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MODEL AND DYNAMIC SIMULATION PROGRAM FOR
VEHICLE ANALYSIS ACCOUNTING SUSPENSION
COMPLIANCE
Ricardo Domingues dos Santos
Thesis to obtain the Master of Science Degree in
Mechanical Engineering
Supervisor: Prof. Luís Alberto Gonçalves de Sousa
Examination Committee
Chairperson: Prof. Luís Manuel Varejão de Oliveira Faria
Supervisor: Prof. Luís Alberto Gonçalves de Sousa
Member of the committee: Prof. Virgínia Isabel Monteiro Nabais Infante
March of 2014
I
RESUMO
O presente trabalho teve como objectivo o desenvolvimento de um modelo dinâmico que
disponibiliza ao engenheiro a possibilidade de incluir a deformação elástica de vários componentes
que compõem o veículo. No caso deste trabalho o estudo incidiu sobre um veículo do tipo Formula
Student. No que toca a veículos de competição é cada vez mais importante a semelhança dos modelos
utilizados com a realidade de maneira a poder compreender o comportamento do veículo em pista
descorando a menor quantidade possível de factores.
Uma revisão bibliográfica de modo a compreender que tipo de abordagens são tomadas para
a inclusão de componentes deformáveis na análise de veículos foi feita em primeiro lugar de modo a
enquadrar este trabalho.
Posteriormente foi desenvolvido o modelo dinâmico do veículo que permite estudar o seu
desempenho. Este modelo utiliza os diferentes subsistemas presentes no protótipo para devolver as
principais variáveis em estudo como a aceleração lateral, longitudinal, velocidades, ângulos de
trajectória e ainda o trajecto do próprio veículo. Para uma modelação mais intuitiva, os controlos do
piloto (volante e pedais) foram escolhidos como os únicos inputs da simulação.
De seguida foi tomada uma decisão quanto à abordagem tomada na inclusão de componentes
deformáveis elucidando os parâmetros que se pretende influenciar com estas deformações. Os
componentes foram escolhidos e são apresentados assim como a revisão do seu projecto para obtenção
das deformações.
Por fim as modificações requeridas ao modelo são efectuadas e são ainda comparadas
diferentes situações de condução para verificar a sua influência.
PALAVRAS-CHAVE
Modelo dinâmico do veículo, suspensão, complacência, simulação de veículos.
II
ABSTRACT
The goal of the present work was to develop a dynamic model to provide the engineer with
the possibility to incorporate the elastic deformation of some suspension components into vehicle
simulations. This work will focus on a particular type of vehicle: a Formula Student prototype. When
it comes to competition vehicles it is becoming more important to perform simulations with a high
level of resemblance to the real performance of the vehicle. This can be done by gradually including
more factors into the models.
An evaluation of the current state of simulations including the effect of compliance in the
dynamics of the vehicle was done first.
The model that allows the engineer to evaluate the vehicle’s performance was developed
next. This model uses the different subsystems of the vehicle to return the main performance variables
like the accelerations, velocities, trajectory angles and the vehicle’s trajectory. The driver inputs
(steering wheel and pedals) are also the main inputs of the model.
Next, a decision was taken in regard to the approach to include compliant components along
with the parameters that would be affected. The compliant components were chosen and presented
along with a revision of their design to obtain the deformations.
Finally these compliant parts were incorporated in the model and simulations were run to
verify the influence of compliant components.
KEYWORDS
Dynamic modelling of vehicles, suspension, compliance, dynamic simulation of vehicles.
III
AGRADECIMENTOS
Antes de mais queria começar agradecendo ao meu orientador, o professor Luís Sousa, pelo
apoio prestado e encorajamento durante a fase de realização deste trabalho. Não só pela ajuda na
abordagem de algumas questões técnicas mas também pela pronta disponibilidade, foi uma
contribuição valiosa para a realização deste trabalho.
Em segundo lugar agradeço à professora Virgínia Infante pela ajuda e disponibilidade para
ajudar o Projecto FST Novabase na realização de ensaios físicos que se tornaram valiosos não só para
o enriquecimento do conhecimento da equipa mas também para esta dissertação.
Quero também fazer um grande agradecimento à equipa do Projecto FST Novabase,
principalmente às equipas que desenvolveram o FST 04e e o FST 05e. Sem dúvida que foram quatro
dos melhores anos da minha vida e onde construímos os sonhos de todos. Sinto-me privilegiado por
tê-lo feito ainda tão novo e foi com enorme prazer que o fiz ao lado dos amigos que conheci neste
projecto e que certamente ficarão para a posteridade.
Quero agradecer ao meu pai e ao meu primo Jorge. O primeiro ensinou-me humildemente
que existe uma ciência por trás do automóvel, algo pelo qual passei grande parte dos últimos anos a
trabalhar e que espero continuar no futuro. O segundo pelo facto de ter perseguido aquilo em que
acredita e por ser uma inspiração diária.
Quero agradecer à minha namorada por acreditar em mim e me encorajar nas alturas mais
difíceis. Por ser uma das bases daquilo que sou hoje e serei no futuro.
Guardo os últimos agradecimentos para a minha família, principalmente para a minha mãe e o
meu avô pelo apoio, carinho e sinceridade que me fazem idolatrá-los.
Por fim quero dedicar este trabalho e todo o meu percurso até este ponto da minha vida à
minha avó Emília por tudo o que me deu e ensinou. Tudo o que faço e farei será não só meu mas
também teu.
IV
V
TABLE OF CONTENTS
RESUMO ................................................................................................. I
PALAVRAS-CHAVE ............................................................................... I
ABSTRACT ............................................................................................ II
KEYWORDS .......................................................................................... II
AGRADECIMENTOS ............................................................................ III
TABLE OF CONTENTS ......................................................................... V
LIST OF FIGURES ............................................................................. VIII
LIST OF TABLES .................................................................................. XI
LIST OF SYMBOLS ............................................................................ XII
1 INTRODUCTION................................................................................. 1
1.1 THE FORMULA STUDENT COMPETITION ............................................................. 1
1.2 THE PROJECTO FST NOVABASE TEAM .............................................................. 1
1.3 WORK CONTRIBUTION AND OBJECTIVE ............................................................. 2
1.4 DOCUMENT STRUCTURE .................................................................................... 3
2 STATE OF THE ART IN SIMULATIONS WITH COMPLIANCE
EFFECTS ................................................................................................. 5
2.1 FORMULA STUDENT EXPERIENCE ....................................................................... 5
2.2 COMPLIANCE MODELING................................................................................... 6
2.3 COMPLIANCE MEASURING ................................................................................. 7
2.4 FRAMEWORK ..................................................................................................... 8
3 VEHICLE DYNAMICS CONCEPTS ................................................... 9
3.1 COORDINATE SYSTEMS ..................................................................................... 9
3.2 SLIP PARAMETERS ........................................................................................... 10
3.2.1 Sideslip angle .................................................................. 10
3.2.2 Slip angle ........................................................................ 11
3.2.3 Longitudinal Slip Ratio ................................................... 11
3.3 VEHICLE GENERAL CHARACTERISTICS ............................................................ 12
3.3.1 Vehicle Dimensions ........................................................ 12
3.3.2 Vehicle mass properties .................................................. 13
3.4 SUSPENSION & STEERING PROPERTIES ............................................................ 13
3.4.1 Wheel Angles .................................................................. 13
3.4.2 Steering axis & properties ............................................... 14
VI
3.4.3 Instant Centers ................................................................ 15
3.5 SUSPENSION SPRINGS AND DAMPERS .............................................................. 16
4 MODEL DEVELOPMENT – WITHOUT COMPLIANCE .................. 19
4.1 DRIVER INPUTS ............................................................................................... 19
4.2 TIRE MODELING .............................................................................................. 19
4.2.1 Tire Lateral Force ........................................................... 19
4.2.2 Tire Longitudinal Force .................................................. 20
4.2.3 Self-Aligning Torque ...................................................... 21
4.2.4 Tire Model – Pacejka’s Magic Formula .......................... 21
4.2.5 Tire Testing ..................................................................... 22
4.2.6 Model Fitting .................................................................. 22
4.3 AERODYNAMICS .............................................................................................. 23
4.4 VEHICLE VIBRATIONAL MODEL ...................................................................... 25
4.4.1 Model layout and components ........................................ 25
4.4.2 Model formulation .......................................................... 25
4.4.3 Mass, Stiffness, Damping Matrices and Force Vector .... 26
4.5 DYNAMICS OF A PARTICLE IN NON-UNIFORM CIRCULAR MOTION ...................... 32
4.6 VEHICLE LONGITUDINAL DYNAMICS ............................................................... 33
4.6.1 Motor .............................................................................. 33
4.6.2 Brake System .................................................................. 34
4.6.3 Wheel rotational dynamics and force calculation ............ 37
4.7 VEHICLE LATERAL DYNAMICS ........................................................................ 38
4.7.1 Steering Mechanism ........................................................ 38
4.7.2 Sideslip and slip angles calculation ................................. 40
4.7.3 Inclination angle calculation ........................................... 41
4.7.4 Force and moment calculation ........................................ 43
4.8 VEHICLE ACCELERATIONS ............................................................................... 43
4.9 VEHICLE MOTION ........................................................................................... 45
4.10 SIMULATIONS .................................................................................................. 46
4.10.1 Straight line acceleration and braking ............................. 47
4.10.2 Skid Pad simulation ........................................................ 49
VII
5 APPROACH FOR INCLUDING COMPLIANT COMPONENTS ....... 55
5.1 PARAMETERS WITH INFLUENCE ON VEHICLE BEHAVIOR ................................... 55
5.1.1 Vertical Load .................................................................. 55
5.1.2 Slip angle ........................................................................ 55
5.1.3 Inclination angle ............................................................. 56
5.2 LOAD CASES CALCULATION ............................................................................. 56
5.3 DEFORMATION ANALYSIS ................................................................................ 59
5.3.1 Steering shaft .................................................................. 59
5.3.2 Wishbones, Tie-rods and Toe-rods.................................. 60
6 MODEL PRESENTATION – WITH COMPLIANCE ......................... 69
6.1 MODEL SCHEMATIC MODIFICATIONS .............................................................. 69
6.1.1 Component loads calculation .......................................... 69
6.1.2 Deformation calculation .................................................. 69
6.1.3 Steering mechanism modifications.................................. 70
6.1.4 IA calculation modifications ........................................... 71
6.2 COMPLIANT SKID PAD SIMULATION................................................................. 72
7 CONCLUSIONS AND FUTURE DEVELOPMENTS ......................... 79
7.1 CONCLUSIONS ................................................................................................. 79
7.2 FUTURE DEVELOPMENTS ................................................................................. 80
8 REFERENCES ................................................................................... 81
APPENDIX A – MAGIC FORMULA EQUATIONS .............................. 82
APPENDIX B – EQUIVALENT LAMINATE PROPERTIES ................ 85
APPENDIX C – FORMULA STUDENT RESULTS ............................... 88
APPENDIX D – MATLAB / SIMULINK DIAGRAMS .......................... 89
VIII
LIST OF FIGURES
Figure 1.1 - FST 04e ....................................................................................................................1
Figure 1.2 – The team with the FST 05e ......................................................................................2
Figure 2.1 - Evolution of vehicle modeling in Projecto FST Novabase ........................................5
Figure 2.2 - Model of a formula student car in MSC ADAMS .....................................................6
Figure 2.3 - K & C Testing Machine from Morse Measurements LLC ........................................8
Figure 3.1 - Vehicle coordinate system ........................................................................................9
Figure 3.2 - Suspension coordinate system ..................................................................................9
Figure 3.3 - Tire coordinate system (SAE-J670) ........................................................................ 10 Figure 3.4 - Sideslip angle representation .................................................................................. 10
Figure 3.5 – Tire slip angle representation ................................................................................. 11
Figure 3.6 - Wheelbase ............................................................................................................... 12
Figure 3.7 – Track ...................................................................................................................... 12
Figure 3.8 - Steering Angles ....................................................................................................... 13
Figure 3.9- Camber and Inclination angles ................................................................................. 14
Figure 3.10 - KPI and Caster angles ........................................................................................... 14
Figure 3.11 - IC Definition (front view) ..................................................................................... 15
Figure 3.12 - IC definition (side view) ....................................................................................... 15
Figure 3.13 - Suspension Corner ................................................................................................ 16
Figure 3.14 - Example of damping curve ................................................................................... 17
Figure 3.15 - ARB location and closer look ............................................................................... 17
Figure 4.1 - Tire Lateral Force (courtesy of Milliken & Milliken) ............................................. 20
Figure 4.2 - Tire Longitudinal Force - Driving (courtesy of Milliken & Milliken) .................... 21
Figure 4.3 - Raw tire data and fitting curves .............................................................................. 23
Figure 4.4 - Aerodynamic reference point .................................................................................. 24
Figure 4.5 - Vehicle vibrational model ....................................................................................... 25
Figure 4.6 - Front Vibrational Model ......................................................................................... 27
Figure 4.7 - Free-body diagram of the vibrational model – Front Right Side ............................. 27
Figure 4.8 - Free-body diagram of the vibrational model – Front Left Side ............................... 27
Figure 4.9 - Free-body diagram of vehicle for moment analysis ................................................ 28
Figure 4.10 - Tire horizontal forces and vehicle reactions .......................................................... 29
Figure 4.11 - Lateral weight transfer .......................................................................................... 29
Figure 4.12 - Anti-Roll effect of tire lateral forces ..................................................................... 30
Figure 4.13 - Anti-Lift, Anti-Squat and Anti-Dive effect of tire longitudinal forces .................. 31
Figure 4.14 - Longitudinal weight transfer ................................................................................. 31
Figure 4.15 - Vehicle motion...................................................................................................... 32
IX
Figure 4.16 - Torque vs Speed Curve ......................................................................................... 34
Figure 4.17 - Brake pedal layout ................................................................................................ 35
Figure 4.18 - Brake pedal free-body diagram ............................................................................. 35
Figure 4.19 - Balance bar free-body diagram ............................................................................. 36
Figure 4.20 - Brake rotor ............................................................................................................ 37
Figure 4.21 - Wheel free-body diagram...................................................................................... 38
Figure 4.22 - Steering Geometry ................................................................................................ 39
Figure 4.23 - Steering system in Simmechanics ......................................................................... 39
Figure 4.24 - Slip angle .............................................................................................................. 40
Figure 4.25 - Wheel velocities schematic ................................................................................... 40
Figure 4.26 - 3-point method schematic ..................................................................................... 41
Figure 4.27 - KPI angle calculation ............................................................................................ 42
Figure 4.28 - Tire forces components ......................................................................................... 43
Figure 4.29 - Calculation loop schematic ................................................................................... 44
Figure 4.30 - Model schematic ................................................................................................... 45
Figure 4.31 - Vehicle in the ground coordinate system .............................................................. 46
Figure 4.32 - Driver inputs ......................................................................................................... 47
Figure 4.33 - Wheel velocities and longitudinal slip ratio .......................................................... 47
Figure 4.34 - Tire longitudinal forces and vehicle acceleration .................................................. 48
Figure 4.35 - Aerodynamic and tire vertical loads ...................................................................... 48
Figure 4.36 - Vehicle velocity .................................................................................................... 49
Figure 4.37 - Distance covered ................................................................................................... 49
Figure 4.38 - Skid Pad layout (FSAE rules) ............................................................................... 50
Figure 4.39 - Driver inputs in Skid Pad ...................................................................................... 50
Figure 4.40 - Wheels steering angles and slip angles ................................................................. 51
Figure 4.41 - Yaw moment ......................................................................................................... 51
Figure 4.42 - Tire vertical loads and lateral forces ..................................................................... 52
Figure 4.43 - Vehicle velocity in the Skid Pad ........................................................................... 52
Figure 4.44 - Vehicle radial acceleration .................................................................................... 52
