Development and analysis of a multi-link suspension for 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 Eindhoven Department Mechanical Engineering Dynamics and Control Group

Eindhoven, May, 2008 Abstract

University teams from around the world compete in the Formula SAE competition with prototype formula vehicles. The vehicles have to be developed, build and tested by the teams. The University Racing Eindhoven team from the Eindhoven University of Technology in The Netherlands competes with the URE04 vehicle in the 2007-2008 season. A new multi-link suspension has to be developed to 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 the suspension 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 centre of 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 the harmonization of camber dependant kinematics which result in the optimal camber angles of the tyres during driving. The suspension is designed to have a steady ride height during cornering which causes the suspension to operate in the intended region. The driver feedback is improved by means of the suspension kinematics and steering wheel forces. The vehicle characteristics are validated with a dynamic vehicle model.

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

ii Preface

The last challenge of my study is the master’s thesis. After careful consideration about the thesis subject, I started late 2006 with the development of a racing suspension for the University Racing Eindhoven team. It is a real challenge to design the complete vehicle dynamic characteristics of a vehicle, one which is not easy to find externally. And even so important: the design will be build and tested 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 is located in Helmond, the Netherlands. Here I had the opportunity to fit the tyre measurement data on the latest tyre model of TNO Automotive: Delft-Tyre. Therefore I want to thank Antoine Schmeitz for helping me doing this.

The next subject was to analyze the steady state vehicle behaviour of a racing vehicle and to find out what the influence of basic vehicle parameters is on the vehicle performance. It appeared to be quite an 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 had a very pleasant time doing this, while working in a truly unique environment: the University Racing Eindhoven team!

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

Willem-Jan Lamers, May 2008

iii Contents

Abstract ii

Preface iii

Sign conventions and symbols 1

1 Introduction 5 1.1 Background ...... 5 1.2 Objective and thesis outline ...... 6

2 Modeling a racing tyre 8 2.1 Introduction ...... 8 2.2 Tyre measurement data ...... 9 2.3 Fitting the measurement data ...... 10 2.4 Validation of the tyre model ...... 12 2.5 Tyre choice ...... 15

3 Steady state vehicle behaviour 19 3.1 Introduction ...... 19 3.2 Load distribution ...... 19 3.3 Two track roll axis vehicle model ...... 21 3.4 Base line vehicle ...... 23 3.5 Model objective ...... 25 3.6 Numerical optimization ...... 26 3.7 Calculation sequence ...... 27 3.8 Other model applications ...... 30 3.9 Pure cornering results ...... 31 3.10 Combined cornering/driving results ...... 34 3.11 Optimization of the steering angles ...... 36 3.12 Optimal tyre inclination angle ...... 38

4 Kinematic suspension design 39 4.1 Introduction ...... 39 4.2 Multi-link layout ...... 39 4.3 Design tools ...... 41 4.4 Kinematic suspension model ...... 41 4.5 Suspension kinematic characteristics ...... 44 4.6 Numerical optimization ...... 53 4.7 Dynamic vehicle model ...... 56 4.8 Suspension collisions ...... 58

iv CONTENTS CONTENTS

5 Suspension design considerations and results 60 5.1 Introduction ...... 60 5.2 Allowable suspension settings ...... 60 5.3 Contact patch pressure fluctuation ...... 61 5.4 Pitch attitude ...... 63 5.5 Roll attitude ...... 64 5.6 Tyre orientation target ...... 66 5.7 Driver feedback ...... 70 5.8 Suspension forces ...... 74 5.9 Initial suspension settings ...... 75 5.10 Double wishbone comparison ...... 76

6 Conclusions and recommendations 79 6.1 Conclusions ...... 79 6.2 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 a complete overview see [1].

on cti ire g d in iv Dr

Figure 1: ISO 8855 sign conventions

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

Throughout the report several angles are used. 12 different rotations sequences can be defined. Only 2 are used, one for the chassis rotation sequence and one for the wheel/tyre rotation sequence. Table 1 shows the rotation sequences for the chassis and wheel starting from the world coordinate system.

Rotation order Produced chassis angle Produced tyre angle First 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 two wheels 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 of an 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 axis definition.

1 CONTENTS CONTENTS

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

The camber angle is defined positive as an angle between the global z-axis and the wheel plane when the 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

2 ax longitudinal acceleration [m/s ]

2 ay lateral acceleration [m/s ] b track width [m] cφ roll stiffness [Nm/rad]

2 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

2 Iy rotational inertia [kgm ]

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 [-]

4 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 mainly founded 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 university engineering design competition.

A FSAE team can apply in three competition classes. The class 3 competition is meant for vehicle designs which exists purely on paper. A physical vehicle has to be developed and build to compete in the class 1 competition. This has to be a newly developed prototype every year. When a team wants to compete with a vehicle which was already used is the class 1 competition one has to apply to the class 2 competition. The competition regulations stimulate the teams to develop and apply new technologies. This involves a lot of team work and together with the short and strict deadlines it is also a management challenge. The Eindhoven University of Technology is competing since the year 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 part of the class 1 competition, which means that it is a newly developed prototype. The team is famous for its application of innovative technology. This expresses itself among other things in the use of exotic materials, such as aluminum honey comb for the chassis, sophisticated vehicle electronics and innovative 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 figure 1.1. Before reliable ball joints became available, independent suspension normally feature a physical kingpin, connecting the upright to the suspension link coupler. The development of durable ball joints

5 1.2. Objective and thesis outline CHAPTER 1. Introduction made it possible to equip vehicles with double wishbone suspensions. These days, most racing cars and also a range of passenger cars are equipped with independent double wishbone suspensions. An evolution of the double wishbone suspension is the multi-link or 5-link suspension. This suspension is 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 vehicle possible. This means that every potential performance gain has to be exploited. Such potential is the application of a multi-link suspension, which simultaneously accommodates the teams vision of developing 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 deliver the highest possible performance in the context of the FSAE competition. The suspension has to be fitted to the URE04 vehicle, competing in the 2007-2008 season. The thesis should result in the kinematic design of the suspension. The spring/damper concept falls beyond the scope of the thesis. The development of the suspension design involves the analysis and understanding of tyres and 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. Thorough knowledge of tyre behaviour is necessary to develop suspension kinematics. The design will be based on the tyre behaviour. Chapter 2 describes an advanced tyre model which is used to represent the non-linear tyre behaviour. Measurement data of commonly used tyres for the FSAE competition will be discussed. The measurement data is fitted on a Magic Formula, [3], based tyre model. The tyre model is used to describe the behaviour of 6 different tyres, and to make a choice for the best suitable tyre for the competition.

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

6 1.2. Objective and thesis outline CHAPTER 1. Introduction fundamental vehicle parameters on the vehicle performance envelope, such as mass and centre of gravity height, are analyzed to define a design strategy. The chapter also discusses other optimization possibilities 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 on a more conventional way with the use of graphical methods and basic linear algebra. The multi-link layout requires a different approach. Chapter 4 discusses the mathematical background of multi-link kinematics 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 suspension aspects are validated with the use of a dynamic vehicle model. Finally a comparison is made the the double wishbone suspension of the URE03 vehicle which was used in the competition in the season 2006-2007.

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

7 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 most important the lateral and longitudinal vehicle performance.

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

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

• 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 relatively small 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, which are important for racing tyres. The tyre fitting is always done at tyre operation temperatures which guarantees a consistent tyre model. The next section gives some background information on the tyre measurement data.

8 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 very expensive, especially for a Formula Student racing team. Therefore Milliken Research Associates has founded the FSAE Tire Test Consortium (FSAE TTC) [5]. This is an organization of Formula Student teams who pool their financial resources to obtain high quality tyre force and moment data. The FSAE TTC’s role is to gather funds from participating formula student teams, organize and conduct tyre force and moment tests and distribute the data to all participating teams. In this way it is possible to obtain this data for a very affordable price.

The FSAE TTC cooperates with the Calspan Tire Research Facility (TIRF) in Buffalo (USA). TIRF performs 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 Range Tyre slip angle (α) 30 deg Tyre inclination angle (γ) 30 deg Tyre slip angle rate 10 deg/s Tyre inclination rate 7 deg/s Tyre load rate 900 kg/s Tyre vertical positoning rate 0.18 m/s Road speed 0-90 m/s Maximal tyre outside diameter 1.19 m Maximal tyre thread width 0.61 m Belt width 0.71 m

Table 2.1: TIRF machine capabilities

9 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 used in the Formula Student competition. This section refers to the second test sequence. Table 2.2 shows a list of the tyres measured.

Brand Dimensions Avon 6.2/20.0-13 Avon 7.2/20.0-13 Goodyear 18x6.5-10 Goodyear 20x6.5-13 Hoosier 20.5x6.0-13 Hoosier 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 round which wasn’t available at the moment of fitting. The following measurements are performed on each tyre, except the Goodyear 10 inch tyre. Due to limitations of the measurement machine it is not possible 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 model fitter which is called MF-Tool [6]. This software fits the measurement data using data files arranged according to the TYDEX standard. The Tyre Data Exchange Format (TYDEX) is developed to make the exchange 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.The fitter has to know which set of data can be used to fit for example the lateral coefficients. Therefore the

10 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 delivers one large file for each tyre containing all measurement data. This file also contains tyre break in and conditioning sweeps, which means that the data has to be sorted out by hand to create the appropriate TYDEX 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 this is not the case. The algorithm of the fitter is adapted to allow the model to go beyond the standard boundary of the camber influence on longitudinal forces. The effect of tyre inclination angle on the longitudinal performance of a tyre can be seen in figure 2.2. This influence in indeed larger then expected, and has the largest deviation on the peak.

Kappa sweep, 12[PSI] inflation pressure 5000 0 [deg] inclination angle 2 [deg] inclination angle 4000 4 [deg] inclination angle

3000

2000

1000 Fx [N] 0

−1000

−2000

−3000

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

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 around the positive z-axis and then around the negative z-axis to perform a measurement. The hysteresis effect is caused by the speed at which the tyre is rotated around its z-axis and by temperature changes during the sweep. This effect is not implemented in the tyre model, but when both sets of measurement data are offered to the fitter the result will be averaged. Figure 2.3 illustrates this.

11 2.4. Validation of the tyre model CHAPTER 2. Modeling a racing tyre

Alpha sweep, 12[PSI] inflation pressure, 0[deg] inclination angle 5000 Fz = 222[N] model Fz = 1112[N] model 4000 Fz = 2000[N] model Fz = 222[N] measurement 3000 Fz = 1112[N] measurement Fz = 2000[N] measurement

2000

1000

0 Fy [N]

−1000

−2000

−3000

−4000

−5000 −15 −10 −5 0 5 10 15 alpha [deg]

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]. Therefore and to do a general check is it important to validate the tyre model in more detail. This is done for one tyre, 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 measurement data and the model for longitudinal forces (2.4a) and the lateral forces (2.4b). The overal fit of the longitudinal forces could be better. The longitudinal slip stiffness at lower normal loads is a bit to low and at higher loads a bit to high. The extrapolation at higher slip values goes quite well. The Magic Formula model is not based on polynomial coefficients, which ensures good extrapolation behaviour outside the measurement range. The lateral fit has some deviation at higher side slip angles where the model should be more digressive, especially for higher normal loads.

12 2.4. Validation of the tyre model CHAPTER 2. Modeling a racing tyre

Kappa sweep, 8[PSI] inflation pressure, 0[deg] inclination angle Alpha sweep, 8[PSI] inflation pressure, 0[deg] inclination angle 4000 4000 Fz = 667[N] model Fz = 667[N] model Fz = 1556[N] model Fz = 1556[N] model 3000 Fz = 667[N] measurement 3000 Fz = 667[N] measurement Fz = 1556[N] measurement Fz = 1556[N] measurement

2000 2000

1000 1000

0 0 Fx [N] Fy [N]

−1000 −1000

−2000 −2000

−3000 −3000

−4000 −4000 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −15 −10 −5 0 5 10 15 kappa alpha [deg]

(a) Longitudinal forces validation (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 illustrates this influence. It can be clearly seen in figure 2.4a that the longitudinal slip stiffness will increase when the inflation pressure increases. This is the case in both the measurement and the model. The forces are decreasing at high slip ratio’s in both the measurement and model. But in reality the effect is higher than the model predicts. The effect should be exaggerated more in the model.

