Gearbox design - MECH 393 final project
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About This Presentation
Project report on double-branch double-reduction gearbox, made for Solar Impulse.
Slide Content
Final Project Report
Winter 2017
MECH 393 – Machine Element Design
April 11th, 2017
Group Number: 2
Group Members:
Georges Matta 260608769
Riad Haissam El Charif 260631084
Stanislav Nemirovsky 260660024
Winter 2017
MECH 393 – Machine Element Design
April 11th, 2017
Group Number: 2
Group Members:
Georges Matta 260608769
Riad Haissam El Charif 260631084
Stanislav Nemirovsky 260660024
2
Table of Contents
Executive Summary ................................................................................................................ 4
Individual Contribution ......................................................................................................... 4
Introduction to the Design Problem ................................................................................. 5
Detailed Design Solution ....................................................................................................... 8
I. Gear Development ................................................................................................................... 9
i. Gearbox Layout ..................................................................................................................................................... 9
ii. Determining the Gear Ratio ............................................................................................................................. 9
iii. Power and Torque Requirements .............................................................................................................. 11
iv. Gear 1 Sample Calculations for Safety Factors ..................................................................................... 11
v. Calculations for Gears 2, 3, 4, 5, 6 ............................................................................................................... 18
II. Shaft Development ............................................................................................................... 19
i. Free Body Diagrams of the Shafts with Gears and Bearings: ................................. 21
ii. Shaft 1- Input Shaft ............................................................................................................... 24
iii. Shafts 2-4 - Reduction Shafts ............................................................................................ 27
iv. Shaft 3 - Output Shafts ......................................................................................................... 28
Bearings ................................................................................................................................... 30
The C parameter is obtained from the manufacturer specification for each
bearing, and so to allow for successful iteration, a large amount bearing
tables were input into our excel sheets to allow for easy selection. The
tables would “iterate” these formulas for many different types of
bearings, until a suitable life limit was found. We should note that was
not compared directly with our , but rather it was multiplied by a
reliability factor of Kr = 0.33 (for a reliability of 98%) for most of our
bearing choices. We now used this new to compare with our ideal life. 31
After a number of bearings of acceptable life were found, the particular
bearing was selected based off geometrical constraints of our shaft. Since
we had a very low stress state, we could afford to add a large shoulder on
the bearing portion of our shaft, and so dout was not a factor. However,
since we used a hollow shaft, din was the limiting parameter. The
selected bearings are presented in later in this report. ................................ 31
Note: Due to our very low loading, the life expectancies of the bearings on
shaft 1 are significantly higher than necessary, however these bearings
were found to be suitable for our geometry , and satisfied our needs, and
Table of Contents
Executive Summary ................................................................................................................ 4
Individual Contribution ......................................................................................................... 4
Introduction to the Design Problem ................................................................................. 5
Detailed Design Solution ....................................................................................................... 8
I. Gear Development ................................................................................................................... 9
i. Gearbox Layout ..................................................................................................................................................... 9
ii. Determining the Gear Ratio ............................................................................................................................. 9
iii. Power and Torque Requirements .............................................................................................................. 11
iv. Gear 1 Sample Calculations for Safety Factors ..................................................................................... 11
v. Calculations for Gears 2, 3, 4, 5, 6 ............................................................................................................... 18
II. Shaft Development ............................................................................................................... 19
i. Free Body Diagrams of the Shafts with Gears and Bearings: ................................. 21
ii. Shaft 1- Input Shaft ............................................................................................................... 24
iii. Shafts 2-4 - Reduction Shafts ............................................................................................ 27
iv. Shaft 3 - Output Shafts ......................................................................................................... 28
Bearings ................................................................................................................................... 30
The C parameter is obtained from the manufacturer specification for each
bearing, and so to allow for successful iteration, a large amount bearing
tables were input into our excel sheets to allow for easy selection. The
tables would “iterate” these formulas for many different types of
bearings, until a suitable life limit was found. We should note that was
not compared directly with our , but rather it was multiplied by a
reliability factor of Kr = 0.33 (for a reliability of 98%) for most of our
bearing choices. We now used this new to compare with our ideal life. 31
After a number of bearings of acceptable life were found, the particular
bearing was selected based off geometrical constraints of our shaft. Since
we had a very low stress state, we could afford to add a large shoulder on
the bearing portion of our shaft, and so dout was not a factor. However,
since we used a hollow shaft, din was the limiting parameter. The
selected bearings are presented in later in this report. ................................ 31
Note: Due to our very low loading, the life expectancies of the bearings on
shaft 1 are significantly higher than necessary, however these bearings
were found to be suitable for our geometry , and satisfied our needs, and
3
thus they were chosen. The same applies for Shaft 2, 4 , however the life
expectancies are not much higher than the ideal requirement. ................. 31
The final spherical thrust bearing was chosen from SKF , and it was chosen for
a suitable its suitable geometry and life expectancies. The bearing is a
Spherical Roller Bearing. .......................................................................................... 31
Design Results ....................................................................................................................... 32
Modified Goodman Diagrams ........................................................................................... 36
Conclusions ............................................................................................................................. 39
Appendix A – All Calculations ........................................................................................... 40
Appendix B – Figures and Tables .................................................................................... 44
Appendix C – Mechanical Drawings of Proposed Design ........................................ 48
thus they were chosen. The same applies for Shaft 2, 4 , however the life
expectancies are not much higher than the ideal requirement. ................. 31
The final spherical thrust bearing was chosen from SKF , and it was chosen for
a suitable its suitable geometry and life expectancies. The bearing is a
Spherical Roller Bearing. .......................................................................................... 31
Design Results ....................................................................................................................... 32
Modified Goodman Diagrams ........................................................................................... 36
Conclusions ............................................................................................................................. 39
Appendix A – All Calculations ........................................................................................... 40
Appendix B – Figures and Tables .................................................................................... 44
Appendix C – Mechanical Drawings of Proposed Design ........................................ 48
4
Executive Summary
NikolaDrive Team
The design report presented here was commissioned by the Solar Impulse initiative for the
design of a gearbox for the titular aircraft. NikolaDrive is a collective of highly motivated
innovative aeronautical engineers, who form a vital subdivision within the Solar Impulse family.
Headed by our chief engineer Mark Driscoll, the team embarked on the proposed design for
a double branch double reduction gearbox, intended for use on the final aircraft. The team
had 3 main design goals: Minimize weight, maximize efficiency and endure the aircraft’s
lifetime.
Integrated design principles were used for the design of Gears, Shafts and Bearings in our
system. A targeted safety factor of 1.5 was chosen for our general design allowing for static,
dynamic, and fatigue failure analysis to be performed on each component. The designs were
iterated until satisfactory results were obtained. All components fall within a safety factor of
1.5. The whole system operates at power losses below 5% as desired. The weight of our
system is 14.4 kg, which did not meet the target criterion of 5.5 kg. Further refinement in
future designs could be considered.
The NikolaDrive Team is proud to present its first and most innovative design: Solar Impulse
Double Branch Double Reduction Gearbox.
Individual Contribution
The NikolaDrive team consists of three engineers, Georges Matta, Stanislav Nemirovsky and
Riad Haissam El Charif, working under the supervision of Mr. Driscoll. With such a tightly knit
and well-functioning unit, all members had significant contributions on each aspect of design.
However, a rough division of individual contribution can nonetheless be made.
Georges Matta, a U3 Mechanical Engineering Student at McGill University, oversaw
spreadsheet production, gear design, as well as material and bearing selection.
Riad Haissam El Charif, also a U3 Mechanical Engineering Student at McGill University
worked on shaft and bearing design, layout, and optimization.
Finally, Stanislav Nemirovsky, U3 Mechanical Engineering, optimized and realized the design
for the bearings, shafts, as well as the gear parameters.
The apparent division of labor provided a rough structure of the team’s organization; however, it
should be reemphasized that NikolaDrive operates on a highly collaborative structure, in which
team members share a significant amount of responsibilities
Executive Summary
NikolaDrive Team
The design report presented here was commissioned by the Solar Impulse initiative for the
design of a gearbox for the titular aircraft. NikolaDrive is a collective of highly motivated
innovative aeronautical engineers, who form a vital subdivision within the Solar Impulse family.
Headed by our chief engineer Mark Driscoll, the team embarked on the proposed design for
a double branch double reduction gearbox, intended for use on the final aircraft. The team
had 3 main design goals: Minimize weight, maximize efficiency and endure the aircraft’s
lifetime.
Integrated design principles were used for the design of Gears, Shafts and Bearings in our
system. A targeted safety factor of 1.5 was chosen for our general design allowing for static,
dynamic, and fatigue failure analysis to be performed on each component. The designs were
iterated until satisfactory results were obtained. All components fall within a safety factor of
1.5. The whole system operates at power losses below 5% as desired. The weight of our
system is 14.4 kg, which did not meet the target criterion of 5.5 kg. Further refinement in
future designs could be considered.
The NikolaDrive Team is proud to present its first and most innovative design: Solar Impulse
Double Branch Double Reduction Gearbox.
Individual Contribution
The NikolaDrive team consists of three engineers, Georges Matta, Stanislav Nemirovsky and
Riad Haissam El Charif, working under the supervision of Mr. Driscoll. With such a tightly knit
and well-functioning unit, all members had significant contributions on each aspect of design.
However, a rough division of individual contribution can nonetheless be made.
Georges Matta, a U3 Mechanical Engineering Student at McGill University, oversaw
spreadsheet production, gear design, as well as material and bearing selection.
Riad Haissam El Charif, also a U3 Mechanical Engineering Student at McGill University
worked on shaft and bearing design, layout, and optimization.
