Rolling Resistance During Cornering - Impact of Lateral Forces for Heavy- Duty Vehicles

Home , Rolling resistance, Slip (vehicle dynamics), Slip angle, Traction (engineering)

DEGREE PROJECT IN MASTER;S PROGRAMME, APPLIED AND COMPUTATIONAL MATHEMATICS 120 CREDITS, SECOND CYCLE STOCKHOLM, SWEDEN 2015

Rolling resistance during cornering - impact of lateral forces for heavy- duty vehicles

HELENA OLOFSON

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

Rolling resistance during cornering - impact of lateral forces for heavy-duty vehicles

HELENA OLOFSON

Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master's Programme, Applied and Computational Mathematics (120 credits) Royal Institute of Technology year 2015 Supervisor at Scania AB: Anders Jensen Supervisor at KTH was Xiaoming Hu Examiner was Xiaoming Hu

TRITA-MAT-E 2015:82 ISRN-KTH/MAT/E--15/82--SE

Royal Institute of Technology SCI School of Engineering Sciences

KTH SCI SE-100 44 Stockholm, Sweden

URL: www.kth.se/sci

iii

Abstract

We consider first the single-track bicycle model and state relations between the tires’ lateral forces and the turning radius. From the tire model, a relation between the lateral forces and slip angles is obtained. The extra rolling resistance forces from cornering are by linear approximation obtained as a function of the slip angles. The bicycle model is validated against the Magic-formula tire model from Adams. The bicycle model is then applied on an optimization problem, where the optimal velocity for a track for some given test cases is determined such that the energy loss is as small as possible. Results are presented for how much fuel it is possible to save by driving with optimal velocity compared to fixed average velocity. The optimization problem is applied to a specific laden truck.

Keywords: Bicycle model, lateral forces, rolling resistance, velocity optimization, fuel consumption iv

Rullmotstånd på kurvig väg - inverkan av laterala krafter för tunga fordon

Sammanfattning

Vi betraktar först den enspåriga cykelmodellen och ställer upp samband mellan däckens sidokrafter och kurvradien. Genom däcksmodellen fås ett samband för hur sidokrafterna beror av slipvinklarna. De extra rullmotståndskrafterna för kurvor fås via linjär approximation som funktion av slipvinklarna. Cykelmo- dellen valideras mot en däcksmodell från Adams. Cykelmodellen tillämpas sedan på ett optimeringsproblem där den optimala hastigheten längs en bana för några givna testfall bestäms så att energiförlus- ten blir så liten som möjligt. Resultat presenteras för hur mycket bränsle det är möjligt att spara genom att köra med optimal hastighet jämfört med ﬁx medelhastighet. Optimeringsproblemet tillämpas på en speciﬁk lastad lastbil.

Nyckelord: Cykelmodell, sidokrafter, rullmotstånd, hastighetsoptimering, bräns- leförbrukning

Acknowledgements

I would like to thank my supervisor Anders Jensen at Scania for the assistance and cooperation and Henrik Wentzel for his continual guidance and support. I would also like to thank Xiaoming Hu at the Division of Optimization and Systems Theory at KTH for the help and encouragement. I am also grateful for all the other people who have, directly or indirectly, supported me.

Helena Olofson Stockholm, 2015

Contents

Contents vi

List of Figures viii

1 Introduction 1 1.1 Motivation ...... 1 1.2 Organization ...... 2

2 Background 3 2.1 Notation ...... 3 2.2 Vehicle coordinate system ...... 5 2.3 Forces ...... 5 2.3.1 Vertical force ...... 5 2.3.2 Longitudinal force ...... 5 2.3.3 Lateral force ...... 7 2.4 Bicycle model ...... 9 2.4.1 Low-speed cornering ...... 10 2.4.2 High-speed cornering ...... 10

3 Bicycle Model 13 3.1 Model ...... 13 3.1.1 Model with aerodynamic drag force ...... 13 3.1.2 Rolling resistance ...... 15

4 Validation of bicycle model 17 4.1 Purpose and method ...... 17 4.2 Truck speciﬁcation ...... 17 4.3 Adams tire model simulation ...... 18 4.4 Comparison bicycle model to Adams tire simulation ...... 19 4.4.1 Lateral force ...... 20 4.4.2 Rolling resistance ...... 21 4.4.3 Weight shift ...... 24 4.5 Summary ...... 25

vi CONTENTS vii

5 Optimization problem 27 5.1 Problem formulation ...... 27 5.2 Method ...... 27 5.3 Solution and simulation ...... 30 5.3.1 Radius ...... 32 5.3.2 Turnlength ...... 34 5.3.3 Average velocity ...... 35

6 Conclusions and discussion 37 6.1 Future work ...... 38

Bibliography 39

A Graphs from the simulations 41 A.1 Validation of bicycle model using Adams tire simulation ...... 41 A.1.1 Slip angle ...... 42 A.1.2 Cornering stiﬀness analysis ...... 46 A.1.3 Longitudinal forces ...... 48 A.1.4 Lateral forces ...... 51 A.1.5 Weight shift ...... 52 List of Figures

2.1 Vehicle coordinate system DIN 70000/ISO 8855 ...... 5 2.2 Tire during forward motion ...... 6 2.3 Slip angle and lateral force of one tire ...... 8 2.4 Camber angle of one tire ...... 8 2.5 Transverse deformation of a tire ...... 9

2.6 Induced aligning moment, Mzi , on a tire from the lateral force, Fyi , acting a distance Pt from the contact point...... 9

2.7 Kamm circle showing the traction limit for the tire forces Fxi and Fyi . 10 2.8 Bicycle model for low-speed cornering ...... 11 2.9 Bicycle model for high-speed cornering, without longitudinal force . . . 11

3.1 Bicycle model, with aerodynamic drag force, FA, and longitudinal force, Fxr...... 14 3.2 Bicycle model, with the rolling resistance forces ...... 15

4.1 Truck with maximum technical weight at front- and rear axle ...... 18

4.2 Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h

and v = 90 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck)...... 20

4.3 Total front lateral force, Fyf , and total rear lateral force, Fyr, versus turning radius from bicycle model and Adams tire model simulation for

v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck)...... 21

4.4 Power loss due to rolling resistance from straight driving, Prr, versus turning radius from bicycle model and Adams tire model simulation for the laden truck. For small radii, the power loss Prr adds to the extra rolling resistance power loss from cornering, Prr,tu+...... 22 4.5 Power loss due to rolling resistance from straight driving, Prr, versus turning radius from bicycle model and Adams tire model simulation for the unladen truck. For small radii, the power loss Prr adds to the extra rolling resistance power loss from cornering, Prr,tu+...... 22

viii List of Figures ix

4.6 Extra rolling resistance power loss from cornering, Prr,tu+, versus turning

radius from bicycle model and Adams tire model simulation using Cαf1 =

4360 N/deg and Cαr1 = 4940 N/deg (laden truck)...... 23 4.7 Extra rolling resistance power loss from cornering, Prr,tu+, versus turning

radius from bicycle model and Adams tire model simulation using Cαf1 =

3320 N/deg and Cαr1 = 1290 N/deg (unladen truck)...... 23 4.8 Weight shift for each tire versus turning radius from Adams tire model simulation for v = 70 km/h (laden truck)...... 24

5.1 The optimization problem track ...... 27 5.2 Track consisting of one straight section and one turn with step length h. 28 5.3 Optimal velocity and fixed average velocity, vm, in each 5 m section along the track. The optimal velocity is lower than vm in the turns. However, at the straight sections, the optimal velocity is higher than vm in order to fulfill the time constraint...... 31 5.4 Power loss in each 5 m section along the track. The power loss for the optimal velocity, is lower in the turns compared to the power loss for the fixed average velocity, vm. However, at the straight sections the power loss is higher for the optimal velocity...... 32 5.5 Optimal velocity and fixed average velocity, vm, in each 5 m section along the track...... 33 5.6 Power loss for the optimal velocity (blue line) and fixed average velocity vm (red line) in each 5 m section along the track...... 33 5.7 Optimal velocity and fixed average velocity, vm, in each 5 m section along the track...... 34 5.8 Power loss for the optimal velocity (blue line) and fixed average velocity vm (red line) in each 5 m section along the track...... 35 5.9 Optimal velocity and fixed average velocity, vm = 80 km/h, in each 5 m section along the track...... 36 5.10 Power loss along the track, for the optimal velocity and vm = 80 km/h. 36

A.1 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 70

km/h using Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck). . 43 A.2 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 50

km/h using Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck). . 43 A.3 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 90

km/h using Cαf1 =4710 N/deg and Cαr1 = 4940 N/deg (laden truck). . . 44 A.4 Best correspondence between bicycle model and Adams tire model sim-

ulation for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck)...... 44 x List of Figures