Kappa sweep, 667[N] normal load, 0[deg] inclination angle Alpha sweep, 667[N] normal load, 0[deg] inclination angle 2500 2000 8[PSI] model 8[PSI] model 8[PSI] measurement 8[PSI] measurement 2000 12[PSI] model 1500 12[PSI] model 12[PSI] measurement 12[PSI] measurement

1500 1000

1000 500

500 0 Fx [N] Fy [N] 0

−500 −500

−1000 −1000

−1500 −1500

−2000 −2000 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −15 −10 −5 0 5 10 15 kappa alpha [deg]

(a) Longitudinal forces validation (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 reason for this is that the moments are quite small and therefore difficult to measure. The second reason is that there a lot a measurement noise exists. Figure 2.6a illustrates the differences between the measurement and the model. At lower loads there appears to be a shift in the measurement, which is probably caused by conicity of the tyre. At higher loads this disappears.

13 2.4. Validation of the tyre model CHAPTER 2. Modeling a racing tyre

The effective rolling radius of the tyre can be seen in figure 2.6b. The measurement is done at 5 fixed loads. 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 tyre inclination angle with respect to the road. Figure 2.7 illustrates the inclination angle behaviour and validates the difference between the measurement and the model. It appears that the inclination angle effect 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.

Overturning moment, 12[PSI] inflation pressure, 0[deg] inclination angle Effective rolling radius, 8[PSI] inflation pressure, 0[deg] inclination angle 100 0.266 Fz = 667[N] model model Fz = 667[N] measurement measurement 80 0.265 Fz = 1556[N] model Fz = 1556[N] measurement 60 0.264

40 0.263

20 0.262

0 0.261 Mx [Nm]

−20 0.26 effective rolling radius [m] −40 0.259

−60 0.258

−80 0.257

−100 0.256 −15 −10 −5 0 5 10 15 0 500 1000 1500 2000 2500 alpha [deg] Fz [N]

(a) Overturning moment validation (b) Rolling radius validation

Figure 2.6: Overturning moment and rolling radius validation

Alpha sweep camber influence, 12[PSI] inflation pressure, 667[N] normal load Alpha sweep camber influence, 12[PSI] inflation pressure, 1556[N] normal load 2000 4000 model, 0[deg] inclination angle model, 0[deg] inclination angle model, 4[deg] inclination angle model, 4[deg] inclination angle 1500 measurement, 0[deg] inclination angle 3000 measurement, 0[deg] inclination angle measurement, 4[deg] inclination angle measurement, 4[deg] inclination angle

1000 2000

500 1000

0 0 Fy [N] Fy [N]

−500 −1000

−1000 −2000

−1500 −3000

−2000 −4000 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 alpha [deg] alpha [deg]

(a) 667N normal load (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 a Formula Student racing car. The lack of combined longitudinal and lateral measurements data means that the tyre model has to generate an estimated combined slip characteristic, which of course will not exactly 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 could be 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 be made 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 measured for longitudinal forces and moments. And a 10" diameter rim has a lot of disadvantages compared with a 13 inch diameter. A larger inner rim diameter gives more freedom in the suspension design because more space is available. This has also the advantage that the connection rod forces are lower because the upright moments can be better transmitted to the chassis. The braking system can also be designed larger, this is especially important in the front, since the braking torques are the largest there. The ratio between the outer tyre diameter and brake disc is of importance. This ratio defines the 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 be advantageous for a 13" rim-tyre combination. The 10" tyre has the advantage of a slightly lower mass and rotational inertia.

The choice of a rim diameter size defines more than only the tyre and rim dimensions. The design of the chassis and powertrain is also dependent of this choice. The chassis has to be higher in the case of 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 and the 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 the other disadvantages the choice is made to use the 13" tyres.

15 2.5. Tyre choice CHAPTER 2. Modeling a racing tyre

Alpha sweep, 0.83[bar] inflation pressure, 0[deg] inclination angle Cornering stiffness, 0.83[bar] inflation pressure, 0[deg] inclination angle 5000 800 Fz = 150 [N] Hoosier 20.5x6.0−13 cornering stiffness Hoosier 20.5x6.0−13 Fz = 450 [N] Hoosier 20.5x6.0−13 cornering stiffness Goodyear 18x6.5−10 4000 Fz = 1050 [N] Hoosier 20.5x6.0−13 Fz = 2100 [N] Hoosier 20.5x6.0−13 700 3000 Fz = 150 [N] Goodyear 18x6.5−10 Fz = 450 [N] Goodyear 18x6.5−10 Fz = 1050 [N] Goodyear 18x6.5−10 2000 Fz = 2100 [N] Goodyear 18x6.5−10 600

1000 500

0 Fy [N] 400 −1000 stiffness [N/%]

−2000 300

−3000 200 −4000

−5000 100 −15 −10 −5 0 5 10 15 0 500 1000 1500 2000 2500 3000 alpha [deg] Fz [N]

(a) Lateral force (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. The competition does not requires a tyre which is resistant against wear. This is because the total amount of 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 and Goodyear 20x6.5-13.

Kappa sweep, 0.83[bar] inflation pressure, 0[deg] inclination angle Alpha sweep, 0.83[bar] inflation pressure, 0[deg] inclination angle 5000 5000 Fz = 150 [N] Hoosier 20.5x6.0−13 Fz = 150 [N] Hoosier 20.5x6.0−13 Fz = 900 [N] Hoosier 20.5x6.0−13 Fz = 900 [N] Hoosier 20.5x6.0−13 4000 Fz = 2100 [N] Hoosier 20.5x6.0−13 4000 Fz = 2100 [N] Hoosier 20.5x6.0−13 Fz = 150 [N] Avon 6.2/20.0−13 Fz = 150 [N] Avon 6.2/20.0−13 3000 Fz = 900 [N] Avon 6.2/20.0−13 3000 Fz = 900 [N] Avon 6.2/20.0−13 Fz = 2100 [N] Avon 6.2/20.0−13 Fz = 2100 [N] Avon 6.2/20.0−13 Fz = 150 [N] Goodyear 20.0x6.5−13 Fz = 150 [N] Goodyear 20.0x6.5−13 2000 Fz = 900 [N] Goodyear 20.0x6.5−13 2000 Fz = 900 [N] Goodyear 20.0x6.5−13 Fz = 2100 [N] Goodyear 20.0x6.5−13 Fz = 2100 [N] Goodyear 20.0x6.5−13

1000 1000

0 0 Fx [N] Fy [N]

−1000 −1000

−2000 −2000

−3000 −3000

−4000 −4000

−5000 −5000 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −15 −10 −5 0 5 10 15 kappa [%] alpha [deg]

(a) Longitudinal force comparison (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 acceleration test [9]. The Avon tyre has the highest peak in most cases, but the Hoosier has the flattest curve after the peak. This has the advantage that the tyre behaves very predictable after this peak. This will be very beneficial, especially with unexperienced drivers. Hoosier’s background can be found at the Nascar competition and other ’drift like’ competitions, which is likely to explain this flat curve. The Hoosier tyre has the largest friction coefficient at the lower loads (figure 2.10a). This region of loads applies

16 2.5. Tyre choice CHAPTER 2. Modeling a racing tyre

Friction coefficient, braking and driving, 0.83[bar] inflation pressure, 0[deg] inclination angle Friction coefficient, lateral, 0.83[bar] inflation pressure, 0[deg] inclination angle 3.2 3 braking Hoosier 20.5x6.0−13 lateral (−) Hoosier 20.5x6.0−13 braking Avon 6.2/20.0−13 lateral (−) Avon 6.2/20.0−13 3 braking Goodyear 20.0x6.5−13 2.9 lateral (−) Goodyear 20.0x6.5−13 driving Hoosier 20.5x6.0−13 lateral (+) Hoosier 20.5x6.0−13 driving Avon 6.2/20.0−13 lateral (+) Avon 6.2/20.0−13 2.8 driving Goodyear 20.0x6.5−13 2.8 lateral (+) Goodyear 20.0x6.5−13

2.6 2.7

2.4 2.6 mu [−] mu [−] 2.2 2.5

2 2.4

1.8 2.3

1.6 2.2

1.4 2.1 0 500 1000 1500 2000 2500 3000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Fz [N] Fz [N]

(a) Longitudinal friction coefficient (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 when this 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 and driving 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 Hoosier tyre 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 performs better. In reality a tyre’s normal load never exceeds half of the vehicles mass (300 [kg] = 1500 [N]). The Hoosier 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 force migrates to very large side slip angles, which aren’t taken into account during both the measurements and the calculation of the friction coefficient.

The values for the longitudinal and lateral friction coefficients appear to be quite large. The reason for this can be found in the tyre temperature during the measurement, the road friction coefficients on 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 friction coefficient. 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 during cornering. During the competition in Italy in 2007 it was proved that the car could really do this during 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 a requirement for the centre of gravity height.

17 2.5. Tyre choice CHAPTER 2. Modeling a racing tyre

Longitudinal stiffness, 0.83[bar] inflation pressure, 0[deg] inclination angle Cornering stiffness, 0.83[bar] inflation pressure, 0[deg] inclination angle 900 800 longitudinal stiffness Hoosier 20.5x6.0−13 cornering stiffness Hoosier 20.5x6.0−13 longitudinal stiffness Avon 6.2/20.0−13 cornering stiffness Avon 6.2/20.0−13 800 longitudinal stiffness Goodyear 20.0x6.5−13 cornering stiffness Goodyear 20.0x6.5−13 700

700

600 600

500 500

400 400 stiffness [N/%] stiffness [N/deg]

300 300

200

200 100

0 100 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 Fz [N] Fz [N]

(a) Longitudinal stiffness (b) Cornering stiffness

Figure 2.11: Longitudinal and cornering stiffness comparison

Combined slip, 0.83[bar] inflation pressure, 0[deg] inclination angle Pneumatic trail, 0.83[bar] inflation pressure, 0[deg] inclination angle 0.12 2000 pneumatic trail Hoosier 20.5x6.0−13 pneumatic trail Avon 6.2/20.0−13 pneumatic trail Goodyear 20.0x6.5−13 1500 0.1

1000 0.08 500

0 0.06 [m] Fy [N]

−500 0.04

−1000

−1500 0.02 Hoosier 20.5x6.0−13 Avon 6.2/20.0−13 −2000 Goodyear 20.0x6.5−13

0 −2500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 3000 Fx [N] Fz [N]

(a) Estimation of the combined performance (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 2.9 and 2.10, the best for the Hoosier tyre, see figure 2.12a. The pneumatic trail is of importance for the feedback to the driver via the steering wheel. The pneumatic trail will decrease when the side slip 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 most important feedback factors. In the past it appeared that steering forces where quite high. These forces are generated with the combination of pneumatic trail and mechanical trail. The later can be influenced by the design of the suspension. The former has to be as low as possible to reduce steering forces 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 slightly better performance in some regions, but this tyre is probably more difficult to drive. The relaxation lengths 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 an obvious 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 which is used to investigate the influence of basic vehicle parameters such as mass, centre of gravity height and 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 with these complex models is that a lot of insight is lost. Another disadvantage is that simulation results often show transient behaviour, which source is not easy to comprehend. Therefore it is desirable to have a model to investigate the pure steady state behaviour and to gain insight in the influence of the basic vehicle parameters.

