Dimensionless analysis of tire characteristics for vehicle dynamics studies

Abstract

This paper demonstrates the use of dimensional analysis for scaled vehicle tires. The motivation for this approach is the understanding of realistic nonlinear tire behavior in scaled vehicle control studies. By examining the behavior of vehicle tires within a dimensionless framework, several key tire parameters are developed that allow for an appropriate relationship between full-sized tires and scaled tires. Introducing these scalings into vehicle dynamics studies allows for the development of scaled vehicles that have a high degree of dynamic similitude with full-sized vehicles but are safer and more economical testbeds on which to develop experimental control strategies. Experimental data is used to compare the nonlinear characteristics for sets of scaled and full-sized tires.

Full text

Abstract— This paper demonstrates the use of dimensional
analysis for scaled vehicle tires. The motivation for this
approach is the understanding of realistic nonlinear tire
behavior in scaled vehicle control studies. By examining the
behavior of vehicle tires within a dimensionless framework,
several key tire parameters are developed that allow for an
appropriate relationship between full-sized tires and scaled
tires. Introducing these scalings into vehicle dynamics studies
allows for the development of scaled vehicles that have a high
degree of dynamic similitude with full-sized vehicles but are
safer and more economical testbeds on which to develop
experimental control strategies. Experimental data is used to
compare the nonlinear characteristics for sets of scaled and
full-sized tires.
I. INTRODUCTION AND MOTIVATION
REVIOUS work at the University of Illinois [1,2] has laid
the groundwork for analyzing the dynamics of vehicles
in a dimensionless framework. The basis for this analysis
approach is the well known Pi theorem [3] that can be
found in many standard textbooks on Fluid Mechanics.
Most of the previous dimensionless vehicles studies
focused on a typical linear 2 degree of freedom vehicle
model known as the “Bicycle Model.” The basic form of
the model is given in Equation (1).
(
)
(
)
()
()
f
z
f
f
z
rf
z
fr
frrf
I
aC
C
m
r
V
UI
bCaC
UI
aCbC
U
mU
aCbC
mU
CC
r
V
δ
α
α
αααα
αααα
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
+−−
−
−+−
=
⎥
⎦
⎤
⎢
⎣
⎡
2
2
22
22
22
&
&
(1)
V Vehicle lateral velocity state
r Vehicle yaw rate state
m Vehicle mass
I
z
Vehicle moment of inertia
a Distance from c.g. to front axle
b Distance from c.g. to rear axle
C
α
f
Front cornering stiffness
C
α
r
Rear cornering stiffness
δ
f
Steering angle
U Vehicle longitudinal velocity
This work was supported in part by the Ford Motor Company.
A. Alleyne (corresponding author) is with University of Illinois,
Urbana-Champaign, 1206 West Green St., Urbana, IL 61801, phone: 217-
244-9993, e-mail: alleyne@uiuc.edu.
M. Polley was with the University of Illinois, Urbana-Champaign. He
is now with MPC Products in Chicago, IL. (e-mail:
polley@illinoisalumni.org)
In [1,2], the equations in (1) were reformulated into a
dimensionless framework with several key advantages
identified. One advantage is the fact that a dimensionless
system representation can elicit underlying parametric
relationships that are difficult to uncover in a dimensional
formulation. The work presented in [1] determined a
duality between the cornering stiffness parameter and the
vehicle speed for a typical passenger vehicle. This duality
manifested itself as an identical effect on open loop system
eigenvalues when examined in a dimensionless setting.
Whereas typical dimensioned representations seemed to
indicate two separate effects on the vehicle’s dynamics, a
dimensionless approach illustrated that cornering stiffness
and vehicle velocity were, in fact, equivalent in the proper
representation. The knowledge of this duality would
greatly simplify any gain-scheduling or robust control
approach to account for parametric variation if these
schemes were formulated in a dimensionless setting.
