Enhancing the MF-Swift Tyre Model for Inflation Pressure Changes I.B.A. op het Veld DCT 2007.144
Master’s thesis
Supervisor:
Prof. Dr. H. Nijmeijer (Eindhoven University of Technology)
Coach(es):
Dr. Ir. A.J.C. Schmeitz (TNO Automotive) Dr. Ir. I.J.M. Besselink (Eindhoven University of Technology / TNO Automotive)
Member of committee: Dr. Ir. J.A.W. van Dommelen (Eindhoven University of Technology)
Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Group Eindhoven, November, 2007
Preface This Master thesis investigation was conducted from July 2006 till December 2007 at the section for Integrated Safety of the Netherlands Organisation for Applied Scientific Research (TNO Automotive) in Helmond. For the supervision and guidance during the investigation I would like to thank Prof. dr. H. Nijmeijer and Dr. Ir. I.J.M. Besselink of the Eindhoven University of Technology and Dr. Ir. A.J.C. Schmeitz of TNO Automotive. Furthermore, I would like to thank my study colleagues, especially Thijs Spijkers, for the fruitful discussions and for their friendship. Last but not least, I am thankful to my family and Annemieke for their non-technical support, patience and encouragements throughout my entire study career.
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Abstract Enhancing the MF-Swift Tyre Model for Inflation Pressure Changes - I.B.A. op het Veld Tyres are a crucial part of the vehicle, they are the only contact between the road and the vehicle. The tyre is the key link in the force transmission between the road surface and the vehicle. The forces generated in the tyre are the result of tyre deflections due to vehicle support and interaction between the tyre and the road. Every steering, braking or driving action is eventually transmitted through the tyre. Understanding the tyre properties is essential to the analysis and design of vehicles and vehicle components. For this purpose different (mathematical) models are developed to analyse the behaviour of pneumatic tyres. At TNO two tyre models are developed: the Magic Formula (an empirical slip model) and the MF-Swift tyre model (a dynamic semi-empirical model, using a rigid ring approach with the Magic Formula slip model). The current version of the MF-Swift tyre model can describe the tyre behaviour for one set of operating conditions. Current research projects aim to a clear, reliable and integrated solution to capture different operating conditions (inflation pressure, temperature, road friction, etc.). In this study, the influence of the inflation pressure on the tyre behaviour is investigated. The subject of this research project is to extend the MF-Swift model for inflation pressure changes. The objectives of this research are: (1) investigating which parts of the MF-Swift tyre model must be made inflation pressure dependent; (2) deriving relations and parameter identification strategies for implementation of inflation pressure influences in the MF-Swift tyre model and (3) investigate the functionality and applicability of the enhanced tyre model using a vehicle simulation model. To achieve these objectives, the MF-Swift tyre model is analysed and a literature study is performed on the inflation pressure dependency of the, in the MF-Swift model, modelled tyre characteristics. Where necessary, additional experiments are conducted and finite element method (FEM) analysis are used. Finally when the developed inflation pressure dependent relations are implemented, a number of vehicle simulations are performed with the enhanced MF-Swift tyre model. The literature survey shows that the inflation pressure influence on the steady-state slip characteristics (Fx , Fy and Mz ) is already implemented in the MF-Swift tyre model. Furthermore the literature survey and the additional experiments, conducted with the Flatplank Tyre Test facility of Eindhoven University of Technology, show that the inflation pressure has the most influence on the characteristic tyre stiffnesses, relaxation behaviour and camber effects (i.e. camber thrust and camber torque). All these tyre characteristics show a (almost) linear relation with the inflation pressure. FEM and analytical analyses make clear that also a linear relation between the inflation pressure and the primary tyre eigenfrequencies (i.e. the first eigenmodes in which the tyre tread band almost retains its circular shape) exists. Next, the measurement and FEM results are used to derive inflation pressure, and if necessary vertical load, dependent relations for implementation in the MF-Swift tyre model. In addition, optimisation strategies are introduced to determine the inflation pressure and vertical load dependent lateral stiffness from measurements or lateral relaxation length experiments. Furthermore, strategies are developed to describe and estimate the sidewall stiffnesses of the rigid ring model for changing inflation pressure. Finally, full vehicle simulations show that the enhanced MF-Swift tyre model can be used to simulate the inflation pressure influence on ride comfort and handling behaviour. It is concluded that the enhancements are a useful addition to the MF-Swift tyre model.
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Samenvatting Het aanpassen van het MF-Swift bandmodel voor bandspanningsveranderingen - I.B.A. op het Veld Banden vormen een cruciaal onderdeel van een voertuig, ze zorgen voor het enige contact tussen voertuig en wegdek. Bij de krachtoverdracht tussen wegdek en voertuig zijn banden dé koppeling. De optredende bandkrachten zijn het gevolg van de bandvervorming door het voertuiggewicht en de interactie tussen de band en het wegdek. Uiteindelijk wordt elke stuur, rem of accelereeractie via de band overgedragen naar het wegdek. Voor het analyseren van het voertuiggedrag, is het van essentieel belang om het gedrag van een band te begrijpen. Om dit te bewerkstelligen zijn er verschillende (wiskundige) modellen ontwikkeld die het gedrag van banden analyseren/beschrijven. TNO heeft twee bandmodellen ontwikkeld: de Magic Formula (een empirisch slipmodel) en het MF-Swift bandmodel (een semiempirisch dynamisch bandmodel, gebaseerd op een starre ring benadering en het Magic Formula slipmodel). De huidige versie van het MF-Swift bandmodel kan het bandgedrag alleen beschrijven voor één set bedrijfscondities. Momenteel is het onderzoek gericht op een heldere, betrouwbare en geïntegreerde oplossing voor het beschrijven van verschillende bedrijfscondities. (bandenspanning, temperatuur, wegdekwrijving, enz.). In dit onderzoek wordt de invloed van de bandenspanning op het bandgedrag onderzocht. Het onderwerp van dit onderzoeksproject is het aanpassen/verbeteren van het MF-Swift bandmodel voor bandenspanningsveranderingen. De doelstellingen van dit onderzoek zijn: (1) het onderzoeken welke delen van het MF-Swift bandmodel bandenspanningsafhankelijk moeten worden gemaakt; (2) het afleiden van relaties en parameter identificatie strategieën voor de implementatie van bandenspanningsafhankelijkheid in het MF-Swift bandmodel en (3) het onderzoeken van de functionaliteit en toepasbaarheid van het nieuwe bandenspanningsafhankelijke MF-Swift bandmodel in voertuigontwikkeling. Om deze doelstellingen te verwezenlijken, is het MF-Swift bandmodel geanalyseerd en is een literatuurstudie uitgevoerd naar de invloed van bandenspanning op de, in het MF-Swift bandmodel, gemodelleerde bandeigenschappen. Waar nodig zijn extra experimenten en eindige elementenmethode (FEM) analyses uitgevoerd. Uiteindelijk, na implementatie van de afgeleide bandenspanningsafhankelijke relaties, is een aantal voertuigsimulaties uitgevoerd met het nieuwe MFSwift bandmodel. Uit het literatuuronderzoek is gebleken dat de bandenspanningsafhankelijkheid van de stationaire slip karakteristieken (Fx , Fy and Mz ) reeds zijn geïmplementeerd in het MF-Swift bandmodel. De resultaten van het literatuuronderzoek en de uitgevoerde experimenten met de Flatplank Tyre Test facility van de Technische Universiteit Eindhoven tonen aan dat bandenspanning de meeste invloed heeft op de karakteristieke bandstijfheden, het relaxatiegedrag en de camber krachten en momenten. Al deze karakteristieke grootheden laten een (bijna) lineaire relatie zien met de bandenspanning. FEM en analytische analyses tonen aan dat er ook een lineaire relatie bestaat tussen de bandenspanning en de primaire eigenfrequenties van de band. De resultaten van de experimenten en de FEM analyses zijn gebruikt om bandspanningsafhankelijke, en zonodig verticale belastingsafhankelijke, relaties af te leiden. Daarnaast is er een optimalisatie strategie geïntroduceerd om de laterale stijfheid te bepalen uit stijfheidsmetingen of relaxatielengte metingen. Verder zijn er strategieën ontwikkeld om de zijwang stijfheden van het starre ring model voor bandenspanningsveranderingen te beschrijven. Tenslotte laten de voertuigsimulaties zien dat het nieuwe bandenspanningsafhankelijke MF-Swift bandmodel de invloed van de bandenspanning op het rijcomfort en het handling gedrag goed kan simuleren. Hieruit v
kan worden geconcludeerd dat de aanpassingen voor bandenspanningsveranderingen een bruikbare uitbreiding zijn van het MF-Swift bandmodel.
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Contents Preface
i
Abstract
iii
Samenvatting
v
List of Symbols
xi
1
General Introduction
1
1.1
Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Objectives and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 The MF-Swift Tyre Model 2.1
The Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2 The MF-Swift tyre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3 3
5
2.2.1
General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.2
Rigid ring dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.3
Contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Summarising this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Literature Review
11
3.1
Force and moment characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1.1
Longitudinal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.1.2
Lateral force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.1.3
Aligning moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.1.4
Vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Empirical relations for aircraft tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2.1
Longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2.2
Lateral stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2.3
Torsional Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2
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viii
CONTENTS 3.2.4
Tyre relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.3
Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.4
Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.5
Summarising this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4 Inflation Pressure Sensitivity Experiments 4.1
Stiffness measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.1.1
Tyre vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.1.2
Tyre longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.1.3
Tyre lateral stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.1.4
Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.2 Lateral relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.2.1
Effective rolling radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Camber thrust and camber torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.4 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.3
4.5 5
21
4.4.1
FEM analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.4.2
Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Summarising this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
39
5.1
Tyre vertical stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.2
Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.3
Lateral relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.3.1
Current model implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.3.2
Proposed enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
5.4
Longitudinal relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.5
Rolling resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.6
Camber thrust and camber torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5.7
Rigid ring dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.7.1
Rotta Membrane Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.7.2
Scaling the nominal pressure sidewall stiffnesses . . . . . . . . . . . . . . . . .
57
5.7.3
Stiffnesses rigid ring model, lateral components . . . . . . . . . . . . . . . . .
58
5.7.4
Stiffnesses rigid ring model, longitudinal components . . . . . . . . . . . . . .
59
Summarising this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5.8
6 Vehicle Behaviour Simulations 6.1
63
Steady-state circular test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.1.1
Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.1.2
Unloaded vs. loaded vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6.2 Random steering test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6.3
73
Ride analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS
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7
Conclusions and Recommendations
75
7.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7.1.1
Inflation pressure influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7.1.2
Enhanced model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
7.1.3
Applicability to the vehicle design . . . . . . . . . . . . . . . . . . . . . . . . .
77
7.1.4
Final conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
7.2
Bibliography
78
A The Flatplank Tyre Tester
83
A.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
A.2 Forces and moments transformations . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
B Measurement Results B.1
87
Characteristic stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
B.1.1
Tyre longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
B.1.2
Tyre lateral stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
B.1.3
Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
B.2 Lateral relaxation behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
B.3 Camber thrust and camber torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
C Modal Analysis C.1 FEM Simulations (ABAQUS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95 95
C.1.1
Free hanging tyre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
C.1.2
non-rolling loaded tyre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
C.2 Analytical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
D Aspects of the introduced inflation pressure dependent relations
101
D.1 Tyre torsional stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 D.2 Rolling resistance force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 D.3 Stiffnesses of the rigid ring model extrapolation properties . . . . . . . . . . . . . . . . 103 D.3.1
Lateral situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
D.3.2
Longitudinal situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
D.4 Camber thrust and camber torque extrapolation . . . . . . . . . . . . . . . . . . . . . . 104 D.5 Overview implemented parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 E Simulation Parameters
109
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CONTENTS
List of Symbols Symbol
Description
Unit
a B C cbγ cbψ cbθ cbx cby cbz CF α CF κ CF x CF y CF z CM z cpx crx cry cu cv cw cx cy D d dx dy E Fax Faz Fa frot ftrans Fx,RR FxW FyW Fy Fz0 Fz
half the contact length stiffness factor in Magic Formula shape factor in Magic Formula torsional sidewall stiffness in x-direction torsional sidewall stiffness in z-direction rotational sidewall stiffness in y-direction translational sidewall stiffness in x-direction translational sidewall stiffness in y-direction translational sidewall stiffness in z-direction cornering stiffness longitudinal slip stiffness tyre longitudinal stiffness tyre lateral stiffness overall vertical stiffness total aligning moment stiffness longitudinal tread stiffness residual longitudinal carcass stiffness residual lateral carcass stiffness tangential direction stiffness (Rotta) lateral direction stiffness (Rotta) radial direction stiffness (Rotta) longitudinal carcass stiffness total carcass stiffness peak factor in Magic Formula outside diameter of unloaded tyre longitudinal deflection lateral deflection curvature factor in Magic Formula longitudinal component of axle force vertical component of axle force axle force rotational eigenfrequency translational eigenfrequency rolling resistance force longitudinal force in contact patch centre lateral force in contact patch centre lateral force nominal vertical load vertical load xi
m Nm/rad Nm/rad Nm/rad N/m N/m N/m N/rad N N/m N/m N/m Nm/rad N/m2 N/m N/m N/m N/m N/m N/m N/m m m m N N N Hz Hz N N N N N N
xii G Ibx Iby Ibz Ir K k1 k2 Kα Kλ kbγ kbψ kbθ kbx kby kbz Kxκ Kx Kyα Kyγ0 ls M ma mb Mx My MzW P p pi,r pi0 pi q r0 Re rl SH SV t V Vx w Z
CONTENTS shear modulus tyre sidewall moment of inertia of the belt about the x-axis moment of inertia of the belt about the y-axis moment of inertia of the belt about the z-axis moment of inertia of the small residual (contact patch) mass stiffness matrix numerical constant longitudinal stiffness (aircraft tyre) numerical constant longitudinal stiffness (aircraft tyre) torsional stiffness lateral stiffness (aircraft tyre) torsional sidewall damping in x-direction torsional sidewall damping in z-direction rotational sidewall damping in y-direction translational sidewall damping in x-direction translational sidewall damping in y-direction translational sidewall damping in z-direction longitudinal slip stiffness in Magic Formula longitudinal stiffness (aircraft tyre) cornering stiffness in Magic Formula camber thrust stiffness in Magic Formula length of sidewall arc mass matrix mass of parts that rotate with the rim belt mass overturning moment rolling resistance moment aligning moment in contact patch center inflation pressure fit parameter in Magic Formula rated inflation pressure nominal inflation pressure inflation pressure fit parameter in Magic Formula unloaded tyre radius effective rolling radius loaded radius horizontal shift in Magic Formula vertical shift in Magic Formula sidewall thickness velocity forward, longitudinal velocity undeflected tyre width vertical load
N/m2 kgm2 kgm2 kgm2 kgm2 Nm/rad N/m Nms/rad Nms/rad Nms/rad Ns/m Ns/m Ns/m N/m N/rad N/m m kg kg Nm Nm Nm bar bar bar bar m m m m m/s m/s m N
CONTENTS
xiii
Greek Symbol
Description
Unit
α βy δ0 δλ γ λ µ µy Ω ω ψ ψr ρz σst σx σy τλ ϕs
sideslip angle effective forward slope vertical deflection vertical sinking at lateral force (aircraft tyre) angle about the x-axis, camber angle scaling factor in Magic Formula peak longitudinal friction coefficient peak lateral friction coefficient tyre angular velocity about the y-axis free vibration frequency, rotational speed steering angle angle between wheel plane and contact patch vertical deflection static relaxation length longitudinal relaxation length lateral relaxation length lateral spring constant coefficient (aircraft tyre) half the angle of tyre sidewall
rad rad m m rad rad rad/s rad/s rad rad m m m m rad
xiv
CONTENTS
Chapter 1
General Introduction This chapter gives a general introduction to the work presented in this thesis. The background, objectives and scope of research are presented and a brief outline of the contents of this Master’s thesis is given.
1.1
Motivation and background
While the wheel may have been one of man’s first inventions, the pneumatic tyre is definitely not one of the simplest components to analyse. The pneumatic tyre is a complex composite consisting mainly of vulcanised rubber and layered reinforcing strands (plies), made of Nylon, Rayon or steel cords which are oriented in a certain configuration. The tyre is a crucial part of the vehicle. Not only to support the vehicle weight and to cushion road irregularities, but also to generate and transmit forces needed to accelerate and decelerate the vehicle and to change the direction of motion of the vehicle. The tyre forces and moments are the result of tyre deflections due to interaction between wheel and road. Tyre vibrations arise through road irregularities, wheel axle motions, and tyre non-uniformities. The complex tyre structure with its compliance and inertia may give rise to attenuation from these irregularities in certain frequency ranges but also to magnification at other frequencies.
Figure 1.1: Construction of radial ply tyre [9].
A modern radial (ply) tyre is characterised by parallel cords running directly across the tyre from one bead to the other, the so called carcass plies. Directional stability of the tyre is supplied by the enclosed pressurised air acting on the sidewall of the carcass and by the stiff belt of fabric or steel, that runs around the circumference of the tyre. The soft carcass provides the tyre with a soft ride and the stiff belt provides the radial tyre with good cornering proporties, by keeping the tread flat on the road despite horizontal deflections of the tyre. The contact between the tyre and the road is established and 1
2
General Introduction
maintained by the tread. The adhesion between tread and road is the main factor for the generation of horizontal forces. Not only the rubber compound and the orientation of the plies (radial or bias ply) can have an effect on these tyre characteristics but also several external parameters, for instance; vertical load, inflation pressure, velocity and (ambient) temperature. Understanding tyre properties is essential for the analysis and design of vehicles and vehicle components. For this purpose different (mathematical) models are developed to describe the behaviour of pneumatic tyres. These models can be distinguished between models based on the physical construction of the tyre (physical models) and models based on experimental data (empirical models). Combinations of both approaches are also used. The most essential requirements for a tyre model for vehicle analyses are: • accurate prediction of forces and moments; • practical in use (e.g. low computational effort); • widely applicable (for many different operating conditions). Through the years various modelling techniques have been developed, but it is still a long way to the complete description of all aspects of tyre behaviour. At TNO two tyre models are developed: the M agic F ormula and the Short W avelength Intermediate F requency T yre model, SWIFT, of Pacejka [28]. The Magic Formula is a empirical tyre handling model, first introduced in 1987, which is capable of dealing with the stationair slip characteristics and dynamics up to about 8 Hz. The SWIFT tyre model, or also called the MF-Swift tyre model, is a dynamic semi-empirical model, based on a rigid ring tyre model combined with the Magic Formula. MF-Swift can describe tyre behaviour for in-plane (longitudinal and vertical) and out-of-plane (lateral and steering) motions up to about 60-100 Hz and it can deal with uneven road surfaces. For more details it is referred to chapter 2. Through the years many revisions have been made. The current research projects aim to a clear, reliable and integrated solution to capture the influence on the tyre behaviour of different operating conditions (road friction, temperature, inflation pressure, etc.) The current model can describe the tyre behaviour for one set of operating conditions only. TNO Automotive already performed a number of research projects on these topics, namely: • A validated tyre temperature model has been developed [38]. • The influence of pavement micro/macro texture on tyre forces and moments data has been investigated. • A brake performance model ("remvermogenmodel") has been developed to predict braking distances and vehicle states during an emergency stop. • Magic Formula parameters are extended with inflation pressure parameters, so that it is possible to describe the stationary forces and moments for a range of tyre inflation pressures [12].
1.2
Objectives and scope
The subject of this Master’s thesis is to extend the MF-Swift model for inflation pressure changes. Taking into account the structure of the model, the following objectives have been defined: 1. Investigate which parts of the model must be made inflation pressure dependent. 2. Derive relations and parameter identification strategies for the implementation of inflation pressure changes in the MF-Swift tyre model.
1.3 Outline of thesis
3
3. Investigate the functionality and applicability of the enhanced tyre model using a vehicle simulation model. To achieve these objectives, a literature study is performed and previous measurements are analysed. Where necessary, additional experiments are conducted. When literature and measurement results do not provide a clear answer, results from a finite element method (FEM) analysis are used. Finally a number of vehicle simulations are executed with the enhanced tyre model to analyse the extended relations on functionality and applicability in vehicle design applications.
1.3
Outline of thesis
The outline of this thesis is as follows. The TNO tyre models are the subject in Chapter 2. The Magic Formula and MF-Swift tyre model are discussed in more detail. In Chapter 3 a literature review on the influence of inflation pressure on the different tyre characteristics is presented and possible trends are identified. This knowledge is used during the analysis and development of the model in the subsequent chapters. Chapter 4 deals with the inflation pressure sensitivity experiments. An overview is given of the experimental investigations and of the tyre characteristics inflation pressure dependency. The subject of Chapter 5 is the enhancement of the MF-Swift tyre model. Inflation pressure depending relations are presented and validated with the results of Chapter 3 and 4. Furthermore methods are presented to describe the influence of inflation pressure on the rigid ring dynamics. Chapter 6 illustrates the influence of inflation pressure on vehicle handling and ride. A number of simulations are performed with different inflation pressure settings to investigate the functionality and applicability of the enhanced relations in vehicle design. Finally, in Chapter 7, conclusions are given and recommendations for further research are formulated.
4
General Introduction
Chapter 2
The MF-Swift Tyre Model As already discussed in Chapter 1, the subject of this Master’s thesis is to extend the MF-Swift model for inflation pressure changes. The present chapter gives an overview of the MF-Swift tyre model, that has been designed to represent the tyre as a vehicle component in a multibody simulation environment. The MF-Swift tyre model is a combination of the Magic Formula and a rigid ring tyre model. The basic modelling approach of the MF-Swift model is termed "semi-empirical", meaning the model is based on measurement data but also contains parts that find their origin in physical models.
2.1
The Magic Formula
The so-called M agic F ormula is a widely used empirical tyre model to calculate steady-state tyre force and moment characteristics for the use in vehicle dynamics studies. The development of the model was started in the mid-eighties, as a cooperative effort of the TU-Delft (Pacejka) and Volvo Car Corporation (Bakker et al.), resulting in a first version in 1987 [3]. In the following years several enhancements were made. Michelin introduced in 1993 a purely empirical method using Magic Formula based functions to describe the tyre horizontal forces at combined slip conditions. This approach was adopted by DVR (Delft Vehicle Dynamics Research Center, a joint venture of TU-Delft and TNO) resulting in the "Delft Tyre" tyre model. All versions of the Magic Formula show the same basic form for pure slip characteristics: a sine of an arctangent. The general form of the Magic Formula reads: y = D sin[C arctan{(1 − E)Bx + E arctan(Bx)}]
(2.1)
Y (X) = y(x) + SV
(2.2)
x = X + SH
(2.3)
Y is the output variable: longitudinal force Fx (= f (κ)) or lateral force Fy (= f (α)). X is the input variable: sideslip angle α or longitudinal slip κ. Cosine of the arctangent versions are used for the description of the aligning torque Mz . The remaining variables of the Magic Formula describe the following coefficients: B : stiffness factor C : shape factor D : peak factor E : curvature factor SH : horizontal shift SV : vertical shift 5
6
The MF-Swift Tyre Model
The coefficients B, C, D, E and the offsets SH and SV characterise the shape of the slip characteristics. Figure 2.1 illustrates the meaning of some of the factors by means of a typical side force characteristic. Each coefficient represents a specific aspect of the slip characteristic. The product of the factors B, C and D determines the slope at the origin. The shape factor C determines the value of Y when x → ∞. The peak factor D influences the maximum value of the characteristic. The curvature factor E influences the characteristic curvature around the peak value and controls the horizontal position of the peak. The two shifts SH and SV make it possible to offset the curve in horizontal and vertical direction with respect to the origin. The slip stiffness K influences the slip stiffness at small values of slip and is determined by the product of BCD. This all results in a new set of coordinates Y (X), as shown in figure 2.1.
Figure 2.1: Curve produced by the Magic Formula; typical side force characteristic [28].
The offsets SH and SV appear to occur when ply-steer and conicity effects cause the lateral force Fy curve not to pass through the origin. Wheel camber may also result in considerable offsets of the Fy versus α curve. To explain the equations describing the Magic Formula coefficients B, C, D and E, the full set of equations for pure lateral slip (pure cornering) will be considered. The Magic Formula for the lateral force Fy reads: Fy = Dy sin[Cy arctan{(1 − Ey )By αy + Ey arctan(By αy )}], (2.4) where the different coefficients are described by: αy Cy Dy Ey Ky SHy SV y
= α + SHy = pCy1 λCy = Fz µy λµy = Fz (pDy1 + pDy2 dfz )(1 − pDy3 γ 2 )λµy = (pEy1 + pEy2 dfz ){1 − (pEy3 + pEy4 γ)sign(αy )}λEy F z (1 − pKy3 γ 2 ) = By Cy Dy = pKy1 Fz0 sin 2 arctan pKy2 Fz0 = (pHy1 + pHy2 dfz )λHy + pHy3 γλKyγ = Fz (pV y1 + pV y2 dfz )λV y + (pV y3 + pV y4 dfz )γλKyγ
(2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11)
The dfz is the normalized load increment: dfz =
Fz − Fz0 , Fz0
(2.12)
where Fz is the actual vertical load, Fz0 the nominal vertical load and γ the camber angle. The nominal vertical load is typically 80% of the load index (LI) of the tyre. Dimensionless parameters p are
2.2 The MF-Swift tyre model
7
introduced to describe the influence of the vertical load and camber angle on the coefficients. These parameters are determined by minimising the difference between the model and measurements using numerical optimisation techniques. Furthermore, a number of scaling factors λ are introduced to scale the formula without changing the dimensionless parameter values. In this way, the tyre characteristics can be adjusted or tuned (e.g. correction for different road types). The set of equations (2.5) till (2.11) shows the basic influence of the vertical load and camber angle on the lateral force. The equations presented here correspond to the latest published version of the Magic Formula [28].
