The Re-Design and Analysis of The Suspension System On The Formula Student Race Car

Department of Mechanical Engineering
BEng Mechanical Design Engineering
David Michael Browne 1007185
Final Year Project 2013-2014
The Re-Design and Analysis of the Suspension System
on the Formula Student Race Car
Supervisor: Dr Christopher Murray

Abstract
Inspired by the FSAE Formula Student competition, a fully fledge working suspension system was
designed for the race car in order to solve the issues that were detected early on in the
troubleshooting stage. All parts except the engine and seat were designed from first principles.
By understanding the basic concepts of suspension systems and mechanical design, this project aims
to give an insight in to re-designing the current car. To fully understand the chassis-suspension
geometry model, a ground foundation in vector theory was established and specifically applied to
the new design in order to determine the positions of all external components relative to the chassis.
Once the geometry and layout of the new chassis-suspension model was determined, the next stage
of the process was to design all the external components and run each of the custom-made parts
through a stress analysis. Ground knowledge of finite element analysis was required in order to
understand the appropriate forces and constraints to apply to the new parts. Each of the custom
parts were run through SolidWorks Simulation and validated.
Knowledge of composites, metals and methods of manufacturing was also required for the design of
the swing arms, pushrods, uprights and bell cranks.
Furthermore, an understanding of sprung and unsprung mass dampener systems was established.
Using basic calculations and MATLAB Simulink to run a spring-dampener simulation, an off-the-shelf
shock absorber was selected for the new race car.
The Re-Design and Analysis of the Suspension System on the Formula Student Race Car 1
Objectives
 Research the various types of suspension systems by giving a brief outline on suspension
theory.
 Troubleshoot any potential design issues on the existing car by carrying out inspections and
speaking to members of the Formula Student team.
 Re-design the suspension system and investigate the use of advanced materials without
compromising structural integrity.
 Determine the most elegant user-friendly interface between the car and race team members.
 Determine the parameters that influence both static and dynamic motion in the chassis-
suspension system by carrying out sprung and unsprung mass calculations. Using these
results, further determine an appropriate shock absorber to purchase ‘off the shelf’.
 Determine appropriate manufacturing methods for each of the individual custom
components.
 Construct a fully-fledged working SolidWorks model demonstrating all the position geometry,
linkages and joining of the components.
Acknowledgements
At this stage I would like to take the opportunity to thank Dr. Christopher Murray of the University of
Glasgow for his support throughout the project. His excellent guidance leant itself to being able to
design a fully-fledged mechanically-working racing car.
I would also like to thank Holly Lockhart from the 2013/2014 Glasgow University Formula Student
team for allowing access to the workshop and for her general guidance about suspension systems.
Lastly I would like to thank Klemen Erzen and Cameron Reid of GRABCAD for providing the Recaro
racing seat and Ducati motorcycle engine respectively.
Contents
1 Introduction ..................................................................................................................................................................... 6
1.1 Types of Suspension Systems ................................................................................................................................. 6
1.1.1 Solid-axle....................................................................................................................................................... 6
1.1.2 Independent wheel suspension .................................................................................................................... 7
1.1.3 Trailing arm (swing arm) independent suspension ....................................................................................... 7
1.1.4 MacPherson strut independent suspension ................................................................................................. 7
1.1.5 Double wishbone independent suspension .................................................................................................. 7
1.1.6 Multi-link double wishbone suspension ....................................................................................................... 8
1.1.7 Short-long-arm double wishbone suspension .............................................................................................. 8
2 Current Design ............................................................................................................................................................... 10
3 Design............................................................................................................................................................................. 11
3.1 Chassis Geometry ................................................................................................................................................. 11
3.2 Chassis Stress Analysis .......................................................................................................................................... 13
3.3 Wheel rim ............................................................................................................................................................. 14
3.4 Tire ....................................................................................................................................................................... 14
3.5 Vehicle Suspension Geometry .............................................................................................................................. 15
3.5.1 Vector Theory: An Outline .......................................................................................................................... 16
3.5.2 Vector Dot Product ..................................................................................................................................... 17
3.5.3 Vector Cross Product .................................................................................................................................. 18
3.5.4 Vector Rotation and Component Vectors ................................................................................................... 19
4 Spherical Joints, Rod Ends and Upright Design .............................................................................................................. 22
4.1 Spherical Joints and Rod Ends .............................................................................................................................. 22
4.2 Uprights ................................................................................................................................................................ 24
4.3 Stress Analysis: Uprights....................................................................................................................................... 25
4.3.1 Forces on wheel .......................................................................................................................................... 25
4.3.2 FEA and SolidWorks Simulation .................................................................................................................. 28
5 Swing Arms and Push Rods ............................................................................................................................................ 30
5.1 Rod Stress Analysis ............................................................................................................................................... 31
5.2 Final rod design .................................................................................................................................................... 33
6 Bell cranks and Shock Absorber Mounts ........................................................................................................................ 34
7 Shock Absorbers, Spring Damping Analysis .................................................................................................................... 36
7.1 Quarter Car Sprung Mass Model .......................................................................................................................... 36
7.2 Results .................................................................................................................................................................. 37
8 Final Designs................................................................................................................................................................... 38
9 Conclusion ...................................................................................................................................................................... 49
10 References................................................................................................................................................................. 49
10.1 Books, journals and lecture notes ........................................................................................................................ 49
10.2 Figures .................................................................................................................................................................. 50
10.3 Equations .............................................................................................................................................................. 50
11 Appendix ................................................................................................................................................................... 51
List of Figures
Figure 1-1: Solid axle suspension 6
Figure 1-2: Trailing arm suspension 7
Figure 1-3: MacPherson strut suspension 7
Figure 1-4: Double wishbone suspension 7
Figure 1-5: Multi-link suspension 8
Figure 1-6: Camber angle 8
Figure 1-7: SLA configuration 8
Figure 1-8: Instant centre 9
Figure 1-9: Roll centre 9
Figure 2-1: Current front suspension design (provided courtesy of FSAE Glasgow University) 10
Figure 2-2: Current rear suspension design (provided courtesy of FSAE Glasgow University) 10
Figure 3-1: Determining new chassis geometry 11
Figure 3-2: Side view properly triangulated chassis frame members 12
Figure 3-3: Top view properly triangulated frame members 12
Figure 3-4: Applying weldments to 3D sketch 13
Figure 3-5: Force applied in the ‘x’ direction showing exaggerated bending stresses 13
Figure 3-6: Tire and wheel assembly 14
Figure 3-7: Hoosier slick tire 14
Figure 3-8: Coordinate systems defining the angle of approach to the chassis hard points 15
Figure 3-9: Position vector P 16
Figure 3-10: Relative position vector 16
Figure 3-11: Vector dot product 17
Figure 3-12: Dot product being used to check for perpendicularity of a universal joint 18
Figure 3-13: Illustrating vector cross product and positive rotation around {C} 1 18
Figure 3-14: Application of vector cross product in the yoke design of a universal joint 19
Figure 3-15: determining the relative reference coordinates for suspension hard points 19
Figure 3-16: 13 Illustrating vector {S}1 as a series of component vectors Sx, Sy and Sz 20
Figure 3-17: Rotation about α from S to S’ with X and Y components 20
Figure 3-18: Points S and T illustrating where the centre points of the uprights 21
Figure 3-19: Axis showing six degrees of freedom 21
Figure 4-1: Adjustable spherical joint at chassis hard points 22
Figure 4-2: Underside view of chassis and associated hard point mountings 23
Figure 4-3: Front outer rod ends fully defined with adjustable thread and spherical joint mount 23
Figure 4-4: Points S and T determining the length of upright 24
Figure 4-5: Front and rear upright attachment points 25
Figure 4-6 26
Figure 4-7 26
Figure 4-8 26
Figure 4-9: (a) front upright mesh, (b) applied constraints and forces, (c) simulation 29
Figure 4-10: (a) rear upright mesh, (b) applied loads and constraints, (c) simulation 29
Figure 5-1: Rod ends linearly connecting to chassis hard points 30
Figure 5-2: Carbon fibre swing arm and push rod profile 30
Figure 5-3: Carbon fibre types 31
Figure 5-4: Newly calculated combined modulus of 22.45GPa in row Elastic Modulus in X 32
Figure 5-5: (a) Rod mesh, (b) simulation of rod ends with applied constraints and forces, (c) carbon rod
simulation 33
Figure 5-6: Rod ends, rods and pre-load nut assembly illustrating thread direction 33
Figure 6-1: Red lines show the sketch lines determining angle for bell crank 34
Figure 6-2: Distance d showing the horizontal component for placement of shock absorber 34
Figure 6-3: Bell crank (B) and shock absorber (A) chassis-mounts 35
Figure 6-4: Final bell crank design 35
Figure 6-5: Bell crank in context with chassis 35
Figure 7-1: Free body diagram of quarter car model 36
Figure 7-2: Simulink template for sprung mass car model 37
Figure 7-3: Results for the newly designed suspension system 37
Figure 8-1: Chassis-suspension assembly 38
Figure 8-2: Chassis-suspension assembly 2. Shock absorbers mounted flush with the chassis 39
Figure 8-3: Hidden chassis-suspension view 40
Figure 8-4: Top view chassis-suspension assembly 41
Figure 8-5: Top view rear chassis-suspension assembly showing repositioning of shock absorber 42
Figure 8-6: Top view front chassis-suspension assembly 42
Figure 8-7: Full chassis suspension assembly 43
Figure 8-8: Full chassis-suspension assembly (wheel steer) 44
Figure 8-9: Full assembly exploded view 45
Figure 8-10: Fully-fledged superimposed model image of front wheel suspension system under load 46
Figure 8-11: Fully-fledged superimposed model image of rear wheel suspension system under load 46
Figure 8-12: Ohlins shock absorber 47
Figure 8-13: Front left upright x 1 Figure 8-14: Rear upright x 2 47
Figure 8-15: Front suspension assembly inc. steering rod (Frameless), fully adjustable rods for camber angle 48
Figure 8-16: Rear suspension assembly (frameless) 48
1 Introduction
The need for high performance suspension systems in race cars lies to do with optimising vehicle
performance when the vehicle is subjected to severe physical and dynamic conditions. This must be
achieved while maintaining control and uncompromising vehicle and driver safety. From initial
concept design to selecting appropriate materials that can withstand environmental and dynamic
forces, there exist a series of technical challenges engineers must consider when designing a
suspension system.
1.1 Types of Suspension Systems

