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A Thesis Submitted for the Degree of PhD at the University of Warwick
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by
Wayne Hall
A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Engineering
School of Engineering, University of Warwick January 2003 To Angie and Patrick Contents
Contents ......
List Figures of ......
List Tables of ...... x
Acknowledgments ...... xi
Declaration ...... xii
Summary ...... xiii
Notation AV ......
Chapter 1: Introduction I ...... 1.1 Automobile Tyres 2 ...... 1.2 'Smart' Tyres 3 ...... 1.3 Thesis Motivation 4 ...... 1.4 Thesis Outline 5 ......
Chapter 2: Automobile Tyres, Tyre Behaviour Modelling 8 and ...... 2.1 Introduction 8 ...... 2.2 Tyre Construction 8 ...... 2.3 Sizes Load Rating 10 and ...... 2.4 Forces Moments 10 and ...... 2.5 Rolling Resistance 12 ...... 2.6 Tractive Properties 13 ......
H 2.7 Comering Characteristics 16 ...... 2.7.1 SHpAngle 17 ...... 2.7.2 CamberAngle 19 ...... 2.7.3 Concity Ply Steer 20 and ...... 2.8 CombinedBraking Cornering 21 and ...... 2.8.1 Brush Model 23 ...... 2.8.2 Magic FonnulaModel 24 ...... 2.9 Finite ElementModels 26 ...... 2.9.1 Implicit Integration 27 ...... 2.9.2 Explicit Integration 30 ...... 2.10Summary 31 ......
Chapter 3: Experimental Work to Characterise Tyre Behaviour in the Contact Patch 32 ...... 3.1 Introduction 32 ...... 3.2 Experimental Tyre 33 ...... 3.3 Testing Equipment 34 ...... 3.3.1 Load-DeflectionMachine 34 ...... 3.3.2 Flat Bed Tyre TestingMachine 35 ...... 3.3.3 Rolling Drum TestingMachine 36 ...... 3.4 Tyre Experiments 37 ...... 3.4.1 Stationary Experiments 37 ...... 3.4.2 Flat Bed Experiments 37 ...... 3.4.3 Rolling Dnim Experiments 38 ...... 3.5 Results Discussion:Stationary Experiments 39 and ...... 3.6 Results Discussion: Flat Bed Experiments 41 and ...... 3.6.1 Free-RoHingCharacteristics 42 ...... 3.6.2 CorneringCharacteristics Shp Angle 46 and ...... 3.6.3 CorneringCharacteristics CamberAngle 48 and ...... 3.7 Results Discussion:Rolling Drum Experiments 50 and ...... 3.7.1 Free-Rolling Characteristics Normal Load 50 and ...... 3.7.2 Free-Rolling Characteristics Speed 52 and ...... 3.7.3 Cornering Characteristics Slip Angle 54 and ...... 3.7.4 Cornering Characteristics Camber Angle 57 and ......
iii 3.8 Summary 57 ......
Chapter 4: Finite Element Models for Simulation of Microscopic Tyre Behaviour 60 0.00.0 0060 4.0...... 0000...... 4.1 Introduction 60 ...... 4.2 Mesh Generation 60 ...... 4.2.1 Tyre Components 63 ...... 4.2.2 SteelWheel 67 ...... 4.3 Material Properties 67 ...... 4.4 Rubber Compounds 67 ...... 4.5 Mooney-Rivlin Equation 70 ...... 4.6 Reinforcements 74 ...... 4.7 Contact Friction 76 and ...... 4.8 Summary 80 ......
Chapter 5: Simulation Stationary (Non-Rolling) Tyre Behaviour 81 of ...... 5.1 Introduction 81 ...... 5.2 StationarySimulations 81 ...... 5.3 Normal Loading Simulation 82 ...... 5.3.1 WheelFit Inflation Phase 84 and ...... 5.3.2 Normal Loading Phase 87 ...... 5.4 Longitudinal Lateral Loading Simulations 88 and ...... 5.4.1 TransverseLoading Phase 88 ...... 5.4.2 RestartFfles ...... 89 5.5 Selection Analysis Results 89 of ...... 5.6 Results Discussion: Normal Loading Simulation 90 and ...... 5.6.1 ParametricStudy ...... 93 5.6.2 Normal PressureDistribution ...... 95 5.7 Results Discussion:Longitudinal Lateral Loading Simulations 98 and and ...... 5.8 Summary 99 ......
Chapter 6: Simulation Rolling Tyre Behaviour 100 of ...... 6.1 Introduction 100 ...... 6.2 ComputationalConsiderations 100 ......
iv 6.3 Rolling Simulations 101 ...... Fit Inflation, Normal Loading Phase 101 6.3.1 Wheel and and ...... 6.3.2 Rolling Phase 102 ...... 6.4 Tyre Damping 103 ...... 6.5 Results Discussion: LS-DYNA 950d. 103 and version ...... Discussion: LS-DYNA 960 104 6.6 Results and version ...... between LS-DYNA 950d 960 108 6.7 Incompatibility version and ...... 6.8 Internal Stresses Strains III and ...... 6.8.1 At Lateral Tyre Centre 113 the ...... from Lateral Tyre Centre 117 6.8.2 At +30mrn and +55mm the ...... 6.9 Summary 122 ......
Work 123 Chapter7: Review, Conclusions and Recommendations for Further ..... 7.1 Thesis Review 123 ...... 7.2 Experimental Investigation 124 ...... 7.3 Modelling Simulation 125 and ...... 7.4 Recommendationsfor Further Work 127 ......
References 129 ......
Appendix A: Cross-Coupling Effects in the Longitudinal and Lateral Shear 137 Stresses...... Appendix B: Stress/StrainData for a Typical Rubber Tyre Compound in Simple 139 Extension ...... 141 Appendix C: Shear Distortion of the Tyre Tread in the Contact Patch ......
V List of Figures
Figure 1.1 Tyre forces 2 and moments ...... Figure 1.2 The 'friction 3 eflipse' concept ......
Figure 2.1 Tyre (a) tyre; (b) bias-ply tyre 9 construction: radial-ply ...... Figure 2.2 Tyre terminology 10 ...... Figure 2.3 Tyre axis II system and nomenclature ...... Figure 2.4 Normal pressure distribution in the contact patch: (a) non-rolling tyre; (b) tyre 12 rolling ...... Figure 2.5 Variation in rolling resistance coefficient of bias- and radial-ply passenger car tyres with speed on a smooth load inflation 13 ground surface at rated and pressure .. Figure 2.6 Contact patch behaviour during braking 14 ...... Figure 2.7 Variation in friction force with longitudinal 15 slip ...... Figure 2.8 Tread deflection in contact 17 patch at a slip angle ...... Figure 2.9 Cornering force response to a in the 18 step change applied slip angle ...... Figure 2.10 Variation in force 18 cornering with slip angle ...... Figure 2.11 Behaviour tyre 20 of a cambered ...... Figure 2.12 Conicity in tyre 21 an automobile ...... Figure 2.13 Variation in braking force with longitudinal 22 slip at various slip angles .... Figure 2.14 Variation in lateral force with longitudinal slip at 22 various slip angles .... Figure 2.15 Normal distribution 24 pressure ...... Figure 2.16 Coefficients used in the magic formula tyre 25 model ...... Figure 2.17 Finite model tyre 28 element of a passenger car ......
Figure 3.1 Experimental tyre 33 cross-section ...... Figure 3.2 Load-deflection 34 machine ...... Figure 3.3 Experimental the flat bed tyre testing 35 set-up on machine ...... Figure 3.4 Sketch drum tyre testing 36 of rolling machine ...... vi Figure 3.5 Contact (a) I kN; (b) 3 kN; (c) 5 kN 37 patchprints; ...... Figure 3.6 Measured load-deflection 39 normal characteristics...... Figure 3.7 Measuredcontact patch dimensions with normalload: (a) length;(b) width 40 ...... Figure 3.8 Measuredcontact stressdistributions with normal load on a horizontal (flat) surface:(a) normal pressure;(b) longitudinalshear stress; (c) lateral shearstress 43 ...... Figure 3.9 Radius 45 changeat the perimeterof the tyre tread; rl > r2 > r3 < r4 < r5 . Figure 3.10 Measured contact stress distributions with slip angle on a horizontal (flat) (a) (b) longitudinal (c) lateral 47 surface: normal pressure; shearstress; shearstress ...... Figure 3.11 Measured contact stress distributions with camber angle on a horizontal (flat) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear stress 49 ...... o..... Figure 3.12 Measured contact stress distributions with normal load on a cylindrical (drum) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear 51 stress ...... o...... o...... Figure 3.13 Measured contact stress distributions with forward speed on a cylindrical (drum) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear stress 53 ...... Figure 3.14 Measured contact stress distributions with slip angle on a cylindrical (drum) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear 55 stress ...... 0...... 0...... o.... Figure 3.15 Measured contact stress distributions with camber angle on a cylindrical (drum) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear 58 stress ...... o...... 0...... oo...... 0....
Figure 4.1 Three-dimensional (non-rolfing) 61 stationary model ...... Figure 4.2 Rolling (a) three-dimensional (b) 62 model: model; cross-section ...... Figure 4.3 Hourglass modes of an eight node element with one integration 65 point ...... Figure 4.4 Deformation of the 'tyre' cross-section when the default (type 1) viscous 65 formulation in LS-DYNA is control employed with the stationary model ...... homogeneous (a) Figure 4.5 Pure strain: unstrained.state; (b) strained state ...... **71 72 Figure 4.6 Particular types of strain: (a) simple extension; (b) simple shear ...... vn Figure 4.7 Comparisonbetween the estimatedstress/strain curve for the rubbertread in derivedfrom Mooney-RivIin function 73 compound tensionand that the strainenergy . Figure 4.8 Sheardistortion of linearsolid elementsin the contactregion: (a) Mooney- Rivlin rubbermodel (LS-DYNA model27); (b) Hyperviscoelasticmodel (model 77).. 75 Figure 4.9 Element in 76 and cord axes the reinforcementof the rubbercomposites .... Figure 4.10 Maximumdiagonal 78 of a shell element...... Figure 4.11 Simulatedfriction coefficient with relative velocity on a 'Safety Walk' 80 surface......
Figure 5.1 Deformation of the 'tyre' cross-section:(a) undeformed;(b) due to inflation pressureand wheel fit; (c) at a normal load of I kN; (d) at a normal load of 5 kN 82 ...... Figure5.2 Deformation of the 'tyre' cross-sectionat a high tyre/wheel impact velocity: (a) undeformed.tyre (t = 0.00 s); (b) tyre/wheel impact (t = 6xlO'3 s); (c) shockwave development(t = 8x 10-3s); shockwave transmissionthrough structure (t 0.00 85 = S); ...... Figure 5.3 Three-dimensional 88 glassplate model ...... Figure 5.4 Normal load-deflection 91 characteristics...... Figure 5.5 Tyre/groundcontact patch dimensionsswith normal load: (a) length; (b) 92 width ...... Figure 5.6 Simulated length load density 93 contactpatch with normal and mesh ...... Figure 5.7 Tyre/groundcontact patch dimensionswith normal load at various 'tyre' componentsstiflnesses (elastic constants) and inflation pressures:(a) length; (b) width 94 ...... Figure 5.8 Simulatednormal pressure distributions: (a) I kN; (b) 3 W; (c) 5W... 96 Figure 5.9 Contactforce on a nodeat the centreof the contactpatch with normalload 97 ...... Figure 5.10 Simulatednormal, longitudinal and lateral load-deflectioncharacteristics 98 ......
Figure 6.1 Three-dimensionaldrum 102 surfacemodel ...... Figure 6.2 Simulatedcontact stressdistributions at a normal load of 3 kN and a velocity of 20 km/h: (a) normal pressure;(b) longitudinalshear stress (c) lateral shear 105 stress ...... viii Figure 6.3 Contactstress distributions at the lateralcentre, of the tyre at a normalload of 3 kN: (a) normalpressure; (b) longitudinalshear stress; (c) lateralshear stress 106 Figure 6.4 Normal load-deflection 109 characteristics...... Figure 6.5 Tyre/ground contact patch dimensionswith normal load (a) length; (b) width ...... o.00...... ollo Figure 6.6 Nodal local to the 112 positions contact patch ...... Figure 6.7 Simulated internal stress distributions at the lateral centre of the tyre with a nonnal load of 3W and a velocity of 20 km/h: (a) vertical; (b) longitudinal;(c) lateral 114 ...... Figure 6.8 Simulatedinternal strain distributions at the lateralcentre of the tyre with a normal load of 3 kN and a velocity of 20 kni/h: (a) vertical; (b) longitudinal;(c) lateral 115 ...... Figure 6.9 Cross-sectional distribution lateral 116 vertical stress at the tyre centre...... Figure 6.10 Simulatedinternal stress distributions at lateralcoordinate of +30 mrn with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical;(b) longitudinal;(c) lateral 118 ...... Figure 6.11 Simulatedinternal strain distributions at lateralcoordinate of +30 mm with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical;(b) longitudinal;(c) lateral 119 ...... Figure 6.12 Simulatedinternal stress distributions at lateralcoordinate of +55 mm with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical;(b) longitudinal;(c) lateral 120 ...... Figure 6.13 Simulatedinternal strain distributions at lateralcoordinate of +55 nun with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical;(b) longitudinal;(c) lateral 121 ......
Figure A. 1 Longitudinal lateral 137 and shear stress cross-coupling ......
Figure C. 1 Simple tyre longitudinal load 141 model of experimental under ......
ix List of Tables
Table 2.1 Typical friction for 16 tyre/gound coefficients a range of surfaces ......
Table 3.1 Measuredcontact patch length of the stationary(non-rolling) and rolling 44 tyre ......
Table 4.1 Mechanical property data for rubber components (Dunlop Tyres Limited) 68 ...... Table 4.2 Mechanical property data for reinforced rubber composites (Dunlop Tyres Limited) 69 ...... Table 4.3 Calculated Mooney-Rivlin 72 elastic constants ......
Table 5.1 Calculated 83 and scaledpart masses......
Table 6.1 Simulated internal 122 strain ranges ......
Table BA Stress/straindata for in 140 a typical rubber compound simple extension ...
x Acknowledgments
I would like to acknowledgethe support provided by colleaguesat Dunlop Tyres Limited and Ove Arup & Partners.A specialmention must be given to Mr. Nigel Nock (Dunlop Tyres) and Dr. Brian Walker (Ove Arup & Partners)for their adviceon the Finite Element(FE) modellingwork, and also Mr. Mike Beesonand Dr. Kim Hardy (Dunlop Tyres) for providing technical information in relation to the experimental aspectsof the research.The financial assistanceprovided by the Engineeringand PhysicalSciences Research Council (research grant GR/M86835)is also acknowledged.
I would like to thank my supervisors Dr. J. Toby Mottram and Dr. R. Peter Jones for their help and encouragement.The guidance they have given me is much appreciated and has no doubt been of the highest professional standard. I would also like to thank my colleague Mr. Daniel Dennehy for his assistancewith the tyre experiments and for the numerous stimulating conversations and discussions. The help and direction provided by other research colleagues at the University of Warwick is also recognised.
Finally, I would like to thank my family and friends for their support during the writing of this thesis. To my beloved wife Angie and son Patrick, I extend my most grateftil appreciation. Their continuous support and encouragementhas made this thesis possible.
xi Declaration
The researchpresented in this them is original work carriedout by the author.Aspects of the work havealso been discussed in the following publications:
W. HALL, J. T. Mottrarn, D. J. Dennehyand R. P. Jones,Tharacterisation of the Contact Patch Behaviour of an Automobile Tyre by Physical Testing,' International Journal of Vehicle Design (subnýtted June 2002).
G. J. Tomka, S. Eaton,J. Milne, W. HALL, R. P. Jonesand J. T. Mottram, 'Foresight Vehicle: Smarter Tires using Advance Sensorsfor Improved Safety,' SAE 2002 World Congress,Paper 2002-01-1871, Arlington, Virginia, USA, June 2002.
J. T. Mottram, W. HALL and R. P. Jones,'Finite ElementModelling and Simulationfor a SmartTire, ' Tire TechnoloV Expo 2002,Hamburg, Germany, February 2002.
W. HALL, R. P. Jonesand J. T. Mottram, 'ModeRing of an Automobile Tyre using LS- DYNA3D, ' Third EuropeanLS-DYNA Conference, Paris, France, June 200 1.
W. HALL, R. P. Jones, J. T. Mottram, N. Nock and K. Hardy, 'Finite Element Simulation of a Vertically Loaded Automobile Tyre, ' 16M International Rubber Conference, Birmingham United Kingdom, June 2001, pp. 536-545.
xii Summary
This thesis presentsan initial Finite Element (FE) basedmodelling investigation aimed at supporting the development of 'smart' tyre or intelligent tyre technologies.Physical tests carried out with a stationary (non-rolling) and rolling experimental tyre are used to enhanceunderstanding of tyre behaviour in the contact patch and validate the modelling methodology. Simulation results with the explicit FE package LS-DYNA are then used to characterise the internal stressesand strains at several positions in the tyre tread.
Two separate FE models are developed to simulate the stationary and rolling tyre behaviour at the macroscopic level. The models differ only with respect to the mesh density in the circumferential direction, the mesh through the cross section is identical. The complex tyre structure is representedas a rubber and reinforced rubber composite, and the mesh specification and the material descriptions used in the models are discussed. The structural behaviour of the stationary experimental tyre under normal load is simulated. The inflation of the tyre, the wheel fit and the normal loading against the horizontal surface are represented. Simulation results are also presented when a subsequentlongitudinal or lateral load is applied to the stationary tyre. These analyses were conducted to determine the longitudinal and lateral tyre stifffiesses, respectively.
The predicted normal load-deflectioncharacteristics and contact patch dimensions (lengthand width) are comparedwith a reasonabledegree of successto thoseobtained in the full-scalephysical tests. The longitudinaland lateral simulationsalso appearto give realistic tyre stiffnesses.The contact patch dimensionsgive a good trend-wise agreement,but the lengthand width are greaterthan the experimentalmeasurements. A parametricstudy is carried out and this disparity is related to a deficiency in the performanceof the contact algorithms.It is concludedthat it not straightforwardto accuratelypredict contactpatch behaviour,and thereforethe internaltransient stresses and strainsin a rolling tyre in absoluteterms. However, the good trend-wiseagreement suggeststhat the modelling methodology should be capable of predicting internal transientresponses which are relatedto the 'actual' deformationsin the contactregion.
To simulate the rolling tyre behaviour on flat bed and drum surfaces, consideration is given to the inflation of the tyre, the wheel fit, the normal loading and the rotation of the tyre. Numerical instabilities are found to occur and these are related to imperfections inherent in version 950d of the code. This version was, at the time, the most up to date release. The current release is version 960 and it does not contain many of the imperfections in the earlier version. Thus, the flat bed simulation is repeated using the current version. The predicted contact patch stressesare presented and a reasonable correlation is achieved with the experimental data. The internal stressesand strains are then characterisedat a number of selected positions in the tread region. These stresses and strains are discussedin context with the development of smart tyre technologies and are useful as a guide to the most appropriate location for an in-tyre sensor (or sensors).
Xifi Notation
Italic Symbols
A areaof contactedsegment B stiffnessfactor C shapefactor; Mooney-RivIinelastic constant (e. g. C,, C2) C, longitudinaltyre stifffiess C, corneringtyre stiffness d depth;maximum diagonal dý decaycoefficient D peakfactor E curvaturefactor A penaltystifffiess factor F resultant force (e. F, FY, Fý) g. ,,, Fya corneringforce
Fyr camberthrust
G shearmodulus I contactpatch half length L length M degreesof freedom M moment(e. g. M, My, MO
N numberof k penaltystiffess K bulk modulus P pressure r. tyre effectiverolling radius
xiv R specific gas constant S slip ratio Sh, Sj S offset (e.g.
t time T temperature
U, V, W displacementcomponents in x-,. ý- and z-directions, respectively
V forward velocity; volume Vle, relative velocity
W contact patch width W strain energy function X, Y, z Cartesiancoordinate system;tyre axis system
Greek Symbols
a slip angle 9 sheardistortion C direct (e. strain g. c, , ey, e. 0 diameter
At time step size yt- averagetime step size
y camber angle; shear strain
extensionratio (e.g. A,, ý, A3) friction (e. coefficient g. pp , /j, direct (e. stress g. a,, ay , ar
shear stress(e. g. r,, r,.,, -r
w natural frequency
Subscripts and Superscripts
c cord e element; effective
xv ex external f friction h horizontal
i labelfor nodes in internal max maximum n nominal n normaldirection or stepnumber P peak s sliding t true v vertical X, Y, Z Cartesiancoordinates; tyre axes 1,2,3 principalcoordinates
Mathematical Symbols
squarematrix
vector, or column matrix
global displacementvector
velocity vector
(8) accelerationvector
(80) elementdisplacement vector IF) globalforce vector (Fe externalforce vector (Fi. internalforce vector
{Pj elementforce vector [K] global/systemstiffness matrix
[K*] elementstiffness matrix [M] global/systemmass matrix
IM01 global/systemmass matrix
xvi Chapter 1 Introduction
In recentyears, there has beena significantincrease in the numberof vehicleson the nation'sroads. An increaseof nearly4 million vehicleshas been observed since 1990 [1] and road traffic is expectedto reach35 mflfion vehiclesby the year 2025 [2], a further increaseof about 10 miffionvehicles. The relationshipbetween road accidentsand traffic flow hasbeen investigated by Dickersonet al. [3] and the accident-flowrelationship is seento substantiallyincrease at high traffic flows. The increasein vehiclesand their habitualuse is causingthe nation's roadsto becomebusier and, thus, more hazardous.
Although the level of traffic congestion has increased, accident statistics published by the Government indicate an improvement in road safety. The statistics show the number of deaths to be nearly half the total of 30 years previously and indicate a considerable reduction in the number of serious injuries. In contrast, the total number of casualties (deaths, injuries) serious and slight has not fallen greatly over the same period - only 12 % since 1967 [1]. Many factors have contributed to the shift from fatal and serious injuries to slight injuries. These factors include compulsory seat belt usage and better highway engineering.A significant contribution has also been made in the area of vehicle safety design were numerous primary and secondary safety features have been introduced. Primary and secondary safety features are defined as vehicle engineering aspects which "as far as possible reduce the risk of an accident", and structural and design features that "reduce the consequencesof accidents", respectively [4]. Primary safety features such as vehicle dynamics control systems, including Anti-lock Braking Systems (ABS) and traction control or Anti-Spin Regulation (ASR) are now standard on many vehicles. Airbags and deformable 'crush zones' that dissipate the energy in the event of an impact are also now standard and are examplesof secondarysafety features.
To sustainthe improvement,ffirther road safetymeasures need to be introducedto cope with the predictedincrease in levelsof traffic congestion.The Governmentis committed
I to road safety issuesand has published a safiety strategy 151.This strategy impacts on design lor vehicle manuthcturersbecause vehicle salýty is considered as .1potential area continuous improvement. Since the greatest progress call be made ill accident prevention, the development ot'new vehicle dynamics control systemsis essential.Tyres play the most crucial role in the support ot'vehicic dynamics and therelbre one approach fiIVOLiredby tyre technologists is to integrate sensor systems into tyres to monitor the contact patch fiorces 16,71. Thus. the 'si-nart' tyre concept may soon become a reality.
1.1 Automobile Tyres
Apart from gravitational and aerodynamic forces. all other fiorces which aIICct tile motion of a ground vehicle are generated at the interface between the tyre and the ground, known as the contact patch. These longitudinal, lateral and vertical forces (see Figure 1.1) are the resultant of normal and shear stresseswhich arise in the contact patch area [8]. They are transmitted through the tyre structure to the vehicle via tile wheel. An overturning moment, a rolling resistance moment and an aligning moment also exist when an offiset in the forces occurs relative to the centre ofthe contact patch.
4 Roling Resistance Moment I Overturning Moment
Lateral Force
Aligning Moment A Longitudinal Force
Vertical Force / Normal Load
Figure 1.1 Tyre forces and moments
It should be noted that there is a limit to the longitudinal and lateral forces that a rolling tyre can develop. This limit is determined by the vertical force (normal load) and the friction coefficient p between the tyre and the ground surface. The normal load is governed by the weight of the vehicle and, thus, at a given load, the friction condition determines the maximum contact patch forces [9], and therefore the manoeuvres a 2 vehiclecan undertake. The friction conditionand the interdependenceof the longitudinal and lateralforces is describedby the 'friction eRipse'[10]. This is shownin Figure 1.2.
LongitudinalForce
Lateral Force
Dry Asphaft Wet Asphaft Snow (hard-packed)
Figure 1.2 The 'friction ellipse'concept
The friction ellipse identifiesthe friction limit for a tyre. The friction can be used to developa longitudinalforce for acceleration/braking,a lateral force for comering/lane changemanoeuvers, or a combinationof the two, but in no casecan the vector total of the two exceedthe limit [8]. The longitudinaland lateral forcesmay developin either a positiveor negativedirection. Adverse driving conditionssuch as on wet, snowyand icy roadssubstantially reduce the friction limit and, as a consequence,the longitudinaland lateralforces which can developare alsoreduced. It is thereforeclear that knowledgeof the friction potential and demandcan help to improve manoeuvrabilityand thereby vehicle safety under slippery road conditions [11]. Despite this fact, current vehicle dynamicscontrols systemssuch as thosementioned earlier do not makeuse of contact patch force measurements.This absenceof force measurementsfor future dynamics control systemshas the potentialto be erasedby the rapid developmentsin 'smart' tyres.
1.2 'Smart'Tyres
The concept of 'smart' tyres or intelligent tyre technologies involves the instrumentation of a tyre by a sensor device embedded in the structure. The sensor device is used to measurethe tyre stresses,strains or deformations and the sensoroutputs are then related to the contact patch forces. Previous work on in-tyre sensor technology has been , concentrated at Darmstadt University of Technology and the tyre manufacturer
3 Continental. The early work at Darmstadt [13,141 used a Hall-effect sensor to detect three-dimensionalmovements of a tread embeddedmagnet. Continental have developed a SideWall Torsion (SWT) sensor [15] that provides an estimate of the longitudinal and lateral forces in the contact patch by monitoring the magnetic field generated by alternate north-south poles embedded in the tyre. A recent collaboration between Continental and Darmstadt [16] has also resulted in the integration of wireless Surface Acoustic Wave (SAW) sensorsto measureinternal stressesin the tread. The smart tyre concept is not a new idea but the technology is yet to advanceinto the mainstream [ 12].
1.3 Thesis Motivation
The work reportedin this thesiswas carried out as part of the ForesightVehicle Link project 'Smarter Tyres using AdvancedSensors for Improved Safety' (STASIS). The long-termaim of the STASISproject is to establishin-tyre sensorsystem technology for monitoringtyre behaviourin road vehicles.The project originallyinvolved a consortium consisting of Dunlop Tyres Limited, QinetiQ (formerly DERA), Avonwood DevelopmentsLimited, Ove Arup & Partners,Rover Group, and the University of Warwick. It should be noted, however,that due to unforeseencircumstances Dunlop Tyreswithdrew supportfrom the project in December2000 and, as a consequence,the project has been continued without input from a tyre manufacturer.The research reported herein is the author's (University of Warwick's) contributionto the project. I
Smart tyre technologieshave the potential to provide information on a vehicle's state via sensors in each tyre. To provide the most direct information, the sensors need to be embedded local to the contact patch and the influence of the tyre behaviour on the sensoroutputs needsto be realised. It appearsto date that little researcheffort has been expended on investigation of contact patch behaviour [9] and therefore the magnitude and frequency of the corresponding internal stresses,strains and deformations are not fully understood. Thus, the most appropriate position for the in-tyre sensorsis still not known. The sensorsmust survive in operation and their position is therefore depend on the transient stressesand strains they will experience.It is therefore clear that in order to support the development of sensor systems it is necessary to gain a greater understanding of tyre behaviour local to the contact region. The work reported in this thesis aims to provide an initial investigation of the complex internal stressesand strains 4 via numerical modelling. The numerical results emerging from the application of the modellingmethodology could then be usedto identify the optimal locationfor a sensor (or sensors)and the transferfunction betweenthe contact stressesand sensoroutputs.
