1. Final Design Project:
Design of a Gear Reducer for a Tractor
MNE-381 Design for Machine Elements
Dr. Afsoon Amirzadeh
Adam York
Karoly Fodor
Thomas Napert
Diarny Fernandes
Fall Semester 2014

2. Introduction
Tractors require robust and reliable components due to the combination of a large engine
torque and the often jerky and uneven loading experienced by the drive train and gears. Such
conditions require careful and methodical design. For this project, teams were given design
requirements and dimensional constraints and expected to design a gear reducer for a tractor that
could feasibly be manufactured. Two gear-pinion pairs were required, and had to be designed in
such a way as to fit inside a 22” x 22” x 25” gearbox while adhering to US standard dimensional
constraints. The reducer has to transmit 22 horsepower and reduce an input velocity of 1800
RPM to a range between 330 and 335 RPM while having an input shaft that is in line with the
output shaft. The design of this gearbox was done to reinforce and prove the concepts learned in
class.
Gear Design
Note: Unless otherwise specified, all references to tables, figures, or appendices refer to the
textbook.
Adequate diameters and amounts of teeth were determined for each gear and pinion with
the use of the residual method. Hunting teeth were avoided by making non-integer velocity
ratios. Diametral pitches for the gears were chosen based on table 8-3 in the book. We ended up
with the first pinion and gear having 55 and 128 teeth, and the second pinion and gear having 44
and 102 teeth, respectively. In accordance with figure 9-24, the first pair was given a Pd of 8
because it receives an input speed of 1800 RPM at 22hp, and the second pair was given a Pd of
10 because it receives 773 RPM at 22hp. Both of these diametral pitches are US standard coarse
pitch values, according to table 8-3. Using table 9-12, a design life of 5000 hours was used,
which is suitable for both automobiles and agricultural equipment. An AGMA quality number of
A6 and a reliability of 99% were used to compliment that design life, according to table 9-3 and
9-11, respectively. We ended up with a minimum hardness of 214.77 HB for the first pair and
318.33 HB for the second. Because of this, we chose SAE 1015 SWQT 350 for the first pair and
SAE 5150 OQT 1000 for the second pair, which we found in Appendix 5. After selecting a
material for each gear, we checked out safety factor (Sf) values to make sure our design could
adequately handle loading. We also solved for the size factor (cs) and S’n values again to ensure

3. that they would be adequate. These calculations can be seen in the appendix of this report. We
also determined that the backlash for the first gear pair would be 0.0095”, and 0.01” for the
second pair, according to table 8-5. The calculations for the gears in this project are shown in
appendix A of this report. Parameters for the gears are shown on tables A, B, and C on the
following pages.
Table A: Geometry Parameters
Parameter Symbol Pinion 1 Gear 1 Pinion 2 Gear 2
Pitch diameter D 5.5" 12.8" 5.5" 12.75"
Outside diameter Do 5.7" 13" 5.75" 13"
Root diameter Dr 5.25" 12.55" 5.188" 12.438"
Base circle diameter Db 5.168" 12.028" 5.168" 11.981"
Addendum a 0.1" 0.1" 0.125" 0.125"
Dedendum b 0.125" 0.125" 0.156" 0.156"
Clearance c 0.025" 0.025" 0.031" 0.031"
Circular Pitch p 0.314" 0.314" 0.393" 0.393"
Whole Depth ht 0.225" 0.225" 0.281" 0.281"
Working Depth hk 0.2" 0.2" 0.25" 0.25"
Tooth Thickness t 0.157" 0.157" 0.196" 0.196"
Center Distance C 9.15" 9.13" 9.15" 9.15"
Fillet radius in basic rack rf 0.03" 0.03" 0.037" 0.037"
Bending Geometry Factor J 0.459 0.479 0.43 0.465
Pitting Geometry Factor I 0.114 0.114 0.112 0.112
Table B: Force and Speed Factors
Parameter Symbol Pair 1 Pair 2
Input Speed np - ng 1800 rpm 773.438 rpm
Output speed ng - np 773.438 rpm 333.64 rpm
Gear ratio mg 2.327 2.318
Quality number Av 6 8
Face Width F 1.2 1.5
Pitch Line Speed vt 2591.814 ft/s 1113.67 ft/s
Tangential Force Wt 280 lbf 651.636 lbf
Normal Force Wn 297.97 lbf 693.457 lbf
Radial Force Wr 101.912 lbf 237.176 lbf
Size Factor Ks 1 1
Load Distribution Factor Km 1.145 1.159
Dynamic Factor Kv 1.116 1.206