The load distribution over the four wheels is one of the fundamental ways to influence the dynamic behaviour of a vehicle. This aspect is discussed in more detail. The layout of the two track roll axis model 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 be best used to increase the maximal longitudinal and lateral vehicle acceleration potential. This chapter ends 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 caused by the tyres producing lateral forces Fy left and Fy right, see figure 3.1. The position of the centre of gravity location is located above the ground. This results in a lateral acceleration during steady state cornering and therefore a reaction force acting on the centre of gravity. In this example the front and rear 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 it is not possible to create a vehicle with the centre of gravity at road height. This decrease is caused by two aspects [11]:

• saturation of the cornering stiffness • degressive friction coefficient with vertical load

Weight transfer effect Hoosier 20.5x7.0−R13 5000

4500

4000

3500

3000

2500 Total axle lateral force [N]

2000

1500 Side slip angle α = −2 [deg] Side slip angle α = −4 [deg] Side slip angle α = −8 [deg] 1000 0 10 20 30 40 50 60 70 80 90 100 weight transfer [%]]

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 side slip angles the change of friction coefficient is of most influence. The effects of both changes are depicted in figure 3.3a and 3.3b. The cornering stiffness (figure 3.3a) is digressively increasing till a certain wheel load, after that it is progressively decreasing. This shape implies a loss of cornering stiffness depending on the amount of load transfer, ∆F . The lateral friction coefficient (figure 3.3b) is decreasing at increasing wheel load. This means that the peak lateral force is not linearly increasing with 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)

Cornering stiffness Average lateral friction coefficient 900 2.55

800 loss 2.5 700 ∆ µ

600

2.45 500

400 mu [−]

stiffness [N/deg] 2.4

300 − ∆ F + ∆ F ∆ + ∆ F 200 − F 2.35

100

0 2.3 0 500 1000 1500 2000 2500 3000 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Fz [N] Fz [N]

(a) Cornering stiffness (b) Lateral friction coefficient

Figure 3.3: Effect of load distribution

These two phenomena are the reason for the decreased performance during cornering. The same happens 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 be investigated 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 as mass and center of gravity height on the performance of the vehcile. Using this model, the influence os 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 and also combinations of these accelerations. This will be called the vehicle potential. The potential is illustrated with a g-g diagram. The x-axis of the diagram represents the longitudinal acceleration of the vehicle, and the y-axis the lateral acceleration. This diagram is often referred to as the performance envelope of a vehicle because is describes the maximum achievable combinations of longitudinal and lateral accelerations possible. Since the vehicle is considered symmetric in a lateral way (left-right cornering) it is sufficient to show only half of the g-g diagram. Comparing the potential of several different vehicle configurations gives understanding in the sensitivity of a vehicle parameter such as its mass.

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

The model incorporates 4 tyre models, rigid axles, a roll axis with front and rear roll stiffness but no pitch dynamics (see figure 3.4). The simplicity of this model ensures that the basic vehicle parameters can be investigated without having the inconveniences of transient behaviour, such as vibrations and rotational 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 distribution of load transfer between the front and rear axle is called roll ratio or roll couple distribution. This mechanism is the most important way to influence under/overstreer (balance) of a vehicle. There are two 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 generated by the tyres is applied to the chassis. This force cannot generate a roll angle. The roll axis couples the front with the rear roll centers. • Roll stiffness. The stiffness which generates a moment at the roll centre depending on the height of 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 and regulations, limits this free choice. A lot of the parameters are chosen by experience and knowledge from 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]). Older vehicles 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 lot can 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 the lower 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 and minimum lateral translation when slaloming. The track is defined measuring from the centre of one contact patch to the other [1]. The lager the track width, the lower the load transfer, the higher the vehicles lateral performance, see section 3.2. But the wider the vehicle the more it has to translate laterally during slaloming and cornering around cones. The choice is made based on these aspects and the experience from the past. The front track is set to 1.225 [m]. The rear track is kept smaller to increases slaloming speed. The rear track has to move less in lateral direction around the cones which increases slalom speed. This reduces lateral translation of the car even more. The rear track is set to 1.175 [m].

The wheelbase is set to 1.6 [m]. Older vehicles showed nervous yaw behaviour with a shorter wheelbase of 1.5 [m]. When the wheelbase is increased to values larger than 1.6 [m] would this influence the time to do a slalom maneuver to much. This would also increase the vehicle yaw inertia which again decreases 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 between performance and stability.

(1.225 + 1.175) /2 WTR = = 0.75 (3.2) 1.6

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 wheelbase would have a positive influence on accelerating! In fact, the acceleration performance will be best when the total vehicle load is supported by the rear tyres of the car. Increasing the centre of gravity height accomplishes this also. Figure 3.5a illustrates the vehicle potential of two identical vehicles is a g-g diagram. One vehicle has a low centre of gravity height and the other a high centre of gravity height. 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 the engine and braking torque is infinite. In reality is the engine the limiting factor for the accelerating side especially at high speeds. This is an illustrative example to show the influence of the centre of gravity 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 with 4 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.

g−g plot two wheel drive center of gravity g−g plot 2 and 4 wheel drive vehicle 1.5 1.5 low CG rear wheel drive high CG 4 wheel drive

1 1

0.5 0.5

0 0 Ay vehicle [g] Ay vehicle [g]

−0.5 −0.5

−1 −1

−1.5 −1.5 −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] Ax vehicle [g]

(a) Center of gravity 2 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 will be kept low to the road surface to reduce jacking (see section 5.5). The roll stiffness’s have been determined 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 unit mass 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 to decrease the amount of lateral force at high side slip angles. This is done to decrease computation time, because a flat curve at high side slip angles causes the model to converge more difficult to a solution. 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 one steady 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 for every combination of lateral and longitudinal acceleration to find the entire performance envelope of a vehicle configuration. The model has, besides the constant vehicle parameters, several inputs which can be used to find a steady-state solution. The model outputs are used to validate if the solution is possible 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 of the vehicle. This can be for example pure cornering. The calculation is started at a rather small lateral acceleration to guarantee that a valid solution can be found. The vehicle is driving on a fixed radius of 100 [m] regardless of the combination of longitudinal and lateral acceleration. This results automatically 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 initiate the calculation of the model.

The input ’steering angle’ results in a certain lateral acceleration level and has to be chosen correct to match the chosen combination of longitudinal and lateral acceleration level. This also yields for the longitudinal slip κ. The input ’lateral vehicle speed’ results together with the calculated longitudinal speed (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 certain choice for a combination of longitudinal and lateral acceleration level.

Depending on the model inputs, the output has to satisfy the chosen combination of longitudinal and lateral 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 of longitudinal and lateral acceleration level, ax and ay, the resulting moment around the z-axis of the vehicle Mz and the wheel loads Fz1−4. Obviously the exact solution can not be found in one step. The reason for this is that the model an algebraic loop contains. The loop exists because each tyre load has to be calculated separately, and this tyre load is dependent of the lateral acceleration, the centre of gravity height, two roll stiffness, the roll centre heights, the wheelbase and track width. The lateral acceleration is again dependent of the generated tyre forces, but these tyre forces are also dependent of 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 solve because the tyre acts as a highly non-lineair element. The next section discusses a technique to solve the model equations.

3.6 Numerical optimization

Numerical optimization is a technique which is widely used in the industry to solve problems and to find 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 to some restrictions on the allowed parameter ranges or other constraint functions. The quantity to be optimized (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 the problem 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 spring windings. The constraint is that the spring has to fit in the engine head. This constraint consists of several maximum and minimum dimension such as its inner and outer diameter and the spring length. The optimization routine has to find a combination of values for the variables to achieve the best solution. The objective function is put into a mathematical equation. The result of this equation has to be as low or high as possible, depending on the optimization definition (minimization or 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 is called the generalized pattern search (GPS, MATLAB : patternsearch) algorithm. The working principle behind 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 lateral potential of the vehicle be investigated. The longitudinal acceleration is chosen 0 [g] and the lateral acceleration 0.5 [g]. This combination is definitely possible given the standard vehicle configuration of 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, tyre models • 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 parameters of the vehicle, which where discussed in section 3.4. Some parameters are calculated when the model is 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 chosen acceleration level. The last group of model inputs is given by the optimization scheduler. These inputs are iteratively changed to find the solution for the roll axis model for the given configuration and acceleration level.

The 4 wheel loads Fz,1−4, for example, are initiated at a quarter of the vehicle mass to start with. The basic model consists of parallel steered front wheels, which can be changed by the algorithm with one variable δ. The lateral vehicle speed, v, can be changed by the algorithm to adjust the vehicle slip angle β. Since the vehicle speed, V , is known, the longitudinal vehicle speed, u, can be calculated also. See appendix A for more details. The basic model consists of one variable for the longitudinal slip. These values 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 braking system. More complex models can also include different steering angle variables for the left and right front tyres to investigate this influence. These models can also include more complex differential and braking systems where different kappa variables for all tyres are used. It is, for example, also possible to 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 the tyre model. This is done according to equation (3.3). This function consists of several quadratic components. This is done to increase the convergence time to find the solution [12]. The reason for this is that the direction coefficient is decreasing while approaching zero on the x-axis. The solver is able to handle this more efficient.

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

The lateral and longitudinal acceleration errors are calculated according to:

 F + F + F + F 2 a = a − x1,chassis x2,chassis x3,chassis x4,chassis (3.4) x,error x m

27 3.7. Calculation sequence CHAPTER 3. Steady state vehicle behaviour

 F + F + F + F 2 a = a − y1,chassis y2,chassis y3,chassis y4,chassis (3.5) y,error y m

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 (3.6) − Fx3,chassis · s3 − Fy3,chassis · a3 + Mz3 2 + Fx4,chassis · s4 − Fy4,chassis · a4 + Mz4)

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

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 some aspect 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 objective function is zero. The optimization inputs of the last iteration yield the solution of the model. At this point 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 lateral acceleration aim has to be increased. This is done by the optimization scheduler which increases the lateral acceleration with 0.5 [g] to 1.0 [g]. The optimization routine tries to find the solution for this state again. At a point the optimization routine tries to solve the model for a vehicle state which is impossible to reach for the given vehicle configuration.

This boundary can be reached via two ways. The first one is the condition when the lateral acceleration is increased and the objective function value is not equal to zero. The boundary is then defined by the previous lateral acceleration level. The second one is when one of the inner tyres will be lifted from the road 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 with a resolution of 0.01 [g]. This is just one point of the g-g diagram which represents the performance envelope of the vehicle. The longitudinal acceleration level is increased by the optimization scheduler with 0.01 [g] and the optimization algorithm tries to find the lateral acceleration boundary again. This is repeated till the longitudinal acceleration boundary is found. This boundary is found when the lateral acceleration boundary is immediately equal to zero. The optimization scheduler start with the positive longitudinal acceleration range. The optimizations scheduler starts with the negative side of the longitudinal acceleration side of the g-g diagram when the positive longitudinal acceleration range is known.

28 3.7. Calculation sequence CHAPTER 3. Steady state vehicle behaviour

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), which arranges the data and saves it to a file. When more vehicle configurations have to be calculated, it will initiate 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 acceleration could take up to several hours. When the whole spectrum of combinations of accelerations has to be calculated would this take several weeks.

There are three ways used to decrease computation time. The first one is the tuning of the weighting factors of (3.7). This directs the optimization algorithm to put more effort on one part of the problem which 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 the tyre 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 decreased the 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 possible to find two or more exact solutions for combinations of input variables. An example is a vehicle which has a initial solution when it would drive neutral steered [11] through a corner. A second solution could be the same vehicle driven with excessive oversteer trough the corner. This could happen when the two tyres on an axle produces the same amount of lateral force while working at different side slip angles. Figure 3.7a illustrates this. This could be achieved in reality by suddenly applying a much larger steering angle to increase the side slip angles of the front tyres. The total lateral force will not change 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 an analogy with the lateral force, is plotted against the average of the side slip angles of the front and rear tyres. 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 of the 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 and rear axles than the first solution (left in figure 3.7b). This means that the second solution describes a much more oversteered vehicle, but still at the same lateral acceleration level.

0 2

1.8

−500 1.6

1.4

−1000 1.2 [N] y

1 Ay [g]

lateral force F −1500 0.8

0.6

−2000 0.4

0.2 Slip angle front axis Slip angle rear axis −2500 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 side slip angle α [deg] axes side slip angle α [deg]

(a) Lateral tyre forces, two states (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 could very easily be identified in the results of the optimization proces by large side slip angles, α, large steer 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 competitions are 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 to investigate the influence of the discussed vehicle parameters but the layout and calculation method makes 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 reached because 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 and display 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 variable such as extra steering angles.

The optimal camber angle for a certain tyre can be calculated. The calculation is performed alongside the 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 user interface.

Figure 3.8: Graphical user interface

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

3.9 Pure cornering results Pure 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. A balanced vehicle shows lineair increasing wheel loads with increasing lateral acceleration, see figure 3.9a. This is only true for a vehicle which has lineair roll spring characteristics between the front and

31 3.9. Pure cornering results CHAPTER 3. Steady state vehicle behaviour

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 understeered vehicle. The reference vehicle reaches of course a higher lateral acceleration of 2.0 [g] where the understeered vehicle reaches 1.8 [g]. See figure 3.9a.