To satisfy the researchaim, Finite Element (FE) based structural analysistechniques are used to simulate the stationary (non-rolling) and rolfing behaviour of an experimental tyre. Only advanced FE based modelling can provide simulations at the desired macroscopic level but such simulations are only as good as the FE program, and the way in which it is employed. Thus, a major aspect of the work presented in the thesis is model validation. Physical tests carried out with the experimental tyre are used to enhanceunderstanding of tyre behaviour in the contact patch and validate the modelling methodology. Simulation results with the explicit software package LS-DYNA [17] are then used to characterisethe internal stressesand strains at severalpositions in the tread.
1.4 Thesis Outline
This chapter has highlighted the importance of tyres in vehicle safety design and has also introduced the smart tyre concept and discussed some relevant work in this area. Chapter 2 reviews the tyre research literature relevant to this study. It describes the modem automobile tyre structure and the forces and moments exerted on it. The chapter discussesprevious work carried out to characterisetyre behaviour in the contact patch and then remarks on the related attempts to simulate this behaviour using simple models. An overview of complex FE tyre models is also presented and the implicit and explicit FE approachesare briefly discussed.The chapter highlights the lack of knowledge about local tyre behaviour in the contact region and the absenceof existing FE tyre models to simulate this behaviour. Tbus, the chapter establishesthe motivation behind the thesis.
Chapters 3 describes an experimental investigation which aims to characterise tyre behaviour in the contact patch with reference to the author's need to validate advanced FE simulations. Physical tests are carried out with a stationary (non-rolling) and rolling experimental tyre. The stationary experiments are conducted on a load-deflection machine and the rolling tyre experiments on flat bed and rolling drum machines, each instrumented with a tri-axial stress transducer. Load-deflection characteristics and
contact patch dimensions are presented for the non-rolling tyre. Many plots giving the 5 normal pressure and shear stress distributions under free-rolling and cornering (slip and camber angle) conditions are presented for a high friction surface, and a comparison is made between the contact patch behaviour when the tyre is rolled on the horizontal (at 0.18 km/h) and drum (10 to 50 krn/h) surfaces.The investigation therefore characterises the contact patch behaviour of the experimental tyre and, by doing so, also contributes novel experimentalmeasurements that correspond to 'actual' tyre contact deformations.
Chapter 4 describestwo state-of-the-artFE tyre models developed to simulate stationaryand rolling tyre behaviour.The models are developedfor analysisby the explicit solver LS-DYNA [17]. They depict the complexstructure of the experimental tyre as a rubber and reinforcedrubber composite.The meshspecification and material descriptionsused in the developmentof the modelsare discussed,and considerationis given to the difficult issueof modellingcontact and friction. The chapteralso discusses salientfeatures of the solutionalgorithms used in the code.The aim of the chapteris to presenta modellingmethodology which is capableof predictingtyre behaviourin the contactpatch and the internalstresses and strainsa sensorwill experiencein operation.
In Chapter5, the structural behaviourof the experimentaltyre during the stationary experimentsis simulatedusing LS-DYNA version 950d. This versionwas, at the time, the most up to date release.The inflation of the tyre, the wheel fit and the normal loading againsta rigid horizontal surfaceare considered.It should be noted that the simulationis representativeof the full-scaletests. A parametricstudy is presentedwhich characterisesthe sensitivityof the analysisresults to changesin the meshdensity of the tyre and surface,and to variationsin the elasticproperties of the 'tyre' components.The chapteralso presents simulation results obtained at severalinflation pressuresand when a subsequentquasi-static longitudinal or lateral load is appliedto the tyre. The aim of the chapteris thereforeto validatethe modellingmethodology for non-rollingbehaviour.
In Chapter 6, rolling tyre behaviour on the flat bed and rolling drum machinesis simulated.Consideration is given to the inflation of the tyre, the wheel fit, the normal loading and the rotation of the tyre. Numerical problems are observed in these simulationsand the problemsare relatedto imperfectionsinherent in version950d of the code. Thus, the flat bed simulationis repeatedusing the current release,version 960. Inconsistenciesin the simulationresults obtained using the two version are found and
6 these are discussedin relation to the earlier model validation work carried out with version 950d. The predicted contact stressesare then comparedwith a reasonable degreeof successto thosefound by full-scalephysical testing, and the internalstresses and strainsare characterisedat a severalpositions in the tread.These stresses and strains are discussedin context with the developmentof smartor intelligenttyre technologies.
The final chapter of this thesis, Chapter 7, summarisesthe work carried out by the author. Each of the individual chaptersare reviewedin turn and the salientpoints are highlighted.The main conclusionsare presentedand the work is discussedin context with to the researchaim. Finally, somerecommendations are given for further work.
7 Chapter 2 Automobile Tyres, Tyre Behaviour and Modelling
2.1 Introduction
Apart from gravitationaland aerodynamicforces, all actionswhich affect the motion of ground vehiclesare generatedat the tyre/groundcontact patch. Thus, it has beensaid that the "critical control forces (and moments)that determinehow a vehicle turns, brakesand acceleratesare developedin four contact patchesno bigger than a man's hand [I]. " As previously discussedin Chapter 1, these forces and momentsare the resultantof normal and shearstresses which arise in the contactpatch area.They are transmittedthrough the tyre structureto the vehiclevia the wheel.It is thereforeclear that a thorough understandingof a tyre's structure and behaviour,particularly the contact patch stressesand their relationshipto the internal tyre deformations,and resultantforces and moments is essentialfor the developmentof smarttyre technologies.
This chapter presents an overview of the tyre research literature. Primarily, it discusses the behaviour of modem radial-ply tyres but the characteristicsof bias-ply tyres are also considered. This is becausemuch of the previous research has been conducted on this type of construction. Finite Element (FE) basedmodelling provides the best opportunity to simulate the internal deformations, so a review of FE tyre models is given, and the implicit and explicit methods are briefly described. The attempts to model tyres using simple empirical and semi-empirical models are also mentioned. The chapter highlights the lack of knowledge about contact patch behaviour and the absenceof existing models to simulate this behaviour. By doing so, it establishesthe motivation behind the thesis.
2.2 Tyre Construction
Two basic tyre constructions are commonly used; radial-ply and bias-ply tyres, as shown in Figure 2.1. Bias-ply tyres were exclusively used in the automotive industry until the
8 advantagesof radial-ply tyres was recognised in the 1960's. Since then, radial-ply tyres have gradually displaced bias-ply tyres on passengercars and have been the standard for a number of years. The use of radial-ply tyres on trucks, however, initially lagged that on passengercars, such that (even as recently as) in the early 1990's both radial- and bias- ply tyres experienced approximately equal use [8]. Today, radial-ply tyres are also standard on trucks [ 18]. Bias-belted tyres were briefly employed as a cross between the radial- and bias-ply constructions during the transition period but are now rarely used.
Tread
Beft Plies
s Plies CarcassPlies Bead
(a) (b)
Figure 2.1 Tyre construction: (a) radial-ply tyre; (b) bias-ply tyre [181
The radial-ply constructionis characterisedby the carcassplies (cords usually of a syntheticmaterial embedded in a rubbermatrix) which extendradially from one beadto the other, i.e. at 90 degreesto the tyre circumference.The beadsform a foundationfor the carcassplies, and anchorthe tyre to the wheel.The constructionprovides a soft ride but little directionalstability. Directional stability is suppliedby the belt (cordsof steelor other high-strengthmaterials embedded in a rubber matrix) that runs circurnferentially aroundthe tyre. The cordsin the belt are usuallyorientated at approximately20 degrees to the tyre circumference.The belt stiffensthe treadregion, keeping the tread in contact with the ground.For passengercar tyres,usually there are two carcassplies of synthetic cords, suchas nylon, rayon or polyester,and the belt comprisestwo plies of steelcords and two plies of syntheticcords, suchas nylon or rayon. For truck tyres, usuallythere is one carcassply of steel cords, and four plies of steel cords in the belt [8,18,191.
In a bias-ply tyre the carcasscomprises two or more plies wMch extend diagonally from one beadto the other with the cords orientatedat an anglebetween 35 and 40 degrees 9 to the tyre circumference.The cords in adjacentplies run in oppositedirections. Thus, the cordsoverlap in a diamond(criss-cross) pattern. In operation,the orientationof the cords causesthe carcassplies to flex and rub. This flexing action producesa wiping motion betweenthe tyre tread and ground surfaceduring rolling which is one of the main causeof higher tyre wear and higher rolling resistancein bias-plytyres [18,19].
2.3 Sizes and Load Rating
Tyre sizes are commonly specified by two dimensions; the section width given in millimeters for radial-ply tyres, and the wheel rim diameter given in inches. The load- carrying capacity (rated load) is primarily dependent upon the tyre size. The outer diameter of the tyre, which is independently variable from the wheel rim diameter, is usually identified by the aspect ratio. This aspect ratio is the ratio between the section height and the section width and is usually expressedas a percentage.The terminology is shown in Figure 2.2. The type and construction of the tyre are identified by letters; P denotes a passengercar tyre and LT a light truck, R, B and D indicate radial-ply, bias- belted and bias-ply constructions, respectively. For example, a P195/65RI5 identifies a passengercar tyre having a section width of 195 mm.and an aspect ratio of 65 percent. The tyre has a radial-ply construction and is mounted on a 15 inch diameter wheel rim.
Se clion Width
Section Height
Wheel Rim Diameter
Figure 2.2 Tyre terminology
2.4 Forces and Moments
To describethe forcesand momentsexerted on a tyre, it is necessaryto definean axis systern. Figure 2.3 shows the tyre axis system recommended by the Society of Automotive Engineers(SAE). The origin of the axis is at the centre of the contact patch. 10 The x-axis, known as longitudinal axis, is the intersection of the wheel plane and the the direction forward. The is ground plane with positive -7-axis or vertical axis perpendicular to the ground plane with the positive direction downward. Thc. v-axis or lateral axis is in the ground plane and is directed to make the axis system right-handed.
The longitudinal fiorce F, is the fi)rce in the x-direction exerted on the tyre by the ground. The force in the y-direction is referred to as the cornering torce when causedby the slip angle a only, camber thrust F, when causedby the carnber angle y
the lateral force F, Slip is fiormed between the only, or more generally . angle the angle direction of travel of the wheel and the line of intersection of the wheel plane and ground plane. Camber angle is the angle formed between the xz-plane and the wheel plane. The normal load (or the vertical force) F is the fiorce in the vertical direction.
The overturning moment M,, the rolling resistance moment M, and the aligning moment M. are moments about the longitudinal, lateral and vertical axes, respectively.
Aligning Moment (M)
Camber Angle Rolling Resistance Moment (Mv) Longitudinal Force (Fý x
Direction of Wheel Travel Slip Angle
Overturning Moment (M) I 'A, y z Lateral Force (Fy)
Normal Load /Vertical Force (F)
Figure 2.3 Tyre axis system and nomenclature [20]
As mentioned, the forces and moments are the resultant of normal and shear stresses distributed in the contact patch. Normal stresses o7, arise as the inflation pressure acts through the tyre the ( ) due to friction onto ground, shear stresses and r,- arise coupling between the tyre tread and the ground surface. There are two primary mechanisms responsible for friction coupling; surface adhesion and hysteresis [21]. Surfaceadhesion is the result of molecularbonding, hysteresis is the energyloss as the tyre deformsduring rolling [22] and is the main causeof rolling resistancein tyres [ 18].
2.5 Rolling Resistance
The pressuredistribution in the contactpatch is not uniform (as was assumedto be the casefor manyyears [23]) it varies in both the longitudinaland lateral directions.In a rolling tyre, the pressuredistribution is also not symmetricalabout the lateralaxis. This is shownin Figure 2.4 and also in the experimentalresults presented in Chapter3. The pressureat the front of the contact patch is slightly greater than at the rear. This is becausea longitudinalforce, referredto as the rolling resistance,exists in the contact patch. The pressuredistribution is shifted in the direction of rolling (to the left) to maintainequilibriurn. This characteristicis discussedin a numberof text books [8,18].
CL
0 z
(a) (b)
Figure 2.4 Norrml pressure distribution in the contact patch: (a) non-rofling tyre; (b) rolling tyre
Numerouspapers have beenpublished that discussthe various factors which influence the rolling resistanceof tyres [24-26]. Thesefactors includethe tyre constructionand materials, surface condition, inflation pressure and speed. The rolling resistance coefficientfor a rangeof radial-plyand bias-plypassenger car tyres (at their rated loads and inflation pressures)with speed is shown in Figure 2.5. The rolling resistance coefficient is ratio of the rolling resistanceto the magnitudeof the normal load. As mentionedearlier, rolling resistanceis higher in bias-plytyres [19]. An increasein the 12 tread depth,the thicknessof the sidewallsor the numberof carcassplies in the structure also tend to increasethe rolling resistance.Tyres made of syntheticand butyl rubber compoundsalso usually have higher rolling resistancethan those made of naturalrubber.
0.025 mC
0.020 co Blas-Ply
4- 0.015 0 ...... 1111,...... 1111,...... Radial-Ply 4) n nin 50 100 150 Speedlkmlhl
Figure 2.5 Variation in rolling resistancecoefficient of bias- and radial-ply passengercar tyres with speedon a smoothground surfaceat rated load and inflation pressure[18]
On smooth surfacesthe rolling resistanceis lower than on rough surfaces.This is shown in research by DeRadd [27] where the author has published experimental results obtained in laboratory and outdoor tests conducted on various road surfaces.The rolling resistancehas also been shown to reduce with an increasein tyre inflation pressure [28]. This reduction, however, is found to be more significant in bias- than in radial-ply tyres.
2.6 Tractive Properties
Under acceleration or braking conditions, longitudinal slip is observed in the contact patch as the tyre tread deflects to develop and sustain a friction force, i. e. an acceleration or braking force, respectively. The contact stresses during braking are shown in Figure 2.6. To the author's knowledge, these contact stresseshave only been investigated by Novop'skii and Nepomnyashchii [29], and Bode [30]. The work by Novop'skii and Nepomnyashchicould not be located, and the publication by Bode was found to be in German (an English translation was not available). The work of the authors' is, however, reviewed by Browne et al. [3 1]. This review is therefore important because it presents the only available experimental data (in English) on the contact tractive stresses.
13 IV ...... v
ýF,
Figure 2.6 Contact patch behaviour during braking [10]
As treadelements enter the contactpatch they are deflectedin the longitudinaldirection. The deflectionoccurs because the tyre is moving faster than the perimeterof the tyre tread. The deflection and the correspondinglongitudinal shearstress build-up as the element moves through the contact patch until the shear stress on the element overcomesthe availablefriction and the elementbegins to slip (slide) noticeablyon the surface.In the sliding region, the deflectionand shearstress diminish, reaching zero as the elementleaves the contactpatch. Integrating the longitudinalshear stresses over the contact patch areayields the longitudinalforce. Thus, accelerationand braking forces are generatedby producing a differencebetween the tyre rolling speedand its linear speedof travel. As a consequence,longitudinal slip is producedin the contact patch.
The longitudinal slip s is usually defined non-dimensionally as a ratio of the forward speed [32,33]. This slip ratio is given by
V:; e-0
where r, is the tyre effective rolling radius, w is angular velocity of the wheel, and V is the forward velocity. It shouldbe notedthat in Equ.(2.1) the slip ratio is positiveduring
14 braking but the longitudinal force is negative according to the SAE tyre axis system shownin Figure 2.3. During acceleration,the ratio is negativebut the force is positive.
The braking force developedby the tyre is known to vary with longitudinalslip, as shownin Figure 2.7. A similar curve is producedin acceleration.As the slip increases, the friction force increasesalong an initial slope which is characterisedby the longitudinalstiffiaess C, of the tyre. The longitudinalstiffness tends to be low when the tyre is new and has full tread depth, and increaseswith age, i.e. as the tyre wears.
LLS! 08
LL u,. F, . Co=tan 9
0 0.2 0.4 0.6 0.8 1.0 Slip Ratio [no units]
Figure2.7 Variation in friction force with longitudinalslip duringbraking
The friction force usuallyreaches a maximumwhen the slip ratio is between0.1 and 0.2 [33,34]. Beyondthe peak value, the friction force reducesas the sliding region at the rear of the contactpatch extends further forward. At a slip ratio of 1.0 the slidingregion has extendedthe entire length of the contactpatch and the wheel is said to be locked.
The friction force is characterised at the peak and sliding conditions by the friction coefficients pp and ji,, respectively. These peak and sliding coefficients are the ratio of the friction forcesto the magnitudeof the normalload. They are influencedby a number of factors includingthe ground surfaceand its condition,the normal load, the inflation pressureand the forward speed.Typical peak and sliding coefficientsfor a variety of surfacesare shownin Table2.1. The grip (surfaceadhesion) is providedwhen the tyre tread deformsaround microscopic asperities in the surface[35]. Rain, ice and snow fill theseasperities and reducetyre grip and thereforethe friction coefficients.In the work conductedby Ervin [36] it is shown that increasingthe normal load decreasesthe friction coefficients;at the rated load of the tyre, the peak and sliding coefficients 15 decreaseby about 0.01 for a 10 Percentincrease in the magnitudeof the normal load. On dry groundsurfaces, the friction coefficientsare known to be slightly affectedby the inflation pressure.On wet surfaces,an increasein the inflation pressuresignificantly increasesboth coefficients [8]. In work by Dugoff and Brown [37], a significant decreasein the peak and sliding coefficientsis shownto occur as the speedincreases.
Surface Peakcoefficient pp Slidingcoefficient a,
Asphalt and concrete (dry) 0.8-0.9 0.75 Asphalt (wet) 0.5-0.7 0.45-0.6 Concrete (wet) 0.8 0.7 Gravel 0.6 0.55
Earth road (dry) 0.68 0.65 Earth road (wet) 0.55 0.4-0.5 Snow (hard-packed) 0.2 0.15 Ice 0.1 0.07
Table2.1 Typical tyre/ground friction coefficients for a range of surfaces[18]
Anti-lock Braking Systems(ABS) mentionedin ChapterI monitor wheelmotion during braking manoeuvers.Sensors detect excessivewheel decelerationand the slip ratio is estimatedfrom this information[9]. The brakingsystem maintains the slip ratio nearthe peakof the friction-slipcurve and doesnot allow the wheelsto lockup. Thus,with ABS the dominant tyre performanceparameter is the peak friction coefficient. As a consequence,the brakingdistances are greatly decreased,reducing the likelihood of an impactwith a vehicleor obstaclein front during braking. Similarly,the peak coefficient is of primary importancewhen traction control or Anti-Spin Regulation (ASR) is employed. By monitoring excessiveaccelerations, the accelerationor hill-climbing perfonnanceof a vehicle is optimised. Such systemsare, as previously mentioned, important to vehicle dynan-ficscontrol but they do not measurecontact patch forces.
2.7 Cornering Characteristics
The lateral forces required during cornering (or lane change maneuvers)are created by lateral slip (slip angle), lateral inclination (camber angle), or a combination of the two. 16 2.7.1 Slip Angle
When a slip angle is applied to a rolling tyre, tread elements are deflected in the lateral direction as they progress through the contact patch, as shown in Figure 2.9 191.The deflection and corresponding shear stress increasesas the element moves through the contact patch until the shear stress on the element overcomes the available friction and sliding occurs. This local behaviour has been investigated by Gough 1381, and Lippillarin and Oblizqjek [39], no other research into the contact stresses could be 16und. Integration of the lateral shear stressesover the contact patch yields the cornering force which is generatedtowards the rear of the contact patch, at a distance korn the centrc of' the contact patch which is usually referred to as the pneumatic trail. At the centrc ofthe contact patch, the cornering force and an aligning moment exist. The magnitude of the aligning moment is equal to the cornering force multiplied by the pneumatic trail.
Fy,,
Contact Patch
m I-
Direction of Wheel Travel Sliding Region
y
Figure 2.8 Tread deflection in contact patch at a slip angle [8]
The development of the cornering force is not instantaneous, it lags the applied slip angle as the tyre sidewalls deflect in the lateral direction [38,40]. The cornering force usually takes approximately one revolution of the tyre to reach the steady-state condition [41]; the distance required to reach the steady-state usually being referred to as the relaxation length. A typical cornering force response to a step change in the applied slip angle is shown in Figure 2.9. Evidently, the time lag depends on the speed of rotation of the tyre. For a passenger car travelling at a speed of 50 km/h, the time lag will be around 0.1 second, which is imperceptible to many motorists. The effect, however, may be noticed by 'expert' drivers as a lag or sluggishness in turning response.
17 - SteadyState --
p
LL0
i E 8
0.25 0.50 0.75 1.00 1.25 1.50 Numberor Revolutons
Figure 2.9 Comering force responseto a step changein the applied slip angle [81
The global corneringproperties of tyres havebeen studied extensively [ 19,22,3 1], and are usuallycharacterised in the steady-statecondition. The generalrelationsMp between the slip angle and the cornering force is shown in Figure 2.10. It should be noted that when the slip angle is positive the cornering force is negative according to the tyre axis system shown in Figure 2.3. A negative slip angle results in a positive cornering force.
LL 03
8
C. =tan 0
0 -15 -30 -45 -60 -75 -90 Slip Angle [degrees]
Figure 2.10 Variation in cornering force with slip angle [8] (note negative slip angles)
The cornering force increaseswith slip angle along an initial slope which is characterised by the comering stiffiiess C,, of the tyre. The comering stffffiess is given by
(dFya Ca =- (2.2) da aO
The cornering stfffness is defined as the negative of the slope, and is therefore positive.
18 The maximum cornering force, which is equal to the peak coefficient of friction multiplied by the normal load, usually occurs when the slip angle is between 15 and 25 degrees [8,18]. Beyond the peak value, the cornering force diminishes as the sliding region grows in the contact patch. At a 90 degree slip angle (beyond normal operating conditions and correspondingto a vehicle sliding sidewayswithout forward motion), the cornering force is equal to the sliding friction coefficient multiplied by the normal load.
The corneringstiflhess of a tyre is dependenton a numberof factors,most notablythe normalload and inflation pressure.The effect of normal load on the corneringstiffness andcornering coefficient is discussedin a numberof text books [8,18] and also in work by Chu [42]. The cornering coefficient is the ratio of the cornering stiffnessto the magnitudeof the normalload. It shouldbe notedthat the corneringstiffness increases to a maximum near the rated load of the tyre [8]. This increasein stiffness is not proportionalto the increasein magnitudeof the normalload and,as a consequence,the corneringcoefficient decreases as the magnitudeof the normal load is increased.The corneringstiffness of a tyre also increaseswith an increasein the inflation pressurebut the increasetends to be lesssignificant than that observedwith normalload [ 18]. This is shownin the papersby Nordeenand Cortese[22], and also Collier and Warchol [28].
Segel[43] hasstudied the relationshipbetween the aligningmoment (sometimes referred to as the aligningtorque) and slip anglefor a bias- and radial-plytruck tyre at various normalloads. Initially, the aligningmoment increases with an increasein slip angle.The momentreaches a maximumat a particularslip angleand then decreaseswith a further increasein slip angle.It shouldbe noted that the maximumaligning moment does not usuallyoccur whenthe corneringforce is at a maximum[38]. This is mainlybecause the sliding of tread elementsat the rear of the contactpatch causesthe point of application of the corneringforce to shift forward. At very high slip anglesthe sliding region can advanceforward sucha distancethat the aligningmoment can becomenegative [8,44].
2.7.2 Camber Angle
Whena camberangle is appliedto a tyre, a camberthrust Fy,, developsorientated in the direction the tyre is inclined. This is shown in Figure 2.11. This camber thrust is generatedbecause the axis of rotation of the tyre is not paraHelto the ground, but the 19 tyre is constrainedto move in a straight-line.The thrust acts in front of the wheelcentre and an aligningmoment is alsoproduced. The aligningmoment due to camberangle for a bias-ply tyre is typically of the order of 10 percent of the magnitudeproduced in responseto an equivalentslip angle,and the value is even lessfor a radial-plytyre [8].
7
p. Fyr
Figure2.11 Behaviourof acwnberedtyre
The relationshipbetween camber thrust and camberangle is discussedin reference[22], is by C, by and characterised a parameterreferred to as the camberstifffiess , given
(LFY,,)I Cy (2.3) = dy r-O
The camber stifffiess is typically between 10 and 20 percent of the cornering stifffiess. Similar to the cornering stiffness, the normal load and inflation pressure have an influence on the camber stiffness. The camber stiffness increaseswith an increasein the magnitude of the normal load [22]. This is shown in work by Segel [43] who considered the variation of stiffness with normal load for three truck tyres. An increase in the stiffiiess is also observed for bias-ply tyres with an increasein inflation pressure.Radial- ply tyres, however, are much less sensitive to changes in the inflation pressure [36].
2.7.3 Conicity and Ply Steer
It is interesting to note that, due to "non-syrnmetries[10]" in the tyre construction, such as conicity and ply steer, a lateral force usually develops even at zero slip and camber 20 angles. Conicity is usually caused by an asymmetrical offset in the position of the belts during fabrication. The tyre has a bias towards a conical shape and, as a consequence, will want to follow an arc centred about the apex of the cone. This is shown in Figure 2.12. The tyre is restrained to maintain a straight-line during rolling and, thus, a lateral force develops. The lateral force does not change direction with reverse rotation.
X Direction of I Travel -Wheel
y
Figure 2.12 Conicity in an automobile tyre
Ply steerarises due to the orientationof the cords in the belts,and is generallya larger effectthan conicity. During free-rolling,the tyre will steer.from its intendedstraight-line course.A lateralforce developswhen the tyre is constrainedto roll in a straight-lineand this force changesdirection with reverserotation. To minimisethe effectof ply steer,the cordsin successivelayers of the belt are orientatedat oppositeangles (see Section 2.2).
2.8 Combined Braking and Cornering
When longitudinal and lateral slip are applied simultaneouslyto a tyre, both the longitudinal and lateral forces differ considerablyfrom the values obtained under independentconditions [8]. Figures2.13 and 2.14 showthe influenceof longitudinalslip and slip angleon the braking force and lateral force, respectively.These measurements were takenfrom the text book by Milliken and Milliken [10]. They showthat applyinga slip angle usually reducesthe braking force at a constantlongitudinal slip. Similarly, applyinga longitudinalslip usuallyreduces the lateralforce at a constantslip angle.The relationshipbetween the braking and lateral forces under various slip conditionshas 21 been investigatedby many researchers[33,45] and it can be shown that the friction ellipseconcept introduced in Chapter I is realisedby envelopingthe resultantcurves.
5
K
m r- 32
I
0 02 OA 0.6 0.8 1.0 Slip Ratio [no units]
Figure2.13 Variation in brakingforce with longitudinalslip at variousslip angles[10]
-18 Slip Angle [degrees] -10
0.2 0.4 0.6 0.8 1.0 Slip Ratio [no units]
Figure 2.14 Variation in lateral force with longitudinal slip at various slip angles [10]
Basedon the experimentalobservations described above, attempts have beenmade to formulatesimple models to predict the longitudinaland lateral forces as functionsof longitudinalslip and slip angle.Two of the most commonmodels are the 'brush model' [33,45,46] and the 'magic formula model' [34,47,48]. The brush model is a semi- empiricaltyre modelwhich derivesa relationshipbetween the slip conditionand the tyre responsebased on a limited understandingof both the tyre structure,and the interaction betweenthe tyre and the ground. The magic formula model is an empiricalmodel and determinesthe complexrelationships based on experimentaldata. A detailedoverview of these tyre models and other empirical and serni-empiricalmodels is provided in reference[34]. The text book by Wong [ 18] also givessome insight into simplemodels.
22 2.8.1 Brush Model
The brushmodel is basedon an idealisedrepresentation of the tyre local to the contact patch. The model consistsof a row of elastic cylindersradially attachedto a circular belt. The cylindersrepresent tread elements and the belt is assumedto deflectonly in the verticaldirection by applyinga normalload. A contactlength arises, and in the caseof a frictionlesscontact surface, the cylinders(tread elements)are assumedto be orientated normal to both the ground and the flattened region of the belt. In the presenceof friction, the cylindersdeflect in the longitudinaland lateraldirection when the tyre/wheel deviates from the free-rolling condition. The deflection of the tread elementsis establishedby consideringthe displacementsat both endsof the cylinder.The tractive contact stressesare acceptablyassumed to be linearly dependentupon the deflections (when no sliding occurs in the contact patch), and are calculated based on the longitudinal and cornering stiffnesseswhich are determinedfrom experimentaldata.