2.2
The MF-Swift tyre model
In this section the MF-Swift tyre model will be discussed in more detail. For a complete description of the model and the latest updates of the model it is referred to the work of Schmeitz [31]. The MF-Swift tyre model is based on the work of Zegelaar [40], Maurice [26] and finally Schmeitz. The research was conducted at Delft University of Technology and supported by TNO Automotive and a consortium of industries. It has been developed first as a tyre model for in-plane dynamics. Later on, the model has been extended for out of plane dynamics and road unevenesses.
2.2.1
General description
The MF-Swift tyre model uses the Magic Formula for the steady-state slip behaviour. This makes it possible to model tyre slip behaviour with empirical relations. To model high frequency dynamic behaviour and obstacle response, a rigid ring is added to model the dynamic behaviour of the tyre (primary or rigid ring tyre modes). A schematic representation of the MF-Swift tyre model is shown in figure 2.2.
Figure 2.2: Schematic representation of the MF-Swift rigid ring tyre model [2].
The rigid ring, that represents the tyre belt, is attached to the rim by springs and dampers. These springs and dampers represent the tyre sidewalls with pressurised air. Furthermore, residual stiffness and damping elements are used to represent the quasi-static tyre stiffnesses correctly. Finally a contact model is added to describe the tyre slip behaviour and enveloping behaviour. The residual spring and damper elements connect the contact model (a combination of a slip model and a model for enveloping obstacles) to the rigid ring. The slip model, based on the Magic Formula, describes the slip behaviour of the contact patch and generates slip forces and moments according to the applied slip. The enveloping model takes the tyre flexibility into account when rolling over road surfaces with short wavelength content. The total tyre mass is distributed between the rigid ring and the rim (typically: 75% of the tyre mass is assigned to the belt and 25% is assigned to the rim according to [28]). As mentioned before, the tyre tread band (belt) is modelled as a circular rigid body with mass mb and inertia Ibx , Iby and Ibz . Because the tyre tread band is modelled as a rigid body, the model is only able to describe the dynamic behaviour for conditions where the tyre tread band retains its circular shape
8
The MF-Swift Tyre Model
(i.e flexible modes of the tyre can not be described). This limits the useable frequency range to 60-100 Hz, depending on the tyre type to be modelled.
2.2.2
Rigid ring dynamics
For the rigid ring, the following equations of motions with respect to the non-rotating belt axis system can be derived. For the complete derivation of the equations of motion reference is made to the work of Schmeitz [31]. a a a b mb V˙ ax − ωaz Vay +x ¨rb + kbx x˙ rb + cbx xrb − kbz Ωzrb = Fbx (2.13) a a a a a b mb V˙ ay + ωaz Vax − ωax Vaz + y¨rb + kby y˙ rb + cby yrb = Fby (2.14) a a a b mb V˙ az − ωax Vay + z¨rb + kbz z˙rb + cbz zrb − kbx Ωxrb = Fbz (2.15) a a b ˙ ˙ Ibx ω˙ ax + γ¨rb − Iby Ω + θrb ωaz + ψrb + kbγ γ˙ rb + cbγ γrb − kbψ Ωψrb = Mbx (2.16) b ¨ ˙ Iby θrb + Ω˙ + kbθ θrb = Mby (2.17) a a b Ibz ω˙ + ψ¨rb − Iby Ω + θ˙rb ω + γ˙ rb + kbψ ψ˙ rb + cbψ ψrb − kbγ Ωγrb = M (2.18) az
ax
bz
mb is the belt mass, Ibx,y,z the belt inertia (x, y, z-direction), cbx,y,z the translational sidewall stiffness (x, y, z-direction) and kbx,y,z the translational sidewall damping (x, y, z-direction). Furthermore, cbγ,ψ is the torsional sidewall stiffness (about x, z-axis), cbθ the rotational sidewall stiffness (about y-axis), kbγ,ψ the torsional sidewall damping (about x, z-axis) and kbθ the rotational sidewall stiffness (about y-axis). Finally, xrb , yrb , zrb , γrb , θrb and ψrb are relative motions of the belt with respect to the wheel centre. ~ b are the forces and moments acting on the belt which are applied at the centre of the belt F~bb and M b body and are obtained from the forces and moments in the residual spring-damper elements and the contact model.
2.2.3
Contact model
The contact model can be divided into an enveloping model and a slip model. When travelling on uneven road surfaces, the tyre acts as a geometric filter which smoothes the response at the axle (i.e the unevenness of the actual road). This is the so-called enveloping behaviour of a tyre. When rolling over a (short) obstacle, the tyre always hits the obstacle at a certain distance before the centre of the axle is above the obstacle (so-called: lengthening). If the centre of the axle is above an obstacle, the (small) obstacle is fully or partially absorbed by the tyre (so-called: swallowing). In figure 2.3 the enveloping phenomena are depicted.
Figure 2.3: Tyre enveloping behaviour [2].
To represent the nonlinear enveloping behaviour, the concept of an effective road profile is used, as developed by Zegelaar [40] and Schmeitz [31]. The wheel states and the effective road surface are the input of the rigid ring tyre model.
2.2 The MF-Swift tyre model
9
The wheel states (position, orientation and velocity), consists of: G G T • Position vector of the wheel centre xG (wrt. global position) w y w zw • Wheel yaw ψaa and camber angle γaa • Absolute velocities of the wheel expressed in the local (axle) axis system: a a a T ~wheel = ~ea T Vax V Vay Vaz a a T a T ω ~ wheel = ~e ωax Ω ωaz The effective road surface is defined by the effective inputs (plane height w, effective forward slope βy , effective road camber angle βx and road curvature dβy /dx) and is the result of the envelopment behaviour on the specific road profile, see figure 2.4. These effective inputs serve as an input for the rest of the tyre model. The effective plane angle is defined as the quasi-statically (measured) longitudinal axle force Fax divided by the vertical axle force Faz : tan βy =
Fax . Faz
(2.19)
When rolling over an obstacle with a constant vertical load, the effective height equals the difference in axle height between the new height.
Figure 2.4: 2D effective road surface; tyre rolls quasi-statically over a step obstacle at a constant vertical load Faz [31].
Schmeitz [31] introduced the so-called tandem − cam model in 2004. The model consists of two elliptical cams that move vertically when travelling over a road surface. The cams represent the outside contour in the area where contact with the obstacle occurs. The distance between the cams is about the contact length ls of the tyre. From the heights of the cams the effective inputs w and βy are determined, see figure 2.5. By using parallel tandem cam models the effective input for 3D road surfaces can be determined, see Schmeitz [31] for details.
10
The MF-Swift Tyre Model
Figure 2.5: The tandem model with elliptical cams [31].
To account for all other flexibilities (i.e. carcass construction), residual springs in longitudinal, lateral, vertical and yaw direction are introduced. They connect the slip model to the rigid ring, see figure 2.6. Furthermore, residual dampers are attached for computational stability reasons. The vertical load is directly applied to the residual vertical stiffness, calculated from the total vertical stiffness and the ring sidewall stiffness. The moments about the x and y-axis are also directly applied to the ring model.
Figure 2.6: MF-Swift contact model representation.
2.3
Summarising this chapter
In this chapter the MF-Swift tyre model and the tyre characteristics that are modelled are discussed in more detail. The most important characteristics of the tyre model are: • steady state slip characteristics (including camber effects), • characteristic tyre stiffnesses (longitudinal, lateral, vertical and torsional), • relaxation behaviour, • eigenfrequencies of the tyre belt. The inflation pressure dependency of these tyre characteristics has to be investigated.
Chapter 3
Literature Review Based on the tyre characteristics that are modelled in the MF-Swift tyre model, a literature survey on the influence of the inflation pressure is performed. In this chapter the results of the literature survey are presented. For the investigation of the inflation pressure influence on passenger car tyres very often only longitudinal force Fx , lateral force Fy and self-aligning moment Mz as result of slip (force and moment characteristics) are taken into account. Because the literature on passenger car tyres does not provide a clear answer for all the tyre characteristics that are modelled, literature on aircraft tyres is also studied.
3.1
Force and moment characteristics
For the description of the force and moment characteristics the work of de Hoogh [12] and the work Schmeitz and de Hoogh [32] is important. In [12] the influence of inflation pressure and velocity effects on the common tyre characteristics are investigated and the Magic Formula is extended to incorporate these effects. For his findings, measurement results are compared with TREADSIM (a discrete "brush" tyre model [28] with flexible carcass) simulations. To include inflation pressure pi , the inflation pressure increment dpi is introduced: dpi
=
pi − pi0 , pi0
(3.1)
where pi and pi0 are the current and nominal inflation pressure.
3.1.1
Longitudinal force
In [12] it is observed that the longitudinal slip stiffness and the peak longitudinal friction coefficient both are depending on the inflation pressure. Since there is no carcass and belt compliance in longitudinal direction the longitudinal slip stiffness of TREADSIM depends solely on the contact length, described by the following equation from [28]: CF κ =
∂Fx = 2cpx a2 , ∂κ κ=0
(3.2)
where, cpx is the tread stiffness in longitudinal direction and a is the half of the contact length. It can be expected that higher inflation pressure results in a lower slip stiffness. With figure 3.1 this behaviour can be confirmed. However, in [12] it is also concluded that other tyres exhibit a different behaviour. 11
12
Literature Review
Measurement
4
x 10
TREADSIM
4
15
Slip Stiffness [N]
Slip Stiffness [N]
15
10
5
x 10
10
5
P = 1.8 [bar] P = 2.1 [bar] P = 2.4 [bar] 0
0
2000
4000 Fz [N]
6000
8000
0
0
2000
4000 Fz [N]
6000
8000
Figure 3.1: Longitudinal slip stiffness: 225/55 R16 tyre [12].
It appeared that a quadratic relation produces significantly smaller fit errors than a linear relation. The following extension for the longitudinal slip stiffness Kxκ is implemented in the Magic Formula: Kxκ = Fz (pKx1 + pKx2 dfz )epKx3 df z (1 + ppx1 dpi + ppx2 dp2i ),
(3.3)
where the p’s are Magic Formula parameters and dfz is the vertical load increment. In [12] it is concluded that for the longitudinal friction coefficient in the measurements an optimal inflation pressure occurs. Furthermore, it is concluded that the behaviour is similar to that of the peak lateral friction. The proposed equation for the longitudinal friction coefficient µx , is as follows: µx = (pDx1 + pDx2 dfz )(1 + ppx3 dpi + ppx4 dp2i ).
(3.4)
The conclusions in [12] are supported by the findings of Marshek and Cuderman [25]. In [25] the effect of inflation pressure on deceleration for a series of emergency braking tests with different vehicles equipped with anti-lock braking systems is investigated. The conclusion in [25] with respect to inflation pressure is that the effect on the braking performance is only slight for large tyres. For smaller tyres there is an optimal inflation pressure, higher and lower inflation pressures lead to a decrease in longitudinal friction coefficient.
3.1.2
Lateral force
In [12], for the lateral slip characteristic, the effect of inflation pressure changes on the cornering stiffness and the peak lateral friction coefficient is investigated. The conclusion with regard to the cornering stiffness is that an increase of inflation pressure has two counteracting effects; a lower cornering stiffness at low vertical loads and a higher cornering stiffness at high vertical loads. These effects are clearly visible in figure 3.2. The first effect is caused by the decreasing contact length as a result of the increased vertical stiffness. A decrease of contact length results in a decrease of cornering stiffness. Secondly, an increase of inflation pressure leads to an increase in carcass stiffness, i.e. a decrease of carcass compliance. This leads to higher lateral force for the same slip angle, which results in a higher cornering stiffness at high vertical loads. These effects have also been observed by van Erp and Verhoeff [14] in the TNO TEK-tyre project.
3.1 Force and moment characteristics
13
TREADSIM 1400
1200
1200 Cornering Stiffness [N/deg]
Cornering Stiffness [N/deg]
Measurement 1400
1000 800 600 400 P = 2.0 [bar] P = 2.4 [bar] P = 2.8 [bar]
200 0
0
2000
4000 F [N] z
6000
1000 800 600 400 200
8000
0
0
2000
4000 F [N]
6000
8000
z
Figure 3.2: Inflation pressure influence on cornering stiffness: 185/60 R14 tyre [12].
To include the inflation pressure effects in the Magic Formula, two linear relations depending on a inflation pressure increment dpi are added: Fz Kyα = pKy1 (1 + ppy1 dpi )Fz0 sin 2 arctan , (3.5) pKy2 (1 + ppy2 dpi )Fz0 where Ky,α is the cornering stiffness, the p’s are Magic Formula parameters, Fz and Fz0 are the actual vertical load and the nominal vertical load respectively. With regard to the peak lateral friction coefficient µy , the force and moment measurements did not show a clear general relation for the inflation pressure. Some tyres show a minimum peak lateral friction coefficient at low vertical loads that shifts to an optimum at high vertical loads. Other tyres show an opposite behaviour. To be able to fit the minimum/maximum a second order polynomial inflation pressure relation was proposed in [12]: µy = (pDy1 + pDy2 dfz )(1 + ppy3 dpi + ppy4 dp2i ). (3.6)
3.1.3
Aligning moment
The aligning moment shows a clear influence of the inflation pressure. A lower inflation pressure shows a higher peak in the aligning moment characteristic, as can be observed in figure 3.3. An increase of inflation pressure results in higher vertical stiffness, reflecting in a shorter contact length and in a smaller pneumatic trail. The pneumatic trail is the relation of aligning moment and lateral force, and is linearly dependent on the contact length [28]. The change in contact length is relatively large compared to the changes of lateral force, which causes the aligning moment to decrease with the inflation pressure. In the Magic Formula, the aligning moment is calculated by multiplying the lateral force with the pneumatic trail [28]. In [12] it is suggested to add a linear relation to the parameter Dt , which determines the magnitude of the pneumatic trail: Dt = Fz
r0 (qDz1 + qDz2 dfz )(1 − qpz1 dpi )sgn(Vcx ). Fz0
(3.7)
In this equation the q’s are parameters, r0 the tyre free radius and Vcx the forward velocity of the contact centre.
14
Literature Review TREADSIM 150
100
100
50
50 Mz [Nm]
Mz [Nm]
Measurement 150
0 −50
−150 −20
−50
P = 2.0 [bar] P = 2.4 [bar] P = 2.8 [bar]
−100
−10
0 Alpha [deg]
10
0
−100
20
−150 −20
−10
0 Alpha [deg]
10
20
Figure 3.3: Aligning moment: 155/65 R15 tyre [12].
3.1.4
Vertical stiffness
In [32] an approach is presented to describe the inflation pressure effects on the vertical stiffness. The vertical stiffness is obtained from the following empirical relation describing the vertical force Fz as a function of the vertical deflection ρz and the inflation pressure increment dpi : Fz = (1 + qF z3 dpi )(qF z1 ρz + qF z2 ρ2z ).
(3.8)
The qF z ’s are fit-parameters. It is assumed that the relation between the inflation pressure and the vertical stiffness is linear. The parameter qF z3 indicates the ratio of the vertical stiffness due to inflation pressure changes with the nominal pressure. The expression for the vertical stiffness can be derived using (3.8): dFz CF z = = (1 + qF z3 dpi )(qF z1 + 2qF z2 ρz ). (3.9) dρz
3.2
Empirical relations for aircraft tyres
In 1958 a study was conducted by Smiley and Horne [33] to derive empirical equations for pneumatic aircraft tyres for landing gear design. The variables used in these equations are in general the tyre width, tyre radius, tyre deflection and inflation pressure. The results give a relevant indication of the effect of inflation pressure on the mechanical properties of pneumatic tyres. However, aircraft tyres have a different construction and are used under different operating conditions than tyres in the automotive industry; this should be taken into account when interpreting these results. It is reasonable to assume that the trends described below are not (completely) corresponding with the behaviour of a passenger car tyre, but still the results of the aircraft tyres give a good perception of the influence of the inflation pressure on the different characteristics of a pneumatic tyre. In [33] the inflation pressure influence is described for a lot of different tyre characteristics. In this section, only the characteristics in which the literature on passenger car tyres does not provide a clear answer are presented. The characteristics described below are: the longitudinal stiffness, the lateral stiffness, the torsional stiffness and the tyre relaxation behaviour.
3.2.1
Longitudinal stiffness
When a constant vertical force Fz is subjected to a longitudinal force Fx , the tyre experiences a corresponding longitudinal deformation and an additional vertical sinking δx besides the initial vertical
3.2 Empirical relations for aircraft tyres
15
deflection δ0 . Also a longitudinal shift of the vertical force centre of pressure location occurs. Tyre data shows that the longitudinal stiffness tends to increase with increasing vertical tyre deflection and to increase only slightly with increasing inflation pressure, see figure 3.4. In according to [33], it seems reasonable to expect that the longitudinal stiffness can be described by an equation of the type: Kx = k1 d(pi + k2 pi,r )f (δ0 /d),
(3.10)
where k1 and k2 are numerical constants and d is the outside diameter of an free tyre. 4
2
x 10
Longitudinal spring constant, Kx [lb/in.]
1.8 1.6 1.4 1.2 1 0.8 0.6 pi/pi,r=0.7 0.4
pi/pi,r=0.8
0.2
pi/pi,r=1.0 pi/pi,r=1.1
0 0
1
2
3 4 Vert. deflection [in.]
5
6
7
Figure 3.4: Longitudinal stiffness variation with vertical deflection and inflation pressure [33].
p It appears from experimental data that f (δ0 /d) = 3 δ0 /d and that the numerical values for k1 =0.5...1.2 and k2 =4 [33]. Upon application of a longitudinal force, a standing tyre sinks vertically through the small distance δx which is according to experimental data approximately 10 percent of the longitudinal deflection. δx = 0.1|λx |. (3.11)
3.2.2
Lateral stiffness
When a lateral force is applied to a vertically loaded tyre with a vertical force Fz , an inflation pressure pi and an initial vertical deflection δ0 , the tyre experiences a corresponding lateral deformation λ0 . Furthermore a vertical sinking δλ and a lateral shifting of the vertical force centre of pressure location will appear. The vertical force and the inflation pressure appear to be the primary variables influencing the variation of lateral stiffness. Furthermore, the lateral stiffness is slightly influenced by the amplitude of the force-deflection hysteresis loop. In figure 3.5 the lateral stiffness versus vertical deflection is depicted for several inflation pressures. The lateral deflection can be predicted by and empirical equation of the type: 0.7δ0 Kλ = τλ w(pi + 0.24pi,r )[1 − ( )]. (3.12) w The quantity 0.24pi,r takes into account that, because of the tyre carcass stiffness, the lateral stiffness at zero inflation pressure is not zero. This quantity may be regarded as an effective lateral carcass pressure. pi,r is the rated inflation pressure (1/4 of the bursting pressure) and w the width of the undeflected tyre. The lateral spring coefficient τλ is estimated from a large number of lateral stiffness data of aircraft tyres. The numerical value of coefficient τλ is approximately between 2 and 3. When applying a lateral force, a tyre sinks vertically through a small distance δλ which can be represented as a fraction, according to experimental data approximately 15 - 21 percent, of the absolute value of the corresponding lateral deflection λ0 : δλ = (0.15...0.21)|λ0 |.
(3.13)
16
Literature Review
Lateral spring constant, Kλ [lb/in.]
3000
2500
2000
1500
1000
pi,r=24 lb/in2 pi,r=32 lb/in2
500
pi,r=40 lb/in2 pi,r=80 lb/in2
0 0
0.5
1
1.5 2 2.5 Vert. deflection, δ0 [in.]
3
3.5
4
Figure 3.5: Lateral stiffness variation with vertical deflection and inflation pressure [33].
3.2.3
Torsional Stiffness
In figure 3.6 an illustration of the effects of vertical deflection and inflation pressure on the torsional stiffness of aircraft tyres [33] is given. The dotted lines represent the inflation pressure trend lines for constant vertical load. The figure indicates that the torsional stiffness increases approximately linearly with both increasing inflation pressure and vertical deflection. In [33] the following type of empirical equation is proposed to derive the torsional stiffness for aircraft tyres: δ 2 0 Kα = (pi + 0.8pi,r )w3 250 2r0
δ0 ≤ 0.03 2r0 (3.14)
δ0 − 0.015 2r0
Kα = (pi + 0.8pi,r )w3 15
δ0 ≥ 0.03, 2r0
where Kα is the torsional spring constant, pi the current inflation pressure, δ0 the vertical deflection of the pure vertical loading conditions and r0 the unloaded tyre radius.
Figure 3.6: Torsional stiffness aircraft tyre [33].
3.3 Rolling Resistance
3.2.4
17
Tyre relaxation behaviour
The static relaxation length (here: the lateral deformation of a non rolling tyre) is depending on the inflation pressure and vertical deflection. In [33] it is concluded that the static relaxation length decreases with increasing inflation pressure and with increasing vertical deflection. For the aircraft tyres the following type of empirical equation is derived: δ0 pi 1.0 − 4.5 , σst = w 2.8 − 0.8 pi,r 2r0
(3.15)
in which σst is the static relaxation length, w is the width of the undeflected tyre, pi is the current inflation pressure and pi,r is the rated inflation pressure (1/4 of the bursting pressure). Furthermore δ0 is the vertical deflection of the pure vertical load conditions and r0 is the unloaded tyre radius. The same type of empirical equation holds for the unyawed rolling relaxation length. For the yawed rolling relaxation length (i.e. the relaxation length of the tyre when a sideslip angle is applied), the tyre builds up a lateral force which exponentially approaches an end-point condition for steady yawed rolling. The lateral force Fy,r builds up with distance x rolled according to a relation of the form: Fy,r = Fy,r,e − A1 e−x/σ , (3.16) where, Fy,r,e is the steady-state force, A1 is a constant which depends on the initial tyre deflection and σ is called the yawed rolling relaxation length (lateral relaxation length in the rest of the report). The rolling relaxation length can be defined with the same parameters as the static relaxation length, by the following type of empirical equations: pi δ0 11 σ = w 2.8 − 0.8 pi,r 2r0 δ 2 pi δ0 0 σ = w 2.8 − 0.8 − 500 − 1.4045 64 pi,r 2r0 2r0 pi δ0 σ = w 2.8 − 0.8 0.9075 − 4 pi,r 2r0
δ0 ≤ 0.053 2r0 0.053 ≤
δ0 ≤ 0.068 2r0
(3.17)
δ0 ≥ 0.068 2r0
For aircraft tyres the relaxation length is approximately within the range of 2a to 3.5a, where a is half the contact length. For passenger car tyres this is quite different. In [19] it is indicated that for passenger car tyres the relaxation length is approximately five times the contact length or more (σ > 10a).
3.3
Rolling Resistance
Collier and Warchol (1980) [10] evaluated Bias, Bias-belted and Radial tyre performance on five different inflation pressures with different vertical load conditions. The results, presented in figure 3.7, indicate that changing the inflation pressure has a lot of influence on the rolling resistance of a tyre. The nominal vertical load in figure 3.7 is Fz0 =6 kN. For a given vertical load condition, the tyre appears to have a lower rolling resistance at increasing inflation pressure. Above 2.75 bar inflation pressure, the rolling resistance levels off rapidly. SAE uses a standard (J2452) to model rolling resistance from measurements into vehicle models. This model is developed and proposed by Grover [17]. The standard requires that the rolling resistance force, applied vertical load and tyre inflation pressure are measured at several velocities and for different load/pressure cases. A minimum of six velocity steps at each load/pressure case is required. With the gathered measurement data, a rolling resistance force model is developed of the form Fx,RR = f (Load, P ressure, V elocity), which serves as a mathematical characterisation of the rolling resistance
18
Literature Review 100% index = 1.65 bar
100% index = 1.65 bar
140
140 1.10 bar 1.65 bar 2.20 bar 2.75 bar 3.30 bar
1.2Fz0 1.0Fz0
130
130
0.8Fz0 0.6F
z0
120 % Rolling Resistance
% Rolling Resistance
120
110
100
110
100
90
90
80
80
70 1
1.5
2 2.5 Inflation pressure [bar]
3
3.5
70 0.5
0.6
0.7
0.8 0.9 % nom. vertical load
1
1.1
1.2
Figure 3.7: Rolling resistance vs. vertical load and inflation pressure [10].
Figure 3.8: Rolling Resistance force: 195/70 R14 tyre (two different tyres) [18].
force in a range of operating conditions (load, pressure, velocity). Two relations, (3.18) and (3.19), can be used with a difference in velocity dependency: Fx,RR = K · P α Z β V γ
(3.18)
Fx,RR = P α Z β (a + bV + cV 2 ),
(3.19)
where Fx,RR is the rolling resistance force, K a constant, P the inflation pressure, Z the vertical load and V the velocity. Furthermore α, β, γ and a, b, c are regression exponents and regression coefficients respectively. (3.18) and (3.19) can be seen as a signature of the rolling resistance performance of the tyre over the operating conditions. Besides that the models can be used for calculating the tyre rolling resistance at other load, pressure and velocity conditions they also can be used for fuel economy calculations and measurements. Typical output is a set of Fx,RR as function of velocity curves for different load/pressure conditions, as shown in figure 3.8. The figure shows a clearly visible influence of inflation pressure on the rolling resistance (see: (1) 536 kg, 2.0 bar and (2) 536 kg, 3.0 bar). It can be observed that a decrease of inflation pressure results in an increase of the rolling resistance with velocity curve.
3.4 Eigenfrequencies
19
Michelin presented in 2005 their own rolling resistance force model [27], based on the SAE model (3.18). The model serves as the rolling resistance characterisation for two different operating conditions, namely; the inflation pressure pi and the vertical load Fz . The velocity effects are left out by Michelin. The model is described as follows: β Fx,RR = K · pα i (Fz ) ,
(3.20)
where Fx,RR is the rolling resistance force, K is a constant for a given tyre and α, β are regression exponents. According to [27], the values of the regression exponents are approximately: α ≈ −0.4 and β ≈ 0.85 for a passenger car tyre, and for a truck tyre designed for motorway use: α ≈ −0.2 and β ≈ 0.9. Furthermore, the following representation of (3.18) is presented in [27] to determine the rolling resistance force at different operating conditions according to the ISO 8767 measurement protocol [20]: p α F β z i , (3.21) Fx,RR = Fx,RR−ISO · pi,ISO Fz,ISO in which, Fx,RR−ISO is the rolling resistance force (measured) at the nominal inflation pressure and nominal vertical load according to ISO 8767, pi,ISO is the ISO nominal inflation pressure (2.1 bar) and Fz0 is the ISO nominal vertical load condition (80% of the maximum vertical load of the tyre Fz,max ). The terms pi and Fz are the actual inflation pressure and the vertical load.
