INVESTIGATION OF LATERAL PERFORMANCE OF AN ATV TIRE ON
NATURAL, DEFORMABLE SURFACES
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
_______________________________
Darrell R. Krueger
Certificate of Approval:
_____________________________
Timothy McDonald
Assistant Professor
Biosystems Engineering
_____________________________
Peter Jones, Chair
Associate Professor
Mechanical Engineering
_____________________________
David Bevly
Assistant Professor
Mechanical Engineering
_____________________________
Robert Jackson
Assistant Professor
Mechanical Engineering
_____________________________
George T. Flowers
Interim Dean
Graduate School
INVESTIGATION OF LATERAL PERFORMANCE OF AN ATV TIRE ON
NATURAL, DEFORMABLE SURFACES
Darrell R. Krueger
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
December 17, 2007
iii
INVESTIGATION OF LATERAL PERFORMANCE OF AN ATV TIRE ON
NATURAL, DEFORMABLE SURFACES
Darrell R. Krueger
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
__________________________________
Signature of Author
__________________________________
Date of Graduation
iv
THESIS ABSTRACT
INVESTIGATION OF LATERAL PERFORMANCE OF AN ATV TIRE ON
NATURAL, DEFORMABLE SURFACES
Darrell R. Krueger
Master of Science, December 17, 2007
(B.S., Auburn University, 2005)
235 Typed Pages
Directed by Peter Jones
A study is undertaken to discover links between soil-rubber interface strength,
camber, pressure and lateral performance of a front ATV tire. New tire and soil test rigs
are also designed and fabricated for the support of this research. The tire is tested at
various geometries and inflation pressures on cohesive and frictional soils. Lateral,
longitudinal and vertical forces are recorded with an electronic data logging system.
Using the results from the tire tests, an exponential tire model is fit to the tire data.
Coefficients of the model are then compared to the inflation pressures, camber angles,
and soil data. The effects on tire performance trends are observed. Data from these
experiments can be used to aid in the programming of autonomous unmanned ground
vehicles, or to create better vehicle dynamics models for small off-road vehicles.
v
ACKNOWLEDGEMENTS
This work is dedicated to all the men and women of War Eagle Motorsports. It is
dedicated to those who believed in me as a freshman, teaching, mentoring, and
motivating me to stick with engineering, and develop the Auburn work ethic.
This work is dedicated to my teammates who gave their all and more throughout
the ups and downs of Baja racing at Auburn, who sacrificed their grades and spare time
for the smell of gasoline, the buzz of the welder, burn of the grinder, and the sight of the
checkered flag.
To Dr. Peter Jones, infinite thanks. He listened to my proposal, believed in it,
found funding, and gave support. As the advisor for War Eagle Motorsports, he has
contributed so much of his time and energy to building great engineers with the right
balance of practical experience and book knowlege.
To Taylor Owens and Marc Jarmulowicz, I give thanks. They are great friends
and great engineers who can solve anything over a plate of barbecue. Marc and Taylor?s
friendship and encouragement through the years have been nothing short of outstanding.
To Randy Whitehead, I thank for numerous discussions of engineering, racing,
Christianity, and life after college, especially when enjoying lunch at Hamilton?s.
A thousand thanks go to the College of Engineering. To Larry Benefield, for his
bold vision and action in making the College the best it can be. To Jim Killian and Cheryl
Cobb for showcasing all the great things that happen within the College, and to Buzz
vi
Powell and his great staff at the National Center for Asphalt Technology. Without their
support and involvement, this research would have been painfully difficult.
And finally, I would like to thank all of my amazing friends and family, including
my parents, Donna and Mark, and brother Christopher, who never doubted my ability,
and encouraged me every step of the way to completion. I could not have done it without
your love and support.
vii
Style of journal used:
ASME Journal of Dynamic Systems, Measurement and Control
Computer software used:
Microsoft Word
viii
TABLE OF CONTENTS
LIST OF TABLES............................................................................................................ xi
LIST OF FIGURES .......................................................................................................... xii
NOMENCLATURE ..........................................................................................................xx
1. INTRODUCTION ..........................................................................................................1
1.1. Motivation and Intent??????????????????????? 1
1.2. Background and Literature Review ?????????????????...3
1.2.1. Vehicle Dynamics ?????????????????????...5
1.2.2. Tires ??????????????????????????..12
1.2.3. Operating Surfaces ?????????????????????16
1.3. Purpose of Thesis and Contribution ?????????????????29
2. RESEARCH TOOLS ????????????????????????..31
2.1. Bevameter ???????????????????????????31
2.1.1. Development???????????????????????. 31
2.1.2. Operations ????????????????????????.40
2.1.3. Assessment ????????????????????????41
2.2. Tire Test Rig ??????????????????????????45
2.2.1. Introduction ???...????????????????????45
2.2.2. Development ???????????????????????.45
2.2.3. Operations ????????????????????????.52
2.2.4. Assessment ????????????????????????53
ix
2.3. Tire Model ??????????????????????????...56
2.3.1. Introduction ???????????????????????...56
2.3.2. Tire Models ???????????????????????...56
2.3.3. Assessment ????????????????????????59
3. RESULTS ????????????????????????????...61
3.1. Introduction ??????????????????????????..61
3.1.1. Test Surfaces ???????????????????????.61
3.1.2. Test Tire ?????????????????????????63
3.2. Bevameter ???????????????????????????65
3.2.1. Shear Response ??????????????????????.66
3.2.2. Normal Response ?????????????????????..81
3.3. Tire Test Rig ??????????????????????????84
3.3.1. Inflation Pressure ?????????????????????..97
3.3.2. Camber ?????????????????????????106
3.3.3. Surface ?????????????????????????113
3.4. Tire Model ??????????????????????????.120
4. CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK???... 144
REFERENCES ................................................................................................................148
APPENDICES .................................................................................................................151
A. Uncertainty Analysis.............................................................................................151
B. Analysis of Tire Vertical Spring Rates and Natural Frequencies........................157
C. Raw Data..............................................................................................................160
x
D. Computer Programs .............................................................................................183
E. Photographs..........................................................................................................212
xi
LIST OF TABLES
Table 3.1 ? Tire rig test matrix indicating three runs per setting ........................................63
Table 3.2 ? Gravel surface shear rates ................................................................................70
Table 3.3 ? Shear rates, Clay soil .......................................................................................77
Table 3.4 ? Average values for various soil-rubber interface properties ............................81
Table 3.5 ? Bekker bevameter sinkage properties ..............................................................84
Table 3.6 ? Exponential Tire Model Coefficient curve fit data .........................................132
Table 3.7 ? Table of normal and shear properties for clay and gravel surfaces .................141
Table 6.1 ? Table of natural frequencies in z-axis for KT 821 tire ...................................166
xii
LIST OF FIGURES
Figure 1.1 ? SAE Vehicle Axis System.............................................................................. 3
Figure 1.2 ? Schematic of tire and forces acting on it ........................................................ 4
Figure 1.3 ? Schematic of simple vehicle model ................................................................5
Figure 1.4 ? Load transfer due to cornering .......................................................................9
Figure 1.5 ? Four wheel vehicle model ............................................................................10
Figure 1.6 ? Sample plot of lateral force v. slip angle ......................................................13
Figure 1.7 ? A cambered tire ............................................................................................15
Figure 1.8 ? Vane shear apparatus ....................................................................................18
Figure 1.9 ? Schematic of a direct shear test ....................................................................18
Figure 1.10 - Schematic of Tri-axial soil testing apparatus ..............................................20
Figure 1.11 ? Cone penetrometer ......................................................................................21
Figure 1.12 ? Schematic of a Bevameter ..........................................................................23
Figure 1.13 ? Mohr-Coulomb strength envelope ..............................................................25
Figure 1.14 ? Nonlinearities of Mohr-Coulomb envelope at high normal stresses
............................................................................................................................................26
Figure 2.1 ? Bevameter shear-head as described by CRREL ...........................................33
Figure 2.2 ? Bevameter as designed by Wong .................................................................34
Figure 2.3 ? Details of shear head instrumentation and control .......................................35
Figure 2.4 ? KT 821 lug contact area ...............................................................................36
xiii
Figure 2.5 ? Bevameter as attached to Yamaha Rhino .....................................................37
Figure 2.6 ? Bevameter as designed in CAD ....................................................................38
Figure 2.7 ? Details of normal load test apparatus ...........................................................39
Figure 2.8 ? Elements of shear head data acquisition system ...........................................40
Figure 2.9 ? Shear head data with curve fit from Equation 1.23 ......................................42
Figure 2.10 ? Shear test data from University of Pretoria ................................................42
Figure 2.11 ? Penetration test data from University of Pretoria .......................................43
Figure 2.12 ? Penetration test data collected by author ....................................................44
Figure 2.13 ? Test rig designed by University of Maryland .............................................47
Figure 2.14 ? Details of data acquisition box and load cells ............................................48
Figure 2.15 ? Details of load cells ....................................................................................48
Figure 2.16 ? 3D model of tire test rig in CAD ................................................................49
Figure 2.17 ? 3D model of tire test rig mounting system .................................................50
Figure 2.18 ? Details of camber and slip angle adjustment ..............................................51
Figure 2.19 ? Lateral force coefficient data from UMD study .........................................54
Figure 2.20 ? Rolling resistance coefficient from ............................................................54
Figure 2.21 ? Example lateral force data generated by the author ................................... 55
Figure 2.22 ? Sample output data. Clay, 10 psi, low camber, 185 lb normal load
........................................................................................................................................... 59
Figure 2.23 ? Sample output data, Gravel, 10psi, low camber, 185 lb normal load
............................................................................................................................................60
Figure 3.1 ? Dunlop KT 821 ATV tire .............................................................................64
Figure 3.2 ? Shear stress v. linear displacement on frictional soil ...................................67
xiv
Figure 3.3 ? Mohr-Coulomb failure envelope for frictional soil ......................................68
Figure 3.4 ? Shear rate for Gravel Test 2 ..........................................................................69
Figure 3.5 ? Exponential shear stress equation fit to shear data .......................................71
Figure 3.6 ? Exponential shear stress equation fit to shear data .......................................72
Figure 3.7 ? Shear modulus v. normal stress, Gravel test 2.............................................. 73
Figure 3.8 ? Shear modulus v. normal stress, Gravel Test 1 ............................................ 73
Figure 3.9 ? Shear stress v. shear displacement, Clay soil test 2 ......................................75
Figure 3.10 ? Mohr-Coulomb failure envelope, Clay soil test 2 ......................................76
Figure ? 3.11 Shear Rate, Clay soil test 2 .........................................................................77
Figure 3.12 Exponential shear stress equation fit to shear data, Clay test 2 .....................78
Figure 3.13 ? Shear modulus v. normal stress, Gravel test 1 ............................................79
Figure 3.14 ? Normalized shear stress v. displacement ....................................................80
Figure 3.15 ? Cohesive soil 3? plate penetration test .......................................................82
Figure 3.16 - Cohesive Soil 4? plate penetration test .......................................................82
Figure 3.17 ? Frictional Soil 3? plate penetration test...................................................... 83
Figure 3.18 ? Frictional Soil 4? plate penetration test...................................................... 83
Figure 3.19 ? Sample raw data from tire test rig ..............................................................85
Figure 3.20 ? Schematic for tire test rig ...........................................................................87
Figure 3.21 ? Sample carpet plot of lateral tire data .........................................................89
Figure 3.22 ? Sample carpet plot of longitudinal tire data ...............................................91
Figure 3.23 ? Peak drag force v. normal load ...................................................................92
Figure 3.24 ? Sample Normalized longitudinal force v. slip angle ..................................93
Figure 3.25 ? Sample Normalized lateral force v. slip angle ............................................94
xv
Figure 3.26 ? Cornering stiffness v. normal load .............................................................95
Figure 3.27 ? Peak lateral force v. normal load ................................................................96
Figure 3.28 ? Lateral force carpet plot for Gravel, 5 psi, Low camber ............................97
Figure 3.29 ? Longitudinal force carpet plot for gravel, 5 psi, low camber .....................98
Figure 3.30 ? Lateral force carpet plot for gravel, 10 psi, low camber ............................99
Figure 3.31 ? Longitudinal force carpet plot for gravel, 10 psi, low camber ...................99
Figure 3.32 ? Cornering stiffness comparison between pressures on gravel ..................101
Figure 3.33 ? Peak lateral force comparison between pressures on gravel
......................................................................................................................................... 102
Figure 3.34 ? Peak longitudinal force comparison between pressures on gravel
..........................................................................................................................................102
Figure 3.35 ?Normalized lateral force comparison between pressures at 95 lbf normal
load, gravel ......................................................................................................................103
Figure 3.36 ? Normalized lateral force comparison between pressures at 140 lbf normal
load, gravel...................................................................................................................... 104
Figure 3.37 ? Normalized lateral force comparison between pressures at 185 lbf normal
load, gravel ......................................................................................................................104
Figure 3.38 ? Normalized lateral force comparison between pressures at 230 lbf normal
load, gravel ......................................................................................................................105
Figure 3.39 ? Normalized lateral force comparison between pressures at 275 lbf normal
load, gravel ......................................................................................................................105
Figure 3.40 ? Normalized lateral force comparison between pressures at 320 lbf normal
load, gravel ......................................................................................................................106
xvi
Figure 3.41 ? Comparison of cornering stiffness v. normal load for camber angle on
gravel ...............................................................................................................................107
Figure 3.42 ? Comparison of peak lateral forces for camber angle, on gravel
..........................................................................................................................................108
Figure 3.43 ? Comparison of peak longitudinal forces for camber angle, on gravel ...........
..........................................................................................................................................109
Figure 3.44 ? Comparison of cambered tire lateral forces at 95 lbf normal load, on gravel
..........................................................................................................................................110
Figure 3.45 ? Comparison of cambered tire lateral forces at 140 lbf normal load, on
gravel ...............................................................................................................................111
Figure 3.46 ? Comparison of cambered tire lateral forces at 185 lbf normal load, on
gravel ...............................................................................................................................111
Figure 3.47 ? Comparison of cambered tire lateral forces at 230 lbf normal load, on
gravel ...............................................................................................................................112
Figure 3.48 ? Comparison of cambered tire lateral forces at 275 lbf normal load, on
gravel ...............................................................................................................................112
Figure 3.49 ? Comparison of cambered tire lateral forces at 320 lbf normal load, on
gravel ...............................................................................................................................113
Figure 3.50 ? Comparison of cornering stiffness v. normal load for different surfaces .114
Figure 3.51 ? Comparison of peak lateral force v. normal load for different surfaces .........
..........................................................................................................................................115
Figure 3.52 ? Comparison of peak longitudinal forces v. normal load for different
surfaces ...........................................................................................................................116
xvii
Figure 3.53 ? Comparison of lateral forces on different surfaces at 95 lbf normal load 117
Figure 3.54 ? Comparison of lateral forces on different surfaces at 140 lbf normal load 117
Figure 3.55 ? Comparison of lateral forces on different surfaces at 185 lbf normal load 118
Figure 3.56 ? Comparison of lateral forces on different surfaces at 230 lbf normal load 118
Figure 3.57 ? Comparison of lateral forces on different surfaces at 275 lbf normal load 119
Figure 3.58 ? Comparison of lateral forces on different surfaces at 320 lbf normal load 119
Figure 3.59 ? Example of exponential model fit to data on Clay, at 10 psi, low camber,
and 185 lbf normal load ..................................................................................................121
Figure 3.60 ? Example of exponential fit for tire data on gravel, 5 psi, high camber, 275
lbf normal load................................................................................................................ 122
Figure 3.61 ? Trace of R-squared values for data on gravel, 5 psi, high camber, 275 lbf
normal load .....................................................................................................................123
Figure 3.62 ? Trace of R-squared values for data on clay, 10 psi, low camber, 90 lbf
normal load .....................................................................................................................123
Figure 3.63 ? A Coefficient plot v. normal load for clay, 10 psi, low camber ...............124
Figure 3.64 ? B Coefficient plot v. normal load for clay, 10 psi, low camber ...............125
Figure 3.65 ? A coefficient v. normal load for gravel, 10 psi, low camber ....................125
Figure 3.66 ? B coefficient v. normal load for gravel, 10 psi, low camber ....................126
Figure 3.67 ? Curve fit to B coefficient data for Clay, 10 psi, low camber ....................127
Figure 3.68 ? Curve fit to A coefficient data for Clay, 10 psi, low camber ...................127
Figure 3.69 ? Curve fit to B coefficient data for Gravel, 5 psi, high camber .................128
Figure 3.70 ? Curve fit to A coefficient data for Gravel, 5 psi, high camber .................128
Figure 3.71 ? Curve fit to B coefficient data for Gravel, 5 psi, low camber ..................129
xviii
Figure 3.72 ? Curve fit to A coefficient data for Gravel, 5 psi, low camber ..................129
Figure 3.73 ? Curve fit to B coefficient for Gravel, 10 psi, high camber .......................130
Figure 3.74 ? Curve fit to A coefficient for Gravel, 10 psi, high camber .......................130
Figure 3.75 ? Curve fit to B coefficient data for Gravel, 10 psi, low camber ................131
Figure 3.76 ? Curve fit to A coefficient data for Gravel, 10 psi, low camber ................131
Figure 3.77 ? Comparison of camber and pressure with tire model at 50 lbf .................133
Figure 3.78 ? Comparison of camber and pressure with Eq. 3.3 at 95 lbf .....................134
Figure 3.79 ? Comparison of camber and pressure with Eq. 3.3 at 140 lbf ...................134
Figure 3.80 - Comparison of camber and pressure with Eq. 3.3 at 185 lbf .................... 135
Equation 3.81 - Comparison of camber and pressure with Eq. 3.3 at 230 lbf ................135
Figure 3.82 - Comparison of camber and pressure with Eq. 3.3 at 275 lbf .................... 136
Figure 3.83 - Comparison of camber and pressure with Eq. 3.3 at 320 lbf ....................136
Figure 3.84 -Comparison of surface with Eq. 3.3 at 50 lbf ............................................ 137
Figure 3.85 - Comparison of surface with Eq. 3.3 at 95 lbf ...........................................137
Figure 3.86 - Comparison of surface with Eq. 3.3 at 140 lbf .........................................138
Figure 3.87 ? Comparison of surface with Eq. 3.3 at 185 lbf .........................................138
Figure 3.88 - Comparison of surface with Eq. 3.3 at 230 lbf ......................................... 139
Figure 3.89 - Comparison of surface with Eq. 3.3 at 275 lbf ......................................... 139
Figure 3.90 ? Comparison of surface with Eq. 3.3 at 320 lbf......................................... 140
Figure 6.1 ? Composite photograph of yaw uncertainty..................................................154
Figure 6.2 ? Graphical Representation of Levels of Uncertainty ...................................154
Figure 6.3 ? Spring ? Mass system .................................................................................155
xix
Figure 6.4 ? Linear fit to Pressure v. Spring rate data for KT 821 tire........................... 156
xx
NOMENCLATURE
a
y,
[G] Lateral Acceleration
a [in] Horizontal distance from CG to front axle
A [-] Exponential tire model coefficient
b [in] Horizontal distance from CG to rear axle
b [in] Pressure plate diameter
B [1/deg] Exponential tire model coefficient
c [-] Exponential tire model coefficient
c [lbf/in
2
] Cohesion
C [1/deg] Exponential tire model coefficient
C
?
[lbf/deg] Tire cornering stiffness
C
?
[lbf/deg] Tire camber stiffness
D [1/deg] Exponential tire model coefficient
F
x
[lbf] Longitudinal tire force
F
y
[lbf] Lateral tire force
F
z
[lbf] Vertical tire force
h [in] CG height above ground
h? [in] CG distance from roll axis
j [in] Soil shear displacement
K [lbf/in] Soil shear deformation modulus
k
?
[lbf/in
n+2
] Frictional sinkage parameter
k
c
[lbf/in
n+1
] Cohesive sinkage parameter
m [-] Exponent for Metz exponential model
M
z
[lbf-in] Moment about vertical (z) axis
n [-] Sinkage exponent
p [lbf/in
2
] Pressure
s [lbf/in
2
] Shear stress
s
max
[lbf/in
2
] Maximum shear stress
t [in] Track width
V [in/s] Velocity
z [in] Sinkage into soil
? [deg] Tire slip angle
? [deg] Vehicle slip angle
? [deg] Tire camber angle
? [deg] Tire steer angle
? [lbf/in
2
] Normal stress
? [lbf/in
2
] Shear stress
? [deg] Friction angle
1
CHAPTER 1
INTRODUCTION
1.1 Motivation and Intent
Tire lateral forces are forces generated by the interaction of the tire and surface in
the ground plane that seek to steer the vehicle and thus induce motion about the yaw axis
and change direction of the vehicle. Lateral forces are particularly important to ground
vehicles as they govern how well the vehicle handles, maneuverability, and the stability
of a vehicle. Design of high performance vehicles (especially autonomous vehicles)
requires significant knowledge of the tire-soil interaction in order to ensure as-designed
operation in all conditions considered.
Tire lateral force information is used by designers for vehicle dynamics
simulations and calculations to assess the handling and maneuvering capabilities of a
vehicle before it is built. By looking at plots of the lateral force data, or using a tire
model, the designer can assess the relative sensitivity to steering input, vertical load, and
the maximum lateral force generated under various conditions of inflation pressure,
camber angle, and operating surface strength.
The intent of the following research is to update the knowledge base regarding
lateral force performance (and some limited longitudinal knowledge) of an ATV tire
when varying camber, inflation pressure, and draw links to the influence of operating
surface normal and shear strength. Furthermore, the data are applied to a tire model to
2
assess the influence of inflation, camber, and operating surface on the coefficients of the
tire model.
Only one other study of this nature was conducted before, and so happened to be
in the late 1980?s at the University of Maryland [25]. However, no quantitative data on
surface, inflation pressure, or camber information was collected. While conducting
experiments similar to those conducted by those at UMD, additional experiments were
conducted on the shear and normal strength of the operating surfaces, also along with the
use of D.A. Crolla?s exponential tire model [15] to compare the performance changes
between surface, inflation pressure, and camber.
An assessment of the problem of measuring ATV tire forces specified a need to
record the several operational variables of the tire as it will be placed in service. Also
required would be strength data of the rubber-soil interface as to make surface
comparisons with. Using the UMD study for initial guidance, a towed test rig designed
and fabricated for use over natural surfaces. In addition, a bevameter was designed and
fabricated for use in testing the strength of the rubber-soil interface. Both pieces of
equipment were outfitted with sensors and transducers as to interface with an
interchangeable data logging system.
