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Original Article
Proc IMechE Part D:
J Automobile Engineering
2015, Vol. 229(10) 1419–1426
ÓIMechE 2015
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DOI: 10.1177/0954407014563734
pid.sagepub.com
The tyre characteristics of the physical
model used to investigate the lateral
stability of a vehicle
Krzysztof Parczewski and Henryk Wne˛k
Abstract
This paper presents a comparison of the characteristics of the tyres of a full-size vehicle with the tyres of a physical
model scaled 1:5. This is a continuation of studies on the use of a scaled vehicle to test the stability of a vehicle. The
results presented are based on analysis of a scaled vehicle and a full-size vehicle on a stand and during road tests. Tests
were carried out involving manoeuvres based on the ISO standard. The effects of the differences in the construction of
the tyres of the scaled vehicle and their impact on the tyre characteristics and its behaviour during testing were com-
pared. This paper presents the results of a comparison of selected parameters of motion for a real vehicle and for a
mobile scale model. These tests allowed a statement to be made about the suitability of the used tyres and the entire
physical model for lateral stability analysis of a full-size vehicle.
Keywords
Lateral stability of a vehicle, scaled vehicle, physical model of a vehicle, tyre characteristics, dynamics of vehicle motion,
similarity
Date received: 29 April 2014; accepted: 17 November 2014
Introduction
During vehicle motion, the transmission of forces
from a vehicle to the ground depends on the condi-
tions of the tyres and the road surface. The wheels,
which are in contact with the road, allow reaction
forces to be generated, which counterbalance the
forces acting on the vehicle in motion or the forces
generated by the powertrain system. The tyres trans-
mit the forces generated in the tyre– road contact
patch, and the behaviour of the vehicle depends on
their properties. The mathematical terms for the reac-
tion between the tyres and the road surface and their
properties describe the tyre models mentioned in the
literature.
1–5
The motion of a car is affected by para-
meters such as the effective wheel radius, the wheel
slip, the rolling resistance and the transmission of the
longitudinal forces and the transverse forces from the
tyre to the road depending on the adhesion condi-
tions, namely the longitudinal, transverse and lateral
slip phenomena. The dynamics of physical models,
regardless of the scale of the vehicle, are studied, and
this shows that the tyre characteristics will influence
the transmission of the forces to the road.
Description of the tests
Ensuring safety in the use of complex technical struc-
tures such as aircrafts, ships or vehicles requires a series
of tests allowing the behaviour of the vehicle to be
checked under the conditions of use. In many cases, it
is possible to test the actual object but, with large items,
these studies are difficult to implement, expensive and
often dangerous. In such cases, the use of smaller-scale
physical models or computer simulations is helpful.
Computer models require determination of and expres-
sions for a large number of parameters characterizing
the vehicle as a whole and its parts. The accuracy of
the calculation depends on the details of the model and
the considered parameters.
Department of Combustion Engines and Vehicles, Faculty of Mechanical
Engineering and Computer Science, University of Bielsko-Biala, Bielsko-
Biala, Poland
Corresponding author:
Krzysztof Parczewski, Department of Combustion Engines and Vehicles,
University of Bielsko-Biala, Willowa 2, Bielsko-Biala 43-309, Poland.
Email: kparczewski@ath.bielsko.pl, hwnek@ath.bielsko.pl
by guest on September 3, 2015pid.sagepub.comDownloaded from
Physical models are based on dimensional analysis
and similarity theory. Preserving similarity allows the
measured motion parameters of the scaled vehicle to be
transposed to the parameters of the real object tested.
The use of the theory of similarity requires preserving
the similarity of the geometric, kinematic and dynamic
models to the real vehicle. This can be achieved by
extracting a number of parameters which, in non-
dimensional notation for the real vehicle and the model,
should be the same.
Fulfilling the criteria of similarity allows interpreta-
tion of the research results on the model and enables
those results to be related to the dynamics of the real
motion of the vehicle. The Buckingham ptheory was
used to determine the scale of the similarity of the indi-
vidual parameters.