A.5 Slip angle versus turning radius for each tire from Adams tire model simulation for v = 70 km/h (laden truck)...... 45 A.6 Front- and rear slip angle (αf and αr) from bicycle model versus turning

radius for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck)...... 45 A.7 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 70

km/h using Cαf1 = 3450 N/deg and Cαr1 = 1320 N/deg (unladen truck). 46 A.8 Best correspondence between bicycle model and Adams tire model sim-

ulation for v = 70 km/h using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck)...... 47

A.9 Variation of Ceff,fi with mass. Horizontal lines denote Cαf1 before (red and blue line) and after change (black and magenta) for the laden- and unladen truck...... 47 A.10 Illustration of which color is associated with which mass. Connected to Figure A.9 above...... 48 A.11 Longitudinal force versus longtudinal slip for each tire obtained from Adams tire model simulation for v = 70 km/h (laden truck)...... 49 A.12 Longitudinal force versus turning radius for each tire from Adams tire model simulation for v = 70 km/h (laden truck)...... 49 A.13 Total longitudinal force for the left- respectively the right tires versus turning radius obtained from Adams tire model simulation for v = 70 km/h (laden truck)...... 50 A.14 Total longitudinal force for the left- and right tires on the front- respec- tively rear axle versus turning radius from Adams tire model simulation for v = 70 km/h (laden truck)...... 50

A.15 Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h

and v = 90 km/h using Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck)...... 51

A.16 Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h

and v = 90 km/h using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck)...... 51 A.17 Total front lateral force, Fyf , and rear lateral force, Fyr, versus turning

radius for v = 70 km/h using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck)...... 52 A.18 Weight shift for each tire versus turning radius from Adams tire model simulation for v = 50 km/h (laden truck)...... 52 A.19 Weight shift for each tire versus turning radius from Adams tire model simulation for v = 90 km/h (laden truck)...... 53

Chapter 1

Introduction

1.1 Motivation

One of the main costs for operating heavy-duty vehicles is the fuel [1]. Two ma- jor factors that have big impact on the fuel consumption for heavy-duty vehicles are the rolling- and aerodynamic resistance. Rolling resistance is bigger than the aerodynamic resistance for speeds below about 72 km/h. For higher speeds, the aerodynamic resistance is bigger [3]. This thesis is focused on rolling resistance. Rolling resistance is defined as the force that resists the motion of the truck and is developed in the contact patch between tire and road due to deformation of the tire. At straight driving rolling resistance arises as a result of deformation in the longitudinal direction. The rolling resistance force from straight driving is stated linearly, by the rolling resistance coefficient, as a function of the vertical load. The rolling resistance coefficient depends on the tire properties and speed. For example an increase of tire load decreases the rolling resistance coefficient and the coefficient increases linearly with speed [12]. During cornering, the tire experience deformation also in the lateral direction. The lateral deformation arises from the lateral forces, that is developed by each tire in response to the lateral acceleration (see Section 2.3.3). Because of the deformation in the lateral direction, the rolling resistance loss increases during cornering [12]. The mathematical modelling of the rolling resistance forces is found in Chapter 3, Section 3.1.2. In this thesis the extra rolling resistance loss from cornering compared to driving straight is quantified in order to see the impact on the total resistance loss. By knowing the increase in rolling resistance at cornering, it is possible for truck drivers to choose route more appropriately such that the resistance loss is as small as possible. The rolling resistance loss is also affected by wet- and uneven roads. In both cases an extra term has to be added to the total rolling resistance loss corresponding to the effect of displacing the water and material in front of the truck [4]. There

1 2 CHAPTER 1. INTRODUCTION is also a rolling resistance term from bearing friction. This term can usually be neglected, however when the truck is started the bearing friction has signiﬁcant impact. The bearing friction increases on unpaved roads, due to vibration [12]. In this thesis the road is smooth and dry.

1.2 Organization

The rest of the thesis is organized as follows. Chapter 2 contains a list of notation for the ﬁrst four chapters of the thesis, thereafter basic concepts (e.g. tire forces, bicycle model) are introduced. Chapter 3 contains all the analysis of the bicycle model. Here, the bicycle model equations are stated and the rolling resistance forces are deﬁned. In Chapter 4, the bicycle model is validated against an Adams tire model simulation. In Chapter 5, an optimization problem using the bicycle model is stated for a test track. Lastly, Chapter 6 contains key results, conclusions and a discussion part. Also, some potential future work is presented. Chapter 2

Background

2.1 Notation

The following notation will be used throughout the thesis.

Parameter Description Unit m total mass of the truck kg mf mass of the front axle kg mr mass of the rear axle kg mi mass of one tire kg Rl loaded radius of a tire m R0 unloaded radius of a tire m

Rei eﬀective rolling radius of a tire m ω angular speed of revolution of a tire rad/s ω0 angular speed of revolution of a free rolling tire rad/s Γi torque acting on a tire Nm Mz aligning moment Nm Pt pneumatic trail m R turning radius m L wheelbase m lf distance from front axle to center of gravity m lr distance from center of gravity to rear axle m t track width m d distance travelled m g gravitational acceleration constant m/s2 ρ density (of air) kg/m3 v velocity at center of gravity m/s v˙ angular velocity at center of gravity rad/s

vxi longitudinal velocity of a tire m/s I yaw inertia kg · m2 r yaw rate s−1

3 4 CHAPTER 2. BACKGROUND

r˙ angular acceleration rad/s2 δ steering angle rad β body slip angle rad κ longitudinal slip - α slip angle rad αf slip angle front tire rad αr slip angle rear tire rad αf,tot total slip angle front axle deg αr,tot total slip angle rear axle deg Cα cornering stiffness N/rad Cαf cornering stiffness front axle N/rad Cαr cornering stiffness rear axle N/rad

Cαf1 cornering stiﬀness of one front tire N/deg

Cαr1 cornering stiﬀness of one rear tire N/deg

Ceff,fi eﬀective cornering stiﬀness front tire N/deg

Ceff,ri eﬀective cornering stiﬀness rear tire N/deg F centrifugal force N

Fzi vertical force of one tire N Fx longitudinal force N Fxr drive force rear axle N Fy lateral force N

Fyfi lateral force of one tire front axle N Fyf lateral force front axle N Fyr lateral force rear axle N Ff friction force N Crr rolling resistance coeﬃcient for straight driving - Cd aerodynamic drag coeﬃcient - A cross sectional area m2

Frri rolling resistance force of one tire, straight driving N Frr,f rolling resistance force of front axle, straight driving N Frr,r rolling resistance force of rear axle, straight driving N ∆Frr,f extra rolling resistance force of front axle, turning N ∆Frr,r extra rolling resistance force of rear axle, turning N Prr power loss due to rolling resistance, straight driving kW Err energy loss due to rolling resistance, straight driving kWh Prr,tu power loss due to rolling resistance, turning kW Err,tu energy loss due to rolling resistance, turning kWh Prr,tu+ extra power loss due to rolling resistance, turning kW Err,tu+ extra energy loss due to rolling resistance, turning kWh Table 2.1: Parameters used throughout the thesis. Parameters speciﬁc for the optimization problem are deﬁned in Chapter 5. 2.2. VEHICLE COORDINATE SYSTEM 5

2.2 Vehicle coordinate system

There exists many standards of the vehicle coordinate system. In this thesis the vehicle coordinate system DIN 70000 or equivalenty ISO 8855 is used. Figure 2.1 shows how this coordinate system is deﬁned [9].

yaw

y pitch

x roll Figure 2.1: Vehicle coordinate system DIN 70000/ISO 8855

2.3 Forces

A tire is characterized basically by three properties; it’s ability to sustain vertical load and to produce longitudinal- and lateral force. Below, the three fundamental forces for a tire are explained.

2.3.1 Vertical force

The vertical force, Fzi , for one tire is obtained according to

Fzi = mig, (2.1) for a smooth road. Here, mi is the mass of one tire. However, the vertical force (and the longitudinal force) change in magnitude if there are discontinuities in the road [12].

2.3.2 Longitudinal force The longitudinal force is developed in the contact patch between tire and road during motion of the vehicle due to e.g. braking, accelerating, rough roads or uneven tires. 6 CHAPTER 2. BACKGROUND

The force is characterized with the help of longitudinal slip. The longitudinal slip of one tire, κi, is deﬁned as

ω0 − ωi vxi − ωiRei κi = − = − (2.2) ω0 vxi

Here, ωi is the angular speed of revolution of a tire, while ω0 is the angular speed of revolution of a free rolling tire. vxi and Rei are the longitudinal speed of a tire respectively the eﬀective rolling radius. They are both depicted in Figure 2.2.

When κi is positive, the longitudinal force of the tire, Fxi , is positive and when κi is negative, Fxi is negative. When κi = −1, wheel lock occurs. However, by formula (2.2), κi is only limited during braking [5, 10]. During motion of the truck, the vehicle feel traction to the road. The traction between a tire and the surface in the longitudinal direction, µxi , is deﬁned as

Fxi µxi = (2.3) Fzi Note that in formula (2.3) the small diﬀerence from the rolling resistance force is neglected. The traction limit between tire and road is denoted µp [5, 12]. Hence, it always holds that

Fxi ≤ µpFzi , (2.4) when no lateral force is present.

