Energy Consumption Estimation in Lap Time Simulation

José Filipe Dias Duque Marques Loureiro

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. Alexandra Bento MoutinhoProf. Luís Alberto Gonçalves de Sousa

Examination Committee

Chairperson: Prof. Paulo Jorge Coelho Ramalho OliveiraSupervisor: Prof. Luís Alberto Gonçalves de Sousa

Member of the Committee: Prof. Ricardo José Fontes Portal

June 2018

ii

Para ti Francisca

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iv

Acknowledgments

I want to thank everyone at Adess-AG for the opportunity of developing this work with the team. Thank

you to Professor Luıs Sousa and Professor Alexandra Moutinho for the availability and Sofia Torres for

the indispensable help in the start of this project.

Obrigado a todos os meus amigos, com quem passei este ciclo de estudos que termina agora com

esta Tese de Mestrado, e a minha famılia, pelo apoio.

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Resumo

Um grande desafio no projeto de veıculos totalmente eletricos e a autonomia. As baterias podem ser

parte significativa no peso total do veıculo e por isso devem ser corretamente dimensionadas para o

armazenamento mınimo. Esta tese tem como objetivo implementar estimacao de consumo energetico

num simulador de tempos por volta (LTS) para veıculos de competicao automovel e compreender como

a complexidade do modelo de veıculo afeta os resultados da simulacao e da estimacao.

Um prototipo de categoria LMP3 e parametrizado em dois simuladores quasi-estaticos. Um e um

software comercial composto por um modelo de veıculo de 4 rodas. O segundo foi desenvolvido no

ambito deste trabalho como uma alternativa mais simples composta por um modelo de 1 roda. Este

trabalho documenta as tomadas de decisao no desenvolvimento de um simulador quasi-estatico e as

simplificacoes feitas num modelo de 1 roda. Foram recolhidos dados num teste no Autodromo do Estoril,

com a melhor volta a servir de validacao para os dois simuladores.

Foi feita uma analise de sensibilidade aos parametros do modelo de 1 roda para validar a sua

utilidade. Comparando com os dados experimentais mostra-se que o modelo de 1 roda, muito menos

complexo e mais facil de implementar, apresenta resultados satisfatorios em relacao ao modelo de 4

rodas, calculando tempos por volta com um erro de 7%. Assim, um modelo de veıculo simples pode

ser facilmente implementado num LTS para estimar consumo energetico em veıculos de competicao. A

maior fonte de discrepancias esta na simplificacao do modelo de pneus.

Palavras-chave: Competicao automovel, Consumo energetico, Quasi-Estatico, Modelo de

1 Roda, Simulacao

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Abstract

A big challenge in the design of a battery electric vehicle is the mileage. Batteries can make up to a

significant part of the vehicle’s weight and so they must be properly designed for the minimum storage

needed. This thesis aims to implement energy consumption estimation in a Lap Time Simulator (LTS)

for motorsport vehicles and to understand how the complexity of the vehicle model affects the simulation

and estimation results.

A LMP3 category prototype racing car is parametrized in two Quasi-Steady-State simulators. One is

a commercial software composed of a 4-wheel vehicle model. The second was developed in the scope

of this work as a simpler alternative composed of a Point-Mass model. This thesis details the decision

making behind the development of a Quasi-Steady-State simulator and the simplifications made in the

Point-Mass model. Data was collected in a test day at Autodromo do Estoril with the best lap serving as

validation for both simulators.

A sensitivity analysis is done to parameters of the Point-Mass model to validate its utility. The re-

sults with the experimental data show that the less complex and much easier to implement Point-Mass

model has satisfactory results when compared to the 4-wheel model, calculating lap time with 7% error.

Therefore, a simple and easy to implement vehicle model can be applied to LTS to estimate energy

consumption in motorsport vehicles. The bigger source of discrepancies lies on the simplification in the

tyres model.

Keywords: Motorsport, Energy Consumption, Quasi-Steady-State, Point-Mass, Simulation

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction 1

1.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Theoretical Overview of Lap Time Simulators . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Energy Consumption of Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Vehicle Dynamics Fundamentals and Terminology 9

2.1 Axis System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Tyre Mechanics Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Slip Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.3 Camber and Inclination Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.4 Pacejka’s Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Energy Consumption Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Implemented Simulations 17

3.1 Point-Mass Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Simulation Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Quasi Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 RaceLap Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Parametrization of a Vehicle in RaceLap . . . . . . . . . . . . . . . . . . . . . . . . 27

xi

4 Point-Mass Simulator Choices 31

4.1 Simulation type: Steady State vs Quasi Steady State . . . . . . . . . . . . . . . . . . . . . 31

4.2 Track Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Results 39

5.1 RaceLap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1.1 Front Grip Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1.2 Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Point-Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.1 PMSim Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.2 Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Point-Mass Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3.1 Drag Force (FD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.2 Longitudinal Acceleration (ax) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3.3 Downforce (Fz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.4 Lateral Acceleration (ay) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Estimator Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4.1 Linear Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4.2 Aerodynamic Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4.3 Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Conclusions 67

6.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Bibliography 69

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List of Tables

5.1 RaceLap main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 PMSim main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 PMSim main results errors with TF = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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List of Figures

1.1 The 7th prototype of FST Lisboa (left) and an Adess AG LMP3 (right). . . . . . . . . . . . 3

1.2 Example of a Plug-to-Wheel Sankey Diagram. . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Vehicle’s Coordinate Axis System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Side slip angle when describing a corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 An hairpin corner with selected points to describe it. . . . . . . . . . . . . . . . . . . . . . 11

2.4 Variation of ay and ψ in a hairpin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 SAE Tyre Axis System. Top view (left) and Rear view (right). . . . . . . . . . . . . . . . . . 12

2.6 Slip angle tyre deformation and pressure distribution in a bottom view. . . . . . . . . . . . 12

2.7 Normal stresses distribution on a tyre’s tread and consequent shift in the normal force

application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Inclination Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.9 Generic form of a Fy(α) curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.10 Forces acting on the vehicle in a hill climb. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Examples of µx(Fz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Engine Torque Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Limiting force and gear selected in a straight line acceleration. . . . . . . . . . . . . . . . 20

3.4 Force Balance on the Point Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Acceleration and Velocity through three segments of a discretized track. . . . . . . . . . . 21

3.6 Velocity profiles from Forward and Reverse Simulations. Resulting lap velocity profile in

dotted black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Flow chart for the main file of PMSim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.8 Flow chart of the maxcorvel function of PMSim. . . . . . . . . . . . . . . . . . . . . . . . . 25

3.9 Example of a g-g Diagram for AD03 at 40m/s . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 Map of a straight followed by a corner with constant radius (in blue) and a corner that

changes radius (in orange). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Velocity profile when entering and describing a corner with constant radius. . . . . . . . . 32

4.3 Velocity profile when entering and describing a corner with decreasing radius. . . . . . . . 33

4.4 AD03 g-g diagram for 34.58 m/s and 59.57 m/s . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Resulting velocity profiles for both type of track input. . . . . . . . . . . . . . . . . . . . . . 35

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4.6 Resulting velocity profiles for the single weighted average tyre method and the two tyres

with weight transfer method for simulating two different tyres in a Point-Mass model. . . . 37

5.1 Velocity profile of the best lap performed in a AD03 at the Estoril track with the lap time of

1:38.295. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Real and RaceLap simulated laps for different front tyres grip factor. . . . . . . . . . . . . 41

5.3 Real and RaceLap simulated laps for 1.1 and 1.4 front tyres grip factor. . . . . . . . . . . 42

5.4 Velocity profiles of both RaceLap simulation (1:39.588) and the real lap (1:38.295). . . . . 43

5.5 Select Track window in PMSim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.6 Results window for the Estoril circuit with the correspondent corner names on the velocity

profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.7 Autodromo do Estoril circuit map. Source: Wikipedia [14]. . . . . . . . . . . . . . . . . . . 46

5.8 Results window for the La Sarthe circuit with the correspondent corner names on the

velocity profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.9 La Sarthe circuit map. Source: Wikipedia [17]. . . . . . . . . . . . . . . . . . . . . . . . . 47

5.10 Velocity profiles of both Point-Mass simulation (1:31.507) and the real lap (1:38.295). . . . 48

5.11 PMSim and RaceLap velocity profiles in a lap at Estoril. . . . . . . . . . . . . . . . . . . . 50

5.12 SCX variation in a lap in RaceLap. Mean value in horizontal line. . . . . . . . . . . . . . . 51

5.13 RaceLap FD profile (blue) and equation (3.5a) applied for RaceLap velocity profile and

constant SCX (orange). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.14 Lap Time sensitivity analysis with drag coefficient. . . . . . . . . . . . . . . . . . . . . . . 52

5.15 Top Speed sensitivity analysis with drag coefficient. . . . . . . . . . . . . . . . . . . . . . 53

5.16 Longitudinal acceleration profile in RaceLap and PMSim. . . . . . . . . . . . . . . . . . . 54

5.17 Lap Time sensitivity analysis with Torque Factor. . . . . . . . . . . . . . . . . . . . . . . . 55

5.18 Top Speed sensitivity analysis with Torque Factor. . . . . . . . . . . . . . . . . . . . . . . 55

5.19 Lap Time sensitivity analysis with longitudinal friction coefficient parameters mx and bx. . 56

5.20 SCZ variation in a lap in RaceLap. Mean value in horizontal line. . . . . . . . . . . . . . . 58

5.21 RaceLap Fz profile (blue) and equation (3.5b) applied for RaceLap velocity profile and

constant SCZ (orange). AD03 static mass in horizontal line. . . . . . . . . . . . . . . . . . 59

5.22 Lap Time sensitivity analysis with downforce coefficient. . . . . . . . . . . . . . . . . . . . 59

5.23 Lateral acceleration profile in RaceLap and PMSim. . . . . . . . . . . . . . . . . . . . . . 61

5.24 Lap Time sensitivity analysis with lateral friction coefficient parameters my and by. . . . . 62

5.25 Energy Consumption Estimator sensitivity analysis with Fx variation in fraction of the de-

fault values of Sections 5.1 and 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.26 Energy Consumption Estimator sensitivity analysis with drag coefficient variation in frac-

tion of the default values of Sections 5.1 and 5.2. . . . . . . . . . . . . . . . . . . . . . . . 64

5.27 Energy Consumption Estimator sensitivity analysis with rolling resistance coefficient vari-

ation in fraction of the default values of Sections 5.1 and 5.2. . . . . . . . . . . . . . . . . 65

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List of Symbols

Convention

a scalar

a vector

Greek symbols

α Tyre slip angle.

β Vehicle side-slip angle.

µ Coefficient of friction.

ω Angular velocity.

φ Roll angle.

ψ Yaw angle.

ρ Air density.

σ Normal stress.

τ Shear stress.

θ Pitch angle.

ζ Road inclination angle.

Roman symbols

Af Frontal Area.

CD Coefficient of drag.

CL Coefficient of lift.

E Energy.

F Force.

FD Drag force.

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Fe Engine propuslsion force.

FL Lift force.

Ft Tyre force.

Fai Angular inertia force.

Fhc Hill climb force.

Fli Linear inertia force.

Frr Rolling resistance force.

I Moment of inertia.

Mrr Rolling resistance moment.

P Power.

SCX Normalized drag coefficient.

SCZ Normalized lift coefficient.

T Torque.

a Acceleration.

d Distance.

frr Rolling resistance coefficient.

g Gravitational acceleration.

gr Gear ratio.

m Mass.

mf Fictive mass of rolling inertia.

r Radius.

rc Corner radius.

rw Wheel radius.

t Time.

v Velocity.

w Weight.

Subscripts

i, j, k Computational indexes.

xviii

x, y, z Cartesian components: Longitudinal, lateral and vertical, respectively.

F Front.

R Rear.

Superscripts

max Maximum.

min Minimum.

rev Reverse.

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xx

Chapter 1

Introduction

Motorsport is used as the benchmark for testing new technologies for the automotive industry. The

goal of any motorsport event is to complete a certain distance of a pre-defined path in the minimum

time possible in a motoring vehicle. Racing tests the vehicle’s limits of performance and allows to

gain empiric knowledge about the components designed to a car or motorcycle that can be put into a

production vehicle to be sold for the general public. The sport component comes from the competition

between teams and drivers to be the fastest in a certain circuit, section or stage.

Ideally, within a certain regulation, a racing car will have adjustable parameters to make it more

efficient in different venues and so the right set-up must be discovered. In the present times, most

championships have restrictions in extra tests outside of a racing weekend to lower the budget needed

to compete. Simulation of competition vehicles comes to suppress the need to know how the variation of

set-up parameters translate to on-track behaviour with limited testing. This way, simulation accelerates

the development of a vehicle either by giving initial inputs to design or to short cut the process of testing.

