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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
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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.
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Table of Contents
List of Figures....................................................................................................................vi
List of Tables....................................................................................................................vii
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
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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
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.
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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
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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
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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.
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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
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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.
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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
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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
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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.
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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.
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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.
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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
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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.
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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.
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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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Analysis. Boca Raton: CRC Press LLC, 1997. [2] Engineers Edge, 2008. [Online] Available: <http://www.engineersedge.com/beari ng_application.htm>. [Accessed: 27 May 2008] [3] STI Field Application Note: Journal Bearings, Sales Technology, Inc., 1999.
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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:
MacMillan Education Ltd, 1991. [9] B. J. Hamrock, Bo Jacobson, and Steven R. Schmid, Fundamentals of Machine
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Horwood Ltd., 1981. [11] J. Y. Jang, and M.M. Khonsari, "Design of Bearings on the Basis of
Thermohydrodynamic Analysis," Proc. Instn Mech. Engrs, Part J: J. Engineering Tribology, vol. 218, pp. 355-363, 2004.
[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
79
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