the prediction of tyre friction on wet roads under braking and ...

THE PREDICTION OF TYRE FRICTION

ON WET ROADS

UNDER BRAKING AND CORNERING

by

M.Sjahdanulirwan, CivE ITB (Indonesia), MSc B' ham (UK)

A thesis submitted in fulfilment of the requirements for the Degree of

Doctor of Philosophy

School of Civil Engineering The University of New South Wales

Australia

February 1993

11

CERTIFICATE or ORIGINALITY

I hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgement is made in the text.

SUPERVISOR'S CERTIFICATE

(Signed)

(Name)

(Date)

M.SJAHDANULIRWAN

r certify that this thesis is, in my opinion, in a form suitable for examination for the degree of Doctor of Philosophy.

(Signed)

(Name) Assoc.Professor W.O.Yandell

(Date)

iii

Abstract

This thesis extends the prediction of tyre-road friction from surface texture and

tread rubber properties developed by Yandell et.al [1983], to cover all ranges of

percent slip (Sx) and slip angles (a) by means of developing an analytical tyre-road

friction model. The proposed model consists of two main components: the locked

wheel Braking Force Coefficient (BFC) or maximum Sideway Force Coefficient

(SFC), and the tyre stiffnesses.

It has been found from early works, that the coefficients of locked-wheel braking

(i.e. at Sx = 100%) and the maximum sideway force (i.e. at optimum a) can be

predicted with reasonably well agreement, in which the factors of surface texture,

water film thickness, tread rubber properties, and certain operating conditions have

been taken into account. The mode of operation for any conditions, such as braking

or cornering, would be the important factor to be considered when dealing with all

ranges of Sx and a.

From previous information it is known that at below the optimum value of Sx (or

a), the relationship between BFC and Sx, or SFC and a, is controlled primarily by

the elastic tyre properties. The effects of "road-contributed" friction (µ) appear

dominantly at and above the optimum value of Sx (or a). It is assumed that the

elastic tyre properties can be represented by the tyre stiffnesses (longitudinal and

lateral), whereas the available friction (µ) will be represented by any value of

coefficient of braking (or cornering) from optimum to maximum value of Sx (or a).

The choice of locked-wheel BFC or maximum SFC as input parameter for the

candidate's tyre model is primarily based on two reasons. First, that the methods for

predicting the locked-wheel BFC or maximum SFC from surface texture

measurements already exist [Yandell et.al (1983)]; hence, the ultimate goal to

develop a theoretical technique for predicting tyre-road friction, from surface texture

data, still be extended. Second, that the equipment for measuring the locked-wheel

BFC or maximum SFC are commonly available (such as the trailer used on

ASTM-E274, and SCRIM); hence, the data from these types of equipment can be

processed directly to obtain the friction for all ranges of Sx and a.

IV

At the present time, the tyre stiffnesses C,. and Cy can be obtained

experimentally by measuring forces and slips (or deformations), either on a road

surface or in a laboratory. Especially for stiffness measurements in a road surface,

the candidate has developed an empirical model for the variation of tyre stiffness

due to vertical load, inflation pressure, and tyre ( or vehicle) speed.

Attempts to predict tyre stiffness directly from other tyre properties or related

parameters are left for future refinement. For stiffness measurements in a laboratory,

the need for applying an empirical model of stiffness variation is probably less

beneficial than the accuracy gained with direct measurement on different variations

(vertical load, inflation pressure). An empirical factor may still be required, however,

if the change of vehicle speed has a significant effect on tyre stiffness.

Field measurements of the coefficients of frictional force are carried out by using

the Multi Mode Friction Test Truck (MMFTT). This vehicle was capable for

measuring pure braking, pure sideway (up to 15 degrees), and braking-in-turning

modes. The use of test wheel motor and gear box allows the accelerating and steady

state braking to be performed.

The analysed data from MMFTT are compared with theoretically predicted

coefficients of friction. The candidate's model is also verified with the laboratory

data obtained from another worker [Sakai (1982)]. The concept of the independence

of tyre stiffness from road conditions, and the difference between the slip stiffness

and the deformation stiffness, have been further examined by additional experiments.

It appeared that the theoretical method of obtaining the coefficient of tyre-road

friction under all conditions, from tyre stiffness and locked-wheel BFC (or maximum

SFC), is feasible. As only one friction parameter is required in the proposed model,

the time and cost associated with the measurements (and calculation) of the friction

parameters can be reduced. In addition, the use of unsymmetric trapezoidal forms of

distribution provides the user with an option and flexibility in representing many

types of actual tyre pressure distribution.

V

Acknowledgements

The candidate wishes to express his thanks to all those who assisted in this

research project and in the preparation of the thesis, including:

Dr.W.O.Yandell, Associate Professor at School of Civil Engineering UNSW, for

his encouragement, guidance, advice, and particularly for the constructive criticism

and helpful supervision throughout the work;

Dr.W.H.Cogill, Researcher, for his critical comments and helpful discussion,

especially on mathematical terms;

Dr.Q.Yang, Research Fellow, for reading certain chapters of the thesis and

p~oviding suggestions for improvements;

Mr.M.J.Kerkhoff, Professional Officer, for his assistance on the electrical matters,

for providing the guidelines on operating the Strain Gauge Amplifier & Acrolog 400,

and for supplying the Program to Read Strain Gauge and Speed Output;

Mr.R.B.McKinnon, Laboratory Officer, for his expert workmanship in the

construction of calibration equipment, and for his help in carrying out the

preliminary test;

Mr.P.J.Gwynne, Technical Officer, for his help during calibration and field

experiment, and especially for operating the Multi Mode Friction Test Truck

(MMFTT);

And to all the other members of the staff in the School and Laboratory Workshop

who have willingly given advice and assistance whenever it was requested.

The experimentation carried out during this research would not have been

possible without the financial support of the Roads and Traffic Authority (RTA) of

New South Wales, and this is gratefully acknowledged.

Finally, the candidate acknowledges his wife, Atu, for her support and patience in

many ways and always standing by him.

Abstract Acknowledgements Table of Contents List of Figures List of Tables

Introduction

Chapter 1. Review of Literature

vi

Table of Contents

1. 1. Nature and Components of Tyre-Road Friction 1.1.1. Hydrodynamic component of friction 1.1.2. Adhesive friction 1.1.3. Hysteretic friction 1.1.4. Abrading friction

1.2. Factors Affecting Tyre-Road Friction 1.2.1. Pavement/lubricant conditions 1.2.2. Tyre factor 1.2.3. Operating conditions

1.3. Contact and Slip Between Tyre and Roadway 1.3.1. Classifications of tyre motions 1.3.2. Types of slip 1.3.3. Sliding and skidding 1.3.4. Contact area 1.3.5. Stress distribution

1.4. Surface Texture Description and Measurement 1.4.1. Description of surface texture 1.4.2. The measurement of surface texture

1.5. Hydrodynamic Effects and Tread Rubber 1.5.1. Reynolds theory of lubrication 1.5.2. Fluid film thickness 1.5.3. Hydroplaning 1.5.4. Models for calculation of water film thickness 1.5.5. Measurement of tread rubber properties

1.6. Theory of Hysteretic Sliding Friction 1.6.1. The mechano-lattice analogy 1.6.2. Mechano-lattice unit 1.6.3. The friction of rubber sliding on an asperity 1.6.4. The hysteretic friction of road surfaces

1.7. Problem Definition and Method of Investigation 1. 7 .1. Problem definition 1.7.2. Method of investigation

Page No. iii

V vi xi

XX

1

3

3 5 5 6 7

8 8

11 15

22 22 23 25 26 27

33 35 39

44 44 45 46 47 50

52 53 54 55 58

59 59 64

vii

Chapter 2. Measurement and Prediction of Tyre-Road Friction 65

2.1. Methods of Measuring Tyre-Road Friction 65 2.1.1. Sideway force measuring method 66 2.1.2. Braking force measuring method 67 2.1.3. Stopping distance method 67 2.1.4. Portable skid resistance testers 68

2.2. Some Theoretical Methods of Predicting Tyre-Road Friction 68 2.2.1. Schonfelds's photo-interpretation of pavement 69 2.2.2. Leu and Henry's model of skid resistance 71 2.2.3. Texture friction meter 72

2.3. Theoretical Prediction of the Locked-Wheel Braking and Sideway Force Coefficients 73

2.3.1. Modification of the theory of hysteretic friction 73 2.3.2. Some assumptions of rubber/water temperature 73 2.3.3. Prediction of the locked wheel braking and sideway force coefficients 75

2.4. The Longitudinal and Lateral Tyre Stiffness 76 2.4.1. Effects of normal (vertical) load 77 2.4.2. Effects of inflation pressure 79 2.4.3. Effects of speed 80 2.4.4. Effects of tyre wear (bald & new) 80 2.4.5. Measurement of tyre stiffness and its related parameters 81 2.4.6. Clarification of tyre stiffness formula and its units 84 2.4.7. The empirical model for tyre stiffness variation 86

2.5. The Existing Tyre Models Under Braking and Cornering 92 2.5.1. Analytical and semi-empirical models 93 2.5.2. Empirical models 97 2.5.3. Miscellaneous models 98

2.6. The Candidate's Model for the Prediction of Tyre-Road Friction Under Braking and Cornering 99

2.6.1. General features of the model 100 2.6.2. Geometry of tyre-road contact 102 2.6.3. Pressure distribution and coefficients of friction 103 2.6.4. Location of sliding boundary 105 2.6.5. The calculation of shear forces 107 2.6.6. Brief procedure using locked-wheel BFC 109 2.6.7. Illustrative example 112 2.6.8. Alternative model using maximum SFC 120

Chapter 3. The Parametric Study of the Model 121

3.1. General 121

3.2. Results of the Main Parameters 122 3.2.1. The effects of longitudinal stiffness 123 3.2.2. The effects of lateral stiffness 126 3.2.3. The effects of locked-wheel BFC 129

viii

3.3. The Significance of All Variables 3.3.1. Dependent and independent variables 3.3.2. Model specification 3.3.3. Regression analysis 3.3.4. Significance of longitudinal force coefficient 3.3.5. Significance of lateral force coefficient 3.3.6. Significance of resultant force coefficient

3.4. Comparison With Other Tyre Models 3.4.1. General description of tyre models 3.4.2. Summary of formulas 3.4.3. Model input data 3.4.4. Model response comparisons

Chapter 4. Field Measurement of Friction

4.1. Objectives of the Friction Measurement

4.2. Design of Measurement 4.2.1. Sample size 4.2.2. Randomization 4.2.3. Choice of levels used

4.3. The Multi Mode Friction Test Truck 4.3.1. The test wheel 4.3.2. The recording device 4.3.3. The road wetting device

4.4. Calibration 4.4.1. Wheel load calibration 4.4.2. Braking force calibration 4.4.3. Sideway force calibration

4.5. Experimentation 4.5.1. Sequence of experiment 4.5.2. Test procedure 4.5.3. Data logging program and corrections

4.6. Possible Variability on Test Results 4.6.1. Fluctuation of voltage supplied 4.6.2. Setting up of slip angle 4.6.3. Test location 4.6.4. Speed 4.6.5. Inflation pressure 4.6.6. Temperature 4.6.7. Rubber properties

133 133 134 135 137 139 142

145 145 146 152 153

158

158

158 158 159 159

163 163 164 164

169 169 169 172

172 172 173 174

175 175 176 176 177 177 177 178

ix

Chapter 5. Analysis of Experimental Results 179

5 .1. General 179

5.2. Presentation of Data 180 5.2.1. Friction coefficient versus time 180 5.2.2. Correction for test wheel speed 184 5.2.3. The actual percent slip on entering the measured section 190

5.3. Transformation into Percent Slip and Slip Angle 191 5.3.1. Correction for wheel speed delay 191 5.3.2. The smoothened technique for wheel speed 193 5.3.3. Transformation of time into percent slip 194 5.3.4. Extrapolations and analysis of variance 203

5.4. Determination of Parameters for Theoretical Prediction 208 5.4.1. Longitudinal stiffness 208 5.4.2. Lateral stiffness 210

5.5. Discussion of Results 211 5.5.1. General results 211 5.5.2. Effects of tyre pressure 215 5.5.3. Effects of the rate of braking 216

Chapter 6. Theoretical Prediction of Tyre-Road Friction Under Braking and Cornering 217

6.1. General 217

6.2. Verification of Tyre Stiffness Variation Model 218

6.3. Input Data for Prediction and Comparison of Tyre-Road Friction 220 6.3.1. From candidate's experimental result 220 6.3.2. From Sakai's laboratory data 221

6.4. Theoretical Prediction and Comparison with Experimental Result 223

6.5. Theoretical Prediction and Comparison with the Sakai's Laboratory Data 232

6.6. Theoretical Prediction of Slippery Road 239

6.7. Theoretical Prediction Using Tyre Stiffness from Load-Deflection Measurements 241

6.6. Viability and Reliability of the Theoretical Prediction of Tyre-Road Friction as a Substitute for Actual Skid Tests 248

X

Chapter 7. Conclusions 251

7 .1. Theoretical Findings 251 7 .1.1. Types of tyre stiffness 251 7 .1.2. Description of friction coefficient 252 7 .1.3. The role of tyre stiffness 252 7 .1.4. Model for tyre stiffness variation 253 7 .1.5. The shape of tyre pressure distribution 253 7 .1.6. Calculation of frictional forces 254 7.1.7. Theoretical effect of some input parameters on the tyre-road friction 255 7 .1.8. Theoretical model response comparisons 256

7.2. Experimental Finding5 257 7.2.1. Rectification of experimental problems 257 7.2.2. The significance of the variation of tyre pressure and rate of braking 259 7 .2. 3. Verification of tyre slip stiffness variation model 260 7.2.4. Comparison between measured and theoretically predicted friction 261 7.2.5. Comparison of Sakai's laboratory data

with the candidate's theoretical prediction of friction 261 7.2.6. The effect of modifying the quantification of lateral stiffness

in the candidate's friction prediction 262 7.2. 7. Independence of tyre stiffness from road conditions 262 7.2.8. The relationship between slip stiffness and deformation stiffness 263

7.3. Benefit of the Model 264

7.4. Summary of Conclusions 266

7.5. Future Work 269

References 270

Appendix A: Basic Computer Program for Calculation of Frictional Forces Using Locked-Wheel BFC A-1

Appendix B: Basic Computer Program for Calculation of Frictional Forces Using Maximum SFC B-1

Appendix C: Procedure to Operate Strain Gauge Amplifier C-1

Appendix D: Procedure to Operate Acrolog 400 D-1

Appendix E: Program to Read Strain Gauge and Speed Output E-1

Appendix F: Results of investigation using MMFfT, with higher speed (20 mph & 30 mph) F-1

Appendix G: Mathematical Derivations of Tyre Stiffness Formula G-1

xi

List of Figures

1.1. Schematic representation of 3 zone concept.

1.2. Components of rubber friction.

1.3. Basic tyre structures.

1.4. Effect of speed and road texture to skid resistance.

1.5. Friction mechanism due to adhesion and hysteresis.

1.6. Lateral force coefficient vs slip angle of 178-15 tyre.

1.7. Braking force coefficient vs slip angle of 178-15 tyre.

1.8. Peak and slide traction envelopes of 178-15 tyre in braking-in-a-tum.

1.9. Model experiments for: (a) free rolling, (b) side slip, (c) braking, (d) driving. Travelling direction from right to left.

1.10. Typical contact area for an automobile tyre.

1.11. Pressure distribution down the length of a tyre with no tread pattern, at constant speed 15 kph, inflation 6.5 atm, load: (a) 1680 kp, (b) 3740 kp.

1.12. Pressure distribution across the width of a tyre with no tread pattern, constant speed 15 kph, load: (a,b,c) 1680 kp, (d) 3740 kp; inflation: (a) 3.5 atm, (b) 5 atm, (c,d) 6.5 atm.

1.13. Pressure distribution down the length of a smooth tyre, load 1680 kp, infl. 6.5 atm, speed: (a) 10.2 kph, (b) 8.4 kph, (c) 8.9 kph; acceleration: (a) 0.6 m/s2,

(b) '1.6 m/s2, (c) 4.1 m/s2•

1.14. Pressure distribution down the length of a smooth tyre, load 1680 kp, inflation 6.5 atm, speed: (a) 10 kph, (b) 35.7 kph, (c) 14.5 kph; deceleration: (a) 4.1 m/s2, (b) 2.7 m/s2, (c) 2 m/s2•

1.15. Pressure distribution of bias-belted H size tyre, 26 psi, 0° slip angle, 100% T and R load = 1580 lb.

1.16. Longitudinal shear stress distribution in the running direction of a tyre under free rolling.

1.17. Longitudinal shear stress distribution in the running direction of a bias-belted H size tyre under traction, 26 psi, 0° slip angle, 100% T and R load= 1580 lb.

1.18. Longitudinal shear stress distribution in the running direction of a smooth tyre under traction, load 1590 kp, inflation 6.5 atm.

1.19. Longitudinal shear stress distribution in the running direction of a smooth tyre under braking force, load 2610 kp, inflation 6.5 atm.

Xll

List of Figures (continued)

1.20. Lateral force intensity along the contact spot, at slip angles up to 12°.

1.21. Lateral force intensity: (a) due to side slip (lg), and (b) due to simultaneous side slip and circumferential slip (tc).

1.22. Mathematical definition for numerical evaluation of surf ace texture.

1.23. Profile ratio and "filtered profile ratio".

1.24. Moore's bearing area method.

1.25. Definition of surface characteristics.

1.26. Influence of the number of inflections.

1.27. Illustration of terms of the road surface texture.

1.28. The texture effect on film thickness as a function of time.

1.29. Measured relationship between the damping factor of filled tread rubber, speed, and rubber temperature.

1.30. Simplified of measured hysteresis loop.

1.31. A unit of the mechano-lattice analogy.

1.32. The assembled units simulating a long section of rubber sliding on an asperity.

1.33. Flow chart of the analogy computer programme.

1.34. Mechano-lattice-determined relationships between hysteretic friction, damping factor of sliding rubber, and average slope of contact of triangular prisms and cylinders.

1.35. Analysis of measured texture profiles into scales.

1.36. Schematically presentation of the coefficient of hysteretic friction of the road surface.

1.37. Measured BFC vs predicted BFC, on many types of pavement (concrete, bituminous) and macro-texture (closed, mild, smooth).

1.38. Measured SFC vs predicted SFC, on many types of pavement (concrete, bituminous) and macro-texture (closed, mild, smooth).

2.1. Sideway Coefficient Routine Investigation Machine (SCRIM).

2.2. Pavement Surface Texture Classification.

2.3. Friction weights of texture parameter E or F.

xiii


2.4. Texture Friction Meter (TFM) vs British Pendulum Number (BPN) and Multi Mode Friction Test Truck (MMFTT).

2.5. Schematic representation of masking of texture.

2.6. Longitudinal stiffness coefficient (CxC) vs tyre load of radial-ply (R), bias-belted (B), and bias-ply (D) tyres.

2.7. Lateral stiffness coefficient (CyC) vs tyre load of radial-ply (R), bias-belted (B), and bias-ply (D) tyres.

2.8. Longitudinal force (Fx) versus travel of flat bed table supporting a radial tyre at the indicated vertical loads.

2.9. Lateral force (Fy) versus travel of flat bed table supporting a radial tyre at the indicated vertical loads.

2.10. The kinematics of tyre motion.

2.11. The distorted equatorial line.

2.12. The equatorial line with adhesion & sliding region.

2.13. Root line and deflection of the profile lugs.

2.14. Simplified diagram for calculation (or prediction) of frictional forces.

2.15. Tyre-road contact geometry, and deformation in nonsliding portion of contact patch.

2.16. Idealized pressure distribution within the contact patch.

2.17. Longitudinal and lateral components of (available) friction coefficient.

2.18. Longitudinal and lateral components of frictional forces coefficient.

2.19. The contribution of tyre stiffness (C) and the available friction (µ) to the frictional forces (F). At slips beyond the critical value, then Jls = µx ( or µa = µy).

2.20. Determination of the available longitudinal friction at zero slip (µ50).

2.21. Determination of the available lateral friction at zero slip (µa0)

2.22. Calculation of shear (frictional) forces.

2.23. -Tyre model (--) and measured data (o o o) of H78-14 tyre (28 psi) with a/21 = 0.06 and b/21 = 0.20. Left: Fx & FY vs Sx at a= 4°. Right: FY vs a at Sx = 0.

xiv


2.24. Tyre model of H78-14 (28 psi) with b/21 = 0.20, using a/21 = 0.11 (--) and a/21 = 0.01 (------). Left: Fx & Fy vs Sx at a= 4°. Right: Fy vs a at Sx = 0.

2.25. Tyre model of H78-14 (28 psi) with a/21 = 0.06, using b/21 = 0.01 (--) and b/21 = 0.40 (------). Left: Fx & Fy vs Sx at a= 4°. Right: FY vs a at Sx = 0.

2.26. The computed relationship between µx and µY at various a (0 ), with a/21 = 0.06 and b/21 = 0.20.

2.27. The computed relationship between µx and Uy at various Sx (% ), with a/21 = 0.06 and b/21 = 0.20.

2.28. Determination of the available lateral friction at zero slip (µa0)

3.2. ex vs µscr• for various µd.

3.3. ex vs µx, a = 0°, µd = 0.6.

3.4. ex vs µx, a = 4°, µd = 0.6.

3.5. ex vs µx, a = 8°, µd = 0.6.

3.6. ex vs µx, a = 32°, µd = 0.6.

3.7. ex vs µy, sx = 0.1, µd = 0.6.

3.8. ex vs µy, sx = 0.2, µd = 0.6.

3.9. ex vs µy, sx = 0.5, µd = 0.6.

3.10. ex vs µx, sx = 0.1, µd = 0.6.

3.11. ex vs µx, sx = 0.2, µd = 0.6.

3.12. ex vs µx, sx = 0.5, µd = 0.6.

3. 13. ey vs a.er> for various µd.

3.14. eY vs µa.er, for various µd.

3.15. ey vs µy, Sx = 0, µd = 0.6.

3.16. ey vs µy, Sx = 0.1, µd = 0.6.

3.17. ey vs µy, Sx = 0.2, µd = 0.6.

3.18. ey vs µy, Sx = 0.5, µd = 0.6.


3.19. Cy vs µx, ex= 4°, µd = 0.6.

3.20. Cy vs µx, ex= go, µd = 0.6.

3.21. Cy vs µx, ex = 32°, µd = 0.6.

3.22. Cy vs µy, ex = 4°, µd = 0.6.

3.23. Cy vs µy, ex = go, µd = 0.6.

3.24. Cy vs µy, ex = 32°, µd = 0.6.

3.25. µd vs SXCI" for various ex.

3.26. µd vs excr, for various CY'

3.27. µd vs µsa• for various Cx-

3.2g. µd vs µacr, for various Cy-

3.29. µd vs µx, ex = (}°, ex = 35.

3.30. µd vs µx, ex = 4°, ex = 35.

3.31. µd vs µx, ex = go, ex = 35.

3.32. µd vs µx, ex = 32°, ex = 35.

3.33. µd vs µy, Sx = 0, CY = 25.

3.34. µd vs µy, Sx = 0.1, Cy = 25.

3.35. µd vs µy, Sx = 0.2, Cy = 25.

3.36. µd vs µy, Sx = 0.5, CY = 25.

3.37. µd vs µx, sx = 0.1, ex= 35.

3.3g_ µd vs µx, sx = 0.2, ex = 35.

3.39. µd vs µx, sx = 0.5, ex= 35.

3.40. µd vs µy, ex = 4°, Cy = 25.

3.41. µd vs µy, ex = go, Cy = 25.

3.42. µd vs µy, ex = 32°, Cy = 25.

xv

3.43. Comparison of longitudinal slip (Sx) versus longitudinal force coefficient (µx = FxfFz) of six tyre models operated at the indicated slip angle (degrees).

xvi


3.44. Comparison of longitudinal slip (Sx) versus lateral force coefficient (µy = F/Fz) of six tyre models operated at the indicated slip angle (degrees).

3.45. The interaction of longitudinal force coefficient (µx = FxfFz) with lateral force coefficient (µy = F/Fz) at the indicated slip angle (degrees).

3.46. Comparison of the straight-ahead longitudinal force response of six tyre models.

3.47. Comparison of the free-rolling lateral force response of the six tyre models.

4.1. Medium-textured bituminous surface.

4.2. Catadioptric targets behind the MMFTT.

4.3. The recording device on cabin.

4.4. Hydraulic and braking controls on cabin.

4.5. Water discharging apparatus & wind shield (in front of test wheel), at near the right side of MMFTT rear axle.

4.6. Infra-red photo detector ( ..1.) and the disc brake for test wheel (i), at near the left side of MMFTT rear axle.

4. 7. Calibration of vertical, longitudinal and lateral forces.

4.8. The MMFTT and the arrangement for calibration.

4.9. Wheel load calibration curve.

4.10. Braking force calibration curves for angles 0°, 10°, and 15°.

4.11. Sideway force calibration curves for angles 0°, 10°, and 15°.

4.12. Testing sequence.

5.1. Experiment result vs Time of treatment A (3 replicates). Legend: -c-(BFC), -<>-(SFC), L-(Wheel Speed), -(Yeh.Speed).

5.2. Results of IP 30 psi & fast braking, for treatments A (0°), I (10°), Q (15°). Legend: -c-(BFC), -<>-(SFC), L-(Wheel Speed), -(Yeh.Speed).

5.3. Results of IP 30 psi & slow braking, for treatments C {0°), K (10°), S (15°). Legend: -c-(BFC), -<>-(SFC), L-(Wheel Speed), -(Yeh.Speed).

5.4. Results of IP 20 psi & fast braking, for treatments E (0°), M (10°), U (15°). Legend: -c-(BFC), -<>-(SFC), L-(Wheel Speed), -(Yeh.Speed).

xvii


5.5. Results of IP 20 psi & slow braking, for treatments G (0°), 0 (100), W (15°). Legend: ---o-(BFC), -<>-(SFC), L-(Wheel Speed), -(Yeh.Speed).

5.6. The comparison between the corrected Wheel Speed (WS) on entering the measured section, and the actual MMFIT speed (VS) averaged over the measured section.

5.7. BFC (---o-) & SFC (-<>-) vs percent slip (S,J; IP 30 psi & fast braking, for treatments A (0°), I (10°), Q (15°).

5.8. BFC (---o-) & SFC (-<>-) vs percent slip (Sx); IP 30 psi & slow braking, for treatments C (0°), K (100), S (15°).

5.9. BFC (---o-) & SFC (-<>-) vs percent slip (Sx); IP 20 psi & fast braking, for treatments E (0°), M (10°), U (15°).

5.10. BFC (---o-) & SFC (-<>-) vs percent slip (Sx); IP 20 psi & slow braking, for treatments G (0°), 0 (10°), W (15°).

5.11. SFC vs slip angle; IP 30 psi & fast braking, from treatments AIQ. Legend: -<>- (SFC at Sx = 0%), -1:,.- (SFC at Sx = 100%).

5.12. SFC vs slip angle; IP 30 psi & slow braking, from treatments CKS. Legend: -<>- (SFC at Sx = 0%), -1:,.- (SFC at Sx = 100%).

5.13. SFC vs slip angle; IP 20 psi & fast braking, from treatments EMU. Legend: -<>- (SFC at Sx = 0%), -1:,.- (SFC at Sx = 100%).

5.14. SFC vs slip angle; IP 20 psi & slow braking, from treatments GOW. Legend: -<>- (SFC at Sx = 0%), -1:,.- (SFC at Sx = 100%).

5.15. Determination of longitudinal stiffness (Cx) from the relationship of BFC and Sx (% ), for treatment A (0°).

5.16. Determination of lateral .stiffness (Cy) from the relationship of SFC and a. (deg), for treatments AIQ.

5.17. The locked-wheel BFC & SFC, measured at the first and the second half time of full braking.

5.18. The locked-wheel RFC, measured at the first and the second half time of full braking.

5.19. The uniformity of measured section (25 m), assessed by steady-state BFC at constant slip (Sx) of 20.8%.

5.20. The wheel load at brake off (average of 1 second before activation and 1 second after release), and at brake on.

xviii


6.1. Input data ( · · c · ·) of Fx versus Sx (at angle 0°), various F2 , from laboratory data [Sakai (1982)].

6.2. Input data ( · · <> · ·) of F Y versus a ( at Sx 0% ), various F 2 , from laboratory data [Sakai (1982)].

6.3. Field experiment (c), and theoretical values of BFe (at angle 0°) by candidate's model: field ex, a/21 = 0.1 (--) & 0.06 (------).

6.4. Field experiment (a), and theoretical values of BFe (at angle 10°) by candidate's model: field ex & ey, a/21 = 0.1 (--) & 0.06 (------).

6.5. Field experiment (<>), and theoretical values of SFe (at angle 10°) by candidate's model: field ex & ey, a/21 = 0.1 (--) & 0.06 (------).

6.6. Field experiment (a), and theoretical values of BFe (at angle 15°) by candidate's model: field ex & eY' a/21 = 0.1 (--) & 0.06 (------).

6.7. Field experiment (<>), and theoretical values of SFe (at angle 15°) by candidate's model: field ex & ey, a/21 = 0.1 (--) & 0.06 (------).

6.8. Field experiment (<>), and theoretical values of SFe (at Sx 0%) by candidate's model: field ey, a/21 = 0.10 (--) & 0.06 (------).

6.9. Field experiment (~). and theoretical values of SFe (at Sx 100%) by candidate's model: field ex & ey, a/21 = 0.10 (--) & 0.06 (------).

6.10. Field experiment (a), and theoretical values of BFe by candidate's model: field ex & ey, a/21 = 0.10, a = 10° (--) & 8° (------).

6.11. Field experiment ( <> ), and theoretical values of SFe by candidate's model: field ex & ey, a/21 = 0.10, a = 10° (--) & 8° (------).

6.12. Field experiment (c), and theoretical values of BFe by candidate's model: field ex & ey, a/21 = 0.10, a = 15° (--) & 12° (------).

6.13. Field experiment (<>), and theoretical values of SFe by candidate's model: field ex & c;,, a/21 = 0.10, a = 15° (-) & 12° (------).

6.14. Sakai's data (c), and theoretical values of Fx (at angle 0°) by candidate's model: parameters from Table 6.5, a/21 = 0.06, various Fz.

6.15. Sakai's data(<>), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5, a/21 = 0.06, various F2 •

6.16. Sakai's data (0 ), and theoretical values of Fx (at angle 0°) by candidate's model: parameters from Table 6.5, a/21 = 0.08, various Fz-

6.17. Sakai's data(<>), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5, a/21 = 0.08, various Fz.

xix


6.18. Sakai's data (c), and theoretical values of Fx (at angle 0°) by candidate's model: parameters from Table 6.5, a/21 = 0.10, various F2 •

6.19. Sakai's data (<>), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5, a/21 = 0.10, various F2 •

6.20. Sakai's data (<>), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5 (modified ey), various a/21 & F2 •

6.21. Field BFe (c) & SFe (<>) on slippery road at angles 0° (X), 100 (Y), and 15° (Z); and theoretical values by candidate's model: from Table 6.9, a/21 = 0.10.

6.22. Arrangement for measuring the laboratory longitudinal tyre stiffness (ex).

6.23. Arrangement for measuring the laboratory lateral tyre stiffness (ey).

6.24. Field BFe (c) on main experiment road (at angle 0°), and theoretical values by candidate's model: a/21 = 0.10, ex from Field(--) & Laboratory(------).

6.25. Field BFe (c) on slippery road (at angle 0°), and theoretical values by candidate's model: a/21 = 0.10, ex from Field(--) and Laboratory (------).

6.26. Field SFe (<>) on main experiment road (at Sx 0%), and theoretical values by candidate's model: a/21 = 0.10, eY from Field (--) and Laboratory (------).

6.27. Field transient BFe (c) on main experiment road (at angle 0°), and theoretical values by candidate's model: a/21 =0.10, ex from Field (--) & Laboratory (------). Field steady-state BFe (*) are also shown in this figure.

7.1. Inputs and outputs of the candidate's model under either braking or cornering. Left~ Input: ex, locked-wheel BFe. Output: Fx vs Sx (at angle 0°). Right~ Input: CY' locked-wheel BFe (or max SFC). Output: FY vs a (at Sx 0%).

7.2. Inputs and outputs of the candidate's model under both braking and cornering. Left~ Input: ex, ey, locked-wheel BFe (or max SFe). Output: FY vs Fx (various Sx). Right~ Input: ex, ey, locked-wheel BFe (or max SFe). Output: Fy vs Fx (various a).

XX

List of Tables

2.1. Assumed rubber/water temperature.

2.2. The decrease of e"e with increasing F2 for 10.00-20/F tyres, at speed 8-88 kph.

2.3. The decrease of eye with increasing F2 of 078-15 tyres, at speed 6 mph, 28 psi.

2.4. Lateral spring rate (Ky) vs inflation pressure (IP) for truck tyres.

2.5. The relationship between eye and F2 at different speed, for 078-15 tyre, 28 psi.

2.6. The e" and eY of truck tyres (10.00-20/F) in 3 states of wear.

2.7. Results of Fx and FY' at a. = 4°, showing contribution from tyre stiffness (ex, ey) and friction coefficient (µ).

2.8. Results of ~. at Sx = 0, showing contribution from tyre stiffness (Cy) and friction coefttcient (µ).

3.1. Detail of Independent Variables.

3.2. a-Coefficient (t-statistics)

3.3. Model Input Data.

4.1. Detailed treatments, showing the sequence of experiment (in bracket).

5.1. Raw data of treatment A (3 replicates).

5.2. Raw data of treatment A (mean).

5.3. The Recorded and Actual speeds (kph), on entering the measured section, for treatment AeEO (0°).

5.4. The speeds of Test Wheel (kph) on entering the measured section, before and after correction, compared with the speeds of MMFTT (kph) over the measured section.

5.5. The wheel speed (WS), vehicle speed (VS), and percent slip (Sx), for each treatment (Tr), on entering the measured section.

5.6. The wheel speed delay, for all treatments.

5.7. The parameters of normal equations of WS vs T, for all treatments.

5.8. Transformation into Sx scale of treatment A.

5.9. Summary of friction coefficients versus slip, for treatments ACEO (0°).

5.10. Summary of friction coefficients versus slip, for treatments IKMO (100).

xxi

List of Tables (contimied)

5.11. Summary of friction coefficients versus slip, for treatments QSUW (15°).

5.12. Representative friction coefficients versus slip, for analysis of variance.

5.13. Analysis of variance of Braking Force Coefficient (BFC).

5.14. Analysis of variance of Sideway Force Coefficient (SFC).

5.15. Longitudinal stiffness, from zero slip angle treatments (A, C, E, G).

5.16. Lateral stiffness, from zero percent slip values of all treatments.

6.1. Measured and predicted values of longitudinal stiffness coefficient (CxC).

6.2. Measured and predicted values of lateral stiffness coefficient (CyC).

6.3. Parameters for theoretical prediction, extracted from field experiment.

6.4. Input data of Fx and FY at various FZ' from laboratory data.

6.5. Parameters for theoretical prediction, extracted from laboratory data.

6.6. Theoretical prediction of BFC & SFC, versus percent slip (Sx), at various a/21, from the candidate's experimental result.

6.7. Theoretical prediction of SFC, versus slip angle (a.), at various a/21, from the candidate's experimental result.

6.8. Theoretical prediction of Fx vs Sx (a. = 0°), Fy vs a. (Sx = 0%), at various a/21, from Sakai's laboratory data.

6. 9. Parameters for theoretical prediction on slippery road.

6.10. Tyre stiffness from the field (main exp.) and laboratory (load-def.).

Introduction

When a road surface becomes wet, its ability to provide friction is greatly

reduced. For safe driving under wet conditions, a minimum level of friction between

the tyre and the road surface must exist. A knowledge of the availability of this

friction is therefore necessary.

It is a common way to express the skid resistance property of a pavement either

by Braking Force Coefficient (BFC) or Sideway Force Coefficient (SFC). Because of

the skid resistance force results from the interaction between tyre and road, the

friction coefficients are not solely the indicator of pavement properties. In addition

to the contribution of the tyre, some other factors are also involved in influencing

the friction: speed, load, water film thickness, temperature etc.

Many pieces of equipment such as ASTM Skid Trailer (for braking force) and

SCRIM (for sideway force), have been widely accepted as standards for measuring

skid resistance. Whereas the braking friction will determine the braking ability of a

vehicle (such as its stopping distance), the sideway friction will determine the

steering ability of a vehicle (such as its cornering speed).

In general, however, and especially under emergency condition braking and

cornering are present together, and the ability of either braking or steering is

modified by the presence of the other. It is then important to know how the

tyre-road friction will behave under all ranges of percent slip (Sx) and slip angles

(a).

Two ranges of S" (or a), in regard to the magnitude of friction, are generally

considered to have the different controlled factors. These are at below the optimum

value of S" (or a) which are controlled primarily by the tyre properties; and at or

above the optimum value of S" (or a) which are controlled by both the tyre and road

properties. It is important to realise this difference, when dealing with the prediction

of the tyre-road friction under braking and cornering.

This study forms part of the continuing research into tyre-road friction with the

ultimate aim of predicting the skid resistance of a road surface. The work therefore

1

includes the prediction and the validation of the coefficient of friction by field

experiment. The fundamental concepts for the prediction of friction had been

developed by Yandell [1970]. Their application to practical problems requires

modification such as works done by Taneerananon [1981]. However, to cover all

ranges of percent slip (Sx) and slip angles (a) it is then necessary to develop further

the method of prediction.

Two main works are done for this thesis: the development of the method of

prediction, and the field measurement of friction. Included in the prediction of

friction are the calculations of tyre stiffness, locked-wheel BFC or maximum SFC.

The field measurement of friction, on the other hand, is intended to serve

simultaneously two purposes: to supply the parameters needed for the prediction, and

to provide some comparisons for the results of prediction.

The literatures relevant to this research are reviewed in Chapter 1, which are

ended by the problem definition and the method of investigation. In Chapter 2 the

measurements and predictions of tyre-road friction, including the proposed

candidate's tyre model, are described. The effects of parameters influencing the tyre

road friction by using the proposed model, and the comparison of response between

this model and other tyre models, are presented in Chapter 3.

The field measurements of friction, including the calibration prior to the

experiment and the possible variability on test results, are explained in Chapter 4. In

Chapter 5 the analysis of experimental results and the determination of parameters

for theoretical prediction are described. Chapter 6 presents the comparison of the

experimental results with the predictive coefficients of friction, the verification of the

candidate's model with Sakai's laboratory data, the examination of the independence

of tyre stiffness from road conditions, and the investigation into the difference

between the slip stiffness and the deformation stiffness.

The theoretical and experimental findings of the research, including the benefit of

the candidate's tyre model, are presented in Chapter 7. The summary of conclusions

and the future work are also included in this chapter.

2

CHAPTER 1. REVIEW OF LITERATURE

1.1. Nature and Components of Tyre-Road Friction

When a vehicle executes maneuvers such as cornering, braking/accelerating, or

combination of two, the handling of this vehicle depends upon the available frictional

force between its tyres and the road surf ace. Frictional forces are highest on dry roads

where coefficient of friction of 0.7 to 1.0 are usual [Allbert and Walker (1965-66)]. If

the water or any lubricants, which act as medium of low tangential shear, wetted the

road surface, the friction available is drastically reduced. The reduction in friction

level depends on many factors, including: pavement/lubricant conditions (e.g. surface

texture, amount of water, temperature), tyre (e.g. tread pattern), and operating

conditions (e.g. speed, mode of operation).

The mechanism of friction development of a tyre rolling, sliding or combined

rolling/sliding on a wet surface, is commonly described by the concept of 3 zones of

contact area [Allbert and Walker (1965-66), Moore (1966), Harris (1968), Keen

(1968)]. The term "contact" in this concept is not necessarily associated with the direct

contact between tyre and road surf ace, since part ( or even all) of the water covering

the road may not be expelled during the time available. In fact, it just indicates the

area of road surface receiving pressure from the tyre, and thus generate resistance to

the tyre movements (see Figure 1.1).

In the forward region or sinkage zone, the tyre contacts the bulk of water covering

the road. The major component of frictional force then is due to the hydrodynamic

drag. Gough [ 1959] also included the hysteresis loss, as a component of frictional

force, although there is no direct contact between the tyre and road asperities. In other

words, tyre deformation can occur either with dry or lubricated contact. It is believed,

however, that the hysteresis friction does not occur if an excess of water exist, in

which the rolling or sliding tyre may plane on the water and prevent the rubber

flowing over the asperities. Nevertheless, the frictional force developed in this zone is

very small in comparison to other zones. When the vehicle velocity increases, the time

for expelling the water diminishes, the boundaries of the zones move backwards.

3

WATER FILM

ROAD SURFACE

DIRECTION OF MOTION

I T·YRE TO GROUND CONTACT I I I • 1

;

Zone I Region of unbroken water film, 'bullt' displacement zone. Zone 2 Region of partial breakdown of water film, 'thin film' zone. (Road surface

asperities penetrating.) Zone 3 Region of dry tyre to road contact, 'dry' zone.

Fig.1.1. Schematic representation of 3 zone concept. [Allbert and Walker (1965-66)].

Along with increasing velocity, the dynamic pressure generated in the water (p = 0.5pV2/g, where p is water density, V is vehicle velocity, and g is gravitation) also

increases. If the water pressure exceeds the tyre vertical pressure then the tyre is

supported by a layer of water film and the forward zone occupies the whole contact

area. This condition is often referred to as hydroplaning.

In the middle region or transition zone, the tyre proceeds to drape over the larger

asperities and make contact with the lesser asperities. The frictional force developed is

the drag of partial breakdown of lubricant film, and the hysteresis loss. Due to the

squeeze effect, the drag in this zone is greater than in the forward region (although

both of them are still very small). The hysteresis loss is built up as more tyre tread

envelops the projecting aggregate of the pavement surface. Additionally, it is possible

however, that the adhesive friction is also being developed in this zone, as there is dry

contact between tyre and road asperities. Clearly, by intimate contact, the molecular

interactions between rubber and pavements will lead to adhesive bonds, having high

shear strength.

In the rear region, the tread rubber and road surface are in dry contact. The drag of

water film is diminished, and replaced by the adhesion, where the surface now is in

molecular contact. The hysteresis loss will continue to occur due to the deformation of

tyre tread, as it is caused by the indentation of surface texture. Actually, even under

dry conditions, there is no such thing as a perfectly dry contact region between tyre

and pavement; fine water films contaminate all surfaces when the humidity is non

zero. Therefore, when reference is made to a dry contact, this implies a region in

which the fluid film thickness has been reduced to that thickness present on what is

casually and technically incorrectly termed dry pavement [Browne et.al (1982)].

4

If the road asperities are sharp, the tearing of rubber is possible, and the friction is

increased [Tabor (1958)]. This additional component is commonly referred to as

abrading friction. This component, together with other components mentioned above,

will be described in details in the following section.

1.1.1. Hydrodynamic Component of Friction

The hydrodynamic drag, buildup from forward region, can be divided into two

components: water drag and viscous drag [Trollope and Wallace (1962)]. The water

drag is derived primarily from the impact of the water on the leading edge of tyre.

This force is a function of film thickness, water viscosity, and impact speed. Although

the pressure developed by impact speed can exceed the tyre vertical pressure (i.e.

results in hydroplaning), the contribution (in tangential direction) to resist the tyre

longitudinal movements is likely to be of negligible magnitude.

The viscous drag is derived from the viscous resistance of the water which is

continuously sheared between tyre and pavement. This component can be calculated

from Newton's law of viscous shear. Roberts [1971] found that the viscosity of water

increased dramatically when the water film thickness was decreased below 500 A (5x10-6 cm). Hence, the thinner the film thickness, the greater the contribution of

viscous drag. Some theoretical calculations, however, showed this friction due to

viscous shear to be insignificant [Moore (1967), Ludema (1975)].

1.1.2. Adhesive Friction

The adhesion component of friction is due to the making and breaking of atomic

junctions between the rubber and pavement surfaces [Meyer (1968)]. There is general

agreement that adhesion has distinct viscoelastic properties. The coefficient of

adhesional friction is known to exhibit a peak value at creep speeds in about 1 cm/sec

[Ludema and Tabor (1966)]. This peak will shift to a lower speed with decreasing

ambient temperature, and to a higher speed with increasing ambient temperature. In

practice, therefore, this friction most often decreases as sliding speed increases.

5

In ideal case, the true area of contact of two atomically smooth surfaces would be

equal to the nominal · area of contact, and adhesive friction would be very high.

However, for the usual surface which is very rough on the atomic scale, contact would

be limited to the highest proturbances on the surfaces.

Even though the mechanism of adhesive friction is so far not fully understood, it is

well known that on well-lubricated surface the adhesion component may be greatly

reduced due to weakening of the shear strength of bond by the water film. As an

approximation, a film thickness of just exceeding the molecular dimension (i.e.

Angstroms), is probably enough to significantly reduce the direct bonding of tyre

rubber to the road material. An experiment by Roberts and Tabor [1968], using the

glass surface (moving at a uniform speed on the rubber hemisphere), showed that the

friction coefficient decrease sharply from 0. 7 to 0.07 by increasing film thickness from

80 A to 110 A, and then decrease gradually to 0.048 at the film thickness of 180 A.

1.1.3. Hysteretic Friction

On a textured surface, when a tyre is sliding or rolling, the tread rubber is subjected

to continous deformation by the asperities. The hysteresis force is a function of

damping energy loss in deformation (so called deformation loss). The rubber in front

of an asperity is compressed when it approaches the asperity and expands when the

rubber flows over the asperity. A certain amount of energy is required to compress the

rubber element in front the asperity but owing to damping losses only a part of this

energy which is stored in the rubber is recovered. The lost energy is converted into

heat. The loss caused by damping results in an incomplete recovery of rubber upon

expansion, thus creating unsymmetrical pressure distribution shown in Figure 1.2. The

net horizontal component of the pressure force is the hysteresis component which acts

in the opposite direction to sliding.

When the textured surface is perfectly lubricated the adhesion component

disappears, but the hysteresis component always exist as the asperities can deform the

rubber surface. The texture of the surface and the damping property of the tread

rubber, therefore, will determine the magnitude of hysteresis friction.

6

AOHESION DU'ORMATION TEARING WEAR

Fig.1.2. Components of rubber friction. [Kummer (1968)].

In general, the road surface is composed of a number of scales of texture ranging

from microscopic sizes to about 3/4 inch. According to Yandell [1974], it is the

scale's absolute slope (rather than the scale's size) which has a direct effect to the

hysteresis friction. The coefficient of hysteresis friction is the sum of the coefficients

generated on each scale. The size of scale, however, will influence the masking effect

of surface lubricant films, and the sliding speed of peak friction.

The damping properties of the rubber is the most important one for the hysteresis

friction, beside other characteristics: resilience, hardness, and elasticity. The increase

in hysteretic friction due to the higher damping properties of the rubber, is well

established by the fact that tyres made of synthetic rubber provide higher skid

resistance than that given by their natural counterparts [Maycock (1965-66), Savkoor

(1966)].

1.1.4. Abrading Friction

This component of friction is contributed by the fine scale texture with sharp

angular projections on the hard (stone) surfaces. The sharp asperities tear or wear

particles from the rubber. According to Schallamach [1954], this process is due to

tensile failure, induced by stress concentration in the rubber surface behind the track

asperities. It seems, however, that the heating of rubber (such as on braking or

cornering) · also participateS it) develop this process. The force exerted on the tyre

tread rubber to abrade it is the measure of this friction.

7

It was found experimentally by Yandell and Gopalan [1976] that the finer scales of

texture are responsible for greater amount of abrasion but participate less in generation

of hysteresis friction as the speed of sliding increases. Abrasion also increases as the

damping factor, the load and the sliding speed of the rubber increase [Yandell

(1971a)]. On well-lubricated surfaces, however, the abrading friction, similar to the

adhesive friction, is believed to be insignificant. Likewise, it could be argued that the

abrading component of friction might have already been accounted for under one of

the other components of friction [Ludema (1975)].

1.2. Factors Affecting Tyre-Road Friction

There are many of such factors, 47 have been listed by Holmes et.al [1972], but

some of these factors have been to have insignificant effect on the skid resistance.

Moyer [1959] identified 15 variables as the major factors contributing to the large

variation in the friction coefficient measured on various road surfaces. In general, the

variation in skid resistance can be attributed to the following groups, as mentioned

before: pavement/lubricant conditions, tyre, and operating conditions.

1.2.1. Pavement/Lubricant Conditions

(1) Road Surface Texture

Road surface texture refers to the distribution and the geometrical configuration of

the individual aggregates on the road surface. The texture is generally divided into two

components, namely: (a) the macro-texture, to refer to the large scale texture of the

pavement which represent the easily visible asperities, and (b) the micro-texture, to

refer to the fine scale texture on the surface of individual pieces of aggregate. In

quantitative study, however, it had been shown that road surface texture can be

divided into more than two scales [Yandell and Gopalan (1976)]. The function of the

texture is to provide a close contact with the tyre surf ace, to secure sufficient

deformation of the tread rubber, and to facilitate the removal of surface water

especially when the tyre surface is smooth.

8

The role of macro-texture in removing bulk water was demonstrated by Wallace

and Trollope [1969]. They found that as the texture depth increased the normal water

force underneath the tyre decreased resulting in the increase of tyre friction. An

experiment on wet surfaces by Schlosser [ 1977] showed that the influence of macro

texture is mostly prominent at high speeds. His results showed, even with worn tread,

that the locked wheel and sideway force coefficients remained almost constant up to

a speed of 100 km/h on the coarse macro-texture surfaces. It is well known that at high

speed, shorter time will be available for the water to drain away, thus the remaining

water film would be thicker if adequate drainage is not provided as with the cases of

smooth macro-texture surfaces.

The effect of micro-texture in reducing the thin water film (hence establishing dry

contact with the tyre surface) was quantitatively studied by Rohde [1976]. He found

that the time of descent of a tread element under constant load decreased significantly

when the micro roughness amplitude was increased. The shape of the micro-texture

also influenced the sinkage time, with the triangular texture pattern taking shorter time

than the square pattern to reach a given minimum film thickness. The increases in the

level of harshness of micro-texture, on the other hand, can cause an increased tyre

wear without a proportional increase in the tyre friction [Yandell and Gopalan (1976),

Lees et.al (1977)].

(2) Aggregate Characteristics

The characteristics of aggregate which influence the skid resistance are the shape,

grading, type, state of wear or polishing. The pressure distribution of the surface of

asperities pressed into tread rubber depends on the shape of asperities rather than their

size, provided that the macro-texture is coarse enough to drain off the surface water

[Tabor (1959)], its spacing is such that there is no texture saturation by the rubber

[Yandell ( 1970, 1971 b)], and visco elastic frequency effects do not intrude.

Laboratory study by Stephens and Goetz [ 1960] found that the mixture made from

finer gradation gave higher values of skid resistance. It is generally agreed that all

types of limestones are highly polishable and result in extremely slippery roads under

wet condition although they give high skid resistance when newly laid [Finney and

Brown (1959), Gray and Renninger (1965)]. Sandstone and quartzite aggregates were

found to posses the highest permanent skid resistance properties [Havens (1959)].

9

According to Lees [1984-85], aggregates which have a distinct difference in hardness

between primary minerals (such as quartz, augite, hornblende) and weathered

secondary minerals (such as kaolinite, chlorite, terpentine, sericite) will also produce

high skid resistance.

With all types of aggregates and methods of construction of roads it is possible to

get high skid resistance when the road surface is new. However, with age, the

aggregates under the action of heavy traffic tend to polish, resulting in lower skid

resistance. The polishing of the road stones is principally due to the continual attrition

of fine abrasive mineral particles found on the road surface caused by the traffic

movements, and it is seen that the finer the detritus, the greater will be the degree of

polishing [Maclean and Shergold (1960)]. Stiffler [1969] concluded that the abrasives

of size less than 10 micron diameter will cause polishing.

(3) Seasonal Effects on Surface Texture

Giles and Sabey [ 1959] reported a marked difference in test results on wet surf aces

for winter and summer. The coefficient of friction is higher in winter than summer.

The above report reveals that the road surf ace was found to be covered with fine dust

during summer. The dust polished the roadstone thus resulting in a lower friction

coefficient. In winter (when the road was wet for 60% of the time), the dust particles

were quickly washed off the tyre and the abrasion patterns tended to disappear. The

polished stones got roughened by weathering in the presence of water.

( 4) Amount of Water

The effect of water film on friction has been found to be the predominant factor

compared to the speed and temperature dependent viscoelastic effect [Clamroth and

Heidemann (1968), Lupton and Williams (1972)]. Giles [1959] and Besse [1972]

found that the friction continue to decrease as the water film increase, until the film

thickness of about 0.02 in (0.508 mm) where the friction tends to level off.

(5) Temperature

It is well known that with increasing temperature (air, surface, tyre tread), the skid

resistance tends to decrease [Giles and Sabey (1959), Grosh and Maycock (1968),

Meyer and Kummer (1969), Meyer et.al (1974)]. Both Giles et.al and Grosh et.al

10

measured an increase in rubber resilience with increased temperature of rubber which

indicates a decrease in hysteresis losses of rubber.

The effect of water temperature was studied by Meyer et.al [1974]. They measured

skid resistance on several pavements with water temperatures of 140° F (6C.f C) and

60° F (15.5° C) and found that the difference was about 1 skid number (0.01 locked

wheel braking force coefficient), the higher temperature gave the lower values. Hence,

it is seen that the temperature dependence of friction is through its effect on rubber

properties rather than the rate of change of water removal caused by viscosity of the

water (in which case the friction would increase and not decrease at higher

temperature).

It is found that the temperature dependence of wet friction is dependent on the

texture of the road surface. On wet coarse textured road surfaces the decrease in

friction is more than that on wet fine textured road surfaces [Giles and Sabey (1959),

Maclean and Shergold (1960)]. The possible explanation is that on the coarse texture,

the temperature will be higher than for the smooth surface due to the greater contact

stresses (as caused by lesser actual contact area), and due to the break down of the

carbon filler [Yandell et.al (1983)].

1.2.2. Tyre Factor

The choice of tyre type (cross ply, radial) for use in a vehicle can have a large

influence on skidding characteristics. The variability in wet skid friction has been

investigated by Allbert and Walker [1965-66], for example up to 4:1 for tread pattern

design, and up to 1.8: 1 for changes in tyre compound. To fulfil its function the tyre

(tread) compound, as being reflected by its physical properties (such as: resilience,

hardness, elasticity, and damping), must meet a number of different tyre requirements.

A compromise in many properties sometimes must be sought, so that a tyre with high

skid resistance as well as high abrasive resistance, can be obtain simultaneously

[Peterson et.al (1974)]. In the following, those most important factors: tyre type, tread

pattern, damping property, and hardness, will be described in details.

11

A .IIAS ,l Y • . IADIAl IElTED C - IIAS HlTED

Fig.1.3. Basic tyre structures. [Davisson (1969)].

(1) Tyre Type

The three basic types of tyres in use nowday are the bias-ply (sometimes called the

cross-ply or conventional), the radial-ply (belted) tyre, and the bias-belted tyre. These

three types are shown in Figure 1.3. Within each of the three basic tyre types, many

variations are possible such as: low or high aspect ratio (i.e. ratio of the section high

to the section width), and tube or tubeless type.

Radial-belted tyres although have lateral spring rates considerably lower than

bias-ply tyres, but usually have higher cornering stiffness properties [Davisson

(1969)]. In other words, for a given slip angle, the lateral force developed in

radial-belted is greater than bias-ply. According to Davisson [1969], belted tyres have

less tread movement in the contact area, and exhibit a slower rate of wear than bias

tyres.

Results from DeVinney [1967] on hydroplaning tests show that radial-belted tyres

have greater coefficient of friction. He explain that the "belt" gives a rigidness to the

tread which serves to keep the grooves open as the tyre rolls through the contact

patch, providing greater water "drainage".

Tests on cornering coefficient by Sakai et.al [1978] found that when the water film

is thin (1 mm) radial-ply tyres are considerably superior to cross-ply tyres. In contrast,

when the water film becomes thick (5 mm) the radial-ply tyres are superior in the low

speed region but inferior in the high speed region.

12

(2) Tread Pattern

From the point of view of removal of water from the ground tyre contact, the tread

pattern and the road surface macro-texture function in a somewhat similar manner.

The tread pattern provides outlet channels for the surf ace water to escape from the

contact area, under the squeezing action of the tyre. By this reasoning, the tread

pattern should have a greater effect on smooth textured surfaces where drainage is

generally poor. Conversely, tread patterns are least effective on rough open textured

surface. There is also a contrast under dry and wet conditions, where best dry skid

resistance is obtained when a tyre has no tread design at all. However, tread design

plays a vital role in wet friction where accidents due to friction mostly occur on wet

surfaces in~tead of on dry surfaces.

The simplest and the most common tread pattern is the provision of longitudinal

grooves on the tread surface which, has shown to increase the friction on wet surfaces

by 20 to 100 percent depending upon the initial value of the friction [Marick (1959)].

The skid resistance was also found to increase with the number of grooves and groove

widths [Allbert and Walker (1965-66), Maycock (1965-66)]. This was attributed to the

greater groove volume in relation to the total drainage area, to the slightly higher

pressure acting in the contact zone as the groove area increases and probably to the

shorter travel paths of water to reach the drainage channels [Maycock (1965-66)].

On the other hand, Kienle [1974] found that the effectiveness of widening grooves

is asymptotic, and there is an optimum tread width for a given number of ribs. A

similar result is obtained by Veith [1977], using the "fractional groove volume" 0

which he defined as the ratio of groove volume to the total tread volume, as a measure

of a tread pattern effect. He found an exponential relation between the locked wheel

coefficient of friction and 0, where the tread pattern effect is seen to reach a limit at 0

of about 0.4 (beyond which the coefficient remain unchanged with increased 0).

In addition to longitudinal grooves, the lateral edges on the tread surface have been

found to provide wiping action over the wet surface. On some rough surfaces, it was

seen that a suction of 10 psi (7 g/mm2) developed behind the transverse grooves

[Wallace and Trollope (1969)]. This agreed with Gough's observation that sharp

transverse edges of the tyre tread increase the skid friction by wiping water from the

13

road surface [Gough (1954-55)]. Slits in the tyre function as pockets to absorb local

lubricant pressure and promote dry contact [Allbert and Walker (1965-66)].

Since the fall of friction coefficient with speed is associated with the greater

difficulty the tyre has in displacing the water beneath it, it is evident that tread pattern

proves more effective at high speeds. Maycock [ 1968] found that the tread pattern,

even of the simplest design such as straight ribs, gave on the smooth surface a large

improvement in both peak and locked wheel braking force coefficients with the speed

range of 30 mph to 60 mph. At low speeds the effects of tread pattern are not

significant especially on coarse textured road surfaces.

(3) Damping Property

The damping factor is usually measured by allowing a weighted pendulum to strike

a rubber sample from a given height. The elasticity is the percentage of potential

energy regained at the first rebound. Therefore a rebound to the original height would

indicate 100 percent elasticity, that is no damping loss. The energy lost in the process

is called damping energy and is dissipated in the rubber as heat. Yandell [1970]

defined the damping factor as the energy lost divided by the energy applied during one

complete loading-unloading cycle. Rubber with high damping losses (low resilience)

will give high values of friction coefficient [Sabey et.al (1970)].

As rubber is partly viscoelastic material, its damping losses are influenced by

temperature and frequency of loading. The damping decreases as temperature

increases, and increases with increasing frequency. With increasing temperature, peak

damping for a given rubber compound decreases and shifts towards higher frequencies

[Kummer and Meyer (1962)]. Furthermore, it was found that the damping for synthetic

rubber is always considerably greater than for natural rubber. The superiority of

natural rubber, however, is for the low heat buildup characteristic [Davisson (1969)].

( 4) Hardness

The hardness is usually measured by Shore Durometer [Bashore (1937)]. Rubber

hardness has been less definitely correlated with friction. There are reports in the

literature that increasing rubber hardness increases friction [Marwick and Starks

(1941), Grime and Giles (1954-55), Giles et.al (1962), Goodwin and Whitehurst

(1962), Sarbach et.al (1965)], or has no effect [Sabey and Lupton (1964), Bassi

14

(1965)]. The general explanation for the increased friction is that the harder rubber

gives a smaller contact area, resulting in higher contact pressure, better drainage and

hence increased friction [Csathy et.al (1968)].

Carr [1967] reported that the relative influence of rubber hardness depends on the

texture on which it slides. Report from Percarpio and Bevilacqua [ 1968] found that on

slippery road surfaces, the increased hardness conversely leads to decreasing friction.

Their report then suggest that friction increases with hardness only on highly abrasive

surfaces.

1.2.3. Operating Conditions

(1) Speed

Numerous tests have been carried out to study the effect of speed on friction

[Hofelt (1959), Sabey (1966), Kem (1967), Allbert and Walker (1968), Maycock

(1968), Sabey et.al (1970)]. Some of these tests were conducted in a laboratory,

however, all the results generally indicated a decrease in friction with increasing

speed. An exception to this trend was reported by Sabey et.al [1970] which indicated a

significant interaction effect between speed and road texture (see Figure 1.4). The

usual explanation given for the decrease in friction on dry surfaces with increasing

speed is that the high temperature generated within the contact patch causes the rubber

to melt, the melted rubber then serves as a lubricant.

Foster [1961] confirmed that the coefficient of adhesive friction increases from a

low value to a maximum value when sliding takes place, then drops as the velocity

further increases. Papenhuyzen [1938] found that the increase in the adhesive friction

is less pronounced on rough tracks, and the velocity-friction curve begins to take a

negative slope at a higher sliding velocity than on a smooth surface. Savkoor [1965]

demonstrated that the adhesive friction of rubber on glass is viscoelastic, and he

concluded that the adhesive friction develops a peak at a velocity which moves

towards a higher velocity when the temperature is increased.

A theory for speed dependence of adhesive friction has been developed by Kummer

and Meyer [1966], in which the exposed atoms of the rubber chains are believed to

15

C 0 ·a--- . .. " .... -· .-c!' ··-__ ... "0 ·- .. s~ 0 u

H

- Smoolh 1urfcac11 0-1 ---- Yer, rough surfocu

0·4

0·2

0 .__ ____ __. __ ...._ _ __._ __ ~-~

20 60 10

0 20 ,o Spnd lm,lt/tll

100

60

120 km/h

I 10

Fig.1.4. Effect of speed and road texture to skid resistance. [Sabey et.al (1970)].

form junctions with the regular array of stone surface atoms. The junctions, according

to their theory, break off after the bulk of rubber moves forward. The rubber

molecules in the process recoil and form another junction. The formation and breaking

of junction dissipates adhesive energy which is a maximum at a particular velocity and

temperature.

The behaviour of the lubricated friction under different sliding speeds up to 60 mph

(100 kph) was studied by Hegmon [1965, 1968], and found that eventually the

lubricated friction reaches a second peak. Kummer and Meyer [1966] proposed that

the hysteresis friction does not vary greatly with speed at low speed but increases

rapidly to a maximum at 100 mph (165 kph). They attributed the increase in friction at

high speeds to the fact that at high speeds the rubber has no time to recover and

separate from the downstream surface of the asperity and the resultant of the

remaining reactions opposes the sliding (see Figure 1.5).

Meyer and Kummer [1969] superimposed their theories of viscoelastic adhesion

and viscoelastic hysteresis to form a unified theory of friction, according to which the

low speed· friction peak is due to adhesion, and high speed friction peak is due to

hysteresis. While they assumed that hysteresis occurs at a particular scale of texture,

Yandell [ 1970] on the other hand maintains that since a road surface consists of

numerous superimposed scales of texture, the resultant hysteresis friction is made up

from a large number of friction-speed curves with each having a peak at a different

speed.

16

,..a 3.0 ,-----,------,---""T"""----.

i-: z l&.I i3 2.0 t-----+---1---,r--#1--'---+-'----I ii:: LI. l&.I 0 u

f a C,

U)

3 ~ C

i U) U)

9

0.010 0.10 10 SLIDING SPEED, mph

EXCITATION FREQUENCY v0 , cps -

LOW SPEED HIGH SPEED

... ~3.0 i-: z l&.I i3 i.: 2.0 LI. l&.I 0 u U)

iii 1.0 l&.I a:: l&.I I-U)

>- 0 J: I 10 100 1000 10,000

SLIDING SPEED, mph

t C

I-z l&.I C, z <t I-U) U) 0 ..J

EXCITATION FREQUENCY, vh, cps -

Fig.1.5. Friction mechanism due to adhesion and hysteresis. [Kummer and Meyer (1969)].

The reduction of friction under wet conditions with increasing speed can be

attributed to the following reasons.

a. Difficulty in squeezing water films from between the tyre road contact within the

available time, which decreases as the speed increases.

b. The higher frequency of loading (due to the higher speed) beyond the rubber's

characteristic peak leads to a lower coefficient of friction [Carrol (1965)].

c. The possible temperature rise in tread rubber (lost energy converted into heat),

when the remaining water is thin, causing a reduction in damping factor of rubber,

hence results in a decrease in hysteresis friction.

17

d. Lack of interasperity drainage at high speeds which results in lower effective load

carried by the asperity [Wallace and Trollope (1969)]. This effect is more severe in

fine textured road surfaces causing steeper friction-speed curves.

(2) Wheel Load

In general, the change in wheel load appears to have a small effect on skid

resistance, and can be regarded as insignificant [Staughton and Williams (1970),

Schlosser (1977)]. However, if the friction is separated into adhesion and hysteresis

components, there is a tendency that the increasing loads leads to a decrease in

adhesive friction [Schallamach (1958), Tabor (1959)], and an increase in hysteresis

friction [Tabor (1952), Greenwood and Tabor (1958), Sabey (1958)].

The influence of load on the adhesive friction is dependent on the range of contact

pressure considered [Thirion (1946), Archard (1953, 1957-58), Denny (1953), Tabor

(1959)]. When pressure is small, the area of actual contact has been found to be

proportional to the load, in which case the friction coefficient is independent of the

loads. For high pressures, however, the adhesive friction is load dependent because the

actual area of contact cannot be made to increase indefinitely with the load.

Schallamach [1958] believed that the adhesion increased with the area of contact and

decreased with the load.

The coefficient of water lubricated friction increased with the load in some tests,

the increase was more rapid on coarse textured road surfaces [Stutzenberger and

Havens (1958), Stephens and Goetz (1961)]. According to Yandell [1974], the

increase in hysteresis friction with increased load is caused by partially unmasking the

particular scale due to the decreased film thickness, and by increasing average

absolute slope for cylindrical asperities. On the other hand, he found that it is also

possible that the hysteresis friction decreased with increased load, in which this is

caused by preventing the stress flux in the rubber from increasing as rapidly as the

normal load (so he called "the stress saturation of rubber by the texture").

(3) Inflation Pressure

Similar to wheel load, the effect of tyre inflation pressure on the friction coefficient

is marginal [Hofelt (1959), Kummer and Meyer (1967)], however there is a general

trend that as inflation pressure increases this coefficient decreases on wet pavements

18

[Moyer (1934), Orchard (1947), Pike (1949), Goodwin and Whitehurst (1962)]. On

dry pavements, there is a little evidence that the friction coefficient increases with

increased inflation pressure [Pike (1949)]. The change in inflation pressure ma~due to

the change in the temperature of the tyre rubber, where the hotter the tyre the higher

the inflation pressure.

The increase in the inflation pressure has been found to increase the pressure under

the central ribs of a pneumatic tyre without appreciably changing the pressure under

the shoulder ribs [Hofelt (1959)] and to decrease the contact area of the tyre with the

road surface. Although the increase in the pressure under the central ribs is good from

the point of view of water removal which may result in better skid resistance, but the

decrease in the contact area may have an opposite effect due to decrease in contact

area per unit load [Moyer (1934)].

