: Dimensionless Load versus Slenderness Ratio Data for the Simplification Models When the Eccentricity Ratio is 0.5

In Figure 2-3, the short and long bearing models are separated by the finite

bearing range. Data obtained using the tabulated numerical solutions [1] may be placed

into this range. Finite-bearing-data points corresponding to ε = 0.5 are shown in Figure

2-4, along with the same straight-line segments from Figure 2-3.

18

Figure 2-4: Generation of the Dimensionless Load versus Slenderness Ratio Curve-fit

When the Eccentricity Ratio is 0.5

In Figure 2-4, a polynomial trend line is overlaid on top of the data for all three bearing

ranges. The shape of this trend line matches the data fairly well. It has also been found

that similar polynomial trend lines overlay the data for the other eccentricity ratios.

Therefore, a polynomial equation may be used in developing the finite bearing model.

The order of the polynomial to use is determined by the number of boundary conditions

required to connect the finite model to both the long and short bearing models.

The finite bearing equation to be developed may initially be expressed as:

XfY , 1/4 < L/D < 3 (2.3.8)

where Y = )log(W and X = log(L/D). The short and long bearing model equations may

also be written in similar forms. Equation (2.3.7) is written as:

XAY S 3 L/D < 1/4 (2.3.9)

19

and (2.3.6) as:

XAY L L/D ≥ 3.0 (2.3.10)

where AS = Sflog and AL = Lflog . Equation (2.3.8) may now be anchored to

the short and long bearing models in these forms, where the lower and upper matching

(mesh) points are assigned as XS = log(1/4) and XL = log(3) respectively. The solutions of

equations (2.3.8) and (2.3.9) must match when X = XS. Similarly, when X = XL, equations

(2.3.8) and (2.3.10) must match. In addition, the equations must connect with equal

slopes at these mesh points to create a smooth curve. Therefore, the boundary condition

equations used to connect the finite and short bearing models are:

SSS XAXf 3, (2.3.11)

3

,

SXXdX

Xdf (2.3.12)

and those to connect the finite and long bearing models are:

LLL XAXf , (2.3.13)

1

,

LXXdX

Xdf (2.3.14)

Between the two mesh points, there are a total of four boundary conditions. Thus, to

obtain a smooth curve, equation (2.3.8) is written as a third order polynomial in the

following form:

oCXCXCXCXfY 12

23

3, 1/4 < L/D < 3 (2.3.15)

Substituting equation (2.3.15) into equations (2.3.11) thru (2.3.14) yields respectively:

SSoSSS XACXCXCXC 312

23

3 (2.3.16)

323 122

3 CXCXC SS (2.3.17)

20

LLoLLL XACXCXCXC 12

23

3 (2.3.18)

and

123 122

3 CXCXC LL (2.3.19)

The coefficients of equation (2.3.15) may now be determined by simultaneously solving

these four equations to give:

SLLSSL

SLLSLS

LSSL

SLLLSS

XXXXXX

XXXXXX

XXXX

XXXAXA

C22

224433

22

3

332

633

2

233

(2.3.20)

SL

SL

XX

XCXCC

2

332 23

23

2 (2.3.21)

SS XCXCC 22

31 233 (2.3.22)

LLLLLo XCXCXCXAC 12

23

3 (2.3.23)

The preliminary analytic-finite-bearing model to aid in calculating the load

capacity may be summarized as:

oCXCXCXCXfY 12

23

3, 1/4 < L/D < 3 (2.3.15)

where:

Y = )log(W

X = log(L/D)

SLLSSL

SLLSLS

LSSL

SLLLSS

XXXXXX

XXXXXX

XXXX

XXXAXA

C22

224433

22

3

332

633

2

233

(2.3.20)

21

SL

SL

XX

XCXCC

2

332 23

23

2 (2.3.21)

SS XCXCC 22

31 233 (2.3.22)

LLLLLo XCXCXCXAC 12

23

3 (2.3.23)

In equations (2.3.20) thru (2.3.23):

XS = log(1/4)

XL = log(3)

22

2

12222

124

43loglog

LL fA (2.3.24)

2

12

22162.0

18loglog

SS fA (2.3.25)

Bearing design systems where X < XS are to be solved using the short bearing model,

equation (2.3.9). Similarly, systems where X > XL are to be solved using the long bearing

model, equation (2.3.10).

The solutions of the preliminary finite-bearing model are plotted in Figure 2-5

along with the results of the short and long bearing models. Combined, the solutions of

each model form a series of curves relating W to L/D. Each curve corresponds to a

unique eccentricity ratio. This family of curves ranges from ε = 0.1 (bottom curve) to 0.9

(top curve) in increments of Δε = 0.1, corresponding to light to heavy loading conditions.

22

Figure 2-5: Dimensionless Load versus Slenderness Ratio Curve Family

The finite bearing model does provide a continuous analytical relationship between

W and L/D across the slenderness ratio range ¼ < L/D < 3. It also bridges the gap

between the short and long bearing models. However, no evidence has been presented

yet to support the validity of this model other than that it has a meaningful trend. An

evaluation is performed next to verify and to refine the model.

23

2.4 Evaluation of the Finite Bearing Model

An evaluation of the preliminary finite model is performed. By extracting L/D

and ε from the design tables [1], the dimensionless load ModelW is solved using the

analytic model summarized in section 2.3 and used in the following calculation:

100

Table

ModelTable

W

WWError (in %) (2.4.1)

where TableW is the result of converting the table value of S using the following equation:

SD

LW1

8

1

(2.4.2)

Equation (2.4.2) is obtained by combining equations (2.3.1) and (A1.1).

The results of equation (2.4.1) are presented in Table 2-1. The short bearing

model is used when L/D = 1/8 and 1/6. At L/D = 1/4, both the short and finite models

yield the same results.

Table 2-1: Error Evaluation of Preliminary Analytic-Finite-Bearing Model

The evaluation shows that the preliminary finite model significantly overestimates the

numerical method. The errors are most severe when L/D ≥ ¾ for lightly-loaded bearings

and between ¼ ≤ L/D ≤ 1.0 for heavily-loaded bearings. It may also be observed, that as

24

the load (or ε) increases, the severity of the error increases for shorter bearings and

reduces for longer bearings. The analytic solutions are reasonably accurate when both

L/D and ε are small and somewhat accurate when both are large.

The error patterns observed in Table 2-1 may be analyzed using basic

hydrodynamic theory. The Reynolds equation contains the pressure variation terms ∂p/∂x

and ∂p/∂z, whose nominal magnitudes vary as the bearing geometry changes. Figure 2-2

illustrates that increasing the load or ε increases the circumferential pressure variation

∂p/∂x. On the other hand, increasing L/D reduces the axial pressure variation ∂p/∂z.

Therefore, the long bearing assumption,

z

p

x

p

(2.2.1)

is more valid when the values of both L/D and ε are high, explaining the reduced errors in

that geometric region. Similarly, the short bearing assumption,

z

p

x

p

(2.2.7)

is more valid when both L/D and ε are low, accounting for the low errors in that

geometric range. Both simplification models have limitations. The short-bearing-model

assumption (2.2.7) does not accurately represent heavily-loaded short bearings, as it loses

its effectiveness due to the higher magnitude of ∂p/∂x generated under heavy loading.

The long-bearing-model assumption (2.2.1), on the other hand, does not accurately

represent lightly-loaded long bearings, as it loses its effectiveness due to the lower

magnitude of ∂p/∂x generated under light loading. These hydrodynamic insights will

help guide the refinement of the finite bearing model.

25

2.5 Refinement of the Finite Bearing Model

The consideration of the limitations existing on both simplification models plays a

key role in determining the mesh point positions of the finite model. If the assumption of

a simplification model produces large errors at the chosen mesh point, the anchored finite

model will carry the inaccuracies. Observations of Table 2-1 suggest that the short

bearing model produces errors as high as 38% for heavily-loaded bearings at the chosen

mesh point of L/D = ¼. But, the errors are reduced when L/D is either 1/6 or 1/8. It

would be more desirable to choose one of these lower slenderness ratios as the new mesh

point. The upper mesh point, on the other hand, is located at L/D = 3, where it is not

possible to compare the analytic and numerical solutions directly. However, moving the

mesh point to a slenderness ratio higher than L/D = 3 theoretically decreases the

magnitude of ∂p/∂z, potentially improving the validity of the long bearing assumption.

Therefore, it is possible to reduce the error for both the finite and simplification models

by simply adjusting the mesh points.

New positions of both mesh points are found using a trial-and-error approach.

The smallest errors are calculated by setting the lower and upper mesh points to

L/D = 1/8 and 4.75 respectively. Choosing these new mesh points does not alter any of

the previously developed analytic equations summarized in section 2.3. Only the

parameters XS and XL are adjusted such that XS = log(1/8) and XL = log(4.75). The error

evaluation of this modified finite model is presented in Table 2-2.

26

Table 2-2: Error Evaluation of Analytic-Finite-Bearing Model with Adjusted Mesh Points

The adjustments made to the mesh point positions have significantly reduced the errors of

the finite model. The maximum error is now less than 14%, as compared to 56% found

in the preliminary model. Two distinct error trends may now be observed. For relatively

short bearings, the errors increase in a somewhat exponential manner as the load (or ε)

increases. This particular trend is most pronounced at L/D = 1/8, the short bearing mesh

point. As the slenderness ratio of the bearing becomes larger, the errors vary in a more

linear fashion as the load increases. The finite bearing model underestimates for heavily-

loaded long bearings and overestimates for lightly-loaded long bearings at L/D = 2.