Finally, Stanislav Nemirovsky, U3 Mechanical Engineering, optimized and realized the design
for the bearings, shafts, as well as the gear parameters.
The apparent division of labor provided a rough structure of the team’s organization; however, it
should be reemphasized that NikolaDrive operates on a highly collaborative structure, in which
team members share a significant amount of responsibilities
5
Introduction to the Design Problem
Solar Impulse’s design objective is simple, yet awe inspiring: Fly around the globe with no
onboard fuel.
To achieve this unique challenge, every engineering sub team involved in the aircraft design
highly optimized parts for maximum efficiency and minimum weight, while maintaining
reasonable reliability constraints. Conversely, the uniqueness of this project has given rise to
some unusual liberties. First, as an experimental aircraft, no specific certifications are mandated
upon our designs, which gave us incredible freedom to pursue our design goals. Second, as a
well-funded experimental aircraft team, Solar Impulse has managed to secure enough funds that
cost is never taken as a constraint. Such freedom allowed for more open-ended innovation.
The gearbox to be designed is of the Double Branched Double Reduction Gearbox type, a sketch
of which is shown in Figure 1.
Figure 1- Double Branched Design
The design allows for the reduction on speed, while increasing torque along two stages.
The double branch design allows for the distribution of loads from the input shaft, to reduce
stresses on the reduction shaft. The input shaft is directly connected to a 5000-rpm brushless
motor, which gets stepdown to a maximum of 525 rpm imparted to the propeller. The propeller
shaft, being the heart of our propulsion system, withstands a 1500lb axial load produced by the
propeller rotation.
Introduction to the Design Problem
Solar Impulse’s design objective is simple, yet awe inspiring: Fly around the globe with no
onboard fuel.
To achieve this unique challenge, every engineering sub team involved in the aircraft design
highly optimized parts for maximum efficiency and minimum weight, while maintaining
reasonable reliability constraints. Conversely, the uniqueness of this project has given rise to
some unusual liberties. First, as an experimental aircraft, no specific certifications are mandated
upon our designs, which gave us incredible freedom to pursue our design goals. Second, as a
well-funded experimental aircraft team, Solar Impulse has managed to secure enough funds that
cost is never taken as a constraint. Such freedom allowed for more open-ended innovation.
The gearbox to be designed is of the Double Branched Double Reduction Gearbox type, a sketch
of which is shown in Figure 1.
Figure 1- Double Branched Design
The design allows for the reduction on speed, while increasing torque along two stages.
The double branch design allows for the distribution of loads from the input shaft, to reduce
stresses on the reduction shaft. The input shaft is directly connected to a 5000-rpm brushless
motor, which gets stepdown to a maximum of 525 rpm imparted to the propeller. The propeller
shaft, being the heart of our propulsion system, withstands a 1500lb axial load produced by the
propeller rotation.
6
The design constraints are presented in table 1. The table presents the specifications given
to us by the Solar Impulse team, as well as the values we calculated and chose for our final
gearbox in bold red.
Powertrain Specification Gearbox Specifications
Motor Brushless +
Sensor less Gear Ratio 7.84
Maximum [rpm] 5000 Total Weight [kg] 14.4
Fuel Consumed [L] 0 L of Fossil Fuel Endurance Life-Gears and
Output Shaft [hrs] 2000
MaximumMotor
Temperature [°C] 135 Temperature Range [°C] -40 to
+40
Propeller Twin Blade
Composite Size limitations [cm] 30 X45
X 45
Propeller Thrust [lbf] 1500 Safety Factor 1.5
Propeller Weight [kg] 160
Propeller-Max RPM [rpm] 525
Table 1- Design Requirments
It should also be noted that the Gears considered in this design are AGMA spur gears of
course diametral pitch, due to their simplicity and versatility. The gears are to be lubricated
with SAE 30W. The team’s target was a fictional loss of less than 5%, and this was successfully
achieved. Finally, the operating conditions considered in our design are shown below.
The design constraints are presented in table 1. The table presents the specifications given
to us by the Solar Impulse team, as well as the values we calculated and chose for our final
gearbox in bold red.
Powertrain Specification Gearbox Specifications
Motor Brushless +
Sensor less Gear Ratio 7.84
Maximum [rpm] 5000 Total Weight [kg] 14.4
Fuel Consumed [L] 0 L of Fossil Fuel Endurance Life-Gears and
Output Shaft [hrs] 2000
MaximumMotor
Temperature [°C] 135 Temperature Range [°C] -40 to
+40
Propeller Twin Blade
Composite Size limitations [cm] 30 X45
X 45
Propeller Thrust [lbf] 1500 Safety Factor 1.5
Propeller Weight [kg] 160
Propeller-Max RPM [rpm] 525
Table 1- Design Requirments
It should also be noted that the Gears considered in this design are AGMA spur gears of
course diametral pitch, due to their simplicity and versatility. The gears are to be lubricated
with SAE 30W. The team’s target was a fictional loss of less than 5%, and this was successfully
achieved. Finally, the operating conditions considered in our design are shown below.
7
Operating Conditions
Operation
Electric power to Motor Driver
[hp]
Electric motor Shaft Torque
[Nm]
RPM
Take- Off 40 70.2 4000
Slow Climb 7 16.4 3000
Steep Climb 13.4 29.6 3180
Descent Glide 0.7 2.2 2225
Horizontal
Flight
4.7 14.8 2225
Table 2 Operating Conditions
Operating Conditions
Operation
Electric power to Motor Driver
[hp]
Electric motor Shaft Torque
[Nm]
RPM
Take- Off 40 70.2 4000
Slow Climb 7 16.4 3000
Steep Climb 13.4 29.6 3180
Descent Glide 0.7 2.2 2225
Horizontal
Flight
4.7 14.8 2225
Table 2 Operating Conditions
8
Detailed Design Solution
The general methodology followed in our design process was “assume and iterate”. We
started process followed for our given points, began with an assumption and tried out
multiple different values until we converged to our desired safety factors and geometry.
One fundamental design issue we wanted to avoid was the “Design Paradox”; as we learned
more about our design and progressed through it, we became more capable of
understanding our design, and its specific requirements. However, as we learn more, we
became increasingly incapable of editing any of our values due to a deep design investment.
To keep a versatile and dynamic system and avoid this crutch, we parametrized all our
values through a master Excel Workbook. This workbook contains all our design, diagrams
and results and allowed us to modify our design drastically multiple times with ease. This
workbook was provided along with the Project Report and will be subsequently referenced
multiple times in the sections below.
Another important design consideration for any shaft based system is the shaft deflection.
This variable typically pushes designers to minimize shaft length ‘L’. However, as our
design requirements do not include a shaft deflection analysis, we decided to use up the full
horizontal length of 30 cm for our layout as an initial guess. This layout provided us with
acceptable results, and so it was chosen for our design with little modification. Our gearbox
layout can be seen in Figure 2, as well as in our design drawings:
Figure 2
The design process then progressed with a logical order, we designed the gears based off
the given inputs and outputs, used the gear data- Face Width and Diameter- to produce a
preliminary design layout for our whole gearbox. The layout was then used along with
Detailed Design Solution
The general methodology followed in our design process was “assume and iterate”. We
started process followed for our given points, began with an assumption and tried out
multiple different values until we converged to our desired safety factors and geometry.
One fundamental design issue we wanted to avoid was the “Design Paradox”; as we learned
more about our design and progressed through it, we became more capable of
understanding our design, and its specific requirements. However, as we learn more, we
became increasingly incapable of editing any of our values due to a deep design investment.
To keep a versatile and dynamic system and avoid this crutch, we parametrized all our
values through a master Excel Workbook. This workbook contains all our design, diagrams
and results and allowed us to modify our design drastically multiple times with ease. This
workbook was provided along with the Project Report and will be subsequently referenced
multiple times in the sections below.
Another important design consideration for any shaft based system is the shaft deflection.
This variable typically pushes designers to minimize shaft length ‘L’. However, as our
design requirements do not include a shaft deflection analysis, we decided to use up the full
horizontal length of 30 cm for our layout as an initial guess. This layout provided us with
acceptable results, and so it was chosen for our design with little modification. Our gearbox
layout can be seen in Figure 2, as well as in our design drawings:
Figure 2
The design process then progressed with a logical order, we designed the gears based off
the given inputs and outputs, used the gear data- Face Width and Diameter- to produce a
preliminary design layout for our whole gearbox. The layout was then used along with
9
torque data from the Gears to produce Torque and Moment diagrams. These were used to
size preliminary diameters for the shaft, which was then completed with the selection of an
appropriate Ball/Roller Bearing which fits our load and lifetime requirements. Finally, the
design was optimized for weight considerations and suitable Safety Factors.
I. Gear Development
i. Gearbox Layout
As shown in Figure 2, a rough sketch of the gearbox was developed in order to
have a general idea about the dimensions of all the components. In particular,
estimations were made for the overall length of the gearbox, as well as the distance
between intermediate gears, and mounting requirements for the shafts and
bearings. All four shafts were represented in the above sketch, with an indication
of how the six gears and eight bearings would be mounted in this gearbox.
ii. Determining the Gear Ratio
As a first step in the design process, an ideal gear ratio must be determined in
order to satisfy the conditions and constraints provided. Maximum input angular
velocities were provided as well as a maximum output RPM. As a result, a
minimum gear ratio will be considered a reference for evaluating the several
iterations studied.