Upon processing the data from numerous runs on the tire test rig, lateral, vertical,
and longitudinal forces can be plotted against each other or against slip angle. A basic
understanding of the operational characteristics of ATV tires is sought, yet further
investigation can lead to more knowledge of the effects of different operating surfaces
and operational parameters of the tire like camber and inflation pressure. By using the
D.A. Crolla?s exponential tire model, one can perform curve fits to the gathered data and
3
deduce coefficients of the tire model. Then the coefficients can be used by vehicle
designers to account for camber, pressure, and operating surface in their vehicle
dynamics simulations with ATV tires, which was previously unavailable.
1.2 Background and Literature Review
Tires are arguably the most important component in a vehicle system. The tires of
a wheeled vehicle are the only link from the body of the vehicle to the ground. All of the
forces required to support, propel, stop, and steer a vehicle are transferred to the surface
through the interface between the tire and surface (except aerodynamic forces). Figure 1
details the Society of Automotive Engineers (SAE) vehicle axis system, while Figure 2
presents the schematic of an uncambered tire as indicated by Pacejka [20]
Figure 1.1 ? SAE Vehicle Axis System [19]
4
Figure 1.2 ? Schematic of tire and forces acting on it [20]
Tires are made to be deformable such that they can assume the shape of the
surface they are in contact with. This deformability provides some level of isolation
between the operator and the environment. The interface is a complex region in which
adhesion and slip occur simultaneously on the microscopic level [20]. Also given the
large monolithic tread blocks of off-road tires, there is mechanical interlocking between
the terrain and these tread elements, provide lateral and longitudinal traction on a
macroscopic level.
While it has been established that the tire is the only link for forces to be imparted
to the vehicle from the ground, just how are these forces used to maneuver a vehicle?
Other key vehicle subsystems are well defined and designers have much information in
specifying their construction and use, including, but not limited to the suspension,
steering, and powertrain subsystems. The tire, and specifically in our interest, the ATV
tire is severely lacking information. To date, only one study regarding lateral forces of
ATV tires has been conducted [25].
5
This study created a test rig to measure tire forces in the lateral, longitudinal, and
vertical directions [25]. The researchers at the University of Maryland only proceeded up
to fifteen degrees of slip angle, and also no terrain measurement was conducted, only
basic descriptions of the surface were made. [25]
1.2.1 Vehicle Dynamics
A simple vehicle model known as the bicycle model can be used to illustrate how
lateral and longitudinal forces are imparted to a vehicle. The bicycle model is shown in
Figure 1.3 as described by Pacejka [20].
Figure 1.3 ? Schematic of simple vehicle model [19]
There are some assumptions Milliken [19] stresses about the bicycle model that
must be made clear before going any further.
? No lateral load transfer
? No longitudinal transfer
6
? No roll or pitch motion
? Linear tire behavior
? Constant forward velocity
? No chassis stiffness
? No aerodynamic loads
First taking the assumption that the tires are linear, an expression can be formed for the
front and rear tire forces with cornering stiffness and geometric parameters.
)( ?? ?+=
V
ar
CY
ff
(1.1)
)(
V
br
CY
rr
+= ? (1.2)
The equations described in 1.1 and 1.2 assume the linear tire model, where the
product of the cornering stiffness and the slip angle equal a tire force in the lateral
direction. The expression within the parenthesis equals the slip angle, by geometric
means. Equation 1.1 has a provision to add steer angle into the calculation of lateral
forces.
Moments can also be summed about the center of gravity (CG) by multiplying the
tire forces by the respective wheelbase dimensions, a and b. The total moment equation
becomes the following equation.
??
frfrf
aCrCbCa
V
bCaCN ?++?= )(
1
)(
22
(1.3)
7
Milliken [19] then describes vehicle movement in derivative notation, where the forces
and moments are, respectively:
() ?
?
?
?
??
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
==
Y
r
r
YY
rfY ,, (1.4)
() ?
?
?
?
??
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
==
N
r
r
NN
rfN ,, (1.5)
Furthermore, Milliken describes various stability derivatives that can be calculated.
rf
CCY +=
?
(1.6)
()
rfr
bCaC
V
Y ?
?
?
?
?
?
?
=
1
(1.7)
f
CY ?=
?
(1.8)
rf
bCaCN ?=
?
(1.9)
()
rfr
CbCa
V
N
22
1
+
?
?
?
?
?
?
= (1.10)
f
aCN ?=
?
(1.1)
The full derivative notation of the equations of motion is presented by Milliken as the
following:
??
??
NrNNrI
rz
++=& (1.12)
???
??
YrYYrmV
r
++=+ )(
&
(1.13)
While this model is simple and workable, one must recall the assumptions placed
on the model. As a tire?s cornering ability changes with normal load and slip angle, an
8
effective model must take this into account. The vertical loads imparted on tires in the off
road world can change significantly, and is thusly another reason to add vertical load
effects to the model. The bicycle model is a good starting point for modeling vehicle
behavior, however further improvements can be made without sacrificing simplicity.
These improvements can be found in the four wheel tire model, and using nonlinear tire
models.
Another improvement to the model is changing the tire forces from their current
linear form to a nonlinear model. The nonlinear model more accurately depicts the force
generation trends and can allow greater slip angles to be simulated. Nonlinear tire models
predict the shape of the transition region, and the shape of the saturation region, which is
where high slip angles occur, and where lateral force either becomes constant or trails off
with increasing slip angle.
Recall that lateral forces are governed in part by the normal load on the tire. The
simple bicycle model assumed that the normal load on the tire was constant. In realistic
conditions, however, there are suspensions which allow for roll motion which in turn
place differing normal loads on each tire of the vehicle. By characterizing the suspensions
roll effects, one can provide a normal load to a tire model or look-up table in a simulation
to generate the required lateral force values.
Observe Figure 1.4 where Milliken provides a look into the forces on a vehicle
when viewed from the front or rear.
9
Figure 1.4 ? Load transfer due to cornering [19]
Note that the centrifugal force that is acting to the left is opposite the direction of
the lateral acceleration which is created by the forces at the tire-surface interface. Also
note that the CG is at a height above the ground and acts as a moment arm for the
centrifugal force to induce a roll moment. If one sums the moments about point O listed
in the figure, they will have:
t
hWA
W
W
y
L
+=
2
(1.4)
Finding the change in weight transfer, one then has:
t
hWA
W
WW
y
L
=?=?
2
(1.15)
Note how no suspension parameters for roll motion have been introduced. This is
to highlight that load transfer can occur even with a very stiff or no suspension
10
whatsoever. Observe Figure 1.5 from Pacejka [20] that depicts a schematic of a simple,
four-wheeled vehicle.
Figure 1.5 ? Four wheel vehicle model [20]
There is a roll axis (also known as the neutral roll axis) defined as a line passing
through the front and rear roll centers. Also, there is a perpendicular distance, h?, that the
CG is located away from the roll axis. The motion of the CG about this axis during
cornering gives rise to roll moments which are resisted by roll stiffness? of the front and
rear suspension setup [19]. The moment about the roll axis created by the motion of the
CG is,
)(' ???=
ys
AhWM (1.6)
11
where the angle ? is the roll angle of the chassis. One can then equate the roll moment to
the roll stiffness and see that:
?
?
K
hWKK
hW
A
sRf
s
y
=
?+
?
=
'
'
(1.7)
Using all information that Milliken and Pacejka have provided so far, full load
transfer equations for the front and rear can be seen as:
WF
F
uF
RF
s
srf
f
f
s
y
f
z
t
W
z
a
hWKK
Kh
t
W
A
W
+
?
?
?
?
?
?
?
?
?
+
?+
=
?
l
l
'
'
'
(1.18)
WR
R
uR
RR
s
srf
r
r
s
y
r
z
t
W
z
a
hWKK
Kh
t
W
A
W
+
?
?
?
?
?
?
?
?
+
?+
=
?
l'
'
'
(1.19)
Again, recall that these load transfer equations give rise to the changing normal
loads that the tire experiences. Simulations need these normal loads along with slip angle
to look up the lateral forces that will be generated by the tire-soil interface.
The last missing piece of information to the vehicle model presented is that of the
lateral forces generated by the tires. One may pick an acceleration, and backtrack to find
what will be expected of the tires, however it is this acceleration we wish to find for a
given vehicle setup. By knowing how the tires behave we can then put together
simulations by Equations 1.12 and 1.13 that find the acceleration, which can then be
integrated to find velocity, and integrated twice to find position.
12
Seeing that these simulations need lateral tire data, how does one acquire the
sought information? One way would be to provide the simulation of look-up tables
created by data from experimentation. Generating these tables for every condition a
particular tire can be put into is time consuming, and the computer storage space required
is tremendous.
An easier way to provide data to a simulation would be with a tire model. A tire
model is a mathematical expression, or set of expressions that, when the proper data is
applied, will return a lateral force. Many tire models exist, however each has their own
advantages and disadvantages. While details of specific tire models will be discussed in
subsequent chapters, the idea is that a tire model is easier and cheaper to implement than
look-up tables.
It is the goal of this research to generate enough empirical data such that tire force
trends in the lateral, vertical, and horizontal directions can be observed and recorded.
Then, a tire model is selected that can appropriately match the experimental results. This
will allow designers to create more effective designs with less physical testing, and thusly
save time and cost.
1.2.2 Tires
To simulations, tire lateral force is simply a value that is fed to equations of
motion. How do tires physically generate these forces? What are the mechanisms that
provide grip? How are tires characterized? One must first step back and understand that it
is not the tire that is simply generating forces, but it is the interaction of the tire with its
13
operating surface. Both the tire and the operating surface have measurable characteristics
that give insight as to what performance one can expect.
Simplistically, tire lateral force is a function of normal load and slip angle. For a
given lateral load, one can expect the shape of a tire curve to look similar to that of
Figure 1.6.
Figure 1.6 ? Sample plot of lateral force v. slip angle [20]
One will observe a linear trend at low slip angles, then entering a transition phase,
and finally to a saturated level, where addition of more slip angle results in no further
lateral force being generated. The saturated region depends on the coefficient of friction
[19], while the linear region is important in comparing the cornering abilities between
tires [19]. The slope of the linear region at zero slip angle is known as the cornering
stiffness [3].
14
0=
?
?
=
?
?
?
y
F
C (1.20)
The shape of this lateral force ? slip angle relationship is governed by several
physical variables, not only of the tire, but of the surface as well. Two main tire variables
to observe are that of the inflation pressure, and the camber angle. The operating surface
has many other variables onto it, and will be discussed further in the text.
Inflation pressure is arguably the easiest thing to change on a vehicle. The
pressure of a tire has several effects on the longitudinal and lateral performance of a
vehicle. Inflation pressure can change the cornering stiffness of a vehicle, i.e. the elastic
properties [19]. A slight increase in pressure can give a slight increase in lateral force
[19]. This knowledge can be used in vehicles to fine tune the balance of a vehicle for
certain conditions [19].
Recall that performance at high slip angles is governed by the effective friction
coefficient. If one reduces inflation pressure, they will realize a larger contact area and
thus a larger friction coefficient. This effect must be balanced such that the inflation
pressure allows the center of the tread and the tire carcass sidewalls to carry a
proportional amount of the vehicle load on that tire [19].
Another effect of altering tire pressure is how the pressure change affects the
motion resistance forces on a vehicle. A decreased pressure will allow a lower spring
rate, and thusly more deformation of the tire [19]. More deformation at speed translates to
more energy going into deforming the tire. This energy is then changed to heat and at
15
great levels, may damage a tire. A higher pressure will produce less deformation, and
thus less motion resistance and increased tire life.
Camber angle also has a significant effect on the performance of a tire. Adding
camber to a tire means that the tire is tilted about an axis that lies in the ground plane, and
is oriented longitudinally to the vehicle. Figure 1.7 depicts this arrangement. Positive
camber is defined as the tilting of the tire such that the top of the tire is moved in the
direction of the lateral force generated by the tire.
Figure 1.7 ? A cambered tire [20]
The increased angle of tilt generates a thrust force in the contact patch. Much like
cornering stiffness, there exists camber stiffness. Camber stiffness is the measure of the
slope of the camber thrust curve at zero slip angle [3].
16
0=
?
?
=
?
?
?
?
y
F
C (1.21)
Camber stiffness is generally less than cornering stiffness, and Milliken reports
that it is around five or six times less than that of cornering stiffness [19]. Furthermore,
Milliken indicates that in wide racing tires, camber effects tend to trail off after about five
degrees [19]. Motorcycle tires however, generate effective lateral force to upwards of
fifty degrees of camber [19]. Other factors influencing the amount of camber a
suspension designer can put into a suspension is the packaging of the vehicle, and the
extra stresses put on wheel set components due to the increased camber angle.
1.2.3 Operating Surfaces
While tire properties have an effect on lateral force generation, the operating
surface also has a large hand in generating lateral forces. The shearing of a soil generates
lateral and longitudinal tractive forces. Normal response of a soil has an effect on the
motion resistance and the vertical support of vehicular loads. Several methods for
evaluating the various soil properties for vehicle operations have been developed over the
last 75 years with some producing more useful results than others. The field of
terramechanics has been devoted to characterizing soil mechanics in relation to vehicular
operations.
The informal beginnings of terramechanics can be linked to World War I where
British Officers walking sticks were reportedly used to prod the ground and determine if
it was safe for passage. These walking sticks are also rumored to be the basis for the cone
penetrometer [2]. Around 1913, Russian scientists proposed an equation relating the
17
sinkage of a plate in soil with a uniformly distributed load. The equation took the
following form:
n
kzp = (1.2)
Where, static sinkage (displacement into soil) is measured by z, and pressure under the
pressure plate by p. Deformation modulus k, and deformation exponent n, was to be
determined by experiment [1]. In that same time frame, Bernstein used a deformation
exponent of n=1/2 in his works on motion resistance of wheels [1]. Goriatchkin and
Bernstein tried to surmount the difficulties seen in using k as a deformation modulus (as
it was directly related to the plate size), however their attempts did not turn the result they
hoped [1]. Many years later, M.G. Bekker would find a way to improve the pressure-
sinkage relationship in his development of the bevameter [27].
Originally, soil measurement has its roots in civil engineering knowledge, and
some soil measurement techniques from civil engineering were imported to be used for
vehicular travel. There are several methods for measuring soil from a civil engineering
approach, and each test must be used carefully in reference to the problem at hand. The
dominant approaches are the vane shear test, tri-axial test, and direct shear test [26].
A vane shear test (Figure 1.8) involves forcing a four-bladed vane into the soil or
a core sample obtained at the test site. It is then rotated at a constant angular velocity and
the torque is recorded [8].
18
Figure 1.8 ? Vane shear apparatus
Care must be used if using vane shear tests as the strength indicated is a function
of angular velocity. This is due to the presence of pore water pressure that may not
dissipate quickly enough based on the grain sizes of the soil and thus fail the soil at a
lower stress. [9]. Note that ground vehicles will apply varying shear rates to the soil, and
not necessarily under the soil, but rather on the top, especially in the case of medium
UGVs, where sinkage is small related to the diameter of the tire.
Direct shear tests (Figure 1.9) comprise of a cylindrical sample of soil placed
between two rings, and a normal load is applied. Then a shear force is applied across the
rings, and then each test is plotted against its normal load to obtain a Mohr-Coulomb
failure envelope [10].
Figure 1.9 ? Schematic of a direct shear test [10]
19
The test specimens are core samples taken from a particular site and then the test
is performed in the lab. On the fastest setting obtainable, the staff at CRREL (Cold
Regions Research and Engineering Laboratory) could only obtain a shear rate of 0.8
cm/s, wheras the typical shear rate of a traction generating tire is between 10 and 40cm/s
[10].
While Shoop and her colleagues have pointed out that there is a discrepancy
between actual and required shear rates for the direct shear test [10], Okello stated, ?The
authors concluded, very uncommittedly, that it may be suspected that the shearing
mechanism of the direct shear test is most closely allied to that of a soil under a
vehicle?? [8] The uncertainty regarding direct shear leads one away from using this test.
Also, like the vane shear test, it indicates the shear stress within a soil, and not at the
interface between a vehicles tire and the soil.
The third and final civil engineering approach to soil measurement for vehicular
travel is the tri-axial test (Figure 1.10). Like the direct shear test, a core sample is
prepared for use in the laboratory. The sample is encased in a rubber membrane with caps
on the top and bottom. It is then placed on a pedestal in a chamber that is filled with
water and a confining pressure is applied to the sample. The pedestal is then set in motion
and normal reaction forces are recorded [8].
20
Figure 1.10 - Schematic of Tri-axial soil testing apparatus [26]
Tests are then compiled by varied normal loads and the Mohr-Coulomb failure
envelope can then be developed [10]. Unfortunately the soil can not be measured in-situ,
nor can the whole field be measured due to the extremely slow and costly nature of the
tri-axial test [8].
While there are several methods available to measure soil, two methods have
found significant support in the field of terramechanics. The cone penetrometer is a
simple, lightweight device which provides limited information, while the bevameter is a
larger, more complex apparatus, yet provides significant insight into the strength
properties of the terrain for vehicular travel.
21
Developed by the Waterways Experiment Station (WES) for the U.S. Military, the
cone penetrometer would originally be used to provide a ?go/no-go? decision on soil
strength for military commanders in the field [3]. The cone penetrometer (Figure 1.11) is
a 30-degree right-circular cone having a ?? base area [3]. The Cone Index is a measure
of force per unit cone base area. At the top is a proving ring or load cell to measure
penetration resistance at various depths when the cone is forced into the ground at 1 cm/s
[3], [4].
Figure 1.11 ? Cone penetrometer [3]
Despite its relative user friendliness and widespread use within the agriculture
world, the cone penetrometer?s effectiveness has been questioned on its ability to deliver
accurate results in mobility studies [5].
While several versions of Turnage?s Mobility number are available (a number
created by dimensional analysis that relates various contact dimensions but only works at
22
zero slip angle), Gee-Cough and Sommer [32] state that ?broadly speaking? the Turnage
Mobility numbers may be used to model the coefficient of side force if soil friction angle
was constant and the cone index was a measure of soil cohesion [29]. With that being
said, however, the ?validity of using soil cone index value to represent soil condition has
always been questionable? and that the soil cone index is ?a compound parameter which
is a function of several soil properties? including soil friction angle and cohesion [18].
The bevameter, also known as the Bekker Value Meter, was developed by M.G.
Bekker in the 1950?s seeing that soil-vehicle properties should be measured under
conditions as close to those presented by a vehicle [3]. The premise is that vehicles apply
normal and shear loads to the soil and therefore soil should be measured when subjected
to normal and shear loads. This is the soil testing apparatus used in this study of ATV
tires. Specifics regarding the bevameter used for the study can be found in subsequent
chapters. The bevameter (Figure 1.12) consists of a two part apparatus with one
measuring normal load response, and the other measuring shear load response. For
normal loading, penetration plates are forced into the ground and the pressure-sinkage
behavior is recorded. The second part is a shear test, typically in the form of an annulus
which measures traction-slip performance [3]. These two tests are indicators of the
strength of a soil for vehicular travel (i.e. rubber-soil friction angle).
23
Figure 1.12 ? Schematic of a Bevameter [3]
As the bevameter characterizes a soil?s strength for vehicular travel, it must be
characterized in such a way that can be connected to existing work. The bevameter
generates data such that a Mohr-Coulomb failure envelope can be generated for a
particular surface. The Mohr-Coulomb failure envelope is a principle used extensively in
soil-mechanics, and much has been written about it. One must be reminded that the
bevameter?s results represent the strength of a soil for vehicular travel.
Before looking into the bevameter parameters, one must have a brief
understanding of the Mohr-Coulomb soil failure envelope, and thus a slice of classic soil
mechanics.
Soils can be viewed as frictional, cohesive, or a combination of both. Frictional
soils (i.e. sand) gain their strength from the intergranular friction forces. The greater the
confinement force, the greater the intergranular friction, and thus a stronger element of
24
soil. Cohesive soils have intermolecular bonds, originating from the positive and negative
electric charges imparted on them [26]. Please note that cohesive soils have much smaller
particle sizes than frictional soils [26]. This small grain size is what allows the
intermolecular forces to dominate the strength of the soil [26].
Both cohesion and friction can be quantified from the Mohr-Coulomb failure
envelope. Normal stresses are plotted on the x-axis, and shear stresses plotted on the y-
axis. A line is laid tangent to the Mohr circles, and the y-intercept is measured as
cohesion, in units of stress, while the angle of the line is the friction angle, in units of
degrees. See Figure 1.13 for a representation of the Mohr-Coulomb envelope.
The Mohr-Coulomb failure theory for soils, states (as reported by Liang, et. al. in
[33]) that the soil at a point will fail if the shear stress meets the conditions of Equation
1.23.
??? tan+= c (1.23)
For Equation 1.23, c is the cohesion, measured in units of stress, ? is the normal
stress on an element of soil, ? is the shear stress experienced on that element of soil, and ?
is the internal friction angle. Equation 1.23 shows that under conditions of normal and
shear stress, cohesion and friction angle govern the strength of the soil [33]. This can be
used with vehicular studies, as vehicles apply shear and normal loads to the surface of a
soil. While the internal friction angle is used for soil mechanics, the rubber-soil friction
angle is used for vehicular studies as the stress is being applied via the tire in this study,
which is rubber.
In the words of Karl Terzaghi, cohesion and friction angle can be quantified as
follows: ?If we dig into a bed of dry or of completely immersed sand, the material at the
25
sides of the excavation slides towards the bottom. This behavior indicates the complete
absence of a bond between the individual sand particles. The sliding material does not
come to rest until the angle of inclination of the slopes becomes equal to a certain angle
known as the angle of repose. The angle of repose of dry sand as well as that of
completely immersed sand is independent of the height of a slope. On the other hand a
trench 20 to 30 feet deep with unsupported vertical sides can be excavated in stiff plastic
clay. This fact indicates the existence of a firm bond between the clay particles. However,
as soon as the depth of the trench exceeds a certain value, dependant upon the intensity of
the bond between clay particles, the sides of the trench fail and the slope of the debris
which covers the bottom of the cut after failure is far from vertical. The bond between the
soil particles is called cohesion. No definite angle of repose can be assigned to a soil with
cohesion, because the steepest slope at which such a soil can stand decreases with
increasing height of the slope. Even sand, if it is moist, has some cohesion.? [13]
Figure 1.13 ? Mohr-Coulomb strength envelope
One must note however that the strength envelope is not linear throughout its
entirety. Particularly associated with granular soils, confinement, and normal stresses can
26
become so great that grain crushing can occur, and thus lower the strength envelope seen
in Figure 1.14 [26]. At the bottom line however, we can see that there are two
mechanisms governing soil strength, cohesion, and friction.