These observations suggest that the accuracy of the finite bearing model may further be

improved at both the long and short bearing mesh points.

27

At the mesh point positions, it may be possible to reduce the errors of the finite

model by modifying both the short and long bearing model equations with respect to ε.

The short-bearing-error trend may be corrected by modifying the eccentricity ratio term

fS(ε) in the dimensionless load equation (2.3.5). The short bearing errors increase in an

exponential trend as load or ε increases, where the error is close to zero at ε = 0.1.

Therefore, the term fS(ε) is modified with an exponential correction factor CS(ε):

2

12

22

2

12

22

162.018

1.00.1

162.018

B

ss

A

Cf

(2.5.1)

By taking the logarithm of equation (2.5.1), the expression for AS of equation (2.3.25) is

modified as:

2

12

22162.0

181.00.1log

BS AA (2.5.2)

The coefficients A and B are unknowns to be determined. Through a trial-and-error

approach, the best fit to minimize the errors is obtained when A = 0.7 and B = 10. Thus,

equation (2.5.2) becomes:

2

12

22

10 162.018

1.07.00.1log

SA (2.5.3)

Similarly, the long-bearing-error trend is corrected by modifying fL(ε) in equation (2.3.3).

Because the long bearing errors vary fairly linearly with respect to ε, the term fL(ε) is

modified with a linear correction factor CL(ε):

22

2

12222

22

2

12222

124

43

124

43

BACf LL (2.5.4)

28

By taking the logarithm of equation (2.5.4), the expression for AL of equation (2.3.24)

becomes:

22

2

12222

124

43log

BAAL (2.5.5)

The best-fit values for A and B are found to be 0.91 and 0.19 respectively, yielding:

22

2

12222

124

4319.091.0log

LA (2.5.6)

With the above modifications made to AS and AL, the finite bearing model continues to be

meshed to the short and long bearing models at L/D = 1/8 and 4.75.

The solutions of this refined model are once again evaluated by comparing them

against the tabulated numerical solutions. The results of the evaluation are presented in

Table 2-3.

Table 2-3: Error Evaluation of Corrected Analytic-Finite-Bearing Model

By refining the eccentricity ratio terms AL and AS, the majority of the finite model errors

are now below 5%. This value is exceeded only at a few of the bearing geometries

considered. The maximum observed error in Table 2-3 is 5.8%, which is reduced from

29

13.7% found in Table 2-2. These observations suggest that the finite bearing model has

been substantially improved using the above correction factors.

The finite bearing model may further be evaluated using the dimensionless film

thickness h , which is defined as [1]:

1min

c

hh (2.5.7)

The model is evaluated by comparing the calculated results of h to that of the tabulated

data:

100

Table

ModelTable

h

hhError (2.5.8)

The evaluation starts by selecting an eccentricity ratio from the design tables of Appendix

A. A dimensionless film thickness Tableh is calculated by converting this table value of ε

to Tableh using equation (2.5.7). The corresponding values of S and L/D are extracted

from the design table and used in the finite model equations to obtain the analytic ε-

solution, which is converted to Modelh also using (2.5.7). The results of the dimensionless-

film-thickness-error analysis are presented in Table 2-4.

Table 2-4: Error Evaluation of the Film-Thickness-Performance Factor

30

It is observed that the film thickness errors in Table 2-4 are small in magnitude. The

values of these errors generally remain below 2% for most of the bearing geometry range.

Only a few heavily-loaded bearings have errors between 2% to 4%. Therefore, this

modified bearing model may be used as the load-capacity design module.

2.6 Summary of the Load Capacity Module

The load-capacity design module is summarized as follows.

1. Calculate a value of the dimensionless load:

WD

cW

4

2

(2.3.1)

2. Determine a bearing eccentricity ratio ε by solving the following equations with a

sound iterative technique:

Y = )log(W

X = log(L/D)

XA

CXCXCXC

XA

Y

L

s

012

23

3

3

75.4/

75.4/8/1

8/1/

DL

DL

DL

(2.6.1)

where:

22

2

12222

124

4319.091.0log

LA (2.5.6)

2

12

22

10 162.018

1.07.00.1log

SA (2.5.3)

31

SLLSSL

SLLSLS

LSSL

SLLLSS

XXXXXX

XXXXXX

XXXX

XXXAXA

C22

224433

22

3

332

633

2

233

(2.3.20)

SL

SL

XX

XCXCC

2

332 23

23

2 (2.3.21)

SS XCXCC 22

31 233 (2.3.22)

LLLLLo XCXCXCXAC 12

23

3 (2.3.23)

XS = log(1/8)

XL = log(4.75)

3. Calculate the minimum film thickness hmin:

1min ch (2.1.2)

An analytic design module is now developed to aid in determining the load capacity of

journal bearings. Another important design aspect is the temperature rise of the lubricant

in the loaded region of the bearing. The development of an analytic-temperature design

module is presented in the next chapter.

32

Chapter 3: The Temperature Module

3.1 Current Temperature Rise Calculation

Another important design aspect to consider is the temperature conditions of the

lubricant flowing through the bearing. Stresses are generated within the lubricant oil

because of shearing of the lubricant film. Work done against these stresses appears as

heat within the lubricant, causing a temperature rise across the bearing and a

simultaneous reduction in the viscosity of the lubricating oil [9]. The current design

methods calculate the bearing performance variables using the average viscosity μavg of

the lubricant at an average temperature Tavg.

The average temperature may be determined using the design aspects of

temperature rise, frictional loss, and oil flow. An equation for Tavg is [9]:

2

TTT inavg

(3.1.1)

In equation (3.1.1), Tin is the temperature of the lubricant entering the loaded region of

the bearing just downstream of the feedhole. The temperature of the oil at this point is

that of the mixture of the supply oil and the recirculating oil [7]. It may be possible to

meaningfully estimate Tin using the ambient temperature of the lubricant supply. An

expression for the temperature rise ΔT may then be developed using the following

equation for heat equilibrium:

fWUTQc avgh (3.1.2)

According to Cameron [10], equation (3.1.2) assumes that the heat is primarily removed

from the bearing by the oil flow. In the equation, ch and ρ are the lubricant specific heat

33

and density, respectively. The coefficient of friction is represented by f. The term Qavg is

the average flow rate of the lubricant and is approximated in equation (A1.7) of

Appendix A [1]:

2leak

inavgQQQ (3.1.3)

The amount of lubricant entering the loaded region of the bearing is represented as Qin.

The side leakage Qleak is the amount of lubricant lost out of the axial ends of the bearing

during normal operation. By first rearranging equation (3.1.2) and then combining it with

(3.1.3), the following equation for ΔT [1] results:

in

leakinh Q

QQc

RNfWT

2

11

2

(3.1.4)

Equation (3.1.4) is similar to (A1.7) of Appendix A. In equation (3.1.4), the journal

rotational speed N is measured in revolutions per second. The performance variables f,

Qin, and Qleak are unknowns to be determined.

The unknown variables in equation (3.1.4) may be solved using dimensionless

performance factors tabulated in the design tables [1] presented in Appendix A. The

tables provide numerical values for the dimensionless factors (R/c)f, inQ and leakQ .

Equations which relate these factors to their respective quantitative performance variables

are also provided. The inflow Qin is related to its dimensionless counterpart inQ by [1]:

inin QDLcN

Q2

(A1.3)

and the side leakage Qleak is related to leakQ by [1]:

leakleak QDLcN

Q2

(A1.2)

34

Combining equation (3.1.4) with (A1.2) and (A1.3) produces:

in

leakin

h

Q

QQ

fc

R

LDc

WT

2

11

4

(3.1.5)

Equation (3.1.5) may be used to calculate a temperature rise in terms of (R/c)f, inQ and

leakQ .

The current design procedure to calculate the lubricant temperature rise is very

similar to the load capacity method described in section 2.2.3. The Sommerfeld number

S of the system is first calculated. The designer locates the correct L/D table from among

the provided tables and searches for the calculated S in the Sommerfeld number column.

The designer then extracts the corresponding values of the performance factors (R/c)f,

inQ and leakQ . If the Sommerfeld number falls in between two of the provided data

points, a linear interpolation technique may be used to extract each of the factors. These

three factors are used in equation (3.1.5) to calculate ΔT.

This design procedure is time-consuming and inconvenient. It is desirable to

develop acceptable analytic equations for the performance factors of (R/c)f, inQ and

leakQ and thus calculate the temperature rise more efficiently. The subsequent sections of

this chapter develop and refine the analytic-performance-factor equations. These

equations are packaged together as an analytical design module for determining the

temperature conditions in the loaded region of the bearing.

35

3.2 Analytic Equation for the Friction Factor

An equation for the friction factor (R/c)f may be developed from the following

expression for f:

c

R

W

NDLf

22 (3.2.1)

Equation (3.2.1) is known as the Petroff equation and is derived assuming that the journal

and bearing are concentric and that the system is fully flooded [12]. Multiplying the ratio

R/c on both sides of the equation results in:

Sfc

R 22 (3.2.2)

where S is the Sommerfeld number defined by the following equation:

W

NDL

c

RS

2

(N is in rev/s) (A1.1)

Equation (3.2.2) provides a simple and convenient analytic expression for (R/c)f.

The results of this equation may now be compared to the numerical data tabulated in

Appendix A using the following evaluation:

100

)/(

)/()/(%

table

equationtable

fcR

fcRfcRError (3.2.3)

The evaluation results are presented in Table 3-1.

