SD Team 09: Torsen Differential for 2016 Formula SAE Race Car

Transcription

EML 4905 Senior Design Project
A B.S. THESIS
PREPARED IN PARTIAL FULFILLMENT OF THE
REQUIREMENT FOR THE DEGREE OF
BACHELOR OF SCIENCE
IN
MECHANICAL ENGINEERING
TORSEN DIFFERENTIAL FOR FSAE 2016
RACE CAR
Final Report
Shivana Mahes
Uber Mantovani
Alessandro Tasca
Advisor: Andres Tremante
November 22, 2015
This B.S. thesis is written in partial fulfillment of the requirements in EML 4905.
The contents represent the opinion of the authors and not the Department of
Mechanical and Materials Engineering.
Ethics Statement and Signatures
ii
Table of Contents
Chapter
Page
Ethics Statement and Signatures ..................................................................................................... ii
Table of Contents ........................................................................................................................... iii
List of Figures ................................................................................................................................ vi
List of Tables ............................................................................................................................... viii
1.
2.
3.
4.
5.
Introduction .......................................................................................................................... 1
1.1.
Problem Statement ........................................................................................................ 1
1.2.
Motivation .................................................................................................................... 1
1.3.
Literature Survey .......................................................................................................... 2
Project Formulation ........................................................................................................... 26
2.1.
Project Objectives ....................................................................................................... 26
2.2.
Design Specifications ................................................................................................. 26
2.3.
Addressing Global Design .......................................................................................... 27
2.4.
Constraints and Other Considerations ........................................................................ 27
Design Alternatives ............................................................................................................ 28
3.1.
Overview of Conceptual Designs Developed ............................................................. 28
3.2.
Design Alternative 1 ................................................................................................... 28
3.3.
3.4.
Project Management .......................................................................................................... 30
4.1.
Overview .................................................................................................................... 30
4.2.
Breakdown of Work into Specific Tasks .................................................................... 30
4.3.
Gantt Chart for the Organization of Work and Timeline ........................................... 31
4.4.
Breakdown of Responsibilities Among Team Members ............................................ 31
Engineering Design and Analysis ...................................................................................... 32
5.1.
Overview .................................................................................................................... 32
5.2.
Kinematic Analysis..................................................................................................... 32
5.3.
Structural Design ........................................................................................................ 34
5.4.
Force Analysis ............................................................................................................ 41
iii
5.5.
Component Selection .................................................................................................. 48
5.6.
Material Selection ....................................................................................................... 48
5.7.
Design Modifications ................................................................................................. 49
5.8.
Housing Design .......................................................................................................... 50
5.9.
Simulations ................................................................................................................. 51
5.10.
6.
Manufacturing ......................................................................................................... 56
Design Experience ............................................................................................................. 57
6.1.
Standards Used in the Project ..................................................................................... 57
6.2.
Cost ............................................................................................................................. 58
7.
Testing and Evaluation ...................................................................................................... 60
7.1.
Overview .................................................................................................................... 60
7.2.
Test Results and Data ................................................................................................. 65
7.3.
Evaluation of Experimental Results ........................................................................... 68
7.4.
Improvement of Design .............................................................................................. 68
7.5.
Discussion ................................................................................................................... 69
8.
Design Considerations ....................................................................................................... 69
8.1.
Health and Safety........................................................................................................ 69
8.2.
Assembly and Disassembly ........................................................................................ 70
8.3.
Manufacturability ....................................................................................................... 70
8.4.
Maintenance of the System......................................................................................... 70
8.5.
Risk Assessment ......................................................................................................... 71
9.
Design Experience ............................................................................................................. 71
9.1.
Overview .................................................................................................................... 71
9.2.
Standards Used in the Project ..................................................................................... 71
9.3.
Professional and Ethical Responsibility ..................................................................... 72
9.4.
Discussion ................................................................................................................... 72
10.
Conclusion...................................................................................................................... 72
10.1.
Conclusion and Discussion ..................................................................................... 72
10.2.
Future Work ............................................................................................................ 73
11.
References ..................................................................................................................... 74
Appendices .................................................................................................................................... 76
iv
Appendix A: Detailed Engineering Drawings of All Parts, Subsystems and Assemblies............ 76
Appendix B: Excerpts of Guidelines Used in the Project: Standards, Codes, Specifications and
Technical Regulations ................................................................................................................... 81
Appendix C: Purchased Components ........................................................................................... 92
Appendix D: Project Photo Album ............................................................................................... 93
v
List of Figures
Figure
Page
Figure 1: Nomenclature of a Gear .................................................................................................. 3
Figure 2: Forces in a Gear Train ..................................................................................................... 5
Figure 3: Example of Torsen T-1 Differential Housing ............................................................... 10
Figure 4: Typical Split Coefficient Performance Curve ............................................................... 11
Figure 5: Layout of Torsen Differential T-2 ................................................................................ 29
Figure 6: Layout of Torsen Differential T-2R .............................................................................. 30
Figure 7: Gearing System, Final Design ....................................................................................... 47
Figure 8: Complete Assembly ...................................................................................................... 50
Figure 9: Complete Assembly of Differential and Sprocket ........................................................ 51
Figure 10: Simulation Results for Differential Housing ............................................................... 52
Figure 11: Safety Factor Analysis in Solidworks ......................................................................... 53
Figure 12: Free Body Diagram of Differential Bracket ................................................................ 54
Figure 13: Simulation Results for Differential Bracket ................................................................ 54
Figure 14: Simulation Results for Housing Lid ............................................................................ 55
Figure 15: Factor of Safety ........................................................................................................... 56
Figure 16: Gear Blanks Machined by Team ................................................................................. 57
Figure 17: Results of 50 psi Braking Pressure on Each Wheel .................................................... 65
Figure 18: Results of 200-100 psi Braking Pressure in Wheels ................................................... 66
Figure 19: Results of 240-150 psi Braking Pressure .................................................................... 67
Figure 20: Drawing for Sun Gear Blanks ..................................................................................... 76
Figure 21: Drawing for Planetary Gear Blanks ............................................................................ 77
Figure 22: Drawing for Left Differential Bracket......................................................................... 78
Figure 23: Drawing for Housing Cylinder .................................................................................... 79
Figure 24: Drawing for Left Housing Lid..................................................................................... 80
Figure 25: SAE Rules for Transmission and Drive ...................................................................... 81
Figure 26: SAE Rules for System Sealing .................................................................................... 81
Figure 27: SAE Rules for Tilt Test ............................................................................................... 81
Figure 28: Table for I and J Factors from the AGMA .................................................................. 82
vi
Figure 29: Pitting Resistance Calculation from the AGMA ......................................................... 83
Figure 30: Bending Strength Formulas from the AGMA ............................................................. 84
Figure 31: Calculation of Dynamic Factor from AGMA ............................................................. 85
Figure 32: Load Distribution Factor Calculation from the AGMA .............................................. 86
Figure 33: Face Load Distribution Factor from the AGMA ......................................................... 87
Figure 34: Face Load Distribution Factor Calculation Continued................................................ 88
Figure 35: Elastic Coefficient Calculation from the AGMA ........................................................ 89
Figure 36: Hardness Ratio Factor Calculation from the AGMA .................................................. 90
Figure 38: CV Joint....................................................................................................................... 92
Figure 39: 3D Printed Components .............................................................................................. 93
Figure 40: Machining of the Housing ........................................................................................... 93
Figure 41: Machining of the Housing ........................................................................................... 94
Figure 42: Housing After In-Shop Machining .............................................................................. 95
Figure 43: Housing After In-House Machining ............................................................................ 95
Figure 44: Differential Housing .................................................................................................... 96
Figure 45: Differential Housing Prior to EDM ............................................................................. 96
Figure 46: Arranging of Gears ...................................................................................................... 97
Figure 47: Gear Arrangement ....................................................................................................... 97
Figure 48: Assembling of Differential .......................................................................................... 98
Figure 49: Assembly of Differential ............................................................................................. 98
Figure 52: Assembled Differential.............................................................................................. 100
Figure 53: Differential Mounted on Car ..................................................................................... 100
Figure 54: Diagram of Components Designed or Selected by Team ......................................... 101
vii
List of Tables
Table
Page
Table 1: Project Gantt Chart ......................................................................................................... 31
Table 2: Values Given From FIU SAE ......................................................................................... 33
Table 3: Calculated Linear Velocities in Turn .............................................................................. 33
Table 4: Calculated Angular Velocities ........................................................................................ 34
Table 5: Calculated Velocity With Respect to the Differential .................................................... 34
Table 6: Assumed Angles Used in Calculating Gear Geometry................................................... 35
Table 7: Calculated Gear Geometry Parameters ........................................................................... 36
Table 8: Physical Constants of Materials Being Considered for Design ...................................... 38
Table 9: Calculated Dynamic Factor ............................................................................................ 39
Table 10: Calculated Elastic Coefficient for Each Material ......................................................... 39
Table 11: Factors Used in Rim-Thickness Factor Calculation .................................................... 39
Table 12: Given Values Used for Force Analysis ........................................................................ 42
Table 13: Force Analysis on the Differential Housing ................................................................. 42
Table 14: Calculated Force per Planet Pair................................................................................... 43
Table 15: Calculated Forces on the Complete Gearing System ................................................... 44
Table 16: Frictional Force Analysis .............................................................................................. 44
Table 17: Frictional Torque at the Sun Gear ................................................................................ 45
Table 18: Calculated Torque Per Side of the Differential ............................................................ 46
Table 19: Calculated Torque at the Wheels in a Right Turn ........................................................ 46
Table 20: Summation of Torques ................................................................................................. 47
Table 21: Cost of Raw Materials .................................................................................................. 59
Table 22: Cost of Parts .................................................................................................................. 59
Table 23: Cost of Manufacturing .................................................................................................. 60
Table 24: Total Cost of Project ..................................................................................................... 60
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1. Introduction
1.1.
Problem Statement
The team chose to design and manufacture a torque sensing differential system, Torsen, and
housing for use in the Florida International University Society of Automotive Engineer’s (SAE)
club racecar. The racecar will compete in the 2016 Formula SAE competition (FSAE). Currently,
the FIU SAE car utilizes a stock Torsen T1 differential. The team chose to build a new differential
to increase vehicle control and stability and to deliver more horsepower from the engine to the
road.
1.2.
Motivation
The FSAE competition was created in 1979 and has expanded ever since. Presently, colleges and
universities from all around the world compete to win the competition. It is a very attractive event
for top automotive companies in their search for future engineers that have experience in design
and manufacturing different components of a vehicle. This event created a unique opportunity for
top industry engineers and judges to inspect and critique the design of the differential based on
other universities designs. The FSAE scoring system, as well as judges’ reviews, gives the team
immediate feedback on how the vehicle performed based on design, functionality, and cost.
The main motivation that drove the team to choose to design a new differential is the fact that the
differential that is currently being used in the racecar was bought directly to a company that massproduces it. In fact, most formula SAE teams buy the Torsen differential type 1, T-1, because of
the discount that the company offers when selling the differential to universities. The original
design of the T-1 was made to handle torques for streetcars. Therefore, the current differential in
the FSAE racecar is not optimized for the car and the tests it will undergo. The team chose to
1
design, manufacture, and assemble a system that is optimized for the car and engine specifications
of FIU’s racecar.
1.3.
Literature Survey
The differential is an essential component of the powertrain of a vehicle. The engine is where
power is generated [1]. The transmission uses gears and gear trains to convert the engine’s power
into speed and torque [1]. The power from the engine is transferred to the differential before it
reaches the output, the wheels of the vehicle. It is the main component that transmits movement
from the engine to the wheels [1].
1.3.1. Gears
One of the main components of any differential are the gears that comprise the differential. Gears
are used for motion and power transmission. Forces between meshed gears create forces and
moments that affect the shafts and bearings it interacts with [2]. There are four prominent types of
gears: spur, helical, bevel, and worm. Spur gears are used to transmit motion between parallel
shafts [2]. Helical gears are used much like spur gears, but they have the ability to reduce noise
[2]. Bevel gears are used for power transmission between intersecting shafts. While worm gears
are used when there is a high speed ratio between two shafts [2]. A small gear is referred to as a
pinion, while a large gear in a set is called the gear [2].
Most calculations dealing with gears reference the pitch circle. The pitch circle is a theoretical
circle, used mainly in calculations [2]. The pitch circles of two mating gears should be tangent [2].