+Γi

vxi

Rl Rei

+Fxi e

+Fzi Figure 2.2: Tire during forward motion

Figure 2.2 shows a driven tire at steady-state motion. In the ﬁgure, Γi is the applied torque from the engine to one tire and Rl the loaded tire radius. As previously 2.3. FORCES 7

noted, vxi denotes the forward (longitudinal) velocity of a tire, Fxi the longitudinal force and Rei the eﬀective rolling radius. As illustrated in the ﬁgure, the vertical load is concentrated at a point located distance e to the right from the center of the contact patch. The moment equation for the tire can be stated as,

Γi − Fxi Rl − Fzi e = 0, (2.5) where Fzi e denotes the rolling resistance moment, Myi , for one tire. During free- rolling of the tire, Γi = 0. This implies that Fxi satisﬁes the formula

Fzi e Fxi = − (2.6) Rl

When Γi = 0, the distance e is constant. Hence, in this case is

Fzi e Frri = , (2.7) Rl where Frri is the rolling resistance force for one tire from straight driving. When Γi > 0,

Γi − Fzi e Fxi = , (2.8) Rl and Fxi and Myi increase until the peak of available traction, µp is reached. Then it drops to the value µs. It is at this point the tire spin inﬁnitely fast. When Γi < 0,

Myi decreases and becomes negative. Hence, Fzi is for this case located to the left of the tire centerline. Fxi also decreases, until µp is reached. After that it increases until wheel lock occurs at κi = −1 [12, 14].

2.3.3 Lateral force

The lateral force, Fyi , is developed in the contact patch between tire and road in response to the lateral acceleration. The lateral acceleration is present during cornering in high speed. For small slip angles α, the lateral force relationship due is linear according to formula (2.9) below [9, 12].

Fyi = Cαα (2.9) Figure 2.3 shows that the lateral force always points in the direction opposite to the slip angle. The slip angle, α, is defined as the angle between the direction of heading, exi , and direction of travel, vi. In Figure 2.3, both the slip angle, α, and the lateral force, Fyi , are defined as positive [12]. The coefficient Cα in the equation (2.9) is called the cornering stiffness and is defined by (2.10) below. Hence the cornering stiffness must be positive [6].

∂Fyi Cα = (2.10) ∂α α=0 8 CHAPTER 2. BACKGROUND

exi vi α

Fyi

Figure 2.3: Slip angle and lateral force of one tire

The cornering stiffness depends on the tire’s properties; for example tire type, tire size, tread design and inflation pressure. It has influence on the handling characteristics of the vehicle [8]. Heavy-trucks usually have solid front- and rear axles. Therefore the contribution of camber angle to the lateral force is not of significance and is not considered [7]. The camber angle, γ, is defined as in Figure 2.4. The camber angle is positive if the top of the tire is aligned outboard compared to the bottom part [9].

ez ezi γ

Figure 2.4: Camber angle of one tire

When the slip angle is created the tire is transversally deformed. The deformation in the transverse direction usually takes the form of a triangle. The total lateral force for one tire is the sum of the diﬀerent single forces developed in the tire for each element deformed in the transverse direction. Thus, Fyi acts at the rear part of the contact patch at a distance known as the pneumatic trail, Pt, from the center of the tire contact area. This implies that the lateral force generates a moment, known as the aligning moment, Mzi , around the center of the tire. At big slip angles the tire elements at the rear part begin to slide on the contact area. Thus, the tire can no longer develop as big lateral forces in this area. Fyi therefore moves towards the center of tire contact area. Hence, the distance Pt becomes smaller and the aligning moment decreases [8, 13].

The traction in the lateral direction, µyi , for one tire is deﬁned in the same way 2.4. BICYCLE MODEL 9

Fyi

Figure 2.5: Transverse deformation of a tire

exi α

Mzi vi

Fyi

Figure 2.6: Induced aligning moment, Mzi , on a tire from the lateral force, Fyi , acting a distance Pt from the contact point. as for the longitudinal force.

Fyi µyi = (2.11) Fzi When both longitudinal- and lateral forces are present, formula (2.4) is revised to formula (2.12).

q 2 2 Fxi + Fyi ≤ µpFzi , (2.12) Inequality (2.12) can be described with the help of a circle, the so called Kamm circle, as is shown in Figure 2.7. According to formula (2.12) is the maximal lateral force for each tire less when Fxi 6= 0. When Fxi = 0 inequality (2.12) simpliﬁes to

Fyi ≤ µpFzi , (2.13) which is on the same form as formula (2.4) but for the lateral force [7, 12]. The peak- and slide coeﬃcient, µp and µs depend e.g. on the climate, road material and vertical load.

2.4 Bicycle model

A truck can be represented with the help of a single-track bicycle model. In this thesis, the single-track bicycle model is denoted bicycle model. An essential assumption in this model is that the height of the center of gravity is positioned at 10 CHAPTER 2. BACKGROUND

µpFzi

Fxi

Fyi

Figure 2.7: Kamm circle showing the traction limit for the tire forces Fxi and Fyi road level. This implies that the weight shift between the right- and left wheel at each axle is not considered, hence each axle is represented by one wheel [9].

2.4.1 Low-speed cornering Figure 2.8 below shows a vehicle during low-speed cornering. The geometry associated with low-speed cornering is known as "Ackerman Geometry" [8]. The steering angles for the front inner- and outer tire can be calculated according to (2.14) and (2.15) below, where L is the wheelbase, t the track width and R the turning radius.

L δ ≈ (2.14) i (R − t/2) L δ ≈ (2.15) o (R + t/2)

Hence, the average steering angle is L/R, which is also called the Ackerman angle [8, 11].

2.4.2 High-speed cornering This thesis is about cornering during high speed. In Figure 2.9 the different angles and the lateral forces for a left hand turn can be seen. As was already noted in Section 2.3.3, the tires develop lateral forces that counteract the centrifugal force, F, during cornering. For a left hand turn, the steering angle δ, the yaw rate r, the angular velocity v˙ and the body slip angle β are all positive in the counterclockwise direction. v is here the truck’s velocity, i.e. the velocity of the center of gravity. The slip angles of the front- and rear wheels, αf and αr, are positive in the clockwise direction. In the literature other definitions exist for the body slip angle [9, 12]. In Figure 2.9, x denotes the direction alongside the truck and y the direction perpendicular to x. The x- and y directions relate to the fix coordinate system 2.4. BICYCLE MODEL 11

t Figure 2.8: Bicycle model for low-speed cornering

x0 and y0 as in formulas 2.16 and 2.17, where Ψ(t) is the yaw angle. In the ﬁx coordinate system, x0 is directed upwards and y0 to the left, perpendicular to x0.

x = x0 cos(Ψ(t)) + y0 sin(Ψ(t)) (2.16)

y = −x0 sin(Ψ(t)) + y0 cos(Ψ(t)) (2.17)

αf

Fyf lf

β v F

r lr mv˙ R αr y

Fyr

Figure 2.9: Bicycle model for high-speed cornering, without longitudinal force 12 CHAPTER 2. BACKGROUND

A cornering maneuver can be divided into transient condition and steady-state condition. The transient condition occurs during initiation and termination of a turn. It also occurs when the radius is varied or when accelerating or braking in a turn. However, if the vehicle is accelerating/braking constantly and if the changes of speed over a small time-interval are small, the vehicle can be considered to be at quasi-steady-state condition. During the transient condition the moment equation (2.18) must be satisﬁed. Here, I is the yaw inertia and r˙ is the angular acceleration [17]. The left-hand side of equation (2.18) is just force balance about the center of gravity.

Fyf cos(δ)lf − Fyrlr = Ir˙ (2.18) During steady state it holds that v˙ = 0, β˙ = 0 and r˙ = 0. The yaw rate is at steady state r = v/R. In this case the moment equation about the center of gravity fulﬁlls equation (2.19) [8, 9, 12].

Fyf cos(δ)lf − Fyrlr = 0 (2.19) This thesis is focused mainly on steady-state cornering during small angles. Generally for an angle, φ, smaller than 4 deg the assumptions speciﬁed in (2.20) can be made.  cos(φ) ≈ 1  sin(φ) ≈ φ (2.20)  tan(φ) ≈ φ

For small slip angles, αf and αr can be derived from the formulas (2.21) respectively (2.22) [12].

l α = δ − β − f r (2.21) f v l α = −β + r r (2.22) r v Hence, using r = v/R, the body slip angle β can be deduced either from formula (2.23) or (2.24) stated below. l β = δ − α − f (2.23) f R l β = −α + r (2.24) r R The steering angle, δ, satisfy equation (2.25) which can be derived from (2.21) and (2.22). Hence, formula (2.25) also only holds for small steering angles. L δ = + α − α (2.25) R f r Chapter 3

Bicycle Model

3.1 Model

In order to setup a model for a truck during high-speed cornering the bicycle model is used. At ﬁrst the bicycle model equations are stated. Thereafter the rolling resistance forces together with the equations for the rolling resistance energy loss respectively power loss are deﬁned.