1500 4 Fz1 [N] β Fz2 [N] β REF Fz3 [N] 3.5 Fz4 [N] Fz1 REF [N] Fz2 REF [N] 3 Fz3 REF [N] Fz4 REF [N] 1000 2.5

2 [deg] β

Fz tires [N] 1.5

500 1

0.5

0

0 −0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ay [g] Ay [g]

(a) Tyre loads (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 reason for this is that one wheel will be lifted of the the ground earlier, which causes the optimization to stop. The performance could be a bit higher when the side slip angle of the remaining tyre would increase, but this would be a small amount. The vehicle is normally chosen neutral steered when the full performance envelope of a vehicle configuration is analyzed.

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

2.6 0 δ front δ front REF 2.4 −0.5

2.2 −1

2 −1.5

1.8 −2 [deg] [deg] δ 1.6 α −2.5

1.4 −3 alpha1 alpha2 1.2 −3.5 alpha3 alpha4 alpha1 REF 1 −4 alpha2 REF alpha3 REF alpha4 REF 0.8 −4.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ay [g] Ay [g]

(a) Steer angles (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 same lateral 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 3.9. Pure cornering results CHAPTER 3. Steady state vehicle behaviour

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

60 3000 Fx1 Fy1 Fx2 Fy2 40 Fx3 Fy3 Fx4 2500 Fy4 Fx1 REF Fy1 REF 20 Fx2 REF Fy2 REF Fx3 REF Fy3 REF Fx4 REF 2000 Fy4 REF 0

1500 −20

Fx tires [N] −40 Fy tires [N] 1000

−60 500

−80

0 −100

−120 −500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ay [g] Ay [g]

(a) Longitudinal tyre forces (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 the driven 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 rear axis. An understeered vehicle would have a larger front axle slip angle which results is a handling diagram pointing to te left, see figure 3.12a. An oversteered vehicle would have a larger rear axle slip angle, see figure 3.12b.

Handling diagram pure lateral Handling diagram pure lateral 2 2

1.8 1.8

1.6 1.6

1.4 1.4

1.2 1.2

1 1 Ay Ay

0.8 0.8 Slip angle front axis Slip angle front axis Slip angle rear axis Slip angle rear axis 0.6 Handling diagram 0.6 Handling diagram l/R l/R Mean steer angle Mean steer angle 0.4 0.4 Slip angle front axis REF Slip angle front axis REF Slip angle rear axis REF Slip angle rear axis REF 0.2 Handling diagram REF 0.2 Handling diagram REF l/R REF l/R REF Mean steer angle REF Mean steer angle REF 0 0 −2 −1 0 1 2 3 4 5 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 α [deg] α [deg]

(a) Handling diagram understeer (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 away from neutral when the lateral acceleration is increased. This means that the vehicle will not switch between 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 they switch 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 sliding behaviour of the vehicle. The lifting of the tyres is caused, as already earlier discussed, by the high performance 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 often since the road surface is often polluted with dust and the tyre temperature is mostly low since the events are short. One exception is the endurance event which takes up to 20 minutes. Here have the tyres time to heat up. During the Formula Student competition 2007 in Italy the URE03 has driven on 2 wheels for a small amount of time at an entry of a slalom. The design of the URE04 vehicle has a much lower centre of gravity height which suppresses this behaviour. But when the friction coefficient during the competition reaches the one of the measurement is it still possible to lift the inner tyres from the road surface.

3.10 Combined cornering/driving results

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

The influence of these two parameters result in a choice for a design strategy. It is obvious that lowering 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 also restricts the design in using innovative systems such as an active differential or a CVT transmission or for example a 4 cilinder engine. When the design should be focussed to lower the centre of gravity as much 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 of the 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 sump system results in a high centre of gravity, because of the larger and higher engine crankcase. The advantage is the lower mass. When a dry sump system is used the engine can be placed lower. But this system will be heavier since there are more components involved. The sensitivity analysis should give an indication for a choice in design strategy.

[g] The mass sensitivity is defined as the sensitivity in acceleration advantage per kg ( [kg] ). The centre of [g] gravity height sensitivity is defined as the sensitivity in acceleration advantage per centimeter ( [cm] ). The choice for a design strategy is not purely dependent on the total vehicle potential, which could be calculated by taking the area beneath the g-g curve. Since the competition is more focussed on the lateral performance of the vehicle this would not be correct. The next table illustrates an estimate for the 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 event longitudinal 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 the lateral performance for 70% since more points can be earned with the autocross and even more with the endurance event. The acceleration and skidpad event weigh even. Both the mass and centre of gravity sensitivities are calculated according to the next equations:

∆a ∆a Mass = x · 0.3 + y · 0.7 (3.8) sensitivity ∆m ∆m

∆a ∆a CG = x · 0.3 + y · 0.7 (3.9) sensitivity ∆h ∆h

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

2.5 mass 280 [kg] 2.5 CG height 0.05 [m] mass 300 [kg] [REF] CG height 0.10 [m] [REF] mass 320 [kg] CG height 0.15 [m]

2 2

1.5 1.5 Ay [g] Ay [g]

1 1

0.5 0.5

0 0 −2 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 Ax [g] Ax [g]

(a) Mass sensitivity (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 by the fact that the vehicle is two wheel driven and brakes on four wheels. The sensitivity calculation is adapted with a separated longitudinal part for braking and driving. The sensitivity values for mass and centre of gravity height are:

35 3.11. Optimization of the steering angles CHAPTER 3. Steady state vehicle behaviour

−3 −1 Masssensitivity = 0.0014 · 0.15 + 0.0012 · 0.15 + 0.003 · 0.7 = 2.49 · 10 [g][kg] (3.10) | {z } | {z } | {z } braking accelerating cornering

−3 −1 CGsensitivity = 0.01 · 0.15 − 0.025 · 0.15 + 0.02 · 0.7 = 11.75 · 10 [g][cm] (3.11) | {z } | {z } | {z } braking accelerating cornering

The values are distracted from the figures 3.13a and 3.13b. The values for the mass sensitivity are normalized in [g][kg]−1 and the values for the centre of gravity height in [g][cm]−1. Lowering the centre of gravity by one centimeter has an almost 5 times higher positive influence than lowering the vehicle mass with one kilogram. Decreasing the centre of gravity is much easier done for the design of 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 of chassis design is maintained. The design strategy is therefore focussed on lowering the centre of gravity height. The design aim is to lower the centre of gravity height with an estimate of 6 [cm] by lowering the engine, driver, the front spring/damper system, the powertrain and the ride height. The vehicle mass can also be decreased by an estimate of 10 [kg] through by more components with the use 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 using this angle, which results in a parallel steering linkage. Most passenger cars and trucks make use of a an Ackerman geometry, which defines the relation between the left and right steering angle. A special case is the Ackerman condition [13]. This condition is true when the left and right steer angle matches the path of the cornering radius of that wheel. This is only valid at very low speeds. This Ackerman condition 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 to generate as large lateral forces as possible. This is almost impossible to calculate by hand. The model has the capability to optimize the front left and front right steer angle independently to optimize the generated lateral forces which should result in a better performing vehicle. It is also possible to optimize the steering angles of the rear wheels also. The next results (calculated for a cornering radius of 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

4 1 δ 1 δ 2 0 3.5 δ REF

−1

3 −2

2.5 −3 [deg] [deg] δ α −4 2

−5 1.5

−6

1 alpha1 −7 alpha2 alpha1 REF alpha2 REF 0.5 −8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ay [g] Ay [g]

(a) Optimal steering angles (b) Handling diagram

Figure 3.14: Side slip angles The dotted lines indicated with REF represent the base line vehicle with parallel steering geometry. It can be clearly seen in figure 3.14a that the steering angle of the outer tyre (δ2) is much larger than the inner one (δ1), see also figure 3.14b which illustrates the difference in side slip angles. This indicates that 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 such mechanical steering linkage. The reason for this is that the difference in steering angles has to be archived at small steering angles, but at some points the steering angle has to be as high as 28 degrees to cop with very narrow and slow corners. The steering angles between the left and right wheel at steering lock would deviate to much and one will collide with the suspension linkage. An electronic actuated steering mechanism could be an answer to this problem, but regulations permit steer by wire systems on the front axle.

Handling diagram pure lateral 2

1.8

1.6

1.4

1.2

1 Ay

0.8 Slip angle front axis Slip angle rear axis 0.6 Handling diagram l/R Mean steer angle 0.4 Slip angle front axis REF Slip angle rear axis REF 0.2 Handling diagram REF l/R REF Mean steer angle REF 0 −3 −2 −1 0 1 2 3 4 5 6 7 α [deg]

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 variables to 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 inclination angles (>40 [deg], see figure 3.16a). The reason for this is that it is numerically beneficial. The used tyre model is transformed to MATLAB code and does not take a maximum valid inclination angle into account. At low side slip angles, between 1.5 and 2.1 [g], is the result valid. When the vehicle reaches it limit the side slip angles would increase and it appears that it would be numerically beneficial to have extremely large inclination angles. It is obvious that these results could not be used to design the suspension kinematics. Therefore the optimal tyre inclination angles will be calculated on a different way.

10

0

0

−2

−10

−4

−20 [deg] [deg] γ γ −6

−30

−8 γ 1 −40 γ 2 γ 3 −10 γ 4 −50 0 0.5 1 1.5 2 2.5 −15 −10 −5 0 Ay [g] side slip angle α [deg]

(a) Optimal tyre inclination angle (optimization) (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 side slip angle and a fixed camber angle. The generated lateral forces are added up. This is done for the whole range of side slip angles and camber inclinations. The curve with the highest cumulative lateral force is decisive for the choice of a camber angle. Figure 3.16b shows the optimal camber angles for the 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 that a camber angle of 0 [deg] is preferable.

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

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 deforms during cornering the contact patch will be stretched out. This deformation restores the contact patch load distribution which increases the friction coefficient. At large slip angles this effect can cause the contact patch pressure distribution to be lower at the outside of the tyre, which can be an explanation for the above results. Restoring the contact patch pressure can then be accomplished by lowering the inclination 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 to design a multi-link suspension. A multi-link suspension is one of the most complex designs and is relatively 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 this suspension. These tools are: a kinematic multi-body model, a collection of mathematical equations to calculate suspension characteristics, a numerical optimization tool to compute the optimal kinematic design, a dynamic vehicle model to validate the suspension design and a tool to visualize suspension collisions. This chapter discusses the design tools in the denoted order. The suspension design targets and 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 often referred to as an evolution of the double wishbone suspension. The layout of a multi-link suspension differs 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 a plane 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 to the chassis with six connection rods. Every connection rod has 2 joints which can be placed arbitrary in space with 3 coordinates each. This results is a total of 6 · 2 · 3 = 36 degrees of freedom on one wheel (assumed that that car is left/right symmetric). The front and rear suspension have in total 72 degrees of freedom. These freedoms are restricted by several design boundary’s, such as the available space. The combination of the number of degrees of freedom and the design boundaries implies a very 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 important reason for this is that the engineers at car manufactory companies have to compete for the available space inside a passenger car chassis. The car is better sellable when more space is available for luggage and an engine. The lack of space demands that the suspension design is capable of delivering good handling performance, comfort, low noise and low cost. A well designed multi-link suspension is capable of doing this. But it is probably one of the most expensive suspension designs also. The low end 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 not used for racing purposes because of its complexity and the engineering time needed to design this suspension.

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 the choice 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 and IPG K&C. Some of these tools are rather simple and can only calculate kinematic behaviour of a suspension. Others can perform fully dynamical vehicle simulations, which include tyre models and non-linear compliance calculations. The problem with these software tools is that they lack insight and flexibility. Most of them are designed as a ’black box’, where most of the mathematics and definitions are 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 such system with manual design iterations. Today a lot of numerical optimization techniques are available to tackle this kinds of problems. Most commercial suspension packages do not incorporate these techniques.