The pitch diameter (d) is the diameter of the pitch circle. Circular pitch (p) is the distance from
one point on a tooth of a gear to a similar point on an adjacent tooth [2]. The module (m) is a ratio
of pitch diameter to number of teeth (N), also known as the gear ratio [2]. Diametral pitch (P) is
just the opposite, it is defined as number of teeth to pitch diameter [2]. They are related by 𝑃 =
2
𝑁
𝑑
,𝑚 =
𝑑
𝑁
,𝑝 =
𝜋𝑑
𝑁
= 𝜋𝑚, 𝑎𝑛𝑑 𝑝𝑃 = 𝜋 [2]. The addendum is the distance between the top land of
one tooth and the pitch circle, while the dedendum is the radial distance from the bottom land to
the pitch circle [2]. Whole depth is the sum of the addendum and the dedendum [2]. The clearance
circle is tangent to the addendum circle of the mating gear. Clearance is how much the dedendum
of one gear exceeds the addendum of the mating gear [2]. Backlash in a gearing system is how
much the tooth width exceeds the tooth thickness of the engaging tooth measured from the pitch
circle [2]. Figure 1 shows all of these terms [2].
Figure 1: Nomenclature of a Gear
Two meshed gears have pitch circles that roll against each other without any slippage [2]. Pitchline velocity is therefore given by 𝑉 = |𝑟1 𝜔1 | = |𝑟2 𝜔2 | [2]. Angular velocities of mating gears are
𝜔
𝑟
related by |𝜔1 | = 𝑟2 [2]. Most dimensions and calculations use the pitch circle.
2
1
Contact ratios (mc) between gears give the average number of pairs of teeth in contact [2]. The
ratio is defined as the length of the path of contact over the base pitch. When tooth profiles are
designed to produce a constant angular velocity they are said to have conjugate action [2]. The
portion of the gears that are not in conjugate is referred to as interference. Interference usually
3
results in one teeth digging into the other, weakening the teeth of the gears [2]. To find the
minimum number of teeth needed on a pinion, Np is calculated by
2𝑘
𝑁𝑝 = 𝑠𝑖𝑛2 𝜑 (1 + √1 + 3𝑠𝑖𝑛2 𝜑),
where k=0.8 for stub teeth and 1 for full-depth teeth and φ is the pressure angle. If the gear ratio is
greater than 1, then the smallest number of teeth that can be on the pinion without causing
interference is
2𝑘
𝑁𝑝 = (1+2𝑚)𝑠𝑖𝑛2 𝜑 (𝑚 + √𝑚2 + (1 + 2𝑚)𝑠𝑖𝑛2 𝜑) [2].
The largest gear that can be used without interference is given by
𝑁𝑝2 𝑠𝑖𝑛2 𝜑−4𝑘 2
𝑁𝐺 = 4𝑘−2𝑁
𝑝 𝑠𝑖𝑛
2𝜑
[2].
There are various methods to manufacture gears. Milling involves cutting gear teeth with a milling
cutter that is shaped to each tooth [2]. A separate set of cutters is necessary for every pitch. When
shaping gear teeth, a pinion cutter or rack cutter may be used [2]. The pinion cutter spins on a
vertical axis and is eased against the gear blank until the desired depth is achieved. A hob is a
worm-shaped cutting tool [2]. The hob and the gear blank have to be rotated at the correct angular
velocity ratio to generate the gear. Finishing of gear teeth reduces errors that may cause additional
dynamic forces in each tooth [2]. Finishing is accomplished by shaving or burnishing the teeth of
gears that have not been heat treated. For gears that are heat treated, grinding and lapping are used
to correct minor errors caused by the manufacturing process [2].
For gear trains, the train value (e) is the product of driving tooth numbers over the product of
driven tooth numbers [2]. The speed of the last gear in a gear train is given by 𝑛𝐿 = 𝑒𝑛𝐹 . The train
value will be positive if the first and last gears rotate in the same direction and negative if they
rotate in opposite directions [2].
4
Two gears that are meshed will exert forces and moments on each other. Each force can be resolved
into 2 components: tangential and radial. The radial component does not transmit power and is
usually ignored in calculations [2]. The tangential load is responsible for all of the power
transmission in this case. Meshed gears have an efficiency of 98%, so power is treated as being
constant throughout the mesh [2]. Figure 2 shows the free body diagram of forces on a simple gear
train.
Figure 2: Forces in a Gear Train
The analysis and design of spur and helical gears rely heavily on making them resistant to bending
failure in the teeth and pitting failure in the tooth surfaces [2]. Bending failure happens when tooth
stress equals or exceeds the yield strength or bending endurance strength. Surface failure occurs
when contact stresses equal or exceed the surface endurance strength [2].
1.3.2. Differentials
Differentials are used in every car, since they are needed in simple operations, like turning a corner
[3]. When a car is turning a corner, the inner wheel spins slower than the outer wheel. Without a
5
differential, there would be wheel slippage or fracturing of the drive shaft due to the differences in
the wheel velocity. The differential allows the wheels to rotate at different speeds, while
transmitting torque to the ground. Differentials also influence vehicle control, stability, and
traction [3].
There are two main types of differentials: open and limited-slip differentials. The open differential
allows equal torque in both wheels, despite the relative speed of the wheels [3]. It does not prevent
the inner wheel from spinning before the outer wheel. Instead, the car starts to understeer and the
engine torque has to be reduced to stop wheel spin—this limits acceleration [3]. In an icy condition,
the open differential will transmit the smallest torque to both wheels, even though one wheel
requires more torque. In this case, the vehicle cannot overcome this situation.
A differential is a device, usually employing gears, capable of transmitting torque and rotation
through three shafts [4]. They are used in one of two ways; first, it receives one input and provides
two outputs--this is found in most automobiles--and secondly, it combines two inputs to create an
output that is the sum, difference, or average, of the inputs. In automobiles and other wheeled
vehicles, the differential allows each of the driving wheels to rotate at different speeds, while
supplying equal torque to each of them [4].
A vehicle's wheels rotate at different speeds, mainly when turning corners. The differential is
designed to drive a pair of wheels with equal torque while allowing them to rotate at different
speeds [4]. In vehicles without a differential, such as karts, both driving wheels are forced to rotate
at the same speed, usually on a common axle driven by a simple chain-drive mechanism. When
cornering, the inner wheel needs to travel a shorter distance than the outer wheel, so with no
differential, the result is the inner wheel spinning and/or the outer wheel dragging [4]. This results
in difficult and unpredictable handling, damage to tires and roads, strain, or possible failure of the
6
drive train. There are various devices for getting more usable traction from vehicles with
differentials. One solution is the limited slip differential (LSD), the most well-known of which is
the clutch-type LSD [4]. With this differential, the side gears are coupled to the carrier via a multidisc clutch, which allows extra torque to be sent to the wheel with high resistance when the limit
of friction is reached at the other wheel. Below the limit of friction more torque goes to the slower
wheel. If there is no load on one wheel, then no torque goes to the other [4]. The LSD provides no
torque except for spring loading, but some extra effect can be obtained by partially applying the
vehicle's parking brake when one wheel is spinning. This can provide some resistance to increase
the overall torque and allow the other driven wheel to move the vehicle [4]. This only works when
the handbrake acts on the driven wheels, as in the traditional rear-wheel drive layout. Naturally,
the handbrake should be released as soon as the vehicle is moving again.
A locking differential, such as ones using a differential along with air or an electrically controlled
mechanical system, allow no difference in speed between the two wheels on the axle when locked.
They employ a mechanism for allowing the planetary gears to be locked relative to each other,
causing both wheels to turn at the same speed regardless of which has more traction [4]. This is
equivalent to effectively bypassing the differential gears entirely. Other locking systems may not
even use differential gears, but instead drive one wheel or both depending on torque value and
direction [4].
A high-friction Automatic Torque Biasing (ATB) differential, such as the Torsen differential,
utilizes friction between gear teeth rather than at added clutches [1]. This applies more torque to
the driven wheel with highest resistance, grip or traction, than is available at the other driven wheel
when the limit of friction is differential is reached at the other wheel. When tested with the wheels
off the ground, if one wheel is rotated with the differential case held, the other wheel will still
7
rotate in the opposite direction like in an open differential, but there will be some frictional losses
and the torque will be distributed differently [1]. Although marketed as being "torque-sensing", it
functions the same as a limited slip differential.
A very high-friction differential, such as the ZF with sliding pins and cams, locks from very high
internal friction [1]. When tested with the wheels off the ground with torque applied to one wheel,
it will lock, but it is still possible for the differential action to occur in use, albeit with considerable
frictional losses, and with the road loads at each wheel in opposite directions rather than the same
(acting with a "locking and releasing" action, rather than a distributed torque).
An additional function of the conventional electronic traction control systems usually use the antilock braking system (ABS) wheel speed sensors to detect a spinning wheel and apply the brake to
that wheel [1]. This progressively raises the reaction torque at that wheel, and the differential
compensates by transmitting more torque through the other wheel, the one with better traction. In
Volkswagen Group vehicles, this specific function is called Electronic Differential Lock (EDL).
In a four-wheel drive vehicle, a viscous coupling unit can replace a center differential entirely, or
be used to limit slip in a conventional 'open' differential [1]. It works on the principle of allowing
the two output shafts to counter-rotate relative to each other, by way of a system of slotted plates
that operate within a viscous fluid, often silicone. The fluid allows slow relative movements of the
shafts, such as those caused by cornering, but will strongly resist high-speed movements, such as
those caused by a single wheel spinning [1]. This system is similar to a limited slip differential.
A four-wheel drive (4WD) vehicle will have at least two differentials (one in each axle for each
pair of driven wheels), and possibly a center differential to distribute torque between the front and
rear axles [1]. In some cases (e.g. Lancia Delta Integrale, 1989 Porsche 964 Carrera 4), the center
differential is an epicyclic differential to divide the torque asymmetrically but at a fixed rate
8
between the front and rear axle. Other methods utilize an Automatic Torque Biasing (ATB) center
differential, such as a Torsen – which is what Audi use in their Quattro cars (with longitudinal
engines). 4WD vehicles without a center differential should not be driven on dry, paved roads in
four-wheel drive mode, as small differences in rotational speed between the front and rear wheels
cause a torque to be applied across the transmission [1]. This phenomenon is known as wind-up,
and can cause considerable damage to the transmission or drive train. On loose surfaces these
differences are absorbed by the tire slippage on the road surface [1]. A transfer case may also
incorporate a center differential, allowing the drive shafts to spin at different speeds. This permits
the four-wheel drive vehicle to drive on paved surfaces without experiencing wind-up.
The gearing arrangement inside the Torsen differential combines the benefits of open, locking, and
torque vectoring differentials [1]. The Torsen differential T-1 utilizes an Invex gearing
arrangement to perform all the previously stated characteristics. The arrangement of worm gears
in the “T-1” makes the differential able to move as a solid unit, transferring equal power to both
axles when moving in a straight line and there is no slippage [1]. The differential housing transfers
the torque to the worm gearing. This can be seen in Figure 3.
9
Figure 3: Example of Torsen T-1 Differential Housing
When there is a difference in rotational speed between the shafts, the gearing inside the differential
creates a relative motion where the slow moving axle rotates backward with respect to the housing
[5]. This creates an increase in the torque that is transmitted to the slow moving axle. The term
torque bias ratio (TBR) is the amount of torque that can be delivered to one shaft instead of the
other. A torque bias ratio of 4:1 means that the differential can deliver four times the amount of
torque delivered to one wheel instead of the other [5]. For example, if the torque at the housing is
100 Newton meter (Nm), the maximum torque at one wheel will be 75 Nm and 25 Nm to the other.
Figure 4 shows a comparative graph between different TBR in the Torsen differential and an open
differential [5]. The TBR is a function of the internal gearing of the differential. Friction in the
Invex arrangement can be increased or decreased by changing the helix angle in the worm gears.
10
Figure 4: Typical Split Coefficient Performance Curve
1.3.3. Brief History of the Differential
There are many claims to the invention of the differential gear, and it is likely that it was known
in ancient times [6]. Some historical milestones of the differential include:
• 1050 BC–771 BC: The Book of Song (which itself was written between 502 and 557
A.D.) makes the assertion that the South Pointing Chariot, which uses a differential gear, was
invented during the Western Zhou Dynasty in China. If the left side gear encounters resistance,
the planet gear rotates about the left side gear, in turn applying extra rotation to the right side gear
[6].
• 227 – 239 AD: Despite doubts from fellow ministers at court, Ma Jun from the Kingdom
of Wei in China, invents the first historically verifiable South Pointing Chariot, which provided
cardinal direction as a non-magnetic, mechanized compass [6].
• 658, 666 AD: Two Chinese Buddhist monks and engineers create South Pointing Chariots
for Emperor Tenji of Japan [6].
11
• 1027, 1107 AD: Documented Chinese reproductions of the South Pointing Chariot by
Yan Su and then Wu Deren, which described in detail the mechanical functions and gear ratios of
the device much more so than earlier Chinese records [6].
• 1720: Joseph Williamson uses a differential gear in a clock [6].
• 1827: Modern automotive differential patented by watchmaker Onésiphore Pecqueur
(1792–1852) of the Conservatoire des Arts et Métiers in France for use on a steam cart [6].
• 1832: Richard Roberts of England patents 'gear of compensation', a differential for road
locomotives [6].
