Theoretical and Experimental Analysis of Strain in a Tire Under Static
Loading and Steady-State Free-Rolling Conditions
by
Vijaykumar Krithivasan
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 6, 2011
Keywords: FEM, Strain, Slip, Tires, Lateral Force, Longitudinal Force
Copyright 2011 by Vijaykumar Krithivasan
Approved by
Robert L. Jackson, Chair, Associate Professor of Mechanical Engineering
Song-Yul Choe, Associate Professor of Mechanical Engineering
Hareesh V. Tippur, Professor of Mechanical Engineering
Dan B. Marghitu, Professor of Mechanical Engineering
Abstract
This main objective of this work is to predict the operating conditions or the state of
a tire based on a wireless sensor suit. First a three dimensional nite element model of a
standard reference test tire (SRTT) was developed to better understand the tire deforma-
tion under separate cases of static loading, steady state free-rolling and steady-state rolling
conditions. A parametric study of normal loading, slip angle and slip ratio was carried out
to capture the in uence of these parameters. The numerical analysis techniques such as the
Fouier analysis, Weibull curve tting and slope curve method were explored to relate the tire
strains to the various loads on the tire. The advantages and disadvantages of the various
methods, mentioned above, for strain analysis is also presented.
A wireless sensor suite comprising of analog devices (strain, pressure and temperature
sensors) was developed to capture the tire deformation under loading conditions similar to
those used in the nite element model. This sensor suite formed the basis for experimentally
verifying the trends captured by the nite element model on a custom built tire test stand
with capabilities of mimicking real-time conditions (under static loading scenario) of a tire
in contact with road and steady state conditions on a FlatTrac test bed. Using the results
from the experiments and the nite element model an empirical model was developed which
demonstrates how the strains measured on the inner surface of the tire could be used to
quantify desired parameters such as slip angle, lateral force, slip ratio, longitudinal force and
normal load. This resulting empirical equations relate measured strains to the normal load,
slip angle, slip ratio, lateral force and longitudinal force.
ii
Acknowledgments
This research work that I commenced four years ago would not have culminated into
this dissertation without the help and constant support of many people. I am grateful to
my advisors, Professor Robert L. Jackson and Professor Song-yul Choe, for providing me
with this wonderful and exciting opportunity. Their deep love and enthusiasm for research
and the passion for life is very infectious. Their e orts towards ensuring the progress and
well-being of their students is truly remarkable and I am a very happy bene ciary of that.
I am immensely proud to have been associated with them and their Smart Tire Research
Group. I would like to thank Hyundai Motor Company, South Korea, for providing support
to my graduate school research.
I would like to express my gratitude to Professor Hareesh V. Tippur, Professor Dan B.
Marghitu, and Professor Asheber Abebe, for serving as members of my Defense Examination
Committee. I would like to thank David Howland of General Motors for the help with my
research work in all possible manner. I am grateful to Dr. Marion Pottinger for sharing
valuable inputs and insights in the eld of tire research.
I would like to thank my colleagues in the Research Group: Dr. Jyoti Ajitsaria, Mr.
Russell Green, Mr. Jeremy Dawkins, Mr. Santosh Angadi, Mr. Everett Wilson, for their
signi cant contributions.
I am grateful to my friends Dr. Vivek Krishnan, Mr. Karthik Narayanan, Mr. Varun
Rupela, Dr. Pradeep Prasad, Mr. Naren Pari, Dr. Anand Sankarraj, Dr. Shankar Balasub-
ramanian, Mr. Raghu Viswananthan, Mr. Kashyap Yellai, Mr. John Polchow and Mr. Jake
Fredrick for their encouragement.
The unwavering support and encouragement from my loving family throughout my
graduate study made this day possible. The sacri ces that my parents Krithivasan and
iii
Sathyabama have made in raising their kids cannot be described in words. I am what I am
today because of them. My brother Dr. Ramkumar Krithivasan has been a pillar of support
in all my endeavors. I am forever indebted to my family.
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Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Types of Tire Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Types of Tire Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Intelligent Tires in the Future . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Static Loading Finite Element Tire Model . . . . . . . . . . . . . . . . . . . . . 19
2.1 Tire Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Material Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Three-Dimensional Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Mesh Convergence and Veri cation of Model Accuracy . . . . . . . . 27
2.5 Preliminary FEA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1 FEM Prediction of Contact Patch Area . . . . . . . . . . . . . . . . . 28
2.5.2 FEM Prediction of Strain on the Tire Innerliner and Footprint . . . . 32
2.5.3 Comparison Between Circumferential Strains and Lateral Strains . . 38
v
3 Strain Analysis Based on Static Loading Finite Element Prediction . . . . . . . 40
3.1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Static Loading Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Tire Test Stand for Experimental Validation . . . . . . . . . . . . . . . . . . 46
4.2 Mechanical Layout of Hardware . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Electrical Layout of Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Strain Analysis Based on Static Loading Experimental Data . . . . . . . . . . . 55
5.1 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Steady-State Free Rolling Finite Element Model . . . . . . . . . . . . . . . . . . 59
6.1 Straight Line Rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Steady-State Free-Rolling at Various Slip Angles . . . . . . . . . . . . . . . . 61
6.3 Strain Pro le Under Steady-State Free-Rolling . . . . . . . . . . . . . . . . . 63
7 Strain Analysis Based on Steady-State Free-Rolling Finite Element Prediction . 69
7.1 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Weibull Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Slope Curve Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.4 Evaluation of Methods for Strain Analysis . . . . . . . . . . . . . . . . . . . 81
7.5 Normal Load Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.6 Slip Angle Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.6.1 Innerliner Deformation Due to Slip Angle . . . . . . . . . . . . . . . . 85
7.6.2 Analytical Relationship Between Strain and Slip Angle . . . . . . . . 91
7.6.3 Relating Strain to the Slip Angle . . . . . . . . . . . . . . . . . . . . 93
7.7 Lateral Force Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.8 Estimating Constants of the Semi-Empirical Model Based on FEM Results . 98
7.8.1 Constants in the Slip Angle Empirical Model . . . . . . . . . . . . . . 98
7.8.2 Constants in the Lateral Force Empirical Model . . . . . . . . . . . . 98
8 Steady-State Free Rolling Experimental Methodology . . . . . . . . . . . . . . . 100
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8.1 Test Bed for Experimental Validation . . . . . . . . . . . . . . . . . . . . . . 100
8.2 Steady-State Rolling at Various Slip Angles . . . . . . . . . . . . . . . . . . 102
8.3 Strain Pro le Under Steady-State Free-Rolling . . . . . . . . . . . . . . . . . 104
9 Strain Analysis Based on Steady-State Free-Rolling Experimental Data . . . . . 110
9.1 Slope Curve Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.2 Normal Load Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.3 Slip Angle Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.4 Lateral Force Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.5 Estimating Constants of the Semi-Empirical Model Based on Experimental
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.5.1 Constants in the Slip Angle Empirical Model . . . . . . . . . . . . . . 119
9.5.2 Constants in the Lateral Force Empirical Model . . . . . . . . . . . . 121
10 Steady-State Rolling Finite Element Model . . . . . . . . . . . . . . . . . . . . . 124
10.1 Steady-State Rolling at Various Slip Ratios . . . . . . . . . . . . . . . . . . . 124
10.2 Strain Pro le Under Steady-State Rolling . . . . . . . . . . . . . . . . . . . . 126
11 Strain Analysis Based on Steady-State Rolling Finite Element Prediction . . . . 130
11.1 Slope Curve Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
11.2 Slip Ratio Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.3 Longitudinal Force Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.4 Estimating Constants of Semi-Empirical Model based on FEM Results . . . 139
11.4.1 Constants in the Slip Ratio Empirical Model . . . . . . . . . . . . . . 139
11.4.2 Constants in the Longitudinal Force Empirical Model . . . . . . . . . 139
12 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A ABAQUS FEA Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B Code for Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
vii
C Code for Weibull Curve Fiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
viii
List of Figures
1.1 Geometry based cord model[17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Two dimensional axisymmetric model for in ation analysis[17]. . . . . . . . . . 4
1.3 Graphical representation of the magic formula[45]. . . . . . . . . . . . . . . . . 6
1.4 Linear model of a single contact point tire model[45]. . . . . . . . . . . . . . . . 7
1.5 Fully non-linear string tire model[45]. . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Types of tire sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Continental?s[56] side wall torsion sensor . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Darmstadt tire sensor[56]: Magnet and Hall sensor. . . . . . . . . . . . . . . . . 11
1.9 Oscillating circuit based wireless strain monitoring[66]. . . . . . . . . . . . . . . 12
1.10 Capacitance change based wireless strain monitoring[66]. . . . . . . . . . . . . . 12
1.11 Flexible capacitance change based wireless strain monitoring[67]. . . . . . . . . 13
1.12 PVDF-based tire tread deformation sensor . . . . . . . . . . . . . . . . . . . . . 14
1.13 Rubber sensor[69] ( also called MetalRubber) by NanoSonic Inc.. . . . . . . . . 14
1.14 Seimens?s[56] SAW sensor with energy generating piezoelectric crystals. . . . . . 15
1.15 Sensor technology road map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 Typical components of a passenger car radial tire. . . . . . . . . . . . . . . . . . 19
2.2 Tire cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Constitutive material models[79]. . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Element types used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Enlarged view of ABAQUS elements used in the FEM. . . . . . . . . . . . . . . 25
ix
2.6 Three dimensional tire model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Mesh convergence and reduction in computational e ort . . . . . . . . . . . . . 27
2.8 FEM results compared to ASTM test data for a SRTT. . . . . . . . . . . . . . . 28
2.9 Contour plot of displacement in the z-direction . . . . . . . . . . . . . . . . . . 29
2.10 Contour plot of displacement in the z-direction . . . . . . . . . . . . . . . . . . 30
2.11 Contour plot of displacement in the z-direction . . . . . . . . . . . . . . . . . . 31
2.12 Contour plot of displacement in the z-direction . . . . . . . . . . . . . . . . . . 32
2.13 Strain prediction flxx for 1600lbs normal load;(a)Innerliner;(b)Contact patch. . . 33
2.14 Strain prediction flyy for 1600lbs normal load;(a)Innerliner;(b)Contact patch. . . 34
2.15 Strain prediction flzz for 1600lbs normal load;(a)Innerliner;(b)Contact patch. . . 35
2.16 Strain prediction flxy for 1600lbs normal load;(a)Innerliner;(b)Contact patch. . . 35
2.17 Strain prediction flyz for 1600lbs normal load;(a)Innerliner;(b)Contact patch. . . 36
2.18 Strain prediction flzx for 1600lbs normal load;(a)Innerliner;(b)Contact patch. . . 37
2.19 Tire deformation and strain prediction fl?? . . . . . . . . . . . . . . . . . . . . . 38
2.20 Strain prediction fl?? and flyy comparison at 1600lbs normal load. . . . . . . . . . 39
3.1 Sensor locations approximated to nodal solutions. . . . . . . . . . . . . . . . . . 40
3.2 Schematic showing the ?=0 rad node position. . . . . . . . . . . . . . . . . . . . 41
3.3 Strain prediction at 1000lbs normal load. . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Strain prediction at 1200lbs normal load. . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Strain prediction at 1400lbs normal load. . . . . . . . . . . . . . . . . . . . . . . 43
3.6 Strain prediction at 1600lbs normal load. . . . . . . . . . . . . . . . . . . . . . . 44
3.7 Strain prediction along sensor 1 for normal loads 1000lbs-1600lbs. . . . . . . . . 45
4.1 AU test stand (a) Pneumatic cylinder; (b) Pressure control unit. . . . . . . . . . 46
4.2 Custom built tire test stand for static loading condition. . . . . . . . . . . . . . 47
4.3 Schematic of the hardware arrangement. . . . . . . . . . . . . . . . . . . . . . . 48
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4.4 Strain sensor placement on the innerLiner of the tire. . . . . . . . . . . . . . . . 49
4.5 Wireless transmitter and signal conditioning circuit arrangement on a bracket. . 50
4.6 Three-Piece Wheel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.7 Final arrangement of bracket on the wheel. . . . . . . . . . . . . . . . . . . . . 51
4.8 Pre-design of signal conditioning circuit for strain measurements. . . . . . . . . 52
4.9 Quarter bridge wheatstone network. . . . . . . . . . . . . . . . . . . . . . . . . 53
4.10 XDA100 and MICA2 wireless module[80]. . . . . . . . . . . . . . . . . . . . . . 54
4.11 Electrical layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.1 Data transmission from sensor to data processing unit. . . . . . . . . . . . . . . 55
5.2 Strain measurements for 800lbs normal load. . . . . . . . . . . . . . . . . . . . . 56
5.3 Strain measurements for 1300lbs normal load. . . . . . . . . . . . . . . . . . . . 57
5.4 FEM results compared to 235/70 R16 tire at 1025lbs. . . . . . . . . . . . . . . . 58
6.1 Braking and Traction simulation for free-rolling angular velocity (!) extraction. 59
6.2 Rolling resistance versus angular velocity . . . . . . . . . . . . . . . . . . . . . . 60
6.3 Lateral tread deformation under steady state cornering. . . . . . . . . . . . . . 61
6.4 Boundary condition for free-rolling at various slip angles. . . . . . . . . . . . . . 63
6.5 Strain pro le 0o 6o slip angle sensor 1; Normal load 1000lbs. . . . . . . . . . . 64
6.6 Strain pro le 0o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . . 65
6.7 Strain pro le 1:5o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . 65
6.8 Strain pro le 3o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . . 66
6.9 Strain pro le 4:5o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . 67
6.10 Strain pro le 6o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . . 68
7.1 Fast Fourier Transform of the strain data (Sensor 1; 1200lbs normal load). . . . 70
7.2 Fast Fourier Transform of the strain data (Sensor 2; 1200lbs normal load). . . . 71
7.3 Fast Fourier Transform of the strain data (Sensor 3; 1200lbs normal load). . . . 72
xi
7.4 Fast Fourier Transform of the strain data (Sensor 4; 1200lbs normal load). . . . 72
7.5 Relating amp. changes to normal load and slip angle 2nd frequency (sensor 1). . 73
7.6 Relating amp. changes to normal load and slip angle 3rd frequency (sensor 1). . 73
7.7 Relating amp. changes to normal load and slip angle 4th frequency (sensor 1). . 74
7.8 Relating amp. changes to normal load and slip angle 5th frequency (sensor 1). . 74
7.9 Weibull distribution for life cycle prediction[81]. . . . . . . . . . . . . . . . . . . 75
7.10 Weibull type function t to strain data (sensor 1). . . . . . . . . . . . . . . . . . 76
7.11 Weibull type function t to strain data (sensor 2). . . . . . . . . . . . . . . . . . 77
7.12 Weibull type function t to strain data (sensor 3). . . . . . . . . . . . . . . . . . 77
7.13 Weibull type function t to strain data (sensor 4). . . . . . . . . . . . . . . . . . 78
7.14 Slope change 0o slip angle (4 sensors and 4 loads). . . . . . . . . . . . . . . . . . 79
7.15 Slope change 1:5o slip angle (4 sensors and 4 loads). . . . . . . . . . . . . . . . . 80
7.16 Slope change 3o slip angle (4 sensors and 4 loads). . . . . . . . . . . . . . . . . . 81
7.17 Procedure for parameter prediction based on methods explored. . . . . . . . . . 83
7.18 FEM prediction of average contact duration. . . . . . . . . . . . . . . . . . . . . 84
7.19 FEM versus semi-empirical model prediction of normal load. . . . . . . . . . . . 85
7.20 Relating lateral tread deformation to the slip angle. . . . . . . . . . . . . . . . . 86
7.21 von-Mises stress distribution on the innerliner 0o slip angle . . . . . . . . . . . . 88
7.22 von-Mises stress distribution on the innerliner 1:5o slip angle . . . . . . . . . . . 88
7.23 von-Mises stress distribution on the innerliner 3o slip angle . . . . . . . . . . . . 89
7.24 von-Mises stress distribution on the innerliner 4:5o slip angle . . . . . . . . . . . 90
7.25 von-Mises stress distribution on the innerliner 6o slip angle . . . . . . . . . . . . 91
7.26 Analytical relationship between slip angle and strain. . . . . . . . . . . . . . . . 92
7.27 Normal load distribution while cornering. . . . . . . . . . . . . . . . . . . . . . 93
7.28 Slip angle estimation from FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . 94
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7.29 The SAE tire axis system[82]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.30 Lateral force prediction from FEM. . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.31 Lateral force estimation from FEM. . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.32 Lateral force comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.33 Load dependent constant Afem estimated from normal load. . . . . . . . . . . . 98
7.34 Load dependent constant Bfem estimated from normal load . . . . . . . . . . . 98
7.35 Load dependent constant C? estimated from normal load. . . . . . . . . . . . . 99
7.36 Load dependent constant D? estimated from normal load. . . . . . . . . . . . . 99
7.37 Load dependent constant E? estimated from normal load. . . . . . . . . . . . . 99
8.1 MTS FlatTracr?test bed courtesy: General Motors. . . . . . . . . . . . . . . . . 101
8.2 Sideview of tread on the belt: MTS FlatTracr?test bed courtesy General Motors. 102
8.3 MTS FlatTracr?test bed courtesy: General Motors. . . . . . . . . . . . . . . . . 103
8.4 Schematic for acquiring test data:(a) 0o phase; (b) 1o phase. . . . . . . . . . . . 104
8.5 Strain pro le 0o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . . 105
8.6 Strain pro le 1:5o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . 106
8.7 Strain pro le 3o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . . 107
8.8 Strain pro le 4:5o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . 108
8.9 Strain pro le 6o slip angle sensors 1, 2, 3,and 4 . . . . . . . . . . . . . . . . . . 109
9.1 Slope change 0o slip angle (4 sensors and 4 loads). . . . . . . . . . . . . . . . . . 110
9.2 Slope change 1:5o slip angle (4 sensors and 4 loads). . . . . . . . . . . . . . . . . 111
9.3 Slope change 3o slip angle (4 sensors and 4 loads). . . . . . . . . . . . . . . . . . 112
9.4 Experimental prediction of average contact duration. . . . . . . . . . . . . . . . 113
9.5 Experimental data versus semi-empirical model prediction of normal load. . . . 114
9.6 Normal load comparison between FEM results and experimental data. . . . . . 114
9.7 Slip angle estimation from experimental data. . . . . . . . . . . . . . . . . . . . 115
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9.8 fi?slip comparison between FEM results and experimental data. . . . . . . . . . . 117
9.9 Measured lateral force for all normal loads and slip angle cases. . . . . . . . . . 117
9.10 Averaged lateral force for cases considered. . . . . . . . . . . . . . . . . . . . . . 118
9.11 Lateral force estimation from experiments. . . . . . . . . . . . . . . . . . . . . . 120
9.12 Lateral force comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.13 Lateral force comparison between FEM results and experimental measurement. 121
9.14 Load dependent constant Aexp estimated from normal load. . . . . . . . . . . . 122
9.15 Load dependent constant Bexp estimated from normal load . . . . . . . . . . . . 122
9.16 Load dependent constant F? estimated from normal load. . . . . . . . . . . . . 122
9.17 Load dependent constant G? estimated from normal load. . . . . . . . . . . . . 122
9.18 Load dependent constant H? estimated from normal load. . . . . . . . . . . . . 123
10.1 Basic variables of tire under steady-state rolling . . . . . . . . . . . . . . . . . . 124
10.2 Boundary condition for the steady-state rolling condition . . . . . . . . . . . . . 125
10.3 Longitudinal force under braking and traction. . . . . . . . . . . . . . . . . . . . 126
10.4 Strain pro le under steady-state rolling 0-20% slip ratio . . . . . . . . . . . . . 127
10.5 Strain pro le under steady-state rolling 0-20% slip ratio . . . . . . . . . . . . . 128
10.6 Strain pro le under steady-state rolling 0-20% slip ratio . . . . . . . . . . . . . 129
10.7 Strain pro le under steady-state rolling 0-20% slip ratio . . . . . . . . . . . . . 129
11.1 Slope change 0% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and 1600lbs. 131
11.2 Slope change 5% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and 1600lbs. 132
11.3 Slope change 10% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and 1600lbs. 133
11.4 Slope change 0%-20% slip ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11.5 Relating strain to the slip ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11.6 Analytical relationship between strain and slip ratio. . . . . . . . . . . . . . . . 135
11.7 Slip ratio plotted as function of flfric. . . . . . . . . . . . . . . . . . . . . . . . . 136
xiv
11.8 Longitudinal force under a driving torque for various normal loads. . . . . . . . 137
11.9 Longitudinal force estimated from the semi-empirical model versus the slip ratio. 138
11.10Longitudinal force estimated from the semi-empirical model . . . . . . . . . . . 138
11.11Load dependent constant Afem estimated from normal load. . . . . . . . . . . . 139
11.12Load dependent constant Bfem estimated from normal load . . . . . . . . . . . 139
11.13Load dependent constant c estimated from normal load. . . . . . . . . . . . . . 140
11.14Load dependent constant d estimated from normal load. . . . . . . . . . . . . 140
11.15Load dependent constant e estimated from normal load. . . . . . . . . . . . . . 140
xv
List of Tables
1.1 Tire embedded sensors (Type I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Tire section hyperelastic material properties[30]. . . . . . . . . . . . . . . . . . . 23
2.2 Tire section elastic material properties. . . . . . . . . . . . . . . . . . . . . . . . 23
9.1 Slip angle constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9.2 Lateral force constants (1st constant term) . . . . . . . . . . . . . . . . . . . . . 123
9.3 Lateral force constants (2nd constant term) . . . . . . . . . . . . . . . . . . . . . 123
9.4 Lateral force constants (3rd constant term) . . . . . . . . . . . . . . . . . . . . . 123
xvi
Nomenclature
?slip slip angle [deg]
?I1 rst invariant of Cauchy-Green deformation tensor
?I2 second invariant of Cauchy-Green deformation tensor
?I3 determinant value of deformation gradient matrix
? weibull shape parameter
fi?slip slip angle strain indicator [mm/mm]
fl?? circumferential strain [mm/mm]
flave1 average value of sensor 1 strains in the contact patch [mm/mm]
flave2 average value of sensor 2 strains in the contact patch [mm/mm]
flave3 average value of sensor 3 strains in the contact patch [mm/mm]
flave4 average value of sensor 4 strains in the contact patch [mm/mm]
flfric strain due to frictional load [mm/mm]
flnorm strain due to normal load [mm/mm]
? bulk modulus [MPa]
weibull scale parameter
? shear modulus [MPa]
! angular velocity [rad/s]
xvii
SR slip ratio or longitudinal slip expressed as %
?k relaxation length [mm]
a Weibull scale parameter
Acontact contact area [mm2]
Aexp load dependent constant t to the experimental data
Afem load dependent constant t to FEM data
B Magic formula: sti ness factor
b Weibull shape parameter
Bexp load dependent constant t to the experimental data
Bfem load dependent constant t to FEM data
C Magic formula: shape factor
c Weibull height parameter
C? load dependent constant t to FEM data
c load dependent constant t to FEM data
Cpq material constants related to distortional response
D Magic formula: peak value
D1 material constant related to volumetric response
D? load dependent constant t to FEM data
d load dependent constant t to FEM data
E Magic formula: curvature factor
xviii
E? load dependent constant t to FEM data
e load dependent constant t to FEM data
Eext bridge excitation voltage [V]
F? load dependent constant t to the experimental data
ffem FEM constant
fs sampling frequency [Hz]
fx longitudinal force [N]
fy lateral force [N]
fz normal load on the tire [lbs]
G? load dependent constant t to the experimental data
Gf gage factor
H? load dependent constant t to the experimental data
Hs tire section height [mm]
pinf tire in ation pressure [MPa]
SH Magic formula: horizontal shift
SV Magic formula: vertical shift
t time [sec]
td contact duration [sec]
v vehicle velocity [mm=s]
v0 vehicle velocity [mm/s]
xix
Vout output voltage [V]
Vo axle speed [mm/s]
vx vehicle velocity in x-direction [mm/s]
vy vehicle velocity in y-direction [mm/s]
W strain energy density function
Ws tire section width [mm]
wtread tread width [mm]
X Magic formula: input variable
Y Magic formula: output variable
xx
Chapter 1
Introduction
1.1 Motivation
Structural Health Monitoring (SHM) of pneumatic tires has evolved into an impor-
tant area of research in the tire industry. The main focus of this research has shifted from
simple pressure and temperature monitoring to wirelessly monitoring the tire deformation
using strain sensing principles. The Bridgestone/Firestone tire recall of 2000 in the US re-
sulted in the TREAD[1] act. The Transportation Recall Enhancement, Accountability and
Documentation (TREAD)[1] act which requires the installation of tire pressure monitoring
systems(TPMS)[2] [6]. This has lead various tire manufacturers investigate strain character-
ization of the tire. Apart from the commercially available TPMS[7] [16], tire manufacturers
are exploring ways in which tire strains can be measured and used to enhance overall vehicle
stability and drivability. In short, tire and car manufactures are working on developing an
intelligent tire sensor suite that will monitor the state of a tire in real-time.
