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INFLUENCE OF TYRE AND VEHICLE CONSTRUCTIVE CHARACTERISTICS ON LINEAR STABILITY OF A FOUR WHEEL VEHICLE MODEL

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The study of vehicle handling behaviour is of fundamental importance in order to improve vehicle safety, especially as concerns the loss of stability in the lateral direction resulting from unexpected lateral disturbances like side wind force, tyre pressure loss, µ-split braking due to different road pavements such as icy, wet, and dry pavement, etc. The interest for vehicle stability has been actually increasing, and consequently the study of the local stability has become a fundamental discipline in the field of vehicle dynamics, being a vehicle a strongly non linear system mainly because of tyres behaviour.
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INFLUENCE OF TYRE AND VEHICLE CONSTRUCTIVE
CHARACTERISTICS ON LINEAR STABILITY
OF A FOUR WHEEL VEHICLE MODEL
Flavio FARRONI, Michele RUSSO, Riccardo RUSSO,
Mario TERZO, Francesco TIMPONE
DiME - Department of Mechanics and Energetics,
University of Naples "Federico II",
Via Claudio 21, 80125 Napoli, Italy
ABSTRACT
The study of vehicle handling behaviour is of fundamental importance in order to improve vehicle safety,
especially as concerns the loss of stability in the lateral direction resulting from unexpected lateral disturbances
like side wind force, tyre pressure loss, µ-split braking due to different road pavements such as icy, wet, and dry
pavement, etc.
The interest for vehicle stability has been actually increasing, and consequently the study of the local stability
has become a fundamental discipline in the field of vehicle dynamics, being a vehicle a strongly non linear
system mainly because of tyres behaviour.
The study of the driving stability of a vehicle travelling on a curve in steady state conditions is the first and most
important step in order to understand the effects of perturbations inevitably occurring during vehicle motion.
Such a study may have a dual application: first of all it can contribute to develop and optimize control logic of
the vehicle subsystems; secondly it may be useful to identify the physical quantities affecting stability and to
find out their optimal set under this point of view.
At this purpose it is fundamental to dispose of suitable stability maps, which typically are constituted by curves
in a phase plane delimiting zones in which the system of equations describing the vehicle dynamics admits stable
and strongly attractive solutions. Outside of such zones only unstable or marginally stable solutions exist.
In the recent years several ways to construct these maps have been proposed. A great number of studies in
literature are mainly oriented to the development of control systems strategies aimed to improve vehicle stability
and so in such papers the authors proposed methods based on vehicle and tyre models extremely simplified.
Typical simplifications consist in considering a bicycle model (i.e. neglecting roll dynamics and the lateral load
transfer) and/or in linearising the tyre-road interaction (i.e. underestimating saturation effects in the tyre lateral
forces).
No studies have been found aimed to find out which vehicle physical characteristics affect significantly its
stability and the influence of their variation on stability. This paper pursues precisely this purpose and hence it is
necessary to use a sufficiently detailed vehicle model allowing to take into account all the physical parameters on
which investigate.
The stability of a complete four wheel, two axle, vehicle model will be discussed, Pacejka magic formula will be
adopted to model the pure tyre-road interaction, lateral load transfer will also be considered.
Two different methods will be presented: a “mathematical” method consisting in the evaluation of the state
matrix eigenvalues and a ”geometrical” method inspired to the so called “handling diagram” construction.
The influence exerted on the stability zone extension in the phase plane by some vehicle parameters, such as tyre
cornering stiffness and front/rear roll stiffness will be evidenced with the aim to suggest solutions able to
increase vehicle stability.
Keywords: phase plane, handling diagram, stability, cornering stiffness, anti-roll stiffness
1. INTRODUCTION
The purpose of the present paper is the study of the influence of tyre cornering
stiffness and axle anti-roll stiffness on the driving stability of a vehicle travelling on a
curve in steady state conditions. Such a study may have a dual application: first of all
it can contribute to develop and optimize control logic of the vehicle subsystems [1];
secondly it may be useful to identify the physical quantities affecting stability and to
find out their optimal set acting in order to generate the necessary corrective yaw
moments [2].