• 1876: James Starley of Coventry invents chain-drive differential for use on bicycles;
invention later used on automobiles by Karl Benz [6].
• 1897: First use of differential on an Australian steam car by David Shearer [6].
• 1913: Packard introduces the spiral-gear differential, which cuts gear noise [6].
• 1926: Packard introduces the hypoid differential, which enables the propeller shaft and
its hump in the interior of the car to be lowered [6].
• 1958: Vernon Gleasman patents the Torsen dual-drive differential, a type of limited slip
differential that relies solely on the action of gearing instead of a combination of clutches and gears
[6].
1.3.4. Types of Differentials
Open differentials distribute equal torque to both wheels, while also allowing them to rotate at
different speeds [4]. Active differentials use clutches to control the distribution of torque to the
wheels [1]. These types of differentials can only transfer torque to the slower wheel.
A traditional open differential is made up of bevel gears [5]. The output shaft can rotate at different
angular velocities while having the same input torque. Active limited slip differentials (ALSD)
12
include a clutch, connecting the output shaft with the rotating differential housing [5]. Torque
transfer is always directed from the faster to the slower shaft, and speed is equal to the average
wheel speed [1]. The ALSD transfers torque from the faster wheel to the slower wheel [5]. Torque
is not transferred if the two outputs have the same speed [6]. ALSDs allows control over the
magnitude of torque transfer, but not its direction. The benefit of an ALSD is that it is simpler and
cheaper to build and maintain than the other types of differentials [6]. The torque vectoring
differential (TVD) used two superpositioned clutches and additional gearing to transfer torque
between slower and faster wheels [1]. The additional gearing can speed up the input shaft of one
clutch and slow down the input speed of the other clutch. Torque can be transferred to the right
wheel by engaging the right clutch and to the left wheel by engaging the left clutch [1].
1.3.5. Materials Used
The team selected several materials for use in the differential. The team chose to use nitrided steel
for the gearing of the differential. Nitrided steel is very common in the manufacture of aircraft,
automotive components, and turbine generation systems [7]. The nitriding process allows the steel
to remain in the ferrite phase for the complete process. The molecular structure of the ferrite
remains in the body-centered cubic lattice and does not grow into the face-centered cubic lattice
[7]. Also, only free cooling takes place in the nitriding process. There is no rapid cooling or
quenching to allow a transformation from austenite to martensite [7]. There is no change in
molecular size, except small changes in volume of the steel surface caused by nitrogen diffusion.
Twisting and bending can be caused by induced surfaces stresses being released by the heat of the
process [7].
Nitriding is a process that diffuses nitrogen into steels and cast irons [7]. This process can occur
because of the solubility properties of nitrogen in these metals. The solubility limit of nitrogen is
13
temperature dependent [7]. Therefore, to successfully nitride steel, several factors must be
carefully controlled. They include furnace temperature, time, gas flow, and gas activity control.
Controlling these factors reduces distortion during the nitriding process [7]. Nitriding steel also
allows the benefit of acting as an additional temper to the steel.
In the early 20th century, Adolph Machlet worked as a metallurgical engineer [7]. Here, he
discovered surface hardening by carburizing led to distortion because of extended periods spent at
high temperatures and rapid quenching. By experimentation, Machlet found the solubility of
nitrogen in iron [7]. The nitrogen diffusion produced a relatively hard surface and produced
corrosion resistance. The process did not subject the steel to high temperatures or rapid quenching
[7]. The specimen was allowed to cool freely in a nitrogen-based atmosphere, reducing distortion,
but still producing a hard, corrosion resistant surface.
One challenge Machlet faced was controlling the decomposition of the ammonia used to liberate
the nitrogen [7]. He discovered that using hydrogen as a dilutant reduced the amount of nitrogen.
Though Machlet applied for a patent on the nitriding process in 1908, it was not granted until 1913
[7]. Machlet’s work in nitriding was technologically important, but he remained unrecognized.
Adolph Fry of Germany was more publicized and he was given the name “father of nitriding [7].”
Adolph Fry headed a parallel research program is German around 1906 [7]. Fry also recognized
the solubility characteristics of nitrogen in iron at high temperatures. Fry also recognized that the
nitrogen source had to be cracked by heat to liberate the nitrogen, but he did not use hydrogen to
dilute the process. Fry had developed the single-stage gas nitriding process [7].
Fry took his research further. He discovered that the nitriding process could only produce a hard
surface on steels that contained chromium, molybdenum, aluminum, vanadium, and tungsten [7].
14
These are known at the stable nitrides. Fry also found that the process depended heavily on process
temperature [7]. If the steel is processed at too high of a temperature, the surface was at risk for
“nitride networks,” which is a saturated solution of nitrogen in the surface of the case. Fry also
developed a group of steels with higher alloy contents, known today as Nitralloy steels [7].
There are differences in the U.S. and German process for nitriding steel. The main differences are:
1. The U.S. process used hydrogen to dilute and control the nitriding potential of the gas
and steel, which controlled the final surface metallurgy.
2. The German process involved alloying the steel, improving on core hardness and
tensile strength.
Machlet’s process did not find much success in the U.S. Fry’s process, however, was used heavily
after World War I [7]. They utilized the process to produce aircraft, textile, and automotive
products. In the 1920’s Fry’s progress has spread to the U.S. [7]. A presentation for the Society of
Manufacturing Engineers brought about the commercialization of Fry’s process in the U.S.
Metallurgists H.W. McQuaid and W.J. Ketcham did further research into the nitriding process [7].
Their experiments focused on the process temperature. They focused on a temperature range that
was lower than those used Machlet. Between 540 and 650° C. They found that higher temperatures
affected the core hardness of alloyed steels but did not affect the ability to nitride [7]. Higher
process temperatures increased the prevalence of nitride networks, due to a higher solubility of the
nitrogen in the iron, which caused cracking and exfoliation. Other studies they ran included:
o Temperature influence on case formation and depth
o Effect of alloying elements in Nitralloy steels
o Effect of temperature on growth and distortion
15
o Effect of time on case depth distortion and growth
o Effect of ammonia and dilution with hydrogen
o Effect of slow and rapid cooling
They found that the nitriding process was easier to control than the carburizing process [7]. They
also found that nitriding improved the corrosion resistance of low-alloy and alloy steels. In
addition, they discovered that most steels could be nitride [7]. They studied the effect of
decarburization on the mechanical strength of the nitrided case. They found that steel should be
free from decarburization if it was to undergo nitriding [7]. A surface that is not free from
decarburization is prone to exfoliation and peeling.
Robert Sergeson also reviewd the work of Fry [7]. He found that. Like McQuaid and Ketcham,
nitriding was a simpler process than carburizing. He found that increasing temperature produced
a better case hardness stability [7]. He also experimented with the temperature and process gas
flow. An increase in process temperature increased case depth but decreased surface hardness [7].
In addition, he worked with varying aluminum and nickel contents. He found that significant
quantities of nickel prevent the nitrogen diffusion process if present in significant quantities.
V.O. Homerberg and J.P. Walsted studied the effects of temperatures up to 750°C and the resulting
case depth [7]. They found that an increase in case depth was produced, but surface hardness was
decreases. They also found that the surface of the steel must be free from decarburization before
undergoing the nitriding process [7].
The Floe Process is a two-stage process. The first part of the cycle is performed like a normal
nitriding cycle at 500°C, this produces a nitrogen-rich compound at the surface [7]. Then, the
16
furnace temperature is increased to 560°C, this produces an atmosphere that is 15 to 25% ammonia.
This purpose serves to reduce the compound zone [7].
Salt bath nitriding uses molten salt as the source for nitrogen [7]. It uses the decomposition of
cyanide to cyanate and nitrogen in the salt for diffusion into the steel.
Ion, or plasma, nitriding uses a plasma discharge of reaction gases to supply nitrogen and to heat
the surface [7]. The importance of this process is that it does not rely on cracking to liberate
nitrogen on the steel surface. It also has a shorter cycle time than other methods [7]. This process
allowed nitriding to be carried out on all steels and cast irons.
Currently, nitriding has become an accepted process [7]. It has become common for a variety of
applications. However, the cutting edge of nitriding is discovering ways to make it more
environmentally friendly [7].
In recent years, aluminum allows have been the focus of many researchers, engineers, and
designers [8]. The 6000 series of aluminums have gotten special attention for their strength,
formability, corrosion resistance, and low cost when compared to other aluminum alloys [8].
Aluminum alloys are expected to replace steels, which will lead to improvements in energy
economy, recycling, and life-cycle cost [8]. In order to utilize the 6000 series of aluminum alloys
in other industries, however, the formability needs to be improved [8]. In order to achieve this,
control of the microstructures and texture is required. 6061 Aluminum alloy is the most commonly
used in the 6000 series, especially in the auto industry and for electrical fittings and connectors
[9].
Aluminum 7075 is predominantly used in the automobile and aerospace industry, due to its high
strength and low weight [10].
17
Aluminum has evolved in the past century. It began as a limited production of alloys and has grown
into high-volume manufacturing [11]. Aluminum production in the US is about ten million tons or
products that are used in building and construction, transportation, and packaging. Though the
company Alcoa was responsible for most of the growth, it could be due to the fact that many
developments took place before many of Alcoa’s competition was founded [11].
The early days of casting aluminum included using 45 kilogram ingots in steel-tilt molds. The
molds produced macrosegregation, porosity, and shrinkage cracking when the alloy content was
increased [11]. To deal with this issue, Alcoa had to scalp the molds to remove the undesired
characteristics.
The direct chill (DC) process was developed by William T. Ennor. He devised the idea of
impinging water on the shell of the ingot as it was cast [11]. Because of this process, it became
possible to drop the ingot and avoid the disturbances of pouring metal into old molds. This
technology was introduced into all of Alcoa’s plants by the 1930s, and it was used during the war
to produce aircraft parts [11]. Using this technology made it possible to fabricate larger aluminum
ingots. In 1950, the largest ingot produced was 3.1 tons, todays sizes of ingots can be as large as
15.5 tons [11]. Not only did the DC process allow for larger ingots, it helped improve
characteristics of the products produced. As the DC became a standard in the aluminum industry,
improved mechanical properties and fatigue-endurance limits were improved [11]. For some
alloys, the process required re-engineering downstream. Alloy 3003, which cooled slowly after
solidifying, needed a small amount of thermal treatment to produce fine-grained products. The
rapid solidification of the ingot resulted in more manganese in solution and a coarser grain size
[11]. W.A. Anderson relieved this problem by applying high-temperature homogenization to the
ingot.
18
The DC process also allowed for the development of new alloys, like high strength alloy 7075,
which was introduced during World War II [11]. In the 1950s, shipbuilding applications required
larger ingots of high magnesium 5000 series alloys. Other alloys that came about from the DC
process include 2000, 3000, 5000, and 7000 series alloys [11].
In 1908, a German researcher named A. Wilm made an accidental discovery about the hearttreatability properties of aluminum [11]. The Germans utilized this technology for 80 airships in
World War I. Shortly after, Alcoa obtained the rights to Wilm’s patent and began researching for
additional alloys [11]. This research led to alloys, like 2025, 2014, and 6051, which were easier to
produce than the method used by the Germans. These types of alloys made the development of
high-performance aircraft possible [11]. The high strength, toughness, and fatigue-resistance of
the 24S series made it ideal for aircraft applications, and it is still used today.
Aluminum 7075 provided high-strength capability not possible with aluminum-magnesiumcopper alloys [11]. Changes in the composition of the base alloys resulted in increased toughness.
Alcoa has continued development of alloys that reduce weight and improve aircraft performance
[11].
Aluminum cans are one of the most recognized consumer package in the world [11]. The demand
for aluminum cans drives the need for continuous improvement in all aspects of the process.
Commercial cans were produced Coors Brewing Company from impact-extruded 1000 series slugs
[11]. They later used thicker 3000 series sheets. Reynolds Aluminum developed a breakthrough
when they found draw-and-iron technology for the use of hard tempers [11]. This process produced
lighter, more economical cans. However, the need to keep improving did not end. High-strength
alloy 5182 was developed in 1967 and helped reduce the lid thickness [11]. This made the
aluminum competitive with steel. The aluminum tabs that stayed attached to the can reduced to
19
litter associated with cans [11]. When aluminum cans became more prevalent, the need to improve
the rolling process was seen. Four-high and six-high rolling mills were designed to produce tighter
tolerances [11]. Tandem rolling mills were developed to reduce the amount of rolling passes.
Advances in rolling lubricants and control technologies enabled sheets to be made faster and with
more consistency than previously thought possible [11]. Highly specialized alloys, like the 3000
and 5000 series, have been developed to fill the needs of the can industry.
The recycling technology has had to keep up with the amount of cans used. Can-collection, baling,
shredding, de-lacquering, and melting technologies have been innovated to keep up with the
demand [11].