Lander and Williams [1968] found that the change in friction coefficient due to

inflation pressure is also influenced by the type of surface texture. Using the sliding

speed of 60 mph they observed a 0.1 reduction of the braking force coefficient on the

fine textured surface by increasing inflation pressure from 40 psi to 260 psi, but little

change on the coarse textured surface. (This may be due to stress saturation of the

rubber on the fine texture [Yandell (1974)]).

( 4) Mode of Operation

The vehicle maneuvers basically can be grouped into four categories: rolling,

cornering, decelerating/accelerating (or braking/driving), and combined cornering with

decelerating/accelerating (such as braking-in-a-tum, or turning-in-driving). It is quite

common to treat skid resistance as a pavement property since road surface properties

influence more than either tyres or the automobile. However, in the following it will

be described that the magnitude of the friction coefficient is also controlled by the

elastic tyre properties and degrees of slip.

In rolling motions, the dissipation of energy due to bending of the tread as it rolls

through the contact patch, causes higher normal pressures in the leading half of the

contact. Hence, the moment of the normal pressures with respect to the axis of rotation

is not zero. In the free-rolling tyre, the applied wheel torque is zero, therefore

frictional forces must exist to maintain equilibrium. The resultant frictional force

19

(called "rolling resistance force") is a result of the normal pressures and zero applied

wheel torque [Nordeen and Cortese (1963)]. However, the magnitude of rolling

resistance coefficient is very small, being in the order of 0.01-0.015 [Hadekel (1952)].

This coefficient increases with increasing speed and load, and with decreasing

inflation pressure [Fuller et.al (1984)].

In cornering motions, the skid resistance characteristics are described by the

relationship between lateral (sideway) force coefficient and slip angle, as typically

shown in Figure 1.6 [Bergman (1977a)]. The lateral coefficient increases steeply in

the range of slip angle between 0° and about 8°. Within this range, the curves for wet

and dry surfaces were identical. This indicated that, within this region, road friction

has practically no effect on the relationship between lateral coefficient and slip angle.

Instead, it is controlled primarily by elastic tyre properties [Bergman and Clemett

(1975)]. The effects of road friction first begin to appear near the critical angle value

which is reached at about 12° on wet surface and at about 300 on dry surface. The

curves then show gradual decline with further increase of slip angle.

Skid resistance characteristics in braking are described by the relationship between

longitudinal (braking) force coefficient and wheel slip, as typically shown in Figure

1.7 [Bergman (1977a)]. These curves show a general similarity to the cornering

coefficient curves for the same tyre in Figure 1.6. The braking force coefficient

increases up to its peak value reached at a critical value of wheel slip and then shows

a gradual decline with further increase of wheel slip. In the region below the critical

slip, the relationship between braking coefficient and wheel slip is also dominated by

the tyre properties [Gough (1974)].

In braking-in-a-tum, skid resistance characteristics are determined primarily by the

peak and slide envelopes of the resultant braking-cornering coefficient, as typically

shown in Figure 1.8 [Bergman (1977a)]. The lateral coefficient versus braking

coefficient curves plotted for individual slip angles describe skid resistance

characteristics as well as elastic properties of braking-in-a-tum tyres. The skid

resistance characteristics at transition from peak-to-slide are described by the solid

portion of these curves located between the peak and slide traction envelope.

20

UTERAL FORCE

COEFFICIENT 0.4

0.3

0.2

0.1

10 20 30 40 50 60 70 80 90 SLIP ANGLE (DEG)

Fig.1.6. Lateral force coefficient vs slip angle of 178-15 tyre. [Bergman (1977a)].

0.8

... z ... u 0.6 ii: ...

0.5 ... 0 u UI 0.4 z ii: 0.3 C • • 0.2

0.1

SKID ZONE (ORY) ----.i i---- SKID ZONE IWETI ----

PEAK VALUE (WET) SN85 DRY SURFACE SN85

WET SURFACE SN30

CRITICAL SLIP (WET) CRITICAL SLIP (DRY)

SN30-.._

0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9 1.0 WHEEL SLIP

Fig.1.7. Braking force coefficient vs slip angle of 178-tyre. [Bergman (1977a)].

0.9.-----------------

o.a 0.7

o.a

SUOE TRACTION ENVELOP£

~TERM. 1.51-i"-.l~~T-~+-;;:s;:~~ FORCE

COEFFICIENT M 1.3

G.2

0.1

SN30

0.1 G.2 1.3 0.4 0.5 0.1 0. 7 0.8 0.9 BRAIUNS COEFFICIENT

Fig. 1.8. Peak and slide traction envelopes of 178-15 tyre in braking-in-a-tum. [Bergman (1977a)].

21

1.3. Contact and Slip Between Tyre and Roadway

For a wheel to produce friction forces, either braking or sideway, there must be

some reduction in wheel speed. A wheel is said to be slipping if it circumferential

velocity is different from its travelling velocity. The friction and normal forces needed

for vehicle support, guidance and maneuvers arise in the tyre contact area. The normal

pressure distribution, for example, does not suddenly rise at the edge of the contact

area from zero to a finite value, but rather increases, gradually or suddenly at a finite

rate. The low normal pressure at the edge of contact patch (hence small adhesion

limits) perhaps leads to a conclusion derived by Novopol'skii and Tret'yakov [1963],

that even when the tyre is rolling, there are slip areas where contact begins and ends.

In the following, the characteristics of tyre motions, their contact area, and the stress

distribution will be described in detail.

1.3.1. Classifications of Tyre Motions

According to Pacejka [ 1966] the motions of a tyre can be distinguished into two

main groups, namely steady-state (stationary) and non steady-state motions. The first

group is associated with constant slip velocity in the rolling direction and

perpendicular to it. For example: a. Tyre rolling with constant braking slip or at

constant slip angle. The second group have a character varying with time for lateral

slip and angular motions. For example [Hegmon (1982)]:

bl. Tyre rolling with fast sweep of slip angle (- 10°/second) at constant braking slip.

b2. Fast braking to wheel lock (- 0.5 second) at constant slip angle.

Another classification is made by including a third group between those two

groups, representing a small time interval of the steady-state behaviour [Uffelmann

(1983)], so called quasi steady-state motions. For example [Hegmon (1982)]:

c 1. Tyre rolling with slow sweep of slip angle (- 3° /second) at constant braking slip.

c2. Slow braking to wheel lock (- 5 second) at constant slip angle.

Most operating conditions can be regarded as the steady-state and quasi steady-state

motions when the automobile is cruising in a normal manner. Under emergency or

panic stops, the non steady-state motion often occurs.

22

1.3.2. Types of Slip

Slip has been defined as the ratio of the difference between the angular wheel

velocity when the tyre is rolling or slipping to the angular wheel velocity when the

tyre is rolling [Kummer and Meyer (1967)]. Slip is commonly expressed as a

percentage by multiplying by 100. It is convenient to define slip as a vectorial quantity

S thus:

v-V S° = (-) X 100 V

where v is the travelling velocity, V is the circumferential velocity, and v is the

magnitude of v.

For a pure circumferential slip, the slip in percentage is given by: v-V S = (-) X 100

V

When braking (v > V), S is positive with the maximum value of 100% for a locked

wheel (V = 0). The maximum braking force is obtained at a slip between 7%-30%

[Giles (1964), Holmes and Stone (1969), Schallamach and Grosch (1982)]. During

acceleration (v < V) is negative and becomes negative infinity when the stationary

wheel spins [Schallamach and Grosch (1982)].

When the wheel rolls in a direction making the slip angle a with its plane, the

situation is one of pure side slip, and the slip is given by:

S = sina x 100

There is a critical value of a where the sideway force developed is a maximum, and

beyond this angle the force begins to decrease. This value is around 5° - 20" [Bradley

and Allen (1930), Holmes and Stone (1969), Veith (1971), Shah and Henry (1978)].

The mechanism of slip produced by braking/driving and cornering was illustrated

by model experiments by Schallamach and Turner [1960], and Schallamach and

Grosch [1982]. A wheel of 2.5 inch diameter and 0.5 inch thick was made to travel

over a Perspex track under different conditions of slip. The wheel surface was marked

with lateral equidistant markings, the orientation of these markings on the slipping

tyre indicated the tangential stress distribution in the contact area.

23

(a)

(b)

(c}

(d)

Fig.1.9. Model experiments for: (a) free rolling, (b) side slip, (c) braking, (d) driving. Travelling direction from right to left. [Schallamach and Grosch (1982)].

The strains set up in the contact area of a slipping wheel are showed in Figure 1.9.

All cases have in common that a circumferential element of the wheel on entering the

contact area adheres to the track at first. As the element moves further into the contact

area, the imposed slip produces a deflection which increases linearly with increasing

distance from the front edge. This is clearly seen in Figure 1.9b, where the deflection

is at right angles to the wheel plane. The corresponding surface stress increases in the

same sense until a limiting frictional stress is reached and the element begins to slide

back towards its undeformed position.

24

A braking force (Figure 1.9c), lengthens an element in the circumferential direction

before entering the contact area, and the element adheres at first to the track in this

state of strain. The deflection of the wheel increases linearly with increasing distance

from the front edge, until sliding commences towards the front of the contact when the

tangential stress exceeds the limiting frictional stress. Under a driving torque,

contraction of an element takes place before entering~ontact region (Figure 1.9d).

1.3.3. Sliding and Skidding

As mentioned above, when appreciable (or even quite small) sideway or braking

forces are applied, sliding begins at the rear of the contact surface in which the

available friction is exceeded. According to Kamm [1938], there is appreciable sliding

at the rear of the contact surface even in straight unbraked rolling, at high speeds. In

the general case, the contact surface can thus be divided into adhesion and sliding

regions, an adhesion region being defined as one in which there is no relative motion

between coincident points of the tyre and ground surfaces.

In extreme cases of course the relative motion is one of pure sliding, for which the

term "skidding" may be reserved. In vehicle scale, skidding as defined by Stonex

[ 1959] is the motion of a vehicle under partial or complete loss of control caused by

(pure) sliding of one or more wheels. Transition between partial sliding and skidding

of a tyre is of particular importance since it corresponds to total loss of tyre directional

stability. Schuster and Weichsler [1935] point out that as long as there is even partial

adhesion, the tyre may be said to have a "virtual guiding rail" as a result of frictional

forces (this "rail" is an elastic one).

The sliding or skidding potential of a road surface is commonly measured by

locked wheel or side slip modes of operation of skid testing machines. The locked

wheel test simulates the longitudinal sliding conditions of a braked vehicle, whereas

the sideway test simulates sliding condition when a vehicle is cornering.

25

8:00 x 14 Automotive Tire Four ply Royon Bloa Ply Construction 24 psi inflation 1.25" Deflection 1350 lbs lood - ~---~ -- - - - - - - -

Fig.1.10. Typical contact area for an automobile tyre. [Browne et.al (1982)].

1.3.4. Contact Area

The shape of the contact area between tyre and roadway depends on the tyre cross

section shape and structure. For example, the contact area between an aircraft tyre and

a flat surface usually appears to be nearly elliptical in shape. For an automotive tyre a

somewhat different set of relationships exists due to the fact that the usual

construction involves the use of a relatively heavy tread, particularly in the shoulder

region. In this case any significant contact spread over the entire width of the tyre

between shoulders so that the contact area tends to have essentially straight parallel

sides, and the width of this contact area is nominally independent of tyre deflection. A

typical contact area is shown in Figure 1.10.

Experimental evidence indicates that tyre deflection is the most important variable

governing the area of contact. Results from Hadekel [1952] show that the relationship

between tyre deflection and gross contact area is nearly linear. If inflation pressure

and load are simultaneously varied so as to maintain constant tyre deflection, the tyre

contact area will remain effectively constant [Browne et.al (1982)]. Using a single

truck tyre,anexperiment by the U.S. Army Engineers Waterway Experiment Station

[1964] showed that in general the contact area of a slowly rolling tyre is slightly lower

than of a standing tyre.

When fluid contaminants such as water, oil, slush, or mud are present on the

pavement surface, (dry) contact area between tyre and pavement is reduced due to the

persistence of a fluid film in portions of the formerly dry contact area [Horne and

26

Dreher (1963), Yeager (1974)]. Just how much the dry contact area will remain

depends on tyre, pavement, fluid, and vehicle factors. According to Browne et.al

[1982], the fractional dry contact area is reduced, with increasing of: speed, fluid

(viscosity, density, depth), size of tread elements; and with decreasing of: inflation

pressure, carcass rigidity, width of grooves (number held constant), number of grooves

(width held constant), and groove depth (width held constant).

1.3.5. Stress Distribution

At the interface between the tyre and the roadway an element of tyre surface area is

acted upon by a force vector which can be expressed as two components, one

perpendicular to the contact surface, called the normal component, and one tangential

to the contact surface. This latter component may be further decomposed into two

components, each lying in the contact plane, but one parallel to the central plane of the

tyre and the other perpendicular to it. These components in the contact plane are

commonly called the shear components. With equal validity they could be expressed

as components parallel and perpendicular to the direction of travel of the wheel. Either

decomposition would be useful for describing the shear effects.

(1) Normal Contact Stresses

Contact pressures between tyre and ground are not necessarily uniform, mainly

because of tyre detail design and partly because of road surface irregularities.

Measurements of contact pressure confirm that since the contact pressure are about

50% greater than the inflation pressure, the casing stiffness plays a part in the

load-carrying, otherwise an infinitely flexible casing would result in contact pressures

equal to inflation pressure [Gough (1958-59)].

Modem instrumentation techniques have allowed measurement of normal pressures

under a variety of conditions, using many different tyre types. Some of this data has

been reviewed briefly by Hadekel, with particular emphasis on the early work by

Martin [1936] and Markwick and Starks [1941] for road vehicle tyres, and by Teller

and Buchanan [1937] and Kraft [1941] for aircraft tyres. The following facts emerge

from the above data [Hadekel (1952)].

27

N E ~ 0 r----.--,----,--"T'-"T""T---.r, .i' 2 t\----t------lf-------'-------l- ............_ ' w 4 a . b...-! a: I ;-}! ::, 6 _________________ ..

a 8t-----+----+--+--+--'---,........! flOO~~-,-~~,........,~_.__....,._,

50 100 150 . 200 250 300 mm LEADING EDGE LENGTH nm. TRAIUNG EDGE

Fig.1.11. Pressure distribution down the length of a tyre with no tread pattern, at constant speed 15 kph, inflation 6.5 atm, load: (a) 1680 kp, (b) 3740 kp. [Bode (1962)].

60 40 20 20 40 60 80 100 mm POSITION ACROSS WIDTH

Fig.1.12. Pressure distribution across the width of a tyre with no tread pattern, constant speed 15 kph, load: (a,b,c) 1680 kp, (d) 3740 kp; inflation: (a) 3.5 atm, (b) 5 atm, (c,d) 6.5 atm. [Bode (1962)].

LEADING EDGt TIWL.ING EDGE

1or--,--,---~-.....---zr-.....--~ ~2~•--t--t-----•-+------l .i'

- 4 .., ~6i-::~~~-ri-~-~+=~~~~-~ en ~8t--+--+---+---+--+------l f .

100 50 100 150 200 250 300mm POSITION IN LENGTH OF CONTACT PATCH

Fig.1.13. Pressure distribution down the length of a smooth tyre, load 1680 kp, inflation 6.5 atm, speed: (a) 10.2 kph, (b) 8.4 kph, (c) 8.9 kph; acceleration: (a) 0.6 m/s2, (b) 1.6 m/s2, (c) 4.1 m/s2• [Bode (1962)].

LEADING EDGE N 0....--..--..--..--........,,.,......,---, e ~ 2 A .. 4 i,j

~ 6 en :a1--+-"---------~-----1 a: ~,00..__...,_-------,,-,--300mm

Fig.1.14. Pressure distribution down the length of a smooth tyre, load 1680 kp, inflation 6.5 atm, speed: (a) 10 kph, (b) 35.7 kph, (c) 14.5 kph; deceleration: (a) 4.1 m/s2, (b) 2.7 m/s2, (c) 2 m/s2• [Bode (1962)].

28

Fig.1.15. Pressure distribution of bias-belted H size tyre, 26 psi, 0° slip angle, 100% T and R load = 1580 lb. [Lippmann and Oblizajek (1974)].

a. There is (as might be expected) a tendency for the pressure to rise from the centre

outwards, up to some point at some distance from the edges, but more particularly

so in transverse direction, the zones of highest pressure being near the sides of the

contact surface. According to Martin, near the centre the stress may be actually less

than the tyre air pres$ure.

b. There is a fairly gradual drop to zero at the edges.

Detailed measurements on truck tyres have been reported by Bode (1962], who

reports data such as shown in Figures 1.11-1.14. Additional data has been reported by

Lippmann and Oblizajek [1974] on car tyres. One set of pressure distribution from

their work is illustrated in Figure 1.15. They found (as expected from Figure 1.9b) that

by introducing steering, on one side of the tyre the length increases, and decreases by

a similar amount on the other side of the tyre.

A number of investigations have examined the role of velocity in modifying the

distributions, including the work of Bode as well as the work of 2.akaharov and

Novopol'skii (1957]. In general, the results seem to show that increasing speed causes

an increasing vertical contact pressure at the forward end of the patch and a decreasing

value at the rear portion of the contact patch.

29

Pos.111on .along contilct length.

Fig.1.16. Longitudinal shear stress distribution in the running direction of a tyre under free rolling. [Novopol'skii and Nepomnyashchii (1967)].

RIB NUMBER

~ : o~,---~ DISTANCE INTO ~ -20 PATCH, inches

Fig.1.17. Longitudinal shear stress distribution in the running direction of a biasbelted H sire tyre under traction, 26 psi, O" slip angle, 100% T and R load = 1580 lb. [Lippmann and Oblizajek (1974)].

(2) Tangential Stresses

According to Browne et.al [1982], the direction and magnitude of the longitudinal

tangential stresses of a driving or braked wheel are determined by the sum of the

stresses created in the rolling of a free wheel and the additional stresses created by the

application of a torque. In driving, immediately before the contact the tread elements

are contracted, and they are stretched at the point at which contact is released, as a

result the shear is opposing travel direction. In braking, the situation is reserved, and

the shear is in the direction of travel

30

Acceleration a: 0.96 m/s1

b: l.28m/s1

c: 1.53 m/s1

d: 2.02 m/sl e: 3.22 m/s1

f: 5.66 m/s1

Velocity a: V:4l.7km/h b: V= 17.0 km/h c V•25.8km/h d: V = 17. 7 km/h e: V = 7.4 km/h f: V= 8.9 km/h

Fig.1.18. Longitudinal shear stress distribution in the running direction of a smooth tyre under traction, load 1590 kp, inflation 6.5 atm. [Bode (1962)].

t - .

deceleration a: -0.4 m/sZ b: - 1.23 m/s2

c: - 1.5 m/s2

d. - 2. 45 m/s2

e. - 2. 78 m/s2

f: -4.3 m/s2

Velocity a: V = 17.6 km/h b: V = 17. 7 km I h c: V= 16.9 km/h d: V = 15.4 km/h e: V • 16.9 km/h f: V = 17.2 km/h

T' 5·0o 50 100 150 200 250mm

L£NGTH ALONG CONTACT

Slip Sb= 1.2 °lo Sb • 3.0% Sb • 4.8% Sb • 6.9% Sb •IQ.I% 5b = 19.3%

Fig.1.19. Longitudinal shear stress distribution in the running direction of a smooth tyre under braking force, load 2610 kp, inflation 6.5 atm. [Bode (1962)].

The form of longitudinal shear stress in a free rolling tyre without steer angle has

been studied, such as by Novopol'skii and Nepomnyaschii [1967] and Lippmann and

Oblizajek [1974]. Their results are shown in Figures 1.16-1.17. Similar results are

obtained by Markwick and Starks [1941], who state that the distribution on a

longitudinal section is somewhat similar for a standing tyre, as it is caused by the

bending of tyre (carcass) with consequent symmetric strain distribution.

Extensive experimental work on the longitudinal component of shear stress has

been published by Bode [1962], such as shown in Figures 1.18-1.19. It can be seen

that the influence of braking or tractive forces is to throw the major part of this

longitudinal component to the rear of the contact patch. Even moderate values of such

forces are sufficient to completely change the stress distribution.

31

0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 0.9 ID I.I 1.2 SIDE SLIP (inchn)

Fig.1.20. Lateral force intensity along the contact spot, at slip angles up to 12°. [Browne et.al. (1982)].

If a tyre is caused to move at a slip angle, that is in a direction different from the

straight ahead, it will develop a ground frictional force resisting that slip. A pneumatic

tyre adheres to the road in the front part of the contact, where it accommodates the

slip angle by lateral deflection. Sliding is confined tQ the rear part of the contact,

where the elastic forces overcome friction. For this reason, the contact area can then

be separated into two regions. As stated by Browne et.al [1982], the first region is one

of essentially static contact between tyre and the road swface, but nevertheless a

region in which secondary slip may exist; whereas the second region, adjoining the

first, is the region of primary slip where the tangential swface stresses exceed the

local frictional stresses available. In the first (adhesion) region, since the stresses in

the contact are simply related to the deflection, it is expected that the lateral stress is

gradually increased until it meets the second (sliding) region, where after that the lateral

stress is decreased following the decrease in the available friction stresses. A typical

of lateral (cornering) force intensity is given in Figure 1.20

Tyres are often required to transmit lateral and longitudinal forces to the ground at

the same time, a common instance being driving wheels in a curve. Simultaneous

lateral and longitudinal slips alter the relation of the lateral and longitudinal forces

with their own slip. This mutual interference is pronounced at large slips when a

substantial part of the contact is sliding because the total available force p must be

shared by the two forces. A simplified example of stress distribution due to

simultaneous slips is shown in Figure 1.21 [Schallamach and Grosch (1982)].

32

X X

a

Totol Traction Lateral Traction

X X (bl

Fig.1.21. Lateral force intensity: (a) due to slip (t.), and (b) due to simultaneous side slip and circumferential slip (fe). [Schallamach and Grosch (1982)].

1.4. Surface Texture Description and Measurement

1.4.1. Description of Surface Texture

The skid resistance which pavement material can provide is strongly dependent on

the textural characteristics of the surface. The term "surface texture" in this context

does not mean the composition of the surface (so much binder, sand, etc), but the

geometrical form of the road surface. The complex and random nature of a road

surface makes it difficult to represent the surface characteristics by a general single

parameter. Posey [1946], for example, suggested that 3 parameters for a representative

length of profile (i.e. histograms of the profile itself and of its slope and curvature),

give sufficient information to permit a complex characterization of the texture;

whereas Moore [1965] attempted to quantify the "feel" of a texture by expressing its

geometrical features as size, spacing and shape factors. In addition, Myers [1962]

listed a new series of single elementary parameters to define texture, but carefully

indicated that depending on the particular application each of the new parameters

might be considered more useful than the others. For example, the RMS (root mean

33

square) of the second derivative of the profile (i.e. its degree of curvature at peaks, or

sharpness) would be most appropriate to determine the degree of wear which a surface

has undergone. In the following, the parameters which are commonly used for

characterization of surf ace texture, will be described in details.

(1) Texture Depth

There is a general trend toward increasing friction coefficient and skid numbers

when deeper surface textures are encountered. However, the typical scatter of the data

about the line of best fit would be of questionable value. Similar results are also

obtained when the texture depth is correlated with speed gradients.

The lack of a definitive relationship between texture depth and both skid number

and speed gradient was attributed, in part, to the possibility of poor test repeatability

[Doty (197 4)]. On the other hand, Lees and Katekhda [197 4] stated that there is no

justification in relating the average texture depth to the drop of friction with speed,

since unconnected voids in the surface are included in the measurement while they

play no part in dissipating the water between the tyre and the road surface. They

argued that disconnected voids act on surface as reservoirs retaining water which aids

the lubrications of nearby particles and prevents full deformation of the tyre into the

road texture.

The average depth, however, is useful for broad classification of surface texture. In

France, for example, texture depth (ID) is classified as: very fine (ID :s; 0.2 mm), fine

(0.2 mm < ID :s; 0.4 mm), medium (0.4 mm < ID :s; 0.8 mm), coarse (0.8 mm < ID :s;

1.2 mm), and very coarse (ID > 1.2 mm). Pavements with very fine-textured are to be

prohibited, while very coarse-textured are used in special cases: danger zones

following a straight line or frequent frost zones [Elsenaar et.al (1977)]. The other

versions of average depth, described in the next section, are shown in Figure 1.22.

(2) Centre Line Average Height (CLA)

The CLA is the measure used in BS 1134: 1972 to specify the fineness of the

surface finish of machined pieces. It is defined as the average value of departure of

the profile from its centre line, whether above or below it. The centre line is defined as

a line conforming to the prescribed geometric shape of the profile and parallel to the

general direction of the profile throughout the sampling length such that:

34

Fig.1.22. Mathematical definition for numerical evaluation of surface texture. [Marian (1962)].

r ydx (y~O) + r ydx (y<O) = o

The Cl.A then is given as:

hCLA = ~f ydx

where y = f(x) is the equation of the profile (see Figure 1.22b).

(3) The Average Depth

The average depth is used (in Europe) as an equivalent of Cl.A in Britain.

Referring to Figure 1.22c the average depth from the crest line is given by:

hAVR = ~f ydx

where y = f(x) is the equation of the profile.

( 4) Root Mean Square

The root mean square height of a profile is given by (see Figure 1.22b):

h_ = j ~fy'dx

This measure accentuates the effect of sharpness and distinguishes between rounded

and sawtooth type textures.

35

Line DI peaks i----.,o~-....... ---#'~--.... -.-= ....... -_:--""-<-~....,_-----,_,...~..----+1s,. ot points -- llnel lf'--___,;iiif,--~1--f--;;.._~ _ _,_ _ _;_.......;:~-,..~~+,-~-+Line C11 depth D bel- peaks

b

Profile ratio • total 11119111 DI prDlile o ..,.glll of base lilw b

·Filtered profile ratoo • w dotllh D n 9o = ...,. of letlpths of profile - deplll D • nf••

,._ of baseline ,_,,..

It,,. 1

Fig.1.23. Profile ratio and "filtered profile ratio". [Sabey (1968)).

8~ s2 s, Reference plane

Fig.1.24. Moore's bearing area method [Moore (1975)).

(5) The Swedish Standard

The Swedish standard is numerically greater than the CLA height, average depth or

root mean square height. The two lines x and y are positioned so that lJ> = K1L and

I.V = KiL (Figure 1.22a). The Swedish standard specifies that K1 = 0.05 and K2 = 0.10. The distance between the parallel line x and y is the measure of surface

roughness.

( 6) Profile Ratio

The measure is the ratio of the profile length to the projected length. This bears

some relation to texture depth and takes into account the shape of the profile [Sabey

(1968)), but does not give any account of the three-dimensional aspect of the surface

texture [Lees and Katekhda (1974)). In addition, the "filtered profile ratio" which is

the profile ratio for the tops of the asperities only, over a different depth D, below the

line of the peaks can be evaluated (see Figure 1.23).

36

(7) Moore 's Bearing Area Method

Moore [1975] used a mathematical expression to represent a single profile. A series

of fiction parallel planes are drawn successively below the reference plane at

distance 61, 6z and so on, so that they intersect different number of asperities N1 (see

Figure 1.24). Two equations may be written:

Ni= Co 6m

Ai= C1 + ½ 6D

where Ni is the number of asperities intersected by the ith plane located at distance

from the reference plane,

Ai is the total contoured area of asperity intersected by this plane,

Ci is the plateau area,

C0, m are constants specifying the height spacing or statistical distribution of the

asperities,

<;, n are constants specifying the mean shape of asperities.

He found, for example, that on concrete road surface: m = 2, n = 3.

(8) Mean Width of Surface Voids

The parameter can be obtained from the asperity density prints. It was found to

correlate with the steepness of the friction speed curve, the closer the surface the

steeper the negative sloped curve [Schulze and Bechmann (1962)].

(9) Mean Hydraulic Radius (MHR)

Moore [1966] measured the drainage capacity of a surface in terms of the mean

hydraulic radius:

MHR = M Vv /tN°·5

where M is the instrument constant,

vis viscosity of water,

t is the recorded time for a fixed volume of water to drain,

N is the number of asperities per square inch of surface texture.

He then used the MHR to predict the wet sliding coefficient of road surf ace.

37

b

m

h

e, f, g & h lie on plane of best fit rtrJ MAXlMUM SLOPE : Angle which triangular planes

a b d , bed ctc make with plane efgh

ABSOWTE SLOPE in DIRECTION of SLIDING :

ldhehael

AREA of SURFACE over· PROJECTED AREA:

abd+ bed efgh

a.

n 0 p

n,o,p & q lie on plane of best fit

CURVATURE: ko-~

no

SIZE of INFLECTION:

ko--~ _Ip-~ no op

b.

Fig.1.25. Definition of surface characteristics. [Yandell (1969)].

Both profiles have similar average absolute Slopes and hysteretic sliding resistances

The lengll'lof drainage paths d & D vary inversely with the r.imb•rs of inflections pH unit length

a.

-- --

Both profiles have similar average absolute slopes

Th• lower profile has fewer inflections per unit length and a lower theoretical hysteret,c lnct,on

b.

Fig.1.26. Influence of the number of inflections. [Yandell (1969)].

( 10). Parameter used by Yandell

Yandell [1969] represented the pertinent characteristics of the

road surface texture by the following parameters:

a. Texture depth.

b. Maximum slope.

38

q

c. Average absolute slope in the direction of sliding.

d. Surface area per unit projected area.

e. Distribution of surf ace.

f. Curvature.

g. Number of inflection.

These parameters can be used for many purposes, such as on the calculation of the

hysteresis friction, the polishing of road stones, and the abrasion of rubber. Some of

them are illustrated in Figures 1.25-1.26. One important parameter, the average

absolute slope, is found to have good correlation with coefficient of hysteresis friction.

It is used with the damping factor of rubber in the mechano-lattice analysis to predict

the coefficient of hysteretic friction.

1.4.2. The Measurement of Surface Texture

The road surface texture is normally categorized by two features, the large scale or

macro-texture which represents the easily visible asperities in the surface and the fine

scale or micro-texture which describes the harshness or state of polish of the stone

surface. (See Figure 1.27).

The method of measuring the surface texture largely depends on the application in

question. Over 20 methods have been used for measurement of surf ace texture, and it

is interesting to note that tests using these methods, have not, on the whole, produced

consistent or reproducible quantitative results [Rose et.al (1973)]. Some of the

important and commonly used methods will be described below; which can be broadly

classified into four categories [Taneerananon (1981)].

a. Volumetric: Sand patch, Grease smear, and Silicone putty.

b. Profile: Texturemeter, Row of needles, Stylus, and Profilograph.

c. Photography: Stereo-photogrammetry, and Stereo-photo interpretation.

d. Miscellaneous: Outflow meter, Surface prints, Laser beam, and Texture friction

meter.

( 1) Sand Patch

It involves spreading a known volume of sand over a circular area until flush with

the tips of the asperities. Average texture depth, the ratio of volume to area, is the

39

road surface macro micro

rough harsh

rough polished

smooth harsh

smooth polished

Fig.1.27. Illustration of terms of the road surface texture. [Sabey et.al (1970), Schlosser (1977)].

measure of surface texture. It is rather simple test, but it evaluates macrotexture only

(the size of the sand used prevents the very fine channels from being measured).

(2) Grease Smear

The principle is same with the sand patch but the grease is applied instead of sand.

The method is used usually for obtaining the texture depth of fine textured road

surfaces, as it has better ability than the sand to fill the narrow channel of the surf ace

[Smith and Fuller (1969)].

(3) Silicone Putty

It is similar in principle to the sand patch and grease smear method. A known

volume of silicone putty is formed into an approximate sphere and placed on the road

surf ace. A recess in a plate is centered over the putty, and the plate is pressed down in

firm contact with the surface. Average diameter of the deformed putty is recorded.

When tested on a smooth flat surface with no texture, the silicone putty will

completely fill the recess. Therefore, a decrease in the measured diameter indicates an

increase in texture depth. This procedure was used by researchers (at Texas

Transportation Institute) to evaluate macrotexture [Rose et.al (1973)].

40

( 4) Texturemeter

The instrument, developed at Texas Transportation Institute, consists mainly of a

series of evenly spaced, vertical, parallel rods mounted in a frame [Rose et.al (1973)].

All except two rods can be moved vertically against spring pressure and independently

of one another. One rod at each end of the device is fixed to the frame for support.

Each movable rod has a hole through which a taut string is passed. One end of the

string is fixed to the frame, and the other is tied to the spring loaded stem of a 0.001

inch dial gauge extensiometer mounted on the frame. When the frame is pressed onto

the road surface, any irregularities in the surface will cause the string to form a

zig-zag line which will give a dial reading. On smooth surface the reading is zero,

therefore the coarser the surf ace the higher will be the dial reading.

(5) Row of Needles

A row, 15 cm long of closely spaced needles guided by a frame is dropped

vertically onto the road surface. The profile measured as depicted by the tips of the

needles is then photographed. The accuracy is limited by the thickness and spacing of

the needles [Astrov (1962)].

(6) Stylus

The method is well known and widely used in mechanical engineering. It gives

more information about a surface than other methods for most surface types [Richards

(1967-68)]. Moore [1966] developed a stylus device to measure the coarse texture of

the road surface. Vertical movement of the stylus are sensed by an electric linear

variable differential transformer and fed into an oscillograph. A similar apparatus was

designed by Yandell to measure the fine texture of the road asperities [Taneerananon

(1981)].

There are, however, some difficulties associated with the stylus method. The

conical shape of the stylus prevents reentrant angles from being detected. If the radius

is large fine crevices will be missed, measurement of troughs and peaks will be

inaccurate. Too fine radius may cause damage to the surface by ploughing. An

optimum angle and sharpness then must be selected, to obtain a compromise between

the conflicting requirements of not ploughing (or sticking) and true reproduction of the

surface profile.

41

(7) Profilograph

This instrument is designed to scribe a magnified profile of road texture as a feeler

probe is drawn across the surface [Rose et.al (1973)]. A mechanical linkage system

magnifies vertical movement of the probe, and the resulting profile is recorded on a

chart. In addition, upward vertical deflection of the probe are recorded on a counter as

the cumulative vertical peak heights of the surface texture through the length

transversed by the probe. Average peak height is obtained by dividing cumulative

peak heights by the number of peaks.

(8) Stereo-Photogrammetry

Two photographs of the macro-texture taken vertically from two distinct points give

sufficient information to produce stereo-photograph which are measured with a

comparator and parallel bar. The height readings are accurate to 0.01 inch [Sabey and

Lupton (1967)]. Yandell and Gopalan [1976] used stereo-pairs from a scanning

electron microscope to measure very fine texture.

(9) Stereo-Photo Interpretation

This method was developed by Schonfeld [1970,1974]. Colour stereo-photographic

transparencies or prints of approximately 6 inch square sections of road surfaces are

obtained and viewed through a micro-stereoscope or mirror-stereoscope. Texture

elements of the surface are classified visually and are rated subjectively according to

an established severity rating for each of several parameters.

( 10) Outflow Meter

Moore [ 1965] described a simple apparatus by which could be measured the

drainage capacity of a surface in terms of MHR (mean hydraulic radius). The

instrument is a transparent cylinder, about 5 inch diameter and 12 inch height with a

rubber ring glued to the bottom face. The cylinder is loaded onto the road surface so

the rubber ring will drape over the aggregate pieces in a way that simulates the

draping of tyre tread. There is no pressure applied on the water except its weight. The

time taken for a known volume of water to drain away is recorded. The short duration

of time or high rate of flow is associated with high macro-texture or high permeability

of the pavement or both.

42

A theoretical relation between MHR and the slope of friction/speed lines was

obtained using stated assumption [Moore (1966)]. The method shows good

discriminating ability [Orchard et.al (1970)]. However, the disadvantages of this

apparatus are: (a) It uses a thin rubber ring which might be affected by an odd particle

or cavity in a manner not representative of drainage in a typical tyre contact patch

[Lees and Katekhda (1974)]; (b) The variability of the readings on very smooth

surfaces is high [Moore (1968)].

A number of modifications have been made to the outflow meter. The high pressure

outflow meter, differs from the original device in that width of the rubber ring has

been increased, time measurement has been automated and the water is pressurized as

opposed to gravity flow [Henry and Hegmon (1975)]. A modification incorporating an

elliptical rubber plate in place of the circular rubber annulus, is aimed to study the

drainage paths lengths for different directions [Lees and Katekhda (1974)].

( 11) Surf ace Prints

Meyer [1964] obtained the asperity density prints by placing an aluminium foil on

the surface and providing a controlled impact onto a rubber disc placed on top of the

foil. The sharp asperities pierce the foil and the number of piercings per unit area is

the measure of texture.

(12) Laser Beam

This method was described by Gee et.al [1975]. The principal elements are laser

source and receiver. Both are off-the-shelf items. Light is emitted from the laser and is

incident on the road surface. The light reflected from the surface is generally scattered

in all directions. The polarization (alignment of the electric field vector) is also

changed after scattering. That is, linearly polarized light will experience

"depolarization", where reflected light is no longer linearly polarized, but is elliptically

polarized. The degree of ellipticity is a function of road surface characteristics, and is

represented by the ratio of the minor and major axes of the polarization ellipse. The

higher ratio indicates coarser texture.

The method appears simple and apparently suited for operation from a moving car.

The correlation coefficient between laser depolarization ratio and skid number,

however, is only about 0.5-0.6 [Gee et.al (1975)]. Another apparatus, the Swedish

43

Laser Road Surface Tester (RST), can measure both fine and rough macro-texture,

where the RST root mean square fine macro-texture is correlated far better with sand

patch texture depth than rough macro-texture. When the result of RST is compared

with the measurement by the Sideways Coefficient Routine Investigation Machine

(SCRIM), no significant correlation was achieved [Jameson et.al (1988)].

(13) Texture Friction Meter

Yandell and Mee have developed a device, that is a small computer controlled

instrument that can accurately sample pavement surface texture to give in seconds the

sideways and locked-wheel friction for several travelling speeds. It analyses a

television image of a sharp knife shaped laser beam which shines on the road surface

as the test vehicle moves along the road. The computer simulates a pneumatic tyre

travelling on that measured wet texture ['Uniken' (1989)].

1.5. Hydrodynamic Effects and Tread Rubber

To study quantitatively the masking effect of the fluid film on tyre-road friction, a

knowledge of the fluid film thickness is necessary. The effective film thickness which

exists between the tyre and road surface during sliding or rolling is different to the

applied film thickness. The Reynolds equation is commonly used as the basis for the

calculation of effective film thickness. According to Yandell et.al [1983], this film

thickness is dependent on the mode of sliding (sideway or locked wheel), the speed of

travel, and the temperature of the water.

1.5.1. Reynolds Theory of Lubrication

Reynolds [1886] applying the Navier-Stokes equations derived what is known as

the Reynolds Equation. In its original form this equation is written as:

!(h':) + !0':) = 6v[ ~0 +U,): + 2v1 ]

where h : fluid film thickness

44

p : fluid film pressure

v : absolute viscosity of fluid

V 1 : velocity of vertical separation

U0, U1 : horizontal velocities of lower and upper surfaces

x, z : coordinate axes on the same plane.

The equation can either be solved by analytical or numerical methods. The

analytical or closed form solution requires the film thickness to be expressed as a

function of x and y. However, as the two dimensional equation can not normally be

integrated directly, one dimensional equation is usually solved by assuming an

infinitely long or short bearing. The pressure p is then obtained as a function of x. The

numerical methods, such as the finite difference and the finite element technique are

common with the aid of digital computers.

1.5.2. Fluid Film Thickness

Various investigators have used different water film thicknesses in their

experimental and theoretical studies. Daughaday and Balmer [1970] used an effective

film thickness of 0.02 inch (0.508 mm) in the thick film region and a thickness of

0.005 inch (0.127 mm) in the semi-dry region. Film thicknesses between 0.03 inch

(0.762 mm) and 0.08 inch (2.03 mm) were used by Allbert and Walker [1968] in their

laboratory test. The commonly used water film thickness in field measurement is 0.02

inch since beyond this thickness the friction tends to level off as was found by Giles

[1959] and Besse [1972]. The internal watering system as specified by the

ASTM-E274 also gives water film thickness of approximately 0.02 inch.

In field tests, however, film thicknesses are difficult to measure with a high degree

of accuracy. This is more difficult on a coarse textured surfaces since the film is non

uniform and the depth varies from top to bottom of asperities and from one asperity to

another. Thus for field tests the water film thickness refers to the applied or nominal

film thickness. The minimum film thickness of 0.0002 inch (0.0051 mm) is assumed

by Taneerananon [1981] in the theoretical calculation of the effective remaining film

thickness between the tyre and road surf ace.

45

1.5.3. Hydroplaning

Hydroplaning can be described as a phenomenon when the tyres of a vehicle

travelling at some critical speed are detached from the wet pavement surface. As a

result, the ability of the tyres to develop braking or cornering forces is drastically

reduced. On a flooded surface if the vehicle speed exceeds the tyre hydroplaning

speed, hydroplaning will occur and control of the vehicle is completely lost. The

hydroplaning speed is a function of the pavement surf ace, the water depth and various

physical tyre parameters. The depth of water on the pavement required for

hydroplaning can vary considerably depending on the particular combination of the

type of road surface texture and the tyre tread design. Patterned tyres operated on

coarse textured surfaces require the greatest depth, while smooth tyres operated on

smooth road surfaces require the least water depth. Hydroplaning can be further

classified into two groups namely dynamic hydroplaning and viscous hydroplaning.

(1) Dynamic Hydroplaning

This situation generally occurs on a flooded surface. The fluid inertia force is the

important factor in this type of hydroplaning. As the moving tyre contacts the

stationary fluid, it causes a sudden change in the momentum of the fluid which in turn

creates hydrodynamic pressure which acts on the surface of the tyre. The

hydrodynamic pressure tends to increase with the square of the tyre speed. A critical

speed is reached when the hydrodynamic pressure force equals the tyre load. This

speed is called "tyre hydroplaning speed" and can be computed by the semi empirical

formula put forwards by Home and Dreher [1963].

This formula ignored the effects of tyre tread design, fluid viscosity and road

surface texture. It was given as:

VP = 10.35./p

where VP = Tyre hydroplaning speed (mph),

p = Tyre inflation pressure (psi).

46

(2) Viscous Hydroplaning

Viscous hydroplaning occurs when there is a thin film of fluid present on the road

surf ace. The internal friction or shear force between the fluid layers prevents the fluid

from escaping from under the tyre footprint. Viscous forces arising from the viscosity

of the thin fluid is dominant in this type of hydroplaning. The pressure force which

develops while the tyre is sliding over the contact patch is calculated by means of the

Reynolds Equation. This pressure force tends to separate the tyre from the road

surface and reduce the amount of friction being generated. The eliminating of the

intervening film is rather difficult. In normal rolling or sliding, it is eliminated by the

squeezing action of the tyre onto the surf ace asperities.

1.5.4. Models for Calculation of Water Film Thickness

Saal [1936] studied the slipperiness of the road by using the sinkage equation

originated by Reynolds. Assuming the contact area of the tyre to be of a flat elliptical

shape, he obtained a reduced water film thickness after a rolling tyre had passed

through the wetted road surface. The duration of time of sinkage was taken as the time

required for the contact segments to travel past the footprint region. The amount of

friction that could be developed was then dependent on whether the height of the

asperities was greater than the residual film thickness. The model, despite its

simplicity gave an insight to the role of the remaining water film in the development

of friction on wet surf ace.

In his study of viscous hydroplaning, Moore [ 1967] used asperities of sinusoidal

shape to represent a pavement surf ace. Individual asperities generate pressure forces

which provide a net uplift tending to separate the sliding rubber from the asperity

surfaces. He divided the film into three region: inlet, central (or foil bearing), and

outlet. Using Reynolds Equation in two dimensional form, the pressure forces were

calculated for the inlet and outlet regions, the central region had constant pressure. He

attributed the main source of friction to the adhesion between microtexture at the

asperity tips and the tyre rubber providing that the micro-texture depths exceed the

thickness of the foil bearing film. The model was applied to a rolling tyre which is

accelerating on a wet surface.

47

Daughaday and Balmer [1970] used a mathematical model to study the rolling and

skidding behaviour of a tyre. The flow of water was divided into: an exterior flow

region, an inlet region, and a footprint region (consists of thick film region and semi

dry region). The amount of thick film penetrate into the footprint region determined

condition of the hydroplaning. In the semi dry region each individual load carrying

asperity was either in dry contact with the tyre surface or the fluid film thickness was

small compared to the depth that the asperity indented the tread surface. For partial

hydroplaning case, the main source of friction was derived from the contact of road

asperity tips with the tyre surf ace in the semi dry region. The other source of friction

was from the viscous forces acting in the thick film region. The model assumed dry

contact exists and thus considered adhesion as the major source of friction. It would

appear that the theoretical friction coefficient depends significantly on the estimation

of the coefficient of friction of the semi dry region.

Taneerananon [ 1981] presented models which using Yandell' s Theory of Hysteretic

Sliding Friction [Yandell (1971 b)] attribute all the friction generated to the hysteresis

process. Adhesion is not considered because of the ever presence of the water film or

contaminants on the texture surf ace. Using Reynolds Equation he developed models

for the sliding tyre (simulation of locked-wheel friction), and for rolling tyre

(simulation of sideway friction). Together with the surface texture measurement and

tread rubber properties, he used these models to predict the locked-wheel BFC, and

(maximum) SFC. Each of these models will be described below.

( 1) Model for Sliding Tyre

The main purpose of this model is to obtain the fluid film thicknesses which are . important in determining the masking of the surf ace texture. The model assumes a one

dimensional flow of fluid in the direction of sliding motion of a tyre. No flow is

assumed to take place in the direction at right angles to the tyre motion. The sinkage

of the tyre takes place simultaneously with the sliding, and the solution has to be

obtained by trial and error method.

The following assumptions were made in the calculation.

a. The tyre surface is parallel to the asperity surface when calculating the film

thickness.

b. The resultant pressure force acts through the centre line of the asperity.

48

c. The flow is laminar and the inertia forces are negligible compared to the viscous

forces.

d. The asperity swface is either smooth or rough.

The first assumption appears reasonable considering the film thickness relative to the

dimension of the asperity. In the second assumption, the actual resultant would act

slightly off the centre line towards the thin end of the film, however owing the relative

smallness of the asperity, the assumption would seem to be justifiable. Taneerananon

examined the third assumption using formula from Pai [1956], and he found that the

inertia force can be ignored. To fulfil the fourth assumption, the effect of micro

texture on the final film thickness then is computed for a range of values of sinkage

time [Taneerananon and Yandell (1981)].

(2) Model for Rolling Tyre

A model of a tyre sinking onto the asperities surface is used to compute the film

thickness. When a wheel is rolling at an angle to its direction of motion, a sideway or

lateral force is developed (due to lateral slip), which acts perpendicular to the wheel

plane. Since there is little or no relative motion between the rolling tyre and the road

swface at the interface, only the sinkage of the upper tyre surface will be considered

in this model. Taneerananon assumed that the hydrodynamic uplift which may be

generated as a result of lateral sliding is small and can be safely ignored. Hence the

model of sinkage is similar to the sliding tyre, except that the tyre motion is sinking

instead of sinking+ shearing. It is clear, that as the speed of travel increases, the time

available for sinkage decreases so that the water films will remain at greater thickness.

One example of relationship between time of sinkage and water film thickness,

derived from Reynolds equations, is shown in Figure 1.28. The effect of tyre tread . (smooth, patterned) is taken into account through the magnitude of fluid film pressure.

During locked-wheel friction, sinkage and shearing are taking place simultaneously.

As the shearing rate increases, an existing water film will increase in thickness for a

given normal contact pressure [Yandell et.al (1983)]. By just referring to this

consideration, it can be expected that the braking coefficient is lower than the sideway

coefficient.

49

mm in

·04

·OJ

·02

01

0

10·' inches

,::;

.. .. • C ,. u

15

i 10 ~

E

E :,

E C

:J: 5 cylindr,coL shape roughness ., depth .ex,, ~ l

:•·;.·.·,-·_,_.

0 ._ ___ ____,..__ ___ ____, ____ ____. ____ ___,.,,---

0 5 ~ ~ ~

Time of sinka.11• ( • 10-1 secs)

Fig.1.28. The texture effect on film thickness as a function of time. [Taneerananon and Yandell (1981)].

1.5.5. Measurement of Tread Rubber Properties

The main tread rubber property to be measured is the damping factor, which is

defined as the area inside the stress-strain hysteresis loop divided by the area under the

loading curve in an unconfined triaxial load-unload test on the tread rubber with

homogeneous stress conditions .

. A number of methods have been used to measure this property, including the

rebound pendulum, which measures the energy returned divided by the energy applied

to the rubber from a hemispherical anvil. Another method involves measuring the

bounce height of a steel ball on the tread rubber. According to Yandell et.al [1983],

most of the methods used are not suitable since (1) they do not load the rubber over

the frequency range encountered by the skidding tyre, and (2) the pulsing stress

pattern is not homogeneous.

50

Temperature

07

! .Q

0·6 2 0

~ u 2

"' C ·o. 05 E 0 0

IO 20 40 50mph

Equivalent slidi119 speed (otter Zonkinl

Fig.1.29. Measured relationship between the damping factor of filled tread rubber, speed, and rubber temperature. [Yandell et.al (1983)].

To overcome those disadvantages above, Zankin [1981] constructed a special

apparatus for measuring the damping factor behaviour of tread rubber for speeds up to

80 kph and temperature up to 15C>°C. Figure 1.29 shows the relationship between

sliding speed, rubber temperature, and the damping factor of a filled 60:40 styrene

butadiene rubber (SBR) - polybutadiene rubber (BR) of tread rubber, with about 33%

high structure carbon black and a Shore A durometer hardness of 66. This figure was

used in road friction prediction.

Another rubber property to be measured, that is when calculating the effect of

rubber mass inertia to friction coefficient, is its density. This effect is significant over

about 40 kph. At low sliding speed the increase of friction coefficient with increased

density is minimal. However, a 3% increase is obtained for 80 kph, when the density

increases from 0.033 lb/cu in to 0.038 lb/cu in [Taneerananon (1981)].

51

1.6. Theory of Hysteretic Sliding Friction

Controversy exists between those attributing the greater part of skidding resistance

on wet surfaces to inter-molecular adhesion and those attributing it to stress-strain

hysteresis in the rubber. Tabor [1952] demonstrated the connection between the

rolling resistance of rigid bodies on rubber and the damping properties of the rubber.

Greenwood et.al [1961] maintained that the increased coefficient of friction observed

as the apex angles of water lubricated cones decreased, was due to the damping

energy to the sliding rubber.

Moore [1967], using the elasto-hydrodynamic theory of lubrication and Tabor's

hysteretic prediction, attempted to demonstrate that hysteresis played only a minor roll

in the skidding resistance of an automobile tyre on a wet road. Kummer [1966]

presented one semi empirical formula for predicting the hysteretic friction and another

for predicting adhesive friction. The adhesion was presented as a hysteresis

phenomenon.

Yandell [1968] presented a mathematical mechano-lattice analogy for predicting the

coefficient of hysteretic friction of rubber sliding on simple geometric asperities. Next

Yandell [1969] proposed and proved a method of applying the results of his

mechano-lattice analogy to the prediction of the coefficients of hysteretic friction of

rubber sliding on lubricated roadstones.

Taneerananon [ 1981] found that in most cases the prediction of hysteretic friction

agrees reasonably well with the measured locked-wheel coefficient when the effects of . the temperature are taken into account. For the sideway force coefficient, the

prediction under estimated the measured coefficient. He explained that the damping

factor of rubber may have increased as a result of the mastication but has not been

accounted for in the prediction.

In the following, brief descriptions of the theory of hysteretic sliding friction will

be presented [Yandell (1971b), Yandell and Holla (1974), Yandell et.al (1983)].

52

1.6.1. The Mechano-Lattice Analogy

Yandell used his mechano-lattice analogy to simulate and analyse the stresses in

and determine the friction of rubber as it slid over triangular prisms or cylinders. The

analogy consisted of an array of connected units designed to simulate the behaviour of

a long section of the sliding rubber in plane stress. In order to give a rigorous

simulation of the rubber the analogy was arranged to allow for large deformations,

possess an appropriate Poisson's ratio and Young's modulus and a damping factor of

a magnitude commensurate with the type of rubber, temperature and speed of sliding.

The analogy was shown to be reasonably accurate when coarse grid predicted

coefficients of friction were compared with those measured by a British Pendulum

friction tester on smooth brass triangular asperities with various apex angles [Yandell

(1969)].

The extent of load-deflection hysteresis, during the single point cyclic loading of a

rubber, is specified by a damping factor. This factor is defined as the energy dissipated

divided by the energy applied in one complete cycle of load, or, referring to Figure

1.30, the dotted area of the hysteresis loop divided by the hatched area under the

loading path of the loop. An applied or measured load-deflection hysteresis loop is

made more mathematically useful if simplified, that is replacing it with a

parallelogram-shaped loop of similar area as shown in Figure 1.30. A model,

consisting of an elastic and a frictional element in parallel, which yields the simplified

hysteresis loop is also shown in Figure 1.30.

The relation between damping factor to the Young's moduli and maximum internal . friction is given by

~ = {(p/Ee)-(p2/EfS)}/{(p/EeS)-(p 2/2EfS)+(l/2EeS)}

where p = internal friction coefficient= F'/(t'emax - t\min)

S = stiffness coefficient of the model.

It can thus be seen that for a given rigidity the damping energy per cycle depends on

the internal friction, be it viscous or Coulomb, and on the maximum load range. The

model described above is the basis for the behaviour of the individual elements

comprising the rubber simulating units.

53

Deflection a. Simplified hysteresis loop

b. Model exhibiting behaviour of simplified loop ~ = damping factor= dotted area/hatched area E0 =elastic Young's modulus a::.a/c £,=frictional Young's modulus a::.b/c F' = maximum internal friction.

Fig.1.30. Simplified of measured hysteresis loop. [Yandell (1971b)].

HU,J) -3-h (I,J)

~ _L ---:,

A = horizontal and vertical elements V =volume elements; S=shear elements E~ = elastic stiffness factor E, = c · E~ frictional stiffness factor; c = a constant.

a. Analogue of mathematical model

VCl,J)~ y- f 0

I,J l,J +1

I 1r~vU,Jl

--0'1-

I -- (;

0 I +1, J I +1,J+1

b. Movement of analogue joints

Fig.1.31. A unit of the mechano-lattice analogy. [Yandell (197lb)].

1.6.2. Mechano-Lattice Unit

I I I

Figure 1.31a shows the mechanical analogue of one of 264 mathematical units

forming the model used for simulating the plane stress behaviour of a long section of

rubber. Each unit is made up of eight elasto frictional elements similar to that shown

in Figure 1.30. There are two horizontal, two vertical, two shear and two volume

elements. The volume and shear behaviour are separated by two crosses to rotate. This

is necessary to allow any Poisson's ratio (a).

54

When volume changes are occurring, the shear cross rotates with no additional

loading on its elements. When shear changes are occurring, the volume cross can

rotate and no additional load is put upon its elements. Shear strains in a unit are

functions of differences in diagonal lengths and volume strains are functions of

changes in the sum of diagonal lengths.

The stiffness factors of the elastic components of the elements are calculated

assuming that no friction is present. McHenry [1943] was one of the original

proposers of the simple lattice analogy for simulating continua with a Poisson's ratio

restricted to 1/3 and obeying the theory of linearised elasticity. He calculated the

stiffness factors of a simple cross braced lattice simulating the behaviour of a

non-buckling plate of unit thickness. The (elastic) stiffness factors are:

(E)))/2(1 +cr) horizontal and vertical elements

{E))V2)/(1-cf) volume elements

(E)))/(l+cr).../2 shear elements

1.6.3. The Friction of Rubber Sliding on an Asperity

A total number of 264 units shown in Figure 1.31 are connected at their joints to

simulate the long section of rubber sliding with plane stress as shown in Figure 1.32.

All boundary joints except those on the lower edge are fixed. About 14 of the central

bottom joints are deflected to the shape of the single fixed smooth asperity. The apex

joint is fixed to but is free to move with the asperity. The joints contacting the asperity

are free to move along its surface. The body joints are all free to move in response to

unbalanced forces acting on them.

As can be seen the units to the left of the asperity have been loaded while those to

the right have not. The extreme right-hand column of units is assumed to be unloaded.

As they move across the asperity corresponding elements go through cycles of load

and deflection to generate hysteresis loops similar to that shown in Figure 1.31. The

area of the hysteresis loop is governed by the damping factor of the rubber and the

maximum elastic stress range experienced by the element. A computer program

obeying the properties of the units ensures this. A simplified flow chart of the

computer program is shown in Figure 1.33.

55

X)C)<)(

)( )c )(

)c )( X )<

)< )< )< )c X )c

X )( ><

I Joints on free surface

Fig.1.32. The assembled units simulating a long section of rubber sliding on an asperity. [Yandell (197lb)].

READ: RUBBER:-!.elastkYoung's modulus E.,. frictional Young's modulus £1; Poisson's ratio u; inter-nal frictional coelT. p. ASPERITY :-slope or a/Rad.; penetration a, surface friction; joints at interface.

½

SET UP: simulating grid (9 x 34); boundary conditions and initial deflections of joints.

i COMPUTE: elastic stiffness factors of horizontal, vertical, volumetric and shear elements.

i ...... COMPUTE: lengths, rotations, elastic and frictional forces in elements taking hysteretic behaviour into ,..

account. The maximum force range experienced by each element.

+ RESOLVE AND SUM: forces in elements horizontally and vertically at joints.

! I MOVE: joints horizontally and vertically in response to any unbalanced forces acting on them. Joints in i contact with the asperity are moved along its surface taking surface friction into account.

NO ! DO forces on free joints approach zero?

YES fvEs

HA VE joints at interface been adjusted.

lNo

WRITE : displacements, boundary reactions, stresses and the coelT. of hyst. friction, /h.

Fig.1.33. Flow chart of the analogy computer program. [Yandell (1971b)].

The program calculates and stores the lengths and rotations of all elements. The

elastic and frictional forces are calculated for each element in each stage of loading.

This is done incrementally working from right to left so that the stress history retains

its continuity. The element has gained a frictional as well as an elastic increment of

force by the move and has commenced to follow the loading path of its hysteresis

loop. After a number of additional sequential moves to the left, the frictional force in

the element reaches its maximum F', the less steep elastic line of the hysteresis loop is

56

0-3

COEFFICIENT

CF HYSTERETIC

FRICTION

0·2

0·1

--- CYLINDERS

-- TRIANGULAR PRISMS

OF of 0·5 / Of BO km/hr // with inertia--,,

,,, /

/

,

DAMPING FACTORS~--"(without inertial effect l

AVERAGE SLOPE OF CONTACT

0 O·I 0·2 0·3 0-4 0·5 0·6

0·5

a 07 R

Fig.1.34. Mechano-lattice-detennined relationships between hysteretic friction, damping factors of sliding rubber, and average slope of contact of triangular prisms and cylinders. [Yandell et.al (1983)].

followed until unloading commences and by a reverse process the loop is completed

and possibly recommenced after the element has travelled across the array.

Subroutines in the computer programme ensure this. As their distance from the free

surface increases the size of the hysteresis loops, predictably, decreases.

. When all the forces in the elements have been calculated, each joint is moved in

response to any unbalanced force transmitted to it from the connecting elements.

However, the joints are moved in one operation and in only small increments so that

stability will be maintained. A damping function which is controlled by the largest

unbalanced force in the array is used for moving the joints. Once the joints have been

moved the forces in the elements are recalculated. The procedure is continued until all

the free joints have moved to stable positions and sustain no significant unbalanced

force. The programme checks periodically to ensure that no joint at the asperity

interface is in tension or penetrates the surface of the asperity.

57

scale 3

Fig.1.35. Analysis of measured texture profiles into scales. [Yandell et.al (1983)].

The coefficient of hysteretic friction fh is determined by dividing the vectorial sum

of the horizontal forces acting on the joints which contact the asperity by the vertical

reactions on these joints. The horizontal, vertical and shear direct stresses at the centre

of each unit are calculated together with the major and minor principal and the

maximum shear stresses. The coefficients of hysteretic friction generated by rubbers,

with a number of damping factors, as they slid over isosceles triangular prismatic and

cylindrical asperities with various average slopes of contact, were calculated by the

mechano-lattice analogy, such as shown in Figure 1.34.

L6.4. The Hysteretic Friction of Road Surfaces

Yandell's theory assumes that the total texture of a road surface can be analyzed

into components or scales ranging from microscopic size up to half or one inch.

Figure 1.35 shows how the texture is analyzed into a number of scales.

Rubber is assumed to "flow" over each scale and generate hysteretic friction. The

coefficient of hysteretic friction at a given speed, was the sum of the coefficients on

each scale of texture, as shown in Figure 1.36.

58

_.c. 2.0

z 0 ~ So! a:: u.

!:! I-l&I a:: 1.0 l&I I-II)

?i: u. 0 o.s ~ l&I

i::i it 0 l&I 0 u

depressed envelope due to masking of some scoles by surface water

Peak fridion on scolel

E:::::::::;:::~:::;:=:.......----,,----::,~----::r=-..:.lc~oa.rses tl 10 20 30 40 so 60 70

mile/h

0 10 20 30 40 so 60 70 80 90 100 110 SLIDING SPEED km/h

Fig.1.36. Schematically presentation of the coefficient of hysteretic friction of the road surface. [Yandell and Holla (197 4)].

1.7. Problem Definition and Method of Investigation

1. 7 .1. Problem Definition

As it will be described in the next section. the field measurement of skid resistance

will require large capital investment, while the test results may not be easy to repeat.

Furthermore. the testing done at high speeds or at difficult locations such as sharp

curves and steep gradients may be dangerous. Therefore. any reliable method of

p~edicting the coefficient of friction from surface texture will be useful and desirable.

Yandell [1970] devised a means for predicting the coefficient of friction between a

rubber slider on road surface. He attributed all the friction to the hysteretic losses in

the rubber. He used the mechano-lattice analogy to calculate coefficient of hysteretic

friction for a range of surface texture and rubber characteristics. From the measured

surface characteristics he obtained the coefficient of hysteretic friction for the divided

scales of texture, the sum of which is the coefficient of friction of the surface. Yandell

was able to verify his theoretical coefficients by comparing with the values measured

with a British Pendulum tester.

59

Holla [1974] used Yandell's method for calculating coefficient of friction and his

empirically constructed damping factor versus frequency curves to predict the sliding

coefficient of friction on a road surface. Assuming the viscoelastic properties of

rubber to be the dominating factor, he computed different maximum coefficients of

friction and different frequencies of loading from the average slope and pitch of the

surface texture. A single wheel trailer was used to measured the locked wheel

coefficient of friction under two arbitrary levels of wetness obtained by external

application of water. Therefore the effect of water film can only be quantitatively

estimated. Holla attributed the difference between the prediction and measurement of

friction coefficient to the abrasion friction, the viscous friction of the water film and

the masking of texture scales which can not be assessed quantitatively without

knowing the water film thickness.

Gopalan [1976] carried out a laboratory study of the abrasion of rubber and

attempted to predict the coefficient of friction of a sliding rubber block. Unlike

Yandell's work in which glycerine was used as lubricant and the amount masking of

fine scales of texture was deduced experimentally, Gopalan used water as lubricant

and therefore required a knowledge of water film thickness in order to determine the

masking effect. He estimated the film thickness by the theoretical method given by

Moore [1967]. He also deduced the same film thickness from the experimental data

produced by Wallace and Trollope [1969]. Also, because much finer scales were left

unmasked by water, Gopalan used electron microscope stereo pairs for fine texture

measurement.

Taneerananon [1981] rigorously established the thickness of water film present

between the tyre and road surfaces. The inertia effect of rubber on the coefficient of

hysteretic friction has been quantitatively studied using the mechano-lattice analogy

evolved by Yandell. He then was able to predict the locked-wheel braking and

sideway force coefficients, by applying his two models for calculating water film

thickness. Taneerananon attributed the discrepancies between the prediction and

measurement of friction coefficient to the under estimation of the temperature of the

tread rubber; and then suggested to introducing an empirical factor to account for

temperature effect.

60

At the same time, Zankin [1981] constructed a special apparatus for measuring the

damping factor behaviour of tread rubber for speed up to 80 kph and temperatures

ranging up to 150°C. He found that the hysteresis behaviour of filled rubber was only

partly viscoelastic. High hysteresis losses at room temperature in carbon-black-filled

rubbers are greatly reduced at high temperatures owing to the temporary breakdown of

filler particle agglomerates. His result seems to agree with Payne's work [1974] who

founded that the tread rubber sliding over the coarse texture will break down the

carbon filler and generate higher temperature simultaneously. It was then suggested

that the introduction of a separate set of rubber temperatures for each road surface

reduces the average absolute error (between prediction and measured value of friction)

to 5 percent [Yandell et.al (1983)]. The SFC and BFC for a simulated pneumatic

passenger car travelling at a number of different speeds were predicted with

reasonable accuracy from surf ace texture measurements and tread rubber properties.

In this work which is a continuation of the effort to predict the coefficient of

friction on road surfaces, it was proposed to investigate the tyre-road friction under

braking and cornering. In other words, the prediction of locked-wheel BFC (at 100%

slip) and maximum SFC (at optimum slip angle) which have already been done

[Yandell et.al (1983)], will be extended to cover all ranges of percent slip and slip

angle.

In developing the model for the theoretical prediction, it was proposed to involve

either the locked-wheel BFC or maximum SFC, as input parameter. It has been found

from Taneerananon's works [1981], that the locked-wheel BFC (i.e. at 100% slip) and

the maximum SFC (i.e. at optimum slip angle) can be predicted with reasonable . agreement, in which the following factors have been taken into account: surface

texture, water film thickness, tread rubber properties and certain operating conditions.

The mode of operation, however, was only represented by the models for sliding and

rolling. Therefore, to extend the prediction, the mode of operation should also be

extended covering all percent slips and slip angles.

It should be noted that although the locked-wheel BFC or maximum SFC are

included in the model, in practical application they can be predicted from surface

texture data, instead of being directly measured. In other words, the prediction of

locked-wheel BFC or maximum SFC could be treated as sub-models of the main

61

model. Hence, the ultimate goal to develop a theoretical technique for predicting

tyre-road friction from surface texture data is still continued.

To assess its validity, it was proposed to conduct experimental tests on road

surfaces, so that the theoretical prediction can be compared with the field results. The

braking and sideway force tests were carried out.

Extensive verification programs for Yandell's theory of hysteretic sliding friction

were carried out in 1981-82 by Yandell and Taneerananon, involving the prediction of

BFC and SFC operating separately of a simulated passenger car pneumatic tyre on

many different types of pavement for a number of travelling speeds. The predictions

were based on the measured pavement surface texture and the measured damping and

stiffness properties of the tread rubber. The University of New South Wales multi

mode friction test truck developed by Yandell and Ferguson was used to directly

measure the friction values. Figures 1.37-1.38 show some of the results.

It is proposed to extend this work further by introducing the concept of tyre

stiffness into the relationship between braking and sideways friction.

This study will involve field friction testing which will require the following

considerations:

a. The measurement of friction on the road to the desired accuracy requires repetition

of measurements, taking into consideration the time required in taking

measurements and doing analyses.

b. The precision of the estimate will increase with increasing number of replications.

However, the number of replicate measurements are limited by the time required

for testing, which may influence the test results through uncontrolled factors such

as change in pavement surface characteristics, temperature (given the long period

of measurement). Thus, the number of replications is determined from a number of

considerations rather than precision alone.

c. The behaviour of a pneumatic tyre is more complex than that simulated by Yandell,

Taneerananon and Zankin. This is partly due to the non uniformity of the normal

stress distribution in the contact area as a result of bending of the carcass. Factors

like the one just mentioned are difficult to include in the prediction, and therefore

some discrepancies in the prediction are to be expected.