Figure 1.14 ? Nonlinearities of Mohr-Coulomb envelope at high normal
stresses
Another soil value that the bevameter can produce is that of the shear deformation
modulus. The shear deformation modulus measures the amount of shear displacement
needed for the development of the maximum shear stress. Denoted by the letter K, the
shear deformation modulus is used in Equation 1.24, where j is the measure of linear
displacement.
?
?
?
?
?
??
?=
K
j
s
s
exp1
max
(1.24)
While the bevameter method has a shear strength component, there is an
associated normal component. This normal component characterized the relationship
27
between normal (vertical) loadings and the associated sinkage (displacement) into the
terrain.
Bekker used the work of Karl Terzaghi for soil bearing capacity in order to
develop a pressure-sinkage relationship [1]. Following the form of the Bernstein
equation, Bekker developed the following equation:
nc
zk
b
k
p )(
?
+= (1.25)
The variables within Equation 1.23 have p as pressure, b as the width or diameter
of the penetration plates, z as the sinkage, and k
?
, k
c
, and n are soil properties. Properties
k
?
, and k
c
, are related to friction angle and cohesion, respectively. The sinkage exponent
is n. As indicated by Wong, significant manipulation of the test data must be performed
in order to generate these properties [3]. First the data must be plotted on a log-log scale,
then, a straight line is fit to the data. No standard method for the fit is observed, and is
generally fit by eye [3].
It must be noted once more that while the bevameter measures various soil
properties, these properties are only of use to vehicular studies. These properties are only
used in describing the strength of a soil for the movement of a vehicle. Soil properties
that are familiar to civil engineers such as void ratio, liquid limit, and water content all
play roles in the values of the bevameter properties, however their effects are
complicated, but manifest themselves in the various values listed previously, and are not
within the scope of this research.
While there were little physical improvements to bevameter equipment over the
years (besides computerization), the use of the data obtained would undergo changes,
28
particularly in the pressure-sinkage relationship. Bekker?s original pressure-sinkage
relations would be modified by A.R. Reece to suggest that pressure increases linearly
with the width of the sinkage plate for a given value of sinkage divided by width [7].
Reece?s work would make the two sinkage coefficients dimensionless and independent of
the sinkage exponent. The new equation was backed up by significant testing, and also fit
into soil behavior that could be predicted by classical soil mechanics as described by Karl
Terzaghi [7].
n
c
bzbkckp )/)((
''
?
?+= (1.26)
In addition to the pressure-sinkage issues, the bevameter came under fire by
Karafiath and Notazwiki, indicating that the true failure surface of a soil did not occur at
the surface of the shear head, but rather on oblique planes due to the lack of lateral
confinement of the soil [11]. This lack of lateral confinement around the shear also
allows the large shear displacements to in turn allow large vertical displacements, known
as slip-sinkage. To alleviate this, Janosi and Karafiath suggest the application of a
surcharge to prevent the shear head from continually sinking [12]. (Note: A surcharge is
the application of an identical normal load to that of what is being applied by a shear
head. This is meant to ensure that the soil immediately outside of that which is being
sheared has the same confining stress as that of adjacent soil.)
J. Okello noted the bevameter method and the cone penetrometer method to be the
most appropriate for vehicular studies as the measurements were made in-situ. He further
states a potential drawback to the use of the bevameter, as it only characterizes the soil on
the surface and does not allow for penetration plates or shear heads to be of the same
contact area as the vehicles under question [23].
29
While these issues of size and depth may seem detrimental to the further use of
the bevameter, these issues are not present in small vehicle studies. This can be realized
by the fact that sinkage of ATV?s or medium sized UGV?s is not enough to cause a
significant change in confinement stress. Okello further notes that the size of the body in
a bevameter is more important than the shape, and should ?reproduce those of the vehicle
ground contact area? [23].
1.3 Purpose of Thesis and Contribution
As described in the preceding text, there are several methods for collecting data
on tire performance and soil strength. Trends in tire pressure and camber changes are
known qualitatively, and changes in performance from a surface change are also
qualitatively known. This research will measure, through load cells and accelerometers,
forces generated by the tire. These forces will be reduced by equations derived from the
geometry of the test rig, or calculated directly in the case of accelerometer data. Soil data
will be measured via bevameter techniques that characterize soil-rubber shear stress
response and soil normal stress response. Soil properties of cohesion, friction angle, and
shear modulus are reduced from the displacement and moment data from the bevameter.
Tire lateral force information is still needed to generate accurate vehicle dynamics
simulations and improve tire performance models. By building test apparatus and using a
tire model for vehicle dynamics, one can gain insight as to how tire parameters, such as
camber, and inflation pressure, and operating surface properties, such as cohesion, shear
modulus, and friction angle, affect the performance of a vehicle. This insight leads to
understanding the links and relationships such that tire model parameters can be
30
formulated, and alleviate the need to do substantial guessing during the design process or
make several expensive performance tests. This work seeks to measure tire forces and
apply a tire model at different settings of camber, pressure, and surface to observe their
effects on the exponential tire models coefficients.
31
CHAPTER 2
RESEARCH TOOLS
The research tools presented in this chapter represent the items used by the author
to either generate or analyze the test data. Much time was invested in the design and
fabrication of the test rigs as nothing meeting the specific need is available commercially.
Chapter 2 will highlight the design and use of these tools in the generation of tire and soil
data.
2.1 Bevameter
As described in Chapter 1, the bevameter is a device which measures the shear
and normal strength of a soil in relation to vehicular travel. Dozens of bevameters have
been built since M.G. Bekker first built them in the 1950?s, and the progression of
technology has certainly aided the effectiveness of the bevameter over time. Highlighted
in this section are some of the more successful designs and their benefits, the goals and
requirements of the bevameter for this study, and the as-built result.
2.1.1 Bevameter Development
The design and development of a bevameter includes a look at existing designs,
highlighting the advantages of each design, and specifying goals for a new design.
Existing bevameter designs were reviewed in order to fine tune design goals for this
work. The result is a field-portable bevameter capable of one-man operation.
32
Bekker?s work at the Land Locomotion Laboratory produced several small, man-
portable bevameters and eventually a large mobile bevameter mounted on a tracked
vehicle [27]. Upon observation, most of the shear heads on Bekker?s bevameters are
those which have grousers (teeth) attached to them. This would seem to indicate testing
that is more for tracked vehicle research than that for wheeled vehicle research.
Several authors indicate the great importance that the pressure plate and shear
head contact area should closely represent those of the vehicle in question [23]. While
this may seem a simple goal to achieve, many of the off-road vehicles studied by Bekker
and Wong were so large that substantial structures to hold the apparatus were required
[27]. This condition limited the mobility of the bevameter that matched pressure plate and
shear head area to the vehicle that was studied.
Shoop, of the U.S. Army Cold Regions Research and Engineering Laboratory
(CRREL) reports on a bevameter shear head designed by Stafford and Tanner (Figure
2.1) [29].
33
Figure 2.1 ? Bevameter shear-head as described by CRREL [10]
While man-portable and easy to use, the shear head did not match the contact area
of the vehicles generally studied by CRREL [10]. The advantage of studying ATV tires is
that the contact area is small, and a bevameter can be designed to be vehicle portable on a
platform that is not excessive in size.
The bevameter constructed by Wong made use of a substantial transport platform,
and electronic sensors [3]. Collecting this data was facilitated by the use of a personal
computer and specifically written software to alleviate the manual data reduction of
earlier strip charts [3] [14].
34
Figure 2.2 ? Bevameter as designed by Wong [3]
In Figure 2.2, note the shear head by the vehicles right track, and the pressure-
sinkage apparatus near the center of the vehicle, closest to the operator. Figure 2.3 shows
a schematic of the shear head.
35
Figure 2.3 ? Details of shear head instrumentation and control [3]
Note that Wong also makes use of hydraulic actuation methods, easily supported
by the transportation platform.
Looking at the systems previously built, three things are clear. First, the
bevameter should have contact areas similar to the vehicle studied. Second, a bevameter
should be portable, either by human, or vehicle. Third, a bevameter should use electronic
sensors with computers to reduce data reduction workload and reduce chances of error.
For the bevameter constructed at Auburn University for this project, all three
design goals were tied into a fourth goal of maintaining low cost. In order to ensure the
proper contact area, a KT 821 tire (the tire used in the study) had its lug-tips coated with
a marking compound (Figure 2.4). The tire was then placed on a vehicle, with the tire on
a piece of paper to depict the contact pattern. Approximately 300 lbf vertical load was
applied, and the tread pattern marked.
36
Figure 2.4 ? KT 821 lug contact area
Each lug marking was measured, and an area for each was calculated. The sum of
the areas totaled 6.933 sq. in. Other tires were tested, and their results were factored into
the decision to make the shear head have a 5 in. outside diameter, and a 3 in. inside
diameter. This results in a 12.5 sq. in. Also, the shear head would be coated in silicone
rubber of the same durometer hardness as the KT 821 tire.
The decision to make the shear head to be 12.5 sq. in. stems from larger contact
patterns from other tires, and also assuming that the KT 821 could sink into the soil
37
beyond than the lug height, with the tire carcass making contact, and effectively doubling
the 6.933 sq. in. contact area by the lug tips.
The pressure plates used for the pressure-sinkage tests were sized to span the area
difference between the lug tips, and the entire carcass making contact. Four inch and
three inch pressure plates were fabricated providing 12.56 sq. in. and 7.06 sq. in. of
contact area, respectively.
As the contact areas were sized to emulate those of ATV tires, the normal loads
must be of a similar magnitude as well. Looking at the man-portable models, carrying
around enough weight to provide a normal load typical to an ATV tire would be quite
cumbersome. According to Weir and Zellner, a typical ATV can have an overall weight
near 600 lbf [28]. Assuming an equally divided weight to all four wheels, this would
result in 150 lbf of normal load at a static condition.
Figure 2.5 ? Bevameter as attached to Yamaha Rhino
38
Figure 2.6 ? Bevameter as designed in CAD
In order to provide the significant normal loads to the shear head and pressure
plates, the bevameter is mounted to a Yamaha Rhino side-by-side ATV (Figure 2.5, 2.6).
The Rhino weighs approximately 1,000 lbf, and with the bevameter mounted on one end,
the Rhino can provide well in excess of 300 lbf of normal load. By looking at Equation
1.5, the load transfer equation, this would generate well over 1 G of lateral acceleration,
which is above the range expected for ATV?s and UGV?s especially on soil surfaces.
Allowing the normal load to be changed as required in experiments, a pneumatic
cylinder was employed to provide the varying normal loads to the pressure plates and the
shear head. A double ended pneumatic cylinder was selected so that applying torque to
the shear head would be easier to control than a hydraulically or pneumatically actuated
system. This manually actuated (torque applied by operator?s strength) system
significantly reduced cost over a hydraulic or pneumatic actuation system.
39
Lastly, to fulfill the goal of simplification of data collection, electronic sensors
were used on the shear test portion of the bevameter. Normal load tests were conducted
(Figure 2.7) by recording the pressure reading on the cylinder, and the displacement into
the soil via a dial indicator.
Figure 2.7 ? Details of normal load test apparatus
The use of the dial indicator allowed much accuracy for the small displacements
found in the stiff test soils. For the shear tests, a torque cell was placed between the
pneumatic cylinder shaft and the breaker bar used by the operator. Displacement was
recorded by a rotary potentiometer as seen just behind the pressure gauge in Figure 2.8.
40
Figure 2.8 ? Elements of shear head data acquisition system
Using design elements from previous, successful bevameters, a custom bevameter
is built for application to ATV tires. Included in the design is field mobility, low cost,
proper sizing, and semi-automated data acquisition.
2.1.2 Operations
Operating the bevameter to collect field data requires some basic knowledge of
soils, computers, and electronics, while also requiring a steady hand. Upon meeting these
requirements, gathering soil data can be completed with relative ease.
As the bevameter is mounted to an ATV, several sites can be tested over a
particular area due to the mobility of the vehicle. Once a particular site is selected, the
vehicle is parked in as a level of a condition as possible to get the best results. Even if the
terrain is not perfectly level at the location, a universal joint mounted between the shear
head and cylinder shaft can allow for slight misalignment.
41
Once positioned, the bevameters compressed gas source is connected to the
cylinder, and the appropriate attachment affixed to the cylinder shaft, be it a pressure
plate or shear head. For the pressure-sinkage tests, the pressure plates are affixed, and the
dial indicator attached. Pressure is applied to the cylinder, and sinkage is measured.
Repeat measurements are taken with higher cylinder pressures.
For the shear head tests, the data acquisition system is initialized, and sensors are
zeroed. A normal load is applied to the shear head by increasing cylinder pressure to the
desired level. The operator then applies a torque about the cylinder shaft, rotating the
shear head through approximately 180 deg. of angular displacement, while the data
recorder collects torque and displacement data. The experiment is repeated with
increasing normal loads.
The operation of the bevameter is convenient due to the mounting on a
transportation platform. A single operator can then collect data from either a normal load
test or a shear head test with relative ease due to the use of a data acquisition system.
2.1.3 Assessment
As any other mechanical device, the bevameter must be evaluated to see that
useful, acceptable results are being generated. Since several bevameters have been built
over the years, there is plenty of data to use for comparisons.
Shear head tests produced data whose shape fit that of equations provided by
Wong [3], and filtered data from Yu [30].
42
Figure 2.9 ? Shear head data with curve fit from Equation 1.23
Figure 2.10 ? Shear test data from University of Pretoria [30]
43
As one can see, Figure 2.9 shows good agreement between the equation presented
by Wong [3] and the data collected for this study. Also, one will see that this data is of
the same shape that was presented by Yu in tests conducted for the University of Pretoria
in Figure 2.10 [30]. Note that the linear displacement is the displacement around the
circumference of the center of the shear head.
Normal load tests also showed the similar trends as recorded previously by Yu.
These trends show an increasing pressure for a proportional increase in sinkage. The tests
conducted by the author used circular pressure plates, while Yu utilized rectangular plates
[30]. The difference of shape is not of concern, as the Bekker equations for sinkage
(Equation 18), do not use an area, but the radius for a circular plate, and the smaller
dimension of a rectangular plate. Figure 2.11 shows Yu?s data, while Figure 2.12 shows
data collected by the author.
Figure 2.11 ? Penetration test data from University of Pretoria [30]
44
Figure 2.12 ? Penetration test data collected by author
45
2.2 Tire Test Rig
2.2.1 Introduction
Just as important collecting soil data, is collecting tire data for the understanding
of the tire-terrain interface. Several methods for testing tires are available, including
rolling drum rigs, belted test rigs, and towed test rigs. Each has their own benefits, yet the
towed test rig is the method of choice for this study for its simplicity, cost, mobility, and
ability to collect field data.
2.2.2 Development
Upon studying the several methods for testing tires, three methods were
considered. These methods included a rolling drum apparatus, an MTS Flat-Trac (rolling
roadway) machine, and developing a towed test rig specifically for this study.
Rolling drum test rigs have an advantage of being less complicated than a Flat-
Trac machine, however they are typically used only in longitudinal studies. A major
disadvantage of the rolling drum test rigs is that soil cannot be directly tested. Drums are
usually made of steel and are much more rigid than soil. While less costly and less
complicated than an Flat-Trac, the rolling drum simply cannot deliver the required data.
Utilization of an MTS Flat-Trac test rig would generate significantly more useful
results than the rolling drum tests, as the Flat-Trac is designed specifically to test tires in
cornering performance and longitudinal performance. The machines themselves are of
tremendous cost and require much care and maintenance. Furthermore, the rolling
roadway utilized is not a soil. The roadway is a belt that is much like sand-paper that has
a given friction coefficient. Vertical stiffness of the roadway can be simulated by
46
changing properties of the water bearing under the belt under the contact patch, and the
tension in the belt, however this still does not mimic the actual conditions in the field.
The lack of being able to simulate field conditions in the laboratory forces the
researcher to develop a test rig to be towed by either a trolley (in soil bins) or by a vehicle
over terrain found in the field. As indicated before, there has been one previous study that
has tested the lateral performance of ATV tires [25]. The researchers at the University of
Maryland constructed an instrumented test rig, towed by a vehicle over various surfaces
[25]. The test rig allowed steer angle, camber angle, and braking slip to be adjusted.
Furthermore the UMD rig had fabricated proving-ring load cells at the reaction
points where the test rig attached to the tow vehicle [25]. These sensors fed into the data
acquisition system where their output was recorded to strip charts. Vehicle speed was
measured by an un-driven wheel, while actual wheel-speed of the test tire was recorded
to gather slip values [25]. Again, this data was fed to the data acquisition system for
recording on strip charts [25].
Normal load was applied through weights which were mounted at a point just
ahead of the contact patch of the tire. Their reaction forces were measured from two load
cells measuring in the vertical direction at the reaction points on the tow vehicle [25].
47
Figure 2.13 ? Test rig designed by University of Maryland [25]
The core design in for the UMD test rig (Figure 2.13) was utilized by the author
for this study. Several improvements were made such that more accurate and reliable data
could be gathered. Improvements included the addition of a fully electronic data
acquisition system, elimination of vertical load cells (and replacement with an
accelerometer at the tire), and an inclinometer to observe true camber angle.
The data acquisition system is centered on a Dataq 710 8-channel data logger. Fed
into the Dataq are lines from three load cells, two measuring longitudinal force, and one
measuring lateral (See Figures 2.14 and 2.15). Also, an accelerometer is mounted under
the weights which are situated directly above the contact patch of the tire.
48
Figure 2.14 ? Details of data acquisition box and load cells
Figure 2.15 ? Details of load cells
49
Mechanical and structural considerations for the test rig (Figure 2.16) were made
such that sufficient camber and steer angle for various sized tires could be induced while
attempting to minimize deflection. CAD software was utilized to design the layout of the
test rig such that it could be affixed to the Yamaha Rhino tow vehicle, and allow 22 in. to
26 in. diameter tires. Limited FEA was conducted to ensure that no major deflections of
the test rig would occur due to significant lateral and vertical loads.
Figure 2.16 ? 3D model of tire test rig in CAD
The towed test rig is designed in three parts: vehicle mount, triangle, and hook.
The vehicle mount allows bolt-on capabilities to the Yamaha Rhino while allowing the
vehicle to be returned to normal use when testing is complete. The mount (Figure 2.17)
also has several settings such that tires of various diameters can be accommodated. The
50
triangle is the simplest part of the test rig, as it is a structural element, bridging the gap
between the separately calibrated load cells and the hook. The triangle also includes the
sector plate, where slip angle selections can be made. The final part of the test rig is the
hook. While the triangle oriented parallel to the ground plane, the hook is in a vertical
plane, which rotates about the z-axis through a bearing hole on the triangle. The hook
also has a tab such that when the hook is rotated within the triangle, a stop-pin can be
inserted when the desired slip angle has been set.
Figure 2.17 ? 3D model of tire test rig mounting system
Camber adjustment is made on the hook, where camber angles from zero degrees
to eight degrees are possible (Figure 2.18). The arc which the camber adjustment follows
is centered about the contact patch of the tire, such that major displacements of the center
51
of vertical loading are avoided. Finally, the hook terminates at a spindle, where a hub and
brake disc mount to the test rig. Since the hook only supports the tire from one side,
changing tires is a short task. The one sided support does raise structural questions,
however.
Figure 2.18 ? Details of camber and slip angle adjustment
In order to allow tires of various diameters and widths, the hook must be of a
certain geometry. Supporting the tire from only one side with this geometry and a
maximum of 300 lbf of static load (and subsequently higher dynamic loads) requires that
proper materials of proper dimensions be employed.
Two design goals were set to limit the flexure of the hook when loaded by forces
seen at the spindle. First, the slip angle must not exceed half of a degree through flexure
of the hook. Second, the camber angle must not exceed one degree through flexure of the
52
hook. With these goals in mind, FEA was applied in order to specify the proper diameter
tubing to make up the hook.
A simple FEA was conducted in ALGOR with a basic beam-truss analysis. The
hook was modeled by x, y, z coordinates, and various properties were changed to model
commonly available tubing sizes. One-and-a-half inch tubing was determined to be
structurally acceptable; however, a decision was made to use 2 in. diameter tubing with a
0.125 in. wall thickness. Two inch tubing of the mentioned wall thickness is a very
commonly available arrangement, and for much less cost than the 1.5 in. diameter tubing.
The use of the 2 in. tubing in the designed arrangement holds the camber angle to 0.5
deg. and the slip angle to 0.27 deg. The loaded condition is assumed to be a 600 lbf
normal load, and a 300 lbf lateral load, which is a worst-case scenario.
Other sources of possible deflection include play at the load cell mounting points,
frame-to-vehicle mounts, and play within the slip angle adjuster. These locations were
checked for excessive movement, and since significant care was taken during
construction, no play of any concern was recorded.
2.2.3 Operations
Like the bevameter, the tire test rig is capable of being run by only one operator.
Operations for collecting data on a particular tire include involvement with the data
acquisition system, mechanical settings on the test rig, and proper driving techniques with
the tow vehicle. Careful observation of all three areas during a test ensure the generation
of useful data.
53
A tire to be tested is affixed to the test rig and the desired camber set into the
hook. The desired weights are then applied to the top of the hook. The operator then
positions the tow vehicle such that a run of approximately 30 to 40 yards can be taken.
Before proceeding, the operator then manually sets the steer angle between the triangle
and the hook. A final check is made to ensure that fasteners are properly tightened and
the proper settings are applied.
Once the settings have been confirmed, the operator accelerates the vehicle to the
test speed, which is nominally 10 mph. Once a steady-state speed is reached, the operator
starts logging data to the computer. Before the end of the run, and decelerating the tow
vehicle, the operator stops data collection. A loop is then made back to the starting point
where runs are repeated to generate enough data to create statistically significant results.
2.2.4 Assessment
Before the test data can be used, the data must be checked to see that acceptable
results are being generated. Checking for acceptable results requires referencing the data
generated by the previous ATV tire lateral force study conducted by the University of
Maryland [25]. Lateral Force and longitudinal force curves are compared for shape and
the magnitudes of forces are compared.
Expectant lateral force data takes on the form of what is presented in Figure 2.19.
54
Figure 2.19 ? Lateral force coefficient data from UMD study [25]
Observe no distinct peak to the data, and how it gradually approaches a constant
value for large slip angles. Also note how slip angles to only 15 degrees were tested.
Figure 2.20 shows longitudinal force data from the UMD study.