3.4
Eigenfrequencies
Yam, Guan and Zhang [39] studied the three-dimensional mode shapes obtained under radial and tangential excitation. In their study they used a free suspension, a single-point excitation and multi-point (16) recorder to eliminate the influences and shortcomings of a fixed support. The natural frequency of the free suspension (1 Hz) is much lower than the first natural frequency of a tyre with free rim (around 100 Hz). Therefore, the influence of the free suspension is negligible. The measurements are performed for four inflation pressures and the tyre is excitated with a swept frequency of 80 - 380 Hz. For the responses in radial, tangential and lateral direction due to a radial excitation, the modes in the different directions have nearly the same modal eigenfrequencies and mode shapes according to [39]. The first obtained mode has an elliptical shape. The following modes have a (multiple) leaves shape, except for mode six which shows a translational mode with the rim moving in opposite direction to the tyre movement. With respect to the inflation pressure influences in [39], it is concluded that for all modes, except mode six, the natural frequencies increase with inflation pressure (figure 3.9). The sixth mode reflects the intrinsic characteristics of the tyre structure and is independent of inflation pressure changes. Furthermore, the vibration amplitudes of mode six are about 25% of the amplitudes of the other modes and the modal damping ratio is less than 10% of the other modes [39]. As can be observed in figure 3.9 the first modal eigenfrequencies (mode 1, 2 and 3) are linearly related with the inflation pressure. The higher modes show a non-linear relation. For the modal damping a non linear relation with the inflation pressure is visible. Furthermore, it can be seen that increasing the inflation pressure results in a decrease of the modal damping. Again for the sixth mode no influence can be observed. Because all the investigated modes are flexible modes, only the trend of the first mode (nearest to the rigid ring modes) is interesting for this research project. As mentioned before, the MF-Swift tyre model can only handle the modes where the tyre tread band retains its circular shape. This means that no flexible modes can be analysed. From [39] it can be concluded that there probably will be a linear trend in the rigid eigenmodes.
20
Literature Review
mode 1 mode 2 mode 3 mode 4 mode 5 mode 6 mode 7 mode 8 mode 9
Modal frequency [Hz]
300
250
200
4 3.5 3
Damping [%]
350
2.5 2 1.5
150 1 100 0.5 50 1.5
2
2.5
3
Inflat. pressure [bar]
0 1.5
2
2.5
3
Inflat. pressure [bar]
Figure 3.9: Influence inflation pressure on modal frequency (left) and modal damping (right) [39].
3.5
Summarising this chapter
In previous chapter the MF-Swift tyre model and the modelled tyre characteristics are discussed in more detail. In this chapter the results are presented of a literature survey on the influence of the inflation pressure on the tyre characteristics that are modelled. The literature survey shows that all the investigated tyre characteristics are inflation pressure dependent. Furthermore, it is shown that the inflation pressure influence on the steady-state slip characteristics (Fx , Fy and Mz ) is already implemented in the MF-Swift tyre model. Although the results of the aircraft tyres give a good impression of the influence of the inflation pressure on different tyre characteristics, still it is reasonable to assume that the trends of passenger car tyres are different due to construction and application area differences. The data available in literature appears to be insufficient to describe the inflation pressure influence of all the tyre characteristics that are modelled, the following characteristics need more investigation: • characteristic tyre stiffnesses: – vertical stiffness, – lateral stiffness, – longitudinal stiffness, – torsional stiffness. • sideslip relaxation behaviour; • rolling resistance; • camber effects; • eigenfrequencies of the tyre belt. These characteristics are investigated with an extensive measurement program with a common production passenger car tyre. The results of this measurement program are presented in the next chapter.
Chapter 4
Inflation Pressure Sensitivity Experiments In the previous chapter an overview of the published knowledge on the influence of inflation pressure is given. It is shown that the inflation pressure has significant influence on several (dynamic) tyre characteristics modelled in the MF-Swift tyre model. In order to get a better understanding of the influence of the inflation pressure on the different tyre characteristics, an elaborate measurement program has been conducted. The results of the measurements are presented in this chapter. In the following chapter these results are used to derive inflation pressure dependent relations to improve the MF-Swift tyre model. In order to be as consistent as possible the measurements within this thesis have been performed with one single type of tyre; a 225/50 R17 normal production passenger car tyre. The experiments are conducted with a relatively new tyre with no wear. TNO Automotive has executed recently an extensive measurement program with this tyre on the new TNO Tyre Test Trailer in order to characterise the steady-state force and moment characteristics. These results are also used within this research project. In addition to the high velocity measurements of the TNO Tyre Test Trailer a measurement program for very low velocity has been set up and conducted on the TU/e Flatplank Tyre Tester. The Flatplank Tyre Tester has been chosen as it operates at very low velocities in comparison to the TNO Tyre Test Trailer. Furthermore the forward velocity and the rotational velocity are measured with a higher accuracy compared to the TNO Tyre Test Trailer. To determine to what extent the inflation pressure has influence on the different tyre characteristics an expression is used to classify the difference between a certain inflation pressure and the nominal inflation pressure. The following expression for the influence level is used: v u N uP u (Apress,i − Apressnom,i )2 u i=1 εA = 100u . (4.1) u N P t (Apressnom,i )2 i=1
In this equation A is the tyre characteristic quantity (e.g. lateral stiffness) and N is the number of measurement points. The subscripts press and pressnom are used to denote the measured inflation pressure and the nominal inflation pressure respectively. The nominal inflation pressure for the tyre is 2.5 bar (cold tyre). The inflation pressure influence level expression (4.1) is a measure for the influence of the inflation pressure on the specific tyre characteristic. A large influence expression implies a large influence of the inflation pressure, a small influence expression implies little influence of the inflation pressure. In general the experiments are performed at three different vertical load conditions (FzW = 3, 5, 7 kN) and five different inflation pressures (pi = 1.9, 2.2, 2.5, 2.7, 3.0 bar). These specific inflation pressures 21
22
Inflation Pressure Sensitivity Experiments
are chosen because earlier measurements performed on the TNO Tyre Test Trailer were carried out with these inflation pressures. The inflation pressure influence levels presented in this chapter are the mean values for all the load cases for an inflation pressure change of 0.5 bar. The test tyre is marked at three (start)positions on its circumference, see figure4.1. Each measurement is performed at all start positions of the tyre to reduce the influence of tyre nonuniformities. The Flatplank is marked so that the measurements are always started at the same point. Furthermore, a digital trigger unit is present at this point, which is used to start the sampling of all measurement channels when the Flatplank is operated. More information on the Flatplank Tyre Tester is presented in Appendix A.
Figure 4.1: Example start positions tyre and Flatplank.
4.1
Stiffness measurements
In the previous chapter it is shown that the literature does not provide an unambiguous answer to the influence of the inflation pressure on the stiffnesses of a passenger car tyre. In order to get a better understanding of the inflation pressure influence, a number of experiments are performed on the Flatplank Tyre Tester. The following stiffnesses are investigated: • tyre vertical stiffness CF z ; • tyre lateral stiffness CF y ; • tyre longitudinal stiffness CF x ; • tyre torsional stiffness CM z .
4.1.1
Tyre vertical stiffness
In order to determine the tyre vertical stiffness a measurement procedure has been set up. The tyre is pressed vertically against the Flatplank till a vertical deflection is reached that corresponds to a vertical load of 9 kN at the tested inflation pressure condition. The experiment is performed for five inflation pressure conditions (pi = 1.9, 2.2, 2.5, 2.7, 3.0 bar). For each inflation pressure, the measurement is repeated three times to reduce the influence of tyre nonuniformities. The main steps in the measurement procedure are: 1. inflate the tyre to the prescribed pressure; 2. rotate the tyre to a start position and place it perpendicular to the start position on the flat plank;
4.1 Stiffness measurements
23
3. start moving the Flatplank; 4. apply maximum vertical load, with constant velocity, till the desired vertical deflection is reached; 5. decrease vertical load, with the same constant velocity, till zero vertical load remains. To obtain a trend in the vertical stiffness, a second order polynomial is fitted through each separate measurement. Finally, the coefficients of the final polynomial are determined by interpolating the coefficients of the three separated polynomials. A graphical representation of the vertical load and the vertical deflection is presented in figure 4.2. 10000 9000 8000
z
Vertical load, F [N]
7000 6000 5000 4000 3000 p19 p22 p25 p27 p30
2000 1000 0 0
5
10
15 20 25 Vertical deflection, ρz [mm]
30
35
Figure 4.2: Vertical load vs. vertical deflection.
With the measured vertical load in the contact patch centre FzW and the vertical deflection ρz the vertical stiffness is obtained with: ∂FzW CF z = . (4.2) ∂ρz To give a graphical representation of the vertical stiffness as function of the inflation pressure, the vertical stiffness is determined for the three general vertical load conditions (FzW = 3, 5 and 7 kN), see table 4.1. In figure 4.3 the results of table 4.1 are graphically presented, and a trend line is added to visualise the relation between the inflation pressure and the vertical stiffness. It can be seen that there is a linear relation between the inflation pressure and the vertical stiffness, the vertical stiffness increases with increasing inflation pressure. The measurement results are in accordance with the findings in [32]; it is concluded that a linear relation between the inflation pressure and the vertical stiffness exists, see subsection 3.1.4 and relation(3.9).
Table 4.1: Tyre vertical stiffness for the different vertical load conditions.
Vertical stiffness CF z [N/m] Inf l. P ressure [bar] FzW = 3 kN FzW = 5 kN 1.9 198400 210700 2.2 217500 232100 2.5 238800 251700 2.7 257000 270400 3.0 271200 285000
FzW = 7 kN 222300 245900 264100 283100 298300
An inflation pressure change of 0.5 bar will lead to a change of approximately 13 percent in vertical stiffness value.
24
Inflation Pressure Sensitivity Experiments 5
3.2
x 10
2.8
Vertical stiffness, C
Fz
[N/m]
3
2.6
2.4
2.2 Fz=3000N Fz=5000N Fz=7000N Trend line
2
1.8 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 4.3: Vertical stiffness vs. inflation pressure.
4.1.2
Tyre longitudinal stiffness
To determine the tyre longitudinal stiffness CF x , the rotation of the test tyre is "locked-up", meaning no rotation is possible around the y-axis. The tyre is pressed vertically against the road surface with 0 degree sideslip angle. The experiment is performed at three different vertical load conditions (FzW = 2, 4, 6 kN) and for five inflation pressure conditions. The vertical load conditions differ from the general load conditions to protect the measurement hub against overloading. The main steps in the measurement procedure are: 1. inflate the tyre to the prescribed pressure; 2. rotate the tyre to a start position, place it perpendicular to the start position on the Flatplank and lock up the measuring hub; 3. apply prescribed vertical load (fix the axle height); 4. start moving the Flatplank. The longitudinal stiffness is determined from the linear part of the relation between the longitudinal force and longitudinal displacement. In figure 4.4 the results are presented for the nominal vertical load (4 kN). In the beginning of the experiments some starting phenomena can arise; the linear part is selected in a way that these starting phenomena are not taken into account. For determining the longitudinal stiffness, a first order polynomial fit is used to characterise the linear part, see figure 4.4 (right). For the results of the remaining vertical load conditions see Appendix B. In general, the longitudinal stiffness is defined as the derivative of the longitudinal force versus the longitudinal tyre deflection dx . Here, the longitudinal stiffness CF x represents the slope of the linear fit function FxW (dx ): ∂FxW CF x = . (4.3) ∂dx dx =0
4.1 Stiffness measurements
25
10000
18000
9000
16000 14000 Longitudinal force, Fx [N]
Longitudinal force, Fx [N]
8000 7000 6000 5000 4000 3000 p19 p22 p25 p27 p30
2000 1000 0 0
0.01
0.02 0.03 0.04 0.05 Longitudinal displacement [m]
0.06
0.07
12000 10000 8000 6000 p19 p22 p25 p27 p30
4000 2000 0 0
0.01
0.02 0.03 0.04 0.05 Rel. longitudinal displacement [m]
0.06
0.07
Figure 4.4: Experimental results at nominal vertical load (left) and linear fits of the selected linear part (right).
Figure 4.4 shows that the slope of the fit increases with increasing inflation pressure. In figure 4.5 the longitudinal stiffness versus the inflation pressure is depicted for the different vertical load conditions. It can be seen that a higher inflation pressure results in a higher longitudinal stiffness and that according to the trend lines a linear relation between the inflation pressure and the longitudinal stiffness exists. This linear relation is also found in the literature of aircraft tyres, see subsection 3.2.2 and relation (3.10). 5
2.5
x 10
Longitudinal stiffness, CFx [N/m]
2.4
Fz=2000N Fz=4000N Fz=6000N Trend line
2.3 2.2 2.1 2 1.9 1.8 1.7 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 4.5: Tyre longitudinal stiffness vs. inflation pressure.
An inflation pressure change of 0.5 bar will lead to a change of approximately 3.5 percent in longitudinal stiffness value. Again, The inflation pressure influence level expression is the mean influence level of all the load cases.
26
Inflation Pressure Sensitivity Experiments
Table 4.2: Tyre longitudinal stiffness for the different vertical load conditions.
Longitudinal stiffness CF x [N/m] Inf l. P ressure [bar] FzW = 2 kN FzW = 4 kN FzW 1.9 173500 214300 2.2 179000 218200 2.5 183400 226600 2.7 188900 230800 3.0 190500 233500
4.1.3
= 6 kN 224800 229700 236200 238600 244300
Tyre lateral stiffness
8000
8000
7000
7000
6000
6000 Lateral force, Fy [N]
Lateral force, Fy [N]
To determine the tyre lateral stiffness, the measuring hub is locked up and the tyre is placed perpendicular to the Flatplank, meaning that the sideslip angle α is 90 degrees. The tyre is pressed vertically against the Flatplank, so no camber angle is applied. The experiment is performed for the general vertical load and inflation pressure conditions as defined at the beginning of this chapter. The steps in the measurement procedure are mainly the same as the steps in the longitudinal stiffness procedure.
5000 4000 3000 2000
p19 p22 p25 p27 p30
1000 0 0
0.01
0.02
0.03 0.04 0.05 Lateral displacement [m]
0.06
0.07
5000 4000 3000 2000
p19 p22 p25 p27 p30
1000 0 0
0.01
0.02 0.03 0.04 0.05 Rel. lateral displacement [m]
0.06
0.07
Figure 4.6: Tyre lateral stiffness, experimental results at nominal vertical load (left) and linear fits of the selected linear part (right).
In figure 4.6 (left) the results of the measurements are presented for the nominal vertical load condition (FzW =5 kN), the results of the remaining vertical load conditions are presented in Appendix B. It can be seen that at several inflation pressures, when the steady state value of the lateral force is reached, fluctuations occur. These fluctuations have no effect on the determination of the lateral stiffness and are the result of stick-slip situations that arose during several measurements. For the determination of the lateral stiffness the linear part of the graphs is used leaving out starting phenomena. A linear polynomial fit is used to characterise the linear part and eliminate possible small fluctuations in the measurement data, see figure 4.6 (right). The lateral stiffness is defined as the derivative of the lateral force versus the lateral tyre deflection dy . This means that the tyre lateral stiffness CF y is represented by the slope of the linear fit function FyW (dy ); ∂FyW . (4.4) CF y = ∂dy dy =0
4.1 Stiffness measurements
27
Figure 4.6 (right) shows that the slope of the linear fit increases when increasing the inflation pressure.
5
1.65
x 10
1.6
Lateral stiffness, CFy [N/m]
1.55
Fz=3000N Fz=5000N Fz=7000N Trend line
1.5 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 4.7: Lateral stiffness vs. inflation pressure.
Table 4.3: Tyre lateral stiffness for the different vertical load conditions.
Lateral stiffness CF y [N/m] Inf l. P ressure [bar] FzW = 3 kN FzW = 5 kN 1.9 116900 124900 2.2 123400 136000 2.5 127200 143200 2.7 132600 148800 3.0 136700 156900
FzW = 7 kN 129200 139100 147400 152800 166000
The relation between the inflation pressure and the lateral stiffness is visualised in figure 4.7. As observed for the longitudinal stiffness, the lateral stiffness and the inflation pressure show a linear relation and increasing the inflation pressure leads to an increase in lateral stiffness. This corresponds with the empirical relation for the lateral stiffness found in [33]. Table 4.3 presents the values of the lateral stiffnesses for the different vertical load conditions. An inflation pressure change of 0.5 bar will lead to a change of approximately 9.5 percent in lateral stiffness value.
4.1.4
Tyre torsional stiffness
To determine the tyre torsional stiffness, parking measurements are executed on the Flatplank. The parking measurements are performed by turning the non-rolling wheel (Vx =0, plank is not moved) around the vertical axis of the tyre from 0 via 20 to -20 and back to 20 degrees. The linear part of the parking measurements can be used to determine the torsional stiffness of the tyre. The torsional stiffness CM z is the stiffness that can be derived from the derivative of the aligning moment MzW versus the steer angle ψ: ∂MzW CM z = . (4.5) ∂ψ ψ=0 The parking experiments are performed for the five defined inflation pressures, but now for a wide range of vertical load conditions; FzW =1, 2, 3, 4, 5, 6 kN.
28
Inflation Pressure Sensitivity Experiments
The development of the aligning moment Mz due to the steering motion is shown in figure 4.8, depicted are the vertical load conditions FzW =2 kN and FzW =4 kN. For the remaining vertical load conditions, see Appendix B. The graphs are plotted at different y-axis scaling for readability and visibility of the differences between the separate inflation pressures. The slope of the linear part is representative for the size of the torsional stiffness. 60
200 p19 p22 p25 p27 p30
100
Aligning moment, Mz [Nm]
Aligning moment, Mz [Nm]
40
p19 p22 p25 p27 p30
150
20
0
−20
50 0 −50 −100
−40 −150 −60 −25
−20
−15
−10
−5 0 5 Steer angle, ψ [deg]
10
15
20
−200 −25
25
−20
−15
−10
−5 0 5 Steer angle, ψ [deg]
10
15
20
25
Figure 4.8: Parking behaviour, experimental results aligning moment at FzW =2 kN (left) and FzW =4 kN (right).
In figure 4.9 the determined torsional stiffness versus the inflation pressure is presented. It can be seen that a more or less linear relation between the inflation pressure and the aligning moment exists. At a vertical load of 1 kN the linear relation has a slightly increasing character. At increasing vertical load the character becomes more and more decreasing with increasing inflation pressure. Table 4.4 shows the numerical values of the torsional stiffness for the different vertical load and inflation pressure conditions. 10000
Fz=1 kN Fz=2 kN Fz=3 kN Fz=4 kN Fz=5 kN Fz=6 kN Trend line
Torsional stiffness, CMz [Nm/rad]
9000 8000 7000 6000 5000 4000 3000 2000 1000 0 1.8
2
2.2 2.4 2.6 Inflation pressure, p [bar]
2.8
3
i
Figure 4.9: Torsional stiffness vs. inflation pressure.
In table 4.4 the influence of the inflation pressure on the torsional stiffness is presented. An inflation pressure change of 0.5 bar results in a change of approximately 12 percent. The accuracy of these values can be questioned. It is expected that the torsional stiffness will decrease with increasing inflation pressure, but for some vertical load conditions the measurement points do not always follow this decreasing trend. By combining the trends predicted in aircraft tyre literature and results obtained
4.2 Lateral relaxation behaviour
29
Table 4.4: Torsional stiffness for the different vertical load conditions
Torsional stiffness CM z [Nm/rad] Inf l. P ress [bar] 1 kN 2 kN 3 kN 4 kN 5 kN 1.9 960 2320 4110 5160 8080 2.2 950 2810 3960 5690 7200 2.5 830 2120 3420 4780 6320 2.7 880 2360 3690 4960 6750 3.0 1180 2030 3300 4930 6970
6 kN 9240 8870 8170 7990 7780
from the Flatplank experiments it is possible to make a verdict about the relation between the torsional stiffness and the inflation pressure in the next chapter.
4.2
Lateral relaxation behaviour
In a physical model, the tyre lateral stiffness CF y and the cornering stiffness CF α of a tyre determine for the lateral relaxation length σy , see (4.6). When performing a first order dynamic system identification, the relaxation length is defined as the distance travelled where 63% of the steady state lateral force value is reached. CF α . (4.6) σy = − CF y In order to determine the lateral relaxation length, measurements are performed with a constant axle height and with 0 and 1 degree sideslip angle. The experiments are performed for the general vertical load and inflation pressure conditions. The main steps of the measurement procedure are: 1. inflate the tyre to the prescribed pressure; 2. rotate the tyre to a start position and place it perpendicular to the start position on the Flatplank; 3. apply the desired sideslip angle first and after that the vertical load (fix the axle height); 4. start moving the Flatplank. The measurements with 0 degree sideslip angle are used as reference measurements to correct for plysteer and conicity influences of the tyre. Furthermore, the forces and moments due to non-uniformities in the tyre structure can be monitored. By subtracting the reference results from the 1 degree sideslip angle results, the tyre (structure) phenomena as mentioned above can be eliminated and the pure relaxation behaviour becomes visible. In figure 4.10 (left) the Flatplank results of the nominal vertical load condition are depicted. The lateral relaxation length from the measurement results is determined using the dynamic system identification method (i.e. 63% of the steady state lateral force). In figure 4.10 (right) the resulting lateral relaxation lengths as function of the inflation pressure for three vertical load conditions are graphically presented. It can be seen that the lateral relaxation length decreases with increasing inflation pressure. The relation of the inflation pressure and the lateral relaxation length can be described with a linear relation. This corresponds rather well with the lateral relaxation length relations (3.17) as described in [33]. Table 4.5 shows the determined relaxation lengths for the different vertical load conditions. It is shown that a change of 0.5 bar in inflation pressure leads to a change of approximately 12 percent in lateral relaxation length.
30
Inflation Pressure Sensitivity Experiments
0
1 p19 p22 p25 p27 p30
Lateral relaxation length, σy [m]
Lateral force, Fy [N]
−500
Fz=3000N Fz=5000N Fz=7000N Trend line
0.9
−1000
−1500
0.8
0.7
0.6
0.5
−2000 0.4
−2500 0
0.5
1 1.5 Track displacment, dx [m]
2
2.5
1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 4.10: Lateral relaxation behaviour, response lateral force to one degree sideslip angle at FzW =5 kN (left) and determined lateral relaxation length (right) for various vertical loads and inflation pressures.
Table 4.5: Lateral relaxation length for the different vertical load conditions.
Lateral relaxation length σy [m] Inf l. P ressure [bar] FzW = 3 kN FzW = 5 kN 1.9 0.51 0.78 2.2 0.44 0.71 2.5 0.43 0.69 2.7 0.38 0.61 3.0 0.36 0.58
4.2.1
FzW = 7 kN 0.89 0.83 0.82 0.77 0.74
Effective rolling radius
An important input parameter of the Magic Formula tyre model is the longitudinal slip. For a large part, the determination of the longitudinal slip κ is governed by the effective rolling radius Re , see (4.7) and figure 4.11. Vx − Re Ω κ=− . (4.7) Vx According to 4.8, the effective rolling radius determines the ratio between the forward velocity Vx and the rotational speed Ω for a freely rolling tyre. So, the effective rolling radius is important for an accurate representation of the rotational speed. Re =
Vx . Ω
(4.8)
For automotive tyres it is shown, in [28], that the effective rolling radius depends on the vertical load FzW . At low vertical load conditions, small increases in vertical load condition result in a decrease of the effective rolling radius. The vertical load will deform the tyre rubber and the circumference of the tyre, so Re , will decrease. As the vertical load further increases the tyre will be compressed, but the high circumferential stiffness of the steel carcass will make that the resulting effect on the effective rolling radius is getting smaller. The tyre still deforms at increasing vertical load, but the circumference, and thus Re , remains almost constant.
4.2 Lateral relaxation behaviour
31
Figure 4.11: Effective rolling radius and longitudinal slip [13].
To investigate the influence of the inflation pressure on the effective rolling radius, the reference measurements of the lateral relaxation length experiments are used. These measurements represent a free rolling tyre, so without longitudinal or lateral slip. The results are shown in figure 4.12. For all the vertical load conditions it can be seen that there is very little influence of the inflation pressure on the effective rolling radius. When increasing the inflation pressure with 1 bar this results in an increase of approximately only 2 mm of the effective rolling radius, which is less than 1 percent. An increase of 0.5 bar inflation pressure leads to an increase of approximately 0.25 percent in effective rolling radius.
0.33 Fz=3000[N] Fz=5000[N] Fz=7000[N] Trend line
0.328
e
Effective rolling radius, R [m]
0.326 0.324 0.322 0.32 0.318 0.316 0.314 0.312 0.31 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 4.12: Effective rolling radius vs. inflation pressure.
32
Inflation Pressure Sensitivity Experiments
Table 4.6: Effective rolling radius for different vertical load and inflation pressure conditions.
Effective rolling radius Re [m] Inf l. P ressure [bar] FzW = 3 kN FzW = 5 kN 1.9 0.3218 0.3205 2.2 0.3217 0.3203 2.5 0.3229 0.3220 2.7 0.3227 0.3210 3.0 0.3238 0.3225
4.3
FzW = 7 kN 0.3205 0.3199 0.3213 0.3205 0.3219
Camber thrust and camber torque
When a camber angle is applied to a free rolling tyre, a lateral force Fyγ (camber thrust), a moment around the vertical axis Mzγ (aligning camber torque) and a moment around the longitudinal axis Mxγ (overturning camber torque) arise, see figure 4.13.