Concerning Formula Student

Formula SAE (FSAE) is a competition organized by SAE International (former SAE, Society of Auto-

motive Engineers) where teams based in universities are challenged to design and build a prototype of

a small size formula race car. Students compete in static and dynamic events, all with specific scoring,

where they must present their work and race their car. The static events include:

• Technical Inspection - scrutineering to determine if the vehicle conforms with the FSAE Rules, the

competition regulations;

• Business Case [75 points] - a presentation of a business plan on why a company can produce and

sell the team’s prototype race car to an amateur, weekend competition market;

• Cost and Manufacturing Event [100 points] - includes a report, discussion and ”real case” sce-

nario to evaluate the cost of the car and the team’s ability to prepare accurate engineering and

manufacturing cost estimates;

• Design Event [150 points] - evaluation of the engineering decisions behind the design of the car.

1

The previous points are brief definitions of the static events evaluated by judges with motorsport and

automotive industry professional experience. In the dynamic events the car must perform on a race track

while maintaining its mechanical integrity. The dynamic events are:

• Acceleration [100 points] - meant to test the car’s longitudinal acceleration, it consists in acceler-

ating in a straight line for 75 meters;

• Skid-Pad [75 points]- meant to test the car’s lateral acceleration, in this event the car must run in

two constant radius turns, twice for each clock direction, ending describing an eight (8) figure;

• Autocross [125 points] - racing alone on a race track, the car must complete a single lap in the

minimum possible time;

• Endurance and Efficiency [275 + 100 points] - comprising two events run at the same time, consists

of completing approximately 22 km in a race track in the minimum possible time (Endurance) at

the same time as competing to be the most fuel efficient possible (Efficiency).

The maximum score of each event can vary between competitions.

The Motivation for this thesis comes from the Tecnico Lisboa FSAE team: FST Lisboa. The team

designs and builds a formula with fully electric powertrain. This means the energy that powers the motors

comes from electric batteries. The weight of the battery pack is a main parameter when designing the

car for currently it makes up for about 30% of the total weight of the vehicle. The energy storage system

must then be designed for maximum performance at the longer event in Formula Student competitions -

the 22 km of the endurance event - and no more than this.

Later I got the opportunity to keep developing my work at Adess AG, presented next, faced with the

challenge of applying this study to a Le Mans Prototype (LMP) race car.

Concerning Le Mans Prototypes and Adess AG

A Le Mans Prototype (LMP) is a type of sports prototype race cars that race in the main endurance

championships in the world: FIA World Endurance Championship (WEC), IMSA WeatherTech SportsCar

Championship (IMSA, based in the USA), European Le Mans Series (ELMS) and Asian Le Mans Series

(ALMS). LMP’s are currently divided in three categories: LMP1, LMP2 and LMP3. LMP1 is the fastest

category of race cars bellow formulas, with LMP2 and LMP3 being of the same type but of decreasingly

performance.

LMP3 is the lower level of endurance prototype racing, aimed at amateur or young professional

drivers as the first step towards the higher categories. Of the previuosly mentioned championships,

LMP3 are currently racing in all except the WEC. The cars must be of closed cockpit with carbon chassis,

tubular steel roll cage and powered by a 420 bhp normally aspirated Nissan V8 engine developed by

Oreca, a french based motorsport company.

Advanced Design and Engineering Systems Solutions (Adess AG) is a company devoted to design

and build racing cars founded by Stephane Chosse in 2012. Past projects of the company include

the Lotus T128 LMP2 that raced in the World Endurance Championship 2013 season, aerodynamic

2

development for the HRT Formula 1 team and the monocoque development of the 2014 Garage 56 1

project in the 24 Hours of Le Mans, the Nissan ZEOD RC. The LMP3 designed by Adess AG is called

the Adess 03 (AD03).

(a) FST07e (b) AD03

Figure 1.1: The 7th prototype of FST Lisboa (left) and an Adess AG LMP3 (right).

1.1 State of the art

A Lap Time Simulator (LTS) is a tool in vehicle design applied to motorsport. The goal of such tool is

to find the best possible lap time of a certain vehicle in a certain track and understand how the variation

of some of its parameters influence this lap time. Siegler [1] describes a LTS as a vehicle handling

model that connects many different manoeuvres all performed at the limit of the vehicle. The vehicle is

simulated to be at its limit at all points of a track: maximum lateral acceleration in corners and maximum

longitudinal acceleration in straights, thus the best possible lap time.

According to Siegler [1] the first try at a LTS was done by Mercedes-Benz in 1954. This first simulation

consisted of individual hand-made calculations for certain sectors of the track and summing the resulting

sector times to get the predicted lap time. Equations of motion were hand derived and increasingly

solved by digital computers until the 1980’s. Meanwhile, in 1971, the first example of a g-g diagram

was published, making way to the development of quasi-steady-state simulations (explained in 1.1.1).

Milliken improved the quasi-steady-state simulations using the ”bicycle” model [2].

While LTS software got more sophisticated, tyre manufacturers also improved their tests to get more

precise data about the behaviour of tyres in different conditions. In order to read and implement equa-

tions that describe tyre behaviour more accurately, tyre models were developed, like the Pacejka Magic

Tyre Formula [3]. Transient Simulations appeared with Casanova in 2001 [4], being the latest major

improvement in LTS techniques.

1An entry at the 24 Hours of Le Mans race for an innovative project with a car that tests new technologies outside of thetechnical, regulations. This idea started in 2012 and the project is selected by the race organization every year.

3

1.1.1 Theoretical Overview of Lap Time Simulators

A race track is modelled in segments that can be corners or straights. The segmentation of the track

can be done previously by knowing the layout and predict an approximation of the real trajectory or

from actual on-track data, getting the radius of trajectory in each sample of the data. In this last case

there is no distinction between straights or corners because all segments will have a radius value but for

increasingly larger values of radius, the simulated car will behave as in a straight line.

From a planar dynamics point of view, when a vehicle is turning there is yaw movement (rotation

about the vertical axis). This rotation is described by equations of motion considering the forces gener-

ated at the tyres. Simulators differ in the way they solve the equations of motion. A cornering vehicle

is said to be in equilibrium when the sum of yaw moment around the CG is null and therefore the yaw

acceleration is also null and the rate of rotation is constant. This equilibrium corresponds to the steady-

state condition. Another way to solve the equations of motion is to consider the change of yaw moment

in which case the simulation is said to be transient. As explained in Chapter 2, in the process of describ-

ing a corner, a real vehicle goes from transient to steady-state back to transient again. The steady-state

condition happens at the appex only.

Ignoring longitudinal forces, the equations of motion are, for the steady-state case

∑Fy = m ay ⇔ FyFL

+ FyFR+ FyRL

+ FyRR= m ay (1.1a)∑

Mψ = 0⇔ (FyFL+ FyFR

) a− (FyRL+ FyRR

) b = 0 (1.1b)

and for the transient case

∑Fy = m ay ⇔ FyFL

+ FyFR+ FyRL

+ FyRR= m ay (1.2a)

∑Mψ = Izz

d2ψ

dt2⇔ (FyFL

+ FyFR) a− (FyRL

+ FyRR) b = Izz

d2ψ

dt2(1.2b)

where a and b are the distances from the front an rear axles to the vehicle’s CG, respectively. Section

2.1 presents the coordinate axis system used in this work.

Neves [5] explains, by the equations of non-uniform circular motion, that when describing a corner

with longitudinal (x-axis or tangent to motion) acceleration different from zero, the yaw acceleration will

also be non-zero. Nevertheless these accelerations can be constant, defining the idea of quasi-steady-

state (QSS). This is the condition when the car has more than one constant acceleration: ax and ay are

both non-zero and constant at the same time. This happens when cornering, where although the lateral

acceleration makes the car turn, the velocity can still be increasing or decreasing. In QSS the car is said

to be momentarily in static equilibrium.

Simulators can vary in the type of the above stated conditions (Steady-State, Quasi-Steady-State,

Transient) it simulates the car but also in the type of vehicle model used. Concerning the vehicle model,

the simpler type is the point-mass, a single point with mass that represents the whole car with 2 degrees-

of-freedom (DOF): longitudinal and lateral. Milliken [2] introduced the more advanced ”bicycle” model

4

where the two front tyres, like the rear ones, are combined in one and the car is simulated has standing

on two different tyres, making a distinction between front and rear. This is more accurate as many race

cars have different tyres in each axle. The mass is said to be concentrated at the vehicle CG that can

vary in the longitudinal (x) axis, like having two points connected through the wheelbase. Yaw movement

(rotation about the vertical axis) is a new DOF in this model, having a correspondent yaw inertia. This

model enables transient simulations for it represents more accurately the car behaviour at the phases of

cornering when the car is not in a steady-state condition, i.e., the yaw velocity is changing with time.

Four-wheel models are an improvement to the ”bicycle” model that can estimate more accurately the

slip angles at each wheel individually and simulate both longitudinal and lateral weight transfer through

suspension mechanics and kinematics calculations. With this model the sprung and unsprung masses

of the car can be distinguished and new DOF can be added: the pitch and roll motions of the sprung

mass - the rotation about the lateral and longitudinal axis, respectively - and a vertical movement to

simulate the car in ride.

1.1.2 Energy Consumption of Electric Vehicles

Energy consumption regarding any kind of vehicle can be quantified in different scopes. The most

enlarged scope is well-to-wheel, the Life Cycle Assessment (LCA) for fuel efficiency in transportation.

A more specific one is the tank-to-wheel, also called plug-to-wheel when regarding electric propulsion

vehicles and more specifically Battery Electric Vehicles (BEV). An example of a plug-to-wheel Sankey

Diagram is depicted in Figure 1.2.

Figure 1.2: Example of a Plug-to-Wheel Sankey Diagram.

In the case of BEV’s, plug-to-wheel energy corresponds to the measurement at the plug, i. e.,

the amount drawn from the supply equipment to recharge [6]. In this work the calculations of energy

5

consumption correspond to the BatteryOut-to-wheels portion of the diagram of Figure 1.2, ignoring aux-

iliaries consumption: the energy required at the wheels to drive the vehicle [7], [8].

To drive a body with constant acceleration there must be an equilibrium of forces acting on said body.

For a road vehicle this corresponds to an equilibrium between the wheels and the road at the contact

patch of the tyres and a total resistance to the movement of the vehicle [9]. This means a force must be

applied at the wheels to overcome the energy losses. The force in question can be divided in 5 terms:

• Tyres rolling resistance;

• Aerodynamic drag;

• Potential energy for hill climb;

• Linear inertia of the whole vehicle;

• Angular inertia of the motor shaft.

The deduction of each of these terms is done in Chapter 2. Summing all the contribution of energy

loss related to a vehicle’s movement at constant acceleration, the force needed at the wheels to keep

this acceleration is

F = m g (frr cos(ζ) + sin(ζ)) +1

2ρ CD Af v

2 + (m+mf ) a (1.3)

where m is the vehicle’s total mass, g is the gravitational acceleration, frr is the tyre’s rolling resistance

coefficient, ζ is the road inclination angle, ρ is the air density, CD is the aerodynamic drag coefficient of

the vehicle, Af is the vehicle’s frontal area, mf is the fictive mass of rolling inertia, v is the velocity and

a is acceleration.

When travelling a distance d this force translates to energy and this will be the amount drawn from

the battery of an electric vehicle.

E =[m g (frr cos(ζ) + sin(ζ)) +

1

2ρ CD Af v

2 + (m+mf ) a]d (1.4)

with E given in joules and d in meters.

Literature review revealed models computed and optimized to predict Battery-to-wheel consumption

for specific electric powered vehicles, using data from experiments and equation (1.4). Baghdadi’s [9]

model predicts this electrical to mechanical power conversion in a dynamometer and real life on-road

condition tests. Dynamometer testing was performed for different fixed velocities while the on-road test

was done for a preset trajectory on public roads. In the dyno were measured resisting forces, mechanical

and battery power to calculate battery-to-wheel efficiencies. The road test gave an overall measure of

the vehicle’s energy consumption. The conclusions taken from the dynamometer test are that efficiency

increases with tractive effort for a fixed velocity and decreases with velocity for a fixed tractive power.

Lebeau [7] solved a fleet size and mix vehicle routing with time windows for electric vehicles (FSMVRPTW-

EV) problem to estimate battery energy consumption for a specific route in public roads. One vehicle

6

was used to collect real data that was fitted to equation (1.4) using non-linear least-square analysis. The

resulting parameters were used as inputs to solve the heuristic optimization FSMVRPTW-EV problem.