What is an intelligent tire? How di erent would it be from the current tire? A modern
pneumatictiredoesnotsensecontactconditionsanddoesnotprovideanydataforcontrolling
vehicle behavior. In contrast, an intelligent tire will contain embedded sensors which will
transmit data wirelessly to the vehicle system. The bene ts of such a sensing system include,
but are not limited to, monitoring road friction, tire forces, tire pressure, tread wear etc.
during normal tire use.
Researchers in the past have developed a wide array of models to extract data from a
tire. Although there are separate models to obtain the slip, normal load, etc., there does
not seem to be a single sensor suite and model capable of predicting the normal load, lateral
force, longitudinal force and wear.
1
1.2 Types of Tire Models
Ring, string and elastic (also viscoelastic) foundation models were widely used in tire
stress and deformation analysis, before computational mechanics became mature enough to
be used in the tire industry. These models were developed extensively by Clark et al.[17].
The equivalent tire parameters required by these models are obtained from full-scale tire
experiments[18]. Given their non complex nature, these models work fairly well for predicting
and understanding certain tire characteristics such as tire vibration, cornering force, braking
and traction. Despite providing an insight into tire behavior, these models had drawbacks
that can?t be overlooked. One of the biggest drawbacks of this type of model[19] is the
need to run extensive experiments to estimate tire parameters. The estimated parameters
only truly apply to the tire for which they were obtained. In the string model[17] the tread
is assumed to be a prestressed membrane. Bending is not considered. The ring model on
the other hand includes bending and the sidewalls are modeled as an elastic foundation
that simply supports the tread. Some more elaborate models where developed where shear
deformations were considered. The character of these models limits their use in the current
tire research.
Figure 1.1: Geometry based cord model[17].
2
With the advent of fast computing more complex models were developed to charac-
terize tire behavior. These models can be broadly categorized as cord and rubber models,
anisotropic ply models and laminate models. This broad classi cation is based on the degree
of complexity that needs to be incorporated into an analysis to obtain a solution.In the cord
and rubber model(see Fig. 1.1) each and every cord is represented by exact dimensions and
the relative positions with respect to the neighboring rubbers. The rubber in this model is
assumed to be isotropic and incompressible. Other components such as the bead wires and
the cords are considered as laments with unique material properties in their normal and
in-plane directions.
Anisotropic ply models are used to grant simplicity. As opposed to the cord model, the
tire structure is represented by plies. The biggest advantage of such a representation is it
allows computation of interply strain, something that is not quite possible in the cord network
model. An added advantage of this model is that is allows non-linear strain distribution over
the tire thickness. The rubber compound is treated as incompressible elements just as in the
cord model. But this model can be used with nite element modeling, as it allows modeling
beads as plies and layups, where the geometrical features and material properties can be
speci ed.
The last set of models are the laminate models (nowadays referred to as nite element
models), which can be viewed as ensembles of shell or solid elements or a combination of the
two. In this type of model plies are grouped together into a laminate and then analysis is
carried out on the entire structure. Mostly isotropic elements are used, but anisotropy and
orthotropy can also be modeled with the necessary computational assistance.
Finite element modeling (FEM) (once known as laminate modeling) has been extensively
employed in the tire research. Continental AG has pioneered FEM[19]of tires since 1970.
Tire FEM in its rst stages(a two dimensional model)[20] [22] was used for simulation of tire
in ation and tire shape change under high rotational speeds. Since then researchers have
used FEM for studying tire footprint, inter-ply shear for tire durability and tire vibrations, all
3
Figure 1.2: Two dimensional axisymmetric model for in ation analysis[17].
based on a three dimensional static model. In the 90?s tire FEM?s were developed for analysis
of tire foot print under rolling conditions[23] [27]. FEM?s were also developed for tire force
and moment predictions[28] [38], which were later combined with theoretical models for the
development of analytical vehicle models. In the recent years tire FEM?s have been developed
using a close to accurate material model, which considers viscoelasticty, hyperelasticity, etc.
These models have accurate geometry with well de ned tread patterns and also consider the
tire model with the metal rim and pressurized air. These models also have the capability
to model accurate loading conditions, which are based on a tire service matrix that can be
used inside the tire simulation[39] [43].
In addition to FEM?s there are models such as the tire brush model[44] and semi-
empirical tire model(also known as the Pacejka model[45]) are in place for modeling tire
behavior under various loading conditions. Pure slip and combined slip conditions can also
be modeled by prescribing the necessary input conditions. The Pacejka model is a dynamic
4
model comprised of equations for lateral force and longitudinal force. These equations are
obtained by tting curves to experimental data for a speci c tire. Although it has come to be
common practice to use these models in tire analysis, the inherent problem is tire speci city.
This implies one global equation cannot be used as the equations pertain to a speci c tire
whose design parameters are taken into consideration. The most famous among the semi-
empirical models is the Pacejka[45] model, from which the magic formula is derived. The
general form of this formula for known values of vertical loads and camber angles is given
by,
y = Dsin[C arctanfBx E(Bx arctanBx)g] (1.1)
with
Y (X) = y(x)+SV (1.2)
x = X +SH (1.3)
where Y: output variable Fx, Fy or Mz , X: input variable tan? or ?, B is the sti ness
factor, C is the shape factor, D is the peak value, E is the curvature factor, SH is the
horizontal shift, and SV is the vertical shift.
The Magic Formula y(x) will produce an asymptotic curve when it passes through the
origin, typically after attaining a maximum peak value. If the necessary coe?cients B, C,
D, and E are known and input into Eq. 1.1, a non-symmetric curve about the abscissa is
obtained with respect to the origin. The parameters SH, and SV are introduced into Eq. 1.1,
to account for the shifts in the curve that are an inherent tire characteristic depending on
the tire loading condition. An example of such a curve which incorporates all the parameters
of the Magic Formula, is shown in Fig. 1.3. The Magic Formula has been shown to provide
tire characteristics, which closely follow data obtained through experiments.
The parameters in the Magic Formula are inferred graphically from Fig. 1.3. The
parameter D is the most obvious as it simply corresponds to the peak value of the curve, the
5
Figure 1.3: Graphical representation of the magic formula[45].
product BCD is claimed to be the value of the slope of the curve at the origin. The shape
factor of the curve C, controls the range of the sine function that will appear in Eq. 1.1.
Knowing the peak and the horizontal asymptote, C is computed as follows,
C = 1?
?
1 2? arcsin yaD
?
(1.4)
The parameters B and C are necessary in computing the parameter E. This allows one to
estimate the occurrence of the peak at say xm and then E is estimated as,
E = Bxm tanf?=(2C)gBx
m arctanfBxmg
(1.5)
Transient tire models are one option for estimating the relaxation length of a tire under
lateral and longitudinal slip. The relaxation length (?k) of a tire is de ned as the ratio of
the lateral sti ness to the cornering sti ness of the tire, assuming an elliptical contact patch.
The relaxation length is given by,
?k = 1(? +a) (1.6)
The transient models can be divided into two categories: 1)Linear model (see Fig. 1.4)
and 2) Fully Non-linear model (see Fig. 1.5). In the linear model the slip characteristics are
restricted to the linear portion of the lateral force versus slip angle curve.
6
In these models the contact patch is treated as a single point that is simply suspended
from the wheel rim by springs in the lateral and the longitudinal directions. The springs are
assumed to represent the tire carcass. With the introduction of tire movement, the contact
patch (point) can have de ections in either the lateral or the longitudinal direction. Based on
the relative motion introduced, lateral force, longitudinal force and the self aligning torque
value is estimated. The forces and moments are estimated by introducing lateral slip and
longitudinal slip parameters. These parameters can then be used in Magic Formula as well
to estimate transient variation from the steady state condition that may exist in the contact
patch.
Figure 1.4: Linear model of a single contact point tire model[45].
Figure 1.5: Fully non-linear string tire model[45].
7
1.3 Types of Tire Sensors
The current tire sensor technology can be broadly classi ed into two categories: I)sensors
embedded in the tire(direct method also noted in Tab. ??) and II)sensors mounted on the
wheel(indirect method).
Table 1.1: Tire embedded sensors (Type I)
Capacitance Magnetic eld Acoustic wave Polymer
Sergio (2003) Side wall torsion
(1999)
Surface acoustic
wave (1998)
PVDF (2008)
Matuzaki (2007) Darmstadt sen-
sor (2000)
Ultrasonic sen-
sor (1998)
MetalRubber
(2008)
The most commonly used TPMS employs a indirect measurement method, where a
wheel-speed based sensor and the anti-lock braking systems (ABS) is used for pressure mea-
surement. Persson et al.[46] proposed an indirect TPMS based on wheel radius and vibration
analysis instead of a pressure sensor. A wheel speed sensor based TPMS was developed by
Kojima et al.[47]. In their work the main focus is on estimating the tire sti ness based on
the observed di erence between the torsional tire sti ness and the tire pressure. Since, the
changes in the road conditions a ect the measured pressure indirectly, the accuracy of an
indirect measurement based system is questionable.
In recent years, direct measurement methods have been developed for TPMS. These
come either as clamp-on rim sensors or valve stem attached sensors for measurering pressure.
The clamp-on type sensor developed by SmartTire System Inc. is simply fastened to the
inner side of the wheel with the help of a clamp. On the other hand the valve-attached
sensor casing is xed to the valve stem. An innovative tire pressure sensor was developed by
Arshak et al.[48]. In that work, a thick- lm capacitor which has an oxide dielectric layer is
used for measuring pressure based on capacitance change.
The operating time or the age of a direct measurement based sensor is an issue because
power is supplied through a battery. In order to overcome this issue, battery-less TPMS?s
have been proposed[49] [54]. In a battery-less system, energy is harvested from the vibrating
8
Figure 1.6: Types of tire sensors.
tire under operation. Some have used piezoelectric reeds in a tire to generate electricity.
While others employ a transponder principle, where the necessary energy is obtained from a
radio signal[55]. The electronic and mechanical robustness of a battery-less TPMS is still in
the research phase. It is to be noted that insu?cient energy could lead to shortened radio
signal range.
The most extensive intelligent tire project ever pursued was the APOLLO tire project
in the European Union. This was setup for the sole purpose of developing an advanced tire
sensor system which can monitor tire deformations [56], temperature[57], etc. in addition
to tire pressure, for occupant safety[57]. These mechatronic tires are commonly known
as "intelligent tires". The strain monitoring in a tire enables one to predict the contact
conditions at the tire-road interface. These predictions can be employed in advanced vehicle
safety systems such as traction control units, vehicle stability assist (VSA), anti-lock braking
system (ABS), etc.
9
For example, an intelligent tire which employs the indirect method, the road friction is
predicted based on the estimated tire force from the di erence in velocities of the driven and
non-driven wheels[58]. Prediction of friction has been shown to improve[59, 60] the e?ciency
of the ABS, by reducing the system response time. Bevly et al. [61] have employed GPS to
estimate wheel-slip, yaw and tire sideslip angle. Control systems based on fuzzy logic[62] [64]
which use data from conventional measurements such as velocity, angular speed and vertical
load on the tire have been shown to correlate to road conditions. Accelerometers mounted
on the tire axle have been used to predict the tire forces. Although many working sensors
are in place, the degree of accuracy based on indirect measurements is questionable.
Figure 1.7: Continental?s[56] side wall torsion sensor. (1) Magnetic eld sensors. (2) Tire
with two magnetized strips on the side wall
In contrast, an intelligent tire which employs the direct method, we can expect to make
more precise measurements of tire operating conditions. The sensors used in the direct
estimation[56] method are either embedded inside the tire or use the tire itself as a sensor to
make precise measurements. The sensing concepts that have been explored are the change
in capacitance, the change in the magnetic eld or the acoustics of the tire.
10
Figure 1.8: Darmstadt tire sensor[56]: Magnet and Hall sensor.
Matsuzaki et al.[65] use a capacitance based method (see Figs. 1.9 and 1.10) to estimate
deformations in a tire. In principle the tire deformation causes a temporary change in the
spacing of the steel wires inside the tread or the carcass causing a change in the measured
impedance or capacitance which can then be related to the desired output quantities. A
exible capacitance change sensor is also proposed, which is of lower sti ness than a typical
strain gage, and is analyzed under only static loading condition (see Fig. 1.11). This leaves
room for the study of inexpensive and reliable resistive strain gages for further exploration.
The array of sensors which are basically magnetic material based sensors, are either embed-
ded in the tire side wall (see Fig. 1.7) or the tread (see Fig. 1.8). In the case of side wall
embedded sensors, the magnetic eld of the individual poles(North and South) are measured.
Whereas, the sensing method for material embedded in the tread works on the Hall-e ect,
where the movement is monitored by a Hall-e ect based sensor[56].
In addition, polymer based sensors have also found a place in characterizing the tire
behavior. Yi et al.[68] propose a polyvinylidene uoride (PVDF) (see Fig. 1.12) based
micro-sensor embedded on the tire sidewall to measure the tread deformation. The PVDF
sensor sits on the base of a exible polyamide substrates and ultra- exible epoxy resin.
This epoxy makes the sensor low in sti ness and high in elongation as a whole structure.
11
Figure 1.9: Oscillating circuit based wireless strain monitoring[66].
Figure 1.10: Capacitance change based wireless strain monitoring[66].
Gondal et al.[69] use a sensor called MetalRubber (see Fig. 1.13). The MetalRubber sensor
has properties that resemble metal (for electrical conductivity) and rubber (for mechanical
exibility). This sensor works very much like a strain gage, where a change in sensor output
voltage can be related to the strains.
A commonly used acoustics based sensor is a surface acoustic wave (SAW) sensor[70]
[75]. A SAW sensor (see Fig. 1.14) uses metallic structures such as interdigital transduc-
ers(IDT) arranged on the surface of a piezoelectric substrate. This then facilitates the
12
Figure 1.11: Flexible capacitance change based wireless strain monitoring[67].
propagation of the SAW over the substrate?s surface. When used in conjunction with a sec-
ond set of IDT?s, the SAW (which is a mechanical signal) can then be converted back into
an electric signal.
Based on the review of the literature on the currently available sensor technologies for
characterizing tire behavior, no resistive strain gage based sensor suites have been proposed
which will be the focus of the current work. A technology road map of the sensor technology
is shown in Fig. 1.15. An array of sensor systems have been proposed in the past. The stark
di erence in making a passive and active smart tire lies in the instrumentation of the tire
itself. Strain measurements form the basis for a passive smart tire and using more advanced
sensor systems with the focus on non contact methods, transient conditions can be predicted.
1.4 Intelligent Tires in the Future
Thetechnologyhasimprovedsincethe rstgenerationtiremonitoringsystems(Fig.1.15).
This generation of tire monitoring systems had a pressure sensor. Termed Tire Pressure
Monitoring System(TPMS) and was a unique solution for monitoring tire in real time. This
TPMS although did not provide the actual pressure instead had a warning signal if the tire
13
Figure 1.12: PVDF-based tire tread deformation sensor[68].(a)Sensor on the side wall. (b)
Output for a periodically applied vertical load
Figure 1.13: Rubber sensor[69] ( also called MetalRubber) by NanoSonic Inc..
was either under in ated or over in ated. The packaging was a multi module package with
an average lifespan of 15 years. The power source is a battery.
The second generation of tire monitoring systems has improved accuracy and a system
level package module. This led to placement issues which were addressed by two methods.
The rst was a valve stem type of arrangement where the pressure sensor was mounted on
to the tire air inlet valve and the pressure transducer had wireless communication enabled.
In the second arrangement the pressure sensor was placed on wheel/hub with a belt type
fastening. This arrangement has a range of 0-70 psi. The current TPMS system is a quad
at pack type of package module.
14
Figure 1.14: Seimens?s[56] SAW sensor with energy generating piezoelectric crystals.
The improvement over the previous array of tire monitoring systems is the shift to
employing low power sensor suites. Wireless communication was integrated with the onboard
sensors thus making this a system on a chip. Although this technology is far better than the
previous one in place, it has its own shortfalls. In future sensor suites should not be limited
by the measurement capability. These sensor suites need to characterize tire in real time.
Major parameters that de ne tire behavior under various conditions are camber, slip, tread
wear, tire pressure and temperature. These sensors may also be extended to predict road
conditions. Current research in this area is led by Pirelli, with a tire integrating bi-axial
accelerometers along with a TPMS.
1.5 Objectives
The purpose of this research is to investigate the feasibility of a resistive strain gage
based sensor suite for tire. Based on the literature concerning development of intelligent tire
sensors, the objectives of this research are:
1. Development of a tire deformation model for static and steady state nite element
analysis.
2. Development of a direct strain based measurement system with wireless data transfer
capability.
15
Figure 1.15: Sensor technology road map.
3. Demonstrate the use of measured strain data in predicting the operating conditions on
the tire.
1.6 Dissertation Organization
This dissertation proposes a theoretical and experimental analysis of strain in a tire
under various operating conditions. A wireless strain monitoring sensor suite based on
resistance changes is proposed. The primary advantage of this sensor suite is the simplicity
of integrating sensors with a tire. The theoretical aspect of this work focuses on developing
a FEM for tire simulation. The dissertation is divided into ten chapters the rst being this
introductory chapter.
Chapter 2 presents the nite element tire deformation model. The tire cross-section
with the various material layers are discussed in this chapter. The underlying assumptions,
the material model, the types of elements used and how a 3D model is developed is also
16
discussed. This 3D model is then veri ed for accuracy based on comparison to experimental
data available in the literature. The nite element model of the tire is fully developed and
the necessary boundary conditions are enforced. The discussion then focuses on reducing
the computational time by performing a mesh convergence.
Chapter 3 presents the results and discussion based on the nite element analysis of the
tire under static loading conditions. The strain pro le under the static loading condition
is investigated. The circumferential strains from FEM are plotted versus the radial node
position and an approximate contact region is estimated from the results.
Chapter 4 presents the experimental setup and the sensor suite used for experimental
validation of the nite element model. The custom built test stand used for running the
experiments is also discussed.
Chapter 5 presents the results and discussion based on experimental analysis of the tire
under static loading conditions. A 235/70 R16 tire is considered in the experiments for the
sole purpose of capturing the strain pro le and in a way validating the nite element model.
Chapter 6 presents the nite element analysis (FEA) for the steady-state free-rolling
condition. Simulations for braking and traction condition are presented. Free-rolling angular
velocity is estimated. Based on the free-rolling angular velocity, parametric cases of slip angle
are considered. An analysis is setup to capture the strain pro le under the steady-state free-
rolling condition.
Chapter 7 presents the strain analysis from the FEM results by various numerical tech-
niques, namely, Fourier Analysis, Weibull Curve Fitting, and Slope Curve Method. Based
on these techniques, the slip angle is estimated. Equations that predict the normal load, slip
angle and lateral force are also presented in this chapter.
Chapter 8 presents the experimental methods for measuring tire strain under the free-
rolling condition. The testing conditions considered are the exactly the same as what were
used in the FEM.
17
Chapter 9 presents an analysis of the experimentally measured strain using the various
techniques (also mentioned in Chapter 7). The validity of each of the techniques is presented.
A quantitative analysis of e ectiveness of each of the numerical techniques is also presented.
Chapter 10 presents the FEA for the steady-state rolling condition. Simulations for
the traction condition are presented. Parametric cases of the slip ratio are considered. An
analysis is setup to capture the strain pro le under the steady-state rolling condition.