Now, an accepted combination of gear and pinion teeth
must result in a gear ratio that is at least equal to this
value. Another alternative to this selection criterion is to
compare the output RPM generated by the gear ratio
selected; this value cannot exceed the given maximum
value of 525 rpm. In addition, the fact that a pressure
angle of 20° is given means that there is also a minimum
number of pinion teeth that could be used in order to
avoid interference, the value of which is 18 teeth, as
shown in Table 12-4.
One of the final constraints when it comes to selecting the number of pinion and
gear teeth is checking that the geometry satisfies the requirements given, as the
sum of diameters of two consecutive gears in the first stage must be equal to that
in the second stage. And since the modulus is taken as constant for all gears, this
means that the sum of teeth between adjacent pinions and gears must be equal
torque data from the Gears to produce Torque and Moment diagrams. These were used to
size preliminary diameters for the shaft, which was then completed with the selection of an
appropriate Ball/Roller Bearing which fits our load and lifetime requirements. Finally, the
design was optimized for weight considerations and suitable Safety Factors.
I. Gear Development
i. Gearbox Layout
As shown in Figure 2, a rough sketch of the gearbox was developed in order to
have a general idea about the dimensions of all the components. In particular,
estimations were made for the overall length of the gearbox, as well as the distance
between intermediate gears, and mounting requirements for the shafts and
bearings. All four shafts were represented in the above sketch, with an indication
of how the six gears and eight bearings would be mounted in this gearbox.
ii. Determining the Gear Ratio
As a first step in the design process, an ideal gear ratio must be determined in
order to satisfy the conditions and constraints provided. Maximum input angular
velocities were provided as well as a maximum output RPM. As a result, a
minimum gear ratio will be considered a reference for evaluating the several
iterations studied.
Now, an accepted combination of gear and pinion teeth
must result in a gear ratio that is at least equal to this
value. Another alternative to this selection criterion is to
compare the output RPM generated by the gear ratio
selected; this value cannot exceed the given maximum
value of 525 rpm. In addition, the fact that a pressure
angle of 20° is given means that there is also a minimum
number of pinion teeth that could be used in order to
avoid interference, the value of which is 18 teeth, as
shown in Table 12-4.
One of the final constraints when it comes to selecting the number of pinion and
gear teeth is checking that the geometry satisfies the requirements given, as the
sum of diameters of two consecutive gears in the first stage must be equal to that
in the second stage. And since the modulus is taken as constant for all gears, this
means that the sum of teeth between adjacent pinions and gears must be equal
10
between the two stages. This is expressed by the equations below and can be seen
in more detail in the Excel sheet provided.
Now, gear ratios can be calculated for several values of N1 through N6, based on
the equation for compound gears:
where N2 = N6 and N3 = N5
The table below includes several iterations
performed before settling on the values shown on
the right, which satisfy all the constraints
provided.
N1 N2 N3 N4 N5 N6
Actual
Gear
Ratio
Satisfie
s
Geomet
ry?
Output
RPM
18 55 18 45 18 55
7.638888
889
523.6363
636
36 110 36 90 36 110
7.638888
889
523.6363
636
18 40 18 60 18 40
7.407407
407 540
20 50 21 64 21 50
7.619047
619
525.0000
0
18 56 18 44 18 56
7.604938
272
525.9740
26
18 54 18 46 18 54
7.666666
667
521.7391
304
18 60 18 41 18 60
7.5925925
93
526.829
2683
35 105 36 91 36 105
7.5833333
33
527.472
5275
20 56 20 56 20 56 7.84
510.204
0816
GEAR TEETH
N1 20
N2 56
N3 20
N4 56
N5 20
N6 56
RATIO 7.84
Output
RPM 510.2040816
between the two stages. This is expressed by the equations below and can be seen
in more detail in the Excel sheet provided.
Now, gear ratios can be calculated for several values of N1 through N6, based on
the equation for compound gears:
where N2 = N6 and N3 = N5
The table below includes several iterations
performed before settling on the values shown on
the right, which satisfy all the constraints
provided.
N1 N2 N3 N4 N5 N6
Actual
Gear
Ratio
Satisfie
s
Geomet
ry?
Output
RPM
18 55 18 45 18 55
7.638888
889
523.6363
636
36 110 36 90 36 110
7.638888
889
523.6363
636
18 40 18 60 18 40
7.407407
407 540
20 50 21 64 21 50
7.619047
619
525.0000
0
18 56 18 44 18 56
7.604938
272
525.9740
26
18 54 18 46 18 54
7.666666
667
521.7391
304
18 60 18 41 18 60
7.5925925
93
526.829
2683
35 105 36 91 36 105
7.5833333
33
527.472
5275
20 56 20 56 20 56 7.84
510.204
0816
GEAR TEETH
N1 20
N2 56
N3 20
N4 56
N5 20
N6 56
RATIO 7.84
Output
RPM 510.2040816
11
iii. Power and Torque Requirements
By considering our system in an ideal scenario, the power generated by the motor
will equal the power provided to the propeller. But because our gearbox is a
double-branch double-reduction type, we can suggest that the power provided by
the motor, which passes through Gear 1, gets divided equally between Gears 2 and
6, remains constant along Gears 3 and 5 (same shaft as 2 and 6 respectively), only
to return to approximately the same initial value through Gear 4, powering the
propeller. In reality however, losses in power exist due to the presence of factors
such as friction; the fact that ball bearings were selected played a role in the
assumption that such losses are considered negligible, as will be explained in
coming sections.
Now, after finding a suitable gear ratio, we can determine the angular velocities
and torques at each of the gears:
iv. Gear 1 Sample Calculations for Safety Factors
iii. Power and Torque Requirements
By considering our system in an ideal scenario, the power generated by the motor
will equal the power provided to the propeller. But because our gearbox is a
double-branch double-reduction type, we can suggest that the power provided by
the motor, which passes through Gear 1, gets divided equally between Gears 2 and
6, remains constant along Gears 3 and 5 (same shaft as 2 and 6 respectively), only
to return to approximately the same initial value through Gear 4, powering the
propeller. In reality however, losses in power exist due to the presence of factors
such as friction; the fact that ball bearings were selected played a role in the
assumption that such losses are considered negligible, as will be explained in
coming sections.
Now, after finding a suitable gear ratio, we can determine the angular velocities
and torques at each of the gears:
iv. Gear 1 Sample Calculations for Safety Factors
12
The modulus m for all the gears was estimated to be 3mm as it was the most
reasonable value that provides relatively good safety factors in bending and
contact, as will be shown.
Thus, the pitch diameter d, the pitch radius r, as well as the addendum a and
dedendum b for Gear 1 can all be found from the value of m:
The pitch line velocity VT and the tangential component of load WT can also be
found:
In addition, the life for input Gear 1 (as well as output Gear 4) is double that of
the other gears because it is a single driving pinion driving two independent gears
causing two fatigue cycles per revolution:
Now, the AGMA approach (SI form) for both bending and contact stress will be
applied to determine suitable gear parameters and safety factors.
The AGMA bending stress equation is given by:
The values of these constants and unknowns will now be calculated.
The Application Factor KA is chosen as 1.25 since the transmitted load cannot be
considered uniform as it fluctuates with time, at least for the driven machine as
opposed to the driving machine (electric motor / turbine). Moderate shock is thus
chosen. (Refer to Table 12-17)
The modulus m for all the gears was estimated to be 3mm as it was the most
reasonable value that provides relatively good safety factors in bending and
contact, as will be shown.
Thus, the pitch diameter d, the pitch radius r, as well as the addendum a and
dedendum b for Gear 1 can all be found from the value of m:
The pitch line velocity VT and the tangential component of load WT can also be
found:
In addition, the life for input Gear 1 (as well as output Gear 4) is double that of
the other gears because it is a single driving pinion driving two independent gears
causing two fatigue cycles per revolution:
Now, the AGMA approach (SI form) for both bending and contact stress will be
applied to determine suitable gear parameters and safety factors.
The AGMA bending stress equation is given by:
The values of these constants and unknowns will now be calculated.
The Application Factor KA is chosen as 1.25 since the transmitted load cannot be
considered uniform as it fluctuates with time, at least for the driven machine as
opposed to the driving machine (electric motor / turbine). Moderate shock is thus
chosen. (Refer to Table 12-17)
13
The Rim Thickness Factor KB is 1 since we are analyzing a solid-disk pinion. This
is not the case for Gears 2, 4, and 6 since rims with spokes are used.
The Idler Factor KI is 1 since a non-idler gear is being analyzed.
The Size Factor Ks is 1 since no drastic changes in size are present.
The Dynamic factor KV attempts to account for internally generated vibration
loads from tooth-tooth impacts induced by non-conjugate meshing of the gear
teeth. These vibration loads are called transmission error and will be worse with
low-accuracy gears. This factor is given by:
where:
These values of A and B were determined for a quality index QV = 10. Thus,
The AGMA Bending Geometry Factor J is determined from Table 12-9 for a
pinion by relating the number of pinion teeth (20) to the number of gear teeth (56)
This value is different when analyzing Gears 2, 4, and 6
The Face Width F is chosen as the minimum in the suitable range in order to get
the best values for safety factors for this gear.
The Rim Thickness Factor KB is 1 since we are analyzing a solid-disk pinion. This
is not the case for Gears 2, 4, and 6 since rims with spokes are used.
The Idler Factor KI is 1 since a non-idler gear is being analyzed.
The Size Factor Ks is 1 since no drastic changes in size are present.