Figure 4.13: Schematic representation of camber thrust and camber torque, rear view [24].
In earlier research on a flat-surface tyre-testing facility [24], it is shown that the contact length is strongly depending on the amount of camber angle, while the contact patch area stays almost unaffected. The camber thrust and the camber torque are depending on the vertical load condition. Increasing the vertical load results in an increase of the camber thrust and aligning camber torque, see figure 4.14. To determine to what extend the inflation pressure has influence on the camber thrust and aligning camber torque, measurements are performed with a free rolling tyre for a wide range of camber angles (γ=-15...15 degrees) and for the general vertical load conditions. To eliminate ply-steer and conicity effects, measurements are also performed with zero camber angle and subtracted from the camber measurements. The main steps of the measurement procedure are: 1. inflate the tyre to the prescribed pressure; 2. rotate the tyre to a start position and place it perpendicular to the start position on the Flatplank; 3. rotate the Flatplank to the desired camber angle; 4. apply vertical load, fix the axle height and start moving the Flatplank.
4.3 Camber thrust and camber torque
33
Figure 4.14: Camber thrust Fyγ and aligning camber torque Mzγ for different vertical load conditions [24].
In figure 4.15 the camber thrust and aligning camber torque versus camber angle for the nominal vertical load condition (FzW =5 kN) are depicted. Because the tested tyre has a symmetrical tread, the amount of camber thrust and aligning camber torque for positive and negative camber angles are expected to be equal. However, figure 4.15 shows that a difference occurs when cambering with a positive or a negative angle. For the camber thrust at nominal inflation pressure and nominal vertical load the difference between +15 degrees camber angle and -15 degrees camber angle is approximately 150 N or 17 percent, for the aligning camber torque this is approximately 5 Nm or 12 percent. When normalising the camber thrust and aligning camber torque with the vertical load the differences still exist, which excludes the influence of the vertical load. Further, conicity and ply-steer effects are eliminated using the reference measurements. The difference can be caused by a deviation or a wrong calibration of the measuring hub or an asymmetry in the tyre carcass/construction. This is not further checked in this research. 1500
60 p19 p22 p25 p27 p30
40 Camber torque, Mzγ [Nm]
Camber thrust, Fyγ [N]
1000
500
0
−500
−1000
−1500 −15
p19 p22 p25 p27 p30
20
0
−20
−40
−10
−5
0 5 Camber angle, γ [deg]
10
15
−60 −15
−10
−5
0 5 Camber angle, γ [deg]
10
15
Figure 4.15: Camber thrust and aligning camber torque for the camber angle range: [-15 15] degrees.
Figure 4.16 and 4.17 show the camber thrust and aligning camber torque respectively as a function of the inflation pressure for a serie of positive and negative camber angles at the nominal vertical load condition. The trend lines show that a linear relation is visible between the inflation pressure and the camber thrust / aligning camber torque. Again, it can be seen that the behaviour for a positive and negative camber angle is not exactly the same. In figure 4.16, small negative camber angles (till -2.5 degrees) show an opposite trend, caused by the measurement results at high inflation pressures (2.7 and 3.0 bar).
34
Inflation Pressure Sensitivity Experiments
The slope of the linear relations is depending on the size of the camber angle. For small camber angles (for instance: 2 degrees) the slope is significantly smaller than the slope for larger camber angles (for instance: 15 degrees). This indicates that the camber thrust stiffness is decreasing with increasing inflation pressure. The same holds for the aligning camber torque stiffness. 1500
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
500
Camber thrust, F
yγ
[N]
1000
0
−500
−1000
−1500 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
Figure 4.16: Camber thrust vs. inflation pressure for various positive and negative camber angles for the nominal vertical load condition.
60
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
20
zγ
Camber torque, M [Nm]
40
0
−20
−40
−60 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
Figure 4.17: Aligning camber torque vs. inflation pressure for various positive and negative camber angles for the nominal vertical load condition.
Furthermore, when analysing the measurement results it appears that changing the inflation pressure of a cambered free rolling tyre has also influence on the overturning moment in the contact patch. It is observed that when increasing the inflation pressure also the amount of overturning camber torque increases (here: further called the overturning camber torque Mxγ ), see figure 4.18. Again, the size of overturning camber torque for positive and negative camber angles is expected to be equal. However, figure 4.18 shows that a difference occurs when cambering with a positive or a negative angle. No influence of the vertical load has been found and conicity and ply-steer effects are eliminated using the reference measurements.
4.4 Eigenfrequencies
35 400
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
300
Camber torque, Mxγ [Nm]
200 100 0 −100 −200 −300 −400 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
Figure 4.18: Overturning camber torque vs. inflation pressure for various positive and negative camber angles for the nominal vertical load condition.
To quantify the influence of the inflation pressure, table 4.7 shows the influence on the camber thrust, aligning camber torque and overturning camber torque. The influence is subdivided in two camber ranges. The first range is limited to -5 and 5 degrees; this is the common range in which passenger car tyres are cambered. To quantify the influence also for the whole measurement range, the second range is limited to -15 to 15 degrees.
Table 4.7: Influence of the inflation pressure on camber effects for the nominal vertical load condition.
Influence inflation Camber range [deg] -5...5 -15...15
pressure εA [%] Fyγ Mzγ Mxγ 12.5 8.0 7.0 15.0 15.0 7.0
When changing the inflation pressure with 0.5 bar the camber thrust increases/decreases approximately with 13 percent of the thrust at nominal inflation pressure (2.5 bar). For the aligning camber torque, 0.5 bar inflation pressure increase leads to a change of approximately 11.5 percent and for the overturning camber torque to a change of approximately 7 percent. Note that these values are determined at, and only hold for, the nominal vertical load condition. In Appendix B the results for the remaining vertical load conditions are presented. When comparing the results in Appendix B with the results for the nominal vertical load condition it is observed that the progressiveness of the linear trend for the camber thrust is decreasing when the vertical load increases, especially in the range [-5 5] degrees. In other words, the influence of the inflation pressure decreases with increasing vertical load. This observation can also be seen in the aligning and overturning camber torque, but to a smaller extent.
4.4
Eigenfrequencies
This section considers the influence of the inflation pressure on the tyre eigenfrequencies. At the start of this project, no measurement data concerning the eigen modes of the measurement tyre was available. Performing dynamic respons experiments is not an option due to costs and the availability of a test facility. Instead, Finite Element Method (FEM) virtual experiments with a tyre model in ABAQUS are executed.
36
Inflation Pressure Sensitivity Experiments
Furthermore to support the FEM analysis, the flexible ring model and the rigid ring model are analysed analytically.
4.4.1
FEM analysis
In the FEM software ABAQUS, a three dimensional model of a passenger car tyre is available. The 3D model is a 175 SR14 passenger car tyre. For more information about the used FEM tyre model it is referred to Appendix C. With the FEM model two situations are investigated: • a free hanging tyre with a fixed rim; • a non-rolling loaded tyre (load: Fz =4 kN) with a fixed rim and fixed contact patch nodes. The experiments are performed for six inflation pressures (pi = 1.5, 2.0, 2.2, 2.5, 2.7, 3.0 bar). The eigenfrequencies are determined with the LAN CZOS method [1]: − ω 2 M M N + K M N φN = 0, (4.9) where, ω is the free vibration frequency, M is the mass matrix (symmetric and positive definite) and K is the stiffness matrix. Furthermore, φ is the eigenvector (mode shape) and superscripts M and N are the degrees of freedom. Because the MF-Swift tyre model consists of a rigid ring model, only the primary tyre modes (rigid belt/ring modes) are evaluated with the FEM experiments. Primary tyre modes are the first eigenmodes of the tyre in which the tyre tread band almost retains its circular shape. These modes can be described with the rigid ring model. Therefore these modes may also be called rigid ring modes. The primary modes that are observed during the FEM experiments are: rotational, lateral, yaw, camber, vertical and longitudinal mode. See Appendix C for a visualisation of the different primary modes for both situations. In figure 4.19 the primary modes as function of the inflation pressure for both situations are depicted. In the free hanging situation (left plot), clear linear trends are visible between the inflation pressure and the eigenfrequencies. Note that, the yaw and camber modes and the vertical and longitudinal modes of the free hanging tyre occur at the same eigenfrequencies. The non-rolling loaded tyre situation is depicted in the right plot of figure 4.19. Again linear trends can be observed. At high inflation pressures (pi =2.7 and 3.0 bar) some deviations occur. This is probably caused by variation in the number of nodes that are in contact with the road. 85 80
70 65 60 55 50
rotational lateral yaw/camber vertical longitudinal
85 80 Natural frequency [Hz]
Natural frequency [Hz]
75
75 70 65 60 55 50
45 40 1.5
90
rotational lateral yaw/ camber vertical/ longitudinal
45 2 2.5 Inflation presssure, pi [bar]
3
40 1.5
2 2.5 Inflation presssure, pi [bar]
3
Figure 4.19: Primary eigenmodes free hanging tyre (left) and non-rolling loaded tyre (right).
4.4 Eigenfrequencies
37
Furthermore, it can be seen that the influence is approximately the same for each primary mode, with the exception of the rotational mode. For the rotational mode of a free hanging tyre, the amount of inflation pressure influence is a bit smaller compared to the other modes, see table 4.8. In table 4.8, the influence is presented for the different primary modes. In general, an increase of 0.5 bar leads to an increase of approximately 5 to 6 percent for both situations. The rotational mode of the free tyre shows little influence, because this mode mainly depends on the (torsional) carcass stiffness. Table 4.8: The influence of the inflation pressure on the primary eigenmodes.
Influence level, increasing pi with 0.5 bar Abaqus P rimary eigen mode εA,0.5bar [%] rotational 2.4 Free tyre lateral 7.1 yaw, camber 6.8 vertical, longitudinal 6.5 rotational 5.0 Loaded tyre lateral 6.4 yaw/camber 5.0 vertical 6.1 longitudinal 5.9
4.4.2
Analytical results
The analytical results of the flexible and rigid ring model are based on the findings of Zegelaar [40] and Gong [16]. In Appendix C.2, the method to determine the primary eigenmodes as a function of the inflation pressure of the rigid ring model using the flexible ring model is discussed in further detail. Only the main results are presented in this section. As set-up a free hanging tyre is analysed. When looking at the plots in figure 4.20 it can be concluded that the rigid ring model shows the same trend as is obtained in the FEM results. Both models show a nearly linear relation between the inflation pressure and the eigenfrequency. 70 ftrans
Natural frequency [Hz]
65
frot
60
55
50
45
40
35 1.5
2 2.5 Inflation pressure, pi [bar]
3
Figure 4.20: Eigenfrequency versus inflation pressure for rigid ring model of a free hanging tyre.
38
Inflation Pressure Sensitivity Experiments
In table 4.9 an overview is given of the influence of the inflation pressure for the rigid ring model. These results correspond reasonably well with the influence levels of table 4.8. It is concluded that the eigenfrequencies of the primary modes are linearly depending on the inflation pressure. Table 4.9: The influence of the inflation pressure on the determined eigenfrequencies.
Influence level, increasing pi with 0.5 bar Eigenmode εA,0.5bar [%] vertical, longitudinal 8.5 rotational 4.3
4.5
Summarising this chapter
In this chapter the measurement results are presented of an elaborate measurement program performed on the TU/e Flatplank Tyre Tester. This measurement program is conducted to get a better understanding of the influence of the inflation pressure on the different tyre characteristics modelled in the MF-Swift tyre model. It is shown that for all the characteristic stiffnesses, i.e. vertical, longitudinal, lateral and torsional, a linear relation with the inflation pressure exists. The lateral relaxation length and the effective rolling radius also show a linear relation with the inflation pressure. The camber thrust and camber torque exhibit a linear relation with the inflation pressure of which the slope is depending on the vertical load condition. Finally, it is concluded that the eigenfrequencies (primary modes) also show a linear relation with the inflation pressure. The influence of the inflation pressure on the different characteristics is determined. An overview of the influence when changing the inflation pressure with 0.5 bar is given in table 4.10 for increasing order of influence. The quantity with the highest influence level will show the largest improvement when made inflation pressure dependent. Table 4.10: Overview influence of the inflation pressure on the different quantities, when the inflation pressure is changed 0.5 [bar].
Influence inflation pressure Quantity εA,0.5bar [%] Re 0.2 CF x 3.5 fnat (primary modes) 6.0 Mxγ (FzW = 5 kN) 7.0 CF y 9.5 CM z 12.0 σy 12.0 CF z 13.0 Mzγ (FzW = 5 kN) 15.0 Fyγ (FzW = 5 kN) 15.0
Chapter 5
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model In this chapter relations will be derived to incorporate the inflation pressure influence in the MF-Swift tyre model. The relations are mainly based on the trends found in the previous chapter and supported by trends described in literature. To describe the influence of the inflation pressure in the relations, the inflation pressure will be expressed in the inflation pressure increment: pi − pi0 dpi = , (5.1) pi0 where pi is the current inflation pressure and pi0 is the nominal inflation pressure. If necessary, the relations will be extended with the vertical load increment. Fz − Fz0 dfz = , (5.2) Fz0 where Fz and Fz0 are the current and nominal vertical load respectively. (5.1) and (5.2) are part of the latest MF-Swift version. As mentioned before, the nominal inflation pressure is 2.5 bar and the nominal vertical load condition is 5 kN in this thesis, unless otherwise indicated. To assess the relations for each characteristic, a fit error is determined. The following expression is used to describe the fit error εA : v u N uP u (Arelation − Ameas,i )2 u i=1 εA = 100u (5.3) u N P t 2 (Ameas,i ) i=1
In this equation A is the tyre characteristic quantity (e.g. lateral stiffness) and N is the number of considered measurement points for a certain inflation pressure. The subscripts relation and meas are used to denote the proposed relations and the measurement data respectively. The error expression (5.3) is similar to the approach used in the MF-Swift tyre model. The error is determined per inflation pressure. Furthermore, a total error (the average of all the inflation pressure errors) is determined. In this way the quality of the fit can be determined, e.g. a low fit error equals a qualitative good fit (the difference between the measurements and the fit is small). In TNO’s Magic Formula software, the optimisation of the Magic Formula parameters is done by finding a minimum of a constrained nonlinear multivariable function (f mincon routine from the Matlab Optimization Toolbox). This function is also used as optimising routine for the new pressure dependent relations. 39
40
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
5.1
Tyre vertical stiffness
As discussed in subsection 4.1.1, a clear linear relation between the inflation pressure and the tyre vertical stiffness exists. Furthermore, the vertical deflection shows a quadratic relation with the vertical force. Schmeitz [32] derived a relation for the vertical load, depending on the vertical deflection ρz and the inflation pressure increment dpi : Fz = (1 + qF z3 dpi )(qF z1 ρz + qF z2 ρ2z ),
(5.4)
where the q’s are the Magic Formula optimisation parameters. The relation (5.4) has been implemented in the MF-Swift tyre model in 2005. From this relation, the tyre vertical stiffness can be derived, reading: dFz CF z = = (1 + qF z3 dpi )(qF z1 + 2qF z2 ρz ). (5.5) dρz To assess the proposed relation, the vertical stiffness obtained with (5.5) is compared with the measurement results in figure 5.1. It can be seen that the proposed relation describes the measurements rather well. The total error is approximately 0.84 percent and per inflation pressure the error stays below 2.1 percent. So by deriving the vertical load relation (5.4) the vertical stiffness can be described well and no further enhancements are necessary. Error: 0.84215 %
5
3.6
x 10
3.4
Vertical stiffness, CFz [N/m]
3.2 3
meas equation p19 p22 p25 p27 p30
Error per inflation pressure Inf l. press [bar] Error [%] total 0.84 1.9 0.50 2.2 1.07 2.5 0.25 2.7 2.07 3.0 0.32
2.8 2.6 2.4 2.2 2 1.8 1.6 0
0.01
0.02 0.03 Vertical deflection, ρz [m]
0.04
0.05
Figure 5.1: Tyre vertical stiffness using (5.5) and measurements for 5 inflation pressures and the three general vertical load conditions.
5.2
Tyre torsional stiffness
The torsional stiffness, shows a more or less linear relation with the inflation pressure. Furthermore, it is shown in Appendix D.1 that a linear relation is visible between the torsional stiffness and the vertical load. Therefore two linear relations are proposed to describe the torsional stiffness as a function of the inflation pressure and the vertical load: CM z = CM z,nom (1 + pCM z1 dpi )(1 + pCM z2 dfz ),
(5.6)
where pCM zi are new Magic Formula parameters. When implementing this equation, it turns out that for small vertical load conditions, in this case
5.2 Tyre torsional stiffness
41
Fz < 500 N, the tyre torsional stiffness can become negative as a result of the linear vertical load relation. To solve this problem two solutions are proposed: 1) 2)
Fz CM z = CM z,nom (1 + pCM z1 dpi ) ; Fz0 Fz Fz 2 CM z = CM z,nom pCM z1 + pCM z2 (1 + pCM z3 dpi ). Fz0 Fz0
(5.7) (5.8)
In figure 5.2 the torsional stiffness for (5.7) and (5.8) are compared with the measurement results. 180
140
160 Torsional stiffness, CMz [Nm/deg]
Torsional stiffness, CMz [Nm/deg]
160
180 equation p19 p22 p25 p27 p30
120 100 80 60 40 20 0 0
140
equation p19 p22 p25 p27 p30
120 100 80 60 40 20
1000
2000 3000 4000 Vertical load, Fz [N]
5000
6000
0 0
1000
2000 3000 4000 Vertical load, Fz [N]
5000
6000
Figure 5.2: Tyre torsional stiffness using (5.7); left plot, and (5.8); right plot, compared with Flatplank measurement results.
It can be seen that the results of (5.8) describe the measurement results more accurately compared with (5.7). In table 5.1 an overview is given of the total error and the error per inflation pressure for both proposed equations. Although, (5.8) shows better results, still (5.7) is chosen to be implemented in the MF-Swift model. The main reason is the more complex implementation of (5.8) (two additional parameters) and the linear trend that is described in literature for aircraft tyres (figure 3.6). Furthermore, the results are based on Flatplank measurements. The friction coefficient of the Flatplank seems to be insufficient to perform reliable torsion stiffness experiments. To solve this problem, the experiments should be repeated on the Flatplank on a high friction surface. Table 5.1: Error results for the proposed torsional stiffness relations; (5.7) and (5.8).
Inf l. press [bar] total 1.9 2.2 2.5 2.7 3.0
Error level [%] Equation (5.7) 9.58 10.48 8.47 7.97 8.16 12.80
Equation (5.8) 5.05 5.15 3.50 7.12 3.07 6.46
42
5.3
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
Lateral relaxation behaviour
As mentioned in section 4.2, the tyre lateral stiffness CF y and the cornering stiffness CF α determine the lateral (or sideslip) relaxation length σy . In this section the current strategy for calculating the lateral relaxation length will be discussed and enhancements will be introduced to achieve more accuracy in the determination of the lateral relaxation length and closely related quantities.
5.3.1
Current model implementation
In the current version of the MF-Swift tyre model, the lateral relaxation length is defined as the relation between the cornering stiffness and the total carcass stiffness cy increased with half the contact length a: CF α CF α σy = − =− + a. (5.9) CF y cy The tyre lateral stiffness CF y reads: 1 a 1 = + . CF y cy CF α
(5.10)
As mentioned in the literature survey, in [32] it is observed that the cornering stiffness depends on the inflation pressure. To include the inflation pressure effects in the MF-Swift tyre model, two linear relations to the cornering stiffness equation have been added: Kyα = pKy1 (1 + ppy1 dpi )Fz0 sin 2 arctan
Fz pKy2 (1 + ppy2 dpi )Fz0
,
(5.11)
where the parameters ppyi are the inflation pressure dependent parameters. A linear inflation pressure relation is added to the factor pKy1 , which determines the maximum value of the stiffness (Kyα /Fz0 ), and to the factor pKy2 , which determines the load at which the cornering stiffness reaches its maximum value. Note that (5.11) only shows the enhanced parts of the cornering stiffness. In the MF-Swift model the half contact length is defined as a function of the vertical deflection, using a relation found by Besselink [6] in 2000. In [6] it is concluded that the contact length is not directly depending on the vertical force, but is primarily a function of the vertical tyre deflection and reads: a = pa1 r0
ρz + pa2 r0
r
ρz , r0
(5.12)
where ρz is the vertical tyre deflection, r0 the unloaded tyre radius and pai the fit parameters. The tyre lateral stiffness is assumed to be constant in the current version of the model, i.e. one average stiffness for all the measured vertical load conditions is used: CF y =
N 1 X CF y,i , N i=1
(5.13)
N is the number of vertical load conditions measured. When no additional relaxation length measurements are available, the relaxation lengths can be determined with equation (5.9). If there are extra relaxation length measurements performed, it is possible to optimise the determined relaxation lengths to the measured relaxation lengths by adjusting the tyre lateral stiffness CF y in a way that the error between the determined and measured relaxation lengths is minimised. Adjusting the tyre lateral stiffness automatically results in a change of the total carcass stiffness cy .
5.3 Lateral relaxation behaviour
5.3.2
43
Proposed enhancements
The tyre lateral stiffness CF y is the quantity where the most progress can be made. Like mentioned in the previous section, the tyre lateral stiffness is currently the average stiffness value for all the measured lateral stiffnesses. This is the best approximation of a vertical load range, when the lateral stiffness is assumed to be a constant value Tyre lateral stiffness In chapter 4 it is shown that the tyre lateral stiffness depends on the inflation pressure and the vertical load. A linear relation between the tyre lateral stiffness and the inflation pressure exists. So it is obvious that a linear relation has to be introduced to describe the relation with the inflation pressure. Currently there is also no relation that describes the influence of the vertical load on the tyre lateral stiffness. In figure 5.3 the lateral stiffness as function of the vertical load is depicted. It is observed that no linear relation exists and that due to the limited measurement points available per vertical load condition only a second order polynomial can be proposed.
5
1.65
x 10
1.6
Lateral stiffness, CFy [N/m]
1.55 1.5
p19 p22 p25 p27 p30 Trend line
1.45 1.4 1.35 1.3 1.25 1.2 1.15 3000
3500
4000
4500 5000 5500 Vertical load, Fz [N]
6000
6500
7000
Figure 5.3: Measured tyre lateral stiffness vs. vertical load.
To describe both influences, the following equation is introduced: CF y = CF y,nom (1 + pCF y1 dfz + pCF y2 dfz2 )(1 + pCF y3 dpi ),
(5.14)
where CF y,nom is the tyre lateral stiffness at the nominal inflation pressure and nominal vertical load condition and pCF yi are the model coefficients.
44
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
In figure 5.4 the lateral stiffness from the proposed equation is compared with the measurements to assess the proposed relation. It can be seen that the proposed relation describes the measurements rather well. The total error is approximately 0.82 percent and per inflation pressure the error stays below 2 percent. Error: 0.81764 %
5
1.7
x 10
Lateral stiffness, CFy [N/m]
1.6
1.5
meas equation p19 p22 p25 p27 p30
Error per inflation pressure Inf l. press [bar] Error [%] total 0.82 1.9 0.53 2.2 1.90 2.5 0.01 2.7 0.28 3.0 1.36
1.4
1.3
1.2
1.1 3000
3500
4000
4500 5000 5500 Vertical load, Fz [N]
6000
6500
7000
Figure 5.4: Proposed lateral stiffness relation (5.14) compared with the Flatplank measurement results.
Relaxation length In figure 5.5 the results are depicted when the lateral stiffness parameters of equation (5.14) are optimised so that the measured relaxation lengths are described as good as possible. This strategy results in an optimised total error of 2.11 percent, so the optimised relaxation lengths correspond rather well with the measurements. For the errors of the separate inflation pressure conditions see figure 5.5. Here, the fit errors stay within 3.5 percent. 1
Lateral relaxation length, σy [m]
0.9
0.8
meas equation p19 p22 p25 p27 p30
Error per inflation pressure Inf l. press [bar] Error [%] total 2.11 1.9 2.18 2.2 1.76 2.5 3.48 2.7 1.92 3.0 1.19
0.7
0.6
0.5
0.4
3000
3500
4000
4500 5000 5500 Vertical load, Fz [N]
6000
6500
7000
Figure 5.5: Resulting relaxation lengths when optimising the parameters of the lateral stiffness using (5.14) compared with the measured relaxation lengths.
So when the parameters of (5.14) are optimised to the relaxation length, the relaxation length can be described rather well. A strange trend occurs when the, for the relaxation length optimised, tyre lateral
5.3 Lateral relaxation behaviour
45
stiffness is observed, see figure 5.6. It might be expected that the trend will be more or less the same as the trend in figure 5.4, but figure 5.6 shows that the stiffness reaches a minimum at the nominal vertical load condition. This is physically hard to explain. The problem is caused by a sign change of parameter pF Cy2 , which is the term of the quadratic vertical force influence. When optimising (5.14) for the tyre lateral stiffness this parameter is negative, whereas optimising (5.14) for the relaxation length results in a positive pF Cy2 . When constraining pF Cy2 ≤ 0 during the relaxation length optimisation sequence, the resulting optimised value is 0.
Figure 5.6: Resulting tyre lateral stiffness when optimised for relaxation length data.
Therefore is has been dicided that the best fit is obtained when the vertical force influence on the tyre lateral stiffness is described by a linear relation. The following equation is proposed to describe the tyre lateral stiffness: CF y = CF y,nom (1 + pCF y1 dfz )(1 + pCF y2 dpi ) (5.15) Performing the relaxation length optimising strategy with (5.15), results in a more plausible optimised tyre lateral stiffness trend. The lateral stiffness increases with increasing vertical load, see figure 5.7.
Figure 5.7: Optimising (5.15) for relaxation length; resulting tyre lateral stiffness.