Figure 2.20 ? Rolling resistance coefficient from UMD [25]
55
Note on the rolling resistance coefficient, it generally increases as slip angle
increases. This is expected as the longitudinal component of the lateral force increases
with increasing slip angle. Also note the rough grouping of data and how it seems to
fluctuate between 0.05 and 0.15.
Lateral and longitudinal force data generated by the tire test rig can be seen in
Figure 2.21. Normalized lateral force (also known as Lateral Force Coefficient, LFC) is
shown on the y-axis. Like the data generated in [25], note the same trends are present,
and the magnitudes of lateral and longitudinal force data are consistent.
Figure 2.21 ? Example lateral force data generated by the author
56
2.3 Tire Model
2.3.1 Introduction
In order to draw connections between the soil and tire performance, a tire model,
a mathematical representation of tire performance, must be employed. Models have
various coefficients which can be linked to variables like tire pressure, camber angle, and
various properties of the operating surface. Several tire models have been developed over
the last few decades, and some are more applicable than others.
According to Metz, there are five basic tire models: Linear, Bi-Linear,
Exponential, Polynomial, and Modified Sinusoidal (Pacejka) [31]. Each model has their
own marks of merit, and shortcomings. This study employs the use of the exponential tire
model as it closely represents off road tire data as observed by Crolla, et. al. [15]
2.3.2 Tire Models
The linear tire model presented by Metz is a mathematically simple model relying
only on cornering stiffness; C?. Calculations are very easy for mathematics packages.
This model is useful only in situations where small slip angles are encountered, and it is
known that the tire is operating in the linear range (Equation 2.1) [31].
?
?yLateral
CF = (2.1)
A bi-linear model is an improvement upon the linear tire model. While retaining
the linear portion, a limiting value is imposed on lateral force. The line of the limiting
value has zero slope and presents a constant lateral force above a particular transition slip
angle. This model can accurately model the linear portions of the tire for low slip angles,
and the saturated region for high slip angles for some tires (Equation 2.2) [31]. The
57
transition region is poorly represented, and has a discontinuous slope, which can be
numerically problematic for vehicle handling simulations.
MaxLateralMaxLateralLateral
MaxLateralLateralyLateral
FFFF
FFCF
,,
,
,
,
?=
<= ?
?
(2.2)
The polynomial tire model (Equation 2.3) is an improvement in accuracy over the
bi-linear and linear models. There are, however, some numerical instabilities and a lack
of camber effect. The coefficients of the polynomial model also do not have any physical
meaning [31].
2
210
3
3
2
210
zzn
Y
FcFccb
bbbbF
++=
+++= ???
(2.3)
The model with the most accurate fit is the Pacejka Modified Sinusoidal tire
model. This model accepts vertical load effects and camber effects, combining them into
a very smooth curve fit. Disadvantages include the fact that the model has six more
equations to compute just to arrive at a lateral force, and the said equations require
significant data for curve fitting [31]. The Pacejka Modified Sinusoidal tire model is as
follows in Equation 2.4.
()??=
?+=
?=
+=
+=
+++?=
+=
12
543
11
2
10
9
7
2
6
2
2
1
1
3.1
))]arctan(sin([
)('
'
))'(arctan()/()')(1(
'))arctan(3.1sin(
a
D
Faaa
B
FaFaS
aS
FaFaE
FaFaD
SBBESE
SBDF
z
zzz
x
zz
yy
xx
zY
???
?
(2.4)
58
Lastly, the exponential model is presented in Equation 2.5. This is the model that
will be utilized in the remainder of the study. The exponential tire model enjoys the
benefits of having vertical load effect, having a continuous slope, and most importantly,
the coefficients have physical meaning. Seeing that the soil shear stress?shear
displacement relationship is of an exponential form, a tire?s behavior on soil assuming an
exponential form should not be surprising. The exponential tire presented by Crolla [15]
and Metz [31] does not, however, include effects for camber. This does raise a question
as to how the coefficients of the exponential model are affected by changes in camber.
))exp(1( ?BAFF
zy
??= (2.5)
One may note the simplicity of the equation developed by Crolla [31]. This allows
for hand calculations to be performed in the requirement of a quick answer. The
simplicity also lends itself to shorter processor times when compared to the Pacejka
model when utilized in vehicle dynamics calculations.
While Crolla presents the simple model in Equation 2.5, Metz offers a
hypothesized improvement on modeling the effect of normal load variations in Equation
2.6 [31]. Metz shows how the ?B? coefficient in the exponential tire model may be
affected by normal loads [31].
ADFFACB
m
zrefz
/)/)(/(
,
+= (2.6)
There are several variable additions to the expanded exponential model. F
z,ref
is a
reference vertical load, and the tire rated vertical load is suggested by Metz to serve as
this value. C, D, and m are all empirical exponents from fit data. While the expanded
exponential model seems to provide an added degree of accuracy, it is the goal of this
study to obtain the basic relationships between soil properties, tire settings, and just the
59
basic A and B coefficients of the exponential tire model. Further analysis of Equation 2.6
may be conducted in subsequent studies.
2.3.3 Assessment
As the exponential tire model was selected for its similarity to soil-rubber shear
characteristics, and ease of implementation in computer models, the true test of any
model is how well it predicts a process or phenomenon. Since tire data collected from the
towed rig was determined to be acceptable and on order with the previous ATV tire study
[25], the data collected would be used to judge how well the exponential model would fit.
Figure 2.22 depicts the application of the tire model on a clay surface. Then, Figure 2.23
shows comparison for a gravel surface. One will note R-squared values exceeding 0.80,
indicating an acceptable fit.
Figure 2.22 ? Sample output data. Clay, 10 psi, low camber, 185 lb normal load
60
Figure 2.23 ? Sample output data, Gravel, 10psi, low camber, 185 lb normal load
61
CHAPTER 3
RESULTS
3.1 Introduction
After satisfactory test equipment were constructed, electronics and data recording
equipment debugged, and several trial runs completed, reliable and repeatable
experiments could be conducted. Presented in the following section are the results from
bevameter tests, empirical tire data, and the curve fits from exponential tire models. Also
included are basic effects of bevameter parameters on key tire performance numbers.
3.1.1 Test Surfaces
Natural surfaces (snow, peat, or soil) can have their mechanical strength described
by two parameters, cohesion and friction angle. Cohesive soils rely on intermolecular
bonds between the particles of the soil. In layman?s terms, it is a measure of the
?stickiness? of the soil. Frictional soils do not have this bond, and their strength is a
function of the confining stress and thus intergranular friction. A soil can be purely
frictional, purely cohesive, or a combination of both. An example of a frictional soil
would be dry sand, whereas a cohesive soil would be a type of clay. The test surfaces for
experimentation were selected to be as close to a purely frictional and purely cohesive
soil as possible. The surfaces have proved to meet requirements, and exhibit the behavior
expected from a frictional and a cohesive soil.
62
Both test surfaces are located at the National Center for Asphalt Technology
(NCAT) on land prepared for use by the 2006 Baja SAE competition hosted by Auburn
University. The terrain is clear of trees, graded nearly level, and grass was planted on
most of the area, while gravel was spread along the approach road and in front of storage
buildings used by NCAT. Photographs of the test lanes can be found in the appendix.
While the gravel, or frictional, surface needed little preparation, the cohesive surface was
to be the clay under the grass. Numerous vehicle passes were made in the early summer
by the test vehicle at various lateral offsets to clear a uniform lane through the grass and
onto the clay surface below.
The frictional surface exists on a lane approximately twenty feet wide, and
approximately thirty to forty yards in length. The loose gravel on top consisted of No. 2
limestone gravel. This upper layer of loose stone ranged in thickness between one and
two inches. Finer particles, resembling the size of beach sand grains found their way to
the bottom layer, which all sat on top of a very stiff layer of compacted fines from the
stones abrading on each other, atop the clay subsoil.
The cohesive test surface shared the same length of thirty to forty yards as the
frictional surface. The lane is approximately twelve to fifteen feet wide, and again, like
the frictional surface was nearly level. During the summer months when testing occurred,
the cohesive soil was hard, dry, and brittle. Care was taken to ensure that the soil was sun
dried for at least three days after a rain so that moisture content would not be a factor in
keeping the soil as uniform between runs as possible. The hard clay surface was also
selected such that it did not allow the tire lugs to penetrate the surface such that
bulldozing forces created by lug interlocking would not affect test data. Allowing the lugs
63
to penetrate the surface would turn the soil much like a disk harrow and alter the soil
strength after each run. This situation would cause the strength to be variable between
runs and would thus negatively affect the experiments certainty. It must also be noted that
a very fine clay dust would continually coat the upper surface of the clay which would be
created by the abrasive action of the tire on the hard, brittle clay soil.
3.1.2 Test Tire
Upon completion of construction of the test rig and soil testing apparatus, it
became clear that hundreds of extensive experiments would necessary on each tire to
gather the data necessary. As this study seeks to gain a general understanding of how
bevameter values affect the lateral performance of small (22? to 26?), off-road tires, and
not tire tread effects, it was decided that one front, steered tire be tested for the sake of
time and uniformity.
The test matrix includes making three runs at each specified normal load and slip
angle combination. The test matrix is seen in Table 3.1.
0? 5? 10? 15? 20? 25? 30? 45?
50 lbf 3 3 3 3 3 3 3 3
95 lbf 3 3 3 3 3 3 3 3
140 lbf 3 3 3 3 3 3 3 3
185 lbf 3 3 3 3 3 3 3 3
230 lbf 3 3 3 3 3 3 3 3
275 lbf 3 3 3 3 3 3 3 3
320 lbf 3 3 3 3 3 3 3 3
Table 3.1 ? Tire rig test matrix indicating three runs per setting
64
Table 3.1 shows three runs for a 7x8 matrix. This yields 168 runs per unique tire
setup. These 168 tests are repeated for two camber settings, two inflation pressures, and
two surfaces. In order to reduce some of the number of runs, camber and inflation
pressure were tested on the gravel surface only. This results in 840 individual passes
down a test surface.
The tire selected for use is the Dunlop KT 821. The KT 821 is a 22 inch diameter
by 8 inch wide front tire designed for use on sport ATV?s. The KT 821 shown in Figure
3.1 accepts 10 inch diameter wheels.
Figure 3.1 ? Dunlop KT 821 ATV tire
Other tires were considered for use in this study; however the KT821 was
ultimately selected as the best solution. The author has over five years experience with
the KT 821 as used on several Baja SAE racing vehicles designed and created by Auburn
University. Numerous test sessions altering camber, inflation pressure, and toe-out angle
65
were conducted over the years, and thus the qualitative performance changes are well
understood. In addition, the 821 is a relatively inexpensive and lightweight tire, fitting
well into the research budget.
Finally, there are several tires of varying states of wear available to observe what
to expect even before any of the tests commenced. The wear of the tire is significant in
initial stages, but settles to a consistent lug shape quickly. The KT 821?s lugs have sharp,
squared edges which deteriorate quickly to a radiused edge of approximately 1/8
th
of an
inch. This radius is maintained through the majority of the life of the tire, whereas the
wear is seen on the tops of the lugs rather than on the edges as seen in the beginning
stages. This consistent wear pattern allowed for the same tire to be used in all tests. No
?chunking? of the lugs occurred, and the tire maintained its integrity throughout the tests.
Seeing as sinkage is insignificant due to the stiff terrains, lug interlocking is also not a
factor in the certainty of lateral force results.
3.2 Bevameter
With the substantial amount of information available regarding bevameter
operations, few issues arose with testing the soil. The vehicle-mounted bevameter proved
to be effective in producing useful data that can be applied to the study at hand. The most
difficult issue to overcome was the wear and structural integrity of the rubber coating on
the shear head. This limited amount of surface testing due to structural concerns of the
shear head do not have negative effects on the data, as it shares the same trends as found
in [3] and [30] Also, the shear rate was on the low end of the range presented by Shoop
that which is acceptable for vehicular studies [10]. Faster shear rates are attainable with a
66
human operator, but the consistency decreases, and risk of tearing the rubber from the
shear head increases. Furthermore, the cost of adapting a motor drive to provide an
appropriate shear rate would be out of range for the budget on this project. Nevertheless,
the bevameter designed for this project has put forth acceptable data to help unlock
knowledge about soil-vehicle interaction.
3.2.1 Shear Response
The shear response of the test surfaces were tested at various points along the
lanes in order to provide overall response characteristics and not soil-rubber properties in
localized spots. Data collected resembles that of other data collected in previous
terramechanics studies. Key values collected from the shear data include shear
deformation modulus, residual shear stress, normal stress, and thusly cohesion and
rubber-soil friction angle.
Frictional Terrain
The frictional terrain proved to be the surface with the most variation in shear
stress. In addition the rocks which made up the soil were sharp and easily gouged and/or
scored the rubber on the shear head. This damage allowed only two test sites on the
gravel surface, however the results were very similar and it was determined that further
tests on gravel not be made as to allow for potentially more testing on the cohesive (clay)
surface.
Unfiltered shear stress versus shear displacement is presented in Figure 3.2. Note
the nearly vertical initial slope, indicating a very stiff terrain. After the initial slope and
throughout the displacement of the shear head, the shear stress changes dramatically. This
67
can be attributed to the sharp rocks interlocking, digging into the substrate and the soft
shear head, failing (releasing), then building up again. The test data does however match
the general trends found by Shoop [10] and Wong [3].
Figure 3.2 ? Shear stress v. linear displacement on frictional soil
Residual shear stresses were determined to be the average of the last 1300 data
points. For comparison, 1800 data points make up each data set, and the time step is
0.002084 s, making for tests around three to four seconds in length. The residual stresses
are collected and stored for use in creating the Mohr-Coulomb failure envelope. The
residual stresses are plotted against their associated normal stresses found under the shear
head. From the Mohr-Coulomb failure envelope, one can determine friction angle, and
68
cohesion (which one would expect to be zero). Figure 3.3 shows the Mohr-Coulomb
envelope associated with Gravel test number two.
Figure 3.3 ? Mohr-Coulomb failure envelope for frictional soil
Data points for the Mohr-Coulomb failure envelope were plotted in MATLAB
and then the polyfit and polyval functions were used to obtain a linear curve fit. As one
can see the fit matches the data quite well. Upon measurement from the slope, a soil-
rubber friction angle of 9.19 degrees is measured. Gravel Test 1 provided a slightly larger
friction angle of 12.16 degrees. This difference could be attributed to the highly irregular
nature of the gravel surface as evidenced in Figure 3.2.
Cohesion in the Mohr-Coulomb plots for both Gravel tests as calculated from the
failure envelope are indicated to be -4.57 psi for Gravel Test 1 and -1.66 psi for Gravel
69
Test 2. While not exactly zero cohesion as one would expect from non-bonded stones, the
apparent negative cohesion is desired more than seeing a positive cohesion for this
situation. Had a positive cohesion been present on a dry, frictional soil, that condition
would cause concern as to the quality of the test being conducted, as there is no physical
bond between particles, as described in [13]. Simply, the gravel tests have proven to be
in-line with expected cohesion values.
As the bevameter designed for this project is human powered, questions of shear-
rate repeatability arise. The human powered shear test has the operator manually apply a
moment to the shear head through a moment arm (i.e. Craftsman cheater bar). Shown in
Figure 3.4 are shear-rate data taken from the Gravel Tests.
Figure 3.4 ? Shear rate for Gravel Test 2
70
Upon first glance, one notices that the data is linear, and of a similar slope
between the three different tests presented. The lines do not overlap due to the difference
in actual start times of rotation versus start times of data recording. The slope of the lines
is the important parameter in Figure 3.4. The author notes that he was the only one that
conducted the shear response tests; therefore variance due to different operators is not a
factor. Shear rates from Gravel Test 2 can be seen in Table 3.1.
Normal Stress, psi Shear Rate, Test 1 Shear Rate, Test 2
42.9 psi 3.367 in/s 4.181 in/s
64.4 psi 2.976 in/s 3.000 in/s
86.9 psi 3.322 in/s 3.217 in/s
Table 3.2 ? Gravel surface shear rates
As one can see, the shear rates average 3.34?0.179 in/s, indicating a very
consistent shear rate conditions for the shear response experiments on the frictional
surface. The shear rate of approximately 3.34 in/s is close to double the shear rate of the
CRREL bevameter; however it is on the bottom end of the range of the rate produced by
the CRREL Instrumented Vehicle [10]. The shear rate of the human-actuated bevameter
could be increased; however repeatability and consistency significantly decrease.
Finally, as friction angle, cohesion, and residual shear stresses are determined, the
final task is to find the shear deformation modulus, which will be represented by K. One
may note that the raw data from the shear stress versus shear displacement graph in
71
Figure 3.2 all shared a very similar initial slope. In Figure 3.5, one may see Equation 1.22
as presented by Wong [3] being fit to shear stress-shear displacement data.
Figure 3.5 ? Exponential shear stress equation fit to shear data
The fit to the shear stress data was generated by creating an array of possible K
values between 0.01 and 0.99. Each value was plugged into a fit equation, and an R-
squared value was calculated. For the largest R-squared value, the index of the K value
was noted, and referenced at the end of the code to display the K value for the best fit.
Note that the residual force in the curve fit is the midpoint between the average true
residual stress and the maximum value of shear stress. This was applied to every fit in
this study. In Figure 3.6, one may see what ?messier? data looks like with this curve fit.
72
Figure 3.6 ? Exponential shear stress equation fit to shear data
As one can see, there is considerable variation in the shear stress, as noted before,
due to the climbing and falling of the stones in the gravel test surface upon each other.
Recall from Equation 1.22 that was used to fit these equations that there is the shear
modulus parameter. As noted before, all tests were fit with the method of checking all
possible K values for the greatest R-squared value, and the shear modulus values were
recorded and plotted against normal stress, as seen in Figure 3.7, and another test in
Figure 3.8.
73
Figure 3.7 ? Shear modulus v. normal stress, Gravel test 2
Figure 3.8 ? Shear modulus v. normal stress, Gravel Test 1
74
Upon further inspection, one can see that the values tend to group about a K-value
of 0.05 in. This indicates that the frictional soil is very stiff and begins to support
substantial shear force with rubber after approximately 0.05 in of displacement.
Despite the significant of fluctuation of shear stress due to stones sliding up and
over each other, consistent shear rates and closely grouped values for shear deformation
modulus were obtained. Values of soil-rubber friction angle ranged between 9 and 12
degrees, yet the cohesion was listed as negative. It is highly improbable for a dry,
frictional soil to have a negative cohesion, especially with no physical bond between
particles. It could be arranged to add an extra data point at the origin for inclusion in the
curve fitting, however the friction angle itself is much more important than cohesion for
this frictional soil, and as long as the cohesion is near zero stress (as has been found) it is
not of concern. Complete results for the frictional terrain may be found in the Appendix.
Cohesive Terrain
As the frictional terrain generated acceptable data, the cohesive data has proven to
be much smoother and easier to fit curves to. The data is not as ?choppy? as the frictional
terrain data as the particles in the cohesive terrain are orders of magnitude smaller and
have a more fluid motion as they slide across each other. The presence of these small
particles also resulted in no damage to the rubber surface on the shear head. Figure 3.9
shows some of the raw data from tests on the cohesive soil.
75
Figure 3.9 ? Shear stress v. shear displacement, Clay soil test 2
After a first glance, notice how tightly grouped the data is in comparison to the
highly variable shear stress values from the rocky frictional terrain. This indicates the
smooth movement of the shear head across the terrain, as the particle size is smaller, so
are the jumps in the graph. Also like the frictional terrain, the cohesive terrain fits the
exponential response described by Wong [3] more closely.
)1(
/
max
kj
ess
?
?= (3.1)
In the same fashion that the frictional terrain was analyzed, the last 1300 points of
the 1800 point data set were averaged and saved as the residual stress. This residual stress
was then used in the construction of the Mohr-Coulomb failure envelope. These residual
stresses are plotted against their associated normal stresses, and line is fit to the data. For
76
Figure 3.10, the soil-rubber friction angle was calculated to be 7.9 degrees, and cohesion
of 0.95 psi. Both values are expected, as a cohesive soil will have a larger cohesion and a
smaller friction angle than a frictional terrain.
Figure 3.10 ? Mohr-Coulomb failure envelope
The other tests made on the cohesive terrain yielded 6.2 degrees and 3.77 psi of
cohesion for the second test, while the third test provided values of 4.30 degrees friction
angle and 5.91 psi of cohesion.
77
Figure ? 3.11 Shear Rate, Clay soil test 2
Following in the path of the frictional terrain tests, the cohesive terrain allowed for very
consistent shear rates. Table 3.1 shows data (from Figure 3.11) from the second shear test
on the cohesive terrain.
Normal Stress, psi Shear Rate, in/s
64.4 psi 2.540 in/s
85.9 psi 2.885 in/s
107 psi 2.704 in/s
Table 3.3 ? Shear rates, Clay soil
These shear rates, averaging 2.70?0.09 in/s, are consistent with those of the other
tests showing that they are very close to those seen during the frictional terrain shear
78
testing, and still above the speed that the CRREL bevameter provided, yet on the low end
of shear rates seen by the CRREL Instrumented Vehicle, which were around 3.8 cm/s
[10].
Furthermore, as other parameters are identified, the shear deformation modulus
(K) is determined by a calculating fits for a table of K values, then calculating R-squared
values, the maximum R-squared value is chosen along with the associated K value. These
curves maximum value are set directly between the maximum shear stress value seen,
and the residual stress recorded for use in the Mohr-Coulomb failure envelope. Figure
3.12 depicts a typical fit seen for the cohesive surface.
Figure 3.12 Exponential shear stress equation fit to shear data, Clay test 2
79
For the three tests conducted on the cohesive surface, the shear modulus values
were found from running curve fits with K values between 0.01 and 0.99, recording the
R-squared value, then based on the maximum R-squared value, the corresponding K
value was selected. Figure 3.13 shows these K values.
Figure 3.13 ? Shear modulus v. normal stress, Gravel test 1
From this plot, one can see that the K-value seems to hover about 0.03 in. Recall
that the frictional terrain supported significant shear stresses in approximately twice the
cohesive surface?s K-value. This can be attributed to the larger cohesion values seen in
the cohesive terrain, ultimately do to the strong bonds between each soil particle.
One final point must be noted regarding shear stress ? shear displacement data.
When the shear stress is normalized with the associated normal stress, it can be seen in
80
Figure 3.14 that the plots collapse upon each other. In the data shown for Dirt Test 2, the
coefficient of normalized shear stress averages 0.185, when using the second half of the
data.