The Dynamic factor KV attempts to account for internally generated vibration
loads from tooth-tooth impacts induced by non-conjugate meshing of the gear
teeth. These vibration loads are called transmission error and will be worse with
low-accuracy gears. This factor is given by:
where:
These values of A and B were determined for a quality index QV = 10. Thus,
The AGMA Bending Geometry Factor J is determined from Table 12-9 for a
pinion by relating the number of pinion teeth (20) to the number of gear teeth (56)
This value is different when analyzing Gears 2, 4, and 6
The Face Width F is chosen as the minimum in the suitable range in order to get
the best values for safety factors for this gear.
14
Since this value of F is below 100mm, the Load Distribution Factor Km is taken
from Table 12-16:
Thus, the value of the AGMA Bending Stress is:
The AGMA Contact Stress equation is given by:
The factors Ca, Cm, Cv, and Cs are equal, respectively, to Ka, Km, Kv, and Ks as
defined for the bending stress equation
The Surface Geometry Factor I is given by:
The Surface Finish Factor CF is used to account for unusually rough surface
finishes on gears, so it can be set as 1 for gears made by conventional methods.
The Elastic Coefficient CP is given by:
where Ep, Eg, vp, and vg are given:
Since this value of F is below 100mm, the Load Distribution Factor Km is taken
from Table 12-16:
Thus, the value of the AGMA Bending Stress is:
The AGMA Contact Stress equation is given by:
The factors Ca, Cm, Cv, and Cs are equal, respectively, to Ka, Km, Kv, and Ks as
defined for the bending stress equation
The Surface Geometry Factor I is given by:
The Surface Finish Factor CF is used to account for unusually rough surface
finishes on gears, so it can be set as 1 for gears made by conventional methods.
The Elastic Coefficient CP is given by:
where Ep, Eg, vp, and vg are given:
15
Thus,
Now, the value of the AGMA Contact Stress is:
The AGMA Bending Fatigue Strength is given by:
where the uncorrected bending strength was chosen for Steel AISI A1-A5
Through Hardened 330 HB (Figure 12-25)
Referring to Figure 12-24, The Life Factor KL is given by:
The Temperature Factor KT is 1 for steel material in oil temperatures up to 250°F
The Reliability Factor KR is taken from Table 12-19 for a reliability of 99%
Thus, the value of the AGMA Bending Fatigue Strength is:
Thus,
Now, the value of the AGMA Contact Stress is:
The AGMA Bending Fatigue Strength is given by:
where the uncorrected bending strength was chosen for Steel AISI A1-A5
Through Hardened 330 HB (Figure 12-25)
Referring to Figure 12-24, The Life Factor KL is given by:
The Temperature Factor KT is 1 for steel material in oil temperatures up to 250°F
The Reliability Factor KR is taken from Table 12-19 for a reliability of 99%
Thus, the value of the AGMA Bending Fatigue Strength is:
16
The AGMA Contact Fatigue Strength is given by:
where the uncorrected contact strength was chosen for Steel AISI A1-A5
Through Hardened 330 HB (Figure 12-27)
Referring to Figure 12-26, The Surface-Life Factor CL is given by:
The Hardness Ratio Factor CF is 1 for the pinion (gear 1)
The Temperature Factor CT is identical to KT
The Reliability Factor CR is is identical to KR
Thus, the value of the AGMA Contact Fatigue Strength is:
Now, we can calculate our safety factors:
The AGMA Contact Fatigue Strength is given by:
where the uncorrected contact strength was chosen for Steel AISI A1-A5
Through Hardened 330 HB (Figure 12-27)
Referring to Figure 12-26, The Surface-Life Factor CL is given by:
The Hardness Ratio Factor CF is 1 for the pinion (gear 1)
The Temperature Factor CT is identical to KT
The Reliability Factor CR is is identical to KR
Thus, the value of the AGMA Contact Fatigue Strength is:
Now, we can calculate our safety factors:
17
18
v. Calculations for Gears 2, 3, 4, 5, 6
Gears 2 and 6 are identical, and so are Gears 3 and 5
When compared to Gear 1, the only factors that change when calculating the safety
factors for the rest of the gears are:
• The life for Gears 2, 3, 5, 6 is half that of Gears 1 and 4 (480000000 cycles
instead of 960000000 cycles)
• The angular velocities, torques, and power differ as shown previously in
Section (iii)
• The face width changed from stage 1 to stage 2 (from 24mm to 45mm)
• The geometry factor J becomes 0.4 instead of 0.34 when analyzing a gear
instead of a pinion.
• A rim thickness factor Kb exists for gears 2, 4, and 6 due to the presence of
spokes. Its value is determined for a certain assumed rim thickness as
follows:
• The Hardness Ratio Factor CH is not 1 when analyzing gears because of a
change in material. Between Gear 2 and Gear 1, it is calculated as follows:
A thorough calculation of safety factors for Gear 2 (and Gear 6 by association) can
be found in the Appendix, and values for the other gears can be found in the results
section of this report as well as in the Excel sheet accompanying it.
v. Calculations for Gears 2, 3, 4, 5, 6
Gears 2 and 6 are identical, and so are Gears 3 and 5
When compared to Gear 1, the only factors that change when calculating the safety
factors for the rest of the gears are:
• The life for Gears 2, 3, 5, 6 is half that of Gears 1 and 4 (480000000 cycles
instead of 960000000 cycles)
• The angular velocities, torques, and power differ as shown previously in
Section (iii)
• The face width changed from stage 1 to stage 2 (from 24mm to 45mm)
• The geometry factor J becomes 0.4 instead of 0.34 when analyzing a gear
instead of a pinion.
• A rim thickness factor Kb exists for gears 2, 4, and 6 due to the presence of
spokes. Its value is determined for a certain assumed rim thickness as
follows:
• The Hardness Ratio Factor CH is not 1 when analyzing gears because of a
change in material. Between Gear 2 and Gear 1, it is calculated as follows:
A thorough calculation of safety factors for Gear 2 (and Gear 6 by association) can
be found in the Appendix, and values for the other gears can be found in the results
section of this report as well as in the Excel sheet accompanying it.
19
II. Shaft Development
The 4 shafts were designed based off the data produced by the gear design process.
Since the gears designs have little dependence on their inner diameters, the shafts
diameters had much sizing freedom. It should be noted that the input shaft, Shaft 1,
was the simplest to design due to its low loading state, and had a similar layout to
our output shaft, Shaft 3. The reduction shafts, Shaft 2 and 4, were completely
identical (albeit operating at a different direction), and as such were designed only
once. We began with the input shaft. As described previously, the shaft layout the
first step in shaft design.
As for the loading conditions on the shafts, all shafts experienced a fully reversed
alternating bending moment, caused by either radial and tangential forces or the
weight of the shaft. The final shaft also experienced a significant axial load, which
was not transferred over to the other shafts. Finally, the torsional loading, due to the
torque, was considered to be a constant mean torsion at a magnitude equal to the
Take-Off conditions of 40 HP and 4000 rpm. The reasoning for this design
consideration was that this steady torque was the largest torque values that would
be applied onto the aircraft under regular flight conditions, and so designing for
those operating conditions provides us with a more conservative and encompassing
failure criterion. Our team could have alternatively chosen the Take-Off conditions
and Descent Glide conditions as a Max and Min for an alternating torsion for much
more conservative results, but we felt that this would be an inappropriate
assumption for our aircraft. Our aircraft is to be flown under highly controlled flight
paths, with a known and limited number of landings and take-offs, which makes
assuming a completely cyclic alternating torsion an unnecessary overly conservative
estimate.
Due to the nature of the symmetric double branch acting upon the inner gears (Gear
1 and Gear 4), radial and tangential forces cancel out and no moments diagrams
need be produced in those planes, on those shafts.
This is shown in the free body diagram below:
Figure 3
Wr Wt
Wr Wt
Tin Tin = 2(WT)(r)
II. Shaft Development
The 4 shafts were designed based off the data produced by the gear design process.
Since the gears designs have little dependence on their inner diameters, the shafts
diameters had much sizing freedom. It should be noted that the input shaft, Shaft 1,
was the simplest to design due to its low loading state, and had a similar layout to
our output shaft, Shaft 3. The reduction shafts, Shaft 2 and 4, were completely
identical (albeit operating at a different direction), and as such were designed only
once. We began with the input shaft. As described previously, the shaft layout the
first step in shaft design.
As for the loading conditions on the shafts, all shafts experienced a fully reversed
alternating bending moment, caused by either radial and tangential forces or the
weight of the shaft. The final shaft also experienced a significant axial load, which
was not transferred over to the other shafts. Finally, the torsional loading, due to the
torque, was considered to be a constant mean torsion at a magnitude equal to the
Take-Off conditions of 40 HP and 4000 rpm. The reasoning for this design
consideration was that this steady torque was the largest torque values that would
be applied onto the aircraft under regular flight conditions, and so designing for
those operating conditions provides us with a more conservative and encompassing
failure criterion. Our team could have alternatively chosen the Take-Off conditions
and Descent Glide conditions as a Max and Min for an alternating torsion for much
more conservative results, but we felt that this would be an inappropriate
assumption for our aircraft. Our aircraft is to be flown under highly controlled flight
paths, with a known and limited number of landings and take-offs, which makes
assuming a completely cyclic alternating torsion an unnecessary overly conservative
estimate.
Due to the nature of the symmetric double branch acting upon the inner gears (Gear
1 and Gear 4), radial and tangential forces cancel out and no moments diagrams
need be produced in those planes, on those shafts.
This is shown in the free body diagram below:
Figure 3
Wr Wt
Wr Wt
Tin Tin = 2(WT)(r)
20
Due to the fact that no other radial or transverse loadings exist, we used the gear weight as
a force in the analysis of Shaft 1. However, due to the small magnitude, its effect was
negligible relative to the much larger torque. This is very evident in the numbers presented
in our EXCEL calculations. Weight was subsequently ignored in all other shafts.
