Gear transmission design
and in-wheel packaging
for a Formula Student race car
T.F. Beernaert
DC 2016.007
Bachelor’s Thesis
Coach:
Supervisor:
Ir. E.C.A. Dekkers
Dr. Ir. I.J.M. Besselink
Technische Universiteit Eindhoven
Department of Mechanical Engineering
Dynamics and Control group
Eindhoven, January 27, 2016
iii
Abstract
In-wheel packaging and gear transmission design for a
Formula Student race car
by Torben Beernaert
This report describes the development of the in-wheel gear
transmission of a Formula Student race car. This final drive has
to reduce the rotational speed of the in-wheel electromotors from
18000 rpm by a factor 12.
Several gear topologies are investigated and, based on the aspects performance, maintainability and cost efficiency, a compound
planetary gearset is chosen. A MATLAB optimisation featuring
basic strength calculations is executed to find minimum-weight
gear dimensions. These results are validated and corrected using
KISSsoft, a calculation software for gear transmissions. Eventually several feasible gear designs are acquired.
After the lubricant selection, the gear designs are expanded
to complete packaging concepts, including sealing, bearing selection and carrying components. A final concept is chosen, based
on weight, assembly procedure and manufacturing considerations. This is supported with the necessary mechanical calculations.
v
Acknowledgements
I take this opportunity to thank Ir. Erwin Dekkers, mechanical
engineer at the Equipment and Prototyping Center at Eindhoven
University of Technology. His technical expertise of mechanics,
dynamics and construction principles were a great support in
making difficult technical decisions. I always looked forward to
our weekly meetings and to discussing newly arised ideas and
problems, of course while enjoying a good cup of coffee.
I would also like to thank Dr. Ir. Igo Besselink, my supervisor during the whole project. He made sure the complete process of the thesis was executed properly, keeping every aspect
in mind. His involvement in the racing team, University Racing
Eindhoven, brought his advises and contributions to a level that
goes beyond the normal boundaries of the thesis.
Finally, I would like to thank the University Racing Eindhoven
team for the encouragement to be innovative and the drive to
keep expanding my knowledge and experience. Special regards
go to Oscar Scholle, Zjelko Parfant and Bjork van der Donk, who
supported me throughout the process of this thesis and the development of the new final drive. Their experience, knowledge
and effort positively influenced the technical result of the project.
- Torben Beernaert
vii
Contents
Abstract
iii
Acknowledgements
v
1
Introduction
1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Analysis tools . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2
Technical requirements and goals
2.1 Requirements . . . . . . . . . . . . . . . . . . . .
2.1.1 Reduction ratio . . . . . . . . . . . . . . .
2.1.2 Packaging . . . . . . . . . . . . . . . . . .
2.1.3 Forces and moments acting on the system
2.1.4 Lifetime . . . . . . . . . . . . . . . . . . .
2.1.5 Brake assembly . . . . . . . . . . . . . . .
2.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
3
3
3
3
4
5
5
6
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7
7
10
12
12
12
14
14
15
16
16
17
17
18
20
3
4
Gear design
3.1 Gear topology . . . . . . . . . . .
3.2 Gear Kinematics . . . . . . . . . .
3.3 Tooth calculations . . . . . . . . .
3.3.1 Material selection . . . . .
3.3.2 Strength calculation . . .
3.4 MATLAB Modeling . . . . . . . .
3.4.1 Reduction ratio sensitivity
3.4.2 Tooth stress . . . . . . . .
3.4.3 Optimisation . . . . . . .
3.4.4 Conclusion . . . . . . . .
3.5 KISSsoft Modeling . . . . . . . .
3.5.1 Implementation . . . . . .
3.5.2 Analysis . . . . . . . . . .
3.5.3 Conclusion . . . . . . . .
Lubrication
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
21
viii
5
6
In-wheel packaging
5.1 Bearing Selection .
5.2 Packaging concepts
5.3 Final design . . . .
5.4 Conclusion . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Conclusions and recommendation
A Gear design
A.1 Topologies . . . . . . . . . .
A.2 Definitions . . . . . . . . . .
A.3 Kinematics . . . . . . . . . .
A.4 Strength analysis . . . . . .
A.5 MATLAB modeling results .
A.6 KISSsoft modeling . . . . .
A.7 Final designs . . . . . . . . .
23
23
25
26
28
29
.
.
.
.
.
.
.
33
33
34
36
39
42
46
47
B Bearing design
B.1 Tire forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Bearing topology . . . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Arrangement selection . . . . . . . . . . . . . . . . . . . . . . .
49
49
50
54
C MATLAB Scripts
C.1 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Wheel bearings . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
61
62
66
D Mechanical calculations
D.1 Planet axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
69
E Technical drawings and datasheets
E.1 Brake assembly . . . . . . . . .
E.2 Wheel bearings . . . . . . . . .
E.3 Planet bearings . . . . . . . . .
E.4 Simmerring . . . . . . . . . . .
71
71
73
75
76
F Illustrations
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
77
ix
List of Abbreviations
CAD
CPGS
DPGS
E.G.
ET
FEM
GCD
HAN
I.E.
PGS
URE
W.R.T.
Computer Aided Design
Compound Planetary Gearset
Double Planetary Gearset
Exempli Gratia
Einpresstiefe
Finite Elements Method
Greatest Common Divisor
Hogeschool van Arnhem en Nijmegen
Id Est
Planetary Gearset
University Racing Eindhoven
With Respect To
xi
List of Symbols
a
b
C
D
Dout
E
F1−2
F2−3
Fa
FN
Fout
Fr
Fres
Ft
Fy,max
Fz,max
i2
itotal
Jplanet
Jring
Jsun
k
L
M
mplanet,L
mplanet,S
mring
msun
N
Nplanets
n
n̄
P
P̄
p
q
r1
r2
r3
r4
Tin
Tres
Tring,reaction
Tout
v1−2
Center distance
Aspect ratio
Bearing basic dynamic load rating
Pitch diameter
Outer diameter
Young’s modulus
Force between sun and planet gear
Force between planet and ring gear
Axial bearing load
Normal tooth force
Effective force on planet gear
Radial bearing force
Resulting force
Tangential tooth force
Maximum lateral tire force
Maximum vertical tire force
Reduction ratio between motor and planets
Total reduction ratio
Moment of inertia of a planet gear
Moment of inertia of the ring gear
Moment of inertia of the sun gear
Gear interactions per revolution
Gear lifetime
Gear modulus
Mass of a large planet gear
Mass of a small planet gear
Mass of the ring gear
Mass of the sun gear
Gear lifetime
Number of planets
Mean bearing rotational speed
Mean rotational speed
Equivalent bearing force
Mean equivalent bearing force
Circular pitch
Bearing life exponent
Pitch radius of sun gear
Pitch radius of large planet gear
Pitch radius of small planet gear
Pitch radius of ring gear
Ingoing motor torque
Resulting torque
Reaction torque from ring gear on housing
Outgoing wheel torque
Pitch line velocity of sun gear
m
−
N
m
m
Pa
N
N
N
N
N
N
N
N
N
N
−
−
kgm2
kgm2
kgm2
−
hours
m
kg
kg
kg
kg
−
−
rpm
rpm
N
N
m
−
m
m
m
m
Nm
Nm
Nm
Nm
ms−1
xii
vout
W
X
Y
YL
z
z1
z2
Absolute velocity of planet gears
Gear facewidth
Radial bearing factor
Axial bearing factor
Lewis bending factor
Number of teeth
Number of teeth on sun gear
Number of teeth on large planet gear
ms−1
m
−
−
−
−
−
−
α
ν
ρ
ρr
ρS
σB
σH
ω
ωp
ωin
ωout
Gear tooth pressure angle
Poisson’s ratio
Radius of curvature
Reduced radius of curvature
Density of steel
Bending stress
Hertzian contact stress
Angular velocity
Angular velocity of planets
Angular velocity of motor
Angular velocity of wheel
degrees
−
m
m
kgm−3
Pa
Pa
rads−1
rads−1
rads−1
rads−1
1
Chapter 1
Introduction
1.1
Description
University Racing Eindhoven (URE) is a team of approximately 50 students from Eindhoven University of Technology, Fontys University of Applied Sciences and HAN University of Applied Sciences. The multidisciplinary team develops an electric race car and
participates in the worldwide Formula Student competition anually.
F IGURE 1.1: URE’s 2015 race car, the URE10 [1].
During the competition, the vehicle is subjected to various disciplines. The Acceleration, a drag race of 75 m, is the benchmark when it comes to straightforward acceleration.
On the Skidpad the vehicles’ cornering ability is graded by driving in a figure of eight.
An Autocross sprint puts the car’s agility and speed to the test and finally the Endurance,
which consists of 22 km over the Autocross track, challenges the vehicle in all possible
ways.
URE is currently designing their newest race car, the URE11. This car will be an evolution of the previous car, the URE10, meaning that the parts that worked properly will be
used, with possible minor adaptions, on the URE11. However, components that do not
fulfill their functions and specifications as planned will be improved or even undergo a
complete redesign.
Based on the knowledge and the experience the team has gained during the last year,
it is decided to improve the concept of the final drive, the in-wheel reduction. The final
drive is a set of gears which transmits the torque from the high-speed electric motor to
2
Chapter 1. Introduction
the wheel by a fixed reduction ratio. This increases the torque applied to the wheel and
reduces its rotational speed.
The URE10 featured a planetary gearset with a stepped sungear. During the season
various oil leakages occured, which are expected to resolve by using a different gear concept. In this design an open motor design is used, which was prone to contaminations of
dust and debris.
F IGURE 1.2: URE10 Hub drive concept; a planetary gearset with a gear
pair input [2].
The goal of this assignment is to design a new final drive that resolves these leakage
problems and enables the use of a sealed electromotor. Furthermore, the performance,
maintainability and cost efficiency has to be improved.
1.2
Analysis tools
MATLAB
Mathworks’ MATLAB (R2015a) will be used to develop various models or calculation
tools throughout the project.
Siemens NX10.0
To develop and visualize the designs, Siemens CAD software NX10.0 will be used. This
software also features several simulation packages, of which the NASTRAN module is
best suited to analyze stresses and strains applied to several components.
KISSsoft
University Racing Eindhoven has full licenses to use KISSsoft, a commercial calculation
software for designing and optimizing rotating machinery parts. KISSsoft will be used
for the complex gear calculations.
3
Chapter 2
Technical requirements and goals
The final drive’s main functionality is the transmission of torque from the electric motor
to the wheel. In this process it will increase the torque and decrease the angular velocity.
The motors itself are relatively high-speed and low-torque motors. Studies show that this
topology, an in-wheel motor with gearset, offers the highest performance [2].
The final drive has to meet certain requirements. Besides these requirements, several
features may be beneficial or harmful. Measures can be taken in the design to achieve
certain goals regarding these features. These features will be discused in Section 2.2.
2.1
Requirements
2.1.1
Reduction ratio
The ratio between in- and outgoing angular velocity is a fixed requirement and defined
as the reduction ratio:
itotal =
ωin
= 12
ωout
(2.1)
where ωin is the angular velocity of the motor and ωout the angular velocity of the wheel,
both in rad/s.
As the reduction ratio is always a result of calculations with integer gear numbers
it is very unlikely that the reduction ratio will be achieved exactly. A margin of 5% is
allowable. Hence, the requirement for the reduction ratio can be stated as 12 ± 0.6.
2.1.2
Packaging
The complete assembly has to fit inside the wheel. The tires used are Apollo 205/50 R10
40P. These are 10 inch tires with a width of 205mm, a sidewall height of 103mm and an
effective radius of 230mm. The rim mounting will have a positive offset of 50.6mm, corresponding to a ET-value of 2inch. These dimensions are illustrated in Figure 2.1.