To this aim, contrary to what usually made in this research field [3] [4] [5], a
quadricycle planar vehicle model has been developed, so that the classical bicycle
approach and the common simplifying of tyre interactions can be overcome.
The proposed vehicle model, equipped with a Pacejka Magic Formula in order to
take into account of the nonlinear tyre behaviour respect to longitudinal and lateral
sliding velocities and to vertical load, allows a detailed study of the physical
parameters on which attention is focused.
Two different methods, both based on nonlinear models, will be discussed: a
numerical method consisting in the evaluation of the state matrix eigenvalues and a
graphical method inspired to the so called “handling diagram” construction [6] [7].
The combined use of the two methods to the vehicle stability analysis allows to get
additional information about the nature of the equilibrium points, extending the
predictive attitude of the handling diagram instrument, and giving a physical meaning
to the not immediate interpretation of the results obtained in the phase plane.
2. VEHICLE MODEL
A quadricycle planar model has been adopted with the aim to study vehicle steady-
state equilibrium in lateral dynamics and its stability conditions. The model is
characterized by two states referred to the in plane vehicle body motions (lateral and
yaw motions).
To describe the vehicle motions two coordinate systems have been introduced: one
earth-fixed (X' ; Y'), the other (x ; y) integral to the vehicle as shown in figure 1.
With reference to the same figure, v is the centre of gravity absolute velocity
referred to the earth-fixed axis system and U (longitudinal velocity) and V (lateral
velocity) are its components in the vehicle axis system; r is the yaw rate evaluated in
the earth fixed system, Fx
ij
and Fy
ij
are respectively longitudinal and lateral
components of the tyre-road interaction forces with reference to tyre middle plane.
The front and rear wheel track are indicated with t
f
and t
r
; the distances from front
and rear axle to the centre of gravity are represented by a and b, respectively. The steer
angle of the front tyres is denoted by δ, while the rear tyres are supposed non-steering.
Steering angle and longitudinal velocity represent manoeuvre parameters.
Figure 1: Coordinate systems.
In the hypothesis of negligible aerodynamic actions along the lateral direction and
around the vertical axis, the in plane motion equations of a rear wheel drive vehicle are
the following:














(1)
where m is the vehicle total mass, J
z
is its moment of inertia respect to z axis.
Tyres have been modeled my means of Pacejka Magic Formula [8], that expresses
the interaction forces as a function of the longitudinal slip ratio, of the slip angle and
of the vertical load by means of non-linear functions. The formula contains a number
of parameters {P}, which have no clear physical meaning, usually identified starting
from experimental data in order to reproduce properly tyre physical behaviours. In the
present paper parameters of common passenger tyres will be employed, analyzing, in
particular, the only lateral dynamics.
Pacejka coefficients implemented in the model provide the lateral force
characteristics shown in figure 2 for three different load conditions.
Figure 2. Lateral force interaction curves for three different vertical loads.
Vertical load on each wheel can be seen as sum of a static load contribution and a
dynamic variable load due to inertia forces on the vehicle body.
With reference to a static approach for vehicle roll motion, normal forces can be
defined as [9]:
( )( )
( )( )
( )( )
( )( )
+
+
+
+
=
+
+
+
=
+
+
+
+
=
+
+
+
=
Urdh
k
k
Ura
ba
d
tba
mga
F
Urdh
k
k
Ura
ba
d
tba
mga
F
Urdh
k
k
Urb
ba
d
tba
mgb
F
Urdh
k
k
Urb
ba
d
tba
mgb
F
z
z
z
z
φ
φ
φ
φ
φ
φ
φ
φ
2
22
2
21
1
12
1
11
)(2
1
)(2
)(2
1
)(2
)(2
1
)(2
)(2
1
)(2
(2)
where g is gravity, d is the height of the intersection point between the roll-axis and
the vertical plane passing by y-axis, h is the height of the centre of gravity, and
are, respectively, front and rear axle roll stiffness (taking into account also roll
stiffness due to suspension elements) and their sum is indicated with   .
