Development and analysis of a multi-link suspension for racing applications

Development and analysisof a multi-link suspensionfor racing applications

W. Lamers

DCT 2008.077

Master’s thesis

Coach: dr. ir. I.J.M. Besselink (Tu/e)

Supervisor: Prof. dr. H. Nijmeijer (Tu/e)

Committee members: dr. ir. R.M. van Druten (Tu/e)ir. H. Vun (PDE Automotive)

Technische Universiteit EindhovenDepartment Mechanical EngineeringDynamics and Control Group

Eindhoven, May, 2008

Abstract

University teams from around the world compete in the Formula SAE competition with prototypeformula vehicles. The vehicles have to be developed, build and tested by the teams. The UniversityRacing Eindhoven team from the Eindhoven University of Technology in The Netherlands competeswith the URE04 vehicle in the 2007-2008 season. A new multi-link suspension has to be developedto improve handling, driver feedback and performance.

Tyres play a crucial role in vehicle dynamics and therefore are tyre models fitted onto tyre measure-ment data such that they can be used to chose the tyre with the best characteristics, and to develop thesuspension kinematics of the vehicle.

These tyre models are also used for an analytic vehicle model to analyse the influence of vehicle pa-rameters such as its mass and centre of gravity height to develop a design strategy. Lowering the centreof gravity height is necessary to improve performance during cornering and braking.

The development of the suspension kinematics is done by using numerical optimization techniques.The suspension kinematic objectives have to be approached as close as possible by relocating the sus-pension coordinates. The most important improvements of the suspension kinematics are firstly theharmonization of camber dependant kinematics which result in the optimal camber angles of thetyres during driving. The suspension is designed to have a steady ride height during cornering whichcauses the suspension to operate in the intended region. The driver feedback is improved by meansof the suspension kinematics and steering wheel forces. The vehicle characteristics are validated witha dynamic vehicle model.

Reference: vehicle dynamics, kinematic suspension design, tyre models, multi-body vehicle models, numericaloptimization

ii

Preface

The last challenge of my study is the master’s thesis. After careful consideration about the thesissubject, I started late 2006 with the development of a racing suspension for the University RacingEindhoven team. It is a real challenge to design the complete vehicle dynamic characteristics of avehicle, one which is not easy to find externally. And even so important: the design will be build andtested in practise which gives a complete picture of the design proces.

The first choice in the proces was to analyse tyre behaviour. Mainly because the tyres are a very fun-damental aspect of vehicle dynamics. I did this partly at the Automotive department of TNO which islocated in Helmond, the Netherlands. Here I had the opportunity to fit the tyre measurement data onthe latest tyre model of TNO Automotive: Delft-Tyre. Therefore I want to thank Antoine Schmeitz forhelping me doing this.

The next subject was to analyze the steady state vehicle behaviour of a racing vehicle and to find outwhat the influence of basic vehicle parameters is on the vehicle performance. It appeared to be quitean extensive job, but a very useful one. The last and major part of the thesis was to design the suspen-sion itself. This was a really interesting subject. When I look back to the thesis I have to say that I hada very pleasant time doing this, while working in a truly unique environment: the University RacingEindhoven team!

Then I want to thank the people who have helped and supported me during my master’s thesis. IgoBesselink for his theoretical input and coaching, Henk Nijmeijer for his supervising role, Roell vanDruten and Hans Vun for participating in the committee and the University Racing Eindhoven teamwhere I could fulfill this challenge. Than I want to thank my family for their support and finally mygirlfriend Patricia van Dongen for her support, patience and the corrections she suggested for thisreport.

Willem-Jan Lamers, May 2008

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Contents

Abstract ii

Preface iii

Sign conventions and symbols 1

1 Introduction 51.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Objective and thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Modeling a racing tyre 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Tyre measurement data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Fitting the measurement data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Validation of the tyre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Tyre choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Steady state vehicle behaviour 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Load distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Two track roll axis vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Base line vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Model objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 Numerical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.7 Calculation sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.8 Other model applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.9 Pure cornering results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.10 Combined cornering/driving results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.11 Optimization of the steering angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.12 Optimal tyre inclination angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Kinematic suspension design 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Multi-link layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Design tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Kinematic suspension model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.5 Suspension kinematic characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6 Numerical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Dynamic vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.8 Suspension collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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CONTENTS CONTENTS

5 Suspension design considerations and results 605.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Allowable suspension settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Contact patch pressure fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.4 Pitch attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.5 Roll attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.6 Tyre orientation target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.7 Driver feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.8 Suspension forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.9 Initial suspension settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.10 Double wishbone comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Conclusions and recommendations 796.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Bibliography 82

A Two track roll axis model equations 84

B Optimization algorithm 87

C Suspension coordinates 91

D Quarter car model derivation 94

v

CONTENTS CONTENTS

Sign conventions

Sign conventions often cause communications problem, therefore will the ISO 8855 [1] sign conven-tion be used throughout this report. The most important definitions are depicted in figure 1, for acomplete overview see [1].

Driving direction

Figure 1: ISO 8855 sign conventions

ISO defines the vehicle axis system as a right-handed orthogonal axis system fixed at the center ofgravity of the vehicle. The x-axis is parallel to the road surface and pointing forwards, the y-axis is alsoparallel to the road surface and pointing to the driver’s left. The z-axis is pointing upwards, normal tothe road.

Throughout the report several angles are used. 12 different rotations sequences can be defined. Only2 are used, one for the chassis rotation sequence and one for the wheel/tyre rotation sequence. Table1 shows the rotation sequences for the chassis and wheel starting from the world coordinate system.

Rotation order Produced chassis angle Produced tyre angleFirst yaw (ψ) steer angle (δ)

Second pitch (θ) inclination angle (γ)Third roll (φ) wheel rotation angle (ω)

Table 1: Rotation sequence

The wheelbase [l] is defined as the distance between the center of the tyre contact point of the twowheels on the same side of a vehicle projected on the x-axis.

The track [b] is defined as the distance between the centers of tyre contact points of the two wheels ofan axle projected on the yz-plane.

The steer angle is defined as the rotation of the wheel around the positive z-axis according to the axisdefinition.

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CONTENTS CONTENTS

The tyre inclination angle [γ] is defined positive when the tyre is inclined by positive rotation aroundthe x-axis of the vehicle.

The camber angle is defined positive as an angle between the global z-axis and the wheel plane whenthe top of the wheel is inclined outward relative to the vehicle body.

More specific definitions used in this report are given when needed.

Symbols

α tyre side slip angle [deg]

β vehicle slip angle [deg]

γ tyre inclination angle [deg]

δ steer angle [deg]

θ chassis pitch angle [deg]

κ longitudinal tyre slip [-]

µ friction coefficient [-]

ξ dimensionless damping coefficient [-]

σ kingpin inclination angle [rad]

τ castor angle [rad]

φ chassis roll angle [deg]

ψ vehicle yaw angle [deg]

ψ yaw velocity [rad/s]

ω wheel rotation angle around the y-axis [deg]

ωi angular velocity [rad/s]

ωδ steering velocity [rad/s]

a vector pointing to the instant center and anti center

ax longitudinal acceleration [m/s2]

ay lateral acceleration [m/s2]

b track width [m]

cφ roll stiffness [Nm/rad]

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CONTENTS CONTENTS

d steering axis vector

ds damping coefficient [Ns/m]

f degrees of freedom [-]

frf ride frequency [Hz]

g gravitational acceleration [9.81 m/s2]

h centre of gravity height [m]

hra height between the roll axis and the centre of gravity [m]

k spring stiffness [N/m]

l wheelbase [m]

m vehicle mass [kg]

n castor offset [m]

nτ castor offset at wheel centre [m]

p brake force distribution [-]

r yaw rate [rad/s]

ri axis vector

rs scrub radius [m]

rc wheel centre offset [m]

t total track width [m]

u longitudinal vehicle speed [m/s]

v lateral vehicle speed [m/s]

vi velocity vector

w wheel load lever arm [m]

Ax longitudinal acceleration [g]

Ay lateral acceleration [g]

ACf front anti center

ACr rear anti center

D point where the virtual steering axis intersects with the road

3

CONTENTS CONTENTS

Fx longitudinal tyre force [N]

Fy lateral tyre force [N]

Fz vertical tyre load [N]

H transfer function

Iy rotational inertia [kgm2]

ICf front instant center

ICr rear instant center

Mx overturning moment [Nm]

My rolling resistance moment [Nm]

Mz self aligning moment [Nm]

MR motion ratio [-]

R cornering radius [m]

V vehicle speed [m/s]

W weighting factor [-]

WTR wheel base - track ratio [-]

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Chapter 1

Introduction

"We’re all on the limit, the car is on the limit, the human being is on the limit,...That’s what it’s all about motor racing."

- Ayrton Senna

1.1 Background

In the early eighties, the Society of Automotive Engineers (SAE) has initiated a formula car competi-tion between universities around the world, named Formula SAE (FSAE). The competition is mainlyfounded to give students the opportunity to gain experience in engineering, manufacturing, testing,racing and managing a formula racing team, next to the normal university curriculum. The competi-tion takes place all around the world and is considered by many to be the most prestigious universityengineering design competition.

A FSAE team can apply in three competition classes. The class 3 competition is meant for vehicledesigns which exists purely on paper. A physical vehicle has to be developed and build to competein the class 1 competition. This has to be a newly developed prototype every year. When a teamwants to compete with a vehicle which was already used is the class 1 competition one has to applyto the class 2 competition. The competition regulations stimulate the teams to develop and applynew technologies. This involves a lot of team work and together with the short and strict deadlinesit is also a management challenge. The Eindhoven University of Technology is competing since theyear 2004 and the team is called University Racing Eindhoven [2]. The team will compete in the UK,Italian and German competition in the 2007-2008 season with URE04 vehicle. The team takes partof the class 1 competition, which means that it is a newly developed prototype. The team is famousfor its application of innovative technology. This expresses itself among other things in the use ofexotic materials, such as aluminum honey comb for the chassis, sophisticated vehicle electronics andinnovative suspension designs.

The majority of vehicles is the FSAE competition is equipped with a double wishbone suspension.The predecessors of the URE04 where also equipped with a double wishbone suspension, see figure1.1. Before reliable ball joints became available, independent suspension normally feature a physicalkingpin, connecting the upright to the suspension link coupler. The development of durable ball joints

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1.2. Objective and thesis outline CHAPTER 1. Introduction

made it possible to equip vehicles with double wishbone suspensions. These days, most racing carsand also a range of passenger cars are equipped with independent double wishbone suspensions. Anevolution of the double wishbone suspension is the multi-link or 5-link suspension. This suspensionis characterized by even more kinematic freedom than the double wishbone suspension, which can,for example, result in better handling, stability, comfort, packaging and driver feedback.

Figure 1.1: Double wishbone suspension URE03

Competing in a race is not about developing a fast vehicle but about developing the fastest vehiclepossible. This means that every potential performance gain has to be exploited. Such potential isthe application of a multi-link suspension, which simultaneously accommodates the teams vision ofdeveloping and using innovative technology.

1.2 Objective and thesis outline

The objective of the master’s thesis is to develop a multi-link suspension which is able to deliverthe highest possible performance in the context of the FSAE competition. The suspension has tobe fitted to the URE04 vehicle, competing in the 2007-2008 season. The thesis should result inthe kinematic design of the suspension. The spring/damper concept falls beyond the scope of thethesis. The development of the suspension design involves the analysis and understanding of tyresand fundamental vehicle behaviour. Both are the fundamental aspects of suspension design.

Already in the early days of vehicle analysis it became clear that tyres strongly influence vehicle dy-namics. The highly non-lineair behaviour of a tyre results in complex vehicle behaviour. Thoroughknowledge of tyre behaviour is necessary to develop suspension kinematics. The design will be basedon the tyre behaviour. Chapter 2 describes an advanced tyre model which is used to represent thenon-linear tyre behaviour. Measurement data of commonly used tyres for the FSAE competition willbe discussed. The measurement data is fitted on a Magic Formula, [3], based tyre model. The tyremodel is used to describe the behaviour of 6 different tyres, and to make a choice for the best suitabletyre for the competition.

The steady-state vehicle behaviour is analyzed in chapter 3. Therefore a steady state analytic vehiclemodel is developed, including the tyre behaviour as discussed in chapter 2. Numerical optimizationtechniques are used to find the steady-state operating conditions of the vehicle. The sensitivity of

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1.2. Objective and thesis outline CHAPTER 1. Introduction

fundamental vehicle parameters on the vehicle performance envelope, such as mass and centre ofgravity height, are analyzed to define a design strategy. The chapter also discusses other optimizationpossibilities such as optimal individual steer angles.

The computational performance of current computers makes it possible to design complex suspen-sion kinematics such as that of a multi-link suspension with the use of numerical optimization tech-niques. Planar suspension layouts, such as the double wishbone suspension, could be designed ona more conventional way with the use of graphical methods and basic linear algebra. The multi-linklayout requires a different approach. Chapter 4 discusses the mathematical background of multi-linkkinematics and the development of suspension design tools. These tools are:

• kinematic multi-body models

• a library of mathematical equations which represent the suspension parameters

• a numerical optimization tool to compute the optimal kinematic design

• a dynamic vehicle model to validate the suspension design

• a tool to visualize suspension collisions in a virtual reality environment

The design considerations and results of the design are discussed in chapter 5. Some suspensionaspects are validated with the use of a dynamic vehicle model. Finally a comparison is made the thedouble wishbone suspension of the URE03 vehicle which was used in the competition in the season2006-2007.

This report ends with conclusions and recommendations for further research as given in chapter 6.

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Chapter 2

Modeling a racing tyre

2.1 Introduction

Due to the complexity of vehicle dynamics and in particular the limit handling behaviour it is nec-essary to understand and quantify the complex non-linear tyre behaviour. The tyres of a vehicle are,besides the air, the only connection to the surrounding world, therefore these devices are very impor-tant for vehicle behaviour. This expresses itself in vehicle stability, comfort, driver feedback and mostimportant the lateral and longitudinal vehicle performance.

To investigate the vehicle performance it is necessary to use an advanced tyre model. The MagicFormula tyre model is able to represent the forces and moments of a tyre very accurately and fast. TNOAutomotive [4] is continuously developing this model, and the current version is able to represent thefollowing main tyre properties:

• all parameters dependent on: vertical load Fz , velocity, road friction coefficient and inflationpressure

• longitudinal forces and moments (Fx and My)

• lateral forces and moment (Fy , Mx and Mz)

• combined longitudinal and lateral forces and moments

• longitudinal, lateral and vertical tyre stiffness

• effective rolling radius

• turn slip

• pneumatic trail

The model is able to make an accurate fit of tyre measurement data in a empirical way with a relativelysmall set of coefficients. Given the model characteristics it is very suitable to use in racing applications.One disadvantage of the model is that it does not incorporate the tyre temperature effects, whichare important for racing tyres. The tyre fitting is always done at tyre operation temperatures whichguarantees a consistent tyre model. The next section gives some background information on the tyremeasurement data.

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2.2. Tyre measurement data CHAPTER 2. Modeling a racing tyre

2.2 Tyre measurement data

Tyre measurment data is required to identify the tyre model parameters. Obtaining these data is veryexpensive, especially for a Formula Student racing team. Therefore Milliken Research Associates hasfounded the FSAE Tire Test Consortium (FSAE TTC) [5]. This is an organization of Formula Studentteams who pool their financial resources to obtain high quality tyre force and moment data. TheFSAE TTC’s role is to gather funds from participating formula student teams, organize and conducttyre force and moment tests and distribute the data to all participating teams. In this way it is possibleto obtain this data for a very affordable price.

The FSAE TTC cooperates with the Calspan Tire Research Facility (TIRF) in Buffalo (USA). TIRFperforms tyre measurements on a flat road surface, see figure 2.1.

Figure 2.1: TIRF flat track measurement machine

The machine has the following capabilities:

Characteristic RangeTyre slip angle (α) 30 degTyre inclination angle (γ) 30 degTyre slip angle rate 10 deg/sTyre inclination rate 7 deg/sTyre load rate 900 kg/sTyre vertical positoning rate 0.18 m/sRoad speed 0-90 m/sMaximal tyre outside diameter 1.19 mMaximal tyre thread width 0.61 mBelt width 0.71 m

Table 2.1: TIRF machine capabilities

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2.3. Fitting the measurement data CHAPTER 2. Modeling a racing tyre

Performed measurements

The FSAE tyre test consortium has performed measurements on several tyres which are widely usedin the Formula Student competition. This section refers to the second test sequence. Table 2.2 showsa list of the tyres measured.

Brand DimensionsAvon 6.2/20.0-13Avon 7.2/20.0-13Goodyear 18x6.5-10Goodyear 20x6.5-13Hoosier 20.5x6.0-13Hoosier 20.5x7.0-13

Table 2.2: Measured tyres

During the second test round TIRF has performed several measurements. These tests are pure lon-gitudinal and pure lateral. TIRF did a combined measurement during the third measurement roundwhich wasn’t available at the moment of fitting. The following measurements are performed on eachtyre, except the Goodyear 10 inch tyre. Due to limitations of the measurement machine it is notpossible to measure the longitudinal characteristics of a tyre with a 10 inch rim.

• Lateral Force I:-Dynamic vertical spring rate test on tyre after break-in-Slip angle sweeps at various loads and inclination angles, 0.83 bar, 0% slip ratio-Inclination angles: 0, 1, 2, 3, 4 deg-Loads: 222, 667, 1112, 1556, 2000 N

• Lateral Force II:-Slip angle sweeps at various loads and pressures, 0 deg inclination angle, 0% slip ratio-Pressures: 0.55, 0.69, 0.83, 0.97, 1.10 bar-Loads: 667, 1556 N

• Longitudinal Force:-Slip ratio sweeps at various loads, 0.55 and 0.83 bar, 0 deg slip angle-Loads: 667, 1112, 1556 N-Inclination Angles: 0, 2, 4 deg

2.3 Fitting the measurement data

The tyre model of TNO Automotive is called Delft-Tyre. TNO has developed an advanced tyre modelfitter which is called MF-Tool [6]. This software fits the measurement data using data files arrangedaccording to the TYDEX standard. The Tyre Data Exchange Format (TYDEX) is developed to make theexchange of tyre measurement data easier [7].

The fitter accepts measurement data with some high frequency noise. Both of the hysteresis measure-ments can also be given as an input for the fitter, which automatically averages the measurements.Thefitter has to know which set of data can be used to fit for example the lateral coefficients. Therefore the

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2.3. Fitting the measurement data CHAPTER 2. Modeling a racing tyre

raw measurement data has to be transformed to the TYDEX format using MATLAB . Calspan deliversone large file for each tyre containing all measurement data. This file also contains tyre break in andconditioning sweeps, which means that the data has to be sorted out by hand to create the appropriateTYDEX files.

The fitter has been developed for passenger car tyres. The camber influence on longitudinal perfor-mance is normally rather small for these kind of tyres, but for the Formula Student racing tyres thisis not the case. The algorithm of the fitter is adapted to allow the model to go beyond the standardboundary of the camber influence on longitudinal forces. The effect of tyre inclination angle on thelongitudinal performance of a tyre can be seen in figure 2.2. This influence in indeed larger thenexpected, and has the largest deviation on the peak.

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−4000

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Fx

[N]

Kappa sweep, 12[PSI] inflation pressure

0 [deg] inclination angle2 [deg] inclination angle4 [deg] inclination angle

Figure 2.2: Tyre inclination angle influence on longitudinal performance

The side slip angle measurement shows a hysteresis effect (see figure 2.3). The tyre is rotated aroundthe positive z-axis and then around the negative z-axis to perform a measurement. The hysteresis effectis caused by the speed at which the tyre is rotated around its z-axis and by temperature changes duringthe sweep. This effect is not implemented in the tyre model, but when both sets of measurement dataare offered to the fitter the result will be averaged. Figure 2.3 illustrates this.

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2.4. Validation of the tyre model CHAPTER 2. Modeling a racing tyre

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alpha [deg]

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Alpha sweep, 12[PSI] inflation pressure, 0[deg] inclination angle

Fz = 222[N] modelFz = 1112[N] modelFz = 2000[N] modelFz = 222[N] measurementFz = 1112[N] measurementFz = 2000[N] measurement

Figure 2.3: Side slip angle sweep hysteresis

2.4 Validation of the tyre model

The tyre model used is a new version which includes the inflation pressure influence [8]. Thereforeand to do a general check is it important to validate the tyre model in more detail. This is done for onetyre, the Hoosier 20.5x6.0-R13.

The most fundamental forces are the lateral and longitudinal forces generated by a side slip angle[α] and a longitudinal slip ratio [κ]. Figure 2.4 illustrates the similarities between the measurementdata and the model for longitudinal forces (2.4a) and the lateral forces (2.4b). The overal fit of thelongitudinal forces could be better. The longitudinal slip stiffness at lower normal loads is a bit to lowand at higher loads a bit to high. The extrapolation at higher slip values goes quite well. The MagicFormula model is not based on polynomial coefficients, which ensures good extrapolation behaviouroutside the measurement range. The lateral fit has some deviation at higher side slip angles wherethe model should be more digressive, especially for higher normal loads.

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(a) Longitudinal forces validation

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Alpha sweep, 8[PSI] inflation pressure, 0[deg] inclination angle

Fz = 667[N] modelFz = 1556[N] modelFz = 667[N] measurementFz = 1556[N] measurement

(b) Lateral forces validation

Figure 2.4: Longitudinal and lateral force validation

The inflation pressure influence on these forces is of course also of importance. Figure 2.5 illustratesthis influence. It can be clearly seen in figure 2.4a that the longitudinal slip stiffness will increasewhen the inflation pressure increases. This is the case in both the measurement and the model. Theforces are decreasing at high slip ratio’s in both the measurement and model. But in reality the effectis higher than the model predicts. The effect should be exaggerated more in the model.

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(a) Longitudinal forces validation

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8[PSI] model8[PSI] measurement12[PSI] model12[PSI] measurement

(b) Lateral forces validation

Figure 2.5: Inflation pressure validation

One of the most difficult things to measure on a tyre is the overturning moment Mx. The first reasonfor this is that the moments are quite small and therefore difficult to measure. The second reasonis that there a lot a measurement noise exists. Figure 2.6a illustrates the differences between themeasurement and the model. At lower loads there appears to be a shift in the measurement, which isprobably caused by conicity of the tyre. At higher loads this disappears.