46
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
In figure 5.8 the results of the relaxation lengths are depicted. The total fit error of the relaxation length is approximately 2.93 percent; this is an increase of 0.8 percent compared to optimising with (5.14). The table in figure 5.8, gives an overview of the relaxation length errors per inflation pressure. 1 meas equation p19 p22 p25 p27 p30
Lateral relaxation length, σy [m]
0.9
0.8
Error per inflation pressure Inf l. press [bar] Error [%] total 2.93 1.9 3.37 2.2 3.44 2.5 4.21 2.7 2.14 3.0 1.48
0.7
0.6
0.5
0.4
3000
3500
4000
4500 5000 5500 Vertical load, Fz [N]
6000
6500
7000
Figure 5.8: Optimising (5.15) for relaxation length; relaxation length results and overview of the error per inflation pressure.
Finally, to assess the new proposed relation purely for the tyre lateral stiffness, figure 5.9 shows the results of (5.15) optimised for lateral stiffness measurements compared with the measurement results. It is obvious that (5.15) describes the measured tyre lateral stiffnesses less accurate than (5.14), see figure 5.4. The total error has increased from approximately 0.81 percent to 3.44 percent. However, (5.15) gives the best compromise for the tyre lateral stiffness of a rolling and non-rolling tyre.
Error: 3.4445 %
5
1.8
x 10
Lateral stiffness, CFy [N/m]
1.7
1.6
meas equation p19 p22 p25 p27 p30
Error per inflation pressure Inf l. press [bar] Error [%] total 3.44 1.9 2.69 2.2 1.96 2.5 3.45 2.7 3.88 3.0 5.23
1.5
1.4
1.3
1.2
1.1 3000
3500
4000
4500 5000 5500 Vertical load, Fz [N]
6000
6500
7000
Figure 5.9: Linear tyre lateral stiffness relation (5.15) compared with the Flatplank measurement results.
5.4 Longitudinal relaxation behaviour
5.4
47
Longitudinal relaxation behaviour
Currently, the tyre longitudinal stiffness is assumed to be constant in the tyre model, being the average stiffness for all the measured vertical load conditions: CF x =
N 1 X CF x,i , N i=1
(5.16)
where N is the number of vertical load conditions measured. With regard to inflation pressure and vertical load influences, the tyre longitudinal stiffness and the tyre lateral stiffness show a similar behaviour as already indicated in section 4.1. It is shown that a linear relation with regard to the inflation pressure exists. As can be seen in figure 5.10, a non-linear relation with the vertical load is visible. Due to the limited measurement points (only three points), the proposed relation is restricted to a second order polynomial. 5
2.5
x 10
Longitudinal stiffness, CFx [N/m]
2.4 2.3
p19 p22 p25 p27 p30
2.2 2.1 2 1.9 1.8 1.7 2000
2500
3000
3500 4000 4500 Vertical load, Fz [N]
5000
5500
6000
Figure 5.10: Measured tyre longitudinal stiffness vs. vertical load.
To describe the inflation pressure and vertical load influences on the tyre longitudinal stiffness, the following equation is introduced: CF x = CF x,nom (1 + pCF x1 dfz + pCF x2 dfz2 )(1 + pCF x3 dpi ).
(5.17)
Here, CF x,nom is the tyre longitudinal stiffness at nominal inflation pressure and nominal vertical load condition. pCF xi are the optimisation parameters. To assess the proposed relation, in figure 5.11 the longitudinal stiffness for the proposed equation is compared with the Flatplank measurements. It can be seen that the proposed relation describes the measurements rather well. The total error is approximately 0.54 percent and per inflation pressure the error stays below 1.0 percent.
48
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model Error: 0.54253 %
5
2.5
x 10
Longitudinal stiffness, CFx [N/m]
2.4 2.3 2.2 2.1 2 meas equation p19 p22 p25 p27 p30
1.9 1.8 1.7 2000
2500
3000
3500 4000 4500 Vertical load, Fz [N]
5000
5500
Error per inflation pressure Inf l. press [bar] Error [%] total 0.54 1.9 0.34 2.2 0.75 2.5 0.01 2.7 0.75 3.0 0.88
6000
Figure 5.11: Proposed tyre longitudinal stiffness relation (5.17) compared with the Flatplank measurements.
The longitudinal relaxation length is determined by: σx = −
CF κ CF κ = + a, CF x cx
(5.18)
where CF κ is the longitudinal slip stiffness (according to (3.2)) and cx is the longitudinal carcass stiffness. Because the tyre longitudinal stiffness shows similar behaviour as the tyre lateral stiffness, the same problems are likely to occur for the relaxation length as is described in the previous subsection. Furthermore, when extrapolating (5.17) for higher vertical load conditions, it is possible that the stiffness decreases at a higher vertical load condition. This is caused by the second order polynomial that is optimised on the measurements. To overcome this problem, a linear relation is used to describe the vertical load influence instead of the second order polynomial. Therefore, the following equation is finally proposed to describe the tyre longitudinal stiffness: CF x = CF x,nom (1 + pCF x1 dfz )(1 + pCF x2 dpi ).
(5.19)
To assess the new proposed relation, figure 5.12 shows the results of (5.19) compared with the measurements. Obviously, it can be seen that (5.19) describes the measured tyre longitudinal stiffness less accurate than (5.17), figure 5.4. The total error is increased from approximately 0.54 percent to 6.35 percent. The influence of equation 5.19 on the longitudinal relaxation length is not further evaluated, because no data of the longitudinal relaxation length σx for the different general inflation pressures is available. the longitudinal relaxation length cannot be measured with the Flatplank.
5.5 Rolling resistance Error: 6.3538 %
5
2.7
x 10
Longitudinal stiffness, CFx [N/m]
2.6 2.5 2.4
49
meas equation p19 p22 p25 p27 p30
Error per inflation pressure Inf l. press [bar] Error [%] total 6.35 1.9 4.38 2.2 5.34 2.5 6.35 2.7 6.93 3.0 8.77
2.3 2.2 2.1 2 1.9 1.8 1.7 2000
2500
3000
3500 4000 4500 Vertical load, Fz [N]
5000
5500
6000
Figure 5.12: Linear tyre longitudinal stiffness relation (5.19) compared with the Flatplank measurements.
5.5
Rolling resistance
Although a lower inflation pressure results in less compression of the tread blocks in the contact patch, it results in an increase in tread bending and shearing. This results in an increase in rolling resistance. The rolling resistance increases rapidly as the inflation pressure decreases. In the MF-Swift tyre model currently the rolling resistance force Fx,RR is determined by the following expression: 4 Vx My Fz 2 Fx 2 +qsy4 Vx Fx,RR = = −Fz λM y qsy1 +qsy2 +qsy3 +q γ +q γ . (5.20) sy5 sy6 r0 Fz0 Vref Vref Fz0 Here, the influence of the vertical load Fz , the longitudinal velocity Vx and the camber angle γ are taken into account by the qsy ’s in this equation. As discussed in section 3.3, Michelin introduced in 2005 a rolling resistance force model based on SAE standard J2452. The Michelin model is described in (3.21) as follows: Fx,RR = Fx,RR−ISO ·
pi pi,ISO
α
Fz Fz,ISO
β
.
The nominal conditions are defined according to the ISO 8767 standard. Therefore the nominal conditions are identified by pi,ISO and Fz,ISO and the rolling resistance force at these conditions is identified by Fx,RR−ISO . When implementing the model of Michelin in the MF-Swift tyre model nominal subscript 0 is used instead of the ISO subscript. The new expression (enhancement underlined) for the rolling resistance force calculation in the MF-Swift model reads: 4 Vx Fx Vx My + qsy3 + qsy4 + qsy5 γ 2 + ... (5.21) Fx,RR = = −Fz0 λM y qsy1 + qsy2 Fz0 Vref Vref r0 qsy7 qsy8 Fz 2 Fz pi γ , qsy6 Fz0 Fz0 pi0 where the new Magic Formula parameters qsy7 and qsy8 refer to the coefficients α and β of (3.21) and cover the influence of vertical load change and inflation pressure change respectively. Furthermore, Fz
50
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
in the original expression (5.20) is replaced by Fz0 to determine the rolling resistance force at nominal vertical load condition with velocity and camber compensation. Note that by using qsy7 =1 and qsy8 =0 as default values in (5.20), the original expression (5.20) is obtained again. Because no suitable rolling resistance data at different inflation pressures is available from the measured tyre in this thesis or another tyre in the TNO database, a selection of the data found in literature [18] will be used to evaluate the quality of the new formula. The measurements to obtain this data were performed according to the ISO 8767 international standard. For an overview of the selected rolling resistance data it is referred to Appendix D.2. In figure 5.13 the results are depicted of the "old" relation (5.20) when the Magic Formula parameters are optimised to the measurement data. In the left plot, the rolling resistance is illustrated as a function of the vertical load. At a constant vertical load condition, the lowest rolling resistance value corresponds to the lowest forward velocity and the highest rolling resistance value corresponds to the highest forward velocity. At the highest vertical load condition (Fz =5360 N) only four values are obtained with (5.20), while eight measurement points exist (2 inflation pressures at 4 velocities). Because (5.20) does not take inflation pressure into account, only four velocity optimised values are generated. In the right plot of figure 5.13 the rolling resistance force versus the velocity is depicted. At a certain velocity, the highest rolling resistance force corresponds with the highest vertical load. The resulting error of (5.20) of all data points is approximately 8.5 percent. −10
−25 −30 −35 −40 −45 −50 −55
Old relation Measurement
−15 Rolling resistance force, Fx,RR [N]
Rolling resistance force, Fx,RR [N]
−20
−60 1500
−10
Old relation Measurement
−15
−20 −25 −30 −35 −40 −45 −50 −55
2000
2500
3000 3500 4000 Vertical force, Fz [N]
4500
5000
5500
−60 5
10
15 20 25 Forward velocity, V [m/s]
30
35
Figure 5.13: "Old" rolling resistance force relation (5.20) optimised for the measurement data, as function of Fz (left) and as function of V (right).
When comparing the results of figure 5.13 with the results of the "new" relation (5.21) in figure 5.14, it is seen that relation (5.21) significantly improves the fit accuracy compared to the old relation. By implementing the two exponential terms, the fit error is approximately 6.7 percent improved to 1.8 percent error over all data points, see table 5.2. The optimised Magic Formula parameters for both situations are given in Appendix D.2. Table 5.2: Total error of the "old" and "new" rolling resistance force relation, (5.20) and (5.21) respectively.
Error of all data points Equation F igure T otal Error [%] (5.20) 5.13 8.50 (5.21) 5.14 1.80
5.6 Camber thrust and camber torque
51
−10
−25 −30 −35 −40 −45 −50 −55
New relation Measurement
−15 Rolling resistance force, Fx,RR [N]
Rolling resistance force, Fx,RR [N]
−20
−60 1500
−10
New relation Measurement
−15
−20 −25 −30 −35 −40 −45 −50 −55
2000
2500
3000 3500 4000 Vertical force, F [N]
4500
5000
5500
−60 5
z
10
15 20 25 Forward velocity, V [m/s]
30
35
Figure 5.14: "New" rolling resistance force relation (5.21) optimised for the measurement data, as function of Fz (left) and as function of V (right).
To create relations as unambiguous as possible, it is tried to replace the vertical load and inflation pressure divisions by the vertical load increment dfz and inflation pressure increment dpi . This creates a problem when Fz = Fz0 and/or pi = pi0 ; the exponential relation becomes zero and eliminates the rolling resistance force. So implementing the vertical load increment and inflation pressure is not an option.
5.6
Camber thrust and camber torque
In section 4.3 it is shown that the inflation pressure has some influence on the camber thrust, the aligning camber torque and overturning camber torque for a large camber angle range. Therefore, it is desirable to implement an inflation pressure dependency. Camber is one of the effects that is deeply embedded in the Magic Formula tyre model. It is not present at one specific position in the equations, but it is present at many different places. Currently, all the implemented inflation pressure dependencies, like in the cornering stiffness Cf α , have very little or no influence on camber effects. Camber thrust As mentioned in section 4.3, the inflation pressure has influence on the camber thrust stiffness Kyγ0 . Currently, the camber thrust stiffness is implemented in the Magic Formula with only a vertical load dependency. To implement inflation pressure dependency in this quantity, the following (underlined) enhancement is proposed: Kyγ0
=
(pKy6 + pKy7 dfz )Fz λKyγ (1 + ppy5 dpi )
(5.22)
The camber thrust stiffness has influence on different parts of the lateral force calculation. (5.23) shows the Magic Formula relation of the lateral force for pure slip conditions. Where for example Kyγ0 has influence on the stiffness factor By . Fyp = Dy sin Cy arctan By αy − Ey (By αy − arctan(By αy )) + SV y , (5.23)
52
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
To assess the enhancement, (5.22) is implemented in the Magic Formula and compared with Flatplank measurements. In figure 5.15, the results are depicted for the nominal vertical load condition (F z=5 kN); the left plot shows the results for negative camber angles and the right plot shows the results for positive camber angles. The lines represent the Magic Formula optimisation results and the markers are the Flatplank measurement results. −15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg
1000
500
0
0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg
−200 −400 Camber thrust, Fyγ [N]
Camber thrust, Fyγ [N]
1500
−600 −800 −1000 −1200 −1400
0 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
−1600 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 5.15: Implemented camber thrust enhancement (lines), equation (5.22), compared with the measurements (markers) for various camber angles at the nominal vertical load condition: negative camber angles (left) and positive camber angles (right).
The results in figure 5.15 are manually tuned to describe the common passenger car camber range (i.e. till γ ≤+/- 5.0 degrees) well. A result of this is that for camber angles larger than +/- 5 degrees a larger deviation occurs. This larger deviation is caused by the Magic Formula. At nominal inflation pressure (pi =2.5 bar) the camber thrust generated by the Magic Formula for large camber angles is too high. The Magic Formula can perfectly optimise pure camber effects, but when slip measurement data is added to the Magic Formula optimisation routine larger deviation occurs for camber angles γ >+/- 5 degrees. Furthermore, the camber measurements are performed on the Flatplank, while the slip data comes from the TNO Tyre Test Trailer. In earlier research, [35], it is already shown that the results of indoor test facilities (e.g. The Flatplank) do not always correspond well with the results of outdoor test facilities (e.g. the TNO Tyre Test Trailer). To exclude this possible cause, the camber measurements have to be repeated on the TNO Tyre Test Trailer. Overturning camber torque The lateral force, determined with (5.23), is used in the determination of the overturning moment Mx . In the Magic Formula, the overturning moment Mx is determined according to the following expression: Fy Mx = R0 Fz qsx1 − qsx2 γy − qsx12 γy |γy | + qsx3 + ... + R0 Fy qsx13 + qsx14 γy , (5.24) Fz0 where Fy is the lateral force determined in the Magic Formula. Because the inflation pressure influence on the lateral force shows almost no influence on the overturning moment, an enhancement of the overturning moment equation is necessary. To implement inflation pressure dependency in the overturning camber torque, the following (underlined) enhancement of (5.24) is proposed: Fy Mx = R0 Fz qsx1 − qsx2 γy (1 + ppM x1 dpi ) − qsx12 γy |γy | + qsx3 + ... + ... (5.25) Fz0 To assess the influence of the enhancement, (5.25) is implemented in the Magic Formula. In figure 5.16 the results of the enhanced relation are compared with Flatplank camber measurements; the left
5.6 Camber thrust and camber torque
53
plot shows the results for negative camber angles and the right plot shows the results for the positive camber angles. The lines represent the Magic Formula results and the markers are the Flatplank measurements. It can be seen that the overturning camber torque describes all measurements rather well.
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg
350
Camber torque, Mxγ [Nm]
300 250 200 150
0
−100 −150 −200 −250
100
−300
50
−350
0 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg
−50
Camber torque, Mxγ [Nm]
400
−400 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 5.16: Overturning camber torque enhancement relation compared with the measurements for various camber angles at the nominal vertical load condition: negative camber angles (left) and positive camber angles (right).
Aligning camber torque In the Magic Formula model the aligning moment due to camber is determined by the lateral force Fy , the pneumatic trail t [28] and the residual aligning moment Mzr . The enhanced relation of the lateral force, (5.22), appears to have almost no influence on the aligning moment Mz . In the Magic Formula the aligning moment is determined by the following expression: Mz Mzr
= Fy t + Mzr , = Dr cos arctan(Br αr,eq ) cos(αM ).
(5.26) (5.27)
To implement inflation pressure influence of the aligning camber torque Mzγ , the peak factor Dr in (5.27) is extended in the following (underlined) way: Dr = Fz R0 (qDz6 + qDz7 dfz )λr + (qDz8 + qDz9 dfz )γλKzγ (1 + ppz2 dpi ) ... (5.28) +(qDz10 + qDz11 dfz )γ | γ | + 1. Again, the enhanced relation, (5.28), is implemented in the Magic Formula to assess the influence of the enhancement. In figure 5.17 the results are depicted for the nominal vertical load condition and compared with the Flatplank measurements; left plot: negative camber angles, right plot: positive camber angles. The aligning camber torque is manually tuned to describes the common passenger car camber angle range (γ ≤+/- 5 degrees) well. It can be seen in figure 5.17 that at larger camber angles a significant difference occurs. As already mentioned for the camber thrust, the cause of this lies in the by the Magic Formula generated too high aligning camber torque values for large camber angles at nominal inflation pressure and the difference between Flatplank measurements and TNO Tyre Test Trailer results. Finally, note that the proposed enhancements in (5.22), (5.25) and (5.28) show that it is possible to describe the influence of inflation pressure on camber effects (Fyγ , Mzγ , Mxγ ). Figures 5.15 till 5.17 are used to illustrate the influence of the enhancements and to confirm the useability of the proposed enhancements. The depicted results are manually tuned for the common passenger car camber angle range (γ ≤+/- 5 degrees), this may result in a bigger deviation for γ >+/- 5 degrees. Furthermore, no optimisation strategy for camber effects is developed yet. In Appendix D.4 the Magic Formula results are presented for two extreme vertical load conditions to assess the robustness of the proposed enhancements.
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
90
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg
80
Camber torque, Mzγ [Nm]
70 60 50 40 30 20
0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg
0 −10 −20 −30 −40 −50 −60 −70
10 0 1.8
10
Camber torque, Mzγ [Nm]
54
−80 2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
−90 1.8
2
2.2 2.4 2.6 Inflation pressure, pi [bar]
2.8
3
Figure 5.17: Aligning camber torque enhancement relation compared with the measurements for various camber angles at the nominal vertical load condition: negative camber angles (left) and positive camber angles (right).
5.7
Rigid ring dynamics
As already described in chapter 2, the rigid ring model consists of a rigid ring that represents the tyre belt. The ring is elastically suspended to the rim by means of spring-damper elements, representing the tyre sidewalls with pressurised air. The contact model consist of two models that are used for describing the contact of the tyre with the road surface, i.e. an enveloping model and a slip model. In addition, residual stiffnesses between the contact model and the rigid ring are used to obtain the correct (quasi-static) tyre stiffness. The rigid ring model can only describe the primary tyre modes, i.e. those motions of the tyre where the shape of the belt remains circular. The flexible eigenmodes of the belt are neglected. In section 4.4, the primary eigenmodes of the tyre are analysed with FEM experiments to determine the influence of the inflation pressure. Currently MF-Swift can handle one inflation pressure condition only. In this section, strategies are proposed to implement the inflation pressure influence in the MF-Swift model, in such a way that the rigid ring tyre dynamics for a certain inflation pressure range can be parameterised. Two possible concepts are discussed based on: 1) the Rotta Membrane Theory and 2) scaling the nominal sidewall stiffnesses.
5.7.1
Rotta Membrane Theory
Rotta introduced T he M embrane T heory f or Cylindrical T yres in 1949 [30]. The theory of Rotta is based on neglecting the circumference curvature of the membrane. In [30] characteristic stiffnesses in the tangential, lateral and radial deformation directions are introduced, see figure 5.18. These tangential cu , lateral cv and radial cw direction stiffnesses depend on the dimensions of the "membrane" and the inflation pressure: Gt 1 cu = + pi ; (5.29) ls tan ϕs sin γ cv = pi ; (5.30) 1 + cos γ − π−γ 2 sin γ cos ϕs + ϕs sin ϕs cw = pi ; (5.31) sin ϕs − ϕs cos ϕs where G is the shear modulus of the tyre sidewall, t is the sidewall thickness and ls is the length of the sidewall arc. Furthermore, ϕs is half the angle of the tyre sidewall, pi is the inflation pressure and γ is
5.7 Rigid ring dynamics
55
the angle between the tangent line of the tyre sidewall swell near the rim and the horizontal surface.
Figure 5.18: Sidewall deformations [5].
The relation between the rigid ring model sidewall stiffnesses and the Rotta directional stiffnesses reads [31]: cbx,z cby cbγ,ψ cbθ
= πr(cu + cw ); = 2πrcv ; = πr3 cv ; = 2πr3 cu .
(5.32) (5.33) (5.34) (5.35)
To describe the sidewall stiffnesses as a function of the inflation pressure, the Rotta direction stiffnesses are implemented in equations (5.32) - (5.35). Assuming that ϕs and γ are constant and not influenced by the inflation pressure, the inflation pressure pi is the only variable in the direction stiffness equations (5.29) - (5.31). Knowing this, the directional stiffnesses can be rewritten in a constant part and an inflation pressure dependent part: cu cv cw
= cu1 + cu2 pi ; = cv1 pi ; = cw1 pi ;
(5.36) (5.37) (5.38)
where cu1 , cu2 , cv1 and cw1 are constants. If (5.36) - (5.38) are implemented in the sidewall stiffness relations (5.32) - (5.35), the sidewall stiffnesses read: cbx,z = πr cu1 + (cu2 + cw1 )pi ; (5.39) cby = 2πrcv1 pi ; (5.40) cbγ,ψ = πr3 cv1 pi ; (5.41) 3 cbθ = 2πr cu1 + cu2 pi . (5.42) In these equations cbx is the overall sidewall stiffness in longitudinal direction, cbz in vertical direction and cby in lateral direction. Furthermore, cbγ is the overall torsion stiffness about the x-axis, cbψ about the z-axis and cbθ about the y-axis. The equations of motion of a free hanging rigid ring model with fixed rim for the in-plane dynamics read: mb x ¨b + kbx x˙ b + cbx xb = 0 mb z¨b + kbz z˙b + cbz zb = 0 Iby θ¨b + kbθ θ˙b + cbθ θb = 0
(5.43) (5.44) (5.45)
mb y¨b + kby y˙ b + cby yb = 0 Ibx γ¨b + kbγ γ˙ b + cbγ γb = 0 Ibz ψ¨b + kbψ ψ˙ b + cbψ ψb = 0
(5.46) (5.47)
and for the out-of-plane dynamics:
(5.48)
56
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
Using the equations of motion and the inflation pressure depending sidewall stiffnesses (5.39) - (5.42), the eigenfrequencies for a free hanging tyre can be determined with: r cbx,y,z ωtrans = ; (5.49) mb r cbγ,θ,ψ ωrot = . (5.50) Ibx,z When the primary eigenfrequencies of a free hanging tyre with fixed rim at the nominal inflation pressure pi0 are known, the nominal sidewall stiffnesses cb0 can be determined using (5.49) and (5.50). With the determined nominal sidewall stiffnesses, cb0x,z and cb0θ , the direction stiffnesses cu and cw can be determined and subsequently the tyre sidewall angle ϕs can be derived using (5.31). Finally, the constants (i.e. cu1 , cu2 , cv1 and cw1 ) can be determined using equations (5.29) and (5.30). In figure 5.19 the eigenfrequencies of a free hanging tyre with fixed rim determined with the FEM simulations and estimated with the Rotta equations (5.39) - (5.42) are depicted for the pressure range 0 - 3.0 bar. The with Rotta estimated results can only be compared in the pressure range 1.5 - 3.0 bar, because FEM simulations are only performed in this pressure range. Note that in this case the nominal inflation pressure pi0 =2.2 bar.
90 80
Error at 1.5 bar inflation pressure P rimary eigenmode Error [%] vertical, radial 5.22 lateral 5.04 rotational 1.90 camber, yaw 5.99
Natural frequency [Hz]
70 60 50 40 30 Abaqus Rotta rotational lateral yaw/camber vertical/radial
20 10 0 0
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
Figure 5.19: FEM primary eigenfrequencies compared with the Rotta estimations.
The eigenfrequencies correspond rather well, showing a maximum difference of approximately 3 Hz (i.e. error of 6 percent) at an inflation pressure of 1.5 bar. Below 1.5 bar the Rotta estimations are robust for both the sidewall stiffnesses and the eigenfrequencies. At pi =0 bar, the sidewall stiffnesses (i.e. cby , cbψ , cbγ ) and the corresponding primary eigenfrequencies become zero. This is caused by the fact that the Rotta theory neglects the influence of the tyre material (rubber) flexibility/stiffness.
5.7 Rigid ring dynamics
5.7.2
57
Scaling the nominal pressure sidewall stiffnesses
A second approach is to scale the nominal inflation pressure sidewall stiffnesses cb0 . I.e. the sidewall stiffnesses are made inflation pressure dependent by scaling the nominal sidewall stiffnesses with a linear inflation pressure relation, meaning: cbx,z cby cbγ,ψ cbθ
= cbx0,z0 (1 + fx,z dpi );
(5.51)
= cby0 (1 + fy dpi );
(5.52)
= cbγ0,ψ0 (1 + fγ,ψ dpi );
(5.53)
= cbθ0 (1 + fθ dpi ).
(5.54)
Here the fi ’s are the inflation pressure fit parameters. In this case no distinction is made between parts that are inflation pressure depending and parts that are not depending on the inflation pressure, like in the Rotta approach. This gives the advantage that it is not necessary to determine certain tyreconstruction and inflation pressure depending parameters first, like the tyre sidewall angle ϕs . The fit parameters fi are determined by optimising (5.51) - (5.54) towards the FEM experiments. In figure 5.20, the FEM eigenfrequency results and the eigenfrequencies estimated with the equations (5.51) (5.54) are depicted for the pressure range 0 - 3.0 bar. 90
Natural frequency [Hz]
80
70
Abaqus Scaling rotational lateral yaw/camber vertical/radial
Optimisation sidewall stiffnesses P res. parameter V alue [−] Error [%] fx , fz 0.65 0.21 fy 0.74 0.27 fγ , fψ 0.49 0.08 fθ 0.69 0.25
60
50
40
30
20 0
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
Figure 5.20: FEM eigenfrequencies compared with optimised scaling estimation.