4
Chapter 2. Technical requirements and goals
F IGURE 2.1: Designspace of the motor (blue) and final drive (red) in the
wheel.
Figure 2.1 also shows the design space for the final drive and the motor in red and
blue. In order to fit between the suspension rods, the allowable design space for the final
drive is a cylinder with diameter and length of 120mm and 70mm, respectively. This still
leaves room for the brake disc and calliper. The brake assembly will be discussed in section 2.1.5. Furthermore, the motor should be placed inside the wheel concentrically w.r.t.
the wheel. It is also desirable to have the motor as much inside the wheel as possible.
2.1.3
Forces and moments acting on the system
The forces and moments acting on the system can be subdivided in tire forces and torques
and moments. These will be discussed seperately in the following sections.
Tire forces
Forces acting on the tires of the vehicle are defined using the coordinate system illustrated
in Figure 2.1. The Z-direction is the vertical, perpendicular to the ground. Its positive direction is upwards; in opposition to the earth’s gravitation. The X-direction is the car’s
longitudinal direction, positive with forward motion of the vehicle. The Y-direction is the
lateral direction, always pointing towards the left of the vehicle.
A multibody simulation provides forces on every tire during an autocross [3]. The
time histories with the vertical and lateral tire forces and angular velocity are shown in
Appendix B.1.
From the time histories can be seen that the lateral forces range between approximately −3000 and 0N for the left wheels and between 0 and 3000N for the right wheels
of the vehicle. The vertical forces range in all cases between 0 and 1800N . For a worst
case scenario, the absolute maximums are multiplied by a safety factor 1.3 and combined
to the loadcases shown in Table 2.1.
TABLE 2.1: Worst case scenario tire forces.
Wheels
Left
Right
Lateral force
-3900
3900
Vertical force
2340
2340
Unit
N
N
2.1. Requirements
5
Torques and moments
The same simulation as mentioned before is used to generate a loadcase of the motor
torques. This holds both tractive torque and regenerative braking torques. As opposed
to the tire loadcase, which actually results in a loadcase per wheel, the motor torques are
averaged over the four wheels. Using one loadcase for every motor will make the gear
calculations clearer and the KISSsoft implementation easier.
The mechanical braking torques are not transmitted through the final drive, hence no
elaborate loadcase is needed. The maximum braking torque of 550N m is only important
for several load-carrying components.
2.1.4
Lifetime
The car’s lifetime is aimed to be 2000km. Using the average driving velocity, 50km/h, the
lifetime in hours can be calculated, which is 40hours. This is also the minimum amount
of working hours for the final drive. A safety factor of 2 will be used, thus the complete
expected lifetime is 80hours.
2.1.5
Brake assembly
The URE11 will make use of a disc brake system. All components for this system have
been selected and should be properly placed inside the wheel. For this project, only the
brake disc and callipers are important. The brake disc is identical in every wheel. A
single piston calliper is used in the rear wheels (CP4226-2S0), whereas a larger double
piston calliper is used in the front wheels (CP4227-2S0). Figure 2.2 shows the components. Relevant dimensions and technical drawings can be found in Appendix E.1.
( A ) Brake disc and floaters.
( B ) AP Racing’s CP4226-2S0.
( C ) AP Racing’s CP4227-2S0.
F IGURE 2.2: In-wheel components of the brake system.
6
2.2
Chapter 2. Technical requirements and goals
Goals
When all design requirements are met, there are still design choices to be made. To make
proper decisions, several goals are stated. The final drive should have high performance
and be easy to assemble and maintain and cost efficient. These goals are vague and difficult
to quantify, so a house of quality [4] is used to analyse the influence of technical features
on them:
F IGURE 2.3: A house of quality is used to determine the importance of
technical aspects.
Each goal is given a priority in the vertical black ribbon. In the tilted part of the table,
quantifiable features are stated. The effect of every feature on every goal is graded using
a one, a three or a nine. Underneath the properties is a row with arrows. An upward arrow
means the goal is to reach a high value of the corresponding property, a downward arrow
indicates that a reduction of the parameter is beneficial. Finally, all grades are multiplied
by the corresponding priority and summed op to a final score, shown in the horizontal
black ribbon.
From the house of quality, Figure 2.3, we can see that the weight and efficiency of the
final drive are of high importance. The complete order of the parameters is shown below:
TABLE 2.2: Priority of several design features.
Priority
1
2
3
4
5
Weighing factor
29
27
15
12
9
Parameter
Weight
Efficiency
Number of components
Size
Component complexity
7
Chapter 3
Gear design
3.1
Gear topology
There are several types of transmissions that can fulfill the stated requirements. The
following solutions are feasible and will be considered:
• Combination of gear pairs
• Double planetary gearset
• Compound planetary gearset
The different topologies will be discussed below. A comparison between the topologies is made, followed by the final choice based on the goals as stated in Section 2.2.
Combination of gear pairs
The torque can be transmitted from the motor to the wheel by using gears in their most
conventional way; simple pairs. A small pinion on the motor axle could interact with a
larger wheel to increase the torque. This sequence can be repeated more often to decrease
the size of the larger wheels and to realise a concentric design. A minimum of two stages
is needed for this. An exmple of a transmission using simple gear pairs is shown in Figure
3.1:
F IGURE 3.1: A combination of spur gear pairs [5].
A major downside of this concept is that every pinion needs to be properly supported.
There will not only be a transfer of torque via a tangential force, but also a radial component on the gears. This component leads to bending of the axle, resulting in a deflection
angle between the centerlines of the interacting gears. In order to support the gears and
reduce this deflection angle, the gears need to have internal bearings. In that case, no
bending moment will occur on the axle. This means, however, that either one bearing
should be removed from the motor and placed in the final drive or three bearings have
to be used on the same axle, this is definitely not desirable due to overconstraining.
8
Chapter 3. Gear design
Furthermore, all gears are loaded quite heavily. The input pinion has to transmit all
torque via one single interaction. The second pinion would have to transfer even more
torque via a single interaction. High tooth forces are expected, as well as large diameters.
This combination will definitely result in large and heavy gears.
Double planetary gearset
A single planetary gearset (PGS) divides the torque over multiple planet gears that rotate
in an internal ring gear. The distribution of forces leads to a relatively high power density
as compared to plain gear pairs.
A reduction can be achieved by using the sun gear as input, fixing the ring gear and
using the planet carrier as output. Unfortunately, the complete reduction ratio of a single
stage is limited to approximately 10 : 1. This limit is due to the size of the planets, which
would be colliding if the ratio was increased more.
( A ) Definitions [6]
( B ) Double PGS
( C ) High reduction ratio [7]
F IGURE 3.2: Planetary gearsets.
Two of these planetary gears in series could fulfill the reduction ratio, see Figure 3.2.
In that case, the first stage’s carrier output will be the second stage’s sun gear input. The
overall input gear of the gearbox will suffer no residual radial force, i.e. the forces are
balanced by the planets. This means that the motor can have two bearings and can be
sealed.
Compound planetary gearset
Another variation on a planetary gearset is the compound planetary gear (CPGS). This
features an additional planet, realising an additional reduction. Again, the sun (red) is
the input gear, distributing force over a number of relatively large planet gears (yellow).
The force is transmitted to smaller planet gears (blue), which at their turn are running in
a ring gear (grey), as shown in Figure 3.3.
3.1. Gear topology
9
F IGURE 3.3: A compound planetary gearset.
A downside of this design is that the axles of the planet gears in Figure 3.3 suffer from
bending, because the radial forces exerted from the sun and ring gear are not alligned,
causing a moment. Proper support is needed, but there are also several variations that
can resolve this problem. All possible topologies are shown in Appendix A.1.
Final concept
The benefits and drawbacks, as discussed in the previous sections, are summarized in
Table 3.1. The following scores are rewarded per criterion and weighed using the factors
given in Section 2.2:
1
3
5
-
for the least performing concept
for the averagely performing concept
for the best performing concept
TABLE 3.1: Scoring table of various gear topologies.
Weight
Efficiency
Number of components
Size
Component complexity
Final score
Priority
29
27
15
12
9
Gear pairs
1
3
5
1
3
224
DPGS
3
3
1
3
3
246
CPGS
5
3
3
5
3
358
Efficiency is graded equally for every topology. This parameter is mainly influenced
by tribological aspects of materials and lubrication, which will not be dealt with in this
project. Thus, no clear distinction can yet be made w.r.t. efficiency. More about the final
drive as a tribological system can be read in Scholle [15].
It can be seen that the compound planetary gearset offers the best combination of
features to meet the design goals.
10
Chapter 3. Gear design
3.2
Gear Kinematics
To get a first estimate of the dimensions and ratios, some calculations need to be done.
Some definitions of parameters are given in the following Figure 3.4 and Table 3.2.
F IGURE 3.4: Section view of a compound planetary gearset with defined
radii.
TABLE 3.2: Gear dimension definitions, as illustrated in Figure 3.4.
Component
Sun gear
Large planet gear
Small planet gear
Ring gear
Color
Red
Yellow
Blue
Grey
Pitch radius
r1
r2
r3
r4 = r1 + r2 + r3
Free body diagrams
Free body diagrams are used to calculate the forces and torques acting on every gear of
the transmission, as illustrated in Figure 3.5.
( A ) Sun
( B ) Planet
( C ) Ring
F IGURE 3.5: Free body diagrams of the CPGS’s gears.
This leads to the following expression of the overall reduction ratio:
itotal =
Tout
r2 + r1 r2 + r3
=
·
Tin
r1
r3
A derivation of this expression can be found in Appendix A.3.
(3.1)
3.2. Gear Kinematics
11
Velocity approach
To check the derivation of the reduction ratio, another approach is taken. Instead of analyzing the forces and moments acting on the gears, the angular and pitch line velocities
are calculated. Knowing the velocities of every gear is also important when selecting the
bearings. We use Figure 3.6 for the calculations.
F IGURE 3.6: Angular and pitch line velocities of the final drive gears.
The following kinematic equations can be found:
v1−2 = ωin · r1
r3
vout = v1−2 ·
r2 + r3
v1−2 − vout
ωp =
r2
vout
ωout =
r1 + r2
Using these equations and the velocity variant of the reduction ratio leads to the following:
itot =
ωin
r1 + r2 r2 + r3
=
·
ωout
r1
r3
(3.2)
where ωin is the motor’s angular velocity and ωout the wheel’s angular velocity. This
is coherent with equation 3.1. Now, another virtual reduction ratio can be introduced to
determine the angular velocity of the planets, defined as:
i2 =
ωin
r2 + r3
=
ωp
r1
(3.3)
with ωp the angular velocity of the planets. Detailed derivations of the expressions
given above can be found in Appendix A.3.
12
3.3
Chapter 3. Gear design
Tooth calculations
In order to determine the size of the gears, we have to take a closer look at the teeth
themselves. Some important parameter definitions are given in Appendix A.2. A proper
material will be selected, wherafter several simple formulae are introduced that lead to
estimations of the strength.
3.3.1
Material selection
The material has much influence on the performance and lifetime of the gears. Important material properties are the surface hardness, Young’s modulus and the bending and
contact fatigue stress.
Steel offers a great combination of these parameters. Another material that might be
interesting is ceramics. This would feature even higher hardness, stiffness and allowable
stress. There is one problem however: ceramics have the ability to cope with incredibly
high compressive stresses, while their tensile strength is way lower. Since gears suffer from
bending, both stresses are present in the gears’ teeth, as shown in Figure 3.7.
Using the steel, seems the safer and more conservative option. Van Eijk Transmissies,
one of URE’s partners, will manufacture the gears. They advise to use the ChromiumNickel-Molybdene (CrNiMo) alloy 17NiCrMo6-4. The mechanical properties of this material can be found in Appendix A.4.