The final design consideration that we applied to all the shafts was the use of a hollow
shaft. This was done primarily to conserve on weight, but also gives us a better stiffness/
mass ratio, which improves our design’s deflection resistance (as deflection was not a
primary constraint, weight was the primary purpose). The hollow shaft choice meant all
our equations had an outer diameter subtracted by an inner diameter.
MN=MOPQN−MSNN ,U=VWXYUVUZ [U \U] ^[_VU V`a\Z[YU
The inner diameter of every shaft was chosen based on an iterative process on the analysis
preformed at the smallest diameter of any given shaft. This insured none of our shaft
shoulders would clash with our bored inner shaft diameter. Furthermore, the MSN was
chosen to be a full height H of the key chosen for our shaft. This insured that no significant
stress concentration would arise between the end of the keyway MSN. These heights were
chosen based off the standard recommendations provided in Norton-Machine Design-Table
10-2.
Due to the fact that no other radial or transverse loadings exist, we used the gear weight as
a force in the analysis of Shaft 1. However, due to the small magnitude, its effect was
negligible relative to the much larger torque. This is very evident in the numbers presented
in our EXCEL calculations. Weight was subsequently ignored in all other shafts.
The final design consideration that we applied to all the shafts was the use of a hollow
shaft. This was done primarily to conserve on weight, but also gives us a better stiffness/
mass ratio, which improves our design’s deflection resistance (as deflection was not a
primary constraint, weight was the primary purpose). The hollow shaft choice meant all
our equations had an outer diameter subtracted by an inner diameter.
MN=MOPQN−MSNN ,U=VWXYUVUZ [U \U] ^[_VU V`a\Z[YU
The inner diameter of every shaft was chosen based on an iterative process on the analysis
preformed at the smallest diameter of any given shaft. This insured none of our shaft
shoulders would clash with our bored inner shaft diameter. Furthermore, the MSN was
chosen to be a full height H of the key chosen for our shaft. This insured that no significant
stress concentration would arise between the end of the keyway MSN. These heights were
chosen based off the standard recommendations provided in Norton-Machine Design-Table
10-2.
21
i. Free Body Diagrams of the Shafts with Gears and Bearings:
Figure 4 – Shaft 1
i. Free Body Diagrams of the Shafts with Gears and Bearings:
Figure 4 – Shaft 1
22
Figure 5 – Shaft 2
Figure 5 – Shaft 2
23
Figure 6 – Shaft 3
Figure 7 – Shaft 4
Figure 6 – Shaft 3
Figure 7 – Shaft 4
24
The next section contains an analysis for each shaft. However, as there is significant
overlap between the analysis methods, Shaft 1 will contain most of the sample calculations
and equations. Further calculated references for each shaft can be found in complete
expansive detail in our appendix and Excel file.
ii. Shaft 1- Input Shaft
Shaft 1 was designed to contain gear 1 as well as bearing 1 and 2. Bearing 2 is to be press fit
onto the end, while Bearing 1 is locked onto the shaft using a clamp and spacer mechanism (the
spacer extending to the motor). The shaft diameter in the bearings is identical, but thinner than
that of the rest of the shaft. 1 mm clearance was added between the shoulder and the bearings to
allow for thermal expansion during regular operation. The gear is supported axially by a
shoulder, and fixed to the shaft using a key. All this is evident in the shaft layout presented in
Figure 4, as well as in our machine drawings in the appendix
Our moment diagrams, considering weight and torque are presented:
A more detailed calculation rundown is provided in our excel file.
We can immediately note that the point of highest Bending Moment and Torque is the same,
which is at the center of our gear. Given the fact that this point will contain our keyway, it is safe
to assume our highest stress concentration will be occurring there, which we named Point B. The
Gear shoulder will have a larger d our gear, and so no further stress calculations are needed
there. We took the shoulder to be 3mm larger than our dB . Finally, our shaft was fit into our
bearings with a d < dB however, due to the low moment and lack of a significant stress
concentration, we do not expect a failure point to exist there. Furthermore, we designed the dB
with a safety factor range of 1.3-1.5, insuring that we are comfortable away from any potential
failure points at the bearings. The d at bearings, henceforth denoted as dbr was sized based off an
appropriate design choice of roller bearing.
Point B: Critical Stress location. For this shaft, since we have no axial forces and constant
Torsion was assumed, the formula dB was:
Figure 8
The next section contains an analysis for each shaft. However, as there is significant
overlap between the analysis methods, Shaft 1 will contain most of the sample calculations
and equations. Further calculated references for each shaft can be found in complete
expansive detail in our appendix and Excel file.
ii. Shaft 1- Input Shaft
Shaft 1 was designed to contain gear 1 as well as bearing 1 and 2. Bearing 2 is to be press fit
onto the end, while Bearing 1 is locked onto the shaft using a clamp and spacer mechanism (the
spacer extending to the motor). The shaft diameter in the bearings is identical, but thinner than
that of the rest of the shaft. 1 mm clearance was added between the shoulder and the bearings to
allow for thermal expansion during regular operation. The gear is supported axially by a
shoulder, and fixed to the shaft using a key. All this is evident in the shaft layout presented in
Figure 4, as well as in our machine drawings in the appendix
Our moment diagrams, considering weight and torque are presented:
A more detailed calculation rundown is provided in our excel file.
We can immediately note that the point of highest Bending Moment and Torque is the same,
which is at the center of our gear. Given the fact that this point will contain our keyway, it is safe
to assume our highest stress concentration will be occurring there, which we named Point B. The
Gear shoulder will have a larger d our gear, and so no further stress calculations are needed
there. We took the shoulder to be 3mm larger than our dB . Finally, our shaft was fit into our
bearings with a d < dB however, due to the low moment and lack of a significant stress
concentration, we do not expect a failure point to exist there. Furthermore, we designed the dB
with a safety factor range of 1.3-1.5, insuring that we are comfortable away from any potential
failure points at the bearings. The d at bearings, henceforth denoted as dbr was sized based off an
appropriate design choice of roller bearing.
Point B: Critical Stress location. For this shaft, since we have no axial forces and constant
Torsion was assumed, the formula dB was:
Figure 8
25
MbO=cd∗ fgh
i∗
jg∗kl
mno.pq∗jgr∗sl
m
tu+
jgw∗kw
mno.pq∗jgrw∗sw
m
txy−MbSc
z
{
For our loading, Ta and Mm are zero. This formula was then solved for Nab, our safety fatigue
safety factor for point B.
|}~=i∗(h?{?h?{)
cd∗
jg∗kl
m
tu+
o.pq∗jgrw∗sw
m
txy
??
Note, both these equations consider din . The first formula provides us with an estimate for dB ,
based off NfBDESIRED , which is an initial target safety factor used to provide us with our first
guess. This was set to a conservative 2.25 initially. Next, this value was chosen, rounded to allow
for ease of manufacturability, and then corresponding din was chosen based off the process
outlined in the first section above. These choices caused equation 2 to produce our new NfB . At
this point we would repeat the process again iteratively until the results were satisfying, and we
have a NfB = [1.3,1.5]. Excel was an excellent tool for this iteration.
A sample calculation is presented here detailing all the parameters and factors chosen for design.
These results were the ones chosen for our Shaft 1. For further reference, please see the appendix
and the excel file.
Plug in with Desired Safety Factor of 2.25
MbO=cd∗(d.dq)
i∗?.?p∗cc.p[f??]m
?q?[k??]+
o.pq∗d.d?∗podcq.?[f??]m
??? [k??]−10[??]c
z
{
MbO=17.8325 [??]
Choose a MbO that is appropriate for machining:
MbO=17 ??
Plug into Safety Factor equation:
|}~=?∗(17c−10c)
32∗1.67∗33.7[|??]d
154[??\]+
0.75∗2.24∗70235.1[|??]d
469 [??\]
??
|}~=1.32
which is acceptable as a target safety factor. Note: The development shown above contained
many iterations, but brevity only the final stage is presented.
MbO=cd∗ fgh
i∗
jg∗kl
mno.pq∗jgr∗sl
m
tu+
jgw∗kw
mno.pq∗jgrw∗sw
m
txy−MbSc
z
{
For our loading, Ta and Mm are zero. This formula was then solved for Nab, our safety fatigue
safety factor for point B.
|}~=i∗(h?{?h?{)
cd∗
jg∗kl
m
tu+
o.pq∗jgrw∗sw
m
txy
??
Note, both these equations consider din . The first formula provides us with an estimate for dB ,
based off NfBDESIRED , which is an initial target safety factor used to provide us with our first
guess. This was set to a conservative 2.25 initially. Next, this value was chosen, rounded to allow
for ease of manufacturability, and then corresponding din was chosen based off the process
outlined in the first section above. These choices caused equation 2 to produce our new NfB . At
this point we would repeat the process again iteratively until the results were satisfying, and we
have a NfB = [1.3,1.5]. Excel was an excellent tool for this iteration.
A sample calculation is presented here detailing all the parameters and factors chosen for design.
These results were the ones chosen for our Shaft 1. For further reference, please see the appendix
and the excel file.
Plug in with Desired Safety Factor of 2.25
MbO=cd∗(d.dq)
i∗?.?p∗cc.p[f??]m
?q?[k??]+
o.pq∗d.d?∗podcq.?[f??]m
??? [k??]−10[??]c
z
{
MbO=17.8325 [??]
Choose a MbO that is appropriate for machining:
MbO=17 ??