Again, comparison between the FEM results and the scaling approach can only be made in the pressure range 1.5 - 3.0 bar, because FEM simulation are only performed in this pressure range. The results correspond rather well, with an error of approximately 0.2 percent over the 1.5 - 3.0 bar pressure range. The trend below 1.5 bar seems plausible, but cannot be verified. When comparing the results of the Rotta approach (figure 5.19) and the scaling approach (figure 5.20), it is assessed that the scaling approach results in more accurate estimations. The scaling approach is easier to implement in the tyre model, and therefore the preference is given to this strategy. However, the eigenfrequencies of the tyre have to be determined experimentally (or with FEM analyses) for at least two inflation pressures.
58
5.7.3
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
Stiffnesses rigid ring model, lateral components
To fully understand what the influence of the inflation pressure is on the stiffnesses of the rigid ring tyre model in lateral direction, an overview of the lateral stiffness components is depicted in figure 5.21.
Figure 5.21: Schematic view of lateral rigid ring model stiffness components.
The tyre lateral stiffness CF y is a relation of the total lateral carcass stiffness cy and the cornering stiffness CF α and reads: 1 1 a = + , (5.55) CF y cy CF α where a is the half contact length. Following figure 5.21, the total carcass stiffness can be described as follows: 1 1 1 r2 = + + l . (5.56) cy cry cby cbγ In this equation, cry is the residual lateral carcass stiffness, cby the lateral sidewall stiffness and cbγ the torsional sidewall stiffness about the x-axis. Furthermore, rl is the loaded radius. In figure 5.22 the lateral stiffnesses are depicted as a function of the inflation pressure for the measured passenger car tyre. The tyre lateral stiffness and the cornering stiffness are determined out of the Flatplank measurements; for pi < 1.9 bar the data is obtained with extrapolation. The sidewall stiffnesses cby and cbγ at the nominal inflation pressure are determined using a MF-Swift tyre property file. The sidewall stiffnesses at the remaining inflation pressures are determined using the scaling strategy. The total lateral carcass stiffness cy is known from (5.55) and the residual lateral carcass stiffness cry is used to balance equation (5.56). The trends in figure 5.22 show that the different lateral stiffness components increase when increasing the inflation pressure. Except for the cornering stiffness, which shows only little influence of the inflation pressure compared to the other lateral stiffness components. The results for pi < 1.9 bar seem plausible but should be handled with some reservations. The sidewall stiffnesses cby and cbγ show the most influence of the inflation pressure.
5.7 Rigid ring dynamics
59 5
12
x 10
10 CFy
Stiffness [N/m]
8
CFα/a cy 2 bγ l
6
c /r cby c
ry
4
Trend line 2
0 0
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
Figure 5.22: Lateral stiffnesses as function of the inflation pressure at nominal vertical load condition.
5.7.4
Stiffnesses rigid ring model, longitudinal components
An overview of the longitudinal stiffness components is depicted in figure 5.23.
Figure 5.23: Schematic view of longitudinal rigid ring model, stiffness components.
The longitudinal relaxation length is described by the following relation: σκ =
CF κ CF κ = + σc CF x cx
(5.57)
where CF κ is the longitudinal slip stiffness, CF x the tyre longitudinal stiffness, cx the total longitudinal carcass stiffness and σc the contact patch relaxation length. When knowing that σc ' a [28], the overall longitudinal stiffness can be expressed as a relation of the longitudinal carcass stiffness and the longitudinal slip stiffness as follows: 1 1 a = + , CF x cx CF κ
(5.58)
60
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
where a is the half contact length. Following figure 5.23, the total longitudinal carcass stiffness can be described as follows: 1 1 1 r2 = + + l cx crx cbx cbθ
(5.59)
In this equation, crx is the residual longitudinal carcass stiffness, cbx the longitudinal sidewall stiffness and cbθ the rotational sidewall stiffness about the y-axis. Furthermore, rl is the loaded radius. In figure 5.24 the longitudinal stiffnesses are depicted as a function of the inflation pressure for the measured passenger car tyre. The tyre longitudinal stiffness is determined out of the Flatplank measurements; for pi < 1.9 bar the data is obtained with extrapolation. The longitudinal slip stiffness is determined using measurement data of the TNO Tyre Test Trailer and processed with MF-Tool that has the equation (3.2) of de Hoogh implemented. The sidewall stiffnesses cbx and cbθ at the nominal inflation pressure are determined using a MF-Swift dataset. The sidewall stiffnesses at the remaining inflation pressures are determined using the scaling strategy. The total longitudinal carcass stiffness cx is known from (5.58). The residual longitudinal carcass stiffness crx is used to balance (5.59). 6
2.5
x 10
CFx CFκ/a cx
2
cbθ/r2l
Stiffness [N/m]
cbx crx
1.5
Trend line
1
0.5
0 0
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
Figure 5.24: Overview longitudinal (carcass) stiffnesses as function of the inflation pressure.
The trends in figure 5.24 show that the different longitudinal stiffness components increase more or less with increasing inflation pressure. The longitudinal slip stiffness shows a optimum at approximately 2.0 bar. The stiffnesses for pi < 1.9 bar seem plausible but should be handled with some reservations. The sidewall stiffnesses cbx and cbθ show the most influence of the inflation pressure. In Appendix D.3 an overview of the longitudinal and lateral (carcass) stiffnesses for 2 extreme vertical load conditions, Fz =0.1 and 10 kN, is presented. The stiffness trends in the appendix show that no negative values occur for extreme vertical load conditions, i.e. the rigid ring model with the inflation pressure enhanced stiffness relations produces robust results for a wide vertical load and inflation pressure range.
5.8 Summarising this chapter
5.8
61
Summarising this chapter
In this chapter relations are derived to describe the inflation pressure influence on different quantities. The relations are mainly based on the trends as found in the previous chapter and supported by trends described in literature. It is shown that not always the equations that describe the inflation pressure influences most accurately (smallest errors) are the most robust solutions for the MF-Swift tyre model fit strategies (e.g. tyre lateral stiffness and torsional stiffness). In the overview below, the final equations that are implemented in the MF-Swift tyre model are presented. Tyre lateral stiffness: CF y = CF y,nom (1 + pCF y1 dfz + pCF y2 dfz2 )(1 + pCF y3 dpi ) Tyre longitudinal stiffness: CF x = CF x,nom (1 + pCF x1 dfz + pCF x2 dfz2 )(1 + pCF x3 dpi ) Note that for the tyre lateral and longitudinal stiffness a quadratic relation is included instead of a more reliable linear relation, as described in this chapter. To create a linear relation, the quadratic parameters pCF y2 and pCF x2 are fixed to zero in normal use. When higher accuracy is desired the quadratic parameters can be used in the optimisation, but results have to be checked carefully. Tyre torsional stiffness: CM z = CM z,nom (1 + pCM z1 dpi )
Rolling Resistance force: Fx,RR = Fx,RR0
Fz Fz0
qsy7
pi pi0
Fz Fz0
qsy8
4 Vx Fx Fz 2 2 + qsy4 Vx Fx,RR0 = −Fz0 λM y qsy1 + qsy2 + qsy3 + q γ + q γ sy5 sy6 Fz0 Vref Vref Fz0
Camber thrust: Kyγ0 = (pKy6 + pKy7 dfz )Fz λKyγ (1 + ppy5 dpi ) Aligning camber torque: Dr = Fz R0 (qDz6 + qDz7 dfz )λr + (qDz8 qDz9 dfz )γλKzγ (1 + ppz2 dpi ) ... Overturning camber torque: Fy Mx = R0 Fz qsx1 − qsx2 γy (1 + ppM x1 dpi ) − qsx12 γy γy + qsx3 + ... + R0 Fy qsx13 + qsx14 γy Fz0
62
Inflation Pressure Relations to Enhance the MF-Swift Tyre Model
Translational sidewall stiffness: cbt = cbt0 (1 + ft dpi ) where, subscript t is the translation direction x, y or z. Rotational sidewall stiffness: cbr = cbr0 (1 + fr dpi ) where, subscript r is the rotation about x, y or z axis, corresponding with the angles θ, γ and ψ respectively. In Appendix D.5 an overview is given of the new implemented parameters, with proposed corresponding initial and boundary conditions.
Chapter 6
Vehicle Behaviour Simulations In this chapter vehicle simulations are performed with the enhanced TNO tyre models to analyse the extended relations on functionality and applicability in vehicle design. To investigate the handling behaviour, a steady-state circular test according to ISO 4138 [21] and a random steering test according to ISO 7401 [23] are executed. The ride comfort is analysed with a vehicle behaviour test on an uneven road. The vehicle model used to analyse the vehicle behaviour is a SimMechanics model of a two track vehicle model with a roll axis, see figure 6.1.
Figure 6.1: Two track vehicle model with a roll axis (ISO).
When going through a curve, the body rolls around the roll axis. This axis is a virtual axis that goes through the front roll centre rc1 and the rear roll centre rc2 . The roll stiffness and damping (which result from suspension springs, dampers and anti-roll bars) are modelled with torsional springs and dampers in the roll centres. Furthermore, translational stiffness and damping elements are used, in the roll centres, to model the vertical stiffness and damping of the suspension when driving over uneven road surfaces. The modelled vehicle is an upper class rear wheel driven sedan with a total unloaded vehicle mass of 1659.5 kg (distribution front-rear 50-50%). In Appendix E an overview is given of the parameters that are used to parameterise the vehicle model.
63
64
Vehicle Behaviour Simulations
6.1
Steady-state circular test
This test is standardised in ISO 4138 with the aim to determine the steady-state behaviour of a vehicle (i.e. no transient effects are considered). The vehicle is driving over a circle with a fixed radius R (R=100 m). This is done with different constant forward velocities V . The steering wheel angle δ is adjusted to maintain the constant radius. To perform a steady-state circular test with the two track model, a steer controller is used to drive over a circle and a cruise control controller is necessary to drive at constant velocities. Within the framework of applicability in vehicle design, two types of analyses are performed: a sensitivity analysis and a loaded vs. unloaded vehicle analysis.
6.1.1
Sensitivity analysis
The inflation pressure sensitivity analyses are performed to investigate what the influence is of the inflation pressure on the handling behaviour of the vehicle. Three inflation pressure approaches are made, with as base inflation pressure pi,f ront = pi,rear =2.3 bar, namely: 1. front and rear tyres have same inflation pressure, with pi,f ront = pi,rear =[ 1.8 2.3 2.8] bar; 2. changing inflation pressure front tyres pi,f ront =[1.9 2.3 2.7] bar, constant inflation pressure rear tyres pi,rear =2.3 bar; 3. changing inflation pressure rear tyres pi,rear =[1.9 2.3 2.7] bar, constant inflation pressure front tyres pi,f ront =2.3 bar. Table 6.1 gives an overview of the different inflation pressure settings. Table 6.1: Inflation pressure settings.
Inflation pressure [bar] Setting F ront tyres Rear tyres 1 1.8 1.8 2.3 2.3 2.8 2.8 2 1.9 2.3 2.3 2.3 2.7 2.3 3 2.3 1.9 2.3 2.3 2.3 2.7
To illustrate the influence of changing the inflation pressure on the steady-state handling behaviour of a vehicle, the lateral vehicle acceleration ay as function of the axle sideslip angle and the handling diagram are assessed. In the handling diagram, the lateral vehicle acceleration ay is described as a function of the difference between the sideslip angle of the rear axle α2 and the front axle α1 . A handling diagram illustrates if a vehicle is understeered, neutral or oversteered. In figure 6.2, the simulation results of the first inflation pressure setting (setting 1) are presented. The handling diagram in figure 6.2 shows that the vehicle has an understeered behaviour. For lateral accelerations below 4.5 m/s2 , the vehicle shows a linear response. For higher lateral accelerations the vehicle shows a nonlinear behaviour, caused by the nonlinear tyre behaviour. The tyres are starting to reach their maximum lateral force.
65
10
10
9
9
8
8 Lateral acceleration, ay [m/s2]
Lateral acceleration, ay [m/s2]
6.1 Steady-state circular test
7 6 5 4 3
front axle, α
2
rear axle, α2
1
p18 p23 p28
1
0 0
2
4
αaxle [deg]
6
8
7 6 5 4 3 2 1
10
0 −4.5
p18 p23 p28 −4
−3.5
−3
−2.5 −2 α2 − α1 [deg]
−1.5
−1
−0.5
0
Figure 6.2: Lateral acceleration vs. axle sideslip angle (left) and handling diagram (right) for inflation pressure setting 1; front and rear tyres have the same inflation pressure 1.8, 2.3 and 2.8 bar.
Lowering the inflation pressure results in a decrease of the axle sideslip angles, due to a change in the axle cornering stiffness CF α,axle , see figure 6.3. Lowering the inflation pressure results in an increase of the axle cornering stiffness. So assuming that 2.3 bar is the nominal inflation pressure, increasing the inflation pressure leads to a decrease in axle cornering stiffness and decreasing the pressure leads to an increase. Furthermore, the lateral axle force necessary for a certain lateral acceleration ay is described by: Fy1 + Fy2 = may ,
(6.1)
where m is the vehicle mass and Fy1 and Fy2 the lateral force of the front and rear axle respectively. The lateral axle force is determined by: Fyi = CF α,i αi ,
(6.2)
where the index i represents the front or the rear axle. Considering (6.1) and (6.2), it is obvious that when the axle cornering stiffness CF α decreases, the axle sideslip angle α has to increase in order to retain the same lateral force. Changing the inflation pressure results, for lateral accelerations below 4.5 m/s2 , in a more or less equal decrease of the front axle sideslip angle and the rear axle sideslip angle (see figure 6.2). Almost no change is visible in the handling diagram. For higher lateral accelerations, the rear axle sideslip angle changes more than the front axle sideslip angle, resulting in a more (for inflation pressure decrease) or lesser (for inflation pressure increase) understeer behaviour.
66
Vehicle Behaviour Simulations
Cornering stiffness (axle), C Fα,axle [N/deg]
3400 Front axle Rear axle Unloaded Loaded
3200 3000 2800 2600 2400 2200 2000 1800 1600 1400 1.8
2
2.2 2.4 Inflation pressure, pi [bar]
2.6
2.8
Figure 6.3: Axle cornering stiffness vs. inflation pressure.
In figure 6.4 and 6.5 the simulation results of settings 2 and 3 respectively are presented. Note that the p-terms in the legend represent the inflation pressure setting (e.g p1923 means the inflation pressure of the front tyres is 1.9 bar and the rear tyres is 2.3 bar). When comparing the simulation results for settings 2 and 3 with the results of setting 1, it can be seen that changing the tyre inflation pressure of one axle has more influence on the steady-state handling behaviour of the vehicle. This is confirmed by the understeer coefficients η, presented in table 6.2. The understeer coefficient, only valid for the linear part of the handling behaviour, is described by the following expression: d(α2 − α1 ) η=− . (6.3) day ay =0
10
10
9
9
8
8 Lateral acceleration, ay [m/s2]
Lateral acceleration, ay [m/s2]
Table 6.2 makes clear that changing the inflation pressure of the front tyres has slightly more influence on the steady-state handling behaviour of the vehicle than changing the pressure of the rear tyres.
7 6 5 4 3
front axle, α1
2
rear axle, α2
1
p1923 p2323 p2723
0 0
2
4
αaxle [deg]
6
8
7 6 5 4 3 2 1
10
0 −4.5
p1923 p2323 p2723 −4
−3.5
−3
−2.5 −2 α2 − α1 [deg]
−1.5
−1
−0.5
0
Figure 6.4: Axle sideslip angles (left) and handling diagram (right) for inflation pressure setting 2; constant inflation pressure rear tyres 2.3 bar, front tyres 1.9, 2.3 and 2.7 bar.
67
10
10
9
9
8
8 Lateral acceleration, ay [m/s2]
Lateral acceleration, ay [m/s2]
6.1 Steady-state circular test
7 6 5 4 3
front axle, α
2
rear axle, α2
1
p2319 p2323 p2327
1
0 0
2
4
αaxle [deg]
6
8
7 6 5 4 3 2 1
10
0 −4.5
p2319 p2323 p2327 −4
−3.5
−3
−2.5 −2 α2 − α1 [deg]
−1.5
−1
−0.5
0
Figure 6.5: Axle sideslip angles (left) and handling diagram (right) for inflation pressure setting 3; constant inflation pressure front tyres 2.3 bar, rear tyres 1.9, 2.3 and 2.7 bar.
Table 6.2: Understeer coefficient for the different inflation pressure settings.
Understeer coefficient η [deg/m/s2 ] Setting Inf lation pressure 1 p18 0.33 p23 0.34 p28 0.35 2 p1923 0.31 p2323 0.34 p2723 0.37 3 p2319 0.36 p2323 0.34 p2327 0.32
6.1.2
Unloaded vs. loaded vehicle
In table 6.3 a small selection of the vehicle specifications distributed by the car manufacturer is given. In this table also the inflation pressures, as recommended by the manufacturer, are presented. As can be seen the loaded vehicle has a different weight distribution than the unloaded vehicle. A change in weight distribution has influence on the (steady-state) handling behaviour of the vehicle. For the loaded vehicle the inflation pressure of the front tyres is increased to 2.3 bar and the inflation pressure of the rear tyres is increased to 2.8 bar. It is expected that the inflation pressures for the loaded vehicle are chosen in such a way that the handling behaviour, compared to the unloaded vehicle, remains the same. This is analysed in this section. To assess the (steady-state) handling behaviour for the loaded vehicle and the unloaded vehicle, a steady state circular test is performed with the following load and inflation pressure situations: • unloaded vehicle with the recommended inflation pressure, pi,f ront =1.9 bar and pi,rear =2.3 bar; • loaded vehicle with the inflation pressure of the unloaded vehicle, pi,f ront =1.9 bar and pi,rear =2.3 bar; • loaded vehicle with the recommended inflation pressure, pi,f ront =2.3 bar and pi,rear =2.8 bar.
68
Vehicle Behaviour Simulations
Table 6.3: Vehicle model parameters (unloaded and loaded).
Vehicle model parameters Quantity U nloaded Total vehicle weight [kg] 1659.5 Weight distribution (F-R) [%] 50 − 50 Inflation pressure (F-R) [bar] 1.9 − 2.3
Loaded 1954 42.5 − 57.5 2.3 − 2.8
10
10
9
9
8
8 Lateral acceleration, ay [m/s2]
Lateral acceleration, ay [m/s2]
In figure 6.6, the lateral acceleration as function of the axle sideslip angle and the handling diagram for the three load and inflation pressure situations is depicted.
7 6 5 4 3
front axle, α1
2
rear axle, α2
1
p1923 (unloaded) p1923 (loaded) p2328 (loaded)
0 0
2
4
αaxle [deg]
6
8
7 6 5 4 3 2 1
10
0 −6
p1923 (unloaded) p1923 (loaded) p2328 (loaded) −5
−4
−3 α2 − α1 [deg]
−2
−1
0
Figure 6.6: Axle sideslip angles (left) and handling diagram (right) for the three situations.
The loaded vehicle shows larger axle sideslip angles, caused by the different weight distribution. The handling diagram makes clear that the loaded vehicle shows approximately the same linear vehicle behaviour (i.e. lateral acceleration below 4.5 m/s2 ) as the unloaded vehicle. This is confirmed by the understeer coefficients in table 6.4. The understeer coefficient of the loaded vehicle with unchanged inflation pressure setting (p1923) is a fraction decreased (i.e. less understeer), increasing the inflation pressure to the recommended level (p2328) results in the same understeer coefficient as an unloaded vehicle. For higher lateral accelerations (i.e. lateral acceleration above 4.5 m/s2 ) the loaded vehicle shows a more understeered behaviour, which means that the front axle sideslip angle is increasing more than the rear axle sideslip angle. By increasing the inflation pressure to the recommended level a part of this more understeer behaviour is eliminated. Table 6.4: Understeer coefficient for unloaded and loaded vehicle.
Understeer coefficient η [deg/m/s2 ] V ehicle Inf lation pressure Unloaded p1923 0.31 Loaded p1923 0.30 Loaded p2328 0.31
For the loaded vehicle the rear axle vertical load increases significantly and consequently the rear axle cornering stiffness increases, see figure 6.3. When increasing the inflation pressure, the axle cornering
6.1 Steady-state circular test
69
stiffness decreases. This causes the axle sideslip angles to increase for the loaded vehicle with the recommended inflation pressure (p2328). Furthermore, the influence of the load and the inflation pressure situation on the vehicle sideslip angle β is depicted in figure 6.7. The vehicle sideslip angle β, see figure 6.1, is defined as: β=
v , u
(6.4)
in which, u is the forward velocity of the vehicle centre of gravity and v is the lateral velocity of the vehicle centre of gravity. The sideslip angle of the rear angle α2 is defined as: α2 =
br v + br =β+ , u u
(6.5)
where b is the position of the vehicle centre of gravity in relation to the rear axle and r is the yaw velocity. When driving over a circle with a fixed radius R, rR = u, than the sideslip angle of the rear angle α2 reads: α2 = β +
b , R
(6.6)
thus: β=−
b + α2 . R
(6.7)
Following (6.2), the sideslip angle of the rear angle α2 can also be described by: α2 =
Fy2 1 a = may , CF α,2 CF α,2 l
(6.8)
where a is the position of the vehicle centre of gravity in relation to the front axle, l is the wheel base, m is the total vehicle mass and CF α,2 is the rear axle cornering stiffness. Finally, when implementing (6.8) in (6.7) the vehicle sideslip angle β reads: β=−
am b + ay , R CF α,2 l
(6.9)
As can be seen in figure 6.7, the vehicle sideslip angle increases when the vehicle is loaded. When also the inflation pressure is increased to the recommended level of the loaded vehicle, the vehicle sideslip angle increases even more. This is caused by the increase of the distance a (15%) and mass m (18%) in relation (6.9), whereas the rear axle cornering stiffness CF α,2 for the loaded vehicle increases just 23% (for p1923) and 15% (for p2328), see figure 6.3. Against all expectations, the vehicle sideslip angle increases with the recommended inflation pressure of the loaded vehicle. It would be expected that the vehicle sideslip angle remains constant (i.e. same vehicle sideslip angle as the unloaded vehicle) with the recommended inflation pressure. To maintain this, in the here presented simulations, the inflation pressure has to be decreased. Note that similar results are observed in measurements with the vehicle but equipped with other tyres (205/60 R15)[37] and [36]. Furthermore note that wheel orientation changes (toe, camber, etc.) due to change in vertical load are not taken into account in the vehicle model. This has also influence on the handling behaviour of the loaded vehicle.
70
Vehicle Behaviour Simulations
0.5
Vehicle side slip angle, β [deg]
0 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 −4 0
p1923 (unloaded) p1923 (loaded) p2328 (loaded) 1
2
3 4 5 6 Lateral acceleration, ay [m/s2]
7
8
9
Figure 6.7: Vehicle sideslip angle versus lateral acceleration for the unloaded and loaded situations.
6.2
Random steering test
The random steering test is standardised in ISO 7410 and ISO/TR 8726. With this test it is possible to identify the transfer function between the steering input and various response variables (here: the lateral acceleration and the yaw velocity). To estimate the transfer function of a dynamic system, the system can be subjected to a random input signal that contains all relevant frequencies, see figure 6.8. The transfer function can be estimated by the quotient of the cross power spectral density (CPSD) of the input and the output and the power spectral density (PSD) of the input. This procedure is implemented in Matlab in a single function, tf estimate, and works only for linear systems. The test is performed at a constant forward velocity V =100 km/h. The (sinus shaped) steering input varies from 0.1 Hz to 5.0 Hz. To avoid nonlinear behaviour, the amplitude of the input has to be small to make sure that the lateral acceleration stays well below 4.5 m/s2 .
Figure 6.8: Example of random steering input.
The same inflation pressure settings are used as in the sensitivity analysis of the steady-state circular test, table 6.1. The simulation results from the three different settings are given in figure 6.9, 6.10 and 6.11. Changing the inflation pressure of both the front and rear tyres, has influence on the magnitude and the phase delay of the lateral acceleration and the yaw velocity transfer function, see figure 6.9. When lowering the inflation pressure of all four tyres (i.e. pi =1.8 bar), the magnitude in the lateral acceleration and the yaw velocity transfer function increases slightly and the peak in magnitude of the yaw velocity occurs at a higher frequency. Furthermore, less phase delay occurs in the lateral acceleration transfer
6.2 Random steering test
71
function for frequencies below 1 Hz. This is not in accordance to the inflation pressure effects on the relaxation behaviour of a tyre, see Section 4.2. Lowering the inflation pressure results in an increase of the lateral relaxation length (i.e. more phase delay). On the other hand, in figure 6.3 it is shown that the axle cornering stiffness increases when the inflation pressure is lowered. This results in a decrease in phase delay. Because the decrease in phase delay as result of the axle cornering stiffness increase has more effect than the phase delay increase as result of the increased lateral relaxation length, less phase delay occurs for frequencies below 1Hz.
yaw velocity to δs
lateral acceleration to δs
3
10
magnitude [1/s]
magnitude [m/s2 ⋅ deg.−1]
Figures 6.10 and 6.11 make clear that changing the inflation pressure of the rear tyres has more influence on the lateral acceleration and yaw velocity response and magnitude level, than changing the inflation pressure of the front tyres. Changing the inflation pressure of the front tyres only shows slight change in magnitude up to 1 Hz and no influence on the phase delay of the lateral acceleration and yaw velocity transfer functions.
2
10
−1
−1
10
0
10
10
−1
10
0
10
0 phase [deg.]
phase [deg.]
