The document provides calculations and analysis of stresses on various shafts and gears used in the system. It calculates maximum stresses and factors of safety for 3 shafts of different diameters made of 1045 steel subjected to bending and torsional loads. It also analyzes a steering shaft made of 303 stainless steel. For the spur and ring gears made of carbon steel and aluminum respectively, it calculates maximum bending stresses and endurance stresses to determine factors of safety above yield strengths for a 5-year operating life.

1. CALCULATIONS AND ANALYSIS
See Stress Calculation Spreadsheet for sources of equations,
sources of constants and material properties, and additional
calculations
Impact Analysis
Direct wheel impact at max speed
 By using the deflection equation,
EI
Fl
s
192
3
= (based upon two fully constrained
rod ends), solving for F, and using a basic kinematic equation (
)(2
22
savv of ∆== ) to solve for s in terms of F, the force of impact can be
determined (227505 N)
 Utilizing shaft stress equations shown below the stress can be determined (400
MPa)
 When comparing this to the shaft’s yield strength, a factor of safety of 1.33 is
calculated

2. Direct pulley impact at max speed
 Utilizing this same force and finding the stress on the shaft due to bending.
I
Mc
=σ =8510 MPa
 This means the shaft will permanently bend due to the moment applied on it
 The way to avoid this catastrophic failure is to ensure the chassis protects these
open gears by extending past its edges or enclosing it completely. While this may
not completely ensure the module’s safety, it will fix nearly every probable
scenario.
Shaft Stress Calculations
Shaft 1 (Diameter=3/8”)
• Material: 1045 Steel, Yield Strength (Sy)= 530 MPa, Ultimate Strength= 625MPa
• Max Stress
o The shaft is keyed for a 3/32” key, thus a close approximation for the actual
yield strength is ¾ the materials yield strength (Keyed Yield Strength=398
MPa)
o Loading is comprised of three components
 Moment-Based on cantilevered distance from bearing and radial load
exerted on shaft from the miter gear (2.1 N-m)
 Force- Based on axial load exerted on shaft from miter gear (156.12
N)
 Torque- Exerted by the stall torque of the motor, through a gear ratio
of 2:1 (9.64 N-m)
o Stress Calculation-

2/122
max ]48)8[(
4
TFdM
d
++
Π
=σ =102 MPa

2/122
3max ]64)8[(
2
TFdM
d
++
Π
=τ = 58.4 MPa
o Factors of Safety-

maxσ
yS
n = = 3.9

max2τ
yS
n = = 3.4
• Fatigue Life
o Infinite Life- 2000RPM (Average operating speed)=33.3 cycles/second
 5 year life @ 1 hour operating time (2 hr per week)-approximately
1,908,000 seconds of use
 33.3*1,908,000=6.4E7 cycles to failure for infinite life
o The endurance strength can be calculated using the stress concentration
factors from the keyway (197 MPa)
o σ’F=Sut+345MPa= 970 MPa

3. o
)2log(
)/'log(
e
eF
N
S
b
σ
−= =-0.109915548
o
b
ut
F
S
f )102(
' 3
⋅=
σ
=.673
o
e
ut
S
Sf
a
2
)( ⋅
= =900 MPa
o Loads are based on typical operating conditions, not max conditions
 Moment-Based on cantilevered distance from bearing and radial load
exerted on shaft from the miter gear (2.1 N-m)
 Force- Based on axial load exerted on shaft from miter gear (156.12
N)
 Torque- Exerted by the operating torque of the motor, through a gear
ratio of 2:1 (2.82 N-m)
o 2/122
3
]48)8[(
4
TFdM
d
a ++
Π
=σ = 39.4 MPa
o b
a
a
N
1






=
σ = 2.25E12 cycles to failure
Shaft 2 (Diameter=1/2”)
• Material: 1045 Steel, Yield Strength= 530 MPa, Ultimate Strength= 625MPa
• Max Stress
o The shaft is keyed for a 1/8” key, thus the actual yield strength can be equated
to ¾ the materials yield strength (Keyed Yield Strength=398 MPa)
o Loading is comprised of three components
 Moment-Based on the axle length between bearings and radial load
exerted on shaft from the miter gear (4.28 N-m)
 Force- Based on axial load exerted on shaft from miter gear (156.12
N)
 Torque- Exerted by the stall torque of the motor, through a gear ratio
of 2:1 (9.64 N-m)
o Stress Calculation-