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The effective rolling radius of the tyre can be seen in figure 2.6b. The measurement is done at 5 fixedloads. This appears to be a more or less linear relation, which is also represented well by the model.This vertical stiffness is especially important during roll of the vehicle, because it results in a tyreinclination angle with respect to the road. Figure 2.7 illustrates the inclination angle behaviour andvalidates the difference between the measurement and the model. It appears that the inclination angleeffect of the measurement certainly larger is than the model predicts. Here is room for improvement.When a new fit would be made in the future this has to be taken into account.

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alpha [deg]

Mx

[Nm

]

Overturning moment, 12[PSI] inflation pressure, 0[deg] inclination angle

Fz = 667[N] modelFz = 667[N] measurementFz = 1556[N] modelFz = 1556[N] measurement

(a) Overturning moment validation

0 500 1000 1500 2000 25000.256

0.257

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0.261

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effe

ctiv

e ro

lling

rad

ius

[m]

Effective rolling radius, 8[PSI] inflation pressure, 0[deg] inclination angle

modelmeasurement

(b) Rolling radius validation

Figure 2.6: Overturning moment and rolling radius validation

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alpha [deg]

Fy

[N]

Alpha sweep camber influence, 12[PSI] inflation pressure, 667[N] normal load

model, 0[deg] inclination anglemodel, 4[deg] inclination anglemeasurement, 0[deg] inclination anglemeasurement, 4[deg] inclination angle

(a) 667N normal load

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alpha [deg]

Fy

[N]

Alpha sweep camber influence, 12[PSI] inflation pressure, 1556[N] normal load

model, 0[deg] inclination anglemodel, 4[deg] inclination anglemeasurement, 0[deg] inclination anglemeasurement, 4[deg] inclination angle

(b) 1556N normal load

Figure 2.7: Inclination angle validation

The overal fit of the tyre model is adequate enough to use for analysis of the dynamic behaviour of aFormula Student racing car. The lack of combined longitudinal and lateral measurements data meansthat the tyre model has to generate an estimated combined slip characteristic, which of course will notexactly match the real world behaviour. Some other tyre characteristics could be better represented.In the mean time the FSAE tyre test consortium has performed new measurements which contain

14

2.5. Tyre choice CHAPTER 2. Modeling a racing tyre

better measurement data and also combined longitudinal and lateral measurements. This data couldbe used in the future improve the tyre fits.

2.5 Tyre choice

Measurement data of six different tyres from three brands is available. The right tyre choice has to bemade according to the following requirements:

• availability of measurement data

• tyre size

• maximum lateral and longitudinal achievable forces

• amount of pneumatic trail

• inclination angle behaviour

• costs

• tyre availability

• tyre mass and inertia

• heat up time

Rim diameter One of the six measured tyres has a 10" rim diameter. These tyres cannot be measuredfor longitudinal forces and moments. And a 10" diameter rim has a lot of disadvantages comparedwith a 13 inch diameter. A larger inner rim diameter gives more freedom in the suspension designbecause more space is available. This has also the advantage that the connection rod forces are lowerbecause the upright moments can be better transmitted to the chassis. The braking system can alsobe designed larger, this is especially important in the front, since the braking torques are the largestthere. The ratio between the outer tyre diameter and brake disc is of importance. This ratio definesthe force that can be generated at the contact patch. The 10" tyre has an 18" outer diameter, and the 13"tyre has a 20.5" outer diamater but the inner space of the rim is 3" larger. Therefore will this ratio beadvantageous for a 13" rim-tyre combination. The 10" tyre has the advantage of a slightly lower massand rotational inertia.

The choice of a rim diameter size defines more than only the tyre and rim dimensions. The design ofthe chassis and powertrain is also dependent of this choice. The chassis has to be higher in the caseof 13" tyres because the connection rods will be mounted higher.

Figure 2.8 compares the lateral performance and cornering stiffness of the Hoosier 20.5x6.0-13 andthe Goodyear 18x6.5-10. Both the peak friction coefficient and the cornering stiffness of the 10"Goodyear tyre is much lower that that of the (less wide!) 13" Hoosier. Given these results and theother disadvantages the choice is made to use the 13" tyres.

15


−15 −10 −5 0 5 10 15−5000

−4000

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0

1000

2000

3000

4000

5000

alpha [deg]

Fy

[N]

Alpha sweep, 0.83[bar] inflation pressure, 0[deg] inclination angle

Fz = 150 [N] Hoosier 20.5x6.0−13Fz = 450 [N] Hoosier 20.5x6.0−13Fz = 1050 [N] Hoosier 20.5x6.0−13Fz = 2100 [N] Hoosier 20.5x6.0−13Fz = 150 [N] Goodyear 18x6.5−10Fz = 450 [N] Goodyear 18x6.5−10Fz = 1050 [N] Goodyear 18x6.5−10Fz = 2100 [N] Goodyear 18x6.5−10

(a) Lateral force

0 500 1000 1500 2000 2500 3000100

200

300

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800

Fz [N]

stiff

ness

[N/%

]

Cornering stiffness, 0.83[bar] inflation pressure, 0[deg] inclination angle

cornering stiffness Hoosier 20.5x6.0−13cornering stiffness Goodyear 18x6.5−10

(b) Cornering stiffness

Figure 2.8: 10 and 13 tyre comparison

Friction characteristics The maximum lateral and longitudinal forces is the next item to compare. Thecompetition does not requires a tyre which is resistant against wear. This is because the total amountof driven kilometers is low and the maximum event time during the competition is only 20 minutes[9]. Therefore the tyre with the highest friction coefficients will be most suitable for the competition.Figure 2.9 and figure 2.10 illustrates the behaviour of the Hoosier 20.5x6.0-13, Avon 6.2/20.0-13 andGoodyear 20x6.5-13.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5000

−4000

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−2000

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0

1000

2000

3000

4000

5000

kappa [%]

Fx

[N]

Kappa sweep, 0.83[bar] inflation pressure, 0[deg] inclination angle

Fz = 150 [N] Hoosier 20.5x6.0−13Fz = 900 [N] Hoosier 20.5x6.0−13Fz = 2100 [N] Hoosier 20.5x6.0−13Fz = 150 [N] Avon 6.2/20.0−13Fz = 900 [N] Avon 6.2/20.0−13Fz = 2100 [N] Avon 6.2/20.0−13Fz = 150 [N] Goodyear 20.0x6.5−13Fz = 900 [N] Goodyear 20.0x6.5−13Fz = 2100 [N] Goodyear 20.0x6.5−13

(a) Longitudinal force comparison

−15 −10 −5 0 5 10 15−5000

−4000

−3000

−2000

−1000

0

1000

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4000

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alpha [deg]

Fy

[N]

Alpha sweep, 0.83[bar] inflation pressure, 0[deg] inclination angle

Fz = 150 [N] Hoosier 20.5x6.0−13Fz = 900 [N] Hoosier 20.5x6.0−13Fz = 2100 [N] Hoosier 20.5x6.0−13Fz = 150 [N] Avon 6.2/20.0−13Fz = 900 [N] Avon 6.2/20.0−13Fz = 2100 [N] Avon 6.2/20.0−13Fz = 150 [N] Goodyear 20.0x6.5−13Fz = 900 [N] Goodyear 20.0x6.5−13Fz = 2100 [N] Goodyear 20.0x6.5−13

(b) Lateral force comparison

Figure 2.9: Longitudinal and lateral force comparison

The longitudinal performance is especially important at corner entry, exit and during the accelerationtest [9]. The Avon tyre has the highest peak in most cases, but the Hoosier has the flattest curve afterthe peak. This has the advantage that the tyre behaves very predictable after this peak. This will be verybeneficial, especially with unexperienced drivers. Hoosier’s background can be found at the Nascarcompetition and other ’drift like’ competitions, which is likely to explain this flat curve. The Hoosiertyre has the largest friction coefficient at the lower loads (figure 2.10a). This region of loads applies

16


0 500 1000 1500 2000 2500 30001.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Fz [N]

mu

[−]

Friction coefficient, braking and driving, 0.83[bar] inflation pressure, 0[deg] inclination angle

braking Hoosier 20.5x6.0−13braking Avon 6.2/20.0−13braking Goodyear 20.0x6.5−13driving Hoosier 20.5x6.0−13driving Avon 6.2/20.0−13driving Goodyear 20.0x6.5−13

(a) Longitudinal friction coefficient

0 200 400 600 800 1000 1200 1400 1600 1800 20002.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Fz [N]

mu

[−]

Friction coefficient, lateral, 0.83[bar] inflation pressure, 0[deg] inclination angle

lateral (−) Hoosier 20.5x6.0−13lateral (−) Avon 6.2/20.0−13lateral (−) Goodyear 20.0x6.5−13lateral (+) Hoosier 20.5x6.0−13lateral (+) Avon 6.2/20.0−13lateral (+) Goodyear 20.0x6.5−13

(b) Lateral friction coefficient

Figure 2.10: Friction coefficients comparison

most to a Formula Student vehicle since the total vehicle mass is approximately 300 [kg] and whenthis loads has to be carried by 2 wheels this would results in a maximum single tyre load of 1500 [N].

Most of the time during the competition the vehicle will produce lateral forces due to cornering anddriving excessive amounts of slaloms and chicanes. The lateral performance is therefore very impor-tant. Figure 2.9b shows the lateral performance of the three tyres. In almost all the cases the Hoosiertyre performs best. Not only will it deliver the highest forces, it has also the highest cornering stiffness(figure 2.11b). Only at very high normal loads (>2000 [N]) and side slip angles the Goodyear performsbetter. In reality a tyre’s normal load never exceeds half of the vehicles mass (300 [kg] = 1500 [N]). TheHoosier tyre has the highest friction coefficient during the whole spectrum of figure 2.10b. The non-linearity of the lateral friction coefficient at the higher loads occurs because the peak of the lateral forcemigrates to very large side slip angles, which aren’t taken into account during both the measurementsand the calculation of the friction coefficient.

The values for the longitudinal and lateral friction coefficients appear to be quite large. The reasonfor this can be found in the tyre temperature during the measurement, the road friction coefficientson the flat track machine and the tyre compound. All the measurements are performed at race oper-ating temperatures (which is monitored during the measurement), this results in the highest frictioncoefficient. The used road surface is 3M Safety-Walk sand paper, which is a representation of asphalt.The tyre compound is one of the softest available and specially designed for the Formula student com-petition [10]. Simulations in the past have shown the car may lift two wheels off the ground duringcornering. During the competition in Italy in 2007 it was proved that the car could really do thisduring cornering. These values of friction coefficients are therefore interpreted as plausible, and be-cause the tyres are measured under the same conditions it is valid to compare them. This also gives arequirement for the centre of gravity height.

17


0 500 1000 1500 2000 2500 30000

100

200

300

400

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600

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800

900

Fz [N]

stiff

ness

[N/%

]

Longitudinal stiffness, 0.83[bar] inflation pressure, 0[deg] inclination angle

longitudinal stiffness Hoosier 20.5x6.0−13longitudinal stiffness Avon 6.2/20.0−13longitudinal stiffness Goodyear 20.0x6.5−13

(a) Longitudinal stiffness

0 500 1000 1500 2000 2500 3000100

200

300

400

500

600

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800

Fz [N]

stiff

ness

[N/d

eg]

Cornering stiffness, 0.83[bar] inflation pressure, 0[deg] inclination angle

cornering stiffness Hoosier 20.5x6.0−13cornering stiffness Avon 6.2/20.0−13cornering stiffness Goodyear 20.0x6.5−13

(b) Cornering stiffness

Figure 2.11: Longitudinal and cornering stiffness comparison

−2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

Fx [N]

Fy

[N]

Combined slip, 0.83[bar] inflation pressure, 0[deg] inclination angle

Hoosier 20.5x6.0−13

Avon 6.2/20.0−13

Goodyear 20.0x6.5−13

(a) Estimation of the combined performance

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

Fz [N]

[m]

Pneumatic trail, 0.83[bar] inflation pressure, 0[deg] inclination angle

pneumatic trail Hoosier 20.5x6.0−13pneumatic trail Avon 6.2/20.0−13pneumatic trail Goodyear 20.0x6.5−13

(b) Pneumatic trail

Figure 2.12: Combines slip and pneumatic trail comparison

The combined slip performance(estimation) is, as already could be expected according to figure 2.9and 2.10, the best for the Hoosier tyre, see figure 2.12a. The pneumatic trail is of importance forthe feedback to the driver via the steering wheel. The pneumatic trail will decrease when the sideslip angles of the front tyres increase. This results in a decreasing moment at the steering wheel.The driver will observe this as a loss of grip of the front of the car, and is therefore one of the mostimportant feedback factors. In the past it appeared that steering forces where quite high. Theseforces are generated with the combination of pneumatic trail and mechanical trail. The later can beinfluenced by the design of the suspension. The former has to be as low as possible to reduce steeringforces and still having proper driver feedback. The Hoosier tyre has the lowest pneumatic trail 5.11b.

Both the Hoosier and Goodyear tyre have the potential to be used. The Goodyear tyre has a slightlybetter performance in some regions, but this tyre is probably more difficult to drive. The relaxationlengths of the tyres are not measured. This parameter is especially of importance during slaloming.Further research is necessary to take this factor also into account. Given all of the above results anobvious tyre choice can be made, namely to make use of Hoosier tyres.

18

Chapter 3

Steady state vehicle behaviour

3.1 Introduction

This chapter discusses the development and analysis of a steady state two track roll axis model whichis used to investigate the influence of basic vehicle parameters such as mass, centre of gravity heightand wheelbase on the maximal achievable longitudinal and lateral acceleration of a vehicle.

Normally a full dynamic vehicle model would be used to analyse this behaviour. The problem withthese complex models is that a lot of insight is lost. Another disadvantage is that simulation resultsoften show transient behaviour, which source is not easy to comprehend. Therefore it is desirable tohave a model to investigate the pure steady state behaviour and to gain insight in the influence of thebasic vehicle parameters.

The load distribution over the four wheels is one of the fundamental ways to influence the dynamicbehaviour of a vehicle. This aspect is discussed in more detail. The layout of the two track roll axismodel is the next item to discuss. It appears that such a model is not easy to solve. The possibilities,model applications and results will be given. Section 3.10 discusses which design strategy could bebest used to increase the maximal longitudinal and lateral vehicle acceleration potential. This chapterends with a discussion on optimal steering angles and tyre inclination angles.

3.2 Load distribution

The distribution of load across the vehicle’s tyres is very important in vehicle behaviour and perfor-mance. Therefore this will be discussed in more detail.

When a vehcile drive to a corner a lateral acceleration is produced. The lateral acceleration is causedby the tyres producing lateral forces Fy left and Fy right, see figure 3.1. The position of the centre ofgravity location is located above the ground. This results in a lateral acceleration during steady statecornering and therefore a reaction force acting on the centre of gravity. In this example the front andrear axle are added together.

19

3.2. Load distribution CHAPTER 3. Steady state vehicle behaviour

Figure 3.1: Load distribution

Figure 3.2 illustrates the cumulative lateral force generated by the two wheels of an axle during corner-ing. Increasing load transfer means a decrease in total lateral force. Load transfer is inevitable since itis not possible to create a vehicle with the centre of gravity at road height. This decrease is caused bytwo aspects [11]:

• saturation of the cornering stiffness

• degressive friction coefficient with vertical load

0 10 20 30 40 50 60 70 80 90 1001000

1500

2000

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3000

3500

4000

4500

5000

weight transfer [%]]

Tot

al a

xle

late

ral f

orce

[N]

Weight transfer effect Hoosier 20.5x7.0−R13

Side slip angle α = −2 [deg]Side slip angle α = −4 [deg]Side slip angle α = −8 [deg]

Figure 3.2: Total lateral axle force, changing load transfer

20

3.3. Two track roll axis vehicle model CHAPTER 3. Steady state vehicle behaviour

At lower side slip angles the change of cornering stiffness has the most influence. At higher sideslip angles the change of friction coefficient is of most influence. The effects of both changes aredepicted in figure 3.3a and 3.3b. The cornering stiffness (figure 3.3a) is digressively increasing till acertain wheel load, after that it is progressively decreasing. This shape implies a loss of corneringstiffness depending on the amount of load transfer, ∆F . The lateral friction coefficient (figure 3.3b) isdecreasing at increasing wheel load. This means that the peak lateral force is not linearly increasingwith increasing wheel load (Fy = Fz · µ). This loss is approximately (2 ·∆F ·∆µ) [11]:

(Fz −∆F ) · (µy + ∆µ) + (Fz + ∆F ) · (µy −∆µ) = 2Fzµy − 2 ·∆F ·∆µ (3.1)

0 500 1000 1500 2000 2500 30000

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400

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600

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Fz [N]

stiff

ness

[N/d

eg]

Cornering stiffness

+ ∆ F− ∆ F

loss

(a) Cornering stiffness

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2.35

2.4

2.45

2.5

2.55

Fz [N]

mu

[−]

Average lateral friction coefficient

− ∆ F + ∆ F

∆ µ

(b) Lateral friction coefficient

Figure 3.3: Effect of load distribution

These two phenomena are the reason for the decreased performance during cornering. The samehappens in the longitudinal direction, during accelerating and braking. Given these results it is ob-vious that the centre of gravity has to be as low as possible. The magnitude of this influence will beinvestigated using the two track roll axis model.

3.3 Two track roll axis vehicle model

A two track roll axis model is used to investigate the influence of basic vehicle parameters such asmass and center of gravity height on the performance of the vehcile. Using this model, the influenceos the following parameters can be investigated:

• vehicle mass

• centre of gravity height

• roll centre height front and rear

• roll stiffness front and rear

• wheel base

• track width front and rear

21

3.3. Two track roll axis vehicle model CHAPTER 3. Steady state vehicle behaviour

• static camber angle of each tyre

• static toe angle of each tyre

• tyre model paramaters

The model will be used to calculate the maximum achievable longitudinal and lateral acceleration andalso combinations of these accelerations. This will be called the vehicle potential. The potential isillustrated with a g-g diagram. The x-axis of the diagram represents the longitudinal acceleration ofthe vehicle, and the y-axis the lateral acceleration. This diagram is often referred to as the performanceenvelope of a vehicle because is describes the maximum achievable combinations of longitudinal andlateral accelerations possible. Since the vehicle is considered symmetric in a lateral way (left-rightcornering) it is sufficient to show only half of the g-g diagram. Comparing the potential of severaldifferent vehicle configurations gives understanding in the sensitivity of a vehicle parameter such asits mass.

The two track roll axis vehicle model is purely based on analytic equations. The steady-state analysiseliminates transient behaviour occurring with simulations in time. Appendix A discusses in detail theequations [11] which describe this model.

The model incorporates 4 tyre models, rigid axles, a roll axis with front and rear roll stiffness but nopitch dynamics (see figure 3.4). The simplicity of this model ensures that the basic vehicle parameterscan be investigated without having the inconveniences of transient behaviour, such as vibrations androtational inertia’s of wheels and engine.

Figure 3.4: Two track analytic vehicle model

As already discussed before, a load transfer to the outside wheel causes a decrease in the total gener-ated lateral force of that axle. This is caused by the height of the centre of gravity. The distributionof load transfer between the front and rear axle is called roll ratio or roll couple distribution. Thismechanism is the most important way to influence under/overstreer (balance) of a vehicle. There aretwo ways to generate this load transfer:

22

3.4. Base line vehicle CHAPTER 3. Steady state vehicle behaviour

• Elastic load transfer via the springs of the suspension system.

• Direct load transfer through the suspension links.

The roll axis model incorporates 3 parameters which influence load transfer:

• Roll centre position. The point in the y-z plane though an axle where the lateral force generatedby the tyres is applied to the chassis. This force cannot generate a roll angle. The roll axis couplesthe front with the rear roll centers.

• Roll stiffness. The stiffness which generates a moment at the roll centre depending on the heightof the centre of gravity relative to the roll axis.

• The height of the centre of gravity relative to the ground and relative to the roll axis.

3.4 Base line vehicle

It is possible to change the discussed vehicle parameters freely, but experience from the past andregulations, limits this free choice. A lot of the parameters are chosen by experience and knowledgefrom the past. Therefore a base line vehicle configuration is defined.

The vehicle mass is set to 300 [kg], which includes the vehicle (225 [kg]) and the driver (75 [kg]). Oldervehicles have a vehicle mass of 280 and 235 [kg]. The tendency is to reduce the mass.

The centre of gravity height of the URE03 (2006-2007) vehicle is measured and equals 0.34 [m]. A lotcan be gained by lowering the engine and driver. The aim for the URE04 vehicle is to reach 0.28 [m].The engine can be lowered by approximately 0.10 [m] because it has a large suction chamber on thelower end of the crankcase. The new engine will be equipped with a much lower dry sump crankcase.The driver will be lowered by stretching the drivers legs and back.

The front and rear track widths are determined by the appraisal of maximum lateral acceleration andminimum lateral translation when slaloming. The track is defined measuring from the centre of onecontact patch to the other [1]. The lager the track width, the lower the load transfer, the higher thevehicles lateral performance, see section 3.2. But the wider the vehicle the more it has to translatelaterally during slaloming and cornering around cones. The choice is made based on these aspectsand the experience from the past. The front track is set to 1.225 [m]. The rear track is kept smaller toincreases slaloming speed. The rear track has to move less in lateral direction around the cones whichincreases slalom speed. This reduces lateral translation of the car even more. The rear track is set to1.175 [m].

The wheelbase is set to 1.6 [m]. Older vehicles showed nervous yaw behaviour with a shorter wheelbaseof 1.5 [m]. When the wheelbase is increased to values larger than 1.6 [m] would this influence the timeto do a slalom maneuver to much. This would also increase the vehicle yaw inertia which againdecreases the slalom and general dynamic cornering performance. The wheelbase-track ratio (WTR)is 0.75. This is a common ratio for a Formula Student vehicle and results in a good balance betweenperformance and stability.