3.1.1 Model with aerodynamic drag force

During high-speed cornering at a speed greater than about 72 km/h, the aerodynamic drag force is greater than the rolling resistance forces. Therefore the aerodynamic drag force also has to be taken into consideration when stating the model equations. Figure 3.1 below shows the bicycle model. The ﬁgure looks the same as Figure 2.9 with the diﬀerence that the drive force on the rear axle, Fxr, and the aerodynamic drag force, FA, also are marked. Further, steady-state cornering is assumed, hence v˙ = 0. The aerodynamic drag force are calculated as

1 F = ρC Av2, (3.1) A 2 d

where Cd denotes the aerodynamic drag coeﬃcient, ρ the air density and A the cross sectional area where the aerodynamic drag force acts. An ideal situation is considered in which there is no wind speed, hence v in formula (3.1) is the vehicle speed. By considering Figure 3.1, the force equations are stated alongside and perpendicular to the truck, i.e. in the x- and y-directions. The moment equation about the center of gravity is also stated.

13 14 CHAPTER 3. BICYCLE MODEL x

αf

Fyf lf

β v F

r lr FA R αr y Fxr

Fyr

Figure 3.1: Bicycle model, with aerodynamic drag force, FA, and longitudinal force, Fxr.

π ← y : F cos(δ) + F − F cos(β) − F cos − β = 0 yf yr A 2 Fyf cos(δ) + Fyr = F cos(β) + FA sin(β) mv2 F cos(δ) + F = cos(β) + F sin(β) yf yr R A

mv2 1 C α cos(δ) + C α = cos(|β|) + ρC Av2 sin(|β|) (3.2) αf f αr r R 2 d

π ↑ x : −F cos − δ + F + F sin(β) − F cos(β) = 0 yf 2 xr A −Fyf sin(δ) + Fxr = −F sin(β) + FA cos(β)

mv2 1 −C α sin(δ) + F = − sin(|β|) + ρC Av2 cos(|β|) (3.3) αf f xr R 2 d

+ x Moment : Fyf cos(δ)lf − Fyrlr = 0

Cαf αf cos(δ)lf − Cαrαrlr = 0 (3.4) 3.1. MODEL 15

3.1.2 Rolling resistance

Figure 3.2 illustrates the bicycle model with the rolling resistance forces from straight driving, Frr,f respectively Frr,r, and the extra rolling resistance forces that appear when driving in turns, ∆Frr,f and ∆Frr,r.

αf

Frr,f ∆Frr,f β

αr y R

∆Frr,r Frr,r Figure 3.2: Bicycle model, with the rolling resistance forces

The rolling resistance forces Frr,f and Frr,r are trivially stated as in formulas (3.5) and (3.6) below. Here, Crr is the rolling resistance coeﬃcient from straight driving and mf respectively mr are the masses of the front- and rear axle. The rolling resistance forces ∆Frr,f and ∆Frr,r, are obtained from formulas (3.7) and (3.8) below, after obtaining the slip angles from the model equations. The formulas hold for small slip angles.

Frr,f = Crrmf g (3.5)

Frr,r = Crrmrg (3.6) 2 ∆Frr,f = Fyf sin(αf ) ≈ Cαf αf (3.7) 2 ∆Frr,r = Fyr sin(αr) ≈ Cαrαr (3.8)

By using the rolling resistance forces, the total rolling resistance energy loss and total rolling resistance power loss from straight driving (Err, Prr) and at cornering (Err,tu, Prr,tu) are obtained. The energy loss and power loss are expressed in kWh and kW. 16 CHAPTER 3. BICYCLE MODEL

Z T d 1 Err = Ff · vdt ≈ (Frr,f + Frr,r) (3.9) 0 1000 3600 v P = (F + F ) (3.10) rr rr,f rr,r 1000

Z T d 1 Err,tu = Ff · vdt ≈ (Frr,f + Frr,r + ∆Frr,f + ∆Frr,r) (3.11) 0 1000 3600 v P = (F + F + ∆F + ∆F ) (3.12) rr,tu rr,f rr,r rr,f rr,r 1000 The extra energy loss and power loss from cornering can thus be stated as below. d 1 E = (∆F + ∆F ) (3.13) rr,tu+ rr,f rr,r 1000 3600 v P = (∆F + ∆F ) (3.14) rr,tu+ rr,f rr,r 1000 Chapter 4

Validation of bicycle model

4.1 Purpose and method

An Adams tire model simulation is used, to test the performance of the bicycle model. The bicycle model and Adams tire model are compared against each other. The results are obtained using a distribution truck speciﬁed below in Section 4.2. In Section 4.5, key results are presented. Further comparisons between the bicycle model and Adams tire model are found in the appendix.

4.2 Truck speciﬁcation

To evaluate the model a distribution truck is used. Two cases are considered, with no load applied and the truck loaded to its maximum technical weight at the front- and rear axle. Table 4.1 shows the data for the unladen truck.

Description Value L 5.3 m lf 1.893 m m 8400 kg mf 5400 kg mr 3000 kg

Table 4.1: Unladen truck data

Figure 4.1 illustrates the truck with maximum technical weight at the front- and rear axle. Some characteristics of this laden truck can be seen in Table 4.2. The truck has a ﬁve cylinder, nine liters motor of 280 horsepower.

17 18 CHAPTER 4. VALIDATION OF BICYCLE MODEL

Figure 4.1: Truck with maximum technical weight at front- and rear axle

Parameter Value L 5.3 m lf 3.208 m m 19000 kg mf 7500 kg mr 11500 kg

Table 4.2: Laden truck data

4.3 Adams tire model simulation

The bicycle model was compared with simulation data obtained from Adams using the PAC 2002 Magic-formula tire model. The magic formula in PAC2002 describes the behaviour under steady-state conditions. Generally, under pure slip conditions the lateral- and longitudinal force are calculated by,

F0(x) = D sin[C arctan{Bx − E(Bx − arctan(Bx))}] + Sv (4.1)

x = X + Sh (4.2) according to the magic formula, where F0(x) denote the lateral- or longitudinal force and X the lateral slip angle or the longitudinal slip. Sv and Sh are the vertical and horizontal shift. D is the peak factor, C the shape factor, B the stiﬀness factor and E the curvature factor. To achieve the formula for the lateral force under combined slip conditions, F0(x) under pure slip must be multiplied with a weightning factor which depends on the slip quantities and vertical load. The weightning factor for the lateral force under combined slip also depends on the inclination angle. In addition a vertical shift must be added to the lateral force. 4.4. COMPARISON BICYCLE MODEL TO ADAMS TIRE SIMULATION 19

According to the equation used for the rolling resistance moment, My, in the magic formula in PAC2002 it is a sum of four terms. All terms depends linearly on the vertical load. The last three terms are the rolling resistance moment depending on longitudinal forces, speed and speed4. Since the coeﬃcients for these factors are deﬁned as zero in the simulation, My is calculated as,

My = R0 · Fz · qSy1 , (4.3)

with qSy1 = 0.008. A rolling resistance coefficient, Crr, between 0.006-0.01 is com- mon for truck tires on concrete and asphalt [18]. Exactly how the parameters, forces and moments are determined can be found in the instruction manual on how to use the PAC2002 tire model [2]. The magic tire formula is a nonlinear model which takes into account weight shift. For small radii and bigger slip angles, there is more weight shift. This is shown below in Section 4.4.3. The simulations are carried out on an ideally flat and smooth road. In the road file, the friction coefficient is specified to µ = 1.0. Generally, a turning radius of 400 m, is above the minimum radius of 370 m for a country road with 5.5 % banking of good standard. A turning radius of 200 m corresponds to a country road with 5.5 % banking of bad standard. For highways, a turning radius of 800 m for a road with 5.5 % banking is of good standard. Generally, less banking corresponds to a road with bigger radius [16].

4.4 Comparison bicycle model to Adams tire simulation

In this section, the lateral forces and the power loss from rolling resistance obtained from the bicycle model and Adams tire model simulation are compared. Also, some analysis of the weight shift is considered. The Adams tire model simulation is not compensated for the aerodynamic force. Therefore, for all comparisons the aerodynamic force is neglected in the bicycle model.