36

Table 3-1: Error Evaluation of the Preliminary Friction-Factor Equation

It is observed that equation (3.2.2) underestimates the numerical solutions across the

entire bearing geometry range. The magnitude of the errors becomes more severe as ε (or

load) increases. Meanwhile, the errors increase modestly as L/D increases. The errors

reach as high as 57% to 67% at ε = 0.9. On the other hand, the errors are less than 1.0%

at ε = 0.1. These results suggest that equation (3.2.2) is a reasonable approximation for

lightly-loaded bearings. The accuracy of this approximation, though, deteriorates as the

load increases.

It is possible to improve the accuracy of equation (3.2.2) for heavily-loaded

bearings. A correction factor is developed using the error trends in Table 3-1. Across the

entire slenderness ratio range, the errors increase in a somewhat exponential manner as

the load (or ε) increases. As the slenderness ratio L/D increases, there is a mild linear

error increase. These observations suggest modifying equation (3.2.2) with a correction

factor Cf which is dependent on both L/D and ε:

SED

LAS

D

LCf

c

R Bf

22 20.12,

(3.2.4)

37

where A, B, and E in the correction factor are unknowns to be determined. By trial and

error, the errors are minimized when A = 0.56, E = 1.93, and B = 4, yielding:

SD

Lf

c

R 24 293.156.00.1

(3.2.5)

With the addition of this correction factor, the friction factor (R/c)f is now made

dependent on both the slenderness ratio and eccentricity ratio of the bearing.

The results of equation (3.2.5) are once again compared against the numerical

solutions in Table 3-2 below.

Table 3-2: Error Evaluation of the Corrected Friction-Factor Equation

As observed in Table 3-2, equation (3.2.5) slightly underestimates the results for bearings

with ε ≤ 0.6 across the entire L/D range. Meanwhile, the solutions are somewhat

overestimated in the range 0.6 < ε ≤ 0.85. A sharp turn in error occurs for bearings with

ε > 0.85, resulting in equation (3.2.5) to once again underestimate the numerical

solutions. Nevertheless, this modified friction-factor equation significantly reduces the

errors of (R/c)f. The maximum observed error is now less than 8.5% as compared to 67%

38

observed with the equation (3.2.2). Equation (3.2.5) is taken to be the analytic equation

for the friction factor (R/c)f.

3.3 Analytic Equation for the Inflow Factor

An analytic expression for inQ may be developed using the equation for the

circumferential inflow of the loaded region Qin. A general expression for the

circumferential flow Qx is expressed as [10]:

L

x

phUhQx

122

3

(3.3.1)

In the equation, 2

Uh is the velocity-induced flow while

x

ph

12

3

is the pressure-

induced flow. By inspection of Figure 2-2.a, it is assumed the loaded region of the

bearing begins at θ = 0. Also illustrated in Figure 2-2.a, the pressure gradient x

p

is

considered very small at θ = 0, allowing it to be neglected as a first-order approximation.

Both assumptions are used to simplify equation (3.3.1), which, when combined with

(2.1.1), produces the following expression for the inlet flow [10]:

1

2

DNLcQin (3.3.2)

Equation (3.3.2) is very similar to (A1.3) of Appendix A. Thus, a first-order

approximation for the dimensionless inflow factor inQ is:

1inQ (3.3.3)

39

Equation (3.3.3) may be evaluated by comparing its results for inQ to the

numerical data tabulated in Appendix A using the following evaluation:

100%

tablein

equationintablein

Q

QQError (in %) (3.3.4)

The results of the evaluation are presented in Table 3-3.

Table 3-3: Error Evaluation of the Preliminary Inflow-Factor Equation

The above evaluation shows that equation (3.3.3) overestimates the numerical solutions

for inQ across the entire bearing geometry range. The errors progressively become more

severe as both L/D and ε increase. The errors are as large as 99% for long bearings with

L/D = 2.0. For short bearings with L/D = 1/8, the errors are all less than 0.25%. These

observations suggest that equation (3.3.3) is a reasonable flow rate approximation for

short bearings under any loading condition. However, its accuracy deteriorates for

heavily-loaded long bearings.

Using these error trends, it is possible to improve the accuracy of equation (3.3.3)

with a correction factor. The errors increase as both L/D and ε increase. As the

40

slenderness ratio varies, the behavior of the error trend is highly exponential in nature.

Meanwhile, as ε varies, the error trend behaves in a more linear manner. These

observations suggest modifying equation (3.3.3) with the following L/D-and-ε-dependent

correction factor Cin:

11.00.11,

B

inin D

LEA

D

LCQ (3.3.5)

The errors are minimized by assigning A = 0.26, E = 0.01, and B = 1.2 in the correction

factor, yielding:

11.001.026.00.1

2.1

D

LQin (3.3.6)

With this modified expression, the inflow performance factor is now dependent on both

the slenderness ratio and eccentricity ratio of the bearing.

The results of equation (3.3.6) are once again compared against the numerical

solutions of inQ in Table 3-4.

Table 3-4: Error Evaluation of the Corrected Inflow-Factor Equation

41

The modified-inflow-factor equation (3.3.6) produces a maximum observed error of

approximately 5.91%, which is significantly lower than 99% calculated using equation

(3.3.3). Furthermore, only a few of the bearing geometries considered contain errors

which exceed 3%. Equation (3.3.6) is taken to be the analytic equation for the inflow

factor inQ .

3.4 Analytic Equation for the Side Leakage Factor

An analytic expression for leakQ may be developed using the equation for side

leakage Qleak. Since side leakage is the amount of lubricant lost out of the axial ends of

the bearing, it is measured by simply taking the difference between the circumferential

flows entering and leaving the loaded region of the bearing. One first-order

approximation is to assume that the loaded region occurs between the bearing

circumferential angles 0 ≤ θ ≤ π. Furthermore, Figure 2-2.a illustrates that the

circumferential pressure gradient x

p

is very small at both the inflow and outflow points

of the loaded region. The pressure gradient x

p

may then be neglected when θ = 0 and

θ = π as another first-order approximation. Therefore, Qleak may be expressed as [10]:

2

21

21

2

DLcNDNLcDNLcQleak (3.4.1)

Comparing equations (3.4.1) and (A1.2), a first-order approximation for leakQ is

developed:

2leakQ (3.4.2)

42

Like the equations for the dimensionless inflow and friction, equation (3.4.2) may

then be evaluated by comparing its results to the data presented in Appendix A using the

following evaluation:

100%

tableleak

equationleaktableleak

Q

QQError (in %) (3.4.3)

The results of this evaluation are shown in Table 3-5.

Table 3-5: Error Evaluation of the Preliminary Side-Leakage-Factor Equation

This evaluation reveals that equation (3.4.2) overestimates the numerical solutions for

leakQ across the entire bearing geometry range. The magnitude of the errors increases

exponentially as L/D increases. For bearings with L/D ≥ 0.75, it is observed that the

errors vary in a somewhat linear manner as the loading (or ε) changes. The errors are

above 100% for long bearings with L/D = 2.0. For short bearings with L/D = 1/8, the

error magnitude is approximately 0.4% for all loading conditions. These observations

43

suggest that equation (3.4.2) is a reasonable flow rate approximation for short bearings

but is ineffective for longer bearings.

It is possible to improve the accuracy of equation (3.4.2) with a correction factor.

The error trends in Table 3-5 behave very similar to those observed in Table 3-3 of the

inflow factor. This observation suggests modifying equation (3.4.2) with a correction

factor identical to that used in equation (3.3.5):

21.00.12,

B

leakleak D

LEA

D

LCQ (3.4.4)

The best-fit values for A, E, and B of the correction factor Cleak are found to be 0.03, 0.23,

and 1.2 respectively, yielding:

21.023.003.00.12.1

D

LQleak (3.4.5)

Similar to the modified inflow-factor equation, the expression for leakQ is now dependent

on both L/D and ε.

The results of equation (3.4.5) are once again compared against the numerical

solutions in Table 3-6.

44

Table 3-6: Error Evaluation of the Corrected Side-Leakage-Factor Equation

The modified-side-leakage-factor equation (3.4.5) produces a maximum observed error

of approximately 8.70%, which is significantly lower than 118% calculated using

equation (3.4.2). The largest errors occur when L/D >1. For all bearings with L/D ≤ 1.0,

the solutions are calculated within 3% of the table data. Equation (3.4.5) is taken as the

analytic equation for the side leakage factor leakQ .

3.5 Summary of the Temperature Module

The temperature module used to calculate both the average temperature and

temperature rise is summarized below.

1. Calculate a Sommerfeld number S:

W

NDL

c

RS

2

(N is in rev/s) (A1.1)

2. Compute the friction and flow factors:

SD

Lf

c

R 24 293.156.00.1

(3.2.5)

45

11.001.026.00.1

2.1

D

LQin (3.3.6)

21.023.003.00.12.1

D

LQleak (3.4.5)

3. Calculate a temperature rise ΔT:

in

leakin

h

Q

QQ

fc

R

LDc

WT

2

11

4

(3.1.5)

4. Calculate an average temperature Tavg:

2

TTT inavg

(3.1.1)

An analytic module has now been developed to calculate both the temperature rise

and average temperature of the loaded region of a bearing. A journal-bearing-design

model may now be developed by integrating this temperature module with the load

capacity module developed in Chapter 2. The integration of these two modules and the

subsequent development of a CAD tool for the bearing design are presented in the next

chapter.

46

Chapter 4: Development of an Analytic Design Tool

4.1 Summary and Integration of the Individual Modules

Two analytic modules have been developed in the previous chapters, each

calculating solutions for different design aspects of a journal bearing. The load capacity

module is summarized below.