3. LOCAL STABILITY ANALYSIS
Studies concerning the stability of complex dynamic nonlinear systems are often
carried out by means of step by step time integration methods; it allows to reach an
easy solution, but not to fully investigate and understand the vehicle behaviour. An
analytical approach, on the contrary, allows to clearly describe each parameter
influence on the system. In the following two different ways to solve the analytical
problem will be discussed.
3.1 Phase plane method
In order to analyze the stability problem in the (β, r) plane, the non linear model (1)
is preliminarily posed in the form:
( )
[ ]
( ) ( ) ( )
+++=
+
++++=
)sin(
2
)cos(
1
1
1
)cos(
1
121122211211
2
22211211
δδ
β
δβ
t
FFbFFaFF
J
r
tg
FFFF
Um
r
yyyyyy
z
yyyy
&
&


being
(
)
U
V
1
tan
=
β


 
in which f is a non linear function of the state
=r
β
x
obtained taking into account,
differently from the widely used approaches, nonlinearities due to tyre-road
interactions and to slip angles, together with the load transfers (2) .

0xf
=
)(

The analysis of the eigenvalues of the linearized system allows to evaluate the
stability properties of the equilibrium points.
In order to find the zeros of Eq. (3) a numerical procedure is recursively adopted
using as initial guesses all the possible pair of values (β, r) with β and r varying in the
range (-5,5) in step of 0.5.
Solutions characterized by a positive value of the vertical load on one or more tyres
are discharged, not being this condition acceptable for the vehicle model (loss of
contact).
Equation (3) is then numerically linearized by computing its jacobian matrix in the
neighbourhood of each found solution and the eigenvalues of the matrix are analyzed
so qualifying the nature of the equilibrium point: stable node, unstable node, stable
focus, unstable focus, saddle point.
With the aim to verify some of the found solutions, the non linear Eq. (3) is then
step by step time integrated, by Runge-Kutta method, starting from suitable initial
conditions; moreover step by step time integration allows to plot phase trajectories.
3.2 Handling diagram method
In normal working conditions of a vehicle performing a steady state lateral
manoeuvre it is possible to suppose little steering angles and longitudinal velocity
much greater than other velocity components ( U >> V and U >> rt/2 ); as a
consequence the slip angles of the two wheels of the same axle can be assumed equal.
In steady state conditions, neglecting the term 

 


as commonly can be
found in literature [9], Eq. (1) become:


where Fy
i
= Fy
i1
+ Fy
i2
. Eq. (7) describe the steady state motion conditions for a non-
linear bicycle vehicle model.
By means of proper equations which take into account suspensions parameters (e.g.
roll stiffness, roll centre) and vehicle geometry [10], it is possible to obtain the lateral
load transfers law of each axle as a function of the correspondent axle lateral load.
At this point, for each value of the axle lateral force, it is possible to find the
corresponding value of the load transfer; two interaction curves (for the two tyres of an
axle) correspond to each load transfer and their sum curve will be considered in the
following.
On this sum curve, the slip angle value of the tyres of an axle that allows to generate
two forces whose sum is equal to the desired value of the lateral axle force, is
detectable.
Slip angle and axle lateral force, estimated by means of the described procedure,
characterize a point of the so called "Effective Axle Cornering Characteristic".
Iterating this process for different values of the axle lateral force it is possible to build,
point by point, the whole Effective Axle Cornering Characteristic (Figure 3). It is
important to highlight that these curves are usually represented in non-dimensional
form obtained dividing them by the vertical static load acting on each axle.