Figure 3.14 ? Normalized shear stress v. displacement
The cohesive terrain has proven to provide much cleaner data than the rough,
rocky, frictional surface shown earlier. Cohesion and friction angle values fit what is to
be expected of a cohesive surface compared to a frictional surface, higher cohesion
values for the cohesive soil and thus lower friction angles on the cohesive soil were
present. Cohesion values ranged between approximately 1 and 6 psi, while soil-rubber
friction angle was between 4.3 and 7.9 degrees. The remainder of the cohesive surface
81
data, full results can be found in the Appendix. Final data from the bevameter is listed in
Table 3.4.
Gravel Surface Clay Surface
Average Friction Angle
?, degrees
10.68? 6.13?
Average Cohesion
c, lb/in
2
0 lb/in
2
4.8 lb/in
2
Shear Deformation
Modulus,
K, lb/in
0.06 lb/in 0.05 lb/in
Table 3.4 ? Average values for various soil-rubber interface properties
3.2.2 Normal Response
Normal load tests were conducted in conjunction with the shear-head tests. The
normal load tests were performed manually, recording displacement from a dial indicator
and pressure from the pressure gauge on the pneumatic cylinder. While results were not
as well grouped as the shear tests, appropriate trends were observed that matched what to
expect of the surfaces.
The following plots (Figures 3.15 - 3.18) are the raw data recorded from the tests.
As noted in Chapter 2, they follow trends (increasing sinkage for increasing pressure)
observed by Yu [30].
82
Figure 3.15 ? Cohesive soil 3? plate penetration test
Figure 3.16 Cohesive Soil 4? plate penetration test
83
Figure 3.17 ? Frictional Soil 3? plate penetration test
Figure 3.18 ? Frictional Soil 4? plate penetration test
84
Looking at the very small displacements on the x-axis, one can immediately discern that
the soil is very stiff. This stiffness does not allow for any appreciable sinkage, and thusly
does not allow lug interlocking. The lack of lug interlocking allows the tire measurements
to be tread-pattern independent, and focus on the soil-rubber interface. It must be noted
that during the normal load tests on gravel, the top layer was not penetrated by the plates.
Evaluating the bevameter data as per [34] and [3], the surface normal properties
are shown in Table 3.2
Gravel (Frictional Surface) Clay (Cohesive Surface)
Sinkage Exponent, n 6.08 2.89
k
?
, lb/in
n+2
44.667 39.33
k
c
, lb/in
n+1
0.0 14
Table 3.5 ? Bekker bevameter sinkage properties
Further analysis with A.R. Reece?s methods were not deemed important, as the soil was
found to be very stiff and thus any sinkage would not give extra lateral force at the tire-
soil interface. This is indicated by the large sinkage exponent found on both soils, and
their stiffness confirmed by numbers compiled by Wong [7].
3.3 Tire Test Rig
A test rig constructed at Auburn University was used to gather raw operational
data of the KT 821 tire in various configurations. Static normal load, static slip angle, and
static camber angle are all set by the operator while dynamic normal load, dynamic
85
camber angle along with reaction forces were recorded by the Dataq model 710 data
recorder through associated sensors. Despite the noisy nature of off-road data, useable
data could be obtained on the effect on lateral performance of camber, tire pressure, and
vehicle-soil interface strength.
Raw data is collected from two accelerometers and three load cells on a typical
run. Other special runs have been made with the inclinometer to measure camber
throughout passes on the test surfaces. In Figure 3.19 is a sample plot of raw data as it
comes from the Dataq 710 to simply illustrate the frequently fluctuating loads imparted
onto the tire. The small voltages coming from the load cells can be seen tightly grouped
against the x-axis.
Figure 3.19 ? Sample raw data from tire test rig
Once processed, several forms of data can be presented. Typical formations of tire
data can be presented as (but not limited to):
86
1) A 3-dimensional plot of Normal Force, Lateral Force, and Slip Angle, also known
as a ?Carpet Plot?
2) A 3-dimensional plot of Normal Force, Longitudinal Force, and Slip Angle. Also
a ?Carpet Plot?
3) Peak Longitudinal Force(in this case, drag force) versus Normal Load
4) Peak Lateral Force versus Normal Load
5) Lateral Force Coefficient (LFC) versus Slip Angle
6) Drag Force Coefficient (DFC) (also can be listed as COT, coefficient of traction)
versus Slip Angle
7) Cornering Stiffness (C
?
, lb/deg) versus Normal Load
A note must be made regarding the uncertainty of the values presented by these plots.
Before going into the uncertainty, insight into the data processing methodology is
required. Once all testing is complete, all runs for a particular configuration are combined
into one data file. Recall that the acceleration and deceleration portions are not part of the
data, so combining the data does not have any ill effects. Three load cells and two
accelerometers are read into the data acquisition system. Once adjusted through the
calibration curves, lateral, longitudinal, and vertical forces can then be deduced.
Vertical load is determined through the accelerometer attached just below the dead
weights. By knowing the acceleration and the mass, a vertical force can be calculated. As
the terrain fluctuates greatly, this can cause wildly fluctuating normal loads. The software
written for this study incorporates a filter that only processes lines of data where the
normal load is within ten percent of nominal. This process manually sets the uncertainty
87
of the vertical load. Brief experiments were conducted to reduce this tolerance, however
further reductions in tolerance significantly limited the number of data points for
analysis. As this is the first excursion into the uncertainty of ATV tire forces it was
determined that a ten percent tolerance was an acceptable starting point, as no analysis of
this type has been conducted.
As data points are accepted by their meeting normal load tolerance, the program
begins calculating lateral and longitudinal forces through the schematic of the tire rig,
presented in Figure 3.20. Lateral and longitudinal forces are calculated as follows:
)sin()sin()cos(
)cos()cos()sin(
8.7.6.
8.7.6.
???
???
ChChChy
ChChChx
fffF
fffF
++?=
++=
(3.4)
Figure 3.20 ? Schematic for tire test rig
88
This process generates lateral and longitudinal force vectors that are associated with
normal loads that are within ten percent of the nominal normal load for that run. As the
vehicle has been driven over as a consistent and level surface as possible, the mean of
these vectors is taken. This average value is then saved as the lateral and longitudinal
forces for that particular run. Standard deviations are also computed for both vectors.
Given the standard deviation, and the number of samples, the standard error of the
mean can be calculated for these mean values of lateral force.
nS
E
/??= (3.5)
For most of the data presented in this paper, standard errors for lateral force and
longitudinal force are approximately ?4 lbs.
The same approach was applied to camber data collected with an inclinometer. Slow
speed tests over each test surface were recorded, and the camber uncertainty was found to
be between around ?5 degrees. Camber values can change even on ?zero? camber runs,
and therefore creates the need to label camber data as ?high? and ?low? camber.
Intermediate values were not fully tested as the statistical difference could not be detected
until the camber in the test rig was placed at the maximum camber value.
The information provided by the plots within this section can allow a vehicle designer
to specify appropriate spring rates as to not saturate the tire, or design for the least
longitudinal resistance for maximum efficiency.
The first plot to be presented is the lateral force carpet plot. It can help put the tire?s
?operational map? in perspective.
89
Figure 3.21 ? Sample carpet plot of lateral tire data
One will notice the trough at 185 lbf. nominal normal load in Figure 3.21. At the
185 lbf. normal load, and 10 psi inflation pressure, an oscillation developed while towing
the tire. The tire can be modeled as a spring-mass system, and a natural frequency
calculated. This natural frequency seemed to be excited by the ten miles-per-hour tow
speed, and the test surfaces. An analysis of tire spring rate and natural frequency can be
found in the Appendix. While the vertical spring rate does not directly correlate to the
towing frequency of the tire, there are other flexibilities of the tire that may contribute to
the oscillations. It is hypothesized that a mechanism very much like the mechanism that
gives rise to relaxation length in tires is causing the trough on Figure 3.21.
90
Relaxation length is the length a tire rolls between when a step steering input is
applied, and when the tire generates its maximum force for that given slip angle and
normal load.
Since the terrain has a highly variable vertical profile, it is hypothesized that there
is a time between when the appropriate normal load is rapidly applied and when the
actual force associated is generated. This may be in part due to the deformable nature of
the operating surface, where on pavement, it is very rigid in comparison to a vehicle tire.
The trough can be seen in other carpet plots, and may vary in size depending on the
surface, inflation pressure, and small fluctuations in the towing speed. Since the data
reduction software written for this study used on all of the data only allows a narrow band
of normal loads about the nominal normal load, the lateral force generated may lag the
application of the normal load by a small amount of time, and thus be rejected by the
selection process in the code. Further exploration of this phenomenon may offer more
insight into the operation of off-road tires on deformable surfaces.
Another version of the carpet plot can be generated with longitudinal forces. Seen
in Figure 3.22, is the same trough about 185 lbf. normal load. As the lateral and
longitudinal forces are derived from the same sensors, the phenomenon described above
will make its way into the longitudinal carpet plot.
91
Figure 3.22 ? Sample carpet plot of longitudinal tire data
Seen in Figure 3.23, while loosely grouped, the relationship between longitudinal
force and slip angle may be linearized within a small band of normal loads.
92
Figure 3.23 ? Peak drag force v. normal load
Another way of visualizing this data is normalizing it with respect to the nominal
normal load imparted on the tire. Shown in Figure 3.24, the first linear section can be
seen, however at increasing normal loads (near 320 lbf. normal load), the curve shape
shifts to that of an exponential form, which matches that presented by Crolla, et al [15].
)1(
max
Ks
eCOTCOT
?
?= (3.2)
93
Figure 3.24 ? Sample Normalized longitudinal force v. slip angle
Similar to Figure 3.24 is the plot of the LFC. Instead of showing longitudinal
force which is normalized, the normalized lateral force is plotted against slip angle for
various normal loads. Normalization is performed by dividing the lateral force by the
nominal normal load for that set of data. Figure 3.25 represents a typical LFC plot.
Note in the LFC plot that the curves do not collapse upon each other. This
showcases what is called tire load sensitivity. Tire load sensitivity has a major effect on
the balance of a vehicle. Breakaway characteristics are also affected, as some tires lateral
force capability may diminish smoothly, whereas some may suddenly drop off. [19].
94
Another entity of note is the data points for 50 lbf. normal load. The lightly loaded tire at
high slip angles found a ?slip-then-grip? mode causing the data points to be less accurate
than for the immediately following normal loads.
Figure 3.25 ? Sample Normalized lateral force v. slip angle
A related plot to Figure 3.25 is that of cornering stiffness, denoted by C
?
, versus
normal load (Figure 3.26). Cornering stiffness is the slope of the linear region of the
lateral force versus slip angle plot, and measured in units of lbs/deg. Seen in Figure 3.25,
there is a trend to have increasing C? with normal load. The values of C? were generated
by the second data point in a lateral force versus slip angle data set, and the origin of the
plot. For example, if the data point at five degrees slip angle is 30 lbf, then the associated
95
C? value is thirty divided by five, and thusly 6 lbf/deg. They follow a general trend
similar to that shown by Pacejka [20].
Cornering stiffness is important in understanding how much slip angle,
(indirectly, steer input), must be provided before the vehicle will generate peak cornering
forces for a given normal load. A balance must be struck between excessive steering
sensitivity and the time and speed required to steer from center to lock.
Figure 3.26 ? Cornering stiffness v. normal load
A similar plot (which was shown before with longitudinal force) is shown as
Figure 3.27, illustrating how peak lateral force changes with increasing normal load.
96
The trend for this type of data will be to increase linearly into a transition period, where
the tire starts to become overloaded vertically and will not produce any more lateral
force.
Figure 3.27 ? Peak lateral force v. normal load
Through the information presented in Figure 3.27, adequate methods have been
used in order to collect and present relevant tire data. These plots can be used to compare
tire performance between the changes in camber, inflation pressure and operating surface.
The following sections will highlight the results of the specific camber, pressure, and
surface tests. One may note that normalizing the values in 3.27 will yield a sort of
?effective friction coefficient? of the tire based on normal load.
97
3.3.1 Inflation Pressure
Utilizing the inflation pressure first in the presentation of results allows the
comparison of trends to other trends observed when pressure was altered. The most
notable change with inflation pressure tends to be that of cornering stiffness, as inflation
pressure directly affects carcass stiffness. Inflation pressure also has an influence on peak
force by way of coefficient of friction, yet is less easily understood [19].
First presented are the carpet plots for both high and low inflation pressures of 10 and 5
pounds per square inch, respectively. As a note, the lateral force carpet plots (Figure
3.28) all share the same axes for ease of comparison, as do the longitudinal force carpet
plots.
Figure 3.28 ? Lateral force carpet plot for Gravel, 5 psi, Low camber
98
Figure 3.28 depicts shows the relation of slip angle, normal load, and lateral force
generated in a three-dimensional format for an operating pressure of 5 psi. Note as slip
angle increases, the lateral force exponentially increases to a more or less constant value
as described by Crolla [18]. Figure 3.29, shows the three-dimensional representation of
longitudinal force in relation to slip angle and normal load. Note now as normal load
increases, the longitudinal force increases exponentially, whereas increasing slip angle
has a semi-linear to exponential relationship with longitudinal force.
Figure 3.29 ? Longitudinal force carpet plot for gravel, 5 psi, low camber
99
Increasing tire inflation pressure alters the carcass stiffness and the contact patch
with the operating surface. Figure 3.30 showcases the carpet plot for lateral force data,
while Figure 3.31 shows the longitudinal data.
Figure 3.30 ? Lateral force carpet plot for gravel, 10 psi, low camber
Figure 3.31 ? Longitudinal force carpet plot for gravel, 10 psi, low camber
100
As was noticed with the low pressure experiment, the same trend of exponential
rise to lateral force with addition of slip angle occurs. The shape of the longitudinal force
carpet plot matches that of the 5 psi tests as well.
One may note a distinct ridge for the 140 lbf normal load plot on the lateral force
carpet plot. Each value is consistently above those of larger and smaller normal loads. By
changing the inflation pressure of the tire, one increases vertical spring rate. Based on
oscillations observed during that particular test, it is believed that the phenomenon
explained previously where there is a lag time between load application and actual
realization of force is taking place. Further work to alleviate potential problems with
unwanted oscillations may focus efforts on the application of a damper to the rig,
properly instrumented as to record the forces at the attachment point of the frame.
As the carpet plots have been presented to align one with the general operating
map of the tire, the following plots compare various data such as peak force in the lateral
and longitudinal direction, cornering stiffness, and normalized lateral force versus slip
angle.
Presented first is the plot of cornering stiffness with respect to normal load in
Figure 3.32. One will note that the 5 psi change (rather large in the scope of ATV tires)
has little effect on the cornering stiffness. Qualitative assessments from KT 821 testing in
a race setting found that inflation pressures below 5 psi negatively affected the handling
of the test vehicle, and added unnecessary rolling resistance to the system. Typical
inflation pressures for the KT 821 are between 5 psi and 10 psi hence the tests were
conducted with these conditions in mind.
101
Figure 3.32 ? Cornering stiffness comparison between pressures on gravel
Note that the closely grouped data sets follow a linear trend, increasing cornering
stiffness with normal load. This relationship allows the KT 821 to undergo pressure
changes due to operating environment change, or from extended use, and not adversely
affect the performance of the vehicle. Controllers do not have to be greatly adjusted, nor
would a driver have to deal with significant changes in the handling of the vehicle. It
must be noted however, that this is only for the linear portion of the tire performance
curve. Peak lateral force may be different between two operating pressures, even if the
cornering stiffness does not change.
That said, Figure 3.33, and Figure 3.34 show how peak longitudinal and lateral
forces are affected by inflation pressure.
102
Figure 3.33 ? Peak lateral force comparison between pressures on gravel
Figure 3.34 ? Peak longitudinal force comparison between pressures on gravel
103
Much like the results found in cornering stiffness, there does not seem to be any
appreciable change in peak lateral and peak longitudinal force between 10 psi and 5 psi
of inflation pressure. Any small differences would be within the error range of the data.
The first analysis of error in ATV lateral performance studies can be found in the
Appendix.
Another way of visualizing the cornering stiffness and peak force data is to
observe these trends from the normalized force plots. Shown in Figures 3.35 ? 3.40 are
comparisons of high and low inflation pressure for the 90, 140, 185, 230, 275, and 320
lbf tests.
One may note that the slope of a line fit to the data in 3.33 will be less than one.
This indicates that lateral force does not increase proportional to normal load.
Figure 3.35 ?Normalized lateral force comparison between pressures at 95 lbf
normal load, gravel
104
Figure 3.36 ? Normalized lateral force comparison between pressures at 140 lbf
normal load, gravel
Figure 3.37 ? Normalized lateral force comparison between pressures at 185 lbf
normal load, gravel
105
Figure 3.38 ? Normalized lateral force comparison between pressures at 230 lbf
normal load, gravel
Figure 3.39 ? Normalized lateral force comparison between pressures at 275 lbf
normal load, gravel
106
Figure 3.40 ? Normalized lateral force comparison between pressures at 320 lbf
normal load, gravel
Note in all figures how the initial slope of each plot is very similar, and the peak
forces between high and low inflation pressure for a given normal load do not change
significantly either. This insensitivity to inflation pressure may prove beneficial to
designers working on control systems for UGV?s, and designers designing vehicles that
do not require intensive driver training or high maintenance requirements to keep check
on air pressure.
3.3.1 Camber
As described in Chapter 2, camber refers to the inclination of the tire if rotated
about the x (longitudinal) axis. Negative camber values are present when the tops of the
107
tires are angled inward, and positive when angled outward. This angle is changed by
static adjustments in the suspension, or by the dynamics of the linkages within a
suspension. Camber is used to fine tune a vehicles performance, and the addition of
negative camber can manifest itself in more lateral force available for cornering [19].
Tests were conducted at varying camber levels to determine the effects on the
generation of tire lateral forces, and longitudinal drag forces. Early tests at each camber
setting (0, 2, 4, 6, and 8 degrees) revealed that differences in camber addition could only
be realized between the 0 and 8 degree settings. This condition, in conjunction with the
camber uncertainty for the given terrains forced ?high? (nominal -8 degrees) and ?low?
(nominal 0 degrees) values to be used in the data gathering process.
A first comparison between both camber settings in Figure 3.41 places a light on that of
the effect of camber on cornering stiffness, C
?.
Figure 3.41 ? Comparison of cornering stiffness v. normal load for camber angle on
gravel
108
Observing Figure 3.41, one sees that increasing camber produces a slight increase in
cornering stiffness. Camber?s effect on peak lateral and longitudinal forces can be seen in
the following Figures, 3.42 and 3.43.
Figure 3.42 ? Comparison of peak lateral forces for camber angle, on gravel
The trends presented in the peak lateral force versus normal load graphs are the same as
noted by Pacejka [20]. The curve is concave down, however the data gathered in this
study did not extend far enough to capture the full trend that Pacejka discusses.
109
Figure 3.43 ? Comparison of peak longitudinal forces for camber angle, on gravel
110
From the plots in Figures 3.42 and 3.43, one can deduce that a slight increase in
peak lateral force can be gained for seemingly little increase in peak longitudinal force.
Presented in Figures 3.44 ? 3.49 are comparisons of the high and low cambered
tires at increasing normal loads, and their lateral force versus slip angle plots.
Figure 3.44 ? Comparison of cambered tire lateral forces at 95 lbf normal load, on
gravel
111
Figure 3.45 ? Comparison of cambered tire lateral forces at 140 lbf normal load, on
gravel
Figure 3.46 ? Comparison of cambered tire lateral forces at 185 lbf normal load, on
gravel
112
Figure 3.47 ? Comparison of cambered tire lateral forces at 230 lbf normal load, on
gravel
Figure 3.48 ? Comparison of cambered tire lateral forces at 275 lbf normal load, on
gravel
113
Figure 3.49 ? Comparison of cambered tire lateral forces at 320 lbf normal load, on
gravel
One will note that the linear region did not change much between high and low
camber, however amongst the lower normal loads (those that are less than static); the
available lateral forces to the vehicle were higher.
3.3.3 Operating Surface
The operating surface of a tire while off-road has just as much as an affect on the
performance of a vehicle as the tire selected to be on the vehicle. The interface between
the surface and the tire is what generates the forces required for propelling and steering a
vehicle. If a typically well performing tire traverses soil which is not strong, it will not
perform well at all. Significant sinkage and / or understeer could occur. As seen from the
bevameter section of the results chapter, both operating surfaces tested were significantly
114
stiff, such that appreciable sinkage would not occur. This allows the rubber-soil interface
to be isolated and studied further.
The first plot on operating surface effects is that of C
?
versus normal load.
Figure 3.50 ? Comparison of cornering stiffness v. normal load for different surfaces
One can observe good agreement up to about 275 lbf, and then the data diverges,
yet retains the same slope at the higher normal loads.
115
Figure 3.51 ? Comparison of peak lateral force v. normal load for different surfaces
As with the previous plot, the peak lateral force of the tire between the two
surfaces does not change much across the frictional and clay surfaces.
116
Figure 3.52 ? Comparison of peak longitudinal forces v. normal load for different
surfaces
This plot of peak longitudinal force shows more scatter after about 95 lbf of
normal load.
The following family of plots tracks the lateral force coefficient for the range of
normal loads while comparing between both clay and gravel surfaces.
117
Figure 3.53 ? Comparison of lateral forces on different surfaces at 95 lbf
normal load
Figure 3.54 ? Comparison of lateral forces on different surfaces at 140 lbf
normal load
118
Figure 3.55 ? Comparison of lateral forces on different surfaces at 185 lbf
normal load
Figure 3.56 ? Comparison of lateral forces on different surfaces at 230 lbf
normal load
119
Figure 3.57 ? Comparison of lateral forces on different surfaces at 275 lbf
normal load
Figure 3.58 ? Comparison of lateral forces on different surfaces at 320 lbf
normal load
120
These plots confirm the findings of the cornering stiffness and peak lateral and
longitudinal force plots. No appreciable difference between the surfaces could be seen.
One may note again, the deterioration of cornering stiffness on the clay surface as higher
normal loads are attainted. This might be explained by the top layer of the soil being
crushed slightly, and adding to the fine dust on top of the clay surface. This barrier
between the structurally sound soil and the tire might delay the tire from building up
force as fast as would be done on gravel.
3.4 Exponential Tire Model
As data have been collected on the strength of soils for vehicular travel, and tire
data collected from the towed test rig, the influence of the two can be tied together by
fitting curves to the tire data with the exponential model. Then, the coefficients of the
model can then be compared to the soil properties to see how the coefficients are
affected. Recall the exponential tire model form as described in Equation 3.3.