WTR =(1.225 + 1.175) /2

1.6= 0.75 (3.2)

A larger wheelbase has a positive influence on braking performance, because a larger wheelbase de-creases load transfer to the front of the vehicle when braking. The reason for this is this same as

23

3.4. Base line vehicle CHAPTER 3. Steady state vehicle behaviour

in the case of cornering, see section 3.2. But as the car is only rear wheel driven, a short wheelbasewould have a positive influence on accelerating! In fact, the acceleration performance will be bestwhen the total vehicle load is supported by the rear tyres of the car. Increasing the centre of gravityheight accomplishes this also. Figure 3.5a illustrates the vehicle potential of two identical vehicles isa g-g diagram. One vehicle has a low centre of gravity height and the other a high centre of gravityheight. The figure illustrates that the vehicle with the high centre of gravity height will have a disad-vantage when braking and cornering but a large advantage when accelerating. It is assumed that theengine and braking torque is infinite. In reality is the engine the limiting factor for the acceleratingside especially at high speeds. This is an illustrative example to show the influence of the centre ofgravity height. The vehicle configuration is not based on the base line vehicle parameters.

When the car has 4 wheel drive a long wheelbase would be beneficial during braking and driving.Figure 3.5b illustrates the performance of a vehicle with rear wheel drive and the same vehicle with4 wheel drive. It is obvious that a 4 wheel drive vehicle is beneficial, but the increased vehicle mass,drive train efficiency loss, complexity and decreased reliability results in choosing rear wheel drive.

−2 −1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Ax vehicle [g]

Ay

vehi

cle

[g]

g−g plot two wheel drive center of gravity

low CGhigh CG

(a) Center of gravity 2 wheel drive

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Ax vehicle [g]

Ay

vehi

cle

[g]

g−g plot 2 and 4 wheel drive vehicle

rear wheel drive4 wheel drive

(b) Rear and 4 wheel drive

Figure 3.5: Vehicle performance

Both the height of the roll centers and the roll stiffness’s can be chosen freely. The roll centers willbe kept low to the road surface to reduce jacking (see section 5.5). The roll stiffness’s have beendetermined to balance the car to neutral steer, where the roll angle is chosen 1 degree (see section 5.5)at the maximal lateral acceleration.

The next table summarizes the parameters of the baseline vehicle used in calculations in the remain-der of this chapter:

24

3.5. Model objective CHAPTER 3. Steady state vehicle behaviour

Vehicle parameter value unitmass 300 [kg]CG height 0.28 [m]CG distance from the front axle 0.8 [m]front track 1.225 [m]rear track 1.175 [m]wheelbase 1.6 [m]roll centre height front 0 [m]roll centre height rear 0 [m]roll stiffness front 54000 [Nm/rad]roll stiffness rear 49000 [Nm/rad]static camber angles 0 [deg]static toe angles 0 [deg]

The tyre model used is the Hoosier 20.5x7.0-R13, which has slightly modified lateral coefficients todecrease the amount of lateral force at high side slip angles. This is done to decrease computationtime, because a flat curve at high side slip angles causes the model to converge more difficult to asolution. The difference is that small that it will not influence the results.

3.5 Model objective

The objective of the analysis is to investigate the influence of the earlier mentioned vehicle parame-ters of the performance envelope of the vehicle. This is illustrated in the g-g diagram of the vehicle.The model equations, as discussed in detail in appendix A, have to be solved to find a solution of onesteady state point in the g-g diagram. This can be for example the maximal achievable lateral acceler-ation during cornering or a combined braking and cornering maneuver. The model has to solved forevery combination of lateral and longitudinal acceleration to find the entire performance envelope ofa vehicle configuration. The model has, besides the constant vehicle parameters, several inputs whichcan be used to find a steady-state solution. The model outputs are used to validate if the solution ispossible in reality. The model inputs are:

• steering angle δ

• longitudinal slip of both rear tyres, κ, when accelerating

• longitudinal slip of all tyres, κ, when braking

• lateral vehicle speed v

• wheel loads Fz,1−4

The calculation starts with a choice of the combination of longitudinal and lateral acceleration ofthe vehicle. This can be for example pure cornering. The calculation is started at a rather smalllateral acceleration to guarantee that a valid solution can be found. The vehicle is driving on a fixedradius of 100 [m] regardless of the combination of longitudinal and lateral acceleration. This resultsautomatically in a longitudinal speed, tangent to the driven radius. It also result in a roll angle, since

25

3.6. Numerical optimization CHAPTER 3. Steady state vehicle behaviour

the static vehicle parameters are known. This information is besides the model inputs used to initiatethe calculation of the model.

The input ’steering angle’ results in a certain lateral acceleration level and has to be chosen correct tomatch the chosen combination of longitudinal and lateral acceleration level. This also yields for thelongitudinal slip κ. The input ’lateral vehicle speed’ results together with the calculated longitudinalspeed (tangent to the driven radius) in the vehicle speed V and therefore in the vehicle slip angle β.The wheel loads are also an input since it is not known on forehand what they will be for a certainchoice for a combination of longitudinal and lateral acceleration level.

Depending on the model inputs, the output has to satisfy the chosen combination of longitudinal andlateral acceleration. The output should also indicate if the solution is valid and possible in reality.This is done by calculating the errors between the inputs and outputs of the chosen combination oflongitudinal and lateral acceleration level, ax and ay , the resulting moment around the z-axis of thevehicle Mz and the wheel loads Fz1−4. Obviously the exact solution can not be found in one step. Thereason for this is that the model an algebraic loop contains. The loop exists because each tyre loadhas to be calculated separately, and this tyre load is dependent of the lateral acceleration, the centreof gravity height, two roll stiffness, the roll centre heights, the wheelbase and track width. The lateralacceleration is again dependent of the generated tyre forces, but these tyre forces are also dependentof the wheel load. This results in the algebraic loop because Fz is required to calculate Fx, Fy and Mz

of the four tyres, but to calculate Fz it requires Fx, Fy and Mz . This loop is especially complex to solvebecause the tyre acts as a highly non-lineair element. The next section discusses a technique to solvethe model equations.

3.6 Numerical optimization

Numerical optimization is a technique which is widely used in the industry to solve problems and tofind optimal solutions for a design. The goal of an optimization problem can be formulated as follows:find the combination of independent variables which minimize a given quantity, possibly subject tosome restrictions on the allowed parameter ranges or other constraint functions. The quantity to beoptimized (maximized or minimized) is termed the objective function.

An example of an optimization problem is the design of an engine valve spring. The objective of theproblem is formulated to reach a low as possible spring mass and a high as possible eigen frequency.The variables are the spring inner and outer diameter, the spring length and the number of springwindings. The constraint is that the spring has to fit in the engine head. This constraint consistsof several maximum and minimum dimension such as its inner and outer diameter and the springlength. The optimization routine has to find a combination of values for the variables to achievethe best solution. The objective function is put into a mathematical equation. The result of thisequation has to be as low or high as possible, depending on the optimization definition (minimizationor maximization). The optimization proces for the roll axis model is defined as a minimization proces.

There are a lot of optimization algorithms developed in the past. To solve this problem a generic opti-mization algorithm is used from the genetic optimization toolbox of MATLAB . The used algorithm iscalled the generalized pattern search (GPS, MATLAB : patternsearch) algorithm. The working principlebehind the algorithm is discussed in detail in appendix B.

26

3.7. Calculation sequence CHAPTER 3. Steady state vehicle behaviour

3.7 Calculation sequence

This section explains the calculation sequence of the roll axis model. Firstly will the pure lateralpotential of the vehicle be investigated. The longitudinal acceleration is chosen 0 [g] and the lateralacceleration 0.5 [g]. This combination is definitely possible given the standard vehicle configurationof a FSAE vehicle. The input for the model consists of three groups, namely:

• vehicle configuration. Examples: vehicle mass, centre of gravity height, roll stiffness’s, tyremodels

• parameters depending on chosen acceleration levels. Examples: roll angle, longitudinal speed,yaw speed

• optimization inputs. Examples: wheel loads, steer angle, longitudinal slip, lateral vehicle speed

The vehicle configuration is chosen before the model is initiated and consists of the basic parametersof the vehicle, which where discussed in section 3.4. Some parameters are calculated when the modelis initiated. This are parameters which can only be derived from the vehicle configuration parameters.This can be for example the roll angle which is dependent of the vehicle configuration and the chosenacceleration level. The last group of model inputs is given by the optimization scheduler. Theseinputs are iteratively changed to find the solution for the roll axis model for the given configurationand acceleration level.

The 4 wheel loads Fz,1−4, for example, are initiated at a quarter of the vehicle mass to start with. Thebasic model consists of parallel steered front wheels, which can be changed by the algorithm with onevariable δ. The lateral vehicle speed, v, can be changed by the algorithm to adjust the vehicle slip angleβ. Since the vehicle speed, V , is known, the longitudinal vehicle speed, u, can be calculated also. Seeappendix A for more details. The basic model consists of one variable for the longitudinal slip. Thesevalues will be applied to both rear wheels when the model is accelerating. This represents the differen-tial. When the model is braking this value will be applied to all four wheels representing the brakingsystem. More complex models can also include different steering angle variables for the left and rightfront tyres to investigate this influence. These models can also include more complex differential andbraking systems where different kappa variables for all tyres are used. It is, for example, also possibleto implement a 4 wheel drive vehicle or a rear wheel steered vehicle.

The correctness of the solution is validated with error function. These acts also as the objective func-tion of the optimization routine which has to be minimized.

The calculated tyre loads Fz are compared with the ones of the model input which are used for thetyre model. This is done according to equation (3.3). This function consists of several quadraticcomponents. This is done to increase the convergence time to find the solution [12]. The reasonfor this is that the direction coefficient is decreasing while approaching zero on the x-axis. The solveris able to handle this more efficient.

Fz,error = (Fz1 input − Fz1)2 +(Fz2 input − Fz2)2 +(Fz3 input − Fz3)2 +(Fz4 input − Fz4)2 (3.3)

The lateral and longitudinal acceleration errors are calculated according to:

ax,error =(ax −

(Fx1,chassis + Fx2,chassis + Fx3,chassis + Fx4,chassis

m

))2

(3.4)

27


ay,error =(ay −

(Fy1,chassis + Fy2,chassis + Fy3,chassis + Fy4,chassis

m

))2

(3.5)

The total moment around the z-axis of the chassis has to be zero to find the steady state solution, see:

Mz,error = (− Fx1,chassis · s1 + Fy1,chassis · a1 +Mz1

+ Fx2,chassis · s2 + Fy2,chassis · a2 +Mz2

− Fx3,chassis · s3 − Fy3,chassis · a3 +Mz3

+ Fx4,chassis · s4 − Fy4,chassis · a4 +Mz4)2

(3.6)

Where s1 and s2 are equal to half of the front and rear track width respectively. a1 and a2 are equalto the distance from the centre of gravity to the front and rear axle respectively. Mz,1−4 are the selfaligning moments generated by the tyres. Appendix A discusses the equations of the model in moredetail.

The objective function consists of the above error functions multiplied with several weighting factors(eq: (3.7)). This is done to balance the solution and to direct the optimizer to put more effort on someaspect of the problem.

Objective function = W1 · Fz,error +W2 · ax,error +W3 · ay,error +W4 ·Mz,error (3.7)

The solution for the vehicle state is found when the error functions equal zero. This results automat-ically in an objective function which is zero. The optimization routine will end when the objectivefunction is zero. The optimization inputs of the last iteration yield the solution of the model. At thispoint is the solution found for a longitudinal acceleration of 0 [g] and a lateral acceleration of 0.5 [g].This is not the maximal achievable lateral acceleration of the vehicle. To find this point the lateralacceleration aim has to be increased. This is done by the optimization scheduler which increases thelateral acceleration with 0.5 [g] to 1.0 [g]. The optimization routine tries to find the solution for thisstate again. At a point the optimization routine tries to solve the model for a vehicle state which isimpossible to reach for the given vehicle configuration.

This boundary can be reached via two ways. The first one is the condition when the lateral accelerationis increased and the objective function value is not equal to zero. The boundary is then defined by theprevious lateral acceleration level. The second one is when one of the inner tyres will be lifted from theroad and the vehicle continues to drive on three wheels. In reality the boundary could be a bit higher,but when the vehicle is correctly balanced this would be a neglectable amount.

When a solution can not be found the scheduler drops back 0.5 [g] and changes the step size to 0.1 [g],the last increment is 0.01 [g]. At his point is the lateral acceleration boundary of the vehicle know witha resolution of 0.01 [g]. This is just one point of the g-g diagram which represents the performanceenvelope of the vehicle. The longitudinal acceleration level is increased by the optimization schedulerwith 0.01 [g] and the optimization algorithm tries to find the lateral acceleration boundary again. Thisis repeated till the longitudinal acceleration boundary is found. This boundary is found when thelateral acceleration boundary is immediately equal to zero. The optimization scheduler start with thepositive longitudinal acceleration range. The optimizations scheduler starts with the negative side ofthe longitudinal acceleration side of the g-g diagram when the positive longitudinal acceleration rangeis known.

28


Test case scheduler

When all combinations of longitudinal and lateral accelerations (the g-g diagram) for one vehicle con-figuration is found the scheduler will send the results to the test case scheduler (see figure 3.6), whicharranges the data and saves it to a file. When more vehicle configurations have to be calculated, it willinitiate to do so and/or plots the results.

Figure 3.6: Calculation sequence optimization solver

Computation time

The computation time needed to calculate one combination of longitudinal and lateral accelerationcould take up to several hours. When the whole spectrum of combinations of accelerations has to becalculated would this take several weeks.

There are three ways used to decrease computation time. The first one is the tuning of the weightingfactors of (3.7). This directs the optimization algorithm to put more effort on one part of the problemwhich decreases computation time.

The second limiting factor is the tyre model, which has to be called four times per model evaluation,and more than 99% of calculation time is spend with evaluating the tyre models. To decrease com-putation time it is obvious to accelerate the tyre model calculation. This is done by implementing thetyre model in MATLAB code. Several unused calculations are omitted to increase speed even more.

The last way to increase speed is to develop a smart optimization scheduler. These measures decreasedthe calculation time for one vehicle configuration from 12 days to several hours.

Uniqueness of the solution

The method does not guarantee that the solution is an unique one. In reality it is sometimes possibleto find two or more exact solutions for combinations of input variables. An example is a vehicle whichhas a initial solution when it would drive neutral steered [11] through a corner. A second solution couldbe the same vehicle driven with excessive oversteer trough the corner. This could happen when thetwo tyres on an axle produces the same amount of lateral force while working at different side slipangles. Figure 3.7a illustrates this. This could be achieved in reality by suddenly applying a muchlarger steering angle to increase the side slip angles of the front tyres. The total lateral force will notchange and the vehicle will have the same lateral acceleration level.

29

3.8. Other model applications CHAPTER 3. Steady state vehicle behaviour

Figure 3.7b illustrates this according tot the handling diagram representation. The handling diagram[3] is used to investigate vehicle behaviour in a non lineair way. The lateral acceleration, which has ananalogy with the lateral force, is plotted against the average of the side slip angles of the front and reartyres. Subtracting these values is a representation of the under/over steer behaviour of the vehicle,meaning a larger rear slip angle than front results in an oversteered vehicle. Again two solutions ofthe vehicle can be constructed. One with small side slip angles and one with large side slip angles.The second solution (right in figure 3.7b) has a larger difference between the slip angle of the front andrear axles than the first solution (left in figure 3.7b). This means that the second solution describes amuch more oversteered vehicle, but still at the same lateral acceleration level.

0 2 4 6 8 10 12 14 16 18 20−2500

−2000

−1500

−1000

−500

0

late

ral f

orce

Fy [N

]

side slip angle α [deg]

(a) Lateral tyre forces, two states

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ay

[g]

axes side slip angle α [deg]

Slip angle front axisSlip angle rear axis

(b) Handling diagram, two states

Figure 3.7: Vehicle states

This first solution is dominant since the optimization scheduler starts at side slip angles of zero de-grees which will be increased slowly, and even when a second solution will be reached this couldvery easily be identified in the results of the optimization proces by large side slip angles, α, largesteer angles, δ, and vehicle slip angles, β. Experience with the model showed that in almost all cases(>99.99%) the first solution is calculated. This the most relevant solution to investigate since the sec-ond solution represents a vehicle state which will not occur often in reality. Yet some drift competitionsare based on this state.

3.8 Other model applications

The vehicle model has potential to be used for different purposes. The most important purpose is toinvestigate the influence of the discussed vehicle parameters but the layout and calculation methodmakes it possible to use it for other applications. The next list shows an overview of the possibilities:

• Find the performance envelope of a vehicle configuration, represented with a g-g diagram

• Do multiple configuration sweeps to make a parameter sensitivity analysis

• Display the vehicle handling diagram to predict vehicle balance

• Show a graphical top view of the vehicle when increasing increments to visualize behaviour

• Optimize steering angles to find optimal steering geometry

• Compute the optimal camber angle for a certain tyre model

30

3.9. Pure cornering results CHAPTER 3. Steady state vehicle behaviour

Finding the maximum vehicle performance is defined by the upper boundary of the lateral and longitu-dinal acceleration levels for a certain vehicle configuration. In realty can this boundary not be reachedbecause rotational inertia’s will lower this boundary. This is discussed in more detail in section 4.7.The task of the vehicle designers is to approach this boundary as close as possible.

It is possible to walk trough the maximum combinations of longitudinal and lateral performance anddisplay a graphical top view of the vehicle showing several important vehicle quantities.

One benefit of the optimization scheduler is the possibility to implement another optimization variablesuch as extra steering angles.

The optimal camber angle for a certain tyre can be calculated. The calculation is performed alongsidethe vehicle model.

GUI

Since it is possible to select a lot af different vehicle configurations and ranges of calculation a graph-ical user interface is made to ensure a user friendly usage. Figure 3.8 visualizes the graphical userinterface.

Figure 3.8: Graphical user interface

The user has the possibility to set the vehicle configurations and selecting several other settings. Thisensures that every user can work with the model. The code is compiled to generate a stand alone toolwhich can run on a PC without the need of MATLAB .

3.9 Pure cornering resultsPure cornering analysis is done to investigate the balance of the vehicle and the pure lateral perfor-mance. The roll stiffness ratio and roll centre heights are important in this phase of the analysis. Abalanced vehicle shows lineair increasing wheel loads with increasing lateral acceleration, see figure3.9a. This is only true for a vehicle which has lineair roll spring characteristics between the front and

31


rear. The slope is defined by the roll stiffness and vehicle geometry. The dotted lines, denoted with[REF], represent the baseline vehicle which is neutral steered. The solid lines are an understeeredvehicle. The reference vehicle reaches of course a higher lateral acceleration of 2.0 [g] where theundersteered vehicle reaches 1.8 [g]. See figure 3.9a.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

Ay [g]

Fz

tires

[N]

Fz1 [N]Fz2 [N]Fz3 [N]Fz4 [N]Fz1 REF [N]Fz2 REF [N]Fz3 REF [N]Fz4 REF [N]

(a) Tyre loads

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Ay [g]

β [d

eg]

ββ REF

(b) Vehicle slip angle

Figure 3.9: Tyre loads and vehicle slip angle

It is clearly shown that an understeered vehicle would have a lower lateral performance. The reasonfor this is that one wheel will be lifted of the the ground earlier, which causes the optimization tostop. The performance could be a bit higher when the side slip angle of the remaining tyre wouldincrease, but this would be a small amount. The vehicle is normally chosen neutral steered when thefull performance envelope of a vehicle configuration is analyzed.

Figure 3.9b depicts the vehicle slip angle. An understeered vehicle should have a lower vehicle sideslip angle which is also shown in the results of the calculation. An understeered vehicle should alsohave a larger steering angle. This can be seen in figure 3.10a.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Ay [g]

δ [d

eg]

δ frontδ front REF

(a) Steer angles

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Ay [g]

α [d

eg]

alpha1alpha2alpha3alpha4alpha1 REFalpha2 REFalpha3 REFalpha4 REF

(b) Tyre slip angles

Figure 3.10: Slip angles

An understeered vehicle causes the tyres to have higher side slip angles on the front to reach the samelateral force, this is depicted in figure 3.10b. This can also be explained when looking at figure 3.11b.The outside tyre forces on the front of the understeered vehicle (Fy2) have to be higher because the

32


inside tyre generates less force. This is caused because the load transfer on the front is higher whichresults in the understeered vehicle. This again shows the importance of load distribution.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−120

−100

−80

−60

−40

−20

0

20

40

60

Ay [g]

Fx

tires

[N]

Fx1Fx2Fx3Fx4Fx1 REFFx2 REFFx3 REFFx4 REF

(a) Longitudinal tyre forces

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−500

0

500

1000

1500

2000

2500

3000

Ay [g]

Fy

tires

[N]

Fy1Fy2Fy3Fy4Fy1 REFFy2 REFFy3 REFFy4 REF

(b) Lateral tyre forces

Figure 3.11: Slip angles

A vehicle is neutral steered when the average of the left and right steering angles corresponds to thedriven radius which equals the wheelbase divided with the driven radius (l/R). The handling diagram[3] can be constructed when the slip angle of the front axis is subtracted from the slip angle of the rearaxis. An understeered vehicle would have a larger front axle slip angle which results is a handlingdiagram pointing to te left, see figure 3.12a. An oversteered vehicle would have a larger rear axle slipangle, see figure 3.12b.

−2 −1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

α [deg]

Ay

Handling diagram pure lateral

Slip angle front axisSlip angle rear axisHandling diagraml/RMean steer angleSlip angle front axis REFSlip angle rear axis REFHandling diagram REFl/R REFMean steer angle REF

(a) Handling diagram understeer

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

α [deg]

Ay



(b) Handling diagram oversteer

Figure 3.12: Handling diagram

33

3.10. Combined cornering/driving results CHAPTER 3. Steady state vehicle behaviour

Both the understeered and oversteered vehicle have a handling diagram which keeps deviating awayfrom neutral when the lateral acceleration is increased. This means that the vehicle will not switchbetween understeer-oversteer or vice versa which results in a predictable behaviour at limit handling.Some vehicles will start for example understeered and when the lateral acceleration increases theyswitch to oversteer behaviour [3].