3.3.2
Strength calculation
The strength of a gear tooth is its resistance against failure. Under load, the typical
stressprofile in two mating gear teeth shows several stress concentrations, see Figure 3.7.
F IGURE 3.7: A typical stressprofile in two mating gears [12].
3.3. Tooth calculations
13
The stresses at the root of the tooth are a result of bending of the teeth and can lead to
cracks. This is a matter of fatigue, so a repetatively alternating force for a certain number
of cycles. Of course, a single extreme static load may also be enough to cause permanent
damage. The stress concentrations at the gear interaction point are a result of Hertzian
contact force. Here, damage occurs due to a single overload, regardless of the number of
loading cycles, and leads to a phenomenon called pitting [13]. Illustrations of both failure
modes can be found in Appendix A.4.
Bending stress: The Lewis equation
A relatively simple and classic method to calculate the bending stress in a gear tooth is
by the use of Lewis bending equation. This method is one of the first gear tooth strength
analysis methods and models the tooth as a cantilever beam.
F IGURE 3.8: A gear represented by a cantilever beam [11].
The tangential force Ft on the gear tooth is the effective force, i.e. the component of the
total force that generates a torque. Since the involute has a pressure angle α, there is also
a radial component Fr . The tangential force is the component that leads to bending of the
tooth, whereas the radial force leads to a uniform normal pressure at the root of the tooth.
The Lewis equation for the bending stress is given by:
σB =
Ft
W M YL
(3.4)
Where σB is the bending stress in P a, W the width of the gear in m, M the gear modulus in mm and YL the so-called Lewis factor. The factor YL depends on the shape of the
involute and the number of teeth. In a gear with a constant pitch diameter, the latter is
only dependent on the modulus.
The relation between the gear number and the Lewis factor YL for different tooth
geometries can be found in Appendix A.4.
14
Chapter 3. Gear design
Contact stress
Two interacting gears can be viewed as a line contact between two cylinders, as illustrated in Figure 3.9.
F IGURE 3.9: Two mating gears, approximated as a line contact between
cylinders [11].
The corresponding formula for the calculation of the Hertzian stress is given by [13]:
s
σH =
r1 + r2
2Ft E
2
πW (1 − ν ) sin(2α) r1 r2
(3.5)
Where σH is the calculated Hertzian stress in P a, Ft the tangential force on the tooth
flank in N (see Figure 3.8), E the Young’s Modulus in P a, W the width of the teeth in
m, ν the dimensionless Poisson’s ratio, α the teeth’s pressure angle in degrees and r the
gears’ pitch radii in m. The indices 1 and 2 refer to the different gears. The derivation of
this formula can be found in Appendix A.4.
3.4
MATLAB Modeling
A MATLAB model has been created based on the previously determined formulae. By
varying the amount of teeth on the pinion and the smaller planetary gear, z1 and z2 ,
respectively, different realisations of the gearbox can be accomplished and evaluated very
quickly.
3.4.1
Reduction ratio sensitivity
Since there are three dimensions that define the reduction ratio, we can investigate the
effect of these free-to-choose variables. The radius of the large and small planetgear, r2
and r3 respectively, and the center distance a, equal to the sum of r1 and r2 , are defining
dimensions for the size and location of surrounding parts. Hence, these will be used to
characterize the final drive.
Calculations are done to isolate the effects of the seperate variables on the overall reduction ratio. From the graphs in Figure A.10, it can be seen that an increase in r2 leads to
an increase in the reduction ratio, whereas an increase in a and r3 decrease the reduction
ratio. These relations are summarized in Table 3.3.
3.4. MATLAB Modeling
15
TABLE 3.3: Influence of several gear dimensions on the overall reduction
ratio.
Dimension
Large planet radius
Small planet radius
Center distance
Symbol
r2
r3
a
Influence
+
-
A simple conclusion can be drawn from these relations. A compound planetary
gearset with a relatively high reduction ratio is characterized by relatively small radii
r1 and r3 , while radius r2 will be rather large.
3.4.2
Tooth stress
From Equation 3.5 it can be concluded that, since the tooth curvature is not depending
on the gear modulus, the contact stress will be constant along the sweep. The bending
stress, however, is not constant in this case. With a decreasing modulus an increasing
Lewis factor is found, however the bending stress is directly negatively influenced by the
modulus (Equation 3.4). The effect of the modulus is best simulated with a sweep over
the gear modulus.
The sweep is executed on a gear with a facewidth of 10mm and a pitch diameter of
20mm, where the modulus ranges between 0.5 and 1.5mm. The opposing gear has a pitch
diameter of 40mm. The used MATLAB script can be found in Appendix C.1.
F IGURE 3.10: Influence of gear modulus on bending and contact stresses.
As can be seen from Figure 3.10, the contact stress is indeed constant over the sweep.
When considering the bending stress in this situation, it is clear that the increase in gear
modulus is dominant over the decrease in Lewis factor.
16
Chapter 3. Gear design
3.4.3
Optimisation
The variation over modulus M , number of planets Nplanets and radius of the sun and
the large planet gear, r1 and r2 , respectively, leads to several sets. These are checked for
feasibility (Appendix C.2). From the initial goals, we can see that the weight of the final
drive is of high importance. As the model calculates different layouts, it will always store
the final drive with the lowest weight. The used script can be found in Appendix C.2.
All final sets feature three planets. The benefit of having more planets is that the
torque is divided over more teeth, hence F1−2 and F2−3 are reduced. It appears that in
this case the effect of the increase in weight is predominant over the decrease in tooth
force.
( A ) Optimised final drive weights.
( B ) Final drive with M = 0.75mm.
F IGURE 3.11: MATLAB optimisation results.
From Figure 3.11 we see that as we expected, with an increasing gear modulus M
the weight of the final drive reduces. However, we also find that in this particular case,
concept 3 with a gear modulus of 0.75mm is the lightest. This final drive is also illustrated
in Figure 3.11. The four variations described in Section 3.1 are also optimised and shown
below in Figure 3.12:
F IGURE 3.12: Four variations of the final drive illustrated in Figure 3.11b.
All results and gear dimensions can be found in Appendix A.5.
3.4.4
Conclusion
Obviously, the difference between concept 1 and 2 is only a matter of packaging, since
their gears are identical. Concept 3 is the lightest, due to the aspect ratio of the small
planet gears being more favorable. The total width of the blue gears can hence be larger,
making them stronger and enabling the reduction of the size of every gear, except the
sun. In total, this is beneficial to the weight. The opposite is what happens in concept
4; due to the minimum aspect ratio, all yellow gears need an enlarged facewidth. These
3.5. KISSsoft Modeling
17
gears are overdimensioned and the weight is thus excessive. This concept has shown too
many cons and will not be analysed any further. From now on, concept 1 and 2 will be
referred to as the ’single ring’ concept and concept 3 the ’double ring’ concept.
3.5
KISSsoft Modeling
The MATLAB model offers a comparison between a large number of possible final drive
dimensions, resulting in a first estimation of weight and size. However, since the MATLAB model features various assumptions, it has several drawbacks and uncertainties.
The commercial software package KISSsoft cannot compare different implementations,
but has a higher degree of detail and uses more extensive calculations. These calculations should lead to a more reliable result.
The most important result missing from the MATLAB model is the calculation of the
gears’ lifetime. KISSsoft can execute a service life calculation using a Wöhler (fatigue)
curve. One can always reduce the size, and weight, of the gears at the cost of lifetime.
Furthermore, the program is capable of using the loadcase described in Section 2.1.3,
analyzing the rolling and sliding contacts between gears more extensively and adjusting
the shape of the teeth by profile shifting. More about the features of KISSsoft can be found
in Appendix A.6.
3.5.1
Implementation
One drawback of KISSsoft is that it doesn’t allow a custom gearbox to be assembled.
Instead it offers various predetermined topologies, such as a single gear, a pinion with
rack, a planetary gear and a gear train of two to four gears. In order to approximate a
compound planetary gearset, the following measures need to be taken.
The compound planetary gearset is viewed as two planetary gears with equal center
distance. One ring and one sun are added to enable this analogy. In Figure 3.13, these
virtual gears are displayed transparantly. From now on, these seperate planetary gears
will be referred to as stage one and stage two. Stage one is the gearset with the virtual
ring gear and stage two the one with the virtual sun. Stage 2 of concept three will be
analysed by two identical planetary gears, which each carry half the torque.
F IGURE 3.13: KISSsoft analogy; two planetary gears are used to approximate the compound planetary gearset.
18
Chapter 3. Gear design
We are interested in the gears’ lifetime, which is expressed in a certain number of
revolutions. This means that the lifetime in hours is proportional to the total acceptable
revolutions per gear and the amount of tooth interactions per gear revolution:
N
(3.6)
60n̄k
Where L is the lifetime in hours, N the total amount of gear interactions, n̄ the mean
rotational speed in rpm and k the amount of gear interactions per revolution. For every
revolution of the planet gears, KISSsoft counts two gear interactions (one with the sun
and one with the ring). Since in both planetary gears one of these two is missing, the
calculated lifetime in hours of the planetary gears can be doubled. Equation 3.6 shows
this mathematically; KISSsoft uses a k of 2 instead of 1 for the planets. As long as the
planet gears are limiting for the total lifetime, KISSsoft’s service life estimation can be
doubled. This even includes an additional safety factor, since the acceptable mechanical
stress increases when less cycles are applied. A typical progression of a Wöhler curve can
be found in Appendix A.4.
L=
3.5.2
Analysis
The previously mentioned concepts 1, 2 and 3 (Section 3.4) are analysed using KISSsoft.
Now, for every concept a lifetime is found, illustrated in Figure 3.14.
F IGURE 3.14: KISSsoft lifetime analysis of MATLAB concepts.
From Figure 3.14 it can be seen that all calculated lifetimes exceed the goal of 80 hours
by far. The total lifetime of the gears, being the minimum lifetime of stage 1 and 2 per
concept, is the highest for a modulus of 0.75 mm for both concepts. Figure 3.11a shows
that this modulus also offers a relatively lightweight result. From now on, only the concepts with a modulus of 0.75 mm will be considered.
The weight of all stages can be further reduced by allowing a reduction in lifetime.
The easiest way to do this is by decreasing the gears’ facewidth. However, because of the
high difference between the current lifetimes and the lifetime goal, this might not result
3.5. KISSsoft Modeling
19
in the lightest gears. First, an attempt is done by only reducing the gears’ facewidth until
the lifetime is at the aimed target. After that, an iterative process is started to reduce
the weight and size by selecting a smaller center distance, keeping in mind the relations
found in Section 3.4.1. This iterative process is described by:
1
2
3
Reduction of a and r3
Calculation of corresponding r2 , r1 and virtual sun and ring dimension
Correction of facewidth until lifetime goal is reached
The results of the different reductions can be found in Appendix A.6. The iterative
process leads to the highest weight reduction, because it has more freedom to alter the
dimensions of the final drive.
The double ring concept is always lighter than the single ring. KISSsoft does not use
the aspect ratio boundaries like the MATLAB model, but calculates effects that result
from the aspect ratio. The most likely advantage of the double ring concept is that due
to its split, small planet gears the effect of torsional twist is reduced. In the double ring
concept, the average distance from small to large planet gear is reduced by a factor two.
This increases the load carrying capacity and thus decreases the total weight and size.
The iterative process not only lead to a reduction of weight, but also in size. This means
even more weight is saved on housing and bearing size.
Another way to save weight is hollowing of the gears. In order to be properly supported, they don’t need the complete diameter to be of solid steel. Too small rims can’t
support the gear teeth and will result in low lifetimes. This hollowing of the gears is
shown below and reduces the weights of 0.461 and 0.449kg of the best concepts to 0.278
and 0.266kg, respectively.