Cauwer constructed three models also based on equation (1.4) using real data to be fitted by linear

regression. Each model has increasingly more inputs to predict final energy consumption. The first

model has as inputs the distance travelled, time of travel and the temperature. The second model

extends the first including the linear inertia component. The third model must be used with real-data

of all kinematic parameters of the energy consumption equation. Models 1 and 2 predicted rolling

resistance (78% for 1 and 55% for 2) and road inclination (23% for positive inclination and -22% for

negative inclination) to be the factors that more contribute to energy consumption. Model 3, that had

a lower and not satisfying value of correlation, predicted the acceleration to be the bigger contributor,

responsible for 59% of all energy consumption.

All these models are specific to a car and need real data from testing. Because of this they are not

suitable for other vehicles either they have been tested or not.

1.2 Objectives

When designing a motorsport vehicle it is important to understand how the variation of the car param-

eters influence it’s performance. These parameters can be general, like the overall mass or drag force

of the whole vehicle, or more specific, like suspension springs stiffness that influence the mass transfer

when cornering.

This thesis objective is to understand if energy consumption estimations can be made through a Lap

Time Simulator (LTS - explained in Section 1.1), how complex should the vehicle model be and what

are the parameters that influence the simulated energy consumption the most. It is proposed to create

a tool that estimates energy consumption in competition vehicles.

1.3 Thesis Outline

This first chapter introduces the motivation and objective of this work. Here is also presented the state

of the art regarding Lap Time Simulators and energy consumption estimation of electric vehicles. The

second chapter gives insight into the vehicle dynamics concepts used throughout the work. On Chapter

3, the two simulators used (PMSim and RaceLap) are presented. PMSim was developed in the scope

of this work and thus a more thorough explanation on the Point-Mass vehicle model is made as well

as describing the approach made to Quasi-Steady-State simulation. Chapter 4 documents some of the

decision making behind the PMSim simulator. Chapter 5 shows the experimental data used for validation

and presents the simulation outputs from both simulators used. Also in this chapter, a sensitivity analysis

is done to the PMSim and the energy consumption estimator implemented. Finally, in the last chapter

are stated the conclusions of this work.

7

8

Chapter 2

Vehicle Dynamics Fundamentals and

Terminology

This Chapter aim is to introduce all the vehicle dynamics concepts that are used in this work. Section

2.1 presents the coordinate axis system of a vehicle, in section 2.2 is explained how the accelerations

of a vehicle change when describing a corner, section 2.3 introduces the terminology regarding tyre

mechanics, including an introduction to the Pacejka’s Magic Formula tyre model, and finally, in section

2.4, all the terms in the energy consumption estimation are explained.

2.1 Axis System

In this work the equations are expressed in a coordinate axis system (Gxyz) with the origin G being the

vehicle’s centre of gravity as show in Figure 2.1. The x-axis defines the longitudinal direction pointing

towards the front of the car while the z-axis defines the vertical direction perpendicular to the ground

when on a flat road, pointing in the direction of gravitational acceleration. In order to form a right-hand

triad, the y-axis defines the lateral direction and is pointing to the right when seen from above as in

Figure 2.1.

9

(a) Side view

(b) Top view

Figure 2.1: Vehicle’s Coordinate Axis System.

The rotation about the x, y and z-axis are, respectively, the roll φ , pitch θ and yaw ψ movements.

When considering only planar dynamics, roll and pitch are not taken into account. This is the case with

the Point-Mass model described in Chapter 3.

When describing a corner, the vehicle is in circular motion and the velocity vector v is tangent to the

trajectory. The angle between v and the vehicle’s x-axis is defined as the side-slip angle (β). This angle,

as well as an example of trajectory, is shown in Figure 2.2.

Figure 2.2: Side slip angle when describing a corner.

2.2 Cornering

When coming from a straight line into a corner, the corner can be divided in three phases: turn-

entry, appex and turn-exit. Figure 2.3 shows an example of an hairpin turn where these phases can be

identified. Turn-entry is A to C, C is the appex point and turn-exit is C to E. At turn-entry (A-C) the lateral

and yaw accelerations start increasing from zero, making the car turn into the corner until it reaches

the appex. The appex (C) is the point where lateral acceleration reaches its maximum value and yaw

10

acceleration is again null: the steady-state condition. In turn-exit (C-E), lateral acceleration decreases

to zero and the yaw acceleration reaches its lower value before going back to zero again. After this,

only the longitudinal acceleration is different from zero going into another straight. Turn-entry and exit

are transient conditions for the yaw moment is changing. These types of conditions (steady-state and

transient) also define the type of simulation. Transient conditions apply when there is no equilibrium in

all of the car forces and so the car is not travelling at a constant condition (accelerations are changing).

Figure 2.3: An hairpin corner with selected points to describe it.

(a) Lateral Acceleration (b) Yaw Acceleration

Figure 2.4: Variation of ay and ψ in a hairpin.

2.3 Tyre Mechanics Terminology

Being the form of contact with the ground, a vehicle’s tyres are the main responsible for its handling

because their compound generate all the lateral and longitudinal forces that cause motion. A car can

only be as fast as the tyres allow, which is why race cars are designed to maximize their performance.

The forces generated by a tyre are assumed to be in a contact point with the ground.

Just like for the vehicle, a tyre can have it’s own coordinate axis system. Different sign conventions

are used in literature, the most common being the SAE and ISO. Figure 2.5 shows a SAE tyre coordinate

axis system.

11

Figure 2.5: SAE Tyre Axis System. Top view (left) and Rear view (right).

From a top view, the slip angle (α) is the angle between the velocity vector and the x-axis. From a

front view, the inclination angle (IA) is the angle between the tyre plane and the z-axis.

2.3.1 Slip Angle

The slip angle defines the angle between the heading of the tyre and the actual direction of its travel.

This is a consequence of a deformation caused in the tyre by applying a lateral force to it. The tyre

deformation gives place to a torsion around the tyre vertical axis, modifying the contact patch shape.

This alteration causes the shear stress distribution (τy) to change, originating the point where Fy is

considered to be applied.

The illustration of this explanation is in Figure 2.6. This way, Fy and α work as a pair action-reaction

because applying a lateral force to the tyre causes a deformation that results on a reaction at the contact

patch, creating lateral force.

Figure 2.6: Slip angle tyre deformation and pressure distribution in a bottom view.

2.3.2 Rolling Resistance

When a tyre is rolling on a road, the contact point is not in fact a single point but a section of the

tyre tread that deforms when in contact with the ground. The energy spent in deforming the tyre is not

12

fully restored in relaxation when the tread is no longer in contact with the ground, causing the contact

pressure distribution in the tyre to change. Due to this, in forward movement, the heading (front) part

of the tread will have higher normal stresses (σz) than the tailing (rear) part, shifting the normal force

applied on the tyre in a horizontal direction [10], as shown in Figure 2.7.

Figure 2.7: Normal stresses distribution on a tyre’s tread and consequent shift in the normal forceapplication.

Rolling resistance comes from the moment about the y-axis that the new application of the normal

force Fz creates in the opposing direction of the wheel’s angular movement. Calling Mrr and Frr to the

rolling resistance moment and force respectively, we have

Frr =Mrr

rw(2.1)

Knowing, from Figure 2.7, that

Mrr = Fz ∆x (2.2)

one gets

Frr =∆x

rwFz (2.3)

The term ∆xrw

is called the rolling resistance coefficient (frr), directly relating the rolling resistance

force to the vertical or normal force exerted in the tyre. This coefficient is in reality not constant, de-

pending on the movement velocity, air pressure inside the tyre and wheel angles like the slip angle and

camber. However, in this work, the rolling resistance coefficient is considered constant and equal to the

Pacejka’s Magic Formula [3] qSy1 coefficient.

2.3.3 Camber and Inclination Angle

Inclination Angle is defined as the angle between the wheel plane and the z-axis of the coordinate

system of Figure 2.5. When discussing this angle related to the vehicle it can be called Camber. This

13

way, Camber and Inclination Angle describe the same thing but in different perspectives. Inclination

Angle is used when talking about tyre mechanics and Camber is used when referring to the angle

relative to the vehicle’s chassis.

Figure 2.8: Inclination Angle.

Camber is said to be negative when the top of the tyre is pointing towards the inside of the car and

positive when pointing to the outside. Inclination angle signs depend on the convention chosen for the

coordinate axis system of the tyre but for both regular SAE and ISO conventions it is considered positive

when inclined to the right in a rear view.

2.3.4 Pacejka’s Magic Formula

The Magic Formula [3] is a semi-empirical Tyre Model to calculate force and moments in a tyre. This is

the most used tyre model in vehicle dynamics and the one adopted by the FSAE tyres data supplier and

Michelin, the tyre supplier at Adess AG. The choice of this model lies on this and the fact that it provides

a simpler alternative for reducing the model to only two friction coefficients, as explained in section 3.1.

Since the start of its development in the 1980’s, the model has been published in 5 versions: Pacejka

’96, Pacejka 2002 (used in this work), Pacejka 2002 With Inflation Pressure Effects [11], Magic Formula

5.2 and Pacejka 2006. The version used in this work, Pacejka 2002, can calculate Lateral Force (Fy),

Longitudinal Force (Fx) and Aligning Torque (Mz) in both pure and combined slip conditions. The novelty

from this version was the inclusion of the calculations of the Overturning Couple (Mx) and the Rolling

Resistance Moment (My). Combined slip conditions correspond to the existence of slip angle and

longitudinal slip at the same time.

The generic form of the formula for a given value of vertical load (Fz) and camber (γ) is

y = D sin[C arctan(Bx− E(Bx− arctan(Bx)))] (2.4)

with

Y (X) = y(x) + SV (2.5a)

x = X + SH (2.5b)

where Y is the output variable (Fx, Fy or Mz), B is the stiffness factor, C is the shape factor, D is the

14

peak value, E is the curvature factor and SH and SV are horizontal and vertical shifts, respectively. X is

the input variable that can be either the tangent of the slip angle (tanα) or the slip ratio (κ).

The shape of Fy(α) and Fx(κ) curves is the same, typically being of the form shown in Figure 2.9.

-15 -10 -5 5 10 15

α (º)

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000 F

y (N

)

Figure 2.9: Generic form of a Fy(α) curve.

In its totality, the Pacejka 2002 model is comprised of 89 coefficients that make up for each of the

formula factors as function of either slip angle, slip ratio or camber.

2.4 Energy Consumption Terms

Section 1.1.2 introduces the force equation that is used on the quantification of energy consumption.

This equation is divided in five terms: Rolling Resistance, Aerodynamic Drag, Potential Energy for hill

climb, linear inertia and angular inertia. All these terms are explained here, with the exception of the

rolling resistance term that is explained in section 2.3.2.

Aerodynamic Drag

Aerodynamic drag corresponds to the drag force generated by the vehicle’s body and aerodynamic

components, like the wings and the undertray. This force opposes the movement and represents the air

resistance acting on the whole body. Its expression is

FD =1

2ρ CD Af v

2 (2.6)

where ρ is the air density, CD is the aerodynamic drag coefficient of the vehicle, Af is the vehicle’s frontal

area, and v is the velocity.

15

Inertia

More forces must be acting on the body to overcome the inertias inherent to movement. This is both

linear in the sense of creating acceleration to describe a trajectory but also angular to make the wheels

turn. For the linear inertia case, Newton’s 2nd law of motion stands

Fli = m a (2.7)

Note that here a is not the actual resulting acceleration of the vehicle’s movement but the one pro-

vided by the force at the wheels, i. e., Fli = Fx, the force that induces positive acceleration, not the

resulting force. Chapter 3 explains that Fx can be either limited by the tyres or the engine.

For the angular inertia case, the same follows

Fai = mf a (2.8)

where mf is the fictive mass of rolling inertia, deduced in [12] as

mf =I gr2

r2w

(2.9)

Here, I is the motor shaft moment of inertia, gr the gear ratio from the gear box to the wheels and rw is

the wheel radius.

Potential Energy for Hill Climb

In the case of inclined roads, there must be potencial energy to overcome hill climb. This is quantified

through the road inclination angle ζ. From Figure 2.10 is observed that the weight of the car creates a

resistance force:

Fhc = m g sin(ζ) (2.10)

The rolling resistance force of the tyres, equation (2.3), also changes with road inclination angle.

Because the resultant of the car’s weight in the tyres decreases in hill climb, the rolling resistance force

will decrease too. This is quantified by the multiplication factor: cos(ζ).

Figure 2.10: Forces acting on the vehicle in a hill climb.