Plug into Safety Factor equation:
|}~=?∗(17c−10c)
32∗1.67∗33.7[|??]d
154[??\]+
0.75∗2.24∗70235.1[|??]d
469 [??\]
??
|}~=1.32
which is acceptable as a target safety factor. Note: The development shown above contained
many iterations, but brevity only the final stage is presented.
26
Material: For our shaft, we arrived at our chosen |}~ by selecting SAE 1020 Cold Rolled
Steel (Table A-9, Norton).
Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the
range of |=10??]??V?. Based on this, we shall design for an endurance limit of ?? using:
??=??S????O???P?}?Q???????S?~???
where
???≅0.5?PQ for steels.
• ??S?? was calculated using ??S??=1.189M?o.o?p, and remained consistent with excel
parametrization.
• ??O?=1 for loading in bending loads, as this shaft as no axial loads. Shaft 3 is the
only shaft in which ??O?=0.70.
• ??P?} was obtained using the relationships ??P?}≅??PQ~ , with our parameters A
and b obtained from Norton Table 6-3, for Machined steel. ??P?}≅0.798.
• ?Q??? =1, as our operating conditions for our gearbox is to be −40℃ ≤? ≤40℃.
• ????S?~=0.814 for a reliability of 99% which seemed like an acceptable reliability
range for the critical application needed for Solar Impulse. It should be noted that all
parts of the aircraft undergo extensive quality testing, and so a higher reliability is
unnecessary.
Finally, our ?} and ?}?? values for our analysis at point B (keyway) were obtained from this
development:
Obtain ?Qand ?Q? from Norton Figure 10-16, an estimate of 2.2 was taken for an r/d ratio of
0.021 for the first iteration. This was later corrected as d was obtained.
?Q=2.2
?Q?=3.0
These was converted a fatigue safety factor using the Neuber equation:
?}=1+a?Q−1
a=1
1+\
?
where \ is obtained from Table 6-6.
Finally, Test ?}????NO?<??
if true then ?}?=?} and ?}??=?}?. This was the case for all the shafts.
Material: For our shaft, we arrived at our chosen |}~ by selecting SAE 1020 Cold Rolled
Steel (Table A-9, Norton).
Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the
range of |=10??]??V?. Based on this, we shall design for an endurance limit of ?? using:
??=??S????O???P?}?Q???????S?~???
where
???≅0.5?PQ for steels.
• ??S?? was calculated using ??S??=1.189M?o.o?p, and remained consistent with excel
parametrization.
• ??O?=1 for loading in bending loads, as this shaft as no axial loads. Shaft 3 is the
only shaft in which ??O?=0.70.
• ??P?} was obtained using the relationships ??P?}≅??PQ~ , with our parameters A
and b obtained from Norton Table 6-3, for Machined steel. ??P?}≅0.798.
• ?Q??? =1, as our operating conditions for our gearbox is to be −40℃ ≤? ≤40℃.
• ????S?~=0.814 for a reliability of 99% which seemed like an acceptable reliability
range for the critical application needed for Solar Impulse. It should be noted that all
parts of the aircraft undergo extensive quality testing, and so a higher reliability is
unnecessary.
Finally, our ?} and ?}?? values for our analysis at point B (keyway) were obtained from this
development:
Obtain ?Qand ?Q? from Norton Figure 10-16, an estimate of 2.2 was taken for an r/d ratio of
0.021 for the first iteration. This was later corrected as d was obtained.
?Q=2.2
?Q?=3.0
These was converted a fatigue safety factor using the Neuber equation:
?}=1+a?Q−1
a=1
1+\
?
where \ is obtained from Table 6-6.
Finally, Test ?}????NO?<??
if true then ?}?=?} and ?}??=?}?. This was the case for all the shafts.
27
iii. Shafts 2-4 - Reduction Shafts
Shaft 2 and 4 are the reduction shafts, and they both have identical designs while rotating in
opposite directions. Unlike Shaft 1, these two shafts have significant bending loads due to the
tangential and radial forces applied at the gears. Bearing 3,4,7 and 8 are to be press fit onto the
ends of both shafts. The shaft diameter in the bearings are identical, but thinner than the rest of
the shaft. Just as Shaft 1, a 1 mm clearance was added between the shoulder and the bearings to
allow for thermal expansion during regular operation. The gears are supported axially by a
shoulder, and both are fixed to the shaft using keys. All this is evident in the shaft layout
presented in Figure 5, as well as in our machine drawings in the appendix. Due to the 3-
dimensional nature of the loading, we produced moment diagrams in two planes, then composed
them into a total Moment diagram.
Our moment diagrams, considering weight and torque:
A more detailed calculation rundown is provided in our excel file.
We can immediately note that the point of highest Bending Moment and Torque is the
same, which is at the center of our Gear 3. Given the fact that this point will contain our
keyway, it is safe to assume our highest stress concentration will be occurring there, which
we named Point B. Similar reasoning for why this point will have the highest stress
concentration is followed as Shaft 1. Since this diameter will be designed for failure, we
shall use an identical diameter for the location at Gear 2.
We followed an identical analysis procedure for fatigue failure as Shaft 1. The diametric
results are shown in the results section, as well as our Excel file.
Note, to size our bearings, we had to choose from a list of standard roller bearings, which
forced us to tailor our bearing diameter to this shaft. To make sure these diameters do not
cause failure, a safety factor analysis was done using shear stress from shear force as the
only loading (as no other loading exists on those points). They were all comfortably within
acceptable range.
Figure 9
iii. Shafts 2-4 - Reduction Shafts
Shaft 2 and 4 are the reduction shafts, and they both have identical designs while rotating in
opposite directions. Unlike Shaft 1, these two shafts have significant bending loads due to the
tangential and radial forces applied at the gears. Bearing 3,4,7 and 8 are to be press fit onto the
ends of both shafts. The shaft diameter in the bearings are identical, but thinner than the rest of
the shaft. Just as Shaft 1, a 1 mm clearance was added between the shoulder and the bearings to
allow for thermal expansion during regular operation. The gears are supported axially by a
shoulder, and both are fixed to the shaft using keys. All this is evident in the shaft layout
presented in Figure 5, as well as in our machine drawings in the appendix. Due to the 3-
dimensional nature of the loading, we produced moment diagrams in two planes, then composed
them into a total Moment diagram.
Our moment diagrams, considering weight and torque:
A more detailed calculation rundown is provided in our excel file.
We can immediately note that the point of highest Bending Moment and Torque is the
same, which is at the center of our Gear 3. Given the fact that this point will contain our
keyway, it is safe to assume our highest stress concentration will be occurring there, which
we named Point B. Similar reasoning for why this point will have the highest stress
concentration is followed as Shaft 1. Since this diameter will be designed for failure, we
shall use an identical diameter for the location at Gear 2.
We followed an identical analysis procedure for fatigue failure as Shaft 1. The diametric
results are shown in the results section, as well as our Excel file.
Note, to size our bearings, we had to choose from a list of standard roller bearings, which
forced us to tailor our bearing diameter to this shaft. To make sure these diameters do not
cause failure, a safety factor analysis was done using shear stress from shear force as the
only loading (as no other loading exists on those points). They were all comfortably within
acceptable range.
Figure 9
28
Material: For our shaft, we arrived at our chosen |}~ by selecting SAE 1050 Cold Rolled
Steel (Table A-9, Norton).
Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range
of |=10??]??V?. Based on this, we shall design for an endurance limit of ?? using:
??=??S????O???P?}?Q???????S?~???
All developments beyond this point are identical to Shaft 1.
iv. Shaft 3 - Output Shafts
Shaft 3 has slightly different considerations than the last two shafts, since it contains a
cantilevered propeller weight, as well as a significant thrust load. The design methodology for
the shaft itself was identical; iterate different values of din and dout until an acceptable safety
factor is achieved. Bearing 5 and 6 are carefully selected to account for the thrust load. This is
further expanded upon in the bearing section. Bearing 5 is to be press fit, while bearing 6 is
attached using a clamp and nut arrangement. No clearance was given between the shaft shoulder
and bearing 6, due to the axial thrust consideration. The shaft diameter in bearing 5 is thinner
than that of the rest of the shaft, while bearing 6’s diameter will be our design diameter. A 1 mm
clearance was added between the shoulder and bearing 5 to allow for thermal expansion during
regular operation. The gear is supported axially by a shoulder, and fixed to the shaft using a key.
Keys are typically non-ideal for attachments were axial loading is present, but our bearing 6 was
designed to absorb all axial loads from the propeller. All this is evident in the shaft layout
presented in Figure 6, as well as in our machine drawings in the appendix
Our moment diagrams, considering weight and torque are presented:
Unlike previous shafts, the point of highest stress is not so immediately clear. Two points are
likely contenders, the keyway at the gear attachment, due to a high torque and moment, and the
second point is at Bearing 6 which has a high torque and maximum moment. Due to this, the
design analysis was repeated on those two points. It is safe to assume that the rest of the shaft
Figure 10
Material: For our shaft, we arrived at our chosen |}~ by selecting SAE 1050 Cold Rolled
Steel (Table A-9, Norton).
Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range
of |=10??]??V?. Based on this, we shall design for an endurance limit of ?? using:
??=??S????O???P?}?Q???????S?~???