F IGURE 3.15: Hollowing of gears.
One final attempt is done to reduce the modulus. As it turns out, even more facewidth
reduction is possible. This somewhat contradicts the results found in Section 3.4.2, but
due to the increased level of detail in the KISSsoft calculations it is explainable.
TABLE 3.4: Weight reduction by modulus decrease.
Concept
Single ring
Double ring
M = 0.75mm
0.278
0.266
M = 0.50mm
0.207
0.198
Unit
kg
kg
As mentioned before, using a different modulus can lead to a different profile shift
coefficient, which greatly influences the strength of the teeth. Furthermore, the Lewis
factor increases (Figure A.8) because more teeth are used on the same pitch diameter.
20
Chapter 3. Gear design
The final result of all gear calculations are the two gearsets shown in Figure 3.16,
whose exact dimensions can be found in Appendix A.7:
( A ) Single ring
( B ) Double ring
F IGURE 3.16: Final two gear topologies.
3.5.3
Conclusion
The MATLAB results from the previous chapter have been refined using KISSsoft. Some
assumptions and additional calculations were needed to enable the analysis of a compound planetary gearset. This was achieved by viewing the CPGS as two seperate planetary gearsets with an equal center distance. The calculated lifetimes were initially well
above the aimed target of 80hours. Reducing the lifetime also lead to a significant weight
reduction. This was done by varying the modulus, facewidth and dimensions of the
gears in a semi-iterative process. Finally, two gearsets with a modulus of 0.50mm have
been found that meet all requirements and goals sufficiently.
21
Chapter 4
Lubrication
Every moving contact needs to be lubricated in order to reach the desired efficiency and
lifetime of the components. Moving contacts are found in bearings and gear interactions.
In this chapter, several lubrication methods will be investigated.
The main functionality of a lubricant is to keep the lubricated objects away from each
other, this can be achieved by using a lubricant with the right viscosity. Ideally, all forces
are transferred through the lubricant and the interacting surfaces never touch. This reduces wear and increases efficiency, since usually the lubricant-to-surface friction is much
lower than the plain surface-to-surface friction.
Lubricants
Two lubricant types will be considered:
• Gear oil
• Grease
It is not recommended to use several lubricants in the same tribological system; accidentally mixing incompatible lubricants could lead to dissolving or other chemical reactions. A brief description of both oil and grease as a lubrication is given in the following
sections, followed by an evaluation and the final selection.
Oil
The most common lubricant in automotive transmissions is gear oil. Gear oil consists of
a mineral or synthetic oil and additives. Oil acts as a fluid and thereby moves through
the gearbox constantly. Transfer of heat, contaminants and debris are some of its main
benefits. However, due to the relatively low viscosity special care should be taken in the
sealing of the lubricated compartment.
The relatively low viscosity has another drawback. The necessary lubrication buildup is determined by the surface smoothness of the lubricated objects. In particular at low
speeds, a high viscosity is needed to reach certain build-up requirements. Using oil could
lead to high surface roughness standards [15].
Grease
Grease is a substance of a higher baseline viscosity, classified as a semisolid. It generally
consists of a soap, emulsified with base oil and additives. When a shear stress is applied
the viscosity drops significantly, due to ’shear thinning’ [14]. Grease will rather stick in
one place during use of the gearbox, thus making it harder to spread heat and particles.
22
Chapter 4. Lubrication
Another downside of this characteristic is that if the grease is cleared from one contact,
there is no guarantee it will flow back. Furthermore, at approximately 150o C the soap
component in standard greases loses its functionality. Although such temperatures are
not expected in the final drive compartment, it is quite uncertain what the local temperature at points of contact will be.
A significant benefit is, however, that due to its solid-like state the risk on leakage is
reduced. As explained in the previous section, grease is very unlikely to move from its
position. It gets even better, as grease itself can function as a seal from water or dust.
Lubricant selection
All previously mentioned risks and benefits are summarized in Table 4.1:
TABLE 4.1: Benefits of different lubricants.
Oil
Grease
• Large amount of heat transfer
• No leakage
• Reaches every part of the gearbox • Better frictional properties
Table 4.1 shows only the benefits of a certain lubricant. The presence of the shown
features of oil implies a lack of these features when using grease and vice versa. The
drawbacks of grease and benefits of oil are difficult to model or quantify theoretically
and thus very risky. The drawbacks of oil on the other hand are easier to overcome. Sealing of oil-filled compartments is a relatively common issue and proper components are
commercially available. The analysis of the frictional properties and proper oil selection
are tougher, but not impossible. Too many risks are seen in the use of grease. Hence, the
choice is made to use oil as a lubricant for the final drive. A seperate study of the lubrication of the gear teeth and bearings of the final drive is conducted by O.J.M. Scholle
[15].
23
Chapter 5
In-wheel packaging
The calculations of Chapter 3 lead to two feasible gear topologies in three different packaging possibilities. In this chapter, the gear topologies will be elaborated into more extensive concepts, featuring component selection, carrying components and ease of assembly.
The three packaging possibilities are displayed in Figure 5.1:
( A ) Concept 1.
( B ) Concept 2.
( C ) Concept 3.
F IGURE 5.1: Three feasible packaging configurations. In all cases will the
motorshaft be entering from the right.
The output of the gearbox, the planet carrier, is directly connected to the wheelhub.
Since this is a rotating component, it should be supported in its housing by bearings.
Several bearing arrangements will be reviewed and combined with the gear topologies.
These combinations will be discussed and evaluated for their functionality, based on
the features stated in Section 2.2.
5.1
Bearing Selection
Various types of bearings are available and different arrangements may have benefits.
Important parameters for the bearing selection are the rotational speed of the axle and
the magnitude, direction and point of application of the forces.
Only a torque is applied on the axle by the final drive. However, tire forces cause both
radial and axial loads in virtually all directions. A bearing type and arrangement needs
to be selected that can cope with this loadcase.
24
Chapter 5. In-wheel packaging
Schaeffler Group’s product catalogue [18] is used to find the options, depicted in Figure 5.2:
( A ) A needle roller - angular contact
ball bearing.
( C ) Two angular contact ball bearings.
( B ) Two crossed roller bearings.
( D ) A double row angular contact ball
bearing.
F IGURE 5.2: Four different wheelbearing types.
The bearing types will be discussed and compared in Appendix B.2, whereafter the
final selection will be made. All used bearing properties and illustrations can be found
in [18].
After an estimation of the weight of all bearings and the stiffness of all resulting construction, it is found that the combination of two angular contact ball bearings, type C in
Figure 5.2, of different sizes in an O-arrangement enables the lightest and stiffest design.
5.2. Packaging concepts
5.2
25
Packaging concepts
For each gear concept, a suitable set of bearings has been selected. These are placed in
possible configurations around the gears, as shown in Figure 5.3.
( A ) Concept 1
( B ) Concept 2
( C ) Concept 3
F IGURE 5.3: Three packaging layouts with in green a possible assembly
seperation line.
It now becomes clear that the double ring has significant drawbacks. Since both ring
gears should be fixed inside the housing, it has to be split in sections in order to assemble
the final drive. Furthermore, it is a challenge to allign both rings in rotation. Absolutely
no difference is allowed, otherwise the planet gears will not be able to run. At this stage,
it appears that the downsides of the double ring have surpassed its benefits. Due to the
manufacturing and assembly reasons stated above, this concept does not seem suitable
anymore.
When the smaller bearing is placed on the motor side of the final drive, the complete
gearbox can be assembled from the outside of the vehicle. Because of the ’stepped’ layout,
assembly will be very easy. Furthermore, nearly all rotating parts can be manufactured
and assembled in one piece. The planet carrier, hub, a rim heart and even the brake disc
carrier can be merged into the same component. This is beneficial to the overall weight,
since no fasteners are needed. Furthermore manufacturing costs will reduce and for the
assembly tolerances (concentricity) are reduced. This because every fit can now be manufactured on the same component in one single fixation. However this concept involves a
longer motor shaft, this is actually not drawback. The longer shaft is relatively compliant
to bending moments and is thus capable of compensating for concentricity allignment
errors between motor and final drive.
Placing the larger bearing on the motor side of the final drive has no obvious benefits. It does not have the opportunity to merge the components, as explained above, and
has the bearings actually configured the wrong way around. The outer bearing is placed
almost directly in line with the vertical tire force (see Figure B.8) and will thus be exerted
to the higher radial forces. This should be the bearing with a higher load capacity, ergo
the larger one.
The benefits of the concept one are very clear, while the apparent advantages of the
double ring concept have been deminished by its manufacturing and assembly issues.
Concept 2 is feasible, but offers no clear advantages. Concept one is found to be the most
suited in terms of weight, number of components and ease of assembly. The bearings
that match the topology are the 71812-B-TVH (inner) and the 71815-B-TVH (outer), their
datasheets can be found in Appendix E.2. The lifetime and strength calculations of these
bearings can be found in Appendix B.3.
26
5.3
Chapter 5. In-wheel packaging
Final design
Hub
Since we want to use oil as a lubricant, the wheelbearings should be open. This means,
however, that an additional seal is needed between the hub and the housing of the final
drive. A simmerring can be used to realise this sealing. This will have a slightly larger
inside diameter than the outer wheelbearing, so the stepped assembly concept is still possible. Furthermore, this simmerring should work on a hardened steel surface (Appendix
E.4). This will be achieved by a thin ring, which is pressed on the hub. The hub assembly
can be seen in Figure 5.4.
F IGURE 5.4: Hub part with several component locations.
This concept leads to two drawbacks. In order to replace the wheel, the rim should
be detached at the six spokes of the rimheart. Furthermore, the brake disc can only be
removed towards the right in Figure 5.4, meaning that the final drive should be dissassembled if a replacement of the brake disc is necessary. Considering the frequency and
extra effort of these actions, the drawbacks are compensated by the benefits regarding
weight and costs.
Planets
The planet gears must be connected to eachother in a stiff setup. A solution to do this is
by using an additional shaft with splines. These splines can than be milled into the inner
surfaces of the gears and assembled. To locate the gears axially, a press fit or C-clips can
be used. However, due to the splines taking in space and the additional shaft, relatively
little room is left for the inner axle. A better solution would thus be to use the gearteeth
as a spline. Extending the small planet gear and cutting its shape out of the large planet
gear enables this. The displayed configuration, Figure 5.5a, offers the easiest assembly
and stiffest design.
Also the planet bearings need to be lubricated by oil, meaning they should be open.
Furthermore, the planet gears experience radial forces only. The axial forces that might
result from irregularities and sideways accelerations of the vehicle are neglected. Small
needle bearings seem to be the only option for the planets (type K7x10x8-TV, see Appendix E.3). Because of the high loads, three bearings are placed back to back on the axle,
as shown in Figure 5.5b.
5.3. Final design
27
( A ) Structural integration of ( B ) Planet bearing arrange- ( C ) Planet axle with lubricathe planet gears.
ment.
tion holes, section view .
F IGURE 5.5: Structural planet topology.
The planets’ axles are subjected to the same high loads as the bearings. Due to their
small dimensions, steel is the only option. The shafts will be hollow with several radial
holes (see Figure 5.5c). These holes are created for lubrication purposes, as explained in
the next section.
The strength analysis of the planet axles can be found in Appendix D.1.
Lubrication
Getting the lubricant to every targeted component is necessary for good functioning of
the gearbox. All gears and the wheel bearings frequently pass through the oil and will
not be at risk. The planet bearings may, however, not have a constant supply of oil. To
solve this, the following measures are taken.
