Project Example 08

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College of Engineering

Design Project #2- Compressor Drive Train

Chris Sanchez

Clay Spencer

Michael (M-P) Spierer

John (J.C.) Stevens

Michael Sullivan

September 23

rd

, 2008MAE 415-003 Analysis for Mechanical Engineering Design

We have neither given nor received any unauthorized assistance on this report


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MAE 415-003 Analysis of ME Design

Mechanical and Aerospace EngineeringDepartment

i NC STATE UNIVERSITY

Abstract

This report proposes a design of the gears, shafts and bearings composing a compressor drivetrain. The main focus of this design is on the gears and bearings of the system. The drive train

converts the output from a 2.5HP gasoline engine operating at 3800RPM via a single gear

reduction of 2.5:1 to the compressor. This system has been designed to operate at 1 shift per dayfor 10 years. Basic gear and bearing analysis techniques were used to determine the type,

dimensions and material of each component within the drive train.

First, an initial set of gears was chosen and resulting stresses from the maximum torque required

by the compressor were calculated. The maximum torque encountered by the system, 595 lbf-in,was used to determine the transmitted load for the gears which was then used to find the bending

and contact stress that the gear teeth experience. These values were compared to allowable

values for the gear material. The initial gears were found to be inadequate and the choice of gearpitch diameters were iterated until a reasonable solution was attained. All gears considered were

production gears that could be bought ready-made.

Deep-groove ball bearings were chosen to account for the radial loads that the shaft encounters

from the gears. The initial bearing chosen met the desired life limit of the system with areliability of 99%. There are no other bearings available through SKF with lower dynamic load

ratings, so the initial bearing is acceptable.


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Mechanical and Aerospace EngineeringDepartment

ii NC STATE UNIVERSITY

Table of Contents

Abstract: ..............................................................................................................................i1. Introduction ....................................................................................................................1

2. Analysis

Gear Analysis ...........................................................................................................1Bearing Analysis ......................................................................................................5

3. Conclusions and Recommendations ..............................................................................7

4. References ......................................................................................................................85. Appendices

Appendix A ..............................................................................................................9

Appendix B ...............................................................................................................10Appendix C ...............................................................................................................10

Appendix D ..............................................................................................................11

Appendix E ...............................................................................................................12

Appendix F ...............................................................................................................12

Appendix G ..............................................................................................................13


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Mechanical and Aerospace Engineering

Department1 NC STATE UNIVERSITY

Introduction

Mechanical systems are often powered by the rotational motion produced from an internalcombustion engine; however, these systems rarely get power directly from the output of an

engines crankshaft. The combustion which takes place in an engine results in an output of a

specific rotational speed which must often be adjusted to attain a desired rotational speed of adriven component in a system. This modification is often achieved through the use of gear trains.

An additional concern for such systems is the issue of mounting rotating components in such a

way that they are secured in place, but also allowed to rotate. This is done through the use ofbearings. In this project a typical system using gears and bearings is evaluated in order to

investigate the critical aspects of transmitting power from a 2.5 horsepower internal combustion

engine to a compressor, as shown below in figure 1. The calculations used to evaluate thesystem contain several symbols which are defined in appendix A.

Figure 1- Layout of proposed system

Analysis

Gear Analysis

Before any calculations were performed, the gears to be used were assumed to be Grade 1through-hardened AISI 4140 steel gears. This is due to this type of gear being widely available

and its characteristics being well documented. The mean properties of AISI 4140 steel were

obtained from eFunda3 and found to be:

Elastic Modulus (mean):

Poisson's Ratio (mean):


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The mean value for hardness was determined from figure 14-2 on p.727 of Shigley2

to beHB=300. Once the material properties are known the allowable bending stress, St, can be

calculated using the correlation found in figure 14-2 on p.727 of Shigley2 for this type of gear:

(1)

Solving equation (1) produces the following:

The allowable contact stress, Sc, can also be found using the correlation in figure 14-5 on p.730

in Shigley2:

(2)

The allowable contact stress found in equation (2) is found at 10 million stress cycles and areliability of 99% for Grade 1 through-hardened steel gears. Solving equation (2) yields:

These strengths will be used to evaluate of the validity of gear choices. The gear evaluation

process begins with choosing a gear pitch and sizes. A gear ratio of 2.5:1 is called for due to

system requirements. There were several gear sets analyzed, some of which yieldedunsatisfactory results. Final design values, which are outlined below, are the result of iteration.

The diametral pitch, P, is chosen to be 6 teeth per inch and the pitch diameters as dp=2 inchesand dg=5 inches for the pinion and gear, respectively. The pitch angle, , is chosen to be thecurrent standard of 20 degrees, which bears loads better than the older standard of 14.5

degrees1,2. Using the Rush Gears part search4, the chosen values above were used to find

production gears with the specifications listed below in table 1.