Selecting a suspension system is dependent on the application and type of motor vehicle concerned.
With current motorcar designs, engineers have to consider everything from chassis design to the
amount of luggage space the car can facilitate. In terms of suspension, vehicle handling is the main
area of focus for many car manufacturers. For high performance car manufacturers and even
Formula 1 teams, it becomes the very heart and soul of vehicle design.
There are various types of suspension systems and selecting one can be a challenge in itself.
Designers are limited to various parameters from vehicle space to cost. As performance and
optimisation was the project’s main area of focus, cost was not considered.
1.1.1 Solid-axle
Arranging the wheels along a solid beam is referred to as a solid axle system. The motion of the
wheels is constrained to allow only a translation in the vertical direction by mounting the springs and
dampeners perpendicular to the axle. If one wheel were to experience bump however, the other
wheel is usually affected. Conventional vehicle designs employed the solid axle due to its simplicity
and ease of maintenance. Shown in Figure 1-1 is an example of a solid axle beam with leaf springs
and dampeners connected perpendicular to the direction of the beam.
Solid-axle suspensions are restricted to the level of adjustment and are therefore usually employed
in conventional vehicles and in lorry trailers where adjustment is not necessary [1a].
Figure 1-1: Solid axle suspension
1.1.2 Independent wheel suspension
Independent wheel suspension systems allow the motion where if one wheel where to experience
bump the other wheels would not necessarily be directly affected. The wheels are connected to the
chassis through a series of control arms that allow vertical motion through a spring-dampener
connected to the arm. This method of connecting the wheels-brake assembly to the chassis is
practised in modern vehicle design to accompany for smoother ride comfort and adjustment. There
are various types of independent suspensions, each displaying their own characteristics and benefits
depending on their application.
1.1.3 Trailing arm (swing arm) independent

suspension
Trailing end is characterised by having the wheel assembly
connected by a control arm that connects through a rotational
joint mounted to the chassis with the spring dampener
mounted above the control arm [1b]. Figure 1-2: Trailing arm suspension
1.1.4 MacPherson strut independent suspension