6–8
The first step in applying this the-
orem is to collect the vehicle parameters into groups of
dimensionless parameters, also known as the ppara-
meters. These groups can be equivalently obtained by
normalizing the mass scale, the length scale and the
time scale by scaling factors dependent directly on the
vehicle’s mass m, the vehicle’s length Land the vehicle’s
longitudinal velocity Urespectively. For the chosen
scaled vehicle, the dimensionless parameters are pre-
sented in Table 1.
In 1937, Huber and Dietz
9
had already considered
the dynamics of motion of a tractor with a semitrailer.
In 1959, Zakin
10
worked on the instability of tractors.
Japanese researchers have used a rolling roadway simu-
lator as a test bench for analysis of the control algo-
rithms for suspensions, which could take into account
the features of the vehicle. The first computer-
controlled roadway simulator was developed by
Brennan and Allyene
11
and Lapapong et al.
12
at Illinois
University at Urbana-Champaign. Polley et al.
13
devel-
oped and manufactured in a testing laboratory the
Pennsylvania University Rolling Roadway Simulator.
There are several advantages of using a mobile vehi-
cle model, made to scale, instead of the full-size vehicle
for experimental testing of the dynamics of motion.
1. The costs of vehicle tests performed on a scale
model are much less than when tests are made on
the full-size vehicle; the same applies to supplies
and spare parts.
2. The testing of a mobile vehicle model requires less
space and is much safer to handle.
3. It is easier to make changes to the vehicle on a
smaller scale.
4. The scope of the research can be wider. A rollover
vehicle model implies a much lower repair cost.
5. The testing time is shorter.
6. The vehicle model made to scale, except for the
similarity parameters analysed in its construction,
is also characterized by the structural similarity of
the whole model and its bands.
Research on scaled vehicle models has also been car-
ried out in the Department of Combustion Engines and
Vehicles, University of Bielsko-Biala. Currently the
Department has a number of different scale models
used for research and for road tests and tests on a
stand. This allows the characteristics of model assem-
blies and components to be achieved.
14,15
During the construction of a physical model, one of
the problems that required solution was the choice of
tyre parameters on the basis of their characteristics.
Tyre characteristics
The tasks of the tyre are the smooth transfer of power
from the vehicle to the road in various traffic conditions
and provision of adequate isolation of the vibrations
generated by a rough road. Providing similarities of the
Table 1. Dimensionless parameter matching.
No pparameters Full-sized Scaled vehicle
p
1
a1
=
Lw0.650 0.648
p
2
a2
=
Lw0.350 0.349
p
3
CafLw
mU20.392 0.513
p
4
CarLw
mU20.434 0.655
p
5
Izz
mL2
w0.214 0.263
p
6
hGC
=
Lw0.350 0.267
p
7
B
=
L0.562 0.560
p
8
Ixx
mL2
w0.045 0.063
p
9
Iyy
mL2
w0.244 0.210
p
10
ms
=
m0.897 0.863
p
11
ðhGC hRCÞ
=
Lw0.243 0.128
p
12
Kfr
mU20.392 0.216
p
13
Df
mULw0.163 0.058
1420 Proc IMechE Part D: J Automobile Engineering 229(10)
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real vehicle and the scaled physical model required choos-
ing comparable isolation parameters of the tyres.
16
Another problem is that, in models with a length
from 0.5 m to 1 m (scale, approximately 1:5), tyres with
polyurethane inserts are usually used. The inserts are
divided into two groups of stiffness: soft and hard.
Pneumatic tyres are not generally produced with these
dimensions. The studies were designed to show the
impact of using tyres of different constructions on the
test results. For comparison we selected 185R15C auto-
mobile tyres (used in the comparable vehicle) and the
tyres used in the physical model (Figure 1) with the fol-
lowing inserts: soft, hard and pumped (inserts used in
place of the inner tube).