−20 0 −50
p18 p23 p28
−100 −1
10
−40 −60 −80 −100
0
10 frequency [Hz]
−120 −1 10
0
10 frequency [Hz]
Figure 6.9: Transfer functions of the simulation model, lateral acceleration (left) and yaw velocity (right), inflation pressure setting 1; front and rear tyres have the same inflation pressure 1.8, 2.3 and 2.8 bar.
Vehicle Behaviour Simulations yaw velocity to δs
lateral acceleration to δs
3
10
magnitude [1/s]
magnitude [m/s2 ⋅ deg.−1]
72
2
10
−1
−1
10
0
10
10
−1
10
0
10
0 phase [deg.]
phase [deg.]
−20 0 −50
p1923 p2323 p2723
−100 −1
10
−40 −60 −80 −100 −120 −1 10
0
10 frequency [Hz]
0
10 frequency [Hz]
yaw velocity to δs
lateral acceleration to δs
3
10
magnitude [1/s]
magnitude [m/s2 ⋅ deg.−1]
Figure 6.10: Transfer functions of the simulation model, lateral acceleration (left) and yaw velocity (right), inflation pressure setting 2; constant inflation pressure rear tyres 2.3 bar, front tyres 1.9, 2.3 and 2.7 bar.
2
10
−1
−1
10
0
10
10
−1
10
0
10
0 phase [deg.]
phase [deg.]
−20 0 −50
p2319 p2323 p2327
−100 −1
10
−40 −60 −80 −100
0
10 frequency [Hz]
−120 −1 10
0
10 frequency [Hz]
Figure 6.11: Transfer functions of the simulation model, lateral acceleration (left) and yaw velocity (right), inflation pressure setting 3; constant inflation pressure front tyres 2.3 bar, rear tyres 1.9, 2.3 and 2.7 bar.
6.3 Ride analysis
6.3
73
Ride analysis
In this section the vehicle behaviour on an uneven road surface is analysed using the vehicle model of figure 6.1. The vehicle model is driving over an uneven road surface of a motorway with a constant vehicle velocity of 120 km/h. Again, two analyses are performed: an inflation pressure sensitivity analysis and an unloaded vs. loaded vehicle analysis. For these analyses the same inflation pressure settings are used as for the steady-state circular tests, see table 6.1 and table 6.3. The ride analysis will be assessed by means of change in ISO ride comfort index (corresponding ISO 2631 [22]), vertical acceleration of the vehicle body az and pitch acceleration of the body. The results for the sensitivity analysis are given in table 6.5. In this table the inflation pressure setting front and rear 2.3 bar (p2323) is the base configuration. Table 6.5: Ride simulation results.
Inflation pressure setting ISO ride comfort body CG az body pitch acc.
[m/s2 ] [m/s2 ] [deg./s2 ]
p2323 0.180 0.265 33.7
p1818 −9.0% −1.0% −0.9%
Influence p2828 +3.5% +6.4% +1.7%
inflation pressure p1923 p2723 p2319 −5.8% +1.1% −2.0% −4.8% +2.4% −3.3% +4.9% −1.3% −4.0%
p2327 +0.7% +2.8% +3.8%
Table 6.5 makes clear that the ride comfort gets worse when increasing the inflation pressure. The most influence occurs when increasing the inflation pressure of both the front and rear tyres, a change of +3.5% is visible. Decreasing the inflation pressure leads to a better ride comfort, increasing the inflation pressure of both the front and rear tyres leads to a change of -9.0% of the ISO ride comfort index. Note that in previous research it is concluded that a 5% change in ISO ride comfort index is noticeable by an average driver [7]. Furthermore, it can be seen that lowering the inflation pressure of the front tyres (i.e. p1923) leads to the largest decrease in vertical acceleration level az . Table 6.5 also shows that changing the inflation pressure of all four tyres has far less impact on the pitch acceleration of the body. The results of the unloaded vs. loaded vehicle analysis are presented in table 6.6. In this case the recommended inflation pressure setting for an unloaded vehicle (p1923) is the base configuration. Table 6.6: Ride simulation results, unloaded vs. loaded vehicle.
Inflation pressure setting ISO ride comfort body CG az body pitch acc.
[m/s2 ] [m/s2 ] [deg./s2 ]
Influence inflation pressure p1923 (unloaded) p1923 (loaded) p2328 (loaded) 0.169 −13.9% +7.8% 0.252 −13.2% +6.0% 35.4 −11.5% −11.5%
The results in table 6.6 make clear that the ride comfort gets better when the sprung mass of the vehicle is increased (loaded vehicle), also the vertical and pitch accelerations decrease. Furthermore table 6.6 shows that increasing the inflation pressure of the loaded vehicle has a negative effect on the ride comfort and corresponding vertical acceleration. The pitch acceleration shows no change when the inflation pressure of the loaded vehicle is increased.
74
Vehicle Behaviour Simulations
Chapter 7
Conclusions and Recommendations This Chapter presents the main conclusions with regard to this Master’s thesis. In addition, some recommendations for further research are formulated.
7.1
Conclusions
The conclusions are split into three parts. The first part comprehends the influence of the inflation pressure on the tyre behaviour. The second part encompasses the enhancements and extension of the MF-Swift tyre model. The third part deals with some example cases of the applicability of the enhanced tyre model in vehicle design.
7.1.1
Inflation pressure influence
The following conclusions have been drawn with regard to the influence of the inflation pressure on the investigated tyre behaviour: • Characteristic stiffnesses It is shown that for all characteristic stiffnesses, an almost linear relation with the inflation pressure exists. The tyre lateral, longitudinal and vertical stiffnesses show an increase with increasing inflation pressure, while the tyre torsional stiffness shows a decrease with increasing inflation pressure. • Lateral relaxation length A linear decreasing relation between the lateral relaxation length and the inflation pressure exists, i.e. the relaxation length becomes shorter when the inflation pressure increases. • Effective rolling radius The influence of the inflation pressure on the effective rolling radius can be described with a linear relation. The influence is however very small. • Camber effects The camber thrust, overturning and aligning camber torque exhibit a linear relation with the inflation pressure. The slope of this linear relation is depending on the vertical load condition and the camber angle. • Eigenfrequencies (primary tyre modes) It is shown that for all primary tyre modes, the influence of the inflation pressure on the eigenfrequencies can be described with a linear relation. The eigenfrequencies increase when the inflation pressure is increased. 75
76
Conclusions and Recommendations
The quantity that shows the highest inflation pressure influence is the camber thrust and the tyre vertical stiffness. The effective rolling radius shows the least influence.
7.1.2
Enhanced model equations
Regarding the inflation pressure enhanced equations, the following conclusions have been drawn: • Tyre vertical stiffness No additional parameter identification work is required to implement inflation pressure dependency of the tyre vertical stiffness. The already implemented inflation pressure dependent vertical force relation can be used to describe the inflation pressure dependency of the tyre vertical stiffness. • Tyre torsional stiffness The enhanced tyre torsional stiffness relation is based on the nominal tyre torsional stiffness and a linear relation that depends on the normalised change in inflation pressure. The enhanced tyre torsional stiffness relation is made vertical load dependent by implementing a normalised vertical load term. • Lateral relaxation behaviour A pressure and load dependent relation for the lateral stiffness has been introduced. In addition, optimisation strategies are introduced to determine this pressure and vertical load dependent lateral stiffness from direct measurements or relaxation length experiments. Considerably more accurate results are achieved. • Longitudinal relaxation behaviour A pressure and load dependent relation for the longitudinal stiffness has been introduced. In addition, optimisation strategies are introduced based on the strategies as described for the lateral relaxation behaviour. • Rolling resistance force The rolling resistance force can be described more accurately by implementing two exponential relations that describe the inflation pressure and vertical load change in combination with the nominal rolling resistance force parameterisation (i.e. the rolling resistance force at nominal inflation pressure and vertical load condition). • Camber thrust and camber torque By implementing inflation pressure dependency in the camber thrust stiffness equation, the overturning moment equation and the aligning moment equation of the Magic Formula model, the influence of the inflation pressure on the camber thrust and camber torque can be described more accurately. For camber angles larger than the common passenger car camber angle range the results are less accurate. The cause is that for a combination of Flatplank pure camber data and TNO Tyre Test Trailer slip data the Magic Formula fitting is less accurate. This can be caused by the deviations between Flatplank data, measured at low velocity on a steel surface, and TNO Tyre Test Trailer data measured at relatively high velocity on an asphalt road. • Tyre dynamics The different sidewall stiffnesses of the rigid ring model can be described and estimated rather accurately for a certain inflation pressure range (i.e. pi,nom ±1 bar) using the Rotta Membrane Theory. It has been shown that the primary eigenmodes of a free hanging tyre can be estimated rather well. At pi =0 bar this approach gives implausible results, because Rotta neglects the influence of the tyre material flexibility. Scaling the pi,nom sidewall stiffnesses with the inflation pressure describes the sidewall stiffnesses for a certain inflation pressure range more accurately. Also outside the mentioned inflation pressure range (e.g. 0 bar), this approach seems to give plausible results.
7.2 Recommendations for future research
7.1.3
77
Applicability to the vehicle design
Within the framework of applicability to the vehicle design it is shown that with the implementation of inflation pressure dependency in the MF-Swift tyre model, the vehicle behaviour (i.e. handling and ride comfort) can be analysed in an accurate way. The inflation pressure sensitivity of the tyre behaviour (i.e. change in: stiffnesses, transient behaviour, enveloping behaviour, etc.) on the ride comfort and (steady-state) handling behaviour can be predicted and the inflation pressure selection for a certain loading condition of the vehicle can be estimated. The inflation pressure enhancements are a useful addition to the MF-Swift tyre model.
7.1.4
Final conclusion
Considering the objectives of this thesis, it is concluded that all objectives formulated at the beginning of this research project have been achieved in a satisfying way. The tyre models have undergone significant improvements, although several topics will require further investigations. These topics are listed in the next section where recommendations for further research are formulated.
7.2
Recommendations for future research
In future research on the area of the inflation pressure influence on tyre behaviour, the following topics require further investigations: • Camber effects For a symmetric tyre, the same camber thrust / torque level is expected when applying a positive or negative camber angle. The Flatplank measurements show a dissimilarity between positive and negative camber angles. Furthermore, the Flatplank results show some deviation when comparing with TNO Tyre Test Trailer results, especially the cornering stiffness shows significant deviations. This was also observed in previous research projects. The camber effects experiments as performed on the Flatplank have to be executed with the TNO Tyre Test Trailer to check the Flatplank measurements and examine if the discovered camber thrust / torque trends can be reproduced. • Flatplank Tyre Tester Within the measurement results of the tyre lateral stiffness and the tyre torsional stiffness some irregularities can be seen. Most probably, these irregularities are stick-slip effects caused by a lack of grip on the Flatplank Tyre Tester. To solve this problem, the road surface on the Flatplank tyre tester can be adapted such that a higher level of friction is obtained. Furthermore, within the measurement results of the pure camber effects (i.e. camber thrust and camber torque) some deviance occurs between positive and negative camber angles. This can be caused by a wrong or off calibration of the measuring hub. To exclude this option, a calibration of the measurement hub is advisable. • Rigid ring tyre dynamics The two proposed approaches to estimate the sidewall stiffnesses (i.e. Rotta membrane theory and scaling nominal sidewall stiffnesses) are only validated for inflation pressures above 1.5 bar, because FEM analyses are only performed for the inflation pressure range 1.5 - 3.0 bar. To investigate the accuracy of the two approaches for inflation pressures below 1.5 bar, it is recommended to perform FEM analyses for inflation pressures below 1.5 bar. The proposed approaches are only validated with FEM simulation data of a free hanging and non-rolling tyre. To create a wider validation support, high speed cleat experiments have to be performed with the tyre used in this thesis. It should be checked that the eigenfrequencies found
78
Conclusions and Recommendations in the cleat experiments at different inflation pressures can be described with the proposed relations. The nominal sidewall stiffnesses scaling strategy is optimised to describe the FEM results as accurately as possible. Currently fixed values for these parameters have been selected and implemented in the MF-Swift tyre model. However, these parameters have to be implemented in the MF-Swift optimisation routine in such a way that they are optimised for the tyre that is being processed. • Validation of the enhanced relations In this thesis the enhanced relations are validated with one single type of tyre. It is recommended to conduct more measurements with different type of tyres and a wider inflation pressure range to investigate if the enhanced relations still hold and to create a wider validation base. • Enhanced tyre models Now all the necessary inflation pressure enhancements are separately validated and implemented in the MF-Swift tyre model, it is interesting to investigate if the accuracy of the model is increased when analysing experiments (e.g. ABS braking on an uneven road) where the complete (dynamical) behaviour of a tyre is important. With the enhanced tyre models it should be possible to describe the tyre behaviour more accurately for a certain inflation pressure range.
Initially, the investigation of the turnslip behaviour was also a part of the research project. Later on, this part is left out because of the extensive inflation pressure part. However, a lead for the investigation of tyre turnslip behaviour will be given. Several tests have already been executed on the Flatplank Tyre Tester, which seem to confirm that improvements to the MF-Swift tyre model are possible. The following topics can be a lead for the investigation: • Parameter identification method The goal is to develop a simple and reliable fitting process, with a focus on an automated parameter identification method and a critical review of the required tests. • Standstill situation The representation of the turnslip forces and moments at standstill and the transition to low forward velocity need to be analysed.
Bibliography [1] ABAQUS. Example Problems Manual. ABAQUS inc., 2006.
ABAQUS Documentation, Version 6.6, Section 3.1,
[2] TNO Automotive. Tyre models users manual; Using the MF-Tyre Model. TNO Automotive, May 2002. [3] E. Bakker, H.B. Pacejka, and L. Lidner. A New Tire Model with an Application in Vehicle Dynamics Studies. SAE Paper No. 890087, 1989. [4] P. Bandel and C. Monguzzi. Simulation Model of the Dynamic Behaviour of a Tire Running Over an Obstacle. Tire Science and Technology, TSTCA, Volume 16, No. 2, 1988. [5] F. B¨ ohm. Zur Mechanic des Luftreifens. Technische Hochschule Stuttgart, 1966. [6] I.J.M. Besselink. Shimmy of Aircraft Main Landing Gears. Dissertation, TU Delft, 2000. [7] I.J.M. Besselink, L.W.L. Houben, and I.B.A. op het Veld. Run-Flat versus Conventional Tyres: An experimental and model based comparison. VDI Berichte, No. 2014, pp. 185-202, 2007. [8] M. Biancolini and R. Baudille. Integrated Multibody / FEM Analysis of Vehicle Dynamic Behaviour. The 29th FISITA World Automotive Congress, Helsinky, 2002. [9] S.K. Clark. Mechanics of Pneumatic Tires. U.S. Dept. of Transportation, National Highway Traffic Safety Administration, Washington D.C., 1982. [10] B. Collier and J. Warchol. The Effect of Inflation Pressure on Bias, Bias-Belted and Radial Tire Performance. SAE Paper No. 800087, 1980. [11] R. Cremers. Investigating Dynamic Tyre Behaviour. Master’s thesis, Technical University Eindhoven, DCT 2005.37, 2005. [12] J. de Hoogh. Implementing Inflation Pressure and Velocity Effects into The Magic Formula Tyre Model. Master’s thesis, Eindhoven University of Technology,2005.46, 2005. [13] N. Dodge, D. Orne, and S. Clark. Fore-and-aft Stiffness Characteristics of Pneumatic Tyres. NASA Contractor Report, CR-900, Washington D.C., 1967. [14] D.J. Erp and L. Verhoeff. The Development of Tyre Modeling at Various Direction Input. TNO Automotive Report, 04.OR.AC.016.1/LV, Helmond, 2004. [15] M. Gipser. BRIT version 2. Documentation, Reading Massachusetts, 1980. [16] S. Gong. A Study of In-plane Dynamics of Tires. Dissertation, TU Delft, 1993. [17] P.S. Grover. Modeling of Rolling Resistance Test Data. SAE Paper No. 980251, 1998. [18] P.S. Grover and S.H. Bordelon. New Parameters for comparing Tire Rolling Resistance. SAE Paper No. 1999-01-0787, 1999. 79
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[19] A. Higuchi. Transient Response of Tyres at Large Wheel Slip and Camber. Dissertation, TU Delft, 2000. [20] ISO. ISO 8767: Passenger car tyres, Methods of measuring rolling resistance. International Organization for Standardization, 1992. [21] ISO. ISO 4138: Passenger Cars, Steady-state Circular Driving Behaviour, Open-loop Test Procedure. International Organization for Standardization, 1996. [22] ISO. ISO 2631: Mechanical vibration and shock, Evaluation of human exposure to whole-body vibration. International Organization for Standardization, 1997. [23] ISO. ISO 7401: Road Vehicles, Lateral Transient Response Test Methods, Open-loop Test Methods. International Organization for Standardization, 2003. [24] I. Kageyama and S. Kuwahara. A study on Tire Modeling on Camber Thrust and Camber Torque. JSAE Review, No. 23, pp. 325-331, 2002. [25] K.M. Marshek and J.F. Cuderman. Performance of Anti-Lock Braking system Equipped Passenger Vehicles-Part III: Braking as a function of Tire Inflation Pressure. SAE Paper No. 2002-01-0306, 2002. [26] J.P. Maurice. Short Wavelength and Dynamic Tyre Behaviour under Lateral and Compined Slip Conditions. Dissertation, TU Delft, 2000. [27] Michelin. Der Reifen. Michelin Reifenwerke KGaA, [ISBN 2-06-711658-4], 2005. [28] H.B. Pacejka. Tyre and Vehicle Dynamics. Butterworth-Heinemann, 2002. [29] A. Popov, D. Cole, D. Cebon, and C. Winkler. Laboratory Measurement of Rolling Resistance in Truck Tyres under Dynamic Vertical Load. Proceedings of the Institution of Mechanical Engineers, Volume 217, No. 12, pp. 1071-1079, 2002. [30] J. Rotta. Zur Statik des Luftreifens. Ingenieur-Archive, Volume 17, pp. 12-141, Berlin, 1949. [31] A.J.C. Schmeitz. A Semi-Emperical Three-Dimensional Model of the Pneumatic Tyre Rolling over Arbitrarily Uneven Road Surfaces. Dissertation, TU Delft, 2004. [32] A.J.C. Schmeitz, J. de Hoogh, I.J.M. Besselink, and H. Nijmeijer. Extending The Magic Formula and SWIFT Tyre Models for Inflation Pressure Changes. In: 10th International VDI Congress, Hannover, 2005. [33] R. Smiley and W. Horne. Mechanical properties of Pneumatic Tires with Special Reference to Modern Aircraft Tires. Technical Report NASA, No. R-64, Langley Research Center, USA, 1960. [34] H.J. Unrau and J. Zamow. TYDEX; Description and Reference Manual. Release 1.1, International Tire Working Group, 1995. [35] P. van der Jagt. The Road to Virtual Vehicle Prototyping. Dissertation, TU Eindhoven, 2000. [36] J. van Honk, C.W. Klootwijk, and F.W. Laméris. Results of handling behaviour test program, Maximum load condition. TNO Automotive Report, 96.MR.VD.038.1/JVH, 1996. [37] J. van Honk, C.W. Klootwijk, and F.W. Laméris. Results of handling behaviour test program, Minimum load condition. TNO Automotive Report,96.MR.VD.039.1/JVH, 1996. [38] V.C. Vlad. THERMAL TYRE: An Analysis Tool for Tire Temperature Modeling. TNO Automotive Report, 2001. [39] L.H. Yam, D.H. Guan, and A.Q. Zhang. Three-dimensional Mode Shapes of a Tire Using Experimental Modal Analysis. Experimental Mechanics, Volume 40, No. 4, pp. 369-375, 2000.
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81
[40] P.W.A Zegelaar. The Dynamic Response of Tyres to Brake Torque Variations and Road Unevennesses. Dissertation, TU Delft, 1998.
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BIBLIOGRAPHY
Appendix A
The Flatplank Tyre Tester A.1
General description
With the Flatplank Tyre Tester measurements can performed to investigate tyre characteristics: (combined) slip characteristics, relaxation lengths, stiffnesses and tyre response to road irregularities. In figure A.1 the Flatplank Tyre Tester is depicted. The Flatplank consists of a flat steel road surface, measuring hub and a turn table. Furthermore, it consists of mechanisms to adjust the axle height, the lateral position and the camber angle. The road surface is positioned upside down and can be moved with a constant velocity. To create a slip angle, the tyre and the measuring hub can be rotated about the vertical axis of the turn table. To change the lateral position of the tyre with respect to the road surface, the measuring hub can be moved in lateral direction. The axle height can be adjusted by a jack (constant axle height) or by an air spring system (constant vertical load). The camber angle can be adjusted by rotating the road surface around the longitudinal axis on the middle of its surface and/or by rotating the measuring hub around its longitudinal axis. Different obstacles (cleats) with a maximum height of 30 mm can be mounted on the road surface to create road irregularities. road camber mechanism
obstacle
strain gauge measuring hub
wheel carrier
road surface (plank)
air spring system
Figure A.1: The Flatplank Tyre Tester [31].
The steel surface of the plank is covered with 3M Safetywalk grid 80 to give certain roughness to the road surface. The road surface velocity can be varied between 0 and 4.75 cm/s. Furthermore, the measuring hub measures reaction forces and moments in the wheel axle. The following forces and moments are measured: longitudinal force Fx , lateral force Fy , vertical force Fz , overturning moment Mx and aligning moment Mz . The measuring hub can measure forces (up to 10 kN) and moments by using five strain gauge bridges, see next section. The angle and velocity of rotation of the wheel, and the displacement and velocity of the road surface are measured with incremental encoders. Furthermore, the slip angle and the vertical displacement of the measuring hub are measured. The measured signals are sent to the signal conditioning system. This system contains an amplifier, power supply, low-pass 83
84
The Flatplank Tyre Tester
filter and an A/D converter. After the A/D conversion the data is sampled by a data acquisition program (in this case LABView) which is triggered by a digital triggering signal.
A.2
Forces and moments transformations
Figure A.2 depicts the measuring hub. As mentioned before, the measuring hub measures the reaction forces and moments by using five strain gauge bridges. The strain gauge bridges sense the longitudinal forces (Gx1 and Gx2 ), the lateral force (Gy ) and the vertical forces (Gz1 andGz2 ). To calculate the moments at the wheel centre the distance a between the forces Gx1 and Gx2 and the distance b between Gx2 and the wheel centre plane are used.
Figure A.2: Measuring hub [19].
The forces K and moments T at the wheel centre are defined as follows: Kx Ky Kz Tx Ty Tz
= = = = = =
Gx1 + Gx2 Gy Gz1 + Gz2 Gz1 (a + b) + Gz2 b 0 Gx1 (a + b) + Gx2 b
(A.1) (A.2) (A.3) (A.4) (A.5) (A.6)
The coordinate system of the measuring hub and at the contact centre is depicted in figure A.3. The forces and moments measured at the measuring hub are transformed to forces and moments at the contact patch (ISO coordinate system). According to figure A.3, the transformations are as follows: α γ Fx Fy Fz Mx My Mz
= = = = = = = =
ψ φwheel + φplank Kx −Ky cos γ − Kz sin γ −Ky sin γ + Kz cos γ −Tx + Ky rl −Tz sin γ + Kx rl cos γ Tz cos γ + Kx rl sin γ
(A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14)
A.2 Forces and moments transformations
85
Note that rl is the loaded radius and consequently has a positive value in these equations. Furthermore φwheel and φplank are the inclination angle of the wheel and the steel surface plank respectively.
Figure A.3: Coordinate systems of measuring hub (no ISO) and contact patch centre (ISO).
In general when studying tyre dynamics, two different types of coordinate systems are used. The coordinate systems are the C-axis system and the W-axis system. In figure A.4 the orientation of these axis systems is depicted.
Figure A.4: C- and W-axis system [2].
These axis systems are in accordance with the standard TYDEX conventions [34]. The C-axis system is the centre axis system with its origin fixed in the wheel centre. The W-system is the standard ISO based axis system with its origin at the wheel intersection point with the road, shown in figure A.3. To transform the measuring hub forces and moments to the C-axis system, the following transformations hold: FxC FyC FzC MxC MyC MzC
= = = = = =
Kx −Ky Kz −Tx 0 Tz
(A.15) (A.16) (A.17) (A.18) (A.19) (A.20)
86
The Flatplank Tyre Tester
Appendix B
Measurement Results As described in chapter 4, the measurements are carried out at three vertical load conditions. The results of the nominal vertical load conditions are discussed in the different sections of chapter 4. In this Appendix the result of the other vertical load conditions are presented. First the results of the characteristic stiffnesses are discussed, after that the remaining results of the lateral relaxation behaviour and the camber influence experiments are presented.
B.1 B.1.1
Characteristic stiffnesses Tyre longitudinal stiffness
10000
10000
9000
9000
8000
8000 Longitudinal force, Fx [N]
Longitudinal force, Fx [N]
This subsection presents the results of the tyre longitudinal stiffness measurements for the following vertical load conditions: 2 and 6 kN. In figure B.1, the experimental results are presented. The linear fits of the selected linear part are depicted in figure B.2.
7000 6000 5000 4000 3000 p19 p22 p25 p27 p30
2000 1000 0 0
0.01
0.02 0.03 0.04 0.05 Longitudinal displacement [m]
0.06
7000 6000 5000 4000 3000 p19 p22 p25 p27 p30
2000 1000 0 0
0.07
0.01
0.02 0.03 0.04 0.05 Longitudinal displacement [m]
0.06
0.07
Figure B.1: Tyre longitudinal stiffness; experimental results for Fz =2 kN (left) and Fz =6 kN (right).
From the graph of Fz =2 kN n figure B.1, it can be seen that at an inflation pressure of 1.9 bar fluctuations occur when reaching the steady state longitudinal force level. These fluctuations are the result of stick-slip due to the insufficient friction level of the Flatplank combined with the low inflation pressure 87
88
Measurement Results
18000
18000
16000
16000
14000
14000 Longitudinal force, Fx [N]
Longitudinal force, Fx [N]
and the low vertical load condition. The stick-slip situation does not occur during the build up of the longitudinal force, so there is no influence on the determination of the longitudinal stiffness.