All comparisons are done for the laden truck using Cαf1 = 4360 N/deg (the cornering stiffness for each front tire) respectively Cαr1 = 4940 N/deg (the cornering stiffness for each rear tire). For the unladen truck, Cαf1 = 3320 N/deg respectively Cαr1 = 1290 N/deg are used. These cornering stiffness values are obtained from the best match of the slip angles from bicycle model and Adams tire model simulation. Slip angle comparisons, cornering stiffness analysis and analysis of the longitudinal forces from the Adams tire model simulation are found in the appendix. As illustrated in the appendix, the cornering stiffness exhibit a complex behaviour. More figures complementing the results presented in this section are found in the appendix. 20 CHAPTER 4. VALIDATION OF BICYCLE MODEL

4.4.1 Lateral force

Figure 4.2 shows the total lateral forces from all tires, Fytot , obtained using the laden truck for v = 50 km/h, v = 70 km/h and v = 90 km/h. As can be seen in the ﬁgure, the total lateral forces from the bicycle model and Adams tire model simulation match well. The total lateral forces for each velocity using the unladen truck are considerably less (about halfed in magnitude, appendix Figure A.16). For large radius |Fytot | → 0.

Figure 4.2: Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h and v = 90 km/h

using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck).

Figure 4.3 illustrates the total lateral forces for the front axle, Fyf , and the total lateral forces for the rear axle, Fyr, versus turning radius for v = 70 km/h. Fyr is here greater than Fyf . This is due to the fact that the mass is bigger at the rear axle. For the unladen truck, Fyf is greater than Fyr for every turning radius (appendix, Figure A.17). The proportion between Fyf and Fyr only depends on lf , lr and δ according to formula (3.4). Section A.1.2 in the appendix illustrates that changing Cαr1 in the bicycle model equations only has a small impact on Fyr. 4.4. COMPARISON BICYCLE MODEL TO ADAMS TIRE SIMULATION 21

Figure 4.3: Total front lateral force, Fyf , and total rear lateral force, Fyr, versus turning radius from bicycle model and Adams tire model simulation for v = 70 km/h

using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck).

4.4.2 Rolling resistance

Figure 4.4 shows the power loss from straight driving, Prr, from the bicycle model compared to the power loss from straight driving obtained from the Adams tire model simulation using Crr = 0.008, for the laden truck. From the Adams tire model simulation, the rolling resistance loss from straight driving is calculated from the rolling resistance moment, My. As can be seen in the ﬁgure is the power loss from straight driving, Prr, constant driving at all radius. When cornering, the extra power loss, Prr,tu+, is added to Prr, as already noted in Section 3.1.2. The power loss increase with the velocity as expected. The power loss from the unladen truck is much lower for each velocity compared to the laden truck. This can be seen in Figure 4.5. Figure 4.6 shows the extra power loss, Prr,tu+, compared to the power loss obtained from the longitudinal forces from the Adams tire model simulation, using the laden truck. In Section A.1.3 in the appendix, the longitudinal forces obtained from the Adams tire model simulation are analysed. As can be seen, correspond the power loss for v = 50 km/h (the red and blue line) well to each other. When the velocity increases, the correspondence for small radii become worse. For large radii the match is good for all velocities. For the unladen truck, the power loss is considerable less than in the laden case, as is shown in Figure 4.7. The power loss from the bicycle model and Adams tire model simulation correspond well for all three velocities. 22 CHAPTER 4. VALIDATION OF BICYCLE MODEL

Figure 4.4: Power loss due to rolling resistance from straight driving, Prr, versus turning radius from bicycle model and Adams tire model simulation for the laden truck. For small radii, the power loss Prr adds to the extra rolling resistance power loss from cornering, Prr,tu+.

Figure 4.5: Power loss due to rolling resistance from straight driving, Prr, versus turning radius from bicycle model and Adams tire model simulation for the unladen truck. For small radii, the power loss Prr adds to the extra rolling resistance power loss from cornering, Prr,tu+. 4.4. COMPARISON BICYCLE MODEL TO ADAMS TIRE SIMULATION 23

Figure 4.6: Extra rolling resistance power loss from cornering, Prr,tu+, versus turning

radius from bicycle model and Adams tire model simulation using Cαf1 = 4360 N/deg

and Cαr1 = 4940 N/deg (laden truck).

Figure 4.7: Extra rolling resistance power loss from cornering, Prr,tu+, versus turning

radius from bicycle model and Adams tire model simulation using Cαf1 = 3320 N/deg

and Cαr1 = 1290 N/deg (unladen truck). 24 CHAPTER 4. VALIDATION OF BICYCLE MODEL

4.4.3 Weight shift

As noted earlier, the magic tire formula is a nonlinear model, which takes into account weight shift. The bicycle model, however, does not consider weight shift. It is interesting to know how much weight shift that is obtained from the Adams tire model simulation. Weight shift is created during turning, because when turning a tire, a slip angle is created and a lateral force is generated. The lateral force generates lateral acceleration, which creates weight shift. Figure 4.8 shows the weight shift for each tire for the laden truck. As can be seen in the figure is the weight evenly distributed on the left- and right tires, when driving straight (big radii). In turns however, weight transfers from the left to the right wheels. In sharper turns (small radii), the weight shift is greater. Because of the weight shift from left- to right side, the lateral forces generated at the right tires are significantly higher than at the left tires. All simulations was made up to a maximum lateral acceleration of 3.7 m/s2. At the end of the simulation only about 0.13 ton less weight at the front right tire was achieved for v = 50 km/h compared to v = 70 km/h. For the front left tire it was about 0.17 ton less weight shift for v = 50 km/h (see appendix, Figure A.18). For v = 90 km/h almost no difference in weight shift was shown compared to v = 70 km/h (appendix, Figure A.19).

Figure 4.8: Weight shift for each tire versus turning radius from Adams tire model simulation for v = 70 km/h (laden truck). 4.5. SUMMARY 25

4.5 Summary

The bicycle model and Adams tire model simulation correspond well to each other for small slip angles. This has been veriﬁed by considering the lateral forces and power loss due to rolling resistance. The front lateral force, Fyf , from bicycle model and Adams tire model simulation match well. The same holds for the rear lateral force, Fyr. The power loss from straight driving, Prr, is constant and the same from bicycle model and Adams tire model simulation. The extra power loss from cornering, Prr,tu+, from bicycle model and Adams tire model simulation correspond well for large radii (for v = 70 km/h approximately R > 380). For smaller radii, the correspondence is not as good. This is due to that the Adams tire model takes weight shift into account, that is not considered in the bicycle model. Furthermore, the bicycle model has been restricted to small slip angles.

Chapter 5

Optimization problem

5.1 Problem formulation

Consider a truck driving along the track in Figure 5.1. The track consists of three straight sections and two turns. Let nst denote the number of straight sections and ntu the number of turns. The length, s, of each straight section are 110 m. Both turns are of circular shape with turn angle, ϕ = 45 deg, and the same radius, R = 400 m. The truck has to drive from start point A to end point B with ﬁxed average speed with as low energy loss as possible. Also, the maximum acceleration/deacceleration is limited. Provided these conditions, optimize velocity in each subsection of the track. Table 5.1 shows a summary of the problem speciﬁc parameters.

A s R R s B s Figure 5.1: The optimization problem track

5.2 Method

Let the track be discretized in totally n points, with step length h. Treat each straight section and turned section separately, such that if any section is not equally divided by h, the last step in that section is shorter than h. This is illustrated in Figure 5.2, which shows a track consisting of one straight section and one turn.

27 28 CHAPTER 5. OPTIMIZATION PROBLEM

Parameter Explanation Starting value ntu Number of turns 2 nst Number of straight Sections 3 R Turn radius of each turn 400 m ϕ Length of each turn in degrees 45 deg s Length of each straight Section 180 m

Table 5.1: Brief explanation of the problem speciﬁc parameters

h h

Figure 5.2: Track consisting of one straight section and one turn with step length h.

An important adoption is that the truck drives with constant velocity, vk, in each subsection k (k = 1, . . . , n). This implies, that the bicycle model steady-state equations stated in Section 3.1.1 can be used to determine the slip angles αf and αr from the given turn radii R1 respectively R2. The slip angles are used to determine ∆Frr,f and ∆Frr,r as explained in Section 3.1.2. The energy loss from each subsection k of the track is summed up and minimized while satisfying specified constraints. The minimization is performed with rolling resistance and airdrag losses. Between each subsection, there is a limit, vdiff , on how much the velocity can change. If the velocity changes too much, there also are other energy losses due to acceleration, braking i.e., that are not considered here. The parameter vdiff is specific for the truck and depends on the step length, h, and the fixed average velocity, vm. vm determines the time limit T for the travel along the track. Furthermore, the velocity in each subsection k has to be greater than 0 km/h and can not exceed vmax = 90 km/h, the speed limit for trucks. The start velocity, v1, is assumed to be greater or equal v1,start = 50 km/h, a velocity where initial energy losses can be neglected. Function f in optimization problem (OPT) denotes the power loss at each subsection of the track and is defined in formula (5.1). Table 5.2 shows a brief explanation of the setup parameters. Here, vini denotes the input velocity to the solver. Note that the velocities are stated in km/h, however in the optimization problem they need to be in m/s.

k 2 2 1 2 P = f(v ,R , α , αr ) = Crrmg + Cα α + Cα α + C ρAv (5.1) tot k k fk k f fk r rk 2 d k 5.2. METHOD 29