Table 2 presents a comparison of the tyres selected.
Comparison of the tyre parameters is required to
carry out a series of tests to identify the radial and longi-
tudinal stiffnesses, the lateral cornering stiffnesses of the
tyres, the rolling resistance, the footprint (the contact
patch area) of the tyres, etc. The study omitted determi-
nation of the coefficient of adhesion because testing was
carried out on a clean dry surface. Because it was impos-
sible to measure the cornering stiffnesses of the automo-
tive tyres, an elastic beam analogy was used.
17
Using the Hewson model, the cornering stiffness of
the tyres was determined on the basis of the
relationship
17
Ca=2Ebw3
(r+wa)2sinfarccos½1swa=(r+wa)g
1
psinfarccos½1swa=(r+wa)g
ð1Þ
Analysis was also used to determine the steering
angle dof the wheel (see equation (2a)), the angle d
A
resulting from the Ackermann relationship (see equa-
tion (2b)), the side-slip angle of the rear wheels (see
equation (3a)), the side-slip angle of the front wheels
(see equation (3b)) and the index of understeer (see
equation (4)). These relations are shown below. Other
parameters of the model motion were measured during
polygon tests. To determine the motion parameters of
the vehicle a bicycle model was used, assuming that the
lateral accelerations do not exceed 4 m/s
2
and that the
roll angle of the model was small. In other cases, for
higher lateral accelerations, the wheel load distributions
for the left wheels and the right wheels will change and
the cornering stiffness has non-linear characteristics. In
addition, there are many other motion parameters
which generate differences between the cornering stiff-
nesses of wheels on one axle, e.g. the flexibility of the
steering system or the lateral slipping of the tyres.
4, 5
The steering angle was determined from the
Ackermann relationship and, after taking into account
the side-slip angles of the wheels, it was found that
dA=R
Lw
ð2aÞ
d=R
Lw
+Wf
Caf
Wr
Car
U2
gR ð2bÞ
The side-slip angle of the rear wheels and the side-
slip angle of the front wheels were determined from the
relations
ar=ba2
_
c
Uð3aÞ
and
af=b+a1
_
c
Udð3bÞ
respectively.
Table 2. Tyres selected for comparison.
Parameter of tyre (units) Value for the following
185R15C SAVA 190/60 Desert
Buster HD
190/60 Desert
Buster HD
190/60 Desert
Buster HD
Aspect ratio (%) 80 50 50 50
Diameter (in) 15 4.5 4.5 4.5
Width (mm) 185 60 60 60
Spring element Air Soft insert Hard insert Air
Air pressure (MPa) Front, 0.32; rear, 0.40 — — 0.06
Type of tread Road Road Road Road
Figure 1. A tyre used in the scaled vehicle.
Parczewski and Wne˛k 1421
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The understeer index Kwas determined from the
relation
K=Wf
Caf
Wr
Car
ð4Þ
The characteristics of the tyres were established during
measurements carried out on testing stands.
Measurements of characteristic
parameters of the tyres
Measurements of the radial stiffness, the longitudinal
stiffness and the lateral stiffness were performed in sta-
tic conditions. The effect of the wheel camber was
omitted. The deformations (vertical, longitudinal and
transverse) of the tyres were measured for real tyre
pressures of 0.33 MPa and 0.43 MPa, and for the mod-
els having tyres with soft, hard and pumped inserts.
Table 3 shows the stiffness of each tyre. The radial stiff-
ness of the vehicle tyre 185R15C was obtained by mea-
surements and was 923 N/mm.
A test bench with a moving path was used to mea-
sure the cornering stiffnesses of the tyres of the model;
the tyre was placed on this bench and performed set
moves at a predetermined angle to the moving tread-
mill. The side-slip angle was measured from the longitu-
dinal direction of the tyre. A lateral force was generated
in these measurements.