The maximum achieved lateral acceleration for the given vehicle configuration is just below 2.0 [g]and is limited by lifting the inside tyres from the road. Lower centre of gravities should show slidingbehaviour of the vehicle. The lifting of the tyres is caused, as already earlier discussed, by the highperformance of the tyres. Which are measured under ideal conditions, such as race tyre temperature,high friction road material and a very soft tyre compound. In reality would this not happen very oftensince the road surface is often polluted with dust and the tyre temperature is mostly low since theevents are short. One exception is the endurance event which takes up to 20 minutes. Here have thetyres time to heat up. During the Formula Student competition 2007 in Italy the URE03 has drivenon 2 wheels for a small amount of time at an entry of a slalom. The design of the URE04 vehicle has amuch lower centre of gravity height which suppresses this behaviour. But when the friction coefficientduring the competition reaches the one of the measurement is it still possible to lift the inner tyresfrom the road surface.

3.10 Combined cornering/driving results

The total vehicle potential is analyzed by calculating all possible combinations of longitudinal andlateral vehicle accelerations and illustrated in a g-g diagram The sensitivity of the vehicle parametersdiscussed can be investigated in this way. Since most of the vehicle parameters are fixed by boundaryconditions (see section 3.4) the focus is put to the influence of the vehicle mass and centre of gravityheight.

The influence of these two parameters result in a choice for a design strategy. It is obvious thatlowering both parameters results in an increase in the vehicle’s acceleration potential. Reliability con-cessions have to be made when the design is focussed on a vehicle mass as low as possible. This alsorestricts the design in using innovative systems such as an active differential or a CVT transmission orfor example a 4 cilinder engine. When the design should be focussed to lower the centre of gravity asmuch as possible all the vehicle’s components should be placed as close to the road surface as possible.This also restricts the design. The spring/dampers can, for example, not be placed above the legs ofthe driver or the differential, which results in using pull rods.

An illustrative example is the choice of a wet or dry sump engine lubrication system. A wet sumpsystem results in a high centre of gravity, because of the larger and higher engine crankcase. Theadvantage is the lower mass. When a dry sump system is used the engine can be placed lower. Butthis system will be heavier since there are more components involved. The sensitivity analysis shouldgive an indication for a choice in design strategy.

The mass sensitivity is defined as the sensitivity in acceleration advantage per kg ( [g][kg] ). The centre of

gravity height sensitivity is defined as the sensitivity in acceleration advantage per centimeter ( [g][cm] ).

The choice for a design strategy is not purely dependent on the total vehicle potential, which could becalculated by taking the area beneath the g-g curve. Since the competition is more focussed on thelateral performance of the vehicle this would not be correct. The next table illustrates an estimate forthe required longitudinal/lateral performance balance for the events in the competition.

34

3.10. Combined cornering/driving results CHAPTER 3. Steady state vehicle behaviour

Event: acceleration event skidpad event autocross event endurance eventlongitudinal requirement 100% 0% 20% 30%lateral requirement 0% 100% 80% 70%

It appears that the vehicle requires more lateral potential than longitudinal. The focus is put on thelateral performance for 70% since more points can be earned with the autocross and even more withthe endurance event. The acceleration and skidpad event weigh even. Both the mass and centre ofgravity sensitivities are calculated according to the next equations:

Masssensitivity =∆ax∆m

· 0.3 +∆ay∆m

· 0.7 (3.8)

CGsensitivity =∆ax∆h· 0.3 +

∆ay∆h· 0.7 (3.9)

Figure 3.13 illustrates the results of the sensitivity analysis for the mass (figure 3.13a) and the centre ofgravity height (figure 3.13b). There is only one side of the lateral acceleration plotted since the vehicleis symmetric in its lateral behaviour. The baseline vehicle is denoted with [REF], but the centre ofgravity height is taken lower than denoted in section 3.4. This is done to prevent the model lifting atyre from the ground which causes the results to be unfair to compare. The curve with a centre ofgravity height of 0.05 [m] in figure 3.13b shows a little deviation around ax = 0. This is caused by aroll stiffness ratio which is not perfectly balanced to neutral steer. The peak would be a little higherwhen this ratio is corrected, but then would a deviation originate at the other curves.

−2 −1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

Ax [g]

Ay

[g]

mass 280 [kg]mass 300 [kg] [REF]mass 320 [kg]

(a) Mass sensitivity

−2 −1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

Ax [g]

Ay

[g]

CG height 0.05 [m]CG height 0.10 [m] [REF]CG height 0.15 [m]

(b) Center of gravity height sensitivity

Figure 3.13: Parameter sensitivity

It appears that the sensitivity for the braking and accelerating side is different, which is caused bythe fact that the vehicle is two wheel driven and brakes on four wheels. The sensitivity calculation isadapted with a separated longitudinal part for braking and driving. The sensitivity values for mass andcentre of gravity height are:

35

3.11. Optimization of the steering angles CHAPTER 3. Steady state vehicle behaviour

Masssensitivity = 0.0014 · 0.15︸︷︷︸braking

+ 0.0012 · 0.15︸︷︷︸accelerating

+ 0.003 · 0.7︸︷︷︸cornering

= 2.49 · 10−3 [g][kg]−1 (3.10)

CGsensitivity = 0.01 · 0.15︸︷︷︸braking

− 0.025 · 0.15︸︷︷︸accelerating

+ 0.02 · 0.7︸︷︷︸cornering

= 11.75 · 10−3 [g][cm]−1 (3.11)

The values are distracted from the figures 3.13a and 3.13b. The values for the mass sensitivity arenormalized in [g][kg]−1 and the values for the centre of gravity height in [g][cm]−1. Lowering thecentre of gravity by one centimeter has an almost 5 times higher positive influence than lowering thevehicle mass with one kilogram. Decreasing the centre of gravity is much easier done for the designof the URE04 vehicle since the engine, driver and spring/damper systems can be placed a lot lower.Reducing the vehicle mass with a large amount is much more difficult when the current concept ofchassis design is maintained. The design strategy is therefore focussed on lowering the centre ofgravity height. The design aim is to lower the centre of gravity height with an estimate of 6 [cm] bylowering the engine, driver, the front spring/damper system, the powertrain and the ride height. Thevehicle mass can also be decreased by an estimate of 10 [kg] through by more components with theuse of FEM analysis and careful picking of vehicle components and materials.

3.11 Optimization of the steering angles

The basic model has just one input variable for the steering angle. Both the front wheels are usingthis angle, which results in a parallel steering linkage. Most passenger cars and trucks make use of aan Ackerman geometry, which defines the relation between the left and right steering angle. A specialcase is the Ackerman condition [13]. This condition is true when the left and right steer angle matchesthe path of the cornering radius of that wheel. This is only valid at very low speeds. This Ackermancondition is, for example, applied on trucks to reduce tyre wear.

The aim when developing a race car suspension is the optimize its performance. All tyres have togenerate as large lateral forces as possible. This is almost impossible to calculate by hand. The modelhas the capability to optimize the front left and front right steer angle independently to optimizethe generated lateral forces which should result in a better performing vehicle. It is also possible tooptimize the steering angles of the rear wheels also. The next results (calculated for a cornering radiusof 100 [m]) depicts the optimization of the individual front steering angles:

36

3.11. Optimization of the steering angles CHAPTER 3. Steady state vehicle behaviour

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.5

1

1.5

2

2.5

3

3.5

4

Ay [g]

δ [d

eg]

δ1

δ2

δ REF

(a) Optimal steering angles

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−8

−7

−6

−5

−4

−3

−2

−1

0

1

Ay [g]

α [d

eg]

alpha1alpha2alpha1 REFalpha2 REF

(b) Handling diagram

Figure 3.14: Side slip angles

The dotted lines indicated withREF represent the base line vehicle with parallel steering geometry. Itcan be clearly seen in figure 3.14a that the steering angle of the outer tyre (δ2) is much larger than theinner one (δ1), see also figure 3.14b which illustrates the difference in side slip angles. This indicatesthat the optimal geometry would not be a parallel steering geometry but reverse Ackerman.

The vehicle is a little more understeered then when parallel steer is used, see figure 3.15. The max-imum lateral performance increases with 0.03 [g]. In reality is it almost impossible to design suchmechanical steering linkage. The reason for this is that the difference in steering angles has to bearchived at small steering angles, but at some points the steering angle has to be as high as 28 degreesto cop with very narrow and slow corners. The steering angles between the left and right wheel atsteering lock would deviate to much and one will collide with the suspension linkage. An electronicactuated steering mechanism could be an answer to this problem, but regulations permit steer by wiresystems on the front axle.

−3 −2 −1 0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

α [deg]

Ay



Figure 3.15: Optimal steering angles and corresponding handling diagram

37

3.12. Optimal tyre inclination angle CHAPTER 3. Steady state vehicle behaviour

3.12 Optimal tyre inclination angle

The model has also the possibility to incorporate the tyre inclination angles as optimization variablesto calculate the optimal tyre inclination angle for every combination of longitudinal and lateral accel-erations. This method is not used since the model generates very large and unrealistic inclinationangles (>40 [deg], see figure 3.16a). The reason for this is that it is numerically beneficial. The usedtyre model is transformed to MATLAB code and does not take a maximum valid inclination angle intoaccount. At low side slip angles, between 1.5 and 2.1 [g], is the result valid. When the vehicle reachesit limit the side slip angles would increase and it appears that it would be numerically beneficial tohave extremely large inclination angles. It is obvious that these results could not be used to design thesuspension kinematics. Therefore the optimal tyre inclination angles will be calculated on a differentway.

0 0.5 1 1.5 2 2.5−50

−40

−30

−20

−10

0

10

Ay [g]

γ [d

eg]

γ1

γ2

γ3

γ4

(a) Optimal tyre inclination angle (optimization)

−15 −10 −5 0

−10

−8

−6

−4

−2

0

side slip angle α [deg]

γ [d

eg]

(b) Optimal tyre inclination angle (single tyre model)

Figure 3.16: Optimal tyre inclination angles

A single tyre model is used to calculate the optimal angles. The tyre is exposed to a range of combina-tions of tyre loads, side slip angles and camber angles. The wheel load range is applied to a fixed sideslip angle and a fixed camber angle. The generated lateral forces are added up. This is done for thewhole range of side slip angles and camber inclinations. The curve with the highest cumulative lateralforce is decisive for the choice of a camber angle. Figure 3.16b shows the optimal camber angles forthe range of side slip angles. It appears that for small slip angles the camber angle should be negative.But at higher slip angles, which in fact are of importance in limit handling situations, it appears thata camber angle of 0 [deg] is preferable.

This is a quite strange result since most commonly used passenger car tyres prefer a slightly negativecamber angle. These results are not verified with another measurement due to time limitations andare assumed valid in the further design of the suspension. The aim is to reach zero camber at alltimes.

The physical explanation why a camber angle should be advantageous lays in the contact patch pres-sure distribution. When the tyre (at rest) would be inclined the tyre contact patch pressure distri-bution would be disturbed which results in a lower friction coefficient. But when the tyre deformsduring cornering the contact patch will be stretched out. This deformation restores the contact patchload distribution which increases the friction coefficient. At large slip angles this effect can cause thecontact patch pressure distribution to be lower at the outside of the tyre, which can be an explanationfor the above results. Restoring the contact patch pressure can then be accomplished by lowering theinclination angle of the tyre.

38

Chapter 4

Kinematic suspension design

4.1 Introduction

This chapter describes the development of a set of design tools and the mathematical background todesign a multi-link suspension. A multi-link suspension is one of the most complex designs and isrelatively new. The design is used in passenger cars since the early eighties. The layout and advan-tages of a multi-link suspension are discussed first. Several design tools are developed to design thissuspension. These tools are: a kinematic multi-body model, a collection of mathematical equations tocalculate suspension characteristics, a numerical optimization tool to compute the optimal kinematicdesign, a dynamic vehicle model to validate the suspension design and a tool to visualize suspensioncollisions. This chapter discusses the design tools in the denoted order. The suspension design targetsand results will be discussed in the next chapter.

4.2 Multi-link layout

A multi-link (or 5-link) suspension is characterized by its large kinematic design freedom and is oftenreferred to as an evolution of the double wishbone suspension. The layout of a multi-link suspensiondiffers from a double wishbone layout by the decoupling of the connection rods at the upright side.Figure 4.1a illustrates a double wishbone layout, where 4.1b illustrates a multi-link layout. The decou-pling of the connection rods implies that the two upper and lower connection rods do not define aplane anymore. This has major consequences for the involved mathematics and kinematic behaviour.

(a) Double wishbone layout (b) Multi-link layout

Figure 4.1: Suspension layout

39

4.2. Multi-link layout CHAPTER 4. Kinematic suspension design

The kinematic behaviour of a multi-link suspension follows from the connection of the upright tothe chassis with six connection rods. Every connection rod has 2 joints which can be placed arbitraryin space with 3 coordinates each. This results is a total of 6 · 2 · 3 = 36 degrees of freedom on onewheel (assumed that that car is left/right symmetric). The front and rear suspension have in total 72degrees of freedom. These freedoms are restricted by several design boundary’s, such as the availablespace. The combination of the number of degrees of freedom and the design boundaries implies avery complex design assignment. Such a complex design demands a thorough approach.

A lot of passenger cars nowadays are equipped with a multi-link suspension. The most importantreason for this is that the engineers at car manufactory companies have to compete for the availablespace inside a passenger car chassis. The car is better sellable when more space is available for luggageand an engine. The lack of space demands that the suspension design is capable of delivering goodhandling performance, comfort, low noise and low cost. A well designed multi-link suspension iscapable of doing this. But it is probably one of the most expensive suspension designs also. The lowend class of passenger cars is therefore often equipped with a more conventional suspension design,such as a Mc Pherson strut design or a twist beam suspension [14].

The most common suspension design for racing cars these days is the double wishbone suspension.This suspension design delivers a lot of kinematic freedom combined with a low (unsprung) weight.Space is much less an issue compared with passenger cars. A multi-link suspension is often notused for racing purposes because of its complexity and the engineering time needed to design thissuspension.

The next list summarizes the advantages and disadvantages of a multi-link suspension:

Advantages

• large kinematic design freedom

• high longitudinal and lateral stiffness

• low weight

• no buckling forces in connection rods

• large brake system design freedom

• larger possible steering angle

• better steering returnability control by screw axis

• possibility to design a negative scrub radius improving µ-split behaviour

Disadvantages

• complex design

• larger number of suspension parts

• expensive

• more difficult to do suspension adjustments

• slightly heavier than a double wishbone design

The advantages of a multi-link suspension above that of a double wishbone suspension result in thechoice of using a multi-link suspension, even when the extra complexity is regarded.

40

4.3. Design tools CHAPTER 4. Kinematic suspension design

4.3 Design tools

The industry has developed several tools to design suspension systems, such as ADAMS Car andIPG K&C. Some of these tools are rather simple and can only calculate kinematic behaviour of asuspension. Others can perform fully dynamical vehicle simulations, which include tyre models andnon-linear compliance calculations. The problem with these software tools is that they lack insight andflexibility. Most of them are designed as a ’black box’, where most of the mathematics and definitionsare invisible. Another problem is that it is not possible to design additional tools around this software.

As already discussed earlier, a multi-link suspension is one of the most complex suspension designs.With its 36 degrees of freedom and a lot of design boundaries, it is virtually impossible to design suchsystem with manual design iterations. Today a lot of numerical optimization techniques are availableto tackle this kinds of problems. Most commercial suspension packages do not incorporate thesetechniques.

The shortcomings of commercial packages require a more radical approach. MATLAB is used to de-velop several suspension design tools, because it has the capability to tackle these shortcomings. Everypossible calculation of suspension parameters can be implemented. This ensures a good insight inthe definitions of suspension parameters. MATLAB is very flexible since it comes with a multi-bodytoolbox. The toolbox can be used to do a kinematic suspension analysis. It is also possible to do afully dynamical vehicle analysis, since the TNO tyre model [6] can be implemented in the multi-bodymodels. MATLAB has also an extensive numerical optimization toolbox. Given these advantages, thechoice is made to use MATLAB to develop the suspension design tools.

To design the suspension several tools can be distinguished. The first one is the kinematic multi-body model. This model is used to investigate the kinematic behaviour. This model is developedin SIMMECHANICS , which is the multi-body toolbox of MATLAB . The second tool is the collectionof mathematical equations which are used to calculate suspension parameters. The third one is thenumerical optimization toolbox, which is used to calculate the optimal suspension geometry givencertain objectives and boundaries. The fourth is a visualization of the movement of the suspension ina virtual reality environment. This tool is used to investigate suspension collisions. The fifth and lasttool is a fully dynamical vehicle model, including the tyre models. This tool is primary used to validatethe vehicle behaviour resulting from the suspension kinematics.

4.4 Kinematic suspension model

The base of the suspension development is the kinematic suspension model. This model is madewith MATLAB SIMMECHANICS . The model has several inputs to actuate the suspension system. Inthe analysis can 2 movements be distinguished, namely the bump movement of the suspension andthe steer movement. Both are needed to evaluate the suspension characteristics. This is discussed inmore detail in section 4.5.

The upright is connected to the chassis with connection rods (suspension links) and ball joints. Thesecomponents restrict (constrain) the movement of the upright. A multi-link suspension is part of thespatial suspension geometries. This means that the movement of the components are individually3D related. It belongs to the independent suspension layouts since it has only one bump degree offreedom, where rigid-axle suspension for example have two bump degrees of freedom (symmetric andasymmetric bump movement).

A multi-link suspension consists of 4 connection rods, 1 steer rod (front) or 1 tie rod (rear), and 1 pull-or push rod. Figure 4.2 illustrates this is a block diagram.

41

4.4. Kinematic suspension model CHAPTER 4. Kinematic suspension design

Figure 4.2: Block diagram of a multi-link suspension layout

Depending of the movement of the upright (bump or steer) one of the connection rods has to beomitted. The multi-link suspension under investigation is connected to the chassis and upright withball-joints. These joints constrain a movement with 3 translations and leave 3 rotation degrees offreedom available [15], see figure 4.3.

Figure 4.3: Spherical joint with 3 rotation degrees of freedom

This results in a kinematic chain which constraints the upright movement. Being all spatial bodies, theupright and each connection rod possess six degrees of freedom in a 3D space. This results in a total of6 bodies: 5 fixed connection rods and 1 upright. The total degrees of freedom equals ftot = 6 · 6 = 36.A spherical joint reduces the total degrees of freedom by fspherical = 3. Each individual rotation of aconnection rod around its own axis, r, does not contribute to the total amount of degrees of freedom.With g joints, the balance of degrees of freedom of the suspension results in:

DOF = ftot − r − g · fspherical = 36− 5− 10 · 3 = 1 (4.1)

Depending on the omitted connection rod this results in 1 degree of freedom for the bump or steermovement [14]. This characterizes an independent suspension.

Bump movement The chassis is fixed to the world when the bump movement is analyzed. One linkbetween the chassis and the upright has to be removed, otherwise will the system be fully determined.Figure 4.4 shows the omitted push/pull rod, which makes it possible to make a bump movement.

42

4.4. Kinematic suspension model CHAPTER 4. Kinematic suspension design

Figure 4.4: Omitted push-pull rod

The wheel can now be actuated in the z-direction. The chassis is fixed to the world and the wheelhas only one degree of freedom left (bump). Since this results is a spatial movement of the wheel inspace it to be connected to an actuator with 5 degrees of freedom with the world. The constraint is thetranslation in z-direction. The rotation around the wheel rotation axis is also blocked to prevent therotation of the wheel around this axis during the bump movement.

Steer movement Another rod has to be removed to analyse the steer movement. Figure 4.5 illustratesthis.

Figure 4.5: Omitted steer rod

This time the steer rod is omitted, which results in the steer degree of freedom. The actuator whichmoves the wheel is also removed because the steer movement is actuated on another way. In fact theupright is actuated by the translating steer rack in y-direction. The steer rod is therefore not reallyomitted but used as the link between the steer rack and the upright. The rear suspension is alsoequipped with a steer rack to analyse the rear suspension.

Since two movements have to be evaluated for the front and rear suspension, four multi-body modelsare required to do a full suspension analysis.

43

4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

4.5 Suspension kinematic characteristics

The second design ’tool’ is a collection of mathematical equations which are used to calculate thesuspension characteristics. Several outputs of the kinematic suspension model are used as the inputfor the equations. This section describes the definitions and equations of the suspension kinematicsin detail. All calculations are done according to the ISO8855 standard [1] and can be applied to thefront and rear suspension.

Most of the discussed suspension parameter calculations are based on the principle of virtual work[14]. Velocity vectors can easily be determined in multi-body models and can often be used to derivesuspension characteristics.

Wheel orientation angles

The orientation angles of a tyre are defined according to a rotation sequence. This has to be taken intoaccount when calculating the exact orientation angles. ISO defines the rotation sequence according totable 4.1.

Rotation order Produced tyre angleFirst steer angle (δ)

Second inclination angle (γ)Third wheel rotation angle (ω)

Table 4.1: Rotation sequence tyre

The rotation of an object in the multi-body model is represented by a rotation matrix consisting of 9values denoted in a matrix form. To calculate the tyre orientation angles the inverse of the rotationmatrix has to be calculated, which is named the direction-cosine matrix [16]. This new matrix consistsof the multiplication of three elementary matrices. Since the rotation sequence for the calculation ofthe tyre orientation angles is 3-1-2 these matrices are denoted by:

A =A43(ω)A32(γ)A21(δ)

=

cosω 0 −sinω0 1 0

sinω 0 cosω

1 0 00 cosγ sinγ0 −sinγ cosγ

cosδ sinδ 0−sinδ cosδ 0

0 0 1

=

cosδcosω − sinδsinγsinω sinδcosω + cosδsinγsinω −cosγsinω−sinδcosγ cosδcosγ sinγ

cosδsinω + sinδsinγcosω sinδsinω − cosδsinγcosω cosγcosω

The tyre orientation angles can be calculated according to:

γ = arcsin (A(2, 3)) (4.2)

ω =arcsin (−A(1, 3))

cosγ(4.3)

δ =arcsin (−A(2, 1))

cosγ(4.4)

44


A same sort of calculation can be used to calculate the chassis orientation angles. The rotation se-quence of the chassis angles are defined different but the principle is the same, see table 4.2. Thesame yields for rotation velocities and accelerations.