Parameter Explanation Starting value vm Fixed average velocity, that deter- 70 km/h mines the time limit T vdiff Deﬁnes the limit of velocity change 0.28 km/h between each subsection vmax Maximum velocity in each subsec- 90 km/h tion v1,start Minimum velocity in ﬁrst subsection 50 km/h vini Input velocity to solver 1 km/h h Length of each subsection (the step 5 m length) n Number of discretization points Not manually along the track set

Table 5.2: Brief explanation of the setup parameters for the optimization problem.

 n  X sk n n n n min f(vk,Rk, αf , αr ) , v ∈ , R ∈ , αf ∈ , αr ∈ , α ,α k k R R R R   f r vk n n   k=1 s ∈ R , t ∈ R     s.t |v − v | ≤ v , ∀ i ∈ 1, ..., n − 1  (OPT )  i i+1 diff   0 ≤ v ≤ v , ∀ i ∈ 1, . . . , n   i max     v1 ≥ v1,start  tn ≤ T (5.2) Note that in optimization problem (OPT), Rk = ∞ for each discretization point corresponding to straight road. For the turns, the radius Rk is given and the same at every discretization point along each turn. For the straight sections, there are no slip angles. 30 CHAPTER 5. OPTIMIZATION PROBLEM

5.3 Solution and simulation

In the evaluation of the optimization problem (OPT) a set of fix parameters are used, that can be seen in Table 5.3. They are specific for the truck and are defined in Section 4.2, 4.3 and A.1.2. The simulations are performed using the laden truck and the best fit for the cornering stiffness values.

Parameter Value m 19000 kg g 9.82 m/s2 L 5.3 m lf 3.208 m

Cαf1 4360 N/deg

Cαr1 4940 N/deg Crr 0.008 ρ 1.293 kg/m3 Cd 0.6 A 2.55 · 3.691 m2

Table 5.3: Brief explanation of the ﬁx parameters specifying the truck (laden) during the simulations.

The optimization problem was solved in matlab with the function fmincon, using the sqp algorithm. Figure 5.3 below shows the optimal velocity (blue line) in each 5 m section for the problem- and setup parameters values specified in Table 5.1 and 5.2. The red line shows the fixed average velocity, vm = 70 km/h. As can be seen in the figure, the first section is straight. Here, it is best to drive with constant velocity, about 73.1 km/h. Thereafter, the truck enters a turn. Because the resistance force is higher in a turn than at a straight section, the velocity decreases. In the turn the velocity stabilize at about 67.6 km/h. When the truck leaves the turn to the second straight section, the velocity increases to about 73.1 km/h. In order to satisfy the time constraint the truck must drive faster than vm at all straight sections. Figure 5.4 shows the power loss in each 5 m section along the track for the fixed average velocity, vm = 70 km/h (red line) and for the optimal velocity (blue line). As can be seen in the figure, the power loss is higher at the turns than at the straight sections for vm = 70 km/h. This is due to the extra rolling resistance power loss from cornering, Prr,tu+, of 4 kW. At the straight sections the power loss is about 57.5 kW. Figure 4.4 in Section 4.4.2 shows that the rolling resistance power loss from straight driving, Prr, is about 29 kW for velocity 70 km/h. Hence, the power loss from airdrag resistance and rolling resistance from straight driving is almost the same. For the optimal velocity, the power loss is less in the turns than at the straight sections. This is because the optimal velocity decreases in the turns. In the transition between each straight section and turn the power loss fluctuate due 5.3. SOLUTION AND SIMULATION 31

straight turn straight turn straight

Figure 5.3: Optimal velocity and ﬁxed average velocity, vm, in each 5 m section along the track. The optimal velocity is lower than vm in the turns. However, at the straight sections, the optimal velocity is higher than vm in order to fulﬁll the time constraint.

to Prr,tu+. The total energy loss for the whole track is about 0.2% less for the optimal velocity than for vm = 70 km/h. Hence, driving with optimal velocity has the potential to save 0.2% fuel. Some of the problem- and setup parameters are changed in order to see the effect on how much fuel that can be saved. Before each simulation in the coming sections all the parameters that have been changed are specified. The rest of the parameters are the same as in Table 5.1 and 5.2. For each simulation (including the simulation above), the length of the straight sections have been set to the distance, s, that maximizes the per cent of fuel that can be saved when driving with optimal velocity compared to fixed average velocity. This analysis are performed with all the other problem parameters and all setup parameters held fix. 32 CHAPTER 5. OPTIMIZATION PROBLEM

straight turn straight turn straight

Figure 5.4: Power loss in each 5 m section along the track. The power loss for the optimal velocity, is lower in the turns compared to the power loss for the ﬁxed average velocity, vm. However, at the straight sections the power loss is higher for the optimal velocity.

5.3.1 Radius Here, a simulation for a track specified by R = 200 m, ϕ = 45 deg and s = 80 m is shown. Furthermore, vm = 50 km/h and vdiff = 0.775 km/h are set. As noted in Section 4.4.2, the bicycle model match well with the Adams tire simulation data for vm = 50 km/h. Figure 5.5 shows the optimal velocity and the fixed average velocity, vm = 50 km/h, in each 5 m section along the track. As can be seen, just like in Figure 5.3, the optimal velocity is higher in each straight section and lower in the turns. In the straight sections the optimal velocity stabilize at about 54.3 km/h and in the turns at 47.3 km/h. Figure 5.6 illustrates the power loss in each 5 m section along the track for the fixed average velocity, vm (red line), and for the optimal velocity (blue line). As can be seen in the figure, the power loss for the optimal velocity is about 5 kW higher at the straight sections compared to the power loss for the fixed average velocity, vm. However, in the turns the power loss for vm is about 3.5 kW higher than for the optimal velocity. For vm = 50 km/h, the rolling resistance power loss from straight driving is about 20.7 kW (see Figure 4.4 in Section 4.4.2), the airdrag power loss 10.4 kW and the extra rolling resistance power loss from cornering 3 kW. The total energy loss for the whole track is about 0.6% less for the optimal velocity than for vm = 50 km/h. Hence, driving with optimal velocity has the potential to save 0.6% fuel. 5.3. SOLUTION AND SIMULATION 33

Figure 5.5: Optimal velocity and ﬁxed average velocity, vm, in each 5 m section along the track.

Figure 5.6: Power loss for the optimal velocity (blue line) and ﬁxed average velocity vm (red line) in each 5 m section along the track. 34 CHAPTER 5. OPTIMIZATION PROBLEM

5.3.2 Turnlength Here, R = 200 m, ϕ = 180 deg and s = 420 m. Just like in the previous simulation, vm = 50 km/h and vdiff = 0.775 km/h. Figure 5.7 shows the ﬁxed average velocity, vm = 50 km/h, and the optimal velocity in each 5 m section along the track. The optimal velocity is approximately 53.6 km/h in each straight section and 46.8 km/h in each turn.

Figure 5.7: Optimal velocity and ﬁxed average velocity, vm, in each 5 m section along the track.

Figure 5.8 illustrates the power loss in each 5 m section along the track for the ﬁxed average velocity vm = 50 km/h and the optimal velocity. As can be seen in the ﬁgure, the power loss at the straight sections is 4 kW higher for the optimal velocity than for vm and in the turns the power loss for vm is about 4 kW higher than for the optimal velocity. The total energy loss for the whole track is about 0.6% less for the optimal velocity than for vm = 50 km/h. Hence, driving with optimal velocity has the potential to save 0.6% fuel. 5.3. SOLUTION AND SIMULATION 35

Figure 5.8: Power loss for the optimal velocity (blue line) and ﬁxed average velocity vm (red line) in each 5 m section along the track.

5.3.3 Average velocity

Here, vm = 80 km/h and vdiff = 0.2125 km/h. The track is speciﬁed by, R = 200 m, ϕ = 180 deg and s = 600 m. Figure 5.9 shows the optimal velocity and vm = 80 km/h in each 5 m section along the track. As previously explained, the maximum time, T, for the travel is determined from the assumption that the truck drives with average velocity vm. Hence, increasing vm, implies that the optimal velocity increases, and decreasing vm that it decreases. Figure 5.9 illustrates that the optimal velocity is 70.4 km/h in both turns and 90 km/h at the straight sections. However, at the second straight section the velocity is not stable for such a long distance. Figure 5.10 illustrates the power loss for the optimal velocity and ﬁxed average velocity, vm = 80 km/h in each 5 m section along the track. For vm, the power loss from straight driving is about 75.7 kW and consists of about 33.2 kW rolling resistance loss from straight driving and 42.5 kW airdrag power loss. The extra rolling resistance power loss from cornering is approximately 31.1 kW. For the optimal velocity, the power loss stabilize at about 97.9 kW at the straight sections and for the two turns at 74.6 kW. In the second straight section, the power loss is not stable for such a long distance (compare with the optimal velocity at the second straight section). The total energy loss for the whole track is about 3% less for the optimal velocity than for vm = 80 km/h. Hence, driving with optimal velocity has the potential to save 3% fuel. 36 CHAPTER 5. OPTIMIZATION PROBLEM

Figure 5.9: Optimal velocity and ﬁxed average velocity, vm = 80 km/h, in each 5 m section along the track.