16

Chapter 3

Implemented Simulations

As stated in Chapter, in a Lap Time Simulator (LTS) a race track is discretized in segments. The

method to solve the vehicle’s motion equations depends on the type of simulation (Steady-State, Quasi-

Steady-State, Transient) and the vehicle model (Point-Mass, ”Bicycle”, 4-Wheel). The lap time is calcu-

lated by summing all the individual segment times, i.e., how long it takes the vehicle from entering to

exiting each segment, equation (3.1),

LapT ime =

n∑i=1

∆ti (3.1)

with n being the number of segments a single lap consists of.

3.1 Point-Mass Vehicle Model

The Point-Mass is the vehicle model implemented in the LTS developed in the scope of this work

(PMSim). This simulator is the subject of Chapter 4 and its results are presented in section 5.2. In

this model the whole vehicle sums up to a point that can be said to be supported on one tyre. The

longitudinal or lateral force produced by this tyre is a consequence of the vertical load (Fz) exerted on it -

the point’s weight plus the downforce generated while moving. Being the simpler method to parametrize

a vehicle, the Point Mass is the model that requires the least parameters.

Here, tyres are simplified to only two parameters defining the longitudinal and lateral coefficients of

friction: µx and µy, respectively. This is due to the fact that the Point-Mass model does not include the

parameters needed to simulate tyre’s slip angles an so the complete Pacejka Magic Formula Tyre Model

[3] can not be implemented. Although this is a simplification, these coefficients still vary with vertical

load as shown in Figure 3.1. The forces generated at the tyres are given by equations (3.2).

Fx = µx Fz (3.2a)

Fy = µy Fz (3.2b)

17

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Vertical load (N)

1.25

1.3

1.35

1.4

1.45

1.5

1.55

Fric

tion

Coe

ffici

ent

FrontRear

Figure 3.1: Examples of µx(Fz)

The friction coefficients can be estimated using equations (3.3) of the Pacejka Magic Formula Tyre

Model resorting to only 5 out of 89 coefficients of the complete model - Pacejka 2002.

µx = (pDx1 + pDx2 dfz) λ∗µx (3.3a)

µy = (pDy1 + pDy2 dfz)/(1 + pDy3 γ∗2

) λ∗µy (3.3b)

where pDx1, pDx2, pDy1, pDy2 and pDy3 are coefficients of the Pacejka 2002 Tyre Model and dfz, γ∗,

λ∗µx and λ∗µy are parameters of the same model.

When taking the engine into account, the longitudinal force (Fx) is no longer only limited by the tyres

because the engine can be the limiting factor. The model of the powertrain system will be the same for

all models in this work and consists of the engine torque curve, as can be seen in Figure 3.2, the gear

ratios of the gearbox and the final reduction at the differential. The propulsion force generated by the

engine is function of the power produced which in turn is a function of torque and so the engine limitation

depends on where in the motor curves the vehicle is operating. It is also needed to simulate when to

change gear, i. e., what gear should be engaged at every point of the track. To achieve this it is assumed

that the gear selected is the one that makes the engine produce more power for a certain linear velocity

of the vehicle and thus producing the most propulsion force of the whole set of gears. Denoting j as the

gear index, the powertrain simulation is done following equations (3.4).

18

Figure 3.2: Engine Torque Curve

ωj =grj diff v

rw(3.4a)

ωjrpm =ωj 60

2π(3.4b)

Pj = Tj ωj (3.4c)

Fe =max(Pj)

v(3.4d)

where ω is the engine rotational speed, gr is the gearbox selected gear ratio, diff is the differential gear

ratio, rw is the wheel radius, P is the engine power output, T is the engine torque output and Fe is the

engine output propulsion force. Figure 3.3 shows an example variation of the engine output propulsion

force and the longitudinal force generated by the tyres in a straight line acceleration.

The aerodynamics in this model sum up to equations (3.5) with the influence of constant drag and lift

coefficients.

FD =1

2ρ CD Af v

2 (3.5a)

FL =1

2ρ CL Af v

2 (3.5b)

where ρ is the air density, CD is the drag coefficient, CL is the downforce or lift coefficient, Af is the

vehicle’s frontal area, FD is the drag force and FL the downforce. CD Af and CL Af can be substituted

by SCX and SCZ, respectively, a normalized alternative of expressing the aerodynamic coefficients in

units of m2.

Figure 3.4 shows a scheme of the forces acting on a Point-Mass model.

19

0 50 100 150 200 250 300

Velocity (km/h)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Lim

iting

For

ce in

Acc

eler

atio

n (N

)

×104

1

2

3

4

5

6

Gea

r

Engine ForceTires ForceGear

Figure 3.3: Limiting force and gear selected in a straight line acceleration.

Figure 3.4: Force Balance on the Point Mass

3.2 Simulation Types

As introduced in section 1.1, LTS can be steady-state, quasi-steady-state or transient. In this work only

the transient type was not implemented. Section 3.2.1 describes the reasoning behind the development

of a steady-state simulator and section 3.2.2 explains how the quasi-steady-state approach can de

adapted from a pure steady-state.

3.2.1 Steady State

The car is said to be in a steady-state condition when it has a constant acceleration and, in the case

of a corner, constant radius as well. This is the most simple approach to lap time simulation. Given that

20

a race track is composed of straights and various types of turns, the steady state condition is simulated

in each segment of the track, i.e, acceleration is constant in each segment.

Figure 3.5: Acceleration and Velocity through three segments of a discretized track.

In Steady State simulation, longitudinal and lateral acceleration do not happen at the same time in

any segment. This means that in a straight line there will be only longitudinal acceleration and at a

corner the car is simulated has having only lateral acceleration.

In order to know where the braking point is when approaching each turn, the simulation is divided

into two phases, the Forward and Reverse simulations, as in [13]. In the forward simulation only positive

acceleration occurs: the velocity increases in a straight line until a corner segment appears where it will

be set for the maximum corner velocity. The reverse simulation does the same but in the opposite way of

the circuit, with the forces moving the car being the tyres braking force, drag force and the tyre’s rolling

resistance, all in the same direction. The final velocity profile of the car on a given track is the minimum

of the two simulations (Forward and Reverse) in each segment, as shown in Figure 3.6.

Forward Simulation

Each segment will have an entering velocity, exit velocity and constant acceleration, as shown in

Figure 3.5. Denoting i as the current segment, its entering velocity is

vi = vi−1 + ai−1 ∆ti−1 (3.6)

where ti−1 is the time that took to travel the previous segment.

The longitudinal acceleration of the segment i is also dependent of the entering velocity vi and is

calculated using equations (3.2), (3.4) and (3.5). The process to calculate ax in the segment i is:

Fz = m g + FL (3.7a)

Fe =P

vi(3.7b)

Frr = frr Fz (3.7c)

Ftx = Fz µx (3.7d)

axi=min(Ftx , Fe)− FD − Frr

m(3.7e)

where m is the vehicle’s mass, g is the gravitational acceleration, frr is the tyre’s rolling resistance

coefficient, Frr is the rolling resistance force and Ftx is the longitudinal force generated by the tyres.

In the case of a corner segment, the maximum corner velocity must be computed. Taking into

21

account only the lateral acceleration:

ayi =v2i

rc↔ vi =

√ayirc (3.8)

where rc is the corner radius.

From equation (3.8) is seen that the velocity in a corner segment is a function of lateral acceleration.

Nevertheless, the opposite is also true and so: vi(ayi) and ayi(vi). To solve this problem an iteration

must be done.

The iteration process to compute the maximum corner velocity uses the following equations in the

given order, starting with vi = 0, until some stopping criteria is met:

Fz = w + FL (3.9a)

Fty = Fz µy (3.9b)

ayi =Ftym

(3.9c)

vi =√ayi rc (3.9d)

where Fty is the lateral force generated by the tyres. In this work, the stopping criteria chosen was

the result of an exponential moving average of the vi values after 300 iterations. This method helps

to find a converging value for cases where the iteration oscillates indefinitely. The flow chart of the

function that implements this iteration in PMSim is presented in Figure 3.8. In this same figure, where

the Exponential Moving Average is described, EMA stands for ”exponential moving average” and SMA

stands for ”simple moving average”. In this application, the SMA is set to be equal to the mean.

The time that takes to travel through a segment is

∆ti =divi

(3.10)

where di is the length of the segment.

Reverse (Braking) Simulation

In this stage, the calculations start from the same point as the Forward simulation but go in the opposite

direction of the track, as stated above. Here the engine propulsion force is not considered and the

longitudinal acceleration is given by:

arevxi=Ftx + FD + Frr

m(3.11)

The method to compute the maximum corner velocity is the same, given that the cornering is only

dependent on the tyre capacity to produce lateral force.

Figure 3.7 presents the flow chart of PMSim, including the reasoning behind the Forward and Re-

22

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

0

100

200

300

400

500

600

700

Vel

ocity

(km

/h)

ForwardReverseFinal

Figure 3.6: Velocity profiles from Forward and Reverse Simulations. Resulting lap velocity profile indotted black.

verse simulations. Note that in this same figure, where the first process block in the reverse simulation

states that vrev(last) = ∞ it means that this segment velocity is set to a unrealistic high value of top

speed so as to not interfere with the forward simulation in that point. This is done to avoid having to run

the reverse simulation twice and assuming that the last segment of the track is not in a braking stage.

Normally the first and last segment of a discretized track are immediately before and after the finish line

that is usually in a straight segment so this assumption should not be a problem.

23

Load Vehicle, Tyres and

Track

Set Torque Factor

Set 𝑣(1) = 0

v(1) =

v(last +1)

FL(1), FD(1), FZ(1), Fe(1),

ax(1), ay(1), Δt(1)

𝑖 = 2

i ≤ last

segment

v(i), FL(i), FD(i), FZ(i), Fe(i),

ax(i), ay(i), Δt(i)

maximum corner velocity (i)

𝑣(𝑙𝑎𝑠𝑡 + 1) = 𝑣(𝑙𝑎𝑠𝑡) + 𝑎𝑥(𝑙𝑎𝑠𝑡)

∙ 𝛥𝑡(𝑙𝑎𝑠𝑡)

𝑣(1) = 𝑣(𝑙𝑎𝑠𝑡 + 1)

𝑣𝑟𝑒𝑣(𝑙𝑎𝑠𝑡) = ∞

𝑗 = 𝑙𝑎𝑠𝑡 − 1

𝑗 ≥ 1

maximum corner velocity (j)

vrev(j), FLrev(j), FDrev(j), FZrev(j),

Ferev(j), axrev(j), ayrev(j), Δtrev(j)

𝑘 = 1

k ≤ last

segment

v(k) = min(v(k), v(k)rev)

Output

True

False

True

False

True

False Forward

Reverse

True

False

Figure 3.7: Flow chart for the main file of PMSim.

24

corner radius - 𝑟𝑐

𝐹𝑧 = 𝑤

𝜇𝑦 = 𝑚𝑦 𝐹𝑧

4+ 𝑏𝑦

𝐹𝑦 = 𝜇𝑦 𝐹𝑧

𝑣(1) = √𝐹𝑦

𝑚 𝑟𝑐

𝑖 ≤ 𝑖𝑡𝑒𝑟

𝑖 = 2

𝐹𝐿 =1

2 𝜌 𝑆𝐶𝑍 𝑣(𝑖 − 1)^2

𝐹𝑧 = 𝑤 + 𝐹𝐿

𝜇𝑦 = 𝑚𝑦 𝐹𝑧

4+ 𝑏𝑦

𝐹𝑦 = 𝜇𝑦 𝐹𝑧

𝑣(1) = √𝐹𝑦

𝑚 𝑟𝑐

𝑣(1: 𝑖𝑡𝑒𝑟 ∙ 0.1) = [ ]

𝑆𝑀𝐴 = 𝑚𝑒𝑎𝑛(𝑣)

𝐸𝑀𝐴(1) = 𝑆𝑀𝐴

𝑗 ≤ 𝑖𝑡𝑒𝑟 ∙ 0.9

𝑗 = 1

𝐸𝑀𝐴 (𝑗 + 1)

= (𝑣(𝑗) − 𝐸𝑀𝐴(𝑗))2

(𝑖𝑡𝑒𝑟 ∙ 0.9) + 1+ 𝐸𝑀𝐴(𝑗)

Exponential

Moving

Average

True

False

True

𝑚𝑎𝑥𝐶𝑜𝑟𝑉𝑒𝑙 =

𝐸𝑀𝐴(𝑒𝑛𝑑)

False

Figure 3.8: Flow chart of the maxcorvel function of PMSim.