Several holes are made in the planetaxles, displayed in Figure 5.5c. Also, the inside
of the hub is lathed into several funnels. Centrifugal forces will press the oil into these
funnels, where small holes line up with the holes in the planet axle. The oil will fill up
the axial hole and be pushed between the planetbearings by the same centrifugal force.
This mechanism is displayed in Figure 5.6.
F IGURE 5.6: Lubrication mechanism based on centrifugal acceleration.
28
Chapter 5. In-wheel packaging
Locking
After assembly, all components need to be positively locked to prevent loosening by, for
instance, vibrations. This can be done by using lockwires or additional retainers. However, a mechanism is used that combines this requirement together with the pretensioning of the wheelbearings. The bearings require this pretension to gain increased stiffness
and reduced backlash.
One custom aluminium locknut will be used to press the bearings together over their
workline, see Figure 5.7.
F IGURE 5.7: The locknut will tighten the wheelbearings and restrain the
assembly.
5.4
Conclusion
The final topology has been selected for its ease of assembly and integration of components. Appropriate wheel- and planet bearings have been selected, having a combined
weight of less than 0.180kg. A lubrication mechanism is integrated and the compartment
is sealed by a simmering, located next to the outer wheelbearing. Finally, the whole assembly is restrained and pretensioned by a single aluminium locknut.
Illustrations of the complete design can be found in Appendix F.
29
Chapter 6
Conclusions and recommendation
A new gear transmission for the URE race car has been developed. Various gear topologies have been compared and the compound planetary gearset was clearly the most suitable for the application in terms of performance, maintainability and cost efficiency. This
topology meets all requirements and goals in a very compact and lightweight design.
Different approaches to derive the gear kinematics have been executed. The gear
kinematics have been combined with simple strength models in a MATLAB optimisation. By varying different aspects and dimensions of the gears, numerous designs could
be calculated and evaluated to find the lightest solution. It appears, however, that these
estimations are very rough. A validation in KISSsoft, a software package for the design
of rotating machinery parts, showed that the earlier derived relations were not entirely
correct for every situation. An iterative process was started to reduce the weight by designing for the minimal required lifetime. The result is a very compact gearset with a
modulus of 0.5mm and a weight of 0.215kg.
A suitable lubricant type, oil, was selected. Furthermore, the use of two angular contact ball bearings in an O-arrangement leads to the stiffest and lightest design. Different
packaging concepts are evaluated. The final concept features a high degree of integration of components and an easy assembly. The 45 components of the final design are
estimated to weigh 1.9kg alltogether, whereas the previous design featured 111 seperate
components with a total weight of 3.8kg.
In order to improve the final drive even more, proper investigation of the tribological
aspects is needed. The selection of the right lubricant can have benefits to the efficiency
and lifetime of the gears. Also the material selection can be examined further. If special
care is taken to polish or finish the gears surface, an increased lifetime can be achieved.
The same counts for the use of hardening coatings.
The used calculation program, KISSsoft, is not able to analyse the custom topology of
a compound planetary gearset. Several assumptions had to be made to achieve proper
results. If another program would be used that can calculate this topology or enables a
free topology input, more accurate results can be retrieved.
While this document describes the design of the final drive, a consecutive study can
be performed to validate the performance. This can lead to future improvement of the
system.
31
Bibliography
[1] <http://universityracing.tue.nl/ure10.html>, visited on 4-1-2016
[2] Bouwman K.R. (2014) Pre-Master’s Assignment: Design of an in-wheel reduction system,
2014.056, Eindhoven University of Technology
[3] Parfant A.G.P. (2015) Design of a formula student multibody simulation model, Eindhoven
Univesity of Technology
[4] John R. Hauser & Don Clausing (1988) The house of quality Harvard Business Review,
May–June, 63-73
[5] <https://woodgears.ca/gear/ratio.html>, visited on 11-1-2016
[6] <https://wikis.engrade.com/planetarygearsetsoperati>, visited on 14-1-2016
[7] <http://www.hexagon.de/zar5_ e.htm>, visited on 14-1-2016
[8] Prof. K.Gopinath & Prof. M.M.Mayuram Machine Design II, Module 2, Lecture 2, Indian
Institute of Technology Madras
[9] Dipl. Ing. I.Boiadjiev, Dr. Ing. J.Witzig, Dr. Ing. T.Tobie and Prof. Dr. Ing. K.Stahl (2015)
Tooth flank fracture - Basic principles and calculation model for a sub-surface-initiated fatigue
failure mode of case-hardened gears, Gear Technology archive
[10] <http://www.novexa.com/en/engrenage-defauts.php>, visited on 4-1-2016
[11] W.H. Dornfeld (2006) Gear tooth strength analysis Fairfield University
[12] <www.lancemore.jp>, visited on 5-1-2016
[13] Harry van Leeuwen (2006) Principes van Mechanische Componenten, Deel 2 Faculty of
Mechanical Engineering, Eindhoven University of Technology
[14] Richard L. Nailen (2002) Grease; What it is; How it works Electrical Apparatus
[15] O.J.M. Scholle (2016) Research to lubrication conditions of interacting gear and bearing
surfaces in an in-wheel final drive of a Formula Student race car Faculty of Mechanical
Engineering, Eindhoven University of Technology
[16] EAssistent Gear Manual Section 7.2.9 <http://www.eassistant.eu/fileadmin/dokumente/eassistant/etc/HTM
_ en/eAssistantHandb_ HTML_ ench7.html# x8-3130007.7>, visited on 10-1-2016
[17] <www.e-education.psu.edu>, visited on 10-1-2016
[18] Schaeffler Medias online product catalogue <http://medias.schaeffler.com/>, visited on
14-1-2016
[19] Wälzlagerpraxis 2015, fourth edition
[20] ERIKS BV online catalogue, visited on 14-1-2016
33
Appendix A
Gear design
A.1
Topologies
Referred to from Section 3.1.
The following figure shows four variations of a compound planetary gearset:
F IGURE A.1: Four variations of the compound planetary gearset.
34
Appendix A. Gear design
A.2
Definitions
Referred to from Section 3.3.
Figure A.2 shows an overview of two mating gears with relevant dimensions.
F IGURE A.2: Basic gear geometry.
While this picture specifies a lot of parameters, only few are relevant for the basic
calculations in this project. The necessary parameters are listed below:
Aspect ratio
The aspect ratio of a gear is the ratio between the facewidth of a gear and its pitch diameter.
W
(A.1)
D
There are basic rules of thumb when it comes to estimating gear sizes. Too long gears,
having a high aspect ratio, could lead to torsional twisting of the gears. This means
that, due to the finite torsional stiffness of the gears, the sections closest to the torque
application are the most effective. A small aspect ratio, on the other hand, is very prone to
alignment and manufacturing tolerances. In the first estimations, the aspect ratio aimed
for is in the range of 0.3 to 1.0.
b=
A.2. Definitions
35
Circular pitch
The circular pitch p is defined as the distance from one point on one tooth to the same
point on the next tooth, measured along the pitch circle (see Figure A.2).
The following relation can be found:
πD
(A.2)
z
Where z is the total number of teeth on the gear. The unit of the circular pitch is [m].
p=
Center distance
The center distance a is the distance between the centers of two mating gears.
Facewidth
Though not displayed in Figure A.2, the facewidth W is a very important gear parameter.
It is the width of a gear at the pitch diameter.
Modulus
To simplify certain calculations, engineers often tend to use a scaled measure for the
pitch, called the modulus, defined as:
D
p
=
(A.3)
π
z
The modulus is a widely accepted measure for the size of the gear teeth and is usually
given in [mm].
M=
In general, a high modulus has a positive effect on the strength of the gears. However, this reduces the smoothness of the transmission. When applied with a constant
ingoing torque, a transmission with a large modulus will show a larger torque rpmple on
its output than a similar transmission with smaller teeth. A small torque ripple has various benefits for the overall vehicle, including higher grip and better controllability for
traction control and other control strategies. As long as the strength calculations show a
small influence of the modulus on the size and weight of the gearbox, reducing the modulus has a positive effect on the performance of the final drive.
Standard gear modules range between 0.3 and 75mm, depending of the application.
The predecessing final drive had a minimum gear modulus of 1.25mm. A decrease in
modulus using standardised 0.25mm increments will be investigated.
Pitch diameter
The pitch diameter D of any gear can be seen as the ’effective’ diameter. It is not a directly measurable dimension, but rather a coordinate location at which other specifications are defined. All torques are defined at the pitch diameter, meaning the effective
radii used throughout this report are half the pitch diameter and vice versa the pitch
diameter equals twice the effective radius.
36
Appendix A. Gear design
Pressure angle
The pressure angle is the angle at which the line of action, the tangent to two gears’ base
circle, intersects the tangent of the pitch circles of both gears. This is also the angle of the
normal tooth force w.r.t. the tangent of the pitch diameter (see Figure 3.8).
A.3
Kinematics
Referred from Section 3.2.
Free body diagrams
In this analysis, a steady-state operation is assumed. Furthermore, any friction losses are
neglected.
Sun gear
Let Tin be the ingoing motor torque and F1−2 the force between the sun and the planet
gear. It is assumed that the torque will be distributed evenly over all planets, making
every force equal. Then, the diagram leads to Figure A.3:
F IGURE A.3: Free body diagram of the sun gear.
Using the moment equilibrium, relation A.4 can be derived:
Tin = Nplanets · F1−2 · r1
(A.4)
A.3. Kinematics
37
Planet gear
The free body diagram of the planet gears is given by Figure A.4.
F IGURE A.4: Free body diagram of the combined planet gear.
With F2−3 the reaction force from the ring gear on the planets, F1−2 the exerted force
from the sun gear (see Figure A.3), and Fout the reaction force.
Now, using the equilibrium of forces and moments more relations can be found. The
following can be determined:
Fout = F1−2 ·
r2 + r3
r3
(A.5)
r2
r3
(A.6)
F2−3 = F1−2 ·
Ring gear
Finally, the ring gear’s free body diagram is illustrated below in Figure A.5.
F IGURE A.5: Free body diagram of the ring gear.
F2−3 is the force on the planet gears and Tring,reaction the reaction torque that follows
from these forces (exerted from the ring’s housing or support frame). The moment equilibrium will be used to derive relation A.7.
Tring,reaction = Nplanets · F2−3 · (r1 + r2 + r3)
(A.7)
38
Appendix A. Gear design
Reduction ratio calculation
The output is generated by the sum of Fout of all planet gears. If again assumed that these
forces are equal for every planet, the following holds:
Tout = Nplanets · Fout · (r1 + r2 )
(A.8)
If equations A.4, A.5, A.6 and A.8 are combined, the outgoing torque can be calculated:
Tout = Tin ·
r2 + r1 r2 + r3
·
r1
r3
(A.9)
With the definition of the reduction ratio itotal as given in section 2.1.1, it can now be
expressed in terms of the gear radii:
itotal =
Tout
r2 + r1 r2 + r3
=
·
Tin
r1
r3
(A.10)
Velocity approach
The calculations are done with the variables displayed in Figure A.6.
F IGURE A.6: Angular and pitch line velocities of the final drive gears
When the sungear is rotating at an angular velocity ωin , the pitch line velocities of the
sun and the large planet gears, v1−2 , can be calculated as:
v1−2 = ωin · r1
(A.11)
Since the small planet gear has zero velocity at the contact point with the ring gear,
the velocities for all points on this component will depend linearly on the distance to
the contact point. Then, the following equation can be used to calculate the outgoing
(translating) velocity vout of the planets:
vout = v1−2 ·
r3
r2 + r3
(A.12)
The angular velocity of the planet ω2 and the outgoing angular velocity ωout can now
be written as:
ωp =
v1−2 − vout
r2
(A.13)
A.4. Strength analysis
39
ωout =
vout
r1 + r2
(A.14)
Using the velocity variant of the reduction ratio and substituting equations A.11, A.12
and A.14 leads to the following:
itot =
ωin
r1 + r2 r2 + r3
=
·
ωout
r1
r3
(A.15)
Now, another reduction ratio can be introduced to determine the angular velocity of
the planets, defined as:
i2 =
A.4
ωin
r2 + r3
=
ωp
r1
(A.16)
Strength analysis
Gear material
Referred to from Section 3.3.1.