62

Measired BFC (locked-wheel) 1.------------------,, Vehicle Speed

0.9 I::.. 10mph

0.8 0 30 mph a 50 mph

0.7

0.6

0.5 a

0.4

0.3

0.2

0.1

0 ,.____.__....____.__......_____. ____ ___.__......___.__......___.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Precl lctecl BFC

Fig.1.37. Measured BFC vs predicted BFC, on many types of pavement (concrete, bituminous) and macro-texture (closed, mild, smooth). [Taneerananon (1982)].

1 Measured SFC ( N maximum)

0.9 Veh le le Speed 90%

I::.. 10 mph Confidence 0.8 0 30 mph

D 50 mph 0.7

0.6

0.5

0.4

0.3

0.2 Regression Line

0.1 Slope = 0.938

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Predicted SFC

Fig.1.38. Measured SFC vs predicted SFC, on many types of pavement (concrete, bituminous) and macro-texture (closed, mild, smooth). [Taneerananon (1982)].

63

In spite of all the difficulties associated with the prediction of the tyre-road

frictional coefficient, it was proposed to investigate the problem in the following

manner.

1. 7 .2. Method of Investigation

1. Study the longitudinal and lateral stiffnesses of tyres in relation to operating

conditions and tyre properties.

2. Develop the model for tyre stiffness variation.

3. Study the theoretical prediction and the existing models for tyre-road friction

under braking and cornering.

4. Develop the model for the prediction of tyre-road friction under braking and

cornering.

5. Study the effects of several factors affecting tyre-road friction by using the

proposed model, and compare the model response with other tyre models.

6. Measure the braking and sideway forces under different operating conditions and

find the corresponding coefficients of friction.

7. Obtain from the measurement in (6) the parameters needed for the model.

8. Predict the tyre-road friction under braking and cornering using the proposed

model.

9. Study the variations of the experimental coefficients of friction under various

operating conditions.

10. Compare the theoretical coefficients of friction in (8) with the experimental

coefficients of friction in (6) . . 11. Modify, where appropriate, the proposed model for prediction by using the

comparison in (10).

12. Study the application and measurements of tyre stiffness.

64

CHAPTER 2. MEASUREMENT AND PREDICTION OF TYRE-ROAD

FRICTION

Notation (Used frequently in Chapter 2) a, b = length of increasing & decreasing pressure zone (unit length) BFC = braking force coefficient

Cx = longitudinal slip stiffness (force/slip, or force/fraction slip), or longitudinal deformation stiffness (force)

Cy = lateral slip stiffness (force/degree, or force/radian), or lateral deformation stiffness (force)

½C = longitudinal slip stiffness coefficient (per slip, or per fraction slip), or longitudinal deformation stiffness coefficient

C,C = lateral slip stiffness coefficient (per degree, or per radian), or lateral deformation stiffness coefficient

Fx, FY = longitudinal & lateral force (force) Fz = vertical load (force)

IP = tyre inflation pressure (psi) 21, w = tyre contact length & width (unit length)

Kx, Ky = longitudinal & lateral carcass spring rate (force/length per unit area) kx, ky = longitudinal & lateral tread element spring rate (force/length per unit area)

P = tyre normal pressure (force/unit area) SFC = sideway force coefficient

Sx = longitudinal slip (%, or per fraction slip) Sy = tancx V = velocity (unit length/unit time) ex = slip angle (degree, or radian)

O'max = tyre-road stress limit (force/unit area) O'x = longitudinal stress (force/unit area) O'y = lateral stress (force/unit area) 0 = arc tan S/Sx (degree, or radian) µ = available friction coefficient

µd = locked-wheel friction coefficient µ;, µY = longitudinal & lateral frictional forces coefficients µ5, µa = available longitudinal & lateral friction coefficients subscript 0,1 = initial (measured), final (measured or predicted)

2.1. Methods of Measuring Tyre-Road Friction

Early works on measuring tyre-road friction have been reported by Agg [1924],

Bradley and Allen [1930-31], and Moyer [1934]. Agg measured the skidding

resistance by using a towed vehicle, Bradley and Allen used a motor cycle and

sidecar, whereas Moyer used a trailer in order to eliminate some dynamic variables.

65

' ri},:··'·" ,dl[

l n l ~

Fig.2.1. Sideways Coefficient Routine Investigation Machine (SCRIM). [Bethune and Read (1974)].

All reported the skid resistance in terms of straight forward braking and sideway

force coefficients. Numerous methods have since been developed to measure skid

resistance. The common examples are sideway force, braking force, stopping

distance (or decelerometer), and portable skid resistance testers.

2.1.1. Sideway Force Measuring Method

This method is based on contention that the critical maneuver with regard to

skidding is cornering. The test wheel which is free to rotate is set at a predetermined

angle to the direction of motion, and the sideway force acting normal to the plane of

the wheel is measured. The frictional resistance is then expressed in terms of the

sideway force coefficient which is the ratio of the sideway force to the vertical load.

The maximum sideway force is obtained at a critical angle of inclination of the

test wheel. This critical angle is a function of pavement surface parameters, and is

usually slightly below 20°. The instrumentation of the test vehicle usually involves

electrical strain gauges for measuring the force and an automatic system to obtain a

continuous graphical representation of the test results. SCRIM (Sideways Coefficient

Routine Investigation Machine) is an example of this measuring method (Figure 2.1).

The sideway force method is accurate (reproducibility ± 1 to 2 percent), yields

continous measurements along the road, the wear of the test tyre is uniform, and

there are no accident hazards involved. Depending on the degree of sophistication

aimed at, the initial equipment cost may be relatively high.

66

2.1.2. Braking Force Measuring Method

A test wheel of a trailer positioned in the normal running mode of a vehicle is

braked and dragged over the surf ace to be tested at a constant velocity, thus

generating a frictional tyre-road force. The wheel may either be locked, giving

sliding friction, or allowed to run at a controlled percentage slip, to find the peak

friction at 5 to 20 percent slip.

Trailer tests can be carried out at normal traffic speeds with minimum

interference to traffic, and the procedure is fast and accurate (reproducibility ± 1 to 2

percent). While the initial cost of the equipment is relatively high, the unit testing

cost (cost per test site) is usually very reasonable [Csathy et.al (1968)].

2.1.3. Stopping Distance Method

The test vehicle is brought to the predetermined initial speed and its brakes are

applied hard instantly to lock the wheels. The stopping distance from the point of

application of the brakes to the point at which the vehicle stopped is measured,

either by means of a stopmeter with a fifth wheel or with a tape from a chalk-mark

made by a brake-activated detonator [Csathy et.al (1968)]. The results of the tests

are reported either as stopping distance at a given speed, or as the average (braking)

coefficient of friction, using the simple formula f = V2/30S, where V is the initial

speed in mph and S is the stopping distance in feet.

This method is relatively inexpensive, closely simulates the critical conditions of

emergency braking. On the other hand, it is normally limited to speeds below 30 to

40 mph, because there is an element of accident hazard involved and elaborate

traffic control measures are required. The long stretch of road is required for testing.

The reproducibility of the measurements varies from 2 to 15 percent.

As an alternative to stopping distance, a decelerometer is used in a similar testing

technique. As the brakes are applied, the deceleration of the vehicle is measured

instead of the stopping distance. The coefficient of sliding friction is numerically

equivalent to the deceleration expressed in g's. The problems at stopping distance

67

method may be minimized by using only very short skids rather than skids to a stop.

Although it is a satisfactorily accurate (typical reproducibility ± 2 to 3 percent), the

testing procedure is rather difficult to standardize.

2.1.4. Portable Skid Resistance Testers

The most widely used among these testers is the British Portable Skid Resistance

Tester. The apparatus measures the frictional resistance between a rubber slider and

the wetted road surface. The slider consists of a block of tread rubber 1/4 inch thick

and 3 inch wide mounted on the end of a pendulum arm. The pendulum arm is

released from the horizontal position so that the slider traverses along the test

surf ace and swings upwards after leaving the test surface. The work done against

friction during sliding is equated to the loss in energy of the pendulum arm thus

enabling the direct calibration of the scale. The reading is given as 100 times the

friction coefficient and known as Skid Resistance Value (SRV) or British Pendulum

Number (BPN).

Good correlation exists between the readings of the pendulum tester and the skid

test results of patterned tyres at speeds of 30 mph. This instrument is relatively

inexpensive, however, there are also some disadvantages, such as: results obtained

on coarse textured surfacings can be misleading, only covered small area of the road

surface, difficulties of measurement in busy traffic, and unable to measure skid

resistance at high speed [Croney (1977), Salt (1977)].

2.2. Some Theoretical Methods of Predicting Tyre-Road Friction

The difficulties associated with direct methods of road friction measurement have

lead many researchers to look for other means of obtaining skid resistance. The high

cost of setting up skid testing vehicles is probably a main consideration. In addition,

particularly with the locked wheel skid testers, the repeatability of measurements by

a tester, according to Meyer et.al [1974], is generally not adequate. The variability of

test results is of course due to a number of factors as discussed in Section 1.2.

68

Surface Texture Qllllllification

Pavement Surface Texture aaBRification

Macro-Texture Of Projections Micro-Texture Of

Density Of

Approximate Distribution

Projections Background Approximate Angulari~y or Projections, Parameter

Height Width Hanhne88 Harshness Number C** As A Percentage E** F A* B Of Total Area

D

Cavity in 0 0mm 16mm 0-12 Percent Surface in

Matrix

Round 13-37 Percent POLISHED

I l/4mm 8mm (No Texture Visible)

SMOOTH (Texture Visible But

2 1/2 mm 4mm Subang. 38-62 Percent Micro-Projections Too Small For Visual Estimate OfHeight)

FINE GRAINED 3 I mm 2mm Angular 63-87 Percent (Micro-Projections Appr.

1/4 mm High) t

COARSE GRAINED-

88-100 Percent SUBANGULAR

4 2mm (Micro-Projections Appr.

Particles leS& 1/2 mm High Or More) t

Than 2mm Wide Are COARSE GRAINED-.

Regarded As ANGULAR 5 4mm

Background (Micro-Projections Appr. 1/2 mm High Or More) t

6 8mm

•A texture element ha, a heiBht dimeruion only if the ,urroundi~ area below it• peak u drained. ••For the u•e of the Photo-Interpretation Chart for A•phalt Pavement• the foUow~ adju.tment u made: if the C-Parameter Number u 2, then the E-Parameter Number u raued by 1; if the C-Parameter Number u 3, then the E-Parameter N.umber u raued by 2. fMicro-projection• mmt protrude by an amount not leu than lu,lf theu width.

Fig.2.2. Pavement Surface Texture Classification. [Schonfeld (1974)].

2.2.1. Schonfelds's Photo-Interoretation of Pavement

The six parameters used to classify the pavement surface are:

1. Parameter A denotes the height above the matrix of projections on the pavement

surface.

2. Parameter B denotes the width of the surface projections at the top of the matrix.

3. Parameter C denotes the shape of the projections.

4. Parameter D denotes density of projection distribution as the proportion of the

69

ic Ill ... c IIC .. Q

it .,, ... Ill

"' :,: ii: Q

.,, ... "' :c .. Cl u 0 iii ... ~ Q

ic z "' 0 .. ;:: z C, u IIC ii 0 ... ... .,, .. z "' :E "' IIC u ! IIC

"' m :E :, z Q

it f!?

70

ASPHALT PAVEMENT ( RANDOM TEXTURE J

60

TEST SPEED 60 mph --TEST SPEED 30 mph ---

50

E or F •o

30

20

10

O -==~10-~20~. -c'30',---A0'---5,'-0-60..._____.,70----l....180--90'---l00.L.....P-ER_CJ...EN_T_O...JF

I I i I SURFACE D-0 D-1 D-2 D-3 D-•

NOTE, IF THE DENSITY PARAMETER IS 0-1. 0-2 0A 0-3. ADD TO THE FRICTION WEIGHT FOR PARAMETER E OBTAINED FROM THIS GRAPH THE FRICTION WEIGHTS FOR PARAMETERS A. B ANO F OBTAINED FROM CHART NO. 2, CHART NO. 3 OR CHART NO. •. AE$PECTIVEL V _

Fig.2.3. Friction weights of texture parameter E or F. [Schonfeld (1974)].

whole surface area occupied by projections.

5. Parameter E denotes the sharpness, angularity or roundness of the microtexture

on the projections' surface.

6. Parameter F denotes the sharpness, angularity or roundness of the micro

projections in the background matrix.

The number given to each parameter together with its description is shown in

Figure 2.2. A pavement surface is thus classified by a set of 6 texture parameter

numbers. Using correlation charts of texture code number and skid test results on

asphalt pavements obtained by an ASTM skid trailer or others, the photo-interpreted

skid number of a pavement is obtained as the sum of the texture parameters' friction

weights which, are given at two speeds of 30 and 60 mph.

70

One disadvantage of the method is that it is rather subjective. For example, with

parameter C which specifies the shape of projections, it is required to be given as a

whole number. Suppose the shape of projections is between subangular (C2) and

angular (C3), and the harshness of the projections is polished (El), then (for asphalt

pavements) it can affect the parameter E by 1 scale depending on whether the

parameter C is rounded to 2 (corresponding with E2), or to 3 (corresponding with

E3). A difference of 1 in E parameter number can give a variation of friction

weights up to 25 at 30 mph. (or up to 20 at 60 mph) which is a large variation,

considering the maximum friction weight for E is only 70 at 30 mph (or 50 at 60

mph). See Figure 2.3.

2.2.2. Leu and Henry's Model of Skid Resistance

Leu and Henry [1978] developed a model to predict the skid number (SN) as a

function of speed (V). The model contains two constants which are functions of

macro-texture and micro-texture. The percentage skid number-speed gradient (PSNG)

which has been shown to be a function of macro-texture is assumed to be

independent of speed in the derivation of the model which is given as:

SN= SN0 exp {(-PSNG/lOO)V}

where SN0 = zero speed intercept and is assumed to be a function of micro-texture.

Using the sand patch method to evaluate macro-texture in terms of mean texture

depth (MD), and British Pendulum Number (BPN) to describe micro-texture, they

found that:

PSNG = 4.l(MDr0·47

SN0 = -31 + 1.38 BPN

Hence the skid number is expressed as:

SN = (-31 + 1.38 BPN) exp {-0.041 V (MD)"0·47 }

Good comparison with measured skid number (ASTM E274-77) was obtained at

speeds of 40 and 60 mph. They also found that skid number at any speed can be

related to a measured skid number at one speed and a macro-texture measurement.

The model's advantage as claimed by the authors is that it clearly distinguishes the

roles of macro-texture and micro-texture. It is interesting to note that the model

71

gives good predicted skid resistance, despite the relatively crude method of obtaining

the macro-texture depth. The effects of water film thickness and other test conditions

are not considered here.

2.2.3. Texture Friction Meter

As mentioned in Section l.4.2 (13), Yandell and Mee (with improvement from

Sawyer) have developed a device that can accurately sample pavement surface

texture. Using Yandell's theory of hysteretic friction, the information from the

surface texture is analysed to give the sideways and locked-wheel friction of a

pneumatic tyre for several travelling speeds. This device is still under trial, however,

some meaningful results have been obtained as part of a pavement management

system [Yeaman (1989)]. Figure 2.4 shows relationships between it and BPN and

MMFTT.

(TFMJ wet EFC (lodced-wt'Bal), 10 mph 1.2-..;.._---'-----'---.;.._--~

(TFM) -t SFC (-maxlmU'Tl), 30 mph 1.2 ..-..;.._---'-----'--__,.;...---~

Rolled asphalt and non textured surface B1tumlno1.B and concrete surface

1 1

0.8 O.B C

C

D 0.6 0.6

0.4 0.4

0.2 0.2 rf = 0.7

o--~-~-__._ _ __._ _ __. _ __. a--~---~-__._ _ __. _ ___. 0 20 40 60 BO 100 120 0 0.2 0.4 0.6 0.8 1 1,2

8"lt1Sh PandUILITl NuTt>ar (BPN) (t,,MFlT) wet SFC (15 deg), 30 mph

Fig.2.4. Texture Friction Meter (TFM) vs British Pendulum Number (BPN) and Multi Mode Friction Test Truck (MMFTT). [Yandell and Sawyer (1992)].

In the following section, the developments of Yandell' s theory of hysteretic

friction for predicting sideways and locked-wheel friction, will be described in

details.

72

2.3. Theoretical Prediction of The Locked-Wheel Braking and Sideway Force

Coefficients

2.3.1. Mcxlification of The Theory of Hysteretic Friction

In its simplest form Yandell' s theory states that the coefficient of hysteretic

friction of rubber on a surface. is the sum of the coefficients of friction on each scale

of texture. With the presence of water film, however, some of the scales of texture

will be masked and rendered less effective in providing friction.

In the original procedure [Yandeil and Holla (1974)], any scale with a texture

depth lower than the water film thickness was considered masked and hence did not

contribute to friction. Taneerananon [1981] devised a more realistic methcxl of

determining masking (see Figure 2.5). He assumed that if the effective coefficient of

friction of scale 1 to be proportional to the ratio of the unmasked texture depth over

the texture depth, and since the fine scale of the texture is superimposed on the next

coarser scale, then it is also to be masked in the same proportion as the coarser

scale. Therefore, al/dl = a2/d2 = a3/d3 ... , and the total coefficient C is then

calculated as follows:

C = (a/d)C1 + (a/d}2Cz + (a/d}3C3 + ... where a = d-h, the unmasked depth of texture.

h is the calculated remaining water film thickness.

d is the depth of the coarsest component of texture.

(2.1)

C1, Cz .. . are the coefficient of hysteretic friction of each scale of the texture.

This procedure automatically causes masking to have the greatest effect on the finest

scales.

2.3.2. Some Assumptions of Rubber/Water Temperature

It is clear from Figure 1.29 that the damping factor of rubber is influenced by its

temperature. The water film thickness, on the other hand, is determined (according

to Reynold Equation) by its viscosity, which is a function of water temperature. To

add to the complexity, there is also an interaction of rubber and water temperatures.

73

Scale 2

Scale 2

h

Scale 1

Fig.2.5. Schematic representation of masking of texture. [Taneerananon (1981)].

The temperature of tyre tread rubber is difficult to ascertain, especially during

skidding (or locked wheel braking). There is however, some analytical work done by

Hegmon and Henry [1973], Hegmon [1975], and Yeow et.al [1978]. They have all

predicted a mean temperature rise of about 150"C under dry conditions of skidding.

The presence of water between the tyre and road surfaces will greatly affect the

temperature of the tyre. Since the heat generated by a sliding tyre is proportional to

the sliding speed, it is reasonable to assume that for the skidding mode, the tyre

temperature increase is also proportional to the sliding speed. Unlike the skidding

tyre where a constant patch of rubber is in contact with the road surface throughout

the skid, the rolling motion of the cornering tyre continuously provides a new patch

of rubber for contact with the road pavement. For this reason the temperature of the

cornering tyre may be assumed constant.

Table 2.1 shows the assumed rubber/water temperature used by Yandell et.al

[1983] for predicting the locked wheel-braking coefficient (BFC), and (maximum)

sideway force coefficient (SFC). Using these assumptions they then are able to

reduce the average absolute error (between predicted and measured value of friction)

to 5 percent.

For BFC on a smooth road, the water is assumed to take the same temperature as

that of the rubber due to the intimate contact and the small texture spaces. A

coarse-textured road, on the other hand, has large pore spaces which can also act as

cool water reservoirs, so it is considered that the water on a coarse surface remain~ at

74

20°C. However, in regard to rubber temperature, the actual contact area will be less

on the coarse texture, the contact stresses greater, and a similar amount of energy

will be dissipated in smaller volumes of rubber so it is assumed that the rubber

temperature will be higher than for the smooth surface.

Table 2.1. Assumed rubber/water temperature. [Yandell (1983)].

Travelling Temperature (°C) for BFC Temp.

Speed Smooth Surface Coarse Surface (OC)

mph (kph) for SFC Rubber Water Rubber Water

10 (16) 40° 40° 80" 20° 20°

30 (48) 600 60° 100° 20" 20°

50 (80) goo goo 120° 20" 20°

For SFC, the main assumption is that the sliding speed of the rubber across the

road surface is less than 10 percent of the travelling speed. For this reason and for

the fact that the rubber in the contact patch being continually changed and

ventilated, the temperature will not rise a great deal and hence the damping factor

will remain constant for 20"C. The surf ace water will also remain at about 20°C.

2.3.3. Prediction of The Locked Wheel Braking and Sideway Force Coefficients

In summary, the prediction process consists of the following steps:

1. Measure the damping factor behaviour of the tread rubber over the required

speed and temperature range (see Section 1.5).

2. Measure the surface texture, analyze it into scales, and determine the average

absolute slope, the texture depth, and mean pitch of each scale (see Section 1.4).

3. Calculate the coefficient of friction for each speed and friction mode (braking or

sideway) of each scale (see Section 1.6).

4. Calculate the average water film thickness on the road (see Section 1.5).

5. Calculate the contact-area film thickness for each speed and friction mode (see

Section 1.5).

75

6. Calculate the total coefficient of friction for each friction mode, speed, and initial

surface water film thickness {using Equation (2.1)).

2.4. The Longitudinal and Lateral Tyre Stiffness

The property termed the longitudinal slip stiffness (Cx), defined by the

relationship,

ex= d.FxfdSX at wheel slip sx = 0 (2.2)

represents the linear behaviour of the tyre under braking. Similarly, the property

termed the lateral slip stiffness (Cy), defined by the relationship,

CY = d.F/da at slip angle a = 0° (2.3)

represents the linear behaviour of the tyre under cornering.

The tyre stiffness (longitudinal, lateral) is considered as one of the most

significant tyre parameters affecting vehicle response properties. Results from

Nordeen [1968], for example, show that in the region from 0-3 degree slip angles an

increase in the lateral stiffness coef. (C,.C ~ Cy/F';z) of 0.005(fetdtgllee)iu significant change

from the standpoint of the performance of the vehicle; that is, it is a readily

detectable change in vehicle behaviour. To provide good vehicle handling response,

it is desiTable to increase a$ far as possible the frictional forces developed at all

percent slips and slip angles by means of, among the other things, achieving high

initial slopes (i.e. longitudinal and lateral stiffness).

Unlike most vehicle components whose mechanical properties do not vary with {e.g. ~tiffnes~)

operating conditions, it is found that the tyre propertiesvaie strongly influenced by

basic operating conditions such as tyre load and inflation pressure. Other variables,

such as speed, tyre pattern (bald & new) are also undoubtedly influential. Both

longitudinal (braking) and lateral (cornering) tyre stiffness can be normalized by

dividing by the vertical load (Fz), to give the s0 called long. stiffness coefficient (CxC) and

lateral stiffness coefficient (CyC).

76

N II.

tJ 200 ... ~ 100

~ 50 II. II. Ill

8 20

! 10 z :1: I t; C, 2 z

Fig. 30 : -

:

a -D,) "'R 11 .n ... 11-R -u~ -U -UBl-u ~-•-o

¥ 1 : 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 ID LOAD/DESIGN LOAD

Fig.2.6. Longitudinal stiffness coef. (CxC) vs tyre load of radial-ply (R), bias-belted (B), and bias-ply (D) tyres. [Schuring et.al (1976)]. Note: CJFz = C/Fz = CxC.

0.3 ,----~--.--~-..----,,--._.--':' r Fig. 26

g IR, u a, ~ 0.2 D::-111

w ... ... 111 D-8 a o. 1 t,.----t---+-......... -+-.;.;.. i iii: Ill z a: 8 o --o~ _____ ....___. _____ ~---------

0.2& 0.50 0.75 1.00 1.25 1.50 1.71 1.00

LOAD/DESIGN LOAD

Fig.2.7. Lateral stiffness coefficient (<;C) vs tyre load of radial-ply (R), bias belted (B), and bias-ply (D) tyres. [Schuring et.al (1976)]. Note: CA/Fz = C/Fz = C7C.

2.4.1. Effects of Normal (Vertical) Load

Figure 2.6 shows the decrease in longitudinal stiffness coefficient (CxC) of

passenger car tyres with increasing tyre load. The decrease is about 25%-30% with

increasing load/design load from 1 to 1.5 (the design load is about 900 lb). The

figure also shows that the CxC are highest for radial tyre and lowest for bias ply

tyres. In Figure 2. 7 the decrease of c;c is about 20% for load/design load of 0.5 to

0.75, and about 40% for load/design load of 1 to 1.5.

77

Table 2.2 shows the decrease of ½C with increasing vertical load (Fz) of truck

tyres. Unlike passenger car tyres which have lower vertical loads, the decrease in

CxC of truck tyres is only 9%-18% by increasing load by 50%. The similar results

(for 12.00-20/G tyres, at speed 1.44 mph) from Tielking et.al [1973] show a

decrease in CxC of about 6% with increasing load from 2000-3000 lb, and a

decrease in CxC of about 20% with increasing load from 6000-9000 lb.

Table 2.2. The decrease of CxC with increasing Fz for 10.00-20/F tyres, at speed 8-88 kph. [Ervin (1976)].

Vertical-load F~ Contact ex CXC

(N) (lb) length (N/slip) (per slip)

(cm)

10000 2250 21.6 132,000 13.2

15000 12.06 (est)

24500 5510 26.7 242,000 9.9

36750 8.14 (est)

40500 9110 31.2 308,000 7.6

Table 2.3. The decrease of CyC with increasing Fz of 078-15 tyres, at speed 6 mph, 28 psi. [Bergman (1977b)].

Test eye (per degree), at vertical load (lb)

Equipment 800 1200 1600 1800* 2000

ealspan Belt 0.1700 0.1358 0.1050 0.0922 0.0795

GM Belt Link 0.1812 0.1462 0.1139 0.0988 0.0858

Gen. Flat Bed 0.1793 0.1374 0.1001 0.0862 0.0723

BFG Flat Bed 0.1699 0.1380 0.1090 0.0963 0.0836

Average 0.1751 0.1394 0.1070 0.0927 0.0803

* estimated (by interpolation).

The decrease of eye with increasing load is shown in Table 2.3. On average, the

data shows a decrease in eye of about 20% with increasing load from 800-1200 lb,

and a decrease of about 34% with increasing load from 1200-1800 lb.

78

2.4.2. Effects of Inflation Pressure

Increasing inflation pressure (IP) reverses the deformation caused by vertical

load. Although a decrease in contact length accompanies an increase in inflation

pressure, the dominant effects of increased pressure are a reduced curvature in the

sidewall and a generally stiffened carcass structure [Tielking et.al (1973)]. The net

result is a lateral spring rate (Ky) that increases with inflation pressure. For

longitudinal stiffness coefficient, the effect of inflation pressure is less significant.

The effects of inflation pressure on lateral stiffness of bias-ply and radial-ply tyre

have been studied by Phillips [1973]. The mathematical model that indicates the

relationship between CY (lb/deg) and IP (psi) of passenger car tyres are as follow

[Sutantra and Harrison (1985)].

For bias-ply tyres,

Cy = 26.4 + 4.32(IP) - 0.0674(IP)2

For radial-ply tyres,

Cy = 33.5 + 5.30(1P) - 0.0916(IP)2

(2.4)

(2.5)

As can be seen, in the range of IP 10-30 psi there is an increase of c;. with

increasing IP, however, above IP 30 psi there is no significant change in Cr

For truck tyres, the increase of lateral spring rate (Ky) with increasing inflation

pressure, is shown in Table 2.4. The results indicate that the increase of KY (hence

also Cy) with increasing IP is more pronounced for wider type tyres.

Table 2.4. Lateral spring rate (Ky) vs inflation pressure (IP) for truck tyres. [Tielking et.al (1973)].

Tyre Fz Ky (lb/inch) at IP

types (lb) 80 psi 90 psi 120 psi

11.00-22/G 6140 1900 1930 1980

12.00-20/G 6140 1900 2100 2390

79

2.4.3. Effects of Speed

There is little information available regarding the effects of speed on longitudinal

stiffness. Results from Holmes and Stone [1969] indicate that there i5 on\y a small or

no decrease in longitudinal stiffness (Cx) with increasing tyre speed (V). It seems

that the effects of speed on longitudinal stiffness can be regarded as insignificant.

For lateral stiffness (Cy), the results from Bergman and Beauregard [1974] show

that on non-steady state motion (- 10 deg/sec) there is an increase of Cy with

increasing speed from 10 mph to 30 mph, and then tends to flatten out with further

increase up to 60 mph. On steady state motion (- 1 deg/sec), no significant change

of Cy is obtained. However, in a laterinvestigation, data from Bergman [1977b], such

as shown in Table 2.5, indicates that there is an increase of CyC, especially at lower

speeds, with increasing speed. The rate of increase is slightly modified by vertical

load. The results indicate that a two-fold increase in load (e.g. 800 lb to 1600 lb, at

30 mph), decrease the CYC by 35 percent, however, a two-fold increase in speed

(e.g. 30 mph to 60 mph, at 1200 lb), increase the CyC by only 1.2 percent.

Table 2.5. The relationship between CyC and Fz at different speed, for 078-15 tyre, 28 psi. [Bergman ( 1977b)].

Speed CyC (per degree) at Fz ·

(mph) Equation 800 lb 1200 lb 1600 lb

6 CyC = 0.235 - 0.0000785 Fz 0.1722 0.1408 0.1094

30 CyC = 0.242 - 0.0000789 Fz 0.1789 0.1473 0.1158

60 CyC = 0.244 - 0.0000791 Fz 0.1807 0.1491 0.1174

2.4.4. Effects of Tyre Wear (Bald & New)

It is often experienced that the stability and controllability of a vehicle varies

during the progress of tyre wear. Considering the area of contact between tyre and

road surface, it can be expected that generally the tyre stiffness is higher for bald

(smooth) tyres than for new (patterned) tyres. Sutantra and Harrison [1985]

80

developed the expressed lateral force (Fy) in pounds in terms of the slip angle (a) in

degrees, as follows.

For new bias-ply: a= 0.0036S(Fy)1·19408

For bald bias-ply: a = 0.0030852S(Fy)1.199158

For new radial-ply: a= 0.00301003(Fy}1·207861

For bald radial-ply: a= 0.0023636(Fy)1·222203

(2.6)

(2.7)

(2.8)

(2.9)

From equations above, it can be calculated that the increase of c;. with tyre wear

from new to bald is about 12% for bias-ply, and about 13% for radial-ply. It is

expected that for longitudinal stiffness (ex), the similar results will occur since e. is

also associated with the area of contact.

Table 2.6. The ex and eY of truck tyres (10.00-20/F) in 3 states of wear. [Tielking et.al (1973)].

Tyre Progress ex (lb/slip) ey (lb/deg)

New 42,000 523.4

Half Worn 52,000 690.S

Fully Worn 60,000 771.S

For truck tyres, an increase of ex and eY with the wear progress is shown in

Table 2.6. The results show that the increase of tyre stiffness is about 45%, from

new to fully worn. Hence, the effects of tyre wear is more pronounced for truck

tyres than car tyres.

2.4.S. Measurement of Tyre Stiffness and Its Related Parameters

The longitudinal slip stiffness (ex) has been defined as the initial slope of the

pure longitudinal force (F x) with respect to the slip ratio (Sx), at zero slip ratio.

Similarly, the lateral slip stiffness (ey) is defined as the initial slope of the pure

lateral force (Fy), with respect to the slip angle (a), at zero slip angle. Therefore, the

measurement of tyre stiffness can be related with the measurement of Fx or FY on

road surface, during braking or cornering. As the tyre forces, even at near zero slip

81

or slip angle, are different at any set of operating conditions (e.g. load, inflation

pressure, speed, and tyre wear), the tyre stiffness may also be measured at any set of

operating conditions.

The slip tyre stiffness has also been related with the tread element spring rate,

and the length and width of tyre contact [Bernard et.al ( 1977), Sakai ( 1981 ), Gim

and Nikravesh (1990)], as follows.

ex = dFxfdSX = 0.5 kx(21)2w (2.10)

eY = dF /da = 0.5 ky(21)2w (2.11)

where: ex = longitudinal slip stiffness (force/slip, or force/fraction slip).

eY = lateral slip stiffness (force/degree, or force/radian).

kx = longitudinal tread element spring rate, or rate of change of longitudinal

force with respect to deformation (force/length per unit area).

Icy = lateral tread element spring rate, or rate of change of lateral force with

respect to deformation (force/length per unit area).

21, w = tyre contact length & width (unit length)

The measurements of carcass spring rates (Kx, Ky), which will be equal to the

tread element spring rates (kx, Icy) if the carcass is rigid, can be done in the

laboratory either at complete sliding [Thieme et.al (1982)] or without/near sliding

[Tielking and Mital (1974)]. If the deformation without sliding is higher than at

complete sliding for a given horizontal force, the spring rate without sliding will be

lower than at complete sliding. If the contact length & width are not changed, then

the deformation stiffness without sliding will also be lower than at complete sliding.

Examples of the measurement of carcass spring rates are presented below [Tielking

and Mital (1974)].

The measurements of Kx are made with the wheel plane in the direction of flat

bed travel. The deflected tyre is. locked against the flat bed table which is manually

moved in increments of 0.1 inch. Fx readings are taken at each displacement until

the tyre begins to slide. The onset of tyre sliding is evident in the representative data

for a radial tyre shown in Figure 2.8. The Kx is the slope of the linear portion of the

longitudinal load-deflection plot. As seen in Figure 2.8, the slope is essentially

constant for the 900, 1100, and 1600 lb vertical loads. The approximate value of Kx = 1000 lb/inch for that tyre's contact area is obtained at these loads.

82

400

300

200

100

Longitudinal Force Fx (lb)

.1

p • 24 psi

! <~\ Ta.le

.2 .3 .4 Table Travel (in)

-·-·- 700 lb. -x-x-900 lb. -A-4-1100 lb.

------- 1600 lb.

.s

Fig.2.8. Longitudinal force <Fx) versus travel of flat bed table supporting a radial tyre at the indicated vertical loads. [Tielking and Mital (1974)].

SOO

400

300

200

100

Lateral Force F y (lb)

24 psi

• 2 .4

-•-•-700 lb.

-X-X-900 lb.

-A-A-1100 lb.

-0-0-1600 lb.

.6 .8 1.0 1.2

Table Travel (in)

Fig.2.9. Lateral force (F1) versus travel of flat bed table supporting a radial tyre at the indicated vertical loads. [Tielking and Mital (1974)].

83

The Ky is obtained from load-deflection measurements made with the wheel plane

perpendicular to the direction of flat bed travel. Lateral force measurements are

made as the flat bed table is moved in small increments until the tyre begins to

slide. The Ky is the slope of the linear portion of the resulting lateral load-deflection

plot. As seen in Figure 2.9, tyre vertical load has little influence on the lateral spring

rate of a radial tyre. The value of Ky = 500 lb/inch for that tyre's contact area is

obtained from the data shown in Figure 2.9.

2.4.6. Clarification of tyre stiffness formula and its units

It can be seen that the left-hand sides of Equations (2.10-2.11) have the unit of

force/unit slip or force/unit angle, whereas the right-hand sides have the unit of

force. The difference on this unit raises some confusions about the origin of the

stiffness formula. On most tyre models, the unit of tyre stiffness must be equal to

the unitsof frictional forces (see Section 3.4).

The mathematical verification of tyre stiffness formula has been shown elsewhere

(e.g. Schallamach and Turner [1960], Livingston and Brown [1969], and Tielking

and Mital [1974]). Basically, the stiffness formulae are derived from the phenomena

of either elastic deformation (in longitudinal direction) in the forward part of the

contact region, or elastic deformation (in lateral direction) following the path of an

element parallel to the direction of motion. Some of their mathematical derivations

are presented in Appendix G.

Graphically, the parameter Cx is the slope at the origin of the Fx versus Sx data

measured in straight-ahead rolling. It is seen, that the angle between this slope and

the horizontal axis is not really a definite angle, since the vertical and horizontal axis

have different units. Analogous to this, the slope of the Fx versus deflection, on

obtaining the parameter kx, is also not a definite angle due to their different units.

Hence, it seems to be meaningless for correlating these angles.

The problem now arises with the practical application of Equation (2.10), such as

the application on the model for predicting/calculating the frictional forces: what

value of slip should be assigned so that the dimension of Cx will be in force similar

84

to the right-hand side of Equation (2.10).

As a first solution, it is useful to rearranged Equation (2.10) in the following

forms.

C1 Cx = 0.5 kx(21)2w (2.10a) (force) (force)

where c1 is a coefficient (from O to 1) representing the value (or fraction) of slip

assigned; and the dimension on the left-hand side of Equation (2.10a) now is force.

On practical application, the selection of c1 will influence the magnitude of frictional

forces at below the critical percent slip, since this region is controlled primarily by

the elastic tyre properties (see Table 2.7). In most tyre models, c1 = 1 was used.

The second problem, is whether the value of ex is quantitatively equal with the

value of 0.5 kx(21)2w. It is then useful to rearranged Equation (2.10a) in the

following forms.

C1 ex = m1 0.5 kx(21)2w (2.10b) (force) (force)

where m1 is a constant to equate the longitudinal tyre stiffness measured from Fx

versus Sx data (i.e. measurement of ex) and from load-deflection data (i.e.

measurement of kx). The value of m1 will be dependent on many factors, such as the

difference in tyre motions (steady-state, quasi steady-state, non steady-state) or

methods of deformation measurement (without/near sliding, complete sliding). This

will be discussed further in Section 6. 7.

Graphically, the parameter eY is the slope at the origin of the FY versus a. data

measured at zero longitudinal slip. Again, it is seen, that the angle between this

slope and the horizontal axis is not really a definite angle, since the vertical and

horizontal axis have different units. Analogous to this, the slope of the F Y versus

deflection, on obtaining the parameter ky, is also not a definite angle due to their

different units. Hence, again it seems to be meaningless for correlating these

angles.

Analogous to the case in longitudinal stiffness, on practical application the value

of angle should be assigned so that the dimension of eY will be in force similar to

the right-hand side of Equation (2.11).

85

As a first solution, the Equation (2.11) then will be rearranged in the following

forms.

C2 Cy = 0.5 k,(21)2w (2.lla) (force) (force)

where c2 is a coefficient (from O to 1) representing the fraction (from a = x/2) of

the angle assigned; and the dimension on the left-hand side of Equation (2.1 la) now

is force. On practical application, the selection of c2 will influence the magnitude of

frictional forces below the critical slip angle, since this region is controlled primarily

by the elastic tyre properties (see Tables 2.7-2.8).

Unlike the selection of coefficient c1 for longitudinal stiffness, the selection of

coefficient ei for lateral stiffness has been different between one investigator and the

others.Most of them used ei = 0.637 (or slip angle assigned a = 57 deg= 1 radian),

whereas some of them used c2 = 0.011 (or slip angle assigned a = 1 deg). Perhaps,

the ones who select a = 1 radian had followed the previous assumption of small slip

angles, that tana = a can be accepted only if a is in radians. (For example, tan4° =

0.0699 = a = 4° = 0.0698 radian). This will be discussed further in Section 6.5.

The second problem, is whether the value of Cy is quantitatively equal with the

value of 0.5 k,(21)2w. Analogous with the case of Cx, the Equation (2.lla) then can

be rearranged in the following form.

c2 CY = m2 0.5 ky(21)2w (2.1 lb) (force) (force)

where m2 is a constant to equate the lateral tyre stiffness measured from FY vs a data

(i.e. measurement of C,) and from load-deflection data (i.e. measurement of le,). Again, the value of m2 will be dependent on many factors, such as the difference m

tyre motions or methods of deformation measurement.

2.4.7. The Empirical Model for Tyre Stiffness Variation

It is desirable (from the point of view of cost and practicalities), if the tyre

stiffness can be predicted instead of being measured. However, by considering the lack

of data and available information, it is not possible in the present study to predict

(from the tyre properties and operating conditions) the stiffness of the tyres. Instead,

86

it is proposed to predict the variation of tyre stiffness. In other words, it is attempted

to construct a model which will be able to predict the change of tyre stiffness at

any set of operating conditions, once the tyre stiffness at one set of operating

conditions has been measured (known).

From previous information, it is found that several factors will influence the

magnitude of tyre stiffness. In regard to the contribution of each factor, it can be

said that the significant factors are vertical load (F2), inflation pressure (IP), tyre

speed (V), and wear progress. Further studies of the effect of wear rate leads to

assumption that the modelling of this factor is considered less important. One reason

is that the time duration taken from one state of wear to another significant state of

wear is considered long enough. Moreover, the change of wear from one state to

another state is less frequent (in comparison with other factors). Hence, each

significant state of wear (e.g. new, half worn, smooth) can be regarded as a separate

type of tyre.

Following information from: Figures 2.6-2.7 [Schuring et.al (1976)], Table 2.2

[Ervin (1976)], Table 2.3 [Bergman (1977b)], Equations 2.4-2.5 [Sutantra and

Harrison (1985)], Table 2.4 [Tielking et.al (1973)], and Table 2.5 [Bergman

(1977b)], the general form of the mathematical model then can be expressed as

follows.

exe1 = exeo + f(Cxeo, F20, IP0, V 0, Fzi• IP 1, V 1)

eye1 = eyeo + g(eyeo, F20, IP0, V0, Fzi• IP1, V 1)

where exe = longitudinal tyre stiffness coefficient

eye = lateral tyre stiffness coefficient

F2 = normal (vertical) load

IP = inflation pressure

V = tyre speed

subscript 1 = predicted (for left-hand side)

subscript 0,1 = measured (for right-hand side)

(2.12)

(2.13)

Three forms of the relation between dependent and independent variables are

considered. Eventually, one is selected on the basis of how well it fits the

experimental data.

y =a+ bx+ cx2 (2.14)

87

Y = axb (2.15)

y = abx (2.16)

In general, the curves obtaining from equations above are not so steep, hence if

necessary, a small extension (extrapolation) outside the range of the model's data,

still can be reasonably applied. The characteristics of each curve are as follows

[Yeomans (197 6)]:

1. Equation (2.14) is most frequently used to fit data having a curve shape.

Depending on the sign of coefficients, the shape can be a concave or convex. A

finite value (a constant) of dependent variable is obtained when the independent

variable equals to zero.

2. Equation (2.15) is simpler than Equation (2.14). The deficiency lies in its near

zero value of independent variable. Depending on the sign of b-coefficient, the

curve can have an assymptote to y positive-infinity (concave downward, b<0) or

to y negative-infinity (convex upward, 0<b<l). Accordingly, any extrapolation, if

required, must be restricted not to cover the near zero values of independent

variable. For b> 1 a finite value (a constant) of dependent variable is obtained

when the independent variable equals to zero.

3. Equation (2.16) is similar to Equation (2.15). However, the sign of b-coefficient

must be positive, otherwise there is no real results for non-integer independent

variable, or there is alternately a positive and negative results for integer variable.

For 0<b<l the curve becomes concave downward, and for b>l the curve

becomes convex upward. This model also has a finite value (a constant) when

the independent variable is zero.

From Figure 2.6 [Schuring et.al (1976)], it is seen that the CxC decreases almost

linearly with increasing load (F2). The Equation (2.16) is selected as best suited for

that particular data. The coefficients are established by statistical method.

For bias belted (B) passenger car tyre,

Cx C = 22 (0.9997l2 (2.17)

Assuming that b.CxCICxCo = (CxC1-CxC0)/CxCo is same for bias belted (B), bias ply

(D), and radial ply (R), then the mathematical model for the effect of vertical load

(for passenger car tyres) is given in the following form.

CC = ((0.9997l21 -(0.9997f"'Jc C = ((0.9997f21 - l} C

b, X F X O F x 0 (0,9997) ZO (0,9997) ZD

(2.18)

88

where Fz = vertical load (lb), in the range of extended model's data 225-1350 lb.

Similarly, for lateral tyre stiffness, from Figure 2.7 [Schuring et.al (1976)]:

eye = 0.31 (O.999fz (2.19)

AC C = ( (0.999f'-(O.999f .. )c C = ((0.999f1 -l !r C (2.20) y l (O.999f zo y O (O.999f zo )'---y O

For truck tyres, see Tables 2.2 [Ervin (1976)] and 2.3 [Bergman (1977)], it is found

that the Equation (2.15) is best suited for longitudinal stiffness, whereas Equation

(2.16) is more suitable for lateral stiffness. Accordingly, using coefficients

established by statistical method, the model for the effect of vertical load (for truck

tyres) is given in the following forms. CxC = 288(Fz>-0.39S

AC C = zl zO C = zl - 1 C ((F t0.39S_(F t0.39S} ((F t0.39S }

x (Fzot0.39S x o (Fzo>-0.39S x o

where Fz = vertical load (lb), within the range of model's data 2250-9110 lb.

eye = O.28(O.9994fz

AC C = ( (O.9994fz,_ (0.9994fzo)c C = ((0.9994fz, _ 1 !r C y l (O.9994fzo y 0 (O.9994fzo ry 0

(2.21)

(2.22)

(2.23)

(2.24)

The model for the inflation pressure (IP) effect is best suited with Equation

(2.14). Using coefficients in Equations (2.4) and (2.5) from Sutantra and Harrison

[1985], the ratio of (Cy1-Cy0)/Cyo is first established. Then by assuming ACyC/CyCo = (CyCcCyC0)/CyCo = (Cy1-Cy0)/Cyo• the mathematical model for the effect of inflation

pressure (for passenger car tyres) is given in the following forms.

For bias-ply passenger car tyre,

AC C = ------------ C (4.32(IP1-IP0)- O.O674{(IP1 ) 2 -(IP0)2}f

Y 26.4 + 4.32(1P0 )- 0.0674(IP0 ) 2 Y 0 (2.25)

For radial-ply passenger car tyre,

(5.3(1Pl -IPO) - 0.0916{(IP.}2-(IP0)2})

ACyc = 2 cyc0 33.5 +5.3(1P0)-O.0916(1P0)

(2.26)

89

where IP = inflation pressure (psi), within the range of model's data 10 psi - 40 psi. Similarly, for truck tyres data in Table 2.4 [Tielking et.al (1973)] the ratio of

(Ky1-l(,o)/Kyo is first established. Then by assuming ACyC/CyCo = (CyCi-C,Co)/CyCo =

(Ky1-Ky0)/Kyo, the mathematical model is given in the following forms.

For normal truck tyre (e.g. 11.00-22/G or smaller),

AC C = ---------- C (-14(IP1-IPO) + 0.1 {(IP/-(IP0)2}}

Y 238O-14(IP0) +O.l(IP0)2 Y 0 (2.27)

For wider truck tyre (e.g. 12.00-2O/G or greater),

AC C = ----------- C (-56.5(IP1 -IPO) +0.45{(IP/-(IP0)2}}

Y 354O-56.5(IP0) +O.45(IP0)2 Y 0 (2.28)

where IP= inflation pressure (psi), within the range of model's data 80-100 psi.

From Table 2.5 [Bergman (1977b)], it can be deduced that CyC increases

(non-linearly) with increasing speed (V). By plotting the data of CyC and V, it is

found that the increase is best fitted with a second-order relation (Equation 2.14).

Accordingly, the equations in Table 2.5 can be formulated as follows.

CyC =a+ bF2 + cV + dV2 (2.29)

The coefficients (a,b,c,d) then can be obtained by least square method.

CyC = 0.2252 - O.0000788F2 + O.00043V - O.0000042V2 (2.30)

Assuming that ACyC/CyCo = (CyC1-CyC0)/CyCo for each car tyre type is the same,

then the mathematical model for the effect of speed is given in the following form.

( O.00043(V1-V0)-O.0000042{(V/-(V0)2} J (2.31)

ACyc = 2 cyc0 0.2252-0.0000788F20 +0.00043V0 -0.0000042(V0 )

where V = tyre speed (mph), within the range of model's data 6-60 mph.

In summary, the empirical models for the variation of tyre stiffness are given as

follows. It should be noted, that the units are: F2 in (lb), IP in (psi), and V in (mph).

1. Longitudinal tyre stiffness,

a. For passenger car tyre,

C C = ((0.9997f• t C X l (0.9997f• rx O

(2.32)

90

b. For truck(;\-o.395 J CC = zi CC

x 1 (Fzof0.395 x o

2. Lateral tyre stiffness,

a. For bias-ply passenger car tyre,

C C - ((0.999fz1Jc C y 1 - (0.999f zo y O

+ ------------ co (4.32(IP1 - IP 0)-O.O674{(IP /- (IP0)2}}

26.4 + 4.32(1P0) - O.O674(1P0)2 Y

( O.00043(V1-V0)-O.0000042{(V.}2-(V0)2} }

+ 2 yCO 0.2252 -O.0000788Fz0 +O.00043V0 -O.0000042(V0)

b. For radial-ply passenger car tyre,

C C = ((0.9997fz·Jc Co y 1 (0.9997f 7.0 y

+ ------------ co (5.3(IP1-IP0)-O.0916{(IP.}2-(IP0)2}}

33.5 +5.3(1P0) -O.0916(1P0)2 Y

+ ----------------....,,.. C C0 ( O.00043(V1-V0)-O.0000042{(V1)2-(V0)2} J

0.2252-0.0000788Fz0 +0.00043V0-0.0000042(V0 ) 2 Y

c. For normal truck tyre,

C C = ((0.9994fz, Jc C y 1 (0.9994fzo y O

91

(2.33)

... (2.34)

... (2.35)

... (2.36)

d. For wider truck tyre,

CC _ ((0.9994lz•)c C y 1 - (0.9994l:r.o y 0

2.5. The Existing Tyre Models Under Braking and Cornering

... (2.37)

The modelling of shear forces is usually started with the model for sideway

(lateral) force between tyre and road surface, generated either by a lateral (elastic)

force at the wheel rim, or by applying a sideslip angle to the tyre rolling motion. The

model then can be extended with the influence of imposed braking force or

longitudinal slip. Figure 2.10 shows the kinematics of tyre motion usually used in

the model.

The complexity of the model ranges from extremely simple functions to relatively

complex alogarithms. From the point of view of easiness, cost and time, the simple

function models are desirable. However, in real cases many factors are actually

involved in determining the magnitude of tyre-road friction (see Section 1.2). In this

regard, the simple models tend to loose their accuracy. On the other side, some

complex models which yield results accurate enough, are often too complex

computationally. For instance, a model developed by Pacejka and Fancher [1972]

required a hybrid (analog and digital) computer to solve the problem.

Most of the models used the friction parameter, which dependson several factors

such as: road surface, lubrication, sliding speed and normal pressure. The friction

parameter, at certain wheel slips and slip angles, are usually obtained by friction

measurement on a road surface, or by prediction from surface texture data such as

using .the method developed by Yandell et.al [1983], which is able to predict the

locked-wheel and (maximum) sideway frictions assuming all to be hysteretic with no

adhesion.

92

'I

FII..IC.TIOIJAL'""---- Fy F'ORC£

Fig.2.10. The kinematics of tyre motion.

In this present study, the existing tyre models are classified into 3 categories:

1. Analytical & semi-empirical models.

Models in this category are developed based on theoretical concepts of a

pneumatic tyre such as: tyre deformation, carcass rigidity, carcass elasticity.

Included in this category are semi-empirical models in which some of the

model's parameters are derived from experiment.

2. Empirical models.

The models are primarily based on the statistical analysis of data collected from

tyre testing.

3. Miscellaneous models.

This category covers models which are not included on the first or second

category, for example models based on the finite element method.

2.5.1. Analytical and Semi-empirical Models

Probably the most fundamental early works regarding lateral elasticity phenomena

1s that developed independently by von Schlippe and Temple [Hadekel (1952)].

Although both of them deduced the lateral elasticity characteristic by considering the

behaviour of the equatorial line of the 1yre ::iurface, they do have a different approach to

calculating lateral force. While von Schlippe integrated the tyre deflection over the

whole circumference, Temple just considered the tyre deformation over the contact

region, as responsible for generating lateral force. In other words, von Schlippe

considered the forces exened by the tyre on the rim, whereas Temple considered the

forces exerted by the ground on the tyre.

93

.......

y

Fig.2.11. The distorted equatorial line. [Hadekel (1952)].

er

WHEEL -,E-NTRE Pi..ANE

According to von Schlippe, the element of force (dFy) applied by the tyre to the

rim is proportional to the deflection (y) of the element of tyre (ds) in the equatorial

line. Temple, on the other hand, considered the element of force applied to the rim

by the element of tyre, deducted by the element of force required to stretch the

equator. According to Temple, the first and the second element are, respectively,

proportional to the deflection and to the curvature of the element of tyre in the

equatorial line. It seems that both approaches by von Schlippe and Temple

essentially will give similar results, since the difference is only in the matter of

choosing the free body.

Following the Temple approach, the element of force applied by the ground to

the tyre is given by (see Figure 2.11):

dFy = Kyds - c(d2y/ds2)ds (2.38)

Assuming that the shape of equator in the free region (near either end of contact

region) is an exponential curve, and by taking c = Ker (where K is a constant), then:

dFy = K(y-<r(d2y/ds2) }ds (2.39)

where a = "relaxation length" (the length of the sub-tangent of the exponential

curve), or the length in which the ordinate of the curve decreases in the ratio e:1.

Integration over the length of contact region, then:

FY = K .f:yds +Ka(y1 +y2 )

For comparison, von Schlippe approach gives:

f.+L FY = K yds

l+L

where L is the total length of the free part of the equator.

If the contact line is straight, then:

94

(2.40)

(2.40a)

C

--- ---- - .____ WHEEL CENTRE PLA~E

A0HE510N

~\..IOtt.JG, RE(.ION (zl- i'nJ

Fig.2.12. The equatorial line with adhesion & sliding region. [Hadekel (1952)].

(2.41)

On substituting the slope of equator at the front of the free region = -yifa = tancx = (y1-yJ/2l, then:

F7 = 2K(a+l)2tana (2.42)

The results derived above can be applied only when the sliding region is negligible,

or in other words when the slip angle is very :;malt.

Basically, an equatorial line is represented structurally as an elastically supported

string or beam under tension. In a string model the "kink" at the rear of the contact

region is permissible, whereas in a beam model the continuity is maintained

(although the curvature of the equatorial line at the rear of the contact print is

relatively high). The concept of stretched beam is based on the assumption that the

tyre carcass possesses a finite bending stiffness and, accordingly, that its equatorial

line has a finite width.

It must be pointed out that the lateral displacement of the free beam (i.e. outside

the contact region), is not only a function of the displacement of the leading and rear

points in the contact region (i.e. y1 and y2 in Figure 2.11) as on stretched string, but

also a function of the slopes of equatorial line at the leading and rear points.

Consequently, two more boundary conditions should be satisfied when dealing with

the stretched beam.

Figure 2.12 shows the extension of the model to the sliding region, neglecting the

bending stiffness of the running band (the zone which comes into contact with the

ground) in its own plane. For portion BC (i.e. sliding region), the F.quation (2.39) is

limited by the product of vertical load (F J to coefficient of (sliding) friction (µ),

95

Region of adhesion Slidiig region

Root surface p,8F I

u • Circumferential line

F• Root line

D• Line of reference

m=Wheel-center plaie

Fig.2.13. Root line and deflection of the profile lugs. [Fromm (1954)].

hence:

dF/ds = K{y-a2(d2y/ds2)} = µFz

Given Fz as function of s, then the lateral force:

F1 = 2K(<J+li2)2 tan<X + f21 µFz(s)ds J.h

(2.43)

(2.44)

As can be seen from Figure 2.11, the parameter <J depends on the tyre deflection

(y). The problem associated with the measurement of tyre deflection, limit the

practical application of Temple (and von Schlippe) theory. Some investigators such

as Segel [1966] and Rogers [1972], employed Equation (2.40) for shimmy and other

vehicle dynamic studies.

Fromm [1954] calculated the lateral stress (in adhesion region) as

~ = qwtana (2.45)

where q is a constant depending on the elastic characteristic of the rubber, w is the

distance of the element from the leading edge of contact region, and a is the side

slip angle (Figure 2.13). It appears that the significant factor assumed by Fromm is

the local distortion of the tread rubber instead of the carcass. In the sliding region,

the lateral stress is determined by the limiting value of the sliding region s, = pp.

The lateral force then is obtained by integrating over both adhesion and sliding

regions.

96

One imponant assumption made by Fromm, and also Julien [Hadekel (1952)], is

that the curvature of the root line (base line) is neglected. They argued that the

"height of camber" (OF - oA) is small in comparison to op. Hence, their theory is

based on what so called a rigid carcass. This carcass rigidity concept has been

employed by several investigators such as Bergman [1961], Livingston and Brown

[1970], Bernard et.al [1977], Golden [1981], and Gim and Nikravesh [1990]. The

analytical model developed by Gim and Nikravesh [1990, 1991] or the semi

empirical model developed by Bernard et.al [1977] was able to calculate the

frictional forces under braking and cornering, from tyre stiffness and two friction

parameters µ0 and µ1 (or Px and µy) obtained experimentally.

As extensions of Fromm's model, which did not consider carcass elasticity, Fiala

[1954] and Freudenstein [1961] developed models in which carcass deformation has

been approximated with a symmetric parabola. Since the carcass deformation itself is

a function of lateral force, the iterative process inevitably must be done to obtain the

lateral force. Such a model has been employed, by among other investigators, Sakai

[1981]. The concept of carcass rigidity, however, has been used by Sakai when he

extended the model for the influence of longitudinal slip.

2.5.2. Empirical Models

According to Maalej et.al [1989], the use of empirical models would result in a

more realistic response in the simulation. This is because empirical models are

developed from dynamic data of rolling tyres, while the other models are mainly

developed from static modelling of tyre structure. However, since many factors are

involved in influencing the frictional forces, any combination of these factors should

also ideally be carried out, in order to establish the empirical constants. Hence, for

every condition new experimentation must be carried out.

The direct derivation of the friction models from experimental data using

statistical approach and regression techniques, generally results in non-linear models

which do not provide any insight into the physical nature of the tyre [Maalej et.al

(1989)]. For example, an empirical model involving 13 coefficients, was developed

by Bakker et.al [1987] in which F Y and F x were approximated by sine function.

97

F Y = Dysin { Cyarctan(ByOy)} +Sv

Fx = Dxsin{Cxarctan(BxOx)}

where Oy = ( 1-Ey)( ex+Sh)+(E/l3y)arctan { By( ex+Sh)}, ex = slip angle,

Ox= (1-Ex)K+(ExfBx)arctan(BxK), K = percent slip.

(2.46)

(2.47)

The parameters (Dy, Cy, By, Sv, Ey, Sb, Dx, Cx, Bx, Ex) are functions of Fz, ex, and 'Y

(camber angle). The empirical coefficients ~ (i=l,13) associated with those functions

are obtained by statistical analysis of experimental data. The mathematical

expressions in Equations (2.46) and (2.47) are valid only for pure cornering or pure

braking.

Polynomial models of different orders and forms were also proposed to describe

the tyre frictional response. For example, a cubic fit for the lateral force as a

function of the slip angle is found to be convenient to present tyre test data. The

cubic fit can compensate for the main trends of variation of the tyre force.

Consequently, the lateral force may be approximated by a third-order polynomial.

2.5.3. Miscellaneous Models

A finite element method of (tyre) contact area has been proposed by Tielking and

Schapery [ 1980] to calculate the shear forces. The procedure employs a friction

coefficient that varies with the normal pressure and the sliding speed distributions

that are calculated in the contact region. The longitudinal and lateral components of

the shear force distribution are integrated to obtain the resultant traction forces

developed by the tyre in response to braking/driving and steering control inputs. As

a consequence of considering only a finite number of points in the contact region (=

19-29 for Fz 600 lb - 2200 lb), the resultant longitudinal force Fx at slip angle ex = 0°

can be a discontinous function of Sx (when Sx is small). A substantially greater

expenditure in computer time is required to include , the effect of tyre stiffness.

Kant et.al [1975] developed a model (able to predict Fx only) which relied on the

prediction of a friction-slip curve. The tyre is replaced by various cantilever beams

fixed at the rim radially and the free ends of the cantilevers are interconnected by

coil springs. The cantilevers represent tyre carcass stiffness, and the coil springs

refer to the contribution of the tread element. The stiffness of each cantilever and

98

coil spring, and the distance between cantilevers are the parameters required. No

comparisons are reported between predicted and measured values. It seems that some

refined models may be needed, to cater for a rather peculiar shape of the friction-slip

relationship resulted from this present model.

A spoke model, where the tyre structure consists of independently acting radial

spokes, has been developed by Sharp and El-Nashar [1986]. The spokes, which only

connected together by virtue of the connection of each one to the wheel hub, are

identical and elastic. Ten parameters were utilised in the later development of this

model [Sharp (1989)]. The capabilities of this multi-radial spoke model, to represent

the shear forces, are presumably much more determined by the way of obtaining its

suitable constant parameters.

2.6. The Candidate's Model For the Prediction of Tyre-Road Friction

Under Braking And Cornering

The frictional forces (or, when divided by wheel load, the friction force

coefficients) of an automobile-tyre travelling over a road surface, are believed to be

dependent upon many factors [Holmes et.al (1972)] which have, on each of them,

various degree of significance. These factors can be grouped into: (a) operational, (b)

tyre, (c) lubricant, and (d) pavement. Consequently, any attempts for modelling such

forces (longitudinal, lateral) must take into account all, or at least some of important

factors, which yield adequate result.

Beside the need of completeness as mentioned above, the following goals, as

stated by Bernard et.al [1977], should also be considered in tyre modelling: (1)

adequacy (i.e. errors within 5% or less when comparing the computed data with the

measured data under all conditions); (2) economical, from the point of view of the

expense involved in the empirical generation of tyre data suitable for deriving the

parameters needed in the computations, the user time required to feed these

parameters into the computer, and the expense involved in performing the

computations.

99

2.6.1. General Features of The Model

In this study, an analytical model for the prediction of friction between a

pneumatic tyre and road surface is presented. The input data needed is minimal and,

importantly, they can be determined or predicted by simple experiments and

measurements. This feature drastically reduces the cost associated with the extensive

experimentation which previously has been considered necessary.

For a known wheel load and pressure distribution the main inputsneeded are: tyre

stiffness, and either locked-wheel BFC or maximum SFC. The model then

determines the longitudinal and lateral forces that act on the tyre, for a given percent

slip and orientation of the tyre relative to the road. Figure 2.14 shows the simplified

diagram for calculation or prediction of the frictional forces. The ability to calculate

frictional forces under all conditions (braking, cornering, and combination of slips)

from tyre stiffness and one of the friction coefficients is a feature of this model.

This model requires one friction parameter less than that required by the

semi-empirical model developed by Bernard et.al [1977] or the analytical model

developed by Gim and Nikravesh [1990., 1991]. The additional (empirical)

parameters such as used by Bernard et.al [1977], are not required in the prediction

model. More importantly, since the locked-wheel BFC (or maximum SFC) can be

predicted from surface texture measurement [Yandell et.al (1983)], this model then

can be used to predict the tyre-road friction under all conditions (including percent

slip and slip angle), from tyre stiffness and surface texture data. As can be seen from

Figure 2.14, another advantage of this concept is that the tyre stiffness (and hence,

its measurement) can be regarded as independent of pavement/lubricant conditions.

The tyre stiffness input, however, must still be obtained experimentally either in

the laboratory or on the road surface. Attempts to predict the tyre stiffness from tyre

properties or other related parameters are left for future refinements. Nevertheless, it

is no longer necessary to experimentally obtain the tyre stiffness for any set of

operating conditions (load, inflation pressure, speed). By using the proposed model

for tyre stiffness variation, they can easily be predicted once the tyre stiffness for

one set of operating conditions has been known.

100

Operational _q h_, Tyre

I stiffness

Tyre

~, Frictional

-> forces

Lubricant h Locked-wheel BFC > > or .___

I > Maximum SFC

Pavement

Fig.2.14. Simplified diagram for calculation (or prediction) of frictional forces.

The wheel pressure P over the contact patch produces the normal force, and its

average value over the contact patch width is approximated by a trapezium, contact

length of 21, and slopes length of a & b. This modelled pressure distribution is

intended to offer flexibility in representing many types of actual pressure

distributions (as found in the literature review), by selecting the appropriate value of

its slope (a/21, b/21). In comparison to the pressure distribution models adopted by

Bernard et.al [1977], which can be used for uniform or symmetric-trapezoidal

distributions, this model provides the user with one additional option of an

unsymmetric trapezoidal distribution.

The weakness of any trapezoidal model, however, lies in the "kink" at points

along the circumference, especially in slope length b (see Figure 2.16) which in turn

may result in a less convex shape of friction-slip curve near the zero slip. On the

other hand, the smooth-shape elliptic or parabolic models of pressure distribution

[Bergman (1961), Livingston and Brown (1969), Sakai (1981), Gim and Nikravesh

(1990)], although considered better than the trapezoidal model in terms of their

continuity, cannot be used to represent unsymmetrical distributions.

In the following sections, the model descriptions, based mostly on the classical

analysis of the rolling tyre by Fiala [1954], are presented in detail. Some of the

general mathematical equations which follow in Sections 2.6.2, 2.6.4 and 2.6.5 are

derived in a similar manner by other investigators such as Dugoff et.al [1969, 1970],

and Bernard et.al [1977]. In Section 2.6.3 the candidate proposed a modification for.

the pressure distribution model, and recognized the concepts of the available friction

coefficient and the frictional force coefficient. The model development, by using the

101

differential approach, and an illustrative example for applying the candidate's tyre

model are given in Sections 2.6.6-2.6.8.

2.6.2. Geometry of Tyre-road Contact

The geometry of the idealized tyre-road contact region is shown in Figure 2.15.

Line 0-1-2 is the longitudinal centerline of the tyre-contact patch. The X-Y

coordinate system in the ground plane has its origin at point 0, the tread touchdown

point, with the X axis passing through point 2, the tread liftoff point. The tyre is

considered to have zero camber angle (although not described here, this effect can

usually be accounted for, approximately, through the introduction of an a equivalent

which is a function of the camber angle and tyre's camber stiffness).

Line 3-4 is the longitudinal centerline of the tyre carcass. Each point on the

carcass centerline is assumed to be elastically connected to the tread (line 0-1-2)

through orthogonal springs producing independent forces in the X and Y directions.

Hence, a point on the tread follows the carcass path as long as no shear force acts

on it. In the present analysis, the carcass centerline deformation (lateral) may be

approximated by a constant, instead of a parabola (such as used in Fiala's original

analysis). Thus line 3-4 lies in the vertical plane passing through the X axis.

At point 0 the local tread deformation is zero so the frictional stress is also zero.

Point 1 of the tyre tread centerline represent the sliding boundary. Points on line

segment 0-1 adhere to the ground surface without sliding. At point 1, the elastic

stresses due to tread deformation reach a value of tyre-road shear stress limit, and

the rubber begins to slide relative to the ground. Accordingly, the shear deformation

along line segment 1-2 is a function of the local sliding friction potential.

The hypothesized deformation condition prevailing at a typical point P (X,Y) on

the nonsliding segment (0-1) of the patch centerline is also shown in Figure 2.15.

The longitudinal coordinate of any contacting point P is equal to the product of the

tyre's longitudinal velocity (Vx) and the time interval (6t) from 0 to P.

102

------- p' CARCASS <t

Fig.2.15. Tyre-road contact geometry, and deformation m nonsliding portion of contact patch. [Dugoff et.al (1969)].

(2.48)

During the same interval At, the base point P' has moved a distance X', as given by

x· = n~At (2.49)

where ~ is the rolling radius of the tyre and n is the wheel spin velocity. The

longitudinal deformation of point P relative to point P' is given by

x -x· = v.,.At - n~At = c1 - ORJV.,.)V.,.At = s.,.x Longitudinal component of stress at point P:

a.,.= k.,.S.,.X

(2.50)

(2.51)

where k.,. is the longitudinal spring constant of tread element (in unit weight per unit

length, per unit width, per unit longitudinal deflection). The lateral deformation of

point P is given by

Y = Xtana = XSy (2.52)

Lateral component of stress at point P:

CSy = k,.(tana)X = k,.S,X (2.53)

where le,. is the lateral spring constant of tread element (in unit weight per unit

length, per unit width, per unit lateral deflection).

2.6.3. Pressure Distribution and Coefficients of Friction

The maximum normal pressure over the contact patch is given by

P mu = F/(21-0.5(a+b) }w (2.54)

where F2 is the normal (wheel) load, w is the width of the contact patch, 21 is the

length of the contact patch, a is the length of increasing pressure zone, and b is the

length of decreasing pressure zone (see Figure 2.16). In lateral direction the pressure

distribution is assumed to be uniform.