MacPherson Strut suspension systems are characterised by having
the spring-dampener mounted directly to the wheel mount (upright)
[1c]. Many modern vehicles employ this design for the ease of access,
adjustment and space saving. Due to its geometry, the strut assembly
can be mounted directly to the chassis therefore saving the number
of parts and weight (Figure 1-3). Adjustment is restricted however so
this system is usually employed in your average family car.
1.1.5 Double wishbone independent suspension Figure 1-3: MacPherson strut suspension
Double wishbone suspension systems are becoming more

common in modern vehicle design due to its higher level of
performance and stability. The wheel assembly is supported by a
top and bottom swing arm. This gives the motion of the wheels
more stability during bump or cornering. It also allows more room
for adjustment depending on the application [1d].
Figure 1-4: Double wishbone suspension
1.1.6 Multi-link double wishbone
suspension
Multi-link double wishbone suspension
systems [1e] are characterised by having
multiple links connecting the wheel
assembly to the chassis. The main reason
for this type of system is its capacity to
change the geometry depending on the
application. Due to its complexity, this
setup is usually found on racing cars where
ride heights and track width may wish to be
changed depending what the track Figure 1-5: Multi-link suspension
conditions are. For these reasons and these

reasons only, a multilink approach will be taken in the design of the race car.
1.1.7 Short-long-arm double wishbone suspension

For the race car, the short-long-arm approach was taken. Short-long-arm suspension systems (SLA)
are a form of double wishbone suspension system (Figure 1-7) however configuration allows for
further adjustment of the wheel assembly and in particular, camber angle.
Camber is the degree of the wheel that is angled from an imaginary perpendicular axis from the
ground (Figure 1-6) [2a]. This is also known as a transversal quadrilateral suspension as can be seen
marked by the red lines with the top swing arm being longer than the bottom swing arm.
Figure 1-7: SLA configuration Figure 1-6: Camber angle
The reason for camber angle is to compensate for the lateral forces that are exerted on the wheels
when cornering at high speed. These lateral forces translate through the tires and wheel assembly to
the suspension swing arms. As a result, these forces act on the chassis and can ‘tilt’ the vehicle. The
severity of tilt can therefore affect the centre of gravity of the vehicle which can ultimately result in
the car flipping. By designing the swing arms so that if there was a theoretical line extended along
Figure 1-8: Instant centre
the swing towards the opposite wheel (‘instant centre’) [3a] , then another line drawn from where
those two lines meet to the patch centre of the tires, a theoretical ‘roll centre’ is determined [3b].
The role centre (R.C) is the important factor as it determines how much the chassis will role about its
axis when lateral forces act on the wheels during vehicle cornering. Having the role centre as close
to the centre of gravity will reduce the overall feeling of roll. In order to achieve this, the angle and
length of swing arms and length of uprights are important factors that determine the position of the
R.C.
Figure 1-9: Roll centre
2 Current Design
The Formula Student race car for the University of
Glasgow was the basis and main drive for the project.
Conducting research through speaking with team
members and inspecting the car itself, there were a
number of aspects of the car that could be improved
for the future. At first glance, the front shock absorbers
were mounted above the shell for a ‘push rod’
configuration (Figure 2-1). Push rods connect from the
base of the lower swing arms to the bell crank
mounted to the chassis. For good reason, they were
mounted for ease of access to quicken the time taken
to adjust depending on race conditions. However they
would affect the aerodynamics of the car. The air
turbulence generated behind the rear of the shock would Figure 2-1: Current front suspension design
(provided courtesy of FSAE Glasgow University)
disrupt the flow of air across the remaining half of the
shell. This was something that was considered in the design analysis by re-positioning the shock
absorbers in-line with chassis members and under the shell for the new design.
At close inspection, the rear shock absorbers are mounted from the bell cranks and run across the
drive train and fixed to a bracket on the rear of the chassis (Figure 2-2). Similar to above, they were
mounted here for ease of access however issues can occur when accessing the drivetrain. The
positioning of the shocks makes accessing the drivetrain difficult. The newly proposed design will
compensate for this while not compromising suspension geometry or the vehicle’s centre of gravity.
Figure 2-2: Current rear suspension design (provided

courtesy of FSAE Glasgow University)
3 Design
Having troubleshot and listed the possible improvements that could be made to the car, a thorough
design approach was taken in order to check off all the ‘things to do’. In order to do this it was not as
simple as taking an existing chassis and mounting fixings for the swing arms and push rods to bolt on
to. Furthermore, having discussed the developments that could be made to improve access to the
vehicle (Section 2: Current Design), it was discovered that re-designing a completely new chassis was
necessary to fulfil all the requirements.
3.1 Chassis Geometry

The initial step in the design process was to define the chassis geometry. This part of the process
was the most crucial as it would determine how all the swing arm fixings and joints were going to
Figure 3-1: Determining new chassis geometry
connect and align correctly to the chassis [Figure 3-1]. A series of 3D sketches were drawn out and
dimensioned with respect to a set of defined sketch planes at a set distance apart from each other.
This allowed ease of visualising 3D when working in 2D. Using the ‘Tab’ button on the keyboard this
made switching between the 3 planes (XY, YZ, ZX) quicker while maintaining accuracy and alignment,
without too much confusion.
Chassis frame geometry was kept under strict control in accordance to the FSAE rule T3.3 where by
it states the chassis should be ‘properly triangulated’ (Figure 3-2 and Figure 3-3). This meant that if
all sketch lines (and therefore frames members) were to be projected on to a plane where a force(s)
were to act in any direction along those projections, the frame members would only experience
tensile or compressive forces. This is of particular importance if the vehicle was subject to impact.
The forces would translate through the chassis and distribute through the triangulated points more
efficiently and as a result, strengthening the chassis.
Figure 3-2: Side view properly triangulated chassis frame members
Figure 3-3: Top view properly triangulated frame members
The next stage of the process was to apply ‘weldments’ to the 3D sketch. This required the tubing to
comply with T3.4.1 where it states the minimum diameter for frame members to be 25.4mm with a
wall thickness of 2.4mm. Shown in Figure 3-4 is the chassis sketch with the applied weldments.
Additionally shown is the suspension geometry in which the swing arms were fixed to the chassis.
What is clear is where the wheels and tires will be placed relative to the chassis. More of this section
will be covered later.
Figure 3-4: Applying weldments to 3D sketch
3.2 Chassis Stress Analysis