Figure 3. The thick curve (Front effective axle cornering characteristic for Kϕ1=10000 Nm/rad) has been built
point by point by means of the thin curves corresponding to the sum of the interaction curves for different load
transfers.
The solution of the motion equations system (7), representing the steady state
equilibrium conditions, can be found by means of a graphic method: the "handling
diagram" [6] which gives the possibility to analyze, in a phase plane, the steady state
equilibrium of a non-linear bicycle model.
To build handling diagram it is necessary to manipulate properly Eq. (7), obtaining:
( ) ( )
[ ]
( ) ( )
mga
ba
F
mgb
ba
F
g
A
bag
U
g
A
yy
y
y
+
=
+
=
+
=
)()(
2211
21
2
αα
ααδ
(8)
In the so called “handling diagram” phase plane, each vehicle equilibrium point
corresponds (Figure 4) to the intersection point between the curves defined in Eq.(8).
The first equation of the (8) is represented in that plane by a straight line with
parameters δ and U setting position and inclination of the line and it is characteristic of
the manoeuvre. The second curve depends on axle effective and on the vehicle
constructive characteristics.
Figure 4. Handling diagram.
3.3 Some considerations about the two methods
In Figure 4 two equilibrium points can be seen, however nothing can be said on the
nature of such points observing the only handling diagram.
Merging the two methods, much more can be said about the found equilibrium
points, as noticeable from Figure 5.
Figure 5. Results of the merge of numerical and graphical method.
The characteristic curves have been reconstructed point by point thanks to the phase
plane method feeding it with a great number of different pairs (U, δ) and saving, for all
equilibrium positions, the quantities (α
1
α
2
) and (a
y
/g).
A good correspondence between numerical and graphic solutions may be noticed;
the little differences between the results provided by the two methods occur only in
correspondence of the greater (α
1
α
2
) values and may be attributed to the further
simplifications introduced by the graphical method in Eq. (7).
Some considerations can be made: handling diagram offers the great advantage to
provide quickly steady state solutions of the motion equations, but analyzing its curves
it is not possible to discriminate univocally between the different equilibrium natures.
Moreover handling diagram allows to synthetically describe by means of a set of
characteristic curves the complete vehicle behaviour independently from the particular
manoeuvre.
The combined use of the two methods allows a physical understanding of the results
obtained in the phase plane highlighting the link between the vehicle parameters and
the number/nature of the equilibrium points. On the other side, such combined use
gives the possibility to find out, thanks to the manoeuvre line, the working conditions
under which a certain equilibrium point has been numerically obtained.
4. RESULTS
The influence of tyre and vehicle parameters will be analyzed in this section
following the combined approach. In particular, the interest will be focused on two
parameters: cornering stiffness and roll stiffness.
Cornering stiffness will be made varying as shown in figure 6, in which lateral force
characteristic of three different tyres in the same conditions of vertical load (3000 N)
and adherence are reported. All the tyres exhibit the same maximum value in terms of
lateral force (F
y
) while they show decreasing cornering stiffness passing from A to C.
Figure 7. Lateral force interaction curves for three different tyres.
Front/Rear Roll Stiffness ratio is the second parameter object of investigation and in
the three proposed cases it will assume values of 0.1, 1 and 10.
4.1 Influence of the tyre cornering stiffness
In order to investigate about the influence of tyres, the first case study refers to a
manoeuvre with δ = 0.0201 rad and U = 10 m/s, performed by a vehicle equipped with
the same front and rear anti-roll stiffness (10000 Nm/rad) and with two different tyres .