12000 10000 8000 6000
10000
p19 p22 p25 p27 p30
4000 2000 0 0
12000
0.01
0.02 0.03 0.04 0.05 Rel. longitudinal displacement [m]
0.06
8000 6000 p19 p22 p25 p27 p30
4000 2000 0 0
0.07
0.01
0.02 0.03 0.04 0.05 Rel. longitudinal displacement [m]
0.06
0.07
Figure B.2: Tyre longitudinal stiffness; linear fit results for the selected linear part for Fz =2 kN (left) and Fz =6 kN (right).
B.1.2
Tyre lateral stiffness
8000
8000
7000
7000
6000
6000 Lateral force, Fy [N]
Lateral force, Fy [N]
Following section 4.1.3, in this section the tyre lateral stiffness measurement results for the vertical load conditions of 3 and 7 kN are presented. Figure B.3 shows the results of the lateral force versus the lateral displacement. The linear fits for the determination of the tyre lateral stiffness are depicted in figure B.4. Again, the fluctuations that occur in several measurements are the result of stick-slip, which occurs due to the low friction level of the Flatplank.
5000 4000 3000 2000
p19 p22 p25 p27 p30
1000 0 0
0.01
0.02
0.03 0.04 0.05 Lateral displacement [m]
0.06
0.07
5000 4000 3000 2000
p19 p22 p25 p27 p30
1000 0 0
0.01
0.02
0.03 0.04 0.05 Lateral displacement [m]
0.06
0.07
Figure B.3: Tyre lateral stiffness; experimental results for Fz =3 kN (left) and Fz =7 kN (right).
89
8000
8000
7000
7000
6000
6000 Lateral force, Fy [N]
Lateral force, Fy [N]
B.1 Characteristic stiffnesses
5000 4000 3000 2000
p19 p22 p25 p27 p30
1000 0 0
0.01
0.02 0.03 0.04 0.05 Rel. lateral displacement [m]
0.06
0.07
5000 4000 3000 2000
p19 p22 p25 p27 p30
1000 0 0
0.01
0.02 0.03 0.04 0.05 Rel. lateral displacement [m]
0.06
0.07
Figure B.4: Tyre lateral stiffness; linear fit results for the selected linear part for Fz =3 [kN] (left) and Fz =7 [kN] (right).
B.1.3
Tyre torsional stiffness
The parking experiments are performed for the five generally defined inflation pressures and for a wider range of vertical load conditions; FzW =1, 2, 3, 4, 5 and 6 kN. In section 4.1.4 the results are presented for the vertical load conditions: FzW =2 and 4 kN. Here, an overview of the remaining Mz results is given in figure B.5. From left to right and from top to bottom the plots refer to the following vertical load conditions: 1, 3, 5 and 6 kN respectively. To improve readability and visibility of the differences in the separate inflation pressures, the graphs are plotted at different y-axis scaling.
90
Measurement Results
25
150 p19 p22 p25 p27 p30
20
100 Aligning moment, Mz [Nm]
Aligning moment, Mz [Nm]
15
p19 p22 p25 p27 p30
10 5 0 −5 −10 −15
50
0
−50
−100
−20 −25 −25
−20
−15
−10
−5 0 5 Steer angle, ψ [deg]
10
15
20
−150 −25
25
250
−15
−10
−5 0 5 Steer angle, ψ [deg]
10
15
20
25
300 p19 p22 p25 p27 p30
200
p19 p22 p25 p27 p30
200 Aligning moment, Mz [Nm]
150 Aligning moment, Mz [Nm]
−20
100 50 0 −50 −100 −150
100
0
−100
−200
−200 −250 −25
−20
−15
−10
−5 0 5 Steer angle, ψ [deg]
10
15
20
25
−300 −25
−20
−15
−10
−5 0 5 Steer angle, ψ [deg]
10
15
20
25
Figure B.5: Parking behaviour; experimental results for different vertical load conditions: 1 kN (top left), 3 kN (top right), 5 kN (bottom left) and 6 kN (bottom right).
B.2 Lateral relaxation behaviour
B.2
91
Lateral relaxation behaviour
This section presents the results of the lateral relaxation behaviour measurements for the following vertical load conditions: 3 and 7 kN. In figure B.6 the experimental results are presented for the 1 degree sideslip lateral relaxation length experiment. Unfortunately afterwards it appeared that the measurement duration of the 7 kN vertical load condition is pretty tight in order to reach a clear steady state Fy,ss level. An extrapolation of the exponential function, used to fit the measurement data, is made to estimate the correct Fy,ss level. 0
0 p19 p22 p25 p27 p30
−500
Lateral force, Fy [N]
Lateral force, Fy [N]
−500
−1000
−1500
−2000
p19 p22 p25 p27 p30
−1000
−1500
−2000
−2500 −0.5
0
0.5 1 1.5 Track displacment, dx [m]
2
2.5
−2500 0
0.5
1 1.5 2 Track displacment, dx [m]
2.5
3
Figure B.6: Lateral relaxation length vs. inflation pressure for FzW =3 kN (left) and FzW =7 kN (right).
B.3
Camber thrust and camber torque
In section 4.4 the camber thrust Fyγ , aligning camber torque Mzγ and overturning camber torque Mxγ in relation to the inflation pressure for the nominal vertical load condition are presented. Here, the results of the remaining vertical load conditions, FzW =3 and 7 kN, are presented. Figure B.7 shows the camber thrust for FzW = 3 and 7 kN respectively. −15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
Camber thrust, Fyγ [N]
600
400
200
0
−200
−400
−600 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
2000
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
1500 1000 Camber thrust, Fyγ [N]
800
500 0 −500 −1000 −1500 −2000 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
Figure B.7: Camber thrust vs. inflation pressure for various positive and negative camber angles for FzW =3 kN (left) and FzW =7 kN (right).
92
Measurement Results
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
Camber torque, Mzγ [Nm]
30
20
10
0
−10
−20
100
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
80 60 Camber torque, Mzγ [Nm]
40
40 20 0 −20 −40 −60 −80
−30 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
−100 1.8
3
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
Figure B.8: Aligning camber torque vs. inflation pressure for various positive and negative camber angles for FzW =3 kN (left) and FzW =7 kN (right).
The resulting aligning camber torque for the remaining vertical load conditions is depicted in figure B.8. Taking a closer look at the aligning camber torque trend for the camber range [-5 5] degrees in figure B.8 an opposite trend is visible compared to figure B.8(right) and figure 4.17. In figure B.8(right) the aligning camber torque increases with increasing inflation pressure while in figure B.8(left) and 4.17 a decreasing trend is visible. As mentioned in section 4.4, the tested tyre has a symmetrical tread so the amount of camber thrust and aligning camber torque for positive and negative camber angles would be expected to be equal. However, figures B.7 till B.8 show that a difference occurs when cambering with a positive or a negative angle. The difference can be caused by a deviation or a wrong calibration of the measuring hub. In figure B.9 the overturning camber torque is presented for FzW = 3 and 7 kN. It is shown that when increasing the inflation pressure also the amount of overturning camber torque increases. Again, an asymmetric behaviour is visible between positive and negative camber angles.
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
200
Camber torque, Mxγ [Nm]
150 100 50 0 −50 −100 −150 −200 −250 1.8
500
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
400 300 Camber torque, Mxγ [Nm]
250
200 100 0 −100 −200 −300 −400
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
−500 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, pi [bar]
3
Figure B.9: Overturning camber torque vs. inflation pressure for various positive and negative camber angles for FzW =3 kN (left) and FzW =7 kN (right).
Finally, to quantify the influence of the inflation pressure, table 4.7 shows the amount of influence on the camber thrust, aligning camber torque and overturning camber torque. The influence is subdivided in two camber ranges. The first range is limited to -5 and 5 degrees (common range for passenger car tyres) and the second range is limited to -15 to 15 degrees (complete measurement range).
B.3 Camber thrust and camber torque
93
Table B.1: The influence of the inflation pressure on camber effects.
Influence inflation pressure εA [%] FzW [kN] Camber range [deg] Fyγ Mzγ 3 -5...5 34.0 13.0 -15...15 15.0 18.0 7 -5...5 14.0 23.0 -15...15 17.0 16.5
Mxγ 9.0 1.0 9.5 2.0
94
Measurement Results
Appendix C
Modal Analysis Finite Element Method (FEM) analysis and analytical analysis of the flexible ring model and the rigid ring model are performed to get a clear view on the influence of the inflation pressure. In this Appendix the FEM simulations with ABAQUS and the analytical analysis are discussed in more detail.
C.1
FEM Simulations (ABAQUS)
In chapter 4, it is shown that the inflation pressure influence is analysed by means of Finite Element Analysis. In the FEM software ABAQUS a 3 dimensional model of a 175 SR14 passenger car tyre is available . In this Appendix a short explanation of the model structure is given, which is derived from the ABAQUS problems manual [1], and the eigenmodes of the free hanging tyre and standing loaded tyre are presented.
Figure C.1: Construction sequence in ABAQUS of the 3D tyre model [1].
95
96
Modal Analysis
The construction sequence of the 3 dimensional model is illustrated in figure C.1. The tyre tread and sidewalls are made of rubber, and the belts and carcass are constructed from reinforced rubber composites. The rubber is modeled as an incompressible hyperelastic material, and the reinforced composites are modeled as a linear elastic material. First a 2 dimensional axisymmetic model is created of the sidewall, tread, reinforced carcass and belts. The 2 dimensional model is rotated 360 degrees in steps of 10 degrees, to create a partial 3 dimensional model (half tyre). Finally the partial 3 dimensional model is mirrored to a full 3 dimensional model. The air inside the tyre is modeled with acoustic elements, which obviously are made of air. In figure C.2 air elements as used in the tyre model are depicted.
Figure C.2: Air elements inside the tyre model [1].
C.1.1
Free hanging tyre
The primary eigenmodes of a free hanging tyre with fixed rim are discussed in chapter 5. In this section a visualisation of the mode shapes is given. Figure C.3 and C.4 show the in-plane and out-of-plane primary mode shapes respectively.
Figure C.3: In-plane primary mode shapes of a free hanging tyre: rotational(left) and radial(right).
C.1 FEM Simulations (ABAQUS)
97
Figure C.4: Out-of-plane primary mode shapes of a free hanging tyre: lateral(left), yaw and camber(right).
C.1.2
non-rolling loaded tyre
In this section a visualisation of the primary mode shapes is given for a non-rolling loaded tyre with Fz =4 kN. Figure C.5 and C.6 show the in-plane and out-of-plane primary mode shapes respectively. Note that the contact patch nodes are modelled to be welded/fixed to the road.
Figure C.5: In-plane primary mode shapes of a non-rolling loaded tyre: rotational(left), vertical(middle) and longitudinal(right).
Figure C.6: Out-of-plane primary mode shapes of a non-rolling loaded tyre: lateral/camber(left) and yaw(right).
98
C.2
Modal Analysis
Analytical analysis
The rigid ring model and the flexible ring model are used to determined the influence of the inflation pressure on the primary eigenmodes of a tyre in an analytical way. The analytical analysis is based on the findings of Zegelaar [40] and Gong [16]. The following situation is used: a free hanging tyre (flexible ring model) with a fixed rim, to analyse the primary modes (i.e. eigenmodes in which the tyre tread band almost retains its circular shape). The flexible ring model is based on the findings of Gong [16], see figure C.7. The parameters used in the analysis are derived from Zegelaar [40] and depicted in table C.1.
Figure C.7: Flexible ring model and carcass cross section [40].
Table C.1: Used parameters derived from [40].
Quantity mb Iby r ls t G ϕs
[kg] [kgm2 ] [m] [m] [m] [N/m2 ] [deg]
V alue 7.1 0.636 0.300 0.121 0.010 1.6e6 62.3
Parameters Description mass of tyre ring moment of inertia that moves with the tyre ring tyre ring radius length of sidewall arc thickness of tyre sidewall shear modules of tyre sidewall half the angle of tyre sidewall
Following Gong and Zegelaar, the tangential sidewall stiffness cbv and the radial sidewall stiffness cbw of the flexible ring model can be determined by: cbv cbw
Gt 1 + pi ls tan ϕs cos ϕs + ϕs sin ϕs = pi sin ϕs − ϕs cos ϕs =
(C.1) (C.2)
Because only the primary modes are analysed, the flexible ring model can be seen as a rigid ring model, see figure C.8.
C.2 Analytical analysis
99
The translational cb and rotational cbθ sidewall stiffness of the rigid ring model, see figure C.8, are related to cbv and cbw of the flexible ring model as follows [40]: cb cbθ
= πr(cbv + cbw ) = 2πcbv r3
(C.3) (C.4)
Figure C.8: Rigid ring model [40].
The translational eigenfrequency ftrans and the rotational eigenfrequency frot of the free hanging rigid ring model can be determined with the following expressions: r 1 cb ftrans = (C.5) 2π mb r 1 cθb frot = (C.6) 2π Iby In figure C.9 the resulting eigenfrequencies are depicted as function of the inflation pressure. Again, a linear trend can be observed between the primary eigenmodes and the inflation pressure. 70 ftrans
Natural frequency [Hz]
65
frot
60
55
50
45
40
35 1.5
2 2.5 Inflation pressure, pi [bar]
3
Figure C.9: Eigenfrequency versus inflation pressure for rigid ring model of a free hanging tyre.
100
Modal Analysis
Appendix D
Aspects of the introduced inflation pressure dependent relations In this Appendix important aspects of the proposed enhancements and introduced relations are treated. First, the relation between the torsional stiffness and the vertical load as derived from the Flatplank measurements will be presented. After that, the rolling resistance force data, which is used to assess the proposed rolling resistance force enhancement, is treated and the robustness of the (carcass)stiffnesses equations of the rigid ring model are presented. Finally, the extrapolation properties of the camber thrust and torque equations are treated.
D.1
Tyre torsional stiffness
In this section, the relation between the determined torsional stiffness and the vertical force load of the Flatplank measurements is presented. The torsional stiffness values are derived from the parking behaviour Flatplank experiments. The results are depicted in figure D.1. It can be seen that a more or less linear relation exists between the torsional stiffness and the vertical load. At the different vertical load conditions, the inflation pressures do not show an unambiguous trend; the maxima and minima fluctuate. For example the highest torsional stiffness at Fz = 4 kN is when pi = 1.9 bar, while at Fz = 2 kN the highest torsional stiffness is shown at pi = 2.2.
Figure D.1: Torsional stiffness as function of the vertical load, Flatplank measurements and linear trend lines.
101
102
D.2
Aspects of the introduced inflation pressure dependent relations
Rolling resistance force
As mentioned in section 5.2, no suitable rolling resistance data at different inflation pressures is available from the measured tyre in this thesis or another tyre in the TNO tyre database. Therefore, a selection of data found in literature [18] will be used to evaluate the quality of the new formula. The measurements in this publication are performed according to the ISO 8767 international standard [20]. Table D.1: Rolling resistance measurement data of a 195/70R14 tyre [18].
Rolling resistance conditions Vx [m/s] pi [bar] Fz [N] Fx,RR [N] 2.5 1800 14 2.75 2380 17 5.6 2.0 3560 27 3.0 5360 35 2.0 5360 41 2.5 1800 15 2.75 2380 18 9.7 2.0 3560 29 3.0 5360 37 2.0 5360 43 2.5 1800 16 2.75 2380 19 15.3 2.0 3560 31 3.0 5360 39 2.0 5360 46 2.5 1800 21 2.75 2380 25 30.6 2.0 3560 40 3.0 5360 48 2.0 5360 59
An overview of the optimised Magic Formula parameters of the "old" relation, equation (5.20), and the "new" relation, equation (5.21), are presented in table D.2. Table D.2: Overview of the Magic Formula parameter values for the "old" rolling resistance relation (5.20) and the "new" rolling resistance relation (5.21).
Rolling resistance Magic Formula parameters P arameter equation (5.20) equation (5.21) qsy1 0.00673 0.00730 qsy2 0.0 0.0 qsy3 0.00144 0.00157 qsy4 0.00008 0.00009 qsy5 0.0 0.0 qsy6 0.0 0.0 qsy7 − 0.9 qsy8 − −0.409
D.3 Stiffnesses of the rigid ring model extrapolation properties
D.3
103
Stiffnesses of the rigid ring model extrapolation properties
In this section an overview of the lateral and longitudinal (carcass) stiffnesses of the rigid ring model for two extreme vertical load conditions, Fz =0.1 and 10 kN, is given. Note that these stiffnesses are estimated using the, in [32] and in this thesis, introduced inflation pressure relations and using extrapolation. The results presented in this section are for robustness interpretation only.
D.3.1
Lateral situation
In figure D.2 an overview of the stiffnesses for the lateral rigid ring model is depicted. The stiffnesses for the extreme load conditions are estimated using MF-Tool, equation (5.15) and the strategy as described in sections 5.7.2 and 5.7.3. The stiffnesses show robust results for the extreme load conditions, no negative values arise. 6
6
5
x 10
14
C
x 10
C
Fy
Fy
CFα/a cy
5
cbγ/r2l
cbγ/r2l c
by
cry Trend line 3
Stiffness [N/m]
Stiffness [N/m]
cy
10
c
4
CFα/a
12
by
cry
8
Trend line
6
2 4 1
0 0
2
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
0 0
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
Figure D.2: Overview lateral (carcass) stiffnesses as function of the inflation pressure for Fz =0.1 kN (left) and Fz =10.0 kN (right).
104
Aspects of the introduced inflation pressure dependent relations
D.3.2
Longitudinal situation
In figure D.3 an overview of the stiffnesses for the longitudinal rigid ring model is depicted. The stiffnesses for the extreme load conditions are estimated using MF-Tool, equation (5.19) and the strategy as described in sections 5.7.2 and 5.7.4. In accordance with the lateral situation, the longitudinal stiffnesses show robust results for the extreme load conditions. 6
2.5
6
x 10
3
C
x 10
C
Fx
2
Fx
CFκ/a
CFκ/a
cx
cx
2.5
cbθ/r2l
cbθ/r2l
c
Stiffness [N/m]
Stiffness [N/m]
c
2
bx
crx
1.5
Trend line
1
bx
crx Trend line 1.5
1 0.5
0 0
0.5
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
0 0
3
0.5
1 1.5 2 Inflation pressure, pi [bar]
2.5
3
Figure D.3: Overview longitudinal (carcass) stiffnesses as function of the inflation pressure for Fz =0.1 kN (left) and Fz =10.0 kN (right).
D.4
Camber thrust and camber torque extrapolation
In this section an overview is given of the camber thrust and both the aligning and overturning camber torque for two extreme vertical load conditions (Fz =0.1 and 10 kN). In table D.3 an overview is given of the manually tuned parameters of the in Section 5.6 proposed equations to describe the camber thrust and camber torque. The parameters are tuned for the nominal vertical load condition (Fz,nom =5 kN) and the common passenger car camber range (i.e. till γ ≤+/- 5.0 degrees). In the figures D.4, D.5 and D.6 the Magic Formula results are presented for various positive and negative camber angles; the markers indicate the evaluated camber angles (i.e. in this case the markers do not represent measurement points). The enhancements, proposed in section 5.6, show robust results for the extreme load conditions. Note that the results for Fz =0.1 and 10 kN are produced with the camber parameters as presented in table D.3. Table D.3: Overview of the manually tuned camber parameters.
P arameter ppy5 ppM x1 ppz2
V alue −0.15 −0.10 −0.60
Equation (5.22) (5.25) (5.27)
D.4 Camber thrust and camber torque extrapolation 30
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.0 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
Camber thrust, Fyγ [N]
20
10
0
−10
−20
−30
−40 1.8
105
2
2.2 2.4 2.6 2.8 Inflation pressure, p [bar]
3
i
4000
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.0 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
3000
Camber thrust, Fyγ [N]
2000 1000 0 −1000 −2000 −3000 −4000 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, p [bar]
3
i
Figure D.4: Camber thrust vs. inflation pressure for various positive and negative camber angles at two extreme vertical load conditions; Fz =0.1 kN (top) and Fz =10 kN (bottom).
106
Aspects of the introduced inflation pressure dependent relations 60
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.0 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
Camber torque, Mxγ [Nm]
40
20
0
−20
−40
−60 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, p [bar]
3
i
400
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.0 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
300
Camber torque, Mxγ [Nm]
200 100 0 −100 −200 −300 −400 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, p [bar]
3
i
Figure D.5: Overturning camber torque vs. inflation pressure for various positive and negative camber angles at two extreme vertical load conditions; Fz =0.1 kN (top) and Fz =10 kN (bottom).
D.4 Camber thrust and camber torque extrapolation 2
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.0 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
1.5
Camber torque, Mzγ [Nm]
1 0.5 0 −0.5 −1 −1.5 −2 1.8
107
2
2.2 2.4 2.6 2.8 Inflation pressure, p [bar]
3
i
150
−15.0 deg −10.0 deg −5.0 deg −2.5 deg −2.0 deg −1.5 deg −1.0 deg −0.5 deg 0.0 deg 0.5 deg 1.0 deg 1.5 deg 2.0 deg 2.5 deg 5.0 deg 10.0 deg 15.0 deg Trend line (neg) Trend line (pos)
Camber torque, Mzγ [Nm]
100
50
0
−50
−100
−150 1.8
2
2.2 2.4 2.6 2.8 Inflation pressure, p [bar]
3
i
Figure D.6: Aligning camber torque vs. inflation pressure for various positive and negative camber angles at two extreme vertical load conditions; Fz =0.1 kN (top) and Fz =10 kN (bottom).
108
D.5
Aspects of the introduced inflation pressure dependent relations
Overview implemented parameters
In this section an overview is given of the implemented parameters with the proposed initial and boundary values, see table D.4. Table D.4: Overview of the implemented parameters with the proposed boundary conditions.
CF x
CF y
Fx,RR CM z Fyγ Mxγ Mzγ
P arameter pCF x1 pCF x2 pCF x3 pCF y1 pCF y2 pCF y3 qsy7 qsy8 pCM z1 ppy5 ppM x1 ppz2
Low −1.0 −0.6 0.0 −1.0 −0.6 0.0 0.5 −1.0 −1.0 −1.0 −1.0 −1.0
Initial 0.23 0.0 0.21 0.17 0.0 0.50 0.85 −0.40 −0.42 −0.15 −0.10 −0.60
U pper 1.0 0.0 1.0 1.0 0.0 1.0 1.0 0.0 1.0 0.0 0.0 0.0
The sidewall stiffnesses are made inflation pressure dependent by implementing the scaling approach of the nominal pressure sidewall stiffnesses (5.51) - (5.54). In table D.5 an overview is given of the fit parameters fi for the different sidewall stiffnesses. The fit parameters are optimised towards the FEM experiments and are currently implemented as fixed parameters. Table D.5: Overview of the implemented parameters of the sidewall stiffnesses.
P arameter fx fy fz fγ fθ fψ
Initial 0.65 0.74 0.65 0.49 0.69 0.49
Appendix E
Simulation Parameters In chapter 6 a two track vehicle model with a roll axis is used to analyse the vehicle handling behaviour. The roll stiffness and damping are modelled with torsional springs and dampers in the roll centres, translational stiffness and damping elements are used to model the vertical stiffness and damping of the suspension. In this Appendix an overview is given of the parameters that are used to parameterise the vehicle model. The parameters of the unloaded vehicle are depicted in table E.1. Table E.2 shows the parameters that change when the vehicle is fully loaded. For confidentiality reasons the numerical values are omitted.
109
110
Simulation Parameters
Table E.1: Vehicle model parameters.
Quantity l hc is mF L mF R mRL mRR munF L munF R munRL munRR m Ixx Iyy Izz h‘ mtyre mrim Ixx Iyy m1 s1 KP I caster γ1 ψtoe,1 rc1 cz,1 dz,1 cϕ,1 dϕ,1 ccomplF y,1 ccomplM z,1 m2 s2 γ2 ψtoe,2 rc2 cz,2 dz,2 cϕ,2 dϕ,2 ccomplF y,2 ccomplM z,2 Pengine Tengine nengine
Parameters unloaded vehicle V alue Description [m] vehicle wheelbase [m] height of vehicle [-] steer ratio [kg] weight front left [kg] weight front right [kg] weight rear left [kg] weight rear right [kg] unsprung mass front left [kg] unsprung mass front right [kg] unsprung mass rear left [kg] unsprung mass rear right [kg] vehicle body mass [kgm2 ] vehicle body inertia about world x-axis [kgm2 ] vehicle body inertia about world y-axis [kgm2 ] vehicle body inertia about world z-axis [m] vertical body centre of gravity z-coordinate [kg] weight tyre [kg] weight rim [kgm2 ] rim inertia about x-axis [kgm2 ] rim inertia about y-axis [kg] weight front axle [m] front axle track width [deg] king pin inclination [deg] caster angle [deg] camber angle [deg] toe angle [m] front roll centre height [N/m] vertical stiffness [Ns/m] vertical damping [Nm/rad] roll stiffness [Nms/rad] roll damping [deg/N] steering compliance due to Fy [deg/Nm] steering compliance due to Mz [kg] weight rear axle [m] rear axle track width [deg] camber angle [deg] toe angle [m] rear roll centre height [N/m] vertical stiffness [Ns/m] vertical damping [Nm/rad] roll stiffness [Nms/rad] roll damping [deg/N] steering compliance due to Fy [deg/Nm] steering compliance due to Mz [kW] max. engine power [Nm] max. engine torque [rpm] max. engine revolutions
Simulation Parameters
111
Table E.2: Changed vehicle model parameters when vehicle is loaded.
Quantity mF L mF R mRL mRR m Ixx Iyy Izz
[kg] [kg] [kg] [kg] [kg] [kgm2 ] [kgm2 ] [kgm2 ]
Parameters loaded vehicle V alue Description weight front left weight front right weight rear left weight rear right vehicle body mass vehicle body inertia about world x-axis vehicle body inertia about world y-axis vehicle body inertia about world z-axis