Other benefits of a dimensionless approach include a
drastic reduction in the parametric uncertainty for a
standard Bicycle Model representation [4]. The
distributions of vehicle dimensionless parameters tend to be
tightly clustered about distinct mean values. This relaxes
the requirements on robust controllers and gives a much
less conservative controller design. A consequence of this
can be seen in the uncertainty representations shown in
Figures 1 and 2. Figure 1 shows a dimensioned
multiplicative uncertainty representation for vehicle
dynamics, using a wide sampling of different vehicles, and
Figure 2 shows a dimensionless version. The overall
uncertainty level is significantly lower in the dimensionless
case because parametric coupling has been inherently taken
into account.
Additionally, the dimensionless parameters not only
cluster about a mean value but they also tend to form
distinct functional relationships between themselves [5]. In
essence, the dimensionless parameter space is relatively
sparsely filled by the different vehicle parameters. For all
vehicles produced, the dimensionless form of the
parameters tends to lie in a subspace of the overall
parameter space. This greatly reduces the burden on any
identification algorithm effectively allowing more
Dimensionless Analysis of Tire Characteristics
for Vehicle Dynamics Studies
Matthew Polley and Andrew G. Alleyne, Senior Member, IEEE
P
Proceeding of the 2004 American Control Conference
Boston, Massachusetts June 30 - July 2, 2004
0-7803-8335-4/04/$17.00 ©2004 AACC
ThP05.6
3411

information to be gained with less system excitation [5].
10
-1
10
0
10
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Freq. (rad/sec)
Mag (dB)
Upper Bound for U = 17 m/s (
π
3
= 0.5)
Fig. 1: Multiplicative uncertainty bounds for
dimensioned form of vehicle dynamics [4].
10
-2
10
-1
10
0
10
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Freq. (rad)
Mag ( dB)
Upper Bound for
π
3
= 0.5
Fig. 2: Multiplicative uncertainty bounds for
dimensionless form of vehicle dynamics [4].
All of the previously mentioned work performed for
dimensionless representations of the vehicle was done on
the linear vehicle model given in Equation (1). This
vehicle representation works well for relatively good road
surfaces and relatively low lateral accelerations (<0.3 g) of
the vehicle [2]. However, it was desired to demonstrate the
ability for scaling and dimensional analysis to extend
beyond the linear Bicycle Model representation and
encompass some of the system nonlinearities. From the
point of view of Vehicle Dynamics, the first major
nonlinearity encountered is the force-producing
characteristics of the vehicle tires. The current work
illustrates how certain tire characteristics can be posed in a
dimensionless framework and can be shown to exhibit
dynamic similitude with full sized tires. Therefore, with
proper understanding and system design, scaled vehicles
can be utilized for testing of controllers in conditions
outside the linear regime.
The rest of this paper is as follows. Section 2
illustrates the experimental hardware system used to
generate the data to be analyzed in subsequent sections.
Section 3 introduces a particular tire model form, termed
the Magic Formula tire model [6] that will be used to
analyze the data obtained by the system of Section 2. It
will be shown that this model form is qualitatively valid for
both scaled and full-sized tires as long as both are
pneumatic. Section 4 discusses the results of the data and
illustrates the dimensionless representation of the tire
characteristics that shows quantitative similarity between
scaled and full-sized tires. Section 5 concludes the paper
with a discussion on some of the key points learned as well
as their impact for scaled vehicle designs.
II. S
CALED TIRE TESTING SYSTEM
Figures 3 and 4 give a front and side view of the
experimental apparatus used to generate the scaled tire data
in this work. It consists of a frame upon which is mounted
a 2 degree of freedom actuation system capable of vertical
motion along the z-axis and rotational motion (α) about the
z-axis. These two degrees of freedom are controlled by
brushless DC motors and used to set the normal force and
slip angle of the tire with respect to the road surface. The
road surface is a high performance commercial treadmill
made by True, Inc. that has its own speed controller
integrated and can be controlled between 0 and 5 m/s in 0.1
m/s increments. The tire forces were sensed by a 6-axis
force transducer and the tire speed was sensed by an optical
encoder that was connected to the wheel axle via a belt and
gear. In all tests, the assumption was that the road speed
was constant as set by the treadmill controller. All control
actions were performed via a DSPACE DS1102 data
acquisition system that also recorded the force and velocity
data for the tire.