25

3.2.2 Quasi Steady State

Quasi Steady State is steady state in the sense that acceleration is still constant through a segment.

The difference to ”pure” Steady State is the possibility of acceleration having both longitudinal and lateral

components happening at the same time. In simulation, the change will be in the corner segments.

Analysing the tyre, one knows that it can produce a certain longitudinal force at a given vertical

load (which depends on the velocity) and the same for the lateral force. In the case of the Point Mass

Model, these forces are give by equations (3.2). If one considers that Fx and Fy in those equations are

the maximum forces the tyre can produce in a pure longitudinal or lateral case, respectively, one can

assume that if when describing a certain corner the driver is not making the car turn at the limit of the

tyre capability - Fmaxy - there is still some longitudinal force to be produced.

Every corner segment has a maximum velocity to be driven by a certain car fitted with certain tyres

and this velocity is known from the maximum Fy the tyre can produce in that car. Not being at the limit

of the tyre capability means the entering velocity of that segment is not the maximum velocity and so the

tyre can still generate a certain amount of longitudinal force to make the car reach the maximum corner

velocity. The amount of longitudinal force that can still be generated is given by the ellipse equation

(3.12): (FtxFmaxtx

)2

+

(Fy

Fmaxy

)2

= 1↔ Ftx =

√1−

(Fy

Fmaxy

)2

Fmaxtx (3.12)

This is the case for both Forward and Reverse simulations, the difference being that in the Forward

case, Fx = min(Fe, Ftx) and in the reverse case Fx = Ftx .

Figure 3.9: Example of a g-g Diagram for AD03 at 40m/s

Figure 3.9 shows the result of applying equation (3.12) to Fy ∈ [0;Fmaxy ] for a specific velocity. This

is a g-g diagram, a plot of ax vs ay that quantifies the vehicle’s maximum accelerations capability. This

26

diagram changes for each value of velocity and only one half of it is computed and plotted because it is

symmetrical for symmetrical cars. Negative ax values denote longitudinal acceleration for braking. The

saturation at the top, for positive ax, is a result of the engine propulsion force being the limiting factor, i.

e., in these points: Fe < Ftx (see Figure 3.3). A g-g diagram can be used as a tool to measure vehicle

performance and behaviour.

The iteration process to find the maximum corner velocity is the same, given that this is only a

function of Fmaxy .

3.3 RaceLap Simulator

In this section is described the simulator available at Adess AG also used in this work. RaceLap

is a Quasi-Steady-State simulator that uses a 4-wheel vehicle model coded in Matlab. This software

calculates the forces acting on the car that give an equilibrium of movement based on 5 degrees of

freedom:

• Suspended mass equilibrium in roll - roll angle;

• Suspended mass equilibrium in pitch - pitch angle;

• Suspended mass vertical equilibrium - CG height;

• Lateral equilibrium of the whole vehicle - steering rack position;

• Equilibrium of the whole vehicle in yaw - sideslip angle.

Differing from the approach in section 3.2.2, the g-g diagrams in RaceLap are computed through

iteration, instead of a simple ellipse equation. Before solving the five equations of motion the simulator

calculates the forces and displacements acting on various modules of the car: Aerodynamic, Suspen-

sions, Tyres and Powertrain. An iterative algorithm then solves the 5 general equations and the lateral

acceleration limit is found for each segment. This process is done for both the forward and reverse

phases of simulation.

On the other hand the velocity calculations are done the same way as described in section 3.2, with

the final velocity profile being the corresponding minimum of the forward and reverse velocities in each

sample.

3.3.1 Parametrization of a Vehicle in RaceLap

Building a 4 wheel model requires a more detailed knowledge of the car’s parameters. In the category

of 4 wheel vehicle models the complexity can be increased or decreased depending on the calculations

the simulator performs. In RaceLap the parametrization of a racing car is divided in six areas or modules:

• Aerodynamics

• Chassis

27

• Suspension Mechanics

• Suspension Kinematics

• Powertrain

• Tyres

Aerodynamics

Like in the Point Mass model, section 3.1, the aerodynamics resume to the lift and drag coefficients

so the resulting forces performed by the aerodynamic components and the bodywork of the car can be

calculated. The increasing complexity comes from the fact that these components are not constant in a

real car. Modelling the suspension behaviour, the car’s ride height (vertical distance from lower point of

the car to the floor) at both front and rear are always changing during the course of a lap. With this, the

drag and lift coefficients are function of front and rear ride heights. Also, the lift is no longer considered

to be the same in the whole car. Distinguishing front and rear axes it is possible to set different lift

coefficients so the front and rear tyres are subjected to different aerodynamic loads.

To model the aerodynamics in RaceLap it is then needed to map the overall drag coefficient and the

front and rear lift coefficients for a range of both front and rear ride heights.

Chassis

In this module it is to be specified the mass of both the sprung and unsprung masses. Giving the

static weight loaded at each wheel, the overall weight of the car is known, as well as the longitudinal and

lateral static weight distributions. Also in the chassis configuration is the vertical position of the vehicle’s

centre of gravity and the moments of inertia along the three (x, y and z) axes.

Here is also provided the information about the brakes. To model the brakes it is needed to know

the maximum pressure at the master cylinder, the relation between this pressure and the torque at the

braking disc and the braking repartition (longitudinal distribution of the braking force).

Suspension Mechanics

The suspension elements displacements and consequent exerted forces at the wheel are used to

calculate weight transfer and ride heights. Suspension Mechanics covers the forces part by providing

the stiffness os the main spring, anti-roll bar and 3rd element spring as well as the static gaps at the

main and 3rd springs, denoting a static displacement of the suspension.

Suspension Kinematics

This module provides the parameters needed to simulate the displacements refereed in the Suspen-

sion Mechanics and is the more complex area in the sense that it needs for precise measures of the

28

behaviour of several suspension parameters. These parameters are motion ratios for several compo-

nents, camber, toe, caster and longitudinal and lateral position of the centre of the wheel. For the front

suspension, these parameters are mapped as function of the wheel centre vertical travel and the steer-

ing rack position. At the rear the same parameters are needed but only as function of the vertical travel

of the wheel.

Powertrain

As stated in Section 3.1, the powertrain model is given by the engine torque curve (Torque(rpm)),

the gearbox ratios for all gears and differential - representing the final reduction - and corresponding

efficiencies. If needed, in RaceLap is also possible to simulate the delay time of changing gears which

is also an input parameter in this module.

Tyres

Like the kinematics this is also a sensitive area to module. As constant parameters, the Tyres Con-

figuration provides the values of the tyres stiffness, static and rolling radius and rolling resistance. The

longitudinal force is given as a function of the vertical load and the lateral force is mapped as a function

of the vertical load and slip angle. The influence of camber on the lateral force is also implemented and

must be provided. RaceLap assumes left and right tyres and wheel angles (toe and camber) are the

same for left and right in each axle, distinguishing only the front from the rear. This is often not the case

in cars that race at tracks with banking, for example.

29

30

Chapter 4

Point-Mass Simulator Choices

In this chapter is described the decision making behind the developing of PMSim, the Point-Mass

Lap Time Simulator (LTS) developed in the scope of this work as a simpler alternative to the 4-wheel

commercial software. Section 4.1 explains the choice in the type of simulation used, section 4.2 presents

two possible ways of modelling the track and section 4.3 analyses further decisions regarding the tyre

model.

4.1 Simulation type: Steady State vs Quasi Steady State

Section 3.2.2 states the difference between quasi steady state and the pure steady state approach.

Here this is illustrated for the specific case of a corner segment with entering velocity lower than the

maximum corner velocity for that segment. In Figure 4.2 is the velocity profile of FST06e in the first

part of a skid-pad. The track for this run is made of a 9 meters straight line, followed by a corner with a

constant 9.125 meters radius, as seen in Figure 4.1. From these profiles the difference from steady-state

and quasi-steady-state can be seen when entering a corner with lower velocity than the actual maximum

velocity that the car can achieve at that radius.

The maximum velocity the car can achieve at a corner with this radius is actually higher than the

velocity the car will reach in a pure Steady State simulation. In the first corner segment (at 9 meters in

Figure 4.2), the entering velocity is lower than the maximum corner velocity so the car, despite being in

a corner, can still go faster. If the velocity was simply set to reach the maximum corner velocity there

would be a discontinuity in the velocity profile representing infinite acceleration. The quasi-steady-state

approach simulates the increase of velocity while cornering because the longitudinal (x-axis) acceler-

ation is non-zero. Knowing the entering velocity of the segment, the lateral acceleration is calculated.

Using the g-g diagram one can know if it is still possible to have linear (not centripetal) acceleration - ax

- and so the exit velocity vi+1 will be different to vi until the maximum is reached. Once the maximum

corner velocity is reached, the vehicle is describing the corner with maximum lateral acceleration.

The same effect can be seen when decelerating in a corner. If after the skid-pad corner there is

another one, in the same direction, but with smaller radius - a corner with non-constant radius - the real

31

-5 0 5 10 15 20 25

Distance (m)

-20

-15

-10

-5

0

Dis

tanc

e (m

)

Constant radius

Changing radius

Corner entry

Entry to smaller radius

Figure 4.1: Map of a straight followed by a corner with constant radius (in blue) and a corner that changesradius (in orange).

0 10 20 30 40 50 60 70

Track Length (m)

0

10

20

30

40

50

60

Vel

ocity

(km

/h)

Steady-StateQuasi-Steady-StateCorner entry

Figure 4.2: Velocity profile when entering and describing a corner with constant radius.

driver will brake or lift the throttle pedal before reaching the smaller radius segment. An example of a

corner like this is also seen in Figure 4.1

In the pure steady-state simulation this is not verified. Because there is no longitudinal acceleration

in a corner, the velocity can not change. This means that in a pure steady-state simulation, a corner

32

0 5 10 15 20 25 30 35 40 45 50

Track Length (m)

0

10

20

30

40

50

60

Vel

ocity

(km

/h)

Steady-StateQuasi-Steady-StateCorner entryEntry to smaller radius

Figure 4.3: Velocity profile when entering and describing a corner with decreasing radius.

with varying radius will be described at the maximum corner velocity for the smaller radius to avoid

discontinuity in the velocity profile. The quasi-steady-state simulation allows to model a more realistic

behaviour. Once again, with the possibility of simulating both lateral and longitudinal accelerations while

cornering, the velocity can change and, in this case, decrease to the maximum value it can reach at the

smaller radius.

This analysis shows how quasi-steady-state is more realistic and therefore more complete than a

simple or pure steady-state approach. The two types have the same result when it is possible to describe

a corner at the maximum corner velocity because lateral acceleration will be maximum: the steady-state

condition. Although this is not always the case because in reality the longitudinal acceleration affects

the velocity at which the vehicle reaches the corner segment. Both steady-state and quasi-steady-state

assume every segment to be an appex (point of the curve where the vehicle is in the steady state

condition) but there can not be discontinuities in the velocity profile. Figures 4.2 and 4.3 illustrate two

cases where in a segment (and so, an appex) the car is not at maximum ay and therefore it is possible

to increase velocity while cornering, if the radius is constant or increasing, or decrease the velocity if

the radius is decreasing. For this reason in the rest of this work the simulation type used will be quasi-

steady-state.

33

4.2 Track Model

In this work, two types of track model were experimented. As expressed in section 1.1.1 these two

types are:

• Type 1 - a succession of straights and corners;

• Type 2 - where all segments are described as corners.

In type 2, straights are simulated by large radius corner segments where the vehicle behaves like

in a straight line. Specifying all straight segments as a corner segment with 1000 meters radius, the

resulting longitudinal acceleration is maximum, as shown in Figure 4.4 for two different cases. The blue

case corresponds to the straight segment at Estoril that is approached with the smallest entry velocity

(34.58 m/s). This value was taken from results of a PMSim simulation with a type 2 track model of the

Estoril circuit. This shows that the specified radius for these segments is enough to get the same result

as in a pure longitudinal simulation. Equation (4.1) shows the vehicle’s lateral acceleration ay when

entering this segment (blue case).

ay =v2

r=

34.582

1000= 1.1958 m/s2 = 0.1219 g (4.1)

Knowing this value, the g-g diagram of Figure 4.4 shows that the corresponding longitudinal acceler-

ation ax is the maximum value, the engine torque limitation, and the same value as when ay is null. The

same effect is shown for a different segment (orange case) that corresponds to the straight segment that

is approached with the higher entry velocity, 59.57 m/s. Once again, this value was taken from results

of a PMSim simulation with a type 2 track model of the Estoril circuit.

Figure 4.4: AD03 g-g diagram for 34.58 m/s and 59.57 m/s

34

This is possible due to the engine torque limitation that creates a saturation in the g-g diagrams.