The used gear steel is 17NiCrMo6-4. The mechanical properties are shown in Table
A.1.
TABLE A.1: Mechanical properties of hardened 17NiCrMo6-4.
17NiCrMo6-4
Density
7950
Surface hardness
60
Tooth root fatigue stress
430
Contact fatigue stress
1500
Tensile strength
1200
Young’s modulus
206
Poisson’s ratio
0.3
kg
m3
HRC
MPa
MPa
MPa
GP a
−
Tooth failure modes
Referred to from Section 3.3.2.
The following figure shows two failure modes, resulting from high bending and contact stress, respectively.
F IGURE A.7: Two gear failure modes; cracking (left) and pitting (right)
[9],[10].
40
Appendix A. Gear design
Lewis form factor
Referred to from Section 3.3.2.
The following figure gives the Lewis form factor YL for different gear geometries and
numbers of teeth.
F IGURE A.8: Relation between gear number and Lewis factor for various
tooth geometries.
Contact stress
Referred to from Section 3.3.2.
Leeuwen [13] states the following formula for contact stress in gear teeth:
σH
v
u
u
=u
t
FN ( ρ11 +
πW (
1−ν12
E1
1
ρ2 )
1−ν 2
+ E2 2 )
(A.17)
where σH is the calculated Hertzian stress in P a, FN the normal force on the tooth
flank in N (see Figure 3.8), ρ the radius of curvature of the respective tooth in m, ν the
dimensionless Poisson’s ratio and E the Young’s modulus in P a. The indices 1 and 2 refer
to the different gears in contact with each other. Since the same material and modulus
will be used for each gear, the equation simplifies to:
s
σH =
FN E
πW ρr (1 − ν 2 )
(A.18)
where ρr is the reduced radius of curvature, given by the pressure angle α and the
pitch radii of the gears r [13]:
ρr = sin(α)
r1 r2
r1 + r2
(A.19)
Finally, we need to add the relation between the tangential and normal tooth force:
FN =
Ft
cos(α)
(A.20)
A.4. Strength analysis
41
Combining Equations A.18, A.19 and A.20, leads to the following expression for the
contact stress on the tooth flank:
s
σH =
r1 + r2
2Ft E
2
πW (1 − ν ) sin(2α) r1 r2
Wöhlers curve
Referred to from Section 3.5.1.
The following figure shows a typical Wöhlers curve.
F IGURE A.9: Typical progression of a material’s fatigue curve [17].
(A.21)
42
A.5
Appendix A. Gear design
MATLAB modeling results
Referred to from section 3.4.3.
This section of the appendix contains the illustrations and gear designs obtained from
the several MATLAB models.
Sensitivity analysis
F IGURE A.10: Sensitivity of major dimensions on the overall reduction
ratio.
A.5. MATLAB modeling results
43
Gear modulus 1.00mm
F IGURE A.11: Final drive with M = 1.00mm, weight optimised by MATLAB modeling.
TABLE A.2: Optimisation results with M = 1.00mm.
M = 1.00mm
Concept
Sun gear
Planet gear (L)
Planet gear (S)
Ring gear
1
9.50
22.0
8.50
40.0
Radius [mm]
2
3
4
9.50 9.50 9.50
22.0 22.0 22.0
8.50 8.50 8.50
40.0 36.4 40.1
Facewidth [mm]
1
2
3
4
13.2 13.2 13.2 26.4
13.2 13.2 13.2 26.4
14.1 14.1 14.1 14.1
14.1 14.1 14.1 14.1
Number of teeth [-]
1
2
3
4
19
19
19
19
44
44
44
44
19
19
19
19
80
80
80
80
44
Appendix A. Gear design
Gear modulus 0.75mm
F IGURE A.12: Final drive with M = 0.75mm, weight optimised by MATLAB modeling.
The four variations are shown in Figure 3.12.
TABLE A.3: Optimisation results with M = 0.75mm.
M = 0.75mm
Concept
Sun gear
Planet gear (L)
Planet gear (S)
Ring gear
1
9.38
22.1
8.63
40.1
Radius [mm]
2
3
4
9.38 9.38 9.38
22.1 19.9 22.1
8.63 7.13 8.63
40.1 36.4 40.1
Facewidth [mm]
1
2
3
4
13.3 13.3 11.9 26.6
13.3 13.3 11.9 26.6
17.1 17.1 19.8 17.1
17.1 17.1 19.8 17.1
Number of teeth [-]
1
2
3
4
25
25
25
25
59
59
53
59
23
23
19
23
107 107 97
107
A.5. MATLAB modeling results
45
Gear modulus 0.50mm
F IGURE A.13: Final drive with M = 0.50mm, weight optimised by MATLAB modeling.
TABLE A.4: Optimisation results with M = 0.50mm.
M = 0.50mm
Concept
Sun gear
Planet gear (L)
Planet gear (S)
Ring gear
1
9.75
24.5
10.3
44.5
Radius [mm]
2
3
4
9.75 9.00 9.75
24.5 19.8 24.5
10.3 7.25 10.3
44.5 36.0 44.5
Facewidth [mm]
1
2
3
4
14.7 14.7 11.9 29.4
14.7 14.7 11.9 29.4
19.6 19.6 26.7 19.6
19.6 19.6 26.7 19.6
Number of teeth [-]
1
2
3
4
39
39
36
39
89
89
79
98
41
41
29
41
178 178 144 178
46
A.6
Appendix A. Gear design
KISSsoft modeling
Referred to from Section 3.5.
Loadcase
Instead of using one nominal torque as an input for the strength calculations, as is the
case with the MATLAB model, KISSsoft is capable of using the loadcase described in
section 2.1.3. The torques will be weighed into an average torque using the corresponding
velocities. Furthermore, an overload and a velocity factor are taken into account, due to
the transient behaviour of the final drive load.
Tooth interaction
A more elaborate model is used to calculate the interactions between two meshing gears.
The MATLAB model assumes that at all times a single interaction takes place. In reality,
the load is constantly divided over several teeth and over different sections of the teeth
flanks. KISSsoft can also analyse the slip occuring between two teeth and use this to
calculate a gear efficiency.
Profile shift
KISSsoft utilizes more tooth geometry parameters than just the modulus and pressure
angle. One particularly interesting parameter is the so-called profile shift. This is an
adjustment in the profile of the teeth that greatly affects the load capacity and sliding
characteristic of the gears. By shifting the profile reference in- or outwards a negative or
positive profile shift can be achieved, while keeping the same modulus.
F IGURE A.14: Three identical teeth with a different profile shift [16].
Figure A.14 shows the effect of profile shift; tooth two is the baseline tooth, while
number one has a positive and three a negative profile shift coefficient. Profile shift can
not be used completely freely; when the pitch circles of two mating gears are tangent,
they must always have an equal opposite coefficient. It can be easily seen that a positive
profile shift leads to a thicker tooth, increasing the gear’s load capacity [16]. KISSsoft can
calculate various profile shift coefficients for different optimisation goals.
A.7. Final designs
47
Weight reduction results
Referred to from Section 3.5.2.
The results of the facewidth reduction and the iterative reduction using KISSsoft are
shown below:
TABLE A.5: Weight reduction results for different concepts and procedures.
Single ring
Double ring
A.7
MATLAB
0.541
0.240
0.781
0.400
0.228
0.628
Stage 1
Stage 2
Total
Stage 1
Stage 2
Total
Facewidth red.
0.365
0.176
0.541
0.303
0.150
0.453
Iterative red.
0.293
0.168
0.461
0.293
0.156
0.449
Unit
kg
kg
kg
kg
kg
kg
Final designs
Referred to from Section 3.5.2.
The dimensions of the two final gear concepts are listed in A.6.
TABLE A.6: Geometrical properties of the final two gear concepts shown
in Figure 3.16.
Modulus
Center distance
Facewidth
Number of teeth
Total weight
Stage 1
Stage 2
Sun
Large planet
Small planet
Ring
Single ring
0.5
25.5
9.5
14
30
72
30
132
0.207
Double ring
0.5
25.5
9.5
13
30
72
30
132
0.198
Unit
mm
mm
mm
mm
−
−
−
−
kg
49
Appendix B
Bearing design
B.1
Tire forces
Referred to from Section 2.1.3.
The tire forces, resulting from a multi-body simulation model, are shown in Figure
B.1:
F IGURE B.1: Tire forces, as resulting from a multi-body simulation model.
50
Appendix B. Bearing design
B.2
Bearing topology
Referred to from Section 5.1.
The following gear topologies are considered:
• A needle roller - angular contact ball bearing
• Two crossed roller bearings
• Two angular contact ball bearings
• A double row angular contact ball bearing
All illustrations and parameters are found in [18].
One needle roller - angular contact ball bearing
F IGURE B.2: A needle roller - angular contact ball bearing.
The needle roller - angular contact ball bearing consists of two sections. The cylinders
(rollers) are very capable of withstanding radial forces, but cannot cope with axial forces.
Therefor, a single row of angular contact balls is added. This bearing can hold forces in
both axial directions.
Only one of these bearings would be needed. A suitable bearing would have the
following dimensions:
TABLE B.1: Dimensions and weight of the NKIB5913 bearing, see Figure
B.2.
Needle roller - angular contact ball bearing
d
65
mm
D
90
mm
B
38
mm
Weight 0.640
kg
B.2. Bearing topology
51
Two crossed roller bearings
F IGURE B.3: A crossed roller bearing.
The crossed roller bearing features rollers, which are oriented at a 45o angle with respect to the axle, with every second roller alternating direction. Each roller can exert
forces in radial and axial direction. Thus, the complete bearing can cope with radial and
axial forces in all directions.
Two of these bearings should be used. A single crossed roller bearing is relatively
compliant when it comes to radial axle forces being applied at a distance from the bearing. Two bearings at a distance from each other can easily handle a moment.
TABLE B.2: Dimensions and weight of the SX011814 bearing, see Figure
B.3.
Crossed roller bearing
di
70
mm
Da
90
mm
H
10
mm
Weight 0.3
kg
52
Appendix B. Bearing design
Two angular contact ball bearings
F IGURE B.4: An angular contact ball bearing.
The angular contact ball bearing has two rings with raceways and a series of steel
balls running between them. These bearings can only exert forces over the displayed angle α (see Figure B.4). This means every force has a radial and an axial (unidirectional)
component. By using two bearings in a mirrored set-up, forces in both axial directions
can be taken in. Also, by creating a distance, moments on the axle can be handled.
This serie offers several feasible types, listed in Table B.3:
TABLE B.3: Dimensions and weight of the 718..-B series, see Figure B.4.
7181.-B-TVH
d
D
B
Weight
1
55
72
9
0.058
2
60
78
10
0.070
3
65
85
10
0.085
4
70
90
10
0.140
5
75
95
10
0.096
6
80
100
10
0.101
Unit
mm
mm
mm
kg
NOTE: The weight of 71814-B-TVH, 0.140kg, seems unrealistic, looking at the other
types. It is treated as a misprint in the catalogue and estimated at 0.090kg.
B.2. Bearing topology
53
One double row angular contact ball bearing
F IGURE B.5: A double row angular contact ball bearing.