Figure 4.45 - Skid Pad completions during simulation ............................................................... 53
Figure 4.46 - Vehicle's CG trajectory path ................................................................................. 53
Figure 5.1 - Scheme of deformation influences .......................................................................... 56
Figure 5.2 - Simmechanics model for dynamic analysis. ........................................................... 56
Figure 5.3 - Detail of corner suspension ..................................................................................... 57
Figure 5.4 - Free body diagram of wheel-upright assembly ....................................................... 58
Figure 5.5 - Steering box assembly ............................................................................................ 58
Figure 5.6 - Rack and pinion free body diagram ........................................................................ 59
X
Figure 5.7 - Steering shaft location ............................................................................................ 59
Figure 5.8 - Steering shaft twist angle ........................................................................................ 60
Figure 5.9 - Formula 1 suspension ............................................................................................. 60
Figure 5.10 - FST 05e suspension .............................................................................................. 60
Figure 5.11 - Wishbone schematic ............................................................................................. 61
Figure 5.12 - Bonding schematic ................................................................................................ 61
Figure 5.13 - Adhesive behavior for rigid adherends ................................................................. 62
Figure 5.14 - Adhesive behavior for flexible adherends ............................................................. 62
Figure 5.15 - Single-lap joint schematic ..................................................................................... 63
Figure 5.16 - Shear stress scenarios in single-lap joint (same loading condition) ....................... 63
Figure 5.17 - Suspension link cut-view ...................................................................................... 64
Figure 5.18 - Example of calculation points and their deformation ............................................ 64
Figure 5.19 - Unit volume of adhesive in shear .......................................................................... 65
Figure 5.20 - Test specimen installed in universal testing machine ............................................ 65
Figure 5.21 - Test specimen ....................................................................................................... 66
Figure 5.22 - Test results for tube ID = 18 mm .......................................................................... 66
Figure 5.23 - Test results for tube ID = 10 mm .......................................................................... 67
Figure 6.1 - Model schematic with compliance modifications ................................................... 69
Figure 6.2 - Deformation calculations schematic ....................................................................... 70
Figure 6.3 - Tie-rod modification ............................................................................................... 70
Figure 6.4 - Three point method sketch with compliance ........................................................... 71
Figure 6.5 - Skid Pad inputs - compliant model ......................................................................... 72
Figure 6.6 - Steering rods compliance and steering shaft twist angle ......................................... 72
Figure 6.7 - Wheels steering angle ............................................................................................. 73
Figure 6.8 - Front right wheel links compliances ....................................................................... 73
Figure 6.9 - Inclination angles during Skid Pad simulation ........................................................ 74
Figure 6.10 - Right tires lateral forces in compliant Skid Pad simulation ................................... 74
Figure 6.11 - Path radius comparison ......................................................................................... 75
Figure 6.12 – Radial acceleration comparison ............................................................................ 75
Figure 6.13 - Vehicle’s tangential velocity comparison ............................................................. 76
Figure 6.14 - Side-slip angle comparison ................................................................................... 76
Figure 6.15 – Comparison between elapsed times to complete one Skid Pad ............................. 77
XI
LIST OF TABLES
Table 5.1 - Test specimen characteristics ................................................................................... 66
Table 5.2 - Results comparison, Tube ID = 18 mm .................................................................... 67
Table 5.3 - Results comparison, Tube ID = 10 mm .................................................................... 67
Table 6.1 - Skid Pad comparisons .............................................................................................. 77
Table 7.1 - Simulation results ..................................................................................................... 79
XII
LIST OF SYMBOLS
First time derivative
Second time derivative
[C] Damping matrix
[K] Stiffness matrix
[M] Mass matrix
[Q] Generalized forces vector
[z] Displacement vector
∂, ∆ Variation
∆l Suspension link elongation
∆ladhesive Adhesive elongation
∆lCFRP CFRP tube elongation
∆lexp Experimental elongation
∆llink Suspension link elongation
µpad,F,R Friction coefficient of brake pads
A Wing area
AC Caliper piston area
Alink Cross-section area of suspension link
AMC Master cylinder piston area
Atube CFRP tube cross-section area
ax Vehicle longitudinal acceleration
aCG Vehicle acceleration vector
ay Vehicle lateral acceleration
CD Drag coefficient
CL Lift coefficient
Cs Effective damper rate
D Drag force
Dy,x, Cy,x, By,x, Ey,x, Sy,x, Gy,x Pacejka’s magic formula coefficients
da-b Distances for 3-point method
e Pedal lever
E Young’s modulus
E1,2 Adherends modulus of elasticity
ECFRP-axial Elasticity modulus of CFRP tube
ER Relative error
eF, R Balance bar position
XIII
FµF,R Friction force exerted with brake pad
FB Balance bar force
FBF,R Master cylinder forces
FCF,R Caliper force
Fclamp,F,R Caliper clamping force
FFRS, FFRUS, FFLS, … Applied forces on masses
FP Force exerted on brake pedal
Frack Steering rack force
Fx Tire longitudinal force
FX Forces in the direction of the x axis of the vehicle
Fy Tire lateral force
FY Forces in the direction of the y axis of the vehicle
G Shearl modulus of elasticity
Gadhesive Adhesive shear modulus of elasticity
hadhesive Bondline thickness
hcgs Sprung mass center of gravity height
hcgus Unsprung mass center of gravity height
IA Inclination angle
IDi Aluminium insert inner diameter
Ix Vehicle roll moment of inertia
Iz Vehicle yaw moment of inertia
J Second moment of area
KARB Anti-roll bar wheel rate
Kch Chassis torsional stiffness
KPI Kingpin inclination angle
Ks Wheel rate
KϕARB Anti-roll bar torsional stiffness
ks Spring rate
kARB Anti-roll bar rate
L Lift force
La-b Length of suspension links
LD Lift distribution
Ljoint Length of bonded joint
l Wheelbase
l0 Suspension link original length
Mch Roll moment
XIV
MR Spring motion ratio
MRARB Anti-roll bar motion ratio
Mz Tire self-aligning torque
MZ Yaw moment
m Vehicle operating mass
ms Sprung mass
mus Unsprung mass
P Load on bonded joint
P% Percentage of throttle pedal depressed
PF, R Pressure in brake system
PR Pedal ratio
SA Slip angle
SL Longitudinal slip ratio
TB, F Braking torque
TDi CFRP tube inner diameter
Tinput Torque delivered by motor
TM Torque available from motor
TR Transmission ratio
Tsteering Steering shaft torque
Twheel Torque delivered to the wheel
t Track
t1,2 Thickness of adherends
v Vehicle velocity
vw Wheel velocity
vx Vehicle longitudinal velocity
vX, Y Vehicle velocities in global coordinate system
vxi,yi Wheel velocity in the vehicle coordinate system
vy Vehicle lateral velocity
w Width of bonded joint
WD Weight distribution
X, Y Vehicle position in global coordinate system
Xi, Yi Wheels position in global coordinate system
x1,7, y1,7, z1,7 Positions for 3-point method
xi, yi Wheels position
xRP Aerodynamic reference point x position
zFRS, zFRUS, zFLS, … Vertical displacements of masses
XV
αy Modified slip angle
β Sideslip angle
βi Wheel sideslip angle
γ Camber angle
δ Steering angle
δ*SW Steering angle input after compliance effect
δSW Steering wheel input
ε Strain
εCFRP Strain of CFRP tube
η Side-view braking anti angle
κx Modified longitudinal slip ratio
λ Front-view anti angle
ν Caster angle
ρ Air density
σ Side-view driving anti angle
τ Pedal actuating angle
τadhesive Adhesive shear stress
φ Heading angle
ϕ Vehicle roll angle
χ Adhesive shear angle
ψ Steering shaft twist angle
ω Vehicle angular velocity around the z axis – Yaw rate
ω0 Reference wheel-spin velocity
ωM Motor rotational speed
ωwheel Wheel angular velocity
1
1 INTRODUCTION
This chapter includes an introduction to the work presented in this thesis.
1.1 THE FORMULA STUDENT COMPETITION
Formula Student is an engineering competition that allows students to put in practice the
engineering and management practices they learn along their scholar path. The students are
encouraged to design, build and race a formula prototype according to the rules published every year.
The design decisions involved in the project and the performance of the vehicle are evaluated during
events all over the world where the students showcase their skills to judges respected in the
engineering sector. The running prototypes are evaluated dynamically in four separate events that try
to push the vehicles to the limits of its capabilities. These events are:
Skid-pad;
Acceleration;
Sprint;
Endurance & Fuel Economy.
The prototypes can be propelled by an internal combustion engine, hybrid engine or electric
motors and it is required that the vehicle is a formula style prototype. An example of a formula student
vehicle is presented in Figure 1.1.
Figure 1.1 - FST 04e
1.2 THE PROJECTO FST NOVABASE TEAM
Projecto FST Novabase is the formula student team from Instituto Superior Técnico.
Founded in 2001, it is the longest-active team in Portugal and the most successful. Constantly
2
renovating its members the team as manufactured five prototypes and is growing in recognition inside
Portugal and internationally.
The fifth prototype of the team, FST 05e, will be the test car in this work. It is the second
electric vehicle of the team and the lightest prototype to date (Figure 1.2).
It uses two AC synchronous motors that control each rear wheel independently through a
two-stage gearbox system developed by the team. The battery uses ion polymer cells, the highest
energy density cells available in the market. The chassis is a carbon fiber reinforced polymer
monocoque, the first monocoque used in a prototype of the team. The suspension system uses carbon
fiber in the wishbones rods. The rim is also totally made of carbon fiber reinforced polymer except the
central nut support. All the systems come together to produce a vehicle weighting only 200 kg,
batteries included.
Figure 1.2 – The team with the FST 05e
1.3 WORK CONTRIBUTION AND OBJECTIVE
In the competitive motorsport world it is important to be one step ahead of the competition in
order to achieve desired results. This competitiveness accentuates even more in Formula Student
because the knowledge of the designers/students is directly evaluated and rewarded by the competition
judges. By having a good understanding of all the relevant parameters influencing the car behavior a
successful design can be achieved and thus better results.
With the development of earlier prototypes, the knowledge in vehicle dynamics grew within
the team and that knowledge has been passed through different generations of team members. The
previous car (FST 04e) was the first prototype to be carefully analyzed using advanced vehicle
dynamics knowledge (Neves 2012). The prototype exhibited very good handling and the judges
3
appreciated the team’s knowledge in vehicle dynamics, meaning that the use of such tools was an
advantage. The FST 05e team looked to carry the legacy of FST 04e as a good handling and
performing car, further improving the tools developed by previous teams and developing its own new
models such as dynamic analysis of the vehicle for torque vectoring design.
This work intends to further develop the dynamic model of the vehicle in order to improve
the vehicle dynamics understanding of the team and to help in the design of the next prototypes.
Up until now, the vehicle dynamics design was based on rigid bodies, but component’s
compliance should be in mind of every designer and vehicle performance analyser. Its effects are often
overlooked, with the designer making sure that the part does not fail and the vehicle performance
analyser despising its effects. It was made popular in the Formula Student community that the majority
of the cars exhibit excessive compliance because of the low awareness and the ambitious designs of
the students when minimizing the car’s weight. This question has raised awareness for this problem
and students are now more focused on minimizing vehicle compliance. In vehicle dynamics studies, in
order to be precise and evaluate compliance in a quantitative manner one must find a way to include
the compliance effects in the simulations. This way an interactive link can be built between the vehicle
performance department and the design and stress department with evidence data to support the design
decisions in what concerns the design of the parts.
The major objective of this work is to provide a methodical approach for the inclusion of
suspension compliance in the dynamic model of the FST 05e prototype. This will provide the engineer
with a better insight in simulations of how the car responds to driver inputs.
The computer model should be able to simulate the car given the inputs of the driver and the
parameters of each subsystem. The model will be implemented in Matlab/Simulink, and its structure
will be presented in this work.
1.4 DOCUMENT STRUCTURE
The outline of this document is as follows:
Introduction;
State of the art in simulations contemplating compliance;
Vehicle dynamics concepts;
Presentation of actual model and simulations – Without compliance;
Methodology for inclusion of compliant components and presentation of compliant
components;
4
Final Model presentation and simulation – With Compliance;
Conclusions and future developments
The second chapter presents a small review about the possibilities available in the market to
include compliance in vehicle dynamics simulation.
In the third chapter, several vehicle dynamics definitions and concepts will be introduced in
order to better understand the modelling of the vehicle as a system. It will be explained the importance
of each concept in the subsystem they belong to.
In the fourth chapter the model development will be presented, with a subchapter for each
subsystem – motors, brakes, suspension (kinematic and vibrational model), steering system and tires.
At the end, with every subsystem presented, the dynamics of the prototype as a full system are
presented divided into two areas – Driving in a straight line and cornering. Simulations will also be
done to evaluate the qualitative validity of the model.
In the fifth chapter the approach taken for the inclusion of compliant components will be
presented. The parameters influencing car performance presented in chapter three will be revisited
with the indication of the parameters that can be affected by the introduction of compliant
components. The components that serve as target for the compliance analysis will be presented and
exemplified ways to obtain the relevant deformations will be suggested.
In the sixth chapter will be presented the changes needed in the initial model to
accommodate for the compliant components. A simulation comparing both the rigid model and the
compliant model will be performed.
Finally, in the seventh chapter, conclusions will be drawn and suggestions will be made for
future developments of the model.
5
2 STATE OF THE ART IN SIMULATIONS WITH COMPLIANCE
EFFECTS
2.1 FORMULA STUDENT EXPERIENCE
Projecto FST Novabase is an experienced team with five vehicles and this experience and
knowledge has been growing every year with each technical development and increasing resources. In
competition, the FST cars have received positive feedback from the judges and obtained good results
in several static and dynamic events. It is with honor that the team’s knowledge in terms of technical
design, vehicle dynamics and suspension tuning has impressed the judges and also proved its
reliability in the dynamic events. Vehicle dynamics and knowledge is vital to a good understanding of
the vehicle and also to achieve a successful design.
Over the years, for several generations of members, a lot of simulation tools were developed
in order to better design the prototype. A natural flow is to start with simple models and increase their
complexity keeping the tools useful for the team. From the famous and simple, bicycle model, that
serves the purpose of analyzing the vehicle in the linear range of the tires without contemplating
weight transfer to more complicated models with the complete vehicle in a steady-state simulation to
assess its limit behavior as in (Neves 2012) were build. Later on, also dynamic models were built to
better assess and model the capabilities of torque vectoring system.
The inclusion of compliance is now modeled, trying to increase the vehicle dynamics
knowledge in order to better understand the vehicle (Figure 2.1).
Figure 2.1 - Evolution of vehicle modeling in Projecto FST Novabase
6
2.2 COMPLIANCE MODELING
Usually in a mechanical model that does not take into account body interactions, the
inclusion of compliant components is very restricted and usually ignored. The designer tries to make
the components stiff enough, sometimes without knowing what is acceptable without compromising
car performance and even safety. This is an acceptable approach but if the know-how is available to do
better designs, it should be used.
In multibody dynamics simulation this occurs more often. The modeling of each individual
component provides the freedom to introduce flexible bodies and measure their effect in vehicle
performance and suspension kinematics. This is widely used in the simulation of commercial road
vehicles where the suspension is mounted to the chassis with bushings, a more flexible component
than the spherical bearings usually used in racing vehicles like the FST 05e.
This multibody simulation is not easy to achieve in a Fomula Student team and commercial
software is usually used for reliability (The preferred choice is usually the famous software MSC
ADAMS). This software also has a vehicle dynamics dedicated environment, making it a favorite not
only of Formula Student teams (see Figure 2.2) but also some recognized automotive brands.
Figure 2.2 - Model of a formula student car in MSC ADAMS
Examples of use of this software can be found in several sources and with several intents.
In (McGuan and Pintar 1994), a cooperation work between a consulting company and Ford
Motor Company combines the capabilities of both MSC ADAMS and MSC NASTRAN to evaluate
the effects of modeling precise flexible suspension linkages. Until the interface was available between
the two MSC environments the compliance could only be modeled by the use of beams, flexible
7
bushings and tires. With this new interaction, complex geometries could be modeled as flexible and a
more accurate simulation could be performed.
The results are evident as the example of a lane-change maneuver results in differences of
30% in camber angles between the rigid and the flexible model. This is an early show of the vast
capabilities of MSC software for flexible vehicle dynamics simulation but also an evidence for the
importance of contemplating compliance effects when analyzing performance and design.
In (Fischer 2001), the contribution of BMW for the development of MSC ADAMS/Car is
noted and the introduction of this environment in the company is also described. Models without
multybody dynamic features are also used due to their speed and easier modeling proving that to
achieve a good perception of reality, one does not necessarily need to go to multibody dynamics. The
development of standardized test methods and standardized test results is also noted. Not giving too
much on how to model a vehicle it gives a good understanding on the chain of thought when leading a
design process.
Finally, in (Antona), a full passenger vehicle is modeled using MSC ADAMS/Car. In this
work, the model of the vehicle contemplates compliance and is used for ride and handling analysis.
Compliance tests were performed to compare and validate the vehicle model. Some full vehicle
dynamic tests were also performed to assess the reliability of the model. Even with some different
magnitude values, the tendencies were found to be similar and explanation about the reason behind
these differences is also given.
A similar context work to the mentioned above is also done in (Wale 2009) even if to a
lesser extent in complexity.