Figure 5.10: Power loss along the track, for the optimal velocity and vm = 80 km/h. Chapter 6

Conclusions and discussion

By using the bicycle model a relation between the front- and rear lateral forces, Fyf respectively Fyr and the turning radius is obtained. From the tire model, the lateral forces are related to the slip angles of the tires. Furthermore, linear approximation provides the relation between the slip angles and extra rolling resistance forces from cornering, ∆Frr,f and ∆Frr,r. The performance of the bicycle model have been tested using the Adams tire model. The validation shows that the bicycle model corresponds well to the Adams tire model simulation for big turning radius where the slip angles are small. For small turning radii the correspondence is not as good as for big turning radii, due to weight shift that has not been considered in the bicycle model. The bicycle model has been applied to an optimization problem, where the optimal velocity along a track is determined. The results show that velocity optimization on curved roads has the potential to save 0 − 2% fuel. The radius and the ﬁxed average velocity are the most important factors for how much fuel that can be saved. For small radii, it is possible to save more fuel when driving with optimal velocity. As noted earlier, it is possible to save 0.2% fuel for turning radii R = 400 m and vm = 70 km/h. For turning radii R = 200 m and vm = 50 km/h it is possible to save 0.6% fuel. The simulation for vm = 80 km/h on a track with R = 200 m shows that it is possible to save more fuel for increased ﬁxed average velocity, vm. However, as shown in Section 4.4.2 the bicycle model has best correspondence for vm = 50 km/h for R = 200 m. For vm = 80 km/h the bicycle model does not match as good with the Adams tire model. This implies, that the more vm is increased, the more impact the model error has. It is always optimal to drive with lower speed in the turns than at the straight sections. This holds because the resistance force, is higher in the turns, due to the extra rolling resistance forces from turning, ∆Frr,f and ∆Frr,r. To satisfy the time constraint for the track, the speed must be higher on the straight sections where the resistance force only consists of rolling resistance from straight driving, Frr,f respectively Frr,r and the airdrag force, FA. However, increasing the speed on the

37 38 CHAPTER 6. CONCLUSIONS AND DISCUSSION straight sections increase the aerodynamic losses. In order for the optimal velocity to vary between the straight sections and turns, it is best if the straight sections are not too small and not too big compared with the length of the turns. In the simulation for R = 200 m, ϕ = 45 deg, the length of each straight section, s, is 51% of the length of each turn. This length maximizes the per cent of fuel that can be saved in this case. The only diﬀerence between the simulations for R = 200 m, ϕ = 45 deg and R = 200 m, ϕ = 180 deg is that the length of the straight sections increase when the length of the turns increase. For both cases, it is possible to save about the same per cent fuel. If the track, is only made of straight sections or only turns, the velocity is not able to adapt because the energy loss is the same along the whole track. Note also that the maximum acceleration/deacceleration limit for the truck along the track is relatively small. In the simulations for vm = 50 km/h the limit is 2 2 2 0.60 m/s , for vm = 70 km/h 0.30 m/s and for vm = 80 km/h, 0.26 m/s .

6.1 Future work

It would be good to perform tests in reality, to measure the impact on the fuel consumption from driving with optimal velocity compared to ﬁxed constant velocity for a particular track. In this way, it would be possible to see if the results are applicable. The bicycle model in this thesis does not consider weight shift. Furthermore, the bicycle model equations do not hold for big slip angles (>4 deg). In the future it would be good to extend the single-track bicycle model considered in this thesis to a two-track bicycle model that separates the left- and right tires on each axle. In this way, the roll dynamics between left- and right side of the truck can be precisely described. The bicycle model equations stated in Section 3.1.1 only hold for steady state. Hence, the bicycle model is not valid at initiation or termination of a turn where transient state occurs (i.e. the moment equation is nonzero see Section 2.4.2). It would be good to extend the bicycle model to hold for the transient state as well. Moreover, banking is also not considered in the bicycle model equations. In turns, the roads are usually banked in order to reduce the lateral forces. A side note is that the cornering behaviour of the truck can be tested for another set of tires. The tires used are radial with tire size 315/70 steer at the front and 315/70 drive at the rear (see Table A.2 in the appendix). Bibliography

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[3] What consumes fuel? http://www.bridgestonetrucktires.com/us_eng/ real/magazines/ra_special-edit_4/ra-special4_pdf_downloads/ra_ special4_fuel-speed.pdf. [Online; accessed 06-Oct-2015].

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[5] Crolla, D.A. Automotive Engineering: Powertrain, Chassis System and Vehicle Body, chapter 11, 12, pages 326, 370–371. Butterworth-Heinemann, 1 edition, 2009.

[6] Duysinx, P. Vehicle dynamics. http://www.ingveh.ulg.ac.be/uploads/ education/meca-0063/notes/AV_VEHDYN_1_2014.pdf. [Online; accessed: 2- Aug-2015].

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[8] Gillespie, T.D. Fundamentals of Vehicle Dynamics. chapter 6, 10, pages 195–203, 347–351, 369–371. 1992. http://www.rmcet.com/lib/Resources/ E-Books/Mech-auto/Fundamentals%20of%20Vehicle%20Dynamics.pdf. [On- line; accessed: 4-Aug-2015].

[9] Heißing, B. and Ersoy, M. Chassis Handbook; Fundamentals, Driving Dynam- ics, Components, Mechatronics, Perspectives, chapter 1, 2, pages 18–19, 22, 89–91. Springer Fachmedien Wiesbaden GmbH, 2011.

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[10] Isermann, R. Fahrdynamik-Regelung: Modellbildung, Fahrerassistenzsysteme, Mechatronik, chapter 2, pages 35–36. Friedr. Vieweg & Sohn Verlag, GWV Fachverlage GmbH Wiesbaden, 2006.

[11] Maxwell, T.T. Steady State Cornering. http://atu587.org/sites/default/ files/Cornering%20Forces%20and%20Geometry.pdf, 2006. [Online; accessed: 13-Aug-2015].

[12] Mitschke, M. and Wallentowitz, H. Dynamik der Kraftfahrzeuge, chapter 2,20, pages 9–47, 613–623. Springer Fachmedien Wiesbaden GmbH, 2014.

[13] Pfeﬀer, P. and Harrer, M. Lenkungshandbuch: Lenksysteme, Lenkgefühl, Fahr- dynamik von Kraftfahrzeugen, pages 24–31. Springer Vieweg, 2013.

[14] Schuring, D.J. The Rolling Loss of Pneumatic Tires, volume 53. Rubber Chemistry and Technology, 1980. No.3.

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[16] Vägverket. Vägar och gators utformning: Linjeföring. VV Publikation, page 60. http://www.trafikverket.se/TrvSeFiler/Foretag/Bygga_och_ underhalla/Vag/Vagutformning/Dokument_vag_och_gatuutformning/ Vagar_och_gators_utformning/Linjeforing/06_horisontalkurvor.pdf. [Online; accessed: 17-Sep-2015].

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[18] Wong, J.Y. Theory of Ground Vehicles, chapter 1, page 18. John Wiley & Sons, Inc., 3 edition, 2001. Appendix A

Graphs from the simulations

A.1 Validation of bicycle model using Adams tire simulation

The model equations defined in Section 3.1.1 contain a lot of parameters. Formula (A.1) and (A.2) show the simple relation between the cornering stiffness of the front- and rear axle to the cornering stiffness of one tire at the respective axles, Cαf1 and

Cαr1 . In Table A.1 the given start values of Cαf1 and Cαr1 are deﬁned. Table A.2 shows tire size and unloaded radius for the tires used.

180 C = 2 · · C (A.1) αf π αf1 180 C = 4 · · C (A.2) αr π αr1

Unladen Laden

Cαf1 (N/deg) 3450 4710

Cαr1 (N/deg) 1320 4940

Table A.1

Front Rear Tire size 315/70 steer 315/70 drive R0 (m) 0.536 0.515

Table A.2

The total slip angle of the front axle, αf,tot, the total slip angle at the rear axle,

αr,tot, and the total lateral force, Fytot , are calculated according to the formulas A.3,

41 42 APPENDIX A. GRAPHS FROM THE SIMULATIONS

A.4 and A.5 below. 180 α = 2 · · α (A.3) f,tot π f 180 α = 2 · · α (A.4) r,tot π r

Fytot = Fyf + Fyr (A.5)

In this section the bicycle model equations speciﬁed in Section 3.1.1 are solved by assuming a front slip angle, αf , between 0.1 deg to 4 deg. Further, by assuming cos(δ) ≈ 1 and cos(β) ≈ 1, the slip angle, αr, and the radius, R, are calculated. Using these values for R, αf and αr respectively the bicycle model equations are iteratively updated by calculating the steering angle, δ, and the body slip angle, β, and again inserting in the model equations. However, the model equations stabilize fast. For example for v = 70 km/h for the slip angle αf = 2.03 deg, both β and δ stabilize in the ﬁrst iteration (to β ≈ −0.625 deg and δ ≈ 2.55 deg). In this case no update for αr is needed and R needs to be updated twice in order to stabilize.