The cornering stiffness of the scaled model tyre was
based on measurements. Figure 2 shows the test stand
for measuring the cornering stiffnesses (in the labora-
tory at the University of Bielsko-Biala).
The results shown in Figure 3 present measurements
(for side-slip angle a’4°of the wheel) for the three
types of tyre used in the model. The cornering stiffness
of the real vehicle tyre was determined from equation
(1) and equals 75,170 N/rad.
After determining the various tyre parameters, they
were converted into the dimensionless values presented
in Table 4. The parameters for two different conditions
of inflation pressure of the real tyres (used on the front
axle and the rear axle) are given.
As can be noted, the comparisons between the
dimensionless parameters of the tyres are quite similar.
The radial stiffnesses of real vehicle tyres are slightly
larger and significantly increase with increasing pres-
sure in the tyre. In the analysis of the moving vehicle
dynamics, the radial stiffnesses are relatively small and
do not affect the results of the tests significantly. The
Table 3. Tyre stiffnesses of the Lublin II vehicle and the scaled model.
Parameter Value for the following
185R15C SAVA 190/60 Desert Buster HD
0.33 MPa 0.43 MPa Load (N) Soft insert Hard insert Pneumatic Load (N)
Radial stiffness (N/mm) 923 1151 — 14.09 17.09 15.4 —
Longitudinal stiffness (N/mm) 376.72 387.12 4340 10.74 15.0 6.06 48
362.51 379.58 8240 11.59 15.5 5.7 88
372.36 384.33 12,400 13.99 15.5 6.27 128
— — — 14.32 16.3 6.6 168
Lateral stiffness (N/mm) 66.89 73.53 4340 3.21 6.05 4.2 48
66.63 88.83 8240 4.5 6.0 4.1 88
68.28 84.14 12,400 5.5 5.8 4.1 128
— — — 5.69 8.02 4.5 168
Figure 2. The test stand for measuring the cornering
stiffnesses of the tyres of the scaled vehicle.
Figure 3. The results of measuring the cornering stiffnesses of
various tyres of the scaled vehicle (own research).
1422 Proc IMechE Part D: J Automobile Engineering 229(10)
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longitudinal stiffnesses of all tyres are similar. The lat-
eral stiffnesses are rarely used during the analysis; the
cornering stiffnesses of the tyres have more influence.
The measurements suggest that the tyre with a soft
insert has the lowest rigidity value. The pneumatic tyre
and the tyre with the hard insert have similar cornering
stiffnesses. A slightly larger cornering stiffness occurred
for full-sized tyres. Because of the different structures
of the tyres with a liner, belt mapping was not possible.
For partial belt stimulation of a pneumatic tyre, an
additional rubber strap was used for further stiffening
of the upper part of the tyre.
Because of the impossibility of accurate representa-
tion of the real tyre by those used in the model, a num-
ber of behaviour comparisons of real vehicles and
model vehicles during motion were made.
Assessment of the influence of the tested
tyres on the motion dynamics of the
vehicle
Road tests were performed in order to compare the
influences of the tyres on the motion parameters of the
vehicle.
A sample chosen for comparison was fixed by driv-
ing on a circular path. The test was performed on a cir-
cle at a fixed speed according to ISO 4138:2012.
18
A
full-size vehicle was moving along a track with a radius
of 21.5 m. The vehicle model was tested on the track
with a radius of 5.61 m. This scaled model, which was
equipped with the investigated tyres, fulfilled the condi-
tions resulting from the behaviour of similarity. The
scaled vehicle during driving tests at a fixed speed along
a circular path is shown in Figure 5 (built in the labora-
tory at the University of Bielsko-Biala).
In both cases, for the scaled vehicle and the full-size
vehicle, the steering angles resulting from the
Ackermann relation were the same, i.e. d
A
= 7.7°
(Figure 6).
Based on the tests, under the conditions of similarity,
the yaw rates, the yaw angles, the side-slip angles of the
vehicles and the side-slip of the wheels were specified, in
the case of installation of each type of tyre in model.