Rotation order Produced chassis angleFirst yaw (ψ)

Second pitch (θ)Third roll (φ)

Table 4.2: Rotation sequence chassis

Instantaneous screw axis

The upright will rotate around an instantaneous virtual axis in space depending on the applied actu-ation on the upright. This axis is an imaginary one and is instantaneous, meaning that it migratesduring the movement of the suspension. Two instantaneous virtual axis can be defined. The first oneis the virtual steer axis, which is linked to the steering degree of freedom of the upright. The secondone is often called the instant axis, which is linked to the bump movement of the upright. Both areused to calculate several suspension characteristics.

These axis’s could be graphically constructed when the suspension layout is a double wishbone sus-pension. But such an axis of a spatial multi-link suspension behaves differently. The geometry cannot be used to construct this axis graphically. The theory mentioned is [14] is used in the remainder ofthis section.

The instantaneous axis can be found using the angular velocity ωK of the upright and the velocityvector vM of a reference point on the upright. The reference point can be arbitrary but it is preferableto use the wheel centre. With the given vectors ωK and vM the state of motion of the spatial suspensioncan be defined. The vector ωK defines the orientation of the instantaneous axis. Equation (4.5) is usedto calculate the vector pointing to the instantaneous axis. Via this method it is possible to calculateboth the instantaneous steering axis and the instantaneous bump axis.

rIA = rM +ωK × vM|ωK |2

(4.5)

The instantaneous axis of a multi-link suspension acts as a screw. It has a shift velocity along that axis.This means that the axis is shifted with a certain velocity along that axis, like a bolt on a thread will do.In contrast with the instantaneous axis of a double wishbone (or even less complex suspensions), thescrew axis of a spatial multi-link suspension is not suitable for a reference axis to calculate momentsand forces. Since the screw axis shows an axial shift coupled with a rotation, forces that act on thescrew axis that are not perpendicular applied to it will introduce a torque on that axis. The intersectionpoint of the instantaneous axis with the side and front plane would normally be used to calculatedifferent suspension parameters. This is not possible here since the screw axis can not be used to doso. Figure 4.6 illustrates the instantaneous screw axis of the front left upright in green.

45


Figure 4.6: Instantaneous screw axis

This screw axis will intersect with the x-z plane and the y-x plane. Both planes cross the wheel contactpoint. In the next discussion the side plane (x-z plane) will be used. The screw axis has an axial shiftvelocity along the axis and a rotation around that axis. The intersection point with the plane wouldtherefore have a velocity in both the x and z direction in that plane. This point can therefore not be usedas a pole to calculate forces and moments around it. However, there is always, in every possible plane,a point that is stationary (meaning: having no velocity) at a particular moment. This is visualized infigure 4.7.

Figure 4.7: Instantaneous screw axis with helix

The axial shift velocity is represented by a constant helix in the figure. This is only done for visualiza-tion purposes because the shift velocity is not constant along the axis. The helix crosses the side planeon a point which is stationary when the screw axis would rotate and shift. Stationary means that it has

46


zero velocity in the plane, in this case in the x and z-direction. This is the point which can be used tocalculate the forces and moments. This pole and the pole in the front plane define a new axis (in red)which is used to calculate the suspension parameters. The vector pointing to the new pole is definedwith ax, ay and az :

vIA = vM + ωK × (rIA − rM ) (4.6)

ax =vIA,zωKy

ay = 0 az =vIA,xωKy

(4.7)

The side pole (termed the anti centre, AC) can be calculated using the vector a which points fromthe instantaneous screw axis pole (green) to that pole (red) [14]. The location of that vector in the y-direction is constant and the velocities of the pole AC in x and z direction have to be zero. This leadsto (4.7). The pole AC can easily be calculated by adding the vector a to the vector of the first pole. Thevelocity of the screw axis can be calculated with 4.6.

The same procedure is used to calculate the pole in the front view, the instant centre (IC) pole:

ax = 0 ay =−vIA,zωKx

az =vIA,yωKx

(4.8)

This principle can also be used to calculate the instantaneous steering axis.

Side view support angles

The side view support angle, θ, controls the pitch motion of the chassis when accelerating or braking,see figure 4.8. The support angle follows from the side view pole which is called the anti centre, AC.This pole is called anti centre because it defines several anti features of the suspension linkage. Thesefeatures are known as anti-dive, anti-rise, anti-squat, anti lift, castor change rate and wheel path.

These features describe the amount of force coupling between the longitudinal (braking/driving) tovertical forces. A part of the forces is transmitted trough the push/pull rod, the elastic part. Theremaining part is transmitted trough the other connection rods and is called the direct part becausethe springs and dampers are not actuated by this part. The longitudinal forces are totally transmittedby the suspension links, and not trough the springs, when the value of an anti feature is 100%.

Anti-dive is defined as the amount of longitudinal force directed trough the suspension links of thefront suspension during braking. A front wheel driven vehicle has also an amount of anti-rise whichis generated during accelerating. Anti-lift is the amount of longitudinal forces that is transmittedthrough the rear suspension links during braking and anti-squat during accelerating.

An extreme example of anti-lift is pro-lift. The amount is then larger than 100%. This is applied to therear swing arm of motorcycles to increases traction during accelerating.

The URE04 vehicle is designed with 4 outboard brakes and rear wheel drive. The feature anti-risecan not be used since the front wheels are not powered by the engine. The anti features anti-dive andanti-lift are dependent on the brake force distribution, p, between the front and rear axles.

Figure 4.8 illustrates the support angles, denoted with θ, which results from the anti centers, ACf andACr.

47


h

l

θad

θasACf

ACr

θal

ACf,x

castor change

wheel path

ACf,y

Figure 4.8: Side view support angles

The anti-dive percentage at the front is calculated with (4.9). The derivation of the equations can befound in [11] and [14].

Anti− dive =tan(θad)h/(lp)

× 100% (4.9)

Anti-lift and anti-squat on the rear are calculated according to:

Anti− lift =tan(θal)

h/(l(1− p))× 100% (4.10)

Anti− squat =tan(θas)h/l

× 100% (4.11)

The wheel path is defined as the amount of longitudinal translation of the tyre contact point during itsbump movement. The amount of wheel patch is calculated according to: tan(θad) and tan(θal) andhas therefore an analogy with the anti features.

The longitudinal coordinate of the anti centre, ACf,x, defines the castor change rate. The castorchange rate is defined as the change of castor angle divided by the change of bump travel, dτdz .

Front view support angle

The front view support angle controls the roll motion of the vehicle during cornering. It relates thelateral force with vertical force transmitted through the spring/damper and the chassis. The supportangle is calculated from the instant centre (IC).

A commonly used tool in the design of suspension systems is the roll centre height. This variablewill not be used in the remainder of the report. It only defines the chassis rotation point at the initialroll movement of the chassis during cornering. At high lateral acceleration levels this point is uselessbecause the generated forces by the left and right tyre differ a lot. Therefore is the design focussed onthe transmission of lateral forces to the chassis and not on a roll centre height.

The support angle will, on the same way as in the side view, generate an anti effect, see figure 4.9.

48


Figure 4.9: Front view support angle

This effect relates the amount of lateral force at the tyre contact point that is transmitted tot the chassisto generate a roll moment and is called anti-roll. It can be calculated according to:

Anti− roll =tan(θar)h/t

× 100% (4.12)

The location of the instant centre pole IC defines, on the same way as the castor change rate andwheel path, the camber change rate and the scrub. Scrub is the amount of lateral translation of thetyre contact point at suspension travel (bump). Both parameters are directly measured in the kinematicsuspension model, rather than calculated according to the location of the instant centre.

Since both tyres transmit a force to the chassis, and both forces are different during cornering, it isdifficult to analyze the anti-roll behaviour. The instant centers will migrate to other positions in spaceduring wheel travel. This amount and direction of migration is interesting since it defines the amountof lateral force which is translated to vertical force during cornering. The vertical force causes thechassis to rise of fall and is called the jacking force. Figure 4.10 illustrates the migration of instantcenters. Both (illustrative) paths are depicted with dash-dotted lines. This migration happens becausethe chassis is rolling and deflects the suspension geometry. The lateral tyre forces have to be be applieddifferently to the chassis when the vehicle is subjected to a lateral acceleration in comparison with theneutral position of the vehicle.

Figure 4.10: Instant centre migration

49


In this example the left instant centre, ICl, migrates beneath the road surface. The resulting vertical(jacking) force will therefore be negative. The right tyre generates a much smaller lateral force in thesame direction. This forces is ’pulling’ on the vector pointing to the instant centre. This contributesalso to a negative jacking force. The combination of instant centre migration paths results in a nettovertical force. Some people in the past claimed to have a method which could be used to calculatethis jacking force and to design a suspension which will not have any jacking effect at all, see [17], [18]and [19]. In reality it is impossible to calculate this because the lateral tyre forces are behaving verynon-lineair at high acceleration levels. Another problem is that the spring/damper configuration isoften behaving in a non-lineair way too.

The migration of instant centers in the suspension design is investigated by the change of the anti-rollpercentage. The amount of jacking is validated with dynamical simulations, this is described in moredetail in section 5.5.

Steering characteristics

The spatial characteristic of a multi-link suspension introduces another way of calculating the conven-tional steering characteristics. The derivations of the equations given in the remainder of this sectioncan be found in [14]. The steer angle, δ, is often used in these equations and is given by 4.4. Figure 4.11illustrates these conventional steering characteristics. In this case of a double wishbone suspension,where the upright is ’fixed’ to the connection rods with single ball joints.

Figure 4.11: Conventional steering characteristics

The kingpin inclination, denoted with σ, is inclined with respect to the vertical z-axis in the y-z plane.The castor angle, denoted with τ , is inclined with respect to the vertical z-axis in the x-z plane. Thisdefinition still holds when the wheel is steered. This means that they are calculated in the cross sec-tional view of the chassis. Both angles are responsible for the camber change relative tot the steeringangle. This will be discussed later.

The kingpin, or steering axis, d, intersects the road surface at point D. The lateral projected distancefrom this point till the point A, the tyre contact point, is defined as the scrub radius, or kingpin offset,rs. In the side view, the distance betweenD andA is defined as the castor offset, n. The correspondingdistances measured at the height of the wheel centre , M , are the wheel centre offset, rc and the castoroffset at wheel centre, nτ . These characteristics refer to the wheel coordinate system, unlike the castorand kingpin angle which are defined is the cross section view of the chassis.

50


The steering characteristics of a spatial suspension system such as a multi-link can not be calculatedby the geometrical projection of the suspension linkage on the cross sectional plane in side and rearviews. The reason for this is, as already discussed before, that the virtual steering axis an instantaneousscrew axis is. This axis cannot be considered as a reference axis for forces and moments in the sameway as the instant axis cannot be used to do so. Therefore an alternative method is used.

The angular velocity vector ωK of the upright and the velocity vector vM of a reference point, M , onthe upright can be used to calculate the steering characteristics.

The kingpin inclination and castor angles can be calculated using the angular velocity of the uprightaccording to:

σ = −arctan(ωKyωKz

)(4.13)

τ = −arctan(ωKxωKz

)(4.14)

The steering velocity, ωδ , around the virtual steer axis is often used to calculate the steering character-istics. It is defined by:

ωδ = ωKz ((tan(τ)sin(δ)− tan(σ)cos(δ)) tan(γ) + 1) (4.15)

It appears that this velocity is only equal to ωKz when the camber angle γ is zero or when the steeringangle is equal to δ = arctan (tan(σ)/tan(τ)). The deviation between the z-component of the uprightangular velocity and the steering velocity is often smaller than 2%. ωKz could be used instead of ωδ ,but for the sake of completeness ωδ is used.

Another velocity vector which is used to calculate the steering characteristics is the virtual velocity ofthe tyre contact point, vA (see figure 4.11). This point is imaginary fixed to the upright and thereforecalled virtual. The velocity at the tyre contact point can be calculated with:

vA = vM + ωK × rMA (4.16)

Where rMA the vector pointing from the wheel centre M to the virtual tyre contact point is.

The scrub radius of a multi-link suspension, which corresponds in magnitude with the scrub radiusof a conventional suspension, can than be calculated with:

rs =− (vAxcos(δ) + vAysin(δ))

ωδ(4.17)

The scrub radius can be used to calculate the influence of braking and driving forces on the steeringwheel. These forces generate a moment around the virtual steer axis which is felt by the driver whenmoment on the left steer axis does not correspond with the moment around the right steer axis.

51


However, the definition is given for the moment around the global z-axis which originates at thepoint where the virtual steering axis (screw axis) crosses the road surface. This is done to correspondto the definition used for conventional suspensions.

When the wheel is driven with a drive shaft, the wheel centre offset rc would be regarded as the leverarm which generates a moment depending on the traction force. The wheel centre offset has a stronganalogy with the scrub radius. It is calculated with:

rc =− (vMxcos(δ) + vMysin(δ))

ωδ(4.18)

The castor offset (sometimes referred as mechanical trail) generates a moment around the steeringaxis during cornering. The pneumatic trail, denoted with nr is figure 4.11, adds up with this value.The castor offset can be calculated according to:

n =(vAxsin(δ) + vAycos(δ))

ωδ(4.19)

The castor offset at wheel centre, nτ can be calculated according to:

nτ =(vMxsin(δ) + vMycos(δ))

ωδ(4.20)

A less common suspension parameter is the wheel load lever arm which is defined as the ability torestore the steering angle at a certain wheel load with the lever arm w:

w = −vAzωδ

(4.21)

The kingpin inclination σ and castor angle τ are, as already mentioned, responsible for the change ofcamber angle when steering. This relation is defined according to:

dγ

dδ=

tan(τ)cos(δ) + tan(σ)sin(δ)tan(γ) · (tan(τ)sin(δ)− tan(σ)cos(δ)) + 1

(4.22)

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4.6. Numerical optimization CHAPTER 4. Kinematic suspension design

4.6 Numerical optimization

The kinematics of a suspension concept are normally designed according to a kinematic design target.This design target consists of an objective for the calculated suspension parameters. This can beaccomplished by relocation the suspension coordinates. A multi-link suspension has, as discussedearlier, 36 degrees of freedom. The design target consists often of less objectives. But the availablespace and vehicle geometric layout restricts the range of adaptation of a degree of freedom. Figure 4.12shows the rear space frame of the vehicle which restricts the placement of the chassis points strongly(red circles).

Figure 4.12: Space restriction rear suspension

Another problem is that the suspension characteristics have mutual dependencies. This complexityrequires a different design approach than used previously to design a double wishbone suspension.The design approach of a double wishbone, or an even more conventional suspension layout, is oftenbased on graphical methods and manual iterations to accomplish the design objectives. Section 3.6already discussed a method to handle more complex problems. Numerical optimization techniquesare a very suitable method to do this.

Unconstrained minimization

The suspension design has a lot of geometrical constraints. For this reason it looks obvious to de-fine a constraint optimization problem [20]. A constraint optimization problem is characterized as aproblem where the optimization variables (inputs) can not be freely and independently chosen. Math-ematical equations restrict the choice of inputs. This can be expressed as equality or inequality con-straints. An equality constraint has to be exactly fulfilled by the optimization algorithm. This couldbe, for example, a constraint which says that the z-coordinates of two suspension points have to beequal. An inequality constraint has to be equal or smaller than zero. An inequality constraint can, forexample, be used to constrain the suspension coordinates in such a way that a connection rod wouldnot intersect with the wheel rim. It sounds obvious to do this, but this would be mathematically verycomplex. Especially since the suspension is moving in 3d space regarding two states of motion (bumpand steer).

53


It is also possible to set a boundary for the suspension coordinates (input variables) when using anunconstrained optimization proces. This simplifies the definition of the optimization proces a lot. Theboundary sets a valid region for the suspension coordinates. This defines a box shaped boundary forevery suspension point since a suspension point consists of 3 degrees of freedom (3 translations in x,yand z direction). This means that every variable can still be chosen independently. This could result ina design which does not satisfies the space restrictions. A new design iteration has to be done whenthis happens.

Two multi-variable algorithms are compared to find the most effective one. The first algorithm iscalled the Broyden-Fletcher-Goldfarb-Shanno (BFGS, MATLAB : fminunc) method and the second thegeneralized pattern search (GPS, MATLAB : patternsearch) algorithm which was also used in section3.6. The last one appears to be in most cases the fastest and most robust algorithm since it does notdepend on local gradients. The working principle behind it is discussed in appendix B.

Optimization proces example

The optimization proces starts by defining the inputs. These are the 36 suspension coordinates withthe corresponding upper and lower boundaries. The starting points are chosen by averaging the upperand lower boundaries.

Figure 4.13 illustrates the calculation sequence of the optimization proces.

Figure 4.13: Optimization calculation sequence

The optimization routine initiates both multi-body kinematic models (bump and steer) and calculatesthe suspension characteristics discussed earlier. These characteristics are used for the objective func-tion. Most suspension characteristics used in the objective function are compared for the whole rangeof the suspension movement and not only as the initial static value. The objective function consists ofthe following suspension characteristics:

54


suspension characteristic objective weighting factor(priority)

bump steer toe curve versus bump travel highcamber inclination angle zero at all times highanti roll slightly negative and progressive with bump high

travel to reduce jacking (validation withdynamic vehicle model)

anti dive (front), or constant value along middleanti squat and rise (rear) the bump travel rangescrub radius constant value along the steering range highcastor offset constant value along the steering range highcastor offset at wheel centre zero at all times lowwheel centre offset zero at all times lowcamber change with high at the static middlesteer angle position and degressive with steer anglesteering ratio 28 deg steer angle by 120 deg high

steering wheel angleinstant center location further than 1 meter low

away from the vehicle center lineto reduce scrub

anti center location depending on the front lowor rear suspension to reduce thecastor change rate and wheel path

wheel load lever arm close to zero to reduce middlesteering wheel torque, validatedwith dynamic vehicle model forsteering returnability

The objective values are not mentioned exactly in the table, and most consist of a curve which isdependent either on the bump travel or steer angle. The objectives and results are discussed in detailin chapter 5.

Not every characteristic is equally important. The progress is monitored during the optimizationproces by evaluating the static values of the characteristics. This is done to make it possible to tunethe weighting factors more easily. The above table includes a priority as the weighting factor. Inreality is the weighting factor a value, but to compare the objectives in a table a priority is showed.Most objectives indicate an optimization direction for the algorithm. It is not possible to fulfill everyobjective exactly since the number of optimization variables (suspension coordinates) is limited andrestricted by boundaries.

The optimization proces changes the input variables iteratively by the proces mentioned in appendixB to accommodate the objectives as close as possible. The end criterium is satisfied when the decreaseof the objective value is to small. The results will be plotted when the routine is ended to validate them.The result of the optimization proces is not guaranteed to be valid to fit both the suspension targetsand the space boundaries. Therefore will it be necessary to adjust the input boundaries, weightingfactors and suspension targets during the design proces.

One of the problems which can occur during the optimization proces is that the kinematic model isaborted because it runs into a mechanical lock. This can happen when a combination of suspensioncoordinates causes the linkage to lock. This is not noticed by the optimization routine because itsimply continues to change the input variables. This problem is solved by adding an objective whichsays that the simulation time performed has to be equal to the simulation time chosen.

55

4.7. Dynamic vehicle model CHAPTER 4. Kinematic suspension design

4.7 Dynamic vehicle model

A dynamic vehicle model is developed to validate the suspension behaviour. The model has also thepossibility to analyse dynamic vehicle behaviour and to determine suspension forces. The suspensionforces can be used to design optimal vehicle components with the use of finite element methods.

The dynamic vehicle model is modeled in the multi-body package of MATLAB which is called SIMME-CHANICS . This means that most elements of the model act as rigid bodies except the spring elements.This does not represent the reality exactly, but since most components are designed with a very highstiffness in mind it is valid to assume infinite stiff bodies.

The concept of the model is kept rather simple since it is primary used to validate the suspensionbehaviour. Aspects such as a driver model and aerodynamics are neglected. The model consists of 7major parts:

• chassis

• 4 wheel/upright subsystems

• drive line subsystem

• visualization subsystem

The model is equipped with several subsystems which are used to control the model. It is possible tochange static model parameters such as suspension settings and spring and damper parameters viaa model data file. The model is equipped with three time dependent inputs: the longitudinal speed,braking and steering.

The speed of the vehicle in controlled with a simple cruise control. The longitudinal speed of thevehicle is measured and compared with the speed input. The difference is fed trough a PI controller.The controller is tuned manually. The output of the cruise control subsystem is used as an input for asimple drive line model. The drive line model consists of a constant gearbox ratio, an engine which islimited by a constant power and a simple differential.

The braking system is the second input. It distributes a brake moment to the front and rear wheel viathe brake moment distribution p. A static brake gain can be defined which is used to apply a negativemoment between the upright and wheel depending on the angular velocity of the tyre.

The steering system is modeled as a single steer rack which is translated is the y-direction. The steerwheel angle is related to the translation of the steer rack via the steering ratio is.

Rotational inertia’s

Most inertia’s of the vehicle are estimated. It appears that the rotational inertia’s of the tyres andengine cause an extra lateral load transfer. This load transfer (see figure 4.14 is dependent on therotational speed of the wheels or engine, ωy , the yaw velocity, ψ and the rotational inertia of the object,Iy :

Mx = Iyωyψ (4.23)

The model is equipped with the tyre model discussed in chapter 2 which includes tyre mass andinertia’s to accommodate the multi-body model. The tyre and rim masses are measured. The tyre and

56

4.7. Dynamic vehicle model CHAPTER 4. Kinematic suspension design

Figure 4.14: Gyroscopic moment

rim inertia’s are measured using the time period for a wheel swinging at the end of a pendulum. Fromthe time period of a swing it is possible to calculate the rotational inertia of the wheel about the pointof rotation of the pendulum. The rotational inertia about the point of rotation of the pendulum canbe transformed into the rotational inertia at the centre of gravity of the wheel. This can be calculatedwith (4.24). The rotational inertia at the rotation axis of the pendulum can then be translated to therotation axis of the wheel centre with (4.25).