Slidingof the tread elementsin the contactpatch is introducedwhen the resultantshear stressexceeds the maximumallowable shear stress r.. determinedby the coefficientof friction a andthe normalpressure q,. In the landmarkpaper by Dugoff et al. [33], the normal pressuredistribution in the contact patch is assumedto be uniform. Although this approximationis not exact,it representsan acceptableestimation for radial-plytyres exceptat the edgesof the contactpatch. The maximumshear stress is givenby
2+ "IF I TYX2)1/2 = Ul a. z; (2.4) T.. 21w
where I is the contactpatch half length,and w is the contactpatch width. It shouldbe noted that this equationis related to the friction ellipse conceptpreviously discussed.
In contrast to Dugoff et al. [33], other researchershave assumeddifferent normal pressuredistributions. In the modelsdeveloped by Bernardet al. [45] and Francheret al. [46], the distribution in the contact patch is assumedto be trapezoidalin form, as shown in Figure 2.15, and is determinedby the ratio a12L A uniform distribution is obtainedwhen the ratio al2l is equalto zero and a triangledistribution is obtainedwhen al2l equals0.5. Bernardet al. [45] discussvarying the ratio al2l and suggesta uniform 23 distributionfor radial-ply tyres and a distribution defMedby the ratio a121equals 0.25 for bias-ply and bias-beltedtyres. Both Zegelaar [41] and Sakai [49] assumeda parabolicdistribution. In the author's opinion, a better methodwould be to use contact stressesobtained by physicaltesting. However, such data hasnot beenreadily available. In Chapter3, data is presentedthat could be used with the brush model to improve understandingof the contact patch stressesand their relationshipto the global forces.
a- 1 I., a.
0 z
21 Longfiudinal Poslion
Figure 2.15 Nonnal pressuredistribution [45]
2.8.2 Magic Formula Model
In the general form, the magic formula model [34,47,48] is based on the observation that global tyre force and moment characteristics under pure slip conditions (either longitudinal slip or slip angle) appear to be sine curves that have been modified by introducing an arctangent function (see Figures 2.7 and 2.10). The magic formula is
y(x) = Dsin(Carctan[Bx - E[Bx - arctan(Bx)])) (2.5) where Y(X)= Y(X)- S, and X+ Sh
Y(X) is either the longitudinalforce F., the lateral force Fy, or the allgningmoment
M,, andX is either the longitudinalslip s or slip angle a. The significanceof the coefficients B, C, and D. and also the offsets Sh and S, is shown in Figure 2.16.
24 sv x
Sý
Figure2.16 Coefficientsused in the magicformula tyre model[44]
The coefficient D representsthe peakvalue and the coefficient C determinesthe shape of the curve.Typical valuesused for the shapefactor C are 1.65,1.30 and 2.40 for the longitudinal force, lateral force and aligning moment curves, respectively[47]. The coefficientsC and D also determinethe asymptoticvalue at large slip anglesy, via
Dsin Irc (2.6) Y. = 2
The coefficient B influences the slope of the curve at the origin BCD. As discussedin Sections 2.6 and 2.7, the initial slope of the longitudinal force-slip and lateral force-slip angle curves is characterisedby the longitudinal stifffiess C, and the comering stiffness
C., respectively. The coefficient E influences the curvature near the peak of the curve and the longitudinal slip or slip angle xM at which the peak value occurs, and is given by
B x. -4 i2CLr)c (2.7) B x. - aretan(Bx. )
The offsetsShand S, accountfor rolling resistance(see Section 2.5) in the longitudinal force curve. In the lateral force and allgning moment curves, they account for non- symmetriesin the tyre construction,such as conicity and ply steer(Section 2.7.3), and may also be usedto representthe offset of the lateral force curve due to camberangle.
25 To expressthe dependenceof the coefficients B, C, D and E and the offiets Sh and S, on the normal load, further functions are provided. The numerous constants (more than 30) in these functions are described by Pacejka and Bakker as the "ultimate parameters in the model [47]. " These parametersare determined by regressiontechniques from tyre experimental data and are usually calculated with specific computer software [50,51].
It is important to reiterate that the magic formula model does not incorporatean understandingof tyre behaviouror the interaction betweenthe tyre and the ground surface,and therefore it has only a limited use in the developmentof smart tyre technologies.Nevertheless, it is noted that manyyears' researchhas beenborne out of this formula and its ability to predict tyre characteristicsunder a wide range of conditions. The formula is routinely used in the automotive industry and was instrumentalin the evolution of tyre modellingas a vehicleand tyre developmenttool.
2.9 Finite Element Models
In parallel with simple models, there has also been a move towards simulating rolling tyres by the Finite Element (FE) method and numerous FE models are presented in the literature. These models have evolved as the software has developed and computational power has increased, and as a consequence, the models vary greatly. Implicit FE software was initially used to simulate tyre behaviour but the complexity of the automobile tyre structure, and limitations associatedwith the implicit method meant the prediction of dynamic tyre responseswas not feasible. Thus, most simulations have been limited to prediction of static behaviour [52-54]. In the early 1970's, Durand and Jankovick [52] and then later (in 1981) Kennedy, Patel and McMinn [53] simulated tyre deformation due to inflation pressure. Trinko [54] attempted to simulate the pressure distribution in the contact patch under normal load but only a reasonablecorrelation was obtained with experimental data. The author suggests the correlation is related to the mesh specification which was limited by the available computational resource. Similar work has been carried out by other researchers[55,56] with varying degreesof success.
In recent years,increases in computationalpower and the developmentof explicit FE software has made the prediction of dynan& (rolling tyre) behaviourattainable, and work in this areahas increased.The work, however,has focusedon the predictionof 26 global tyre behaviour[57-60]. Koa and Muthukrishnan[57] have simulatedthe global responsesof a tyre impactinga cleat and a reasonablecorrelation was obtainedbetween the dynamicforces at the wheel hub with those obtainedexperimentally. Further work has also beencarried out by Kamoulakosand Koa [58], and Hanley [59]. At selected inflation pressures,Koishi, Kabe and Shiratori [60] havesimulated the corneringforces of a radial-plypassenger car tyre. To the author's knowledgeno other work has been performedto specificallysimulate local tyre behaviourin the contact region. And it is this categoryof modelwhich is essentialto the developmentof smarttyre technologies.
To develop a FE model, a physical understanding of the tyre structure, particularly the geometry and material properties, is required. The tyre structure is divided into small discrete subregions(elements) joined only at specific points (nodes) on their boundaries to form a mesh. This is shown in Figure 2.17 [61]. The forces are transmitted by the nodes from one element to the next. The elements may be one-, two, or three- dimensional as described in the text book by Mottram and Shaw [62], and the model may consist of a few axisymmetric elements [63] or many thousands of solid and shell/membraneelements [57,64]. The most frequently used elementsare based on the stiffness method and the stresses,strains and displacementsare determined by simple displacementfunctions; the unknown in these functions being the nodal displacements.
2.9.1 Implicit Integration
In traditional implicit integration, the responseof an element in the FE mesh is given by
(2.8)
where IF') is the elementforce vector, [K*] is the elementstiffness matrix, and IS') is the elementdisplacement vector. It should be noted that for an eight-nodesolid element,nodal displacementscomprise three independenttranslational displacements. Similarly,three independentforces are assumedto act at eachnode. These independent forcesand displacementsare definedbased on a Cartesiancoordinate systen-4 and have componentsin the x-, y- and z-directions,such that at node i the nodal displacements are u,, v, and w,, and the nodal forcesare F Fyj and F.,. The elementvectors IF') ,ýj,
27 and IS') havetwenty four terms (degreesof freedomm) and thus [W] is a 24 by 24 matrix. In a whole structure,the numberof degreesof freedomis a combinationof all of the elementdegrees of freedomand are typically in the order of hundredsof thousands.
UCross-Sectlon
Figure 2.17 Finite element model of a passengercar tyre
In non-linearanalysis which is necessaryfor the simulationof tyre behaviour,the global force vector {F) and stifffiessmatrix [K] are non-linearlydependent upon the global displacementvector {8) and the complexityand computationaltime is increased.To determinethe non-linearresponse, the force-displacementrelationship is usuallybroken down into a seriesof linearsteps, and a combinationof load incrementationand iteration methodsare employedto obtainthe final solution.A generaloverview of thesemethods is providedby Becker [65]. Further detailson the algorithmsetc. are presentedin the text book by Crisfield [66]. The methodsare usuallyautomated in the softwarepackage and require little or no interaction from the FE analyst.Thus, the accuracyof these numericalmethods has a significantinfluence on the reliability of the simulationresults.
Implicit solution methods can also be used to progress simulations in the time domain, but the global equilibrium must first be achieved by iteration before the local element variables are calculated. In contrast, explicit methods calculate local variables directly without the need for global equilibrium calculations. Thus, the explicit method is better suited for predicting highly transient deformations [67]. This is the reason the explicit package LS-DYNA [17] is used for the simulation work herein. The fundamental difference between explicit and implicit time integration methods is demonstrated by
28 comparisonof the solutionof Newton's SecondLaw usingeach method [68]. Newton's SecondLaw can be expressedin matrix format (ignoring damping)for FE purposesas
[M] (ä) + [K] (8) = {F. ) (2.9)
where [M] is the massmatrix and {9), and {F,. ) are the nodal accelerationsand extemal force vectors, respectively.
In implicit integration,a solution is incrementedfrom step n to step n+I in time. The implicit programiterates to determineEqu. (2.9) is satisfiedby the equilibriumof interrial forces(left-hand side) with externalforces (right-handside) and the implicit solution is
(Fý ,)», +[K]{8). +I = 1
The time integration is typically calculated using a finite difference method based on averageacceleration and, thus, the nodal accelerationsand displacementsare relatedvia
2 {8). ++2 (8),, + {S) At24
By substituting Equ.(2.11) into Equ. (2.10), the displacementscan be solved directly:
4[M] I 4[M] + [K] (8).., + (2 {8). (2.12) At, At 2 -
[M](2 18), + {8
where At is the time step size. The displacementsat step n +I are a function of the displacementsand accelerationsat the previous time step and, thus, Equ.(2.12) requires the formation of the mass/stifffiessmatrix terms on the left-hand side before equilibrium calculations can commence. If equilibrium is not achieved, the matrix coefficients need to be recalculated with a different time step at a considerablecomputational cost.
29 2.9.2 Explicit Integration
In explicit integration,Equ. (2.9) is first rearrangedinto the form:
(8) = (fý (F ,) - (Fi. ) (2.13) where (Fi. ) is the internal nodal force vector and the accelerationscan be calculated directly via a diagonal(lumped) massmatrix that is trivial to solve. Thus, the nodal velocitiesand displacementscan then be calculatedvia the central-differencemethod as
18}n-112 + At (8). (2.14(a))
{8)n+l ý- {8)n + Atn+l {8)n+112 (2.14(b)) where Atn+l +At" 2
It shouldbe notedthat in Equ.(2.14(a)) the velocitiesare calculatedat the half time step position but this is usually trivial becausethe time step size is small [68]. More significantlyis the fact that (for stability of the central-differencemethod) the time step sizeis limited by the systemfrequency w. The stabletime step sizecan be shownto be
ü)
Thus, the maximum time step size is governed by the highest system natural frequency. A good estimate of the highest frequency of the system can be obtained by considering the properties of the individual elements in the mesh. This method has a significant computational benefit [68] and is therefore used in explicit software programs. In the program, the time step size for each element is calculated to determine the controlling value. Increasing the mass (via the element mass matrix [M]) or reducing
30 the stiffness (via the element stiffness matrix [K']) of the controlling elementsincreases the stable time step by reducing the highest frequency of the mesh. Obviously, reducing the mass or increasing the stifffiess has the opposite effect. An increasein the time step size reduces the computational time, and visa versa This method to reduce simulation time is used by the author in Chapters 5 and 6. In non-linear analysis, there is not a significant increase in computational time because the small time step size effectively allows the development of numerical methods that linearise the non-linear response[67]. A comprehensivedescription of these methods is provided by Belytschko et al. [69].
2.10 Summary
A review of the tyre researchliterature has beencompleted. The chapterhas described the modemautomobile tyre structureand the normal and shearstresses, and resultant forces and momentsexerted on it. Previous work carried out to characterisetyre behaviourin the contactpatch has beendiscussed and the relatedattempts to simulate this behaviourusing simpleempirical and semi-empiricalmodels have been highlighted. An overviewof complexFE tyre modelshas also beenpresented and the implicit and explicit methodshave been discussed. Thus, the informationreported in this chapteris essentialto aid understandingof the researchmethodology and discussionspresented in subsequentchapters. The chapterhas highlighted the lack of knowledgeabout local tyre behaviourin the contactpatch and the absenceof existingtyre modelsto simulatethis behaviour.By doing so, the chapterhas establishedthe motivation behindthis thesis.
31 Chapter 3 Experimental Work to Characterise Tyre Behaviour in the Contact Patch
3.1 Introduction
Numerouspapers exist which presentsimple modelsthat predict resultantforces and momentsin a rolling tyre. As discussedin Section 2.8, the models may be broadly categorisedinto two groups;empirical models, notably the 'magic formula model' [34, 47,48], and sen-d-empiricalmodels such as the 'brush model' [33,45,46]. Empirical modelsderive a relationshipbetween the an input conditionand the tyre responseusing experimentaldata. Semi-empiricalmodels combine experimentaldata with a limited understandingof the tyre structureand the interactionbetween the tyre and the ground, but in doing so, imply the nature of the contact normal and shearstress distributions.
In parallel with these simplified treatments there has also been a move towards simulatingrolling tyres by the Finite Element(FE) method(see Section 2.9). Although FE tyre modelshave the potentialto show internal structuraldeformations local to the contactpatch, the modelshave tended to be usedto simulateglobal tyre behaviour[57, 59,60]. Advancedsimulations of rolling tyres are importantto the tyre industrysince if contactpatch stresses could be continuallymeasured in eachof the tyres under motion, that tyre could becomea key sensorin future vehicle control technology[7,70]. To assessthe performanceof such models,there is a needto have physicaltest data that correspondsto 'actual' contact deformations.Thus, the purposeof this chapter is to enhanceunderstanding of tyre behaviourin the contact patch with referenceto the author's need to validate FE simulations.The investigationalso contributesessential new physicaltest data that could be usedby tyre technologistsand/or vehicle dynamic analyststo evaluatethe assumedcontact stressesin existing semi-empiricalmodels.
32 3.2 Experimental Tyre
A P195/65R15 automobiletyre mountedon a standardwheel with a S.SJ rim contour was providedby Dunlop Tyres Limited for the purposeof the investigation.The same experimentaltyre was also used in the related project carried out by Dennehy[9]. A labelledand dimensionedsketch of the tyre cross-sectionis shown in Figure 3.1.
Bead Seat Bead Seat Width 185mm Dia 378.3mm 2.0 COnch 21.0 Bead 1 35.0jý Apexex 65.0 4.1 I CDCl)
3.0 a Uner C Pbes arcass'as
39.0
List -nrBretakers Bandages 0.0 Od, Tread
agetBreaker Width 155mm N.B. ANDimensions in mrn Section Width 209mm Not to scale
Figure 3.1 Experimental tyre cross-section
The construction of the experimental tyre was considered to be typical of a radial-ply passengercar tyre (see Section 2.2) and consisted of a number of rubber components (tread, sidewall, liner, apex and clinch), and a number of reinforced rubber composites (bead, carcassplies and belt). The carcasscomprises two plies of nylon cords orientated at 90 degreesto the tyre circumference and the belt comprises two plies of steel cords known as breakers, and two plies of nylon cords referred to in this thesis as bandages. This terminology is common in the tyre industry. The steel cords in the belt are orientated at approximately 20 degreesto the tyre circumferenceand the nylon cords are oriented circumferentially around the tyre. The tyre has a simple plain tread consisting of two circumferential grooves. This tread pattern was selected to eliminate the complex
33 but small scale local effects which are caused by modem tread designs. The grooves were cut in the experimental tyre to represent the approximate ratio of contacting to non-contacting patch area, the 'land/sea ratio', observed in a typical automobile tyre.
3.3 Testing Equipment
The stationary (non-rolling) behaviour of the experimental tyre was investigated using a load-deflection machine and the rolling behaviour using flat bed and rolling drum tyre testing machines at the research laboratories of Dunlop Tyres Ltd., Birmingham, UK.
3.3.1 Load-Deflection Machine
The load-deflection machine shown in Figure 3.2 is used to measure the structural behaviour of a stationary tyre under normal load. The machineconsists of two adjustable carriages running on rails, and a rigid glass plate against which the tyre is loaded. The specimen is mounted on a wheel unit and subjected to quasi-statical loading. The rear carriage contains an electrically operated loading system which allows normal loads to 50 kN to be applied. The position of the carriage is set by meansof four pegs which fit into holes to give a coarse adjustment to enable a range of tyre sizes to be tested. The second carriage contains the load cell and the unit to which the tyre/wheel is attached. The glass plate can also be adjusted to give camber angles between ± 45 degrees.
Static Glass Plate
-
Adjustable Carriages
Figure 3.2 Load-deflection machine
34 3.3.2 Flat Bed Tyre Testing Machine
The rolling tyre experiments on the flat bed tyre testing machine were carried out by a colleague at the University of Warwick [9] with the assistanceof the author. The flat bed tyre testing machine is shown in Figure 3.3. It should be noted that the physical testing data has not previously been analysed in the detail presented in this thesis.
screw Jacks
Experimental Tyre Horizontal Surface
Figure 3.3 Experimental set-up on the flat bed tyre testing machine [9]
The flat bed tyre testing machine is used to roll a tyre at a low-speed over a 1.83 m long horizontal surface. On top of the steel flat bed there is a thin surface layer of Safety Walk', a coarse glasspaper material. The tyre is normally loaded against the surface which is then electrically driven in the longitudinal direction at a constant speed of 0.05 m/s (0.18 km/h) causing the tyre to rotate. The tyre rolls over a stress transducer embedded in the horizontal surface (see reference [9]). This transducer measuresthe normal and shear stresses at the surface, and is regularly calibrated to ensure its accuracy. The measurementsare sampled at a frequency of 100 Hz, processed by an adjoining computer and then stored electronically in ASCII-format computer data files. These files are loaded into MATLAB [71] for analysisand visualisation. Adjustment on a number of electrically operated screw jacks allows normal loads to 35 kN, and slip and camber angles between ±15 and ±25 degrees, respectively, to be applied to the tyre. Lateral offsets can also be applied to change the position of the tri-axial stress
' Trade mark of 3M United Kingdom PLC (http: //www. 3m.com)
35 transducer relative to the tyre's centre-line. A program executed on the adjoining computer is used to calculate the screw jack settings for the required normal load, lateral offset, and slip and camber angles. The jack settings are then applied manually.
3.3.3 Rolling Drum Tyre Testing Machine
The rolling drum tyre testing machine is used to roll a tyre on a 2.39 m diameter drum, as shown in Figure 3.4. The tyre is mounted on a stub axle, which is connected to a frame structure and supported to ground. The drum has the sameSafety Walk surface as the flat bed tyre testing machine. A computer controlled hydraulic system allows normal loads to 10 kN, and slip and camber angles between ± 14 degreesto be represented.The rotation of the drum is electrically driven and causes the tyre to rotate in the opposite senseto that of the drum. The tangential speed of the drum surface (0 to 230 km/h) is selectedvia a control panel and regulated by a computer control system. The drum and tyre are acceleratedto the desired speed, and then maintained constant for the duration of the test. The tyre rolls over a stress transducer embeddedin the surface of the drum. This transducer is similar in construction and operation to that on the flat bed tyre testing machine. The normal and shear stressesare sampled at a constant frequency of 1024 Hz, processedby an adjoining computer and then stored electronically in ASCII- format computer data files. MATLAB [71] is again used for analysis and visualisation.
Frame Strudure
Hydraulic
Drum
Figure 3.4 Sketch of rolling drum tyre testing machine
36 3.4 Tyre Experiments
The experiments carried out oil tile load-deflection machine are referred to lici-Cas the stationary experiments. Those carried out oil tile flat bed and rolling druin tyrc testing machineare reterred to as the flat bed and drum experiments, respectivcly. It should be noted that, in all the experiments, the inflation pressureoftlic tyre was kept constant at 200 kPa (29 psi). This is because the inflation pressure is knovvil to Influence tyre characteristics(see Sections 2.5 and 2.7). To ensure a meaningful comparison between the flat bed and drurn experiments, similar environmental conditions were also observed.
3.4.1 Stationary Experiments
The stationary experiments were conducted to determine the vertical stiffincssand tile growth of the contact patch tinder normal load. Fxperiments were conducted at normal loads between I kN and 5M in increments of I kN. The lyre's surface was coatcd in carbon black and a sheet of card was fixed to the glass plate. A print of the contact patch was produced at each normal load. The contact patch prints at 1,3 and 5 kN are shown in Figure 3.5. Tile contact patch dimensions were measured from these prints using a mm ruler. The contact length was measuredat tile lateral centre of the tyre (see Figure 3.1) and the width was measuredat the longitudinal centre, i.e. along the.v-axis.
li)-ø. v (b)
(a)Figure 3.5 Contact patch prints: (a) I kN; (b) 3 kN; (c)1Ii-py 5(c) kN
3.4.2 Flat Bed Experiments
The objective of the flat bed experiments was to characterise the normal pressure and shearstress distributions in the contact patch when the experimental tyre was rolled on a
37 horizontalsurface. The contactpatch behaviourwas investigatedunder free-rollingand corneringconditions (see Section 2.5 and 2.7), and the experimentswere conducted with the following parametersas independentvariables: " normalloads between I kN and 5 kN, in incrementsof I kN; " slip anglesto 2 degrees,in 0.5 degreeintervals; and " camberangles to 6 degrees,in 2 degreeintervals.
During the slip and camberangle experiments,the normal load was kept constantat 3 kN. This value was selectedbased on the assumptionthat the weight of a typical salooncar is 12 kN (i.e. 3 kN per wheel).A constantnormal load was appliedbecause the tyre/ground friction coefficientsand cornering characteristicsare known to be influencedby normal load. This has previouslybeen mentioned in Sections2.6 and 2.7.
To compensatefor conicity and ply steer effects, a slip angle correction was applied to the experimental tyre. The correction was approximately -0.4 degrees and the application of a slip angle was additional to this correction. Thus, an experiment conducted at zero slip angle was actually performed at a slip angle of -0.4 degreesand at a slip angle of I degree the test was performed at 0.6 degrees etc. [9]. The slip angle correction ensured that the tyre was in the free-rolling condition, i. e. the lateral force exerted on the tyre was about zero when zero slip and camber angles were applied.
3.4.3 Rolling Drum Experiments
The rolling drum experiments were conducted to characterisethe normal pressure and shear stress distributions in the contact patch when the tyre was rolled over a cylindrical surface. The drum experiments were conducted with the following parameters as independentvariables:
" normal loads between I kN and 5 kN, in incrementsof 1 kN; " speedsbetween 10 km/h and 50 km/h, in incrementsof 10 knV14- " slip anglesto 4 degrees,in I degree increments;and " camber anglesto 4 degrees,in I degree increments.
It should be noted that the slip angle correction describedin relation to the flat bed experimentswas againapplied. The normalload exertedon the experimentaltyre during 38 the speed, slip and camber angle experiments was also again kept constant at 3 kN-
3.5 Results and Discussion: Stationary Experiments
The load-deflectioncharacteristics of the experimentaltyre are shown in Figure 3.6. In the figure, the normal load is given in kN and the deflection in mm. The load was measuredby the load cell on the rear carriage.This load cell was calibratedto comply with the requirementsof Dunlop Tyres Ltd. in-houseprocedures in accordancewith ISO 9001. An approximatelylinearly relationshipis observedbetween the load and deflection.This relationshipis typical and has beenconfirmed by othersresearches [35, 72]. The stiffnessof the experimentaltyre is estimatedto be approximately180 kN/m.
36
30
25
20
15
10
5
3 Noffnal Load
Figure3.6 Measurednomml load-deflectioncharacteristics
As an undeflected tyre is inflated, it takes up an equilibrium shape. This inflated shape causestension in the cords of the carcassplies and belt (casing cords), and in the cords of the bead, referred to as the bead coil. To carry a normal load, the tyre deforms and a contact region develops at the interface between the tyre and ground surface. The growth of this contact patch in the experimental tyre is shown in Figure 3.7. The inflation pressure acts through the tyre and onto the surface. As the normal load increases,the contact patch becomeslonger and wider and changesfrom an oval shape to a rectangular one. This has been shown for the experimental tyre in Figure 3.5. The tension in the casing cords in the contact region is determinedby the tension in the cords
39 in the defonned'free' regions(adjacent to the contactregion) not the inflationpressure. In the defortnedfree regionsthere is a reductionin the curvatureand, as a consequence, the tensionin the cords is reduced.However, the tensionin the beadcoil is rnaintained by the load transmittedfrom the wheelto the tyre. In effect, "the wheelrim hangsin the beadcoil which in turn hangsin the tyre walls away from the deflectedregion [72]."
180
160
140
120
1100
80
60
40
20
3 456 1 Normal Load DdA (a)
ISO
160
'r 140 _E t 120 ?. I oo
80
u 60
40
20
34 NormWLoad M
(b)
Figure3.7 Measuredcontact patch dimensions with normalload: (a) length;(b) width
To provide a meaningfulcomparison, the contactpatch dimensions(length and width) are presentedon identicalaxes. The experimentalmeasurements are representedby the circular points and a line of 'best-fit' is also included. The possible errors are
40 representedwith 'error bars'. The normal load is equal to the inflation pressure multiplied by the contact areaplus a small contribution due to the tyre structure [72]. Sincethe inflation pressureremains approximately constant, the contactpatch areaand, hence,the contactlength and width grow as the load increases.An approximatelylinear relationshipbetween load and contact patch length, and a non-linear relationship betweenload and contactpatch width are observed(for loadsbetween I kN and 5 kN). The rate of increaseof the contactpatch width reducesas the normal load is increased.
3.6 Results and Discussion: Flat Bed Experiments
In Figures3.8,3.10 and3.11, and Figures3.12 to 3.15, plots arepresented for the stress distributionsmeasured using the transducersin the flat bed and rolling drum testing machines,respectively. The two transducerswere again calibratedto comply with the requirementsof Dunlop Tyres Ltd. in-houseprocedures in accordancewith ISO 9001. The componentsmeasured are the normal Pressure,and the longitudinaland lateral shearstresses. The normal engineeringsign conventionis used [73] and the units for stressare kPa. To provide a basisfor a meaningfulcomparison (as was the casefor the contact patch length and width), the normal pressureand shear stress ranges are identical in each of the figures. It should be noted that the stressvalues are those measuredby the stresstransducer as the contactingtread elementprogresses through the contactpatch. These normal pressure and shearstress measurements are assumedto be the sameas the instantaneousstresses in the contact patch. In what follows the contactpatch starts on the left-sideof the plot and the first part of the stressdistribution is referred to as the 'front'. The contactingelement progresses through the contact patchfrom left to right. The 'rear' of the contactpatch is the regionwhere the contactis graduallylost. The region betweenthe front and the rear is referredto as the 'centre'.
It is recognisedby the tyre industry that no measurementsshould be taken from a rolling tyre until it has revolved approximately one revolution. This relaxation distance (see Figure 2.9) for the experimental tyre is approximately 1.8 m. The ordinates in Figures 3.8,3.10 and 3.11, and Figures 3.12 to 3.15 give the distance travelled by the tyre and are different on the two tyre testing machines. In the case of the flat bed tyre testing machine, the distance is limited by the travel of the horizontal surface. This limit is approximately 0.9 revolutions of the tyre. The same restriction does not apply to the
41 rolling drum tyre testing machine and the tyre was therefore allowed to roll significantly further than the relaxationdistance. For the stressdistributions shown in Figures3.8, 3.10 and 3.11, and Figures3.12 to 3.15, the distancerepresented by the plots is 0.2 m.