Figures 8 and 9 show the phase portraits (β, r) respectively for A and C tyres
equipped vehicle. Stable (node) and unstable (saddle) equilibrium points can be
noticed together with system trajectories delimiting the stable regions. The results
provided by handling diagram method in the same conditions are shown in figure 10 in
which it is possible to distinguish the straight line representing the manoeuvre and two
set of curves for the two different tyres. Figure 11 represents a zoom of figure 10
focused on the linear region of the handling curves. Once again, in figures 10 and 11
the equilibrium points have been marked in accordance with the results provided by
the numerical method for what concerns their nature.
Both figures 8 and 9 show three equilibrium points (i.e. two saddle points and one
stable node). The same equilibrium points can be seen in the handling as intersections
between the manoeuvre straight line and the handling curves. In particular saddle
points are located in the non linear part of the handling curves, while the nodes are
located in its linear part and are clearly visible in figure 11.
Figure 8. Phase portrait of the vehicle equipped with A tyre.
Figure 9. Phase portrait of the vehicle equipped with C tyre.
Figure 10. Handling diagram for the vehicle equipped with A and C tyre.
Figure 11. Handling diagram for the vehicle equipped with A and C tyre (zoom of fig. 10).
Vehicle equipped with C tyre exhibits a phase portrait characterized by a wider
stable region (figure 9) than the one exhibited (figure 8) by the vehicle equipped with
A tyre. This can also be observed in the handling diagram (figure 10) observing the
greater distance between the two saddle points relative to the same handling curve. At
the same time, vehicle equipped with A tyre is characterized by a more attractive
stable node as can be seen comparing the eigenvalues reported for both cases. This
attractive property is due to the major cornering stiffness that, in presence of a
perturbation of the system state and consequently of the side slip angles, makes the
tyre A able to generate a higher reaction force variation, as also noticeable in the
handling diagram (figure 10), and so a more effective return to the starting stable
condition.
Tyres Equilibrium nature
β
(rad)
r (rad/s) Eigenvalues
21
αα
(rad) A
y
/g
A Stable node 0,0021 0,0940 -19,3494; -13,8389 0,0292 0,0958
Saddle point 1 -0,0704 0,4716 6,4159; -20,1166 4,7677 0,4810
Saddle point 2 0,1062 -0,4453 5,5526; -19,4436 -6,7245 -0,4542
C Stable node -0,0075 0,0980 -8,0553; -4,7140 0,0806 0,1000
Saddle point 1 -0,1394 0,5552 4,6176; -8,5780 5,7324 0,5663
Saddle point 2 0,1780 -0,5414 4,4662; -8,2141 -7,8055 -0,5522
The results of this first case study confirm that, also for what concerns stability, the
tyre should be characterized by high cornering stiffness and by an increasing trend of
the lateral force up to high saturation values.
4.2 Influence of the anti-roll stiffness
A second case study deals with the investigation about the effect of the anti-roll
stiffness on vehicle stability; in particular, figure 12 shows in the already described (β,
r) plane the effect of changes in the front/rear stiffness distribution. With reference to a
manoeuvre characterized by U = 10 m/s and δ = 0.1 rad, three different front/rear
stiffness ratios have been considered; in particular, figure 12a refers to Kϕ
1
=5000
Nm/rad - Kϕ
2
=50000 Nm/rad, figure 12b refers to Kϕ
1
=27500 Nm/rad - Kϕ
2
=27500
Nm/rad, figure 12c refers to Kϕ
1
=50000 Nm/rad - Kϕ
2
=5000 Nm/rad.
Analyzing the figure 12a or equivalently 13, only one equilibrium point can be
noticed, distinguishable by negative value of r and then of A
y
. It represents an unstable
analytical solution, very hard to be physically reproduced by an actual vehicle; in fact
the only intersection found in the handling diagram plane belongs to a branch of the
vehicle curve not coherent with the manoeuvre inputs.
Increasing the front/rear roll stiffness ratio three different equilibrium points can be
found: a stable node and two saddle points. The stable node condition is always
coherent with manoeuvre inputs and moves toward the central zone of the stable
region (figures 12b and 12c) confirming the positive effect of the increasing of the
front axle stiffness.