Chapter 11 presents the strain analysis from the FEM results by the Slope Curve
Method. Based on this the slip ratio is estimated. Equations that predict the slip ratio
and longitudinal force are also presented in this chapter.
Chapter 12 concludes the dissertation with inferences from the research on the analysis
of strains in a tire and relating them to tire conditions. A suggestion for future work and
improvements for the current sensor technology is also presented.
18
Chapter 2
Static Loading Finite Element Tire Model
The modern tire geometry is a complex structure to model. The complexity arises from
the various sections making up the composite structure of the tire. A typical tire section
(shown in Fig. 2.1)has in all about sixteen sections composing of di erent materials. From
this, it is clear that a close to accurate model is essential to analyze the tire deformation.
Figure 2.1: Typical components of a passenger car radial tire.
Following are the assumptions made in the current tire nite element model:
1. Simpli ed tire cross-section(only Tread, Side wall, Innerliner, Bead ller, Carcass,
Steelbelt and Cord plies are modeled).
2. Combination of elastic and hyperelastic material models.
3. Modeled under isothermal and isobaric conditions.
4. Only static and steady state free-rolling conditions are analyzed.
19
5. FEM nodal outputs assumed to be same as sensor outputs from the experiments.
2.1 Tire Cross-Section
In the current work the cross-section (see Fig. 2.2) of a Standard Reference Test Tire
(SRTT) is modeled. A digital pro le of the tire is imported into SolidEdge for developing
the 2D solid model. The surfacing option in SolidEdge is used to smooth out rough edges in
the digital pro le. This pro le is then imported into ABAQUS(Ver.6.7.1) for developing the
FEM by de ning the sections associated with the SRTT. As mentioned in the assumptions
before, this is a simpli ed tire cross-section and only important sections are modeled. These
Figure 2.2: Tire cross-section.
sections can be divided into solid and surface type sections. The tread, sidewall, innerliner
and bead ller form the solid sections. Whereas the cord ply, steelbelt and carcass which
have elastic properties are modeled as surface sections with the rebar option. The rebar
option in ABAQUS provides for a way to add to structural rigidity to the material model.
The material properties of these sections are discussed in the following section.
20
2.2 Material Model
It has been shown in the past (see Fig. 2.3), based on tensile test data, that the stress-
strain relationship of tire rubber compounds is non-linear but still elastic. The rubber
compounds used in a SRTT exhibits hyperelastcity. A hyperelastic material is de ned as a
material, which derives the stress-strain relationships from a strain energy density function.
In the area of tire research it has become a common practise to employ Mooney[76]-
Rivlin[77] for modeling tire rubber compounds. The strain energy density function for a
Mooney-Rivlin type of material is
W = C01 ?I1 3?+C10 ?I2 3?+D1 ?I3 1?2 (2.1)
where C01 and C10 are empirically determined material constants, D1 is a material constant
related to the volumetric response, and ?I1, ?I2, ?I3 are the rst, second and third invariants of
the strain tensor. The rst invariant which is the dilation, and is also known as the trace of
the strain tensor invariant, is given by,
?I1 = "11 +"22 +"33 (2.2)
The second invariant which is magnitude of the strain tensor is given by,
?I2 = "11 ?"22+"22 ?"33 +"11 ?"33 "212 "223 "213 (2.3)
The third invariant is the determinant value of the strain tensor and is given by,
?I3 = det(fl) (2.4)
For this constitutive material model to be consistent with linear elasticity in the region
where small strains are most important, constraints are applied to the value of the empirical
constants as,
? = 2?D1 (2.5)
21
and
? = 2?(C01 +C10) (2.6)
where ? is the bulk modulus and ? is the shear modulus.
In Fig. 2.3 a stress-strain curve is shown for a rubber compound typically used in
automobile tires. It can be seen that a linear Hooke?s law based material model does not t
well to the tensile test data. The Mooney-Rivlin material model is a better t to the data.
A Neo-Hookian material model, which is also a material model sometimes used for tires, is
a special case of the Mooney-Rivlin model, (it is obtained by setting the C01 constant to
zero).
Figure 2.3: Constitutive material models[79].
22
The material properties of the tire sections modeled are tabulated in Tables 2.1 and 2.2.
These material properties are available in the tire literature[30]. The Mooney-Rivlin con-
stants used in the current work are that of a rubber compound commonly used in automobile
tires.
Table 2.1: Tire section hyperelastic material properties[30].
Tire sections Mooney-Rivlin constants (MPa)C
10 C01 D1
Tread 0.8061 1.8050 0.0191
Sidewall 0.1718 0.8303 0.0498
Inner liner 0.1404 0.4270 0.0881
Bead ller 14.140 21.260 0.0014
The cord plies, steel belt and the carcass sections of the tire are the anisotropic section
composites made up of mostly steel, rubber and ber reinforcements. The primary composi-
tion of the tire carcass varies with each tire, but for the most part the carcass can be assumed
to be made of rubber and organic bers like nylon, polyester etc. The belt sections of the
tire (some parts included in the carcass) also has rubber for the most part and steel wires in
a meshed fashion are embedded in the rubber compound. The anisotropy nature of the steel
wires and organic bers is introduced due to the rebar orientation angle that is speci ed as
an input option while de ning the material model in ABAQUS. The steel orientations were
set at 118o while the cord ply orientations were set at 70o. The coulomb friction between the
tire and the road is modeled as frictionless for the static nite element model and is later
changed t0 ?=0.85 for the steady state free-rolling nite element model.
Table 2.2: Tire section elastic material properties.
Tire
sections
Area
(mm2)
Poisson?s
ratio
Young?s
Modulus
(GPa)
Cord ply 0.28 0.25 3.97
Steel belt 0.30 0.33 200
Carcass 0.30 0.30 2.48
23
2.3 Three-Dimensional Tire Model
The three-dimensional tire model is obtained by revolving the two-dimensional cross
section area about the tire axis of rotation. The symmetric model generation technique,
which is a convenient technique available in ABAQUS[78], is utilized to generate the sym-
metric tire deformation model. The types of elements (see Fig. 2.4) used in the nite element
model are as follows,
1. The solid sections are modeled as a combination of 8-node brick elements and 6-node
triangular prism elements.
2. The embedded sections (carcass, steel belt and cord ply) are modeled using special
elements in ABAQUS. One such special element available in ABAQUS is called the
surface element. This is simply a 4-node membrane element with zero thickness. These
surface elements also come with a rebar option, which can be speci ed to add structural
rigidity to the model.
Figure 2.4: Element types used.
24
Figure 2.5: Enlarged view of ABAQUS elements used in the FEM.
25
2.4 Finite Element Analysis
2.4.1 Boundary Conditions
The boundary conditions for the normal load case that were enforced on the tire are
depicted in Fig. 2.6. The hub is constrained in all directions throughout the length of the
simulation. Contact interaction property is speci ed at the tire/rigid surface interface. The
Lagrange multiplier algorithm is used in the contact property de nition. Interference is
applied to the rigid surface (in the z-direction) and the reaction force obtained at the hub is
equal to the normal load that is acting on the tire. This method is employed because there
is a considerable reduction in the simulation time and it converges more easily than other
methods. In addition, the tangential behavior on the rigid surface is modeled as frictionless
(this is changed later). The in ation pressure is set to 0.2 MPa (29 psi) for all cases.
Figure 2.6: Three dimensional tire model.
26
2.4.2 Mesh Convergence and Veri cation of Model Accuracy
The next step is to perform a mesh convergence for the nite element model developed
from the three-dimensional solid model of the tire. The nal mesh is chosen by comparing
the highest nodal stress at a particular node to the stress value at the integration point (some
times referred to as a Gauss point). This also serves as a way to reduce the computational
e ort by reducing the number of elements (see Fig. 2.7) used in a simulation.
The mesh convergence and element reduction technique is applied to the current model
by reaching an optimal value of the tire sti ness. The sti ness of a SRTT is about 240
Nmm 1[39]. The sti ness value of the tire modeled in the current work is obtained from
the ratio of the reaction force on the hub to the applied interference. The model accuracy
is then veri ed by plotting the FEM results against standard ASTM test data (Fig. 2.8). It
can be seen that the FEM predictions are fairly close to the test data, showing reasonable
accuracy.
Figure 2.7: Mesh convergence and reduction in computational e ort for coarse mesh opti-
mization.
27
0 2 4 6 8 10 12 14 16 18 20
0
1000
2000
3000
4000
5000
6000
7000
Reaction Force (N)
Deflection (mm)
FEM Prediction
ASTM Test Data for SRTT
Figure 2.8: FEM results compared to ASTM test data for a SRTT.
2.5 Preliminary FEA Results
2.5.1 FEM Prediction of Contact Patch Area
The preliminary FEA results in this section pertain to the contact patch region. In order
to develop theoretical models, which will predict the contact patch area, it is important to
analyze how the contact patch dimensions vary with the normal load.
Figure 2.9 shows the contour results for the displacement in the normal direction, which
is the same direction in which the load of 1000lbs is applied as de ection on the tire control
node. The parametric cases performed have the same boundary conditions as mentioned
before in section 2.4.1. It can be seen that for this static condition the contact patch width
(simply the tread width) and the length appear to be straight lines. Hence the contact area
would simply be the product of the tread width and the contact length.
28
Figure 2.9: Contour plot of displacement in the z-direction (a)1000lbs Normal
load;(b)Contact patch at 1000lbs.
29
Next the load is increased to 1200lbs(equivalent to approximately 15mm de ection), the
contour results of the z-direction displacements results are shown in Fig. 2.10. As expected
the contact patch area still appears to be a rectangle. This means that the contact patch
length for the most part can be assumed to be invariant of the axial location. The load is
then increased to 1400lbs(equivalent to approximately 17.5mm de ection). The displacement
results is shown in Fig. 2.11. The contact patch at this load predicted by the FEM is similar
to the earlier loads that were considered. Therefore the contact patch area can be easily
predicted by multiplying the width times the contact length. This will be used later when
correlating strains to normal load.
Figure 2.10: Contour plot of displacement in the z-direction (a)1200lbs Normal
load;(b)Contact patch at 1200lbs.
The load is then increased to 1600lbs(equivalent to approximately 20mm de ection).
This load is also very close to the maximum load rating of a SRTT(which is about 1653lbs).
At this load the contact patch under static loading still resembles a rectangle. The contour
plot of displacement results is shown in Fig. 2.12.
30
Figure 2.11: Contour plot of displacement in the z-direction (a)1400lbs Normal
load;(b)Contact patch at 1400lbs.
31
Figure 2.12: Contour plot of displacement in the z-direction (a)1600lbs Normal
load;(b)Contact patch at 1600lbs.
2.5.2 FEM Prediction of Strain on the Tire Innerliner and Footprint
The strain predictions from the nite element model is analyzed in this section, with
special focus on the tire innerliner and the tire rigid surface contact interface, sometimes
referred to as the tire footprint or contact patch.
The strains (flxx) along the longitudinal direction (x-direction) is shown in Fig. 2.13. For
the sake of consistency, strain predictions from only the 1600lbs normal loading condition
are considered. In Fig. 2.13(a), the paths connecting the nodes from which the strain values
are extracted, in the contact region, are illustrated by dotted lines. The footprint strains in
the contact patch region are shown in Fig. 2.13(b). The strains on the inside and the outside
of a tire are studied in order to understand which strain tensor component or combination of
components correlates with the applied load. As expected, the strains in the footprint have
higher values compared to strains in the innerliner. From the contour plot, it becomes clear
32
Figure 2.13: Strain prediction flxx for 1600lbs normal load;(a)Innerliner;(b)Contact patch.
that the strains along paths (illustrated by dotted lines) vary signi cantly with applied load.
The outer nodal path predicts same trends but is signi cantly di erent from what nodes
along the inner nodal paths predict. One reason for this behavior arises from the anisotropy
of the steel belts and cord plies. Upon in ation the tire expands asymmetrically about the
plane on which the rigid surface is created. This leads to an edge e ect where the tire bends
along this plane causing a di erence in trend prediction by the outer and the inner nodal
paths.
The strains along the lateral direction (flyy) are shown in Fig. 2.14. This component of
strain is believed to have little to no e ect on the circumferential strains that will eventually
be measured on the surface of the tire innerliner. Even though the strain predictions appear
to be signi cant in this contour plots, the actual values from the FEM are fairly low on
the contact patch region and the tire innerliner itself. The reason being when the hoop
is decomposed into its necessary components, there is no lateral component of the strain
33
Figure 2.14: Strain prediction flyy for 1600lbs normal load;(a)Innerliner;(b)Contact patch.
tensor. A qualitative comparison is presented in the following section where circumferential
strains are compared to the lateral strains.
The strains along the normal load or z-direction are shown in Fig. 2.15. The strains on
the innerliner (Fig. 2.14(a)) and the strains in the contact patch region (Fig. 2.14(b)) show
a signi cant value change in the normal direction. This is the same direction in which the
normal load (an equivalent interference of 20mm) is applied on the tire control node. At this
load it can be seen that all four nodal paths (illustrated by dotted lines) show signi cant
variation in their trends in response to the applied load. The edge e ect due to sidewalls is
evident from the trends predicted by the nodal paths on the inside and outside. This also
causes uneven deformation along the longitudinal direction. This is one reason (addressed
in chapter 4) why the nodal paths on the outside predict similar trends and the nodal paths
on the inside predict similar trends.
34
Figure 2.15: Strain prediction flzz for 1600lbs normal load;(a)Innerliner;(b)Contact patch.
Figure 2.16: Strain prediction flxy for 1600lbs normal load;(a)Innerliner;(b)Contact patch.
35
The static nite element model does not account for friction behavior (this changed
later in the steady state model) since the tangential direction material behavior is modeled
as frictionless. The strains in the xy-direction is shown in Fig. 2.16. Nothing signi cant to be
noted occurs in this direction and hence the strains along this direction is ignored as it does
not form part the hoop direction. The same is also true for the strains in the yz-direction
as it as no in uence in the hoop direction. The strains from the nite element model for the
yz-direction is shown in Fig. 2.17.
Figure 2.17: Strain prediction flyz for 1600lbs normal load;(a)Innerliner;(b)Contact patch.
The strain in the xz-direction is not dependent on the friction behavior in the tangential
behavior but should be accounted for in the strains in the hoop direction. The contour plot
of strains in the xz-direction is shown in Fig. 2.18. In e ect of this component of strain
can be visualized by assuming a mid-line that separates the footprint (Fig. 2.18(b)) into
two. From the contour it is evident that the strains in this direction varies signi cantly
along each of the nodal paths from the inside to the outside along the lateral direction. The
36
in uence of each of the strain tensor component is crucial in establishing the strain along
the circumferential direction.
Figure 2.18: Strain prediction flzx for 1600lbs normal load;(a)Innerliner;(b)Contact patch.
37
2.5.3 Comparison Between Circumferential Strains and Lateral Strains
When a tire is loaded in the direction normal to the rigid surface at the rigid surface/tire
interface, a tensile strain distribution occurs on the innerliner of the tire in the hoop direction
(see Fig. 2.19), and the region outside the contact patch experiences a compressive strain
distribution.
Figure 2.19: Tire deformation and strain prediction fl?? due to a normal load:(a) Normal
load;(b)Strain pro le
When only the normal load acts on the tire and the rigid surface is assumed to be
frictionless, a symmetric strain distribution (Fig.. 2.19(b)) appears about the abscissa.
38
It is now clear that if strain sensors were to be mounted on the innerliner of the tire,
the strain sensed would be the circumferential or the hoop strains. In order to make the case
for the choice of hoop strain, a comparison between the hoop strain fl?? and lateral strain flyy
is shown in Fig. 2.20. The FEM results from the 1600lbs normal load case is considered in
this example. Now it becomes clear that the hoop strains predict a trend which is consistent
with the strain distribution that will occur when a symmetric elastic member is deformed
about the axis of symmetry. Despite the deformation that occurs in the lateral direction,
lateral strains (flyy) show no clear trend with the applied load. For this simple reason, trends
based on the circumferential strains is adopted for developing empirical models for slip angle
and normal load in the later chapters of this dissertation.
?0.5 ?0.4 ?0.3 ?0.2 ?0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.005
0.01
0.015
0.02
0.025
Node Position (rad)
strain (mm/mm)
e
qq
e
yy
Sensor 2
Sensor 3
Sensor 1
Sensor 4
Figure 2.20: Strain prediction fl?? and flyy comparison at 1600lbs normal load.
39
Chapter 3
Strain Analysis Based on Static Loading Finite Element Prediction
The location of strain gages is crucial with respect to capturing the contact patch
dimensions. In this work the nodal solutions of the nite element model used approximately
to predict the circumferential strains that will be measured from experiments. The sensor
location (shown in Fig. 3.1) correspond to the nodes that equidistant are from each other
and run circularly around the tire. From here on forth sensors 1 through 4 will represent
node paths from the nite element model. The node path comprises of nodes which contain
strain results from the integration points approximated at nodes, and extracted to make the
necessary predictions.
Figure 3.1: Sensor locations approximated to nodal solutions.
The nodal solutions correlate to sensors as the following:
1. All sensors are equidistant from each other
2. Sensor 1 is on the edge of contact patch (tire outside)
40
3. Sensors 2 and 3 are on the mid region
4. Sensor 4 is on the edge of contact patch (tire inside)
3.1 Results and Discussion
Figure 3.2: Schematic showing the ?=0 rad node position.
The hoop strains predicted from the FEM are plotted versus the radial node position
(see schematic shown in Fig. 3.2) around the tire for parametrically varied cases of normal
load. The strain predictions from this analysis are an approximate indicator of the contact
patch length for a given case of normal load. In Fig. 3.3, the strain pro le for the normal
load case of 1000lbs is shown. The nodes that go in and out of contact experience a gradual
increase and decrease in strain. A maximum strain of approximately 2.25% strain occurs at
the edges of contact and is almost the same value throughout the contact.
When the normal load is increased to 1200lbs (Fig. 3.4) it can be seen that more nodes
enter the contact patch zone. This is an indicator that the contact patch length increases
with the applied load. When the nodes enter and exit the contact patch, the strain at these
two extremities are in compression. The compressive strains are in reality realized as the
41
?0.5 ?0.4 ?0.3 ?0.2 ?0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.005
0.01
0.015
0.02
0.025
Node Position (rad)
e
q
q
(mm/mm)
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 3.3: Strain prediction at 1000lbs normal load.
strains in the circumferential or the hoop direction. A similar observation can be be made
from the 1400lbs(Fig. 3.5) and the 1600lbs(Fig. 3.6) load cases.
It is interesting to notice that sensors 1 and 4 and sensors 2 and 3 predict very similar
trends but middle gages produce smaller magnitudes than the outer gages. The reason for
the non-uniform behavior could be attributed to some sort of an edge e ect[17] that comes
into play when the tire is loaded in the normal direction. In Fig. 3.7, the contact length
progression along sensor 1 for all loads considered in the FEM is shown. It can be seen that
the contact length increases with an increase in the normal load. This is a direct indication
that the contact patch area (which is simply the product of contact length and the tread
width) increases as well.
Since the model in the current work approximates the tire geometry by considering a
reduced number of tire sections, the predicted trends need to be validated experimentally.
In order to validate the the nite element model, experiments with very similar loading
42
?0.5 ?0.4 ?0.3 ?0.2 ?0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.005
0.01
0.015
0.02
0.025
Node Position (rad)
e
q
q
(mm/mm)
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 3.4: Strain prediction at 1200lbs normal load.
?0.5 ?0.4 ?0.3 ?0.2 ?0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.005
0.01
0.015
0.02
0.025
Node Position (rad)
e
q
q
(mm/mm)
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 3.5: Strain prediction at 1400lbs normal load.
43
?0.5 ?0.4 ?0.3 ?0.2 ?0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.005
0.01
0.015
0.02
0.025
Node Position (rad)
e
q
q
(mm/mm)
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 3.6: Strain prediction at 1600lbs normal load.
conditions have to conducted, to try and see if similar trends are captured. This forms the
primary discussion in the next chapter.
44
Figure 3.7: Strain prediction along sensor 1 for normal loads 1000lbs-1600lbs.
45
Chapter 4
Static Loading Experimental Set-up
4.1 Tire Test Stand for Experimental Validation
Experiments for the validation of the nite element model and evaluation of the new
sensor suite and resulting strain/condition relations under static loading condition (also
discussedinthepreviouschapter)areperformedonacustombuiltteststandattheAdvanced
Propulsion Research Laboratory, Auburn University, Auburn. The tire test stand has two
important features that is used to replicate the static loading condition used in the nite
element model. The rst feature is the rigid at surface that is restrained in all directions
and which also acts as the tire-road interface. The second feature is the capability of varying
the normal load exerted on the tire. This is achieved by simply changing the air pressure
of a pneumatic cylinder (see Fig. 4.1), which is mounted inline with the tire?s rotational
axis. A load cell is also mounted inline with the cylinder so that the load on the tire can
be measured accurately. This fairly simple test setup (see Fig. 4.2) replicates the normal
loading condition modeled using nite elements.
Figure 4.1: AU test stand (a) Pneumatic cylinder; (b) Pressure control unit.
46
Figure 4.2: Custom built tire test stand for static loading condition.
47
4.2 Mechanical Layout of Hardware
Figure 4.3: Schematic of the hardware arrangement.
The strain sensors are mounted on the innerliner of the tire as shown in Fig. 4.4. The
sensors are placed equidistant from each other (similar to the placement mentioned in chapter
3). Sensor 1 is on the edge of contact patch (tire outside). Sensors 2 and 3 are mounted
in the mid region over the width of the tread. Sensor 4 is on the edge of contact patch
(tire inside). The sensor suite for the current work is packaged onto an aluminum bracket
speci cally built for holding all the necessary components. The mechanical components in
the layout are,
1. Aluminum bracket (see Fig. 4.5) for holding the wireless transmitter and the signal
conditioning circuit.
48
Figure 4.4: Strain sensor placement on the innerLiner of the tire.
2. A single array of strain sensors (see Fig. 4.4) are placed equidistant from each other,
while covering the entire span of the tread width. This is done so in hopes that for
every revolution of the tire the entire contact patch strains are captured.
3. A three-piece wheel (see Fig. 4.6) is used for reducing tire mounting e ort once the
sensor suite is secured to the wheel.