Although, when braking this is not verified. In the negative ax part of the diagrams there is no saturation

and so the lower half follows the ellipse equation (3.12). The result of both track models differ in a full

lap because the reverse simulations - see section 3.2.1 - that represent the braking stages will have

differences.

Comparing both track models on a full lap the result is then not the same, with the differences only

appearing in segments where the vehicle is braking. As explained above for the positive ax case, type 2

tracks have a radius associated to every segment and this means there is ay at all times. In the braking

stages, because there is no saturation in the g-g diagram, the resulting ax is not always the same as

when ay is null.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

Type 1 - 1:32.074Type 2 - 1:31.507

Figure 4.5: Resulting velocity profiles for both type of track input.

Figure 4.5 shows that the difference in a full lap at Estoril is small: subtracting the lap times displayed

in the figure’s legend, the difference gives 0.567s that account for a 0.577% error comparing to the real

lap time that will be introduced and discussed next. Observable in Figure 4.5 is also the confirmation

that the difference in the velocity profiles only appear in the braking stages of the lap.

Because the final velocity profile and lap time differences are small, the choice for the track model

rests in other factors. Type 2 is the chosen model since the inputs are the same as in RaceLap (see

section 3.3) and also, regarding the Point-Mass simulator developed in this work, the choice is justified

by the relative simplicity of the code.

35

4.3 Tyres

Assuming that all the weight of the car is standing on one single tyre can be a bad assumption

because most motorsport prototypes run different tyres in the front and rear axles. In such a case there

are different tyre friction coefficients for the front and rear tyres in the same vehicle. In order to see

the influence of this assumption in a Point-Mass model, three solutions to model different front and rear

tyres were tested:

• 1 tyre based on the weighted average of the friction coefficient of the two different real tyres;

• Discriminate 2 tyres for the front and rear axles;

• Discriminate 2 tyres for the front and rear axles with longitudinal weight transfer.

Considering the linear functions that describe tyre’s coefficient of friction for two different tyres, one

gets four relations: µxF, µxR

, µyF , µyR , where F stands for front and R for rear. Writing a generic friction

coefficient linear function as

µ = m Fz + b (4.2)

The relation between µ and Fz relies on two parameters, m (the slope) and b (the intercept). The

first method tested relies on a weighted average of m and b for each coefficient of friction, µx and µy.

This results in a combination of the two tyres in a single one with different characteristics. Taking µx as

example

mx = WDF mxF+WDR mxR

(4.3a)

bx = WDF bxF+WDR bxR

(4.3b)

µx = mx Fz + bx (4.3c)

where WD stands for the static weight distribution, with

WDF +WDR = 1 (4.4)

The second method to simulate two different tyres in a Point-Mass model consists on directly dividing

the vertical load Fz in front and rear components, FzF and FzR respectively, like in a ”bicycle” model.

Without taking mass transfer into account, the division of Fz is always done through the static weight

distribution. This method results in two different components for both Fx and Fy that are summed to give

the total longitudinal and lateral components of the force exerted by the tyre.

The lap time difference between the two methods (weighted average single tyre and two different

tyres) is 0.02% when compared with the real lap presented next in Chapter 5. The second method,

though, introduces the possibility of implementing longitudinal weight transfer. This way the vertical load

36

can be divided in front and rear components according to

FzF = Fz WDF −hcglFz

axg

(4.5a)

FzR = Fz WDR +hcglFz

axg

(4.5b)

where hcg is the CG height and l is the wheelbase. In each of equations (4.5), the first and second

terms correspond, respectively, to the static and dynamic weight distributions.

Comparison between the single weighted average tyre and the two tyres with weight transfer is done

through the resulting velocity profiles in a lap of the Estoril circuit in Figure 4.6.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

1 Tyre - 1:31.5072 Tyres w/ WT - 1:31.960

Figure 4.6: Resulting velocity profiles for the single weighted average tyre method and the two tyres withweight transfer method for simulating two different tyres in a Point-Mass model.

It is observed that the differences are mainly on the braking stages, with the weight transfer model

resulting in earlier braking points and consequently a lower lap time. The resulting lap times have a

difference of 0.46% when compared with the real lap registered in testing. Considering this, the single

tyre method can be used in order to simplify the code.

37

38

Chapter 5

Results

This chapter presents the outputs of both simulators used: the Point-Mass simulator developed (PM-

Sim) and RaceLap, the simulator available at Adess AG. PMSim is a Quasi-Steady-State simulator that

implements the vehicle model presented in section 3.1 using 11 parameters while for RaceLap a com-

prehensive and detailed list of 94 parameters is needed. This means the two compared simulators have

very different complexities. The main results – velocity profiles and lap times – are presented first in

sections 5.1 for RaceLap and 5.2 for PMSim. In sections 5.3 and 5.4 are the sensitivity analysis applied

to PMSim and the Energy Consumption Estimator, respectively. In sensitivity analysis is shown how the

variables that account for the main results vary in the course of a lap, including all that affect energy

consumption.

The validation for simulation comes from the results of a test day that an AD03 did at Autodromo do

Estoril [14] (Estoril). The best lap is used as reference to compare the simulation results. Data saved

from the test includes lap time, velocity profile and fuel consumption. Plotted in Figure 5.1 is the velocity

profile of the best lap done in the test by a professional racing driver. The lap time, as represented,

was 1:38.295 (min:sec). In simulations, the vehicle was parametrized to approximate the real car set-up

configuration.

The main difference from the real to simulated vehicle lies on the front tyres. Despite being the correct

model, the parametrized front tyres do not correspond to the real compound used. LMP3 category cars

usually race medium compound tyres at the front and hard at the rear but the manufacturer only provides

the Pacejka [3] model of the hard compound. This means that the simulated grip from the front tyres

does not correspond to the real ones. For this reason, an adjustment of the grip level of the front tyres

was added in order to achieve the minimum corner velocities verified in the real lap. Further explanation

on tyre compounds is in section 5.1.

Due to impossibility of using test data to design the trajectory described in the real lap, the Estoril

circuit trajectory used in simulation was taken from the OptimumG Track Database [15]. It is to be noted

that the real lap trajectory was 100 meters shorter than this track used in simulation.

39

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

Real lap - 1:38.295

Figure 5.1: Velocity profile of the best lap performed in a AD03 at the Estoril track with the lap time of1:38.295.

Energy Consumption

In the test day, not only in the fastest lap but also in all laps done in the same pace (1:38’s an 1:39’s)

the fuel consumption registered was of 2 litres/lap. The conversion from litres to kWh will be assumed

the same as presented in [12] where the energy density is considered 9.6111 kWh/litre. Using this

conversion, the real energy consumption for a lap of the Estoril track is 19.22 kWh. In estimation, the hill

climb and angular inertia terms of equation (1.4) are neglected because there is no information about

the track inclination and the engine shaft moment of inertia.

5.1 RaceLap

5.1.1 Front Grip Factor

Racing tyres are usually available in different compounds. Each tyre model provided by the manufac-

turer is divided in categories that correspond to the grip level of the tyre, called compounds or treads.

The categories are usually separated in Soft, Medium and Hard, with decreasing level of grip. A Soft

tyre has more grip but a shorter life-span than a Hard tyre. Different compounds also correspond to

distinct optimal operation conditions like pressure and temperature.

As stated in the introductory text of this chapter, in order to approximate the tyre model to the medium

40

compound used in the front axle, the grip of such tyres in the simulation must be multiplied by a factor.

The first choice criteria is the smallest factor that gives the same minimum corner velocities verified in

the real lap. These points are located before mid-turn (or before the appex) after the deceleration phase

when entering the corner and before the acceleration phase when exiting [16]. This is analysed here

for these are the points of a lap where the car’s behaviour is fully dependent on the tyre’s capability to

generate Fy and therefore other parameters are not a source of error.

Figure 5.2 shows the resulting velocity profile for five different grip factors as well the real lap for

comparison.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

RealSimSim×1.1Sim×1.2Sim×1.4Sim×1.5

Figure 5.2: Real and RaceLap simulated laps for different front tyres grip factor.

The results for a 1.1 and 1.2 factor (also for 1.3 but not shown for simplicity) are close to be identical,

with the same happening for 1.4 and 1.5. All of the results shown get identical appex velocities with the

main difference being the velocity slope right before the finishing line. Considering this, the final decision

for the grip factor will be between 1.1 and 1.4 because those are the smaller values within each group

and a second choice criteria must be met.

The 1.1 and 1.4 grip factors, highlighted in Figure 5.3 profiles diverge in the beginning and end of

the lap which correspond to points in the main straight line. The 1.4 velocity slope at the start of the

lap is identical to the slope of the real lap but the top speed is more accurate for the 1.1 profile. This

could mean that when testing, the real car was not in ideal conditions in terms of straight line behaviour

(powertrain related issues, for example). To analyse it further, the same simulation was performed for

the Le Mans track. Also called the Circuit the La Sarthe [17], the famous 24 Hours of Le Mans semi-

permanent race track is characteristic for very long straights. This particularity of the french circuit makes

this track ideal to assess the impact of the two grip factors in straight line behaviour. It was observed

from the Le Mans simulation with different vehicles that the 1.4 grip factor results, for some cases, in an

unusual behaviour in the straights, with the velocity profile suffering of several slope changes during a

41

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

RealSim*1.1Sim*1.4

Figure 5.3: Real and RaceLap simulated laps for 1.1 and 1.4 front tyres grip factor.

straight line. This lead to achieving lower speeds in longer straights. It was concluded that using a 1.4

grip factor might induce in unrealistic simulations and so the grip level chosen to simulate the medium

tyre compound for the front tyres was set to be 1.1.

5.1.2 Final Results

Finally are presented the final main results from the RaceLap simulation in Figure 5.4. Despite always

achieving higher velocities, the simulation gives a slower time. This can be justified by the extra 100

meters of the trajectory simulated. Taking the mean of the RaceLap velocity, 46.59 m/s, the extra

100 meters account for an extra 2.146 seconds, giving a lap time prediction of 1:37.442 for the same

trajectory that was done in the best lap of the real test. This result is given in Table 5.1 under ”Predicted

real trajectory time”. The top speed registered in the real lap was 257 km/h, with the simulation giving

256.6 km/h. The main difference between the profiles is in the positive acceleration stages whereas in

the braking stages the profiles are more similar. All main results are presented in Table 5.1.

Applying equation (1.4), the amount of energy needed at the wheels to perform a lap as simulated

in RaceLap is 5.62 kWh. Assuming 100% efficiency for an electric powertrain, this is would be the

calculated amount of energy consumed from a battery if AD03 was an electric vehicle. Because this

energy actually comes from a combustion engine, this value has to be divided by the efficiency of the

powertrain (tank-to-wheel) that is far from the electric vehicles values that come close to 100%. Advised

42

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

Real - 1:38.295RaceLap - 1:39.588

Figure 5.4: Velocity profiles of both RaceLap simulation (1:39.588) and the real lap (1:38.295).

by the engine manufacturer company, the engine efficiency is considered to be 30%. The gearbox must

also be accounted for, in this case with an efficiency of 96.04%. The total energy consumption of the

simulated lap is then

E =5.62

0.3 · 0.9604= 19.51 kWh (5.1)

Because there is no information about the track elevation and the engine shaft moment of inertia,

the hill climb and angular inertia terms of equation (1.4) were not accounted for. The resulting energy

consumption calculated value is 73.18% from the linear inertia term, 24.52% from aerodynamic drag

and 2.3% from tyres rolling resistance.

Table 5.1: RaceLap main resultsReal RaceLap Error (%)

Lap Time (min:sec) 1:38.295 1:39.588 1.32Predicted real trajectory time1(min:sec) - 1:37.442 -0.87Average Speed (km/h) 164.5 167.7 1.95Top Speed (km/h) 257 256.6 -0.16Energy Consumption (kWh) 19.22 19.51 1.5

1Prediction for the real trajectory by subtracting time corresponding to travelling 100 meters at the simulation average velocityfrom the simulation Lap Time.

43

5.2 Point-Mass

5.2.1 PMSim Demonstration

In this section is presented a demonstration of the Point-Mass model, Quasi-Steady-State simulator

developed in the scope of this work. Here, PMSim will be applied to two different tracks, Estoril [14] and

La Sarthe [17], and the resulting velocity profiles will be compared with the respective circuit maps. A

further analysis on the Estoril results is done in section 5.2.2, comparing with the validation data.