Having built a chassis with all the correct geometry and complying with FSAE rules and regulations, it
was critical to test for stress. This was performed using SolidWorks simulation software. FSAE rules
state the chassis should be able to withstand a force acting in the ‘x’ direction of up to 80kN before
Figure 3-5: Force applied in the ‘x’ direction showing exaggerated bending stresses
plastic deformation shown in Figure 3-5. The chassis was modelled using shelled elements for the
pipe sections due to their computational effectiveness (and processing time), compared to
modelling in solid elements.
3.3 Wheel rim
FSAE rule T6.3.1 states “The wheels of the car must be 203.2 mm (8.0 inches) or more in diameter”.
Taking this in to consideration it allowed quite a bit of freedom in the physical design of the wheel
and the choice of materials. A simple wheel and tire was designed (Figure 3-6) for the purpose of the
suspension system and to fulfil the requirements of the analysis mentioned later in the report. In
order to reduce the vehicle’s weight, improve acceleration and handling, forged magnesium alloy
wheels (similar to kind used in Formula 1 and high performance motorsport) were used rather than
the standard aluminium alloy found on the average car. Another benefit to using forged magnesium
alloy wheels is its value for the principal moment of inertia. This is defined as the amount of torque
required for a desired angular acceleration about an axis of rotation. For this particular case, it was
taken about the centre of mass (CoM) of the wheel, Pz = 0.1 kgm2.
Table 3-1: Wheel specification
Material Wheel Mass Principal Moments

(kg) of Inertia at CoM
(kgm2)
Aluminium alloy 7.66 Px = 0.09

Py = 0.09
Pz = 0.15
Magnesium alloy 5 Px = 0.06
Py = 0.06
Figure 3-6: Tire and wheel assembly Pz = 0.10
SolidWorks was used to calculate the mass of the rim and the principal moment of inertia about the
centre of gravity and it was found magnesium alloy was the better choice (Table 3-1).
3.4 Tire
For tyre selection, a standard Hoosier racing slick tyre was chosen for the new rim design. The
reason for this choice was Hoosier have strong existing links
with the FSAE Formula Student organisation.
Table 3-2: Tire specification (courtesy of Hoosier)
Tire Tire Tire Tire Tire spring rate @

diameter width weight pressure 200lb static load
18” 6” 8kg 16psi 645.16lbs/in
Figure 3-7: Hoosier slick tire
3.5 Vehicle Suspension Geometry
In order to determine the suspension geometry, a series of hard points with known coordinates in
space were required. Hard points are the theoretical centres of the spherical joints where the swing
arms connect to the chassis. Due to chassis geometry design and FSAE rules compliance, these hard
points had to coincide with existing chassis members in order to avoid non-triangulation. Due to the
asymmetrical nature of the swing arms – mentioned in section 1.1.7 on short long arm suspension –
positioning the hard points required a series of sketches extending out from the chassis fixed points
(points Q, R, U and V) to coordinate systems (points S and T), and marking a centre point where the
theoretical centre of the spherical joints would be placed – demonstrated by the red circles in Figure
3-8. In order to progress any further, a background in vector theory was required.
Figure 3-8: Coordinate systems defining the angle of approach to the chassis hard points
3.5.1 Vector Theory: An Outline
For any given vector P, its position is referenced relative to an origin O. The position vector can be
described to have a set of three components with magnitudes PX, PY and PZ relative to the axis of the
reference frame O (Figure 3-9) [2b].
Figure 3-9: Position vector P
The position vector P can be expressed in matrix notation where by denotes the vector being
measured from the centre origin O.
[ ] (3.1)
Magnitude can be determined for by simply applying Pythagoras’ theorem yielding a term for
.
√ (3.2)
These will come in to use when determining positions of spherical joints and distances from the
chassis fixed points to the coordinate systems outside of the chassis. Because suspension geometry
will vary from the global reference frame for all four wheels, it is important to set up a series of
relative position vectors to allow for dimensions to be made relative to each and every hard point
along the chassis.
Figure 3-10: Relative position vector
Figure 2.8 illustrates position P relative to O. It also represents position P relative to Q. Position
vector Q is represented by which is the summation of both and .
(3.3)
Determining the magnitude of relative position vector by using Pythagoras’ method similar in
(3.2) will obtain
√ (3.4)
3.5.2 Vector Dot Product

Obtaining the dot product of two vectors is an important factor in the application of designing the
joints that will be used to connect the swing arms to the chassis, and in particular, checking for
perpendicularity – which will be explained shortly.
Taking the dot product of two vectors and this will result in a scalar C. The magnitude of
this new scalar product C will therefore have a magnitude equal to the product of the magnitude of
the two coincident vectors and the cosine angle between them (Figure 3-11) [2c].
Figure 3-11: Vector dot product
(3.5)
By simply rearranging (3.5) will yield an expression for the angle between the two vectors
(3.6)
For example: the use of a universal joint in a drive shaft that is subject to bump on the rear wheels.
In this application demonstrating the joint’s perpendicularity when the vehicle is subject to static
load i.e. when the vehicle’s vertical velocity and acceleration is zero, which then can be compared to
when the vehicle experiences a vertical force and therefore changing the value of the angle.
Figure 3-12: Dot product being used to check for perpendicularity of a universal joint
In the case of the universal joint (Figure 3-12) scalar dot products are used to enforce
perpendicularity by making:
(3.6)
3.5.3 Vector Cross Product

The vector product of two coincident vectors and produces a vector .The vector
is perpendicular to the plane of both vectors and . This can be illustrated below in
Figure 3-13 where it also shows the direction of vector as a rotation about in the positive
direction from to [2d].
Figure 3-13: Illustrating vector cross product and positive rotation around {C}1
Similar to the dot product, the magnitude of the vector product is simply the product of the two
magnitudes of the two coincident vectors multiplied by the sine angle between them (3.7).
(3.8)
The direction of the resultant vector is determined by calculating X . Firstly has to be

arranged in skew-symmetric form resulting in
[ ] [ ][ ] [ ] (3.9)
Figure 3-14: Application of vector cross product in the yoke design of a universal joint
Figure 3-14 demonstrates a practical application of the vector cross product. It illustrates the
rotation around – say the rotation of a drive axel – on the plane.
For the purpose of this project and in particular determining the suspension geometry and its
changes in position due to road undulations and vehicle dynamics, it is important to consider vector
rotations in order to fully understand the role of vectors in suspension system applications. Figure 3-
15 illustrates part of the chassis model where a set of coordinate systems have been put in place to
mark the theoretical centres for the ball joints. Note the coordinates are relative to the global
reference frame for the entire vehicle – this meaning all coordinates of hard points, spherical joints,
lengths and dimensions of swing arms will be taken relative to the global reference frame from
hereon.
Figure 3-15: determining the relative reference coordinates for suspension hard points
3.5.4 Vector Rotation and Component Vectors