2.3 COMPLIANCE MEASURING
In regards to measuring the effects of compliance in vehicle kinematics the most common
and accepted way to do it is by performing a full vehicle Kinematics and Compliance (K & C) test.
Such tests are performed in specific machines like the Suspension Parameter Measurement Machines
of Anthony Best Dynamics, Ltd or the K & C Testing Machine from Morse Measurements LLC
(Figure 2.3).
These machines provide the engineers with the capability of applying relative displacements
between the suspension and the chassis, forces at the contact patch and combined situations to better
recreate vehicle steady-state conditions and evaluate kinematic relationships for more accurate vehicle
modeling. By performing such tests the engineer is able to build the vehicle model with the real
kinematic relationships accounting compliance instead of the perfectly rigid kinematic relationships
obtained from purely kinematic multibody simulation.
8
Figure 2.3 - K & C Testing Machine from Morse Measurements LLC
Some works are available showing the potential use of K & C testing.
In (Holdmann, Köhn et al. 1998), a very good overview of K & C concepts is presented.
From the basic considerations when designing a K & C Testing Machine, through the process of
performing simulations and comparisons between computer-built models in MSC ADAMS and testing
data. The work covers, even if in a resumed way, the process of designing, building and using a K& C
Testing Machine in Aachen University.
In (Morse 2004), Phillip Morse (from Morse Measurements LLC) takes on the task of using
K & C testing data for practical use in suspension tuning. In more common suspension tuning, the
effects of some old-established geometric characteristics of the vehicle are used together with stiffness
and damping adjustments. Trying to add to this old and well established knowledge, the use of K & C
data is interpreted and used for suspension tuning. This work is valuable in the way that it shows more
ways in which the testing data can be used besides its already valuable contribution to vehicle
modeling.
2.4 FRAMEWORK
This work positions itself in the middle of the two dynamic modeling approaches. It models
the vehicle behavior by directly using motion equations to describe the vehicle motion, kinematic
relationships to model wheel orientations, solid mechanics and even physical testing to obtain the
components displacements.
Kinematics and Compliance testing is an expensive effort for a Formula Student team, and
there is no such machines in Portugal to perform those tests. This work aims to provide ways of
including the effects of compliant components in vehicle simulation and open this once closed door.
9
3 VEHICLE DYNAMICS CONCEPTS
This chapter presents relevant vehicle dynamics concepts and definitions as the coordinate
systems used, angles and important variables for future model development.
3.1 COORDINATE SYSTEMS
In this work a set of four coordinate systems are used: two for vehicle motion modelling and
two for modelling of specific subsystems. All of these coordinate systems are in accordance with
(SAE 2008). The first coordinate system is the vehicle coordinate system and in this work it is
attached to the body centre of gravity (xyz). The equations of motion of the vehicle are expressed in
this coordinate system (Figure 3.1).
Figure 3.1 - Vehicle coordinate system
The other main coordinate system is the ground coordinate system (XYZ). This ground is
coincident with the vehicle coordinate system at the beginning of the simulation and serves the
purpose of measuring the vehicle motion, in other words, the position and orientation of the vehicle
coordinate system in respect to the ground coordinate system.
The third coordinate system (see Figure 3.2) is used to specify the pickup points of the
suspension in the chassis and at the wheel (xsyszs). This coordinate system has the same orientation as
the coordinate system of the vehicle but it is located at ground level, in the intersection of the xz plane
of the car and the line uniting the two contact patches of the front tires.
Figure 3.2 - Suspension coordinate system
10
The fourth coordinate system used is the tire coordinate system (xtytzt). The tire coordinate
system is attached to the contact patch of each tire and accompanies the rotation of the z axis of the
wheel. The purpose of this system is to represent the orientation of the wheels in order to calculate the
forces developed by the tire (see Figure 3.3).
Figure 3.3 - Tire coordinate system (SAE-J670)
The rotations of the vehicle coordinate system with respect to the ground coordinate system
in x, y and z are called the roll, pitch and yaw angles. The yaw angle and the x and y positions will be
used to represent the vehicle trajectory in the XY ground plane.
3.2 SLIP PARAMETERS
3.2.1 Sideslip angle
In vehicle dynamics three main slip quantities are defined to study the vehicle behaviour.
The first slip quantity is the side-slip angle, β (Figure 3.4). This is the angle between the
vector velocity and the x axis of the vehicle coordinate system.
Figure 3.4 - Sideslip angle representation
11
The next two slip quantities are important in tire dynamics and their influence in force
generation in the contact patch will be explained later. As of now, they will be presented as they are
defined.
3.2.2 Slip angle
The first slip quantity is the Slip Angle, SA. This is the angle measured between the
direction in which the wheel is heading and the direction of the velocity of that wheel projected onto
the xtyt plane (Figure 3.5). It represents a measure of the distortion that generates or is generated by
lateral force in the tire contact patch.
Figure 3.5 – Tire slip angle representation
3.2.3 Longitudinal Slip Ratio
The other slip quantity is a percentage of the slip of the wheel in the xtzt plane. To
understand this variable one must be aware of the notion of a free-rolling tire. A free-rolling tire is a
tire rolling with certain velocity and without sliding on a road with no applied torque (corrections
might be made to contemplate the rolling resistance of the tire).
Taking a wheel with no slip angle and a certain velocity, vw, one can define the reference
wheel-spin velocity of this wheel as the angular velocity for a free-rolling condition, ω0.
By defining the wheel-spin velocity as the actual angular velocity of the wheel, ωwheel, one
can define the tire longitudinal slip velocity as the difference between the two velocities.
Finally the tire longitudinal slip ratio, SL, is defined as a percentage by dividing the tire
longitudinal slip velocity by the reference wheel-spin velocity:
0
0
wheelSL
(3.1)
12
3.3 VEHICLE GENERAL CHARACTERISTICS
There are some general dimensions and weight properties that can be said as being a vehicle
characteristic and not of some specific area of the vehicle. In the following paragraphs the ones of
most importance for this work are presented.
3.3.1 Vehicle Dimensions
The wheelbase, l, is the distance measured parallel to the x direction of the vehicle between
the front and rear contact patch on the same side (Figure 3.6). It can be different from side to side, but
in the case of the model described it is considered to be equal.
Figure 3.6 - Wheelbase
The track, t, is the distance measured parallel to the y direction of the vehicle between the
left and right contact patch of the tire (Figure 3.7). It can also be different between the front and the
rear of the vehicle and it is a more common practice so it is contemplated in the model.
Figure 3.7 – Track
13
3.3.2 Vehicle mass properties
In this work, three main vehicle masses will be considered. They are the vehicle operating
mass, the unsprung mass and the sprung mass.
The difference between these weights is described below:
Vehicle operating mass (m) – The total mass of the car plus a conventional 68 kg driver;
Unsprung mass (mus) – The mass of the car that is not carried by the suspension, being directly
supported by the tire. This includes the tires, the wheels, the parts that move directly with the
wheel and a percentage of the mass of suspension linkages that in the case of this work will be
considered 50%. In this work this mass is evaluated for each corner of the car.
Sprung mass (ms) – The mass of the car that is carried by the suspension being calculated as
the subtraction between the vehicle operating mass and the total of the unsprung masses. It is
also evaluated at the front and the rear of the vehicle.
Finally, of the three principal moments of inertia of the vehicle, two of them will be used for
the model:
Vehicle Roll Moment of Inertia (Ix) – The moment of inertia around the x axis of the vehicle.
Vehicle Yaw Moment of Inertia (Iz) – The moment of inertia around the z axis of the vehicle
3.4 SUSPENSION & STEERING PROPERTIES
The suspension of a vehicle is one of the systems with the most parameters and definitions.
The most relevant ones for the developed model will be presented in the next sections.
3.4.1 Wheel Angles
The steering angle, δ, is the angle defined for each wheel as the angle between the x axis of
the vehicle and the wheel plane measured about the z axis of the vehicle. It is usually different
between the two different steerable wheels because of the steering geometry implemented (Figure
3.8). The static steering angle of the wheels is called the static toe angle.
Figure 3.8 - Steering Angles
14
The camber angle, γ, is the angle between the z axis of the vehicle and the wheel plane
measured about the x axis of the vehicle. The convention is that it is positive when the top of the
wheel leans toward the vehicle. An important angle in tire dynamics is the inclination angle, IA, which
is similar to the camber angle. The difference is that the inclination angle is the angle measured with
respect to the axis perpendicular to the ground plane instead of the z axis of the vehicle and is positive
when the top of the wheel is leaning towards the right (Figure 3.9).
Figure 3.9- Camber and Inclination angles
3.4.2 Steering axis & properties
The steering axis is defined by the pickup points at the wheel of the upper and lower
wishbone for a double-wishbone suspension. The two important angles are the kingpin inclination
angle, KPI, and the caster angle, ν (Figure 3.10). These angles contribute to the camber change while
the wheel is steered (because it is steered around an inclined axis) and also to the torque at the steering
wheel required to maintain the wheels steered.
Figure 3.10 - KPI and Caster angles
15
An important steering characteristic for the modeling approach taken for the steering system
is the c-factor of the steering system. The definition of this characteristic for a linear steering box is
the rack displacement per revolution of the input shaft.
3.4.3 Instant Centers
A relevant point in the design process and simulation is the instantaneous center of rotation
of the suspension in 2D.
In front view it is defined for each wheel and can be useful to define the path of the wheel
and also to be used as a point on which the jacking forces produced by the lateral tire forces are
reacted, producing the Anti-Roll effect. The variable used in the model to define this instant center is
the angle λ (Figure 3.11).
Figure 3.11 - IC Definition (front view)
In side view the instantaneous center is found following the same methodology. This time,
for the reaction of the jacking forces that constitute the anti-dive, anti-squat and braking or
acceleration anti-lift of the suspension the angle used in the model is found depending if the force
acting on the axle is a driving force or a braking force. In case of driving producing anti-squat (rear
suspension) or acceleration anti-lift (front suspension) the angle used is the one defined as σ. For
braking anti-lift (rear suspension) or anti-dive (front suspension) the angle η is used (Figure 3.12).
Figure 3.12 - IC definition (side view)
16
3.5 SUSPENSION SPRINGS AND DAMPERS
To provide suspension displacement for better handling and comfort, vehicles are usually
equipped with springs and dampers. In the FST 05e, the suspension layout has a damper and spring
tandem actuated by a bellcrank which gives freedom to select the shock absorber displacement as a
function of the wheel center displacement (Figure 3.13).
Figure 3.13 - Suspension Corner
In order to know the change in vertical load applied at the wheel center in function of the
displacement of the wheel one needs to easily translate the wheel movement to spring movement. By
using kinematic relations, the rotation of the bellcrank and consequently the spring displacement in
function of the displacement of the wheel can be calculated. This function is called the Motion Ratio,
MR.
SpringTravelMR
Wheel CenterTravel (3.2)
In the case of the FST 05e, this ratio can be considered linear as it was designed with this
purpose by the team. With this function the effect of the springs installed can be evaluated by
calculating an equivalent stiffness of a spring mounted between the wheel center and the tire, KS.
2
S SK k MR (3.3)
In this equation, kS is the spring stiffness actually installed in the vehicle.
This approach facilitates the calculations during simulation by letting the complicated
kinematic equations represented by a single ratio, MR.
The same methodology can be assumed for the effective damper mounted between the wheel
center and the tire, CS.
17
A specific approach is taken here because of the nature of most dampers. Usually, dampers
have four damping ratios, low speed compression and rebound and high speed compression and
rebound (Figure 3.14). Because the main purpose of the model is to give a hint on changes in vehicle
performance the normal cornering situations will be considered. This normal cornering is considered
as low-speed behavior of the damper as in (Kasprzak).The high speed characteristics are used for the
treatment of bumps and inspection of suspension performance with road irregularities. Also, for easier
modeling the rebound and compression ratio are considered the same.
Figure 3.14 - Example of damping curve
Regarding roll behavior, usually some race cars are equipped with anti-roll bars (ARB),
which is the case of the FST 05e. The anti-roll bar connects opposite side wheels and acts only when
there is a difference in displacement between the two sides (Figure 3.15).
Figure 3.15 - ARB location and closer look
This mechanism contributes to the suspension roll stiffness which is a combination of the
stiffness provided by the springs and the anti-roll bar to resist the roll moment induced during a turn.
To calculate the equivalent spring between the wheel center and the tire that represents the
ARB one needs to first convert the equivalent torsional stiffness of the bar to a linear stiffness, kARB,
and then apply the motion ratio of the wheel/ARB couple to get the equivalent stiffness, KARB.
18
2
ARBARB
Kk
actuating arm
(3.4)
2
ARB ARB ARBK k MR (3.5)
In (3.4), KϕARB represents the torsional stiffness of a bar as in (Beer, Jr. et al. 2011).
19
4 MODEL DEVELOPMENT – WITHOUT COMPLIANCE
In the present chapter the model will be presented. A description of the drivers inputs will be
done first, then the several subsystem models will be presented separating longitudinal dynamics and
lateral dynamics of the vehicle for better organization.
4.1 DRIVER INPUTS
In a normal vehicle the direct driver interface are the throttle and brake pedals, the steering
wheel and the gear stick. Since the FST 05e is a fully-electric powered vehicle and the gearbox has a
single gear ratio the only inputs considered from the driver will be the pedals and the steering wheel.
The inputs will be received in the following format:
Throttle pedal: Percentage of pedal deflection from 0% to 100 %;
Brake pedal: Input force at the pedal in kg;
Steering wheel: Input steering wheel angle in degrees.
4.2 TIRE MODELING
The main forces that help the vehicle in its motion around the track arise from the tire and
the nature of this component to deform. By combination or isolated action of sliding of the tread and
distortion of the carcass, the tire presents a capability to withstand forces and moments in all three
coordinate directions. The mechanics of an operating tire is a complicated matter and in this work only
the main principles relevant to the modeling executed will be presented. Usually the normal operation
of a tire has a combination of distortion and sliding as opposed to when the tire is said to have “broken
away” and it is only sliding as described in (Milliken and Milliken 1995).
4.2.1 Tire Lateral Force
When applying a rising lateral force, Fy, to a tire it suffers a lateral distortion until the point
it begins to slide (the tire has “broken away”). Thus one can say that by applying a distortion to the
tire, it generates a lateral force because of its elastic nature (a characteristic of its construction).
The slip angle, as described in the third chapter, is usually the parameter that is used to
describe the lateral distortion of the tire and the behavior of the tire print when measuring forces
(Figure 4.1).
20
Figure 4.1 - Tire Lateral Force (courtesy of Milliken & Milliken)
4.2.2 Tire Longitudinal Force
In the case of a rolling tire at a certain velocity without slip angle if a torque is applied to the
wheel, and consequently a variation in angular velocity of the wheel, a distortion arises in the tire print
because a relative velocity between the tire print and the road must be opposed by the adhesion of the
tire to the road. This adhesion produces a longitudinal force, Fx, in the print that counter-acts the
torque input.
The most common measure of this distortion used in vehicle dynamics is the longitudinal
slip ratio as described in chapter three (Figure 4.2). As with slip angle, if the longitudinal slip ratio is
too high (in absolute value) the tire is said to slide. In the case of braking for example, a value of -1
describes a wheel with no angular velocity. In fact, and supported by experimental data, a longitudinal
slip ratio outside of the range of -0.2 and 0.2 can be describe a sliding tire and the forces fall quickly.
21
Figure 4.2 - Tire Longitudinal Force - Driving (courtesy of Milliken & Milliken)
4.2.3 Self-Aligning Torque
The self-aligning torque, Mz, is a moment measured around the z axis of the tire coordinate
system that tries to steer the tire back to a situation without lateral force. This can be experienced by a
normal driver and is actually an intuitive characteristic. This self-aligning torque arises because of the
distribution of tire lateral force in the contact print. As one can see in Figure 4.1, the centroid of the
tire lateral force is located behind the tire contact patch center. The distance to this center is called the
pneumatic trail and multiplied by the tire lateral force results in the self-aligning torque.
4.2.4 Tire Model – Pacejka’s Magic Formula
In order to be able to simulate the vehicle behavior one must have a way to obtain the tire
forces to calculate the car’s accelerations. One way to do this is to use a function that simulates the tire
performance. This function must accept certain inputs and return the output forces and moments
presents in the contact patch. Hans Pacejka was able to evaluate tire data and trace the typical behavior
of a tire with the variation of certain conditions of operation. With this evaluation, Pacejka was able to
create is well-reputed Magic Formula where each output is a function of the operating conditions. This
formulation is described in (Pacejka 2005).