4. The nal arrangement of the sensor suite before the tire is mounting the tire is shown
in Fig. 4.7.
5. A CN strain gage adhesive and RTV black silicone is used for securing the strain gages
to the innerliner of the tire.
6. A printed circuit board of the signal conditioning circuit is also built to considering
various issues pertaining to tire mounting.
49
Figure 4.5: Wireless transmitter and signal conditioning circuit arrangement on a bracket.
Figure 4.6: Three-Piece Wheel.
50
Figure 4.7: Final arrangement of bracket on the wheel.
4.3 Electrical Layout of Hardware
The predesign of the electrical circuit used as a signal conditioning circuit is shown
in Fig.4.8. A quarter wheatstone bridge is employed in the current work. The bridge has
a nominal resistance of 120 and an excitation voltage of 3.3 V. The strain gage acts as
the variable resistor in the quarter bridge wheatstone network, as shown in Fig. 4.9. The
measured strains is in ?v, this small voltage measured is ampli ed with the help of a signal
conditioning circuit. The signal conditioning circuit has a instrumentation ampli er. The
gain resistor in the instrumentation ampli er is modi ed accordingly to amplify the measured
voltage.
The output from the quarter bridge, where one of the member is a strain gage, is related
to the measured strain. This change in output from when the bridge is balanced is related
to the resistance change of the strain gage. A resolution of .05% can be achieved with the
current amplifying circuit. Knowing the gage factor of the strain gage, the measured output
51
Figure 4.8: Pre-design of signal conditioning circuit for strain measurements.
voltage is converted to the strain fl by using the following equation:
Vout
Eext =
Gf ?"?10 3
(4+2?Gf ?"?10 6) (4.1)
where Vout is measured output voltage, Eext is the bridge excitation voltage, fl is the strain
and Gf is the gage factor, typically found to be 2.1. The gage factor is given by the following
equation:
Gf = R=R L=L (4.2)
A CrossBow XDA100 wireless/MICA2 DAQ(Fig. 4.10) system is used in the current
work for wireless data transmission and collection. The wireless data transmitter used in
52
Figure 4.9: Quarter bridge wheatstone network.
the current work has a sampling time of 0.1 s (fs=10Hz). The XDA100 has a 51-pin pro-
grammable prototyping area and comes with an onboard temperature sensor. The electrical
layout (Fig. 4.11) is comprised of the wireless transmitter/receiver, signal conditioning for
voltage ampli cation of outputs from strain sensors. The size of this circuit is signi cantly
reduced by building a PCB. The size reduction helped reduce the mechanical e ort in mount-
ing the tire onto the wheel. The reduced e ort in turn helps avoid damage to the strain
gages as a result of the tire mounting process. The custom built sensor suite in it?s current
form is not ready for industrial use. A higher sampling frequency, smaller package size of
the sensing module, and the over all integration with the wheel needs to be addressed before
this sensor suite can become commercially available.
53
Figure 4.10: XDA100 and MICA2 wireless module[80].
Figure 4.11: Electrical layout.
54
Chapter 5
Strain Analysis Based on Static Loading Experimental Data
An approach similar to the one discussed in the previous chapter is adopted in exper-
iments for static loading cases. The strain gages are placed (see Fig. 4.4) along the lateral
direction capturing the width of the tread. A 235/70 R16 (di erent from FEM tire) is con-
sidered in experiments. The goal of the experiments for the static loading condition is to
validate the trends predicted by the FEM, demonstrate the created hardware and to develop
relationships between normal load and strain. A SRTT does not have to be necessarily used
for this purpose since the analysis is qualitative in nature.
Figure 5.1: Data transmission from sensor to data processing unit.
55
5.1 Results and Discussion
The strains from the normal load case of 800lbs are shown in Fig. 5.2. A maximum
strain of approximately 2% strain is measured for this load case. The strains vary gradually
as the strain gages go in and out of contact and remain fairly constant outside the contact
patch region. A similar trend is noticed for the higher load case of 1300lbs (see Fig. 5.3).
In this load case a maximum value of 2.5% strain is reached. It can also be seen that the
contact patch length increases with an increase in load. The trends from the experiments
match fairly close to the trends predicted by the FEM. A similar conclusion can be made
about the edge e ect. It can be seen that sensors 1 and 4 measure approximately the same
magnitude of strains. Sensors 2 and 3 behave in a similar fashion.
Figure 5.2: Strain measurements for 800lbs normal load.
56
The load on the tire is then increased to 1300lbs and the circumferential strain pro le is
shown in Fig. 5.3. As expected at this load the maximum strain value in the contact patch
is more than what was measured from the 800lbs normal load case. This simply means
that the applied load is directly proportional to tire de ection in the normal direction. The
same trend is observed in the FEM where the simulation is displacement controlled and the
reaction force at the wheel hub is the same as the normal load.
Figure 5.3: Strain measurements for 1300lbs normal load.
A comparison can be made between the trends predicted by the FEM and the experi-
ments. In Fig. 5.4, for the normal load case of 1000lbs, strain predictions from the FEM is
compared to the strains from the experiments. It can be seen that trends are similar although
quantitatively di erent. Although a di erent tire is considered in the experiments, the strain
pro le, which is of importance in this exercise, seems to qualitatively follow closely with the
57
FEM predictions. The experimental comparison is simply used to validate the FEM before
more complicated conditions involving slip and surface friction are modeled. The reasons for
di erence in magnitude of strains between FEM and experiments are as follows:
1. Di erent tires are considered (the tire size in FEM is 225/60 R16 & the tire size in the
experiments is 235/70 R16).
2. Model inaccuracies due to di erent material compounds in the tire structure.
3. Neglected tire sections in the FEM.
Figure 5.4: FEM results compared to 235/70 R16 tire at 1025lbs.
58
Chapter 6
Steady-State Free Rolling Finite Element Model
6.1 Straight Line Rolling
Figure 6.1: Braking and Traction simulation for free-rolling angular velocity (!) extraction.
A straight line rolling analysis is a combination of a braking and traction simulation (see
Fig. 6.1). The purpose of this analysis is to obtain the free rolling equilibrium solution (that
is the torque on the axle is zero) for a velocity of 10 kph on a rigid at surface. To achieve
59
this equilibrium condition a full braking and full traction condition is required. In the rst
step for the full braking condition the corresponding angular velocity is estimated as follows.
A free rolling tire will travel farther in one revolution than determined by the loaded tire
radius, but less than determined by the unloaded tire radius. The unloaded tire radius is
339.38 mm and for a maximum vertical de ection of 20 mm, the wheel center height=319.38
mm. Now using the unloaded radius and applied de ection, it is estimated that free rolling
occurs somewhere between an angular velocity !=8.18 rad/s and 8.69 rad/s. It is clear that
smaller angular velocities will result in braking and larger angular velocities would result in
traction. Hence an angular velocity !=8.0 rad/s is used to run the braking simulation to
ensure a steady-state solution is in the braking condition region.
Figure 6.2: Rolling resistance versus angular velocity for free-rolling angular velocity (!)
extraction.
60
In the next step the angular velocity is gradually increased to !=9.0 rad/s while keeping
the wheel velocity constant. Since the solution at the end of each time step is a steady-state
solution each of the time step would correspond to a steady state solution under the full
braking and traction condition.
To better understand when free-rolling occurs, rolling resistance is plotted (Fig. 6.2)
versus the angular velocity. From Fig. 6.2, it can be seen that free rolling occurs when the
torque on the axle is zero and thus providing an estimate of the free rolling angular velocity.
A partial traction analysis (see inset Fig. 6.2), which is performed in the neighborhood of
the estimated angular velocity, is next performed to obtain a more re ned solution of the
free rolling angular velocity.
6.2 Steady-State Free-Rolling at Various Slip Angles
Figure 6.3: Lateral tread deformation under steady state cornering.
61
Once the free-rolling angular velocity is estimated it is used in all the steady state
simulations involving various slip angles. By de nition, slip angle is the angle between the
vehicle heading direction and the tire heading direction. Slip generally occurs while the
vehicle is negotiating a corner or turn. This is oftentimes referred to as cornering. The
e ect of cornering induces a force in the lateral direction. The lateral force is self induced
by the tire in response to the direction in which the tire is being steered. The lateral tread
deformation that occurs while cornering is pictorially depicted in Fig. 6.3.
The boundary conditions for the slip angle simulation are shown in Fig. 6.4. For this
simulation it is required that both the translational and rotational velocities be supplied. The
translation velocity, which in the free-rolling case is the wheel velocity, is decomposed into
longitudinal and lateral components. And the rotational velocity is the free-rolling angular
velocity estimated in the braking and traction simulations. To incorporate slip angles from
0o to 6o, the longitudinal and lateral components of the vehicle velocity are supplied to the
FEA code as,
vx = v0cos(?) vy = v0sin(?) (6.1)
where v0 is the vehicle velocity and vx and vy are longitudinal and lateral components
respectively.
All other boundary conditions are kept intact for the parametric normal load(1000lbs,
1200lbs, 1400lbs and 1600lbs) cases. Each case simply involves restarting the whole sim-
ulation for a particular load that needs to be modeled. The simulation time for each the
cases considered varies. The low normal load (1000 lbs) cases have a shorter simulation time
while the high normal loads (1600lbs) have a longer simulation times. The average time to
completion of each simulation varies from 20 mins to 2 hrs on a computer with 4 GB RAM
and 3 GHz speed.
62
Figure 6.4: Boundary condition for free-rolling at various slip angles.
6.3 Strain Pro le Under Steady-State Free-Rolling
In Fig. 6.5, the strains predicted along the path of sensor 1 for the 1000lbs normal load
and all the slip angles modeled (0o 6o) is shown. Fig. 6.5 shows that with an increase in
slip angle the predicted strains around the tire decreases. At 0o slip angle, which is also the
straight line rolling condition, the strains around the tire are symmetric about the abscissa.
The strains predicted from the 1:5o slip angle case, as can be seen, are also asymmetric about
the x-axis. The asymmetry becomes more pronounced with an increase in the slip angle for a
given normal loading condition. From this unsymmetrical strain prediction it can been seen
that a set of leading and trailing edges are formed. The tread deforms more in the lateral
direction in the leading edge region caused by the tire induced lateral force. The region in
which the tread does not deform as much, leads to the formation of the trailing edge. From
this Fig. 6.5 it can also be seen that contact duration, which is an indicator of the contact
length, remains practically constant with change in slip angle for a given particular load.
63
In the rst set of gures (see Fig. 6.6), the strain pro les of normal loads from 1000lbs to
1600lbs in increments of 200lbs and 0o slip angle are shown. When the nodes go in and out
of contact, depending on the tire headed direction, a leading and trailing edge is formed (see
Fig. 6.3). For this particular case where a slip angle of 0o exists, the strain distribution is
fairly symmetric about the x-axis. This is consistent with the fact that tire is not cornering
and hence a lateral force will not be generated.
0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53
0
0.005
0.01
0.015
0.02
0.025
0.03
time (sec)
e
q
q
(mm/mm)
0
o
Slip angle 1.5
o
Slip angle 3
o
Slip angle 4.5
o
Slip angle 6
o
Slip angle
Leading Edge
Contact
Length
0
o
6
o
Trailing Edge
Normal load 1000 lbs
Figure 6.5: Strain pro le 0o 6o slip angle sensor 1; Normal load 1000lbs.
When tire starts to undergo cornering (ie. ?slip > 0o), like in Fig. 6.7, it can be seen
that the leading and trailing edges start to di er and the strain pro le is not symmetric
about the x-axis anymore. This is because the tire induces a force in the lateral direction to
counter the e ect of slip angle. In doing so the tread deforms in the lateral direction. This
phenomenon becomes fairly clear when the slip angle is further increased. It is interesting
to note that the average contact duration of these sensors is not a function of slip angle. It
appears to be dependent only on the load carried by the tire.
With a su?cient increase in the slip angle, the cornering force produced by the tire
tends to dominate the tire deformation. At a slip angle of 3o (see Fig. 6.8) it should be
64
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Figure 6.6: Strain pro le 0o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Trailing
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Figure 6.7: Strain pro le 1:5o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
65
noted that the circumferential strains measured on the tire innerliner become the same for
all values of normal load. At this slip angle the lateral deformation seems to be dominating.
This can be seen from how the strain curves for sensors 2-4 shift in the direction of lateral
deformation. To a certain extent the same e ect can be seen for the slip angle case of 4:5o
(Fig. 6.9 for normal loads of 1000lbs-1400lbs for all the sensors and then when the normal
load is increased to 1600lbs sensor 2 suddenly drops while sensors 3 and 4 still deform in the
lateral direction.
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Figure 6.8: Strain pro le 3o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
The lateral tread deformation peaks for the 6o slip angle (see Fig. 6.10) relative to the
lower slip angle values. Once this value of slip angle is reached it can be seen that there is
not much di erence between the 4:5o slip angle case and the 6o slip angle case (Also shown
in Fig. 6.5). This is because once the peak value of lateral force is attained (as limited by
the friction coe?cient), a force saturation occurs and the force no longer increases with slip
66
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Trailing
Edge
Leading
Edge
Trailing
Edge
Leading
Edge
Trailing
Edge
Leading
Edge
Trailing
Edge
Leading
Edge
Figure 6.9: Strain pro le 4:5o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
angle. This would mean the tread can deform no more in the direction in which the lateral
force is acting.
67
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
Edge
Trailing
Edge
Leading
Edge
Trailing
Edge
Leading
Edge
Leading
Edge
Trailing
Edge
Trailing
Edge
Figure 6.10: Strain pro le 6o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
68
Chapter 7
Strain Analysis Based on Steady-State Free-Rolling Finite Element Prediction
The strain predictions from the FEM are analyzed using various numerical techniques.
The biggest challenge lies in developing relationships between tire operating conditions and
the measured or the predicted strain. Hence this chapter evaluates several of di erent meth-
ods that might be used.
7.1 Fourier Analysis
The Fourier analysis is a subject area in mathematics which came into existence from
analyzing a Fourier series. The Fourier formula of a typical Fourier series involving 2?-
periodic functions with sines and cosines for a function f(x) that is integratable over the the
period [ ?;?] is given by,
F (s) = a02 +
NX
n=1
[an cos(nx)+bn sin(nx)] (7.1)
where,
a0 = 1?
8;?
?
f (x)dx (7.2)
an = 1?
8;?
?
f (x)cos(nx)dx (7.3)
bn = 1?
8;?
?
f (x)sin(nx)dx (7.4)
The Fourier transform, is a technique that involves transforming the data from the
time domain to the frequency domain. One way to e?ciently implement this technique is
to use Fast Fourier Transform (FFT) to compute the fourier transform. This method is
69
implemented in the current work by taking the FFT of the strain data over the contact
patch region. The FFT of the strain data yields (see Fig. 7.1) the frequency spectrum. In
Fig. 7.1 the strain data from sensor 1 is considered for the Fourier analysis for various slip
angles and a normal load of 1200lbs. It can be seen that the amplitude changes occur with
di erent slip angles, but remain fairly constant for upto 100Hz.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.01
0.02
time (sec)
Hoop strain
e
q
q
10
0
10
1
10
2
0
0.05
0.1
0.15
Frequency (Hz)
Amplitude
Sensor1@ 0
o
SA
Sensor1@ 1.5
o
SA
Sensor1@ 3
o
SA
Sensor1@ 4.5
o
SA
Sensor1@ 6
o
SA
Figure 7.1: Fast Fourier Transform of the strain data (Sensor 1; 1200lbs normal load).
Fig. 7.2 shows the strain pro le along the path of sensor 2 and the FFT of the strain data.
It can be seen that the frequencies from 1 to 20 or so all contribute to the entire spectrum.
An FFT of sensor 3 (see Fig. 7.3) and sensor 4 (see Fig. 7.4) show no variation with normal
load. Amplitude change alone is not su?cient to quantify the necessary parameters.
The strain pro le and the FFT over the data from sensors 3 is shown in Fig. 7.3. For
the rst few frequencies the amplitude change is noticed when the slip angle is varied. The
amplitude change is maximum for the 0o slip angle case and with changes in the slip the
amplitude too changes. The length of contact however is not captured even though the
strain data is for the contact patch region. This missing information can be a factor that is
70
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.01
0.02
0.03
time (sec)
Hoop strain
e
q
q
10
0
10
1
10
2
0
0.1
0.2
Frequency (Hz)
Amplitude
Sensor2@ 0
o
SA
Sensor2@ 1.5
o
SA
Sensor2@ 3
o
SA
Sensor2@ 4.5
o
SA
Sensor2@ 6
o
SA
Figure 7.2: Fast Fourier Transform of the strain data (Sensor 2; 1200lbs normal load).
in uenced by the resolution, in this case the number of elements, being insu?cient for this
method to work. Next the strains along sensor 4 (see Fig. 7.4) is analyzed. The same trend
as seen with strains along the other sensors is realized. The amplitude change is noticed
when the slip angle varies, but the contact length is not to be noticed with change in the
normal load. This issue is further analyzed by extracting the rst few frequencies and then
all the parameters necessary, which are the normal load and slip angle, are plotted in uniform
three dimensional array.
An FFT is then performed over the parametric loading cases. The amplitudes from the
rst few frequencies is then analyzed. Figs. 7.5 7.8 are the plots of the parametric cases of
normal load and slip angles. Normal load of 1000lbs, 1200lbs, 1400lbs and 1600lbs and slip
angles of 0o-6o are considered. It can be seen that the extracted amplitude changes with slip
angle but does not exhibit any signi cant change with the normal load. From this method
only slip angle can be predicted from all the contributing frequencies, the other parameter
(normal load) cannot be estimated from this method.
71
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.01
0.015
0.02
0.025
time (sec)
Hoop strain
e
q
q
10
0
10
1
10
2
0
0.1
0.2
Frequency (Hz)
Amplitude
Sensor3@ 0
o
SA
Sensor3@ 1.5
o
SA
Sensor3@ 3
o
SA
Sensor3@ 4.5
o
SA
Sensor3@ 6
o
SA
Figure 7.3: Fast Fourier Transform of the strain data (Sensor 3; 1200lbs normal load).
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.01
0.02
0.03
time (sec)
Hoop strain
e
q
q
10
0
10
1
10
2
0
0.05
0.1
0.15
Frequency (Hz)
Amplitude
Sensor4@ 0
o
SA
Sensor4@ 1.5
o
SA
Sensor4@ 3
o
SA
Sensor4@ 4.5
o
SA
Sensor4@ 6
o
SA
Figure 7.4: Fast Fourier Transform of the strain data (Sensor 4; 1200lbs normal load).
72
0
1
2
3
4
5
6
1000
1200
1400
1600
0
0.05
0.1
0.15
0.2
Slip angle (deg)
Normal load (lbs)
Amplitude
FEM data 1000 lbs
FEM data 1200 lbs
FEM data 1400 lbs
FEM data 1600 lbs
Figure 7.5: Relating amp. changes to normal load and slip angle 2nd frequency (sensor 1).
0
1
2
3
4
5
6
1000
1200
1400
1600
0
0.05
0.1
0.15
0.2
Slip angle (deg)
Normal load (lbs)
Amplitude
FEM data 1000 lbs
FEM data 1200 lbs
FEM data 1400 lbs
FEM data 1600 lbs
Figure 7.6: Relating amp. changes to normal load and slip angle 3rd frequency (sensor 1).
73
0
1
2
3
4
5
6
1000
1200
1400
1600
0
0.05
0.1
0.15
0.2
Slip angle (deg)
Normal load (lbs)
Amplitude
FEM data 1000 lbs
FEM data 1200 lbs
FEM data 1400 lbs
FEM data 1600 lbs
Figure 7.7: Relating amp. changes to normal load and slip angle 4th frequency (sensor 1).
0
1
2
3
4
5
6
1000
1200
1400
1600
0
0.05
0.1
0.15
0.2
Slip angle (deg)
Normal load (lbs)
Amplitude
FEM data 1000 lbs
FEM data 1200 lbs
FEM data 1400 lbs
FEM data 1600 lbs
Figure 7.8: Relating amp. changes to normal load and slip angle 5th frequency (sensor 1).
74
7.2 Weibull Curve Fitting
The Weibull distribution is typically used in reliability studies where life cycle predic-
tions are made based on the data distribution. It is a versatile and easy to use distribution
which can take the characteristics of other distributions, simply based on the shape param-
eter ?. An example of Weibull distribution is shown in Fig. 7.9. A function that is used for
making predictions is as follows,
f (x) =
(
?
x
?? 1 e (x
)
?
x>=0
0 x<0
(7.5)
where ? is the shape parameter and is the scale parameter. And as the name suggest the
way the shape and scale of the curve changes, the parameters that in uence these changes
can be predicted.
Figure 7.9: Weibull distribution for life cycle prediction[81].
The strain data from the FEM predictions appears to follow the shape of a Weibull
type function. Therefore one might be able to t to the data in order to relate strain to
normal load and slip angle. In Fig. 7.10, which is the strain pro le for varying slip angles
75
from sensor 1 for 1200lbs normal load, a Weibull type function is t to the strain data. This
function is given by,
"?? =
c
a
?(1
a)
x
a
?(b
a 1) e(
x
a )
b
c (7.6)
where a is the scale parameter, b is the shape parameter and c is the height parameter. Now,
this method simply involves tting this equation to all the strain data and then relating
the function parameters to the loading conditions. The biggest drawback of this method
is that it involves tting too many constants. Also multiple ts are possible for the same
strain data and so the Weibull parameters then do not relate directly to changes in load
conditions. Hence the prediction model will become highly tire speci c. Weibull curve ts
to sensors 2, 3, and 4 are shown in Figs. 7.11 7.13 respectively. It is clear that good ts
can be generated for observed strains in a tire but the underlying fact is that this is a very
cumbersome process and the number of constants involved is too many.
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0
0.005
0.01
0.015
0.02
time (sec)
Hoop strain (
e
q
q
)
FEM data?>Sensor 1 @ 0 SA
FEM data?>Sensor 1 @ 1.5 SA
FEM data?>Sensor 1 @ 3 SA
FEM data?>Sensor 1 @ 4.5 SA
FEM data?>Sensor 1 @ 6 SA
Weibull fit?>Sensor 1 @ 0 SA
Weibull fit?>Sensor 1 @ 1.5 SA
Weibull fit?>Sensor 1 @ 3 SA
Weibull fit?>Sensor 1 @ 4.5 SA
Weibull fit?>Sensor 1 @ 6 SA
Figure 7.10: Weibull type function t to strain data (sensor 1).