The presenceof the Safety Walk surface provides a high friction contact which is believednecessary to obtain shearstress measurements that are comparableto how contactand friction is modelledby FE simulation.In the relatedwork by Dennehy[9], a series of tests with the flat bed tyre testing machinehas been conductedand the contactingsurface was either steel or 'highly' greasedsteel. It is the author's opinion that with thesetwo surfaceconditions, the stresstransducer output was adverselyaffect by slippage(low friction). Thus, simulationof theseresults was consideredunfeasible.
3.6.1 Free-Rolling Characteristics
The normalpressure and shearstress distributions in the contactpatch at normal loads betweenI kN and 5 kN are shown in Figure 3.8. Thesemeasurements were obtained alongthe lengthof the contactpatch at the lateralcentre of the tyre (seeFigure 3.1). As discussedin Section3.4.2, a slip anglecorrection was appliedto the experimentaltyre.
The normal pressureat the front of the contact patch is shown in Figure 3.8(a) to be slightly greaterthan that at the rear. This characteristichas been previously discussed in Section2.5. The distribution is not symmetricalbecause a longitudinalforce, usually referredto as the rolling resistance,exists in the contactpatch. This rolling resistanceis also evidentin Figure 3.8(b) where the magnitudeof the longitudinalshear stresses at the rear of the contactpatch are greaterthan thoseat the front. As discussed,the rolling resistancewill changeon different surfaces,and with various tyre constructionsand sizes. This point is worthy of considerationif the normal and shear contact stress distributionspresented in this thesis are to be used (herein or by others) to validate advancedFE modelsor evaluatethe assumedcontact stresses in semi-empiricalmodels.
As a tread elemententers the contactpatch, the normalpressure (Figure 3.8(a)) exerted on the horizontalsurface rapidly increasesto a maximum,and then steadilyreduces as the element progressesthrough the contact patch. The maximum pressure in Figure 3.8(a) doesnot changewith normal load and it is approximately320 kPa. This
42 600
500
400
300
a- 200 ii
0ý
i- -1001.4 1.42 1.44 146 1-48 1.6 1.52 154 1 56 1 58 16 Distance Travelled [m]
(a)
200
CLW 160
100 (4
50
0
i3 -50
rz 0Cn-loo -" -150
-200 L- 1.4 1.42 1.44 1.46 1-48 1.5 1.52 1.54 1 56 1.58 1.6 Distance Travelled (m]
(b)
500
400-
300- 0
2001
U) 100 CD
0
-100 1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 Distance Travelled [m]
(C)
Figure 3.8 Measured contact stress distributions with normal load on a horizontal (flat) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear stress
33 exceedsthe tyre inflation pressureby a factor of 1.6. DcBecr [23] suggeststhe maximum pressurein the contact patch exceedsthe inflation pressureby a factor between1.5 and 2.0. Before the elementexits the contactpatch a slight increasein the normalpressure is observedand this increasebecomes more evidentas the normalload increases.The normal pressurerapidly reducesto zero as the elementexits contact.
A reduction in the normal pressureat the centre of the contact patch is observed (Figure3.8(a)) as the normalload increases.At the sametime there is an increasein the contactlength from about0.06 to 0.16 m. The lengthis similarto that determinedunder stationary(non-rolling) condition as shownin Figure 3.7(a). A comparisonbetween the static and rolling contactlengths is given in Table 3.1. The measurementsalso confirm the observationby Browneet aL [3 1] that there is a reductionin the normalpressure as the tyre centre'buckles' upwardsunder increasingnormal load. A larger proportion of the pressuredistribution must be shifted to the tyre shouldersas the pressureat the lateralcentre reduces [9,23]. To verify such a redistributionin pressureit is necessary to move the transducerlaterally across the contact patch width and repeatthe flat bed experiments.This hasbeen confirmed for a flat steelsurface in the work by Dennehy[9].
Normal load Stationarycontact length Rolling contactlength [kN] [niml [niin] 1 56 59 2 87 88 3 113 114 4 133 136 5 155 156
Table 3.1 Measured contact patch length of the stationary (no-rolling) and rolling tyre
In Figure3.8(b), the longitudinalshear stress exerted on the horizontalsurface is shown to follow a sinusoidal-stylesweep as a tread elementprogresses through the contact patch. The magnitudeof the negativeand positive longitudinalshear stresses increases as the normal load increases.At a normal load of 5 kN, the maximaare -120 and 165 kPa, respectively.These characteristics are mentionedby Dennehyet aL [74], but the longitudinal(x-) axis is incorrectlyidentified as the lateral (y-) axis, and vice versa.
44 This highlightsthe lack of knowledgeof contact patch behaviourand gives additional supportto the motivation behindthe thesis.The increasein the magnitudeof the shear stressesat the rear comparedto thoseat the front is also evidentbut is not discussed.
The sinusoidal-stylesweep in the longitudinal shearstresses shown in Figure 3.8(b) developsbecause the radiusat the perimeterof the tyre tread changes[10] as shownin Figure 3.9. As a tread clemententers contact, the tangentialvelocity decreasesas the radius reducesand a negative longitudinal shear stress is exerted on the horizontal surface;a positive shearstress must be exerted on the tyre. A positive shearstress developson the surfaceat the rear as the tyre radius recovers.A small kink in the longitudinalshear stress is usuaUyobserved at the trailing edgeof the contactpatch and is moreevident as the magnitudeof the normalload increases.This kink is also shownin the work publishedby Lippmannand Oblizajek[39] and by Seitz and Hussman[751. It is statedin the paperby Browne et al. [3 1] that no clear explanationfor the behaviouris at hand. However, in a more recent paper by Lippmann [76] the kink is related to 'sfippagc' in the contact patch. The author believesthis explanationto be correct.
Figure 3.9 Radius changeat the perimeter of the tyre tread; A> r2 > r3 < r4 < r5
The lateral shearstresses (Figure 3.8(c)) are small in comparisonto the longitudinal shearstresses. This is to be expectedsince the slip anglecorrection was appliedto the tyre. If the slip angle correction was exact, the lateral shearstresses would be zero; conicity and ply steereffects would be negligible.Evidently this is not the casebut the lateral shearstresses are not consideredsignificant and, thus, are not discussedfurther.
45 3.6.2 Cornering Characteristics and Slip Angle
The normalpressure and shearstress distributions in the contact patch under slip angle (a) variationsto 2 degrees(in incrementsof 0.5 degrees)are shown in Figure 3.10. Further measurementsat higher slip angles(up to 4 degrees)were also taken but the datawas foundto exhibit obviouserrors and thus, the contactstresses are not presented herein.These errors were due to a problem associatedwith the stresstransducer. The normalload exertedon the tyre during the slip angletests was 3 kN and the appliedslip anglewas additionalto the slip anglecorrection. It shouldbe noted that by applyinga slip angle,the orientationof the tyre axes(x-, y-) changesrelative to the fixed transducer axes.A cross-couplingeffect occurs, and the longitudinal (x-) and lateral (y-) shear stressesneed to be resolved[9]. At small slip angles(<10 degrees),the cross-coupling effect is not significant,and its presencewas neglected.This is shown in AppendixA.
In Figure3.1 O(a), a reduction(maximum 15 percent)in the normalpressure exerted on the SafetyWalk surfaceis shownto occur as the slip angleincreases. A slight reduction in the contactpatch length is also evident.These characteristics are probablyrelated to treaddistortion occurring under slip angleconditions. A reductionof up to 30 percentis observedin the longitudinalshear stresses (Figure 3.10(b)) at the rear of the contact patchas the slip angleincreases. A muchsmaller reduction occurs at the front wherethe stressis lower. Thesereductions are relatedto the friction ellipseconcept described in Section1.1. The friction ellipseassumes that the friction limit for a tyre is determinedby the coefficient of friction and the normal load, and that the friction can be used to developa longitudinal force, a lateral force or a combinationof the two. Since the longitudinaland lateralresultant forces are relatedto the longitudinaland lateral shear stressdistributions, the friction limit determinesthe shearstress that can develop.The availablefriction is dominatedby the lateral shearstresses in Figure 3.10(c). As Figure 3.10(b) shows, an accompanyingreduction in the longitudinal shearstresses follows.
In Figure 3.10(c), the lateral shearstress exerted on the surfaceis shown to increase approximatelylinearly as a tread elementprogresses through the contactpatch. This is a commonassumption in physicalmodels [33,45,46]. The elemententers the contact patch and remainsin the original position of contact with the surface.The longitudinal axis (direction of heading) differs from the direction of travel and the element is
46 600-
500-
400
300
200- E 0 100 z
100 -Iý 1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 Distance Travelled [m]
(a)
200
150
100
50
0
-50
-100 r_
-j -150 1 -20 1.42 1.44 1.46 1.48 1.5 1.52 1-54 1.56 1.58 1.6 Distance Travelled [rn]
(b)
500 0. 0.51 400 ------1.0, to ...... o 1.5'' ! zý-c 300- -o2.0'' ID
200
100 ------0------
00111111111 -1 1.4 1.42 1.44 1-46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 Distance Travelled [m]
(C)
Figure 3.10 Measured contact stress distributions with slip angle on a horizontal (flat) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear stress 47 deflectedlaterally. This has been discussedin Section 2.7.1. A positive slip angle is shown in Figure 3.11(c) to produce positive lateral shear stresseson the surface; negativeshear stresses are exertedon the tyre. As expected,the lateral shearstresses increaseas the slip angle increases.At 2 degreesslip angle the maximumstress is 290 kPa Oustbelow the maximumnormal pressureof 320 kPa at 0 degreesslip angle).
3.6.3 Cornering Characteristics and Camber Angle
The normal pressureand shearstress distributions in the contact patch under camber angle(, v ) variationsto 6 degrees(in incrementsof 2 degrees)are shownin Figure3.11. The normalload exertedon the tyre was 3 kN and a slip anglecorrection was appliedto the tyre. The measurementswere obtainedat the lateralcentre of the experimentaltyre.
A small increasein the normalpressure and a reductionin the contactpatch lengthare shownto occurin Figure3.11 (a) as the camberangle increases. This is becauseone side of the experimentaltyre is 'lifted' from the surface,while the other side is compressed more againstthe surface.The shapeof the contactpatch changes[74] and the contact patch length at the lateral centre of the tyre reduces.It is postulatedthat the contact patcharea also reduces and, as a consequence,a generalincrease in the normalpressure occurs.This is the reasonfor the increasein normal pressureat the lateral tyre centre.
In Figure 3.11(b), the longitudinal shear stress on the surface is generally shown to follow the usual sinusoidal-style sweep. At the front the shear stress remains approximately constant at 80 kPa as the camber angle increases.The magnitude of the recovering positive shear stressesat the rear significantly reduces and for higher camber angles it even becomeshighly negative just prior to the transducer exiting the contact patch. These characteristicsare consistent with those observed when a driving torque is applied to a tyre [31]. Their relationship to camber angle is not yet ftilly understood.
A positive camberangle is generallyseen in Figure 3.11(c) to producenegative lateral shearstresses on the SafetyWalk surface;positive shear stresses are exertedon the tyre. The magnitudeof the lateralshear stresses are significantlylower (< 100kPa) thanthose obtainedunder the equivalentslip angleconditions (see Figure 3.1 O(c)). This is because the camberstiffness of the tyre is significantlyless than that of the corneringstiffness.
48 600
50
tv cL 40 gn (1) 300
200
z0 10
T4 ---L -10 - --142 -- 1.44- --14i6- I-Aý-8-- -1-.-5-2 --1.-514 -- 1.516- 1.158 1,6 Distance Travelled [m]
(a)
200 F
150
100
((D D 50
0
-50
(M-100 a 0 -j - 150
TA -20 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 Distance Travelled (m]
(b)
ID
1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 Distance Travelled [m)
(c)
Figure 3.11 Measured contact stress distributions with camber angle on a horizontal (flat) surface: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear stress
49 As a tread elemententers contact the lateral shear stressexerted on the horizontal surfaceincreases negatively over a short distanceof travel. The lateral shear stress remainsapproximately constant as a tread progressesthrough the contactpatch, until a smallkink is observedas the elementexits contact.This kink is againunderstood to be due to the slippagein the contact patch. As expected,the magnitudeof the lateral shearstresses in Figure 3.11(c) are found to increaseas the camberangle is increased.
3.7 Results and Discussion: Drum Experiments
Figures3.12 to 3.15 presentthe results from the rolling drum tyre testing machine. When comparedwith the stressplots in Figures3.8,3.10 and 3.11 from the flat bed machineit is straightforwardto see similarities and differences.The discontinuous appearanceof the plots in the drum experimentsis a resultof the constantsample rate of 1024Hz. The numberof measurementscollected in the contactpatch therefore reduces as the speedincreases and this samplingeffect gets more prominent(see Figure 3.13). At 10 km/h the numberof individual points is 75 but this reducesto 15 at 50 km/h.
3.7.1 Free-Rolling Characteristics and Normal Load
The normalpressure and shearstress distributions in the contactpatch at normal loads between I kN and 5 kN (in incrementsof I kN) are shown in Figure 3.12. The tangentialspeed of the drum and tyre (no relative slip) was 20 km/h. Measurements were obtainedat the lateral centreof the tyre and a slip anglecorrection was applied.
The shapeof the normal pressuredistributions in Figure 3.12(a) are similar to those obtainedon a horizontalsurface (see Figures 3.8(a)). However,the maximumvalue is usuallysignificantly higher and exceedsthe inflation pressureof 200 kPa by a factor of between1.75 (at 5 kN) and 2.25 (at I W). This indicatesthat on the drum the contact patcharea must be smaller.As the normalload increasesthere is a reductionin pressure at the centre and an increasein contact patch length. Thesecharacteristics were also observedon the flat bed. However, on the drum surfacethe stressreduction at the centreis more significant.This is due to the tyre bucklingphenomenon being magnified by the drum surface.As a tread elementexits the contactpatch the normal pressureis shown to slightly oscillatebefore returning to the zero state. It is believedthat such 50 600 1W 500 AN RN UN (L 400- aL 5kN 300
200-
0 luo -
01 %0- - 1.. i- -L-- -- II---- -I 4.9 4.92 4.94 4.96 4.98 5 5W 5D4 5.06 5,08 51 -iooL--, ý--, Distance-- -Travelled [m] (a)
200 -T--- T lkN 150 ---- AN RN looi ON 5kN 50'ý
0
-501 100 - 1-
- 150
-2004.9 4.92 4.94 4.96 4.98 5 5.02 5. (W 5.06 5.08 5 .1 Distance Travelled [m]
( h)
500
400
300
200
100
-1001 111111 4.9 4.92 4.94 4.96 4.98 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled [m]
(C)
Figure 3.12 Measured contact stress distributions with normal load on a cylindrical (drum) surface: (a) normal pressure,(b) longitudinal shear stress; (c) lateral shear stress
51 secondaryoscillation is a dynamicresponse of the stresstransducer. This oscillationis not evident in the flat bed experimentsas the surfacewas driven at too low a speed.
In Figure 3.12(b), the longitudinalshear stress exerted on the drum surfaceis seento follow the usual sinusoidal-stylesweep. The plots are similar to those found on the horizontalsurface (see Figure 3.8(b)). The maximumnegative and positivestresses from the two testingmachines are not significantlydifferent. An increasein the magnitudeof the stressat the rear comparedto that at the front is again evident, as well as the secondarykink in the distributionas the tread elementexits contact.This suggeststhe curvature of the drum does not significantly effect the longitudinal shear stress distribution.The changein the radiusat the circumferenceof the tyre tread increaseson the drum surfacebut this increaseis smallin comparisonto the changealready exhibited when the tyre is rolled on the horizontal surface.Since only a small additionalchange in the radius occurs, the changein the longitudinal shearstress distribution is small.
Lateral shear stresses are shown in Figure 3.12(c) to be generally low compared to longitudinal shear stressesin Figure 3.12(b). This characteristic has been discussedin Section 3.6.1 with regards to the free-rolling behaviour observed during the flat bed is experiments. The exception is the lateral shear stresses observed at I kN load. It postulated here that the slip angle correction was not accurately applied in this instance.
3.7.2 Free-Rolling Characteristics and Speed
The normal pressureand shear stress distributions in the contact patch at speeds between10 km/h and 50 km/h (in incrementsof 10 krn/h) are shownin Figure3.13. The normalload was 3 kN and againa slip anglecorrection was applied.The measurements were obtainedat the lateralcentre of the experimentaltyre. As the speedincreases, the numberof experimentaldata points obtainedalong the contactpatch lengthreduces and this samplingeffect makesthe contact stressdistributions appear more discontinuous.
A significantchange in the nommlpressures is observedin Figure3.13 (a) at the different speeds.No logical trend is evident, with the ascendingorder for maximumstresses shownto be 20,50,30,10 and 40 km/h; at speedsof 10,30 and 50 km/h the stress peaksat about 500 kPa. A changein the longitudinalshear stresses is also evidentin
52 600
500
0 ci. 400 m 300
Cl- 200
z0 10
L-. i--- II11 -100 --- - -i --- --L ---L------i 4.9 4.92 4.94 4.96 4.98 5 5ý02 5D4 5.06 508 5.1 Distance Travelled [m]
(a)
200-
W 150-
100-
50-
0
50
cn-loo c 0 -150
-200[4.9 4.92 4.94 4.96 4.98 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled [m]
(b)
0 gl-e 0
ic
-IVV4.9 4.92 4.94 4.96 4.98 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled (m]
(c)
Figure 3.13 Measured contact stress distributions with forward speed on a cylindrical (drum) surface: (a) normal pressure;(b) longitudinal shear stress; (c) lateral shear stress
53 Figure 3.13(b), but again there appearsto be no discerniblepattern to this change.A numberof investigationsby other researchershas examined the effectsof speed[3 1] but thesestudies have mostly been carried out using bias-plytyrcs. Generally,the results suggestan increasein speedcauses an increasein the normalpressure at the front of the contactpatch and a decreaseat the rear. Sincesuch a changein the pressuredistribution is causedby an increasein the rolling resistance(as discussedin Section3.6.1), there mustbe an increasein the longitudinalshear stress towards the rear of the contactpatch.
This type of change in the contact stress distributions is not confwmed by the measurementsplotted in Figure 3.13 or in the work by Seitz and Hussmann[751 where the authorssuggest that no effect wiU be evidentfor radial-plytyres up to approximately 130km/h, an opiniongiven support by Figure2.5. Furtherwork is requiredto determine the true influenceof speedon the contactpatch behaviour of the experimentaltyre. This could not be donedue to the closureof the experimentalfacilities at Dunlop Tyres Ltd. Based on the availableinformation, it is postulatedin this thesis that no noticeable differencewill occur in the free-rollingcharacteristics tested here (speedsto 50 km/h).
In Figure3.13(c), a smallchange in the lateral shearstresses is observedat the different speeds.The measurementsobtained at 10 km/h are significantly higher than those obtainedat speedsbetween 20 km/h and 50 knV1LSuch a changecannot be related to the physical behaviourof the experimentaltyre and, thus, it is again postulated that the slip anglecorrection was not accuratelyapplied at the lowest speedof 10 km/h.
3.7.3 Cornering Characteristics and Slip Angle
The normal pressure and shear stress distributions in the contact patch under slip angle (a) variations to 4 degrees (in increments of I degree) are shown in Figure 3.14. The ASCII-format data files are unavailable at 2 degrees slip angle and therefore the results cannot be presented. The normal load was 3 kN and the tangential speed of the drum was 20 kni/h. The applied slip angle was additional to the slip angle correction.
In Figure 3.14(a),the norrml pressuredistribution is shownto differ considerablywith slip angle from that obtainedusing the flat bed machine(see Figures 3.10(a)). The pressureis much higher with the maximum about 1.5 times that recorded on the
54 a
'I (I,
a E 0 z
-IWU 4.9 4.92 4.94 4.96 4.99 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled (m]
(a)
200 F-
150
looý
50: -
0 0- U).r-
-50:,
(M-100, 0c -150
-2004.9 4.92 4.94 4.96 4.98 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled [m]
(b)
CLw ,?e
cn
4.9 4.92 4.94 4.96 4.98 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled [m]
(c)
Figure 3.14 Measured contact stress distributions with slip angle on a cylindrical
(drum) surflace:(a) normal pressure;(b) longitudinal shear stress; (c) lateral shear stress
55 horizontal surface. Another notable difference is the contact patch length. On the flat bed surface it is fairly constant at 0.11 rn (up to the maximum slip angle of 2 degrees). The length increasesfrom about 0.1 to 0.13 rn on the drum surface as the slip angle increasesfrom I to 3 degrees. To the author's knowledge this phenomenon has not previously been identified and a clear explanation for the behaviour is not at hand. Thus, further measurementsobtained across the contact patch width (or simulation results from a validated FE model) are required to explain the phenomenon. Again the work could not be done due to the closure of the experimental facilities at Dunlop Tyres Ltd.
The longitudinal shear stressesshown in Figure 3.14(b) again follow a sinusoidal-style
sweep. A significant reduction in the stressesis shown to occur towards the rear as the slip angle increasesfrom I to 3 degrees.This is related to the friction ellipse limiting the maximum shearstresses that can develop. As discussedin Section 3.6.1, the lateral shear stressesdominate the friction and a reduction in the longitudinal shear stressesoccurs.
The lateral shear stressesin Figure 3.14(c) increasewith slip angle from approximately 0 kPa at zero degrees slip angle to approximately 350 kPa at I degree slip angle. At 4 degreesslip angle, this further increasesto approximately 450 kPa. These values are much higher than those measured on the horizontal surface (see Figure 3.10(c)) at an equivalent slip angle. At a slip angle of I degree the difference is greater than three times. A smaller contact patch area on the cylindrical surface is one possible reason for the difference. On the drurn, the shear stress increasesmore rapidly as a tread element
progressesthrough the contact patch. Growth continues until the shear stress overcomes the available friction (static coefficient of friction p, multiplied by the normal pressure)
and sliding occurs. For slip angles3 degreesand higher, the lower stressplateau towards the rear of the plots gives a measure of the dynamic friction. Interaction of the developing longitudinal and lateral shearstresses in this sliding region is governed by the dynamic (sliding) coefficient of friction a, and the normal pressure. The static and
dynamic coefficients of friction are estimated from Figure 3.14 to be close to 1.2 and
1.0, respectively. It is reasonablyassumed that the friction coefficients are the sameat all positions across the contact patch. This is the consistent with the approach adopted in semi-empirical tyre models [33,45,46]. It should be noted that these friction coefficients are significantly higher than those observed on typical road surfaces (see Table 2.1). The values are relevant to the development of the rolling FE tyre model 56 discussedin Chapter 4 and to the subsequentrolling tyrc simulations using this model.
3.7.4 Cornering Characteristics and Camber Angle
The normal pressureand shearstress distributions in the contact patch with camber angle(y ) variationsto 4 degrees(in incrementsof I degree)are shownin Figures3.15. The normal load exertedon the tyre was 3 kN and the tangentialspeed of the drum was 20 km/h. The measurementswere obtainedalong the lengthof the contactpatch at the lateral centre of the experimental tyre and a slip angle correction was again applied.
The normalpressure and shearstress characteristics shown in Figure 3.15 are similarto thosepreviously highlighted from the flat bed experimentsin Figure3.11. The exception to this is the distinct reduction in the normal pressuresat non-zero camberangles. Figure 3.15(a) showsthat the reduction is approximatelyconstant with camberangles I degreeand higher.The author cannotoffer an explanationfor this observationbased on the physical behaviourof the experimentaltyre. However, it is known that the measurementsat camberangles between I and 4 degreeswere not taken on the same day as the 0 degreeresults; they were taken much later. Thus, it is postulatedhere that the stresstransducer was not in calibrationwhen the normal pressuremeasurements at camberangles between I and4 degreeswere obtained.Calibration was the responsibility of DunlopTyres Ltd. andwas beyondthe control of the author.The authorbelieves that the longitudinaland lateralshear stresses are accurateand representativeof the 'actual' tyre behaviour.Only the normalpressure measurements appear to be effected.The shear stressdistributions in Figures3.15(b) and 3.15(c) correlatewell with their corresponding distributionsfrom the flat bed experimentsin Figures3.11 (b) and 3.11(c), respectively.
3.8 Summary
A comprehensiveexperimental investigation into the contact patch behaviourof an automobiletyre hasbeen completed. The stationarybehaviour of the tyre under normal load, and the behaviour under free-rolling and cornering (slip and camber angle) conditionshas beencharacterised. The load-deflectioncharacteristics and contactpatch dimensionsof the stationarytyre havebeen presented, and a reasonablycomplete picture is given on how the normal and shearstress distributions might changeunder driving 57 600 Oý 500- yI y 2" 30 400 4"
300-
200-
o 100-
0
-1004.9 492 4.94 4.96 4.99 5 502 5,04 506 508 51 Distance Tn"lled (m)
(a)
ol
150 - 1. 2, 3' luu - 4' 50
0
-50 -
a)-ioo c 0 -1-150 -
-2004 4.92 4.94 4.96 4.98 5 5.02 5. G4 5.06 508 51 .9 e Travelled [m)
(b)
4' 300
200-
U) 100
0
L -100494.92 4.94 4.96 4.98 5 5.02 5.04 5.06 5.08 5.1 Distance Travelled [m)
(C)
Figure 3.15 Measured contact stress distributions with camber angle on a cYlindrical (drum) surface: (a) normal pressure-,(b) longitudinal shear stress-,(c) lateral shear stress
58 conditions. The results have been analysedto provide better understandingof the physical behaviour of tyres in the contact patch. A comparison between stress measurementsobtained on a horizontaland a drum surfacehas also beenpresented and significant differencesbetween the two surfaces arc reported. The measurements presentedin this chapterprovide a valuablesource of physicaltest datathat can be used to evaluate the performanceof mathematicalmodels such as those presentedin Chapter4. Simulationresults obtained using thesemodels are comparedin Chapters5 and 6 to the stationaryand free-rollingphysical test data presentedin this chapter.The contactpatch behaviour during corneringconditions is not usedin this thesisbut maybe usedby other researchersto evaluatesimilar FE tyre modelsor semi-cmpiricalmodels.
59 Chapter 4 Finite Element Models for Simulation of Macroscopic Tyre Behaviour
4.1 Introduction
Finite Element (FE) tyre models presented in the literature vary greatly (see Section2.9). The meshmay consistof only a few axisymmetricelements [63] or many thousandsof solid, membrane/shelland beam elements[57,64]. This is becausethe modellingapproach depends not only on the requirementsof the simulationbut also on the availablecomputational resource. To satisfy the aims of the researchoutlined in Section 1.3, it is necessaryto establisha mesh refinementsufficiently adequateto simulatemacroscopic tyre behaviourlocal to the contact patch, but which is not so refined that the computational cost becomes intolerable with currently available computationalresources. The material constitutive equationsare important since no numericalsimulation can reliably predict the structural responsewhen the material property descriptionsare incorrect.An acceptablerepresentation of contactand friction is also importantto simulatethe actualtyre/wheel contact, and the interactionbetween the tyre and the contacted'ground' surface.This chapter addressesthese important aspectsof the FE modelling,and presentstwo modelsdeveloped to simulatestationary and rolling tyre behaviour. The chapter also highlights the author's modelling experiencesin relation to the explicit solver LS-DYNA [17] and identifies important areaswhich must be consideredby analystsundertaking tyre simulations.It shouldbe notedthat, unlessotherwise stated, version 950d of the codehas been used in this thesis.
4.2 Mesh Generation
Two separatetyre models were developed for analysis using LS-DYNA [17]. ,141,01. !;, -jN,- HyperMesh[77] was usedto generatethe LS-DYNA model descriptions.One model was specifically developedto simulate the structural behaviour of the stationary experimentaltyre (seeSection 3.4.1), while the secondmodel was developedto predict 60 the rolling tyre response (Sections 3.4.2 and 3.4.3). Fliese models are referred to here as the stationary and rolling models. respectively. The stationarv and (lie rolling ineshes are shown in Figures 4.1 and 4.2. The two meshes difler onlý N\ith respect to (lie element density in the circumferential direction, the niesh through (lie cross-scolon is Identical.