Io = mgr

(t

2π

)2

(4.24)

Ic = Io −mr2 (4.25)

Where:

t = period of one oscillation of the swingIo = rotational inertia of the wheel around the axis of rotation of the pendulumIc = rotational inertia of the wheel around the wheel centrem = mass of the wheel (tyre + rim)g = gravitational accelerationr = distance from the rotational axis of the pendulum to the centre of the wheel

This method gives an indication of the rotational inertia and is definitely not an exact method since ameasurement error will occur in the period measurement, even when the period period is averaged onmultiple swings. The rotational inertia around the y-axis off a Hoosier 20.5x7.0-R13 tyre with a Keizerrim is estimated on 1.0552 kgm2 and 4.950 kg. The rotational inertia around the x and z-axis is notthat important sine the rotational speeds around those axis’s will be low.

Equation 4.23 implies that the generated roll moment of the tyres adds up with the lateral load transfer.This means that the rotational inertia of the tyres and rims have to be as low as possible. This is bothbeneficial for the accelerating and decelerating of the wheels and the lateral load transfer.

57

4.8. Suspension collisions CHAPTER 4. Kinematic suspension design

The engine inertia in measured on a engine test bench and is appropriately equal to 0.03kgm2. Thisinertia is quite small but is of large influence since the rotational speed of the engine can be quitehigh. The rotational speed of the used engine has the same sign as the wheels. This implies that acounter rotating engine would have a positive influence on the lateral load transfer!

Spring-damper concept

The spring-damper concept consists of rocker actuated spring/dampers at every wheel. The spring-damper configuration is a damper with a coil-over spring. The left and right wheels are coupled withan anti-roll bar to assist the roll stiffness. The system is designed with a motion ratio which is aslineair as possible. This results in a more predictable roll behaviour. A progressive motion ratiocauses the chassis to rise more during cornering (jacking). The is caused because the outside wheelhas a higher stiffness in comparison with a linear motion ratio during cornering, the inner wheel hasa lower stiffness. This results in a chassis that moves less to the ground at the outside an lifts more onthe inside. A degressive motion ratio results in an decrease of the jacking effect. A linear motion ratiois also beneficial for the calculation of the damper curve which defines the generated damper forcedepending on the shaft velocity. The motion ratio is defined as:

MR =vA

vshaft(4.26)

The dampers are designed such that they can handle a motion ratio up to 0.5. This means that thespeed of the damper is half of the speed of the tyre contact patch. Lowering the shaft speed of thedamper reduces inertia forces and also shortens the damper since the required shaft travel is halved.This requires a rigid damper design and damper seals which can cope with higher pressures in thedamper chambers. The motion ratios are designed at 0.5 for both the front and rear suspension.

The dynamic vehicle model is also used to find the initial suspension settings. The will be discussedin section 5.9.

4.8 Suspension collisions

The last tool in the kinematic suspension design is a method to visualize suspension collisions. Theoptimization proces in only restricted with upper and lower boundaries of the optimization variables.This results in a box shaped boundary for every suspension point. The calculated suspension locationscould cause a suspension collision with, for example, the connection rods and the rim.

A potential collision could be analyzed by drawing the suspension in a CAD package but this has twodrawbacks, namely that all extreme suspension movements are difficult to draw and that it would takea lot of time to do this for every design iteration. The solution is found in a virtual reality world whichis coupled to the dynamic vehicle model. The most important CAD designs of the vehicle are used tovisualize it in a virtual reality world. The visualization consists of the next items:

1. chassis

2. 4 rims

3. 4 tyres

4. 4 uprights

58

4.8. Suspension collisions CHAPTER 4. Kinematic suspension design

5. suspension connection rods

6. tyre force arrows

Most objects are static, but the connection rods are automatically generated depending on the sus-pension design. The model is actuated to analyse all possible extremes of the suspension movement.Potential collisions are easily visualized. A collisions would result in a new iteration of the design byadjusting the boundaries of the optimization proces.

Figure 4.15: Suspension collision

Figure 4.15 illustrates a collision with the rim. The upper front connection rods collides with the rimand tyre during a combined steer and bump movement. Such a constraint is difficult to describe ina mathematical equation. In reality will the tyre (transparent in the figure) deflect under an infla-tion pressure and tyre forces which could cause a collision also. The clearance between the tyre andconnection rods is kept large enough to incorporate tyre deflections.

The visualization is also used to visualize the dynamical behaviour of the vehicle during the performedmaneuvers such as accelerating, braking, cornering and slaloming. The tyre forces in longitudinal, lat-eral and vertical direction are shown as 3d arrows which change in length depending on the generatedforces.

59

Chapter 5

Suspension design considerationsand results

5.1 Introduction

The chapter discusses the suspension design considerations and results. The numerical optimizationproces is used to accomplish the design targets. The mutual dependencies of the suspension charac-teristics are an important reason to use optimization techniques to design the suspension kinematics.Some of the discussed behaviour is part of an iteration process where the dynamic vehicle model isused to validate or analyse parameters.

Firstly the allowable suspension settings will be given. Then the design considerations will be dis-cussed in detail. This part of the chapter is dived in the pitch attitude, roll attitude, tyre orientationangles and driver feedback. The depicted figures in this part are a result from the optimization procesand calculated with the theory discussed in chapter 4.

The calculated suspension forces are discussed in section 5.8. This section also discusses the impactforces exerted on the suspension when driving over a bump with the use of an estimated SWIFT(Short Wavelength Intermediate Frequency Tire) model.

The dynamic vehicle model is also used to find the initial bump and roll spring stiffness’s, which isdiscussed in section 5.9. This reduces test time and cost and gives a good initial vehicle balance. Thelast section compares the performance of a double wishbone suspension with the designed multi-linksuspension. The double wishbone suspension was used on the URE03 vehicle in the 2006-2007season.

All the depicted results in the chapter are plotted for the left tyres, meaning that a negative steer anglerepresents the outside tyre during cornering.

5.2 Allowable suspension settings

The suspension layout has the ability to adjust some suspension settings for alignment purposes orto change the vehicle behaviour. This has to be done with precaution because every adjustment couldruin the intended kinematics. This will be shown later in some examples.

60

5.3. Contact patch pressure fluctuationCHAPTER 5. Suspension design considerations and results

The only allowable settings are the static camber angles and the static toe angles. The static camberangle is changed by placing material between the upper two connection rods and the upright. Theupright is designed to accommodate as less as possible kinematic change when changing the staticcamber angle. The static toe angles can be adjusted also. This is done by changing the length ofthe steer rods (front) and the tie rods (rear). This setting will also have a minimal influence on thekinematic behaviour.

On the front it is possible to make small adjustments to the steering linkage. The steering linkage isdesigned as parallel steering geometry where the steer angles of the left and right tyre will be identical.It is possible to change this to a situation where the inner tyre will be steered more or less than theouter. This is often called the Ackerman geometry [14]. The adjustment of this setting will have anegative influence on the overal kinematic behaviour and will probably also have very little influenceon the driving behaviour. This will be discussed in more detail in section 5.6.

The ride height can also easily be changed, but this has to discouraged because this causes the sus-pension to operate in a totally different region than intended.

5.3 Contact patch pressure fluctuation

An important choice for the design strategy is the choice of chassis movement, which is analog tothe suspension stiffness’s. This section explains the influence of suspension spring stiffness’s on thecontact patch pressure fluctuation.

A stiffer suspension is often interpreted by the driver as a better handling vehicle which gives the bestfeedback. Softer springs result in a lower contact patch pressure fluctuation which again is advanta-geous for the generated tyre forces. Hence, a lower fluctuation of the vertical tyre load results in more’grip’. The vertical tyre load fluctuation has a direct relation to the contact patch pressure fluctuation bythe contact patch area. Both terms are used in this chapter. The influence of the bump spring stiffnessof the vertical tyre load fluctuation is investigated with a simple quarter car vehicle model, see figure5.1. This model covers the basic dynamics of a tyre and chassis mass. This makes this model verysuitable for the understanding of the relation between the spring stiffness and tyre load fluctuation.

Sprung mass ms

Unsprung mass mu

ks

kt

ds

xs

xu

zr

Figure 5.1: Quarter car layout

This simple model consists of 2 degrees of freedom, namely the vertical displacement of the sprungmass, the vehicle body ms, and the unsprung mass consisting of the tyre, rim, upright and brakesystem mu. The bump spring stiffness is represented with ks and the bump damping with ds. Thetyre stiffness is represented with kt and is measured while fitting the tyre model in chapter 2 and is

61

5.3. Contact patch pressure fluctuationCHAPTER 5. Suspension design considerations and results

equal to approximately 136000 N/m. The equations of motion are:

msxs + ds(xs − xu) + ks(xs − xu) = 0 (5.1)

muxu − ds(xs − xu)− ks(xs − xu) = −kt(xu − zr) (5.2)

The relation between the road excitation and the tyre load fluctuation is interesting since this definesthe road holding ability. The input of the system is the road excitation zr. The output is the dynamictyre load Fz = kt(xu − zr) which in analog to the contact patch pressure fluctuation. The differentialequations can be denoted in a transfer function via a Laplace transformation. The derivation of thetransfer function can be found in appendix D.

H(s) =Y (s)Zr(s)

=−(mumss

4 + (mu +ms)dss3 + (mu +ms)kss2)(

mumss4+(mu+ms)dss3+(mu+ms)kss2

kt+mss2 + dss+ ks

) (5.3)

Figure 5.2a illustrates the influence of the bump stiffness on the transmissibility between the roadexcitation and the fluctuation of the vertical tyre force.

100

101

102

0

2

4

6

8

10

12

14

16

18x 10

4

excitation frequency [Hz]

tran

smis

sibi

lity

∆ F

z/zr

ks = 30000 [N/m] [REF]

ks = 25000 [N/m]

ks = 35000 [N/m]

(a) Bump stiffness effect on tyre load fluctuation

100

101

102

0

2

4

6

8

10

12

14

16

18x 10

4

excitation frequency [Hz]

tran

smis

sibi

lity

∆ F

z/zr

mu = 6 [kg] [REF]

mu = 4 [kg]

mu = 8 [kg]

(b) Unsprung mass effect on tyre load fluctuation

Figure 5.2: Bode plot of a quarter car vehicle model

Figure 5.2a clearly shows that a lower spring stiffness results is less fluctuation of the vertical tyre loadwhich results in more ’grip’, especially in the region between 3 and 20 Hz. But a lower stiffness alsocompromises the driver feeling. Section 5.9 discusses this in more detail.

The quarter car vehicle model shows the importance of a low unsprung mass in figure 5.2b. A 2 kglower unsprung mass has already a large influence on the transmissibility which also reaches to quitehigher excitation frequencies of 100 Hz.

62

5.4. Pitch attitude CHAPTER 5. Suspension design considerations and results

5.4 Pitch attitudeThe pitch attitude of the vehicle is defined as the pitch motion during accelerating or braking. Thesuspension travel has to be at least 25.4 mm bump and 25.4 mm rebound according to the regulations[9]. This does not mean that the suspension has to travel 25.4 mm in bump and rebound duringdriving. It only implies that the suspension can move that distance. The maximum bump travel ischosen 20 mm. The ground clearance is chosen 30 mm to keep 10 mm ground clearance left at alltimes.

The motion is controlled with the spring stiffness, the anti-pitch features and the brake moment distri-bution. The effect of the damper force is neglected here since the maximum wheel travel during brak-ing and accelerating is of importance. The suspension will not move anymore when the maximumachievable longitudinal acceleration or deceleration is reached. This means than the shaft velocity ofthe damper is zero and therefore are the dampers forces zero.

The anti-dive percentage on the front is of importance when a brake maneuver is considered. Thepercentage relates the amount of longitudinal brake force which is transmitted trough the suspensionlinks and the bump springs. The braking force is totally transmitted trough the suspension linksand not trough the suspension springs when 100% anti-dive is applied. However, this will never beapplied to racing vehicles since it will disturb the vertical tyre load because road irregularities causean excitation of the vertical tyre dynamics trough the relative stiff and undamped suspension links.Lowering the stiffness felt at the contact patch results in a better vertical tyre load fluctuation andtherefore more grip. Anti-dive percentages lower than 50% are common for racing applications tocompromise ’grip’ and the amount of dive at the front.

The amount of dive at the front results also in a decrease of the centre of gravity height which isagain beneficial for the maximum achievable longitudinal deceleration. This was already discussed insection 3.10. But this amount also influences the feeling of the driver. A larger amount of dive is ofteninterpreted by the driver as a vehicle which gives less feedback. This is another trade off which has tobe taken into account. The low speed setting of the damper is also partly responsible for this feeling.

The anti-dive percentage is designed to accomplish a compromise between the trade-offs. It is setto a static value of 31% and is made progressively, see figure 5.3a. The figure is a result from theoptimization proces and is plotted against the wheel travel. The progressiveness results in a bettercontact patch pressure fluctuation when road irregularities are attacking the suspension in its neutralposition and also in a better driver feel during braking without losing the effect that the centre ofgravity is lowered during braking.

−0.03 −0.02 −0.01 0 0.01 0.02 0.0315

20

25

30

35

40

45

50

wheel travel [m]

Ant

i div

e [%

]

(a) Anti-dive

−0.03 −0.02 −0.01 0 0.01 0.02 0.039

9.5

10

10.5

11

11.5

12

12.5

13

13.5

14

wheel travel [m]

Ant

i lift

[%]

(b) Anti-lift

Figure 5.3: Anti features while braking

63

5.5. Roll attitude CHAPTER 5. Suspension design considerations and results

The anti-lift percentage on the rear is also of importance when braking. It defines the amount oflongitudinal braking force which is transmitted trough the suspension links which suppresses the liftof the rear of the vehicle. Figure 5.3b depicts the anti-lift percentage. The static value is approximately12% and quite linear. A higher value is desirable but this is not possible since the anti-lift percentageis directly linked to the anti-squat value (except for the brake moment distribution).

The suspension does not have an anti-rise effect at the front when accelerating since the vehicle is rear-wheel driven. The anti-squat percentage at the rear is of importance when accelerating. In section 3.4 itwas already discussed that the centre of gravity height should be as high as possible when accelerating.A high anti-squat value is beneficial, but this will disturb the contact patch pressure fluctuation tomuch. The anti-squat percentage is designed to approximately 32%.

5.5 Roll attitude

The roll attitude is defined as the roll behaviour during cornering. Preceding on the suspensiondesign a maximum roll angle has to be defined. The roll angle results from the suspension geometry,the centre of gravity height and the roll stiffness’s at the front and rear suspension. The tyre is also ofimportance since the vertical stiffness contributes to the roll angle.

A lower roll stiffness results in a larger roll angle which is beneficial for the maximum achievablelateral acceleration. This reason for this is that a low roll stiffness results in a lower vertical stiffness feltat the tyre during cornering. A lower vertical stiffness is beneficial for the vertical tyre force fluctuationas discussed in section 5.3. And a lower vertical tyre force fluctuation results in higher transmittablelateral forces by the tyre, hence a higher lateral acceleration. But a large roll angle results also is a largerlateral translation of the centre of gravity, meaning that the load transfer to the outside of the vehicleis larger during cornering. A larger roll angle gives also less driver feel during cornering. The lastdisadvantage of a large roll angle is that it results in larger positive camber change during cornering.The maximum roll angle is set to 1 degree given the above considerations.

As discussed earlier, the anti-roll feature of the suspension will be used to investigate the roll behaviourinstead of a roll centre height. The anti-roll percentage is defined as the amount of lateral force whichis transmitted to the chassis via the suspension links. A larger anti-roll percentage will result in a lowroll stiffness. This improves the contact patch pressure fluctuation. But it also generates larger jackingforces, resulting is an increase of the centre of gravity height. The increase of centre of gravity heightis not only disadvantageous for the load transfer to the outside of the vehicle during cornering but itaffects also the region of operation of the suspension. This means that the vehicle will operate in adifferent than intended region of the suspension kinematics which results obviously in an unwantedvehicle behaviour. Therefore the design target is to develop a suspension which has no jacking effectat all.

The migration of instant centers (IC) is of importance in the jacking behaviour. This was alreadydiscussed in section 4.5. The dynamic vehicle model showed that the anti-roll percentage of the outsidewheel should be a little negative during roll and the inside percentage a little positive. Both tyres willgenerate a negative jacking force in theory. Figure 5.4a and 5.4b depict the anti-roll behaviour of thefront and rear suspension. The curve is plotted against the bump travel of the tyre which is related tothe roll angle of the vehicle.

64

5.5. Roll attitude CHAPTER 5. Suspension design considerations and results

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−10

−8

−6

−4

−2

0

2

4

6

8

wheel travel [m]

Ant

i rol

l [%

]

(a) Anti-roll front

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−8

−6

−4

−2

0

2

4

6

8

10

12

wheel travel [m]

Ant

i rol

l [%

](b) Anti-roll rear

Figure 5.4: Anti-roll

The vehicle will roll in reality around an unknown axis. The amount of bump travel of the outsidewheel and the amount of rebound travel of the inside wheel will not be the same in magnitude. Asalready mentioned, it is very complex to calculate a zero jacking suspension since the generated forcesat the tyres behave very non-linear. The roll behaviour is also very complex. The jacking behaviouris therefore validated with dynamical simulations. The depicted anti-roll values (calculated from theinstant centre migrations) result in a zero-jacking suspension. Figure 5.5a depicts the axle height atthe front and rear axle at the maximum lateral acceleration. Figure 5.5b depicts the centre of gravityheight.

0 2 4 6 8 10 12 14−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

time [sec]

axle

hei

ght [

m]

(a) Front and rear axle heights

0 2 4 6 8 10 12 140.27

0.272

0.274

0.276

0.278

0.28

0.282

0.284

0.286

0.288

time [sec]

CG

hei

ght [

m]

(b) Center of gravity height

Figure 5.5: Jacking

The axle height changes between 0 and 3 seconds are caused by the acceleration of the vehicle to reacha steady vehicle speed. The steer angle is slowly increased between 6 and 9 seconds to achieve themaximal lateral acceleration. It is clearly shown that the axle heights changes are very low. The rearaxle height change is lower than 0.5 mm. The front axle has a small axle height change of approxi-mately 2.7 mm. This is caused because the motion ratio is slightly progressive which means that the

65

5.6. Tyre orientation target CHAPTER 5. Suspension design considerations and results

vertical stiffness at the tyre is slightly progressive and causes the chassis to rise by the exerted lateraltyre force of the outside tyre. The source of the progressive motion ratio is the rocker geometry of thespring/dampers.

The changes are that small that it results in a very predictable suspension behaviour where the sus-pension will operate in its intended region. The centre of gravity height change during accelerationis almost zero and during cornering smaller than 1.6 mm which ensures a constant ride height and asmall as possible load transfer caused by the height of the centre of gravity. The driver also experiencesa natural roll motion.

5.6 Tyre orientation target

The tyre orientation angles (camber angle, γ and steer angle, δ) result from the suspension kinematics.The camber angle γ influences both the lateral and longitudinal tyre forces. The steering angle, andmore particular the difference between the left and right steering angle, influences the behaviour whilecornering. It influences the maximal achievable lateral acceleration but also the driver feel and vehiclebalance.

Section 3.12 discussed the optimal tyre inclination angle. It appeared that a camber angle of 0 degreesis best at higher side slip angles. This is also true in the longitudinal direction of the tyre. Hence,the camber angles should be 0 at all times to achieve the best performance. There are three ways toaccomplish a camber angle. The first way is to design a camber change rate with bump travel. Thecamber angle results from the location and migration of the instant centre (IC). The second way togenerate a camber angle results from the combination of a castor and kingpin inclination angle. Thesuspension has to be steered, either by applying a steering angle via the steering wheel or with anamount of bump steer, to generate a camber angle. The third way is to apply a static camber angleat the suspensions neutral position. The static camber angle is a suspension setting which can bechanged on the vehicle.

Both suspension degrees of freedom (bump and steer) could be used to generate a camber angle.The bump degree of freedom is used during braking/driving and cornering. The pitch motion ofthe chassis does not contribute to a camber angle, but the roll motion (during cornering) does. Theroll motion generally generates a positive camber angle at the outside wheels and a negative camberangle at the inside wheels. This could be compromised by designing an instant centre location andmigration which results in zero camber during roll. The problem with this is that this camber changealso applies to the suspension during braking and driving. The solution is a well designed combinationof the mentioned three ways to apply a camber angle.

This is especially usable at the front suspension, since the rear suspension is not steered. The bumpsteer amount at the rear it that small that it does not contribute to the camber change. The frontsuspension has a rather large steering deflection which could be used to compromise the camberchange rate and static camber angle.

The results from the calculations on the analytic vehicle model can be used to indicate the steeringangle at the maximum lateral acceleration. The steering angle at maximum lateral acceleration isstrongly dependent on the balance of the vehicle and the vehicle speed and therefore cornering radius.The steer angle is equal to approximately 3 degrees when the vehicle is neutral steered and drives ona corner radius of 100 m. A camber objective could be defined using this value and a maximum rollangle of 1 degree. The optimization proces tries to find a solution to achieve a 0 camber angle duringbraking/driving and cornering. This only applies to the front suspension since the rear suspension isnot steered.

66


The castor and kingpin inclination angles and migration could be chosen freely by the optimizationproces. This also yields for the instant centre location and migration. The latter is also used in theroll attitude objective. Figures 5.6a and 5.6b depict the castor and kingpin inclination angles duringsteering.

−30 −20 −10 0 10 20 30 404.5

5

5.5

6

6.5

7

7.5

steer angle [deg]

cast

or [d

eg]

(a) Castor angle

−30 −20 −10 0 10 20 30 40−4

−3

−2

−1

0

1

2

3

4

5

6

steer angle [deg]

king

pin

incl

inat

ion

[deg

]

(b) Kingpin inclination angle

Figure 5.6: Virtual steer axis geometry

Both the castor and kingpin inclination result in a camber change during steering. A positive castorangle contributes to a negative camber change during steering. A positive kingpin inclination con-tributes to a positive camber change during steering. The combination of both results in the camberangle during steering. This is depicted in figure 5.7a. The rate of change during steering is depictedin figure 5.7b. The highest camber change rate during steering is applied to the neutral position ofthe vehicle. This is done to prevent large camber angles at steering lock. Such result is very difficult toachieve when manual iterations will be used instead of an optimization proces, since the combinationof the castor and kingpin migration is responsible for this.