2/122
max ]48)8[(
4
TFdM
d
++
Π
=σ = 47.2 MPa

2/122
3max ]64)8[(
2
TFdM
d
++
Π
=τ = 26.5 MPa
o Factors of Safety-

maxσ
yS
n = = 8.4

max2τ
yS
n = = 7.5
• Fatigue Life
o Infinite Life- 1000RPM=16.67 cycles/second

4.  5 year life @ 1 hour operating time (2 hr per week)-approximately
1,908,000 seconds of use
 16.67*1,908,000=3.2E7 cycles to failure for infinite life
o The endurance strength can be calculated using the stress concentration
factors from the keyway (197 MPa)
o σ’F=Sut+345MPa= 970 MPa
o
)2log(
)/'log(
e
eF
N
S
b
σ
−= =-0.109915548
o
b
ut
F
S
f )102(
' 3
⋅=
σ
=.673
o
e
ut
S
Sf
a
2
)( ⋅
= =900 MPa
o Loads are based on typical operating conditions, not max conditions
 Moment-Based on the axle length between bearings and radial load
exerted on shaft from the miter gear (4.28 N-m)
 Force- Based on axial load exerted on shaft from miter gear (156.12
N)
 Torque- Exerted by the operating torque of the motor, through a gear
ratio of 2:1 (2.82 N-m)
o 2/122
3
]48)8[(
4
TFdM
d
a ++
Π
=σ = 25.6 MPa
o b
a
a
N
1






=
σ = 1.15E14 cycles to failure
Shaft 3 (Diameter=3/4”)
• Material: 1045 Steel, Yield Strength= 530 MPa, Ultimate Strength= 625MPa
• Max Stress
o The shaft is keyed for a 3/16” key, thus the actual yield strength can be
equated to ¾ the materials yield strength (Keyed Yield Strength=398 MPa)
o Loading is comprised of three components
 Moment-Based on the axle length between bearings and the force
exerted by the weight of the system (21.53 N-m)
 Force- Based on axial load exerted on the shaft from turning forces
(235.44 N)
 Torque- Exerted by the stall torque of the motor, through a gear ratio
of 8:1 (38.56 N-m)
o Stress Calculation-

2/122
max ]48)8[(
4
TFdM
d
++
Π
=σ = 59.0 MPa

2/122
3max ]64)8[(
2
TFdM
d
++
Π
=τ = 32.7 MPa
o Factors of Safety-

5. 
maxσ
yS
n = = 6.7

max2τ
yS
n = = 6.1
• Fatigue Life
o Infinite Life- 500RPM=8.34 cycles/second
 5 year life @ 1 hour operating time (2 hr per week)-apprx 1,908,000
seconds of use
 8.34*1,908,000=1.6E7 cycles to failure for infinite life
o The endurance strength can be calculated using the stress concentration
factors from the keyway (197 MPa)
o σ’F=Sut+345MPa= 970 MPa
o
)2log(
)/'log(
e
eF
N
S
b
σ
−= =-0.109915548
o
b
ut
F
S
f )102(
' 3
⋅=
σ
=.673
o
e
ut
S
Sf
a
2
)( ⋅
= =900 MPa
o Loads are based on typical operating conditions, not max conditions
 Moment-Based on the axle length between bearings and the force
exerted by the weight of the system (21.53 N-m)
 Force- Based on axial load exerted on the shaft from turning forces
(235.4 N)
 Torque- Exerted by the operating torque of the motor, through a gear
ratio of 8:1 (11.28 N-m)
o 2/122
3
]48)8[(
4
TFdM
d
a ++
Π
=σ = 35.6 MPa
o b
a
a
N
1






=
σ = 5.7E12 cycles to failure
Steering Shaft (Diameter=1/4”)
• Material: 303 Stainless Steel, Yield Strength= 240 MPa, Ultimate Strength= 620 MPa
• Max Stress
o Loading is based on torque alone (0.745 N-m)
o Stress Calculation-