22
In this work, the formulation presented in the book referenced above is used. This
formulation makes use of the following operating conditions: Fz and IA. The variables of the
formulations are SA, SL or both.
The general form of the equations is shown below:
1 1
0 sin[ tan { ( tan ( ))}]y y y y y y y y y y vyF D C B E B B S (4.1)
1 1
0 sin[ tan { ( tan ( ))}]x x x x x x x x x x vxF D C B k E B k B k S (4.2)
1 1
0 0. cos[ tan { ( tan ( ))}].cosz yo t t t t t t t t t zrM F D C B E B B M (4.3)
1
0 cos tanzr r r r rM D C B (4.4)
It is a normal operating condition to have combined slip situations where slip angle and
longitudinal slip ratio are present in the tire. Since theoretically the maximum force developed by the
tire in the horizontal plane must be fixed it only makes sense that by trying to raise the tire
longitudinal force its capability to generate lateral force drops and vice-versa. The approach taken
when formulating this combined slip condition is by applying a multiplying factor to the formulas that
represents the drop in force generation:
0.x x xF G F (4.5)
0.y yk y vykF G F S (4.6)
The parameters B, C, D, E, S and G in the equations define the shape of the resultant tire
curve. These parameters are functions of the operating conditions earlier defined and must be found
through curve fitting of tire testing data.
The symbols α and k are transformations of the slip angle and the longitudinal slip ratio.
The full set of equations have been evolved over the years and have suffered improvements
and new versions, the ones used here are based on the P2002 as in (Pacejka 2005) and can be found in
Appendix A – Magic Formula equations.
4.2.5 Tire Testing
Tire testing is one of the single most important things for vehicle dynamics. Fortunately,
FSAE teams can have access to tire data from the TTC. TTC stands for Tire Test Consortium, an
organization that provides the teams with data of tires usually used for this competition. The team has
the responsibility of using this data for analysis and model fitting.
4.2.6 Model Fitting
The fitting of the tire lateral force in pure slip conditions (slip angle only) is presented below.
The methodology is the same for the fitting of the other functions.
23
First, the data obtained from the TTC testing is organized in order to obtain the raw data tire
curves.
Then, and after implementing a Matlab function with the equations present in Appendix A –
Magic Formula equations, one defines the initial value for the coefficients and using a predefined
Matlab least squares algorithm for curve fitting problems one can obtain the resulting coefficients that
best fit the set of raw data (Figure 4.3):
Figure 4.3 - Raw tire data and fitting curves
With these coefficients one can easily calculate the forces and moments present in the tire by
specifying the operating conditions.
, , ,y zF f F IA SA SL (4.7)
4.3 AERODYNAMICS
With the increased awareness about the benefits of using aerodynamic devices in FSAE, the
amount of teams displaying an aerodynamic design for their cars is also rising. Projecto FST Novabase
was also quick to predict the increased performance of the vehicle and the FST 05e has an
aerodynamic package of front and rear wings and undertray.
To treat the aerodynamic forces, the approach taken in this work requires the knowledge of
the drag and lift coefficients, the area of the wings and also the position of the aerodynamic devices
with respect to the center of gravity.
In order to calculate the forces in the wings the following fluid mechanics expressions are
used:
24
21
. . . .2
DD C Av (4.8)
21
. . . .2
LL C Av (4.9)
D and L are the drag and lift forces and CD and CL are the drag and lift coefficients. The
forces are calculated for each wing according to the velocity of the vehicle.
In order to translate the aerodynamic forces to the wheels the approach taken in this work is
to define an aerodynamic reference point to which all the forces are translated. This follows the
principle that a force can be translated to any point if the moment the force makes around that point is
properly accounted for. The point location is at the center of gravity height for convenience and the
longitudinal position is found in order to make the sum of moments caused by all the forces is equal to
zero (Figure 4.4). This results in a point that has all the drag and lift forces and no moment.
Figure 4.4 - Aerodynamic reference point
The calculation of this point is as follows:
0 . .
. . 0
RP F FW F FW RP
R RW R RW RP
M D z L x x
D Z L x x
(4.10)
With this reference point defined one can add the acceleration caused by the drag force to the
acceleration caused by the tire forces and distribute the lift force between the four wheels according to
the xRP position.
This defines the lift distribution for use in the vibrational model:
RPx
LD WDl
(4.11)
25
4.4 VEHICLE VIBRATIONAL MODEL
In order to calculate the vertical forces in the wheels a vibrational model of the car’s
suspension will be implemented due to the fact that it provides the possibility to contemplate the
transient part of weight transfer, include migrating instant centers and better represent the anti-
features. These possibilities are not available when using traditional weight transfer equations which
only serve for steady-state analysis.
4.4.1 Model layout and components
This model includes the relevant energy accumulating and dissipating components. The
suspension will contribute with the coil springs, anti-roll bars and dampers. The tire is also taken into
account by using its spring rate and an estimation of its damping rate. Finally, in order to not
compromise the suspension work, the chassis needs to be sufficiently stiff. This stiffness is also taken
into account by dividing the chassis into two torsional springs (Figure 4.5). This stiffness is obtained
from the FEA model and is courtesy of the Projecto FST Novabase team.
Figure 4.5 - Vehicle vibrational model
In the model presented in Figure 4.5 the green components are considered infinitely stiff and
the black are the ones with the stiffness or damping quantified. The blue and red ones are the sprung
and unsprung masses of the system. The purple object represents the vehicle’s moment of inertia in the
x axis of the vehicle as described in chapter three.
4.4.2 Model formulation
To solve this kind of dynamic problems one needs to define the operating conditions (forces)
and the system parameters (stiffness, mass and damping matrices). First, the degrees of freedom
(DOF) of the model are identified:
26
FRS
FRUS
FLS
FLUS
RRS
RRUS
RLS
RLUS
z
z
z
z
z z
z
z
z
(4.12)
As one can see, there are 9 independent DOF in this system, the vertical displacement of
each mass system and the roll of the chassis.
The applied forces vector should also be formulated in the beginning of the problem:
FRS
FRUS
FLS
FLUS
RRS
RRUS
RLS
RLUS
ch
F
F
F
F
Q F
F
F
F
M
(4.13)
The equation to be solved is then:
M z C z K z Q (4.14)
4.4.3 Mass, Stiffness, Damping Matrices and Force Vector
The mass, stiffness and damping matrices are found by combining the equations that result
from the free-body diagram analysis of each mass component (Figure 4.6 and Figure 4.7).
27
Figure 4.6 - Front Vibrational Model
Figure 4.7 - Free-body diagram of the vibrational model – Front Right Side
Using Newton’s second law one can write the equations of the bodies:
2
FRS FRS FRS s FRUS FRS arb FRUS FRS FLUS FLS
chf chf
s FRUS FRS FLS FRS
f f
m F K z z K z z z z
K KC z z z z
t
z
t
(4.15)
FRUS FRUS FRUS s FRS FRUS arb FRUS FRS FLUS FLS
s FRS FRUS t FRUS t FRUS
m F K z z K z z z z
C z z K z z
z
C
(4.16)
The same can be done for the front left side of the vehicle:
Figure 4.8 - Free-body diagram of the vibrational model – Front Left Side
28
2
FLS FLS FLS s FLUS FLS arb FLUS FLS FRUS FRS
chf chf
s FLUS FLS FLS FRS
f f
m F K z z K z z z z
K KC z z z z
t
z
t
(4.17)
FLUS FLUS FLUS s FLS FLUS arb FLUS FLS FRUS FRS
s FLS FLUS t FLUS t FLUS
m z F K z z K z z z z
C z z K z C z
(4.18)
The same type of equations can be written for the rear end of the vehicle.
For the inertia that represents the vehicle resistance to body roll, the free-body diagram is
also drawn and the analysis is made again but this time accounting for the moments (Figure 4.9):
Figure 4.9 - Free-body diagram of vehicle for moment analysis
FRS FLSF
f
z z
t
(4.19)
RRS RLSR
r
z z
t
(4.20)
x ch f r
ch chf f chr r
FRS FLS RRS RLSch chf chr
f r
I M M M
M K K
z z z zM K K
t t
(4.21)
With all the equations formulated one can identify the mass, stiffness and damping
parameters and organize them in matrices. The force and moment vector can also be identified in the
expressions.
To compute the force vector one makes use of the knowledge of the loads of the vehicle in
motion. When running, a car is subjected to many forces in the horizontal plane. These forces arise
mainly from the tires and aerodynamics devices. By Newton’s law, the sum of these forces results in
an inertia force at the vehicle’s center of gravity (Figure 4.10). This resultant force (divided in its
components along the vehicle axis) is the origin of weight transfer.
29
Figure 4.10 - Tire horizontal forces and vehicle reactions
Each force and moment identified in Eq. (4.13) must be defined and there are several
contributions to this formulation. Weight transfer, aerodynamic lift, static vertical loads and anti-
characteristics must be taken into account as identified next.
First, the moment around the x axis of the vehicle caused by the fact that the center of
gravity of the sprung mass is above the ground and experiences a lateral acceleration is defined. It is
called the roll moment (see Figure 4.11):
ch S y CGSM m a h (4.22)
Figure 4.11 - Lateral weight transfer
The unsprung mass experiences the same acceleration and an equal approach is taken using
the appropriate parameters and is performed by axle:
. .US y CGUS
z
m a hF
t (4.23)
30
In chapter three, the notion of instant center is explained. Along with the roll stiffness of the
suspension (springs and anti-roll bars) this instant centers can help counteract roll if placed above the
ground. Defined as the instantaneous center of curvature of the contact patch it is the pivot of the
rotation of the wheel and it can be treated as equivalent to a bar pinned to the instant center and always
connecting the ground. Drawing a free-body diagram (Figure 4.12) one can write the equations that
describe the effect that the tire lateral force has on the IC, causing the appearance of a vertical force
between the sprung and the unsprung mass (dashed lines represent the reactions at the IC).
tan z yF F Right Side (4.24)
tan z yF F Left Side (4.25)
Figure 4.12 - Anti-Roll effect of tire lateral forces
The action of braking and driving forces is treated equally (Figure 4.13) but with the
difference in the reference angle depending of the direction of the force:
tan z xF F Braking Front axle (4.26)
tan z xF F Braking Rear axle (4.27)
tan z xF F Driving Rear axle (4.28)
31
Figure 4.13 - Anti-Lift, Anti-Squat and Anti-Dive effect of tire longitudinal forces
The direction of the forces in the formulas is as they are applied to the unsprung mass
meaning that the same force is applied to the sprung mass with the opposing direction.
The drag force, as explained previously is moved to a point at the same height of the center
of gravity of the vehicle so its effect on weight transfer can be added as a component of the
acceleration of the sprung mass to the components caused by braking and driving.
Longitudinal weight transfer is treated as the lateral weight transfer, separating the sprung
and unsprung accelerations but instead of using the moment produced by the acceleration, the forces
are taken directly to the wheels.
To define longitudinal weight transfer, one can follow the free-body diagram of Figure 4.14:
Figure 4.14 - Longitudinal weight transfer
Equation (4.29) gives the change in vertical load in each axle due to the longitudinal
acceleration of the sprung mass. The same applies for the unsprung mass but using the appropriate CG
height.
. .S x CGS
z
m a hF
l (4.29)
32
Finally the static weight is applied to each body and also the contribution of the total lift
force by properly dividing its value by the sprung masses (where the aerodynamic devices are
mounted) using the lift distribution.
. zF L LD Front Axle (4.30)
. 1 zF L LD Rear Axle (4.31)
Adding all the contributions shown above, the force and moment vector defined in Eq. (4.13)
can be build.
This vibrational model is implemented in Matlab’s Simulink environment. This way one can
easily obtain solution to Eq. (4.14) using a feed-back system. The velocities and positions are obtained
via integration of the acceleration and fed back in the system that comprises the equation.
From this system one can obtain the roll of the vehicle, the vertical displacements of the
sprung and unsprung mass and also the vertical forces at the tires.
4.5 DYNAMICS OF A PARTICLE IN NON-UNIFORM CIRCULAR MOTION
A vehicle moving around a track can be described as an object with a certain linear velocity
of its center of gravity and also an angular velocity around this same point. Below a description of the
moving object will be made and the equations of motion will be obtained.
As described in (Beer, Johnston et al. 2006) one can define an instantaneous center of
curvature for an object and use the rotational velocity around the center of gravity around this same
point (Figure 4.15):
Figure 4.15 - Vehicle motion
Using kinematics of moving particles with the particle being at the vehicle’s center of
gravity, one can write the equation for the acceleration of the particle:
33
CG
dva r v
dt (4.32)
Being a non-uniform circular motion, the first term of the equation is the variation in
magnitude of the velocity vector and the second term represents the variation of the vector velocity in
direction. Since it is more common to use the accelerations and velocities in components of the vehicle
coordinate axes. One can divide the above equation in two planar components:
.cos . .sin .x x ya v v v v (4.33)
.sin . .cos .y y xa v v v v (4.34)
In this model only planar motion with three degrees of freedom will be taken into account
when defining the horizontal motion of the vehicle. For simplicity reasons the effect of the rotation
around the vehicle x and y axis will not be taken into account when defining the vehicle’s motion.
This said one must be able to find the lateral and longitudinal forces acting on the vehicle’s center of
gravity and also the yaw moment (moment around the vehicle z axis). The equations of motion for an
object in planar motion are then defined as follows:
. .
. .
.
X x y
y xY
Z zz
F m v v
F m v v
M I
(4.35)
4.6 VEHICLE LONGITUDINAL DYNAMICS
4.6.1 Motor
The FST 05e is powered by two electric AC synchronous motors with a peak power of 62.8
kW. The model of the motor will control this device by evaluating the rotational speed of the output
shaft, ωM, and making available the maximum torque, TM, according to a Torque vs Speed
characteristic curve. This characteristic curve will be obtained by using the maximum torque and
speed the motor can deliver and also the cutoff speed.
In general, electric motors have their maximum torque at zero speed. In this case the motor
maintains the maximum torque or a similar value until it reaches the cutoff speed. When the speed
increases above this value the torque decreases linearly until it reaches zero for maximum speed (see
Figure 4.16).
Although this is not the most accurate motor modelling it is the most simple and can be used
for the purposes of this work. Nevertheless, the model of the vehicle is done in such a way that the
motor model can be developed in a more detailed way.
34
Figure 4.16 - Torque vs Speed Curve
The following characteristics are identified:
Maximum Torque: 24 N/m
Maximum Speed: 40000 RPM
Cutoff Speed: 15000 RPM
The torque delivered to the transmission is then controlled by the percentage of the throttle
pedal, P%, depressed by the driver.
The transmission of the vehicle is a gearbox with only one input to output ratio, this is called
the transmission ratio, TR. For the FST 05e this fixed ratio is of 21.
The torque to the wheels is then delivered as follows:
M MT f (4.36)
%input MT P T (4.37)
wheel inputT T TR (4.38)
Since the FST 05e has a motor and gearbox for each one of the rear wheels, the above
equations are used for both wheels.
4.6.2 Brake System
Today the vast majority of racing cars uses a brake system that presents a braking rotor on
each one of the four wheels of the vehicle. This is also the case of the FST 05e. The brake system used
and modelled in this work is a hydraulic system with no energy regeneration. Such systems are
reliable and can be easily modelled for simulation purposes.
The purpose of the braking system is to provide the driver with a possibility to quickly slow
down the car. Generally this is done by exerting a force on the brake pedal, which the brake system
0
5
10
15
20
25
30
0 10000 20000 30000 40000 50000
Torq
ue -
N.m
Output Shaft Rotational Speed - RPM
Torque vs Speed Curve
35
transforms in torques acting at the wheels to cause an angular deceleration and the consequent
decrease in vehicle velocity.
In a racing car, when the driver presses the brake pedal he exerts a force on two hydraulic
cylinders – These are called master cylinders. In the FST 05e one of these cylinders actuates the front
brakes and the other the rear brakes allowing two independent systems. The force of the pedal is
divided by the two master cylinders by a device called balance bar (Figure 4.17). This balance bar
serves the purpose of tuning the front to rear balance of the braking force to fit specific desires of the
driver or the engineer.
Figure 4.17 - Brake pedal layout
The position of the balance bar and its orientation is also chosen in order to give the driver a
mechanical advantage. This means that the force exerted by the driver in the pedal is mechanically
multiplied in order to be able to brake effectively without excessive force. This advantage is called
pedal ratio, PR (see Figure 4.18).
Figure 4.18 - Brake pedal free-body diagram
0 ( sin ) 0sin
PR P B B
FM F e F e F
(4.39)
36
So the pedal ratio is:
1
sinPR
(4.40)
To divide by the two cylinders the following free-body diagram is used (Figure 4.19).