4. Table C: Additional Force and Speed Factors
Force/Speed Factor Symbol Pinion 1 Gear 1 Pinion 2 Gear 2
Rim Thickness Factor Kb 1.292 1.292 1.649 1.649
Number of Load Cycles Nc 6.48E+08 2.78E+08 2.78E+08 1.20E+08
Bending Stress Cycle Factor Yn 0.945 0.959 0.959 0.974
Pitting Stress Cycle Factor Zn 0.909 0.926 0.926 0.944
Expected Bending Stress St 16775.995 psi 16089.089 37251.553 psi 34444.64 psi
Expected Contact Stress Sc 70924.55 psi 70924.55 psi 102272.472 psi 102272.472 psi
Allowable Bending Stress # Sat 15092.376 psi 14258.403 psi 33012.912 psi 30071.942 psi
Allowable Contact Stress # Sac 66354.946 psi 65078.242 psi 93842.157 psi 92044.87 psi
Shaft Design
Note: Unless otherwise specified, all references to tables, figures, or appendices refer to the
textbook.
Shaft 1 and 3 were modeled hypothetically in for a range between 6” and 11” long. The
6” shaft was determined to be the best choice because its short length minimized bending and
deflection while maintaining a large enough clearance between shaft ends to account for the
bearing mounts. Shaft 2 was made to be 15” long to match the length of shafts 1 and 3. After
some analysis and several checks, these shaft lengths turned out to be satisfactory. All of the
shaft diameters were set to US standard sizes as shown in appendix A2-1, except for the shaft
ends. The shaft ends were given diameters equal to standard bearing sizes according to table 14-
3. The material selected for all three of the shafts was SAE 4140 OQT 1300 steel because of its
high tensile strength and ductility. The material properties in question were found in appendix 3.
The bulk of each shaft had a constant diameter, with the exception of the shaft ends. The
reasoning behind this was two-fold: keeping a constant diameter on either side of the gear saves
material while making manufacturing easier, since the gear can be slid onto the shaft from either
end. This meant that each gear would have to be held in place by two retaining rings. Because of
the loading requirements, a pair of appropriately sized CR heavy duty spiral rings would be used
to hold each gear in place. A design factor (N) of 4 was used for the shafts, which we checked

5. after selecting standard sizes. The calculations for the dimensions, loading and safety factors of
each shaft are described in appendix B of this report. Parameters for shaft 1, 2 and 3 are shown
on tables D, E and F, respectively, on the following pages. Diagrams (not to scale) of each shaft
are provided in figures A, B and C to clarify the information on the tables.
Figure A: Shaft 1
Table D-1:
Tangential Force (lb) Radial Force (lb) Torque (lb -in) Reaction-y (lb) Reaction-x (lb)
280.00 101.91 770.00 50.96 140.00
Table D-2:
Section (D) Stress Factor (Kt) Moment [M] (lb-in) Torque [T] (lb-in)
1 2.50 0.00 770.00
2 3.00 446.95 770.00
3 2.00 446.95 770.00
4 3.00 446.95 0.00
5 2.50 0.00 0.00
Table D-3:
Section
(D)
Minimum Diameter
[Dmin] (in)
Min. Dia. with
Ring Groove
Standard
Diameter
Cs Check New S'n
1 0.65 - 1.1811 0.860 30652.75
2 1.25 1.32 1.4 0.844 30084.79
3 1.09 - 1.4 0.844 30084.79
4 1.24 1.32 1.4 0.844 30084.79
5 0.00 - 1.1811 0.860 30652.75