103

p ~le c--- •---> c-----------------~ x·

21 ----------------->

Fig.2.16. Idealized pressure distribution within the contact patch.

The normal pressure in each of 3 regions of the contact patch:

{pmax(21-X')/b 21-b~X'~21

p = P a~X'~21-b max

PmaxX'/a 0~x·~a

Tyre-road stress limit:

crmax = µP

(2.55)

(2.56)

Two types of friction coefficient are deduced from the literature (e.g. Gim and

Nikravesh [1990, 1991]): the available (or nominal) friction coefficient (µ) which

can be resolved into the longitudinal and lateral components (Ps, µJ; and the

frictional force coefficients (µx, µy). The available friction coefficient may be

assumed to be a linear function of sliding speed or slips, while the frictional force

coefficient is a non-linear function of slip and is zero at zero slip. At slips beyond

the critical value (i.e. complete sliding without any elastic deformation), the

frictional force coefficient equals the available friction coefficient. At below

critical slip the frictional force coefficient is less than the available friction

coefficient.

The concepts of the available friction coefficient and the frictional force

coefficient, are shown in Figures 2.17-2.18. The available friction coefficients at zero

slips (Pso, µao) can be regarded as static (adhesion) friction coefficients, which are ;;::

the locked-wheel coefficient (µd). One important feature of this model is the friction

coefficient when a = 1t/2 is assumed to equal µd [Bergman (1977a)]. The expressed

functions of the available friction coefficients are shown as follows.

Ps = Pso-(Pso-µd)Sx

Pa = ~-(~-µd)2a/1t

µ = Ps+(µa-Ps)20/7t

tana = SY

tan0 = S/Sx

104

(2.57)

(2.58)

(2.59)

.__ _________ Sx 0 i

(Jc(, :: 1.( o<o -({,(

«o - "") 2« ii'" .Uci--------------

1--------------,-......_ o( lt "i

Fig.2.17. Longitudinal and lateral components of (available) friction coefficient.

t.£so - ---

0 '---'-1 ------......,.5x

S"r 1

.U.y

tl.

:u.ccr I I

o<cr

Fig.2.18. Longitudinal and lateral components of frictional forces coefficient.

2.6.4. Location of Sliding Boundary

Using Equation (2.56) the position of the sliding boundary (point 1 in Figure

2.15) can be determined when the resultant elastic deformation stress just equals the

tyre-road shear stress limit, i.e.

(cr,.2+cr/)°'5 = crmax = µP (2.60)

Let "5 denote the longitudinal coordinate of the sliding boundary point. On

substituting Equations (2.51) and (2.53) into Equation (2.60), the "5 can be obtained

"5 = µP{(kxSi/+(k,.Sy)2J-0·5 (2.61)

Let "5' denote the longitudinal coordinate of the point on the tyre carcass associate

with point X5 , as in Figure 2.15. Then from Equation (2.50):

"5' = (l-Sx)"5 = µP(l-Sx){(kxSx)2+(kyS,}2}-0.S (2.62)

The resultant of the shear stresses in the adhesion region may now be computed

by (a) locating the sliding boundary "5' and (b) integrating the shear stresses over

the area from O to "5'. The integration over the area from "5' to 21 will yield the

shear force in the sliding region.

105

Due to the assumed trapezoidal form of the pressure distribution, there are 3

separate regions in which X5' may be located:

(1). A point in decreasing pressure zone (associated with small area of sliding

region, hence low Sx or Sy).

(2). A point in the central region of the pressure distribution.

(3). A point in increasing pressure zone (associated with large area of sliding

region, hence high Sx or Sy).

Case (1): 21-b S X5'

From Equations (2.62) and (2.55):

21-b s µPmaxC21-X')(l-Sl[)/b{(kl[Sl[)2+(kySy)2} 0.5

On inserting the limit value of X' into Equation (2.63), namely

X' = 21-b

and on substituting for P max as defined by Equation (2.54), then:

21-b S µFz<l-Sx)/{21-0.5(a+b)}w{(kxSx)2+(kySy)2} 0·5

sx < µFZSX

1-Sx - {21-0.5(a+b)}{21-b}w{(k S )2+(k S )2}0·5 X X y y

S µF S {(C S )2+ (C S )2}-0.5 X< ZX XX yy

1-Sx - {2-3(b/21)-(a/2l)+(a/2l)(b/2l)+(b/21)2}-

where: ex= 2I2wkx = longitudinal stiffness = dFxCSx)/dSX (at sx = 0%).

Cy = 212wky = lateral stiffness = dFy(a)/da (at a= 0°).

a/21, b/21 = the shapes of pressure distribution.

Case (2): a S X,' S 21-b


a S µP max(l-Sx)/{ (kxSx)2+(kySy)2} 0·5 S 21-b

Using the lower limit, namely

Xs' = a


a S µFz(l-Sx)/{21-0.5(a+b)}w{(kxSx)2+(kySy)2} 0·5

S µF S {(C S )2+(C S )2}-0·5 X< ZX XX yy 1-Sx - {2(a/21)-(a/21)2-(a/2l)(b/21)}

Using the upper limit, namely

~· = 21-b

106

(2.63)

(2.64)

(2.65)

(2.66)

(2.67)

(2.68)

(2.69)

(2.70)

(2.71)

(2.72)

and on substituting for P max as defined in Equation (2.54), then:

21-b ~ µFz<l-Sx)/{21-0.5(a+b)}w{(kxS,/+(kySy)2 }0.s

S µF S {(C S )2+ (C S )2}-0·5 X> ZX XX yy

1-Sx {2-3(b/21)-(a/21) +(a/2l)(b/2l)+(b/21)2}

Combining Equations (2.71) and (2.74), then: µF S {(C S )2+ (C S )2}-0·5 S µF S {(C S )2 + (C S )2 }-0.S

ZX XX yy < "< ZX XX yy 2-3(b/21)-a/21 +(a/2l)(b/21) +(b/21)2 - 1-S" - 2(a/21)-(a/21)2-(a/2l)(b/21)

Case (3): a~ Xs'


a~ µP max(X')(l-Sx)/a{ (kxSx)2+(kySy)2} 0·5

On inserting the limit value of X' into Equation (2.76), namely

X' = a


a~ µFzCl-Sx)/{21-0.5(a+b)}w{(kxSx)2+(kySy)2} 0·5

S µF S {(C S )2+(C S )2 }-0·5 X> ZX XX yy

1-S" {2(a/21)-(a/21)2-(a/2l)(b/21)}

2.6.5. The Calculation of Shear Forces

(2.73)

(2.74)

(2.75)

(2.76)

(2.77)

(2.78)

(2.79)

The shear forces (longitudinal and lateral) produced at the tyre-road interface are

calculated using the following procedure:

1. Inequalities (2.67), (2.75), and (2.79) yield the sliding region.

2. Equation (2.62) yields~· (sliding boundary).

3. The longitudinal (braking) force Fx, and lateral (sideway) force Fy, are obtained by

integrating the shear stress over the contact patch, viz:

Fx = fs'wcrxCX')dX' + cos8 f'.wcrm,x(X')dX' s

F = r··wcr (X')dX' + sin8 fl1wcr (X')dX' Y y .Jx_, max

s

Using Equations (2.50), (2.51), and (2.53), then:

(Jx = (kxSxX')/( 1-Sx)

cry = (kySyX')/( 1-Sx)

107

(2.80)

(2.81)

(2.82)

(2.83)

as limited by Equation (2.56), viz: crmax = µP

The integrations expressed by Equations (2.80) and (2.81) are summarized below

for Cases 1 through 3. The contribution of tyre stiffness and the available friction to

the frictional forces is typically shown in Figure 2.19. It may be deduced that the

critical slips (Sxcv Cler) are located at the boundary between Case 2 and Case 3.

Case (1): 21-b $ X5'

Combining Equations (2.62) and (2.55), then:

Xs' = µP ma/1-Sx)(2}-X5')/b{ (kxSx)2+(kySy)2 } 0·5

X5' = µFz(l-Sx)(2l-X5')/{21-0.5(a+b) }bw{ (kxSx)2+(kySy)2 }0.5

Equation (2.85) may be rewritten as:

Xs' µF/1-Sx) ------------------which further simplifies to: X' s 1 =-----------------...,... 21 {2(b/21)-(a/2l)(b/21)-(b/21)2H(C S )2 +(C S )2}0·5

1+ X X y y

µF/1-Sx)

Using Equations (2.80) and (2.81 ), then:

CS (X 'J F X X s

x = 1-Sx 21 + (µFzcos0){ 1-2(X5'/21 )+(X5'/21 )2}

{2(b/21)-(a/2l)(b/21)-(b/21)2 }

F = CYSY (Xs'J + (µFzsin0){1-2(X 5'/2l)+(X5'/21)2 }

Y 1-Sx 21 {2(b/21)-(a/2l)(b/21)-(b/21)2 }

Case (2): a $ Xs' $ 21-b

Combining Equations (2.62) and (2.55), then:

Xs' = µP maxCl-Sx)/{ (kxSx)2+(kySy)2 } 0·5

X5' = µFz(l-Sx)/{21-0.5(a+b)}w{(kxSx)2+(kySy)2} 0.5

Equation (2.91) may be rewritten, and further simplifies to:

Xs' µF/1-Sx) --------------21 {2-(a/21)-(b/2l)H(C S )2+(C S )2 }0·5

X X y y

108

(2.84)

(2.85)

(2.86)

(2.87)

(2.88)

(2.89)

(2.90)

(2.91)

(2.92)

f.

1,2 Case 3 Case 3 ._ _ __._ _____ ___._s~ 0 Sxcr 1

._ _ ____, _____ ___..._ o(

00 ~,r

Fig.2.19. The contribution of tyre stiffness (C) and the available friction (µ) to the frictional forces (F). At slips beyond the critical value, then J.1s = Jlx (or µa = µy).

Using Equations (2.80) and (2.81), then:

_ CxSx(X1'J F----- + x 1-Sx 21

(µF z cos0) { 1-0.5 (b/21 )-(X1'/2l)}

h-0.5(a/21)-0.5(b/21)}

_ CYSY ( X1' J (µFzsin0){1-0.5(b/21)-(X 5'/21)}

FY - 1-Sx l 21 + h-0.5(a/21)-0.5(b/21)}

Case (3): a ;;:: X5'

(2.93)

(2.94)

From Equations (2.55) and (2.56), it is found that since both pressure and stress are

linear functions of~·. sliding will occur at all points OS~· s a. Hence, Equations

(2.80) and (2.81) reduce to:

Fx = cos0 r wcrmu(X')dX' (2.95)

FY = sin0 f1wcrmu(X')dX' (2.96)

On carrying out these integrations, the following results are obtained:

Fx = µFzcos0 (2.97)

F y = µFzsin0 (2.98)

It is seen that for Case 3 there is no direct contribution of Cx and Cy-

2.6.6. Brief Procedure Using Locked-Wheel BFC

From Equations (2.57) and (2.58), the available longitudinal & lateral friction

coefficients at zero slip (J.1s0 and ~) must be obtained. This can be solved by using

the equations available for the frictional force coefficients.

109

Data: a/21, b/21, ex, Cy, µd, Fz Variable: Sx, a

Setting: a= O"-+ 8 = 0° then from Eq.(2.57): µ = µ, = f1(JI.o, SJ (la)

Using Eq.(2.93), Xa'/21 = fz{JI.o, SJ, and Eq.(la) then Fx = f3(JI.o, SJ (lb)

Differentiate Eq.(lb) and equalize to zero then dFxfdSx = fiJI.o, SJ = 0 (le)

Using Eq.(2.92) and Xa'/21 = a/21 then µ50 = fs(SJ

Substitute Eq.(ld) to Eq.(lc) then dF xfdSx = fiSx) = 0

obtain sx from f6, named sx = sxcr

Substitute Eq.(le) to Eq.(ld) then µ.o = fs(SJ = fs(Sxcr)

(le)

(lt)

Fig.2.20. Determination of the available longitudinal friction at zero slip (Ps0).

Data: a/21, b/21, ex, Cy, µd, Fz Variable: Sx, a

Setting: Sx = 0 -+ 8 = 1t/2 then from Eq.(2.58): µ = µa = g1(Jlao, a) (2a)

Using Eq.(2.94), Xa'/21 = gz<µa0, a), and Eq.(2a) then F = g3(µa0, a) (2b)

Differentiate Eq.(2b) and equalize to zero then dF da = &(µa0, a)= 0 (2c)

Using Eq.(2.92) and Xa'/21 = a/21 then µa0 = gs( a) (2d)

Substitute Eq.(2d) to Eq.(2c) then dF/da = gia) = 0

obtain a from g6, named a = <la (2e)

Substitute Eq.(2e) to Eq.(2d) then µa0 = gs(a) = g5(<Xcr) (2t)

Fig.2.21. Determination of the available lateral friction at zero slip (µao).

110

Data: a/21, bill, Cx, Cy, Pd• Fz Variable: Sx, a

Using procedure (1) and (2), detennine: Ps0, Pa0

J,

Calculate: Ps = Ps0-(µs0-Pd)Sx Pu = Pa0-Wa0-Pd)2<X/1t p = p.+(µ«-pJ28/1t

Calculate X.'/11 using Eq.(2.87) l

Yes J,

Case (1): Calculate Fx, Fy Eqs.(2.88, 2.89)

X,' ~ 21-b

No J,

Calculate X.'/ll using Eq.(2.92)

Yes J,

Case (2): Calculate F x• F y

Eqs.(2.93, 2.94)

Fig.2.22. Calculation of shear (frictional) forces._

1

No J,

Case (3): Calculate F x• F Y

Eqs.(2.97, 2.98)

The boundary conditions to be considered in this model are (Figs. 2.17 and 2.18):

a. The relationship between the available friction coefficients (Jls, µJ and slips (Sx,

a.) is assumed in this study to be linear (Equations (2.57) and (2.58)).

b. The frictional force coefficients (µx, µy) are assumed to equal the available

friction coefficients (Jls, µJ from the locations of the maximum value (at d.FxfdSx

= 0, or d.F/da. = 0) up to the locked-wheel position (at Sx = 100%, or a.= 1t/2).

As mentioned before the locked-wheel BFC is assumed to equal the frictional

force coefficient at a. = 1t/2, whereas the maximum BFC (at d.FxfdSx = 0) is not

necessarily equal to the maximum SFC (at dF /da. = 0). For a given locked-wheel

BFC (µd) the determination of the available friction coefficients at zero slip (µ50, Pa0)

are presented in Figures 2.20 and 2.21. The frictional forces then are calculated

using procedure shown in Figure 2.22. The derivation of mathematical equations will

be given in an illustrative example.

111

2.6.7. Illustrative Example

The following data was taken from Bernard et.al [1977]: Cx = 10000 lbs/slip, Cy

= 220 lbs/deg = 12605 lbs/rad, Fz = 1100 lbs. By using the relationship of

longitudinal force <Fx) as a function of percent slip (Sx), the locked-wheel braking

coefficient (µd) of H78-14 tyre (at 28 psi) is obtained from their curve as 0.87.

A good match between measured data and tyre model will be influenced by an

appropriate choice of pressure distribution (a/21, b/21). From previous information

[Bode (1962), Lippmann and Oblizajek (1974)], it is found that for braking/cornering

mode the ranges of a/21 and b/21 are normally about 0.05-0.20 and 0.10-0.30,

respectively. These values are dependent also on tyre type and operational factors

(e.g. speed, inflation pressure, and load). In this example, the a/21 and b/21 are

selected as 0.06 and 0.20 which are found to give closer agreement between

measured data and the tyre model.

1. Determination of µso• µscr• and Sxcr-

Since Sxcr is located at the boundary between Case 2 and Case 3, equations in Case 2

can be used for the determination of the required parameters.

a. From Equation (2.57):

µ = µs = µso-(µs0-µd)Sx

b. From Equations (2.93), (2.92) and (la):

F = -(CxSx>-1(1-Sx)(µsO-µsOsx +µdSx)2F;

X {2-(a/21)-(b/21)}2

F z { 4-2(a/21) + (a/21 )(b/21) + (b/21 )2- 4(b/21 )} +---------------(µs0-µsOsx +µdSx>- 1 {2- (a/21)- (b/21)}2

c. Differentiate Equation (1 b) to Sx and equalize to zero:

(CxSS2Cx{µso-~0Sx +µdSx)2Fz + 2(½Sx)°1( 1-Sx)(~o-µd)FzCµso-~oSx +µdSx)

(la)

... (lb)

+ (µd-µsoH 4-2(a/2l)+(a/2l)(b/2l)+(b/21)2-4(b/21)} = 0 ... (le)

d. Using Equation (2.92) and Xs'/21 = a/21, then:

µ = (a/21){2-(a/21)-(b/21) }CxSxF/(1-Sx)"1

µso = (µ-µdSx)(l-Sxf1

= (a/21){2-(a/21)-(b/2l)}CxSxF/(1-Sx}"2-µdSx<l-Sx)" 1

112

... (ld)

e. Substitute Equation (ld) to Equation (le):

(CxSS2Cx{ACxSxC1-Sx)°1 }2 + 2A{ACxSxC1-Sx)"2-pdSxFz<l-Sx)"1-p~z}

+ 2B{p~z-ACxSxC1-Sx)°2+pdFzSx(l-Sx)°1} = 0

And rewritten as:

S = c;1pdFz(B-A)+0.5A 2

x c;1pdFz(B-A)+AB-A2

where: A = (a/21){2-(a/21)-(b/21)}

B = {2-(a/2l)+o.5(a/2l)(b/21)+0.5(b/21)2-2(b/21)}

f. Inputing the values of a/21, b/21, Cx, Pd and Fz to Equations (le) and (ld):

sxcr = 10000-1(0.87)1100(1.566-0.104)+0.5(0.104)2 = 0.4969 10000-1(0.87)1100(1.566-0.104)+0.164-0.011

Pscr = ACxSxF/(1-Sx)"1 = 0.9374

Pso = ACxSxF/(1-Sxf2-µdSx(l-Sx}" 1 = 1.0039

(le)

It is seen that the above parameters are governed by Cx. It may be deduced that the

steepness of Fx versus Sx beyond the critical slip will also be determined by Cx.

2. Determination of Pa0, Pacr, and a.cr-

Similar to step ( 1 ), equations in Case 2 then will--be used:

a. From Equation (2.58):

P = Pa = Pa0-(Paa-µd)2a/7t

b. From Equations (2.94), (2.92) and (2a):

-(CY tana.t1(Pao-Pa02a./1t + pd2a./1t )2F; F =------------

Y {2- (a/21 )- (b/21 )P

F { 4-2( a/21) + (a/21 )(b/21) + (b/21 }2- 4(b/21 )} + z

(pa0-Pa0 2<X./7t + pd2<X./1t t 1 {2-(a/21 )-(b/21)}2

c. Differentiate Equation (2b) to a. and equalize to zero:

(sina.)·2cy·1(pa0-Paa2a/1t+pd2a/1t}2Fz

+2(tana.)"1Cy·1(pa02/1t-pd2/1t)(pa0-Pa02a/1t+pd2a/1t)F2

+(µd2/7t-µa02ht){ 4-2(a/2l)+(a/2l)(b/2l)+(b/21)2-4(b/21)} = 0

d. Using Equation (2.92) and X5'/21 = a/21, then:

p = (a/21){2-(a/21)-(b/21) }Cytana.Fz·l

Pa0 = (µ-µd2a/1t)(l-2a/1t)" 1

= { (a/21)(2-(a/21)-(b/2l))Cytana.F/-pd2a/1t }( l-2a/1t)"1

113

(2a)

... (2b)

... (2c)

... (2d)

e. Substitute Equation (2d) to Equation (2c):

(sina.)·2cy·1(ACytana.)2Fz·1

+2(tana.)·1c/ { (ACytana.-pd2a./1t)(2/7t)( 1-2a./1t)"1-pd2/1t}

+2B {pd2/1t-(ACytana.F/-pd2a./1t)(2/1t)(l-2a./1t)·1 } = 0

And rewritten as:

a.= 1t/2-2sina.cosa.(B/A-1)+2(cosa.)2C/pdFzCB/A-l)A1

where: A = (a/21){2-(a/21)-(b/21)}

B = {2-(a/2l)+o.5(a/2l)(b/21)+0.5(b/21)2-2(b/21)}

The value of a. which satisfied Eq. (2e) can be obtained by trial and error.

f. Inputing the values of a/21, b/21, Cy, Pd and Fz to Equations (2e) and (2d):

a.er= 1.5708-13.5761+12.6674 = 0.6621 radians

Paa= (0.06)(1.74)(12605)(0.7794)(1100)"1 = 0.9324

Pa0 = (0.9324-0.3667)(1-0.4215)"1 = 0.9779

(2e)

Similar to step (1), it may be deduced that the steepness of FY versus a. beyond the

critical slip angle will also be determined by Cy.

3. Calculation of frictional forces.

The calculation of frictional forces (F x• F y) is carried out using Equations (2.88,

2.89) for Case 1, Equations (2.93, 2.94) for Case 2, and Equations (2.97, 2.98) for

Case 3. One example is given here, that is for Sx = 0.1 and a. = 4°.

Note: 4° = 0.0698 radians, SY= tana. = 0.0699.

8 = arctan(S/Sx) = 34.95° = 0.61 radians.

a. Calculations of Ps, Pa and p :

Ps = Pso-(Pso-Pd)Sx = l.0039-(1.0039-0.87)(0.1) = 0.9905

Pa= PaO-(PaO-Pd)2a./1t = 0.9779-(0.1079)2(0.0698)/1t = 0.9731

P = p5+(pa-Ps)28/1t = 0.9905+(-0.0174)2(0.61)/7t = 0.9837

b. Using Equation (2.87): X'

- 5 - ______ l ----- = 0.6774 21 1 + {0.348H(1000)2+(881.1)2}0-5

(0.9837) (1100) (0.9)

1-b/21 = 0.8 > 0.6774 ~ Not Case 1

c. Using Equation (2.92):

Xs' (0.9837)(1100)(0.9) - - -,----,,-,----------- = 0.42

21 { 1.74}{(1000)2 + (881.1)2}0-5

a/21 = 0.06 < 0.42 < 1-b/21 = 0.8 ~ Case 2

114

d. For Case 2, Fx and FY are calculated using Equations (2.93) and (2.94):

F x = 196+489 = 685 lbs

Fy = 173+342 = 515 lbs

Using the same procedure as above, any forces at other combinations of Sx and a.

can be calculated. Some of the results are presented in Tables 2. 7 and 2.8, and

plotted in Figure 2.23. The relationships between braking force coefficient (µx) and

sideway force coefficient (µy) are presented in Figures 2.26 and 2.27.

Table 2.7. Results of Fx and Fy, at a. = 4°, showing contribution from tyre stiffness

sx (½) (µ)

Fx

(Cy)

(µ)

Fy

(C C) and friction coefficient (µ) X' 'V

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 196 128 80 52 0 0 0 0 0 0 489 785 907 963 1024 1013 1001 987 974 0* 685 913 987 1015 1024 1013 1001 987 974

430 173 56 24 11 0 0 0 0 0 248 342 275 211 169 143 118 100 86 75 678 515 331 235 180 143 118 100 86 75

* For Sx = 0 ~ cos0 = 0 ~ F x = 0 + 0 = 0 lbs. § For Sx = 1 ~ µx = µdcosa. ~ Fx == µdFzcosa. = 955 lbs. # For Sx = 1 ~ µY = µdsina. ~ FY == µdFzsina. = 67 lbs.

1.0 0

955 955§

0 67 67#

Table 2.8. Results of Fy, at Sx = 0, showing contribution from tyre stiffness (Cy) and friction coefficient (µ).

a. (0)

0 4 8

12 16 20 24 28 32 36 40 44

(~) (µ) Fv a. (0) (CJ 0 0 0* 48 0

430 248 678 52 0 211 679 890 56 0 138 820 958 60 0 101 888 989 64 0 79 927 1006 68 0 64 952 1016 72 0 53 968 1021 76 0 45 979 1024 80 0 38 988 1026 84 0 0 1023 1023 88 0 0 1018 1018 90 0

*Fora.= (f ~SY= 0 ~ sm0 = 0 ~ FY = 0 lbs. § For a. = 9(f ~ µY = µd ~ FY = µdFz = 957 lbs.

115

(µ) Fv 1012 1012 1007 1007 1002 1002 997 997 991 991 986 986 981 981 975 975 970 970 965 965 960 960 957 957§

Force [x1000 lbs) 1.2

1

0.8

0.6

0.4

0.2

fx

D +-""'T""-..--r---r----r-----r-r---r----r---.

D 10 20 30 40 50 60 70 BO 90 100

5x (%)

1.2 Force (x1000 lbs)

1

F, 0.8

0.6

0.4

0.2

0 -t--"""'T"'-r---r----r-~"""'T"'-r---r---.

D 10 20 30 40 50 60 70 8D 90

ex (deg)

Fig.2.23. Tyre model(--) and measured data (o o o) of H78-14 tyre (28 psi) with a/21 = 0.06 and b/21 = 0.20. Left: Fx & FY vs Sx at a.= 4°. Right: Fy vs a. at Sx = 0.

As can be seen from Tables 2.7 and 2.8, the contribution from tyre stiffness (Cx,

Cy) decreases with increasing Sx (or a.), whereas the contribution from friction

coefficient (µ) continues to increase. At Sxcr (or a.er) there is no direct effect of tyre

stiffness. However, the frictional forces are still affected indirectly, through the

influence on the steepness of the friction coefficient, from Pso ( or Pao) to µd. At Sx = 100% (or a. = 900) the magnitude of frictional forces coefficient then equals to µd

which is independent of tyre stiffness.

A BASIC computer program (see Appendix A) has been written for the routine

calculations of frictional forces. The computing time for obtaining a.er has been

minimised without loosing accuracy of the result. This can be done by providing two

loops for achieving the convergence. The first loop with large increments is intended

to quickly bring the position into the region of convergency. The second loop then

allows smooth increments to be executed within this region. As a result, about 20

seconds CPU time ( using PC A T-286) are required to run the program.

Figure 2.23 shows the comparison of tyre model and measured data of H78-14

tyre (28 psi). This figure demonstrates the extent to which the tyre model agrees

with experimental data.

116

Force (x1000 lbs) 1.2

1

o.a

0.6

0.4

0.2

fx

D +--........ --r-----r--.,--,...........----r-----..-----r--.

0 10 20 30 40 50 60 70 80 90 100

Sx (%)

1.2 Force (x1000 lbs)

1

/-------- F, o.e

: I I

0.6 I I ' ' I

0.4 '

0.2

D +-----r-----r-......... ""'T""----r-r---r--,----,

D 10 20 30 '10 50 60 70 BO 90

0< (deg)

Fig.2.24. Tyre model of H78-14 (28 psi) with b/21 = 0.20, using a/21 = 0.11 (--) and a/21 = 0.01 (------). Left: Fx & FY vs Sx at ex= 4°. Right: FY vs ex at Sx = 0.

Force (x 10 0 0 lbs) 1.2

1

D.B

0,6

0.4

0.2

D +--........ --r-----r--.,--..--....----r-----..-----r----.

D 10 20 30 40 50 60 70 80 90 100

sx (%)

Force (x1000 lbs) 1.2

1

D.B

0.6

' ' 0.4 ' ' I ' ' I

0.2 ' '

D +-----r-----r-......... ""'T""----r-r---r--,----.

D 10 20 30 'ID 50 60 70 BO 90 0< (deg)

Fig.2.25. Tyre model of H78-14 (28 psi) with a/21 = 0.06, using b/21 = 0.01 (--) and b/21 = 0.40 (------). Left: Fx & Fy vs Sx at ex= 4°. Right: Fy vs ex at Sx = 0.

Figure 2.24 shows the sensitivity of using other values of a/21, where the tyre

model using a/21 = 0.01 is compared with a/21 = 0.11. As can be seen, a big

difference is found near optimum value of Sx (for Fx vs Sx at ex = 4°), and near

optimum value of ex (for FY vs ex at Sx = 0). For the curve of FY versus Sx, a small

difference is found near the low and middle values of Sx.

117

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Uy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

U X

Fig.2.26. The computed relationship between Px and Py at various a. (0 ), with a/21 = 0.06 and b/21 = 0.20.

Unlike a/21, the change of b/21 is found to be nearly small, such as shown in

Figure 2.25. However, a significant difference still occurs for the curve of FY vs Sx,

near the low values of Sx. These results suggest an optional model with unsymmetric

pressure distribution is justified.

Figure 2.26 shows the computed relationship between braking force coefficient

(Px) and sideway force coefficient (µy) at various slip angles (a.). This figure

represents the braking-in-a-tum condition, obtained from braking the tyre up to

wheel lock up. A curve (dashed line) fitted to the ends of. the curves plotted for the

individual slip angles, describes the slide values of the resultant braking-cornering

coefficient. Since the friction coefficient at a. = 900 is assumed equal to the

locked-wheel coefficient, this curve is approximately in the form of a circle through

points Py= µd and µx = µd.

Figure 2.27 shows the computed relationship between braking force coefficient

(Px) and sideway force coefficient (py) at various percent slips (Sx). This figure

represents the turning-in-braking condition, obtained from turning the braked tyre.

As can be seen from this figure, when the tyre is steered until 90° (in theoretical

situation), the Py will reach the value equal to Pd and will be independent from Sx-

118

1 Uy

0.9 l<lllli:i=~~~~-;~-::: 0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

D (o/o !.lip)

0 -+-----r-~"--T---r---,---.----,1---r-.......... i-'--i

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

U X

Fig.2.27. The computed relationship between Px and µY at various Sx (% ), with a/21 = 0.06 and b/21 = 0.20.

If a/21 = 0 is applied in this model, the curve of the relationship between braking

and cornering at Sx = 100% (or dashed line in Figure 2.26) will be exactly in the

form of a circle through points µY = µd and µx = µd. Referring to equations in Section

2.6.7 (Equations le and 2e) with a/21 = 0, the sxcr = 100% and a.er = x/2, hence both

µscr and µacr equal to µd. All other values of µx and µY will be less than µd. In other

words, all other curves of the relationship between braking and cornering will be

inside the circle of Sx = 100%.

The application of b/21 = 0 (or complete adhesion), on the other hand, will lead

to very high frictional forces at below Sx = 100%. Referring to equations in Case 1

(Equations 2.88-2.89 for b/21 = 0) with no sliding region, except in the vicinity of 21

[Livingston and Brown (1969)], the frictional forces will be independent from the

available friction (µ). For current practice, such as the use of a uniform pressure

distribution, a slightly greater b/21 than zero (say, b/21 "" 0.001) will yield more

realistic results.

119

Data: a/21, b/21, Cx, Cy, Paa• Fz Variable: Sx, a.

Setting: Sx = 0 ~ 0 = rr./2 then from Eq.(2.58): p =Pu= g1(Puo, Pd• a.)

Using Eq.(2.92) and "Xs'/21 = a/21 then a.er = &(Paa) and Pu0 = gipJ

Using Eqs.(2.94, lk), and "Xs'/21 = &(Pu0, Pd• a.)

(lk)

(11) (lm)

then F = gs(pu0, Pd• a.) (ln)

Differentiate Eq.(ln) and equalize to zero then dF/da. = giPu0, Pd• a.)= 0 (lo)

Substitute Eqs.(11, lm) to Eq.(ln) then dF /du = gipd) = 0

obtain pd from g7 (lp)

Substitute Eq.(lp) to Eq.(lm) then Pu0 = gipd) (lq)

Fig.2.28. Determination of the available lateral friction at zero slip (µa0),

2.6.8. Alternative Model Using Maximum SFC

With a little modification in procedure, an alternative model using maximum SFC

can be built similar to the general model (using locked-wheel BFC). Another BASIC

computer program (see Appendix B) has been written for the routine calculations of

frictional forces. In contrast to the locked-wheel BFC, no iteration process for acr is

needed in this program. Hence, by using the maximum SFC as an input parameter,

the CPU time required is only about 2 seconds.

Figure 2.28 shows the determination of nominal lateral friction coefficient at zero

slip (µao), When obtaining µd from this procedure (Equation 1 p ), the determination

of the available (nominal) longitudinal friction at zero slip (µ50) is then the same as

that derived from the model of locked-wheel BFC (see Figure 2.20). The calculation

of frictional forces is also the same with the procedure for locked-wheel BFC, by

knowing µd as above (see Figure 2.22).

120

CHAPTER 3. THE PARAMETRIC STUDY OF THE MODEL

Notation (Used frequently in Chapter 3) a, b = length of increasing & decreasing pressure zone (unit length)

½ = longitudinal slip stiffness (force/slip, or force/fraction slip), or longitudinal deformation stiffness (force)

<;. = lateral slip stiffness (force/degree, or force/radian), or lateral deformation stiffness (force)

= longitudinal force (force) = lateral force (force) = vertical load (force) = tyre contact length (unit length)

Sx = longitudinal slip (%, or per fraction slip) Sxcr = critical longitudinal slip(%, or per fraction slip), see Fig.2.18

SY (l

= tana. = slip angle (degree, or radian)

a.er = critical slip angle (degree, or radian), 5ee Fig.2.18 = available friction coefficient p

Pd = locked-wheel friction coefficient Px, Py = longitudinal & lateral frictional forces coefficients Ps, Pa = available longitudinal & lateral friction coefficients

Pscr• Paa- = critical longitudinal & lateral friction coefficients, see Fig.2.18

3.1. General

The main purpose of this study is to provide a better explanation for the effects

of the parameters influencing tyre-road friction using the proposed model, and to

compare the response of the model with other tyre models.

In the first attempt, the data needed for this study are calculated using the model

explained in Section 2.6. The ranges of the main parameters to be investigated are

taken as follows.

ex = 10000 lb/slip - 60000 lb/slip

Cy = 10000 lb/radian - 40000 lb/radian

Pd = 0.3 - 0.9.

121

The ranges above are considered to represent the normal values which commonly

occur in practice. The other two parameters used (i.e. pressure distribution, and

wheel load) are keep constant, and their values are chosen arbitrarily. In this case

their values are made similar to those in the illustrative example presented in Section

2.6, as follows.

a/21 = 0.06, b/21 = 0.20, Fz = 1100 lbs.

The effects of the main parameters on critical slips (Sxcr, acr), critical friction

coefficients (Pscr, p<XCI'), and frictional force coefficients (Px, Py) are then calculated.

The variables which significantly affect the tyre-road friction are identified by

means of regression analysis. All parameters (except b/21) are then varied to form a

2x2x2x2x2x3x3 factorial, and their effects on frictional force coefficients (Px, py) and

on resultant of frictional force coefficient (PrsJ are calculated. This statistical method

is usually used to study the effects of several independent variables on a dependent

variable [Box et.al (1978), Draper (1981)].

In the second attempt, the responses of this model are compared with other tyre

models. The behaviour of five tyre models (i.e. HSRI-I, HSRI-11, HSRI-ill, Good

Year, Sakai), which have been summarised by Tielking and Mital [1974], will be

compared with the behaviour of the candidate's model.

3.2. Results of The Main Parameters

The results of the effects of the main parameters are presented in the following

sections through Figures 3.1-3.42. In those figures the subscripts of 10, 35, 60 are

used for longitudinal stiffness (Cx) = 10000, 35000, 60000 lb/slip; while the

subscripts of 10, 25, 40 are used for lateral stiffness (Cy) = 10000, 25000, 40000

lb/radian.

122

3.2.1. The Effects of Longitudinal Stiffness (ex)

The degree of signification:

a. The effects of longitudinal stiffness ex seem very significant on both critical

longitudinal slip (Sxcr) and critical longitudinal friction coefficient (Pscr), such as

shown in Figures 3.1 and 3.2.

b. The effects of ex on frictional force coefficients (µx, µy) are shown in Figures

3.3-3.12, where the change of ex seems to have little effect to the Px and Py·

However, at low values of longitudinal slip (Sx) and slip angle (a.), the effect of

ex on Px seem significant, such as shown in Figures 3.3-3.4 and 3.10-3.11.

The form of the effects:

1. The critical longitudinal slip (Sxcr) will decrease as the longitudinal stiffness ex

increases. The greater drop of Sxcr with increasing ex is found at the higher

friction surface (represented by the higher locked-wheel coefficient µd).

2. The critical longitudinal friction coefficient (Pscr) will increase as the ex increases.

3. The frictional force coefficients (µx, µy) will increase as the ex increases. The rate

of increasing (µx, µ,) with increasing ex will be greater at low values of percent

slip (Sx) and slip angle (a.).

The results above support the statement previously mentioned, that the

contribution from tyre stiffness decreases with increasing Sx (or a.), whereas the

contribution from the friction coefficient continue to increase.

123

0.6 S.cr

o.s

0.1

0.3

0.2 ·····, ...

0.1

.. .... ............ -.......... __ ------

ud 0.9 0.6 0.3

0 ,_____,J.____. _ __. _ ___., _ __._ _ _.._ _ __. 0 10 20 30 -10 50 60 70

c. (1000 lb/slip)

Fig.3.1. ex vs SXCT' for various µd.

1.2 U,...ax"'---------------.

1

0.B

0.6

0.-1

0.2

s.

---~~---------- ~:i :: ...... ----

D .....____, ___ ___. _ __. _ __._ _ __._ _ __._ _ _.

D 10 20 30 10 50 60 70 C1 (1000 lb/shp)

Fig.3.3. ex vs µx, a = 00, µd = 0.6.

1.2 Ux

1

D.B

0.6

0.1

0.2

D D

c, = 25

Sx

-=-==-==----_-_-_-_-_-_-_---=-=-= 0.5 0.2

-·------- .. --------------0.1

1D 20 30 -10 50 60 70

c, (1000 lb/slip)

Fig.3.5. Cx vs µ1 , a = 8°, µd = 0.6.

124

1,2 u,cr

1

0.B 0.6

0.6 ---- 0.3 ____ ....... --

...... ------____ .... 0.-1

0.2

D D 10 20 30 10 so 60 70

C1 (1000 lb/shp)

Fig.3.2. ex vs µscr• for various µd.

1.2 .... u1----------....,,,...-~ c, = 25

1

O.B

0.6

0.-1

0.2

- ~--.-;;;;;~:::;;;;;:: 0.2 0.5 =2 --------------- 0.1 -----.... --___ ..

D ....._ ______ ___, _ ___. _ __._ _ __._ _ __._ _ __.

0 10 20 3 0 10 50 60 70

ex (10 0 0 lb/ slip)

Fig.3.4. ex vs µ1 , a = 4°, µd = 0.6.

1.2 .... u•------------~ c, = 25

1

D.B

0.6

0.1

0.2

Sx 0.5

0.2

-------------------------------- 0.1 D .__ ___ .____. ___ __. _ ___., _ __._ _ _.._ _ __.

0 10 20 3 0 10 50 60 70

c. (1000 lb/shp)

Fig.3.6. C1 vs µ1 , a= 32°, µd = 0.6.

1.2'"'u''-----------=------=-:, c, = 25

1

O.B

0.6

0.4

0.2

0(

32 8

-------- 4 ... --- -- ---- .. ------------

D...._____, _ __.._ _ __._ _ __.__....._ __ ~

0 10 20 30 40 so 60 70

C 1 ( 10 0 0 lb / slip)

Fig.3.7. e. vs µy, S1 = 0.1, µd = 0.6.

u 1.2 -----------------. c, = 25

0.8

0.6

O.'I

0.2

0(

32

B ••••••••••••••••••••••••••••• - - - 4

0 ...._____,.....___.._ _ _._ _ __.__....._ __ ~

0 10 20 30 4 0 S D 60 70

c. (1000 lb/shp)

Fig.3.9. ex vs µY' Sx = 0.5, Pd = 0.6.

1.2 """Ux"--------------~ c, = 25

1

D.B

0.6

0,4

0.2

0(

----- 0 ------::: ..... --- 4

•... :::::---•·•••· --··· B

32

0 ~___,.....___.._ _ __._ _ __.__....._ __ ~

D ID 2D 3D 4D 5D &D 70

c. (1000 lb/sllp)

Fig.3.11. ex vs µ1 , Sx = 0.2, Pd = 0.6.

125

u 1·2 ,..,_ __________ c-=--, -= -=2-=5

1

O.B

D.6

0.4

D.2

X

32

8

····················-······---- 4

0 ._____. _ __,__ ......... _ _._ _ _.__..&.-____, D 10 2D 3 D 'ID 5D 60 70

ex [1000 lb/sllp)

Fig.3.8. e. vs µy, Sx = 0.2, µd = 0.6.

1.2 ... u•---------------=--c, = 25

0.8

0.6

0.4

D.2

0(

0

....... ------------------ -- -- --·· 4

_____ .. ----------------B

---------- 32 0 ...._____,.....___.._ _ _._ _ __.__....._ _ _._____,

D 1D 2D 3D 10 50 6D 7D

c. (1000 lb/slip)

Fig.3.10. ex vs µ1 , S1 = 0.1, µd = 0.6.

u 1,2 ... x-------------------=-

C, = 25

0.8

D.6

0.4

D.2

=-=------------· ------------·-0(

0 B

32

o~~-__.._ _ __._ _ __._ ____ ~ D ID 2D 3D 'ID SD 6D 7D

C1 [1000 lb/sllp)

Fig.3.12. ex vs µ1 , S1 = 0.5, µd = 0.6.

3.2.2. The Effects of Lateral Stiffness (Cy)


a. Similar to Cx, the effects of lateral stiffness (Cy) seem very significant on both

critical slip angle (acr) and critical lateral friction coefficient (Paa), such as shown

in Figures 3.13 and 3.14.

b. The effects of Cy on frictional force coefficients (µx, µy) are shown in Figures

3.15-3.24, where the change of Cy seems to have little effect to the µx and µY. The

effect of Cy on µy, however, seem significant at low values of longitudinal slip

(Sx) and slip angle (a), such as shown in Figures 3.15-3.16 and 3.22-3.23.


1. The critical slip angle (acr) will decrease as the lateral stiffness (Cy) increases.

The greater drop of acr with increasing Cy is found at the higher friction surface

(represented by the higher locked-wheel coefficient µd).

2. The critical lateral friction coefficient (µaa-) will increase as the CY increases.

3. The frictional force coefficients (µx, µy) will increase as the CY increases. The rate

of increasing (µx, µy) with increasing Cy will be greater at low values of percent

slip (Sx) and slip angle (a).

The results above again support the statement previously mentioned that the

contribution from tyre stiffness decreases with increasing Sx (or a), whereas the

contribution from the friction coefficient continuesto increase.

126

oc., {raaJ 1.2~....:....---'-----------,

1

0.B

0.6

0.4

0.2

D.__ _ __,_ __ ...__ _ __. __ __.__~

0 ID 20 30 10 50

c, (1000 lb/rad)

Fig.3.13. Cy vs ex.er, for various Pd·

u 1,2 .-L---------------,

0.8

0.6

0.4

0.2

0( ~ 3 2 ::::::----==:. ~,, --B ____ •••••• 4 ••

o~-~----~~---~ 0 10 20 30 10 50

c, (1000 lb/rad)

Fig.3.15. Cy vs Py, Sx = 0, Pd = 0.6.

1.2 ,...u,,.__ _________ =-----=-= c. = 35

1

0.8

0.6

0.4

0.2 -- ........ --------- .. --.. --------

0(

32

8

D.__ _ __._ __ ...._ _ __.. __ _._ _ ____.

0 10 20 30 40 50 c, (1000 lb/rad)

Fig.3.17. Cy vs Py, Sx = 0.2, pd = 0.6.

127

u 1 _2 acer

1

0.8

0.6

0.4

0,2

_ 0.9 ---------0.6 --------........ ----- 0.3 ----........

--------- u d

D.___......_ __ ...__ _ __. __ _._ _ ___,

0 10 20 30 40 50 c, (1000 lb/rad)

Fig.3.14. CY vs Pa.er, for various Pd·

u 1.2 y c.= 35

1

0.8

0.6

0.4

0.2

D

.. ........................ ------- ----

()(

32 8

0 ID 20 30 40 50

c., (1000 lb/rad)

Fig.3.16. CY vs Py, Sx = 0.1, Pd= 0.6.

u 1.2 ~---------......,,.C-1 -="""'3,--=,5

1

0.B

0.6

0.4

0.2

0(

32

8

--------------------------- 4 0 .__ _ __._ __ _,__ _ __..'--_ __._ _ __,

0 10 20 30 10 50 c., (1000 lb/rad)

Fig.3.18. Cy vs Py, Sx = 0.5, Pd= 0.6.

1.2 ....;u•=-------------=-----:-:, c. = 35

1

D.B

D.6

D.4

D.2

----------------- -

s. 0.5 0.1

D.__ _ __._ __ ....._ _ ___. __ ......._ __

D 1D 2D 3D 1D so c, (1000 lb/rad)

Fig.3.19. Cy vs µ1 , a= 4°, µd = 0.6.

1.2-u·=---------------"'=" c.= 35

1

O.B

0.6

0.4

D.2

s. 0.5

0.2

-------------------------- 0.1 D.__ _ __._ __ ....__ _ __,'---_ __._ _ ____.

D 1D 2D 3D 4 D 5D c, (1000 lb/rad]

Fig.3.21. Cy vs µ1 , a= 32°, µd = 0.6.

1.2 ...,.UY=---------------=--C x = 35

O.B

0.6

0.2

----------s. 0 --

--- ----------- ---------- 0 .1 ---------------0.2

o.s

D...__ _ __._ __ ......._ _ __,'------'------' D 10 20 30 40 50

c, (1000 lb/rad)

Fig.3.23. Cy vs µy, a= 8°, µd = 0.6.

128

1.2 ..,u•=-----------=------=,-:, c. = 35

1

D.B

D.6

D.4

D.2

s. 0.5 D.2

------------------------- - 0.1

D.__ _ ___._ __ ......._ _ __,'------'------'

D ID 2D 3D 4D so Cy (1000 lb/rad)

Fig.3.20. Cy vs µ1 , a = 8°, µd = 0.6.

1.2 ,--Uy=------------=---::::-= c.= 35

D.B

D.6

D.4

0.2

__ , .. ----------------

s. 0

___ , ....

--------------------------- 0.1 0.2

0.5 0 .__ _ __._ __ ....__ __ .___ _ ___._ _ ____,

D ID 20 30 40 50

Cv (1000 lb/rad)

Fig.3.22. Cy vs µy, a = 4°, µd = 0.6.

u 1·2 ,-L-----------c---. -='""3..,,.,5

1

D.B

0.6

D.4

D.2

---------............

s. D

0.2

0.5

0 .__ _ ___._ __ ......._ _ __,'------'------'

0 ID 20 30 4D 5D c, (1000 lb/rad)

Fig.3.24. CY vs µy, a= 32°, µd = 0.6.

3.2.3. The Effects of Locked-Wheel BFC (pd)


a. The effects of locked-wheel BFC (pd) on critical longitudinal slip (Sxcr), critical

slip angle (a.er), critical friction coefficients (Psa• Paa), and frictional force

coefficients (Px, Py) are shown in Figures 3.25-3.42. As can be expected from the

contribution of tyre stiffness and friction coefficient to frictional forces, the

effects of Pd seem very significant. Generally, the higher the Sx, the more

significant the effects of Pd on Px· Similarly, the higher the a., the more significant

the effects of Pd on Py·


1. The critical longitudinal slip (Sxcr) will increase as the locked-wheel BFC (pd)

increases. The rate of increasing Sxer with increasing pd will be greater at low

longitudinal stiffness (Cx).

2. The critical slip angle (a.er) will increase as the pd increases. The rate of

increasing a.er with increasing Pd will be greater at low lateral stiffness (Cy).

3. The critical friction coefficients (Psa• Paa) will increase as the pd increases.

4. The frictional force coefficients (Px, Py) will increase as the Pd increases. The rate

of increasing (Px, py), however, will be reduced for Px at low values of Sx

(particularly at high a.), and for Py at low values of a. (particularly at high Sx).

The results above again are in agreement with the statement previously

mentioned, that the contribution from the friction coefficient continue to increase

with increasing Sx (or a.).

129

0.6 s.,,

0.5 --c.

-- 10

0.4

0.3

---------------· -· -· -· -· ---· 0.2

0,1

0 .____,__......__.......__.__.......,_......__.......__.

0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9

ud

Fig.3.25. Pd vs Sxcr• for various ex.

u 1.2 ,er 50 35

1

o.e

-· 10 .,

____ .,. c.

0.6

0.4

0.2

-· --------------

------- .,.. .,. ,,.-

0 .____._ _ ___._ ___ ~ ........ _ ....... __ _.

0.2 o.3 0.1 o.s o.6 0.1 o.e o.e

uo

Fig.3.27. Pd vs Pscr• for various ex.

1.2 u.

1

O.B

0.6

0.4

0.2

0.2 0.1 0.5

s.

0 .____._ _ ___._ ___ ~__._......__.......__.

02 0~ 0.4 0~ 0~ 0.7 0~ 0~

ud

Fig.3.29. Pd vs Px, a. = O", ex = 35.

130

1.2<Xc, LraaJ

-· --------------

c, --- 10

1

0.8

0.6

0.1

0.2 :::::/:::::::== ~~ -o .__ ........ _ ___._ ___ ...._ ........ _ ....... __ __.

0.2 0.3 0.1 0.5 0.6 0.7 0.8 0.9

Uo

Fig.3.26. Pd vs a.er• for various ey.

1.2 U ... ,

1

0.8

0.6

0.4

0.2

.-----------· ---· ---·

---· -·

40 25

_,,- 10 _ _.... C

----- ,

0 .____._ _ ___._ ___ ...._ ........ _ ....... __ __.

0.2 o.3 o.4 o.5 o.6 0.1 o.e o.9

ud

Fig.3.28. Pd vs Pacr, for various er

1,2 --'Ux"---------------C, = 25

1

O.B

0.6 s.

---_, ___ ,

------· -----------0 4 0.2 -----·

· 0.5 --· 0.1

0.2

0 ,____,__......__.......__....____._.......,_.....___,

0.2 O.J 0.4 0.5 0.6 0.7 O.B 0.9

uo

Fig.3.30. Pd vs Px, a. = 4°, ex = 35.

1.2-u•;;.__ ___________ .....,

C,=25 Sx 1 0.5

0.2

0.8

0.6

0.4

0.2

.. --· _______ ... -----------·· .. --- 0.1

D ..__.__......_____,.___.__..._____. _ _._____.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ud Fig.3.31. µd vs µ., a = 8°, c. = 35.

u 1,2 ..,....... ____________ ()(--,

32 1 8

---- 4 0.8

0.6

0.4

0.2

-------.. .. -· ... ----,.

o-~--~---~----01 0~ OA OS 0~ 0.7 0~ 0~ u.,

Fig.3.33. µd vs µy, s. = 0, Cy = 25.

u 1.2 ..,.,...__ ___________ __,

c.= 35 0(

1 32

0.8

B 0.6

0.4

------------ 4

0.2 ---....... ---------------

0 .._____.__.....___ __ ..__.......__..._____._......._____.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

uo Fig.3.35. µd vs µy, s. = 0.2, Cy = 25.

131

1.2 ,....;Ux.;.__ ___________ .....,

1

0.8

0.6

0.4

0.2

c, = 25

_____ ....... ------------

s. 0.5

0.2

----------- 0 .1 D ..__.......__...._____,...__.......__..._____. _ _.______,

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

u d

Fig.3.32. µd vs µ., a = 32°, C. = 35.

1.2 .-'u,.__ ___________ _

1

0,8

0.6

0.4

0.2

c. = 35 0(

32

8

-- 4

. __ ... ---------------------

0 ..__.......__...._____, _ _.__..._____._.......______,

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Uo

Fig.3.34. µd VS µY' s. = 0.1, Cy = 25.

u 1.2 ......... -------------,

c.= 35 1 0(

0.8

0.6

0.4

0.2

-------------

32

B

------------- -- 4 o-~--~---~----0.2 0,3 0,1 0,5 0,6 0,7 0,8 0,9

u.,

Fig.3.36. µd vs µy, s. = 0.5, Cy = 25.

1.2 .... u·=----------------, Cy = 25

-_, ..,,,, ...

()(

0 1

D.B

D,6

0,1

0.2

.,,, .. ,-' .. ,- 411

.,.--- ,--' -- , .. .. -- ., .. ----- __ .... ,.-.,.- ---- .. --------- ------------

8

------- 32 o.._~_......__....._ ____ ~---oi 03 0~ 05 0~ 0~ 0~ OJ

ud Fig.3.37. Pd vs Px, Sx = 0.1, ex = 35.

1.2 ,...ux _______________ ---,

Cy = 25 ~ B

O.B

0.6

0.1

0.2

32

0 .______._.....__.....___._---L.._.....______..____,

0.2 0.3 0.1 0.5 0.6 0.7 O.B 0.9

ud

Fig.3.39. pd vs Px, Sx = 0.5, ½ = 35.

1.2.--u .... , --------------,

1

D.B

0,6

0,1

0.2

Cx = 35

0.2

_ 0.5 -=----------0 .______._.....__...______. _ __._ _ _.______..____,

0.2 0.3 0.1 0.5 0.6 0.7 O.B 0.9

Uo

Fig.3.41. Pd vs Py, a = 8°, ey = 25.

132

1.2 ,...ux=---------------, ()(

O.B

D.6

0,1

0.2

Cy = 25 o ,,--:: 4

..... -::......... B -- ., -- ,-_ .. -:-;-.,,,-

.-::::::::----- _, ,, .,.-

::=-=-=--32

0 .______._.....__...____. _ __._ _ _.__..____.

0~ 03 OA 05 0~ 0~ 0~ 0~

u. Fig.3.38. Pd vs Px, Sx = 0.2, ex = 35.

u 1,2 .-'---------------,

s. 1

O.B --.... 0

D.6

D.1

0.2

----,-' ,,- ----

-------- ----- 0.1 .. --- ---

---~~~---------------------- 0.2

=--------- 0.5 0 ._____._ _ _.__...____..____.__...,___..____,

0.2 0.3 0.1 0.5 0.6 0.7 O.B 0.9

ud Fig.3.40. Pd vs Py, a = 4°, ey = 25.

1

O.B

0.6

0.4

0.2

c. = 35

D .______.__......__....._____._---L.. _ _.______..____,

0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9

ud

Fig.3.42. Pd vs Py, a = 32°, ey = 25.

3.3. The Significance of All Variables

The significance of variable can be identified by using one of the statistical

analyses, called the regression analysis. The goal of this analysis is to express the

response (dependent) variable as a function of the predictor (independent variables).

Once such an expression is obtained, the relationship also can be utilized to predict

values of the response variable or to verify hypothesized causal models of the

response.

3.3.1. Dependent and Independent Variables

The dependent variables considered for this analysis are frictional force

coefficients (µx, µy) and their resultance: ~ 1 = (µ/+µ/)°" 5•

The independent variables considered here are the increasing zone of pressure

distribution (a/21), stiffnesses (½, Cy), locked-wheel BFC (µd), wheel load (F2),

percent slip (Sx), and slip angle (ex). The decreasing zone of pressure distribution

(b/21) will not be investigated again here, since its effect on the frictional forces

has been found to be nearly small (see Figure 2.25), and only significance to the

lateral force at low values of Sx. Moreover, by keeping the b/21 as constant, the

amount of the data will decrease, resulting in a significant effort for obtaining a

manageable size of data.

Two levels of variables are assigned for a/21, Cx, Cy, µd, and F2 • With primary

objective to study the tyre-road friction under braking and cornering, an additional

level of Sx and ex in middle values was assigned. Summing up, this experimentation

would be termed as a 2x2x2x2x2x3x3 factorial, or with total data of 288. The

corresponding t-statistic at 5% level is found (from statistical tables) as 1.65.

The range of level for each independent variables will be chosen so that it will

represent the normal ranges which commonly occur in practice. For standardization,

the lowest and the highest values of the designed variable levels were assigned as -1

and 1, giving the range of experiment as two e.u. (experiment unit). See Table 3.1.

133

Table 3.1. Detail of Independent Variables.

"-- Experimental Unit (e.u.)

No Variable -1 0 1

Unit

1 a/21 0.05 - 0.20

2 ex 10000 - 60000 lb/slip

3 Cy 10000 - 40000 lb/radian

4 µd 0.3 - 0.9

5 Fz 200 - 1800 lbs

6 sx 0 0.5 1

7 a 0 20 40 degree

3.3.2. Model Specification

Model specification becomes important in a regression analysis whose objective is

the investigation of the relative value of individual predictor variables on the

prediction of the response. The following points were considered.

1. The (statistical) model is not intended to replace the analytical model. Since the

statistical model is based on a certain range of data, it could be dangerous for

extrapolations outside the range of data base. Even in the range itself, the

abi I ity of the statistical model to conform with th.e analytical model is still questionable,

due to the limited level used in the range considered (e.g. 2 or 3 levels only).

Several mathematical functions such as are used in analytical model (e.g. sine

function) will not be involved in this simplified model.

2. Knowledge of which variables are important predictors and how they vary, may

be sufficient. Hence, it is not of paramount importance that the regression

coefficients are estimated precisely. In statistical terms, the coefficient of multiple

determination (R2) of about 75% is probably sufficient.

3. Since there is always a difficulty in interpretating the higher interactions (for

example, the practical meaning of a 4-factor interaction or greater), this statistical

model then will be limited to a 3-factor interaction.

134

With some points considered above, the 78 a-coefficients in the following

equation were calculated.u.sin9 -/.he cc.mclidate $ cp·,cdy t ic:Q \ mode\, 2 2

Y = 3o + a1X1 + aix2 + ~X3 + a4X4 + a5X5 + 3t;X6 + ~X7 + ~x6 + a,,x, + a12X1X2

+ a13X1X3 + a14X1X4 + a15X1X5 + a16X1X6 + a17X1X7 + 3i3XzX3 + 8z4XzX4 + 3isXzX5

+ 3i6XzX6 + 3i,XzX7 + ~X3X4 + a35X3X5 + a36X3X6 + a37X3X7 + a45X4X5 + a46X4X6

2 2 2 2 + a47X4X7 + as6XsX6 + a57X5X7 + 3ti7X6X7 + a166X1X6 + a177X1X7 + 3i66XzX6 + 3i,,X2X7

2 2 2 2 2 2 2 + a366X3X6 + 1½11X3X7 + a466x4x6 + a477X4X7 + as66XsX6 + a577X5X7 + 3ti77~X7

+ ~66x,x/ + a123X1X2X3 + a124X1X2X4 + a125X1X2X5 + a126X1X2X6 + a127X1X2X7

+ a134X1X3X4 + a135X1X3X5 + a136X1X3X6 + a137X1X3X7 + a145X1X4X5 + a146X1X4~

+ a147X1X4X7 + a1S6X1XsX6 + a157X1X5X7 + al61X1X6X7 + 3i34XzX3X4 + 3i35XzX3X5

+ 3i36X2X3X6 + 3i37XzX3X7 + 8z45XzX4X5 + 8z46X2X4X6 + 8z47XzX4X7 + 3is6X2XsX6

+ 3is1X2X5X7 + au1X2X6X7 + a345X3X4X5 + ~X3X4X6 + a347X3X4X7 + a356X3X5~ /

+ a357X3X5X7 + ~67X3X6X7 + a456X4X5X6 + a457X4X5X7 + a467x4x6x7 + a567X5~X7

where Y is the frictional force coefficient at any chosen combination of variable

levels x1, x2, x3, x4, x5, x6, and x7•

Variables x1, x2, x3, x4, x5, x6, and x7 are the increasing zone of pressure

distribution (a/21), longitudinal stiffness (Cx), lateral stiffness (Cy), locked-wheel BFC

(µd), wheel-load (Fz), percent slip (Sx), and slip angle (a), respectively in

experimental units (e.u.). The 78 a-coefficients consist of: 1 constant, 9 main effects

(7 linear effects, 2 quadratic effects), 33 two-factor interactions (21 linear x linear

interactions, 12 linear x quadratic interactions), and 35 three-factor interactions (35

linear x linear x linear interactiom). See Table 3.l for 288 data u5,ed (2x2"2"2i<2><3x3).

3.3.3. Regression Analysis

The stepwise regression procedure was used when selecting variables in

regression. In principle, any variable which provides a non significant contribution

was removed from the model.

Table 3.2 shows the calculated a-coefficients for the response equation. The R2

and the S (estimated standard deviation about the regression line) give an indication ~ of how well fitted equation explains the variation in the data.

135

Table 3.2. a-Coefficients (t-statistics).

a Px Pv Prst

3o 1.785 (20.94) 1.464 (17.08) 2.721 (18.29) a1 1.192 (13.98) 0.970 (11.32) 1.850 (12.45)

3i 0.622 (7.30) - 0.728 (6.31)

~ - 0.529 (6.17) 0.599 (5.20) a4 0.176 (3.57) 0.154 (3.11) 0.279 ( 4.19) as -0.015 (-11.91) -0.852 (-9.94) -1.594 (-10.71) c1<; 0.271 ( 4.49) -1.268 (-12.08) -1.071 (-7.59) a, -0.567 (-5.43) 0.937 (8.93) -

'¼6 -1.514 (-14.50) - -1.182 (-8.38) a,, - -0.730 (-6.96) -0.637 (-4.51) a12 0.190 (3.86) - 0.222 (3.34) a13 - 0.299 (6.05) 0.329 (4.94) a1s -0.309 (-6.27) -0.482 (-9.74) -0.687 (-10.33) a16 - -0.551 (-9.10) -0.551 (-6.77) a11 -0.129 (-2.13) 0.433 (7.15) -

3is -0.167 (-3.39) - -0.195 (-2.93) 3i, -0.155 (-2.57) - -~5 - -0.266 (-5.38) -0.293 (-4.40)

~6 - -0.366 (-6.04) -0.366 (-4.49) ~7 - 0.237 (3.91) 0.251 (3.08) a46 0.135 (2.24) - -a56 - 0.487 (8.04) 0.487 (5.98) as1 - -0.377 (-6.22) -'¼7 - -0.439 (-5.92) -0.536 (-5.37) a166 -1.192 (-11.42) - -0.978 (-6.93) a111 - -0.537 (-5.12) -0.483 (-3.42) 3i66 -0.622 (-5.96) - -0.728 (-5.16) a311 - -0.292 (-2.79) -0.348 (-2.47) as66 1.015 (9.72) - 0.818 (5.80) as11 - 0.474 (4.52) 0.432 (3.06) '¼77 - 0.829 (6.45) 0.835 (4.83) a166 0.532 (4.16) -0.305 (-2.37) 0.536 (5.37) a125 -0.153 (-3.10) - -a121 -0.142 (-2.35) - -a13s - -0.240 (-4.85) -0.263 (-3.96) a136 - -0.327 (-5.40) -0.327 (-4.02) a131 - 0.213 (3.51) 0.225 (2.77) a1s6 - 0.436 (7.19) 0.436 (5.35) a1s1 - -0.340 (-5.62) -a161 - -0.345 (-4.65) -0.345 (-3.46) 3is1 0.124 (2.06) - -a3s6 - 0.290 (4.79) 0.290 (3.56) ~57 - -0.191 (-3.16) -0.203 (-2.49) a361 - -0.227 (-3.06) -0.227 (-2.28) a<,.., - 0.303 (4.08) 0.303 (3.03)

I S (R2) I 0.835 (74.12) I 0.840 (85.26) I 1.130 (81.46) I

136

3.3.4. Significance of Longitudinal Force Coefficient

a. Increasing zone of pressure distribution (a/21)

From Table 3.2:

Aµx = l.192(a/21) + 0.190(a/2l)(ex) - 0.309(a/2l)(F2 ) - 0.129(a/2l)(Sx)

- l.192{a/2l){Sx)2 - 0.153(a/2l){ex)(F2) - 0.142(a/2l)(ex)(CX.)

From equation above, it can be concluded that:

1. Main effect -+ µx will increase with increasing a/21.

2. Two factor interaction -+ µx will increase with increasing a/21 at high values of ex,

and at low values of F2 • This increase of µx with increasing a/21 will be reduced

at extreme values of Sx, particularly at high Sx.

3. Three factor interaction -+ µx will increase with increasing a/21 at high values of

ex & low values of F2 (or vice versa), and at high values of ex & low values of ex. (or vice versa).

b. Longitudinal stiffness (~)

From Table 3.2:

Aµx = 0.622(ex) + 0.190(ex)(a/21)- 0.167(ex)(F2) - 0.155(ex)(CX.)- 0.622(ex)(Sx)2

- 0.153(Cx)(a/2l)(F2) - 0.142(ex)(a/2l)(CX.) + 0.124(ex)(F2)(CX.)


1. Main effect -+ Px will increase with increasing ex.

2. Two factor interaction -+ µx will increase with increasing ex at high values of

(a/21), at low values of F2 , and at low values of ex.. This increase of µx with

increasing ex will be reduced at extreme values of Sx.

3. Three factor interaction -+ µx will increase with increasing Cx at high values of

a/21 & low values of F2 (or vice versa), at high values of a/21 & low values of ex.

(or vice versa), and at high values of F2 & ex. (or low values of F2 & ex.).

c. Later-al stiffness (ey)

From Table 3.2 it is found that there is no significant effect of eY on the ~-

d. Locked-wheel BFe (µd)

From Table 3.2:

Aµx = 0.176(µd) + 0.135(µd)(Sx)


137

1. Main effect -+ Px will increase with increasing Pd·

2. Two factor interaction -+ Px will increase with increasing Pd at high values of Sx-

e. Wheel-load (F2)

From Table 3.2:

.6.Px = -l.015(Fz) - 0.309(Fz)(a/21) - 0.167(Fz)(ex) + l.015(Fz)(Sx)2

- 0.153(F2)(a/2l)(ex) + 0.124(Fz)(Cx)(CX)


1. Main effect-+ Px will increase with decreasing Fz.

2. Two factor interaction -+ P>< will increase with decreasing Fz at high values of

a/21, and at high values of ex. This increase of Px with decreasing Fz will be

reduced at extreme values of Sx-

3. Three factor interaction -+ Px will increase with decreasing Fz at high values of

a/21 & ex (or low values of a/21 & ex), and at high values of ex & low values of

a (or vice versa).

f. Percent slip (Sx)

From Table 3.2:

.6.Px = 0.271(Sx) - l.514(Sx)2 + 0.135(Sx)(pd) - l.192(Sx)2(a/21) - 0.622(Sx)2(ex)

+ l.015(Sx)2(F2) + 0.532(Sx)2(a)


1. Main effect -+ Px will increase with increasing Sx from low values, and then

decrease with further increase of Sx to high values.

2. Two factor interaction -+ Px will increase with increasing Sx from low values and

then decrease with further increase of Sx to high values, where the rate of

increasing Px will be greater at high: pd, a/21, ex, and at low: Fv a.

g. Slip angle (a)

From Table 3.2:

.tt.px = -0.567(a) - 0.129(a)(a/21) - 0.155(a)(ex) + 0.532(a)(Sx)2 - 0.142(a)(a/ll)(ex)

+ 0.124(a)(ex)(Fz)


1. Main effect -+ Px will increase with decreasing a. 2. Two factor interaction -+ Px will increase with decreasing a at high values of a/21,

and at high values of ex. This increase of Px with decreasing a will be reduced at

138

extreme values of Sx.

3. Three factor interaction ~ Px will increase with decreasing a at high values of a/21

& Cx (or low values of a/21 & ½), and at high values of Cx & low values of Fz

( or vice versa).

3.3.5. Significance of Lateral Force Coefficient


From Table 3.2:

APy = 0.970(a/21) + 0.299(a/2l)(Cy) - 0.482(a/2l)(Fz) - 0.55l(a/2l)(Sx) + 0.433(a/2l)(a)

- 0.537(a/2l)(a)2 - 0.240(a/2l)(Cy)(Fz) - 0.327(a/2l)(Cy)(Sx) + 0.213(a/2l)(Cy)(a)

+ 0.436(a/2l)(Fz)(Sx) - 0.340(a/2l)(Fz)(a) - 0.345(a/2l)(Sx)(a)


1. Main effect ~ Py will increase with increasing a/21.

2. Two factor interaction ~ Py will increase with increasing a/21 at high values of Cy,

at low values of Fz, and at low values of Sx. This increase of Py with increasing

a/21 will be reduced at extreme values of a, particularly at low a.