))exp(1( ?BAFF
zy
??= (3.3)
Curve fitting the exponential tire model to the data generated by the towed test rig
is done by maximizing the R-squared value of a particular fit. An array of possible values
for the A and B coefficients of the tire model are generated. Possible situations for A and
B values are run through the exponential tire model. For each pass, an R-squared value is
calculated and stored with an index. The highest R-squared value is selected, and the
associated index is used to retrieve the A and B values for that particular fit. While not
computationally efficient, the speeds of modern computers and the low volume of data
processing do not hinder the process of data reduction.
121
One interesting thing to note is that for every single run through checking for the
best R-squared value, the values for A and B came out to be exactly the same. The cause
of this phenomenon is unknown, and may be the basis of a further study. Subsequent
plots will illustrate this phenomenon.
The following plots (Figure 3.59, 3.60) are representative of those found for each
run on a particular slip angle, normal load, surface, camber, and inflation pressure.
Figure 3.59 ? Example of exponential model fit to data on Clay, at 10 psi, low
camber, and 185 lbf normal load
122
Figure 3.60 ? Example of exponential fit for tire data on gravel, 5 psi, high camber,
275 lbf normal load
An inspection of the plots will show a fit curve by a purely dashed line. The R-
squared value is listed directly in Figure 3.60. Error bars calculated from the standard
error of the mean are found at each data point for lateral and longitudinal data. Dozens of
tests like these were run, and the associated A and B coefficients calculated.
As mentioned previously, the plots for finding the best R-squared value are shown
in Figures 3.60 and 3.61. Note that the index that the x-axis is referring to is that within
the array which stores thousands of combinations of A and B values for fitting to the data.
Many plots assume the shape of Figure 3.60, while the few instances of low R-squared
values take on the shape of Figure 3.61.
123
Figure 3.61 ? Trace of R-squared values for data on gravel, 5 psi, high camber,
275 lbf normal load, High R
2
value
Figure 3.62 ? Trace of R-squared values for data on clay, 10 psi, low camber,
90 lbf normal load. Low R
2
value
124
The key data, however, for this section is that of the A and B values versus normal
loads. The shape of these curves confirms Metz?s suggestion that the B coefficient should
be exponential in nature when comparing to normal loads. Note, values of A and B of
zero indicate the lack of computable data due to corrupt data.
Figure 3.63 ? A Coefficient plot v. normal load for clay, 10 psi, low camber
125
Figure 3.64 ? B Coefficient plot v. normal load for clay, 10 psi, low camber
Figure 3.65 ? A coefficient v. normal load for gravel, 10 psi, low camber
126
Figure 3.66 ? B coefficient v. normal load for gravel, 10 psi, low camber
A careful look at these graphs for the gravel and clay surfaces will see the A and B
coefficients converge to around 0.3 for higher normal loads.
Noting that the data for A and B coefficients (which have been found to be
identical, A=B), fall off and assume a relatively constant value with increasing normal
load, one can begin to apply curve fits to this data to uncover the relationship between
model coefficient and normal load. After experimenting with various curve fits, the
following equation provided the most appropriate fits.
m
z
cFBA == (3.2)
This Equation 3.2 was applied to fit the data for A and B coefficients that were
presented in Figures 3.64 and 3.65. Very good agreement was reached with R-squared
values ranging from 0.72 to 0.97.
127
Figure 3.67 ? Curve fit to B coefficient data for Clay, 10 psi, low camber
Figure 3.68 ? Curve fit to A coefficient data for Clay, 10 psi, low camber
128
Figure 3.69 ? Curve fit to B coefficient data for Gravel, 5 psi, high camber
Figure 3.70 ? Curve fit to A coefficient data for Gravel, 5 psi, high camber
129
Figure 3.71 ? Curve fit to B coefficient data for Gravel, 5 psi, low camber
Figure 3.72 ? Curve fit to A coefficient data for Gravel, 5 psi, low camber
130
Figure 3.73 ? Curve fit to B coefficient for Gravel, 10 psi, high camber
Figure 3.74 ? Curve fit to A coefficient for Gravel, 10 psi, high camber
131
Figure 3.75 ? Curve fit to B coefficient data for Gravel, 10 psi, low camber
Figure 3.76 ? Curve fit to A coefficient data for Gravel, 10 psi, low camber
132
Tabulated data for the coefficients shown in Figures 3.67 ? 3.77 can be seen in
Table 3.2. The values, m, and C are those which belong to Equation 3.2
Setup
For Model
Coefficient
m C R
2
Clay, 10psi,
Low Camber
B -0.45 4.439 0.92
A -0.465 4.542 0.92
Gravel, 10psi,
Low Camber
B -0.58 6.627 0.79
A -0.58 6.727 0.80
Gravel, 10psi,
High Camber
B -0.627 10.812 0.83
A -0.634 10.967 0.83
Gravel, 5psi,
Low Camber
B -0.583 8.366 0.97
A -0.595 8.68 0.97
Gravel, 5psi,
High Camber
B -0.469 4.333 0.72
A -0.473 4.334 0.75
Table 3.6 ? Exponential Tire Model Coefficient curve fit data
A quick glance at Table 3.2 will reveal m-values between 0.4 and 0.7. C-values
spanned a range of 4.33 to 10.96. One may note that only one clay run is available on
Table 3.6. In order to make the test matrix more digestible, the camber and pressure
changes were evaluated on the gravel surface. This would save from having to make 252
extra runs.
133
Modification of the exponential tire model by Crolla [15] to accept the capability
to change A and B values for changing normal loads would look like that of Equation 3.3.
)1)((
)( ?
m
z
CFm
zzy
eCFFF
?
?= (3.3)
To check Equation 3.3 and thus the effectiveness of the curve fits to A and B
coefficient data, situations involving camber change, pressure change, and surface change
are checked.
As shown in Figure 3.42, small additions of camber will result in a small increase
in the amount of lateral force available. Changes in pressure between 5 and 10 psi do not
seem to make noticeable changes as evidenced by the data in Figure 3.33. Surface
changes in Figure 3.51 do not make an appreciable difference in lateral force either. The
following figures depict how the tire model described by Equation 3.3 recognizes the
differences between camber, surface, and inflation pressure.
Figure 3.77 ? Comparison of camber and pressure with Eq. 3.3 on gravel at 50 lbf
134
Figure 3.78 ? Comparison of camber and pressure with Eq. 3.3 on gravel at 95 lbf
Figure 3.79 ? Comparison of camber and pressure with Eq. 3.3 on gravel at 140 lbf
135
Figure 3.80 - Comparison of camber and pressure with Eq. 3.3 on gravel at 185 lbf
Figure 3.81 - Comparison of camber and pressure with Eq. 3.3 on gravel at 230 lbf
136
Figure 3.82 - Comparison of camber and pressure with Eq. 3.3 on gravel at 275 lbf
Figure 3.83 - Comparison of camber and pressure with Eq. 3.3 on gravel at 320 lbf
137
Figure 3.84 -Comparison of surface with Eq. 3.3 at 50 lbf
Figure 3.85 - Comparison of surface with Eq. 3.3 at 95 lbf
138
Figure 3.86 - Comparison of surface with Eq. 3.3 at 140 lbf
Figure 3.87 ? Comparison of surface with Eq. 3.3 at 185 lbf
139
Figure 3.88 - Comparison of surface with Eq. 3.3 at 230 lbf
Figure 3.89 - Comparison of surface with Eq. 3.3 at 275 lbf
140
Figure 3.90 ? Comparison of surface with Eq. 3.3 at 320 lbf
The clay and gravel surfaces in this study are again defined as having the
properties in Table 3.7
141
Clay (Cohesive Surface) Gravel (Frictional Surface)
Sinkage exponent, n 2.89 6.08
Frictional sinkage
coefficient, k
?
39.33 lb/in
n+2
44.67 lb/in
n+2
Cohesive sinkage
coefficient, k
c
14 lb/in
n+1
0 lb/in
n+1
Average friction angle, ? 6.13 deg. 10.68 deg.
Average cohesion, c 4.8 lb/in
2
0 lb/in
2
Soil-rubber shear modulus,
k,
0.05 in. 0.06 in.
Table 3.7? Table of normal and shear properties for clay and gravel surfaces tested
Analysis of Figures 3.77 through 3.90, and Table 3.7 will show the following for
the KT 821 tire:
1. At higher normal loads, cohesive surfaces will hold higher lateral loads
than frictional soils.
2. At 10 psi, increasing the normal load on an uncambered tire will not
generate as much as a cambered tire at the same pressure, or a tire at 5 psi,
at high or low camber.
3. Camber effects are more significant at lower normal loads.
4. As inflation pressure increases, the addition of camber is advantageous in
order to develop more lateral force.
142
5. A small difference in available lateral force is noticeable between 5 psi
and 10 psi at low normal loads.
6. Dry frictional soils will produce less lateral force for increasing loads than
a dry cohesive soil.
7. Soils with higher friction angles will produce less lateral force at
increasing normal loads than a dry cohesive soil
8. Dry soils with elevated cohesion values will produce more lateral force
than a dry soil with less cohesion
9. Shear deformation modulus seems to not be dependent on the type of
surface (frictional or cohesive) but perhaps on the water content of the
soil, where the build up of pore water pressure fails the surface faster than
a soil which does not have any pore water pressure.
10. Normal load properties do not seem to have an effect on traction when the
soil is very stiff. Had the soils been less stiff, the lugs of the tire would be
able to penetrate and use the edges of the lugs for an extra bearing surface
for generating more lateral traction.
These observations all make sense when considering the behavior of tires in
general when working in concert with the mechanics of the soil-rubber interface. For the
difference between surface effects, imagine a cylinder of gravel, with forces being
applied to the top and bottom of the cylinder. What holds this gravel in a cylindrical
shape? Since there are no molecular bonds between the stones, something must confine
the gravel on the sides, as it is being loaded from the top and bottom. On a gravel surface,
there is no confining stress, so the stones are free to shift laterally under the tire. At
143
higher normal loads, more gravel is forced out and away from the contact patch as these
higher loads are increasing only in the vertical direction, and not the directions to confine
the movement of the stones.
When increasing inflation pressure, especially on stiff surfaces, this can reduce
the contact area that the tire has with the surface. A reduction in contact area will result in
a reduced ability to generate lateral forces. Essentially, the setup tries to make less
material to do more work, which it cannot do. Increasing negative camber on a tire will
add to the amount of lateral force available, and counter the effects of a high tire pressure,
as evidenced in Figure 3.82.
Significant amounts of data have been presented in the results chapter. Bevameter
data for measuring the soil showed that the gravel (frictional) and cohesive (clay)
exhibited trends that are expected to be found in those types of soil. Tire data from the
tire rig provided insight into the noisy world of off-road testing. However, acceptable
data could be generated, and thus verified the acceptance of the exponential tire model.
Upon curve fitting with the exponential model, it was found that for the best fit, both
model coefficients turned out to be the same value. Collecting this data for each run
showed a consistent trend of the coefficients from test to test. The coefficients of the
curve fit were then used to modify a tire model, which generated results that mimicked
the expected behavior of tires on off-road surfaces, as evidenced by other researchers,
and the data collected by this study.
144
CHAPTER 4
CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH
Substantial amounts of data have been presented in the results chapter. Bevameter
data describes the soil-rubber shearing behavior, and the vertical stiffness of the terrain.
Tire rig data shows the lateral and longitudinal force data for a steered tire in various
configurations. Finally the exponential tire model was fit to the data generated by the tire
rig. Changes in the model coefficients versus changes in the tire configuration were
recorded and analyzed. The data presented is of use to vehicle designers who employ
ATV tires in their vehicle designs.
The bevameter data collected for this study matched trends found from other
researchers. Shear behavior matched that of [3] and [30]. Normal loading trends were
similar in the way of increasing sinkage for increasing pressure; however the soils tested
in this work were substantially stiffer than the soils used for the works of [3] and [30].
The tire test rig generated acceptable data amidst the long and difficult process of
collection. Tire data for the KT 821 tire used in this study showed behavior consistent
with [19], [20], and especially [25] where ATV tires were specifically studied.
The use of the exponential tire model, chosen as the soil reacts in an exponential
fashion, fit the tire data generated by the tire test rig with good results. Least squares
methods were used to determine the best fit, and then the model coefficients were
recorded. This was repeated for all various combinations of inflation pressure, nominal
145
camber angle, and surface. A note of interest is that the two coefficients of the tire model
tended to become identical for the best fit. The coefficients are compared to the values of
camber, pressure, and surface properties to determine basic correlations between the tire
settings and model coefficients.
Upon the completion of this work, there is now a further understanding of the
basics of lateral force generation of ATV tires and the effect of camber angle, inflation
pressure, and surface properties. Further improvements to this work are within the
capabilities of current technology and are highly encouraged.
While by no means this study is complete, there are several other paths to take
from this point forward. There are two primary routes to take, one being centered on
different surfaces or various aspects of the strength of surfaces. The remaining, and
obvious, path is that of the tire. The tire is a complex and highly non-linear part of the
vehicle, and to understand its operation, one must start from the most basic of models and
move onward from there. Neither direction has more merit than the other and can
generate equally weighted information towards solving problems of off-road mobility.
Researchers electing to investigate the operating surface may dive into problems
with measuring the terrain. By no means is the method used in this study the most
accurate or technologically advanced. The method used in this study borne out of the
requirement for low cost and simplicity.
A person wishing to create a better bevameter could improve the methods for
electronically collecting data through sensors. The displacement sensor for the shear test
was connected to the rotating shaft by means of a pulley and belt. Certainly there could
be some play in that system. Also, the actuation method of the bevameter, by human
146
power, could be improved with the addition of a servo motor system. Remembering to
specify the motor such that shear rates match that of actual vehicles will save headache,
and improve accuracy.
Furthermore, studies centered about the soil could be positioned to test several
surfaces to varying degrees of accuracy. The soil tests for this study were not as
substantial in number as was planned due to structural insufficiencies with the shear-
head. Care must be taken to properly design the link between the rubber coating and the
shear head body.
The route of further investigating the tire finds several opportunities available to
researchers. While this study was conducted at zero longitudinal slip, Lateral forces can
be investigated when the tire is subjected to traction (accelerative forces) or braking
(decelerative forces). This would require retrofitting the hook on the current test rig to
accept a traction motor, be it hydraulic or electric. The results of researching the
traction/braking problem would result in an entire operational map of the tire on a
particular surface.
Other studies could be conducted with different tires, comparing tread patterns.
Even on one tire, various tread modifications could be studied, such as decreasing lug
height, applying siping to the tire, or removing more material from the lugs by way of a
tire grooving iron. These tread modifications change how the tire interacts with the
surface, and thus changes the performance of the tire. The item of interest to the
researcher is the magnitude of the change. Still in the tread modification category is the
modification of the profile of the tire. Some tires have a squared profile, whereas some
147
have a very rounded profile, i.e. motorcycle tires. The interaction of profile and surface
can have an effect on the way camber affects the performance of the tire.
Researchers building a similar test rig are advised to add a hydraulic damper
between the vehicle and the triangle with a load cell on the end. This will tame some of
the vertical oscillations of the tire that are excited by the ground at a steady speed. The
equations to break down the lateral and longitudinal forces may be modified to accept the
addition of a fourth load cell. The modification to the test rig to include a damper is not a
significant investment and has the chance of improving the quality and accuracy of the
data collected.
Finally, a problem of pure mathematics is the phenomenon observed in this study
where the A and B coefficients of the exponential tire model were found to be the same
value when run through software to check these parameters with the R-squared value to
measure goodness of fit. In every case, the values for A and B became identical. No ill
effects from this phenomenon were observed on the modification of the tire model to
accept the curve fits to A and B coefficients.
148
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151
APPENDICES
A. Uncertainty Analysis
The uncertainty of the data presented in this study is addressed in this following
section. Data processing methods and data analysis paired with structural deflections
provide levels of uncertainty of the measurement required. Tire forces in three directions,
lateral, vertical, and longitudinal are analyzed. Also, slip angle and camber are reviewed
for the ranges they assume during a test.
As the test vehicle goes over an inherently uneven surface, the camber of a tire
changes with position down the test track. The vertical perturbations in the terrain give
rise to roll motions in the test vehicle, and these roll angles back and forth are directly
seen at the test rig, as the mounting apparatus is very stiff.
For each surface, a slow-speed (>5 mph) was conducted with an inclinometer affixed
to the hook, near where camber adjustments are made. Over 6,000 data points were
collected, and analyzed. This was determined using the t-distribution tables and the
following equations.
?
=
?=
N
i
i
xx
N
1
2
)(
1
? (6.1)
N
t
CL
?
?= (6.2)
Equation 6.1 is the standard deviation of a population of N samples. Equation 6.2 is
the confidence limit for samples in the population. For the camber data, a confidence
limit of 0.104 degrees was calculated at 99%. The standard deviation, which is a measure
152
of the spread of the values about a mean, was recorded to be 3.47 degrees. This spread is
what gives rise to the selection of ?high? and ?low? cambers. Seeking an average value of
zero degrees for ?low? camber, one will see that this value can fluctuate about zero. The
same goes for the ?high? camber, where the nominal value is 8 degrees, however from
the analysis, the spread of the values from a desired average of 8 degrees is around 3.5
degrees as well. This leaves the intermediate camber values to be shrouded in statistical
shadows, as the effect of intermediate values cannot be discerned.
As camber angle has uncertainties, so does the slip angle. As indicated in the text, a
simple FEA was conducted on the hook part of the tire test rig. A normal load of 600 lbf,
and a lateral load of 300 lbf were applied. From the tire data generated in this study, there
appears to be a factor of safety of two. The deflections recorded in the FEA showed a
camber deflection of approximately one-half to three-quarters of a degree. Similarly, the
slip angle had less deflection, on the order of one-half of a degree. By looking at the force
diagram in Figure 3.20, one will see that addition of the lateral and longitudinal vectors
will seek to decrease the slip angle.
Also a factor in the slip angle uncertainty is the ground-holding ability of the test
vehicle. Even though the Yamaha Rhino weighs in excess of 1,000 lbf, at extreme normal
loads (275 and 320 lbf) a yawing effect was noticed, and slight counter-steer was
required to maintain a path parallel to the length of the test surface. The counter-steering
effect to balance out the yaw effect would result in a slightly decreased slip angle than
what was set on the test rig. A simple analysis was conducted to generate a yaw angle
that would reduce the slip angle at extreme normal loads.
153
On both surfaces, tests were run at 320 lbf of normal load and 20 degrees of slip
angle to generate significant lateral forces, creating the most pronounced yaw effect. A
camera was placed at an elevated position in line with the direction of travel, which is
capable of taking 3 frames per second. Marks on the roll cage of the vehicle were made to
facilitate high contrast in the photographs. The photographs were compiled together in
photo editing software, and by using the marks on the roll cage, each frame could be
analyzed to trace the path of the center of the roll cage. Pixel measurements were
manually taken from the images to calculate a rotation about the center of the roll cage.
This rotation is assumed to be the yaw angle of the vehicle, as the center of the top of the
roll cage is situated very near the yaw axis, as indicated by weight split calculations for
the longitudinal position of the center of gravity. Figure 6.1 shows a trace of the
movement of the vehicle. While more advanced image processing techniques could be
employed at the cost of more analysis time, the manual analysis has generated an
acceptable first cut for the uncertainty in slip angle, which turns out to be approximately
3.5 degrees at high normal loads and slip angles. This phenomenon could be assumed to
be linearly varying with normal load and slip angle. Therefore the worst deflections occur
at the highest normal loads and slip angles, and the insignificant deflections would be at
zero slip angle, and the lowest normal load, 50 lbf.
154
Figure 6.1 ? Composite photograph of yaw angle uncertainty
Tire force certainty is handled by software written in MATLAB. Once all testing
is complete, all runs for a particular configuration are combined into one data file. Recall
that the acceleration and deceleration portions are not part of the data, so combining the
data does not have any ill effects. Three load cells and two accelerometers are read into
the data acquisition system. Once adjusted through the calibration curves, lateral,
longitudinal, and vertical forces can then be deduced.
Vertical load is determined through the accelerometer attached just below the dead
weights. By knowing the acceleration and the mass, a vertical force can be calculated. As
the terrain fluctuates greatly, this can cause wildly fluctuating normal loads. The software
written for this study incorporates a filter that only processes lines of data where the
normal load is within ten percent of nominal. This process manually sets the uncertainty
of the vertical load. Brief experiments were conducted to reduce this tolerance, however
further reductions in tolerance significantly limited the number of data points for
analysis. As this is the first excursion into the uncertainty of ATV tire forces it was
155
determined that a ten percent tolerance was an acceptable starting point, as no analysis of
this type has been conducted.
As data points are accepted by their meeting normal load tolerance, the program
begins calculating lateral and longitudinal forces through the schematic of the tire rig,
presented in Figure 3.20. Lateral and longitudinal forces are calculated as follows:
)sin()sin()cos(
)cos()cos()sin(
8.7.6.
8.7.6.
???
???
ChChChy
ChChChx
fffF
fffF
++?=
++=
(6.3)
This process generates lateral and longitudinal force vectors that are associated with
normal loads that are within ten percent of the nominal normal load for that run. As the
vehicle has been driven over as a consistent and level surface as possible, the mean of
these vectors is taken. This average value is then saved as the lateral and longitudinal
forces for that particular run. Standard deviations are also computed for both vectors.
Given the standard deviation, and the number of samples, the standard error of the
mean can be calculated for these mean values of lateral force.
nS
E
/??= (6.4)
For most of the data presented in this paper, standard errors of the mean for lateral
force and longitudinal force are approximately ?4 lbs.
All of these uncertainty values can be visualized graphically in Figure 6.2. In three
dimensions, the box shows the levels of uncertainty for a given data point in a set of tire
data.
156
Figure 6.2 ? Graphical Representation of Levels of Uncertainty
Average Lateral or
Longitudinal Load
? 4 lbs
Slip Angle,
-3.5
degrees
Normal
Load
Fz ? 10%
157
B. Analysis of Tire Spring Rates and Natural Frequencies
As a tire a flexible composite structure, this flexibility gives rise to varying
deflections for varying loads. The proportion of load to deflection is viewed as a spring
rate, in units of lb/in. A point raised in Chapter 2, was that the tire could be modeled as a
spring-mass system in the vertical direction against the question that the tow speed
excites the bounce frequency of the tire test rig. The basic system is described in Figure
6.3.
Figure 6.3 ? Spring ? Mass system
Equations associated with this one degree-of-freedom system are:
)2cos()( tfAtx
kxxm
nat
?=
?=&&
(6.5)
Also noting that the natural frequency of this system in Hertz is:
m
k
f
nat
?2
1
= (6.)