76
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
time (sec)
Hoop strain (
e
q
q
)
FEM data?>Sensor 2 @ 0 SA
FEM data?>Sensor 2 @ 1.5 SA
FEM data?>Sensor 2 @ 3 SA
FEM data?>Sensor 2 @ 4.5 SA
FEM data?>Sensor 2 @ 6 SA
Weibull fit?>Sensor 2 @ 0 SA
Weibull fit?>Sensor 2 @ 1.5 SA
Weibull fit?>Sensor 2 @ 3 SA
Weibull fit?>Sensor 2 @ 4.5 SA
Weibull fit?>Sensor 2 @ 6 SA
Figure 7.11: Weibull type function t to strain data (sensor 2).
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
time (sec)
Hoop strain (
e
q
q
)
FEM data?>Sensor 3 @ 0 SA
FEM data?>Sensor 3 @ 1.5 SA
FEM data?>Sensor 3 @ 3 SA
FEM data?>Sensor 3 @ 4.5 SA
FEM data?>Sensor 3 @ 6 SA
Weibull fit?>Sensor 3 @ 0 SA
Weibull fit?>Sensor 3 @ 1.5 SA
Weibull fit?>Sensor 3 @ 3 SA
Weibull fit?>Sensor 3 @ 4.5 SA
Weibull fit?>Sensor 3 @ 6 SA
Figure 7.12: Weibull type function t to strain data (sensor 3).
77
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
time (sec)
Hoop strain (
e
q
q
)
FEM data?>Sensor 4 @ 0 SA
FEM data?>Sensor 4 @ 1.5 SA
FEM data?>Sensor 4 @ 3 SA
FEM data?>Sensor 4 @ 4.5 SA
FEM data?>Sensor 4 @ 6 SA
Weibull fit?>Sensor 4 @ 0 SA
Weibull fit?>Sensor 4 @ 1.5 SA
Weibull fit?>Sensor 4 @ 3 SA
Weibull fit?>Sensor 4 @ 4.5 SA
Weibull fit?>Sensor 4 @ 6 SA
Figure 7.13: Weibull type function t to strain data (sensor 4).
7.3 Slope Curve Method
In the slope curve method the slope or gradient of the data is predicted by the change in
the strain between two consecutive nodes. The basic idea behind this method is that maxima
and minima of slope changes would typically occur at the edges of contact. This is because
when the sensors go in and out of the contact patch, they will sense maximum strains at the
entry and exit points of the patch. And so this would then correspond to two spikes in the
slope change pro le, one each for a maxima and a minima. The contact duration which is
the time di erence between the instances of these spikes can then be estimated. The slope
change is calculated using the center di erence method which is done iteratively over all the
nodes under consideration. The center di erence for slope change calculation is as follows,
? "
??
t
?
i
= ("i+1 "i 1)(t
i+1 ti 1)
(7.7)
78
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1600 lbs
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 7.14: Slope change 0o slip angle (4 sensors and 4 loads).
In order to estimate the contact area under the cornering condition (ie. ?slip > 0o), it is
imperative to know if the contact patch area changes with slip angle. Intuitively speaking,
for a given normal load without any cornering, if the load is symmetric and the tread and
road are at, then the entire width of the tread should be in contact with road. To better
understand the e ect of cornering on a contact patch area, the slope change method is
applied to parametric cases of slip angle and normal load. The free-rolling condition with 0o
slip angle (see Fig. 7.14) is a good starting point to see how the slope change method can be
used e ectively. For the 0o slip angle case the slope change prediction is shown in Fig. 7.14.
It can be seen that every sensor as it goes in and out of contact generates two peaks which
correspond to a maxima and a minima. The di erence in time between the occurrences of a
maxima and a minima is simply the contact duration.
Next, the slope change for the 1:5o slip angle case is shown in Fig. 7.15. From this gure
it can be seen that the spikes corresponding to the maximum and minimum change in the
79
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1600 lbs
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 7.15: Slope change 1:5o slip angle (4 sensors and 4 loads).
slope with respect to time remains constant with the change in slip angle and only changes
when the normal load is varied.
The 3o slip angle case shown in Fig. 7.16, also exhibits a similar trend, where the
peaks (for a particular case of normal load) occur at the same time indicating that the
maximum strain values are realized at the edges of the contact. It should be noted that
the magnitude of the maximum and minimum slope change decreases with increases in slip
angle. A possible reason for this occurrence at higher slip angles could be attributed to
high values of the lateral force which could cause the lateral tread deformation making it
slightly more pronounced than when there is no slip angle. An interesting thing to note from
the slope curve method is the prediction of the contact duration. It is clear that contact
duration, which the time di erence between the maximum and minimum values of the slope
change, remains a constant with change in slip angle. The contact duration appears to be
only a function of the load on the tire. This suggests that the average of the contact duration
predicted by all four sensors can be used to predict the load on the tire.
80
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.04
?0.02
0
0.02
0.04
time (sec)
D
e
q
q
/
D
t
1600 lbs
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 7.16: Slope change 3o slip angle (4 sensors and 4 loads).
7.4 Evaluation of Methods for Strain Analysis
The various methods for strain analysis mentioned in the previous sections have their
own advantages and disadvantages. To help one narrow down on using the above mentioned
techniques, the following observations might prove useful,
Notes on the Fourier analysis are
1. Is a powerful tool, but requires a high sampling frequency to e ectively capture trends.
2. Fairly simple procedure once the amplitudes and frequency are extracted from the
data.
3. Computationally expensive method.
4. May not be feasible for real-time data processing.
Notes on the Weibull curve tting are
81
1. Multiple functions can be made to follow strain data.
2. Too many constants involved, making the model very tire speci c.
3. Weibull parameters don?t relate to changes in load conditions.
Notes on slope curve method are
1. Fairly simple procedure for parameter estimation.
2. Does not require a high sampling frequency.
3. Slope change could be ampli ed if the data is noisy. As an alternative something like
"total-variation regularization" technique (a method for derivative calculation of noisy
data) could be used if the data appears too noisy.
These observations are purely based on the FEM, and since the slope change method works
best with the strain data, it is further explored to relate the necessary parameters (normal
load, slip angle and lateral force). The steps involved in predicting the necessary parameters
based on slope change method are as follows:
1. Collect the strain data.
2. Employ slope curve method.
3. Determine the maximum and minimum change in slope
4. Determine the average contact duration over all four sensors (underlying assump-
tion:contact patch area does not change with slip angle).
5. Estimate the contact area based on average contact duration (this would be a function
of the vehicle speed).
6. Predict normal load based on estimated contact area.
Based on the methods explored using the FEM data the procedure shown in Fig. 7.17
will be adopted.
82
Figure 7.17: Procedure for parameter prediction based on methods explored.
7.5 Normal Load Estimation
The normal load estimation is based on the fundamental equation relating force and
pressure when two objects are in contact. This stems primarily from the fact that the load
is carried mostly by the in ation pressure rather than the structure of the tire carcass[17].
The equation is given below,
fz = pinf ?Acontact (7.8)
where fz is the normal load, pinf is in ation pressure and Acontact is the contact area. Eq. 7.8
is a fairly good estimate of the normal load on the tire. In practice the pressure term in the
above equation needs an additional factor for the equation to work because some of the load
is actually carried by the carcass.
83
0.04 0.045 0.05 0.055 0.06 0.065
1000
1100
1200
1300
1400
1500
1600
Average Contact Duration (t
d
) (sec)
Normal Load (lbs)
0 1 2 3 4 5 6
0.04
0.045
0.05
0.055
0.06
0.065
a
slip
t
d
(sec)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
FEM Prediction
Figure 7.18: FEM prediction of average contact duration.
The predictions from the FEM show that contact area is independent of the slip angle
and is only dependent on the normal load applied to the tire. This prediction is used to
estimate the contact area, as it can be simply approximated as follows,
Acontact = v ?td ?wtread (7.9)
where v is the velocity of the vehicle, td is the average contact duration time and wtread is
the width of the tread. The above mentioned equation can also be inferred from Fig. 7.18.
In this gure an FEM prediction of the average contact duration for varying normal loads is
shown. Once the contact duration is known (this is also an indirect measure of the contact
length), the contact area(Acontact) can be quickly estimated with Eq. 7.9.
Now Eq. 7.8 can be modi ed to incorporate the in ation pressure factor by introducing
a additional factor, and so,
fz = ffem ?(pinf)?Acontact (7.10)
where the additional factor ffem is t to FEM data.
84
Next the normal load estimated from Eq. 7.10 is plotted (see Fig. 7.19 versus the true
normal load (reaction force at the hub). A good correlation is present and the average error
between the theoretical model and the nite element model is less than 5%.
4000 4500 5000 5500 6000 6500 7000 7500
4000
4500
5000
5500
6000
6500
7000
7500
Normal Load FEM (N)
Normal Load Semi?Empirical Model (N)
Figure 7.19: FEM versus semi-empirical model prediction of normal load.
7.6 Slip Angle Estimation
7.6.1 Innerliner Deformation Due to Slip Angle
The tire innerliner deformation ideally should follow the tread deformation while the
vehicle is cornering. If this is true, then a strain sensor mounted on the innerliner will
measure the lateral tread deformation for a given normal load. The relationship that might
exist between slip angle and the deformation (which is nothing but the strain) becomes fairly
simple to develop.
The top-down view of a tire contact patch is shown in Fig. 7.20. For the sake of
visualization, sensors are represented as S1, S2, S3, and S4. In the current work these labels
refer to strain gages. Under the straight line rolling condition, the tread deforms only in
85
Figure 7.20: Relating lateral tread deformation to the slip angle.
86
the longitudinal direction. And since it is assumed that the tread and the innerliner deform
simultaneously, the sensors should measure the same strains. When cornering occurs, the
tread will deform in the direction in which lateral force acts. The sensors on the innerliner
now measure the lateral deformation as strains due to the slip angle. The contact patch can
be divided into two sections (see Fig. 7.20), section A and section B, in order to relate the
lateral deformation to the slip angle. Section A consists of sensors 1 and 2 and section B
consists of sensors 3 and 4. The relationship between slip angle and the lateral deformation
can be mathematically expressed as a sum of the di erences of the strains in section A and
section B. This is as follows,
?slip = ("4 +"3) ("2 +"1) (7.11)
The physical meaning behind Eq. 7.11 can be inferred by analyzing the changes in the
stress distribution in the innerliner due to slip angle. The von-Mises stress from the 0o slip
angle and 1200 lbs normal load (this normal load case, for the sake of argument is used in
the rest of this section) condition, which is the straight line rolling condition, is shown in
Fig. 7.21. The stress distribution about the xz plane is symmetric as predicted by the FEM.
This means that when the sensors go in and out of the contact patch, the lateral deformation
is not sensed, as there is only deformation along the longitudinal direction. From Eq. 7.11
it can be quickly concluded that the sum of strain di erence between section A and section
B is zero.
The stress distribution for the 1:5o slip angle case is shown in Fig. 7.22. It can be seen
that when the slip angle is enforced, that the stress distribution is no longer symmetric about
the xz plane. The stresses in section B are more than in section A. This di erence is a result
of the lateral deformation in response to the slip angle. So when the strain sensors go in
and out of contact, depending on the direction in which the tread is deforming, a di erence
in deformation is sensed by all the sensors. Going back to the section assignment it can be
seen that the di erence of the sum of strains in each section is simply a function of the slip
angle and Eq. 7.11 is just one way to understand the relationship.
87
Figure 7.21: von-Mises stress distribution on the innerliner 0o slip angle and 1200 lbs normal
load.
Figure 7.22: von-Mises stress distribution on the innerliner 1:5o slip angle and 1200 lbs
normal load.
88
Figure 7.23: von-Mises stress distribution on the innerliner 3o slip angle and 1200 lbs normal
load.
Next the stress distribution for the 3o slip angle case is analyzed. The von-Mises stress
distribution is shown in Fig. 7.23. With the increase in slip angle, the stresses in the tire
increase as well. As the slip angle increases a signi cant amount of stress is developed in
section B due to the lateral tread deformation. Again when the sensors trace the contact
patch, a di erence in the measured strain can be related to the slip angle. As this slip angle
the di erence in stress values between each individual sensor itself appears to be signi cant.
Since the strains are of more importance to this work, the stress distribution is simply looked
at as realizing a physical signi cance to the proposed Eq. 7.11.
The von-Mises stress distribution for the 4:5o slip angle case is shown in Fig. 7.24. At
this slip angle condition the intensity of the stress distribution in section B becomes more
pronounced. The intense distribution extends almost over the entire contact length in section
B. Sensors 1 and 2 for every revolution of the tire will sense the strain in response to the
lateral deformation. This is also an implication that the strain di erence between section
B and section A can be related to the slip angle itself for a given load. To put things
in perspective, what is important to note is that even though the circumferential strains
89
predicted by the FEM show an asymmetric trend, the stress distribution appears to be fairly
symmetric.
The e ect of the friction at the tire road interface also has to be accounted for, as shear
stress is induced in the contact patch region. The stress values predicted by the FEM is
slightly larger in magnitude than one might expect. The reason for the elevated values is
due to the ?=0.85 value used. This a ects the lateral deformation as well.
Figure 7.24: von-Mises stress distribution on the innerliner 4:5o slip angle and 1200 lbs
normal load.
In Fig. 7.25, the von-Mises stress distribution for the 6o slip angle is shown. At this slip
angle it can be seen that there is a signi cant distribution of stress is over the entire contact
length in section B. This is an indication that the strains in the innerliner are a ected as
a result of the slip angle. This simply means that the di erence in strains between section
A and section B enlarges with increase in slip angle. The variations that may exist even
under controlled conditions can be easily accounted for by simply averaging the strains over
the entire contact patch. These average values of strains can then be used to develop the
necessary equations that relate the tire strains to the slip angle.
90
Figure 7.25: von-Mises stress distribution on the innerliner 6o slip angle and 1200 lbs normal
load.
7.6.2 Analytical Relationship Between Strain and Slip Angle
In this section an analytical relationship between the strain and the slip angle is pre-
sented. The lateral deformation of the tire innerliner practically follows the lateral tread
deformation under the steady state free-rolling condition. The geometry of the contact
patch (see Fig. 7.26) can be used to derive an analytical expression which will be used to
support the claims in the previous section. The contact patch can be divided into two sec-
tions A and B (this is also discussed in the previous section). The outer and the middle edges
and the middle and inner edges as shown in Fig. 7.26 form sections A and B respectively.
Assuming an original length L, the strains along the outer edge Louter and the inner edge
Linner can be analytically expressed as follows,
flouter = LouterL ;flinner = LinnerL (7.12)
Where the outer edge Louter is given by,
Louter =
h
Rave + wtread2
i
??patch (7.13)
91
Figure 7.26: Analytical relationship between slip angle and strain.
and the inner edge Linner is given by,
Louter =
h
Rave wtread2
i
??patch (7.14)
In the above mentioned equations (Eq. 7.13 and Eq. 7.14), wtread is the width of the tread,
Rave is the radius of curvature given by,
Rave = j1+(x
0)2j32
jx00j (7.15)
and ?patch is the contact patch angle given by,
?patch ? LaveR
ave
(7.16)
Now, from Eqs. 7.13, 7.14 and 7.12 the following conclusion is attained,
flouter > flinner (7.17)
It is clear from the above relationship that the strains measured in section A will be greater
than that of the strains measured in section B ( this is also consistent with the claims made
in section 7.6.1).
92
7.6.3 Relating Strain to the Slip Angle
Theestimationoftheslipangleundersteadystatefree-rollingconditionwillbediscussed
in this section. It is a fact that load distribution occurs between the tires on a vehicle when
negotiating a turn. For example, when a car with front wheel drive is negotiating a turn (see
Fig. 7.27), the front tires and the rear tires react from cornering in a di erent fashion. The
front wheels (because of the steering angle), tend to have a larger value of slip angle(?slip)
than the rear wheels. An equilibrium is then reached by simply redistributing the normal
load, which occurs instantaneously as a reaction to the lateral force. This is one reason why
lateral force induced under such a scenario is termed as a self-induced tire force.
Figure 7.27: Normal load distribution while cornering.
The estimation of slip angle based on the strain becomes rather straight forward once
the load on the tire is estimated. The slip angle can be related to the strains by the following
variable, delta?slip
fi?slip = (("ave4 +"ave3) ("ave2 +"ave1)) (7.18)
where flave1, flave2, flave3 and flave4 are the average value of the strains in the contact patch. This
average value is calculated from knowing the average contact duration, which indirectly gives
93
an estimate of the contact length. And, fi?slip is a new term introduced in this estimation
process and is referred to as the slip angle strain indicator.
Next, the slip angle is plotted versus fi?slip (see Fig. 7.28)to obtain an equation relating
the slip angle to the measured strain. The t to the strain data results in an equation, which
is of the form,
?slip = AfemeBfem?fi?slip +Cfem (7.19)
where Afem, Bfem, and Cfem are load dependent constants t to the FEM data, Hs is tire
section height, Ws is section width and Scv is the suspension comfort value. The estimation
of the constants from the resulting ts is discussed in section 7.8.1. It is to be noted that
these equations serve as a tool for initial starting point while analyzing the experimental
data.
1 2 3 4 5 6 7 8 9 10
x 10
?4
0
1
2
3
4
5
6
d
a
slip
(mm/mm)
Slip Angle (deg)
FEM Data 1000lbs
FEM Data 1200lbs
FEM Data 1400lbs
FEM Data 1600lbs
EM 1000lbs
EM 1200lbs
EM 1400lbs
EM 1600lbs
Figure 7.28: Slip angle estimation from FEM.
94
7.7 Lateral Force Estimation
The tire forces and moments in reality are a complex non-linear function of tire variables
which are established based on inputs and vehicle output responses. The primary focus of
this work is to establish a relationship between slip angle and cornering force under steady
state free-rolling condition. The SAE tire axis system shown in Fig. 7.29 is used in de ning
the necessary parameters. According to the SAE de nition, the cornering force is simply
the lateral force when a side slip in the form of slip angle exists. For the sake of consistency
lateral force will be used and the condition of steady state free-rolling under various slip
angles will be referred to as cornering.
Figure 7.29: The SAE tire axis system[82].
The lateral force for all the normal load slip angle cases considered is shown in Fig. 7.30.
Since every load step in the FEM corresponds to a steady state solution, the lateral force
for a particular slip angle of interest is extracted from the FEM. It can be seen that with
the increase in load the slip at which the peak force occurs also changes accordingly. For
example the 1000lbs normal load case when considered, the lateral force peaks out at about
95
0 1 2 3 4 5 6
0
1000
2000
3000
4000
5000
Slip Angle (deg)
Lateral Force (N)
FEM data 1000 lbs
FEM data 1200 lbs
FEM data 1400 lbs
FEM data 1600 lbs
Figure 7.30: Lateral force prediction from FEM.
4:5o of slip angle and tends to remain until a peak value corresponding to 6o slip angle is
attained.
The lateral force acting on a tire is a function of the slip angle (?slip). The FEM
prediction of lateral force versus slip angle is shown in Fig. 7.31. The current model predicts
a peak lateral force at 6o slip angle. This suggests that a non-linear polynomial would be
su?cient to develop an empirical model. A quadratic polynomial function is then t to the
FEM data. The resulting equation is as follows,
fy = C? ?(?slip)2 +D? ?(?slip)+E? (7.20)
where C?, D?, E? are load dependent constants t to the FEM data. Equations relating
the FEM constants to the normal load are shown in section 7.8.2. Eq. 7.20 should not used
outside the range of the data considered here.
In Fig. 7.32, the lateral force estimated from the semi-empirical model versus the lateral
force measured through experiments is shown. An average error of 5% error exists between
the predicted and the measured values of the lateral force.
96
0 1 2 3 4 5 6
0
1000
2000
3000
4000
5000
Slip Angle (deg)
Lateral Force (N)
FEM data 1000 lbs
FEM data 1200 lbs
FEM data 1400 lbs
FEM data 1600 lbs
EM 1000 lbs
EM 1200 lbs
EM 1400 lbs
EM 1600 lbs
Figure 7.31: Lateral force estimation from FEM.
1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
Lateral Force FEM (N)
Lateral Force Semi?Empirical Model (N)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
Direct relationship
Figure 7.32: Lateral force comparison between the semi-empirical model prediction and the
FEM prediction.
97
7.8 Estimating Constants of the Semi-Empirical Model Based on FEM Results
7.8.1 Constants in the Slip Angle Empirical Model
1000 1100 1200 1300 1400 1500 1600
8
8.5
9
9.5
10
Normal Load (lbs)
A
fem
A
fem
Eq.(7.10)
Figure 7.33: Load dependent constant
Afem estimated from normal load.
1000 1100 1200 1300 1400 1500 1600
?13000
?12000
?11000
?10000
?9000
?8000
?7000
?6000
Normal Load (lbs)
B
fem
B
fem
Eq.(7.11)
Figure 7.34: Load dependent constant
Bfem estimated from normal load
Constants (Afem) (also shown in Fig. 7.33) obtained from the slip angle empirical model
for the parametric normal loading cases follows a linear relationship with the load and is
estimated graphically by tting an equation. A linear function t to the Afem values results
in Eq.7.21.
Afem = 0:004363?(fz)+3:425 (7.21)
Next the constant terms Bfem are plotted versus the normal load and is shown in Fig. 7.34.
These values are also only functions of the normal load. A quadratic equation is t to the
Bfem values and this results in Eq. refeq711.
Bfem = 0:02153?(fz)2 +67:86?(fz) 5:953e4 (7.22)
7.8.2 Constants in the Lateral Force Empirical Model
The next load dependent constant term C?, is estimated for the lateral force empirical
model (Eq. 7.20). C? is rst plotted versus the load (see Fig. 7.35) and an equation is t.