The double row angular contact ball bearing is basically a combination of two single
row angular contact ball bearings. These are mirrored into an O-arrangement and placed
in one inner and one outer ring. More information on X- and O-arrangements can be
found in Section B.3.
The weight of the double row bearings is approximately equal to an equivalent set-up
of single row bearings. A benefit of the double row bearing is that the part of the axle
on the other side of the load engagement will not suffer from bending. A drawback is,
however, that it would always result in a more compliant construction, since two single
bearings can be placed at a distance from each other, increasing stiffness.
TABLE B.4: Dimensions and weight of the 38..-B-2Z-TVH series, see Figure
B.5.
381.-B-2Z-TVH
d
D
a
Weight
1
55
72
13
0.90
2
60
78
14
0.140
4
70
90
15
0.190
6
80
100
15
0.230
Unit
mm
mm
mm
kg
54
B.3
Appendix B. Bearing design
Arrangement selection
Based on the information presented in the previous sections it becomes clear that the needle roller & angular contact ball bearing and the two crossed roller bearings will result in
a heavy design, leaving only the angular ball bearings as a viable option. As discussed
previously, seperating the bearings will result in a stiffer design. The stiffness of the design is very important, since it greatly affects the contact between the gear teeth. If the
gears aren’t properly alligned, especially under load, the efficiency will drop due to an
uneven lubrication film. Also wear will increase, because of the uneven loading. This
must be prevented, so two single row angular contact ball bearings are selected for the
final drive.
Bearing arrangement
The angular contact ball bearings need to be mirrored to be able to withstand axial forces
in both directions. There are two possible ways to do this. In the O-arrangement the
contact cones are pointing outwards, whereas they point inwards when configured in an
X-arrangement. This is illustrated in Figure B.6:
F IGURE B.6: Bearings configured in the O- (left) and X- (right) arrangement
[18].
The support base H is always larger in the O-arrangement and will thus always lead
to less tilting. A benefit of the X-arrangement is that the axial clearance will be reduced
in the case of thermal expansion, decreasing free play. This benefit is not obvious in the
O-arrangement. It is only present when the roller cones overlap, see illustration (2) in
Figure B.7 [18].
B.3. Arrangement selection
55
F IGURE B.7: Intersecting (left) and overlapping (right) roller cones in an
O-arrangement[18].
The length of the smallest possible bearing’s (71811-B-TVH) roller cone is 50.6mm
[18]. Since the total distance between the bearings is approximately 25mm, given by the
length of the gears, every combination of angular contact ball bearings will have overlapping roller cones, see (2) in Figure B.7. This results in the increase of tension, due to
thermal expansion. No extreme temperatures are expected in the final drive. It is, however, good to know that during an increase of temperature the bearings will not loosen.
The final configuration of the wheel bearings will be two angular contact ball bearings in an O-arrangement. Additionally, using two different diameters for the bearings
can ease the assembly of the final drive.
56
Appendix B. Bearing design
Bearing calculations
Referred to from Section 5.2.
In order to calculate the lifetime of the wheelbearings, the exerted forces on the wheel
should be converted to radial and axial forces on the bearings itself. The tire forces,
presented in Section 2.1.3 will be used as input. The MATLAB model can be found in
Appendix C.3.
The smaller wheel bearing (71812-B-TVH) is placed on the vehicle side of the wheel
and the larger wheel bearing (71815-B-TVH) is placed on the outer side of the wheel. The
position of these bearings is given in Figure B.8:
F IGURE B.8: Estimation of the wheel bearings’ location.
Using this location, the application point of the tire forces with respect to the bearings
is known. Now, the forces and moments can be used to calculate the radial and axial
forces on every bearing. Figure B.9 shows the bearings’ pressure points and the radial
and axial forces acting on them.
F IGURE B.9: Radial and axial forces working on the wheel bearings.
B.3. Arrangement selection
57
Lifetime
Based on a simulation using a multi-body model the axial and radial force and angular
velocity for every bearing in every wheel can be determined (Section 2.2). One of these
eight load maps has been displayed below:
F IGURE B.10: Load map of the inner bearing in the front left wheel, showing angular velocity and axial and radial forces.
To calculate the lifetime of the bearings based on these loadcases, an equivalent bearing force needs to be introduced, which combines the radial and axial forces, Fr and Fa ,
respectively, into one equivalent force P :
P = XFr + Y Fa
(B.1)
with X and Y factors given by the bearing’s manufacturer. For the dynamic loading
of angular ball bearings, the values of Table B.5 apply.
TABLE B.5: Values for bearing factors X and Y in different situations [18].
if FFar
≤ 1.14
> 1.14
X
1
0.57
Y
0.55
0.93
The mean equivalent force over an autocross can be calculated by using the time histories of every force, which are proportional to the speed at which they occur [19]. Because
every data point from the histories is sampled in a fixed time, the speed gives a relative
amount of revolutions at that point.
P̄ =
1 X
Pi ωi
ωtot i=1
(B.2)
Where Pi is the equivalent load and ωi the angular velocity at every data point and
ωtot the sum of all angular velocities. The estimated lifetime of the gears can now be
derived by [19]:
16666 C q
( )
(B.3)
n
P̄
with L the lifetime in hours, C the basic dynamic load rating in N , n the average
operating speed in rpm and q a dimensionless life exponent. This life exponent depends
on the type of bearings and is equal to 3 for all ball bearings.
L=
58
Appendix B. Bearing design
F IGURE B.11: Equivalent bearing force of the front left bearings over an
Autocross run.
The blue line in Figure B.11 represents the equivalent load and the green line the rotational speed. The red lines represent the average value, needed for Equation B.3. Now,
the lifetimes can be calculated, see Table B.6:
TABLE B.6: Calculated lifetimes of every bearing on the vehicle.
Wheel
FL
FR
RL
RR
Inner
48913
56744
47312
47141
Outer
159990
188060
101650
106030
Unit
hours
hours
hours
hours
The lifetime of the bearings is very high compared to the goal of 80hours. Smaller
bearings might be considered from this point of view, but they would not fit in the current
design of the final drive.
Worst-case scenario loadcase
As described in Section 2.1.3, the maximum loads that can occur on the tires are:
Fz,max = 2.06 · 103 N
Fy,max = 3.30 · 103 N
These loads can occur in three different loadcases; pure vertical load, pure lateral load
and combined load. Using the same equilibrium equations as in Section B.3, the radial
and axial loads on the bearings can be calculated:
TABLE B.7: Worst-case loadcases translated to bearing forces.
Loadcase
Fr,inner
Fa,inner
Fr,outer
Fa,outer
Vertical force
480
0
1580
0
Lateral force
6860
3300
6860
3300
Combined force
7340
3300
8440
3300
Unit
N
N
N
N
B.3. Arrangement selection
59
The maxmimum occuring forces are below the limits, specified by the manufacturer
[18].
61
Appendix C
MATLAB Scripts
C.1
Stresses
Referred to from section 3.4.2.
Visualization script available on request.
load('LewisFactor.mat')
% Material properties
rho = 7800;
E
= 200e9;
nu = 0.3;
% Material density [kg/m^3]
% Elasticity modulus [N/m^2]
% Poisson ratio [-]
% Force input
Ftang = 1.5e3;
% Applied force [N]
% Gear geometry
phi_n = 20*(2*pi/360);
Fw = 0.01;
r1 = 0.0075;
r2 = 0.0200;
%
%
%
%
Mod = 0.0005:0.00025:0.0015;
Pressure angle [rad]
Gear facewidth [m]
Radius of pinion gear [m]
Radius of wheel gear [m]
% Modulus vector
Y = interp1(TeethNrs,LewisF,round(2*r1./Mod));
% Interpolate Lewis Factor [-]
sigma_B = Ftang1./(Fw*pi*Mod.*Y);
sigma_H = ones(1,length(Mod))*sqrt(Ftang1/cos(phi_n)*E*(r1 + r2)/..
..(pi*Fw*sin(phi_n)*r1*r2*(1-nu^2)));
62
Appendix C. MATLAB Scripts
C.2
Optimisation
Referred to from Section 3.4.3.
Boundary conditions
The design iterations have been done considering the following requirements:
Outer dimension
The maximum diameter of the gearbox is 120mm (see section 2.1.2). This dimension is
given by:
Dout = r1 + 2r2
(C.1)
During the iterations of the model, all results with a larger outside diameter will be
discarded.
Collisions
When too many or too large planets are chosen (i.e. high r2 or high Nplanets ) it is possible
that these collide with eachother. A spacing of 10mm between the gears is minimum, as
there should still be space room for a carrier. If any planets are touching, the result is not
feasible and will be discarded.
Splanets =
2π
sin( Nplanets
)
sin( 12 π −
π
Nplanets )
· (r1 + r2 ) − 2r2 < 10mm
(C.2)
Tooth number
During the iterations, it can happen that a gear is calculated to have less than seventeen
teeth. This is the minimum acceptable number of teeth on any gear with a pressure angle
of 20◦ [8]. If less teeth are chosen, gear interference is guaranteed and the transmission
will not work. Solutions with less than seventeen teeth will be discarded.
Greatest common divisor
The greatest common divisor between any two interacting gears is a measure for the
amount of tooth pairs that are able to interact. The following illustrates an example:
Example
Two interacting gears have the following amount of teeth:
z1 = 16[−], z2 = 32[−]
(C.3)
In this case, the greatest common divisor is sixteen:
GCD(16, 32) = 16
The result can be seen in Figure C.1.
(C.4)
C.2. Optimisation
63
F IGURE C.1: Illustration of the effect of the greatest common divisor.
It can be seen that the red tooth of the smaller gear only interacts with the two red teeth on the
larger gear. Imperfections and deviations on a single tooth can cause wear on the specific paired
teeth of another gear. When the greatest common divisor is equal to one, every tooth will be in
contact with every other tooth of the opposing gear, spreading the wear over the complete gear
instead of concentrating it on fewer teeth.
It would be preferrable that the greatest common divisor between any two gears is
equal to 1, since every tooth will have the largest number of opposing teeth when in
operation.
Weight estimation
The gears’ weight is calculated by the following equations:
msun
mplanet,L
mplanet,S
mring
=
=
=
=
ρS πW (r12 − (r1 − trim )2 )
2
2
2
Nplanets ρS π W
3 (r2 − trim ) + Nplanets ρS πW (r2 − (r2 − trim ) )
2
2
Nplanets ρπW (r3 − (r3 − trim ) )
2ρπr4 trim W
where ρS is the density of the steel. Hereby it is assumed that the gears are hollow
and have a rim thickness trim of 5mm. The large planet gear is supported by a flange of
a third of its width.
Script
Referred to from section 3.4.3.