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. Every possible calculation of suspension parameters can be implemented. This ensures a good insight in the definitions of suspension parameters. MATLAB is very flexible since it comes with a multi-body toolbox. The toolbox can be used to do a kinematic suspension analysis. It is also possible to do a fully dynamical vehicle analysis, since the TNO tyre model [6] can be implemented in the multi-body models. MATLAB has also an extensive numerical optimization toolbox. Given these advantages, the choice 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 developed in SIMMECHANICS , which is the multi-body toolbox of MATLAB . The second tool is the collection of mathematical equations which are used to calculate suspension parameters. The third one is the numerical optimization toolbox, which is used to calculate the optimal suspension geometry given certain objectives and boundaries. The fourth is a visualization of the movement of the suspension in a virtual reality environment. This tool is used to investigate suspension collisions. The fifth and last tool is a fully dynamical vehicle model, including the tyre models. This tool is primary used to validate the 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 made with MATLAB SIMMECHANICS . The model has several inputs to actuate the suspension system. In the analysis can 2 movements be distinguished, namely the bump movement of the suspension and the steer movement. Both are needed to evaluate the suspension characteristics. This is discussed in more detail in section 4.5.

The upright is connected to the chassis with connection rods (suspension links) and ball joints. These components restrict (constrain) the movement of the upright. A multi-link suspension is part of the spatial suspension geometries. This means that the movement of the components are individually 3D related. It belongs to the independent suspension layouts since it has only one bump degree of freedom, where rigid-axle suspension for example have two bump degrees of freedom (symmetric and asymmetric 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 be omitted. The multi-link suspension under investigation is connected to the chassis and upright with ball-joints. These joints constrain a movement with 3 translations and leave 3 rotation degrees of freedom 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, the upright and each connection rod possess six degrees of freedom in a 3D space. This results in a total of 6 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 a connection 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 steer movement [14]. This characterizes an independent suspension.

Bump movement The chassis is fixed to the world when the bump movement is analyzed. One link between 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 wheel has only one degree of freedom left (bump). Since this results is a spatial movement of the wheel in space it to be connected to an actuator with 5 degrees of freedom with the world. The constraint is the translation in z-direction. The rotation around the wheel rotation axis is also blocked to prevent the rotation 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 illustrates this.

Figure 4.5: Omitted steer rod

This time the steer rod is omitted, which results in the steer degree of freedom. The actuator which moves the wheel is also removed because the steer movement is actuated on another way. In fact the upright is actuated by the translating steer rack in y-direction. The steer rod is therefore not really omitted but used as the link between the steer rack and the upright. The rear suspension is also equipped 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 models are 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 the suspension characteristics. Several outputs of the kinematic suspension model are used as the input for the equations. This section describes the definitions and equations of the suspension kinematics in detail. All calculations are done according to the ISO8855 standard [1] and can be applied to the front 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 derive suspension characteristics.

Wheel orientation angles

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

Rotation order Produced tyre angle First 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 9 values denoted in a matrix form. To calculate the tyre orientation angles the inverse of the rotation matrix has to be calculated, which is named the direction-cosine matrix [16]. This new matrix consists of the multiplication of three elementary matrices. Since the rotation sequence for the calculation of the tyre orientation angles is 3-1-2 these matrices are denoted by:

A =A43(ω)A32(γ)A21(δ)  cosω 0 −sinω   1 0 0   cosδ sinδ 0  =  0 1 0   0 cosγ sinγ   −sinδ cosδ 0  sinω 0 cosω 0 −sinγ cosγ 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)

arcsin (−A(1, 3)) ω = (4.3) cosγ

arcsin (−A(2, 1)) δ = (4.4) cosγ

44 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

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. The same yields for rotation velocities and accelerations.

Rotation order Produced chassis angle First 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 migrates during the movement of the suspension. Two instantaneous virtual axis can be defined. The first one is the virtual steer axis, which is linked to the steering degree of freedom of the upright. The second one is often called the instant axis, which is linked to the bump movement of the upright. Both are used 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 can not be used to construct this axis graphically. The theory mentioned is [14] is used in the remainder of this section.

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

ωK × vM rIA = rM + 2 (4.5) |ωK |

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), the screw axis of a spatial multi-link suspension is not suitable for a reference axis to calculate moments and forces. Since the screw axis shows an axial shift coupled with a rotation, forces that act on the screw axis that are not perpendicular applied to it will introduce a torque on that axis. The intersection point of the instantaneous axis with the side and front plane would normally be used to calculate different suspension parameters. This is not possible here since the screw axis can not be used to do so. Figure 4.6 illustrates the instantaneous screw axis of the front left upright in green.

45 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

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 contact point. In the next discussion the side plane (x-z plane) will be used. The screw axis has an axial shift velocity along the axis and a rotation around that axis. The intersection point with the plane would therefore have a velocity in both the x and z direction in that plane. This point can therefore not be used as 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 in figure 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 plane on a point which is stationary when the screw axis would rotate and shift. Stationary means that it has

46 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design zero velocity in the plane, in this case in the x and z-direction. This is the point which can be used to calculate 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 defined with ax, ay and az:

vIA = vM + ωK × (rIA − rM ) (4.6)

vIA,z vIA,x ax = ay = 0 az = (4.7) ωKy ωKy

The side pole (termed the anti centre, AC) can be calculated using the vector a which points from the 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 leads to (4.7). The pole AC can easily be calculated by adding the vector a to the vector of the first pole. The velocity 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:

−vIA,z vIA,y ax = 0 ay = az = (4.8) ωKx ωKx

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. These features 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) to vertical forces. A part of the forces is transmitted trough the push/pull rod, the elastic part. The remaining part is transmitted trough the other connection rods and is called the direct part because the springs and dampers are not actuated by this part. The longitudinal forces are totally transmitted by 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 the front suspension during braking. A front wheel driven vehicle has also an amount of anti-rise which is generated during accelerating. Anti-lift is the amount of longitudinal forces that is transmitted through 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 the rear 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-rise can not be used since the front wheels are not powered by the engine. The anti features anti-dive and anti-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 and ACr.

47 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

castor change AC r

θas AC f h θal

AC f,y θad

wheel path AC f,x l

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 be found in [11] and [14].

tan(θ ) Anti − dive = ad × 100% (4.9) h/(lp)

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

tan(θ ) Anti − lift = al × 100% (4.10) h/(l(1 − p))

tan(θ ) Anti − squat = as × 100% (4.11) h/l

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

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

Front view support angle

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

A commonly used tool in the design of suspension systems is the roll centre height. This variable will not be used in the remainder of the report. It only defines the chassis rotation point at the initial roll movement of the chassis during cornering. At high lateral acceleration levels this point is useless because the generated forces by the left and right tyre differ a lot. Therefore is the design focussed on the 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 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

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 chassis to generate a roll moment and is called anti-roll. It can be calculated according to:

tan(θ ) Anti − roll = ar × 100% (4.12) h/t

The location of the instant centre pole IC defines, on the same way as the castor change rate and wheel path, the camber change rate and the scrub. Scrub is the amount of lateral translation of the tyre contact point at suspension travel (bump). Both parameters are directly measured in the kinematic suspension 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 is difficult to analyze the anti-roll behaviour. The instant centers will migrate to other positions in space during wheel travel. This amount and direction of migration is interesting since it defines the amount of lateral force which is translated to vertical force during cornering. The vertical force causes the chassis to rise of fall and is called the jacking force. Figure 4.10 illustrates the migration of instant centers. Both (illustrative) paths are depicted with dash-dotted lines. This migration happens because the chassis is rolling and deflects the suspension geometry. The lateral tyre forces have to be be applied differently to the chassis when the vehicle is subjected to a lateral acceleration in comparison with the neutral position of the vehicle.

Figure 4.10: Instant centre migration

49 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

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 the same direction. This forces is ’pulling’ on the vector pointing to the instant centre. This contributes also to a negative jacking force. The combination of instant centre migration paths results in a netto vertical force. Some people in the past claimed to have a method which could be used to calculate this 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 very non-lineair at high acceleration levels. Another problem is that the spring/damper configuration is often behaving in a non-lineair way too.

The migration of instant centers in the suspension design is investigated by the change of the anti-roll percentage. The amount of jacking is validated with dynamical simulations, this is described in more detail 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 section can be found in [14]. The steer angle, δ, is often used in these equations and is given by 4.4. Figure 4.11 illustrates 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. This definition 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 steering angle. This will be discussed later.

The kingpin, or steering axis, d, intersects the road surface at point D. The lateral projected distance from 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 between D and A is defined as the castor offset, n. The corresponding distances measured at the height of the wheel centre , M, are the wheel centre offset, rc and the castor offset at wheel centre, nτ . These characteristics refer to the wheel coordinate system, unlike the castor and kingpin angle which are defined is the cross section view of the chassis.

50 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

The steering characteristics of a spatial suspension system such as a multi-link can not be calculated by the geometrical projection of the suspension linkage on the cross sectional plane in side and rear views. The reason for this is, as already discussed before, that the virtual steering axis an instantaneous screw axis is. This axis cannot be considered as a reference axis for forces and moments in the same way 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, on the upright can be used to calculate the steering characteristics.

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

ω  σ = −arctan Ky (4.13) ωKz

ω  τ = −arctan Kx (4.14) ωKz

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 steering angle is equal to δ = arctan (tan(σ)/tan(τ)). The deviation between the z-component of the upright angular 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 of the tyre contact point, vA (see figure 4.11). This point is imaginary fixed to the upright and therefore called 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 radius of a conventional suspension, can than be calculated with:

− (vAxcos(δ) + vAysin(δ)) rs = (4.17) ωδ

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

51 4.5. Suspension kinematic characteristics CHAPTER 4. Kinematic suspension design

However, the definition is given for the moment around the global z-axis which originates at the point where the virtual steering axis (screw axis) crosses the road surface. This is done to correspond to 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 lever arm which generates a moment depending on the traction force. The wheel centre offset has a strong analogy with the scrub radius. It is calculated with:

− (vMxcos(δ) + vMysin(δ)) rc = (4.18) ωδ

The castor offset (sometimes referred as mechanical trail) generates a moment around the steering axis 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:

(v sin(δ) + v cos(δ)) n = Ax Ay (4.19) ωδ

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

(vMxsin(δ) + vMycos(δ)) nτ = (4.20) ωδ

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

v w = − Az (4.21) ωδ

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

dγ tan(τ)cos(δ) + tan(σ)sin(δ) = (4.22) dδ tan(γ) · (tan(τ)sin(δ) − tan(σ)cos(δ)) + 1

52 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 be accomplished by relocation the suspension coordinates. A multi-link suspension has, as discussed earlier, 36 degrees of freedom. The design target consists often of less objectives. But the available space and vehicle geometric layout restricts the range of adaptation of a degree of freedom. Figure 4.12 shows 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 complexity requires 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 often based on graphical methods and manual iterations to accomplish the design objectives. Section 3.6 already discussed a method to handle more complex problems. Numerical optimization techniques are 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 a problem 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 could be, for example, a constraint which says that the z-coordinates of two suspension points have to be equal. An inequality constraint has to be equal or smaller than zero. An inequality constraint can, for example, be used to constrain the suspension coordinates in such a way that a connection rod would not intersect with the wheel rim. It sounds obvious to do this, but this would be mathematically very complex. Especially since the suspension is moving in 3d space regarding two states of motion (bump and steer).

53 4.6. Numerical optimization CHAPTER 4. Kinematic suspension design

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

Two multi-variable algorithms are compared to find the most effective one. The first algorithm is called the Broyden-Fletcher-Goldfarb-Shanno (BFGS, MATLAB : fminunc) method and the second the generalized pattern search (GPS, MATLAB : patternsearch) algorithm which was also used in section 3.6. The last one appears to be in most cases the fastest and most robust algorithm since it does not depend 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 with the corresponding upper and lower boundaries. The starting points are chosen by averaging the upper and 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 calculates the 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 range of the suspension movement and not only as the initial static value. The objective function consists of the following suspension characteristics:

54 4.6. Numerical optimization CHAPTER 4. Kinematic suspension design

suspension characteristic objective weighting factor (priority) bump steer toe curve versus bump travel high camber inclination angle zero at all times high anti roll slightly negative and progressive with bump high travel to reduce jacking (validation with dynamic vehicle model) anti dive (front), or constant value along middle anti squat and rise (rear) the bump travel range scrub radius constant value along the steering range high castor offset constant value along the steering range high castor offset at wheel centre zero at all times low wheel centre offset zero at all times low camber change with high at the static middle steer angle position and degressive with steer angle steering ratio 28 deg steer angle by 120 deg high steering wheel angle instant center location further than 1 meter low away from the vehicle center line to reduce scrub anti center location depending on the front low or rear suspension to reduce the castor change rate and wheel path wheel load lever arm close to zero to reduce middle steering wheel torque, validated with dynamic vehicle model for steering returnability

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

Not every characteristic is equally important. The progress is monitored during the optimization proces by evaluating the static values of the characteristics. This is done to make it possible to tune the weighting factors more easily. The above table includes a priority as the weighting factor. In reality 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 every objective exactly since the number of optimization variables (suspension coordinates) is limited and restricted by boundaries.