The resulting equation is a quadratic equation of the form,
C? = 3:313?10 5 ?(fz)2 +0:1191?(fz) 45:86 (7.23)
98
1000 1100 1200 1300 1400 1500 1600
?150
?148
?146
?144
?142
?140
?138
?136
?134
?132
Normal Load (lbs)
C
a
C
a
Eq.(7.12)
Figure 7.35: Load dependent constant C?
estimated from normal load.
1000 1100 1200 1300 1400 1500 1600
1450
1500
1550
1600
1650
1700
1750
1800
Normal Load (lbs)
D
a
D
a
Eq.(7.13)
Figure 7.36: Load dependent constant D?
estimated from normal load.
The D? constant term from Eq. 7.20 is plotted versus the normal load. This is shown in
Fig. 7.36. A quadratic polynomial is t to the data and the resulting equation is of the form,
D? = 0:00015?(fz)2 +1:086?(fz)+48:42 (7.24)
The last set of constant terms E? from Eq. reflafeq for the parametric normal load case
1000 1100 1200 1300 1400 1500 1600
?70
?60
?50
?40
?30
?20
Normal Load (lbs)
E
a
E
a
Eq.(7.14)
Figure 7.37: Load dependent constant E? estimated from normal load.
is shown in Fig. 7.37. This constant term follows a quadratic trend and hence a quadratic
polynomial it t to the data. The resulting equation is of the form,
E? = 0:0001294?(fz)2 +0:2378?(fz) 121:3 (7.25)
99
Chapter 8
Steady-State Free Rolling Experimental Methodology
8.1 Test Bed for Experimental Validation
The steady-state free-rolling experiments were conducted on a MTS FlatTrac test bed
(see Fig. 8.1). The important features of the FlatTrac test bed are as follows,
1. Actuating pneumatic cylinders that can apply a vertical load of up to 5000lbs.
2. Uniform belt speed of 7kph.
3. Capability of incremental testing, where slip angle and normal load can be varied in
increments independently.
4. Sweep testing where normal load increases incrementally and then sweeps slip angle
from negative to positive.
5. Lateral tread movement through laser based optical sensor.
6. External force sensors on the wheel unit.
7. Fully automated system with high speed DAQ boards.
The same hardware used in the static loading experimental study to measure strain in the
tire is used here. Following is the list of components:
1. Crossbow wireless DAQ: Mica2 reciever & XDA100 wireless transmitter.
2. PCB of signal conditioning circuit.
3. Strain sensors mounted on the tire innerliner.
100
Figure 8.1: MTS FlatTracr?test bed courtesy: General Motors.
101
8.2 Steady-State Rolling at Various Slip Angles
The steady state response to slip angle can be carried out by free-rolling cornering with
an incremental test. In this test (which was conducted on the FlatTrac test bed(see Fig. 8.3))
the slip angle (?slip) is increased incrementally and then the normal load is also increased,
to create the necessary test conditions that were considered in the FEM. The following are
the speci c test conditions used,
1. Both slip angles and normal loads have xed values at every test condition.
2. Ample rolling distance is given to tire at each test point once slip angle and normal
load reach the speci ed value.
3. Ample rolling distance to ensure attainment of steady state equilibrium condition.
4. Uniform belt speed of 7kph.
5. Force data averaged to suppress tire non-uniformity.
Figure 8.2: Sideview of tread on the belt: MTS FlatTracr?test bed courtesy General Motors.
102
Figure 8.3: MTS FlatTracr?test bed courtesy: General Motors.
103
8.3 Strain Pro le Under Steady-State Free-Rolling
The steady state free rolling experiments were performed at a belt speed of 7kph. For
this belt speed, the strain measurements around the tire for one revolution per second is
considered. Normal loads were varied from 1000lbs to 1600lbs in increments of 200lbs and
slip angles from 0o-6odeg. Due to limitations in the sampling rate of the wireless transmitter
(10Hz), only a few data points could be taken during each revolution. The initial position
of the tire (from 0o phase with respect to the tire axis) is incremented by 1o in each test to
map the strain at a higher resolution around the tire. Each of the test was run over a period
of 20 seconds. Thus allowing to capture 360 data points over 36 tests by superimposing
the results. The method adopted for taking the data is pictorially shown in Fig. 8.4. The
hoop strain predictions from experiments does not show a gradual strain pro le as seen in
the predictions from the FEM. Notes on possible reasons that might be responsible for this
behavior is listed at the end of this section.
Figure 8.4: Schematic for acquiring test data:(a) 0o phase; (b) 1o phase.
Figure 8.5 shows the hoop strain prediction for all the strain gages under the pure rolling
condition with varying normal load. It can be seen that the contact duration predicted by
104
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
edge
Leading
edge
Leading
edge
Leading
edge
Trailing
edge
Trailing
edge
Trailing
edge
Trailing
edge
Figure 8.5: Strain pro le 0o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
all the sensors is the same for a given load. And the contact duration increases with an
increase in normal load. The strains seem to jump to a maximum value of about .022?strain
while in contact and fall down to a smaller value when out of contact. This happens over
all the loads for which the testing was done.
In the next test case the slip angle is incremented to 1:5o and the normal loads are
varied from 1000lbs to 1600lbs with gradual increments of 200lbs. The strain pro le of this
test case is shown in Fig. 8.6. At 1000lbs and 1:5o slip angle a gradual formation of the
leading and trailing edges can be noticed (also see Fig. 6.3). The cornering e ect is more
pronounced in the FEM as compared to experiments because the tire used in experiments
could be slightly sti er. This however cannot be veri ed because the exact rubber compounds
of the test tire is not known and at this time only a hypothesis can be made to understand
the experimental results. The same trend appears to be occurring with the increase in load
and when a normal load value of 1600lbs is reached the trends momentarily disappear. One
105
reason could be because the maximum load rating of the test tire is 1653lbs and at such high
loads the tire might be experiencing bending and compression and not exactly undergoing
cornering.
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Figure 8.6: Strain pro le 1:5o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
In the third test case the slip angle is increased to 3o and the testing is carried out for all
the normal loads which are of interest. The strains from this test are shown in Fig. 8.7. From
the gure it can be seen that the e ect of cornering is captured to a reasonable level. All
the sensors tend to exhibit a strain pro le which can be deduced into a leading and trailing
edge. A 225/60 R16 (SRTT) tire is a popular choice for an sport?s utility vehicle who?s curb
weight is in the neighborhood of 1200lbs. At this load and a 3o slip angle a clear shift in
the curve is produced indicating cornering under the steady state free rolling condition. The
lateral force generated by the tire at this slip angle starts to dominate the tire deformation
by de ecting the tread in the lateral direction. Surprisingly, at this slip angle and a normal
106
load of 1600lbs it can be seen that a trailing and leading edge in fact is formed, another
indication that the generated lateral force tends to deform the tread laterally.
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Figure 8.7: Strain pro le 3o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
The slip angle is then increased to 4:5o and a parametric variation of the normal loading
is performed over this case. The strain pro le is shown in Fig. 8.8. In this test case scenario
the tire is nearing its peak saturation point in terms of the lateral force. It is interesting to
note that the lower normal load at this slip angle barely forms a leading and a trailing edge
and is more prevalent when the normal is at higher values of 1200lbs, 1400lbs and 1600lbs.
This occurs primarily due to the lateral force domination at higher slip angles until a peak
value of lateral force is reached as limited by the friction coe?cient.
The last test case scenario of 6o slip angle and the parametric variation of normal loads
is performed to capture the lateral force saturation e ects on the lateral tread deformation.
The strains from this test case are shown in Fig. 8.9. The 1000lbs normal load case at 6o
predicts a strain pro le from which neither the leading edge nor the trailing edge can be
107
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Leading
Edge
Leading
Edge
Leading
Edge
Leading
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Figure 8.8: Strain pro le 4:5o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
noticed. This is an indication that the lateral force generated by the tire has most certainly
reached a peak value and the hence the lateral tread deformation is almost absent. The
same conclusion cannot be reached for the higher loads at this slip angle because the strain
pro le predict that some amount of lateral tread deformation still exists at this slip angle.
This drastic shift in the trend raises a question. Has the lateral force really reached its peak
value? Since this current work was focused on developing models for the linear portion of
the side force versus slip angle curve, higher slip angles were not considered to quantify the
conditions beyond the peak force value.
Some notes on the strain pro le from the steady state free rolling experiments:
1. A higher or matching resolution than the FEM could not be achieved.
2. Lesser than 1o angular position could not be achieved as this is limited by the test
machine and the sampling rate or wireless transmitter.
108
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1000 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1200 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1400 lbs
0.46 0.48 0.5 0.52
0
0.01
0.02
time (sec)
e
q
q
(mm/mm)
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Trailing
Edge
Leading Edge
Leading Edge
Leading Edge
Leading Edge
Trailing
Edge
Trailing
Edge
Trailing
Edge
Figure 8.9: Strain pro le 6o slip angle sensors 1, 2, 3,and 4; Loads 1000lbs, 1200lbs, 1400lbs,
and 1600lbs.
3. Based on the experimental data it can be concluded that this method is capturing only
the contact patch conditions.
4. The adhesives used in this current work might be restraining the strain gages more
than they should and result in possible sti ening.
5. The strain gages themselves are much more sti er than the tire, Matsuzaki et al.[67]
have shown that an ultra exible strain sensor works better with tire strain monitoring.
109
Chapter 9
Strain Analysis Based on Steady-State Free-Rolling Experimental Data
9.1 Slope Curve Method
The slope curve method discussed in Chapter 7(section 7.3), is used to analyze the
experimental data of free-rolling from Chapter 8 to estimate the contact duration. The
slope change is estimated from Eq. 9.1. In order to validate the slope curve method ex-
plored in Chapter 7, gures similar to Fig. 7.14, Fig. 7.15, Fig. 7.16 are generated using the
experimental data. The center di erence for slope change calculation is as follows,
? "
??
t
?
i
= ("i+1 "i 1)(t
i+1 ti 1)
(9.1)
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Figure 9.1: Slope change 0o slip angle (4 sensors and 4 loads).
110
The analysis process starts with the 0o slip angle case(i.e the tire is freely rolling) while
the load on the tire is varied from 1000lbs to 1600lbs. The slope change for this test condition
is shown in Fig. 9.1. The curve shows two distinct peaks that occur over all the 4 sensors.
As expected the time between the occurrences of the two peaks increases as the normal load
is increased on the tire. This can also be clearly seen from the slope change plot in Fig. 9.2.
Later, this observation could be used to relate strain to normal load measurement.
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Figure 9.2: Slope change 1:5o slip angle (4 sensors and 4 loads).
In the next test condition a slip angle of 1:5o is applied. The slope change plot when
the slip angle is held constant and the normal load is varied is shown in Fig. 9.2. As shown,
the contact length does not change signi cantly with slip angle.
The 3o slip angle test is next considered. The slope change with respect to time is shown
in Fig. 9.3. It can be seen that all 4 sensors predict the exact same time of occurrence of
peak values corresponding to maximum sensed strain by the strain gages. Note that there
might be a change too small to recognize with the current resolution. A thing to be noted
here is that the magnitude change of the slope with respect to time is not as pronounced as
111
the magnitude change predictions by the FEM. However as long as the predictions are within
the region of maximum allowable error, the experimental predictions should be acceptable.
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?2
0
2
time (sec)
D
e
q
q
/
D
t
1600 lbs
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 1 Sensor 2 Sensor 3 Sensor 4
Figure 9.3: Slope change 3o slip angle (4 sensors and 4 loads).
9.2 Normal Load Estimation
Now Eq. 7.8 (same method was adopted in chapter 7) is modi ed to incorporate the
in ation pressure factor by introducing an additional factor, and so Eq. 7.8 becomes,
fz = fexp ?(pinf)?Acontact (9.2)
where the additional factor fexp is constant t to the experimental data.
The above developed model (also discussed in Chapter 7) in an ideal case scenario of
FEM should work with the experimental results. The average contact duration, which is
also an indicator of the contact length, is practically a function of the load and the vehicle
speed and not the slip angle experienced by a tire while cornering. Fig. 9.4 is a veri cation
of this claim. In this gure the normal load on the tire is plotted versus the average contact
112
duration and it can bee seen that slip angle has little to no in uence on the contact duration.
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085
1000
1100
1200
1300
1400
1500
1600
Average Contact Duration (t
d
) (sec)
Normal Load (lbs)
0 2 4 6
0.05
0.06
0.07
0.08
0.09
a
slip
t
d
(sec)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
EXP Prediction
Figure 9.4: Experimental prediction of average contact duration.
Once the average contact duration is veri ed to be not signi cantly in uenced by the
slip angle, Eq. 9.2 can be used to estimate the normal load from the experimental data.
The estimated normal load from Eq. 9.2 is then plotted versus the normal load from the
experiments, which is shown in Fig. 9.5. An average error of less than 2% exists between the
theoretical model and experimental results, which is a promising result and suggests that
the ts are well within an acceptable margin of error. Finally, a comparison between the
FEM prediction of the normal load and the experimental measurement of the normal load is
shown in Fig. 9.6. From this it can been seen that the FEM predictions and the experimental
measurements are in good agreement. The average error is less than 5%.
113
4000 4500 5000 5500 6000 6500 7000
4000
4500
5000
5500
6000
6500
7000
7500
Normal Load EXP (N)
Normal Load Semi?Empirical Model (N)
Figure 9.5: Experimental data versus semi-empirical model prediction of normal load.
4000 4500 5000 5500 6000 6500 7000 7500
4000
4500
5000
5500
6000
6500
7000
7500
Normal Load EXP (N)
Normal Load FEM (N)
Figure 9.6: Normal load comparison between FEM results and experimental data.
114
9.3 Slip Angle Estimation
In order to validate the slip angle/strain relationship developed in section 7.6, a similar
approach is taken to estimate the slip angle from the average of the strains from all the
sensors in the contact patch region. The average contact duration, which is an indicator of
the contact length, is used for the purpose of collecting all the strains in the contact patch
region. Now, a relationship (similar to Eq. 7.18) between slip angle and measured strains is
developed from the following,
fi?slip = (("ave4 +"ave3) ("ave2 +"ave1)) (9.3)
where flave1, flave2, flave3 and flave4 are the average value of the maximum strains in the contact
patch.
0 1 2
x 10
?4
0
1
2
3
4
5
6
d
a
slip
(mm/mm)
Slip Angle (deg)
EXP Data 1000lbs
EXP Data 1200lbs
EXP Data 1400lbs
EXP Data 1600lbs
EM 1000lbs
EM 1200lbs
EM 1400lbs
EM 1600lbs
Figure 9.7: Slip angle estimation from experimental data.
The slip angle is plotted versus fi?slip (see Fig. 9.7) and a curve is experimentally t to
obtain an equation relating the slip angle to measured strain. The equation is of the form
115
(the same as the FEM),
?slip = AexpeBexp?fi?slip (9.4)
where Aexp and Bexp are load dependent constants t to the experimental data. The only
di erence between Eq.(9.2) and Eq.(7.7) is the additional constant term that was used while
tting an equation to FEM data is ignored and still a good t to experimental data is
obtained (average error less than 5%). From Fig. 9.7 it can be seen that the fi?slip value
decreases with an increase in the slip angle and reaches a saturation point at the highest slip
angle considered. This trend is very similar to what can be seen in Fig. 9.10, where a lateral
force saturation occurs at higher values of slip angle.
A comparison between the fi?slip, based on FEM results, and the fi?slip, based on experi-
mental data is shown in Fig. 9.8. The comparison pertains to all the normal load cases that
were considered. From this it can be seen that the FEM predictions are almost an order of
magnitude di erent from the experimental data. One reason could be that the FEM tire is
less sti than the tire used in experiments. This would simply mean that the lateral tread
deformation in the FEM is signi cantly more than the experiments. Friction coe?cient could
also be responsible for the mismatch between FEM strains and the experimental strains. As
is shown, later, it could also be due to the lateral force in the FEM being higher than the
experiments.
The estimation of the constants Aexp and Bexp, which independent of the slip angle and
only a function of the normal load, by tting equations is shown in section 9.5.1.
9.4 Lateral Force Estimation
The tire forces and moments in reality are a complex non-linear function of tire variables
which are established based on inputs and vehicle output responses. The primary focus of
this work is to establish a relationship between slip angle and cornering force under steady
state free-rolling condition. The SAE tire axis system shown in Fig. 7.29 is used in de ning
116
0 0.5 1 1.5 2 2.5
x 10
?4
0
0.2
0.4
0.6
0.8
1
x 10
?3
d
a
slip
EXP (mm/mm)
d
a
slip
FEM (mm/mm)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
Figure 9.8: fi?slip comparison between FEM results and experimental data.
Figure 9.9: Measured lateral force for all normal loads and slip angle cases.
117
the necessary parameters. According to the SAE de nition, the cornering force is simply
the lateral force when a side slip in the form of slip angle exist. For the sake of consistency
lateral force will be used and the condition of steady state free-rolling under various slip
angles will be referred to as cornering.
The raw data of the lateral force from the experiments is shown in Fig. 9.9. The test was
conducted in such a way that that the normal load cases loop within the slip angle cases. For
example say for 3o slip angle case, the load was varied while keeping the slip angle constant.
This method is employed while keeping in mind the amount of time required to complete
all the testing. From Fig. 9.9, the necessary lateral forces for the slip angles considered is
extracted and this is shown in Fig. 9.10.
0 1 2 3 4 5 6
0
1000
2000
3000
4000
5000
6000
Slip Angle (deg)
Lateral Force (N)
EXP Data 1000lbs
EXP Data 1200lbs
EXP Data 1400lbs
EXP Data 1600lbs
Figure 9.10: Averaged lateral force for cases considered.
The lateral force acting on a tire is a function of the slip angle (?slip). The experimental
prediction of lateral force versus slip angle is shown in Fig. 9.11. The current model predicts
a peak lateral force at 6odeg slip angle. A quadratic polynomial function is then t to the
experimental data. The resulting equation that is t to the experimental data is as follows,
fy = F? ?(?slip)2 +G? ?(?slip)+H? (9.5)
118
where F?, G?, H? are load dependent constants t to the experimental data. The estimation
of the constants F?, G? and H?, which are independent of the slip angle and only a function
of the slip, is shown in section 9.5.2.
In Fig. 9.12, the lateral force estimated from the semi-empirical model versus the lateral
force measured through experiments is shown. An average error of 4% error exists between
the predicted and the measured values of the lateral force. Finally, the lateral force from the
FEM results is compared to the lateral force from the experimental data for all the normal
load cases that were considered are shown in Fig. 9.13. From this it can be seen that the FEM
predictions of the lateral forces are higher in comparison to the experimentally measured
lateral forces. The average error between the FEM prediction and the experimental data is
about 12%. One reason for this mismatch could be due to the sti ness of the tire. The tire
sti ness directly a ects the cornering sti ness, which is the ratio of cornering (or lateral )
force to the slip angle. This simply means that a tire with low vertical sti ness (as witnessed
by the FEM tire), produces a high force in the lateral direction. The mismatch could also be
due to the friction coe?cient used in the FEM (?=0.85) which is di erent than the friction
coe?cient of the FlatTrac tire testing machine?s belt (?=0.63). The boundary conditions
that are used in the FEM and experiments are not exactly the same, this could also be a
reason for the mismatch.
9.5 Estimating Constants of the Semi-Empirical Model Based on Experimental
Results
9.5.1 Constants in the Slip Angle Empirical Model
The Aexp constant terms from all the normal load cases that were considered in de-
veloping Eq. 9.5 is collected and plotted versus the normal load. This results in a linear
relationship between Aexp and the normal load (shown in Fig. 9.14). A linear polynomial is
119
Figure 9.11: Lateral force estimation from experiments.
1000 2000 3000 4000 5000 6000 7000
1000
2000
3000
4000
5000
6000
7000
Lateral Force EXP (N)
Lateral Force Semi?Empirical Model (N)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
Direct relationship
Figure 9.12: Lateral force comparison between semi-empirical model prediction and experi-
mental measurement.
120
1000 2000 3000 4000 5000 6000 7000
1000
2000
3000
4000
5000
6000
7000
Lateral Force EXP (N)
Lateral Force FEM (N)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
Direct relationship
Figure 9.13: Lateral force comparison between FEM results and experimental measurement.
then t to the data and the resulting equation is given by,
Aexp = 0:004343?(fz)+3:505 (9.6)
Fig. 9.15 shows the plot of Bexp (a constant term in Eq. 9.5) versus the normal load. A non-
linear tread is observed and hence a quadratic polynomial is t to the data. The resulting
equation is given by,
Bexp = 0:03181?(fz)2 60:02?(fz) 1:211e4 (9.7)
9.5.2 Constants in the Lateral Force Empirical Model
The rst constant term in Eq. 9.5, F?, is plotted versus the normal load and is shown
in Fig. 9.16. A polynomial of order three best ts to the data. The resulting cubic equation
is given by,
F? = 1:479?10 7 ?(fz)3 +0:000675?(fz)2 1:03?(fz)+345:4 (9.8)
121
1000 1100 1200 1300 1400 1500 1600
8
8.5
9
9.5
10
10.5
Normal Load (lbs)
A
exp
A
exp
Eq.(9.4)
Figure 9.14: Load dependent constant
Aexp estimated from normal load.
1000 1100 1200 1300 1400 1500 1600
?4
?3.8
?3.6
?3.4
?3.2
?3
?2.8
x 10
4
Normal Load (lbs)
B
exp
B
exp
Eq.(9.5)
Figure 9.15: Load dependent constant
Bexp estimated from normal load
1000 1100 1200 1300 1400 1500 1600
?180
?175
?170
?165
?160
Normal Load (lbs)
F
a
F
a
EXP
Eq.(9.6)
Figure 9.16: Load dependent constant F?
estimated from normal load.
1000 1100 1200 1300 1400 1500 1600
1500
1600
1700
1800
1900
2000
2100
Normal Load (lbs)
G
a
G
a
Eq.(9.7)
Figure 9.17: Load dependent constant G?
estimated from normal load.