In the handling diagram plane (figure 13) this effect can be highlighted considering
the intersection between the manoeuvre line and the linear part of the vehicle curve.
Such intersection occurs for lower values of lateral acceleration with the increasing of
the front/rear roll stiffness ratio. This kind of result confirms that the vehicle loss of
stability manifests itself with the sliding of the rear axle and suggests that it is possible
to vary the whole vehicle tendency to instability acting properly on front and rear axle
stiffness. For example adopting active control systems, such as active anti-roll bars and
active suspensions, it would be possible to change properly the front/rear roll stiffness
ratio, so assuring a major stability margin [11].
For what said above, the front/rear roll stiffness ratio can be seen as a saddle node
bifurcation parameter. In figure 14, the bifurcation diagram has been reported with
reference to the two states (β, r). The front/rear roll stiffness ratio corresponding to the
bifurcation point would make, in handling diagram plane, the manoeuvre straight line
tangent to the vehicle curve.
Figure 12. Manoeuvre: δ = 0.1 rad, U = 10 m/s. Saddle point (square) and Stable node (circle) for three different
Roll Stiffnes set : a) Kϕ
1
= 5000 Nm/rad, Kϕ
2
= 50000 Nm/rad; b) K ϕ
1
= K ϕ
2
=27500 Nm/rad; c) K ϕ
1
= 50000
Nm/rad, K ϕ
2
= 5000 Nm/rad.
Figure 13. Handling Diagram for three different values of the ratio between front and rear axle roll stiffness.
Figure 14. Manoeuvre: δ = 0.1 rad, U = 10 m/s. Saddle (•) – Node () Bifurcation diagram.
5. CONCLUSIONS
A fully non-linear quadricycle planar vehicle model has been discussed, taking into
account lateral load transfers and non-linear tyre road interactions. Two different
analytical methods, the phase plane and the handling diagram, have been employed
and their results have been combined to better analyse the in curve vehicle lateral
stability. The combined use of the two methods allows a more satisfying
understanding of the results obtained in the phase plane, highlighting their physical
meaning and establishing a link between vehicle parameters and the number/nature of
the equilibrium points.
The analysis of the influence of parameters such as tyre cornering stiffness and
front/rear roll stiffness ratio, has been carried out. The obtained results confirm
theoretical expectations about the effect of their variations in the working range and
show the possibility that in certain conditions they can become node bifurcation
parameters.
Concerning with future works, the investigated influence of the front/rear roll
stiffness, not widely studied in literature from an analytical point of view, suggests the
employment of active/semi-active anti roll bar and suspensions not only for enhancing
vehicle handling but also for stability improvement.
6. BIBLIOGRAPHY
[1] De Rosa, R.., Russo, M., Russo, R., Terzo, M., Optimisation of Handling and
Traction in a Rear Wheel Drive Vehicle by Means of Magneto-Rheological Semi-
Active Differential. Vehicle System Dynamics, 47, 533-550, 2009.
[2] Russo, R., Terzo, M., Timpone, F., Software-in-the-loop development and
validation of a cornering brake control logic, Vehicle System Dynamics, 45, 149-163,
2006.
[3] Taeyoung, C. and Kyongsu, Y., Design and Evaluation of Side Slip Angle-Based
Vehicle Stability Control Scheme on a Virtual Test Track, IEEE Transactions on
control systems technology, 14, 2006.
[4] Ono, E., Hosoe, S., Tuan, H.D., Doi, S., Bifurcation in Vehicle Dynamics and
Robust Front Wheel Steering Control, IEEE Transactions on control systems
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Lateral Dynamics of Road Vehicles -3rd ICTS Seminar on "Advanced Vehicle Systems Dynamics
  • H B Pacejka
Pacejka H. B., Lateral Dynamics of Road Vehicles -3rd ICTS Seminar on "Advanced Vehicle Systems Dynamics", 1986.