3. Three factor interaction ~ Py will increase with increasing a/21 at high values of

Cy & low values of Fz (or vice versa), at high values of Cy & low values of Sx (or

vice versa), at high values of Cy & a (or low values of Cy & a), at high values of

Fz & Sx (or low values of Fz & Sx), at high values of Fz & low values of a (or

vice versa), and at high values of Sx & low values of a (or vice versa).

b. Longitudinal stiffness (½)

From Table 3.2 it is found that there is no significant effect of Cx on the Py-

c. Lateral stiffness (Cy)

From Table 3.2:

APy = 0.529(Cy) + 0.299(Cy)(a/21) - 0.266(Cy)(Fz) - 0.366(Cy)(Sx) + 0.237(Cy)(a)

- 0.292(Cy)(a)2 - 0.240(Cy)(a/2l)(Fz) - 0.327(Cy)(a/2l)(Sx) + 0.213(Cy)(a/2l)(a)

+ 0.290(Cy)(Fz)(Sx) - 0.191(Cy)(Fz)(a) - 0.227(Cy)(Sx)(a)


1. Main effect ~ Py will increase with increasing Cr

2. Two factor interaction ~ Py will increase with increasing CY at high values of a/21,

139

at low values of Fz, and at low values of Sx. This increase of Py with increasing

Cy will be reduced at extreme values of a, particularly at low a. 3. Three factor interaction ---+ Py will increase with increasing CY at high values of

a/21 & low values of F z ( or vice versa), at high values of a/21 & low values of Sx

(or vice versa), at high values of a/21 & a (or low values of a/21 & a), at high

values of Fz & Sx (or low values of Fz & Sx), at high values of Fz & low values

of a (or vice versa), and at high values of Sx & low values of a (or vice versa).

d. Locked-wheel BFC (pd)

From Table 3.2:

b.Py = 0.154(pd)

From equation above, it can be concluded that: Py will increase with increasing pd

(main effect). There is no significant interaction effect between Pd and other

parameters in influencing Py-

e. Wheel load (F z)

From Table 3.2:

b.py = -0.852(Fz) - 0.482(Fz)(a/21) - 0.266(Fz)(Cy) + 0.487(Fz)(Sx) - 0.377(Fz)(a)

+ 0.474(Fz)(a)2 - 0.240(Fz)(a/2l)(Cy) + 0.436(Fz)(a/2l)(Sx) - 0.340(Fz)(a/2l)(a)

+ 0.290(Fz)(Cy)(Sx) - 0.191(Fz)(Cy)(a) + 0.303(Fz)(Sx)(a)


1. Main effect ---+ Py will increase with decreasing Fz.

2. Two factor interaction ---+ Py will increase with decreasing Fz at high values of a/21,

at high values of Cy, and at low values of Sx. This increase of Py with decreasing

Fz will be reduced at extreme values of a, particularly at low a. 3. Three factor interaction ---+ Py will increase with decreasing Fz at high values of

a/21 & Cy ( or low values of a/21 & Cy), at high values of a/21 & low values of Sx


values of Cy & low values of Sx (or vice versa), at high values of Cy & a (or low

values of Cy & a), and at high values of Sx & low values of a (or vice versa).

140


From Table 3.2:

.t..µy = -l.268(Sx) - 0.551(Sx)(a/21) - 0.366(Sx)(Cy) + 0.487(Sx)(Fz) - 0.439(Sx)(<X)

+ 0.829(Sx)(a)2 - 0.305(Sx)2(a) - 0.327(Sx)(a/2l)(Cy) + 0.436(Sx)(a/2l)(Fz)

- 0.345(Sx)(a/2l)(<X) + 0.290(Sx)(Cy)(Fz) + 0.303(Sx)(Fz)(<X)


1. Main effect --+ µY will increase with decreasing Sx.

2. Two factor interaction --+ µY will increase with decreasing Sx at high values of a/21,

at high values of Cy, and at low values of Fz. This increase of µY with decreasing

Sx will be reduced at extreme values of a, particularly at low a. 3. Three factor interaction --+ Py will increase with decreasing Sx at high values of

a/21 & Cy (or low values of a/21 & Cy), at high values of a/21 & low values of Fz


values of Cy & low values of Fz (or vice versa), and at high values of Fz & low

values of a (or vice versa).

g. Slip angle (a)

From Table 3.2:

.t..py = 0.937(a) - 0.730(a)2 + 0.433(a)(a/21) + 0.237(a)(Cy) - 0.377(a)(F2)

- 0.439(a)(Sx) - 0.537(a)2(a/21) - 0.292(a)2(Cy) + 0.474(a)2(Fz) + 0.829(a)2(Sx)

- 0.305(a)(Sx)2 - 0.213(a)(a/2l)(Cy) - 0.340(a)(a/2l)(Fz) - 0.345(a)(a/2l)(Sx)

- 0.19l(a)(<;,)(F2 ) - 0.227(a)(Cy)(Sx) + 0.303(a)(F2)(Sx)


1. Main effect --+ Py will increase with increasing a from low values, and then

decrease with further increase of a to high values.

2. Two factor interaction --+ Py will increase with increasing a from low values and

then decrease with further increase of a to high values, where the rate of

increasing µY will be greater at high: a/21, Cy, and at low: F2 , Sx.

3. Three factor interaction --+ Py will increase with increasing a at high values of a/21

& low values of Cy (or vice versa), at high values of a/21 & low values of F2 (or

vice versa), at high values of a/21 & low values of Sx (or vice versa), at high

values of CY & low values of F2 (or vice versa), at high values of <;, & low

values of Sx (or vice versa), and at high values of F2 & Sx (or low values of F2 &

Sx).

141

3.3.6. Significance of Resultant Force Coefficient


From Table 3.2:

AJlnt = 1.851(a/21) + 0.222(a/2l)(Cx) + 0.329(a/2l)(Cy) - 0.687(a/2l)(Fz)

- 0.551(a/2l)(Sx) - 0.978(a/2l)(Sx)2 - 0.483(a/2l)(a)2 - 0.263(a/2l)(Cy)(FJ

- 0.327(a/2l)(Cy)(Sx) + 0.225(a/2l)(Cy)(a) + 0.436(a/21)(Fz)(Sx)

- 0.345(a/21)(Sx)(a)


1. Main effect --+ Jlnt will increase with increasing a/21.

2. Two factor interaction --+ Jln1 will increase with increasing a/21 at high values of

Cx, at high values of Cy, and at low values of F z- This increase of Jlnt with

increasing a/21 will be reduced at extreme values of Sx, particularly at high Sx,

and at extreme values of a. 3. Three factor interaction --+ µrs1 will increase with increasing a/21 at high values of

Cy & low values of Fz (or vice versa), at high values of Cy & low values of Sx (or

vice versa), at high values of Cy & a (or low values of Cy & a), at high values of

Fz & Sx (or low values of Fz & Sx), and at high values of Sx & low values of a ( or vice versa).

b. Longitudinal stiffness (Cx)

From Table 3.2:

AJlnt = 0.728(Cx) + 0.222(½)(a/21) - 0.195(Cx)(Fz) - 0.728(Cx)(Sx)2


1. Main effect --+ Jlnt will increase with increasing Cx.

2. Two factor interaction --+ Jln1 will increase with increasing Cx at high values of

a/21, and at low values of Fz. This increase of Jln1 with increasing Cx will be

reduced at extreme values of Sx.

c. Lateral stiffness (Cy)

From Table 3.2:

.t.Jlnt = 0.599(Cy) + 0.329(Cy)(a/21) - 0.293(Cy)(Fz) - 0.366(Cy)(Sx) + 0.251(Cy)(a)

- 0.348(Cy)(a)2 - 0.263(Cy)(a/2l)(Fz) - 0.327(Cy)(a/2l)(Sx) + 0.225(Cy)(a/2l)(a)

+ 0.290(Cy)(Fz)(Sx) - 0.203(Cy)(Fz)(a) - 0.227(Cy)(Sx)(a)


142

1. Main effect ~ Jlrs1 will increase with increasing Cy.

2. Two factor interaction ~ Pni will increase with increasing Cy at high values of

a/21, at low values of Fz, and at low values of Sx. This increase of Jlrs1 with

increasing Cy will be reduced at extreme values of a, particularly at low a. 3. Three factor interaction ~ Jlrs1 will increase with increasing Cy at high values of

a/21 & low values of Fz (or vice versa), at high values a/21 & low values of Sx (or

vice versa), at high values of a/21 & a (or low values of a/21 & a), at high values

of Fz & Sx (or low values of Fz & Sx), at high values of Fz & low values of a (or

vice versa), and at high values of Sx & low values of a (or vice versa).

d. Locked-wheel BFC (µd)

From Table 3.2:

AJlrs1 = 0.279(µJ

From equation above, it can be concluded that: Pni will increase with increasing µd

(main effect). There is no significant interaction effect between µd and other

parameter in influencing Pnr

e. Wheel load (F z)

From Table 3.2:

AJlrs1 = -l.594(Fz) - 0.687(Fz)(a/21) - 0.195(Fz)(Cx) - 0.293(Fz)(Cy) + 0.487(Fz)(Sx)

+ 0.818(Fz)(Sx)2 + 0.432(Fz)(a)2 - 0.263(Fz)(a/2l)(Cy) + 0.436(Fz)(a/2l)(Sx)

+ 0.290(Fz)(Cy)(Sx) - 0.203(Fz)(Cy)(a) + 0.303(Fz)(Sx)(a)


1. Main effect~ Pni will increase with decreasing Fr

2. Two factor interaction ~ Pni will increase with decreasing Fz at high values of

a/21, at high values of Cx, and at high values of Cr This increase of Jlrs1 with

decreasing F z will be reduced at extreme values of Sx, particularly at high Sx, and

at extreme values of a.

3. Three factor interaction ~ Pnt will increase with decreasing Fz at high values of

a/21 & Cy ( or low values of a/21 & Cy), at high values of a/21 & low values of Sx

(or vice versa), at high values of Cy & low values of Sx (or vice versa), at high

values of CY & a (or low values of Cy & a), and at high values of Sx & low

values of a ( or vice versa).

143


From Table 3.2:

Lt.Jlnt = -1.071(Sx) - l.182(Sx)2 - 0.551(Sx)(a/21) - 0.366(Sx)(Cy) + 0.487(Sx)(Fz)

- 0.536(Sx)(<X) - 0.978(Sx)2(a/21) - 0.728(Sx)2(Cx) + 0.818(Sx)2(Fz)

+ 0.835(Sx)(a)2 + 0.536(Sx)2(a) - 0.327(Sx)(a/2l)(Cy) + 0.436(Sx)(a/2l)(Fz)

- 0.345(Sx)(a/2l)(<X) + 0.290(Sx)(Cy)(Fz) - 0.227(Sx)(Cy)(<X) + 0.303(Sx)(Fz)(<X)


1. Main effect ~ Pnt will increase with decreasing Sx from high values, and then

decrease with further decrease of Sx to low values.

2. Two factor interaction ~ Prst will increase with decreasing Sx from high values

and then decrease with further decrease of Sx to low values, where the rate of

increasing Pnt will be greater at high: a/21, Cy, and at low Fz. This increase of Prst

with decreasing Sx will be reduced at extreme values of a.

3. Three factor interaction ~ Prst will increase with decreasing Sx at high values of

a/21 & Cy (or low values of a/21 & Cy), at high values of a/21 & low values of Fz


values of Cy & low values of Fz (or vice versa), at high values of Cy & a (or low

values of Cy & a), and at high values of Fz & low values of a (or vice versa).

g. Slip angle (a)

From Table 3.2:

.tt.Jlnt = -0.637(a)2 + 0.251(a)(Cy) - 0.536(a)(Sx) - 0.483(a)2(a/21) - 0.348(a)2(Cy)

+ 0.432(a)2(Fz) + 0.835(a)2(Sx) + 0.536(a)(Sx)2 + 0.225(a)(a/2l)(Cy)

- 0.345(a)(a/2l)(Sx) - 0.203(a)(Cy)(Fz) - 0.227(a)(Cy)(Sx) + 0.303(a)(Fz)(Sx)


1. Main effect ~ Prst will increase with increasing a from low values and then

decrease with further increase of a to high values.

2. Two factor interaction ~ Prst will increase with increasing a from low values and

then decrease with further increase of a to high values, where the rate of

increasing Prst will be greater at high: Cy, a/21, and at low: Sx, a/21, Fz.

3. Three factor interaction ~ Jln1 will increase with increasing a at high values of

a/21 & Cy (or low values of a/21 & Cy), at high values of a/21 & low values of Sx

(or vice versa), at high values of CY & low values of Fz (or vice versa), at high

values of Cy & low values of Sx (or vice versa), and at high values of Fz & Sx (or

low values of Fz & Sx).

144

3.4. Comparison With Other Tyre Models

This investigation is intended to present all of the models (five tyre models and

the candidate's model) in a common format, and to compare the response of each

model to the others, with the objective of identifying the effects of the various

assumptions made during derivation. The features of five tyre models: HSRI-1

[Dugoff et.al (1969)], HSRI-11 [Fancher (1972)], HSRI-m [Tielking and Mital

(1974)], Good Year [Livingston and Brown (1970)], and Sakai [1969]; have been

described by Tielking and Mital [1974], and will be presented in the condensed

forms.

3.4.1. General Description of Tyre Models

a. HSRI-1

This model presumes elastic tread blocks on a rigid wheel, uniform contact

pressure, uniform deformation in sliding region, tyre-road friction decreasing linearly

with slip speed.

b. HSRI-11

This model presumes elastic tread blocks on a foundation allowed to translate

uniformly, finite transition between adhesion and fully developed sliding, uniform

contact pressure, distinction between static and dynamic friction which decrease

linearly with slip speed.

c. HSRI-111

This model presumes parabolic contact pressure, parabolic distribution of tread

element displacements in the transition region, otherwise identical to HSRI-11 model.

d. Good Year

This model presumes elastic tread blocks on rigid wheel, parabolic or

experimentally determined contact pressure, deformation in sliding region decreases

with shear force, constant friction. Deformation discontinuity avoided by assuming

sliding shear force to act in same direction as shear force in the adhesion region.

145

e. Sakai

This model presumes elastic tread blocks on flexible elastically-supported beam

carcass (subs.equently reduced to rigid beam with uniform translation) with an

artificial mechanism connecting longitudinal stiffness with lateral stiffness (but not

vice-versa), parabolic contact pressure, distinction between static and sliding friction

which is considered to be orthotropic.

f. Candidate

This model presumes elastic tread blocks on a foundation allowed to translate

uniformly, unsymmetric trapezoidal contact pressure, the available friction (Ps, µJ

decreases linearly with sliding speed or slips (longitudinal, lateral), and equals to the

frictional force coefficient (µx, µy) at slips beyond the critical value.

3.4.2. Summary of Formulas

a. HSRI-1

1. Adhesion limit: X'

s µFz(l-Sx) =--------

2{(C s )2+(C s )2}0·5 X X y y

where

µ = ~(1-A5V5); V5 = (S/+S/)°"5 IV I cos<X

2. Friction modes:

X1' ~ 21 complete adhesion

adhesion and sliding 0 < ~· < 21

Xs' = 0 complete sliding (only at wheel lock)

3. Friction forces:

F = X

{C2 + (C s )2}0,S X . y y

146

(3.1)

(3.2)

(3.3)

(3.4)

X/ = 0 (3.5)

F = y

{C2 + (C S )2}0.S X y y

b. HSRI-11

1. Adhesion limit:

X.' PoFz(l -Sx) = 21 -2-{(_C_S _)2_+_(_C_S_)2_}0-.s

X X y y

where

µ = Po(l-A5V5); V5 = (S/+S/)°-5 IV I cosa

3. Friction modes:

x: ~ 21 complete adhesion

adhesion and transition

adhesion and transition and sliding

adhesion and sliding

X/ ~ 21

0 < X5 ' < 21

X/ =0

Xa' < Xs' ~ 21

Xa' < Xs' < 21

Xs'::;; Xa' < 21

x: =0 complete sliding ( only at wheel lock)

4. Friction forces:

Fx = Fxa + Fxt + Fxs

Fy = Fya + Fyt + Fys

where

CS (X '} F _ x x a

xa - 1-Sx 21

F- xx a zx s a (Cs X' µF s )(x· x·) xt - 1-Sx 21 + 2(S; + s:)o.s 21 - 21

147

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.2)

(3.11)

(3.12)

(3.13)

(3.14)

F _ µ z x 1- s F S ( X 'J xs - (S; +S:)°-s 21

(3.15)

cysy (x:J Fya = 1-Sx 21

(3.16)

(CSX' µFS J(X' X'J F YY a zy s a

yt = 1-Sx 21 + 2(S; +s:)°·5 21 - 21

F S ( X 'J F = Pzy 1 __ s ys cs; + s:)o.s 21

(3.17)

(3.18)

Notes:

Fxa, Fya are terms contributed by adhesion region, exists when O < X.' ~ 21.

Fxt• Fyt are terms contributed by transition region, exists when O < X.' < ~· ~ 21.

Fxs• Fys are terms contributed by sliding region, exists when X.', X5' < 21.

c. HSRI-111

1. Adhesion limit:

x: = 1 - {(CXSX)2+(CYSY)2}0,S

21 3µ0Fz(l-Sx)

where

µ = Po0-AsVs); Vs = (S/+S/)°"5 IV I cosa.

3. Friction modes:

X.' < X5' < 21

~· ~ x: < 21

x: =0

4. Friction forces:

adhesion and transition and sliding

adhesion and sliding

complete sliding ( occurs before wheel lock)

Fx = Fxa + Fxt + Fxs

F y = F ya + F yt + F ys

where

148

(3.19)

(3.20)

(3.2)

(3.11)

(3.12)

Fn = [ ~~::b- !(~;'J- !(~;·)}{(~;')!(1 -~;')}

F s { (X I) (X I J (X I )(x I)}] [x I X I] µ z x 3 s -2 s _ a s s _ a

+ <s: + s:ts 21 21 21 21 21 21

F S { (X 'J (X 'J} F - µ z X 1-3 s +2 s

xs - (S; + s:)o.s 21 21

F ~ = [ ~~ !: { ~· -~ ( ~;'}- ! ( ~;' ) }{ ( ~')! ( I -~;')}

F = ys

Notes:

F s { (X-I ) (X I J (X I )(x I )}] [x I X I] µ z y 3 s -2 s _ a s s _ a

+ (S; + s:)o.s 21 21 21 21 21 21

F s { [x·J [x·J} µ z y 1-3 s +2 s

(S; + s:)o.s 21 21

Fxa, Fya are terms contributed by adhesion region, exists when O < X.' < 21.

(3.13)

... (3.21)

(3.22)

(3.16)

... (3.23)

(3.24)

F"" Fyt are terms contributed by transition region, exists when O < X.' < Xs' < 21.

Fxs, Fys are terms contributed by sliding region, always exists when a parabolic

pressure distribution is presumed.

149

d. Good Year

1. Adhesion limit (for parabolic contact pressure):

x.· = 1 - {(CXSJ/ +(Cysy)2}o.s

21 3µ0Fz(l -Sx)

2. Friction modes:

adhesion and sliding 0 < X.' < 21

X 1' = 0 complete sliding (occurs before wheel lock)

3. Friction forces:

F = X

F = y

e. Sakai

1. Adhesion limit:

Xs' = 1 - {(Cx Sx)2 + (Cy Sy)2 }0.5

21 3µ0Fz(l - Sx)

2. Friction modes:

0 < X.' < 21 adhesion and sliding

X11 = 0 complete sliding (occurs before wheel lock)

3. Friction forces:

F = X

150

(3.25)

0 < X.' < 21 (3.26)

X.' = 0 (3.27)

O<X,'<21 (3.28)

(3.29)

(3.25)

(3.30)

X5' = 0 (3.31)

F = y

f. Candidate

1. Adhesion limit: X' s 1

= 21 ----,-{2-(-b/-21_)_--(a/_2_1)-(b-/2-l)---(-b/-21~)2~}{~(C_x_S_x)~2 +_(_C_YS-Y"""'.'.)2=}0·~5

1 + µFz(l -S,)

--------------21 {2-(a/21)-(b/2l)H(CxS)2 +(CyS)2}0·5

where

tan8 = S/Sx

0<Xs'<21 (3.32)

Xs' = 0 (3.33)

Xs' ~ 21-b (2.87)

X5' S 21-b (2.92)

(2.59)

µs = µ,o-(~o-µd)Sx (2.57)

µ0 = µ1X0-(µ1XO-µd)2a/1t tana = SY (2.58)

Within those equations µd is locked-wheel coefficient; µso• µ.0 are longitudinal and

lateral components of (available) friction coefficient at zero slips, determined by

using procedure in Figures 2.20 and 2.21; and a/21, b/21 are the shapes of

increasing and decreasing zone of pressure distribution (unsymmetric trapezoidal).

2. Friction modes:

X,' ~ 21 complete adhesion (only if b/21 = 0)

a S X,' S 21 adhesion and sliding

X,' S a complete sliding

3. Friction forces:

F = X

(CxSx)/(1-Sx)

CxSx Xs' + (µFzcos8)H -2(X,'/21) + (X,'/21)2}

1 -Sx 21 {2(b/21) - (a/2l)(b/21) - (b/21)2}

CS X' X X S

(µFzcos8){1-0.5(b/21)-(X '/21)} + s

{1-0.5(a/21)-0.5(b/21)}

151

(Case 1)

(Case 1, 2)

(Case 3)

Xs' ~ 21 (2.88a)

(2.88)

(2.93)

(2.97)

(CYSY)/(1-Slt) ~· ~21 (2.89a)

C, S, ( X,' J + (µF. sinO) 11-2(X,'/21) + (X,'/2l)'l 21-b S ~· S 21 (2.89)

F = 1- slt 21 {2(b/2I) - (a/2I)(b/2I) - (b/2I)2l

y

c, s, ( x; J + (p F. sinO) { l -0.5(b/21)- (X,'/21) l a S ~· S 21-b (2.94) 1-Sx 21 {1-0.5(a/21)-0.5(b/21)}

~·sa (2.98) µFzsin8

3.4.3. Model Input Data

The following model input data values were used. These values, especially for 5

tyre models (a,b,c,d,e), are based on data from Tielking and Mital [1974].

Table 3.3. Model Input Data.

Application in Model

Parameters Values b d f a C e

Contact Length 21 = 7.4 inch - - - - - -

\ Uniform * * - - - -

Pressure Distr. Parabolic - - * * * -

Trpzd. (a/21=0.06, b/21=0.2) - - - - - * Longtdl. Stiffness ex = 16000 lb/slip * * * * * * Lateral Stiffness CY = 8000 lb/radian * * * * * * Static Friction Coef. Po= 1.0 * * * * * -

Isotrp. Sliding Frie. Px = µY = 0.9 - - - - * -Locked-Wheel Frie. µd = 0.9125 § - - - - - * Travelling Velocity V = 25 ft/sec * * * - - -Speed Senst. Factor As= 0.0035 * * * - - -Tyre Load Fz = 1000 lb * * * * * *

Notes: (a), (b), (c), (d), (e), and (f) refers to, respectively, tyre models of: HSRI-I, HSRI-11, HSRI-III, Good Year, Sakai, and the Candidate's. § estimated from: µ = µ0(1-~ V).

152

3.4.4. Model Response Comparisons

The tyre models were compared over a range of S" and Sy values covering braking

at slip angles varying from zero to 16 degrees.

Figure 3.43 presents the longitudinal friction force response of six models

operated at five slip angles. The candidate's model shows a significant difference

with HSRI-1 and Good Year models, and a small difference with HSRI-11, HSRI-111

and Sakai models. In Figure 3.44 the lateral friction force response of six models

operated at four slip angles are presented. The candidate's model shows nearly ihe

same result a$ the Sakai model.

Figure 3.45 compares the interaction of longitudinal force with lateral force. The

candidate's model shows a similar result to the Sakai model in terms of the slight

increase of Py with increasing Px at low slip angles. However, the candidate's model

differs significantly in terms of the separation of the endpoints of the friction curve,

with the Sakai and Good Year models which assume a constant coefficient of sliding

friction.

Figure 3.46 compares the straight-ahead longitudinal friction force response of the

six models. The candidate's model which assumes a friction coefficient decreasing

with slip (or slip speed), shows a decreasing Px as S" increases after the peak, while

the Sakai and &ood Year models show flat responsesafter the peak. Probably, these flat

response are suitable for very rough surf aces, which show a slight increase of Px

with increasing speed, as found in literature review [Sabey et.al (1970)].

Figure 3.47 compares the free-rolling lateral force response of six models. The

candidate's model produces nearly the same curve as the Sakai model, and nearly

the same curve at smaller slip angles as the HSRI-111 and Good Year models.

153

O.B

0.6

0.4

(a) HS Rl-1 0.2

0 ,__......____.___._,.__......____.___,....__......_~-

0 10 20 30 40 SO 60 70 BO 90 100 s. (%)

0.8

0.6

0.4

(c) HSRl-111 0.2

0 ,__......_____.___._,.__......____.___,_......_~-

0 10 20 30 40 SO 60 70 BO 90 100 s. (%)

0.8

0.6

0.1

(e) Sakai

0.2

0 ,__......____.___.,_.,___.......___.___._..._____.___, 0 10 20 30 40 50 60 70 BO 90 100

s. (%)

u. 1

0.8

0,6

0.4

0.2 (b) H SRI-II

0 ....._......___.___, ____ ...__~~--........ --0 ID 20 30 40 50 &O 70 BO 90 100

S,c(,i;)

0.8

0.6

0.4

(d) Good Year 0.2

0 .__......___.___,_...____._~ __ ....._____.__

0 ID 2 0 3 0 4 0 5 0 & 0 7 0 B D 9 0 10 0 s. (9')

O.B

0.6

0.1

[ f) Ca nd !date 0.2

0 ,__......____.___._,._____.____.__.____......____.____,

0 ID 20 30 40 50 60 70 BO 90 100 s. (%)

Fig.3.43. Comparison of longitudinal slip (Sx) versus longitudinal force coefficient (Jlx = FxfFz) of six tyre models operated at the indicated slip angle (degrees).

154

u, 1

O.B

0.6

0.4

0.2

(a) HSRl-1

D .____....,__......____.....__.....____.____. _ _.___.___,____, 0 10 20 30 40 50 60 70 BO 90 l00

s. (%)

u, 1

O.B (c) HSRl-111

0.6

0.4

0.2

D ..__.......___.__.__...._......____.....__....____.____.___. 0 ID 20 30 40 5D 60 70 BO 90 100

s. (%)

O.B (e) Sakai

0.6

0.4

0.2

0 ..__.......___.____..__....___,____._...._____.____.___.

0 ID 20 30 10 SO 60 70 BO 90 l00 s. (%)

1 u,,

0.B (b) HSRl-11

0.6

0.4

0.2

0 ..__.......___.__.__.......___,____. ____ ......___,____.___.

D ID 20 30 40 50 60 70 BO 90 100

s. [%)

1 u,,

0.B (d) Good Year

0.6

0.4

0.2

D ..__.......___.____..__....___,____. ____ ......___,____.___.

D ID 20 30 40 50 60 70 BD 90 100

Uy 1

O.B

0.6

0.4

0,2

s. [%)

(f) Candidate

0 .__.......___,____. ____ ......_ _ ___._....__ __ ___.

0 10 20 30 40 50 60 70 BD 90 100

s. (%)

Fig.3.44. Comparison of longitudinal slip (Sx) versus lateral force coefficient (µy = F/Fz) of six tyre models operated at the indicated slip angle (degrees).

155

1 u, 1 u,

(a) HSRl-1 (b) HSRl-11

0.8 0.8

0.6 0.6

4 4

M 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 0 0.2 D.4 0.6 0.8

u. u.

1 u, 16

1 u, 16

(c) HSRl-111 (d) Good Year 1

0.8 O.B 8

0.6 0.6

0.4 0,4

0.2 0.2

0 0 0 D.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

u. u.

1 u, 1 u, 1& (e) Sakai 16 (f) Candidate 12 12 O.B 0.8

0.6 0.6

4 0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

u. u.

Fig.3.45. The interaction of longitudinal force coefficient (Px = FxfFz) with lateral force coefficient (}ly = F/Fz) at the indicated slip angle (degrees).

156

1 Ux

0.9

o.e

0.7

0.6

0.5

0.4

0,3

0.2

0.1

0 0 10

C

e

(ab) HSRl-1/11 (c) HSRl-111 (d) Good Year (e) Sakai (f) Candidate

20 30 40 50 60

Sx (%) 70 BO 90 100

Fig.3.46. Comparison of the straight-ahead longitudinal force response of six tyre models.

1

0.9

O.B

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Uy

[ a b) H SR 1-1/ I I (cd) HSRl-111/Gd Year

[e) Sakai [f) Candidate

0 _______ __._ _____ ......._ ____ __,_ _____ _.

0 B 12 16

Slip Angle (deg)

Fig.3.47. Comparison of the free-rolling lateral force response of six tyre models.

157

CHAYfER 4. FIELD MEASUREMENT OF FRICTION

Notation (Used frequently in Chapter 4) BF = braking force (force) SF = sideway force (force) WL = wheel load (force) VBr = voltage of braking force (voltage) VCr = voltage of sideway force (voltage)

VWL = voltage of wheel load (voltage)

4.1. Objectives of The Friction Measurement

The Multi Mode Friction Test Truck (MMFTT) was used to measure the braking

and sideway forces of the tyres on road surfaces. See also Section 4.3.

The main objectives for carrying the friction measurement are:

1. To obtain the parameters needed (e.g. tyre stiffness) for the prediction of

tyre-road friction using the proposed model.

2. To validate the theoretically obtained values of coefficient of friction, by

comparing with the measured data.

In addition, some investigations into the effect of certain operational factors (i.e.

tyre pressure, and the rate of braking) will also be carried out.

4.2. Design of Measurement

4.2.1. Sample Size

Repeatability in skid testing is always difficult to achieve. This is due to, among

the other things, the intrinsic variability of the experimental material (such as road

surface and tread rubber) and the accuracy of the experimental work.

158

From Taneerananon's works [1981], it was found that by using a sample size of

two replicates, the 90% confidence limits averaged for all conditions of friction

coefficient tests is ± 0.035, which represents ± 8.4% of the average Braking Force

Coefficient (BFC) and± 4.9% of the average Sideway Force Coefficient (SFC).

In this present experiment, it is decided that an error of ± 5% is to be tolerated.

Hence, the sample size must be increased. By using the sample size of three

replicates, the 90% confidence limits averaged for all conditions of friction

coefficient becomes ± 0.016 (assuming the same standard deviation), which will

represent ± 3.9% of the average BFC and ± 2.3% of the average SFC.

4.2.2. Randomization

One purpose of randomization is to take into account the possible systematic

variations in the experiment, and thus transform them into independent and random

variations. If randomization is not employed, the quoted measure of reliability may

be biased. Further, any inference would be unsupported by a meaningful probability

statement.

There are, of course, situations in which complete randomization is either

impossible or uneconomical. Whereas, on the other hand, a completely systematic

design will have no statistical back up for proper analysis. Therefore, some

intermediate position between the two extremes of complete randomization and a

strictly systematic design is more realistic. In other words, some degree of

randomization is required for the valid application of statistical analysis.

4.2.3. Choice of Levels Used

In general, when choosing levels for testing, consideration should be given to the

practical side of tests, that is whether the results of the investigation can be extended

or applied to practical cases. On the other hand, the implementation of the chosen

levels may be limited by the available resources or the capacity of the test

equipment, in such cases then some compromise has to be made.

159

For a factorial experiment, such as the investigation into the effect of certain

operational factors, the number of levels to be used should also be considered. A

minimum two levels for each factor is possible, where this will give an estimate of

the average change of the response parameter over the range investigated. If it is

desired to have some idea of the shape of response curve, then of course, at least

three levels would be required.

By considering that the factorial experiment is only the secondary objective in

this friction measurement, it was decided to use only two levels for such an

investigation, in which the quantitative conclusions are considered sufficient.

a. Speed

The normal speed of vehicles travelling on wet road surf aces is about 10 mph -

60 mph. The choice of high speed was fixed by the limitation of test vehicle as well

as safety considerations.

The length of test road section is another consideration in choosing the level of

speed. For example, if a 30 mph (50 kph) is chosen, and a 4 second observation is

needed, then the length required for the measured section is 55 m, which should be

in homogenous condition. By adding at least a 30 m length for the starting section

(to reach the desired speed) and another 30 m length for the stopping section, then

the minimum length of the test road section would be 115 m, which ideally should

be in a straight ahead condition and free from disturbing access.

In this work, for the purpose of reducing the effect of vibrations, a test speed of

10 mph wasused.

b. Type of pavement

The common type of road pavement will be used in this work. That is a

medium-textured bituminous road surface, with maximum aggregate size of 0.5 inch

(see Figure 4.1). The length of the measured section was 25 m. This length enables

around 5 secondsobservation with 10 mph of test speed.

160

c. Rate of braking

The MMFIT has a capability to continously measure the frictional forces (about

10 measurements per second), from zero percent slip until 100 percent slip

(locked-wheel). The effect of braking rate (time interval from brake application to

full braking) will be investigated by using two levels of the braking rate, that is a

fast braking (= 0.5 second) and a slow braking (= 2 second).

d. Slip angle

For the purpose of obtaining the lateral tyre stiffness (Cy), the shape of the

response curve, particularly at low slip angles,

levels of slip angle is a necessity.

is required. A minimum three

It wa~ found from the literature review, that the lateral forces (for mode without

braking) increased drastically from a slip angle of 0° to lQ°, while from 10° to 25°

the lateral forces increased gradually, reaching its peak, and then decreasing

gradually. Therefore, it is expected that between 0° and lQ° the high variability of

lateral forces will be encountered. Hence, if the variability will be reduced into an

acceptable level, the setting up of test wheel into the designated angle must be

carried out with a high degree of accuracy for this range of slip angles.

Considering also the maximum slip angle that can be set up using this

MMFIT, slip angles of Q°, 10°, and 15° were used in this experiment.

e. Wheel load

The normal wheel loads of vehicles are about 500 lb - 1000 lb for cars, and 800

lb - 6000 lb for trucks. Unfortunately, the MMFIT (with the car tyre size of test

wheel) was unable to provide such normal range of wheel loads.

The total weight of engine, on which the test wheel is attached, was about 400 lb

(181.4 kg). However, at the time of experiment, the maximum load that can be

applied to the test wheel, using the available hydraulic force, was only about 200 lb.

Probably, the only advantage of using this low wheel load (200 lb) is its

lower influence on the operation of the test vehicle, such as the influence on the

vehicle speed during the application of the test wheel brake.

161

f. Tyre type

As found in the literature review, many types of car tyre are available for normal

use. In this present study, the 5.20 x 10 inch cross-ply tyres with smooth tread (no

tread pattern) were used.

The choice of smooth tyre was much more influenced by its high contribution to

the traffic accidents on wet roads. Furthermore, it is considered that the use of

smooth tyre will eliminate the effect of tyre wear (due to testing) on the frictional

force results.

The tread rubber of these smooth tyres is made of TR 457 compound which is a

mixture of Styrene Butadine Copolymer and Poly Butadine in the proportion (60/40

SBR/BR).

g. Tyre pressure

The common inflation pressure range for passenger car tyres is 20 psi to 30 psi.

In this present work, the effect of inflation pressure will be investigated using two

levels of pressure, that is 30 psi (207 kN/m2) and 20 psi (138 kN/m2).

h. Water film

The range of water film occurring in practice varies from just wet (shower/light

rain) to flooded (long period of rain). Test results given by Meyer et.al [1974]

showed that there was little or no change of the skid number for water films of 0.02

inch (0.508 mm) or thicker (flooded), for test speed up to 50 mph.

The effect of inaccurate setting up of pump pressure (due to the changing

temperature of pump engine, fluctuation of power voltage, or variation in the test

truck speed) on producing the water film, hence on the results of friction, will be

less significant by using the flooded water level.

The use of this flooded level, will not cause dynamic hydroplaning for 10 mph

test speed, where according to Home and Dreher [1963] this phenomena generally

occurs, on a flooded surface, above the following speeds:

For 30 psi: V = (10.35)~30 = 56.7 mph

For 20 psi: V = (10.35)~20 = 46.3 mph.

162

4.3. The Multi Mode Friction Test Truck

The vehicle was designed by Yandell primarily for use in measuring tyre-road

friction to validate the Texture Friction Meter. The water discharging system was

designed by Taneerananon [1981]. For this present work, a new computerised data

logging system has been installed, to replace the old ultra-violet chart recorder.

The test truck consists of a six tonne Dodge chassis with the retractable test

wheel mounted on the back .. Water tanks for road wetting are mounted behind the

driver's cabin. A 120 cubic ft (3.4 cubic meter) per minute high pressure pump

delivers the water to the road surface. An instrument and operator's cabin is

mounted on the rear of the truck behind the water tanks (see Figures 4.3-4.4). A 2

kilowatt 240 volt generator provides electric power for instruments and for the

hydraulic power unit.

As indicated earlier, despite the capability to measure braking and sideway

forces, there are 3 main limitations of the MM F T T :

1. Inability to turn (continously) the slip angle during the measurement. Hence, it is

not possible to simulate the turning-in-braking condition. Instead, it can only

simulate the braking-in-turning condition.

2. The maximum slip angle is limited up to 15°-20°.

3. The maximum wheel load is lower than that usually used on the passenger car.

4.3.1. The Test Wheel

The transverse engine, gear box, locked differential and right hand front driving

wheel of a Morris Mini Minor is mounted on the rear of the truck on a hydraulic

lowering device. The steerable driving wheel acts as the friction test wheel. The left

hand wheel has been removed and replaced by a disc brake for braking the test

wheel through the locked differential.

The brake is actuated by a solenoid through an adjustable needle valve so that

braking can, if necessary, be applied slowly enough to allow peak braking to be

detected.

163

The Morris Mini suspension was redesigned with no castor angle to allow the

accurate measurement of the forces. The longitudinal, lateral and vertical forces on

the test wheel are sensed by electric transducers and their amplified values are

recorded by a laptop computer.

4.3.2. The Recording Device

An ACRO-400 Data Acquisition and Control Unit system is used for continously

( ~ 10 measurements per second) recording the loads (vertical, longitudinal and

lateral) on the test wheel. Amplified bridge outputs from a six channel amplifier are

fed into the recorder. The test wheel speed sensed as pulses per second, is converted

to voltage and read directly into the recorder. The procedure to operate the Strain

Gauge Amplifier and ACROLOG-400 are attached (see Appendix C and D). The

configuration of the recording device is shown in Figure 4.3.

A device is used to tum the brake on at the commencement of the test strip and

to tum it off at the end of the test strip. Catadioptric targets (see Figure 4.2) are

placed at each end of the test strip facing the passing test truck. A modulated

infra-red photo detector (see Figure 4.6) turns a relay on the first target and off at

the second. The relay operates the on and off position of the solenoid.

4.3.3. The Road Wetting Device

The MMFTT has the capability to produce a belt of water about 6 inch (15 cm)

wide, laid immediately in front of the test tyre. The water is deposited by means of

one of a number of different sized water jets. The water pressure and the jet size are

adjusted so that the water is emerging from the jet at the test truck speed, so there is

no relative velocity between the water and the road, and consequently no splashing.

The flow rate is also adjusted so that the desired water film thickness is laid for a

particular speed. Hence, the flow rate has to be increased as the truck speed is

increased for a given film thickness. A wind shield is fitted around the water

discharging apparatus (see Figure 4.5) to ensure that all water reaches the road in the

desired strip.

164

Fig.4.1. Medium-textured bituminous surface.

Fig.4.2. Catadioptric targets behind the MMFIT.

165

Fig.4.3. The recording device on cabin.

Fig.4.4. Hydraulic and braking controls on cabin.

166

Fig.4.5. Water discharging apparatus & wind shield (in front of test wheel), at near the right side of MMFIT rear axle.

Fig.4.6. Infra-red photo detector ( ..L) and disc brake for test wheel (i), at near the left side of MMFIT rear axle.

167

Fig.4.7. Calibration of vertical, longitudinal and lateral forces .

Fig.4.8. The MMFIT and arrangement for calibration.

168

4.4. Calibration

4.4.1. Wheel Load Calibration

The wheel load was calibrated by using a special designed rig and winch as

shown in Figures 4.7-4.8. The deflections were converted into voltage and recorded

as the load being applied to the test wheel through a spring balance. A wheel load

calibration curve is given in Figure 4.9.

W L (lbs) 400

350

300

250

200

150

1 DO

50

WL = 375VWL

D i.:::;_ _ __._ __ --'---.&......----'-----'---.&......-----1.-----L---_.__ _ __.

0 D .1 0.2 0.3 0.4 0.5 0.6 VWL (voltage]

Fig.4.9. Wheel load calibration curve.

4.4.2. Braking Force Calibration

0.7 0.8 0.9

For calibration of braking force, the test wheel was locked from rotating by the

application of the brake. A belt was wound around the wheel, the loose end of

which was connected to a spring balance (see Figures 4.7-4.8). The horizontal load

(in longitudinal direction) applied to the wheel was noted from the spring balance

reading and the corresponding voltage were recorded on the computer.

169

BF (lbs) 350 BF = 235V8r + 30VBr .Ver + 250VCr 300 (Angle 0)

250

200

150

100

50 01--~....-::;._ ____________________ _

-50 L-_ __._ __ .___....._ _ ____...__ _ _._ _____ ...._ _ ___. __ ...._ _ __,

-0.1

350

300

250

200

150

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

BF = S00VBr + 200VBr(0.001/Er - 400VBr .VCr(0.001)ver -23DVBr 2 - 730VBr .VCr + 460VBr 2.VCr + 110VCr

(Angle 10)

50 ..... ~-

0.8 0.9

VCr = 0

0---------------------------SOL-...___._ __ .___ ......... _ ____..___ _ ___._ _ ____. __ __._ _ ____. __ __._ _ __,

-0.1

350

300

250

200

150

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

BF = 590VBr + 1370VBr(0.001) va- - 2740VBr .VCr(0.001)va-280VBr 2 - 1060VBr.VCr + 560VBr 2.VCr + SOVCr

(Angle 15)

0.8 0.9

VCr = 0.5

soL_j_~----------0----+------------------------so.______........._ _ ____..___ _ ___._ _ ____. __ ....__ _ __._ __ .....______,__._ __ ....__ _ _. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

VBr (voltage)

Fig.4.10. Braking force calibration curves for angles 00, 10°, and 15°.

170

0.8 0.9

SF [lbs) 350 SF = 245VCr + 5SVCr 2 - 340VCr .VBr 300 + 385VCr 2.VBr - 20V8r

[Angle 0) 250

200

150

100

50

0

-50 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

350 SF = 180VCr + 110VCr 2 - 300VCr.VBr

300 + 490VCr 2 .VCr - 40VBr VBr = 0.5 [Angle 10)

250

200

150

100

so 0

-so -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

350 SF = BOVCr + 200VCr 2 - 145VCr .VBr

300 + 415VCr 2.VBr - 65VBr VBr = 0.5

(Angle 15) 250

200

150

100

so

0

-50 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

VCr [voltage)

Fig.4.11. Sideway force calibration curves for angles er, 10°, and 15°.

171

Due to the geometrical position of transducers used, there was a significant

interaction between longitudinal and lateral forces. This will be taken into account,

by providing a series of curves with different sideway forces (voltages). The braking

calibration was then carried out for each designated slip angle. The curves of

braking force calibration are given in Figure 4.10.

4.4.3. Sideway Force Calibration

The sideway force calibration was carried out with the test wheel in the free

rolling condition. In this case the belt was wrapped around the wheel at right angles

to the wheel plane, the lateral load was thus applied in the perpendicular direction to

the wheel plane (see Figures 4. 7-4.8).

Similar to braking force calibration, there was also (with a lesser significance) an

interaction between lateral and longitudinal forces. This will be taken into account,

by providing a series of curves with different longitudinal forces (voltages). The

sideway force calibration curves, for each slip angle, are given in Figure 4.11.

4.5. Experimentation

4.5.1. Sequence of Experiment

The total number of treatments with three replications is 36 (2 tyre pressure x 2

rate of braking x 3 slip angle x 3 replication). Complete randomization of the series

of 36 treatments, even though not impossible, was quite impractical (such as, the

change of slip angle after each replication).

To provide some degree of randomization, it was proposed:

1. To arrange the treatments into blocks of convenient treatments so that the

execution time was kept to minimum. This is done by choosing one tyre pressure

and one slip angle in a block.

2. To randomize the rate of braking within a block.

172

3. To execute consecutively the three replications following each treatment.

4. To randomize the execution of various blocks.

5. To control or take into account the variations which might affect the test results,

such as the change in tyre temperature, the change in testing technique or testing

instruments.

Numbers in bracket listed in Table 4.1 are the sequence of experiment.

Table 4.1. Detailed treatments, showing the sequence of experiment (in bracket).

Slip angle oo 100 15°

Tyre pressure (psi) 20 30 20 30 20 30

Rate of Fast (~ 0.5 sec) E[4] A[l] M[5] 1[8] U[12] Q[9] Braking Slow (~ 2 sec) G[3] C[2] 0[6] K[7] W[ll] S[lO]

Sequence of block II I III IV VI V

Date of testing Day 1 Day 2 Day 3

4.5.2. Test Procedure

The test procedure for the purpose in this present experiment is outlined as

follows:

1. Warm the test tyre prior of commencement of a test series (one day) by running

over the test section for about 2 minutes.

2. Check the tyre inflation pressure.

3. Record the ambient air temperature.

4. Check and adjust the signal amplifier to zero position before each test.

5. Check and adjust the rate of braking to the desired value.

6. Record the static values of signals (test wheel in up position) for use as base

lines before and after test.

7. Lower the test wheel, and start run.

8. Start the measurement (recording) at a distance approximately 10 meter from the

beginning of the measured section.

9. Apply the water at a distance of about 6 m from the beginning of the measured

section.

173

10. Turn off the water and stop the recording after leaving the measured section.

A test sequence over the test section is shown in Figure 4.12.

-------------> T 25 m T J..----- L -----J..

DD-o-o-- - - - - - - - - D D-- - -

(1) Start

run

( 2) Recorder

activated

(3) (4) Water Brake

applied activated

(5) Brake

released

L = length of measured section (25m). T = catadioptric target.

Fig.4.12. Testing sequence.

4.5.3. Data Logging Program and Corrections

(6) Water/

Recorder stopped

The ACR0-400 recorder system has been provided with a Program to read Strain

Gauge and Speed Output (see Appendix E). Using this program, the signals

(deflections) from the transducers were converted and recorded as voltage in the

output. The Time (in seconds) and the Status of the braking (on/off) were also

recorded. The recorded test wheel speed (km/h) is obtained from the rotation of the

test wheel (revolving/second), and its rolling radius.

Actually, the speed of the test vehicle over the measured section can be

calculated by dividing the length of the measured section (25 m), with the time

interval from the brake on (activated by the first catadioptric) to the brake off

(released by the second catadioptric ).

It is known that the rolling radius can be affected by many factors, such as the

tyre size. Therefore, the recorded test wheel speed is considered as an approximation

only, so needs correction. This was done as follows:

174

1. Calculate the speed of the test vehicle over the measured section.

2. Assuming that the speed of the test vehicle is constant over the measured section,

then this speed is assumed the same as the speed at the first catadioptric target

(braking start being activated).

3. Assuming that at zero slip angle, there is no slip of the test wheel until passing

the first catadioptric target, then the recorded test wheel speed at approaching this

catadioptric target must be equal with the speed of the test vehicle over the

measured section.

4. A correction factor then can be obtained by equalizing the speed of the test

wheel with the speed of the test vehicle.

For slip angles other than zero degree, another correction can be carried out.

First, the correction factor at zero slip angle as obtained above is applied to the

recorded test wheel speed, to obtain the corrected test wheel speed in the direction

of wheel plane. Second, this corrected test wheel speed then is divided by coscx

(where ex is slip angle), to obtain the corrected test wheel speed in the direction of

travel.

4.6. Possible Variability on Test Results

As mentioned earlier, variability of test results can occur due to the intrinsic

variability of the experimental material itself (road surface). Other possibilities are

errors or inaccuracies during experiments. Some of these errors are of a systematic

type (the residual mean is not zero) which leads to a difference from the true mean,

and some are of a random type (the residual mean is zero) which leads to

increasing variation.

4.6.1. Fluctuation of Voltage Supplied

The fluctuations of voltage supplied to the recording device can cause variation

in the signal recorded. This variation is believed to be random.

175

Another source is the difference between the stationary values of the signals at

the beginning and end of test run. This is due to the gradual increase ( or decrease)

of voltage supplied. This error can be considered as systematic. The problem can be

brought under control by corrected the values recorded into a skew base line

(connecting the stationary values before and after testing).

4.6.2. Setting Up of Slip Angle

The setting up of slip angle for markings, has been done as accurately as

possible, even though using only simple tools (e.g. ruler). The error on this stage, if

it is occurs, will lead to an incorrect magnitude of angle used, which can produce

different result (i.e. differ with the true mean). However, this does not contribute to

the variability. The effect of this error will be of more significance for slip angles

between 00-10° (where a small change on slip angle will lead to a significant change

on lateral force).

The random variations begin to appear if the factor of slip angle is not be placed

in one series of testing. In other words, a random variation occurs if the

experiment of one slip angle is not finished at once, but is interrupted by other

testing with another slip angle.

4.6.3. Test Location

The friction coefficient varies laterally as well as longitudinally on a road

surface. Hence, when repeated testing of a particular stretch of pavement is desired,

the test should be made at the same location on the pavement.

In selecting the test sections, a stretch of road which appeared to be

longitudinally uniform (at least along the measured section) was chosen, thus

ensuring minimum variation of skid resistance along the pavement. The lateral

uniformity is of secondary importance, since the truck driver had a reasonably good

control of the vehicle position, using the clearly visible edge of pavement as a guide.

176

4.6.4. Speed

In general, the measured friction on wet pavements decreases with increasing

speed. The rate of decrease depends on tyre type and surface texture of pavements.

The gradient of friction coefficient is generally steeper at lower speeds and also

the gradient is steeper on smooth textured surfaces.

A variation in speed can be caused by the random speed deviations of the

driver. In most cases, the driver of the truck was able to keep the desired speed

within the limits of ± 1.6 km/h (± 1 mph), and since the measured section is only 25

meter (82 feet) long, the speed was easily maintained over the measured length.

4.6.5. Inflation Pressure

The change of tyre temperature during testing is the main source for the variation

in inflation pressure. Taneerananon (1981] found that the maximum increase in

inflation pressure (with increasing temperature) was about 2 psi (13.8 kN/m), but in

most cases the increase was less than this amount. Kummer and Meyer [1962]

suggested that if the inflation pressure was held constant ± 2-5 percent, then no

effect of this fluctuation should be detectable on normal pavements.

The tyre pressure in this experiment was set at the test pressure in the laboratory

before going out to the test location. It was again checked before the test. During a

series of tests, the pressure was regularly checked at the end of the run. Since the

inflation pressuresused in the experiment are 20 and 30 psi (137.9 and 206.8 kN/m2),

it is then reasonable to say that the effect of inflation pressure variations on the

measured coefficient is negligible.

4.6.6. Temperature

The water temperature used during experiment was fairly constant, and thought to

have no effect on the measured friction. During this experiment (3 days), the range

177

of air temperature measured was between 17°C to 23°C, with average variation was

± 3°C. Over a test period (1 day), the air temperature rarely varied more than± 2°C.

According to Meyer et.al [1974] the effect of temperature on the skid resistance

was slight, the correction if. carried out, was of the order of 2% per 10°F (5.5°C),

where the dry temperature of the rolling tyre measured just before skid should be

used for correction. In view of this and the lack of information on the tyre tread

temperature, it was reasonable to assume that the effect due to temperature variation

was insignificant.

4.6.7. Rubber Properties

According to Meyer et.al [1974] the variations of rubber properties (hardness and

damping factor) were small compared to the over-all variability of skid testing and

had no significant effect on the skid number.

Taneerananon [1981] found that the hardness of tread rubber used in his

experiment, measured (by hand held durometer) when the tyre was cold, was 64, and

then after some preliminary test it decreased and varied between 62-63. During

experiment, he observed that there was no appreciable change in the hardness.

Accordingly, he believed that the damping factor of the test tyres also remained

relatively unchanged.

For this present experiment, it can also be concluded that it is not likely that the

rubber properties changed significantly, particularly after the new tyres had lost their

initial stiffness.

178

CHAPTER 5. ANALYSIS OF EXPERIMENTAL RESULTS

Notation (Used frequently in Chapter 5) BF = braking force (force)

BFC = braking force coefficient ½ = longitudinal slip stiffness (force/slip, or force/fraction slip), or

longitudinal deformation stiffness (force) Cy = lateral slip stiffness (force/degree, or force/radian), or

lateral deformation stiffness (force) C"C = longitudinal slip stiffness coefficient (per slip, or per fraction slip), or

longitudinal deformation stiffness coefficient CyC = lateral slip stiffness coefficient (per degree, or per radian), or

lateral deformation stiffness coefficient Fz = vertical load (force) IP = tyre inflation pressure (psi)

SF = sideway force (force) SFC = sideway force coefficient

S" = longitudinal slip (%, or per fraction slip) T = time (unit time) V = velocity (unit length/unit time)

VS = vehicle speed (km/h) VBr = voltage of braking force (voltage) VCr = voltage of sideway force (voltage)

VWL = voltage of wheel load (voltage) WL = wheel load (force) WS = wheel speed (km/h)

a = slip angle (degree, or radian)

5.1. General

With the total treatments of 12 (as shown in Table 4.1), and each treatment

consisting of 3 replicates, 36 computer outputs were obtained. For each output there

were 6 values being recorded: time (T), voltage of wheel load (VWL), voltage of

lateral force (VCr), voltage of longitudinal force (VBr), wheel speed (WS), and

braking position of test wheel (B).

For the 25 m length of measured section and 10 mph of test truck speed, about 5

seconds of observations were obtained. Taking about 1 second (i.e about 10 data)

before and 1 second after the measured section, from which 7 seconds of data were

179

to be produced. Summing up, in this experiment the total amount of data was around

12x3x6x7x10 = 1512.

The friction coefficients were obtained by converting the voltage to forces, using

the appropriate calibration charts. The results, are presented in tables and figures for

one treatment (treatment A). Whereas, the data from the remaining 11 treatments are

presented in figures. The friction coefficient versus time can be transformed into the

friction coefficient versus slip (or slip angle); and the results for all treatments are

presented in tables and figures.

5.2. Presentation of Data

5.2.1. Friction Coefficient versus Time

Table 5.1 presents the raw data of treatment A (3 replicates), obtained from the

computer output. Using the appropriate calibration chart, the friction coefficients

(SFC and BFC) can be calculated. Figure 5.1 presents the graphical form of the

friction coefficients, the vehicle speed (averaged over the measured section), and the

wheel speed (after correction, see Section 5.2.2) of treatment A.

Table 5.2 presents the raw data of treatment A (mean), calculated from Table 5.1.

For simplicity, the mean value of friction coefficient was obtained by averaging the

corresponding voltage values, and then the friction coefficient was obtained based on

this average value (instead of obtaining the friction coefficient for each replicate, and

then averaging the values of this friction coefficient).

Using the same procedure as above, the friction coefficient for other treatments

can be obtained. The recorded wheel speeds (see Section 4.5.3) have been corrected

as described in Section 5.2.2. The percent slip on entering the measured section have

also been corrected, as described in Section 5.2.3. The graphical forms of friction

coefficients (SFC and BFC), and the wheel speed and vehicle speed, versus time, are

presented in Figures 5.2-5.5.

180

Table 5.1. Raw data of treatment A (3 replicates). Al A2 A3

T VWL VCr VBr ws B T VWL VCr VBr WS B T VWL VCr VBr WS B

19 0.55 -0.10 0.14 17.73 0 37 0.57 -0.09 0.11 19.43 0 48 0.54 -0.11 0.12 17.21 0 19 0.55 -0.10 0.14 17.73 0 37 0.58 -0.11 0.10 19.43 0 48 0.55 -0.12 0.12 17.21 0 19 0.53 -0.08 0.11 17.73 0 37 0.57 -0.11 0.12 19.43 0 48 0.55 -0.11 0.12 17.21 0 19 0.57 -0.11 0.12 17.73 0 37 0.57 -0.11 0.11 19.43 0 48 0.53 -0.10 0.10 17.21 0 19 0.55 -0.09 0.12 17.73 0 37 0.56 -0.10 0.11 19.43 0 48 0.54 -0.12 0.13 17.21 0 19 0.53 -0.08 0.13 17.73 0 37 0.55 -0.10 0.12 19.43 0 48 0.57 -0.11 0.11 17.21 0 20 0.55 -0.10 0.11 17.73 0 37 0.56 -0.12 0.12 19.43 0 48 0.58 -0.15 0.14 17.21 0 20 0.57 -0.12 0.12 19. 93 0 37 0.59 -0.14 0.13 19.43 0 49 0.56 -0.14 0.15 16.25 0 20 0.56 -0.10 0.14 19. 93 0 37 0.60 -0.15 0.14 19.43 0 49 0.56 -0.14 0.16 16.25 0 20 0.58 -0.14 0.15 19. 93 0 38 0.58 -0.14 0.15 19.43 0 49 0.59 -0.16 0.15 16.25 0 20 0.58 -0.13 0.13 19. 93 1 38 0.61 -0.17 0.17 20.43 1 49 0.57 -0.15 0.15 16.25 1 20 0.57 -0.10 0.13 19. 93 1 38 0.59 -0.14 0.17 20.43 1 49 0.57 -0.01 0.51 16.25 1 20 0.57 0.00 0.56 19.93 1 38 0.59 -0.02 0.55 20.43 1 49 0.56 0.01 0.43 16.25 1 20 0.53 0.00 0.55 19. 93 1 38 0.57 o.oo 0.69 20.43 1 49 0.56 0.03 0.45 16.25 1 20 0.53 0.00 0.59 19.93 1 38 0.55 0.00 0.58 20.43 1 49 0.54 0.02 0.43 16.25 1 21 0.51 0.02 0.57 19.93 1 38 0.53 -0.01 0.56 20.43 1 49 0.53 -0.01 0.61 16.25 1 21 0.50 0.06 0.35 10.07 1 38 0.52 0.03 0.46 20.43 1 50 0.54 0.02 0.41 7.52 1 21 0.51 0.07 0.31 10.07 1 38 0.54 0.06 0.42 20.43 1 50 0.53 0.03 0.46 7.52 1 21 0.51 0.09 0.25 10.07 1 39 0.56 0.05 0.30 20.43 1 50 0.53 0.06 0.39 7.52 1 21 0.52 0.06 0.23 10.07 1 39 0.53 0.04 0.46 4.44 1 50 0.53 0.05 0.45 7.52 1 21 0.53 0.06 0.24 10.07 1 39 0.54 0.05 0.40 4.44 1 50 0.55 0.02 0.49 7.52 1 21 0.52 0.03 0.26 10.07 1 39 0.54 0.05 0.48 4.44 1 50 0.56 0.02 0.44 7.52 1 21 0.55 0.03 0.35 10.07 1 39 0.56 0.02 0.44 4.44 1 50 0.55 0.03 0.41 7.52 1 21 0.56 0.05 0.30 10.07 1 39 0.55 0.04 0.40 4.44 1 50 0.54 -0.01 0.55 7.52 1 21 0.56 0.02 0.38 10.07 1 39 0.55 0.03 0.45 4.44 1 50 0.55 -0.01 0.59 7.52 1 22 0.50 0.01 0.55 1.13 1 39 0.56 0.03 0.40 4.44 1 51 0.53 0.01 0.52 7.52 1 22 0.49 -0.01 0.58 1.13 1 39 0.55 0.03 0.40 4.44 1 51 0.55 -0.01 0.56 0.00 1 22 0.50 0.00 0.57 1.13 1 40 0.55 0.04 0.39 4.44 1 51 0.55 0.00 0.54 0.00 1 22 0.48 0.02 0.44 1.13 1 40 0.55 0.05 0.37 0.06 1 51 0.53 0.00 0.52 0.00 1 22 0.47 -0.01 0.45 1.13 1 40 0.55 0.05 0.37 0.06 1 51 0.54 0.00 0.53 0.00 1 22 0.48 0.02 0.43 1.13 1 40 0.55 0.06 0.36 0.06 1 51 0.54 0.01 0.45 0.00 1 22 0.49 0.05 0.36 1.13 1 40 0.54 0.05 0.41 0.06 1 51 0.53 0.01 0.50 0.00 1 22 0.53 0.03 0.35 1.13 1 40 0.55 0.04 0. 4 6 0.06 1 51 0.54 0.01 0.50 0.00 1 22 0.53 0.03 0.29 1.13 1 40 0.56 0.04 0.39 0.06 1 51 0.54 0.03 0.38 0.00 1 23 0.54 0.04 0.35 1.13 1 40 0.56 0.04 0.36 0.06 1 51 0.53 0.04 0.51 0.00 1 23 0.50 0.02 0.50 1.19 1 40 0.55 0.04 0.40 0.06 1 52 0.54 0.02 0.42 0.00 1 23 0.50 0.05 0.35 1.19 1 40 0.55 0.05 0.35 0.06 1 52 0.54 0.02 0.44 0.00 1 23 0.53 0.02 0.34 1.19 1 41 0.54 0.05 0.31 0.06 1 52 0.54 0.00 0.57 0.00 1 23 0.53 0.00 0.39 1.19 1 41 0.55 0.06 0.35 0.03 1 52 0.55 0.00 0.47 0.00 1 23 0.50 0.03 0.34 1.19 1 41 0.54 0.06 0.34 0.03 1 52 0.54 0.01 0.48 0.00 1 23 0.51 0.02 0.37 1.19 1 41 0.55 0.03 0.33 0.03 1 52 0.54 0.02 0.37 0.00 1 23 0.53 0.01 0.37 1.19 1 41 0.56 0.04 0.32 0.03 1 52 0.54 0.02 0.44 o.oo 1 23 0.52 0.01 0.44 1.19 1 41 0.54 0.05 0.37 0.03 1 52 0.54 0.02 0.42 0.00 1 24 0.53 0.01 0.43 1.19 1 41 0.55 0.04 0.34 0.03 1 52 0.54 0.01 0.38 0.00 1 24 0.51 0.02 0.42 0.09 1 41 0.55 0.05 0.34 0.03 1 53 0.54 0.01 0.42 0.00 1 24 0.52 0.01 0.46 0.09 1 41 0.55 0.04 0.44 0.03 1 53 0.55 0.01 o. 49 0.00 1 24 0.53 0.02 0.40 0.09 1 42 0.56 0.03 0.33 0.03 1 53 0.54 0.02 0.36 0.00 1 24 0.52 0.03 0.34 0.09 1 42 0.55 0.06 0.34 0.06 1 53 0.54 0.01 0.51 o.oo 1 24 0.52 0.03 0.33 0.09 1 42 0.55 0.05 0.41 0.06 1 53 0.54 0.00 0.53 0.00 1 24 0.51 0.05 0.30 0.09 1 42 0.54 0.06 0.30 0.06 1 53 0.55 0.00 0.46 0.00 1 24 0.51 0.04 0.38 0.09 1 42 0.54 0.05 0.32 0.06 1 53 0.54 0.00 0.54 0.00 1 24 0.51 0.05 0.30 0.09 1 42 0.55 0.07 0.26 0.06 1 53 0.56 0.00 0.41 0.00 1 24 0.51 0.06 0.28 0.09 1 42 0.56 0.07 0.21 0.06 1 53 0.54 0.02 0.34 0.00 1 25 0.51 0.06 0.33 0.00 1 42 0.53 0.08 0.40 0.06 1 53 0.54 0.02 0.51 0.00 1 25 0.52 0.04 0.35 0.00 1 42 0.55 0.04 0.38 0.06 1 54 0.55 0.02 0.37 0.00 1 25 0.52 0.03 0.32 0.00 1 42 0.55 0.04 0.36 0.06 1 54 0.54 0.05 0.42 0.00 1 25 0.52 0.07 0.26 0.00 1 43 0.55 0.06 0.24 0.03 1 54 0.54 0.04 0.33 0.00 1 25 0.51 0.06 0.29 0.00 1 43 0.54 0.08 0.29 0.03 1 54 0.54 0.06 0.25 0.00 1 25 0.52 0.06 0.29 0.00 1 43 0.54 0.07 0.37 0.03 1 54 0.53 0.06 0.32 0.00 1 25 0.51 0.05 0.34 0.00 1 43 0.55 0.05 0.42 0.03 1 54 0.54 0.00 0.46 0.00 1 25 0.52 0.00 0.39 0.00 0 43 0.55 0.01 0.48 0.03 0 54 0.54 0.02 0.38 0.00 1 25 0.52 0.01 0.37 0.00 0 43 0.55 -0.02 0.45 0.03 0 54 0.54 0.03 0.35 0.00 1 26 0.52 -0.07 0.18 0.00 0 43 0.53 -0.07 0.08 0.03 0 54 0.54 0.06 0.25 0.00 1 26 o. 49 -0.07 0.10 2.58 0 43 0.54 -0.10 0.13 0.03 0 55 0.53 0.07 0.33 0.00 1 26 0.48 -0.08 0.12 2.58 0 43 0.52 -0.08 0.13 0.03 0 55 0,54 0.01 0.48 0.00 1 26 0.52 -0.10 0.14 2.58 0 44 0.55 -0.12 0.15 0.03 0 55 0.55 0.00 0.48 0.00 0 26 0.55 -0.12 0.15 2.58 0 44 0.58 -0.14 0.16 8.10 0 55 0.55 0.00 0.43 0.00 0 26 0.55 -0.12 0.15 2.58 0 44 0.56 -0.12 0.16 8.10 0 55 0.53 -0.09 0.05 0.00 0 26 0.53 -0.10 0.12 2.58 0 44 0.56 -0.11 0.15 8.10 0 55 0.53 -0.09 0.10 0.00 0 26 0.51 -0.08 0.11 2.58 0 44 0.55 -0.11 0.13 8.10 0 55 0.52 -0.10 0.11 0.00 0

55 0.54 -0.12 0.14 0.00 0 55 0.57 -0.14 0.15 0.00 0 56 0.57 -0.12 0.15 0.00 0 56 0.57 -0.12 0.14 11. 64 0 56 0.57 -0.13 0.14 11. 64 0

Notes: T = times (seconds); VWL, VCr, VBr = voltages of wheel load, sideway force, and braking force; WS = wheel speed (km/h); B = brake position (0 = off, 1 = on).

181

BFC & SFC Speed (Km/ h) o.s ,--;;;:;;===i~~~~;;;~~~~~~~~~~J]s.[_=;--_11.5-s_.:s:a_[------:A~1,11s 0,7 16

0.6 14

0.5 12

0.4 10 0.3

8 0.2

6 0.1 0 4

-0,1 2

-0.2 o O 10 20 30 40 50 60 70 BO

0.8 .-----------------------.V-;;=S.-=--;;1~6n.s~s-----A2----, 18

0.7 16

0.6 14

0,5 12

0.4 10 0.3

8 0.2

6 0.1 D 4

-0.1 2

-0.2 0 0 10 20 30 40 50 60 70 80

0.8 ..---------------------------, 18 A3

0.7 16

0.6 14

0.5 12

0.4 10 0.3

8 0.2

6 0.1 O 4

-0.1 2

-0.2 D 0 10 20 30 40 50 60 70 80

Time (x 0.107 sec)

Fig.5.1. Experiment result vs Time of treatment A (3 replicates). Legend: --o-- (BFC), -<>- (SFC), L-- (Wheel Speed), - (Veh. Speed).