The optimization proces changes the input variables iteratively by the proces mentioned in appendix B to accommodate the objectives as close as possible. The end criterium is satisfied when the decrease of 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 targets and the space boundaries. Therefore will it be necessary to adjust the input boundaries, weighting factors and suspension targets during the design proces.

One of the problems which can occur during the optimization proces is that the kinematic model is aborted because it runs into a mechanical lock. This can happen when a combination of suspension coordinates causes the linkage to lock. This is not noticed by the optimization routine because it simply continues to change the input variables. This problem is solved by adding an objective which says 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 the possibility to analyse dynamic vehicle behaviour and to determine suspension forces. The suspension forces 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 high stiffness 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 suspension behaviour. Aspects such as a driver model and aerodynamics are neglected. The model consists of 7 major 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 to change static model parameters such as suspension settings and spring and damper parameters via a 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 the vehicle 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 a simple drive line model. The drive line model consists of a constant gearbox ratio, an engine which is limited 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 via the brake moment distribution p. A static brake gain can be defined which is used to apply a negative moment 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 steer wheel 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 and engine cause an extra lateral load transfer. This load transfer (see figure 4.14 is dependent on the ˙ rotational 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 and inertia’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. From the time period of a swing it is possible to calculate the rotational inertia of the wheel about the point of rotation of the pendulum. The rotational inertia about the point of rotation of the pendulum can be transformed into the rotational inertia at the centre of gravity of the wheel. This can be calculated with (4.24). The rotational inertia at the rotation axis of the pendulum can then be translated to the rotation axis of the wheel centre with (4.25).

 t 2 I = mgr (4.24) o 2π

2 Ic = Io − mr (4.25)

Where: t = period of one oscillation of the swing Io = rotational inertia of the wheel around the axis of rotation of the pendulum Ic = rotational inertia of the wheel around the wheel centre m = mass of the wheel (tyre + rim) g = gravitational acceleration r = 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 a measurement error will occur in the period measurement, even when the period period is averaged on multiple swings. The rotational inertia around the y-axis off a Hoosier 20.5x7.0-R13 tyre with a Keizer rim is estimated on 1.0552 kgm2 and 4.950 kg. The rotational inertia around the x and z-axis is not that 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 both beneficial 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. This inertia is quite small but is of large influence since the rotational speed of the engine can be quite high. The rotational speed of the used engine has the same sign as the wheels. This implies that a counter 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 with an anti-roll bar to assist the roll stiffness. The system is designed with a motion ratio which is as lineair as possible. This results in a more predictable roll behaviour. A progressive motion ratio causes the chassis to rise more during cornering (jacking). The is caused because the outside wheel has a higher stiffness in comparison with a linear motion ratio during cornering, the inner wheel has a lower stiffness. This results in a chassis that moves less to the ground at the outside an lifts more on the inside. A degressive motion ratio results in an decrease of the jacking effect. A linear motion ratio is also beneficial for the calculation of the damper curve which defines the generated damper force depending on the shaft velocity. The motion ratio is defined as:

v MR = A (4.26) vshaft

The dampers are designed such that they can handle a motion ratio up to 0.5. This means that the speed of the damper is half of the speed of the tyre contact patch. Lowering the shaft speed of the damper 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 the damper 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 discussed in section 5.9.

4.8 Suspension collisions

The last tool in the kinematic suspension design is a method to visualize suspension collisions. The optimization 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 locations could 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 two drawbacks, namely that all extreme suspension movements are difficult to draw and that it would take a lot of time to do this for every design iteration. The solution is found in a virtual reality world which is coupled to the dynamic vehicle model. The most important CAD designs of the vehicle are used to visualize 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 by adjusting 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 rim and tyre during a combined steer and bump movement. Such a constraint is difficult to describe in a 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 and connection rods is kept large enough to incorporate tyre deflections.

The visualization is also used to visualize the dynamical behaviour of the vehicle during the performed maneuvers 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 generated forces.

59 Chapter 5

Suspension design considerations and results

5.1 Introduction

The chapter discusses the suspension design considerations and results. The numerical optimization proces 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 is used 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 orientation angles and driver feedback. The depicted figures in this part are a result from the optimization proces and calculated with the theory discussed in chapter 4.

The calculated suspension forces are discussed in section 5.8. This section also discusses the impact forces 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 is discussed in section 5.9. This reduces test time and cost and gives a good initial vehicle balance. The last section compares the performance of a double wishbone suspension with the designed multi-link suspension. The double wishbone suspension was used on the URE03 vehicle in the 2006-2007 season.

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

5.2 Allowable suspension settings

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

60 5.3. Contact patch pressure fluctuationCHAPTER5. Suspension design considerations and results

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

On the front it is possible to make small adjustments to the steering linkage. The steering linkage is designed 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 the outer. This is often called the Ackerman geometry [14]. The adjustment of this setting will have a negative influence on the overal kinematic behaviour and will probably also have very little influence on 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 to the suspension stiffness’s. This section explains the influence of suspension spring stiffness’s on the contact patch pressure fluctuation.

A stiffer suspension is often interpreted by the driver as a better handling vehicle which gives the best feedback. 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 by the contact patch area. Both terms are used in this chapter. The influence of the bump spring stiffness of the vertical tyre load fluctuation is investigated with a simple quarter car vehicle model, see figure 5.1. This model covers the basic dynamics of a tyre and chassis mass. This makes this model very suitable for the understanding of the relation between the spring stiffness and tyre load fluctuation.

xs Sprung mass ms

ks ds

xu Unsprung mass mu

kt zr

Figure 5.1: Quarter car layout

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

61 5.3. Contact patch pressure fluctuationCHAPTER5. Suspension design considerations and results equal to approximately 136000 N/m. The equations of motion are:

msx¨s + ds(x ˙ s − x˙ u) + ks(xs − xu) = 0 (5.1)

mux¨u − ds(x ˙ s − x˙ u) − ks(xs − xu) = −kt(xu − zr) (5.2)

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

Y (s) − m m s4 + (m + m )d s3 + (m + m )k s2
H(s) = = u s u s s u s s (5.3)  4 3 2  Zr(s) mumss +(mu+ms)dss +(mu+ms)kss 2 + mss + dss + ks kt

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

4 4 x 10 x 10 18 18

16 16

14 14

12 12 r r /z /z z z F F ∆ 10 ∆ 10

8 8 transmissibility transmissibility 6 6

4 4

k = 30000 [N/m] [REF] m = 6 [kg] [REF] s u 2 k = 25000 [N/m] 2 m = 4 [kg] s u k = 35000 [N/m] m = 8 [kg] s u 0 0 0 1 2 0 1 2 10 10 10 10 10 10 excitation frequency [Hz] excitation frequency [Hz]

(a) Bump stiffness effect on tyre load fluctuation (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 load which results in more ’grip’, especially in the region between 3 and 20 Hz. But a lower stiffness also compromises 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 kg lower unsprung mass has already a large influence on the transmissibility which also reaches to quite higher excitation frequencies of 100 Hz.

62 5.4. Pitch attitude CHAPTER 5. Suspension design considerations and results

5.4 Pitch attitude The pitch attitude of the vehicle is defined as the pitch motion during accelerating or braking. The suspension 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 during driving. It only implies that the suspension can move that distance. The maximum bump travel is chosen 20 mm. The ground clearance is chosen 30 mm to keep 10 mm ground clearance left at all times.

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 maximum achievable longitudinal acceleration or deceleration is reached. This means than the shaft velocity of the 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. The percentage relates the amount of longitudinal brake force which is transmitted trough the suspension links and the bump springs. The braking force is totally transmitted trough the suspension links and not trough the suspension springs when 100% anti-dive is applied. However, this will never be applied to racing vehicles since it will disturb the vertical tyre load because road irregularities cause an 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 and therefore more grip. Anti-dive percentages lower than 50% are common for racing applications to compromise ’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 is again beneficial for the maximum achievable longitudinal deceleration. This was already discussed in section 3.10. But this amount also influences the feeling of the driver. A larger amount of dive is often interpreted by the driver as a vehicle which gives less feedback. This is another trade off which has to be 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 set to a static value of 31% and is made progressively, see figure 5.3a. The figure is a result from the optimization proces and is plotted against the wheel travel. The progressiveness results in a better contact patch pressure fluctuation when road irregularities are attacking the suspension in its neutral position and also in a better driver feel during braking without losing the effect that the centre of gravity is lowered during braking.

50 14

13.5 45 13

40 12.5

12 35

11.5 Anti lift [%]

Anti dive [%] 30 11

25 10.5

10 20 9.5

15 9 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 wheel travel [m] wheel travel [m]

(a) Anti-dive (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 of longitudinal braking force which is transmitted trough the suspension links which suppresses the lift of the rear of the vehicle. Figure 5.3b depicts the anti-lift percentage. The static value is approximately 12% and quite linear. A higher value is desirable but this is not possible since the anti-lift percentage is 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 it was 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 to much. 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 suspension design 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 of importance 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 achievable lateral acceleration. This reason for this is that a low roll stiffness results in a lower vertical stiffness felt at the tyre during cornering. A lower vertical stiffness is beneficial for the vertical tyre force fluctuation as discussed in section 5.3. And a lower vertical tyre force fluctuation results in higher transmittable lateral forces by the tyre, hence a higher lateral acceleration. But a large roll angle results also is a larger lateral translation of the centre of gravity, meaning that the load transfer to the outside of the vehicle is larger during cornering. A larger roll angle gives also less driver feel during cornering. The last disadvantage 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 behaviour instead of a roll centre height. The anti-roll percentage is defined as the amount of lateral force which is transmitted to the chassis via the suspension links. A larger anti-roll percentage will result in a low roll stiffness. This improves the contact patch pressure fluctuation. But it also generates larger jacking forces, resulting is an increase of the centre of gravity height. The increase of centre of gravity height is not only disadvantageous for the load transfer to the outside of the vehicle during cornering but it affects also the region of operation of the suspension. This means that the vehicle will operate in a different than intended region of the suspension kinematics which results obviously in an unwanted vehicle behaviour. Therefore the design target is to develop a suspension which has no jacking effect at all.

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

64 5.5. Roll attitude CHAPTER 5. Suspension design considerations and results

8 12

6 10

8 4

6 2

4 0 2 −2 Anti roll [%] Anti roll [%] 0

−4 −2

−6 −4

−8 −6

−10 −8 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 wheel travel [m] wheel travel [m]

(a) Anti-roll front (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 outside wheel and the amount of rebound travel of the inside wheel will not be the same in magnitude. As already mentioned, it is very complex to calculate a zero jacking suspension since the generated forces at the tyres behave very non-linear. The roll behaviour is also very complex. The jacking behaviour is therefore validated with dynamical simulations. The depicted anti-roll values (calculated from the instant centre migrations) result in a zero-jacking suspension. Figure 5.5a depicts the axle height at the front and rear axle at the maximum lateral acceleration. Figure 5.5b depicts the centre of gravity height.

0.02

0.015 0.288

0.286 0.01

0.284 0.005

0.282 0 0.28 −0.005 CG height [m] axle height [m] 0.278

−0.01 0.276

−0.015 0.274

−0.02 0.272

−0.025 0.27 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 time [sec] time [sec]

(a) Front and rear axle heights (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 reach a steady vehicle speed. The steer angle is slowly increased between 6 and 9 seconds to achieve the maximal lateral acceleration. It is clearly shown that the axle heights changes are very low. The rear axle 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 lateral tyre force of the outside tyre. The source of the progressive motion ratio is the rocker geometry of the spring/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 acceleration is almost zero and during cornering smaller than 1.6 mm which ensures a constant ride height and a small as possible load transfer caused by the height of the centre of gravity. The driver also experiences a 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, and more particular the difference between the left and right steering angle, influences the behaviour while cornering. It influences the maximal achievable lateral acceleration but also the driver feel and vehicle balance.