Figure 4.19 - Balance bar free-body diagram
0 0
1
FF B F BR F R BR B
F R
FBF B BR B
F R
eM F e F e e F F
e e
eF F F F
e e
(4.41)
In the hydraulic cylinders this force is transformed into a pressure that is held constant along
the brake lines (losses will be despised). This pressure is transformed back into force in the brake
calipers, two at each end of a master cylinder. These brake calipers proceed to use this force to clamp
the brake rotors. To transform force into pressure and vice-versa, the master cylinders and calipers
have in their interior a cylinder for which the area is known.
In the case of the master cylinders, one cylinder is present. For the calipers that equip the
FST 05e, two opposed cylinders are used. This said, the clamping forces at the front calipers are
calculated as follows (equivalent for the rear):
BFF
MC
FP
A (4.42)
BFCF F C CF C
MC
FF P A F A
A (4.43)
, 2clamp F CFF F (4.44)
The caliper is assembled with a component called brake pad that produces a friction force in
the surface of the rotor:
37
, , F pad F clamp FF F (4.45)
This friction force can be replaced by a torque around the center of the wheel proportional to
the radius of the brake rotor (Figure 4.20):
Figure 4.20 - Brake rotor
, , wheel B F F ROTOR FT T F R (4.46)
This braking torque is then reacted at the contact patch by the tire longitudinal force which
generates a deceleration in order to slow the vehicle’s speed.
4.6.3 Wheel rotational dynamics and force calculation
Since the input torque at each wheel is known through the models described above, one must
find the tire longitudinal forces present at the contact patch.
, ,x ZF f SL SA F (4.47)
The vertical load at each tire is known through the vibrational model leaving the longitudinal
slip ratio to be defined in order to calculate the tire longitudinal force. The slip angle calculation will
be described ahead in section 4.7.2.
For this calculation, the velocity of the vehicle is used to calculate the reference wheel-spin
velocity as described in chapter 3. By drawing a free-body diagram of the wheel and writing the
equations that define the angular motion of the wheel one can define its behavior (Figure 4.21).
38
Figure 4.21 - Wheel free-body diagram
wheel center wheel wheel wheel x wheel wheel wheelM I T F R I (4.48)
wheel x wheelwheel
wheel
T F R
I
(4.49)
By calculating the angular acceleration of the wheel when subjected to the moments
described above, one can once again use the Matlab Simulink environment advantages of looped
systems, differentiation and integration. By differentiating the angular acceleration one obtains the
wheel angular velocity, which serves as an input to the longitudinal slip ratio calculation as described
in chapter 3.
4.7 VEHICLE LATERAL DYNAMICS
4.7.1 Steering Mechanism
The most common racing cars have front steerable wheels, which does not mean that the rear
wheels are not steered by default (static toe). The steering mechanism of the FST 05e is a rack and
pinion assembly that translates the angular motion of the steering wheel to linear motion of the tie-
rods.
The steering assembly as a parameter called c-factor that is usually used to represent the gear
ratio of the rack and pinion:
factor
rack travel mc
Input shaft revolutionr t tm
o ro (4.50)
The steering rods are then connected to the uprights at a specifically defined position in order
to turn the wheels by the amount desired.
Usually the two front wheels are not steered by the equal amount. This is done because of
the different radius of travel of each wheel and also in racing applications to achieve a desired steering
39
behavior accounting to develop the slip angle that provides the maximum tire lateral force in specified
conditions of load and inclination.
To achieve this condition the tie-rod pickup point of the upright is positioned with an offset
in the y axis relative to the intersection of the steering axis with the wheel center plane in top view
(Figure 4.22):
Figure 4.22 - Steering Geometry
The solid lines represent the nominal position of the steering system with no wheels steered.
The dashed lines represent a linear displacement of the rack.
In order to model this, advantage was taken of the Matlab Simulink environment once again.
Simulink presents in its package a variant that allows the modelling of multi-body systems for
dynamic and kinematic analysis. This is the Simmechanics library.
Since the driver input in the model is the steering wheel angle, δSW, one can first of all
transform it in rack displacement by using the c-factor:
factor SWrack displacement c (4.51)
A multibody system contemplating the front suspension, the steering rack and tie-rods, is
modelled in Simmechanics. The rack is assembled in the model using a prismatic joint with one
degree of freedom for the rack displacement to be introduced – done using a joint actuator (Figure
4.23). A joint sensor is then placed at the wheel centers to measure the rotation relative to the z axis.
Figure 4.23 - Steering system in Simmechanics
40
4.7.2 Sideslip and slip angles calculation
As defined in chapter two, the sideslip angle of the vehicle can be calculated as follows:
1tan
y
x
v
v
(4.52)
This variable is useful for vehicle performance analysis but for the calculation of slip angles
the side-slip angles of each wheel will be used (Figure 4.24), as defined in (Jazar 2008).
Figure 4.24 - Slip angle
i i iSA (4.53)
1tan
yi
i
xi
v
v
(4.54)
The value of the steering angle is obtained from the Simmechanics model but the sideslip
angle of each wheel is calculated as in Eq. (4.54).
Figure 4.25 - Wheel velocities schematic
Knowing the velocities of the vehicle’s center of gravity (Figure 4.25) and according to
(Beer, Johnston et al. 2006) one can calculate the velocities at the wheels (xi and yi are the position of
the wheels measured in the vehicle coordinate system):
41
yi y i
xi x i
v v x
v v y
(4.55)
4.7.3 Inclination angle calculation
The inclination angle as defined in chapter three will be calculated using the displacement of
the wheel and also the wheel orientation. As the steering axis is usually not vertical, the steering of the
wheel causes the inclination angle to suffer changes according to mentioned geometry. The inclination
angle also changes with the wheel travel due to the suspension design. These two contributions happen
at the same time but as described in (Neves 2012) they can be treated independently and added to give
the final contribution.
The commonly known inclination angle gain with vertical motion is considered in static
conditions equal to the camber gain, and also to the variation of the KPI angle. In order to calculate the
KPI angle change, the method described in (Blundel and Harty 2004) as the 3-point method will be
used. This method states that it is possible to find a valid position of a point if the position of three
other points is known and also the distance from these points to the unknown point. This is useful
when treating a suspension wishbone where we know the chassis pickup-points and the length of the
wishbone arms.
Figure 4.26 - 3-point method schematic
Looking at Figure 4.26, the goal of this method is to calculate points 1 and 2 knowing points
4, 5, 6, 7 and 3 and also their distance to the unknown points.
2 2 22
1 3 1 3 1 3 1 3
2 2 22
1 4 1 4 1 4 1 4
2 2 22
1 5 1 5 1 5 1 5
2 2 22
2 3 2 3 2 3 2 3
2 2 22
2 6 2 6 2 6 2 6
2 2 22
2 7 2 7 2 7 2 7
2 2 2
1 2 1 2 1 2 1 2. .
d x x y y z z
L x x y y z z
L x x y y z z
d x x y y z z
L x x y y z z
L x x y y z z
S t x x y y z z d
(4.56)
42
Solving of the first six equations always finds two solutions, so, to filter this, the last
equation is introduced as a constraint. The distance, d1-2, is known as it is the distance between the
pickup-points of the upright.
By solving these equations for several positions of the point 3, which represents the wheel
center, one finds the successive positions of the upright pickup-points through the wheel travel. Using
trigonometry one can find the gain in KPI angle and caster angle (Figure 4.27).
Figure 4.27 - KPI angle calculation
1 2 1
1 2
tany y
KPIz z
(4.57)
As the suspension of the FST 05e is designed with nearly linear and symmetric KPI angle
gain one can define a coefficient to be multiplied by the wheel travel in order to know the KPI angle
gain. For this the equations are only solved for the static position and then with a wheel travel of 10
mm.
10
10
z static z static mm
wheel wheel
KPI KPIIA KPI
z z
(4.58)
The same can be done to calculate the gain in caster angle.
The contribution of the steering axis rotation for the inclination angle gain is presented next
and also the calculation of the inclination angle by adding all the contributions:
(1 cos ) sin
(1 cos ) sin
FR FR
FR
FL FL
FL
IAKPI
IAKPI
(4.59)
43
, ,
, ,
, ,
, ,
( )
( )
( )
(
FR static FR wheel FR static
wheel FR
FR static FL wheel FL static
wheel FL
RR static RR wheel RR static
wheel
RL static RL wheel
wheel
IA IAIA IA z z
z
IA IAIA IA z z
z
IAIA IA z z
z
IAIA IA z
z
)RL staticz
(4.60)
4.7.4 Force and moment calculation
The tire lateral forces and self-aligning moments are calculated using the Pacejka model
developed, all the parameters present in the calculation were described above:
, , ,yF f SA SL FZ IA (4.61)
, ,zM f SA FZ IA (4.62)
4.8 VEHICLE ACCELERATIONS
Since the tire lateral forces calculated are perpendicular to the wheel, it is needed to convert
these forces into the vehicle coordinate system in order to calculate the vehicle accelerations. The
same happens for the tire longitudinal forces. This has a major effect on the front wheels since these
are the steered wheels.
Figure 4.28 - Tire forces components
It is obvious from Figure 4.28 that a tire lateral force on a steered wheel causes a
longitudinal force on the vehicle. And the opposite happens for a tire longitudinal force.
44
cos sin
sin cos
Yi y i x i
Xi y i x i
F F F
F F F
(4.63)
To calculate the vehicle longitudinal acceleration and lateral accelerations, the tire and
aerodynamic forces are needed. Since the tire forces depend on the longitudinal slip ratio and the slip
angle, which depends on the car velocities and thus also in its accelerations, one can see the loop
created (see Figure 4.29) and the necessity to use the Simulink environment.
Figure 4.29 - Calculation loop schematic
Following the equations given in the section 4.5:
X x yF m v v (4.64)
.XFR XFL XRR XRL
x y
F F F F Dv v
m
(4.65)
Y y xF m v v (4.66)
YFR YFL YRR YRL
y x
F F F Fv v
m
(4.67)
y y xa v v (4.68)
x x ya v v (4.69)
To calculate the vehicle angular acceleration, the yaw moment around the vehicle’s center of
gravity needs to be calculated.
, ,2 2
f rZ z Y Front Y Rear XFR XFL XRR XRL
t tM M a F b F F F F F (4.70)
45
Z zzM I (4.71)
Z
zz
M
I (4.72)
To obtain the vehicle velocity, the magnitude variation of the velocity vector is integrated.
The longitudinal, lateral and angular velocities can then be used in the calculation of the longitudinal
slip ratio and slip angle.
A schematic of the whole model is presented in Figure 4.30 where the different interactions
between the several subsystems can be seen:
Figure 4.30 - Model schematic
4.9 VEHICLE MOTION
To inspect the vehicle trajectory one must have the motion variables in the ground
coordinate frame, XYZ, see Figure 4.31. For this, one must transform the vehicle velocities using the
yaw angle.
0 dt (4.73)
cos sin
sin cos
X x y
Y x y
v v v
v v v
(4.74)
Integrating the vehicle velocities in the ground coordinate system the vehicle planar
trajectory is obtained.
46
X
Y
X v dt
Y v dt
(4.75)
It is also possible to describe the trajectory of the vehicle wheels using the relative position
of the wheels with respect to the vehicle coordinate system.
i
CG i
i
Xd r
Y
(4.76)
Figure 4.31 - Vehicle in the ground coordinate system
Using this positions one can plot the trajectory of the vehicle’s CG and that of its wheels in
the XY plane.
4.10 SIMULATIONS
In order to evaluate the simulations, a Matlab script was developed to show the important
measured variables within the vehicle model during the simulation running time. Since the Simulink
environment is not the friendliest for showing data, the normal Matlab environment was used thinking
about one of the goals of this work: its use in Formula Student car development.
Any variable within the vehicle model shown above can be extracted from the simulation,
making this a useful tool for vehicle simulation analysis.
Next, some simulations will be presented together with some comments about the results of
some variables.
47
4.10.1 Straight line acceleration and braking
A simulation was done for the vehicle accelerating in a straight line and braking until it is
completely stopped. This aims to simulate the vehicle behavior in the acceleration event from the
formula student competition and also to evaluate its performance in braking (Figure 4.32). Also, some
observations are drawn about the qualitative validity of the model.
Figure 4.32 - Driver inputs
The brakes are applied as they commonly are for a car with heavy downforce. The pedal
force is decreased along with the decrease in velocity (Figure 4.33).
Figure 4.33 - Wheel velocities and longitudinal slip ratio
When accelerating, the motor provides the wheel with a torque causing an angular
acceleration that increases the wheel angular velocity. At the beginning of the simulation wheel-spin
occurs and the longitudinal slip ratio is very high. As soon as the wheels gain sufficient traction from
the road to counter-act the imposed torque, the wheel spin decreases and the wheels present a smaller
longitudinal slip ratio, when equilibrium is reached between motor torque and tire longitudinal force.
The front wheels, being non-driven wheels, present an angular velocity proportional to the vehicle’s
velocity.
The vehicle then brakes without locking the wheels until it is completely stopped around the
7.5 second mark, see Figure 4.34.
48
Figure 4.34 - Tire longitudinal forces and vehicle acceleration
Above one can see the gain in traction when the wheel-spin ends and also check the
longitudinal acceleration capability of the FST 05e. The aero devices in the FST 05e allows it to brake
at almost 3 G’s of deceleration due to the drag at high velocities and also the downforce available at
the wheels. In Figure 4.35, the evolution of the aerodynamic and tire vertical loads during simulation
can be seen.
Figure 4.35 - Aerodynamic and tire vertical loads
The vehicle’s velocity is also tracked and can be seen below, having a 0-100 kph time of
2.94 seconds, similar to the best formula student prototypes and a very good value for a rear-wheel
drive only vehicle (Figure 4.36).
In Figure 4.37, the distance traveled can be seen. The vehicle crosses the 75 meters with a
time of 5.073 seconds. Since it starts to accelerate at the one second mark, the total acceleration time is
of 4.073 seconds.
49
Figure 4.36 - Vehicle velocity
Figure 4.37 - Distance covered
These results can be compared to similar formula student prototypes in formula student
competitions through the results available at the several competitions websites. These results are also
available in Appendix C – Formula student Results. One can see that the FST 05e positions himself
along similar cars in terms of weight and aerodynamic devices such as the AMZ 2012 umbrail.
4.10.2 Skid Pad simulation
The Skid Pad is the formula student event that aims to measure the vehicle’s cornering
capability. By performing turns around a similar corner radius the vehicles are compared (Figure
4.38). The measured time is the average time between the right circumference and the left
circumference. Each run includes two laps around the right circumference and two laps around the left
circumference.
50
Figure 4.38 - Skid Pad layout (FSAE rules)
To simulate the vehicle in the Skid Pad and extract a time around one circumference only
one lap is completed. The inputs found to provide a good Skid Pad simulation can be seen in Figure
4.39.
Figure 4.39 - Driver inputs in Skid Pad
Due to the non-parallel steer introduced in the FST 05e design, the front wheels are steered
to different angles (Figure 4.40).
Because the steering of the wheels is a transient maneuver, one can expect the slip angles of
the front wheels to rise to very high values. This effect is due to the vector velocity of the vehicle
being longitudinal at the moment the wheels are steered. This effect is quickly counteracted by the
development of lateral velocity.
51
Figure 4.40 - Wheels steering angles and slip angles
The lateral force at the front of the vehicle causes a yaw moment which produces a lateral
velocity at the rear of the vehicle. This lateral velocity causes the appearance of slip angles and
consequently a lateral force. This rear force tries to balance the yaw moment created by the front
wheels. This balance happens in a steady-state maneuver as the Skid Pad. During corner entry or exit
and also during mid-corner the yaw moment contributions are not balanced causing changes in the
path radius of the vehicle.
In Figure 4.41, the yaw moment of the vehicle can be seen and the steady-state is achieved to
maintain a constant radius and velocity turn.
Figure 4.41 - Yaw moment
The weight transfer can also be identified by inspecting the vertical loads on the wheels.
These different vertical loads are the main cause for the difference between the tire lateral forces in the
same axle generated at the contact patches (Figure 4.42).
52
Figure 4.42 - Tire vertical loads and lateral forces
The throttle pedal position was controlled manually at the beginning of the simulation to
maintain a nearly constant vehicle velocity (Figure 4.43).
Figure 4.43 - Vehicle velocity in the Skid Pad
The resulting radial acceleration of the vehicle is also nearly constant (Figure 4.44).
Figure 4.44 - Vehicle radial acceleration
53
The Skid Pad time is a very good result comparing to the competition results. The vehicle is
able to complete it in 4.722 seconds (Figure 4.45).