182

Table 5.2. Raw data of treatment A (mean). Al+ A2 + A3 Treatment A

T' l:VWL :Ever l:vBr l:ws l:B VWL vcr VBr ws B WL SFC BFC WS'

1 1.66 -0.30 0.37 54.37 0 0.5533 -0.1000 0.1233 18.12 0 207.5 -0.105 0.017 15.94 2 1.68 -0.33 0.36 54.37 0 0.5600 -0 .1100 0.1200 18.12 0 210.0 -0 .113 0.001 15.94 3 1.65 -0.30 0.35 54.37 0 0.5500 -0.1000 0.1167 18.12 0 206.2 -0.106 0.010 15.94 4 1. 67 -0.32 0.33 54.37 0 0.5567 -0.1067 0.1100 18.12 0 208.8 -0.111 -0.006 15.94 5 1.65 -0.31 0.36 54.37 0 0.5500 -0.1033 0.1200 18.12 0 206.2 -0.109 0.010 15.94 6 1. 65 -0.29 0.36 54.37 0 0.5500 -0.0967 0.1200 18.12 0 206.2 -0.103 0.018 15.94 7 1. 69 -0.37 0.37 54.37 0 0.5633 -0.1233 0.1233 18.12 0 211.2 -0.123 -0.011 15.94 8 1. 72 -0.40 0.40 55.61 0 0.5733 -0.1333 0.1333 18.54 0 215.0 -0.127 -0.012 16. 30 9 1. 72 -0.39 0.44 55.61 0 0.5733 -0. 1300 0.1467 18.54 0 215.0 -0.123 0.006 16.30

10 1. 75 -0.44 0.45 55.61 0 0.5833 -0.1467 0.1500 18.54 0 218.8 -0.133 -0.009 16.30 11 1. 76 -0.45 0.45 56.61 3 0.5867 -0.1500 0.1500 18.87 1 220.0 -0.135 -0.013 16.60 12 1. 73 -0.25 0.81 56. 61 3 0.5767 -0.0833 0.2700 18.87 1 216.2 -0.079 0.194 16.60 13 1. 72 -0.01 1.54 56. 61 3 0.5733 -0.0033 0.5133 18.87 1 215.0 -0.049 0.557 16.60 14 1. 66 0.03 1. 69 56.61 3 0.5533 0.0100 0.5633 18.87 1 207.5 -0.052 0.651 16.60 15 1. 62 0.02 1. 60 56.61 3 0.5400 0.0067 0.5333 18.87 1 202.5 -0.051 0.628 16.60 16 1.57 0.00 1. 74 56.61 3 0.5233 0.0000 0.5800 18.87 1 196.2 -0.059 0.695 16.60 17 1.56 0.11 1.22 38.02 3 0.5200 0.0367 0.4067 12.67 1 195. 0 -0.020 0.539 11.15 18 1.58 0.16 1.19 38.02 3 0.5267 0.0533 0.3967 12.67 1 197.5 -0.007 0.543 11.15 19 1. 60 0.20 0.94 38.02 3 0.5333 0.0667 0.3133 12.67 1 200.0 0.019 0.455 11.15 20 1. 58 0.15 1.14 22.03 3 0.5267 0.0500 0.3800 7.34 1 197.5 -0.007 0.518 6. 46 21 1. 62 0.13 1.13 22.03 3 0.5400 0.0433 0.3767 7.34 1 202.5 -0.010 0. 493 6. 46 22 1. 62 0.10 1.18 22.03 3 0.5400 0.0333 0.3933 7.34 1 202.5 -0.019 0.500 6. 46 23 1. 66 0.08 1. 20 22.03 3 0.5533 0.0267 CJ.4000 7.34 1 207.5 -0.024 0.487 6.46 24 1. 65 0.08 1.25 22.03 3 0.5500 0.0267 0.4167 7.34 1 206.2 -0.026 0.509 6.46 25 1. 66 0.04 1.42 22.03 3 0.5533 0.0133 0.4733 7.34 1 207.5 -0.040 0.553 6. 46 26 1. 59 0.05 1. 47 13.09 3 0.5300 0.0167 0.4900 4.36 1 198.8 -0.042 0.602 3.84 27 1. 59 0.01 1. 54 5.57 3 0.5300 0.0033 0.5133 1. 86 1 198.8 -0.050 0. 611 1. 63 28 1. 60 0.04 1.50 5.57 3 0.5333 0.0133 0.5000 1. 86 1 200.0 -0.045 0.605 1. 63 29 1.56 0.07 1. 33 1.19 3 0.5200 0.0233 o. 3967 0.40 1 195.0 -0.034 0.566 0.35 30 1.56 0.04 1. 35 1.19 3 0.5200 0.0133 0.4500 0.40 1 195.0 -0.040 0.560 0.35 31 1.57 0.09 1. 24 1.19 3 0.5233 0.0300 0.4133 0.40 1 196. 2 -0.025 0.535 0.35 32 1. 56 0.11 1. 27 1.19 3 0.5200 0.0367 0.4233 0.40 1 195. 0 -0.023 0.560 0.35 33 1. 62 0.08 1. 31 1.19 3 0.5400 0.0267 0.4367 0.40 1 202.5 -0.030 0.541 0.35 34 1.63 0.10 1.06 1.19 3 0.5433 0.0333 0.3533 0.40 1 203.8 -0.013 0.450 0.35 35 1. 63 0.12 1.22 1.19 3 0.5433 0.0400 0.4067 0.40 1 203.8 -0.017 0.521 0.35 36 1.59 0.08 1.32 1.25 3 0.5300 0.0267 0.4400 0.42 1 198.8 -0.031 0.556 0.37 37 1.59 0.12 1.14 1. 25 3 0.5300 0.0400 0.3800 0.42 1 198.8 -0.013 0.502 0.37 38 1. 61 0.07 1.22 1.25 3 0.5367 0.0233 0.4067 0.42 1 201.2 -0.027 0.505 0.37 39 1. 63 0.06 1.21 1.22 3 0.5433 0.0200 0.4033 0.41 1 203.8 -0.029 0. 491 0.36 40 1. 58 0.10 1.16 1.22 3 0.5267 0.0333 0.3867 0.41 1 197.5 -0.019 0.504 0.36 41 1. 60 0.07 1.07 1. 22 3 0.5333 0.0233 0.3567 0.41 1 200.0 -0.021 0.449 0.36 42 1. 63 0.07 1.13 1. 22 3 0.5433 0.0233 0.3767 0.41 1 203.8 -0.023 0.464 0.36 43 1. 60 0.08 1.23 1.22 3 0.5333 0.0267 0.4100 0.41 1 200.0 -0.026 0.517 0.36 44 1. 62 0.06 1.15 1.22 3 0.5400 0.0200 0.3833 0.41 1 202.5 -0.026 0. 471 0.36 45 1. 60 0.08 1.18 0.12 3 0.5333 0.0267 0.3933 0.04 1 200.0 -0.024 0.497 0.04 46 1. 62 0.06 1. 39 0.12 3 0.5400 0.0200 0.4633 0.04 1 202.5 -0.037 0.564 0.04 47 1. 63 0.07 1. 09 0.12 3 0.5433 0.0233 0.3633 0.04 1 203.8 -0.021 0.449 0.04 48 1. 61 0.10 1.19 0.15 3 0.5367 0.0333 0.3967 0.05 1 201. 2 -0.020 0.507 0.04 49 1. 61 0.08 1. 27 0.15 3 0.5367 0.0267 0.4233 0.05 1 201.2 -0.028 0.529 0.04 50 1. 60 0.11 1.06 0.15 3 0.5333 0.0367 0.3533 0.05 1 200.0 -0. 011 0.463 0.04 51 1. 59 0.09 1.24 0.15 3 0.5300 0.0300 0.4133 0.05 1 198.8 -0.025 0.528 0.04 52 1. 62 0.12 0.97 0.15 3 0.5400 0.0400 0.3233 0.05 1 202.5 -0.004 0.427 0.04 53 1. 61 0.15 0.83 0.15 3 0.5367 0.0500 0.2767 0.05 1 201.2 0.012 0.387 0.04 54 1.58 0.16 1.24 0.06 3 0.5267 0.0533 0.4133 0.02 1 197.5 -0. 011 0.563 0.02 55 1. 62 0.10 1.10 0.06 3 0.5400 0.0333 0.3667 0.02 1 202.5 -0.015 0.468 0.02 56 1. 61 0.12 1.10 0.06 3 0.5367 0.0400 0.3667 0.02 1 201. 2 -0.011 0.480 0.02 57 1. 61 0.17 0.83 0.03 3 0.5367 0.0567 0.2767 0.01 1 201. 2 0.018 0.396 0.01 58 1. 59 0.20 0.83 0.03 3 0.5300 0.0667 0.2767 0.01 1 198.8 0.026 0.414 0.01 59 1.59 0.19 0. 98 0.03 3 0.5300 0.0633 0.3267 0.01 1 198.8 0.013 0.469 0.01 60 1. 60 0.10 1.22 0.03 3 0.5333 0.0333 0.4067 0.01 1 200.0 -0.022 0.522 0.01 61 1. 61 0.03 1.25 0.03 1 0.5367 0.0100 0.4167 0.01 0.3 201. 2 -0.036 0.500 0.01 62 1. 61 0.06 1.17 0.03 1 0.5367 0.0200 0.3900 0.01 0.3 201. 2 -0.027 0.481 0.01 63 1. 59 -0.08 0.51 0.03 1 0.5300 -0.0267 0.1700 0.01 0.3 198.8 -0.042 0.167 0.01 64 1. 56 -0.10 0.56 2. 61 1 0.5200 -0.0333 0.1867 0.87 0.3 195.0 -0.049 0.181 0.77 65 1. 54 -0.15 0.73 2. 61 1 0.5133 -0.0500 0.2433 0.87 0.3 192.5 -0.065 0.230 0.77 66 1. 62 -0.22 0.77 2.61 0 0.5400 -0.0733 0.2567 0.87 0 202.5 -0.078 0.205 0.77 67 1. 68 -0.26 0.74 10.68 0 0.5600 -0.0867 0.2467 3.56 0 210.0 -0.085 0.170 3.13 68 1. 64 -0.33 0.36 10.68 0 0.5467 -0.1100 0.1200 3.56 0 205.0 -0.115 0.001 3.13 69 1. 62 -0.30 0.37 10.68 0 0.5400 -0.1000 0.1233 3.56 0 202.5 -0.107 0.018 3.13 70 1. 58 -0.29 0.35 10.68 0 0.5267 -0.0967 0 .1167 3.56 0 197.5 -0.108 0.015 3.13 71 0.54 -0.12 0.14 0.00 0 0.5400 -0.1200 0.1400 0.00 0 202.5 -0.123 0.012 0.00 72 0.57 -0.14 0.15 0.00 0 0.5700 -0.1400 0.1500 0.00 0 213.8 -0.131 -0.002 0.00 73 0.57 -0.12 0.15 0.00 0 0.5700 -0.1200 0.1500 0.00 0 213.8 -0 .115 0.022 0.00 74 0.57 -0.12 0.14 11. 64 0 0.5700 -0.1200 0.1400 11. 64 0 213.8 -0.117 0.011 10.24 75 0.57 -0.13 0.14 11. 64 0 0.5700 -0.1300 0.1400 11. 64 0 213.8 -0.125 -0.001 10.24

Notes: T' = time (x 0.107 seconds); SFC, BFC = = sideway and braking force coefficients; WS' = wheel speed (km/h), after correction (see Section 5.2.2).

183

5.2.2. Correction for Test Wheel Speed

The test wheel speed of treatment A on passing the first catadioptric target are

recorded in Table 5.1 as follows: 19.93 kph (Al), 19.43 kph (A2), and 16.25 kph

(A3). It can be found that the actual time required by the vehicle to pass the

measured section, as marked by brake position B = 1, are as follows: (24 + 7/9) -

(19 + 4/9) = 5.33 seconds (Al), (42 + 4/9) - (37 + 1/9) = 5.33 seconds (A2), and

(54 + 2/9) - (48 + 3/9) = 5.89 seconds (A3). The actual vehicle speeds for these

treatments can be obtained by dividing the 25 m with the actual time, viz: 16.88 kph

(Al), 16.88 kph (A2), and 15.28 kph (A3).

Using the same procedure as above, the actual vehicle speeds for treatments C, E,

and G can be obtained. By equalizing the actual vehicle speed over the measured

section with the recorded wheel speed on entering the measured section, for these

zero slip angle treatments, the correction factor then can be obtained.

Table 5.3 presents the recorded wheel speed and the actual vehicle speed. The

correction factor required for the test wheel speed is obtained by simply dividing the

mean of the actual vehicle speed with the mean of the recorded wheel speed.

Table 5.3. The Recorded and Actual speeds (kph), on entering the measured section, for treatment ACEG (0°).

Treatment A Treatment C Treatment E Treatment G (30 psi, fast brk) (30 psi, slow brk) (20 psi, fast brk) (20 psi, slow brk)

Ree. Act. Ree. Act. Ree. Act. Ree. Act.

19.93 16.88 19.44 16.53 18.02 16.07 15.87 15.52 19.43 16.88 18.23 16.20 18.11 16.53 18.43 15.79 16.25 15.28 18.72 15.00 18.05 15.57 18.72 16.53

Notes: Mean of Recorded speed= 18.27 kph. Mean of Actual speed= 16.07 kph. Correction factor= (16.07)/(18.27) = 0.8796.

Table 5.4 presents the test wheel speed (WS), before and after correction for all

treatments, and the comparison with the actual vehicle speed (VS). This comparison

between the corrected WS and the actual VS, are also displayed in Figure 5.6.

184

Table 5.4. The speeds of Test Wheel (kph) on entering the measured section, before and after correction, compared with the speeds of MMFIT (kph) over the measured section.

Treatment ACEG (D°)

Bef. i Aft. i Act. i (WS) (WS) (VS)

Treatment A: 19.93 17.53 16.88 19.43 17.09 16.88 16.25 14.29 15.28 18.54 16.30 16.35

Treatment C: 19.44 17.10 16.53 18.23 16.04 16.20 18.72 16.47 15.00 18.80 16.54 15.91

Treatment E: 18.02 15.85 16.07 18.11 15.93 16.53 18.05 15.88 15.58 18.06 15.89 16.06

Treatment G: 15.87 13.96 15.52 18.43 16.21 15.79 18.72 16.47 16.53 17.67 15.55 15.95

Treatment ACEG: 18.27 16.07 16.07

Treatment IKMO ( 10°)

Bef.,,. Aft.,,. Aft. i Act. i (WS) (WS) (WS) (VS)

Treatment I: 16.02 14.09 14.31 16.63 17.99 15.82 16.06 17.61 17.06 15.01 15.24 17.53 17.02 14.97 15.20 17.26

Treatment K: 18.60 16.36 16.61 18.08 17.09 15.03 15.26 16.98 17.61 15.49 15.73 17.65 17.77 15.63 15.87 17.57

Treatment M: 16.12 14.18 14.40 16.36 16.39 14.42 14.64 16.36 17.67 15.54 15.78 16.53 16.73 14.71 14.94 16.42

Treatment 0: 17.82 15.67 15.92 16.53 16.83 14.80 15.03 17.09 17.84 15.69 15.93 15.76 17.50 15.39 15.63 16.46

Treatment IKMO: 17.25 15.17 15.41 16.93

Treatment QSUW (15°)

Bef.,,. Aft.,,. Aft. i Act. i (WS) (WS) (WS) (VS)

Treatment Q: 16.18 14.23 14.73 15.58 16.05 14.12 14.62 16.20 16.28 14.32 14.82 16.73 16.17 14.22 14.72 16.17

Treatment S: 15.55 13.68 14.16 16.01 17.90 15.74 16.30 16.67 18.45 16.23 16.80 17.01 17.30 15.22 15.75 16.56

Treatment U: 18.02 15.85 16.41 17.61 15.06 13.25 13.71 15.20 15.38 13.53 14.00 16.36 16.15 14.21 14.71 16.39

Treatment W: 14.06 12.37 12.80 15.58 17.32 15.23 15.77 16.23 15.35 13.50 13.98 16.36 15.58 13.70 14.18 16.06

Treatment QSUW: 16.30 14.34 14.84 16.30

Notes: Correction factor for 0° (ACEG) = 0.8796 (see Table 5.3). Correction factor for 100 (IKMO) = 0.8796/cosl0° = 0.8932. Correction factor for 15° (QSUW) = 0.8796/cosl5° = 0.9106.

185

BFC &. SFC Speed (Km/ h) 0.8 .--------------------------, 18

1 __ ,.. ... .., ________ ..... v_s_= ..... 1_s.iiii,,io35 A 0.7 r. 16

0.6 14

0.5 12

0.4 10 0.3

8 0.2

6 0.1 o~~~ 4

-0.1 ~~l!J!i.A,/ 2 - 0.2 L__ _ ___J_ __ ...L,__~~~~~-..&...-.....L....L.=-......11,,J.___J 0

0 10 20 30 40 50 60 70 80

0 .9 .-----~~~~~~~~~~~~~~~~~~~~;;;;~~~;;;;:;~:;--------, 18 0.8 vs = 17.26 16

0.7 14

0.6 12

0.5 10

0.4 8

0.3 6 0.2 4 0.1

O 2

-0.1 0 o 10 20 30 40 50 60 10 so

0.9 .--------------------------, 18

0 8 vs = 16.17 Q · 16

0.7 14

0.6 12

0.5 10 0.4

8 0.3

6 0.2

0.1 4

0 2

-0.1 0 0 10 20 30 40 50 60 70 80

Time (x 0.107 sec)

Fig.5.2. Results of IP 30 psi & fast braking, for treatments A (D°), I (10°), Q (15°). Legend: --0- (BFC), -<>- (SFC), 1-- (Wheel Speed), - (Veh. Speed).

186

BFC & SFC Speed (Km/h) 0.8 C 18 0. 7 VS = 15.91 16

0.6 14

0.5 12

0.4 10

0.3 e 0.2

6 0.1

0 4

-0.1 2 -0.2 L---.:.......JL.,__ _ ____J __ _.L. ____________ -L-1_.____. 0

0 10 20 30 40 50 60 70 80

0.9 ,----;=========;;;:;:;:;;;;;;;;;;;;;;,;;;;;;;;-------;:;--, 18 VS = 17.57 K

0.8 16

0.7 14

0.6 12

0.5 10 0.4

8 0.3

6 0.2 4 0.1

O 2

-0.1 0 0 10 20 30 40 so 60 70 80

0.9 ,----------------------------, 18 VS = 16.56 s

0.8 16

0.7 14

0.6 12

o.s 10 0.4

8 0.3

6 0.2 0.1 4

0 2

-0.1 0 0 10 20 30 40 50 60 70 80

Time (x 0.107 sec)

Fig.5.3. Results of IP 30 psi & slow braking, for treatments C (0°), K (100), S (15°). Legend: -a- (BFC), -<>- (SFC), L- (Wheel Speed), - (Yeh. Speed).

187

BFC & SFC Speed (Km/ h) 0.8 E 18

0.7 16

0.6 14

0.5 12

0.4 10

0.3 9

0.2 6

0.1 oll'FB~~ 4

-0.1 2 -0. 2 L_ _ ___J __ ___J, __ ___s., _______ ....i.,. __ ...... _,__....1..-...JL..-----1 0

0 10 20 30 40 50 60 70 BO

0.9 .--------------------------, 18 VS = 16.42 M

0.8 16

0.7 14

0.6 12

0.5 10 0.4 8 0.3

6 0.2

4 0.1

D 2

-0.1 0 D 10 20 30 40 50 60 70 80

0.9 -----------------------u--, 18 VS = 16.39

0.8 16

0.7 14

0.6 12

0.5 10 0.4

8 0.3

6 0.2 4 0.1

0 2

-0.1 0 0 10 20 30 40 50 60 70 80

Time (x 0.107 sec)

Fig.5.4. Results of IP 20 psi & fast braking, for treatments E (0°), M (HY'), U (15°). Legend: -o- (BFC), -<>- (SFC), L- (Wheel Speed), - (Veh. Speed).

188

BFC & SFC Speed (Km/ h) 0.8 G 18 0. 7 VS = 15.95 16

0.6 14

0.5 12

0.4 10

0.3 B 0.2

6 0.1 ,r:aa=Ao.11'"""'1'.1

0 4

-0.1 2

-0.2 L------''-------L-----'-'----------'--.....__ _ ___, 0 0 10 20 30 40 50 60 70 80

0.9 -------------------------, 18 vs = 16.46 0

0.8 16

0.7 14

0.6 12

0.5 10 0.4

8 0.3

6 0.2

4 0.1

O 2

-0.1 0 0 10 20 30 40 50 60 70 80

0.9 -------------------------, 18 w 0.8 vs = 16.06 16

0.7 14

0.6 12

0.5 10 0.4

8 0.3

6 0.2

0.1 4

0 2

-0.1 0 0 10 20 30 40 50 60 70 80

Time (x 0.107 sec)

Fig.5.5. Results of IP 20 psi & slow braking, for treatments G (D°), 0 (10°), W (15°). Legend: -o- (BFC), -<>- (SFC), L-- (Wheel Speed), - (Veh. Speed).

189

Speed (Km/h) 20.-----------------------------, 101-------------=---------------i 16 14 12 10

8

6

4 2

0 A C E G I K M 0

Treatments Q s

- WS (Wheel Speed) g VS (Vehicle Speed)

u w

Fig.5.6. Comparison between the corrected Wheel Speed (WS) on entering the measured section, and the actual MMFIT speed (VS) averaged over the measured section.

5.2.3. The Actual Percent Slip on Entering the Measured Section

Due to the lesser values of test wheel speed in comparison with the vehicle speed

(especially at slip angles HY' and 15°, as shown in Figure 5.6), it is assumed that

some slips have occurred, even though the brake was not activated. The exact causes

of this slip are not known. One possible explanation is that the lateral forces due to

slip angle create a drag force through the wheel axle, and then increase the frictions

at the joints with the axle shaft. In addition, it is possible that with a low wheel

load, such as used in this experiment, some local sliding can occur on a freely

rolling test tyre drawn by a vehicle.

Furthermore, the changes of assumed constant vehicle speed may also have an

effect on creating a slip, as the test wheel was not rotated by its own power. For

example (see Figure 5.6), at zero slip treatments (A, C, E, G), it can be expected

that the lower values of treatments A, E and G are probably due to an increase

(accelerating) of the test vehicle speed on entering/across the measured section;

190

whereas the higher value of treatment C may be due to a decrease of the test vehicle

speed (decelerating). Ideally, this could be confirmed if the history of the vehicle

speed (particularly when approaching the catadioptric targets) had be recorded. With

only the test wheel speed being recorded, it is rather difficult to reach a conclusion.

Also the test wheel speed itself is based only on 1 second duration, instead of an

instant speed or a shorter duration.

The actual percent slip on entering the measured section can be found by

assuming that the decrease in speed is linearly proportional to the increase in

percent slip. As an example, for treatment A (see Table 5.4), the actual percent slip

was = (16.35-16.30)/(16.35) x 100% = 0.3%. For the test wheel speed greater than

the vehicle speed (i.e. treatment C) it is assumed that no slip occurred, and the

percent slip on entering the measured section is assumed to be zero. The actual

percent slip for all treatments are presented in Table 5.5.

Table 5.5. The wheel speed (WS), vehicle speed (VS), and percent slip (Sx), for each treatment (Tr), on entering the measured section.

Tr ws vs sx Tr ws vs sx Tr ws vs sx A 16.30 16.35 0.3% I 15.20 17 .26 11.9% Q 14.72 16.17 9.0%

C 16.54 15.91 0.0% K 15.87 17.57 9.7% s 15.75 16.56 4.9%

E 15.89 16.06 1.1% M 14.94 16.42 9.0% u 14.71 16.39 10.2%

G 15.55 15.95 2.5% 0 15.43 16.86 8.5% w 14.18 16.06 11.7%

5.3. Transformation Into Percent Slip and Slip Angle

5.3.1. Correction for Wheel Speed Delay

From Figures 5.2-5.5, it can be seen that generally some time has elapse before

the wheel speed begins to decrease, following the significant changes of friction

coefficients. One possible source of this delay is

wheel speed.

191

inaccuracy of the recording

The magnitude of the wheel speed delay (for shifting purposes) will be estimated

as follows.

a. Drawing a line of the average rate of decreasing Wheel Speed (WS) with Time

(T). The values of this average rate, however, are dependent upon the range of

the Time chosen. This uncertainty will be eliminated, by establishing the

following conditions:

- The range of the Time should cover the dominant range of WS. In this case the

range of T which covers ~ 85% of the range of WS is considered appropriate.

- The representative line will be determined by the least squares method. In this

case the coefficient of determination (R2) ~ 0. 70 is regarded as sufficient.

Several lines which satisfied those conditions can be drawn. The line which gives

the minimum range of Time is selected. This line is T = f(WS).

b. Obtaining the last Time at which the percent slip (Sx) is still zero. This can be

done by intercepting the line of (a) above with the line of the Vehicle Speed

(VS). In other words, by substituting WS = VS into T = f(WS), the Time where

the wheel speed is assumed begin to decrease can be obtained.

c. The wheel speed delay then is the difference between the Time in (b) above and

the Time= 10 (when brake is activated).

Table 5.6. The wheel speed delay, for all treatments.

Ranges of data Tr

T ws (kph)

A 16-27 16.60 - 1.63 C 20-32 16.19 - 0.08 E 15-28 15.60 - 1.01 G 18-29 14.97 - 1.14 I 18-22 14.98 - 1.55 K 20-31 14.53 - 0.79 M 19-22 14.91 - 1.81 0 16-30 14.96 - 0.74 Q 15-27 14.43 - 0.10 s 19-29 15.47 - 1.38 u 17-28 13.76 - 1.07 w 15-25 13.99 - 1.64

(%)

(90.2%) (97.5%) (91.9%) (89.0%) (88.4%) (86.6%) (87.7%) (91.2%) (95.3%) (89.4%) (86.3%) (87.0%)

Equations of Shift T = f(WS) R2 T

T = 27.916 - 0.817WS 0.82 4.56 T = 32.017 - 0.769WS 0.94 9.78 T = 28.224 - 0.878WS 0.91 4.12 T = 29.749 - 0.900WS 0.74 5.39 T = 22.675 - 0.266WS 0.71 8.08 T = 30.338 - 0.796WS 0.89 6.35 T = 22.218 - 0.224WS 0.88 8.54 T = 29.659 - 0.869WS 0.93 5.36 T = 27.834 - 0.862WS 0.94 3.90 T = 28.512 - 0.712WS 0.84 6.72 T = 26.702 - 0.812WS 0.80 3.39 T = 27.940 - 0.790WS 0.78 5.25

Average delay (x 0.107 sec)= 5.95 (== 6)

192

Table 5.6 presents the wheel speed delay for each treatment, with average value

of 6 (x 0.107 sec). Generally, it is also found that on average the delay is lower (T = 5.5) for fast braking treatments (AEIMQU), and is higher (T = 6.5) for slow braking

treatments (CGKOSW). While the inaccuracy of the wheel speed recording is the

only possible source known for this delay, it is reasonable to assume that a shift of

less than the speed accuracy (one second duration) can be justified. Hence, a shift of

T = 6 (x 0.107 sec) is used for all treatments.

5.3.2. The Smoothened Technique for Wheel Speed

As can be seen from Figures 5.2-5.5, the decrease of Wheel Speed (WS) with

increasing Time (T) does not follow a smooth relationship. This is due to the fact, as

mentioned in Section 5.2.3, that the Wheel Speed in this experiment is based only

on 1 second duration, instead of an instant speed or a shorter duration.

Many techniques are available for smoothing the data, ranging from the free hand

curves, moving averages, or by using mathematical curves. To minimize an error

due to subjectivity, and to obtain a smooth curve, a mathematical curve then will be

used for the smoothing technique of Wheel Speed vs Time curve.

The form of the mathematical equation will be selected in accordance with the

general trends of the Wheel Speed (WS) vs Time (T) curve. From past experience,

and from the results of WS vs T curve in Figures 5.2-5.5, it can be said that the WS

vs T curve may be approximated by a bell shaped curve. Accordingly, the normal

curve (Gaussian) equation will be used for the smoothing process:

y = ae -o.s(X)2 .... WS = ae --O.S{(T-u)/sl2 (5.1)

The magnitude of (a) in Equation (5.1) is the maximum value of WS. In this case

(a) is selected as the average value of WS from T=lO to T=15. The position of (u)

is the commencement of Time of the WS vs T curve after shifted (hence, u = 16).

The magnitude of standard deviation (s) will be determined from the available WS

data, starting from T = 16, with a number of T data (n) about 14-16 is considered

enough to represent the WS. The equation for obtaining (s) is as follows:

193

l:ln(WS) = n.ln(a) - 0.5(s)"2l:(T-16)2 (5.2)

An example of this smoothing technique is given here for treatment A. From

Table 5.2, it can be found that the average WS from T=lO to T=15 is 16.55 (km/h).

From Figure 5.2, it is seen that the WS has approached the zero value at T=30. The

magnitude of (s) then will be obtained from the WS data, ranging from WS at T=16

until T=30. Hence, the value of (n) is 15. The l:ln(WS) can be obtained as 21.460.

Then ln(a) = ln(16.55) = 2.806. Using Equation (5.2), the magnitude of (s) is found

as 4.959. The normal (Gaussian) equation for treatment A then is as follows: WS = 16.55--0.S{(f-16)/4.959>2 (5. la)

Using the same procedure as above, the normal equations for other treatments

can be found. Table 5.7 presents the parameters of normal equations of Wheel Speed

(WS) versus Time (T) curves for all treatments, in which: (n) is the number of T

data, (a) is the maximum value of WS, and (s) is the standard deviation of WS data.

Table 5.7. The parameters of normal equations of WS vs T, for all treatments.

Tr T (n) (a) (s) Tr T (n) (a) (s)

A 16-30 (15) 16.55 4.959 M 16-30 (15) 14.92 4.494 C 16-31 (16) 16.26 7.949 0 16-31 (16) 15.28 5.948 E 16-30 (15) 15.65 4.841 Q 16-29 (14) 14.53 4.249 G 16-30 (15) 15.16 6.089 s 16-30 (15) 15.70 5.484 I 16-30 (15) 15.09 4.480 u 16-29 (14) 14.23 4.260 K 16-31 (16) 15.82 6.414 w 16-30 (15) 14.15 4.924

5.3.3. Transformation of Time into Percent Slip

As it found in the literature review, percent slip (S,i:) will increase proportionally

as Wheel Speed (WS) decreases. Hence the transformation of Time into Sx can be

done using a simple equation: (VS-WS)NS = (1-Sx)100%, where VS is the vehicle

speed (in which Sx is assumed to be zero), and Sx = 0% is given for WS ~ VS.

Table 5.8 presents the transformation of time into longitudinal percent slip (Sx)

scale for treatment A. The summary of friction coefficients versus Sx, for all

treatments, are presented in Tables 5.9-5.11, and Figures 5.7-5.10.

194

Table 5.8. Transformation into Sx scale of treatment A

TB SFC BFC ws sx T B SFC BFC ws sx 01 0 -0.105 0.017 39 1 -0.029 0.491 0.00 100 02 0 -0.113 0.001 40 1 -0.019 0.504 0.00 100 03 0 -0.106 0.010 41 1 -0.021 0.449 0.00 100 04 0 -0.111 -0.006 42 1 -0.023 0.464 0.00 100 05 0 -0.109 0.010 43 1 -0.026 0.517 0.00 100 06 0 -0.103 0.018 44 1 -0.026 0.471 0.00 100 07 0 -0.123 -0.011 45 1 -0.024 0.497 0.00 100 08 0 -0.127 -0.012 46 1 -0.037 0.564 0.00 100 09 0 -0.123 0.006 47 1 -0.021 0.449 0.00 100 10 0 -0.133 -0.009 16.55 0.0 48 1 -0.020 0.507 0.00 100 11 1 -0.135 -0.013 16.22 0.8 49 1 -0.028 0.529 0.00 100 12 1 -0.079 0.194 15.26 6.7 50 1 -0.011 0.463 0.00 100 13 1 -0.049 0.557 13.78 15.7 51 1 -0.025 0.528 0.00 100 14 1 -0.052 0.651 11.95 26.9 52 1 -0.004 0.427 0.00 100 15 1 -0.051 0.628 9.96 39.1 53 1 0.012 0.387 0.00 100 16 1 -0.059 0.695 7.96 51.3 54 1 -0.011 0.563 0.00 100 17 1 -0.020 0.539 6.11 62.6 55 1 -0.015 0.468 0.00 100 18 1 -0.007 0.543 4.50 72.4 56 1 -0.011 0.480 0.00 100 19 1 0.019 0.455 3.19 80.5 57 1 0.018 0.396 0.00 100 20 1 -0.007 0.518 2.17 86.7 58 1 0.026 0.414 0.00 100 21 1 -0.010 0.493 1.41 91.4 59 1 0.013 0.469 0.00 100 22 1 -0.019 0.500 0.89 94.6 60 1 -0.022 0.522 0.00 100 23 1 -0.024 0.487 0.53 96.7 61 0.3 -0.036 0.500 24 1 -0.026 0.509 0.31 98.1 62 0.3 -0.027 0.481 25 1 -0.040 0.553 0.17 99.0 63 0.3 -0.042 0.167 26 1 -0.042 0.602 0.09 99.4 64 0.3 -0.049 0.181 27 1 -0.050 0.611 0.05 99.7 65 0.3 -0.065 0.230 28 1 -0.045 0.605 0.02 99.9 66 0 -0.078 0.205 29 1 -0.034 0.566 0.01 99.9 67 0 -0.085 0.170 30 1 -0.040 0.560 0.00 100 68 0 -0.115 0.001 31 1 -0.025 0.535 0.00 100 69 0 -0.107 0.018 32 1 -0.023 0.560 0.00 100 70 0 -0.108 0.015 33 1 -0.030 0.541 0.00 100 71 0 -0.123 0.012 34 1 -0.013 0.450 0.00 100 72 0 -0.131 -0.002 35 1 -0.017 0.521 0.00 100 73 0 -0.115 0.022 36 1 -0.031 0.556 0.00 100 74 0 -0.117 0.011 37 1 -0.013 0.502 0.00 100 75 0 -0.125 -0.001 38 1 -0.027 0.505 0.00 100

. . Notes: T = time (x 0.107 sec); B = brake pos1t1on (0 = off, 1 = on) .

WS = wheel speed (km/h), after smoothing.

195

Table 5.9. Summary of friction coefficients versus slip, for treatments ACEG (O").

T WL SFC BFC sx I T WL SFC BFC sx I Treatment A: Treatment E: lO 218.8 -0.133 -0.009 0.0 lO 203.8 -0.057 0.031 2.6 11 220.0 -0.135 -0.013 0.8 11 203.8 -0.054 0.054 4.6 12 216.2 -0.079 0.194 6.7 12 202.5 -0.034 0.544 10.5 13 215.0 -0.049 0.557 15.7 13 190.0 -0.004 0.650 19.6 14 207.5 -0.052 0.651 26.9 14 187.5 0.027 0.585 30.7 15 202.5 -0.051 0.628 39.1 15 187.5 0.004 0.668 42.8 16 196.2 -0.059 0.695 51.3 16 188.8 0.016 0.618 54.8 17 195.0 -0.020 0.539 62.6 17 186.2 0.050 0.556 65.7 18 197.5 -0.007 0.543 72.4 18 186.2 0.059 0.553 75.1 19 200.0 0.019 0.455 80.5 19 190.0 0.063 0.496 82.7 20 197.5 -0.007 0.518 86.7 20 186.2 0.071 0.536 88.5 21 202.5 -0.010 0.493 91.4 21 191.2 0.047 0.567 92.6 22 202.5 -0.019 0.500 94.6 22 191.2 0.069 0.501 95.5 23 207.5 -0.024 0.487 96.7 23 191.2 0.048 0.563 97.4 24 206.2 -0.026 0.509 98.1 24 191.2 0.079 0.477 98.5 25 207.5 -0.040 0.553 99.0 25 190.0 0.070 0.560 99.2 26 198.8 -0.042 0.601 99.4 26 192.5 0.054 0.581 99.6 27 198.8 -0.050 0.611 99.7 27 191.2 0.074 0.502 99.8

28-29 197.5 -0.040 0.586 99.9 28 190.0 0.060 0.551 99.9 30-60 200.4 -0.017 0.493 100 29-61 190.0 0.082 0.509 100

(204.4) (191.6)

Treatment C: Treatment G: l0-11 218.l -0.157 -0.032 0.0 lO 206.2 -0.053 0.061 5.0 12 217.5 -0.152 -0.014 1.0 11 200.0 -0.048 0.071 6.2 13 213.8 -0.129 0.067 4.8 12 207.5 -0.063 0.134 10.0 14 207.5 -0.100 0.201 10.0 13 200.0 -0.030 0.412 15.8 15 216.2 -0.070 0.411 16.1 14 193.8 0.014 0.678 23.4 16 215.0 -0.062 0.588 23.1 15 193.8 0.066 0.562 32.2 17 203.8 -0.023 0.508 30.6 16 197.5 0.041 0.500 41.5 18 198.8 -0.025 0.484 38.4 17 193.8 0.044 0.568 50.9 19 198.8 -0.018 0.425 46.2 18 191.2 0.068 0.544 59.9 20 201.2 -0.032 0.477 53.7 19 190.0 0.077 0.510 68.1 21 205.0 -0.031 0.394 60.8 20 188.8 0.082 0.513 75.3 22 200.0 -0.031 0.476 67.3 21 191.2 0.081 0.507 81.4 23 202.5 -0.031 0.422 73.2 22 190.0 0.052 0.601 86.4 24 202.5 -0.034 0.473 78.3 23 193.8 0.042 0.563 90.3 25 201.2 -0.042 0.522 82.8 24 193.8 0.061 0.528 93.2 26 203.8 -0.045 0.476 86.5 25 192.5 0.057 0.552 95.4 27 205.0 -0.038 0.443 89.6 26 193.8 0.064 0.532 97.0 28 202.5 -0.032 0.454 92.1 27 192.5 0.057 0.569 98.1 29 202.5 -0.035 0.465 94.1 28 193.8 0.038 0.580 98.8 30 201.2 -0.034 0.480 95.7 29 195.0 0.074 0.484 99.3 31 200.0 -0.034 0.451 96.9 30 191.2 0.074 0.536 99.6 32 201.2 -0.028 0.445 97.8 31 191.2 0.092 0.482 99.8 33 198.8 -0.037 0.514 98.4 32-33 191.2 0.082 0.540 99.9 34 202.5 -0.034 0.453 98.9 34-61 192.6 0.080 0.491 100 35 203.8 -0.037 0.462 99.3 (194.4) 36 203.8 -0.041 0.457 99.5 37 203.8 -0.024 0.398 99.7 38 203.8 -0.019 0.410 99.8

39-40 205.0 -0.027 0.404 99.9 41-61 202.9 -0.030 0.426 100

(204.7)

196

T bl 5 10 S a e . . ummary o f fri . ffi . li £ ctton coe 1c1ents versus s lP, or treatments IKMO (100) .

T WL SFC BFC sx T WL SFC BFC sx

Treatment I: Treatment M:

10 175.0 0.525 0.445 12.6 10 162.5 0.533 0.498 9.1

11 170.0 0.534 0.394 14.7 11 161.2 0.577 0.508 11.4

12 177.5 0.438 0.495 20.9 12 185.0 0.225 0.634 17.7

13 190.0 0.055 0.686 30.1 13 197.5 0.020 0.727 27.3

14 190.0 0.060 0.616 41.3 14 186.2 0.022 0.738 38.9

15 201.2 0.023 0.667 53.1 15 192.5 0.024 0.698 51.1

16 203.8 0.029 0.610 64.3 16 192.5 0.042 0.602 62.7

17 198.8 0.053 0.573 74.2 17 191.2 0.070 0.540 73.0

18 196.2 0.063 0.555 82.2 18 193.8 0.058 0.564 81.4 19 197.5 0.072 0.526 88.4 19 192.5 0.062 0.576 87.8 20 197.5 0.062 0.543 92.8 20 193.8 0.067 0.528 92.4 21 201.2 0.054 0.539 95.7 21 193.8 0.076 0.516 95.5 22 198.8 0.047 0.575 97.6 22 193.8 0.071 0.546 97.4 23 201.2 0.054 0.570 98.7 23 191.2 0.071 0.567 98.6 24 200.0 0.060 0.541 99.3 24 195.0 0.042 0.611 99.3 25 200.0 0.060 0.584 99.7 25 195.0 0.065 0.562 99.7 26 202.5 0.042 0.592 99.8 26 192.5 0.093 0.525 99.8 27 201.2 0.046 0.589 99.9 27 192.5 0.067 0.579 99.9

28-58 200.7 0.064 0.534 100 28-61 194.4 0.075 0.529 100 (194.9) (189.3)

Treatment K: Treatment 0: 10 173.8 0.484 0.486 10.0 10 161.2 0.457 0.563 7.2 11 168.8 0.491 0.491 11.0 11 161.2 0.469 0.537 8.5 12 170.0 0.515 0.500 14.2 12 161.2 0.488 0.557 12.3

13 172.5 0.421 0.540 19.3 13 168.8 0.381 0.604 18.3 14 185.0 0.244 0.635 25.9 14 183.8 0.238 0.593 26.0 15 210.0 0.104 0.600 33.5 15 203.8 0.094 0.516 34.8 16 217.5 0.077 0.502 41.8 16 207.5 0.055 0.579 44.2 17 215.0 0.115 0.380 50.3 17 207.5 0.082 0.470 53.6 18 216.2 0.120 0.382 58.6 18 203.8 0.104 0.456 62.4 19 217.5 0.109 0.395 66.3 19 205.0 0.125 0.432 70.5 20 217.5 0.113 0.380 73.3 20 203.8 0.097 0.462 77.4 21 218.8 0.107 0.374 79.3 21 205.0 0.088 0.488 83.2 22 215.0 0.109 0.426 84.3 22 205.0 0.100 0.454 87.9 23 217.5 0.075 0.491 88.4 23 206.2 0.099 0.433 91.5 24 218.8 0.097 0.425 91.7 24 205.0 0.090 0.446 94.2 25 215.0 0.082 0.475 94.1 25 203.8 0.094 0.494 96.1 26 220.0 0.091 0.401 96.0 26 205.0 0.074 0.487 97.5 27 216.2 0.097 0.434 97.3 27 205.0 0.082 0.512 98.4 28 216.2 0.099 0.416 98.2 28 206.2 0.095 0.448 99.0 29 218.8 0.112 0.354 98.9 29 205.0 0.119 0.418 99.4 30 216.2 0.115 0.401 99.3 30 203.8 0.111 0.417 99.7 31 216.2 0.112 0.401 99.6 31 207.5 0.081 0.461 99.8 32 217.5 0.109 0.408 99.7 32-33 205.6 0.102 0.445 99.9

33-34 220.0 0.071 0.458 99.9 34-59 203.4 0.106 0.416 100 35-56 216.4 0.100 0.403 100 (197.2)

(208.3)

197

Table 5.11. Summary of friction coefficients versus slip, for treatments QSUW (15°).

I T WL SFC BFC sx II T WL SFC BFC sx I Treatment 0: Treatment U: 10 142.5 0.544 0.338 10.1 10 137.5 0.622 0.289 13.2 11 141.2 0.559 0.323 12.6 11 135.0 0.641 0.280 15.5 12 152.5 0.423 0.382 19.6 12 158.8 0.217 0.565 22.2 13 177.5 0.095 0.558 30.0 13 171.2 0.118 0.585 32.2 14 178.8 0.085 0.574 42.3 14 170.0 0.144 0.500 44.1 15 176.2 0.099 0.593 55.0 15 172.5 0.105 0.619 56.4 16 180.0 0.068 0.644 66.8 16 175.0 0.085 0.637 67.8 17 180.0 0.099 0.580 76.9 17 172.5 0.144 0.534 77.5 18 175.0 0.144 0.482 84.7 18 170.0 0.164 0.464 85.1 19 176.2 0.164 0.365 90.5 19 168.8 0.170 0.428 90.7 20 172.5 0.144 0.495 94.4 20 168.8 0.149 0.520 94.5 21 176.2 0.118 0.536 96.9 21 170.0 0.157 0.502 96.9 22 175.0 0.132 0.456 98.3 22 171.2 0.149 0.494 98.4 23 173.8 0.115 0.569 99.2 23 168.8 0.150 0.526 99.2 24 177.5 0.143 0.369 99.6 24 171.2 0.154 0.497 99.6 25 172.5 0.145 0.452 99.8 25 168.8 0.140 0.551 99.8 26 177.5 0.110 0.552 99.9 26 172.5 0.132 0.557 99.9

27-60 175.3 0.135 0.464 100 27-58 169.7 0.164 0.449 100 (171.1) (166.2)

Treatment S: Treatment W: 10 147.5 0.654 0.218 5.2 10 138.8 0.589 0.311 11.9 11 146.2 0.490 0.364 6.8 11 133.8 0.438 0.481 13.7 12 140.0 0.647 0.254 11.3 12 132.5 0.590 0.355 18.9 13 146.2 0.515 0.350 18.4 13 143.8 0.406 0.464 26.8 14 158.8 0.321 0.446 27.3 14 158.8 0.215 0.555 36.7 15 177.5 0.119 0.599 37.4 15 170.0 0.088 0.671 47.4 16 182.5 0.106 0.579 47.9 16 173.8 0.084 0.645 58.1 17 180.0 0.131 0.502 58.0 17 175.0 0.111 0.509 67.9 18 180.0 0.163 0.439 67.3 18 172.5 0.138 0.511 76.5 19 180.0 0.177 0.385 75.3 19 172.5 0.126 0.478 83.4 20 176.2 0.142 0.492 82.0 20 172.5 0.144 0.470 88.8 21 178.8 0.157 0.386 87.3 21 171.2 0.116 0.555 92.7 22 177.5 0.162 0.418 91.3 22 172.5 0.135 0.501 95.5 23 178.8 0.137 0.476 94.3 23 175.0 0.118 0.533 97.3 24 178.8 0.145 0.487 96.4 24 171.2 0.133 0.493 98.5 25 176.2 0.155 0.460 97.7 25 172.5 0.115 0.520 99.1 26 178.8 0.148 0.464 98.7 26 172.5 0.126 0.516 99.6 27 177.5 0.144 0.496 99.2 27 168.8 0.116 0.586 99.8 28 178.8 0.152 0.467 99.6 28-29 172.5 0.114 0.546 99.9 29 177.5 0.157 0.428 99.8 30-61 170.1 0.142 0.443 100

30-31 176.9 0.161 0.430 99.9 (164.5) 32-59 177.5 0.163 0.410 100

(171.4)

198

BFC & SFC 0.8

A 0.7

0.6

0.5 0.4

0.3

0.2

0.1 0

-0.1

-0.2 0 10 20 30 40 so 60 70 80 90 100

0.9

0.8 0.7

0.6 o.s ---------·

0.4 ' ' ' /

0.3 ' ' ' ' ' ' 0.2 ' / , 0.1 ' ' ' ,

' 0 ,

-0.1 0 10 20 30 40 50 60 70 80 90 100

0.9 Q

0.8

0.7 0.6 ......... _

0.5 0.4

0.3 ' ' ' ,

0.2 ,

/ , , 0.1 , '

' ' ' 0 '

-0.1 0 10 20 30 40 50 60 70 80 90 100

Sx (96)

Fig.5.7. BFC (-o-) & SFC (-<>-) vs percent slip (Sx); IP 30 psi & fast braking, for treatments A (Q°), I (10°), Q (15°).

199

BFC & SFC 0.8

C 0.7

0.6 0.5 0.4

0.3 0.2

0.1

0

-0.1 -0.2

0 10 20 30 40 50 60 70 80 90 100

0.9

0.8 K

0.7

0.6

0.5 ......... ___

. 0.4 .

' ,' . 0.3 ' / . 0.2 ,'

,'

0.1 ,' I

I I 0 ,

-0.1 0 10 20 30 40 50 60 70 80 90 100

0.9 s 0.8

0.7 0.6

0.5 0.4

0.3 0.2 ,'

I

0.1 /,' I 0 ,

-0.1 0 10 20 30 40 50 60 70 80 90 100

Sx (96)

Fig.5.8. BFC (-o-) & SFC (-<>-) vs percent slip (Sx); IP 30 psi & slow braking, for treatments C (C>°), K (10°), S (15°).

200

BFC & SFC 0.8

E 0.7

0.6

o.s 0.4 0.3

0.2

0.1

0 -0.1 -0.2

0 10 20 30 40 so 60 70 80 90 100

0.9

0.8 M

0.7

0.6 ......... __ 0.5 , 0.4 ,' , , , 0.3

, , , ' ' 0.2 ,' , , ,

0.1 , , ,'

o· -0.1

0 10 20 30 40 50 60 70 80 90 100

0.9 u 0.8

0.7 .......... _ 0.6

0.5 0.4

0.3 , ,

0.2 , , , , , , 0.1 , , , , , ,

0 ,

-0.1 0 10 20 30 40 50 60 70 80 90 100

Sx (96)

Fig.5.9. BFC (-o-) & SFC (-<>-) vs percent slip (Sx); IP 20 psi & fast braking, for treatments E (0°), M (HY'), U (15°).

201

BFC & SFC 0.8

G 0.7

0.6 0.5 0.4 0.3

0.2 0.1

0 , ,

-0.1 -0.2

0 10 20 30 40 50 60 70 80 90 100

0.9 0

0.8 0.7 0.6

o.s 0.4

0.3 . . . . 0.2

. . . . . 0.1 .

0

-0.1 0 10 20 30 40 50 60 70 80 90 100

0.9 0.8

w

0.7 0.6 __ ........ ----0.5 0.4

0.3 ,, . 0.2 , . , , . , 0.1 , , , , ,

D •

-0.1 0 10 20 30 40 50 60 70 80 90 100

Sx (96)

Fig.5.10. BFC (-o-) & SFC (-<>-) vs percent slip (Sx); IP 20 psi & slow braking, for treatments G (00), 0 (10°), W (15°).

202

5.3.4. Extrapolations and Analysis of Variance

To obtain the friction coefficients at 0% slip, some extrapolations are needed

(except treatment C); the results are shown in Figures 5.7-5.10 (dashed lines). The

SFC versus slip angle, at 0% and 100% slip, are presented in Figures 5.11-5.14. It is

expected that SFC will reach the value of locked wheel BFC at angle 90°.

SFC (at Sx = 0% & 100%) 0.8 ...-----------------------A-IQ--. 0.7

0.6 o.s 0.4

0.3

0.2

0.1

'"'-- ............................. ...

---------------------

... .................................... __________ _

-------··--·------------------·--_____ ........

---------------

0J.:,,,r=------------------------~ -0.1 -0 .2 .__ _ __,_ __ ___._ __ .....__ __ ..____ _ __,_ __ _,__ __ .....__ __ ..____ _ __,

D 10 20 30 40 50 60 70 80 SI ip Angle (deg)

Fig.5.11. SFC vs slip angle; IP 30 psi & fast braking, from treatments AIQ. Legend: -<>- (SFC at Sx = 0%), -ll,- (SFC at Sx = 100%).

90

SFC (at Sx = 0% & 100%) 0.8 ~------------------------

CKS 0.7

0.6 o.s 0.4

0.3

0.2 0.1

------ ----- ............ _

----------................

----------- .......... __

__ ...... ---------___ ..

........... __ _ ---·----- .... .. ____ _ --------- ..... ----------------......... -----........ ---------------·-------

OJ,.,,,,..t,=:;;__ ______________________ ---J

-0.1 -0 .2 .__ _ ___., __ __._ __ _._ __ 1,__---1.. __ _,L. __ ....1...,__,___Ja___ _ ___,J

D 10 20 30 40 50 60 70 80 Sllp Angle (deg)

Fig.5.12. SFC vs slip angle; IP 30 psi & slow braking, from treatments CKS. Legend: -<>- (SFC at Sx = 0%), -ll,- (SFC at Sx = 100%).

203

90

SFC (at Sx = 096 & 10096) o.9 ....----------------------E-M-U--, 0;7

0.6 0.5 0.4

0.3 0.2

0 .1 llr+--«

-------------------

------------..... --

------------- ------------------------------------------- ----.

--------

0-------------------------"""1 -0.1 -0.2 ,...__ _ ____. __ __._ ___________________ _

0 10 20 30 40 50 60 70 80

Slip Angle (deg)

Fig.5.13. SFC vs slip angle; IP 20 psi & fast braking, from treatments EMU. Legend: -<>- (SFC at Sx = 0%), -11- (SFC at Sx = 100%).

90

SFC (at Sx = 096 & 10096) o.9 r------------------------G-O_W__, 0.7

0.6 0.5

0.4

0.3

0.2 0.1~--ca.

........ ___ ...... ----

.. --__ ....

-------------------------------· ------------------------

0---------------------------1 -0.1

-0.2 ~-------------------------0 10 20 30 40 50 60 70 80

Slip Angle (deg)

Fig.5.14. SFC vs slip angle; IP 20 psi & slow braking, from treatments GOW. Legend: -<>- (SFC at Sx = 0%), -11- (SFC at Sx = 100%).

90

For the purpose of analysis of variance, the results of friction coefficients versus

slip are represented by the values of friction coefficient at 0, 10, 20, ... 100 percent

slips (obtained by extrapolation/interpolation where appropriate). Table 5.12 presents

the representative friction coefficients versus slip. Results of analysis of variance are

given in Tables 5.13-5.14.

204

Table 5.12. Representative friction coefficients versus slip, for analysis of variance.

I sx (%) 11 SFC BFC I SFC BFC I SFC BFC I SFC BFC I Treatment A: Treatment C: Treatment E: Treatment G:

0 -0.133 -0.009 -0.157 -0.032 -0.058 0.000 -0.054 -0.040 10 -0.068 0.327 -0.100 0.201 -0.036 0.502 -0.063 0.134 20 -0.050 0.593 -0.066 0.510 -0.003 0.648 -0.006 0.559 30 -0.052 0.645 -0.026 0.514 0.025 0.589 0.053 0.591 40 -0.052 0.633 -0.024 0.472 0.009 0.649 0.045 0.510 50 -0.058 0.688 -0.025 0.451 0.011 0.638 0.044 0.561 60 -0.029 0.575 -0.031 0.403 0.032 0.588 0.068 0.544 70 -0.010 0.542 -0.031 0.451 0.054 0.555 0.078 0.511 80 0.017 0.460 -0.037 0.492 0.062 0.516 0.081 0.508 90 -0.009 0.500 -0.037 0.445 0.062 0.547 0.043 0.566

100 -0.017 0.493 -0.030 0.426 0.082 0.509 0.080 0.491

Treatment I: Treatment K: Treatment M: Treatment 0: 0 0.540 0.000 0.520 0.000 0.560 0.000 0.500 0.000 10 0.528 0.353 0.484 0.486 0.550 0.502 0.476 0.545 20 0.452 0.480 0.402 0.550 0.176 0.656 0.349 0.602 30 0.059 0.684 0.168 0.616 0.020 0.730 0.173 0.558 40 0.059 0.621 0.083 0.523 0.022 0.734 0.072 0.551 50 0.033 0.654 0.114 0.384 0.024 0.702 0.072 0.512 60 0.027 0.632 0.118 0.384 0.038 0.624 0.098 0.460 70 0.043 0.589 0.111 0.387 0.062 0.558 0.124 0.433 80 0.060 0.560 0.107 0.381 0.060 0.560 0.093 0.474 90 0.068 0.532 0.086 0.459 0.064 0.553 0.099 0.442 100 0.064 0.534 0.100 0.403 0.075 0.529 0.106 0.416

Treatment 0: Treatment S: Treatment U: Treatment W: 0 0.580 0.000 0.610 0.000 0.660 0.000 0.560 0.000 10 0.544 0.335 0.602 0.286 0.631 0.219 0.584 0.261 20 0.410 0.389 0.480 0.367 0.356 0.471 0.564 0.370 30 0.095 0.558 0.267 0.487 0.140 0.581 0.344 0.493 40 0.087 0.571 0.116 0.594 0.135 0.529 0.176 0.591 50 0.093 0.586 0.111 0.563 0.125 0.557 0.087 0.665 60 0.086 0.615 0.138 0.488 0.099 0.625 0.089 0.619 70 0.078 0.624 0.168 0.421 0.098 0.614 0.118 0.509 80 0.117 0.541 0.152 0.460 0.151 0.511 0.132 0.494 90 0.162 0.375 0.160 0.408 0.169 0.432 0.135 0.496

100 0.135 0.464 0.163 0.410 0.164 0.449 0.142 0.443

205

Table 5.13. Analysis of variance of Braking Force Coefficient (BFC).

Source of Sum of

Variation Squares

Angle 00:

PS 1.3084446

IP 0.0182458

RB 0.0837818

PSxIP 0.0063672

PSxRB 0.0554192

IPxRB 0.0021560

Error 0.0305270

Total 1.5049416

Angle HY":

PS 1.1699857

IP 0.0194881

RB 0.1124131

PSxIP 0.0202594

PSxRB 0.0929314

IPxRB 0.0001681

Error 0.0175362

Total 1.4327820

Angle 15°:

PS 1.2272858

IP 0.0034039

RB 0.0087646

PSxIP 0.0172538

PSxRB 0.0387541

IPxRB 0.0063120

Error 0.0099628

Total 1.3117370

Notes: ** Significant at 1 % level * Significant at 5% level NS Not Significant

Degrees of Mean

Freedom Squares

10

1

1

10

10

1

10

43

10

1

1

10

10

1

10

43

10

1

1

10

10

1

10

43

206

0.1308445

0.0182458

0.0837818

0.0006367

0.0055419

0.0021560

0.0030527

0.1169986

0.0194881

0.1124131

0.0020259

0.0092931

0.0001681

0.0017536

0.1227286

0.0034039

0.0087646

0.0017254

0.0038754

0.0063120

0.0009963

PS = Percent Slip IP = Inflation Pressure RB = Rate of Braking

F ratio

42.86 ** 5.98 *

27.45 ** 0.21 NS

1.82 NS

0.71 NS

66.72 ** 11.11 ** 64.10 ** 1.16 NS

5.30 ** 0.10 NS

123.18 ** 3.42 NS

8.80 * 1.73 NS

3.89 * 6.34 *

Table 5.14. Analysis of variance of Sideway Force Coefficient (SFC).

Source of Sum of

Variation Squares

Angle Q°:

PS 0.0689424

IP 0.0606808

RB 0.0000153

PSxIP 0.0035077

PSxRB 0.0050932

IPxRB 0.0012233

Error 0.0012882

Total 0.1407509

Angle IQ°:

PS 1.3705144

IP 0.0038766

RB 0.0172419

PSxIP 0.0251657

PSxRB 0.0276614

IPxRB 0.0005182

Error 0.0138410

Total 1.4588192

Angle 15°:

PS 1.5575144

IP 0.0021142

RB 0.0139339

PSxIP 0.0068741

PSxRB 0.0471744

IPxRB 0.0032302

Error 0.0133840

Total 1.6442252

Notes: ** Significant at 1 % level * Significant at 5% level NS Not Significant

Degrees of Mean

Freedom Squares

10

1

1

10

10

1

10

43

10

1

1

10

10

1

10

43

10

1

1

10

10

1

10

43

207

0.0068942

0.0606808

0.0000153

0.0003508

0.0005093

0.0012233

0.0001288

0.1370514

0.0038766

0.0172419

0.0025166

0.0027661

0.0005182

0.0013841

0.1557514

0.0021142

0.0139339

0.0006874

0.0047174

0.0032302

0.0013384

PS = Percent Slip IP = Inflation Pressure RB = Rate of Braking

F ratio

53.53 ** 471.12 **

0.12 NS

2.72 NS

3.95 * 9.50 **

99.03 ** 2.80 NS

12.46 ** 1.82 NS

2.00 NS

0.37 NS

116.37 ** 1.58 NS

10.41 ** 0.51 NS

3.52 * 2.41 NS

5.4. Determination of Parameters for Theoretical Prediction

As described in Section 2.6, there are 6 parameters needed for this theoretical

prediction of tyre-road friction, under all ranges of percent slip and slip angle: a/21,

b/21, Cx (force/slip), Cy (force/radian), Fz (force), and either locked-wheel BFC or

maximum SFC.

The values of a/21 and b/21 can not be obtained from this experiment, so they

will be assumed. The values of Fz (wheel load) can be obtained from this

experiment, such as shown in Tables 5.9-5.11 (i.e. the average value of wheel loads

during braking). The values of locked-wheel BFC can also be obtained from Tables

5.9-5.11 (i.e. the BFC at Sx = 100%). The values of longitudinal and lateral stiffness

(Cx, and Cy) will be obtained as follows.

5.4.1. Longitudinal Stiffness

Referring to Equation 2.2, the (field) longitudinal stiffness (Cx) can be obtained

from the relation between braking force (BF) and percent slip (Sx) at zero slip angles

(treatments A, C, E, G). Since the vertical axis of Figures 5.7-5.10 is in BFC =

(BF)/(Fz) the tangent of these slopes at/near Sx = 0% is the longitudinal stiffness

coefficient (~C). The longitudinal stiffness then is obtained by simply multiplied the

CxC with the Fz (wheel load).

One example is given here, that is for treatment A, as shown in Figure 5.7. The

lines connecting the BFC data at low percents slip clearly give rough estimates of

the slope. To be more accurate, a second-degree curve (y = a + bx + cx2) connecting

the BFC data at this low percent slips then can be drawn by using a least square

method. Since the longitudinal stiffness is not influenced by the BFC value at high

percent slips, the BFC data at up to Sx = 50% are considered sufficient to represent

the curve. The slope of this curve is the coefficient of (b). On this example, the

curve equation for treatment A is: BFC = -0.019 + (0.0388)Sx - (0.00051)S/. (See

Figure 5.15). The non-zero coefficient of (a) indicates that this line is not precisely

passing the zero value of BFC at zero percent slip.

208

BFC (at Angle 0) O.B ------------------------A---. 0.7

0.6

0.5

0.4

0.3

0.2

0.1

D

D

D []

D

FC = -0.019 + (0.0388)Sx - (0.00051)5/

0nfr---------------------------, -0.1 -0.2 '-----'-----''-----'------L--_.__-__._ __ ......._ _ __._ __ ..__ _ _.

0 10 20 30 40 50 Sx (%)

60 70 80 90 100

Fig.5.15. Determination of longitudinal stiffness (Cx) from the relationship of BFC and Sx (% ), for treatment A (0°).

The longitudinal stiffness coefficient (½C) is equal to the coefficient of (b)

multiplied by 100. Hence, for treatment A the CxC = 0.0388 (per % slip) x 100 = 3.88 (per 100% slip). Using the WL data in Table 5.9 the longitudinal stiffness (½)

is equal to (3.88 x 204.4) = 793 lb/100% slip.

Using the similar approach as described above, the longitudinal stiffness for other

treatments can be obtained. Table 5.15 presents the results of longitudinal stiffness,

from zero slip angle treatments (A, C, E, G).

Table 5.15. Longitudinal stiffness, from zero slip angle treatments (A, C, E, G).

Treatment µd Fz CXC ex (lb) (per 100% slip) (lb/100% slip)

A 0.493 204.4 3.88 793

C 0.426 204.7 3.60 737

E 0.509 191.6 4.29 822

G 0.491 194.4 3.94 766

Average 0.480 198.8 3.93 780

209

SFC (at Sx = 0%) 0.8 ..---------------------A-IQ--, 0.7

0.6

0.5

0.4 0.3

0.2 0.1

FC = -0.133 + (0.1068)0< - (0.00395)0< 2

0----------------------------t -0.1 - 0.2 ,__ __ ....._ _ ___, _____ ____. _____ ~--~---------

0 10 20 30 40 50 60 70 80 90 Slip Angle (deg)

Fig.5.16. Determination of lateral stiffness (ey) from the relationship of SFC and a (degree), for treatments AIQ.

5.4.2. Lateral Stiffness

Anologous with ex, the (field) lateral stiffness (ey) can be obtained from the

relation between SFe and slip angle (a), at zero percent slips. The tangent of the

slope at/near a = (}° is the lateral stiffness coefficient (eye). The lateral stiffness

then is obtained by simply multiplied the eye with the Fz (wheel load).

One example is given here, that is for treatments AIQ, as shown in Figure 5.11.

With the absence of SFe data between (}° and 10°, the lines connecting the SFe

from 0°, 10° and 15° then can be regarded as to representing the slope. A

second-degree curve (y = a + bx + cx2) connecting the SFe data, is considered

appropriate for smoothening the data. The slope of this curve is the coefficient of

(b ). On this example, the curve equation for treatment AIQ is: SFe = -0.133 +

(0.1068)a - (0.00395)a.2• (See Figure 5.16). Again, the non-zero coefficient of (a)

indicates that this curve is not precisely passing the zero value of SFe at zero slip

angle.

210

The lateral stiffness coefficient (CyC) is equal to the coefficient of (b) multiplied

by 57.3 Hence, for treatment AIQ the eye= 0.1068 (per deg) x 57.3 = 6.12

(per radian). Using the WL data in Tables 5.9-5.11, the lateral stiffness (Cy) is equal

to (6.12 x 190.1) = 1163 lb/radian.

Using the similar approach as described above, the lateral stiffness for other

treatments can be obtained. Table 5.16 presents the results of lateral stiffness, from

zero percent slip values of all treatments.

Table 5.16. Lateral stiffness, from zero percent slip values of all treatments.

I Treatment II IP (psi) I V (kph, mph) I F2 (lb) I eye (per rad) II Cy (lb/rad) I

AIQ 30 16.56 (10.28) 190.1 6.12 1163

CKS 30 16.68 (10.36) 194.8 5.78 1126

EMU 20 16.20 (10.06) 182.8 5.13 936

GOW 20 16.16 (10.04) 185.4 4.83 895

Average 25 16.40 (10.18) 188.2 5.47 1030

S.S. Discussion of Results

5.5.1. General Results

By using 3 replicates, the scatter of data, although still high, has been reduced.

This can be seen, for example, by comparing the friction coefficients data (BFC,

SFC) in Figure 5.1 with the corresponding results in Figure 5.2 (treatment A).

It is found from Figures 5.2-5.5 that the locked-wheel BFC tends to decrease

with increasing time. Clearly, this can be seen on those figures where most of the

BFC decrease significantly at approximately the second half of time during fully

braking. It could be argued that the decrease of BFC can be due to a certain change

(a decrease) of slip angle during full braking. The reason is that there is also a

corresponding increase of SFC with decreasing BFC. See Figure 5.17 below.

211

0.9 0.8 0.7 0.6 0.5 0.4

0.3 0.2 0.1

0

-0.1

BFC 8t SFC

I'

" ' " ~ I' ~ ~

l't- ' ' " ~ " ~

' ' " " ~ -I' I'

~ ~ ' ~ I' ~~

I' ~ ' ' ~ ' -j ~ PI ~ ~ 81 ~ ' I' I' I'

" " " - ,.,_ I I I I I I I I I I

A C E G I K M 0

Treatments

~ 1st BFC g 2nd BFC ~ 1st SFC

I'

I' I' ~

I' I' I' - .__ ~

' ' I' .__ - -:;II~ I ~ I'

I ,I I'

tl' I ,I I'

'"~ I' ~

" I I I I

Q s u w

- 2nd SFC

Fig.5.17. The locked-wheel BFC & SFC, measured at the first and the second half time of full braking.

0.9 0.8 0.7

0.6 0.5 0 .4

0.3 0.2

0.1

0

-0.1

Resultant of Fr1ct. Coef. (RFC)

- --

- - - - - -

- - - - - -- - - - - -

I I I I I I I I I I

A C E G I K M 0

Treatments

filffl1st RFC 02nd RFC

,- -- ,....

,... ,... -

- - -,... - -

I I I I

Q s u w

Fig.5.18. The locked-wheel RFC, measured at the first and the second half time of full braking.

212

From another view, however, if the resultant of friction coefficients, RFC = (BFC2+SFC2)°"5, are compared, it is found that the locked-wheel RFC also

decreases significantly (instead of remaining constant) with increasing time. This can

be seen in Figure 5.18. Hence, it can be concluded, that the decrease of slip angle is

only a minor source for decreasing the locked-wheel BFC during the full braking,

whereas the decrease of slip angle is seen to be the primary source for increasing the

locked-wheel SFC.