Visualization script available on request.
close all; clear all; clc;
load('LewisFactor.mat')
i
= 12;
% Desired reduction ratio [-]
Tin = 15;
% Ingoing maximum motor torque [Nm]
omega_in = 300*2*pi;
% Ingoing maximum motor speed [rad/s]
SpaceReservationPlanets = 0.01; % Space reservation between planet gears [m]
SpaceReservationGears
= 0.002;% Space reservation between gear stages [m]
Rtotmax = 0.06;
% Maximum total radius [m]
% Tooth properties
64
Appendix C. MATLAB Scripts
t_rim = 0.005;
% Estimated thickness of gear rims [m]
phi_n = 20*(2*pi/360);
% normal profile angle [rad];
% Material properties
rho = 7800;
E
= 210e9;
nu = 0.3;
sigma_H_max = 1500e6;
sigma_B_max = 430e6;
%
%
%
%
%
Material density [kg/m^3]
Elasticity modulus [N/m^2]
Poisson ratio [-]
Maximum allowable Hertzian stress [N/m^2]
Maximum allowable bending stress [N/m^2]
%% Start iterations
Mold = Inf;
for concept = 1:4
Mold = Inf;
for Mod = 0.0005:0.00025:0.001
for N_planets = 3:3;
for z1 = 14:80
for z2 = 14:100
j = j + 1;
r1 = z1*Mod/2;
r2 = z2*Mod/2;
% Radius of sungear (1) [m]
% Large gear radius of planet gear (2) [m]
%% Calculate 2D gear dimensions
% Calculation of r3
z3 = round(z2/(i*z1/(z2 + z1) - 1));
r3 = z3*Mod/2;
% Ring gear (3)
r4 = r1 + r2 + r3;
z4 = z1 + z2 + z3;
% Tooth number of small planet gear (2) [-]
% Small gear radius of planet gear (2) [m]
% Ring gear radius [m]
% Number of teeth on ring gear [-]
[F12,F23,Fout,T_p] = Calculateforces(r1,r2,r3,Tin,N_planets);
Tout = Fout*N_planets*(r1 + r2);
i2 = (r2 + r3)/r1;
% Determine outgoing torque [Nm]
% Ratio between ingoing and planet angular velocity (i2 =
% Tangential tooth forces
Ftang1 = cos(phi_n)*F12;
Ftang2 = cos(phi_n)*F23;
%% Determine minimal face width
% Determine minimum width by Hertzian contact pressure
Fw1vec(1) = 2*Ftang1*E*(r1 + r2)/(pi*sigma_H_max^2*(1 - nu^2)*sin(2*phi_n)*r1*r2);
Fw2vec(1) = 2*Ftang2*E*(r3 + r4)/(pi*sigma_H_max^2*(1 - nu^2)*sin(2*phi_n)*r3*r4);
% Determine bending stress
Y1 = interp1(TeethNrs,LewisF,z1);
Y2 = interp1(TeethNrs,LewisF,z3);
Fw1vec(2) = Ftang1/sigma_B_max/Mod/Y1;
Fw2vec(2) = Ftang2/sigma_B_max/Mod/Y2;
% Correct for minimum aspect ratio
if concept == 4
Aspmax1 = 2;
Aspmin1 = 0.6;
else
Aspmax1 = 1;
Aspmin1 = 0.3;
C.2. Optimisation
65
end
if concept == 3
Aspmax2 = 2;
else
Aspmax2 = 1;
end
Fw1vec(3) = Aspmin1*2*r2;
%% Select final face width
Fw1 = max(Fw1vec);
Fw2 = max(Fw2vec);
%% Determine gear mass;
M(1) = rho*pi*(r1^2 - (r1 - t_rim)^2)*Fw1;
% Mass of sungear [kg]
M(2) = 0.6*rho*pi*(r2 - t_rim)^2*Fw1/3*N_planets + rho*pi*2*r2*t_rim*Fw1*N_planets;
% Mass of large planet gears [kg]
M(3) = rho*pi*(r3^2 - (r3 - t_rim)^2)*Fw2*N_planets;
% Mass of small planet gears [kg]
M(4) = rho*pi*2*r4*t_rim*Fw2;
% Mass of ring gear [kg]
%% Iteration checks
if Fw1/2/r1 > Aspmax1
else if Fw2/2/r3 > Aspmax2
else if r3 < 0.007
else
Planetspacing = sin(2*pi/N_planets)/sin((pi - 2*pi/N_planets)/2)*(r1 + r2) - 2*r2;
if Planetspacing < SpaceReservationPlanets
else if (r1 + 2*r2) > Rtotmax
else if min([z1 z2 z3]) < 17;
else if gcd(z1,z2) ~= 1 || gcd(z3,z4) ~= 1
else
icalc = Tout/Tin;
if abs(icalc - i) > 0.6
else if sum(M) < Mold
Mold = sum(M);
r1opt = r1;
r2opt = r2;
r3opt = r3;
z1opt = z1;
z2opt = z2;
z3opt = z3;
Mopt = M;
N_planetsopt = N_planets;
Fw1opt = Fw1;
F1 = Ftang1;
F2 = Ftang2;
Fw2opt = Fw2;
Fw1vecopt = Fw1vec;
Fw2vecopt = Fw2vec;
iopt = Tout/Tin;
i2opt = i2;
Mod1opt = Mod;
Mod2opt = Mod;
Foutopt = Fout;
end end end end end end end end end end end end end
% Optimal solution
r1 = r1opt; r2 = r2opt; r3 = r3opt;
66
Appendix C. MATLAB Scripts
z1 = z1opt; z2 = z2opt; z3 = z3opt;
Fw1 = Fw1opt; Fw2 = Fw2opt;
Fw1vec
Fw2vec
Fout = Foutopt;
N_planets = N_planetsopt;
omega_2 = omega_in/i2;
end
C.3
Wheel bearings
Referred to from Section B.3.
Visualization script available on request.
% Bearing and tire dimensions
r_wheel = 0.23;
d1 = 0.085;
w1 = 0.01;
h1 = 0.01;
% Outer bearing mean diameter [m]
% Outer bearing width [m]
% Outer bearing height [m]
d2 = 0.069;
w2 = 0.01;
h2 = 0.009;
% Inner bearing mean diameter [m]
% Inner bearing width [m]
% Inner bearing height [m]
x1 = 0.010;
x2 = x1 + 0.0365;
alpha = 40/360*2*pi;
% Bearing pressure angle
% Positions of bearings
X11 = [x1 (r_wheel + d1/2)];
X12 = [x1 (r_wheel - d1/2)];
% Outer bearing
X21 = [x2 (r_wheel + d2/2)];
X22 = [x2 (r_wheel - d2/2)];
% Inner bearing
X1
X2
= [(x1 - d1/2*tan(alpha)) r_wheel];
= [(x2 + d2/2*tan(alpha)) r_wheel];
%% Load data
load('suspensionmodel_newdriver_baseline.mat')
%%
Corners = {'FL' 'FR' 'RL' 'RR'};
Bearings = {'Inner' 'Outer'};
Mmax = 0;
for i = 1:length(Corners)
for j = 1:length(Bearings)
Bearing.(Corners{i}).(Bearings{j}).Speed = data.(Corners{i}).tire.omega.data;
if i == 2 || i == 4
Fy = -data.(Corners{i}).tire.Fy.Data;
else
Fy = data.(Corners{i}).tire.Fy.Data;
C.3. Wheel bearings
67
end
Fz = data.(Corners{i}).tire.Fz.Data;
end
% Moment around (2)
F1A
F1R
M
F2A
F2R
=
=
=
=
=
zeros(length(Fy),1);
-(Fz*X2(1) - Fy*r_wheel)/(X2(1) - X1(1));
max(Fz*X2(1) - Fy*r_wheel);
zeros(length(Fy),1);
-(Fz + F1R);
for j = 1:length(Fy)
if Fy(j) > 0
F1A(j) = -Fy(j);
elseif Fy(j) < 0
F2A(j) = -Fy(j);
else
end
end
if Mmax < M
Mmax = M;
else end
Bearing.(Corners{i}).Inner.Frad = abs(F2R);
Bearing.(Corners{i}).Outer.Frad = abs(F1R);
Bearing.(Corners{i}).Inner.Faxi = abs(F2A);
Bearing.(Corners{i}).Outer.Faxi = abs(F1A);
end
Bearing = open('Bearing.mat');
Corners = fieldnames(Bearing);
C = [12.3e3 16.2e3];
p = 3;
% Basic dynamic load carrying number ([Inner Outer]) [N]
% Life exponent (for ball bearings: p = 3) [-]
z = 0;
for i = 1:numel(Corners)
Bearings = fieldnames(Bearing.(Corners{i}));
for j = 1:numel(Bearings)
z = z + 1;
Frad = Bearing.(Corners{i}).(Bearings{j}).Frad;
Faxi = Bearing.(Corners{i}).(Bearings{j}).Faxi;
X =
1*ones(length(Frad),1);
Y = 0.55*ones(length(Frad),1);
for k = 1:length(Frad)
if Frad(k)/Faxi(k) > 1.14
X(k) = 0.57;
Y(k) = 0.93;
else end
end
68
Appendix C. MATLAB Scripts
Bearing.(Corners{i}).(Bearings{j}).P = X.*Frad + Y.*Faxi;
P = sum(Bearing.(Corners{i}).(Bearings{j}).P .* Bearing.(Corners{i}).(Bearings{j}).Speed)
Bearing.(Corners{i}).(Bearings{j}).Pmean = P;
n = mean(Bearing.(Corners{i}).(Bearings{j}).Speed)/2/pi*60;
% Mean rotational speed of
Bearing.(Corners{i}).(Bearings{j}).Lifetime = 16666/n*(C(j)/P)^p;
end
end
end
69
Appendix D
Mechanical calculations
D.1
Planet axle
Referred to from Section 5.3.
The planet axles are subjected to the loads shown in Figure D.1. The magnitude and
direction of the forces under full motor torque, 33N m, follow from the gear kinematics.
F IGURE D.1: Loadcase of the planet gears and axle.
The forces from Figure D.1 and their magnitude are calculated from the moment and
force equilibrium equations and shown in Table D.1:
TABLE D.1: Name and magnitude of parameters, shown in Figure D.1.
Parameter
Tangential force from sun on planet
Radial force from sun on planet
Tangential force from ring on planet
Radial force from ring on planet
Outer tangential reaction force on axle
Outer radial reaction force on axle
Inner tangential reaction force on axle
Inner radial reaction force on axle
Length of axle segment 1
Length of axle segment 2
Length of axle segment 3
Abbrevation
F1−2 , t
F1−2 , r
F2−3 , t
F2−3 , r
Fro,t
Fro,r
Fri,t
Fri,r
Lp,1
Lp,2
Lp,3
Magnitude
1520
550
3640
1330
2280
-280
2890
1050
0.0055
0.012
0.0075
Unit
N
N
N
N
N
N
N
N
m
m
m
70
Appendix D. Mechanical calculations
The loadcase is applied to a FEM (Finite Element Method) model. To realistically
simulate the effects of bearing backlash and compliance, these are modelled as a shaft of
PVC.
F IGURE D.2: FEM Loadcase of a planetaxle.
The model is solved and the following stresses are found:
F IGURE D.3: Strength simulation results of a planetaxle.
Steel has a maximum bending stress of 430M P a in its fatigue limit, see Table A.1.
The applied forces are an extreme loadcase and the resulting stresses are well beneath
the limit.
71
Appendix E
Technical drawings and datasheets
E.1
Brake assembly
Brake Disc
Referred to from Section 2.1.5.
F IGURE E.1: Technical drawing of the brake disc showing relevant dimensions.
72
Appendix E. Technical drawings and datasheets
Brake Callipers
Referred to from Section 2.1.5.
F IGURE E.2: Technical drawing of AP Racing’s CP4226-2S0.
F IGURE E.3: Technical drawing of AP Racing’s CP4227-2S0.
E.2. Wheel bearings
E.2
Wheel bearings
Referred to from Section 5.1.
71812-B-TVH
F IGURE E.4: Datasheet of the inner wheel bearing, 71812-B-TVH [18].
73
74
Appendix E. Technical drawings and datasheets
71815-B-TVH
F IGURE E.5: Datasheet of the outer wheel bearing, 71815-B-TVH [18].
E.3. Planet bearings
E.3
Planet bearings
Referred to from Section 5.3.
F IGURE E.6: Datasheet of a planet bearing, K7x10x8-TV [18].
75
76
E.4
Appendix E. Technical drawings and datasheets
Simmerring
Referred to from Section 5.3.
F IGURE E.7: Technical specifications of ERIKS FPM Simmerring [20].
77
Appendix F
Illustrations
Referred to from Section 5.4.
78
Appendix F. Illustrations
Appendix F. Illustrations
79
Declaration of Scientific Conduct