Hydaulic extrusion was founded in the 19th century [11]. It was not until the 1900’s that Alcoa
made the first aluminum extrusions. In aluminum extrusion, the aluminum is solidified and forced
downward through a die [11]. With increase of size and pressure, a horizontal press became
necessary. Today, aluminum extrusions are common in building, construction, and aerospace [11].
Almost all alloys are used in extrusion, and extrusion can be used for tubes as small as microvoids
in heat exchangers to wings for airplanes.
The developments that led to larger ingots also called for improvements in quality requirements
[11]. As the 5000 series were hot rolled, cracking had to be prevented by lowering the amount of
sodium and calcium. Lowering hydrogen levels prevented blisters in other alloys [11]. Early in the
development process, the furnace was fluxed with chlorine to remove hydrogen, but this was
inefficient and bad for the environment. Therefore, other methods were needed for treating the
metal [11].
Deep-bed filtration used aluminum balls and chat to capture oxide inclusions while the metal
flowed from the furnace to the casting pit [11]. Another process, called the Alcoa 181 process,
20
used argon to filter out hydrogen. Internally heated bed filters was an important development that
allowed the size of the units and rate of metal flow to be increased [11]. Alcoa 622 was a process
that used a spinning nozzle to inject argon-chlorine gas bubbles into the motel metal to remove
impurities. This process reduced emissions, since the amount of chlorine used was very small [11].
A side benefit of in-line metal treatment was continuously fed grain refiners. Large filter boxes
allowed grain refiners to be fed at a desired rate without settling out in the furnace [11]. Continuous
grain refiners led to a reduction in ingot cracking and improved forgeability. Rigid, ceramic foam
filters allowed the metal to be cleaned before casting, which was an option over more expensive,
larger bed filters [11]. The new in-line metal treatments led to increased fatigue strength of
aerospace parts, less pinholes in aluminum foil, aluminum sheets that would not fracture during
forming, and higher surface quality [11].
Aluminum 6061 was introduced in 1935, and it filled the need for a medium strength, heattreatable alloy with good corrosion resistance [11]. Aluminum 6062 was introduced in 1947 and
had a finer grain size when cold-worked. 6061 and 6062 was easily fabricated through extrusion,
rolling, and forging [11]. Their mechanical properties allowed them to compete against with steel.
6061 continued to evolve until about 1963, when it was combined with 6062 [11]. Alloy 6061
keeps its corrosion resistance after welding, which made it popular in railroad and marine
applications.
J.W. Hoopes began to investigate an electric-conductor cable [11]. He began developing alloys
with conductivity and low strength. He achieved this by reinforcing softer aluminum wire with
steel [11]. The aluminum conductor steel-reinforced wire performed better than copper, cost less,
and withstood extreme temperatures. He obtained a patent in 1908, and by 1929 almost 500,000
km of this wire spanned the U.S [11].
21
Continuous casting was made possible in 1948 by wheel/belt caster that was used to produce lowcost electrical-conductor wire [11]. Rigamonti developed the first slab caster to cast narrow strips.
Pechiney, Alcan, and Hunter Douglas also developed casters for small-volume niche products [11].
Coors still uses narrow casters to produce impact-extrusion slugs. Hazlett developed a twin-belt
casting process that enabled a manufacturing of wider products [11]. It used mild steel belts and
water cooling to produce a slab that is continuously hot rolled. The Hunter twin-roll caster,
developed in the 1940s, produces strips from two water-cooled steel rolls [11]. Currently, casters
can produce widths greater than 2100 mm. Continuous casting has allowed the industry to avoid
the high capital costs seen in ingot/hot rolling facilities [11]. Nearly a quarter of sheet and foil
volume is produced by roll or slab coasters in North America. Continuous casting sheets are
popular for building, foil, and formed containers [11].
Some of the most important innovations of the aluminum industry have been in an alloy with good
fluid flow properties and desirable mechanical properties once cooled. In 1921, Archer and Jeffries
developed alloy 195, a beat-treatable, sand-casting alloy with various uses [11]. One famous use
of castings produced by Alcoa was the exterior of the Empire State Building [11]. By 1928, 11,300
tons of heat-treated cast products were being produced.
Competition from wood and plastic for building products resulted in developing cheaper aluminum
products [11]. Now, the largest market for aluminum extrusions is the construction market. In
1999, U.S. production was 635,000 tons [11]. Low extrusion pressure for the 6000 series of alloys
make them ideal for complex shapes and simplifies joining and assemblies. Press-quenching
eliminates the need for a separate solution heat-treatment step and produces a low-cost product
with reasonable strength [11]. At Alcoa, this process began in the 1930s. Alloy 2117 extrusions
that were longer than heat-treating furnaces were water-quenched with hand-held hoses [11]. At
22
the same time, alloy 6053-T5 was developed. It cooled in ambient air on a runout table [11]. It was
also discovered that forced-air cooling allowed the alloy to achieve the strength levels. 6063 was
developed in 1944 for extruded products [11]. It could be extruded at high rates and still harden to
good strength. It could also be anodized and easily colored and had better corrosion resistance than
6061 [11]. It had a low quench sensitivity allowed it to be press heat treated with moderate cooling
rates and minimal quench distortion.
Today, there are various alloys and processing methods to meet the demand in the building and
transportation industries [11]. Extrusions can be cooled in various ways to ensure the final
product’s geometry and needs.
1.3.6. History of the Formula SAE Competition
In 1979, the SAE Mini-Indy was held at the University of Houston [12]. Started by Dr. Kurt M.
Marshek, the competition was inspired by a how-to article that appeared in Popular
Mechanics magazine. A small, "Indy-style" vehicle made out of wood, and powered by a fivehorsepower Briggs and Stratton engine was the inspiration [12]. The Mini Baja competitions were
used as a guide. Engineering students had to design and build small, "Indy-style" vehicles using
the same stock engine used in the Popular Mechanics article [12]. Thirteen schools entered and
eleven competed, The University of Texas at El Paso won the overall competition.
Although Dr. William Shapton suggested hosting a similar competition in 1980, no one stepped
up to organize another competition [12].
Three students at the University of Texas at Austin saw potential, and with support from Dr. Ron
Matthews, contacted the SAE Educational Relations Department to propose a new mini-Indy with
new rules [12]. The new rules kept restrictions to a minimum, therefore any four-stroke engine
with a 25.4 mm intake restriction could be used. Students were to design a racing car which could
not cost over a set amount [12]. The teams were required to show receipts for their purchases. To
23
reflect better the road-racing nature of the event and its increased engineering content, the Formula
SAE name was adopted [12].
The University of Texas at Austin hosted the competition through 1984 [12]. In 1985, the
competition was hosted by The University of Texas at Arlington. There, Dr. Robert Woods, with
guidance from the SAE student activities committee, changed the concept of the competition from
one where students built a pure racing car, to one that mirrored the SAE Mini-Baja competitions,
where they were to design and build a vehicle for limited series production [12]. General Motors
hosted the competition in 1991, Ford Motor Co. in 1992, and Chrysler Corp. in 1993. After the
1992 competition, the three formed a consortium to run Formula SAE [12].
At the end of the 2008 competition, the consortium disbanded. The event is now funded by SAE
through company sponsorships and donations along with the teams' enrollment fees [12].
Formula SAE is a competition for university undergraduate and graduate students [12]. Teams that
compete must design, fabricate and compete with small, formula style autocross cars. Teams
typically spend between eight and twelve months designing, building and testing the vehicles for
competition [12]. Cars are judged on cost, design, marketability and performance [13].
The cost event is comprised of three sections. The entries are judged based on the lowest cost, the
presentation of a cost report, and a manufacturing presentation. The cost of a car is determined
based on the value of raw stock, machine time, labor, and commercial parts purchased at retail
value [13]. Therefore, designers must focus on minimizing cost throughout the design process.
The design portion of the competition consists both of an evaluation of the car’s design and an
evaluation of how well the team can justify design decisions [13]. Teams give presentations to a
panel of judges made up of engineers from automotive companies and the racing industry. Top
24
teams are selected from the initial round of presentations and those teams advance to a final round
of competition where the designs are further scrutinized [13].
In addition to the cost and design portions of the competition, teams can earn points for their
performances in five dynamic and three static events designed to thoroughly test each vehicle [13].
Approximately two thirds of all potential points come from dynamic events.
The cars are judged in a series of static and dynamic events, including technical inspection, cost,
presentation, and engineering design, solo performance trials, and high performance track
endurance [13]. The dynamic events are scored to determine how well the car performs [13]. Each
dynamic event has specified minimum acceptable performance levels that are reflected in the
scoring equations [13]. The following points are possible: Static events: presentation-75,
engineering design-150, cost analysis-100, and the dynamic events, acceleration-75, skid-pad-50,
autocross-150, efficiency-100, endurance-300. This is results in a total of 1,000 points [13].
The first dynamic event is a skip pad. The skid pad tests the car’s cornering ability by measuring
elapsed time around a figure eight [13]. The time directly corresponds to the maximum lateral
acceleration of the car.
The second dynamic event is an acceleration run [13]. The performance of a car in this event is
directly related to the power plant, the vehicle weight, suspension geometry and traction.
Typically, cars achieved 0-60 miles per hour times in the mid to upper three second range.
The third dynamic test is autocross [13]. Cones are laid out to form a low speed course with tight
corners. Fast times are a function of vehicle agility and driver skill. Normally, the driver is involved
in extensive training.
The fourth and final dynamic event is an endurance race [13]. The course is approximately 22
kilometers long comprised of numerous laps on a track laid out in an enormous parking lot. The
25
course is made up of many tight turns and several long straightaways [13]. Two drivers must
complete the event with a driver change half way through the endurance run. During the driver
change, the car must be turned off and then restarted by the second driver [13]. Judges scrutinize
the car, looking for any fluid leaks, evidence of which is grounds for removal from the race. Teams
that finish all 22 kilometers are awarded points based on their overall time and its ratio of the
fastest recorded time [13].
2. Project Formulation
2.1.
Project Objectives
The team designed and manufactured a fully functional differential and differential mounts to be
used in the Formula SAE racecar built for 2016 competition. The differential and its mounting
parts outperformed the current differential used by the following criteria; performance and weight.
2.2.
Design Specifications
The design was custom-made based on specifications given by the engine and chassis and
suspension teams. From the engine team, the maximum chain tension is 900 N. From the chassis
and suspension team, the coefficient of friction between tires and road was found to be between
1.7 and 2. The differential was mounted outside the chassis. This is known as a “floating
differential” in the auto industry. For this reason, the team designed a special mounting system.
The differential that was used in 2015 was analyzed for performance and comparison to the new
differential. The Torsen T-1 is capable of a TBR of 4:1. However, with an excessive weight of 13
lbs. it was very inefficient for the job in hand. Also, because of the Invex gearing arrangement, the
T-1 has a backlash that causes a system delay in the torque vectoring operation. This backlash also
makes the gearing collide whenever the direction of the torque vectoring is changing, making the
transition rough. Based on this information, the team designed a new differential with the hope of
26
reducing weight without reducing the TBR and cancelling the backlash that the Invex arrangement
creates.
2.3.
Addressing Global Design
For this project, the team hoped to create a differential that is capable of being used in different
climatic conditions, countries, and vehicles. The team designed a differential that can be installed
with ease. The team also hoped to design a differential that can be placed in any vehicle as long as
the torque produced by the vehicle does not exceed the maximum torque that the differential can
hold. The issue of components rusting was solved by sealing the differential as a requirement for
the leakage test, making it suitable for using in weather conditions where rusting is an issue.
2.4.
Constraints and Other Considerations
The biggest constraints that the team faced was to incorporate the differential in the rest of the
components that have direct interaction with and that were already part of the drivetrain. Since the
2015 differential was taken out of a production car, the shafts that transmit power from the
differential to the wheels were in accordance to the dimensions of the differential. For this reason,
if the gear that is connected to the shaft is designed and results to be smaller than the shaft, the
team will have to redesign that gear in order to accommodate for the shaft. On the other hand, if
the advantage of having smaller gears is of significant importance, then the team would design and
manufacture new axle shafts.
Another significant constraint was the distance from the chassis that the differential had to be
mounted. Since the engine sprocket is inside the chassis and the differential outside, the mounting
of it had to be well thought out. The differential sprocket that was designed dictated the minimum
27
distance from the differential and the chassis. The clearance so the chain does not hit the chassis
and other components had to be considered for the differential to be mounted correctly.
The team faced another constraint, which was the tight packaging of the differential. The mounting
of the gears inside is crucial for the differential to operate as prescribed. Any errors inside the
assembly can create catastrophic failure of the system.
Another constraint that the team faced is the leakage test. At competition the judges will tilt the
vehicle at a 45° angle in search for any leaking fluid. For this, the team will have to design a sealed
differential, so no oil will come out at any point.