2/122
max ]48)8[(
4
TFdM
d
++
Π
=σ = 25.7 MPa

2/122
3max ]64)8[(
2
TFdM
d
++
Π
=τ = 14.8 MPa
o Factors of Safety-

maxσ
yS
n = = 9.4

6. 
max2τ
yS
n = = 8.1
• Fatigue Life
o σ’F=Sut+345MPa=965E6 MPa
o
)2log(
)/'log(
e
eF
N
S
b
σ
−= =-0.07772
o
b
ut
F
S
f )102(
' 3
⋅=
σ
=.862
o
e
ut
S
Sf
a
2
)( ⋅
= =914 MPa
o Load is comprised of torque alone (.745 N-m)
o 2/122
3
]48)8[(
4
TFdM
d
a ++
Π
=σ = 25.7 MPa
o b
a
a
N
1






=
σ = 9.3E19 cycles to failure
Spur Gears (Calculated using ANSI standards)
Driving Spur
 Material- Carbon Steel, Yield Strength=76900 psi, Modulus of Elasticity=30E6
psi, Poisson’s Ratio=.29, Brunell Hardness 179
 Max Bending Stress
o
V
H
W t ⋅
=
33000
= 306.8 lbf
o Ko= 1.25 - Overload Factor, based on light shocks encountered
o Kv= 1.15 - Dynamic Factor, based on quality and velocity of gears
o Ks= 1 - Size Factor
o Pd= .833” – Pitch diameter
o F= .25” – face width
o Km= 1.20 – Load-Distribution factor, based on geometry
o KB= 1 – Rim Thickness factor, based on geometry
o J= .325- Geometry factor, based on number of teeth of gears
o
J
KK
F
P
KKKW Bmd
svo
t
=σ =5357.1 psi
o
maxσ
yS
n = = 9.0
 Endurance Stress
o
2/1
22
)
11
(
1














−
+
−
=
G
G
p
p
p
E
v
E
v
C
π
=2284.7 lbf/in2
o Cf=1
o I=0.08- Geometry Factor

7. o 2/1
)(
I
C
FP
K
KKWC
f
d
m
so
t
p=σ =56972.4 psi
o Sc= 180000 psi- Repeatedly applied contact strength @ 107
cycles,
material property
o Zn=.59 - Stress cycle life factor, based on hardness and number of cycles
o CH=1 -Hardness ratio factor
o KT= 1- Temperature factor
o KR= 1 – Reliability factor
o
σ
)/( RTHNc
H
KKCZS
S = =1.9
o Comparable factor of safety= SH
2
=3.5
Driven Spur
 Material- Carbon Steel, Yield Strength=76900 psi, Modulus of Elasticity=30E6
psi, Poisson’s Ratio=.29, Brunell Hardness 179
 Max Bending Stress
o
V
H
W t ⋅
=
33000
= 306.7 lbf
o Ko= 1.25 - Overload Factor, based on light shocks encountered
o Kv= 1.15 - Dynamic Factor, based on quality and velocity of gears
o Ks= 1 - Size Factor
o Pd= 1.667” – Pitch diameter
o F= .25” – face width
o Km= 1.19 – Load-Distribution factor, based on geometry
o KB= 1 – Rim Thickness factor, based on geometry
o J= .389- Geometry factor, based on number of teeth of gears
o
J
KK
F
P
KKKW Bmd
svo
t
=σ =9011.0 psi
o
maxσ
yS
n = = 5.4
 Endurance Stress
o
2/1
22
)
11
(
1














−
+
−
=
G
G
p
p
p
E
v
E
v
C
π
=2284.7 lbf/in2
o Cf=1
o I=0.08- Geometry Factor
o 2/1
)(
I
C
FP
K
KKWC
f
d
m
so
t
p=σ = 40010.7 psi
o Sc= 180000 psi- Repeatedly applied contact strength @ 107
cycles,
material property
o Zn=.60 - Stress cycle life factor, based on hardness and number of cycles
o CH=1 -Hardness ratio factor