Thc models represent the tyre as a complex arrangement of' rubber components and reinfiorced rubber composites. The wheel is considered as a separate entity. Many FF tyre models in the literature 178-801 simply represent the \%liccl using constraint equations but this approach is only acceptable il'the purpose ol'the model is to simulate behaviour under static or typical driving conditions. During atypical driving conditions,
slippage or, in extreme cases, seperation can occur between the tyrc all(] \kllccl. To
enable simulations to be acceptable under atypical loading SitUatiOnS.an independent is Thus, wheel model necessary. the tyre and wheel are modelled separately in this thesis and the tyre/wheel contact (wheel tit) is considered as in additional houndary condition.
Tyre
Wheel
Contact Region
Figure 4.1 Three-dimensionalstationary (non-rolling) model
61 Tyre
VJheel
(a)
Wheel /
Tyre
vi
(h)
Figure 4.2 Rolling model: (a) three-dimensional model, (b) cross-section
62 The stationary model in Figure 4.1 is refmed local to the intended contact region and the elements extend every 3 degrees around the circumference. Further from the contact region, the mesh density is reduced and elements extend every 6, and then 12 degrees. Reducing the mesh density reduces the number of nodes and elementsin the model and, hence,the computational time. A total of 56,400 solid and membraneelements are used to model the stationary (non-rolfing) tyre and 1,400 shell elements to model the wheel.
In Figure4.2, the rolling model is shownto havea uniform meshdensity with elements extendingevery 3 degreesaround the tyre circumference.This mesh specificationis essentialin a Lagrangiananalysis because the region of contact betweenthe 'tyre' and gground' continually changes during rolling. In an Eulerian analysis, the mesh specificationin Figure4.1 would be appropriate[69] but to the author's knowledgeit is not possible to implement this approach in LS-DYNA [17] for solid mechanics problems,only those in fluid mechanics[81]. In the rolling model there are 135,360 solid and membraneelements in the tyre and 3,360 shell elementsin the wheel. As a result of the necessarycircumferential mesh refinement there is therefore2.4 times the numberof nodal displacementunknowns (degrees of freedom) in the rolling model.
The meshspecification used in the stationaryand rolling modelswas consideredto be the maximumacceptable in practice.A more refinedmesh (greater than 3 degreeslocal to the contact region) was consideredto be computationallytoo expensive.Details of the computationaltimes in relation to stationary and rolling tyre simulationsare provided in Chapters 5 and 6, respectively. In the stationary simulations, the computationaltimes are in excessof 200 h (8 days)and a significantadditional cost is evident in the rolling simulations. An increase in computational power and improvementsin the code will reducethese computational times or alternativelyallow greatermesh refinement in the future. It is noted that the meshdensity can influence the simulationresults and is thereforeimportant when simulatinglocal tyre behaviour.
4.2.1 Tyre Components
To allow separatestresses and strainsto be determinedin the tyre components,the - rubber components(tread, sidewalls,liner, apexesand clinches)and reinforcedrubber composites(carcass plies, bandagesand breakers)need to be individuaUyrepresented.
63 The rubbercomponents are modelledhere as deformablecontinua using linear (constant stress)solid elements[17]. Volume integration is carried out with one-pointGaussian quadrature.A significant cost saving is obtained by using the one-point integration (default) elementinstead of the fully integrated(8-point) solid clement.In simulations conductedwith the stationaryand rolling models,using the fully integratedelements was foundto approximatelytreble computational time. It shouldalso be notedthat onc-point integrationhas another advantage in addition to cost. Fully integratedelements used in plasticity problemsand other problemswhere a material's Poisson'sratio approaches 0.5 (rubber typically has a Poisson'sratio >0.49) can lock-up in constant volume bendingmodes. This elementlocking is undesirableand can causenumerical instability.
The biggestdisadvantage to one-pointintegration is the needto control the zero energy modeswhich arise, known as hourglassingmodes (see Figure 4.3). Hourglassmodes typically give a zig-zag appearanceto a mesh usually referred to as hourglass deformation.Such unwarranted deformation modes can swampthe actual deformation which is sought and, as a consequence,are commonly resistedby viscosity which is automaticallycalculated in the code.The viscousform of hourglasscontrol is the default in LS-DYNA [17] and is acceptablein most situations.In some situations,however, numerical problems can occur and it is beneficial to select a stiffness method of hourglasscontrol insteadof a viscousone. A detaileddescription of the viscousand stiffnessmethods of hourglasscontrol is given in the book by Jacoband Goulding[681.
An investigationwas conductedby the author to determinethe most acceptablecontrol method for the stationaryand rolling models.It was found that the viscousmethods availablein the LS-DYNA code (types I to 3) did not adequatelycontrol hourglass deformationin the models.This is clearlyshown in Figure4.4 wherethe defaultviscous formulation(type 1) hasbeen used with the stationarymodel. In the figure, the wheelfit and a inflation pressureof 200 kPa are represented.The standardFlanagan-Belytschko stiffnessformulation (type 4) also did not perform well. Thus, the author prefers the Flanagan-Belytschkostiffness formulation with exact volume integration (type 5) to control hourglassmodes. A stiffness formulation based on the Flanagan-Belytschko formulation is also used in a LS-DYNA tyre model developed by Kao and Muthukrishnan[57] but the type is not clearlyspecified. This leadsthe authorto believe that for tyre simulationsusing explicit FE software,such as LS-DYNA [17], a stiffness
64 1\
k
i:L:: "N. I
iI
1,-igure 4.3 Hourglass modes of an eight node element %N,lith one IIintegration point 1681
Figure 4.4 Deformation of the 'tyre' cross-section when the detýult (type 1) viscous hourglass control formulation in I, S-DYNA is employed with the stationarý, model
65 formulation to the hourglass control should be the principal method for consideration.
It shouldbe noted that, when usingthe stiffnessapproach, care is neededto ensurethe hourglasscontrol does not causeartificial stiffening of the model [57]. To check for artificial stiffening, the one-point integration solid elementswere replacedwith fully integratedelements and the predictedload-deflection characteristics of the tyre under normal load were determined.The approachadopted to determinethe load-dcflection characteristicsis discussedin Chapter5. The resultsobtained using the one-pointand the fully integratedelements were compared,and the differencewas found to not be significant.Based on theseresults it was assumedthat artificial stiffeningdid not occur.
The carcassplies and belt are modelledusing 'discretereinforcement' techniques. The rubbermatrix is modelledas a deformablecontinuum using linear solid elementsand the reinforcement(cords in a layer of rubber matrix, the thicknessof reinforcementbeing determinedby the nominal diameter of the cords) is discretely modelled using quadrilateralmembrane elements based on the Belytschko-Lin-Tsayshell element[82, 83]. The approachallows separatestresses and strainsto be obtainedin the matrix and reinforcement,but not in individual cords. To predict stressesand strainsin individual cords would require each of the cords to be individually representedand would significantlyincrease the number of elementsin the model, and hencecomputational time. The interfaces between adjacent componentsand between the matrix and reinforcementof the carcassplies, bandagesand breakersare modefledusing shared nodes. In doing this, it has been assumedthat the interfacesbetween the various materialsare fully bonded.Although delarninationhas had a substantialamount of attention recently [84] it rarely occurs in the field. Thus, the assumptionthat the interfacesare fully bondedis consideredacceptable here. An alternativeapproach would be to model theseinterfaces as separatecontacts, with the nodestied to the adjacent surfaceuntil a failure criterion is reached.This would requirea considerableamount of additionalcomputational cost and was thereforediscounted as an option in this thesis.
The beadsare modelledusing linear solid elements.A detailedrepresentation of the beadswas not consideredto be importantbecause they are a significantdistance from the areaof particularinterest, i. e. the contactpatch. Only the overall behaviour(usually referred to as the global material behaviour) of the beads is modelled and, as a
66 consequence,separate stresses and strainsmay not be obtainedin the cords and rnatrix.
4.2.2 Steel Wheel
The steel wheel is coarselymodelled in two halves using Belytschko-Lin-Tsayshell elements[82,83]. To allow simulationof the interactionbetween the tyre andthe wheel, thesetwo halvesare initially positionedeither sideof the tyrc (seeFigure 4.2). Sincethe steel wheel is much stiffer than the contactingtyre clinch (>100 times) and no stress output is required,it was assumedto be rigid. The rigid materialmodel (model 20) in LS-DYNA [17] is used.This model requiresthe user to definea modulusof elasticity, which is usedin any contactmodel (see Section 4.7), and also the densitywhich is used to determinethe massof the wheel.A Young's modulusof 200 GPa and a densityof 7860 kghiý are assumedin this thesis.These valuesare representativeof bulk steel.
4.3 Material Properties
The tyre consistsof a number of material components.Their mechanicalproperties requiredas input data to LS-DYNA [17] are given in Tables4.1 and 4.2. Densitiesare essentialsince a dynamic(rolling) analysisrequires knowledge of the massassociated with eachdegree of freedom(see Section 2.9). The densityfor all rubbercompounds are shown in Tables4.1 and 4.2. To determinedensities for the reinforcements,the wen known Rule of Mixtures was used with the constituentdensities given in Table 4.2 and their calculatedvolume fractions.The volume fraction of the cords V, is given by
(; r 0.2 /4) V. =Nx 0.
where N is the numberof cords per unit length, 0. is the nominal diameterof the cords and 0. is the effectivediameter. The volume fraction of rubber matrix is I-V,.
4.4 Rubber Compounds
A material'selastic behaviour is describedin terms of its stress/strainrelationships. For
67 RubberComponent SecantModulus (100 %) Density Poisson'sRatio [MPa] [kg/ný] [no units] Tread 1.25 1150 >0.49 Sidewalls 1.07 1120 >0.49 Liner 1.79 1140 >0.49 Apexes 7.13 1180 >0.49 Clinches 3.30 1160 >0.49
Table4.1 Mechanicalproperty data for rubbercomponents (Dunlop Tyrcs Limited) many materials,such as mfld steel, theseconstitutive relationships are assumedto be linearto the elasticlimit and a correspondingmodulus of elasticityis defined[62]. The tensile stress/strainrelationship is non-linear for rubber compoundsand therefore a single modulus of elasticity can only refer to a very small portion of the elastic behaviour.The secantmodulus is usually quoted (it is the instantaneousmodulus at a specific strain value). For rubbers,it is sometimesquoted at strains of 50,100 and 300 percent[85] but is often givenonly at the 100percent strain value. In FE modelling, this is usefulonly when rubberis highly strainedand the resultscan be approximatedin those regionswhere the strainsare much lower. Secantmoduli data is thereforenot usefulto the tyre FE analyst,as internalrubber compound strains are typically lessthan 10 percent(this valuewas obtainedfrom FE analysesusing the modelsdescribed here). As a consequenceof this, and the reluctanceof tyre manufacturersto releaseproperty datafor their preferredrubber compounds, relevant stress/strain data for FE modellingis not easyto obtain. Thus, methods-to approximatethe elastic behaviourare needed.
A simple method to characterisethe fundamentalstress/strain relationships for the rubber compoundsin the experimentaltyre (see Section 3.2) is used in this thesis. Considerationis given to the five rubber components,and the three rubber matrices associatedwith the carcassplies, bandages and breakers.The relationshipsare estimated using experimentaldata provided by Dunlop Tyres Limited for a typical 'unknown' rubbercompound subjected to simpleextension. The stress/straindata was obtainedin accordancewith BS 903: Part A2: 1995(ISO 37: 1994).The engineeringstress against engineeringstrain data is providedin AppendixB. Considerationis givento strainsup to
68 P-"
"CI = C) ONýý o- N "0
0 C) C) tr) 4n -8 9 C) C)
"0 Clq CD tn tn I C; C; C;
(Z CD (D
c; c; d
W2 rZ . - 0Goý +ý * ":CD -1:Q *1
N. * " --
r-" cl %D en 2 en Cý 40. -ci Cf) CA
110>b lZ 0
ON O's CN Itr IT IT C; C; C; A A A (L) ýo .
"N C) C) C) Ei CN C14 1ý v"
ýo ýc ý I "0 CD Q) 0 c> iý
rA Iciu A-4
.E*
z 50 percent. The secant modulus at 100 percent strain was also provided and was 3.19 MPa. To estimate the stress/strain relationship for the rubber compounds in the experimental tyre, the engineering stresses in Appendix B were scaled based on the specific 100 percent secant modulus listed in Tables 4.1 and 4.2, and that of the unknown tyre compound. This approximation assumes the stress/strain relationships for each of the rubber tyre compounds is similar to that of the unknown compound.
4.5 Mooney-Rivlin Equation
Numerousconstitutive models exist in the literatureto simulatethe elasticbehaviour of rubber compounds [86,87]. Seven different rubber models are available in LS-DYNA [17]. The Mooney-Rivlinequation is commonlyused in FE tyre models[57, 88] and it is seenas the industry standard.The model mathematicallyrepresents the elastic behaviourin terms of a strain energy function. This strain energy function is I), basedon the acceptableassumption that rubber is incompressible(i. e. A, 'ý A3= sincethe bulk modulusis severalorders of magnitudegreater than the shearmodulus, and isotropic in the unstrainedstate. The Mooney-Rivlin strain energyfunction W is
C2(I2 (4.2) W= CI(Ij - 3) + -3)
C2 where C, and are elasticconstants related to the material'sshear modulus, and the independentvariables I, and 12are relatedto the principal extensionratios and 2 dI= I/A2 + IIA22 + IIA These A,3yI, b II =A +'ý +A an 213. extension ratios are related to the engineering strain c by A=I+e, and the relationships between the principal true stresses q,, a,,' and q2', and the corresponding strains (Figure 4.5) are
(A, 222 ý C2) uj - ci =2 -, ) (CI +A3 (4.3(a»
tt 'C2) (4.3(b)) qý -aj =2(A2'-A3')(Cl +A, and
tt222 (4.3(c)) aj - ai = 2(.ý -A, )(Cl +, ý C2)
70 Thus, if a rubber compound is subjected to simple extension or uniaxial compressionin 0; the A, -direction then: A, = A, X"; the principal true stressesal23 = a, =
and the principal stress a,' can be obtained from Equ. 4.3(a). Equs. 4.3(b) and 4.3(c) is are used when extension is in the A2- and A. -directions, respectively. The true stress
2 A)(Cl 1 A) all =2 (A -11 +C2 (4.4)
and the engineering stress a, (force per unstrained cross-sectional area) is given by
)(Cl + C2 / A) (4.5) a, = 2(A -I/ A2
It shouldbe noted that for the caseof simpleextension A>I (seeFigure 4.6(a)), while for uniaxial compressionA< 1. Thus, a positive stress is obtained in tension and a
negative stress in compression. This is consistent with standard sign convention [731.
If the same compound is subjected to simple shear as shown in Figure 4.6(b): A, = A; A2 I; A, The 0 is = and =I/A. shear strain y=A-I/A= tan and the shear stress r,,y
2(Cl C2)y (4.6) r=XY +
a3l -t
x3 U all
.73,
(a) (b)
Figure4.5 Purehomogeneous strain: (a) unstrainedstate; (b) strainedstate
71 y
A-IAw 0 rxy all '44- all -1/2 Vxy
(a) (b)
Figure 4.6 Particular types of strain: (a) simple extension; (b) simple shear
Herein,the Mooney-Rivlinmodel is usedto simulatethe elasticbehaviour of the eight rubber compoundsin the experimentaltyre (i.e. the five rubber componentsand the rubbermatrices of the carcassplies, bandages and breakers).The Mooney-Rivlinelastic constants C, and C2 were calculated from the estimated tensile stress/strain relationshipsusing a curve-fitting program basedon the least squaresmethod [891. Theseelastic constants are shownin Table4.3 wherethe rubbermatrices of the carcass plies,and belt (bandagesand breakers)are groupedas one compoundreferred to as the topping.This is acceptablesince their mechanicalproperties are similar (seeTable 4.2).
Rubber Components Mooney-Rivlin Constants
C, C2
[MPa] [MPa] Tread 0.07 0.32 SidewaHs 0.06 0.27 Liner 0.10 0.45 Apexes 0.38 1.80 Clinches 0.18 0.84 Topping 0.17 0.80
Table 4.3 Calculated Mooney-Rivlin elastic constants
The estimatedstress/strain curve for the rubber tread compoundsubjected to simple extensionand the correspondingMooney-Rivhn curve derived from Equ. (4.5) are shown in Figure 4.7. Similar stress/straincurves can also be producedfor the other rubbercompounds. A favourablecomparison is observedup to an engineeringstrain of
72 45 percent and in the most important region up to 10 percent strain, the correlation is excellent. The gradient of the Mooncy-Rivfin curve at zero strain is similar to (lie
estimated curve and, thus. the stress/strain relationship in compression is also likely to be
accurately represented. The shear modulus In Equ. (4.6) was Ibund to be 0.77 MPa. Tlic
elastic behaviour in simple shear has been shown in Chapter 3 (see Figures 3.8 and
3.10 to 3.15) to be particularly relevant to modelling tyre behaviour in the contact patch.
0.7 ýj ýstimated' , (4 5) 0.6 --- Derived from Equ.
CL 0.5ý
(n 2? 0.4
0.3 -s
0.2
0.1
ý5 05 10 15 20 2ý 35 4'0 45 50 Engineering Strain [%]
Figure 4.7 Comparison between the estimated stress/strain Curve lbr the rubber tread compound in tension and that derived from the Mooney-Rivlin strain energy function
The Mooney-Rivlin expressionin Equ. (4.2) can be representedin LS-DYN A[ 171using either the incompressible Mooney-Rivlin rubber material model (model 27) or the I lyperviscoelast ic rubber material model (model 77). The tyre FE analysts Kao and Muthukrishnan [57] have simulated the Mooney-Riviin expressionbut do not specify the LS-DYNA material model used in their work. It is also not clear whether the Mooney- Rivlin or Hyperviscoelastic models are employed by C. Lee el al. [88]. 1lerein. to simulate a rolling tyre, the author prefers the Hypervisoelastic model and believes that. to control numerical instability, the actual model selection is important. This observation has been given support in private correspondencewith Dunlop Tyres Ltd. FE analystsat Dunlop Tyres have used the Mooney-Rivlin model in tyre analysesand have found it to
exhibit "some odd behaviour [90]". The material constitutive model appears not to accurately simulate the deformation of linear solid elements under shear deformation.
73 To verify the poor performance, an investigation was conducted and the main result is shown in Figure 4.8. In the figure, the 'tyre' is subjected to a3 kN normal load followed by the imposition of a longitudinal load (acting in the negative x-direction). An unrealistic shear distortion is observed in the tread with the Mooney-Rivlin rubber model. This is shown in Figure 4.8(a) when the longitudinal load is I kN. As Figure 4.8(b) shows, the excessive localized tread distortion disappears when the Hypervisoelastic model is employed. A shear distortion of approximately 2.3 mm is shown to occur at a2 kN longitudinal load. This value corresponds to the 1.8 mm distortion calculated in Appendix C using classical elasticity theory. A constant shear modulus of 0.77 MPa (see above) was used in the calculation. Based on this information, the author suggests that care should be taken in any analysis with the Mooney-Rivlin rubber model and advises against using the model for simulation of rubber deformation in a transversely (longitudinally and/or laterally) loaded or rolling tyre where shear stressesare known to arise in the contact patch (see Section 3.6). However, the author believes that the model can be used to realistically simulate tyre behaviour under normal loading and, thus, it is used in the normal loading simulations described in Section 5.3. Only a3 percent Merence (Section 5.4) is observed in the load-deflection behaviour using these two material models to simulate a load of 3M
4.6 Reinforcements
The reinforcementsof the carcassplies, bandagesand breakersare representedin the stationaryand rolling modelsusing an orthotropic elasticmaterial model. Singlevalue elasticconstants were calculatedusing volume fractions(see Equ. (4.1)) and the elastic constantdata for the constituentsshown in Table 4.2 by employingthe Halpin-Tsai micromechanicalequation [91]. This semi-empiricalequation was originally derivedfor continuousfibre reinforcedplastic composites.An alternativeapproach might be to use the Rule of Mixture equationswhich hasbeen employed by FE analystsat Dunlop Tyres Ltd. but the Halpin-Tsaiapproach is preferred in this thesis becauseit is known to give more accuracywhen determiningthe transversemoduli of a unidirectionallamina.
The OrthotropicElastic material model (model 3) in LS-DYNA [ 17] requiresthe analyst to define the elastic constantsin the elementaxes (x-, y-, z-), while the Halpin-Tsai equationcalculates them in the cord axes(1-, 2-, 3-). The orientationof the cord axes
74 fl
(a)
(h)
Figure 4.8 Shear distortion of linear solid elements in the contact region: (a) Mooney- Riviin rubber model (LS-DYNA model 27), (b) llyperviscoelastic model (model 77) for the carcassplies and breakersdiffers from the element axes (see Figure 4.9) and, as a consequence,the elastic constants used as inputs need to be resolved. This complication can be avoided by using the Composite Failure material model (model 22) available in the code. This Composite Failure model simulates an orthotropic material with optional brittle failure. If the failure option is not used, orthotropic elastic behaviour is represented. The model allows the elastic constants to be defined in the cord axes instead of the element axes but the orientation of the cords also need to be defined (see Table 4.2). No noticeable difference was found when analyseswere carried out using the Orthotropic Elastic and the Composite Failure material models and therefore the simpler approach using the Composite Failure model is preferred and used in this thesis.
75 Cord
Element
Figure 4.9 Element and cord axes in the reinforcementsof the rubber composites
To determinethe Halpin-Tsaiconstants, a single value elastic modulusof 3.16 MPa (correspondingto the 100 percentsecant modulus) was assumedfor the rubbermatrix of the carcassplies, bandages and breakers(topping compound).Elastic moduli of 3.36 and 2.96 GPa(see Table 4.2) were assumedfor the nylon cords in the carcassplies and bandages,respectively. A Poisson'sratio of 0.4 was used for the cords in both these components.For the steel cords in the breakers,the Young's modulusand Poisson's ratio were simplyassumed to correspondto thoseof bulk steel (i.e. 200 GPa and 0.3).
The beadswere simplyrepresented using an isotropicelastic material model (model I in LS-DYNA [17]) with the elastic constantsassumed to correspondto those of bulk steel. This simplificationwas again consideredto be acceptablebecause the beads are a significantdistance from the area of particular interest (as mentionedearlier).
4.7 Contact and Friction
There are two contactregions that needto be modelled:contact betweenthe tyre and wheel;and contactbetween the tyre and 'ground' surface(i. e. the contact patch). The groundsurface may be the glassplate usedin the stationaryexperiments, or the flat bed and drum surfacesused in the rolling experiments(see Sections 3.3). Thus, the ground surfacemodel is simulationdependent and is thereforenot discussedin detail here,only salientfeatures of the contact algorithmsare consideredrelevant. The ground surface modelsare describedin detail in relation to specificsimulations in Chapters5 and 6. It shouldbe noted, however,that in each of the ground surfacemodels, the contacting surfaceis assumedto be rigid (materialmodel 20 in LS-DYNA [17]) and is modelled
76 using Belytschko-Lin-Tsay shell elements [82,83]. This acceptable approximation simplifiesthe complex contact problem and hence reducesthe computationaltime.
LS-DYNA [17] has numerous(>30) contact models incorporatedin its code. These contact options are used, for example,to model contact of deformablebodies, single surfacecontact in deformablebodies, ajqd deformableto rigid body contacts.They require the analystto input numerousparameters (16 mandatoryinputs are typically neededand further optionalinputs can also be used)which influencethe operationof the contact algorithms. The influence of many of these parametershas not been characterisedand there is a lack of benchmarkproblems [17] to provide standardisation. Thus, contactmodelling often relieson the experienceand expertiseof the FE analyst.
To simulatethe tyre/wheelcontact and the contactpatch behaviour, versatile surface-to- surface contact models are used here. A surface-to-surfacecontact model allows compressionto be carriedbut not tension,thereby allowing two bodiesto be separateor in contact. Master nodes correspondingto the tyre/wheelcontact are defined on the wheel and the contactingslave nodes on the tyre clinch (see Figures3.1). The master nodesat the contactpatch are definedon the 'ground' surfaceand the slavenodes on the 'tyre' tread.The masternode nearest to eachslave node is determinedautomatically by the codeand the mastersegment (a 'segment'is definedto be a four-nodeelement of the surface)attached to the masternode is the first to be checkedfor contact. The processis repeatedfor all the relevant masternodes and slave segments.Thereafter, movementin the contactregion may causenodes to move from one segmentto another.
The standard surface-to-surface option available in LS-DYNA [17] (contact type 3) tracks contact between two surfaces using 'mesh connectivity'. Instead of checking all segmentsfor contact, only those segmentsattached to nodes of a previously contacted segment are checked. As a consequence,contact cannot be tracked across free edges. This standard option is suitable to model the contact between the tyre and the ground surface. It is, however, not suitable for modelling the tyre/wheel interaction becausein this instance contact occurs across a free edge of the tyre. Thus, the automatic option (contact type a3) is required. This option checks for contact using a search method known as a 'bucket sort' which divides the three-dimensional space occupied by the contact surface into cubes (these cubes are referred to as buckets). The idea behind the
77 bucket sort is to perform somegrouping of the nodesso that the sort operationsneed only calculatethe distanceof the nodes in the nearestgroups. Nodes can contact any segmentin the samebucket or an adjacentbucket. The bucketsort methodis robustand between 100 to 1000 times faster than the old direct sort method employedin the code [17]. This sort method,however, is still an expensivepart of the contactalgorithm.
Sliding with closure and separationis analysedin LS-DYNA [17] by the Penalty stifffiessmethod [92]. Oncethe contactingsegments have been determine for eachnode, the amountof penetrationof the slavenode into the mastersegment is calculatedand resistedby a penaltystiffness. For shell elements,the penaltystiffness (k) is given by
AK (4.7) d
where f, is the penalty factor (default 0.1), A is the area of the contactedsegment, K is the bulk modulusof the contactedelement and d is the maximumdiagonal of the shell.The maximumshell diagonalis shown in Figure 4.10. Equ. (4.7) is modified for a solid elementby replacing(I / d) with AV, where W is the element'svolume.
Figure 4.10 Maximum diagonal of a shell element
LS-DYNA [17] must calculate both the solid element penalty stifffiess corresponding to the 'tyre' tread and clinches, and also the shell element penalty stifffiess analogous with the steel wheel and ground surface models. These calculations are necessarybecause the code must factor both the master and slave stiffnesses for the contact algorithm.
It shouldbe notedthat a significantaniount of unwarrantedpenetration can occur using the Penaltystiffness method. To avoid this, an optional SOFT flag is availablein the surface-to-surfacecontact models. By settingthe flag to I (i.e. on), the 'soft constraint' 78 contact formulation is used instead of the basic Penalty stiffnessmethod. The soft constraint formulation is an advancedPenalty method where the maximum stable contact stiffiess is chosenfor each penetratingnode. Thus, unwantedpenetration is minimised.The approachis preferred by the author becausein less complex rolling simulations(carried by FE analystsat Dunlop Tyres Ltd.), the basic Penaltystifffiess method was found to cause contact problems. In these simulations,nodes on the perimeterof the 'tyre' tread were observedto penetratethe elementsused to represent the 'ground' surface.The penetratingnodes became constrained on the oppositeside of the definedtyre/ground contact and, as a consequence,the simulationbecame unstable.