Fig. 9.17 shows the second term in Eq. 9.5, G?, plotted versus the normal load. A quadratic
polynomial best ts to the data. The resulting equation is given by,
G? = 0:001119?(fz)2 1:76?(fz)+2088 (9.9)
Fig. 9.18 shows the second term in Eq. 9.5, H?, plotted versus the normal load. A cubic
polynomial best ts to the data. The resulting equation is given by,
H? = 2:708?10 8 ?(fz)3 7:65e 5 ?(fz)2 +0:0734?(fz)+65:4 (9.10)
Table 9.1: Slip angle constants
Type A B
FEM Afem = 0:004363?(fz)+3:425 Bfem = 0:02153?(fz)2 +67:86?(fz) 5:953e4
EXP Aexp = 0:004343?(fz)+3:505 Bexp = 0:03181?(fz)2 60:02?(fz) 1:211e4
122
1000 1100 1200 1300 1400 1500 1600
90
91
92
93
94
95
96
97
98
Normal Load (lbs)
H
a
H
a
Eq.(9.8)
Figure 9.18: Load dependent constant H? estimated from normal load.
Table 9.2: Lateral force constants (1st constant term)
Type 1st constant term
FEM C? = 3:313?10 5 ?(fz)2 +0:1191?(fz) 45:86
EXP F? = 1:479?10 7 ?(fz)3 +0:000675?(fz)2 1:03?(fz)+345:4
Table 9.3: Lateral force constants (2nd constant term)
Type 2nd constant term
FEM D? = 0:00015?(fz)2 +1:086?(fz)+48:42
EXP G? = 0:001119?(fz)2 1:76?(fz)+2088
Table 9.4: Lateral force constants (3rd constant term)
Type 3rd constant term
FEM E? = 0:0001294?(fz)2 +0:2378?(fz) 121:3
EXP H? = 2:708?10 8 ?(fz)3 7:65?10 5 ?(fz)2 +0:0734?(fz)+65:4
123
Chapter 10
Steady-State Rolling Finite Element Model
10.1 Steady-State Rolling at Various Slip Ratios
In this section the steady-state rolling nite element model is developed to estimate
the slip ratio (also known as the longitudinal slip). The slip ratio (according to the SAE
convention) is de ned as a di erence between the tire speed in the longitudinal direction and
the axle speed relative to the road. This is represented by the following equation.
SR =
8>
<
>:
Vo R!
Vo ?100% if Vo > R!
R! Vo
R! ?100% if R! > Vo
(10.1)
where SR is the slip ratio, R is the radius of the wheel, ! is the angular velocity (rad/s),
and Vo is the axle speed relative to the road (illustrated in Fig. 10.2). The slip ratio is
expressed as a % and is limited as such by j SRj <= 100%. For full-braking (sometime
referred to as a locked wheel), the axle speed (Vo) is used in the denominator so that the
slip ratio is 1 when ! is zero. The slip ratio has the opposite sign for the full-traction when
tractive force is generated.
Figure 10.1: Basic variables of tire under steady-state rolling
124
Figure 10.2: Boundary condition for the steady-state rolling condition
The simulation methodology described in Chapter 6 (section 6.1) for the straight line
rolling condition is used to develop the steady-state rolling nite element model. The straight
linerollingsimulationinvolvesobtainingsolutionsforthebrakingandthetractionconditions.
The free-rolling angular velocity is estimated from the straight line rolling simulation and is
braking and traction simulations. The braking simulation results are obtained at a rotational
velocity (!) of 8 rad/sec (starting from the free-rolling angular velocity) and a translational
velocity (uo) of 10 kph. The traction simulation results are obtained at a rotational velocity
(!) of 9 rad/sec (starting from the free-rolling angular velocity) and a translational velocity
(uo) of 10 kph. The velocities considered in this Chapter are the same as what was considered
in Chapter 6 (section 6.1) while estimating the free-rolling angular velocity. A coe?cient of
friction of ?=0.85 is used in the steady-state rolling FEM. The constraints set on the hub
are the same as those used in Chapter 6.
The SAE tire conventions are used in this work. The traction slip ratio is negative and
the corresponding longitudinal force is positive and vice-versa for braking. The longitudinal
125
forces from the FEM results from the braking and traction simulations for various normal
load cases is shown in Fig. 10.3. From this it can be seen that the force curves are symmetric
about the x-axis. Owing to the symmetry, it is su?cient to consider either traction or braking
for analysis purposes. Hence, only the results from the traction simulations are considered.
?0.2 ?0.15 ?0.1 ?0.05 0 0.05 0.1 0.15 0.2
?8000
?6000
?4000
?2000
0
2000
4000
6000
8000
Slip Ratio (W
SR
)
Longitudinal Force (N)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
TRACTION BRAKING
Figure 10.3: Longitudinal force under braking and traction.
10.2 Strain Pro le Under Steady-State Rolling
The strains in the contact patch under traction for the various cases of the normal load
and the slip ratio (longitudinal slip) are analyzed in this section.
In the rst set of gures (see Fig. 10.4), the circumferential strains (fl?? as predicted by
the FEM along sensors 1-4) when a normal load of 1000 lbs is applied are shown. From this,
it can be seen that sensors 2 and 3 show no variations with increases in slip ratio from 0-
20%. In contrast, sensors 1 and 4 exhibit variations with the increases in the slip ratio. The
strain distribution starts as symmetrical about the center of contact, but asymmetry can be
observed along all the four sensors with increases in the slip ratio up to 20%. However, the
126
asymmetric strain distribution is absent when the slip ratio is 0% and no driving torque is
applied on the wheel. It is also evident from the strain predictions (especially sensors 1 and 4
) that the force saturation that occurs beyond the 10% slip ratio limits the longitudinal tread
deformation. This is the reason that the strain curves to fall on top of each other beyond a
10% slip ratio. i.e. no change in the strains for slip ratio?s >10%. But, the saturation limit
at 10% might change if a di erent coe cient of friction value is used. Next the normal load
is increased to 1200 lbs and the FEM strain predictions are shown in Fig. 10.5. From this it
can be quickly concluded that the strain predictions are similar to the 1000 lbs normal load
case.
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 1
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 2
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 3
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 4
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
Figure 10.4: Strain pro le under steady-state rolling 0-20% slip ratio; Normal load 1000lbs
(Sensors 1-4).
The strain pro le for the 1400 lbs normal load case of the sensors 1-4 is shown in
Fig. 10.6. From this, it can be seen that the trailing edge, which is under tension for the
traction case, deforms asymmetrically about the x-axis (sensors 1 and 4 clearly exhibit this
trend). The sensors 2 and 3 on the other hand show no variation with the increases in the slip
127
ratio. However, it is interesting to note that the sensor 2 and 3 strain curves almost atten
out while going in and out of contact. At the edges of contact, the strains predicted by
sensors 1 and 4 clearly exhibit a dependency on slip ratio (up to 10%). A similar conclusion
can be obtained for the 1600 lbs normal load case (see Fig. 10.7). Further, from all the
normal load cases that were considered, it can be concluded that the time of contact is only
a function of the applied normal load. This simply implies that the equation developed in
Chapter 7, for predicting the normal load should work for the steady-state rolling condition.
(However this needs to be con rmed with additional tests).
In conclusion, it is evident from the strain pro les that the strains at the edges of contact
can be used when developing equations for the slip ratio. The symmetric and the asymmetric
strain distribution should provide a way for relating the strains to the slip ratio. Therefore
the focus of the next Chapter will be on deriving an analytical relationship between the slip
ratio and the strain due to the frictional load (longitudinal slip).
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 1
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 2
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 3
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 4
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
Figure 10.5: Strain pro le under steady-state rolling 0-20% slip ratio; Normal load 1200lbs
(Sensors 1-4).
128
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 1
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 2
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 3
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 4
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
Figure 10.6: Strain pro le under steady-state rolling 0-20% slip ratio; Normal load 1400lbs
(Sensors 1-4).
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 1
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 2
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 3
0.46 0.48 0.5 0.52
0
0.005
0.01
0.015
0.02
time (sec)
e
q
q
(mm/mm)
Sensor 4
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
0% SR 5% SR 10% SR 15% SR 20% SR 0% SR 5% SR 10% SR 15% SR 20% SR
Figure 10.7: Strain pro le under steady-state rolling 0-20% slip ratio; Normal load 1600lbs
(Sensors 1-4).
129
Chapter 11
Strain Analysis Based on Steady-State Rolling Finite Element Prediction
In this chapter the strain results from the steady-state rolling nite element model are
analyzed. Owing to the symmetric longitudinal force as a function of slip ratio about the free-
rolling case results from the braking and traction simulations, the strain results from either
the braking or the traction FEM is su?cient to predict the slip ratio and the longitudinal
force. In this work only the traction simulation results are considered for analysis purposes
but can then be ?reversed? to obtain the braking results. The slope curve method is used to
estimate the contact duration from the FEM results (see section 7.3). A relationship between
the strains predicted from the FEM and the slip ratio is developed. An empirical equation
relating the slip ratio to the longitudinal force is also developed. The longitudinal force
predictions from the FEM is then compared to the predictions from the empirical model.
11.1 Slope Curve Method
The slope curve method developed in Chapter 7 is used to analyze the strain data from
the FEM. In this method the slope change is calculated by employing the center di erence
method. The equation for slope change based on center di erence method is as follows,
? "
??
t
?
i
= ("i+1 "i 1)(t
i+1 ti 1)
(11.1)
The slope change curves for the steady-state rolling without longitudinal slip (0% slip
ratio) as a function of time is shown in Fig. 11.1. It can be seen that the time of contact along
the paths of the four sensors increases with increases in the normal load. It can also be seen
that every sensor as it goes in and out of contact generates two peaks which correspond to a
130
maxima and a minima slope and approximately make the edge of contact. The di erence in
the time between the occurrences of a maxima and a minima slope is the contact duration.
0.46 0.48 0.5 0.52 0.54
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1600 lbs
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 11.1: Slope change 0% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and 1600lbs.
Next, the slope change for the 5% slip ratio is shown in Fig. 11.2. From this gure it can
be seen that the maximum and minimum changes in slope with respect to slip ratio remains
constant with the change in slip ratio and only changes when the normal load on the tire is
varied.
The 10% slip ratio case shown in Fig. 11.3, also exhibits a similar trend, where a maxima
and a minima occur at the same time indicating that the maximum strain values are realized
at the edges of the contact. It is interesting to note that the magnitude of the maximum
and minimum changes in the slope remain almost constant with further increase in slip ratio
(this can be clearly seen in Fig. 11.4). A possible reason could be that the longitudinal force
reaches saturation for a given load as limited by the friction coe?cient. It is clear that the
contact duration, which is the di erence in time between the occurrence of a maxima and
a minima, is practically a function of normal load only. This implies that the magnitude
131
of the strain due to friction is predominant at the edges of the contact and the magnitude
of the strains due to normal load almost vanish. This would then lead to an asymmetric
strain distribution. Hence, at the edges of the contact, the slip ratio can be correlated to
the frictional strains.
0.46 0.48 0.5 0.52 0.54
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1600 lbs
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 11.2: Slope change 5% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and 1600lbs.
11.2 Slip Ratio Estimation
The slip ratio estimation follows a rather straight forward approach with an underlying
assumption that the tread deforms only in the longitudinal direction (see Fig. 11.5). Based on
this assumption, it can be quickly concluded that the contact patch deforms in the direction
in which the tire rotates (clockwise for traction). When the tire rotates in the clockwise
direction and a driving torque is applied on the wheel, a shear deformation of the tread
occurs. This shear deformation creates an asymmetric strain distribution about the center
point of contact. In other words, an additional tensile strain results on one side of the contact
132
0.46 0.48 0.5 0.52 0.54
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.2
?0.1
0
0.1
0.2
time (sec)
e
q
q
/
D
t
1600 lbs
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Sensor 1
Sensor 2
Sensor 3
Sensor 4
Figure 11.3: Slope change 10% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and 1600lbs.
0.46 0.48 0.5 0.52 0.54
?0.2
0
0.2
time (sec)
e
q
q
/
D
t
1000 lbs
0.46 0.48 0.5 0.52
?0.2
0
0.2
time (sec)
e
q
q
/
D
t
1200 lbs
0.46 0.48 0.5 0.52
?0.2
0
0.2
time (sec)
e
q
q
/
D
t
1400 lbs
0.46 0.48 0.5 0.52
?0.2
0
0.2
time (sec)
e
q
q
/
D
t
1600 lbs
SR 0%
SR 5%
SR 10%
SR 15%
SR 20%
SR 0%
SR 5%
SR 10%
SR 15%
SR 20%
SR 0%
SR 5%
SR 10%
SR 15%
SR 20%
SR 0%
SR 5%
SR 10%
SR 15%
SR 20%
Figure 11.4: Slope change 0%-20% slip ratio; Normal loads 1000lbs, 1200lbs, 1400lbs and
1600lbs along sensor 1.
133
and a compressive on the other. Which is on the leading or the trailing edge depending on if
it is braking or traction. When the sensors go in and out of contact (as shown in Fig. 11.5),
the longitudinal slip for a particular load can be estimated.
Figure 11.5: Relating strain to the slip ratio.
In order to theoretically explain this change in strain at the leading and the trailing edge
the following analysis is provided. The circumferential strain in the tire can be expressed as
follows,
fl?? = flnorm +flfric (11.2)
where flnorm is the strain due to the normal load and flnorm is the strain due the frictional
load on the tire. At the edges of contact, points A and B (see Fig. 11.6) respectively, the
134
Figure 11.6: Analytical relationship between strain and slip ratio.
circumferential strain is given as follows,
"??A = "norm +"fric;"??B = "norm +( "fric) (11.3)
The symmetric and asymmetric characteristics of the tire strains at the edges of contact
can be exploited to obtain a relationship between frictional strain and the slip ratio. The
primary reason for the asymmetry in frictional strain is because point A (for clockwise tire
rotation) is in tension (?+?) and point B is under compression (?-?). In short, the frictional
strain is related to the circumferential strain as follows,
"??A "??B = ("norm +"fric) ("norm +( "fric)) (11.4)
"fric = ("??A "??B)2 (11.5)
Eq. 11.5 derived above will be a function of slip ratio (and longitudinal force). In Fig. 11.7,
the slip ratio as a function of the frictional strain flfric from the FEM results for various
135
normal load cases considered, is shown. A linear polynomial is then t to the FEM data and
an empirical equation is obtained. The equation is given as follows,
SR = Afem ?("fric)+Bfem (11.6)
where Afem and Bfem are load dependent constants. The equations for these two constants
is given in section 11.4.1.
1.5 2 2.5 3 3.5 4 4.5
x 10
?3
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
e
fric
(mm/mm)
Slip Ratio (
W
SR
)
FEM Data 1000 lbs
FEM Data 1200 lbs
FEM Data 1400 lbs
FEM Data 1600 lbs
EM 1000 lbs
EM 1200 lbs
EM 1400 lbs
EM 1600 lbs
Figure 11.7: Slip ratio plotted as function of flfric.
11.3 Longitudinal Force Estimation
The steady-state rolling condition with a negative slip ratio will capture full-traction.
The force exerted on the tire under such a condition is known as longitudinal force (also
known as rolling resistance). According to the SAE de nition, the longitudinal force is
simply the force exerted on the tire by the road when a longitudinal slip as characterized by
the slip ratio exists.
136
In Fig. 11.8, the longitudinal force versus the slip ratio (0-10%) from the FEM results
is shown. It can be seen that for a given load the longitudinal force increases when the slip
ratio increases. The longitudinal force is of course limited by the friction coe?cient (?=0.85
in this case). A quadratic polynomial of the form shown in Eq. 11.7 is t to the FEM results.
The curve t to the FEM results is shown in Fig. 11.9.
fx = c ?( SR)2 +d ?( SR)+e (11.7)
where c , d , and e are load dependent constants t to the FEM results. The equations
for these constants is given in the following section.
The strain based longitudinal force prediction labeled (EM) is shown in Fig. 11.10 in
comparison to that predicted from the FEM. First, the slip ratio is estimated from the
empirical model by Eq. 11.6. Then, the estimated slip ratio is input into Eq. 11.7 and the
longitudinal force from the semi-empirical model is obtained. An average error less than 5%
exists between the FEM results and the strain-based model (Eqs. 11.6 and 11.7).
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Slip Ratio (W
SR
)
Longitudinal Force (N)
FEM Data 1000 lbs
FEM Data 1200 lbs
FEM Data 1400 lbs
FEM Data 1600 lbs
Figure 11.8: Longitudinal force under a driving torque for various normal loads.
137
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Slip Ratio (W
SR
)
Longitudinal Force (N)
FEM Data 1000 lbs
FEM Data 1200 lbs
FEM Data 1400 lbs
FEM Data 1600 lbs
EM 1000 lbs
EM 1200 lbs
EM 1400 lbs
EM 1600 lbs
Figure 11.9: Longitudinal force estimated from the semi-empirical model versus the slip
ratio.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
1000
2000
3000
4000
5000
Longitudinal Force EM (N)
Longitudinal Force FEM (N)
1000 lbs
1200 lbs
1400 lbs
1600 lbs
Direct relationship
Figure 11.10: Longitudinal force estimated from the semi-empirical model compared to the
FEM results.
138
11.4 Estimating Constants of Semi-Empirical Model based on FEM Results
11.4.1 Constants in the Slip Ratio Empirical Model
1000 1100 1200 1300 1400 1500 1600
60
65
70
75
80
85
Normal Load (lbs)
A
W
FEM data
EM (A
W
)
Figure 11.11: Load dependent constant
Afem estimated from normal load.
1000 1100 1200 1300 1400 1500 1600
?0.25
?0.2
?0.15
?0.1
Normal Load (lbs)
B
W
FEM data
EM (B
W
)
Figure 11.12: Load dependent constant
Bfem estimated from normal load
The Afem constant terms from all the normal load cases that were considered in devel-
oping Eq. 11.6 are collected and plotted versus the normal load. A curve is t to obtain a
relationship between Afem and the normal load (shown in Fig. 11.11). The curve represents
a linear polynomial and the resulting equation is given by,
Afem = 0:03888?(fz)+24:95 (11.8)
Figure 11.12 shows the plot of Bfem (a constant term in Eq. 11.6) versus the normal load. A
non-linear tread is observed and hence a linear polynomial is t to the data. The resulting
equation is given by,
Bfem = 0:0003441?(fz)+0:2739 (11.9)
11.4.2 Constants in the Longitudinal Force Empirical Model
The rst constant term in Eq. 11.7, c , is plotted versus the normal load and is shown
in Fig. 11.14. A polynomial of order one best ts to the data. The resulting linear equation
is given by,
c = 155?(fz) 31340 (11.10)
139
1000 1100 1200 1300 1400 1500 1600
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
x 10
5
Normal Load (lbs)
c
W
FEM Data
c
W
Figure 11.13: Load dependent constant
c estimated from normal load.
1000 1100 1200 1300 1400 1500 1600
1.9
2
2.1
2.2
2.3
2.4
2.5
x 10
4
Normal Load (lbs)
d
W
FEM data
EM d
W
Figure 11.14: Load dependent constant
d estimated from normal load.
Figure 11.14 shows the second term in Eq. 11.7, d , plotted versus the normal load. A linear
polynomial best ts to the data. The resulting equation is given by,
d = 11:2?(fz)+7085 (11.11)
Figure 11.15 shows the third term in Eq. 11.7, e , plotted versus the normal load. A linear
1000 1100 1200 1300 1400 1500 1600
?50
0
50
100
150
Normal Load (lbs)
e
W
FEM data
EM e
W
Figure 11.15: Load dependent constant e estimated from normal load.
polynomial best ts to the data. The resulting equation is given by,
e = 0:4145?(fz)+570:4 (11.12)
140
Chapter 12
Conclusions and Future Work
This dissertation contributes to the research in the area of advanced tire sensor tech-
nology using a nite element modeling technique for monitoring the state of the tire under
various operating conditions. The deformation characteristics of a Standard Reference Test
Tire under static loading and steady-state loading conditions are investigated. The numeri-
cal analysis techniques such as the Fourier analysis, the Weibull curve tting and the slope
curve method are explored to relate strains to the loads experienced by a tire under the static
and steady-state rolling conditions. The various numerical analysis techniques explored have
their own advantages and disadvantages. The Fourier analysis is a powerful tool, but requires
a high sampling frequency to e ectively capture the trends. This is a fairly simple procedure
once the amplitudes and the frequencies are extracted from the data. Although, this is a
computationally expensive method. Also, this method may not be feasible for real-time data
processing. The Weibull curve tting method is a simple method to generate good ts to
the strain data. However, multiple functions can be t to the same strain data. The number
constants involved are too many, which would make this method very tire speci c. Also, the
Weibull parameters don?t relate to changes in the load conditions. The slop curve method
involves a simple procedure for parmeter estimation, which does not require a high sampling
frequency. The notable problem with this method is the error ampli cation if the aquired
data is noisy. However, this could be overcome by employing an alternative technique like
the "total-variation regularization" technique (a method for derivative calculation of noisy
data).
In summary the slope curve method appears to be the best method for estimating the
necessary parameters. Hence, this method is used in this work. Based on the FEM results,
141
a semi-empirical model relating normal load to the strains is proposed. An empirical model
relating the slip angle to the tire strains, based on the steady-state free-rolling FEM results,
is also proposed. An empirical model relating the lateral force to the slip angle is proposed.
The strain based prediction of the lateral force is then compared to the FEM results. An
equation relating the slip ratio to the tire strains, based on the steady-state rolling FEM
results, is proposed. An equation relating the longitudinal force to the slip ratio is also
proposed. The strain based prediction of the longitudinal force is then compared to the
FEM results. These models can be used in contact patch strain sensing modules.
This dissertation contributes to the research in the area of advanced tire sensor tech-
nology using an experimental technique for monitoring the state of the tire under various
operating conditions. A wireless tire sensor suite is developed and it?s use is demonstrated
in aquiring the tire strains in real time. The tire sensor suite is comprised of strain sensors,
a signal coditioning circuit and a wireless data transmitter/reciever. The test setup for the
static loading and the steady state rolling are discussed in Chapter 3 and Chapter 8 respec-
tively. A semi-empirical model for each of the tire loads considered above (only static and
steady-state free- rolling) is developed based on the experimental data.