A.1.1 Slip angle The slip angle comparisons between the bicycle model and Adams tire model simulation are initially done, using the given values in Table A.1 of Cαf1 and Cαr1 for the laden- and unladen truck.

Case I: Laden

Figure A.1 shows αf,tot and αr,tot versus turning radius for v = 70 km/h. As can be seen, there is good correspondence for radii greater than 300 meters between the bicycle model and Adams tire model simulation for both αf,tot and αr,tot. However, the correspondence is a bit better for αr,tot. For radii less than 300 meters (for bigger slip angles), the match is not as good between the bicycle model and Adams tire model simulation. Also here it less deviation for αf,tot. For v = 50 km/h, the match is better between the bicycle model and Adams tire model simulation (Figure A.2) and for v = 90 km/h the match is worse (Figure A.3). For large turning radius, the slip angles approach zero. To obtain a better correspondence of the total front slip angle versus radius between the bicycle model and Adams tire model simulation, Cαf1 is changed (for v = 70 km/h). It turned out the best match between the green line and the red line is for Cαf1 = 4360 N/deg (Figure A.4). The least radius obtained by the bicycle model for v = 70 km/h is R = 81.6 m. Figure A.5 shows the slip angle for each tire versus turning radius for v = 70 km/h obtained from the Adams tire model simulation. As can be seen the slip angles are the same for the left- and right tires. This is due to the connection through the steering system. For small radii, the slip angles are big. The slip angles A.1. VALIDATION OF BICYCLE MODEL USING ADAMS TIRE SIMULATION 43

Figure A.1: Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 70 km/h using

Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck).

Figure A.2: Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 50 km/h using

Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck). 44 APPENDIX A. GRAPHS FROM THE SIMULATIONS

Figure A.3: Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 90 km/h using

Cαf1 =4710 N/deg and Cαr1 = 4940 N/deg (laden truck).

Figure A.4: Best correspondence between bicycle model and Adams tire model sim-

ulation for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck). A.1. VALIDATION OF BICYCLE MODEL USING ADAMS TIRE SIMULATION 45

Figure A.5: Slip angle versus turning radius for each tire from Adams tire model simulation for v = 70 km/h (laden truck).

Figure A.6: Front- and rear slip angle (αf and αr) from bicycle model versus turning

radius for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck).

from the Adams tire model simulation are about double the size of the slip angles obtained from the bicycle model (Figure A.6). 46 APPENDIX A. GRAPHS FROM THE SIMULATIONS

Case II: Unladen

Figure A.7 shows αf,tot and αr,tot versus turning radius for v = 70 km/h using

Cαf1 = 3450 N/deg and Cαr1 = 1320 N/deg. As can be seen in the ﬁgure, the bicycle model corresponds pretty good to the Adams tire model simulation, both for αf,tot and αr,tot. However, a better correspondence is achieved using Cαf1 = 3320

N/deg and Cαr1 = 1290 N/deg (see Figure A.8).

Figure A.7: Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 70 km/h using

Cαf1 = 3450 N/deg and Cαr1 = 1320 N/deg (unladen truck).

A.1.2 Cornering stiffness analysis The cornering stiffness vary with load. To investigate the variation of cornering stiffness with load, one tire is investigated at a time for different vertical load. By dividing the lateral force, Fyfi , from the Adams tire model simulation with the slip angle αf for each mass, the variation of the "cornering stiffness" is obtained. As the cornering stiffness is defined at zero slip, the effective cornering stiffness is obtained [15]. The same holds for the rear axle. In order for the lateral force from the bicycle model to match with the lateral force from the Adams tire model simulation, the effective cornering stiffness must be used instead of the cornering stiffness and also the load variation has to be considered.

Figure A.9 illustrates the effective cornering stiffness of the front tire, Ceff,fi , for different mass. The colored lines match with the mass shown in the same color in Figure A.10. The red and blue horizontal lines denote the cornering stiffnesses

Cαf1 = 4710 N/deg and Cαf1 = 3450 N/deg set from the start for the laden- and unladen truck. The black and magenta horizontal lines denote the best match, A.1. VALIDATION OF BICYCLE MODEL USING ADAMS TIRE SIMULATION 47

Figure A.8: Best correspondence between bicycle model and Adams tire model sim-

ulation for v = 70 km/h using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck).

Cαf1 = 4360 N/deg and Cαf1 = 3320 N/deg (laden- and unladen truck). The eﬀective cornering stiﬀness of the rear tire, Ceff,ri , shows the same behaviour as

Ceff,fi .

Figure A.9: Variation of Ceff,fi with mass. Horizontal lines denote Cαf1 before (red and blue line) and after change (black and magenta) for the laden- and unladen truck. 48 APPENDIX A. GRAPHS FROM THE SIMULATIONS

Figure A.10: Illustration of which color is associated with which mass. Connected to Figure A.9 above.

The sensitivity of the cornering stiﬀnesses Cαf1 and Cαr1 in the bicycle model equations are tested, by changing Cαf1 and Cαr1 one at a time from the best match

Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck). This is done for the front slip angle αf = 2.03 deg.

Changing the front cornering stiﬀness to Cαf1 = 2920 N/deg, yields approximately an increase of the turning radius by 79 meters, a decrease of Fyf by 5.8 kN and Fyr by 8.9 kN (v = 70 km/h). αr decreases about 0.458 deg. However, for Cαr1 = 3500 N/deg (same decrease as above for Cαf1 ), the radius is almost the same (0.09 m decrease), Fyf the same and Fyr only 0.01 kN increase. This is due to that the eﬀects of changing Cαr is cancelled by the increase of αr of about 0.561 deg.

A.1.3 Longitudinal forces

The extra rolling resistance power loss from cornering, Prr,tu+, from the Adams tire model simulation is derived from the longitudinal forces as already noted in Section 4.4.2. Figure A.11 and A.12 show that the longitudinal forces for each tire from Adams tire model simulation increase with |κ| and for decreasing radius (except front right tire). In Figure 4.8, Section 4.4.3, the weight shift from the Adams tire model simulation is shown. As can be seen, weight is transferred to the right side of the truck. Hence, it is a left-hand turn. According to Figure A.11, κ is only limited during braking, just like explained in Section 2.3.2. Also, the ﬁgure shows that the rear left tires have higher longitudinal slip than the rear right tires. This can be due to weight shift from the left to the right side. Also, as the rear wheels are driving A.1. VALIDATION OF BICYCLE MODEL USING ADAMS TIRE SIMULATION 49

Figure A.11: Longitudinal force versus longtudinal slip for each tire obtained from Adams tire model simulation for v = 70 km/h (laden truck).

Figure A.12: Longitudinal force versus turning radius for each tire from Adams tire model simulation for v = 70 km/h (laden truck). wheels, they develop considerably high longitudinal forces, compared to the front wheels that only develop longitudinal forces due to rolling resistance. Figure A.13 shows that the total longitudinal force from the left tires are greater in magnitude compared to the total longitudinal force from the right tires. This can also be explained due to weight shift. It is also shown that the longitudinal forces from the left tires at each axle are greater than the longitudinal forces from 50 APPENDIX A. GRAPHS FROM THE SIMULATIONS the right tires (Figure A.14).

Figure A.13: Total longitudinal force for the left- respectively the right tires versus turning radius obtained from Adams tire model simulation for v = 70 km/h (laden truck).

Figure A.14: Total longitudinal force for the left- and right tires on the front- respec- tively rear axle versus turning radius from Adams tire model simulation for v = 70 km/h (laden truck). A.1. VALIDATION OF BICYCLE MODEL USING ADAMS TIRE SIMULATION 51

A.1.4 Lateral forces

Figure A.15: Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h and v = 90 km/h

using Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck).

Figure A.16: Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h and v = 90 km/h

using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck). 52 APPENDIX A. GRAPHS FROM THE SIMULATIONS

Figure A.17: Total front lateral force, Fyf , and rear lateral force, Fyr, versus turning

radius for v = 70 km/h using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck).

A.1.5 Weight shift

Figure A.18: Weight shift for each tire versus turning radius from Adams tire model simulation for v = 50 km/h (laden truck). A.1. VALIDATION OF BICYCLE MODEL USING ADAMS TIRE SIMULATION 53

Figure A.19: Weight shift for each tire versus turning radius from Adams tire model simulation for v = 90 km/h (laden truck).

TRITA -MAT-E 2015:82 ISRN -KTH/MAT/E--15/82-SE