1. Calculate the dimensionless load:

WD

cW

avg 4

2

(2.3.1)

where the lubricant average viscosity μavg in the loaded region of the bearing is evaluated

at the average lubricant temperature Tavg.

2. Determine a bearing eccentricity ratio ε by solving the following equations with a

sound iterative technique:

Y = )log(W

X = log(L/D)

XA

CXCXCXC

XA

Y

L

s

012

23

3

3

75.4/

75.4/8/1

8/1/

DL

DL

DL

(2.6.1)

where:

22

2

12222

124

4319.091.0log

LA (2.5.6)

47

2

12

22

10 162.018

1.07.00.1log

SA (2.5.3)

SLLSSL

SLLSLS

LSSL

SLLLSS

XXXXXX

XXXXXX

XXXX

XXXAXA

C22

224433

22

3

332

633

2

233

(2.3.20)

SL

SL

XX

XCXCC

2

332 23

23

2 (2.3.21)

SS XCXCC 22

31 233 (2.3.22)

LLLLLo XCXCXCXAC 12

23

3 (2.3.23)

XS = log(1/8)

XL = log(4.75)

The temperature module, developed in Chapter 3, is summarized below.

1. Calculate a Sommerfeld number S:

W

NDL

c

RS

2

(N is in rev/s) (A1.1)

2. Compute the friction and flow factors:

SD

Lf

c

R 24 293.156.00.1

(3.2.5)

11.001.026.00.1

2.1

D

LQin (3.3.6)

21.023.003.00.12.1

D

LQleak (3.4.5)

48

3. Calculate a temperature rise ΔT:

in

leakin

h

Q

QQ

fc

R

LDc

WT

2

11

4

(3.1.5)

4. Calculate an average temperature Tavg:

2

TTT inavg

(3.1.1)

5. Determine the lubricant average viscosity μavg:

μavg = f(Tavg) (4.1.1)

where any available viscosity-temperature chart or equation may be used.

The load capacity and temperature modules are interrelated, as each is dependent

on the other. The load capacity module calculates the bearing eccentricity ratio using the

lubricant average viscosity, requiring the value of the corresponding average temperature.

The temperature module, on the other hand, calculates the value of the average

temperature using the bearing eccentricity ratio. Therefore, these two modules need to be

integrated to form a basic design model. This integrated model should simultaneously

satisfy all of the equations comprising both modules and determine the bearing

performance variables from a set of input design parameters. The integration of the two

modules is schematically illustrated in the flow chart in Figure 4-1.

49

Figure 4-1: Schematic of Integrated Model

50

The bearing design parameters to input into the integrated model include the bearing

length, journal diameter, clearance ratio, applied load, journal speed, and inlet viscosity

of the lubricant. Other parameters to input include the specific heat, density, and inlet

temperature of the lubricant. The designer then provides a meaningful initial estimate of

the lubricant average temperature Tavg. Using a viscosity-temperature relation, the

corresponding average viscosity is calculated as a function of the average temperature,

inlet temperature, and inlet viscosity of the lubricating oil. Then, the dimensionless load

and slenderness ratio values are calculated and the eccentricity ratio determined in the

load capacity module using a reasonable iterative technique. Next, the temperature

module is called to calculate the friction and flow factors and subsequently the lubricant

temperature rise ΔT and average temperature Tavg. The difference between the calculated

Tavg and the estimated value is determined. This temperature difference is compared to

an error tolerance for the average temperature that is selected by the designer. If the

temperature difference is greater than this error tolerance, the average of the calculated

and estimated temperature values is taken to be the new average temperature and the

calculations are repeated. This iterative process continues until the temperature

difference is below the required tolerance, satisfying the equations for both the load

capacity and temperature modules. Then, the values for the bearing performance

variables such as hmin, Qleak, Qin, f, and ΔT are calculated. These variables help evaluate

the load capacity, frictional loss, heat generation, and lubricant flow conditions of the

bearing.

51

4.2 Implementation of the Core Design Calculations

The basic design model outlined in section 4.1 is written as a computer program.

The program is divided into six routines, including:

1. Program Driver Routine

2. Input Routine

3. Viscosity-Temperature Relation Routine

4. Load Capacity Routine

5. Temperature Routine

6. Output Routine

The function of the ‘program driver’ is to coordinate the other five routines in the order

dictated by the flowchart in Figure 4-1. These five routines, or subroutines, then perform

their specific calculations in the design process.

The driver routine first calls the input subroutine, which supplies all of the

necessary input parameters for the calculations. The designer accesses the input file and

assigns estimated values to the following design parameters:

1. Bearing Length L – in meters

2. Journal Diameter D – in meters

3. Clearance Ratio c/R – unitless

4. Applied Load W – in Newtons

5. Journal Rotational Speed N – in revolutions per minute

6. Lubricant Inlet Viscosity μin – in Pascal seconds

52

The designer then supplies the estimate of the lubricant inlet temperature Tin (in °C) as

discussed in Chapter 3, as well as the density (kg/m3), specific heat (J/(kg-°C)), and, if

used, the viscosity-temperature coefficient β (1/°C) of the lubricating oil. In addition, the

designer provides an initial estimate of the lubricant average temperature Tavg (in °C).

Furthermore, the designer provides acceptable error tolerances for both the bearing

eccentricity ratio Δε and the average temperature. It is recommended to set the

temperature tolerance within 1 - 2 °C. Finally, the designer specifies whether to use the

default lubricant viscosity-temperature relation of the program or supply one’s own

model.

The driver routine then calls the viscosity-temperature subroutine to calculate the

average viscosity of the lubricant. The viscosity is calculated using either the default or

user-specified viscosity-temperature relation. The default relation is a non-linear

equation known as the Barus model and is presented in the references [13] and [11]. The

equation is written as:

inavg TTinavg e (4.2.1)

If the user wishes to use one’s own relation, the expression and associated parameters or

factors are to be written into the viscosity-temperature subroutine, overriding equation

(4.2.1). The designer is free to use any equation which calculates a lubricant viscosity as

a function of the corresponding temperature, lubricant inlet viscosity, and inlet

temperature.

Next, the driver routine calls the load capacity subroutine, which uses the

bisection method [14] to iteratively solve the load capacity equations for a bearing

eccentricity ratio. First, values for the maximum eccentricity ratio εhigh and minimum

53

eccentricity ratio εlow of the bearing are assigned as 1- Δε and Δε respectively, where the

error tolerance Δε is assigned in the input subroutine. The dimensionless load W of the

bearing is calculated using equation (2.3.1). After supplying the slenderness ratio L/D to

this load capacity subroutine, a function f(ε) is defined by moving all of the terms in

equation (2.6.1) to the right-hand side. This function is equal to zero only when an

eccentricity ratio solution satisfying (2.6.1) is found. Calculating f(ε) using both εlow and

εhigh yields two residuals f(εlow) and f(εhigh). If the product of these two residuals is

negative, the eccentricity ratio solution falls between εlow and εhigh. This range is bisected,

and the value of either εlow or εhigh is updated depending on which half of the range

contains the solution. The updating process continues to narrow the range of εlow to εhigh

until εhigh – εlow ≤ Δε. Then, the calculations stop, and the solution is assigned as εhigh with

an error less than Δε.

The eccentricity ratio solution may also theoretically fall below εlow (= Δε) or

above εhigh (= 1 - Δε), even though such occurrences are practically impossible. When the

eccentricity ratio is less than εlow = Δε, the bearing load is approximately zero [5]. By

inspection of equation (A1.1), the resulting Sommerfeld number is extremely high.

Likewise, when the eccentricity ratio is greater than εhigh = 1 – Δε, the bearing load is

nearly infinite [5], and the Sommerfeld number is extremely low. The occurrence of

either situation is detected by the bisection method when the product of the residuals

f(Δε) and f(1- Δε) is positive. The program then calculates the Sommerfeld number. If

the value is greater than 1.0, the solution is assigned as Δε; otherwise, it is assigned as

1- Δε with an error less than Δε.

54

After a bearing eccentricity ratio is calculated, the driver routine calls the

temperature subroutine to determine the lubricant average temperature and temperature

rise using the temperature module equations outlined in section 4.1. The Sommerfeld

number is calculated using the lubricant average viscosity determined by the viscosity-

temperature subroutine. The friction and flow factors are calculated using ε determined

by the load capacity subroutine. After the average temperature Tavg of the lubricant is

determined, the temperature subroutine calculates the difference between this current

value and its previous estimate. If this temperature difference is greater than the

specified error tolerance, the average of these two temperature values is taken to be the

new ‘estimated’ average temperature. Then, the viscosity-temperature, load capacity, and

temperature module calculations are repeated. Once the difference between the

calculated and estimated average temperatures is below the specified error tolerance, the

final calculated Tavg is taken to be the average-temperature solution of the bearing.

Once a solution which satisfies all of the model equations is obtained, the driver

routine calls the output subroutine to calculate and present the bearing performance

variables of design interest. These variables include:

1. Minimum Film Thickness hmin – in meters

2. Coefficient of Friction f

3. Lubricant Side Leakage Qside – in m3/s

4. Lubricant Inflow of Bearing Loaded Region Qin – in m3/s

5. Lubricant Temperature Rise ΔT – in °C

55

4.3 Packaging the Basic Design Tool

By expanding the core design program described in section 4.2, the development

of a basic CAD tool is now in order. This design tool (CAD package) is developed to

assist the bearing designer in generating solution curves of the bearing performance

variables. The performance variables are plotted with respect to a selected design

parameter, so that the curve trends may aid the designer in determining the best value of

that parameter. The flow chart of Figure 4-2 schematically illustrates how this design

tool uses the core program to generate these solution curves. In addition, a sample code

written in MATLAB [15] is provided in Appendix C.