The tire slip angle [7] is the difference between the
direction of motion of the tire center and the orientation of
the tire longitudinal axis. The tire’s longitudinal slip [7] is
defined as the difference between the road speed and the
tangential speed of the tire’s outer surface, which is then
normalized by the speed of the road for the braking studies
presented here. The tire slip angle is generated by rotating
the tire assembly with respect to the direction of treadmill
travel. The longitudinal slip is generated by application of
a pressure to a small hydraulic brake caliper that is coupled
to a steel brake disc attached to the wheel as shown in
Figure 4. The braking pressure is controlled by a manually
adjusted pressure regulator that was sufficient for the
steady state braking tests performed.
Both the z-axis and the slip angle were under closed loop
control to maintain accuracy in the data measurements.
The z-axis controller was critical to maintain a constant
normal force on the tire throughout a particular test. As
detailed in [8] a hybrid position/force controller was
designed and implemented for the z-axis motion.
3412

z
x
y
z
x
y
z
x
y
Fig 3: Front view of scaled tire tester showing frame,
treadmill, and actuation.
Fig. 4: Side view of tester showing caliper, brake, force
transducer, and encoder.
Fig 5: Four sizes of the scaled tires used for data
generation.
The chosen tire type for the testing was the DuBro TV
model large scale treaded wheel that was originally
designed for model airplanes. This model came in four
different sizes ranging in diameter from 4.5 inches (0.114
m) to 6.0 inches (0.152 m), which were all used for the tire
testing shown in Section 4. A picture of the four sizes of
test tires is shown in Figure 5. This tire type resembles full
sized automobile tires in construction, due to a rubber tread
design and the fact that the tire is pneumatic, but there are
some differences in the tire construction [8]. One difference
is the relative shape of the tire. The ratio of the wheel
diameter to the outside diameter of the tire is significantly
larger for the full sized vehicles. This is attributed to the
smaller values of aspect ratio that the full sized tires exhibit
(65-70) versus the scaled tires (110). Aspect ratio is
defined as the (100*tire height/tire width). The cross-
section shown in Figure 6 gives a better idea of the tread
pattern and aspect ratio for the scaled tires. Section 4
further details the impact of varied aspect ratio.
1 ¾ “
2 “
¼“
1 ¾ “
2 “
¼“
Fig. 6: Cross sectional view of scaled tire used.
III. T
IRE MODELING
To define steady state tire response a number of tire models
have been developed. Such models have utilized a variety
of methods to describe the complex nonlinear phenomenon
that a tire exhibits. Models utilizing finite element methods
have been developed, as shown in [9], breaking the tire into
small 3 dimensional nodes. These models are attractive
because they are quite accurate in describing tire response.
Although these models can accurately define tire behavior
they are computationally intensive and are usually not the
best option for real-time vehicle simulations. Other types
of models utilize a physical tire model to define tire
behavior. An example of such a model is the ‘brush’ tire
model described in [10]. This model defines the tire as a
group of elastic/spring-like cylinders positioned radially
outside of a circular belt. The resulting model is analytical
in nature. The classification of tire models considered for
the current research employs the use of empirical models
that require testing data for characterization. These models
are defined by fitting tire test data to a developed formula.
Such functions have been developed using polynomial fits
and fits based off of the physical nature of the tire [6].
After examining several types of models, the tire model
chosen for the fit of the scaled tire data was the Magic
Formula [6]. This model is an empirical model that is
widely used due to its ability to accurately define a tire
response over its linear and nonlinear regions, without
requiring intensive computation. The model parameters
include terms that define physical characteristics of the tire.
The models for the lateral force, longitudinal force and self
aligning moment are summarized as follows.