Another possible source of the decreasing locked-wheel BFC with increasing time

is through the uniformity condition of the measured section itself. A picture of the

uniformity of the measured section then should be obtained. This was carried out by

measuring the steady-state BFC along the measured section at a constant slip below

the locked-wheel, with the idea of eliminating the effect of time on the tyre surface

due to locked-wheel condition. The test wheel motor then was used to rotate the test

wheel (against the direction of motion), while the MMFIT was running at constant

speed without applying the brake.

Steady-state BF C 1...---------------------------0. 9 - - - - - - - - - - - 0, 815 - - - - - - - - - - - ><- - - - - 0 . 7 5 7 - - - - -><- - - - - 0 . 7 4 0 - - - - - >

0.8 0.7 0.6 0.5

0.4 0.3 0.2 0.1

0 LL.L. .......... LL.L..1....1.,,;~l<J..l<Cla..l,.4.L.l.4.L..L.LLJl.4.4..llltl..Ll'-I...Ja.4.LLJI....La<:~l..L..W~4...k:IL4.L.a...LU-J:LL.J.44.,;[Ll...[LJJ

0-1 21-25

Measured Section (m)

Fig.5.19. The uniformity of measured section (25 m), assessed by steady-state BFC at constant slip (Sx) of 20.8%.

213

Figure 5.19 presents the steady-state BFC along the measured section, at constant

slip (Sx) of 20.8% and overall average steady-state BFC of 0. 778. For the purpose of

analysis the measured section (25 m) can be divided into 3 areas: prior to locked

wheel (- 11 m), first half of fully braking (- 7 m), and second half of fully braking

(- 7 m). The average steady-state BFC for each area is: 0.815, 0.757, and 0.740,

respectively. As can be seen, on average the skid resistance property in the third area

is slightly lower than in the second area.

As mentioned in Section 5.2.3, the wheel speed on entering the measured section

is considerably lower than the average speed of the vehicle over the measured

section. This is particularly obvious for slip angles 10° and 15°. One possible cause

is the occurence of local sliding by the freely rolling tyre (before the brake is

activated). The magnitude of wheel load before and after brake activation will also

give an indication on whether the test tyre is locally sliding as it is drawn by the vehicle.

As can be seen in Figure 5.20 below, the wheel load does have a lower value at

brake off than at brake on.

350

300

250

200

150

100

50

0

W L (lbs)

~ ._-

,_

._

-._ '-'-

A C

-~ -- - -,_ ,_ ,_ ..... ,_ '-= ~ ,___ -

._ ._ I- I- I- I- I- I-

,_ ._ ,_ ~ ,_ I- ,_ ,_

'-'- --~ I ,__ '-'- ~- I ~ ._ ~~ ~-

E G I K M 0 Q s u w Treatments

D W L (brake off) - W L (brake on)

Fig.5.20. The wheel load at brake off (average of 1 second before activation and 1 second after release), and at brake on.

214

5.5.2. Effects of Tyre Pressure

· (1) Braking Force Coefficient (BFC)

From Table 5.13 it can be seen that the main effect of tyre pressure is significant

at 5% level for angle Cl°, and significant at 1 % level for angle 10°, but insignificant

for angle 15°. The BFC decreases with increasing tyre pressure.

The interaction effects of tyre pressure and percent slip is found to be

insignificant.

The interaction effects of tyre pressure and the rate of braking is insignificant for

angles Cl° and 10°, but significant at 5% level for angle 15°. The BFC decreases with

increasing tyre pressure, in which the percentage of decrease is greater at the slow

rate of braking.

(2) Sideway Force Coefficient (SFC)

It is seen from Table 5.14 that the main effect of tyre pressure is highly

significant for angle 0°, and insignificant for angles 10° and 15°. The SFC decreases

with increasing tyre pressure for angle Cl°. It should be noted, that in ideal conditions

there will be no influence of tyre pressure on SFC at angle Cl°, since the SFC is

ideally zero. The result therefore, should be read as a mechanism of the deviation

from the ideal condition. Probably, the tyre pressure has an effect on the

symmetrical or vertical position of the tyre relative to the pavement.

The interaction effects of tyre pressure and percent slip is found to be

insignificant.

The interaction effects of tyre pressure and the rate of braking is significant at

1 % level for angle 0°, and insignificant for angles 100 and 15°. The SFC decreases

with increasing tyre pressure for angle 0°, in which the percentage of decrease is

greater at the slow rate of braking. Again, for angle Cl° any change of SFC with the

change of tyre pressure should be read as a mechanism of the deviation from the

ideal condition.

215

5.5.3. Effects of the Rate of Braking

( 1) Braking Force Coefficient (BFC)

From Table 5.13 it can be seen that the main effect of the rate of braking is

significant at 1 % level for angles er and 10°, and significant at 5% level for angle

15°. The BFC increases with increasing the rate of braking.

The interaction effects of the rate of braking and percent slip is significant at 1 %

level for angle 1er, and significant at 5% level for angle 15°. The BFC increases

with increasing the rate of braking, in which the percentage of increase is greater at

the middle percent slip.

The interaction effects of the rate of braking and tyre pressure (as mentioned in

Section 5.5.2) is insignificant for angles 0° and 10°, but significant at 5% level for

angle 15°. The BFC increases with increasing the rate of braking, in which the

percentage of increase is greater at the high tyre pressure.

(2) Sideway Force Coefficient (SFC)

From Table 5.14 it can be seen that the main effect of the rate of braking is

significant at 1 % level for angles 10° and 15°, and insignificant for angle er. The

SFC decreases with increasing the rate of braking.

The interaction effects of the rate of braking and percent slip is significant at 5%

level for angles er and 15°. The SFC decreases with increasing the rate of braking,

in which the percentage of decrease is greater at the middle percent slip.

The interaction effects of the rate of braking and tyre pressure is significant at

1 % level for angle er. However, as explained in Section 5.5.2, for angle 0° ideally

there is no such effect on the SFC.

216

CHAPTER 6. THEORETICAL PREDICTION OF TYRE-ROAD FRICTION

UNDER BRAKING AND CORNERING

Notation (Used frequently in Chapter 6) a, b = length of increasing & decreasing pressure zone (unit length)

BFC = braking force coefficient Cx = longitudinal slip stiffness (force/slip, or force/fraction slip), or

longitudinal deformation stiffness (force) Cy = lateral slip stiffness (force/degree, or force/radian), or

lateral deformation stiffness (force) ½C = longitudinal slip stiffness coefficient (per slip, or per fraction slip), or

longitudinal deformation stiffness coefficient CyC = lateral slip stiffness coefficient (per degree, or per radian), or

lateral deformation stiffness coefficient Fx, FY = longitudinal & lateral force (force)

Fz = vertical load (force) 21, w = tyre contact length & width (unit length) ~. Ky = longitudinal & lateral carcass spring rate (force/length per unit area) ~. ky = longitudinal & lateral tread element spring rate (force/length per unit area) SFC = sideway force coefficient

Sx = longitudinal slip (%, or per fraction slip) a = slip angle (degree, or radian)

µd = locked-wheel friction coefficient subscript 0,1 = initial (measured), final (measured or predicted)

6.1. General

In this Chapter, the results from the candidate's experiment will be used for two

purposes. The first is to supply the parameters needed for the theoretical prediction

of tyre-road friction under braking and cornering. The second is to provide field

measurements for comparison with the candidate's theoretical prediction.

By considering some limitations of the experiment performed, such as the low

values of wheel load and the limited ranges of slip angle, another comparison will

also been done. That is to compare the laboratory data, obtained from other worker's

data [Sakai (1982)], with the candidate's theoretical prediction, where the parameters

needed are also supplied by this laboratory data.

217

The verification of tyre stiffness variation model will be shown in Section 6.2.

The input data for theoretical prediction, from both the candidate's and Sakai's data,

will be presented in Section 6.3. The theoretical prediction and its comparison, will

be presented in Section 6.4 for the candidate's experimental results, and in Section

6.5 for Sakai's laboratory data. Section 6.6 presents the theoretical prediction of a

slippery road, whereas the theoretical prediction using the tyre stiffness from load

deflection measurements will be shown in Section 6.7. Finally, the viability and

reliability of the theoretical prediction of tyre-road friction, as a subtitute for actual

skid tests, will be described in Section 6.8.

6.2. Verification of Tyre Stiffness Variation Model

As presented in Section 2.4.5, the empirical model for tyre stiffn~ss variation

consists of several equations (Equations 2.32-2.37), which can be selected according

to the tyre type. In this Section, the Equations (2.32) and (2.34) will be verified with

the data from experiment.

According to Equation (2.32), the predicted longitudinal tyre stiffness coefficient

(CxC1) will depend on: the initial longitudinal tyre stiffness coefficient (CxC0), the

initial wheel load (Fz0), and the present wheel load (Fz1). Referring to Table 5.15, the

average values from field measurements are assumed to be the initial (actual) values,

and the predicted values of CxC1 for treatments A, C, E, and G are calculated based

on their present values of wheel load.

Table 6.1 presents the comparison between the longitudinal tyre stiffness

coefficient (CxC) from field measurements and its predicted values using the

candidate's empirical model. The measured values in columns 2 and 3 are obtained

by using MMFIT, and their average values in the first row are assumed to be the

initial value. The predicted longitudinal stiffness coefficients (CxC1) in column 4 are

obtained based on the initial value, and the corresponding present values of wheel

load (Fz1).

218

Table 6.1. Measured and predicted values of longitudinal stiffness coefficient (CxC).

Treatment Fz (lb) Measured CxC Predicted cl(c +/-(per slip) (per slip)

Average 198.8 <Fzo) 3.93 (CxC0)

A 204.4 <Fz1) 3.88 (CxC1) 3.924 (CxC1) + 1.1% C 204.7 <Fz1) 3.60 (CxC1) 3.923 (CxC1) +9.0% E 191.6 <Fz1) 4.29 (CxC1) 3.939 (CxC1) -8.2% G 194.4 <Fz1) 3.94 (CxC1) 3.935 (CxC1) -0.1%

The results from Table 6.1 shows a good agreement between the predicted and

the measured values of longitudinal tyre stiffness, with discrepancies less than 10%.

The experimental values are also qualitatively in agreement with the prediction in

terms of the trend of the CxC, where the measured value of CxC decreases as the

wheel load increases (or vice versa). In addition, it seems that the factor of rate of

braking could be incorporated in the model. As mentioned in Section 5.5.3, the BFC

increases with an increasing the rate of braking (at all ranges of percent slip). Hence,

the CxC would be higher at the faster rate of braking (treatments A and E) than at

the slow rate of braking (treatments C and G).

According to Equation (2.34 ), the predicted lateral tyre stiffness coefficient

(CyC1) will depend on: the initial lateral tyre stiffness coefficient (CxCo), the initial

and present value of wheel load <Fzo, Fz1), the initial and present value of tyre

pressure (IP0, IP1), and the initial and present value of tyre (travelling) speed (V0,

V1). Referring to Table 5.16, the average values from field measurements are

assumed to be the initial (actual) values, and the predicted values of CYC1 for

treatments AIQ, CKS, EMU, and GOW are calculated based on their present values

of wheel load, tyre pressure, and travelling speed.

Table 6.2 presents the comparison between the lateral tyre stiffness coefficient

(CyC) from field measurements and its predicted values using the candidate's

empirical model. The measured values in columns 2-5 are obtained by using

MMFIT, and their average values in the first row are assumed to be the initial

value. The predicted longitudinal stiffness coefficients (CyC1) in column 6 are

obtained based on the initial value, and the corresponding present values of wheel

load (Fz1), tyre pressure (IP 1), and travelling speed (V 1).

219

Table 6.2. Measured and predicted values of lateral stiffness coefficient (CyC). '

Treat- Fz IP V Measured CYC Predicted eye +/-ment (lb) (psi) (mph) (per radian) (per radian)

Average 188.2 25 10.18 5.47 (CyC0)

AIQ 190.1 30 10.28 6.12 (CyC1) 5.642 (CyC1) -7.8% CKS 194.8 30 10.36 5.78 (CyC1) 5.617 (CyC1) -2.8% EMU 182.4 20 10.06 5.13 (CyC1) 5.120 (CyC1) -0.2% GOW 185.4 20 10.04 4.83 (CyC1) 5.103 (CyC1) +5.7%

The results on Table 6.2 shows a good agreement between the measured and the

predicted values of lateral tyre stiffness, with discrepancies less than 10%. For the

range of data above, the tyre pressure is found to be the most influencial factor,

while the travelling speed being the least. As can be seen, the measured CyC

increases as the tyre pressure increases (or vice versa).

6.3. Input Data for Prediction and Comparison of Tyre-Road Friction

6.3.1. From Candidate's Experimental Result

As described in Chapter 5, there were 12 treatments used in the main experiment.

For the purpose of prediction and comparison, the data will be reduced (averaged)

into 3 treatments only: Cl° (ACEG), 100 (IKMO), and 15° (QSUW). Therefore these 3

treatments can be considered as representing the conditions of: 30/20 psi tyre

pressure, fast/slow braking, smooth tyre, flooded water, 10 mph average speed, and

at medium textured asphaltic concrete.

The field data for comparison can be obtained from the summary of friction

coefficient versus slip in Tables 5.9-5.11 (i.e for angles 0°, 10°, and 15°). Whereas

the parameters needed for prediction, can also be obtained and calculated from those

tables above (for Fz, and µd), and from the calculation of slip stiffness in Tables

5.15-5.16 (for Cx, and Cy).

Table 6.3. presents the results of calculation of wheel load (Fz), locked-wheel

BFC (µd), longitudinal slip stiffness (Cx), and lateral slip stiffness (Cy) from the field

220

data. These parameters then will be used in Section 6.4 to enable the candidate's

experimental result to be compared with his theoretical predictions. The parameter of

b/21 is assumed to be 0.20. Three values of a/21 will be used: 0.06, 0.08, and 0.10.

Table 6.3. Parameters for theoretical prediction, extracted from field experiment.

Treatment Fz µd ex ey

(lbs) (lb/100% slip) (lb/1 radian)

All 188.2 0.480 780 1030

6.3.2. From Sakai's Laboratory Data

The laboratory data for comparison with predicted values are taken from Sakai

[1982], as shown in Table 6.4 and Figures 6.1-6.2. The parameters required for

prediction (µd, F2, ex, and ey) can also be obtained or calculated from these data.

Table 6.4. Input data of Fx and FY at various F2 , from laboratory data [Sakai (1982)].

ex= 0°

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90 100

Fx (kg) at various F2 (kg)

100 200 300 400 500 600

0 0 0 0 0 0 037 076 120 164 226 278 068 136 214 302 393 485 090 183 277 382 479 574 112 214 319 425 528 626 123 230 341 447 548 650 129 249 356 463 567 667 130 250 357 465 567 668 131 248 354 458 557 657 130 243 347 449 547 646 127 236 339 440 535 634 121 228 331 430 524 622 117 222 324 421 513 610 115 216 317 412 503 599 113 211 310 405 492 589 113 207 304 396 484 579

221

cx(deg)

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

F Y (kg) at various F2 (kg)

100 200 300 400 500 600

0 0 0 0 0 0 056 105 149 169 186 200 101 184 263 333 359 382 123 226 321 417 483 546 131 241 343 445 536 625 133 248 352 450 545 644 130 245 351 453 554 655 127 239 346 446 551 652 126 235 340 441 546 647 126 233 337 435 542 643 126 231 334 431 538 639 126 229 332 427 533 635 125 227 328 421 526 629 124 224 323 413 516 617 119 219 315 402 500 597

fx (kg), at Angle 0

700 OD

,a·a ·a ·a Fz (kg) ·a .. a 600 a .. ·D .. D 600 a .a·a ·O •D aa ' ·a .. a a SOO . ' ·a .. D 500 Ca aa .a·a ·O ·D ·a .. a a .. ·a .. D 400 400 Ca ao ·D·C ·a ·a ·D .. a a .. ·a .. D 300 300 a 0 a aD

,D•O . ·C ·D ·a .. a 200 Do a .. ·D .. a 200

a a aD aD ·a·a ·a ·D ·D .. a . . a . . ·D . . D 100 100 aaa a

0 0 10 20 30 40 50 60 70 80 90 100

Sx (%)

Fig.6.1. Input data ( · · a · ·) of Fx versus Sx (at angle 0°), various Fv from laboratory data [Sakai (1982)].

Fy (kg), at Sx = 096

700,.. F2 (kg) 00

,O ·O 0 0 0 ·O 600 ._ ·<> ·O 600 ,

OoO ,0 ·O 0 0 0 ·O ·<> 500 ._ . <> 500 , 0 oo ·O ·O 0 0 0 ·O 0 400 ._ 0 . <> 400 ,

~ 0 oo ·O . <> 0 0 ·O ·O . <> ·O 300 300,.. l

0 0 00 ·O ·O 0 0 0 ·O . <> ·O 200 200 ... ,0

l

100 ... oo oOO . 0 . <> 0 0 0 ,0 . <> ·O 100 0

0 0 I I I I I I I I

0 10 20 30 40 50 60 70 BO 90 Angle (deg)

Fig.6.2. Input data ( · · <> · ·) of FY versus a. (at Sx 0% ), various F2 , from laboratory data [Sakai (1982)].

Table 6.5. presents the calculated wheel load (F2), locked-wheel BFC (µJ, and

slip tyre stiffness (Cx, Cy) from Sakai's laboratory data, which then will be used in

Section 6.5 to be compared with the candidate's model. The parameter of b/21 is

assumed to be 0.20. Three values of a/21 will be used: 0.06, 0.08, and 0.10.

222

Table 6.5. Parameters for theoretical prediction, extracted from laboratory data [Sakai (1982)].

Fz Pd ex CV (kg) (kg/100% slip) (kg/1 deg) (kg/1 rad)*

100 1.130 1850 28 1604 200 1.035 3800 52.5 3008 300 1.013 6000 74.5 4269 400 0.990 8200 84.5 4841 500 0.968 11300 93 5329 600 0.965 13900 100 5730

Note: *) 1 radian = 57 .3 degree

6.4. Theoretical Prediction and Comparison With Experimental Result

Tables 6.6-6.7 present the theoretical prediction of Braking Force Coefficient

(BFC) and Sideway Force Coefficient (SFC) using the input data in Table 6.3, and

the procedure in Section 2.6.5. The comparisons between the candidate's measured

field data and theoretical prediction are presented in Figures 6.3-6.9.

BFC (at Angle 0) 0.8 ..--------------------------~ 0,7

0.6

0.5

0.4

0.3

0.2

0.1

D D

D

____ .. .IJ-----cru-------o-·······o ..... D D D

o------------------------------1 -0.1 - 0.2 ....._ _ __.__ _ ____. __ ...,__ _ __._ __ ....__ _ ___._ __ ..___--'---L---.....1

0 10 20 30 40 50 Sx (96)

60 70 80 90 100

Fig.6.3. Field experiment (c), and theoretical values of BFC (at angle 0°) by candidate's model: field Cx, a/21 = 0.10 (--) & 0.06 (------).

223

Table 6.6. Theoretical prediction of BFC & SFC, versus percent slip (Sx), at various a/21, from the candidate's experimental result.

I a/21 ~ 1 0.06 I 0.08 I 0.10 II a121 ~ I 0.06 I 0.08 I 0.10 I AnJ le D°:

S,. (%) BFC S,. (%) BFC 0 0 0 0 30 0.498 0.521 0.548 2 0.080 0.081 0.081 40 0.510 0.532 0.558 4 0.156 0.157 0.158 50 0.513 0.532 0.551 6 0.227 0.229 0.231 60 0.510 0.522 0.537 8 0.293 0.297 0.300 70 0.502 0.511 0.522

10 0.351 0.359 0.366 80 0.495 0.501 0.508 15 0.428 0.445 0.464 90 0.487 0.490 0.494 20 0.465 0.486 0.511 100 0.480 0.480 0.480

AnJ;?Je 10°: S,. (%) BFC sx (%) SFC

0 0 0 0 0 0.453 0.474 0.500 2 0.049 0.051 0.054 2 0.454 0.475 0.502 4 0.097 0.101 0.106 4 0.450 0.471 0.498 6 0.143 0.150 0.157 6 0.441 0.463 0.490 8 0.186 0.195 0.205 8 0.430 0.451 0.478

10 0.225 0.236 0.249 10 0.416 0.437 0.463 15 0.306 0.321 0.339 15 0.375 0.394 0.418 20 0.365 0.383 0.406 20 0.333 0.350 0.372 30 0.435 0.457 0.483 30 0.262 0.275 0.292 40 0.470 0.493 0.520 40 0.211 0.221 0.233 50 0.488 0.507 0.528 50 0.174 0.179 0.186 60 0.493 0.507 0.524 60 0.145 0.149 0.154 70 0.491 0.503 0.517 70 0.124 0.127 0.130 80 0.488 0.497 0.508 80 0.108 0.109 0.112 90 0.484 0.489 0.497 90 0.095 0.096 0.097

100 0.479 0.482 0.486 100 0.084 0.085 0.086

Angle 15°: sx (%) BFC sx (%) SFC

0 0 0 0 0 0.483 0.507 0.537 2 0.035 0.037 0.039 2 0.484 0.509 0.539 4 0.070 0.073 0.078 4 0.483 0.508 0.538 6 0.104 0.109 0.116 6 0.479 0.504 0.534 8 0.137 0.144 0.153 8 0.473 0.498 0.528

10 0.169 0.177 0.188 10 0.465 0.490 0.520 15 0.240 0.252 0.267 15 0.440 0.463 0.491 20 0.298 0.313 0.332 20 0.409 0.430 0.457 30 0.380 0.400 0.423 30 0.346 0.364 0.386 40 0.429 0.450 0.475 40 0.292 0.306 0.318 50 0.459 0.475 0.495 50 0.246 0.254 0.265 60 0.470 0.484 0.502 60 0.210 0.216 0.224 70 0.475 0.486 0.501 70 0.182 0.186 0.192 80 0.476 0.485 0.497 80 0.159 0.162 0.166 90 0.474 0.481 0.489 90 0.141 0.143 0.146

100 0.472 0.476 0.482 100 0.126 0.128 0.129

224

Table 6.7. Theoretical prediction of SFC, versus slip angle (ex.), at various a/21, from the candidate's experimental result.

I a/21 ~ 1 0.06 I 0.08 I 0.10 I a/21 ~ 0.06 0.08 0.10

sx = 0% sx = 100%

ex. (0) SFC ex. (0) SFC

0 0 0 0 0 0 0 0 2 0.170 0.171 0.172 2 0.017 0.017 0.017 4 0.306 0.310 0.314 4 0.034 0.034 0.034 6 0.389 0.402 0.417 6 0.051 0.051 0.051 8 0.429 0.448 0.470 8 0.068 0.068 0.068

10 0.453 0.474 0.500 10 0.084 0.085 0.086 15 0.483 0.507 0.537 15 0.126 0.128 0.129 20 0.497 0.522 0.553 20 0.168 0.170 0.172 30 0.508 0.532 0.561 30 0.247 0.250 0.255 40 0.510 0.528 0.549 40 0.318 0.323 0.329 50 0.506 0.518 0.535 50 0.379 0.385 0.392 60 0.499 0.509 0.521 60 0.427 0.433 0.440 70 0.493 0.499 0.508 70 0.461 0.465 0.472 80 0.486 0.490 0.494 80 0.478 0.481 0.485 90 0.480 0.480 0.480 90 0.480 0.480 0.480

From Figures 6.4 and 6.6 which show results for the Braking Force Coefficient

(BFC), it can be seen that the predicted values are generally lower than the

experiment values at the middle-lower percent slips, but show a good agreement at

the middle and high percent slips.

For the SFC in Figures 6.5 and 6. 7, on the other hand, the predicted values show

a good agreement just for high percent slip, but the predicted values are higher than

the experiment for middle percent slips, and are lower than the experiment for low

percent slips.

There are at least two reasons for discrepancies mentioned above:

a. It is possible that the slip angle decreased, when the tyre is braked up to the

locked-wheel condition (say, for Sx;::: 30%).

b. The correction of wheel speed delay as described in Section 5.3.1 seems to be

under estimated.

225

0.9 BFC (at Angle 10)

0.8 0.7 a a

a DC

0.6 ~ D a aa a D ClJ D

D cP D 0.5 q§ D D

D 0.4 D

0.3

0.2 0.1

0 -0.1

0 10 20 30 40 50 60 70 80 90 100 Sx (96)

Fig.6.4. Field experiment (c), and theoretical values of BFC (at angle 10°) by candidate's model: field½ & Cy, a/21 = 0.10 (--) & 0.06 (------).

SFC (at Ang le 10) 0.9 ~-------------------------, 0.8 0.7

0.6

0.5 -----0.4 - ............. -.. _ 0.3 - ............ _ A---..

0 V ----0,2 ------------0.1 -~n~~~~-4-A.d

0 _________ __._ __ ......... ..._ __________ --1

-0.1 --~------~--....__ _ __._ __ ..__ _ _.._ __ ...__ _ _. 0 10 20 30 40 50 60 70 80 90 100

sx (96)

Fig.6.5. Field experiment (<>), and theoretical values of SFC (at angle 100) by candidate's model: field Cx & Cy, a/21 = 0.10 (--) & 0.06 (------).

226

BFC (at Angle 15) 0.9 -------------------------, 0.8

0.7 0.6

0.5

0.4

0.3

0.2 0.1

D

D

0 Cl , .... ,-c¥i .,, Cl

,/

D .-"' .. ,,,' ..

.. .,, .,,

Cl

----------------- D

0---------------------------t - 0.1 ..___ _ __.__ _ __._ __ ...__ _ __._ _ ___. __ ....._ ___ ~~-......_ _ ___.

0 10 20 30 40 50 Sx (96)

60 70 80 90 100

Fig.6.6. Field experiment (a), and theoretical values of BFC (at angle 15°) by candidate's model: field Cx & CY' a/21 = 0.10 (--) & 0.06 (------).

SFC (at Angle 15) 0.9 ..--------------------------, 0.8

0.7 ~ o0 o

0.6 i o i---~~ 0

0.5 ---~ -----o--.. ~ 0.4

0.3

0.2 0.1

-----~-----------------

01---------------------------1 -0.1 ~--------------__.__ _ __._ __ ...__ _ __._ _ ___. __ ....__ _ ___.

0 10 20 30 40 50 Sx (96)

60 70 80 90 100

Fig.6.7. Field experiment (<>), and theoretical values of SFC (at angle 15°) by candidate's model: field½ & Cy, a/21 = 0.10 (--) & 0.06 (------).

227

SFC (at Sx = 0%) 0.8 ..-------------------------,

0.7

0.6

0.5 0.4 0.3

0.2 0.1

01----------------------------1 -0.1 -0.2 ..___ _ __._ __ ...._ __ .___ _ __._ __ _.__ _ ___..____~ __ _.__ _____

D 10 20 30 40 50 60 70 80 90 Slip Angle (deg)

Fig.6.8. Field experiment (<>), and theoretical values of SFC (at Sx 0%) by candidate's model: field c;,, a/21 = 0.10 (--) & 0.06 (------).

SFC (at Sx = 100%) 0.8 .---------------------------, 0.7

0.6

0.5 0.4

0.3 0.2

0.1

...... ....... _ .... ------

------------------------

----

0JtC-------------------------l

-0.1 -0.2 .____~ __ _.__ _____ ~-~--.....___ _ ____, __ __._ __ .....___ _ __,

D 10 20 30 40 50 60 70 80 90 Slip Angle (deg)

Fig.6.9. Field experiment (A), and theoretical values of SFC (at Sx 100%) by candidate's model: field Cx & Cy, a/21 = 0.10 (--) & 0.06 (------).

228

A looseness of the wheel position was observed during the experiment. The

resultant of the shear forces will act a distance t (called pneumatic trail) behind the

geometric centre of the contact area, causing a tyre to generate a self-aligning

moment. With increasing the proportion of tyre to slide as Sx increases, the position

of a smaller slip angle then was continuously maintained. Decreasing the slip

angle by 4°, for example, the SFC can be decreased by 0.1 (see Figure 3.44); while

the BFC is increased by a lesser amount at low Sx, but only has a small effect at

high Sx (see Figure 3.43).

Figures 6.10-6.13 present the comparisons between the candidate's measured field

data of BFC & SFC, and the theoretical prediction after assuming the decreasing slip

angles from 100 to 8°, and from 15° to 12°, due to the above looseness.

A greater correction of wheel speed delay, say 7 x 0.107 second, will certainly

also reduce the discrepancies. Referring to the characteristic of normal equation

(Section 5.3.2), the greater the correction of (u) the smaller the value of (s). Hence,

the curve of Wheel Speed (WS) = f(T) then tends to become leptocurtive.

Accordingly, the same T will reach smaller WS (or higher Sx) by using a greater

correction. For example (see Figures 5.7-5.10), the "drops" of SFC at around Sx

30%-40% will be moved forward to the higher Sx, as the standard deviation (s)

decreases.

The two factors described above: the looseness of the wheel position and the

delay of the wheel speed, can be regarded as badly controlled factors in this

experiment. The first factor can be eliminated only by improving the surrounding

construction of the test wheel (and its calibration equipment) and angling the wheel

in the correct sense. The second factor is associated with the recording device. One

source of delay: the accuracy of wheel speed, can be eliminated by reducing the

duration for obtaining "instant" wheel speed. A shorter duration of say 0.1 second,

for example, means a requirement for increasing by up to 10 times of the device

present capacity. Other source of the delay, if it still occurs, can be eliminated

empirically either by a method of estimation described in Section 5.3.1 or by

providing a calibration apparatus for measuring the delay. The results then can be

used as a feed back for improving the program supplied.

229

0.9 BFC (at Angle 10 & 8)

0.8 0.7 D D

D CD

0.6 l2i a D aa D a D cP (b D 0.5 iHi' a D D D D a 0.4 a D D a D D a 0.3

0.2 0.1

0

-0.1 0 10 20 30 40 50 60 70 80 90 100

Sx (96)

Fig.6.10. Field experiment (c), and theoretical values of BFC by candidate's model: field ½ & Cy, a/21 = 0.10, a = 10° (--) & g0 (------).

SFC (at Angle 10 & 8) 0.9 ------------------------0.8 0.7

0.6 0.5 0.4 0.3 ............ ... ........

~:! <> -----------------~-;---~~--o~-~-~)":.:'j __ ~--[-~-~~~=%~-Oo ~

0 1--------------""'----.&...&...--...;._---------o. 1 ~-~-~----~-~--......._ _ __._ _ ___, ____ ....._ _ __.

0 10 20 30 40 50 Sx (96)

60 70 80 90 100

Fig.6.11. Field experiment ( <> ), and theoretical values of SFC by candidate's model: field ½ & Cy, a/21 = 0.10, a = 10° (-) & g0 (------).

230

0.9 BFC (at Angle 15 & 12)

O.B 0.7 D

r:flD QJ 0.6 DO D D D a D D o.s ---------0.4 D

0.3 0.2 0.1

0 -0.1

0 10 20 30 40 so 60 70 80 90 100 Sx (%)

Fig.6.12. Field experiment (c), and theoretical values of BFC by candidate's model: field½ & Cy, a/21 = 0.10, a= 15° (--) & 12° (------).

SFC (at Angle 15 & 12) 0.9 ,---------------------------, 0.8

0.7

0.6 0.5

0.4

0.3

0.2

0.1

0

-0.1 0

0

0 ---

10 20

-- ... ... , 0 ,, __

o O ··:··--:~·-;·····;i"·-... -;~-=--:::. --~-i~-==--~~~-

30 40 50 Sx (%)

60 70 80 90 100

Fig.6.13. Field experiment (<>), and theoretical values of SFC by candidate's model: field Cx & Cy, a/21 = 0.10, a = 15° (--) & 12° (------).

231

Referring now to the candidate's model itself, it is seen that the theoretical

prediction of BFC versus slip (at a = Q°) in Figure 6.3 shows a good agreement with

the experimental results, whereas the predictive values of SFC versus slip angle (at

Sx = 0%) in Figure 6.8 under estimate the experimental results. It seems that a small

modification is needed for the candidate's model, by means of modifying the input

values of the lateral tyre stiffness (Cy). This will be discussed later following the

results from Sakai's laboratory data.

6.5. Theoretical Prediction and Comparison With the Sakai's Laboratory Data

Table 6.8 presents the theoretical prediction of frictional forces Fx and FY using

the input data of Sakai's laboratory data in Table 6.5, and the procedure in Section

2.6.5. The comparisons between Sakai's laboratory data and the candidate's

theoretical values are presented in Figures 6.14-6.19.

From Figures 6.14-6.19, it is seen that the predicted values are generally in good

agreement with the laboratory data, especially for the data of longitudinal force (F x)

versus longitudinal slip (Sx). In most cases, the predictive values with a/21 = 0.06-0.1

can be used for lower wheel load (F2), whereas a/21 = 0.06-0.08 are suitable for

higher F2 • It seems that the selection of a/21 is more sensitive with increasing the

tyre stiffness.

For the data of lateral force (Fy) versus slip angle (a), particularly for lower a,

the predictive values under estimated the laboratory data. As this phenomenon is

also found in the candidate's experimental results (Figure 6.8), a modification of the

predictive model seems to be a necessity. It can be done by introduced a higher

lateral stiffness ( Cy) into the model. As shown in Figure 2.19, the contribution of Cy

to the FY is in the area of below critical slip angle (a.er).

232

Table 6.8. Theoretical prediction of Fx vs Sx (a=(}°), FY vs a (Sx = 0%), at various a/21, from Sakai's laboratory data.

s (%)

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

100

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

100

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

100

Fx (kg) at various Fz (kg)

100 200 300 400 500 600

a/21 = 0.06 0 0 0 0 0 0

034 070 109 149 202 248 064 129 201 272 362 441 087 170 258 343 440 532 100 190 286 378 478 577 107 202 303 399 501 602 116 217 323 424 528 634 121 224 332 435 539 647 124 228 338 442 546 654 124 228 336 438 538 644 122 225 330 431 529 634 120 221 325 424 520 623 119 218 320 417 511 612 117 214 315 410 502 601 115 211 309 403 493 590 113 207 304 396 484 579

a/21 = 0.08 0 0 0 0 0 0

035 070 110 150 205 251 065 131 204 277 373 456 090 177 271 362 468 567 105 202 304 404 514 621 114 216 324 428 541 652 124 233 347 457 572 688 128 240 357 469 584 702 132 244 362 474 585 701 130 240 354 463 570 684 127 234 346 452 556 666 124 229 337 440 542 649 122 223 329 429 527 631 119 218 321 418 513 614 116 212 312 407 498 596 113 207 304 396 484 579

a/21 = 0.10 0 0 0 0 0 0

035 071 111 152 207 254 066 133 208 282 382 468 093 185 285 382 500 608 111 215 326 434 558 675 121 232 349 463 590 713 133 252 377 498 628 757 139 260 388 511 641 772 141 263 389 510 634 761 138 255 377 494 612 735 134 247 365 478 591 709 130 239 353 461 570 683 125 231 341 445 548 657 121 223 328 429 527 631 117 215 316 412 505 605 113 207 304 396 484 579

233

a (deg)

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

0 2 4 6 8

10 15 20 30 40 50 60 70 80 90

FY (kg) at various Fz (kg)

100 200 300 400 500 600

a/21 = 0.06 0 0 0 0 0 0

049 091 129 149 166 181 084 156 224 267 300 329 099 184 266 330 386 438 107 197 287 361 428 495 112 205 300 379 453 528 117 215 315 403 485 571 120 220 322 413 500 591 121 223 326 421 511 607 121 222 325 423 514 612 119 219 321 417 510 610 118 216 317 412 503 602 116 213 312 407 497 594 115 210 308 401 490 587 113 207 304 396 484 579

a/21 = 0.08 0 0 0 0 0 0

049 092 131 150 167 182 086 160 230 271 304 333 104 193 279 343 398 450 113 209 303 378 446 513 118 218 317 398 474 549 124 229 333 424 508 596 127 233 340 435 524 617 128 235 344 442 534 633 126 230 337 437 532 634 123 226 331 429 522 623 121 221 324 420 513 612 118 216 317 412 503 601 116 212 311 404 494 590 113 207 304 396 484 579

a/21 = 0.10 0 0 0 0 0 0

050 093 132 152 168 183 089 165 237 275 308 336 110 204 294 358 413 462 120 222 322 399 467 534 126 233 338 422 499 575 133 245 357 450 537 627 136 250 364 462 554 650 135 249 363 467 563 665 132 242 354 455 552 657 128 235 344 443 539 641 124 228 334 432 525 626 120 221 324 420 511 610 117 214 314 408 498 595 113 207 304 396 484 579

Fx (kg), at Angle 0 a/ 21 = 0.06

700 D a

600

500

400

300 a

200

100

0 0 10 20 30 40 50 60 70 80 90 100

Sx (%)

Fig.6.14. Sakai's data (c), and theoretical values of Fx (at angle 00) by candidate's model: parameters from Table 6.5, a/21 = 0.06, various Fz.

700

600

SOO

400

300

200

100

Fy (kg), at Sx = 096

0

0

0

0

0

a/21 = 0.06

Fz (kg) 0

0

0 __ __. ____ ___._ __ _._ __ ...,__ __ .__ _ ___._ __ ___._ __ ....._ _ ____.

0 10 20 30 40 50 60 70 80 90 Angle (deg)

Fig.6.15. Sakai's data (<>), and theoretical values of s, (at Sx 0%) by candidate's model: parameters from Table 6.5, a/21 = 0.06, various l'~.

234

700

600

500

400

300

200

100

Fx (kg), at Angle 0

a a 0

D 0

a/21 = 0.08

0 a a F1 (kg)

D a

o--___._ __ ..__ _ __._ __ ....._ _ _._ __ .__ _ _._ _ __. __ ......_ _ ___, 0 10 20 30 40 50 60 70 80 90 100

Sx (%)

Fig.6.16. Sakai's data (c), and theoretical values of Fx (at angle 00) by candidate's model: parameters from Table 6.5, a/21 = 0.08, various Fz.

700

600

500

400

300

200

100

a 21 = 0.08

F1 (kg) ~

~ s

0 O-----'----'-----'-----'---___,J'-----'-----'---_.__ _ __.

0 10 20 30 40 50 60 70 80 90 Angle (deg)

Fig.6.17. Sakai's data (<>), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5, a/21 = 0.08, various Fz

235

700

600

500

400

300

200

100

Fx (kg), at Angle 0

D

D

D

D

D

D

D

D

21 = 0.1

D D

D D D D

D D D

D D

o 1'1-----'---....___ _ ___., __ _.__ _ __.'---_ _._ __ L...-_ __._ __ _.__ _ __.

o 10 20 30 40 50 60 70 80 90 100 Sx (%)

Fig.6.18. Sakai's data (a), and theoretical values of Fx (at angle 0°) by candiadate's model: parameters from Table 6.5, a/21 = 0.10, various Fz.

700

600

500

400

300

200

100

Fy (kg), at Sx = 096 a 21 = 0.1

6

40

o------~--~--__._ _________ ......_ _____ ~ 0 10 20 30 40 50 60 70 80 90

Angle (deg)

Fig.6.19. Sakai's data (o), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5, a/21 = 0.10, various Fz.

236

From the candidate's original model, the input of lateral stiffness (Cy) is force/1

radian, whereas the input of longitudinal stiffness (Cx) is force/100% slip. These

units are the same as used by Dugoff et.al [1969, 1970], where the candidate's

model was derived from. Other investigators have al~o used these inputs in their

models [Tielking and Mital (1974)].

It is known that the normal full range of longitudinal percents slip is between 0%

and 100% ( or O S Sx S 100% slip), while the normal full range of slip angles is

between 0° and 9Q° ( or O S a S 1.57 radians). Hence, the previous input of CY is

based only on a part of the range of slip angles, whereas the input of Cx is based on

a full range of percents slip. To fulfil the analogy, it seems reasonable to modify the

input of Cy into its full range, i.e. the input of Cy becomes force/1.57 radians (or

force/90 degrees), hence with reference to Eq.(2.11a): v~ing coefficient of c2 -= 1.

The effect of increasing CY (::::: 57%), on frictional forces Fx and FY can be studied

from Section 3.2.2 (see Figures 3.15-3.24). One thing is clear, that there is no effect

of increasing Cy into Fx versus Sx at angle 0. Hence the predictive values of Fx

versus Sx at angle O such as shown on Figures 6.14, 6.16, and 6.18 are not

influenced by this modification.

Figure 6.20 shows the predictive values of FY versus a (at Sx = 0%) using the

modified input (i.e. Cy in force/1.57 radians). From this figure, it is seen that a closer

agreement between the predictive values and Sakai's laboratory data can be

achieved. Moreover, it is also consistently similar with the predictive values of Fx

versus Sx (at a= Q°) of Sakai's laboratory data in Figures 6.14, 6.16, and 6.18.

237

Fy (kg), at Sx = 096 a 21 = 0.06

700 Fz (lc:g) 0 <> 0 <> 0 0 0

600 0

0 0 0 0 0 <> 500

400

300

200

100

0 D 10 20 30 40 50 60 70 80 90

a 21 = 0.08 700

Fz (kg)

600 0 <>

6

500 0 5

400 400

300 3

200

100

0 0 10 20 30 40 50 60 70 BO 90

a/ 21 = 0.1 700

0 0 0 Fz (kg)

600 0 0 0

500 0 <> 0 0

400 400 <> 0 0 0

300 <> 0 <> 200

100 <>

0 D 10 20 30 40 so 60 70 80 90

Angle (deg)

Fig.6.20. Sakai's data (<>), and theoretical values of FY (at Sx 0%) by candidate's model: parameters from Table 6.5 (modified Cy), various a/21 & F2•

238

6.6. Theoretical Prediction of Slippery Road

As shown in Figure 2.14 the tyre stiffness (and hence, its measurement) can be

regarded as independent of pavement/lubricant conditions. This concept, although it

already has a literature background (such as Bergman [1977a]), will be further

examined by an additional experiment. Another road pavement, different from the

one used in the main experiment, will be chosen. This bituminous pavement has a

lower skid resistance (i.e. slippery road).

The tyre and operational conditions in this additional experiment were chosen in

accordance with the main experiment. Hence, the same smooth tyre was used with

flooded water. The tyre pressure was set up at 25 psi (to represent 30/20 psi tyre

pressure). The rate of braking was set up at a moderate position (to represent

fast/slow braking), and with 10 mph average speed of vehicle.

Table 6.9 presents the parameters for prediction, where the tyre stiffness (Cx, Cy)

are taken from the candidate's main experiment (Table 6.3). The measured vertical

load (Fz) was 189.6 lb, which is close to 188.2 lb as in the main experiment. The

different parameter is the locked-wheel BFC (µd) which was 0.375 for this slippery

road (compare to 0.480 in the main experiment). The values of a/21 = 0.10 and b/21

= 0.20, such as used in Figures 6.10-6.13, will also be used in the prediction.

Table 6.9. Parameters for theoretical prediction on slippery road.

Treatment Fz µd ex Cy (lbs) (lb/slip) (lb/rad) (lb/l.57rad)

XYZ 189.6 0.375 780 1030 1617

Figure 6.21 presents the comparisons between the candidate's measured field data

and the theoretical values. This figure shows a reasonably good agreement between

the experiment values and the predicted values. Some small discrepancies are

believed to be due to vibration and the difficulties in maintained the straight ahead

position on that slippery road. It can be concluded that the concept of the

independence of tyre stiffness from pavement/lubricant conditions can be applied.

239

BFC & SFC 0.8 0.7 0.6 0.5 a a 0.4 D D D D 0.3

a D

0.2 0.1

0 -0.1 -0.2

0 10 20 30 40 50 60 70 80 90 100

0.8 y 0.7

0.6

0.5 D

0.4 D D a

0.3 D

0.2

0.1 0

0 -0.1

-0.2 0 10 20 30 40 50 60 70 80 90 100

0.8 z 0.7

0.6 0.5

0.4 oa

0.3 D D

0.2 0

0.1

0 0

-0.1

-0.2 0 10 20 30 40 50 60 70 BO 90 100

sx (96)

Fig.6.21. Field BFC (c) & SFC (<>) on slippery road at angles 0° (X), 10° (Y) & 15° (Z); and theoretical values by candidate's model: from Table 6.9, a/21 = 0.10.

240

6.7. Theoretical Prediction Using Tyre Stiffness from Load-Deflection

Measurements

As mentioned in Section 2.4.5, the tyre stiffness has also been related with the

spring rate, and the length and width of tyre contact (Equations 2.10-2.11). It seems

that the first stiffness (d.FxfdSx and d.F/da) and the second stiffness (0.5 kx(21)2w and

0.5 ky(21)2w) may have a different concept, where the first is from slip measurement

and the second is from deformation measurement However, it can be argued that the

limit of the first concept (i.e. near zero slips) may be equal with the limit of the

second concept (i.e. near slips).

To investigate the discrepancy of those measurements and its effect on theoretical

prediction, the tyre used in the main experiment was tested in the laboratory. The

procedure was similar with Tielking and Mital [1974], except that the tyre (instead

of the table) was moved. The vertical load and the tyre pressure were keep constant

at the value used in the candidate's field experiment. The arrangements for

measuring the laboratory tyre stiffness are shown in Figures 6.22-6.23.

For measuring the laboratory longitudinal stiffness, the horizontal load in

longitudinal direction applied to the test wheel (at the middle of tyre vertical height)

was noted from the spring balance reading. Hence, the (shear) load on both the top

and the bottom of the tyre was a half of the spring balance reading. The horizontal

load was manually increased in increments of 50 lb. The corresponding deflections

were noted from a gaugemeter.

Similar with the longitudinal stiffness, for lateral stiffness the horizontal load was

applied to the test wheel in lateral direction (i.e. perpendicular to the wheel plane) at

the middle of tyre vertical height. The horizontal load was noted from the spring

balance and manually increased in increments of 50 lb. The (shear) load on the top

and the bottom of the tyre was a half of the spring balance reading. The

corresponding deflections were noted from a gaugemeter.

The contact length (21) and contact width (w) of the tyre were obtained by

measuring the tyre print on the paper (inserted between the tyre and the table), with

the same vertical load and tyre pressure used in the field.

241

Fig.6.22. Arrangement for measuring the laboratory longituilinal tyre stiffness (Cx)·

,/ I

I

Fig.6.23. Arrangement for measuring the laboratory lateral tyre stiffness (Cy).

242

From the (shear) load vs deflection curves and the contact area, the values of

longitudinal and lateral carcass spring rates (Kx, Ky) can be calculated, which are

equal to the longitudinal and lateral tread element spring rates (kx, ky) by assuming

rigid carcass. The laboratory longitudinal and lateral stiffness (½, Cy) can be

obtained by using the formula in Equations (2.10-2.11).

Table 6.10 presents the comparison between the tyre slip stiffness obtained from

the field and the tyre deformation stiffness obtained from the laboratory. The

theoretical predictions using the candidate's model are shown in Figures 6.24 and

6.26 for main experiment road, and in Figure 6.25 for slippery road.

Table 6.10. Tyre stiffness from the field (main exp.) and laboratory (load-def.).

Source IP (psi) Fz (lb) ex CV

Field (slip) 20/30 188.2 780 (lb/slip) 1617 (lb/l.57rad) Lab. (deformation) 25 188.2 1650 (lb) 640 (lb)

From Table 6.10, it is found that there are some discrepancies between the tyre

stiffness from the laboratory and from the field, which can be attributed into the

following factors.

1. The accuracy of measurement.

The stiffness from the field rely on a few points in the relationship between BFC

(or SFC) and Sx (or a), while the relationship itself was derived from the

transformation of time into percent slip. The accuracy of transformation will

depend on the smoothened technique for wheel speed, and on the correction for

wheel speed delay. The measurement of stiffness from the laboratory, however,

more straightforward, can exhibit scatter results if there are some inconsistensies

in the experiment methods used to obtain such data [Loeb et.al (1990)].

2. The difference on tyre motions.

It is clear that the data for obtaining field Cx (Figures 6.24-6.25) were derived

from the tyre motion which can be regarded as quasi steady-state or non steady

state motions (i.e. slow or fast braking to wheel lock), whereas the data for

243

obtaining field Cy (Figure 6.26) were derived from the tyre motion which can be

regarded as steady-state motion (i.e. tyre rolling with constant braking slip and

constant slip angle). On the other hand, the load-deflection stiffness (both ½ and

Cy) were derived from the static measurements which will be, presumably, close

to steady-state motion and higher than transient motion. A possible explanation is

the effect of non-homogeneous shear resistance between the tyre and road

surface during transient sliding.

3. The different methodsof deformation measurement.

As mentioned in Section 2.4.5, the laboratory measurement of tyre stiffness

without sliding will be lower than the measurement with complete sliding.

Referring to Equations 2.10-2.11, it is more likely that the slip measurement will

be close to the deformation measurement at the condition of complete sliding

[Thieme et.al (1982)] instead of without sliding [Tielking and Mital (1974)]. In

other words, the stiffness derived from the slip measurements (of steady-state

motion) will be close to load-deflection stiffness with complete sliding, but will

be higher than load-deflection stiffness without sliding.

4. The assumption of carcass rigidity.

The tread element spring rates (kx, ky) for calculating the deformation stiffness

(Cx, Cy) are obtained from measuring the carcass spring rates (Kx, Ky), which are

only equal under the condition of rigid carcass. In actual case the carcass was

not perfectly rigid, hence a higher value of tread element spring rates (kx, ky)

should be used. The degree of carcass rigidity in lateral direction is expected to

be lower than in longitudinal direction, hence a greater correction (i.e. addition)

will be required for lateral deformation stiffness (Cy) than for longitudinal

deformation stiffness (Cx).

244

0.8 BFC (at Angle 0)

0.7 D D a

a a 0.6 D ------cr-- -------------y·-o 0.5 a D D

,l 0.4

, a D I

I

0.3 : I . .

0.2 . a a .

I . 0.1 .

: alP 0

-0.1

-0.2 0 10 20 30 40 50 60 70 80 90 100

Sx (%)

Fig.6.24. Field BFe (a) on main experiment road (at angle 0°), and theoretical values by candidate's model: a/21 = 0.10, ex from Field(--) & Laboratory(------).

BFC (at Angle D) 0.0 .--------------------------=-x.,--, 0.7

0.6

0.5

0.4

0.3 0.2

0.1

D

D D

a -------------

D a a a

0t----------------------------1 -0.1 -0.2 .__ _ __._ __ ....._ _ _.__ __ .,_ _ __,__ __ ..___--L,. __ ..,___ _ __._ _ ___,

0 10 20 30 40 50 60 70 80 90 100 Sx (%)

Fig.6.25. Field BFe (a) on slippery road (at angle 0°), and theoretical values by candidate's model: a/21 = 0.10, ex from Field(--) & Laboratory(------).

245

SFC [at Sx = 096) 0.8 .-----------------------------,

0.7

0.6

0.5 0.4

0.3 0.2 0.1

, , I

/ /

/ I

I I

0

/------------------ ---------------

/

01--------------------------------1 -0.1 -0.2 .__ _________ .__ _ ___. __ __. __ ~--~----------~

0 10 20 30 40 50 60 70 . 80 90 Slip Angle (deg)

Fig.6.26. Field SFe ( <>) on main experiment road (at Sx 0% ), and theoretical values by candidate's model: a/21 = 0.10, <; from Field (--) & Laboratory (------).

From Figures 6.24-6.25 it is seen, that the laboratory-predicted BFe generally

overestimates the candidate's measured field data. Results from Bergman and

Beauregard [1974] show that under steady-state conditions, the lateral slip stiffness

(at a = 0) is greater than that for the transient conditions. It may be expected, that

this phenomenon is also valid for longitudinal slip stiffness (ex). In other words,

under steady-state conditions the slip ex (at Sx = 0) will be greater than that for

transient conditions. Hence, when the deformation ex is close to steady-state

conditions, the laboratory-predicted values will also be higher than that for transient

conditions. Presumably, in this case factor (2) has a dominant effect over factor (3),

From Figure 6.26 it is seen, that the laboratory-predicted SFe generally

underestimates the candidate's measured field data. Since the field eY was derived

from steady-state motion (which is closer to static laboratory measurement), so the

effect of factor (2) is less relevant, and the discrepancy is mainly due to factor (3), i.e

the '5tiffnes5 without 5\iding (in laborator_y) will be lower than the stiffness with complete

5liding (in the field).

246

BFC (at Angle 0) 0.8 * 0.7

0.6

0.5 0.4

0.3

0.2

-0.1

* D D

D

a a D a

-0.2 .__ _ __._ __ ....._ _ __. __ __._ __ ....__ _ __._ __ ~-~----~

0 10 20 30 40 50 Sx (96)

60 70 80 90 100

Fig.6.27. Field transient BFC (c) on main experiment road (at angle 0°), and theoretical values by candidate model: a/21 = 0.10, Cx from Field (--) & Laboratory (------). Field steady-state BFC (*) are also shown in this figure.

Figure 6.27 presents the comparison between theoretical BFC and transient field

BFC. The steady-state field BFC, which were obtained by using the test wheel

motor, are also shown in this figure. The rotating test wheel (against the direction of

motion) created slips when the MMFTT was running at constant speed without

applying the brake.

It can be seen from Figure 6.27 that the steady-state BFC is generally higher than

the transient BFC. It should be noted that these steady-state values are on the

conservative side, since they are based on overall averages along the measured

section (25 m), while the skid resistance property at the area of below the locked

wheel braking (- 11 m) is actually higher than at the area of full braking (- 14 m).

(See Figure 5.19).

Another possible effect due to the difference on tyre motions, i.e. factor (2)

above, is through the shapes of the pressure distribution (a/21, b/21). Probably at

constant slip (steady-state) the a/21 will be higher (i.e. flatten pressure distribution)

than at slow braking (quasi steady-state) or fast braking (non steady-state). In other

247

words, at steady-state conditions the frictional forces become higher due to the

flatten pressure distribution.

Results from Bode [1962] confirm that the pressure distribution is rather flatter at

low deceleration (see Figure 1.14). On the contrary, results from the candidate's

main experiment show that the slip stiffness at slow braking is rather lower than at

fast braking. So, even if the slip stiffness is slightly decreased in the case of

candidate's results, the frictional forces may be still high if the effect of the flatten

pressure distribution is more dominant than the effect of decreasing slip stiffness

(see Figure 2.24).

Providing that the tread element spring rates (kx, ky) can be measured (obtained),

it can be concluded that for steady-state conditions both the tyre stiffness from slips

and load-deflection measurements with complete sliding can be applied for

predicting the frictional forces. For quasi steady-state or non steady-state motions,

however, a lower value of tyre stiffness or a lower value of a/21 may be required if

the tyre stiffness is derived from load-deflection measurements. For load-deflection

measurements without sliding, a higher value of tyre stiffness may be required.

The introduction of an empirical factor (see Section 2.4.6) for using load

deflection stiffness, to account for: i) other than steady-state motions, ii) other than

measurements with complete sliding, iii) other than measurements of tread element

spring rates, appears to be a practical solution.

6.8. Viability and Reliability of the Theoretical Prediction of Tyre-Road Friction

As a Substitute for Actual Skid Tests

Yandell and Gopalan [1976] used the mechano lattice analogy and the theory of

hysteretic friction to predict the coefficient of friction of a sliding rubber blocks at

one constant speed. A modified portable skid resistance tester was used to measure

the actual coefficients.

248

Taneerananon [1981] and Yandell et.al [1983] extended the precision of the

method used by Holla [1974] for estimating the coefficient of locked wheel friction

of road surfaces at various speeds. In addition he predicted the (maximum) sideways

force coefficients as well as the locked wheel coefficients from surface texture

measurements. The actual coefficients were also mea~ured by the Multi Mode Friction

Test Truck (MMFTT).

The candidate has extended the prediction of frictional force coefficients to cover all

ranges of percent slip and slip angles encountered by motor vehicles. Even though

some discrepancies (due to less controlled factors) were found to exist between the

predictive and the experiment values (measured by MMFTT), the method gave a

generally good agreement and correct trend of the coefficients. Furthermore, using

his modified model, an even c.\01,er agreement was found when the predictive values are

compared with the laboratory data obtained b.Y 5akai [1<182 ], in which all factors are

presumably assumed to be in fully controlled conditions.

The present experiment involved a common type of road pavement (i.e.

medium-textured bituminous surface). Referring to the literature survey, it is most

likely that the friction of the other types of road pavement (with an exceptional of

very rough surface) can also be predicted with reasonably good agreement, as their

behaviour (in respect to percent slip and slip angle) will also have a similar trend

with this present pavement. Recent investigations (see Appendix F), using higher

speeds (20 mph & 30 mph), al,o showed a similar trend to those with the 10 mph

speed used.

One possible difficulty in obtaining a reliable estimation of tyre-road friction, is

the determination of the shape of the tyre pressure distribution, particularly the input

value of a/21. An empirical model for determining this distribution, or a list of

expected pressure distributions under a variety of conditions, may be the practical

solution. This can be achieved by gathering more information about the effects of

the relevant factors (such as: tyre types, wheel load, inflation pressure, and wheel

speed), into the shape of the distribution.

At the present time, the tyre stiffnesses Cx, and Cy can be obtained

experimentally by measuring forces and slips (or deformations), either in a road

249

surface or in a laboratory [Tielking and Mital (1974), Thieme et.al (1982)].

Especially for stiffness measurements on road surfaces, the candidate has developed

an empirical model for the variation of tyre stiffness due to: vertical load, inflation

pressure, and tyre speed.

Attempts to predict tyre stiffness directly from other tyre properties or related

parameters are left for future refinement. For stiffness measurements in the

laboratory, the need for applying empirical models is probably less beneficial than

the accuracy gained with direct measurement on different variation (vertical load,

inflation pressure). An empirical factor may still be required, however, if there is a

sufficient change of vehicle speed so that it has a significant effect on tyre stiffness.

The theoretical method of obtaining the coefficient has the advantages of low

capital cost and relatively more safety when compared with measurements made

by a skid testing vehicle. More advantages will be gained if the method of

obtaining the locked-wheel braking coefficient (or maximum sideway coefficient),

from surface texture data [Taneerananon (1981) and Yandell et.al (1983)], has been

further improved and established. The Yandell-Mee Texture Friction Meter Mark II

[Yandell and Sawyer (1992)] is at least as reliable as direct measuring methods.

Above all, the theoretical method would yield values of coefficient of friction which

have less degree of difficulty and potential errors associated with the field

measurement by actual skid tests.

250

CHAPTER 7. CONCLUSIONS

7 .1. Theoretical Findings

7 .1.1. Types of Tyre Stiffness

For both longitudinal and lateral directions, there are two types of tyre stiffness:

force versus slip stiffness (i.e. Cx = dFxfdSx at Sx = 0, and Cy = dF/da at a = 0),

and force versus deformation stiffness (i.e. Cx = 0.5 kx(21)2w, and Cy = 0.5 ky(21)2w).

The equality between the slip stiffness and the deformation stiffness has been

mathematically proved elsewhere (see Appendix G).

Two problems were found, however, with the practical application of tyre

stiffness. The first problem is the value of slip or slip angle to be used in the

calculation of slip stiffness, so that the dimension of slip stiffness will be in force,

similar to the dimension of deformation stiffness and the dimension of frictional

forces. The second problem is that the quantitative relation between the slip stiffness

and the deformation stiffness may show the two to be unequal.

On the first problem, it was found that the assignment of 100% longitudinal slip

and 9()° slip angle into the slip stiffness calculation, generally gives satisfactory

results in terms of the agreement between the measured and the predicted values of

frictional forces (see also Section 7 .2.10). Other workers used empirical and lesser

values of slip stiffness (see Section 2.4.6).

On the second problem, it was found that the difference in tyre motions (steady

state, quasi steady-state, non steady-state) will influence the magnitude of slip

stiffness (see Figure 6.27); while it was expected that the methods of deformation

measurement (without/near sliding, complete sliding) will influence the magnitude of

deformation stiffness. The candidate suggested the introduction of an empirical

factor to account for different types of tyre stiffness when comparing either slip

stiffnesses or deformation stiffnesses (see also Section 7.2.12). For example it can be

seen in Figure 6.27 that steady-state slip stiffness is greater than transient slip

stiffness.

251

7 .1.2. Description of Friction Coefficient

The candidate deduced from the literature two types of friction coefficient: the

available (or nominal) friction coefficient (µ) which can be resolved into the

longitudinal (µJ and lateral (µJ components (see Equation 2.59), and the frictional

force coefficients (µx, µy).

The available friction coefficients (Jls, µJ has been assumed to be linear functions

of sliding speed or slips (see Equations 2.57-2.58), while the frictional force

coefficient is zero at zero slip (see Figures 2.17-2.18). At slips beyond the critical

value, i.e. complete sliding without any elastic deformation, the frictional force

coefficient equals the available friction coefficient. The available friction coefficient

when a = rc/2 is assumed to equal the locked-wheel coefficient (µd).

7 .1.3. The Role of Tyre Stiffness

Simulation using the candidate's tyre model suggests that the magnitude of the

tyre-road frictional force is significantly influenced by the stiffness of the tyre, both

in longitudinal (½;) and lateral (Cy) direction. In general, it means that the skid

resistance is not only the property of the road pavement. The tyre stiffness can also

be regarded as independent of pavement/lubricant conditions. This applies to both

types of stiffness: slip stiffness and deformation stiffness. (Available friction

influences frictional force for all percentages of slip; see Figure 2.19).

It was found ( see Figure 2.19, Tables 2. 7-2.8), that the direct contribution of tyre

stiffness to the frictional forces is in the area of below the critical percent slip (Sxcr)

and critical slip angle (Cler). The contribution from tyre stiffness decreases with

increasing percent slip or slip angle, whereas the contribution from friction

coefficient (µ) continues to increase.

The candidate has shown algebraically (see Section 2.6. 7) that beyond the critical

percent slip or slip angle, the tyre stiffness still contributes, indirectly, to the

frictional forces. The effect is through the steepness of decreasing frictional forces

beyond the critical percent slip or slip angle area. He also showed that the

252

magnitude and the position of critical percent slip and slip angle are also determined

by the tyre stiffness. The role of tyre stiffness then is diminished when the condition

of 100% slip (locked-wheel) or 90 degree slip angle was achieved.

7.1.4. Model for Tyre Stiffness Variation

Unlike most vehicle components whose mechanical properties do not vary with

operating conditions, it is found that the tyre stiffness is strongly influenced by basic

operating conditions such as: wheel load, inflation pressure, speed, and tyre type.

From the point of view of cost and practicability, it is desirable that the tyre stiffness

be predicted using empirical formula, instead of being measured at each condition.

Due to the lack of data and available information, however, it is not possible in

the present study to predict tyre stiffness (from tyre properties and operating

conditions). Instead, the candidate has predicted the variation of tyre stiffness. In

other words, the candidate attempted to construct a model (see Equations 2.32-2.37)

which will be able to predict the change of tyre stiffness for any set of operating

conditions, once the tyre stiffness for one set of operating conditions has been

measured (known).

The proposed empirical model (see Section 2.4.7), incorporates the significant

factors influencing the tyre stiffness for each tyre type: wheel load (Fz), inflation

pressure (IP), and tyre speed (V). Data for this model were gathering from the

literature survey. The factor of wear-rate is not included in the model, due to the fact

that the time duration, taken from one state of wear to another significant state of

wear, is considered long. Hence, each significant state of wear (e.g. new, half worn,

smooth) is regarded as a separate type of tyre.

7.1.5. The Shape of Tyre Pressure Distribution

Contact pressures between tyre and ground are not necessarily uniform, mainly

because of the tyre detail design and partly because of the road surface irregularities.

Based on the measured data from the literature, however, only the pressure

253

distribution across the width of a tyre is normally treated as uniform. To represent

the pressure distribution down the length of a tyre, some forms are available, such

as: elliptical, hyperbolic, and symmetric trapezoidal.

Certain factors are found to significantly influence the shape of pressure

distribution. Among the other things are: tyre type, wheel load, inflation pressure,

tyre speed, and mode of operation (braking, driving, cornering). It is also found that

the shape of pressure distribution is not necessarily symmetrical, due to the

influences of tyre speed and mode of operation.

The candidate has proposed a model for the shape of tyre pressure distribution in

a form of a trapezium, with contact length of 21, and slopes length of a and b. This

modelled pressure distribution is intended to offer flexibility in representing many

types of actual pressure distribution (as found in the literature review), by selecting

the appropriate value of its slope (a/21, b/21). In comparison to symmetric trapezoidal

[Bernard et.al (1977)], this model provides the user with one additional option of an

unsymmetric trapezoidal distribution. The maximum normal pressure over the

contact patch then is given by: P max = Fj{21-0.5(a+b) }w, where Fz is the vertical

(wheel) load, and w is the width of the contact patch. By taking a = b, the

symmetric trapezoidal distribution is obtained; whereas by taking a = b = 0, there

will be a uniform pressure distribution.

7 .1.6. Calculation of Frictional Forces

The frictional forces (Fx and Fy) of an automobile-tyre travelling over a road

surface, are believed to be dependent upon many factors [Holmes et.al (1972)] which

have, on each, various degrees of significance. These factors can be grouped into: (a)

operational, (b) tyre, (c) lubricant, and (d) pavement. By incorporating those factors,

an analytical model has been developed by the candidate, which enables the

calculation of frictional forces over all ranges of percent slip and slip angles.

The proposed model is based mostly on the classical analysis of the rolling tyre

by Fiala [1954]. Some of the general mathematical equations are derived in a similar

manner lo other investigators such as Dugoff et.al [1969, 1970], and Bernard et.al

254

[1977]. By using the differential approach, however, the candidate was able to

reduce the friction parameters required for predicting the frictional forces. The

benefit of the candidate's model will be shown in Section 7.3.

7.1.7. Theoretical Effect of Some Input Parameters on the Tyre-Road Friction

The effect of some input parameters (½, ey, a/21, b/21, µd, and Fz) are studied

theoretically using the candidate's model explained in Section 2.6. The ranges of

parameter were selected to represent the normal values which commonly occur in

practice. The variables which significantly affect the tyre-road friction were identified

by means of regression analysis.

(1) Effect of longitudinal stiffness (ex)

The main effect is the increase of BFe with increasing ex- The rate of increasing

BFe will be greater at low values of Sx and a. In addition, the magnitude of critical

longitudinal friction coefficient (Pscr) will also increase with increasing ex, and the

position of critical percent slip (Sxcr) will move into the lower value. The increase of

SFe with increasing ex is found to be insignificant. At a = D° there is no effect of

Cx on SFe.

(2) Effect of lateral stiffness (ey)

Similar to ex, the main effect is the increase of SFe with increasing eY. The rate

of increasing SFe will be greater at low values of a and Sx. Additionally, the

magnitude of critical lateral friction coefficient (µacr) will also increase with

increasing Cy, and the position of critical slip angle ( acr) will move to a lower

value. The increase of BFe with increasing eY is found to be insignificant. At Sx =

0% there is no effect of eY on BFe.

(3) Effect of increasing-zone of pressure distribution (a/21)

The main effect is the increase of BFe and SFe with increasing a/21 (decreasing

leading steepness, as shown in Figure 2.16). The rate of increasing BFe will be less

at extreme values of Sx, particularly at high values of Sx. Similarly, the rate of

increasing SFe will be less at extreme values of a, particularly at high values of a.

255

(4) Effect of decreasing-zone of pressure distribution (b/21)

The effect of varying b/21 on the BFC and SFC has been found to be nearly

small. The significant effect is obtained for SFC at low values of Sx and a., where

the SFC decreases with increasing b/21 (decreasing trailing steepness, Figure 2.16).