Last, the team faced cost decisions. It was important for the design and manufacturing to be in
budget. If the best design was out of the cost/benefit range, the team would have to redesign in
order to accommodate the budget.
3. Design Alternatives
3.1.
Overview of Conceptual Designs Developed
The designs that will be described next are three design choices. The main difference between the
alternatives is the amount of TBR that can be achieved with each design. However, to gain a higher
TBR, the team sacrificed ease of manufacturing and assembly. Another factor is the replacement
time of the components inside the differential. Because friction plates wear out, replacing them
will be necessary at a specified amount of time.
3.2.
Design Alternative 1
The first design consists of only meshing gears inside the differential. This arrangement consists
of worm gears mating in parallel, where the locking effect characteristic of worm gears is present.
This
arrangement
can
be
seen
in
Figure
4.
The worm gear that is connected to the right axle is represented in Figure 1 as “6, 7”. The difference
28
between the two is the helix angle direction. The “6” is a left hand and the “7” is a right hand helix
angle. “6” drives a worm pinion that is meshed to a worm pinion.
1. Cap
2. Housing
3,4,5. Friction plates
6,7. Sun gears
8. Differential gears
Figure 5: Layout of Torsen Differential T-2
3.3.
This design alternative is an improvement of alternative 1. The difference is the addition of friction
plates between gears 6 and 7. Moreover, friction discs are also be placed between the gears and
the housing. The addition of the friction discs will not increase the overall weight of the differential
by a significant amount.
3.4.
The final alternative for the design is an improvement on the friction that can be generated inside
the differential. For this, the addition of spring loaded fiction plates that will be placed between
gears “6, 7”. This arrangement is represented by Figure 3. This upgraded Torsen differential is
29
known as the T-2R because of its performance in racing vehicles. This differential setup is capable
of achieving the highest TBR of the three design alternatives. However, its disadvantage is the
complexity of assembly, initial cost, and serviceability. Because it is equipped with springs and
several friction plates that wear out, the cost of doing service to the differential far exceeds the
cost of the previous two mentioned. Another downside is the added weight of the extra
components. Figure 6 shows the components and set-up of the T-2R.
Figure 6: Layout of Torsen Differential T-2R
4. Project Management
4.1.
Overview
To successfully complete this project, the team needed to delegate certain tasks to each member
of the group. Though one person was given a specific task, they were able to rely on their team
members when they needed help.
4.2.
Breakdown of Work into Specific Tasks
The division of responsibilities among team members was as follows:
30
Shivana Mahes was responsible for Solidworks modeling, assembly, and simulation, calculations
and design of housing for differential and various spacers, and written documents, including those
to submit to possible sponsors.
Uber Mantovani was responsible for Solidworks modeling, assembly, and simulation, and
calculations and design of sprocket, friction plates, and springs.
Alessandro Tasca was responsible for calculations and design of gears, and prototyping and
manufacturing.
As a team, everyone contributed to the research and the search for sponsors.
4.3.
Gantt Chart for the Organization of Work and Timeline
Dec. 2015
Nov. 2015
Oct. 2015
Sept. 2015
Aug. 2015
Jul. 2015
Jun. 2015
May 2015
Apr. 2015
Mar. 2015
Feb. 2015
Jan. 2015
Table 1: Project Gantt Chart
Research
Design, Analysis & Simulation
Prototyping
Manufacturing
Assembly
Testing
Presentation and Report Preparation
4.4.
Breakdown of Responsibilities Among Team Members
Shivana Mahes-written reports (major task), Solidworks modeling, assembly, and simulation
(support).
Uber Mantovani- Solidworks modeling, assembly, and simulation (major task), calculations and
design (support).
Alessandro Tasca- calculation and design of gears (major task), prototyping and manufacturing
(support).
31
5. Engineering Design and Analysis
5.1.
Overview
To begin designing the T2 system, the team began with a kinematic analysis of the FIU SAE car.
Once a design was planned, force and stress analysis was performed to ensure that the system
could withstand the required forces and stress. From here, the team chose a material for the
differential. Once a material was chose, the team performed static and fatigue failure calculations,
and deflection analysis. After, the team designed and selected the necessary components to
complete the design.
5.2.
Kinematic Analysis
When a particle moves on a curved path, the equations of motion are written in terms of tangential,
normal, and binormal directions [14]. Since the particle must move in a path, there is no motion in
the binormal direction. The team chose to model the FIU SAE car as a particle with motion in two
dimensions, since the car will not leave the road. The analysis was done under the most extreme
condition for the differential, which would be the skidpad test.
The angular velocity of an object moving on a curved path can be described as how fast the radian
measure of the angle of the car is changing as it moves on the circular path [14]. Radians is always
used as the unit of measure for angular velocity. There are other units used to describe angular
velocity that give a better understanding of how fast the angle is changing, like revolutions per
minute or degrees per second [14].
To begin designing, the team needed to ascertain the linear and angular velocity of the gearing
system. Using information from FIU SAE, the team calculated the necessary values. The tire radius
(RT) was given as 0.75 feet, the car width (W) given was 4.5 feet, and the tire width (WT) was 0.75
feet. Also, the team needed the pertinent radii for the turns the car would be making on the skid
32
pad course. The inner radius of the turn (IR) was 25 feet, the midpoint radius (MR) was 27.25 feet,
and the outer radius (OR) of the turns were found to be 29.5 feet. This information was used to
find the linear and angular velocity of the car at each radius on the turn. Table 2 shows the values
that were given to the team from the FIU SAE team.
Table 2: Values Given From FIU SAE
The team assumed a linear velocity of 20 miles per hour at the middle of the turn. Therefore, the
linear velocity of the car at the inner radius was found by VL,IR =
IR*VL,M
MR
. This gave the team a linear
velocity of 18.3 miles per hour or 1614. 68 feet per minute. For the linear velocity in the outer
radius, the team used VL,OR =
OR*VL,M
MR
, which gave a linear velocity at the outer radius of 21.6 miles
per hour or 1905.3 feet per minute. Table 3 summarizes the calculated values for the linear
velocities.
Table 3: Calculated Linear Velocities in Turn
The angular velocity of the sun gear was found by dividing the linear velocity in feet per minute
by the radius of the sun gear. Given that tire radius of 0.75 feet, the angular velocity of the tire at
the inner radius was found to be 2152.9 radians per minute, the angular velocity at the middle
33
radius was found to be 2346.67 radians per minute, and at the outer radius 2540.4 radians per
minute. Table 4 summarizes the calculated values for angular velocity in the turn.
Table 4: Calculated Angular Velocities
From here, the team calculated the tire velocity with respect to the differential. The angular
velocity of the tire with respect to the differential was found by subtracting the outer angular
velocity by the angular velocity in the middle of the turn. The velocity of the tire with respect to
the differential at the outer radius of the turn was found to be 193.8 radians per minute. The inner
angular velocity was found similarly, and the team obtained a value of -193.8 radians per minute.
Table 5 summarizes the calculated velocity with respect to the differential.
Table 5: Calculated Velocity With Respect to the Differential
5.3.
Structural Design
Once the team determined the velocities the car would have in turns, they began calculating the
geometry of the gears. To begin designing the gears, several parameters had to be assumed. The
team assumed values for the normal pressure angle (ΦN), helix angle (ψ) , pitch diameter of the
sun gear (ds), pitch diameter of the planet gear (dp), transverse diametral pitch (PT), spline diameter,
and face width (F). The face width of the planet and sun were assumed to be equal. Table 6 shows
34
the assumed values for the various angles used for the gear geometry calculations. Initially, the
team selected other values for these angles. After speaking with the gear manufacturer, Southern
Gears, the team had to adjust these values. The team also had to adjust these values when it was
found that the angles and geometry would result in undercut teeth in the smaller gear.
Table 6: Assumed Angles Used in Calculating Gear Geometry
tan(Φ )
The transverse pressure angle (ΦT) was calculated by ΦT =arctan ( cos(Ψ)N ). The number of teeth on
the sun gear was calculated by multiplying the sun pitch diameter by the transverse diametral pitch.
The number of teeth on the planet was similarly found by multiplying the planet pitch diameter by
π
the transverse diametral pitch. The transverse circular pitch was found by pt = P . The normal
T
p
t
circular pitch was found by Pn =pt *cosΨ. The axial pitch was found by px = sinΨ
. The normal
P
t
diametral pitch was calculated by Pn = cosΨ
. The transverse diametral pitch was calculated from
teeth on sun
Pt =Pn cosΨ. The gear ratio is found by m= teeth on planet. For full-depth teeth, the addendum is
1
calculated by a= P , while the dedendum is found by d=
n
1.25
Pn
. The clearance is c=d-a. For stub teeth,
0.8
1
n
n
the formulas are similar. The addendum is given by a= P , the dedendum is d= P . The tooth height
is found by adding the addendum and dedendum. The sun root diameter was found by
droot,s =(ds -2)*d. The tooth height was found by subtracting the spline diameter from the sun root
diameter. The planet root diameter was found by droot, p =(dp -2)*d. The complete diameter of the
planet was found by adding the planet pitch diameter and the addendum. The complete diameter
35
of the sun was found by adding the sun pitch diameter and the addendum. The center to center
distance was found by adding the planet pitch diameter and the sun pitch diameter and dividing
this value by two. Table 7 summarizes the calculated values for the gear geometry.
Table 7: Calculated Gear Geometry Parameters
Other values that needed to be calculated were the gear ratio and face contact ratio. The face
contact ratio was needed to verify that the gearing system was a standard gearing system and not
a low contact ratio helical gear (LACR). In order to calculate the safety factor for the gear sets, the
team first needed to calculate various modifying factors for the gears. The overload factor (KO)
was taken to be 1, since the system would not experience an applied load that would be in excess
of the nominal tangential load. The Temperature factor (KT) was taken to be 1, since the
36
temperature in the differential was not expected to exceed 250°F (120°C). The reliability factor
(KR) was taken to be 1, since the team did not account for the effect of the statistical distribution
of material fatigue failures in this step. The hardness ratio factor (CH) was taken to be 1, because
both the pinion and the gear would be made from the same material. Stress-cycle factors (YN and
ZN) are taken to be 1, since the team assumed the life of the differential would be 10 7 cycles. The
dynamic factor (KV) was found by the formula
A+√V
KV= (
)
A
B
, since the team’s velocity was calculated in feet per minute. With a quality number (QV) of 5, B
was found from
B=0.25(12-QV )
2⁄
3
, and A was found from
A=50+56(1-B)
This gave the team a velocity factor of 1.20. Using the material properties of Aluminum 6061
T6, steel, Aluminum 7075 T6, and titanium, the elastic coefficient was calculated by
1⁄
2
CP = [
1
]
1-ν2 1-ν2
π( P + G )
EP EG
The load distribution factor (Km) was found from
K m =1+Cmc (Cpf Cpm +Cma Ce )
where Cmc is 1 for uncrowned teeth,
37
Cpf =
F
-0.0375+0.0125F
10d
Cpm is 1, Ce is 1 for gearing that is not adjusted at assembly, or compatibility is improved by lapping
or both. The reliability factor (Kr) was taken to be 1, since the team used a reliability of 0.99. The
rim-thickness factor (KB) was found from
K B =1.6ln
2.242
mB
t
where mB = hR. In this equation, t R is the rim thickness and ht is the tooth height.
t
To calculate these factors, the team first needed to obtain the physical constants of the materials
they were considering for the design. Table 8 shows the modulus of elasticity, Poisson’s ratio,
modulus of rigidity, and Brinell Hardness for the materials the team considered for the gearing
system inside the differential.
Table 8: Physical Constants of Materials Being Considered for Design
38
Table 9 shows the variables used to calculate the dynamic factor and the resulting KV of 1.058.
Table 9: Calculated Dynamic Factor
The elastic coefficient was then calculated for the materials being considered for the gearing
system. Table 10 shows the calculated elastic coefficient for the materials.
Table 10: Calculated Elastic Coefficient for Each Material
The rim-thickness factor was calculated next. Table 11 shows all the variables used to calculate
the rim-thickness factor, along with the final value for the rim-thickness factor of 1.070.
Table 11: Factors Used in Rim-Thickness Factor Calculation
For steel 4340 nitrided, the Brinell Hardness was 432. The allowable bending stress number was
found by σc, all =82.3HB +12150. The allowable contact stress was found to be 150000 psi, and
the allowable bending stress was 47703.6 psi.
After finding these variables, the team needed to calculate the bending safety factor and gear wear
safety factor. The team chose to start with gear bending. The tangential force found in the force
analysis was used. Several of the modifying factors found earlier was also used, including Ko , K v ,
39
Ks , Km , and K B . The J factor utilizes a modified value of the Lewis form factor. However, the team
utilized graphs to determine the J factor for the gearing set. Using the AGMA standard, the team
found a J factor of 0.58. The stress-cycle factors, YN and ZN, were assumed to be one earlier. KT
was taken to be one since the operating of the differential is assumed to stay below 250° F. KR,
the reliability factor, was taken to be one. To determine the gear bending stress, the team used the
formula σ=Wt Ko Kv Ks
Pd Km KB
F
J
. This produced a value of 27365.68 psi. The team then calculated
the gear bending endurance strength. The formula used was
σall =
St YN
SF KT KR
This produced a value of 23851.80 psi. The bending safety factor was calculated from
SF =
St
YN
⁄(K K )
T R
σ
A value of 1.7 was obtained, which the team was satisfied with for the gearing system.