Lateral Force Model
:
(
)
[
]
vFy
SBCDF
y
∆+Φ=
−1
tansin
(2)
(
)
(
)
(
)
()
[]
hhF
SB
B
E
SE
y
∆++∆+−=Φ
−
αα
1
tan1 (3)
3.1
=
C (4)
DCBC
⋅
⋅
=
α
(5)
Self Aligning Moment Model
:
3413

(
)
[
]
vMz
SBCDM
z
∆+Φ=
−1
tansin (6)
()( )
(
)
()
[]
hhM
SB
B
E
SE
z
∆++∆+−=Φ
−
αα
1
tan1 (7)
4.2=C (8)
DCBC
M
⋅⋅=
α
,
(9)
Longitudinal Force Model
:
(
)
[
]
x
Fx
BCDF Φ=
−1
tansin
(10)
()
(
)
()
λλ
B
B
E
E
x
F
1
tan1
−
+−=Φ (11)
65.1=C (12)
The lateral cornering stiffness is given in Equation (5) and
the slip angle is α and the longitudinal slip is λ. As can be
seen in the equations describing the Magic Formula the
model utilizes six basic parameters. Two of these
parameters,
∆
S
v
and
∆
S
h
, are shifts that compensate for
intrinsic tire characteristics such as conicity and ply steer.
These will not be considered in detail here. The other four
terms were all involved in shaping the resulting fit, each
having a specific role that is appropriately described by
their names [6].
B = Stiffness Factor
C = Sha
p
e Factor
D = Peak Factor
E = Curvature Factor
-20 -15 -10 -5 0 5 10 15 20
-100
-50
0
50
100
Lateral Force vs Slip Angle for 5 Loads at a Speed of 6.0 mph
Lateral Force (N)
F
z
= 50 N
F
z
= 60 N
F
z
= 70 N
F
z
= 80 N
F
z
= 90 N
-20 -15 -10 -5 0 5 10 15 20
-0.4
-0.2
0
0.2
0.4
0.6
Self-Aliging Moment vs Slip Angle
Slip Angle (degrees)
Self-Aligning Moment (N-m)
Fig. 7: Lateral force and aligning moment data.
The shape factors used in Equations (4), (8), and (12) are
for full sized vehicles [6]. To determine the values of the
various parameters for the scaled tires, the experimental
system of Section 2 was utilized to determine the lateral
force and aligning moment values for different values of
slip angle as well as longitudinal force as a function of
longitudinal slip. Figure 7 shows the lateral force and
aligning moment data for a given tire and loading
conditions. Extensive amounts of additional data,
including longitudinal force data, can be found in [8] and
are not reproduced here due to space limitations. A
comparison with data presented in [6] illustrates a high
degree of qualitative similarity between the scaled tire data
and full sized tire data.
1.3 20 0.132 1.30 21.30 0.04 0.06 -0.59
1.3 30 0.112 1.23 31.14 -0.62 1.44 -0.67
1.3 40 0.104 1.15 38.77 -0.99 3.15 -0.90
1.3 50 0.079 1.18 50.98 -1.71 3.64 -0.99
1.3 60 0.065 1.35 57.45 -1.05 3.66 -0.92
1.3 70 0.057 1.43 62.86 -1.17 3.06 -0.82
1.3 20 0.114 1.67 25.27 1.04 2.48 -0.68
1.3 30 0.104 1.55 35.99 0.99 2.83 -0.62
1.3 40 0.081 1.73 44.73 0.90 4.11 -0.76
1.3 50 0.109 0.96 56.29 -1.22 6.25 -1.08
1.3 60 0.078 1.25 62.36 -1.10 7.41 -1.32
1.3 70 0.064 1.22 74.17 -2.24 8.98 -1.54
1.3 20 0.138 1.68 23.77 1.04 3.41 -0.92
1.3 30 0.125 1.30 41.91 1.05 5.78 -1.09
1.3 40 0.110 1.30 51.09 -0.05 6.73 -1.10
1.3 50 0.119 0.97 59.23 -1.45 8.35 -1.37
1.3 60 0.096 1.06 69.44 -1.62 9.83 -1.62
1.3 70 0.075 1.33 78.63 -0.79 10.38 -1.57
∆
S
h
BCD
(
N
)
E
4.5
5.0
5.5
∆
S
v
Diam.