Friction modellingin LS-DYNA [17] is basedon the Coulombformulation where it is assumedthat no sliding occurs below a specific load level. The maximum friction force Ff that canbe developedis aF. (seeSection 1.1), where F,, representsthe force
in the normaldirection. A transverseforce exceedingthis maximumfriction level causes sliding to occur. It shouldbe noted, however,that the force necessaryto causesliding differs from that to maintainit and, as a consequence,a distinctionis madebetween the static (peak) pp and the dynamic(sliding) p, coefficientsof friction (Section2.6). This
transitionis smoothedin the LS-DYNA code by an interpolation.This interpolationis
l'-Il (4.8) p =, u., + (pp - g) e-d,
where d, is a decaycoefficient and v,,,, is the relative velocity betweenthe slavenode and the master segment.A higher decay coefficient increasesthe decay rate of the coefficientof friction and a lower valuereduces it. Sincethe frictional force is relatedto the coefficientof friction, the force necessaryto maintainsliding is influencedby d,
The peak and sliding friction coefficientsfor a numberof typical ground surfacesare presentedin the literature(see Table 2.1), but to the author'sknowledge no tyre/ground datais availablein relationto the decaycoefficient in Equ.(4.8). There is also very little informationin relation to tyre/wheelinteraction and its correspondingdecay rate. It is knownthat FE analystsat Dunlop TyresLtd. typically use a decaycoefficient of 0.5 and thereforethis value is assumedby the author in this thesis. It has been used for all tyre/groundand tyre/wheelcontacts herein. The peak and sliding friction coefficients
79 betweenthe tyre and wheelhave been estimated to be 0.3 and 0.1, respectively.Those at the contactpatch are simulationdependent but, on the 'Safety Walk' surfaceused in the rolling experiments,are 1.2 and 1.0 (Section3.7.3). The changein the tyre/ground friction coefficienton the this surfacewith relative velocity is shown in Figure 4.11.
1.3
12
C w 1.1 ;w 0 U 0c u I . r_ LL
O.9o 1234567a9 lo ReladveVelocky Im/sl
Figure 4.11 Simulated friction coefficient with relative velocity on a 'Safety Walk' surface
4.8 Summary
Two FE tyre models, developed to simulate the stationary and rolling behaviour of an automobile tyre have been described in detail. These models have been developed for simulations using the explicit FE software package LS-DYNA. The results from these analysesare shown in Chapters 5 and 6. The modelling methodology has been discussed and related to the author's experiencesusing the code. The 'tyre' has been represented as a composite material and the 'wheel' has been separately modelled. The interaction between the tyre and the wheel has been consideredas an additional boundary condition. The chapter has addressedimportant aspects of the FE modeffing, such as the mesh specificationand the materialdescriptions, and the lack of experimentaldata neededto characterisethe materialproperties of tyre rubber compoundshas beenhighlighted. A simplemethod to estimatethe elasticbehaviour has then beenpresented. The method used to model the wheel fit and tyre/ground interaction has also been described.
80 Chapter 5 Simulation of Stationary (Non-Rolling) Tyre Behaviour
5.1 Introduction
The purposeof the Finite Element(FE) modellingwork is to simulaterolling (dynamic) tyre behaviourat the microscoPiclevel. However, to accuratelypredict dynamictyre characteristics,it is clearlynecessary to be ableto simulatethe lesscomplex problem of stationary(non-rolling) tyre behaviour.Thus, this chapteraddresses this importantissue. Numericalresults are presentedthat can be comparedto thoseobtained in the stationary experiments(see Section 3.5) andto informationavailable in the literature.These results are used to validatethe stationarymodelling methodology described in Chapter4. A parametricstudy is then usedto characterisethe sensitivityof the numericalresults to changesin the mesh density of the 'tyre' and contacting 'ground' surface, and to changesin the elasticproperties of the tyre components.The developmentof a relevant andreliable modelling methodology is a major aspectof the work reportedin this thesis.
5.2 Stationary Simulations
As discussedin Section 3.4.1, stationary experiments were conducted to determine the vertical tyre stiffness and the growth of the contact patch under normal load. In the present chapter, these experiments are simulated in a single LS-DYNA [17] analysis using the stationary model. This analysis is referred to here as the normal loading simulation. The inflation of the tyre, the wheel fit (interaction between the tyre and the wheel) and the normal loading of the tyre against the glass plate are considered. The 'tyre' deformation during the simulation is shown in Figure 5.1. Figure 5.1(a) shows the tyre cross-section at the start of the simulation and Figures 5.1(b) to 5.1(d) show the cross-section as the simulation evolves with time. In other stationary (non-rolling) simulations referred to as the longitudinal and lateral loading simulations, a3 kN normal load was followed by the imposition of a longitudinal or lateral load. These simulations 81 were carried out to determine the longitudinal and lateral tyre stillnesses. respectively. Fhestiffinesses are important to a tyrc's tractive propcrtic,, and conicring characteristics.
I
(a) (b)
(c) (d)
Figure 5.1 Deformation of the *tyre' cross-section: (a) undeformed, (b) due to inflation pressure and wheel fit; (c) at a normal load of' I kN, (d) at a normal load of 5 kN
5.3 Normal Loading Simulation
An initial test simulation was carried out over a 'real time' of'O. 8 s with tile time stcp size controlled by LS-DYNA [ 17]. This mesh dependenttime step was 1.86 x1 0-7s and was governed by the solid elements in the beads local to the contact region (see Figure 4.1). The simulation time was found to be nearly 1200 h (50 days) basedon the available computational resource, a Sun Ultra 60,360 MHz workstation. This computational cost was considered intolerable for a quasi-static analysis and therefore modelling optimisation techniques [68] were employed to refine the analysis such that the most efficient use was made of the resource. As a consequence,the analysis time
82 was reduced to about 200 h, approximately one sixth of the initial simulation time.
The real time was halved by optimisationof the wheel fit and inflation, and normal loading phases,and the time step size was also 'forced' to minimisethe numberof calculations.The optimisationof the wheelfit and inflation,and normal loading phases is discussedin Sections5.3.1 and 5.3.2, respectively.A forced 'mass scaled' minimum time step of 5x 10-7s was selected(see Section 2.9) and resultedin an increasein model mass of approximately 13 percent. It should be noted that the controlling elementsof the meshwere locatedin the beadsand, as a consequence,the beadmass was increased.No changein the mass of the other parts, i.e. the tyre components (rubbercompounds and reinforcements), the two halvesof the wheeland the glassplate, was observed.This was becausemass is only addedto elementswhose time step size would otherwisebe lessthan the forcedvalue. The calculatedand scaledpart massesare shownin Table5. L In the table,the matricesof the carcassplies, bandages and breakers are againgrouped as a single rubber compoundwhich is referredto as the 'topping'.
Part CalculatedMass [kg] ScaledMass [kg] T,ge 12.87 14.99 Tread 3.46 3.46 Sidewalls 0.74 0.74 Liner 0.82 0.82 Apexes 0.36 0.36 Clinches 0.52 0.52 Beads 0.85 2.97 Topping 3.35 3.35 Reinforcementsof carcassplies 1.17 1.17 Reinforcementsof bandages 0.45 0.45 Reinforcementsof breakers 1.15 1.15 Wheel 1.84 1.84 GlassPlate 1.18 1.18 15.89 18.01
TableS. I Calculatedand scaledpart masses
83 Mass scaling (via modification to a material's density) is a very useful and simple method to increasethe minimum time step of an analysis. However, it should be noted that a significant change in material density can influence the dynamic response of the model by increasing inertia forces through added mass and modifications to inertial properties, and the overall energy balance via changesin kinetic energy levels [681. The analysis carried out here simulates the stationary experiments and, thus, a dynamic response does not exist. Similarly, a dynamic response is also not present in the longitudinal and lateral loading simulations (see Section 5.4) because the loading is quasi-static. The increasein the bead mass is therefore not important. It is postulated by the author that this increase in bead mass is also insignificant in the rolling tyre simulations presented in Chapter 6. This is becausethe beads are in contact with the wheel, at a significant distance from the area of particular interest, i. e. the contact patch.
To improve the model response,the tyre componentswere critically damped.The fundamentalfrequency of a typical passengercar tyre in isolation(and thereforeof the tyre components)is approximately40 Hz [35]. In LS-DYNA [17], dampingcan be appliedat both the materialand systemlevels. At a materiallevel, dampingis usually controlledby applyinga factor proportional to the massand/or stiffnessterm This is corrunonlyreferred to as Rayleighdamping [68] and a different factor can be appliedto eachmaterial. In the mass-proportionalmethod, a retardingforce is appliedto eachnode proportionalto its velocity, while in the stiffnessmethod a restoringstress is appliedat an elementallevel basedon the elementstrain and its constitutiveproperties. System dampingis similar to material dampingbut on a global scale,i. e. the samedamping factor mustbe appliedto all materials.To the author's knowledge,no notableadvantage is obtainedby usingone methodover the other methods.Mass-proportional damping at a materiallevel is employedherein because it is easilyapplied in the LS-DYNA code.
5.3.1 Wheel Fit and Inflation Phase
The wheelfit andtyre inflation (seeFigure 5.1) were carriedout simultaneously,initially over a period of 0.05 s. This real time is not representativeof an actual situationbut does not invalidatethe analysis,since the transient results during the wheel fit and inflation phaseare not relevantto the stationaryexperiments. Only the steady-statetyre deformationis important.Minimising the real time reducesthe computationalcost but,
84 at the extreme. may cause numerical instability, such as contact problems which are known to occur at high impact velocities. Thus. the initial real time was chosen conservatively to mininlise simulation time while ensuring a stable solution. It was reduced to 0.03 s and the simulation was repeated without a disccrnibic change in the steady-statedeformation. A flurther reduction was fiound to cause numerical instability in the analysis.This is shown in Figure 5.2 with a real time ol'O. 01 s. In (he figure, a high velocity impact occurs between the tyre and wheel, and a shock wave develops. This shock wave is transmitted through the structure and causesan unplannedtermination of' the simulation. This type of' termination is usually referred to as in analysis 'crash'.
High Velocity Impact
(a) (b)
I (c) (d)
Figure 5.2 Deformation of 'tyre' cross-section at a high tyre/wheel impact velocity: (a)
undeformed tyre (I=0.00 s); (b) tyre/wheel impact (I=6x 10- s); (c) shock wave
development (f =8x 10-' s); (d) shock wave transmissionthrough structure (i = 0.01s)
A constant pressureof 200 kPa (see Section 3.4) was applied normal to the inner-faceof the *tyre' liner and the two halves of the 'wheel' were displaced laterally (in the y-direction) until the actual fit between the tyre and wheel was represented.One half of the wheel was displaced in the positive y-direction while the other was displaced in the
85 negativedirection. The two halveswere constrainedin the longitudinal(x-) and vertical (z-) translationsand in rotationsabout the x-, y- and z-axesvia body constraints.They were merged to create a single part (referred to as the wheel) once the fit was achieved.
It shouldbe notedthat a constantinflation pressureis often assumedin tyrc analysis[57, 61,931 eventhough a slight changein the pressureis known to occur when a vehicleis running on the road [88]. This changeis mainly due to tyre operating temperature variationsand also deformation(air volumechange), and for an ideal gas like air [94] is
PV T
where p is the pressure, V is the volume, T is the temperature, and R is a constant.
It is postulatedhere that the temperaturechange will not be significantin the stationary experimentssince they were carried out for only a short time period. Similarly, the changeis alsonot consideredsignificant in the rolling experiments.Thus, Equ. (5.1) can be usedwith constanttemperature and only the volume changeneeds to be considered. Gough [72] statesthat "pressurechanges during tyre deflectionare small" and quotesa 2 percentpressure rise from the undeflectedstate to normal operatingdeflection, under isothermalconditions. Errors associatedwith FE simulationsare commonlyfound to be 10 percentor more and,thus, a constantpressure is consideredacceptable in this thesis.
An alternative approach is to simulate tyre inflation using an 'airbag' [59]. Airbags are represented in LS-DYNA [171 as an enclosed mesh of membrane elements. These elements define a 'control volume'. A mass flow rate for the inflation gas and a parameter to control the venting also need to be defined. In tyre analysis, the inflation gas is air and venting is usually set to zero. This assumesthat there is no loss of inflation gas. Thermodynamic relationships (see Equ. (5.1)) are then used to determine the internal pressure in the control volume based on the mass and temperature of the inflation gas present and the volume enclosed. This internal pressure is then applied to the membrane elements causing the airbag to inflate. Thus, airbags can be used to simulate tyre inflation and the subsequent changes in the air volume, and the correspondingpressure change as the tyre deforms. Since the volume change is small (as
86 mentionedearlier), it is the author's opinion that this approachonly addsan unnecessary complication to tyre analysis and therefore it is not used in the work reported herein.
5.3.2 Normal Loading Phase
After the 'tyre' was allowed to achievea state of equilibrium,it was normally loaded. The glassplate model was displacedvertically until a normalload greaterthan 5 kN was appliedto the tyre. The displacementwas initially carriedout at a rate of 0.05 mls (over a real time of 0.6 s). This rate is significantlylower than that selectedby others [88] in sirailar analyses but was conservatively chosen to ensure quasi-static loading, representativeof the stationaryexperiments. The rate was increasedand the simulation was repeated.The normalload exertedon the tyre was comparedto the reactionat the wheel and at rates less than, or equal to, 0.125 m/s the load and reaction did not significantlydeviate. At the maximumrate the reaction was 4.95 kN, being I percent lower thanthat applied.The kinetic energywas alsochecked at this rate and foundto be negligiblein the normal loading phasewhen comparedto the total energy.Thus, at a displacementrate of 0.125 m/s, the loadingcould still be consideredto be quasi-static.
It shouldbe noted that the glassplate is representedby a 0.34.5 ni horizontalsurface coarselymodelled using 15 quadrilateralshell elementsbased on the Belytschko-Lin- Tsay shellelements [82,83]. As mentionedin Section4.7, the surfacewas taken to be rigid. The glassplate model is shown in Figure 5.3. Significantchanges in the mesh density of the surface (up to 1500 elementshave been used) were found not to noticeablyaffect the simulationresults. No changein the predictedvertical tyre stiffness or contact patch dimensions(length and width) was observedwith the more refined mesh.The peak pp and sliding p, coefficientsof friction at the contact patch were
estimatedbased on the valuesin Table 2.1 to be 0.3 and 0.2, respectively.However, it was consideredunlikely that changing these values would significantly alter the simulationresults. This was later confirmedin an investigationcarried out to determine the sensitivityof the analysisto variationsin the friction levels.No changein the results was found with variationsin the peak and sliding friction coefficientsbetween 0.15 and 1.2, and 0.1 and 1.0, respectively.These coefficientsare consideredto envelopethe possiblefriction level betweenthe tyre and the glassplate in the stationaryexperiments.
87 ppwý IT mmol,
vý-,
Figure 5.3 Three-dimensional glass plate modcl
5.4 Longitudinal and Lateral Loading Simulations
In the longitudinal and lateral loading simulations. consideration x\as gken to a normal load of 3 kN followed by a subsequentquasi-static transverse (longitudinal or lateral) load. The normal load was selectedbased on the assumption that tile \,\-eight ofa typical saloon car is 12 kN (i. e. 3 kN per wheel). This load was exerted on the 'tyre' by displacing the glass plate model vertically. The displacementnecessary to create the tyre load was calculated using the predicted quasi-static load-deflection characteristics obtained from the normal loading simulation. At the selected displacement, the normal load was 3.1 M about 3 percent too high. This was considered acceptable becauseas mentioned earlier, errors inherent in FE simulations are typically 10 percent or more.
5.4.1 Transverse Loading Phase
The 'tyre' structure was allowed to achieve a state of equilibrium and the glass plate model was displaced either longitudinally or laterally at a rate of 0.05 M/s until sliding was evident. Sliding was assumedto occur when an increase in the displacementof the plate did not result in a corresponding increasein either the longitudinal or lateral load. The point when sliding occurs is governed by the normal load and the peak coefficient of friction. Therefore to ensure an adequate amount of data prior to sliding, the peak coefficient was specified to be artificially high, i.e. 0.7. The longitudinal and lateral tyre stiff-hesseswere assumedto not be influenced by this friction coefficient. It should be noted that the longitudinal and lateral loads are developed by displacement of the glass plate and therefore the simulation time is dependent upon the load-deflection characteristics. A 10 mm displacement (0.2 s) corresponds to a computational time of 88 approximately 100 h. Thus, the simulation time for the longitudinal and lateral loading phasescan be shownto be around 230 h (10 days) and 115 h (5 days), respectively.
5.4.2 Restart Files
Since the longitudinal and lateral loading simulationsonly representa subsequent loadingphase in additionto the wheel fit and inflation, and normal loadingphases (see Section5.3), restart files can be usedto reducethe total computationaltime. A restart file is effectivelya databasethat representsthe completedescription of a modelat a time specifiedby the FE analyst.This method allows modificationsto be madeto a model without havingto return to the initial stageof data initialisation.The simulationcan be repeatedfrom the restarttime with the modificationsincluded in subsequenttime steps.
Two typesof restartfiles are availablein explicit softwareand theseare referredto here as smalland full-deck restart files. This terminologyis commonlyused by FE analysts. The smallrestart files are usuallycreated more frequentlythan the full-deck restartfiles becausethey use lessstorage space. However, they are useful only when smallchanges needto be madeto a model.An exampleis an increasein the real time of an analysis.A full-deckrestart file is necessarywhen major changesare needed,such as the deletionor additionof parts of a modeland/or the re-specificationof loads.They are often created when an analysisis carried out in distinct loading phases.This is becausea restart file createdat the end of an analysisphase can be used as an input file for subsequent loadingphases. A benefitis thereforeachieved if a numberof subsequentcases need to be assessed.In the normal loadingsimulation reported in Section5.3, full-deck restarts can be createdafter the wheelfit and inflation phase,and when the normal load exerted on the tyre is 3 kN. To reducecomputational cost, this later restart file could then be usedas the input file for the longitudinaland lateralloading simulations. At the time the normalloading simulation was carriedout (Nov. 2000), the author was not awarethat restart files could be createdin LS-DYNA [17] and, thus, they are not used in this thesis.The approachwould havebeen used had the author beenaware of the method.
5.5 Selection of Analysis Results
The selectionof analysisresults for output and their output frequencyis an finportant 89 aspect of any explicit FE simulation. A careful selection avoids the unnecessary generation of large results files that contain superfluous data. These large data files cause an increase in computational cost and storage space. Thus, only analysis results needed to validate the modelling methodology in Chapter 4 have been created herein.
In LS-DYNA [17], analysis results can be output as field data (stresses, strains, displacementsetc. ) in binary or ASCII-format, or as X-Y data. The field data is typically used in this thesis for graphical animation (for example see Figure 5.1), and verification and debugging purposes, and the X-Y data is used for plotting key results (Figure 5.4). This method is recommended in the text book by Jacob and Goulding [68]. The X-Y data is stored electronically in ASCII-format and loaded into MATLAB [71] for analysis and visualisation. This is consistent with the method used to analyse the experimental data in Chapter 3. Here, the sampling frequency of the field data and X-Y data are 2x 10-3s and 5x 10-4s, respectively. These sampling frequencieswere chosen to give an adequatedata resolution while ensuring an efficient use of storage space. It should be noted that the X-Y data can be stored more efficiently than the field data. The field data and X-Y data at each sampling point are taken to be the instantaneous results at the end of the time step that coincides with the time output is desired [951.
5.6 Results and Discussion: Normal Loading Simulation
The load-deflection characteristicsobtained from the normal loading simulation describedin Section5.3 are shownin Figure5.4. Figure5.5 showsthe predictedcontact patch dimensions(length and width). In the figures, the normal load is given in kN, and the deflection and contact patch dimensionsare given in mm. To provide a meaningfulcomparison, the contactpatch length and width are plotted on identicalaxes. The correspondingphysical test data taken from Figures3.6 and 3.7 is also included. The normalload was determinedfrom the vertical contactforce and the deflectionfrom the vertical displacementof the 'glass-plate'. The contact patch dimension were calculatedfrom the co-ordinatesof the nodes on the perimeter of the tyre tread; assuminga symmetricalcontact patch only a quarterof the tyre nodeswere considered. These nodes were assumedto be in contact when their vertical co-ordinates correspondedto thoseof the nodeson the glassplate. Sincelinear solid elementswere use to model the tyre tread it was assumedthat the contact patch dimensionsremain 90 constantbetween successive nodal contactsand, as a consequence,a 'staircase'plot is thereforeproduced. These nodal contacts are representedin Figure5.5 by squarepoints.
36
30
25
20
15
10
6 Physical Test Deft (from Figure 3.6) Normal Loading Simulabon n 01234 Normal Load [kM
Figure5.4 Nonnal load-deflectioncharacteristics
The predictedload-deflection characteristics are shown in Figure 5.4 to be in excellent agreementwith those determinedfrom the stationaryexperiments. Both show a near linearrelationship between load and deflection,and indicatethe vertical tyre stiflhessto be approximately180 kN/m. Basedon this information,it is postulatedthat global tyre behaviour is accuratelyrepresented using the modelling methodology describedin Chapter4. This is given further support by the resultsobserved in the longitudinaland lateral loading simulationsin Section 5.7. These simulationsappear to give realistic longitudinal and lateral tyre stiffinesses.As will be discussedlater, however, these stiffiesseshave only beenvalidated by typical tyre data and not by physicaltest data.
The numericalcontact patch dimensionsin Figure 5.5 yield a staircaseplot. In reality, the growth of the contactpatch is continuousand a trend-line (representedby closely spaceddots) is thereforealso indicated.One approachwould have been to draw the trend-line mid-way betweenthe extreme values of the staircaseplot. However, the authorhas chosen to representthe contactpatch at the higherextreme. It is the author's opinion that refiningthe meshwill only displacethe valuesat the lower extremeto the higher values;a coarsermesh will further reducethe lower values.The higher values, which correspondto the distinct points where node-to-surfacecontacts occur will 91 ISO
160
-f 140 E 120
100
80 CL 60
40 -e- PhysicalTest Date (from Figure 3.7(a)) 20 -a- Normal Loading Simulation ------Normal Loading SimulationTrend-Line 0 3 456 Normal Load
(a)
ISO
160 lp- ...... -g 140 ...... L 120 t ?. 100
90
Ig 60 0Q ý) 40 PhysicalTest Data (from Figure 3.7(b)) 20 Normal Loading Simulation Normal Loading Simulatior Cd 3 456 Normal Load
(b)
Figure 5.5 Tyre/ground contact patch dimension with normal load: (a) length; (b) width remainunchanged. This behaviouris confirmedin Figure 5.6. To createthe figure, the mesh density of the stationary model local to contact region was reduced in the circumferentialdirection and the simulationwas repeated.No changewas madeto the meshthrough the tyre cross-sectionand thereforea significantvariation in the contact patch width was not expected.Thus, only the contact patch length is presentedhere.
In terms of the contact patch dimensionsshown in Figure 5.5, a good trend-wise agreementis evidentbetween the simulationresults and thoseobtained in the stationary
92 I 60-
40-
20-
CW i 00-
80-
60-
40-
20- Original Mesh Density (from Figure 5.5(a)) Reduced Mes ýý nI 345 Normal Load Pq
Figure5.6 Simulatedcontact patch length with normalload and meshdensity experiments.Both setsof dimensionsshow a similar linear relationshipbetween normal load and contact patch length, and non-linearrelationship between load and contact patch width (for loadsbetween I kN and 5 kN). Basedon the assumedFE trend-line, however,the numericalcontact patch lengthand width are typically 30 mm and 20 nun higher,respectively. To find out why the predictedcontact patch dimensionsdiffer from those obtainedin the full-scalephysical tests, a parametricstudy has beencarried out.
5.6.1 Parametric Study
In the parametricstudy, the elasticconstants of the tyre components,the meshdensity of the tyre and glass plate models, and the contact algorithm were all identified as possible reasonswhy the contact patch dimensionsdiffered. As mentionedearlier, changesin the meshdensity of the tyre (seeFigure 5.6) and glassplate (Section5.3.2) do not significantlyinfluence the contact length and width. Variations in the elastic constantsof up to ±25 percentof the nominalvalues were also unableto accountfor the difference.This is clearlyshown in Figure 5.7. In the figure, the physicaltest datais shownas a benchmark.To illustratethe fact that tyre behaviourunder normal load is governedby the inflation pressureand not the tyre structure,the effectsof an identical changein the pressureis also includedin the contactpatch lengthplot (Figure 5.7(a)).
By disregardingthe elasticconstants, and the meshdensity of the tyre and glassplate models,it is postulatedthat the differenceis relatedto the performanceof the contact 93 180 160
140
120
100 A
80
60 -u, mTysjc&ji est wata (Trom1-jgurej. i tap 40 El ElasticConstants +25% ElasticConstants -26% 20 Tyre InflationPressure +25% Tyre InflationPressure -25% Co 2346 NormalLoad [kNj
(a)
180 160
-0140 13 120 ......
El- loo - I-- 13-- 80
u 60
0 40 e Physical Test Deft (from Figure 3J(b)) 20 -m- Elastic Constants +25 % Ei-- Elastic Constants % OL - -26 0 3 456 Normal Load Dq
(b)
Figure 5.7 Tyre/ground contact patch dimensions with normal load at various 'tyre' component stiffnesses(elastic constants) and inflation pressures: (a) length; (b) width algorithm. Numerous parametersinfluence the contact algorithm (16 mandatory and 14 optional inputs are needed in the standard surface-to-surface model) and therefore, to verify the author's opinion, it is necessaryto characterise the effect of each of these. This is beyond the scope of the work reported here and, thus, this has not been possible.
It should be noted, however, that a deficiency with the contact algorithm has been given support by way of private communication with a FE analyst at Ove Arup & Partners, the distributors of LS-DYNA [17] in the United Kingdom. Dr B. Walker [96] has strongly
94 indicatedthat a one-to-onecorrelation is unrealistic for such a complex non-finear contactproblem and that a trend-wiseagreement should be consideredto be the norm.
5.6.2 Normal Pressure Distribution
The predictednormal pressure distributions in the contactpatch at normal loadsof 1,3 and 5 kN are shown in Figure 5.8. Thesepressure distributions! were calculatedusing nodal force data at the contactedtyre nodes;again only a quarter of the nodeswere considered.An alternativeand (in hindsight) more direct method is to use the nodal stressfield data.This methodcan be shownto yield the samecontact pressures but was not usedhere because as mentionedin Section5.5, field dataneeds more storagespace. At the time the analysiswas carried out, this storagespace was not available.At the centreof the contactpatch, the longitudinaland lateralpositions are 0 mm. It shouldbe noted that the 'channels'evident across the contactpatch (seenat a lateral position of ±20 mm) do not correspondprecisely to the grooves of the tyre model shown in Figure4.1. This is becausethe contactpressure at the very edgeof the grooveswas not considered.The channelsare thereforewider than the groovesof the experimentaltyre.
In Figure 5.8, a reductionin the normal pressureat the centre of the contact patch is seento occur as the normal load increases.The simulationresults therefore confirm the observationby Browne et aL [31] that the tyre centre 'buckles' upwards under increasingnormal load. This phenomenonhas previouslybeen discussed in relation to the flat bed experiments(see Section 3.6.1). At the sametime there is an increasein the contactpatch size and the shapechanges from an oval to a rectangularone (Figure3.5). Thus, it is evidentthat the modellingmethodology can be usedto simulatethe 'actual' contact deformations experiencedin a stationary tyre. However, a one-to-one correlationin the normal pressuresis not realisedbecause, as discussed,the contact patch lengthand width (and thereforealso area) differ from those obtainedin the full- scalephysical tests. This is clearlyobserved when the pressuredistribution is compared to the rolling contact stressesat the lateral tyre centre, given in Chapter 3, and also those presentedacross the contact patch width, in work carried out by Dennehy[9].
It shouldbe noted that a significantamount of 'noise' was evident in the nodal force data usedto createFigure 5.8. This is shown for a node at the centre of the contact
95 (a)
(b)
600
100 'low i0o 50 -inn -50 00 Lateral Position [mm] -50 50 Longitudinal Position [mm] -100 100
(c)
Figure 5.8 Simulated normal pressuredistributions: (a) I kN, (b) 3)M (c) 5 kN
96 patch in Figure 5.9. In tile figure, a smooth trend-fine (represcntcd by a I-voldline) is also indicated to illustrate the trend. Similar nodal force data can tv presented I'm- the other contacted tyre nodes. The normal pressures were cý11culatedbased oil tile smoothed nodal forces oil the contacted nodes and the corresponding 'area ofinflucilce'. The tread is modelled using linear elements and therefibre the area ofinfluericc was taken to be a quarter ofthe area ofthe adjoining segments. The definition ol'a segment has previously been given in Section 4.7. Since tile niesh is unifiorm in the contact region (see
Figure 4.1), this area was approximately tile same fior each contacted flodc. The arca ý was found to be 8.5x 10-. rn2-,a segment can be shown to he approximatcly 5x 17 rnin.