To put vector rotation in to context of the suspension system and its coordinates, take coordinate
system 2 (CS2) in Figure 3-15. This will provide the design with a chassis hard point for the spherical
joint to be fixed to.
In the context of the chassis-suspension system, vector rotation is arguably the most relevant and
important factor when determining the dimensions for suspension geometry. For the sake of the
analysis the swing arms will be represented by vectors. Figure 3-16 illustrates a vector with
reference to a local reference frame denoted CS2. As can be seen, the initial position vector S has
component vectors along the X, Y and Z axis denoted by , and .
Figure 3-16: 13 Illustrating vector {S}1 as a series of component vectors Sx, Sy and Sz
When the vehicle experiences bump the vectors will rotate through angle α and in turn, their X and Y
components will also change. To describe what is happening a rotation has been set around Z so
that the z component of vector remains unchanged and leaving the z axis normal to the page
(Figure 3-17).
Figure 3-17: Rotation about α from S to S’ with X and Y components
Mathematically this is expressed as
(3-10)
(3-11)
(3-12)
The same approach was used in order to determine the vector rotations of point T (Figure 3-18) and
for the other three wheels. The next stage of the process was to design the rod ends. These are
machined pieces of alloy steel that connect the swing arms to the uprights. Having acknowledged
the vector rotations of the swing arms for all four wheels, it was essential to design the rod ends to
compensate for the vertical motion the wheels will undergo, while maintaining the wheels’
verticality as much as possible.
Figure 3-18: Points S and T illustrating where the centre points of the uprights
In order to achieve this, an understanding of degrees of freedom and constraints of parts was
required. For any free-floating body in three-dimensional space, it will have six degrees of freedom
[2e]. Figure 3-19 demonstrates an axis system on the 2014 Mercedes AMG Petronas Team car
where motion of the vehicle translates longitudinally along the x axis, laterally along the y axis and
vertically along the z axis. Along these axis are the rotational motions associated with the axis – in
this case, roll about the x axis, pitch around the y axis and yaw around the z axis.
Figure 3-19: Axis showing six degrees of freedom
For the purpose of the suspension design for the current car, degrees of freedom (DoF) are
applicable at all part levels – including the design and analysis of the rod ends and the joints that will
connect the uprights to the wheels.
In order to understand the motion of the swing arms, the Gruebler equation was used to determine
the total number of degrees of freedom in the suspension system.
(3-13)
The reduction in the number of parts in equation (3-13) accounts for the ground as a non-moving
part while Table 4 represents the translational and rotational constraints of the various types of
joints.
Table 3-3: Joint translations, rotations and constraints
Constraint Type Translational Rotational Constraints Total Constraints

Contraints (X,Y,Z) (pitch, roll, yaw)
Cylindrical joint 2 2 4
Fixed joint 3 3 6
Revolute joint 3 2 5
Spherical joint 3 0 3
Universal joint 3 1 4
4 Spherical Joints, Rod Ends and Upright Design
4.1 Spherical Joints and Rod Ends

To allow the swing arms to rotate through angle α in Figure 3-17, spherical joints were implemented
in the design. The reason for adjustable spherical joints over cylindrical joints was to allow for a
greater degree of flexibility and to compensate for camber adjustment.
Figure 4-1: Adjustable spherical joint at chassis hard points
Figure 4-1 illustrates the spherical joint attached to the rod end designed for the racing car. They are
made from cast stainless steel then bored and machined to precision to meet the 20 ± 0.05mm inner
spherical diameter. The bolt size used for the adjustable shaft was an M12 with a 1.5mm fine pitch
thread in order to accurately adjust camber angle. With respect to the suspension geometry, the
spherical joints were fixed to the chassis hard points via a bolt in accordance to FSAE rule T3.3 where
chassis members had to be properly triangulated. This is illustrated by the red circles in Figure 4-2
where it shows the underside of the chassis.
Figure 4-2: Underside view of chassis and associated hard point mountings
The next stage in the process was to design the rod ends that will connect the uprights to the swing
arms. Referring back to Figure 3-8 showing the sketch lines from the chassis to the coordinate
systems, the angles were measured using SolidWorks ‘measure’ tool to determine the angle
between triangles ∆QTV and ∆RSU. This gave angles at and respectively. These angles
were translated on to a custom-made part to create constrained rod ends (Figure 4-3). The initial
Figure 4-3: Front outer rod ends fully defined with adjustable thread and spherical joint mount
step in the manufacturing process for the outer rod ends differed from the stainless steel rod ends
mentioned previously. Because one of the aims was to reduce weight while maintaining structural
integrity, it was decided to manufacture the rod ends from a hot forged aluminium 6061-T6 billet
(slow pressed in a die) then machined and threaded to an M12 bolt with a 1.5mm fine pitch thread.
The forging process requires the aluminium alloy billet to be heat up between 412˚C and 468˚C and
then slow pressed in a hot die to be quenched in a solution. This changes the mechanical properties
of the alloy making it stronger and being able to withstand the cyclic forces transmitted from the
road surface.
4.2 Uprights
The next piece to be designed, tested, and manufactured were the uprights. In order to carry out the
process, the suspension sketch geometry was used to determine the distance between the upper
and lower rod ends marked S and T in Figure 4-4.
Figure 4-4: Points S and T determining the length of upright
The front uprights of the vehicle were designed to encounter two degrees of freedom - vertical
motion along the Y direction and spin around the Y axis for steering. They also had to compensate
for when the race engineers wished to change the camber angle of the wheels without
compromising the original suspension geometry. In order to do this a series of spherical joints were
designed to fit concentrically within the rod ends of points T and S shown previously in Figure 4-3.
The rear uprights were designed with two purposes in mind, one of which was to house the drive
shaft for the rear wheels and the other to allow for travel in the vertical Y direction. To do this a
number of constraints were implemented in the swing arm and rod end designs in order to limit the
motion of the rear wheels strictly to the vertical direction.
The distance between centre points M and N was 120mm. Similar in approach to the front uprights,
the intersection of the swing arms to the centre points of the spherical joints were designed to
coincide. Spherical joints were then bolted at S, T, M1, M2, N1 and N2 (Figure 4-5)
Figure 4-5: Front and rear upright attachment points
4.3 Stress Analysis: Uprights