56

Figure 4-2: Schematic of Basic Design Tool

57

As illustrated in Figure 4-2, the basic CAD package is composed of three parts:

an input routine, a calculation routine, and an output routine. The input routine of the

package is an expanded version of the input subroutine of section 4.2. In addition to the

routine allowing the designer to provide nominal values of the bearing input parameters,

it also allows the designer to select one of the parameters to vary from a list of options.

For the purposes of designing an efficient and reliable journal bearing, the parameters of

bearing length, clearance ratio, and inlet lubricant viscosity may be selected by the

designer to be varied. The parameters of applied load, rotational speed, and journal

diameter are typically governed by the engineering system the bearing is being designed

to fit. However, this package is designed to allow these parameters to be varied too. The

designer is also free to choose the minimum, maximum, and increment values of the

selected design parameter. Next, the calculation routine iteratively solves both the

average temperature and eccentricity ratio solutions using the viscosity-temperature, load

capacity, and temperature subroutines as described in section 4.2. The solutions of the

performance variables are then stored in a column of a solution matrix for later use in the

output routine. To begin compiling data for the curves, the calculation routine is set-up

to initially calculate the performance variables using the minimum value of the selected

design parameter. The results are saved in the first column of the solution matrix. The

selected design parameter is then updated using the specified increment, and new

performance variables are then calculated and stored in the next column of the solution

matrix. The calculation routine is developed to repeat this process until the maximum

value of the selected design parameter is reached and the corresponding results stored.

The CAD package then calls the output routine to plot a series of continuous solution

58

curves, each representing a performance variable. The solution curves are plotted by

extracting the generated data from the solution matrix and plotting it with respect to the

selected design parameter.

This CAD package may be used to generate solution curves for multiple

performance variables of the bearing system. It yields solutions for the basic

performance variables of:

1. Minimum Film Thickness hmin – in meters

2. Eccentricity Ratio ε

3. Coefficient of Friction f

4. Lubricant Side Leakage Qside – in m3/s

5. Lubricant Inflow of Bearing Loaded Region Qin – in m3/s

6. Lubricant Temperature Rise ΔT – in °C

It then calculates the frictional power loss Ploss for the given design using [1]:

NDfWPloss (in Watts) (4.3.1)

7.745

NDfWPloss

(in horsepower (hp)) (4.3.2)

where N is in rev/s. Another performance variable that is calculated is the maximum

lubricant temperature Tout:

Tout = Tin + ΔT (4.3.3)

which is the lubricant temperature at the end of the loaded region of the bearing.

The trends of the solution curves may aid the designer in identifying the best

combination of the design parameters for the bearing system. For example, a design

scenario provides estimated values for each of the six design parameters, as well as a

range for each. The designer selects to vary one of these parameters and supplies its

59

minimum, maximum, and increment values into the input routine. The CAD package

calculates as directed, outputs the solution curves with respect to the selected parameter,

and saves each plot for future reference. Then, the designer goes back to the input,

selects another design parameter to vary, adjusts the minimum, maximum, and increment

values accordingly, and plots and saves these new curves. The designer repeats this

process, generating curves of every performance variable for each design parameter in a

relatively short amount of time. The designer may then use the combined trends of these

solution curves to select more optimal values of the design parameters.

4.4 Demonstrations of the Design Tool

4.4.1 Example 1

The benefits of using the CAD package in place of the design table method are

illustrated by comparing the two methods using the following example. A practical

design problem is used, where the clearance ratio is varied in the range

0.0002 < c/R < 0.005. Table 4-1 provides a list of nominal values of the parameters that

are used. The Barus equation is used to relate the viscosity and temperature parameters.

The solution curves are presented in Figure 4-3.

60

Table 4-1: List of Input Parameters Used in the Clearance Ratio Verification Test

Input Parameter Value Units

Journal Rotational Speed N 3600 rpm

Applied Load W 6650 N

Lubricant Inlet Temperature Tin 49 ° C

Length of Bearing L 0.05 m

Diameter of Journal D 0.05 m

Inlet Viscosity μin 0.028 Pa-s

Lubricant Specific Heat ch 1760 J/kg-°C

Lubricant Density ρ 880 kg/m3

Viscosity-Temperature Coefficient β 0.033 1/°C

Minimum Clearance Ratio c/R 0.000125 --

Maximum Clearance Ratio c/R 0.005 --

Clearance Ratio Increment 0.00005 --

Estimated Average Temperature Tavg 80 ° C

Average Temperature Tolerance 1 ° C

Eccentricity Ratio Tolerance 0.0001 --

61

62

In Figure 4-3, solutions for both the CAD package and the manual design-table

method are plotted for the performance variables of film thickness, frictional power loss,

lubricant side leakage, and outlet temperature. The CAD-generated solutions form

continuous curves which span the selected range of the clearance ratio. On the other

hand, the design table solutions consist of discrete data points scattered across the same

range of c/R. The handful of table solutions presented in Figure 4-3 took a few hours of

design work to manually calculate, as each data point required multiple iterations to

converge to the solution. Using the CAD package, it took only a few minutes to fill in

the input routine, run the calculation routine, and generate all of the solution curves.

Some differences exist between the design table and CAD-generated solutions though.

The magnitude and trends of these differences are observed in the error analysis tables of

the film thickness (Table 2-4), friction factor (Table 3-2), and side leakage factor (Table

3-6) when L/D = 1. None of these error evaluations contain errors which exceed 10%,

suggesting the CAD package produces reasonable approximations for the solutions

obtained by the design table method.

The solution curves presented in Figure 4-3 are then used to aid the designer in

selecting the best value of the clearance ratio. By observation of the performance

variable trends, the journal bearing operates fairly efficiently and reliably when

c/R = 0.002. At this value, the film thickness peaks at a sufficient magnitude while the

maximum (outlet) temperature is relatively low. In addition, the lubricant side leakage is

neither very low nor high. Using this selected clearance ratio, the designer simply

extracts the corresponding performance variables from each curve as needed.

63

4.4.2 Example 2

A second example is provided to showcase the steps a designer would take to

solve a design problem while using the bearing CAD package. For this example, the

designer is to explore the effects that the lubricant inlet viscosity has on the performance

of a bearing system. The nominal values of the bearing length, radial clearance, journal

diameter, journal rotational speed, and applied load are provided in Table 4-2 for this

system. An estimated inlet temperature is also provided.

64

Table 4-2: List of Input Parameters Used in the Inlet Viscosity Verification Test

Input Parameter Value Units

Journal Rotational Speed N 1000 rpm

Applied Load W 20000 N

Lubricant Inlet Temperature Tin 40 ° C

Length of Bearing L 0.06 m

Diameter of Journal D 0.0762 m

Clearance Ratio c/R 0.00075 --

Lubricant Specific Heat ch 1900 J/kg-°C

Lubricant Density ρ 850 kg/m3

Viscosity-Temperature Coefficient β 0.036 1/°C

Minimum Inlet Viscosity μin 0.005 Pa-s

Maximum Inlet Viscosity μin 0.9 Pa-s

Inlet Viscosity Increment 0.01 Pa-s

Estimated Average Temperature Tavg 50 ° C

Average Temperature Tolerance 1 ° C

Eccentricity Ratio Tolerance 0.0001 --

RMS Surface Roughness of Journal 0.7 μm

RMS Surface Roughness of Bearing 0.7 μm

65

In Table 4-2, the lubricant properties of density, specific heat, and the viscosity-

temperature coefficient are estimated nominal values. Furthermore, the Barus equation is

used in the viscosity-temperature subroutine. For the purposes of this demonstration, a

rather large range for the inlet viscosity is selected, where the viscosity varies from

0.005 ≤ μin ≤ 0.9 Pa-s. The solution curves generated by the CAD program are presented

in Figures 4-4 thru 4-6.

66

Figure 4-4: Design Solution Curves of the a) Eccentricity Ratio and b) Minimum Film Thickness with Respect to the Inlet Viscosity

67

Figure 4-5: Design Solution Curves of the a) Friction Coefficient and b) Power Loss with Respect to the Inlet Viscosity

68

Figure 4-6: Design Solution Curves of the Lubricant a) Flow and b) Temperature Conditions with Respect to the Inlet Viscosity

69

The designer uses these generated curves to extract an optimal value of the inlet

viscosity and subsequently select the corresponding SAE grade. During the selection

process, however, the following design limits are to be considered:

1. The maximum lubricant temperature is to be no more than 120 ºC.

2. The eccentricity ratio is to remain within the range 0.3 ≤ ε ≤ 0.7.

3. The minimum film thickness should be greater than or at least equal to 20 times

the combined RMS values for the surfaces [1].

If the first two design limitations are met, the designer is free to choose any lubricant

inlet viscosity from within the range of 0.025 < μin < 0.21 Pa-s. The bounds of this range

are presented in each plot. Using the SAE viscosity-temperature chart [12] in Appendix

B, it is observed that the grades SAE 20, 30, and 40 fall within this viscosity range when

Tin = 40 ºC. The inlet viscosities of these grades are 0.03, 0.05, and 0.12 Pa-s

respectively. Considering the third limit, the allowable minimum film thickness for this

bearing is approximately 14 μm. By inspection of Figure 4-4.b, the oil grade which best

fits this film thickness is SAE 40. By following the lines traced on each plot, the

corresponding values for the performance variables are extracted from each of the

solution curves:

ε = 0.5

hmin = 0.0000145 m

f = 0.00425

Ploss = 350 W

Qside = 0.0000029 m3/s

Qin = 0.0000047 m3/s

70

ΔT = 65 ºC

Tout = 105ºC

As illustrated by this example, these solutions are obtained without tedious

manual calculations. The results are presented in convenient curve plots which may be

used to extract the performance variables of the bearing system. Therefore, the CAD

package provides bearing design solutions which are obtained in an efficient, user-

friendly manner.

