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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

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Traction in a Rear Wheel Drive Vehicle by Means of Magneto-Rheological Semi-

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[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

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technology, 6, 1998.

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[8] Pacejka H. B., Tyre and Vehicle Dynamics, Butterworth-Heinemann, 2007.

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[11] Farroni, F., Russo, M., Russo, R., Terzo, M. and Timpone, F., On the

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