All developments beyond this point are identical to Shaft 1.
iv. Shaft 3 - Output Shafts
Shaft 3 has slightly different considerations than the last two shafts, since it contains a
cantilevered propeller weight, as well as a significant thrust load. The design methodology for
the shaft itself was identical; iterate different values of din and dout until an acceptable safety
factor is achieved. Bearing 5 and 6 are carefully selected to account for the thrust load. This is
further expanded upon in the bearing section. Bearing 5 is to be press fit, while bearing 6 is
attached using a clamp and nut arrangement. No clearance was given between the shaft shoulder
and bearing 6, due to the axial thrust consideration. The shaft diameter in bearing 5 is thinner
than that of the rest of the shaft, while bearing 6’s diameter will be our design diameter. A 1 mm
clearance was added between the shoulder and bearing 5 to allow for thermal expansion during
regular operation. The gear is supported axially by a shoulder, and fixed to the shaft using a key.
Keys are typically non-ideal for attachments were axial loading is present, but our bearing 6 was
designed to absorb all axial loads from the propeller. All this is evident in the shaft layout
presented in Figure 6, as well as in our machine drawings in the appendix
Our moment diagrams, considering weight and torque are presented:
Unlike previous shafts, the point of highest stress is not so immediately clear. Two points are
likely contenders, the keyway at the gear attachment, due to a high torque and moment, and the
second point is at Bearing 6 which has a high torque and maximum moment. Due to this, the
design analysis was repeated on those two points. It is safe to assume that the rest of the shaft
Figure 10
29
will have lower stresses than those two points, just as we did in previous shafts. We started with
point B at our bearing.
This formula :
MbO=cd∗ fgh
i∗
jg∗kl
mno.pq∗jgr∗sl
m
tu+
jgw∗kw
mno.pq∗jgrw∗sw
m
txy−MbSc
z
{
Is not applicable for this shaft, as it was derived from the Case 3 loading of the modified
Goodman diagram, with the assumption of no axial load. To remedy this, the original equations
were used for the calculations instead. Namely:
1
|?=??
??+???
?PQ
where
??=?}32??
?(MbOc−MbSc)
???=?? ??S??d+3??do.q
??=?}??16??
?(MbOc−MbSc)
?? ??S??=?}?34???S??
?(MbOd−MbSd)
We then proceeded in a fashion similar to the previous shafts, iterating diameters until our target
safety factor was achieved.
We then repeated this process for the keyway point, and found it to be less critical. We chose a
diameter 5mm larger than that of the bearing point.
Material: For our shaft, we arrived at our chosen |}~ by selecting SAE 1050 Cold Rolled Steel
(Table A-9, Norton).
Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range
of |=10??]??V?. Based on this, we shall design for an endurance limit of ?? using:
??=??S????O???P?}?Q???????S?~???
where ??O?=0.70, as opposed to 1 in our other shafts.
Note, our ?Q and ?Q? were obtained from Norton Tables C-1 – C-3 for our bearing analysis , as
it has a shoulder stress concentration point at that area. Otherwise, ?} and ?}?? were obtained
identically to Shaft 1.
will have lower stresses than those two points, just as we did in previous shafts. We started with
point B at our bearing.
This formula :
MbO=cd∗ fgh
i∗
jg∗kl
mno.pq∗jgr∗sl
m
tu+
jgw∗kw
mno.pq∗jgrw∗sw
m
txy−MbSc
z
{
Is not applicable for this shaft, as it was derived from the Case 3 loading of the modified
Goodman diagram, with the assumption of no axial load. To remedy this, the original equations
were used for the calculations instead. Namely:
1
|?=??
??+???
?PQ
where
??=?}32??
?(MbOc−MbSc)
???=?? ??S??d+3??do.q
??=?}??16??
?(MbOc−MbSc)
?? ??S??=?}?34???S??
?(MbOd−MbSd)
We then proceeded in a fashion similar to the previous shafts, iterating diameters until our target
safety factor was achieved.
We then repeated this process for the keyway point, and found it to be less critical. We chose a
diameter 5mm larger than that of the bearing point.
Material: For our shaft, we arrived at our chosen |}~ by selecting SAE 1050 Cold Rolled Steel
(Table A-9, Norton).
Subsequently, due to our Lifetime of 2000 hour ,our number of cycles for shaft 1 is in the range
of |=10??]??V?. Based on this, we shall design for an endurance limit of ?? using:
??=??S????O???P?}?Q???????S?~???
where ??O?=0.70, as opposed to 1 in our other shafts.
Note, our ?Q and ?Q? were obtained from Norton Tables C-1 – C-3 for our bearing analysis , as
it has a shoulder stress concentration point at that area. Otherwise, ?} and ?}?? were obtained
identically to Shaft 1.
30
Bearings
Early on in the process of bearing selection the team decided it would be best to choose
roller/ball bearings for the entire system rather than journal bearings. Ball bearings have
many desirable attributes, primarily an very operating friction, which reduces frictional
loses in our overall system, that would have otherwise been generated by the viscosity in a
journal bearing. Rolling bearings also have no transient start-up speeds, which makes them
ideal for a critical application such as an aircraft. Finally , they can handle axial and radial
loads, which is desirable if the aircraft happens to operate under unexpected conditions.
Roller bearings are a lot less sensitive to any potential interruptions with their lubrication
as well.
We used ball bearings for most of our system, where axial loads were not present. The
selection of these bearings was fairly straight forward. For our axial load on the output
shaft, a spherical roller bearing was chosen to absorb all the axial loads from the propeller.
This choice was made so that no stress concentrations may arise in the remainder of Shaft
3, shielding the gear and preventing the transfer of any axial loads to the rest of the
gearbox in the case of an unexpected fluctuation in the propeller.
The Ball bearing selection process can now be outlined.
For every one of our shafts we used the Free Body Diagram analysis outlined in the Shaft
section (Figure 4,5, 6 and 7) to obtain the reaction forces necessary for our bearings. This
allowed us to obtain reaction forces FR and FA , for radial and axial respectively.
We then combined those loads using:
?=????+???
where X, V and Y are bearing load factors obtained from Norton Figure 11-24.
Now, using ??o=?
?
c
we may obtain the expected ??o life in millions of revolutions.
This ??o life can then be compared to the ideal number of cycles provided as a design
constraint, to appropriately select a bearing that will exceed this limit.
Our ?S???=?ooo ??∗?o∗??
?o?
Where ?N is the angular velocity of each shaft.
Bearings
Early on in the process of bearing selection the team decided it would be best to choose
roller/ball bearings for the entire system rather than journal bearings. Ball bearings have
many desirable attributes, primarily an very operating friction, which reduces frictional
loses in our overall system, that would have otherwise been generated by the viscosity in a
journal bearing. Rolling bearings also have no transient start-up speeds, which makes them
ideal for a critical application such as an aircraft. Finally , they can handle axial and radial
loads, which is desirable if the aircraft happens to operate under unexpected conditions.
Roller bearings are a lot less sensitive to any potential interruptions with their lubrication
as well.
We used ball bearings for most of our system, where axial loads were not present. The
selection of these bearings was fairly straight forward. For our axial load on the output
shaft, a spherical roller bearing was chosen to absorb all the axial loads from the propeller.
This choice was made so that no stress concentrations may arise in the remainder of Shaft
3, shielding the gear and preventing the transfer of any axial loads to the rest of the
gearbox in the case of an unexpected fluctuation in the propeller.
The Ball bearing selection process can now be outlined.
For every one of our shafts we used the Free Body Diagram analysis outlined in the Shaft
section (Figure 4,5, 6 and 7) to obtain the reaction forces necessary for our bearings. This
allowed us to obtain reaction forces FR and FA , for radial and axial respectively.
We then combined those loads using:
?=????+???
where X, V and Y are bearing load factors obtained from Norton Figure 11-24.
Now, using ??o=?
?
c
we may obtain the expected ??o life in millions of revolutions.
This ??o life can then be compared to the ideal number of cycles provided as a design
constraint, to appropriately select a bearing that will exceed this limit.
Our ?S???=?ooo ??∗?o∗??
?o?
Where ?N is the angular velocity of each shaft.
31
The C parameter is obtained from the manufacturer specification for each bearing, and so
to allow for successful iteration, a large amount bearing tables were input into our excel
sheets to allow for easy selection. The tables would “iterate” these formulas for many
different types of bearings, until a suitable life limit was found. We should note that was
not compared directly with our , but rather it was multiplied by a reliability factor of Kr =
0.33 (for a reliability of 98%) for most of our bearing choices. We now used this new to
compare with our ideal life.
After a number of bearings of acceptable life were found, the particular bearing was
selected based off geometrical constraints of our shaft. Since we had a very low stress state,
we could afford to add a large shoulder on the bearing portion of our shaft, and so dout was
not a factor. However, since we used a hollow shaft, din was the limiting parameter. The
selected bearings are presented in later in this report.
Note: Due to our very low loading, the life expectancies of the bearings on shaft 1 are
significantly higher than necessary, however these bearings were found to be suitable for
our geometry , and satisfied our needs, and thus they were chosen. The same applies for
Shaft 2, 4 , however the life expectancies are not much higher than the ideal requirement.
The final spherical thrust bearing was chosen from SKF , and it was chosen for a suitable its
suitable geometry and life expectancies. The bearing is a Spherical Roller Bearing.
The C parameter is obtained from the manufacturer specification for each bearing, and so
to allow for successful iteration, a large amount bearing tables were input into our excel
sheets to allow for easy selection. The tables would “iterate” these formulas for many
different types of bearings, until a suitable life limit was found. We should note that was
not compared directly with our , but rather it was multiplied by a reliability factor of Kr =
0.33 (for a reliability of 98%) for most of our bearing choices. We now used this new to
compare with our ideal life.
After a number of bearings of acceptable life were found, the particular bearing was
selected based off geometrical constraints of our shaft. Since we had a very low stress state,
we could afford to add a large shoulder on the bearing portion of our shaft, and so dout was
not a factor. However, since we used a hollow shaft, din was the limiting parameter. The
selected bearings are presented in later in this report.