8. o KT= 1- Temperature factor
o KR= 1 – Reliability factor
o
σ
)/( RTHNc
H
KKCZS
S = = 2.70
o Comparable factor of safety= SH
2
=7.2
Ring/Pinion Gears (Calculated using ANSI standards)
Steering Spur
 Material- 2024-T4 Aluminum, Yield Strength=47000 psi, Modulus of
Elasticity=10.4E6 psi, Poisson’s Ratio=.333
 Max Bending Stress
o
V
H
W t ⋅
=
33000
= 39.3 lbf
o Ko= 1.25 - Overload Factor, based on light shocks encountered
o Kv= 1.10 - Dynamic Factor, based on quality and velocity of gears
o Ks= 1 - Size Factor
o Pd= .4375” – Pitch diameter
o F= .125” – face width
o Km= 1.20 – Load-Distribution factor, based on geometry
o KB= 1 – Rim Thickness factor, based on geometry
o J= .24- Geometry factor, based on number of teeth of gears
o
J
KK
F
P
KKKW Bmd
svo
t
=σ =951.8 psi
o
maxσ
yS
n = = 44.1
Steering Ring
 Material- 2024-T4 Aluminum, Yield Strength=47000 psi, Modulus of
Elasticity=10.4E6 psi, Poisson’s Ratio=.333
 Max Bending Stress
o
V
H
W t ⋅
=
33000
= 39.3 lbf
o Ko= 1.25 - Overload Factor, based on light shocks encountered
o Kv= 1.10 - Dynamic Factor, based on quality and velocity of gears
o Ks= 1 - Size Factor
o Pd= 3.125” – Pitch diameter
o F= .125” – face width
o Km= 1.8 – Load-Distribution factor, based on geometry
o KB= 1 – Rim Thickness factor, based on geometry
o J= .4- Geometry factor, based on number of teeth of gears
o
J
KK
F
P
KKKW Bmd
svo
t
=σ = 3996.0 psi

9. o
maxσ
yS
n = = 10.5
Miter Gears (Calculated using ANSI standards)
Both Miters (At max torque)
 Material- Medium Carbon Steel, Yield Strength=76900 psi
 Max Bending Stress
o Pd= 1.25” – Pitch diameter
o
o
d
t
P
T
W
⋅
=
2
= 84.7 lbf
o Ko= 1.25 - Overload Factor, based on light shocks encountered
o Kv= 1 - Dynamic Factor, based on quality and velocity of gears
o Ks= .5 - Size Factor
o F= .27” – face width
o Km= 1.10 – Load-Distribution factor, based on geometry
o J= 0.175- Geometry factor, based on number of teeth of gears
o Kx= 1, Lengthwise curvature factor
o
JK
KK
KKP
F
W
x
ms
vod
t
=σ =14922.2 psi
o
maxσ
yS
n = = 5.15
Both Miters (At max speed)
 Material- Medium Carbon Steel, Yield Strength=76900 psi
 Max Bending Stress
o Pd= 1.25” – Pitch diameter
o
o
d
t
P
T
W
⋅
=
2
= 4 lbf
o Ko= 1.25 - Overload Factor, based on light shocks encountered
o Kv= 1.28 - Dynamic Factor, based on quality and velocity of gears
o Ks= .5 - Size Factor
o F= .27” – face width
o Km= 1.10 – Load-Distribution factor, based on geometry
o J= 0.175- Geometry factor, based on number of teeth of gears
o Kx= 1, Lengthwise curvature factor
o
JK
KK
KKP
F
W
x
ms
vod
t
=σ =901.6 psi
o
maxσ
yS
n = = 85.3
 Forces

10. o Knowing max torque on miter (9.63 N-m), we can find the max tangential
force by dividing by half the pitch diameter=> Ftan=606.6 N
o α= 20 degress -pressure angle
o d= 45 degrees
o
αcos
tanF
Fn = =645.5 N
o
αsin
1
nF
F = =220.8 N
o dFFF radialaxial sin1== =156.1 N
Retaining Rings
 On 3/8” shaft
o Ring can withstand 542.7 N of axial force
o Miter gear provides axial load= 156.1 N
o Factor of safety= 3.48
 On 1/2” shaft
o Ring can withstand 542.7 N of axial force
o Miter gear provides axial load= 156.1 N
o Factor of safety= 3.48
 On 3/4” shaft
o Ring can withstand 631.6 N of axial force
o Axial load is from turning
 Assume wheel instantaneously turns 90 degrees, the max force that
can be applied axially would be equivalent to the frictional force
 µ⋅== WFF frictionaxial =235.4 N (assuming µ=.6)
o Factor of safety= 2.68
Mechanical Brake
Max Temperature
 Assuming all kinetic energy is converted directly into heat energy,
TCmvm vplatevehveh ∆=
2
2/1
 Assume emergency brake will not be used continuously, but rather for one cycle
during the emergency
 Assume initial temperature of 23 ° Celcius
 Solving the above equation for Tfinal we find it to be 38.9 °Celcius
Heat Dissipation
 Assuming Free Convection, the time required for heat dissipation can be
calculated
 Utilizing the properties of air at room temperature, the Rayleigh number, Nusselt
number, and convection heat transfer coefficient can be calculated