The wear factor of safety was found next. For this equation, variables found earlier needed to be
used. Those values were Cp , Wt , Ko , Kv , Ks , Km , and Cf . I, the pitting-resistance geometry factor
by the AGMA, was found from graphs. The value used was 0.195. The formula used to calculate
the gear contact stress was
1⁄
2
Km Cf
σc =Cp (Wt Ko Kv Ks
)
dp F I
This resulted in a value of 73797.36 psi. The gear contact endurance strength was found from
σc, all =
Sc ZN CH
SH KT KR
This was found to be 100000 psi. The wear factor of safety was found from
40
SH =
Sc ZN
CH
⁄(K K )
T R
σc
This resulted in a wear safety factor of 2.03, which the team felt was sufficient for the differential.
After calculating these values, the team had to calculate internal friction inside the differential and
frictional torque. From these numbers, the team could determine if the differential would lock
under the appropriate conditions.
5.4.
Force Analysis
The team performed force analysis based on information provided by the FIU SAE team. The
coefficient of friction between the wheel and the road was previously found to be 1.13. The car
weight was 415 pounds, and the weight of the person driving was assumed to be 190 pounds. This
gave a total weight of the car and driver of 605 pounds. The maximum torque that the wheels can
support before slipping was found by adding the 50/50 weight distribution, longitudinal weight
transfer, and weight added by developing and multiplying that number by the coefficient of friction
and the wheel radius. The longitudinal weight transfer is the amount of weight transferred during
a turn, and the weight added by developing is an estimate used to account for any changes made
to the car design by the SAE team, like the addition of a wing. Table 12 summarizes the assumed
and given values used for the force analysis.
41
Table 12: Given Values Used for Force Analysis
This gave the team a maximum torque before slippage of 328 foot pounds. The torque on the
sprocket was assumed to be the maximum torque before slippage. Using an applied torque of 350
foot pounds, the team found the force on the planet gears from the housing to be 1039 pounds. The
formula used to calculate this was
Fplanet gears, housing
Tmax
DP,sun *(
DP,planet +DP, sun
)
2
Table 13 summarizes the values the team calculated for the forced on the differential housing.
Table 13: Force Analysis on the Differential Housing
From here, the team calculated the force on a set of planetary gears inside the differential housing.
The radial component of the force (WR) was found by multiplying the total force (W) by the sine
of the normal pressure angle (φn). The tangential component (WT) was found by multiplying W by
42
the cosine of φn and the cosine of the helix angle (ψ). The axial component (WA) was found by
multiplying W by the cosine of φn and the sine of ψ. From these formulas, the total force was 1039
pound force, the radial component was 439 pound force, the tangential component was 885 pound
force, and the axial component was 322 pound force. Table 14 summarizes the calculated values
for the force on a pair of planet gears.
Table 14: Calculated Force per Planet Pair
The total force on the housing is four times the total force on each planet pair, since there are four
planet pairs in the design. Therefore, the total force was found to be 4154 pounds. Similarly, the
tangential component, radial component, and axial component were found as described above. The
total force was found to be 4154 pound force, with a distribution of 2077 pound force on the right
and left side of the differential. The radial component was found to be 1756 pound force, with a
distribution of 878 pound force on each side of the differential. The tangential component was
found to be 3538 pound force, with a distribution of 1769 pound force on each side of the
differential. Finally, the axial component of the total force was found to be 1288 pound force, with
a distribution of 644 pound force on each side of the differential. Table 15 shows the calculated
values for forces on the complete gearing system inside the differential.
43
Table 15: Calculated Forces on the Complete Gearing System
Next, the team calculated the friction and frictional torque the differential would experience. The
team began by obtaining a theoretical coefficient of friction of 0.31. Based on the results of the
calculations using this coefficient of friction, the team could determine if the friction is suitable to
make the differential perform correctly. The total friction on the planetary gears from the housing
was found to be 544 pounds when the total force was multiplied by the coefficient of friction. This
results in a 272 pound force being distributed to both sides of the differential. The radial component
of friction exerted on the planetary gear by the housing was found by multiplying the radial
component of total force by the coefficient of friction. This gave the team a force of 1288 pounds,
and a distributed force of 644 pounds to each side of the differential. The axial frictional force was
found to be 399 pounds, which is a distributed force of 200 pounds on each side of the differential.
The calculated values are presented in Table 16.
Table 16: Frictional Force Analysis
44
The frictional torque at the sun gear due to the planetary gear was found by adding the friction on
the planetary gears from the housing and the radial component of friction on the planetary gear by
the housing and multiplying that number by the pitch diameter of the sun gear. The team obtained
a value of 53 pounds on each side of the differential. The frictional torque due to friction of the
sun gear was found by multiplying the axial component of the friction exerted on the sun gear by
the housing by the difference between the root diameter of the sun and the spline diameter. The
team obtained a value of 112 pounds on each side of the differential. The total frictional torque at
the sun gear was found by adding the two torques together, giving the team 165 pounds at the right
and left sides of the differential. The calculated values are presented in Table 17.
Table 17: Frictional Torque at the Sun Gear
The team then calculated the torque per side of the differential. To calculate the torque on the sun
gear, the team used the following formula
Tsun gear =
WT, complete *ds
2
Similarly, the torque on the planetary gear on one side of the differential was found from
Tplanet gear =
WT, complete *dp
2
From these formulas, the torque per side on the sun gear was found to be 102 foot pounds, and the
torque per side on the planetary gear was found to be 47 foot pounds. Table 18 shows the calculated
torque per side of the differential on each type of gear.
45
Table 18: Calculated Torque Per Side of the Differential
The torque at the wheels during a right turn on the skid pad were found by multiplying the weight
at the right and left wheel by the coefficient of friction between the tire and road. At the right
wheel, the team found a torque at the wheel of 184 foot pounds, and 192 foot pounds at the left
wheel. Table 19 summarizes the values the team calculated for the torque at the wheels in a right
turn.
Table 19: Calculated Torque at the Wheels in a Right Turn
Next, the team had to determine if the differential would perform correctly under the required
conditions. When the vehicle is turning, the sum of the torques at both wheels should be positive.
The team determined this by using a right turn. The team used the following formula to calculate
the torque in the right wheel during a right turn
Tright =|Tsun -Twheel |-Tsun, frictional, right
This gave a torque at the right wheel of -83 foot pounds. For the left wheel, the team used the
formula
Tleft =|Tsun +Twheel |-Tsun, frictional, left
This gave the team a value of 129 foot pounds at the left wheel during a right turn. The team could
conclude that the differential performed correctly, since the sum of the two torques were a positive
number, which would result in forward motion. Therefore, the team determined that the friction
46
coefficient and applied torques were sufficient to make the differential work. Table 20 shows the
calculated values for the summation of torques at the right and left wheel during a turn.
Table 20: Summation of Torques
Figure 7 shows the final design for the gearing system.
Figure 7: Gearing System, Final Design
Finally, the team needed to calculate the pressure on the bearings that would seal the differential.
To do that, the team used the formula
p=
δ
d d2o +d2
d d2 +d2i
(
+v
)
+
o
Eo d2o -d2
Ei ( d2 -d2i -vi )
47
where δ is interference between the shaft and hub, d is nominal diameter, E is modulus of elasticity,
and ν is Poisson’s ratio. In this formula, the o subscript represents the shaft, and the i subscript
represents the hub. The shaft is the housing that is aluminum 6061, and the hub is the bearing that
is stainless steel. The team utilized metric units for this calculation, since those units were easier
to work with in this case. At the end of the calculation, the conversion was made to English units.
This resulted in a pressure of 8887.40 kpsi.
5.5.
Component Selection
There were several components the team had to design or purchase in order to integrate the
differential into the SAE car. First, the team purchased a new CV joint. The team chose a new CV
joint because the team needed a longer shaft spline than the current CV joint provided. The added
benefit to this new CV joint was a reduced axle angle. This meant more power would be
transmitted from the differential to the wheels. The team purchased the CV joint from a company
that produces motorcycle parts.
The team also needed to design new differential brackets to mount the differential to the chassis
of the car. The team chose to modify the existing brackets, only changing the size to suit the new
differential.
The team also needed to purchase a sprocket to transfer the power from the engine, through a
chain, to the differential.
5.6.
Material Selection
For the gearing system, the manufacturer suggested steel 4340 nitrided. Therefore, the team
included that material in their calculations to determine if it was a suitable material for the system.
After completing the calculations with steel 4340 nitrided, aluminum 6061 T6, aluminum 7075
48
T6, and titanium alloy, the team was able to conclude that the steel 4340 nitrided was the best
material for the gearing system.
For the housing, the team chose aluminum 7075 T6 for its light weight and strength. The team
chose this material with the wear of the housing in mind.
For the differential brackets, that mount the differential to the chassis of the car, the team chose
aluminum 6061, because it is easy to manufacture and cheap. It is also strong and not very brittle.
5.7.
Design Modifications
Once the team had a final design, they consulted with the gear manufacturer, Southern Gears. The
team originally had a value of 1.5 inches for the pitch diameter of the gears. The shaft of the CV
joint gave the team the rim thickness for the gear. However, after speaking with the manufacturer,
the team had to adjust the normal diametral pitch of the gears, since no cutter was available for the
team’s original design. The team changed the design based on the available cutters that was
provided by the manufacturer.
The team also had to adjust the helix angle and normal pressure angle in order to achieve the
necessary friction, a high safety factor, and the reduce undercut in the teeth. The team chose a face
width that resulted in a good safety factor, but also had the constraint of needing the new
differential to fit into the same space as the old differential. Therefore, the team could not infinitely
expand the face width.
The purpose of the planetary gears is to transmit power from the sun gear and between each planet
gear. Therefore, the face width of the planetary gears are larger than the sun gear to produce a
larger contact area between planetary gears. This was done to transmit maximum torque. The
49
contact area between sun and planet gears do not need to be as large as the planet to planet contact
area, since the planet gears only transmits between planets during differentiation.
5.8.
Housing Design
The team then focused on the design of the differential housing. The internal geometry of the
housing was chosen to withstand the radial force component of the gears. Since there should be no
movement in the sun gear, the radial force component of the forces in the gears should cancel. The
material for the housing was chosen to be aluminum 7075 T6, for its strength. The team was
concerned about wear on the housing.
Figure 8 shows the final assembly with the housing and lids.
Figure 8: Complete Assembly
50
Differential
Sprocket
Figure 9: Complete Assembly of Differential and Sprocket
5.9.
Simulations
To verify the calculated values, the team ran simulations in Solidworks. First, simulations were
run on the housing of the differential. For the housing, a force of 353.18 pounds was applied in
each bolt hole, this led to a total force of 3531.8 pounds. The team used this amount, since the total
tangential force from the gearing system was found to be 3538 pounds. Also, the team applied a
distributed force of 1756 pounds to the holes for the gears. This was found from the radial
component of force found in the force analysis.
Figure 10 shows the results of the simulation. The yield strength of the material is 5.050 *108 Pa,
while the maximum Von Mises stress on the housing is 7.534*107 Pa.
51
Figure 10: Simulation Results for Differential Housing
52
The team also conducted a safety factor analysis in Solidworks using the same parameters as
above. Figure 11 shows the results of the safety factor analysis. The minimum factor of safety was
6.7.
Figure 11: Safety Factor Analysis in Solidworks
For the differential brackets, the team had to conduct a static analysis in order to find the forces to
apply to the part in Solidworks. Figure 12 shows the free body diagram the team formulated to
calculate the force on the bracket. The force from the chain was found to be 983 pounds. Using
the diagram, the team calculated the following: FA +FB =983 lbf. The moment about point A was
found.
MA =983 lbf* (
1.213"
5.300"+1.213"
) -FB (
)
12"
12"
FB =
9936
=183 lbf
0.5428
53
983 lbf
FB
FA
A
1.213”
5.300”
Figure 12: Free Body Diagram of Differential Bracket
The team found that the force at point A should be 183 lbf. This was the value used to conduct
the simulation in Solidworks. Figure 13 shows the results of the Solidworks simulation.
Figure 13: Simulation Results for Differential Bracket
The yield strength is 2.75*108 Pa, while the highest Von Mises stress on the bracket is 8.529*107
Pa.
54
For the lid, the team used the axial force found in the complete force analysis. Figure 14 shows
the results of the simulation. Here, the yield strength of the material is 39.9 ksi, but the maximum
stress on the lid is 21.4 ksi, well within the strength of the material.