Section 3.12 discussed the optimal tyre inclination angle. It appeared that a camber angle of 0 degrees is 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 to accomplish a camber angle. The first way is to design a camber change rate with bump travel. The camber angle results from the location and migration of the instant centre (IC). The second way to generate a camber angle results from the combination of a castor and kingpin inclination angle. The suspension has to be steered, either by applying a steering angle via the steering wheel or with an amount of bump steer, to generate a camber angle. The third way is to apply a static camber angle at the suspensions neutral position. The static camber angle is a suspension setting which can be changed 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 of the chassis does not contribute to a camber angle, but the roll motion (during cornering) does. The roll motion generally generates a positive camber angle at the outside wheels and a negative camber angle at the inside wheels. This could be compromised by designing an instant centre location and migration which results in zero camber during roll. The problem with this is that this camber change also applies to the suspension during braking and driving. The solution is a well designed combination of 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 bump steer amount at the rear it that small that it does not contribute to the camber change. The front suspension has a rather large steering deflection which could be used to compromise the camber change rate and static camber angle.

The results from the calculations on the analytic vehicle model can be used to indicate the steering angle at the maximum lateral acceleration. The steering angle at maximum lateral acceleration is strongly 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 on a corner radius of 100 m. A camber objective could be defined using this value and a maximum roll angle of 1 degree. The optimization proces tries to find a solution to achieve a 0 camber angle during braking/driving and cornering. This only applies to the front suspension since the rear suspension is not steered.

66 5.6. Tyre orientation target CHAPTER 5. Suspension design considerations and results

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

7.5 6

5

7 4

3 6.5

2

6 1 castor [deg] 0 kingpin inclination [deg] 5.5 −1

−2 5

−3

4.5 −4 −30 −20 −10 0 10 20 30 40 −30 −20 −10 0 10 20 30 40 steer angle [deg] steer angle [deg]

(a) Castor angle (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 castor angle 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 camber angle during steering. This is depicted in figure 5.7a. The rate of change during steering is depicted in figure 5.7b. The highest camber change rate during steering is applied to the neutral position of the vehicle. This is done to prevent large camber angles at steering lock. Such result is very difficult to achieve when manual iterations will be used instead of an optimization proces, since the combination of the castor and kingpin migration is responsible for this.

4 0.13

0.12 3 0.11

2 0.1

0.09 1

0.08

camber [deg] 0 0.07

−1 0.06 camber change with steering angle [−]

0.05 −2 0.04

−3 0.03 −30 −20 −10 0 10 20 30 40 −30 −20 −10 0 10 20 30 40 steer angle [deg] steer angle [deg]

(a) Camber angle during steering (b) Camber change rate with steering angle

Figure 5.7: Camber angle with steering

67 5.6. Tyre orientation target CHAPTER 5. Suspension design considerations and results

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

0.6

0.4

0.2

0

camber [deg] −0.2

−0.4

−0.6

−0.8 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 wheel travel [m]

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 right steer angles to achieve the maximum lateral acceleration. It appears that the outside tyre needs to have a larger steer angle than the inner tyre. This could not be achieved with a mechanical steering linkage. It could be done for small steer angles required for cornering at maximum lateral acceleration but this would interfere with the large steering angles needed for tight corners. For this reason the design target is set to a parallel steering linkage. The suspension design has the possibility to adjust the steering kinematics to positive or negative Ackerman. The adjustment would probably have no noticeable 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 operate in another region then intended. This would especially the case for the bump steer amount of the suspension.

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

As basic rule to follow is to design as less as possible bump steer because the vehicle will steer (and therefore 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. These forces 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 for accelerating.

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

68 5.6. Tyre orientation target CHAPTER 5. Suspension design considerations and results

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 this happens.

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 angle for the outside wheel stays approximately zero and has the tendency to generate a small toe in angle when 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 but this is not possible due to the space constraints of the suspension rack location. In fact it is not that important since the inside wheel has a very low wheel load and can not generate large lateral forces at all. The generated lateral force will not cause an under steer behaviour and could only cause the tyre to operate more in the saturated lateral region.

0.1 0.1

0.08 0

0.06

−0.1 0.04

−0.2 0.02

toe [deg] toe [deg] 0 −0.3

−0.02 −0.4

−0.04

−0.5 −0.06

−0.6 −0.08 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 wheel travel [m] wheel travel [m]

(a) Bump steer on the front suspension (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 not be located elsewhere. This is in fact beneficial for the camber compliance of the suspension. The suspension will always have some sort of compliance effect since the suspension can not be infinite stiff. The amount will be small and is neglected in the design. The lower front location causes the suspension to have very low steer compliance since both the lower connection rods and the steer rod will be loaded with the same order of magnitude of lateral force. The suspension compliance could generate a very small toe in angle resulting in less under steer tendency because the steer rod in located in front of the virtual steer axis.

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

The static toe angles should be very close to zero. A small toe in angle could be applied to overcome play in the suspension under tyre drag. This should be tested since play is not taken into account in the 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-link suspension causes the suspension to move in all directions. A numerical optimization proces is a very suitable 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 in another region when, for example, the ride height would be adjusted. This can be seen in figure 5.10 where the bump steer curve is compared with the original curve when the ride height is adjusted with 1 cm. The bump steer amount is very different and causes an unwanted vehicle behaviour.

0.1

0

−0.1

−0.2

−0.3

−0.4 toe [deg]

−0.5

−0.6

−0.7

−0.8 standard ride height adjusted ride height +0.01 [m] −0.9 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 wheel travel [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 subsections and especially the steering wheel torques during driving, braking and cornering. These torques are generated with the tyre forces acting on the steering rack. They are transmitted to the steering wheel via 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 moment around the z-component of the virtual steering axis. The castor offset is calculated in such a way (see

70 5.7. Driver feedback CHAPTER 5. Suspension design considerations and results

subsection 4.5) that it corresponds to the equivalent used in planar suspensions, such as a double wishbone suspension. The sum of the castor offset and the pneumatic trail of the tyre define a lever arm 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 causes a torque on the steering wheel.

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

0.04 0.035

0.03 0.03

0.02 0.025

0.01 0.02

0 0.015

−0.01 0.01 castor offset [m] pneumatic trail [m]

−0.02 0.005

−0.03 0

−0.04 −0.005

−0.05 −0.01 −30 −20 −10 0 10 20 30 40 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 steer angle [deg] side slip angle [deg]

(a) Castor offset front suspesnion (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 to be coupled to the steer wheel torque to give a desired driving impression to the driver. However, the steering wheel torque should not increase progressively to the lateral acceleration because this would lead to high forces at the steering wheel and hence to cramped hands and arms and loss of sensitivity for steering corrections.

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

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

71 5.7. Driver feedback CHAPTER 5. Suspension design considerations and results opposite to the desired one. The simulation does not show a negative steer wheel force, but a problem can occur when driving trough tight corners at low speeds and large steering angles. Tests at the test track can result in the necessity to change the castor offset curve.

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

120

100

80

60 steering wheel force [N]

40

20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 lateral acceleration [g]

Figure 5.12: Steering wheel force

The wheel load lever arm causes also a steering wheel force during cornering. This parameter is not mentioned in the foregoing discussion since it is rather small compared to the effects of the castor offset and pneumatic trail. The wheel load lever arm causes a steering returnability torque which causes the steering wheel to return to its initial straight line position and therefore generating some straight line stability. This returnability is more applicable to passenger cars since those will drive more 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 position under 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.

−3 x 10 1.5

1

0.5

0

−0.5

−1 wheel load lever arm [m]

−1.5

−2

−2.5 −30 −20 −10 0 10 20 30 40 steer angle [deg]

Figure 5.13: Steering returnability with wheel load lever arm

72 5.7. Driver feedback CHAPTER 5. Suspension design considerations and results

The steering returnability is analyzed by performing a cornering maneuver with the dynamic vehicle model. A steering force is applied to perform a cornering maneuver at 72 km/h, see figure 5.14a. The steering force is removed at 1.75 sec. to investigate the steering returnability. Figure 5.14b illustrates the returning of the steering wheel to its initial position. A small overshoot can be seen. The time needed to return the steering wheel to zero is quite small. Here has to be denoted that friction is not modeled. The plot ends in a flat curve indicating the straight line stability.

1 0 0.5

0 −20

−0.5

−40 −1

−1.5 −60

steer angle [deg] −2 steering wheel force [N]

−80 −2.5

−3 −100 −3.5

−120 −4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time [sec] time [sec]

(a) Applied steering wheel force (b) Steering returnability

Figure 5.14: Steering returnability Braking will also induce a steering wheel torque since the scrub radius acts as a lever arm to generate a moment around the z-component of the virtual steering axis. Braking on a straight line will not induce any steering wheel torque since the left and right tyre scrub radius are identical. However, braking at corner entry results in a steering torque since the outer tyre will be loaded more. A multi-link suspension has the ability to design a negative scrub radius. This results in a decrease of the steering wheel torque which gives the driver a strange feedback since he or she expects an increase. The scrub radius is kept positive at all times to accomplish this. This is depicted in figure 5.15a. Figure 5.15b illustrates the increased steering wheel force when braking at corner entry when the simulation time is 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 sensitive feel for braking.

0.035 140

120 0.03

100 0.025

80 0.02

60

0.015 scrub radius [m] 40 steering wheel force [N]

0.01 20

0.005 0

0 −20 −30 −20 −10 0 10 20 30 40 0 2 4 6 8 10 12 14 steer angle [deg] time [sec]

(a) Scrub radius (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 during several 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 curb stones or other obstacles results in one of the highest occurring suspension loads. The tyre model discussed in chapter 2 is based on the Magic Formula tyre model which is not very suitable to simulate driving over bumps or cleats. TNO Automotive has developed an extension to the tyre model which incorporates high frequency tyre behaviour (up to 60 Hz) and tyre envelopment properties. The Short Wavelength Intermediate Frequency Tyre Model (SWIFT, [21]) makes it possible to do an impact force analysis 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 the ground clearance is only 30 mm. The larger beam is used to simulate a worst case scenario. Figure 5.16a illustrates the 3D SWIFT enveloping model. The coefficients are not fitted on measurement data but are estimated by the tyre model to give an indication of the behaviour. Figure 5.16b depicts the difference in the obstacle force between the Magic Formula (dotted line) and the SWIFT (solid line) model.

9000 SWIFT tyre model Magic Formula tyre model 8000

7000

6000

5000

4000

pull rod force front left [N] 3000

2000

1000

0 98 100 102 104 106 108 110 distance [m]

(a) 3D SWIFT 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 the SWIFT model. This is obvious since the tyre rolls onto the obstacle with the SWIFT model. The Magic Formula model gets a step input to the tyre contact point when it hits the obstacle. The SWIFT model shows some more oscillation after the obstacle which is probably caused by the rigid ring dynamics of the 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 rod material 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 a monocoque 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 spring stiffness’s and the static camber angles. Four spring parameters can be distinguished, namely the bump stiffness front and rear and the roll stiffness front and rear. The bump springs are displaced when the vehicle is accelerated both in the longitudinal and lateral direction whereas the anti-roll bar is only displaced in the lateral direction. The choice of the roll stiffness’s is primary dependent on the design choice of a maximum roll angle of 1 degree. The bump stiffness is a compromise between driver feel and contact patch pressure fluctuation.

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

d ξ = √ s 2 msks (5.4) p ds = ξ · 2 msks

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

r s 1 ks 1 30000 frf = = = 3.2 Hz (5.5) 2π ms 2π 300/4

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 designing racing 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 motion such 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 and is chosen much lower. This results in a well balanced compromise between driver feedback and tyre load fluctuation.

The roll stiffness is easier to choose because a maximum roll angle of 1 degrees is allowed to keep the suspension in its intended operation range. The ratio between the front and rear roll stiffness is responsible for the tyre load distribution during cornering and therefore the balance of the vehicle as already discussed earlier. The ratio is chosen to result in a neutral steered vehicle. Tests at the test track should be used to fine tune the setting which can be done in infinite small steps. The motion ratio 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 the remaining camber angles while cornering when the initial camber angles are zero.