Below one can see the vehicle’s center of gravity trajectory path to assure that the vehicle is
within the limits of the track (Figure 4.46). It was considered that the driver turns the steering wheel
when he enters the circumference.
Figure 4.45 - Skid Pad completions during
simulation
Figure 4.46 - Vehicle's CG trajectory path
The result of this simulation is a very good Skid Pad time but not too far off the best results
seen in competition. It is important to notice that this event demands a very good and constant driver
with good notions and sensibility for vehicle performance limits.
The nearly constant radial acceleration at a fairly high value is difficult to achieve so this
result is expected to be near the peak performance of the FST 05e.
Nevertheless the model shows a good representation of the vehicle expected optimal
behavior and can be used for vehicle performance analysis.
Several vehicle performance indicators can be calculated with the results of the several
variables present in the model.
54
55
5 APPROACH FOR INCLUDING COMPLIANT COMPONENTS
5.1 PARAMETERS WITH INFLUENCE ON VEHICLE BEHAVIOR
There are several vehicle parameters that can affect vehicle behavior as can be seen from the
model description in chapter four. The predominant system in a formula student car influencing the
dynamic of the vehicle is the suspension system, and this will be the target of this work.
The contact patches of the tires are the major sources of forces and moments that define the
motion of the vehicle. These forces and moments are influenced by a lot of known and unknown
physical quantities because the mechanics of a pneumatic tire are not understood in full detail and are
too complex. As shown in chapter four, the most important parameters are used to formulate a semi-
empirical model of the tire. The parameters used in this work are:
Vertical load;
Slip angle;
Inclination angle;
Longitudinal slip ratio.
Changes in these parameters result in a difference in tire horizontal forces and tire moments.
Thus this work attempts to quantify deformations in suspension components that can influence these
parameters and have an effect on tire forces and moments.
5.1.1 Vertical Load
The vertical load on a tire is affected by the downforce provided by the aerodynamic devices
and also the effects of weight transfer.
The most influence could arise from bellcrank deformation which causes a difference in
motion ratios and consequently wheel rates. This could modify the weight transfer distribution in
contrast with a non-compliant model. Since the bellcrank mechanism deformation is difficult to
model, this will not be target of analysis. The deformation of aerodynamic devices is also not a part of
the suspension so its analysis will be exclusively with the non-compliant configurations. From above,
the vertical load will not be directly influenced by compliant components in this work.
5.1.2 Slip angle
The slip angle is a parameter directly influenced by the steering system. The deformations in
the steering system from the steering wheel to the tie-rod influence the slip angle compliance. The tie-
rod can be treated as a deformable link and the changes in steering angle can be modeled. The twist
angle in the steering shaft is also an easy parameter to take into account when calculating the rack
displacement.
56
5.1.3 Inclination angle
The inclination angle is mainly influenced by the suspension components and also the wheel
deformation.
By modeling the wishbones as deformable components, one can calculate new KPI angle
gains that directly influence the inclination angle (Figure 5.1).
Figure 5.1 - Scheme of deformation influences
5.2 LOAD CASES CALCULATION
Deformation is caused by loads and these loads must be calculated in order to better model
the compliant components. The loads on the suspension components arise from direct reactions to
what happens at the contact patches of the tires.
While it is obvious that the wheel load cases can be obtained by direct application of the tire
forces, a way must be found to calculate the loads on the remaining suspension components.
To calculate the loads on wishbones and tie-rods a Matlab Simulink model of the suspension
is built (Figure 5.2). As stated in the previous chapter, the Simmechanics library of Simulink is able to
model multibody system for dynamic analysis.
Figure 5.2 - Simmechanics model for dynamic analysis.
57
By introducing an extension of the contact patch to the upright body, the tire forces and
moments can be easily introduced in order to extract the loads in every suspension linkage (Figure
5.3).
Figure 5.3 - Detail of corner suspension
Since the wishbones and tie-rods are assembled with spherical joints, they are treated as links
with only axial loading from the mechanical point of view.
The tire loads enter the Simmechanics system and are filtered in order to be introduced in the
correct points of application – contact patch or wheel center. Matlab then performs a static analysis of
the system and returns the suspension loads.
In order to correctly analyze the system, one must correctly place the tire loads in order to
generate a correct result as stated above. This is of major importance when longitudinal tire forces
come into play.
The free-body diagram of a wheel can be used to evaluate the constraints acting on this
component (Figure 5.4). Being attached to the hub and the brake rotor, the three components can be
considered as a rigid body for this static analysis. This rigid body is constrained by the upright in all
three directions but also in two rotations, about the x axis of the wheel and the steering axis.
A tire lateral force must be applied to the contact patch extension of the upright in order to
reflect the reaction moment because of the rotational constraint around the x axis of the wheel.
In the case of a longitudinal force, the contact patch force is not reacted by any rotational
constraint, thus in simulation must be directly placed at the upright center point.
58
Figure 5.4 - Free body diagram of wheel-upright assembly
Although this is true, when braking, the brake caliper exerts a moment in the upright
equivalent to the braking moment that results in the longitudinal force as seen in section 4.6.2. This
being said, to represent the load exerted by the caliper on the upright, the longitudinal braking force
must be placed in the contact patch extension of the upright, the same point as the lateral force. For a
driving force there is no moment exerted on the upright and the force can be correctly applied in the
upright center point.
The explanation above for the loads in the suspension linkages is also useful to calculate the
load on the steering column (Figure 5.5). By using the calculated forces on the tie-rods one can find
the force present on the steering rack.
Figure 5.5 - Steering box assembly
This force can be converted into a moment reacted by the steering pinion. Considering this to
be the torque reacted by the driver at the steering wheel, this is also the torque applied that causes the
twist of the steering shaft.
59
Figure 5.6 - Rack and pinion free body diagram
2
factor
steering rack
cT F
(5.1)
5.3 DEFORMATION ANALYSIS
This section intends to present a way to quantify the deformation of the components listed
above. It will present the approach taken in this work and also suggestions for different approaches.
5.3.1 Steering shaft
As described above the torque applied to the steering shaft (Figure 5.7) will result in a twist
angle of this component, ψ, which can be subtracted from the final steering input.
Figure 5.7 - Steering shaft location
If the steering shaft is produced as a single component, with a length lsteering-shaft, one can
apply Eq. (5.2) :
steering
steering shaft
J GT
l
(5.2)
In this equation, J is the second moment of area and G is the shear modulus of elasticity.
Taking the steel steering shaft from the FST 04e as an example for this work one can see the
influence of the steering torque in the twist angle (Figure 5.8).
60
Figure 5.8 - Steering shaft twist angle
5.3.2 Wishbones, Tie-rods and Toe-rods
The suspension links as these parts will be called from here on are usually made of steel. In
Formula Student it was not long ago that most teams were running a full set of steel suspension links.
With competition growing and the desire to learn more the teams started to turn their attention to
CFRP (carbon fiber reinforced polymer) links (Figure 5.10). This can be mainly attributed to their
very successful use in Formula 1 (Figure 5.9), the highest category of motorsport innovation.
Figure 5.9 - Formula 1 suspension
Figure 5.10 - FST 05e suspension
The design of the CFRP suspension differs in some aspects to the design of the steel
suspension, starting in the material characteristics – isotropic vs orthotropic.
61
As described in the previous section, being mounted with spherical bearings makes the
loading of the wishbone arms easy to predict since it will only withstand axial loads – compression or
tension (Figure 5.11).
Figure 5.11 - Wishbone schematic
After obtaining the loads present in each link one can calculate its deformation using the
elastic properties of the component. In the case of a steel suspension – or any other isotropic material
suspension – the Young’s modulus, E, is the only material characteristic needed.
link
F
E A
(5.3)
Where F is the force present in the link and Alink is the cross-section area of the suspension
link. Considering the deformation to be symmetric in tension and compression one can use the strain
to calculate the change in the length ( l ) of the suspension link:
0l l (5.4)
In the case of the CFRP suspension the behavior is not so easily predicted because the link is
not made from the same material. Beginning with the construction of the wishbone, this is made by
bonding an aluminum insert into the CFRP tube using high-strength structural adhesive (Figure 5.12).
Figure 5.12 - Bonding schematic – side view
This bonding area is cylindrical and due to the axial nature of the loads present in the
suspension link, the bonding area is able to sustain these loads in shear – one of the most reliable ways
to load the adhesive.
62
This said one must find a way to study the combined behavior of the tube and the adhesive in
order to correctly address the deformation of the links. The first approach taken during the design of
the suspension was to consider the adhesive as the fuse of the joint, producing a cohesive failure.
Since it is the first time that a Formula Student car from the team uses adhesive in the
suspension links, not much knowledge was present within the team know-how to design the bonded
joint. A previous work was successfully done to design the drivetrain half-shafts in (Valverde 2007), a
composite assembly with the shear area loaded by the action of a torque. In this work, reference is
made to the Volkersen Shear Lag theory which provides a way to analytically study the stresses along
a bonded joint in shear loading.
As explained in (Adams, Comyn et al. 1996), a single-lap joint can present different
behaviors in terms of adhesive shear stress. This is dependent on the stiffness of the adherends used.
If the adherends are considered to be infinitely stiff one can see that the shear stress is
uniform along the joint (Figure 5.13):
Figure 5.13 - Adhesive behavior for rigid adherends
This is not the case when the stiffness of the adherends is quantified and even more
important when the stiffness of the two adherends is different – which is the case for a CFRP-
Aluminum joint. The evolution of the shear stresses in the adhesive is now dependent on the stiffness
of the adherends (Figure 5.14):
Figure 5.14 - Adhesive behavior for flexible adherends
This behavior was first approached by Volkersen in 1938 and consists on the oldest theory to
analyze single-lap joints taking into account the stiffness of the adherends (Figure 5.15). Due to its
63
simplicity, vast literature sources and the available adhesive properties it was the technique used by
the team to design the bonded joints.
Figure 5.15 - Single-lap joint schematic
The shear stress evolution is formulated as follows:
int int
( ) cosh sinh
2 sinh 2 cosh2 2
adhesive
jo jo
PW PWx Wx w M Wx
WL WLw w
(5.5)
2 2 1 1
2 2 1 1
E t E tM
E t E t
(5.6)
2 2 1 1
2 2 1 1
adhesive
adhesive
G E t E tW
h E t E t
(5.7)
In these equations, Ei, represents the Young’s Modulus of each adherend and Gadhesive is the
shear modulus of the adhesive. In Figure 5.16, Volkersen’s Theory is employed and one can see the
differences between different joint designs, concerning the thickness and material of the adherends and
the bondline thickness of the adhesive:
Figure 5.16 - Shear stress scenarios in single-lap joint (same loading condition)
64
To use this theory for the design of the suspension links the approach taken was to use the
cylindrical shape of the bonding area and treat it as a single-lap joint with the following characteristics
(Figure 5.17):
22 2
i adhesiveTD hw
(5.8)
22 2
i iadhesive
TD IDt h (5.9)
linkP F (5.10)
Figure 5.17 - Suspension link cut-view
The classic laminate theory as described in (Reddy 1997) is used to obtain the equivalent
elasticity modulus of the CFRP tube in the axial direction. A complete display of the equations is
present in Appendix B – Equivalent laminate properties.
The Volkersen equation is used along several discrete points defined along the joint giving
the stress distribution as depicted above (Figure 5.18).
Figure 5.18 - Example of calculation points and their deformation
Using solid mechanics one can calculate the shear strain and the consequent displacement of
the bonded joint (Figure 5.19).
65
Figure 5.19 - Unit volume of adhesive in shear
tan( )adhesive adhesivel h (5.11)
tan( ) adhesive
adhesiveG
(5.12)
To this elongation one must add the carbon fiber tube deformation also. This is done using
the normal stress equation and the Hooke’s Law – the same procedure as for isotropic materials.
link
tubeCFRP
CFRP axial
FA
E
(5.13)
0CFRP CFRPl l (5.14)
In the equation above, l0, is the original length of the CFRP tube.
This said one can compute the total elongation of the assembled link in the following way:
2link CFRP adhesivel l l (5.15)
During the design of the FST 05e, experimental tests were done to make sure that the
suspension links would be able to sustain the loads they would be subjected to (Figure 5.20). Even
though the testing was done with the intent of verifying the maximum load supported by the link, one
can compare the theoretical results with experimental data to draw some conclusions to help model the
compliance of the suspension links.
Figure 5.20 - Test specimen installed in universal testing machine
66
Since the FST 05e uses only two types of tubes for the entire set of suspension links, these
were the target of the physical testing.
It is important to note that during the design of the FST 05e, several tests were performed to
evolve and better understand the importance of surface preparation and adhesive application. In this
work only the results of the final set of tests will be shown and considered for further analysis (Figure
5.21 and Table 5.1).
Figure 5.21 - Test specimen
Table 5.1 - Test specimen characteristics
Bondline thickness -
hadhesive Overlap length - Ljoint CFRP tube length
Tube Inner Diameter
0.1 mm 20 mm 70 mm 10 mm and 18 mm
The test specimen has the same bonded area and bondline thickness as in the suspension
links. The only difference is the length of the CFRP tube used.
Using the experimental data and extracting a sufficient load range of data points, one can
find the force versus strain curve for the entire test specimen (Figure 5.22 and Figure 5.23).
Figure 5.22 - Test results for tube ID = 18 mm
y = 4087,5x
R² = 0,9993
-1
0
1
2
3
4
5
6
7
0 0,0005 0,001 0,0015 0,002
Forc
e -
kN
Strain
Force vs Strain - 18 mm CFRP tube
67
Figure 5.23 - Test results for tube ID = 10 mm
Using the equation of the fitted line to the experimental data and Eq. (5.4) one can calculate
the elongation of the test specimen and compare the results with the theoretical result using the method
described above. Results are presented in Table 5.2 and Table 5.3.
As can be seen the error is large and conclusions can be tough to draw. One explanation is
that only the displacement of the machine’s crosshead and the load exerted are measured. This makes
it impossible to separate the deformation of the carbon tube from the deformation of the bonded joint.
Tube ID = 18 mm
Load (kN) Strain Experimental Δl (mm) Theoretical Δl (mm) Relative Error
1 0,000245 0,01712 0,00594
65%
2 0,000489 0,03425 0,01188
3 0,000734 0,05137 0,01783
4 0,000979 0,06850 0,02377
5 0,001223 0,08562 0,02971
Table 5.2 - Results comparison, Tube ID = 18 mm
Tube ID = 10 mm
Load (kN) Strain Experimental Δl (mm) Theoretical Δl (mm) Relative Error
1 0,000365 0,02554 0,01198
53%
2 0,00073 0,05108 0,02396
3 0,001095 0,07662 0,03593
4 0,00146 0,10216 0,04791
5 0,001824 0,12770 0,05989
Table 5.3 - Results comparison, Tube ID = 10 mm
In what the adhesive is concerned some explanations to these errors may come from the use
of the Volkersen theory allied to the differences between a theoretical joint and a real joint.
y = 2740,6x
R² = 0,9878
-1
0
1
2
3
4
5
6
7
-0,0005 0 0,0005 0,001 0,0015 0,002 0,0025 0,003
Forc
e -
kN
Strain
Force vs Strain - 10 mm CFRP tube
68
The Volkersen theory is fairly old but can be hold accountable to some extent to treat bonded
joints. The problem with this and almost every theory is that surface condition can’t be accounted for
when doing the calculations.
Also because of the manufacturing process of the specimens, the specific bondline thickness
cannot be totally guaranteed, in part because of machining imperfections. The tube manufacturing
presents an imperfect inner surface and also the increase in thickness due to manual sanding of the
surfaces during the surface preparation stage. All this contributes to the increase of bondline thickness
that can alter the results of the experimental test. Looking at Eq. (5.11), one can see that an increase in
bondline thickness increases the distortion angle. Even though the stresses are lower, this fact can’t
prevent the distortion from rising.
Concerning the tube, since it is made from CFRP, being an orthotropic material also gives
room for some uncertainty in its equivalent properties, even if in a lesser extent.
Since the error is calculated to the full extent of the specimen and the deformation of the tube
and the adhesive cannot be separated the decision was to take the error and apply it to the theoretical
value, this means considering the error as a multiplication factor to Eq. (5.15).
exp
exp
theoretical
R
l lE
l
(5.16)
2
1
CFRP adhesive
link
R
l ll
E
(5.17)
As seen above, the methodologies considered for the suspension links compliance were
presented, either when using steel or aluminum wishbones and for hybrid wishbones using bonded
joints. This represents the examples and methodologies used in this work which means that if a more
precise way to estimate Δllink is available it should be used for the vehicle compliant model since its
inclusion in the model is independent of the method used to obtain Δllink..