Figure 7 presents the results of the research and
analysis of the scaled vehicle tyres with a soft insert,
scaled vehicle tyres with a hard insert, pneumatic tyres
Table 4. The dimensionless parameters of the tyres.
Wheel parameter Value for the following
185R15C SAVA x190/60 Desert Buster HD
0.33 MPa 0.43 MPa Soft insert Hard insert Pneumatic
Aspect ratio 0.8 0.8 0.5 0.5 0.5
Effective radius 1.66 1.66 1.50 1.50 1.50
Tyre width 0.97 0.97 1.17 1.17 1.17
Radial stiffness 39.77 49.98 19.40 20.27 15.82
Longitudinal stiffness 14.56 15.21 13.11 17.04 9.74
Lateral stiffness 3.04 3.56 5.35 9.10 5.23
Cornering stiffness 10.10 11.14 3.05 7.20 6.79
Length of contact with the road 1.26 1.26 1.04 1.04 1.04
Rolling resistance coefficient 0.10 0.10 0.14 0.12 0.12
Figure 5. View of the scaled vehicle during driving tests at a
fixed speed along a circular path.
Figure 4. Comparison of the dimensionless stiffness values for
various types of tyre (own research).
Parczewski and Wne˛k 1423
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and real vehicle tyres (dimensions, 185R15C). The fol-
lowing parameters were compared: the side-slip angles
bof the vehicle’s centre of mass, the side-slip angles
a
f
of the front wheels, the side-slip angles a
r
of the
rear wheels, the average steering angles dof the front
wheels, the Ackermann steering angle d
A
and the dif-
ferences between the side-slip angles of the front
wheels and the rear wheels. The comparison shows
that the real vehicle is characterized by a larger side-
slip angle of the vehicle’s centre of mass and larger
side-slip angles of the wheels. The measured steering
angles and the Ackermann steering angles were very
close.
The larger differences in the slip angles were due to
the high aspect ratio (h
tyre
/b
tyre
= 0.8) for the tyres used
in the real vehicle. The tyres used in the scaled vehicle
had a lower aspect ratio (0.5). The increased rigidity of
the angle for the real tyres (approximately 30%) com-
pensated for this difference.
The coefficient K, which was determined during the
tests at the proving ground, is shown in Figure 8. A
negative value of the coefficient means that the tested
vehicle has understeer characteristics. The use of tyres
with a soft insert in the model increases the doubled
understeer. The tyre model with a hard insert and the
pumped-tyre model reduce that coefficient to –0.07.
The understeer coefficient for the real vehicle was
–0.095 and is slightly larger than for the scaled vehicle.
Summary and conclusions
Based on the analysis the following can be concluded.
1. As in the real vehicle, the tyres significantly affect
the motion parameters of the scaled vehicle.
Providing similarity of the tyres is one of the condi-
tions for preserving the similarity of the dynamics
of the vehicle motion during testing.
2. Difficulties in obtaining the compliance of the
dimensionless parameters of the full-size vehicle
and the reduced-scale vehicle result from the differ-
ent production technologies of the tyres. The con-
struction of the tyre on a smaller scale is
significantly different from the construction of a
radial tyre. Therefore, it was decided to carry out
several tests to obtain answers to how the differ-
ences in the tyre construction influenced the
dimensionless parameters of vehicle motion.
3. Pneumatic tyres and tyres with hard inserts
resulted in similar motion parameters of the scaled
vehicle. In driving tests along a circular path using
a scaled vehicle and a real vehicle, the steering
angles of the front wheels are similar and the dif-
ference does not exceed 0.2°.
Figure 6. Schematic diagram for driving tests at a fixed speed
along a circular path.
19
Figure 7. Selected parameters of the vehicle’s motion, during
the driving tests at a fixed speed along a circular path for the
scaled vehicle and the full-size vehicle (own research).