6. Table D-3:
SAE 4140 OQT 1300
S'n 28512
Sn 44000
Cm 1
Cst 1
CR 0.81
Cs 0.8
Figure B: Shaft 2
Table E-1
Gears
[Left to
Right]
Max Bending
Moment
[Mx] (lb-in)
Max Bending
Moment
[My] (lb-in)
Max Bending
Moment [M]
(lb-in)
Tangential
Force (lb)
Radial
Force
(lb)
Torque
(lb -in)
Reaction
-y (lb)
Reaction
-x (lb)
Gear 1 1131.20 1062.98 386.89 280.00 101.91 1792.00 128.96 354.33
Gear 2 1843.08 1731.93 630.37 651.64 237.18 1873.45 210.12 577.31
Table E-2
Diameter [D] Stress Factor
(Kt)
Moment [M]
(lb-in)
Torque [T] (lb-
in)
Minimum Diameter
[Dmin] (in)
1 2.50 0.00 0.00 0.00
2 3.00 1131.20 1792.00 1.70
3 2.00 1131.20 1792.00 1.49
4 3.00 1843.08 1792.45 1.99
5 2.00 1843.08 1792.45 1.74
6 3.00 1843.08 1792.45 1.99
7 2.50 0.00 0.00 0.00

7. Table E-3
Diameter [D] Min. Dia. with Ring
Groove
Standard
Diameter
Cs Check New S'n
1 - 1.1811 0.860 30652.75
2 1.80 2.2 0.803 28625.60
3 - 2.2 0.803 28625.60
4 - 2.2 0.803 28625.60
5 - 2.2 0.803 28625.60
6 2.11 2.2 0.803 28625.60
7 - 1.1811 0.860 30652.75
Figure C: Shaft 3
Table F-1
Tangential Force (lb) Radial Force (lb) Torque (lb -in) Reaction-y (lb) Reaction-x (lb)
651.64 237.18 4154.18 118.59 325.82
Table F-2
Section [D] Stress Factor (Kt) Moment [M] (lb-in) Torque [T] (lb-in) Minimum Diameter [Dmin] (in)
1 2.50 0.00 4154.18 1.14
2 3.00 1040.19 4154.18 1.67
3 2.00 1040.19 4154.18 1.49
4 3.00 1040.19 0.00 1.65
5 2.50 0.00 0.00 0.00

8. Table F-3
Section [D] Min. Dia. with Ring Groove Standard Diameter Cs Check New S'n
1 - 1.57 0.83 29697.93
2 1.77 1.80 0.82 29264.50
3 - 1.80 0.82 29264.50
4 1.74 1.80 0.82 29264.50
5 - 1.57 0.83 29697.93
Table F-4
SAE 4140 OQT 1300
S'n 28512
Sn 44000
Cm 1
Cst 1
CR 0.81
Cs 0.8
Because of the compact nature of the shafts, bending moments and shaft deflections were
minimized. This is important since large deflections could possibly result in improper gear
meshing or even shaft failure. Deflection information is shown on table G below, where
negatives denote downward deflections. Since shaft 2 would be subjected to moments and
deflection at two different locations, worst case loading was used in our calculations. All
pertinent formulas are from A14-1 in the textbook.