Note: Due to our very low loading, the life expectancies of the bearings on shaft 1 are
significantly higher than necessary, however these bearings were found to be suitable for
our geometry , and satisfied our needs, and thus they were chosen. The same applies for
Shaft 2, 4 , however the life expectancies are not much higher than the ideal requirement.
The final spherical thrust bearing was chosen from SKF , and it was chosen for a suitable its
suitable geometry and life expectancies. The bearing is a Spherical Roller Bearing.
32
Design Results
In summary, the resulting gear specifications are:
Design Results
In summary, the resulting gear specifications are:
33
As for shafts, the specifications are as follows:
As for shafts, the specifications are as follows:
34
35
The results summarized above are the output of a long iterative design process. As we can
see from the tables above, the chosen design provides satisfactory safety factors, in
concordance with our initial estimations. For the gears, the factors of safety for both
bending and pitting are acceptable. Also, for shafts, the safety factors for both failure and
yield are acceptable, and finally, for keys, the safety factors for bearing stress and those for
shaft stress are good. This ensures the viability of the design. Moreover, the life constraints
are also respected for the bearings. The life calculated for the bearing chosen is higher than
the life requirements given, which proves that the design is enduring as well.
Unfortunately, the mass requirement could not be respected with the current design
choices. Alternatives could be discussed in the conclusion. A detailed drawing of each part
is present in the appendix and supplementary calculations are provided in the excel file.
The results summarized above are the output of a long iterative design process. As we can
see from the tables above, the chosen design provides satisfactory safety factors, in
concordance with our initial estimations. For the gears, the factors of safety for both
bending and pitting are acceptable. Also, for shafts, the safety factors for both failure and
yield are acceptable, and finally, for keys, the safety factors for bearing stress and those for
shaft stress are good. This ensures the viability of the design. Moreover, the life constraints
are also respected for the bearings. The life calculated for the bearing chosen is higher than
the life requirements given, which proves that the design is enduring as well.
Unfortunately, the mass requirement could not be respected with the current design
choices. Alternatives could be discussed in the conclusion. A detailed drawing of each part
is present in the appendix and supplementary calculations are provided in the excel file.
36
Modified Goodman Diagrams
After compiling all the data in the tables above, we can now illustrate the stress and safety
factors using modified Goodman Diagrams.
The equations below will help us to determine the different values that construct the axis and
lines.
To
obtain the safety factor, we also have to choose
for
which failure case we are solving. A standard
choice would be case 3 where we assume
that Ơm and Ơa will increase in a constant ratio.
Modified Goodman Diagrams
After compiling all the data in the tables above, we can now illustrate the stress and safety
factors using modified Goodman Diagrams.
The equations below will help us to determine the different values that construct the axis and
lines.
To
obtain the safety factor, we also have to choose
for
which failure case we are solving. A standard
choice would be case 3 where we assume
that Ơm and Ơa will increase in a constant ratio.
37
Shaft 1
Shaft 2/4
Safety Factor Case 3
1.418797553
Safety Factor Case 3
1.31817201
Shaft 1
Shaft 2/4
Safety Factor Case 3
1.418797553
Safety Factor Case 3
1.31817201
38
Shaft 3
Safety Factor Case 3
1.48390892
Shaft 3
Safety Factor Case 3
1.48390892
39
Conclusions
NikolaDrive's vision for the future and passion for innovation drove the team to give
it's best in the design of this gearbox., Using the specifications of the plan such as the power
input, the mass of the propeller and respecting the dimensional and mass constraints.
The design process for the double-brunch double-reduction gearbox quickly proved
itself as an incredibly challenging one. As the team tackled the variety of problems and
constraints, often times, there was no visible solution, as different variables and factors are
dependent on each other. To overcome these emerging difficulties, conceptual
understanding of machine element design was required for laying true and quality
assumptions that helped narrowing down the range of possible solutions.
The gearbox design presented in this report is a result of considerable amount of
experimental iterations based on stress analysis knowledge and the team's ability to
manipulate the provided data into a feasible well-performing solution. The design achieves
most of its initial goals except for the weight reduction requirement. This parameter could
also be improved with more iterative experimentation. Specifically, with the use of lighter
and stronger metals such as titanium, magnesium and others. In addition, there is still
room for reducing the spacing between the components which may improve the
performance in terms of lowering the bending stresses as well as cutting on weight.
For conclusions, the rigorous process that led the team to its final design was an
extremely challenging and fruitful process. All team members had to join forces, thoughts
and sacrifice precious sleeping hours to achieve this impressive solution. It is these kind of
challenges, that encourages us to constantly improve and makes us better future engineers.
Conclusions
NikolaDrive's vision for the future and passion for innovation drove the team to give
it's best in the design of this gearbox., Using the specifications of the plan such as the power
input, the mass of the propeller and respecting the dimensional and mass constraints.
The design process for the double-brunch double-reduction gearbox quickly proved
itself as an incredibly challenging one. As the team tackled the variety of problems and
constraints, often times, there was no visible solution, as different variables and factors are
dependent on each other. To overcome these emerging difficulties, conceptual
understanding of machine element design was required for laying true and quality
assumptions that helped narrowing down the range of possible solutions.
The gearbox design presented in this report is a result of considerable amount of
experimental iterations based on stress analysis knowledge and the team's ability to
manipulate the provided data into a feasible well-performing solution. The design achieves
most of its initial goals except for the weight reduction requirement. This parameter could
also be improved with more iterative experimentation. Specifically, with the use of lighter
and stronger metals such as titanium, magnesium and others. In addition, there is still
room for reducing the spacing between the components which may improve the
performance in terms of lowering the bending stresses as well as cutting on weight.
For conclusions, the rigorous process that led the team to its final design was an
extremely challenging and fruitful process. All team members had to join forces, thoughts
and sacrifice precious sleeping hours to achieve this impressive solution. It is these kind of
challenges, that encourages us to constantly improve and makes us better future engineers.
40
Appendix A – All Calculations
• Full Calculations for Gear 2 (and Gear 6 by Association)
Appendix A – All Calculations
• Full Calculations for Gear 2 (and Gear 6 by Association)
41
42
43
44
Appendix B – Figures and Tables
0
20000
40000
60000
80000
100000
120000
050100150200250300
T2
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
050100150200250300
V2 yz
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
050100150200250300
V2 xy
0
10000
20000
30000
40000
50000
60000
050100150200250300
M2 yz
-20000
0
20000
40000
60000
80000
100000
120000
140000
050100150200250300
M2 xy
0
20000
40000
60000
80000
100000
120000
140000
160000
050100150200250300
M total
-180000
-160000
-140000
-120000
-100000
-80000
-60000
-40000
-20000
0
050100150200250300
M3 yz
-2000
-1500
-1000
-500
0
500
1000
1500
2000
050100150200250300
V3 yz
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
050100150200250300
M total
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
050100150200250300
V4 yz
0
20000
40000
60000
80000
100000
120000
050100150200250300
T4
0
10000
20000
30000
40000
50000
60000
050100150200250300
M4 yz
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
050100150200250300
V4 xy
-20000
0
20000
40000
60000
80000
100000
120000
140000
050100150200250300
M4 xy
0
20000
40000
60000
80000
100000
120000
140000
160000
050100150200250300
M total
-80000
-70000
-60000
-50000
-40000
-30000
-20000
-10000
0
020406080100120140
T1 Nmm
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
020406080100120140
V1 (yz plane)
0
5
10
15
20
25
30
35
40
020406080100120140
M1 yz (Nmm)
0
5
10
15
20
25
30
35
40
020406080100120140
M total
Appendix B – Figures and Tables
0
20000
40000
60000
80000
100000
120000
050100150200250300
T2
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
050100150200250300
V2 yz
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
050100150200250300
V2 xy
0
10000
20000
30000
40000
50000
60000
050100150200250300
M2 yz
-20000
0
20000
40000
60000
80000
100000
120000
140000
050100150200250300
M2 xy
0
20000
40000
60000
80000
100000
120000
140000
160000
050100150200250300
M total
-180000
-160000
-140000
-120000
-100000
-80000
-60000
-40000
-20000
0
050100150200250300
M3 yz
-2000
-1500
-1000
-500
0
500
1000
1500
2000
050100150200250300
V3 yz
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
050100150200250300
M total
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
050100150200250300
V4 yz
0
20000
40000
60000
80000
100000
120000
050100150200250300
T4
0
10000
20000
30000
40000
50000
60000
050100150200250300
M4 yz
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
050100150200250300
V4 xy
-20000
0
20000
40000
60000
80000
100000
120000
140000
050100150200250300
M4 xy
0
20000
40000
60000
80000
100000
120000
140000
160000
050100150200250300
M total
-80000
-70000
-60000
-50000
-40000
-30000
-20000
-10000
0
020406080100120140
T1 Nmm
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
020406080100120140
V1 (yz plane)
0
5
10
15
20
25
30
35
40
020406080100120140
M1 yz (Nmm)
0
5
10
15
20
25
30
35
40
020406080100120140
M total
45
46
47
48
Appendix C – Mechanical Drawings of Proposed Design
Isometric View
Appendix C – Mechanical Drawings of Proposed Design
Isometric View
49
Top View
Top View
50
Courtesy of Engineering edge website.
http://www.engineersedge.com/mechanical,045tolerances/general_iso_tolerance_.htm
Courtesy of Engineering edge website.
http://www.engineersedge.com/mechanical,045tolerances/general_iso_tolerance_.htm
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