71

Chapter 5: Summary and Recommendations

Reliable and efficient journal bearings operate with sufficient load capacity, low

frictional loss, low heat generation, and a sufficient supply of lubricant oil. To design

such bearings, values of the bearing design parameters are selected and used to calculate

bearing performance variables. The performance variables are then used to describe the

operational state of the bearing, aiding the designer in deciding whether the selected

values of the design parameters are optimal or not. In the current design methods, these

performance variables are often solved by interpolating and extracting solutions from

design tables. This technique may become tedious and time-consuming, especially when

trying to compile a large number of results to generate solution curves of the performance

variables.

This research develops an improved design method for journal bearings. The

solution is a design tool which uses analytic design modules to calculate the different

design considerations of the bearing system. Each module focuses on a single design

aspect and calculates the associated performance variables using a series of analytic

equations. This analytic method is implemented into a computer program and forms a

basic CAD package capable of generating solution curves of the performance variables.

These curves aid the designer in selecting the best design parameters for the system.

Thus, this analytic design tool produces similar results to the current manual approach but

in a much more convenient and timely manner.

The basic design tool is structured as two analytic design modules, the load

capacity and temperature modules. Each module consists of a series of analytic equations

72

which use the bearing design parameters to yield a set of performance factors. The

design parameters used in these equations include the bearing length, journal diameter,

clearance ratio, applied load, rotational speed, and lubricant inlet viscosity. The resulting

performance factors are used to calculate quantitative solutions for the performance

variables of film thickness, lubricant side leakage, lubricant inflow, coefficient of

friction, and temperature rise. The film thickness is calculated using the load capacity

module. The temperature rise, coefficient of friction, and flow variables are calculated

using the temperature module. These two design modules are interrelated. The load

capacity module calculates the eccentricity ratio using the lubricant average temperature

and corresponding viscosity. The temperature module yields both the lubricant

temperature rise and average temperature using the bearing eccentricity ratio. The

modules are integrated together to develop a basic design model which simultaneously

satisfies all of the module equations and determines the bearing performance variables

from a set of input parameters. This design model is implemented as a computer program

and expanded to form a basic CAD tool capable of generating solution curves for

numerous performance variables of the journal bearing system with respect to any one of

the six design parameters.

The major advantage that the bearing CAD tool presented here has over the

current methods is its efficiency. The design tool contains a series of analytic equations

which calculate the performance factors. On the other hand, the current methods

manually extract the same solutions from design tables. The analytic equations of the

CAD tool are continuous and are capable of calculating for journal bearings of any size

or loading condition. The design tables, on the other hand, contain discrete data points

73

and require interpolation steps to extract the performance factor values. The CAD tool is

also more user-friendly, in that once the input parameters are supplied to the program, the

temperature and load capacity solutions are iteratively solved in an automated fashion. In

the current method, these same iteration steps are performed manually, increasing the

possibility of introducing human errors into the design process. Furthermore, to develop

a solution curve using the current methods, it would take hours of design work to

calculate enough data to generate a meaningful trend. Meanwhile, the same work is

completed in a few minutes using the CAD package. Reduced design time cuts the cost

spent on design work, making the journal bearing more cost-effective. Finally, a

computer program allows for globalization. The solutions of the CAD tool may be saved

into computer files which may easily be transferred via the internet to other designers and

individuals involved in the manufacturing process such as project managers, customers,

site engineers, machinists, etc. The ability to transfer data using the internet is especially

useful at the present time, as it is common that the various design and manufacturing

stages of a product may take place in different locations around the world.

The future of engineering design lies with the development of computer-aided

design tools. It is recommended that the bearing CAD tool presented here be integrated

with other CAD programs. This tool may be written as its own computer program,

calculating solutions solely for journal bearing systems. The solutions may then be saved

directly into computer files which may be accessed by other computer programs to use as

input. The bearing CAD tool may also be included in a larger design program. One

possibility is to develop a CAD package where each component of a large engineering

system is designed using a separate design module. These different design modules are

74

then integrated together, similar to how the design modules of the journal bearing design

tool are integrated with one another. Each module focuses on a single design aspect and

may be dependent on output from the other modules.

By modernizing the current design methods of journal bearings, a potentially

powerful design tool will result. It will be able to generate numerous solution curves of

the performance variables for journal bearings of almost any geometry. The design tool

will be accurate, time-efficient, reliable, and user-friendly. It may even be integrated

with other existing CAD programs to aid in the design of broader engineering problems.

The CAD tool provided here not only has the potential for improving the design process

of journal bearings, but it may motivate future designers to develop similar CAD tools to

replace manual design procedures of other engineering systems as well.

75

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[Online]. Available: <http://www.stiweb.com/appnotes.jb>. [Accessed: 27 May 2008]

[4] J. A. Williams, Engineering Tribology. Oxford: Oxford University Press, 1994. [5] B. J. Hamrock, Fundamentals of Fluid Film Lubrication. New York: McGraw-

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Butterworth-Heinemann, 2005. [7] Y. Hori, Hydrodynamic Lubrication. Tokyo: Springer-Verlag, 2006. [8] R. D. Arnell, Tribology: Principles and Design Applications, 1st ed. London:

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[12] J. E. Shigley, Mechanical Engineering Design, 7th ed. New York: McGraw-Hill, Inc., 2004.

[13] J. C. P. Claro, L. Costa, A. S. Miranda, and M. Fillon, "An Analysis of the Influence of Oil Supply Conditions on the Thermohydrodynamic Performance of a Single-Groove Journal Bearing," Proc. Instn Mech. Engrs, Part J: J. Engineering Tribology, vol. 217, pp. 133-144, 2003.

[14] R. W. Hornbeck, Numerical Methods. New York: Quantum, 1975. [15] MATLAB. Edition 7.6.0. Natick, Ma.: The MathWorks, Inc., 2008.

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Appendix A: The Numerical Method Design Tables

Table A-1: The Finite-Bearing Design Tables as Compiled in Booser [1]

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78

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Source: M. M. Khonsari, Tribology Data Handbook: Journal Bearing Design and Analysis. Boca Raton: CRC Press LLC, 1997. p. 671 - 674. Book Compiled by E. Michael Booser.

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Appendix B: Sample Viscosity-Temperature Chart

Figure B-1: A Common Viscosity-Temperature Chart. This particular chart may be used

for the SAE oil grades 10, 20, 30, 40, 50, 60, and 70.

Source: Shigley, Joesph E. Mechanical Enginering Design. 7th ed. New York: McGraw-Hill, Inc., 2004

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Appendix C: MATLAB Code for the Bearing CAD Tool

Routine: MainDesign.m clear all; %Journal Bearing Design Calculator %By: Richard K. Naffin %2009 Penn State University Graduate Student %Advisor: Dr. Liming Chang %The following CAD tool may be used to aid in the design of full journal %bearings. %This program has not been developed to solve for partial journal bearings. %Note: This program may be used if the following parameters %are predefined, or whose values are being optimized. %-Load on Bearing %-Rotational Speed of Journal %-Bearing inlet conditions %-Bearing Length %-Journal Diameter %-Clearance Ratio %-Lubricant Type or Grade %This program will output values for the following design considerations: %-Minimum Film Thickness %-Power Loss of Bearing System %-Average Flow Temperature and Temperature Rise %-Leakge Flow Rate %-Outlet Flow Temperature %Before running this program, go to the m-file Input.m and supply the %necessary input parameters. Input %Fill this out! %Once the correct parameters are supplied, run the program to calculate %through the following code. Current_Range_Value=Range_Min; i=0;

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while Current_Range_Value<Range_Max i=i+1; Current_Range_Value = Current_Range_Value + increment; if BigChoice==1 L=Current_Range_Value; elseif BigChoice==2 Clearance_ratio=Current_Range_Value; elseif BigChoice==3 P=Current_Range_Value; elseif BigChoice==4 RPM=Current_Range_Value; elseif BigChoice==5 D=Current_Range_Value; elseif BigChoice==6 viscinlet=Current_Range_Value; else %N/A. end Temp_Residual=2; while Temp_Residual>Tavg_tol Relationship_Module %This subroutine assigns the viscosity-temperature relationship %based on selection in input subroutine Epsilon_Iterative %This subroutine solves the eccentricity ratio TempChange_Output %The subroutine first calculates the performance parameters of %the friction, side leakage, and inflow. %This allows for temperature change to be solved, %Which may be used to calculate 'T_average_calc' end Performance_Calculation end Output Subroutine: Input.m

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%Please assign a value for the variable 'BigChoice' %To calculate design output with respect to the bearing length, select 1') %To calculate design output with respect to the clearance ratio, select 2') %To calculate design output with respect to the applied load, select 3') %To calculate design output with respect to the journal's speed, select 4') %To calculate design output with respect to the bearing diameter, select 5') %To calculate design output with respect to the inlet viscosity, select 6') %If you already know what to select for each input parameter, select 7') BigChoice=2; Verify=2; %No, select 1. Yes, select 2 %Bearing Input Parameters %Begin by filling in the following input about your bearing. %If one of these parameters is selected to vary over a range, %Set it equal to zero in this list. RPM=3600; %Journal Rotational Speed in rotations per minute P=6650; %Applied Radial Load in Newtons Tinlet=49; %Estimation of Bearing Inlet Temperature in Celsius. %Estimation Based on Ambient Temperature of Supplied %Lubricant. L=0.05; %Bearing Length in meters D=0.05; %Journal Diameter in meters Clearance_ratio=0; %The clearnance ratio, or c/R %Please supply the Lubricant Viscosity at the bearing inlet which %corresponds to the inlet temperature supplied previously. viscinlet=0.028; %in Pa*s. Set equal to zero if varied %Please select the viscosity-temperature relationship to use.