−30 −20 −10 0 10 20 30 40−3

−2

−1

0

1

2

3

4

steer angle [deg]

cam

ber

[deg

]

(a) Camber angle during steering

−30 −20 −10 0 10 20 30 400.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

steer angle [deg]

cam

ber

chan

ge w

ith s

teer

ing

angl

e [−

]

(b) Camber change rate with steering angle

Figure 5.7: Camber angle with steering

67


The camber change with the bump travel is kept rather small to ensure small camber angles duringbraking and cornering. This can be seen in figure 5.8. Results from dynamical simulations indicate theremaining static camber angles. This is especially the case on the rear suspension since the steeringgeometry can not be used here to achieve a camber angle.

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

wheel travel [m]

cam

ber

[deg

]

Figure 5.8: Camber angle with wheel travel

In section 3.11 a method was discussed to calculate the optimal steer angles for the front left and rightsteer angles to achieve the maximum lateral acceleration. It appears that the outside tyre needs tohave a larger steer angle than the inner tyre. This could not be achieved with a mechanical steeringlinkage. It could be done for small steer angles required for cornering at maximum lateral accelerationbut this would interfere with the large steering angles needed for tight corners. For this reason thedesign target is set to a parallel steering linkage. The suspension design has the possibility to adjustthe steering kinematics to positive or negative Ackerman. The adjustment would probably have nonoticeable effect since the adjusting range is rather small. The adjustment could be used for test-ing purposes. Changing this steering geometry has a drawback since the suspension would operatein another region then intended. This would especially the case for the bump steer amount of thesuspension.

The amount of bump steer is defined as the change in steer angle of a tyre at wheel travel. The amountof bump steer has influence on the cornering behaviour and the driver feel. The bump steer on thefront and rear suspension interact with each other as will the bump steer on the inside and outsidetyre.

As basic rule to follow is to design as less as possible bump steer because the vehicle will steer (andtherefore generate a lateral force) when it hits a bump. The chassis will also pitch under braking.This generates a bump movement and the tyres will generate counteracting lateral forces. Theseforces cause the tyres to operate in a smaller region of the friction ellipse of the tyre (see figure 2.12a)which results in a decrease of the maximum achievable braking force. The same principle is valid foraccelerating.

Bump steer at the front is especially of effect at corner entry since the front tyres are loaded more inthat case. For the same reason is bump steer at the rear of most importance at corner exit. To muchtoe in at the outside wheel could be dangerous at high lateral acceleration since the chance exists thatthe lateral tyre limit is already saturated. A small toe in at the front during roll is preferable since

68


it causes the vehicle to follow its path more when the tyre hits a bump. The tyre will operate in its,rather flat, lateral saturation range. A toe out would cause the vehicle to tend to under steer when thishappens.

Figure 5.9a illustrates the bump steer behaviour at wheel travel on the front suspension. The max-imum wheel travel for the outside wheel is approximately 0.11 [m] when cornering. The toe anglefor the outside wheel stays approximately zero and has the tendency to generate a small toe in anglewhen the wheel travel is increased beyond the maximum wheel travel of 0.11 [m] during cornering.The inside wheel generates a small toe in angle. It is preferable to have the toe angle even smaller butthis is not possible due to the space constraints of the suspension rack location. In fact it is not thatimportant since the inside wheel has a very low wheel load and can not generate large lateral forces atall. The generated lateral force will not cause an under steer behaviour and could only cause the tyreto operate more in the saturated lateral region.

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

wheel travel [m]

toe

[deg

]

(a) Bump steer on the front suspension

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

wheel travel [m]

toe

[deg

]

(b) Bump steer on the rear suspesnion

Figure 5.9: Bump steer

The steer rod location is restricted to the lower front quadrant of the tyre since the steer rack can notbe located elsewhere. This is in fact beneficial for the camber compliance of the suspension. Thesuspension will always have some sort of compliance effect since the suspension can not be infinitestiff. The amount will be small and is neglected in the design. The lower front location causes thesuspension to have very low steer compliance since both the lower connection rods and the steer rodwill be loaded with the same order of magnitude of lateral force. The suspension compliance couldgenerate a very small toe in angle resulting in less under steer tendency because the steer rod in locatedin front of the virtual steer axis.

Figure 5.9b illustrates the rear bump steer behaviour which is close to zero. The outside wheel willgenerate a very small toe out angle. This angle is applied to reduce the under steer tendency seenin earlier designs of the suspension on FSAE vehicles in Eindhoven. The braking force causes thesuspension to deflect under play and compliance to generate a toe out angle. This causes the vehicleto turn in more easy. At corner exit will the toe out angle decrease because a positive longitudinal forcecauses the suspension to deflect to a toe in angle. These effects are probably not noticeable since theyare very small.

The static toe angles should be very close to zero. A small toe in angle could be applied to overcomeplay in the suspension under tyre drag. This should be tested since play is not taken into account inthe kinematic design.

69

5.7. Driver feedback CHAPTER 5. Suspension design considerations and results

The design of bump steer is very difficult to do manually since the spatial characteristic of a multi-linksuspension causes the suspension to move in all directions. A numerical optimization proces is a verysuitable method to use here.

In section 5.2 was already discussed that it is important to be careful with suspension adjustments.The bump steer curve on the front is a good example for this because the suspension will operate inanother region when, for example, the ride height would be adjusted. This can be seen in figure 5.10where the bump steer curve is compared with the original curve when the ride height is adjusted with1 cm. The bump steer amount is very different and causes an unwanted vehicle behaviour.

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

wheel travel [m]

toe

[deg

]

standard ride heightadjusted ride height +0.01 [m]

Figure 5.10: Influence on bump steer when ride height is adjusted

5.7 Driver feedback

The feedback to the driver consists of behaviour of the vehicle described in the foregoing subsectionsand especially the steering wheel torques during driving, braking and cornering. These torques aregenerated with the tyre forces acting on the steering rack. They are transmitted to the steering wheelvia a steering ratio.

The steering wheel torques are generally caused by the following parameters:

• castor offset n

• pneumatic trail nr

• scrub radius rs

• wheel load lever arm w

• self aligning moment Mz

In lateral direction the castor offset and pneumatic trail are of importance. Both generate a momentaround the z-component of the virtual steering axis. The castor offset is calculated in such a way (see

70


subsection 4.5) that it corresponds to the equivalent used in planar suspensions, such as a doublewishbone suspension. The sum of the castor offset and the pneumatic trail of the tyre define a leverarm which has to be multiplied with the lateral tyre force to calculate the moment around the z-component of the virtual steering axis. The moment introduces a force on the steer rod which causesa torque on the steering wheel.

The castor offset is dependent of the steer angle of the front suspension. This is depicted in figure5.11a. The pneumatic trail is dependent on the tyre load and the side slip angle, see figure 5.11b.

−30 −20 −10 0 10 20 30 40−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

steer angle [deg]

cast

or o

ffset

[m]

(a) Castor offset front suspesnion

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

side slip angle [deg]

pneu

mat

ic tr

ail [

m]

(b) Pneumatic trail

Figure 5.11: Lateral lever arm

Both curves imply a relation with the lateral acceleration of the vehicle. The lateral acceleration has tobe coupled to the steer wheel torque to give a desired driving impression to the driver. However, thesteering wheel torque should not increase progressively to the lateral acceleration because this wouldlead to high forces at the steering wheel and hence to cramped hands and arms and loss of sensitivityfor steering corrections.

The pneumatic trail is caused by the tyre and can obviously not be influenced. This means that thecastor offset and steering ratio define the maximum steering torque. Experience in the past showedthat a steering wheel force felt and the hands of the driver should not exceed 110 [N]. The steering ratiois chosen to accommodate a large enough lever arm at the upright side and a small and light steer rackand steering wheel while maintaining a maximum steering wheel angle of 120 [deg] to the left andright.

The resulting steering wheel force is dependent on more factors than only the mentioned ones. It isalso difficult to link the steering angle to the side slip angle. The harmonization of lateral accelerationand steering wheel torque is therefore done using the dynamical vehicle model. Figure 5.12 illustratesthe steering wheel force felt at the hands of the driver when the lateral acceleration is increased. Thisis done for a high speed and a low speed corner. It depicts an almost linear increase with the lateralacceleration which gives the driver a perfect indication of the lateral acceleration. The steering wheelforce at the vehicle limit will saturate and give the driver a good indication when reaching the vehicleslimit. This degressive end is caused by the combination of castor offset and pneumatic trail. Themaximum steering force is 122 [N] at the high speed corner which is approximately 10% higher thanthe target of 110 [N] but this is still acceptable. This steering force can only be reached by designinga high degressive castor offset curve. The problem here is that both the castor offset and pneumatictrail migrate to negative values at high side slip and steering angles. This implies a steering torque

71


opposite to the desired one. The simulation does not show a negative steer wheel force, but a problemcan occur when driving trough tight corners at low speeds and large steering angles. Tests at the testtrack can result in the necessity to change the castor offset curve.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140

lateral acceleration [g]

stee

ring

whe

el fo

rce

[N]

speed 72 km/h, corner radius 22.5 mspeed 36 km/h, corner radius 6 m

Figure 5.12: Steering wheel force

The wheel load lever arm causes also a steering wheel force during cornering. This parameter is notmentioned in the foregoing discussion since it is rather small compared to the effects of the castoroffset and pneumatic trail. The wheel load lever arm causes a steering returnability torque whichcauses the steering wheel to return to its initial straight line position and therefore generating somestraight line stability. This returnability is more applicable to passenger cars since those will drivemore straight line kilometers. But a racing car still needs a small amount of steering returnability.

The wheel load lever arm is defined as the lever arm which causes the tyre to go to its neutral positionunder a vertical load of the vehicle. Figure 5.13 depicts the wheel load lever arm of the front suspension.The parameter is kept very low to reduce steering forces and axle height changes.

−30 −20 −10 0 10 20 30 40−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

steer angle [deg]

whe

el lo

ad le

ver

arm

[m]

Figure 5.13: Steering returnability with wheel load lever arm

72


The steering returnability is analyzed by performing a cornering maneuver with the dynamic vehiclemodel. A steering force is applied to perform a cornering maneuver at 72 km/h, see figure 5.14a. Thesteering force is removed at 1.75 sec. to investigate the steering returnability. Figure 5.14b illustratesthe returning of the steering wheel to its initial position. A small overshoot can be seen. The timeneeded to return the steering wheel to zero is quite small. Here has to be denoted that friction is notmodeled. The plot ends in a flat curve indicating the straight line stability.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−120

−100

−80

−60

−40

−20

0

time [sec]

stee

ring

whe

el fo

rce

[N]

(a) Applied steering wheel force

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

stee

r an

gle

[deg

]

time [sec]

(b) Steering returnability

Figure 5.14: Steering returnabilityBraking will also induce a steering wheel torque since the scrub radius acts as a lever arm to generate amoment around the z-component of the virtual steering axis. Braking on a straight line will not induceany steering wheel torque since the left and right tyre scrub radius are identical. However, brakingat corner entry results in a steering torque since the outer tyre will be loaded more. A multi-linksuspension has the ability to design a negative scrub radius. This results in a decrease of the steeringwheel torque which gives the driver a strange feedback since he or she expects an increase. The scrubradius is kept positive at all times to accomplish this. This is depicted in figure 5.15a. Figure 5.15billustrates the increased steering wheel force when braking at corner entry when the simulation timeis equal to 10 seconds. The increase in steering wheel force felt at the drivers hand is approximately 10[N]. This force is small enough to counteract with the drivers muscles and to give the driver a sensitivefeel for braking.

−30 −20 −10 0 10 20 30 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

steer angle [deg]

scru

b ra

dius

[m]

(a) Scrub radius

0 2 4 6 8 10 12 14−20

0

20

40

60

80

100

120

140

time [sec]

stee

ring

whe

el fo

rce

[N]

(b) Steering wheel torque at corner entry

Figure 5.15: Braking feedback

73

5.8. Suspension forces CHAPTER 5. Suspension design considerations and results

5.8 Suspension forces

The dynamic vehicle model is also very suitable to calculate the resulting suspension forces duringseveral maneuvers. These forces can be used to design suspension components such as the upright,connection rods, braking system and chassis pickup points.

The suspension has to be designed to cope with the maximum occurring forces. Driving over curbstones or other obstacles results in one of the highest occurring suspension loads. The tyre modeldiscussed in chapter 2 is based on the Magic Formula tyre model which is not very suitable to simulatedriving over bumps or cleats. TNO Automotive has developed an extension to the tyre model whichincorporates high frequency tyre behaviour (up to 60 Hz) and tyre envelopment properties. The ShortWavelength Intermediate Frequency Tyre Model (SWIFT, [21]) makes it possible to do an impact forceanalysis when the vehicle is driven over an obstacle on the road surface.

The obstacle is a 40x40 mm beam. In reality it is not possible to drive over this obstacle since theground clearance is only 30 mm. The larger beam is used to simulate a worst case scenario. Figure5.16a illustrates the 3D SWIFT enveloping model. The coefficients are not fitted on measurement databut are estimated by the tyre model to give an indication of the behaviour. Figure 5.16b depicts thedifference in the obstacle force between the Magic Formula (dotted line) and the SWIFT (solid line)model.

(a) 3D SWIFT model

98 100 102 104 106 108 1100

1000

2000

3000

4000

5000

6000

7000

8000

9000

distance [m]

pull

rod

forc

e fr

ont l

eft [

N]

SWIFT tyre modelMagic Formula tyre model

(b) Pull rod forces

Figure 5.16: Obstacle impact forces using SWIFT

The impact forces on the pull rod of the front suspension appear to be significant lower with theSWIFT model. This is obvious since the tyre rolls onto the obstacle with the SWIFT model. The MagicFormula model gets a step input to the tyre contact point when it hits the obstacle. The SWIFT modelshows some more oscillation after the obstacle which is probably caused by the rigid ring dynamics ofthe SWIFT model.

The other forces can be measured on the same way. The resulting connection rod forces, for exam-ple, are used as an input for finite element analysis design software and to choose the connection rodmaterial and rod end bearings which can cope with these loads. The anti-roll bar torques and rota-tion angles are used to design the anti-roll bars. In the future will these forces be used to design amonocoque chassis and composite rims.

74

5.9. Initial suspension settings CHAPTER 5. Suspension design considerations and results

5.9 Initial suspension settings

The dynamic vehicle model is also suitable to find the initial suspension parameters such as springstiffness’s and the static camber angles. Four spring parameters can be distinguished, namely thebump stiffness front and rear and the roll stiffness front and rear. The bump springs are displacedwhen the vehicle is accelerated both in the longitudinal and lateral direction whereas the anti-roll baris only displaced in the lateral direction. The choice of the roll stiffness’s is primary dependent onthe design choice of a maximum roll angle of 1 degree. The bump stiffness is a compromise betweendriver feel and contact patch pressure fluctuation.

The bump spring stiffness is 30000 N/m and is chosen with the dynamic vehicle model to result ina vehicle which has a maximum axle height change of 20 mm during braking and accelerating. Theremaining ground clearance will be 10 mm. Therefore is 30000 N/m the lowest possible stiffnesswhich can be applied to the vehicle. A higher stiffness can be applied but would result in a vehiclewhich generates less grip on road irregularities. The lowest possible bump stiffness of 30000 N/m istherefore applied to the vehicle.

ξ =ds

2√msks

ds = ξ · 2√msks

(5.4)

The choice for the lowest possible bump spring stiffness results in more grip but also compromisesthe driver feel. The ride frequency is still approximately 3.2 Hz, according to (5.5), which is reasonablyhigh for a vehicle with no down-force measures.

frf =1

2π

√ksms

=1

2π

√30000300/4

= 3.2 Hz (5.5)

To give the, mostly unexperienced drivers, a better feel and confidence the damping coefficient is cho-sen slightly overdamped by the damper manufacturer Koni. Koni has a lot experience in designingracing dampers and claims better lap times by compromising driver feel and optimal tyre load fluctu-ation by doing this. The low speed setting of the damper is mostly used to control the chassis motionsuch as the roll and pitch motion during braking/accelerating and cornering which is chosen over-damped. The high speed setting of the damper is mostly used to control the tyre load fluctuation andis chosen much lower. This results in a well balanced compromise between driver feedback and tyreload fluctuation.

The roll stiffness is easier to choose because a maximum roll angle of 1 degrees is allowed to keepthe suspension in its intended operation range. The ratio between the front and rear roll stiffness isresponsible for the tyre load distribution during cornering and therefore the balance of the vehicle asalready discussed earlier. The ratio is chosen to result in a neutral steered vehicle. Tests at the testtrack should be used to fine tune the setting which can be done in infinite small steps. The motionratio and anti-rol bar geometry result in the next list of bump spring stiffness’s and roll bar stiffness’s:

• front bump spring stiffness: 61000 N/m

• rear bump spring stiffness: 61000 N/m

• front anti-roll bar stiffness 1800 Nm/rad

• rear anti-roll bar stiffness 800 Nm/rad

75

5.10. Double wishbone comparison CHAPTER 5. Suspension design considerations and results

The dynamic model can be used to make a choice for the static camber angles. Figure 5.17 depicts theremaining camber angles while cornering when the initial camber angles are zero.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

lateral acceleration ay [g]

cam

ber

[deg

]

camber angle front left [deg]camber angle front right [deg]camber angle rear left [deg]camber angle rear right [deg]

Figure 5.17: Static camber angles

The remaining camber angles on the front are obviously much smaller because of the steering kine-matics. The remaining camber angles on the rear are slightly higher. To accommodate zero camberangles at the outside wheels during cornering a static camber angle of -0.3 degrees on the front and-0.85 degrees on the rear has to be applied.

5.10 Double wishbone comparison

The new suspension design is compared to the double wishbone suspension of the older URE03vehicle to validate the new design considerations and kinematics. In the comparison is chosen for thesame track widths, wheelbase, centre of gravity location, tyres, masses and inertia’s. The only variablein the comparison is the kinematic difference.

One of the important suspension design targets was to develop a zero jacking suspension. Figure5.18a and 5.18b illustrates the difference between the jacking attitude during cornering of the oldURE03 double wishbone suspension and the new multi-link design.

76


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

0.02

0.025


axle

hei

ght [

m]

axle height front URE04 [m]axle height rear URE04 [m]axle height front URE03 [m]axle height rear URE03 [m]

(a) Axle height comparison

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.28

0.282

0.284

0.286

0.288

0.29

0.292

0.294

0.296


CG

hei

ght [

m]

center of gravity height URE04 [m]center of gravity height URE03 [m]

(b) Center of gravity height comparison

Figure 5.18: Jacking comparison

The double wishbone design shows a large jacking effect during cornering, which is of course dis-advantageous by the increasing centre of gravity height and moreover, the suspension will operate inanother than intended region. The jacking effect is also quite progressive when the lateral accelerationis increased. This also causes the vehicle to reach a lower maximum achievable lateral acceleration.The jacking effect is still a little present in the new design. But this would not influence the intendedoperating range of the suspension since the values are quite small. Figure 5.18b illustrates the centreof gravity height change which is almost 7 times higher in the old case. This is a good improvementwhich is rather difficult to realize.

The use of optimization techniques makes it also possible to optimize the camber change duringcornering, which is as already mentioned earlier, dependent on the camber change rate, the steeringkinematics and the static camber angle. Figure 5.19a and 5.19b illustrate the camber angles duringcornering. Both vehicle are set with static camber angles of zero degrees which makes the comparisoneasier. Both the front and rear camber angles of the outside (left) wheels of the old suspension arequite larger and increase progressively in comparison with the new suspension. This is also causedby the jacking of the suspension. The inner tyre camber angles of the old suspension stay close tozero which is advantageous but this is a result from the large jacking effect. On the rear suspensionthe camber change of the new suspension is lineair, which is mostly caused by the migration of theinstant centre and not by the steering geometry since the rear wheels are not steered.

77


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


cam

ber

[deg

]

camber angle front left URE04 [deg]camber angle front right URE04 [deg]camber angle front left URE03 [deg]camber angle front right URE03 [deg]

(a) Front camber angles during cornering

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2


cam

ber

[deg

]

camber angle rear left URE04 [deg]camber angle rear right URE04 [deg]camber angle rear left URE03 [deg]camber angle rear right URE03 [deg]

(b) Rear camber angles during cornering

Figure 5.19: Camber angle comparison

Another main target of the new suspension design is a well designed feedback to the driver. The steer-ing wheel force is one of the most important feedback methods. Figure 5.20 illustrates the differencebetween the steering wheel force of the old and new suspension. It is clearly shown that the old sus-pension has a progressively increasing steering wheel force. The driver gets an improper indicationof the lateral acceleration and more important no indication that the vehicle limit is reached. The newsuspension indicates that the vehicle limit is reached via the saturation of the steering wheel force.The new steering wheel force is also slightly lower to compromise cramped muscles and a sensitivefeel for steering corrections.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140


stee

ring

whe

el fo

rce

[N]

steering wheel force URE04 [N]steering wheel force URE03 [N]

Figure 5.20: Steering wheel force comparison

78

Chapter 6

Conclusions and recommendations

6.1 Conclusions

The objective of this master’s thesis is to develop a new multi-link suspension suitable for the URE04FSAE vehicle. Four research areas of vehicle dynamics are addressed in the thesis: tyre behaviour,steady state vehicle behaviour and analysis, suspension development and validation with a dynamicvehicle model. The tyre models are crucial for the basic understanding of vehicle dynamics. Severaltyres are modeled to make a substantiated choice. The fit accuracy could be better but is accurateenough to be used in the suspension design.

An analytic steady state vehicle model gives insight in the steady state vehicle behaviour and the sen-sitivity of vehicle parameters on the vehicle performance envelope. The analysis of the performanceenvelope shows that the vehicle design strategy should primary be focussed on lowering the centre ofgravity height and secondly on vehicle mass. The model has also other applications, such as findingoptimal steering angles. The model certainly has potential for further research. The tyre model showsthat the optimal camber inclination angle at high side slip angles would be zero. This is question-able since the behaviour is quite different from normal tyres, especially regarding the relative soft sidewalls. Future research should indicate if this is true.

The kinematics of the multi-link suspension are based on the results from the analytic vehicle modeland the tyre models. New design tools are developed to design the suspension kinematics. The useof numerical optimization techniques makes it possible to design superior kinematics even with thetight space boundaries, especially on the rear. The next list summarizes the improvements in thesuspension design:

• Harmonization of the camber change rate (caused by the instant center location and migration),the camber change with steering angle (caused by the steering axis kinematics) and the staticcamber angles which results in optimal camber angles.