In order to design the uprights with as high strength as possible but cutting the weight down without
compromising the strength of the part deems itself to being one of the main challenges in
automotive engineering – indeed engineering as a whole. Due to the constant forces acting on the
wheels and car chassis itself, choosing an effective design and material that could withstand these
extreme forces over extended periods of time was of paramount importance for driver safety.
4.3.1 Forces on wheel

Having carried out the physical design of the uprights (Figure 4-5), it was essential in testing them
through a series of loading conditions. These loads would essentially be a result of the car travelling
at high speed over bumps and holes. In order to do this, tire data was required from section 3.4 in
order to calculate the forces acting on the rim of the wheel for a high speed bump.
Assume a vehicle of mass 250kg (approximately the wet weight of a FSAE Formula Student race car
including its driver) travelling at 60mph (approx 26.82m/s) hits a bump 4cm high and 8cm at base
width (Figure 4-6).
Figure 4-6
In order to determine the forces involved, the wheel rim was assumed a rigid body and the mass of
the car was neglected for the purpose of purely determining the forces in the wheel going over a
bump. A free body quarter-model diagram of the system illustrates the system.
Figure 4-8 Figure 4-7
This system is referred to as the unsprung mass. It is where only the mass of the tyre and wheel are
taken in to consideration. This was of particular interest in calculating the forces exerted on the
uprights due to the fact a simple undamped rigid body system did not take spring and damping rates
in to consideration – which would inadvertently yield inaccurate and unrealistic results.
Table 4-1: Wheel and tire specifications
Wheel weight Tire weight Tire pressure Tire spring rate @ 200lbs static
(rigid) loading
5kg 8kg 16psi 645.16lbs/in
A quick conversion from imperial to metric was required:
⁄ ⁄
The following equation was used to calculate the forces exerted on the wheel going over a 4cm
bump
̈ ( ̇ ̇ ) (4.1)
[ ]
[ ]
⁄ [ ]
[ ]
The first step was to calculate the time taken for the vehicle to cover half the width the bump the
highest point).
( ⁄ )
⁄
The time taken to cover half the width of the bump equals the time taken to reach the highest point
of the bump.
̇ ⁄ ⁄
̇ ⁄ ⁄
The next stage was to calculate the tire damping coefficient. This was not given in any
documentation or journals as its value depends on the mass of wheel – which can vary from team to
team. In this case, a custom-made wheel was designed so its specific mass was used.
As far as Hoosier are concerned, their tires are designed so that critical damping is achieved (ᵹ=1).
Rearrange ⁄ (4.2)
√
√ (4.3)
⁄ ⁄ (4.4)
Substituting these values in to equation 2.16 yields
(4.5)
This is the force experienced on the wheel in the vertical direction as a result of a vehicle travelling
at 26m/s over a bump of 4cm high and 8cm wide with both tire spring and damping constants
considered. Note this would be an extreme scenario as spring and dampening of the shock absorbers
have not yet been considered for this model.
4.3.2 FEA and SolidWorks Simulation

Because this was not an off-the-shelve part there was a need perform finite element analysis (FEA)
to test for stress and strain on the uprights. This was carried out using SolidWorks Simulation
software. The first part was to create a mesh and model the parts in solid elements (Figure 4.9 [a]).
The next stage was to apply constraints and forces acting on the model. Using the force calculated in
(4.5) this was applied to the surface of the inner hub along the vertical direction in order to simulate
the forces exerted from the road bump to the part. For the simulation to work, the upright was
converted in to a fine mesh to create a series of nodes and elements. Due to where the spherical
joints and rod ends connected to the uprights - and for the purpose of the model - the top and
bottom hole fixings were constrained and fixed in space (Figure 4.9 [b]).
Figure 4-9: (a) front upright mesh, (b) applied constraints and forces, (c) simulation
A simulation was carried out (Figure 4.9 [c]) and it was found using a titanium-alloy (Ti-6Al-4V
solution and annealed) gave the most effective results. It was found that the stresses flowing
through the part (σ) fell significanlty less than the yield stress of the material itself (σ <
827,370,880N/m2). This also allowed the part to be designed using less material totalling a mass of
only 0.47kg. The same approach was used in order to test for stresses in the rear uprights (Figure 4-
10) and it was found it gave similar results in that the stresses flowing through the part were less
than the yield strength of the material, totalling a weight of only 0.4kg.
Figure 4-10: (a) rear upright mesh, (b) applied loads and constraints, (c) simulation
5 Swing Arms and Push Rods
Having determined the suspension geometry and the positioning of the uprights for all four wheels,
the next phase of the project was to design the swing arms. These had to retain the uprights at a set
distance from the chassis while allowing room for adjustment. A simple ‘bottle-screw’ mechanism
was used where at one end of the rod the thread ran clockwise and the other counter clockwise. The
threading was exactly the same as the rod ends specified in 4.1 - M12 diameter and 1.5mm fine
pitch. Shown in Figure 5-1 is where the rod ends coincide.
Figure 5-1: Rod ends linearly connecting to chassis hard points
The next stage was to sketch the profile of the rod ends then extrude to the required length (marked
by the red crosses in Figure 5-1). This proved to be an interesting part to design due to the fact vast
amounts of research is undertaken in Formula 1 designing swing arms that are strong, lightweight
and aerodynamic as possible. Due to the complexity of the subject, it is a project within itself and a
job left for the aerodynamicists. For the purpose of the design, the push rods were based purely on
Euler’s Buckling Theory and designed to withstand the bending and compressive forces exerted from
varying loading conditions with a basic aerodynamic profile. Figure 5-2 shows the profile and shape
of the proposed swing arms and push rods.
Figure 5-2: Carbon fibre swing arm and push rod profile
At both ends of the rods there are stainless steel bushings for the adjustable screw threads. The
material used in the manufacturing process is high modulus plain weave carbon fibre with an
adhesive epoxy resin (known as the matrix) [4a] to bond the layers together. The part was then
autoclaved at 120˚C at a pressure of 100psi. The reason for using plain weave fibres is that the
material yields better drape characteristics especially when forming round complex shapes and
being able to withstand lateral forces acting across the direction of the main fibres. Figure 5-3
illustrates the use of various types of carbon fibres and highlighted in red was the fibre chosen
specifically for this design.
Figure 5-3: Carbon fibre types
5.1 Rod Stress Analysis