To run PMSim it is needed to have the PMSim main script, the maxcorvel function file to calculate

the maximum corner velocity for a certain corner radius and the motorcurves function file that calculates

the power, torque and gear output of the powertrain for a specific velocity and torque factor. The torque

factor is introduced in section 5.3.2. Vehicle, tyres and track data files are also needed. Only for this

demonstration, extra figure and function files of the final results GUI are necessary as well. The vehicle

data file is composed by:

• mass;

• SCX - normalized drag coefficient;

• SCZ - normalized lift coefficient (positive downwards);

• percentage of static weight distribution at the front axle;

• wheel radius;

• differential gear ratio;

• a vector with all gear ratios in the gearbox;

• a vector of engine rpm values that correspond to the torque values vector;

• a vector of engine torque values that correspond to the rpm values vector;

The tyres data file is composed by the parameters that make for the longitudinal and lateral friction

coefficients, µx and µy, respectively. These parameters are mx, bx, my and by.

The track data file is made of three column vectors with the following information about each segment

of the trajectory: distance from the finish line, trajectory radius of each point and road inclination angle.

Calling PMSim on the command window, the program asks for a track file to be selected as shown

in Figure 5.5.

44

Figure 5.5: Select Track window in PMSim.

The end product of the simulator is shown in Figure 5.6. Here is presented the velocity profile, lap

time, top speed and energy consumption for one lap around the selected circuit. Also in this Figure, the

circuit corners can be identified and corresponded to the circuit map of Figure 5.7.

Figure 5.6: Results window for the Estoril circuit with the correspondent corner names on the velocityprofile.

45

Figure 5.7: Autodromo do Estoril circuit map. Source: Wikipedia [14].

Making the same analysis applied to the Le Mans 24 Hours circuit, La Sarthe, the GUI with the

respective final results is seen in Figure 5.8. Again, the corners names can be identified in the velocity

profile and corresponded to the circuit map of Figure 5.9. Note that for this track there is no real test

data to compare with the results.

Figure 5.8: Results window for the La Sarthe circuit with the correspondent corner names on the velocityprofile.

46

Figure 5.9: La Sarthe circuit map. Source: Wikipedia [17].

47

5.2.2 Final Results

The final main results from the Point-Mass simulation are presented in Figure 5.10.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

Real - 1:38.295PMSim - 1:31.507

Figure 5.10: Velocity profiles of both Point-Mass simulation (1:31.507) and the real lap (1:38.295).

This model resulted in a more optimistic simulated vehicle behaviour. Even with the extra 100 meters

in the simulation trajectory, the lap time is still considerably lower than the real case. This corresponds

to a larger error than the RaceLap results with minimum corner velocities higher than the real case.

PMSim’s lower lap time is a consequence of higher top speeds in straights and a significant corner

velocity difference in turns 2, 8, 11 and the entry of 13 (see Figure 5.7), like in the final RaceLap results.

The average speed of 49.87 m/s gives a prediction of 1:29.502 for the real trajectory. This prediction, as

explained in the previous section, corresponds to subtracting the time equivalent to travelling 100 meters

at the average velocity of the simulation results and is given in Table 5.2 under ”Predicted real trajectory

time”. The top speed is 267.6 km/h.

Regarding fuel consumption, equation (1.4) results in 6.62 kWh of required energy to complete a lap.

As explained in section 5.1.2, this value has to be divided by the powertrain efficiency that accounts for

both the gearbox and the combustion engine.

E =6.62

0.3 · 0.9604= 22.98 kWh (5.2)

Once again, because there is no information about the track elevation and the engine shaft moment

of inertia, the hill climb and angular inertia terms of equation (1.4) were not accounted for. The resulting

48

energy consumption calculated value is 73.16% from the linear inertia term, 24.69% from aerodynamic

drag and 2.15% from tyres rolling resistance.

Table 5.2: PMSim main resultsReal PMSim Error (%)

Lap Time (min:sec) 1:38.295 1:31.507 -6.9Predicted real trajectory time2(min:sec) - 1:29.502 -8.9Average Speed (km/h) 164.5 179.5 9.1Top Speed (km/h) 257 267.6 4.1Energy Consumption (kWh) 19.22 22.98 19.56

2Prediction for the real trajectory by subtracting time corresponding to travelling 100 meters at the simulation average velocityfrom the simulation Lap Time.

49

5.3 Point-Mass Sensitivity Analysis

In this section, PMSim is used to apply sensitivity analysis on parameters of the Point-Mass model.

In the process, PMSim and RaceLap are compared through common outputs in both simulators. The

sensitivity analysis gives insight into the utility of the Point-Mass model by changing input parameters

different from the nominal used in section 5.2. Together, both the simulations outputs comparison and

the sensitivity analysis done on the PMSim lead to conclusions regarding the complexities of the two

vehicle models. The mentioned common outputs between both simulators are: ax, ay, Fz, FD and v. In

Figure 5.11 is the comparison of the velocity profiles from both simulators discussed in sections 5.1 and

5.2.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

50

100

150

200

250

300

Vel

ocity

(km

/h)

PMSim - 1:31.507RaceLap - 1:39.588

Figure 5.11: PMSim and RaceLap velocity profiles in a lap at Estoril.

The Point-Mass model results in an optimistic simulation in the sense that the overall lower lap time

results from higher velocities in both corners (lateral behaviour) and straights (longitudinal behaviour).

The sensitivity analysis will provide insight into what parameters most contribute to the error of PMSim

and if the higher straight-line velocities are a consequence of high corner velocities or the two are

independent of one another.

50

5.3.1 Drag Force (FD)

A Point-Mass model has a fixed value for SCX while on RaceLap, with a 4-wheel vehicle, this value

varies with the front and rear ride heights. In reality, the drag coefficient does change with variations

of ride height and so a correct approximation must be made to parametrize this in a Point-Mass model.

Figure 5.12 shows the variation of this coefficient in the course of a lap in Estoril. The default SCX value

chosen for the PM model was the mean of this variation also represented in this same figure.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

TrackLength (m)

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

SC

X

RaceLap SCX outputPMSim SCX nominal value

Figure 5.12: SCX variation in a lap in RaceLap. Mean value in horizontal line.

It is only in the 3 longer straights that the drag coefficient has a value lower than the average, achiev-

ing it’s higher values in the corners. In Figure 5.13 is the variation of the drag force (FD) in RaceLap for

one lap. In order to compare the influence of the variation of SCX in the drag force with the case of a

constant drag coefficient, the RaceLap velocity output was used to compute a FD profile with constant

SCX equal to the mean following equation (3.5a). The result shows that the profiles only diverge in the

sections of the lap where the RaceLap SCX is lower than the mean constant value. These sections

correspond to the 3 main straights, with the difference being higher the longer the straight is.

Aerodynamic drag mainly affects the straight line speed. In order to see the influence of SCX on

the overall lap performance, the following sensitivity analysis was done to this coefficient. The range of

values for this analysis is based on the RaceLap SCX vector output of Figure 5.12 and on references.

In the RaceLap simulation SCX varies roughly from 0.97 to 1.04. Smith [18] states as an example that

an old Formula 5000 has a SCX of 1.027 m2 in the end of a straight while Guiggiani [19] says that a

Formula 1 car, at the time of his writing, had an SCX between 0.91 and 1.3 m2. For Prototypes like the

AD03, [20] gives the example of a lower value of 0.5 m2. For context, passenger cars can vary from

51

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

0

50

100

150

200

250

300

350

FD (

kgf)

Varying SCXConstant SCX

Figure 5.13: RaceLap FD profile (blue) and equation (3.5a) applied for RaceLap velocity profile andconstant SCX (orange).

approximately 0.5 to 2.5 m2 [21].

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

SCX

89

90

91

92

93

94

95

Lap

Tim

e (s

)

Figure 5.14: Lap Time sensitivity analysis with drag coefficient.

The results of lap time analysis of Figure 5.14 make for a linear relation between the two variables

52

within the presented range. The slope of this linear relation is 0.4 s/0.1 SCX. Even for SCX values

higher than the maximum value found in literature, the simulation lap time is still far lower than the real

case or the RaceLap results. From this, it is concluded that the drag coefficient is not the parameter that

causes the less realistic, low PMSim lap times.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

SCX

230

240

250

260

270

280

290

300

310

Top

Spe

ed (

km/h

)

Figure 5.15: Top Speed sensitivity analysis with drag coefficient.

Figure 5.15 shows the impact of the drag coefficient on top speed. Although the variation in the

whole range analysed is considerable, the top speed is not sensible enough to the SCX. Approximating

the variation to a linear function, the rate of change is -5.25 km/h / 0.1 SCX. Knowing from Figure 5.12

that the maximum variation of SCX in a RaceLap is 0.07 is concluded that the nominal value used in

the Point-Mass model is not the cause of simulation error in top speed.

53

5.3.2 Longitudinal Acceleration (ax)

In section 3.1 of Chapter 3 is discussed that the longitudinal acceleration can be limited either by the

powertrain or the tyres longitudinal grip. For this reason, this section’s analysis lies on a torque factor

(introduced next) and on the tyre’s longitudinal coefficient of friction µx.

Figure 5.16 shows the variation of ax for both simulators in one lap. The main difference between

both profiles is in the braking stages, with PMSim achieving higher negative acceleration.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

ax (

g)

PMSimRaceLap

Figure 5.16: Longitudinal acceleration profile in RaceLap and PMSim.

Torque Factor

As explained in section 5.1.2 when discussing RaceLaps’s final results, the higher top speeds can

be a consequence of the real powertrain running conditions having a smaller efficiency than predicted.

To evaluate this possibility, a sensitivity analysis was applied to a factor that impacts the engine per-

formance. The torque output from the engine is multiplied by this factor in order to simulate either a

powertrain lower efficiency or the throttle pedal not being actuated in full capacity. Figures 5.17 and 5.18

show the impact of the torque factor on lap time and top speed, respectively.

It is observable that this factor has bigger impact in top speed. Because the corner speeds are

maintained, the lap time is still far lower than the real case but the longitudinal behaviour is considerably

affected. Considering a factor of 0.9, the error values tabulated in Table 5.2 lower to the ones presented

in Table 5.3.

54

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

Torque Factor

91.5

92

92.5

93

93.5

94

94.5

95

95.5

96

Lap

Tim

e (s

)

Figure 5.17: Lap Time sensitivity analysis with Torque Factor.

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

Torque Factor

235

240

245

250

255

260

265

270

Top

Spe

ed (

km/h

)

Figure 5.18: Top Speed sensitivity analysis with Torque Factor.

55

Table 5.3: PMSim main results errors with TF = 0.9Error (%)

Lap Time -5.74Predicted real trajectory time3 -1Average Speed 7.04Top Speed 0.41Energy Consumption 2.85

Longitudinal Coefficient of Friction (µx)

When describing the Point-Mass model in Chapter 3, it is explained that ax can be limited by the

tyres capacity to generate longitudinal force. As stated in the same Chapter, the coefficient of friction

is composed of two parameters that regulate it’s linear relation with the vertical load exerted on the tyre

(Fz): the slope (mx) and the intercept (bx).

In the case of a simple straight line acceleration from zero, as shown in Figure 3.3, the tyres are the

limiting factor until the car achieves around 110 km/h. This means that µx has a bigger impact at the exit

of slow corners. The longitudinal coefficient of friction is also responsible for the braking capacity of the

vehicle.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

bx

90.5

91

91.5

92

92.5

93

93.5

94

94.5

95

Lap

Tim

e (s

)

bx

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

mx ×10-5

90.5

91

91.5

92

92.5

93

93.5

94

94.5

95

mx

Figure 5.19: Lap Time sensitivity analysis with longitudinal friction coefficient parameters mx and bx.

Figure 5.19 shows that the simulation is more sensitive to bx than mx which is expected because

the slope (mx) is 5 units of magnitude lower. The maximum decrease of the intercept value resulted in

a lap time still considerably lower than the real case. For b = 1.1 the top speed was only lowered to3Prediction for the real trajectory by subtracting time corresponding to travelling 100 meters at the simulation average velocity

from the simulation Lap Time.

56

266.191 km/h, corresponding to a decrease of 0.52 %. This leads to the conclusion that lowering the

tyre longitudinal grip can contribute to more realistic lap times but not on it’s own. The higher values of

ax in the braking stages are a consequence of modelling the longitudinal behaviour of the tyres through

the friction coefficient but its impact in lap time is not enough to be considered a main source of error

in the Point-Mass model. Also concluded is that µx has no significant impact on top speed, which is

realistic.