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%To use the Barus Equation (default), set 'Tv_relate' equal to 1. %To use the Other Relationships, set 'Tv_relate' equal to 2') Tv_relate=1; %If you know the lubricant grade that you wish to use, please supply %the following values. If you don't know the actual grade, please supply %nominal values of these material properties. specific_heat=1760; %Specific Heat of the Lubricant in J/(Kg/C). %Typically, it is 1900 J/(Kg/C) density=880; %Nominal Density of Supplied Lubricant in Kg/m^3 %If you choose to use the default temperature-viscosity relation please %fill in the following: B=0.033; %Baurus Constant of the Lubricant in 1/Celsius. %This factor is needed if you want to use the %default viscosity-temperature relation or if it %used in YOUR CHOSEN relation. %If this factor is not needed in YOUR CHOSEN %viscosity-temperature relation, set it to zero. %Now, if are varying one of the input parameters, select the range %to calculate output for. %Supply the values in the corresponding units as specified in above list. Range_Min=0.000125; %set value to 0 if not varying an input Range_Max=0.005; %set value to 1 if not varying an input %Please supply the increment change of the varied input. %It is recommended to select an increment change which will result %in the program only increasing the parameter approximately 100 times %across the range. increment = 0.00005; %set value to 1 if not varying an input %Please supply an estimated value of the average temperature T_average_guess=80; %in deg Celsius %Please supply acceptable error tolerances for the following: Tavg_tol=1; %Tolerance of average temeperature in deg C eta_tol=0.0001; %Tolerance of eccentricity ratio

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%This completes the input file Subroutine: Relationship_Module.m %Chooses the Temperature-Viscosity Relationship if Tv_relate == 1 %This is the default equation. DO NOT CHANGE! visc=viscinlet*exp(-B*(T_average_guess-Tinlet)); else %If you do not want to use the default equation, please supply %your own in the space provided before the 'end' statment. %Your equation must follow the following format: % %-You must use the the variables 'viscinlet', 'Tinlet', and % 'T_average_guess'as input into the equation. % %-The equation must output the variable 'visc', which is the % average viscosity of the system end Subroutine: Epsilon_Iterative.m %Preliminary Set-up Code %Due to time-constraints, the bisection method is not programmed into this %module. Instead, a simpler iterative method is used that produces %reasonable results for bearings of eccentricity ratios greater than %0.0001 %Conversion of Shaft Speed from rpm to rev/s N=RPM/60; %Conversion of Shaft Speed from rev/s to physical units (m/s or in/s) U=N*(D*pi); %Conversion of speed from rev/s to rad/s omega=N*2*pi; %The following code defines the boundary conditions of the %Derived Finite Curve-Fit %L/D Input SR=L/D; loSR=log10(SR);

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%Curve-Fit Boundary Conditions XS=log10(1/8); XL=log10(4.75); %Radial Clearance Calculation c=(0.5*D)*Clearance_ratio; %Start of Load Capacity Module Code to find Epsilon %Known Dimensionless Force on Bearing Wbarknown=(P*c^2)/(visc*omega*D^4); %Initial Epsilon eta=0; Residual=1; %The following while loop will run until the known dimensionless bearing %force matches that calculated from the Analytical Curve-Fit Equation while Residual>0 %Code that creates the incremental values of epsilon eta=eta+0.0001; %The correctiion factors used to keep error below 5% Correctionfactorshort=1-(0.7*(eta-.1)^10); Correctionfactorlong=0.91+(0.19*eta); %Solving for Long Bearing eccentricity function EL=(3*eta*sqrt((4*eta^2)+pi^2-(pi^2*eta^2)))/(4*(2+eta^2)*(1-eta^2)); %Solving for Short Bearing eccentricity function ES=(pi*eta*sqrt(1+0.62*(eta^2)))/(8*(1-(eta)^2)^2); if SR>4.75 Yl=log10(Correctionfactorlong*EL)+loSR; Wbarcurve=10^Yl; elseif SR<0.125 Ys=log10(Correctionfactorshort*ES)+(3*loSR); Wbarcurve=10^Ys; else %Defines the Dimensionless loads at the boundary conditions %Equations include any correction factors Wlong=log10(Correctionfactorlong*EL)+XL; Wlongp=1; Wshort=log10(Correctionfactorshort*ES)+(3*XS); Wshortp=3; %The curve-fit constants for Wbar

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%Below define the 4 constants that make-up of the curve-fit equation %These constants are functions of the dimensionless loads at the %boundary conditions and the eccentricity function equations. C3d=XS^3-XL^3+(((3*XS^4)+(3*XL^4)-(6*XL^2*XS^2))/(2*(XL-XS)))+(3*XS^2*XL)-(3*XL^2*XS); C3w=(Wshort-Wlong+(Wshortp*XL)-(Wlongp*XS)-((Wlongp-Wshortp)/(2*(XL-XS)))*(XS^2-XL^2))/C3d; C2w=(Wlongp-Wshortp-(3*C3w*XL^2)+(3*C3w*XS^2))/(2*(XL-XS)); C1w=Wshortp-(3*C3w*XS^2)-(2*C2w*XS); Conw=Wlong-(C3w*XL^3)-(C2w*XL^2)-(C1w*XL); %The final form of the curve-fit eqaution Yw=(C3w*loSR^3)+(C2w*loSR^2)+(C1w*loSR)+Conw; Wbarcurve=10^Yw; end %When residual equals zero, the loop will end and the current %value of epsilon will output Residual=Wbarknown-Wbarcurve; end %Computing Sommerfeld Number Sommerfeld=SR/(8*pi*Wbarcurve); Subroutine: TempChange_Output.m %Computing coefficient of frction dim_friction=(1+((0.56*SR+1.93)*(eta)^4))*(2*pi^2*Sommerfeld); f1=(dim_friction*c)/(0.5*D); %Attitude Angle Angle=atan((pi*sqrt(1-eta^2))/(4*pi)); %in rad %Calculating the Side Leakage Qside_factor=(1.0-((0.03*eta+0.23)*(SR-0.1)^1.2)); Qside_bar=2*eta*Qside_factor; Qside=0.5*pi*D*c*N*L*Qside_bar; %Calculating the Flow Rate Entering the Loaded Region Qin_factor=(1.0-((0.26*eta+0.01)*(SR-0.1)^1.2)); Qin_bar=(eta+1)*Qin_factor; Q_in=0.5*pi*D*c*N*L*Qin_bar;

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%Calculating the Temperature Rise Temperature_Change=(4*pi*P*dim_friction)/(density*specific_heat*L*D*pi*Qin_bar*(1-(0.5*(Qside_bar/Qin_bar)))); T_average_calc=Tinlet+(0.5*Temperature_Change); %Residul and Looping Calculations Temp_Residual=abs(T_average_calc-T_average_guess); T_average_guess=0.5*(T_average_calc+T_average_guess); Subroutine: Performance_Calculation.m %Computing Minimum Film Thickness hmin=c*(1-eta); %Computing Power loss in Watts Power_Loss=f1*P*U; %Computing Power loss in HP Power_Loss_HP=Power_Loss/745.7; %Computing the Outlet Temperature T_out=Tinlet+Temperature_Change; Vary_hmin(i)=hmin; Vary_eta(i) = eta; Vary_Qside(i) = Qside; Vary_Q_in(i) = Q_in; Vary_SR(i) = SR; Vary_f1(i) = f1; Vary_Power_Loss(i) = Power_Loss; Vary_T_average_calc(i) = T_average_calc; Vary_Temperature_Change(i) = Temperature_Change; Vary_T_out(i)=T_out; Vary_Clearance_ratio(i) = Clearance_ratio; Vary_P(i) = P; Vary_RPM(i) = RPM; Vary_viscinlet(i) = viscinlet;

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Subroutine: Output.m if BigChoice==1 figure(1) plot(Vary_SR,Vary_eta) title('Eccentricity Ratio') xlabel('Slenderness Ratio, L/D') ylabel('Epsilon') grid on figure(2) plot(Vary_SR,Vary_hmin) title('Minimum Film Thickness') xlabel('Slenderness Ratio, L/D') ylabel('hmin (in meters)') grid on figure(3) subplot(2,1,1); plot(Vary_SR,Vary_Qside) title('Lubricant Side Leakage') xlabel('Slenderness Ratio, L/D') ylabel('Flow Rate (in m^3/s)') grid on subplot(2,1,2); plot(Vary_SR,Vary_Q_in) title('Lubricant Inflow') xlabel('Slenderness Ratio, L/D') ylabel('Flow Rate (in m^3/s)') grid on figure(5) plot(Vary_SR,Vary_Power_Loss) title('Power Loss') xlabel('Slenderness Ratio, L/D') ylabel('Power Loss (in W)') grid on figure(6) plot(Vary_SR,Vary_f1) title('Coefficient of Friction') xlabel('Slenderness Ratio, L/D') ylabel('f (dimensionless)') grid on figure(7) plot(Vary_SR,Vary_T_average_calc,Vary_SR,Vary_T_out) title('Temperature') xlabel('Slenderness Ratio, L/D') ylabel('Tavg (in deg C)') legend('Average Temperature','Outlet Temperature') grid on figure(8)