Following the analysis of the uprights, a similar approach was taken in that a full stress analysis was
required due to the custom nature of the rods. Because the of the rod’s composition (plain weave
carbon fibre and epoxy resin), a calculation was required to determine the composite modulus (EC).
The composite modulus takes in to consideration the Elastic Modulus of both materials and assumes
perfect adhesion between the layers of carbon fibre and the epoxy resin matrix to formulate a ‘net
modulus’ for the component. In the analysis it was found that the SolidWorks Simulation software
was limited in the list of properties it had stored in the database. As a result a manual calculation
was required in order to modify the properties in the ‘Edit Materials’ field. In order to do this, the
‘Rule of Mixtures’ was used:
(5.1)
where,
= fibre modulus; = fibre volume fracture; = fibre efficiency factor; = matrix modulus. For
‘fibre efficiency factor’, A = 1 for unidirectional fibre, 0.5 for bi-directional fibre and 0.4 for random
fibre.
Table 5-1: Materials Data (Taken from Performance Composites)
Material Young’s modulus Poisson’s ratio Ultimate tensile Volume fracture

(GPa) strength (MPa)
High modulus 85 0.1 300 50%
carbon fibre
Epoxy resin 2.4
Substituting these values in to the Rule of Mixtures equation yields
(5.2)
This new value for composite modulus was then entered in to SolidWorks materials editor in row
“Elastic Modulus in X”:
Figure 5-4: Newly calculated combined modulus of 22.45GPa in row Elastic Modulus in X
(a)
(b)
(c)
Figure 5-5: (a) Rod mesh, (b) simulation of rod ends with applied
constraints and forces, (c) carbon rod simulation
Similar to 4.3.2 in performing FEA on the uprights, careful attention was taken as to how the rod
ends were attached and where to apply the loads. Because of the ‘bottle-screw’ set-up which
allowed for length adjustment, there was a clockwise and counter clockwise nut at each end to apply
a preload on the contact threads (Figure 5-6).
Figure 5-6: Rod ends, rods and pre-load nut assembly illustrating thread direction
5.2 Final rod design

The preload essentially fixes the ends allowing only the carbon fibre to yield under load rather than
having the end of the aluminium bushings ‘pinned’. This changes the analysis with respect to
buckling. The SolidWorks simulation yielded a positive result by not significantly deforming under a
static load of 20,000N (approximately the forces experienced under extreme road bump conditions).
Under compressive loads, the yield stresses in the rod equated to approximately 660MPa according
to the colour chart (Figure 5.5 [c]) which fell under the ultimate compressive strength of the material
of 850MPa. This confirmed and validated the design allowing the rods to be used for the race car.
6 Bell cranks and Shock Absorber Mounts
As far as being meticulous in designing correct suspension goes, it hinged on designing a bell crank
that could transfer the loads from one direction to another. It also depended on designing chassis
mounts that ran perpendicular along the direction of the push rods. In order to create this, a series
of temporary sketches were dimensioned along the length of the push rods then angled off at height
‘h’ and depth ‘s’ then drawn back distance ‘d’ shown in red in Figure 6-1 and Figure 6-2.
Figure 6-1: Red lines show the sketch lines determining angle for bell crank
Figure 6-2: Distance d showing the horizontal component for placement of shock absorber
Having determined the push rod-to-chassis geometry a bell crank and suspension mount was
designed on to the chassis shown by A and B (Figure 6-3).
Figure 6-3: Bell crank (B) and shock absorber (A) chassis-mounts
The final design for the bell cranks is illustrated in Figure 6-4. It was designed to tolerate the forces
exerted by the push rods while at the same time reduce weight. Figure 6-5 shows the crank in
context.
Figure 6-4: Final bell crank design
Figure 6-5: Bell crank in context with chassis
7 Shock Absorbers, Spring Damping Analysis
The final step in the design and analysis of the suspension system was the most important aspect of
the project. Designing a specific spring dampening system for the race car required all the previous
design steps. The reason why – as you can see – that the summation of parts resulted in an overall
net mass. There are three parameters when referring to suspension systems: mass, spring constants
and damping coefficients. As previously mentioned in section 4.3.1 on forces on a wheel, a free body
diagram was drawn to illustrate a quarter model of the system. However, this was only reflecting the
‘unsprung mass’ model of the system. Now that all the suspension and chassis components have
been designed it was important to carry out a ‘sprung mass’ model calculation. This was to
determine the spring constant and damping factor for the spring-shock absorber that was to be
purchased and connected to the mounts described in the previous chapter.
7.1 Quarter Car Sprung Mass Model
Figure 7-1: Free body diagram of quarter car model
In order to carry on with the analysis, MATLAB was used to test for damping. The total mass of the
car including driver and engine was 230kg. The spring and damping constant for the wheel and tire
(K1 and b1) were taken previously in 4.3.1. Using a basic template on MATLAB Simulink (Figure 7.2)
and data previously calculated, the car was modelled and a basic spring mass dampener model was
run.
Figure 7-2: Simulink template for sprung mass car model
7.2 Results
Figure 7-3: Results for the newly designed suspension system
Bearing mind the spring and damping rates for the wheel and tire system, it was found an additional
spring-dampener of 20000N/m and 120N.s/m respectively, was required.
For the selection process, an Ohlins shock absorber was chosen for the race car. Reason for this
choice was Ohlins currently already have links with Formula Student and are heavily involved in high
performance motorsport.
8 Final Designs
Figure 8-1: Chassis-suspension assembly
The Re-Design and Analysis of the Suspension System on the Formula Student Race Car
39
Figure 8-2: Chassis-suspension assembly 2. Shock absorbers mounted flush with the chassis
Figure 8-3: Hidden chassis, suspension view
Figure 8-4: Top view chassis-suspension assembly
Figure 8-6: Top view front chassis-suspension assembly
The Re-Design and Analysis of the Suspension System on the Formula Student Race Car
42
Figure 8-5: Top view rear chassis-suspension assembly showing repositioning of shock absorber
Figure 8-7: Full chassis suspension assembly
Figure 8-8: Full chassis-suspension assembly (wheel steer)
Figure 8-9: Full assembly exploded view
Figure 8-10: Fully-fledged
superimposed model image of
front wheel suspension
system under load
Figure 8-11: Fully-

fledged superimposed
model image of rear
wheel suspension
system under load
Figure 8-12: Ohlins shock absorber
Figure 8-13: Front left upright x 1 Figure 8-14: Rear upright x 2
Figure 8-15: Front suspension assembly inc. steering rod (frameless), fully adjustable rods for camber angle
Figure 8-16: Rear suspension assembly (frameless)
9 Conclusion
Having step-by-step run through a complete re-design and analysis for each of the suspension
components for the new racing car, a full working assembly was constructed. Attached in the report
is a USB stick with all 145 parts, along with a full assembly model. Included in the Appendix are the
technical drawings for the uprights also.
10 References
10.1 Books, journals and lecture notes