(
in
)
Speed
(
m/s
)
~Load
(
N
)
Table 1: Scale tire Magic Formula params. (F
y
vs. α)
To adapt the resulting data to the Magic Formula tire
model, a fit for each tire response was made using a least
squares approach. Table 1 gives parameter fits for a lateral
force generation test operating at different normal loads.
Other fits can be found in [8]. As evidenced in the table,
the shape factor C is approximately the same (~1.3) as for
full sized vehicles. Other qualitative similarities were
determined for the aligning moment and longitudinal force
data thereby leading to the conclusion that the force-
generating characteristics of scaled pneumatic tires are
qualitatively similar to their full-sized counterparts.
IV.
SCALED TIRE CHARACTERISTICS
Due to space limitations, the following analysis will
focus on a comparison in lateral force generation.
Reference [8] gives additional detail on aligning moment
comparisons and, to a lesser extent, longitudinal force
comparisons. Using the fits described in the previous
section, comparisons can be made for various sizes of tires.
This can be done using values of cornering stiffness and
cornering coefficient. Cornering stiffness is well known to
be the slope at the origin of the lateral force versus slip
angle curve. Typically, cornering coefficient is defined as
a normalization of the cornering stiffness with respect to
normal load [7]. This is defined as:
z
F
C
CC
α
α
≡ (13)
Due to the changing value of friction coefficient under
the varying tests conditions between scaled and full-sized
tires it was decided to use a slightly altered version of the
cornering coefficient. To compensate for a possible
3414

difference in tire/road conditions between scaled and full-
sized tires a modified cornering coefficient was calculated
by dividing the cornering stiffness by the maximum lateral
force generated, defined as D in the Magic Formula,
thereby giving:
D
C
CC
D
α
α
≡
,
(14)
This modified value of cornering coefficient was
implemented to compensate for the varying value of the
coefficient of friction from test to test, depending on the
size, normal load and speed of the tire. This technique is
similar to those introduced in the non-dimensionalization
methods shown in [11], where tire model fits are developed
to represent tire response independent of normal load and
friction conditions.
Definition Variable Dimension
Aspect Ratio AR [-]
Tire Diameter d
t
[L]
Wheel Base L [L]
Cornering Coefficient
CC
α,
D
[-1/rad]
Table 2: Variables used for defining tire scaling
The key terms chosen to represent the dimensionless
cornering response of the tire are shown in Table 2. Only
two of the particular terms have any dimensions: the wheel
base of the vehicle, and the diameter of the tire. These two
can be transformed into a dimensionless parameter by
simply dividing the tire diameter by the length of the wheel
base:
L
d
t
=Π
1
(15)
The term in Equation (15) is considered to be the first pi
term for the dimensional analysis of a tire. It is essentially
the dimensionless size of the tire, relative to the vehicle
size. The remaining variables considered are already
dimensionless, if radians are considered to not be a
dimension. Both of these variables can be considered to be
a pi term on their own, but from previous analysis [8] it
was noted that there was a relationship between these two
variables. The aspect ratio for full sized tires and the
scaled tires was significantly different. As a result, the
higher aspect ratio scaled tires had proportionately lower
cornering coefficients than their full sized counterparts
even when tire/road interaction effects were accounted for.
The reason why high aspect ratios lead to low cornering
coefficients is that the high aspect ratios indicate high
sidewalls which, in turn, indicate greater flexibility. The
greater flexibility leads to lower load carrying capacity at a
given slip angle. To account for this, a second pi term was
created that included both the cornering coefficient with
respect to the maximum lateral load, CC
α
,D
, and the aspect
ratio (AR). This second pi term is defined as:
ARCC
D
⋅
≡
∏
,2
α
(16)
The effect of the dimensionless representation of
cornering coefficient and aspect ratio can be seen in
Figures 8 and 9. Figure 8 illustrates the normalized
cornering coefficient of Equation (14) for several scaled
tires and full sized tires [12]. The entire set of full sized tire
data used for Figures 8 and 9 can be found in [8] and is not
reproduced here due to space constraints.