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plot(Vary_SR,Vary_Temperature_Change) title('Temperature Change') xlabel('Slenderness Ratio, L/D') ylabel('delt T (in deg C)') grid on elseif BigChoice==2 figure(1) plot(Vary_Clearance_ratio,Vary_eta) title('Eccentricity Ratio') xlabel('Clearance Ratio, c/R') ylabel('Epsilon') grid on figure(2) plot(Vary_Clearance_ratio,Vary_hmin) title('Minimum Film Thickness') xlabel('Clearance Ratio, c/R') ylabel('hmin (in meters)') grid on figure(3) subplot(2,1,1); plot(Vary_Clearance_ratio,Vary_Qside) title('Lubricant Side Leakage') xlabel('Clearance Ratio, c/R') ylabel('Flow Rate (in m^3/s)') grid on subplot(2,1,2); plot(Vary_Clearance_ratio,Vary_Q_in) title('Lubricant Inflow') xlabel('Clearance Ratio, c/R') ylabel('Flow Rate (in m^3/s)') grid on figure(5) plot(Vary_Clearance_ratio,Vary_Power_Loss) title('Power Loss') xlabel('Clearance Ratio, c/R') ylabel('Power Loss (in W)') grid on figure(6) plot(Vary_Clearance_ratio,Vary_f1) title('Coefficient of Friction') xlabel('Clearance Ratio, c/R') ylabel('f (dimensionless)') grid on figure(7) plot(Vary_Clearance_ratio,Vary_T_average_calc) title('Average Temperature') xlabel('Clearance Ratio, c/R') ylabel('Tavg (in deg C)') grid on

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figure(8) plot(Vary_Clearance_ratio,Vary_Temperature_Change) title('Temperature Change') xlabel('Clearance Ratio, c/R') ylabel('delt T (in deg C)') grid on figure(9) plotyy(Vary_Clearance_ratio,Vary_eta,Vary_Clearance_ratio,Vary_hmin) title('Load Capacity Conditions') xlabel('Clearance Ratio, c/R') legend('Film Thickness','Eccentricity Ratiio') grid on elseif BigChoice==3 figure(1) plot(Vary_P,Vary_eta) title('Eccentricity Ratio') xlabel('Load (in N)') ylabel('Epsilon') grid on figure(2) plot(Vary_P,Vary_hmin) title('Minimum Film Thickness') xlabel('Load (in N)') ylabel('hmin (in meters)') grid on figure(3) subplot(2,1,1); plot(Vary_P,Vary_Qside) title('Lubricant Side Leakage') xlabel('Load (in N)') ylabel('Flow Rate (in m^3/s)') grid on subplot(2,1,2); plot(Vary_P,Vary_Q_in) title('Lubricant Inflow') xlabel('Load (in N)') ylabel('Flow Rate (in m^3/s)') grid on figure(5) plot(Vary_P,Vary_Power_Loss) title('Power Loss') xlabel('Load (in N)') ylabel('Power Loss (in W)') grid on figure(6) plot(Vary_P,Vary_f1) title('Coefficient of Friction')

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xlabel('Load (in N)') ylabel('f (dimensionless)') grid on figure(7) plot(Vary_P,Vary_T_average_calc) title('Average Temperature') xlabel('Load (in N)') ylabel('Tavg (in deg C)') grid on figure(8) plot(Vary_P,Vary_Temperature_Change) title('Temperature Change') xlabel('Load (in N)') ylabel('delt T (in deg C)') grid on elseif BigChoice==4 figure(1) plot(Vary_RPM,Vary_eta) title('Eccentricity Ratio') xlabel('Rotation (in RPM)') ylabel('Epsilon') grid on figure(2) plot(Vary_RPM,Vary_hmin) title('Minimum Film Thickness') xlabel('Rotation (in RPM)') ylabel('hmin (in meters)') grid on figure(3) subplot(2,1,1); plot(Vary_RPM,Vary_Qside) title('Lubricant Side Leakage') xlabel('Rotation (in RPM)') ylabel('Flow Rate (in m^3/s)') grid on subplot(2,1,2); plot(Vary_RPM,Vary_Q_in) title('Lubricant Inflow') xlabel('Rotation (in RPM)') ylabel('Flow Rate (in m^3/s)') grid on figure(5) plot(Vary_RPM,Vary_Power_Loss) title('Power Loss') xlabel('Rotation (in RPM)') ylabel('Power Loss (in W)') grid on figure(6)

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plot(Vary_RPM,Vary_f1) title('Coefficient of Friction') xlabel('Rotation (in RPM)') ylabel('f (dimensionless)') grid on figure(7) plot(Vary_RPM,Vary_T_average_calc) title('Average Temperature') xlabel('Rotation (in RPM)') ylabel('Tavg (in deg C)') grid on figure(8) plot(Vary_RPM,Vary_Temperature_Change) title('Temperature Change') xlabel('Rotation (in RPM)') ylabel('delt T (in deg C)') grid on elseif BigChoice==5 figure(1) plot(Vary_SR,Vary_eta) title('Eccentricity Ratio') xlabel('Slenderness Ratio, L/D') ylabel('Epsilon') grid on figure(2) plot(Vary_SR,Vary_hmin) title('Minimum Film Thickness') xlabel('Slenderness Ratio, L/D') ylabel('hmin (in meters)') grid on figure(3) subplot(2,1,1); plot(Vary_SR,Vary_Qside) title('Lubricant Side Leakage') xlabel('Slenderness Ratio, L/D') ylabel('Flow Rate (in m^3/s)') grid on subplot(2,1,2); plot(Vary_SR,Vary_Q_in) title('Lubricant Inflow') xlabel('Slenderness Ratio, L/D') ylabel('Flow Rate (in m^3/s)') grid on figure(5) plot(Vary_SR,Vary_Power_Loss) title('Power Loss') xlabel('Slenderness Ratio, L/D') ylabel('Power Loss (in W)') grid on

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figure(6) plot(Vary_SR,Vary_f1) title('Coefficient of Friction') xlabel('Slenderness Ratio, L/D') ylabel('f (dimensionless)') grid on figure(7) plot(Vary_SR,Vary_T_average_calc) title('Average Temperature') xlabel('Slenderness Ratio, L/D') ylabel('Tavg (in deg C)') grid on figure(8) plot(Vary_SR,Vary_Temperature_Change) title('Temperature Change') xlabel('Slenderness Ratio, L/D') ylabel('delt T (in deg C)') grid on figure(9) subplot(2,1,1); plot(Vary_SR,Vary_eta) title('Eccentricity Ratio') xlabel('Slenderness Ratio, L/D') ylabel('Epsilon') grid on subplot(2,1,2); plot(Vary_SR,Vary_hmin) title('Minimum Film Thickness') xlabel('Slenderness Ratio, L/D') ylabel('hmin (in meters)') grid on elseif BigChoice==6 figure(1) plot(Vary_viscinlet,Vary_eta) title('Eccentricity Ratio') xlabel('Inlet Viscosity, Pa*s') ylabel('Epsilon') grid on figure(2) plot(Vary_viscinlet,Vary_hmin) title('Minimum Film Thickness') xlabel('Inlet Viscosity, Pa*s') ylabel('hmin (in meters)') grid on figure(3) subplot(2,1,1); plot(Vary_viscinlet,Vary_Qside) title('Lubricant Side Leakage')

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xlabel('Inlet Viscosity, Pa*s') ylabel('Flow Rate (in m^3/s)') grid on subplot(2,1,2); plot(Vary_viscinlet,Vary_Q_in) title('Lubricant Inflow') xlabel('Inlet Viscosity, Pa*s') ylabel('Flow Rate (in m^3/s)') grid on figure(5) plot(Vary_viscinlet,Vary_Power_Loss) title('Power Loss') xlabel('Inlet Viscosity, Pa*s') ylabel('Power Loss (in W)') grid on figure(6) plot(Vary_viscinlet,Vary_f1) title('Coefficient of Friction') xlabel('Inlet Viscosity, Pa*s') ylabel('f (dimensionless)') grid on figure(7) plot(Vary_viscinlet,Vary_T_average_calc) title('Average Temperature') xlabel('Inlet Viscosity, Pa*s') ylabel('Tavg (in deg C)') grid on figure(8) plot(Vary_viscinlet,Vary_Temperature_Change,Vary_viscinlet,Vary_T_out) title('Temperature Conditions') xlabel('Inlet Viscosity, Pa*s') ylabel('Temperature (in deg C)') legend('Temperature Change','Outlet Temperature') grid on else fprintf('The eccentricity ratio is %f.', eta); disp(' ') fprintf('The minimum film thickness is %f m.', hmin); disp(' ') fprintf('The coefficicent of friction is %f.', f1); disp(' ') fprintf('The Power loss of this bearing is %f Watts.', Power_Loss); disp(' ') fprintf('Or the power loss can be written in the form of %f HP.', Power_Loss_HP); disp(' ') fprintf('The leakage flow rate of this bearing is %f m^3/s.', Qside); disp(' ') fprintf('The lubricant flow rate into the loaded region of bearing is %f m^3/s.', Q_in);

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disp(' ') fprintf('The temperature rise of the loaded region of bearing is %f deg C.',Temperature_Change); disp(' ') fprintf('The outlet temperature of the loaded region of bearing is %f deg C.',T_out); disp(' ') end