[1…] Genta, G. (1997). Motor Vehicle Dynamics: Modeling and Simulation. Series on Advances in
Mathematics for Applied Sciences, vol. 43. World Scientific Publishing Co. Pte. Ltd Sharp RS,
Casanova D, Symonds P (2000) Mathematical model for driver steering control, with design, tuning
and performance results. Veh Syst Dyn, 33(5), 289-326.
[…a] Page 328
[…b] Page 338
[…c] Page 338
[…d] Page 336
[…e] Page 341
[2…] Blundell, M., & Harty, D. (2004). The Multibody Systems Approach to Vehicle Dynamics. Elsevier.
[…a] Page 7, 163
[…b] Page 23
[…c] page 26
[…d] Page 26
[…e] Page 99
[3…] Y C Chen, H H Huang and J B Lin, Proceedings of the Institution of Mechanical Engineers, Part C:
Journal of Mechanical Engineering Science, November 2011; vol. 225, 11: pp. 2586-2596., first
published on September 14, 2011.
[…a] Page 2
[…b] Page 2
[4…] Hashim,S, University of Glasgow, Advanced Materials Technology P4 (ENG4004), 2014, Fibre
reinforced plastics (FRP) and sandwich construction.
[…a] Page 6
10.2 Figures
Figure 1-1: www.ultimate-design.org/2014/03/zis-5-cargo.html
Figure 1-2: www.ultimate-design.org/2014/03/zis-5-cargo.html
Figure 1-3: http://projectzerog.com/archives/ZeroGversion1/suspension.html
Figure 1-4: http://suv-buster.blogspot.co.uk/2006/10/bmw-x5-e70-to-get-double-wishbone.html
Figure 1-5: www.autoevolution.com/news-image/how-multi-link-suspension-works-7804-4.html
Figure 2-1: Courtesy of FSAE University of Glasgow
Figure 2-2: Courtesy of FSAE University of Glasgow
Figure 3-12: www.huco.com/products.asp?p=true&cat=512
Figure 3-19: www.mercedesamgf1.com/en/car/f1-w05/
Figure 5-3: Hashim,S, University of Glasgow, Advanced Materials Technology P4 (ENG4004), 2014,
Fibre reinforced plastics (FRP) and sandwich construction, page 6.
10.3 Equations
(3.1) – (3.13): Blundell, M., & Harty, D. (2004). The Multibody Systems Approach to Vehicle Dynamics.
Elsevier
(4.1):
http://ctms.engin.umich.edu/CTMS/index.php?example=Suspension&section=SystemModeling
(5.1): Hashim,S, University of Glasgow, Advanced Materials Technology P4 (ENG4004), 2014, Fibre
reinforced plastics (FRP) and sandwich construction, Page 11.
11 Appendix
Roll axis close to centre of gravity validating new design, satisfying section 1.1.7 on roll centre.
New design showing adjustable swing arms for camber angle Ɵ
Instructions to view file full car assembly:
>Computer > DB(D:) > SolidWorks Files > Assembly
MATLAB Simulations:
>Computer > DB(D:) > MATLAB Simulations > Assembly, OR, Assembly 2, OR, GIF

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Department of Mechanical Engineering

BEng Mechanical Design Engineering

David Michael Browne 1007185

Final Year Project 2013-2014

The Re-Design and Analysis of the Suspension System

on the Formula Student Race Car

Supervisor: Dr Christopher Murray

1.1 Types of Suspension Systems

Figure 1-1: Solid axle suspension

1.1.3 Trailing arm (swing arm) independent

1.1.4 MacPherson strut independent suspension

Double wishbone suspension systems are becoming more

Figure 1-4: Double wishbone suspension

conditions are. For these reasons and these

1.1.7 Short-long-arm double wishbone suspension

Figure 1-7: SLA configuration Figure 1-6: Camber angle

Figure 1-8: Instant centre

Figure 1-9: Roll centre

Figure 2-2: Current rear suspension design (provided

3.1 Chassis Geometry

Figure 3-1: Determining new chassis geometry

Figure 3-2: Side view properly triangulated chassis frame members

Figure 3-3: Top view properly triangulated frame members

Figure 3-4: Applying weldments to 3D sketch

3.2 Chassis Stress Analysis

Table 3-1: Wheel specification

Material Wheel Mass Principal Moments

Aluminium alloy 7.66 Px = 0.09

Table 3-2: Tire specification (courtesy of Hoosier)

Tire Tire Tire Tire Tire spring rate @

Figure 3-9: Position vector P

Figure 3-10: Relative position vector

3.5.2 Vector Dot Product

Figure 3-11: Vector dot product

3.5.3 Vector Cross Product

The direction of the resultant vector is determined by calculating X . Firstly has to be

3.5.4 Vector Rotation and Component Vectors

Figure 3-17: Rotation about α from S to S’ with X and Y components

Mathematically this is expressed as

Figure 3-19: Axis showing six degrees of freedom

Table 3-3: Joint translations, rotations and constraints

Constraint Type Translational Rotational Constraints Total Constraints

4 Spherical Joints, Rod Ends and Upright Design

4.1 Spherical Joints and Rod Ends

Figure 4-1: Adjustable spherical joint at chassis hard points

Figure 4-4: Points S and T determining the length of upright

Figure 4-5: Front and rear upright attachment points

4.3 Stress Analysis: Uprights

4.3.1 Forces on wheel

Figure 4-8 Figure 4-7

Table 4-1: Wheel and tire specifications

5kg 8kg 16psi 645.16lbs/in

Substituting these values in to equation 2.16 yields

4.3.2 FEA and SolidWorks Simulation

Figure 5-1: Rod ends linearly connecting to chassis hard points

Figure 5-3: Carbon fibre types

5.1 Rod Stress Analysis