11.  Using this information the heat transfer rate is determined
 q
E
t
∆
=∆ gives the time to dissipate the heat (4.8 minutes)
 This resultant was later verified by the manufacturer of the brake
Keys
On 3/8” Shaft
 Key is High carbon steel, Yield Strength 427 MPa, 3/32” square
 Knowing the diameter of and torque on the shaft, the shear force on the key can
be calculated (2024.1 N)
 Assuming a factor of safety of 4, the required length of the key is calculated
(.63”)
On 1/2” Shaft
 Key is High carbon steel, Yield Strength 427 MPa, 1/8” square
 Knowing the diameter of and torque on the shaft, the shear force on the key can
be calculated (1518.1 N)
 Assuming a factor of safety of 4, the required length of the key is calculated
(.35”)
On 3/4” Shaft
 Key is High carbon steel, Yield Strength 427 MPa, 3/16” square
 Knowing the diameter of and torque on the shaft, the shear force on the key can
be calculated (4048.3 N)
 Assuming a factor of safety of 4, the required length of the key is calculated
(.63”)
Keyways
• Keyway analysis was done using Cosmos FEA software
• By utilizing shaft diameters and torques, forces on keyway surfaces were
calculated and input into the program
• Factor of Safety
o Driving Miter=16
o Driven Miter=15
o Driving Pulley=3.4
o Driven Pulley=8.4
o Driven Spur=8.1
o Wheel=1.8, but in reality, failure would result in the slip of a pressed
insert, rather than physical failure of the wheel
Set Screws
To connect spur gear to 5/16” drive motor shaft
 By choosing a screw size and quantity (2- #8’s), the maximum force at the shaft
surface can be calculated (3425.1 N)

12.  The torque and diameter of the shaft is used to determine the actual force seen at
this shaft surface (1214.5 N)
 By comparing these two values the factor of safety is determined (2.82)
To connect spur gear to 8mm steering motor shaft
 By choosing a screw size and quantity (2-#6’s), the maximum force at the shaft
surface can be calculated (2224.1 N)
 The torque and diameter of the shaft is used to determine the actual force seen at
this shaft surface (234.6 N)
 By comparing these two values the factor of safety is determined (9.5)
Timing Belt and Pulleys
• Utilizing MITCalc simulation software and inputting various parameters
including distance between centers, power applied to belt, operating
speeds, and other operating conditions a belt type and specific model was
selected
• From this the 5M Powergrip GT2 belt was chosen and matched with
pulleys of 18 and 72 teeth
• The selection of these parts was also verified with an engineer at the
supplier sdp-si.com
Bearings
C10 = Catalog Load Rating (lbf)
LR = Rating Life (hrs)
nR = Rating Speed (RPM)
FD = Desired Radial Load (lbf)
LD = Desired Life (hrs)
nD = Desired Speed (RPM)
FR = Radial Force (lbf)
FA = Axial Force (lbf)
Fe = Equivalent Radial Load (lbf)
C0 = Static Load Rating (lbf)
X2 = Factor dependent on bearing geometry
Y2 = Factor dependent on bearing geometry
V = Rotation Factor
a = 3 (ball bearing)
e = abscissa
Lower Drive Bearing
C10 (lbs) 1171 Upper Drive Bearing Center Bearing
LR * nR 1.0E+06 C10 (lbs) 1187 C10 (lbs) 691
FD (lbs) 44.125 LR * nR 1.0E+06 LR * nR 1.0E+06
nD (RPM) 5.0E+02 Fe (lbs) 84.6 Fe (lbs) 70.6
a 3 nD (RPM) 2000 nD (RPM) 4000