9. Table G-1: Shaft Deflections
Shaft Diameter
[D] (in)
Area Moment of Inertia
[I] (in4
)
Load [P]
(lb)
Modulus of Elasticity [E] (psi)
1 1.40 0.19 101.91 30458000
2 2.20 1.15 237.18 30458000
3 1.80 0.52 237.18 30458000
Table G-2: Shaft Deflections (cont.)
Shaft Total Length [L]
(in)
Length from Edge to Load [a]
(in)
Max Deflection [ymax] (in)
1 6.00 3.00 -7.98E-05
2 15.00 3.00 -5.41E-04
3 6.00 3.00 -6.80E-05
Keys
Note: Unless otherwise specified, all references to tables, figures, or appendices refer to the
textbook.
All of the keys were designed to be made out of SAE 1018 steel, according to table 11-4.
The height and width of the keys were set according to the US standard sizes found in table 11-1,
while the proper lengths of the keys were found in appendix A2-1. Standard fillet radii were
found in table 11-2. Because the keys are composed of a weaker material, they will fail before
the gears or shafts are damaged in an overload situation. This is desirable, since the keys are
significantly less expensive to produce or replace, and their failure would serve to preserve the
gears and shafts. Selected information about the keys is provided in table H on the following
page.

10. Table H: Key Parameters
Parameter Key 1 (shaft 1) Key 2 (shaft 2) Key 3 (shaft 2) Key 4 (shaft 3)
Diameter of Shaft [D] (in) 1.4 2.2 2.2 1.8
Height [H] (in) 0.375 0.625 0.625 0.375
Width [W] (in) 0.375 0.625 0.625 0.375
Yield Strength [Sy] (psi) 54000 54000 54000 54000
Torque [T] (lb-in) 770 1792 1792 4154
Length [Torque] (in) 0.326 0.29 0.29 1.368
Length [Normal Stress] (in) 0.326 0.29 0.29 1.368
Minimum Length [L] (in) 0.326 0.29 0.29 1.368
Closest Standard [L] (in) 0.375 0.3125 0.3125 1.5
Chordal Height [Y] 0.03 0.05 0.05 0.02
Depth of Key seat [S] 1.19 1.84 1.84 1.59
T-Value [T] 1.56 2.47 2.47 1.97
Standard Fillet Radii 0.031 0.063 0.063 0.031
Bearings
Note: Unless otherwise specified, all references to tables, figures, or appendices refer to the
textbook.
With the exception of the bearing pair on which shaft 3 rotates, all of the bearings in this gearbox
are US standard 6006 bearings. The bearings holding shaft 3 are US standard 6008 bearings. All
of the standard bearing sizes and parameters were found in table 14-3 in the textbook. The
process was simplified by designing the shaft ends to have a diameter equal to the inner diameter
of a standard bearing. The bearings we selected are very strong and far exceed the design
requirements, but this was unavoidable since their size had to be taken into consideration as well.
However, this also ensures that the bearings will not have to be replaced throughout the useful
life of the gearbox. The parameters for each bearing assembly are shown on table I on the
following page.

11. Table I: Bearing Parameters
Parameter
Bearing Pair
(shaft 1)
Bearing 3
(shaft 2)
Bearing 4
(shaft 2)
Bearing Pair
(shaft 3)
Design Load [Pd] (lb) 50.96 128.96 210.12 118.59
Design Load [Ld] (h) 5000 5000 5000 5000
Bearing Coefficient [k] 3 3 3 3
Rotation Speed [ω] (RPM) 1800 773.6 773.6 332.48
Basic Dynamic Load [C] (lb) 414.95 792.53 1291.28 549.48
Standard Bearing 6006 6006 6006 6008
Conclusion
Through the surprisingly complex design process of a mechanically simple gear reducer,
we came to appreciate the considerable amount of thought that goes into engineering mechanical
systems. The design process took numerous hours and multiple failed attempts, but successfully
enlightened our group on what it is like to design machine elements, however simple they may
be. We learned about the importance of precise calculation, the integration calculations and
mechanical drawings through CAD software, and about the convenience of industry standard
dimensions and factors. But perhaps the most important takeaway of this project was an
appreciation for the engineering techniques we used. There are people in the world who use the
very methods we used in this project to design things of astounding complexity, such as
automatic transmissions or large scale manufacturing equipment. Even though we are at a novice
level when it comes to engineering, the knowledge that the material we are learning today could
someday help us design the machines that enable our modern society to flourish made the
relative tedium of this project well worth it.