2 (D p 0 Lj-
'a 0 z
013 6 Normal Load
Figure 5.9 Contact force on a node at the centre of the contact patch with normal load
The nodal force characteristics shown in Figure 5.9 illustrate tile reduction in load pressure observed at the centre of the contact patch. This provides evidence relating to the onset of tyre 'buckling. In the figure, this buckling effect appears to occur when the normal load exceeds 0.8 kN. Since the tyre construction is typical of that used oil a standard road vehicle, this value is likely to be representative of when buckling occurs in many tyres. This is of particular interest to tyre technologists and vehicle dynamic analysts involved in the implementation of semi-empirical tyre models because it indicates a clear point when the normal pressure distribution changes. Thus, it is suggested here that the parabolic normal pressure distribution sometimes assumed [41,
49] is only representative at low normal loads, e.g. to I kN. At slightly higher loads a uniform or trapezoidal distribution [33,45] is better and. at typical operating loads
(>2 kN), an alternative distribution which represents the tYre buckling etTect is needed.
97 5.7 Results and Discussion: Longitudinal and Lateral Loading Simulations
The load-deflectioncharacteristics obtained from the longitudinal and lateral loading simulationsare shown in Figure 5.10. The loads are again given in kN and the deflectionsin mm. The normal load-deflectioncharacteristics (see Figure 5.4) are also includedin the figure to allow a simple visual comparisonto be made betweenthe normal, longitudinal and lateral tyre stiffnesses.These have been found to be approximately180 kN/m, 200 kN/m.and 100 kN/m, respectively.It shouldbe notedthat the longitudinaland lateral loadsexerted on the 'tyre' were againdetermined from the contactpatch forces, and the deflectionsfrom the displacementof the glassplate model.
30
25
T 20 E c Is .0
co lo
6
2L 2.6 3 3'6 Load Pq
Figure 5.10 Simulated normal, longitudinal and lateral load-deflection characteristics
Due to the closure of the experimental facilities at Dunlop Tyres Limited, the author does not have longitudinal and lateral load-deflection measurementsfor the experimental tyre. As an alternative, however, Dunlop Tyres have provided longitudinal and lateral stiffnessvalues for a similar 195/60 R15 tyre with an inflation pressure of 200 kPa [97]. These stifffiesseswere 300 kN/m and 170 Min, respectively. It should be noted that
although the tyres are of similar construction, these stiffness values are likely to be significantly higher than those for the experimental tyre. This is because sidewall. deformations influence the longitudinal and lateral stiffnesses. These sidewall deformations are affected by the rubber properties and the tyre's aspect ratio (see
98 Section2.3). The aspectratio is different for the two tyres and thereforethe stiffnesses are only useful as a guide. It is noted that for a 195/70R15 tyre, the lateral stiffnessis assumedin the work by Lee et al. [88] to be about90 kN/m. Basedon this information, it is postulatedthat the predicted longitudinal and lateral stifffiessof 200 kN/m and 100 Min, respectively,are within the range expected.The simulationsappear to give realistic longitudinaland lateral tyre stiffnesseswhich are necessaryto simulatetyre behaviourduring acceleration/braking, and cornering(slip and camberangle) conditions.
5.8 Summary
The structuralbehaviour of the experimentaltyre during the stationaryexperiments (see Chapter3) hasbeen successfully simulated. Numerical results have been compared with a reasonabledegree of successto experimentalmeasurements. An excellentcorrelation was found for the normal load-deflectioncharacteristics. Longitudinal and lateral load- deflectioncharacteristics have also beenanalysed and theseappear to give realistictyre stiffnesses.These simulation results therefore suggest that the modellingmethodology presentedin Chapter4 could be employedto accuratelysimulate global tyre behaviour.
The predictedcontact patch dimensions(length and width) were found to give a good trend-wiseagreement, but the contactpatch lengthand width were typically 30 nim and 20 mrn greaterthan the physicaltest data, respectively.As a consequence,the normal pressuresin the contact patch were found to be lower than those expected.To determinethe reason(s)for the difference,a parametricstudy has been carried out. Basedon the results of the study and on information from Ove Arup & Partners,it appearsthat the differenceis likely to be relatedto the contactalgorithm inherent in the code. This suggestsit is not straightforward to accurately predict the contact deformation, and correspondinginternal transient stressesand strains in the tyre structure,in absoluteterms. However, the good trend-wiseagreement suggests that the modelling methodologyshould be capableof predicting internal transient responses which are, at least,related to the 'actual' contact patch deformationsin a rolling tyre.
99 Chapter 6 Simulation of Rolling Tyre Behaviour
6.1 Introduction
It has alreadybeen mentioned that the aim of the Finite Element(FE) modellingwork is to provide an initial investigationof the complex internal stressesand strains in a rolling tyre. Furthermore,it hasbeen stated that the accuracyof any simulationis only as good as the FE programand the way in which it is employed.Stationary (non-rolling) tyre simulationshave beencarried out in Chapter5, using a Sun Ultra 60,360 MHz workstation. The computationaltime for simulatinga tyre under normal loading (to 5 kN) was found to about 200 h. This time can be approximatelyhalved utilising the current resource,a Sun Blade 1000,750 MHz workstation. All simulationsin the chapterwere carried out using LS-DYNA [17] version 950d. This versionwas, at the time, the most up to date release.It shouldbe noted that the current releaseis version 960. The stationaryresults suggest the modellingmethodology is capableof predicting internaltransient stresses and strainswhich are relatedto those in a rolling tyre. Thus, the rolling behaviouron a flat bed and a rolling drum (seeChapter 3) is consideredhere.
6.2 Computational Considerations
Onefull revolutionof a rolling tyre is requiredat the desiredspeed (see Section 3.6). To simulateone revolution at a longitudinaltyre velocity correspondingto that in the flat bed experiments(0.18 km/h) would require a 'real time' in excessof 10 s. The computationalcost for sucha simulationis estimatedto be greaterthan 12,000h (500 days)using LS-DYNA version 950d with a 'forced' minhnum.time step of 5X 10-7S. This estimationis basedon the current computationalresource, a Sun Blade 1000.The computationaltime is doubled using the 'double precision' option now availablein version960 of the code. As will becomeclear later in this chapter,double precision is neededto successfullysimulate a rolling tyre and the simulation the is therefore 100 obviously unacceptable.It was, however, recognisedthat a significantly reduction in the computational cost can be achievedby increasingthe rolling tyre velocity. Consideration has been given here to a velocity of 20 kmIL This choice of speed was deemed acceptablebased on the assumption that no discernible change occurs in the contact patch stress distributions up to a velocity of approximately 130 km/h [75]. Thus, the rolling simulationsare carried out over a real time of approximately I s, about one tenth of the computational time originally estimated if the tyre velocity was only 0.18 km/h.
6.3 Rolling Simulations
The rolling model describedin Chapter 4 is used to simulate tyre behaviour during free- rolling conditions on the flat bed and rolling drum surfaces.In the simulations,the wheel fit and inflation, the norrnal loading to 3 kN and the rolling of the tyre arc represented.
6.3.1 Wheel Fit and Inflation, and Normal Loading Phases
The wheelfit and inflation phasewas carriedout in an identicalmanner to that in the normal loading simulationdescribed in Section5.3. The Ityre' was then allowed to achievea stateof staticequilibrium and the 'ground' surfacewas displacedvertically (in the z-direction)at a rate of approximately0.4 m/s. This rate was chosento minimisethe simulationtime while ensuringa stable solution. Contact problems,at high impact velocities,are known to occur with LS-DYNA [17] and thesehave beendiscussed in Section5.3.1. The rateof 0.4 mIs is not representativeof the rolling experimentsand, as a consequence,it introducesa numericaldynamic effect into the normalloading phase. This doesnot invalidatethe analysisbecause transient deformations during this phaseare not relevantto the rolling simulationresults, only the steady-statetyre deformationis important.To simulatethe deformation,the surfacedisplacement was again initially selectedbased on the quasi-staticnormal load-deflection characteristics shown in Figure 5.4. This hasbeen discussed in Section5.4. As will becomeclear later, it wasnecessary to modify this displacementto accuratelysimulate the deformation.The tyre was allowedto achievea stateof equilibriumafter the specifieddisplacement was reached.
The flat bed is representedby a 7.75 m long x 1.15 m wide horizontal surface coarsely modeHedusing 3,565 quadrilateral rigid sheU,elements. The mesh is uniform and the
101 elements are based on the Belytschko-Lin-Tsay shell element 182,83 1. Similarly, tile 2.39 m diameter drum surface is modelled using 8,280 shell elements.The drum model is shown in Figure 6.1. It should be noted that this model is more than two times as refined circumferentially as the flat bed model is in the longitudinal (x-) direction. This is necessaryto adequately represent the curvature of the drum using flat quadrilateral elements.Significant changesin the mesh density do not noticeably affect the stationary simulation results (see Section 5.3.2) and it is therefore acceptablehere to assumethat similar mesh density changeswill also not significantly influence the rolling simulations.
V-- x-
Fhýure 6.1 Three-dimensional drum surface model
6.3.2 Rolling Phase
To simulate the rolling action, the wheel rotational constraint about the y-axis is released
and a velocity is then applied to the 'ground' surface. This is representative of the
situation in the flull-scale physical tests carried out on the flat bed and rolling drum
machines, respectively. This velocity was applied in the longitudinal (x-) direction in the
flat bed simulation. In the drum simulation, the drum was rotated about its centroid
causing the contacting tyre to rotate. The velocity was increased linearly to the desired
speed of 20 km/h over a real time of 0.4 s. This real time was chosen because excessive slip was found, by trial and error procedures, to occur at higher surface accelerations.
102 6.4 Tyre Damping
Mass-proportional damping has again been applied at a material level to optimise the model responsein the wheel fit and inflation, and normal loading phases.This damping was removed prior to the rolling phasebecause retarding forces applied to eachnode are known to eliminate realistic transient responses.Appropriate damping levels are an area of constant debate and, like contact modelling, rely heavily on the experience and expertise of the FE analyst. Jacob and Goulding [68] have presented some representativeviscous damping levels for selected structures but no recommendationis given for an automobile tyre (or similar structure). The recommendedlevels are typically less than 3 percent of the critical damping level. Based on this information, it was consideredacceptable to discount damping in the rolling phase. This issue is discussed further in Section 6.5 where the damping level is linked to the rolling resistance characteristics.Rolling resistance has a constant but a relatively minor effect on the contact patch stress distributions and it is therefore postulated that the assumptionto neglect dampingdoes not invalidate the modelling methodology. To simulatetyre rolling resistanceeffects, further work is neededto characterisea more realistic damping level.
6.5 Results and Discussion: LS-DYNA version 950d
The flat bed and drum simulation were initially analysedusing LS-DYNA version 950d. These analyseswere found to experience numerical instabilities and in both cases a , floating point exception' was found to occur. This numerical problem arises because real numbersare stored as words of finite length (i. e. fixed numbers of digits), usually referred to as either single or double precision with a word length of 32 bits and 64 bits, respectively.A floating point exception is said to occur when a number is too large or too small to be representedand, as a consequence,the analysis'crashes'. This problem is known to occur in the code and has also been found by other FE analysts[98] but to the author's knowledge it has not previously been reported in relation to tyre analysis. The flat bed and drum simulations were carried out in July 2001 when only single precision was available.Double precision was not availableuntil February2002 with the current release(version 960) and, thus, a significant time period (almost 8 months) was spent by the author unsuccessfullytrying to provide a modelling solution to a numerical instability problem inherent in the code. For example, various tyre acceleration rates 103 between 7 mls' and 24 m/s2 were tested. Based on this experience, it is the author's opinion that rolling tyre simulations using single precision are not to be attempted becausethe analysisis likely to crashbefore the requirednumerical results are available.
6.6 Results and Discussion: LS-DYNA version 960
The flat bed simulation was reanalysedusing version 960 and the simulation time was found to be approximately 2,400 hours (100 days), double that needed for a single precision analysis.This penalty is deemed acceptablein terms of the project's aim but, for future analyses, the simulation time could be significantly reduced by model optimisation of the rolling phase.For example, the tyre accelerationcould be increased by applying a rotational velocity to the wheel which corresponds to that given to the horizontal surface. This wheel rotational velocity could be removed once the desired speedis reachedand the tyre would then be driven solely by the surface. Since the tyre accelerationis not relevant to the rolling simulations,such model optimisation would not invalidate the results. In a feasibility investigation carried out by the author, it was found that the desired speed could be reached in a real time of 0.1 s. This correspondsto a computationalsaving of about 720 h and thereby reducesthe simulation time to 70 days.
After the tyre, travelling at 20 km/h, had rotated one full revolution, the contact patch behaviourwas examined.The predicted contact stressdistributions at the normal load of 3 kN are shown in Figure 6.2. Figure 6.3 shows a comparison between the stressesat the lateral tyre centre and those obtained in the flat bed experiments(again at 3 kN).
In Figure 6.2, the contact stressesare given in kPa, and the distancetravelled by the tyre and the lateral tyre position are given in m and mm, respectively. The stressvalues are those obtained as a contacting node progressesthrough the contact patch and 17 nodes acrossthe width are considered.The distancetravelled in Figure 6.3 correspondsto that found in the flat bed simulation and not that found in the physical tests where the tyre travelled about 1.6 m. The physical test data is plotted on top of the simulation results for comparisonpurposes. In both figures, the numerical stresseswere obtained directly from the nodal stressfield data, not via the nodal force data, at an interval of 5X 10-4 s. As explainedin Section 5.6.2, this does not influencethe simulation results. The stresses in the figures are those exerted on the tyre and not on the horizbntal surface. The shear
104 600 500 400 300 0 c9 200 gi ii 100 E 00 Z - 100 100
0 3.4 Lateral Position [mm] -50 3.35 Distance Travelled [m] -100 3.3
(a)
w cl- 200 M
(b)
(c)
Figure 6.2 Simulated contact stress distributions at a normal load of 3 kN and a velocity of 20 km/h: (a) normal pressure; (b) longitudinal shear stress; (c) lateral shear stress
105 600-
500-
400-
300-
200-
z look
-10 3.32 3.34 3.36 3.38 3A 3.42 3.44 3A8 3AS 3.6 Distance Travelled (m]
(a)
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CL 100-
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0- 10m
M r
-2OgL- 3.32 3.34 3.36 3.39 3A .3 3A2 3.44 3.48 3.48 3.6 Distance TrmHed lm)
500 PhysicalTest Data (from Rý7ureU) Flat Bad Sirrvulationvorsion 1960 960 400 C 7 CL 300
U) ;i 200 ID Co 2 100
C
T3 1 -10 j- -L L 3.3'2 3.3'4 3.36 3.38 3L4 3A2 3ti 3"46 3.4'8 3.5 DistanceTravelled [m]
(c)
Figure6.3 Contactstress distributions at the lateral centreof the tyre at a normal load of 3 kN: (a) normal pressure;(b) longitudinal shear stress; (c) lateral shear stress
106 stressesare thereforeopposite in senseto the experimentalmeasurements presented in Figures3.8, and 3.10 to 3.15. The normal and shearcontact stressesare plotted on similar axes to provide a simple meaningfulcomparison, and the length depicted in the plots is again0.2 m. At the lateralcentre of the tyre, the lateralcoordinate is 0 nun.
The normalpressure exerted on the contactingnodes in Figures6.2(a) tendsto be high at the front of the contact patch, lowest at the centrc and then high againat the rear. This characteristicis relatedto the 'buckling' effect of the tyre which hasbeen discussed in Section3.6.1. The pressuresat the front are approximatelythe sameas thoseat the rear suggestingthat the rolling resistanceeffect is not simulated.This is also suggested by the longitudinalshear stress distribution in Figure6.2(b). The authorbelieves that the rolling resistancecan be introduced into the numerical results by damping the tyre components.However, rolling resistanceis a minor cffect (typically about 50 N) and, thus, neglectingdamping in the rolling phaseis consideredacceptable in this thesis.
At the lateraltyre centre(see Figure 6.3(a)), the maximumpredicted pressure is found to be about450 kPa and at the centreof the contactpatch the pressureis 325 kPa. The correspondingnormal pressuremeasurements obtained in the flat bed experimentsare about 320 kPa and 275 kPa, respectively.The discrepancyin these numerical and measuredvalues can be relatedto a differencein the pressuredistributions across the width of the contactpatch. Interestingly,the predictedpressure distribution also differs significantlyfrom that experiencedin the stationary(non-rolfing) numericalresults (at 3 kN) shownin Figure 5.8(b). This result is discussedfurther in Section6.6 where the disparity is relatedto changesin the code betweenLS-DYNA version 950d and 960.
In Figures6.3(b) and 6.3(c), the predictedlongitudinal and lateral shearstresses at the lateral tyre centreare shownto exhibit similar characteristicsto those observedin the flat bed experiments.The longitudinalshear stress follows a sinusoidal-stylesweep as the contactingnode progresses through the contactpatch, and an excellentcorrelation is evidentin the peak numericaland measuredstresses at the front (--80 kPa). In contrast to the physicaltest data, however,the predictedlongitudinal shear stresses at the rear are significantlylower than those at the front, the peak value having a magnitudeof about30 kPa.Again this disparitycan be relatedto a differencein the stressdistribution acrossthe width. Zero net longitudinalforce is found in the rolling simulationbecause
107 the shear stressespredicted under the tyre shoulders are significantly grcater at the rcar of the contactpatch than at the front. At a lateraltyre co-ordinateof ±55 mm, the peak stressesat the front and rear of the contact patch are 20 kPa and -50 kPa, respectively.
The lateral shear stressesacross the width (see Figure 6.2(c)) exhibit similar characteristicsto thoseobserved in the experimentalwork by Dennehy[9] and DeBecr [23]. On one sideof the tyre, positive lateral shearstresses arc developed,whilst on the other sidethese stresses are negative.The stressesare about equal in magnitudeand, as a consequence,the overall effect is a zero net lateral force. The lateral shearstresses at the lateralcentre of the tyre are approximatelyzero. This is to be expectedunder free- rolling conditionsbecause conicity and ply steer effects (Section 2.7.3) are negligible.
The numericalresults show characteristics wl-iich are similarto thoseobserved in the flat bed rolling experiments,and also mentionedin the literature [9,31,75]. The buckling effect in the normal pressuredistribution and the sinusoidal-stylestyle sweep in the longitudinalshear stress distribution are both evident.The lateral stresscharacteristics are alsorealistically simulated [9,23]. Whilst the resultsare not a one-to-onecorrelation with the physical test data they again confirm the potential of the modelling methodologyto predict contact deformationswhich are experiencedby a rolling tyre.
6.7 Incompatibility between LS-DYNA version 950d and 960
The normalpressures in the contactpatch shownin Figure 6.2(a) are significantlyhigher than those observedfrom the stationary(normal loading) simulationin Figure 5.8(b) usingversion 950d. This differencecannot be relatedto the 'real' physicalbehaviour of the tyre during the rolling mechanismand thereforemust be related to changesin the coding. This was confirmedwhen the normal loading simulationwas repeatedusing version 960. The norinal load-deflectioncharacteristics and contact patch dimensions (lengthand width) are shownin Figures6.4 and 6.5, respectively.To provide a simple visual comparison,the simulationresults obtained using version 950d (see Figures5.4 and 5.5) and the physicaltest data (Figures3.6 and 3.7) are also includedin the plots. To discountthe possibilitythat the rolling tyre simulationresults were influencedby the analysisprecision, both single and double precision analyseswere carried out with version 960. Precisioncan affect the results through truncation of real numbers,or 108 round off-errors in repeatednumerical calculations. No discerniblechange was found herebetween the two precisions.The differencesshown in Figure 6.4 and 6.5 bctwccn the two versionsare thereforedue to algorithm changesin the LS-DYNA [17] coding.
25- 9 E 20- r
PhysicalTest Data(from Figure 3.6) NormalLoading Simulation version 960d (from Figure5.4) Normal LoadingSimulation version 960 012346 NormalLoad M
Figure6.4 Normal load-deflectioncharacteristics
The load-deflectioncharacteristics shown in Figure 6.4 are seento differ using the two different versionsof the code. Version 960 again gives a linear relationshipbetween normalload and deflectionbut the correlationwith the physicaltest data is now not as good as observedwith version 950d. The new predicted vertical tyre stiffness is approximately200 kN/m. This stiffnessis II percent higher than that obtainedusing version950d and that measuredin the stationaryexperiments. It is, however,within the error limits usuallyexpected for FE simulationsand is thereforeconsidered acceptable.
The contact patch area is more accurately represented using version 960 (see Figure 6.5). The contact dimensions (length and width) again show a good trend-wise agreement with the physical test data but the difference in the width dimension has significantly reduced. Based on the distinct points when the node-to-surface contacts occur (represented by the square points), the predicted width from version 950d is typically reduced by approximately 15 mm. The contact patch length remains approximately the sameto a load of 4 kN, i. e. typically 30 nun greater than the physical test data and, as a consequence,the contact patch area is reduced. This reduction in contact patch area correlates to the increasein the normal pressuresusing version 960.
109 10
160-
140 -
120 - loo
80-
so- .0 0C L) 40ý Ee Physical Test Data (from Figure 3.7(a)) 20 --13- Normal Loading Simulaton version 950d (from Figure 5.5(a)) -Uý- Normal Loading Simulation version 060 Co 235 Normal Load
(a)
ISO 160
-f 140 _E 9 120 9 loo 80 u 60
0 40 --e- PhysicalTest Date(from Figure3.7(m)) 20 --Ei- NormalLoading Simulation version 950d (from Figure5.5(a)) -E3-. NormalLoading Simulation version 960 co 23 456 NormalLoad PcNj
(b)
Figure 6.5 Tyre/ground contact patch dimension with norrml load: (a) length; (b) width
It should be noted that as LS-DYNA [17] develops,'bugs' found in the code are removed.A numberof problemswith version950d have beenidentified by others [991 and changeshave been made to the two subsequentreleases, i. e. versions950e and 960. Thesechanges, as well as new additionalfeatures, are summarisedin the Update and ReleaseNotes [100,101]. The changesare numerous(>100 bugs were removedfrom version950d prior to releasing950e) and the differencesshown in Figures6.3 and 6.4 from are unlikely to be relatedto any one specificmodification. Possible causes range changesin elementformulations to the performanceof the contact algorithm. Analysis resultscan also differ from machineto machine[99]. Researchplays a major role in FE
110 softwaredevelopment and this thesishas identified someproblems with version 950d. For example,the performanceof the incompressibleMooncy-Rivlin rubber material model (materialmodel 27) under sheardeformation. This problem is not mentionedin the subsequentUpdate and ReleaseNotes [ 100,101] and must thereforeremain an issue for the code developers.The problemis beyondthe scopeof the work reportedherein.
It is the author'sopinion that the differencein the resultsfrom the two versionsdoes not underminethe earlier model validation work when only version 950d was available. Thesenumerical results, shown in Figures6.3 and 6.4, only reemphasizethe difficulty in predictingtyre behaviourlocal to the contactpatch. A good trcnd-wiscagreement in the contact patch dimensions(length and width) is still observedand becausethe contact patch width is more accuratelyrepresented, it is likely that, the internal transient responsesin the tyre will be more accuratelypredicted. Thus, the author believesan improvedunderstanding of the transienttyre stressesand strains local to the contact patch can still be achievedvia numericalsimulation using version 960. The resultsalso suggeststhat a FE analyst should demonstrateextreme caution before using results obtainedfrom differentversions of the samecode to validatea modellingmethodology.
6.8 Internal Stresses and Strains
In Section 1.3 it was stated that to support the development of in-tyre sensor system technologies, it is necessaryto gain a greater understanding of the internal transient stressesand strains in a rolling tyre. These are important becausethey can be used to determine the most appropriate location for an in-tyre sensor (or sensors) and the dynamic relationship (transfer function) between the sensor outputs and the contact patch stresses. It should be noted, however, that the location of a sensor will be influenced by its performance in terms of strain range and cyclic endurance, as well as tyre operating temperature. The cyclic endurance is a function of the strain level [1021.
A numberof technologiesare potentiallyavailable for in-situ measurementof transient elastic deformations in rolling tyres. There are a number of advantagesand disadvantageswith each potential method, and a significant amount of development work is neededbefore any solution becomespractical. Some of thesetechnologies are detailedin the work by Tomka et aL [103], and include surfaceacoustic wave (SAW)
III clectroactive materials and magnetoactive soil magnetic materials. Vhjs %%ork has focused on one of the key potential sensor technologies in(] this has resulted in Ilic development of a magnetic field insensitive stress sensor based on amorphous magnetic materials [104]. This sensor technology is Rely to bc attractive in the ineasurcinciii of' the longitudinal, and lateral stresses at. or near to. the contact patch. Displacciliclit sensors such as those developed by Darnistadt I Iniversity 111.141 are l1kcly to better suited to the measurement of vertical stresses due to changes in flic týrc thickness.
To provide an improved understanding ol'intcrnal b1-haViOUrin a rolling tyrc, the N-crtical longitudinal and lateral ( ) firoin the flat bed cT-, (T, , stresses (T, and strains i.. , i.-, and i,-, simulation (using version 960) are shown in Figures 6.7 and 6.8. and 6.10 to 6.13. I'llesc results provide information which could be used to identify the best location for ill-tvrc sensors. The stresses and strains at six nodal positions local to the contact patch are considered. Figure 6.6 shows that the selected six nodes are located at three diflerclit lateral coordinates, i. e. at the lateral centre ofthe tyre. and at 130 and f 55 111111firoill the centre. The height positions are 5 nim and 10 min tioni the contacting hori/ontal surface. Nodal positions 1,3 and 5 are in the tread, half-way bLtA'ccn the contact patch and belt (bandages and breakers), while those at positions 2.4 and 6 arc situated at the interface between the tread and bandages. The internal behaviour at other positions call easily be obtained, and the stresses and strains are approximately symmetrical about the lateral Thus, lateral tyre centre. the values at a position of' -30 nini and -55 film are approximated by those shown here at +30 nim and +55 nim. In the figures. the stresses and strains were again obtained directly from the nodal stress field data and were taken as the nodes progress through the contact region at a sampling firequency of 5X 10-' s.
1 MM- 1.4 -- 55mm -1
Figure 6.6 Nodal positions local to the contact patch
112 Typically, strain gaugesmeasure ranges between ±2 percent, and the cyclic enduranceis ±2000 micro-strain for between 10' and 108cycles [ 102]. A tyre is commonly used over approximately 50,000 krn (>25 x 106revolutions/cycles) under acceleration/brakingand comering conditions, and sometimes in misuse situations [59]. These conditions are likely to increasethe internal stressesand strains observed in the tyre structure beyond the values observed under free-rolling conditions. The strains may be experiencedover short time periods of less than 1/200ths and at temperatures in excessof 100 T. These values were estimatedbased on a linear velocity of 100 km/h and a contact patch length of 0.1 m. They should be considered by the tyre technologist when choosing a position for an in-tyre sensor package. Furthermore, it should be noted that the strains experiencedby a sensor wM be influenced by its mechanical properties. The relatively stiff sensor will cause a redistribution of stressesand strains in the surrounding rubber. Thus, the stressesand strains presentedherein are useful as a initial guide to the location of an in-tyre sensor(or sensors).To determine the outputs with specific types of sensor, the simulation must be repeated with the sensor package representedin the tyre model.
6.8.1 At the Lateral Tyre Centre
The internalstresses and strainsat nodal positions I and 2 (at the lateral centre of the tyre) are shown in Figures 6.7 and 6.8. The stresses and strains at position A, the corresponding lateral coordinate in the contact patch, are also shown as a comparison.
The vertical, longitudinal, and lateral stressesare shown in Figure 6.7 to exhibit sirnilar characteristics.As the nodes enter the contact region (at the front) the stressesrapidly increase. The contact patch starts at a distance of approximately 3.33 m and ends at 3.47 m. The stressesreduce at the centre,and then increaseagain towards the rear. The corresponding stresses differ by less than 10 percent in the three directions and a change in the shapeof the stress distribution is not evident through the tread thickness.