• Zero jacking of the chassis during cornering trough migration of the instant centers.

• Optimization of the bump steer curves which is especially difficult at the front with the givenspace boundaries.

• Harmonization of the castor offset curve, scrub radius curve and steering geometry resultingin steering forces which gives the driver a good indication of the lateral acceleration, the brak-ing forces at corner entry and the reaching of the vehicles lateral limit by the saturation of thesteering force.

79

6.2. Recommendations CHAPTER 6. Conclusions and recommendations

• Steering geometry results in large steering angles for tight corners.

• Degressive camber change rate when the suspension is steered to the left and right which pre-vents large camber angles at the maximum steering angle

• Good steering returnability (results also in straight line stability) with low vertical movement ofthe chassis and low steering forces.

• Sensitive feel for braking at corner entry trough the increase on steering force caused by themigration of the scrub radius

The above list of improvements is possible by using numerical optimization techniques for the designof a multi-link suspension. The use of a multi-link suspension has also the advantage that there are nobuckling forces present in the connection rods and that more space is created for the braking system.

Most results are validated with a dynamic vehicle model. The kinematics show also an improvementcompared with older designs. A potential problem could be that the steering forces get negative atlarge steering wheel angles because of the castor offset migration at large steering angles. Simula-tions do not show this behaviour during normal driving, but tests at the test rack should validatethis. Another conclusion which that can be drawn from the analysis of the suspension kinematics isthat changing suspension settings such as the ride height, steering rack location, suspension pick-uppoints or ackerman settings results in different than the intended suspension kinematics. These set-tings can not be changed without a thorough analysis of the suspension kinematics. The static camberangles and static toe angles can be changed. Both have little influence on the suspension kinematics.

The overall conclusion of the thesis is that the objective to design a multi-link suspension for theUREO4 vehicle is reached. Physical test at the test track can be used to make more improvements forthe future. The next section discusses recommendations based on this thesis.

6.2 Recommendations

This section is used to discuss some recommendations for future research.

To develop the multi-link kinematics for future vehicles it is advisable to do thorough test at a test trackwith suitable hardware. The next list gives an indication of measurements which can be performed todevelop the suspension:

• Linear pot meters mounted on the suspension springs can be used to analyze the chassis be-haviour during driving. This measurement can be used to analyze the jacking behaviour of thevehicle. The measurement data can be used to change the migration of the instant centers toaccommodate even less jacking. The data can also be used to tune the anti features such as anti-dive, anti-squat and anti-lift. Together with the bump spring stiffness it is possible to change thepitch behaviour of the vehicle, either to achieve better vertical tyre force fluctuations or to givethe driver a move confident feel.

• It is also advisable to purchase a steering torque sensor. These measurements can be used toadjust the castor offset curve, scrub radius curve and steering linkage to optimize steering forces.

• Optical sensors which measure lateral ground speed can be used to investigate the balance ofthe vehicle, especially at corner entry and exit.

• Infrared tyre temperature sensors placed across the tread area on the outside, middle and innerside can be used to analyse the contact patch pressure distribution at corner entry, apex, corner

80

6.2. Recommendations CHAPTER 6. Conclusions and recommendations

exit and during braking and accelerating. The temperature gradient should be as constant aspossible. This measurement can be used to adapt the suspension kinematics to optimize thecamber angles during the mentioned maneuvers.

• The influence of tyre temperature is not known. Two tests, one with cold and one with hot tyres,can give information about this behaviour. Based on this information a choice can be made touse different tyres which perform better when cold.

The tyre models could be improved with newly performed measurements by Calspan and with testsat the Technical University in Eindhoven. The tyre model of the chosen tyre shows that the tyre hasan optimal camber inclination angle of zero degrees at high slip angles. This has to be validated sinceit plays an important roll in the kinematics of the suspension. The effect of tyre temperature is alsovery important since all the dynamic events in the competition have to be started on cold tyres. Theregulations prohibit influencing tyre temperature with external equipment.

The developed analytic steady state vehicle model is very suitable to be extended to analyze moredetailed performance envelopes. A drive line model incorporating engine, and differential could beimplemented to investigate the longitudinal performance in more detail. One interesting aspect is toanalyze in more detail the longitudinal, lateral and combined requirements for the tracks driven in thecompetition. By applying weighting factors to parts of the performance envelope the design strategycan be fine tuned.

A tyre can be used to apply a longitudinal and lateral force to the chassis. A longitudinal force canobviously be applied by the braking system or the drive line system. A lateral force could be appliedby the front steer angles resulting in tyre side slip angles. If one would have perfect control of theacceleration direction and yaw behaviour of the vehicle it would be very interesting to design a vehiclecapable of delivering individual longitudinal and lateral forces on every wheel. Longitudinal forcescan be applied with ESP and active differential like systems. Current research is performed on thissubject which is called ’torque vectoring’. Lateral forces on the rear can be applied with active rearwheel steering since this is allowable by regulations. Especially the latter method is very interestingsince it is relatively easy to do. Even more interesting is when the front wheels are also individuallysteered. The development of a control algorithm is a challenge. Active (individual) steering is theaddition to the vehicle which is of most use and can result in the largest increase in performanceand innovativeness. The system can be used to achieve better handling, larger lateral performancebecause the tyre slip angles can be optimized, lower yaw-oscillations [22] or to prevent spinning out ofthe track.

The dynamic vehicle model could be extended with an accurate drive line model to make it possibleto do detailed dynamic analysis such as driving laps by a prescribed path or even more sophisticatedwith an intelligent driver model.

The chassis and in fact every vehicle component has a stiffness which is disregarded in the performedsimulations. A superior dynamic model could be made when finite element analysis is coupled withmulti-body simulations. In the future software packages become available where finite element com-ponents, multi-body and tyre models are combined. These packages combined with the computationalcapacity of these days make is possible to do even more detailed vehicle dynamic analysis.

81

Bibliography

[1] ISO 8855 Road vehicles Vehicle dynamics and road-holding ability Vocabulary.

[2] University Racing Eindhoven. www.universityracing.nl,.

[3] H.B. Pacejka. Tyre and vehicle dynamics. Elsevier, ISBN 0-7506-5141-5, 2002.

[4] TNO Automotive. MF-Tyre and MF-Swift 6.1 equation manual, Document revision 0.1. TNO Auto-motive, Delft, The Netherlands, 2007.

[5] FSAE Tire Test Consortium. http://www.millikenresearch.com/fsaettc.html.

[6] TNO Automotive. MF-Tool User Manual. TNO Automotive, Delft, The Netherlands, 2005.

[7] TYDEX Tyre Data Exchange Format. www.kfzbau.uni-karlsruhe.de/de/inhalt/gruppen/kfzbau/forschung/tydex/TydexFrame.

[8] J. de Hoogh. Implementing inflation pressure and velocity effects into the Magic Formula tyre model,DCT-2005.46. 2005.

[9] Formule SAE Rules. http://students.sae.org/competitions/formulaseries/fsae/.

[10] Formula Student competition tyres Hoosier. www.hoosiertire.com/rrtire.htm.

[11] I. Besselink. Vehicle dynamics, Lecture notes course 4L150. Eindhoven University of Technology,2006.

[12] I.Besselink. Numerical optimization of the Linear Dynamic Behaviour of Commercial vehicles, VehicleSystem Dynamics 23. Butterworth-Heinemann, 1994.

[13] W.F. Milliken and D.L. Milliken. Race car vehicle dynamics. SAE, ISBN 1-56091-526-9, 1995.

[14] Wolfgang Matschinsky. Road Vehcile Suspensions. Professional Engineering Publishing, London,ISBN 1-86058-202-8, 2000.

[15] the Mathworks; Matlab language for technical computing software. www.mathworks.com. Natick,Massachusetts.

[16] Nathan van de Wouw. Multibody dynamics lecture notes. 2005.

[17] William C. Mitchell. Force-based roll centers and an enhanced kinematic roll center, sae. 2006.

[18] Phillip Morse. A force-based roll center model for vehicle suspensions, sae 962536. 1996.

[19] M. B. Gerrad. Roll centers and jacking forces in independent suspensions - a first principlesexplanation and a designer’s toolkit, sae 1999-01-0046. 1999.

[20] Singiresu S. Rao. Engineering Optimization - Theory and Practice, 3rd ed. Wiley, ISBN 978-0-471-55034-1, 1996.

82

BIBLIOGRAPHY BIBLIOGRAPHY

[21] A.J.C. Schmeitz. A semi-empirical three-dimensional model of the pneumatic tyre rolling over arbitrar-ily uneven road surfaces. 2004.

[22] T. Veldhuizen. Yaw rate feedback by active rear wheel steering. DCT 2007.080.

83

Appendix A

Two track roll axis model equations

The speed of the vehicle V tangent to the driven radius is equal to the square root of the lateralacceleration ay times the driven radius R :

V =√ay ·R (A.1)

Here the yaw velocity r is equal to:

r =V

R(A.2)

The vehicle is driving around on a fixed driving radius. Increasing the lateral acceleration is done byincreasing the driving speed.

The arm which generates the moment around the roll axis is hra (figure A.1) and is equal to (A.3),where h is the height of the centre of gravity, h1 and h2 the roll centre heights, a1 the distance fromthe front track to the centre of gravity and l the wheelbase.

hra =(h− a1 ·

(h2 − h1

l+ h1

))· cos

(atan

(h2 − h1

l

))(A.3)

The roll angle is dependent on a component generated by the lateral acceleration ay and roll stiffness’s(cϕ1 + cϕ2) and a component of gravity (m · g · hra):

ϕ =ay ·m · hra

cϕ1 + cϕ2 −m · g · hra(A.4)

The following equations are given for one wheel/tyre only. The speed in longitudinal (u) and lateraldirection (v) is given by (A.5) and (A.6), where s1 is half of the track and a1 the distance from the front

84

CHAPTER A. Two track roll axis model equations

x

CG

RC front

RC rear

h1

h2h

hra

l

a

RA

ε

= Roll Axis

= CG at height and perpendicular distance from the RA

= Front and rear roll center heights

CGh1 h2

h hraRA

1 a2

Figure A.1: CG - roll axis arm

axle till the centre of gravity (rear tyre: a2 and s2). When the wheel is steered by an angle δ the speedsneed to be corrected with (A.7) and (A.8).

u1 = u− r · s1 (A.5)

v1 = v + r · a1 (A.6)

Vx1 = u1 · cos(δ1)− v1 · sin(δ1) (A.7)

Vsy1 = u1 · sin(δ1) + v1 · cos(δ1) (A.8)

The longitudinal slip ratio κ is a model input. The side slip angle is calculated using the longitudinaland lateral speeds:

α1 = atan

(Vsy1Vx1

)(A.9)

The tyre forces and moments are calculated using the Magic Formula tyre model of TNO Automotive[4]. The model uses the longitudinal slip (κ), the side slip angle (α), the tyre inclination angle (γ), theamount of turnslip (ϕt1) which is kept zero, and the longitudinal speed (Vx) as the input variables. The

85

CHAPTER A. Two track roll axis model equations

model outputs the longitudinal tyre force (Fx), the lateral tyre force (Fy) and the self aligning moment(Mz), see:

[ Fx1 Fy1 Mz1] = MF [ Fz1 κ1 α1 γ1 ϕt1 Vx ] (A.10)

The forces and moments of the four tyres are used to calculate the resulting forces on the chassis (eq:(A.11) till (A.15)).

Fx1 chassis = Fx1 · cos(δ1)− Fy1 · sin(δ1) (A.11)

Fy1 chassis = Fx1 · sin(δ1) + Fy1 · cos(δ1) (A.12)

Fx total = Fx1 chassis + Fx2 chassis + Fx3 chassis + Fx4 chassis (A.13)

Fy total = Fy1 chassis + Fy2 chassis + Fy3 chassis + Fy4 chassis (A.14)

Mztotal =− Fx1 chassis · s1 + Fy1 chassis · a1 +Mz1

+ Fx2 chassis · s2 + Fy2 chassis · a2 +Mz2

− Fx3 chassis · s3 − Fy3 chassis · a3 +Mz3

+ Fx4 chassis · s4 − Fy4 chassis · a4 +Mz4

(A.15)

These forces and moments are then needed to calculate load transfer in the lateral and longitudinaldirection according to (A.16) and (A.17).

∆Fz roll front =((Fy1 chassis + Fy2 chassis) · h1 + cϕ1 · ϕ)

2 · s1(A.16)

∆Fz brake/drive =h

2l· Fx total (A.17)

The normal loads of the tyres are calculated according to:

Fz1 =a2

2l·m · g + ∆Fz roll front −∆Fz brake/drive (A.18)

86

Appendix B

Optimization algorithm

The generalized pattern search algorithm is part of the ’Genetic Algorithm and Direct Search Toolbox’of MATLAB [15]. The next discussion is based on the documentation which comes with MATLAB .

Direct search is a method for solving optimization problems that does not require any informationabout the gradient of the objective function. Unlike more traditional optimization methods that useinformation about the gradient or higher derivatives to search for an optimal point, a direct searchalgorithm searches a set of points around the current point, looking for one where the value of the ob-jective function is lower than the value at the current point. The direct search algorithm can be usedto solve problems for which the objective function is not differentiable, stochastic, or even continuous.The Genetic Algorithm and Direct Search Toolbox implements two direct search algorithms called thegeneralized pattern search (GPS) algorithm and the mesh adaptive search (MADS) algorithm. Bothare pattern search algorithms that compute a sequence of points that get closer and closer to an opti-mal point. At each step, the algorithm searches a set of points, called a mesh, around the current pointthe point computed at the previous step of the algorithm. The mesh is formed by adding the currentpoint to a scalar multiple of a set of vectors called a pattern. If the pattern search algorithm finds apoint in the mesh that improves the objective function at the current point, the new point becomesthe current point at the next step of the algorithm. The MADS algorithm is a modification of theGPS algorithm. The algorithms differ in how the set of points forming the mesh is computed. TheGPS algorithm uses fixed direction vectors, whereas the MADS algorithm uses a random selection ofvectors to define the mesh.

Example

The pattern search begins at the initial point x0 that is provided by the user. In this example, x0 =[2.1 1.7

].

Iteration 1: At the first iteration, the mesh size is 1 and the GPS algorithm adds the pattern vectors tothe initial point x0 =

[2.1 1.7

]] to compute the following mesh points:

[1 0

]+ x0 =

[3.1 1.7

][0 1

]+ x0 =

[2.1 2.7

][−1 0

]+ x0 =

[1.1 1.7

][0 −1

]+ x0 =

[2.1 0.7

]

87

CHAPTER B. Optimization algorithm

The algorithm computes the objective function at the mesh points in the order shown above. Thefollowing figure shows the value of an example objective function at the initial point and mesh points.

1 1.5 2 2.5 3 3.50.5

1

1.5

2

2.5

3

4.6347 4.782

5.6347

4.5146

3.6347

Objective function values at initial points and mesh points

optimization variable 1

optim

izat

ion

varia

ble

2

Figure B.1: Initial points and mesh points

The algorithm polls the mesh points by computing their objective function values until it finds onewhose value is smaller than 4.6347, the value at x0. In this case, the first such point it finds is[1.1 1.7

], at which the value of the objective function is 4.5146, so the poll at iteration 1 is successful.

The algorithm sets the next point in the sequence equal to x1 =[1.1 1.7

]. By default, the GPS

pattern search algorithm stops the current iteration as soon as it finds a mesh point whose objectivefunction value is smaller than that of the current point. Consequently, the algorithm might not pollall the mesh points. It is possible to adjust the algorithm settings to do a complete poll.

After the first successful poll, the algorithm multiplies the current mesh size by 2. Because the initialmesh size is 1, at the second iteration the mesh size is 2. The mesh at iteration 2 contains the followingpoints:

2 ∗[1 0

]+ x1 =

[3.1 1.7

]2 ∗[0 1

]+ x1 =

[1.1 3.7

]2 ∗[−1 0

]+ x1 =

[−0.9 1.7

]2 ∗[0 −1

]+ x1 =

[1.1 −0.3

]The following figure shows the point x1 and the mesh points, together with the corresponding valuesof the example objective function.

The algorithm polls the mesh points until it finds one whose value is smaller than 4.5146, the value atx1. The first such point it finds is

[−0.9 1.7

], at which the value of the objective function is 3.25, so

the poll at iteration 2 is again successful. The algorithm sets the second point in the sequence equal

88


−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5146 4.7282

6.5416

3.25

3.1146

Objective function values at x1 and mesh points


optim

izat

ion

varia

ble

2

Figure B.2: Second mesh points

to: x2 =[−0.9 1.7

]. Because the poll is successful again, the algorithm multiplies the current mesh

size by 2 to get a mesh size of 4 at the third iteration.

By the fourth iteration, the current point is x3 =[−4.9 1.7

]and the mesh size is 8, so the mesh

consists of the points:

8 ∗[1 0

]+ x3 =

[3.1 1.7

]8 ∗[0 1

]+ x3 =

[−4.9 9.7

]8 ∗[−1 0

]+ x3 =

[−12.9 1.7

]8 ∗[0 −1

]+ x3 =

[−4.9 −1.3

]The following figure shows the mesh points and their objective function values.

89


−14 −12 −10 −8 −6 −4 −2 0 2 4 6−2

0

2

4

6

8

10

−0.2649 4.7282

7.7351

64.11

4.3351

Objective function values at x3 and mesh points


optim

izat

ion

varia

ble

2

Figure B.3: Unsuccessful poll

At this iteration, none of the mesh points has a smaller objective function value than the value at x3,so the poll is unsuccessful. In this case, the algorithm does not change the current point at the nextiteration. That is, x4 = x3.

At the next iteration, the algorithm multiplies the current mesh size by 0.5, so that the mesh size atthe next iteration is 4. The algorithm then polls with a smaller mesh size.

This proces continues till the algorithm is ended by the user or when a stopping criterium is reached.

90

Appendix C

Suspension coordinates

The mentioned coordinates are given for the front left and rear left suspension. The vehicle is symmet-ric over the x-axis. The coordinates of the right suspension are found by multiplying the y-coordinateswith -1.

Coordinates labels

front = front left suspensionrear = rear left suspensionchassis = chassis coordinateupright = upright coordinatewc = wheel center coordinate

The chassis and upright coordinates are labeled with a number, see the next figure:

Figure C.1: Suspension coordinates labels

91

CHAPTER C. Suspension coordinates

d.front.chassis.p1 =[1.3852 0.0522 0.1079

]d.front.chassis.p2 =

[1.8446 0.1366 0.1191


[1.8905 0.2315 0.3643


[1.3915 0.2478 0.2837


[1.7619 0.2431 0.1613


[1.6000 0.1290 0.1100

]

d.front.upright.p1 =[1.5595 0.5415 0.1405

]d.front.upright.p2 =

[1.6205 0.5411 0.1376


[1.6122 0.5415 0.3588


[1.5378 0.5614 0.3463


[1.6495 0.5536 0.1762


[1.5594 0.5305 0.3557

]d.front.upright.wc =

[1.6000 0.6125 0.2550

]

d.rear.chassis.p1 =[−0.1800 0.0495 0.1299

]d.rear.chassis.p2 =

[0.5117 0.2579 0.1820


[0.4993 0.2818 0.3510


[−0.1800 0.1902 0.3198


[−0.1800 0.1336 0.2228


[−0.1805 0.1592 0.4104

]

d.rear.upright.p1 =[−0.0473 0.5150 0.1600

]d.rear.upright.p2 =

[0.0634 0.5200 0.1600


[0.0156 0.5161 0.3595


[−0.0537 0.5200 0.3600


[−0.0715 0.5175 0.2580


[0.0000 0.5175 0.1600

]d.rear.upright.wc =

[0.0000 0.5875 0.2550

]Spring stiffness’sFront and rear mono-shock spring: 61000N/mFront anti-roll bar: 1800 Nm/radRear anti-roll bar: 800 Nm/rad

Initial suspension alignmentFront static camber: -0.3 degFront static toe IN: 0.1 deg

92

CHAPTER C. Suspension coordinates

Rear static camber: -0.9 degRear static toe OUT: 0.1 deg

93

Appendix D

Quarter car model derivation

Sprung mass ms

Unsprung mass mu

ks

kt

ds

xs

xu

zr

Figure D.1: Quarter car layout

Vertical tyre force:

Fz = kt(xu − zr) (D.1)

Equations of motion:

msxs + ds(xs − xu) + ks(xs − xu) = 0 (D.2)

muxu − ds(xs − xu)− ks(xs − xu) = −kt(xu − zr) (D.3)

Laplace transform:

(mss2 + dss+ ks)Xs − (dss+ ks)Xu = 0 (D.4)

94

CHAPTER D. Quarter car model derivation

(mus2 + dss+ ks)Xu − (dss+ ks)Xs = −kt(Xu − Zr) (D.5)

Xs =dss+ ks

mss2 + dss+ ksXu (D.6)

(mus2 + dss+ ks)Xu −

((dss+ ks)(dss+ ks)mss2 + dss+ ks

)Xu = −kt(Xu − Zr) (D.7)

(mus

2 + dss+ ks −(dss+ ks)(dss+ ks)mss2 + dss+ ks

)Xu = −kt(Xu − Zr) (D.8)

(mumss

4 + (mu +ms)dss3 + (mu +ms)kss2

mss2 + dss+ ks

)Xu = −kt(Xu − Zr) (D.9)

Y = kt(Xu − Zr) (D.10)

(mumss


mss2 + dss+ ks

)Xu = −Y (D.11)

Xu =Y

kt+ Zr (D.12)

(mumss


mss2 + dss+ ks

)(Y

kt+ Zr

)= −Y (D.13)

(mumss


(mss2 + dss+ ks) kt+ 1)Y =

−(mumss


mss2 + dss+ ks

)Zr

(D.14)

(mumss


kt+mss

2 + dss+ ks

)Y =

−(mumss

4 + (mu +ms)dss3 + (mu +ms)kss2)Zr

(D.15)

H(s) =Y (s)Zr(s)

=−(mumss

4 + (mu +ms)dss3 + (mu +ms)kss2)(

mumss4+(mu+ms)dss3+(mu+ms)kss2

kt+mss2 + dss+ ks

) (D.16)

95

Development and analysis of a multi-link suspension for racing applications

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