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
0.00 0.20 0.40 0.60 0.80
Diameter (m.)
Cornering Coefficient (N/N/rad)
Scaled
Full Sized
Fig. 8: Cornering coefficients for scaled and full sized
[8] tires.
Figure 9 illustrates the
Π
2
values for the same distribution
of scaled and full sized tires. As evidenced by the results
of Figure 9, the distributions of the dimensionless
parameter are nearly identical for full sized and scaled tires.
Figures 8 and 9 are even more relevant when compared
with existing dimensional data shown in Figure 10.
Although the values given in Figure 10 are in terms of
(1/deg), a conversion from (1/deg) to (1/rad) will indicate
that the range of cornering coefficient, and subsequently
Π
2
, for Figures 8 and 9 exactly matches that of previously
established data. The range for of about 4 to 18
1
/
rad
for
CC
α
,D
corresponds to 0.1 to 0.3
1
/
deg
for the cornering
coefficient shown in Figure 10. This indicates a
quantitative match in tire characteristics between scale and
full-sized tires to further support the qualitative match
shown in Section 3.
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
0.00 0.20 0.40 0.60 0.80
Diameter (m.)
2
Scaled
Full Sized
Fig. 9:
Π
2
values for scaled and full sized [8] tires
Π
2
3415

Fig. 10: Frequency distribution of cornering coefficient
values for full sized tires [7].
V. DISCUSSION AND CONCLUSION
The results in Section 3 and 4 demonstrate that the force-
producing characteristics of scaled pneumatic tires can
exhibit similitude with full-sized tires. The shapes of the
force curves are very similar, as shown in Section 3. This
was illustrated by nearly identical shape factors between
scaled and full sized tires. By appropriate usage of
dimensionless descriptions, the nonlinear scale tire
characteristics can be made to match those of a full-size
tire. The implications for scaled vehicle dynamics and
control are as follows. One can use these results to design
a scaled vehicle such that it exhibits dynamic similitude for
both the linear bicycle model of Equation (1) as well as
nonlinear behavior characteristic of low friction or high
lateral acceleration maneuvers. As detailed in [8] one
could start with a target value of
Π
1
and
Π
2
for available
scaled pneumatic tires and surfaces. From these, one could
then determine other key dimensions of the vehicle using
the dimensionless analysis and
Π
parameters developed in
[1,2]. This would result in a scaled vehicle designed
around the appropriate tire/road interface for dynamic
similitude with full sized vehicles.
One interesting insight gained from the analysis of
Sections 3 and 4 is as follows. Suppose the lateral force
response is crudely simplified into two linear regions that
represent the original linear portion and the nonlinear
portion of the tire response. This simplification assumes
that the tire lateral force linearly approaches its maximum
value, with slope C
α
, where its response saturates and
remains with zero slope for high slip angles. It is known
from Equation (14) that the modified cornering coefficient
used for this analysis is the ratio of C
α
to D and from the
discussion of this value it is known that it remains
approximately the same for the scaled and full sized
automobile tires for tires with similar aspect ratios.
Therefore, the slip angle at which the maximum lateral load
will occur remains the same for varying tire sizes.
Defining α
max
to be the slip angle of peak lateral force, this
invariance with respect to tire size is deduced from the
simplified tire response model since:
max
α
α
D
C ≈
(17)
max
max
,
1
α
α
α
α
≈
⎟
⎠
⎞
⎜
⎝
⎛
≈=
D
D
D
C
CC
D
(18)
This relationship is an approximation from the model
simplification, but it shows that the value of the cornering
coefficient, CC
α
,D
, will be approximately equal to 1/
α
max
.
Since the cornering coefficient remains about the same, so
will the value of slip angle at which the maximum load
occurs. The value of slip angle (
α
max
) represents the
approximate angle at which the tire behavior transitions
from a linear to a nonlinear regime. Therefore, the basic
response of a tire remains the same as it transitions from
partial to full sliding for both full size and scaled tires; even
for varying types of tire construction.
R
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, Warrendale, PA.
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