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6 MODEL PRESENTATION – WITH COMPLIANCE
In order to include compliance in the model developed in chapter four, one must find a way
to employ the knowledge of chapter five. By modifying Figure 4.30 of chapter four, one can easily
present the areas of modification, and the inclusion of new subsystems.
6.1 MODEL SCHEMATIC MODIFICATIONS
Figure 6.1 - Model schematic with compliance modifications
As can be seen in Figure 6.1, the modifications in the model schematic contemplate the
addition of new subsystems in red and also the modification of two existent subsystems, in dashed
lines.
6.1.1 Component loads calculation
This subsystem contains the Simmechanics model with the suspension system and calculates
the loads present in each link. The wheel loads are applied as described in chapter five and the loads in
the links are calculated in components of the vehicle coordinate system. These components are then
summed to give the final axial load on the link and these loads are the output of this subsystem
6.1.2 Deformation calculation
The deformation calculation subsystem receives the load in each suspension links and
proceeds to calculate its elongation. It also includes the necessary variables to calculate the twist angle
in the steering shaft (Figure 6.2).
70
Figure 6.2 - Deformation calculations scheme
6.1.3 Steering mechanism modifications
The steering mechanism is modified to receive the twist angle and also the elongation of the
tie-rods. The twist angle is used by subtracting its value to the input steering angle.
*
SW SW (6.1)
To use the elongation of the tie-rods, the capabilities of Simmechanics are used by adding a
sliding joint in the link, dividing the link into two separate components that are able to slide with
respect to each other along the axial direction.
Figure 6.3 - Tie-rod modification
The joint actuator uses the elongation of the tie-rods calculated in the deformation
calculation subsystem to apply a displacement to the joint.
71
6.1.4 IA calculation modifications
As seen in chapter four, the calculation of the inclination angle is done using the KPI angle
gain through the three point method. This method uses the length of the suspension links and in the
vehicle model without compliance, this length is fixed, so the inclination angle is independent of the
forces present in the links.
In the compliant model, the equations are modified to use the elongation of the wishbone
links in the three point method routine (Figure 6.4).
Figure 6.4 - Three point method sketch with compliance
2 2 22
1 3 1 3 1 3 1 3
2 2 2 2
1 4 1 4 1 4 1 4
2 2 2 2
1 5 1 5 1 5 1 5
2 2 22
2 3 2 3 2 3 2 3
2 2 2 2
2 6 2 6 2 6 2 6
2 2 2
2 7 2 7 2 7 2
1 4
1 5
2 6
2 7
static
static
static
static
d x x y y z z
L x x y y z z
L x x y y z z
d x x y y z z
L x x y y z z
L x y
l
x
l
y z
l
l
2
7
2 2 2
1 2 1 2 1 2 1 2. .
z
S t x x y y z z d
(6.2)
The compliance modified steering angle of the wheels also enters in the IA calculation, but is
directly introduced in the calculations as in chapter four.
72
6.2 COMPLIANT SKID PAD SIMULATION
As in section 4.10.2, a Skid Pad simulation with the same exact input parameters was
performed in order to compare the results of both simulations.
The driver inputs are reminded in Figure 6.5:
Figure 6.5 - Skid Pad inputs - compliant model
When steering the wheels, the driver imposes a deformation on the tire, causing a lateral
force (and consequently slip angle) to appear. This lateral force gives rise to a self-aligning torque at
the contact patch, forcing the driver to resist this with a torque at the steering wheel. The forces
involved come to the driver through the steering components. Those components compliant behavior
can be seen on Figure 6.6 and Figure 6.7:
Figure 6.6 - Steering rods compliance and steering shaft twist angle
73
Figure 6.7 - Wheels steering angle
As one can see, the steering shaft twist angle is not very noticeable. On the other side, the
most loaded wheel in the front axle forces its steering rod to stretch 0.15 mm. This causes the steering
angle of that wheel to decrease by 0.2º.
The forces and moments on the wheels also load the wishbones, causing deflections in these
components. In Figure 6.8, the links compliances from the most loaded wheel (front right wheel) are
shown:
Figure 6.8 - Front right wheel links compliances
As one can see, the displacement of the most loaded link is around 0.45 mm in compression.
These compliances cause the inclination angle of the wheels to change from the nominal values of the
rigid simulation.
74
Figure 6.9 shows the inclination angles of all wheels:
Figure 6.9 - Inclination angles during Skid Pad simulation
The variables shown above are the ones with most meaning in the difference in net force at
the contact patch as explained in chapter 5. The lateral forces at the tires, suffered changes and from
the obvious look at the data above, the front right wheel was the most affected one.
Figure 6.10 plots a closer look at the tire lateral force from the two most loaded wheels and
consequently most affected ones:
Figure 6.10 - Right tires lateral forces in compliant Skid Pad simulation
As one can see, the forces take a longer time to reach a similar value to the ones from the
rigid model simulation and this is not related to the variables above, because they preserve roughly the
75
same value along the simulation (Figure 6.11). It is the effect of a small change in trajectory and
acceleration.
Figure 6.11 - Path radius comparison
Figure 6.12 – Radial acceleration comparison
As seen in the figures above, the trajectory suffers a considerable change from the rigid
model simulation and the vehicle’s acceleration is also reduced, even though it begins to increase as
the simulation goes on.
This two combined effects result in the vehicle’s tangential velocity data shown on Figure
6.13:
76
Figure 6.13 - Vehicle’s tangential velocity comparison
The velocity increases as the simulation goes on because of the rise in acceleration and path
radius. This goes hand to hand, as an increase in velocity causes an increase in downforce which then
provides more tire horizontal force (longitudinal and lateral) available.
The side-slip angle also decreases during simulation, a result of the larger radius and higher
velocity (Figure 6.14):
Figure 6.14 - Side-slip angle comparison
All the changes shown above capitalize into a slower time around the Skid Pad even if by not
a big difference (Figure 6.15):
77
Figure 6.15 – Comparison between elapsed times to complete one Skid Pad
The comparison between the most notable variables between the two simulations is
presented in Table 6.1:
Table 6.1 - Skid Pad comparisons
Rigid Model Compliant Model
1.796 G Maximum radial acceleration 1.790 G
45 kph Maximum tangential velocity 45.1 kph
4.722 s Elapsed Time 4.735 s
78
79
7 CONCLUSIONS AND FUTURE DEVELOPMENTS
7.1 CONCLUSIONS
The objective of building a vehicle model including compliant components was fulfilled.
The vehicle model was developed and the methodology for the inclusion of compliant components
was exposed.
An overview of several works contemplating compliance in the vehicle dynamics world was
done and this work was placed in that spectrum of engineering activities.
The compliant components were included recurring to classical solid mechanics knowledge
and also the use of previously performed physical testing.
A comparison with the same input data between the rigid and compliant model was
performed to evaluate the differences in the vehicle behavior but fundamentally the difference between
the elapsed times to complete the specified circuit.
Table 7.1 - Simulation results
Rigid Model Compliant Model
Skid Pad 4.722 s 4.735 s
Acceleration 4.073 s -
As can be seen from the results in Table 7.1, the result of the inclusion of compliant
components was a small difference in elapsed time due to small changes in trajectory. This difference
is not very significant and the conclusion is that the modeled compliant components may not have a
very preponderant influence in the FST 05e behavior around the Skid Pad. This may not be true for
other simulations, but this is not the mainly intent of this work.
The acceleration event was not performed because the model does not feel an influence of
the compliant components on the tire forces in purely longitudinal dynamics.
One must notice that the rigid model simulation of the Skid Pad is a best case scenario in
terms of elapsed time. This event requires more driving skills than the acceleration event and the
inputs in the model are ideally set at the start of the simulation. One should expect a lower
performance from the real FST 05e, depending on the driver and track conditions but for comparison
purposes the simulation is valid.
It must be noted from the simulations that even though the driver inputs were held constant
for purposes of comparison, the small changes in trajectory and velocity could cause the driver to
change the inputs so the elapsed times could have a larger difference if a Skid Pad simulation with a
real driver was performed.
80
Validation of results with further physical testing, preferably with an instrumented FST 05e,
would have been helpful and it is certainly the missing piece of this work. Besides that fact, this work
can contribute to the design process of the next prototypes for Projecto FST Novabase and serve as a
building ground for further improvements in vehicle dynamics simulation performed by the team.
7.2 FUTURE DEVELOPMENTS
To further improve this work the inclusion of other vehicle’s components is the task at hand .
Several other suspension parts could be implemented with the use of FEA, be it via an
interface between Matlab and a FEA software, as ANSYS for example, or even by previously
performing the simulations and by mapping the displacements that arise from several load cases. The
use of a previously built map of displacements could be the best choice but only if the superposition of
results was validated via a combined load condition. If this superposition of results was valid, relevant
savings in simulation time could be achieved. The other way would involve a FEA software setup for
each added component and the inclusion of a bridge between the vehicle model simulation and the
FEA analysis.
The chassis behavior could also be included in a simpler way. By having the stiffness of
every suspension pickup point and with the loads in the suspension links, the compliance of the
chassis could be directly introduced in the three-point method described in this work.
The methods used for the calculation of suspension links compliance could also be revised
and the work done in (Ferreira 2013) to model the bonded joint with finite elements could be used.
If opportunity exists for K & C testing of the FST 05e, validation of the modeled compliant
components could be achieved. This testing, as stated earlier is very expensive and other ways to
validate the models may be investigated.
The instrumentation of the FST 05e with strain gauges in the suspension links, could give an
insight to the forces present and also the compliance of each link.
Finally the tire model may be revised to introduce more parameters with the first target being
the influence of the inclination angle in the longitudinal force to check the difference in the
acceleration times. The equations for this model were used in (Neves 2012) but not in this work.
81
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(SICETE).
82
APPENDIX A – MAGIC FORMULA EQUATIONS
The equations used in this work to fit the tire experimental data were adapted from (Pacejka
2005) and the final set of implemented equations will also be presented next.
The equations have some minor simplifications and the majority of the scaling factors are set
equal to one. The only factor used that has a value different from one is λµx,y and this is only for
purposes of vehicle simulations. For tire modelling, one is used.
First, the lateral slip is defined to account for the case of large slip angles:
* tan( )SA (A.1)
For the spin due to camber angle the following is also introduced:
* sin( )IA IA (A.2)
The normalized change in vertical load is presented next:
z zoz
zo
F Fdf
F
(A.3)
Longitudinal Force (pure longitudinal slip):
1 2( )Vx z Vx Vx zS F p p df (A.4)
1 2( )Hx Hx Hx zS p p df (A.5)
1 2 3( ) exp( )x z Kx Kx z Kx zK F p p df p df (A.6)
1 2( ) 0x Dx Dx z xp p df (A.7)
0x x zD F (A.8)
1 0x cxC p (A.9)
xx
x x
KB
C D
(A.10)
x HxSL S (A.11)
2
1 2 3 41 sgn( ) 1x Ex Ex z Ex z Ex xE p p df p df p (A.12)
sin arctan arctan( )xo x x x x x x x x x VxF D C B E B B S (A.13)
Lateral Force (pure side slip):
*
1 2 3 4( ) ( )Vy z Vy Vy z Vy Vy zS F p p df p p df IA (A.14)
*
1 2 3( )Hy Hy Hy z HyS p p df p IA (A.15)
83
1
2
sin 2arctan zy o Ky zo
Ky zo
FK p F
p F
(A.16)
2*
31y y o KyK K p IA (A.17)
2*
1 2 3( ) 1 0y Dy Dy z Dy yp p df p IA (A.18)
y y zD F (A.19)
1 0y cyC p (A.20)
y
y
y y
KB
C D
(A.21)
*
y HyS (A.22)
*
1 2 3 21 sgn( ) 1y Ey Ey z Ey Ey yE p p df p p IA (A.23)
sin arctan arctan( )yo y y y y y y y y y VyF D C B E B B S
(A.24)
Longitudinal Force (Combined Slip):
1Hx HxS r (A.25)
1 2 1x Ex Ex zE r r df (A.26)
1x CxC r (A.27)
1 2cos arctan 0x Bx BxB r r (A.28)
*
S HxS (A.29)
cos arctan arctan( )x o x x Hx x x Hx x HxG C B S E B S B S (A.30)
cos arctan arctan( )
( 0)x x S x x S x S
x
x o
C B E B BG
G
(A.31)
x x xoF G F (A.32)
Lateral Force (Combined Slip):
1 1Hy Hy Hy zS r r df (A.33)
* *
1 2 3 3( ) cos arctan( )Vy y z Vy Vy z Vy VyD F r r df r IA r (A.34)
5 6sin arctan( )Vy Vy Vy VyS D r r (A.35)
1 2 1y Ey Ey zE r r df (A.36)
84
1y CyC r (A.37)
*
1 2 3cos arctan ( ) 0y By By ByB r r r
(A.38)
S HySL S (A.39)
cos arctan arctan( )y o y y Hy y y Hy y HyG C B S E B S B S
(A.40)
cos arctan arctan( )
( 0)y y S y y S y S
y
y o
C B E B BG
G
(A.41)
y y yo VyF G F S (A.42)
Aligning Torque (pure side slip):
VyHf Hy
y
SS S
K
(A.43)
*
t HtS (A.44)
10r Bz y yB q B C (A.45)
1rC (A.46)
*
6 7 8 9r z wheel Dz Dz z Dz Dz z yD F R q q df q q df IA (A.47)
cos arctan[ ]zro r r r rM D C B (A.48)
*
1 2 3 4( )Ht Hz Hz z Hz Hz zS q q df q q df IA (A.49)
*
r HfS (A.50)
2 * *
1 2 2 4 21Bz Bz z Bz z Bz Bz
t
y
q q df q df q IA q IAB
(A.51)
1( 0)t CzC q (A.52)
1 2( )wheelto z Dz Dz z
zo
RD F q q df
F
(A.53)
2* *
3 31t to Dz DzD D q IA q IA (A.54)
2 *
1 2 3 4 5
21 arctan ( 1)t Ez Ez z Ez z Ez Ez t t tE q q df q df q q IA B C
(A.55)
cos arctan arctan( )o t t t t t t t t tt D C B E B B (A.56)
0'z o yoM t F (A.57)
'zo zo zroM M M (A.58)
85
APPENDIX B – EQUIVALENT LAMINATE PROPERTIES
The carbon fiber tubes used in the FST 05e are purchased from Easy Composites. The fiber
layup and the properties of the fiber and matrix used are known.
Besides being known as a CFRP tube, it also uses glass fiber together with carbon fiber. The
tubes in this category are roll-wrapped pre-preg made.
The layup of the tube is as follows:
[ 0 90 0 90 0]
The fibers at 0 degrees are oriented parallel to the longitudinal axis of the tube and the 90
degrees fibers are oriented parallel to the radial direction of the tube.
At 0 degrees, carbon fiber from Toray is used: 300 gsm Toray T700. At 90 degrees, the
reinforcement of glass fibers is used: 300 gsm E-glass.
The carbon fiber pre-preg is unidirectional and the glass fiber pre-preg can also be
considered unidirectional.
To calculate the equivalent modulus of elasticity in the longitudinal direction, micro-
mechanics and classic laminate theory are used.
To start one can approximate the thickness of each fiber layer by dividing the total thickness
of the tube by the number of layers:
tubelayer
layers
hh
n (B.1)
To calculate the fiber and matrix volume fraction in the layer the following equation is used,
where g symbolizes the weight per area of the fiber and the density is represented as ρfiber.
1
f
fiber layer
m f
gV
h
V V
(B.2)
These volume fractions will be used to compute the rule of mixtures and obtain the elastic
properties of the layer by knowing the properties of the matrix and fiber in the following way:
L m m f fE E V E V (B.3)
LT m m f fV V (B.4)
1 fm
T m f
VV
E E E (B.5)
86
1 fm
LT m f
VV
G G G (B.6)
LT TL
L TE E
(B.7)
The elasticity matrix in the fiber coordinate system is then computed as:
01 1
01 1
0 0
L LT T
LT TL LT TL
L L
TL L TT T
LT TL LT TL
LT LT
LT
E E
E E
G
(B.8)
In a laminate, every layer is not usually with the same orientation, so a transformation to the
above properties must be made to obtain the said properties in the laminate coordinate system. In the
following, ξ, symbolizes the angle at which the layer is oriented with respect to the laminate
coordinate system.