The FEM work serves as a template for the experimental work in relating the various
tire operating conditions to the strains. The empirical model based on the FEM results for
estimating the normal load on the tire is in good agreement with the empirical model based
on the experimental data. An order of magnitude di erence in the fislip values exists between
the FEM results and the experimental data in the slip angle empirical model. Despite the
di erence, similar functions t well to the FEM results and the experimental data. The FEM
over predicts the lateral force in comparison to the experiments. The lateral force empirical
model based on the FEM results is of the same order as that of the empirical model based
on the experimental data. The resulting constants are a function of the normal load on the
tire. The equations that relate the various constants to the load are of the same order, but
the values of these constants are di erent. Since the FEM work is viewed more as a tool
142
for developing a template for the various tire operating conditions, a qualitative comparison
between the models and the resulting load dependent constants may not be required.
In conclusion, the slope curve method used in the current work, is shown to work well
with the FEM results and the experimental data under the steady-state free-rolling conditon.
This method also works well with the FEM results for the steady-state rolling condition
(for estimating slip ratio and longitudinal force). However, further testing is required to
understand the implications of the slope curve method on the experimental data.
In the future, the FEM work can be extended by modeling all the sections of the
tire for better predictions. The material dependent properties such as viscoelasticity and
hysteresis can be included in the material model to better understand their in uence on
the tire deformation. An actual strain sensor can be modeled instead of depending on the
nodal solution for strain measurement. The debonding e ects can also be studied if an
actual strain sensor were to be modeled. In addition, dynamic loading conditions such as
tire vibration, impact loading etc. can be considered. These considerations will extend the
validity of equations and models to the non steady-state conditions.
The experimental work can be extended by considering novel contact patch strain sens-
ing modules, such as rubber based sensors or MEMS based devices or optical sensors. The
driving and braking torque tests can be done to aquire wireless strain data. The experiemen-
tal data can be used to develop models relating the slip ratio to the tire strains and the
longitudinal force to the slip ratio. The testing can also be extended by running the tire
tted on a real vehicle. A comprehensive set of components can be achieved by incorporating
temperature sensors, pressure sensors, accelerometers etc., in the tire sensor suite.
143
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150
Appendices
151
Appendix A
ABAQUS FEA Code
In ation analysis (2D axisymmetric model)
?HEADING
?RESTART;WRITE;FREQ = 1
?NODE;NSET = TIRENODE;INPUT = TIRENODE:INP
?NODE;NSET = RIM
799;0;0;0
?ELEMENT;TYPE = CGAX3;INPUT = TIRECGAX3:INP
?ELEMENT;TYPE = CGAX4RH;INPUT = TIRECGAX4RH:INP
?ELEMENT;TYPE = CGAX4R;INPUT = TIRECGAX4R:INP
?ELEMENT;TYPE = SFMGAX1;INPUT = SURFELEM:INP
?NSET;NSET = HUB
298;297;296;295;20;294;293;292;19;51
221;222;223;224;12;225;226;227;13;45
?ELSET;ELSET = CORDPLY;GENERATE
560;598;1
?ELSET;ELSET = STEELBELT;GENERATE
599;637;1
?ELSET;ELSET = MEMCARCASS;GENERATE
638;725;1
?ELSET;ELSET = TREAD;GENERATE
1;153;1
291;437;1
?ELSET;ELSET = SW;GENERATE
154;284;1
?ELSET;ELSET = BEAD
285;286;287;288;289;290;438;439;440;441
?ELSET;ELSET = IL;GENERATE
442;559;1
?ELSET;ELSET = CARCASS
SW;IL
?SURFACE;NAME = IPIL
IL
??
?RIGIDBODY;TIENSET = HUB;REFNODE = RIM
??
??SECTION : TREAD
152
?SOLIDSECTION;ELSET = TREAD;MATERIAL = TREAD
1:;
??SECTION : SW
?SOLIDSECTION;ELSET = SW;MATERIAL = SIDEWALL
1:;
??SECTION : BEAD
?SOLIDSECTION;ELSET = BEAD;MATERIAL = BEAD
1:;
??SECTION : IL
?SOLIDSECTION;ELSET = IL;MATERIAL = UNDERTREAD
1:;
??SECTION : CORDPLY
?SURFACESECTION;ELSET = CORDPLY
?REBARLAYER
CP;0:28;1:;;"CORDPLY";70:;1
??SECTION : STEELBELT
?SURFACESECTION;ELSET = STEELBELT
?REBARLAYER
SB;0:3;1:;;"STEELBELT";118:;1
??SECTION : CARSCASS
?SURFACESECTION;ELSET = MEMCARCASS
?REBARLAYER
CARCASS;0:420835;1:;;CARCASS;0:;1
?EMBEDDEDELEMENT;HOST = TREAD;ROUNDOFFTOL = 1:E 6
CORDPLY
?EMBEDDEDELEMENT;HOST = TREAD;ROUNDOFFTOL = 1:E 6
STEELBELT
?EMBEDDEDELEMENT;HOST = CARCASS;ROUNDOFFTOL = 1:E 6
MEMCARCASS
??MATERIALS
??
?MATERIAL;NAME = BEAD
?HYPERELASTIC;MOONEY RIVLIN
14:14;21:26;0:001412
?DENSITY
1100000000000;
?MATERIAL;NAME = "CORDPLY"
?ELASTIC;TYPE = ISO
3970:;0:3
?DENSITY
1500000000000;
?MATERIAL;NAME = SIDEWALL
?HYPERELASTIC;MOONEY RIVLIN
0:1718;0:8303;0:0498
?DENSITY
153
5900000000000;
?MATERIAL;NAME = "STEELBELT"
?ELASTIC;TYPE = ISO
200000:;0:3
?DENSITY
5900000000000;
?MATERIAL;NAME = TREAD
?HYPERELASTIC;MOONEY RIVLIN
0:8061;1:805;0:0191
?DENSITY
1100000000000;
?MATERIAL;NAME = UNDERTREAD
?HYPERELASTIC;MOONEY RIVLIN
V0:1404;0:427;0:0881
?DENSITY
1100000000000
?MATERIAL;NAME = CARCASS
?ELASTIC;TYPE = ISO
9870;0:3
?DENSITY
1500000000000
??
??
??
??STEP : INFLATION
??
?STEP;NAME = INFLATION;NLGEOM = YES;INC = 10000
?STATIC
:1;1:;:1;1:
?BOUNDARY
RIM;1;2
RIM;5;6
??
??LOADS
??
??NAME : INFLATIONTYPE : PRESSURE
?DSLOAD
IPIL;P;0:2
??
??OUTPUTREQUESTS
??
?RESTART;WRITE;FREQUENCY = 0
??
??FIELDOUTPUT : F OUTPUT 1
154
??
?OUTPUT;FIELD;VARIABLE = PRESELECT;FREQUENCY = 99999
??
??HISTORYOUTPUT : H OUTPUT 1
??
?OUTPUT;HISTORY;VARIABLE = PRESELECT;FREQUENCY = 99999
?ENDSTEP
Normal load analysis (3d symmetric model generation)
?HEADING
?NODE;NSET = ROAD
99999;0;0:252958298; 338:200012
?RESTART;WRITE;FREQ = 1
?SYMMETRICMODELGENERATION;REVOLVE;ELEMENT = 800;NODE =
800
0;0;0;0;1;0
0;0;1
90;3
70;3
15;7
10;4
15;7
70;3
90;3
?ELSET;ELSET = FOOT;GEN
5151;18751;800
5150;18750;800
5149;18749;800
5148;18748;800
5147;18747;800
5128;18728;800
5146;18746;800
5200;18800;800
5126;18726;800
5144;18744;800
5190;18790;800
5194;18794;800
5120;18720;800
5140;18740;800
5119;18719;800
5192;18792;800
5142;18742;800
5117;18717;800
5111;18711;800
5181;18781;800
5140;18740;800
155
5107;18707;800
5110;18710;800
5183;18783;800
5135;18735;800
5182;18782;800
5136;18736;800
5137;18737;800
5138;18738;800
5204;18804;800
5233;18833;800
?SURFACE;TYPE = CYLINDER;NAME = SROAD
0;0:252958298; 338:200012;3;0:252958298; 338:200012
0;3:252958298; 338:200012
START; 1000;0:252958298
LINE;1000;0:252958298
?RIGIDBODY;REFNODE = ROAD;ANALYTICALSURFACE = SROAD
?SURFACE;NAME = STREAD
FOOT
?CONTACTPAIR;INTERACTION = SRIGID;TYPE = SURFACETOSURFACE
STREAD;SROAD
?SURFACEINTERACTION;NAME = SRIGID
?FRICTION
0:0
?NSET;NSET = FOOT;ELSET = FOOT
?NSET;NSET = PATH1;GENERATE
5173;19573;800
?NSET;NSET = PATH2;GENERATE
5183;19583;800
?NSET;NSET = PATH3;GENERATE
5193;19593;800
?NSET;NSET = PATH4;GENERATE
5204;19604;800
?FILEFORMAT;ZEROINCREMENT
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
1 : INFLATION
?STATIC
:25;1;:25;1
?BOUNDARY
RIM;1;6
ROAD;1;2
ROAD;4;6
?DSLOAD
IPIL;P;0:2
?OUTPUT;FIELD;FREQUENCY = 1
156
?NODEPRINT;NSET = ROAD;FREQ = 1
U;
RF;
?ELPRINT;FREQ = 0
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?PRINT;SOLVE = YES
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
2 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;2:5
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
3 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
157
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;5:0
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
4 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;7:5
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
5 : FOOTPRINT(Displacementcontrolled)
158
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;10:0
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
6 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;12:5
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
159
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
7 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;15:0
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
8 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;17:5
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
160
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
?STEP;INC = 100;NLGEOM = YES
9 : FOOTPRINT(Displacementcontrolled)
?STATIC
1e 9;1;1e 9;
?BOUNDARY;OP = NEW
RIM;1;6
ROAD;1;2
ROAD;4;6
ROAD;3;;20:0
?OUTPUT;FIELD;FREQ = 1
?NODEOUTPUT
U;
?OUTPUT;FIELD;FREQ = 1
?ELEMENTOUTPUT;POSITION = INTEGRATIONPOINTS
S;
LE;
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = ROAD
U3;RF3
?CONTACTOUTPUT;MASTER = SROAD;SLAVE = STREAD
CAREA;
?print;solve = yes
?ENDSTEP
??????????????????????????????????????????????????
Steady state analysis (braking and traction analysis)
?HEADING
STEADY STATEROLLINGANALYSISOFATIRE :
?RESTART;READ;STEP = 9;INC = 51
??????????????????????????????????????????
?STEP;INC = 300;NLGEOM = YES;UNSYMM = YES
1 : STRAIGHTLINEROLLING(Fullbraking)
?STEADYSTATETRANSPORT
0:5;1:0
?CHANGEFRICTION;INTERACTION = SRIGID
?FRICTION;SLIP = 0:017
:85
?TRANSPORTVELOCITY
TIRENODE;8:0
?MOTION;TYPE = VELOCITY;TRANSLATION
TIRENODE;1;;2777:8
?NODEPRINT;FREQ = 1
161
?ELPRINT;FREQ = 1
?OUTPUT;FIELD;OP = NEW;FREQ = 1
?ELEMENTOUTPUT
S;LE
?NODEOUTPUT
U;V;VR
?OUTPUT;HISTORY;FREQ = 1
?NODEOUTPUT;NSET = RIM
U;RF
?NODEOUTPUT;NSET = ROAD
U;RF
?ENDSTEP
???????????????????????????????????????????
?STEP;INC = 300;NLGEOM = YES;UNSYMM = YES
2 : STRAIGHTLINEROLLING(FULLTRACTION)
?STEADYSTATETRANSPORT
0:1;1:0;;0:1
?RESTART;WRITE;FREQ = 1
?TRANSPORTVELOCITY
TIRENODE;9
?ENDSTEP
???????????????????????????????????????????
Steady state analysis (partial traction analysis)
?HEADING
?RESTART;READ;STEP = 11;INC = 4;ENDSTEP;WRITE;FREQ = 1
?FILEFORMAT;ZEROINCREMENT
???????????????????????????????????????????????????
?STEP;INC = 300;NLGEOM;UNSYMM = YES
1 : STRAIGHTLINEROLLING
?STEADYSTATETRANSPORT
1:0;1:0;
?TRANSPORTVELOCITY
TIRENODE;8:42
?ENDSTEP
???????????????????????????????????????????????????
?STEP;INC = 300;NLGEOM;UNSYMM = YES
2 : STRAIGHTLINEROLLING(Partialtraction)
?STEADYSTATETRANSPORT
0:25;1:0;;0:25
?TRANSPORTVELOCITY
TIRENODE;8:43
?ENDSTEP
???????????????????????????????????????????????????
Steady state analysis (free-rolling at various slip angles analysis)
?HEADING
162
STEADY STATEROLLINGANALYSISOFATIRE :
?RESTART;READ;STEP = 5;INC = 1;ENDSTEP;WRITE;FREQ = 1
?FILEFORMAT;ZEROINCREMENT
??????????????????????????????????????????
?STEP;INC = 300;NLGEOM;UNSYMM = YES
1 : STRAIGHTLINEFREEROLLING
?STEADYSTATETRANSPORT
1:0;1:0;
?TRANSPORTVELOCITY
TIRENODE;8:4225
?ENDSTEP
??????????????????????????????????????????
?STEP;INC = 10000;NLGEOM;UNSYMM = YES
2 : SLIP(6degrees)
?STEADYSTATETRANSPORT;LONGTERM
0:25;1:0;;
?MOTION;TYPE = VELOCITY;TRANSLATION
TIRENODE;1;;2762:5829
TIRENODE;2;;290:3591
?ENDSTEP
163
Appendix B
Code for Fourier Analysis
Xcord = [:263157895
:315789474
:368421053
:421052632
:473684211
:526315789
:578947368
:631578947
:684210526
:736842105
:789473684
:842105263
];l = length(Xcord);
FFTProcedure
Sensor1
Samplingfrequency
Fs = 1024;
nfft = 1024;
F = [0 : Fs 1]?(Fs=nfft);
figure(1)
subplot(211);
plot(Xcord;s01;0 ok0;Xcord;s11;0 dk0;Xcord;s21;0 sk0;Xcord;s31;0 xk0;Xcord;s41;
0 pk0;0LineWidth0;4);
xlabel(0Contactduration(sec)0);ylabel(0Hoopstrainfl0??);
z = legend(0Sensor1@0oSA0;0Sensor1@1:5oSA0;0Sensor1@3oSA0;0Sensor1@4:5oSA0;
0Sensor1@6oSA0;5);
set(z;0Interpreter0;0tex0)
Xejw01 = fft(s01;nfft);
Xejw11 = fft(s11;nfft);
Xejw21 = fft(s21;nfft);
Xejw31 = fft(s31;nfft);
Xejw41 = fft(s41;nfft);
amplitudeextraction
164
FF = [F(3 : 7)];
Ampls10deg = [abs(Xejw01(2 : 6))];
Ampls11deg = [abs(Xejw11(2 : 6))];
Ampls12deg = [abs(Xejw21(2 : 6))];
Ampls13deg = [abs(Xejw31(2 : 6))];
Ampls14deg = [abs(Xejw41(2 : 6))];
subplot(212);
semilogx(F;abs(Xejw01);0 ok0;F;abs(Xejw11);0 dk0;F;abs(Xejw21);0 sk0;F;abs(Xejw31);
0 xk0;F;abs(Xejw41);
0 pk0;0LineWidth0;4);
xlabel(0Frequency0);ylabel(0Amplitude0);axis([05120max(Xejw01)]);
z = legend(0Sensor1@0oSA0;0Sensor1@1:5oSA0;0Sensor1@3oSA0;
0Sensor1@4:5oSA0;0Sensor1@6oSA0;5);
set(z;0Interpreter0;0tex0)
subplot(313);holdon;boxon;
set(gca;0XTick0;2 : 1 : 6)
set(gca;0XTickLabel0;020;030;040;050;060)
stem(FF;Ampls10deg;0r0;0LineWidth0;2);xlabel(0Frequency0);ylabel
(0Amplitude0);
stem(FF;Ampls11deg;0g0;0LineWidth0;2);
stem(FF;Ampls12deg;0b0;0LineWidth0;2);
stem(FF;Ampls13deg;0c0;0LineWidth0;2);
stem(FF;Ampls14deg;0k0;0LineWidth0;2);
z = legend(0Sensor1@0oSA0;0Sensor1@1:5oSA0;0Sensor1@3oSA0;
0Sensor1@4:5oSA0;0Sensor1@6oSA0;5);
set(z;0Interpreter0;0tex0)
165
Appendix C
Code for Weibull Curve Fiting
functioncreateFit1(t;s01;s11;s21;s31;s41)
f=clf;
figure(f);
set(f0;Units0;0Pixels0;0Position0;[445129688485]);
legh=[];
legt=;
xlim=[Inf Inf];
ax=axes;
set(ax0;Units0;0normalized0;0OuterPosition0;[0011]);
set(ax0;Box0;0on0);
axes(ax);
holdon;
t = t(:);
s01 = s01(:);
h=line(t;sm:y1;0Parent0;ax0;Color0;0r0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@0SA0;
s11 = s11(:);
h=line(t;sm:y2;0Parent0;ax0;Color0;0g0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@1:5SA0;
s21 = s21(:);
h=line(t;sm:y3;0Parent0;ax0;Color0;0b0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
166
legtend+1 =0 FEMdata > Sensor1@3SA0;
s31 = s31(:);
h=line(t;sm:y4;0Parent0;ax0;Color0;0c0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@4:5SA0;
s41 = s41(:);
sm:y5 = smooth(t;s41;5;0moving0;0);
h=line(t;sm:y5;0Parent0;ax0;Color0;0k0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@6SA0;
ifall(isfinite(xlim))
xlim=xlim+[ 11]?0:01?diff(xlim);
set(ax0;XLim0;xlim)
else
set(ax0;XLim0;[0:41684210568999996;0:84631578930999996]);
end
ok=isfinite(t)isfinite(sm:y1);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0dependent0;0y0;
0independent0;0x0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y1(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0r0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@0SA0;
ok=isfinite(t)isfinite(sm:y2);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;:::
0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;
0coefficients0;0a0;0b0;0c0);
167
cf=fit(t(ok);sm:y2(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0g0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@1:5SA0;
ok=isfinite(t)isfinite(sm:y3);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0dependent0;0y0;
0independent0;0x0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y3(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0b0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@3SA0;
ok=isfinite(t)isfinite(sm:y4);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0dependent0;0y0;
0independent0;0x0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y4(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0c0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@4:5SA0;
ok=isfinite(t)isfinite(sm:y5);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;:::
functioncreateFit1(t;s01;s11;s21;s31;s41)
f=clf;
figure(f);
set(f0;Units0;0Pixels0;0Position0;[445129688485]);
legh=[];
legt=;
xlim=[Inf Inf];
168
ax=axes;
set(ax0;Units0;0normalized0;0OuterPosition0;[0011]);
set(ax0;Box0;0on0);
axes(ax);
holdon;
t = t(:);
s01 = s01(:);
h=line(t;sm:y1;0Parent0;ax0;Color0;0r0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@0SA0;
s11 = s11(:);
h=line(t;sm:y2;0Parent0;ax0;Color0;0g0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@1:5SA0;
s21 = s21(:);
h=line(t;sm:y3;0Parent0;ax0;Color0;0b0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@3SA0;
s31 = s31(:);
h=line(t;sm:y4;0Parent0;ax0;Color0;0c0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@4:5SA0;
s41 = s41(:);
sm:y5 = smooth(t;s41;5;0moving0;0);
h=line(t;sm:y5;0Parent0;ax0;Color0;0k0;:::
0LineStyle0;0none0;0LineWidth0;1;:::
0Marker0;0:0;0MarkerSize0;15);
xlim(1) = min(xlim(1);min(t));
xlim(2) = max(xlim(2);max(t));
169
legh(end+1) = h;
legtend+1 =0 FEMdata > Sensor1@6SA0;
ifall(isfinite(xlim))
xlim=xlim+[ 11]?0:01?diff(xlim);
set(ax0;XLim0;xlim)
else
set(ax0;XLim0;[0:41684210568999996;0:84631578930999996]);
end
ok=isfinite(t)isfinite(sm:y1);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0dependent0;0y0;
0independent0;0x0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y1(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0r0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@0SA0;
ok=isfinite(t)isfinite(sm:y2);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;:::
0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y2(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0g0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@1:5SA0;
ok=isfinite(t)isfinite(sm:y3);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0dependent0;0y0;0independent0;0x0
cf=fit(t(ok);sm:y3(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0b0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
170
legtend+1 =0 Weibullfit > Sensor1@3SA0;
ok=isfinite(t)isfinite(sm:y4);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;0dependent0;0y0;0independent0;0x0
cf=fit(t(ok);sm:y4(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0c0;0LineStyle0;0 0;0LineWidth0;4;0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@4:5SA0;
ok=isfinite(t)isfinite(sm:y5);
if all(ok)
warning(0GenerateMFile : IgnoringNansAndInfs0;0IgnoringNaNsandInfsindata:0);
end
st=[0:505528006113628250:974681752559468540:52991989837207953];
ft=fittype(0(c=a)(1=a)?(x=a)(b=c 1)?exp( x=a)(b)=c0;:::
0dependent0;0y0;0independent0;0x0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y5(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0k0;:::0LineStyle0;0 0;0LineWidth0;4;:::0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@6SA0;
holdoff;
leginfo=0Orientation0;0vertical0;0Location0;0NorthEast0;
h=legend(ax;legh;legt;leginfo:);
set(h0;Interpreter0;0none0);
xlabel(ax00; );
ylabel(ax00; );
0dependent0;0y0;0independent0;0x0;0coefficients0;0a0;0b0;0c0);
cf=fit(t(ok);sm:y5(ok);ft0;Startpoint0;st);
h=plot(cf0;fit0;0:95);
set(h(1);0Color0;0k0;:::0LineStyle0;0 0;0LineWidth0;4;:::0Marker0;0none0;0MarkerSize0;6);
legendoff;
legh(end+1) = h(1);
legtend+1 =0 Weibullfit > Sensor1@6SA0;
holdoff;
leginfo=0Orientation0;0vertical0;0Location0;0NorthEast0;
h=legend(ax;legh;legt;leginfo:);
set(h0;Interpreter0;0none0);
xlabel(ax00; ); ylabel(ax00; );
171