Figure 14: Simulation Results for Housing Lid
Next, the team used Solidworks to calculate the factor of safety for the housing lids. Figure 15
shows the resulting safety factors. The minimum safety factor is 1.86, which told the team that
the lids will not fail with the applied axial load from the gearing system.
55
Figure 15: Factor of Safety
5.10.
Manufacturing
The team was able to procure sponsorship for the manufacturing of the gearing system. Southern
Gears, who works with the FIU SAE team, agreed to manufacture the gears. However, since the
gears were more complex, the team did need to begin the machining process in the FIU SAE
machine shop. The team created gear blanks that had the basic shape of the gears, without the
teeth. These blanks were then sent to Southern Gears for them to machine the teeth.
Figure 16 shows the eight planetary gear blanks that the team machined in the FIU SAE machine
shop. Equipment used to machine the blanks included the horizontal band saw and lathe. Once
the team completed the gear blanks, they were sent to Southern Gears, who machined the teeth
onto the blanks.
56
Figure 16: Gear Blanks Machined by Team
For the housing, the team had a more complicated time. The team sent out for several quotes at
local machine shops, including Mr. Zicarrelli’s company. The team was told by Mr. Zicarrelli
that the housing might not be able to be machined by CNC machines. The team would need to
look into electrical discharge machining (EDM). The team then allowed Mr. Zicarrelli to
forward the housing design to an EDM shop he was familiar with.
6. Design Experience
6.1.
Standards Used in the Project
The team used various standards for this project. When selecting materials, the team referred to
the American Society for Testing and Materials (ASTM). ASTM A291/A291M-05(2010) is titled
“Standard Specification for Steel Forgings, Carbon and Alloy, for Pinions, Gears and Shafts for
Reduction Gears.” This standard covers normalized and tempered carbon steel, liquid quenched,
and tempered alloy steel forgings for use in pinions, gears, and shafts. ASTM B427-09 details the
standards for bronze alloy castings used in gears. ASTM D7450-13 deals with the testing and
57
acceptable criteria for gear oils and lubricants. These are a sampling of the ASTM standards the
team must follow. Once the team determined a suitable material for the differential and all the
components, the ASTM standards had to be consulted to make sure a suitable material was being
used.
The American Gear Manufacturers’ Association (AGMA) has set gearing standards since 1916. It
began with the goal of creating standards for non-metallic gears. Their standards address all critical
topics, like load capacity and lubrication to accuracy and inspection. These standards apply to
gears, couplings, and power transmission components. It includes top gearing experts from all over
the world. Therefore, in order for the differential design to be acceptable by engineering standards,
the AGMA standards must be incorporated into the design.
The Society of Automotive Engineers (SAE) standards will also be utilized if necessary. These
standards deal with oils and lubricants, specifically their viscosity [15].
6.2.
Cost
The cost of this project was broken down into three categories, raw materials, parts, and
manufacturing. Raw materials were the aluminum and steel used to manufacture the differential,
housing, and brackets. The parts included the PLA plastic used in rapid prototyping the system,
the purchased sprocket, and parts used in testing, like the pressure gauges. Manufacturing costs
were those incurred from sending the parts out to vendors to be manufactured. Some vendors used
for this project were Southern Gear and Machining, CNC Technology, and Braddock Metallurgic.
58
Table 21: Cost of Raw Materials
Cost of Raw Materials
Material
Aluminum Plates
4340 Steel
4130 Steel and
Aluminum
Total
45.26
10.23
178.07
Table 22: Cost of Parts
COST OF PARTS
Part
Total
Oil seal
8.56
CV joint
165.99
PLA plastic
43.43
Sprocket
33.1
Ball bearings
70.31
Lathe cutters
11.21
Bolts
32.25
Oil
19.24
Washers
8.03
Pressure gauge 36.36
Fitting
9.62
Magnets
9.98
Restocking fee
30
59
Table 23: Cost of Manufacturing
COST OF MANUFACTURING
Component
Housing
Planetary
gears
Sun gears
Shafts
Heat
Treatment
Brackets
Vendor
CNC Technology
Total
350
Southern Gears
Southern Gears
Southern Gears
Braddock
Metallurgic
Mr. Zicarelli
1859
1683
957
150
195
Tables 21, 22, and 23 show the breakdown of all the costs of this project. The team was able to
have several sponsors who donated most of the manufacturing costs. Also, since the team worked
in conjunction with the FIU SAE team, some of their funding went into the team’s project. Table
24 shows the total cost for the project, excluding sponsor donated items.
Table 24: Total Cost of Project
TOTAL COST
Parts
478.08
Raw materials
379.52
Manufacturing
5194
TOTAL
$6,051.60
7. Testing and Evaluation
7.1.
Overview
The goal of testing the differential was to see how the differential would perform in various
scenarios:
1. When both rear wheels have an equal amount of grip and the car is going straight,
2. When both rear wheel have an equal amount of grip and the car is going straight,
3. When one of the rear wheels has grip and one does not.
60
7.1.1. Mechanical Components and Variables
The engine of the car generates the power and torque that goes to the engine. The engine can
generates different amounts of horsepower and torque depending on the RPM it is putting out. The
RPM is controlled by the driver thought the accelerator. Therefore, the team needed to understand
how the differential behaved with changes in RPM.
The RPMs of the engine was one of the controlled variables during testing. Therefore, the team
conducted testing on several ranges of the engine’s RPM. In order to do this, a holding mechanism
needed to be attached to the car’s accelerator, so the RPMs of the engine was known. The RPM of
the engine was read from the car’s tachometer.
The transmission is a critical component of the powertrain of the car. It transfers power from the
engine to wheels. The transmission uses different gear ratios to determine or alter the relation
between the speed of the car’s engine and the speed of the wheels. In order for the team to be sure
of the angular velocity and torque going to the differential, the team fixed the gear of the
transmission.
The differential is the component under test, therefore the team needed to find out how it
differentiates speed and torque going to the rear wheels in each scenario. Therefore, the angular
velocity coming from the engine was fixed. The team used an angular speed sensor on each rear
wheel to find the angular velocity at each wheel. The two angular velocities at the wheel are the
dependent variable. The team gathered data for each scenario to see how the differential performed.
Once the ratio of angular speeds into the differential and out to the wheels is known, a
mathematical relation can be used to find the percentage of torque that is delivered to each wheel.
61
Since the tests were performed on a static car for safety, the brakes were essential in conducting
the tests. The team needed to control the amount of grip each rear tire was receiving by using the
brakes to simulate driving conditions. This made the differential behave as it would under driving
conditions.
To control the brakes, the front brakes were disconnected, so that the valve that regulates the
distribution of hydraulic pressure in the rare and front brakes were used to regulate the rear brakes
alone. Then, a hydraulic pressure gage was attached to the main system to find the hydraulic
pressure on each wheel. The braking torque on each wheel was an independent variable that was
fixed in order to find out how the differential will behave. Also, a similar mechanism was attached
to the braking pedal to fix the amount of braking.
The team began the mathematical calculations at the engine, since it was performing at a fixed
angular velocity. The transmission gear ratio is also known. So, the team was able to find the
angular velocity of the chain connecting the transmission and sprocket with the formula
ωchain =Rt *ωengine
where, ωchain is the angular velocity of the chain in radians per second, Rt is the transmission gear
ratio, and ωengine is the angular velocity of the engine in radians per second.
Since the angular velocity of the chain is known, the team assumed that the angular velocity into
the differential is equal to the angular velocity of the chain.
Power was another important relationship in the powertrain. The team measured how much speed
the differential sent to each wheel to see how torque was being differentiated. The team used the
following formulas
62
P=Tin *ωin
P=Tout *ωout
Therefore,
Tin *ωin =Tout *ωout
Tout =Tin *
ωin
ωout
In these formulas, Tin is the torque coming into the differential, Tout is the torque leaving the
differential, ωin is the angular velocity into the differential, and ωout is the angular velocity out of
the differential.
The team assumed the torque into the differential was 1. This allowed the team to use this formula
Tout =Tin
ωin
engine RPM*gear ratio
=1*
ωout
measured wheel speed
Finally, the team needed to calculate braking pressure. Since braking pressure would be used to
simulate grip in a tire, this was an important consideration. The team had to conduct the testing
with the wheels of the car off the ground, so the pressure applied by the brakes simulated the grip
in a turn. When brakes are applied, a different amount of torque is applied to each wheel. This was
used to accurately simulate the car going into a turn. By sending most of the braking power to one
wheel, a loss of grip is simulated.
The design of the differential was such that when there is a loss of traction in one wheel, the
differential should lock itself and send an equal amount of angular velocity and torque to both
63
wheels. In order to conduct the tests, the team needed to know how much braking torque was being
sent to each wheel. The team used the formula
Fclamp =P*Acaliper
where Fclamp is the clamping force of the brake, P is pressure, and Acaliper is the area of the brake
caliper.
In the above equation, the clamping force was calculated by multiplying pressure obtained from
the pressure gauge connected to the brake system of the car by the known area of the caliper.
Once the clamping force was found, the team used the equation
Tbrake =Fclamp *Reffective
where Tbrake is the braking torque exerted on the wheel, Fclamp is the clamping force on the brake,
and Reffective is the effective radius of the brake rotor.
7.1.2.
Design of Experiments
In scenario one, the team wanted to find the approximate losses on the differential. To achieve
that, the team needed to simulate the car on a straight path. Therefore, no braking torque was
applied to the wheels. The differential was expected to send half of the output speed and torque to
both wheels. To verify this, the team needed to check how much velocity and torque was going
into and coming out of the wheels.
In scenario two, the team needed to test how the differential would behave in a turn. In this test,
the team expected the differential to send different amounts of torque and velocity to both wheels.
The team needed to test the differential under different RPMs and braking torque on each wheel.
64
In scenario three, the team needed to teat if the differential locked itself when there was a major
difference in torque between the two rear wheels. This test utilized a similar setup as scenario two,
except more extreme braking torques were used in this test.
7.2.
Test Results and Data
From the bench test, the team was able to determine if the differential did distribute torque to each
wheel under different conditions. The first test performed determined how torque would be
distributed if both wheels received the same amount of torque, or if they have the same amount of
traction. Each brake caliper was given 50 psi of pressure with the engine providing the power. The
results are summarized in Figure 17.
12.00
10.00
8.00
6.00
4.00
2.00
0.00
-2.00
Channel 0
Channel 1
0
0.0276
0.0552
0.0828
0.1104
0.138
0.1656
0.1932
0.2208
0.2484
0.276
0.3036
0.3312
Voltage
50 psi Braking Pressure on Each Wheel
Time (s)
Figure 17: Results of 50 psi Braking Pressure on Each Wheel
To find the amount of revolutions per minute of each wheel, the team used the following formula:
RPM=
60
Period
where the period was found from the MyDaq software. This gave an angular velocity of 394
revolutions per minute in the left wheel, and 398 revolutions per minute in the right wheel. This
was expected, since when both wheels have the same amount of traction, they should receive the
same amount of torque.
65
For the second case, the team tested when one wheel has more traction that the other to see how
the differential would distribute the torque. In this case, one brake caliper was given 200 psi of
braking pressure, while the other was given only 100 psi of pressure. The results of this testing is
summarized in Figure 18.
200 - 100 psi Braking Pressure
12
10
Voltage
8
6
Channel 0
4
Channel 1
2
-2
0.1984
0.232
0.2656
0.2992
0.3328
0.3664
0.4
0.4336
0.4672
0.5008
0.5344
0.568
0.6016
0.6352
0.6688
0.7024
0.736
0.7696
0
Time (s)
Figure 18: Results of 200-100 psi Braking Pressure in Wheels
For this test, the team calculated the RPMs to each wheel. Under this scenario, one wheel had an
RPM of 139, while the other had an RPM of 172. Since the values were different, the team was
able to confirm that the differential successfully distributed the torque to each wheel.
In the final test that the team was able to conduct, a larger difference in traction was simulated.
One wheel received a braking pressure of 240 psi, while the other only received 150 psi. This
was done to see how the greater difference in traction would affect the differential. Figure 19
summarizes these findings.
66
240 - 150 psi Braking Pressure
12
10
Voltage
8
6
channel 0
4
channel 1
2
-2
0
0.052
0.104
0.156
0.208
0.26
0.312
0.364
0.416
0.468
0.52
0.572
0.624
0
Time (s)
Figure 19: Results of 240-150 psi Braking Pressure
The team found that one wheel had an angular velocity of 149 RPM, while the other was 172 RPM.
Again, the differential was able to distribute more torque to one wheel.
The final test that the team conducted was to find the difference in RPM that would cause the
differential to lock. To do this, one wheel received all the braking pressure, while one wheel
received none. This simulated a situation where one wheel had a total loss of traction. The engine
of the FIU SAE car was revved to its maximum RPM. However, the locking effect, where the
differential locks and both wheels receive the same amount of torque, was not seen.