57

5.3.3 Downforce (Fz)

Just as when discussed about drag, the downforce (or lift) coefficient, SCZ, changes with front and

rear ride heights in a real vehicle. RaceLap replicates this. Figure 5.20 shows the variation of the SCZ

coefficient in RaceLap. The mean value is also represented through a constant line, chosen as the

nominal value in the Point-Mass model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

TrackLength (m)

2.8

2.9

3

3.1

3.2

3.3

3.4

SC

Z

RaceLap SCZ outputPMSim SCZ nominal value

Figure 5.20: SCZ variation in a lap in RaceLap. Mean value in horizontal line.

Figure 5.21 presents the RaceLap Fz output and the profile generated from applying the aerodynamic

downforce equation (3.5b) to the RaceLap velocity vector with constant SCZ equal to the nominal Point-

Mass model value.

Analysing both Figures 5.20 and 5.21 is shown that the points where there is a bigger difference in Fz

between the two simulators correspond to where the SCZ coefficient in RaceLap is bellow the nominal

mean value of the Point-Mass model. These points correspond to the corners of the track. Assuming

most of the lower PMSim lap time is due to the higher corner velocities, approximating the Point-Mass

model SCZ to the values registered when cornering in RaceLap could lead to closer results between

both simulators.

The range for SCZ values sensitivity analysis is based on the results of Figure 5.20 and on refer-

ences. Guiggiani [19] states that a Formula 1, at the time of his writing, had a downforce SCZ of 5.2 m2

while [20] gives the example of 3.2 m2 for an SCZ value of a Prototype class car.

The lap time analysis of Figure 5.22 shows an overall linear relation with SCZ. The slope is approx-

imately -0.2 s/0.1 SCZ which is lower than the LapTime(SCX) relation in module. The decrease until

SCZ = 2 still holds a lap time far from the RaceLap result and the real case, leading to the conclusion

58

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

1000

1200

1400

1600

1800

2000

2200

Fz (

kgf)

Varying SCZConstant SCZStatic mass

Figure 5.21: RaceLap Fz profile (blue) and equation (3.5b) applied for RaceLap velocity profile andconstant SCZ (orange). AD03 static mass in horizontal line.

2 2.5 3 3.5 4 4.5 5 5.5

SCZ

87

88

89

90

91

92

93

94

Lap

Tim

e (s

)

Figure 5.22: Lap Time sensitivity analysis with downforce coefficient.

that the SCZ nominal value used is also not the parameter mainly responsible for the error of simulation.

The main influence of the downforce is in corner velocity for it impacts the tyre grip. Recalling

59

equation (3.2b), Fz is the argument of both Fy and µy. This directly impacts ay and so this analysis

must be done together with the results from section 5.3.4.

60

5.3.4 Lateral Acceleration (ay)

The lateral acceleration of the vehicle resorts on the tyres lateral grip. From equation (3.2b) is known

that the lateral force Fy the tyre produces is a linear function of the vertical force exerted on it - the

downforce Fz. Figure 5.23 shows the comparison of the registered values of ay in a lap in Estoril for

both simulators.

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Track Length (m)

0

0.5

1

1.5

2

2.5

ay (

g)

PMSimRaceLap

Figure 5.23: Lateral acceleration profile in RaceLap and PMSim.

It is evident that the higher corner velocities in PMSim are a consequence of higher lateral acceler-

ation. This can have two causes: the downforce (SCZ) or the friction (µy) coefficients. Section 5.3.3

presents the sensitivity analysis applied to the SCZ coefficient. Regarding µy is recalled equation (4.2)

that has two parameters: m and b. These are the slope and the intercept of the grip linear function,

respectively, here mentioned as my and by. The sensitivity analysis to the tyres Fy is done on these two

grip parameters.

The analysis of Figure 5.24 shows that the intercept is more sensitive than the slope. This is ex-

pected as the slope is 5 units of magnitude lower. This way, the intercept variation is more suitable to

describe the sensitivity of the lateral friction coefficient µy. These results show that µy is a very sensitive

parameter and together with the analysis of Figure 5.23 is concluded that it is a major responsible for the

bigger error of the Point-Mass simulation. The drop of top speed for by = 1.1 was only 1.26 %, meaning

that the lowering of corner velocities did not contribute for a significant drop in top speed.

61

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

by

88

90

92

94

96

98

100

102

104

106La

p T

ime

(s)

by

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

my ×10-5

88

90

92

94

96

98

100

102

104

106

my

Figure 5.24: Lap Time sensitivity analysis with lateral friction coefficient parameters my and by.

62

5.4 Estimator Sensitivity Analysis

The energy consumption estimation is common between the two simulators. As stated in the above

Chapters, this estimation is done through equation (1.4) without the angular inertia and hill climb terms.

In this section, a sensitivity analysis is done to this estimator seeking to conclude if it is rightly applied to

both simulators.

5.4.1 Linear Inertia

For this analysis, the variation of the propulsion force is done through a multiplication factor applied

to equation (5.3). The range for the factor in this analysis is [65,165]%. The results of the sensitivity

analysis of the linear inertia term of the estimator are shown in Figure 5.25.

ELI = Fx d (5.3)

70 80 90 100 110 120 130 140 150 160

Fx fraction (%)

70

80

90

100

110

120

130

140

150

Ene

rgy

Con

sum

ptio

n fr

actio

n (%

)

PMSimRaceLap

Figure 5.25: Energy Consumption Estimator sensitivity analysis with Fx variation in fraction of the defaultvalues of Sections 5.1 and 5.2.

The results show a big similarity on the estimation variation applied to both simulator outputs.

63

5.4.2 Aerodynamic Drag

In the simulation models, higher SCX values result in lower longitudinal acceleration and subse-

quently lower top speeds. These two variables impact the energy consumption through equation (1.4).

Nevertheless, SCX is also a variable in this equation. This way, the drag coefficient is present in both

the simulation models and the estimator. The sensitivity analysis to the aerodynamic term of the energy

consumption equation is done to the SCX in the estimator, by applying a multiplication factor to equation

(5.4).

ED =1

2v2 ρ SCX d (5.4)

The range of drag coefficient values in this analysis is the same as in section 5.3.1. The range

corresponds to approximately 40% to 180% of the nominal value used in the Point-Mass model. The

results are plotted in percentage of the nominal energy consumption result in both simulations: 5.62

kWh for RaceLap and 6.62 kWh for PMSim.

40 60 80 100 120 140 160 180

SCX fraction (%)

85

90

95

100

105

110

115

120

Ene

rgy

Con

sum

ptio

n fr

actio

n (%

)

PMSimRaceLap

Figure 5.26: Energy Consumption Estimator sensitivity analysis with drag coefficient variation in fractionof the default values of Sections 5.1 and 5.2.

Figure 5.26 shows that the estimator is approximately equally sensitive for both models. Estimation

applied to RaceLap outputs resulted in a slope of 0.245 energy variation / SCX variation while the same

applied to the PMSim outputs resulted in a slope of 0.247.

64

5.4.3 Rolling Resistance

The tyres rolling resistance, like the aerodynamic drag, contribute to lowering the longitudinal acceler-

ation in the simulation models. This factor (frr) is also present in the energy consumption equation (1.4).

Here, a sensitivity analysis is applied to frr in the estimator using the outputs from both simulators, by

applying a multiplication factor to equation (5.5).

Err = Fz frr d (5.5)

The analysis is done in terms of fraction of the nominal values of sections 5.1 and 5.2. Figure 5.27

shows the results of this sensitivity analysis.

20 40 60 80 100 120 140 160 180 200

frr fraction (%)

97.5

98

98.5

99

99.5

100

100.5

101

101.5

102

102.5

Ene

rgy

Con

sum

ptio

n fr

actio

n (%

)

PMSimRaceLap

Figure 5.27: Energy Consumption Estimator sensitivity analysis with rolling resistance coefficient varia-tion in fraction of the default values of Sections 5.1 and 5.2.

The results show that the RaceLap outputs are more sensitive to the energy consumption estimation

as function of the rolling resistance with a slope of 0.0230 energy variation / frr variation. The PMSim

outputs are less sensitive, with a slope of 0.0215.

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Chapter 6

Conclusions

Regarding energy consumption estimation, results show that the linear inertia of the vehicle is the

main contributor of energy loss, followed by the aerodynamic drag and a relatively small contribution

of the tyres rolling resistance, not considering the contribution of the angular inertia of the wheels and

engine shaft and the road inclination variations of the track. The estimator has identical variations when

applied to different simulators outputs, working for both a commercial 4-wheel model and a Point-Mass

model developed in the scope of this work with about 88% less input parameters.

A Point-Mass model, Quasi-Steady-State simulator (PMSim) was developed as a simpler and easier

alternative to predict the vehicle performance in a race track. In this development, different approaches

were tested to simplify simulation the most. The Point-Mass is composed of a powertrain model con-

taining the engine torque curve and all gear ratios, an aerodynamic model made from constant drag and

downforce coefficients, a chassis model that includes the vehicle weight and static weight distribution

and a tyre model with the wheel radius and the longitudinal and lateral friction coefficients of the Pacejka

2002 as function of vertical load. Regarding track modelling, two approaches were tested. It is showed

that making a distinction between straights and corners segments or modelling all segments as corners

give similar results because specifying a large enough radius on a corner segment gives the same re-

sult as in a straight. To solve the problem of modelling different front and rear tyres in a Point-Mass

model, it was found that combining two tyres into one by weighted average of the corresponding friction

coefficients gives marginally the same results has making a distinction between the two different tyres.

Modelling the AD03 and the Estoril track in RaceLap, the commercial 4-wheel model, Quasi-Steady-

State simulator, resulted in satisfactory simulation results validated by a real test lap. Adjusting the grip

level to approximate the hard compound tyres model to the medium compound used in the real car, it

was found that increasing the grip factor more than 10% only affects the longitudinal behaviour of the

car, inducing in unrealistic simulations.

In the Point-Mass simulation it was found that neither of the aerodynamic coefficients have enough

impact in lap time and top speed to be a considerable source of error in the model. The same happens

when reducing the longitudinal friction coefficient. On the other hand, the variation of the torque factor

applied to the engine torque output significantly impacts the resulting top speed. From this was con-

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cluded that the source of longitudinal acceleration that mainly impacts the top speed of the vehicle is

the engine propulsion force. Regarding the lateral behaviour of the vehicle, results show that the lateral

coefficient of friction has a considerable impact in lap time, leading to the conclusion that this is the main

source of discrepancies in the Point-Mass model. This agrees with the main difference between the

two simulators used in this work being in the lateral acceleration output. Because the lateral coefficient

of friction has no significant impact in top speed and the torque factor reduction alone does not impact

enough the lap time, it is concluded that the errors in longitudinal and lateral behaviour are independent

of each other.

6.1 Achievements

An energy consumption estimator was successfully implemented in two different lap time simulators

with very different vehicle model complexities and so the outcome of this work can be applied in the

design of the energy storage system of an electric competition vehicle.

When compared with the real test lap, the final results of the Point-Mass model are a -6.9% error

for lap time, 4.1% error in top speed and an estimation error of 19.56% for energy consumption. In

RaceLap, with a much more complex vehicle model, these results are a 1.32% error in lap time, -0.16%

in top speed and 1.5% in energy consumption estimation. This shows that the Point-Mass model is a

valid alternative to make vehicle dynamics simulation when only basic vehicle parameters are known

and that energy consumption estimation can be made in Lap Time Simulation.

6.2 Future Work

Because the main source of error in the Point-Mass model comes from the simplification of the tyres

model, a recommendation for improving this work is implementing a ”bicycle” model. The ”bicycle”

model stands somewhere in the middle of the vehicle models scope, with the Point-Mass and 4-wheel

models being in the extreme ends. This vehicle model includes a distinction between the front and rear

axles of the car and calculates tyre slip angle and vehicle side slip. Knowledge of these angles in each

track segment gives the opportunity to implement a full Pacejka tyre model to calculate longitudinal and

lateral tyre forces with more precision than just using the coefficients of friction. Also as a consequence

of this more precise tyre model, longitudinal weight transfer simulation might have a bigger impact in the

results. Simplifying the vehicle model helps to better understand in a broader sphere the areas of the

car that mainly affect energy consumption and a step to a more complex vehicle model or simulation

type must be done only when the previous is understood to its full extent.

The implementation of simulation tools must always be accompanied by testing for validation pur-

poses. To validate a Lap Time Simulator it is needed to model the trajectory of a track where the vehicle

will be testing and certify that the driver can operate at maximum accelerations. It is also important

to know the evolution of the efficiency of the energy storage system throughout several laps and pos-

sibly implementing this variation in the Lap Time Simulator. This can be done by making different lap

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simulations for each value of efficiency to understand how the performance and the energy required to

complete a lap change in the course of an event.

69

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