158
Given the data in Figure 6.3 for spring rate versus pressure that was
experimentally determined, natural frequencies for various pressure and mass
combinations can be seen in Table 6.1.
Figure 6.4 ? Linear fit to Pressure v. Spring rate data for KT 821 tire
f
n
, Hz 5 psi 6 psi 7 psi 8 psi 9 psi 10 psi
50 lb 0.369 0.372 0.382 0.389 0.395 0.402
95 lb 0.268 0.272 0.277 0.282 0.287 0.291
140 lb 0.220 0.225 0.228 0.232 0.236 0.240
185 lb 0.192 0.195 0.199 0.202 0.205 0.209
230 lb 0.172 0.175 0.178 0.181 0.184 0.187
275 lb 0.157 0.160 0.163 0.166 0.168 0.171
320 lb 0.146 0.148 0.151 0.153 0.156 0.158
Figure 6.1 ? Table of natural frequencies in z-axis for KT 821 tire
159
One will notice that when Table 6.1?s data is compared to the 2.0 Hz ? 3.12 Hz
range of excitation frequencies, that there is a significant difference. When observed from
the perspective of wavelength, the wavelengths associated with the frequencies in Table
6.1 are on the range of 30 ft to 75 ft. This would suggest that certain undulations in the
terrain or the acceleration period of the test run would have an impact on the excitation of
the natural frequencies of the test rig. Another theory to the oscillations is that of a
natural frequency in the lateral direction. As the carcass is flexible in the vertical
direction, it is also flexible in a shearing sense, with lateral movement between the wheel
and the surface. Coupled with the traction trends in the lateral direction, this phenomenon
may be studied further yet is out of the scope of this work as no lateral spring rate
measurements were taken.
160
C. Raw Data
Clay, 10psi, Low Camber
161
162
0
100
200
300
400
0
20
40
60
0
50
100
150
Normal Load, lbs
Longitudinal Forces
Slip Angle, deg
Clay, 10psi, Low Camber
Longi
t
u
di
na
l
For
c
e
,
l
b
s
0
100
200
300
400
0
20
40
60
-50
0
50
100
150
Normal Load, lbs
Lateral Forces
Slip Angle, deg
Clay, 10psi, Low Camber
L
a
te
r
a
l F
o
r
c
e
, lb
s
163
164
165
166
167
168
Gravel, 5 psi, Low Camber
169
170
171
172
173
Gravel, 5 psi, High Camber
174
175
176
177
178
Gravel, 10 psi, High Camber
179
180
181
182
183
D. Computer Programs
Bevameter2.m ? Bevameter data reduction program
clear
clc
clf
addpath('C:\DATAQMATLABExamples')
ring_od = 5; %Inches
ring_id = 3; %Inches
annulusarea = (pi*(ring_od/2)^2) - (pi*(ring_id/2)^2); %sq. in.
ringradius = 2/12; %ft
rval = 'R^2 = ';
mmkay = 'K = ';
%50psi
data = (ReadWinDaqFile('50psi.wdq',0,1800,1))'; % Read in Dataq file
shear50 = (((data(:,1)+.0004)/.0002)/ringradius)/annulusarea;
angle50 = (data(:,2)-data(1,2))/(5/360);
angle50(1168:1583) = angle50(1385:end)+34.63+685.2; % Adjusted because
pot ran over break point
angle50(1584:end) = angle50(1584:end)+104+658.7
% Adjusted because pot ran over break point
inch50 = angle50*((pi/180)*ringradius*12);
j=500;
k = 1;
while j <1800
beaners50(k,1) = shear50(j);
k = k+1;
j= j+1;
end
residual50 = mean(beaners50);
%75psi
data = (ReadWinDaqFile('75psi.wdq',0,1800,1))'; % Read in Dataq file
shear75 = (((data(:,1)+.0004)/.0002)/ringradius)/annulusarea;
angle75 = (data(:,2)-data(1,2))/(5/360);
inch75 = angle75*((pi/180)*ringradius*12);
j=500;
k = 1;
while j <1800
beaners75(k,1) = shear75(j);
k = k+1;
j= j+1;
end
residual75 = mean(beaners75);m
%100psi
data = (ReadWinDaqFile('100psi.wdq',0,1800,1))'; % Read in Dataq file
shear100 = (((data(:,1)+.0004)/.0002)/ringradius)/annulusarea;
184
angle100 = (data(:,2)-data(1,2))/(5/360);
inch100 = angle100*((pi/180)*ringradius*12);
j=500;
k = 1;
while j <1800
beaners100(k,1) = shear100(j);
k = k+1;
j= j+1;
end
residual100 = mean(beaners100);
%125psi
data = (ReadWinDaqFile('125psi.wdq',0,1800,1))'; % Read in Dataq file
shear125 = (((data(:,1)+.0004)/.0002)/ringradius)/annulusarea;
angle125 = (data(:,2)-data(1,2))/(5/360);
inch125 = angle125*((pi/180)*ringradius*12);
j=500;
k = 1;
while j <1800
beaners125(k,1) = shear125(j);
k = k+1;
j= j+1;
end
residual125 = mean(beaners125);
%Plot the shear displacement data
figure(1)
plot(inch50, shear50, '-k', 'linewidth',2)
hold on
plot(inch75, shear75, ':k', 'linewidth',2)
hold on
plot(inch100, shear100, '-k', 'linewidth',2)
hold on
plot(inch125, shear125, ':k', 'linewidth',2)
xlabel('\bf \fontsize{12}Displacement, in')
ylabel('\bf \fontsize{12}Shear Stress, psi')
title('\bf Shear Stress v. Displacement, Dirt Test 1')
legend('\sigma = 42.9 psi', '\sigma = 64.4 psi', '\sigma = 85.9 psi',
'\sigma = 107 psi')
print -djpeg dirt1raw
cylpress = [50;75;100;125];
cylinderarea =(pi*2^2)-(pi*.75^2); %2" OD, .75" ID on the pneu cyl
honeydew = cylpress.*cylinderarea;
normals = honeydew./annulusarea;
shears = [residual50;residual75;residual100;residual125];
fit = polyfit(normals, shears, 1);
fitting = polyval(fit,normals);
phi = atan(fit(1))*(180/pi)
%Plot Mohr-Coulomb Failure Envelope
figure(2)
185
plot(normals, shears, 'kx', 'linewidth', 2,'markersize', 15)
hold on
plot(normals, fitting, '-k', 'linewidth', 2)
title('\bf Mohr-Coulomb Failure Envelope, Dirt Test 2')
xlabel('\bf \fontsize{12}Normal Stress, psi')
ylabel('\bf \fontsize{12}Shear Stress, psi')
axis([30 120 0 20])
text(90, 18, '\bf \fontsize{12} \phi = 6.20 deg')
print -djpeg dirt1mohrcoulomb
% Shear Rate Plots
shish = [1:1:1800]';
sheartime = shish*.002084; % .002 is the time step
slopedy = sheartime(1000:1550); %indexes when sliding happened
figure(3)
plot(slopedy, inch100(1000:1550), '-k', 'linewidth',2)
hold on
plot(slopedy, inch75(1000:1550), ':k', 'linewidth',2)
hold on
plot(slopedy, inch125(1000:1550), '--k', 'linewidth',2)
title('\bf \fontsize{12} Shear Rate, Dirt Test 1')
xlabel('\bf \fontsize{12} Time, s')
ylabel('\bf \fontsize{12} Displacement, in')
legend('\sigma = 85.9 psi', '\sigma = 64.4 psi', '\sigma = 107 psi',
'Location','SouthEast')
axis([2 3.5 0 10])
print -djpeg dirt1shearrate
% Create plots to show fitting of shear stress data
%50 psi cylinder pressure
figure(4)
plot(inch50, shear50, '-k', 'linewidth',2)
K50=.01:.01:.99;
cheese = 1;
for cheese = 1:length(K50)
shearfit50 = (((max(shear50)-residual50)/2)+residual50).*(1-exp(-
inch50./K50(cheese)));
rsqd(cheese) = 1- sum((shearfit50-shear50).^2)/sum((shearfit50-
mean(shear50)).^2);
if rsqd(cheese) >= max(rsqd)
index50 = cheese;
end
end
shearfit50 = (((max(shear50)-residual50)/2)+residual50).*(1-exp(-
inch50./K50(index50)));
hold on
plot(inch50, shearfit50, '--k')
title('\bf \fontsize{12}Exponential Shear Equation Fit, Dirt Test 1')
xlabel('\bf \fontsize{12} Displacement, in')
ylabel('\bf \fontsize{12} Shear Stress, psi')
legend('\sigma = 42.9 psi, Test 1', 'Exponential Curve Fit',
'location', 'southeast')
axis([-.5 6 0 15])
k50rsqd = num2str(rsqd(index50));
text(1,12, strcat(rval, k50rsqd))
mmkay50 = num2str(K50(index50));
186
stereoshift50 = strcat(mmkay, mmkay50);
text(1, 11, stereoshift50)
print -djpeg dirt1fit43
%75 psi cylinder pressure
figure(5)
plot(inch75, shear75, '-k', 'linewidth',2)
K75=.01:.01:.99;
cheese = 1;
for cheese = 1:length(K75)
shearfit75 = (((max(shear75)-residual75)/2)+residual75).*(1-exp(-
inch75./K75(cheese)));
rsqd(cheese) = 1- sum((shearfit75-shear75).^2)/sum((shearfit75-
mean(shear75)).^2);
if rsqd(cheese) >= max(rsqd)
index75 = cheese;
end
end
shearfit75 = (((max(shear75)-residual75)/2)+residual75).*(1-exp(-
inch75./K75(index75)));
hold on
plot(inch75, shearfit75, '--k')
title('\bf \fontsize{12}Exponential Shear Equation Fit, Dirt Test 1')
xlabel('\bf \fontsize{12} Displacement, in')
ylabel('\bf \fontsize{12} Shear Stress, psi')
legend('\sigma = 64.4 psi, Test 1', 'Exponential Curve Fit',
'location', 'southeast')
axis([-.5 6 0 15])
k75rsqd = num2str(rsqd(index75));
text(1,11, strcat(rval, k75rsqd))
mmkay75 = num2str(K75(index75));
stereoshift75 = strcat(mmkay, mmkay75);
text(1, 10, stereoshift75)
print -djpeg dirt1fit64
%100 psi cylinder pressure
figure(6)
plot(inch100, shear100, '-k', 'linewidth',2)
K100=.01:.01:.99;
cheese = 1;
for cheese = 1:length(K100)
shearfit100 = (((max(shear100)-residual100)/2)+residual100).*(1-
exp(-inch100./K100(cheese)));
rsqd(cheese) = 1- sum((shearfit100-shear100).^2)/sum((shearfit100-
mean(shear100)).^2);
if rsqd(cheese) >= max(rsqd)
index100 = cheese;
end
end
shearfit100 = (((max(shear100)-residual100)/2)+residual100).*(1-exp(-
inch100./K100(index100)));
hold on
plot(inch100, shearfit100, '--k')
title('\bf \fontsize{12}Exponential Shear Equation Fit, Dirt Test 1')
187
xlabel('\bf \fontsize{12} Displacement, in')
ylabel('\bf \fontsize{12} Shear Stress, psi')
legend('\sigma = 85.9 psi, Test 1', 'Exponential Curve Fit',
'location', 'southeast')
axis([-.5 6 0 20])
k100rsqd = num2str(rsqd(index100));
text(1,4, strcat(rval, k100rsqd))
mmkay100 = num2str(K100(index100));
stereoshift100 = strcat(mmkay, mmkay100);
text(1, 5, stereoshift100)
print -djpeg dirt1fit86
% %125 psi cylinder pressure
% figure(7)
% plot(inch125, shear125, '-b', 'linewidth',2)
% K125=.02;
% shearfit125 = (((max(shear125)-residual125)/2)+residual125).*(1-exp(-
inch125./K125));
% hold on
% plot(inch125, shearfit125, '-r')
% title('\bf \fontsize{12}Exponential Shear Equation Fit, Dirt Test 2')
% xlabel('\bf \fontsize{12} Displacement, in')
% ylabel('\bf \fontsize{12} Shear Stress, psi')
% legend('\sigma = 107 psi, Test 2', 'Exponential Curve Fit',
'location', 'southeast')
% axis([-.5 6 0 25])
% Shear modulus versus normal stress
shearmodulus = [K50(index50); K75(index75); K100(index100)];
figure(8)
plot(normals(1:3), shearmodulus, 'ko', 'linewidth', 2, 'markersize',
10')
title('\bf \fontsize{12} Shear Modulus v. Normal Stress, Gravel Test
1')
xlabel('\bf \fontsize{12} Normal Stress, psi')
ylabel('\bf \fontsize{12} Shear Modulus, in')
axis([40 110 0 .1])
print -djpeg dirt1shearmod
% See if the shear-displacement graph can collapse
normshear50 = shear50./42.9;
normshear75 = shear75./64.4;
normshear100 = shear100./85.9;
normshear125 = shear125./107;
norm_coeff_array =
[mean(normshear50(800:end));mean(normshear75(800:end));mean(normshear10
0(800:end));mean(normshear125(800:end))];
norm_coeff = mean(norm_coeff_array); % last half of data was analyzed,
average coeff for each data set was taken, then those were averaged
figure(9)
plot(inch50, normshear50, '-k', 'linewidth', 2)
hold on
plot(inch75, normshear75, '-.k', 'linewidth', 2)
hold on
188
plot(inch100, normshear100, ':k', 'linewidth', 2)
hold on
plot(inch125, normshear125, '--k', 'linewidth', 2)
title('\bf \fontsize{12} Normalized Shear Stress v. Displacement, Dirt
Test 2')
xlabel('\bf \fontsize{12} Displacement, in')
ylabel('\bf \fontsize{12} Normalized Shear Stress')
text( .5,.05, '\bf \fontsize{12} Mean Normalized Shear Stress = 0.185')
print -djpeg dirt1collapsed
dataout(:,1) = normals;
dataout(:,2) = shears;
csvwrite('dirt1.csv', dataout)
189
reduction.m ? Tire rig data reduction program
clear
clc
clf
addpath('C:\DATAQMATLABExamples')
% 1 - vert accel
% 2 - lat accel
% 3 - hall effect
% 4 - clinometer
% 5 - f6 load cell
% 6 - f7 load cell
% 7 - f8 load cell
%Test Details
pressure = 5; %PSI
fz = 140; %lbs
surface = 1 %Gravel
info = zeros(8,1) %Generate blank vector
info(1,1) = pressure; %Add pressure to info vector
info(2,1) = fz; %Add normal load to info vector
info(3,1) = surface; %Add surface to info vector
tolerance = .1; % normal load tolerance
upperbound = fz+(fz*tolerance); %Set upper normal load bound
lowerbound = fz-(fz*tolerance); %Set lower normal load bound
%Setup slipangle vector
slipangle = [0;5;10;15;20;25;30;45]; %Generate slip angle vector
%Now, individually go through each file, and calculate appropriate data
% Alpha 0--------------------------------------------------------------
---
data = (ReadWinDaqFile('alpha0.wdq',100,2000,1))'; % Read in Dataq file
fvert = fz*(1.0256*data(:,1)-2.5331); %take normal load voltage and
convert to lbs
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound) %if the
normal load is within tolerance, save that index of data
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
190
end
glat = 1.0256*newdata(:,2)-2.5331; % take lateral accel voltage and
convert to G
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5); %Load cell voltage and convert to lbs
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180)); %Longitudinal force
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180)); %Lateral Force
avgfx(1,1) = mean(fx0) %Average value of longitudinal force vector
stdfx(1,1) = std(fx0)/sqrt(length(fx0)) %Standard error
avgfy(1,1) = mean(fy0) %Average value of lateral force vector
stdfy(1,1) = std(fy0)/sqrt(length(fx0)) %Standard error
%======================================================================
==
%
% % Alpha 5 -----------------------------------------------------------
-----
data = (ReadWinDaqFile('alpha5.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
191
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
avgfx(2,1) = mean(fx0)
stdfx(2,1) = std(fx0)/sqrt(length(fx0))
avgfy(2,1) = mean(fy0)
stdfy(2,1) = std(fy0)/sqrt(length(fx0))
%
%======================================================================
===
%
% % Alpha 10 ----------------------------------------------------------
-----
data = (ReadWinDaqFile('alpha10.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
avgfx(3,1) = mean(fx0)
stdfx(3,1) = std(fx0)/sqrt(length(fx0))
avgfy(3,1) = mean(fy0)
stdfy(3,1) = std(fy0)/sqrt(length(fx0))
192
%
%======================================================================
===
%
% % Alpha 15 ----------------------------------------------------------
-----
data = (ReadWinDaqFile('alpha15.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
avgfx(4,1) = mean(fx0)
stdfx(4,1) = std(fx0)/sqrt(length(fx0))
avgfy(4,1) = mean(fy0)
stdfy(4,1) = std(fy0)/sqrt(length(fx0))
%
%======================================================================
===
%
% % Alpha 20 ----------------------------------------------------------
-----
data = (ReadWinDaqFile('alpha20.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
193
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
avgfx(5,1) = mean(fx0)
stdfx(5,1) = std(fx0)/sqrt(length(fx0))
avgfy(5,1) = mean(fy0)
stdfy(5,1) = std(fy0)/sqrt(length(fx0))
%
%======================================================================
====
%
% Alpha 25 ------------------------------------------------------------
----
data = (ReadWinDaqFile('alpha25.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
194
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
avgfx(6,1) = mean(fx0)
stdfx(6,1) = std(fx0)/sqrt(length(fx0))
avgfy(6,1) = mean(fy0)
stdfy(6,1) = std(fy0)/sqrt(length(fx0))
%======================================================================
====
%
% % Alpha 30 ----------------------------------------------------------
-----
data = (ReadWinDaqFile('alpha30.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
195
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
avgfx(7,1) = mean(fx0)
stdfx(7,1) = std(fx0)/sqrt(length(fx0))
avgfy(7,1) = mean(fy0)
stdfy(7,1) = std(fy0)/sqrt(length(fx0))
%======================================================================
====
% % Alpha 45 ----------------------------------------------------------
------
data = (ReadWinDaqFile('alpha45.wdq',100,2000,1))';
fvert = fz*(1.0256*data(:,1)-2.5331);
n=1;
j=1;
while n <= length(data)
if (fvert(n) >= lowerbound) & (fvert(n) <= upperbound)
newdata(j,1) = data(n,1);
newdata(j,2) = data(n,2);
newdata(j,3) = data(n,3);
newdata(j,4) = data(n,4);
newdata(j,5) = data(n,5);
newdata(j,6) = data(n,6);
newdata(j,7) = data(n,7);
j=j+1;
end
n=n+1;
end
glat = 1.0256*newdata(:,2)-2.5331;
halleff = newdata(:,3);
f6 = newdata(:,5)/(7*10^-5);
f7 = newdata(:,6)/(7*10^-5);
f8 = newdata(:,7)/(7*10^-5);
fx0 =
f6*sin(slipangle(1,1)*(pi/180))+f7*cos(slipangle(1,1)*(pi/180))+f8*cos(
slipangle(1,1)*(pi/180));
fy0 = -
f6*cos(slipangle(1,1)*(pi/180))+f7*sin(slipangle(1,1)*(pi/180))+f8*sin(
slipangle(1,1)*(pi/180));
196
avgfx(8,1) = mean(fx0)
stdfx(8,1) = std(fx0)/sqrt(length(fx0))
avgfy(8,1) = mean(fy0)
stdfy(8,1) = std(fy0)/sqrt(length(fx0))
%======================================================================
====
% "It doesn't hook up to the internet, it doesn't brush your teeth for
you,
% it just makes phone calls. Damnit I miss that phone."
% "Yep, ya cross them at Jaspermatater"
% JL Scott
clf
%Curve Fitting
%Seek to minimize the sum of absolute value of the errors
alphafit = [0;5;10;15;20;25;30;45];
%generate all the possible combinations
rickybobby = makecomb(100,2);
rickybobby = (rickybobby/100)';
% Check every possible combination and note the lowest error's index
% Yes, it's inefficient, but it's easy to program, and there's not that
% many numbers to go through.
for r=1:length(rickybobby)
fyfit = fz*rickybobby(r,1)*(1-exp(-rickybobby(r,2)*alphafit));
rsqd(r) = 1-(sum((fyfit-avgfy).^2)/sum((fyfit-mean(avgfy)).^2));
if rsqd(r) >= max(rsqd)
index = r;
end
end
% Take calculated values for A and B and apply them to a curve fit.
A = rickybobby(index,1);
B = rickybobby(index,2);
fyfit = fz*A*(1-exp(-B*alphafit));
%Put A and B values into the information vector for transfer
info(4,1) = A;
info(5,1) = B;
%Put error value into info vector
%Recall the error figure here is the sum of the absolute value of the
errors
info(6,1) = rsqd(index);
figure(1)
errorbar(slipangle, avgfy, stdfy, '-kx')
hold on
errorbar(slipangle, avgfx, stdfx, '-.k*')
hold on
plot(alphafit,fyfit,'--k')
axis([0 45 -5 150]);
title('\bf \fontsize{12}Tire Forces')
xlabel('\bf \fontsize{12} Slip Angle, deg')
197
ylabel('\bf \fontsize{12} Force, lbs')
legend('Lateral Force', 'Drag Force', 'Lateral Force Fit','location',
'southeast')
turd = 'Fit R Squared = ';
dash9 = num2str(rsqd(index));
text(2,120, strcat(turd, dash9))
% title('Tire Forces');
% xlabel('Slip Angle, deg');
% ylabel('Force, lbs');
grid on
% figure(2)
% plot(slipangle, avgfy, '-r')
% hold on
% plot(alphafit, fyfit)
% axis([0 45 -5 150]);
% title('Tire Forces');
% xlabel('Slip Angle, deg');
% ylabel('Force, lbs');
% grid on
% Need to add camber angle certainty stuff
% Need to add vertical load certainty?