(5) Effect of locked-wheel BFC (µd)

The main effect is that the BFC and SFC, as well as the magnitude of critical

longitudinal friction coefficient (PsCI') and critical lateral friction coefficient (µa.a-),

will increase with increasing µd. The rate of increasing BFC will be less at low

values of Sx (and high values of a.). Similarly, the rate of increasing SFC will be less

at low values of a. (and high values of Sx). In addition, the position of critical

percent slip (Sxcr) and critical slip angle (a.er) will move into the higher value with

increasing Pd·

(6) Effect of wheel load (Fz)

The main effect is that the BFC and SFC will decrease with increasing Fz. The

rate of decreasing BFC will be less at extreme values of Sx. Similarly, the rate of

decreasing SFC will be less at extreme values of a..

7 .1.8. Theoretical Model Response Comparisons

In order to identify the effects of various assumptions made during derivation, the

candidate's model has been compared with other tyre models: HSRI-I, HSRl-11,

HSRI-III, Good Year, and Sakai. All tyre models were compared over a range of Sx

and Sy values covering braking at slip angles varying from zero to 16 degrees; and

by using the same input data for tyre stiffness and wheel load, approximately the

same friction coefficient, and different forms of pressure distribution (see Table 3.3).

In general, the candidate's model was found to be give median results relative to the

other model's results.

For the percent slip (Sx) versus braking force coefficient (µx) at five slip angles

(see Figure 3.43), the candidate's model shows a significant difference with HSRI-1

and Good Year models, and a small difference with HSRl-11, HSRI-ID, and Sakai

models. For the Sx versus Px at angle O (see Figure 3.46), the candidate's model

256

which recognises the possibility that a friction coefficient decreases with slip or slip

speed, shows a decreasing Px as Sx increases after the peak, while Sakai and Good

Year models show flat response after the peak. Probably these flat responses are

suitable for very rough surfaces, which show a slight increase of Px with increasing

speed, as found in literature review [Sabey et.al (1970)].

For the percent slip (Sx) versus sideway force coefficient (µy) at five slip angles

(see Figure 3.44), the candidate's model shows nearly the same result as the Sakai

model. For slip angle (a) versus µY at Sx = 0% (see Figure 3.47), the candidate's

model produces nearly the same curve as the Sakai model, and nearly the same

curve at smaller slip angles as the HSRI-111 and Good Year models.

For interaction between braking force coefficient (Px) and sideway force

coefficient (µy) at five slip angles (see Figure 3.45), the candidate's model shows a

similar result to that of Sakai's model in terms of the slight increase of µY with

increasing Px at low slip angles. However, the candidate's model differs significantly

in terms of the separation of the endpoints of the friction cur,we, wit.tit.he Sakai and

Good Year models which assume a constant coefficient of sliding friction.

7.2. Experimental Findings

7.2.1. Rectification of Experimental Problems

(1) Correction for Test Wheel Speed

It is known that the rolling radius, on which the recorded wheel speed of the

MMFIT (Multi Mode Friction Test Truck) was based, can be affected by many

factors, such as tyre size. Therefore, the recorded test wheel speed -..as considered as

an approximation only, and need of correction. A procedure has been set up for the

correction. The basic principle in obtaining the correction factor was that the speed of

the MMFIT over the measured section, was assumed to be e1ual with the wheel speed at

approaching the first catadioptric target (at zero slip angle, free rolling, and start of

brake activation).

257

For slip angles other than zero degree, another correction was also carried out.

First, the correction factor at zero slip angle (as obtained above) was applied to the

recorded test wheel speed, to obtain the corrected test wheel speed in the direction

ofthewheelplane. Second, this corrected test wheel speed then is divided by cosa

(where a slip angle), to obtain the corrected test wheel speed in the direction of

travel.

(2) Correction for Wheel Speed Delay

It was found from experiment that generally some time had elapse before the

wheel speed began to decrease, following a significant change of friction

coefficient. One possible source of this delay was the inaccuracy of the recording

wheel speed (the speed is based on 1 second duration, instead of an instant time or a

shorter duration). A procedure then was set up to estimate the correction needed

(Section 5.3.1). The result of the correction factor, however, was found to be too

small. A slightly greater correction then is seen to be a practical solution.

(3) The Locked Wheel Braking Force Coefficient of Friction

It was found that the locked-wheel BFC tended to decrease with increasing

duration of application (see Figures 5.2-5.5). Also a small increase of the

locked-wheel SFC was found with increasing duration of brake application. It has

been concluded, that the decrease of slip angle due to the looseness of wheel angle

wa&only a minor cause for decreasing the locked-wheel BFC during the full braking,

whereas the decreasing slip angle was seen to be the primary source for increasing the

locked-wheel SFC. A lower skid resistance in that particular section of road

contributed slightly to the decrease of locked-wheel BFC.

(4) Local Sliding During Free Rolling

It was found for slip angles other than zero, that the wheel speed (after

correction) on entering the measured section was lower than the average speed of the

258

vehicle over the measured section. One possible cause was the occurrence of local

sliding during free rolling (before brake activation). This may be due to the fact that

the wheel load during brake off (before brake activation and after brake release) was

significantly lower than the wheel load during brake on (see Figure 5.20).

With the difference in speed mentioned above, some slip by definition had

occurred, even though the brake was not yet activated. One possible explanation for

this slip was that the lateral forces due to slip angle creates a drag force through the

wheel axle, and then increases the frictions at the joints in the axle shaft.

7 .2.2. The Significance of the Variation of Tyre Pressure and Rate of Braking

The significance of the following effects was determined by analysis of variance

as described in Section 5.3.4. All the data was gathered by the candidate using the

Multi Mode Friction Test Truck (MMFTT) on a medium-textured bituminous road

surface (see Figure 4.1), with the conditions of: 30/20 psi tyre pressure, smooth tyre,

flooded water, and 10 mph average speed.

(1) Effect of Tyre Pressure

a. Braking Force Coefficient (BFC)

The main effect is the decrease of BFC with increasing tyre pressure. This effect

is significant for angles 0° and 10°, but insignificant for an angle of 15°. The

interaction effect of tyre pressure and percent slip was found to be insignificant. The

interaction effect of tyre pressure and the rate of braking is insignificant for angles

0° and 10°, but significant for angle 15°. The percentage of decrease of BFC with

increasing tyre pressure is greater at a slow rate of braking.

b. Sideway Force Coefficient (SFC)

Theoretically, there will be no effect of tyre pressure on SFC for angle Q°. The

result of the analysis variance for angle Q° therefore should be read as a mechanism

of deviation from ideal conditions (such as the symmetrical or vertical position of

the tyre to the pavement). For angles lQ° and 15° the effect of tyre pressurewasfound

to be insignificant. The interaction effect of tyre pressure and percent slip, and the

interaction effect of tyre pressure and the rate of braking were also found to be

insignificant.

259

(2) Effect of the Rate of Braking

a. Braking Force Coefficient (BFC)

The main effect was the increase of BFC with increasing rate of braking. This

effect was significant for angles 0°, 10°, and 15°. The interaction effect of the rate of

braking and percent slip was also significant for angles 00, 10°, and 15°. The

percentage of increase of BFC with increasing rate of braking is greater at the

middle percent slip. As mentioned before, the interaction effect of the rate of braking

and tyre pressure is insignificant for angles 0° and 100, but significant for an angle

of 15°. The percentage of increase of BFC with increasing rate of braking is

greater at high tyre pressure.

b. Sideway Force Coefficient (SFC)

The main effect was the decrease of SFC with increasing rate of braking. This

effect was significant for angles 100 and 15°, but insignificant (as theoretically

expected) for angle 00. The interaction effect of the rate of braking and percent slip

was significant for angles 00 (should be negligible) and 15°. The percentage of

decrease of SFC with increasing rate of braking is greater at the middle percent

slip. The interaction effect of the rate of braking and tyre pressure is significant for

angle 0° (should be negligible), and insignificant for angles 100 and 15°.

Similar results regarding the effect of the rate of braking could not be found in

the literature. It may be concluded that increasing the rate of brake application will

increase braking and decrease sideway force for the above conditions.

7.2.3. Verification of Tyre Slip Stiffness Variation Model

The resultsu5ingthecandidate's measured field data (Tables 6.2-6.3) show a good

agreement between the candidate's predicted and the measured values, either for

longitudinal or lateral tyre stiffness, with discrepancies less than 10%. In addition, it

seems that for even closer agreement, a factor involving of the rate of braking could

be incorporated in the longitudinal tyre stiffness variation model.

260

7.2.4. Comparison Between Measured and Theoretically Predicted Friction

It was found that for BFC the predicted values w-ere generally lower than the

experimental values at the middle-lower percents slip, but show a good agreement at

the middle and high percents slip. For SFC, on the other hand, the predicted values

show a good agreement only for high percents slip, but the predicted values are

higher than the experiment values for middle percents slip, and lower than the

experimental values for low percents slip.

Some of the above discrepancies may be due to some poorly controlled

experimental conditions such as: the looseness of the test wheel position, and the

delay of wheel speed measurement response. It was thought that closer agreement

could be achieved by using a smaller slip angles after a certain percent slip (say,

after Sx ~ 30%) on theoretical prediction, and by using a greater correction of the

wheel speed delay (say, 7 x 0.107 second) on the measured data.

The above improvements, however did not affect the discrepancies of the SFC

versus slip angle at zero percent slip. In other words, the lateral stiffness was only

slightly affected by this improvement, and so the theoretical values of SFC versus

slip angle at zero percent slip remain lower than the measured values. A

modification of the prediction model is thus required for the input values of lateral

stiffness. The effect of this will be discussed later.

7.2.5. Comparison of Sakai's Laboratory Data With the Candidate's Theoretical

Prediction of Friction

It was found that the predicted values, especially for data of Fx versus Sx, are

generally in a good agreement with the laboratory data. In most cases, the predictive

values with a/21 = 0.06-0.10 can be used for low wheel loads (Fz), whereas a/21 =

0.06-0.08 are suitable for higher wheel load. The selection of a/21 was seen to be

more sensitive with increasing the tyre stiffness (both longitudinal and lateral). For

the data of FY versus slip angle (ex), particularly for lower values of slip angle, the

predictive values under estimated the laboratory data. This is seen in Figures

261

6.14-6.19. The zero and 100% slip values of friction in the prediction were made equal to

Sakai's laboratory values.

7.2.6. The Effect of Modifying the Quantification of Lateral Stiffness in the

Candidate's Friction Prediction

The candidate has used the laboratory data of Sakai [1982.], in which all factors

are assumed to be in fully controlled conditions, to examine his model.

The predicted values of lateral forces (Fy) are less than both the candidate's

experimental results and Sakai's laboratory data. A modification of the candidate's

model seemed to be a necessity. This has been done by using a higher lateral tyre

stiffness (Cy) in the prediction.

In the modified model, the input of lateral stiffness (Cy) is force/1.57 radians (or

force/ 90°), instead of force/radian (or force/57.3°) as in original model. This

modification did not influence the predicted values of longitudinal force (Fx) versus

Sx at angle 0°. However, by using the modified model, a closer agreement between

the predicted and the measured values was achieved.

This modified model also produced more consistent or similar results between FY

versus Cl at Sx = 0 (see Figure 6.20) and Fx versus Sx at ex = 0° (see Figures 6.14,

6.16, and 6.18). In addition, the full range of ex (00 S ex S 90°), as a result of

modification, has an analogy with the full range of Sx (0% S Sx S 100%) within the

model.

7.2. 7. Independence of Tyre Stiffness from Road Conditions

It was found, that the slip type tyre stiffness obtained from the main experiment

can be used to predict the BFC and SFC on other pavements (i.e. slippery road) with

reasonably good agreement to the experimental data provided that the locked-wheel

BFC or maximum SFC are measured on the other road. The experimental data on

this slippery road were gathered by Multi Mode Friction Test Truck (MMFTT) with

262

the same tyre and operational conditions used in the previous pavement experiments

(main experiment).

7 .2. 8. The Relationship Between Slip Stiffness and Deformation Stiffness

It was found that there are some discrepancies between the values of tyre

stiffness obtained from the field (i.e. slip measurement) and from the laboratory (i.e.

deformation measurement). The discrepancies have been attributed to the factors of:

accuracy of measurement (number of points, corrections, and inconsistencies),

difference on tyre motions (steady-state, quasi steady-state, and non steady-state),

different methods of deformation measurement (without sliding, and with complete

sliding), and the inclusion of the carcass spring rates instead of the tread element

spring rates. It should be noted that the two types of stiffness have been proved

theoretically equal at zero friction force (see Appendix G).

Providing that the tread element spring rates (kx, ky) can be measured (obtained),

it has been suggested that for steady-state conditions both the tyre stiffness measured

from slip and deformation (with complete sliding) are nearly equal, and can be

applied for predicting the frictional forces. For quasi steady-state or non steady-state

behaviour, however, while slip stiffness gives an accurate prediction of friction

measurement the use of deformational stiffness over predicts measured friction. A

possible explanation is the effect of non-homogeneous shear resistance between the

tyre and road surface during transient sliding. Hence, for other than steady-state

motions, a lower value of deformation stiffness is required to predict the frictional

forces. For deformation measurement without sliding, a higher value of deformation

stiffness is required to predict the frictional forces accurately. (See also Section

2.4.5).

The introduction of an empirical factor for deformation stiffness, to account for:

i) other than steady-state motions, ii) other than measurements with complete sliding,

iii) other than measurements of tread element spring rates, appears to be a practical

solution.

263

7 .3. Benefit of the Model

The candidate's tyre model for calculating the tyre-road frictional forces has the

following main benefits.

1. The model requires one friction parameter less than that required by the semi

empirical model developed by Bernard et.al [1977] or the analytical model

developed by Gim and Nikravesh [1990, 1991]. Hence, the time or cost

associated with the measurements (and calculation) of the friction parameters can

be reduced.

2. The model is able to calculate frictional forces from tyre stiffness (as

representing the factors of operation and of tyres), and either locked-wheel BFC

or maximum SFC (as representing the factors of operation, of tyres, of lubricant,

and of the pavement). (See Figures 7.1-7.2). For all calculations of frictional

forces, the secondary inputs needed are: the assumed shape of pressure

distribution (a/21, b/21), and the applied wheel load (Fz).

3. The candidate's model, with a higher lateral stiffness, is more precise than others

since the true stiffness at zero frictional force is used rather than an empirical

and lesser value. (See Section 2.4.6).

4. Since the locked-wheel BFC (or maximum SFC) can be predicted from surface

texture measurements [Yandell et.al (1983)], this model can be used to predict

the tyre-road frictional forces for various degrees of slip and slip angle (including

critical or peak friction), merely from tyre stiffness and surface texture data.

5. Since the equipment for measuring the locked-wheel BFC or maximum SFC is

commonly available (such as the trailer used on ASTM-E274, and SCRIM), the

data from this equipment can be processed directly to obtain the tyre-road

frictional forces for all ranges of percent slip (Sx) and slip angle (a). That is the

behaviour of combined braking and cornering friction can beeasily predicted, as

shown in Figure 7 .2.

264

D

I predicted from

the model from c., locked-w hee I BF C

locked-wheel BFC [measured)

~ Slip (5.c) 100% 0

c, (measured) predicted from

the model /

max SFC (alternative input)

SFC at 90° = locked-wheel BFC

[measured)

Slip angle (oc) 90°

Fig. 7 .1. Inputs & outputs of the candidate's model under either braking or cornering. Left~ Input: ex, locked-wheel BFe. Output: Fx vs Sx (at angle 00). Right~ Input: ey, locked-wheel BFe (or max SFe). Output: FY vs a (at Sx 0%).

Input: c., c,, locted-wheel BFC [or max SFC)

predicted from / the model

Input: Cx, c,, locked-wheel BFC (or m11x SFC)

predicted from / the model

~ ~ Fig.7.2. Inputs & outputs of the candidate's model under both braking and cornering. Left~ Input: Cx, Cy, locked-wheel BFe (or max SFe). Output: FY vs Fx (various Sx). Right~ Input: ex, ey, locked-wheel BFe (or max SFe). Output: FY vs Fx (various a).

265

7.4. Summary of Conclusions

1. It was found that the application of slip stiffness (i.e. ex = dF xfdSx at Sx = 0, and

Cy = d.F/da at a= 0) using 100% longitudinal slip and 90° slip angle, generally

gives satisfactory results in terms of the agreement between the measured and the

predicted values of frictional forces. The slip stiffness used is the tangent of the

slope of the friction versus slip curve at zero friction and not some empirical

lesser value used by others.

It was expected that the values of deformation stiffness (i.e. ex = 0.5 kx(21)2w,

and Cy = 0.5 ky(21)2w) will be influenced by the methods of deformation

measurement (without/near sliding, complete sliding). It was determined

experimentally that the equality between the deformation stiffness and the slip

stiffness was also influenced by the tyre motions (steady-state, quasi steady-state,

non steady-state).

2. The available friction coefficient (µ), which can be resolved into the longitudinal

and lateral components (p5, µJ, has been expressed as a function of slip. At slips

beyond the critical value, the frictional force coefficients (µx, µy) are equal to the

available friction coefficients. At a = rt/2 the friction coefficient is assumed to

equal the locked-wheel coefficient (µd).

3. The candidate deduced mathematically from hi~ predic:tion model, with inconclusive

evidence from field measurement, that the direct contribution of tyre stiffness to

the frictional forces is in the region of below the critical percent slip (Sxcr) and

critical slip angle (acJ Beyond this area, the tyre stiffness still contributes

indirectly, and diminishes when the condition of 100% slip (locked-wheel) or 90°

slip angle was achieved. See al&o Figure 2.19 and Tables 2.7-2.8.

Using the data from the literature survey, the candidate proposed an empirical

model which will be able to predict the change of tyre stiffness for any set of

operating conditions, once the tyre stiffness for one set of operating conditions

has been measured (known). The equation~ (Eq~.2.~2-2.37) are ~et out in Section 2.4.7. The

results using the candidate's measured field data confirm a good agreement between

the predicted and the measured values of tyre stiffness.

266

4. The candidate has proposed a modification to an existing model for the shape of

tyre pressure distribution in a form of trapezium, with contact length of 21, and

slope length of a and b. This model offers flexibility in representing many types

of actual pressure distribution, by selecting the appropriate value of its slope

(a/21, b/21). It was found using Sakai's laboratory data that the selection of a/21 is

more sensitive with increasing tyre stiffness.

5. An analytical model has been developed by the candidate, which enables the

calculation of frictional forces over all ranges of percent slip and slip angle. The

ability to calculate frictional forces from tyre stiffness and either locked-wheel

BFC or maximum SFC, is a feature of the candidate's model. In general, the

candidate's model is found to be in the median position when the model

response was compared with 5 tyre models: HSRI-I, HSRI-11, HSRI-III, Good

Year, and Sakai. The way of u~ing the candidate's model and its example are presented in Sections 2.6.6-2.6.8.

Since the locked-wheel BFC (or maximum SFC) can be predicted from surface

texture measurement [Yandell et.al (1983)], this model can be used to predict the

frictional forces, from tyre stiffness and surface texture data. With the availability

of equipment for measuring the locked-wheel BFC or maximum SFC, the data

from this equipment can be processed directly to obtain the frictional forces for

all ranges of percent slip (Sx) and slip angles ( ex). See al~o Figures 7.1 - Z Z.

6. The effect of some input parameters (Cx, CY' a/21, b/21, µd, and Fz) has been

studied using the candidate's model. Almost all parameters have significant

effects on the BFC and SFC, except for b/21 which is found to be small.

7. A procedure has been set up for correcting the test wheel speed of Multi Mode

Friction Test Truck (MMFTT) by using: the calculated vehicle speed (over the

measured section), the recorded test wheel speed (at approaching the first

catadioptric target), and the slip angle used.

Another procedure has also been set up to estimate the correction needed for the

wheel speed delay, by using the data of the decreasing wheel speed with time. A

slightly greater correction than when using this procedure was required to

improve the estimation.

267

8. It was found that during the fully braking period the magnitude of locked-wheel

BFC tends to decrease with time.

9. The effect of tyre pressure and the rate of braking have been studied by means

of analysis of variance. The BFC decreased with increasing tyre pressure,

whereas the effect of tyre pressure on SFC was found to be insignificant. The

BFC increased with increasing the rate of braking, whereas the SFC decreased

with increasing the rate of braking.

10. In general, the results from the candidate's experiment and Sakai's laboratory

data, show a good agreement between the measured and predicted values.

While two lesser-controlled factors were involved in the candidate's

experiment, a closer agreement can be achieved by using a smaller slip angle

(after certain percent slip) on the theoretical prediction, and by using a greater

correction (for wheel speed delay) on the candidate's field measured data.

For both candidate's experimental results and Sakai's laboratory data, the

predicted values of SFC (versus slip angle) at zero percent slip, was found to

under estimate the measured values. A closer agreement can be achieved by

using the modification in the input of the candidate's tyre model, so the input

values of Cy is force/1.57 radians (or force/90°) instead of force/radian (or

force/57.3°) as in the original input of the candidate's tyre model.

11. It was found that the magnitude of field tyre stiffness (i.e. slips measurement) is

practically independent of road conditions, and gives an accurate prediction of

friction measurement.

268

7.5. Future Work

1. Gathering more information on the factors influencing the tyre stiffness. The

results then can be used for further refinements of the model or empirical factor

for tyre stiffness variation. It is equally important, to develop the model for the

prediction of tyre stiffness (from other tyre properties or related parameters).

2. Gathering more information on the factors influencing the shape of tyre pressure

distribution. The results then can be used for selecting the appropriate shape of

pressure distribution (a/21, b/21) for each condition.

3. Carry out extensive field tests on different types of pavement at different speeds,

to further examine the model of frictional forces calculation. The results then can

be used for establishing the model, or (where appropriate) to derive a set of

empirical factors in theoretical prediction.

4. Extend the research on predicting the tyre-road friction under braking and

cornering, from tyre stiffness and locked-wheel BFC (or maximum SFC), in

which the latter parameter is obtained from the surface texture data.

269

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180. Tabor, D. (1952), "The Mechanism of Rolling Friction", Philosophical Magazine, Serial 7, Vol. 43, No. 345, pp 1055-1059.

181. Tabor, D. (1959), "The Importance of Hysteresis Losses in the Friction of Lubricated Rubber", Proc. 1st Int. Skid Prevention Conf., pp 211-218.

182. _Taneerananon, P. (1981), "The Analysis of the Mechanism of Tyre Friction on Wet Roads", Ph.D Thesis, The University of New South Wales.

183. Taneerananon, P. (1982), Confidential Report. (unpublished).

184. Taneerananon, P. and Yandell, W.O. (1981), "Microtexture Roughness Effect on Predicted Road-Tyre Friction in Wet Conditions", Wear, Vol. 69, pp 321-337.

185. Teller, L.W. and Buchanan, J.A. (1937), "Determination of Variation of Unit Pressure over the Contact Area of Tires", Public Roads, Dec., pp 195-198.

186. (van Eldik) Thieme, H.C.A., Dijks, A.J. and Bobo, S. (1982), "Measurement of Tire Properties", Mechanics of Pneumatic Tires, pp 542-613.

281

187. Thirion, P. (1946), "Coefficients of Adhesion of Rubber", Rev. Gen. Caoutch, Vol. 23, p 101.

188. Tielking, J.T. and Mital, N.K. (1974), "A Comparative Evaluation of Five Tire Traction Models", Report No. UM-HSRI-PF-74-2, Univ. of Michigan, pp 1-162.

189. Tielking, J.T. and Schapery, R.A. (1980), "Calculation of Tire-Pavement Shear Force", ASME Symp. Proc., The General Problem of Rolling Contact, ASME AMO, Vol. 40, pp 19-39.

190. Tielking, J.T., Fancher, P.S. and Wild, R.E. (1973), "Mechanical Properties of Truck Tires", SAE Paper, No. 730183, pp 1-11.

191. Trollope, D.H. and Wallace, K.B. (1962), "The Nature of Skid Resistance", Proc. ARRB, Vol. I, pp 779-791.

192. Uffelmann, F. (1983), "Automobile Stability and Handling Dynamics in Cornering and Braking Maneuvers", Vehicle System Dynamics, Vol. 12, pp 203-223.

193. 'Uniken' (1989), "Friction Meter Device: A Boon for Road Safety", UNSW News, No. 274, pp 1-2.

194. U.S. Army Engineer Waterways Experiment Station, (1964), "Stresses Under Moving Vehicles - Report 4", Technical Report No. 3-545.

195. Veith, A.G. (1971), "Measurement of Wet Cornering Traction of Tires", Rubber Chemistry and Technology, Vol. 44, pp 962-995.

196. Veith, A.G. (1977), "Tire Wet Traction Performance: The Influence of Tread Pattern", Skidding Accidents, TRR, No. 621, pp 113-125.

197. Wallace, K.B. and Trollope, D.H. (1969), "Water Pressure Beneath a Skidding Tyre", Wear, Vol. 13, pp 109-118.

198. Yandell, W.O. (1969), "The Effect of Surface Geometry on the Lubricated Sliding Friction and Polishing of Roadstones", Aust. Rd. Res., Vol. 3, No. 10, pp 50-68.

199. Yandell, W.O. (1970), "Measurement of Surface Texture of Stones with Particular Regard to the Effect on the Frictional Properties of Road Surfaces", Ph.D Thesis, The University of New South Wales.

200. Yandell, W.O. (1971a), "The Use of Mechano Lattice Analogy for Determining the Abrading Stresses in Sliding Rubber", Rubber Chemistry and Technology, Vol. 44, pp 758-770.

201. Yandell, W.O. (1971b), "A New Theory of Hysteretic Sliding Friction", Wear, Vol. 17, pp 229-244.

282

202. Yandell, W.O. (1974), "The Relation Between the Stress Saturation of Sliding Rubber and the Load Dependence of Road Tyre Friction", The Physics of Tire Traction - Theory and Experiment, Eds. D.F.Hays and AL.Browne, pp 311-323.

203. Yandell, W.O. and Gopalan, M.K. (1976), "The Relation Between Surface Texture of Roads and Friction and Abrasion of Tyre Tread Rubber", ARRB Proceedings, Vol. 8, pp 8-14.

204. Yandell, W.O. and Holla, L; (1974), "The Prediction of the Coefficient of Friction from Surface Texture Measurement", Proc. ARRB, Vol. 7, Pt. 7, pp 354-363.

205. Yandell, W.O. and Sawyer, S. (1992), Report to Roads and Traffic Authority of NSW. (unpublished).

206. Yandell, W.O., Taneerananon, P. and Zankin, V. (1983), "Prediction of Tire-Road Friction from Surface Texture and Tread Rubber Properties", Frictional Interaction of Tire and Pavement, ASTM, STP 793, pp 304-322.

207. Yeager, R.W. (1974), "Tire Hydroplaning: Testing, Analysis, and Design", The Physics of Tire Traction - Theory and Experiment, Eds. D.F. Hays and A.L. Browne, pp 25-64.

208. Yeaman, J. (1989), "Pavement Maintenance Management for Local Government", Pavement Management Service Pty. Ltd., Dynatest PMS.

209. Yeomans, K.A. (1976), "Statistics for the Social Scientist: 2 Applied Statistics", Penguin Education, Great Britain.

210. Yeow, S.H., El-Sherbiny, M. and Newcomb, T.P. (1978), "Thermal Analysis of a Tyre During Rolling or Sliding", Wear, Vol. 48, pp 157-171.

211. Zakaharov, S.P. and Novopol'skii (1957), "Distribution of Specific Pressure of a Tire on the Road at High Velocities", Trudy Tauchno-Issledovatel'skoga Instituta Shinnoi Pronyschlennosti, Sbomik 3, Methody Rascheta i Ispaytania Automobil'nykh Shin: U.S.S.R. Nauchno, pp 139-153.

212. Zankin, V.G. (1981), "Determination of Damping Behaviour of Viscoelastic Substrates from High Speed Rolling Friction Tests", Ph.D Thesis, The University of New South Wales.

283

A-1

Appendix A

100 REM BASIC COMPUTER PROGRAM FOR CALCULATION OF 110 REM FRICTIONAL FORCES USING LOCKED-WHEEL BFC 120 REM CONSTANT: A a/21 B = b/21 C = Cx (force/slip) 130 REM D Cy (force/rad), or D (mod.) = Cy (force/1.57rad) 135 REM E µd F = F z (force) 140 REM VARIABLE: G Sx H = a (rad) 150 REM**************************************************************

160 READ A, B, C, D, E, F 170 LET P A*(2-A-B) 180 LET N = 2-A+A*B*0.5+B*B*0.5-2*B 190 LET G = (E*F*(N-P)/C+P*P*0.5)/(E*F*(N-P)/C+P*N-P*P) 195 IF A= 0 THEN 213 200 LET IA= P*C*G/(F*(l-G)) 210 LET L = (IA-E*G)/(1-G) 211 GOTO 216 213 LET IA= E 214 LET L = E 216 PRINT "a/21 = ";A, "b/21 = ";B, "F.= ";F 218 PRINT "Cx = ";C, "Cy = ";D, "µd = ";E 22 0 PRINT "Sxcr = "; G, "µscr = "; IA, "µ. 0 = "; L 225 IF A= 0 THEN 304 230 FOR R OTO 1.57 STEP 0.001 240 LETS= 1.5708-2*SIN(R)*COS(R)*(N/P-1)

+2*COS(R)*COS(R)*E*F*(N/P-1)/(D*P)-R 250 IF ABS(S) < 0.1 THEN 270 260 NEXT R 270 FOR H R-0.0005 TO 1.57 STEP 0.00001 280 LETO= l.5708-2*SIN(H)*COS(H)*(N/P-l)

+2*COS(H)*COS(H)*E*F*(N/P-l)/(D*P)-H 290 IF ABS(O) < 0.001 THEN 310 300 NEXT H 302 GOTO 310 304 LETH= 1.5708 305 LET IB = E 306 LET M = E 308 GOTO 330 310 LET IB = P*D*TAN(H)/F 320 LET M = (IB-E*H*.636619772)/(1-H*.636619772) 330 PRINT "<lcr 340 PRINT"

= ";H,

F x'', " "~r = ";IB, "~0 = ";M

Fy", " µx", " ~", " 350 FOR H = .06981317 TO 1.57 STEP .06981317 360 LET K = M-(M-E)*H*.636619772 370 LET YA= l/(1+(2*B-A*B-B*B)*D*TAN(H)/(K*F)) 380 IF YA>= (1-B) THEN 520 390 LET YB = K*F/((2-A-B)*D*TAN(H)) 400 IF YB >= (1-B) THEN 420 410 IF YB >= A THEN 470 420 LET RC 0 425 LET CR RC/F 430 LET SC K*F

435 LET CS= SC/F 440 LET RSC= SQR(CR*CR+CS*CS) 450 PRINT RC, SC, CR, CS, RSC 460 GOTO 560

LET RB 0 LET BR RB/F

A-2

470 475 480 485 490

LET SB D*TAN(H)*YB*YB+K*F*(l-.5*B-YB)/(1-.5*A-.5*B) LET BS SB/F LET RSB = SQR(BR*BR+BS*BS)

500 PRINT RB, SB, BR, BS, RSB 510 GOTO 560 520 LET RA 0 525 LET AR RA/F 527 IF B = 0 THEN 532 530 LET SA D*TAN(H)*YA*YA+K*F*(l-2*YA+YA*YA)/(2*B-A*B-B*B) 531 GOTO 535 532 LET SA= D*TAN(H) 535 LET AS= SA/F 540 LET RSA= SQR(AR*AR+AS*AS) 550 PRINT RA, SA, AR, AS, RSA 560 NEXT H 570 FOR G 580 590 600 610 620

FOR LET LET LET LET

H J K

I XA =

.1 TO .99 STEP .1 0 TO 1.57 STEP 0.06981317 L-(L-E)*G M-(M-E)*H*.636619772 J+(K-J)*ATN(TAN(H)/G)*.636619772

l/(1+(2*B-A*B-B*B) *(SQR(C*C*G*G+D*D*TAN(H)*TAN(H)))/(I*F*(l-G)))

630 IF XA >= (1-B) THEN 735 640 LET XB = I*F*(l-G)/((2-A-B)*(SQR(C*C*G*G+D*D*TAN(H)*TAN(H)))) 650 IF XB >= (1-B) THEN 660 655 IF XB >= A THEN 700 660 LET PC I*F*COS(ATN(TAN(H)/G)) 665 LET CP PC/F 670 LET QC I*F*SIN(ATN(TAN(H)/G)) 675 LET CQ QC/F 680 LET PQC = SQR(CP*CP+CQ*CQ) 685 PRINT PC, QC, CP, CQ, PQC 690 GOTO 770 700 LET PB

705 LET BP 710 LET QB

715 LET BQ

C*G*XB*XB/(1-G)+I*F*COS(ATN(TAN(H)/G)) *(1-.5*B-XB)/(1-.5*A-.5*B) PB/F D*TAN(H)*XB*XB/(1-G)+I*F*SIN(ATN(TAN(H)/G)) *(1-.5*B-XB)/(1-.5*A-.5*B) QB/F

720 LET PQB = SQR(BP*BP+BQ*BQ) 725 PRINT PB, QB, BP, BQ, PQB 730 GOTO 770 735 IF B = 0 THEN 743 740 LET PA= C*G*XA*XA/(1-G)+I*F*COS(ATN(TAN(H)/G))

*(1-2*XA+XA*XA)/(2*B-A*B-B*B)

741 743 745 747

GOTO 745 LET PA LET AP IF B = 0

C*G/ (1-G) PA/F THEN 753

A-3

750 LET QA D*TAN(H)*XA*XA/(1-G)+I*F*SIN(ATN(TAN(H)/G)) *(1-2*XA+XA*XA)/(2*B-A*B-B*B)751 GOTO 755

753 LET QA D*TAN(H)/(1-G) 755 LET AQ QA/F 760 LET PQA = SQR(AP*AP+AQ*AQ) 765 PRINT PA, QA, AP, AQ, PQA 770 NEXT H 780 NEXT G 785 FOR H 0 TO 1.57 STEP 0.06981317 787 LET K = M-(M-E)*H*.636619772 788 LET I= E+(K-E)*ATN(TAN(H)/G)*.636619772 790 LET PD= I*F*COS(H) 795 LET DP PD/F 800 LET QD I*F*SIN(H) 805 LET DQ QD/F 810 LET PQD = SQR(DP*DP+DQ*DQ) 815 PRINT PD, QD, DP, DQ, PQD 820 NEXT H 830 STOP 840 DATA 0.06, 0.2, 10000, 12605, 0.87, 1100 850 END

B-1

Appendix B

100 REM BASIC COMPUTER PROGRAM FOR CALCULATION OF 110 REM FRICTIONAL FORCES USING MAXIMUM SFC 120 REM CONSTANT: A= a/21 B = b/21 C = Cx(force/slip) 130 REM D = Cy (force/rad), or D (mod.) = Cy (force/1.57rad) 135 REM IB = µ._,,r F = Fz (force) 140 REM VARIABLE: G = Sx H = a (rad) 150 REM************************************************************** 160 READ A, B, C, D, IB, F 170 LET P = A*(2-A-B) 180 LET N = 2-A+A*B*0.5+B*B*0.5-2*B 185 IF A= 0 THEN 216

ATN (F*IB/ (P*D)) 190 LET R 200 LET E = (R-1.5708+2*SIN(R)*COS(R)*(N/P-1))

*0.5*D*P/(COS(R)*COS(R)*F*(N/P-1)) 210 LET M (IB-E*R*.636619772)/(1-R*.636619772) 211 GOTO 216 212 LET R 1.5708 213 LET E = IB 214 LET M = IB 216 PRINT "a/21 = ";A, "b/21 = ";B, Fz = ";F 218 PRINT "Cx = ";C, "Cy = ";D, "µacr = "; IB 220 PRINT "acr = ";R, "µd = ";E, "µ..0 = ";M

230 LET G = (E*F*(N-P)/C+P*P*0.5)/(E*F*(N-P)/C+P*N-P*P) 235 IF A= 0 THEN 253 240 LET IA= P*C*G/(F*(l-G)) 250 LET L = (IA-E*G)/(1-G) 251 GOTO 260 253 LET IA= IB 254 LET L = IB

260 PRINT "Sxcr = ";G, "µscr = ";IA, "µso= ";L 340 PRINT" F " " X I F " " y , µ " " X I

350 FOR H = .06981317 TO 1.57 STEP .06981317 360 LET K = M-(M-E)*H*.636619772

~", "

370 LET YA= 1/(1+(2*B-A*B-B*B)*D*TAN(H)/(K*F)) 380 IF YA>= (1-B) THEN 520 390 LET YB = K*F/((2-A-B)*D*TAN(H)) 400 IF YB >= (1-B) THEN 420 410 IF YB >= A THEN 470 420 0 LET RC 425 430 435 440

LET LET LET LET

CR SC CS RSC=

RC/F K*F SC/F

SQR(CR*CR+CS*CS) 450 PRINT RC, SC, CR, CS, RSC 460 GOTO 560 470 LET RB 475 LET BR 480 LET SB 485 LET BS

0 RB/F D*TAN(H)*YB*YB+K*F*(l-.5*B-YB)/(1-.5*A-.5*B) SB/F

490 LET RSB = SQR(BR*BR+BS*BS)

500 PRINT RB, SB, BR, BS, RSB 510 GOTO 560 520 LET RA= 0525 LET AR= RA/F 527 IF B = 0 THEN 753

B-2

530 LET SA= D*TAN(H)*YA*YA+K*F*(l-2*YA+YA*YA)/(2*B-A*B-B*B) 531 GOTO 535 532 LET SA= D*TAN(H) 535 LET AS= SA/F 540 LET RSA= SQR(AR*AR+AS*AS) 550 PRINT RA, SA, AR, AS, RSA 560 NEXT H 570 FOR G 580 FOR H 590 LET J 600 LET K 610 LET I

.1 TO .99 STEP .1 0 TO 1.57 STEP 0.06981317 L-(L-E)*G M-(M-E)*H*.636619772 J+(K-J)*ATN(TAN(H)/G)*.636619772

620 LET XA = l/(1+(2*B-A*B-B*B) *(SQR(C*C*G*G+D*D*TAN(H)*TAN(H)))/(I*F*(l-G)))

630 IF XA >= (1-B) THEN 735 640 LET XB = I*F*(l-G)/((2-A-B)*(SQR(C*C*G*G+D*D*TAN(H)*TAN(H)))) 650 IF XB >= (1-B) THEN 660 655 IF XB >= A THEN 700 660 LET PC= I*F*COS(ATN(TAN(H)/G)) 665 LET CP PC/F 670 LET QC I*F*SIN(ATN(TAN(H)/G)) 675 LET CQ QC/F 680 LET PQC = SQR(CP*CP+CQ*CQ) 685 PRINT PC, QC, CP, CQ, PQC 690 GOTO 770 700 LET PB

705 LET BP 710 LET QB

C*G*XB*XB/(1-G)+I*F*COS(ATN(TAN(H)/G)) *(l-.5*B-XB)/(1-.5*A-.5*B) PB/F D*TAN(H)*XB*XB/(1-G)+I*F*SIN(ATN(TAN(H)/G)) *(1-.5*B-XB)/(1-.5*A-.5*B)

715 LET BQ QB/F 720 LET PQB = SQR(BP*BP+BQ*BQ) 725 PRINT PB, QB, BP, BQ, PQB 730 GOTO 770 735 IF B = 0 THEN 743 740 LET PA= C*G*XA*XA/(1-G)+I*F*COS(ATN(TAN(H)/G))

*(1-2*XA+XA*XA)/(2*B-A*B-B*B) 741 GOTO 745 743 LET PA 745 LET AP 747 IF B = 0 750 LET QA

C*G/(1-G) PA/F THEN 753 D*TAN(H)*XA*XA/(1-G)+I*F*SIN(ATN(TAN(H)/G)) *(l-2*XA+XA*XA)/(2*B-A*B-B*B)

751 GOTO 755 753 LET QA = D*TAN(H)/(1-G) 755 LET AQ = QA/F 760 LET PQA = SQR(AP*AP+AQ*AQ)

765 PRINT PA, QA, AP, AQ, PQA 770 NEXT H 780 NEXT G

B-3

785 FOR H = 0 TO 1.57 STEP 0.06981317 787 LET K = M-(M-E)*H*.636619772 788 LET I= E+(K-E)*ATN(TAN(H)/G)*.636619772 790 LET PD 795 LET DP 800 LET QD 805 LET DQ

I*F*COS(H) PD/F I*F*SIN(H) QD/F

810 LET PQD = SQR(DP*DP+DQ*DQ) 815 PRINT PD, QD, DP, DQ, PQD 820 NEXT H 830 STOP 840 DATA 0.06, 0.2, 10000, 12605, 0.9324, 1100 850 END

C-1

Appendix C

PROCEDURE TO OPERATE STRAIN GAUGE AMPLIFIER

1 . Power Supply

a. When operating from 24V power supply, set small power switch to down position (large power switch can be in any position).

b. When operating from 240V main supply, set small power switch in upward position and use large power switch. The red power indicator should come in either case.

2. Balancing Strain Gauge Bridges

a. The strain gauge bridges must be balanced before use. To do this, first select the channel to be balanced (1 through 6).

b. On the appropriate channel select the required attenuation (topmost control) before going on to the next step. Notes: channel 1 [WHEEL LOAD] should be set to 21 (i.e. position between 18 and 24), channel 2 [SIDEWAY FORCE] set to 3 (i.e. position between O and 6), and channel 3 [BRAKE FORCE] set to 12.

c. Select R on the function switch (rotary switch at bottom of panel), then return control marked R so that a zero reading on the meter is read.

d. Set the function switch to Rx and turn the control marked X so that a "null" reading is obtained, that is, the point at which the needle swings closest to the zero position, note that it doesn't have to actually read zero in this case.

e. Switch back to R position on the function switch and rebalance as in step (c).

f. This completes the balancing procedure for one channel, leave the function switch in the R position for operation. Each channel being used should be balanced in this way.

3 . Operation

a. To start reading, ensure all channel function switches are set to the R position.

b. Make sure that the channel select switch (far right hand side) is set to an unused, or to the SET TO MARK position.

D-1

Appendix D

PROCEDURE TO OPERATE ACROLOG 400

1. Starting logging program

a. Switch computer on (red button on rear), and note that there is no disk in the A drive (floppy disk).

b. The C:\> prompt should now appear. c. Type in 'cd acrolab\term' <return>, note the space. d. C:\ACROLAB\TERM> should now appear. e. Type in 'term' <return>. f. Hit <return> twice. g. The Ready should now appear. h. Hit Fl. i. Type in 'truck.bas' <return>. j. Hit F3. k. At the prompt type in 'a:\filename.dat' <return>.

This tells the logger that data is to be recorded in this file on the specified drive (in this case drive A and file filename.dat). Ignore the above two steps (i.e. F3 etc.) if a disk log is not required.

Note: Wherever possible, avoid using drive C (i.e. the hard disk drive) as writing to this drive whilst in motion may cause damage to the drive.

1. Type in 'run' <return>, the logging program now starts.

2. Logging and writing data

a. Read the screen for instructions. b. Enter in a test run name at the prompt and <return>.

The data logger is now logging data and writing data to the floppy disk.

c. To stop logging hit 'CTRL C' the Ready prompt will now appear. At this point the data file on the floppy disk is still open, that is, further data samples taken on subsequent runs will be appended to this file.

3. Temporary interrupting of logging activity

a. If it is required to increase the pressure in the hydraulic system, then hit 'CTRL C' to stop the program. Now increase the hydraulic pressure.

b. The logging program now has to be reloaded into the ACROLOG, that is, hit Fl, then type in 'truck.bas' as described above. The data file is still open and should still record data, that is, F3 does not have to be hit again.

D-2

4. Returning to DOS and restarting

a. To return to DOS hit 'ALT X'. On return to DOS the data file on disk will be closed, that is, if any subsequent test runs are made logging into the same file name, the existing file will be overwritten.

b. Before travelling to another location, hit 'CTRL C' then 'ALT X' then switch the computer, the instruments and the brake controller off.

c. On restarting make sure that the data file is a different name (e.g. truckl.dat, truck2.dat) otherwise existing data will be overwritten; this must be done whenever 'ALT X' has been hit (i.e. to return to DOS) as this actually closes the data file.

E-1

Appendix E

10 REM 20 REM 30 REM 40 REM 50 REM 60 REM 70 REM 80 REM 90 REM 100 REM 110 CLS

+------------------------------------------------------------+ U.N.S.W./R.T.A. ROAD FRICTION TEST VEHICLE

PROGRAM TO READ STRAIN GAUGE AND SPEED OUTPUT VERS 2.2 MAY 1991 M.J.KERKHOF

CONVERTS, PRINTS AND LOGS ANALOG AND DIGITAL INPUTS ACRO-400 MUST BE IN SIMULATION MODE.

SET SWITCH 1-1 ON. +------------------------------------------------------------+

130 RESOLVE=13:REM SET RESOLUTION OF A-TO-D CONVERSION TO 13 BITS 140 RANGE(l)=l0:REM SET RANGE CHANNEL 1 TO+/- 10v 150 RANGE(2)=10:REM SET RANGE CHANNEL 2 TO+/- 10v 160 RANGE(3)=10:REM SET RANGE CHANNEL 3 TO+/- 10v 170 DCONFIG(3) = 0:REM SET DIGITAL LINES 17 TO 24 AS INPUT 180 COUNTER= 0:REM DISABLE COUNTER 190 COUNT= 0:REM RESET COUNTER 200 S = VAL(TIME$(7,8)) :REM CONVERT SECONDS TO A NUMBER 210 LOCATE 2,25:REM LOCATE CURSOR 220 PRINT "ACROLOG 400 DATA LOGGING PROGRAM" 230 LOCATE 5,1 240 PRINT "CTRL-C TO STOP LOGGING" 250 PRINT "CTRL-C THEN ALT-XTO RETURN TO DOS" 260 PRINT "F3 TO LOG DATA TO DISK"; 270 LOCATE 9,2 280 INPUT "ENTER TEST RUN NAME: ",N$ 290 LOCATE 9,2 300 PRINT" 310 LOCATE 11, 2

"· REM REMOVE LINE

320 PRINT "FRICTION TEST RECORDED ON ";DATE$;" AT ";N$ 330 LOCATE 13,2 340 PRINT "TIME"; TAB ( 15), "LOAD"; TAB (25), "S. W. F."; TAB (35), "BRAKE";

TAB(45),"SPEED";TAB(58),"BRAKE ON/OFF" 350 PRINT TAB(15),"Volts";TAB(26),"Volts";TAB(36),"Volts";

TAB(47),"km/h" 360 G = VAL(TIME$(7,8)) :REM READ SECONDS FROM R.T. CLOCK, CONVERT

370 380 390 400 410 420 430

SECONDS TO A NUMBER IF G = S THEN GOTO 450 ELSE GOTO 380 S = G COUNTER= 0:REM STOP COUNTER R = COUNT:REM READ VALUE IN COUNTER COUNT= 0:REM RESET COUNTER COUNTER= 1:REM START COUNTER Rl R/200:REM CALCULATE NUMBER OF WHEEL REVS/SEC

440 R2 ((Rl*l.612)*3600)/1000:REM CALCULATE SPEED IN KM/H 450 Vl VIN(l) :REM READ INPUT FROM CHANNEL 1 460 V2 VIN(2) :REM READ INPUT FROM CHANNEL 2 470 V3 VIN(3) :REM READ INPUT FROM CHANNEL 3 480 IF DIN(l7) THEN B$ = "OFF" ELSE B$ = "ON":REM READ BRAKE STATE 490 LOCATE 16,1:REM LOCATE CURSOR 500 REM NOW CONVERT AND PRINT INPUTS 510 PRINT TIME$;%8F2,TAB(12),Vl;TAB(23),V2;TAB(33),V3;TAB(43),R2;

TAB(63),B$ 520 LOCATE 20,1:REM SEND CURSOR NEAR END OF PAGE 530 GOTO 360 540 END

E-2

10 REM 20 REM 30

+------------------------------------------------------------+

40 50 60

REM REM REM REM

70 REM 80 REM 90 REM 100 REM 110 CLS

U.N.S.W./R.T.A. ROAD FRICTION TEST VEHICLE PROGRAM TO READ STRAIN GAUGE AND SPEED OUTPUT

VERS 3.2 MAY 1991 M.J.KERKHOF CONVERT AND PRINT ANALOG INPUT

ACRO-400 MUST BE IN SIMULATION MODE. SET SWITCH 1-1 ON.

+------------------------------------------------------------+

130 RESOLVE=13:REM SET RESOLUTION OF A-TO-D CONVERSION TO 13 BITS 140 RANGE(l)=l0:REM SET RANGE CHANNEL 1 TO+/- 10v 150 RANGE(2)=10:REM SET RANGE CHANNEL 2 TO+/- 10v 160 RANGE(3)=10:REM SET RANGE CHANNEL 3 TO+/- 10v 170 DCONFIG(3) = 0:REM SET DIGITAL LINES 17 TO 24 AS INPUT 180 COUNTER= 0:REM DISABLE COUNTER 190 COUNT= 0:REM RESET COUNTER 200 REM T$ = TIME$(7,8) :REM READ SECONDS FROM R.T. CLOCK ON STARTUP 210 S = VAL(TIME$(7,8)) :REM CONVERT SECONDS TO A NUMBER 220 LOCATE 2,25:REM LOCATE CURSOR 230 PRINT "ACROLOG 400 DATA LOGGING PROGRAM" 240 LOCATE 4,1 250 PRINT "CTRL-C TO STOP LOGGING" 260 PRINT "CTRL-C THEN ALT-XTO RETURN TO DOS" 270 PRINT "F3 TO LOG DATA TO DISK"; 280 LOCATE 9,2 290 INPUT "ENTER TEST RUN NAME: ",N$ 292 LOCATE 9,2 295 PRINT:REM REMOVE LINE 300 LOCATE 11,2 310 PRINT "FRICTION TEST RECORDED ON ";DATE$;" AT ";N$ 320 LOCATE 13,2 330 PRINT "TIME";TAB(15),"LOAD";TAB(25),"S.W.F.";TAB(35),"BRAKE";

TAB(45),"SPEED";TAB(58),"BRAKE ON/OFF" 340 G = VAL(TIME$(7,8)) :REM READ SECONDS FROM R.T. CLOCK, CONVERT

SECONDS TO A NUMBER IF G = S THEN GOTO 440 ELSE GOTO 360 S = G COUNTER= 0:REM STOP COUNTER R = COUNT:REM READ VALUE IN COUNTER REM COUNT= 0:REM RESET COUNTER COUNTER= 1:REM START COUNTER Rl R/200:REM CALCULATE NUMBER OF WHEEL REVS/SEC

350 360 370 380 390 400 410 420 430 R2 ((Rl*l.595)*3600)/1000:REM CALCULATE SPEED IN KM/H 440 Vl VIN(l) :REM READ INPUT FROM CHANNEL 1 450 V2 VIN(2) :REM READ INPUT FROM CHANNEL 2 460 V3 = VIN(3) :REM READ INPUT FROM CHANNEL 3 470 IF DIN(17) THEN B$ = "OFF" ELSE B$ = "ON": REM READ BRAKE STATE 480 LOCATE 15,1: REM LOCATE CURSOR 490 PRINT TIME$;%8F2,TAB(12),Vl;TAB(23),V2;TAB(33),V3;TAB(43),R2;

TAB(63),B$:REM CONVERT AND PRINT INPUT 500 LOCATE 18,1:REM SEND CURSOR NEAR END OF PAGE 510 GOTO 340 520 END

F-1

Appendix F

RESULTS OF INVESTIGATION USING MMFTT, WITH HIGHER SPEED (20 MPH & 30 MPH). Extracted from Dermoredjo (1992].

1. Example for 20 mph (± 32 kph) vehicle speed, 10° slip angle, 0.5 second rate of braking, 20 & 30 psi tyre pressure.

Coefl. of Frlotlon 1

0.8

o.e

OA

0.1

0

-0.2 0

-

\

\ ./

-

+ ......

{ \ .

I I - - - c.. -

.,_ -. -,,.

.

--

,.

-

- -- -

T

-

- .. - . .

I _.I!

-.

~

21

20

11

10

I

lpNd(ka/la) as

ao

21

20

11

10

a -0.2 ..__...__..___..___..1...-_ _.___.J__.J__..J__..J...._..J 0

0 10 20 ao 40 so eo ·-- 70 eo 80 100

......

Alllle10

.,... ...

F-2

2. Bxample ~or 30 mph (± 48 kph) vehicle speed, 10° slip angle, 0 .5 second rate o~ braking, 20 6i 30 psi tyre prassura.

Ooeft. of ,rtet1on 1.2 I

- -

\ \

\ -

RateoflwaltllW.O.r ,..........,. .... , I - - -

I I i

- - -1

0.8

0.1

OA

0.2

1~-1 T .,..._ .

I i ..L .

I /.

./

0 0

I I

Ooetf. of ,r1e11on 1.2

1

0.8

0.1

0.4

0.2

0 0

I I - -

"' -·

' .... -+

i I

j !

I I I -~

~x, I ·, I

- llcle--, , __ Coetf

~ lpNdoftMIDle.

- I '--

i

I

Rate or 1wa1 .. _o.r ,.... ............. I I I i ! I I

i i I I I

~ ! l __.

I r .

I !

-I

- -

+ ,__ ~""'--k....

I I

-+- ....... ,__ Ooetf.

-

½ .

.....

I I . - -I-· -· ! I I i I ,A!' r-:+"

eo

ao

20

10

eo

80

20

10

Alllle10

Data 002

G-1

Appendix G

MATHEMATICAL DERIVATIONS OF TYRE STIFFNESS FORMULA

After Schallamach and Turner [1960], Dugoff et.al [1969], Livingston and Brown [19691, Tielking and Mital [1974].

__;,...__.!..,j,l~L.......L--...:;:'vll;.;..;H.:.:£ E:.:L;_P;..;;L;.;..;A.;.;.;NE;;.____.._ X

FRIC.TIOWI.L'"'------'I r-,, F'ORCE

Vx

Fig.G.1. The kinematics of tyre motion.

-contact leng¼h-

p' CA/tCASS ~

Fig.G.2. Left: tyre-road contact geometry. Right: deformation in adhesion region of contact patch. [Dugoff et.al (1969)].

EI

Fig.G.3. Rectangular tread elements attached to an elastically supported deformable ring of bending stiffness EI. [Tielking and Mital (197 4)].

G-2

Notation

Cx = longitudinal slip stiffness (force/slip, or force/fraction slip), or longitudinal deformation stiffness (force)

~ = lateral slip stiffness (force/degree, or force/radian), or lateral deformation stiffness (force)

= longitudinal & lateral force (force) = tyre contact length & width (unit length) = longitudinal & lateral carcass spring rate (force/length3)

= longitudinal & lateral tread element spring rate (force/length3)

= longitudinal slip (%, or per fraction slip) = tana = resultant of tyre velocity (unit length/unit time) = longitudinal & lateral tyre velocity (unit length/unit time) = free-rolling velocity (unit length/unit time) = resultant of slip velocity (unit length/unit time) = longitudinal & lateral slip velocity (unit length/unit time) = slip angle (degree, or radian) = longitudinal & lateral stress (force/unit area) = arc tan S/Sx (degree, or radian)

1. Pure longitudinal slip under braking

Consider the case of pure longitudinal slip under braking (see Figure G.1). Since the slip angle (a) is equal to zero, then the direction of motion is parallel to the wheel plane.

When the tyre is free-rolling, the rolling velocity (Vr) is equal to the longitudinal velocity (Vx) of the wheel center.

Vr = Vx (free-rolling) (G.1)

When braking is applied, the longitudinal velocity (Vx) and the rolling velocity (Vr) are reduced by a certain amount each instant in time. Elastic deformation and sliding in the contact region affect the reduction in longitudinal velocity and rolling velocity. The longitudinal velocity decreases until equal to the longitudinal slip velocity at the condition of a lockedwheel, whereas the rolling velocity decreases until equal to zero when the the wheel is locked. The difference is the longitudinal slip velocity (V sx> which is given by:

VSX = vx - vr (G.2)

It is convenient to define a parameter Sx as a fraction of longitudinal velocity which indicates the amount of braking.

sx = VsxfVx (G.3) This parameter may exhibit values from O in the case of a free-rolling wheel to 1 in the case of a locked-wheel. By Equation (G.2), the definition of Sx may be written as

G-3

Sx = 1 - VJVx Eliminating Vx from Equations (G.2) and (G.3), one obtains

VsxfVr = SJ(l-Sx)

(G.4)

(G.5)

As the angle of slip is equal to zero, the magnitude of the slip velocity Vs = I Vs I is the component of Vs in the direction of wheel plane, ~=~ ~~

Consider now the deformation in the contact patch (see Figure G.2). Since the slip angle (a) is equal to zero, the typical point P (X, Y) lies on the wheel plane.

In time at, the base point P' will move into the contact region a longitudinal distance X' determined by the rolling velocity Vr.

X' = Vrat (G.7)

If there is braking, the contacting point P, which moves relative to Y with the wheel center velocity V x, will cover an additional distance (X - X') as indicated in Figure G.2. The displacement of point P from the axis Y is

X = Vxat (G.8) Eliminating time from Equations (G.7) and (G.8) gives an expression for the longitudinal displacement (X - X').

X - X' = (Vx/Vr - l)X' (G.9) or, in terms of the longitudinal slip speed given by Equation (G.2),

X - X' = (VsxfVr)X' (G.10)

In adhesion region, the deformation (X - X') is produced by static friction with a limiting coefficient of friction µ. The static friction force per unit area of element (ax) required to produce this displacement depends upon the longitudinal stiffness (~) of the element (see Figure G.3),

ax = ~(X - X') (G.11) and, using Equation (G.10),

(Jx = ~(VsxfVr)X' (G.12)

It is convenient to define the non-slip contact length 21 on the equator of the tyre carcass to which the X,Y coordinates are fixed. (Note: 21 is the contact length of the free-rolling tyre at zero slip angle). If it is assumed that the behaviour is uniform in the Y direction, the distributed · contact shear stress CJx is defined as function of X only. The resultant force Fx, transmitted to the tyre mounting rim, is now given by the following integral.

F = w P1a dx (G.13) X Jo X

where w is the contact width.

If the rolling and slip velocities are such that X' ~ 2t there is complete adhesion over the entire contact region and from Equations (G.12) and (G.13):

G-4

Fx = 0.5 w I<x<VsxfVr)(21)2 (G.14) The use of Equation (G.5) permits the adhesive forces, given by Equation (G.14), to be written in the following way:

Fx = 0.5 w kx{Sx/(1-Sx)}(21)2 (G.15)

The longitudinal stiffness for complete adhesion, Cx is given by ex= dFJdSX (at sx = 0) (G.16)

Differentiating Equation (G.15) into Sx (at Sx = 0) yields, Cx = 0.5 kx(21)2w (G.17)

It should be noted from Equation (G.17) that the dimension of½ is force per unit slip, whereas the dimension of 0.5 kx(21)2w is force. The right hand side of Equation (G.16) is the slip stiffness, whereas the right hand side of Equation (G.17) is the deformation stiffness.

It can be concluded that at zero slip the longitudinal slip stiffness is mathematically equal to the longitudinal deformation stiffness (but not apply for other values of slip).

2. Pure lateral slip under cornering

For the case of pure lateral slip, the longitudinal slip Sx is equal to zero, and the direction of motion make an angle a with the plane of the wheel (see Figure G.1).

Analogous to the longitudinal slip parameter, the lateral slip parameter SY is defined to be:

Sy = Vs/Vx = tana (G.18) This parameter may exhibit values from 0 (free rolling, at a = D°) to oo

("locked-wheel", at a. = 1t/2). With Sx = 0, then Vx = Vr (G.19)

and hence, Sy = Vs/Vr (G.20)

As the longitudinal slip is equal to zero, the magnitude of the slip velocity V s = I V s I is the component of V s perpendicular to the plane of the wheel,

Vs = Vsy (G.21)

Consider now the deformation in the contact patch (see Figure G.2). The typical contact point P (X, Y) lies in the direction of motion relative to the origin (point 0).

In time at, the contact point P is displaced laterally a distance Xtana Xtana = V5yat (G.22)

In adhesion region, the deformation Xtana is produced by static friction with a limiting coefficient of friction µ. The static friction force per unit area

G-5

of element (cry) required to produce this displacement depends upon the lateral stiffness (ky) of the element (see Figure G.3).

cry = kyXtana (G.23) and, using Equations (G.18) and (G.19),

cry = ky(Vs/Vr)X (G.24)

The resultant force Fy, transmitted to the tyre mounting rim, is now given by the following integral.

F =W~~ ~~ Y Jo Y

where w is the contact width, and 21 is the non-slip contact length.

The integration of Equation (G.25) under the condition of complete adhesion yields

Fy = 0.5 w ky(Vs/Vr)(21)2 (G.26) The use of Equation (G.20) permits the adhesive force, given by Equation (G.26), to be written in the following way:

FY = 0.5 w kytana(21)2 (G.27) And for small slip angle, tana "" a, hence

Fy = 0.5 w kya(21)2 (G.28)

The lateral stiffness for complete adhesion, Cy is given by CY = dF/da (at a= 0) (G.29)

Differentiating Equation (G.28) into a (at a = 0) yields, CY = 0.5 kyC21)2w (G.30)

It should be noted from Equation (G.30) that the dimension of CY is force per unit angle, whereas the dimension of 0.5 ky(21)2w is force. The right hand side of Equation (G.29) is the slip stiffness, whereas the right hand side of Equation (G.30) is the deformation stiffness.

It can be concluded that at zero slip angle the lateral slip stiffness is mathematically equal to the lateral deformation stiffness (but not apply for other values of slip angle).

EXPLANATORY NOTES

A. Modification of Summary of Conclusions

7.4. Summary of Conclusions

1. Using the data from the literature survey, the candidate proposed an

empirical model which will be able to predict the change of tyre

stiffness (C , C ) for any set of operating conditions (F , IP, V)., X y Z

once the tyre stiffness for one set of operating conditions has been

measured (known). The equations (Eqs. 2.32-2.37) are set out in

Section 2.4.7.

2. The results using the candidate's measured field data confirm a good

agreement between the predicted and the measured values of tyre

stiffness. This can be seen, for example, in Table 6.2 where one set

of operating conditions is presented in the first row, and any set

of operating conditions are presented in columns 2-4 of rows 2-4. The

predicted values in column 6 are calculated using Eguation 2.34. The

measured values in column 5 of rows 2-4 are obtained by using MMFTT.

3. The candidate has proposed a modification to an existing model for

the shape of tyre pressure in a form of trapezium, with contact

length of 21, and slope length of a and b. This model offers

flexibility in representing many types of actual pressure

distribution, by selecting the appropriate value of its slope (a/21,

b/21). The results from the literature survey (Section 1.3.5) can be

used as a guidance for choosing the slope. It was found using Sakai's

laboratory data that the selection of a/21 is more sensitive with

increasing tyre stiffness.

4. An analytical model has been developed by the candidate, which

enables the calculation of frictional forces over all ranges of

1

percent slip and slip angles. The ability to calculate frictional

forces from tyre stiffness and either locked-wheel BFC or maximum

SFC, is a feature of the candidate's model. In general, the

candidate's model is found to be in the median position when the

model response was compared with 5 tyre models: HSRI-I, HSRI-II,

HSRI-III, Good Year, and Sakai.

5. The candidate deduced mathematically from his prediction model, with

inconclusive evidence from field measurement, that the direct

contribution of tyre stiffness to the frictional forces is in the

region of below the critical percent slip (S ) and critical slip xcr

angle (a ). Beyond this area, the tyre stiffness still contributes er

indirectly, and diminishes when the condition of 100% slip (locked-

wheel) or 90° slip angle was achieved. See Figure 2.19 and Tables

2.7-2.8 for contribution of C, C andµ to frictional forces. X y

6. The way of using the candidate's analytical model for predicting the

frictional forces are presented in Sections 2.6.6-2.6.8. By knowing

(or measuring) the input of C , C , locked-wheel BFC, a/21, b/21, and X y

F , the values of F and F at various S and a can be calculated. This Z X y X

can be seen, for example, in Section 2.6.7 where ex= 10000 lbs/slip,

c = 12605 lbs/rad, locked-wheel BFC = 0.87, a/21 = 0.06, b/21 = y

0.20, and Fz = 1100 lbs. By using the candidate's analytical model,

it was found that for Sx = 0.1 and a = 4°, the Fx = 685 lbs and FY =

515 lbs. The values of Fx and FY at other values of Sx and a are

tabulated in Tables 2.7-2.8.

7. Since the locked-wheel BFC (or maximum SFC) can be predicted from

surface texture measurement (Yandell et.al (1983)], the candidate"s

analytical model can be used to predict the frictional forces, from

tyre stiffness and surface texture data. With the availability of

equipment for measuring the locked-wheel BFC or maximum SFC, the data

from this equipment can be processed directly to obtain the

frictional forces for all ranges of percent slip (Sx) and slip angles

(a). See also Figures 7.1-7.2.

2

8. The effect of some input parameters (C , C , a/21, b/21, µd and F ) on X y Z

frictional forces has been studied using the candidate"s analytical

model. Almost all parameters have significant effects on the BFC and

SFC, except for b/21 which is found to be small.

9. In general, the results from the candidate's experiment and Sakai's

laboratory data, show a good agreement between the measured values

and the predicted values (using the candidate"s analytical model).

However, for both candidate's experimental results and Sakai's

laboratory data, the predicted values of SFC (versus slip angle) at

zero percent slip, was found to under estimate the measured values.

A closer agreement can be achieved by using the modification in the

input of the candidate"s tyre model, so the input values of cy is

force/1.57 radians (or force/90°) instead of force/radian (or

force/57.3°) as in the original input of the candidate's tyre model.

B. Explanation of Section 3.3

a. There are 288 data available for the development of regression

model. These data are created from the independent variables in

Table 3.1 (2x2x2x2x2x3x3 factorial). For example:

Data 1: a/21 = 0.05, C = 10000 lbs/slip, C = 10000 lbs/rad, X y

0 µd = 0. 3, F = 200 lbs, S = 0, a = 0 •

Z X

Data 2: a/21 = 0.20, ex= 10000 lbs/slip, CY = 10000 lbs/rad,

µd = 0. 3, F = 200 lbs, S = 0, a = o0 • Z X

Data 288: a/21 = 0.20, C = 60000 lbs/slip, C = 40000 lbs/rad, X y

µd = 0.9, F = 1800 lbs, S = 1, a = 40°. Z X

The values of the dependent variables (µx, µY and µrst) for each of

288 data above can be calculated by using the candidate's

analytical model. (Note: the b/21 was taken constant 0.20). The

288 sets of the dependent and independent variables are then used

to form the regression model.

3

b. It should be noted that Section 3.3 occupies only 11 pages (pages

133-144) in this thesis. This section forms a part in providing a

better explanation for the effects of the parameters influencing

tyre-road friction by using the candidate's analytical model. The

variables which significantly affect the tyre-road friction are

presented in Table 3.2. The results of signification are concluded

in Sections 3. 3. 4-3. 3. 6, and are summarised in the Theoretical

Findings (under Section 7.1.7: Theoretical Effect of Some Input

Parameter on the Tyre-Road Friction).

C. Explanation of Section 7.2.3 (formerly Section 7.2.6)

a. The candidate has proposed an empirical model which will be abl~

to predict the change of tyre stiffness for any set of operating

conditions, once the tyre stiffness for one set of operating

conditions has been measured (known). The equations (Eqs. 2.32-

2. 37) are set out in Section 2. 4. 7. The results between the

measured and the predicted values of tyre stiffness are considered

as a good agreement when the discrepancies are less than 10%.

b. The input values to the candidate's empirical model above (F , IP, z

V, C and C ) can be measured without knowing the value of locked-x y

wheel BFC (or maximum SFC). See Figures 5.15-5.16, for example, in

determining the values of ex and CY.

c. It should not be confused between the candidate's empirical model

mentioned above, and the candidate's analytical model for

predicting the frictional forces. In the latter, it is true that

the locked wheel BFC (or maximum SFC) is one of the input values,

where its predicted values are all of the BFC outside 100% slip

(or all of the SFC outside the maximum value).

4

the prediction of tyre friction on wet roads under braking and ...

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