Figure 8. The vehicle understeer coefficient Kfor a real
vehicle and a scaled vehicle determined during driving tests at a
fixed speed along a circular path (own research).
1424 Proc IMechE Part D: J Automobile Engineering 229(10)
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4. Real vehicle tyres have a higher aspect ratio of 0.8
and the tyres tested on a scaled vehicle have a lower
aspect ratio of 0.5. This resulted in the occurrence
of larger side-slip angles of wheels for the real vehi-
cle during the tests. When similar aspect ratios are
maintained for the tyres, the side-slip angles should
be similar.
5. The increased stiffness of the tyre angle reduces the
real value of the average steering angle of the front
wheels. The change is small; however, its impact
on the measurement results should be taken into
account.
6. Preserving similarities during the driving tests on a
circular track at a fixed velocity requires adjust-
ments of the steering angle of the wheels. In the
case of the tyres with a soft insert on the scaled
vehicle, the correction is the highest. Tyres with a
hard insert and pumped tyres require significantly
less correction. The average steering angle of the
front wheels of a real vehicle is intermediate
between that of a scaled vehicle with tyres with a
hard insert and that of a scaled vehicle with tyres
with a soft insert.
7. The scaled model and the real vehicle were under-
steering. For tyres with a soft insert, the coefficient
Kis more than twice the Kvalues for the tyres with
a hard insert and for tyres with a tube. Kfor a real
vehicle is somewhat higher than that for the tyres
with a hard insert.
The analysis shows that preserving the similarity of
the tyres allows the use of the model test scaled vehicle
with tyres with a hard insert to assess the real vehicle
behaviour, especially in terms of the lateral stability.
The tests on scaled vehicles allow the vehicle’s lateral
stability to be evaluated and can be used when the test-
ing of full-size vehicles is not practicable to carry out.
Declaration of conflict of interest
The authors declare that there is no conflict of interest.
Funding
This work was partially supported by the project R&D
NCBR ‘Evaluation of the stability of the vehicle on the
basis of the scaled vehicle’ (grant number PB 5478/B/
T02/2011/40). The material presented in our paper has
not been previously published; the diagrams and figures
are new and original.
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Appendix 1
Notation
aratio of the height to the width of the tyre
a
1
distance from the centre of gravity to the
front axle (m)
a
2
distance from the centre of gravity to the
rear axle (m)
bthickness of the belt material (m)
Bwheel track (m)
C
a
cornering stiffness of the tyres (Nms)
C
af
cornering stiffness of the front tyres (N/rad)
C
ar
cornering stiffness of the rear tyres (N/rad)
D
f
rotational damping coefficient (Nms/rad)
Emodulus of elasticity of the material of the
belt (N/m
2
)
h
GC
height of the centre of mass (m)
h
RC
height of the centre of rotation (m)
I
XX
roll moment of inertia of the vehicle (kg m
2
)
I
YY
pitch moment of inertia of the vehicle (kg m
2
)
I
ZZ
yaw moment of inertia of the vehicle (kg m
2
)
K
f
rotational stiffness coefficient (Nm/rad)
Llength of the tyre contact with the road (m)
L
w
wheelbase = a
1
+a
2
(m)
mmass of the vehicle (kg)
m
s
sprung mass of the vehicle (kg)
rradius of the wheel rim (m)
Rradius of the track (m)
R
A
radius of the track on the Ackermann
vehicle (m)
slateral deformation of the tyre under a
load (%)
Ulongitudinal speed of the vehicle (m/s)
wwidth of the belt (m)
W
f
load on the front axle (N)
W
r
load on the rear axle (N)
a
f
side-slip angle of the front wheels (rad)
a
r
side-slip angle of the rear wheels (rad)
bside-slip angle of the centre of mass of the
vehicle (rad)
daverage steering angle of the front wheels
(rad)
d
A
Ackermann steering angle (rad)
pdimensionless parameter
_
cyaw rate (rad/s)
1426 Proc IMechE Part D: J Automobile Engineering 229(10)
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