1 camber angle front left [deg] camber angle front right [deg] 0.8 camber angle rear left [deg] camber angle rear right [deg] 0.6

0.4

0.2

0 camber [deg] −0.2

−0.4

−0.6

−0.8

−1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 lateral acceleration a [g] y

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 camber angles 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 URE03 vehicle to validate the new design considerations and kinematics. In the comparison is chosen for the same track widths, wheelbase, centre of gravity location, tyres, masses and inertia’s. The only variable in the comparison is the kinematic difference.

One of the important suspension design targets was to develop a zero jacking suspension. Figure 5.18a and 5.18b illustrates the difference between the jacking attitude during cornering of the old URE03 double wishbone suspension and the new multi-link design.

76 5.10. Double wishbone comparison CHAPTER 5. Suspension design considerations and results

0.025 axle height front URE04 [m] center of gravity height URE04 [m] axle height rear URE04 [m] center of gravity height URE03 [m] axle height front URE03 [m] 0.296 axle height rear URE03 [m] 0.02 0.294

0.292

0.015 0.29

0.288 CG height [m] axle height [m] 0.01

0.286

0.284 0.005

0.282

0 0.28 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 lateral acceleration a [g] lateral acceleration a [g] y y

(a) Axle height comparison (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 in another than intended region. The jacking effect is also quite progressive when the lateral acceleration is 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 intended operating range of the suspension since the values are quite small. Figure 5.18b illustrates the centre of gravity height change which is almost 7 times higher in the old case. This is a good improvement which is rather difficult to realize.

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

77 5.10. Double wishbone comparison CHAPTER 5. Suspension design considerations and results

0.8 2 camber angle front left URE04 [deg] camber angle rear left URE04 [deg] 0.7 camber angle front right URE04 [deg] camber angle rear right URE04 [deg] camber angle front left URE03 [deg] camber angle rear left URE03 [deg] camber angle front right URE03 [deg] camber angle rear right URE03 [deg] 1.5 0.6

0.5

1 0.4

0.3 0.5 0.2 camber [deg] camber [deg]

0.1 0

0

−0.1 −0.5

−0.2

−1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 lateral acceleration a [g] lateral acceleration a [g] y y

(a) Front camber angles during cornering (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 difference between 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 indication of the lateral acceleration and more important no indication that the vehicle limit is reached. The new suspension 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 sensitive feel for steering corrections.

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

120

100

80

60 steering wheel force [N]

40

20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 lateral acceleration a [g] y

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 URE04 FSAE 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 dynamic vehicle model. The tyre models are crucial for the basic understanding of vehicle dynamics. Several tyres are modeled to make a substantiated choice. The fit accuracy could be better but is accurate enough 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 performance envelope shows that the vehicle design strategy should primary be focussed on lowering the centre of gravity height and secondly on vehicle mass. The model has also other applications, such as finding optimal steering angles. The model certainly has potential for further research. The tyre model shows that 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 side walls. Future research should indicate if this is true.

The kinematics of the multi-link suspension are based on the results from the analytic vehicle model and the tyre models. New design tools are developed to design the suspension kinematics. The use of numerical optimization techniques makes it possible to design superior kinematics even with the tight space boundaries, especially on the rear. The next list summarizes the improvements in the suspension 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 static camber 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 given space boundaries. • Harmonization of the castor offset curve, scrub radius curve and steering geometry resulting in 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 the steering 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 of the chassis and low steering forces. • Sensitive feel for braking at corner entry trough the increase on steering force caused by the migration of the scrub radius

The above list of improvements is possible by using numerical optimization techniques for the design of a multi-link suspension. The use of a multi-link suspension has also the advantage that there are no buckling 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 improvement compared with older designs. A potential problem could be that the steering forces get negative at large 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 validate this. Another conclusion which that can be drawn from the analysis of the suspension kinematics is that changing suspension settings such as the ride height, steering rack location, suspension pick-up points 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 camber angles 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 the UREO4 vehicle is reached. Physical test at the test track can be used to make more improvements for the 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 track with suitable hardware. The next list gives an indication of measurements which can be performed to develop 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 the vehicle. The measurement data can be used to change the migration of the instant centers to accommodate 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 the pitch behaviour of the vehicle, either to achieve better vertical tyre force fluctuations or to give the driver a move confident feel.

• It is also advisable to purchase a steering torque sensor. These measurements can be used to adjust 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 of the vehicle, especially at corner entry and exit. • Infrared tyre temperature sensors placed across the tread area on the outside, middle and inner side 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 as possible. This measurement can be used to adapt the suspension kinematics to optimize the camber 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 to use different tyres which perform better when cold.

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

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

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

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

The chassis and in fact every vehicle component has a stiffness which is disregarded in the performed simulations. A superior dynamic model could be made when finite element analysis is coupled with multi-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 computational capacity 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, Vehicle System 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 principles explanation 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 lateral acceleration ay times the driven radius R :

p V = ay · R (A.1)

Here the yaw velocity r is equal to:

V r = (A.2) R

The vehicle is driving around on a fixed driving radius. Increasing the lateral acceleration is done by increasing 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 from the front track to the centre of gravity and l the wheelbase.

 h − h   h − h  h = h − a · 2 1 + h · cos atan 2 1 (A.3) ra 1 l 1 l

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):

a · m · h ϕ = y ra (A.4) cϕ1 + cϕ2 − m · g · hra

The following equations are given for one wheel/tyre only. The speed in longitudinal (u) and lateral direction (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

CG

RC rear hra

RC front RA h h2

h x 1

ε

a1 a2 l

RA = Roll Axis CG = CG at height h and perpendicular distance h ra from the RA h1 h2 = Front and rear roll center heights

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 speeds need 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 longitudinal and lateral speeds:

  Vsy1 α1 = atan (A.9) Vx1

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), the amount 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 (A.15) − Fx3 chassis · s3 − Fy3 chassis · a3 + Mz3

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

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

((Fy1 chassis + Fy2 chassis) · h1 + cϕ1 · ϕ) ∆Fz roll front = (A.16) 2 · s1

h ∆F = · F (A.17) z brake/drive 2l x total

The normal loads of the tyres are calculated according to:

a F = 2 · m · g + ∆F − ∆F (A.18) z1 2l z roll front z brake/drive

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 information about the gradient of the objective function. Unlike more traditional optimization methods that use information about the gradient or higher derivatives to search for an optimal point, a direct search algorithm 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 used to 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 the generalized pattern search (GPS) algorithm and the mesh adaptive search (MADS) algorithm. Both are 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 point the point computed at the previous step of the algorithm. The mesh is formed by adding the current point to a scalar multiple of a set of vectors called a pattern. If the pattern search algorithm finds a point in the mesh that improves the objective function at the current point, the new point becomes the current point at the next step of the algorithm. The MADS algorithm is a modification of the GPS algorithm. The algorithms differ in how the set of points forming the mesh is computed. The GPS algorithm uses fixed direction vectors, whereas the MADS algorithm uses a random selection of vectors 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 to   the 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. The following figure shows the value of an example objective function at the initial point and mesh points.

Objective function values at initial points and mesh points 3

5.6347

2.5

2

4.5146 4.6347 4.782

1.5 optimization variable 2

1

3.6347

0.5 1 1.5 2 2.5 3 3.5 optimization variable 1

Figure B.1: Initial points and mesh points

The algorithm polls the mesh points by computing their objective function values until it finds one whose 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 objective function value is smaller than that of the current point. Consequently, the algorithm might not poll all 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 initial mesh size is 1, at the second iteration the mesh size is 2. The mesh at iteration 2 contains the following points:

    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 values of the example objective function.

The algorithm polls the mesh points until it finds one whose value is smaller than 4.5146, the value at   x1. 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 CHAPTER B. Optimization algorithm

Objective function values at x and mesh points 1 4

6.5416 3.5

3

2.5

2

3.25 4.5146 4.7282 1.5 optimization variable 2 1

0.5

0

3.1146 −0.5 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 optimization variable 1

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 CHAPTER B. Optimization algorithm

Objective function values at x and mesh points 3

10 7.7351

8

6

4 optimization variable 2

2 4.7282 64.11 −0.2649

0

4.3351 −2 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 optimization variable 1

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 next iteration. That is, x4 = x3.

At the next iteration, the algorithm multiplies the current mesh size by 0.5, so that the mesh size at the 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-coordinates with -1.

Coordinates labels front = front left suspension rear = rear left suspension chassis = chassis coordinate upright = upright coordinate wc = 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 d.front.chassis.p3 = 1.8905 0.2315 0.3643 d.front.chassis.p4 = 1.3915 0.2478 0.2837 d.front.chassis.p5 = 1.7619 0.2431 0.1613 d.front.chassis.p6 = 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 d.front.upright.p3 = 1.6122 0.5415 0.3588 d.front.upright.p4 = 1.5378 0.5614 0.3463 d.front.upright.p5 = 1.6495 0.5536 0.1762 d.front.upright.p6 = 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 d.rear.chassis.p3 = 0.4993 0.2818 0.3510 d.rear.chassis.p4 = −0.1800 0.1902 0.3198 d.rear.chassis.p5 = −0.1800 0.1336 0.2228 d.rear.chassis.p6 = −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 d.rear.upright.p3 = 0.0156 0.5161 0.3595 d.rear.upright.p4 = −0.0537 0.5200 0.3600 d.rear.upright.p5 = −0.0715 0.5175 0.2580 d.rear.upright.p6 = 0.0000 0.5175 0.1600 d.rear.upright.wc = 0.0000 0.5875 0.2550

Spring stiffness’s Front and rear mono-shock spring: 61000N/m Front anti-roll bar: 1800 Nm/rad Rear anti-roll bar: 800 Nm/rad

Initial suspension alignment Front static camber: -0.3 deg Front static toe IN: 0.1 deg

92 CHAPTER C. Suspension coordinates

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

93 Appendix D

Quarter car model derivation

xs Sprung mass ms

ks ds

xu Unsprung mass mu

kt zr

Figure D.1: Quarter car layout

Vertical tyre force:

Fz = kt(xu − zr) (D.1)

Equations of motion:

msx¨s + ds(x ˙ s − x˙ u) + ks(xs − xu) = 0 (D.2)

mux¨u − ds(x ˙ s − x˙ u) − ks(xs − xu) = −kt(xu − zr) (D.3)

Laplace transform: 2 (mss + dss + ks)Xs − (dss + ks)Xu = 0 (D.4)

94 CHAPTER D. Quarter car model derivation

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

dss + ks Xs = 2 Xu (D.6) mss + dss + ks

  2 (dss + ks)(dss + ks) (mus + dss + ks)Xu − 2 Xu = −kt(Xu − Zr) (D.7) mss + dss + ks

  2 (dss + ks)(dss + ks) mus + dss + ks − 2 Xu = −kt(Xu − Zr) (D.8) mss + dss + ks

 4 3 2  mumss + (mu + ms)dss + (mu + ms)kss 2 Xu = −kt(Xu − Zr) (D.9) mss + dss + ks

Y = kt(Xu − Zr) (D.10)

 4 3 2  mumss + (mu + ms)dss + (mu + ms)kss 2 Xu = −Y (D.11) mss + dss + ks

Y Xu = + Zr (D.12) kt

 4 3 2    mumss + (mu + ms)dss + (mu + ms)kss Y 2 + Zr = −Y (D.13) mss + dss + ks kt

m m s4 + (m + m )d s3 + (m + m )k s2  u s u s s u s s + 1 Y = (m s2 + d s + k ) k s s s t (D.14)  4 3 2  mumss + (mu + ms)dss + (mu + ms)kss − 2 Zr mss + dss + ks

 4 3 2  mumss + (mu + ms)dss + (mu + ms)kss 2 + mss + dss + ks Y = kt (D.15) 4 3 2
− mumss + (mu + ms)dss + (mu + ms)kss Zr

Y (s) − m m s4 + (m + m )d s3 + (m + m )k s2
H(s) = = u s u s s u s s (D.16)  4 3 2  Zr(s) mumss +(mu+ms)dss +(mu+ms)kss 2 + mss + dss + ks kt

95