The team tested the ability of the differential to withstand the 45 degree tilt test. The differential
was filled with oil to lubricate the gears. When the differential was turned 45 degrees, the
differential did not leak, which satisfies the tilt test.
The old stock differential weighed 15.65 pounds when assembled with brackets. The newly
designed Torsen differential weighed 12.35 pounds. This resulted in a weight reduction of 3.3
pounds or about 21.09%.
67
The backlash on the old stock differential was 8 degrees. The newly designed differential had a
backlash of 3.5 degrees. This was a difference of 4.5 degrees or about a 56.25% reduction in the
backlash.
Finally, the team measured the CV joint angle between the ole setup and the new setup. The team
found that the old CV joint angle was 13.5 degrees. The new design reduced the CV joint angle to
4.5 degrees, a reduction of 66.7%. This led to better power transmission through the system.
7.3.
Evaluation of Experimental Results
On the bench testing, the team was able to see the differential action, which resulted in different
RPMs on each wheel when they experience different traction forces. However, the team did not
see the locking effect, which is an important characteristic of this type of differential.
The team was able to reduce the weight of the differential by 21.09%, which is more than the goal
of 15% the team set at the beginning of the semester. The team was also able to reduce the degree
of backlash by more than 50%, this improved the amount of tooth damage and prevents
overheating.
Once the team took apart the differential after testing, there was an unusual wearing of the lid. This
could be due to the gears experiencing an axial force that is greater than the one calculated.
7.4.
Improvement of Design
To improve the design, the team would recommend changing the material of the purchased
sprocket from steel to aluminum for a greater weight reduction. Weight can also be reduced by
using lighter CV joints.
68
To improve the differential action and bring about the locking effect, the team would recommend
trying a gear oil with different viscosity. Also, changing the material and size of the spacers may
increase friction inside the differential.
To reduce wearing on the lid, the team recommends exploring how much axial force is felt on the
lid further. The lid could also be manufactured from a more tough material to prevent unnecessary
wearing.
7.5.
Discussion
The team was able to accomplish the weight reduction they set out to achieve at the beginning of
this project. In addition, they were able to reduce the backlash from the current differential.
The members of the team were able to utilize the skills they learned throughout the mechanical
engineering curriculum to complete this project. In addition, the team was able to turn theories
learned into practical use. The team was able to practice extensive machining on multiple
components of the powertrain.
By doing this project, the team gained real-world experience in design, manufacturing, and
assembling. To successfully complete it, the team members needed to effectively manage time,
make informed decisions, and work together.
8. Design Considerations
8.1.
Health and Safety
Since the car is being occupied by a person and encountering relatively high speeds, the team made
sure to utilize safety factors over one. Also, as required by SAE rules, the differential is sealed.
This prevents components from becoming debris hazards if the differential were to fail. The
housing of the differential will keep the gears enclosed if something were to go wrong.
69
8.2.
Assembly and Disassembly
The team designed the differential to be easy to assemble and disassemble. To assemble the
differential, the user would have to locate the correct gears and slip them into the housing. Using
bolts, the user could tighten the lid. Bolts are also used to mount the sprocket to the housing and
to mount the differential brackets to the care body.
To disassemble the differential, the user would simply need to remove the bolts that mount the
differential to the car. Then, the user would remove the bolts that seal the lid and housing cylinder.
8.3.
Manufacturability
The differential designed was manufactured using several methods. Simple machining tools, like
the mill and lathe, were used. The team also needed more complex techniques, like EDM, for the
differential housing. Therefore, this is not an easy part to manufacture. However, should the FIU
SAE team need to replace or reproduce the gearing used in the differential, the team has provided
drawings and Solidworks files.
8.4.
Maintenance of the System
8.4.1. Regular Maintenance
For regular maintenance of the differential, the team recommends the gears be properly lubricated
before each use. The team used full synthetic 75W-90 oil for the gears. Failure to lubricate the
gears can result in excessive noise and damage to the teeth.
8.4.2. Major Maintenance
For major replacements of the differential, the team would recommend the gears be replaced when
excessive wear is seen on the teeth.
70
8.5.
Risk Assessment
The risk with this differential is relatively low. Should the differential fail, power would not be
transferred from the engine to the wheels, and the wheels would stop spinning. Therefore, the risk
of an accident from a failed differential is low. Also, should a part of the differential come loose,
the housing is capable of containing the debris until the driver can stop the car.
9. Design Experience
9.1.
Overview
The design experience for the team was a successful one. The team was able to perform the
necessary research and calculations to design a functioning differential. The team began by
conducting a thorough literature study to understand all aspects of differentials and power transfer.
Next, the team carried out methodical calculations, while applying the theoretical knowledge
learned in classes.
To design the differential, the team utilized Solidworks to perform simulations. The simulations
allowed the team to test various configurations of the parts to see how certain parameters would
be affected without having to build a prototype.
Once the team was satisfied with the Solidworks results, the components were sent to various
manufacturers. Here, the team learned how to deal with manufacturers, and even received feedback
about the design.
9.2.
Standards Used in the Project
Various standards were used in this project. Some standards used are the American Gear
Manufacturer’s Association, the American Society of Testing and Materials, and Society of
71
Automotive Engineers. The team heavily utilized the AGMA standards for calculations relating to
gear geometry. Excerpts of the standards used can be found in the appendix.
9.3.
Professional and Ethical Responsibility
Knowing that this differential would be used in a car, the team felt a need to make it absolutely
safe. Also, since the FIU SAE team contributed a portion of the funding for this project, the team
also made sure to keep track of all the money being spent on the project.
9.4.
Discussion
After completing this project, the team has learned valuable skills that could not be taught in a
classroom setting. The team was able to communicate with vendors, possible sponsors, and other
engineers regarding this project. Manufacturers and engineers would let the team know if an idea
was feasible or how to go about solving a potential problem.
From this project, the team was able to see a complete development of an idea. It began with a
simple idea and progressed into the final prototype. Along the way, the team had to make various
decisions that impacted the final product. Sometimes, the decision-making process involved
getting in touch with people who had more expertise in certain areas than the team members.
10. Conclusion
10.1.
Conclusion and Discussion
Overall, the team gained invaluable experience doing this project. Team members were able to
utilize and refine skills, like Solidworks modeling, machining, and Excel calculations. Though
these skills were taught in classes, the team was able to apply them to their own problems.
In carrying an idea from design the implementation, the team was able to experience engineering
concepts that are not easily taught in a class. The team had to make decisions based on cost and
72
time, factors that are usually overlooked in theoretical calculations. The team also worked in a
world where things are not ideal, which is what is assumed in most classes. Therefore, dimensions
were sometimes out of tolerance or gears were not perfectly made. The team had to figure out
solutions to these imperfections.
10.2.
Future Work
For future work, the team recommended further weight reduction by changing the material of the
sprocket from steel to aluminum and choosing lighter CV joints. Also, as machining technology
becomes more sophisticated, it may be possible to achieve higher gear ratios in the same amount
of space. Gear backlash can also be reduced if more precise gearing is used.
73
11. References
[1] R. Morselli, R. Zanasi and G. Sandoni, "Detailed and Reduced Dynamic Models of Passive
and Active Limited-Slip Car Differentials," Mathematic and Computer Modelling of
Dynamical Systems, vol. 132.
[2] R. Budynas, J. K. Nisbett and J. E. Shigley, Shigley's Mechanical Engineering Design,
New York: McGraw-Hill, 2011.
[3] "Electronic Limited Slip Differential," Automotive Engineer, p. 46, July-August 2010.
[4] J. Koo, C. Choi, Y. Huh and C. Seok, "Development of a Hydraulic Limited Slip
Differential System Using a Pressure Generator," Internation Journal of Automotive
Technology, pp. 323-327, 9.
[5] H. Bayrakceken, "Failure Analysis of an Automobile Differential Pinion Shaft,"
Engineering Failure Analysis, vol. 13, pp. 1422-1428, 2006.
[6] A. Sorniotti, "All Wheel Drive Components Modeling and Their Influence on Vehicle
Dynamics," International Journal of Mechanics and Control, vol. 6, no. 1, pp. 33-44, 2005.
[7] J. Deur, V. Ivanovic, M. Hancock and F. Assadian, "Modeling and Analysis of Active
DIfferential Systems," Journal of Dynamic Systems, Measurement, and Control, vol. 9, no.
3, pp. 347-362, 2006.
[8] A. Tremlett, F. Assadian, D. Purdt, N. Vaughn, A. Moore and M. Halley, "Quasi-Steady
State Linearisation of the Racing Vehcle Acceleration Envelope: A Limited Slip
Differential Example," Vehicle System Dynamics, vol. 52, no. 11, pp. 1416-1442.
[9] D. Pye, Practical Nitriding and Ferritic Nitrocarburizing, ASM International, 2003, pp. 111.
[10] S. Lee, Y. Saito, T. Sakai and H. Utsunomiya, "Microstructures and Mechanical Properties
of 6061 Aluminum Alloy Processed by Accumulative Roll-Bonding," Materials Science
and Engineering, pp. 228-235, 2002.
[11] W. Dai, S. Xue, J. Lou and S. Wang, "Development of Al-Si-Zn-Sr Filler Metals for
Brazing 6061 Aluminum Alloy," Materials and Design, pp. 395-402, 2012.
[12] A. Mukherjee, M. Ghosh, K. Mondal, P. Venkitanarayanan, M. A.P. and A. Varshney,
"Study of Mechanical Properties, Microstructures and Corrosion Behavior of Al 7075 T651
74
Alloy with Varying Strain Rate," in 4th National Conference on Processing and
Characterization of Materials, 2015.
[13] R. E. J. Sanders, "Technology Innovation in Aluminum Products," 2001. [Online].
Available: www.tms.org/pubs/journals/jom/0102/sanders-0102.html.
[14] Society of Automotive Engineers, "Formula SAE History," [Online]. Available:
http://students.sae.org/cds/formulaseries/about.htm.
[15] Society of Automotive Engineers, "2015 Formula SAE Rules," [Online]. Available:
http://students.sae.org/cds/formulaseries/rules/2015-16_fsae_rules.pdf.
[16] R. Hibbeler, Engineering Mechanics: Dynamics, Prentice Hall, 2012.
[17] Society of Automotive Engineers, "SAE International Automotive," [Online]. Available:
http://www.sae.org/automotive.
[18] D. Rubin and S. Arogeti, "Vehicle Yaw Stability Control Using Active-Limited Slip
Differential Via Model Predictive Control Methods," International Journal of Vehicles
Mechanics and Mobility: Vehicle System Dynamics, vol. 53, no. 9, pp. 1315-1330.
75
Appendices
Appendix A: Detailed Engineering Drawings of All Parts, Subsystems and Assemblies
Figure 20: Drawing for Sun Gear Blanks
76
Figure 21: Drawing for Planetary Gear Blanks
77
Figure 22: Drawing for Left Differential Bracket
78
Figure 23: Drawing for Housing Cylinder
79
Figure 24: Drawing for Left Housing Lid
80
Appendix B: Excerpts of Guidelines Used in the Project: Standards, Codes, Specifications
and Technical Regulations
Figure 25: SAE Rules for Transmission and Drive
Figure 26: SAE Rules for System Sealing
Figure 27: SAE Rules for Tilt Test
81
Figure 28: Table for I and J Factors from the AGMA
82
Figure 29: Pitting Resistance Calculation from the AGMA
83
Figure 30: Bending Strength Formulas from the AGMA
84
Figure 31: Calculation of Dynamic Factor from AGMA
85
Figure 32: Load Distribution Factor Calculation from the AGMA
86
Figure 33: Face Load Distribution Factor from the AGMA
87
Figure 34: Face Load Distribution Factor Calculation Continued
88
Figure 35: Elastic Coefficient Calculation from the AGMA
89
Figure 36: Hardness Ratio Factor Calculation from the AGMA
90
Figure 37: Allowable Bending Stress Graph from the AGMA
91
Appendix C: Purchased Components
Figure 38: CV Joint
92
Appendix D: Project Photo Album
Figure 39: 3D Printed Components
Figure 40: Machining of the Housing
93
Figure 41: Machining of the Housing
94
Figure 42: Housing After In-Shop Machining
Figure 43: Housing After In-House Machining
95
Figure 44: Differential Housing
Figure 45: Differential Housing Prior to EDM
96
Figure 46: Arranging of Gears
Figure 47: Gear Arrangement
97
Figure 48: Assembling of Differential
Figure 49: Assembly of Differential
98
99
Figure 52: Assembled Differential
Figure 53: Differential Mounted on Car
100
Brackets
Shafts
Sprocket
Differential
CV Joints
Figure 54: Diagram of Components Designed or Selected by Team
101

SD Team 09: Torsen Differential for 2016 Formula SAE Race Car

Transcription

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