Table 1- Gear Specifications

Pinion Gear

Part # F612 F630

Pitch, P 6 6 teeth/inch

Number of teeth, N 12 30 teeth

Pitch diameter, d 2 5 inch

Pressure angle, 20 20 degrees

Face width, F 1.5 1.5 inches

The diameter of the base circle of each gear, used for forming involute curves for the gears

teeth, is found by modifying equation 13-6 from Shigley2 into:

(3)

The addendum, a, and dedendum, b, distances are defined by equations from table 13-1 on p.676

of Shigley2:

, (4)

(5)

The above information was used to model the pitch, base, addendum, and dedendum circles inSolidWorks. This resulted in the information presented below in figure 2.


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Figure 2- Gear Layout

See appendix B for a model of the actual gears. To see the actual meshing of the teeth betweenthe pinion and the gear, appendix C can be referenced. An additional factor describing the gears

is the tooth form factor, Y. The value ofYfor the gear and pinion is obtained from Boston Gear1

to be:

From a modified version of Shigley equation 13-33 (p. 686)2, the maximum transmitted load is

calculated by:

, (6)

where dis the pitch diameter of the gear in question and is the maximum torque applied tothe gear. Therefore for the gear:

This is the maximum transmitted load experienced by the gear, and thus also the coupled pinion.

The bending stress in a spur gear according to the Lewis Bending Equation is defined by Shigley

(p.717 equation 14-2)2

as:

, (7)


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where P is the diametral pitch and F is the face width. Therefore substituting the above values forthe gear into equation (7) gives:

and for the pinion:

These are the stresses associated with direct loading between the gears. However, there are a

number of dynamic factors associated with the noise encountered during operation. The effects

of the dynamic factors are dependent on the pitch-line velocity which is defined by equation 13-34 of p.687 of Shigley2 to be:

, (8)

where n is the number of gear rotations per minute (RPM). Thus substituting the values for the

pinion into equation (8) yields:

Since the pinion and gear are connected, their pitch line velocities are the same

(Vgear=Vpinion=1989.7 ft/min). The effects of dynamic factors are approximated with a Kfactor

using equation 14-4b from p. 719 of Shigley2:

(9)

Solving equation (9) for the pinion yields the following results:

As the velocity is the same for each gear, the Kfactor for the gear will be equal to that of the

pinion (KV,gear=KV,pinion=2.66). The K factor is used to find the dynamic stress, dynamic, in

equation 14-7 from p.719 of Shigley

2

:(10)

Solving equation (10) by substituting the above values for the pinion leads to:

and, for the gear:

A safety factor, SF, can be calculated by comparing allowable stresses to the maximum stresses

components see, as defined below:

(11)

The dynamic stresses found using equation (10) prove to be maximum stresses, and thus can be

used with equation (11) to calculate the following factors of safety:


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Aside from the bending loads applied to the gear, the compressive load must also be taken intoaccount. To accomplish this, the elastic coefficient, Cp, of the gear must be calculated. Equation

14-3 from p. 724 of Shigley2 defines this coefficient as:

(12)

Substitution of Poissons Ratio and Youngs Modulus into equation (12) yields:

The compressive stress, c, in the gears can be found using this coefficient along with equation14-14 from p.724 of Shigley2:

, (13)

where is the pitch angle as given in Table 1 and r1 and r2 are the radii of curvature for bothgears at the pinch point, defined as:

(14)

(15)

Substituting values into equations (14) and (15) yields:

Once the values for r1 and r2 are known equation (13) can be solved to find the compressivestress:

These values can be compared with the allowable contact stress calculated earlier using equation(11) to find the factor of safety for contact stress as:

Both the bending and contact safety factors are sufficient to conclude that the gears defined in

table 1 are appropriate choices for this drive train. The calculations performed on the gears were

done in Excel and the actual Excel sheet can be seen in appendix D.

Bearing Analysis

The bore diameter for both gears is defined by Rush Gears4 to be 1 inch. Deep-grove ballbearings were chosen because no axial thrust load was present in this system. The required life of

the bearings as specified in the problem statement is 10 years of operation at 1 shift per day, with

an assumed shift length of 8 hours. The desired reliability of 99% is also specified in the problemstatement. Shigley2 uses the symbolL for bearing life in number of revolutions. It is assumed

that the loads will be shared equally among the bearings in each gear, so both bearings for the

pinion will receive the same loads as each other. The bearings for the gear will also share the


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same loads, though different from those for the pinion. The desired lives for the Pinion bearingsare:

Similarly, for the gear bearings:

In order to rate bearings on a common scale, the design life is compared to the manufacturers

specified catalog life using the dimensionless life variate, xD. SKF was chosen as themanufacturer for this project. The catalog life used by SKF is 106rev and on p.558 Shigley

definesxD as:

(16)

Using equation (16) to calculate the dimensionless life variate for the pinion yields:

and for the gear:

ThesexD values are used to calculate the catalog load rating, which is used to select

commercially-available bearings based on their loads and life requirements. This catalog loadrating is defined by Shigley (p.557, eq11-6)2 as:

, (17)

where , , and are known as the Weibull parameters. For a deep-groove ball bearing from

SKF p.558 of Shigley2

defines these parameters to be:

Also from equation (17), is the desired reliability rating, in this case 0.99, and for a ball

bearing is 3 (Shigley, p.558)2. is the desired radial load. This must be calculated from the

loads experienced by the shaft. The problem statement gives the average torque of the

compressor shaft (most suitable for bearing design) as 92 lbf-in. Modifying equation (6) by using

the average torque value translates into an average transmitted load of:

The radial component of the average loading on the gears can be calculated using the pressure

angle of 20 degrees and the transmitted (tangential) load:

The overall load vector on the shaft elicits an equal overall reaction from the bearings holding it.

This load is essentially the resultant vector from the tangential and radial gear loads as follows:

Note that for each shaft and thus each gear there are two bearings. For this analysis the shaft

loads are assumed to be shared equally by each of the bearings supporting the shaft. Thus the

desired radial load per bearing is:


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Substituting this and the values stated above into the equation (17) allows the catalog load ratingfor the bearings on the pinion shaft and for the bearings on the gear shaft to be calculated as:

All calculations that were performed in order to choose a proper bearing were done in Excel andthe actual Excel sheet can be seen in appendix E. With these bearing load ratings, the SKF

bearing catalog5 was consulted to find the appropriate bearings with a 1 inch bore diameter and

the ability to handle 3800 RPM. It was found that the same bearing best satisfied therequirements of this system. The actual catalog entry from SKF for the chosen bearings can be

seen in appendix F. The specifications of the bearing are as follows in table 2:

Table 2- Bearing Specifications

Bearings

SKF Part # YAT 205-100Dynamic load rating, C10 (lbf) 3150

Diameter, d (in) 1

Max RPM 7000

Conclusions and Recommendations

For the purpose of this design, the gears and bearings selected meet the defined criteria. Throughthe calculations and analysis outlined in this report, the proposed design can be recommended as

the components presented meet all requirements and specifications. The design process also

yielded other designs that met these requirements, but these designs were excessive considering

the system given. The final design can be seen in better detail in the exploded view of theassembly shown in appendix G. The proposed gear sizes were minimized in order to provide for

a compact design while still obtaining an appropriate factor of safety. The factor of safety may

still appear excessive for the bending stress in the gears; however, the factor of safety for theallowable contact stress cannot be decreased further without becoming unsatisfactory. Therefore,

it can be concluded that the bending and contact stress safety factors are within acceptable

ranges. It is also recommended that before this design is utilized, a cost analysis be performed todetermine whether or not the proposed design is cost effective.


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References

1Boston Gear. Open Gearing Catalog. 20 Oct. 2008

. 2Budynas, Richard, and J. Keith Nisbett. Shigley's Mechanical Engineering Design. New York,

NY: McGraw-Hill Science, Engineering & Mathematics, 2006.3eFunda. AISI 4140 Material Properties. 20 Oct. 2008

.4Rush Gears. 20 Oct. 2008 .5SKF Group. Bearing Product Tables. 20 Oct. 2008

.


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Appendices

Appendix A- List of symbols

Shape Parameter that Controls the Skewness

Bearing Catalog Load Rating

Gear Elastic Coefficient

Gear Pitch Diameter (in)

Youngs Modulus (psi)

Gear Face Width (in)

Brinell Hardness Rating

Dynamic Effect Velocity Factor

Bearing Life (rev)

Rotation Speed (RPM)

Gear Pitch (teeth/in)

Radius of Curvature at Gear Pinch Point (in)

Reliability

Compressive Strength (psi)

Bending Strength (psi)

Factor of Safety

Torque (lbf-in)

Pitch-Line Velocity (ft/min)

Radial Gear Load (lbf)

Transmitted Gear Load (lbf)

Minimum Value of Bearing Life Variate

Bearing Life Measure Dimensionless Variate

Tooth Form Factor

Characteristic Parameter Corresponding to the 63.2121Percentile Value of the Bearing Life Variate

Poissons Ratio

Stress (psi)

Gear Pitch Angle (deg)


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Appendix B- SolidWorks model of gear and pinion

Appendix C- Detail of meshing involute gear teeth


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Appendix D- Gear analysis Excel file


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Appendix E- Bearing analysis Excel file

Appendix F- SKF bearing product sheet


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Appendix G- Exploded view of gearbox assembly

Project Example 08

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