13. ARe
D
RR
a
D
D
FYVFXF
n
nL
F
C
L
22
10 *
+=












=
LD (hrs) 3.74E+07 FA (lbs) 35.1 FA (lbs) 35.1
LD (years) 4267 FR (lbs) 35.1 FR (lbs) 35.1
a 3 a 3
Steering Bearing V 1 V 1
C10 (lbs) 300 e 0.24 e 0.3
LR * nR 1.0E+06 X2 0.56 X2 0.56
FD (lbs) 10 Y2 1.85 Y2 1.45
nD (RPM) 340 LD (hrs) 1.38E+06 LD (hrs) 2.34E+05
a 3 LD (years) 158 LD (years) 27
LD (hrs) 7.94E+07
LD (years) 9065
Screws
Bolts connecting Yoke to Turntable
Bolt type: 4 * 10-32 (SAE)
Torque applied to the turntable
T= 2.38 N-m
Converted ASTM
T = 21 lb-in
Resultant load on each bolt
lb
in
inlb
r
T
V 108.8
59.2
1
*21 =−==
inlbTM −== 21
Primary Shear Load per Bolt is
lb
n
V
F 027.2
4
108.8
' ===
Since the secondary shear Forces are equal we have
lb
r
M
r
Mr
F 027.2
59.2*4
21
44
'' 2
====
The resultant force is

14. lbFr 867.2=
Fr = Fa = Fb = Fc = Fd = 2.867lb
Maximum Shear Stress
As = 0.155
psi
A
F
s
r
497.18
155.
867.2
===τ
Bolts connecting Brake to Brake plate
Bolt type: 4 * 8-32 (SAE)
Torque applied to the turntable
T = 15 lb-in
Resultant load on each bolt
lb
in
inlb
r
T
V 33.13
125.1
1
*15 =−==
inlbTM −== 15
Primary Shear Load per Bolt is
lb
n
V
F 33.3
4
33.13
' ===
Since the secondary shear Forces are equal we have
lb
r
M
r
Mr
F 33.3
125.1*4
15
44
'' 2
====
The resultant force is
lbFr 714.4=
Fr = Fa = Fb = Fc = Fd = 4.714 lb

15. Maximum Shear Stress
As = .0992
psi
A
F
s
r
9.28
0992.
867.2
===τ
Yoke
• Yoke stress analysis was done using Cosmos FEA software
• Loading
o Weight vertically loads lower bearing holes (196.2 N each)
o Turning force loads inside wall (158.3 N)
o Driven Miter axial force loads inside wall (156 N)
o Driven Miter radial force loads upper bearing holes
o Driving Miter axial force loads upward on top plate

16. • Minimum factor of safety= 20
Brake Plate
• Brake plate stress analysis was done using Cosmos FEA software
• Loading
o Outside edge was fixed, as it is welded to the motor mount assembly
o Each brake mounting hole was loaded with a force corresponding to the
brake’s torque output and the holes distance from center
• Minimum factor of safety= 200
Turntable
• Capable of withstanding 750 lbs, or 340 kg
• Actual weight is about 40 kg per module
• Factor of Safety= 8.5

17. Figure X: Shaft Stress and Fatigue Strength Calculations
Figure X: Drive Motor Spur Gear Stress Calculation

18. Figure X: Miter Gear Stress and Force Calculation
Figure X: Retaining Ring Calculations Figure X: Impact Calculations
Figure
X:
Brake
Temperature and Heat Dissipation Calculations

19. Figure X: Key and Set Screw Analysis

20. Figure X: Timing Belt and Pulley Analysis

21. Figure X: Driven Miter Gear Keyway Analysis
Figure X: Driving Miter Gear Keyway Analysis

22. Figure X: Driven Pulley Keyway Analysis
Figure X: Driving Pulley Keyway Analysis (Representation)

23. Figure X: Driven Spur Gear Keyway Analysis
Figure X: Wheel Keyway Analysis (Representation)
Note: Actual wheel utilizes press fit keyway insert, thus failure during stall will result in
slip of this insert, rather than the physical failure of a mechanical part

24. Figure X: Yoke Displacement