A decreasein the vertical stressesis evident in Figure 6.7(a) as the distancefrom the contacting surface increases.This must occur since the inflation pressure is only 200 kPa. The vertical stressat the tread perimeter(node A) is much higher (between 300 kPa and 450 kPa). Throughthe tyre thickness,the stressreduces from the valueat node A to the inflation pressureat the inner-faceof the liner. The stressreduction is 113 7IL Cm- up
-13 3.32 3.34 3.36 3.38 3A 3A2 3.44 3.46 3AS 3.5 Distance Travelled [m)
(a)
'001ý
. 1000
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(b)
0. I. (A I
3.32 3.34 3.36 3.39 3A 3A2 3.44 3A6 3AS 3.5 DistanceTramfled [m)
(C)
Figure 6.7 Simulatedinternal stress distributions at the lateral centreof the tyre With a normal load of 3 kN and a velocity of 20 knVh: (a) vertical; (b) longitudinal;(c) lateral
114 -IIIIIII -- - tiodsA 4, --Nod. 1 Node 2 3.
C
U
3.3 3.32 3.34 3.36 328 3A 3.42 3.44 3.46 3AS 3.6 Distance Travelled [m]
(a)
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(b)
IIIII N- ''I'll. '' 'ý I-11, V. - '- ---=-
J.,
Distance TraveRed[m)
(c)
Figure 6.8 Simulatedinternal strain distributionsat the lateral centreof the tyre with a normal load of 3 kN and a velocity of 20 km/h: (a) vertical; (b) longitudinal;(c) lateral
115 approximatelylinear, and is 20 kPa every 5 mm.of tread depth. Beyond the interface betweenthe tread and bandages,however, there must be a changein the distributionas the pressureat the inner-faceof the liner is uniforrn.This changeis shownin Figure6.9.
Key
1 -350 kPa 2 -300 kPa 3 -250 kPa 4 -200 kPa InflaUonPressure m 200 kPa 6 -150 kPa 6 -100 kPa 7 -50 kPa
Tread Thickness
Figure6.9 Cross-sectionalvertical stressdistribution at the lateraltyre centre
The linear reductionthrough the tread does not occur with the longitudinaland lateral stressdistributions shown in Figures6.7(b) and 6.7(c). No logical trend can be identified as the distancefrom the contacting surfaceincreases. A reduction of approximately 25 kPa is evidentin the longitudinalstresses from node A to node I and node I to 2, while the typical reductionsin the lateral stressesare 20 kPa and 0 kPa. It is postulated that the internallongitudinal and lateral stressesat nodes I and 2 are influencedby the behaviourof the adjoiningbelt. This influenceis more significantat node 2 becauseit is situatedat the interfacebetween the tread and belt. Sincethe cords in the breakersare orientatedat an angleof ±20 degreesto the tyre circumference,the overlappinglayers stretchcausing the rubbermatrix to flex. This local flexing action is transmittedto the rubbertread via the bandagesand diminishes as the distancefrom the breakersincreases.
In Figure 6.8, the direct strainsat the lateral tyre centre are shown to be less than 2 percentfor the flat bed simulation.The magnitude(and variation) of these strains tends to reduce through the tread thickness. At node 1, the strains range is approximately1.4 percent in the vertical direction, and 2.2 and 1.5 percent in the longitudinal and lateral directions, respectively.The correspondingstrain ranges at node2 are 1.5,1.2 and0.5 percent.The reductionthrough the tread is to be expectedas the reinforcingcords in the belt stiffen the tyre and, thus, reducethe local deformation.
116 6.8.2 At +30 mm and +55 mm from the Lateral Tyre Centre
The internal stressesand strainsat the two positions +30 mm (nodes 3 and 4) and +55 mm (nodes5 and6) from the lateraltyre centreare shownin Figures6.10 and 6.11, and 6.12 and 6.13, respectively.The stressesand strains at positions B and C, the correspondinglateral coordinates in the contactpatch, are also shownas a comparison.
In Figures6.10 and 6.12, the three stressesare shownto exhibit similarcharacteristics to thoseat the lateraltyre centre(see Figure 6.7) as the distancefrom the surfaceincreases. The stressdistributions obviously differ becausethe contact stressesare not the same acrossthe width of the contact patch. The vertical stressdistributions again suggesta linear reductionthrough the thicknessof the tread. At +30 mm and +55 nun from the lateralcentre, the reductionis about45 kPa and 10 kPa every5 nun, respectively.Again a logical trend cannotbe identifiedin the longitudinaland lateral stresses.From node B to node 3 and from node 3 to node 4, the longitudinalstresses reduce by approximately 45 kPa and 10 kPa, respectively.The lateral stressesreduce by 15 kPa and 20 kPa. The reductionin the longitudinalstresses from node C to nodes5 and 6 is typicafly 20 kPa, and those in the lateral stressesare 20 kPa and 0 kPa. As mentionedearlier, this behaviouris influencedby the action of the belt, primarily the cords in the breakers.
The direct strains(at nodes3 to 6) are shownin Figures6.11 and 6.13 to vary between +5 and -4 percent.The strainranges at nodes3 and 4 are 1.7,1.8 and 2.0 percent,and 0.8,1.1 and 0.8 percentin the vertical, longitudinaland lateral directions,respectively. At nodes5 and 6 theseranges are 3.1,1.5 and 2.0 percent,and 2.0,0.7 and 1.6 percent. The simulatedstrain characteristicsat nodal positions I to 6 are given in Tables6.1.
The numerical results suggest that an in-tyre sensor package will experience high strain variations even under free-rolling conditions. These strain ranges coupled with the cyclic endurance and temperature requirements raise questions about the survivability of in- tyre sensors.Further work is neededto fiffly characterisethe internal stressesand strains in a rolling tyre under typical driving scenarios, such as acceleration/braking and cornering (slip and camber angle) conditions, and in specific sensors embedded in the tyre. The work presentedhere has suggestedthat magnitude of the internal strains tend to reduce through the tread depth since the reinforcing cords of the bandages and
117 0
4
U
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(a)
I uu
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T 0- 100- dL 6, 00-
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C» 0 00- i 00- Nods a -5 Nods 3 Nod94 --3.3 3.32 3.34 3.36 3.38 3A 3A2 3A4 3AS 3AS 3.5 Distance Tmelled [m]
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(c)
Figure6.10 Simulatedinternal stress distributions at lateral coordinateof +30 mm with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical;(b) longitudinal;(c) lateral
118 Nod* B 4- Node 3 Node4 3-
2- I .3 A 5.3 3.32 3.34 3.36 3.38 3A 3A2 3.44 3.46 3AS 3.6 Distance Travelled [m]
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IZR:2-
ä 2 v' .ii-00
-13 3.32 3.34 3.36 3.38 3.4 3A2 3A4 3AS 3A8 3.5 DistanceTripmIled [m)
(c)
Figure6.11 Simulatedinternal strain distributions at lateral coordinateof +30 nun with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical; (b) longitudinal;(c) lateral 119 i0o
0
-Z -100
-200 U) " ;;o ,,,
ý' -40C Node C -50C Node 6 Node S -r,nr 3.34 3.36 3.38 3A 3A2 3A4 3AS 3AB 3.6 .33.32 Distance Travelled (m) (a)
Iz
co c 0 -j
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(b)
100
-100 -
-200- / 1/ .. 300- '. . -" t '1 -400- Nods C 11 -500 Node 6 Node 6 9.3 ý L -60 3.-3L2 3.; 336 3.38 3A 3A2 3A4 3AS 3AS 3.5 Distance Traveled [m]
(c)
Figure6.12 Simulatedinternal stress distributions at lateralcoordinate of +55 nun with 20 km/h: (a) (b) longitudinal; lateral a normalload of 3 kN anda velocity of vertical; (c) 120 5
CD i
-3.3 3.32 3.34 326 3.38 3.4 3.42 3.44 3AS 3.48 3.6 Distance TraveRed [m)
(a)
IIII -NodeC 4 -- Nod. 5 NodeS 3 FS
V) I0 C
C- 0 -i
Distance Twelled [m]
(b)
IIIII -NodeC 4. -- Node5 -"-Nod. 6 3. 2-
0 I.1
DistanceTmveRed [m] (C)
Figure 6.13 Simulatedinternal strain distributions at lateral coordinateof +55 nun with a normalload of 3 kN and a velocity of 20 km/h: (a) vertical; (b) longitudinal;(c) lateral
121 Node Number StrainRanges (Max/Min) 11cightPosition 1%] [mm] Vertical Longitudinal Lateral 1 0.1/-1.3 0.7/4.5 1.3/-0.2 5 2 0.7/-0.8 0.3/-0.9 0.7/0.2 10 3 0.8/-0.9 0.8/4.0 1.2/-0.8 5
4 1.0/0.2 0.5/-0.6 -0.1/-0.9 10 5 3.2/0.1 1.01-0.5 -1.0/-3.0 5 6 2.5/0.5 0.8/0.1 -1.0/-2.6 10
Table6.1 Simulatedinternal strain ranges breakersstiffen the local region. The resultsalso suggestthat the strainsare highernear the tyre shoulders.Based on this inforination, it is postulatedthat considerationshould be given to positioningin-tyre sensorsnear to, or in, the belt. By doing so, the author believesthe strainsexperienced by in-tyre sensorscould be within acceptablelevels.
6.9 Summary
Rolling tyre experimentson the flat bed and rolling drwn machines(see Chapter 3) have been simulated using LS-DYNA version 950d. These were found to experience numericalinstabilities which havebeen related to imperfectionsinherent in the code.The flat bed simulationwas successfullyrepeated using the current release,version 960. Inconsistencieswere highlightedin simulationresults observed using the two versions and thesehave been investigated. Based on the resultsit has beenconcluded that these inconsistenciesdo not underminethe earlier model validation work carried out using version 950d. The contact stressesobtained using the current release have been comparedwith a reasonabledegree of successto thosefound in full-scalephysical tests. The internalstresses and strainsin the tyre structurehave then beenextracted at selected positionslocal to the contactingsurface. By doing so, the chapter provides valuable informationto the tyre engineerwishing to developrobust in-tyre sensors.The results suggestthat the most appropriateposition for in-tyre sensorsis near to, or in the belt.
122 Chapter 7 Review, Conclusions and Recommendations for Further Work
The automobiletyre is often consideredto be a simple and reliable componentof a vehicle.A closerinspection, however, shows that the tyre is subjectedto severestresscs anddeformation whose quantities must be determinedin order to accuratelypredict tyre behaviour.The aim.of this thesiswas to provide an initial investigationinto the internal stressesand strains in a rolling tyre via numerical simulation.An understandingof internaltyre behaviouris essentialto aid the developmentof 'smart' tyre technologies. Tyresplay a crucialrole in the supportof vehicledynamics and the integrationof in-tyre sensorscould allow the contact patch stressesto be monitored in each tyre under motion. Tbus, the tyre could becomea key sensorin future vehiclecontrol technology.
7.1 Thesis Review
The thesisis broadly categorisedinto three main areas:a review of the relevant tyre researchliterature; an experimental investigation into the contact patch behaviour of an automobile tyre; and a Finite Element (FE) modelling investigation using LS-DYNA.
The review of the literaturecarried out in ChaptersI and 2 set the backgroundto the work, and also highlightedthe lack of knowledgeon contact patch stressdistributions and the absenceof existing models to simulatethis behaviour. Chapter 3 described physicaltests conducted on an experimentalautomobile tyre. The resultshave provided a better understandingof the physical behaviourof tyres in the contact patch. The measurementsalso provideda valuablesource of data that can be used to validatethe modelling methodology.Finally, and most crucially, Chapters 4 to 6 describe the investigation of macroscopic tyre behaviour via numerical simulation. The two LS-DYNA modelsdescribed in Chapter4 were developedto simulatethe stationary (non-rolling)and rolling behaviourof the experimentaltyre. Resultsfrom the modelsare presentedin Chapters5 and 6, and have beenvalidated against the physicaltest data.
123 The internal stressesand strainsin the tyre under free-rolling conditions (at 20 km/h) were characterisedat severalpositions in the tread region. Thesenumerical results have beendiscussed in context with the developmentof in-tyre,sensor system tcchnologics.
7.2 Experimental Investigation
Full-scalephysical tests have been performed to characterisethe behaviour of the stationarytyre under normal load, and the rolling tyre behaviourunder free-rollingand cornering(slip and camberangle) conditions. The rolling experimentswere carried out on flat bed and drum surfaces.The load-deflectioncharacteristics and the contactpatch dimensionsof the experimentaltyre have been discussed,and a reasonablycomplete picture is presentedon how the normal and shearcontact stressesmight changeunder driving conditions.A comparisonbetween the stressmeasurements obtained on the two surfaceshas also been given. No sinilar comparisonis reportedin the literatureand the investigationtherefore provides unique measurementsin relation to the interfacial stressesexperienced by a rolling tyre on two geometricallydifferent ground surfaces.
The normal and shearcontact stressdistributions exhibit distinct characteristics.The normalpressure at the front of the contactpatch is higherthan at the rear and, thus, the distributionis not symmetricalabout the lateralaxis. This is becausea longitudinalforce, usually referred to as the rolling resistance,exists at the tyre/ground interface.The maximumpressure on the flat bed exceedsthe inflation pressure(200 kPa) by a factor of about 1.6,and a tyre 'buckling' effect is also evidentat the centreof the contactarea.
A sinusoidal-stylesweep is observed in the longitudinal shear stress distributiorL This is because there is a change in radius at the perimeter of the tyre tread. The rolling resistanceis evident in this distribution as a difference in the magnitude of the maximum positive and negative stresses. The lateral shear stresses are shown to increase approximately linearly through the contact patch under slip angle conditions. At 2 degrees slip angle, the maximum stress was found to be 290 kPa on the horizontal surface Oust below the maximum normal pressure measurement (320 kPa)). The magnitude of the lateral shear stressesat the same camber angle are significantly lower (<200 kPa) since the camber stiffness of the tyre is less than the cornering stiffness.
124 Normal pressuresat the tyre/drum.interface are higherthan thoseexperienced on the flat bed. The maximumvalue exceedsthe inflation pressureby a factor between1.75 and 2.25. This indicatesthe contact patch on the drum surface is smaller than on the horizontal surface.The longitudinal and lateral shear stressesare similar under free- rolling and camberangle conditions. Under slip angleconditions, the contactstresses on the drum surfacediffer considerablyfrom thoseobserved on the flat bed. The maximum pressureis muchhigher and the contactpatch length (at the lateraltyre centre)increases with slip angle.This phenomenonhas not previouslybeen identified and no explanation is currentlyat hand.The lateralshear stresses on the drum surfaceare also higher.At I degreeslip angle,the differencein the maximumshear stress is greaterthan threetimes.
7.3 Modelling and Simulation
The advancedFE models,developed to simulatemacroscopic tyre behaviour,represent the structureas a rubberand reinforcedrubber composite.Two-dimensional membrane elementshave been used to model the fibre and steel reinforcements,and three- dimensionalsolid elementsto representthe thick rubber sections.The behaviourof the reinforcementswas simulated using the Halpin-Tsai equation and the rubber was modelledbased on the Mooney-Rivlin strain energy function. The modelshave been developedfor simulationusing LS-DYNA and the model descriptionswere created using HyperMesh.The modelsrepresent a distinct improvementfrom those described in the literaturewhich have tendedto be developedto simulateglobal tyre behaviour.
The rubber industry characterisesthe elastic behaviourof a compoundbased on the secantmodulus. This method,combined with the reluctanceof tyre manufacturersto release property data for their preferred compounds, had meant that material stress/straindata for rubber componentswas not easyto obtain. This issuehas been addressedherein and a simple method has been developedto characterisethe elastic constantsof rubbermaterials used in a tyre. The methodis basedon the secantmodulus of the rubberand on the stress/straindata for a typical tyre compound.It is validatedby the numericalresults and, as a consequence,is of particularinterest to FE tyre analysts.
Other model simplifications,such as the assumptionsthat the elastic constantsof the
125 reinforcingcords in the steelbreakers (and also thoseof the beads)correspond to those of bulk steel, have been implementedbecause this mechanicalproperty data %N2snot available.Furthermore, the inflation pressurehas been assumedto remain constant during typical tyre deflections.Tyre analysisis much more complexif inflation pressure variationsdue to volumechanges are introduced.These simplifications have only a small influenceon the numericalresults and, thus, do not adverselycfIcct the simulations.
The stationary (non-rolling) behaviour Of the experimentaltyrc %,,-as successfully simulated.The normal load-deflectioncharacteristics and the contact patch dimensions havebeen compared with a reasonabledegree of successto those obtainedin the full- scalephysical tests. Consideration was givento the inflation of the tyre, the wheelfit and the normal loading of the tyre. Simulation results were also presented%Vhcn a subsequentlongitudinal or lateral load was applied. The longitudinaland lateral tyrc stiffnessesappear to be realisticallyrepresented. The contact patch dimensionsgave a good trend-wiseagreement, but the length and width were found to be greaterthan the experimentalmeasurements. A parametricstudy was carried out and this disparity is related to a deficiencyin the performanceof the contact algorithm. It is therefore observedthat it far from straightforwardto accuratelypredict contact patch behaviour and the internal stressesand strainsin absoluteterms. However, the good trcnd-wisc agreementdoes suggest that the modellingmethodology is capableof predictinginternal responseswhich are related to the 'actual' tyre deformationsat the contact patclL
To simulatethe rolling tyre behaviouron flat bed and drum surfaces,in addition to developingmodelling methodology to inflate the tyre, fit it to the wheel and apply the normal loading, it was necessaryto consideraccelerating the tyre to reach a desired constantvelocity. Despiteconsideration of thesesalient modelling features,numerical instabilitieswere found to occur. Theseinstabilities have beenrelated to imperfections inherentin version950d of the code.This versionwas, at the time (July 2001), the most is it up to daterelease. The currentrelease version960 and doesnot containmany of the imperfectionsin the earlierversion. Thus, the flat bed simulationwas repeatedusing the free-rolling current release.The predictedcontact patch stressesunder conditionshave beenpresented and theseare shownto exhibit similarcharacteristics to those evidentin the experimentaldata. The internalstresses and strainshave then beencharacteriscd at a These severalpositions in the tread region. stressesand strainsare useful as an initial
126 guide to the location of in-tyre sensors.The 'actual' d6ormations cxpcrienccdby a sensorwill be influencedby its mechanicalproperties. The numericalresults suggcst that the tread could experiencehigh strain ranges (>3 percent) cvcn under frcc-rolling conditions. Such high ranges, coupled with the cyclic enduranceand tcnipcraturc requirements,raise questions about the survivabilityof in-tyre sensors.Furthermore, the simulationresults suggest that considerationshould initially be givcn to positioningthe sensorsnear to, or in, the belt. The internalstrain ranges can be more than halvedas the distanceform the contactingsurface increases (and that from the stiff belt reduces).
7.4 Recommendations for Further Work
Attention hasbeen drawn to the fact that there appearsto be a lack of knowlcdgeabout tyre behaviourin the contact patch and thereforethe internal stressesand strainsin a in in rolling tyre. The researchwork presented this thesisis thereforenovel naturc and, hence,offers significantscope for further investigation.It is hoped that the modcLs developedherein, together with the comprehensiveexperimental work, can be uscd to improve understandingof tyre behaviourand its relationshipto vehicle dynamics.
The airn of the thesiswas to provide an initial investigationof the internalstresses and strainsin an automobiletyre via numericalsimulation. This hasbeen achieved for a tyrc, on a horizontal surfaceat a speedof 20 km/h under free-rolling conditions.Further simulationsof rolling tyre behaviourhave not been possiblein this thesis due to the limitationsin the availablecomputational resource and also time constraints.The next logical step is to investigatethe internal stressesand strains under typical driving conditions,i. e. at variousnormal loads, speeds,slip and camberangles and slip ratios, investigationis and on different surfaces.Such an neededto fully chamctcriscthe internal stressfield in a tyre structure. This information would further aid the tyre in-tyre for engineerwishing to develop sensors monitoring tyre behaviour.The work investigation by could alsoprogress to the of stressesand strain experienced specificin- tyre sensors.This would require the mechanicalproperty data of the sensorpackage.
herein The contact and friction models employed have addressedthe macroscopic interfaces Coulombformulation. characteristicsat the tyre/groundand tyre/wheel via the Since the main reason for the differencebetween the simulation results and those 127 observedexperimentally is the performanceof the contact algorithm used in the code, the issue of contact modelling is an obvious area for further consideration.I'lic microscopicasperities in the groundsurface havc not beenconsidered herein but may be importantin the simulationof rolling resistance.Rolling resistanceis kno%Nmto be higher on rough than on smooth surfaces.Also important to the numericalrolling resistance characteristicsis the assumeddamping level. Damping%vas acceptably neglected in this thesisbecause roffing resistancecffects arc small(typically lessthan 50 N). Furtherwork could be usedto determineappropriate damping levels for simulatingrolling resistance.
The experimentalwork presentedin this thesishas addressedthe lack of knowlcdgcof contact patch behaviourfor frce-rolling and comcring (slip and cambcr) conditions. Thesestudies could be extendedto includeacccIcration/braking characteristics and also larger data ranges,and cross-contactpatch mcasurcrncnts.This could be achicvcdvia experimentalmeasurements (or using the FE models).This thesishas considcrcdsmall slip and camber angles,but these do not envelop the possibleranges cxpcricnccd. Furthermore,the influenceof speedon the contact stressesneeds grcatcr considcration.
Other obviousareas for future work are the simulationof rolling tyrcs on dcformablc surfacessuch as mud, sandand snow, and the dclaminationcffccts of qTcs.To simulatc tyre behaviour on deformable surfaces would simply require a more dctailcd representationof the ground surface wMe the dclaminationcffccts could casily be incorporatedinto the current models.The interfacebct%Nccn the adjoiningcomponcnts, and betweenthe rubber matrix and reinforcements,could be modelled as separate contacts,with the nodestied to the adjacentsurface until a failurecriterion A-asreached.
Finally, this rolfing tyre study has been limited to the investigationof tyrC behaviour under experimentalsituations represented in the laboratory.These simulation have not beeninfluenced by tyre temperaturevariations because the tyre rolling cxperimentswere carriedout for only a short time period. This is a major considerationfor tyrcs operating in 'real' driving conditionsand, thus, must be consideredas an area for further work.
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136 Appendix A Cross-Coupling Effects In the Longitudinal and Lateral Shear Stresses
In the rolling tyre experiments(see Chapter 3), the shcarstresses exerted on the ground surfacewere measuredin the transduceraxes (. V-, Y-) and not tile tyrc axes(x-, )--). As discussedin Section 3.6.2, applying a slip angle changesthe orientation or the tyrc axesrelative to the transduceraxes and a cross-couplingcffect occurs.I'his is shownin FigureA. I. Thus,the longitudinal r. and lateral r, slicar stressesneed to be resolved.
ria (2): x
I y
FigureA. 1 Longitudinal and latcral shcar strcss cross-coupling
The shearstress measurement in the X-axis ( r.,, ) can be rcsolvcd into the longitudinal lateral These by and shearstress components r. (,) and componentsarc given
rxz cos(a) (A. 1)
and (A.2) r ),Z(I) %sin(a)
Similarly,the shearstress measurement in the Y-axis( rrz ) can also be rcsolvcdinto the longitudinal lateral These by and components r.,-(, ) and r,.,(, ) . components arc given
T., (2) = rrzsin(a) (A. 3)
137 and r,., rrzcos(a) (, ) = (A. 4)
The total longitudinalslicar stressis obtaincdfrom Equ. (A. I) and (A. 3), and is givcn by
r,. r., 2= rxz(l) - (2) = r,, cos(a)- rrzsin(a) (A. 5)
Similarly,the total lateral shcarstress obtained from Equ. (A.2) and (A. 4) is gi%-cnby
+ ry: ý rx. (I) ryz(2) (A. 6) = r,,.zsin(a) + r,, cos(a)
At srnallslip anglessuch as those consideredin this thesis(. -52 degreeson the flat bed tyre testing machine),it should be noted that cos(a) ft I and sin(a) m0. Ilius, an acceptableapproxirnation. of the longitudinaland latcral shcar stressesis obtainedvia
(A. 7) r.. -=r,, and -r rz (A. 8)
138 Appendix B Stress/Strain Data for a Typical Rubber Tyre Compound In Simple Extension
The stress/straindata provided by Dunlop Tyrcs Linuitedfor a typical 'unkno%Wrubber tyre compoundsubjected to simpleextension is given here(see Section 4.4). The secant modulusat 100 percentstrain was also providedand was 3.19 NIPa.ncsc strcss/strain measurementswere obtainedin accordancewith BS 903: Part Al 1995(ISO 37: 1994).
139 Engineering Strain Engineering Stress Engineering Strain Engineering Stress 1%] [MPa] 1%] [MPa] 0.0 0.00 26.1 0.96 2.0 0.12 27.1 0.99 2.9 0.16 28.2 1.01 3.9 0.22 29.2 1.04 4.7 0.27 30.1 1.06 5.6 0.32 31.3 1.08 6.5 0.37 32.3 1.11 7.5 0.42 33.4 1.13 8.5 0.46 34.3 1.16 9.5 0.50 35.4 1.18 10.5 0.54 36.4 1.21 11.5 0.58 37.4 1.23 12.6 0.61 38.5 1.26 13.6 0.64 39.5 1.29 14.6 0.68 40.5 1.32 15.7 0.71 41.3 1.34 16.7 0.74 42.4 1.38 17.8 0.77 43.4 1.41 18.8 0.79 44.3 1.44 19.8 0.82 45.3 1.48 20.9 0.84 46.3 1.52 21.9 0.87 47.2 1.56 23.0 0.89 48.2 1.60 24.0 0.92 49.1 1.65 25.0 0.94 50.0 1.70
TableB. I Stress/straindata for a typical rubbertyre compoundin simpleextension
140 Appendix C Shear Distortion of the Tyre Tread in the Contact Patch
The poor performanceof the Mooney-Rivlin model (model 27) in the LS-DYNA code [17] is discussedin Section4.5. Excessivelocalised shear distortion in the tyrc tread is observedunder longitudinalloading. This excessivedistortion disappearswhen the Hyperviscoelasticmodel (model 77) is usedand, thus, the author recommendsthis model to simulatea transversely(longitudinal and/or lateral) loadedor rolling tyre. To aid validation of the performanceof the Hyperviscoelasticmodel, a simple hand calculation is performed here using classical elasticity theory. It is based on the assumptionthat the tyre's belt is rigid sinceits stiffnessis severalorders of magnitude greaterthan that of the rubber tread. This simplemodel is shown in Figure C.I. The assumptionis usedin serni-empiricaltyre models,such as the brushmodel [33,45,461.
Riald Beft
t= lo mm T Horizontal Surface 8 Tread
Figure CA Simple model of experimentaltyre under longitudinal loading
For the particularcase of simpleshear, rubber exhibits a linear stress/strainrelationship (see Equ. (4.6)). This relationship is characterisedby the shear modulus G, given by
(C.!) "xx
where is the shearstress and y. is the shearstrain. The averageshear stress over the contactpatch is F,,/A, where F,, is the appliedlongitudinal force and A is the area.
141 Assumingthe shearmodulus of the tread is 0.77 MPa, the longitudinalforce is 2 kN, and the contactpatch length and width are 115 mm and 122 mrn, y,, can be estimated to be approximately18 percentusing Equ. (C. 1). The sheardistortion 5 is then givenby
8=y,,.. (C.2) where d is the depth of the tread and, thus, the distortion is estimatedto be 1.8 nim. This valuecorresponds to the 2.3 mm obtainedin the advancedFE simulationusing the samemodulus and longitudinal force. It should be noted, however, that the contact patchlength and width usedwere thoseobtained in the stationaryexperiments discussed in Section3.4.1. The calculatedshear distortion increasesto approximately2.6 mm.if the simulatedcontact patch dimensions(length and width) from Figure 5.5 are utilised.
142