%Output Tire Data
tiredata = [slipangle,avgfy,stdfy,avgfx,stdfx,info]
csvwrite('C:\DATAQ\446CE22C\dunlop821\gravel\5psi\high\output140.csv',
tiredata)
figure(2)
plot(rsqd, 'k')
xlabel('Index number')
ylabel('R^2 Value')
title('Trace of R^2 values')
print -dtiff rsquared
198
compare.m ? Program to compare results of camber, pressure, and surface tire data
% This file will be used to generate comparison plots for camber,
% inflation pressure and operating surface
clear
clc
clf
% The key plots to compare each situation will be:
% 1) Calpha v. Fz
% 2) Peak Fy v. Fz
% 3) Peak Fx v. Fz
% 4) Normalized Fy v. alpha
% Firstly, lets load everything
grav10highfz =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\10psi\high\grav10highfz');
grav10highfy =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\10psi\high\grav10highfy');
grav10lowfz =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\10psi\low\grav10lowfz');
grav10lowfy =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\10psi\low\grav10lowfy');
grav5highfz =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\5psi\high\grav5highfz');
grav5highfy =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\5psi\high\grav5highfy');
grav5lowfz =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\5psi\low\grav5lowfz');
grav5lowfy =
csvread('C:\DATAQ\446CE22C\dunlop821\gravel\5psi\low\grav5lowfy');
clay10lowfz =
csvread('C:\DATAQ\446CE22C\dunlop821\clay\10psi\low\clay10lowfz');
clay10lowfy =
csvread('C:\DATAQ\446CE22C\dunlop821\clay\10psi\low\clay10lowfy');
%======================================================================
====
%Camber effects
%======================================================================
====
%On Calpha
figure(1)
plot(grav10highfz(:,1), grav10highfz(:,2), 'k*', 'linewidth', 2,
'markersize',10) %High Camber
hold on
plot(grav10lowfz(:,1), grav10lowfz(:,2), 'kx', 'linewidth', 2,
'markersize',10) %Low Camber
title('\bf \fontsize{12} Cornering Stiffness v. Normal Load')
xlabel('\bf \fontsize{12} Normal Load, lbs')
199
ylabel('\bf \fontsize{12} Cornering Stiffness, C\alpha, lbs/deg')
legend('High Camber', 'Low Camber', 'location', 'southeast')
axis([0 350 0 20])
print -dtiff calphacamber
%On Peak Fx
figure(2)
plot(grav10highfz(:,1), grav10highfz(:,3), 'k*', 'linewidth', 2,
'markersize', 10) %High Camber
hold on
plot(grav10lowfz(:,1), grav10lowfz(:,3), 'kx', 'linewidth', 2,
'markersize', 10) %Low Camber
title('\bf\fontsize{12} Peak Longitudinal Force v. Normal Load')
xlabel('\bf\fontsize{12} Normal Load, lbs')
ylabel('\bf\fontsize{12} Peak Longitudinal Force, lbs')
legend('High Camber', 'Low Camber', 'location', 'southeast')
axis([0 350 0 150])
print -dtiff peakfxcamber
%on Peak Fy
figure(3)
plot(grav10highfz(:,1), grav10highfz(:,4), 'k*', 'linewidth', 2,
'markersize', 10) %High Camber
hold on
plot(grav10lowfz(:,1), grav10lowfz(:,4), 'kx', 'linewidth', 2,
'markersize', 10) %Low Camber
title('\bf\fontsize{12} Peak Lateral Force v. Normal Load')
xlabel('\bf\fontsize{12} Normal Load, lbs')
ylabel('\bf\fontsize{12} Peak Lateral Force, lbs')
legend('High Camber', 'Low Camber', 'location', 'southeast')
axis([0 350 0 150])
print -dtiff peakfycamber
%on normalized fy v. alpha
%split the first few normal loads to some graphs
figure(4)
plot(grav10highfy(1:8,1), grav10highfy(9:16,1), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav10lowfy(1:8,1), grav10lowfy(9:16,1), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
axis([0 45 -.01 .8])
legend('95 lbs, High Camber', '95 lbs, Low Camber','location',
'southeast')
print -dtiff fyValpha95camber
figure(5)
plot(grav10highfy(1:8,2), grav10highfy(9:16,2), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav10lowfy(1:8,2), grav10lowfy(9:16,2), '--k', 'markersize', 10,
'linewidth', 1.5)
200
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
axis([0 45 -.01 .8])
legend('140 lbs, High Camber', '140 lbs, Low Camber','location',
'southeast')
print -dtiff fyValpha140camber
figure(6)
plot(grav10highfy(1:8,3), grav10highfy(9:16,3), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav10lowfy(1:8,3), grav10lowfy(9:16,3), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
axis([0 45 -.01 .8])
legend('185 lbs, High Camber', '185 lbs, Low Camber','location',
'southeast')
print -dtiff fyValpha185camber
%now do the remaining normal loads
figure(7)
plot(grav10highfy(1:8,4), grav10highfy(9:16,4), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav10lowfy(1:8,4), grav10lowfy(9:16,4), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('230 lbs, High Camber', '230 lbs, High
Camber','location','southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha230camber
figure(8)
plot(grav10highfy(1:8,5), grav10highfy(9:16,5), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav10lowfy(1:8,5), grav10lowfy(9:16,5), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('275 lbs, High Camber', '275 lbs, Low
Camber','location','southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha275camber
figure(9)
plot(grav10highfy(1:8,6), grav10highfy(9:16,6), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
201
plot(grav10lowfy(1:8,6), grav10lowfy(9:16,6), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('320 lbs, High Camber','320 lbs, Low
Camber','location','southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha320camber
%======================================================================
====
%Pressure Effects
%======================================================================
====
%On Calpha
figure(10)
plot(grav10lowfz(:,1), grav10lowfz(:,2), 'k*', 'linewidth', 2,
'markersize',10) %High pressure
hold on
plot(grav5lowfz(:,1), grav5lowfz(:,2), 'kx', 'linewidth', 2,
'markersize',10) %Low pressure
title('\bf \fontsize{12} Cornering Stiffness v. Normal Load')
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} Cornering Stiffness, C\alpha, lbs/deg')
legend('10 psi', '5 psi', 'location', 'southeast')
axis([0 350 0 20])
print -dtiff calphapressure
%On Peak Fx
figure(11)
plot(grav10lowfz(:,1), grav10lowfz(:,3), 'k*', 'linewidth', 2,
'markersize', 10) %High pressure
hold on
plot(grav5lowfz(:,1), grav5lowfz(:,3), 'kx', 'linewidth', 2,
'markersize', 10) %Low pressure
title('\bf\fontsize{12} Peak Longitudinal Force v. Normal Load')
xlabel('\bf\fontsize{12} Normal Load, lbs')
ylabel('\bf\fontsize{12} Peak Longitudinal Force, lbs')
legend('10 psi', '5 psi', 'location', 'southeast')
axis([0 350 0 150])
print -dtiff peakfxpressure
%on Peak Fy
figure(12)
plot(grav10lowfz(:,1), grav10lowfz(:,4), 'k*', 'linewidth', 2,
'markersize', 10) %High pressure
hold on
plot(grav5lowfz(:,1), grav5lowfz(:,4), 'kx', 'linewidth', 2,
'markersize', 10) %Low pressure
title('\bf\fontsize{12} Peak Lateral Force v. Normal Load')
xlabel('\bf\fontsize{12} Normal Load, lbs')
ylabel('\bf\fontsize{12} Peak Lateral Force, lbs')
legend('10 psi', '5 psi', 'location', 'southeast')
axis([0 350 0 150])
202
print -dtiff peakfypressure
%on normalized fy v. alpha
figure(13)
plot(grav10lowfy(1:8,1), grav10lowfy(9:16,1), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav5lowfy(1:8,1), grav5lowfy(9:16,1), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('95 lbs, 10 psi', '95 lbs, 5 psi','location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha95pressure
figure(14)
plot(grav10lowfy(1:8,2), grav10lowfy(9:16,2), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav5lowfy(1:8,2), grav5lowfy(9:16,2), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('140 lbs, 10 psi', '140 lbs, 5 psi','location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha140pressure
figure(15)
plot(grav10lowfy(1:8,3), grav10lowfy(9:16,3), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav5lowfy(1:8,3), grav5lowfy(9:16,3), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('185 lbs, 10psi', '185 lbs, 5 psi','location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha185pressure
figure(16)
plot(grav10lowfy(1:8,4), grav10lowfy(9:16,4), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav5lowfy(1:8,4), grav5lowfy(9:16,3), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('230 lbs, 10 psi', '230 lbs, 5 psi', 'location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha230pressure
203
figure(17)
plot(grav10lowfy(1:8,5), grav10lowfy(9:16,5), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav5lowfy(1:8,5), grav5lowfy(9:16,5), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('275 lbs, 10 psi', '275 lbs, 5 psi', 'location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha275pressure
figure(18)
plot(grav10lowfy(1:8,6), grav10lowfy(9:16,6), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(grav5lowfy(1:8,6), grav5lowfy(9:16,6), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('320 lbs, 10 psi', '320 lbs, 5 psi','location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha320pressure
%======================================================================
====
%Suface Effects
%======================================================================
====
%On Calpha
figure(19)
plot(grav10lowfz(:,1), grav10lowfz(:,2), 'k*', 'linewidth', 2,
'markersize',10) %High Camber
hold on
plot(clay10lowfz(:,1), clay10lowfz(:,2), 'kx', 'linewidth', 2,
'markersize',10) %Low Camber
title('\bf \fontsize{12} Cornering Stiffness v. Normal Load')
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} Cornering Stiffness, C\alpha, lbs/deg')
legend('Gravel', 'Clay', 'location', 'southeast')
axis([0 350 0 20])
print -dtiff calphasurface
%On Peak Fx
figure(20)
plot(grav10lowfz(:,1), grav10lowfz(:,3), 'k*', 'linewidth', 2,
'markersize', 10) %High Camber
hold on
plot(clay10lowfz(:,1), clay10lowfz(:,3), 'kx', 'linewidth', 2,
'markersize', 10) %Low Camber
title('\bf\fontsize{12} Peak Longitudinal Force v. Normal Load')
xlabel('\bf\fontsize{12} Normal Load, lbs')
204
ylabel('\bf\fontsize{12} Peak Longitudinal Force, lbs')
legend('Gravel', 'Clay', 'location', 'southeast')
axis([0 350 0 150])
print -dtiff peakfxsurface
%on Peak Fy
figure(21)
plot(grav10lowfz(:,1), grav10lowfz(:,4), 'k*', 'linewidth', 2,
'markersize', 10) %High Camber
hold on
plot(clay10lowfz(:,1), clay10lowfz(:,4), 'kx', 'linewidth', 2,
'markersize', 10) %Low Camber
title('\bf\fontsize{12} Peak Lateral Force v. Normal Load')
xlabel('\bf\fontsize{12} Normal Load, lbs')
ylabel('\bf\fontsize{12} Peak Lateral Force, lbs')
legend('Gravel', 'Clay', 'location', 'southeast')
axis([0 350 0 150])
print -dtiff peakfysurface
%on normalized fy v. alpha
figure(22)
plot(grav10lowfy(1:8,1), grav10lowfy(9:16,1), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(clay10lowfy(1:8,1), clay10lowfy(9:16,1), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
axis([0 45 -.01 .8])
legend('95 lbs, Gravel', '95 lbs, Clay','location', 'southeast')
print -dtiff fyValpha95surface
figure(23)
plot(grav10lowfy(1:8,2), grav10lowfy(9:16,2), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(clay10lowfy(1:8,2), clay10lowfy(9:16,2), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
axis([0 45 -.01 .8])
legend('140 lbs, Gravel','140 lbs, Clay', 'location', 'southeast')
print -dtiff fyValpha140surface
figure(24)
plot(grav10lowfy(1:8,3), grav10lowfy(9:16,3), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(clay10lowfy(1:8,3), clay10lowfy(9:16,3), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
205
axis([0 45 -.01 .8])
legend('185 lbs, Gravel', '185 lbs, Clay', 'location', 'southeast')
print -dtiff fyValpha185surface
%other normal loads
figure(25)
plot(grav10lowfy(1:8,4), grav10lowfy(9:16,4), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(clay10lowfy(1:8,4), clay10lowfy(9:16,3), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('230 lbs, Gravel','230 lbs, Clay', 'location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha230surface
figure(26)
plot(grav10lowfy(1:8,5), grav10lowfy(9:16,5), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(clay10lowfy(1:8,5), clay10lowfy(9:16,5), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('275 lbs, Gravel','275 lbs, Clay', 'location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha275surface
figure(27)
plot(grav10lowfy(1:8,6), grav10lowfy(9:16,6), '-k', 'markersize', 10,
'linewidth', 1.5)
hold on
plot(clay10lowfy(1:8,6), clay10lowfy(9:16,6), '--k', 'markersize', 10,
'linewidth', 1.5)
title('\bf\fontsize{12} Normalized Lateral Force v. Slip Angle')
xlabel('\bf\fontsize{12} Slip Angle, deg')
ylabel('\bf\fontsize{12} Normalized Lateral Force, lbs')
legend('320 lbs, Gravel', '320 lbs, Clay', 'location', 'southeast')
axis([0 45 -.01 .8])
print -dtiff fyValpha320surface
206
norm_dirt.m ? Program to process data from normal load tests on the bevameter
% Bevameter Normal Load Tests
% Dirt Surface
clear
clc
clf
% Small Penetration Plate is 3 inches diameter
% Large Penetration Plate is 4 inches diameter
b1 = 3.0/2;
b2 = 4.0/2;
load big1.csv;
load big2.csv;
load big3.csv;
load small1.csv;
load small2.csv;
load small3.csv;
kphi(1) = (b2*big1(1,2) - b1*small1(1,2))/(b2-b1);
kphi(2) = (b2*big2(1,2) - b1*small2(1,2))/(b2-b1);
kphi(3) = (b2*big3(1,2) - b1*small3(1,2))/(b2-b1);
kc(1) = ((small1(1,2)-big3(1,2))*(b1*b2))/(b2-b1);
kc(2) = ((small2(1,2)-big3(1,2))*(b1*b2))/(b2-b1);
kc(3) = ((small3(1,2)-big3(1,2))*(b1*b2))/(b2-b1);
keq = (mean(kc)/b1)+mean(kphi);
n(1) = log((small1(3,2)*1000)/keq)/(log(small1(3,1)*1000));
n(2) = log((small2(3,2)*1000)/keq)/(log(small2(3,1)*1000));
n(3) = log((small2(3,2)*1000)/keq)/(log(small3(3,1)*1000));
mean(kphi)
mean(kc)
mean(n)
figure(1)
plot(big1(:,1), big1(:,2).*.3125, '-kx', 'linewidth', 2, 'markersize',
10)
hold on
plot(big2(:,1), big2(:,2).*.3125, ':kx', 'linewidth', 2, 'markersize',
10)
hold on
plot(big3(:,1), big3(:,2).*.3125, '-.kx', 'linewidth', 2, 'markersize',
10)
title('\bf \fontsize{12}Dirt Penetration Test, 4" Plate')
xlabel('\bf \fontsize{12}Sinkage, in.')
ylabel('\bf \fontsize{12}Pressure, psi')
207
figure(2)
plot(small1(:,1), small1(:,2).*.21, '-kx', 'linewidth', 2,
'markersize', 10)
hold on
plot(small2(:,1), small2(:,2).*.21, ':kx', 'linewidth', 2,
'markersize', 10)
hold on
plot(small3(:,1), small3(:,2).*.21, '-.kx', 'linewidth', 2,
'markersize', 10)
title('\bf \fontsize{12}Dirt Penetration Test, 3" Plate')
xlabel('\bf \fontsize{12}Sinkage, in.')
ylabel('\bf \fontsize{12}Pressure, psi')
208
nicedata.m ? Program to compile all data from various normal loads for a given
setup
clear
clc
clf
ident = '\bf Gravel, 10psi, High Camber';
%Define vectors for z-plane in surface plot
normal = [50;95;130;185;230;275;320];
slip = [0; 5; 10; 15; 20; 25; 30; 45];
load('output50.csv');
load('output95.csv');
load('output140.csv');
load('output185.csv');
load('output230.csv');
load('output275.csv');
load('output320.csv');
fy50 = output50(:,2);
fx50 = output50(:,4);
fy95 = output95(:,2);
fx95 = output95(:,4);
fy140 = output140(:,2);
fx140 = output140(:,4);
fy185 = output185(:,2);
fx185 = output185(:,4);
fy230 = output230(:,2);
fx230 = output230(:,4);
fy275 = output275(:,2);
fx275 = output275(:,4);
fy320 = output320(:,2);
fx320 = output320(:,4);
fy = [fy50 fy95 fy140 fy185 fy230 fy275 fy320];
fx = [fx50 fx95 fx140 fx185 fx230 fx275 fx320];
figure(1)
surf(normal, slip,fy )
title('\bf \fontsize{12} Lateral Forces')
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} Slip Angle, deg')
209
zlabel('\bf \fontsize{12} Lateral Force, lbs')
text(0, 60, 100, ident)
print -dtiff grav10highfysurf
figure(2)
surf(normal, slip,fx)
title('\bf \fontsize{12} Longitudinal Forces')
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} Slip Angle, deg')
zlabel('\bf \fontsize{12} Longitudinal Force, lbs')
text(0, 60, 100, ident)
print -dtiff grav10highfxsurf
% Peak Fx vs Fz
figure(3)
peakfx = [max(fx50); max(fx95); max(fx140); max(fx185); max(fx230);
max(fx275); max(fx320)];
plot(normal, peakfx, 'kx', 'linewidth', 2, 'markersize', 15)
axis([0 350 0 175])
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} Peak Longitudinal Force (Drag)')
title('\bf \fontsize{12} Peak Drag Force v. Normal Load')
text(100,10, ident)
print -dtiff peakfxfy
% Peak Fy vs Fz
figure(4)
peakfy = [max(fy50); max(fy95); max(fy140); max(fy185); max(fy230);
max(fy275); max(fy320)];
plot(normal, peakfy, 'kx', 'linewidth', 2, 'markersize', 15)
axis([0 350 0 175])
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} Peak Lateral Force, lbs')
title('\bf \fontsize{12} Peak Lateral Force v. Normal Load')
text(100,10, ident)
print -dtiff peakfyfz
% C_alpha vs Fz - use 0,0, and the 5degree fy value for C alpha
calpha = [fy50(2)/5; fy95(2)/5; fy140(2)/5; fy185(2)/5; fy230(2)/5;
fy275(2)/5; fy320(2)/5];
figure(5)
plot(normal, calpha, 'kx', 'linewidth', 2, 'markersize', 15)
axis([0 350 0 12])
title('\bf \fontsize{12} Cornering Stiffness (C\alpha) v. Normal Load')
xlabel('\bf \fontsize{12} Normal Load, lbs')
ylabel('\bf \fontsize{12} C\alpha, lb/deg')
text(50, 1,ident)
print -dtiff normalcalpha
% Normalized Lateral Force Coefficient v. Alpha
;
figure(6)
plot(slip, fy50./50, ':ko', 'linewidth', 2, 'markersize', 10)
hold on
210
plot(slip, fy95./95, '-kx', 'linewidth', 2, 'markersize', 10)
hold on
plot(slip, fy140./140, ':k+', 'linewidth',2,'markersize', 10)
hold on
plot(slip, fy185./185, '-k*', 'linewidth',2,'markersize', 10)
hold on
plot(slip, fy230./230, ':ks', 'linewidth',2,'markersize',10)
hold on
plot(slip, fy275./275, '-kd', 'linewidth',2,'markersize',10)
hold on
plot(slip, fy320./320, ':kv', 'linewidth', 2,'markersize',10)
legend('50 lb', '95 lb', '140 lb', '185 lb', '230 lb', '275 lb', '320
lb', 'location', 'best')
title('\bf \fontsize{12}Normalized Lateral Force v. Slip Angle')
xlabel('\bf \fontsize{12} Slip Angle, deg')
ylabel('\bf \fontsize{12} Normalized Lateral Force')
text(10, 0, ident)
print -dtiff normlatalpha
% Normalized Longitudinal Force Coefficient v. Alpha
figure(7)
plot(slip, fx50./50, ':ko', 'linewidth', 2, 'markersize', 10)
hold on
plot(slip, fx95./95, '-kx', 'linewidth', 2, 'markersize', 10)
hold on
plot(slip, fx140./140, ':k+', 'linewidth',2,'markersize', 10)
hold on
plot(slip, fx185./185, '-k*', 'linewidth',2,'markersize', 10)
hold on
plot(slip, fx230./230, ':ks', 'linewidth',2,'markersize',10)
hold on
plot(slip, fx275./275, '-kd', 'linewidth',2,'markersize',10)
hold on
plot(slip, fx320./320, ':kv', 'linewidth', 2,'markersize',10)
legend('50 lb', '95 lb', '140 lb', '185 lb', '230 lb', '275 lb', '320
lb', 'location', 'best')
title('\bf \fontsize{12}Normalized Longitudinal (Drag) Force v. Slip
Angle')
xlabel('\bf \fontsize{12} Slip Angle, deg')
ylabel('\bf \fontsize{12} Normalized Longitudinal (Drag) Force')
text(10, .05, ident)
print -dtiff normdlongalpha
%Get the tire coefficient data together
A = [output50(4,6), output95(4,6), output140(4,6), output185(4,6),
output230(4,6), output275(4,6), output320(4,6)];
B = [output50(5,6), output95(5,6), output140(5,6), output185(5,6),
output230(5,6), output275(5,6), output320(5,6)];
rsqd = [output50(6,6), output95(6,6), output140(6,6), output185(6,6),
output230(6,6), output275(6,6), output320(6,6)];
figure(8)
plot(normal, A, 'xk', 'linewidth',2,'markersize', 10)
title('\bf \fontsize{12} A Coefficient v. Normal Load')
211
xlabel('\bf \fontsize{12}Normal Load')
ylabel('\bf \fontsize{12} A Coefficient')
print -dtiff coeffA
figure(9)
plot(normal, B, 'dk', 'linewidth',2,'markersize', 10)
title('\bf \fontsize{12} B Coefficient v. Normal Load')
xlabel('\bf \fontsize{12}Normal Load')
ylabel('\bf \fontsize{12} B Coefficient')
print -dtiff coeffB
normlat95 = [slip;fy95./95];
normlat140 = [slip;fy140./140];
normlat185 = [slip;fy185./185];
normlat230 = [slip;fy230./230];
normlat275 = [slip;fy275./275];
normlat320 = [slip;fy320./320];
comparefzdata = [normal, calpha, peakfx, peakfy];
comparefydata = [normlat95, normlat140, normlat185, normlat230,
normlat275, normlat320];
csvwrite('grav10highfz', comparefzdata)
csvwrite('grav10highfy', comparefydata)
212
E. Photographs
Photograph depicting yaw motion at high normal loads and high slip angles.
Photograph of test rig operating on clay surface.
213
Photograph detailing the camber adjustment
214
Left side view of test rig.
Right side view of test rig and tow vehicle.
215
Detail view of mounting frame and data acquisition box. Note video camera
mounted within trailer hitch.
1987