2 2
2 2
2 2
2
2
c s cs
R s c cs
cs cs c s
(B.9)
cos
sin
c
s
(B.10)
The equations (B.9) and(B.8) are combined in the following way:
1
01 1
1 0 0
0 0 1 01 1
0 0 20 0
L LT T
LT TL LT TL
X X
TL L TY Y
LT TL LT TL
XY XY
LT
E E
E ER R
G
(B.11)
This results in the following matrix:
87
1
01 1
1 0 0
0 0 1 01 1
0 0 20 0
L LT T
LT TL LT TL
TL L T
LT TL LT TL
LT
E E
E EK R R
G
(B.12)
To compute the final stiffness matrix of the laminate one needs to assemble the several
stiffness matrixes from the several layers present. This is done by writing the constitutive equations of
the laminate. Since the only objective is to obtain the extensional properties of the laminate and since
the laminate is symmetric one only needs to write the constitutive relations for a tensile force:
0
11 12 13
0
21 22 23
0
31 32 33
X X
Y Y
XY XY
N A A A
N A A A
N A A A
(B.13)
The A matrix is obtained with the knowledge of the components of matrix K:
1
layersn
ij ij layer ji
t
A K h A
(B.14)
To obtain the final stiffness matrix the following is done:
tube
AA
h (B.15)
The stiffness variable that is used in this work is then obtained:
11CFRP axialE A (B.16)
88
APPENDIX C – FORMULA STUDENT RESULTS
Figure C.1 – Acceleration Results from Formula Student Electric 2012
Figure C.2 – Skid Pad results from Formula Student 2013 competition in the UK
89
APPENDIX D – MATLAB / SIMULINK DIAGRAMS
Signal 1Group 1
Throttle Pedal
Signal 1Group 1
Steering Wheel
Steering_wheel
Steering Displacements
Twist Angle
Steer
Steering Mechanism
vy
vx
w
Steer
Slip angle
vi
Vehicle and Wheel Angles
Steer
Vertical Displacements
L_rods
IA
IA calculator
Throttle
Brake
w
vx
Torque available
Torque to wheels
Motor / Brake System
SA
Torque to wheels
vi
Torque from Fx
FZ
SL_raw
Wheels Angular Velocity
Wheel Rotational DynamicsFZ
SL_r
aw
Torq
ue to
Whe
els SA IA
Fx Fy Mz
Tire DynamicsSteer
Mz
Fy
Fx
Vertical Displacements
w
vy
vx
v
Torque from Fx
FZ
FX
FY
MZ
Vehicle Dynamics
Signal 1Group 1
Brake Pedal
FZ
FX
FY
MZ
F_rods_FR
F_rods_FL
F_rods_RR
F_rods_RL
F_steering_rods
Force Calculator
F_rods_FR
F_rods_FL
F_rods_RR
F_rods_RL
F_steering_rods
L_rods
Steering Displacements
Twist angle
Compliance Calculator
Memory
Memory1
Memory2
Memory3
1
Zero 2
1
Zero 1
Memory4
0
Zero 2
Signal 1Group 1
Throttle Pedal1
FY
w
FX
vy
DOWNFORCE
DRAG
vx
ay
dvy
FZ
dvx
z
9DOF
steer
Fy
Fx
FY
FX
fcn
Fy/Fx -> FY/FX
[a;a;-b;-b]
Gain
Sum ofElements
1/76
Gain1
1s
Integrator3
Yaw Velocity
Yaw Acceleration
1s
Integrator4
Yaw Position
yaw.mat
To File1
[tf/2;-tf/2;tr/2;-tr/2]
Gain2
1s
Integrator
Lateral Velocity
1/9.81
Gravity3Lateral Acceleration
dvy
1Steer
3Fy
4Fx
7FZ
1Vertical Displacements
Linear Acceleration
1/9.81
Gravity
1s Integrator1
u2
Velocity squared
0.91
Drag Coefficient
-2.12
Lift Coefficient
Linear Velocity
3.6
m/s -> km/h
2w
3vy
4vx
r
Wheel radius
Torque from FX
6Torque from Fx
yaw
vx
vy
VX
VYfcn
Visual Preparation
1s
Integrator6
1s
Integrator7
X.mat
To File5
Y.mat
To File6u2
vx squared
u2
vy squared
sqrt(u)
Fcn
5v
Yaw moment from FX
Yaw moment from FY
Yaw moment
8FX
9FY
2Mz
Yaw moment from Mz
10MZ
Ay.mat
Lateral Acceleration to file - Ay
vy.mat
Lateral Velocity to file - vy
vx.mat
Longitudinal Velocity to file - vx
Ax.mat
Longitudinal Acceleration to file - Ax
Yaw_m.mat
Yaw Moment to file - Yaw_m
Yaw_a.mat
YawAcc to file - Yaw_a
Yaw_v.mat
Yaw vel to file - Yaw_v
FZ.mat
FZ to file - FZ
Velocity
XY Graph
X position
Product7
1/9.81
Gravity1
Radial Acceleration
u2
vx squared1Divide1
Path Radius
v.mat
Tangential Velocity to file - v
ar.mat
Radial Acceleration to file - ar
R.mat
Path Radius to file - R
Vertical Force on tire
Applied Forces
MatrixMultiply
Product2
MatrixMultiply
Product3
MatrixMultiply
Product4
Product5
-K-
Gain2
-1
Constant1
-Kt/1000
Constant2
C
Damping
Ax
FX
Az
Ay
FY
FORCESfcn
MATLAB Function
Minv
Mass - Inverse
K
Stiffness
Displacements
Velocity Plot
Add
1s
Position
1s
Velocity
-502.3
-241.4
-591.4
-24.48
-466.6
-306.8
-602.8
-25.66
993.3
Display
-44.53
-6.85
-24.3
-1.224
-36.63
-7.141
-16.26
-1.187
1.021
Display1
1153
206.1
1202
199.9
Display2
3FX
Sum ofElements m
Mass
Divide
4dvx
3FZ
6DRAG
5DOWNFORCE
Divide1
m*9.81
Mass1
1FY
Sum ofElements1 Divide2
m
Mass2
2dvy
7vx
2w
Product6
1ay
4vy
Product7
5z
z_nom.mat
To File
1
Gain
1/9.81Gain3
1/9.81
Gain1
Iy_roda
Wheel Inertia
Divide
Divide1
Subtract
Total Torque
Subtract1
1
Constant ~= 0
Switch2
[0;0;0;0]
SR for v = 0
T
Angular Acc - raw
v
FZ
w
Angular Acceleration and Velocity Product1
2 Torque to wheels
4 Torque from Fx
5FZ
1SA 1
SL_raw
3vi
f(u)
COS(SA)
2Wheels Angular Velocity
Wheels Velocity - angular
Velocity of wheels in the body CS x-y
SL raw plot
~= 0
Switch1
r
Wheel radius 1
Product2
wi.mat
Wheel Angular Velocity to file - wi
1/r
Wheel radius 2
SL raw plot1
Product
Torque from the motors to the wheels
w
E_T_W
E_T_M
w_Mfcn
MOTOR
21
Gear Ratio Motor Torque
Motor Speed
> 0
Switch Brake/Throttle
>= 0
Switch1
[0 0 0 0]
Brakes if v = 0
4vx
1Throttle
3w
2Torque to wheels
1Torque available
Throttle Pedal
T_to_wheels.mat
Torque to wheels to file - T_to_wheel
Pedal_in.mat
Pedal input to file - Pedal_input
Memory1
2Brake
F_Brake_Pedal Tfcn
Brake System
Steering Input
Steering Displacements - Right
Steering Displacements - Left
Steering Right
Steering Left
Steering Mechanism
Steering_wheel
Twist_angleRack_d
fcn
Rack Displacement
0
Rear Right Steering
0
Rear Left Steering
ROT_FR
ROT_FL
ROT_RR
ROT_RL
steerfcn
Rotation matrix to steering angle
Steering wheel input
1Steering_wheel
1Steer
Wheels Steering
2Steering Displacements
1
Gain1
Gain1
delta.mat
Steer to file - delta
delta_in.mat
Steering input to file - delta_in
3Twist Angle
~= 0
Switch1
[0 0 0 0]Steering if steering wheel = 0
vy
vx
w
xi
yi
B
Bi
vi
fcn
Sideslip angle calculation
[a;a;-b;-b]
x Positions
[-tf/2;tf/2;-tr/2;tr/2]
y Positions
x_positions.mat
To File3
y_positions.mat
To File4
Subtract2
1vy
2vx
3w
4Steer
1Slip angle
Sideslip angle
2vi
Slip angles
B.mat
Sideslip to file - B SA.mat
Slip angles to file - SA
> 0
Switch1
[0 0 0 0]
SA if v = 0
Steer
z
L_rods
up_f
Camber_nom
KPI_nom
Caster_nom
z_NOM
up_r
IAfcn
IA calculation
up_fFront Upright
Camber_nom
Camber_nom
z_NOMNominal Positions
KPI_nom
KPI_nom
Caster_nom
Caster_nom
up_rRear Upright
1Steer
2Vertical Displacements
1IA
IA plot
3L_rods
IA.mat
IA to file - IA
T
SL_raw
FZ
SA
IA
COEFSXX
COEFSXY
COEFSYY
COEFSYX
COEFSMZY
Fx
Fy
Mz
SL
fcn
P2002XY
COEFSX
COEFSX COEFSXY
COEFSXY
Longitudinal Slip
TFXSL
FX_effcn
Limit FX
Longitudinal Force - Raw
COEFSYY
COEFSYY COEFSYX
COEFSYY
3Torque to Wheels 2
SL_raw 1FZ 4
SA
1Fx
2Fy
5IA
Lateral Force
Longitudinal Force
Self-Aligning Torque
COEFSMZY
COEFSMZY
3Mz
Fx.mat
Fx to file - FxFy.mat
Fy to file - Fy
Mz.mat
Mz to file - Mz
SL.mat
SL to file - SL
2FX
3FY
1FZ
[1;1;1]
Gain
[1;1;1]
Gain1
[1;1;1]
Gain2
[1;1;1]
Gain3
CG
CS4
CS6
CS9
CS1
CS7
CS2
CS8
CS3
CS5
Front Left Upright
Ground
EnvMachine
Environment
B F
Spherical Ground1
Ground2
B F
Spherical3
Ground3
Ground4
Ground5
CS1 CS2
Tie-Rod
B F
Spherical7
BF
Spherical1
BF
Spherical4
Body Actuator
FLUF Sensor
CS1 CS2
Pullrod
B F
Spherical6 Ground6
B F
Six-DoF
CS1 CS2
FL Upper A-Arm Front
B F
Spherical2
CS1 CS2
FL Upper A-Arm Rear
BF
Spherical5
CS1 CS2
FL Bottom A-Arm Rear
CS1 CS2
FL Bottom A-Arm Front
BF
Spherical8
B F
Spherical9
BF
Spherical10
B F
Spherical11
FLUR Sensor
FLBF Sensor
FLBR Sensor
CG
CS4
CS6
CS9
CS1
CS7
CS2
CS8
CS3
CS5
Front Right Upright
Ground7
EnvMachine
Environment1
Body Actuator1
BF
Spherical12Ground8
Ground9
BF
Spherical17Ground10
Ground11
Ground12
CS1CS2
Tie-Rod1
BF
Spherical21
BF
Spherical13
BF
Spherical18
FRUF Sensor
CS1CS2
Pullrod1
BF
Spherical20
Ground13
CS1CS2
FR Upper A-Arm Front
BF
Spherical16
CS1CS2
FR Upper A-Arm Rear
BF
Spherical19
CS1CS2
FR Bottom A-Arm Rear
CS1CS2
FR Bottom A-Arm Front
BF
Spherical22
BF
Spherical23
BF
Spherical14
BF
Spherical15
FRUR Sensor
FRBF Sensor
FRBR Sensor
BF
Six-DoF1
CG
CS4
CS6
CS9
CS1
CS7
CS2
CS8
CS3
CS5
Rear Left Upright
Ground14
EnvMachine
Environment2
B F
Spherical24 Ground15
Ground16
B F
Spherical29
Ground17
Ground18
Ground19
CS1 CS2
Tie-Rod2
B F
Spherical33
BF
Spherical25
BF
Spherical30
RLUF Sensor
CS1 CS2
Pullrod2
B F
Spherical32
Ground20
B F
Six-DoF2
CS1 CS2
RL Upper A-Arm Front
B F
Spherical28
CS1 CS2
RL Upper A-Arm Rear
BF
Spherical31
CS1 CS2
RL Bottom A-Arm Rear
CS1 CS2
RL Bottom A-Arm Front
BF
Spherical34
B F
Spherical35
BF
Spherical26
B F
Spherical27
RLUR Sensor
RLBF Sensor
RLBR Sensor
CG
CS4
CS6
CS9
CS1
CS7
CS2
CS8
CS3
CS5
Rear Right Upright
Ground25
EnvMachine
Environment3
BF
Spherical36Ground26
Ground27
BF
Spherical41Ground21
Ground22
Ground23
CS1CS2
Tie-Rod3
BF
Spherical45
BF
Spherical37
BF
Spherical42
RRUF Sensor
CS1CS2
Pullrod3
BF
Spherical44
Ground24
CS1CS2
RR Upper A-Arm Front
BF
Spherical40
CS1CS2
RR Upper A-Arm Rear
BF
Spherical43
CS1CS2
RR Bottom A-Arm Rear
CS1CS2
RR Bottom A-Arm Front
BF
Spherical46
BF
Spherical47
BF
Spherical38
BF
Spherical39
RRUR Sensor
RRBF Sensor
RRBR Sensor
BF
Six-DoF3
F_rods_raw
up_f
up_f_s
up_r
up_r_t
ch_f
ch_r
F_rods_FR
F_rods_FL
F_rods_RR
F_rods_RL
F_rods_toe_tie
verif ica
fcn
MATLAB Function
up_f
UP Front
up_r
UP Rear
ch_f
Chassis Front ch_r
Chassis Rear
2
MatrixConcatenate
-1158
-295.9
-1002
-2356
Display
-477.2
-207.2
485
167.3
Display1
669.7
742.4
-1195
-735.1
Display2
-178
-178.7
72.35
722.2
Display3
Body Actuator4
>= 0
Switch
[0;0;0]
Constant
[1;0;0]
Gain4
>= 0
Switch1
[0;1;1]
Gain5
[1;0;0]
Gain6 >= 0
Switch2
Body Actuator5
[0;0;0]
Constant1
>= 0
Switch3
[0;1;1]
Gain7
Body Actuator6
[1;0;0]
Gain8 >= 0
Switch4
Body Actuator7
[0;0;0]
Constant2
>= 0
Switch5
[0;1;1]
Gain9
Body Actuator3
Body Actuator8
>= 0
Switch6
[0;0;0]
Constant3
[1;0;0]
Gain10
>= 0
Switch7
[0;1;1]
Gain11
1F_rods_FR
2F_rods_FL
3F_rods_RR
4F_rods_RL
4MZ
Body Actuator2
Body Actuator9
Body Actuator10
Body Actuator11
[0;0;1]
Gain12
[0;0;1]
Gain14
[0;0;1]
Gain15[0;0;1]
Gain13
Tie-L SensorTie-R Sensor
Toe-R Sensor
Toe-L Sensor
up_r_t
UP Rear1
up_f_s
UP Front2
660.3
-0.5335
-389.2
30.46
Display4
5.292e-14
660
-21.18
Display5
5.69e-16 1.485e-15 4.927e-16 2.429e-16 6.661e-16 4.441e-16 1.665e-16 2.394e-16 8.327e-17 9.992e-16 1.409e-17 3.574e-15 6.939e-16 3.331e-16 2.312e-15
-1.54e-17 1.912e-16 -7.914e-14 5.077e-15 8.571e-15Display6
5F_steering_rods
F_rods_FR plot
F_rods_FL plot
F_rods_RR plot
F_rods_RL plot
F_rods_toe_tie plot
F_FR.mat
F_FR to file - F_FR
F_rods_FR
F_rods_FL
F_rods_RR
F_rods_RL
F_steering_rods
L_F_nom
L_R_nom
L_rods
displacements_steering
twist_angle
displacements
T_steering
fcn
MATLAB Function
L_R_nomRod lenghts-R
L_F_nomRod lenghts-F
1F_rods_FR
2F_rods_FL
3F_rods_RR
4F_rods_RL
1L_rods
displacements
1000
Gain
5F_steering_rods
2Steering Displacements
displacements_steering
1000
Gain1
3Twist angle
twist angle plot
Torque on steering wheel
1
Gain2
L_rods.mat
L_rods to file - L_rods
delta_steering.mat
delta_steering to file - delta_steering
twist_angle.mat
twist_angle to file - twist_angle
delta_w.mat
delta_w to file - delta_w